Central European Journal of Mathematics

Welcome Message from the Editor-in-Chief We are presenting the very ¯rst issue of Central European Journal of Mathematics. I would like to thank all the authors of submitted papers, the members of the Editorial Board as well as the editors. This ¯rst issue is the ¯rst step towards building a journal which will be helpful for the entire community of mathematicians, demanding but friendly for the authors, interesting and valuable for the readers, organised and distributed in a XXI century manner. This ¯rst issue is only our ¯rst success, we will welcome the next issues of CEJM on a quarterly basis. Although located in Central Eastern Europe and close to authors and readers from this region, it is also willing to serve and to be close to all mathematicians throughout the world. Thank you for your support, help and encouragement. Warsaw, January 2003 Andrzej BiaÃlynicki-Birula Editor-in-Chief Central European Journal of Mathematics

CEJM 1 (2003) 1{35

Free and non-free subgroups of the fundamental group of the Hawaiian Earrings Andreas Zastrow¤ Instytut Matematyki, Uniwersytet Gda¶ nski ul. Wita Stwosza 57, 80-952 Gda¶ nsk, Poland

Received 5 July 2002; revised 9 August 2002 Abstract: The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called \The Hawaiian Earrings". The fundamental group of this space is known to be a subgroup of the inverse limit of the ­ nitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains in­ nitely huge \virtual powers", i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of `natural generators’ can be proven to be not contained in any free basis of this free group. c Central European Science Journals. All rights reserved. ® Keywords: Fundamental group of the Hawaiian Earrings, word sequences, almost free structure MSC (2000): Primary 20E18; Secondary 55Q52

1

Introduction

¤

The purpose of this paper is to continue a discussion that was started in [17], namely of understanding the obstruction the prevents a fundamental group like that of the Hawaiian Earrings from being free. Although the Hawaiian Earrings Y resemble many asE-mail: [email protected]

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A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

pects of a graph (see Fig.2) and although their fundamental group can be described by a combinatorial concept (cf. [16].x2, [17].x1) that does not contain other relations than come from cancelling, their fundamental group is known since [11] not to be free (cf. also [15], [16].4.2/4.5). This fact has inspired some groups of authors even to suggest alternate algebraic concepts to be used instead of the classical concept of groups (\Omega-Groups" by Bogley and Sieradski in [14], [1] & [2], \Big Groups" by Cannon and Conner in [4]) in order to reinstall a free structure for the (generalized) fundamental group of the Hawaiian Earrings | however in this paper only the classical concepts of groups and of freeness of groups is used. In the course of this discussion it is not so hard to see (e.g. cf. [16].4.0(ii)) that all ¯nitely generated subgroups of º 1 (Y ) have to be free. Apart from that, so far two totally di®erent types of huge free subgroups have been found in this group: Namely in [17], that a subgroup that is characterized by a certain boundedness condition on the number of appearances of the combinatorial describers is free ([17].(Thm.0.1)); and Cannon & Conner have proved the fact that if the number of limit points of the cut-points of the combinatorial describers is not too big, this also characterizes a huge free subgroup ([3].(Thm.5.25) , cf. also [6].(Thm.2.1) and [6].(Thm.2.2) for shortenings of these proofs, resp.). In this context \Not too big" means that it is scattered on the sense of the order topology. In this paper now another uncountable subgroup is proven to be free that fails both characterizations as mentioned above. In addition it contains arbitrarily high \virtual powers" (cf. [17].2.27/2.22), i.e. special elements that can be constructed with the property that a ¯nite number of changes can turn them into arbitrarily high powers. Such elements are in one or the other way used in all proofs (known to the author) that the fundamental group of the Hawaiian Earrings is not free. However, for this free group some properties are di®erent with respect to the other two free groups. E.g., its free basis does not contain the letters (i.e. the homotopy classes of paths running around one of the loops of the Hawaiian Earrings only) as generators (7). It is also proven, that this system cannot be extended to a basis of this free group (Thm.6.1), but it is a subsystem of the bases of the other two free groups (as follows directly from the methods of proof in [6], [17] and [3]). In addition, although the system of cut-points of this group is not too wild (though it is not scattered), this system does not characterize freeness. Two other groups are also introduced in this paper which are constructed according to a similar scheme (in one case just one generation of isolated cut-points is added), and it is shown that they are not free. One of them even satis¯es that by restricting to blocks of combinatorial describers a countable non-free subgroup of the fundamental group of the Hawaiian Earring arises. In a preliminary version of [3] (Question 2.5.1 in the 1997-preprint) it had been asked whether such subgroups exist.

2

Intuitive description of the subgroups under discussion

In order to be able to state the results of this paper before having to run through technical de¯nitions, we only introduce intuitivly the combinatorial description of paths through

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3

w1 = a 1 w2 = a 1 a2 a 2 w3 = a 1 a2 a 3 a3 a 2 a3 a3 w4 = a 1 a2 a 3 a4 a 4 a3 a4 a4 a2 a3 a4 a4 a3 a 4 a4 Fig. 1 (belonging to Sect.2): This ­ gure shows the ­ rst four elements of a word sequence and illustrates its insertion structure as described in Sect.2/3.1

the Hawaiian Earrings in this chapter; more precise de¯nitions will follow in Sect. 3 (3.1{3.4): Essentially a path running through the Hawaiian Earrings cannot do more than run over various loops in some fancy order. Describing this fancy order is the hard part, because a continuous path can run over a sequence of in¯nitely many paths (which are getting smaller and smaller) and after that do something else. Hence, whatever combinatorial concept one uses in order to describe such paths, it has to cope with the phenomenon of \in¯nity in the interior". Ordinary sequences and biin¯nite sequences do not su±ce, since they are only \in¯nite towards their end(s)". Hence we use the concepts of sequences of words (\word sequences", cf. 3.1) that have an \insertion structure". I.e. the ith word of such a sequence combinatorially describes the moves that our path performs on the i biggest loops of the Hawaiian Earrings, and for the (i + 1)st level these letters are copied and those letters are inserted that in addition describe where and in what direction our path runs over the (i + 1)st loop. This concept is similar to the concept of an inverse limit of groups, but since here we are considering sequences of words (and not of elements of free groups) those elements of the inverse limit of the ¯nitely generated free groups that do not correspond to an element in the fundamental group of the Hawaiian EarQ1 rings (e.g. i=2[¬ 1 ; ¬ i ], cf. [9].(3.51), [13].x1(Thm), [16].2.1, [17].1.15) are automatically excluded by our combinatorics. With this understanding of word sequences the examples in this paper will be all given or derived from the word sequence ! = (!i )i2N which together with their insertion structure is pictured in (Fig.1) It will be proven that (i) The group generated by all substrings of these word sequence is free, uncountable, and neither scattered nor bounded (3.9(i), (v) & (vi)). A basis cannot be explicitly named, it has to be chosen by an uncountable selection mechanism that is developed below (Sect.5, in particular (7)). But we will also show that, although this group contains the entire system of letters as elements, this system cannot be extended to give a free basis for this group (6.1). (ii) The group that is generated by that word sequence where every letter in (Fig.1) is replaced by its square (and where hence only isolated points are added to the set of cut-points) is not free. In addition, it is possible to restrict this construction to

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A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

Fig. 2 (belonging to Sect. 1): This ­ gure shows the Hawaiian Earrings Y , i.e. a topoloical space Y which consists of in­ nitely many circles that are embedded into the plane in that way as drawn above, in the same way the ­ rst four of them are drawn in the above ­ gure. P is the name of the common tangent point of all of these rings.

a group which is generated by certain special \blocks" of the corresponding word sequence, and when then the letters are added to the generators, a countable non-free subgroup is obtained (3.9(iii)). (iii) Finally we take a look at that subgroup that is generated by all substrings that are constructed in a similar way than the one pictured in (Fig.1), but where instead of inserting two letters ¬ i+1 after each letter ¬ i , for each i we may choose between two or three letters (3.6(i)). We point out that (with a method of proof from [16].x4) it follows that neither this group nor the analogous construction where we restrict to subblocks are free (3.9(ii)). The reader might observe that for these kind of groups it thus does not need that much di®erence in construction to make a free group non-free or vice versa. This paper is organized as follows: The ¯rst section is devoted to giving the technical de¯nitions that are needed to state the above results precisely. The corresponding statement of these results will follow at the end of this section (Thm.3.9). The proofs of these claims will then be spread over the Sections 4{7.

3

Technical statements of the results of this paper.

3.1 On the De¯nition of word sequences and their motivation: Word sequences are used by the author as a self-contained combinatorial concept that is able to completely describe the group º 1 (Y ). Since [16].x2 and [17].x1 contain a complete de¯nition of word sequences, we can restrict ourselves here to vaguely describing the idea in the following: Since by [9], [10] and [13] º 1 (Y ) can be described as a subgroup of the inverse limit of the ¯nitely generated free groups, i.e. º 1 (Y ) » lim Á¡ (Fn ) where Fn = h¬ 1 ; ¬ 2 ; : : : ; ¬ n j |i, in principle each element of º 1 (Y ) can be described by a sequence of group elements, the ith element of which is contained in F i . Word sequences are now based on the idea that it may be more suitable than reducing every word individually to choose a word

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5

!i in f¬ 1 ; ¬ 2 ; : : : ; ¬ i g as the ith entry of such a sequence in such a way that !i+1 can be obtained from !i by merely inserting letters ¬ i+1 and ¬ ¡1 i+1 , only. ¡1 ¡1 Example: If the ¯nite word ¬ 1 ¬ 2 ¬ 3 ¬ 2 ¬ 1 should be described as a word sequence (!1 ; !2 ; : : :), then despite the fact that the ¯rst and second entry of the corresponding lim Á¡ (F n )-sequence are the neutral elements in F1 and F2 , respectively, !1 and !2 cannot be choosen as empty words. Instead of this we have to de¯ne !1 = ¬ 1 ¬ 1¡1 , !2 = ¬ 1 ¬ 2 ¬ 2¡1 ¬ ¡1 1 and !3 = ¬ 1 ¬ 2 ¬ 3 ¬ 2¡1 ¬ ¡1 . Since our example is a word which is already contained in F 3, 1 for this example the !i for i ¶ 3 will be equal to !3 . Hence using word sequences often implies that one has to resist working with reduced words. However for certain tasks this concept has some advantages with respect to the classical idea of describing each element of an lim Á¡ (Fn )-sequence by a reduced word, since in case that some substring of !j reads \¬ k ¬ j ¬ ¡1 k " the classical concept implies that, if k < j, within the sequence (!1 ; !2 ; : : :) the letters ¬ k and ¬ ¡1 k show up the ¯rst time for the index j, and it might be that j ¾ k ! Hence we will call a sequence (!i )i2N where each !i is a word in f¬ 1§1 ;...; ¬ §1 i g and where §1 each !i follows from !i+1 be merely deleting the letters ¬ i+1 and ¬ i+1 a word sequence, provided that it in addition is reduced in the sense 3.3(i). Based on this de¯nition word sequences can be introduced and discussed as a self-contained combinatorial concept. Alternately, they can be introduced as an alternative concept for denoting those elements of lim Á¡ (F n ) that belong to that subgroup that is naturally isomorphic to º 1 (Y ). Since in this paper mainly combinatorial arguments will be used, we will usually regard º 1 (Y ) as \the group of word sequences", and we will use the letter \G" to denote this group.

3.2 Remark: Assume that w gives a parametrization of a path through the Hawaiian Earrings Y 3 P = w(0) = w(1) such that w has neither constant nor any other nullhomotopic segments and let ² = (² i )i2N be the word sequence describing w. Then P = w ¡1 (fP g) corresponds to the set of possibilities for splitting w up in such two subcurves w1 and w2 that both represent elements of º 1 (Y ) = º 1 (Y; P ). Hence each of these possibilities corresponds to one possibility of splitting up of each of the words of ² into two parts such that the sequence of the ¯rst parts gives the word sequence describing w1 and the sequence of the second parts gives the word sequence describing w2 . Assume that in the course of splitting up ² the ¯rst ki letters of ² i go into the ¯rst part. Then the sequence of integers µ = (k i )i2N appears to be the suitable information to determine our splitting up of ² . The set of all such \position sequences", i.e. of sequences of integers which gives such valid split-ups of a word sequence ² will be denoted by P(² ) (for a precise de¯nition see [16].2.6{2.7/[17].1.3{1.4). Based on the insertion structure of ² the set P(!) can completely be characterized by purely combinatorial means. If ` denotes the length (i.e. the number of letters of a word), then the sequences (0; 0; 0; : : :) and (`(² 1 ); `(² 2 ); : : :) always belong to P(² ), since they are regarded as giving the trivial splitups where the ¯rst or the second split-up sequence is empty, respectively. 0N and `N will be used as abbreviations for these particular position sequences. If µ = (ki )i2N 2 P(² ), then

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P(² ¡1 ) 3 µ¡1 := (`(² i ) ¡ ki )i2N , since this is the sequence that points at the corresponding place as µ, if ² is inverted. Based on their relations to the legitimate split-ups of word sequences, position sequences can be used to denote sub-word sequences of word sequences, i.e. ² [0N ;·] and ² [·;`N ] will be our notation for the two word sequences obtained from splitting up ² as above. In general any two position sequences µ1 ; µ2 2 P(² ) with µ1 < µ2 can be put into such an expression \² [·1;·2]" (1) to describe some subword-sequence of ² . Observe that in [16] the notation \¼ [·1;·2] (² )" was used instead of \² [·1;·2] ", where ¼ was to be understood as a \substring operator". Note that P(² ) is totally ordered, but not well-ordered (the bijective correspondence between P(² ) and P(² ) can be used to establish this order), hence the existence of maxima and minima within P(² ) is not trivial. However it can be proven (see references below) that for any two word sequences ² and ! there exists a maximal position sequence µ within P(² ) \ P(!) such that ² [0N ;·] = ! [0N ;·]. The corresponding word sequence which can equally be denoted as ² [0N ;·] or as ! [0N ;·] will be called the \greatest comm on initial word sequence" of ² and ! (cf. [16].2.9/(On 2.1), [17].1.6/1.7).

