8, 2003

Volume 32 Number 7/8 2003

ISBN 0-86176-866-3

ISSN 0368-492X

Kybernetes The International Journal of Systems & Cybernetics Some new theories about time and space Guest Editors: Leon Feng, B. Paul Gibson and Yi Lin

Selected as the official journal of the World Organisation of Systems and Cybernetics

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Kybernetes

ISSN 0368-492X

The International Journal of Systems & Cybernetics

Volume 32 Number 7/8 2003

Some new theories about time and space Guest Editors Leon Feng, B. Paul Gibson and Yi Lin

Access this journal online _________________________

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Editorial advisory board __________________________

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Abstracts and keywords __________________________

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Preface __________________________________________

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Guest Editorial ___________________________________

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The mathematical logic failure of Einstein’s special relativity Arthur Bolstein ________________________________________________

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An impossible physical reality in Einstein’s relativity Arthur Bolstein ________________________________________________

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Scanning the structure of ill-known spaces: Part 1. Founding principles about mathematical constitution of space Michel Bounias and Volodymyr Krasnoholovets ______________________

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CONTENTS

CONTENTS continued

Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space Michel Bounias and Volodymyr Krasnoholovets ______________________

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Scanning the structure of ill-known spaces: Part 3. Distribution of topological structures at elementary and cosmic scales Michel Bounias and Volodymyr Krasnoholovets ______________________ 1005

The theory of cold quantum: a counter theory of gravitation Cao Junfeng __________________________________________________ 1021

Matter and vacuum. A new approach to the intimate structure of the universe Marcelo A. Crotti ______________________________________________ 1035

Panrelativity laws and scale relativity – against Einstein with Einstein’s Tao Xiangjun (Leon) Feng___________________________________________ 1043

A modification of the special theory of relativity B. Paul Gibson ________________________________________________ 1048

The light path in three-dimensional space B. Paul Gibson ________________________________________________ 1083

Relativistic coordinate shifting within three-dimensional space B. Paul Gibson ________________________________________________ 1099

Time-lapsed reality visual metabolic rate and quantum time and space John K. Harms ________________________________________________ 1113

The space-time equation of the universe dynamics Peter Kohut___________________________________________________ 1129

Asymmetry of uniform motion Peter Kohut___________________________________________________ 1134

The measurement of speed of gravitational wave Andrej Madac and Kamil Madac _________________________________ 1138

Relativity, contradictions, and confusions Cameron Rebigsol ______________________________________________ 1142

Regular Journal Sections Contemporary systems and cybernetics ____________ 1163 News, conferences and technical reports ___________ 1170 Internet commentary Alex M. Andrew _______________________________________________ 1178

Book reviews C.J.H. Mann __________________________________________________ 1182

Book reports _____________________________________ 1195 Announcements __________________________________ 1198 Special announcements ___________________________ 1201

CONTENTS continued

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EDITORIAL ADVISORY BOARD A. Bensoussan President of INRIA, France V. Chavchanidze Institute of Cybernetics, Tbilisi University, Georgia A.B. Engel IMECC-Unicamp, Universidad Estadual de Campinas, Brazil R.L. Flood Hull University, UK F. Geyer The Netherlands Universities Institute for Co-ordination of Research in Social Sciences, Amsterdam, The Netherlands A. Ghosal Honorary Fellow, World Organisation of Systems and Cybernetics, New Delhi, India R. Glanville CybernEthics Research, UK R.W. Grubbstro¨m Linko¨ping University, Sweden Chen Hanfu Institute of Systems Science, Academia Sinica, People’s Republic of China G.J. Klir State University of New York, USA Yi Lin International Institute for General Systems Studies Inc., USA

K.E. McKee IIT Research Institute, Chicago, IL, USA M. Ma˘nescu Academician Professor, Bucharest, Romania M. Mansour Swiss Federal Institute of Technology, Switzerland K.S. Narendra Yale University, New Haven, CT, USA C.V. Negoita City University of New York, USA W. Pearlman Technion Haifa, Israel A. Raouf Pro-Rector, Ghulam Ishaq Khan (GIK) Institute of Engineering Sciences & Technology, Topi, Pakistan Y. Sawaragi Kyoto University, Japan B. Scott Cranfield University, Royal Military College of Science, Swindon, UK D.J. Stewart Human Factors Research, UK I.A. Ushakov Moscow, Russia J. van der Zouwen Free University, Amsterdam, The Netherlands

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The mathematical logic failure of Einstein’s special relativity Arthur Bolstein Keywords Cybernetics, Logic The proof of the mathematical logic failure of Einstein’s special theory of relativity has been made. An impossible physical reality in Einstein’s relativity Arthur Bolstein Keywords Cybernetics, Theory A physical process, the course of which according to the Lorentz transformations is impossible in the frame of reference in inertial motion, can be created in some inertial frame of reference.

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Scanning the structure of ill-known spaces: Part 1. Founding principles about mathematical constitution of space Michel Bounias and Volodymyr Krasnoholovets Keywords Space, Structures, Theory, Topology, Cybernetics Some necessary and sufficient conditions allowing a previously unknown space to be explored through scanning operators are reexamined with respect to measure theory. Some generalized concepts of distances and dimensionality evaluation are proposed, together with their conditions of validity and range of application to topological spaces. The existence of a Boolean lattice with fractal properties originating from non-wellfounded properties of the empty set is demonstrated. This lattice provides a substratum with both discrete and continuous properties from which existence of physical universes can be proved, up to the function of conscious perception. Space-time emerges as an ordered sequence of mappings of closed 3D Poincare´ sections of a topological four-space provided by the lattice, and the function of conscious perception is founded on the same properties. Self-evaluation of a system is possible against indecidability barriers through anticipatory mental imaging occurring in biological brain systems; then our embedding universe should be in principle accessible to knowledge.

The possibility of existence of spaces with fuzzy dimension or with adjoined parts with decreasing dimensions is raised, together with possible tools for their study. The work presented here provides the introductory foundations supporting a new theory of space whose physical predictions (suppressing the opposition of quantum and relativistic approaches) and experimental proofs are presented in detail in Parts 2 and 3 of the study. Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space Michel Bounias and Volodymyr Krasnoholovets Keywords Structures, Theory, Cybernetics An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Space-time is represented by ordered sequences of topologically closed Poincare´ sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one section to the next involve morphisms of the general structures, representing a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides space-time with the features of a non-linear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as the mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then, the fractal kernel stands for a ‘‘particle’’ and the reduction of its volume (together with an increase in its area up to infinity) is compensated by morphic changes of a finite number of surrounding cells. Quanta of distances and quanta of fractality are demonstrated. The interactions of a moving particle-like deformation with the

surrounding lattice involves a fractal decomposition process, which supports the existence and properties of previously postulated inerton clouds as associated to particles. Experimental evidence of the existence of inertons is reviewed and further possibilities of experimental proofs proposed. Scanning the structure of ill-known spaces: Part 3. Distribution of topological structures at elementary and cosmic scales Michel Bounias and Volodymyr Krasnoholovets Keywords Structures, Theory, Cybernetics The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of setdistance function in such spaces provides a particular kind of space-time-like structure which favors the aggregation of such deformations into fractal forms standing for massive objects. A correlative dilatation of space appears outside the aggregates. At large scale, this dilatation results in an apparent expansion, while at submicroscopic scale the families of fractal deformations give rise to families of particle-like structure. The theory predicts the existence of classes of spin, charges and magnetic properties, while quantum properties associated with mass have previously been shown to determine the inert mass and the gravitational effects. When applied to our observable space-time, the model would provide the justifications for the existence of the creation of mass in a specified kind of void, and the fractal properties of the embedding lattice extend the phenomenon to formal justifications of big-bang-like events without any need for supply of an extemporaneous energy. The theory of cold quantum: a counter theory of gravitation Cao Junfeng Keywords Cybernetics, Gravity, Space Through many years of study, we have found that cold quantum is the most important force in nature. Under the pressure of coldness on hotness, various materials are formed. Under the pressure of cold quantum, these materials

are provided with gravity, and celestial bodies start to move. The pressure of cold quantum exists in space and materials. It is the pressure of cold quantum that huge changes between the four seasons on the earth begin to appear. The whirlpool, produced from the cold quantum pressure, pushes all the celestial bodies making them turn and change. The coldness converts frozen water into ice, which could not be achieved by any other force. The extreme and powerful strength of cold quantum has been wellknown. Therefore, we claim that the cold quantum pressure is the greatest force which ever existed in the universe. Matter and vacuum. A new approach to the intimate structure of the universe Marcelo A. Crotti Keywords Cybernetics, Space In order to describe and simplify the properties of material systems and their interactions, a simple model, based on linear oscillators, is presented. These oscillators define the space and time framework from which the length and time properties of material systems are derived. Matter and energy are postulated as the physical result of grouping and interaction among primary oscillators. The length of the material systems and the time required for the information to travel both ways (back and forth) change with the system’s motion. The derived formulas coincide with the special relativity transformations for space and time. Based on this model, the speed of light seems constant for all inertial systems. There is no contradiction with the special relativity theory in the usual case of the experimental results that imply two-way trips of the electromagnetic signals, but differences arise when only one-way phenomena are considered. Panrelativity laws and scale relativity – against Einstein with Einstein’s Tao Xiangjun (Leon) Feng Keywords Cybernetics, Theory, Quantum mechanics People have been led objectively to the directions of delusion and confusion since Einstein published his special relativity in 1905. Basically, Einstein’s special relativity is against the truth of the universe – The cause

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effect law. The fatal mistake of Einstein was, as the ‘‘collapse of the wave function’’ in modern quantum theory, not separating the physical reality from the observation of instruments and human being. This paper attempts to bring back science from inside the ‘‘Ha – Ha’’ mirror to the common senses and rational study. This paper reflects some important viewpoints of International Pansystems School. A modification of the special theory of relativity B. Paul Gibson Keywords Cybernetics, Theory Light, when constructed in terms of the elementary quanta of light, may be viewed in particle-like or wave-like terms. The elementary quanta of light, when placed in motion through space/time at a speed of a constancy of c forms a light path through the space or reference frame viewed. The light path formed is curved, as space/time is curved. The curvilinear light path formed is a function of the gravitational potential within the viewed frame of reference. The linear description of this light path, termed the geodesic (Riemannian), does not describe the curvilinear light path, but rather the chord of the curvilinear path described by the inscribed arc. This linear description of the light path is the manner in which we describe the coordinate system involved, and is the same manner in which we determine the ‘‘speed of light’’. The arc length of the light path, compared to the lesser value as described by the chord length, allows for a displacement to be determined, if both measures are applied to a linear measure. A displacement of linear coordinates then occurs, with this displacement a result of the gravitational potential occurring within the frame viewed. This displacement, derived via observation and predictions of the quantum model, resolves Maxwell as well as Newton. The theory concludes that the Special Theory of Relativity, suitably modified to account for gravitational displacement within one particular frame, derives a precise relative model of gravitation within the special frame. This model satisfies Newton, as the model arrives at an exact description of the threebody problem.

The light path in three-dimensional space B. Paul Gibson Keywords Cybernetics, Space, 3D Light, when viewed as a particle, reacts in a determinable manner with reference to the gravitational potential existing within the reference frame viewed. The elementary quanta of light, expressed under the terms of Planck, and as derived via the expressions of Einstein as a particle, may not reach a speed exactly equating to the speed (electromagnetic) of light of c. Here c is viewed as an electromagnetic constancy in any gravitational frame of reference. The theory is that a relative particle of mass may not achieve the speed of light, for the energy of that particle would then equate to infinity or in that the force required allowing the relative particle to reach c would then be infinite. The theory is then totally reliant upon the tenants of what has become to be known as the Special Theory of Relativity. As per the General Theory, light would be ‘‘bent’’, more or less, from one gravitational reference frame as compared to another gravitational reference frame. The theory then evolves that light, when viewed as a particle, forms a curvilinear light path through the gravitational reference frame viewed. However, until now, the light path has been solely described on a linear basis. It is the result of the theory that the light path may be described on a curvilinear basis, under the method of Lagrange. This method, or model, allows a particle of light (viewed as a projectile of mass under a constant velocity, therefore under a constant acceleration) to achieve Newton’s description of the path of a projectile. Note that the following paper is applicable to a previous paper, which proposes a displacement of light within the gravitational field.

Relativistic coordinate shifting within three-dimensional space B. Paul Gibson Keywords Cybernetics, Space, 3D Let us consider that light, when viewed as a particle, forms a conic arc segment inscribed within the space viewed. The space (or frame)

viewed is considered to exhibit a gravitational potential, and it is thus this potential that deforms the light path from a Euclidean/ Newtonian derivation of a straight line to that of a relativistic curvilinear nature. Given a distance over this conic arc segment (assumed to form a parabolic arc segment) and a given time (considering the given distance involved), one derives a constancy of the speed of light of c, where c is considered as a constant regardless of the gravitational potential exhibited by the frame viewed. If we further consider that the Special Theory requires that light propagate on a linear measure as the velocity v (of necessity v being less than c on a comparable linear measure) between the axes concerned; then a displacement (in linear measure equal to c 2 v) occurs. The displacement evolved is then assumed to agree with the form of Maxwell. We assume that this linear displacement of c 2 v occurs upon the y-axis of the frame viewed. Of necessity, a relative displacement must occur upon the x-axis of the frame viewed. From the calculus, the dot products derived must vary in concept, in order to derive the totality of relative coordinate shifts occurring within any threedimensional space. One displacement is linear in nature, while the other is trigonometric in nature. We consider the displacement of Maxwell, Lorentz, Compton, and de Broglie to be linear in nature. Based on the principle of the Special Theory (and the other forms as mentioned), we consider the total displacement to be mechanically derivable. That derivation, once allowed, results the physics to agree with the observations complete to this moment in time. The paper concludes that the error in coordinate positioning shown by the global positioning system (GPS satellite platform) is resolvable.

Time-lapsed reality visual metabolic rate and quantum time and space John K. Harms Keywords Time, Cybernetics, Entropy, Space This text proposes that time is essentially related to one’s visual metabolic rate. Metabolic rate is regulated by the speed of the intake of energy, the rate of the production

of adenosine triphosphate (ATP) to the visual areas. The author’s working hypothesis is that the visual area of the brain known as human V5, the region involved in motion detection, may be the region most responsible for time perception. Our time sense from all the senses is, thus, compiled in the visual region, the speed of the human perception of reality. Time is the relationship of the human perception of reality and the rate that the reality itself is taking place (given by light waves in the environment). Hence, vision (and the visual cortex area V5) may be the vitally important aspects in answering the question: what is time? When we are not looking at a clock, time may be governed by our rate of metabolism; rate of the production of ATP by the mitochondria in V5. For example, when general human metabolism (and V5) is fast, time runs slow. When metabolic rate is relatively slow, time runs relatively faster. Many factors enter into the speed of metabolism such as age, sex, drug effects, velocity compared to speed c, states of boredom or excitement, darkness or light and mental states such as sleep. The relationship between time and space is discussed with the metabolic rate of V5 in mind. Because the uncertainty principle and the quantum picture of reality are adopted, this model qualitatively quantizes space and time, showing why they must forever be connected i.e. space-time. This idea is discussed in relation to Zeno’s paradox, which suggests that space and time are indeed quantized. Events, instants and entropy are defined. Reality can be understood in terms of the speed of the processing of instants. The arrow of time is pictured as caused by long-term potentiation of synaptic neurons within the brain. Minkowski-Einstein space-time is analyzed and compared with the visual metabolic rate. The probable consequences of this model are proposed.

The space-time equation of the universe dynamics Peter Kohut Keywords Cybernetics, Dynamics, Space, Time Space and time are the forms of material being of the Universe. They have a quantum

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character as a result of a dialectic relation between attraction and expansion, continuity and discreetness. Time and space consist of their elementary parts – quantums. The question is: does the Universe have its own time, despite the fact that according to the theory of relativity, the local parts of the Universe with different gravity or inertia have different local times? Yes, it does! The time etalon for the entire history of the Universe is represented by the cosmic quantum jump, in which the Universe realizes its transition from one quantum level to the next. The time quantifying enables the cosmic jump from the expansion to the contraction to be made and on the other hand, from the contraction to the expansion without getting to singularities. The Universe pulsates. The knowledge of the nature of time and space quantifying allowed us to find the basic equation of the space dynamics of the Universe. Its analysis gives the answer to the question: what is the nature of gravity, speed of light and gravitational constant.

Asymmetry of uniform motion Peter Kohut Keywords Cybernetics, Motion A new theory on asymmetry of uniform motion is presented.

The measurement of speed of gravitational wave Andrej Madac and Kamil Madac Keywords Cybernetics, Gravity This paper introduces an hypothesis concerning the reason of gravitation. Preliminary results of the experiment, not finished yet, are presented.

Relativity, contradictions, and confusions Cameron Rebigsol Keywords Cybernetics, Theory, Time, Space The derivation leading to the formulation of Lorentzian transformation in special relativity is actually a duplication of an ancient ‘‘miracle’’ in algebra: 2x 2 x ¼ 0, 2x ¼ x, 2 ¼ 1. Dominated by such a mathematical confusion, relativity displays fundamental uncertainty in understanding physics. As such, with equations, it claims to have ‘‘discovered’’ two speed limits in nature: the speed of light in the vacuum space and the speed of light at the mass center of a material body. Needless to say, these two speed limits repel each other, not to mention that the second speed limit is even against nature. Relativity then further extends this confusion and uncertainty in physics to make up many self-contradicting concepts. These concepts include the so-called homogeneous gravitational field and the idea of having (circumference/diameter) . 3.1415926. . . for a spinning circle. With the same mathematics guiding to its ‘‘success’’, however, relativity presents no homogeneous gravitational field, but a monster that must be called a homogeneously inhomogeneous field for its appropriation. Based on the same erroneous mathematics, relativity must force itself to have (circumference/diameter) , 3.1415926. . . for a spinning circle. With the idea of a homogeneous gravitational field, relativity believes that it can establish the validity of the so-called Principle of Equivalence for the legitimacy of general relativity. However, Newtonian mechanics, supported by the close orbital movements of numerous heavenly objects, must witness the nonexistence of such a ‘‘principle’’ in nature.

Preface

Preface

Special double issue: Some new theories about time and space Guest Editors: Leon Feng, B. Paul Gibson and Yi Lin This special double issue is concerned with some of the new theories of space and time. We would wish to express our thanks to the Guest Editors, Leon Feng, B. Paul Gibson and Yi Lin for accepting the invitation of the Editorial Advisory Board (EAB) of this journal to prepare it. The invitation follows the success of workshops and seminars organised by them, particularly, at the recent 12th Congress of Systems and Cybernetics (WOSC, 2002). It was at that gathering, they brought together some of the leading experts in the field and arranged presentations, which caught the imagination of the Congress participants. They have carefully selected contributors for this issue whose work, our referees believe, provides new and exciting advances in the study of these fascinating concepts. It has to be said, however, that not all members of the EAB of this journal consider the chosen subject to be relevant to what they regard as ‘‘true’’ cybernetics. Others quickly responded that without cybernetics and systems venturing into new fields that reflected current interests, there would be no progress in its studies and applications in the 21st century. Indeed, they pointed out that without the new approaches and application of cybernetics and systems by Norbert Wiener as exemplified by, for example, Wiener’s Cybernetics (Wiener, 1961) where he affirms that ‘‘. . . if a scientific subject has a real vitality, the center of interest in it must and should shift in the course of years’’ there may have been little change from Ampe`re’s ‘‘cybernetique’’ of the art of governing in general (Ampe`re, 1838). Dr Grey Walter made the comment (Walter, 1969) that:

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So often as a cybernetical analysis merely confirmed or described a familiar phenomenon in biology or engineering, so rarely has a cybernetical theorem predicted a novel effect or explained a mysterious one

It may be well that this issue, by presenting some of the new theories on space and time, will go some way towards achieving Wiener’s hopes and contradicts Grey Walter’s comment. Without accepting innovative ideas, cybernetics would not have embraced, for example, the works of such important figures of our evolving field as Stafford Beer and Heinz von Foerster. Whilst the traditional researches and our current interests still continue to flourish, there is still a need for the injection of new and exciting ideas into our endeavours in systems and cybernetics. We believe that this specially compiled double issue will contribute to the way forward. As is our current practice, we have also included our ‘‘regular journal sections’’ in addition to the specially contributed papers. Readers are reminded

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of the section ‘‘Communications and forum’’, which appears in our regular issues. It is designed to encourage reader’s comments and alternative viewpoints on all matters pertaining to cybernetics and systems, particularly those raised in issues such as this one, that is devoted to a specially selected area of interest. Brian H. Rudall Editor-in-Chief References Ampe`re, Andre´-Marie (1838), Essai sur la philosophie des sciences ou exposition analytique d’une classification naturelle de toutes les connaissances humaines, Paris, Bachelier, premie`re partie; 1843, seconde partie. Walter, W. Grey (1969) Neurocybernetics in Rose, J. (Ed.) Survey of Cybernetics, Gordon and Breach, New York, pp. 93-108. Wiener, A. (1961), Cybernetics, The MIT Press and Wiley, New York, paperback edition. The MIT Press, 1965 (this is the second edition of Cybernetics or Control and Communication in the Animal and Machine published in 1948). WOSC (2002), World Organisation of Systems and Cybernetics (WOSC) 12th Congress and 4th Workshop of the International Institute of General Systems and Cybernetics (IIGSS), Pittsburgh, Pensylvania, March 2002. (See also Kybernetes, Vol. 31 Nos 9/10, 2002 and Vol. 32 Nos 5/6, 2003 for selected contributions.)

Guest Editorial Time and Space, One Hot Topic of Our Modern Time Recently, Mr Yong Wu and I published a book entitled ‘‘Beyond Nonstructural Quantitative Analysis: Blown-Ups, Spinning Currents and Modern Science’’, (World Scientific, 2002, ISBN 981-02-4839-3). One of the main results proved in this volume is that the general form of motion of materials in the universe is eddy motion. And, all materials’ movements are resulted from unevenness of materials. With eddy motions as the general form of materials’ movements in the universe, we argued that time is nothing but a measurement of rotation and space the relative location in various whirlpools of materials. With the help of eddy motions, we have successfully coined the two concepts: black holes and big bangs, into one multi-dimensional entity as follows (Figure 1). By pondering over this model of our universe, where a great number of such structures co-exist, one can draw conclusions about and provide explanations for many interesting phenomena. For example, with this model in place, one can answer such question as: What is gravitation? For more details, the reader is advised to check our book out. What we argued is not just a coined structure for our universe, but also the very existence of time and space is embedded in materials, which exist first. If we combine our results described here with those derived in another publication by Ren, Z.Q., OuYang, S.C. and me (Conjecture on Law of Conservation of Informational Infrastructure, Kybernetes, Vol. 27, pp. 543-552), one can see another interesting phenomenon: time and space expand, then contract, and then expands, then contracts. This cyclic process goes on and on forever. Historically, the concepts of time and space have been felt and mentioned in a great many scientific works. Recently, Isaac Newton and his followers widely applied these intuitive concepts. Then, it was Albert Einstein whose work pointed to the need for a more in-depth study of these concepts. If one is interested in learning about more recent progress along this line, you only need to do a simple search on the web and you will get an abundant source of either new publications or existing works. It is truly our fortune that Dr Robert Vallee and I had the opportunity and honor to organize the joint conference in Pittsburgh, Pennsylvania, USA, during 24-26 March 2002. There, Dr Leon Feng organized a special workshop on new theories of space-time. Later on, he and Dr Paul Gibson chaired the sessions jointly. With our joint effort (Leon, Paul and me), we now present all the papers delivered at the conference either in person or by title, suitably updated, to our readers in a much greater audience through Kybernetes. So, at this special place and moment, on behalf of Leon and Paul, I would like to take this opportunity to express our heartfelt appreciation to Dr Brian Rudall,

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Figure 1.

Editor-in-Chief of Kybernetes, Drs Robert Vallee and Alex Andrew of World Organisation of Systems and Cybernetics (WOSC), whose participation in our Pittsburgh event had surely made it more exciting, and all other officials of the WOSC, for their support and encouragement. Yi Lin Department of Mathematics, Slippery Rock University, Slippery Rock, PA, USA E-mail: [email protected]

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The mathematical logic failure of Einstein’s special relativity Arthur Bolstein Bolstein Research Team, Brectanova, Kosice, Slovakia

The mathematical logic failure 943

Keywords Cybernetics, Logic Abstract The proof of the mathematical logic failure of Einstein’s special theory of relativity has been made.

Introduction Inertial frame S 0 is moving in the inertial frame S along its x-axis that coincides with the x 0 -axis of S 0 . The clock C0 in the origin of S and the clock C 00 in the origin of S 0 display zero in both the frames of reference S and S 0 at the moment when the origins coincide. There is the clock C 01 synchronized in the frame of reference S 0 with C 00 in the point x 0 on the right from the origin of S 0 . There is also another inertial frame S 10 in the rest to S 0 with the origin located in x 0 of S 0 . The clock C 10 in the origin of S 01 and the clock C1 in the origin of the inertial frame S1 located in some point x of S in the rest to S, display zero in both frames of the references S 01 and S1 at the moment when the origins coincide. Proof According to the Lorentz transformations (Feynman et al., 1966) as it is given in the “Introduction”, C0 with C 00 and C 01 with C1 coincide simultaneously in the frame of the reference S 0 , so C0 with C 00 and C 01 with C1 do not coincide simultaneously in the frame of reference of S. Consequently, C0 displays zero not simultaneously with C1 in the frame of S. According to the “Introduction”, C0 displays zero in the frame of the reference S, when simultaneously C 00 displays zero in the frame of the reference S 0 , when simultaneously C 01 displays zero in the frame of reference S 10 (S 01 is in the rest to S 0 ), simultaneously C1 displays zero in the frame of the reference S1. Consequently, C0 displays zero in the frame of the reference of S, simultaneously C 1 displays zero in the frame of the reference of S (S is in the rest to S1). Conclusion The mathematical logic failure of Einstein’s special theory of relativity has been proved. References Feynman, R.P., Leighton, R.B. and Sands, M. (1966), The Feynman Lectures on Physics, 4th edn, Addison-Wesley, Reading, MA, USA.

Kybernetes Vol. 32 No. 7/8, 2003 p. 943 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483108

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An impossible physical reality in Einstein’s relativity Arthur Bolstein Bolstein Research Team, Brectanova, Kosice, Slovakia Keywords Cybernetics, Theory Abstract A physical process, the course of which according to the Lorentz transformations is impossible in the frame of reference in inertial motion, can be created in some inertial frame of reference.

Introduction A real physical body, for example, steel is in inertial motion along a flat surface. The velocity of the body is close to the velocity of light. There is a hole interrupting the flat surface in front of the body. Take an inertial frame of reference that is attached to the flat surface that we call the rest inertial frame and one that is attached to the body while it is in inertial motion that we call the moving inertial frame. Let the length of the hole in the rest inertial frame be extreme, and let the object, for example, be 1 km long in the moving inertial frame. Problem According to the theory, consider the body to be extremely contracted in the rest inertial frame from initial position to accelerate perpendicularly into the hole with constant acceleration at the moment when the back of the body is at the closer edge of the hole in the rest inertial frame, and let the front of the body in this rest inertial frame be at the opposite wall of the hole, several meters below the surface. However, if the velocity of the moving inertial frame is sufficiently close to the velocity of light, the length of the hole in the moving inertial frame is close to zero according to the theory. At some sufficient velocity the hole can be, for example, 1 m long there. Therefore, in the moving inertial frame the above given physical process, until the front of the object is at the opposite wall several meters below the surface, is possible only if there is an impossible deformation of the object in this moving inertial frame. Conclusion There is some physical process in which the Lorentz transformations used have impossible course in some frame of reference. Kybernetes Vol. 32 No. 7/8, 2003 p. 944 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483117

Reference Feynman, R.P., Leighton, R.B. and Sands, M. (1966), The Feynman Lectures on Physics, 4th ed., Addison-Wesley, Reading, MA, USA.

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Scanning the structure of ill-known spaces Part 1. Founding principles about mathematical constitution of space

Structure of ill-known spaces Part 1 945

Michel Bounias BioMathematics Unit (University/INRA), Domain of Sagne-Soulier, Le Lac d’Issarle`s, France

Volodymyr Krasnoholovets Institute of Physics, National Academy of Sciences, UA, Kyı¨v, Ukraine Keywords Space, Structures, Theory, Topology, Cybernetics Abstract Some necessary and sufficient conditions allowing a previously unknown space to be explored through scanning operators are reexamined with respect to measure theory. Some generalized concepts of distances and dimensionality evaluation are proposed, together with their conditions of validity and range of application to topological spaces. The existence of a Boolean lattice with fractal properties originating from non-wellfounded properties of the empty set is demonstrated. This lattice provides a substratum with both discrete and continuous properties from which existence of physical universes can be proved, up to the function of conscious perception. Space-time emerges as an ordered sequence of mappings of closed 3D Poincare´ sections of a topological four-space provided by the lattice, and the function of conscious perception is founded on the same properties. Self-evaluation of a system is possible against indecidability barriers through anticipatory mental imaging occurring in biological brain systems; then our embedding universe should be in principle accessible to knowledge. The possibility of existence of spaces with fuzzy dimension or with adjoined parts with decreasing dimensions is raised, together with possible tools for their study. The work presented here provides the introductory foundations supporting a new theory of space whose physical predictions (suppressing the opposition of quantum and relativistic approaches) and experimental proofs are presented in detail in Parts 2 and 3 of the study.

1. Introduction Starting from perceptive aspects, experimental sciences give rise to theoretical descriptions of hidden features of the surrounding world. On the other hand, the mathematical theory of demonstration reveals that any property of a given object, from a canonical particle to the universe, must be consistent with the characteristics of the corresponding embedding space (Bounias, 2000a). In short, since what must be true in abstract, mathematical conditions should also be fulfilled upon application to the observable world, whether the concept of “reality” has a meaning or not. Conversely, since similar predictions can infer when classical properties of space-time are tested against various theories, e.g. an unbounded or a bounded (non-Archimedean) algebra (Avinash and Rvachev, 2000), the most general features should be

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accessible without previous assumptions about peculiarities of the explored spaces. Indeed, even the abstract branch of sciences, e.g. pure mathematics, show a tendency in going to a form of “experimental research”, essentially under the pressure of some limitations that metamathematical considerations raise about the fundamental questions of decidability (Chaitin, 1998, 1999). However, algorithmic information theory does not embrace the whole of mathematics, and the theory of demonstration escapes the limitations of arithmetical axiomatics, in particular through anticipatory processes (Bounias, 2000b, 2001). To which extent the real world should obey just arithmetics rules remains to be supported, and instead, the foundations of the existence and functionality of a physical world have been shown to be more widely provided by general topology (Bounias, 2000a). On the other hand, how to explore a world considered as a system has been the matter of thorough investigation by Lin (1988, 1989). The goal of the present work is to examine in depth some founding mathematical conditions for a scientific scanning of a physical world to be possible through definitely appropriate tools. In this respect, it is kept in mind that human perceptions play a part, at least, since humans (which includes scientists) are self-conscious of their existence through their perception of an outside world, while they believe in the existence of this outside world because they perceive it. This “judge and party” antinomy will be addressed in the present paper. The first part of this paper deals with the notions of measure and distances in a broad topological sense, including the assessment of the dimensionality of a space whose detached pieces (i.e. the data collected through apparatus from a remote world) are scattered and displayed on the table of a scientist, in view of a reconstruction of the original features. The existence of an abstract lattice will be deduced and shown to stand for the universe substratum (or “space”). The second part will focus on the specific features of this lattice and a confrontation of this theoretical framework with the corresponding model of Krasnoholovets and Ivanovsky (1993) about quantum to cosmic scales of our observed universe will be presented, along with the experimental probes of both the theory and the model. The third part will further present the structures predicted for elementary particles whose existence derives from the described model, and lead to a confrontation of the predictive performances of the various theories in course. 2. Preliminaries 2.1 About the concepts of measure and distances Whatever be the actual structure of our observable space-time, no system of measure can be operational if it does not match the properties of the measured objects. Scanning a light-opaque world with a light beam, by ignorance of

the fundamental structure of what is explored, though a caricatural example, Structure of illustrates the principle of a necessarily failing device, whose results would raise ill-known spaces discrepancies in the knowledge of the studied world. An example is given by the Part 1 recent development of UV astronomy: when the sky is scanned through UV instead of visible radiation, the resulting extreme UV astrophysical picture of our surrounding universe becomes different (Malina, 2000). 947 One of the problems faced by modeling unknown worlds could be called “the syndrome of polynomial adjustment”. In effect, given an experimental curve representing the behavior of a system whose real mechanism is unknown, one can generally perform a statistical adjustment by using a polynomial system like M¼

N X

ai x i :

i¼0

Then, using an apparatus deviced to test for the fitting of the N þ 1 parameters to observe data will require increasingly accurate adjustment, so as to convincingly reflect the natural phenomenon within some boundaries, while if the real equation is mathematically incompatible with the polynomial, there will remain some irreducible parts in the fitting attempts. This might well be what occurs to the standard cosmological model and its 17 variables, with its failure below some quantum scales (Arkani-Hamed et al., 2000). Furthermore, Wu and Li (2002) have demonstrated how the approximations of solutions of equations describing non-linear systems mask the real structures of these systems. It may be that what is tested in accelerators is a kind of self-evaluation of the model, which poses a problem with respect to the indecidability of computed systems as successively raised by Church (1936), Tu¨ring (1937), Go¨del (1931), and more recently by Chaitin (1998, 1999). However, mathematical limits in computed systems can be overpassed by the biological brain’s system, due to its property of self-decided anticipatory mental imaging (Bounias, 2001). It will be shown how this makes eventually possible a scanning of an unknown universe by a part of itself represented by an internal observer. 2.1.1 Measure. The concept of measure usually involves particular features such as the existence of mappings and the indexation of collections of subsets on natural integers. Classically, a measure is a comparison of the measured object with some unit taken as a standard ( James and James, 1992). However, sets or spaces and functions are measurable under various conditions which are cross-connected. A mapping f of a set E into a topological space T is measurable if the reciprocal image of an open of T by f is measurable in E, while a set measure on E is a mapping m of a tribe B of sets of E in the interval [0, 1], exhibiting denumerable additivity for any sequence of disjoint subsets (bn) of B, and denumerable finiteness, i.e. respectively:

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  X 1 1 m < bn ¼ mðbn Þ n¼0

ð1:1Þ

n¼0

’ An; An [ B; E ¼
ð1:2Þ

Now, coming to the “unit used as a standard”, this is the part played by a gauge ( J ). Again, a gauge is a function defined on all bounded sets of the considered space, usually having non-zero real values, such that (Tricot, 1999): (1) a singleton has measure naught: ;x; J ð{x}Þ ¼ 0; (2) ( J ) is continued with respect to the Hausdorff distance; (3) ( J ) is growing: E , F ) J ðEÞ , J ðFÞ; (4) ( J ) is linear: F(r · E ) : r · J(E ). This implies that the concept of distance is defined: usually, a diameter, a size, or a deviation are currently used, and it should be pointed that such distances need to be applied on totally ordered sets. Even the Caratheodory measure (m*) poses some conditions which again involve a common gauge to be used: (1) A , B ) m* ðAÞ # m* ðBÞ; P (2) For a sequence of subsets ðEiÞ : m* < ðRiÞ # m* ðRiÞ; (3) /ðA; BÞ; A > B ¼ B : m* ðA < BÞ ¼ m* ðAÞ þ m* ðBÞ (in consistency with equation (1.1)); (4) m* ðEÞ ¼ m* ðA > EÞ þ m* ð›E A > EÞ: The Jordan and Lebesgue measures involve respective mappings (I ) and (m*) on spaces which must be provided with > , < and ›. In spaces of the R n type, tessellation by balls are involved (Bounias and Bonaly, 1996), which again demands a distance to be available for the measure of diameters of intervals. A set of measure naught has been defined by Borel (1912) first as a linear set (E ) such that, given a number (e) as small as needed, all points of E can be contained in intervals whose sum is lower than (e). Remark 2.1. Applying Borel intervals imposes that appropriate embedding spaces are available for allowing these intervals to exist. This may appear as in explicitly formulated axiom, which might involve important consequences (see later). 2.1.2 A corollary on topological probabilities. Given a set measure (P ) on a space E ¼ ððX; AÞ; ’Þ; (A) a tribe of parts of X, then (Chambadal, 1981): for a [ A; PðaÞ ¼ Probability of (a) and PðXÞ ¼ 1; for A, B in X, one has PðA < BÞ ¼ PðAÞ þ PðBÞ; for a sequence {An} of disjoint subspaces, one has: lim PðAnÞ ¼ 0:

n!1

A link can be noted with the Urysohn’s theorem: let E, F be two disjoint parts Structure of of a metric space W: there exists a continuous function f of W in the real ill-known spaces interval [0, 1] such that f ðxÞ ¼ 0 for any x [ E; f ðxÞ ¼ 1 for any x [ F; and Part 1 0 , f ðxÞ . 1 in all other cases. This has been shown to define conditions providing a probabilistic form to a determined structure holding for a deterministic event (Bonaly and Bounias, 1995). In effect, if W is the 949 embedding space and S a particular state of universe in W as recalled in Section 3 below. Let C be the complementary of S in W, and x an object in the set of closed sets in W. Let Ix(x) an indicative function such that I S ðxÞ ¼ 1 if x [ S and I S ðxÞ ¼ 0 if x [ C: Writing I S ðxÞ ¼ PðXÞ: x [ S ) PðxÞ ¼ 1; x [ X ) PðxÞ ¼ 0 and 0 , PðxÞ , 1 in all other cases. A probabilistic adjustment, as accurate as it can show, is thus not a proof that a phenomenon is probabilistic in its essence. 2.1.3 Distances. Following Borel, the length of an interval F ¼ ½a; b is: X LðFÞ ¼ ðb 2 aÞ 2 LðCnÞ ð1:3Þ n

where Cn are the adjoined, i.e. the open intervals inserted in the fundamental segment. Such a distance is required in the Hausdorff distances of sets (E ) and (F ): Let E(e) and F(e) be the covers of E or F by balls B(x, e), respectively, for x [ E or x [ F; distH ðE; FÞ ¼ inf{e: E , FðeÞ ^ F , EðeÞ}

ð1:4aÞ

distH ðE; FÞ ¼ ðx [ E; y [ F : inf dist ðx; yÞÞ

ð1:4bÞ

Since such a distance, as well as most of the classical ones, is not necessarily compatible with the topological properties of the concerned spaces; Borel provided an alternative definition of a set with measure naught: the set (E ) should be Vitali-covered by a sequence of intervals (Un) such that: each point of E belongs to a infinite number of these intervals, and the sum of the diameters of these intervals is finite. However, while the intervals can be replaced by the topological balls, the evaluation of their diameter still needs an appropriate general definition of a distance. A more general approach (Weisstein, 1999b) involves a path w(x, y) such that wð0Þ ¼ x and wð1Þ ¼ y: For the case of sets A and B in a partly ordered space, the symmetric difference DðA; BÞ ¼ ›A BÞ has been proved to be a true distance also holding for more than two sets (Bounias, 1997-2000; Bounias and Bonaly, 1996). However, if A > B ¼ B; this distance remains D ¼ A < B; regardless of the situation of A and B within an embedding space E such that ðA; BÞ , E:

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A solution to this problem will be derived below in terms of a separating distance versus an intrinsic distance. 2.2 On the assessment of space dimensions One important point is the following: in a given set of which members structure is not previously known, a major problem is the distinction between the unordered Nuples and the ordered Nuples. This is essential for the assessment of the actual dimension of a space. 2.2.1 Fractal to topological dimension. Given a fundamental segment (AB) and intervals Li ¼ ½Ai; Aði þ 1Þ; a generator is composed of the union of several such intervals: G ¼ <ði[½1;nÞ ðLiÞ: Let the similarity coefficients be defined for each interval by ri ¼ distðAi; Aði þ 1ÞÞ=distðABÞ: The similarity exponent of Bouligand is (e) such that for a generator with n parts: X ð2:1Þ ðriÞe ¼ 1 ði[½1;nÞ

When all intervals have (at least nearly) the same size, then the various dimension approaches according to Bouligand, Minkowsky, Hausdorff and Besicovitch are reflected in the resulting relation: nðrÞe < 1

ð2:2aÞ

e < log n=log r

ð2:2bÞ

that is

When e is an integer, it reflects a topological dimension, since this means that a fundamental space E can be tessellated with an entire number of identical balls B exhibiting a similarity with E, upon coefficient r. 2.2.2 Parts, ordered N-uples and simplexes. 2.2.2.1. Parts. A set is composed of members of which some are themselves containing further members. In solving the Russell and Burali-Forti paradoxes by making more accurate the definition of a set, Mirimanoff (1917) classified members in nuclei (i.e. singletons or “atoms”) with no members inside themselves, and parts containing members. Then, a set E is said to be of first kind (E I) if it is isomorphic to none of its members, and second kind (E II) if it is isomorphic to at least one of its members. Hence, E ¼ {a; b; c; ðd; e; f Þ} is first kind since X ¼ ðd; e; f Þ is not isomorphic to {a; b; c; ðXÞ}: F ¼ {a; ðb; ðc; ðZ ÞÞ} is of second kind since by posing H ¼ {c; ðZ Þ} and G ¼ {b; ðH Þ}; it appears that F ¼ {a; ðGÞ} is isomorphic to G ¼ {b; ðH Þ}; as well as to H, and eventually further to members of H. Mirimanoff called “descent” a structure of the following form, where (E ) denotes a set or part of set, and (e) a nucleus:

Structure of ill-known spaces Part 1 A descent is finite if none of its parts are infinitely iterated. Then, a second kind E ðnÞ ¼ {e ðnÞ ; ðE ðnþ1Þ Þ}

ð3:1Þ

set is ordinary if its descent is finite and extraordinary if its descents include some infinite part. One can recognize a fractal feature in an extraordinary second kind set. Remark 2.2. A finite number (n) of iterations provide a form of measure called the length of the descent. In the above example of set F, assuming Z is not isomorphic to F, the sequence {F, G, H } is bijectively mapped on segments {1, 2, 3} of natural integers. 2.2.2.1 Ordered N-uples. Let the members of the above set E be ordered into a structure of the type of F, for example: F 0 ¼ {a; ðb; ðc; ðd; e; f ÞÞÞ}: The length of the descent of the exemplified F 0 is LðF 0 Þ ¼ 3; and the last part is not isomorphic to F 0 . Now, suppose that members ða; b; c; . . .Þ are similar, that has no particular structure nor essential feature allow one to be distinguished from the other. Then, F 0 could be written in the alternative form: F 00 ¼ {{a}; {a; {a; . . .}}}: This indicates the availability of an order to hold on F 00 . Usually, a part (a, b) or {a, b} is not ordered until it can be written in the form: ðabÞ ¼ {{a}; {a; b}}: The non-empty part of (ab) owns a lower boundary: {a}, and a upper boundary: (ab) itself. Similarly: ðabcÞ ¼ {{a}; {a; ða; bÞ}; {a; ða; bÞ; ða; b; cÞ}}; etc:

ð3:2Þ

Stepping from a part ða; b; c; . . .Þ to a N-uple (abc) needs that singletons are available in replicates. Two cases are met: if members ða; b; c; . . .Þ are not identical, replicates can be found in the set of parts of the set. Otherwise, if members are similar singletons, then the set is just isomorphic with a segment of the set of natural integers. In both cases, for any pair of members (i.e. as subset members or singletons), the cartesian product will give a set of ordered pairs. Repeating the operation in turn gives ordered N-uples. A formal distinction of (xy), (x, y) and {x, y} will appear below. 2.2.2.2 Simplexes. A simplex is the smaller collection of points that allows the set to reach a maximum dimension. In a general acception, it should be noted that the singletons of the set are called vertices and ordered N-uples are ðN 2 1Þ faces A N 21 : A set of ðN þ 1Þ members can provide a structure of dimension at most (N ), that is: a connected “ðN þ 1Þ 2 object” has dimension d # N: The number of 1-faces A1 will be ðN þ 1ÞN =2!; the number of 2-faces A2 will be ðN þ 1ÞN ðN 2 1Þ=3!; nðA3 Þ ¼ ðN þ 1ÞN ðN 2 1ÞðN 2 2Þ=4!; and finally nðA k Þ ¼ ðN þ 1ÞNðN 2 1Þ. . .ðN 2 k þ 1Þ=ðk þ 1Þ! (Banchoff, 1996). One question emerging now with respect to the purpose of this study is the following: given a set of N points, how to evaluate the dimension of the space embedding these points?

951

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3. Distances and dimensions revisited 3.1 The relativity of a general form of measure and distance Our approach aims at searching for distances that would be compatible with both the involved topologies and the scanning of objects not yet known in the studied spaces. No such configuration is believed to be an exception or a general case. 3.1.1 General case of a not necessarily ordered topological distance. Proposition 3.1. A generalized distance between spaces A, B within their common embedding space E is provided by the intersection of a path-set w(A, B) joining each member of A to each member of B with the complementary of A and B in E, such that: w(A, B) is a continuous sequence of a function f of a gauge ( J ) belonging to the ultrafilter of topologies on {E; A; B; . . .}: Proof. (i) A measure comes to reality when it maps a perceiving system (e.g. A) to a perceived one (e.g. B) so that B is measured by A within E. It has been formerly shown that A and B should be topologically closed, since Jordan’s points are needed for the characterization of a path joining any point of the interior of B to a point of the interior of A. The non-empty intersection b of the path with the frontier ›B of B leads to the intersection a of the path with the frontier ›A of A, and a sequence of mappings u( a ) of ( a ) to a fixed point f ðaÞ ¼ a in A provides the mathematical foundation of a mental image (Bounias and Bonaly, 1997a, b) (Figure 1).The path w (A, B) is a set composed as follows: (i) w ðA; BÞ ¼ <w ða; bÞ; all defined on a sequence of interval ½0; f n ðxÞ; x [ E: a[A; b[B

Then, for any closed D situated between A and B, f n(m) intersects the frontiers of B, D, and A: thus, the sequence f n(x) has some of its points identified with b; ðdi; dj; . . . [ ›DÞ; and a. Therefore, the relative distance of A and B in E, noted LE (A, B) is contained in w (A, B): LE ðA; BÞ # wðA; BÞ

Figure 1. Some features of the measure of a distance between closed sets (A, B) in an embedding space E. Space A is the perceiver and space B is the perceived one. Points a, b, d1, d2 are Jordan’s points

ð4:1Þ

Eventually, one may have in the above case: min{LE ðA; BÞ} ¼ {b; di; dj; a}; Structure of while in all cases: inf{LE ðA; BÞ} ¼ { b _ a }: ill-known spaces Denote by E8 the interior of E, then: Part 1 min {wðA; BÞ > E8} is a geodesic of space E connecting A to B ð4:2aÞ max {w ðA; BÞ > ›E8 ðA < BÞ} is a tessellation of E out of A and B

ð4:2bÞ

It is noteworthy that the relation (4.2a) refers to dimL ¼ dimw; while in relation (4.2b) the dimension of the probe is that of the scanned sets. (ii) Let J ¼ f n ðmÞ; such that: wð0Þ ¼ b and wð f n ðmÞÞ ¼ a: Then, m [ UðEÞ; with U(E ) the ultrafilter on topologies of E. Suppose that m  UðEÞ: Then, there exists a filter FA holding on A and a filter FB holding on B, such that FA – FB : Let x [ FA : there exists y [ FB with x – y and y  FA : Therefore, if x [ wð f n ðmÞÞ; y  wð f n ðmÞÞ and the path-set does not measure B from a perception by A. That m [ UðEÞ is a necessary condition. (iii) Let (O) be a open set of E. Then, a member of w(A, B) joining B to A through O meets no frontier other than ›B and ›A, and the obtained LE (A, B) ignores set O. As a consequence, only closed structures can be measured in a topological space by a path founded on a sequence of Jordan’s points: this justifies and generalizes the Borel measure recalled earlier. In contrast, if a closed set D contains closed subsets, e.g. D 0 , D; then there is a member of w(A, B) which intersects D 0 . If in addition the path is founded on f n(m), m [ UðEÞ; there exists non-empty intersections of the type d 0 i, d 0 j [ f n ðmÞ > ›D0 : Therefore, LE (A, B) will include d0 i and d0 j to the measured distance. This shows that wðA; BÞ > E8 is a growing function defined for any Jordan point, which is a characteristic of a Gauge. In addition, the operator LE (B, A) defined by this way meets the characteristics of a Fre´chet metrics, since the proximity of two points a and b can be mapped into the set of natural integers and even to the set of rational numbers: for that, it suffices that two members wð f n ðxÞ; f n ð yÞÞ are identified with a ordered pair {wð f n ðxÞÞ}; {wð f n ðxÞ; f n ð yÞÞ}: (iv) Suppose that one path w(a, b) meets an empty space {B}. Then a discontinuity occurs and there exists some i such that: wð f i ðbÞÞ ¼ B: If all w(A, B) meets {B}, then no distance is measurable. As a corollary, for any singleton {x}, one has: w{ f ðxÞ} ¼ B: The above properties meet two other characteristics of a gauge. Remark 3.1. Given closed sets {A; B; C; . . .} ¼ E; then a path set wðE; EŒÞ exploring the distance of E to the closure (Œ ) of E meets only open subsets, so that wðE; EŒÞ ¼ B: This is consistent with a property of the Hausdorff distance. Similarly, given A; B , E; one can tentatively note: inf{wðA; {x}Þ > E8} 7 ! distHausdorff ð{x}; AÞ

x[E

as reported by Choquet (1984) and Tricot (1999b).

ð4:3aÞ

953

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sup{wðA; BÞ > E8} 7 ! distHausdorff ðA; BÞ

ð4:3bÞ

as reported by Tricot (1999b). inf {wða; bÞ > E8} 7 ! distðA ^ BÞ

a[A; b[B

954

ð4:3cÞ

as reported by Choquet (1984) for (E, d ) a metric space. max{ðA; BÞ , EjLE ðA; BÞ} 7 ! diamHausdorff ðEÞ

ð4:3dÞ

in all cases. (v) LE ðA; BÞ ¼ LE ðB; AÞ and LE ð{x; y}Þ ¼ B , x ¼ y: If the triangular inequality condition is fulfilled, then LE (B,A) will meet all the properties of a mathematical distance. LE ðA; BÞ < LE ðB; CÞ may contain members of LE (A, C) since the latter are contained in neighborhoods of A, B, C. Thus, LE ðA; CÞ ¼ {’B; ðA > B – B; C > B – BÞ; LE ðA; CÞ # LE ðA; BÞ < LE ðB; C Þ}

ð4:3eÞ

This completes the proof of Proposition 3.2. 3.1.2 The particular case of a totally ordered space. Let A and B be disjoint segments in space E. Let E be ordered by the classical relations: A,B,AaB

ð5:1aÞ

ðA; BÞ , E , E s A; E s B

ð5:1bÞ

Then, E is totally ordered if any segment owns an infimum and a supremum. Therefore, the distance (d) between A and B is represented as shown in Figure 2 by the following relation: dðA; BÞ # distðinf A; inf BÞ > distðsup A; sup BÞ

ð5:2Þ

with the distance evaluated through either classical forms or even the set-distance D(A, B) which will be revisited below. 3.1.3 The case of topological spaces. Proposition 3.2. A space can be subdivided in two main classes: objects and distances.The set-distance is the symmetric difference between sets: it has been proved that it owns all the properties of a true distance (Bounias and Bonaly, 1996) and that it can be extended to manifolds of sets (Bounias, 1997). In a topologically closed space, these distances are the open complementary of

closed intersections called the “instances”. Since the intersection of closed sets Structure of is closed and the intersection of sets with non-equal dimensions is always ill-known spaces closed (Bounias and Bonaly, 1994), the instance rather stands for closed Part 1 structures. Since the latter have been shown to reflect physical-like properties, they denote objects. Then, the distances as being their complementaries will constitute the alternative class: thus, a physical-like space may be globally 955 subdivided into objects and distances as full components. This coarse classification will be further refined in Part 2. The properties of the set-distance allow an important theorem to be now stated. Theorem 3.1. Any topological space is metrizable as provided with the set-distance (D) as a natural metrics. All topological spaces are kinds of metric spaces called “delta-metric spaces”. Proof. Conditions for a space X (generally, belonging to the set of parts of a space W ) to be a topological space are threefolds (Bourbaki, 1990a, b): firstly, any union of sets belonging to X belongs to X. If A and B belong to X, then DðA; BÞ ¼ ›A BÞ # ðA < BÞ , X: Secondly, any finite intersection of sets belonging to X belongs to X. Let ðA; BÞ [ X: Since max DðA; BÞ ¼ ðA < BÞ and min DðA; BÞ ¼ B; and that B [ X (Schwartz, 1991), then necessarily DðA; BÞ [ X: The symmetric distance fulfills the triangular inequality, including in its generalized form, it is empty if A ¼ B ¼ . . .; and it is always positive otherwise. It is symmetric for two sets and commutative for more than two sets. Its norm is provided by the following relation: kDðAÞk ¼ DðA; BÞ:Therefore, any topology provides the set-distance which can be called a topological distance and a topological space is always provided with a self mapping of any of its parts into any one metrics: thus any topological space is metrizable. Reciprocally, given the set-distance, since it is constructed on the complementary of the intersection of sets in their union, it is compatible with the existence of a topology. Thus, a topological space is always a “delta-metric” space. Remark 3.2. Distance D(A, B) is a kind of an intrinsic case ½LðA;BÞ ðA; BÞ of LE ðA; BÞ while LE ðA; BÞ is called a “separating distance”. The separating

Figure 2. The distance of two disjoint segments in a totally ordered space

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distance also stands for a topological metrics. Hence, if a physical space is a topological space, it will always be measurable. 3.2 A corollary about intrinsic vs separating distances 3.2.1 Introduction. The previously proposed set distance as the symmetric difference between the two or more sets is independent of any embedding space. It should thus be considered as an intrinsic one. However, the measure based on a path provided with a gauge pertaining to the common filter on A, B in E seeks for an identification of what is between A and B within E. Thus, a particular application can be raised: 3.2.2 Results. Proposition 3.2. Let spaces ðA; BÞ , E: Then a measure of the separating distance of A and B is defined if there exists a space X with non-empty intersections E, A, B, such that, X belongs to the same filter F as A and B, and: LE ðA; BÞ : E > {DðA; XÞ > DðB; XÞ}

ð6Þ

Preliminary proof. Since filter F holds on E, and A; B [ F the three properties of a filter state the following (Bourbaki, 1990b, I.36): (1) X [ F implies X must contain a set G [ F: (2) Since any finite intersection of sets of F must belong to F, one has: G [ {Ai; Aj [ F; i – j : Ai > Aj}: (3) The empty part of X does not belong to F. Therefore, since G [ X; and A > B ¼ B; then there must be G [ A > X and G [ B > X; G – B; which proves (4.1). Hence, this example (Figure 3) further provides evidence that a definition is just a particular configuration of the intersection of two spaces of magmas of which one is the reasoning system (eventually a logic) and the other one is a probationary space. 3.2.3 Particular case: measuring open sets. As pointed earlier, even a continuous path cannot, in general, scan an open component of the separating distance between the two sets, since a path has, in general, no closed intersection with a open with same dimension. This is consistent with the exclusion of open adjoined intervals in the Borel measure. Hence, a primary topology is a topology of open sets, since a primary topological space cannot be a physically measurable space. However, an intersection of a closed (C ) with a path (w) having a non-equal dimension than (C ) owns a closed intersection with C provided this intersection is non-empty. This implies that the general conditions of filter membership are fulfilled. Remark. An open 3D universe would not be scanned by a 3D probe. But in a closed Poincare´ section, the topologies are distributed into the closed parts

and their complementaries as open subparts. Therefore, there may be open Structure of parts in our universe that would not be detectable by the 3D probes. This ill-known spaces problem might be linked with the still pending problem of the missing dark Part 1 matter (see also Arkani-Hamed et al., 2000). 3.2.4 An alternative perspective. Owing to the case in which there exists no intersection of spaces A and B with one of the other spaces contained in E, in 957 order to define a common surrounding of A and B in an appropriate region of E, the following proposition would hold: Proposition 3.3. A “surrounding distance” of spaces A and B in an embedding space E is given by the complementary of A and B in the interior of their common closure. This distance is denoted @E (A, B): @E ðA; BÞ ¼ ½{ðA; BÞ8}E w ðA < BÞ . {A} < {B}

ð7Þ

Corollary 3.3.1. A condition for the surrounding distance to be non-zero is that A and B must be dense in E. The closure of A < B is different from, and contains, the union of the respective closures of A and B. This important property clearly delimitates a region of space E where any object may have to be scanned through a common gauge in order to allow a measure like LE (A, B). Hence, the concept of the surrounding distance is more general; since it induces that of the separating distance and belongs to a coarser topological filter. 3.3 About dimensionality studies A collection of scientific observation through experimental devices produce images of some reality, and these images are further mapped into mental

Figure 3. A path scanning the separating part between a closed space A and another closed space B must own a non-empty intersection with the objects situated between A and B. Sets A and G, and sets B and G own their symmetric difference as an intrinsic set-distance. The intersection of these two measures is the separating distance

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images into the experimentalist’s brain (Bounias, 2000a). The information from the explored space thus stands like parts of an apparatus being spread on the worker’s table, in view of a further reconstitution of the original object. We propose to call this situation an informational display, likely composed of elements with dimensions lower than or equal to the dimensions of the real object. The next sections will thus deal with this particular problem. 3.3.1 Analysis of unordered vs ordered pairs in an abstract set. A robust definition of an ordered N-uple is given by the following: Lemma 3.1. An expression noted ðabc. . .zÞ is a ordered N-uple iff: ðabc. . .zÞ ¼ ðx1; x2; x3; . . .; xnÞ , a ¼ x1; b ¼ x2; c ¼ x3; . . .; z ¼ xn ð8:1aÞ In the construction of the set N of natural integers, Von Neumann provided an equipotent form using replicates of B: 0 ¼ B; 1 ¼ {B}; 2 ¼ {B; {B}}; 3 ¼ {B; {B}; {B; {B}}}; 4 ¼ {B; {B}; {B; {B}}; {B; {B}; {B; {B}}}} A Von Neumann set is Mirimanoff-first kind since it is isomorphic with none of its parts. However this construction, associated with the application of Morgan’s laws to (B), allowed the empty set to be attributed to an infinite descent of infinite descents and thus to be classified as a member of the hypersets family (Bounias and Bonaly, 1997b). Proposition 3.4. In an ordered pair {{x}; {x; y}}; the paired part {x; y} is unordered. A classical acception (Schwartz, 1991) states that {a; a} ¼ {a}: This may introduce a confusion, which can then be treated from relation (5.1a): ðaaÞ ¼ ðxyÞ , a ¼ x; y

ð8:1bÞ

This just means that for any x, y, a pair (a, a) is not ordered. In effect, it becomes ordered upon rewriting: A ¼ ða; ða; aÞÞ: Comparing (aa) and A through relation (8.1a), it comes: ðaaÞ ¼ {a; {a; a}} , a ¼ a; and a ¼ {a; a} Thus, {a; b} ¼ {b; a}; while ðabÞ – ðbaÞ and ½{a; a} ¼ a – ½{a; ðaÞ} ¼ ða; aÞ where (a) is a part, for which a < ðaÞ – {a} < {a}: Consequently, ðb; aÞ ¼ ða; bÞ and ða; aÞ ¼ !ða; aÞ: In an abstract set (in the sense of a set of any composition), one can find parts with non-empty intersections as well as parts or members with no common members. For instance, the following E and F are, respectively, without order and at least partly ordered: E ¼ {a; b; {c; d}; G; X} with a – b; c – d; G – X

ð8:2Þ

Structure of ill-known spaces Part 1 In particular cases, in a set like E or F, it may be that two singletons, e.g. F ¼ {a; b; c; {a; d}; {b; c; d}; {d; {a; e; f }}}

ð8:3Þ

{a},{b}, though different, have the same weight with respect to a defined property. In such cases one could write that there exists some (m) such that {a} ; {m}; and {b} ; {m}; so that {a; d; G} > {c; {b; c}; X} ¼ ða ; bÞ: Proposition 3.5. An abstract set can be provided with at least two kinds of orders: one with respect to the identification of a max or a min, and one with respect to ordered N-uples. These two order relations become equivalent upon additional conditions on the nature of involved singletons. (i) A Mirimanoff set of the (2.2) type and derived forms can be provided a order. Let E ¼ {e; { f ; G}}: Then as seen earlier, there exists some m such that {e} ; { f } ; {m}; and E is similar to E 0 ; {m; {m; G}}: Then, for any G – B; m , {m; G} and thus m is a minimal member of {m, G}. Pose {m; G} ¼ M ; then E 0 becomes E 0 ¼ {m; M} of which m is the minimal and M is the maximal of the set. Then, one can rewrite {m; {m; G}} in the alternative form {m; ðm; GÞ} and E ¼ {e; ð f ; GÞ} since ( f, G) is in some sort ordered. These notations will be respected later. Note that /ð f ; GÞ ) ’{ f ; {G}} while the reciprocal is not necessarily true. The necessary condition is the following. Suppose { f ; {G}} ) ’ð f ; GÞ: Then, since { f ; {G}} ¼ {{G}; f }; one should be allowed to write f ; G and G ; f ; that is there is neither minimal nor maximal in the considered part. Now, writing ( f, G) imposes a necessary condition that there exists some m and n such that ð f ; GÞ ; {m; {m; n}} ¼ {m; {n; m}}: This can then be (but only speculatively) turned into a virtual ðm; ðm; nÞÞ: When comparing two sets, parts ordered by this way will have to be compared two by two: {e; { f ; G}} ¼ {m; ðn; Z Þ} ) e ¼ m; and since m – M : f ¼ n; G ¼ Z (otherwise { f ; g} ¼ ðn; zÞ could give alternatively f ¼ z; n ¼ g). This drives the problem to the identification of orders in the set of parts of a set, as compared with components of a simplex set (Table I). (ii) A set can be ordered through the rearrangement of its exact members and singletons in a way permitted by the structure of the set of parts of itself. In effect, existence of a set axiomatically provides existence to the set of its parts (Bourbaki, 1990a, p. 30) though an axiom of availability has been shown to be required for disposal of the successive sets of parts of sets of parts (Bounias, 2001). Now, how to identify preordered pairs in a set? The set of parts P(E ) of a set E provides the various ways members of this set can be gathered in subsets, still not ordered. Therefore, an analysis of the members of a set can involve a rearrangement of the singletons contained in the set in a way permitted by the arrangements allowed by P(E ), including when a same singleton is present several times in the set. Then, this may let emerge configurations that can be identified with a structure of ordered N-uples.

959

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Remark 3.3. In N-uples as well as in the set of parts, singletons gathered in any subpart are unordered. Let (E 0 ) be a set reordered by using P(E ). Then: E 0 > PðEÞ ¼ PartitionðEÞ: Application to simplicial structures. Table I illustrates for the first four simplexes, the comparison of the ordered pairs of the simplexes with the remaining part of the set of parts, which is constituted of ð2N 2 N 2 1Þ unordered non-empty members for a simplex of N vertices. The remaining parts can in turn be arranged into appearing ordered subparts. Hence for E3: {{x2}; {x2; x3}} ¼ ðx1x3Þ and the “in fine” ordered {{x3}; {x1; x3}} ¼ ðx3x1Þ since {x1; x3} ¼ ðx3; x1}: For E4, one gets {{x2}; {x2; x3}; {x2; x3; x4}} ¼ ðx2x3x4Þ;{{x3}; {x3; x4}} ¼ ðx3x4Þ; {{x1; x3}; {x1; x3; x4}} ¼ ððx1; x3Þx4Þ; and the “in fine” ordered: {{x4}; {x1; x4}} ¼ ðx4x1Þ; {{x2; x4}; {x1; x2; x4}} ¼ ððx2; x4Þx1Þ: This further justifies that for a set like (En), the set of its (2n) parts has dimension D # DðE n Þ: In fine, a fully informative measure should provide a picture of a space allowing its set component to be depicted in terms of the ordered N-uples ðN ¼ 0 ! n; n , N Þ: These considerations are intended to apply when a batch of observational data and measurement are displayed in a scattered form on the laboratory table of a scientist, and must then be reconstructed in a way most likely representative of a previously unknown reality. 3.3.3 Identification of a “scanning” measure on an abstract set. Several kinds of measure of a set, including various forms of its diameter infer from Section 3.1. Proposition 3.6. A set can be scanned by the composition of an identity function with a difference function. Let E ¼ {a; b; c; . . .} be a set having N members.

Sets E 1 ¼ {x1} E 2 ¼ {x1, x2} E 3 ¼ {x1, x2, x3}

Table I. Sets and simplexes: N and D are the numbers of vertices and the maximum dimension, respectively

Vertices

Ordered pairing (N)

dim.

N¼1 N¼2 N¼3

{x1} {x1},(x1, x2) ¼ (x1x2) {x1},(x1, x2),(x1, x2, x3) ¼ (x1x2x3) {x1},(x1,x2),(x1,x2,x3), (x1, x2,x3,x4) ¼ (x1x2x3x4)

D¼0 D¼1 D¼2

Remaining of P(En) (2N2 N 2 1)

2N2 1

none 1 {x2} 3 {x2},{x3},{x1,x3}, 7 (x2,x3} 15 D¼3 {x2},{x3},{x4}, E 4 ¼ {x1, x2, N¼4 {x1,x3},{x1,x4},{x2,x3}, x3, x4} {x2,x4},{x3,x4} {x1,x2,x4},{x1,x3,x4}, {x2,x3,x4} Notes: P(En) denotes the set of non-empty parts of each En considered. Each class of subsets of k members appears in C kN forms. One of each is used in a ordered N-uple, so that C kN 2 1 are remaining. Finally, Sn¼1!N ðC kn 2 1Þ þ N ¼ 2N

(i) An identity function Id maps any members of E into itself: ;x [ E; Structure of IdðxÞ ¼ x: ill-known spaces Thus, /(a or b or c,. . .) this provides one and only one response when Part 1 applied to E. (ii) A difference function is f such that: ;x [ E; f n ðxÞ – x: The exploration function is a self-map M of E : M : E 7 ! E; M ¼ Id ’ f :

961

/x [ E; M ðxÞ ¼ f ðIdðxÞÞ; ;n : M n ðxÞ ¼ f n ðIdðxÞÞ – x

ð9:1Þ

Proof. (a) Suppose M ¼ IdðxÞ then, each trying maps a member of E to a fixed point and there is no possible scanning of E. (b) Suppose one poses just f ðIdðxÞÞ – x : then, given f ðIdðxÞÞ – x; say f 1 ðIdðxÞÞ ¼ y; since y – x; then one may have again f 2 ðIdðxÞÞ ¼ x: Therefore, there can be a loop without further scanning of E, with probability ðN 2 1Þ21 : (c) Suppose one poses M ¼ f ; such that f n ðxÞ – x: Then, since f 0 ðxÞ – x; there can be no start of the scanning process. If in contrast one poses a modification of the function f then /x; f 0 ðxÞ ¼ x; this again stops the exploration process, since then f 0 [ Id: This has been shown to provide a minimal indecidability case (Bounias, 2001).The sequence of functions M n ðxÞ ¼ f n ðIdðxÞÞ – x; ;n; is thus necessary and sufficient to provide a measure of E which scans N 2 1 members of E. The sequence stops at the Nth iterate, if in addition: f n ðIdðxÞÞ – { f i ðIdðxÞÞ}ð;i[½1;N Þ

ð9:2Þ

The described sequence represents an example of a path as described earlier in more general terms. Now, some preliminary kinds of diameters can be tentatively deduced for the general case of a set E in which neither a complete structure nor a total order can be seen. 3.3.4 Tentative evaluations of the size of sets with ill-defined structure and order. Since, in this case, any two members of E are of similar weight, regarding the definition of a diameter (4.3d), the following definition proposed. Definition 3.1a. Given a non-ordered set E, Id the identity self map of E, and f the difference self map of E, a kind of diameter is given by the following relation: diamf ðEÞ < {ðx; yÞ [ E : max f i ðIdðxÞÞ > max f i ðIdðyÞ}

ð10:1Þ

This gives a ðN 2 2Þ members parameter. Subdefinition 3.1b. If E is well-ordered, i.e. at least one, e.g. the lower boundary can be identified among members of E, such as a singleton {m}, then an alternative form can be written. Since in this case, the set E can be represented by two members: E ¼ {m; Z } with Z ¼ ›E ðmÞ the complementary of m in E, relation (4.3d) results in a measure MHD: M H D ðEÞ ¼ {m ¼ minðEÞ; Dðm; Z Þ} with D the symmetric difference.

