American Mathematical Monthly, volume 117, June July 2010

MONTHLY THE AMERICAN MATHEMATICAL

Volume 117, Number 6

James T. Smith Hartmut Logemann Eugene P. Ryan Stan Gudder David Borwein Jonathan M. Borwein Isaac E. Leonard NOTES John H. Elton Brian S. Thomson Marco Manetti Timothy W. Jones

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June–July 2010

Definitions and Nondefinability in Geometry Volterra Functional Differential Equations: Existence, Uniqueness, and Continuation of Solutions Finite Quantum Measure Spaces L p Norms and the Sinc Function

Indefinite Quadratic Forms and the Invariance of the Interval in Special Relativity Monotone Convergence Theorem for the Riemann Integral A Proof of a Version of a Theorem of Hartogs Discovering and Proving that π Is Irrational

PROBLEMS AND SOLUTIONS REVIEWS Charles R. Hampton

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Mathematical Modeling: A Case Studies Approach. By Reinhard Illner, C. Sean Bohum, Samantha McCollum, and Thea van Roode

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The Moore Method: A Pathway to Learner-Centered Instruction Charles A. Coppin, Ted Mahavier, E. Lee May, and Edgar Parker, Editors

That student is taught the best who is told the least. —R. L. Moore, 1966 The Moore Method: A Pathway to Learner-Centered Instruction offers a practical overview of the method as practiced by the four co-authors, serving as both a “how to” manual for implementing the method and an answer to the question, “what is the Moore method. Moore is well known as creator of The Moore Method (no textbooks, no lectures, no conferring) in which there is a current and growing revival of interest and modified application under inquiry-based learning projects. Beginning with Moore’s Method as practiced by Moore himself, the authors proceed to present their own broader definitions of the method before addressing specific details and mechanics of their individual implementations. Each chapter consists of four essays, one by each author, introduced with the commonality of the authors’ writings. Topics include the culture the authors strive to establish in the classroom, their grading methods, the development of materials and typical days in the classroom. Appendices include sample tests, sample notes, and diaries of individual courses. With more than 130 references supporting the themes of the book the work provides ample additional reading supporting the transition to learner-centered methods of instruction.

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MONTHLY THE AMERICAN MATHEMATICAL

Volume 117, Number 6

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Definitions and Nondefinability in Geometry1 James T. Smith

Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which handled some basic concepts awkwardly and imprecisely. They would introduce precision required for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By following this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics.

1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for generalization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the question, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal. The story is about • • • •

Euclidean geometry as an axiomatic system, definitions and definability in geometry, and emphasizes contributions of Mario Pieri and Alfred Tarski.

The story follows a thread of related mathematical ideas and personalities for more than a century, to the present, and includes a glimpse of ongoing historical work. With it the author hopes to kindle readers’ interest not only in a fascinating area of geometrical and logical research, but also in studies of a particularly exciting era in the history of mathematics. doi:10.4169/000298910X492781 paper was adapted from and expands on material in [23]. A shorter version was presented to the October 2008 congress, Giuseppe Peano and His School between Mathematics, Logic, and Interlingua, in Turin. Some of the text in Section 2 was adapted from [46, Chap. 2]. 2 In his 1960/1961 paper [6] and its translated version [7], Hans Freudenthal discussed controversies that seethed throughout the nineteenth century. 1 This

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2. THE AXIOMATIC METHOD. For centuries, mathematicians agreed that geometry should be an applied science, the study of certain aspects of physical space. They regarded as valid those propositions commonly believed to follow from Euclid’s postulates and definitions: the theorems of Euclidean geometry. This was based on their deep trust in the deductions, and the strong agreement between these theorems and observable phenomena that they described. Euclidean geometry was so successful in explaining and predicting properties of the physical world that for two millenia no alternative system was considered. Euclidean geometry was considered correct, not simply accurate and useful. A geometry whose theorems contradicted Euclid’s would be regarded as incorrect and therefore useless. Analyzing Euclid’s use of the parallel postulate and related issues, mathematicians eventually began to consider alternatives that did contradict that postulate. This generated controversy, as mentioned earlier, and scientists wrestled with the idea of a geometric postulate system as a correct description of physical space. Late in the nineteenth century, to facilitate that study, they began to reexamine the axiomatic method that Euclid used to construct his system. The method’s most memorable aspect is its derivation of theorems from others proved earlier, which were derived from others yet more basic, which were. . . , and so on. The most basic principles, necessarily left unproved, are called axioms. This deductive organization was described by Aristotle [1] around 350 B . C . E . Euclid famously but imperfectly employed it in his Elements [3] around 300 B . C . E . Filling the gaps required investigation of steps that Euclid and his successors for two millennia evidently regarded as not needing proof. In some cases that amounted to formulating axioms that they assumed but did not state. In particular, Euclid’s reasoning about the sides of lines in a plane left so many gaps that one can imitate him closely and derive nonsense.3 The theorems of an axiomatic theory are about concepts, some defined in terms of others considered earlier, which were defined in terms of others yet more basic, which were. . . , and so on. The most basic concepts, necessarily left undefined, are called primitive. Aristotle also described this organization. But Euclid didn’t employ it fully. He did define precisely most of the concepts he used, in terms of a few he introduced at the beginnings of his presentations. However, his “definitions” of those most basic concepts were not definitions in our sense, but vague, circular discussions. Nevertheless, Euclid’s work gained such prestige that Aristotle’s definitional technique was rarely mentioned again until the late 1800s. At that time the relationship of various geometric theories was becoming involved and confusing. Mathematicians studied hyperbolic non-Euclidean geometry as deeply as Euclidean, and showed how to extend each to a projective geometry by introducing “ideal points at infinity.” They could establish the use of homogeneous coordinates in the projective geometry and employ them with a polar system to measure distance. (To understand this story, readers need not know details of projective geometry—but only that sophisticated mathematical devices were used for this purpose.) Hyperbolic or Euclidean geometry could be reconstructed from the projective by using different polar systems. Felix Klein provided a comparative framework with his 1872 Erlanger program [18], which regarded projective geometry as the most basic. He also emphasized the study of the automorphisms of a geometry: the transformations that preserve its basic notions. That complexity and the new projective methods used in algebraic geometry, the study of solutions of systems of multivariate algebraic equations, led to a new emphasis on the axiomatic method, in Aristotle’s original form. 3 David

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Hilbert noted this in 1897/1898: see [53, Sec. 4.2]. For an English discussion, see [46, Sec. 2.2].

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3. PASCH. Moritz Pasch (1843–1930) was born to a merchant family in Breslau, Prussia (now Wrocław, Poland). He studied at the university there, earned the doctorate in 1865, and pursued an academic career in Giessen, Germany. Pasch wrote research papers and some influential texts, and served with great distinction as an administrator. His early research was in algebraic geometry, but he soon turned to foundations of analysis and geometry. He saw the need to repair faults in classical Euclidean geometry, particularly in reasoning about the sides of a line in a plane, and to firmly ground the powerful projective methods presented in 1847 by G. K. C. von Staudt [47]. To this end, Pasch published in 1882 the first completely rigorous axiomatic presentation of a geometric theory.4 Pasch noted that, in contrast to earlier practice, he would discuss certain concepts without definition. Determining which he actually left undefined requires close reading. They are • • • •

point, segment between two points, coplanarity of a point set, congruence of point sets.

He defined all other geometric notions from those. For example, Pasch called three points collinear if they are not distinct or one lies between the other two, and defined the line determined by two distinct points to be the set of points collinear with them. Those definitions rely on logic: the words not, distinct, or, and set appear there. Arithmetic is not really required: one, two, and three could be avoided through use of variables and more logic. As example axioms, consider a simple one, that there exist three noncoplanar points, and the famous, more complicated, Pasch axiom illustrated by Figure 1. These involve even more logic: if, any, and exist. The Pasch axiom directly implies transitivity of the relation satisfied by points A and B when they lie on the same side of a given line DF. (Let this mean that no point on DF should lie between A and B. If A and B lay on different sides of DF, then by the Pasch axiom, either A and C, or else B and C, would lie on different sides.) C

A

F D

B

Figure 1. The Pasch axiom: If A, B, and C are distinct coplanar points and F is a point between A and B, then there exists on any line DF some point of segment AC or of BC.

Pasch developed the geometry of incidence, betweenness, and congruence, extended that to projective space, and then showed how to select a polar system to 4 Heinrich Schr¨ oter supervised Pasch’s doctoral study. In his autobiography [32, pp. 9–12], Pasch wrote that he could not continue working in higher mathematics until he settled questions about its foundations. The traditional presentations provided him no support, so he wrote his 1882 books [29] and [30] on foundations of analysis and geometry.

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develop Euclidean or hyperbolic geometry. In spite of the abstract nature of his presentation, Pasch clearly indicated, as he introduced his axioms, that he regarded his system as a description of the physical world: In contrast to the propositions justified by proofs. . . there remains a group of propositions from which all others follow. . . based directly on observations. . . .5 4. PEANO. The next character in this story stemmed from a very different milieu: a small farm near Cuneo in the Piedmont region of the Kingdom of Sardinia. Giuseppe Peano (1858–1932) was schooled in Cuneo and in the regional capital, Turin. By then this region was part of a unified Italy. Peano earned the doctorate in 1880 at the University of Turin and spent his career there and at the adjacent military academy. As a junior professor Peano undertook to reformulate with utmost precision all of pure mathematics! His 1889 booklet [33] on foundations of geometry contained some technical improvements over Moritz Pasch’s book [30].6 More importantly, Peano departed from Pasch’s then prevalent approach by divorcing that discipline from the study of the physical world: Depending on the significance attributed to the undefined symbols. . . the axioms can be satisfied or not. If a certain group of axioms is verified, then all the propositions that are deduced from them will be equally true. . . [33, §1, note]. This freedom to consider various interpretations of the undefined, or primitive, concepts, and the distinction between syntactic properties of symbols and their semantic relationships to the objects they denote, were essential for all later studies of definability. Moreover, in that publication Peano was introducing logical symbolism that quickly became famous for its precision and infamous for its opacity. Over decades it was transformed into the familiar notation used later in the present paper. Peano became a center of mathematical controversy. He pointed out and corrected errors and oversights in various texts, and introduced a number of our now standard delicate arguments in analysis. In 1891 he published a polemical interchange with the noted Turin algebraic geometer Corrado Segre about rigor in geometry. Segre favored leniency, to foster intuitive discovery and formulation of general results, leaving exceptional cases to later study. Peano roared that a seemingly general statement that is sometimes false cannot be a mathematical theorem. Bertrand Russell reported that Peano always got the best of the arguments following papers presented at the 1900 Paris congress of philosophers. In the introduction to his 1894 paper [35], to be discussed next, Peano railed about geometry texts that began with vague descriptions of numerous concepts such as space, homogeneity, and unboundedness. There and in closing, Peano indicated the relevance of his work for improving elementary instruction.7 In [35] Peano introduced the use of direct motion to replace congruence as a primitive concept in Euclidean geometry. A geometric transformation, this sort of motion 5 [30,

p. 17]; see also [14, §6].

6 The identity of the supervisor of Peano’s doctoral study seems uncertain. In [33] Peano defined coplanarity

in terms of collinearity, and thus avoided listing it as a primitive notion. He introduced notation and terminology that enabled him to analyze such ideas more deeply than Pasch. 7 See [16, Chaps. 3–5], [22], [13], and [43, pp. 232–233]. Russell wrote that his encounter with Peano changed his intellectual life; later he achieved worldwide renown in logic and philosophy. Pasch also complained in his 1894 celebratory address [31, pp. 12, 17] that most textbooks were still inadequate. He emphasized their vague treatment of concepts such as space and dimension. In 1899, Peano’s associate Giuseppe Ingrami published an elementary text [12] based on Peano’s approach to geometry.

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does not involve time: only the initial and final positions of a figure are relevant. Motions are isometries: they preserve distance. Those that also preserve orientation are called direct. Figures can be defined as congruent if some direct motion maps one to the other. (In contrast, indirect motions reverse orientation; figures related by indirect motions can be called anticongruent.) Some of Peano’s axioms stated geometric properties of motions: for example, should A and B be distinct points, P a point between them, and m a motion, then the image mP must lie between mA and mB. Others, however, would now be called group-theoretic: the identity should be a motion, motions should be bijective, and their inverses should be motions. 5. PIERI AND MOTIONS. Mario Pieri was the third of eight children of a lawyer in Lucca, Italy. He was schooled there and in Bologna. Pieri attended university in Bologna and Pisa, where he earned the doctorate in 1884 in algebraic and differential geometry, under the supervision of Luigi Bianchi. Two years later, Pieri became professor of projective and descriptive geometry at the Royal Military Academy in Turin. In 1888 he was also appointed assistant at the nearby university. Soon after, Corrado Segre suggested that Pieri translate G. K. C. von Staudt’s fundamental 1847 work [47] on the projective geometry that underlies those subjects. Pieri published that in 1889. Evidently he, like Pasch, became intrigued with its logic: he returned to study it again and again. Pieri also continued to work in algebraic geometry, and became particularly noted in that field for his methods of enumerating solutions of systems of algebraic equations. Pieri’s senior colleague Giuseppe Peano was already investigating deep questions in foundations, and in an 1890 paper [34] repaired a lapse in Staudt’s work. During the next two decades, in several major studies in foundations of geometry, Pieri used, refined, and publicized Peano’s logical techniques. A series of Pieri’s papers culminated in the 1898 memoir [37], the first full axiomatization of projective geometry. This employed as primitive only the concepts of point and of the line joining two points, and it covered all dimensions.8 Pieri presented that work and later axiomatic studies as hypothetical-deductive systems. He and Peano’s collaborator Alessandro Padoa formally introduced in 1900 this explicit formulation of Peano’s idea that the primitive concepts may be interpreted arbitrarily, as long as the interpretations satisfy the axioms. That framework was adopted by postulate theorists during 1900–1925, particularly Edward V. Huntington, who precisely echoed the Italians’ formulation. Recent historical analysis has confirmed that this approach turned into today’s version of the axiomatic method.9 Following Peano’s lead, Pieri pursued deeply the use of direct motion as a primitive concept. His 1900 Point and Motion memoir [39] axiomatized a large part of three-dimensional geometry common to the Euclidean and hyperbolic. He employed only two primitive concepts, point and direct motion. The following definitions were central: 8 Segre had suggested in 1892 that an axiomatic study was needed to establish a synthetic approach to higher-dimensional projective geometry. Pieri’s colleague Gino Fano indicated then [4, p. 106] that it was necessary to determine what properties of flat subspaces were critical for that development. Pieri started an 1896 paper [36] with the comment that higher-dimensional projective geometry was still a controversial subject. 9 See [39, Preface], [40, §III], [24], [11, pp. 288–290], [14, §6], and [44, §2]. Born in Venice, Padoa (1868– 1937) earned the doctorate from the University of Turin in 1895 under Peano’s supervision, with a dissertation on foundations of geometry. An especially effective expositor, Padoa remained one of Peano’s closest collaborators. His career included various positions in middle schools, universities, and the naval institute, mostly in Genoa.

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Figure 2. Mario Pieri (1860–1913).

• • • •

Three points are called collinear if they are fixed by some nontrivial direct motion. (Think of an axial rotation.) Points M and P are called equidistant from a point O if some direct motion maps M to P but leaves O fixed. A point is said to lie midway between two others if it is collinear with and equidistant from them. A point Q is said to lie somewhere between two points P and R if it is collinear with them and lies midway between two points M and N such that the pairs M, P and N , P are equidistant from a point O midway between P and R. Figure 3 displays this ingenious definition. M

P

O

Q

R

N Figure 3. Pieri’s 1900 definition of Q lies somewhere between P and R.

Pieri adapted his definition of collinearity from the 1679 work of G. W. Leibniz [20, p. 147], which was then under serious study by scholars in Peano’s group. Pieri formulated complicated axioms in terms of these primitive concepts, and provided full proofs of all his theorems. He did not include axioms about completeness of lines, and 480

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thus stopped short of proving that the points on a given line enjoy all the properties of the real numbers.10 6. INTERLUDE. Mario Pieri’s Point and Motion memoir [39] was published almost simultaneously with David Hilbert’s 1899 masterpiece, Foundations of Geometry [10], the most famous treatment of this subject. The two were independent and very different. With a single introductory sentence, “. . . [this work] is tantamount to the logical analysis of our intuition of space,” Hilbert claimed that his underlying philosophy agreed with that of Moritz Pasch, as described here in Section 3. But the abstract approach he actually followed was like the Italians’. Reviewing Hilbert’s book, Hans Freudenthal wrote, “What was expressly formulated by [Pieri and] Padoa was tied up in Hilbert’s work with the mathematically important facts. . . .” Freudenthal lauded “the convincing power of a philosophy that is not preached as a program, but is only the silent background. . . . This thoroughly and profoundly elaborated piece of axiomatic workmanship was infinitely more persuasive than programmatic and philosophical speculations on space and axioms could ever be.” Hilbert formulated some axioms in terms of defined concepts, which permitted simple exposition. He presented proofs in familiar style, with much left unsaid—readers could supply that if they wished. Pieri showed how to formulate axioms, no matter how complicated, solely in terms of the primitive concepts, and how to provide all details of proofs. Hilbert’s presentation popularized the subject. Pieri’s never attracted great acclaim but, as you will see, it provided a path for significant later research.11 In 1904 Oswald Veblen proposed an axiomatization [54] of Euclidean geometry that regarded only point and betweenness as primitive concepts. His axioms were simpler than Pieri’s or Hilbert’s. Veblen followed Pasch [30] in using a projective polar system to define Euclidean congruence and then equidistance and motion. In 1907, however, Federigo Enriques noted [2, §6] that Veblen’s polar system was not uniquely determined: it seemed also to be undefined, and thus a very complicated primitive concept. In 1911, Veblen published another axiomatization [55] that avoided this problem.12 7. PIERI’S POINT AND SPHERE MEMOIR. After a long struggle Mario Pieri obtained appointment as university professor in 1900 at Catania on the Italian island of Sicily. There he completed his 1908 Point and Sphere memoir [41], a full axiomatization of Euclidean geometry based solely on the primitive concepts point and equidistance of two points N and P from a third point O, written ON = OP. He 10 Introducing [39], and in his 1900 Paris address [40, §III], Pieri quoted Pasch’s 1894 comment. Pieri contrasted Pasch’s and Peano’s incisive work with the proliferation of unexplained concepts in Wilhelm Killing’s 1893–1898 text [17], including space, occupying a space, and so on. Pieri displayed considerable interest in reforming elementary instruction, particularly in his review [38] of Giuseppe Ingrami’s text and in the introduction and appendix to his 1908 Point and Sphere memoir [41]. 11 [10, Intro.], [7, pp. 618–621]; see also [53, Sec. 6.1]. Hilbert (1862–1943) was born near K¨ onigsberg, Prussia (now Kaliningrad, Russia); his father was a judge. Hilbert earned the Ph.D. at the university there in 1884 under the supervision of Ferdinand von Lindemann. By 1899 Hilbert was a senior professor at G¨ottingen and had achieved worldwide acclaim. Freudenthal reported that Hilbert’s book did attract some controversy, but not for long. 12 Veblen (1880–1960) was born in Decorah, Iowa; his father was a professor. Veblen earned the Ph.D. at the University of Chicago in 1903 under the supervision of Eliakim H. Moore, the leader of the American postulate theorists. Veblen’s 1904 paper stemmed from his dissertation. Later, Veblen would play a leading role in the development of American mathematics.

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had suggested this possibility already in 1900 [39, Preface].13 Pieri used the following definitions (letters N to R may refer to points in Figure 4): • • • • • • •

N is said to lie on the sphere PO through P about O if ON = OP. If O  = Q, then P is called collinear with O and Q if PO intersects PQ only at P. (Pieri adapted this definition from G. W. Leibniz [20, Part IV, pp. 185, 189].) Q is called a reflection of O over P, and P is said to lie midway between O and Q, if PO = PQ and Q is collinear with O and P. Two spheres are called congruent if the points on one are related to those on the other by reflection over some single point. Point pairs O, P and Q, R are called congruent if R lies on a sphere about Q congruent to PO . An isometry is a point transformation that preserves congruence of pairs. A direct motion is the composition of an isometry with itself.14

Pieri then proceeded as in [39]. In particular, he adopted verbatim his earlier definition of betweenness, thus achieving one solely in terms of equidistance of two points from a third. His axioms were frightfully complicated. But except for his Archimedean and continuity axioms, they would now be called first-order: they can be phrased solely in terms of the primitive concepts without using any set theory. Further, Pieri again published all details of his proofs: the translation in [23, Chap. 3] is 111 pages long! But his memoir, overshadowed by David Hilbert’s book [10], received little acclaim, not even one review.

R

N

O

P

Q

R

Figure 4. Pieri’s 1908 definition of collinearity: P is collinear with O and Q; R is not.

Pieri’s untimely death in 1913 and the following decades of war and economic and political turmoil probably contributed to this neglect. Point and Sphere shined through that fog once in 1915, as a Polish translation, and a decade later in the work of the 13 Using ternary equidistance ON = OP as primitive rather than quaternary segment congruence MN = OP occurred to Alessandro Padoa independently at about the same time. He understood Pieri’s intention only in Paris, just before his own talk [25] to the International Congress of Mathematicians. That preliminary report explicitly mentioned only segment congruence. But close reading reveals that Padoa employed ternary equidistance wherever he could. In a 1901 letter [26], Padoa explained this situation to Pieri with great deference. In effect, Padoa ceded this research area to Pieri. 14 In the intended interpretation, an isometry is a direct motion just when it is a “square.” To see this, recall that some isometries may be indirect, reversing orientation; their “squares” must be direct. Moreover, every direct motion is a screw—a composition of a rotation about an axis and a translation along it—and hence is the “square” of the screw with half the angle and half the vector.

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Polish logician Alfred Tarski. The present author does not know exactly how the thread of this story wound in that direction. Ongoing historical research suggests that this was due to the influence of Giuseppe Peano among some Polish mathematicians. 8. TARSKI’S SYSTEM OF GEOMETRY. Alfred Tarski was born in Warsaw, Poland, then part of Russia. The family was Jewish and secular, involved in business. There were two children: Alfred had a younger brother. Schooled during World War I, Alfred became a Polish nationalist. He entered the university in Warsaw afterward, and in 1924 earned the doctorate in logic under the supervision of Stanisław Le´sniewski. The economy and antisemitism made it hard to find a job. In 1926, starting on the path to become one of the world’s top logicians, Alfred Tarski was a university assistant and high-school teacher in Warsaw.15 At this time, Tarski’s research emphasis included application of logical techniques to geometric problems. Studying axiomatic presentations of geometry, he adopted the refinements of the axiomatic method introduced by Giuseppe Peano, Alessandro Padoa, and Mario Pieri.16 Tarski was also beginning to emphasize first-order logic. This framework employs just the logical symbols for equality, Boolean connectives, variables ranging over individuals in a specific domain, the existential and universal quantifiers ∃ and ∀, and selected constants, relations, and operations in and on that domain. Thus it minimizes the use of set-theoretic notions in the logical framework. Pieri’s 1908 Point and Sphere memoir fit into that framework. Reporting a conversation with Tarski, Steven Givant wrote, Tarski was critical of Hilbert’s axiom system [10]. . . [and] preferred Pieri’s system [41], where the logical structure and the complexity of the axioms were more transparent. Tarski developed and presented his own axiomatization in a 1926–1927 course at Warsaw University. According to his later collaborator Lesław Szczerba, “the system that Tarski presented in this course was designed after” Pieri’s 1908 memoir.17 Tarski’s primitive concepts were point and two relations among points, the ternary relation expressing betweenness and the quaternary one expressing congruence of point pairs. Oswald Veblen had completed an axiomatization [55] based on those three concepts in 1911. With these slightly more complex primitives, Tarski was able to greatly simplify Pieri’s 1908 axioms. Tarski’s axioms were two-dimensional, but he noted that they could be easily modified for use in three or higher dimensions without loss of simplicity.18 As continuity axioms, Tarski used all first-order instances of Pieri’s second-order axiom. All Tarski’s axioms except those for continuity had ∀ ∃ form, with all quantifiers at the beginning, universal preceding existential. Their total length was less than that of Pieri’s single most complicated axiom. That one, Pieri’s version of the Pasch axiom, had form ∀ ∃ ∀ ∃. (A standard procedure will convert any first-order sentence to a logically equivalent form with all quantifiers at the 15 For biographical information, consult [5]. Before 1936, Tarski published several works about high-school geometry or addressed to high-school teachers [23, Sec. 5.2]. Their relationship to Tarski’s mathematical research is under investigation by Andrew McFarland and the present author. 16 See [14] and [44]. 17 See [8, p. 50] and [48, p. 908]. Whether Tarski consulted Pieri’s original memoir [41] or its 1915 Polish translation is an open historical question. In the latter case, Tarski’s work would be linked directly to the Polish Mianowski foundation, which supported the translation, and probably to Peano’s contacts with Polish intellectuals during 1900–1915. 18 See [50, Sec. 3.6], and [51, footnote 5].

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Figure 5. Alfred Tarski (1901–1983).

beginning. The number of alternations of quantifier types then provides a measure of complexity.19 ) Tarski proved that the structures satisfying his axioms are those isomorphic with coordinate planes over real-closed ordered fields—ordered fields in which every polynomial with odd degree has a root, and whose nonnegative elements are all squares. Because of economic and political turmoil and war, Tarski’s system was not broadly publicized until his 1959 summary [51], What Is Elementary Geometry? In lectures, Tarski followed Pieri’s practice of full disclosure of all proofs, but they remained unpublished commercially until the appearance of [45] in 1983.20 The earlier formulation of the system, though, enabled much deeper research into provability, decidability, and definability in geometry. 9. NONDEFINABILITY. Consider elementary geometry based on congruence δ of point pairs and betweenness β of triples, like Alfred Tarski’s system. Formulas such as βABC and δABCD should be read, “B lies between A and C” and “pairs A, B and C, D are congruent.” As described here, Mario Pieri had shown in [41] how to construct a formula ϕABC involving just δ, and to prove (∀ A, B, C)[βABC ⇔ ϕABC]. That is, ϕ characterizes β, and β is definable from δ; it could thus be eliminated from the list of primitive concepts. In [54] Oswald Veblen had claimed the reverse: that he had defined δ from β. But in [2] Federigo Enriques had objected: Veblen had not. Could δ be defined from β at all? Settling this question required a precise definition of definition. That was achieved by first adopting as standard some axiom system such as Tarski’s, assumed consistent. If ν is a concept and  a family of concepts defined in that system, then a first-order phrase mentioning only the concepts in  should be called a definition of ν in terms 19 See [19]. In 2008 Victor Pambuccian [28] showed how to construct an axiom system equivalent to Pieri’s, with the same two primitives, in which all axioms except those for continuity have ∀ ∃ form. He also referred there to others’ recent work on equivalent systems with those primitives. 20 Many of Tarski’s students and colleagues contributed to proofs published there.

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of  if it provably characterizes ν in the standard system. In the 1935 study [49, §1], after considering definitions in general, Tarski noted that indeed, betweenness β cannot serve as the sole primitive relation in an axiomatization of elementary geometry with variables ranging over points. His discussion suggests the following argument, based on a technique introduced in 1900 by Alessandro Padoa [24, Sec. 16]: any affine transformation that is not a similarity would preserve β, and thus also any concept defined by a first-order phrase solely in terms of β, but it would not preserve the congruence relation δ. To make that argument precise, consider such a transformation A  A that is represented by a formula α A A of the standard system. A convenient choice is the shear that would be described by the equations x  = x + y and y  = y using a Cartesian coordinate system based on an origin O and some unit points X and Y . The formula α should include the clause δOXOY and the requirement that ∠XOY be a right angle. (The simplest α might describe a construction of A from A using XOY and various other triangles.) These formulas can then be proved (see Figure 6): αO O  & α X X  & αY Y  ⇒ O = O  & X = X  & δOXOY & ¬ δOXOY 

(1)

α A A & α B B  & αCC  ⇒ (βABC ⇔ β A B  C  ).

(2)

Now suppose δ were definable from β: for some formula ϕ involving just β the sentence ∀ ABCD(ϕABCD ⇔ δABCD) should be provable. Rules of logic would yield a proof of the analog of (2) with ϕ in place of β, and thus of α A A & α B B  & αCC  & α D D  ⇒ (δABCD ⇔ δ A B  C  D  ).

(3)

Substituting O, X, O, Y for A, B, C, D would then yield a formula that contradicts (1). Y

Y

OX  =/ OY 

O = O

X = X

Figure 6. Shear α: O, X, Y  O  , X  , Y  .

That same year, Adolf Lindenbaum and Tarski presented a related argument that has the following consequence: no family , however large, of binary relations definable in the standard system can serve as the family of all primitive concepts. Thus Pieri’s selection of ternary equidistance as the sole primitive concept was in a sense optimal. The argument proceeded by reasoning about the real Euclidean plane R2 , with the usual betweenness and congruence relations. It is a model of the standard system of geometry. The same symbols are used for formulas in the system and for their interpretations in the model. The argument is based on the automorphism group, June–July 2010]

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emphasized by Felix Klein, as mentioned in Section 2; the group consists of the similarity transformations.21 First, consider a binary relation on R2 represented by a formula ρ AB. It must be invariant under any similarity A  Aσ of R2 , because the betweenness and congruence relations are invariant. That is, for any A, B ∈ R2 , statements ρ AB and ρ Aσ B σ are equivalent: either both true or both false. Now, given two pairs A, B and A , B  of points in R2 , either both coincident or both distinct, there is a similarity σ such that Aσ = A and B σ = B  . Consequently, statements ρ A A and ρ A A are equivalent, as are ρ AB and ρ A B  . Thus, either (1) ρ A A is true for all A ∈ R2 or (2) it is false for all such A; and moreover, either (a) ρ AB is true for all pairs A, B of distinct points in R2 , or (b) it is false for all such pairs. In the four cases (1a), (1b), (2a), and (2b), ρ represents the universal, equality, inequality, and empty relation, respectively. Therefore, a set  of representable binary relations on R2 can contain only those four. But they are invariant under every transformation. Since neither the betweenness nor the congruence relation has that property, these cannot be defined solely in terms of the formulas representing relations in . Tarski’s work has led to related studies, many reported in [45]: for example, what other ternary relations suffice as the sole primitive relation? More recently, Victor Pambuccian has investigated the effect of strengthening the underlying logic to permit conjunctions &m,n ϕm,n of infinite families of open sentences ϕm,n that depend on natural numbers m, n. In 1990 he discovered a startling fact [27]: with that logic, a single binary relation υ can be used as the sole primitive relation for a system of geometry very closely related to Tarski’s! This relation v P Q holds for points P and Q just when the distance PQ is equal to 1. Incorporating υ into Tarski’s system adds theorems about specific distances, and restricts the automorphisms to the group of isometries.22 Pambuccian considered for each m, n the auxiliary relation vm,n PQ that says PQ = m/2n . This relation can be defined solely in terms of v P Q by a complicated firstorder formula that describes some familiar geometric constructions. He then proved that P, Q is congruent to another point pair R, S just when   &m,n ∃ T [vm,n P T & vm,n QT ] ⇒ ∃ U [vm,n RU & vm,n SU ]   & ∃T [vm,n RT & vm,n ST ] ⇒ ∃ U [vm,n PU & vm,n QU ] . The first implication fails for some m, n just when PQ < RS: see Figure 7. The formula just displayed is a definition, in the strengthened logical system, of congruence of point pairs. That yields a definition of Pieri’s single primitive relation, ternary equidistance, solely in terms of v. Therefore, with the new logic, v can also serve as the single primitive relation. But the geometry is not new: the constructions that Pambuccian employed in 1990 to define vm,n in terms of v had already been used by Pieri in 1908 to analyze the continuity of a line!23 21 See [21, §1]. The argument in the following paragraph is adapted from [45, pp. 285–287]. This result would still hold should  also contain singulary relations σ , or properties: just replace each such σ by a new binary relation ρ such that ρ P Q stands for σ P. 22 Thus, the preceding paragraph’s argument, that the primitive relations cannot all be binary, fails for the augmented system. But Raphael M. Robinson proved in [42], by a different method, that the result still holds, provided the underlying logic remains standard. 23 See [41, §IV, proposition P18, and §VIII, proposition P21ff.]. For information on logic with infinite conjunctions, consult [15]. The displayed sentence says that the following statement holds for all natural numbers m and n: whenever there is a point T at distance m/2n from each of P, Q then there is a point U at that distance from each of R and S, and conversely. Victor Pambuccian (1959– ) earned the doctorate from the University of Michigan in 1993 under the supervision of Andreas Blass, with a dissertation on foundations of geometry. He is the author of many axiomatic studies related to questions in the present paper.