3.3 Convention for our notation: (i) In [16].2.3/[17].1.12 word sequences are de¯ned in such a way that any element of º 1 (Y ) is described by one and only one word sequence. Consequently, there are not any equivalence operations like elementary cancellations and insertions on the set of word sequences. This uniqueness has been achieved by requiring that, wherever in its ith word the letters ¬ i and ¬ i¡1 follow each other, then there has to exist j such that in the (i + j)th word other letters have been inserted between this ¬ i and ¬ ¡1 i , forming a subword which cannot be reduced to the empty string by elementary cancellations any more. Due to that property (\reducedness") word sequences must be regarded as reduced representatives of their º 1 (Y )-elements and hence the grouptheoretical multiplication within º 1 (Y ) cannot be described by simply concatenating the word sequences; instead the system of words obtained from having concatenated the words of our word sequence has to be transformed into a reduced word sequence again. This is done by the following formula ! ¢ ² := ! [0N ;·¡1] ²

[·;`N] ;

(2)

where µ is the greatest common initial word sequence of ! ¡1 and ² , the concatenation of word sequences is denoted by simply juxtaposing the corresponding symbols, while \¢" denotes the multiplication in º 1 (Y ). (ii) We further agree that the substring notation as introduced in (1) shall be used regardless whether the position sequences a; b 2 P(!) satisfy a < b or not. If a > b then this notation shall be understood as ! ¡1 [b¡1;a¡1] , and if a = b this notation describes the empty word sequence. Note that with this convention we

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

7

have ! [a;b] = ! [a;c] ¢! [c;b] for any a; b; c 2 P(!), regardless whether c is smaller, bigger or equal with respect to a or to b. (iii) Recall that by (2) we have to make a distinction between the \juxtaposition" or \concatenation" (we will be using these terms synonymously for word sequences) on the one hand, and the \product" or \multiplication" of word sequences on the other hand. For that purpose we will resist omitting our \¢"-symbol in all products, unless we want to stress that this is a place where juxtaposition and product do coincide, i.e. where cancellations are not to be performed. However, note that standard mathematical notation does not provide one with the possibility to make such a distinction if one has to write powers or to use big operators; in particular we will Q have to use the symbol \ " for both purposes, juxtaposition and product.

3.4 De¯nitions: Let g 2 lim Á¡ (Fn ) be a word sequence, then bi (g) denotes the number of occurrences of the letter ¬ §1 in a sequence describing g. For G as de¯ned at the end of 3.1, if g 2 G i and thus can be described by a word sequence !, this number is ¯nite and hence can be read from number of occurrences of ¬ §1 in each word !j with j ¶ i. If g is just i described by an lim Á¡ (F n )-sequence (g i )i2N , this number can only be read from looking at all g i simultaneously, and it can be in¯nite. Let b1 (g) := maxi (bi (g)). This number may be in¯nite, even for g 2 G. If g 2 G and if it is ¯nite, then g and a word sequence describing g are called \b ounded". The bounded word sequences form a subgroup in G which shall be denoted by G1 .

3.5 On alternate combinatorial concepts in order to describe the fundamental group of the Hawaiian Earrings: It appears that the various groups that are presently working on similar problems all developed their own combinatorial concepts and terminology in order to describe this group. For the reader’s convenience it is pointed out here, that most of the other concepts start to de¯ne that type of in¯nite products of groups that correspond to how countable many Z-factors describe the fundamental group of the Hawaiian earrings (called \Weak join" by Morgan & Morrison in [13], \¼ -products" by Eda in [7] & [8], \topologists’ product" by Gri±ths in [10].x4 and it is incorporated in the \Big Free" structure in [4]). In such de¯nitions, it is usually the ¯rst step to consider the compact totally linearly ordered spaces (e.g. in [3],x5.2, [4].x4, and called \order types" in [2] and [14], [7].Def.1.1{1.3) that correspond to our \set of position sequences" or to a dual object (cf. 3.2 or [17].1.3{1.4). However, in most of these other concepts considering these spaces is the ¯rst step of de¯nition, and based on that the in¯nite products are constructed by associating letters (or generators or group-elements) with points in these ordered spaces. With that other concept of in¯nite products it is also possible to extend such group-theoretical constructions to uncountable products with uncountable many factors (e.g. [6]).

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For describing the examples to be discussed in this paper the following notation will be used:

3.6 De¯nition: (i) ! (a1 ;a2;a3;:::), i.e. the symbol ! with a sequence of positive integers (a1 ; a2 ; a3 ; : : :) attached, denotes the word sequence de¯ned as follows: !1 = ¬

1

!2 = ¬

1

¢¬

!3 = ¬

1

¢¬

¢::: ¢¬ | 2 {z }2 a1 times

2

¢¬

¢ : : : ¢ ¬ ¢¬ | 3 {z }3

2

¢¬

¢ ::: ¢ ¬ ¢:::::: ¢ ¬ | 3 {z }3 a2 times {z

a2 times

|

2

a1 times

¢¬

¢::: ¢¬ | 3 {z }3 a2 times }

In general, the ith word of ! = ! (a1 ;a2 ;a3;:::) is de¯ned by ai¡1 times inserting the letter ¬ i after each occurrence of ¬ i¡1 in !i¡1 . In (Fig.1) ! (2;2;2;:::) had been pictured. (ii) Let ! be a word sequence and let ² be the word sequence obtained from ! by replacing all letters ¬ i , wherever they occur in !, by ¬ i+j . Then we call ² \the j th shift of !", or in formulae: ² = shj (!) (iii) Note that each of the word sequences ! (a1;a2;a3;:::) as de¯ned for 3.6(i) has an in¯nite number of subpatterns which satisfy a similar construction scheme. The following sequence of equations giving obvious product decompositions of ! (a1;a2;a3 ;:::) re°ects this fact: ! (a1 ;a2 ;:::) = ¬

¢ (sh1 (! (a2;a3 ;:::) ))

1

a1

=¬

1

¢ (¬

a2 a1

2

¢ (sh2 (! (a3 ;a4 ;:::) )) )

= :::

(3)

Each of these word sequences shj (! (aj+1;aj+2;:::)) is called \(sub)block of !", and this terminology was chosen to express the fact that this pattern occurs as a very special substring within ! = ! (a1;a2;:::) . Note that zero is permitted as an index of the sh-operator giving that the entire word sequence is also considered as a block. (iv) Let ! = ! (a1;a2;a3 ;:::) and let ² be the word sequence resulting from ! by replacing each letter ¬ i by its square ¬ i2 = ¬ i ¬ i . We de¯ne the \squaring-op erator sq" so as to perform this replacement for each letter within the given word sequence. Example: Let ! = ! (3;2;:::) and let ² = sq(!). Then the ¯rst three words of ² are: ²

1

=¬

1¬ 1 ;

²

2

=¬

1¬ 1

¢¬

2 ¬ 2 ¬ 2¬ 2 ¬ 2 ¬ 2 ;

²

3

=¬

1¬ 1

¢¬

2¬ 2

¢¬

3¬ 3 ¬ 3 ¬ 3

¢¬

2¬ 2

¢¬

3 ¬ 3 ¬ 3¬ 3

¢¬

2¬ 2

¢¬

3 ¬ 3 ¬ 3 ¬ 3:

(v) In analogy to 3.6(iii) we de¯ne the \blocks of squarings of the word-sequences" ! (a1 ;a2;:::)". Let ² = sq(! (a1 ;a2;:::)), then the following substrings of ² are considered as blocks: sq(shj (! (aj+1;aj+2;:::) )):

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Observe that by construction all word sequences ! (a1;a2;a3;:::) or sq(! (a1 ;a2 ;a3 ;:::) ) (with ai ¶ 2 8i) are virtually arbitrary high powers in the sense this expression had been used in the introduction. This can be seen by deleting the 1 + a1 + a1 ¢a2 + a1 ¢a2 ¢a3 + . . . . . . + a1 ¢ : : : ¢ai¡1 letters ¬ j with j µ i, since this turns the word sequences ! (a1 ;a2;a3;:::) into an (a1 ¢ : : : ¢ai )th power.

3.7 List of Examples under discussion: G1 := group of bounded word sequences as de¯ned in 3.4, H1 := group of all word sequences whose set of position sequences is scattered with respect to the order topology G2 := group generated by all substrings of ! (2;2;2;:::), H2 := group generated by the subblocks of ! (2;2;2;:::) , G3 := group generated by all substrings of all word sequences ! (a1;a2 ;a3 ;:::) with ai 2 f2; 3g, H3 := group generated by all the subblocks of these word sequences, G4 := group generated by all substrings of sq(! (2;2;2;:::)), H4 := group generated by the subblocks of this word sequence and by the letters ¬ i .

3.8 Technical phrasing of related recent results: As already mentioned in the introduction, in [17].0.1/x2 and [6].(Thm.1) G1 has been proven to be an uncountable free subgroup. In [3].(Thm.5.25) and [6].(Thm.2.2) H1 has also been proven to be an uncountable free subgroup.

3.9 THEOREMS of this paper: (i) (ii) (iii) (iv)

H2 is countable, G 2 is uncountable, but both of them are free (cf. 4.5 /5.4/5.9{5.12). G3 and H 3 are both uncountable, and both are not free (cf. 4.4). H4 is countable and G4 uncountable, and both of them are not free (cf. 4.3). f¬ 1 ; ¬ 2 ; ¬ 3 ; : : :g » H 2 » G 2 , but there does neither exist a free basis for G2 nor for H2 that entirely contains all of the ¬ i (Thm.6.1) (v) Except G 1 , none of the groups as mentioned in 3.7 is bounded (cf. 4.1(i)). (vi) Except H 1 , none of the groups as mentioned in 3.7 is scattered (cf. 4.1(ii)). The remainder of this paper is devoted to proving the various statements of this Theorem. The proofs of those claims that can be given rather quickly will be gathered together in Sect.4, while the remaining Sections 5, 6 and 7 will be devoted to showing how those statements can be obtained whose proofs rely on some auxiliary statements.

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Short proofs of some claims of Theorem 3.9

4.1 On the (un)boundedness and (un)scatteredness results of Theorem 3.9: Q1 (i) 3.9(v) can be seen as follows: H1 is not bounded because i=1 ¬ ii is a simple sequence of letters, so that the cut-points of this sequence form a monotone sequence of points accumulating only towards the ends. Except for the endpoint, every cutpoint is isolated, and after having removed these isolated points, the endpoint is also isolated. Hence this gives an easy example of a scattered, but unbounded word sequence. Further observe, that within ! (a1 ;a2;a3;:::) the letter ¬ i occurs a1 ¢a2 ¢ : : : ¢ai¡1 times. Hence this is an unbounded word-sequence, and since G 2 ; H 2 ; G3 ; H3 ; G 4 and H4 contain similar word sequences, it follows that all of these groups are also unbounded. (ii) 3.9(vi) can be seen as follows: An example for a bounded but not scattered wordsequence can be constructed as follows: !n1 := !1 := ¬ 1 , !n2 := !3 := ¬ 2 ¬ 1 ¬ 3 , !n3 := !7 = ¬ 4 ¬ 2 ¬ 5 ¬ 1 ¬ 6 ¬ 3 ¬ 7 . More precisely, this sequence is inductively constructed such that !ni+1 follows from !ni by inserting letters ¬ j that have not been used so far, before the ¯rst letter of !ni , between any two letters of !ni and after the last letter of !ni . I.e., the insertion structure of this word sequence is given by ¯rstly removing the middle open interval from the unit interval and then iteratively removing the middle open interval from the remaining connected components. That way one sees that the set of cut-points of this word sequence corresponds to the Cantor-set and hence is a perfect set, i.e. the opposite of being scattered. Let us consider a word sequence of type ! (a1;a2;a3;:::) and let us associate a path with this word sequence according to the standard scheme, i.e. by iteratively reserving intervals such that the associated path is supposed to run in each of those intervals through one of the loops of the Hawaiian Earrings according to the associated letter, while choosing these intervals adjacent to the ones that have been chosen before if no insertions are going to take place at this position, and leaving correspondingly space in the opposite case. This scheme of construction has been more thoroughly described in [17].1.14. By applying it one can see that the sequence of positions, whose ith element is the start point of the last block that is starting with ¬ i , accumulates towards the terminal position `N . However, each of these start points is at the same time the end-point of the preceding block, and since according to 3.6(iii) each of these blocks is just a shift of the entire word sequence and has the same construction scheme, it follows that for each of these start- & endpoints of blocks another sequence of start- & endpoints of ¯ner blocks can be constructed that accumulates to this points, and analogously for each of the positions in those sequences another sequence can be constructed and so on. Since this construction scheme for ¯nding accumulating position sequences can be in¯nitely many times iterated, it follows that, when trans¯nitely inductively removing isolated points in order to

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11

separate the scattered part from the perfect kernel, none of these positions will have ever to be removed. Hence they all belong to the perfect kernel of positions.

4.2 On the (un)countability claims of Theorem 3.9: Note that in this context only ¯nite products are de¯ned, hence it follows that a ¯nite or a countable generator system can only give a countable group, and on the other hand it is evident that an uncountable generating system of distinct elements gives an uncountable group. Based on these facts most of the (un)countability claims of 3.9 follow from the observation that the word sequences ! (a1 ;a2;a3;:::) have countably many subblocks but uncountably many distinct substrings, (provided that in¯nitely many of the ai are at least \2"). It is evident that the subblocks give a countable system, since they are indexed by the shift-index. In order to see the uncountability of the set of the substrings, take a look at (6). For the word sequence ! (2;2;2;:::) this formula introduces a system for denoting some position sequences by 0-1-sequences. Since any 0-1-sequence can occur and gives a di®erent position sequence p, and since the substrings ! (2;2;2;:::) [0N ;p] give de¯nitely distinct word sequences for distinct p, we get the uncountability of the group generated by all substrings. The uncountability of H 3 can be seen since there are uncountably many sequences (ai )i2N , if there are at least two choices for the ai , and they all give distinct word sequences. From these arguments all the (un)countability claims of 3.9 follow.