ð10:2Þ

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This gives a ðN 2 1Þ member parameter which in turn provides a derived kind of diameter diamHD by repeating the measure for two members m0 and m00 as on Figure 4. Let E 0 ¼ {m0 ; ›E ðm0 Þ}; and E 00 : {›E ðm00 Þ}; then: diamðEÞ # {m0 ; m00 [ E : max DðE 0 ; E 00 Þ}

ð10:3Þ

Subdefinition 3.1c. If both upper and lower boundaries can be identified, i.e. the set E is totally ordered, then the distance separating the two segments A and B of E is: ðA; BÞ , E; distE ðA; BÞ ¼ {distðinf A; inf BÞ > distðsup A; sup BÞ} ð10:4Þ where a set distance D is again provided by the symmetric difference or by a n 2 D Borel measure. A separating distance is a extrinsic form of the set-distance as an intrinsic form. A diameter is evaluated on E as the following limit: diamðEÞ ¼ {inf A ! inf E; sup A ! min E; inf B ! max E; ð10:5Þ sup B ! sup Ej lim distE ðA; BÞ} These preliminary approaches allow the measure of the size of tessellating balls as well as that of the tessellated spaces, with reference to the calculation of their dimension through relations (2.1)-(2.2) and (11.1)-(11.2) derived later. A diagonal-like part of an abstract space can be identified with and logically derived as a diameter. Remark 3.4. If a measure is obtained each time from a system, this means that no absolutely empty part is present as an adjoined segment on the trajectory of the exploring path. Thus, no space accessible to some sort of measure is strictly empty in both mathematical and physical sense, which supports the validity of the quest of quantum mechanics for a structure of the void. Figure 4. A preliminary approach of the diameter of an abstract set in a partly ordered space: this examples shows the sup(diamE ) as included in the complementary of c and d in E

3.3.5 The dimension of an abstract space: tessellating with simplex k-faces. Structure of A major goal in physical exploration will be to discern among the detected ill-known spaces objects, which are equivalent with abstract ordered N-uples within their Part 1 embedding space. A first of further coming problems is that in a space composed of members identified with such abstract components, it may not be found tessellating balls 963 all having identical diameter. Also a ball with two members would have no such diameter as defined in (10.1) or (10.4). Thus a measure should be used as a probe for the evaluation of the coefficient of size ratio ( r) needed for the calculation of a dimension. Some preliminary proposed solutions hold on the following principles. (i) A three-object has dimension 2 iff the longer side of A1max fulfills the condition, for the triangular strict inequality, where M denotes an appropriate measure: M ðA1max Þ , M ðA12 Þ þ M ðA13 Þ

ð11:1aÞ

Similarly, this condition can be extended to higher dimensions (Figure 5): M ðA2max Þ , M ðA22 Þ þ M ðA23 Þ þ M ðA24 Þ

ð11:1bÞ

Then, more generally, for a space X being a N object: o   N 21n M Akmax , < M ðAki Þ

ð11:1cÞ

i¼1

with N ¼ number of vertices, i.e. eventually of members in X, k ¼ ðd 2 1Þ ¼ N 2 2; and Akmax the k-face with maximum size in X. This supports a former proposition (Bounias, 2001). Remark 3.5. One should note that, according to relation (11.1c), for N ¼ 2 (a “2-object”), X ¼ {x1; x2} has dimension 1 iff x1 , ðx1 þ x2Þ; that is iff x1 – x2 (Figure 5). This qualifies the lower state of an existing space X 1. (ii) Let the space X be decomposed into the union of balls represented by D-faces AD proved to have dimension Dim ðA D Þ ¼ D by relation (11.1c) and size M(A1) for a 1-face. Such that a D-face is a D-simplex Sj whose size, as a ball, is evaluated by M ðA1max ÞD ¼ S Dj : Let N be the number of such balls that can be filled in a space H, so that: N

< {S Dj } # ðH < Ldmax Þ

ð11:2aÞ

i¼1

with H being identified with a ball whose size would be evaluated by Ld, L the size of a 1-face of H, and d the dimension of H. Then, if ;Sj; Sj < So; the dimension of H is:

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Figure 5. The first three steps of the N-angular strict inequality for the assessment of the dimensionality of a simplex. In the lower right picture, the larger side standing for Akmax is S1 such that in a 2D space one has exactly: S1 ¼ S2 þ S3 þ S4

dðH Þ < ðD:log S 0 þ log NÞ=log L1max

ð11:2bÞ

Remark. Relation (11.2b) stands for a kind of interior measure in the Jordan’s sense. In contrast, if one poses that the reunion of balls covers the space H, then d(H) represents the capacity dimension, which remains an evaluation of a fractal property. Definition 3.2b. A n-(simplex-ball) is the topological unit ball circumscribed to a n-simplex. (iii) Extension to an ill-defined space E. The problem consists of identifying first a 1-face component of E from k-faces ðk . 1Þ; which implies to identify ordered N-uples. Then, components of E, whatever be their nature, should be analogically decomposed into appropriate simplexes. The number of these simplexes will then be evaluated in E, and relations (11.2) will be finally applied (Figure 6). Remark. There is no other condition on simplexes Sj than the need that one face is maximal: this is the face on which the others will be projected, so that the generalized inequality will be applied. Depending on N, non-integer d(H) can be obtained for a fractal or a fractal-like H. Some applications of the above protocol will be illustrated in further communications. Lemma 3.2. A singleton or member (a) is putatively available in the form of a self-similar ordered N-uple: {a} ¼ {a < a < . . . < a}h – {a; ðaÞ; . . .}: In effect, the set theory axiom of the reunion states that A < A ¼ A as well as the axiom of the intersection states that A > A ¼ A: Corollary 3.2. In the analysis of a abstract space H ¼ L x ; of which dimension x is unknown, the identification of that members can be identified with N-uples supposedly coming from a putative Cartesian product of members of H, e.g.: G , H : /a [ G; ða; aa. . .aÞh [ G h ; is allowed by an anticipatory process.

Preliminary proof. At time (t) of the analysis of the formal system involved, Structure of there is no recurrent function that can “imagine” the existence of an abstract ill-known spaces component not existing in the original data and parameters, and not directly Part 1 inferring from a computation of these data and parameters. Devicing ðaaa. . .aÞh [ G h ; implies making a mental image at ðt þ teÞ and further confronting the behavior of the system with “anticipatively recurrent” images 965 succeeding to those at t, that is computed from ðt; t þ eÞ conditions within ðt þ 1; t þ 2; . . .; up to t þ e 2 1Þ: Let E ¼ {a; {b; {; B; C}}} then one anticipates on the putative existence of an unordered pair {a; a} ¼ a in a former writing of E. Therefore, ðaÞ ¼ {{a; {a; a}}j{a; a} ¼ {a}}: The presence of singletons can be identified with a putative former reduction of ordered parts owning the same nucleus (a). A nucleus thus appears as an ordered form of a singleton, that is the only case where an ordered form is identical with an unordered one. Application 3.2. Given ðE en Þ composed of N-uples denoted ðPei Þ in their ordered state, and K ei ; K ej ; . . .; in their unordered acceptation: an approach of the identification of the diagonal, and thus of a further measure of the respective diameters of ðE en Þ and ðPei Þ is given by the following propositions: Diag ðE en Þ # min <{ðK ei ; K ej Þ [ ðK en Þ : DðK ei ; K ej Þi–j }

ð11:3aÞ

Diag ðPei Þ # max <{ðK ei ; K ej Þ [ ðPei Þ : DðK ei ; K ej Þi–j }

ð11:3bÞ

An example is given in Appendix as a tentative application case. Regarding just a Cartesian product, the set of parts with empty or minimal intersections stands for the diagonal and the diagonal of a polygon stands for a topological diameter. Figure 6. The symmetric difference (D) between two or more sets provides a kind of distance which is consistent with a topology. This distance therefore stands for a topological distance and provides topological spaces with a general form of metrics. By this way, any topological space may be treated like a (delta-)metric space

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4. Physical inferences 4.1 Defining a probationary space A probationary space (Bounias, 2001) is defined as a space fulfilling exactly the conditions required for a property to hold, in terms of: (1) identification of set components; (2) identification of combinations of rules; (3) identification of the reasoning system. All these components are necessary to provide the whole system with decidability (Bounias, 2001). Since lack of mathematical decidability inevitably flaws also any physical model derived from a mathematical background, our aim is to access as close as possible to these imposed conditions in a description of both a possible and a knowable universe, in order to refine in consequence some current physical postulates. The above considerations have raised a set of conditions needed to obtain some knowledge about a previously unknown or ill-known space, upon pathwise exploration from a perceiving system A to a target space B within an embedding space B. In the absence of preliminary postulate about the existence of the so-called “matter” and related concepts, it has been demonstrated that the existence of the empty set is a necessary and sufficient condition for the existence of abstract mathematical spaces (W n) endowed with topological dimensions (n) as great as needed (Bounias and Bonaly, 1997). Hence, the empty set appears as a set without members though containing empty parts. The reasoning intermediately proved that the empty set owns the properties of a non-wellfounded set and exhibit self-similarity at all scales and nowhere derivability, that is: two characteristics of fractal structures. These properties have been shown to clear up some antinomies remained unaddressed about the empty set properties. These findings will now be shown to provide additional features of physical interest. In first, the empty set can provide at least an intellectual support in existence of some sort of space. 4.2 The founding element It is generally assumed in textbooks of mathematics (see, for instance, Schwartz, 1991, p. 24), that some set does exist. This strong postulate has been reduced to a weaker form reduced to the axiom of the existence of the empty set (Bounias and Bonaly, 1997). It has been shown that providing the empty set (B) with ([, , ) as the combination rules (that is also with the property of complementarity (›)) results in the definition of a magma allowing a consistent application of the first Morgan’s law without violating the axiom of foundation iff the empty set is seen as a hyperset, that is a non-wellfounded set (Aczel, 1987, Barwise and Moss, 1991). A further support of this conclusion emerges

from the fact that several paradoxes or inconsistencies about the empty set Structure of properties are solved (Bounias and Bonaly, 1997). These preliminaries now ill-known spaces drive us to the formulation of the following theorem, which will be established Part 1 using several Lemmas. 4.3 The founding lattice Theorem 4.1. The magma BB ¼ {B; ›} constructed with the empty hyperset and the axiom of availability is a fractal lattice. Remark. The notation BB has been preferred to say (EB) or others. In effect, in Bourbaki notation, E F means the space of functions of set F in set E. Therefore, writing (BB) denotes that the magma reflects the set of all self-mappings of (B), which emphasizes the forthcoming results. Lemma 4.1. The space constructed with the empty set cells of EB is a Boolean lattice. Proof. (i) Let <ðBÞ ¼ S denote a simple partition of (B). Suppose that there exists an object (e ) included in a part of S, then necessarily ðe Þ ¼ B and it belongs to the partition. (ii) Let P ¼ {B; B} denote a part bounded by sup P ¼ S and inf P ¼ {B}: The combination rules < and > provided with commutativity, associativity and absorption hold. In effect: B < B ¼ B; B > B ¼ B and thus necessarily B < ðB > BÞ ¼ B; B > ðB < BÞ ¼ B: Thus, space {PðBÞ; ð<; >Þ} is a lattice. The null member is B and the universal member is 2B which should be denoted by :B. Since in addition, by founding property ›B ðBÞ ¼ B; and the space of (B) is distributive, then S(B) is a boolean lattice. A Lemma 4.2. S(B) is provided with a topology of discrete space. Proof. (1) The lattice S(B) owns a topology. In effect, it is stable upon union and finite intersection, and it contains (B). (2). Let S(B) denote a set of closed units. Two units B1, B2 separated by a unit B3 compose a part {B1, B2, B3}. Then, owing to the fact that the complementary of a closed is a open: ›{B1 ;B2 ;B3 } {B1 ; B3 } ¼ B2 ; and B2 is open. Thus, by recurrence, {B1, B3} are surrounded by open ]B[ and in parts of these open, there exists distinct neighborhoods for (B1) and (B3). The space S(B) is therefore Hausdorff separated. Units (B) formed with parts thus constitute a topology (TB) of discrete space. Indeed, it also contains the discrete topology (BB, (B)) which is the coarse one and of much less mathematical interest. Lemma 4.3. The magma of empty hyperset is endowed with self-similar ratios. The Von Neumann notation associated with the axiom of availability, applying on (B), provide existence of sets (N B) and (Q B) equipotent to the natural and the rational numbers (Bounias and Bonaly, 1997). Sets Q and N can thus be used for the purpose of a proof. Consider a Cartesian product

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En £ En of a section of (Q B) of n integers. The amplitude of the available intervals ranges from 0 to n, with two particular cases: interval [0, 1] and any of the minimal intervals ½1=n 2 1; 1=n: Consider now the open section ]0, 1[: it is an empty interval, denoted by B1. Similarly, note Bmin ¼ ½0; 1=nðn 2 1Þ: Since interval ½0; 1=nðn 2 1Þ is contained in [0, 1], it follows that Bmin , B1 : Since empty sets constitute the founding cells of the lattice S(B), the lattice is tessellated with cells (or balls) with homothetic-like ratios of at least r ¼ nðn 2 1Þ: The absence of unfilled areas will be further supported in Part 2 of this study by the introduction of the “set with no parts”. Definition 4.1. Such a lattice of tessellation balls will be called a “tessellatice”. Lemma 4.4. The magma of empty hyperset is a fractal tessellatice. Proof. (1) From relations (2.3) one can write ðBÞ < ðBÞ ¼ ðB; BÞ ¼ ðBÞ: (2) It is straightforward that ðBÞ > ðBÞ ¼ ðBÞ: (3) Last, the magma ðBB Þ ¼ {B; ›} represents the generator of the final structure, since (B) acts as the “initiator polygon”, and complementarity as the rule of construction. These three properties stand for the major features, which characterize a fractal object ( James and James, 1992). Finally, the axiom of the existence of the empty set, added with the axiom of availability in turn provide existence to a lattice S(B) which constitutes a discrete fractal Hausdorff space and the proof is complete. 4.4 Existence and nature of space-time Lemma 4.5. A lattice of empty sets can provide existence to at least a physical-like space. Proof. Let B denote the empty set as a case of the whole structure and {B} denote some of its parts. It has been shown that the set of parts of B contains parts equipotent to sets of integers, of rational and of real numbers, and owns the power of continuum (Bounias and Bonaly, 1994, 1997). Then, looking at the inferring spaces (W n), (W m), . . . thus formed, it has been proved (Bounias and Bonaly, 1994) that the intersections of such spaces having non-equal dimensions give raise to spaces containing all their accumulation points and thus forming closed sets. Hence: {ðW n Þ > ðW m Þ}m.n ¼ ðQn Þ closed space

ð12Þ

These spaces provide collections of discrete manifolds whose interior is endowed with the power of continuum. Consider a particular case (Q4) and the set of its parts P(Qn): then any of the intersections of subspaces ðE d Þd,4 provide a d-space in which the Jordan-Veblen theorem allows closed members to get the status of both observable objects and perceiving objects (Bounias and

Bonaly, 1997b). This stands for observability, which is a condition for a space Structure of to be in some sort observable, that is physical-like (Bonaly’s conjecture, 1992). ill-known spaces Finally, in any (Q4) space, the ordered sequences of closed intersections Part 1 {ðE d Þd,4 }; with respect to mappings of members of {ðE d Þd,4 }i into {ðE d Þd,4 }j ; provides an orientation accounting for the physical arrow of time (Bounias, 2000a), in turn embedding an irreversible arrow of biological 969 time (Bounias, 2000b). Thus the following proposition: Proposition 4.1. A manifold of potential physical universes is provided by the (Qn) category of closed spaces. Our space-time is one of the mathematically optimum ones, together with the alternative series of {ðW 3 Þ > ðW m Þ}m.3 : Higher space-times ðQn Þn.3 could exist as well. Now, some new structures which can pertain to a topological space as described earlier will be briefly examined, and deserves specific attention. 5. Towards “fuzzy dimension” and “beaver spaces” 5.1 Introduction The exploration of nature raises the existence of strange objects, such as living organisms, whose anatomy suggests that mathematical objects having adjoined parts with each having different dimensions could exist. This will then introduce the idea that the dimensions of some objects may even not be completely established. The existence of such strange objects implies that appropriate tools should be prepared for their eventual study. This is sketched here and will be a matter of further developments. 5.2 From hairy spaces to “beaver spaces” A “hairy space” is a ðn $ 3Þ-ball having 1D lines (also called “hair” of “grass”) planted on it (Berger, 1990). It is interesting that for such spaces (n $ 3 only), the volume and the area do not change with the insertion of 1-spaces on them. Consider a simplex S nn : the last segment allowing set E nþ1 to be completed from En of S n21 is ðxn ; xnþ1 Þ: Consider the simplex F n21 such that one of its facets A n22 has its last segment ðyn21 ; yn22 Þ ; ðxn ; xnþ1 Þ: Repeat this operation for descending values: one gets a space having n- and ðn 2 1Þ-adjoined parts with their intersections having lower dimension: dim{S n > S n21 } # ðn 2 2Þ: With respect to a beaver, having a spheric body, with a flat tail surrounded with hair, these spaces will be denoted as “beaver spaces”. Other possibilities include unordered appositions of parts with various dimensions. The existence of Beaver spaces implies some specific adjustment of the methods used for their scanning. In this respect, it should be recalled that a new mode of assessment of coordinates has been formerly proposed (Bounias and Bonaly, 1996): it consists of studying the intersection of the unknown space with a probe composed with an ordered sequence of topological balls of decreasing dimensions, down to a point ðD ¼ 0Þ: This process has been shown

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to wear the advantage of being able to define coordinates even in a fractal space. This may be of particular interest with components provided by and embedded in the lattice S(B). 5.3 “Fuzzy dimensioned” spaces Proposition 5.1. There exist spaces with fuzzy dimensions. Proof. Consider a simplex S nn ¼ {ðE nþ1 Þ; ð’ nÞ}; and its two characteristic structures, namely L1n ¼ Si¼1!n ðdistðxi21 ; xi ÞÞ and ¥1nþ1 ¼ Si¼1!n ðdistðxnþ1 ; xi ÞÞ: Then, a condition for the assessment of simple dimensionality is given by the structure defining a simple space (Bounias, 2001) in consistency with a (1D)-probe for a ðD . 1Þ-space: ð’ nÞ ¼ ð¥1nþ1 . kðn; d ÞL1n , d . nÞ

ð13Þ

Let the last segment of the simplex ðxn ; xnþ1 Þ be such that, in consistency with Zadeh and Kacprzyk (1992), one has: dist ðxn ; xnþ1 Þ [ ½0; 1: Then, the expression of ¥1nþ1 reaches a value fuzzily situated between the assessment of d ¼ n 2 1 and d ¼ n at least for n , 4: Thus, the simple set E nþ1 is of the fuzzy type and the simplex is a fuzzy simplex. Such a space will therefore be having a “fuzzy dimension”. This provides a particular extension of the concept of fuzzy set into that of “fuzzy space of magma”, since the set is unchanged while it is the rule that provides the magma with a fuzzy structure. These problems will be the matter of further developments, since they belong to the kind of situations that could really be encountered during the exploration of universe, and as pointed by Klir and Wierman (1999): “Knowledge about the outcome of an uncertain event gives the possessor an advantage”. 6. Discussion 6.1 Epistemological assessment It has been stressed in the introduction that computable systems can in principle not be self-evaluated, due to incompleteness and indecidability theorems. It will be shown here on a simple example why this does not apply to the self-evaluation of our universe structure by one of its components, when this component is a living subpart provided with conscious perception functions, that is a brain. Theorem 6.1. A world containing subparts endowed with consciously perceptive brains is a self-evaluable system. The proof will be found in two main Lemmas. Lemma 6.1. There exists a minimal indecidable system (Bounias, 2001). Proof. The former theorem of Go¨del provided the most sophisticated proof, while the systems devised by Chaitin described the most giant examples of known indecidable systems. In contrast, a recurrent reasoning would state that

there should exist most simpler cases. Consider the set E ¼ {ðAÞ} where (A) is Structure of a part, eventually composed of nuclei or singletons. Try an exploration of E by ill-known spaces itself: without prior knowledge of the inside structure of E, the exploration Part 1 function f n ðIdðxÞÞ is described in relations (2.3.1)-(2.3.5). Applying this function to E, one gets IdðAÞ ¼ A and f n ðAÞ – A does not exist from the definitions (this applies to A ¼ {a} a singleton, as well). Hence, the system returns no 971 result: not a zero result, but literally no answer. Now, let A be composed of singletons, the same procedure must be applied to each detected singleton, with the same failure. Finally, let A be (BB): for any singleton {B}, one will get IdðBÞ ¼ B and f n ðBÞ again returns no response. Hence, the lattice S(B) is not self-evaluable as it stands at this stage. Since {B} is the minimum of any non-empty set, the system stands for a minimum of an indecidable case of the classical set theory (though a new component will be considered in Part 2). Lemma 6.2. A consciously perceptive biological brain is endowed with anticipatory properties (Bounias and Bonaly, 2001). The function of conscious perception has been shown to infer from the same conditions as those allowing a physical universe to exist from abstract mathematical spaces: A path connecting a Jordan’s point of an outside closed B to the inside of another closed A is prolonged in a biological brain into a sequence of neuronal configurations which converges to fixed points. These fixed points stand for mental images (Bounias, 2000a, b; Bounias and Bonaly, 1996). The sequence of mental images owns fractal properties, which can provide additional help to the construction by the brain of mental images of an expected future state. These mental images will in turn be used as a guide for the adjustment of further actions to the expected goal (Bounias, 2000a, b), which is also a way by which the organism returns molecular information to the brain, making unconscious (autonomous) mental images which are used in turn for the control of the homeostasis of the organism. Now, consider a space F ¼ {ðAÞ; ðBÞ} where (A) is a living system endowed with anticipatory properties. Then, A is able to analyze some of its components ða1; a2; . . .Þ through a way similar to the exploration function. By an anticipatory process, it is able to construct the power set of parts (Pn) of at least a part of (A) or (B). Consequently, there will always be a step (n) in which (Pn) i.e. the set of parts of the set of parts of . . . (n times) the set of parts of part of (A) or (B) will have a cardinality higher than (A) or (B), since (A) and (B) are finite and (Pn) is infinitely denumerable. Then (A) will be able to construct a surjective map of (Pn) on either (A) or (B). This completes the proof. 6.2 About the assessment of probationary spaces There remain enormous gaps in the present-day knowledge of what universe could really be. For instance, current cosmological theories remain contradictive with astronomical observation (Bucher and Spergel, 1999; Krauss, 1999; Mitchell, 1995 and many other articles). The inconsistency of

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Lorentz covariance used by Einstein, Minkowski, Mach, Poincare´, Maxwell etc., with Lorentz invariance used by Dirac, Wigner, Feynman, Yang, etc. remains unsolved (Arunasalam, 1997). Quantum mechanics still failing to account for macroscopic phenomena, could be – and has been – interpreted through classical physics (Wesley, 1995), while flaws have been found in the so-called “decisive experimental evidence” or classical observations about as fundamental parameters as the Bell inequalities (Wesley, 1994), the velocity of light (Driscoll, 1997), the red shift meaning (Meno, 1998), and others. More, whether space is independent from matter or matter is deformation of space remains questioned (Krasnoholovets, 1997; Kubel, 1997; Rothwarf, 1998). These discrepancies essentially come from the fact that probationary spaces supporting a number of explicit or implicit assessments have not been clearly identified. At cosmological scales, the relativity theory places referentials in a undefined space, with undefined gauges nor substrate for the transfer of information and the support of interactions. That matter exists and is spread into this undefined medium is just implicitly admitted without justifications. Here, distances are postulated without reference to objects. At quantum scales, a probability that objects are present in a certain volume is calculated. But again, nothing is assessed about what are these objects, and what is their embedding medium in which such “volumes” can be found. Furthermore, whether these objects are of a nature similar or different to the nature of their embedding medium has not been addressed. In this case, objects are postulated without reference to distances. About quantum levels, justifications have been mathematically produced in order to cope with some unexplainable observations, but this does not constitute a proof, per se, since the proof is not independent from the result to be supported. Last, neither the Big Bang energy source nor others have been justified, which led to the theory of an energy of the void: but then, this precludes the existence of a true “void”. These remarks support the need for finding ultrafilter properties which would be provided to any object and distance from microscopic to cosmic scales in our universe, assuming that it is not composed of separate component with discontinuity or break of arcwise connectivity. The next part of this work will provide some logical answers to such problems and derive physical properties of an inferring space-time, with particular reference to the derivation of cosmic scale features from submicroscopic characteristics.

References Aczel, P. (1987), Lectures on Non-Well-Founded Sets. CSLI Lecture-notes 9, Stanford, USA. Arkani-Hamed, N., Dimopoulos, S. and Dvali, G. (2000), “Les dimensions cache´es de l’univers”, Pour la Science (Scientific American, French Edition), Vol. 276, pp. 56-65.

Arunasalam, V. (1997), “Einstein and Minkowski versus Dirac and Wigner: covariance versus invariance”, Physics Essays, Vol. 10 No. 3, pp. 528-32. Avinash, K. and Rvachev, V.L. (2000), “Non-Archimedean algebra: applications to cosmology and gravitation”, Foundations of Physics, Vol. 30 No. 1, pp. 139-52. Banchoff, T. (1996), The Fourth Dimension, Scientific American, French Edition, Pour La Science-Berlin, Paris, pp. 69-80. Barwise, J. and Moss, J. (1991), “Hypersets”, Math. Intelligencer, Vol. 13 No. 4, pp. 31-41. Bonaly, A. (1992), Personal communication to M. Bounias. Bonaly, A. and Bounias, M. (1995), “The trace of time in Poincare´ sections of topological spaces”, Physics Essays, Vol. 8 No. 2, pp. 236-44. Borel, E. (1912), Les ensembles de mesure nulle. Oeuvres comple`tes. Editions du CNRS, Paris, Vol. 3. Bourbaki, N. (1990a), The´orie des ensembles Chapters 1-4, Masson, Paris, p. 352. Bourbaki, N. (1990b), Topologie Ge´ne´rale Chapters 1-4, Masson, Paris, p. 376. Bounias, M. (1997), “Definition and some properties of set-differences, instans and their momentum, in the search for probationary spaces”, J. Ultra Scientist of Physical Sciences, Vol. 9 No. 2, pp. 139-45. Bounias, M. (2000a), “The theory of something: a theorem supporting the conditions for existence of a physical universe, from the empty set to the biological self”, in Daniel, M. Dubois (Ed.) CASYS’99 Int. Math. Conf., Int. J. Comput. Anticipatory Systems, Vol. 5, pp. 11-24. Bounias, (2000b), “A theorem proving the irreversibility of the biological arrow of time, based on fixed points in the brain as a compact, delta-complete topological space”, in Daniel M. Dubois (Ed.) CASYS’99 Int. Math. Conf. American Institute of Physics, CP Vol. 517, pp. 233-43. Bounias, M. (2001), “Indecidability and incompleteness in formal axiomatics as questioned by anticipatory processes”, in (Daniel, M. Dubois (Ed.) CASYS’2000 Int. Math. Conf., Int. J. Comput. Anticipatory Systems, Vol. 8, 259-74. Bounias, M. and Bonaly, A. (1994), “On mathematical links between physical existence, observability and information: towards a “theorem of something”, J. Ultra Scientist of Physical Sciences, Vol. 6 No. 2, pp. 251-9. Bounias, M. and Bonaly, A. (1996), “On metrics and scaling: physical coordinates in topological spaces”, Indian Journal of Theoretical Physics, Vol. 44 No. 4, pp. 303-21. Bounias, M. and Bonaly, A. (1997a), “The topology of perceptive functions as a corollary of the theorem of existence in closed spaces”, BioSystems, Vol. 42, pp. 191-205. Bounias, M. and Bonaly, A. (1997b), “Some theorems on the empty set as necessary and sufficient for the primary topological axioms of physical existence”, Physics Essays, Vol. 10 No. 4, pp. 633-43. Bounias, M. and Bonaly, A. (2001), “A formal link of anticipatory mental imaging with fractal features of biological time”, Amer. Inst. Phys. CP, Vol. 573, pp. 422-36. Bucher, M. and Spergel, D. (1999), “L’inflation de l’univers”. Pour la Science (Scientific American, French Edition), Vol. 257, pp. 50-7. Chaitin, G.J. (1998), The Limits of Mathematics, Springer-Verlag, Singapore, Vol. 17, pp. 80-3. Chaitin, G.J. (1999), The Unknowable, Springer, Singapore, p. 122. Chambadal, L. (1981), Dictionnaire de mathe´matiques, Hachette, Paris, pp. 225-6.

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Choquet, G. (1984), Cours de Topologie, Masson, Paris, pp. 64-5. Church, A. (1936), “An unsolvable problem of elementary number theory”, Am. J. Math., Vol. 58, pp. 345-63. Go¨del, K. (1931), “On formally undecidable propositions of Principia Mathematica and related systems I”, Monatshefte fu¨r Mathematik und Physik, Vol. 38, pp. 173-98. James, G. and James, R.C. (1992), Mathematics Dictionary, Van Nostrand Reinhold, New York, pp. 267-8. Klir, G.J. and Wierman, M.J. (1999), “Uncertainty-based information”, Studies in Fuzziness and Soft Computing, Springer-Verlag, Berlin, New York, Vol. 15, p. 168. Krasnoholovets, V. and Ivanovsky, D. (1993), “Motion of a particle and the vacuum”, Physics Essays, Vol. 6 No. 4, pp. 554-63, (Also arXiv.org e-print archive quant-ph/9910023). Krasnoholovets, V (1997), “Motion of a relativistic particle and the vacuum”, Physics Essays, Vol. 10 No. 3, pp. 407-16, (Also quant-ph/9903077). Krauss, L. (1999), “L’antigravite´”, Pour la Science, Vol. 257, pp. 42-9. Kubel, H. (1997), “The Lorentz transformation derived from an absolute aether”, Physics Essays, Vol. 10 No. 3, pp. 510-23. Lin, Y. (1988), “Can the world be studied in the viewpoint of systems ?”, Math. Comput. Modeling, Vol. 11, pp. 738-42. Lin, Y. (1989), “A multirelational approach of general systems and tests of applications. Sybthese”, Int. J. Epistemol. Methodol. and Philos. of Sci., Vol. 79, pp. 473-88. Malina, R. (2000), “Exploration of the invisible cosmos”, in Exploration: Art, Science and Projects, Institute of Ecotechnics Conference, Aix-en-Provence, France, 27-30. Meno, F. (1998), “A smaller bang?”, Physics Essays, Vol. 11 No. 2, pp. 307-10. Mirimanoff, D. (1917), “Les antinomies de Russell et de Burali-Forti, et le proble`me fondamental de la the´orie des ensembles”, L’ Enseignement Mathe´matique, Vol. 19, pp. 37-52. Mitchell, W.C. (1995), The Cult of the Big-Bang, Cosmic Sense Books, Carson City, USA, p. 240. Rothwarf, A. (1998), “An aether model of the universe”, Physics Essays, Vol. 11 No. 3, pp. 444-66. Schwartz, L. (1991), Analyse I: the´orie des ensembles et topologie, Hermann, Paris, pp. 30-5. Tricot, C. (1999), Courbes et dimension fractale, Springer-Verlag, Berlin, Heidelberg, (See also: 1999a, pp. 240-60; 1999b, p. 51, 110). Turing, A.M. (1937), On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc., Series 2, Vol. 42, pp. 230-65 and Vol. 43, pp. 544-6. Weisstein, E.W. (1999), CRC Concise Encyclopedia of Mathematics, Springer-Verlag, New York, (See also: 1999a, pp. 1099-1100; 1999b, pp. 473-4; 1999c, p. 740). Wesley, J.P. (1994), “Experimental results of Aspect et al. confirm classical local causality”, Physics Essays, , Vol. 7, p. 240, (See also: 1998, Vol. 11 No. 4, p. 610). Wesley, J.P. (1995), “Classical quantum theory”, Apeiron, Vol. 2 No. 2, pp. 27-32. Wu, Y. and Li, Y. (2002), “Beyond non-structural quantitative analysis”, in, Blown Ups, Spinning Currents and Modern Science, World Scientific, New Jersey, London, p. 324. Zadeh, L. and Kacprzyk, J. (Eds) (1992), Fuzzy Logic for the Management of Uncertainty, Wiley, New York.

Further reading Abbott, E.A. (1884), Flatland: A Romance of Many Dimensions, publ. 1991, Princeton University press, Princeton. Dewdney, A.K. (2000), “The planiverse project: then and now”, The Mathematical Intelligencer, Vol. 22 No. 1, pp. 46-51. Hannon, R.J. (1998), “An alternative explanation of the cosmological redshift”, Physics Essays, Vol. 11 No. 4, pp. 576-8. Ko¨rtve´lyessy, L. (1999), The electrical universe, Effo Kiado e´s Nyomda, Budapest, p. 704. Krasnoholovets, V. and Byckov, V. (2000), “Real inertons against hypothetical gravitons. Experimental proof of the existence of inertons”, Ind. J. Theor. Phys., Vol. 48 No. 1, pp. 1-23, (Also quant-ph/0007027). Krasnoholovets, V. (2000), “On the nature of spin, inertia and gravity of a moving canonical particle”, Ind. J. Theor. Phys., Vol. 48 No. 2, pp. 97-132, (Also quant-ph/0103110). Lebesgue, H. (1928), Lec¸ons sur l’ inte´gration, Ed. Colle`ge de France, Paris, p. 179. Lester, J. (1998), “Does matter matter ?”, Physics Essays, Vol. 11 No. 4, pp. 481-91. Loewenstein, W. (1999), The Touchstone of Life: Molecular Information, Cell Communication, and the Foundations of Life, Oxford University Press, Oxford, p. 368. Meno, F. (1997), “The photon as an aether wave and its quantized parameters”, Physics Essays, Vol. 10 No. 2, pp. 304-14. Verozub, L. (1995), “The relativity of space-time”, Physics Essays, Vol. 8 No. 4, pp. 518-23. Watson, G. (1998), “Bell’s theorem refuted: real physics and philosophy for quantum mechanics”, Physics Essays, Vol. 11 No. 3, pp. 413-21. Appendix A specimen case of calculation of the dimension of a set whose members are put on the physicist’s table like detached parts. Let ðE ne Þ ¼ {aa; ba; ca; bc; ab; cc; ac; bb; cb}: Applying (11.3a) gives several results such as: {a; b; bb; ca; cc; ab; ac} or {aa; cc; bb; ca; bc; b; a; }; etc., and {aa; bb; cc}: Thus, Diam ðE ne Þ # {aa; bb; cc} matches with the diagonal of the cartesian product {a; b; c} £ {a; b; c}: Then, applying (11.3b) gives either {B} or {ab}, or {bc}, or {ac}, that is cobbles having one member of two nuclei as the max representing Diag ðPei Þ: Since Diag ðE en Þ has three members of two nuclei each, the size ratio is r ¼ 1=3; while the number of cobbles tessellating the set is 9. Hence, applying either equations (2.1)-(2.2) or equations (11.2) gives 9:ð1=3Þe ¼ 1; that is e ¼ ln9=ln3 ¼ 2 or e ¼ ðln9 þ 2ln2Þ=ln6 ¼ 2: The dimension of ðE 13 Þ £ ðE 13 Þ ¼ ðE 23 Þ has been, therefore, correctly estimated. Had the set been alternatively composed differently, as for instance: ðE 0 ne Þ ¼ {aa; ba; ca; bc; ab; cc; ac; bb} having a lesser number of heterogeneous cobbles, then one would have found: 8:ð1=3Þe ¼ 1; that is e ¼ 1:89; a non-integer dimensional exponent, indicating a space with some fractal-like feature.

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Scanning the structure of ill-known spaces Part 2. Principles of construction of physical space Michel Bounias BioMathematics Unit (University/INRA), France

Volodymyr Krasnoholovets Institute of Physics, Natl. Acad. Sciences Kyı¨v, Ukraine Keywords Structures, Theory, Cybernetics Abstract An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Space-time is represented by ordered sequences of topologically closed Poincare´ sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one section to the next involve morphisms of the general structures, representing a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides space-time with the features of a non-linear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as the mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then, the fractal kernel stands for a “particle” and the reduction of its volume (together with an increase in its area up to infinity) is compensated by morphic changes of a finite number of surrounding cells. Quanta of distances and quanta of fractality are demonstrated. The interactions of a moving particle-like deformation with the surrounding lattice involves a fractal decomposition process, which supports the existence and properties of previously postulated inerton clouds as associated to particles. Experimental evidence of the existence of inertons is reviewed and further possibilities of experimental proofs proposed.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 976-1004 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483135

1. Introduction Part I of this study deals with some founding principles about how to assess more accurately, though in a general way how one can define the space of magmas (that is the sets, combination rules and structures) in which a given proposition can be shown to be valid (Bounias, 2001; Bounias and Krasnoholovets, 2001). Such a space, when identified, is called a probationary space (Bounias, 1997; 2001). Here, it will be presented with the formalism, which leads from the existence of abstract (e.g. purely mathematical) spaces to the justification of a distinction between parts of a physical space that can be said empty and parts which can be considered as filled with particles. This question thus deals with a possible origin of matter

and its distribution, and changes in the distribution give rise to motion Structure of (in physical terms, in the sense pointed by de Broglie and Dirac, Rothwarf ill-known spaces (1998)). Experimental evidence and propositions for further verifications will Part 2 then be presented and discussed. Recent findings (Krasnoholovets, 1997; Krasnoholovets and Ivanovsky, 1993) in the realm of fundamental physics support the prediction that an 977 abstract lattice whose existence originates from the existence of the empty set, is able to correctly account for various properties of the observed space-time at both the microscopic and cosmic scales. The models of Krasnoholovets (2000) and Krasnoholovets and Byckov (2000) suggest a new research methodology based on some practical standpoints. Specifically (Okun, 1988), the values of the constants of electromagnetic, weak and strong interactions as functions of distances between interacting particles converge at the same point on a scale of about 102 28 cm. This suggests that a violation of space homogeneity took place at this size. The model proceeds from the assumption that all quantum theories (quantum mechanics, electrodynamics, chromodynamics and others) are in fact only phenomenological. Accordingly, for the understanding of real processes occurring in the real microworld, one needs a submicroscopic approach which, in turn, should be available for all peculiarities of the microstructures of real space. In other terms, gauges for the analysis of all components of the observable universe should belong to an ultrafilter, as shown in Part 1 of this study. The investigations about the model of inertons (Krasnoholovets, 1997, 2000; 2001a-c) suggested that a founding cellular structure of space shares discrete and continuous properties, which is also shown to be consistent with the abstract theory of foundations of existence of a physical space (Bounias and Bonaly, 1997a, b). 2. Preliminaries 2.1 About gaps in former assessment of probationary spaces 2.1.1 Quantum mechanics. Quantum mechanics is founded on the calculation of the probability that a particle is present in a given volume of space. This theoretical approach postulates the existence of undefined objects called corpuscles, and does not describe the structure and properties of any embedding medium, which is considered as forbidden (Blokhintsev, 1981). Only recently, however, was raised the need that this medium, sometimes called the void, should be a space allowing the formation of pairs of particles and antiparticles (see Boyer, 2000, for review), so as to justify the existence of a material world. However, this postulate just displaces the question of the corresponding embedding medium, which is supposed not to exist independently from the photons but is often considered as if it was independent from at least large matter of masses. Parameter time is not basically, but implicitly present in the foundations of quantum physics. The concept of velocity of wave propagation and its expressions in the uncertainty

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principle, the Bell’s inequalities, etc. emerge from further developments of the theory. 2.1.2 Relativity. The relativistic theory postulates the existence of frames of references and the validity of some particular cases of measure, used as classical metrics, still without consideration for the embedding medium (Blokhintsev, 1976). It also postulates the primary existence of parameter time and the consistency of the possibility of motion in an undefined space, sometimes identified with “void”, with the properties of this “void”. However, the limitation found for the velocity of light up to the cosmological constant through electromagnetism, and the proposition of curvature of space implicitly impose some conditions on some relevant embedding medium (Einstein, 1920; Keilman, 1998; Marinov, 1996, and many others). 2.2 Assessment of existence of a space-time-like structure A former conjecture (Bonaly, 1992) stated that a characteristic of a physical space is that it should be in some way observable. This implies that an object called the “observer” should be able to interact with other objects called the “observed”. In order to avoid confusion with the usual vocabulary of systems theory, we will instead refer to the “perceiver” and the “perceived” objects. The conjecture implied that perceived objects should be topologically closed; otherwise they would offer no frontier to allow a probe to reflect their shape. Therefore, the first step of the work was to assess the existence of closed topological structures, and a proof was given that the intersection of two spaces having non-equal dimensions owns its accumulation points and is therefore closed. We propose here a shorter alternative proof. Theorem 2.2. The intersection of two connected spaces with non-equal dimension is topologically closed. Alternative proof. Let E n and E m be two spaces with topological dimensions m, n ðm . nÞ embedded in W1, a compact connected space. Let S n denote the intersection E n > E m and X n the complementary of S in E n. Consider the continuity of mappings in W1 inducing continuity to E n and E m: the neighborhood of any point in E m is the mapping of a neighborhood of a point in X n. Suppose S n is open, then because the union of open sets is open, the entire E m is neighborhood of any point in S n. Thus, there would exist a bijective mapping of opens of X m on opens of S n. In particular, an open subset of (n+2) points in S n could be homeomorphic to a (n+1)-simplex in E m. This is impossible because two spaces with non-equal dimensions cannot be A homeomorphic. Thus S n is closed. The closed 3D intersections of parts of a n-space (with n $ 3) own the properties of Poincare´ sections (Bonaly and Bounias, 1995). Then, given a manifold of such sections, the mappings of one into another section provides an ordered sequence of corresponding spaces in which closed topological structures are to be found: this accounts for a time-like arrow. As the

Jordan-Veblen theorem states that any path connecting the interior of a closed Structure of system to an outside point has a non-empty intersection with the frontier of the ill-known spaces closed system, interactions between closed objects are allowed: this accounts Part 2 for physical interactions. Furthermore, if such a path is connected to a converging sequence of mappings, the fixed points (of Banach type) will stand for perceptions of the outside. Moreover, as the Brouwer’s theorem states that 979 in a closed system, all continuous mappings have a fixed point, and that the brain represents a compact complete space in which mappings from a topological into a discrete space are continuous, there exists an associate set of fixed points (of the Brouwer’s type) representing the self (Bounias and Bonaly, 1997). Finally, spaces of topologically closed parts account for interaction and for perception, thus they meet the properties of physical spaces (Figure 1). In a former conjecture, Bonaly and Bounias (1993) proposed that the fundamental metrics of our space-time should be represented by a convolution product where the embedding part U4 would be described by the following relation:  Z Z U4 ¼ ðdx~:d~y:d ~zÞ * dðwÞ ð1Þ dS

where dS is an element of space-time and dc(w) is a function accounting for the extension of 3D coordinates to the fourth dimension through convolution (*) with the volume of space. Formal proofs of this structure will be provided later. 3. On foundations of space-time 3.1 Space-time as a topologically discrete structure How two Poincare´ sections are mapped is assessed by using a natural metrics of topological spaces: the set-distance, first established for two sets (Bounias and Bonaly, 1996) and further generalized to manifolds of sets (Bounias, 1997). In brief, let DðA; B; C; . . .Þ be the generalized set distance as the extended symmetric difference of a family of closed spaces: DðAi Þi[N ¼ › Aj Þ

ð2:1Þ

<{Ai }

The complementary of D, that is Aj Þ in a closed space is closed. It is also closed even if it involves open components with non-equal dimensions. Thus, in this system, mh {Ai }i ¼ Aj Þ has been called the “instans”, that is the state of objects in a timeless Poincare´ section (Bounias, 1997). As distances D are the complementaries of objects, the system stands as a manifold of open and closed subparts. Mappings of these manifolds from one to another section which preserve the topology represent a reference frame in which the “analysis situ” (the original name for “topology”) will allow us to characterize the eventual changes in the configuration of some components:

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Figure 1.

if morphisms are observed, then this will be interpreted as a motion-like Structure of phenomenon when comparing the state of a section with the state of the ill-known spaces mapped section. Part 2 It should be noted that the spaces referred earlier can exist upon acceptance of the existence of the empty set as a primary axiom (Bounias and Bonaly, 1997a, b), with consequences which will be addressed below. 981 Lemma 3.1. The set-distance provides a set with the finer topology and the set-distance of non-identical parts provides a set with an ultrafilter. Proof. The set-distance is founded on { >< } and it suffices to define a topology as union and intersection of set-distances are distances, including D(A, A)¼Ø. The latter case must be excluded from a filter, which is non-empty. Then, since any filter and topology are founded on { ><[  . }; it is provided with D. Conversely, regarding a topology or a filter founded on any additional property (’), this property is not necessarily provided to a D-filter. The topology and filter induced by D are thus, respectively, the finer topology and an ultrafilter. A The mappings of both distances and instans from one to another section can be described by a function called the “moment of junction”, because it has the global structure of a momentum. Consider the particular case of the homeomorphic sequence of mappings of the general topology of the system, this provides a kind of reference frame, in which it will become possible to assess the changes in the situation of points and sets of points eventually present within these structures. The origin of such points will be addressed in the next section. Here, just consider point (x, y, z,. . .) belonging to either of closed or open parts. For any x belonging to a set Ei in a section S(i), an indicatrix function l(x) is defined by the correspondence of x with some c(x) in S(i+1): x [ EðiÞ; 1Ei ðxÞ ¼ 1 iff x [ Ei; 1Ei ðxÞ ¼ 0 iff x  Ei cðxÞ [ Eði þ 1Þ; 1ðEiþ1Þ ðxÞ ¼ 1 iff cðxÞ [ Eði þ 1Þ; 1ðEiþ1Þ ðxÞ

ð2:2Þ

¼ 0 iff cðxÞ  Eði þ 1Þ Then, a function f ðEi;Eiþ1Þ (more shortly noted as fE ) is defined as: f E ¼ 1 iff : 1ðEiÞ ¼ 1ðEiþ1Þ

ð2:3Þ

f E ¼ 0 iff : 1ðEiÞ – 1ðEiþ1Þ Summing over all points x in the whole of {E} provides fE (E) which accounts for a distribution of the indicatrix functions of all points out of the maximum number of possibilities, which would be 2E for the set of parts of set E. This finally leads to the expression of the proportion of points involved in the mappings of parts of E(i ) into E(i +1):

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f E ðEÞ ¼ f E ðEÞ=2E

0 , f E ðEÞ , 1

ð2:4Þ

Two species of the moment of junction are represented by the composition (’) of f EE (E ) with either the set-distance of the instans, since E $ DðEÞ < mh E i and the distribution of points in the complementary structures is not the complementary of their distributions. Hence: MJ D ¼ DðEÞ ’ f EE ðEÞ

ð2:5aÞ

MJ m ¼ mh E i ’ f EE ðEÞ

ð2:5bÞ

Generally, one will have MJ D – MJ m As a composition of variables with their distribution, relations (2.5a, b) actually represent a form of momentum. 3.2 Space-time as fulfilling a non-linear convolution relation The “moments of junctions” (MJ) mapping an instans (a 3D section of the embedding four-space) to the next one apply to both the open (the distances) and their complementaries the closed (the reference objects) in the embedding spaces. But points representing physical objects able to move in a physical space may be contained in both there reference structures. Then, it appears that two kinds of mappings are composed with one another. Theorem 3.2. A space-time-like sequence of Poincare´ sections is a non-linear convolution of morphisms. Proof. The demonstration involves the following four steps. (1) One kind of mapping (M) connects a frame of reference to the next: here, the same organization of the reference frame-spaces must be found in two consecutive instants of our space-time, otherwise, no change in the position of the contained objects could be correctly characterized. However, there may be some deformations of the sequence of reference frames, on the condition that the general topology is conserved, and that each frame is homeomorphic to the previous one. Mappings (M) will thus denote the corresponding category of morphisms. (2) The other kind of mapping ( J) connects the objects of one reference cell to the corresponding next one. Mappings ( J) thus behave as indicatrix functions of the situation of objects within the frames, and therefore, they are typically relevant from the “Analysis situs”, that is the former name for topology, originally used by Poincare´ himself (Bottazzini, 2000). These morphisms thus belong to a complementary category. Then, each section, or timeless instant (that is a form of the above more general “instans”) of our space-time, is described by a composition (W) of these two kinds of morphisms:

Structure of ill-known spaces (3) Stepping from one to the next instant is finally represented by a mapping T, Part 2 Space-time instant ¼ MWJ

ð2:6Þ

such that the composition (MWJ) at iterate (k) is mapped into a composition (M’J) at iterate (k + i): ðM’JÞkþi ¼ T ’ ðMWJÞk

ð2:7Þ

Hence, mapping (T ’ ) appears like a relation of the type Rðk þ iÞ similar to that denoted below by RðkþjÞ , which maps a function Fi+k into F 0jþk : F 0jþk ¼ RðkþjÞ F iþk

ð2:8Þ

(4) The above relation represents a case of the generalized convolution that is a non-linear and multidimensional form of the convolution product, which was first described by Bolivar-Toledo (1985). The authors proposed this concept as a tool for computing the behaviour of visual perception. The demonstration that relation (2.8) is a form of convolution is achievable by considering the following example. Let að j 2 kÞ be a particular form of RðkþjÞ , then equation (2.8) becomes: F 0k ¼ Sa ð j 2 kÞF j that is, for the case of an integrable space: Z 0 F ðXÞ ¼ aðX 0 2 XÞFðX 0 ÞdðX 0 Þ

ð2:9Þ

ð2:10Þ

a relation exhibiting a great similarity with a distribution of functions, in the Schwartz sense (Schwartz, 1966): X k f ; wl ¼ wðxÞf ðxÞ dx ð2:11Þ or a convolution product: Z f ðX 2 uÞFðuÞdðuÞ ¼ ð f * FÞðXÞ

ð2:12Þ

E

Thus, the connection from the abstract universe of mathematical spaces and the physical universe of our observable space-time is provided by a convolution of morphisms, which supports the conjecture of relation (1). Interestingly, the distributions were primarily considered by Schwartz as an invention, in contrast with other concepts, which he considered as the discovery of preexisting foundations of the total universe (Schwartz, 1997). Now, the present work could provide the distributions with the status of a discovery. A

983

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Remarks 2.2. Our observable space-time should possess non-linear properties which necessarily involve specific features (Lin and Wu, 1998): therefore, peculiar approaches are needed through appropriate mathematical concepts, otherwise incorrect descriptions of the real world result as a consequence of approximate treatment of non-linear model resolution (Wu and Lin, 2002). 4. Relative scales in the empty-set lattice It has been demonstrated by Bounias and Krasnoholovets et al. (2001) that the antifounded properties of the empty set provide existence to a lattice involving a tessellation of the corresponding abstract space with empty balls. This structure has thus been called a “tessellatice”. Its formalism will be completed in Section 4.2 up to Corollary 3.2.4. 4.1 Quantum levels at relative scales Inside any of the above spaces, properties at micro-scale are provided by properties of the spaces whose members are empty-set units. It will be shown here that particular levels of a measure of these units can be discerned. Lemma 4.1.2. The Cartesian product of a finite beginning section of the integer numbers provides a variety of non-equal empty intervals. Proof. Let AðN þ1Þ ¼ 0; 1; 2; . . .; N denote a beginning section, that is the set of all members of a part ðM ; dÞ of the natural integers (N ) provided with an order relation (d), which are lower than (N+1). A set {Ø, a,b,. . .} is equipotent with {0,1,2,. . ., n} and will be denoted by (En). Then, since any set contains Ø, one has ðE n Þ ; AðN þ1Þ . Consider (En)2¼(En) £ (En). The resulting set contains ordered pairs including the diagonal (aa, bb,. . .) and the heterologous pairs (ab, ac, bd,. . .) accounting for rational numbers (aa 7 ! 1; ab 7 ! a=b; dc 7 ! d=c, etc.). Since all the members of the diagonal are mapped into one, there remains n 2 ðn 2 1Þ distinct pairs. A rational can be represented by a 2-simplex or facet, whose small sides are the corresponding integers. Jumping to three-dimensional conditions with (En)3¼(En)2 £ (En), each new rational is represented by a 3-simplex. This representation offers the advantage over the usual square to cube representation of avoiding several squares or cubes to share common edges or facets. Then, the number of 3-simplexes reflects the number RðE n Þ3 of rational numbers available from any initial beginning segment A(N+1). A Consider the cuttings or segments, represented by intervals between any two of these rational numbers. These segments represent the whole of available distances in the corresponding subpart of a 3-space. Remind that such a subpart is involved in some part of an ordered sequence, which is in a segment of an observable space-time. Let [0, n ] be the larger interval of (En). Denote by mE]0, n[ a measure of the open part or interior ]0, n[ of this interval in space (En): if E is a segment of N, one has:

mE 0; n½ ¼ ØN

ð3:1Þ

This is also encountered in the Cartesian product (En)3 when it includes (Ø). Consider any of the smaller intervals (s) in (En)3 and denote its measure by mE ½ðs ðE n Þ3 . By definition, mE ½ðs ðE n Þ3  , mE ½0; n ð3:2Þ Q 3 Consider Ø any of the open or interior ](s (En) [, then since G , H ) ðg , GÞ , ðH Þ and that the distance of the interior of a set to its frontier is naught (Tricot, 1999): ØQ , ØN

ð3:3Þ

This imposes an order relation holding on empty sets constructed on various segments of a finite product of finite beginning sections of the set of natural integers or equipotent to such a section. Corollary 4.1.2. A finite set of rational numbers inferring from a Cartesian product of a finite beginning section of integer numbers establishes a discrete scale of relative sizes. Proof. (i) Intervals are constructed from mappings G : N D 7 ! Q of ðN £ N £ N £ . . .Þ in Q. For example, with D ¼ 2, the smaller ratios available are 1/n and 1/(n 2 1), so that their distance is the smaller interval: 1/n(n 2 1). Consider now the few smaller intervals (s) in (En)3. One can observe, in the following order of increasing sizes: ;n . 1 : ðsðiÞ Þ ¼ 1=n 2 ðn 2 1Þ , ðsðiiÞ Þ ¼ 1=nðn 2 1Þ2 ð3:4Þ 2

, ðsðiiiÞ Þ ¼ ðn 2 1Þ=n ðn 2 1Þ

2

Consider the maximal of the ratios of larger (n) to smaller (1/n 2(n21)) segments: then one gets maxðsÞ=minðsÞ ¼ n 3 ðn 2 1Þ

ð3:5Þ

One possibility of a scaling progression covering integer subdivisions (n) consists in dividing a fundamental segment (n ¼ 1) by 2, then each subsegment by 3, etc. Thus, the size of structures is a function of iterations (n). At each step (nj) the ratio of size in dimension D will be: (Pnj)D, so that the maximal will be, following equation (3.5):

r / {ðPnj ÞD ðPnj 2 1Þ}

ð3:6Þ

j¼1!n

The manifold (Pnj) is a commutative Bourbaki-multipliable indexed on the integer section I ¼ [1, n ]. In practice, values can be written as rj ¼ aj.10xj, where in base 10, one will take aj belonging to the (always existing) neighborhood of

Structure of ill-known spaces Part 2 985

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unity, i.e. aj : [ 1½, and look at the corresponding integer exponents xj as the order of sizes of structures constructed from the lattice L ¼ ðPnj ÞD . Regarding distances (D ¼ 1) to areas and volumes (D ¼ 2 and 3), equation (3.6) consistently provides the predicted orders of scales listed in Table I. The latter

986 r ¼ Pj(vj) x

Table I. Range and intermediate levels of the scale of size of objects composing a universe constructed as described in relation (3.6). Intervals constructed with powers of ten on the neighbourhood of unity (a]1 [ ) are confronted through dimensions: (i) D ¼ 1 with the simple multipliable set Pj (vj); (ii) D ¼ 2 involving intervals 1/n(n 2 1); (iii) D ¼ 3 involving intervals 1/n 2(n2 1). Here the choice ]0.7, 1.3[ just reflects the case of a normal distribution quantile sufficiently close to unity as the mean

0 2 10 11 17 21 26 28 29 31 33 40 42 56 59 61 ......

(1 ^ 0.3)10

x

0

r ¼ {Pj(vj) (Pj (vj)21)} (v)

1.10 1.20 £ 102 0.87 £ 1010

(1) (5) (14)

1.22 £ 1017 1.12 £ 1021

(19) (22)

1.09 £ 1028

(27)

v

(1 ^ 0.3)10x 0

1.20 £ 1056

(v)

v

(1)

29

(27)

1.26 £ 1042 1.18 £ 1056 0.93 £ 1059

(22) (27) (28)

1.27 £ 10 0.88 £ 1031 0.87 £ 1033 1.03 £ 1040

r ¼ {Pj (v j )2 (Pj (vj)2 1)}

(29) (33) (35) (45)

(1 ^ 0.3)10x

(v)

0

(1)

1.28 £ 1011

(7)

1.10 £ 1026

(12)

0.91 £ 1040

(16)

1.33 £ 1059 (21) 1.24 £ 1061 ..........................................................................

(a) 82 84

0.83 £ 1082

99 100

1.20 £ 10100

(60) 1.29 £ 1084

(27)

1.12 £ 1099 (70)

(45) 112 1.25 £ 10112 115 1.13 £ 10115 117 0.89 £ 10117 (79) 120 1.10 £ 10120 (35) ................................................................................ (b) 139 142 150 163 171

142

1.25 £ 10 1.00 £ 10150 1.00 £ 10163 1.24 £ 10171

0.85 £ 10139 (92) (96) (103) (107)

(39)

(c) 0.92 £ 10171

(62)

Notes: (a) and (b) suggest further levels of higher scale universes; (c) a continued cluster from 10142 to 10171, suggesting a quite different organization of matter.; Predictable orders of size, from the Planck scale, roughly comply with quark-like size (1010 – 11), particle to atoms (1011 – 17), molecules (1021), human size (1028), stars and solar systems (1040 – 42), up to the estimated upper limit (1056) which could be bounded by a “anti-Planck” scale at (1060 – 61)

represents quantic-like levels of clusters of objects sharing successive orders of Structure of sizes while constructed from the others. ill-known spaces (ii) There is a finite number of segments or intervals, since their supremum is Part 2 the number of 3-simplexes, which is finite, and contains redundant terms. This number is the number of pairwise combinations of distinct rationals. Therefore: supðNðE n Þ3 Þ ¼ C 2n 2 2nþ1 ¼ nðn 2 1Þðn 2 2 n þ 1Þ=2

ð3:7Þ

3

A general formula for exactly (N(En) ) is not readily available since it involves prime numbers occurring in (En), but this does not change the meaning of the reasoning. Finally, there is a finite number of ratios of segment sizes imposing on a subpart of a space-time sequence a limited number of relative scales for any of the objects represented by closed subspaces in (En)3. A Corollary. The axiom of availability (stating that a rule provided to a set must be considered as explicitly applying to the set’s members and parts) is necessary to an exploration of an unknown space, either mathematical or physical. The following simple counter-example provides the proof. Suppose the axiom of availability is not stated: then, a complete subset of the rational numbers may not be provided in all bases by the Cartesian product of a segment of natural numbers. Let En¼{1,. . ., n} with n , 9, and let two integers, p; q [ E n : Then, the pair ðp; qÞ [ ðE 2n Þ ¼ E n £ E n : Usually, ( p,q) accounts for the ratio p/q, so that the set of pairs ( p, q) is equipotent to the set of rational numbers noted as fractions: (ei.d1d2. . .di) where ei. stands for the entire part and d1d2. . .di stands for the decimal part. Let n ¼ 4, and take p ¼ 1, q ¼ 4. Then the ordered pair (1, 4) stands for the ratio 1/4. However, writing 1/4 ¼ 0.25 needs digit 5 to be available, whereas one has just 1, 2, 3, 4 available and not 5. Therefore, in this system, since digit 5 does not exist unless the additional axiom of the addition is introduced, the mapping of ordered pairs to the writing in base 10 of the corresponding rational numbers is not valid. The availability of the power set of parts, i.e. the infinitely iterated sets of parts (Bounias, 2001) is enough to break this barrier. 4.2 About boundaries Converging sequences of rational numbers are known to provide the set of real numbers. However, in a space of finite dimension, real numbers cannot infer by this way. In contrast, infinitely descending sequences (in the Mirimanoff sense) of pairs of the ØQ and ØN types can be found inside each part {Ø}. Therefore, infinitely smaller intervals could always be found in the lower range of scales. These infinitely decreasing sizes are of a different nature in that they fill each discrete part {Ø}. However, this does not mean that these structures necessarily are the ultimate ones. Besides the empty set as the set with no members has:

987

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Proposition 4.2.1. There exists a set with no parts. Preliminary proof. The space of (Ø) provided with the complementarity property gives rise to abstract sets equipotent to sets of numbers, so that for instance (W m) owns parts equipotent to R m. Consider the union U ¼ <ðW Þ of all possible W m spaces, and its structural complementary in the resulting fundamental space F(U): the structural complementary of a space with parts is a space with no parts. Since it has no parts, it cannot have members. In effect, if its members were non-empty, this would be a non-empty space, which is excluded by definition, while if it had empty parts, it would just be Ø. Thus, there exists a set denoted here by (¢) which has neither members nor parts. Furthermore, ¢ is contained in none of the existing sets: otherwise, it would be the complementary of Borel sets and therefore it would include parts of itself. This provides a set of possible structures with a lower boundary. A The fundamental set embedding space U can thus be written as FðU Þ ¼ { <ðW Þ} < ð¢Þ

ð3:8Þ

In particular, given a partition of <ðW Þ into WX and WY, the separating distance between WX and WY in <ðW Þ is naught iff it does not belong to the filter F holding on W, that is since ¢ and only ¢  F: L<ðW Þ ðW X ; W Y Þ ¼ ð¢Þ

ð3:9Þ

The empty hyperset can no longer be treated as it was formerly considered the well founded empty set: in particular, Ø  F whereas Øø has no status. Corollary 4.2.1. The set with neither members nor parts is strictly unique. Proof. Let two universes FðUiÞ ¼ { <ðWi Þ} < ð¢iÞ and FðUjÞ ¼ {<ðWj Þ} < ð¢jÞ as in equation (3.8). Then, ð¢iÞ < ð¢jÞ ¼ ð¢Þ since the reunion of no parts and no member is just identical with (¢). However, this would also allow us to write: (¢)¼{(¢j), (¢j)}, and (¢j) would be composed of two parts, which is contradictory. A Set (¢) can thus be denoted as the “nothingness singleton” {¢}. Corollary 4.2.2. A set equipotent to the set of natural integers N is intrinsically of non-zero measure, and a segment E of N cannot be of measure naught even relatively to the correspondingly available segment of Q. A member of N x is never of measure naught. Preliminary proof. (i) Given N and only N as the fundamental set, one cannot insert each member in an interval as small as needed, since there exists no segment available with size lower than 2 units, able to contain each point. For this to be achieved, it is necessary to provide the system with at least N £ N so as to generate Q. The former case is called the intrinsic measure on N, while the latter is the measure on N relative to Q. (ii) Let E(Ø) be equipotent to a beginning section of (N ): the result presented earlier states that there exists a finite number of rational inferring from the Cartesian product of this space. Therefore, it is not

possible to insert the members of E in a sum of intervals of E £ E as small Structure of as needed. Thus E(Ø) can be neither intrinsic nor Q-relative measure ill-known spaces naught. Part 2 (iii) Let finally E x ¼ ðN £ N £ . . . £ N Þfinitely x times x,1 : each member of the equipotent set is an ordered pair. Even if an unordered N-uple {a, b, . . .}, whatever the nature of a and b, could be eventually of measure zero, except 989 intrinsically, an ordered N-uple (ab. . .) owns a dimension $1 and cannot be inserted in an interval as small as needed. A Corollary 4.2.3. A finite set is not of measure zero, if it contains members and parts that are representative of a space of dimension D . 1: Corollary 4.2.4. Due to the uniqueness of (¢), the “tessellatice” is correctly tessellated since no gap can subsist between any two or more of its empty tessellation balls. Furthermore, ¢ provides the tessellatice with an infimum, and thus with a partial order. 5. Particles in a lattice universe 5.1 Introduction Let space be represented by the lattice FðU Þ ¼ <ðW Þ < ð¢Þ as from relation (9.1) in Bounias and Krasnoholovets (2001), where ¢ is the set with neither members nor parts. This accounts for both relativistic space and quantic void, since: (i) the concept of distance and time have been defined, and (ii) this space holds for a quantum void since on one hand, it provides a discrete topology, with quantum scales, and on the other hand it contains no “solid” object that would stand for a given provision of physical matter. Relation (2.2) involves the mapping of a frame of reference into its image frame of reference in the next section of space-time. Without such continuity, there would be no possibility of assessing the motion of any object in the perceived universe. This is exactly a case of “analysis situs”, in the original meaning used by Poincare´. Now, continuity in the perception of a space-time is provided iff the frames of references are conserved through homeomorphic mappings. This means that there is no need for exact replication: just topological structures should be conserved. Hence, the realization of varieties if allowed, even in a space of different dimension. This supports the following: Proposition 5. The sequence of mappings of one into another structure of reference (e.g. elementary cells) represents an oscillation of any cell volume along the arrow of physical time. However, there is a case in which a threshold may exist, precluding the conservation of homeomorphisms: let a transformation of a cell involve some iterated internal similarity (see Figure 2 for simplified example). Then, if N similar figures with similarity ratios 1/r are obtained, the Bouligand exponent (e) is given by N ð1=rÞe ¼ 1

ð4:1Þ

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and the image cell gets a dimensional change from d to d0 ¼ lnðN Þ=lnðrÞ ¼ e . 1: Then, the putatively homeomorphic part of the image cell is no longer a continued figure and the transformed cell no longer owns the property of a reference cell. This transformation stands for the formation of a “particle” also called “particled cell” or more appropriately “particled ball”, since it is a kind of topological ball B[Ø, r(ø)]. Thus, the following: Statement 5. A particled ball is represented by a non-homeomorphic transformation in a continuous deformation of space elementary cells. 5.2 On quanta of fractality and fractal decomposition Before examining the interactions of particled balls with the degenerate space lattice and further with other particled balls, it is necessary to demonstrate some mathematical preliminaries. 5.2.1 Quanta of fractality. A minimum fractal structure is provided by a self-similar figure whose combination rule includes an initiator and a generator, and for which the similarity dimension exponent is higher than unity. (i) Initiator. Due to self-similarity of $ Ø, each time one considers the complementary of itself in itself, one gains: Ø 7 ! {ðØÞ; Ø}: That is, one ball gives two identical balls. This is continued into a sequence of {1/2, 1/4,. . ., 1/2n} numbers at the nth iteration. Thus, the series X ðI Þ ¼ {1=2i } ði¼1!1Þ

Figure 2. The continuity of homeomorphic mappings of structures is broken once if deformation involves an iterated transformation with internal self-similarity, which involves a change in the dimension of the mapped structure

stands for the initiator, providing the needed iteration process. The terms of (I ) Structure of are indexed on the set of natural numbers, and thus provide an infinitely ill-known spaces countable number of members. Part 2 Interestingly, 2n also denotes the number of parts from a set of n members. (ii) Generator. Let an initial figure (A) be subdivided into r subfigures at the first iteration. The similarity ratio is thus r¼1/r. Let N ¼ (r + a) be the 991 number of subfigures constructed on the original one. Then, one has e ¼ Ln(r + a)/Ln N. The value of e is bounded by unity if r is extended to infinity. For any r finite (likely the case in a physical world), the exponent e is above unity. Then: {minðeÞje . 1} ¼ LnðmaxðrÞ þ 1Þ=LnðmaxðrÞÞ

ð4:2Þ

This completes the description of a quantum of fractality. 5.2.2 The fractal decomposition principle. Let a fractal system be denoted by G, such as G= {(Ø), (r + a)}. More complex systems just need to be incorporated in several different subfigures to which the following reasoning could be extended. At the nth iteration, the number of additional subfigures is Nn ¼ (r + a)n. The similarity ratio becomes r n ¼ 1/r n. Owing to subvolumes (vi) constituted at each iteration, in the simplest case vi ¼ vi21 ð1=rÞ3 . Since at the ith iteration, as many as Ni ¼ (r + a)i such subvolumes are created, the total volume occupied by the subvolumes formed by the fractal iteration to infinity is the sum of the series: X vf ¼ ð4:3aÞ {ðr þ aÞi · vi21 ð1=rÞ3 } ði¼1!1Þ

which can be developed into: X {½ vf ¼ ði¼1!1Þ

P ðr þ aÞi21 =ðrÞ3i }

ði¼1!nÞ

ð4:3bÞ

This leads to the following definition: Definition 5.2. A fractal decomposition consists in the distribution of the members of the set of fractal subfigures: 8 9 < X = G. {ðr þ aÞi :vi21 ð1=rÞ3 } :ði¼1!1Þ ; constructed on one figure, among a number of connected figures (C1, C2,. . ., Ck) similar to the initial figure (A). If k reaches infinity, then all subfigures of A are distributed and (A) is no longer a fractal. Figure 3 shows in a very schematic way the fractal decomposition process.