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1 PQ 2

< m/2n < 12 RS

∃ intersection T

m/2n

P

¬ ∃ intersection U

U/

T

Q

R

S

Figure 7. Pambuccian’s 1990–1991 definition of P Q < RS.

10. CONCLUSION. The pioneering work of Giuseppe Peano during 1889–1894 on the logic underlying geometry and on the use of direct motion as a fundamental geometric idea led from the systems of Euclid and Moritz Pasch straight to Mario Pieri’s detailed 1900 and 1908 axiomatizations of geometry. Pieri formalized his presentations as hypothetical-deductive systems, which became our familiar setting for axiomatic studies. His choice of primitive concepts was spare, he relied on set theory only for continuity considerations, and he published all details of his proofs. During the 1920s Alfred Tarski followed Pieri’s approach to achieve a surprisingly efficient first-order axiomatization of Euclidean geometry, which has become a standard of comparison for work in foundations of geometry. Tarski formulated in the 1930s a theory of first-order definitions, with which he showed that Pieri’s choice of primitives was in a sense optimal. In the 1990s, Victor Pambuccian, using geometry that would have been familiar to Pieri, showed that some greater economy could be achieved, but only by strengthening the underlying logic and slightly changing the geometry under consideration. Research in this field continues today. ACKNOWLEDGMENTS. The author gratefully acknowledges inspiration by Elena Anne Marchisotto and many suggestions and corrections from Victor Pambuccian and two referees. Figure 2 appears through the courtesy of the Biblioteca Matematica “Giuseppe Peano” of the Department of Mathematics of the University of Turin; Figure 5, from S. R. Givant, Unifying threads in Alfred Tarski’s work, Math. Intelligencer 21, no. 1 (1999) 58, with the kind permission of Springer Science+Business media.

REFERENCES 1. Aristotle, Posterior Analytics (trans. J. Barnes, with a commentary), 2nd ed., Clarendon Press, Oxford, 1994. 2. F. Enriques, Prinzipien der Geometrie, in Encyklop¨adie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 3, Geometrie, part 1, half 1, art. III A, B 1, F. Meyer and H. Mohrmann, eds., B. G. Teubner, Leipzig, 1907, 1–129. 3. Euclid, The Thirteen Books of Euclid’s Elements (trans. T. L. Heath from the text of Heiberg, with introduction and commentary), three vols., 2nd ed., Dover, New York, 1956. 4. G. Fano, Sui postulati fondamentali della geometria proiettiva in uno spazio lineare a un numero qualunque di dimensioni, Giornale di matematiche 30 (1892) 106–132. 5. A. B. Feferman and S. Feferman, Alfred Tarski: Life and Logic, Cambridge University Press, Cambridge, 2004. 6. H. Freudenthal, Die Grundlagen der Geometrie um die Wende des 19. Jahrhunderts, MathematischPhysikalische Semesterberichte 7 (1960/1961) 2–25. , The main trends in the foundations of geometry in the 19th century, in Logic, Methodology and 7. Philosophy of Science: Proceedings of the 1960 International Congress, E. Nagel, P. Suppes, and A. Tarski, eds., Stanford University Press, Stanford, 1962, 613–621. 8. S. R. Givant, Unifying threads in Alfred Tarski’s work, Math. Intelligencer 21(1) (1999) 47–58. doi: 10.1007/BF03024832

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9. L. Henkin, P. Suppes, and A. Tarski, eds., The Axiomatic Method with Special Reference to Geometry and Physics: Proceedings of an International Symposium Held at the University of California, Berkeley, December 26, 1957–January 4, 1958, North-Holland, Amsterdam, 1959. 10. D. Hilbert, Grundlagen der Geometrie, Verlag von B. G. Teubner, Leipzig, 1899; English trans. E. J. Townsend, Foundations of Geometry, reprint ed., Open Court, LaSalle, IL, 1959. 11. E. V. Huntington, Sets of independent postulates for the algebra of logic, Trans. Amer. Math. Soc. 5 (1904) 288–309. doi:10.2307/1986459 12. G. Ingrami, Elementi di Geometria per le scuole secondarie superiori, Tipografia Cenerelli, Bologna, 1899. 13. International Congress of Philosophy, Biblioth`eque du Congr`es International de Philosophie, four vols., Librairie Armand Colin, Paris, 1900–1903. 14. I. Jan´e, What is Tarski’s common concept of consequence? Bull. Symbolic Logic 12 (2006) 1–42. doi: 10.2178/bsl/1140640942 15. C. R. Karp, Languages with Expressions of Infinite Length, North-Holland, Amsterdam, 1964. 16. H. C. Kennedy, Peano: Life and Works of Giuseppe Peano, definitive ed., Peremptory Publications, Concord, CA, 2006; this has the footnotes omitted from the original edition, Studies in the History of Modern Science, vol. 4, D. Reidel, Dordrecht, 1980. 17. W. Killing, Einf¨uhrung in die Grundlagen der Geometrie, two vols., Druck und Verlag von Ferdinand Sch¨oningh, Paderborn, Germany, 1893–1898. 18. F. Klein, Vergleichende Betrachtungen u¨ ber neuere geometrische Forschungen, Andreas Deichert, Erlangen, 1872; English trans. M. W. Haskell, with additional footnotes by the author, A comparative review of recent researches in geometry, Bulletin of the New York Mathematical Society 2 (1892–1893) 215–249. doi:10.1090/S0002-9904-1893-00147-X 19. M. Krynicki and L. W. Szczerba, On simplicity of formulas, Studia Logica 49 (1990) 401–419. doi: 10.1007/BF00370372 20. G. W. von Leibniz, Characteristica geometrica, in Leibnizens mathematische Schriften, part 2, vol. 1, Olms paperback, no. 45, K. I. Gerhardt, ed., Georg Olms Verlag, Hildesheim, Germany, 1971, 141–211; originally published by A. Asher, Berlin, 1849–1863. ¨ 21. A. Lindenbaum and A. Tarski, Uber die Beschr¨ankteit der Ausdrucksmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums 7 (1936) 15–22; English trans., On the limitations of the means of expression of deductive theories, in [52, 384–392]. 22. C. F. Manara and M. Spoglianti, La idea di iperspazio: Una dimenticata polemica tra G. Peano, C. Segre, e G. Veronese, Memorie della Accademia Nazionale di Scienze, Lettere e Arti di Modena (series 6) 19 (1977) 109–129. 23. E. A. Marchisotto and J. T. Smith, The Legacy of Mario Pieri in Geometry and Arithmetic, Birkh¨auser, Boston, 2007. 24. A. Padoa, Essai d’une th´eorie alg´ebrique des nombres entiers, pr´ec´ed´e d’une introduction logique a` une th´eorie d´eductive quelconque, in [13, vol. 3, 309–365]; partial English trans., Logical introduction to any deductive theory, in From Frege to G¨odel: A Source Book in Mathematical Logic, 1879–1931, J. van Heijenoort, trans. and ed., Harvard University Press, Cambridge, MA, 1967, 118–123. , Un nouveau syst`eme de d´efinitions pour la g´eom´etrie euclidienne, in Compte rendu du Deuxi`eme 25. Congr`es International des Math´ematiciens tenu a` Paris du 6 au 12 Aout 1900: Proc`es-verbaux et communications, E. Duporcq, ed., Gauthier–Villars, Paris, 1902, 353–363. , Letter 80 (6 January 1901), in Lettere a Mario Pieri (1884–1913), Quaderni P.RI.ST.EM 6 per 26. l’archivio della corrispondenza dei matematici italiani, G. Arrighi, ed., ELEUSI, Sezione P.RI.ST.EM., Milan, 1997. 27. V. Pambuccian, Unit distance as single binary predicate for plane Euclidean geometry, Zeszyty Nauk. Geom. 18 (1990) 5–8; Correction to: Unit distance as single binary predicate for plane Euclidean geometry, 19 (1991) 87. 28. , Universal-existential axiom systems for geometries expressed with Pieri’s isosceles triangle as single primitive notion, Rend. Sem. Mat. Univ. Politec. Torino 67 (2009) 327–339. 29. M. Pasch, Einleitung in die Differential- und Integralrechnung, Druck und Verlag von B. G. Teubner, Leipzig, 1882. , Vorlesungen u¨ ber neuere Geometrie, Druck und Verlag von B. G. Teubner, Leipzig, 1882. 30. , Ueber den Bildungswerth der Mathematik: Akademische Festrede zur Feier des Jahresfestes 31. der Grossherzoglich Hessischen Ludewigs-Universit¨at am 2. Juli 1894, Grosshandlung Hof- und Universit¨ats-Druckerei Curt von M¨unchow, Giessen, Germany, 1894. , Eine Selbstschilderung, M¨unchow’sche Universit¨ats-Druckerei Otto Kindt, Giessen, Germany, 32. 1931. 33. G. Peano, I principii di geometria logicamente esposti, Fratelli Bocca, Turin, 1889. 34. , Sopra alcune curve singolari, Atti della Reale Accademia delle Scienze di Torino 26 (1890–1891)

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35. 36. 37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47.

48. 49.

50. 51. 52.

53.

54. 55.

299–302; English trans., On some singular curves, in Selected Works of Giuseppe Peano, H. C. Kennedy, trans. and ed., with a biographical sketch and bibliography, University of Toronto Press, Toronto, 1973, 150–152. , Sui fondamenti della Geometria, Rivista di matematica 4 (1894) 51–90. M. Pieri, Un sistema di postulati per la geometria projettiva astratta degli iperspazˆı, Rivista di matematica 6 (1896) 9–16. , I principii della geometria di posizione composti in sistema logico deduttivo, Memorie della Reale Accademia delle Scienze di Torino (series 2) 48 (1898) 1–62. , review of [12], Revue de math´ematiques (Rivista di matematica) 6 (1899) 178–182; summarized in [23, pp. 394–395]. , Della geometria elementare come sistema ipotetico deduttivo: Monografia del punto e del moto, Memorie della Reale Accademia delle Scienze di Torino (series 2) 49 (1900) 173–222. , Sur la g´eom´etrie envisag´ee comme un syst`eme purement logique, in [13, vol. 3, 367–404]. , La geometria elementare istituita sulle nozioni di “punto” e “sfera,” Memorie di matematica e di fisica della Societ`a Italiana delle Scienze (series 3) 15 (1908) 345–450; Polish trans. S. Kwietniewski, Gieometrja elementarna oparta na poj˛eciach “punktu” i “kuli,” Bibljoteka Wektora A3, Skład Głowny w Ksi˛egarni Gebethnera i Wolffa, Warsaw, 1915; English trans., Elementary geometry based on the notions of point and sphere, in [23, pp. 160–270]. R. M. Robinson, Binary notions as primitive relations in elementary geometry, in [9, pp. 68–85]. B. Russell, The Autobiography of Bertrand Russell, 1872–1914, Little, Brown, Boston, 1951. M. Scanlan, American postulate theorists and Alfred Tarski, Hist. Philos. Logic 24 (2003) 307–325. doi: 10.1080/01445340310001599588 W. Schwabh¨auser, W. Szmielew, and A. Tarski, Metamathematische Methoden in der Geometrie, Springer-Verlag, Berlin,1983. J. T. Smith, Methods of Geometry, John Wiley, New York, 2000. G. K. C. von Staudt, Geometrie der Lage, Verlag der Fr. Korn’schen Buchhandlung, Nuremberg, Germany, 1847; annotated Italian trans. M. Pieri, with a study of the life and works of Staudt by C. Segre, Geometria di posizione, Biblioteca matematica, vol. 4, Fratelli Bocca Editori, Turin, 1889. L. W. Szczerba, Tarski and geometry, J. Symbolic Logic 51 (1986) 907–912. doi:10.2307/2273904 A. Tarski, Einige methodologische Untersuchungen u¨ ber die Definierbarkeit der Begriffe, Erkenntnis 5 (1935) 80–100; English trans., Some methodological investigations on the definability of concepts, in [52, pp. 296–319]. doi:10.1007/BF00172286 , The Completeness of Elementary Algebra and Geometry, Centre National de la Recherche Scientifique, Institute Blaise Pascal, Paris, 1967. , What is elementary geometry? in [9, pp. 16–29]. , Logic, Semantics, Metamathematics: Papers from 1923 to 1938, J. H. Woodger, trans. and ed., 2nd ed., J. Corcoran, ed., with introduction and analytical index, Hackett Publishing, Indianapolis, IN, 1983. ¨ M.-M. Toepell, Uber die Entstehung von David Hilberts “Grundlagen der Geometrie,” Studien zur Wissenschafts-, Sozial-, und Bildungsgeschichte der Mathematik, vol. 2, Vandenhoeck & Ruprecht, G¨ottingen, Germany, 1986. O. Veblen, A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904) 343–384. doi:10.2307/ 1986462 , The foundations of geometry, in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field, J. W. A. Young, ed., Longmans, Green, London, 1911, 3–54.

JAMES T. SMITH received the A.B. from Harvard College in 1961, and the Ph.D. from the University of Saskatchewan, Regina, in 1970, in foundations of geometry, under the supervision of H. N. Gupta. At his thesis defense, Alfred Tarski told him that he should study the work of Mario Pieri. Since then Smith has worked primarily at San Francisco State University, mostly teaching and writing, and in software development. He is now retired, but fully occupied with history, especially the legacies of Pieri and Tarski. Department of Mathematics, San Francisco State University, 1600 Holloway, San Francisco, CA 94132 [email protected]

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Volterra Functional Differential Equations: Existence, Uniqueness, and Continuation of Solutions Hartmut Logemann and Eugene P. Ryan Abstract. The initial-value problem for a class of Volterra functional differential equations— of sufficient generality to encompass, as special cases, ordinary differential equations, retarded differential equations, integro-differential equations, and hysteretic differential equations— is studied. A self-contained and elementary treatment of this over-arching problem is provided, in which a unifying theory of existence, uniqueness, and continuation of solutions is developed. As an illustrative example, a controlled differential equation with hysteresis is considered.

1. INTRODUCTION. Initial-value problems for systems of differential equations permeate many areas of mathematics: such problems arise naturally in modelling the evolution of dynamical processes in economics, engineering, and the physical and biological sciences. As a starting point, consider an ordinary differential equation in RN : u  (t) = f (t, u(t)),

(1.1)

where f is some suitably regular function, and the independent variable t carries the connotation of “time.” Loosely speaking, such a formulation is appropriate in applications wherein the forward-time, or future, behaviour of the process under investigation depends only on the current state u(t) (at current time t) and, in particular, is independent of its past u(s), s < t, and future u(s), s > t, states. Adopting the standpoint that the processes under investigation are “real-world” phenomena, it is reasonable to assume independence with respect to future states (and this we do throughout, via an assumption of causality or non-anticipativity); however, there are many situations wherein the process may “remember” the past and so its future behaviour depends, not only on the current state, but also on its past states. The simplest example of such dependence on the past is a differential equation with a point delay of length h > 0: u  (t) = f (t, u(t), u(t − h)).

(1.2)

System (1.2) may be embedded in the class of retarded differential equations of the form u  (t) = f (t, u t ),

(1.3)

where, for a continuous function u on some interval I containing [−h, 0] and t ∈ I with t ≥ 0, u t denotes the continuous function on [−h, 0] defined by u t (s) := u(t + s). Another commonly encountered class of systems which exhibit dependence on the past is comprised of integro-differential equations of the form  t  u (t) = k(t, s)g(s, u(s)) ds. (1.4) 0

doi:10.4169/000298910X492790

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As a final class of systems with memory, consider hysteretic differential equations of the form u  (t) = f (t, u(t), (H (u))(t)),

(1.5)

where H is a hysteresis operator (that is, an operator which is causal and rate independent in a sense to be made precise in due course). As a prototype, consider a scalar nonlinear mechanical system with hysteretic restoring force y  (t) + g(t, y(t), y  (t))y  (t) + (P(y))(t) = 0, where P is the play or backlash operator, the action of which is captured in Figure 1, wherein z = P(y) (we will return later to such an operator, with full details). z

−σ

y

σ

Figure 1. Play hysteresis.

Writing u(t) = (u 1 (t), u 2 (t)) := (y(t), y  (t)) and defining f : R+ × R2 × R by f (t, v, w) = f (t, (v1 , v2 ), w) := (v2 , −g(t, v1 , v2 )v2 − w), this system takes the form (1.5) with H (u) = P(u 1 ). Notwithstanding an outward appearance of diversity, the above examples (1.1)– (1.5) can be subsumed (as we shall see) in a common formulation, expressed as an initial-value problem for a functional differential equation of the form u  (t) = (F(u))(t),

t ≥ 0,

u|[−h,0] = ϕ,

(1.6)

with h ≥ 0 and ϕ continuous. The operator F is assumed to be causal or nonanticipative (loosely speaking, F is causal if, whenever functions u and v are such that their values u(t) and v(t) coincide up to t = τ , (F(u))(t) and (F(v))(t) also coincide up to t = τ : a precise definition is contained in hypothesis (H1) in Section 3 below). This paper provides an elementary, self-contained, and tutorial treatment of this over-arching problem: a unifying theory of existence, uniqueness, and continuation of solutions is developed which, when specialized, applies in the context of each of the systems (1.1)–(1.5) outlined above. For clarity of exposition, proofs of only the main results (viz., Proposition 3.5, Theorem 3.6, Theorem 3.7, and Corollary 3.8) are contained in the main body of the text: proofs of auxiliary technical propositions and lemmas are provided in the appendix. We close the introduction with a brief list of some standard references: for ordinary differential equations, see [1, 15]; for functional equations with causal operators, see [3, 4, 8]; for retarded differential equations of the form (1.3), see [5, 6, 9]; for integrodifferential equations of the form (1.4), see [3, 8]; for systems with hysteresis of the form (1.5), see [2, 12]. June–July 2010]

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Notation. The vector space of continuous functions defined on an interval I with values in R N is denoted by C(I ). If I is compact, then, endowed with the norm uC(I ) := sup{u(t) : t ∈ I },

(1.7)

where  ·  denotes the Euclidean norm in R N , C(I ) is a Banach space. For u ∈ C(I ), define gr u, the graph of u, by gr u := {(t, u(t)) : t ∈ I } ⊂ R × R N . Finally, if I is an interval, then I+ := I ∩ R+ , where R+ := [0, ∞). 2. PRELIMINARIES: ABSOLUTELY CONTINUOUS FUNCTIONS AND THE SOBOLEV SPACE W 1,1 . Let a < b. A function u : [a, b] → R N is said to be absolutely continuous if, for all ε > 0, there exists δ > 0 such that, for any finite collection of disjoint open intervals (a1 , b1 ), . . . , (an , bn ) contained in [a, b], n  i=1

(bi − ai ) ≤ δ

⇒

n 

u(bi ) − u(ai ) ≤ ε.

i=1

The importance of the concept of absolute continuity stems from the fact that absolutely continuous functions are precisely the functions for which the fundamental theorem of calculus (in the context of Lebesgue integration) is valid: a function u : [a, b] → R N is absolutely continuous if, and only  t if, u is differentiable at almost every t ∈ [a, b], u  ∈ L 1 [a, b], and u(t) = u(a) + a u  (s) ds for all t ∈ [a, b]; see, for example, [7, 11]. We define   W 1,1 [a, b] := u : [a, b] → R N | u is absolutely continuous . It is well known that, endowed with the norm uW 1,1 := u L 1 + u   L 1 , the space W 1,1 [a, b] is complete, and hence a Banach space. Usually, W 1,1 [a, b] is referred to as a Sobolev space. An alternative, and sometimes more convenient, norm on W 1,1 [a, b] is the BV-norm (where BV stands for bounded variation) defined by u BV := u(a) + u   L 1 . n N The total variation of a function u : [a, b] → k=1 u(tk ) −  R is given by sup u(tk−1 ) : n ∈ N, a = t0 < t1 < · · · < tn = b and, if this quantity is finite, u is said to be of bounded variation. As is well known, an absolutely continuous function u is of bounded variation and its total variation is equal to u   L 1 : this fact motivates the name “BV-norm.” Proposition 2.1. There exist k1 , k2 > 0 such that k1 uW 1,1 ≤ u BV ≤ k2 uW 1,1

∀ u ∈ W 1,1 [a, b],

that is, the norms  · W 1,1 and  ·  BV are equivalent. Consequently, W 1,1 [a, b] is complete with respect to the BV-norm. The proof of Proposition 2.1 can be found in the appendix. For an arbitrary (not necessarily compact) interval I ⊂ R and arbitrary p ∈ [1, ∞], we define p

L loc (I ) := {u : I → R N | u|[a,b] ∈ L p [a, b] for all a, b ∈ I , a < b} 492

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

and 1,1 (I ) := {u : I → R N | u|[a,b] ∈ W 1,1 [a, b] for all a, b ∈ I , a < b}. Wloc p

1,1 Clearly, if I is compact, then L loc (I ) = L p (I ) and Wloc (I ) = W 1,1 (I ).

3. VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS. The focus of our study is an initial-value problem of the form u  (t) = (F(u))(t), t ≥ 0;

u|[−h,0] = ϕ ∈ C[−h, 0], gr ϕ ⊂ ,

(3.1)

with h ≥ 0 and  ⊂ [−h, ∞) × R N , and where F is an operator acting on suitable spaces of functions u defined on intervals of the form [−h, η] or [−h, η) (where η > 0) and with gr u ⊂ . Throughout, we assume that • •

 is open relative to [−h, ∞) × R N , that is, there exists an open set D ⊂ R N +1 such that  = D ∩ ([−h, ∞) × R N ); {u ∈ C[−h, 0] : gr u ⊂ }  = ∅.

The following convention is adopted: in the case h = 0, C[−h, 0] should be interpreted as R N and the second of the above assumptions is equivalent to 0 := {x ∈ R N : (0, x) ∈ }  = ∅. We proceed to describe the nature of the operator F in (3.1). For each interval I (possibly singleton), we define 1,1 (I+ ), gr u ⊂ }. W (I ) := {u ∈ C(I ) : u| I+ ∈ Wloc

Note that W [−h, 0] = {u ∈ C[−h, 0] : gr u ⊂ }, and, by convention, in the case h = 0 we have W ({0}) = 0 . As will shortly be precisely defined, a solution of (3.1) is a function in W (I ) for some interval I of the form [−h, η] (where 0 < η < ∞) or [−h, η) (where 0 < η ≤ ∞). In the following, I denotes the set of all such intervals I with the property that W (I )  = ∅. Thus, I contains all possible domains of solutions of the functional differential equation. From the assumptions imposed on , it follows that [−h, α] ∈ I for all sufficiently small α > 0, or equivalently, T := sup{α > 0 : W [−h, α]  = ∅} > 0. We are now in a position to make precise the nature of the operator F in (3.1) and the concept of a solution of the initial-value problem. We assume that, for every I ∈ I, the operator F (in general, nonlinear) maps W (I ) to L 1loc (I+ ). In particular, the domain of F is ∪ I ∈I W (I ) and the range of F is contained in ∪ I ∈I L 1loc (I+ ). We say that u : I → R N is a solution of (3.1) (on the interval I ) if I ∈ I, u ∈ W (I ), u|[−h,0] = ϕ, and u satisfies the differential equation in (3.1) for almost every t ∈ I+ . If w ∈ W [−h, α] is a solution, then it is natural to ask if this solution can be extended to a solution u on [−h, β] with β > α. We will be especially interested in extensions u which are “well behaved” in the sense that, for given γ > 0, they satisfy June–July 2010]

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

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β

u  (s) ds ≤ γ (so that, in particular, u(s) − w(α) ≤ γ for all s ∈ [α, β]). This and other technical reasons lead to the consideration of a suitable space of extensions of functions w ∈ W [−h, α]. Specifically, let 0 ≤ α < T , β > α, γ > 0, and w ∈ W [−h, α], and define α

W(w; α, β, γ ) :=   1,1 u ∈ C[−h, β] : u|[−h,α] = w, u|[α,β] ∈ W [α, β],

α

β



u (s) ds ≤ γ .

It is clear that, for all u ∈ W(w; α, β, γ ), u|[0,β] ∈ W 1,1 [0, β]. We equip the space W(w; α, β, γ ) with the metric μ given by  μ(u, v) =

β α





u (s) − v  (s) ds.

A routine argument, invoking Proposition 2.1, yields the following lemma (see appendix for details). Lemma 3.1. For every α ∈ [0, T ), β > α, γ > 0, and w ∈ W [−h, α], the metric space W(w; α, β, γ ) is complete. Observe that some elements of W(w; α, β, γ ) may not be in W [−h, β]: such elements do not qualify as candidate solutions extending the solution w. This observation motivates the following definition. Given α ∈ [0, T ) and w ∈ W [−h, α], we define A(w; α) := {(β, γ ) ∈ (α, T ) × (0, ∞) : W(w; α, β, γ ) ⊂ W [−h, β]}. It is clear that (β, γ ) ∈ (α, T ) × (0, ∞) is in A(w; α) if and only if gr u ⊂  for all u ∈ W(w; α, β, γ ). Important properties of the set A(w; α) are given in the following lemma, the proof of which is relegated to the appendix. Lemma 3.2. If α ∈ [0, T ) and w ∈ W [−h, α], then the following statements hold. (1) The set A(w; α) is nonempty. (2) If (β, γ ) ∈ A(w; α), then (b, c) ∈ A(w; α) for all b ∈ (α, β] and all c ∈ (0, γ ]. If α ∈ [0, T ), w ∈ W [−h, α], (β, γ ) ∈ A(w; α), b ∈ (α, β], and c ∈ (0, γ ], then it follows from Lemma 3.2 that W(w; α, b, c) ⊂ W [−h, b], implying that F(u) is well defined for all u ∈ W(w; α, b, c). This fact will be used freely throughout. We assemble the following hypotheses on F which will be variously invoked in the theory developed below. (H1) Causality: if I, J ∈ I, then, for all τ ∈ (I+ ∩ J+ ) \ {0}, all u ∈ W (I ), and all v ∈ W (J ), u|[−h,τ ] = v|[−h,τ ]

⇒

F(u)|[0,τ ] = F(v)|[0,τ ] .

(H2) Local Lipschitz-type condition: for every α ∈ [0, T ) and every function w ∈ W [−h, α], there exists λ ∈ (0, 1) and (β, γ ) ∈ A(w; α) such that F(u) − F(v) L 1 [α,β] ≤ λμ(u, v) ∀ u, v ∈ W(w; α, β, γ ). 494

(3.2)

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(H3) Integrability condition: for every I ∈ I and every u ∈ W (I ) such that gr u is compact and contained in , the function F(u) is integrable, that is, F(u) ∈ L 1 (I+ ). The causality condition in (H1) is also referred to as non-anticipativity or as the Volterra property. Furthermore, in the literature, the term Volterra operator is sometimes used for a causal (or non-anticipative) operator. If (H1) holds, then the differential equation in (3.1) is often referred to as a Volterra functional differential equation or an abstract Volterra integro-differential equation (see, for example, [3, 8]). The essence of (H3) is that it encompasses all I ∈ I (including noncompact intervals): if I is compact, then obviously, even without (H3) being satisfied, F(u) ∈ L 1 (I+ ) for all u ∈ W (I ), since L 1loc (I+ ) = L 1 (I+ ). The following lemma records a particular consequence of hypotheses (H1) and (H2), which will be invoked in the later analysis. A proof of the lemma is provided in the appendix. Lemma 3.3. Assume that (H1) and (H2) hold. Then, for every α ∈ [0, T ) and every w ∈ W [−h, α], there exist λ ∈ (0, 1) and (β, γ ) ∈ A(w; α) such that, for every b ∈ (α, β] and every c ∈ (0, γ ],  F(u) − F(v) L 1 [α,b] ≤ λ

b α

u  (s) − v  (s) ds,

∀ u, v ∈ W(w; α, b, c).

On first encounter, it may seem that, in hypothesis (H2), the requirement that λ < 1 is quite restrictive. We proceed to show that this is not the case. To this end, assume p that, for every I ∈ I, F(W (I )) ⊂ L loc (I+ ) for some p ∈ (1, ∞] and consider the following hypothesis. (H2 ) For every α ∈ [0, T ) and every w ∈ W [−h, α], there exists ρ > 0 and (β, γ ) ∈ A(w; α) such that F(u) − F(v) L p [α,β] ≤ ρμ(u, v) ∀ u, v ∈ W(w; α, β, γ ).

(3.3)

Note that, in (H2 ), the Lipschitz constant ρ is not required to be smaller than 1. p

Proposition 3.4. Assume that there exists p ∈ (1, ∞] such that F(W (I )) ⊂ L loc (I+ ) for every I ∈ I. If (H1) and (H2 ) are satisfied, then (H2) holds. A proof of this proposition is contained in the appendix. In order to study the problems of existence, uniqueness, and continuation of solutions, it is convenient to consider the following initial-value problem which is slightly more general than (3.1): u  (t) = (F(u))(t), t ≥ α;

u|[−h,α] = ψ ∈ W [−h, α], α ∈ [0, T ).

(3.4)

Trivially, the original initial-value problem (3.1) can be recovered from (3.4) by setting α = 0. We say that u : I → R N is a solution of (3.4) (on the interval I ) if I ∈ I with sup I > α, u ∈ W (I ), u|[−h,α] = ψ, and u satisfies the differential equation in (3.4) for almost every t ∈ I ∩ [α, ∞). We now arrive at the first of three core results. June–July 2010]

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Proposition 3.5. Assume that the operator F satisfies (H1) and (H2), and let α ∈ [0, T ) and ψ ∈ W [−h, α]. There exists η > α such that (3.4) has precisely one solution u on the interval [−h, η]; moreover, for every I ∈ I such that α < sup I ≤ η, the function u| I is the only solution of (3.4) on I . Proof. Let α ∈ [0, T ) and ψ ∈ W [−h, α]. We proceed in two steps. Step 1. Existence and uniqueness in W(ψ; α, β, γ ) for small β − α > 0. First, we convert (3.4) into an integral equation. To this end, for each β ∈ (α, T ), define an operator G β on W [−h, β] by  ψ(t), t ∈ [−h, α] t (G β (u))(t) := ψ(α) + α (F(u))(s) ds, t ∈ (α, β]. It is clear that u ∈ W [−h, β] is a solution of (3.4) if and only if G β (u) = u. We claim that there exist β ∗ > α and γ > 0 such that (3.4) has a solution u ∈ W(ψ; α, β ∗ , γ ), and moreover, for every β ∈ (α, β ∗ ], u|[−h,β] is the only solution of (3.4) in W(ψ; α, β, γ ). Invoking the completeness of W(ψ; α, β, γ ) (guaranteed by Lemma 3.1) and the contraction-mapping theorem, it is sufficient to show that there exist (β ∗ , γ ) ∈ A(w; α) and λ ∈ (0, 1) such that G β (W(ψ; α, β, γ )) ⊂ W(ψ; α, β, γ ) ∀ β ∈ (α, β ∗ ]

(3.5)

and, moreover, μ(G β (u), G β (v)) ≤ λμ(u, v) ∀ u, v ∈ W(ψ; α, β, γ ),

∀ β ∈ (α, β ∗ ].

(3.6)

We proceed to establish (3.5) and (3.6). Using Lemma 3.3, we conclude that there exist λ ∈ (0, 1) and (β  , γ ) ∈ A(w; α) such that, for every β ∈ (α, β  ], F(u) − F(v) L 1 [α,β] ≤ λμ(u, v)

∀ u, v ∈ W(ψ; α, β, γ ).

(3.7)

Let β ∈ (α, β  ]. To show that G β (W(ψ; α, β, γ )) ⊂ W(ψ; α, β, γ ), let u ∈ W(ψ; α, β, γ ) and note that, by the definition of G β , we have that G β (u) ∈ C[−h, β], (G β (u))|[−h,α] = ψ, and (G β (u))|[α,β] ∈ W 1,1 [α, β]. It remains to show that β (G β (u)) (s) ds ≤ γ . To this end, define a function ψ˜ : [−h, β] → R N by α  ψ(t), ˜ ψ(t) = ψ(α),

t ∈ [−h, α] t ∈ (α, β].