4.3 Proof that G 4 and H 4 are both non-free: We give the proof for H4 , the result for G4 follows since by Schreier’s Theorem free groups can only have free subgroups. On H4 now the following argument of Kurosh-Dunwoody can be applied (cf. the precise references at the end of this paragraph). Assume that H 4 is free. Since by an obvious shift of indices we have that ¹ H4 = h¬ 1 i ¤ H4 = Z ¤ H4 , it follows that H 4 cannot be ¯nitely freely generated. Using that hence H4 could only be in¯nitely freely generated, our assumption gives us the existence of a non-trivial free product-decomposition H4 = A¤B with ­ 0 := sq(! (2;2;2;:::) ) 2 A. Such a free product-decomposition can be constructed by letting A be generated by all those generators of our free H4 -basis, that do occur in the basis-representation of ­ 0 (which can only be ¯nitely many), and by letting B be generated by the remaining ones. Use ­ i (i ¶ 0) to denote the subblocks of ! (2;2;2;:::) , i.e. ­ i := sq(shi (! (2;2;2;:::) )) and note that (3)/3.6(iv) gives that ­

i¡1

=¬

2 i

¢ ­ i2

for all i 2 N. On products of that type one can apply the \Main Technical Lemma" 7.1, which states that if a product g = h2 ¢h0 2 is contained in A, then h and h0 are elements

12

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of A, also. Since ­ 0 2 A by construction, this lemma can be iteratively applied and gives that all ¬ i (i > 0) and all ­ i (i ¶ 0), i.e. that all generators of H 4 have to be contained in A, contradicting the assumption that we have a free product-decomposition with a non-trivial factor B, also. This is the desired contradiction telling us that H 4 cannot be free. Hence, what remains to be done is to prove the Main Technical Lemma. This proof shall only be outlined at this place (more details will be given in Section 7): One is using the normal form for representing group elements within a free product and the fact that within this normal form precisely the elements of A and of B are represented by words of length one, and that all other elements are represented by more lengthy products with its factors taken alternately from A and from B. When multiplying two of these normal forms with each other, a cancellation can only take place at the attaching point, and once the cancelling process is stuck, one has directly obtained a new normal form without any further adaptations. In the main part of the proof one assumes that least one of our factors h or h0 has a lengthy normal form, considers the corresponding product h ¢ h ¢h0 ¢h0 and discusses the possibilities for cancellation. A su±ciently careful distinguishing of all cases gives that there is enough symmetry in this product structure so that, if the cancellations process is not stuck before the last two letters of our product meet, then these last two letters do cancel completely against each other, also. A full workout of such an argument can be found in the original paper of Kurosh ([12]) in the situation where we have a commutator instead of a product of two squares (\g = h2 ¢h0 2 "). The observation that a similar argument also works for the double-square-structure is due to Dunwoody, but not worked out there ([5].(End of x3)). Hence we will discuss some details in Section 7 of this paper.

4.4

Proof that G 3 and H 3 are both non-free:

As in 4.3, we can restrict our considerations to showing that H 3 is non-free. This argument was given in [16].(4.3{4.6) as a proof that a certain subgroup of the fundamental group of the Hawaiian Earrings which was de¯ned by some boundedness on the order of growth of the word sequences is not free. However, in principle only the group here denoted by t u H3 was considered.

4.5 Proof that H 2 is free: The free basis for this group can be explicitly named, since the subblocks B0 := fshi (! (2;2;2;:::) ) j i 2 N0 g

(4)

as de¯ned as a generating set by 3.7/3.6(iii) turn out to be free already. Since i is precisely the lowest index of a letter that is contained in the ith subblock, the system of (4) ¯ts into a classical structure for obtaining a free system. Observe that single letters ¬ i , even if not de¯ned as generators within our free basis, are elements of this group. They can

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

13

be represented in our basis as ¬

i

= sh i¡1 (! (2;2;2;:::)) ¢ (sh i (! (2;2;2;:::) ))¡2

(5) t u

5

Proof that G 2 is free

This proof is a bit more complicated and hence spread over the entire section. As a preparation for e±ciently representing it we start by de¯ning an annotation system for the position sequences of ! (2;2;2;:::) and introducing the concept of coincidence patterns.

5.1 The notation system for the position sequence of ! (2;2;2;:::) : Our elements of P(! (2;2;2;:::) ) will be denoted by expressions of type \­ + k". Here ­ is a 0-1-sequence, k is a non-negative integer, and \+" is merely used as a separator. If k = 0, then the notation \­ " is permitted as an abbreviation for \­ + 0". We write down the ­ -sequences the same way like binary representations of the reals, i.e. with no space or separator between the digits, apart from the binary point. We also use the overbar to denote periods in the usual way. Note that \1:0" and \0:1 = 0:1111 : : :" de¯nitely denote di®erent position sequences. I.e. the ­ -sequences may only be understood as digit sequences and must not be confused with binary representations of reals. However, we hereby de¯ne a map by letting µ(­ ) be the real whose binary representation is ­ . We will call these ­ -sequences \labels" and regard those position sequences, which can be denoted as \­ + 0" as \labelled". Even, if our system of notations is strong enough to describe all position sequences of ! (2;2;2;:::) , those positions which need a non-zero k-component will consequently be called \unlabelled". It turns out that there are enough labelled position sequences within ! (2;2;2;:::), so that for an arbitrary element of P(! (2;2;2;:::) ) the nearest labelled position on the left hand side is only a ¯nite number of letters apart. This number of letters is what gives the k-component for our notation system. In order to understand the rules according to which our ­ -labels are constructed, recall that a position sequence in P(!) was originally de¯ned as a sequence, namely such a sequence where the ith entry is a place between the letters of the ith word of ! and where this sequence of places is compatible with the insertion structure of ! (cf. 3.2). Hence the in¯nite 0-1-sequences giving ­ -labels can in principle be de¯ned by labelling the positions between the letters of the ¯nite words by ¯nite 0-1-sequences, such that the labels of any two positions in two adjacent words which are related to each other by the insertion structure have labels which result from each other by appending an additional digit. Thus our in¯nite ­ -labels can be constructed by continuing the system the ¯rst four of

14

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

which steps are given by the formula below:

(6) In words this system can be described as follows: Recall that ! (2;2;2;:::) was de¯ned by inserting two letters ¬ i+1 after each letter ¬ i . Hence each of the corresponding positions of the ith word gives rise to three positions within the (i + 1)st word. Our labelling scheme is based on the principle that the last two of these three positions are described by appending the digits \0" and \1" to the corresponding label of our position in the ith word, while the ¯rst of these three positions remains unlabelled. Since insertions will not be taking place between this position and the next labelled position on the left hand side, we will also be able to denote this ¯rst position with help of a non-zero k-component. Note that position sequences which have to be denoted by a non-zero k-component can only occur in connection with a ­ -component that has period \1". For e±ciently comparing word-sequences that are similarly built and for discussing word-sequences with that many repetitions of letters as has ! (2;2;2;:::) we introduce the following terminology:

5.2 De¯nition of coincidence patterns: (cf. [17].2.1) Let !1 ; !2 be two word sequences and let a < b and c < d be position sequences of !1 and !2 , respectively. Then we say that !1 [a;b] is a \coinciding pattern" with !2 [c;d] if either !1 [a;b] = !2 [c;d] or !1 [a;b] = (!2 [c;d])¡1 , and if the intervals [a; b] and [c; d] are maximal in the sense that neither does exist a position sequence a0 < a nor some b0 > b such that for suitable c0 < c and d 0 > d we could get one of the following equations: !1 [a0;b] = !2 [c0;d] , !1 [a0;b] = (!2 [c;d0] )¡1 , !1 [a;b0] = !2 [c;d0] or !1 [a;b0] = (!2 [c0;d] )¡1 : Remark: Recall that in 3.2 /[16].2.9/[17].1.6{1.7 when de¯ning the \greatest common initial word sequences" we developed a technique of tracing coinciding domains of word sequences and showed that the boundary of such coinciding domains does always exist as a well-de¯ned position sequence. Hence, wherever one letter occurs in both word sequences !1 and !2 , this occurrence is part of a coinciding pattern between !1 and !2 , since the techniques of [16].(On 2.1(iii))/[17].1.7 can be used to uniquely determine the initial and the terminal position sequences a, b, c and d that meet the maximality properties as above required for a coincidence pattern. However, the shortest well-de¯ned coincidence pattern consists of one letter only.

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15

5.3 De¯nition: In analogy to in 5.2 we de¯ne \internal coincidence patterns", i.e. we similarly mark those patterns which occur at di®erent places in the same word sequence (cf. [17].2.2)).

5.4 Construction of a free basis for G 2 : Let B be the additive subgroup of those reals which can be described by such fractions whose denominator is a power of 2. Consider a system of coset-representatives of this subgroup of (R; +) which satisfy to be bigger than 1 and are smaller than 2; however, do not take any representative for the trivial coset B, but precisely one for each other coset. Note that each of these representatives has a unique representation within the binary system, hence by applying µ¡1 (cf. 5.1) our system of coset-representatives is lifted to a system of position sequences of ! (2;2;2;:::) =: !. For any of those position sequences p consider the substring ! [0N ;p] and denote the corresponding system of substrings as B20 . Q1 Let B2 := f i=1 ¬ i g [ B20 and let 0

B := B0 [ f

1 Y

¬ i g [ B20

| i=1 {z B2

(7)

}

Q1 where B0 is the free basis of H2 as de¯ned in (4). Here we use \ i=1 ¬ i " as an alternate notation for ! [0N ;1:0] . The proof that B 0 is actually a free basis for G 2 follows in 5.4/5.9{5.12. Corresponding to the decomposition of B 0 into B 0 and B 2 in Formula (7), we now introduce the subsets P0 (!) and P1 (!) for obtaining the analogous decomposition of P(!). For that purpose we let P0 (!) := f­ + k 2 P(!) j ­ has period \1"; k 2 Ng; P2 (!) := (P(!) ¡

N

P0 (!)) [ f0 g:

and

(8) (9)

As an immediate consequence we have that P0 (!) [ P2 (!) = P(!) ;

P0 (!) \ P2 (!) = f0N g;

! [p;q] 2 H 2 () p; q 2 P0 (!):

(10) (11)

5.5 Proof that (7) gives a generating system of G 2 : It su±ces to show that an arbitrary substring ! (2;2;2:::) [0N ;p] can be represented as a product of B 0 -word-sequences. In the following we give the corresponding arguments for each of the possible types of position sequences p 2 P(! (2;2;2;:::) ) separately. (i) Assume that p = ­ + 0 where ­ is periodical with period \1". Then ! (2;2;2:::) [0N ;p] has a ¯nite decomposition into subblocks of ! = ! (2;2;2;:::) , and hence in this case ! [0N ;p] already is contained in H 2 . Thus it can be represented by B0 -elements already.

16

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(ii) Now assume that p = ­ + k with non-zero k. By 5.1 ­ must have period \1". Use the decomposition ! [0N ;¯+k] = ! [0N ;¯] ¢! [¯;¯+k] where the ¯rst factor can be represented according to 5.4(i) and the second factor is a ¯nite word. Hence by (5) it can be represented, also. (iii) Now let p = ­ + 0, where ­ has period \0". Let ­ 0 be the binary sequence obtained from ­ by replacing all digits \0" of its period by \1", and the preceding digit \1" by \0". Then use the decomposition ! [0N ;¯] = ! [0N ;¯ 0] ¢ ! [¯0;¯] , where the ¯rst factor can be represented according to 5.4(i) Q1 and the second factor must be some product j=1 ¬ j according to Formula (6). Such a product can then be represented as Q1 Q1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 j=i ¬ j = ¬ j ¢¬ j¡1 ¢ : : : ¢¬ 1 ¢ j=1 ¬ j = ¬ j ¢¬ j¡1 ¢ : : : ¢¬ 1 ¢! [0N ;1:0] which by (5) and (7) all are B 0 -elements. (iv) Now assume that p = ­ + 0 where ­ has neither period \0" nor period \1". Hence µ(­ ) 2 = B (cf. 5.1). Let ­ 0 2 P(!) be the digit sequence which according to (6)/5.4 represents the ((R; +)=B)-coset containing µ(­ ). By de¯nition ­ 0 and ­ can only di®er at the ¯rst ¯nitely many digit positions and have to coincide for all the remaining ones. De¯ne position sequences ® and ® 0 by copying from ­ and ­ 0 , respectively, the ¯rst ¯nitely many possibly non-coinciding digits, and by putting all the in¯nitely many coinciding digits to \0". Hence ® and ® 0 are both essentially ¯nite digit sequences or, equivalently, digit sequences with period \0". Then de¯ne ¯ and ¯ 0 by similarly as in 5.4(iii) replacing within ® or ® 0 , respectively, their period \0" by a period \1" and by interchanging the one preceding digit, also. From the subblock-structure of ! (2;2;2;:::) and the corresponding repetition of patterns within ! = ! (2;2;2;:::) we get that ! [±;¯] is very similar to ! [±0;¯ 0] . However, equality does not hold if ¯ and ¯ 0 mark the beginning of subblocks of di®erent shift-indices, in this case both word sequences di®er within a ¯nite subproduct at their beginning. But, in any case there exist integers k and k 0 such that coincidence holds between ! [±+k;¯] and ! [±0+k 0;¯ 0] . This equality is used in the following computation: ! [0N ;¯] = ! [0N ;±+k] ¢ ! [±+k;¯] = ! [0N ;±+k] ¢ ! [±0+k0;¯ 0] = ! [0N ;±+k] ¢ ! [±0+k0;0N ] ¢ ! [0N ;±0+k 0] ¢ ! [±0+k0;¯ 0] = ! [0N ;±+k] ¢ ! [± 0+k 0;0N ] ¢ ! [0N ;¯ 0] . Here the last factor is a B2 -element by 5.4 and the ¯rst two factors can be represented within our B 0 -basis by 5.4(ii).