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Corollary 5.2. Reciprocally, a fractal figure can be recomposed from an infinitely enumerable set of self-similar figures whose numbers and sizes are distributed as in relation (5.2.2a). 5.3 Interactions involving exchanges of structures 5.3.1 Scattering from friction. Consider a ball (A) containing a fractal subpart on it, as shown in Figures 2 and 3. Deformations can be transferred from one to another ball with conservation of the total volume of the full lattice (which is constituted by a higher scale empty set). If a fractal deformation is subjected to motion, it will collide with surrounding degenerate balls. Such collisions will result in fractal decompositions at the expense of (A) whose exponent (eA) will decrease, and to the profit of degenerate cells: if k is finite, one will have (e1) . 1, (e2) . 1,. . ., (ek) . 1. A fractal decomposition gives rise to a distribution of coefficients f(ek), whose most ordered form is a sequence of decreasing values: f ðek Þ ¼ {ðei Þði[k;1Þ }

ð4:4Þ

5.3.2 Boundary conditions. From relation (4.4), it follows that the remaining of fractality decreases from the kernel (i.e. the area adjacent to the original particled deformation) to the edge of the inerton cloud. At the edge, it can be conjectured that, depending on the local resistance of the lattice, the last decomposition (denoted as the nth iteration) can result in (en) ¼ 1. Since in all cases, one has ðen21 Þ . 1, eventhough the corresponding remaining deformation is a fragment of the original fractal structure, the resulting non-fractal deformations can be theoretically distributed up to infinite distance.

Figure 3. A canonic ball is represented as a triangle, figuring three dimension, in a metaphoric form. A degenerate ball keeps the same dimension, in contrast with a particled ball endowed with a fractal substructure. A complete decomposition into one single ball (k ¼ 1) conserves the volume without keeping the fractal dimension. The von Koch-like fractal has been simplified to three iterates for clarity

Therefore, while central inertons exhibit decreasing higher boundaries, edge Structure of inertons are bounded by a rupture of the remaining fractality. ill-known spaces 5.4 Discussion and alternative hypothesis 5.4.1 About infinitesimal elements. One would wonder if an infinite iteration could not physically take an infinite time and thus would be non-physical. (i) One answer is that all from the first iteration would take a time decreasing to infinitely small values to occur. (ii) Another would state that a quantum of fractality with the simplest geometrical generator and initiator would also stand for a quantum of time corresponding to the maximum velocity for the corresponding particle, and that the velocity of a non-fractal deformation would stand for the maximum velocity of non-massive corpuscles. 5.4.2 Incomplete fractality hypothesis. However, one could also conjecture that mass could be in some way proportional to the number of iterations on the way to a fractal whose completion would be only a theoretical (likely hyperbolic) limit. Here again, the number of iterations provides an alternative kind of qualitative jump, where n ¼ 1 iteration would stand for non-massive corpuscle, and n $ 2 for massive particles. 5.4.3 Topological alternatives. Some other features about massive vs non-massive properties remain to be explored. Among other candidates, one could list dense vs non-dense, compact vs not compact, complete vs not complete subspaces. Conjecturally, one could envision that the space of particles could be dense everywhere in that of superparticles while non-dense in the total space, and the space of superparticles could be dense in the total space. This is a matter of work in progress. 6. Practical predictions from the inertons theory 6.1 Preliminaries A particled ball, as described earlier, provides a formalism describing the elementary particles proposed by Krasnoholovets (1997, 2000) and Krasnoholovets and Ivanovsky (1993). In this respect, mass is represented by a fractal reduction on volume of a ball, while just a reduction of volume as in degenerate cells it is not sufficient to provide mass. Accordingly, if vo is the volume of an absolutely free cell, then the reduction of volume resulting from a fractal concavity is the following: V part ¼ vo 2 vf that is, according to relation (4.3b): 0 V part ¼ vo @1 2

X h ði¼1!1Þ

that is, since ðr þ aÞ ¼ ðrÞe :

1  i. P ðr þ aÞi21 ðrÞ3i A

ði¼1!nÞ

ð5:1aÞ

Part 2 993

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V part

0 X X  @ ¼ vo @1 2 ðnÞ

994

ði¼1!1Þ

1 1   P ðrn Þe n i21 =ðr n Þ3i AnA

ði¼1!nÞ

ð5:1bÞ

where (n) denotes the possibly several fractal concavities affecting the particled ball. This point will be the matter of an examination of the various kinds of particles predicted by the model in part 3 of this study. Relation (5.1b) relates the volume of particled balls to the fractal dimensional change (e), which can be expressed as the following. Proposition 6.1. The mass Ma of a particled ball A is a function of the fractal-related decrease of the volume of the ball: Ma/ ð1=V part Þ · ðen 2 1Þen $ 1

ð5:2Þ

where (e) is the Bouligand exponent and (e2 1) is the gain in dimensionality given by the fractal iteration. Just a volume decrease is not sufficient for providing a ball with mass, since a dimensional increase is a necessary condition. 6.2 Foundations of the inertons theory Two interaction phenomena are considered: first, the elasticity (g) of the lattice favours an exchange of fragments of the fractal structure between the particled ball and the surrounding degenerate balls. In a first approach, the resulting oscillation has been considered homogeneous. Second, if the particled ball has been given a velocity, its fractal deformations collide with neighbouring degenerate balls and exchanges of fractal fragments occur. Proposition 6.2. The velocity of the transfer of deformations is faster for non-fractal deformations and slower for fractal ones, at lowering rates varying as the residual fractal exponent (ei). Justifications. A fractal subvolume owns an infinite surface, which imposes a more important transfer than the progression of a non-fractal volume, which involves a finite surface. More generally, each iteration step involves a proper quantum of transfer time. Proposition 6.3. The motion of the system constituted by a particled ball and its inerton cloud provides the basis for de Broglie and Compton wavelengths. Justifications. During the progression of a particled ball, its mass is progressively transferred to a cloud of surrounding balls which get fragments of the particle mass. These new quasi-particles are called “inertons”. The velocity of inertons is faster due to lower fractal dimension, and the cloud migrates forward up to the state where the residual mass of the particle is low enough. At this step, the particle progressively lost its velocity due to the collisions: then, since collision with the degenerate lattice balls stop, no more

inertons are produced. At this time, the elasticity of the lattice starts to reinject Structure of the fragments of deformations and progressively restores the initial fractal in ill-known spaces the particled ball, which is reconstructed. Similarly, the equivalent in Part 2 momentum (that is (kml.kvl), where kml and kvl denote the average mass and velocity of the system composed with (particle + inerton) and further denoted J) is progressively retransferred to the particled ball, which gets back its initial 995 velocity. Then, the trajectory of a particle is represented by a complex oscillating system, which drives from a state (denoted as initial) where the particle owns its full mass and velocity to a state where its velocity is minimal (eventually null) and part of its mass has been transferred to the inerton cloud (Table II).

Phase state

Start point

]0, l/2[

l/2

]l/2, l[

l

Particled Maximum ball velocityv0 Maximum mass M0 Maximum tenseness Maximum fractal (e)

Mass and velocity losses Tenseness decrease

Minimum mass and minimum velocity Minimum tenseness balance with cloud

Reincrease of Return to initial mass and state with velocity maximum mass and velocity

Inertons Minimum cloud mass M0 ¼ 0 Not yet resistance to the motion of particled deformation

Collisions with lattice: emission of inertons with high speed (low lattice tenseness)

Maximum dispersion of inertons by the degenerate lattice Maximum mass Maximum reaction Maximum tenseness as opposed to particle motion

Reaction decrease Return of mass Decrease of tenseness

Whole system

Balance in the Tenseness respective tenseness balance and and center of mass flexibility of lattice facilitate inertons reaction

No inerton reaction Resistance from degenerate lattice

Disappearance of the inertons cloud

de Broglie Reverse period reached change in localization of the center of mass

Note: The velocity of transmission of the deformations from the particled ball to the degenerate balls is a non-linear function of the following variable: V ¼ F(g, K, t, v, R; f(e)) where: g ¼ elasticity factor; K ¼ tenseness of the lattice; t ¼ transmittivity of deformations; v ¼ resistance of the lattice; R ¼ reaction to the emission of deformations; f(e) ¼ fractal characteristic

Table II. The sequence of events occurring during a cycle of the behaviour of the system composed of a moving particle deformation and its inertons cloud

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This represents a cycle with a period (l) of the path, such that min{M, V part} occurs at (l/2): this defines a parameter identified with the de Broglie wavelength. Then, the system (J), which exhibits similarities with a crystallite ~ 0 Þ and identified with the Compton wavelength has an oscillating size called ðlv (Krasnoholovets, 1997, 2000, 2001a-c). The inertons cloud itself is characterized by a specific amplitude (L) which has been related with the other two by the following relations: ~ o Þ · c^ 2 =ðvo Þ2 ðLÞ ¼ ðlÞc^=vo ¼ ðlv

ð5:3Þ

where vo is the initial velocity of the particled ball and cˆ is the velocity of emitted inertons. Table I summarizes the main steps and features of the particle to inertons cycle. Remark 6.1. The center of mass of the system (J) permanently oscillates between the original particle and the inerton cloud. This introduces an additional parameter whose various forms provide a solid support to the concept of spin (Krasnoholovets, 2000). Remark 6.2. The system composed of the particle and its inertons cloud is not likely to be of homogeneous shape. Therefore, their relative motion will expectably exhibit an eddy-like form (Wu and Lin, 2002). This property will be accounted for spin-related properties of observable matter. Proposition 6.4. The fractality of particle-giving deformations gathers its space parameters (wi)i and velocities (v) into a self-similarity expression which provides a space-to-time connection. Justifications. Let (wo) and (vo) be the reference values: then the similarity ratios are rðwÞ ¼ ðwi =wo Þ, and rðvÞ ¼ ðv=vo Þ: then

rðwÞe þ rðvÞe ¼ 1

ð5:4aÞ

Since (wi)i ¼ {distances (L), and masses (m)}, one can write: mo/m ¼ L/Lo, so that: ðmo =mÞe þ ðv=vo Þe ¼ ðL=Lo Þe þ ðv=vo Þe ¼ 1

ð5:4bÞ

While coefficient (e) gets a value above unity, the geometry outside e ¼ 1 escapes the usual (3-D + t) space-time and, owing to the previously demonstrated necessity of an embedding 4-D (timeless) space, the coefficient (e) must reach e ¼ 2. Hence, the boundary conditions provide the following results: ðmo =mÞ2 þ ðv=vo Þ2 ¼ 1 , m ¼ mo =ð1 2 ðv=vo Þ2 Þ1=2

ð5:4cÞ

ðL=Lo Þ2 þ ðv=vo Þ2 ¼ 1 , L ¼ Lo ð1 2 ðv=vo Þ2 Þ1=2

ð5:4dÞ

Remark 6.3. The Lagrangean (L) should obey a similar law and (L/Lo) Structure of should fulfill relation (5.4b) as a form of r(w)e. Then, ðL=Lo Þe þ ðv=vo Þ2 ¼ 1 ill-known spaces and analogically one could take Lo ¼ 2mv 2o , thus finally: Part 2 L ¼ 2mv2o ð1 2 ðv=vo Þ2 Þ1=2 . By analogy with special relativity, m, L, v, are the parameters of a moving object, while vo ; c, celerity of light. This supports some requirements pointed 997 by Krasnoholovets (2001a-c). 6.3 Wave mechanics analysis Work in progress (Krasnoholovets, 2001a) has provided the derivation of wave equations from the inerton system. In short, given a set {p} of two parameters describing the behaviour of, respectively, the mass of the particle and the total inertons cloud (called the “rugosity” of the surrounding space, due to the distribution of scattered fragments of fractal deformations), the equations take the following general form: p € 2 cp · Lp ¼ 0

ð5:5Þ

Solutions of equation (5.5) provide a real macroscopic wave function, allowing us to determine the equivalent of the Schro¨dinger equation. The same works finally also derive a mass field accounting for gravitation, and proves that the inert and the gravitational mass are the same. Hence, cosmic scale properties can be inferred from particle scale characteristics. In Part 3 of this work, further corollaries will be derived about the origin and classification of families of particled balls, and in connection with the former, an explanation on the mechanics underlying the origins of our observable space-time will be deduced. To which extent the motion of the {particle + inertons} system should follow the eddy-like motion that non-homogeneous systems should follow (Lin and Wu, 1998) is a matter of further investigation. 7. Experimental assessment of the inertons existence Some preliminary experimental verifications have already been achieved, and some protocols for further proofs can be proposed. 7.1 Former evidence The prediction of collective behaviour of atoms in solid matter from the existence of the (particle + inerton) system has been tested on both physical and chemical systems. 7.1.1. Moving electrons emit inerton clouds which can be detected in the form of anomalous photoelectric effects (Krasnoholovets, 2001b). 7.1.2. The impact of inertons on the collective behaviour of atoms in various metals has been evaluated and then experimentally observed by high resolution electron microscopy scanning (Krasnoholovets and Byckov, 2000).

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7.1.3. The existence of inertons clouds has been calculated for hydrogen atoms clustering in crystals of d-KH(IO3)2, and proton dynamic study has verified the theory (Krasnoholovets et al., 2001). 7.2 Further experimental protocols Former experiments have confronted the recording of signals from informational fields other than electromagnetic ones, like the Kozyrev effect and others (Kozyrev and Nazonov, 1978). Krasnoholovets and Strokach (2001) have proposed a series of protocols for testing the emission of “inerton radiation” in various conditions. The measurements will be performed by using pyroelectric sensors constructed for this purpose. The project includes the following three cases. 7.2.1. Prediction of inerton emission by the sun, and observation of fluctuations of their field. 7.2.2. Prediction and measurement of the velocity of inerton waves emitted from distant stars whose parameters are sufficiently well-known. 7.2.3. Prediction and measurement of inerton flows emitted from satellites of solar planets. 8. Discussion and conclusions The most important problem faced by most of the physical theories is that they pose some assumptions, which are founded on observations and speculations whose validity can hardly be supported except by indirect testing. About speculations, for instance, classical theories pose the existence of elementary particles, string theory supposes the existence of free strings and of masses whose linear density gives the tension of the strings (Maldacena, 2000), while the rolling up of a membrane into a cylinder of infinitesimal diameter has been identified with a 1D line (Duff, 1998), which is topologically wrong; lower boundaries are hypothesized for a Hamiltonian, if any, to account for a still ill-defined void state, while upper limits are required for the number of quantum states which are not postulated, but resulting from the statistical developments, and should be smoothed at large scales so as to let a discrete system be replaced by a continuous limit (t0 Hooft, 1999). The latter point implicitly suggests that the observable universe is infinite, which remains an open question. At observational levels, it remains difficult to obtain decisive answers for essentially two reasons. First, experimental data can often be interpreted in several ways: de Broglie noted that the same mathematical equations can have several (physical) explanations (Rothwarf, 1998), and Maldacena (2000) points that an inconsistent theory could agree with experiment. Second, even the measures of physical phenomena returns uncertain data: while the homogeneity and the isotropy of universe is required around each point for applicability of the FRW metrics (Smoller and Temple, 2000), the uniformity of

universes appears as paradoxical (Bucher and Spergel, 1999) and the claim for Structure of anisotropy-supporting data (Ralston et al., 1997) consistently raises polemic ill-known spaces reactions (Ralston and Nodland, 1997). Part 2 Between these extreme positions, there unfortunately, lies the situation in which no appropriate measure can decide between alternative hypothesis: this is the case of whether the expansion works for nearby galaxies only, or for the 999 whole universe (Smoller and Temple, 2000), and what would be expand between two independent objects (Bucher and Spergel, 1999; Walker, 1996). That objects can be independent, and why the expansion of “space” would not affect the microscopic world remains as many postulates, since the measuring devices may not be independent from the measured phenomena. Our approach aimed at trying to pose as few postulates as possible, and to rather examine which kind of probationary space(s) and mathematical properties would fulfill the conditions required to support a proposition such as: there exists a physical universe embedding a self-perceived phenomenon that we use to call life. This drove to the identification of a primary axiom as the existence of the empty set, in turn providing existence of abstract mathematical spaces. Then, spaces of topologically closed objects give rise to physical-like spaces, up to the function of self-conscious perception (Bounias, 2000a, b). Several main aspects of this model will now be examined with respect to usual requirements for space-time structures and properties. 8.1 Space-time continuity and quantum structures Such a physical-like space is, therefore, composed of discrete cells endowed with quantumwise defined relative scales whose interior is potentially provided with the power of continuum. An important consequence is that this property lays a bridge exactly on the gap separating the so-far discrete nature of microscopic world and the apparent continuity of the macroscopic universe. The moments of junction map a timeless Poincare´ section representing a state of the involved spaces into another state. Each Poincare´ section may present some relationship with what t0 Hooft (1999) called Cauchy surfaces of equal time The moments of junction represent the interval between two successive states (each timeless) of a universe. Let E(i ) be a Poincare´ section like S(i ) defined earlier: if it is an identity mapping, MJ ¼ IdðSÞ; then there is no time interval from S(i ) to S(i + 1). In all other cases, MJ represents two important parameters: first, it accounts for a differential time interval, and then for a differential element of the geometry of the corresponding space. In this sense, it neither has “thickness” nor duration. There is no “distance” in the Hausdorff sense between S(i) and S(i + 1), just a change in the topological situation. Since the step from S(i ) to S(i + 1) is a discrete one, it follows that: (i) the corresponding space owns discrete quantum properties and (ii) these discrete properties are valid whatever the scales, since they are founded on the set difference which is not dependent on any scale or the size of phenomena. It is

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noteworthy that these properties meet some requirement for space, time and matter, as suggested by t0 Hooft (1999). 8.2 Space-time and motion The moment of junction formalizes the topological characteristics of what is called motion, in a physical universe, that is what has been considered as needed for the understanding of physics (Rothwarf, 1998). While an identity mapping denotes an absence of motion, that is a null interval of time, a non-empty moment of junction stands for the minimal of any time interval: this meets a proposition of Sidhart (1999), stating that there should exist a minimum space-time interval and that “one cannot go to arbitrarily small space-time intervals or points. In our sense, there is no such “point” only instans which per se do not reflect timely features. The need for morphisms of the topological structures as frame of reference as combined with the morphisms of objects interestingly meets with a requirement hypothesized by t0 Hooft (1999): “beables” as commutable operators might be represented in morphisms of frames and the moments of junction, while “changeables” as non-commutable components might be related to morphisms of objects, up to the not commutative biological components, which makes the transition to the next points. Furthermore, morphisms are compatible with rotations that are to be expected in eddy-motion, which stands at the basis of physical motion in a non-linear universe (Wu and Lin, 2002). It should be noted that discontinuities may occur from changes occurring when a structure partly escapes the observable space-time through dimensional plunging into the embedding 4-space and re-emerges in the perceived 3-space. 8.3 Space-time reversibility properties It is noteworthy that the moment of junction is provided with reversibility, which accounts for the temporal reversibility of physical phenomena, as postulated by cosmologists (Smoller and Temple, 2000). In contrast, the biological arrow of time exhibits some irreversible features, since the sequence of brain mental images is founded on surjective mappings, which do not have the same fixed points in the reverse sense (Bounias, 2000b): this means that even if a biological system is physically reversible, the correspondence of mental images associated to outside perceptions with the self of the perceiver would be changed. 8.4 Topological constraints on space-time Seriu (2000) proposed a very interesting study on metric properties of a space of spaces. The author reaches the conclusion that physical constraints suggest the existence of drastic topological fluctuations at Planck scales. These observations would result in two correlative hypotheses: first, there would be sets of spaces with various topologies, and second, there would be

scale-dependent topologies. Seriu acknowledges the risk that such Structure of considerations depart from the very foundations of topology as a ill-known spaces mathematical concept. However, a mathematical space can give rise to Part 2 several topologies, which range from coarser to finer forms, in an ordered relation (Bourbaki, 1990a, b). In contrast with the smoothing at large scale recalled by t0 Hooft (2000), here a smoothing of topologies at low scale would be 1001 needed. These apparent contradictions vanish with the properties of the empty hyperset providing discrete features at all scales, but also own the power of continuum, that is physical “continuity” inside each fundamental cell. Note that continuity in the mathematical sense does not require smoothing. Concerning the problem of scale-related topological changes, it should be pointed that the set-distance is a scale-independent measure, and would thus fulfill a requirement formerly raised in Part 1 of this study (Bounias and Krasnoholovets, in preparation), in the form of an ultrafilter of the topologies required by Seriu. No contradiction seems to lurk in these approaches. The spectra D1 proposed by Seriu should be invariant by spatial diffeomorphism: this implies continued differentiability, a property, which is fulfilled by the convolution structure derived from our model. In addition, while the Laplacian used by Seriu as a probe for the geometry of the explored universe accounts for linear properties, the non-linear convolution provides a generalization to non-linear properties. While scale-dependent topologies may appear contradictory with fractal properties, our approach instead is consistent with such a structure. 8.5 Fractal space-time features Following the pioneer remarks of Mandelbrot (1989), some fractal properties have been found for the distribution of galaxies at rather small scales. However, the existence of a lower cut-off, as shared by the DN spectra of Seriu (2000) seen earlier, precludes that the observed autocorrelations reflect a general fractality of the entire universe. In our model, the lattice which provides space and matter with their properties has been shown to meet the properties of a true fractal. This means that some fractal features should be shared by objects at all scales of our space-time. The contradiction of uniformity of Hubble law of isotropy of the background waves with the heterogeneity of a fractal geometry exhibiting voids and structures is just apparent. When Mandelbrot drew fractal galaxies on a sheet of paper, the “void” parts were represented by some lines connecting groups of points on the sheets. This would pose a problem only if an absolute void would be postulated between objects in the universe. In contrast, our approach suggests that a common fractal structure may hold on the whole of space as an embedding (non-material) medium. It has been sketched here how this fractal geometry can account for the formation of matter corpuscles: due to all-scales self-similarity ratios, the same property should be extended to large scale

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formations. This would predict that similar fractal parameters should be found for both particle and cosmic scales. Lastly, that objects are constructed in and from a fractal lattice suggests that fractality if provided by the embedding medium: this answers a question raised by Cherbit (1987) and supports a former proposition of Feynman and Hibbs (1965) that the quantum trajectory of a particle is continued, but not derivable. 8.6 Predictions of fundamental physical parameters In the absence of “given” knowledge of what an object could be, Krasnoholovets (1997, 2000a, 2001a-c) and Krasnoholovets and Byckov (2000) and Krasnoholovets and Ivanovsky (1993), proposed that a corpuscle could be represented by a local change in the geometry of a lattice. Independently, Bounias and Bonaly (1995, 1996, 1997a, b) studied why could a mathematical space exist and how could it provide existence to a physical-like space. It was found that the results support both the hypothesis of existence of a founding lattice of Krasnoholovets et al., and the prediction of emergence of the phenomenon of self-conscious perception, which could stand for a major characteristic of life (Bounias, 2000a, b; Bounias and Bonaly, 1997b). Since the model makes useless the discrimination between relativistic and quantum approaches, we modestly expect that it might be fruitfully thought-provoking to the community, with emphasis on the fact that various theories harbour correct elements which presently are diluted in a complex network of scattered hypothesis. The next part of this study (Part 3) will provide a formal description of the construction and structures of the various fundamental particles allowed by the model and examine some implications up to cosmic scales and the origins and behaviour of universe. References Blokhintsev, D. (1976), Principles of Quantum Mechanics, Russian, French edition 1981, MIR, Moscow, p. 683. Bolivar-Toledo, O., Candela Sola, S. and Mun˜oz Blanco, J.A. (1985), “Nonlinear data transforms in perceptual systems”, Lecture Notes in Computer Sciences, Vol. 410, pp. 1-9. Bonaly, A. (1992), Personal communication to M. Bounias. Bonaly, A. and Bounias, M. (1995), “The trace of time in Poincare´ sections of topological spaces”, Physics Essays, Vol. 8 No. 2, pp. 236-44. Bounias, M. (1997), “Definition and some properties of set-differences, instans and their momentum, in the search for probationary spaces”, J. Ultra Scientist of Physical Sciences, Vol. 9 No. 2, pp. 139-45. Bounias, M. (2000a), “The theory of something: a theorem supporting the conditions for existence of a physical universe, from the empty set to the biological self”, in Daniel M. Dubois (Ed.) CASYS ’99 Int. Math. Conf. Int. J. Comput. Anticipatory Systems, Vol. 5, pp. 11-24.

Bounias, (2000b), “A theorem proving the irreversibility of the biological arrow of time, based on fixed points in the brain as a compact or delta-complete space”, Am. Inst. Phys. Conf. Proc. CP517, pp. 233-43. Bounias, M. (2001), “Indecidability and incompleteness in formal axiomatics as questioned by anticipatory processes”, in Daniel M. Dubois (Ed.) CASYS”2000 Int. Math. Conf. Int. J. Comput. Anticipatory Systems (in press). Bounias, M. and Bonaly, A. (1996), “On metrics and scaling: physical coordinates in topological spaces”, Indian Journal of Theoretical Physics, Vol. 44 No. 4, pp. 303-21. Bounias, M. and Bonaly, A. (1997a), “The topology of perceptive functions as a corollary of the theorem of existence in closed spaces”, BioSystems, Vol. 42, pp. 191-205. Bounias, M. and Bonaly, A. (1997b), “Some theorems on the empty set as necessary and sufficient for the primary topological axioms of physical existence”, Physics Essays, Vol. 10 No. 4, pp. 633-43. Bourbaki, N. (1990a), The´orie des ensembles, Masson, Paris, Chapters 1-4, p. 352. Bourbaki, N. (1990b), Topologie Ge´ne´rale, Masson, Paris, Chapters 1-4, p. 376. Boyer, T. (2000), “The infinitely empty does not exist”, Pour La Science (Scientific American, French edition), Vol. 278, pp. 128-37. Bucher, M. and Spergel, D. (1999), “L’inflation de l’ univers”, Pour La Science (Scientific American, French edition), Vol. 257, pp. 50-7. Cherbit, (1987), Dimension locale, quantite´ de mouvement et trajectoires, Fractals, Masson, Paris, pp. 340-52. Duff, M. (1998), “The new string theory”, Pour La Science (Scientific American, French edition), Vol. 246, p. 68. Einstein, A. (1920), Aether and the theory of relativity, Leyde University Lecture. French translation: Gauthier-Villars, Paris, pp. 1-12. Feynman, R. and Hibbs, A. (1965), Quantum mechanics and path integrals, McGraw Hill, NY, p. 176. Keilman, Y. (1998), “On the breakdown of the principle of relativity”, Physics Essays, Vol. 11 No. 2, pp. 325-9. Kozyrev, N.A. and Nasonov, V.V. (1978), Astronometry and Celestian Mechanics, Akademia Nauk SSSR, Moscow, Leningrad, pp. 168-79. Krasnoholovets, V. (1997), “Motion of a relativistic particle and the vacuum”, Physics Essays, Vol. 10 No. 3, pp. 407-16 (Also quant-ph/9903077). Krasnoholovets, V. (2000a), “On the nature of spin, inertia and gravity of a moving canonical particle”, Indian J. Theoretical Physics, Vol. 48 No. 2, pp. 97-132 (Also quant-ph/0103110). Krasnoholovets, V. and Byckov, V. (2000), “Real inertons against hypothetical gravitons. Experimental proof of the existence of inertons”, Ind. J. Theor. Phys., Vol. 48 No. 1, pp. 1-23 (Also quant-ph/0007027). Krasnoholovets, V. (2001a), “On the theory of anomalous photoelectric effect stemming from a substructure of matter waves”, Ind. J. of Theor. Phys., Vol. 49 No. 1, pp. 1-32 (Also arXiv.org e-print archive quant-ph/9906091). Krasnoholovets, V. (2001b), “Space structure and quantum mechanics”, Space-time and Substance, Vol. 1 No. 4, pp. 172-5 (Also quant-ph/0106106). Krasnoholovets, V. (2001c), “On the way to submicroscopic description of nature”, Ind. J. of Theor. Phys., Vol. 49 No. 2, pp. 81-95 (Also quant-ph/9908042).

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Krasnoholovets, V. and Ivanovsky, D. (1993), “Motion of a particle and the vacuum”, Physics Essays, Vol. 6 No. 4, pp. 554-63 (Also arXiv.org e-print archive quant-ph/9910023). Krasnoholovets, V., Lev, B., Gavrilko, T., Puchkovska, G. and Baran, J. (2001), “Clustering caused by crystal lattice inerton field. H-clusters in the KIO3.HIO3 crystal”, (Cond-mat/0108417). Lin, Y. and Wu, Y. (1998), “Blown-ups and the concept of whole evolutions in systems science”, Problems of Nonlinear Analysis in Engineering Systems, Vol. 4, pp. 16-31. Maldacena, J. (2000), “Gravity, particle physics and their unification”, eConf C990809, pp. 840-52 (Also arXiv.org e-print archive hep-ph/0002092). Marinov, S. (1996), “Cosmological aspects of the absolute space-time theory”, Physics Essays, Vol. 9 No. 3, pp. 357-67. Okun, L.B. (1988), Physics of Elementary Particles (in Russian), Nauka, Moscow, p. 92. Ralston, J.P. and Nodland, B. (1997), “An update on cosmological anisotropy in electromagnetic propagation”, in Donnelly, T.W. (Ed.) Proc. 7th Internat. Conf. on the Intersection of Particle and Nuclear Physics (Big Sky, Montana, 1997), Amer. Inst. Phys. CP (Also astro-ph/978114). Ralston, J., Jain, P. and Nodland, B. (1997), “The corscrew effect”, Phys. Rev. Lett., Vol. 816, pp. 26-9. Seriu, M. (2000), “Space of spaces as a metric space”, Commun. Math. Phys., Vol. 209, pp. 393-405. Sidhart, B.G. (1999) The fractal universe: from the Planck scale to the Hubble scale (Also quant-ph/9907024). Smoller, J. and Temple, B. (2000), “Cosmology with a shock-wave”, Commun. Math. Phys., Vol. 210, pp. 275-308. t0 Hooft, G. (1999), “Quantum gravity as a dissipative deterministic system”, Class. Quantum Grav., Vol. 16, pp. 3263-79. Walker, F.L. (1996), “The expanding space paradox: can the galaxies really recede?”, Physics Essays, Vol. 9 No. 2, pp. 209-15. Wu, Y. and Lin, Y. (2002), “Beyond non-structural quantitative analysis. Blown ups, spinning currents and modern science”. World Scientific, New Jersey, London, pp. 324. Further reading Bounias, M. and Bonaly, A. (1994), “On mathematical links between physical existence, observability and information: towards a “theorem of something”, J. Ultra Scientist of Physical Sciences, Vol. 6 No. 2, pp. 251-9. Hales, T.C. (2000), “Cannonballs and honeycombs”, Notices of the AMS, Vol. 47 No. 4, pp. 440-9. Hannon, R.J. (1998), “An alternative explanation of the cosmological redshift”, Physics Essays, Vol. 11 No. 4, pp. 576-8. Havard, G. and Zinsmeister, M. (2000), “Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets”, Commun. Math. Phys., Vol. 210, pp. 225-47. Joyce, M., Anderson, P.W., Montuori, M., Pietronero, L. and Sylos Labini, F. (2000), “Fractal cosmology in an open universe”, Europhysics Letters, Vol. 49, pp. 416-22. Lavrentiev, M.M., Eganova, I.A., Lutset, M.K. and Fominykh, S.F. (1990), “About distance influence of stars on resistor”, Proc. Acad. Sci. USSR, Vol. 314 No. 2, pp. 352-5 (in Russian).

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Scanning the structure of ill-known spaces Part 3. Distribution of topological structures at elementary and cosmic scales

Structure of ill-known spaces Part 3 1005

Michel Bounias Biomathematics Unit, (University/INRA), Domain of Sagne-Soulier, Le Lac d’Issarle`s, France

Volodymyr Krasnoholovets Institute of Physics, National Academy of Sciences, Pr. Nauky, Kyı¨v, Ukraine Keywords Structures, Theory, Cybernetics Abstract The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of set-distance function in such spaces provides a particular kind of space-time-like structure which favors the aggregation of such deformations into fractal forms standing for massive objects. A correlative dilatation of space appears outside the aggregates. At large scale, this dilatation results in an apparent expansion, while at submicroscopic scale the families of fractal deformations give rise to families of particle-like structure. The theory predicts the existence of classes of spin, charges and magnetic properties, while quantum properties associated with mass have previously been shown to determine the inert mass and the gravitational effects. When applied to our observable space-time, the model would provide the justifications for the existence of the creation of mass in a specified kind of void, and the fractal properties of the embedding lattice extend the phenomenon to formal justifications of big-bang-like events without any need for supply of an extemporaneous energy.

1. Introduction Despite the striking progress in the present-day research, from corpuscle physics (Fritzsh, 2000) to astrophysics (Bo¨rner, 2000), many fundamental questions remain unsolved and often contradictory (Krasnoholovets, 2001). In previous papers (Bounias and Krasnoholovets, 2003a, b) formal demonstrations have identified mass with a disruption in homeomorphic mappings of reference medium, from one to the next Poincare´ section whose ordered sequence stands for a space-time-like structure (Bonaly and Bounias, 1995). In an attempt to identify the forlam conditions of existence of a physical-like world, the existence of the empty set as the founding space along with the theory of sets and topology, extended to nonwell-founded sets as

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1005-1020 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483144

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the combination rules, was found as necessary and sufficient conditions (Bounias, 2000; Bounias and Bonaly, 1997). This paper starts from the lattice of empty elements which are balls constituted from the empty set and its successive complementaries, all exhibiting self-similarity and fractal properties (Bounias and Krasnoholovets, 2003a, b). Elementary balls within a given range of size were attributed to a virtual volume at the free state. These volumes, in fact, are reference frames in which the position of objects is assessed by an operator called the “moment of junction”, since it connects one to the next Poincare´ sections and owns the structure of a moment (Bounias, 1997). These volumes belong to the space of distances (topologically open), as a topological complementary of the space of objects (topological closed), and they will belong to this class as far as their morphisms are homeomorphic. In contrasts, balls exhibiting dimensional changes (here through fractal shaping) no longer fulfill this condition, and they have been attributed to the class of objects (Bounias and Krasnoholovets, 2003b). However, at this stage, no rationale was yet provided for the justification of existence of such structures: this point will therefore be addressed first in this paper. 2. Preliminaries The distribution of variables X, Y is a density function h(x, y) which admits margin densities for each variable in the following (Ru¨egg, 1985): Z Z f ðxÞ ¼ hðx; yÞ dx and gðyÞ ¼ hðx; yÞ dy ð1:1Þ If E is a probabilized space, which is the case of the topological spaces in which we are working (Bonaly and Bounias, 1995; Bounias, 2000), and the variables are continued, then the probability that x, y belong to E is: ZZ Pðx; y [ EÞ ¼ PðEÞ ¼ hðx; yÞ dx dy ð1:2Þ E

For discrete variables, the integral is replaced by a union or a sum. The repartition function is represented by a summation (again in either sense) with boundaries: H ðx; yÞ ¼ P{ðX # xÞ > ðY # yÞ}

ð1:3Þ

More generally, one may consider x [ [u, v ] within a domain [a, b ] of E. Then the probability of finding x in the closed segment [u, v ] is Pðu # x # vÞ ¼ ðv 2 uÞ=ðb 2 aÞ if E is totally ordered. In other cases, alternative solutions have been examined by Bounias and Krasnoholovets (2003a). Then the process will be extended to y [ [q, r ], and so on. In a discrete space, probabilities are

multiplicative, while in a continued space the repartition functions are Structure of multiplicative: H ðx; y; . . .Þ ¼ FðxÞ GðyÞ ð. . .Þ: ill-known spaces Now, let X; Y ; . . . be random objects defined on the same probabilized space, Part 3 and Z ¼ FðX; Y ; . . .Þ a real function in E. Then the moment, including the expected value, wears the form: ZZ 1007 EðZ Þ ¼ FðX; Y ; . . .Þ hðx; y; . . .Þ dx dy dð. . .Þ: ...

Whatever the form of a distribution, it owns a family of moments mck of order k and centered on c: the expected value is E ¼ m01 ; and the variance Var ¼ mE2 : One particular case is the covariance CovðX; Y Þ ¼ EðX; Y Þ 2 EðXÞ EðY Þ; so that if one has FðX; Y ; . . .Þ ¼ X þ Y ; then: VarðX þ Y Þ ¼ VarðXÞ þ VarðY Þ þ 2 CovðX; Y Þ

ð1:4Þ

This brings the question of the dependence of variables X and Y: Cov(X, Y) is bounded by zero for independent X, Y and by a maximum if X and Y are completely self-similar, like in any subpart of a fractal structure. Finally, the distribution K(z) as the probability to get the sum ðX þ Y þ · · ·Þ # z is given by the derivative of the repartition: ZZ PðX þ Y þ · · · # zÞ ¼ . . . hðx; y; . . .Þ dx dy dð. . .Þ ð2:1aÞ xþyþ· · ·#z

The summation on one variable, e.g. y is bounded by z 2 ðx þ · · ·Þ 0 z2ðxþ· · ·Þ 1 þ1 Z Z gð yÞ dyA dx PðX þ Y þ · · · # zÞ ¼ KðzÞ ¼ . . .ð f ðxÞ. . .Þ@ 21

ð2:1bÞ

21

that is in terms of distribution: kðzÞ ¼

þ Z1

. . .ð f ðxÞ. . .Þ ðgðz 2 ðx þ · · ·ÞÞÞ dx

ð2:2aÞ

21

which is a convolution function. Remark 2.1. We have shown in Part 2 of this study that the morphisms of distances and objects already fulfill a nonlinear form of generalized convolution: ðM ’ J Þkþi ¼ T ’ ðM WJ Þk

ð2:2bÞ

where M and J are morphisms of distances and objects, respectively, and T an operator mapping a Poincare´ section (Si) into (Si+1), on the basis of the moment

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of junction, MJ, that is a composition function of either the set distances or their complementaries (the “instans”) with a distribution function (Bounias, 1997). The operator T translates a composition rule (here: W) into (’). Redundancy will be considered in either active or with commutation forms. The latter involves multiple convolution of densities:

1008

f n* ¼ f 1 * f 2 * · · ·f i * · · ·* f n :

ð2:3Þ

3. Main results 3.1 On the law determining the sequence of Poincare´ sections Let Si be a closed intersection of topological dimension n produced by the intersection of a n-subspace with a m subspace ðm . nÞ belonging to the set of parts of the embedding v-space (W v). A universe will thus be constructed in a space ðW v Þ ¼ {X; ’} with X [ {X 3 > X 4 Þ a set and (’) a combination rule determining the choice of Si+1 from Si. Remark 3.1. It has been argued (Bonaly and Bounias, 1995) that v ¼ 4 and n ¼ 3 provide a optimal situation, in terms of mathematical organizational properties, which would place our space-time among the most efficient universe configurations. Thus, throughout this study, it will be sufficient to consider n ¼ 3 coming from v ¼ 4: Remark 3.2. There exists as many universes as there are laws (’). However, one particular case deserves particular attention. Proposition 3.1. Let Si denote a 3D Poincare´ section of W 4. Continuity in mappings of members of Si in the sequence {S i }i is favored if the successors S iþ1 are such that: {S i }i ¼ ð;S i ; S i > S iþ1 ¼ max{S i > S iþk }k . iÞ: Proof. Some lemmas of continuity of set-distance functions will first be demonstrated, and the proposition will then be deduced. A 3.1.1 Continuity of set distance functions. The following definition recalls the generalized distance provided by topologies as it has been presented in Part 1 (Bounias and Krasnoholovets, 2003a). Definition 3.1. Let E be a topological space E ¼ {X; T}; and A, B, C, . . ., G, . . . subspaces constituted from the set of parts of set X composing E. Then, the separating set-distance between A and B within E is denoted by LE (A, B) and identified by: LE ðA; BÞ ¼ min{ðG [ EÞ; ðA > G – Ø; B > G – ØÞ : DðA; GÞ > DðB; GÞ} ð3:1Þ where D denotes the simple set-distance as the symmetric difference: DðA; BÞ ¼ › ðA > BÞ ðA
ð3:2Þ

The generalized set-distance is given by the following relation: DE ðA; BÞ ¼ min{ðG [ EÞ; ðA > G – Ø; B > G – ØÞ : DðA; B; GÞ}

Structure of ð3:3Þ ill-known spaces

Part 3 with DðA; B; GÞ ¼

›

ððA > BÞ > ðA > GÞ > ðB > GÞÞ

ð3:4Þ

ðA
If G ¼ Ø; then relation (3.3) reduces to (3.2) and DE reduces to D. Lemma 3.1. The mapping f : D 7 ! R of the set distance (D) on the set of real numbers (R) is continuous. Proof. Let A, B in E be mapped into f ðAÞ ¼ a and f ðBÞ ¼ b in R. If a and b are cuttings, the proof is trivial. If a and b are initial segments (like simple numbers) then, take the case where a , b; and consider e as small as needed, such that a0 ¼ a þ e: For any e, there exists x in E such that e ¼ f ðxÞ: When the distance D{DðA; BÞ; ðA0 ; BÞ} is decreased by x, then the difference (b2a) becomes ((b 2 a) 2 e), i.e. it is decreased by e. A Lemma 3.2. Let A, B, G in E and f ðAÞ ¼ a; f ðBÞ ¼ b; f ðGÞ ¼ g in R. (i) The mapping £: LE 7 ! R of the separating distance on the set of real numbers is continuous if a, b, g are cuttings. (ii) If a, b, g are initial segments, the mapping remains continuous if E is totally ordered, while if E is only partly ordered by inclusion or intersection, then the mapping £ is continuous for any e , a or e . b: Proof. The first case is trivially inferring from the continuity of D. In the second case, if a , e , b; then dist(a, g) and dist(b, g) have a null difference only if g ¼ ka; bl or if they were to be considered as adjacent cuttings, then their intersection would always be null. However, in these two particular cases, the mapping £ remains correct if E is totally ordered, so that A , G , B and LE ¼ Ø: Then continuity is proved for any g. A 3.1.2 Continuity in ordered Poincare´ sections of space. Let (Si) be one 3D timeless section in W 4, ai be a member or a part of (Si) and V(ai) a neighborhood of (ai) in (Si). Call ðai Þiþk and ðV ðai ÞÞiþk the homeomorphic projections of ai and V(ai) on ðS iþk Þ: Proposition 1 states that DððS i Þ; ðS iþ1 ÞÞ must be minimal and that for the same reason, DððV ðai ÞÞiþ1 ; ðV ðaiþ1 ÞÞÞ is minimal, which is consistent with the clause of continuity. If, in contrast, there exists a section ðS iþh Þ whose distance with (Si) is smaller than DððS i Þ; ðS iþ1 ÞÞ; then the neighborhood ðV ðaiþ1 ÞÞ may be contained in DððS i Þ > ðS iþh Þ; ðS iþ1 ÞÞ. In particular, one may have DE ððV ðai ÞÞiþ1 ; ðV ðaiþ1 ÞÞÞ . DððV ðai ÞÞiþh ; ðV ðaiþh ÞÞÞ; and the condition of continuity is no longer necessarily fulfilled. This achieves the justification. 3.2 Distribution of the deformations of lattice balls 3.2.1 Introduction. Sections {S i }i are composed as pointed in Part 1 of distance (D and L) and objects ðmklÞ: The former are open and the second are closed.

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The space of distances provides the reference frame from which the topological changes of objects localization will be observed. This space has been shown in Part 2 to be basically constituted of elementary cells represented by free forms C free and degenerate forms C deg. A putative volume V free ¼ V o is attributed to free cells which are devoid of any deformations and thus described by the identity mapping (Id) from (Si) to ðS iþ1 Þ: In contrast, degenerate cells result from homeomorphic transformations, which involve some change in their volumes (in 3D sections) without dimensional alteration. Then, if dVo canonically denotes such a change in the volumes, then V deg ¼ V o ð1 ^ dÞ: From sections (Si) to ðS iþ1 Þ one has dVo(i ) mapped into dV oðiþ1Þ : Within each deg section, the set of all such deformations will be
ð4:1Þ

and with N deg the cardinal (Card) of set {C deg i }: deg deg Varð
ð4:2Þ

3.2.3 Boundaries. Gather relations (4.1) and (4.2) in: Varð<ðdÞÞ ¼
ð4:3Þ

These variances are subjected to boundary conditions, depending on the level deg of dependence or independence of ðddeg i Þ and ðdi Þ. First kind. If the variables are totally independent, like in a completely random space, one will get Covð{d; }Þ ¼ 0: Thus, the variance of the sum is minimal. Second kind. In contrast, maxCovð{d; }Þ is attained if the components {dVoðiÞ } exhibit the maximum of similarity. This condition is achieved through fractal properties of the lattice, whose cells are self-similar balls composed with the empty hyperset {Øø}. 3.2.4 Theorem of the distribution of volumes. Then, owing to Proposition 1, the selection of ðS iþ1 Þ from (Si) will preferably retain a first kind distribution. Lemma 3.3. The degenerate lattice contains a nondenumerable infinity of subdeformations.

Proof. It has been previously proved that the empty hyperset provides Structure of existence of a n-space, n as great as needed, endowed with the power of ill-known spaces continuum (Bounias and Bonaly, 1997). Each empty set unit gives a empty Part 3 complementary in itself, so that each unit provides a sequence of structures fitted one into the other, which can be indexed on a sequence of the {1=2ni }i type (Part 2). Thus, the distribution of volumes {dV oðiÞ } in the degenerate space 1011 contains infinitely many times the collections of deformations required for constituting a quantum of fractality. Hence, each time these quanta are available in the topological neighborhood of a cell in (Si), the law of selection of the next section will select {S iþ1 } in (W 4) such that the same set of deformation is organized into one single structure, that is a fractal. A This now allows the following founding statement. Corollary 3.1. The combination rule of a continued space-time-like sequence of Poincare´ sections fulfilling the option stated in Proposition 1 exhibits a trends to collapsing random distributions of degenerate cells into massive objects. Justification. Continuity associated with the condition of maximum intersection (Proposition 1) favors the collapse of scattered deformations into one single aggregate forming a fractal structure: this results in a change of dimensionality of the affected cells. The latter are no longer homeomorphic images, and therefore, they get a mass, in the sense defined in Part 2. Therefore, these cells escape the class of “reference frame” or distances, and fall into the class of “objects”. They become “particled balls”, denoted C part and their volumes are V part as described in Part 2. 3.3 Predicted structural classes of particled cells 3.3.1 Predicted particle-like components. 3.3.1.1 Mass-equivalent nonmassive corpuscles. Denote by u ¼{r, a, I } a quantum of fractality where X 1 I¼ i i¼1!1 2 is the initiator, r the self-similarity ratio and a the additional number of subfigures inserted in the (1/r) fragments of the initial figure. The corresponding fractal structure is denoted (G). It has been shown in Part 2 that (G) can be decomposed in a sequence of elementary components {C 1 ; C 2 ; . . .; C k ; . . .}: If all these elementary deformations are gathered on one single ball, then this ball contains all the quantum of fractality, though its dimension is not changed. It is, therefore, nonmassive as it stands and its motion is determined by the velocity of transfer of nonmassive deformations, that is the maximum permitted by the elasticity of the space lattice. Since the deformations are ordered and distributed in one particular structure, it owns a stability through mappings of Poincare´ sections. Such particles are

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likely to correspond to bosons, that is to pseudoparticles representing just transfer of packs of deformations in a isolated form. Hence, photon-like corpuscles will carry the equivalent of various quanta of fractality {ui }i ; that is their equivalent in mass in a decomposed form. This represents as many deformations of the lattice, and finally of equivalent in energy. 3.3.1.2 Families of massive particles. Any single ball carrying a group {Gi }i of quantum fractals will represent a class of massive particles. Depending on both the number and the mode of association of these fractal quanta, various symmetries will result and provide these classes with specific properties. Hadron-like families will thus be represented by the following common structures: Hi;k ¼ ð{ai ; ri ; ei }i;k Þ

ð5:1Þ

Simpler particles made from one single quantum of fractality {r, a} would likely correspond with lepton-like structures, such that: Li ¼ ð{N i · ðri ; Þei }Þ

ð5:2Þ

3.3.2 Spins for hadron-like balls. 3.3.2.1 Fermion-like cases. Moving massive balls have been shown to carry a cloud of deformations transferred to degenerate balls of the surrounding space, with periodic exchange between this “inertons” cloud and the original particle (Part 2). The period of this pulse has been identified with the de Broglie wavelength. Hence, the center of mass ( y) of the system composed of the particle and the inerton cloud permanently undergoes a movement forward and backward along the trajectory of the system. Two canonical positions are possible, with respect to the particle: (1) y is centered on the particled ball, and (2) t is no longer centered. The probability of state of x is thus Pð yÞ ¼ 1=2: 3.3.2.2 Boson-like cases. Consider a ball carrying quanta of masses in the decomposed form. Then, such a system opposed the minimum resistance by the surrounding degenerate balls, which are of the same nature, excepted that their individual densities of deformation are much smaller. Therefore, boson-like particles do not generate a cloud similar to that of a massive particle, and their center of mass ( y) owns only one main state: thus Pð yÞ ¼ 1: 3.3.2.3 Spin module. The state of the center of mass is assessed by the expected moment of junction kMJ l of its components, so that the spin-like system is described by P(x, y). kM l; standing for s · h/2, that is the classical spin module expression. This parameter would likely be summable over an association of particles into a more complex system, which is consistent with the additivity of spins.

In all cases, eddy-like components of the motion concern the relative Structure of behavior of the particle and of its inertons cloud, respectively. These relative ill-known spaces rotation movements could likely be of opposite sense, and at least in some cases Part 3 under current investigation, the whole {particle + inertons} system may either escape rotation, or get a resulting rotation axis and speed, depending on the rotation parameters of the most massive part of the system. Then, a rotation 1013 pulse with reversion of direction can be expected in some conditions. 3.3.3 Charges. 3.3.3.1 Opposite kinds of particle deformations. When a quantum of fractal deformations collapses into one single ball, two adjacent balls exhibit opposite forms: one in the sense of convexity and the other in the sense of concavity. Hence, there occurs a pair instead of a single object. The paired structures hold the same fractal dimension, and they will retain the same masses if they get the same volumes. This is realized if the member of the pair whose deformation is in the convex sense looses an equivalent volume in a nonfractal form, as schematically shown in Figure 1. The progression of such structures in the degenerate space will generate several kinds of inerton cloud equivalent, depending on convexity trends (J) and symmetry properties (C) of the corresponding structures. The properties generated by Q ¼ {J; C} will be called “charge effects”. 3.3.3.2 Electric and magnetic charges. The motion of a particled ball in the space lattice exhibits some similarity with a cutwave, which would be produced by a boat made with water. The shape of the waves depend on the shape of the moving object, as shown in Figure 2. Component {J} of Q produces two kinds of inerton clouds (Figure 2), which can be identified with electric fields, since they depart from the “neutral” inerton cloud by a symmetric kind of deformation. These shapes are

Figure 1. Schematic representation of the paired deformations produced by the collapse of a quantum of fractal deformations into a ball. One of the two complementary topologies could be called a positive charge and the other a negative one

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complementary and can be associated in an oriented field, providing the corresponding area of the lattice with vectorial properties. Component {C} appears when the fractal clusters are no longer symmetric in a massive particle. This case corresponds to well characterizable parametrizations. For instance, a sufficient condition is that the fractal quanta of masses have the following form, where at least some Ni are odd numbers: X N i ðri Þei ¼ 1Þ ð5:3Þ ðGÞ ¼ ð{N i ; ai ; ri }: i

In such cases, shown by Figure 3, the asymmetry will provide the cloud of inertons with an additional deformation represented by a torsion. Neutrality may not be so simple. In a strict sense, it can be reflected by the absence of deformation. However, it can also be represented by symmetry and homogeneity of the distribution of convex and concave components present

Figure 2. Changes imprinted in the shape of the inerton clouds by the shape of the particles deformation of symmetric type

Figure 3. An asymmetric component in the deformation of a particle induces a torsion of the inerton cloud

simultaneously on the same edges of particles, while the volume reduction Structure of associated to mass would remain fulfilled. The latter case would stand for a ill-known spaces pseudo-neutrality worth to be taken into consideration. Figure 4 illustrates Part 3 these features in a quasi-metaphoric sense. 3.3.4 Predicted expansion and the “quintessence”. 3.3.4.1 Introduction. Each time the distribution of degenerate deformations collapses into particled balls, 1015 there occurs a corresponding increase of volume in the surrounding balls, which compensates the reduction of volume in the particled cell. The motion of the particle can likely be provided by the reaction to the creation of this kind of “anti-inerton” cloud, which behaves in a opposite way than the resistance of the inerton cloud to the motion of the particle. These increases of volume are then progressively scattered by transfer of the corresponding deformations to an expanding cloud of “dilatation” quanta. This phenomenon operates a gain of space volume away from the particles. Therefore, it represents a kind of force acting in a way opposite to the gravitation. This suggests two main corollaries. 3.3.4.2 Quintessence. The existence of a “fifth cosmic element”, somewhat related with Einstein’s “cosmological constant” has been thoroughly discussed (see Bo¨rner, 2000; Krauss, 1999; Ostriker and Steinhardt, 2001, etc. for review). In our model, this factor appears strictly in connection with the creation of matter from the degenerate lattice, which may stand for the cosmic form of a void. Its quantitative expression is directly correlated with the density of fractal deformations, that is of energy, and its range will be shown below to be of the long type. Basically, the above theorem of distribution shows that it appears independently of any previous presence of matter nor radiations. Last,

Figure 4. Illustration of canonical particles differing only by their charge

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it seems not to be braked by gravitational forces, since it appears as a by-product of the same event which produces gravity. All these points make this compensatory dilatation phenomenon a candidate for the “quintessence”. 3.3.4.3 Space expansion. The transfer of elementary volumes released by the formation of the massive particles occurs within the frame of the degenerate space, since it is processed in a nonfractal state, just for the compensation of lost volumes, without the need for dimensional change. Consider (a) as an element of volume: first, it is finite. While the area of the iterated self-similar transform is theoretically infinite, its volume in 3D sections is not. Therefore, while a part of the volume of the transformed cell is reduced by a finite value, the volume of the surrounding cells is increased by a corresponding finite value. The homogeneity of the lattice is partly restored through progressive transfer of the additional volume to neighbor cells. Remark 3.4. A degenerate ball constitutes a “superparticle”. Proof. The oscillations of an elementary cell have been considered by Krasnoholovets (1997, 2000) as the “degenerate” state of space. It is a state without formation of a particle, though potentially able to provide particles upon proposition 3 and statement 3. One elementary ball thus constitutes the putative generator of particles, what has been called a “superparticle” in previous attempts for unification of theories. A Remark 3.5. In any one Poincare´ section, representing a timeless instantaneous state (an instans) of universe, the lattice of space is represented by a stacking of balls with nonidentical shape. Conjecture 3.1. Elementary balls exhibit increasing volumes from the center to the periphery of a 3D stacking. Justifications. Three arguments concur to the same proposition. (i) Oscillating deformations excess in one cell can be partly compensated by transfer to neighboring balls, like an equivalent to the inerton cloud surrounding a particled ball. However, in central parts, the volume available is limited by the density of the stacking, and this limit is likely decreasing while going to the outer coats of the lattice. In a simple estimation, we denote by (a) the radius of the canonical (smallest) volume which can be transfered from a ball to another. Assuming that each cell forwards a volume (a) to another situated closer to the periphery, in the stacking, then the radius of a ball in the nth coat is approximated by: r n ¼ r 1 þ ðn 2 1Þa

ð5:4Þ

(ii) While the above considerations are valid for a particleless lattice, if the lattice is filled with particled balls, then there results a kind of pressure due to the inerton clouds. Hence, relation (4.2) is affected by a corrective quantitative term to (a) and its distribution is determined by the distribution of particled balls in the considered space.

(iii) In contrast with the finiteness of volume to be compensatively Structure of distributed in the surrounding cells, the area of a particled cell is infinite, and ill-known spaces the needed area cannot be compensated by a finite number of the surrounding Part 3 cells. Thus, an influence of any particle is likely to be found up to the most remote parts of the lattice. The last two points will be further examined more in detail in the third part 1017 of this study, through involvement of the concept of quantum of fractality in relation with the mass of particled cells. Corollary 3.2. Since elementary balls can be found at various scales, due to the quantic ratios which characterize the lattice, as shown above, this means that elementary particles are not of one unique size. Corollary 3.3. Transfers of nonfractal elementary volumes between balls are operated without dimensional increase. Proof. At each given scale, the corresponding increments (a) are represented by similar topological features. In effect: following relation (3), we have for n ¼ 2 : r2 ¼ r 1 þ a; and for n ¼ 1 : r1 ¼ r 0 þ ðn 2 1Þa ¼ r0 : Since then r0 ¼ r2 2 a; r0 stands for a founding ball. Let r0 ¼ ðØÞ; a empty set. Then, r2 ¼ r0 þ a can be represented by (Ø, {Ø}), where {Ø} is the frontier of ball r2. The element {Ø} is what is exchangeable, and since it is a frontier, it has a dimension lower than the dimension of the interior, that is: dimðaÞ , dimðrÞ: Thus, exchanges do not modify the dimensionality of involved balls. A However, mass transfers from a particled cell to its surrounding degenerate balls involves a distribution of quanta of fractality through the concept of fractal decomposition described earlier. Remark 3.6. Consequently, it may be considered that these exchanges apply to the frontiers of the balls, which will result in changes in the density of their internal structures. It is noteworthy that the density has been used as a probe for the identification of the packing of balls, though in this case only solid balls are considered (Hales, 2000). Otherwise, the adjunction of (a) to (r) may result in the reunion of two spaces having nonequal dimensions, which can result in a structure of the “beaver space” type as described in Part 1. Corollary 3.4. A measure on such a lattice space by using a scanning function as described in Part 1 will not scan the same components in elementary balls situated at various distances from its origin. Since there likely occurs an increase of the composition of balls from this origin, then the gauge will decrease with increasing distances: in effect, a larger set of scanned structures will appear at farther distances. Then, remote distances will be overestimated by a measure using a local gauge. This might account for the phenomenon known as the Doppler effect, in turn usually involving the Hubble constant. It should be noted here that the interpretation of the redshift has been a matter of diverging treatments (Hannon, 1998).

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3.3.5 Towards a formalism to big-bangs. In Part 1 of this study, it has been proved: (i) that the lattice existing from empty hyperset units provide a manifold of quantic scales represented by a set of defined integer ratios (Bounias and Krasnoholovets, 2003a), and (ii) that there exists empty set units of various size, with integer vs rational similarity ratios. One universe U j ¼ {{S i }i ; ’ }j ; represented by one particular sequence of Poincare´ sections selected through a particular combination rule is nothing but a manifold of organized empty set units, and since the lattice in which it is embedded is strictly fractal, the reunion of these empty set units is a higher scale empty set. Thus, U j ¼ Øj : Now, consider the part of the embedding lattice in which Øj owns just the size of a free ball. This “over universe” denoted Øþj will behave like described through Proposition 1. The distribution collapse of degenerate balls of Øþj will result in the formation of a particle whose subparts contain potentially as many quanta of fractality as Uj contains massive objects. Thus, what represents a creation of a particle inside Uj is a primordial condensation of a ball into Uj inside U þj : This suggests that such kinds of “big-bangs” may have occurred, occur and will occur in at least denumerably many balls of the embedding lattice, without need for an “outside” provision of energy. However, these “big-bangs” fulfill some conditions. In effect, it has been specified in the first part of this study that universe is definitely constructed in a specified space ðW v Þ ¼ {X; ’ } with set X [ {X 3 > X 4 Þ and combination rule (’ ) determining the choice of S iþ1 from Si. Hence, relations between different universes and past-to-present successive universes can exist only through the same law (’). 4. Discussion and conclusions The law ( ’ ) proposed as the operator of the selection of successive Poincare´ sections constituting a space-time presents the interest, besides providing continuity of this space-time, of keeping inert or low-moving structures (like mountains, landscapes, etc.) stable. Furthermore, it brings as a corollary that events will basically follow the shorter path between two steps, which is consistent with both the least action principle and the geodesic trajectory principle. This suggests that the kind of universe that we have described is consistent with our observable space-time, even if our description of submicroscopic events, through the formalism of set theory extended to nonwell-founded sets (a consistent extension) may be considered in some sort as a metaphoric description. The components of the {particle + inertons} system are likely inhomogeneous as topological balls do not need to be strictly spherical

(the latter case is just a particular one). Therefore, their coexistence in a single Structure of system representing the dual {wave/particle} system deserves special ill-known spaces attention, since spin-related properties could reflect the eddy-like motion that Part 3 inhomogeneity should impulse to the components and finally, in a resulting manner, to the system. Such properties have been described by Lin and OuYang (1996, 1998) and Wu and Lin (2002), while Lin (1988) explored the 1019 compatibility of world exploration with the theoretical study of systems. These goals are well converging with our objective of mathematical exploration of an unknown world, as developed in Part 1 of this study. The theoretical reasoning presented in this study, following the basis developed in Parts 1 and 2, sheds some light on the question of the hypothetic “origins” of universe. In fact, there is no need for beginning for end. Even the expansion might not induce the consequences expressed through other approaches in terms of forever expansion and progressive immobilization, nor cyclic contraction and collapse. Our approach, basically founded on a formal justification of existence of “something”, and then on corpuscular description and properties, turns to introduce some insights about cosmic scales and cosmic-size properties. Interestingly, Andreı¨ Linde pioneeringly suggested that a “Grand-Universe” could be composed of bubbles of universes that could form and disappear in various parts in an independent fashion. Though, it was not primarily our aim to treat these questions, it turns out that the development of our model from defined initial points comes to support Linde’s hypothesis. Furthermore, the former hypothesis raised long ago by Feynmann about particle trajectories which would be infinite and not derivable is consistent with our proposition that particles are distinguished from the degenerate space by a shift of dimensional properties, that is with a fractal organization. The next part of this study will aim to examine more in detail the peculiarities of the various kinds of corpuscles predicted by our approach.

References Bonaly, A. and Bounias, M. (1995), “The trace of time in Poincare´ sections of topological spaces”, Physics Essays, Vol. 8 No. 2, pp. 236-44. Bo¨rner, G. (2000), “The infinitely large”, Pour La Science (Scientific American, French edition), Vol. 278, pp. 120-7. Bounias, M. and Bonaly, A. (1997), “Some theorems on the empty set as necessary and sufficient for the primary topological axioms of physical existence”, Physics Essays, Vol. 10 No. 4, pp. 633-43. Bounias, M. and Krasnoholovets, V. (2003a), Scanning the structure of ill-known spaces: Founding principles about mathematical constitution of space (this issue). Bounias, M. and Krasnoholovets, V. (2003b), Scanning the structure of ill-known spaces: Part 2. Principles of construction of physical space (this issue). Frizsh, H. (2000), “The infinitely small in physics”, Pour La Science (Scientific American, French edition), Vol. 278, pp. 112-19.

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Krasnoholovets (2001), “On the way to submicroscopic description of nature”, Indian Journal of Theoretical Physics (in press). Krauss, L. (1999), “Antigravity”, Pour La Science (Scientific American, French edition), pp. 42-9. Lin, Y. and OuYang, S.C. (1996), “Exploration of the mystery of nonlinearity”, Research of Natural Dialectics, Vols 12-13, pp. 34-7. Lin, Y. (1988), “Can the world be studied in the viewpoint of systems ?”, Math. Comput. Modeling, Vol. 11, pp. 738-42. Lin, Y. and OuYang, S.C. (1998), “Invisible Tao and realistic nonlinearity propositions”, Kybernetes: The International Journal for Systems and Cybernetics, Vol. 27, pp. 809-22. Ostriker, J. and Steinhardt, P. (2001), “The fifth cosmic element”, Pour La Science (Scientific American, French edition), Vol. 281, pp. 44-53. Ru¨egg, A. (1985), Probabilities and Statistics, Presses Polytechniques Romandes, Mausanne, Switzerland, pp. 52-87. Wu, Y. and Lin, Y. (2002), “Beyond nonstructural quantitative analysis”, Blown Ups, Spinning Currents and Modern Science, World Scientific, New Jersey, London, p. 324. Further reading Caldwell, R., Dave, R. and Steinhardt, P. (1998), “Comological imprint of an energy component with general equation of state”, Phys. Rev. Lett., Vol. 80 No. 8, pp. 1582-5. Krauss, L. (1998), “The end of age problem, and the case for a cosmological constant revisited”, Astrophysical Journal, Vol. 501 No. 2, pp. 461-6. Schwartz, L. (1997), Un mathe´maticien aux prises avec le sie`cle, Odile Jacob, Paris, p. 250. Wang, L., Caldwell, R., Ostriker, J. and Steinhardt, P. (2000), “Cosmic concordance and quintessence”, Astrophysical Journal, Vol. 530 No. 1, pp. 17-35.

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The theory of cold quantum: a counter theory of gravitation Cao Junfeng 170-1-5-1, Guihua Street, ShujiaTun District, Shenyang City, People’s Republic of China

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Keywords Cybernetics, Gravity, Space Abstract Through many years of study, we have found that cold quantum is the most important force in nature. Under the pressure of coldness on hotness, various materials are formed. Under the pressure of cold quantum, these materials are provided with gravity, and celestial bodies start to move. The pressure of cold quantum exists in space and materials. It is the pressure of cold quantum that huge changes between the four seasons on the earth begin to appear. The whirlpool, produced from the cold quantum pressure, pushes all the celestial bodies making them turn and change. The coldness converts frozen water into ice, which could not be achieved by any other force. The extreme and powerful strength of cold quantum has been well-known. Therefore, we claim that the cold quantum pressure is the greatest force which ever existed in the universe.