Clearly, ψ˜ ∈ W(ψ; α, β, γ ). By Lemma 3.2, (β, γ ) ∈ A(w; α), so that ψ˜ ∈ ˜ is well defined, and, furthermore, invoking (3.7), W [−h, β]. Consequently, F(ψ)  α

β





(G β (u)) (s) ds ≤

α

˜ (F(u))(s) − (F(ψ))(s) ds +



≤λ 496

β

α

β

u (s) − ψ˜  (s) ds + 

 α

β

 α

β

˜ (F(ψ)(s) ds

˜ (F(ψ))(s) ds.

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

Since ψ˜  (s) = 0 for all s ∈ [α, β], we obtain  α

β





(G β (u)) (s) ds ≤ λγ +

α

β

˜ (F(ψ))(s) ds.

Now choose β ∗ ∈ (α, β  ] such that 

β∗

α

˜ (F(ψ))(s) ds ≤ γ (1 − λ).

β Then α (G β (u)) (s)ds ≤ γ for every β ∈ (α, β ∗ ], so that (3.5) follows. Furthermore, (3.6) is an immediate consequence of (3.7) (in which λ < 1) and the fact that, for all β ∈ (α, β ∗ ],  μ(G β (u), G β (v)) =

α

β

(F(u))(s) − (F(v))(s) ds

∀ u, v ∈ W(ψ; α, β, γ ).

Step 2. Uniqueness in W [−h, β] for small β − α > 0. By Step 1, there exists β ∗ > α such that (3.4) has a solution u ∈ W(ψ; α, β ∗ , γ ), and moreover, for every β ∈ (α, β ∗ ], u|[−h,β] is the only solution of (3.4) in W(ψ; α, β, γ ). Choosing η ∈ (α, β ∗ ] such that  η u  (s) ds < γ , (3.8) α

let I ∈ I be such that α < sup I ≤ η and let v ∈ W (I ) be a solution of (3.4). Setting σ := sup I , we claim that  σ v  (s) ds ≤ γ . (3.9) α

Seeking a contradiction, suppose that this is not true. Then there exists τ ∈ (α, σ ) such that  τ v  (s) ds = γ . (3.10) α

In particular,  τ v ∈ W(w; α, τ, γ ), so that, by Step 1, v[−h,τ ] = u|[−h,τ ] . Consequently, by (3.8), α v  (s) ds < γ , contradicting (3.10). We conclude that (3.9) holds, and thus, for every β ∈ (α, σ ), v ∈ W(w; α, β, γ ). Invoking Step 1 again, we obtain that v[−h,β] = u|[−h,β] for every β ∈ (α, σ ), implying that v = u| I . We now use Proposition 3.5 to prove the following result relating to the initial-value problem (3.1). Theorem 3.6. Assume that F satisfies (H1) and (H2). (1) The initial-value problem (3.1) has a solution u : [−h, η] → R N for sufficiently small η > 0. (2) Given an interval I ∈ I, there exists at most one solution u : I → R N of the initial-value problem (3.1). June–July 2010]

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Proof. (1) The claim follows immediately from an application of Proposition 3.5 with α = 0. (2) Let I ∈ I. Suppose that u 1 and u 2 are solutions of (3.1) defined on I . Define τ := sup{t ∈ I+ : u 1 (s) = u 2 (s) ∀s ∈ [0, t]}. Note that an application of Proposition 3.5 with α = 0 shows that τ > 0. It is sufficient to show that τ = sup I = sup I+ . Seeking a contradiction, suppose that τ < sup I . Defining ψ : [−h, τ ] → R N by  ϕ(t), −h ≤ t ≤ 0 ψ(t) = u 1 (t), 0 < t ≤ τ, it is clear that ψ ∈ W [−h, τ ] and u 1 (t) = u 2 (t) = ψ(t) ∀ t ∈ [−h, τ ]. Applying Proposition 3.5 again, now with α = τ , yields the existence of an ε > 0 such that u 1 (t) = u 2 (t) for all t ∈ [0, τ + ε], contradicting the definition of τ . Let I ∈ I and assume that u ∈ W (I ) is a solution of the initial-value problem (3.1). We say that the interval I is a maximal interval of existence, and u is a maximally defined solution, if u does not have a proper extension which is also a solution of (3.1), that is, there does not exist I˜ ∈ I and a solution u˜ ∈ W ( I˜) of (3.1) such that I ⊂ I˜, I  = I˜, and u(t) = u(t) ˜ for all t ∈ I . Theorem 3.7. Assume that F satisfies (H1) and (H2). Then there exists a unique maximally defined solution u of (3.1). The associated maximal interval of existence is of the form [−h, τ ), where 0 < τ ≤ ∞. If τ < ∞ and, in addition, F satisfies (H3), then, for every compact set  ⊂  and every σ ∈ (0, τ ), there exists t ∈ (σ, τ ) such that (t, u(t))  ∈ . If F satisfies (H1)–(H3) and if u : [−h, τ ) → R N is the unique maximally defined solution of (3.1), then the last assertion of the above theorem implies the following two statements: (i) if τ < ∞, then there exists a sequence (tn ) in [0, τ ) such that tn → τ as n → ∞ and at least one of the following two properties holds: (a) limn→∞ u(tn ) = ∞, (b) (tn , u(tn )) approaches ∂ as n → ∞; (ii) if u is bounded and gr u ⊂ , then τ = ∞. Proof of Theorem 3.7. Set τ := sup{η ≥ 0 : there exists a solution of (3.1) on [−h, η]} By Theorem 3.6, τ > 0, and, for every η ∈ (0, τ ), there exists a unique solution u η ∈ W [−h, η] of (3.1). We define u : [−h, τ ) → R N by u(t) = u η (t) −h ≤ t ≤ η,

for every η ∈ (0, τ ).

It follows from statement (2) of Theorem 3.6 that u is well defined. Moreover, it is clear that u ∈ W [−h, τ ) and that u solves (3.1). We claim that u is a maximally defined 498

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solution on the maximal interval of existence [−h, τ ). If τ = ∞, then there is nothing to prove. So without loss of generality assume that τ < ∞. Seeking a contradiction, suppose that there exists an extension u˜ of u, u˜ ∈ W [−h, τ˜ ] with τ˜ ≥ τ , and such that u˜ solves (3.1). From the definition of τ it follows immediately that τ˜ = τ . An application of Proposition 3.5, with α = τ = τ˜ and ψ = u, ˜ shows that there exist τ ∗ > τ and a (proper) extension u ∗ ∈ W [−h, τ ∗ ] of u˜ which also solves (3.1); this contradicts the definition of τ . Uniqueness of u follows from statement (2) of Theorem 3.6. Now assume that τ < ∞ and F satisfies (H3). Seeking a contradiction, suppose there exist a compact set  ⊂  and σ ∈ (0, τ ) such that (t, u(t)) ∈  for all t ∈ (σ, τ ). Then gr u ⊂ {(t, u(t)) : t ∈ [−h, σ ]} ∪ , and consequently, gr u is compact and contained in . Invoking (H3), together with the identity  t (F(u))(s) ds, ∀ t ∈ [0, τ ), u(t) = u(0) + 0

shows that the limit l := limt↑τ u(t) exists. Obviously, (τ, l) ∈ gr u, and thus, (τ, l) ∈ . This implies that the function u˜ : [−h, τ ] → R N defined by  u(t), −h ≤ t < τ u(t) ˜ = l, t =τ is in W [−h, τ ], is a solution of (3.1), and is a proper extension of the solution u. This contradicts the fact that u is a maximally defined solution. Example: A Controlled Differential Equation with Hysteresis. Consider a forced system with forcing input (control) v subject to hysteresis H : my  (t) + cy  (t) + ky(t) + (H (v))(t) = 0,

y(0) = y 0 , y  (0) = y 1 .

(3.11)

In a mechanical context, y(t) represents displacement at time t ∈ R+ , m > 0 and c are the mass and the damping constant, and k is a linear spring constant: the function v is interpreted as a control (which is open to choice and may be generated by feedback of y) and the operator H models hysteretic actuation. Such hysteretic effects arise in, for example, micro-positioning control problems using piezo-electric actuators or smart actuators, as investigated in, for example, [13]; general treatments of hysteresis phenomena can be found in, for example, [2], [12], and [14]. We deem an operator H : C(R+ ) → C(R+ ) to be a hysteresis operator if it is both causal and rate independent. By rate independence we mean that H (y ◦ ζ ) = H (y) ◦ ζ for every y ∈ C(R+ ) and every time transformation ζ : R+ → R+ (that is, a continuous, nondecreasing, and surjective function). The control objective is to generate the input function v in such a way that the displacement y(t) tends, as t → ∞, to some desired value r ∈ R. In view of this objective, it is natural to seek to generate the input by feedback of the error y(t) − r . Proportional-integral-derivative (PID) feedback control action is ubiquitous in control theory and practice and takes the form  t v(t) = k p (y(t) − r ) + ki (3.12) (y(τ ) − r ) dτ + kd y  (t), 0

June–July 2010]

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

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the control objective being reduced to that of determining the parameter values k p , ki , kd ∈ R so as to cause the variable y to approach asymptotically the prescribed constant value r . This methodology, applied in the context of (3.11), is depicted in Figure 2. Introducing the variables  t u 1 (t) = (y(τ ) − r ) dt, u 2 (t) = y(t) − r, u 3 (t) = y  (t), 0

the feedback system given by (3.11) and (3.12) may be expressed as ⎫ u 1 (t) = u 2 (t), u 2 (t) = u 3 (t), ⎪ ⎬  mu 3 + cu 3 (t) + k(u 2 (t) + r ) + (H (ki u 1 + k p u 2 + kd u 3 ))(t) = 0, ⎪ ⎭ u 1 (0) = 0, u 2 (0) = y 0 − r, u 3 (0) = y 1 .

(3.13)

Therefore, central to any study (see [10], for example) of the efficacy of the PID feedback structure is the initial-value problem (3.13) which we proceed to show is subsumed by (3.1). Writing u := (u 1 , u 2 , u 3 ) and defining an operator F by   F(u) = u 2 , u 3 , −cm −1 u 3 − km −1 (u 2 + r ) − m −1 H (ki u 1 + k p u 2 + kd u 3 ) , (3.14) we see that the initial-value problem (3.13) may be expressed as u  = F(u),

u(0) = ϕ := (0, y 0 − r, y 1 ),

(3.15)

which has the form (3.1) with h = 0 and  = R+ × R3 . In particular, I consists of all intervals of the form [0, η] (where 0 < η < ∞) or [0, η) (where 0 < η < ∞). Clearly, 1,1 for every I ∈ I, the operator F maps W (I ) = Wloc (I ) to L 1loc (I ). v

H

z

my  + cy  + ky + z = 0

y

y −r

PID

−

r

Figure 2. System (3.11) under PID control action.

Many commonly-encountered hysteresis operators H satisfy a global Lipschitz condition in the sense that there exists a Lipschitz constant L > 0 such that sup |(H (y1 ))(t) − (H (y2 ))(t)| ≤ L sup |y1 (t) − y2 (t)| ∀ y1 , y2 ∈ C(R+ ). t∈R+

t∈R+

Corollary 3.8. Assume that the hysteresis operator H satisfies a global Lipschitz condition. Then the operator F given by (3.14) satisfies (H1)–(H3) and, for every initial condition ϕ, the equation (3.15) has a unique maximally defined solution with maximal interval of existence R+ . Proof. Since H is a hysteresis operator, H is causal, whence causality of F follows. Therefore hypothesis (H1) holds. The global Lipschitz condition for H clearly implies that F is globally Lipschitz and hence, (H2) is satisfied. Moreover, it follows easily 500

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

from the rate independence of H that H (0) (where 0 denotes the zero function on R+ ) is a constant function, implying that the function F(0) is constant. Denoting this constant by c and using the global Lipschitz property of F, a routine argument yields 1,1 (F(u))(t) ≤ c + L F u(t) ∀ u ∈ Wloc [0, η),

∀ η ∈ (0, ∞),

(3.16)

where L F denotes the Lipschitz constant of F. It follows that (H3) also holds. Consequently, Theorem 3.7 applies. Let u : [0, τ ) → R3 be a maximally defined solution of (3.15). To complete the proof, we have to show that τ = ∞. Seeking a contradiction, suppose that τ < ∞. Since  = R+ × R3 , it then follows from Theorem 3.7 that u is unbounded. Integrating (3.15) from 0 to t and invoking (3.16) leads to  t u(s) ds ∀ t ∈ [0, τ ). u(t) ≤ u(0) + ct + L F 0

An application of Gronwall’s lemma (see, for example, [1, Lemma 6.1]) now yields that   u(t) ≤ u(0) + cτ e L F τ ∀ t ∈ [0, τ ), which is in contradiction to the unboundedness of u. There follows an illustrative example of hysteresis with the requisite Lipschitz property. Play and Prandtl Hysteresis. A basic hysteresis operator is the play operator (already alluded to in the introduction). A detailed discussion of the play operator (also called the backlash operator) can be found in, for example, [2, 12, 14]. Intuitively, the play operator describes the input-output behaviour of a simple mechanical play between two mechanical elements as shown in Figure 3, where the input y is the position of the vertical component of element I and the output z is the the position of the midpoint of element II. The resulting input-output diagram is shown in Figure 1. The output value z(t) at time t ∈ R+ depends not only on the input value y(t) but also on the past history of the input. To aid in the characterization of this dependence, it is convenient to restrict initially to piecewise monotone input functions y. We seek an operator Pσ,ζ such that, given a piecewise monotone function y, the corresponding output function is given by z = Pσ, ζ (y). Here the parameter ζ ∈ R plays the role of an “initial state,” determining the initial output value z(0) ∈ [y(0) − σ, y(0) + σ ]. To give a formal definition of the play operator, let σ ∈ R+ and introduce the function pσ : R2 → R given by   pσ (v1 , v2 ) := max v1 − σ, min{v1 + σ, v2 } ⎧ ⎪ ⎨v1 − σ, if v2 < v1 − σ = v2 , if v2 ∈ [v1 − σ, v1 + σ ] ⎪ ⎩v + σ, if v > v + σ. 1 2 1 Let Cpm (R+ ) denote the space of continuous piecewise monotone functions defined on R+ . For all σ ∈ R+ and ζ ∈ R, define the operator Pσ, ζ : Cpm (R+ ) → C(R+ ) by  for t = 0, pσ (y(0), ζ ) (Pσ, ζ (y))(t) = pσ (y(t), (Pσ, ζ (y))(ti )) for ti < t ≤ ti+1 , i = 0, 1, 2, . . . , June–July 2010]

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

501

y I 2σ

II z

Figure 3. Mechanical play.

where 0 = t0 < t1 < t2 < · · · , limn→∞ tn = ∞, and u is monotone on each interval [ti , ti+1 ]. It is not difficult to show that the definition is independent of the choice of the partition (ti ). It is well known that Pσ, ζ extends to a hysteresis operator on C(R+ ), the so-called play operator, which we shall denote by the same symbol Pσ, ζ ; furthermore, Pσ, ζ is globally Lipschitz with Lipschitz constant L = 1 (see [2] for details). We are now in a position to model more complex hysteretic effects (displaying, for example, nested hysteresis loops) by using the play operator as a basic building block. To this end, let ξ : R+ → R be a compactly supported and globally Lipschitz function with Lipschitz constant 1 and let m ∈ L 1 (R+ ). The operator Pξ : C(R+ ) → C(R+ ) defined by  ∞ (Pσ, ξ(σ ) (y))(t)m(σ ) dσ ∀ y ∈ C(R+ ), ∀ t ∈ R+ , (Pξ (y))(t) = 0

is called a Prandtl operator. It is clear that Pξ is a hysteresis operator (this follows from the fact that Pσ, ξ(σ ) is a hysteresis operator for every σ ≥ 0). Moreover, Pξ is gobally Lipschitz with Lipschitz constant L = m L 1 (see [2]). For ξ = 0 and m = I[0,5] (where I[0,5] denotes the indicator function of the interval [0, 5]), the Prandtl operator is illustrated in Figure 4.

40

P0 (y)

40

P0 (y) y 0 −20

0

0

t

10

−20 −5

y

10

Figure 4. Example of Prandtl hysteresis.

4. RETARDED DIFFERENTIAL EQUATIONS. Let h ≥ 0 and let I be an interval containing [−h, 0]. For u ∈ C(I ) and t ∈ I+ , u t ∈ C[−h, 0] is defined by u t (s) := u(t + s) for all s ∈ [−h, 0]. Let  ⊂ [−h, ∞) × R N be a relatively open set with the property that the set {u ∈ C[−h, 0] : gr u ⊂ } is nonempty. As before, let I be the set 502

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

of all intervals I of the form [−h, η] (with 0 < η < ∞) or [−h, η) (with 0 < η ≤ ∞) such that W (I )  = ∅ and recall that T := sup{α > 0 : W [−h, α]  = ∅} > 0. Define D := {(t, u) ∈ R+ × C[−h, 0] : (t + s, u(s)) ∈  ∀ s ∈ [−h, 0]}. It is readily verified that D is nonempty and is open relative to R+ × C[−h, 0], where C[−h, 0] is endowed with the topology induced by the supremum norm (1.7). Moreover, for I ∈ I and u ∈ C(I ), we have gr u ⊂ 

⇔

(t, u t ) ∈ D ∀ t ∈ I+ .

If  is a Cartesian product, that is,  = [−h, α) × G, where 0 < α ≤ ∞ and G ⊂ R N is open, then D is also a Cartesian product, namely D = [0, α) × C([−h, 0], G), where C([−h, 0], G) is the subset of all functions in C[−h, 0] with values in G. Consider the initial-value problem u  (t) = f (t, u t ), u|[−h,0] = ϕ ∈ C[−h, 0], gr ϕ ⊂ ,

(4.1)

where f : D → R N . By a solution of (4.1), we mean a function u ∈ W (I ), with I ∈ I, such that u|[−h,0] = ϕ and u  (t) = f (t, u t ) for almost every t ∈ I+ . We impose the following hypotheses on f . (RDE1) For every (t, w) ∈ D there exist a relatively open interval I ⊂ R+ containing t, an open ball B ⊂ C[−h, 0] containing w, and a function l ∈ L 1 (I ) such that I × B ⊂ D and  f (s, u) − f (s, v) ≤ l(s) u − vC[−h,0]

∀ s ∈ I, ∀ u, v ∈ B.

(RDE2) For every I ∈ I and every u ∈ W (I ) such that gr u is a compact subset of , the function I+ → R N , t  → f (t, u t ) is in L 1 (I+ ). For each I ∈ I, define the operator F on W (I ) by (F(v))(t) := f (t, vt ) ∀ t ∈ I+ . Let I ∈ I and v ∈ W (I ) be arbitrary. Let τ > 0 be such that J := [−h, τ ] ⊂ I and write u := v| J . Then gr u is a compact subset of  and so, by (RDE2), the function F(u) is in L 1 (J+ ). It follows that the function F(v) is in L 1loc (I+ ). Therefore, for each I ∈ I, F maps W (I ) to L 1loc (I+ ) and, in view of (RDE2), F satisfies the integrability hypothesis (H3). It is evident that the causality hypothesis (H1) is also valid. We proceed to show that F satisfies the local Lipschitz hypothesis (H2). Let α ∈ [0, T ) and w ∈ W [−h, α] be arbitrary. By (RDE1), there exist a relatively open interval I ⊂ R+ containing α, an open ball B ⊂ C[−h, 0] containing wα , and l ∈ L 1 (I ) such that I × B ⊂ D and  f (t, x) − f (t, y) ≤ l(t)x − yC[−h,0]

∀ t ∈ I, ∀ x, y ∈ B,

(4.2)

A routine argument (see appendix for details) then yields the following. Lemma 4.1. There exists γ > 0 such that  α+γ λ := α l(t) dt < 1, vt ∈ B June–July 2010]



∀ v ∈ W(w; α, α + γ , γ ), ∀ t ∈ [α, α + γ ].

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

(4.3)

503

By Lemma 4.1, there exists γ > 0 such that (4.3) holds. Set β := α + γ and let u, v ∈ W(w; α, β, γ ) be arbitrary. Then, by (4.2) and (4.3), we have (F(u))(t) − (F(v))(t) =  f (t, u t ) − f (t, vt ) ≤ l(t)u t − vt C[−h,0] ≤ l(t)u − vC[α,β]  β ≤ l(t) u  (s) − v  (s) ds ∀ t ∈ [α, β]. α

Integrating from α to β = α + γ yields  F(u) − F(v) L 1 [α,β] ≤ λ

α

β

u  (s) − v  (s) ds.

Therefore, the local Lipschitz hypothesis (H2) holds. We may now infer that, if (RDE1) and (RDE2) hold, then the assertions of Theorems 3.6 and 3.7 are valid in the context of the initial-value problem (4.1). 5. ORDINARY DIFFERENTIAL EQUATIONS. Let G ⊂ R+ × R N be a relatively open set with G 0 := {x ∈ R N : (0, x) ∈ G}  = ∅. In this section, we consider the initial-value problem for an ordinary differential equation of the form u  (t) = f (t, u(t)),

u(0) = u 0 ∈ G 0 ,

(5.1)

where f : G → R N . Let I be the set of all intervals I of the form [0, α] (with 0 < α < 1,1 ∞) or [0, α) (with 0 < α ≤ ∞) such that WG (I ) = {u ∈ Wloc (I ) : gr u ⊂ G}  = ∅. By a solution of (5.1), we mean a function u ∈ WG (I ), with I ∈ I, such that u(0) = u 0 and u  (t) = f (t, u(t)) for almost all t ∈ I . We impose the following hypothesis on f . (ODE) For every (t, z) ∈ G, there exists a relatively open interval I ⊂ R+ containing t, an open ball B ⊂ R N containing z, and a function l ∈ L 1 (I ) such that I × B ⊂ G,  f (s, x) − f (s, y) ≤ l(s)x − y

∀ s ∈ I, ∀ x, y ∈ B,

and, moreover, the function I → R N , s  → f (s, x) is measurable for all x ∈ B and is in L 1 (I ) for some x ∈ B. The following proposition shows that, under this hypothesis, the initial-value problem (5.1) is subsumed by the theory of retarded differential equations developed in the previous section specialized to the situation h = 0, in which case C[−h, 0] = R N ,  = D = G, and the initial-value problems (4.1) and (5.1) coincide. Proposition 5.1. If (ODE) is satisfied, then (RDE1) and (RDE2) hold with h = 0. Proposition 5.1 (a proof of which may be found in the appendix), together with the conclusion of Section 4, imply that, if hypothesis (ODE) is satisfied, then Theorems 3.6 and 3.7 apply to the initial-value problem (5.1). 6. INTEGRO-DIFFERENTIAL EQUATIONS. In this section, we apply the theory developed in Section 3 to initial-value problems associated with integro-differential equations (also called Volterra integro-differential equations), that is,  t  u (t) = k(t, s)g(s, u(s)) ds, t ≥ 0; u(0) = u 0 . (6.1) 0

504

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Here k ∈ L loc (J × J, R N ×M ) and g : G → R M , where p ∈ [1, ∞], J = [0, a) for some a satisfying 0 < a ≤ ∞, R N ×M is the set of all N × M matrices with real entries, and G ⊂ R+ × R N is relatively open. It is assumed that G 0 := {x ∈ R N : (0, x) ∈ G}  = ∅ and u 0 ∈ G 0 . To apply the theory developed in Section 3 to the initial-value problem (6.1), we set h := 0 and  := (J × R N ) ∩ G. Note that 0 = G 0 and, moreover, since h = 0, I+ = I for all I ∈ I. Trivially, if I ∈ I is such that W (I )  = ∅, then I ⊂ J and W (I ) = WG (I ). By a solution of  t (6.1), we mean a function u ∈ W (I ), with I ∈ I, such that u(0) = u 0 and u  (t) = 0 k(t, s)g(s, u(s)) ds for almost every t ∈ I . Defining, for each I ∈ I, the operator F on W (I ) by p



t

k(t, s)g(s, v(s)) ds

(F(v))(t) =

∀ t ∈ I,

(6.2)

0

the initial-value problem (6.1) can be written in the form (3.1). Set q := p/( p − 1). We impose the following hypothesis on g. (IDE) For every (t, z) ∈ G, there exist a relatively open interval I ⊂ R+ containing t, an open ball B ⊂ R N containing z, and a nonnegative function l ∈ L q (I ) such that I × B ⊂ G, the function I → R M , s  → g(s, x) is measurable for all x ∈ B and is in L q (I ) for some x ∈ B, and moreover, g(s, x) − g(s, y) ≤ l(s)x − y

∀ s ∈ I,

∀ x, y ∈ B.

(6.3)

Proposition 6.1. Assume that (IDE) holds and that F is given by (6.2). Then, for all I ∈ I, F maps W (I ) to L 1loc (I ) and F satisfies (H1), (H2), and (H3). Proposition 6.1, a proof of which may be found in the appendix, implies that if hypothesis (IDE) is satisfied, then Theorems 3.6 and 3.7 apply to the initial-value problem (6.1). 7. APPENDIX. Proof of Proposition 2.1. Let u ∈ W 1,1 [a, b]. Since u(t) = u(a) + t ∈ [a, b], it is clear that

t a

u  (s) ds for all

u∞ := max u(t) ≤ u BV . t∈[a,b]

Consequently,   uW 1,1 ≤ (b − a)u∞ + u   L 1 ≤ (b − a) + 1 u BV ,  −1 and so, setting k1 := (b − a) + 1 , it follows that k1 uW 1,1 ≤ u BV for all u ∈ W 1,1 [a, b]. Furthermore, denoting the components of u by u j , the mean value theorem for integrals guarantees the existence of c j ∈ [a, b] such that 1 u j (c j ) = b−a June–July 2010]



b

u j (s) ds

j = 1, . . . , N .

a

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

505

Consequently, for j = 1, . . . , N ,  |u j (a)| ≤ |u j (a) − u j (c j )| + |u j (c j )| ≤

b

|u j (s)| ds

a

1 + b−a



b

|u j (s)| ds, a

and thus, by a routine calculation,  √  N  (1 + b − a) N  2 uW 1,1 . |u j (a)| ≤ u(a) = b−a j =1 √ Therefore, setting k2 := 1 + (1 + b − a) N /(b − a), it follows that u BV ≤ k2 uW 1,1 for all u ∈ W 1,1 [a, b]. Proof of Lemma 3.1. It is clear that μ is a metric on W(w; α, β, γ ). Let (u n ) be a Cauchy sequence in W(w; α, β, γ ) and set vn = u n |[α,β] . Then vn ∈ W 1,1 [α, β], vn (α) = w(α), and  β u n (s) − u m (s) ds = μ(u n , u m ), vn − vm  BV = α

showing that (vn ) is a Cauchy sequence in W 1,1 [α, β] with respect to the norm  ·  BV . By Proposition 2.1, W 1,1 [α, β] is complete with respect to the BV-norm and so there exists v ∈ W 1,1 [α, β] such that  β v  (s) − vn (s) ds = v − vn  BV → 0 as n → ∞. v(α) − w(α) + Thus, v(α) = w(α), Therefore, defining

β α

α

v  (s) − vn (s) ds → 0 as n → ∞, and

β α

v  (s) ds ≤ γ .

 w(t), t ∈ [−h, α] u(t) := v(t), t ∈ (α, β], it follows that u ∈ W(w; α, β, γ ) and μ(u, u n ) → 0 as n → ∞, completing the proof. Proof of Lemma 3.2. (1) Since (α, w(α)) ∈ , it follows from the assumptions imposed on  that there exists β > α and γ > 0 such that [α, β] × Bγ ⊂ , where Bγ denotes the closed ball of radius γ > 0 centered at w(α) ∈ R N . We claim that (β, γ ) ∈ A(w; α). To this end, let u ∈ W(w; α, β, γ ). Then,  β u(t) − w(α) = u(t) − u(α) ≤ u  (s) ds ≤ γ ∀ t ∈ [α, β]. α

Consequently, (t, u(t)) ∈ [α, β] × Bγ ⊂  for all t ∈ [α, β], implying that gr u ⊂ . Since u ∈ W(w; α, β, γ ) was arbitrary, it follows that (β, γ ) ∈ A(w; α). (2) Let (β, γ ) ∈ A(w; α), b ∈ (α, β], and c ∈ (0, γ ], and let u ∈ W(w; α, b, c). It is sufficient to show that gr u ⊂ . To this end, define an extension u˜ : [−h, β] → R N of u by  u(t), t ∈ [−h, b] u(t) ˜ := u(b), t ∈ (b, β]. 506

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Clearly, u˜ ∈ W(w; α, β, γ ), and hence, by the admissibility of (α, β), gr u˜ ⊂ . Since u˜ is an extension of u, we have that gr u ⊂ gr u, ˜ and so gr u ⊂ . Proof of Lemma 3.3. Let α ∈ [0, T ) and w ∈ W [−h, α]. Then, by (H2), there exist λ ∈ (0, 1) and (β, γ ) ∈ A(w; α) such that (3.2) holds. Let b ∈ (α, β], c ∈ (0, γ ], and u, v ∈ W(w; α, b, c). Define u, ˜ v˜ ∈ W(w; α, β, γ ) by   u(t), t ∈ [−h, b] v(t), t ∈ [−h, b] u(t) ˜ := v(t) ˜ := u(b), t ∈ (b, β], v(b), t ∈ (b, β]. By (H1), F(u)| ˜ [0,b] = F(u)|[0,b] and F(v)| ˜ [0,b] = F(v)|[0,b] , so that, by (3.2),  F(u) − F(v) L 1 [α,b] ≤ F(u) ˜ − F(v) ˜ L 1 [α,β] ≤ λ The claim now follows, since

β α

u˜  (s) − v˜  (s) ds =

b α

α

β

u˜  (s) − v˜  (s) ds.

u  (s) − v  (s) ds.

Proof of Proposition 3.4. Let α ∈ [0, T ) and w ∈ W [−h, α]. Since (H2 ) is satisfied, there exists ρ > 0 and (β, γ ) ∈ A(w; α) such that (3.3) holds. Invoking (H1), the argument employed in the proof of Lemma 3.3 applies mutatis mutandis to yield that, for every b ∈ (α, β],  F(u) − F(v)

L p [α,b]

≤ρ

α

b

u  (s) − v  (s) ds

∀ u, v ∈ W(w; α, b, γ ).

Hence, setting q := p/( p − 1) ∈ [1, ∞), it follows from H¨older’s inequality that, for every b ∈ (α, β] and all u, v ∈ W(w; α, b, γ ), F(u) − F(v) L 1 [α,b] ≤ (b − α)1/q F(u) − F(v) L p [α,b]  b ≤ ρ(b − α)1/q u  (s) − v  (s) ds. α

Choosing b ∈ (α, β] sufficiently close to α, so that ρ(b − α)1/q < 1, and noting that, by Lemma 3.2, (b, γ ) ∈ A(w; α), we conclude that (H2) holds with λ = ρ(b − α)1/q . Proof of Lemma 4.1. To show that there exists γ > 0 such that (4.3) holds, recall that α ∈ [0, T ), w ∈ W [−h, α], I ⊂ R+ is a relatively open interval containing α, and B ⊂ C[−h, 0] is an open ball containing wα . Without loss of generality we may assume that B is centered at wα . Let ρ denote the radius of B. By Lemma 3.2, the set A(w; α) is nonempty. Let (β ∗ , γ ∗ ) ∈ A(w; α) and define w˜ ∈ C[−h, β ∗ ] by  w(t), t ∈ [−h, α] w(t) ˜ := w(α), t ∈ (α, β ∗ ]. By uniform continuity of w, there exists δ > 0 such that s, t ∈ [−h, α], June–July 2010]

|s − t| < δ ⇒ w(s) − w(t) <

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

ρ . 2 507

Now choose γ > 0 sufficiently small so that γ < min{β ∗ − α, γ ∗ , δ, ρ/2},

 [α, α + γ ] ⊂ I,

λ :=

α

α+γ

l(t) dt < 1

and define β := α + γ . By Lemma 3.2, (β, γ ) ∈ A(w; α). Let t ∈ [α, β] and v ∈ W(w; α, β, γ ) be arbitrary. Observe that s ∈ [−h, 0],

t + s ≤ α ⇒ |(α + s) − (t + s)| = t − α ≤ β − α = γ < δ

and so s ∈ [−h, 0], t + s ≤ α ⇒ w(α + s) − v(t + s) = w(α + s) − w(t + s) <

ρ . 2

Furthermore, s ∈ [−h, 0],

t + s > α ⇒ |(α + s) − α| = |s| < t − α ≤ β − α = γ < δ

and so, for s ∈ [−h, 0] such that t + s > α it follows that w(α + s) − v(t + s) ≤ w(α + s) − w(t ˜ + s) + w(t ˜ + s) − v(t + s)  t+s



= w(α + s) − w(α) + w(α) − v(α) − v  (σ ) dσ

<

ρ + 2

 α

α

β

v  (σ ) dσ ≤

ρ ρ ρ + γ < + = ρ. 2 2 2

Thus, s ∈ [−h, 0], t + s > α ⇒ w(α + s) − v(t + s) < ρ. We have now shown that w(α + s) − v(t + s) < ρ

∀ v ∈ W(w; α, β, γ ), ∀ s ∈ [−h, 0], ∀ t ∈ [α, β].