5.6 Remark: The proof that Formula (7) describes a free system of elements of G2 relies on similar techniques as the corresponding proof in [17].2.14{2.25 for our group of bounded word sequences G 1 . After two short preparatory steps we will be giving this proof in 5.9{5.12. For stressing the analogy, the proof contains references to [17], but since we do not need to quote lemmata or other results from [17], the proof as given below is essentially complete.

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17

5.7 Lemma: Let ! = ! (2;2;2;:::) and let p and q be elements of P(!) such that there exist p0 ; q 0 2 P(!) with ! [p;p0] = ! [q;q 0] (either p0 < p and q 0 < q, or p0 > p and q 0 > q). Then with respect to the indexing as introduced in 5.1 the ­ -labels of p and q have to coincide up to the ¯rst ¯nitely many digits. Proof: The equation ! [p;p0] = ! [q;q 0] implies that ! [p;p0] and ! [q;q 0] are a part of an (internal) coincidence pattern of ! (cf. 5.3 ). Since any coincidence pattern contains letters and since our word sequence does only contain countably many letters, and since by 5.2(Rem.) any coinciding letter only belongs to one (maximal) coincidence pattern for !, there exists a countable list of maximal coinciding internal coincidence patterns and [p; p0 ], [q; q 0 ] has to be a pair of matching substrings of one of the entries of this list. Now, this list can be composed together by looking at the repeated occurrences of the letters ¬ 1 , ¬ 2 , ¬ 3 ; : : : . ° ¬ 1 occurs only once within !. ° ¬ 2 occurs twice within !, and the corresponding coincidence pattern are ! [0:1+1;1:01] = ! [1:01;1:1] . ° ¬ 3 occurs four times within !, and the coincidence patterns are ! [0:1+2;1:001] = ! [1:001;1:01] = ! [0:1+1;1:101] = ! [1:101;1:1] , where the ¯rst and the third coincidence pattern might be extended to the one as listed in the preceding item. Since the construction principle iterates, we see that all coincidence patterns are bounded by position sequences that only di®er within the ¯rst ¯nitely many digits.

5.8 Remark: The moral of the above proposition is that any coincidence pattern within ! (2;2;2;:::) is essentially compatible with the subblock-structure of !.

5.9 On the scheme of proof that Formula (7) describes a free system of elements of G 2 : We consider an arbitrary reduced product of basis elements and their inverses, and we investigate the possibilities for cancellations to such an extend that we see that all cancellations get stuck fairly soon | in any case soon enough so that one can rule out that the ¯rst cancellations could give way to other ones that one might not be able to control any longer. As in [17].2.14{2.25 the proof will consist of one Adaptation Step 5.11 before it can be concluded with ruling out cancellation facilities in 5.12. In that context it is essential to recall ([17].1.16) that all phenomena of performing the multiplication of word sequences (i.e. of in¯nite objects) essentially reduce to phenomena of ¯nitely generated free groups, provided one establishes the corresponding subdivision for each of the word sequences (in di®erence to the case of ¯nitely generated free groups there are just in¯nitely many possibilities for such subdivisions!). More precisely the following lemma

18

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was proven in [17]:

5.10 Lemma: (cf. [17].1.16) Let !1 ; : : : ; !k denote word sequences. Then, if [0N ; `Nj ] denotes the interval of position sequences of !j , there exists a ¯nite subdivision 0N = aj;0 < aj;1 < : : : < aj;jk = `Nj of each of these intervals such that the following procedure leads to the word sequence that describes the product !1 ¢ : : : ¢ !k : Denote each of the word sequences !i [ai;j¡1;ai;j ] by a symbol using the same symbols for identical word sequences and inverse symbols for inverse ones. Then replace the product !1 ¢ : : : ¢ !k by the corresponding product of those symbols. Now start to reduce the latter product according to the calculus used for free generators of groups. Replace the symbols that still occur in the reduced product by the corresponding word sequences and concatenate these word sequences. If the aj;k have been appropriately chosen, the result is the reduced word sequence that describes !1 ¢ : : : ¢ !k . Sketch of Proof: Recall that the product of two word sequences ! and ² as described in 3.3(i) was obtained by concatenating ! [0N ;a] and ² [b;`N ] , where a and b have been de¯ned so that the greatest common initial word sequence of ² and ! ¡1 is assumed to be ² [0N ;b] = (! [a;`N ] )¡1 . Hence the case of a two-factor product is immediate from Formula (2), the other cases then result by carefully investigating which of these phenomena of computing the product between two word sequences persist in the course of an iterative t u execution of a longer multiplication.

5.11 The Adaptation Step (cf. 5.9) Let ! = ! (2;2;2;:::) and ² 10 ¢ ² 20 ¢ : : : ¢ ² k0 0¡1 ² k0 0 be our reduced product with ² i00 2 B 0 §1 . We now perform a similar adaptation step as we did in [17].2.14(¤) for G 1 . This adaptation process consists of two types of adaptations: (i) For each maximal subproduct of B0 -factors ² i00 ¢ ² i00+1 ¢ : : : ¢ ² j0 0 (maximal means that either i0 = 1 or ² i00¡1 2 B2 and that either j 0 = k 0 or ² j0 0+1 2 B2 ) we compute the reduced word sequence representing this product. This word sequence must be a concatenation of substrings of ! by 5.10. The corresponding substrings are then de¯ned as new factors ² i ; ² i+1; : : : ; ² j of our adapted product. If due to some coincidence patterns of ! the corresponding position sequences that give the break points between our ² º -factors (i µ ¸ µ j) should not be uniquely de¯ned, we require that they should be chosen in such a way that the ¯rst factor ²

i

and the last factor ²

j

are de¯ned

(12)

so that they come out as long as possible. Note that, since our initial product was reduced and since B0 is a free basis of H2 by 4.5, the resulting product ² i ¢ : : : ¢ ² j can never be void.

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

19

(ii) The second type of adaptation process is similar as the one performed in [17].2.14(¤); however it is only applied to such adjacent factors ² i00¢² i00+1 that are both taken from B2 : If these adjacent factors should be of type ! [a;0N ] ¢! [0N ;b], they are replaced by one new factor which is de¯ned as ! [a;b]. We then can conclude the proof that Formula (7) describes a free system of elements of G2 by investigating the possibilities for cancellation within a product of word-sequences that has been adapted in the sense of the preceding paragraph 5.11:

5.12 Concluding the proof that G 2 is free: (i) Cancellations cannot take place between adjacent factors that are both taken from B0 , since by 5.11(i) these factors have been de¯ned as adjacent substrings of a reduced word sequence. (ii) Cancellations cannot take place between two adjacent factors from B2 , since by 5.11(ii) these factors can only be of type ! [?;p] ¢ ! [q;?] where p and q are not the same and in addition are either coset representatives of non-trivial (R; +)=B-cosets or 1:0 or 0N . If some cancellation could take place, it would re°ect an internal coincidence pattern of !, i.e. there would exist position sequences p0 and q 0 such that ¡1 ! [p0;p] = (! [q;q 0] ) . However, note that one can prove with similar arguments as in 5.7{5.8 that the entire structure of coincidence patterns within ! is given by the subblock-structure of !. In addition, we have that corresponding positions in di®erent subblocks are always given by such ­ -labels that only di®er within the ¯rst ¯nitely many digit positions, possibly with some slight shift in the k-component. Hence knowing that p and q are di®erent representatives of non-trivial (R; +)=B-cosets implies that these position sequences cannot satisfy the latter condition. This argument also extends to the positions 1:0 and 0N as some analysis of the patterns of ! in the neighbourhood of the latter positions and at any non-trivial (R; +)=B-coset shows. (iii) Hence the only situation which still needs discussion is given when B 0 -factors and B 2 -factors meet each other, but even in this case for most combinations of such factors any possibilities for cancellations can be ruled out by the arguments of 5.12(i) and 5.12(ii). Indeed, we are left with the need to discuss one situation only, and this is where our corresponding adjacent factors are ! [p;0N ] ¢ ! [0N ;q] , i.e. where both position sequences of the attaching point are 0N . In this case, where we assume that these two factors are taken from the two sets B 0 and B 2 , a similar property holds for the bounding positions p and q (cf. (7) ). Without loss of generality we can assume that p 2 P2 (!) and q 2 P0 (!) (cf. (8){(9)). As in 5.11(ii)/[17].2.14(¤) we replace our two adjacent factors by ! [p;q]. Note that such a replacement does imply a \reversion" at one of the two bounding position sequences p or q. Saying that this reversion takes place at q means pointing out that, while the end of the factor ! [0N ;q] is modelled from those letters of ! that are found immediately before q, the end of the word sequence ! [p;q] is modelled from the letters of ! ¡1 , i.e. it is modelled from those inverse letters as they are found within ! immediately after q.

20

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Note that the main argument for non-cancellation as given in 5.12(ii) is not a®ected by such a reversion, but the argument as given in 5.12(i) might be. This is, why we have in particular to discuss the case where p > q and hence we get an inversion at q. We now investigate the possibilities for further cancellations that could take place between ! [p;q] and the following factor which shall be denoted by ! [r;s]. In the case where ! [r;s] 2 B2 , we get by 5.12(ii) that there is none. In the other case, when we assume that ! [r;s] is bounded by two P0 (!)-position sequences we ¯nd, taking the reversion of the ¯rst factor into account, that any kind of cancellation at the attaching point of our two position sequences would imply that our original factor ! [0N ;q] had not been de¯ned maximal | contradicting to what was assumed in (12). Hence we see that we are bound to get stuck with our possibilities for cancellation in this case also. (iv) The purpose of this item is to close a gap that was hidden in the arguments above. Namely, note that the demand of (12) to de¯ne the ¯rst and the last factor of a B0 -product in a maximal way might be contradictory in the case of a two-factorproduct. On the other hand, note that this maximality was only needed at one place within the above arguments, namely at the end of 5.12(iii). And hence it is only important for one particular type of factor. If now we assume that both factors of a two-factor-product are of that relevant type, our product would read ! [0N ;q] ¢ ! [q0;0N ]. Since in the second factor all letters are taken to its inverse and in the ¯rst factor they are not, there is no freedom of choice for breaking up the corresponding product-word-sequence into two substrings. Hence we ¯nd that the demand for choosing the ¯rst and the last factor of a B0 -subproduct maximal is not contradictory in all relevant cases, so that the above arguments all have been t u essentially correct.

6

Proof that the letters are not good as generators for G 2 or for H 2

6.1 Theorem: The system of letters (¬ i )i2N can neither be extended to a free basis of G 2 , nor to such a basis of H 2 (cf. 3.9(iv)). Proof: In the following we will restrict our language to the more di±cult case of G 2 ; the statement for H2 can either be obtained by accordingly simplifying the arguments and constructions below, or as a corollary of the Theorem by an indirect argument: Although not explicitly stated there, the arguments of Sect.5 (cf. in particular (7)) gave for an arbitrary free basis of H2 how to extend this basis to G 2 . By (7)/(4) the system of blocks (­ i )i2N is part of a free basis of G2 . We denote this basis by f­ i j i 2 Ng [ f® j j j 2 Jg, i.e. we use di®erent symbols for denoting that part of the basis of G 2 that are not subblocks of ! (2;2;2;:::) . The relation between our

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

subblock-generators ­ i and the letters ¬ ¬

i

i

21

is according to (5) given by ¡2 = ­ i ­ i+1 :

We want to use this relation for showing that the system of the ¬ i cannot be used the generate G 2 freely. The corresponding contradiction will be found by considering the homomorphism given by 1 ­ i 7! i ® j 7! ® j (13) 2 where these formulae are used to describe a homomorphism ’ : G 2 7! B ¤ h® j j j 2 Ji, where B := f 2pk j p 2 Z ; k 2 N0 g is the additive group of all binary fractions. It is not hard to see that this homomorphism is onto and that ’ takes on the value zero on all letters ¬ i . Hence according to the isomorphism theorem for groups we have that G2 =ker(’) ¹= B ¤ F jJj . In the next proposition we will show that the kernel of this homomorphism consists precisely of all conjugate products of the letters ¬ i . Hence, if we denote the (obviously free) subgroup that is generated by the letters by H 20 and used doubleangles (\hh ¢ ii") to denote the normal closure, we have that G2 =ker(’) = G2 =hhH 20 ii. On the other hand, if the system (¬ i )i2N could be extended to a free basis, we would, denoting the free subgroup that is generated by the complementary part of the free basis by F! , have that G 2 = H 20 ¤ F ! and hence that G 2 =hhH 20 ii = F ! , i.e. that our quotient group would be a free group. This is a contradiction, because B ¤ FjJ j is not free, it contains B as a subgroup and B contains arbitrarily high powers (1 = 12 + 12 = 14 + 14 + 14 + 14 = : : :). Hence the proof of this theorem will be completed after having proved in 6.10 that

every word of the kernel of ’ with ’ as constructed in (13) can

(14)

be represented as a product of conjugates of the letters ¬ i . This proof is elementary but lengthy and hence essentially requires all the remaining paragraphs of this section. Hence we continue by ¯rstly explicitly explaining

6.2 The Scheme of Proof of (14) Let g be an arbitrary element of ker(’) » G 2 represented in its free basis f­ i ; ® j j i 2 N ; j 2 Jg. We regard the corresponding word as a concatenation of subwords alternately composed from the letters of f­ i j i 2 Ng and of f® j j j 2 Jg. Because the image under ’ can according to the normal form of the free product B¤F jJj be represented by an analogous product of factors that are alternately taken from B and from F jJj , and since we can without loss of generality assume that none of the (® j )j2J -words is trivial, we can conclude that at least one of the ­ j -subwords has to be mapped to zero. In paragraphs 6.3{6.9 we will restrict ourselves to showing how such a product of ­ i that is under ’ mapped to zero can be replaced by conjugates of ¬ , and ¯nally in 6.10 we give an analogous rewriting method for the situation where the ­ i - and ® j -factors are mixed.