1. Existence of cold and hot quantum pressures Space consists of cold quantum pressures. There is no vacancy in the space. All the celestial bodies, their light and heat occupy only about 10 per cent of the total space. The remaining nearly 90 per cent of the space is taken by the cold quantum. Cold quantum exists in the space and materials. Hot quantum surrounded by cold quantum constitutes the material world. The remaining 90 per cent cold quantum fills the rest of the universe. There does not exist any hot quantum left behind after having formed various materials. Therefore, it can be said that the universe is the one consisting of cold quantum where cold quantum flows to areas not yet occupied by them. Each area, not occupied by cold quantum, is occupied by hot quantum. That is, cold quantum flows to hot quantum areas. At the same time, the spatial hot quantum also flows to areas not yet occupied by hot quantum. So, hot quantum also flows into cold quantum areas. As the space is mainly occupied by cold quantum, through mutual interaction of flows, cold quantum cuts hot quantum into small pieces and is surrounded by these individual pieces. As soon as hot quantum is cut by cold quantum into as small pieces as dots, the so-called hot photons are formed. Due to the fact that there is way more cold quantum than hot quantum, there is no way for hot quantum to cut cold quantum into small dots. This explains why cold quantum particles do not exist in the universe. Since the ancient times, the existence of cold quantum has been known. However, due to whatever reasons, no attempt has been made in order to understand the cold quantum. The impact of coldness has been huge on the earth. Its impact in the universe is even greater. Since the significance of the coldness in the universe has not been well recognized and understood, it explains why we have not

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1021-1034 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483153

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achieved a more complete understanding of the materialistic universe and why no laws of nature have been found. Cold and hot quantum fight against each other constantly, no matter whether one quantum exists in space or materials. Through constant fight against each other between huge amounts of cold quantum and a small amount of hot quantum, a materialistic world appears. Through their continued fight in materials, changes and movement in materials appear. When cold quantum pressures on human beings, we feel the sensation of coldness. When hot quantum pressures on human beings, we feel the sensation of hotness. Both coldness and hotness stand for a quantum instead of a particle. Only when the other cuts one into small dots, they are called particles. Each quantum possesses elasticity and so, each quantum can apply pressure. When in motion, the phenomenon of waves appears as soon as a quantum meets a resistance. For example, Shenyang, China, is located at the middle of the northern hemisphere. Northwestern winds bring lower temperature to the region, because the northwestern winds are very cold. In this example, it is the cold winds that cold quantum is expressed in atmospheric motions. When winds blow on the surface of a pond, the surface is under pressure to move forward. However, the water in the front constitutes a resistance to the forward motion. In this fashion, the force in the forward motion creates the phenomenon of waves. When the amount of hot quantum is greater than the cold quantum in the atmosphere, hot winds will be created. Conversely, cold winds would be created. The movement of cold and hot winds is an expression of moving cold quantum in the atmosphere. Cold quantum can pressure frozen water into ice, while hot quantum melts ice back into water. Without cold and hot quantum, nothing in the universe can transform water into ice and ice into water. Similarly, cold quantum can pressure materials into celestial bodies. This is what is widely believed in China that the rules are the same, no matter whether one is in the heavens or on the earth. Our analysis also shows that the greatest forces in nature are the cold and hot quantum pressures. The formation of the nature and the movement of celestial bodies are all caused by the pressure of cold quantum. All materialistic changes are mainly produced by the pressure of hot quantum. The pressures of cold and hot quantum on materials and human beings are expressed as pressures of particles. Cold quantum can squeeze atoms smaller, so that the phenomenon of material contraction under cold temperature appears. When the atomic field is influenced by external hot quantum, the atomic field expands. This explains why each material expands with heat. Each pressure of cold quantum on microscopic particles leads to the phenomenon of freezing of macroscopic materials. This is the reason why we have changed the term of cold freezing to cold quantum. With such a terminology change, one can better understand the great effects of cold quantum in the universe and materials and how cold quantum exists in materials. For example, there would be no

coldness in a refrigerator without the substances of cold quantum. It is the compressor of the refrigerator that chases out some of the hot quantum in the materials placed inside the refrigerator so that more of the cold quantum is left within the materials, which appear to be colder than before. When the cold and hot quantum reach a certain balance in water, an amount of liquid water appears. When some of the cold quantum is removed, the liquid water becomes warmer and starts to evaporate. When enough hot quantum is removed, the liquid water will become colder, start to freeze, and take the form of ice. This example explains that both cold and hot quantum exist in materials in varying quantities. The morphology of a material changes with varying quantities of the cold and hot quantum contained in the material. Pressures can also be called pushing forces. Human power, mechanical forces, water power, wind power, and the turning power of a celestial body are all caused by pressures and pushing forces. Cold quantum constantly pushes the celestial bodies around continuously. It is the cold quantum, which permeates the entire universe, making the celestial bodies move instead of the so-called inertia in the vacuum, as there is no vacuum in space. The so-called inertia is the pushing force remaining in the moving object after the acting force ceases to exist. When an object moves forward, it always faces resistance. When the remaining pushing force equals the resistance, the forward-moving object will stop its motion. This is the reason why when the remaining pushing force in an object equals the resistance, the inertia disappears. Under the effect of resistance, an object stops its original motion. Objects with greater density retain more pushing force after the force ceases to exist than those with lesser density. This explains why the former objects move further than the latter under the effects of inertia. There are two kinds of magnetic forces, one pushing force and the other pulling force. This is due to the special characteristics of the atomic structures of magnetic objects. A general atomic nucleus radiates heat in all directions, forming a spherical atomic field with electrons circulating around the nucleus within the field. In terms of magnetic materials, their atomic nuclei radiate heat in one direction with a high concentration of hot quantum. Such a high concentration of hot quantum pushes the electrons away into the space forming a pushing force. In the direction opposite to the pushing force of the atomic nuclei, no hot quantum is released so that no pushing force exists. So, without electrons, a vacuum field is formed in this area. At the same moment, the cold quantum, existing outside the magnetic materials, forces free electrons in space into this area of the magnetic atomic nuclei. Since the electrons orbiting along the outer atomic field of a piece of iron are more likely to be forced into the vacuum area of magnetic atomic nuclei, guided by the movement of the electrons, the piece of iron moves to the magnetic material and is attracted to the magnetic material. In fact, such a phenomenon is the result of the pushing forces of the cold quantum existing outside the iron and the magnetic material,

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even though we cannot see these forces. For example, from a distance an observer sees a person being blown against a wall by a strong wind. If the observer could not feel the wind, it would be very likely for the observer to believe that the wall has a pulling force or an attraction. Here, for the same reason, the attraction or pulling force of a magnetic material on iron is in fact the result of cold quantum pressures. Each object cannot give off two opposing forces at the same time. If it gives off a pushing force, the other, working on the object, must be a force imposed on the object from the exterior world. Obviously, this object possesses two opposite forces. What is the most significant and the greatest force in the universe? It is the pressure of cold quantum. As it is the pressuring force of cold quantum which moves celestial bodies in the universe, it can be said that there is no vacuum in the universe. 2. Material particles formed by cold quantum encircling hot quantum Where are electrons from? And, how are they produced? There have been very few people who have asked these questions. When electrons meet with obstacle(s) in their movement, we call it (electric) resistance. Electricity can be transformed into light and run away. Positive and negative electrons can also be transformed into light and run away through collisions. Here, light is a characteristic representation of heat. Now, a natural question is: where is the light produced out of electrons? When the light heat is surrounded by cold quantum, a dot, the electron is formed. No light is seen unless the cover of cold quantum is broken, because the light has been surrounded and well covered by the coldness. Each cold quantum is a quantity instead of a particle. Similarly, each hot quantum is a quantity instead of a particle with light being its characteristic representation. So, light is also a quantity instead of a particle or particles. When light is cut and surrounded by cold quantum, dots, named photons, are formed. After cold quantum pressure cuts light into small dots, cold quantum further employs pressures on photons. A whirlpool consisting of cold quantum is formed, surrounding and spinning around each photon. This tiny whirlpool has the photon encircled in the middle so that to our human eyes, the photon becomes invisible. This moving tiny whirlpool is the so-called electrons. This is how the electrons are formed. When positive and negative electrons collide, some photons are released. This fact shows that as soon as the covering of a photon, consisting of cold quantum, is broken, the photon is revealed and the broken whirlpool melts back into the cold quantum. As our universe is a space of cold quantum pressures, electrons are also called negative electrons. It is because the outer fit of each electron consists of cold quantum without any heat. So, there is no heat ever coming out of an electron. They are called negative electrons, because they are cold. There rarely

exists any space for positive electrons. If the amount of light heat contained in an electron is relatively high, the electron is called positive. It is possible for heat to be released out of a positive electron. The electron is called a positive electron, because heat exists outside the electron. Or, an electron may appear to be positive when it gains enough heat from its movement. Generally, it is always the man who makes an electron move to gain heat on its outside covering so that the electron carries positive charge. Here, coldness is seen as negative and heat positive. It is the electronic charge that represents for the so-called electronic power, which is an expression of the power of cold quantum. Electrons are produced in this way and new electrons are produced constantly so that the universe is filled with electrons. How are the so-called protons, the smallest particles of materials in the universe, produced? Let us note that protons cannot be produced by themselves. Instead, protons and neutrons must be produced at the same time. Protons, which surround neutrons, and neutrons, appear in the universe together. This is what is called hydrogen nuclei, the smallest particle of the universe. Now, the natural question is where are the hydrogen nuclei produced in the universe? It is through electronic fusion reactions that produce hydrogen nuclei. The tiny whirlpool surrounding each electron is very weak. That is, each electron is easily broken due to its low density. When the number of electrons in the universe increases, electronic clouds are formed. The external pressure of cold quantum produces high temperature and high pressure within each electronic cloud. Under such high temperature and high pressure, electrons are forced to break. Such a process is called an electronic fusion. Each electronic fusion creates a strong shock pressure. It compresses all other electrons together. Each cold quantum whirlpool surrounding the compressed electrons forms a proton. The compressed photons surrounding the middle of the proton form a neutron. This is how a hydrogen nucleus is formed in nature. A proton is not a particle. Instead, it only represents the amount of protons in a large element’s nucleus after the first generation of protons of hydrogen nuclei were produced. Each amount of protons is a multiple of the proton quantum of the hydrogen nucleus. So, such a quantity also represents the amount of pressure of cold quantum, the weight of an object, and the mass of the object. Under the effect of cold quantum, materialistic particles are formed in space when electrons are acted upon by pressure. This explains why electrons and materialistic particles always appear together. Even though cold and hot quantum seem to be the two opposite sides, they have worked together to form electrons. Even though atomic nuclei are relatively steady, electrons are fragile. Reactions in electrons can occur anytime when some forces are applied on them. For example, each electronic reaction, occurring during the thunder, lightning of a thunderstorm or a northern light, has the possibility to form new hydrogen nuclei, which are currently called protons. Cold quantum, when combined with hot quantum, produces electrons. Through electronic reactions,

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materialistic particles are formed. As over 90 per cent of the universe is filled with cold quantum and hot quantum only permeates in a small fraction of the remaining space, the cold quantum cuts the hot quantum into small pieces and creates the material world. There is still a great amount of cold quantum left behind from forming the material world, while no hot quantum is left behind, all of which have been used to form the material world. This explains why the current natural world is cold with cold quantum constantly pressuring on the material world so that all materials possess both strength and hardness. This is the so-called nuclear power. Through dashes from an accelerator, a materialistic particle can be broken into smaller pieces, so that the components of a materialistic particle can be seen clearly. After many repeated experiments, it is discovered that each materialistic particle is made up of over ten different types of smaller particles. Consequently, each one of these smaller particles is named formally. For example, some are called meson, quark, etc. We do not agree with such a work. In the infinitely great universe, how can these secondary particles be combined together to form a materialistic particle in such a wonderful way that no more and no fewer secondary particles are applied? In the universe, as of today, we have found that no single secondary particle was left behind from the combination of materialistic particles. This understanding about how materialistic particles are made up is incredibly impossible for us to fathom. We do not think this is how materials are formed. For example, in order to see how a watermelon is formed, we throw a watermelon to the ground. When it is broken, the watermelon becomes small pieces of different sizes. Some pieces are red, some green, and some seem to be the seeds. Can we conclude from this experiment, which can be repeated as many times as one wishes, that each watermelon is made up of these broken pieces? Each broken watermelon deteriorates quickly. Similarly, all the secondary particles of a materialistic particle, obtained through dashes of an accelerator, disappear quickly. Stable particles are truly particles. Unstable secondary particles are not truly particles. The outer shell of each particle consists of protons surrounding neutrons in the center. These secondary particles can only be parts of the protons and neutron being broken. Therefore, the belief that these secondary particles make up the particles of materials is incorrect. It is only the cold and hot quantum that make up all the materialistic particles. 3. Proportional relationship between neutrons and protons and fission and decay In a nuclear fusion, several smaller atomic nuclei are fused into larger atomic nuclei. That is, through fusion reactions, atomic nuclei become larger. Initially, each of the smaller atomic nuclei consists of the same number of protons and neutrons. As newer and larger nuclei are formed, the number of protons contained in one nucleus decreases, while the number of neutrons increases.

As the size of nuclei continues to increase, eventually the protons can no longer wrap the neutrons completely. So, the heat pressure given off from the neutrons bursts the control of the protons. That is, a fission of the large nuclei occurs, giving off neutrons, called radioactive elements. Now, the natural question is why the number of neutrons is proportionally increasing compared to that of protons when nuclei grow in their sizes? Let us use an example to answer this question. Let us fill up two 1 m3 boxes with potatoes. If these potatoes are moved to a 2 m3 box, then the larger box needs less surface material than the two smaller boxes. If the surface material of these boxes is seen as the protons and the potatoes the neutrons in our previous discussion, then larger boxes need less material for their surfaces than smaller (but more in numbers) boxes. This is the reason why the larger a nucleus, fewer protons are needed to enclose the neutrons. Each atom is a field with the nucleus being surrounded by several electrons. Therefore, the greater the nucleus, the larger is the field with more electrons circulating. Now, what is a field? Where is a field from? Each atomic field is the area filled with the heat released from the nucleus, which is confined in the area around the nucleus by the external cold quantum. As the pressure of the external cold quantum is greater than the heat pressure released by the nucleus, a whirlpool, spinning around the nucleus, is formed. At the same time, the cold quantum also pushes some of the free electrons floating in the space into the whirlpool. So, some electrons start to circulate around the nucleus. And, this is how an atomic field is formed and each such field is simply called an atom. When a nucleus increases its size, more heat is released so that more electrons can be allowed to circulate in the whirlpool field. When the number of neutrons in the nucleus increases, more heat is released. As soon as the number of neutrons in the nucleus reaches a certain threshold, some neutrons burst out of the nucleus directly. Such a process is called atomic fission. When the number of neutrons inside a nucleus increases to a certain limit, the protons, forming the outer shell of the nucleus, can no longer hold the heat pressure from the neutrons so that some of the neutrons rush out of the nucleus with their own heat. This process is called the nuclear fission. As soon as some of its neutrons are released into the space, the nucleus loses some of its energy and mass and starts to shrink. So, a large nucleus through fission becomes smaller. Such a process is called decay. The more the heat released from a nucleus, the greater is its field and the more electrons are allowed to circulate in the field. The more the heat released and the longer the nucleus releases heat, the fewer neutrons are left in the nucleus. Such a process of large nuclei shrinking to small nuclei is called a nuclear decay. When the number of neutrons reaches its maximum and the protons can no longer restrain the neutrons from bursting outward, it is called fission. If the number of neutrons is not high, the protons are able to inclose the neutrons in the center so that heat is released slowly. As soon as the heat released reaches the quantity of one neutron, the nucleus starts to decay and becomes smaller.

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The faster and more an atom releases heat, the faster is the decay. Conversely, the less the heat released the longer it takes for the atom to decay. As the number of neutrons and protons are roughly the same in small atoms, cold quantum proton shell is relatively steadier. So, the heat pressure from within is relatively small and the process of decay is very slow. If the smallest nucleus did not have any heat released, there would be no whirlpool field formed so that there would be no electrons circulating around the nucleus. In this case, there would be no decay. Each cold quantum proton shell has a certain thickness pressure. So, if the thickness pressure of a cold proton shell is too great, it would be impossible for such a shell to enclose too many neutrons. On the other hand, the thickness pressure of a proton shell can only be finite, because, if the thickness pressure of a cold proton shell is greater than that of the external cold quantum, then the forces with more protons will rush out into the cold quantum. This also explains why the size of nuclei has an upper limit, because when too many neutrons are enclosed by a cold proton shell, the limited strength of the shell will be broken by the heat released from the neutrons. Thus, as soon as an atom starts fission, its life span reaches an end. At the time when the number of neutrons in the nucleus equals that of protons of the nuclear shell, the process of decay is very slow. This slowness of decay represents the youth of the nucleus. As soon as the number of neutrons is greater than the protons in a nucleus, the fission process speeds up so that the nucleus reached its middle age. When there are too many neutrons in the nucleus, the cold proton shell can no longer prevent neutrons from escaping from the nucleus, the nucleus starts to experience an accelerated decay. This is the last period of the nucleus’s life. Through fission and decay, eventually, the nucleus ceases to exist. 4. Celestial bodies pushed by cold quantum Cold quantum produces pressures on hot quantum. All celestial bodies give off heat of various magnitudes. As the sun gives off heat, cold quantum imposes pressures on the sun. The entire cold quantum pressing towards the sun flows out of the star from the two solar poles. As the area of cold quantums pressing toward the sun is huge, while the areas of the cold quantum flowing away from the sun are much smaller, more cold quantum is pressing toward the sun with less cold quantum flowing away from the solar poles. Some resistance is created around the sun along with some pressures of the accumulated cold quantum. The cold quantum pressure faces with the resistance in the front and space on the back. The push caused by the cold quantum pressure on the accumulated cold quantum produces a high speed, horizontal movement of the cold quantum around the sun in order to give off some of the accumulated forces. This horizontal cold quantum pressure circulating around the sun is the cold quantum whirlpool circulating the sun. Other than the rotation of the sun, this cold quantum whirlpool also makes the entire solar system spin. Each

whirlpool is created by accumulated and gathered forces and is an expression of the spreading of the accumulated forces. The whirlpool, which makes the sun rotate, also pushes the nine planets in the solar system to travel around the sun. The vortical force is a product of the difference in speed of the outer and inner areas of the whirlpool, where the outer speed is slower than the inner speed. This fact explains why the planets, which are further away from the sun travel at a slower speed than those which are closer to the sun. As each galaxy also gives off heat, cold quantum also applies pressures on the galaxy. So, in the surrounding area outside the galaxy and areas inside the galaxy, a whirlpool, making the galaxy rotate, is created so that the galaxy takes the shape of a whirlpool. All galaxies, observable as of now, take the form of a whirlpool. What is given earlier is an explanation on how each galaxy takes its whirlpool like shape. As each planet gives off heat and reflects some of the solar lights, a planet whirlpool of cold quantum is created which makes the planet rotate. For example, the rotation of our earth is created in this way. The reason why outer planets rotate more slowly than the inner ones is that the outer planets experience more cold quantum pressures than the inner ones, where the solar heat helps to block some of the cold quantum. Then, why is the rotation speed of Jupiter, located at the midway between the outer and inner planets, especially high? The orbit, along which Jupiter travels, is located in the area where the solar heat and the outer space coldness meet and form a strong convection of cold and hot quantum. Under the influence of this convection, the rotation of Jupiter speeds up. Among the nine planets, Mercury is located closest to the sun compared to other planets. The surrounding space of Mercury is very hot, as hot as the nearby solar whirlpool heat. Therefore, no whirlpool of cold quantum of Mercury’s own can be formed. Without such a whirlpool, Mercury experiences no force to rotate. This explains why Mercury does not rotate. As the amount of heat around the moon is the same as that of the earthly whirlpool, it explains why the moon does not have its own cold quantum whirlpool and why the moon does not rotate. The greater a planet’s self changes, the larger is the planet’s orbit. The smaller a planet’s self changes, the smaller is its orbit. For example, when experiencing heat, the volume of a planet increases with its decreasing density. Under the effect of the solar wind, such a planet will be pushed further away from the sun. When a planet is cooled by cold quantum and the volume of the planet shrinks, the density of the planet increases so that under the pressure of cold quantum, such a planet will fall to the sun and be located near the sun. This example explains why each planet travels around the sun along an elliptic orbit. When the planet is near the sun, it is under the influence of the solar winds and the solar whirlpool. When the planet travels away from the sun, it flies away from the solar whirlpool plane. Then, a greater dip-angle is formed. This describes the relationship between the planetary dip-angles and orbital forces of its elliptic orbit.

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Let us now look at the relationship between the dip- and cross-angles. For example, Uranus experiences very small self changes so that the ellipticity of its orbit is very small, and its dip-angle is also very small. All the planets with small dip-angles have very large cross-angles. For example, Uranus has a very large cross-angle. Now, the natural question is why such a relationship between the dip- and cross-angles exist. The explanation lies in the relationship between the large whirlpool of the sun and the small whirlpool of the planet. When these whirlpools spin in the same direction, these whirlpools are located in the same plane so that the dip-angle of the planet is very small. Under the influence of the force of the large whirlpool, the plane of the small whirlpool becomes tilted which forms an angle with the plane of the large whirlpool. This angle is called the cross-angle and is very large. If a small planetary whirlpool is far away from the plane of the large solar whirlpool, the tilt angle of the small whirlpool, caused by the large whirlpool, is small, while far away from the plane of the large whirlpool implies large dip-angle. Here, the angle tilting away from the solar whirlpool’s zodiac plane is called a dip-angle and the planetary angle, formed by the solar whirlpool from the solar plane, is called a cross-angle. For Uranus, its dip-angle is less than 18, while its cross-angle is 828. For the earth, its dip-angle is nearly 28, while its cross-angle is 238. These are two examples showing the inverse proportional relationship between the dip- and cross-angles. With the existence of cross-angles, there are four different seasons on the earth. Without a cross-angle, there would not be any seasonal differences on the earth. Let us look at this experiment about the relationship between cross-angles and dip-angles. Let us place a small magnetic ring over a large magnetic ring along a line with the positive pole facing the positive pole. Now, the plane of the small magnetic ring is tilted under the effect of the repelling force of the large magnetic ring. When the small ring is moved further and further away from the line, the slantness of the small ring becomes smaller and smaller. Here, the large magnetic ring is like the solar whirlpool, the small magnetic ring, a planetary whirlpool, and the straight line, the plane of whirlpools. When the small magnetic ring starts to tilt, it is like the cross-angle of the planet moving further away from the straight line so that the dip-angle increases. Both the magnetic field and whirlpool follow the same rule with the same model. In fact, a whirlpool is a large magnetic field and a small magnetic ring is a small whirlpool. Now, let us look at the relationship between the density of a planet and its distance from the sun. All planets are constantly under the hot pushing influence of solar winds. Planets with smaller densities are pushed further away from the sun by solar winds, while planets with higher densities are more difficult for solar winds to influence. So, planets with high densities stay closer to the sun. The surface temperature of 6,0008C of the sun represents a hot pushing force. For the sake of convenience for computation, let us write the solar surface temperature as 6 and each 100 million kilometers of distance as 1.

Then, the ratio between the solar surface temperature and the distance of the earth from the sun is 6 4 1:5 ¼ 4 g; Mercury 6 4 0:6 ¼ 10 g; Uranus 6 4 60 ¼ 0:1 g: Thus, Mercury might be made up of heavy metals, while Pluto of hydrogen gas. This is the reason why more distant planets have less densities than nearby planets. All comets have a long tail which is also made up of materials. They fly away from the sun no matter how far they are from the sun. This fact explains the fact that solar winds are a powerful pushing force, which pushes planets with lighter densities further away from the sun. Weights and masses are all composed of hot quantum encircled by cold quantum. All attributes of materials are expressed by the outer proton shell of cold quantum. The so-called mass of an object stands for the amount of cold quantum carried by the object. The so-called gravity of the object represents the cold quantum pressure of the object. And, the so-called weight of the object is a measure of the amount of cold quantum carried by the object. Here, cold winds are representations of moving cold quantum in the atmosphere. Now, let us turn our attention to explain why the ellipticity of planetary orbits can be different. A smaller ellipticity appears when the planet is closer to the sun. For example, when the earth travels close to the sun, the rotation speed of the solar whirlpool is faster. So, the movement of the earth is accelerated, causing the earth to travel away from the sun. When the earth is further away from the sun, the slower rotation speed of the solar whirlpool does not push the earth much. Under the influence of outer space cold quantum pressures, the earth falls back to the sun. As soon as the earth gets closer to the sun, it starts to experience increased solar whirlpool speed such that its travelling speed is accelerated and consequently, it travels away from the sun again. This is how each time when the earth travels around the sun once, it travels along an elliptic orbit. For the planets with large elliptic orbits, such as Jupiter and Pluto, their large elliptic orbits are formed by their bodily changes with their locations in terms of the sun. For instance, when either Jupiter or Pluto travels near the sun, their volumes increase and their densities decrease dramatically under the influence of the solar heat so that they are blown far away from the sun. When they travel far away from the sun, the influence of the solar heat drops such that under the cold pressure, their volumes decrease and densities increase dramatically. So, pushed by the cold pressure, they fall back to the sun. This is the reason why when Jupiter and Pluto travel around the sun once, their elliptic orbits seem to be larger than that of the earth. Each comet is composed of a pile of rocks, hence there are lots of holes in the comet. These holes make the density of the comet small, while the volume still seems big. Under solar winds, the comet is blown away from the sun. When travelling in the outer space, the holes of the comet are filled with water, ice and other materials. With the holes filled up, the density of the comet increases to such a degree that it becomes heavier than some of the distant planets. So, under the influence of cold quantum pressures, the comet falls back to

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the sun linearly. Now, the natural question is why the comet cannot fall straight on to the sun? It is because as soon as the comet reaches the nearby region of the sun, powerful solar winds keep the comet from falling onto the sun. On its trip to the sun, the ice and water filling the holes of the comet are blown away and off the comet by the heat of solar winds, forming the tail of the comet. Through the phenomenon of comets’ tails, one understands the strength of the repelling force of solar winds. The materials, constituting the tails of comets, are similar to those making up distant planets. That is why we say that these distant planets are repelled away from the sun by solar winds. As soon as a comet loses the materials filling up its holes, all the holes reappear in the comet so that the density of the comet becomes lighter and it is once again pushed away into the distant space from the sun. The orbit of the comet travelling around the sun is large and is called a curvilinear trajectory. Let us see how ocean tides are formed. When the moon is pushed by the earthly whirlpool which forces the earth to rotate, the moon travels around the earth so that the moon blocks some of the whirlpool pressure pressing towards the earth. Because of this blocking effect, the ocean water under the moon experiences less pressure than the water of other areas. Consequently, the area under the moon experiences a tide with rising sea level. This is how ocean tides are formed. The so-called lunar calendar is also a moon calendar. Why is there a high tide on the first and the 15th of the lunar month? It is because on these days, the solar winds increase its pressure on the earth plus the pressure of the outer space cold quantum. So, the earth is squeezed in the middle between the solar wind force and the pressure of the cold quantum about 150 million kilometers from the sun. The repelling force of solar winds and outer space cold quantum pressure line up linearly with the earth located in the middle. When the moon travels to the same line on the side of the sun, the moon blocks some of the repelling force from the solar winds. Because of the blocking effect of the moon, the ocean under the moon experiences a huge tide. This is the first of the lunar month. When the moon travels along the opposite side of the earth, it blocks some of the cold quantum pressure from the outer space for the earth. So, the ocean under the moon once again experiences another major tide. This is the 15th of the lunar month. This is how the ocean tides on the first and the 15th of the lunar month appear. The earthly whirlpool pushes the earthly atmosphere, which in turn applies pressure on the earth. The speed of sound in the atmosphere is 340 m/s. The rotation speed of the earth in the region of Cancer and Capricorn is also 340 m/s. At the equator, it is faster than 340 m/s. At the two poles, it is slower than 340 m/s. If the rotation speeds at the equator and the poles are averaged, it is also roughly 340 m/s. The speed of the earthly rotation is the same as that of the speed of sound in the atmosphere. This is in fact the pressure speed of the atmosphere on the earth.

Nuclear reactions on the star sun release heat of both high temperature and high pressure so that small and light nuclei are pressed together to form large and heavier nuclei. This is the so-called nuclear fusion. When the number of heavy nuclei increases on the sun, the solar nuclear fusion slows down. As partial decay period of heavy nuclei is brief and each heavy nucleus has more heat to release, it is more difficult for proton shells, consisting of cold quantum, to inclose neutron heat inside. So, the fusion process of solar nuclei slows down. As greater pressure is needed for heavy nuclei to fuse, when the number of heavy nuclei increases drastically in the sun, the volume of the sun shrinks with dropping fusion pressure. Because the needed pressure diminishes, the appearance of even larger and heavier nuclei become impossible. With decreased speed of nuclei fusion, the solar temperature also drops so that the intensity of the solar light radiation weakens. As a consequence, all the planets travel more closely to the sun. Because the sunlight is now confined to a small region by cold quantum pressure, the force of solar winds becomes too smaller and the planets fall to the sun. At the moment, when the solar heavy nuclear fusion is the slowest, the surface gas of the sun becomes liquid with some solids formed. As soon as more solar gas is transformed into solids, the sun shrinks much further than before so that a high pressure is formed within the sun. This high pressure helps the solar heavy nuclei go through their last fusion reaction. However, this intensive nuclear fusion creates a tremendous outward repelling force breaking the newly formed solid shell of the sun. Large broken pieces of the solar shell becomes such large planets as the earth with small pieces being the small planets. The existence of small planets explains the last solar nuclear fusion through which the solidified sun experienced a major explosion. This is how a star lives through its life. The broken pieces of the dead star fly to the whirlpools of the nearby stars and become the planetary members of these whirlpools. This is how our earth is formed. All celestial entities must be visible and tangible objects. All materials whose existence is created by cold quantum are eternal. Due to the existence of galaxies, the heat released from many celestial bodies which are not far apart from each other connects into a whole. This is the so-called background microwave radiation discovered lately. No heat can exist in the space for hundreds of billions of years. All heat in space is released constantly by stars. Even though each bit of heat in space is weak, as soon as small bits of heat are connected together, external cold quantum starts to pressure on this combined heat so that a huge whirlpool is created within the heat. This huge whirlpool pushes all the celestial bodies located within the heat to rotate together. This is how a galaxy is formed. There are no special forces existing at the center of a galaxy. Each galaxy is also created by cold quantum pressures. That is to say, each cold quantum pressure is a piece of hidden invisible material.

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In terms of light, we do not think it can live for over hundreds of billions of light years while travelling through the space. First, we reason if the speed of light in space is nearly several hundreds of million kilometers per second. Secondly, or alternatively, as soon as light enters the space filled with cold quantum, it is transformed into electrons. As soon as these electrons leave the space of cold quantum, they are converted back to light. This is how we can see the light from the stars which are hundreds of billions of light years away from us. The thought about light still needs to be polished or modified. We will write about it in a forthcoming paper. 5. Some final words Our discussion in this paper shows that there are only two things which truly exist in nature. One is the huge cold quantum pressure, and the other hot quantum pressure. When the cold and hot quantum combine, the so-called nature appears. The entire nature is dominated by cold quantum pressures. And, there is no other form of forces in nature. Also, all forces existing in nature are pushing or repelling pressures. Nature does not contain any other directional forces. Nature is a world of forces caused by cold and hot quantum. Without forces, there would be no nature in which we live.

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Matter and vacuum. A new approach to the intimate structure of the universe

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Marcelo A. Crotti Inlab S.A., Quilmes, Argentina Keywords Cybernetics, Space Abstract In order to describe and simplify the properties of material systems and their interactions, a simple model, based on linear oscillators, is presented. These oscillators define the space and time framework from which the length and time properties of material systems are derived. Matter and energy are postulated as the physical result of grouping and interaction among primary oscillators. The length of the material systems and the time required for the information to travel both ways (back and forth) change with the system’s motion. The derived formulas coincide with the special relativity transformations for space and time. Based on this model, the speed of light seems constant for all inertial systems. There is no contradiction with the special relativity theory in the usual case of the experimental results that imply two-way trips of the electromagnetic signals, but differences arise when only one-way phenomena are considered.

Introduction In order to describe and simplify the properties of material systems and their interactions, a simple model based on linear oscillators is hereby presented. These oscillators define the space and time framework from which the length and time properties of material systems are derived. Matter and energy are postulated as the physical result of grouping and interaction among primary oscillators. The length of the material systems and the time required for the information to travel both ways (back and forth) change with the system’s motion. The derived formulas coincide with the special relativity transformations for space and time. Based on this model, the speed of light seems constant for all inertial systems. There is no contradiction with the special relativity theory in the usual case of the experimental results that imply two-way trips of the electromagnetic signals, but differences arise when only one-way phenomena are considered. Oscillatory model Consider the phenomena to be taking place in the ocean, where the entire assembly of water molecules may be considered as a frame of reference. Waves, currents and tides take form within this frame. Simultaneously, sound waves, thermal phenomena, etc., are being transmitted. Despite this internal molecular structure, when swimming, we usually consider the ocean waves as independent entities, over imposed on the huge body of water. It is possible to develop a good characterization of ocean waves

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1035-1042 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483162

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analyzing them as independent phenomena. But only through the kinetic theory, which does consider molecules, we can understand the transmission characteristics of sound waves and, in general, all transportation phenomena occurring in water. In the same line of thoughts, let us assume for a moment that there is a frame of reference in the universe. As water in the ocean, this frame of reference need not be stationary. Let us also assume that the frame of reference is formed by entities with oscillatory properties. These entities join together to create the basic constituents of what we generically call matter, waves, energy, etc. Let us assume, in addition, that this connection is governed by interactions transmitted (in average) with the speed of light in a vacuum (c). The basic constituents of this primordial medium are postulated as one-dimensional oscillators (like superstrings theory does (Green et al., 1987)). The basic difference with superstring theory is that the oscillators are not considered as the basic constituents of particles. Linear oscillators are postulated as filling (and defining) all the universe, while particles are only the physical manifestation of their coordinate interaction. From the above-mentioned basic assumptions, the concept of subjective time for any observer, either stationary or moving with respect to this frame of reference, can be conjectured. From this subjective concept of time, the value of the speed of light, measurable for each observer, can be derived. As suggested by our experience in other fields, the speed of interactions can be considered as constant within the frame of reference, as with sound waves in water or in air. At this point, it becomes very important to clarify the concept of subjective time. Once postulated that any material or energetic manifestation is no more than a grouping of oscillatory interacting constituents, if by any reason these oscillators change their coupling frequency, all the processes associated to matter or energy will change correlatively. Take, for example, the variation of the speed of sound in the air at different temperatures. This phenomenon should affect not only the biological clocks (e.g. aging), but also the mechanical ones. As expressed, the assumptions for the model are: . all material or energetic manifestations result from the interaction of linear oscillators, which not only cover entirely the space under study, but also define it; . the average transmission speed of the information, between oscillators, is constant in the frame determined by the oscillators. The question of how many oscillators are required to form an elemental particle (i.e. an electron) is similar to asking how many molecules are required to form a wave or a whirlpool. As envisioned in this model, a particle or an energetic manifestation is simply a stable grouping of the primary entities, which

originate a differentiated phenomenon within the frame of reference. It is irrelevant to know how many molecules of water or which, among all those available, form part of a wave. The wave is a stable and differentiated phenomenon within the body of water, even if the molecules that originate such a wave do not move along with it. “Stable” means that the phenomenon propagates during a certain period of time, maintaining its identity. In the following discussion, length L refers to the “observable” physical results of the groupings and interactions of a large number of primary oscillators. Similarly, the changes in the observable length L or in the internal oscillation period T for the material system is the “macroscopic” representation of the changes among primary oscillators. No attempt is made to define the physical concepts as mass. It is only assumed that the macroscopic results or particle properties are derived from the intrinsic properties of primary entities and their coupling characteristics. Defined in such a way that any “particle” is the result of the coupling of primary oscillators and must be considered as a macroscopic “observable” oscillator, which represents the result of the overall coupling. In accordance with such assumptions, the measurable properties of particles are, as with classical oscillators, the result of the equilibrium between the internal restorative forces and the kinetic energies derived from the coordinated displacement of primary oscillators. Similar to waves, the oscillators do not move with the particle. At all times, the particle propagates its identity, incorporating and abandoning the oscillators as required. During length measurements or any other interaction, any particle can be considered as a simple linear oscillator that is the result of the summation of a huge number of elementary oscillators. Further references to “particles” in the following paragraphs must be considered as a group of coupled primary oscillators. In the proposed model, it is not possible to talk about particles without primary oscillators in the same manner that it is not possible to obtain water waves without water molecules. First case: stationary particle The equations for linear oscillators that are manifested as a stationary material particle, are derived in the following paragraphs. In this stationary case, sub-index “0” is employed. During the following mathematical derivation, mass m is used only as an auxiliary variable, canceled at the end of the process. Mass m must be included in order to employ the classical physical formulas that describe the behavior of linear oscillators. For simplicity, the mass m of the particle is assumed as the sum of the individual masses of the oscillators. As primary oscillators are postulated as interacting at speed c, also mass m, or the sum of primary oscillators mass, is displaced at speed c in one way or another. Plus (+) and

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minus (2) signs will be used to reflect opposite directions of the absolute magnitude, c. Within this explanation, mass m is not necessarily identified as the measurable mass of the particle. It must only be considered as an “internal” mass reflecting the addition of individual oscillator’s mass. As previously explained, no attempt is made to define the concept of mass associated with individual oscillators. For a material particle (of “internal” mass m), assumed as a linear oscillator which reflects the coupling of a group of primary linear oscillators, the “internal” kinetic force (F0), in equilibrium with the “internal” elastic restorative force, is: F 0 ¼ Momentum variation=Oscillation period where, with the “internal” mass m changing its speed from +c to 2c, it results in Momentum variation ¼ 2mc

ð1Þ

and the oscillation period T0 is related to the length of the particle L0 by 2L0 c

ð2Þ

2mc ð2L0 =cÞ

ð3Þ

T0 ¼ leading to F0 ¼

and L0 is determined through the elastic constant k, in accordance with: F 0 ¼ kL0

ð4Þ

So, equating (3) and (4): 2mc ¼ kL0 ð2L0 =cÞ and

 L0 ¼

mc 2 k

0:5 ð5Þ

Second case: particle in motion The case of a particle moving at a speed v (sub-index “1”) with respect to the frame, can be calculated similarly. Once again, it must be emphasized that the fundamental linear oscillators do not move with the particle, as water molecules do not travel with the waves.

Matter and vacuum

The time of a complete oscillation (T1) can be determined as T1 ¼

L1 L1 þ ðc þ vÞ ðc 2 vÞ

which, after regrouping the terms, leads to 2L1 c T1 ¼ 2 ðc 2 v 2 Þ

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and the average force can be calculated as F1 ¼

2mc ð2L1 c=ðc 2 2 v 2 ÞÞ

ð7Þ

The equations are always expressed in accordance with the observations collected from the stationary system. The variation of the momentum is the same as in the first case, since any elementary oscillator has an average speed +c or 2c, because it is stationary in the reference system (the oscillators are themselves the reference system). However, the system length L1 cannot be determined beforehand. As in the previous case, the elastic restorative force is related to L1 according to F 1 ¼ kL1

ð8Þ

and from equations (7) and (8):  L1 ¼

mðc 2 2 v 2 Þ k

0:5 ð9Þ

while dividing equations (9) and (5):  L1 ¼ L0

v2 12 2 c

0:5

and dividing equations (2) and (6):    L1 c2 T1 ¼ T0 L0 ðc 2 2 v 2 Þ

ð10Þ

ð11Þ

where, after replacing the length quotient, based on equation (10), we obtain T1 ¼

T0 ð1 2 v 2 =c 2 Þ0:5

ð12Þ

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Equations (10) and (12) express the relationship between the lengths and period, for a system in motion, with respect to the same values obtained for a stationary system. Both measurements are referred to the stationary system. The results obtained are the same equations that govern coordinate transformation in special relativity (Einstein, 1905), but the consequences differ notably. In general, the “it seems as if. . .” of special relativity becomes “it actually occurs that. . .” and vice versa. As a consequence of this model, the moving clocks are in fact slower with respect to the stationary clocks. However, this is not evident in the moving system when carrying out experiments that involve length and cyclic time measurements. The explanation for this is that the observer’s clocks and lengths are all affected in the same way as any material manifestation in the system. The above analysis is a consequence of the fact that, indirectly, in the measuring process we use the property which is, in fact, what we intend to measure. Even if the system velocity varies, the time for the light to cover the two-way path of a given distance, as measured by the observer, remains unchanged. This is not the case with the time employed for a one-way trip. It should be noted that at this point, all accurate measurements of c (leading to the postulate of constancy of light speed) have been performed using two-way trips of the electromagnetic waves. One way measurements like Ro¨mer’s study on the eclipses of the moons of Jupiter and Bradley’s determinations on aberration of light, are not accurate enough to detect the absolute movement of the order of the Solar System local speed. Even if the postulate of a frame of reference in the universe may be taken as a reference to the obsolete theory of “ether”, it actually involves a basically different concept. This supporting media is not thought of as a fluid in which waves propagate and which is crossed – with or without friction – by the material particles. According to this model, matter is an abstraction of the interaction of primary oscillators. In this model, the concept of “corpuscle” associated with material manifestations is meaningless. If such “corpuscles” were mere stable groupings of the supporting medium’s components, it would not be convenient to think of them as being constituted by any other material, since this requires thinking that matter travels through the supporting medium (the old ether theory). In this model, corpuscles propagate through the supporting media. In a similar way, ocean waves do not travel across the water, but propagate through it. Still waves are fairly stable groupings of the supporting medium particles (water molecules). A practical difference between this model and the special relativity may be found when measuring the propagation speed of light by using one-way travels or the effect of one-way interactions. Such measurements are possible, but not

easily made. In fact, the “Dipole Anisotropy” on the Cosmic Microwave Background (CMB) (Smoot et al., 1977) is in better accordance with this model than with special relativity. The “Dipole Anisotropy” as determined by the NASA’s cosmic background explorer (COBE) satellite, is easily explained as a classical Doppler effect originated in Earth’s movement through the background reference system. Doppler effect is the result of a typical one-way interaction (there is no need for the return of the wave).

Conclusions The transformation formulas for length and time, which are obtained through the special relativity theory, can be derived using a “classical” model of the universe. The model presented here is simple and leads directly to the previously mentioned equations. The speed of light, in accordance with the proposed oscillatory model, would actually be a fundamental property of the universe. It would not, however, be in the sense given by the special relativity theory, but because of being a manifestation of the intrinsic properties of what we call matter and/or energy. It would accomplish a role similar to that of a molecule speed in the kinetic theory of gases. It would be, in this case, the speed at which the primary components of the universe interact. Within this model, the concept of “corpuscles” and “waves” as independent entities are meaningless. They are only the result of interactions between the primordial components. The conception of wave-particle duality would turn meaningless as a consequence that, according to this model, differences in internal structure of waves and particles vanish. Absolute time would coexist with the times proper (other than absolute) of each inertial system. Each system would have its own “true” time, whatever the meaning of this expression may be. Light speed would become a practical limit for the material particles we know, but it would not be an absolute limit. As previously expressed, the “it seems as if. . .” of special relativity will become “it actually occurs that. . .” and vice versa. Based on this model, although there may be a reference system under conditions equivalent to absolute repose, it cannot be detected using experiments involving two-way travels of the electromagnetic waves. This model does not imply the existence of an “absolute” reference system and mobile systems would only exist with respect to the basic frame that defines the space. This concept is analogous to the movement of waves and currents with respect to the extended mass of water of which they form part. There are no water molecules that can claim to be at absolute rest. Still the velocity of transportation phenomena becomes meaningful only when compared to the “stationary” local water extension.

Matter and vacuum

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The differences with special relativity would be evident only if performing experiments involving accurate measurements of the speed of light, using single way travels. The “Dipole Anisotropy” on the CMB is in better accordance with this model than with special relativity. References Einstein, A. (1905), “Zur elektrodinamyk bewegter Ko¨rper”, Annalem der Physik, Vol. 17. Green, M., Schwartz, J. and Witten, E. (1987), Superstring Theory, Cambridge University Press, Cambridge. Smoot, G.F., Gorenstein, M.V. and Muller, R.A. (1977), “Detection of anisotropy in the cosmic blackbody radiation”, LBL Report 6468, Physical Review Letters, Vol. 39, p. 898.

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Panrelativity laws and scale relativity – against Einstein with Einstein’s Tao

Panrelativity laws

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Xiangjun (Leon) Feng A Ideas International SanJose, CA, USA Keywords Cybernetics, Theory, Quantum mechanics Abstract People have been led objectively to the directions of delusion and confusion since Einstein published his special relativity in 1905. Basically, Einstein’s special relativity is against the truth of the universe – The cause effect law. The fatal mistake of Einstein was, as the “collapse of the wave function” in modern quantum theory, not separating the physical reality from the observation of instruments and human being. This paper attempts to bring back science from inside the “Ha – Ha” mirror to the common senses and rational study. This paper reflects some important viewpoints of International Pansystems School.

1. Various axioms related with space-time 1.1 Guy Jamarie’s axioms (A1) A system S along is meaningless, and it makes sense only with respect to its environment S* with which it exchanges information. (A2) A system (S, S*) is defined only with respect to a given observer R, say (S/R) and (S*/R). (A3) A system is characterized by the amount of uncertainty H1 (S/R) and H0 (S/R) such that the observer R has about its internal structure and the internal structure of its environment, respectively. And the knowledge of R about (S, S*) will evolve in such a manner that the following equation holds, H 12 ðS=RÞ 2 G2 * H02 ðS=RÞ ¼ constant; where G is a positive constant which depends upon the measurement units. Guy said that axiom 3 was based on his analysis about most real systems. 1.2 Lorentz’s axiom There is an ether. Relative to the ether, things physically change with the velocity. 1.3 Einstein’s axioms (a) Relativity Principle – in any inertia system, the physical laws have no change for things that really changed. (b) The light speed in vacuum is a constant to any inertia observer. 1.4 Newton’s axiom Things are in motion in an absolute space and absolute time.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1043-1047 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483171

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1.5 Leon’s axioms (a) The space and time are objective and things are in a motion relative to this objective space and objective time following the cause-effect law (CEL). The objective universe in a motion following the CEL is very meaningful instead of being meaningless. (b) The absolute observed things are meaningless and they make sense only relative to the super complicated Sheng-Ke Pansystems of object, subject, medium or generalized instrument and environments. The super complicated Sheng-Ke Pansystems determine the scale to observe the observed things. It is so clear that Leon’s axioms are related and also different from all those “axioms” of Guy, Lorentz, Einstein and Newton. Leon thinks that his axioms are more comprehensive and are without delusions. As any other people’s axioms, Leon’s axioms system cannot escape from the philosophy behind it. 2. Panrelativity laws Law I. The CEL will not change no matter the body is in a motion or at rest, no matter the object to be studied is a micro world or a macro universe, and no matter it is at ancient time, present or in the future. Any observation or measurement ignoring the CEL is a delusion and must be corrected by a factor of scale. The CEL is a pansystem CEL ¼ ðH ; SÞ where S ¼ There will be no change if there is no Yin Yuan force’s action. H ¼ Things keep changing with the Yin Yuan forces. Yin Yuan forces are the Pansystems which include any physical force and any cause and condition which could bring about an effect. Law II. Everything in this universe shares the same original nature or the nature before the action of any Yin Yuan forces. The losing of the original nature is the origin of delusion in measurement and observation. Law III. Yin Yuan force will bring about a change. PMðts þ dtsÞ ¼ Gð f ðtsÞÞPMðtsÞ þ f ðtsÞ Where PM is a Pansystem, G is the generalized transform related with f, and f is the Yin Yuan force, ts is the space-time, and dts is the change of the space-time. 3. Einstein’s special relativity is incompatible with CEL (A) Einstein’s C axiom is incompatible with CEL. Purely from C axiom,
ðd AÞ · ðd BÞ ¼ I 2 1 2 ðV =CÞ2

where dA and dB are the objective length for a body in a world moving with V relative to the body. The body has an objective length I in a world at rest relative to it. From the CEL

dA ¼ I dB ¼ I So that dA· dB is not changing with velocity V. Therefore, Einstein’s C axiom alone is enough to break the CEL. (B) Einstein’s rigid rod is against CEL. From CEL ðdÞ* S 0 ¼ lS where d is the observed length from a moving observer with a relative velocity V to a body with an observed length l from the observer at rest relative to it, S 0 is the unit used by the moving observer and S is the unit used by the observer at rest. S 0 ¼ S=d Since experience has told us that d as an observed thing will change with velocity, we always have, for V . 0 S0 – S Therefore, to use the same unit as observer at rest to observe things for an observer in motion is also against the CEL. (C) Newton followed CEL. In Newton’s theory X 0 ¼ X 2 VT T0 ¼ T For two points a and b Xa 0 ¼ Xa 2 VT Xb 0 ¼ Xb 2 VT Let Length AB ¼ Xa 2 Xb ¼ I . We immediately have Length AB 0 ¼ Length AB ¼ I The objective length has no change in Newton’s theory relative to the world in motion. Therefore, Newton followed CEL.

Panrelativity laws

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Unfortunately, Newton did not discuss the “Ha– Ha” mirror – observed thing can change in any way, so that Leon’s scale relativity feels necessary to follow Newton only because he followed CEL, and explains Einstein’s delusion as a theory of “Ha– Ha” mirror. Leon’s scale relativity just tells a simple story: Things will not change without an action of Yin Yuan force. Observed things keep changing because the scale used to observe things keeps changing, and the scale is determined by the super complicated ShengKe Pansystems: subject, object, medium or generalized instrument and environment. 4. Einstein’s special relativity and a “Ha– Ha” mirror Let us use CEL to discuss a “Ha– Ha” mirror. We have DS 0 ¼ lS where l is the observed length without a “Ha–Ha” mirror and S 0 is the scale used with a “Ha– Ha” mirror to observe things. Since “Ha– Ha” mirror makes S0 – S We can have D,1 and we also could have D$1 Einstein’s theory is just a theory for the “Ha– Ha” mirror of velocity V. In Einstein’s theory, just a look from a moving body could make a thing physically change, which has been well “explained” by his follower Guy Jumarie as the result of “information exchange”. Another look from an observer at rest could physically change things back. No matter how far away the distance between the “information source” and “information destination”, the results of “information exchange” are all the same as long as the relative velocity V is the same. It is really amazing that such an absurd theory could last for almost one century. 5. Scale relativity applies for any case When there is an Yin Yuan force, things will objectively change. However, for a specific moment t1, the dynamics issue changes to be an observation issue. It could be approximately considered that there is no change for any specific time t1 instead of for a time interval t22t1. Therefore, any observation result should be calibrated by the cause effect law taking into consideration a scale factor S 0 ¼ S=dðV Þ; where V is the changed velocity caused by Yin Yuan force.

6. Conclusions Panrelativity laws and scale relativity are put forward. Any theory against CEL will only lead to confusion and delusion. Einstein’s special relativity is incompatible with CEL so that it objectively led science into the direction of delusion and confusion. Modern science needs hard work to calibrate the measurement results with the CEL. Further reading Bohm, D. (1996), The Special Theory of Relativity, 1st published in 1965, W.A. Benjamin, Inc., Reading, MA. Einstein, A. (1916), Relativity. Jumarie, G. (2000), Maximum Entropy, Information Without Probability and Complex Fractals, Kluwer Academic Publisher, Dordrecht. Xuemou, W. (1990), The Pansystems View of the World, Chinese People University Press.

Panrelativity laws

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A modification of the special theory of relativity B. Paul Gibson Davis and Elkins College, Parkersburg, West Virginia, USA Keywords Cybernetics, Theory Abstract Light, when constructed in terms of the elementary quanta of light, may be viewed in particle-like or wave-like terms. The elementary quanta of light, when placed in motion through space/time at a speed of a constancy of c forms a light path through the space or reference frame viewed. The light path formed is curved, as space/time is curved. The curvilinear light path formed is a function of the gravitational potential within the viewed frame of reference. The linear description of this light path, termed the geodesic (Riemannian), does not describe the curvilinear light path, but rather the chord of the curvilinear path described by the inscribed arc. This linear description of the light path is the manner in which we describe the coordinate system involved, and is the same manner in which we determine the “speed of light”. The arc length of the light path, compared to the lesser value as described by the chord length, allows for a displacement to be determined, if both measures are applied to a linear measure. A displacement of linear coordinates then occurs, with this displacement a result of the gravitational potential occurring within the frame viewed. This displacement, derived via observation and predictions of the quantum model, resolves Maxwell as well as Newton. The theory concludes that the Special Theory of Relativity, suitably modified to account for gravitational displacement within one particular frame, derives a precise relative model of gravitation within the special frame. This model satisfies Newton, as the model arrives at an exact description of the three-body problem.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1048-1082 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483180

1. Concerning gravitation and the transmission of light Let us consider light in terms of the elementary quanta of light, according to the derivation of Planck. We further consider that Planck was ambiguous in his description of quanta, and that Einstein agonized over the term. Einstein proposed the particle nature of this quanta, based on the work of Planck. From Einstein, de Broglie derived an apparently ad hoc description of the quanta in wave-like terms, based on Einstein’s particle description. Based on the de Broglie hypothesis, Compton showed us that the electron could be described in wave-like and particle-like terms. We therefore consider that the elementary quanta, described as a wave, agrees with the Maxwell’s description of the wave nature of light. This paper concerns Einstein’s Special Theory relevant to the hypothesis of de Broglie and Newton’s derivation of gravitation. Relevant to the de Broglie hypothesis, where de Broglie originally found that if the quanta of light were assigned a small mass derived from the Special Theory and Planck’s constant, then that quanta would only be able to linearly move at a constant speed slightly less than that assigned to the so-called constant of the speed of light of c. Here c is relative to the Special Theory. This speed derived by de Broglie is then a velocity v0, where de Broglie solves an apparent contradiction between Planck’s and Einstein’s form.

Later, in his Comptes Rendus note, de Broglie found that there was a Special theory of contradiction between the quantum principle of Bohr/Sommerfeld and the relativity relativistic principle of Einstein, if there were not to be a frequency shift in the internal periodic phenomenon. To solve the problem, de Broglie introduced a fictitious pilot wave that would be in phase with the standing wave at the first Bohr radius, but this pilot wave would necessarily be forced to travel slightly 1049 faster than c or a velocity v1. de Broglie speed of v1 has been treated as the actual speed of light c, where v1 equals c. In order to account for the Compton shift, the fictitious velocity v0 of de Broglie is used to determine the wavelength l of the particle based on Planck’s constant h and mass m.

l ¼ h 4 {m £ v0 }

ð1Þ

f ¼c4l

ð2Þ

Then frequency f is:

In the Special Theory, we propose that c is then replaced (in one aspect) by v, where the Special Theory requires that light always propagate along the axes concerned, at the velocity v. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 4 c 2 £ v20 ð3Þ The property v is then considered as the measured (actual) linear speed of light on the surface of the earth, in vacuum. As a consequence of defining v in this manner, we find that the form of Maxwell does not derive the value of c used herein, but derives the quantity 1 in numerical form. Based on the permeability and permittivity of the vacuum, the measurements yield a close approximation of c as used herein, while the best measurements (laser) of the speed of light (v) yields 299,792,457.4 mps, a close approximation of the value used in this paper. Towards the Special Theory, the mass/energy equivalency becomes, as per Planck’s constant and the frequency involved; E0 ¼ m £ v2 ¼ h £ f

ð4Þ

In equation (4), if we substitute the Planck’s constant h for E0, and where the frequency of the elementary quanta is exactly 1 H, we may derive the theoretical mass of the elementary quanta, applicable to the hypothesis of de Broglie. The assumption in the aforementioned is that de Broglie properly identified the speed of light c, based on a curvilinear light path. The light path is viewed as being bent as a result of gravitation (as per the General Theory) causing the light path to form a parabolic arc segment between the two points A and B. The time involved (1 s) in generating c over the curvilinear path

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results in the linear value of v (the geodesic of Riemann) between the same two points. The de Broglie concept is linear from a coordinate standpoint, as is the Special Theory. Therefore, there is a displacement in coordinates of d1, in linear measure of:

1050

d1 ¼ c 2 v

ð5Þ

In Maxwell, a particle travelling in vacuum at speed c from A to B would suffer a displacement of d1. In one full Maxwell circulation of the particle from A to B and returning from B to A, the particle suffers a total displacement of d2. d2 ¼ 2 £ {c 2 v}

ð6Þ

This Maxwell displacement results in a proper Lorentz transformation as regards to the wavelength of the elementary quanta. We consider that as the quanta, generated at A, travels to B, the time, t considered equals zero. At this point of time, A and B are connected by the linear distance v; corresponding to a wavelength. If A and B are considered to be constantly connected by a continual stream of light, any change in the gravitational potential would affect the light involved relative to v, while c remains constant. If the gravitational potential is changed, then v would change to v*. The time of transmission of the information would not be instantaneous, but would result from the difference of v compared to v* divided by c. We find the following. (1) The measurable, or actual, linear speed of light of v is identified by the Special Theory, relevant to the mass of particles. (2) The constancy of the speed of light of c, relevant only to waves, identified under de Broglie, achieves Maxwell, and resolves the General Theory that light is affected, or bent, by the gravitational potential of the frame viewed, and a displacement of wavelength results in de Broglie’s v0 (an arc length to chord length comparison or the Riemannian geodesic or condition). (3) The speed v0, solves for Maxwell’s displacement from c, results in a proper Lorentz transformation, and resolves the Compton shift of wavelength. If the Special Theory is slightly modified to account for v, while evolving c, then the Special Theory is an account of gravitation in one special, or local, frame of reference. The Special Theory must then derive Newton in that frame. The following requires numerical values, which are known and verifiable. Those values are attached at the end of the brief. In the following Einstein/Lorentz transformation, the quantity x is the distance from an observer placed exactly on the equatorial radius of the earth of r, to a position above the equator identifying a body in geosynchronous orbit.

At a geosynchronous orbit described by the quantity x, the period of one Special theory of revolution of a body at this orbit exactly agrees with the period of one rotation relativity of the earth about its axis as noticed at the equatorial radius of the earth. The paper will show that this period resolves a relative transfer of gravitation, as described by Newton. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1051 x 0 ¼ ½x 2 {v £ t} 4 1 2 {v 2 4 c 2 } ð7Þ The time t: t ¼x4c

ð8Þ

b ¼ x 4 x0

ð9Þ

The quantity b:

where 1 4 b ¼ ½1 2 {v £ ð1 4 cÞ} 4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 {v 2 4 c 2 }

ð10Þ

Time dilation of t from t is not a matter here. Since time and distance are intrinsically involved, a resolution of the relative distance between the frames resolves the time differentiation between those frames. In equation (10), b is not defined as a magnitude, but as a distance. Under the theory, we assume that the distance x may be considered as a special case, where gravitation is not affecting the light path viewed on a linear basis. We assume that b is the distance that light would see as a result of gravitation affecting a particle of light within a relativistic frame as compared to the special frame. Here, it is stressed that b results, or is resolved, from the totality of all the gravitational forces affecting the special frame viewed. Electrical properties may be viewed as positive (attractive), negative (repulsive) or void of charge (neutral). On the other hand, gravitational properties are viewed solely as attractive. This may only be true if a body within the frame is isolated from any other gravitational forces. Any other gravitational force introduced into the frame is a counter force to the gravitation exerted by the body present, and the introduced forces, counter in nature, may be viewed as repulsive to the force present. The earth exerts a force of gravity at its surface, and one could state that this force (regardless of its magnitude), is compressive (attractive). Yet, the moon exerts its own force of gravity, and this force acts on the earth’s surface. In fact, the force exerted by the moon against the earth literally lifts the entire crust of the earth at about 2/3 of a meter of lift. This lunar lifting of the earth’s surface is counter to the compressive force of the earth acting against its surface. Therefore, one may consider the earth’s force of gravity to be attractive, while the lunar force is counter-attractive or repulsive. In terms of magnitude, the maximum force of gravity formed from the earth’s mass at its surface,

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is reduced to some lesser value, as a result of other gravitational forces existing within the frame viewed, and countering the force exerted by the earth. So-far as the measurement of the force of gravity on the earth’s surface is concerned, one may only measure and observe the totality of the attractive and repulsive forces involved within a vector approach. The quantity b is then assumed to derive a distance that, when applied to Newton, reduces the totality of the earth’s gravity from some maximum to a lesser value. The application of beta then reduces the force of gravity of g from a value of 9.89 m/s2, to the known value of 9.80 m/s2. The theory is that the earth resolves the known parameters identifiable under Newton and that light, once gravitation is accounted for in the curvature of the light path, may derive the modifications required to resolve Newton. This modification, or application, solves the three-body problem in an exact manner, as Newton and Hamilton may only provide approximations, based on the study conducted by Poincare´. Beta also allows for the determination of the maximum velocity of a satellite in orbit according to Newton’s laws. Then, in the following, where g 0 ¼ standard acceleration of gravity, r ¼ equatorial radius of earth, R ¼ mean separation distance between the earth and moon, and P ¼ mean orbital period of lunar body. g 0 ¼ ½4p 2 {R þ b}3  4 ½{r þ b}2 £ P 2 

ð11Þ

where the mean radius of the earth is rm and the mean polar radius rp, and as the earth is an oblate spheroid, then: r m ¼ ½{r £ 2} þ rp  4 3

ð12Þ

The General Theory requires that c be the acceleration, and that the maximum acceleration of gravity becomes c. Newton requires that the mass of the earth be assumed to be concentrated about the center of the earth, at some unspecified radius. We then predict Einstein’s radius of RE, where the mass of the earth is considered as concentrated about this radius. g 0 ¼ c 4 ½{r m þ RE } 4 RE 2

ð13Þ

The property g 0 then defines a radius r0 from a fixed earth center. g 0 ¼ c 4 ½{r0 4 RE }2 

ð14Þ

r 0 ¼ r m þ RE

ð15Þ

Then,

where the mass of the earth is ME and Newton’s constant of universal gravitation is G:

  g 0 ¼ M E 4 r20 £ G

ð16Þ Special theory of

  c ¼ M E 4 R2E £ G

ð17Þ

relativity

Deriving,

In this sense, we have satisfied both Newton and Einstein. With the mass of the earth, ME from equation (16), and Einstein’s radius, RE from equation (17), we derive the mass of the sun, MS, where this mass must conform to Newton’s Laws. hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i M S 4 M E £ 2 4 RE ¼ M E 4 ½{RE £ ðRE 4 2Þ} 4 2 ð18Þ where the mean separation radius between the earth and the moon is R and conforms to equation (11), and the mass of the sun conforms to equation (18), then: M S ¼ {R 4 RE } £ M E

ð19Þ

The earth would exert its own force of gravity of g1 at the mean lunar separation radius of R. pffiffiffiffiffiffiffiffiffiffiffiffiffi c 4 g 1 ¼ M S 4 M E ¼ R 4 RE ð20Þ Then, g1 ¼ {M E 4 R 2 } £ G

ð21Þ

From the above, at any radius r 0 from a fixed earth center; the force of gravity per Newton: g ¼ c 4 ½{r 0 4 RE }2 

ð22Þ

At a geosynchronous orbit identified by the linear distance x from the equatorial radius of r, we find the following: if T is the mean period of one rotation of earth about its axis. {4p 2 } 4 {G £ M E } ¼ T 2 4 ½{x þ r} 2 {42 £ RE }3

ð23Þ

The mass of the lunar body of ML then conforms to the following two equations. g 0 ¼ ½{2pR} 4 P 4 ½{ðR 4 M L Þ £ M E } 4 c

ð24Þ

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 M E 4 M L ¼ RE 4 {M L £ G} 4 c

ð25Þ

and

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The value of gravity on the lunar surface, corresponding to g0 on the earth, becomes gL, as per the Maxwell displacement of c 2 v: ½1 4 {c 2 v} £ p 2 ¼ ½{gL 4 M L } 4 {g 0 4 M E }

1054

ð26Þ

The lunar radius of rL, equating to r0 on the earth. g L ¼ {M L 4 r 2L } £ G

ð27Þ

hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i {gL 4 M L } 4 {g 0 4 M E }

ð28Þ

Then, r0: r0 ¼ rL £

For the sun, a value of gravity of gS, corresponding to g 0 on the earth: where x 0 from equation (7) of the Einstein/Lorentz transformation is derived: x 0 ¼ ½{1 4 M E } 4 {gS 4 M S } The solar radius of rS, corresponding to r0 on the earth;   gS ¼ M S 4 r2S £ G Then, r0: rS ¼ r0 £

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi {g 0 4 M E } 4 {gS 4 M S }

According to the speed of light;

n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio 2 0 g 4 gL ¼ rL 4 M L 4 ðc 4 GÞ 4 r0 4 M E 4 ðc 4 GÞ

ð29Þ

ð30Þ

ð31Þ

ð32Þ

and 0

g 4 Sg ¼

n

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio 2 r0 4 M E 4 ðc 4 GÞ 4 RS 4 M S 4 ðc 4 GÞ

ð33Þ

A mean distance of one astronomical distance of AU separates the earth and sun. ½{M S 4 M E } 4 {AU 4 R}2  ¼ ½ð{2pR} 4 PÞ 4 {ð2prÞ 4 T}

ð34Þ

where ½{ð2pRÞ 4 P} 4 {ð2prÞ 4 T} ¼ ½{ðM S 4 AU2 Þ £ G} 4 {ðM E 4 R 2 Þ £ G}

ð35Þ

From the values used, equation 35 results in that the equatorial radius of the Special theory of earth, as per the value attached as defined by Merritt, holds to an uncertainty of relativity 0.02 m, unless modified according to Note 1 of the brief. Where the period of one earth orbit about the sun is Y, then for the earth/sun: ½{4p 2 AU3 } 4 {r 2S £ Y 2 } ¼ gS £ ½{ð4p 2 Þ} 4 {G £ M S } 4 {Y 2 4 AU3 } ð36Þ The basis of this paper is that a linear displacement occurs relevant to the wavelength of the elementary quanta, resulting in the displacement termed c 2 v: In equation (20), the value of the earth’s force of gravity g1, found at the mean separation distance of R between the earth and moon systems is determined by the proportion of the mass of the sun compared to the mass of the earth. In equation (26), we find that the displacement c 2 v is determined by the proportion of the mass of the earth compared to the mass of the lunar body. We then find the following, relevant to the solar/earth separation distance of AU. AU ¼ ½{M S 4 M E }2 £ {1 4 ðc 2 vÞ} þ {R 4 2}

ð37Þ

In reference to equation (36), we find for the earth/moon system; ½{4p 2 £ R 3 } 4 {r20 £ P 2 } ¼ g 0 £ ½ð4p 2 4 {G £ M E }Þ 4 {P 2 4 R 3 }

ð38Þ

One of the hallmarks of Newton’s laws is the determination of angular momentum. A curiosity results if one examines the momentum w of a satellite’s velocity, based on the distances from the moon ever decreasing to the surface of the earth. The velocity of the satellite increases on travelling from the moon to the earth’s surface, yet dramatically decreases from some point just above the earth’s surface to the velocity of the earth surface as it rotates. During the process, one reaches a maximum satellite velocity of Wmax. The distance b determines the maximum orbital velocity. As per Newton in terms of motion (which satisfies Lagrange and Hamilton), we find that the force of gravity at b is gb, then, g b ¼ ½M E 4 {r þ b}2  £ G

ð39Þ

gb ¼ W 2max 4 {r þ b}

ð40Þ

Then,

where from Laplace: W max ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi {2GM E } 4 {r þ b} 4 2

ð41Þ

Based on the aforementioned, a form problem results. Schwartzschild developed a radius from the field equations of the General Theory, where a

1055

K 32,7/8

1056

singularity would be described. The Schwartzschild radius, defining a small radius where the mass of the earth would be supposedly concentrated within this radius, defines an event horizon where light could not escape from this mass/radius, since the speed of light of c must be exceeded. At this radius, the escape velocity would necessarily exceed c, and based on the General Theory, this is not possible. Then, a singularity would result, an idea that Einstein never accepted in total or even in concept. Then, the Schwartzschild radius of rs (where the equation is a derivative of the formulation of the Laplace equation (from Newton) that derives the escape velocity at the surface of the earth) is the Laplace equation. rs ¼ {2GM E } 4 c 2

ð42Þ

If the above equation is true, then this brief is an error, as Newton is in error, for the Schwartschild radius is minutely smaller than the Einstein radius of RE. In equation (17), if we replace Einstein’s radius with the Schwartzschild radius, we find the resulting acceleration to be 5.0718 mps, or 1.6910 times the speed of light. Laplace originally determined from his derivation of the escape velocity that his concept of the “speed of gravity” would be a minimum of 110 times the speed of light. If gravity is viewed as acceleration, and the Schwartzschild/Laplace construct has any physical meaning within the interior of the earth, then a particle may be accelerated to speeds far in excess of light speed. Perhaps, it would be more prudent to state that the physics of the moment do not exist to explain these phenomena. Yet, rs ¼ {R2E £ 2} 4 c

ð43Þ

This brief requires a linear value of the speed of light, denoted by v, be verifiable. Based on laser measurements, the utilized value agrees with the measurement within an uncertainty of ^0.03 mps. The displacement c 2 v then determines c or the curvilinear speed of light. Therefore, the displacement c 2 v is of importance. The values attached to the times, or periods, of P, T and Y were determined during 1988. Those values are affected by the displacement c 2 v: Yet, c 2 v results in a value only accurate to four decimal places (based on the values used in the paper), and that is not acceptable, considering our current ability to maintain time. Currently, the method to maintain time is based on the quantum frequency exhibited by the radioactive cesium 133 atom, under extremely controlled conditions. This standardized frequency of f133 (as per the second of time) is determined at 9,192,631,770^2 H. For simplicity sake, we derive the exact value of the Maxwell displacement of D in two equations, where the quantum factor Q is derived. h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 Q ¼ 1 4 { f 133 4 c} 4 p ð44Þ

The exact value of the Maxwell displacement D, equating to c 2 v; then Special theory of becomes: relativity 1 4 D ¼ {D 2 2 Q} 4 {1 4 2}

ð45Þ

Equation (43) then derives equation (44), where equation (44) is no more than a statement of Pythagoras. The Maxwell displacement D, regardless of the values attached to c and v, solves the congruent calculus of Maxwell, Lorentz, and Compton. By inference, de Broglie and Einstein are resolved. In terms of gravitation, we return to the radius RE. The radius RE is purely theoretical, and in no sense could either be measured or detected, if in fact a body such as described by this radius could exist. However, the speed of light of c, of necessity, would be the acceleration of gravity at the surface of this spherical body as described by the radius RE. We then assume that the proton, or the nucleus of the hydrogen atom, has a spherical form and exhibits a verifiable radius of r *. On the surface of the proton, described by this radius r *, we then assume that the speed of light of c is the acceleration of gravity on that surface. We further assume that the Newton’s universal constant of gravitation of G is identical in both frames of reference, as c is assumed to be identical. If Newton’s Laws are applicable to the earth body, which they essentially appear to be in terms of gravitation, those same laws could not be directly applicable to the proton. We state so, because Newton’s Laws apply to a mass external to the body as described by the radius RE, while any mass found within the radius as described by RE would be as theoretical as the radius itself. Any density of this body would, of necessity, be just theoretical. However, because we assume that the force of gravity exactly equals c on the surface of the radius RE and on the surface of the proton, we have one theoretical similarity. We then hypothetically state that the quantum model of the hydrogen atom derives Newton in one sense. The angular momentum w of the electron at the first Bohr radius of a0 compared to the mass of the electron of M1, results in the Newton’s definition of the kinetic energy of the particle involved, where the ionization energy of hydrogen of IE is known. Then, the elementary charge on the electron is ec; IE ¼ ½{1 4 2} £ M 1 £ w 2  4 ec

ð46Þ

Therefore, we have two similarities, both based on a derivation of Newton’s Laws. We then assume that the proton is comparable, in one sense, to the body in the interior of the earth as described by the radius RE. However, we assume that the Newton’s constant of G is not applicable to the proton, for G is related not to the internal mass within RE, but to the external mass outside that body. With the mass of the proton of Mp, and the radius RE known, we may

1057

K 32,7/8

determine the radius of the proton of r* by the following equation, where the value agrees with the known radius of the proton. RE ¼ M p 4 r* 2

1058

ð47Þ

Knowing the mass and radius of the proton, one may derive the density of the proton, which appears to be a density roughly constant across the atomic nuclei spectrum. We then assume that the body described by the radius RE contains within itself roughly the same density. This leads to a theoretical mass of the body contained within the structure determined by the radius RE to be roughly 245.4 times the mass of the earth as defined by the densities involved, as compared per Newton. The magnitude 245.4, termed X0, can be derived as: X 0 ¼ ½M p 4 {ð4 4 3Þpr* 3 } 4 ½M E 4 ð{4 4 3}pR3E Þ

ð48Þ

In equation (48), we assume that the known mass of the earth is contained within the body as described by RE. If we assume that this same body has exactly the same density as the proton, and hence a mass equal to roughly 245.4 times the known mass of the earth, we find, in percentage form, that there would be a large difference in mass. Then: 98:59 percent ¼ ½100 2 {100 4 X 0 } 2 1

ð49Þ

Therefore, we find that one would be “missing” 98.59 percent of the nuclear totality mass of the earth, and hence as per Newton, the mass of the Universe. This missing mass, when applied to known mechanics derived via the General Theory, offers one (at this point no other solution exists) solution for the most horrendous problem facing the physics at the moment. A nuclear model of the earth, on a similar basis to that theory as applied by Lamaitre, allows for the current closure of the Universe as per the General Theory and the beginning of the Universe as per Guth, describes the Schwartzschild radius problem as the General Theory does not account for the “missing mass” (and hence does not define mass relative density). The value of X0 assumes that the mass of the earth is concentrated within the radius defined by RE. We further assume that the speed of light of c is exactly the same in the vacuum of the gravitational field of the earth and within hydrogen. However, this may not be the case, as the index of refraction in hydrogen is approximately that of the index of refraction for “air”, where the “speed of light” in hydrogen would be c divided by the index of refraction for “air”. Since c would “change” in hydrogen, the product c 2 v (relative to the vacuum described by the earth) is not directly transferable. Therefore, the following equations do not lead to exact equivalencies, yet very good approximations are resulted under the assumptions that follow.