Consequently, vt ∈ B for all v ∈ W(w; α, α + γ , γ ) and for all t ∈ [α, α + γ ], completing the proof. To facilitate the proofs of Propositions 5.1 and 6.1, we state and prove the following lemma. Lemma 7.1. Let G ⊂ R+ × R N be a relatively open set and let 1 ≤ q ≤ ∞. Assume that g : G → R M is such that, for every (t, z) ∈ G, there exist a relatively open interval I ∈ R+ containing t, an open ball B ⊂ R N containing z, and a function l ∈ L q (I ) such that I × B ⊂ G, g(s, x) − g(s, y) ≤ l(s)x − y

∀ s ∈ I, ∀ x, y ∈ B,

(7.1)

and the function I → R M , s  → g(s, x) is measurable for all x ∈ B and is in L q (I ) for some x ∈ B. Then the following statements hold. (1) If I ⊂ R+ is an interval and v ∈ C(I ) is such that gr v ⊂ G, then the function I → R M , s  → g(s, v(s)) is measurable. 508

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(2) For every (t, z) ∈ G, there exist a relatively open interval I ∈ R+ containing t, an open ball B ⊂ R N containing z, and a function b ∈ L q (I ) such that I × B ⊂ G and g(s, x) ≤ b(s) for all (s, x) ∈ I × B. (3) If I ⊂ R+ is an interval and v ∈ C(I ) is such that gr v is a compact subset of G, then the function I → R M , s  → g(s, v(s)) is in L q (I ). Proof. (1) It is sufficient to show that, for every compact subinterval K ⊂ I , the function K → R M , s  → g(s, v(s)) is measurable. To this end, let K ⊂ I be a compact subinterval. Then gr(v| K ) is compact and contained in G. It follows from the hypothesis that there exist finitely many relatively open intervals I1 , . . . , Im in R+ and m open balls B1 , . . . , Bm in R N such that gr(v| K ) ⊂ ∪i=1 Ii × Bi and, for i = 1, . . . , m, M v(Ii ) ⊂ Bi , Ii × Bi ⊂ G, and the function Ii → R , s  → g(s, x) is measurable for m every x ∈ Bi . Setting K i := Ii ∩ K it follows that K = ∪i=1 K i . Let i ∈ {1, . . . , m} be arbitrary, let J1 , . . . , Jn be any finite disjoint family of subintervals of K i such that K i = ∪nj =1 J j , and let x1 , . . . , xn ∈ Bi . Since K i → R M , s  → g(s, x j ) is measurable for each j, it follows that the function K i → RM , s →

n 

g(s, x j )I J j (s)

(7.2)

j =1

is measurable, where I J j denotes the characteristic function of J j . Since, for fixed s ∈ K i , the function Bi → R M , x  → g(s, x) is continuous and v| K i is continuous with v(K i ) ⊂ Bi , it follows that the function K i → R M , s  → g(s, v(s)) is the pointwise limit of functions of the form (7.2) and hence is measurable. Since i ∈ {1, . . . , m} is m arbitrary and K = ∪i=1 K i , the function K → R M , s  → g(s, v(s)) is measurable. (2) Let (t, z) ∈ G. Then there exist a relatively open interval I ⊂ R+ containing t, an open ball B ⊂ R N containing z, and l ∈ L q (I ) such that I × B ⊂ G, (7.1) holds, and, moreover, the function I → R M , s  → g(s, y) is in L q (I ) for some y ∈ B. Therefore, g(s, x) ≤ g(s, x) − g(s, y) + g(s, y) ≤ l(s)x − y + g(s, y) ≤ l(s) sup x − y + g(s, y) =: b(s) ∀ s ∈ I, ∀ x ∈ B. x∈B

Since l ∈ L q (I ) and g(·, y) ∈ L q (I ), it follows that b ∈ L q (I ). (3) By compactness of gr v and statement (2), there exist finitely many relatively open intervals I1 , . . . , Im in R+ , open balls B1 , . . . , Bm in R N , and functions bi ∈ m L q (Ii ), i = 1, . . . , m, such that Ii × Bi ⊂ G, i = 1, . . . , m, gr v ⊂ ∪i=1 Ii × Bi , and g(s, x) ≤ bi (s) for all (s, x) ∈ Ii × Bi , i = 1, . . . , m. Defining b˜i : I → R+ by  ˜bi (s) := bi (s), s ∈ Ii ∩ I 0, s ∈ I \Ii ,   it follows that g(s, x) ≤ i b˜i (s) for all (s, x) ∈ gr v. Therefore, since i b˜i ∈ L q (I ) and since the function s  → g(s, v(s)) is measurable (by statement (1)), the result follows. Proof of Proposition 5.1. We make the following connections with the notation of Section 4: for h = 0, we set C[−h, 0] = R N and  = D = G. Then the initialvalue problems (4.1) and (5.1) coincide. Furthermore note that, for all I ∈ I, I+ = I . June–July 2010]

VOLTERRA FUNCTIONAL DIFFERENTIAL EQUATIONS

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By (ODE), we see that (RDE1) holds (with h = 0). It remains only to show that (RDE2) also holds (with h = 0). Let I ∈ I and v ∈ WG (I ) be such that gr v is a compact subset of G. In view of (ODE), we see that the hypotheses of Lemma 7.1 hold with M = N , q = 1, and g = f . By assertion (3) of Lemma 7.1, we may infer that I → R N , t  → f (t, v(t)) is in L 1 (I ). Therefore, (RDE2) holds. Proof of Proposition 6.1. In view of (IDE), we see that the hypotheses of Lemma 7.1 hold. Let I ∈ I and v ∈ W (I ). Invoking statement (3) of Lemma 7.1, we conclude q that the function s  → g(s, v(s)) is in L loc (I ). It follows now from a routine argument based on Fubini’s theorem (see, for example, [7, 11]) that F(v) ∈ L 1loc (I ). Similarly, under the additional assumption that gr v is compact and contained in , statement (3) of Lemma 7.1 guarantees that the function s  → g(s, v(s)) is in L q (I ) and the same routine argument based on Fubini’s theorem yields F(v) ∈ L 1 (I ), showing that (H3) is valid. Moreover, it is trivial that (H1) holds, and therefore, it only remains to show that (H2) is satisfied. To this end, let α > 0, w ∈ W [0, α], and (β0 , γ0 ) ∈ A(w; α). Then (α, w(α)) ∈  ⊂ G and, by (IDE), there exist β1 ∈ (α, β0 ], an open ball B ⊂ R N , and l ∈ L q [α, β1 ] such that w(α) ∈ B, [α, β1 ] × B ⊂  ⊂ G, and g(s, x) − g(s, y) ≤ l(s)x − y

∀ s ∈ [α, β1 ],

∀ x, y ∈ B.

Let Bγ denote the closed ball of radius γ centered at w(α). Choose β ∈ (α, β1 ] and γ ∈ (0, γ0 ] such that  λ :=

β α



t

k(s, t)l(s) ds dt < 1

Bγ ⊂ B.

and

0

Let u, v ∈ W(w; α, β, γ ). Then, u(s), v(s) ∈ Bγ ⊂ B for all s ∈ [α, β], and, moreover,  β t k(s, t)g(s, u(s)) − g(s, v(s)) ds dt F(u) − F(v) L 1 [α,β] ≤ α

 ≤

β

α



0

t

k(s, t)l(s)u(s) − v(s) ds dt 0

≤ λ sup u(s) − v(s) ≤ λ s∈[α,β]

 α

β

u  (s) − v  (s) ds,

completing the proof. ACKNOWLEDGMENT. This work was supported in part by the U.K. Engineering & Physical Sciences Research Council (EPSRC) Grant GR/S94582/01.

REFERENCES 1. 2. 3. 4. 5.

H. Amann, Ordinary Differential Equations, Walter de Gruyter, Berlin, 1990. M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. , Functional Equations with Causal Operators, Taylor & Francis, London, 2002. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer, New York, 1995. 6. R. D. Driver, Ordinary and Delay Differential Equations, Springer, New York, 1977. 7. G. B. Folland, Real Analysis, 2nd ed., John Wiley, New York, 1999.

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c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

8. G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integral Equations and Functional Equations, Cambridge University Press, Cambridge, 1990. 9. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. 10. B. Jayawardhana, H. Logemann, and E. P. Ryan, PID control of second-order systems with hysteresis, Internat. J. Control 81 (2008) 1331–1342. doi:10.1080/00207170701772479 11. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York, 1975. 12. M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Berlin, 1989. 13. X. Tan, J. S. Baras, and P. S. Krishnaprasad, Control of hysteresis in smart actuators with application to micro-positioning, Systems Control Lett. 54 (2005) 483–492. doi:10.1016/j.sysconle.2004.09. 013 14. A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. 15. W. Walter, Ordinary Differential Equations, Springer, New York, 1998. HARTMUT LOGEMANN received his Ph.D. at the University of Bremen (Germany) under the guidance of Diederich Hinrichsen. He teaches and conducts research at the University of Bath (in the southwest of England). His research interests are in mathematical systems and control theory with particular emphasis on the control of infinite-dimensional systems. His outside interests include jazz and literature. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK [email protected]

EUGENE P. RYAN received his Ph.D. from the University of Cambridge (England). He teaches and conducts research at the University of Bath, England. His research interests are in mathematical systems and control theory with particular emphasis on nonlinearity. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK [email protected]

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Finite Quantum Measure Spaces Stan Gudder

Abstract. Quantum measure spaces possess a certain “quantum weirdness” and lack some of the simplicity and intuitive nature of their classical counterparts. Much of this unusual behavior is due to a phenomenon called quantum interference, which is a recurrent theme in the present article. Because of this interference, quantum measures need not be additive but satisfy a more general condition called grade-2 additivity. Examples of quantum measure spaces such as “quantum coins” and particle-antiparticle pairs are considered. Even more general spaces called super-quantum measure spaces are discussed. You don’t need quantum mechanics or measure theory to understand this article.

1. INTRODUCTION. Measure and integration theory is a well-established field of mathematics that is over a hundred years old. The theory possesses many deep and elegant theorems and has important applications in functional analysis, probability theory, and theoretical physics. Measure theory can be applied whenever you are measuring something, whether it be length, volume, probabilities, mass, energy, etc. Although finite measure theory, in which the measure space has only a finite number of elements, is much simpler than the general theory, it also has important applications to probability theory, combinatorics, and computer science. In this article we shall discuss a generalization called finite quantum measure spaces. Just as quantum mechanics possesses a certain “quantum weirdness,” these spaces lack some of the simplicity and intuitive nature of their classical counterparts. Although there is a general theory of quantum measure spaces, we shall consider only finite spaces to keep technicalities to a minimum. Nevertheless, these finite spaces still convey the flavor of the subject and exhibit some of the unusual properties of quantum objects. Much of this unusual behavior is due to a phenomenon called quantum interference which is a recurrent theme in the present article. 2. CLASSICAL AND QUANTUM WORLDS. We first discuss finite measure theory in the classical world. Let X = {x1 , . . . , xn } be a finite nonempty set and denote the power set of X , consisting of all subsets of X , by P (X ). For A, B ∈ P (X ) we use the notation A ∪· B for A ∪ B whenever A ∩ B = ∅. Denoting the set of nonnegative real numbers by R+ , a measure on P (X ) is a map ν : P (X ) → R+ satisfying the additivity condition ν(A ∪· B) = ν(A) + ν(B)

(2.1)

for all disjoint A, B ∈ P (X ). No matter what we are measuring, the reason for (2.1) is intuitively clear. We call the pair (X, ν) a finite measure space. It immediately follows from (2.1) that ν(∅) = 0 and  ν

m  · Ai i=1

 =

m 

ν(Ai ).

(2.2)

i=1

doi:10.4169/000298910X492808

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c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

If ν : P (X ) → R satisfies (2.1) we call ν a signed measure and if ν : P (X ) → C satisfies (2.1) we call ν a complex measure. For all types of measures we use the shorthand notation ν(xi ) = ν ({xi }). Denoting the complement of a set A by A , since A = (A ∩ B) ∪· (A ∩ B  ), we have that ν(A ∩ B  ) = ν(A) − ν(A ∩ B) for all of the previous types of measure. Also, A ∪ B = (A ∩ B  ) ∪· (A ∩ B) ∪· (B ∩ A ), so we obtain the inclusion-exclusion formula ν(A ∪ B) = ν(A) + ν(B) − ν(A ∩ B).

(2.3)

A probability measure is a measure ν that satisfies ν(X ) = 1. In this case, the elements xi ∈ S are interpreted as sample points or elementary events and the sets A ∈ P (X ) are interpreted as events. Then ν(A) is the probability that the event A occurs. For example, suppose we flip a fair coin twice. Denoting heads and tails by H and T , respectively, the sample space becomes X = {H H, H T, T H, T T } . The probability measure ν satisfies ν(H H ) = ν(H T ) = ν(T H ) = ν(T T ) = 1/4. The event that at least one head occurs is given by A = {H H, H T, T H } and it follows from (2.2) that ν(A) = 3/4. We conclude from (2.2) that a measure ν is determined by its values ν(xi ), i = 1, . . . , n. In fact, we have ν



xi1 , . . . , xim



=

m 

ν(xi j ).

j =1

Conversely, given any nonnegative numbers pi , i = 1, . . . , n, we obtain a measure ν given by  { pi : xi ∈ A} . ν(A) = This same observation applies to signed and complex measures. Up to this point we have been discussing the classical world, in which life is simple and intuitive. But along comes quantum mechanics and our desire to explain it mathematically. It turns out that quantum measures need not satisfy additivity (2.1) and are therefore not really measures. But if (2.1) is so intuitively clear, how can it not hold in a physical theory like quantum mechanics? The reason is because of a phenomenon called quantum interference. If the points of X represent quantum objects, they can interfere with each other both constructively and destructively. For example, suppose x1 and x2 represent subatomic particles and μ is a measure of mass. Then we could have μ(x1 ) > 0 and μ(x2 ) > 0, but x1 and x2 could be a particle-antiparticle pair that annihilate each other producing pure energy (fission). Taken together we would June–July 2010]

FINITE QUANTUM MEASURE SPACES

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have μ ({x1 , x2 }) = 0. Hence, μ ({x1 , x2 }) = μ(x1 ) + μ(x2 ) and additivity fails. On the other hand, two particles colliding at high kinetic energy can convert some of this energy to mass and combine (fusion) to form a single particle, in which case μ ({x1 , x2 }) > μ(x1 ) + μ(x2 ). For another example, suppose a beam of subatomic particles such as electrons or photons impinges on a screen containing two closely spaced narrow slits. The particles that pass through the slits hit a black target screen and produce small white dots at their points of absorption. It is well known experimentally that this results in a diffraction pattern consisting of many light and dark strips. Why do the particles accumulate in the light regions and not in the dark regions? It seems as though the particles communicate with each other to conspire to land along the white strips. Let X = {x1 , . . . , xn } represent this set of particles and let R be a region of the target screen. For A ⊆ X , let μ(A) measure the propensity for the particles in A to hit a point in R. Now it can happen that μ(x1 ), μ(x2 ) > 0 and μ ({x1 , x2 }) = 0. More generally, we can have μ ({x1 , x2 }) > μ(x1 ) + μ(x) or μ ({x1 , x2 }) < μ(x1 ) + μ(x2 ). We then say that the particles interfere constructively or destructively, respectively. In a deeper analysis, the points of X represent particle paths, and it is the paths that interfere. In this case, a single particle results in two possible paths, one through each of the slits, and these paths interfere. The standard explanation for this phenomenon is called wave-particle duality. The diffraction pattern is easily explained for waves. Two waves interfere constructively if they combine with crests close together and destructively if they combine with a crest close to a trough. In wave-particle duality, an unobserved subatomic particle behaves like a wave. When the wave impinges upon the first screen, it divides into two subwaves each going through one of the two slits. These subwaves combine, interfere, and then hit the target screen. It is then observed (as a small white dot) at which point it acts like a particle. Whether you like this explanation or not (and some don’t), the mathematics of quantum mechanics accurately describes the diffraction pattern. 3. QUANTUM MEASURES. We have seen in Section 2 that quantum measures need not be additive. To find the properties that they do possess, we examine some of the mathematics of quantum mechanics. Let X = {x1 , . . . , xn } be a set of quantum objects. In various quantum formalisms an important role is played by a decoherence function D : P (X ) × P (X ) → C [3, 4]. This function (or at least its real part) represents the amount of interference between pairs of subsets of X and has the following properties: D (A ∪· B, C) = D(A, C) + D(B, C);

(3.1)

D(A, B) = D(B, A);

(3.2)

D(A, A) ≥ 0;

(3.3)

|D(A, B)| ≤ D(A, A)D(B, B). 2

(3.4)

In (3.2), the bar is complex conjugation, and (3.1) and (3.2) imply that D is additive in one of the arguments when the other argument is fixed. An example of a quantum measure is defined by μ(A) = D(A, A), and μ is a measure of the inference of A with itself. A simple example of a decoherence function is D(A, B) = ν(A)ν(B), where ν is a complex measure on P (X ). In this case ν is called an amplitude (which comes from the analogy with waves) and we have μ(A) = |ν(A)|2 . In fact, quantum probabilities are frequently computed by taking the modulus squared of a complex 514

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

amplitude. This example illustrates the nonadditivity of μ because

 μ(A ∪· B) = |ν(A ∪· B)|2 = |ν(A) + ν(B)|2 = μ(A) + μ(B) + 2Re ν(A)ν(B) .

 Hence, μ(A ∪· B) = μ(A) + μ(B) if and only if Re ν(A)ν(B) = 0. In this case we say that A and B do not interfere or A and B are compatible. Theorem 3.1. Let D : P (X ) × P (X ) → C be a decoherence function and define μ(A) = D(A, A). Then μ : P (X ) → R+ has the following properties: μ(A ∪· B ∪· C) = μ(A ∪· B) + μ(A ∪· C) + μ(B ∪· C) − μ(A) − μ(B) − μ(C);

(3.5)

if μ(A) = 0, then μ(A ∪· B) = μ(B) for all B ∈ P (X ) with B ∩ A = ∅; if μ(A ∪· B) = 0, then μ(A) = μ(B).

(3.6) (3.7)

Proof. To prove (3.5), let R be the right side of (3.5) and apply (3.1) and (3.2) to obtain R = D(A ∪· B, A ∪· B) + D(A ∪· C, A ∪· C) + D(B ∪· C, B ∪· C) − μ(A) − μ(B) − μ(C) = 2[D(A, A)+ D(B, B)+ D(C, C)+ReD(A, B)+ReD(A, C)+ReD(B, C)] − μ(A) − μ(B) − μ(C) = D(A, A) + D(B, B) + D(C, C) + 2 [ReD(A, B) + ReD(A, C) + ReD(B, C)] = D(A ∪· B ∪· C, A ∪· B ∪· C) = μ(A ∪· B ∪· C). To prove (3.6), apply (3.1) and (3.2) to obtain μ(A ∪· B) = D(A ∪· B, A ∪· B) = μ(A) + μ(B) + 2ReD(A, B). By (3.4) if μ(A) = 0, then D(A, B) = 0 so that μ(A ∪· B) = μ(B). To prove (3.7), applying (3.1)–(3.4) we have μ(A ∪· B) = μ(A) + μ(B) + 2ReD(A, B) ≥ μ(A) + μ(B) − 2 |D(A, B)|

2 ≥ μ(A) + μ(B) − 2μ(A)1/2 μ(B)1/2 = μ(A)1/2 − μ(B)1/2 . Hence, μ(A ∪· B) = 0 implies that μ(A) = μ(B). Condition (3.5) is a generalized additivity that we call grade-2 additivity. The usual additivity (2.1) is called grade-1 additivity. Of course, grade-1 additivity implies grade-2 additivity, but the converse does not hold. Conditions (3.6) and (3.7) do not follow from (3.5), and a map satisfying (3.6) and (3.7) is called regular. A grade-2 additive map μ : P (X ) → R+ is a grade-2 measure, and a regular grade-2 measure is a quantum measure (or q-measure, for short) [6, 7]. If μ is a q-measure we call (X, μ) a q-measure space. We have seen that if μ(A) = D(A, A) for a decoherence June–July 2010]

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function D, then μ is a q-measure. We will later exhibit more general q-measures that do not have this form. Simple examples of q-measures are μ(A) = ν(A)2 , where ν is a signed measure. It follows from (3.5) that any q-measure μ satisfies μ(∅) = 0. Example 1. Let (X, ν) be the probability space of our fair coin example. But now we have a quantum coin with q-measure μ(A) = ν(A)2 . Then the sample points have “quantum probability” 1/16 and the certain event X has “quantum probability” 1 as it should. The event A that at least one head appears has “quantum probability” 9/16. Example 2. Let X = {x1 , x2 } and define μ(x1 ) = μ(x2 ) = 1, μ(∅) = 0, and μ(X ) = 6. Then (X, μ) is a q-measure space, but μ does not have the form μ(A) = D(A, A) for a decoherence function D. Indeed, if such a D existed we would have 2D(x1 , x2 ) + D(x1 , x1 ) + D(x2 , x2 ) = D(X, X ) = μ(X ) = 6. Hence, D(x1 , x2 ) = 2. But then (3.4) is not satisfied, which is a contradiction. Example 3. Let X = {x1 , x2 , x3 } with μ(∅) = μ(x1 ) = 0 and μ(A) = 1 for all other A ∈ P (X ). Then (X, μ) is a q-measure space. Example 4. Let X = {x1 , . . . , xm , y1 , . . . , ym , z 1 , . . . , z n } and call (xi , yi ), i = 1, . . . , m, destructive pairs (or particle-antiparticle pairs). Denoting the cardinality of a set B by |B|, we define μ(A) = |A| − 2 |{(xi , yi ) : xi , yi ∈ A}|

(3.8)

for every A ∈ P (X ). For instance μ ({x1 , y1 , z 1 }) = 1 and μ ({x1 , y1 , y2 , z 1 }) = 2. We now check that μ is a q-measure on X . If μ(A) = 0, then A = ∅ or A has the form   A = xi1 , yi1 , . . . , xi j , yi j . If in addition B ∈ P (X ) with A ∩ B = ∅, then μ(A ∪· B) = |A| + |B| − 2 |{(xi , yi ) : xi , yi ∈ A}| − 2 |{(xi , yi ) : xi , yi ∈ B}| = |B| − 2 |{(xi , yi ) : xi , yi ∈ B}| = μ(B). Hence, (3.6) holds. To show that (3.7) holds, suppose μ(A ∪· B) = 0. Then z i ∈ A ∪· B, i = 1, . . . , n. If xi ∈ A then yi ∈ A ∪· B, and if yi ∈ A then xi ∈ A ∪· B. Hence, μ(A) = |{xi ∈ A : yi ∈ B}| + |{yi ∈ A : xi ∈ B}| = |{yi ∈ B : xi ∈ A}| + |{xi ∈ B : yi ∈ A}| = μ(B). We conclude that μ is regular. To prove grade-2 additivity (3.5), let A1 , A2 , A3 ∈ P (X ) be mutually disjoint. If xi ∈ Ar and yi ∈ As , r, s = 1, 2, 3, we call (xi , yi ) an r s-pair. We then have μ(A1 ∪· A2 ) + μ(A1 ∪· A3 ) + μ(A2 ∪· A3 ) − μ(A1 ) − μ(A2 ) − μ(A3 ) = |A1 | + |A2 | − 2 |{r s-pairs, r, s = 1, 2}| + |A1 | + |A3 | − 2 |{r s-pairs, r, s = 1, 3}| + |A2 | + |A3 | − 2 |{r s-pairs, r, s = 2, 3}| − |A1 | + 2 |{11-pairs}| − |A2 | + 2 |{22-pairs}| − |A3 | + 2 |{33-pairs}| 516

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

= |A1 ∪· A2 ∪· A3 | − 2 |{r s-pairs, r, s = 1, 2, 3}| = μ(A1 ∪· A2 ∪· A3 ). We conclude that (X, μ) is a q-measure space. The next result shows that grade-2 additivity is equivalent to a generalization of (2.3). The symmetric difference of A and B is A B = (A ∩ B  ) ∪ (A ∩ B). Theorem 3.2. A map μ : P (X ) → R+ is grade-2 additive if and only if μ satisfies μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B) + μ(A B) − μ(A ∩ B  ) − μ(A ∩ B). (3.9) Proof. If μ is grade-2 additive, we have

μ(A ∪ B) = μ (A ∩ B  ) ∪· (A ∩ B) ∪· (A ∩ B) = μ(A B) + μ(A) + μ(B) − μ(A ∩ B  ) − μ(A ∩ B) − μ(A ∩ B), which is (3.9). Conversely, if (3.9) holds, then letting A1 = A ∪· C, B1 = B ∪· C we have μ(A ∪· B ∪· C) = μ(A1 ∪ B1 ) = μ(A1 ) + μ(B1 ) − μ(A1 ∩ B1 ) + μ(A1 B1 ) − μ(A1 ∩ B1 ) − μ(A1 ∩ B1 ) = μ(A ∪· C) + μ(B ∪· C) − μ(C) + μ(A ∪· B) − μ(A) − μ(B), which is grade-2 additivity. We now show that grade-2 additivity can be extended to more than three mutually disjoint sets [5]. Theorem 3.3. If μ : P (X ) → R+ is grade-2 additive, then for any m ≥ 3 we have   m m m    · Ai = μ μ(Ai ∪· A j ) − (m − 2) μ(Ai ). (3.10) i=1

i< j =1

i=1

Proof. We prove the result by induction on m The result holds for m = 3. Assuming the result holds for m − 1 ≥ 3 we have   m  · Ai = μ [A1 ∪· · · · ∪· (Am−1 ∪· Am )] μ i=1

=

m−2 

μ(Ai ∪· A j ) +

i< j =1



− (m − 3)

m−2 

μ [Ai ∪· (Am−1 ∪· Am )]

i=1 m−2 



μ(Ai ) + μ(Am−1 ∪· Am )

i=1

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FINITE QUANTUM MEASURE SPACES

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=

m−2 

μ(Ai ∪· A j ) +

i< j =1

m−2 

μ(Ai ∪· Am−1 ) +

i=1

m−2 

μ(Ai ∪· Am )

i=1 m−2 

+ (m − 2)μ(Am−1 ∪· Am ) −

μ(Ai ) − (m − 2)μ(Am−1 )

i=1

− (m − 2)μ(Am ) − (m − 3)

 m−2 

 μ(Ai ) + μ(Am−1 ∪· Am )

i=1

=

m 

μ(Ai ∪· A j ) − (m − 2)

i< j =1

m 

μ(Ai ).

i=1

The result follows by induction. Notice that Theorem 3.3 also holds for signed and complex grade-2 additive measures. 4. QUANTUM INTERFERENCE. Unlike a measure on P (X ), a q-measure μ is not determined by its values on singleton sets. However, by Theorem 3.3, μ is determined by its values on singleton and doubleton sets. Thus, if pi ≥ 0 and qi j ≥ 0, i, j = 1, . . . , n, satisfy qi j = q j i and 

qi j − (|A| − 2)

i, j ∈A i< j



pi ≥ 0

i∈A

for A ⊆ {1, . . . , n} with |A| ≥ 3, then  there exists  a unique q-measure μ on X = {x1 , . . . , xn } such that μ(xi ) = pi and μ xi , x j = qi j , i, j  = 1, . . . , n. Conversely, given a q-measure μ on X , then pi = μ(xi ), qi j = μ xi , x j have these properties. We now introduce a physically relevant parameter called quantum interference that can also be used to determine a q-measure. For a q-measure μ on X = {x1 , . . . , xn } we define the quantum interference function Iμ : X × X → R by Iμ (xi , x j ) = μ



xi , x j



− μ(xi ) − μ(x j )

if i = j and Iμ (xi , xi ) = 0, i, j = 1, . . ., n. The function Iμ gives the deviation of μ from being a measure on the sets xi , x j and hence is an indicator of the interference between xi and x j . Notice that Iμ can have positive or negative values. For instance, in Example 3, Iμ (x2 , x3 ) = −1, while in Example 2, Iμ (x1 , x2 ) = 4. By Theorem 3.3, μ is determined by the numbers μ(xi ) and Iμ (xi , x j ), i, j = 1, . . . , n. We extend Iμ to a signed measure λμ on P (X × X ) by defining λμ (B) =



 Iμ (xi , x j ) : (xi , x j ) ∈ B .

Since Iμ (xi , x j ) = Iμ (x j , xi ) it follows that λμ is symmetric in the sense that λμ (A × B) = λμ (B × A) for all A, B ∈ P (X ). The classical part of μ is defined to be the unique measure νμ on P (X ) that satisfies νμ (xi ) = μ(xi ), i = 1, . . . , n. The next result shows that we can always decompose μ into its classical part and its interference part. 518

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Theorem 4.1. If μ is a q-measure on X = {x1 , . . . , xn }, then for any A ∈ P (X ) we have 1 μ(A) = νμ (A) + λμ (A × A). 2

(4.1)

Proof. We first prove that δ(A) = λμ (A × A) is a grade-2 signed measure on P (X ). To show this we compute δ(A ∪· B) + δ(A ∪· C) + δ(B ∪· C) − δ(A) − δ(B) − δ(C) = λμ (A ∪· B × A ∪· B) + λμ (A ∪· C × A ∪· C) + λμ (B ∪· C × B ∪· C) − λμ (A × A) − λμ (B × B) − λμ (C × C) = λμ (A × A) + 2λμ (A × B) + λμ (B × B) + λμ (A × A) + 2λμ (A × C) + λμ (C × C) + λμ (B × B) + 2λμ (B × C) + λμ (C × C) − λμ (A × A) − λμ (B × B) − λμ (C × C) = λμ (A × A) + λμ (B × B) + λμ (C × C)

+ 2 λμ (A × B) + λμ (A × C) + λμ (B × C) = λμ (A ∪· B ∪· C × A ∪· B ∪· C) = δ(A ∪· B ∪· C). Hence, νμ (A) + 12 λμ (A × A) is a grade-2 signed measure. Now 1 1 νμ (xi ) + λμ ({xi } × {xi }) = νμ (xi ) + Iμ (xi , xi ) 2 2 = νμ (xi ) = μ(xi ), and for i = j we have νμ



   1  + λμ x i , x j × x i , x j 2 = νμ (xi ) + νμ (x j ) + Iμ (xi , x j )   = μ(xi ) + μ(x j ) + μ xi , x j − μ(xi ) − μ(x j )   = μ xi , x j .

xi , x j



Since μ and A  → νμ (A) + 12 (A × A) are both grade-2 signed measures that agree on singleton and doubleton sets, it follows from Theorem 3.3 that they coincide. Notice that (4.1) can be written μ(A) = νμ (A) +

 1  Iμ (xi , x j ) : xi , x j ∈ A . 2

(4.2)

We shall now illustrate (4.2) in some examples. In Example 1 we have that νμ (xi ) = 1/16 and Iμ (xi , x j ) = 1/8 for i = j. By (4.2) we have for all A ∈ P (X ) that μ(A) = June–July 2010]

1 1 1 |A| + |A| (|A| − 1) = |A|2 . 16 16 16

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In Example 4 we have νμ (xi ) = 1, Iμ (xi , yi ) = Iμ (yi , xi ) = −2, and Iμ vanishes for all other pairs. Hence, (4.2) agrees with (3.8). We can use (4.2) to construct q-measures. For example, letting ν(xi ) = 0 for all i and I (xi , x j ) = 1 for i = j we conclude from (4.2) that   |A| 1 μ(A) = = |A| (|A| − 1) . 2 2 For another example, let X = {x1 , . . . , x2n+1 }, ν(xi ) = n for all i, and I (xi , x j ) = −1 for all i = j. Applying (4.2) gives   |A| 1 μ(A) = n |A| − = |A| (|X | − |A|) . 2 2 5. COMPATIBILITY AND THE CENTER. Let (X, μ) be a quantum measure space. We say that A, B ∈ P (X ) are μ-compatible and write AμB if μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B). Recalling (2.3) we see that μ acts like a measure on A ∪ B, so in some weak sense A and B do not interfere with each other. For example, {x} and {y} are μ-compatible if and only if Iμ (x, y) = 0. This analogy is not completely accurate because AμA for all A ∈ P (X ) and certainly points of A can interfere with each other. It follows from (3.9) that AμB if and only if μ(A B) = μ(A ∩ B  ) + μ(A ∩ B).