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In 6.5{6.8 we will describe an inductive process that step by step replaces some original given product of ­ i by a product of ­ i and ¬ i such that it is just a product of conjugates of letters ¬ i . We will put parentheses around the parts that already have been accordingly replaced, and hence in following talk of parts of our word that appear \inside parentheses" and \outside parentheses". Nesting of parentheses will not occur. Hence in order to technically prove (14) even in the case of a pure ­ i -product, we will ¯rst have to extend it to a statement on \parenthesized words":

6.3 Proposition: Let w be a ¯nite word in the letters ¬ i and ­ i , in which some parts appear inside not-nested pairs of parentheses. Assume that these parts are products of conjugates of letters ¬ i and assume, that the ¬ i -letters do not appear outside parentheses. Further, assume that the entire word, when interpreted as an element of G2 , lies in ker(’). As an G2 -element the entire word can be transformed into a product of conjugates of the letters ¬ i . The Proof of this proposition is lengthy and hence spread over the paragraphs below up to 6.9. It follows from the inductive Replacement Process 6.4{6.8 in the course of which the parts of the actual word of consideration that appear under parentheses will be systematically extended. However, our proof of 6.3 ¯rst starts with the de¯nition of an auxiliary function:

6.4 De¯nition: 1 §" 2 k (i) Let be given a word ­ i§" ­ i2 : : : ­ i§" ("k 2 f¡ 1; +1g) that is not of length zero, 1 k §"k §"1 but satis¯es that ’(­ i1 : : : ­ ik ) = 0, where ’ has been de¯ned in (13). We consider

à : f0; 1; 2; : : : ; kg ! B

;

1 m 7! ’(­ i§" :::­ 1

§"m im )

(15)

as a function on a discrete set, which by de¯nition takes on the value zero at zero. (ii) The de¯nition can be immediately extended to words whose letters are either of type ­ i§1 or ¬ §1 i . For the beginning of a word, the end of a word and for each place between such letters the Ã-value is de¯ned. (iii) Remark: By de¯nition à jumps at each ­ -letter, but it does not jump at an ¬ -letter. If applied to a word of ker(’), it has also to take on the value 0 at the end of a word. Hence if applied to a non-empty word that also contains ­ -letters it must have a non-zero maximum or a non-zero minimum. In the following we will in particular use these places as places where we start our replacement process. (iv) Remark: In addition, observe that if we consider parenthesized words as described in 6.2, at the start- and endpoint of each parenthesized segment à takes on the same value. In that context we will mainly ask for the maximum or minimum that is taken on outside the parentheses, and often consider the two endpoints of a parenthesized segment as one place \outside parentheses".

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23

6.5 Some Basics for the replacement process: By 6.2, if our replacement process is not yet ¯nished, we have to have ­ -letters outside parentheses, so that by 6.4(iii) jÃj takes on a non-trivial maximum outside parentheses. By 6.4(iv) this can either be at some place that is obtained from identifying the endpoints of a parenthesized place or an ordinary place (i.e. without a parenthesis appearing at this place). If an absolute maximum of jÃj is given at a place of the ¯rst type, the letters that occur as the last letter before and the ¯rst letter after the parenthesized segment have to have di®erent exponents. If they should have the same index, they are conjugating factors of the entire pair of parentheses, and hence we can extend our parenthesized segment over these two letters and loose our extremum that way. If not, then we insert a cancelling pair of the ­ -letter with the bigger index at the other side of our pair of parentheses, so that again our parenthesized domain can to each of its sides be extended by one letter and that that way one letter of the cancelling pair disappears under these parenthesis. Although by this extension of parentheses our old extremum has disappeared under them, it has by our insertion process been created a second time. This has been the desired e®ect of this insertion process: not to lower the extremum, but just to shift it to an ordinary place (where there are no parentheses around). Hence we can in following restrict all considering to extrema that are taken at ordinary places. At such a place the two adjacent letters are ­ -letters with di®erent exponents and, because we can assume that our word was given in a reduced form, letters with di®erent indices. There are in principle four \basic situations" how our word at such an extremum might look like: according to which of the two letters has the exponent ¡ 1 and which of the two letters has the smaller index. Observe that we did not write \reduced" into the assumptions of Proposition 6.3. We did so, because in the course of the replacement process 6.5{6.8 we want to be able to perform insertions without having to worry whether they might overturn the property of our word of being reduced. However, in spite of this we need not have to discuss the situation of a maximum or of minimum of à that occurs at a place of type \­ i¨1 ­ i§1 " as an extra basic situation, since simple process of cancelling these two letters against each other does also reduce the complexity in that sense it will be de¯ned in 6.9.

6.6 The replacement mechanism in the easy case: Such an \easy case" is given when the two letters have adjacent indices i ; i+1, and if the letter with the higher index occurs two times (i.e. as a square) at this place. In this case ¡2 in one of the four principle situations as described at the end of 6.5 our letters read ­ i ­ i+1 , and they can be directly replaced by the letter ¬ i which is to be put in parentheses. In ¡2 ¡2 ¡2 another principal situation we have ­ i+1 ­ i = [­ i¡1 ­ i ]­ i+1 ­ i = ­ i¡1 [­ i ­ i+1 ]­ i so that with this tricky insertion we get a conjugate of the letter ¬ i , namely a conjugate of what appeared in the last term in square brackets. That way again all of these letters and with it the extremum disappear under parentheses. In the third and forth basic situation the

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corresponding words are precisely the inverses of these words, and hence these situations can be correspondingly treated. The four basic situations do also occur in the forthcoming cases 6.7 and 6.8, and the reduction mechanism is in each of the these cases the same. Hence in the cases 6.7{6.8 we will restrict our language to the ¯rst basic situation.

6.7 The replacement mechanism in the more general case: In \the m ore general case" and in the ¯rst basic situation a maximum occurs at a subword ¡2¸ of type ­ i ­ i+º . I.e. in this case we do not longer assume that the letters that occur at an extremum have adjacent indices, but we assume that the letter with higher index is repeated precisely that often that at the beginning and at the end of the corresponding subword our function à must take on the same value. In the case ¸ = 2 the word can be as follows replaced by a product of conjugates of letters: ¡4 ­ º ¢ ­ º+2 = ­ º ¢ (­

= (­ º ¢ ­

¡2 ¡2 ¡1 ¡2 +2 +1 º+1 ¢ ­ º+1 ) ¢ ­ º+2 ¢ (­ º+1 ¢ ­ º+1 ) ¢ ­ º+2 = ¡2 ¡2 ¡1 +1 º+1 ) ¢ (­ º+1 ¢ (­ º+1 ¢ ­ º+2 ) ¢ ­ º+1 ) ¢ (­ º+1

(16) ¢­

¡2 º+2 )

where after the ¯rst insertion step the parenthesizing has been set to make stress the insertion structure and then be regrouped that every element either occurs inside some product that could be replaced by 6.6 already, or as conjugating factor of such a product. For the ¸ = 3 an analogous extension trick can be performed, namely ¡8 ­ º ¢ ­ º+3 = ­ º ¢ (­

= (­ º ¢ ­

¡2 ¡4 ¡1 ¡4 +2 +1 º+1 ¢ ­ º+1 ) ¢ ­ º+3 ¢ (­ º+1 ¢ ­ º+1 ) ¢ ­ º+3 = ¡2 ¡4 ¡1 +1 º+1 ) ¢ (­ º+1 ¢ (­ º+1 ¢ ­ º+3 ) ¢ ­ º+1 ) ¢ (­ º+1

(17) ¢­

¡4 º+3 ):

¡4 Taking into account that according to our preceding ¯nding expressions of type \­ º+1 ¢­ º+3 " can be replaced by a product of conjugates of letters, it is clear that the whole of the Formula (17) also just represents a product of conjugates of letters. Then from using the analogous structure of Formulae (16) and (17), it is clear that the desired result follows inductively.

6.8 The replacement mechanism in the general case: ¡k In \the general case" we have to assume that the corresponding subword is of type ­ i ­ i+º . I.e. we can assume nothing but the fact that the ­ -letters have di®erent exponents, and we take the liberty that, in case the letter with the bigger index should occur more than once at this place, to put this information also into our formula. However we cannot rule out the case k = 1. If k should be bigger that 2º we can proceed as in 6.7, using the ¯rst ¡1 2º ­ j+º -factors for the corresponding replacement mechanism, and leaving the others in the unchanged part of our word. If k should be smaller than 2º , we insert at this place k¡2¸ 2¸ +k the subword ­ j+º ­ j+º , and proceed as in the sentence before.

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25

6.9 Concluding the proof of Proposition 6.3: As pointed out before, 6.3 follows from an iterated replacement process which has been de¯ned in 6.5{6.8. Its proof can be concluded by in the following arguing why this replacement process must stop: For that purpose we consider complexity tuples with the following entries: The ¯rst entry is the maximal value of the jÃj-function outside parentheses. The second entry is the number of times that maximum is taken on, and third, forth, and all further entries are the number of letter ­ 1§1 ; ­ 2§1 ; : : :, that occur outside parentheses. Because we are only considering ¯nite words in this context, only ¯nitely many di®erent indices can occur at our ­ -letters, and hence this tuple is ¯nite. These complexity-tuples are regarded as being lexicographically ordered. Assume that N is the biggest index of a ­ -letter that occurs in our word. Although in general our function à has an in¯nite dense set of possible values, for this problem we can think of our function à as a function that takes on values on the discrete set of all fractions that can be written with denominator 2N . Hence any process of reducing the maximum of the absolute value of this function has to stop after a ¯nite number of steps, because none of our insertion processes required to insert ­ -letters with an index bigger than N into our word. With the lexicographical order of our complexity tuples, any of our our replacement processes reduces the complexity, because the former absolute extremum and usually all newly inserted letters disappear under parentheses. This makes sure that either the ¯rst entry of our tuples are diminished (if this particular value of our maximum of jÃj should have only taken on once), or the least the second entry is reduced, i.e. the number of times this maximum has been taken on. Only case 6.8 deserves closer attention, because here not all inserted letters disappear inside parentheses. However, even in this case we have made sure that the range of Ã-values at the newly inserted letter lie within the range of Ã-values that have been taken on before, namely, if n is the place number of our maximum, i.e. the number of the ¡k place between the letters \­ j " and \­ j+º ", all of these values Ã-values lie within the range of values between Ã(n ¡ 1) and Ã(n). However, there is one bad case in which neither the absolute value of the maximum nor the number of times it is taken on is reduced, namely when jÃ(n ¡ 1)j = jÃ(n)j, i.e. if n ¡ 1 is also a place where the absolute maximum of jÃj is taken on, namely as extremum of à with the opposite sign. In this case our insertion of k¡2¸ 2¸ ¡k \­ j+º ­ j+º " created between the ­ ¡1 -letters and the ­ +1-letters a new place where our absolute maximum of jÃj is taken on. That way the second entry of our complexity has not become bigger (because the old maximum has disappeared under parentheses), but is has also not become smaller due to this new place. On the other hand, if this special case should be given, we also see that ­ j is a letter that makes à jump from its absolute maximum to its absolute minimum, and hence it is de¯nitely the biggest letter that can occur outside parentheses in our word. Since according to our process only letters with higher indices have been inserted and one ­ j -letter has disappeared under parentheses, we see that even this case our complexity gets smaller. Since lexicographically ordered tuples of this type form a well-ordered set, we see that our replacement-process must

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stop, which technically is only possible if everything has disappeared under parentheses, t u i.e. has become a product of conjugates of ¬ i -letters.