We assume that the hydrogen atom is earth-like, and may be described under Special theory of the terms of Newton. We then compare the two values of gravity relative to the relativity earth, with those values previously described as g0 and g1; to two values comparable within the hydrogen atom. A value of g2 at the first Bohr radius will be derived, as we assume that the earth-based value of g1 will describe the same value within hydrogen at a radius of the hydrogen atom defined in the free state. 1059 We assume that the maximum force of gravity is c, and not a lesser value. We assume that this force of gravity of c occurs on the surface of the proton, as radius is described by r*. The ad hoc value of X0 of 245.4 derived as X0, assumes the magnitude of 255, or the product X1, where: c ¼ M p 4 ½r* 4 {X 1 £ 2}2 

ð50Þ

Under the assumptions, we may determine a value of gravity at the first Bohr radius of a0, resulting in a value of gravity of g2. g 2 ¼ M p 4 ½a0 4 {X 1 £ 2}2 

ð51Þ

where k is the Coulomb’s constant, the attractive electrical force Fe at the first Bohr radius becomes: F e ¼ k £ ½{ec 4 a0 }2 

ð52Þ

We know that the attractive electrical force at the first Bohr radius (the electron in the ground state) may be defined. That force, as compared to a comparitive gravitational force as defined by Newton, dwarfs the Newtonian force by a magnitude of the order of 2.339. However, we state that the force of gravity at this Bohr radius of g2, is relative to the attractive electrical force of Fe, at least where mass is concerned. Stating that hydrogen, in the Newtonian sense, is earth-like, we find the following three equations, where M1 is the mass of the electron. A ¼ ½{g2 4 F e } 4 {M p 4 M 1 } 4 {X 1 4 2}2 B¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g0 4 g2

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 4 b ¼ {4 £ 2} 4 4 A 2 £ B 2 2 1

ð53Þ ð54Þ ð55Þ

Bohr identified the theoretical ionization energy of hydrogen. Referring to equation (46), we find:   IE ¼ {k £ ec} 4 {2 £ a0 } ð56Þ Referring again to equation (46), we find the kinetic energy, KE of the electron to derive the second Newton’s laws, where w is the angular momentum of the electron as predicted by Bohr.

K 32,7/8

1060

KE ¼ {1 4 2} £ w 2 M 1

ð57Þ

The last equation, in terms of kinetic energy and the elementary charge, then satisfies equation (46). Hydrogen, in the “free state”, has a definable radius of rH. Again, assuming that the speed of light is constant in all gravitational frames of reference, the radius is defined by the product of c 2 v: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4 {c 2 v} ¼ IE 4 {rH 4 a0 } ð58Þ Assuming that hydrogen is earth-like, we find that the force of gravity exerted by the earth at the mean lunar separation distance of R, results in g1 from equation (21), and that force may describe the hydrogen atom in terms of Newton. g1 ¼ M p 4 ½rH 4 {X 1 £ 2}2

ð59Þ

Referring to equation (26) in terms of the lunar body, we find in terms of c 2 v as per the hydrogen body. ½1 4 {c 2 v} £ p ¼ {g 2 4 g1 } 4 IE

ð60Þ

If the speed of light is constant in all gravitational frames of reference, and the angular momentum of the electron of w from equation (45) is correct, then according to Bohr and Sommerfeld, the fine structure constant of a derives:

a¼w4c

ð61Þ

We may then apparently derive an earth-like model of hydrogen from the Bohr model that satisfies both Newton and Einstein, as long as one reduces the mass of the proton by a magnitude of 255, or if one just considers that more than 99 percent of the hydrogen atom mass is missing. The electron has been described in terms of mass, which has resulted the so-called “classical radius” of the electron. Yet, all attempts to discover a form (size) of the electron have failed to this point. In the particle (mass) sense, if one considers that the electron does possess ponderable mass in the particle sense of the term, and a definable radius, it is believed that radius would be on the order of 12 17 m. Then, for the radius of the electron of r1, where the mass of the electron is M1: c ¼ M 1 4 ½r1 4 {X 1 £ 2}2 

ð62Þ

{r* 4 r1 }2 ¼ M p 4 M 1

ð63Þ

where

If one assumes that the proton and electron are ponderable bodies of mass, with Special theory of a definable form, then their stability is a direct result of the force of gravity at relativity their surface radius being exactly equal to the speed of light of c. A gain in mass by either body would increase the body radius, leading to a decrease in g at the new surface being less than c. Therefore, matter (in the form of the electromagnetic waves) could escape the surface since the escape velocity of 1061 c would be greater than the surface gravity of the body. Hence, the electron and proton may provide electromagnetic emissions during a mass gain from a definable radius and mass, where g is less than c. Regardless of the proton and its characteristics, the General Theory renders it highly probable that gravitational forces hold the electrical masses of an electron together. In either case, neither the proton nor the electron could be noted to decay under the decay process; as long as the force of gravity at the surface of the bodies in question remained equal to c. Note 1. In equation (35), one may either assume that the equatorial radius of r does possess a small uncertainty in value, or one may modify r to re, where: r1 ¼ r 2 ½r £ {1 4 c}

ð64Þ

The replacement of r by re in equation (35) then satisfies that equation. Rather than apply the statement as an ad hoc one, equation (64) then reaches an agreement with equation (10), where equation (64) is derived from the upper part of equation (10). Note 2. Equations (1)-(4) are based on the critical identity of v, where the speed v is derived from (or conforms to) the de Broglie hypothesis, hence the Special Theory, and fulfills Maxwell. It remains critical that v be an identifiable, therefore measurable, entity. The addition of beta to the radii involved in equation (11) could, in one sense, appear to be ad hoc. Equations (7)-(11) may become more intelligible, if one considers that the property v derives the angular momentum ve of the earth in its mean orbit about the sun, with ve derived from the Einstein/Lorentz transformation. From Newton/Kepler, the angular momentum of ve (on a mean basis) and the relative Laplace velocity: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ve ¼ ½{2pAU} 4 Y  < {2GM S } 4 AU 4 2 ð65Þ Let us then consider that Lorentz (and hence Fitzgerald) was attempting to prove the motion of the earth through space, with that motion being a proof of the concept of Maxwell. The Einstein/Lorentz transformation, regarding the motion of the earth through space, is then no more than a validation of Maxwell. Then, as per equation (65), relative to equation (7): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 2 1 2 {v 4 c } ¼ {v2e 4 c 2 } 4 2 £ d ð66Þ

K 32,7/8

1062

The use of the quantity d in equation (66) may appear to be arbitrary. However, d is defined in equation (87) to follow, and the entity is a relativistic quantity, resolving gravitation. The results of equation (87) show that the use of d in equation (66) is not required from the relativistic results of Chandrasekhar. The addition of beta in equation (11) is then necessarily a de facto consideration. Equation (66) then apparently defines the quantity c, where the velocity of the earth through the electrodynamic medium of space is determinable, where the quantity v is identifiable and/or determinable. The allowance of the Einstein/Lorentz transformation to achieve Maxwell, allows equation (66) to become an expression of Maxwell, where Maxwell describes the field potential in terms of the electrodynamics. Note 3. The description of a “singularity”, so abhorrent to Einstein and popular with current cosmology is totally dependant on the field equations of the General Theory deriving Newton in the form of the escape velocity derivation (noting that Laplace derived the escape velocity equation from Newton prior to Schwartzschild). Schwartzschild derived Laplace (Newton) from the General Theory field equations, and thus apparently provided a proof of the validity of the General Theory. The derived Schwartzschild radius, applicable to equations (42) and (43), is in conflict with the attached theory. Let us consider that the theory of Newton creates the “missing mass” problem as currently understood and stated within the current literature of cosmology. Let us consider that the theory of Newton is indeed correct, within the limited narrow scope of its focus. Let us further consider that the General Theory does indeed derive Newton (in the weak field sense) and that the General Theory provides a minor modification to the theory of Newton, allowing Newton to achieve reality in a narrow focus. We therefore consider that the theories of both Newton and Einstein are, an error evidently requiring modification, relevant to the “missing mass”. In the General Theory, the entity Omega is required to result 1 (unity), in order to allow for the “closure” of the Universe. If Einstein’s conceptualization of Omega is correct, and if Omega is indeed 1, then current cosmology is apparently complete as a preliminary description of the Universe. Within the General Theory, only one “special density” is noted, relevant to the Universe as proposed by Einstein. This special (critical) density arrives at an Omega of 0.02 rather than exactly 1 (unity). The result is that 98 percent of the mass of the Universe is missing, if closure is indeed to theoretically result, and as Omega achieves unity. Based on Einstein’s radius of RE, the Schwartzschild radius of rs, and the missing mass quantity of X1 as derived herein; then the aforementioned problem concerning Omega (unity) is addressed by the following: 1 ¼ ½{RE 4 r s } 4 X 21  4 2 where

ð67Þ

X1 £ 2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 4 RE

ð68Þ Special theory of

relativity

Deriving: 1 ¼ ½{c 4 RE } 4 {RE 4 rs } 4 2

ð69Þ

The aforementioned does not verify the Universe. Yet, if Omega is indeed exactly 1, then a flat Universe results. Of the consequences of Guth’s inflationary Universe, the prediction that the Universe is indeed flat is the most striking. Flat space then corresponds to the prediction that Omega, the ratio of the critical density of the Universe to its actual density, must equal 1. The General Theory predicts a flat Universe, but is unable to derive an Omega of 1. Equations (67)-(69) then apparently allow for a General Theory Omega of 1 (therefore predicting a flat Universe in line with observation), if the method of Schwartzschild is modified to account for the “missing mass” or cold dark matter. Let us consider that the galaxy is observable in its totality. That space then exists outside the galaxy one could then state as being the “Universe”. However, the totality of the Universe itself is not observable. Rather, we may only view small portions of that Universe. Being able to observe the galaxy, in all its aspects, allows one to determine the characteristics of that galaxy. In terms of mass, we may determine that the mass of the galaxy does not, and may not, conform to the sense of mass as described by Newton. Rather, to “close” the galaxy, one must derive a mass many times greater than the mass proposed by Newton. Within this galaxy, and others, we may observe that 20 percent of the Newtonian mass of the system may be found in the form of dust and gas interspersed within the system. This 20 percent is in addition to the predicted Newtonian mass of the system. Therefore, the mass of closure for the galaxy of Mx is predicted to be: M x ¼ {X 1 £ 2} £ {1 þ 0:2} £ 1

ð70Þ

Therefore, the mass of the galaxy, as predicted by Newton, would be required to be a magnitude of Mx greater than the Newtonian mass prediction. From equation (70), the magnitude increase would equate to a magnitude of 612, the mass predicted by Newton. From observation, this is the mass increase required to close the galaxy, considering the massive halo of matter that does encompass the galaxy. Note 4. Equations (11) and (23) do not follow Newton’s standard form, as we are assuming that Newton’s form is essentially correct if no other gravitational forces are affecting the earth. However, the application of equations (11) and (23) proportionally applied to the sun then derives distances greater than those

1063

K 32,7/8

1064

allowed by Newton, specifically in the derived distances for the AU and the solar radius. If we assume that no other gravitational forces are affecting the sun, then Newton is essentially correct. Assuming that the masses predicted are correct, we find the following, where a comparable Einstein radius for the sun of RE2 Sol is derived. c ¼ {M S 4 R2E2Sol } £ G

ð71Þ

The solar radius of rSol, conforming to observation, then becomes: rSol ¼ ½r S 2 {42 £ RE2Sol } þ ½2 £ RE2Sol 

ð72Þ

The mean separation distance between the earth and the sun, conforming to observation, then becomes AU2N, as per Newton. {4p 2 } 4 {G £ M S } ¼ Y 2 4 AU 2 N 3

ð73Þ

As per Newton, the value of gravity at rSol becomes gSol. g Sol ¼ ½M S 4 r2Sol  £ G

ð74Þ

The solar radius and the mean separation radius between the earth and the sun derived in the aforementioned are in agreement with the currently accepted values. We find agreement by: i h h i pffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ {g Sol 4 r Sol } £ { r0 2 b} £ 2 4 {1 4 b} þ 1 ð75Þ At no point in the paper, a mass has been determined that could be found within the radius, we have termed Einstein’s radius of RE. Yet, if Newton is to be essentially correct, that mass must closely approximate the Newtonian mass of the earth, in order to preserve the necessity of Newton. Where the mass within Einstein’s radius is MRE, we find: 1 X 1 ¼ M RE £ ½{RE 4 c 2 } 4 c ¼ ½M RE £ {ðrs 4 c 2 Þ 4 RE } 2

ð76Þ

The paper derives a mass of the earth of 5.96724 kg, while the mass derived for the interior of the Einstein’s radius becomes 5.96224 kg, per equation (76). Therefore, Newton was correct in assuming that the mass of the earth could be considered to be concentrated about the center of the earth, and where that concentration of mass describes a radius. Einstein is then apparently correct in assuming that the maximum force of gravity is the speed of light of c. Neither theory is correct in predicting the mass of large-scale structures (galaxies, clusters of galaxies, etc.), unless the missing mass factor of X1 is applied.

This application apparently completes the theories of both Newton and Special theory of Einstein as general descriptions of gravitation. relativity Let us consider a stellar object (a white dwarf) with a mass at the Chandrasekhar limit of 1.44 solar masses. The stellar object becomes a supernova, leaving behind a pulsating neutron star. The star, prior to becoming a supernova, is found with a radius of r+ and, after the explosion, a radius of 1065 r2 , roughly where rþ ¼ 3; 500 km and r2 ¼ 10 km: The theory behind the above is that the stellar object finds that the interior of the star reaches nuclear density, and the resulting bounce causes the supernova, since electron degeneracy can support a white dwarf mass of not more than the Chandrasekhar limit of Climit. For sun, we assume that the radius RE2 Sol finds a nuclear density contained within this radius. Then, pffiffiffi C limit ¼ 3 3 ð77Þ However, the original Chandrasekhar limit has been modified (by some) to the following. pffiffiffiffiffiffiffiffiffiffi W limit ¼ C limit ð78Þ We determine the Schwartzschild radius Rs of the neutron star, after the supernova affects the white dwarf. Rs ¼ C limit £ 3 £ 1; 000

ð79Þ

The radius r2 of the neutron star, from the following three equations. H ¼ ½{M S £ C limit } 4 Rs  4 {c 4 G}

ð80Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 4 C limit

ð81Þ

r2 ¼ I 4 W limit

ð82Þ

I¼

Based on the derivations of RE2 Sol and rSol, we find the radius of r+. r2 ¼ 3 £ ½{r þ £ RE2Sol } 4 rSol  The Chandrasekhar limit, based on the radii r2 and r+. hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i C limit ¼ 4 r þ þRs 4 r2 4 3

ð83Þ

ð84Þ

where C limit £ 3 £ 1;000 ¼ ½RE2Sol 4 r22 The Chandrasekhar mass MCh of the stellar object.

ð85Þ

K 32,7/8

1066

M Ch ¼ M S £ C limit

ð86Þ

Since Chandrasekhar derived his limit from the tenants of the Special Theory, and considering that the Einstein radius predicted for the sun is derived under those same tenants; then apparently, one may then derive these known parameters by assuming that our sun does possess a nuclear density body within its interior. If the neutron star does result this radius of r2 after the supernova, then the radius must be in excess of r2, if the star is to irradiate. The comparable solar radius derived as Einstein’s radius for the sun, would not radiate as the value of gravity at this radius would be equal to c. Therefore, the actual radius of the neutron star would be required to be in excess of r2, as the surface gravity at the stars surface would be less than c. In equation (66), we found that the entity d was required to satisfy equation (66). From equation (30), we originally defined a solar radius of rS, but the assumption had to be made that no other gravitational forces were affecting the sun in order to derive the radius. From equation (72), we defined a modified solar radius of rSol, where this radius is derived from Chandrasekhar and the resulting radius agrees with observation. We define the entity d by equation (87).

d ¼ r S 4 r Sol

ð87Þ

In equation (65), the mean orbital velocity of the earth about the sun must then be modified, by subtracting the product of rS from RE2 Sol from the AU in equation (65). The resulting velocity then satisfies equation (66) without the requirement of d. Note 5. We may ionize the hydrogen atom by introducing 13.6 eV of energy into that atom (where the energy as denoted herein is labeled as IE). By the introduction of IE into the hydrogen atom, the electron is stripped from the atom, and the field surrounding the hydrogen proton apparently ceases to exist. Under the theory of Bohr, the kinetic energy of the electron at the first Bohr orbital within hydrogen equates to the energy IE. The energy IE, constructed in equations (46) and (54), is then considered to remain a quantized state of energy. According to the Special Theory, if a body of mass emits an energy E, the mass of the body will decrease (relatively) by that E divided by c squared. We consider that Planck’s constant of h represents the quantized energy contained within the elementary quanta of light. We find IE £ 2 ¼ h 4 ½{IE 4 c 2 } £ ec

ð88Þ

h ¼ 2 £ IE2 £ {1 4 c 2 } £ ec

ð89Þ

Deriving

Considering that light has been quantized from the form of Planck on the order Special theory of of Planck’s constant of h, we assume from equations (88) and (89) that the relativity electron itself has been quantized, in terms of the kinetic energy of the electron under the terms of Bohr. Equation (89) then apparently results that the quantization of light and of the same of the electron results in one description of the hydrogen atom. 1067 This description is necessarily limited, in that it does not describe the field comprising the hydrogen atom, and that the electron and light both coexist and interact within this field. The field description of the hydrogen atom is then assumed to be the result of the existence of the proton, embedded within the hydrogen atom itself. If the proton representing the base construct of the hydrogen atom fuse with other protons to form elements of larger mass (to within the observational limits), while the characteristics of the elementary quanta and of the electron remain constant, then equation (88) is a description of the atomic spectrum itself. Here, we consider that the spectrum may be quantized (without a corresponding quantization of the field involved), in that this spectrum may be quantized in terms of light and electron interaction. A description of the field encompassing the hydrogen atom is then a field theory that describes the proton. The results of equation (89) do not describe the field, although the relationship in equation (89) is obviously a result of such a field being in existence. A field description of the hydrogen atom is therefore apparently in order, and that the description must resolve the relationship as identified in equation (89). At this point of time, there appears to be no such field theory available. Note 6. Equation (41) gives the orbital velocity of a body at a distance b directly above the equatorial radius of the earth. Equation (41) is no more than a derivation of the escape velocity equation of Laplace. Assuming that the earth may be treated as a rigid body and that any radius from the equator to the earth’s center would rotate exactly as per the time T defining the mean period of the earth’s rotation about its axis as measured at the earth’s equator, we find that equation (41) will not hold in defining the velocity of a body located at RE. Where this rotational velocity is rv, we find; rv ¼ ½2pRE } 4 T

ð90Þ

We find the rotational velocity of a body at RE to be 0.084 mps. However, we know from observation that the inner core of the earth rotates at a rate roughly 0.63 s less than that of the rotational period of T. We describe an initial period difference of 0.095 s as T 0 . In the following three equations, we define the quantity Z, which is essentially the displacement of c 2 v; and essentially the same as the Maxwell displacement of D. Then:

K 32,7/8

1068

K ¼ 2pRE

ð91Þ

K 0 2 T ¼ ½p 2 4 2 £ ½c 4 {rv £ c}

ð92Þ

K 4 K 0 ¼ ½{Z 4 c} £ 42  4 c

ð93Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T þ T 0 ¼ 3 {c 2 v} 4 Z £ T

ð94Þ

We find essentially the same result for T 0 by using the Maxwell displacement of D in the place of the displacement c 2 v: pffiffiffiffiffiffiffiffiffiffiffiffiffi 12 T þ T0 ¼ D 4 Z £ T ð95Þ The period T 0 , when multiplied by 2p, resolves the 0.63 s in rotational difference as noted by the observation from the comparison of rotational periods taken at the earth surface and at the solid inner core. From equation (41), we define a pseudo rotational velocity v*, which may not exist in reality. The velocity v*, like the rotational velocity rv, is defined at the radius RE. pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð96Þ v* £ 2 ¼ ½2GM E  4 RE We then conclude that equation (41), and the Laplace/Schwartzschild escape velocity derivation of equation (42), will not hold as one nears the center of the earth. The reality rotational velocity of rv is then not the pseudo rotational velocity of v*, because of the mass involved describing the radius RE. This mass is then not the Newtonian mass of the earth, but the mass concerned is a nuclear density mass. Where the missing mass factor of X1 was derived herein, we find, X 1 £ 2 ¼ c 4 v*

ð97Þ

X 1 £ 2 ¼ v* 4 RE

ð98Þ

c ¼ ½X 1 £ 22 £ RE

ð99Þ

where

One may then determine why the velocity of a body increases as the body comes closer to the earth from space, reaching a maximum velocity at a radius b (measured directly above the earth’s equatorial radius); and then dramatically decreases orbital velocity as per the rotation of the earth. In equation (41), we determined this maximum orbital velocity of Wmax. Then, where r is the radius of the earth’s equator:

rþb¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v* 4 W max £ RE

ð100Þ Special theory of

Again, from equation (9), the quantity b defines the Maxwell displacement of c 2 v: From the above, we report that the use of the Laplace escape velocity equation being used to determine a Schwartzschild radius relative to the interior of the earth is then apparently physically erroneous, based on the physical evidence in hand. However, Laplace did provide us with what he termed as the rule of determinism. The advent of quantum physics, and its subsequent postulate resolving the uncertainty principle, has apparently disallowed the determinism of Laplace to be realized as a legitimate physical process. This paper suggests that the determinism of Laplace does, in fact, hold real meaning as a valid physical process. Note 7. Regardless of the intent or result of this paper, one counter-argument may be presented that will not allow this paper or its results. The following addresses the obvious counter-argument. The basis of this paper is that a displacement of coordinates, in linear measure, occurs when the elementary quanta (in terms of Planck) are viewed as forming or inscribing a conic arc segment (i.e. the light path in terms of the General Theory) within any gravitational field viewed. The linear measure of the speed of light within the viewed frame has been termed v, while the curvilinear light path has been resolved as the constancy of the term c. Here, c is viewed as constancy regardless of the degree of curvature of the light path resolving the gravitational potential of the frame viewed. The proposed solution towards the questions arising from the observation is delivered primarily as an algebraic solution herein, related to the following historical antecedents. Omar Khayim solved the general conic to the third order circa 1070 A.D. The solution he proposed was algebraic. Roughly 500 years later (i.e. the 16th century), Tartaglia, and others of his ilk, solved the general conic as a geometric solution, to the fourth order. Tartaglia, re Khayim, therefore constructed as what has been historically viewed as the unification of geometry and algebra. This perceived unification apparently allowed the genius of Liebnitz/Newton to mechanically bear fruit. Newton’s solution of gravitation remains geometric, relative to the General Theory that derives gravitation as pure geometry as per the condition of Riemann. Yet, if the Special Theory may derive a description of gravitation, and that description derives Newton, then one must consider that an algebraic solution tendered towards gravitation is necessary, and even tenable, based on the historical precedence. We state such considering that the Special Theory is no more than a simple statement of Pythagoras, relevant to his undeniable logic relative to the Hindu/Arabic derivation of the algebra. We have tendered the values related to c and v, and these values represent a small discrepancy over the distance concerned. Therefore, one is describing,

relativity

1069

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1070

in the rough sense, a flat plane rather than a noticeably different curvilinear plane. The displacement in linear coordinate positioning has been considered as a result of the entirety of the gravitational potential involved. Obviously, a problem arises, in the form of a counter-argument, relative to the position of this paper. A lone counter-argument is enough to destroy the conjecture of the displacement as defined by c 2 v herein. The enclosed paper has deduced that the displacement c 2 v resolves the force of gravity of g 0 upon the surface of the earth. However, the total gravitational force is not deemed as solely dependant on g 0 , but also in that the gravitational force concerns the angular momentum of the body in question in conjunction with g 0 . The angular momentum of the body (and the force of gravity related to the body) is then construed as being a description of the Newton’s laws of relative motion. This notation, to this point, has apparently ignored angular momentum as reference to the displacement of c 2 v: The enclosed paper has theorized that a body exerts a force of gravity of g that is reduced from some maximum level (per Newton) to a lesser level (per measurement) via the result of an opposing body of mass exerting a counter-force towards the body concerned. Yet, we apparently continue to ignore the angular momentum of the bodies concerned. The displacement c 2 v is theorized as describing the gravitation on the surface of the earth, while in the very near presence of a body of great mass. Immediately, one must realize that the theory requires the displacement of c 2 v to be consistent throughout the solar system itself. Yet, the theory requires that the light path be bent from a flat plane to a curvilinear plane due to the nearby presence of a great body of mass, in the Newtonian sense of mass. Therefore, as one approaches the sun from the outer-most planets, the property c 2 v should inherently change rather abruptly, particularly as one arrives near the surface of the body observed. This apparent requirement does not occur, as per the myriad observations taken during the later part of the 20th century. The galaxy exists, and we are a part of that galaxy. Obviously, the galaxy exerts a huge gravitational potential, as least as juxtaposed towards the same relative potential exerted by our sun within the space of the galaxy viewed on the minute level concerned. Since the total gravitational force as exerted by the galaxy must relatively dwarf the same potential as exerted by the solar body, it is apparently not possible that the solar body exert a potential towards the galaxy, whereby the two potentials basically cancel each other out (i.e. c 2 v resulting in a value of zero, apparently describing a pure, flat plane). We then propose a series of values relative to the term Omega y, with reference to the angular momentum rather than simply psi (especially in view of the Schrodinger notation of the term). For the solar system, we find Omega to

be cs. For the galaxy, we find Omega to be cg. For the space outside the galaxy, Special theory of or the Universe, we find Omega to be cu. Relative to the solar system, we derive relativity Omega as; Cs ¼ 1 þ ½{c 2 v} 4 c

ð101Þ

1071 The value achieved describes a flat plane to eight decimal places. If the galaxy does exert a huge gravitational potential relative to the distances involved, then v in the last equation becomes far less in the confines of the galaxy as compared to the relative confines of the solar system. We then assume that the similar values may be derived for the galaxy and for the space outside the galaxy, where we consider this space to resolve the Universe. The numerical values are as follows, and we resolve those values in the following equations: Cg ¼ 1:1746; Cu ¼ 1:2563: Since we assume that the intensity of light falls per the square of the distance involved, and since the time involved is based on the observation and nature of light, we assume that an observed distance taken towards an object within the reaches of the galaxy be reduced from the recorded distance by the factor of cg. Therefore, if the galaxy were observed to resolve an apparent diameter of 100,000 light years, then that diameter would be reduced by cg. Conversely, the best measurements of the Hubble constant of Ho, taken under two differing methods, will not agree beyond 25.63 percent, according to cu as developed by the theory. We find that an upper limit of Ho (per one set of observations) equates to 74 Mpc/km; while the lower limit records 59 Mpc/km, per the opposing set of observations. The observations and methods deriving the two limits are apparently indisputable, and the proportional difference between the two resolve cu is disclosed. The aforementioned does not allow for the formulation of this paper, nor its proposed results. An object of immense mass must lie in near proximity to the solar system in order to allow for the properties as described by Omega. We then predict the existence of such a huge mass in near proximity to the solar body. The object we name for the first initials of the persons Jeinay Marie Gibson and Jessica Leigh Gibson, where this object of immense mass is termed J2. From observation, J2 appears to exist. Via observation, a huge halo of matter exists about the galaxy, engulfing the galaxy itself. Other galaxies show a similar halo, and the halo observed apparently does verify an apparent concentration of mass. This halo signifies a gravitational field in existence, where this field/mass may only be described by Newton’s laws, however inadequately. The observations of the halo in question have been verified by the technology supplied/applied by the Hubble Space Telescope (HST).

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Within 17 light years of the solar system, our sun finds 31 other companion stars. Many observations confirm that this local group of stellar objects does move in tandem through the space of the galaxy. This movement is not predictable and is not describable unless the group is gravitationally bound together. HST has observationally confirmed that this local group is enshrouded by an immense halo of matter, where the halo extends itself to a diameter of roughly 300 light years. The local group, including our solar system, lies at the periphery of the halo; at a radius of 128 light years from the center of this halo.We further assume that this halo of matter signifies a concentrated center of mass, under the terms of Newton. We have termed this system, which may only be described as a gravitational system under the terms of Newton, as J2. Since the solar system is engulfed by the observable halo, we therefore conclude that our solar system gravitationally reacts in a defined manner within this system termed J2. Although we may record the various values of the force of gravity of g rather accurately in our local space, we may not perform such direct observations of g of any object/point at far reach from our planetary position. Yet, we may measure the angular momentum of any body within the galaxy, directly, as long as the body is observable within our favored frame. These observations of angular momentum are then recorded as follows, concerning the body in question. We provide terms of A through F, where B through F are the velocities of the objects concerned through the space viewed or described. A is defined as a constant. We then concern ourselves with the observable motions of the solar body itself through the space viewed. We must require that the solar system is not orbiting the center of the galaxy. Rather, we require that the system J2 is orbiting the galaxy, and as our solar system is herein considered as orbiting J2, we achieve the appearance that our solar system is orbiting in the center of the galaxy. From observation, we record that the solar body resolves two major motions through the space of the galaxy, and one minor motion. The two major motions are recorded as D and E; we record E as the apparent velocity of the solar system about the center of the galaxy. The motion D is in opposition to the motion E, and is generally in a direction away from the center of the galaxy. The minor motion is ignored at this point, but we note that this motion is in opposition to the major motion D. Hence, one may consider the minor motion (340 mps) of the solar body to be cancelled out by the major solar motion D. The motion labeled B is the velocity of the lunar body about the earth. The motion C is the velocity of the earth about the sun. Lastly, the motion F is the velocity of the galaxy through the space of what may only be considered as the Universe. B to F are verifiable and observational mean velocities. We then determine the constant A.

A¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 0 4 ½1 4 {c 2 v}

ð102Þ Special theory of

relativity

With A as a theoretical construct labeled as a constant, we then list the observational values of B through F in conjunction with A as follows: A ¼ 2:70148; B ¼ 1023:22 mps; C ¼ 29; 870 mps; D ¼ 19; 160 mps; E ¼ 219; 000 mps; F ¼ 601; 200 mps: We then proscribe that the solar system is orbiting the system J2 in a counter-clockwise direction. J2 is then proscribed as orbiting the center of the galaxy in a clockwise direction. We find the solar system itself to be found on a line drawn between the center of the galaxy and the center of the halo of matter as defined by the observations of HST, where this center is defined as J2 and where we note that the solar system is in actuality slightly offset from the line we have arbitrarily constructed. The direction of rotation concerning the motion D is important, as D must be applied with opposing directions in mind during the following three equations. In these equations, we are not concerned with either the distances or masses involved, but we are only concerned with the proportionalities of the angular momentums involved. We derive that the constant A is a constant relative to the force of gravity g that is applicable towards the observed motions involved. We derive that this is possible solely due to the constancy of the speed of light represented by c in all gravitational frames of reference, and that the displacement of c 2 v may be proportionally applied to any relative frame if the motions of the bodies involved (relative to one another and relative to one preferred frame) are resolved via observation. Then, we find, pffiffiffiffiffiffiffiffiffiffiffiffiffi A¼ C4B42 ð103Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi A ¼ E 4 ½C þ D

ð104Þ

A ¼ ½F 2 {D 4 2} 4 E

ð105Þ

In the sense of mass as defined by Newton (while ignoring the distances or radii involved), then we derive: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1{c 2 v} £ A 2 ¼ ½M sun 4 M earth  4 ½M galaxy 4 MJ 2 c2v

ð106Þ

Concerning the bodies in question, we then predict: LUNA: Radius from earth ¼ 384; 400; 000 m; Orbital velocity ¼ 1023:22 mps; Orbital period ¼ 27:396 days, Mass ¼ 7:3322 kg: Earth: Radius from sol ¼ 1:49611 m; Orbital velocity ¼ 29; 870 mps; Orbital period ¼ 365:242 days, Mass ¼ 5:9724 kg: Sol: Radius from J2 ¼ 128 light years, Orbital velocity ¼ 19; 160 mps;

1073

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1074

Orbital period ¼ 12:5 million years, Mass ¼ 1:9930 kg: J2: Radius from the galaxy center ¼ 8; 551 parsecs, Orbital velocity ¼ 219; 000 mps; Orbital period ¼ 240 million years, Mass ¼ 6:836 kg: Milky way galaxy: Orbital velocity ¼ 601; 200 mps; Mass ¼ 241 kg: Note that the predicted mass of the bodies concerned is the Newtonian mass, and that the missing mass factor of X1 still must be applied to the above masses in order to close the galaxy in line with observation. We then resolve that the body J2 exerts a force of gravity from its center location in the space of the galaxy to a radius towards the center of the galaxy that may be defined as roughly 150 light years, resolving a 300 light year diameter of the halo of matter as observed by HST. The force of gravity exerted by J2 at this point is roughly equal to (hence negating) the force of gravity as exerted by the center of mass located at the center of the galaxy and defined as a radius roughly 8; 500 pc from the center of the galaxy. The resulting gravitational potential at the point described is then viewed as nearly equal to zero. The linear speed of light is then forced to nearly equal the curvilinear speed of light, and the Special Theory results as per this one special frame of reference. We then deduce the displacement of light at this point and resolve the product c 2 v as disclosed herein. Then, at the radii, the solar system holds from the center of J2 and jointly from the center of the galaxy; we find rough values of the force of gravity of g relative to the solar system position in space. gJ2 ¼ ½M J2 4 {128 £ 9:4615 }2  £ G

ð107Þ

gJa ¼ 3:094210 m=s2 : For the galaxy and its force at the distance concerned; g galaxy ¼ ½M galaxy 4 {8; 512 £ 3:26 £ 9:4615 }2  £ G

ð108Þ

ggalaxy ¼ 1:936210 m=s2 : Then, in the consideration of unity 1 ¼ d½ gJ2 4 ggalaxy  4 ½1 4 {c 2 v} e 4 4 2

ð109Þ

Cu ¼ dA 4 ½ gJ2 4 ggalaxy  4 ½1 4 {c 2 v}e

ð110Þ

pffiffiffiffiffiffi Cg ¼ d½{gJ2 4 ggalaxy } 4 Cu  4 Cg e

ð111Þ

Then, and

Observing the galaxy, we find the solar system to lie on a limb of the galaxy at roughly two-thirds of the proportional distance of the radius of the galaxy from the center of the galaxy. Given that we describe the distance from the center of the galaxy to the solar system as being measured as 8,512 pc, then the radius of the galaxy would equate to a distance of 8,512 pc divided by two-thirds or

41,600 light years. Multiplied by 2, and with the product increased by Cg, then Special theory of the diameter of the galaxy results as 97,800 light years. relativity Viewing the galaxy, based on the intensity of light falling at the square of the distance involved, we then resolve the diameter of the galaxy at 98,000 light years. According to the theory, this is an apparent (curvilinear) distance, and the linear (actual) distance is lessened by the factor Cg. 1075 In the manner expressed under the terms of this paper, we rely on the postulate of the Special Theory in that the speed of light is constant within all gravitational frames of reference. Based on the success of the quantization effort of the current physics, we then assume the Planck’s constant h and the elementary charge on the electron of ec are both constant in all frames of the references. Therefore, the energy of any particle is assumed to be relative towards all frames of the reference. However, as per equation (4), we have defined relativistic energy relative not to the curvilinear speed of light of c, but to the linear measure of the speed of light of v. Under the terms of Planck, where these terms were made relative to energy within Einstein’s production of the photoelectric effect, we find the energy of the elementary quanta (expressed in electron volts), to conform to the following. In the following, the entity Rg is held as an expression (derivative) of the molar gas constant of R. Here, Rg is forced within the equation, and results in a value slightly less than as published by CODATA, under SI terms. In the following, the entity Na is an expression of Avogadro’s number or Loschmidt’s number as one prefers. h 4 ec ¼ ½Rg £ c 4 Na

ð112Þ

We are now assuming that the elementary charge on the electron, Planck’s constant, the speed of light, and the mass of the electron are all theoretically constant within all frames (gravitational) of reference. In equation (4), we state that energy is relatively defined not by c, but by v. Here, v is relative to the one preferred, special, or favored frame of reference. The description of that frame, in one particular sense relative to the particle nature of light, has been relatively resolved by the reference entity cu. Then, we determine cu, from equation (110), relative to the mass of the electron. C2u ¼ dec2 4 ½h £ ce 4 dM 1 £ v 2 e

ð113Þ

In accordance with the energy relative to the elementary quanta of equation (112) and the electron, we find: ½M 1 £ v 2  4 ec ¼ Na 4 ½Rg £ C2u £ c 2 

ð114Þ

Within the aforementioned, we have attempted to resolve the relative relationship between energy as found in the special frame observed, and as

K 32,7/8

compared to the preferred (Universal) frame. However, the galaxy itself (in the gravitational sense) lies juxtaposed between the special frame and the preferred frame. Our attempt to describe this frame as exhibited by the galaxy then follows as the entity cg. Then:

1076

C2u ¼ ½Cg 4 c 4 ½{c 2 v} 4 c

ð115Þ

We therefore assume that gravitation (in the Newtonian sense and not the Riemannian sense of gravitation as pure geometry) and energy are then directly made relative as per the aforementioned. For the definition of Omega to hold, in the aforementioned, cs must become a description of Omega within the spherical conic encompassing the system J2, where that is considered as a celestial object initially described/observed by Messier. Then, cs describes Omega not only within the confines of the solar system, but also roughly within the confines (boundary) of the entity of the system we have described as J2. In equation (102), the arbitrary constant of A is derived via the coordinate displacement termed c 2 v: This displacement is arbitrarily termed as resolving the force of gravitation, with that force including the vectors of the forces existing in the frame of which we term the earth/sun frame. D, E, and F are the velocities of the bodies and g 0 is the force of gravity upon the surface of the earth in conjunction with gSol representing that same force upon the surface of the solar body; we then find; gSol 4 g0 ¼ d½D 4 M sun  4 ½E 4 MJ 2e 4 d½E 4 MJ 2 4 ½F 4 M galaxy e ð116Þ In order to further clarify J2 from observation, we consider the halo observed by HST as to identify a globular cluster, of the type as identified by Messier. According to the theory, all the objects within this cluster are orbiting J2 as per Newton’s laws. These objects, observed from the earth, would appear to be either blue-shifted or red-shifted. Roughly 2,000 of these stellar objects have been catalogued relative to the light shift. Objects within this system may then be predicted to occupy a position in relative space as per the orbital position they hold relative to the solar spatial position in real time. In order to account for the observed light shifts, in conjunction with the observed motions of the solar system through space, we submit the following. The minor observed motion of the solar system through space, equivalent to 340 mps, is in apparent opposition to the major motion of 19 km/s. The stellar object Centauri B appears to be blue-shifted, from the earth bound observations. Extrapolating, Centauri B will find its closest approach to Sol roughly 25,000 years from the present, at a separation of 2.2 light years.

We then assume that the two stellar objects of Centauri A and B, with a Special theory of combined mass of 1.44 stellar masses, form a trinary orbital system in relativity conjunction with our solar system. Centauri B orbits Centauri A once every 500,000 years. We further assume that our solar system orbits the combined center of mass developed by the Centauri system, with the direction of rotation of the sun about the Centauri system being in a counter-clockwise direction. 1077 We then predict that the combined masses of Centauri A (0.9 solar masses) and Centauri B (0.5 solar masses), create a combined and concentrated center of mass about which the solar system may be viewed as orbiting. The mean radial separation distance is viewed as 2.2 light years, with the mean orbital velocity held as 340 mps. Then, Centauri system: Radius from sol to Centauri mean system ¼ 2:2 light years, Orbital velocity ¼ 340 mps; Orbital period ¼ 118; 000 years, Mass ¼ 1:44 solar masses (Chandrasekhar limit). The mean separation distance between the sol and Centauri A would be roughly constant as per the above, with the radius being approximately 4.1 light years. During the 118,000 year orbital period, Centauri B would change position in space, relative to its distance from sol. If all the three stellar objects are gravitationally bound together, the force exerted by the Centauri system towards the solar system and the planets involved would decrease and increase. The elliptical orbits of the planets, including the earth orbit, would become more and less elliptical over time. The perturbations of these orbits from that as predicted by Kepler/Newton would be as predictable as they are measurable. We conclude that the motions of the solar system through space have real physical meaning, as per Newton’s laws.

2. Conclusion We conclude that the speed of light, designated as c, is constant in all the gravitational frames of the reference. All observers will measure c as the metric of 1 within their individual frames of reference. This conclusion requires that the base (metric) unit of time (the second) be the metric of 1 in all frames of reference. The base unit of length (the meter) is then required to be the metric of 1 in all frames of the reference. The time t for light to travel exactly 1 m is then 1/c. We conclude that light, when viewed as a particle, forms a curvilinear light path as the light travels through vacuum. The resulting linear value, or the linear speed of light, is designated as v. In any frame of reference, v will remain less than c and may reach 0, but remain not less than 0. A measurement of the linear speed of light in any gravitational reference frame, by any means, will resolve v, not c. The speed of light of c must be inferred from v, based on the gravitational potential involved within the frame viewed. Only in the absence of the gravitation c will equate v.

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From this conclusion, two bodies of exactly the same mass will exert exactly the same potential at the square root of the distance between the two bodies. As both potentials are attractive, the net force of the two exerts at this distance is exactly 0. Therefore, the speed of light of c equates to v in this theoretical instance. Under these conditions, we conclude that the Special Theory of Relativity results reality. These conditions then resolve Einstein’s special case, resulting in the principle of relativity. We conclude that there may not be any two reference frames (when viewed in reality) that exhibit exactly the same gravitational potential. Therefore, v will vary in all frames of reference, in the real observable sense. As the curvilinear light path that constantly defines c is viewed as inscribing a parabolic arc segment within the frame of reference, then a linear determination of v within these two frames will find this linear comparison as being a non-parallel nature. This comparison of linear coordinate positioning between the two frames of the reference that exhibit differing gravitational potentials leads to the condition of Riemann. The Special Theory of Relativity requires uniform parallel translational motion, and as the last conclusion eliminates this possibility, then one may not describe gravitation, on a translational basis, under the conditions of the Special Theory, if the condition of Riemann is met. We conclude that the displacement of light, in linear terms under the conditions of Maxwell, results in a linear displacement equal to c 2 v: The determination of this displacement in one special frame of reference is then able to resolve gravitation within this frame of reference, once the displacement is applied to the Special Theory and consequently, to the gravitational theory as defined by Newton. We conclude that the theory of Newton resolves measurement in this one special frame, and describes a local aberration, not a Universal solution. We conclude that c is a constant and that v is a local aberration. We conclude that Newton defines this local aberration. We conclude that Newton proportionally defines mass within this local frame of reference, and that Newton does not either define, or determine, the concept of nuclear mass existing within the Universe. This nuclear consideration of the term mass, derived from the observable densities of the nuclei of the elements, derives a missing mass quantity of approximately 255 times that of the mass as identified by Newton. Applying this nuclear missing mass quantity (dark matter) to the form of Newton resolves gravitation on the galactic and larger scales, as well as resolving the characteristics of the base unit of nuclear matter, i.e. the proton. We conclude that the General Theory of Relativity is not Newtonian in the sense of defining gravitation, and that the General Theory is therefore much more than suspect as a complete theory, as the General Theory does not take into consideration the missing mass (as originally ignored by Newton). Therefore, the General Theory may describe gravitation in a very weak field,

and will only describe a local aberration, as does Newton. The General Theory Special theory of must then be modified severely, or else abandoned, as a description of relativity large-scale gravitational fields. The inability of the General Theory to account for, or to take into consideration, the Newtonian missing mass, then leads to the determination of the Schwartzschild, i.e. the Schwartzschild radius. This radius describes a 1079 mass/density that may not exist under the terms of Newton. However, the Special Theory does resolve that nuclear density may be achieved within the Universe, and such an achievement may be observed under the form of neutron stars. Chandrasekhar derived the prediction of these bodies being in evidence from the tenants of the Special Theory of Relativity. We therefore conclude that the Special Theory of Relativity is an appropriate theory of gravitation, within local limits being applied relatively to other proportional masses. The field equations of the General Theory do, in fact, result the Schwartzschild radius. The metric of Schwartzschild does resolve the Laplace equation in terms of escape velocity. The Laplace equation is a direct derivation of the form of Newton, and resolves the escape velocity at the surface of the earth (11,174 mps). We conclude (based on prediction and observation) that any particle may not exceed, or reach, the speed of light of c; if in fact that particle is viewed as being constructed of mass. We conclude that Newtonian gravitation is a resolution of mass being accelerated. We therefore surmise that the maximum acceleration of any theoretical body of mass, construed in wave-like terms, then becomes c. The maximum acceleration of the force of gravity, then becomes c. Under these considerations, the Laplace equation is considered as exact at the earth surface, but fails as one approaches the center of the earth, if the maximum acceleration of the force of gravity does indeed equate to c. For the earth, a determination of the force of gravity (under the terms of Newton) at the Schwartzschild radius results in an acceleration 1.6910 times that of c. Laplace resolved the same value of the “speed of gravity” as he resolved the escape velocity equation. However, since the Schwartzschild radius is derived from the field equations of the General Theory, and as this theory does not account for the missing mass, then the resulting equation and radius are erroneous. Under these same terms, the equation (and indeed the subsequent results) of Laplace is not applicable within the interior of a body, unless one modifies Newton’s form to account for this missing mass, thereby negating Laplace within close proximity to the earth center. Schwartzschild (and the General Theory) as a complete description of gravitation in near presence of a massive body with nuclear density, is then concluded to be erroneous. To this point, no observation of the Universe has defined an object existing with a density greater than that of an atomic nucleus. Until the observation of such a body confirms this existence, we conclude that such a body does not exist, and we

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therefore hold the General Theory to become untenable in that regard in its present condition. The concept derived from the General Theory of that gravitation is not a force, but geometry is then held to be mechanically untenable. Under the tenants of the General Theory, the Special Theory is not held to be simply falsifiable, but falsified. The conclusion herein reached that the Special Theory is tenable (thereby measurable, predictable and provable), and does not conclude that the General Theory is falsifiable, but falsified, with regard to gravitation as geometry rather than a force. We conclude that the theory of Newton, once suitably modified, resolves Universal gravitation. We conclude that Newton’s concept of acceleration is limited to particles of mass and particles of zero mass, and the subsequent accelerations are limited to accelerations equal to, or less than, the speed of light of c. If the equations relative to Laplace, Schwartzschild, and Einstein are to remain tenable, then the force that describes the acceleration of gravity must be able to instantaneously allow for the transfer of information, at “speeds” in excess of that denoted by c. If this is allowable, then the objection to the theory of Newton (i.e. action at a distance), is overcome. However, this description may not, and cannot, be delivered in terms of particle-like, or wave-like terms. We conclude that Einstein falsified the 2,500-year-old concept of the lumineferous ether, and replaced that concept with a new form of ether. This concept of Einstein is resolved as being the concept of the field, gravitational in nature. We conclude that there apparently exists no mechanically sound derivation as to the nature of this field at this time. We conclude that such a theory is of great importance to the physics, and we further conclude that this paper does not resolve that theory. We conclude that the greatest physical achievement of the 20th century (i.e. QED) is not a field theory, but a statistical evaluation of the quantization of light relativistically compared to the quantization of the electron, where both exists within a gravitational field as in some unknown manner it is defined by the presence of an atomic nucleus (i.e. the proton). We conclude that the various forms deriving the particle/wave nature of light do indeed resolve the form of Newton, and hence QED is a description of the form of Newton. However, the form of Newton does not describe the field. Rather, the form of Newton simply predicts the existence of such a field, and not its nature. 3. Values For the physical constants, the values containing ( ), are current CODATA values, and the brackets denote the uncertainty involved as defined by NIST. c ¼ 299; 792; 458:2141 mps: v ¼ 299; 792; 457:4699 mps ð^0:03 m=s uncertainty). v0 ¼ 299; 792; 456:7257 mps: x ¼ 35; 789; 948:42502 m:

x0 ¼ 1259:767041151 m: b ¼ 28; 409:97363474 m: r ¼ 6; 378; 136:5 m ðMerritt; 1985: ^ 1 m uncertaintyÞ: r0 ¼ 6; 372; 160:986 m: rp ¼ 6; 356; 752:51245 m: P ¼ 2; 360; 591:556808 s: Y ¼ 31; 556; 925:79593 s: G ¼ 6:67259211 Nm: M S ¼ 1:99041379925230 kg: AU ¼ 1:49678748780311 m: rL ¼ 1; 749; 772:953994 m: gS ¼ 264:7638451889 ms2 : rs ¼ 8:86089278574123 m: Q ¼ 0:1024544230523: ec ¼ 1:602176462ð63Þ219 C: h ¼ 6:62606876ð52Þ234 Js: a0 ¼ 5:291772083ð19Þ211 m: X 1 ¼ 255:0135802925: r* ¼ 1:204708017634215 m: IE ¼ 13:60561078473 eV: F e ¼ 8:23867279886128 C: KE ¼ 9:964227894611225 ergs: gb ¼ 9:701548660622 ms2 : ve ¼ 29; 801:99404497 mps: AU 2 N ¼ 1:496313123311 m: gSol ¼ 271:8706260996 ms2 : C limit ¼ 1:442249570307: r2 ¼ 10; 118:76987192 m: Rs ¼ 4; 326:748710921 m: d ¼ 1:013332108142: T 0 ¼ 9:52422522 s: Z ¼ 0:7441975321802 m: cs ¼ 1:00000002482: cu ¼ 1:256339081004:

t ¼ 0:1193824175505 s: g 0 ¼ 9:806531191727 ms2 : r e ¼ 6; 378; 136:478725 m: r m ¼ 6; 371; 008:50415 m: R ¼ 384; 399; 678:9632 m: T ¼ 86; 164:09029838 s: RE ¼ 1152:481850232 m: M E ¼ 5:96752781967924 kg: M L ¼ 7:33381827193422 kg: g 1 ¼ 2:69477717378523 ms2 : g L ¼ 1:598310621749 ms2 : RS ¼ 708; 254; 866:3783 m: D ¼ 0:7442073555562 m: k ¼ 8; 987; 498; 319:512 C=m2 : M 1 ¼ 9:10938188ð72Þ231 kg: M p ¼ 1:67262158ð13Þ227 kg: a ¼ 7:29733081957923 : X 0 ¼ 245:3918448651: r H ¼ 3:98747988417210 m: g 2 ¼ 0:1553753224523 ms2 : w ¼ 2; 187; 684:744803 mps: r 1 ¼ 2:811430857588217 m: W max ¼ 7; 883:742953562 mps: RE2Sol ¼ 665; 592:7082209 m: r Sol ¼ 698; 936; 568:4632 m: M RE ¼ 5:96199110403924 kg: W limit ¼ 1:200936955176: rþ ¼ 3; 541; 894:917633 m: M Ch ¼ 2:87067344670430 kg: rv ¼ 8:40403119636622 mps: v* ¼ 587; 797:0456983 mps: Na ¼ 6:02214199ð47Þ23 mol21 : cg ¼ 1:174636265102: Rg ¼ 8:307605775334 mol21 K21 :

Special theory of relativity

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Note that the derivation of Rg is relative to the molar gas constant of R. The CODATA concise form of the molar gas constant is presently 8.314472(15) J mol2 1 K2 1. Further reading Chandrasekhar, S. (1931), Astrophysical Journal, Vol. 2, pp. 74-81. Compton, A. (1923), Physical Review, Vol. 21, pp. 483-502. de Broglie, L. (1922), Journal de Physique, Vol. 3, 6th Series, pp. 422. de Broglie, L. (1923), Comptes rendus de l’Academie des Sciences, Vol. 177, pp. 507-10. Einstein, A. (1905a), Annalen der Physik, Vol. 17, pp. 132-48, 549-560, 891-921. Einstein, A. (1905b), Annalen der Physik, Vol. 18, pp. 639-41. Einstein, A. (1905c), Annalen der Physik, Vol. 19, p. 891. Einstein, A. (1907), Annalen der Physik, Vol. 22, pp. 180-90. Hamilton, W.R. (1833), “On a general method expressing the paths of light and of the planets by the coefficients of a characteristic function”, Dublin University Review, pp. 795-826. Hamilton, W.R. (1834), “On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function”, Philosophical Transactions of the Royal Society, Part II, pp. 247-308. Hamilton, W.R. (1837), “Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time”, Transactions of the Royal Irish Academy, Vol. 17, pp. 293-422. Hamilton, W.R. (1839), “On the propagation of light in vacuo”, British Association Report, Part 2, pp. 2-6. Hamilton, W.R. (1841), “Researches on the dynamics of light”, Proceedings of the Royal Irish Academy, Vol. 1, pp. 245, 267-270. Hamilton, W.R. (1844), “On the composition of forces”, Proceedings of the Royal Irish Academy, Vol. 2, pp. 166-70. Hamilton, W.R. (1847), “On theorems of central forces”, Proceedings of the Royal Irish Academy, Vol. 3, pp. 308-9. Hamilton, W.R. (1864), “On a general centre of applied forces”, Proceedings of the Royal Irish Academy, Vol. 8, p. 394. Hamilton, W.R. (1867), “On a new system of two general equations of curvature”, Proceedings of the Royal Irish Academy, Vol. 9, pp. 302-5. Laplace, P. (1977), Mechanique Celeste, Vols. 1798-1825, Translation (English), Chelsea Publications, NY, USA. Oppenheimer, J. and Snyder, H. (1939), Physical Review, Vol. 56, pp. 455-9. Planck, M. (1900a), Annalen der Physik, Vol. 1, pp. 69-99. Planck, M. (1900b), Annalen der Physik, Vol. 1, pp. 719-37. Planck, M. (1901), Annalen der Physik, Vol. 4, p. 561. Stachel, J. (1998), Einstein’s Miraculous Year, Princeton University Press, Princeton, New Jersey. Stachel, J. (1998), “Five Papers that Changed the Face of Physics”, 1998.

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The light path in three-dimensional space B. Paul Gibson Davis and Elkins College, Parkersburg, West Virginia, USA

Light path in 3D space

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Keywords Cybernetics, Space, 3D Abstract Light, when viewed as a particle, reacts in a determinable manner with reference to the gravitational potential existing within the reference frame viewed. The elementary quanta of light, expressed under the terms of Planck, and as derived via the expressions of Einstein as a particle, may not reach a speed exactly equating to the speed (electromagnetic) of light of c. Here c is viewed as an electromagnetic constancy in any gravitational frame of reference. The theory is that a relative particle of mass may not achieve the speed of light, for the energy of that particle would then equate to infinity or in that the force required allowing the relative particle to reach c would then be infinite. The theory is then totally reliant upon the tenants of what has become to be known as the Special Theory of Relativity. As per the General Theory, light would be “bent”, more or less, from one gravitational reference frame as compared to another gravitational reference frame. The theory then evolves that light, when viewed as a particle, forms a curvilinear light path through the gravitational reference frame viewed. However, until now, the light path has been solely described on a linear basis. It is the result of the theory that the light path may be described on a curvilinear basis, under the method of Lagrange. This method, or model, allows a particle of light (viewed as a projectile of mass under a constant velocity, therefore under a constant acceleration) to achieve Newton’s description of the path of a projectile. Note that the following paper is applicable to a previous paper, which proposes a displacement of light within the gravitational field.

1. Concerning the path of light through space under the terms of Lagrange Let us consider that light, when viewed as a particle, forms a curvilinear light path as it travels through a space viewed, and where a gravitational potential exists within that frame. According to the field equations, and the tenants of the General Theory, the light path formed is curvilinear in nature. We consider that a conic arc segment is produced describing the light path over a given period of time. We further consider that, in a general nature, the light path describes a parabola. We consider the parabola, for a parabolic segment produces the quadratic, which may be used to describe the consideration within classical terms. We do not eliminate the consideration of the hyperbolic path, but we do eliminate the consideration of the circular and elliptical paths. Previously, we have shown that the light path is curvilinear in nature, considering one special frame viewed. The curvilinear nature of light, compared to the Riemannian geodesic consideration, then leads to a displacement of the light path under the terms of Maxwell. Under these terms and considerations, we then visit the formulation of the light path as described or evolved by Einstein within the Special Theory. It must be considered that the light path, as described by the special theory, in no

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1083-1098 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483199

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manner contradicts the same path as described under the general theory. Both are linear in nature, and both follow exactly the same equation as delineated within the special theory. Note that the light path is derived on a linear basis between the two points A and B, and this derives a linear velocity of the speed of light of v, but not c, if c is formed on a curvilinear basis. Therefore, the linear speed of light must propagate at v between the axes of the coordinates concerned. v ¼ ½2 £ AB 4 ½t 0 A 2 tB

ð1Þ

In equation (1), the special theory results uniform translational parallel motion. However, the general theory does not allow equation (1) (although the general theory apparently accepts equation (1) without offering any appropriate acceptable equation) since Riemannian geodesics forego uniform parallel motion. Then, the Special theory states that every light ray moves in the “rest” coordinate position with a fixed velocity v, independently of whether this ray of light is emitted by a body at rest or in motion. However, this in no manner (conceptually or mechanically) identifies the constancy of c based upon a curvilinear nature of light. Hence, on a linear basis, we may state linear velocity, v, as: linear light path ð2Þ linear time interval where t is the actual time light takes to travel a given curvilinear distance over any given period of time, we find c to be described over the curvilinear light path, over a given period of time. Hence, Linear velocity ¼ v ¼

Curvilinear velocity ¼ c ¼

curvilinear light path given time interval

ð3Þ

While the light path in equation (2) has been described, the light path in equation (3) has neither been described nor addressed in any adequate sense of the term. Therefore, we assume that one may describe the light path as described in equation (3) under the terms of Newton. To do so, or to allow for such, we turn to the method of Lagrange. The history of the following dates to Poincare´, who studied the solution to the three-body problem as presented to us by Newton. Therefore, the following is no more than the summarization of Poincare´. 2. The Lagrangian In Lagrangian mechanics we start by writing down the Lagrangian of the system: L¼T 2U

ð4Þ

where T is the kinetic energy and U is the potential energy. Both are expressed Light path in 3D in terms of coordinates ðq; q_ Þ where q [ R n is the position vector and q_ [ R n space is the velocity vector. 3. Equations of motion In Lagrangian mechanics, the equations of motion are written in the following universal form:
 d ›L ›L ð5Þ ¼ dt ›q_ ›q 4. The meaning of dot We emphasize that u_; or q dot in applicable terms, has a dual meaning. It is both a coordinate and derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation. We state so in order to resolve the terms of displacement, where displacement under Maxwell (and hence Lorentz and Compton) resolves as a linear coordinate shift. A trigonometric displacement involves an angle, where there is not a linear displacement of coordinates. Rather, we resolve a “deflection angle”. Under the hypothesis of de Broglie, the displacement is said to vary per the sin of 2p, which apparently signifies a deflection angle. Yet, the de Broglie hypothesis must be viewed as a description of a linear coordinate displacement or else the hypothesis gains no validity from a mechanical viewpoint. The matter of displacement, in terms of linear coordinates, then becomes a matter of significance considering the forms of Maxwell, Lorentz, Einstein, Compton, and de Broglie, where displacement is considered as a shift of linear coordinate measure. 5. Lagrangian vs Newtonian mechanics In Newtonian mechanics we represent the equations of motion in the form of the second of Newton’s laws m€q ¼ f ðq; tÞ

ð6Þ

where f (q, t) is the force applied to the particle. This equation is identical to the equation obtained from the Lagrangian representation if f (q, t) is considered as a conservative field (i.e. the field exists with a potential). A potential (including gravitational potential) is a function U (q, t) such that: f ðq; tÞ ¼ 2 Indeed, the Lagrangian can be written as

›U ›q

ð7Þ

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L ¼ 1=2 mð_qÞ2 2 U ðq; tÞ

ð8Þ

According to equation (5) the equations of motion reduce to equation (6).

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6. The variational principle It is an important realization that solutions of the Lagrange equation (5) then solves an extreme path problem between the two points in the configuration space considered (under the potential held by the frame viewed). The problem can be stated as that of the finding path; qðtÞ [ R n ; t 0 # t # t1 ; such that the integral S¼

Z

t1

LðqðtÞ; q_ ðtÞ; tÞ dt

ð9Þ

t0

must be minimal. The classical variational calculus studies the variation of this integral under perturbations of the path q(t). We then substitute the initial path q(t) with the new (minimal) path: q1 ðtÞ ¼ qðtÞ þ 1dqðtÞ

ð10Þ

where we realize dq(t) is an arbitrary vector-valued function on the segment [t0, t1]. We, therefore, define the variation of integral (9) under the perturbation dq(t) to be (Figure 1):  Z t1 d  dS ¼  Lðq1 ðtÞ; q_ 1 ðtÞ; tÞ dt d1 1¼0 t0 The classical calculation yields:

Figure 1. The variation of a path

ð11Þ

 Z t1 Z t1  d  d  Lðq1 ðtÞ; q_ 1 ðtÞ; tÞ dt ¼   Lðq1 ðtÞ; q_ 1 ðtÞ; tÞ dt d1 1¼0 t0 t 0 d1 1¼0 ¼

Z t0

t1


 ›L ›L dqðtÞ þ dq_ ðtÞ dt ›q ›q_

Light path in 3D space ð12Þ

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Using the fact that d ðdqðtÞÞ dt and integration of the second term by parts yield
 Z t1  ›L d ›L 2 dS ¼ dqðtÞ dt ›q dt ›q_ t0   ›L ›L ðqðt1 Þ; q_ ðt1 Þ; t1 Þ 2 ðqðt 0 Þ; q_ ðt 0 Þ; t0 Þ dqðt0 Þ þ ›q_ ›q_

dq_ ðtÞ ¼

ð13Þ

ð14Þ

This equation implies that if the ends of the perturbation path are clamped at the ends, (i.e. dqðt 0 Þ ¼ dqðt1 Þ ¼ 0) and then the second summed drops out of necessity. Moreover, if dS ¼ 0 is for all perturbations; then the Lagrange equation (5) must be satisfied. The above extreme property of the solutions of the Lagrange equation (5) shows that the invariance of these equations under coordinate changes must be classically invariant. If we use a time-dependent substitution q ¼ FðQ; tÞ; where F : R n £ R ! n R is a change of variables then the new Lagrangian with respect to the coordinates Q is: _ tÞ ¼ LðFðQÞ; DFðQ; tÞQ _ þ FðQ; _ KðQ; Q; tÞ; tÞ

ð15Þ

where DF(Q, t) is the derivative (Jacobi matrix) of F at (Q, t) with respect to Q and F ¼ ›F=›t: This formula allows us to choose coordinates in a convenient manner, for instance, to express the motion of a body in a rotating coordinate system. 7. Generalized momentum and the Hamiltonian In Hamiltonian mechanics we use generalized momentum in place of velocity as a coordinate. The generalized momentum is defined in terms of the Lagrangian and the coordinates ðq; q_ Þ : p¼

›L ›q_

The Hamiltonian is defined in terms of the Lagrangian as follows:

ð16Þ

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H ð p; qÞ ¼ p_q 2 Lðq; q_ Þ

ð17Þ

where in the above equation q_ is replaced with a function of ( p, q) by solving equation (16) with respect to q_ : We note that this task may be rather difficult if the dependence of L on q_ is complicated (i.e. QED). Fortunately, in most of the interesting situations (Newtonian), L is quadratic in q_ (i.e. T, the kinetic energy is a quadratic function of the velocity). Thus, the equation for q_ in terms of ( p, q) is linear. 8. Equations of motion The Lagrangian equation of motion (5) becomes a pair of equations known as the Hamiltonian system of equations. 9 ›H > > q_ ¼ = ›p > ð18Þ ›H > > p_ ¼ > ›q ; The second equation of this system is easy to explain. It is simply the Lagrangian found in equation (5). The second equation can be explained by using duality. For the sake of this argument we need to assume that Lðq; q_ Þ is strictly convex as a function of q_ : It is sufficient to assume that the Hessian  2  ›L ð19Þ _ ›qi ›q_ j is positive definite. We note that for Newtonian equations this matrix is I. The function H in this case is the Legendre transform of L, i.e. H ð p; qÞ ¼ inf ½ p_q 2 L ðp; q_ Þ p

ð20Þ

where the infimum is taken over all p [ R n : One can show that L is a Legendre transform of H as well, i.e. Lðq; q_ Þ ¼ inf ½ p_q 2 H ðp; qÞ q_

ð21Þ

In particular, the minimum is attained for q_ ¼

›H ›p

This is exactly the second equation of the system shown in (18).