(5.1)

The μ-center of P (X ) is Z μ = {A ∈ P (X ) : AμB for all B ∈ P (X )} . The elements of Z μ are called macroscopic sets because they behave like large objects at the human scale [7]. Lemma 5.1. (i) (ii) (iii) (iv)

If A ⊆ B, then AμB. If AμB, then A μB  . ∅, X ∈ Z μ . If A ∈ Z μ , then A ∈ Z μ .

Proof. (i) If A ⊆ B, then μ(A ∪ B) = μ(B) = μ(A) + μ(B) − μ(A ∩ B). Hence, AμB. (ii) If AμB, then by (5.1) μ(A B  ) = μ(A B) = μ(A ∩ B  ) + μ(A ∩ B)



= μ (A ) ∩ B  + μ A ∩ (B  ) . Hence, by (5.1), A μB  . (iii) follows from (i), and (iv) follows from (ii). 520

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A set A ∈ P (X ) is μ-splitting if μ(B) = μ(B ∩ A) + μ(B ∩ A ) for all B ∈ P (X ). Lemma 5.2. A is μ-splitting if and only if A ∈ Z μ . Proof. Suppose A is μ-splitting. Then for every B ∈ P (X ) we have

μ(A ∪ B) = μ [(A ∪ B) ∩ A] + μ (A ∪ B) ∩ A = μ(A) + μ(B ∩ A ) = μ(A) + μ(B) − μ(A ∩ B). Hence, A ∈ Z μ . Conversely, suppose A ∈ Z μ . Then for every B ∈ P (X ) we have

μ(A ∪ B) = μ A ∪· (B ∩ A ) = μ(A) + μ(B ∩ A ). Thus, μ(B) = μ(A ∪ B) − μ(A) + μ(A ∩ B) = μ(B ∩ A) + μ(B ∩ A ), so A is μ-splitting. A Boolean subalgebra of P (X ) is a collection of sets A ⊆ P (X ) such that X ∈ A, A ∈ A implies A ∈ A, and A, B ∈ A implies A ∪ B ∈ A. A measure on A is defined just as it was on P (X ). Theorem 5.3. Z μ is a Boolean subalgebra of P (X ) and the restriction μ | Z μ of μ to Z μ is a measure. Moreover, if Ai ∈ Z μ are mutually disjoint, then for every B ∈ P (X ) we have  μ [∪· (B ∩ Ai )] = μ(B ∩ Ai ). Proof. By Lemma 5.1, X ∈ Z μ , and A ∈ Z μ whenever A ∈ Z μ . Now suppose A, B ∈ Z μ and C ∈ P (X ). Since A is μ-splitting we have

μ [C ∩ (A ∪ B)] = μ [(C ∩ A) ∩ (A ∪ B)] + μ (C ∩ A ) ∩ (A ∪ B) = μ(C ∩ A) + μ(C ∩ A ∩ B). Hence, since B is μ-splitting we conclude that μ(C) = μ(C ∩ A) + μ(C ∩ A ) = μ(C ∩ A) + μ(C ∩ A ∩ B) + μ(C ∩ A ∩ B  )

= μ [C ∩ (A ∪ B)] + μ C ∩ (A ∪ B) . It follows that A ∪ B is μ-splitting, so A ∪ B ∈ Z μ . Hence, Z μ is a Boolean subalgebra of P (X ). Moreover, μ | Z μ is a measure because if A, B ∈ Z μ with A ∩ B = ∅, since AμB we have μ(A ∪· B) = μ(A) + μ(B). To prove the last statement, let Ai ∈ Z μ be r mutually disjoint, i = 1, . . . , m, and let Sr = ∪· i=1 Ai , r ≤ m. We prove by induction on r that for B ∈ P (X ) we have μ(B ∩ Sr ) =

r 

μ(B ∩ Ai ).

i=1

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The case r = 1 is obvious. Suppose the result is true for r < m. Since Sr ∈ Z μ we have μ(B ∩ Sr +1 ) = μ(B ∩ Sr +1 ∩ Sr ) + μ(B ∩ Sr +1 ∩ Sr ) = μ(B ∩ Sr ) + μ(B ∩ Ar +1 ) =

r 

μ(B ∩ Ai ) + μ(B ∩ Ar +1 ) =

i=1

r +1 

μ(B ∩ Ai ).

i=1

By induction, the result holds for r = m, so that 

 m m   μ · (B ∩ Ai ) = μ(B ∩ Sm ) = μ(B ∩ Ai ). i=1

i=1

We now illustrate these ideas in Example 4. All the results in the rest of this section apply to the quantum measure space (X, μ) of Example 4. Theorem 5.4. AμB if and only if xi ∈ A ∩ B  implies that yi ∈ B ∩ A and yi ∈ A ∩ B  implies that xi ∈ B ∩ A . Proof. The condition is equivalent to the following: if {xi , yi } ⊆ A B, then {xi , yi } ⊆ A ∩ B  or {xi , yi } ⊆ B ∩ A . Suppose the condition holds. We may assume without loss of generality that {x1 , y1 , . . . , xr , yr } ⊆ A ∩ B  , {xr +1 , yr +1 , . . . , xs , ys } ⊆ B ∩ A , and there are no other destructive pairs in A B. Then     μ(A B) = |A B| − 2s =  A ∩ B   − 2r +  B ∩ A  − 2(s − r ) = μ(A ∩ B  ) + μ(B ∩ A ). By Theorem 3.2, AμB. Conversely, suppose AμB. Again, without loss of generality we can assume that {x1 , y1 , . . . , xr , yr } are all the destructive pairs in A ∩ B  and {xr +1 , yr +1 , . . . , xs , ys } are all the destructive pairs in B ∩ A . Assume that S = {xs+1 , ys+1 , . . . , xt , yt } ⊆ A B. Then |A B| − 2t = μ(A B) = μ(A ∩ B  ) + μ(B ∩ A )     =  A ∩ B   − 2r +  B ∩ A  − 2(s − r ). It follows that t = s so that S = ∅. Hence, all the destructive pairs in A B are in A ∩ B  or B ∩ A . Corollary 5.5. A ∈ Z μ if and only if for i = 1, . . . , m, either {xi , yi } ⊆ A or {xi , yi } ⊆ A . 522

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Proof. If A ∈ Z μ , then AμA . By Theorem 5.4, if xi ∈ A then yi ∈ A and hence yi ∈ A, and similarly if yi ∈ A then xi ∈ A. Conversely, suppose the condition holds and B ∈ P (X ). Then μ(B ∩ A ) + μ(B ∩ A) = |B ∩ A| − 2 |{(xi , yi ) : {xi , yi } ⊆ B ∩ A}|     +  B ∩ A  − 2  (xi , yi ) : {xi , yi } ⊆ B ∩ A  = |B| − 2 |{(xi , yi ) : {xi , yi } ⊆ B}| = μ(B). By Lemma 5.2, A ∈ Z μ . Corollary 5.6. The following statements are equivalent: (i) AμA . (ii) A ∈ Z μ . (iii) μ(X ) = μ(A) + μ(A ). Proof. (i) ⇒ (ii) follows from Theorem 5.4 and Corollary 5.5. (ii) ⇒ (iii) ⇒ (i) is trivial. It follows from Theorem 5.3 that μ | Z μ is a measure. In fact, by Corollary 5.5 we have for every B ∈ Z μ that μ(B) = |{z i : z i ∈ B}|, and this is clearly a measure. 6. QUANTUM COVERS. A q-measure μ on X = {x1 , . . . , xn } is called a qprobability if μ(X ) = 1. Of course, a q-probability would give a very strange probability because it need not be additive and we could have μ(A) > 1 for some A ∈ P (X ). Nevertheless, q-probabilities have been studied and have been useful for certain applications. If μ is a q-measure on X for which μ(X ) = 0, then μ can be “normalized” by forming the q-probability μ1 = μ/μ(X ). Another reason for wanting to know whether μ(X ) = 0 is that this would mean that “X happens.” Why can’t we just check to see whether μ(X ) = 0 or not? This may be difficult when X is a large, complicated system. For example, in some applications of this work, X represents the entire physical universe! In this case, q-measures are used to study the evolution of the universe going back to the big bang [1, 2, 6, 7]. More specifically, X is the set of possible “histories” of the universe and for A ∈ P (X ), μ(A) gives the “propensity” that the true history is an element of A; this is studied in the field of quantum gravity and cosmology. To check whether μ(X ) = 0 we could test simpler subsets of X to see if they have zero q-measure. If many of these sets have zero q-measure it would be an indication (but not a guarantee) that μ(X ) = 0. The quantum covers that we shall consider give a guarantee. A collection of sets Ai ∈ P (X ) is a cover for X if ∪Ai = X . If ν is an ordinary measure, then applying additivity we conclude that ν(X ) = 0 if and only if X does not have a cover consisting of sets with ν-measure zero. But this doesn’t work for q-measures. For example, let X = {x1 , x2 , x3 } and define μ ({x2 , x3 }) = 4 and μ(∅) = μ ({x1 , x2 }) = μ ({x1 , x3 }) = 0, μ ({x1 }) = μ ({x2 }) = μ ({x3 }) = μ(X ) = 1. It is easy to check that μ is a q-measure on X . Now the sets {x1 , x2 } and {x1 , x3 } cover X and have μ-measure zero, but μ(X ) = 0. We call a cover {Ai } for X a quantum cover if μ(Ai ) = 0 for all i implies that μ(X ) = 0 for every q-measure μ on X [8]. June–July 2010]

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Notice that a quantum cover applies to all q-measures. This is because in quantum mechanics, the q-measures correspond to physical states of the system and one frequently needs to consider many states simultaneously. A cover {Ai } for X is a partition if Ai ∩ A j = ∅ for i = j. An arbitrary partition {A1 , . . . , Am } for X is an example of a quantum cover. Indeed, let μ be a q-measure on X and suppose that μ(Ai ) = 0, i = 1, . . . , m. Then by regularity we have μ(X ) = μ(A1 ∪· · · · ∪· Am ) = μ(A2 ∪· · · · ∪· Am ) = · · · = μ(Am ) = 0. We now show that there are other types of quantum covers. A subset A ⊆ X is a k-set if |A| = k. The k-set cover for X is the collection of all k sets in X . Thus, the 1-set cover is the collection of singleton sets in X and the 2-set cover is the collection of all doubleton sets in X . The next result appears in [8] and uses a nice combinatorial argument. Theorem 6.1. The k-set cover is a quantum cover. Proof. The result is true for k = 1 since the 1-set cover is a partition. Let 2 ≤ k ≤ n and assume that every k-set has μ-measure zero. By Theorem 3.3 for any distinct i 1 , . . . , i k ∈ {1, . . . , n} we have (2 − k)

k 

μ(xi j ) +

j =1

k 

μ



x ir , x i s



=μ



xi1 , . . . , xik



= 0.

r <s=1

    k-sets and xir xis is a Adding up the nk possible k-sets, since xi j appears in n−1 k−1 n−2 subset of k−2 k-sets we have 

  n  n  n−2  n−1 μ(x j ) + (2 − k) ({xr , xs }) = 0. k − 2 r <s=1 k−1 j =1

Since (k − 2)

   n−2 (n − 1)(k − 2) n−1 = k−2 k−1 k−1

we conclude that n 

μ ({xr , xs }) =

r <s=1

n (n − 1)(k − 2)  μ(x j ). k−1 j =1

Since k ≤ n we have by Theorem 3.3 that μ(X ) = μ ({x1 , . . . , xn }) =

n 

μ ({xr , xs }) − (n − 2)

r <s=1

=

n 

μ(x j )

j =1

n k−n  μ(x j ) ≤ 0. k − 1 j =1

Hence μ(X ) = 0, so the k-set cover is a quantum cover. 524

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We now briefly consider a generalization of the k-set cover that has physical significance [8]. An antichain in P (X ) is a nonempty collection of sets {A1 , . . . , Am } in P (X ) that are incomparable. That is, Ai ⊆ A j for i = j, i, j = 1, . . . , m. An antichain {A1 , . . . , Am } is maximal if it is not contained in a strictly larger antichain. That is, for any B ∈ P (X ) either B ⊆ Ai or Ai ⊆ B for some i. Notice that a maximal antichain {A1 , . . . , Am } forms a cover for X . Indeed, for any x ∈ X there is an Ai such that {x} ⊆ Ai . Thus ∪Ai = X . The k-set cover Ck is an example of a maximal antichain. To see this, first notice that two different k-sets are incomparable, so Ck forms an antichain. To show maximality, let B ∈ P (X ). If |B| < k, then there exists an A ∈ Ck such that B ⊆ A, while if |B| ≥ k, then there exists an A ∈ Ck such that A ⊆ B. We conclude that maximal antichains generalize the k-set cover, k = 1, . . . , n. It is conjectured in [8] that every maximal antichain is a quantum cover. This is an interesting unsolved problem that the reader is invited to investigate. 7. SUPER-QUANTUM MEASURE SPACES. We say that a map μ : P (X ) → R+ is a grade-m measure if μ satisfies the grade-m additivity condition μ(A1 ∪· · · · ∪· Am+1 ) m+1 

=

m+1 

μ(Ai1 ∪· · · · ∪· Aim ) −

i 1 <···
μ(Ai1 ∪· · · · ∪· Aim−1 )

i 1 <···
+ · · · + (−1)m+1

m+1 

μ(Ai ).

(7.1)

i=1

Grade-m measures for m ≥ 3 correspond to super-quantum measures and these may describe theories that are more general than quantum mechanics. It can be shown by induction that a grade-m measure is a grade-(m + 1) measure [5]. Thus, we have a hierarchy of measure grades with each grade contained in all higher grades. Instead of giving the induction proof we will just check that any grade-2 measure μ is also a grade-3 measure. Indeed by Theorem 3.3 we have 4 

μ(Ai ∪· A j ∪· Ak ) −

i< j
4  i< j =1

=2

4 

μ(Ai ∪· A j ) − 3

i< j =1

=

μ(Ai ∪· A j ) +

4  i< j =1

4 

4 

μ(Ai )

i=1

i=1

μ(Ai ∪· A j ) − 2

4 

μ(Ai ) −

4 

μ(Ai ∪· A j ) +

i< j =1

4 

μ(Ai )

i=1

μ(Ai ) = μ(A1 ∪· A2 ∪· A3 ∪· A4 ).

i=1

Of course, if μ : P (X ) → R is grade-m additive, we call μ a grade-m signed measure. We denote the Cartesian product of a set A with itself m times by Am . As in Section 4, we say that a signed measure λ on P (X 2 ) is symmetric if λ(A1 × A2 ) = λ(A2 × A1 ) for all A1 , A2 ∈ P (X ). More generally, we say that a signed measure λ on P (X m ) is symmetric if  λ(A1 × · · · × Am ) = λ Aπ(1) × · · · × Aπ(m) for any A1 , . . . , Am ∈ P (X ) and any permutation π : {1, . . . , m} → {1, . . . , m}. The next result gives a general method of generating grade-m signed measures. The proof June–July 2010]

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for the case m = 2 is given in Theorem 4.1 and we leave the proof for general m to the reader. Theorem 7.1. If λ is a symmetric signed measure on P (X m ), then μ(A) = λ(Am ) is a grade-m signed measure on P (X ). Let μ be a grade-3 measure on P (X ). We define the classical part λ1μ = νμ and the two-point interference function Iμ2 = Iμ just as we did in Section 4. We now define the three-point interference function by         Iμ3 (xi , x j , xk ) = μ xi , x j , xk − μ xi , x j − μ xi , xk − μ x j , xk + μ(xi ) + μ(x j ) + μ(xk ) if i, j, and k are distinct, and Iμ3 = 0 otherwise. Of course, Iμ3 = 0 for quantum measures and so this term introduces a new type of phenomenon that does not seem to occur in quantum mechanics. We next define the symmetric signed measures λ2μ , λ3μ on P (X 2 ) and P (X 3 ), respectively by   Iμ2 (xi , x j ) : (xi , x j ) ∈ B , λ2μ (B) =   Iμ3 (xi , x j , xk ) : (xi , x j , xk ) ∈ B . λ3μ (B) = The next result extends Theorem 4.1. Theorem 7.2. If μ is a grade-3 measure, then for every B ∈ P (X ) we have μ(B) = λ1μ (B) +

1 2 2 1 λμ (B ) + λ3μ (B 3 ). 2! 3!

(7.2)

Proof. It follows from Theorem 7.1 that the right side of (7.2) is a grade-3 signed measure. As in Theorem 4.1, we are finished if we show that this grade-3 signed measure coincides with μ for k-sets, k = 1, 2, 3. They clearly agree for 1-sets, and they agree on 2-sets as in the proof of Theorem 4.1. Finally, if B = {x1 , x2 , x3 }, we have λ1μ (B) +

1 2 2 1 λμ (B ) + λ3μ (B 3 ) = μ(x1 ) + μ(x2 ) + μ(x3 ) + Iμ2 (x1 , x2 ) 2! 3! + Iμ2 (x1 , x3 ) + Iμ2 (x2 , x3 ) + Iμ3 (x1 , x2 , x3 ).

Expanding Iμ2 and Iμ3 in terms of their definitions, the right side becomes μ({x1 , x2 , x3 }). Of course, Theorem 7.2 can be generalized to grade-m measures to obtain μ(B) =

m  1 i i λμ (B ). i! i=1

(7.3)

In this last paragraph, let your imagination take flight and don’t worry about technicalities. If we consider the vector space V of complex-valued functions on X = {x1 , . . . , xn } with the natural inner product, then V is isomorphic to Cn . In a similar way, the vector space of complex-valued functions on X × X is isomorphic to Cn ⊗ Cn . We now form the inner product space H = Cn ⊕ (Cn ⊗ Cn ) ⊕ · · · ⊕ (Cn )⊗m , 526

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where the last summand is the m-fold tensor product. The measure λiμ in (7.3) provides a linear functional on (Cn )⊗i via its integral and hence is itself a member of (Cn )⊗i . The q-measure μ in (7.3) is given by some kind of “collapse” of these vectors. For those who know about these sorts of things, this is beginning to look like a Fock space in quantum field theory. But this is a new story that has not yet been told. In any case, the subject of quantum and super-quantum measure spaces may usher in a whole new world for mathematicians to explore. REFERENCES 1. M. Gell-Mann and J. B. Hartle, Classical equations for quantum systems, Phys. Rev. D 47 (1993) 3345– 3382. doi:10.1103/PhysRevD.47.3345 2. R. B. Griffiths, Consistent histories and the interpretation of quantum mechanics, J. Stat. Phys. 36 (1984) 219–272. doi:10.1007/BF01015734 3. S. Gudder, A histories approach to quantum mechanics, J. Math. Phys. 39 (1998) 5772–5788. doi:10. 1063/1.532592 4. O. Rudolf and J. D. Wright, Homogeneous decoherence functionals in standard and history quantum mechanics, Commun. Math. Phys. 204 (1999) 249–267. doi:10.1007/s002200050645 5. R. Salgado, Some identities for the quantum measure and its generalizations, Mod. Phys. Lett. A 17 (2002) 711–728. doi:10.1142/S0217732302007041 6. R. Sorkin, Quantum mechanics as quantum measure theory, Mod. Phys. Lett. A 9 (1994) 3119–3127. doi: 10.1142/S021773239400294X 7. , Quantum mechanics without the wave function, J. Phys. A 40 (2007) 3207–3231. doi:10.1088/ 1751-8113/40/12/S20 8. S. Surya and P. Wallden, Quantum covers in quantum measure theory (2008), available at http://arxiv. org/abs/0809.1951. STAN GUDDER received his B.S. from Washington University in 1958 and his Ph.D. from the University of Illinois at Champaign-Urbana in 1964. He taught at the University of Wisconsin–Madison from 1964 to 1969. Since 1969 he has taught at the University of Denver where he is currently the John Evans Emeritus Professor of Mathematics. He is the author of four books and over 250 research articles most of which concern the foundations of quantum mechanics. His other interests include mathematical art, skiing, primary school mathematics enrichment, and spending time with his grandchildren and the rest of his extended family. Department of Mathematics, University of Denver, Denver, CO 80208 [email protected]

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L p Norms and the Sinc Function David Borwein, Jonathan M. Borwein, and Isaac E. Leonard

Abstract. It’s everywhere! It’s everywhere! . . . In this note we give elementary proofs of some of the striking asymptotic properties of the p-norm of the ubiquitous sinc function. Based on experimental evidence we conjecture some enticing further properties of the p-norm as a function of p. See, for example, http: //www.carma.newcastle.edu.au/~jb616/oscillatory.pdf.

1. INTRODUCTION. The sinc function is a real-valued function defined on the real line R by the following expression: ⎧ sin x ⎪ ⎨ x sinc(x) = ⎪ ⎩ 1

if

x  = 0,

otherwise.

This function is important in many areas of computing science, approximation theory, and numerical analysis. For example, it is used in interpolation and approximation of functions and approximate evaluation of transforms (e.g., Hilbert, Fourier, Laplace, Hankel, and Mellon transforms as well as the fast Fourier transform). It is used in finding approximate solutions of differential and integral equations, in image processing (it is the Fourier transform of the box filter and central to the understanding of the Gibbs phenomenon [12]), and in signal processing and information theory. Much of this is nicely described in [7]. The first explicit appearance of the sinc function in approximation theory was probably in the use of the Whittaker cardinal functions C( f, h) to approximate functions analytic on an interval or on a contour. Given a function f which is defined on the real line R, the function C( f, h) is defined by ∞ 

C( f, h) =

f (kh)S(k, h)

k=−∞

whenever the series converges, where the step size h is positive and where S(k, h)(x) =

sin (π(x − kh)/ h) , π(x − kh)/ h

that is, S(k, h)(x) = sinc (π(x − kh)/ h) . See, for example, [11]. The object of this note is to study the behavior and properties of the function I ( p) =

√

 p 0

∞

   sin x  p    x  dx

doi:10.4169/000298910X492817

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∞ for 1 < p < ∞. Note that this function is only defined for p > 1, since 0 (sin x)/x d x is conditionally convergent. Indeed   ∞  ∞  sin x  π sin x   dx = while  x  d x = +∞; x 2 0 0 see [12]. This integral arises, for example, in the L p -approximation of real-valued functions by Whittaker cardinal functions, and is important in estimating the error made in the approximation. It also arises in many other computational problems, and it is surprising that so little is known about it. Various properties of the function I ( p) are known. For example, the behavior of I ( p) for large p is known:

  ∞  sin x  p √   d x = 3π . lim I ( p) = lim p  p→∞ p→∞ x  2 0 This result, obtained independently by A. Meir and I. E. Leonard, is in principle not new (see equation (3)). We provide a self-contained proof below as part of our more general result in Theorem 1. Also, for integer p, the integral  ∞ sin x p dx x 0 can be calculated explicitly. In fact, for n ≥ 1 we have 



∞ 0

sin x x

n

 

n−1 π n/2 1 k n · n · dx = (−1) . n − 2k k (n − 1)! 2 k=0

This result is most definitely not new; it can be found in Bromwich [4, Exercise 22, p. 518], where it is attributed to Wolstenholme, and in many other places—including two relatively recent articles on integrals of more general products of sinc functions [2, 3]. Thus, if p is an even integer, then we have a closed-form expression for I ( p), and in this case the values of I ( p) can be calculated exactly: I ( p) =

√

 p 0

∞



sin x x

p dx =

√

 p/2 π  1 k p · p· · (−1) ( p − 2k) p−1 . k ( p − 1)! 2 p k=0 (1)

√ √ In particular I (2) = π/ 2, I (4) = 2π/3, and I (6) = 11 6π/40. That said, this sum is very difficult to use numerically for large p. Not only are the rational factors growing rapidly, but it contains extremely large terms of alternating sign and consequently dramatic cancelations. For example I (36) =

731509401860533204925821188658871713 π, 1063081066500632194410149314560000000

and I (10) = Q 100 π, where Q 100 is a rational number whose numerator and denominator both have roughly 150 digits. Similarly I (12) = Q 144 π, where Q 144 is comprised of 240-digit integers. We also note that numerical integration of I ( p) even to single June–July 2010]

L p NORMS AND THE SINC FUNCTION

529

precision is not easy and so (1) provides a very good confirmation of numerical integration results. We challenge the reader to numerically confirm the limit at infinity to 8 places. The behavior of I ( p) for intermediate values of p is not fully established. It had been conjectured that I ( p) has a global minimum at p = 4. However, (very) recent computations using both Maple and Mathematica suggest that the global minimum, and unique critical point, is at approximately p = 3.36 . . . , as illustrated in Figure 1. 2.20 2.18 2.16 2.14 2.12 2.10 2

3

4

5

6

7

8

9

10

Figure 1. The function I on [2, 10].

 Although it is known that lim p→1+ I ( p) = +∞, and that lim p→∞ I ( p) = 3π2 ,  is approached, although both it is not known precisely how the asymptote y = 3π 2 numerical and graphical evidence strongly suggest the following conjecture: Conjecture. I is increasing for p above the conjectured global minimum near 3.36 and concave for p above an inflection point near 4.469.  . Moreover, in This is shown in Figure 2, in which the dashed line has height 3π 2 Theorem 2 we shall prove



 3π 2 p 3π 1 > 1− , (2) I ( p) > 2 2p + 1 2 2p for all p > 1. We conclude this introduction by observing that one can derive the existence of an asymptotic expansion for I ( p) from a general result of Olver [10] on asymptotics of integrals using critical point theory and contour integration. Specialized to our case, [10, Theorem 7.1, p. 127] (with q = 1 and p = log(sin(x)/x) on [−π, π]) establishes the existence of real constants cs such that   π  sin(x)  p 1√   p I ( p) ∼  x  dx 2 −π



∞ 3 3π 1  3π 1 − + ∼ cs s + · · · (3) 2 20 2 p s=2 p 530

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2.20 2.18 2.16 2.14 2.12 2.10 10 20 30 40 50 60 70 80 90 100 Figure 2. The function I and its limiting value on [2, 100].

as p → ∞. From this one may deduce that I ( p) is concave and increasing for sufficiently large values of p—consistent with our stronger conjecture—as (3) may be differentiated termwise. 2. OUR MAIN RESULTS. In order to study the properties of the function I ( p), we consider first the functions     ∞   sin x  n  sin x  p   dx   ϕn ( p) = · log  x  x  0 for p > 1 and n a nonnegative integer. We write   ∞  sin x  p   ϕ( p) = ϕ0 ( p) =  x  d x. 0 In Lemma 1 below we confirm that ϕ( p) is analytic in a region containing (1, ∞) and that its nth derivative for p > 1 is given by ϕ (n) ( p) = ϕn ( p). Then in Theorem 1 we shall use induction to prove the following result for n a nonnegative integer:

 1 3 1 n+ (n) n lim p 2 ϕ ( p) = (−1)  n+ . p→∞ 2 2 The base case, n = 0, for our induction is established in Lemma 2 below. It uses Laplace’s method for determining asymptotic behavior of an integral for large values of a parameter p; see, e.g., [6, p. 60]. Lemma 1. For p − 1 > z > 1 − p, ϕ( p − z) =

∞  zn (−1)n ϕn ( p) . n! n=0

In particular, ϕ( p) is analytic in a region containing (1, ∞) and its nth derivative for p > 1 is given by ϕ (n) ( p) = ϕn ( p). June–July 2010]

L p NORMS AND THE SINC FUNCTION

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Proof. We have       ∞  ∞ ∞    sin x  p−z  sin x  n  sin x  p z n   ·   ϕ( p − z) = dx = dx − log   x   x  n! x  0 0 n=0     ∞  ∞ ∞    sin x  n  sin x  p z n zn n  ·  d x = , = (−1) ϕ ( p) − log  n  x  n! x  n! n=0 0 n=0

(4)

(5)

the inversion of sum and integral in (4) being justified as follows: Case i. p − 1 > z ≥ 0. All the terms involved are nonnegative. Case ii. 0 > z > 1 − p. By Case i    ∞ ∞    sin x  n  dx − log  ϕ( p − |z|) = x  0 n=0

   sin x  p |z|n  < ∞. ·  x  n!

Thus (5) yields the Taylor series for ϕ( p − z) at z = 0, and the final conclusion follows. Lemma 2. lim I ( p) = lim

p→∞

√

p→∞

p ϕ( p) =

3π . 2

(6)

Proof. Let a > 0. Then for p > 1 we have     ∞  a  ∞  sin x  p  sin x  p  sin x  p √ √ √       I ( p) = p  x  dx = p  x  dx + p  x  d x. 0 0 a We show first that lim

√

p→∞

 p a

∞

   sin x  p    x  d x = 0.

(7)

It suffices to consider the case 0 < a < 1, since for a ≥ 1 we have   ∞  b √  sin x  p p 1 1 √ √   · p−1 −→ 0 p lim p dx =  x  d x ≤ b→∞ p p−1 a a a x as p → ∞. Now, for a < x < 1, we have 0<

sin a sin x < < 1, x a

and it follows that     ∞  1  ∞  sin x  p  sin x  p  sin x  p √ √ √       0< p  x  dx ≤ p  x  dx + p  x  dx a a 1 p  √ p √  sin a  −→ 0 ≤ (1 − a) p  +  a p−1 as p → ∞. This establishes (7). 532

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

We next use the following easily proved results [9, 8]: 1−

x2 sin x x2 x4 ≤ ≤1− + 6 x 6 120

and 

for all real x,

√ π ( p + 1) ,

2  p + 32

1

(1 − u 2 ) p du = 0

(8)

(9)

where the equality is a special case of a beta-function evaluation (see also [12, Theorem 7.69]). It follows from (8) and (9) that

  √6  √  1  sin x  p 3π ( p + 1) 2 p  

, (10) (1 − u ) du =  x  dx ≥ 6 2  p + 32 0 0 and hence that

lim inf I ( p) ≥ lim p→∞

p→∞

3π 2

√

p ( p + 1)

 .  p + 32

(11)

Now, in order to get an appropriate inequality for the limsup, we note that for any w > 1, if we let  √ 1 W =2 5 1− , w then we have x2 x4 x2 sin x ≤1− + ≤1− x 6 120 6w √ If, in addition, w ≤ 10/7, then W ≤ 6, whence 

W 0

  √   sin x  p   d x ≤ 6w  x 

√W 6w

for

0 < x < W.

(12)

3πw ( p + 1)

. 2  p + 32

(13)

(1 − u 2 ) p du

0

√  1 ≤ 6w (1 − u 2 ) p du = 0



It follows from (7) and (13) that

lim sup I ( p) ≤ lim p→∞

p→∞

√ 3πw p ( p + 1)  ,

2  p + 32

and therefore from (11) and (14), for w ∈ (1, 10/7] we have

√ 3π p ( p + 1)

 ≤ lim inf I ( p) ≤ lim sup I ( p) lim p→∞ p→∞ 2  p + 32 p→∞

√ 3πw p ( p + 1)  .