6.10 Completing the proof of Theorem 6.1 by proving (14): In principle (14) follows from an analogous iterative replacement process as did the proof of the restricted statement Prop. 6.3 that was described in 6.4{6.9. Denote the free group h® j j j 2 J i by ¡ and represent our preimage of ’ in G2 in a normal form of factors that are alternatingly taken from the subgroups h­ i j i 2 Ni and h® j j j 2 J i. We denote factors of the ¯rst type by bº and factors of the second type by cº . When assuming that such an element is mapped to the neutral one under the homomorphism ’ as constructed in (13), we have to observe that ’ is invariant to this alternating product structure and, apart from that, leaves the cº -factors unchanged and just replaces the bº by some binary fractions 2 B that we will denote by bº . If none of the bº is zero, than the image under ’ is again a normal form and hence is not the neutral element. This gives us the start-point for our inductive process: By Prop.6.3 we can replace that bº that has bº = 0 by a product of conjugates of ¬ i . Assuming that ’ of our element of consideration is trivial implies assuming that from our alternate product of bº and cº , when starting with those bº that are zero, an iterated cancellation is possible and ¯nally deletes the whole product. Along this cancellation process we have iteratively to replace our preimage by conjugates of ¬ i . Within this cancellation process there are essentially two types of cancellations possible: (i) If a subproduct cº ¢cº+1 ¢cº+2 ¢ : : : ¢cº+· (18) cancels (because all b? -factors that stood between these c? -factors are zero), we can extend the preimage cº ¢ bº+1 ¢ cº+1 ¢ bº+2 ¢ cº+2 ¢ bº+2 ¢ bº+3 ¢ cº+3 ¢ bº+4 ¢ : : : ¢ bº+·¡1 ¢ cº+·¡1 ¢ bº+· ¢ cº+· as follows: ¡1 ¡1 ¡1 ¡1 cº ¢ bº+1 ¢ (c¡1 º ¢ cº ) ¢ cº+1 ¢ bº+2 ¢ (cº+1 ¢ cº ¢ cº ¢ cº+1 ) ¢ cº+2 ¢ bº+3 ¢ (cº+2 ¢ cº+1 ¢ ¡1 ¡1 ¢ c¡1 º ¢ cº ¢ cº+1 ¢ cº+2 ) ¢ cº+3 ¢ bº+4 ¢ : : : : : : ¢ bº+·¡1 ¢ (cº+·¡2 ¢ cº+·¡3 ¢ : : : ¢ ¡1 ¢ c¡1 º+1 ¢ cº ¢ cº ¢ cº+1 ¢ : : : ¢ cº+·¡3 ¢ cº+·¡2 ) ¢ cº+·¡1 ¢ bº+· ¢ cº+· :

From the above parenthesizing it was visible that the product was merely extended and, when changing this parenthesizing as follows ¡1 ¡1 ¡1 ¡1 (cº ¢ bº+1 ¢ c¡1 º ) ¢ (cº ¢ cº+1 ¢ bº+2 ¢ cº+1 ¢ cº ) ¢ (cº ¢ cº+1 ¢ cº+2 ¢ bº+3 ¢ cº+2 ¢ cº+1 ¢ ¡1 ¡1 ¢ c¡1 º ) ¢ (cº ¢ cº+1 ¢ cº+2 ¢ cº+3 ¢ bº+4 ¢ : : :) ¢ : : : ¢ (: : : ¢ bº+·¡1 ¢ cº+·¡2 ¢ cº+·¡3 ¢ : : : ¢ ¡1 ¢ c¡1 º+1 ¢ cº ) ¢ (cº ¢ cº+1 ¢ : : : ¢ cº+·¡3 ¢ cº+·¡2 ¢ cº+·¡1 ¢ bº+· ¢ cº+· );

it is for all bº+¸ -factors with ¶ < µ visible that they have just been conjugated by an appropriate product of cº+? -factors. However, such a property does also follow

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27

for the last bº+· -factor, if we use that (18) was a cancelling product. Quoting that by Prop.6.3 each of our bº+¸ -factors is just a conjugate of ¬ -letters, we get the result for such subproducts, also. (ii) Vice versa we then have to assume that the cancellation process a®ects two factors bi1 and bi2 which appear in our product according to : : : ¢bi1 ¢(: : : : : : : : :)¢bi2 ¢ : : :

(19)

where the parentheses in the above formula replace some subproduct which is known to be mapped to zero and has in the course of the iterative process already been replaced by a product of conjugates of letters ¬ i . For the sake of simplicity we assume that bi1 = ¡ bi2 ; the case where only some longer product of bi? cancels can be treated by an analogous extension argument as given in Case 6.10(i). In the simple case of Formula (19) we extend our given product to bi1 ¢bi2 ¢b¡1 i2 ¢(: : : : : :)¢bi2 ¡1 so that the subproduct bi2 ¢(: : :)¢bi2 is also a clear product of conjugates of ¬ i -letters and so that that subproduct bi1 ¢bi2 has also this form by Prop.6.3. From the arguments of 6.10(i) and 6.10(ii) we see that any element of G2 that by ’ t is mapped to zero can be iteratively rewritten as a product of conjugates of ¬ i -letters. u

7

The main technical lemma for the proof that G 4 and H 4 both are not free.

The section is solely devoted to give a full workout of the subsequent lemma that we needed to quote in 4.3. Inspired by Dunwoody’s appropriate remark ([5].(End of x3)) it can be achieved by imitating a proof-strategy of Kurosh ([12]):

7.1 Statement of the Lemma: Let G = A ¤ B be a group with a ¯xed decomposition into a free product of two free groups A and B and let 1 6= g 2 A be an element which has a product decomposition g = h ¢h ¢h0 ¢h0 with h; h0 2 G. Then, in fact, h; h0 2 A.

7.2 On the strategy of the proof: Essentially the proof relies on using the normal form of free products of groups, i.e. we can write h = a1 ¢b1 ¢a2 ¢b2 ¢a3 ¢ : : : ¢ak ¢bk and h0 = a01 ¢b01 ¢ 0 a2 ¢b02 ¢a03 ¢ : : : ¢a0k 0¢b0k0 with ai 2 A , bi 2 B. We will have to distinguish cases according to whether this product decomposition of h and h0 starts or ends with an a-letter or with a b-letter, and whether h or h0 appears in this product decomposition cyclically reduced or not. In each of these cases we get di®erent words for the squares h2 and h02 . We then can start the possible cancellations that can take place when we concatenate these two words to form one word representing g = h2 ¢h02 . There is only one place within this word where cancellations

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Table 1 (belonging to 7.2) This table shows the enumeration of the ten cases that are discussed in 7.3{7.7

may take place, namely the place of concatenation of the two given factors h2 and h02 . If the two letters following each other both are from the same factor A or both from B and if they are inverse to each other, then they cancel, and the reduction process may continue, if the second letter of h0 2 and the penultimate letter of h2 , (which are now in adjacent positions) then satisfy a similar condition. Once this process is stuck, we end up with a normal form and if this normal form should still contain more than one letter we know that h2 ¢h02 is not contained in A or in B. Hence we see that h2 ¢h02 can only be contained in one of our free factors A or B if the lengths of h2 and of h02 di®er at most by one. If they di®er by one, it could happen that all letters that meet each other at the concatenation place cancel, and the last letter of the longer factor remains; if they are the same it could happen that all letters meeting at the concatenation place cancel, except when the initial letter from h2 and the last letter of h02 meet, in this case we have to assume that these letters are not inverse to each other and they give g 2 A. We discuss in the following by distinguishing the according cases that neither of this possibilities can happen, i.e. that the symmetries that imposed by the assumption that h2 and h02 are both squares make sure that the process cannot get stuck at the time when the last two letters meet. Essentially, there will be t e n c a s e s to be distinguished: Instead of making a distinction, whether the ¯rst and/or the last letters are factors taken from A or from B, it su±ces to make a distinction whether the corresponding factors are of odd or of even length, which is equivalent saying that the initial and the terminal letter of the corresponding factors are of the \same" or of \di®erent typ e" i.e. are from the same of our factors A or B, or are from the two of them, respectively. We think of each word w as decomposed into \cyclically reduced kernel" K (w) and a \conjugacy-part" C (w) according to w = C(w)¢K (w)¢C (w)¡1 . Note that if C (w) is not empty, then K (w) needs to be of odd length, since its ¯rst and its last letter must be of the same type, because

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29

both of them must be of the other type than the terminal letter of the conjugacy part C(w). We will discuss each of the cases in Table 2 schematically, but we give verbal descriptions only by gathering distinct cases with similar arguments together. Table 1 describes our enumeration system for the ten cases to be distinguished; the numbers in the table give the number of the according cases.

7.3 Instructions for how to read the Tables 1 and 2: In Table 2 essentially each word is represented by line segments with markers at the end. For each case ¯rstly a line with markers describing the substrings of h¡2 is drawn, below that line another line with markers illustrating the corresponding substrings of h02 is placed, so that the corresponding factors which have to cancel if the process is not stuck are pictured precisely below each other, with the cancellation starting from left side. If one of the ends of the line segments is labelled with a or b, this is just to indicate that this factor is an element of A or of B, respectively, but di®erent a- or b-labels, even in the same word, can represent di®erent factors. Only, if indices are attached to some letter a or b, then the corresponding letter is de¯ned as a variable which may be used later in calculations. In this case, using the same index at di®erent places implies that the corresponding letter has to be the same. If according to the over- and underbraces the domain of di®erent factors overlap, this indicates that we had to attach factors which begin and end with a letter from the same of our two free factors, and hence that according to the rules of normal form, the terminal letter of the ¯rst letter and the initial letter of the second factor had to be gathered together to give one new letter for the product. The operator \`" denotes the length of each word, and C and K have been introduced in 7.2 already. Some of the cases can be discussed by computing the length of both factors only, because, if they have to di®er by more than one, as explained above (7.2) the cancellation can never go through. Note that in the course of these calculations the denominator 2 was only used, if the nominator is an even number, so that any of these fractions represents an integer. Based on these explanations it should now be clear how to read the entries of Table 2 with the arguments how to rule out each of the cases explained below:

7.4 Discussion of the Cases 1,2 & 5: If one is considering products where h and h0 are both cyclically reduced, the arguments in all three cases are essentially as follows: If our cancellation process should go through, the letters which meet at last, do have met in the middle of the word also, and if the cancellation process had gone through there, it has to go through at the end, as well. Note that, of course, this case contains the one subcase which can actually be given, but we are regarding this case as part of the special case of \words with extreme short length" and the discussion of these cases shall be postponed to the end of proof this section (cf. 7.7).

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Table 2 (First half )

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Table 2 (Second half )

31

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7.5 Discussion of the Cases 8{10: If both factors, h and h0 are not cyclically reduced, the length of the squares can only be odd, so that some calculation of lengths either gives a contradiction from the very beginning, or the result that both squares have to be of the same length if our cancellation process should go through. In the latter case this means that right at the end of cancellation process the same letters are meeting that have already met the beginning of the cancellation process, namely the initial letters of the conjugacy-parts of our words. Having made this observation also prevents us from making further distinction in these cases, e.g. if it could happen that factors h and h0 may have conjugacy parts of di®erent length.

7.6 Discussion of the Cases 3, 4, 6 & 7: The common structure of the arguments in these cases is due to the fact that our factors h and h0 are and are not assumed to be cyclically reduced, respectively. This gives us an equation of type h¡1 ¢h¡1 = C (h0 )¢K (h0 )¢ K (h0 )¢C (h0 )¡1 , or C (h0 )¢K (h0 ) º h¡1 º K (h0 )¢C (h0 )¡1 , if we compare the ¯rst half and the second half of our coinciding word sequences directly. The fact that reduction mechanisms at the attaching point might make it necessary to except this point from comparing the two halves, and that due to mismatching lengths between h and h0 there might be a oneletter shift around is acknowledged by using the symbol \º" instead of \=". These phenomena are then discussed in a symbolic language in Table 2. Essentially any of these cases needs to be split up into two subcases, depending on whether C(h0 ) µ K (h0 ) 0

0

C(h ) > K (h );

or whether

(20) (21)

respectively. The second of these cases is the easier one, since in this case one can conclude that for some middle position of h a letter has to coincide with its own inverse, since this letter occurs at some position of C (h0 ) and at the corresponding position of C(h0 )¡1 . Of course it is vital for this argument to know that a middle position with such a symmetry is actually the position of a letter, not just the position of a gap between two adjacent letters. However, the latter follows since all our factors are alternately taken from the groups A and B and thus any coincidence can never take place between two factors that are not taken from the same groups A and B. In the case \C (h0 ) µ K (h0 )" we have to observe that our approximate equation C(h0 ) ¢ K (h0 ) º K (h0 ) ¢ C(h0 )¡1 gives that within the same word the same substring K (h0 ) occurs at di®erent, but adjacent or even overlapping positions. This implies that the word has to have some periodicity inside, i.e. that, with perhaps one position excepted, our word is just composed of periodically repeating C (h0 ). Observe that there is no demand that the length of the whole word is just a multiple of C(h0 ), i.e. the last period might not be complete. However, for similar reasons C (h0 )¡1 gives a substring that is

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

33

periodically repeated from the end. By having observed this periodicity, essentially the same conclusion can be applied as in the case discussed before: Although there is not any overlap between C (h0 ) and C (h0 )¡1 at their original positions, by having noticed that they periodically reappear in our word, we can ¯nd some position where a letter has to coincide with its inverse one. This is a contradiction since all our factors are non-trivial elements of a free group. Of course, in this case it still must be discussed that we actually do get a contradiction, i.e. that those positions which are excepted from periodicity do not mess up the desired chain of conclusions. In Table 2 these positions have been marked by a \?". The remaining positions are marked by xi and yi with xi , yi belonging to C (h0 ), C(h0 )¡1 , respectively. I.e. for all i we have that xi = yi¡1 , and we are looking for some i where xi = yi also, to get the desired contradiction. The key-observation for the latter purpose is that all our words alternatingly consist of A-factors and B-factors, and that all question marks are either at A-positions or at B-positions. This especially implies that not all entries can be question marks only; we at most could have every second entry a question mark giving a clash at all positions between the question marks automatically. In the general case we can argue as follows: Either both question marks (i.e. the one from the x-series and the one from the y-series) clash at the same place. In this case, half a period apart from that place we get a direct clash of the type xi = yi . If the question marks of the x-period and of the y-period do not occur at the same place, we then have at half way between their occurrences again a clash of type \xi = yi ". The fact that all question marks do either occur at A-positions or at B-positions ensures, that \half way between the appearances of question marks" is de¯nitely a position of letters and not just at the gap between two adjacent letters. Having found a clash that way in each case ensures that we get a contradiction also in the last of our ten main cases that had to be discussed (cf. 7.2). However, in the above discussion 7.4{7.6 the special cases of extreme short length have been ignored. On the other hand, we are discussing these cases in the following:

7.7 Discussion of the special cases with at least one short factor: When arguing in 7.4{7.6 with the diagrammes of Table 2, we did implicitly use that the ¯rst and the last letter of a word are di®erent letters. This is not true for words of length one, and hence these cases need an extra-consideration. However, computing the length of a square gives `(w 2 ) = 2 ¢`(C (w)) + 2¢`(K (w)) for even `(K (w)) or `(w 2 ) = 2¢`(C(w)) + 2 ¢`(K (w)) ¡ 1, if `(K (w)) is of odd length. Hence the result can never be two, and so the di®erence between the length of a square of a one-letter word (which is one) and the length of the square of a longer word is at least two. In that case by 7.2 the length of h2 ¢h0 2 can also not be one. Thus ¯nally only one case remains, namely when h and h0 are both of length one. In that case h2 ¢h0 2 can be of length one, namely if h and h0 belong to the same group, both to A or both to B. Having found a contradiction in any of our ten Main Cases and having discussed the \short-word-cases" separately in 7.7 ensures that we cannot have h2 ¢ h0 2 = g, except, if h and h0 are both of

34

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

length one and taken from the same factor A or B as g. This gives the Main Technical t u Lemma 7.1. By now, all claims of Thm.3.9 have been proven.