ð22Þ

9. Time-dependent, linear change of variables Light path in 3D It will be convenient to first consider arbitrary linear changes of variables. We space then consider to let Lðq; q_ ; tÞ be a Lagrangian, which then allows us to perform the change of variables we have noted: q ¼ BðtÞQ

ð23Þ

where BðtÞ: R n ! R n becomes any linear transformation (matrix), including the matrices of Lorentz and Heisenberg. 9.1 Newtonian case Let us first consider what happens to Newton’s equation ½m€q ¼ f ðqÞ under the change of variables noted in equation (23). We have _ _ þ BQ q_ ¼ BQ

ð24Þ

_ so that the following It will be convenient to introduce the matrix ½V ¼ B 21 B; first-order matrix differential equation is satisfied: B_ ¼ BV

ð25Þ

_ ¼ BðQ _ þ VQÞ q_ ¼ BVQ þ BQ

ð26Þ

Using this equation, we may write

Differentiating second time, we obtain: € þ VQ _ þ VQÞ _ þ VQÞ _ þ Bð _ Q q€ ¼ BðQ € þ VQ _ þ VQÞ _ þ VQÞ ¼ BðQ € þ 2VQ _ þ VQ _ þ BVðQ _ þ V2 QÞ ð27Þ ¼ BðQ By multiplying this equation by m, we obtain: € þ 2mVQ _ þ mVQ _ þ mV2 QÞ f ðqÞ ¼ BðmQ

ð28Þ

We introduce the new “force” FðQÞ ¼ B 21 f ðBQÞ

ð29Þ

(This formula means that the force is actually a vector, i.e. it transforms under coordinate changes as above.) We multiply equation (28) by B2 1 and obtain: € þ 2mVQ _ þ mVQ _ þ mV2 Q FðQÞ ¼ mQ

ð30Þ

9.2 Newton’s equations of motion in new coordinates Summarizing the previous mechanics, the Newtonian equations of motion ½m€q ¼ f ðqÞ (after a linear time-dependent change of coordinates q ¼ BðtÞQ) becomes (equation (30)):

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€ ¼ FðQÞ 2 2mVQ _ 2 mVQ _ 2 mV2 Q mQ

ð31Þ

_ where V ¼ B 21 B:

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9.3 Rotating coordinate system This is a special case (noting the special case and the principle of relativity) when n ¼ 3 the matrix B is orthogonal of determinant 1. In this case, the matrix V is skew-symmetric. There is a vector v, which represents the angular velocity of the rotating coordinate system, such that for all Q [ R n we have VQ ¼ v £ Q

ð32Þ

where £ denotes the usual cross product of vectors. Indeed, if v ¼ ðv 1 ; v 2 ; v 3 Þ then the matrix of the linear transformation Q ! v £ Q is V. On the other hand, 32 3 2 Q1 0 2v3 v2 76 7 6 v 0 2v1 76 Q2 7 ð33Þ v£Q¼6 54 5 4 3 Q3 2v2 v1 0 Thus, we have 2

0

6 v V¼6 4 3

2v2

2v3 0

v1

v2

3

7 2v1 7: 5 0

ð34Þ

9.4 Newton’s equations in rotating coordinates According to the previous derivation, for n ¼ 3; the equations of motion are: € ¼ FðQÞ 2 2mv £ Q _ 2 mv £ Q 2 mv £ ðv £ QÞ mQ

ð35Þ

The interpretation of these equations is the fact that in a rotating coordinate system there are additional forces acting upon the body, represented by the three additional terms in the right-hand side of this equation. They have their _ is called the Coriolis force; names in classical mechanics: 2mv £ Q mv £ ðv £ QÞ is simply the centripetal force; mv_ £ Q is the force of inertia; this force is 0 if the rotation is uniform, i.e. V is constant. 9.5 Lagrangian case Let Lðq; q_ ; tÞ be a Lagrangian. According to equation (15) the change of variables q ¼ BðtÞQ leads to motions described by the new Lagrangian

_ tÞ ¼ LðQ; BQ _ þ BQ; _ þ VQÞ; tÞ _ tÞ ¼ LðQ; BðQ KðQ; Q;

ð36Þ Light path in 3D

space This formula illustrates the benefits of the Lagrange formalism when dealing with coordinate changes. It is interesting to see how the Coriolis, centripetal and inertia forces can be derived from the Lagrangian formalism. Let L ¼ T 2 U where T ¼ ð1=2Þmð_qÞ2 : If B is orthogonal, then kBx; Byl ¼ kx; yl where k; l is the dot product (so ð_qÞ2 ¼ k_q; q_ l). Hence, the new Lagrangian is: _ tÞ ¼ 1=2 mðQÞ _ 2 þ mðVQ; QÞ _ þ 1=2 mðVQ; VQÞ 2 U ðBQ; tÞ KðQ; Q;

ð37Þ

The extra two terms translate into the additional terms in the equations of motion, which are called Coriolis, centripetal and inertia forces. 9.6 Equations of motion We find the generalized momentum:

›K _ þ mVQ ¼ mQ _ ›Q

ð38Þ

~ ›K _ 2 mV2 Q 2 ›U ¼ 2mVQ ›Q ›Q

ð39Þ

and

~ where UðQ; tÞ ¼ U ðBQ; tÞ is the new potential. After identifying; FðQÞ ¼ 2

›U~ ›Q

ð40Þ

(use B 2 1 ¼ BT here!) We obtain exactly the same result as in equation (35). 9.7 The Hamiltonian Having calculated the generalized momentum in rotating coordinates, then, P¼

›K _ þ mVQ ¼ mQ _ ›Q

We may find the Hamiltonian in rotating coordinates:

ð41Þ

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2 P 1 P _ _ ~ H ðP; QÞ ¼ P Q 2 KðQ; QÞ ¼ P tÞ 2 VQ 2 2 VQ þUðQ; m 2 m ¼

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1P2 ~ þ kVP; Ql þ UðQÞ 2m

ð42Þ

The Hamiltonian equations of motion are: _ ¼ P 2 VQ Q m

9 =

~ P_ ¼ 2 ››UQ 2 VP ;

In three dimensions ðn ¼ 3Þ these equations could also be written as; 9 _ ¼ P 2 VQ Q = m ~ P_ ¼ 2 ››UQ 2 v £ P ;

ð43Þ

ð44Þ

9.8 Note on the direction of angular velocity It is easy to make the sign mistake in determining the direction of v. A simple rule eliminates this mistake: v is opposite to the angular velocity of a point at rest in the original coordinate system expressed in the rotating coordinate system. For example, if we use a coordinate system rigidly attached to the earth then a point on the surface of the earth resting in non-rotating coordinates appears to be moving west for an observer in the rotating coordinate system. Thus, v is pointing towards the north pole with length equal to 2p 24 £ 3600

ð45Þ

From the above equation we find that the period of revolution about the earth on its axis becomes 86,400 s. The mean period of that rotation is known to be 86,164.0902 s. Equation (45) must then be modified in view of the observation, added to the observational fact that the solid inner core of the earth rotates 0.63 s faster than the same period as noted at the equatorial radius of the earth. 10. Restricted (circular) three-body problem This illustrates the formalism introduced herein and provides an example with a history dating back to (or actually from) Poincare´. Here we interject historical notation, relative to mechanical derivation as compared to actuality. Poincare´, prior to Einstein, had developed the mechanics Einstein utilized in producing

what we know now as the special theory. For all the formalism of Newton, Light path in 3D and considering the genius of Hamilton, the three-body problem remains just space that i.e. a problem. Poincare´ studied that problem, but was not able to eliminate the difficulty of the Newtonian mechanics, even after the elucidation of Hamilton. Let us consider that Einstein, in his elucidation of the special theory, in no 1093 manner violates Poincare´. Let us then consider that the special theory derives the de Broglie hypothesis. We must then consider that Schrodinger, in his genius, developed the wave function from the hypothesis as developed by de Broglie. At this stage in time, the ad hoc nature of de Broglie’s hypothesis must be considered as a moot point, considering the results of QED. Indeed, the particle nature of the de Broglie hypothesis may achieve QED results, if brought forth in the manner shown to us by Bohm. Importantly, Dirac showed us that the Schrodinger model was in need of correction, as it remained in its infancy, since the model was not Lorentz invariant. By the relativistic correction of the form of Schrodinger, Dirac allowed QED to be borne. Just as importantly, Dirac showed us that the Schrodinger form (the wave function) was no more than an ingenious application of the form of Hamilton. This fact distressed Schrodinger greatly, but the proof of Dirac may not be denied. Dirac also proved that Heisenberg’s matrix mechanics and Schrodinger’s wave function were indeed equal to each other. One derived the other, a fact that was greatly distressing to Heisenberg. Therefore, both Schrodinger and Heisenberg derive Hamilton in the statistical sense of the Hamiltonian operator. As it is well-known, the form of Hamilton derives Lagrange and the form of Lagrange derives Newton concerning motion. As it is well-known, neither Newton nor Hamilton were able to alleviate the three-body problem, in an exact formulation (hence the three-body problem), as shown to us by Poincare´. As it is well-known, QED begins to fail the three-body problem. Indeed, when multiple electrons are involved, QED becomes so statistically difficult that solutions are impossible to render at this technological point of time. Concerning Newton, the entire theory elucidates the principle of gravitation, on the whole frame view. Individual gravitational reference frames, where variances within potentials compared to viewed frames, are not considered. The special theory, for all its genius, does not concern itself with varying gravitational potential at all. Rather, in line with Newton, the special theory considers all frames as a single frame or a special case. Previously, we have shown that this must not be the case, from a physical standpoint, and that varying potentials must, and can, be accounted for. In the same vein, we have shown that the form of Chandrasekhar, itself derived from the special theory, derives observational effect if varying gravitational

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potential is accounted for. Therefore, the following solution, well versed in classical mechanics, is erroneous in its conceptualization and its outputs, although agreeing with the form derived by Poincare´. In a restricted three-body problem (Figure 2), one of the three masses is negligible as compared to the other two masses. In the circular version we assume that the remaining two masses move along circular orbits about their center of mass. It is natural to introduce a coordinate system in which the two heavy masses are at rest, say at points A and B and that the center of mass is at 0. The potential energy in this coordinate system is UðQÞ ¼ 2

mA mB 2 kQ 2 Ak kQ 2 Bk

ð46Þ

where mA and mB are suitable constants. We assume that the lightest mass is equal to 1. Thus, the equations of motion are: 9 _ ¼P 2v£Q = Q ð47Þ P_ ¼ 27U ðQÞ 2 v £ P ; where P, Q [ R 3 : Of course, we may perform the differentiations and obtain a completely explicit system of ODE with four free parameters. Indeed, we may assume that A and B are on the x-axis and A ¼ ða; 0; 0Þ; B ¼ ð2b; 0; 0Þ: We must have m1 a ¼ m2 b; which is the only relation between the four parameters. We may also assume that v ¼ ð0; 0; oÞ and that o is another parameter.

Figure 2. Restricted three-body problem

11. Testing the proposed theory Light path in 3D The proposed theory rests solely upon the principle of relativity as proposed by space Einstein. Referring to Figure 1, the theory proposes the following. In Figure 1, we find two points. The first point was designated as q(t0). The second point was designated as q(t1). The first point we now designate as A. The second point is now designated as B0 . The following becomes crucial concerning A and B0 . 1095 Concerning time (relative to a congruent coordinate system), the end points must be considered to be clamped. In other words, the light (as a particle) leaves the point A and travels to the point B0 . The light leaves the point B0 and travels back to the point A. The coordinates of A are concrete and do not vary during the flight time of the light, where the light is viewed as a particle. The coordinates of B0 are defined upon the arrival of the light at B0 . The sense of time begins only when the light identifies the coordinates of B0 . Time, in the mechanical sense, is then identified by the distance the light travels (on a linear basis) from B0 to A. From the fixed point A, relative to the light defining B0 and returning to A, we copy Figure 1, and the end result will be Figure 2, as follows. The theory then requires that the points A and B0 be defined in real time, and in the physical sense. We assume that A is a fixed point upon the surface of the earth (the earth’s equatorial radius), where the point A defines an exact radius from a fixed earth center. This radius is exactly a definition of the equatorial radius of the earth, and that radius is defined by r. We further assume that B0 is a point rigidly connected to A, and that the point A is rigidly connected to a fixed earth center designated by the entity Q. We further assume a point B, based upon the Cartesian coordinate system. From a fixed earth center of Q, a straight line is extended from Q to A, and is further extended into space. On this straight line, B is identified under the form of Descartes. B is then at a distance x, measured from A, that identifies a body above the equatorial radius of the earth where the body is exactly in a geosynchronous orbit. Physically we must realize that the earth is in motion about its axis. For time being, we assume that this rotation of the earth about its axis is not present (in terms of the coordinate system of the light), but in that we are viewing a static system at precisely one moment in static time, concerning the light. From Descartes, and indeed theoretically, we may define the entity B. Yet the theory holds that the light may not define B on a light path that defines a straight line. Rather, the light would evolve a curvilinear path in its travel. Therefore, in the perspective of the light, the point B0 would be defined on a coordinate basis, rather than the point B. Here we assume that the direction of rotation of the earth about its axis is a result of gravitation, and the resulting direction of rotation is considered to be from the direction of B0 to B.

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In this consideration, we deduce the age-old question of “Who owns the light?”. The theory renders this question a moot point, for the theory derives that the coordinate system evolved is relevant only to the light, in a nonrotating system. Therefore, the light evolves an inertial (or static) system, in terms of the coordinates relative to the light only, and the light resolves those coordinates within a static system relative to the light itself. The point A is a coordinate held by the observer from a fixed earth center Q at a distance r equating to the equatorial radius of the earth. The point B is satisfied by a body in geosynchronous orbit exactly above the observer at A, on a line drawn from the fixed earth center Q, through A and deriving B. The observer at A sends out the light as a signal. However, due to the curvature of space, the light does not identify B, but rather identifies B0 . In this case, the observer owns the light, relative to the time and distances involved (since the observer is defining the parameters), but the observer’s ownership of the light is, and remains, only relative to the point A, which continually exists (theoretically) in static time. The light identifies B0 and begins the return path to A. During the flight time from B0 to A, the observer is influenced by the rotation of the equatorial radius of the earth. As the light returns to A, the observer finds himself at the new coordinate A0 . In the reference frame of the light, the light sees the observer not at A0 , but at the coordinate C. The distance from A to A0 is dilated to the distance from A to C, in terms of light. The observer, in order to account for the curvature of space, and the motion of the observer during the time involved, must shift his original coordinates of A to C 0 , and of B to B0 , at the time the observer initiates the light from A. The light path is then derived by applying the Lagrangian of Figure 1 (equation (5)) to the points A and B0 . The light path will be deformed from a pure parabola, as the light does not travel in vacuum from A to B0 (since layers of atmosphere are involved during the flight), and the variational principle is graphically shown, if exaggerated, in Figure 1. The test of the theory is then applied to an existing platform, where it is known that the present physics applied acknowledge that there remains a constant error in coordinate positioning. By applying the previous theory that light does suffer a displacement caused by the gravitational potential within the frame viewed to the results of Figure 3, we arrive at the predicted coordinate shifts. These predicted shifts account for the known coordinate error within the existing platform or the global positioning satellite system. The current error in positioning is termed circular error probable (CEP), and the theory derives this error, as the theory derives actual coordinate positioning of any object above the observer at any moment in real time. We report that the physics are now available to provide the coordinate positioning at any moment in time of any object in space, located above a fixed point in time identified on the surface of the earth. From observation, the predictions are exact, and render a total error in spatial coordinate positioning of less than ^0.1 m at any

Light path in 3D space

1097 Figure 3. Coordinate shifting of bodies comprising the global positioning satellite system platform. (Relative three-body problem)

moment in real time, applied towards any body in motion within the frame viewed. One must note that within Figure 3, the extreme path of Lagrange identified from A to C 0 is on a curvilinear basis (considering the end points of A and B0 are indeed clamped), reflecting an assumed circular nature of the equatorial radius of the earth, rather than the linear nature as depicted by Figure 3. In terms of the Lagrangian extreme path perturbation of Figure 1, we may note that the path may be rudimentarily determined under Newton’s “classical” terms of the path of a projectile. The equations are arduous and rudimentary. The initial equations must assume the projectile under a constant velocity. Then under the terms of Newton, we may fire a projectile from point A to point B0 , where both points are considered as earth based ground points. The terms of Newton do not allow a return path of the projectile from B to C 0 . From the form of Lagrange, we are able to circumnavigate Newton and arrive at mechanically realistic results. The Lagrange perturbation of the light path relevant to Figure 1 applies from the point A to the point B0 , where B0 is identified by clamping the ends of the light path. Obviously, in order to identify C 0 under the Lagrange coordinate system (by means of Figure 1) the Lagrange path perturbation must be applied from B to C 0 , in order to identify C 0 under the Lagrange coordinate system. In reference to Figure 3, it may appear that a faux pas is apparently committed, in that there appears to be no allowance for the Einstein/Lorentz transformation to be applied to the graphic illustration (which is by no means shown at scale). For the conjecture relative to Figure 3 to remain mechanically sound, the projection must remain Lorentz invariant. In the case as shown, the illustration derives a coordinate system neither on the basis of solely the observer nor on the basis of solely the light. Rather, the perspective is derived simultaneously, from either the coordinate perspective of the observer to

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the light or of the light to the observer. The theory requires that the coordinates of B 0 and C 0 be derived in the manner of the form of Lorentz, but a dual transformation resulted resolves the distance dilation between A0 as compared to A, and equally of A as compared to C 0 . These results are complimentary to the coordinate shift of B to B 0 , with that coordinate shift being derived as Lorentz invariant. This transformation is derived from the previously proposed Maxwell displacement of light, equating to c 2 v, and upon the phenomena known as the Sagnac effect. The theory derives that the Sagnac effect is a result of the Maxwell displacement of c 2 v, based upon the distance involved between A and B 0 , and the aforementioned CEP. We report the mechanical results in the following paper.

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Relativistic coordinate shifting within three-dimensional space

Relativistic coordinate shifting 1099

B. Paul Gibson Davis and Elkins College, Parkersburg, West Virginia, USA Keywords Cybernetics, Space, 3D Abstract Let us consider that light, when viewed as a particle, forms a conic arc segment inscribed within the space viewed. The space (or frame) viewed is considered to exhibit a gravitational potential, and it is thus this potential that deforms the light path from a Euclidean/Newtonian derivation of a straight line to that of a relativistic curvilinear nature. Given a distance over this conic arc segment (assumed to form a parabolic arc segment) and a given time (considering the given distance involved), one derives a constancy of the speed of light of c, where c is considered as a constant regardless of the gravitational potential exhibited by the frame viewed. If we further consider that the Special Theory requires that light propagate on a linear measure as the velocity v (of necessity v being less than c on a comparable linear measure) between the axes concerned; then a displacement (in linear measure equal to c 2 v) occurs. The displacement evolved is then assumed to agree with the form of Maxwell. We assume that this linear displacement of c 2 v occurs upon the y-axis of the frame viewed. Of necessity, a relative displacement must occur upon the x-axis of the frame viewed. From the calculus, the dot products derived must vary in concept, in order to derive the totality of relative coordinate shifts occurring within any three-dimensional space. One displacement is linear in nature, while the other is trigonometric in nature. We consider the displacement of Maxwell, Lorentz, Compton, and de Broglie to be linear in nature. Based on the principle of the Special Theory (and the other forms as mentioned), we consider the total displacement to be mechanically derivable. That derivation, once allowed, results the physics to agree with the observations complete to this moment in time. The paper concludes that the error in coordinate positioning shown by the global positioning satellite system (GPS satellite platform) is resolvable.

1. Concerning the coordinate shifting of spatial positions within 3D space based on the nature of light resolving those coordinate shifts as determined (measured) by the global positioning satellite system Let us consider the Einstein/Lorentz transformation as a possible derivation resulting from Newton’s laws. Previously, we had considered this transformation and apparently found that Newton’s laws may be derived (under a special case) under the Lorentz transform, with consideration given to a modified Special Theory. Let us consider that light, constructed in particle-like terms, does form a curvilinear light path (in a given period of time and over a given distance) This paper is derived from two previous papers and those papers are essential in understanding that which follows. Since the paper is a matter of physics, numerical values must be attached in order to agree with known observation. Those values are attached to the end of this brief.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1099-1112 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483207

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through the frame viewed. Previously, we apparently derived that the curvilinear speed of light (resulting in the constancy of c) would vary compared to the linear speed v, where v is a direct result of the Special Theory predicting the linear speed of light traveling between the axes concerned during the time concerned. Here, it is far easier to assume v as a distance (traveled under a given time as required by the speed of light of c), rather than a speed. Let us consider that a displacement of wavelength (a linear measure) results as a difference signified by the property c 2 v, and that this difference may be resolved under the terms of Maxwell. Previously, we had apparently derived the displacement under the consideration of the forms presented. That displacement, under the terms of Maxwell and the forms given, applied to the Einstein/Lorentz transformation, was then apparently able to derive the form of Newton. Let us consider the curvilinear aspect of the nature (propagation) of light through space. Under the consideration, we find that the theory of relativity mechanically predicts the linear light path between the axes concerned, and not the curvilinear path of light. Let us consider that the available literature does not predict (mechanically) the curvilinear light path involved within any gravitational frame of reference, whether that frame is construed to be concerned as special or otherwise. Previously, we have apparently shown that the proposed curvilinear light path may be described (under the theory tendered) in the form of a Lagrangian (where that form describes an extreme path perturbation between two given points over the time concerned). Let us consider that if light shows a displacement within the special frame viewed (equal to c 2 v), then a linear shift of coordinates may be predicted. To this point, the shift has only been considered under a linear nature, not under a trigonometric nature. We consider that linear displacement predicts a shift upon the y-axis, while a corresponding trigonometric shift predicts a change in coordinates along the x-axis. Previously, we derived that the calculus, when considering coordinate displacement, results in a dot product. Theta dot and q dot may then predict either a linear or trigonometric shifting of coordinates. Considering the linear distance involved, a shift on the x and y-axes results in a shift upon the z-axis, under the method of Pythagoras. We predict those shifts. We begin by considering a point B directly above a point A, where A is found on the surface of the earth. B is found from A at a distance x, signifying a body in geosynchronous orbit above A. A is found exactly on the equatorial radius of the earth, at a distance r measured from a fixed earth center Q to A.

To define the geostationary distance x, we have shown that a modification to Newton’s laws is required, based on the results of the Special Theory and where the displacement of light is considered in one special frame of reference. We then define a radius RE (Einstein’s radius), derived from the mass of the earth and considering the maximum force of the earth’s gravity to exactly equal the speed of light of c. Then 2

c ¼ ½ME 4 RE  £ G

ð1Þ

where G is Newton’s value describing the universal constant of gravitation and ME represents the mass of the earth. A and B are then considered to be rigidly attached to one another, as they are rigidly attached to the fixed earth center Q. The time (period) of one full earth rotation about its axis at the equatorial radius is defined as T, where the period T is exactly equal concerning the points A and B, during one full rotation of the earth about its axis. We then derive the geosynchronous (geostationary) distance x, where the equatorial radius of the earth is r and the mass of the earth is ME. {4p 2 } 4 {G £ ME} ¼ T 2 4 ½{x þ r} 2 {42 £ RE}3

ð2Þ

Previously, we derived (from the de Broglie hypothesis) that the speed of light of c (considered constant within all gravitational frames of reference), was slightly greater than the linear value attached to the speed of light (via the Special Theory) of v. Here, as v is considered representing the linear speed of light one would note upon the surface of the earth, where the notation is made in vacuum, and in that c is displaced (linearly) to v as a result of gravitation. With the entities derived, we consider the Einstein/Lorentz transformation within this special case. The time t, t ¼x4c The entity x 0 , where relativistic distance dilation is considered. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 0 ¼ ½x 2 {v £ t} 4 1 2 {v 2 4 c 2 }

ð3Þ

ð4Þ

Previously, we had not considered time dilation under the Special Theory, resolving t 0 from t. We do so now, in order to maintain Lorentz invariance concerning that which follows. Then the time dilation of t0 relative to t. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 0 ¼ ½{1 2 ðv 4 cÞ} £ t 4 1 2 {v 2 4 c 2 } ð5Þ We now note the use of the expression used herein as the Einstein/Lorentz transformation, rather than as the Lorentz transform. Within the Einstein/Lorentz transformation, we find:

Relativistic coordinate shifting 1101

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1102

c ¼ x0 4 t0

ð6Þ

Previously, we derived the entity b, which may be viewed either as a magnitude, or as a proper distance. The entity b, so crucial to the theory as we have presented it, becomes only of critical value, if the Einstein/Lorentz transformation is correct. As a magnitude, b derives (in line of the transform considering time dilation as shown by Einstein):

b ¼ x 4 x0 ¼ t 4 t 0 Therefore, where the entity b is considered to be a distance: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4 b ¼ ½1 2 {v £ ð1 4 cÞ} 4 1 2 {v 2 4 c 2 }

ð7Þ

ð8Þ

The Einstein/Lorentz transformation may be shown in the standard matrix form. As we have shown previously, the equations of motion may be defined by the calculus to derive any matrix, under the forms as shown to us by Hamilton and Lagrange, and where (importantly), these matrices derive Newton. Where Newton derives motion, the calculus of Hamilton defines the generalized momentum. Therefore, a matrix consideration of Einstein/Lorentz (or of Heisenberg for that matter), must resolve what has become to be known as Lorentz invariancy. The matrix consideration requires the well-known quantity gamma g. Then g, relative the Einstein/Lorentz transformation. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 1 4 1 2 {v 2 4 c 2 } ð9Þ Previously, we considered the curvilinear nature of light through the space viewed to be a predictable result of the method of Lagrange, where an extreme path perturbation is considered. We considered two points, q(t0) and q(t1). We considered that these end points (coordinates) must be considered as clamped. In order for the formalism to result reality, we require that the coordinates of both points be derived at a relative moment in space/time, with the known coordinates being derived from nonrelativistic coordinates derived from real time measurement. The previous formalism is shown in Figure 1. 2. The Lagrangian In Lagrangian mechanics, the equations of motion are written in the following universal form:   d ›L ›L ð10Þ ¼ dt ›q_ ›q The classical calculation yields:

Relativistic coordinate shifting 1103 Figure 1. The variation of a path

 Z t1 Z t1  d  d  Lðq1 ðtÞ; q_ 1 ðtÞ; tÞ dt ¼   Lðq1 ðtÞ; q_ 1 ðtÞ; tÞ dt d1 1¼0 t0 t 0 d1 1¼0  Z t1  ›L ›L _ ¼ dqðtÞ þ dqðtÞ dt ›q ›q_ t0

ð11Þ

Using the fact that: d_qðtÞ ¼

d ðdqðtÞÞ dt

And integration of the second term by parts yields:   Z t1  ›L d ›L dS ¼ 2 dqðtÞ dt þ d ›q dt ›q_ t0

ð12Þ

ð13Þ

This equation implies that if the ends of the perturbation path are clamped at the ends (i.e. dqðt 0 Þ ¼ dqðt1 Þ ¼ 0) and then the second summand drops out, of necessity. Moreover, if dS ¼ 0 for all perturbations, then the Lagrange equation (10) must be satisfied. The above extreme property of the solutions of the Lagrange equation (10) shows that the invariance of these equations under coordinate changes must be classically invariant. If we use a time-dependent substitution q ¼ FðQ; tÞ; where F : R n £ R ! n R is a change of variables, then the new Lagrangian with respect to coordinates Q is: _ tÞ ¼ LðFðQÞ; DFðQ; tÞQ _ þ FðQ; _ KðQ; Q; tÞ; tÞ

ð14Þ

where DF (Q, t) is the derivative ( Jacobi matrix) of F at (Q, t) with respect to Q and F_ ¼ ›F=›t: This formula allows us to choose coordinates in a

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Figure 2. Coordinate shifting of bodies comprising the GPS platform (relative the three-body problem)

convenient manner, for instance, to express the motion of a body in a rotating coordinate system. Previously, we derived a graphical consideration of the proposed theory, where that graphical representation was supposedly offered as a test of the proposed theory, related to the results found within coordinate positioning as derived by the global positioning satellite system (GPS) platform. In Figure 2, we present that illustration. Note that the two coordinate positions of Figure 1 now become points A and B0 . The point A is a coordinate held by the observer from a fixed earth center Q at a distance r equating to the equatorial radius of the earth. The point B is satisfied by a body in geosynchronous orbit exactly above the observer at A, on a line drawn from the fixed earth center Q, through A and deriving B. The observer at A sends the light as a signal. However, due to the curvature of space, the light does not identify B, but rather identifies B0 . In this case, the observer owns the light, relative to the time and distances involved (since the observer is defining the parameters), but the observer’s ownership of the light is, and remains, only relative to the point A, which continually exists (theoretically) in static time. The light identifies B0 and begins the return path to A. During the flight time from B0 to A, the observer is influenced by the rotation of the equatorial radius of the earth. As the light returns to A, the observer finds himself at the new coordinate A0 . In the reference frame of the light, the light sees the observer not at A0 , but at the coordinate C. The distance from A to A0 is dilated to the distance from A to C, in terms of the light. The observer, in order to account for the curvature of space, and the motion of the observer during the time involved, must shift his original coordinates of A to C 0 , and B to B0 , at the time the observer initiates the light from A. The light path is then derived by applying

the Lagrangian of Figure 1 (equation (10)) to the points A and B0 . The light path will be deformed from a pure parabola, for the light does not travel in vacuum from A to B0 (since layers of atmosphere are involved during the flight), and hence the variational principle is graphically shown, if exaggerated, in Figure 1. The test of the theory is then applied to an existing platform, where it is known that the present physics applied acknowledge that remains a constant error in coordinate positioning. By applying the previous theory that light suffers a displacement caused by the gravitational potential within the frame viewed to the results of Figure 2, we arrive at the predicted coordinate shifts. These predicted shifts account for the known coordinate error within the existing platform, or the GPS. The current error in positioning is termed circular error probable (CEP), and the theory derives this error, as the theory derives actual coordinate positioning of any object above the observer at any moment in real time. We report that the physics is now available to provide the coordinate positioning at any moment in time of any object in space, located above a fixed point in time identified on the surface of the earth. From observation, the predictions are exact, and render a total error in spatial coordinate positioning of less than ^0.1 m at any moment in real time, applied towards any body in motion within the frame viewed. One must note that within Figure 2, the extreme path of Lagrange identified from A to C 0 is on a curvilinear basis (considering the end points of A and B0 are indeed clamped), reflecting an assumed circular nature of the equatorial radius of the earth, rather than the linear nature as depicted in Figure 2. In terms of the Lagrangian extreme path perturbation of Figure 1, we may note that the path may be rudimentarily determined under Newton’s “classical” terms of the path of a projectile. The equations are arduous and rudimentary. The initial equations must assume the projectile does assume a constant velocity. Then under the terms of Newton, we may file a projectile from point A to B, where both points are considered as earth-based ground points. The terms of Newton do not allow a return path of the projectile from B to A. From the form of Lagrange, we are able to circumnavigate Newton, and arrive at mechanically realistic results. In reference to Figure 2, it may appear that a faux pas is apparently committed, in that there appears to be no allowance for the Einstein/Lorentz transformation to be applied to the graphic illustration (which is by no means shown at scale). For the conjecture relative to Figure 2 to remain mechanically sound, the projection must remain Lorentz invariant. In the case as shown, the illustration derives a coordinate system neither on the basis of solely the observer, nor the light. Rather, the perspective is derived simultaneously, from either the coordinate perspective of the observer to the light, or of the light to the observer. The theory requires that the coordinates of B 0 and C 0 be derived in the manner of the form of Lorentz, but a dual transformation is resulted that resolves the distance dilation between A as compared to C, and equally of A as

Relativistic coordinate shifting 1105

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compared to C 0 . These results are complimentary to the coordinate shift of B-B0 , with that coordinate shift being derived as Lorentz invariant. This transformation is derived from the previously proposed Maxwell displacement of light, equating to c 2 v, and upon the phenomena known as the Sagnac effect. The theory derives that the Sagnac effect is a result of the Maxwell displacement of c 2 v, based on the distance involved between A and B0 , and the aforementioned CEP. Within Figure 2, we consider that a linear distance termed r 0 is derivable, with that distance determining the coordinate shift between points A and C, and equally between points A and C 0 . The fixed point A is derived, in the coordinate sense, as lying at the equatorial radius of r from a fixed earth center Q. Both C and C 0 lie at that same distance r from Q, but in varying time. Therefore, the distance r 0 becomes a linear distance, or a chord distance, rather than an arc length. In real time, based on the consideration of the observer at A (who exists in a frame where motion is a fraction of light speed), we derive a time interval of st 0 , related to the distance r 0 and relative to the motion of the observer as noted within the frame viewed. From the relativistic quantity b, we derive the observer quantity (distance) of s. s¼b42 ð15Þ The product gamma, from the matrix result applied to the Einstein/Lorentz transformation, does not equal b divided by 2. Therefore, we derive gamma prime of g 0 , where the entity represents distance dilation in terms of the light verifying the position of the observer during the time allotted. The gamma prime of g 0 then allows for a reverse proper Lorentz transformation from the viewpoint of the light in a relativistic frame, to that of the observer in his “inertial” frame.

b ¼ {g þ g 0 } £ 2

ð16Þ

Then, the quantity of time st 0 ;

g ¼ t 4 st 0

ð17Þ

s ¼ g þ g0

ð18Þ

g ¼ 2 £ ½s 4 {st 0 4 t 0 }

ð19Þ

g 0 ¼ b 4 {st 0 4 t 0 }

ð20Þ

then

and

and

where the above derive:

g 0 ¼ x 4 {st 0 £ c}

ð21Þ

As the purpose of this paper is to define the existing coordinate positioning error identified by observation of the GPS platform, we require terminology relevant to this system. The Sagnac effect is crucial concerning the GPS platform. The Sagnac effect may be noted as the Sagnac time difference. The Sagnac effect may be derived from the Special Theory, but we may consider that the Special Theory does not explain the Sagnac effect. We submit that the Sagnac effect is a result of the displacement of light previously noted as the quantity c 2 v. The Michelson– Morley experiment of 1887 was conducted in a farm field in Ohio. By the use of mirrors and the reflection of light, the experiment was meant to confirm the lumineferous aether theory as presented by Maxwell. Instead, this experiment apparently disproved Maxwell’s concept, and the results were based on the classical predictions of the time and were limited by the technology employed by Michelson – Morley. This experiment was expecting a large shift in wavelength, based on the classical mechanical predictions. That shift (via noted interference) was not observed by Michelson – Morley. A century later, and with huge leaps in technological capability being resulted over the time involved, Alain Aspect in Paris conducted a similar, technologically superior, experiment compared to Michelson– Morley. This experiment proves interference occurring during the transmission of light, yet the noted interference is on such a miniscule level as to have been nondetectable as observed (perhaps expected) by Michelson – Morley. On the basis of the Aspect experiment (and of many others like it), we suffice it to state that this experiment apparently shows that the Sagnac effect exists, as it is real (based on the fringe interference patterns observed). The Aspect experiment then is not considered as explaining the Sagnac effect, or that the consideration is given to the plausibility of the Special Theory predicting the Sagnac time difference. The consideration is tendered that the Aspect experiment identifies an observable phenomena outside the technological capability of Michelson – Morley to measure, or to prove or disprove. Essentially, the Michelson– Morley experiment was a verification of the one-way speed of light, on a linear basis. A two-way experiment, where light is simultaneously emitted and sent in two opposing directions, shows interference fringe patterns, while the one-way experiments do not. The fringe patterns in the two-way experiments (Aspect/Sagnac) verify the existence of the displacement of light equal to c 2 v. Concerning both experiments, with the use of mirrors, a frequency of light is sent out in one circular direction. At the same time of this initial propagation of light, the same frequency is sent out in the opposite direction. When the two rays of the same frequency are received at the initial starting point, we find that

Relativistic coordinate shifting 1107

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fringe interference has somehow occurred (i.e. the Sagnac effect, or time difference). Therefore, the Sagnac effect notes a change in wavelength when the two frequencies are compared. The displacement c 2 v is then considered to be an effect shown by the Sagnac effect. We begin the description with the following equation, where the period S is the expression of the number of days the earth results in one full orbit about the sun. The period Y is that same time expressed in seconds, while the period T is the time expressed in seconds that resolves one full earth rotation about its axis. S ¼Y 4T

ð22Þ

1 ¼ ½{1 4 g 0 } 4 ½{ðc 2 vÞ 4 c} 4 p 4 S 3 

ð23Þ

The quantity g 0 is then

The error in coordinate positioning concerning the GPS platform is, in one sense, termed CEP. At a geosynchronous orbit identified by the distance x, the CEP.

g 0 ¼ {CEP 4 x} £ c

ð24Þ

CEP ¼ {g 0 £ x} 4 c

ð25Þ

where

Deriving, relative to equation (23): CEP ¼ ½{x 4 c} 4 ½{c 2 v} 4 p 4 S 3

ð26Þ

The Sagnac time difference at the distance x T Sagnac ¼ {1 4 CEP} 4 c

ð27Þ

The Sagnac time difference increases as one approaches the earth surface, and decreases as the distance from the earth surface increases. The CEP decreases as one approaches the earth surface, and increases as the distance from the earth surface increases. The values predicted above are known values from the GPS platform at the distance involved. B 0 is displaced from B as a function of CEP as shown in Figure 2. The point A will then be displaced to C 0 , per the CRE, or circular return error, per the time involved. According to the Lagrange path perturbation of Figure 1, path must be additionally applied from B to C 0 as per the manner shown, in order to derive a Lagrangian coordinate system applicable to the objects viewed within the given frame. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CRE ¼ CEP2 £ {x 4 c} þ 1 ð28Þ

From Figure 2, we now algebraically determine the coordinates of the points B0 and C 0 . For B 0 , the displacement of coordinates will be defined by the distances d on the y-axis and by e on the x-axis. For point C 0 , the displacement of coordinates will be defined by the distances p on the y-axis and by q on the x-axis. The fixed earth centre Q is x ¼ 0; y ¼ 0. In the following, the distance x is the distance from the earth’s equator to a body in geosynchronous orbit and the distance r is measured from the equator to a fixed earth center Q ( Table I). Therefore, in the following the points B-B 0 form a right triangle from the distances d, e, b. For the points A-C 0 , the distances p, q, r 0 form a right triangle. Then: a ¼ CEP 4 2

ð29Þ

c 0 ¼ ½a £ {c 2 v} 4 ½{1 4 2} 4 {c 2 v}

ð30Þ

b2 ¼ c 0 2 2 a2

ð31Þ

d ¼ {b 4 c 0 } £ a

ð32Þ

e2 ¼ b2 2 d 2

ð33Þ

Relativistic coordinate shifting 1109

The shift of coordinate positioning (considered as a displacement of B-B0 ), is assumed to be a result of the displacement of light (signified as the product of c 2 v), in conjunction with the predicted CEP (itself herein considered as a formulation of the Sagnac effect). However, the predicted coordinate shift from B to B0 , relative to one moment in static time reference the relativistic light frame, deduces the aforementioned CEP. If the theory is to remain congruent, a shift in coordinates from A to C 0 must be accounted for, and the CRE must allow that accommodation. Concerning the following, we refer Figure 2. The light is assumed to be propagated at A, then leave A and allowed to travel to B 0 . The light is assumed then to be allowed to leave B 0 and travel back to A. Since the point A is Points Q A B B0 C0

x-axis 0 r r+x {r + x} 2 e r2q

y-axis 0 0 0 2d 2p

Table I.

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considered to be statically invariant during the time concerned , the observer (originally at A) moves in a coordinate manner per the rotation of the earth to the point A0 (during the time concerned as defined by x/c). Within Figure 2, additional displacements of coordinate positioning are shown as points C and C 0 , depend on the frame of reference relative from either A or A0 , or to C or C 0 . From A, the displacement to C or C 0 is exactly equal. From Figure 2, the point A relates to the point q(t0) in Figure 1. The point B 0 relates to the point q(t1) of Figure 1. Considering that the coordinates of A are static during the time period involved for the transmission of light in a single direction (or in that the end points are considered clamped under the limitations as defined by Lagrange); then the light path in one full Maxwell circulation derives the Lagrangian path perturbation of Figure 1. Then p ¼ {d £ CRE} 4 CEP

ð34Þ

q ¼ {e £ CRE} 4 CEP

ð35Þ

r 0 2 ¼ p2 þ q2

ð36Þ

b ¼ {CEP £ r 0 } 4 CRE

ð37Þ

CEP ¼ { p 4 d} 4

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ {x 4 c}

ð38Þ

From equation (38), where b finds b ¼ {st 0 4 t 0 } £ {CEP 4 x} £ c: Then: {st 0 £ c} 4 x 0 ¼ st 0 4 t 0

ð39Þ

The quantity st 0 was derived as a unit of time. According to the velocity vo of the observer, this unit of time derives the displacement p of the light from the point A. vo ¼ {2pr} 4 T

ð40Þ

p ¼ vo £ st0

ð41Þ

Then:

The observer, during the measured time t in his frame, has shifted per the rotation of the earth, the distance l.

l ¼ t £ vo

ð42Þ

We conclude this paper considering that, based on the use of light to determine spatial coordinate positioning, a definable error exists between the observers determination of his coordinates and the coordinates of the position derived by the use of the light. If an observer determined his coordinates at A (by the use of a satellite in geosynchronous orbit) and wished to verify his future coordinates based on the time of the transmission of light from A to the satellite and back to his future position; then we derive that the positional error occurs as a result of the static nature of the light coordinate positioning, where the Lagrangian requires the end points to be clamped. We allow the observer to be at A, send the light signal to a geosynchronous satellite station, and to receive the signal at a position held by the observer at the point Z. The observer’s station is fixed at exactly a point on the earth’s equator, and we allow the observer to shift from A to Z as the earth rotates about its axis. Where the arc distance from the observer originally at A to the displaced coordinate C 0 is the distance p, we find the relative total error in coordinate positioning to be EP. EP ¼ ½vo £ {x 4 c} £ 2 þ p

ð43Þ

Therefore, the observer would determine an error in linear measure of coordinate positioning of 113.20 m. By adjusting the original coordinates of the observer in order to account for this displacement of coordinate positioning, actual real-time coordinates may be determined by the use of the light, where the light is construed as a particle. If the error EP is distributed between the x and y axes, we find an error of ye on the y-axis and an error of xe on the x-axis. xe ¼ {EP 4 4} 2 ½{q þ e} 4 2

ð44Þ

ye ¼ xe 2 { p £ 2} 2 {q þ e}

ð45Þ

p ¼ ½{EP 4 4} 2 {l 4 2} £ 2

ð46Þ

ye þ xe ¼ {EP 4 2} 2 {2 £ p} 2 ½2 £ {q þ e}

ð47Þ

where

These are the coordinate positioning errors as currently defined by the GPS platform, where the predicted horizontal error (xe) becomes 27.63 m and the predicted vertical error (ye) becomes 21.99 m. The known horizontal error exhibited by the GPS platform is given as 27.7 m, while the known vertical error is given as 22.2 m, where the total

Relativistic coordinate shifting 1111

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uncertainty is given as 0.1 m. We report the solution, considered to be exact to within 4 decimal places. 3. Values c ¼ 299; 792:458:2141 mps: CEP ¼ 3:075502859677 ms: a ¼ 1:537751429839 m: c 0 ¼ 1:703316943605 m: e ¼ 0:315037757603 m: q ¼ 1:025104130045 m: S ¼ 366:2421977259 days: Y ¼ 31; 556; 925:79593 s: t 0 ¼ 4:202130529419 £ 1026 s: x 0 ¼ 1259:767041151 m: st 0 ¼ 4:634091486978 £ 1023 s: RE ¼ 1152:481850232 m: G ¼ 6:67259 £ 10211 Nm: g ¼ 14; 179:22504331 m: g 0 ¼ 25:76177399302: EP ¼ 113:2017033258 m: l ¼ 55:5248942301 m: ye ¼ 21:9863832687 m:

v ¼ 299; 792; 457:4699 mps: CRE ¼ 10:00740580243 ms: b ¼ 0:7325361085987 m: d ¼ 0:6613322626981 m: p ¼ 2:151914865641 m: r 0 ¼ 2:383605686015 m: T ¼ 86; 164:09029838 s: t ¼ 0:1193824175505 s: x ¼ 35; 789; 948:42502 m: r ¼ 6; 378; 136:5 m: b ¼ 28; 409:97363474 m: ME ¼ 5:967527819679 £ 1024 kg: p ¼ 3:14159254359: s ¼ 14; 204:98681736 m: TSagnac ¼ 1:084583920676 £ 1029 s: vo ¼ 465:1011042444 mps: xe ¼ 27:63035488763 m:

Further reading Hoffmann-Wellenhof, Lichtenegger, B. and Collins, J. (1997), GPS: Theory and Practice, 3rd ed., Springer-Verlag, New York. Parkinson, B. and Spilker, J. (1997), Global Positioning System: Theory and Practice, Vols I and II, American Institute of Aeronautics and Astronautics, Inc., Washington, DC.

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Time-lapsed reality visual metabolic rate and quantum time and space

Visual metabolic rate

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John K. Harms Tandy Corporation, Austin, Texas, USA Keywords Time, Cybernetics, Entropy, Space Abstract This text proposes that time is essentially related to one’s visual metabolic rate. Metabolic rate is regulated by the speed of the intake of energy, the rate of the production of adenosine triphosphate (ATP) to the visual areas. The author’s working hypothesis is that the visual area of the brain known as human V5, the region involved in motion detection, may be the region most responsible for time perception. Our time sense from all the senses is, thus, compiled in the visual region, the speed of the human perception of reality. Time is the relationship of the human perception of reality and the rate that the reality itself is taking place (given by light waves in the environment). Hence, vision (and the visual cortex area V5) may be the vitally important aspects in answering the question: what is time? When we are not looking at a clock, time may be governed by our rate of metabolism; rate of the production of ATP by the mitochondria in V5. For example, when general human metabolism (and V5) is fast, time runs slow. When metabolic rate is relatively slow, time runs relatively faster. Many factors enter into the speed of metabolism such as age, sex, drug effects, velocity compared to speed c, states of boredom or excitement, darkness or light and mental states such as sleep. The relationship between time and space is discussed with the metabolic rate of V5 in mind. Because the uncertainty principle and the quantum picture of reality are adopted, this model qualitatively quantizes space and time, showing why they must forever be connected i.e. space-time. This idea is discussed in relation to Zeno’s paradox, which suggests that space and time are indeed quantized. Events, instants and entropy are defined. Reality can be understood in terms of the speed of the processing of instants. The arrow of time is pictured as caused by long-term potentiation of synaptic neurons within the brain. Minkowski-Einstein space-time is analyzed and compared with the visual metabolic rate. The probable consequences of this model are proposed.

Introduction It is commonly asserted by physicists that either that time does not exist or it is absolute in a given reference frame. Indeed, there are a wide variety of opinions among physicists when it comes to the topic of time. It was also previously thought that before 1905, the space was absolute. Einstein showed us that space and time are personal quantities relative to how one’s reference frame is moving. In Einstein’s space-time continuum, the apparent linearity of events depends upon the observer. The questions still remain: what precisely is time? What is the nature of the relationship between space and time? Why must changes in time also affect space? Why does time only flow in one direction – into the future? I wish to thank the brain-behavioral researcher, Gene Johnson, for his input about the synaptic processes of long-term potentiation and long-term depression. Comments by Rohn Roth concerning instants were also useful.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1113-1128 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483216

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It is proposed by this text that time is essentially one’s personal metabolic rate, the rate of production of adenosine triphosphate (ATP) by the cells in our visual system. Time, hence, is related to our visual sense, a process associated with the visual cortex region V5 within the brain. Moreover, it will be shown that time has an effect on space.

1114 Time-lapsed reality Reality is pictured here as analogous to a moving picture camera. When the camera takes pictures at high speed (a high-speed camera) and this in-turn is played back and projected on a screen, the motion on the screen will appear to run slow. Conversely, if the pictures are taken in a time-lapsed mode (say one or two frames per minute), during projection the motion on the screen appears very fast i.e. time-lapsed. Reality is pictured as very similar to this analogy. When one’s metabolic rate (as defined by the changes in voltage of the receptors in the visual system) is very fast, the rate at which we perceive reality may be slowed. However, when this metabolic rate slows down, the rate at which we perceive reality (our consciousness speed) may be comparatively faster. Hence, time may be the relationship between the speed of reality itself (given by the frequency and speed c of the light waves in the surrounding environment) and the speed of our perception of reality i.e. cognition (as determined by our personal V5 metabolic rate). Subsequently, this can be understood as the rate of the “instants” that our visual system (the V5 area) can process. What we call perception is analogous to the almost instantaneous playback of this reality (in our visual cortex) – our consciousness. Therefore, there may be an inverse relationship between the metabolic rate in our visual cortex and time. Time, therefore, may be essentially one’s own visual metabolic rate. Hence, the rate that time passes ¼ 1/Cortex V5 metabolic rate. Thus, as the metabolic rate of our visual sense increases, the rate at which time passes may become slower. Time is visual metabolic rate Einstein showed us that time and space are personal (observer-dependent) quantities related to our states of motion. Our internal clock is relative not only to our state of motion, but also the metabolic rate of the visual sense. Again, this is closely related to the operational speed that our cells can process visual information (or instants). The author believes that time, thus, is associated with the visual cortex located at the rear of the brain. The author’s working hypothesis is specifically the area V5 of the brain, the area involved largely in motion detection. Respiratory metabolism consists of a multitude of chemical steps where food is broken down and the energy released is utilized for the synthesis of ATP.

ATP is the fuel of the cell, the driving force for the electrical functions of the Visual metabolic visual system (Loewy and Siekevitz, 1969). rate Hence, the higher the quantity of ATP fabricated by the cellular mitochrondria in the V5 region, the higher the electrical voltage potential and visual processing speed of instants (Barrett et al., 1986). As we will subsequently see, if time appears to run fast, there may be a greater distance 1115 between instants than if time appears to run slowly. These separations may be associated with the operational speed of the V5 region. It may be the case that at high velocities (in a spacecraft, for example), there is an increase in the visual metabolic rate, hence, slowing the rate at which time passes. It is not argued that the perception of time is the visual metabolic rate, but that time itself is the metabolic rate of V5 area. Time is something that humans create and measure, and not something external to us. So, time exists in our conscious psyche. While physical objects may exist in “reality”, time is a far less concrete notion. Hence, time does not exist in what we call external reality, but lies and is coordinated with the senses within the brain itself. Drugs and time An accurate measurement of the overall metabolic rate does not exist yet. As we age, our general metabolic rate (and that of the V5 area) slows, so time passes at a faster rate. This is a quite common experience. Hence, people of the same age (in years) may lead lives of a different duration. Men oxidize their food approximately 5-7 percent faster than woman of the same age (Anthony and Thibodeau, 1979). Perhaps, this may be related to the shorter life spans on average of men versus that of women. The length of a person’s life may fundamentally depend upon one’s overall metabolic rate. This may demonstrate, as proposed by this text, that time is related also to the overall metabolic rate of the cells in the body. Sleep may be a state of the extreme slowing of one’s metabolic rate, so time may pass very fast when we are asleep. Perhaps, V5 minimizes itself when we are sleeping. When our metabolic rate (perhaps, also the V5 region) is artificially slowed by drugs, for example, the rate at which we perceive reality may be sped-up. So, drugs that increase the V5 metabolic rate may slow down the perception of reality. Hence, within the V5, there may be, in general, more instants. This is discussed in greater detail subsequently. Hyperactive children are sometimes given speed related drugs to slow down their thoughts. Psychiatric patients are sometimes given tranquilizers to slow their metabolic rates to match the rates at which they perceive reality, which may be very fast in some cases. A very long frightening period, just before our automobile crashes, demonstrates that time can run in slow motion when the rate at which we perceive reality is very fast. So, when we are in a period of extreme fear, our

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visual system and consciousness may operate at a very high speed, slowing down our reality. Similarly, a racecar driver may perceive typical highway speeds of around 80 km/h as being very slow. A racecar driver’s visual metabolic rate, when driving, may be very fast. Thus, reality may slow to a crawl at normal highway speeds for this type of person. This is the notion of “adaptation” and this concept may obey the rules of organic relativity. Organic relativity There is an old joke with a rumor that it was once told to Einstein. It said that relativity is the passage of time when one sits on a hot stove; a minute seems like an hour. But when a beautiful woman sits on your lap; an hour seems like a minute. This suggests that the relativity of time may be inherently perceptual. How can this be so? Time is absolute? It is usually argued that time is absolute. Our perceptions of time may differ, but time is the same for everybody (in the same reference frame). But what precisely is time as measured by a clock? How do we know that time is only measured by a clock? Why should we believe a clock or other measurement of time such as the movement of the Sun caused by the Earth’s rotation? How do we know that clocks are the true measures of time? Perhaps, time perception is rather analogous to subatomic particle experiments. The fact that we are looking at the clock affects it. Hence, humans and the measurement of time by clocks are interrelated with each other in the same complimentary reality. As in quantum mechanics, there is a difference in the result when you are looking at an experiment versus when you are not. Time in essence is a wave function that collapses when we actualize it by looking; or in another way, time exists, as subatomic particles do, in a number of random possibilities until we look at the clock. The act of looking at a clock in essence is a measurement and this act changes the result! The actual measure of time may be our own personal visual metabolic rate. Time may not be a constant when we are not looking at the clock. When we are not looking, time may be governed by our own personal visual V5 metabolic rate. So, each of us have a different visual metabolic time. What determines what time will be the next time when we look at the clock, is the metabolic rate of the V5 region of our brain. When one is not looking at a clock and, hence, making a measurement, our personal metabolic time governs the rate at which time flows and this can only be in the forward direction. For example, when one is bored, visual V5 metabolic rate may tend to increase. Therefore, one looks at one’s watch more often and time appears to run slower. When one is relaxed and having a “good time”, visual metabolic rate may tend to slow and time passes at a faster rate.

Thus, one looks at one’s watch less-often in this case. This is a quite common Visual metabolic experience. Hence, the common expression “time flies when you are having rate fun”. It is interesting to note that a person shut-up in a dark cave for 5 days will invariably guess that he or she has been entombed for a lower number of days than five. This implies that human beings are inherently incompetent when it 1117 comes to estimating time. Thus, all these hours, minutes, seconds, this so-called official time we have constructed for ourselves, has nothing to do with “real” time, the time as defined by our visual metabolic rates (Waugh, 1999). Indeed, this may also suggest that the state of light or darkness may have an effect on our metabolic rate and, thus, the rate of time. In the dark, we may inaccurately guess the passage of time. The late physicist, Richard Feynman performed some tests concerning time in which he practiced counting seconds to himself to 60. Feynman practiced counting at a standard rate and tried to determine what would affect that rate – his relative time. He tried heart rate, running up and down stairs while counting and also differences in temperature with no apparent effect on his rate of (relative) counting. In fact, Feynman found that no activity, he could think, affected his counting rate (Feynman, 1999). So, it must be the case that these activities must not be connected with the visual metabolic functioning of V5. It is interesting to note that when one has a rising fever, then he tends to count to 60 more rapidly, suggesting that a rising fever in some way may affect visual metabolic functioning. When the fever falls again, the rate of counting slows as well. Hence, a rising fever may have an effect upon visual metabolic functioning (Feynman, 1999). However, despite a somewhat accurate relative rate of counting, when compared to a clock, a person counting to himself rarely counts 1 min as 60 s – the absolute time as measured by a clock. Indeed, counting to oneself may not actually be able to be sustained with any kind of precision for periods of over 10 min or so. This is because, in general, the greater the amount of time spent in counting, the greater may be the tendency for inaccurate counting. As mentioned earlier, people in dark caves guess quite poorly at the length of time that they have been in the cave. Even if the cave’s occupants were able to count the entire time they were in the cave, it is doubtful that their absolute time would be anywhere near accurate. Visual metabolic functioning may, therefore, be a longer-term effect, affecting time lapses of above 10 min or so; or, perhaps, darkness is related to metabolic functioning. How can this be so? It is well-known among neuroscientists that the photoreceptors located on the retina are “on” in darkness, but “off” in visible light. Surprising as this may seem, the voltage potential sent across the neuron receptor is actually reduced, when a light is shown in the eye and increases in darkness. This is known as the “dark current” (Hubel, 1995).

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Perhaps, this is one of the reasons that in the dark, people may estimate the passage of time so poorly. The dark current, an increase in voltage by the photoreceptors in the darkness, may, therefore, be closely related to the processing of “instants” within the V5 region. Perhaps, the excess voltage sent to the V5 area in the dark may tend to increase the number of instants produced, causing an apparent slowing down of the passage of time. Quantum time and motion Time is related to space, distances and motion. This is because if one is traveling in a car at an uniform speed and direction, then one’s distance perception is related also to the rate at which we perceive reality (as time). Since this quantity of space may also be affected by visual metabolic rate (and the consciousness playback speed) as time, the judgement of distance and space measurement may also be related to the time element. Thus, when one is in uniform motion, the relationship between time and space can be better understood. Moreover, time and distance are related to speed. So, as time slows at high speeds close to the speed of light (c), distances (space) Lorentz-Fitzgerald must contract in the direction of motion. This must also be true as one’s visual V5 metabolic rate speeds-up. Can it be that very high velocities (or the location of a body in a gravitational field) speeds-up the visual metabolic processing rate of human beings? This is one of the predictions of this model. If one assumes that visual V5 metabolic rate is affected by motion, then time and space must be joined. It is generally taught in universities that one can never find a mechanism or a reason why time dilates and space contracts at speeds close to c. While observers in a spacecraft will not notice the relative slowing of clocks and contraction of space, their relative visual V5 metabolic states may be sped-up, compared to observers in other frames at lower speeds relative to c. Hence, an increase in metabolic rate may describe the physical process of time dilation and space contraction. By following the uncertainty relation and quantum mechanical principles this may, therefore, lead to a quantum description of space and time. Visual reality is essentially composed of light waves i.e. light quanta. Space and time, which are both not possible without visual reality, must, therefore, also be quantized. Thus, both time and space are quantum effects and related in this proposal to the human visual sense, the processing speed of V5. Zeno’s paradox and quantum time and space The ancient Greek thinker Zeno proposed an insightful paradox, which demonstrates that space and time are unavoidably quantum systems. While Zeno’s Paradox (as it is called) was merely a puzzle in its day, the idea actually

strikes right at the heart of relativity and quantum thinking more than 2,000 Visual metabolic years later. Below is a short description. rate To set the problem. A male athlete runs a race of 100 m; how is he to cross the finish line at the far end? For him to get there, he need to pass the halfway point of 50 m first. Once he has arrived at the 50 m mark, he will still have an equal distance to run; yet to complete the last 50 m, he must first pass the 75 m point. 1119 Wherever he is, however far from or close to the finish line, the athlete has a remaining distance to travel which he cannot complete before he has first run half the distance. As we keep dividing the remaining distances into two, he will get closer and closer to the finish without ever reaching it! Hence, our athlete will never finish the race (Waugh, 1999). This is known as Zeno’s paradox. Moreover, by inverting this same line of logic, our athlete can never even start the race. For the athlete to reach the finish line, he must get to the 50 m mark, but to get there he must have already passed the 25 m point. But how will he reach the 25 m mark if he has not already arrived at the 12.5 m point? And so on and on. He cannot go 1 mm until he has gone half a millimeter (Waugh, 1999). So, the athlete can not even start the race much-less finish it! What is fundamentally wrong with this picture of reality? Why does not this description agree with our common sense notions? It must be said that this is not some sort of trick either with words or with mathematical reasoning (or some combination thereof); it essentially indicates that time and space must inherently be quantum systems. This paradox is only a paradox if space and time are continuous systems – which is assumed to be true by most people. If the athlete succeeds in reaching the finish line, it must be that at some stage he reaches a point so close to the end line that the remaining distance between him and the finish is so minute that it cannot be divided any further – so, he crosses the finish line (Waugh, 1999). Common sense, therefore, is saved by quantum space and time! Zeno’s paradox is resolved. It is generally thought by physicists that space and time can be quantum systems, if one invents a quantum world based upon Max Planck’s work. For example, a Planck space (or length) is 102 33 cm or 102 20 the size of a proton. A Planck time is the time it takes for light to travel the Planck length or 102 43 of a second (Waugh, 1999). These are physical inventions to understand the world and may have no physical meaning. We do not, yet, have the technology to probe matter on these scales or measure time with this kind of accuracy – so we actually do not know at this point. The external viewpoint has been the invention of the Planck length and time, but what if the actual quantum system exists (as has been the ongoing theme of this text), not external to us, but lie entirely within the brain itself? What if what we deduce as Planck space and time are actually the inner-workings of our own internal quantum based system, the human brain? Indeed, not only is the visual system, a quantum system because it absorbs light quanta, but also the brain itself is an electrochemical firing mechanism – the brain is not continuous, but

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a discontinuous quantum apparatus. Hence, if we perceive reality as “quantum”, space and time also must be quantum systems. Reality may, therefore, be divided into instants!

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Events, instants and motion Since time is unavoidably connected to space (as described earlier) as Einstein showed us, the characteristic that best defines time is what is called “events” in physics. Time is called a byproduct of events. Events give us a sense of the before and after. Hence, if there are no events, there can be no time (Waugh, 1999). An event takes energy. An event is essentially the correction of an imbalance of energy in the Universe. The Universe is always seeking an equilibrium state of energy or space-time flatness. The event that defines time here is the firing of neurotransmitters within the brain; the brain always seeks flatness and equilibrium. The brain is, without any doubt, a quantum system that regulates the flow of time, operating through the increasing entropy and the firing of neurotransmitters. The other approach to divide reality into a quantum system may be to use the “instant” methodology. If one takes still pictures of a body in motion, the picture can often be blurry. Increasing the shutter speed until the picture has become absolutely clear gives one an instant in time. This instant must be related to the frequencies of the visible light that is entering the eye. At that instant, there can be no motion and (as described earlier) no time. Hence, motion and time simply do not exist at that unique instant. So, the question arises, what precisely is motion? Motion must be the perceptual differences that exist from one instant to the next. Therefore, motion may only take place between the instants in time. Here we are not talking about the instants as given to us by the frequency of light entering the eye, but the processing speed of these instants by the V5 area. The V5 area may be the region where all the senses compile the data associated with time, our internal clocks so to speak. Since motion occurs between instants, perhaps, it is the case that motion is added to what we call “reality” by us; by our brains. Therefore, motion is what happens when we do not actually perceive visual reality! What we see as the flow of time, thus, may be simply a sequence of instants, the quantized snapshots of what we call “reality”. These snapshots are only the limited perceptual versions, which are given to us by the external world. Hence, to the brain, an instant in time may be inherently visual, so an instant has partially to do with the speed of light and its frequency. This is given to us by our visual experience. How our metabolic rates fit into this scenario is that our perceptual speed (fast or slow) can change the rate of time. For example, if time flows fast to us, then there may be more distance between instants

(the instants are farther apart from each other). So, we gather less information Visual metabolic in the form of instants from the visual world. rate If, on the other hand, time runs slow to us, the instants produced within the visual cortex are relatively closer together. We, therefore, take in a greater amount of “instant” information. Within the visual system at V5, essentially this is what is meant by perceptual speed. 1121 Therefore, when time appears to run slow, visual metabolic rate and perceptual speed are fast and reality (for us) consists of relatively more instants. However, when time is fast, visual metabolic rate and perceptual speed are slow, then there are comparatively fewer visual instants produced in that “reality” from our perspective. So, it is the metabolic rate of the visual cortex V5 area that affects the distance between our instants i.e. the speed at which these snapshots are presented to our consciousness. So, why is the human V5 area of the visual cortex involved in time? This is because the V5 area is the region of the brain that processes motions of all kinds. Any damage to V5 can destroy the ability to perceive the direction or coherence of motion (Zeki, 1992). As stated earlier, motion is related to the processing of instants (motion occurs between instants). Since instants can be understood as the quanta of time, motion and instants are unavoidably connected with each other. Therefore, this is the reason for the author’s hypothesis that the V5 area must be involved in time. In cases of extreme tiredness or drunkenness, the V5 metabolic rate may actually be slowed down. So, time flows at a somewhat faster rate and fewer instants on average present themselves to our perceptual consciousness. One can understand, therefore, why reaction time can (for example, while driving an automobile) be slowed either by being overly tired or when intoxicated with alcohol. Reaction time may be slowed because our “reality” consists of on average fewer instants of time with which to react and make any kind of rapid decisions. As mentioned previously, during sleep, metabolic functioning of V5 may slow to a crawl (perhaps, because no visual information is then being processed) and time may in turn speed-up. It is common to wake from a sleep and find that much time has passed. Where did it go? So, in the above viewpoint, instants become the quanta of time; this provides the resolution to Zeno’s paradox. Time and entropy The arrow of time (that time has a direction) may be related to increasing entropy. This is also the view taken by many physicists. As suggested in the author’s other texts, increasing entropy is increasing space-time flatness. The brain is constantly increasing flatness through the transmission of electrical signals (for example, within the visual system). These processes only take place in one direction; the direction that time flows into the future.

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Electrical impulses (potentials) increase flatness in the same direction that time flows – the arrow of time. It is ironic that these impulses do increase the order and complexity of the brain, locally violating entropy. However, on a somewhat wider scale, the production of heat by the intake of chemical energy (food) by the body increases entropy more than the smaller quantity of complex ordering taking place in the brain. However, what if the arrow of time does not define the entropy of the system? The Nobel Prize for Physiology or Medicine for 2000 was awarded to Arvid Carlsson, Eric R. Kandel and Paul Greengard for their contributions to neuroscience. Among their discoveries was how the nerve cells communicate with each other. Neurons send signals to each other using dopamine, a synaptic messenger that may regulate mood, movement and even the way the brain responds to drugs and alcohol. Kandel’s work investigated how synapses change shape and function producing memory and learning (Travis, 2000). Changes in synaptic function, known as “long-term potentiation”, are required for learning and memory. This discovery may be the key to giving a direction to time within the brain. Like entropy, one cannot learn something before a change in synaptic function takes place (thus, one can never remember events in the future). Therefore, it is the brain that gives a direction to time and it may also be that the collective processing speed of instants by the visual cortex V5 area that determine the speed at which reality happens; an important component of the rate of the passage of time as described earlier. Consciousness (being the continuous playback of instants), however, travels only in the forward-in-time direction. What we call “visual” consciousness must be dependent upon the speed of light. Experiments have demonstrated that when a photon exceeds the speed of light, it may travel backward in time. Hence, the direction of visual reality must in some sense depend upon the speed of the photons entering our visual field. Therefore, if faster-than-light photons do enter our visual field, we may see a backward-in-time reality! Since (as far as we know) most photons on average do not exceed the speed of light, we experience largely a cause and effect forward-in-time reality. The direction of time, thus, is based not only on the events within the brain, but also on the speed of light in the visual field. Time, motion and relativity Recall that the visual metabolic rate (hence, also time) is affected by motion. Changes in one’s motion may alter time, so time and space must be connected. Hence, V5 metabolic functioning and time may be equivalent to the Einstein relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t ¼ t £ 1 2 v 2 =c 2

It is important to recognize that all the objects in one’s own reference frame Visual metabolic (if one sees them) are essentially in resonance with one’s own visual system. rate Otherwise, these objects (living or nonliving) would not at all be observed. It can, therefore, be understood in Einstein’s equation that as one approaches the speed of light, one’s personal time slows down and metabolic V5 functioning may speed up. So, it can be deduced that if time is affected by 1123 motion and motion (as mentioned earlier) happens between instants, then time itself must take place within the visual system between instants! Therefore, the speed of the V5 area actually processes instants (and the number of these instants given to us effects time), but time itself physically takes place between these instants. What we observe as our external “reality”, therefore, is when there is an alignment (or resonance) of our bodies with our surroundings i.e. within the same frame of reference, at the same time. Other individuals, as mobile wavelike units glued to the Earth as we all are, may pass through this same “reality” as we do. We may encounter these individuals or objects, if at some point, these bodies pass through our space and time and come into resonance with our own experience during our lifetime. Time is relative and strictly personal It is important to note that time is only personal, from our own point of view. Thus, we cannot actually ever know what other points of view are doing. While other people may pass through our “reality” with us, we cannot know any other information about them than from our own perspective. So, time is relative. In special relativity, observer #1 in one frame of reference may observe that observer #2’s (in some other reference frame) time is running slow, while observer #2 may observe the same of #1 – that their time is running slow relative to their own. This appears in this context, and does so in special relativity also, to be somewhat of a paradox. However, in this model, this type of actualization of time simply cannot be allowed as a legitimate observation! Therefore, this model prohibits measurements of time across different frames of reference, either freely moving or accelerated. Time is, therefore, only personal. This also appears to be an important theme in quantum mechanics as well. So, for this to be a viable model, the author has realized that certain ground rules for allowable observations of time must be laid down. Perhaps, it is these kinds of necessary prohibitions that may also make the paradoxes in special relativity somewhat more understandable. So, widely separated observers moving at different velocities with varying visual V5 metabolic rates may carry on board with them, clocks to measure relative time. Hence, observers present in the same entangled reality may affect these clocks. When it was mentioned previously that the observance of a clock affects it, what this meant was that moving clocks close to the speed of light

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run slow because of the presence of an observer. So, as in quantum mechanical experiments, the functioning of a clock is affected by an observation of it. One can, therefore, deduce that time exists as a series of random possibilities, until someone actualizes a clock or some other measure of time by observing it. This being the case, what does it mean when we bring together and compare two previously synchronized clocks that have been in relative motion at high speeds, and find that their times are somewhat different? Hence, what is the relationship between the time sense (in the brain) and the movement of hands on a clock? Since it is argued by this text that the actual notions of time are determined within our visual cortex V5 region, the clock experiment with two clocks moving in different ways can have no physical meaning whatsoever! Hence, any experiment involving clocks alone (utilizing whatever high technology that is available) can have no meaning without the presence of a human observer. So, for this model to be viable, two observational conditions must be met. (1) As mentioned earlier, measurements across accelerated or freely moving reference frames are prohibited. Any technology to measure relative time in this fashion, therefore, violates the rules for an acceptable observation or actualization of time. (2) Measurements strictly utilizing clocks alone can have no physical meaning (when the clocks are brought together again and in the same reference frame). We, thus, can draw no meaningful conclusions from any of these types of observations. Therefore, these observational rules tend to limit somewhat the physical meaning of the time-based experiments in special and general relativity. It should be noted that it is not the author’s intention here to doubt the findings of either special or general relativity. Rather, it is to more precisely define and understand the meaning of time. Quantum mechanics does suggest a very personalized approach to observe the Universe. Relativity suggests that time is a personal quantity, relative to the person who measures it. The personalized nature of time (as described earlier) brings together the ideas of quantum mechanics and relativity into a single unified system.