≤ lim p→∞ 2  p + 32 June–July 2010]

L p NORMS AND THE SINC FUNCTION

(14)

(15) 533

Since we have



√ a  a + 12 =1 lim a→∞ (a + 1)

from [8, Problem 2, p. 45] or (23), letting p → ∞ in (15), we obtain



3π 3πw ≤ lim inf I ( p) ≤ lim sup I ( p) ≤ , p→∞ 2 2 p→∞

(16)

for all w ∈ (1, 10/7]. Finally, letting w → 1+ , we get the desired equation (6). We are now ready for our more general result. Theorem 1. For all natural numbers n we have   ∞  1 1  sin x  n n+ n+ (n)   lim p 2 ϕ ( p) = lim p 2 log  p→∞ p→∞ x  0

 1 3 n  n+ = (−1) . 2 2

   sin x  p  dx  · x  (17)

Proof. The first equality was noted above. We proceed to establish equation (17) by induction. The proof of the base case was given in Lemma 2. For the inductive step of the proof, we assume that for a given nonnegative integer n, we have

 1 3 1 n+ (n) n 2 lim p  n+ . ϕ ( p) = (−1) p→∞ 2 2 It is easily verified that x < − log(1 − x) <  sin t  p   , that

x 1−x

for 0 < x < 1, and setting x = 1 −

t

 p      sin t  p  sin t  p 1 −  sint t   < − log   1 −   t  <  sin t  p t  t for all but countably many values of t. For q > p + 1, multiplying these inequalities by the nonnegative term       sin t  n  sin t q n   ,   (−1) log  · t  t  we have        sin t  n  sin t q  sin t  p+q       0 ≤ (−1) log   t  − t  t        sin t  n+1  sin t q    · < −(−1)n p log   t  t           sin t  n  sin t q− p  sin t q n       < (−1) log  −  t  t  t  

n

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for the same values of t, and integrating over (0, ∞) yields  n (n) (n) (−1) ϕ (q) − ϕ ( p + q) ≤ −(−1)n p ϕ (n+1) (q)  (n) (n) ≤ (−1) ϕ (q − p) − ϕ (q) , n

and hence

⎛

(−1)

1

n+ 2 ϕ (n) (q) n ⎝q 1

p q n+ 2 ≤ −(−1)

nq

n+1+

1

p ( p + q)n+ 2

1 2 ϕ (n+1) (q) 1

⎛

≤ (−1)

−

⎞ 1 ( p + q)n+ 2 ϕ (n) ( p + q) ⎠

q n+1+ 2 1

n+ 2 ϕ (n) (q − p) n ⎝ (q − p) 1

p (q − p)n+ 2

−

⎞ 1 q n+ 2 ϕ (n) (q) ⎠ 1

p q n+ 2

.

(18)

Now let q = kp, where k > 2 is fixed; then (18) becomes ⎛ ⎞ 1 n+ (n) 1 k ( p + q) 2 ϕ ( p + q) ⎠ (−1)n ⎝k q n+ 2 ϕ (n) (q) − 1 (1 + 1k )n+ 2 1

≤ −(−1)n q n+1+ 2 ϕ (n+1) (q) ⎛ ⎞ 1 n+ (n) 1 (q − p) 2 ϕ (q − p) − k q n+ 2 ϕ (n) (q)⎠ . ≤ (−1)n ⎝k 1 (1 − 1k )n+ 2 Next let q → ∞, keeping k > 2 fixed, so that p → ∞ and q − p = (k − 1) p → ∞. It follows from the inductive hypothesis that 1

1

1

lim q n+ 2 ϕ (n) (q) = lim ( p + q)n+ 2 ϕ (n) ( p + q) = lim (q − p)n+ 2 ϕ (n) (q − p)

q→∞

q→∞

= (−1) and therefore ⎛ ⎝k −

k

n+ 1 2 1 + 1k

⎞

⎠

n

q→∞

 1 3  n+ , 2 2

 1 1 3  n+ ≤ lim inf (−1)n+1 q n+1+ 2 ϕ (n+1) (q) q→∞ 2 2 1

≤ lim sup (−1)n+1 q n+1+ 2 ϕ (n+1) (q) q→∞

⎛ ≤⎝

June–July 2010]

k

n+ 1 2 1 − 1k

⎞

− k⎠

L p NORMS AND THE SINC FUNCTION

 1 3  n+ . 2 2

(19)

535

Since ⎛ lim ⎝k −

k→∞

⎞ k

n+ 1 2 1 + 1k

⎛

⎠ = lim ⎝ k→∞

⎞ k

n+ 1 2 1 − 1k

− k⎠ 1

1 1 − (1 + t)−n− 2 =n+ , = lim t→0 t 2 it follows from (19) that lim

q→∞

1 (−1)n+1 q n+1+ 2 ϕ (n+1) (q)

  

1 3 3 1 1  n+1+ = n+  n+ = , 2 2 2 2 2

and this completes the proof of the inductive step. 3. FINAL REMARKS. Our proof of Theorem 1 shows both that 1

lim p n+ 2 ϕ (n) ( p) = an

(20)

p→∞

exists and determines the value of an . If we know in advance that the limit exists for every nonnegative integer n, then we can use Lemmas 1 and 2 to write lim

p→∞

√

p ϕ( p(1 + x)) = lim

p→∞

∞ 

p

n+

√ 3π/2 =√ n! 1+x

1 xn 2 ϕ (n) ( p)

n=0

for 1 − 1p > x > 1p − 1, and then justify the exchange of limit and sum, and expand the final term to obtain

∞ ∞  

 xn xn 3 = (−1)n  n + 12 . an n! 2 n! n=0 n=0 Comparing coefficients of the above two exponential generating functions yields the desired valuation

 3 (−1)n  n + 12 . (21) an = 2 In fact, to justify the exchange by means of the series version of Lebesgue’s theorem on dominated convergence one needs to establish something like     n+ 1 (n)  p 2 ϕ ( p)  ≤M    n!   with M a positive constant independent of n and p, and this requires an inequality such as the right-hand side of (19) (with q replaced by p and n by n − 1) used in the given proof of Theorem 1. 536

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Another way of determining the value of an in (20), if we know it exists for every n, is to proceed via L’Hospital’s rule as follows: an−1 = lim

p→∞

ϕ (n−1) ( p) 1 p −n+ 2

= lim

p→∞

ϕ (n) ( p) −(n −

1 1 ) p −n− 2 2

=−

an , n − 12

whence, by Lemma 2,

n   

3 1 (−1)n  n + 12 , k− an = (−1) a0 = 2 2 k=1 n

which is (20) again. One advantage of our explicit proof of Lemma 2 over Olver’s asymptotic result in (3) is that it is easily exploited to establish (2). Theorem 2. For all p > 1 we have



 3π 2 p 3π 1 > I ( p) > 1− . 2 2p + 1 2 2p

(22)

Proof. For x > 0 and 0 < s < 1, Abromowitz and Stegun [1] records (as (5.6.4) in the new web version) that x 1−s <

(x + 1) < (x + 1)1−s . (x + s)

(23)

Hence, from (10) and (23) we obtain for p > 1 that



√ 3π ( p + 1) 3π 2 p ( p + 1) √  

= I ( p) > p 2  p + 32 2 2 p + 1  p + 12



 3π 2 p 3π 1 > > 1− . 2 2p + 1 2 2p Here, for the penultimate inequality, we have used the left-hand inequality in (23) with x = p, s = 1/2. Note that (22) implies that  √ 1/ p 2 6p π

sinc p > 2p + 1 when sinc is viewed as a function in L p ([−∞, ∞]). We finish by observing that the lower bound is asymptotically of the correct order, and leave as an open question whether similar explicit techniques to those in Theorem 1 can be used to establish the second-order term in the asymptotic expansion (3) or the concavity properties conjectured in the introduction. Finally, we note that a much more accurate computation of the critical and inflection points can be found at http://www.carma.newcastle.edu.au/~jb616/oscillatory.pdf, and the values are as shown below: June–July 2010]

L p NORMS AND THE SINC FUNCTION

537

•

•

•

p at critical point (conjectured minimum): 3.36354876022451532816334301553541106982340973010200 93393024274526853624322808822111780630522743546839 65168546672961485462827077846841786411218613089950 8745727158152731 I ( p) at critical point (conjectured minimum): 2.09002860269180412254956491550781177353834974949186 75161558946115770419271274624491776411344314758189 93461306711846030747363223735023118868888017902470 29802232734781888386061734850631082243846394257215 38511911622108100945818827513170410889481080593453 364388301851618971531246883340068963419076 p at inflection point: 4.46987788658564578917780820674988693171596919867299 11634253975525983837941459705451646979509928424279 4233718363336416486397093

ACKNOWLEDGMENTS. We wish to thank Amram Meir and David Bailey for very useful discussions during the preparation of this note.

REFERENCES 1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972; also available at http://dlmf.nist.gov/. 2. D. Borwein and J. M. Borwein, Some remarkable properties of sinc and related integrals, Ramanujan J. 5 (2001) 73–90. doi:10.1023/A:1011497229317 3. D. Borwein, J. M. Borwein, and B. Mares, Multi-variable sinc integrals and volumes of polyhedra, Ramanujan J. 6 (2002) 189–208. doi:10.1023/A:1015727317007 4. T. J. Bromwich, Theory of Infinite Series, 2nd ed., Blackie & Sons, Glasgow, 1926. 5. H. S. Carslaw, An Introduction to the Theory of Fourier’s Series and Integrals, 3rd revised ed., Dover, New York, 1952. 6. N. G. de Bruijn, Asymptotic Methods in Analysis, 2nd ed., North-Holland, Amsterdam, 1961. 7. W. B. Gearhart and H. S. Schultz, The function sin(x) x , College Math. J. 21 (1990) 90–99. doi:10.2307/ 2686748 8. P. Henrici, Applied and Computational Complex Analysis, Volume 2, Wiley, New York, 1977. 9. I. E. Leonard and J. Duemmel, More—and Moore—power series without Taylor’s theorem, American Mathematical Monthly 92 (1985) 588–589. doi:10.2307/2323175 10. F. W. J. Olver, Asymptotics and Special Functions, 2nd ed., A K Peters, Natick, MA, 1997. 11. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Computational Mathematics, vol. 20, Springer–Verlag, New York, 1993. 12. K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Belmont, CA, 1981.

DAVID BORWEIN obtained two B.Sc. degrees from Witwatersrand University, one in engineering in 1945 and the other in mathematics in 1948. From University College London (UK) he received a Ph.D. in 1950 and a D.Sc. in 1960. He has been at the University of Western Ontario since 1963 with an emeritus title since 1989. His main area of research has been classical analysis, particularly summability theory. Department of Mathematics, The University of Western Ontario, London, ONT, N6A 5B7, Canada [email protected]

JONATHAN M. BORWEIN received his B.A. from the University of Western Ontario in 1971 and a D.Phil. from Oxford in 1974, both in mathematics. He currently holds a Laureate Professorship at University of Newcastle and until recently a Canada Research Chair in the Faculty of Computer Science at Dalhousie University. His primary current research interests are in nonlinear functional analysis, optimization, and experimental (computationally-assisted) mathematics.

538

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School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia and Faculty of Computer Science and Department of Mathematics, Dalhousie University, Halifax NS, B3H 2W5

ISAAC E. LEONARD received his B.A. and M.A. from the University of Pennsylvania in 1961 and 1963, both in physics. He received his M.Sc. and Ph.D. from Carnegie-Mellon University in 1969 and 1973, both in mathematics; and a B.Sc. in Computer Science from the University of Alberta in 1994. He currently holds a position as a Sessional Lecturer with the Department of Mathematical and Statistical Sciences at the University of Alberta, and has taught courses in the Computing Science and Electrical and Computer Engineering Departments at the University of Alberta. His current research interests are in analysis of algorithms and numerical analysis. Department of Mathematical and Statistical Sciences, The University of Alberta, Edmonton, AB, T6G 2G1, Canada [email protected]

Most Unlikely Breakfast Table Conversation by a 13-Year-Old High School Freshman and his Mother The following exchange occurs in the 2008 Australian movie “Hey Hey It’s Esther Blueburger,” written and directed by Cathy Randall: Jacob Blueburger: “Mom, do the trigonometric functions form a complete basis for the space of all continuous functions?” Grace Blueburger: “Yes, but the functions must have compact support.” No dialogue before or after this in the movie is even remotely like the above exchange, which seems to come out of the blue (no pun intended). —Submitted by Frederick G. Schmitt, College of Marin, Kentfield, CA

June–July 2010]

L p NORMS AND THE SINC FUNCTION

539

NOTES Edited by Ed Scheinerman

Indefinite Quadratic Forms and the Invariance of the Interval in Special Relativity John H. Elton

Abstract. In this note, a simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in special relativity from Einstein’s postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert’s Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizability of semi-definite real quadratic forms is given.

1. INTRODUCTION. In the special theory of relativity, an event is a point in spacetime whose coordinates with respect to an inertial reference frame correspond to some point (t, x, y, z) in R4 . Coordinates of events in different inertial reference frames are assumed to be connected by linear transformations, based on the assumption of homogeneity and isotropy of space-time. A famous postulate of Einstein is the universality of the speed of light: the speed of light in a vacuum is the same in all inertial reference frames, independent of the motion of the source. One can use the postulate of the universality of the speed of light, together with the assumption that changes of coordinates are linear, to determine what changes of coordinates are possible. The idea is to use this postulate to directly show the invariance of a certain quadratic function of the coordinates, which can in turn be used to determine the linear transformations connecting the coordinates (called Lorentz transformations). Defining the Lorentz transformations as the group of linear transformations which leave this quadratic function invariant is geometrically very appealing. To be most satisfying, and not circular, the invariance of the quadratic function should be shown to be a simple and immediate consequence of the postulates; the Lorentz transformations should only then be developed after that. Suppose points in space-time are specified by (t, x, y, z) in one inertial reference frame K , and by (t  , x  , y  , z  ) in a second inertial reference frame K  whose origin coincides with the first (that is, t = 0, x = 0, y = 0, z = 0 in K corresponds to the same event as t  = 0, x  = 0, y  = 0, z  = 0 in K  ). Let a pulse of light be emitted at this common event. Then events on the wave front have coordinates satisfying x 2 + y 2 + z 2 − c2 t 2 = 0 in system K , and also x 2 + y 2 + z 2 − c2 t 2 = 0 in system K  , where c, the speed of light, is the same in both systems. This is from Einstein’s postulate. In 1966 the author was taking a course in “modern” physics, and remembers being puzzled by the next step taken in the text [8, p. 58]. The text simply assumed without further ado that x 2 + y 2 + z 2 − c2 t 2 = x 2 + y 2 + z 2 − c2 t 2 for all events (not just doi:10.4169/000298910X492826

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those on the wave front of the pulse, when both expressions are zero) and proceeded to use that for a derivation of the form of the Lorentz transformations. Looking in some other texts, we found the same “unconscious” assumption of the invariance of the interval x 2 + y 2 + z 2 − c2 t 2 . In [6, p. 90], it is even stated that “(3.27) x 2 + y 2 + z 2 − c2 t 2 = 0”; “(3.28) x 2 + y 2 + z 2 − c2 t 2 = 0”; and then the amazing statement “. . . equating lines 3.27 and 3.28, we conclude x 2 + y 2 + z 2 − c2 t 2 = x 2 + y 2 + z 2 − c2 t 2 .” So our confusion remained unresolved for the moment, puzzled by the logic of “things that are equal when zero are always equal” that seemed to be used in these books. Next semester the author took a course in classical mechanics using the text by J. B. Marion [2]. Appendix G of that book has a demonstration of the invariance of the interval arguing directly from Einstein’s postulates, acknowledging the issue that concerned us. (This text is still popular today.) Here is the beginning of the proof given in Appendix G, p. 558, of that book: (∗) The wave front is described by x 2 + y 2 + z 2 − c2 t 2 = s 2 = 0 in K , and x 2 + y 2 + z 2 − c2 t 2 = s 2 = 0 in K  . “The equations of the transformation that connect the coordinates (t, x, y, z) in K and (t  , x  , y  , z  ) in K must themselves be linear. In such a case the quadratic forms s 2 and s 2 can be connected by, at most, a proportionality factor: s 2 = κs 2 .” (It is then shown by further arguments using homogeneity, isotropy, and continuity that in fact κ = 1.) We are of the opinion that the statement above about the reason for the proportionality of the quadratic forms would be misleading to many readers. It is not generally true that if one quadratic form is the result of making a linear change of variables in another quadratic form, and the two quadratic forms have the same zero set, then they must be proportional (even when this zero set has infinitely many points). Here is a somewhat arbitrary example with three variables: Let s 2 = 2x 2 + 2y 2 + z 2 − 2x z − 2yz, and let s 2 = 2x 2 + 2y 2 + z 2 − 2x  z  − 2y  z  , where x  = −2x − 2y + z, y  = 2y − 2z, and z  = −2z, so the coordinates are connected by a linear transformation. Algebra shows that s 2 = 8x 2 + 16y 2 + 10z 2 + 16x y − 16x z − 24yz, which is clearly not proportional to s 2 . Yet both quadratic forms are zero on the same infinite set {(x, y, z) ∈ R3 : z = x + y, x = y}, which is apparent after we reveal that actually s 2 = (x + y − z)2 + (x − y)2 and s 2 = 8(x + y − z)2 + 2(2y − z)2 , noting that if x + y = z, then x = y if and only if 2y = z. For another sort of example (not really related to the statement in Marion but relevant later in this paper), in two variables, let s 2 = x 2 + y 2 − 2x y and s 2 = x 2 − y 2 . Then s 2 = 0 ⇒ s 2 = 0, yet these quadratic forms are not even simultaneously diagonalizable. So it would seem the statement about proportionality of the quadratic forms could use further explanation. The author fashioned a proof for himself, but remained puzzled why the books seemed unconcerned about the logical gap. Fast-forwarding 43 years, we recently had occasion, after not thinking about physics since being an undergraduate, to come upon this topic again. The 1985 text on general relativity by Schutz [7, p. 32] gives a logically correct argument for the proportionality of the quadratic forms in (∗). But this does not seem to have been propagated to the community of physics students and textbook writers. From the 2006 relativity text [3], we find on page 10 essentially the same puzzling statements that occurred in the 1964 text [6] mentioned above: “c2 t 2 − x 2 − y 2 − z 2 = 0”; “c2 t 2 − x 2 − y 2 + z 2 = 0”; “These are equal, so c2 t 2 − x 2 − y 2 − z 2 = c2 t 2 − x 2 − y 2 + z 2 .” And we have evidence, from the Physics Forums [5], that indeed other physics students are still finding themselves confused by exactly the same thing that we found unexplained so long ago! The answers we saw given by other students there were unfortunately June–July 2010]

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not correct and were essentially on the level of the “unconscious” proofs of some of those texts, along with some rather arrogant statements about the students who didn’t understand the “proofs” they saw in their books. So we decided this time to fill in the gap, for the benefit of others who might be confused, by stating and proving a more general but very simple result about indefinite quadratic functions that settles the matter. This result about containment of zero sets suggests a more general result, proved using Hilbert’s Nullstellensatz. Also we prove a simple result about simultaneous diagonalization of semidefinite quadratic forms and containment of zero sets. 2. A THEOREM ON INDEFINITE QUADRATIC FORMS. A function q : Rn → R is a real quadratic form if there is a symmetric bilinear function q˜ : Rn × Rn → R such that q(x) = q(x, ˜ x). In matrix language, this means there is a symmetric n × n matrix Q = [Q ] of real numbers such that q(x1 , . . . , xn ) = q(x) = i j n n t n i=1 j =1 Q i j x i x j , i.e., q(x) = x Qx for x ∈ R . The elements of the matrix Q are the components of q˜ in the standard basis. A real quadratic form q is indefinite if it takes both positive and negative values; this is equivalent to the matrix Q having at least one positive eigenvalue and at least one negative eigenvalue. See [4] for example, or any book on linear algebra. For a real quadratic form q, define Z q = {x ∈ Rn : q(x) = 0}; this is the zero set of q. Theorem 1. Let q be an indefinite real quadratic form on Rn , and let r be a real quadratic form on Rn such that Z q ⊂ Z r ; that is, q(x) = 0 ⇒ r (x) = 0. Then r is proportional to q; that is, there exists a real number α such that r (x) = αq(x) for all x. If α is not zero, then r is also indefinite and has the same zero set as q. Proof. There exists a basis {v1 , . . . , vn } for Rn such that the matrix Q i j = q(v ˜ i, vj) representing q in this basis is diagonal, with only 1’s, −1’s, and 0’s on the diagonal, and Q ii = 1 for 1 ≤ i ≤ k; Q ii = −1 for k + 1 ≤ i ≤ k + m; Q ii = 0 for k + m + 1 ≤ i ≤ n; and Q i j = 0 for i = j. The numbers k and m here are unique: k is the number of positive eigenvalues and m is the number of negative eigenvalues of any matrix representing q (Sylvester’s law of inertia; see [4, p. 202]). Since q is indefinite, k > 0 and m > 0. So without loss of generality, in the proof which follows we will just assume that Q is a diagonal matrix with k ones and m negative ones and the rest (if any) zeroes on the diagonal, in order, as described above. (In the application to invariance of the interval which motivated this discussion, Q is already of this form, but we wanted to treat the general case.) Let R be the symmetric matrix representing r in this basis. The idea is to make judicious choices of points where q is zero, and to conclude that R must also be diagonal and that the on-diagonal elements of R are a common multiple of those of Q. To that end, let j be an integer such that k + 1 ≤ j ≤ k + m. Let x have components x1 = 1, x j = 1, and all other components zero. Then q(x) = Q 11 x12 + Q j j x 2j = 1 − 1 = 0, so r (x) = R11 x12 + R j j x 2j + 2R1 j x1 x j = R11 + R j j + 2R1 j = 0, by hypothesis. Now change the sign of the jth component of x so that x j = −1 but leave the other components of x unchanged; then q(x) = 0 still, so r (x) = R11 x12 + R j j x 2j + 2R1 j x1 x j = R11 + R j j − 2R1 j = 0 also. These two equations together imply R1 j = 0 and then R j j = −R11 . Then for 1 < i ≤ k, using i in place of 1 in the argument above shows Rii = −R j j = R11 , and Ri j = 0. 542

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

Next let j be an integer (if any) such that m + k + 1 ≤ j ≤ n. First let x be the vector with x j = 1 and all other components zero. Then q(x) = Q j j = 0, so r (x) = R j j = 0. Next let 1 < i ≤ k, k + 1 ≤ l ≤ k + m, and let x be the vector with components xi = 1, xl = 1, x j = 1, and all other components zero. Then q(x) = Q ii + Q ll + Q j j = 1 − 1 + 0 = 0, so r (x) = Rii + Rll + R j j + 2Ril + 2Ri j + 2Rl j = 2Ri j + 2Rl j = 0 also. Changing x so that xi = −1 and x is otherwise unchanged leads to −2Ri j + 2Rl j = 0. This implies that Ri j = 0, and then Rl j = 0. Suppose that k ≥ 2. Let 1 ≤ i < j ≤ k, k + 1 ≤ l ≤ k + m, and let x be the vector with components xi = 3, x j = 4, xl = 5, and all other components zero. Then q(x) = Q ii xi2 + Q j j x 2j + Q ll xl2 = 9 + 16 − 25 = 0, so r (x) = Rii xi2 + R j j x 2j + Rll xl2 + 2Ri j xi x j + 2Ril xi xl + 2R jl x j xl = R11 (9 + 16 − 25) + 2Ri j (12) = 0 also (note we have already shown that Ril = R jl = 0 and the proportionality of the diagonal elements). This proves Ri j = 0. Similarly, if m ≥ 2 or n − (k + m) ≥ 2, the corresponding off-diagonal terms of R are zero. This completes the proof that R = R11 Q, and the proof of the theorem. 3. AN ALTERNATE PROOF USING HILBERT’S NULLSTELLENSATZ, AND A STRONGER RESULT. The containment of zero sets in the hypothesis of Theorem 1 suggests Hilbert’s Nullstellensatz [1, p. 254], of importance in algebraic geometry. We can also prove Theorem 1 using this theorem rather than using diagonalization and bases as we did above; and although the proof above is certainly simple enough, there is some insight to be gained from this alternate proof, and a more general result can be proved this way as well. The Nullstellensatz concerns zero sets of ideals in the ring of polynomials in several variables over an algebraically closed field. For our application the ideal in question will be simply the principal ideal generated by a single polynomial q. If the reader is not familiar with ideal theory and the Nullstellensatz, it will not matter because we shall use only the following immediate consequence of Hilbert’s theorem: If q(x) and r (x) are complex polynomials in n variables such that x ∈ Cn and q(x) = 0 implies r (x) = 0, then r p (x) = q(x)s(x) for some polynomial s(x) and positive integer p. If q is square-free (that is, the irreducible factors of q occur only to the first power), p can be taken to be one. In Theorem 1, q and r are quadratic forms with real coefficients, q is indefinite, and the real zeroes of q are assumed to be zeros of r by hypothesis. If q were not squarefree, it would be the square of a linear polynomial or the negative of such a square, contrary to the indefiniteness of q, so we can take p to be one in our application (it is easy to see that q is actually irreducible when its rank exceeds two, but we don’t need that). Thus all we need to do is to show that, as a consequence of the indefiniteness of q, the complex zeroes of q are also zeroes of r , and the conclusion of Theorem 1 will follow from the Nullstellensatz, since the degrees of q and r being two requires s to be constant. To that end, suppose q(x + iy) = 0, for some x, y ∈ Rn , so q(x) − q(y) = 0 and q(x, ˜ y) = 0. If q(x) = 0 (hence q(y) = 0) then q(x + y) = 0, so r (x) = r (y) = r (x + y) = 0, which implies r˜ (x, y) = 0 and so r (x + iy) = 0. Suppose then that q(x) > 0 (the opposite case would be handled similarly); by rescaling assume q(x) = 1. Since q is indefinite, there is u ∈ Rn such that q(u) < 0. Let w = u − q(u, ˜ x)x − q(u, ˜ y)y, so q(w, ˜ x) = 0 and q(w, ˜ y) = 0 (a “GramSchmidt” construction). Now q(w) = q(w, ˜ u) = q(u) − q(u, ˜ x)2 − q(u, ˜ y)2 < 0. By rescaling we may assume that q(w) = −1, q(w, ˜ x) = 0, and q(w, ˜ y) = 0. Thus q(w + αx + βy) = −1 + α 2 + β 2 = 0 whenever α 2 + β 2 = 1, so by hypothesis r (w + αx + βy) = r (w) + α 2r (x) + β 2r (y) + 2αr˜ (w, x) + 2β r(w, ˜ y) + 2αβ r˜ (x, y) = 0 June–July 2010]

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also for α 2 + β 2 = 1. Taking α = ±1, β = 0, we conclude that r˜ (w, x) = 0, and similarly r˜ (w, y) = 0. Then choosing α = 2−1/2 , β = ±α, we conclude that r˜ (x, y) = 0. Then choosing α = 1, β = 0 and α = 0, β = 1, we see that r (x) = r (y), and thus r (x + iy) = 0, concluding the proof. This proof from the Nullstellensatz is perhaps slightly cleaner than the first proof of Theorem 1. But also one can prove more this way, with a little more work. The quadratic form r is a polynomial in n variables in which each term has degree 2. In general, a polynomial in n variables for which each term has the same degree d is called a homogeneous polynomial of degree d. Theorem 2. Suppose r is a homogeneous real polynomial in n variables, not necessarily a quadratic form, with the other hypotheses of Theorem 1 unchanged. Then q is a factor of r ; that is, r (x) = q(x)s(x) for some polynomial s(x). Proof. We only need to show that any complex zeroes of q are zeroes of r . Suppose q(x + iy) = 0, and suppose that q(x) > 0. As above, we can assume that q(x) = q(y) = 1, q(x, ˜ y) = 0, and there is w such that q(w) = −1, w is q-orthogonal to x and y, and q(w + αx + βy) = 0, so r (w + αx + βy) = 0 also, whenever α 2 + β 2 = 1. Suppose that r has even degree 2m. Now r (γ w + αx + βy) is a homogeneous polynomial of degree 2m in the variables α, β,and γ that is zero when γ = 1 and α 2 + β 2 = 1. We may write r (γ w + αx + βy) = j +k≤2m α j β k γ 2m− j −k c( j, k) where the indices j and k are nonnegative. By changing the signs of α and β separately, and then together, we see that for γ = 1 and α 2 + β 2 = 1,  α j β k c( j, k) = 0, j +k≤2m, j odd, k even



α j β k c( j, k) = 0,

j +k≤2m, j even, k odd



α j β k c( j, k) = 0,

j +k≤2m, j and k odd

and



α j β k c( j, k) = 0.

j +k≤2m, j and k even

Consider the last expression above (with both indices even), which can be rewritten with a change of indices as  (α 2 ) j (1 − α 2 )k c(2 j, 2k) = 0 j +k≤m

for α 2 ≤ 1. This is a polynomial of degree 2m in α; the coefficient of the highest power term α 2m must be zero because of the constancy of the polynomial on an infinite set, so  (−1)m− j c(2 j, 2m − 2 j) = 0. j ≤m

Next consider the next-to-last expression (with both indices odd) which can be rewritten  αβ (α 2 ) j (1 − α 2 )k c(2 j + 1, 2k + 1) = 0, j +k≤m−1

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so for 0 < α 2 < 1,



(α 2 ) j (1 − α 2 )k c(2 j + 1, 2k + 1) = 0.

j +k≤m−1

Setting the coefficient of the highest power term in the polynomial in α (which occurs when k = m − j − 1) to zero, we get  (−1)m− j −1 c(2 j + 1, 2m − 2 j − 1) = 0. j ≤m−1

Since r is homogeneous of degree 2m, r (x + iy) = r (0w + 1x + iy)  (−1)m− j c(2 j, 2m − 2 j) = j ≤m



+i

(−1)m− j −1 c(2 j + 1, 2m − 2 j − 1),

j ≤m−1

because i 2m−2 j = (−1)m− j and i 2m−2 j −1 = (−1)m− j −1 i. The results just proved show this is zero, completing the proof of the theorem when the degree of r is even and q(x) > 0. Now suppose r has odd degree 2m − 1. Then  r (γ w + αx + βy) = α j β k γ 2m−1− j −k c( j, k), j +k≤2m−1

and this breaks into four sums equaling zero when γ = 1 and α 2 + β 2 = 1 as before, depending on the parities of the indices. Consider first the sum corresponding to j odd and k even; this can be rewritten  α (α 2 ) j (1 − α 2 )k c(2 j + 1, 2k) = 0 j +k≤m−1

for α 2 ≤ 1. Setting the coefficient of the highest power to zero gives  (−1)m− j −1 c(2 j + 1, 2m − 2 j − 2) = 0. j ≤m−1

Now consider the sum corresponding to j even and k odd, which can be rewritten  β (α 2 ) j (1 − α 2 )k c(2 j, 2k + 1) = 0, j +k≤m−1

so 

(α 2 ) j (1 − α 2 )k c(2 j, 2k + 1) = 0

j +k≤m−1

for α 2 < 1, which implies 

(−1)m− j −1 c(2 j, 2m − 2 j − 1) = 0.

j ≤m−1

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But r (x + iy) = r (0w + 1x + iy)  = (−1)m− j −1 c(2 j + 1, 2m − 2 j − 2) j ≤m−1

+i



(−1)m− j −1 c(2 j, 2m − 2 j − 1),

j ≤m−1

so this is zero, and the proof is concluded when r is of odd degree and q(x) > 0. The case q(x) < 0 is handled in a similar way. Finally, if q(x + iy) = 0 and q(x) = 0, then since q(y) = 0 and q(x, ˜ y) = 0, q(x + αy) = 0 and thus r (x + αy) = 0 for all real numbers α. Now r (x + αy) is a polynomial in α which is identically zero, so all its coefficients are zero, and this clearly implies that r (x + iy) = 0, which concludes the proof of Theorem 2. 4. SIMULTANEOUS DIAGONALIZATION OF QUADRATIC FORMS. Theorem 1 implies a result on simultaneous diagonalizability: if q is an indefinite real quadratic form on Rn and r is a real quadratic form on Rn such that Z q ⊂ Z r , then q and r are simultaneously diagonalizable (meaning there is a basis in which the matrices representing q and r are both diagonal). However, if q is a semi-definite real quadratic form on Rn (semi-definite means q(x) ≥ 0 for all x or q(x) ≤ 0 for all x), and r is a real quadratic form on Rn such that Z q ⊂ Z r , then q and r are not necessarily simultaneously diagonalizable. For an example in R2 , let q(x) = (x − y)2 and let r (x) = x 2 − y 2 ; this example (already mentioned in the introduction) satisfies the conditions and it is easy to see these are not simultaneously diagonalizable. But if q and r are both assumed semi-definite, there is a similar (and similarly easy) result on containment of zero sets implying simultaneous diagonalizability. Theorem 3. Let q and r be semi-definite real quadratic forms on Rn such that Z q ⊂ Z r . Then r and q are simultaneously diagonalizable. Proof. Without loss of generality assume they are both positive semi-definite. First observe that the zero sets are subspaces: Let q(x) = 0 and q(y) = 0; then q(ax + by) = a 2 q(x) + b2 q(y) + 2abq(x, ˜ y) = 2abq(x, ˜ y) ≥ 0 for all real numbers a, b implies q(x, ˜ y) = 0, so q(ax + by) = 0. This is quite different from the indefinite case where the zero sets are cones and not subspaces. There is a subspace M such that Rn = M ⊕ Z q (choose any basis for Z q and extend it to a basis for Rn , and M is the span of those added-on basis vectors). Let x = y + z with y ∈ M and z ∈ Z q . Then q(y + αz) = q(y) + 2α q(y, ˜ z) + α 2 q(z) = q(y) + 2α q(y, ˜ z) ≥ 0 for all real α implies q(x, ˜ y) = 0, so q(y + z) = q(y). Similarly, r (y + z) = r (y) for y ∈ M and z ∈ Z q , since Z q ⊂ Z r . Thus q and r may be considered as positive semi-definite quadratic forms on M, and in fact q is positive definite on M, because if y ∈ M and q(y) = 0, then y ∈ Z q by definition, so y = 0. By a well-known theorem [4, p. 218], this implies q and r are simultaneously diagonalizable on M, and they are then simultaneously diagonalizable on Rn = M ⊕ Z q , with zeroes on the diagonal corresponding to the basis vectors for Zq . 546

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5. APPLICATION TO THE PROOF OF INVARIANCE OF THE INTERVAL. Suppose the coordinates x = (t, x, y, z) in K and x = (t  , x  , y  , z  ) in K  are connected by a linear transformation, so x = Lx for some 4 × 4 matrix L. Let q(x) = −c2 t 2 + x 2 + y 2 + z 2 = xt Qx, where Q is the diagonal matrix with diagonal entries (−c2 , 1, 1, 1). Let r (x) = −c2 t 2 + x 2 + y 2 + z 2 = (Lx)t Q Lx = xt (L t Q L)x, so r (x) = xt Rx, where R = L t Q L. Now q is indefinite, and r (x) = 0 precisely when q(x) = 0, from (∗) above. So the conditions of Theorem 1 are in force, and we may conclude that r is proportional to q, which is equivalent to the statement from (∗) that we wanted to prove, namely that s 2 is proportional to s 2 . ACKNOWLEDGMENT. We would like to thank Michael Loss for suggesting looking at Hilbert’s Nullstellensatz for a connection with the topic in this note.