Acknowledgment A ¯rst draft of this manuscript had been prepared between February 1999 and Spring 2000 when the author was formally visiting Andrzej Szczepa¶ nski in Gda¶nsk while attempting to move from Germany to Poland and learning the language. These attempts were supported by the Polish Academy of Sciences by granting o±ce-resources and later also part-time employment. The writing up of the submitted version was supported by BW UG 5100-5-0156-1

References [1] Bogley, W.A. and Sieradski, A.J.: \Universal Path Spaces", preprint, Oregon State University and University of Oregon, Corvallis and Eugene (Oregon, USA), May 1997 [2] Bogley, W.A. and Sieradski, A.J.: \Weighted Combinatorial Group Theory for wild metric complexes". In \Groups|Korea ’98", Proceedings of a conference held in Pusan (Korea) 1998, de Gruyter, Berlin, 2000, pp. 53{80 [3] Cannon, J.W. and Conner, G.R.: \The Combinatorial Structure of the Hawaiian Earring Group", Top. Appl. 106, No. 3 (2000), pp. 225{272 [4] Cannon, J.W. and Conner, G.R.: \The Big Fundamental Group, Big Hawaiian Earrings, and the Big Free Groups" Top. Appl. 106, No. 3 (2000), pp. 273{291 [5] Dunwoody, M. J.: \Groups acting on protrees", J. London Math. Soc. (2), Vol. 56, no. 1 (1997), 125{136 [6] Eda, Katsuya: \Free subgroups of the fundamental group of the Hawaiian Earring", J. Alg. 219 (1999), pp. 598{605 [7] Eda, Katsuya: \Free ¼ -products and non-commutatively slender groups", J. Algebra pp. 243{263 [8] Eda, Katsuya: \Free ¼ -Products and fundamental groups of subspaces of the plane", Topology Appl. 84 (1998), no. 1-3, pp. 283{306 [9] Gri±ths, H. B.: \The Fundamental group of two spaces with a common point",, Quart. J. Math. Oxford (2), 5 (1954), pp. 175{190 (Correction: ibid. 6 (1955), pp. 154{155) [10] Gri±ths, H. B.: \In¯nite Products of semi-groups and local connectivity", Proc. London Math. Soc. (3), 5 (1956), pp. 455{80 [11] Higman, Graham: \"Unrestricted free product, and varieties of topological groups", J. London Math. Soc. (2), 27 (1952), pp. 73{88 [12] Kutosh, A.G.: \Zum Zerlegungsproblem der Theorie der freien Produkte", Rec. Math. Moscow (New Series), 2 (1944), pp. 995{1001

A. Zastrow / Central European Journal of Mathematics 1 (2003) 1{35

35

[13] Morgan, John and Morrison, Ian: \A van-Kampen Theorem of weak Joins", Proc. London Math. Soc (3), 53 (1986), 562{76 [14] Sieradski, Alan: \Omega-Groups", preprint, University of Oregon, Eugene (Oregon, U.S.A.) [15] de Smit, Bart: \The fundamental group of the Hawaiian Earrings is not free", Internat. J. Algebra. Comput., Vol. 3 (1992), 33{37 [16] Zastrow, Andreas: \Construction of an in¯nitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an R-tree", Proc. R. Soc. Edinb., 128A (1998), pp. 433{445 [17] Zastrow, Andreas: \The non-abelian Specker-Group is free", J. Algebra 229, 55{85 (2000)

CEJM 1 (2003) 36{60

Error Autocorrection in Rational Approximation and Interval Estimates. [A survey of results.] Grigori L. Litvinov¤ Independent University of Moscow B. Vlasievskii per., 11, Moscow, 121002 Moscow, Russia

Received 21 August 2002; revised 15 September 2002 Abstract: The error autocorrection e¬ect means that in a calculation all the intermediate errors compensate each other, so the ­ nal result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the e¬ect which appears in rational approximations to functions. The error autocorrection e¬ect occurs in all e¯ cient methods of rational approximation (e.g., best approximations, Pad´e approximations, multipoint Pad´e approximations, linear and nonlinear Pad´e-Chebyshev approximations, etc.), where very signi­ cant errors in the approximant coe¯ cients do not a¬ect the accuracy of this approximant. The reason is that the errors in the coe¯ cients of the rational approximant are not distributed in an arbitrary way, but form a collection of coe¯ cients for a new rational approximant to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The e¬ect of error autocorrection indicates that variations of an approximated function under some deformations of rather a general type may have little e¬ect on the corresponding rational approximant viewed as a function (whereas the coe¯ cients of the approximant can have very signi­ cant changes). Accordingly, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation is not stable under small deformations of the approximated functions. This property is \individual", in the sense that it holds for speci­ c functions. ® c Central European Science Journals. All rights reserved.

¤

Keywords: error autocorrection, rational approximation, algorithms, interval estimates. MSC (2000): Primary 41A20; Secondary 65Gxx

E-mail: [email protected] and [email protected]

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

1

37

Introduction

The paper contains a brief survey description of some, but not all, results published in [1], [2], [3], [4], [5], [6], [7] and some new material on the general form of the error autocorrection e®ect and its relation to interval analysis in the spirit of [8], [9], [10]. Some old results are presented in a new form. The treatment is partly heuristic and based on computer experiments. The error autocorrection e®ect means that in a calculation all the intermediate calculating errors compensate each other, so the ¯nal result is much more accurate than the intermediate results. In this case standard interval estimates (e.g., in the framework of Yu. Matijasevich’s a posteriori interval analysis) are too pessimistic. The error autocorrection e®ect appears in some popular numerical methods, e.g., in the least squares method. In principle this e®ect is already known to experts in interval computations. We shall discuss a very strong form of the e®ect. For the sake of simplicity let us suppose that we calculate values of a real smooth function z = F (y1 ; ¢ ¢ ¢ ; yn ) where y1 ; ¢ ¢ ¢ ; yn are real, and suppose that all round-o® errors are negligible with respect to input data errors. In this case the error ¢F of the function F can be evaluated by the formula ¢F =

N X @F i=1

@yi

¢ ¢yi + r;

P

@F where r is negligible. The sum N i=1 @yi ¢ ¢yi in this formula can be treated as a scalar @F g and the vector of errors f¢yi g. For all the standard product of the gradient vector f @y i interval methods the best possible estimate for ¢F is given by the formula

j¢F j µ

N X @F

j

i=1

@yi

j ¢ j¢yi j:

@F g and f¢yi g are large but the scalar This estimate is too pessimistic if both vectors f @y i product is small enough for these vectors to be almost orthogonal. That is the case of the error autocorrection e®ect. To calculate the values of a function we usually use four arithmetic operations (applying to arguments and constants) and this leads to a rational approximation to the calculated function. The author came across the phenomenon of error autocorrection in the late 1970ies while developing nonstandard algorithms for computing elementary functions on small computers. It was desired to construct rational approximants of the form

R(x) =

a0 + a 1 x + a 2 x2 + : : : + a n xn b0 + b1 x + b2 x2 + : : : + bm xm

(1)

to certain functions of one variable x de¯ned on ¯nite segments of the real line. For this purpose a simple method (described in [1] and below) was used: the method allows to determine the family of coe±cients ai , bj of the approximant (1) as a solution of a certain system of linear algebraic equations. These systems turned out to be ill conditioned, i.e.,

38

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

the problem of determining the coe±cients of the approximant is, generally speaking, ill-posed and small perturbations of the approximated function f (x) or calculation errors lead to very signi¯cant errors in the values of coe±cients. Nevertheless, the method ensures a paradoxically high quality of the obtained approximants. In fact the errors are close to the best possible [1], [2] For example, for the function cos x the approximant of the form (1) on the segment [¡ º =4; º =4] obtained by the method mentioned above for m = 4, n = 6 has the relative error equal to 0:55 ¢ 10¡13 , and the best possible relative error is 0:46 ¢ 10¡13 [11]. The corresponding system of linear algebraic equations has the condition number of order 109 . Thus we risk to lose 9 accurate decimal digits in the solution because of calculation errors. Computer experiments show that this is a serious risk. The method mentioned above was implemented in Fortran code. The calculations were carried out with double precision (16 decimal places) on two di®erent computers. These computers were very similar in architecture, but when passing from one computer to another the system of linear equations and the computational process are perturbed because of calculation errors, including round-o® errors. As a result, the coe±cients of the approximant mentioned above to the function cos x experience a perturbation at the sixth{ninth decimal digits. But the error in the rational approximant itself remains invariant and is 0:4 ¢ 10¡13 for the absolute error and 0:55 ¢ 10¡13 for the relative error. The same thing happens for approximants of the form (1) to the function arctan x on the segment [¡ 1; 1] obtained by the method mentioned above for m = 8, n = 9. The relative error is 0:5 ¢ 10¡11 and does not change while passing from one computer to another although the corresponding system of linear equations has the condition number of order 1011 , and the coe±cients of the approximant experience a perturbation with a relative error of order 10¡4 . Thus the errors in the numerator and the denominator of a rational approximant compensate each other. The e®ect of error autocorrection is connected with the fact that the errors in the coe±cients of a rational approximant are not distributed in an arbitrary way, but form the coe±cients of a new approximant to the approximated function. It can be easily understood that all standard methods of interval arithmetic (see, for example, [6], [8], [9], [10]) do not allow us to take into account this e®ect and, as a result, to estimate the error in the rational approximant accurately (see section 12 below). Note that the application of standard procedures known in the theory of ill-posed problems results in this case in a loss of accuracy. For example, if one applies the regularization method, then two thirds of the accurate ¯gures are lost [12]; in addition, the number of calculations required increases rapidly. The matter of import is that the exact solution of the system of equations in the present case is not the ultimate goal; the aim is to construct an approximant that is precise enough. This approach allows to \rehabilitate" (i.e., to justify) and to simplify a number of algorithms intended for the construction of the approximant, and to obtain (without additional transformations) approximants in a form that is convenient for applications. Professor Yudell L. Luke kindly drew the author’s attention to his papers [13], [14] where the e®ect of error autocorrection for the classical Pad¶e approximants was described

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

39

and explained at a heuristic level. The method mentioned above leads to the linear Pad¶e{ Chebyshev approximants if the calculation errors are ignored. In the present paper the error autocorrection mechanism is considered for a very general situation (including linear methods for the construction of rational approximants and nonlinear generalized Pad¶e approximations). The e±ciency of the construction algorithms used for rational approximants is due to the error autocorrection e®ect (at least in the case when the number of coe±cients is large enough). Our new understanding of the error autocorrection mechanism allows us, to some extent, to control calculation errors by changing the construction procedure depending on the form of the approximant. It is shown that the use of a control parameter allowing to take into account the error autocorrection mechanism ensures the decrease of the calculation errors in some cases. Construction methods for linear and nonlinear Pad¶e{Chebyshev approximants involving the computer algebra system REDUCE (see [4]) are also brie°y described. Computation results characterizing the comparative precision of these methods are given. We analyze the e®ect described in [2] ewith regard to the error autocorrection phenomenon. This e®ect is connected with the fact that a small variation of an approximated function can lead to a sharp decrease in accuracy of the Pad¶e{Chebyshev approximants. The error autocorrection e®ect occurs not only in rational approximation but appears in some other cases - approximate solutions of linear di®erential equations, the method of least squares, etc. In this more general situation relations between the error autocorrection e®ect and standard methods of interval analysis are also discussed in this paper. The author is grateful to Y.L. Luke, B.S. Dobronets, and S.P Shary for stimulating discussions and comments. The author wishes to express his thanks to I. A. Andreeva, A. Ya. Rodionov and V. N. Fridman who participated in the programming and organization of computer experiments.

2

Error autocorrection in rational approximation

Let f’0 ; ’1 ; : : : ; ’n g and fÃ0 ; Ã 1 ; : : : ; Ãm g be two sets of linearly independent functions of the argument x belonging to some (possibly multidimensional) set X . Consider the problem of constructing an approximant of the form R(x) =

a 0 ’ 0 + a 1 ’ 1 + : : : + an ’ n b0 à 0 + b1 à 1 + : : : + bm à m

(2)

to a given function f (x) de¯ned on X . If X coincides with a real line segment [A; B] and if ’k = xk and Ãk = xk for all k, then the expression (2) turns out to be a rational function of the form (1) (see the Introduction). It is clear that expression (2) also gives a rational function in the case when we take Chebyshev polynomials T k or, for example, Legendre, Laguerre, Hermite, etc. polynomials as ’k and Ãk . Fix an abstract construction method for an approximant of the form (2) and consider the problem of computing the coe±cients ai , bj . Quite often this problem is ill-conditioned

40

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

(ill-posed). For example, the problem of computing coe±cients for best rational approximants (including polynomial approximants) for high degrees of the numerator or the denominator is ill-conditioned. The instability with respect to the calculation error can be related both to the abstract construction method of approximation (i.e., with the formulation of the problem) and to the particular algorithm implementing the method. The fact that the problem of computing coe±cients for the best approximant is ill-conditioned is related to the formulation of this problem. This is also valid for other construction methods for rational approximants with a su±ciently large number of coe±cients. But an unfortunate choice of the algorithm implementing a certain method can aggravate troubles connected with ill-conditioning. Let the coe±cients ai , bj give an exact or an approximate solution of this problem, and let the ~ai , ~bj give another approximate solution obtained in the same way. Denote by ¢ai , ¢bj the absolute errors of the coe±cients, i.e., ¢ai = ~ai ¡ ai , ¢bj = ~bj ¡ bj . These errors arise due to perturbations of the approximated function f (x) or due to calculation errors. Set P (x) = ¢P (x) =

Pn

i=0

Pn

i=0

ai ’i ;

Q(x) =

¢ai ’i ;

¢Q(x) =

Pe (x) = P + ¢P;

Pm

j=0 bj à j ;

Pm

j=0 ¢bj à j ;

e Q(x) = Q + ¢Q:

It is easy to verify that the following exact equality is valid: P + ¢P ¡ Q + ¢Q

Ã

P ¢Q ¢P ¡ = Q Q ¢Q

!