Philosophical considerations Let us discuss briefly the philosophical considerations of the subject of time. While it is generally considered true that (what we call) “physical reality” is sending signals from the environment at speed of light to us that we measure as time, for example, the tick of a clock on the wall. This is what Immanuel Kant calls the noumena (or the things-in-themselves) sending signals to become what we experience as the phenomenal world – the world of the phenomena.

There is, thus, some physical reality (or wave energy field etc.) “out-there” Visual metabolic that is sending signals to us at the speed of light, that we experience as our own rate personal reality. What essentially is proposed by this document is that the “out-there” noumenal world of the things-in-themselves are the photons we experience with our eyes – our visual sense. It should, therefore, turnout to be correct that events are regulated by the 1125 processing speed of our visual sense – the rate at which V5 processes instants. This processing speed and cortex metabolic rate was the primary focus of this paper, as well as the effect of the speed of light. Events may happen all at once in “external reality”, but the processing speed of our visual sense can never be over-stimulated in this manner, so the brain itself creates a separation or a barrier (a flow of time) between the different instants. These events (or instants) give us little time, so as not to overload the neurological system. This is what we experience as time. If everything did indeed happen at once, our lives would be in such utter chaos – so our brains give us only what we can safely handle at each unique moment. The speed of this reality coming to us as well as the direction of time is governed by the speed of light. However, the speed of the reception of this reality as based upon our cortex V5 metabolic rate must be strictly a personal experience, hence, it is not the same for everyone. So, as mentioned previously, time is strictly personal. Recent experiments (as of 2000) support the idea that everything actually happens at once. The faster-than-light photon experiment demonstrated that when a photon travels faster than c, it arrived at the detector before it was emitted. While this appears to be logically impossible, it can be made sense, only if the cause (the emission of the photon) and the effect (the photon’s arrival at the detector) both exist side by side happening all at once (existing “out-there”) before the experiment. The faster-than-light velocity simply reversed the order of our visual sense of this perceived reality. Hence, events may indeed happen all at once, but the time direction that we experience is in a real sense governed by the visual speed of the visible light illuminating this reality for us. Therefore, special relativity is essentially correct in its prediction that time becomes reversed at above the speed of light c. So, there must essentially be no time at the level of Kant’s noumena. Reality at the noumenal level, therefore, must happen all at once. However, the noumena grants us that reality (in the form of signals) in small separate quantum-sized chunks; a quantum reality that our brain and visual system can tolerate at each unique moment. The physical separation of these signals (or instants) by our V5 receptor consciousness playback system is the phenomenon of time. The V5 compiles all the sensory data associated with time from each sense in one location. If the signal is given to us faster-than-light, we then view it in the reverse order (as mentioned earlier).

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Minkowski-Einstein space-time The Minkowski-Einstein space-time theory suggests that all time is out-there coexisting (as suggested earlier). When we look at the stars, we are looking backward in time. Indeed, our future is contained in the light waves now on their way to us. What we are seeing now in the dark sky is what it was like millions of years in the past (Waugh, 1999). Any single atom, which is deemed by us to exist in a certain place, also exists at other times and at other places, if viewed from a different perspective. Hence, all times coexist (as mentioned earlier), although on a separate plane, our deaths, our lives, every decision we have made, including the beginning and the end of the Universe, all are out-there inhabiting their own present (Waugh, 1999). The rate at which we are given this reality is limited in the Universe by the speed of light. What we call time is the relationship of the speed of this reality c, to our individual V5 visual cortex processing speed of instants – our consciousness. Time and free will It is worth to mention that if all time exists somewhere “out-there”, then our lives must all be predetermined, hence, we indeed have no free will – no freedom to choose our own destinies! This apparent paradox of free will can be resolved by assuming that all the possible future events of the Universe exist “out-there” also, so, only when we make some sort of personal choice (our freedom of choice) about some event, does this future event itself become actualized. A human, hence, has free will because he or she selects one of the many possibilities to become the actual history of the Universe. The wave function of our personal Universe collapses, actualizing what we experience as “now”. Humans are the “actualizers” of one’s own future lifetime events, despite the fact that one has only limited control over some events in one’s own life. The actual amount of control that a person has over these events will remain a matter of debate and unfortunately is beyond the scope of this text. It is also worth to note that Einstein’s relativity makes the idea of a universal “now” completely untenable, even as a logical possibility (Barbour, 1999). The actualization of what we experience as “now”, therefore, is a completely personal experience. Thus, another person’s experience of “now” cannot be the same as yours! Conclusion This “Time” model leads to the following probable predictions. (1) Time is compiled in one location as the visual cortex’s (in particular, the V5 region involved in motion) metabolic rate. When one is not looking at a clock, this personal metabolic time governs the rate at which time

(2)

(3)

(4)

(5) (6)

(7)

(8)

(9)

flows. Clocks and humans are an integral part of the same reality. They, Visual metabolic therefore, affect and are affected by each other. rate Time is related to the speed of reality as light waves and the perception of this reality by the V5 visual system. Perception (consciousness) is the almost instantaneous playback of this reality as instants. Hence, the V5 region, an area of the brain involved in motion that controls for us the 1127 speed at which instants are given to our consciousness. So, the V5 visual area is involved in time. It is a quantum system. Drugs may increase or decrease the V5 metabolic rate and change the rate at which time flows. So, drugs may also affect the rate of a person’s counting. Age and gender differences may change the overall metabolic rate (thus, also V5) and the rate at which time flows. Time flows faster when one is older because the overall metabolic rate slows down with age. Women may have different overall metabolic rates than men. Boredom may be an increase in the V5 metabolic rate, hence, time may slow. One may check one’s watch more often in this case. Having a “good time” or being interested in something may be the slowing of the V5 metabolic rate, thus, time may appear to pass more rapidly. In this situation, one may glance at the clock less-often. Electrical impulses (as increasing space-time flatness) in the brain may give a direction to time – the arrow of time. That synapses change shape to learn something new i.e. short-term memory, may give a direction to time. This long-term potentiation process in the synapses may be the actual arrow of time mechanism. A backward in time direction for light (for example, if light traveled faster than c) would tend to reverse the time arrow. High velocities close to c may speed-up the V5 metabolic rates of human beings and other animals. The relative metabolic rate increase for frames close to c may be noticeable. Slow moving frames may have observers with slower metabolic rates. Speed, therefore, increases visual metabolic rate and slows time for moving observers. This concept joins time and space. This must also be true for accelerated frames of reference, for example, in a gravitational field. Hence, gravitating bodies’ speed-up metabolic rates depending upon the location within the field. The closer a person is to the massive body, the faster may be their visual V5 metabolic rate. Both motion and time may take place in the brain in between instants. This is where change may occur. The processing of instants may take place in the V5 region of the cortex. So, the location V5 may be where time happens in the brain.

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(10) Time is personal. The observational rules for time measurements are that one cannot measure time across different frames of reference, in either accelerated or freely moving frames. In addition, time measurements with clocks alone have an ambiguous physical meaning, so cannot be relied upon. So, concerning time, one cannot say what is taking place with any accuracy in some other frame of reference (or with only clocks alone). When the moving frames do come together again, there may be a comparison of clocks and elapsed time – if and only if an observer was present. (11) The rates of clocks are affected by the presence of observers. Like quantum mechanical systems, the presence of an observer may change the result. This strictly “personal time” unites relativity and quantum mechanical concepts of time. References Anthony, C.P. and Thibodeau, G.A. (1979), Anatomy and Physiology, 10th ed., C. V. Mosby Co., St Louis, p. 526. Barbour, J. (1999), The End of Time, Oxford University Press, New York, p. 28,142. Barrett, J.M., Abramoff, P., Kumaran, K.A. and Millington, W.F. (1986), Biology, Prentice-Hall Inc., NJ, p. 128. Feynman, R.P. (1999), The Pleasure of Finding Things Out, Perseus Books, Cambridge, Massachusetts, pp. 218-22. Hubel, D.H. (1995), Eye, Brain and Vision, Sci. Am. Library, New York, pp. 48-9. Loewy, A.G. and Siekevitz, P. (1969), Cell Structure and Function, 2nd ed., Holt, Rinehart and Winston, New York, pp. 26-7. Travis, J. (2000), Science News: Pioneers of Brain-Cell Signaling Earn Nobel, A Science Service Publication, Vol. 158 No 16, p. 247. Waugh, A. (1999), Time: Its Origin, Its Enigma, Its History, Carroll and Graf Publishers Inc., New York, pp. 8-11, 14, 208, 226. Zeki, S. (1992), Scientific American, New York, pp. 69-76.

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The space-time equation of the universe dynamics Peter Kohut

The space-time equation

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Volgogradska, Presov, Slovak Republic Keywords Cybernetics, Dynamics, Space, Time Abstract Space and time are the forms of material being of the Universe. They have a quantum character as a result of a dialectic relation between attraction and expansion, continuity and discreetness. Time and space consist of their elementary parts – quantums. The question is: does the Universe have its own time, despite the fact that according to the theory of relativity, the local parts of the Universe with different gravity or inertia have different local times? Yes, it does! The time etalon for the entire history of the Universe is represented by the cosmic quantum jump, in which the Universe realizes its transition from one quantum level to the next. The time quantifying enables the cosmic jump from the expansion to the contraction to be made and on the other hand, from the contraction to the expansion without getting to singularities. The Universe pulsates. The knowledge of the nature of time and space quantifying allowed us to find the basic equation of the space dynamics of the Universe. Its analysis gives the answer to the question: what is the nature of gravity, speed of light and gravitational constant.

Introduction The physics is now in a deep crisis as a result of a contradiction between quantum theory and Einstein’s theory of gravity. The findings of the common base of these theories is one of the most important problems in physics. The Universe equation According to Kohut’s Integral theory of the Universe (ITU) published in Slovak Republic (Kohut, 1993a, b), all interactions – strong, weak, electromagnetic and gravitational – have the same base. It is the quantum of space-energy (basic elementary interaction). This basic elementary interaction is an etalon of space. The nature of space-time quantifying is described in Kohut (1993a, b) ITU. Now, we confine to note that the space and time of the Universe are quantified in such a way that the next basic space-time equation of the Universe is valid: V ¼ t2

ð1Þ

where V is the volume of space and t is the time of space expansion. dV ¼ 2t dt

ð2Þ

d2 V ¼2 dt 2

ð3Þ

In these equations, the space-volume and time are dimensionless quantities.

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dV dt

2 ¼

2V d2 V dt 2

ð4Þ

This relation stays valid, if we use actual dimensional units. The basic space-time equation (1) changes into:

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V ¼ kt 2 ;

ð5Þ

where k ¼ ðd2 V=dt 2 Þ=2 The quantity d2V/dt 2 is the fixed constant during the whole evolution of the Universe.
2  dV dV t ð6Þ ¼ dt dt 2 All these equations express the space-time unity of the Universe. The speed of expansion of the space volume is in direct proportion to the time of expansion. It accelerates unceasingly and this acceleration is constant. The Universe as a three-dimensional surface of four-dimensional sphere is closed with no space limit, but with the final volume: V ¼ 2p 2 r 3

ð7Þ

where r is the radius of space curvature. After substituting o with 2p r, we obtain: V¼ dV ¼ dt 2

dV ¼ dt 2




o3 4p

 3o 2 do 4p dt

h     i do 2 2 d2 o 3 2o dt þo dt 2 4p

From relations (8)-(10) and basic space-time equation (5), we obtain:
2 do 22o d2 o ¼ dt dt 2

ð8Þ

ð9Þ

ð10Þ

ð11Þ

The relations between the space circumference and time are: o ¼ kt 2=3

ð12Þ

do ¼ dt


 2 kt 21=3 3


 d2 o 2 ¼2 kt 24=3 2 dt 9

ð13Þ

ð14Þ

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where k ¼ ð2p d2 V =dt 2 Þ1=3 : These equations show that the space circumference is expanding in time, but the speed of this expansion decelerates. So, the acceleration is negative. It means that the speed of Universe expansion decelerates. Hubble’s constant is:

 do 2 21 H¼ ð15Þ o¼ t dt 3 The time of cosmic expansion is:
 2 t¼ H 21 3

ð16Þ

Equation (11) is the base for its modifications. If we explain the nature of gravity and the real sense of such cosmological constants like speed of light c and gravitational constant k, we obtain the most famous physical laws.

2   o
do2 2 d o 4 ¼ M 2o 8M ; 4 dt dt 2

ð17Þ

where M is the mass of the Universe, k ¼ ð2o 2 d2 o=dt 2 Þ=8M ¼ 2p 2 r 2 p ðd2 r=dt 2 Þ=M is the gravitational constant, c ¼ ðdo=dtÞ=2 ¼ p dr=dt is the speed of light. The light is emitted from one point to all directions, so the speed of its spreading in a single direction equals one half of the speed of the increase of space circumference. g ¼ 2ðd2 o=dt 2 Þ=2 ¼ 2p d2 r=dt 2 is the gravitational acceleration (deceleration of the space expansion). The mass of the Universe is affected by gravity from all directions; therefore, the gravitational acceleration in a single direction equals one half of the deceleration of the increase of space circumference. The speed of light expresses the speed of cosmic expansion. On the other hand, gravity expresses the deceleration of this expansion. The relation between light speed and gravitational acceleration is: g¼

2dc dt

The space-time equation

ð18Þ

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The gravitational acceleration of the Universe equals the deceleration of light speed during cosmic evolution. This is a very important result. The basic cosmological constants c, k, g expresses the basic characteristics of the cosmic expansion and they are changing during the cosmic evolution. The fixed cosmological constants are space energy E and the acceleration d2V/dt 2. From equation (17), we obtain: o ð19Þ c2 ¼ Mk 4


 pc 2 M¼ r 2k The mass density of the Universe is:

rM ¼

M pH 2 : ¼ V 4k

Then equation (11) has the form: c 2 ¼ go

ð20Þ

After multiplication of this equation by the mass of the Universe, M, we have: Mc 2 ¼ Mgo ¼ Go ¼ E;

ð21Þ

where G ¼ Mg is the force of gravity and E is the whole energy of the Universe. The force of gravity is equal to the force of expansion in the opposite direction. So, gravity force is a reaction to the space expansion (the law of action and reaction). This is Einstein’s famous equation for equivalence of mass and energy. On the other hand, it shows that the base of gravity and acceleration is the same. After multiplication of equations (19) and (21) we obtain: G ¼ Mg ¼

c4 ¼ 3; 036 £ 1043 N 4k

ð22Þ

So, we know the exact value of the gravity force of the Universe for the present time. This value decreases during the space expansion according to the following relation:
21=3 2p d2 V G ¼ E=o ¼ E t 22=3 ð23Þ dt 2 From the relation for gravitational constant k ¼ gðo=2Þ2 =M ; we obtain:

kM g ¼  2 o

ð24Þ

2

The whole mass of the Universe is affected by gravity. After multiplying a/m relation by M:

kM 2 G ¼  2 o

ð25Þ

2

It is Newton’s classical law of gravity for the whole Universe that may be written as follows:

  M M o 2 ð26Þ G¼k 2 2 4 In this modification, the whole mass of the Universe is divided into two equal parts. The average distance between matter objects of the Universe (the average length of elementary quantum of space-energy) is in reality o/4 as an average value between maximum distance o/2 and minimum distance 0. This important result confirms the way from the basic equation of the Universe to Newton’s law of gravity with assumptions about the nature of gravity, speed of light c, gravitational constant k and gravitational acceleration g is correct. Conclusion Gravity is quantified in such a way that every elementary quantum of space reacts to space expansion immediately. So, gravity is not spread by limited speed, but immediately. The EPR experiment and other ones just prove the immediate particle communication. Everything is connected with everything! The gravity has a quantum character. After the transition from expansion to contraction by the quantum jump, the gravity changes into antigravity and time turns in the opposite direction. Universe pulsation means the pulsation of time and space. References Kohut, P. (1993a), What is the Universe as the Whole Like? (Integral Theory of the Universe), Datapress Presov. Kohut, P. (1993b), The Nature of the Universe and Gravity, Technical News No. 37, Bratislava, p. 15.

The space-time equation

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Asymmetry of uniform motion Peter Kohut Bolstein Research Team, Presov, Slovak Republic

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Keywords Cybernetics, Motion Abstract A new theory on asymmetry of uniform motion is presented.

Introduction The basic problem of Einstein’s special theory of relativity (STR) is the absolutization of light velocity and the wrong application of its changeless value to all the systems with relative uniform motion including four-dimensional space-time invariant. Based on this consideration, Einstein (Minkowski) has built-up the incorrect conception of space-time that leads to the next absurd and incorrect result: “Two events synchronized in one frame need not be synchronized in other one”. This result is in contradiction with the unity and quantum character of the universe, which transits by quantum jump from one quantum level to the next one during its expansion. The actual quantum state of the universe defines the simultaneity not only for a whole space, but also for its every system. Such understanding of the universe leads to the bright principle: “If two events are simultaneous in one system, they are at the same time simultaneous in all systems”. It is interesting, that rather than to consider, whether this theory does not include some system errors, the absurd result of the STR was accepted. The well-known Twin Paradox shows very clearly the inconsistency in logical structure of the STR. This paradox can be removed only if two systems with mutual uniform motion originally considered like relative and equivalent are finally declared as asymmetrical in order to obtain time dilation in one of them.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1134-1137 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483234

Asymmetry It is clear that light velocity is a boundary value in vacuum. But it is also clear that light velocity depends on the medium in which the light propagates. This medium with its permittivity and permeability defines the velocity of electromagnetic wave propagation. Light is a form of energy propagating in material medium. The vacuum is also a material medium with physical characteristics (permittivity and permeability). Thus, vacuum is not a coarse void, deadness, nothing. All moving material objects are firmly connected with it, influence it and create together with it the material being of the universe. Therefore, the negative result of Michelson’s experiment is a consequence of contemporary motion of vacuum and the earth near its surface due to their mutual firm connection. Vacuum, as a link of all the objects till the level of elementary particles, is a

medium for their motion and underlies their influence. It represents the base of motion of material objects (systems). The mutual relation between vacuum and objects with high velocities is manifested by physical changes like deceleration of processes in moving objects (the so-called time dilation), length contraction and mass increase. These effects occur not because of mutual uniform motion of two systems, but just the other way, due to asymmetry of their uniform motion concerning their different speeds towards the material medium – vacuum. The so-called relativistic effects followed from the Lorentz transformations indicate the asymmetry of uniform motion of systems and confirm an existence of physical base – material medium (vacuum), which defines the measure of their asymmetry. Especially relict radiation, equally filling in our space, together with vacuum represent the base (modern ether), towards which velocities of other systems like earth, sun and galaxies refer. Existence of this base as well as effects followed from the Lorentz relations confirm asymmetry between systems with high velocities towards this base and systems with slow or negligible speed, even though the mutual motion of these systems is uniform. From the viewpoint of the systems at rest, the processes in systems with high velocities slowdown and also at the same time light velocity slowdowns from c to c 0 . Let us make one assumption. From the viewpoint of the system S deceleration of light velocity in the system S0 is: c 02 ¼ c 2 2 v 2 : This relation leads to the Lorentz transformations. As the change of time flux expresses the change of velocity of processes, so the next relations are valid: cDt ¼ c 0 Dt 0 ; vDt ¼ v 0 Dt0 ; where c=c 0 ¼ Dt 0 =Dt ¼ v=v 0 : Let us explain the sense of a/m parameters and show their difference in comparison with parameters in Einstein’s STR: . c – light velocity in the frame S, . c 0 – light velocity in the frame S 0 , . Dt – the duration of the certain process in the frame S, . Dt 0 – the duration of the same process in the frame S0 from the viewpoint of the frame S,

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.

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v – the difference in velocities of frames S and S0 towards the base from the viewpoint of the frame S, or the mutual speed of frames from the viewpoint of the frame S if they move in the same direction, or if the frame S is at rest towards the base, v 0 – the difference in velocities of frames S and S0 towards the base from the viewpoint of the frame S0 , or the mutual speed of frames from the viewpoint of the frame S0 if they move in the same direction, or if the frame S is at rest towards the base.

We can see that Einstein omitted to apply duplicity viewpoint to this parameter as well as he forgot to apply it to the parameter c, too. This is the basic mistake of the STR. The light in the frame S0 moves away from the observer in the frame S with the velocity: . c 0 þ v ¼ ðc 2 2 v 2 Þ1=2 þ v; if light and S0 travel in the same direction, . c 0 2 v ¼ ðc 2 2 v 2 Þ1=2 2 v; if light and S0 travel in mutually opposite directions, . ðc 02 þ v 2 Þ1=2 ¼ c; if light and S0 travel in mutually perpendicular directions. For the observer in the frame S0 , the light velocity does not change because its deceleration to c 0 is accompanied with time dilation Dt 0 . After light leaves the frame S0 and comes into the frame S, its velocity becomes c. In fact, light velocity does not depend on moving radiation source, if light travels out of this source. After substitution of c 02 ¼ c 2 2 v 2 in the relation cDt ¼ c 0 Dt 0 we obtain the Lorentz transformation for time dilation: Dt 0 ¼ Dtð1 2 v 2 =c 2 Þ21=2 : The difference in momentums of frames towards the base stays equal from the viewpoints of S and S0 , so mv ¼ m 0 v 0 and we get the Lorentz transformation for mass increase and from the relation l=v ¼ l 0 =v 0 we get the Lorentz transformation for length contraction, where: m, m 0 – masses of the same object in frames S and S0 , l, l0 – lengths of the same object in frames S and S0 . The high speed of the frame causes not only mass increase but also the space curvature, which this frame takes along. From the relation cDt ¼ c 0 Dt 0 we have deceleration of processes and light velocity not only in the direction of the frame motion but also in all directions. The relation lDt ¼ l 0 Dt0 shows that length contractions act in all directions, as well as time is not a vector and cannot flux in various directions by different ways. These effects follow from

the frame motion towards the base and so they do not represent the result of relative motions of equivalent frames. It means that logical system reflections based on classic understanding of physics lead also to the Lorentz transformations without logical contradictions and mistakes being made in the STR. In accordance with the classic approach, simultaneity must be simultaneity everywhere. Supporting argument against Einstein is the EPR experiment, which Einstein created to show inaccuracy of quantum physics, from which the immediate particle communication follows irrespective of their distance. According to Einstein it cannot happen. But these results are confirmed in other experiments, so simultaneity must be simultaneity everywhere. Einstein was mistaken and space-time conception in special relativity is defective. Conclusions The big irony of Einstein’s STR is that the Lorentz transformations used in order to make the uniform motion of moving systems equivalent and to refuse the ether, just confirm asymmetry of their mutual uniform motion and at the same time the existence of material base, which defines the measure of their asymmetry according to velocities of their motion towards this base. Indeed, the biggest irony is that official science despite the evident system and logical failure of the STR does not allow its deep revaluation. But the truth must be accepted and Einstein’s STR will be replaced by the theory of asymmetry of uniform motion of systems, which can be fully described by the next simple system of relations: c 02 ¼ c 2 2 v 2 c Dt ¼ c 0 Dt 0 v Dt ¼ v 0 Dt 0 ; m v ¼ m0 v0 1Dt ¼ 10 Dt 0 From these relations all the Lorentz transformations follow.

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The measurement of speed of gravitational wave Andrej Madac MaM, Hroncova, Kosice, Slovakia

Kamil Madac Technical University in Kosice, Kosice, Slovakia Keywords Cybernetics, Gravity Abstract This paper introduces an hypothesis concerning the reason of gravitation. Preliminary results of the experiment, not finished yet, are presented.

Introduction Gravitation is the universal attraction of every particle (or object) of matter for every other, wherever located (Duncan, 1955). The present physics does not know the real reason of gravitation. One of the hypotheses says about gravitons, not captured yet.

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1138-1141 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483243

Hypothesis In this paper, we present a hypothesis that gravitation is caused by the flow of an unknown kind of matter into every object. This flow causes an under-pressure between the objects and “pressure” from the object’s outer side. The flow motion of the matter is caused due to suck into the object mass. The reason of suck is not clear. We have some ideas but at present that is not the case. Because we consider the flow of the matter (like a wind in the atmosphere), the speed of the gravitational wave is less than the speed of the electro-magnetic wave. Besides, we consider the “law of volumes”, similar to the “law of areas” (Kepler) for flow speed. It means that the flow into the object mass has constant volume speed. The speed value depends on the object mass and distance from the object. The present physics speaks about the potential of gravitational field, which by our hypothesis is the flow speed. Every mass object creates the gravitational field with equipotential surfaces around the object. In the case of a ball, the surfaces are around its center. This is the case for stars and planets. When these objects are in motion, the equipotential surfaces are deformed. The value of deformation depends on the object speed and distance from the object. Diagram of deformation for constant object speed is shown in Figure 1. These ideal conditions are never fulfilled in nature, since every object is “attracted” by all others.

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Figure 1. The deformation of gravity field due to object motion

The experiment To verify the hypothesis we carried out the following experiment “The measurement of weight motion”. The weight hangs up on two wires (Figure 2). Two wires make the weight to move in one plane, which moves from south to north. We assumed that the weight motion through deformed gravitational field reached the maximum and minimum value in time of tangent direction to flow motion. The forces affected on weight are: . static attractive force of deformed gravity field . dynamic force due to accelerated motion of weight (mass object) – in deformed gravity field The resultant force makes the weight to move in the plane from south to north, due to rotation and orbital earth motion around the sun and moon motion. The next diagram shows the measured weight deflection during 21 h on 13th March. The position of experiment place, Kosice, is 218150 E 488430 N. Used devices do not allow us to get the absolute values of deflection in millimeter as well as the precise time of maximum and minimum values of weight deflection. The time values in Figure 3 are approximated only. It should be kept in mind that the gravity field close to the earth is deformed also by the position of the moon. From the data analysis, we determined the first approximation of speed of gravitation wave to be 15,000 km/s.

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Figure 2. The diagram of the experiment

Figure 3. The measured values of weight deflection

If the hypothesis is correct the time of the maximum and minimum values of the weight deflection will occur at noon or midnight two times a year as it is clear from Figure 4. Probably it will occur in the beginning of July and January. We are continuing in measuring and also we are searching for cooperation to get better results and analyse the data.

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Figure 4. The diagram of probable current earth’s position in the gravitational field of the sun

Conclusion One must use more sophisticated measurement equipment to get better results. For example, Laser Scan Interferometers. Also the mass of weight must be much higher and the place of the experiment should be at a quieter place. We have continuous problems with dumping the weight vibrations due to daily works in the building. If the hypothesis is confirmed, it is possible to find out the direction and speed of the solar system motion and substitute the Michelson-Morley experiment. Reference Duncan, J.C. (1955), Astronomy, Harper and Row, NY, USA.

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Relativity, contradictions, and confusions Cameron Rebigsol San Francisco, CA, USA Keywords Cybernetics, Theory, Time, Space Abstract The derivation leading to the formulation of Lorentzian transformation in special relativity is actually a duplication of an ancient “miracle” in algebra: 2x 2 x ¼ 0, 2x ¼ x, 2 ¼ 1. Dominated by such a mathematical confusion, relativity displays fundamental uncertainty in understanding physics. As such, with equations, it claims to have “discovered” two speed limits in nature: the speed of light in the vacuum space and the speed of light at the mass center of a material body. Needless to say, these two speed limits repel each other, not to mention that the second speed limit is even against nature. Relativity then further extends this confusion and uncertainty in physics to make up many self-contradicting concepts. These concepts include the so-called homogeneous gravitational field and the idea of having (circumference/diameter) . 3.1415926. . . for a spinning circle. With the same mathematics guiding to its “success”, however, relativity presents no homogeneous gravitational field, but a monster that must be called a homogeneously inhomogeneous field for its appropriation. Based on the same erroneous mathematics, relativity must force itself to have (circumference/diameter) , 3.1415926. . . for a spinning circle. With the idea of a homogeneous gravitational field, relativity believes that it can establish the validity of the so-called Principle of Equivalence for the legitimacy of general relativity. However, Newtonian mechanics, supported by the close orbital movements of numerous heavenly objects, must witness the nonexistence of such a “principle” in nature.

About special relativity General failure of special relativity in mathematical terms The crucial role of x ¼ ct and x0 ¼ ct 0 in the derivation of the Lorentzian equations must evidence the indisputable presence of the following equation set in special relativity: x 2 vt x 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc   t 2 x cv2 t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc 0

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1142-1162 q MCB UP Limited 0368-492X DOI 10.1108/03684920310483252

x 2 ¼ ðctÞ2 x 02 ¼ ðct 0 Þ2

ðset AÞ

Given v and c as constants, either of the two Lorentzian equations (set A) can be obtained by the other three equations within the set by the following simple operation: First, with the first Lorentzian equation (set A), we have x 2 vt cx 2 vx ffi¼x x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc þ vÞðc 2 vÞ 12 c 0

rffiffiffiffiffiffiffiffiffiffiffi c2v cþv

ðA-1Þ

Second, with the second Lorentzian equation, we have   t 2 x cv2 t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc 0

v ct 2 cx c2 ct 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 12 c rffiffiffiffiffiffiffiffiffiffiffi c2v 0 x ¼x cþv

ðA-2Þ

The exactness between equations (A-1) and (A-2) duplicates one Lorentzian transformation equation from the other back and forth without losing any mathematical equivalence between them. This further allows one to decide that equation (set A) can be reduced to rffiffiffiffiffiffiffiffiffiffiffi c2v x ¼ ·x cþv 0

x 2 ¼ ðctÞ2

ðset BÞ

x 0 2 ¼ ðct 0 Þ2 The first equation in (set B) means that there is always only one x that can be found matching x0 , no matter how time might have advanced. This can happen only if there exists no movement, or, v ¼ 0; between the two axes. Any nonzero value of speed found in the Lorentzian equations in equation (set A), such as v ¼ 0:2c; for example, must be ridiculed, unless v ¼ 0:2c ¼ 0 is accepted. If anyone applies the two Lorentzian transformation equations with some numerical constants assigned to two of the four variables, such as x ¼ m and t ¼ n; for example, where m and n can be any nonzero constants, he must have organized an equation set that reads as:

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x 2 vt x 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc   t 2 x cv2 t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc 0

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ðset B-IÞ

x¼m t¼n As mentioned before, x ¼ ct and x 0 ¼ ct 0 are inseparable from the two Lorentzian equations, one who establishes equation (set B-I) has actually established the following equation set: x 2 vt x 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc   t 2 x cv2 t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc 0

x 2 ¼ ðctÞ2

ðset B-IIÞ

x 02 ¼ ðct 0 Þ2 x¼m t¼n With six equations and four unknowns, (set B-II) is easily seen as containing no solution. Any nonzero solution of equation (set B-I) turns out to be a mirage arrived only because x ¼ ct and x 0 ¼ ct 0 have been omitted from (set B-II) without valid reason. Failure of Lorentzian transformation equations in studying movement The constant speed v0 of the x axis along the x0 axis can be expressed as dx 0 =dt0 ¼ v0 : Now, if we take the derivative of the first three equations of equation (set A) with respect to t0 , we have

  dx 0 dtdx0 2 v dtdt0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 dt 0 1 2 vc    dt 0 dtdt0 2 dtdx0 cv2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ¼ 1 dt 0 1 2 vc

Relativity, contradictions, and confusions ðset CÞ

dx dt ¼c 0 dt 0 dt Within equation (set C), substituting the third equation into the second equation leads to rffiffiffiffiffiffiffiffiffiffiffi dt cþv ¼ 0 dt c2v Then, this new relationship and the third equation in (set C) together will lead to dx 0 =dt 0 ¼ c; instead of dx 0 =dt 0 ¼ v0 ; out of the first equation in the same set. dx 0 =dt 0 ¼ c is a relationship that can be displayed by the fourth equation in equation (set A). Clearly, this means that the Lorentzian transformation equations cannot distinguish c, the speed of light, from v0 , the speed that is supposedly possessed by the x axis with respect to the x0 axis. Given that v and v0 must be of equal magnitude, setting aside their directions, the Lorentzian transformation equations must obviously force such a relationship between the two axes: c ¼ dx 0 =dt 0 ¼ v 0 ¼ v ¼ dx=dt: Invalidity of the length contraction Length contraction from relativity means that a segment of length of ðx2 2 x1 Þ is worth rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ðx2 2 x1 Þ 1 2 c when this length is cut from the x axis, but attached to x 0 axis and viewed by the observer on the x axis. Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 ðx2 2 x1 Þ 1 2 c is an apparent length of ðx2 2 x1 Þ on the x0 axis to the observer on the x axis. Conversely, for a length cut from the x axis, but moving with the x0 axis to cover a stationary length of ðx2 2 x1 Þ on the x axis, the length to be cut from the x axis must be

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ðx2 2 x1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc

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ðx2 2 x1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc be the equivalent length on the x0 axis to the observer staying on the x axis. So, for the observer on the x axis, if it takes ðx2 2 x1 Þ from his own frame to obtain speed v, it is well justified to say that it takes an equivalent length from the other, but moving the axis for him to obtain speed v. The equivalent length, if cut from the moving axis, but made stationary to the observer on the x axis, is measured as ðx2 2 x1 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc All these inevitably lead the observer on the x axis to conclude ðx2ffiffiffiffiffiffiffiffiffiffi 2x1 Þ p 2 12ðvcÞ ðx2 2 x1 Þ v¼ ¼ ðt 2 2 t 1 Þ ðt2 2 t 1 Þ

This relationship must lead to v ¼ 0: Once again, any nonzero speed written in the Lorentzian equations is ridiculed. Mathematics prohibits the promise of length contraction to the observer. Invalidity about time dilation The second equation in equation (set A) advocates that a single time instant registered by a clock on the x axis can identify different time instants between different clocks on the x 0 axis. Subsequently, this allows a zero time interval quoted on the x axis to match a nonzero time interval quoted on the x0 axis. Relativity must let the above principle to hold conversely so that a zero time interval quoted on the x 0 axis can correspond to a nonzero time interval quoted on the x axis. Now, relativity has the following options. (1) It allows ðt 02 2 t 01 Þ ¼ ðt2 2 t 1 Þ; where ðt 02 2 t 01 Þ is zero, while ðt 2 2 t1 Þ is nonzero. This option, of course, cannot be tolerated by any mathematical rule. (2) Corresponding to t20 and t10 quoted on the x 0 axis, two time instants at two locations on the x axis are identified:

  t 01 þ x 01 cv2 t1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc

  t 02 þ x 02 cv2 t2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc

Relativity, contradictions, and confusions

Then, a time lapse between the two axes must match each other according to the following:   ðt 02 2 t 01 Þ þ ðx 02 2 x 01 Þ cv2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt2 2 t1 Þ ¼  2 1 2 vc This further leads to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v2 ¼ ðt 02 2 t 01 Þ þ ðx 02 2 x 01 Þ 2 ðt2 2 t1 Þ 1 2 c c

ðDÞ

For a nonzero time interval shown by a clock on the x axis, we must have t2 and t1 not equaling each other. With these two time instants, a clock must always identify two locations x 20 and x 10 on the x 0 axis because of the movement by this clock with respect to the x 0 axis during the said time interval. For a single time instant, or a zero time interval, on the x0 axis, we must have t 02 ¼ t 01 : With t 02 ¼ t 01 ; any pair of x 02 and x 01 that are identified in the above manner must obey the truth of ðx 02 2 x 01 Þ ¼ vðt 02 2 t 01 Þ ¼ 0: So, if ðt 2 2 t1 Þ must be nonzero, the only choice left for relativity by the relationship shown in equation D is to have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 12 ¼ 0; c inevitably, v ¼ c: With v ¼ c; Lorentzian transformation equations must fail. If we first pick two random locations such as x30 and x10 at a single time instant on the x 0 axis, special relativity must claim that a nonzero time lapse of ðt 3 2 t1 Þ will result, to be registered by clocks on the x axis. At the completion of any fraction of this time lapse of ðt3 2 t1 Þ; a single clock on the x axis that has identified the location of x10 , must identify a new location x 20 between x 30 and x 10 on the x0 axis because of its movement. Subsequently, being between x 30 and x 10 , x 20 can only be picked within the same time interval, or at the same time instant, shown by clocks on the x 0 axis that x 30 and x 10 are picked. Then, all the preceding refutation against the concept of time dilation repeats, ending up with v ¼ c: Another popular conclusion, i.e. time dilation, from relativity is then unacceptable in mathematical terms. It is now apparent that special relativity must encourage the v term of any value in the Lorentzian transformation equations to take this form: 0 ¼ v ¼ c:

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About general relativity Speed of light and the limits of speed Besides the speed limit asserted by special relativity, general relativity announces another speed limit in nature: For measuring time. . . relativity states, . . .at a place which, relatively to the origin of the coordinates, has the gravitation potential F, we must employ a clock which – when removed to the origin of co-ordinates – goes (1+F/c2) times more slowly than the clock used for measuring time at the origin of co-ordinates. If we call the velocity of light at the origin of co-ordinates c0, then the velocity of light c at a place with the gravitation potential F will be given by the relation c ¼ c0 (1+F/c2) (Einstein, 1911).

This statement has obviously placed the mass center of a gravity body at the origin of the coordinates. With the gravitational field so arranged, F, the gravitational potential must be zero at the origin, but negative elsewhere. So, general relativity, with an equation as shown in the above statement, must hereby assert that it has “discovered” that light can travel through the mass center of a material body with a speed and this speed can even be concluded as being the highest in nature. In addition to ridiculing the speed limit advocated in special relativity, the “newly found” speed of light also expels a concept that is extremely important to general relativity: the so-called homogeneous gravitational field. According to general relativity, no measurement made in a homogeneous gravitational field is supposed to be varied between different coordinates. The above equation for the different speed of light at different location obviously claims that such a homogeneous field is unfound in nature. A homogeneously inhomogeneous field In the explanation of homogeneous gravitational field by relativity, one can find the following: In the article by Einstein (1911), relativity states that . . .relatively to K, as well as relatively to K0 , material points which are not subjected to the action of other material points, move in keeping with the equations . . ., d2 x ¼ 0; dt 2

d2 y ¼ 0; dt 2

d2 z ¼ 2g dt 2

where K represents a coordinate system that is at rest in a gravitational field, and K0 represents a coordinate system that is (mechanically) accelerated. Since relativity has apparently compared accelerations in all three dimensions between the two systems, relativity must allow X==X0 ; Y==Y0 and Z==Z0 as the only orientation between all the axes of the two systems. In another article, relativity states that Let K0 be a second system of reference which is moving relatively to K in uniformly accelerated translation. (Einstein, 1916). In this statement, K is referred to as an inertial system. In order not to confuse with the K that represents an inertial system in this statement with

the K that represents a system’s rest in the gravitational field in the previous paragraph, let us use K0 to represent the inertial system mentioned in this paragraph. The axes of K0 will then be named Xo, Yo, and Zo. In the movement comparison presented in this paragraph, relativity of course must have restricted the orientation of all axes between K0 and K0 in such a way that X0 ==Xo; Y0 ==Yo and Z0 ==Zo: Putting together all of the above restrictions regarding the orientation of axes, relativity must have stressed the overall relationship between the axes in all three systems aforementioned as X==X0 ==Xo; Y==Y0 ==Yo; and Z==Z0 ==Zo: With the geometrical orientation that is specified above, we immediately recognize that relativity has forced an impossibility upon the gravitational field. While we agree with relativity that the mechanically accelerating field does provide point in its entire field for any satisfaction to the mathematical relationship d2 x ¼ 0; dt 2

d2 y ¼ 0; dt 2

d2 z ¼ 2g; dt 2

we must stress that such a satisfaction is unfound in the gravitational field. If we happen to have found a certain point in the gravitational field, where the material movement obeys this group of equations, at any other point of the same field the material movement must disagree with this group of equations. The reason is simple: while the value and direction of acceleration in the entire mechanically accelerating field is coordinate-independent, as shown by the above equations, it is absolutely not so in the gravitational field. We all know that, in describing the dependence of the acceleration on the coordinates in the gravitational field, Newtonian physics has a highly accurate expression, which is d2 r M ¼ 21 2 ; 2 dt r where r¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ z2

and M is the mass of the body causing the gravity. Newton’s equation allows no homogeneity for the gravitational field, but matches the reality with such an accuracy that human beings so far can confirm no universal data for a dispute. Homogeneity, by definition in physics, is the quality of coordinate independence. So, before relativity can remove the inhomogeneous characteristics from the gravitational field, i.e. the coordinate dependence of

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the gravitational acceleration in naming a homogeneous gravitational field, general relativity has actually created a term that must be read as a homogeneously inhomogeneous field. As a matter of fact, relativity even fails itself in recognizing a gravitational field that is homogeneous. In the preceding analysis regarding the speed of light in the gravitational field, relativity apparently finds no homogeneous gravitational field for itself. In another statement concerning a gravitational field, relativity states: The unit measuring-rod thus appears a little shortened in relation to the system of co-ordinates by the presence of the gravitational field, if the rod is laid along a radius, . . .With the tangential position,. . . the gravitational field of the point of mass has no influence on the length of a rod. (Einstein, 1916)

Relativity does hereby predict the variation of measuring results at different locations (radii can have different orientations and lengths) in the gravitational field. Now, is the gravitational field homogeneous, as what relativity imagines, or inhomogeneous, as what relativity fits itself in? Principle of equivalence If the gravitational field and the mechanically accelerating field are really equivalent so that material movement can only follows the equations d2 x ¼ 0; dt 2

d2 y ¼ 0; dt 2

d2 z ¼ 2g dt 2

then calculation will lead us to believe that the free movement path of all objects can only have one type of trajectory: parabolic curves in the coordinate system with respect to which acceleration is detected. However, the movement of the abundant heavenly objects must disagree with this belief. Many of their moving paths demonstrate as close loops. Relativity fails to provide equations to match this kind of movement. In contrast to the inability of relativity, Newtonian physics can provide us an equation to describe those moving paths. This can be evidenced by the following conic section equation that, as shown in Figure 1, depicts the path of an object, named as B, flying by a massive object, named as A, in this figure: R¼

12

Rve vRve vTve sin u

where R is the instantaneous distance between the mass centers of A and B, ve is the abbreviation of designation for a point in space known as the virtual equilibrium point, where the gravitational force between A and B precisely cancels out the centrifugal force produced by the movement of object B around body A, Rve is the distance between the mass center of A and B when mass center of B coincides with the ve point, vRve is the moving speed of B at the ve

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Figure 1.

point, but resolved along Rve, vTve is the moving speed of B at the ve point, but resolved along a line that is tangential to Rve. We can always orient the coordinate system X-O-Z in such a way that the ve point coincides with the x axis, then u is the angle between R and the x axis. The three types of curve described by the above conic section equation will be: (1) a hyperbolic path if vRve . 1; v Tve (2) a parabolic path if vRve ¼ 1; v Tve (3) an elliptical path if vRve , 1: v Tve In case of vRve ¼ 0; of course, R ; Rve ; the ellipse is actually a perfect circle. The more detailed development of the aforementioned conic section equation will appear in the later text of this article. Since the free moving path of an object in the gravitational field shows three types of paths, but only parabolic

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curve in a mechanically accelerating field (its equation is omitted in this paper), the equivalence drawn by relativity about two different fields can only be explained as a product of immature consideration in physics. (Circumference/diameter).3.1415926. . . for a spinning circle

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Einstein (1916) states that In a space which is free of gravitational fields we introduce a Galilean system of reference K (x, y, z, t), and also a system of coordinates K 0 (x 0 , y 0 , z 0 , t 0 ) in uniform rotation relatively to K. With a measuring-rod at rest relatively to K 0 , the quotient (circumference/diameter, noted by this author) would be greater than p. This is readily understood. . .that the measuring-rod applied to the periphery undergoes a Lorentzian contraction,. . . Hence Euclidean geometry does not apply to K 0 .

The paragraph is quite long, but can be condensed to say that, according to general relativity, an observer can conclude that (circular circumference/diameter) . p for a circle that is spinning with respect to him. Without offering any kind of proof to support such an assertion, general relativity believes that, based purely on its imagination, it can even remove the universal validity of Euclidean geometry from anything that is moving. Unfortunately, using the very same idea it advocates, general relativity leads to a conclusion that is the exact opposite of its assertion, i.e. (circular circumference/diameter) , p. If we construct equilateral polygons of sides of 3, 4, 5,. . .n to circumscribe and inscribe a circle in the manner as shown in Figure 2. We must have ðn l i , L , n l o Þ; where li is the length of each side of the inscribing polygon, lo is the length of each side of the circumscribing polygon, n is the total number of sides of the corresponding polygons and can approach infinity, L is the length of circumference of the circle. Now, let us rotate both the circumscribing and inscribing polygons about the center of the circle with the same angular velocity, leaving the circle stationary relative to an observer who is at the center of the circle. Disregarding the magnitude of the angular velocity, we must accept the following. (1) Once defined, n, the number of sides of each polygon, remains unchanged. (2) Once n is defined, the polygon circumscribing the circle can neither expand nor shrink. That it cannot expand is obvious. If it shrinks,

Figure 2.

however, it would contradict a statement that can be found in the above quotation: “. . .while the one (the measuring-rod) applied along the radius does not (undergo a Lorentzian contraction).” (3) The above analysis can also be applied to the polygon inscribing the circle. Further, each side of the inscribing polygon can serve as a measuring-rod along the circle’s periphery, for, each of them always has both of its endpoints on the circle. Please note that each of these rods is now moving. (4) Basic geometry tells us that as n increases, the total sum of the length of the circumscribing polygon decreases while that of the inscribing polygon increases, but both approach the same limit, which is the periphery of the circle. Because of the assumed effect of the Lorentzian contraction on each side of the rotating polygons, as specified by relativity in the above statement, all of the above deductions would further lead us to have ðn l 0i , L , n l 0o Þ; where the primes denote the new respective length of each of the polygon’s sides that undergoes Lorentzian contraction. Subsequently, we must have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2ffi v 2 o i nl i 1 2 , L , nl o 1 2 c c where vi is the linear speed of the sides of the inscribing polygon and vo is the linear speed of the sides of the circumscribing polygon. If the angular velocity of the rotation increases, both vi and vo must increase and result in smaller and smaller values out of both the left and the right terms in the above inequality, forcing the value of L to decrease and even approach zero. Any circle that is between these inscribing and circumscribing polygons and spins with them together, according to the above mathematical deduction, must also have its circumference decrease and finally approach zero. With a decreasing value of L, we are naturally unable to have L=D . p; where D is the diameter of the circle and, as predicted by relativity, will not undergo a Lorentzian contraction, but stays constant. So, a decreasing L and constant D must result in L=D , p ! What makes general relativity believe that it can remove the validity of Euclidean geometry while offering absolutely no proof of any kind? Roots of relativity’s mistakes Those that lead to the formulation of special relativity In order to keep things simple, let us examine the following equation set: x 0 ¼ a11 x þ a12 t t 0 ¼ a21 x þ a22 t

ðSet 1Þ

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1154

The task of (set 1) is to pursue the numerical relationship between the variable elements of (x, t, x 0 , t 0 ). Obviously, this task cannot be realized unless (a11, a12, a21, a22) are defined with numerical values. However, if we must apply (set 1) in an effort of defining (a11, a12, a21, a22), we would have naturally placed ourselves in a position of solving equation set in which (a11, a12, a21, a22) are regarded as variables. Subsequently, the other elements, i.e. (x, t, x 0 , t 0 ), must be regarded as playing the role of known constant coefficients in the same set. In doing so, beginning with (set 1), the derivation of Lorentzian transformation equations normally go through the following steps: Step one. By supplying the conditions x 0 ¼ 0 and x ¼ vt to (Set 1), the following set is reached: x 0 ¼ a11 x 2 va11 t

t 0 ¼ a21 x þ a22 t

ðSet 2Þ

Step two. By substituting (set 2) into x 0 2 ¼ ðct0 Þ2 and, after some term rearrangement, by having the new equation, equate the equation x 2 2 ðctÞ2 ¼ 0; a solution set for the a’s is reached: 1 a11 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc v a12 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc v=c 2 a21 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc

ðSolution 1Þ

1 a22 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 1 2 vc Step three. Finally, equation (set 1) is rewritten as x vt x 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 12 c 1 2 vc ðv=cÞ2 x t t 0 ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 12 c 1 2 vc

ðSolution 2Þ

Solution 2 is then the so-called Lorentzian transformation equations. Apparently, upon the performance of step one, the following equations must have been bundled together to form a set:

x 0 ¼ a11 x þ a12 t t 0 ¼ a21 x þ a22 t x0 ¼ 0

ðSet 3Þ

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x ¼ vt However, with this set, we must determine whether (a11, a12, a21, a22) or (x, t, x0 , t0 ) to be the variables. If (a11, a12, a21, a22) are the variables, what x 0 ¼ 0 and x ¼ vt play in the set is to provide an explanation of how a constant has been constructed, and therefore is redundant to the task of seeking solution of the a’s. Who, with any equation set, needs extra expressions such as 0 ¼ 0; 21 ¼ 3 £ 7. . .; as what x 0 ¼ 0 and x ¼ vt are playing in (set 3) when the a’s are variables? If (x, t, x0 , t0 ), are the variables, then (a11, a12, a21, a22) must be constants. Subsequently, a12 ¼ 2va11 ; as indicated by (set 2), is reached as a result of an illegal operation in algebra! To prove this point, let us examine the following equation set that is analog to (set 30 ), but with numerical constants: x 0 ¼ 3x þ 5t t 0 ¼ 7x þ 9t 0

Relativity, contradictions, and confusions

ðSet 30 Þ

x ¼0 x ¼ 4t From (set 3 0 ), in an operation parallel to those that lead to the first equation of (set 2), we have 0 ¼ 3ð4tÞ þ 5t 2 3ð4t Þ ¼ 5t 23£4¼5 So, by accepting a12 ¼ 2va11 ; as indicated in (set 2), we must also accept 5 ¼ 23 £ 4 here. This is the “key” operation where all the “miracle” of relativity begins. Upon its performance, step two has actually bundled (set 2) and x 0 2 ¼ ðct0 Þ2 and x 2 2 ðctÞ2 ¼ 0 together to form the following set:

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x 0 ¼ a11 x 2 va11 t t 0 ¼ a21 x þ a22 t x 0 2 ¼ ðct 0 Þ2

1156

ðSet 4Þ

x 2 ¼ ðctÞ2 Given how (set 2) has established its “validity”, then (set 4) is actually a collection of all of the following equations together before further derivation begins: x 0 ¼ a11 x þ a12 t t 0 ¼ a21 x þ a22 t x0 ¼ 0 x ¼ vt

ðSet 5Þ

x 0 2 ¼ ðct 0 Þ2 x 2 ¼ ðctÞ2 Now, which elements are the variables in this set? If (a11, a12, a21, a22) are the variables, as pointed out previously, this set has provided insufficient equations for them to reach a finite solution. If (x, t, x0 , t0 ) are the variables, (Set 5) apparently has no solution other than (0, 0, 0, 0). The acquisition of solution 2, or the so-called Lorentzian transformation equations, is actually the result of: (1) the inconsistency in distinguishing the roles between the variables and constants in equation sets, (2) repetition of violation of mathematical rules. The above works, demonstrated from Sets (1-5), are done based on the analysis on the mistakes that are popular in textbooks. In the original paper of special relativity concerning the derivation of Lorentzian transformation equations, one can easily find that the “success” of the derivation of special relativity also depends on the derivative taken with respect to a constant instead of a variable; a critical rule in calculus is not respected. But, the author does not intend to pursue further on this matter. Those that lead to the formulation of general relativity What serves as the fundamental support to the general relativity is: . special relativity and . the principle of equivalence.

That special relativity is unable to offer support to anything is obvious, because, as what has been shown so far, special relativity itself is a product of fallacious derivation in mathematical terms. As to the principle of equivalence, its invalidity is evidenced in two folds. (1) A gravitational field that needs to be homogeneous is indispensable for the existence of this “principle”. However, as we learnt previously, even general relativity itself presents enough ideas to have the concept of homogeneous gravitational field rejected. (2) More detailed derivation guided by Newtonian mechanics will show that an object must demonstrate a significant difference, instead of equivalence, by its free moving path between a gravitational field and a mechanically accelerating field. Let us refer to Figure 1 again. In the gravitational field, abbreviated as GF, produced by the massive object A, we attach a coordinate system X-O-Z to A, with the origin O of the system located at the mass center of object A. Assume that at a certain time instant, in the vicinity of A, of mass MA, we found a projectile B, of mass MB, having a distance of R from the mass center of A. With respect to the inertial frame X-O-Z, this projectile is found moving with velocity vB, which forms angle b with R. This velocity can be resolved into two components: a tangential component vB=T ð¼ vB sin bÞ; and a radial component vB=R ð¼ vB cos bÞ: We can always find the gravitational force F between objects A and B according to the following formula: F¼2

GM A M B R2

ð3-1Þ

The total mechanical energy of B with respect to A is 1 GM A M B E ¼ M Bv 2 2 2 R

ð3-2Þ

If there is no foreign interference, E is a constant. We can divide both sides of equation (3-2) by MB to get 1 2 GM A v 2 ¼e 2 R

ð3-3Þ

where e represents the total mechanical energy per unit mass of projectile B; e is also obviously a conserved quantity. It can be shown that in the course of B’s movement there always exists one point where the centrifugal force, Fc, developed by B’s sideway movements with respect to A, precisely cancels the gravitational force, Fg, between A and B, leaving us with F c þ F g ¼ 0: We will call this point the virtual equilibrium

Relativity, contradictions, and confusions 1157

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point (VEP). Naturally, at the VEP, we should also have an equation similar to equation (3-3) for B’s mechanical energy per unit mass with respect to A: 1 2 GM A vve 2 ¼e 2 Rve

1158

ð3-4Þ

where the subscript ve stands for the quantity at the virtual equilibrium point. At VEP, F c þ F g ¼ 0 will lead us to have Fc þ Fg ¼ 0 M B v2B=Tve Rve

þ

2

GM A M B

!

R2ve

¼0

where vB/Tve is the tangential component of the B’s speed at the VEP. This equation further leads us to have Rve ¼

J2 GM A M 2B

ð3-5Þ

where J ¼ Rve M B vB=Tve is the angular momentum of B with respect to A, a quantity that must always be conserved. (In the upcoming text, we will replace MB with m for simplicity of notification.) If J is a constant, Rve must then also be a constant by equation (3-5) once all the initial moving status are defined, including the speed with which B was found. At a point other than VEP, we can set R ¼ ð1=f Þ · Rve ; with f being any positive number. Then equation (3-3) can be rewritten as 1 2 1 2 GM A v þ v 2 Rve ¼ e 2 R 2 T f

ð3-6Þ

where vR is the component of B’s speed along R and vT is the tangential component of B’s speed. With mvT R ¼ mvTve Rve ;

vT ¼

vTve Rve ¼ f ·vTve ; R

equation (3-6) becomes 1 2 1 2 2 fGM A vR þ f vTve 2 ¼e 2 2 Rve equations (3-4) and (3-7) together lead us to have

ð3-7Þ

1 2 1 2 2 fGM A 1 2 1 GM A ¼ vRve þ v2Tve 2 v þ f vTve 2 2 R 2 2 2 Rve Rve

ð3-8Þ

Relativity, contradictions, and confusions

Because at VEP, F c þ F g ¼ 0; we have: mv2Tve GM A m ¼ Rve R2ve v2Tve ¼

1159 ð3-9Þ

GM A Rve

Substituting equation (3-9) into equation (3-8), we have 1 2 1 2 2 1 1 v þ f vTve 2 f v2Tve ¼ v2Rve þ v2Tve 2 v2Tve 2 R 2 2 2

ð3-10Þ

equation (3-10) further leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vR ¼ ^ v2Rve 2 v2Tve ð f 2 1Þ2

ð3-11Þ

The positive direction of R is assumed to be pointing away from A. vR is along the radial line and in the direction of decreasing R, so we take the negative sign for vR, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vR ¼ 2 v2Rve 2 v2Tve ð f 2 1Þ2 or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dR ¼ 2 v2Rve 2 v2Tve ð f 2 1Þ2 dt

ð3-12Þ

If we express B’s sideway movements with angular speed v ¼ du=dt; then, because of conservation of angular momentum, we would have mvT R ¼ mvTve Rve ðvRÞR ¼ vTve Rve du vTve Rve ¼ dt R2 Dividing both sides of equation (3-13) by equation (3-12), we have

ð3-13Þ

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du 2vTve Rve ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dR R 2 v2 2 v2 ð f 2 1Þ2 Rve

T ve

or

1160 du ¼

2vTve Rve dR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 v2Rve 2 v2Tve ð f 2 1Þ2

ð3-14Þ

Since R¼

Rve ; f

dR ¼ 2

Rve df ; f2

equation (3-14) becomes vTve df du ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2Rve 2 v2Tve ð f 2 1Þ2 Z Z vTve df u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þC v2Rve 2 v2Tve ð f 2 1Þ2

u ¼ sin21 sinðu 2 CÞ ¼

ð3-15Þ

vTve ð f 2 1Þ þ C vRve

vTve ð f 2 1Þ vRve

At VEP, if vRve takes the negative sign, the ratio of vTve =vRve is a negative value. Besides, at VEP, R ¼ Rve ; thus f ¼ 1: The continuous movement of B serves to further decrease R, or, to further increase f. These conditions enable us to assume an initial condition of 1808 for C. So, equation (3-15) gives us vTve ð f 2 1Þ vRve

 vTve Rve 21 2sin u ¼ vRve R

sinðu 2 1808Þ ¼

R¼

12

ð3-16Þ

Rve vRve vTve sin u

Equation (3-16) is a conic section formula. In this conic section formula, Rve is a constant according to equation (3-5). Once Rve is obtained, vTve can be

calculated from equation (3-9). Then, vRve can be obtained from equation (3-7) by setting f ¼ 1 at VEP. This conic section equation can predict three types of paths for a free moving object in the gravitational field. Can we find any compatible equation in general relativity? The answer is, NO! If the principle of equivalence is ever valid, parabolic curve is the only curve to be shown by relativity for the path of a free moving object.

Relativity, contradictions, and confusions 1161

Questions that relativity is obliged to answer What is speed? Before the debut of special relativity, speed was always a simple and clear concept: completion of linear displacement during a unit time interval. However, by introducing the concept of length contraction and time dilation, relativity has confused the concept of speed. Let us refer to Figures 3 and 4. Assuming the position of frames AB and A0 B0 in both figures they are recorded by an observer riding with frame AB with a clock that is stationary to him. Which of the following expressions will relativity accept as a speed expression that the observer can use for the speed term v in the Lorentzian transformation equation? (1) v ¼ length CB=ðt2 2 t 1 Þ (2) v ¼ length C0 B0 =ðt 2 2 t1 Þ Given that relativity can also instruct the observer that there is always a dilated time interval, say ðt 02 2 t 01 Þ; recorded by clocks on the A0 B0 frame to match the time interval ðt 2 2 t1 Þ; the observer must also have the following expressions (1) v ¼ length CB=ðt 02 2 t 01 Þ (2) v ¼ length C0 B0 =ðt 2  t 1 Þ:

Figure 3. At time instant t1

Figure 4. At time instant t2

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Can relativity answer whether to accept any one of these four expressions but reject the others, or to accept them all, or reject them all in determining a v value for the Lorentzian transformation equations? Why are the methods different? If it is said that the measurements concluded by special relativity such as length contraction, time dilation, and mass escalation are observation-dependent, the same measurements concluded by general relativity are no longer observation-dependent, but absolute. In other words, for example, special relativity asserts that, in an environment free of gravity, the determination of the length of an object depends on how fast an observer is moving with respect to the object that he is measuring. The speed no longer plays any crucial role in general relativity when the same measurement is made. Once the mass quantity of the object that causes the gravitational field is determined, the other factor that governs the result of the measurement would be the location with respect to the massive object. (Not to mention that this is another violation brought about by general relativity against its own concept of homogeneous gravitational field, with which relativity draws the principle of equivalence. The principle of equivalence, however, can tolerate no difference between locations when measurements between these locations are made.) Why is the method in determining certain measurements in general relativity absolute but once acceleration is absent, as stressed by special relativity, must the method of making the same measurements be observation-dependent? How can this fit into what Einstein said: “The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion”? (Einstein, 1916). What more is needed from relativity to convince people about its fallacious nature, both in mathematics and physics? References Einstein, A. (1911), On the Influence of Gravitation on the Propagation of Light. Einstein, A. (1916), The Foundation of the General Theory of Relativity. Further reading The Principle of Relativity, a collection of original memories on the special and general theory of relativity, Dover Publications, Inc. New York, USA (Library of Congress Catalog Card Number: A52-9845). Robert, R. (1968), Introduction to Special Relativity, Wiley, New York, USA (Library of Congress Catalog: 67-3111).

Regular Journal Sections Contemporary systems and cybernetics

Contemporary systems and cybernetics

Keywords Automation, Cybernetics, R&D, Technical innovation Abstract Gives reports and surveys of selected current research and developments in systems and cybernetics. They include: Interdisciplinary initiatives; Artificial intelligence; Innovative research; Cybernetics and Robotics; Supergen programme.

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Interdisciplinary initiatives Project to use human body as electronic circuit A project has been described by the Nippon Telegraph and Telephone corporation in Japan, to harness the human body’s capacity to conduct electrical signals. The technology, being currently researched, aims to use the human body to improve communication. The body would be used to function at broadband speeds and allow signals to circulate through it. As a result, no wires would be needed for the transmission of any signal that may be required for communication between humans. A scenario is presented, where a mere handshake could be enough to pass information such as identification exchange of telephone numbers, e-mail addresses or other messages. Other suggestions for applying the body’s electrical conducting properties include entrance to security areas, access to computer systems as well as many more activities where data have to be exchanged. Such systems could well be used in conjunction with embedded electronic devices which are currently being tested worldwide. Using bacteria as a database A novel solution to the problems of storing data was suggested in the New Scientist (January, 2003). Data could be encoded as artificial DNA and stored within the genomes of multiplying bacteria and then, it claims, accurately retrieved. The main concern of the scientists developing such systems is that nearly all of the current ways of storing data, such as paper or electronic, allow it to be lost or destroyed. The users of some electronic systems are well aware that valuable business, industrial data can be corrupted and files destroyed, often at the press of the wrong button. Hence, there is a need to create new types of memory systems. Research in the US has already shown that data stored as artificial DNA can be stored and processed. Dr Pak Chung Wong of the Pacific Northwest National Laboratory, Washington State, US, is an information technologist, who has expressed concern about the protection of valuable information in the case of a nuclear catastrophe. We are told that bacteria may be an inexpensive and stable long-term means of data storage. The laboratory conducting this research was set up as a nuclear energy research institute so that it has a

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1163-1169 # MCB UP Limited 0368-492X

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particular need to find alternative and more secure data storage. A report on the laboratories work describes the researchers approach and experimentation. The scientist, Dr Wong says: .......... took the words of the song It’s a Small World and translated it into a code based on the four ‘‘1etters’’ of DNA. They then created artificial DNA strands recording different parts of the song.

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and explains: These DNA messages, each about 150 bases long, were inserted into bacteria such as E. coli and Deinococcus radiodurans. The latter is especially good at surviving extreme conditions. It can tolerate high temperatures, desiccation, ultraviolet light and ionising radiation doses 1,000 times higher than would be fatal to humans. The beginning and end of each inserted message have special DNA tags devised by the scientists. These ‘‘sentinels’’ stop the bacteria from identifying the message as an invading virus and destroying it. ‘The magic of the sentinel is that it protects the information, so that even after a hundred bacterial generations we were able to retrieve the exact message.’

The researchers tell us that once the DNA message is in the bacteria, it is well protected and can survive. It has been estimated that just 1 ml of liquid can store as many as billion bacteria. If this innovative system of storage proves viable, then the potential for the storage of data in a secure environment is enormous. New secure storage systems based on bacteria will then be able to offer an unlimited capacity for protected data. A computer to help reveal the secrets of life A supercomputer installed at a laboratory in Daresbury, Warrington, Chesire, UK is reported to be able to carry out some three trillion calculations for a second. Called The High Performance Computer (HPCx), the project for its implementation has already cost £53 million and it is promised that its power will be doubled every 2 years. A UK government Minister for Science and Innovation says that: The range of applications will potentially enable scientists to answer some of the greatest questions about the world in which we live. A major task of the project is to run software that will be able to boost scientist’s studies such as gene research and improve the detection of breast cancer. Professor Paul Durham of the Daresbury Laboratory believes that one day the machine could provide answers to the very secrets of life itself. Cyberneticians and systemists will recognise that this supercomputer in its design is a fairly standard IBM system. The difference appears in its configuration which instead of having one processor has 1,280. It means that highly complex systems can be modelled and tested.

The project leaders say that they will be able to use the new supercomputer, which has been named The Brain, on a number of important researches which include: . areas of gene research – now that the human genome code has been ‘‘cracked’’, . investigations on how the cell and the various organs of the body function, . material research into smart alloys, superconductivity and nanostructures, . investigate the structure of the earth and how its core impacts on our lives, . drug design – ability to screen molecules faster by hitting the exact chemical compounds to treat disease, . the area of research into air turbulence from aircraft and, for example, a knowledge of how it is formed could speed-up the traffic in airports. These are just some of the areas that can be tackled by a machine that can boast such power. When it was installed its calculation time of three trillion calculations for a second (it has some million million bytes of memory) made it one of the most powerful academic computers in the world. It took 3 weeks to install and had the task of serving all of the 121 UK universities. Artificial intelligence Whilst researchers in artificial intelligence continue to ponder the future directions of their endeavours and what they have achieved in the past decades, it was encouraging to read the positive applications of AI to ‘‘real’’ problems. As usual, it often takes a competition to publicise the work of many research teams and the Progress Towards Machine Intelligence Prize from the British Computer Society (BCS) Artificial Intelligence Specialist Group provided the incentive. The winner of the award was Lars Nolle from Nottingham Trent University, UK who developed a system that automates an industrial process in seconds currently carried out only by experienced scientists. Indeed, the system is claimed to perform much better than the scientists. The system is called X-WOS and it is claimed that: it could revolutionise processes that use a gas called plasma – including the manufacture of semiconductors. Normally highly qualified scientists are needed to manually adjust 14 parameters, often taking several minutes at a time. X-WOS enables the task to be completed in seconds, and with much better accuracy. It makes an intelligent search of possible solutions to home in on the best one.

What is interesting about this competition is that it attracted entries across the world and showed the level of interest and development in this important area.

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Some of the other finalists for the AI award also exhibited live practical systems: .

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.

Wouter Teepe (Groningen Univerity, The Netherlands) demonstrated a system to help voters establish, which political party best reflected their views. Patrick Wong (Open University, UK) presented and demonstrated a new approach to identify hidden faults in metal using ultrasonic images which is designed to prevent plane, rail and other disasters.

All the entries and indeed the finalists who demonstrated their systems brought exciting new developments in machine intelligence with genuinely useful applications in the real world. There is little doubt that the great number of people now involved in artificial intelligence projects worldwide, will bring even more exciting entries in future years of the competition. The award is sponsored by the company Electrolux and also suppported by the Applied Knowledge Research Institute. Further details of the award scheme can be obtained by readers from: [email protected] or [email protected]. See also the Web site www.bcs.org for details of the Artificial Intelligence Specialists Group of the BCS. Innovative research A number of innovative proposals with potentially far-reaching consequences are underway at a number of British research establishments. A programme of funding called ‘‘Basic Technology Research’’ has been undertaken by the government’s Research Councils, with the aim of creating fundamentally new capabilities for pursuing futuristic research. The idea being that anyone who is able to demonstrate how their research would contribute to a generic technology base that can be adapted to a diverse range of research problems and challenges that spans the interest of all the research councils should be encouraged to apply for funding. Those who have been awarded funding range from optical biochips to very fast genome sequencing technologies with common threads of solid-state and nanotechnologies. Recent funding has been given to the following projects at British Universities (the grant, project leader and university are included, for reader reference, in brackets after the project title – for further details of the Basic Technology Programme see [1] ). Attosecond technology – light sources, metrology and applications (£3.6m, Tisch, Imperial College London) A proper understanding of the way in which the motion of electrons affect the fundamental processes of chemistry, biology and materials science requires probes into attosecond timescales. This project aims to deliver a source of isolated attosecond optical pulses that will provide the scientist with the tools to study these fundamental processes directly for the first time.

Putting the quantum into information technology (£3.6 m, Stoneham, University College London) Quantum computing and information technology exploit universal gates. This project centres on a new concept for J-gates, which control the entanglement of two spins. The aim is to make this new class of quantum gates in a system which is silicon-compatible, and to demonstrate the operation of these gates in a representative computation at a useful temperature (liquid nitrogen or above).

Optical biochips (£2.1 m, Smith, University of Wales, Cardiff) This project aims to bring down to a microscale all of the main components used to analyse biological samples in a modern life sciences laboratory, including lasers the size of a human cell, to create an optical lab-on-a-chip. Potential benefits include increasing the success rate of drug discovery, genomics research, disease diagnosis, and operation of ultrafast computers.