REFERENCES 1. 2. 3. 4. 5.

N. Jacobson, Lectures in Abstract Algebra, vol. III, Van Nostrand, New York, 1964. J. B. Marion, Classical Dynamics of Particles and Systems, Academic Press, New York, 1965. D. McMahon, Relativity Demystified, McGraw-Hill, New York, 2006. G. D. Mostow and J. H. Sampson, Linear Algebra, McGraw-Hill, New York, 1969. Physics Forums, Proving invariance of spacetime interval, available at http://www.physicsforums. com/showthread.php?t=115451. 6. W. G. V. Rosser, An Introduction to the Theory of Relativity, Butterworths, London, 1964. 7. B. F. Schutz, A First Course in General Relativity, Cambridge University Press, Cambridge, 1985. 8. R. T. Weidner and R. L. Sells, Elementary Modern Physics, Allyn and Bacon, Boston, 1960. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 and Accusoft-Pegasus Imaging, 4001 N. Riverside Drive, Tampa, FL 33603 [email protected]

Monotone Convergence Theorem for the Riemann Integral Brian S. Thomson

Abstract. The monotone convergence theorem holds for the Riemann integral, provided (of course) it is assumed that the limit function is Riemann integrable. It might be thought, though, that this would be difficult to prove and inappropriate for an undergraduate course. In fact the identity is elementary: in the Lebesgue theory it is only the integrability of the limit function that is deep. This article shows how to prove the monotone convergence theorem for Riemann integrals using a simple compactness argument (i.e., invoking Cousin’s lemma). This material could reasonably and appropriately be used in classroom presentations where the students are indoctrinated on this antiquated, but still popular, integration theory.

The monotone convergence theorem is usually stated and proved for the Lebesgue integral, but there is little difficulty in formulating and proving a version for the Riemann integral. doi:10.4169/000298910X492835

June–July 2010]

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547

5. APPLICATION TO THE PROOF OF INVARIANCE OF THE INTERVAL. Suppose the coordinates x = (t, x, y, z) in K and x = (t  , x  , y  , z  ) in K  are connected by a linear transformation, so x = Lx for some 4 × 4 matrix L. Let q(x) = −c2 t 2 + x 2 + y 2 + z 2 = xt Qx, where Q is the diagonal matrix with diagonal entries (−c2 , 1, 1, 1). Let r (x) = −c2 t 2 + x 2 + y 2 + z 2 = (Lx)t Q Lx = xt (L t Q L)x, so r (x) = xt Rx, where R = L t Q L. Now q is indefinite, and r (x) = 0 precisely when q(x) = 0, from (∗) above. So the conditions of Theorem 1 are in force, and we may conclude that r is proportional to q, which is equivalent to the statement from (∗) that we wanted to prove, namely that s 2 is proportional to s 2 . ACKNOWLEDGMENT. We would like to thank Michael Loss for suggesting looking at Hilbert’s Nullstellensatz for a connection with the topic in this note.

REFERENCES 1. 2. 3. 4. 5.

N. Jacobson, Lectures in Abstract Algebra, vol. III, Van Nostrand, New York, 1964. J. B. Marion, Classical Dynamics of Particles and Systems, Academic Press, New York, 1965. D. McMahon, Relativity Demystified, McGraw-Hill, New York, 2006. G. D. Mostow and J. H. Sampson, Linear Algebra, McGraw-Hill, New York, 1969. Physics Forums, Proving invariance of spacetime interval, available at http://www.physicsforums. com/showthread.php?t=115451. 6. W. G. V. Rosser, An Introduction to the Theory of Relativity, Butterworths, London, 1964. 7. B. F. Schutz, A First Course in General Relativity, Cambridge University Press, Cambridge, 1985. 8. R. T. Weidner and R. L. Sells, Elementary Modern Physics, Allyn and Bacon, Boston, 1960. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 and Accusoft-Pegasus Imaging, 4001 N. Riverside Drive, Tampa, FL 33603 [email protected]

Monotone Convergence Theorem for the Riemann Integral Brian S. Thomson

Abstract. The monotone convergence theorem holds for the Riemann integral, provided (of course) it is assumed that the limit function is Riemann integrable. It might be thought, though, that this would be difficult to prove and inappropriate for an undergraduate course. In fact the identity is elementary: in the Lebesgue theory it is only the integrability of the limit function that is deep. This article shows how to prove the monotone convergence theorem for Riemann integrals using a simple compactness argument (i.e., invoking Cousin’s lemma). This material could reasonably and appropriately be used in classroom presentations where the students are indoctrinated on this antiquated, but still popular, integration theory.

The monotone convergence theorem is usually stated and proved for the Lebesgue integral, but there is little difficulty in formulating and proving a version for the Riemann integral. doi:10.4169/000298910X492835

June–July 2010]

NOTES

547

Monotone Convergence Theorem. Let { f n } be a nondecreasing sequence of Riemann integrable functions on the interval [a, b]. Suppose that f (x) = lim f n (x) n→∞

for every x in [a, b]. Then, provided f is also Riemann integrable on [a, b],  b  b f (x) d x = lim f n (x) d x. n→∞

a

(1)

a

This theorem should have been useful in many calculus presentations, but it does not appear in any of the usual textbooks. Perhaps the reason is that, because the Lebesgue version of the theorem is deep, it might follow that this version too is at some deeper level than the students should be taken. But it is not the identity (1) that is deep in Lebesgue’s theory, but his conclusion that such a function must be integrable. Here we are assuming integrability so the theorem is entirely elementary. Teaching this theorem offers the instructor some real opportunities. First is the chance to introduce a major theorem of integration theory at an elementary level and discuss its importance and how it must be improved. Second is the occasion (always tempting) to launch a polemic against the Riemann integral. The unfortunate hypothesis that the limit function is integrable is essential here, but reduces the theorem to a curiosity: in most applications we would know nothing more about the limit function than that it is a pointwise limit of integrable functions and would have serious difficulty finding some property that would assure Riemann integrability. The proof is nothing but some manipulations of Riemann sums and surely as accessible as any of the other theorems proved in Riemann integration theory. We need a few preliminaries. By a partition of an interval [a, b] we mean a collection π = {([u i , vi ], wi ) : i = 1, 2, . . . , n} of interval-point pairs for which each wi ∈ [u i , vi ] and the intervals form a collection of nonoverlapping intervals whose union is [a, b]. Any subset of a partition is a subpartition. The use of the Greek letter π to denote a partition will, no doubt, distress a calculus class but I am addicted to it. The Riemann integral, defined as a limit of Riemann sums, possesses also this apparently stronger property: () If the function f is integrable in the Riemann sense on an interval [a, b] then, for every  > 0, there is a δ > 0 so that     v   < f (x) d x − f (w)(v − u) (2)   ([u,v],w)∈π

u

whenever π is a partition or subpartition of the interval [a, b] such that v − u < δ for every pair ([u, v], w) ∈ π. This well-known property is seldom proved in calculus courses although it is only a simple computation using Riemann sums. It should be proved in any case since, from (2), one immediately deduces  ω f ([u, v])(v − u) < 2 (3) ([u,v],w)∈π

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c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

which is Riemann’s famous characterization of integrability expressed in terms of the oscillation ω f of the function f on subintervals. That, in turn, allowed Lebesgue to easily formulate his more famous characterization. Thus there is a lot of narrative potential in (). While the proof of () is elementary it is sufficiently subtle that a student might have trouble attempting it without coaching. The method in [2, p. 77] works here and can be attributed to Saks [3] who used it in his study of the Burkill integral. Now we may present the proof of the monotone convergence theorem. For each integer n, let gn = f − f n . The sequence of integrable functions {gn } is nonnegative and monotone decreasing with limn→∞ gn (x) = 0 at each x. Let  > 0 and write η = /(b − a + 1). For each integer n, use () to choose a positive number δn so that  ([u,v],w)∈π

   

v

u

  gn (x) d x − gn (w)(v − u) < η2−n

whenever π is a partition of the interval [a, b] such that v − u < δn for every pair ([u, v], w) ∈ π. Choose, for each x ∈ [a, b], the first integer N (x) so that gn (x) < η for all integers n ≥ N (x) and, for j = 1, 2, 3, . . . , let E j = {x ∈ [a, b] : N (x) = j}. We use these sets to define δ(x) = δ j whenever x belongs to the corresponding set E j . Take any partition π of the interval [a, b] for which v − u < δ(w) for every pair ([u, v], w) ∈ π. That such partitions exist is the conclusion of Cousin’s lemma. That lemma plays the same role as, and is equivalent to, the nested interval property on the real line. Many of the theorems of the calculus can conveniently use either argument. (See the discussions in [1], [4], and [5].) Let N be the largest value of N (w) for the finite collection of pairs ([u, v], w) in π. We carve the partition π into a finite number of disjoint subsets by writing π j = {([u, v], w) ∈ π : w ∈ E j } for integers j = 1, 2, 3, . . . , N . Note that π = π1 ∪ π2 ∪ · · · ∪ π N and that these collections are pairwise disjoint. Now let m be any integer greater than N . We compute 

b

0≤

gm (x) d x = a

=

N  j =1

≤

N  j =1

⎛ ⎝ ⎡

([u,v],w)∈π



([u,v],w)∈π j

⎣







v



v

 gm (x) d x

u

⎞ ⎛  N  ⎝ gm (x) d x ⎠ ≤

u

j =1

⎤

 ([u,v],w)∈π j



v

⎞  g j (x) d x ⎠

u

g j (w)(v − u) + η2− j ⎦

([u,v],w)∈π j

June–July 2010]

NOTES

549

<

N 

⎡



⎣

j =1

⎤ η(v − u) + η2− j ⎦ < η(b − a + 1) = .

([u,v],w)∈π j

The identity 



b

f (x) d x − lim a

n→∞



b

f n (x) d x = lim

n→∞

a

b

gn (x) d x = 0 a

follows. REFERENCES 1. R. G. Bartle, Return to the Riemann integral, American Mathematical Monthly 103 (1996) 625–632. doi: 10.2307/2974874 2. , A Modern Theory of Integration, Graduate Studies in Mathematics, vol. 32, American Mathematical Society, Providence, RI, 2001. 3. S. Saks, Sur les fonctions d’intervalle, Fund. Math. 10 (1927) 211–224. 4. B. S. Thomson, Rethinking the elementary real analysis course, American Mathematical Monthly 114 (2007) 469–490. 5. B. S. Thomson, J. B. Bruckner, and A. M. Bruckner, Elementary Real Analysis, 2nd ed., CreateSpace, Scotts Valley, CA, 2008. Mathematics Department, Simon Fraser University, B.C., Canada V5A 1S6 [email protected]

A Proof of a Version of a Theorem of Hartogs Marco Manetti Abstract. It is proved that a formal power series in s complex variables is convergent, if it is convergent on each line through the origin.

A formal power series in s variables  i f = ai1 ,... ,is z 11 · · · z iss i 1 ,... ,i s ≥0

with complex coefficients ai1 ,... ,is ∈ C is called convergent if is absolutely convergent in a neighbourhood of 0. This means that there exists a positive real number r such that   +∞    i 1 +···+i s f = |ai1 ,... ,is |r = |ai1 ,... ,is | r n < +∞. i 1 ,... ,i s ≥0

n=0

i 1 +···+i s =n

The aim of this note is to prove that the convergence of a formal power series can be established by checking convergence only on the lines passing through the origin of Cs . doi:10.4169/000298910X492844

550

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

<

N 

⎡



⎣

j =1

⎤ η(v − u) + η2− j ⎦ < η(b − a + 1) = .

([u,v],w)∈π j

The identity 



b

f (x) d x − lim a

n→∞



b

f n (x) d x = lim

n→∞

a

b

gn (x) d x = 0 a

follows. REFERENCES 1. R. G. Bartle, Return to the Riemann integral, American Mathematical Monthly 103 (1996) 625–632. doi: 10.2307/2974874 2. , A Modern Theory of Integration, Graduate Studies in Mathematics, vol. 32, American Mathematical Society, Providence, RI, 2001. 3. S. Saks, Sur les fonctions d’intervalle, Fund. Math. 10 (1927) 211–224. 4. B. S. Thomson, Rethinking the elementary real analysis course, American Mathematical Monthly 114 (2007) 469–490. 5. B. S. Thomson, J. B. Bruckner, and A. M. Bruckner, Elementary Real Analysis, 2nd ed., CreateSpace, Scotts Valley, CA, 2008. Mathematics Department, Simon Fraser University, B.C., Canada V5A 1S6 [email protected]

A Proof of a Version of a Theorem of Hartogs Marco Manetti Abstract. It is proved that a formal power series in s complex variables is convergent, if it is convergent on each line through the origin.

A formal power series in s variables  i f = ai1 ,... ,is z 11 · · · z iss i 1 ,... ,i s ≥0

with complex coefficients ai1 ,... ,is ∈ C is called convergent if is absolutely convergent in a neighbourhood of 0. This means that there exists a positive real number r such that   +∞    i 1 +···+i s f = |ai1 ,... ,is |r = |ai1 ,... ,is | r n < +∞. i 1 ,... ,i s ≥0

n=0

i 1 +···+i s =n

The aim of this note is to prove that the convergence of a formal power series can be established by checking convergence only on the lines passing through the origin of Cs . doi:10.4169/000298910X492844

550

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 



i Theorem 1. A formal power series f = i1 ,... ,is ai1 ,... ,is z 11 · · · z iss is convergent if and only if for every α1 , . . . , αs ∈ C, the formal power series in one variable    i1 f (α1 t, . . . , αs t) = ai1 ,... ,is α1 · · · αsis t n n

i 1 +···+i s =n

is convergent in some interval containing 0. Proof. The “only if” part is clear; we prove the “if” part of the theorem. According to

i Abel’s lemma [4, 1.5.8], a power series f = i1 ,... ,is ai1 ,... ,is z 11 · · · z iss is convergent if and only if there exists a positive real number m such that |ai1 ,... ,is | ≤ m i1 +···+is for every i 1 , . . . , i s ≥ 0 such that i 1 + · · · + i s > 0. Equivalently, the series f is not convergent if and only if for every m > 0 there exists i 1 , . . . , i s ≥ 0 such that i 1 + · · · + i s > 0 and |ai1 ,... ,is | > m i1 +···+is . Denote by f n the homogeneous component of degree n of f , i.e.,   i fn , fn = ai1 ,... ,is z 11 · · · z iss . f = i 1 +···+i s =n

n≥0

Since the function Cs → R, z  → | f n (z)|, is continuous for every n, for every integer m > 0 the subset  {z ∈ Cs | | f n (z)| ≤ m n } Cm = {z ∈ Cs | | f n (z)| ≤ m n for every n > 0} = n>0

is closed in Cs ; clearly Cm ⊂ Cl for every l ≥ m. We claim that if the set Cm has nonempty interior part for some m > 0, then f is convergent. To see this assume that for some point v = (v1 , . . . , vs ) ∈ Cs and some real positive number t we have {z ∈ Cs | max |z i − vi | ≤ t} ⊂ Cm . Fix a sequence of nonnegative integers i 1 , . . . , i s , let n = i 1 + · · · + i s , choose a prime number

p such that p > n, and denote by μ p ⊂ C the cyclic group of pth roots of 1. Since ξ ∈μ p ξ k = ξ ∈μ p ξ −k = 0 for every k such that 0 < k < p, for every sequence of integers j1 , . . . , js ≥ 0 such that j1 + · · · + js ≤ n we have  s  ξ j1 · · · ξ js  p s if i 1 = j1 , . . . , i s = js , 1 s jh −i h = ξ = i1 is 0 otherwise. ξ ,... ,ξs ∈μ p ξ1 · · · ξs h=1 ξ ∈μ p 1

Moreover f n (v1 + tξ1 , . . . , vs + tξs ) is a polynomial in ξ1 , . . . , ξs of degree n whose i coefficient of ξ11 · · · ξsis is t n ai1 ,... ,is . Therefore 

f n (v1 + tξ1 , . . . , vs + tξs )

ξ1 ,... ,ξs ∈μ p

ξ11 · · · ξsis

i

= t n p s ai1 ,... ,is ,

and then, by the triangle inequality, |ai1 ,... ,is | ≤ June–July 2010]

1 t n ps



| f n (v1 + tξ1 , . . . , vs + tξs )| ≤

ξ1 ,... ,ξs ∈μ p

NOTES

 m n t

.

551

We are now ready to prove the theorem: more precisely we show that if f is not convergent then there exists a dense subset U ⊂ Cs such that for every (α1 , . . . , αs ) ∈ U the series f (α1 t, . . . , αs t) is not convergent. Assume that f is not convergent; under this assumption the closed sets Cm , m > 0, have no interior part, and by the Baire theorem [3, Ch. 8] the subset  Cm m∈N, m>0

has no interior part in Cs and its complement U = Cs − ∪m Cm is dense. For every vector (α1 , . . . , αs ) ∈ U and for every m > 0 there exists n such that | f n (α1 , . . . , αs )| > m n ; since f n (α1 , . . . , αs ) is the coefficient of t n in the formal power series f (α1 t, . . . , αs t), this series is not convergent. As an application of Theorem 1 we give an elementary proof of the following result. Corollary 2. The ring of convergent power series is integrally closed in the ring of formal power series. In other words, Corollary 2 says that if p(z 1 , . . . , z s , t) is a monic polynomial in t with coefficient in the ring C{z 1 , . . . , z s } of convergent power series, and φ(z 1 , . . . , z s ) is a formal power series such that p(z 1 , . . . , z s , φ(z 1 , . . . , z s )) = 0, then φ is convergent. Notice that if φ(0) is a simple root of the polynomial p(0, . . . , 0, t) then the convergence of φ is a easy consequence of the Weierstrass preparation theorem. The corollary is also a rather easy consequence of the (nontrivial) theorem of M. Artin on the solution of analytic equations [1]. Proof. By Theorem 1 it is sufficient to consider the case s = 1. Assume therefore that p(z, t) ∈ C{z}[t] is a monic polynomial in t and let φ(z) be a formal power series such that p(z, φ(z)) = 0. According to the Newton-Puiseux theorem [2] there exists a positive integer N such that every root of the polynomial p(z N , t) belongs to C{z}. Therefore φ(z N ) is convergent and then also φ(z) is convergent. REFERENCES 1. M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968) 277–291. doi:10.1007/ BF01389777 2. E. Brieskorn and H. Kn¨orrer, Plane Algebraic Curves, Birkh¨auser, Basel, 1986. 3. J. R. Munkres, Topology, 2nd ed., Prenctice-Hall, Upper Saddle River, NJ, 2000. 4. V. Scheidemann, Introduction to Complex Analysis in Several Variables, Birkh¨auser, Basel, 2005. Dipartimento di Matematica Guido Castelnuovo, Sapienza Universit`a di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy [email protected] http://www.mat.uniroma1.it/people/manetti/

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Discovering and Proving that π Is Irrational Timothy W. Jones Abstract. Ivan Niven’s proof of the irrationality of π is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of π . This simplifying concept yields a more motivated proof of the irrationality of π and π 2 .

Charles Hermite proved that e is transcendental in 1873 using a polynomial that is the sum of derivatives of another polynomial [7]. Ivan Niven in 1947 found a way to use Hermite’s technique to prove that π is irrational [12]. Lambert in 1767 had proven this result in a twelve-page article using continued fractions [10]. Niven’s half-page proof, using only algebra and calculus, is frequently cited and sometimes reproduced in textbooks [14, 9, 15, 4, 6]. Although his proof is brief and uses ostensibly simple mathematics, it begins by defining functions as in the technique of Hermite without any motivation. In this article a simplifying concept is used that provides a more motivated and straightforward proof than Niven’s. Using this concept, we, as it were, discover that π might be irrational and then confirm that it is with a proof. 1. A MOTIVATED APPROACH. We seek to combine a known falsity with a known truth and then to derive a contradiction from the combination. If π is assumed to be rational, π = p/q with p and q natural numbers, then the maximum of sin x occurs at p/2q. The quadratic −qx 2 + px = x( p − qx) will have its maximum at the same point, as will the product of the two functions. If we have a blender that allows inferences from this statement we might be able to derive a contradiction. Such a blender exists in a definite integral. A definite integral allows for evaluations that might contradict upper or lower bounds. We have  p/q p3 p2 p 0< · = 2, x( p − qx) sin x d x ≤ (1) 4q q 4q 0 where the lower bound holds as the integrand is always positive,1 and the upper bound is formed from the length of the interval of integration multiplied by the maximum value of the integrand [16, Property 8, p. 389]. For a polynomial f (x), repeated integration by parts2 gives the indefinite integral pattern  f (x) sin x d x = − f (x) cos x − f  (x) sin x + f  (x) cos x + f  (x) sin x − · · · . (2) doi:10.4169/000298910X492853 see that the inequality is strict, consider:  3 p/4q   p/4q x( p − qx) sin x d x + x( p − qx) sin x d x +

1 To

0

p/4q

p/q

x( p − qx) sin x d x.

3 p/4q

2 Tabular integration by parts (see [11, p. 532] and [5]) is especially well suited for integrals of the type given in (1).

June–July 2010]

NOTES

553

For the function f (x) = x( p − qx), as f (k) (x) = 0 for k ≥ 3, we have  p/q  p/q  f (x) sin x d x = {− f (x) cos x − f  (x) sin x + f  (x) cos x} 0

0

(3)

and the odd term drops out (sin p/q = sin 0 = 0) leaving an alternating sum of even derivatives of f (x) evaluated at the endpoints:  p/q f (x) sin x d x = f ( p/q) + f (0) − f  ( p/q) − f  (0). (4) 0

This sum is 4q. Combining (1) and (4) we have 0 < 4q ≤

p3 . 4q 2

(5)

2. DISCOVERING π IS IRRATIONAL. 2.1. Candidate π Values. The inequalities in (5) show π does not equal 1 or 2. For π = 7/2, this n = 1 case of the general polynomial x n ( p − qx)n does not give a contradiction. We will try the n = 2 case and see if it works for this rational. This is possible as the same reasoning about x( p − qx) applies to x n ( p − qx)n : it is symmetric like sin x on [0, p/q] and x n ( p − qx)n sin x when integrated in that interval should have a value consistent with the integral’s upper and lower bounds. 2.2. The n = 2 Case. With f (x) = x 2 ( p − qx)2 , repeated integration by parts gives  p/q f (x) sin x d x = f (0) ( p/q, 0) − f (2) ( p/q, 0) + f (4) ( p/q, 0), (6) 0

where f (k) ( p/q, 0) = f (k) ( p/q) + f (k) (0). Multiplying out f (x), we have f (x) = x 2 ( p − qx)2 = q 2 x 4 − 2 pqx 3 + p 2 x 2 .

(7)

Derivatives for this function are easily computed. The values of these derivatives at the endpoints 0 and p/q are given in Table 1. Table 1. Derivatives of x 2 ( p − qx)2 .

k

f (k) (0)

f (k) ( p/q)

0

0

0

1

0

2

2! · p

3

−3! · 2 pq

3! · 2 pq

4

4! · q 2

4! · q 2

0 2

2! · p 2

Using Table 1, with the same logic used for the inequalities in (5), we form the inequality  2 p p2 2 2 0 < −4 p + 48q ≤ (8) q 4q 554

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

and letting p = 7 and q = 2 we get −4 p 2 + 48q 2 = −4, a contradiction of the lower bound. 2.3. The n = 3, 4 Cases. Similar calculations can be carried out for the n = 3 and n = 4 cases. The inequalities for each are p 0 < −144 p q + 1440q ≤ q 2



3

p2 4q 2

3 (9)

and p 0 < 48 p − 8640 p q + 80640q ≤ q 4

2 2

4



p2 4q

4 ,

(10)

respectively.3 For the n = 3 case, when p/q equals 3/1, 13/4, 16/5, and 19/6 the upper or lower bound of (9) is contradicted. We discover that 22/7 is not π using (10), the n = 4 case. We have evidence that our method can be used to prove π is irrational. 3. PROVING π IS IRRATIONAL. 3.1. The General Case. Referring to Table 1, it is likely that f (x) = x n ( p − qx)n will be such that the alternating sum of its even derivatives evaluated at the endpoints 0 and p/q will be divisible by n!. If the integral in 

p/q

0<

p x ( p − qx) sin x d x ≤ q n

0

n



p2 4q 2

n < p 2n+1

(11)

is divisible by n!, then the upper bound in (11) can be used to prove π is irrational. This follows as the integral is increasing with n factorially, but the upper bound has polynomial growth. We know factorial growth exceeds polynomial—see [16, Equation 10, p. 764]; [3, Example 2, p. 86] gives a direct proof of this result. 3.2. Proving the General Case. The lower and upper bounds of (11) follow from the properties of the integrand. Repeated integration by parts establishes that 

p/q

x n ( p − qx)n sin x d x = 0

n  (−1)k f (2k) ( p/q, 0).

(12)

k=0

Consequently, we need only prove that the right-hand side of (12) is divisible by n!. First, symmetry of f (x) allows us to consider only the left endpoint in this sum. This follows as the equation f (x) = f ( p/q − x), differentiated repeatedly, gives f  (x) = − f  ( p/q − x), f  (x) = f  ( p/q − x), and, by induction, f (k) (x) = (−1)k f (k) ( p/q − x). So f (k) (0) = (−1)k f (k) ( p/q). For the even derivatives, with which we are concerned, we have f (2k) (0) = f (2k) ( p/q). 3 Leibniz’s formula [1, Problem 4, p. 222] gives a means of calculating nth derivatives of a product of two functions. In the case of the product of two polynomials, all derivatives can be calculated by placing the derivatives of one polynomial along the top row of a table, the derivatives of the other polynomial along the left column, and forming a Pascal’s triangle in the interior of the table. After forming products of these row and column entries with the binomial coefficients of Pascal’s triangle, all derivatives are given by sums along interior diagonals of the table.

June–July 2010]

NOTES

555

Next, f (x) when expanded will have the form an x 2n + · · · + a0 x n . For k < n, f (0) = 0, and for k ≥ n, f (k) (0) is divisible by k! and therefore n!. We have established that the sum in (12) is divisible by n! and that π must be irrational. (k)

4. CONCLUSION. Niven gives two proofs of the irrationality of π. One has been cited in the introduction. The other occurs in his book on irrational numbers [13]; there he shows the irrationality of π 2 . We will re-examine these proofs. Looking at Hermite’s transcendence of e proof [8, p. 152], one sees definitions of two functions f (x) and F(x) with the derivatives of f (x) being used in the definition of F(x). An integral is then used with the integrand having e−x in it. In Niven’s π and π 2 proofs he defines one function as the sum of derivatives of the other, as Hermite does. The manipulations Niven performs are to obtain forms like Hermite’s. In both articles the integral of one function equals an expression involving the other. To someone unsteeped in Hermite’s technique the motivation for the proof must be unclear. In this note a concept motivates the introduction of the polynomial Niven defines. The concept is that if π is rational then the evaluation of a definite integral comprised of the product of two functions symmetric about x = π/2 should be consistent with bounds for the integral. This being shown not to be the case, a contradiction occurs and π is proven irrational. The graphs of sin x, x( p − qx), and their product give the concept—visually. The same logic used for π can be applied to π 2 . Assume a/b = π 2 . We have  n  a/b a a2 x n n dx ≤ x (a − bx) sin √ , (13) 0< b 4b a/b 0 with the same reasoning as before: the integrand by assumption is a symmetric function with its maximum at x = a/2b. The integral, using repeated integration by parts, evaluates to n   (−1)k ( a/b)2k+1 ( f (2k) ( p/q) + f (2k) (0))

(14)

k=0

where f (x) = x n (a − bx)n . With some factoring, this sum is n π  (−1)k bn−k a k ( f (2k) ( p/q) + f (2k) (0)). bn k=0

With a multiplication by bn /π to clear π/bn from this sum, we have then  n  x bn a a 2 bn a/b n d x = n!Rn ≤ x (a − bx)n sin √ < a 3n+1 , 0< π 0 π b 4b a/b

(15)

(16)

which gives a contradiction. Note: reproductions of older articles by Hermite [8] and others can be found in [2]. ACKNOWLEDGMENTS. I would like to thank E. F. for helping me to believe that one can spell π without an e. Thanks also go to Richard Foote of the University of Vermont for his patience with me over the years.

REFERENCES 1. T. Apostol, Calculus, vol. 1, 2nd ed., John Wiley, New York, 1967. 2. L. Berggren, J. Borwein, and P. Borwein, Pi: A Source Book, 3rd ed., Springer, New York, 2004.

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3. G. Chrystal, Algebra: An Elementary Textbook, vol. 2, 7th ed., American Mathematical Society, Providence, RI, 1964. 4. P. Eymard and J.-P. Lafon, The Number π, American Mathematical Society, Providence, RI, 2004. 5. K. W. Folley, Integration by parts, Amer. Math. Monthly 54 (1947) 542–543. doi:10.2307/2304674 6. G. H. Hardy, E. M. Wright, R. Heath-Brown, J. Silverman, and A. Wiles, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, London, 2008. 7. C. Hermite, Sur la fonction exponientielle, Compt. Rend. Acad. Sci. Paris 77 (1873) 18–24, 74–79, 226– 233, 285–293. , Oeuvres Compl´etes, vol. 3, Hermann, Paris, 1912. 8. 9. I. N. Herstein, Topics in Algebra, 2nd ed., John Wiley, New York, 1975. 10. J. Lambert, M´emoire sur quelques propri´et´es remarquables des quantiti´es transcendentes circulaires et logarithmiques, Histoirie de l’Acad´emie Royale des Sciences et des Belles-Lettres der Berlin 17 (1761) 265–276. 11. R. Larson and B. H. Edwards, Calculus, 9th ed., Brooks/Cole, Belmont, CA, 2010. 12. I. Niven, A simple proof that π is irrational, Bull. Amer. Math. Soc. 53 (1947) 509. doi:10.1090/ S0002-9904-1947-08821-2 13. , Irrational Numbers, Carus Mathematical Monographs, no. 11, Mathematical Association of America, Washington, DC, 1985. 14. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. 15. G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, Mathematical Association of America, Washington, DC, 2007. 16. J. Stewart, Calculus: Early Transcendentals, 5th ed., Thomson Brooks/Cole, Belmont, CA, 2003. Naples, FL 34108 [email protected]

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PROBLEMS AND SOLUTIONS Edited by Gerald A. Edgar, Doug Hensley, Douglas B. West with the collaboration of Itshak Borosh, Paul Bracken, Ezra A. Brown, Randall Dougherty, Tam´as Erd´elyi, Zachary Franco, Christian Friesen, Ira M. Gessel, L´aszl´o Lipt´ak, Frederick W. Luttmann, Vania Mascioni, Frank B. Miles, Bogdan Petrenko, Richard Pfiefer, Cecil C. Rousseau, Leonard Smiley, Kenneth Stolarsky, Richard Stong, Walter Stromquist, Daniel Ullman, Charles Vanden Eynden, Sam Vandervelde, and Fuzhen Zhang.