Four billion bases a day – practical individual genome sequencing (£4.8 m, Bradley, Southampton) A fundamentally new method of synthesising, screening and sequencing DNA at a rate that is thousands of times faster than existing methods will allow over four billion bases to be sequenced in less than a day on one instrument. If successful, the technique will forever change the landscape of biological and medicinal sciences.

Hyperpolarised technologies for medical and materials science (£1.9m, Morris, Nottingham) This project will bring together established UK research communities in magnetic resonance, semiconductors and neutron scattering to develop and enhance technologies for hyperpolarising materials and for storage and transport of these materials. The possibility of devising novel techniques for transferring the polarisation to other materials of interest will be explored.

Cryogenic instrumentation for quantum electronics (£3.1 m, Briggs, Oxford) Many of the most exciting developments in quantum devices need to operate at extremely low temperatures, supported by several stages of electronics operating at successively lower temperatures. This project will develop generic electronics platforms and very low power circuits so that candidate nanoelectronic technologies can be measured and complete circuit designs characterised.

Control and prediction of the organic solid state (£2.3 m, Price, University College London) This proposal seeks to understand the mysterious phenomenon of organic polymorphism, by developing a range of experimental and computational techniques to provide a complete atomistic description of the polymorphs of several organic molecules. These include the use of automated crystallisation techniques to find all likely polymorphs, diffraction technology, nuclear magnetic resonance, and computer simulation to predict possible structures and their physical properties.

For cyberneticians and systemists, the importance of these individual projects is their interdisciplinary nature and the desire of all the UKs Research councils to combine in such a programme that is aimed at embracing such a range of basic technologies. The projects chosen undoubtedly exhibit all the features of what may herald futuristic research not only in the UK, but also worldwide. Note 1. The Basic Technology Programme has a Manager – Dr Alasdair Rose – E-mail: alasdair. [email protected]

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Cybernetics and robotics Biorobotic vision A report from the Australian National University Biorobotic Vision Group describes their innovative researches and their developments in the field of robotic vision. The group has been researching the principles by which insects see, control flights and navigate, in order to transfer their findings to their work in robotic vision. They believe that this work will help them in their aim to devise algorithms for machine vision and autonomous visually-guided robots. In particular, this work is in reference to using optic flow to measure image motion. The researchers have already successfully developed a hover controller that can hover over an arbitrary set of landmarks with no manual intervention.They have also designed an autonomous helicopter that will hover and fly without using GPS. Motion-sensitive visual systems Both these prototypes use a motion-sensitive visual system that determines the apparent angular rate of the ground plane in the longitudinal and lateral directions. Height is determined from the speed of the craft and the measurement of optical flow. The autonomous helicopter is said to have successfully flown for a distance of 2 km without the use of GPS and was guided by ‘‘looking around’’ for itself using this innovative system. Panoramic imaging system Research on panoramic imaging systems which use a standard video camera viewing a specially shaped reflective surface is also in hand. This system is being developed for use in surveillance systems and for visual guidance and the control of autonomous helicopter. A number of projects to produce such devices are being carried out throughout the world at a time when ‘‘drone systems’’ are deemed to be preferable to human controlled ones. A great deal of work in this direction is being carried out at defence establishments. Further robot vision projects The Biorobotic Group are also developing a number of other systems. They include: . corridor following robot, . a terrain following autopilot for an autonomous helicopter, . a robot that measures visual image deformation to estimate its own motion in the environment, and . an obstacle-avoiding robot. These cannot be achieved without progress by the biorobotics researchers into the construction of algorithms for using image deformation to estimate selfmotion in two and three dimensions and for the estimation of self-motion from

optical flow. There is an enormous scope for the use of these types of systems over a whole range of applications. In particular, the development of intelligent ‘‘drone’’ aircraft in both the military and other applications. The need for robots that are more ‘‘intelligent’’ remains and their use in applications that range from domestic and industrial to defence and civil disasters are ever present.

Contemporary systems and cybernetics

Supergen programme A UK initiative called Supergen for sustainable power generation and supply was launched in the Spring of 2003 by the Engineering and Physical Sciences Research Council (EPSRC). Supergen will invest over £25 m during a five-year cycle to address the broad challenges of sustainable power generation and supply. A report ‘‘Powering the Future’’ issued by the EPSRC says that:

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There is a general recognition that the first few decades of the 21st century will witness a significant change in focus of energy supply in the UK. The main drivers for this include political, social, environmental, and economic factors covering climate change, fossil fuel extraction rates, emissions control, and public awareness of environmental concerns.

The UKs Royal Commission on Environmental Pollution – ‘‘Energy – The Changing Climate’’ highlighted the need to consider a sustainable approach to power generation in terms of low- or zero carbon technologies. This is a view that is supported in Britain by a number of recent studies including the government’s Foresight initiative and the work of the UK’s Cabinet Office Performance and Innovation Unit Energy Review. The first awards given under this initiative have supported: . Marine energy – energy from the seas around the coastline . Hydrogen – the fuel of the future? . Biomass – that is using fast growing crops as a renewable fuel supply . Networks – to ensure a reliable supply of power to the UK. To help this endeavour, EPSRC is already supporting consortia of universities, industry users, and stakeholders to tackle what are seen as strategically important areas. It is also working with other UK Research Councils in its initiative. Obviously, this is an initiative which should not be confined to one country and it is one which reflects the challenges that other nations worldwide are also tackling. Further details of the Supergen initiative can be obtained from: EPSRC’s Edward Clarke – E-mail: [email protected] B.H. Rudall Norbert Wiener Institute and University of Wales (UK)

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Appointment to the Kybernetes editorial advisory board Dr Ranulph Glanville, one of the leading cyberneticians, has accepted an one of our invitation to join this journal’s editorial advisory board. We are delighted that he will be bringing with him his wide-ranging expertise in cybernetics and systems that will help in to the compilation of these pages. Ranulph Glanville was educated at Bryanston School and the Architectural Association (AA) School in London, where he spent much of his time learning contemporary music, that included electronic music and early electronic performance. On deciding that he did not want to spend his life converting terrace houses, he accepted the offer of a studentship to study for a PhD in Cybernetics with Gordon Pask (his PhD is one of the foundation works for second-order cybernetics). Meanwhile, he was invited to join the teaching staff at the AA where he discovered that his efforts in contemporary music had left him with little knowledge of architecture (space). Subsequently, he began to research how we appreciate and understand space in order to be able to teach. This led him to a second PhD (with Laurie Thomas) in Human Learning, in which he argued that we perceive space wholistically rather than by using a check list (of variables). His main positions have been at the AA and at Portsmouth Polytechnic School of Architecture (now the University of Portsmouth). For eight years he took a leading role in the Dutch Government funded research programme “Support, Survival and Culture,” a major improvement. He took early retirement, some years ago because of changes in the aspirations of British Universities, and has since been a freelance academic with a continuing position as a professor in the Faculty of the Constructed Environment at Royal Melbourne Institute of Technology, University of Australia, dividing his time between his home in the UK and a flat in Melbourne. His special area of interest is in the development of a masters and doctoral programme in design based on reflective practice, where his cybernetic investigations are seen as central. He has a position at the Bartlett School of Architecture in London’s University College. He has held visiting posts in several continents and has lectured in all bar Antarctica. Most (but not all) of his teaching has been in the areas of design, architecture and art. In contrast, Glanville’s main area of research has been in cybernetics, in particular the second-order cybernetics, which he understands as the consideration (in cybernetics terms) of cybernetics itself. He regards this area as both the heart and the conscience of cybernetics, and sees his task as maintaining a coherence and clarity in the way we understand the basic

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concepts of cybernetics, and their consequences and connections, by News, reconsidering them. His other main areas of research are design (an action conferences and which he sees as an embodiment of cybernetics), how we understand space, technical reports interaction and the space between (related to the new electronic media and to the understanding of interface) and how design and research fit together: he holds that research is a special and specially limited form of design, and that 1171 we design our concepts together to make up our world views. He is on the board of several societies, regular conferences and journals. The various awards conferred on him include the Distinguished Service and Special Scholarship Awards of the International Institute for Advanced Study in Systems Research and Cybernetics. He has exhibited art works and installations, and had performed music internationally – most recently, an audio piece with accompanying video composed with his son Severi, which has been well received in hearings around the world. Glanville’s publications exceed 200 in number, and include his contribution both as author and editor (with Bernard Scott) to the commemorative volumes of Kybernetes published in honour of Gordon Pask. He writes short stories and has a small publishing house. Ranulph Glanville is married to the Dutch therapist Aartje Hulstein. His son Severi lives in Helsinki and practices as a digital artist in the time-based arts.

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Conferences Celebration of Stafford Beer’s Life and Work School of Economics, London UK – 3 March 2003 Following Stafford Beer’s death last August 2002, there was an unanimous desire amongst his family and friends to organise, in general, a celebration of his life and in particular, his contributions to knowledge and society. This journal has already reported the many tributes paid to him and the special events that honoured him. In particular, in an earlier article (Leonard, 2003) we published the text that was distributed at the celebration of his life, held at the Royal Society for the Arts in London (UK) on 11 October, 2002. Written by Dr Allenna Leonard it provided us with a fascinating account of his life’s endeavours and one that needed to be consulted before attending this symposium and gathering of his friends and colleagues. It was arranged by Professor Raul Espejo after discussions with Professor Beer’s partner and colleague, Dr Allenna Leonard, family and friends. In his invitation Dr Espejo wrote that: Working in the traditions of Warren McCulloch, Norbert Wiener and Ross Ashby, Stafford Beer created and developed organisational cybernetics. He grounded his inventions, among others the Viable System Model and Team Syntegrity, in his practice as a management scientist, most notably in the political project of the Chilean Government of the early 1970s. He offered a cogent, compelling and enlightened account of this practice in writings such as Decision and Control, Brain of the Firm, Designing Freedom, Platform of Change, The Heart of Enterprise and Beyond Dispute. For us the challenge is to transform his legacy into a growing body of knowledge, contributing to the production of fair societies and organisations and sustainable environments. In this meeting, we want to debate and clarify the challenge of this transformation.

The event was hosted by the Complexity Research Group of the London School of Economics (UK) and held in the Vera Anstey (VAR) suite of the old building at the school. Although arranged to last one day the gathering of some of the most distinguished personalities working in Management Science could not be constrained to mere 24 h. Even so, the 12 sessions held on that day had a limited duration each. This resulted, inevitiably, in further gatherings of the participants in nearby hotels, restaurants and of course, in the pubs of Fleet Street. Stafford Beer is regarded as the founder of Management Cybernetics and was undoubtedly among the world’s most provocative, creative, and profound thinkers on the subject of management (Ackoff, 1974). This event also displayed his many talents and interests and those who were invited to attend came from a variety of backgrounds and included not only academics and practitioners working in organisational cybernetics and applied epistemology, but also poets, writers and artists. Some 50 invited participants were welcomed at the London School of Economics (LSE) by the organiser and chairman, Professor Rau´l Espejo

and Dr Eve Mitleton-Kelly of the Complexity Research Programme at LSE. This News, was followed by Dr Allenna Leonard’s address on “Equilibrium and Values: conferences and Pluralism or Competition among fundamentalists”. Dr Leonard was Stafford technical reports Beer’s partner and colleague for over 30 years and there was noone better equipped to discuss and interpret his innovative ideas and aspirations. She described some of the concerns that he raised (Beer, 2002) at the University of 1173 Valladolid, Spain when he asked the question – What is cybernetics ? On this important occasion he discussed popular notions of cybernetics and genuine difficulties in understanding its origin, derivations and definitions. In particular, he related cybernetics to the current world situation at a time when we had just experienced the horror of the 11 September 2001. Dr Leonard added to our understanding of these ideas. Accept complexity, she said and make problem solutions work. She also reminded us of the Stafford Beer Archive and invited those present to contribute to it. The archive is held at the John Moores University at Liverpool and materials should be sent to Dr Maurice Yolles. The presentations that followed were concerned with the many and varied facets of Stafford Beer’s life and his contributions to cybernetics and systems and to our society. Trevor Hilder of Cavendish Software talked about “Viability vs Tribalism – why Cybernetic Interventions usually fail, and what we can do about it”. Most people agree that tribalism can destroy the validity of organisations. Maurice Yolles of the John Moores University at Liverpool dicussed “The System-Metasystem Dichotomy”. He examined some of Stafford Beer’s ideas and the relevance of the legacy of his work. Markus Schwaninger (St. Gaullions, Switzerland), followed with a presentation on “City Planning” – Dissolving “Urban Problems: Insights from an Application of Management Cybernetics”. He described Professor Beer’s links with his university and his interest in its projects. Two speakers, German Bula and Roberto Zarama spoke about “Embodying and Grounding Distinctions: Revisiting Intervention and Research Archives”. With details of projects carried out in Columbia and Chile they elaborated their theme with discussions about: The construction of a Viable State and Education as a variety amplifier. Throughout the meetings, both videos and DVD relevant to the work of Stafford Beer were made available on monitor displays. It was of enormous interest to the gathering to see the filmed sequences of various stages of his life displayed on the screen. Later sessions emphasised his wide interests. His poetry and the style in which it was written was introduced by the Canadian publisher – David Whittaker. Several poems were read and the Welsh “cynghanedd” method of writing he used was discussed. Stafford Beer lived in Wales and had a great empathy with his Welsh neighbours, and the language and culture of the country. David Weir, who is well-known worldwide for his contributions to business matters and to management sciences and is now based in France, took us

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through a discussion about the “Sequence of Failure in Complex Socio-Technical Systems”. He was followed by Werner Schuhmann, who is an Emeritus Professor at Mannheim University. He recounted his special links with Stafford Beer illustrating his thesis around five main headings. Amongst them he highlighted the establishment of the German Society of Social Cybernetics (GWS), observing experiences with VSM, Syntegration and second-order cybernetics. Jonathan Rosenhead spoke of Professor Beer’s past initiatives in Operational Research (OR). He described the setting up and the development of Science in General Management (SIGMA) in the 1950s. This was the first OR consultancy in the United Kingdom. Professor Rosenhead is a Professor of OR at LSE and his contribution was aptly titled “Stafford Beer – the conscience of OR”. Raul Espejo, who is a Professor of Information at the University of Lincolnshire and Humberside Department of Management Systems, was a longtime friend and colleague of Stafford Beer. We were all indebted to Professor Espejo for arranging this meeting, which served both as a celebration and a working symposium, his address emphasised future directions and tied Stafford Beer’s important and innovative work to organisations and problem-solving. The final formal presentation was by Gerard de Zeeuw (The Netherlands), who once again showed us that a clear and incisive approach to considerations of “knowledge and knowledge systems” even after eight hours of presentations and discussions, could be both stimulating and exciting. It was at this point that the formal meeting to celebrate Stafford Beer’s life and work ended and the discussions, anecdotes, memories about him resumed in the hostelries of London. Editor’s note: We are grateful to both Raul Espejo and Allenna Leonard for accepting the invitation of the Kybernetes Editorial Advisory Board to be the Guest Editors of a Special Double issue of the journal that will be based on the presentations given by the invited distinguished speakers at this event. The issue is scheduled to be published in Vol. 33 Nos. 3/4, 2004. We are also particularly pleased that permission has now been granted for Professor Beer’s name to appear in Kybernetes as “Founder Patron (In memorium)”. He was, readers will recall, the President of the World Organisation of Systems and Cybernetics (WOSC) and a founder patron of this journal, which is the official publication of WOSC. References Ackoff, D.L. (1974), Comment in Praise of “Designing Freedom”, by Stafford Beer, Wiley, Chichester, Reprinted by Stafford Beer Classic Library 1994. Beer, S. (2002), “What is Cybernetics?”, Kybernetes, Vol. 31 No. 2, pp. 209-19. Leonard, A. (2003), “Stafford Beer, 25 September -23 August 2002” Stafford Beer – Celebration of a life, Kybernetes, Vol. 32 No. 4, pp. 459-61.

Conference reports Systems, Man and Cybernetics: Principles and Applications Workshop The “Systems, Man and Cybernetics” Chapter of the United Kingdom and Republic of Ireland Section of the IEEE was formed in June 2002, and held its first meeting in the City University, London, in the new Chapter. Details can be found on the Web site: http://www.cyber.rdg.ac.uk/people/R.Mitchell/ieeesmc/. Following a resolution made at the first meeting, a one day inaugural workshop was held in London on 26 November, with title as above. The venue was the Royal Statistical Society. With just over a dozen participants and a programme allowing time for discussion, the atmosphere was relaxed and constructive. The Keynote Speech was given by Professor D. Linkens of Sheffield University on: “Intelligence in Anaesthesia: Integrated Concepts for Patients, Clinicians and Machines”. He reviewed the requirements of medical anaesthesia, dealing separately with considerations of patient awareness, analgesia, and relaxation, and went on to discuss means by which depth of anaesthesia can be monitored and controlled. An operational automatic control system was described, using feedback from EEG responses to auditory stimuli as well as from blood pressure and heart rate. It was emphasised that the system operated under close supervision with ample opportunity for manual override if needed. This was followed by “An Overview of Cybernetics” by Dr Richard Mitchell of Reading University, who is also the Newsletter Editor of the new Chapter and responsible for publicity. He described the origins of the subject and its subsequent development in various contexts, from elementary principles of feedback to modern applications in Robotics and Virtual Reality. Lovelock’s Gaia Hypothesis was included, as well as the extension to social and business systems by Stafford Beer, and second-order cybernetics was mentioned. There were six further papers, three before the lunch break and three after. One was on “Principles of Vision Systems” by Dr B. Amavasai of Sheffield Hallam University, who is responsible for professional activities of the new Chapter. The focus was on artificial vision systems and their applications ranging from robot control and industrial inspection to remote surveillance and astronomy. Specific processing techniques were described, particularly in the context of robot control. Another paper was on “Human System Dependability” by Dr M. Oussalah of City University, who is responsible for educational activities of the new Chapter. The dependability of complex systems in which humans and computers interact raises many important questions. The other paper before lunch was on “Principles of Systems Assurance” by Dr A. Hessami of the consultancy firm, Atkins Global, with previous experience in British Rail and then Railtrack. Dr Hessami is the Chairman of the new Chapter.

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The first paper after lunch was on “Robots, from Workhorses to Social Beings” by Dr B.R. Duffy who is affiliated to the Media Lab Europe, Dublin, in association with the famous Media Laboratory of MIT. His research is into sociable robots and he gave an account of the development of robots from Vaucanson’s duck, through Grey Walter’s tortoise and industrial robots and experimental AI machines such as “Shakey”, to recent innovations stressing the social aspect. He explored motivations and deprecated the suggestion of robot take-over. Another paper was on “Systematic Decision Support” by Professor B. Lehany of Coventry University. He defended the use of computer simulation as a tool for improving organisation and referred to a successful application to a hospital outpatient department. A key to success was close co-operation with the medical and administrative staff, which was achieved by exhibiting simulation results that agreed with their experience. The final paper was on “Principles of Systems Reliability and Integrity” by Professor N. Sinnadurai of the Bookham Technology company. Attention was given to ethical and legal aspects. The organisation of the presentations was admirable. All the speakers made good use of a computer-linked projector. Participants were given a folder that included abstracts of most of the papers and biographical notes on most of the speakers. In addition, for all but one of the papers presented, the folder contained pages on which all of the projected images of the presentation were reproduced in black-and-white, with a lined area alongside for note-taking. No definite plans for future activities of the chapter were announced, but this workshop with its interesting mix of topics was certainly a good start. Alex M. Andrew Symposium honouring D.A. Pospeloc On 11 February 2003, a symposium was held in the Polytechnical Museum, Moscow, to mark the 70th birthday of Professor Dmitri Alexandrovich Pospelov, a foremost Russian worker in the field of Artificial Intelligence and an Academician of the Russian Academy of Natural Science. The meeting was hosted jointly by the Russian Association for Artificial Intelligence and the Polytechnical Museum, with both of which D.A. Pospelov has been associated. In a Press Release announcing the event, Professor Vadim Stefanuk referred not only to the immediate contributions of Pospelov to the subject area, but also to his wide interests which attracted around him a group with diverse specializations, from computing technology to psychology, spanning applied as well as human sciences. He also had a long standing connection with the Polytechnical Museum and was an enthusiastic populariser of science and author of numerous books. He, and his students, gave important inputs to numerous symposia held in the museum. Professor Pospelov was not able to join in all of the day’s activities as he was severely injured in an accident about 4 years ago and is still confined to a

wheelchair. He was able to attend the first part of the proceedings which was in News, a fairly small auditorium and consisted of relatively short addresses mostly conferences and from students and former colleagues. I was invited to participate, and referred technical reports to the very first beginnings of AI with Ada Lovelace and Charles Babbage and the enormous developments since then in which Dmitri Pospelov played a major part. His work included analysis of fairy tales and fantasy, and so carries 1177 to new limits the departure from plain numerical computing first foreseen in the writings of Lovelace. The second part of the day’s activities was in a large auditorium and included longer presentations. The proceedings were open to anyone interested and were intended to be intelligible to non-specialists. At the end, participants were invited to an adjoining room where a generous buffet had been set out by the museum staff. As well as tea and coffee, Russian wine and vodka were provided in amounts that amply allowed for numerous toasts to Dmitri Pospelov and colleagues. A number of the major contributions also appear in a special issue of AI News, the journal of the Russian Association for Artificial Intelligence. The issue (No. 6 for the year 2002) has the theme of “development of ideas on AI in our country” and is prompted by the birth anniversary of Dmitri Pospelov. The papers are in Russian with brief abstracts in English and refer to Pospelov’s theories of situation control and the related field of semiotics. The five papers in the issue, using the English-language versions of titles, are as follows: “From situation control to applied semiotics” by G.S. Osipov, pp. 3-7. “Knowledge in intelligent systems” by V.N. Vagin, pp. 8-18. “Natural-language text processing from language understanding models to knowledge extraction technologies” by V.V. Khoroshevsky, pp. 19-26. “Introduction to applied semiotics. Chapter 5: Operations in semiotic knowledge bases” by D.A. Pospelov and G.S. Osipov, pp. 28-35. “Logical Linguistic Control as an introduction to Knowledge Management” by T.A. Gavrilova, pp. 36-40. The presentations were in Russian (except for my few words, which were translated by Vadim Stefanuk) and I cannot report on the contents in detail. It is clear, though, that there is a vigorous school of thought built around the innovations and specific viewpoint of Dmitri Pospelov. Fortunately, for those who are poor linguists a good deal can be gleaned from his many publications in English. We congratulate him on his birthday and send best wishes for an early recovery from the effects of his accident. Alex M. Andrew

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Internet commentary Keywords Semantic Web, Search engines, ERCIM, Professor Edsger Dijkstra

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Abstract The Semantic Web is introduced as an emerging development from the World Wide Web, promising enormous improvements in performance and ease of use. Sources of information include a free journal. The death of the computing pioneer Professor E.W. Dijkstra is noted.

Semantic Web The vast amount of information accessible on the World Wide Web is certainly not arranged for the convenience of access. The penetration of the Web, virtually into all areas of knowledge is only possible, because it has grown without overall constraint or direction, but a consequence is that there is no prearranged structure or indexing system. The difficulty of navigating has been alleviated by the introduction of search engines, but it is easy to see that they are not an ideal solution. A special issue of the journal ERCIM News is devoted to the next development, referred to as the Semantic Web. Euzenat (2002) uses a simple illustration to show the deficiency of the existing search facilities. He points out that looking for a book about a prolific author, say Agatha Christie, is not easy because a search engine given the name will list the many works that are by Agatha Christie as well as those that are about her, presumably including her autobiography, which is in both categories. If the word “about” is included among the keywords inserted, it will have no effect on the search unless one of the titles happens to include it. A language devised to improve the search capability is termed resource description framework (RDF) and one of its main features is attention to relational information as conveyed by the terms like “is about” and “is author of”. This is part of the “Semantic Web” initiative, first proposed by the pioneer of the World Wide Web, Tim Berners-Lee. A valuable Web site, with numerous links, is: ,http://www.semanticweb. org . . This includes links to sites where research from different groups is reported, and a “news” section with notices of relevant scientific meetings. There is an invitation to become involved by founding a Semantic Web community and leading an effort to develop a vocabulary for a specific domain, and anyone with this in mind is invited to communicate with Stefan Decker at: ,[email protected] . . “Challenges” or suggestions for software that might usefully be developed can also be submitted, and sponsors are sought. Suggestions and offers can be sent to Terry Payne at: ,[email protected] . . There is an opportunity to subscribe to an interest group, run by Yahoo, or to join it, by sending an empty e-mail to: , semanticweb-subscribe@egroups. com . and then responding to the invitation that is sent. Anyone who

subscribes to the group receives periodic e-mail updates on progress, and members additionally have access to certain group resources and can contribute. The aim of the developments is to let the semantics of the subject matter on the web be available to the machines, so that much more can be done to assist the user. Work on knowledge-based systems, and hence AI, will play a part, but it is not intended that the Web should become autonomous and independent of humans. According to Berners-Lee and Miller (2002) and Berners-Lee et al. (2001), the new version of the Web will evolve from the old one, and will have the same decentralised character. The main tools already exist, one of them being the language RDF, and the other XML or Extensible Markup Language. The decentralised character means that the system will be incomplete, in a mathematical sense meaning that it may fail to give an answer. Berners-Lee et al. (2001) suggest a rule, reminiscent of Go¨del’s theorem, to the effect that any system complex enough to be useful also encompasses unanswerable questions. The language RDF allows the setting up of triples that are like the subject, verb and object of an elementary sentence. The “verb” part indicates a relationship like “is author of”, and types of such verbs are stored as Universal Resource Identifiers (URIs) similar to the familiar URLs or Uniform Resource Locators. As with URLs, new URIs can be defined anywhere by anyone and so the semantics of the Web will become continually more comprehensive. The Extensible Markup Language allows tags to be attached to hypertext that are not represented in the screen display of ordinary browsers. They are semantic links for machine use. Scenarios Both Euzenat (2002) and Berners-Lee et al. (2001) illustrate the expected power of the developments by describing the hypothetical situations. Euzenat describes a software “agent” acting as an efficient travel agent and planning a complex trip involving several modes of transport as well as hotel stays and conference and entertainment registrations, at minimum cost. The choices would be subject to constraints of acceptable hotels with vacancies, eating requirements, frequent-flyer affiliated car rental, and so on. Berners-Lee et al. refer to a brother and sister, arranging to take their mother to clinics for a series of physiotherapy sessions, where both have full appointment diaries and there is a choice of clinics. The personal “agent” of one of them is able to search for clinics within a stated distance and to verify that they are suitably accredited and covered by the mother’s insurance plan. It then tries to devise a schedule of sessions, where each slots into a vacancy at one of the clinics and also into free time in the appointments diary of one or other of the siblings.

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Agents Operation depends on the setting up of software “agents” able to communicate. As well as the personal agents postulated, agents would be set up by hotels and airlines and car-hire companies, and also by clinics and insurers and accreditation authorities. Not all information would be freely accessible, and for example, the personal agent (that of the sister, say) forming the plan for visits to clinics would only be able to access the appointments diary of the brother once he had indicated to his agent that the sister’s agent was to be trusted. A “verb” indicating “is trusted by” can be represented as an URI, though in practice, something more complex defining the contexts in which access is permitted would probably be wanted in even the most friendly of family relationships. The possibilities of the Semantic Web are most readily illustrated in terms of these tasks of a scheduling nature, but more significant applications can be visualised for academic research and discussion and the amassing of human knowledge. Ontologies Certain information about the structure of knowledge is needed. For example, it needs to be known that a postal address normally contains a house name or number, street name, zip code, etc. The terms “zip code” and “postal code” should be recognised as equivalent, and “autobiography” must be seen to imply both “is about” and (in reverse direction) “is author of”. The word “ontology” is borrowed from philosophy and used to denote a facility for the formal representation of relations among terms, usually by a set of inference rules. The effectiveness of the Semantic Web depends critically on the construction of suitable ontologies. ERCIM News The journal ERCIM News is published by the European Research Consortium for Informatics and Mathematics, and is very generously distributed free of charge. Details can be found on the Web site: , http://www.ercim.org/ publication/Ercim_News/ . . Besides offering the possibility of registering to receive the journal, this accepts requests for back issues that can be supplied in either printed or electronic form. A search facility is also provided. Back issues are held from October 1994. Professor Edsger Wybe Dijkstra In the same issue of ERCIM News, the death of E.W. Dijkstra, on 6 August 2002, after a long struggle with cancer, was announced. He was the most famous member of the Mathematisch Centrum in Amsterdam (now Centrum voor Wiskunde en Informatica) and among his many achievements was the introduction of the stacking principle, which was the basis of the world’s first

ALGOL60 compiler. From 1984 to his retirement in 1999, he worked at the University of Texas in Austin. Details can be found at: , http://www.cs.utexas.edu/users/EWD/ . , including proceedings of a symposium held in honour of his birthday in May 2000 (occupying 10 MB) and a 25 min video. He was the author of more than 1,300 papers and all of them have been made available online, many of them not previously published. There are also warm tributes to his warm and generous character. Alex M. Andrew References Berners-Lee, T. and Miller, E. (2002), “The Semantic Web lifts off ”, ERCIM News, No. 51, pp. 9-11. Berners-Lee, T., Hendler, J. and Lassila, O. (2001), “The Semantic Web”, Scientific American, Vol. 284 No. 5, pp. 35-43. The paper can be downloaded free of charge from: , http://www. scientificamerican.com/2001/0501issue/0501berners-lee.html . . Euzenat, J. (2002), “A few words about the Semantic Web and its development in the ERCIM institutes”, ERCIM News, No. 51, pp. 7-8.

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Book reviews Vie artificielle Ou` la biologie recontre l’informatique Illustre´ ave Java Jean-Philippe Rennard (http://www.rennard.org) Vuibert (http://www.vuibert.com) Paris 2002 408 pp. ISBN 2-7117-8694-3 Review DOI 10.1108/03684920310483261 Every generation of scientists is eager to gain immortality. The easiest way to do it is to find a novel field of science. If you lack creativity to construct the new field – just rename the old one. What was called cybernetics in 1950s became bionics and bio-cybernetics in 1970s and then was subsequently renamed to artificial life in 1980s. Due to stronger flux of scientists between disciplines and the emerging trend towards academic Renaissance, this newborn nomen was lucky enough to accommodate computer science, biology, physics, chemistry and engineering. Rapidly artificial life is becoming a theory of everything, and therefore, newcomers feel themselves completely disoriented in this soup of weird ideas, concepts, algorithms and implementations. This is why a good introductory book would be highly appreciated. Jean-Philippe Rennard undertook a painful task of systematizing and popularising basic concepts of the artificial life. His efforts resulted in the very accessible and ‘‘politically correct’’ overview of the field. Right from the title page – Vie artificielle – and then all over the text, Dr Rennard keeps us reminding that ‘‘artificial life’’ is first of all ‘‘life’’ and only then ‘‘artificial’’. The text is subdivided into seven chapters – that logically introduce fundamental concepts of artificial life – and two appendices – that provide biological foundations of evolution and basics of computer graphics. Essential concepts of artificial life are unfolded in the first part (chapters 1-4), from defining ‘‘artificial’’ to emergence to self-replication and recursions. Second part (chapters 5-7) deals with bio-inspired computations and natural distributed computing. First chapter tries to answer the question ‘‘What is artificial life?’’ It starts with a short and pleasurable excursion in the history of artificial life, from Golem and Frankenstein to mechanical duck. Then, the entelechy is derived from thermodynamics and Maturana-Verala’s autopoietic systems, selfmaintaining and self-reproducing autonomous processes. The notion of life is considered in the context of universality, thus universal Turing machine is discussed in detail. To the end of the chapter, reader finds himself reading

about Java classes and methods, which will be used in simulation examples all over the book. Second chapter is about emergence and its role in the framework of complexity and non-linearity. Cellular automata are motherhood of emergence. This is why rest of the chapter deals with these one- and two-dimensional lattices of locally connected finite-state machines. After briefly mentioning Ulam and von Neumann, author introduces Conway’s game of life. This species of cellular automata exhibits remarkably rich spatio-temporal dynamics and yet bounded growth of its configurations, and thus instantiates a non-trivial (i.e. emergence-based) mathematical model of artificial life. A reader will not only enjoy walking in the zoo of famous patterns – gliders and glider guns, oscillators and breathers, but also will be given a chance to implement his own cellular-automaton models based on Java classes described by the end of the chapter. Universality and self-replication are two essential attributes of biological life. They are looked at through a prism of cellular-automaton lattices in third chapter of the book. A computational universality, i.e. an ability to implement any logical function, is introduced in the chapter using game of life cellular automata. Author shows how to represent truth-values in states of mobile compact patters, or gliders, travelling on the lattice. Then basic logical gates are technically implemented via collisions of glider streams, and cellular-automaton model of universal Turing machine is exemplified. Wolfram and Heudine phenomenological classifications are introduced to demonstrate emergence of non-trivial (now coined as ‘‘computation at the edge of chaos’’) properties. Rest of the chapter, lavishly illustrated, is devoted to Neumann’s self-replicating automaton, its version simplified by Codd and Langton’s loops. Typically, for this book we can take a delight in studying Java implementation of Morita-Imai’s cellular-automaton selfreplicating system. Life is recursive is the thesis of fourth chapter. The chapter introduces recursive algorithms and mathematics of self-similarity, which then exploited constructing biomorphs and L-systems. Biomorphs give us rather aesthetic insight of ‘‘would-life’’ life forms invented by Dawkins, while Lindenmayer’s systems offer a formal-language-based approach to imitation of growth and form. Customary, we can also get hands on L-systems Java applet, analysed in full details. Fifth chapter shows how to do optimisation using genetics and evolution. The chapter starts with detailed informal introduction to genetic algorithms followed by excursion in the field of evolutionary programming and natureinspired parallel optimisation and genetic programming. A brief discussion of advances in evoware, including evolutionary electronics, programmable field array, nicely indicates promising ways of evolving in silica. A detailed study of Java classes gives us a chance to implement our own computer experiments in natural optimisation.

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Most up-to-date concepts of bio-inspired distributed intelligence – from social insects to robotic collectives – are condensed in sixth chapter. This is the chapter about swarms, natural and artificial. The chapter starts (a concept of emergence in collectives is introduced) and finishes (a Java applet is vivisected) with Reynold’s boids. Author shows how self-organization and stigmergy (interaction via environment) can be applied to design sensible algorithms for optimisation on graphs. This is exemplified by the approximation of shortest path in a colony of ants. Animats and Brainteberg vehicles serve as a prelude to overview collective and evolutionary robotics. Author, quite reasonably selected two demonstrations of collective robots: pioneer Beckers-HollandDeneubourg experiments in collectives sorting in robot swarms and KubeZhang findings in cooperative activities in robotic collectives. All usual suspects are presented in the sections in evolutionary robotics – Sims’s virtual creatures, Pollack’s evolved robots, and Yim modular self-configurable robots. The last, seventh, chapter is about the origination of virtual life. Initially, readers are warmed up by Core Wars, where tiny assembler programs fight for computer memory resources. Then, an exciting concept of programmable matter is introduced. Unfortunately, we could not find any references to Toffoli-Margolus ideas of programmable matter; however, we have enjoyed a brief introduction to Rasmussen’s self-assembling automata and Dittrich’s artificial chemistry. There are two appendices. The first one introduces the basics of biological evolution, from Lamarck and Darwin to Kimura and Kauffman, and informal excurse into elementary genetics and cell biology. Second appendix teaches how to write Java applets for visualisation. Another advantages of the book include quite comprehensive glossary and exhaustive list of references. Extensive bibliography will give a quick source of references to professionals. The book is self-consistent, accessible to amateurs and comprehensive in subject representation. Those thinking to enter this exciting field of crossdisciplinary research will appreciate the text. The book is a lovely piece of work – fun to browse through and pleasure to read. It is a must, if you need an informal introduction to artificial life, digital biology and bio-inspired computation. Frenchmen have everything that others lack – healthy food, comfortable climate and wonderful wines – now they have got Jean-Philippe Rennard’s book on artificial life. Let us hope that the book will be promptly translated into English so that the rest of the world will enjoy it. Andrew Adamatzky Computing, Engineering and Mathematical Sciences, University of the West of England E-mail: [email protected]

Probabilistic Reasoning in Multiagent Systems: A Graphical Models Approach Yang Xiang Cambridge University Press Cambridge 2002 ISBN 0-521-81308-5 xii + 294 pp. hardback, £45.00 Review DOI 10.1108/03684920310483270 This is about intelligent decision support systems to operate with uncertain data, where the decisions supported may be in such areas as business management or medical diagnosis. This is not a new field and methods depending on graphical models known as Bayesian or belief networks have been studied for two decades and are treated in a number of texts including a standard one by Judea Pearl. These methods are for processing at one location, within a single intelligent agent. The present book extends the methods to allow multiagent operation. As it is expressed in the notes on the back cover: The framework developed in the book allows distributed representation of uncertain knowledge on a large and complex environment embedded in multiple cooperative agents and effective, exact, and distributed probabilistic inference. The treatment is inevitably mathematical, but the presentation is designed to make it as readily accessible as possible. Each of the ten chapters except the first (‘‘Introduction’’) and last (‘‘Looking into the Future’’) starts with a relatively readable ‘‘guide’’ or ‘‘roadmap’’ section. These chapters are also followed by sets of examples and the book is suitable either for self-study or as the basis of a taught course. Use of belief networks is not the only method that could be used for decision support. They are used to set-up and maintain a set of beliefs about the environment to be managed and to allow decisions to be made. An alternative would be to go directly from the observations to recommended actions without an explicit model, and this is allowed by rule-based systems with their ‘‘if-then’’ rules. Another possibility is the use of neural nets, which can be effective without any semantic labelling at all. The use of belief networks has the immense advantage of transparency, such that it is possible to trace back and analyse any decision. The treatment of multiagent systems is developed from the established methods for the single-agent case, but the book is written so as to be selfsufficient, and the first five chapters deal with the single-agent situation, but with emphasis on aspects that will be needed in the extension to multiple agents in the second half. To introduce the methods, it is necessary to choose a specific sample environment about which beliefs are to be maintained. Reference to a chemical

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plant or an area of medical diagnosis would not be suitable because most readers would not have the necessary specialised knowledge. The type of environment that has been chosen is digital electronic circuitry because it seems fairly safe to assume that all readers will have acquaintance with this. (I have to admit that I had forgotten the significance of the symbols and had to check elsewhere that the shape with a straight line on its input side is an AND gate and the concave one is the OR. A very small amount of this elementary knowledge is all that is needed.) Where the possibility of faulty operation is acknowledged, a two-input gate has four binary variables associated with it, namely its two inputs, its output, and the choice between normal and abnormal. The probability of a given output can be deduced or estimated for any permutation of the other three variables. (It can be deduced as 1 or 0 for a normal component, and values can be guessed for an abnormal one, where abnormality implies erratic rather than consistently wrong operation.) Application of Bayes’ rule then allows estimation, from observations, of the probability that the component is faulty. Probability estimates would be stored for the 16 possible permutations of the four binary variables. In practice, it would not be feasible to monitor such a simple component as a single gate in such detail, but this is the basis of the method. However, the number of variables observed in a real application will certainly be such that the number of states for which beliefs must be updated is unmanageably large. The method is rendered tractable by processing smaller sets of related variables and then combining the results, using graph-theoretic methods. The research leading to the book was stimulated by a project for medical diagnosis called the PainULim project. (Strangely, PainULim does not appear in the subject index of the book – the description is on page 139.) The aim was to develop a system for diagnosing patients suffering from painful or impaired upper limbs due to diseases of the spinal cord, nervous system, or both. The need for a multiagent approach seems to have arisen, because the diagnosing neurologists focused attention on one system at a time. Other application areas for belief networks, mentioned in the first chapter, include advanced robotic devices such as an automatic driver of a vehicle, analysis of operation of a firm to assist business decisions, and similar analysis in the context of intelligent tutoring. Another application area is to ‘‘intelligent houses’’ where the operation of the services, including perhaps the reordering of supplies, is under central control. An important application area is in systems to give warning of dangerous situations in many contexts, including earthquake prediction, and an occurrence in China is mentioned where successful prediction based on many great indicators was responsible for saving many lives. In the final chapter, possible extensions of the theory are considered, including that to dynamic, rather than fixed, environments. The extension to operate on continuous variables is also discussed, since the theory in the book assumes discrete variables. Other work on belief networks with continuous

variables, but only for single agents, is referred to and it seems clear that a forthcoming project for somebody will be to extend the principles to the multiagent case. The enhancements of the general methods, prompted by the multiagent requirement, appear to be valuable even apart from this special aspect, as suggested by the references to ‘‘effective’’ and ‘‘exact’’ in the back-cover excerpt. My strong impression is that this is a valuable and welcome comprehensive guide to the state-of-the-art in applying belief networks. Alex M. Andrew

Spiking Neuron Models: Single Neurons, Populations, Plasticity Wulfram Gerstner and Werner Kistler Cambridge University Press Cambridge 2002 ISBN 0-521-89079-9 xiv + 479 pp. Paperback, £24.95 (also hardback, ISBN 0-521-81384-0, £65.00) Review DOI 10.1108/03684920310483289 Developments in the use of artificial neural nets, in the last few decades, have encouraged the view of the neuron as a continuous device with a sigmoid response function. Such a view is required for both the backpropagation method of learning and the one due to Hopfield. The transmission of information, at least in peripheral nerves, appears to depend on pulse frequency coding and hence a continuous signalling function. This is in contrast to the earlier theory of McCulloch and Pitts, which has been the basis of much speculation in cybernetics, where the all-or-none character of neural excitation was associated with the true-or-false of formal logic. The McCulloch-Pitts model was put forward as a basis for discussion and not as a detailed representation of real neural activity. Real neurons (except some in the mammalian retina) do ‘‘fire’’ or ‘‘spike’’ and a variety of neural phenomena can only be explained by taking this into account. It is easy to show that neural channels conveying continuous data by frequency modulation do not account for observed behaviour because the reception of an analogue value requires a certain time of integration, and the responses of people and animals are faster than this would allow. A faster response is possible, if the integration is instantaneous over many parallel channels with appropriate randomness, and the analysis of this requires consideration of spiking neurons. Another demonstration that the spiking behaviour is significant is in connection with stereophony, where location of a source of sound depends on extremely accurate estimation of the difference in times of arrival of a sound at

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the two ears, or phase differences in a sustained input. For barn owls, the time difference can certainly be estimated within 5 micro seconds. The treatment is divided into three parts indicated in the second part of the title. In the first part, the structure of a neuron is described, and the nature of its excitability is explained in terms of the well-known Hodgkin-Huxley equations that refer to the flow of different ions through the membrane. As the equations have four variables, they are intractable for incorporation in population models, and several more manageable approximations are considered. Synaptic transmission is treated in considerable detail, with accounts of the changes in permeability of postsynaptic membrane. It is acknowledged that neurons may be considered as compartmented models, since for example, an inhibitory synapse can be at the base of a particular branch of the dendritic tree and then it is more effective in nullifying excitation arising in that branch than in countering excitation generally. Possible sources of noise affecting the output of the neuron are discussed. In the second part of the book, a number of approaches to the modelling and analysis of neuron populations are treated, and it is interesting that early papers by Wilson and Cowan are still highly relevant, where Jack Cowan worked at one time with Warren McCulloch. It is shown that the output of an entirely deterministic net can appear highly irregular, and also that information can be transmitted through an active net much more rapidly than would be expected from the time courses of individual neural responses. This can be attributed to the fact that at any moment, there are neurons about to fire and whose firing can be precipitated by the input. Special attention is given to oscillatory behaviour and synchronisation, where one reason for special interest is the theory developed by W. Singer and reviewed by Andrew (1995) which suggests that precise synchronisation of impulses is part of the means by which sensory stimuli are grouped or ‘‘bound’’ to allow recognition of objects. A theory of complex reverberatory behaviour, first analysed in connection with interactions of fireflies, is shown to be applicable also to neural populations. The third part of the book, on plasticity and hence possible mechanisms of learning, is specially interesting. The mechanism at the single-cell level is assumed to be essential as postulated by Hebb, but its implications in neural structures and populations are developed in considerable detail. It is shown how ‘‘learning to be fast’’ can occur, so that a response is triggered by the earliest of the events associated with it, an effect that is illustrated by the classical conditioned reflex. A less obvious effect that can also be accounted for is ‘‘learning to be precise’’, where the response comes to be triggered by the event with least time variability. In this part, there is discussion on binaural sound localisation and the localisation abilities of electric fish, and other special features of perception. The book is essentially mathematical and there are few pages that are free of equations. The authors insist that only fairly elementary mathematics is used

and that a diligent reader or researcher should not be deterred. Each chapter ends with a summary of its main points and a discussion on the relevant literature. The latter is specially welcome in some of the chapters, where it is easy to get the impression that the emphasis is on the mathematical model, and wider reading is needed to appreciate the biological relevance. It is difficult to see how this could have been avoided, while keeping the volume manageable in size. Despite all the good work that has been done, in the ‘‘decade of the brain’’ and earlier, it is clear that the nervous system has yielded only some of its secrets. Further advances will certainly depend on accurate modelling of neural activity, for which the review provided here is a much needed basis. Alex M. Andrew Reference Andrew, A.M. (1995), ‘‘The decade of the brain – some comments’’, Kybernetes, Vol. 24 No. 7, pp. 25-34.

Imitation in Animals and Artifacts Kerstin Dautenhahn and Chrystopher L. Nehaniv (Eds) MIT Press (Bradford Book) Cambridge, Mass 2002 ISBN 0-262-04203-7 xv+607 pp. hardback, £44.95 Review DOI 10.1108/03684920310483298 This substantial volume is based on a symposium with the same title, held in Edinburgh in April 1999, under the auspices of the Society for the Study of Artificial Intelligence and Simulation of Behaviour (AISB). It has 22 chapters by different authors from different countries. The papers are all ‘‘meaty’’ and well prepared and because the average length is about 26 pages, there is a great deal of material here. In addition, in an Appendix, it is mentioned that the symposium also prompted a special double issue of the journal Cybernetics and Systems (Vol. 32 Nos 1/2, 2001) on ‘‘Imitation in natural and artificial systems’’, with another ten relevant papers. A list of presentations and speakers at the symposium is on the Web at: http://homepages.feis.herts.ac.uk/~nehaniv/ aisb.html (which is not exactly the address given in the book itself, which does not work!). One aspect that is not covered is the genetically-determined imitative marking shown particularly by insects and fishes. The interest, if the AISB is

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in behavioural imitation, gives plenty to be going on with. A rather pleasant feature is that imitation is invariably viewed as something positive; the terms ‘‘deception’’, ‘‘intimidation’’, ‘‘warning’’ and even ‘‘caricature’’ do not appear in the subject index. A feature of the high standard of presentation is that there is a good overall subject index. In the Preface and the first two chapters, the editors claim that the symposium broke new ground by bringing together researchers from a broad range of disciplines ranging from animal behaviour to artificial intelligence and from computer science, software engineering and robotics to experimental and comparative psychology, neuroscience, primatology and linguistics. They also show that the importance and complexity of the topic are often underestimated. Imitation has tended to be dismissed as a rather low-grade activity having, almost by definition, no innovative content, but this is wrong. For one thing, even apparently simple imitation of a physical action by another individual involves complicated processing from the observation, probably visual, to the corresponding muscular activity. In many cases, especially when a skill is transferred by example, the learned behaviour is a subtle mixture of imitation with adaptive finetuning by the learner and incorporation of the learner’s prior capabilities for balance, etc., and modifications necessitated by the learner’s different situation. Despite the necessary complexity of its mechanism, the human ability to imitate seems to be innate and to be displayed at an early age, as shown by the readiness of very young children to play imitation games, some involving facial expressions. In the papers of the volume, variations in the character of the imitation are studied; for instance, the degree to which the precise ‘‘style’’ of the exemplar is copied. Where a sequence of actions achieves a goal, it may be copied in a goal-centred way, without regard to style. A difference between autistic and other children in this respect is noted by one author, with the autistic subjects more goal-directed, and there are several references to autism throughout the volume. Various studies on imitation in animals are reported. One paper is on imitation by dolphins, and on the dust cover, there is a very sweet series of illustrations showing a dolphin rather exactly copying postures assumed by a human. Primates are extensively studied, and the title of one paper draws attention to the fact that the readiness of monkeys to imitate is enshrined in the verb ‘‘to ape’’. A study on imitation by parrots is reported showing that their production of human language has much more structure and relevance to tasks and goals than is generally supposed, and is not at all, what is generally referred to as parrot-like repetition. It has been mentioned that the imitation of human movements by another human involves complex processing. The nature of this can be studied by building robots able to imitate humans, and there is a chapter in the book

describing work to this end, in the Artificial Intelligence Laboratory of MIT. As well as interest in imitation for its own sake and for its social function, there is the aim of finding convenient ways of training robots in new tasks. The experimental anthropomorphic robots, Cog and Kismet are referred and shown in illustrations. Another chapter reports a very interesting example of machine learning, with the title ‘‘Learning to fly’’. It is by a group in the Turing Institute in Glasgow that includes Professor Donald Michie, and the experiments are designed to train an artificial system to become an effective autopilot, and particularly an autolander for an aircraft. The training has been done using a flight simulator. It is claimed that existing autopilots are inadequate in some respects, particularly when responding to unexpected gusts of wind close to a landing. Since human pilots capable of better performance are available as models, there is the possibility of developing better autopilots by imitation of the human performance, and experiments towards this end are described. An extremely interesting and significant study is reported in a chapter by Michael Arbib, the longest in the book with 52 pages. The chapter has the title: ‘‘The mirror system, imitation, and the evolution of language’’. Observations of the brain reactions of chimpanzees have shown that, in a certain brain area, the neural response is the same when an action is observed and when it is actually performed by the animal. This seems to be consistent with the observation that humans have an innate capacity for imitation, and a neural structure showing the effect is termed a ‘‘mirror system’’. An additional suggestive observation is that the mirror system in chimpanzees corresponds to an area associated with language in humans. These observations have led to a theory of the origin of language in which it is suggested that language developed from the mechanism allowing imitation of actions, initially not constituting intentional communication, but later evolving into it and finally into speech. Arbib also points that the analysis of a complex action into constituents has some correspondence to the parsing of a linguistic communication. The theory is developed in detail and this is an important contribution both because it relates the observations to the neurophysiology and if correct, it clarifies the puzzling phenomenon of the emergence of language. The 22 chapters inevitably contain much important material that has not been mentioned here, including several attempts to devise formalisms for the systematic study of imitation and findings from a great deal of experimentation. I think, this account of a selection of topics will leave no doubt that the book is an extremely valuable and is well presented with up-to-date source of information on the many facets of its topic. Alex M. Andrew

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Intelligent Information Systems 2002: Proceedings of the IIS 2002 Symposium, Sopot, Poland, 3-6 June 2002 Mieczysław A. Kłopotek, Sławomir T. Wierzchon´ and Maciej Michalewicz (Eds) Physica-Verlag (A Springer-Verlag Company) Heidelberg 2002 ISBN 3-7908-1509-8, ISSN 1615-3871 x + 468 pp. Paperback, £46.00 Advances in Soft Computing Series Review DOI 10.1108/03684920310483306 This is the proceedings of a symposium held in a resort location on the Gulf of Danzig. It is the 11th in the general area of artificial intelligence organised by the Institute of Computer Science of the Polish Academy of Sciences, the first being in 1992. The ‘‘Advances in soft computing’’ series is designed to allow short publication time and worldwide distribution, and the former has undoubtedly been achieved here. The 49 papers in the present volume are grouped under five headings. (1) Decision Trees and Other Classifier Systems (14 papers), (2) Neural Network and Biologically Motivated Systems (seven papers), (3) Clustering Methods (seven papers), (4) Handling Imprecision and Uncertainty (11 papers), and (5) Deductive, Distributed and Agent-based Systems (ten papers). The great majority of the papers are by workers in Poland, with four having joint Polish and American authorship. One paper shows only American affiliations and another only British. There is one paper from Belarus and another with joint authors from Britain, France and Ukraine. In one of the papers with joint American authorship, the author having an American affiliation also has a Polish one and is R.S. Michalski, a well-established authority on machine learning. Two of the papers having Polish authorship have authors from a Polish-Japanese Institute of Information Technology in Warsaw. The subject matter obviously ranges widely. In their Preface, the editors express their satisfaction that, since the previous events, there is an increase in the diversity of practical applications. The first paper in the book has an important medical application, since it refers to the use of data mining techniques to help diagnose melanoma. The paper by R.S. Michalski as a joint author, under heading (1), is about the modelling of computer user behaviour. Other papers that mention specific applications refer to e-commerce and deductions from financial data, and another is on the extraction of information from audio signals, where the information includes features relevant to speaker and mood identification.

The seven papers under heading (2) show a fine diversity of approach, since the first refers to a computing technique inspired by an ant colony, the next two are about neural nets, but treating them in quite different ways, and the next is about an artificial immune system. These are followed by two papers dealing with genetic algorithms. The final paper under the heading is about an approach using quantitative property structure relations (QSPRs) to infer physical properties of chemical compounds, in this case, their boiling points. The first paper under heading (3) is the one, whose authors have British affiliations and is on the intriguing topic of the use of AI techniques for the synthetic generation of crowd scenes in TV or film productions. Another paper in this section discusses double clustering, which is essentially the two-stage process in which ordinary clustering is performed first and then the centres of the clusters are themselves clustered. This paper has a medical application, since the use of the method is illustrated with reference to a study relating cardiac arrhythmia to meteorological data. The papers under heading (4) range widely in topic and refer to special algebras and logics and developments in knowledge representation, data mining, reinforcement learning and Bayesian belief networks. Those under heading (5) also range widely and two of them make specific reference to the Internet, one of them proposing an expert system applied to intelligent web search that should permit powerful worldwide market analysis. This review of some contents of the symposium is inevitably patchy and biased towards the contributions mentioning immediate practical applications, though the more abstract treatments may well be even more significant. The symposium was undoubtedly, productive of many new ideas and methods and these proceedings are a valuable account of them. Alex M. Andrew

Computational Models for Neuroscience – Human Cortical Information Processing Robert Hecht-Nielsen and Thomas McKenna (Eds) Springer-Verlag London, Berlin, Heidelberg 2003 ISBN 1-85233-593-9 i-xix+299 pp. Hardcover, Eur 89.95:SFR 149.50:GBP 55.00 US $ 99.00 (The Euro price is the net price, subject to local VAT) The editors of this book, Drs Hecht-Niesen and McKenna are from HCN Software Inc., San Diego, CA, USA and the Office of Naval Research, Arlington, VA, USA, respectively. They have provided a Preface in which they explained their view on the formal study of neuroscience and the specific purposes of their book. They say that:

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Amongst neuroscientists, there is a general anticipation that somebody (not soon) will fit enough pieces of the puzzle together to discern a key concept that will trigger rapid and sustained progress in assembling the rest of the existing pieces and discovering the still missing pieces. The purpose of this book is to try to hurry this timetable by describing several different conceptual frameworks that indicate how some of the pieces of the neuroscience puzzle may fit together.

The book has been written to inspire readers to create this ‘‘trigger concept’’. The book’s aim are indeed laudable. But, will they achieve these goals? In this field, it is very much a matter of labouring away, at attempting to understand and analyse what we know and there is no doubt that this text helps us to identify key concepts, because the information included in the contributions by some of our leading experts provide new and stimulating overviews. These indicate perhaps, where we should be looking for some of the pieces of the neuroscience puzzle. The invited authors and the editors, who also provide contributions, have their papers published in chapters that are organised alphabetically by author name. This immediately suggests that there is no intention by the editors to provide any sort of continuity in the text. I see no reason for this, unless precedence is more important than content. Even so, treating each chapter as an entity is acceptable, because many contributions are not linked with each other and there is no discernible attempt at an integrated approach to the task of solving the puzzle of this science. The editors are also reluctant to tell us in any sort of final summary, what they think of the claims of their participating authors. Each author is left to claim their own place in the ‘‘big picture’’ that the editors ambitiously, introduced in their preface. Each of the chapters, however, in its own right, contain a great deal of information that has been compiled by an expert. Each chapter could, therefore, be a book in itself and I have no hesitation in listing their titles, if not, for reasons of space, naming the individual contributor. They are as follows. – The NeuroInteractive Paradigm: Dynamical Mechanics and the Emergence of Higher Cortical Function. – The Cortical Pyramidal Cell as a Set of Interacting Error Backpropagating Dendrites: Mechanism for Discovering Nature’s Order – Performance of Intelligence Systems Governed by Internally Generated Goals – A Theory of Thalamocortex – Elementary Principles of Non-Linear Synaptic Transmission. – The Development of Cortical Models of Enable Neural-Based Cognitive Architectures – The Behaving Human Neocortex as a Dynamic Network of Networks – Towards Global Principles of Brain Processing – The Neural Networks for Language in the Brain: Creating LAD – Cortical Belief Networks Index.

As to whether the book can be recommended, there is no question that the information on computational models for neuroscience and the detail about human cortical information processing makes it a necessary book for anyone involved in the field, either in its research or in following a formal course in neuroscience. Some of the chapters will be of special interest to cyberneticians and systemists who will be interested in the human brain and the various approaches now being taken in the study of neuroscience. D.M. Hutton

Book reports Social Fuzziology – Study of Fuzziness of Social Complexity Vladimir Dimitrov and Bob Hodge Physica-Verlag A Springer-Verlag Company, Heidelberg, New York 2002 i xxiv+ 188 pp. ISBN: 3-7908-1506-3 Price: Euro 54.95 (Net), £38.5, sFr 94.50, $72.00 (Hardcover)

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This is a contribution to the series ‘‘Studies in fuzziness and soft computing’’ Editor-in-Chief: Professor Janusz Kacprzyk), Polish Academy of Science, Warsaw). Details of the other volumes in this series can be found at: http:// www.springer.de/cgi-bin/search_book.pl?series=2941. Whenever Fuzzy Logic is applied in cybernetics and systems, many potential readers are disconcerted by what appears to be a complex mathematical process. This book does not frighten non-mathematicians or those who dislike using symbols and the corresponding notation. The subject of the text is Social Complexity and the book applies the principles and insights of Fuzzy Logic to its study. The publishers say that: It draws on the full range of the social sciences, including some of the most up-to-date work on the escalating complexities of post modern life, e.g. new information technologies in a global world, new forms of consciousness and identity, new relationships between humans, machines, nature and environment. The volume is interdisciplinary, combining expertise from Fuzzy Logic and the Postmodern Social Sciences, synthesising and applying concepts and methods of Fuzzy Logic in innovative ways to the world of social life.

The authors aim to give its readers a comprehensive and practical guide to this application of Fuzzy Logic and they do so by including consideration of many social problems. The contents list gives some indication of the way in which they have approached their task: . .

. . .

. . .

Introduction to Social Fuzziology Bridging the Study of Complexity with Social Fuzziology Understanding Fuzziness of Ourselves Understanding Fuzziness of Society Case Studies: Understanding Fuzzy Social Categories Fuzziness of the West and the East Key Terms in the Language of Fuzziology References

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Embedded Systems and Computer Architecture Graham Wilson Newnes Press St Louis 320 pp. ISBN: 0-7506-5064-8 Price: $56.95 Cyberneticians and systemists need to have a knowledge of electronics to a reasonable level to use this text, although nowadays, computer users are well-versed in the subject. The publishers say that the author uses a ‘‘design-oriented’’ approach in introducing microprocessors and computer architecture. The topics sound interesting and cover a wide range of well-known ‘‘building-blocks’’. These include: Binary numbers and logic expressions Computer design issues (Instruction set and code assembly) Program structures Input/Output methods Design of larger computers (e.g. cache memory and memory management).

The book provides a program suite that includes: interactive animations of digital circuits; Integrated IDE system (includes microprocessor graphical simulator). We are told that using the microprocessor simulator, readers are able to run and test their own system designs on their own PC. This could mean that the user has the equivalent of a full microprocessor system laboratory at his/her fingertips. More details of the text and accompanying program suite can be obtained from: http://www.newnespress.com

Fundamentals of Data Warehouses Matthias Jarke, Maurizio Lenzerini, Yannis Vassiliou and Panos Vassiliadis Springer-Verlag London, Berlin, Heidelberg 2003 2nd ed. i-xiv, 207 pp. ISBN: 3-540-42089-4 Eur. 39.95 (net price): £28.00: sFr 68.50, & 44.95 (Hardcover) This is an extended and revised second edition concerned with a subject that is now demanding a great deal of attention in the interdisciplinary fields of cybernetics and systems. It attracts both practitioners/researchers and those

with an interest that extends to matters involving the uses of data warehouses. The publishers report: . . .the design and optimization of data warehouses remains an art rather than a science. This book presents the first comparative review of the state-of-the-art and best current practice of data warehouses. It covers source and data integration, multidimensional aggregation, query optimization, update propagation, metadata management, quality assessment, and design optimization. Also, based on results of the European Data Warehouse Quality project, it offers a conceptual framework by which the architecture and quality of data warehouse efforts can be assessed and improved using enriched metadata management combined with advanced techniques from databases,

A full review of this edition will be included in the later issues of this journal. Although compiled for researchers and database professionals who work in academia and industry, it promises to be a widely acceptable book and offers an introduction to the issues of quality and metadata usage in the context of data warehouses. Further details are available from: http://springer.de/cgi/svcat/ bag_generate. pl?ISBN=3-540-42089-4 C.J.H. Mann Book Reviews and Reports Editor

Book reports

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September 2003 (From 19th only) C & T 2003 – Communities and Technologies International Conference, Amsterdam, The Netherlands, September 19-21 Contact: Workshops Erik Andriessen, TU Delft, The Netherlands. Papers Volker Wulf, University of Siegen and Fraunhofer FIT, Germany. E-mail: [email protected]; [email protected]; For further information and latest updates, see Web site: www.feweb.vu.nl/C&T2003 6th Hellenic European Research on Computer Mathematics and its Applications – HERCMA 2003, Athens, Greece, September 25-27 Contact: E.A. Lipitakis, Department of Informatics, Athens University of Economics and Business, 76 Patision Street, Athens 10434, Greece. E-mail: eal@ aueb.gr; For further information, see conference Web site: www.aueb.gr/ conferences/hercma2003/ Cybernetics Society Annual Conference, London, UK, Venue and Date to be announced Power – Gen Asia 2003 HIECC, Ho Chi Minh City, Vietnam, September 23-25 Contact: Zeph Landers. E-mail: [email protected]; For further information, see Web site: www.powergenasia.com ASC International conference: Impact of Technology on the Survey Process, Warwick University, Warwick UK, September 17-19 Contact: Diana Elder. Tel/Fax: 01494 793033; E-mail: [email protected]; For further Information, see Web site: http://www.asc.org.uk October 2003 WI 2003 – IEEE/WIC International Conference on Web Intelligence Beijing, China, October 13-17 Contact: See Web site: http//www.comp.hkbu.edu.hk/WI03/ Compsac 2003 – 27th Annual International Computer Software and Applications Conference, Hong Kong, September 30-October 3 Contact: Stephen S. Yau, Computer Science and Engineering Department, Arizona State University, P.O. Box 875406, Tempe, AZ85287-5406 USA. E-mail: [email protected]; For further information, see Web site: http://se.kaist. ac.kr/compsac2003

Kybernetes Vol. 32 No. 7/8, 2003 pp. 1198-1200 # MCB UP Limited 0368-492X

2003 International Conference on Computer Networks and Mobile Computing – ICCNMC-03, Shanghai, China, October 20-23 Contact: Zhongzhi Shi. E-mail: [email protected]; Ming T. Liu. E-mail: liu@ cis.ohio-state.edu; For further information, see Web site: http://www. iccnmc2003.edu.cn/ or http://www.cis.ohio-state.edu/~iccnmc

IEEE IRI-2003 – The 2003 IEEE International Conference on Information Reuse and Integration, Luxor Hotel, Resort, Las Vegas, Nevada, USA, October 27-29 Contact: Waleed W. Smari, Department of Electrical and Computer Engineering, University of Dayton, 300 College Park Dayton, OH 45469-02226 USA. Tel: (937) 229-2795; Fax: (937) 229-4529; E-mail: Waleed. [email protected]; For further Information, see Web site: http:// parks.slu.edu/IRI2003/ ADMOS 2003 – Special International Conference on Adaptive Modelling and Simulation – IACM/ECCOMAS Conference, Goteborg, Sweden, September 29-October 1 Contact: Nils-Erik and Pedro Diez; Conference Chairmen. E-mail: admos03@ cimne.upc.es; For further information, see Web site: http://www.cimne.upc.es/ congress/admos03 ICCPP-2003-32nd International Conference on Parallel Processing, Kaohsiung, Taiwan, ROC, October 6-9 Contact: Mike Liu. E-mail: [email protected] for general information and for paper submissions: P. Sadayappan. E-mail: [email protected],edu; For further information, see Web site: http://www.cis.ohio-state.edu/~icpp2003 November 2003 FIE 2003 – Frontiers in Education Conference, Boulder, Colorado, USA, November 5-8 Contact: see Web site: http://fie.engrng.pitt.edu/ Moving Boundaries 2003 – 7th International Conference on Computational Modelling Problems, Santa Fe, New Mexico, USA, November 4-6 Contact: Conference Secretariat, Wessex Institute of Technology Ashurst Lodge, Ashurst, Southampton S040 7AA, UK. Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853; E-mail: [email protected]; For further information, see Web site: www.wessex.ac.uk/conferences/2003/movingboundaries03/ December 2003 Cyberworlds 2003 – International Conference on Cyberworlds, Singapore, December 3-5 Contact: CW2003 Secretariat, Confer. Management Centre, NTU, Nanyang Avenue, Singapore 639798. Tel: +65 6790 4723; Fax: + 65 6793 0997; E-mail: [email protected]; For further information, see Web site: http:// www.ntu.edu.sg/sce/cw2003

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Crypography and Coding IX – Institute of Mathematics and its Applications, Royal Agricultural College, Cirencester, UK, December 16-18 Contact: Carole Harp, Conference Office, The Institute of Mathematics and its App., Catherine Richards House, 16 Nelson Street, Southend-on-Sea, Essex SS1-1EF., UK. Tel: (01702) 356113; Fax: (01702) 354111; E-mail: [email protected] SEEM4-conference: Statistics in ecology and environmental monitoringpopulation dynamics: the interface between models and data, Dunedin, New Zealand, December 9-13 Contact: E-mail: [email protected]; For further information, see Web site: www.maths.otago.ac.nz

Special announcements

Special announcements

Conference reminders

32nd Annual Conference 2003 International Conference on Parallel Processing (ICPP-2003) Kaohsiung, Taiwan, ROC 6-9 October 2003 Sponsored by: The International Association for Computers and Communications (IACC). In cooperation with: The Ohio State University, USA; The National Center for HighPerformance Computing, Taiwan; National Sun Yat-Sen University, Taiwan; National Cheng Kung University, Taiwan. The conference provides a forum for engineers and scientists in academia, industry and government to present their latest research findings in any aspects of parallel and distributed computing. For further information, please contact: Professor Mike Liu. E-mail: liu@cis. ohio-state.edu For general information, see Web site: http://www.cis.ohio-state.edu/~icpp2003, http://it.cse.nsysu.edu.tw/~icpp2003

Seventh International Conference on Computational Modelling of Free and Moving Boundary Problems

Moving Boundaries 2003 Santa Fe, New Mexico, USA 4-6 November 2003 One of the main objectives of the conference is the establishment of interdisciplinary links between scientists and engineers who tackle similar problems from different research perspectives, and scientists who work on different applications, but use similar numerical methods. The meeting will look at the computational modelling of a continuum, where the positions of its borders or interphase boundaries have to be determined as part of the solution. The transient character leads to the so-called moving boundary problems and the steady one to free boundary problems. Contact: Conference Secretariat. E-mail: [email protected] For more information, see Web site: www.wessex.ac.uk/conferences/2003/ movingboundaries03/

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2003 International Conference on Computer Networks and Mobile Computing

ICCNMC-03 Shanghai, China 20-23 October 2003 Sponsored by: China Computer Federation In cooperation with: IEEE Computer Society, Beijing Center; IEEE Computer Society Technical Committee on Distributed Processing; Shanghai Computer Society Scope The conference provides a forum for engineers and scientists in academia, industry and government to present their latest research findings in any aspect of computer networks and mobile computing. Topics of interest Those that will be of interest to scientists working in academia, business, industry and government includes: . Multimedia Systems . Internet and Web Applications . Admission/Congestion/Flow Control . Routing and Scheduling . Network Security and Privacy . Protocol Analysis and Design In addition, the main themes of Wireless Networks and Mobile Computing/Optical Networks are also important to researchers who work in systems and cybernetics. Other topics will also be featured. Call for papers Call for contributions have been made, but see Web pages for the latest dates and details. Contacts: E-mail: [email protected], [email protected]; Web site: http://www. iccnmc2003.edu.cn/