Proposed problems and solutions should be sent in duplicate to the MONTHLY problems address on the inside front cover. Submitted solutions should arrive at that address before October 31, 2010. Additional information, such as generalizations and references, is welcome. The problem number and the solver’s name and address should appear on each solution. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.

PROBLEMS 11496 (April, 2010, p. 370) Correction: On the left, square s(A A T ) and s(B B T ). 11509. Proposed by William Stanford, University of Illinois-Chicago, Chicago, IL. Let m be a positive integer. Prove that  2  m 2 −m+1 m −2m+1 1 k−m m 2  = 2m−1 . m m k k k=m 11510. Proposed by Vlad Matei, student, University of Bucharest, Bucharest, Romania. Prove that if I is the n-by-n identity matrix, A is an n-by-n matrix with rational entries, A  = I , p is prime with p ≡ 3 (mod 4), and p > n + 1, then A p + A  = 2I . 11511. Proposed by Retkes Zoltan, Szeged, Hungary. For a triangle ABC, let f A denote the distance from A to the intersection of the line bisecting angle B AC with edge BC, and define f B and f C similarly. Prove that ABC is equilateral if and only if f A = f B = fC . 11512. Proposed by Finbarr Holland, University College Cork, Cork, Ireland. Let N be a nonnegative integer. For x ≥ 0, prove that   N −m+1 k m  1 N x ≥ 1 + x + · · · + xN. m! k m=0 k=1 11513. Proposed by P´al P´eter D´alyay, Szeged, Hungary. For a triangle with area F, semiperimeter s, inradius r , circumradius R, and heights h a , h b , and h c , show that 5(h a + h b + h c ) ≥

10r (5R − r ) 2s F + 18r ≥ . Rr R

doi:10.4169/000298910X492862

558

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

11514. Proposed by Mihaly Bencze, Brasov,  Romania. Let k be a positive integer, and n let a1 , . . . , an be positive numbers such that i=1 aik = 1. Show that n 

1 ai + n i=1

i=1

ai

≥ n 1−1/k + n n/k .

11515. Proposed by Estelle L. Basor, American Institute of Mathematics, Palo Alto, CA, Steven N. Evans, University of California, Berkeley, CA, and Kent E. Morrison, California Polytechnic State University, San Luis Obispo, CA. Find a closed-form expression for ∞ 

  4n sin4 2−n θ .

n=1

SOLUTIONS An Old Four-Squares Chestnut 11374 [2008, 568]. Proposed by Harley Flanders and Hugh L. Montgomery, University of Michigan, Ann Arbor, MI. Let a, b, c, and m be positive integers such that abcm = 1 + a 2 + b2 + c2 . Show that m = 4. Solution by Afonso Bandeira and Joel Moreira, Universidade de Coimbra, Portugal, and Jo˜ao Guerreiro, Instituto Superior T´ecnico, Portugal. Viewing the equation modulo 4 shows that 4 divides m. Let n = m/4. Now suppose there is a solution with n > 1. Let (a, b, c) be such a solution where a + b + c is minimal. Name the values so that a ≥ b ≥ c. Now a is a solution to the quadratic equation x 2 − x(4bcn) + (b2 + c2 + 1) = 0. By Vieta’s formula, another solution is a  , where a  = 4bcn − a. If a  ≥ a, then a 2 + b2 + c2 + 1 = 4abcn ≥ 2a 2 , and so a 2 ≤ b2 + c2 + 1 ≤ 2b2 + 1. Now a 2 < a 2 + 1 ≤ 2b2 + 2 ≤ 4b2 , so a < 2b. This yields 4abcn > 2a 2 cn ≥ 4a 2 ≥ a 2 + b2 + c2 + 1, which contradicts (a, b, c) being a solution. Thus (a  , b, c) is a solution that contradicts the minimality of a + b + c. We conclude that n > 1 is impossible, so n = 1 and m = 4. Editorial comment. We print this proof because of its brevity. A. Hurwitz showed in ¨ Uber eine Aufgabe der unbestimmten Analysis, Arch. Math. Phys. 3 (1907) 185–196, that x12 + x22 + · · · + xn2 = kx1 x2 . . . xn has no solution in positive integers if k > n, from which the present claim follows directly. This reference was supplied by each of S. Gao, W. C. Jagy, J. H. Jaroma, and J. P. Robertson. A new proof of Hurwitz’s theorem may be found in S. Gao, C. Caliskan, and S. Rong, Some properties of ndimensional generalized Markoff equation, Congr. Numer. 177 (2005) 217–221. Also solved by R. Chapman (U.K.), J. Christopher, P. Corn, S. Gao, H. S. Hwang & K. J. Kim (Korea), ´ Pit´e (France), I. M. Isaacs, W. C. Jagy, J. H. Jaroma, O. Kouba (Syria), O. P. Lossers (Netherlands), E. C. R. Pranesachar (India), J. P. Robertson, B. Schmuland (Canada), N. C. Singer, R. Stong, H. T. Tang, M. Tetiva (Romania), Fisher Problem Group, Szeged Problem Solving Group “Fej´ental´altuka” (Hungary), GCHQ Problem Solving Group (U.K.), Microsoft Research Problems Group, NSA Problems Group, and the proposers.

Perpendicular Half-Area 11392 [2008, 855]. Proposed by Omran Kouba, Higher Institute for Applied Science and Technology, Damascus, Syria. Let the consecutive vertices of a regular n-gon P June–July 2010]

PROBLEMS AND SOLUTIONS

559

be denoted A0 , . . . , An−1 , in order, and let An = A0 . Let M be a point such that for 0 ≤ k < n the perpendicular projections of M onto each line Ak Ak+1 lie interior to the segment (Ak , Ak+1 ). Let Bk be the projection of M onto Ak Ak+1 . Show that n−1  k=0

Area( (M Ak Bk )) =

1 Area(P). 2 y

Solution by P´al P´eter D´alyay, Szeged, Hungary. Select as unit of length the X radius of the circumcircle of Y the regular n-gon. Use the coordinate system x O y in the plane so that vertex Ak has M coordinates (xk , yk ) with xk = cos(2kπ/n) and x O yk = sin(2kπ/n) for A0 0 ≤ k ≤ n. Let M have coordinates (ρ cos φ, ρ sin φ). Fix one index k. If the axes of Ak Bk coordinate system x O y are Ck rotated by the   angle (2k + 1)π /n − π, then we obtain the axes of a new Ak+1 coordinate system  X OY . Note   X O M = φ − (2k + 1)π /n + π. Let H be the point where B M crosses OY , and k k let Ck be the midpoint of the segment Ak Ak+1 . Since the axes of the coordinate system X OY are parallel to Bk M and Ak Ak+1 , respectively, we have

1 (2k + 1)π +π Ak Bk = Ak Ck − Bk Ck = Ak Ak+1 − ρ sin φ − 2 n

π (2k + 1)π = sin + ρ sin φ − , n n

(2k + 1)π +π Bk M = Ck O + Hk M = Ck O + ρ cos φ − n

π (2k + 1)π = cos − ρ cos φ − . n n Therefore,   2 Area (M Ak Bk ) = Ak Bk · Bk M 



π π (2k + 1)π (2k + 1)π = sin + ρ sin φ − cos − ρ cos φ − n n n n





2π 2(2k + 1)π 1 2(k + 1)π ρ2 sin 2φ − = sin + ρ sin φ − − . (1) 2 n n 2 n Recall that for α, β ∈ R and β  = 2sπ with s ∈ Z,

n−1  1 sin(nβ/2) sin α + (n − 1)β . sin(α + kβ) = sin(β/2) 2 k=0 560

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

Thus, since n ≥ 3 implies that 2π/n  = 2sπ with s ∈ Z, and since φ − 2(k + 1)π/n = (φ − 2π/n) − 2kπ/n, we have



n−1 n−1   2(k + 1)π 2(2k + 1)π sin φ− sin 2φ − =0= . n n k=0 k=0 Summing both sides of (1) over k, we obtain the required result:

n−1    n 2π Area (M Ak Bk ) = sin , 2 2 n k=0 and this last expression gives the area of P. Also solved by M. Bataille (France), D. Beckwith, R. Chapman (U.K.), C. Curtis, J. Freeman, D. Grinberg, J.-P. Grivaux (France), K. Hanes, E. A. Herman, S. Hitotumatu (Japan), E. J. Ionascu, L. R. King, P. T. Krasopoulos (Greece), J. H. Lindsey II, O. P. Lossers (Netherlands), V. Mihai (Canada), C. R. Pranesachar (India), M. A. Prasad (India), R. A. Russell, A. Stadler (Switzerland), R. Stong, M. Tetiva (Romania), J. Vinuesa (Spain), A. Vorobyov, Z. V¨or¨os (Hungary), M. Vowe (Switzerland), GCHQ Problem Solving Group (U.K.), Microsoft Research Problems Group, and the proposer.

Concurrent Lines 11393 [2008, 856]. Proposed by Cosmin Pohoata, student, National College “Tudor Vianu,” Bucharest, Romania. In triangle ABC, let M and Q be points on segment AB, and similarly let N and R be points on AC, and P and S, points on BC. Let d1 be the line through M, N , d2 the line through P, Q, and d3 the line through R, S. Let ρ(X, Y, Z) denote the ratio of the length of XZ to that of XY. Let m = ρ(M, A, B), n = ρ(N , A, C), p = ρ(P, B, C), q = ρ(Q, B, A), r = ρ(R, C, A), and s = ρ(S, C, B). Prove that the lines (d1 , d2 , d3 ) are concurrent if and only if mpr + nqs + mq + nr + ps = 1. Solution by Michel Bataille, Rouen, France. We use barycentric coordinates relative to (A, B, C), and accordingly we write U (u 1 , u 2 , u 3 ) as an abbreviation for “U = (u 1 A + u 2 B + u 3 C)/(u 1 + u 2 + u 3 ).” (When u 1 + u 2 + u 3 = 0 we obtain a “point at infinity”). With this convention we have M(m, 1, 0), N (n, 0, 1), P(0, p, 1), Q(1, q, 0), R(1, 0, r ), and S(0, 1, s). The equation of line d1 is

x m n

y 1 0 = 0, that is, x = my + nz.

z 0 1 Similarly, the equation of line d2 is y = pz + qx, and the equation of line d3 is z = r x + sy. These three lines are parallel (concurrent at a point at infinity) or concurrent (literally) if and only if

−1 q r

m −1 s = 0.

n p −1 This can be rewritten as mpr + nqs + mq + nr + ps = 1,

(∗)

so this is a necessary condition for concurrence of d1 , d2 , d3 . Conversely, suppose that (∗) holds. If d1 , d2 , d3 were parallel, then the point at infinity on d1 , namely (n − m, −1 − n, 1 + m), would also lie on d2 and d3 . This means June–July 2010]

PROBLEMS AND SOLUTIONS

561

mq = 1 + n + p + mp + qn and nr = 1 + s + m + sn + r m. Since m, n, q, r, s are nonnegative, it follows that mq + nr ≥ 2. But mq + nr ≤ 1 follows from (∗). This contradiction shows that d1 , d2 , d3 cannot be parallel, and must instead be concurrent. Also solved by R. Chapman (U.K.), P. P. D´alyay (Hungary), M. Goldenberg & M. Kaplan, D. Grinberg, J. Grivaux (France), S. Hitotumatu (Japan), B.-T. Iordache (Romania), O. Kouba (Syria), J. H. Lindsey II, R. Nandan, C. R. Pranesachar (India), R. Stong, M. Tetiva (Romania), R. S. Tiberio, A. Vorobyov, Z. V¨or¨os (Hungary), J. B. Zacharias, GCHQ Problem Solving Group (U.K.), and the proposer.

Jensenoid Inequalities 11399 [2008, 948]. Proposed by Biaggi Ricceri, University of Catania, Catania, Italy. Let (, F , μ) be a measure space with finite nonzero measure M, and let p > 0. Let f be a lower semicontinuous function on R with the property that f has no global minimum, but for each λ > 0, the function t → f (t) + λ|t| p does have a unique global minimum. Show that exactly one of the two following assertions holds: (a) For every u ∈ L p () that is not essentially constant,  

1/ p   1 p < |u(x)| dμ f (u(x)) dμ, Mf M   and f (t) < f (s) whenever t > 0 and −t ≤ s < t. (b) For every u ∈ L p () that is not essentially constant,  

1/ p   1 < |u(x)| p dμ f (u(x)) dμ, Mf − M   and f (−t) < f (s) whenever t > 0 and −t < s ≤ t. Solution by Julien Grivaux, student, Universit´e Pierre et Marie Curie, Paris, France. First note that we may assume that p = 1. Indeed, let θ : R → R be defined by θ(t) = f (t) = f (θ −1 (t)) and  u (t) = θ(u(t)). Then signum(t)|t| p , and let    

1/ p   

    p   f  f u(t) and f ±  |u| u = f ± . u (t) = 







We may also assume without loss of generality that M = 1. For λ > 0, let φ(λ) be the unique value where the function t → f (t) + λ|t| reaches its minimum. Lemma 1. The function φ is continuous on (0, ∞). Proof. Let λ be positive and let λn  be a sequence of positive numbers converging to λ. Letting tn = φ(λn ), we have f (t) + λn |t| ≥ f (tn ) + λn |tn |. Let λ0 be such that 0 < λ0 < λ and m = infR ( f (t) + λ0 |t|). Now f (tn ) + λn |tn | = f (tn ) + λ0 |tn | + (λn − λ0 )|tn | ≥ m + (λn − λ0 )|tn |. This proves that for all t, (λn − λ0 )|tn | ≤ f (t) + λn |t| − m, so that for n large enough that λn − λ0 > 12 (λ − λ0 ), taking t = 0 gives |tn | < 2( f (0) − m)/(λ − λ0 ). Thus tn  is bounded. Let t  be a limit point of tn . There exists a subsequence tψ(n)  which converges to t  . For all t in R, f (t) + λn |t| ≥ f (tn ) + λn |tn |. By lower semicontinuity, for all t,

    f (t) + λ|t| = lim inf f (t) + λψ(n) |t| ≥ lim inf f (tψ(n) ) + λψ(n) tψ(n) = lim inf f (tψ(n) ) + λ|t  | ≥ f (t  ) + λ|t  |. 562

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

By the uniqueness of the minimum, t  = φ(λ). Since tn  is bounded we conclude that tn  converges to φ(λ). This shows that φ is continuous. Lemma 2. limλ→+∞ φ(λ) = 0 and limλ→0+ |φ(λ)| = +∞. Proof. Let λn  be a sequence such that limn→∞ λn = +∞, and let tn = φ(λn ). For t ∈ R, we have f (tn )/λn + |tn | ≤ f (t)/λn + |t|, and in particular f (tn )/λn + |tn | ≤ f (0)/λn . Let λ0 be a fixed positive value, and let m = infR [ f (t) + λ0 |t|]. Now f (tn ) ≥ m − λ0 |tn |, so (1 − λ0 /λn )|tn | ≤ f (0) − m /λn . Therefore limn→∞ tn = 0. For the other claim of the lemma, let λn  be a positive sequence that tends to zero, let tn = φ(λn ), and let t  be a limit point of tn  (if one exists). The argument of Lemma 1 proves that for any real t, f (t) ≥ f (t  ). That makes f (t  ) a global minimum for f , contrary to the hypothesis. Since tn  has no limit point, limn→∞ |tn | = +∞. From these two lemmas, we see that the range of φ contains (0, ∞) or (−∞, 0) (but not both). We will show that in the first case conclusion (a) holds. Similarly, the second case leads to (b).   Assume the range contains (0, ∞), and let m(λ) = inf R f (t) + λ|t| . Now f (t) ≥   supλ m(λ) − λ|t| . If t = φ(λ), then f (φ(λ)) = m(λ) − λ|φ(λ)|. Thus f is the pointwise supremum of a family of affine functions on (0, ∞), so f is convex there. We claim that f is actually strictly convex. Indeed, if f is affine on some interval [a, b] with 0 < a < b, then we can choose λ such that the function f λ given by f λ (t) = f (t) + λ|t| reaches its infimum at a point of (a, b). Since f λ is is affine on this interval, it is minimized at an interior point only if it is constant on that interval, which contradicts the uniqueness of the minimum point. Let s, t be given with t > 0 and −t ≤ s < t. There exists λ such that t = φ(λ). Thus f (s) + λ|s| > f (t) + λ|t| ≥ f (t) + λ|s|. We obtain f (s) > f (t). (If −t ≤ s ≤ t, we obtain f (s) ≥ f (t).)  For the integral inequality, we have −|u(x)| ≤ u(x) ≤ |u(x)|. So f (u(x)) ≥ f |u(x)| . Since f is convex, Jensen’s inequality yields 

    f (u) ≥ f |u| ≥ f |u| . 





It is a strict inequality since u is not essentially constant and f is strictly convex. Also solved by R. Stong.

Squares On Graphs 11402 [2008, 949]. Proposed by Doru Catalin Barboianu, Infarom Publishing, Craiova, Romania Let f : [0, 1] → [0, ∞) be a continuous function such that f (0) = f (1) = 0 and f (x) > 0 for 0 < x < 1. Show that there exists a square with two vertices in the interval (0,1) on the x-axis and the other two vertices on the graph of f . Solution by Byron Schmuland and Peter Hooper, University of Alberta, Edmonton, AB, Canada. Extend f by letting f (x) = 0 for x ≥ 1. Define g(x) = f (x + f (x)) − f (x) for x ≥ 0. If there exists x ∈ (0, 1) with g(x) = 0, then a square as required exists with vertices (x, 0), June–July 2010]

(x + f (x), 0),

(x, f (x)),

(x + f (x), f (x)).

PROBLEMS AND SOLUTIONS

563

Now g is continuous, so to show that such x exists we will show that y, z ∈ (0, 1) exist with g(y) ≥ 0 and g(z) ≤ 0. Let z be a value where f takes its maximum. Then f (z) ≥ f (z + f (z)), so that g(z) ≤ 0. Since 0 + f (0) = 0 < z < z + f (z), by continuity there is a value y ∈ (0, z) so that y + f (y) = z. Hence g(y) = f (y + f (y)) − f (y) = f (z) − f (y) ≥ 0. Editorial comment. P´al P´eter D´alyay (Hungary) noted a generalization: Given any p > 0, there exists a rectangle with base-to-height ratio p having two vertices on the x-axis and the other two vertices on the graph of f . ´ Also solved by B. M. Abrego & S. Fern´andez-Merchant, F. D. Ancel, K. F. Andersen (Canada), R. Bagby, N. Caro (Brazil), D. Chakerian, R. Chapman (U.K.), B. Cipra, P. Corn, C. Curtis, P. P. D´alyay (Hungary), C. Diminnie & R. Zarnowski, P. J. Fitzsimmons, D. Fleischman, T. Forg´acs, O. Geupel (Germany), D. Grinberg, J. Grivaux (France), J. M. Groah, E. A. Herman, S. J. Herschkorn, E. J. Ionascu, A. Kumar & C. Gibbard (U.S.A. & Canada), S. C. Locke, O. P. Lossers (Netherlands), R. Martin (Germany), K. McInturff, M. McMullen, ´ Plaza & S. Falc´on (Spain), K. A. Ross, M. D. Meyerson R. Mortini M. J. Nielsen, M. Nyenhuis (Canada), A. T. Rucker, J. Schaer (Canada), K. Schilling, E. Shrader, A. Stadler (Switzerland), R. Stong, B. Taber, M. Tetiva (Romania), T. Thomas (U.K.), J. B. Zacharias & K. Greeson, BSI Problems Group (Germany), GCHQ Problem Solving Group (U.K.), Lafayette College Problem Group, Microsoft Research Problems Group, Missouri State University Problem Solving Group, Northwestern University Math Problem Solving Group, NSA Problems Group, and the proposer.

A Trig Series Rate 11410 [2009, 83]. Proposed by Omran Kouba, Higher Institute for Applied Sciences and Technology, Damascus, Syria. For 0 < φ < π/2, find   ∞  1 (−1)n−1 sin2 (nx) 2 −2 log cos φ + lim x sin (nφ) . x→0 2 n (nx)2 n=1 Solution by Otto B. Ruehr, Michigan Technological University, Houghton, MI. We begin with three elementary identities. The first is ∞ 

r n sin2 nφ =

n=1

r (r + 1) sin2 φ  . (1 − r ) (1 − r )2 + 4r sin2 φ

(i)

This is derived by writing sin2 nφ in terms of exponentials and summing the resulting geometric series. Now divide (i) by r and integrate with respect to r to get   ∞  (1 − r )2 + 4r sin2 φ rn 1 2 sin nφ = log . (ii) n 4 (1 − r )2 n=1 Differentiate (i) with respect to r to obtain ∞ 

nr n−1 sin2 nφ =

n=1

  1 1 (r − 1)2 − 2(r 2 + 1) sin2 φ − . 2(1 − r )2 2 [(1 − r )2 + 4r sin2 φ]2

(iii)

The limit at r = −1 in (ii) gives us ∞  (−1)n−1 n=1

n

1 sin2 nφ = − log cos φ. 2

Now we can write the requested limit as lim x

x→0

564

−2

lim

r →−1+

 ∞  rn n=1

n

 sin2 nx 1 − 2 2 sin2 nφ. n x

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 

Here we have anticipated the divergent series that would result if the limx→0 were taken directly. Since the series as written is convergent, by the regularity of the Abel summation process it is equal to its Abel sum. Now, for |r | < 1, we can bring the outer limit under the sum, which yields lim

r →−1+

From (iii) we obtain

1 24

∞ 1 nr n sin2 nφ. 3 n=1

tan2 φ as the desired limit.

Also solved by R. Bagby, D. H. Bailey & J. M. Borwein (Canada), D. Beckwith, P. Bracken, R. Chapman (U.K.), H. Chen, P. P. D´alyay (Hungary), J. Grivaux (France), F. Holland (Ireland), K. L. Joiner, G. Keselman, A. Stadler (Switzerland), R. Stong, E. I. Verriest, and the proposer.

A Minimum Determinant 11415 [2009, 180]. Proposed by Finbarr Holland, University College Cork, Cork, Ireland. Let (A1 , . . . , An ) be a list of n positive-definite 2 × 2 matrices of complex numbers. Let G be the group of all unitary 2 × 2 complex matrices, and define the function F on the Cartesian product G n by   n  ∗ Uk Ak Uk . F(U ) = F(U1 , . . . , Un ) = det k=1

Show that minn F(U ) =

U ∈G

n  k=1

σ1 (Ak ) ·

n 

σ2 (Ak ),

k=1

where σ1 (A j ) and σ2 (A j ) denote the greatest and least eigenvalue of A j , respectively. Solution by Roger A. Horn, University of Utah, Salt Lake City, UT. It suffices to assume that the matrices and therefore Hermitian. Let A =  n Ai are positive semidefinite n n ∗ U A U , α = σ (A ), and β = σ i i i i=1 i i=1 1 i=1 2 (Ai ). Note that α ≥ β ≥ 0 and (α + β)/2 ≥ β. Let λ = σ (A) and μ = σ (A), so that λ ≥ μ and λ + μ = tr(A) = 2 n n 1 n ∗ tr(U A U ) = tr A = (σ (A ) + σ2 (Ai )) = α + β. i i i i i i=1 i=1 i=1 1 For Hermitian matrices C and D, Weyl’s inequality ensures that σ2 (C) + σ2 (D) ≤ σ with the definition of A it follows that μ = σ2 (A) ≥ 2 (C + D). From this  along n n ∗ σ (U A U ) = σ (A i i i ) = β. Since det A = λμ, we want to determine i i=1 2 i=1 2 min{λμ : λ + μ = α + β and λ ≥ μ ≥ β}. That is, for f (μ) = (α + β − μ)μ, we require min{ f (μ) : β ≤ μ ≤ 12 (α + β)}. Clearly, f  (μ) = α + β − 2μ ≥ 0 for μ ∈ [β, 12 (α + β)], so the minimum value of f (μ) is f (β) = αβ. If the unitary matrices are chosen such that Ui∗ Ai Ui = diag(σ1 (Ai ), σ2 (Ai )) for i = 1, . . . , n, then A = diag(α, β), and it follows that det(A) = αβ. Also solved by R. Chapman (U.K.), M. J. Englefield (Australia), J.-P. Grivaux (France), E. A. Herman, O. Kouba (Syria), J. H. Lindsey II, O. P. Lossers (Netherlands), R. Stong, M. Tetiva (Romania), E. I. Verriest, GCHQ Problem Solving Group, and the proposer.

June–July 2010]

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565

REVIEWS Edited by Jeffrey Nunemacher Mathematics and Computer Science, Ohio Wesleyan University, Delaware, OH 43015

Mathematical Modeling: A Case Studies Approach. By Reinhard Illner, C. Sean Bohum, Samantha McCollum, and Thea van Roode. American Mathematical Society, Providence, RI, 2005, xvi + 196 pp., ISBN 978-0-8218-3650-7, $35.

Reviewed by Charles R. Hampton Over the past twenty-five years or so courses in mathematical modeling have entered the curricula of many colleges and universities. Such a course provides us with a distinct way of demonstrating to our students the usefulness of mathematics in solving problems from a wide variety of sources. Whether it be in modeling the spread of disease, the flight of a golf ball, or the pricing of a financial derivative, mathematics leaves the ivory tower to get its hands on “real” problems. Unlike calculus or linear algebra, such a course does not have a small collection of well-defined syllabi. Local conditions determine the content and even the student level for which it is designed. Is the course for students prior to calculus or for those who have had the basic courses for the math major? Does the curriculum already include a course in operations research or mathematical programming? The wide variety of mathematical modeling courses has led to a correspondingly wide variety of textbooks designed to meet the demand. For courses aimed at students without calculus or linear algebra background we have a book like Hadlock [3]. For upper-level undergraduates we find books such as Giordano, Weir, and Fox [2], Beltrami [1], and the volume from the MAA by Mooney and Swift [6]. At a higher level there are options such as Mesterson-Gibbons [5]. It is for the middle of these categories that we find the most options for textbooks including Illner, Bohun, McCollum, and van Roode (IBMvR), which takes an unusual approach to the topic—more about it later. Although many books had been published on the applications of particular parts of mathematics and on the uses of mathematics in particular social and life sciences (as well as the physical sciences), the first text on the process of mathematical modeling for upper-level undergraduates was Maki and Thompson [4], published in 1973. This volume included a wide variety of applications to the life, social, and management sciences, and for several years it was the paradigm for subsequent texts. Each instructor would ask questions such as: Does the new text fit my course and students better than Maki and Thompson? Does it contain the topics from Maki and Thompson and add others not there that I wish to consider? Are its examples and problems more up to date? My own experience with a junior/senior-level mathematical modeling course began about 1980, although the course was labeled Applied Mathematics until the mid 1980s. Since my background was in the hard sciences, physics in particular, I looked for texts which included topics from the social and biological sciences. My desire was to give a course which included a wide variety of models taken from as wide a range of subjects as possible. For at least a decade I used the volume by Michael Olinick [7] as it most doi:10.4169/000298910X492871

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nearly met my preferences. Alas it went out of date and out of print. Caveat: I have for several years encouraged the author (an acquaintance from graduate school) to produce a second edition and have recently reviewed his proposed updated and revised edition for a publisher. In the last decade or so I have used several other texts, including most recently the one under review. For years at the January Joint Math Meetings I perused the exhibit area looking for texts to use in my course. I have collected quite a few over the years but by no means all of the options. Thus I have sources from which to find treatment of topics that I can use to supplement whatever text I select or to use to construct my course in place of a text. The latter option is cheaper for the students but is not as helpful to them or as easy on me in preparation. I much prefer to use a text around which to build my modeling course and to supplement it as I think appropriate. It is clear that each of us who teaches such a junior/senior-level modeling course has his or her own ideas of just what should be included. The breadth of material from which to choose is nearly overwhelming. One could design a course just from the applications of differential equations, or linear programming, or probability. Also one could concentrate on applications to one field outside of mathematics, such as biology or management science. For what it is worth, my preference is to include a wide range of types of models and sources of applications. It seems to me that in addition to illustrating the modeling process, one ought to include as many as possible of the following types: probabilistic and deterministic, discrete and continuous, qualitative and quantitative, as well as illustrating descriptive, predictive, and axiomatic models. Now let’s turn to the IBMvR text published by the American Mathematical Society to see how it might fit our course preferences. The subtitle of IBMvR, A Case Studies Approach, indicates the style chosen for the text. Written with the assistance of two of Illner’s students (McCollum and van Roode), who took his course at the University of Victoria, the nine chapters each consider a modeling problem and some of its extensions. Each chapter begins with a list of those mathematical ideas employed and ends with a short selection of exercises. I found the paucity of exercises in IBMvR to be a flaw, but one easily remedied with selections from other sources. IBMvR includes a chapter on dimensional analysis that is then illustrated via an interesting problem: the estimation of the strength of the 1945 nuclear explosion at Los Alamos based solely on the film released two years after the event. Here is a quick catalog of the problems considered: the dynamics of crystallization, hydrostatics, annuities, dimensional analysis applied as noted above, prey-predator, fishery management, determination of fair wage scales, and traffic flow both at the macro and micro level. Each of these cases has the potential to engage the student and each provides a vehicle to illustrate the modeling process. Several are applications not found in other modeling texts and thus make available useful examples for the instructor even if he/she chooses a different text. The authors’ goal was “to produce a text limited to what can comfortably be covered in one term. . . for a third-year audience.” They have succeeded admirably. While it is highly unlikely a single text will include those topics and only those topics preferred by any instructor (except the author of course), this one contains enough interesting ones to base a course upon. I used a bit more than half of the text in my course and supplemented it with student presentations of their own modeling projects and also with topics such as voting theory with Arrow’s Theorem, models of the spread of disease, and fair division problems. IBMvR is a welcome addition to the collection of texts for modeling courses. It fits well its intended audience, all of whom appreciate its small size and reasonable price. June–July 2010]

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I would use it again as the text for my junior/senior-level course, and it should be on anyone’s shortlist for such a course. REFERENCES 1. E. Beltrami, Mathematical Models for Society and Biology, Academic Press, San Diego, CA, 2002. 2. F. R. Giordano, M. D. Weir, and W. P. Fox, A First Course in Mathematical Modeling, 2nd ed., Brooks/Cole, Pacific Grove, CA, 1997. 3. C. Hadlock, Mathematical Modeling in the Environment, Mathematical Association of America, Washington, DC, 1998. 4. D. P. Maki and M. Thompson, Mathematical Models and Applications, Prentice-Hall, Englewood, Cliffs, NJ, 1973. 5. M. Mesterson-Gibbons, A Concrete Approach to Mathematical Modeling, John Wiley, New York, 1995. 6. D. Mooney and R. Swift, A Course in Mathematical Modeling, Mathematical Association of America, Washington, DC, 1999. 7. M. Olinick, An Introduction to Mathematical Models in the Social and Life Sciences, Addison-Wesley, Reading, MA, 1978. The College of Wooster, Wooster, OH 44691 Calvin College, Grand Rapids, MI 49546 [email protected]

Mathematics Is . . . “Mathematics is an experimental science, and definitions do not come first, but later on.” Oliver Heaviside, On operators in physical mathematics. Part II, Proceedings of the Royal Society of London 54 (1893) 121. —Submitted by Carl C. Gaither, Killeen, TX

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What’s Luck Got to Do with It? The History, Mathematics, and Psychology of the Gambler’s Illusion

Joseph Mazur “This is a fascinating book. It’s a fresh, funny, philosophical look at gambling by a mathematician who knows what he’s talking about, and who has quite obviously thought about gambling for a long time. Mazur isn’t afraid to make provocative, opinionated statements. I have not seen a gambling book like this before.” —Paul J. Nahin, author of Digital Dice Cloth $29.95 978-0-691-13890-9

Numbers Rule The Vexing Mathematics of Democracy, from Plato to the Present

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An Imaginary Tale The Story of Ɖ-1

Paul J. Nahin “[An Imaginary Tale] can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry.” —William Thompson, American Scientist “A book-length hymn of praise to the square root of minus one.” —Brian Rotman, Times Literary Supplement Princeton Science Library Paper $16.95 978-0-691-14600-3

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A Certain Ambiguity A Mathematical Novel

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