P : Q

(3)

As mentioned in the Introduction, the fact that the problem of calculating coe±cients is ill-conditioned can nevertheless be accompanied by high accuracy of the approximants e are close to the approxiobtained. This means that the approximants P=Q and Pe =Q mated function and, therefore, are close to each other, although the coe±cients of these e = ¢Q=(Q + ¢Q) of approximants di®er greatly. In this case the relative error ¢Q= Q the denominator considerably exceeds in absolute value the left-hand side of equality (3). This is possible only in the case when the di®erence ¢P =¢Q ¡ P=Q is small, i.e., the function ¢P =¢Q is close to P =Q, and, hence, to the approximated function. Thus the function ¢P =¢Q will be called the error approximant. For a special case, this concept was actually introduced in [13]. For "e±cient" methods, the error approximant provides e indeed a good approximation for the approximated function and, thus, P =Q and Pe = Q di®er from each other by a product of small quantities in the right-hand side of (3). Usually the following \uncertainty relation" is valid: Ã

P ¢ Q

!

=¯ Q¢

Ã

¢P ¡ ¢Q

P Q

!

º¯ Q¢

Ã

¢P ¡ ¢Q

f

!

¹

";

¢Q ¡ f is where ¯ Q = Q+¢Q is the relative error of the denominator Q, the di®erence ¢P ¢Q ¢P the absolute error of the error approximant ¢Q to the function f , and " is the absolute \theoretical" error of our method; the argument x can be treated as ¯xed.

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

41

The function f (x) is usually treated as an element of a Banach space with a norm P k ¢ k. The absolute error ¢ of the approximant (2) is de¯ned as ¢ = kf ¡ Q k. Its relative P P P error ¯ is the de¯ned as ¯ = k(f ¡ Q )=f k or ¯ = k(f ¡ Q )= Q k. In what follows, we shall consider the space C [A; B] of all continuous functions de¯ned on the real line segment [A; B] with the norm kf (x)k = max jf (x)j: A·x·B

Below we discuss examples of the error autocorrection e®ect for linear and nonlinear methods of rational approximation.

3

Error autocorrection for linear methods in rational approximation

Several construction methods for approximants of the form (2) are connected with solving systems of linear algebraic equations. This procedure can lead to a large error if the corresponding matrix is ill-conditioned. Consider an arbitrary system of linear algebraic equations Ay = h; (4) where A is a given square matrix of order N with components aij (i; j = 1; : : : ; N ), h is a given column vector with components hi , and y is an unknown column vector with components yi . De¯ne the vector norm by the equality kxk =

N X

jxi j

i=1

(this norm is more convenient for calculations than norm is determined by the equality kAk = max kAyk = max kyk=1

1·j·N

q

x21 + : : : + x2N ). Then the matrix

N X

kaij k:

i=1

If a matrix A is nonsingular, then the quantity cond(A) = kAk ¢ kA¡1 k

(5)

is called the condition number of the matrix A (see, for example, [15]). Since y = A¡1 h, we see that the absolute error ¢y of the vector y is connected with the absolute error of the vector h by the relation ¢y = A¡1 ¢h, whence k¢yk µ kA ¡1 k ¢ k¢hk and k¢yk=kyk µ kA ¡1 k ¢ (khk=kyk)(k¢hk=khk): Taking into account the fact that khk µ kAk ¢ kyk, we ¯nally obtain k¢yk=kyk µ kAk ¢ kA ¡1 k ¢ k¢hk=khk;

(6)

42

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

i.e., the relative error of the solution y is estimated via the relative error of the vector h by means of the condition number. It is clear that (6) can turn into an equality. Thus, if the condition number is of order 10k , then, because of round{o® errors in h, we can lose k decimal digits of y. The contribution of the error of the matrix A is evaluated similarly. Finally, the dependence of cond(A) on the choice of a norm is weak. A method of rapid estimation of the condition number is described in [15], Section 3.2. Let an abstract construction method for the approximant of the form (2) be linear in the sense that the coe±cients of the approximant can be determined from a homogeneous system of linear algebraic equations. The homogeneity condition is connected to the fact that, when multiplying the numerator and the denominator of fraction (2) by the same nonzero number, the approximant (2) does not change. Denote by y the vector whose components are the coe±cients a0 ; a1 ; : : : ; an , b0 ; b1 ; : : : ; bm . Assume that the coe±cients can be obtained from the homogeneous system of equations H y = 0;

(7)

where H is a matrix of dimension (m + n + 2) £ (m + n + 1). The vector y~ is an approximate solution of system (7) if the quantity kH y~k is small. If y and y~ are approximate solutions of system (7), then the vector ¢y = y~ ¡ y is also an approximate solution of this system since kH ¢yk = kH y~ ¡ H yk µ kH y~k + kH yk. Thus it is natural to assume that the function ¢P =¢Q corresponding to the solution ¢y is an approximant to f (x). It is clear that the order of the residual of the approximate solution ¢y of system (7), i.e., of the quantity kH ¢yk, coincides with the order of the largest of the residuals of the approximate solutions y and y~. For a ¯xed order of the residual the increase in the error ¢y is compensated by the fact that ¢y satis¯es the system of equations (7) with greater \relative" accuracy, and the latter, generally speaking, leads to the increase in the accuracy of the error approximant. To obtain a particular solution of system (7), one usually adds to this system a normalization condition of the form n X

¶ i ai +

i=0

m X

· j bj = 1;

(8)

j=0

where ¶ i , · j are numerical coe±cients. As a rule, the relation b0 = 1 is taken as the normalization condition (but this is not always successful with respect to minimizing the calculation errors). Adding equation (8) to system (7), we obtain a nonhomogeneous system of m + n + 2 linear algebraic equations of type (4). If the approximate solutions y and y~ of system (7) satisfy condition (8), then the vector ¢y satis¯es the condition n X i=0

¶ i ¢ai +

m X

· j ¢bj = 0:

(9)

j=0

Of course, the above reasoning is not very rigorous; for each speci¯c construction method for approximations it is necessary to carry out some additional analysis. More

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

43

accurate arguments are given below for the linear and nonlinear Pad¶e{Chebyshev approximants. The presence of the error autocorrection mechanism described above is also veri¯ed by numerical experiments (see below). It is clear that classical Pad¶e approximations, multipoint Pad¶e approximations, linear generalized Pad¶e approximations in the sense of [16] (e.g., linear Pad¶e{Chebyshev approximations) give us good examples of linear methods in rational approximation. From our point of view, the methods for obtaining the best approximations can be treated as linear. Indeed the coe±cients of the best Chebyshev approximant satisfy a system of linear algebraic equations and are computed as approximate solutions of this system on the last step of the iteration process in algorithms of Remez’s type (see [7], [17] for details). Thus, the construction methods for the best rational approximants can be regarded as linear. At least for some functions (say, for cos((º =4)x), ¡ 1 µ x µ 1) the linear and the nonlinear Pad¶e{Chebyshev approximants are very close to the best ones in relative and absolute errors, respectively. The results that arise when applying calculation algorithms for Pad¶e{Chebyshev approximants can be regarded as approximate solutions of the system which determines the best approximants. Thus the presence of the e®ect of error autocorrection for Pad¶e{Chebyshev approximants gives an additional argument in favor of the conjecture that this e®ect also takes place for the best approximants. Finally, note that the basic relation (3) becomes meaningless if one seeks an approximant in the form a0 ’0 + a1 ’1 + : : : + an ’n , i.e., the denominator in (2) is reduced to 1. However, in this case the e®ect of error autocorrection (although much weakened) is also possible. This is connected to the fact that the errors ¢ai approximately satisfy certain relations. Such a situation can arise when using the least squares method.

4

Linear Pad¶e{Chebyshev approximations and the PADE program

Let us begin to discuss a series of examples. Consider the approximant of the form (1) Rm;n (x) =

a0 + a 1 x + a 2 x2 + : : : + an xn b0 + b1 x + b2 x2 + : : : + bm xm

(10)

to a function f (x) de¯ned on the segment [¡ 1; 1]. The absolute error function ¢(x) = f (x) ¡

Rm;n (x)

obviously has the following form: ¢(x) = ©(x)=Qm (x); where ©(x) = f (x)Q m (x) ¡

Pn (x):

(11)

44

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

The function Rm;n (x) = P n (x)=Q m (x) is called the linear Pad¶ e{Chebyshev approximant to the function f (x) if Z1

©(x)Tk (x)w(x) dx = 0;

k = 0; 1; : : : ; m + n;

(12)

¡1

p where Tk (x) = cos(n arccos x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 . This concept allows a generalization to the case of other orthogonal polynomials (see, e.g., [16], [18], [19], [20]). Approximants of this kind always exist [18]. The system of equations (12) is equivalent to the following system of linear algebraic equations with respect to the coe±cients ai , bj : m X

j=0

bj

Z1

¡1

xj T k (x)f (x) p dx ¡ 1 ¡ x2

n X i=0

ai

Z1

¡1

xi T (x) p k dx = 0: 1 ¡ x2

(13)

The homogeneous system (12) can be transformed into a nonhomogeneous one by adding a normalization condition. In particular, any of the following relations can be taken as this condition: b0 = 1;

(14)

bm = 1;

(15)

am = 1:

(16)

The sources [1], [2] brie°y describe the program PADE, written in Fortran, with double precision) which constructs rational approximants by solving the system of equations of type (13). The complete text of a certain version of this code and its detailed description can be found in the Collection of Algorithms and Codes of the Research Computer Center of the Russian Academy of Sciences [3]. For even functions the program looks for an approximant of the form R(x) =

a0 + a1 x2 + : : : + an (x2 )n ; b0 + b1 x2 + : : : + bm (x2 )m

(17)

a0 + a1 x2 + : : : + an (x2 )n ; b0 + b1 x2 + : : : + bm (x2 )m

(18)

and for odd functions it is R(x) = x

The program computes the values of coe±cients of the approximant, the absolute and the relative errors ¢ = maxA·x·B j¢(x)j and ¯ = maxA·x·B j¢(x)=f (x)j, and gives the information which allows us to estimate the quality of the approximation (see [7] and [3] for details). Using a subroutine, the user introduces the function de¯ned by means of any algorithm on an arbitrary segment [A; B], introduces the boundary points of this segment, the numbers m and n, and the number of control parameters. In particular, one can choose the normalization condition of type (14){(16), look for an approximant in

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

45

the form (17) or (18) and so on. The change of the variable reduces the approximation on any segment [A; B] to the approximation on the segment [¡ 1; 1]. Therefore, we shall consider the case when A = ¡ 1, B = 1 in the sequel unless otherwise stated. For the calculation of integrals, the Gauss{Hermite{Chebyshev quadrature formula is used: Z1 s ³ ’(x) º X 2i ¡ 1 ´ p dx = ’ cos º ; (19) 2 s i=1 2s ¡ 1 x ¡1 where s is the number of interpolation points. For polynomials of degree 2s¡ 1 this formula is exact, so the precision of formula (19) increases rapidly as the parameter s increases and depends on the quality of the approximation of the function ’(s) by polynomials. To calculate the values of Chebyshev polynomials, the well-known recurrence relation is applied. If the function f (x) is even and of the form (17) is desired, then the system (13) is transformed into the following system of equations: n X i=0

ai

Z1

¡1

x2i T 2k (x) p dx ¡ 1 ¡ x2

m X

j=0

bj

Z1

¡1

x2j T2k (x)f (x) p dx = 0; 1 ¡ x2

(20)

where k = 0; 1; : : : ; m + n. If f (x) is an odd function and an approximant of the form (18) is desired, then one ¯rst determines an approximant of the form (17) to the even function f (x)=x by solving the system (20) complemented by one of the normalization conditions. Then the obtained approximant is multiplied by x. This procedure allows us to avoid a large relative error for x = 0. This algorithm is rather simple; for its implementation only two standard subroutines are needed (for solving systems of linear algebraic equations and for numerical integration). However, the algorithm is e±cient. The capabilities of the PADE code are demonstrated in Table 1. This table contains errors for certain approximants obtained by means of this program. For every approximant, the absolute error ¢, the relative error ¯ , and (for comparison) p the best possible relative error ¯ min given in [11] are indicated. The function x is approximated on the segment [1=2; 1] by an expression of the form (1), the function cos ¼4 x is approximated on the segment [¡ 1; 1] by an expression of the form (17), and all the others are approximated on the same segment by an expression of the form (18).

5

Error autocorrection for the PADE program

The condition numbers of systems of equations that arise while calculating, by means of the PADE program, the approximants considered above are very large. For example, for calculating the approximant of the form (18) on the segment [¡ 1; 1] to sin ¼2 x for m = n = 3, the corresponding condition number is of order 1013 . As a result, the coe±cients of the approximant are determined with a large error. In particular, a small perturbation of the system of linear equations arising when passing from one computer

46

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

Function p x p x

m

n

¢

¯

¯

2

2

0:8 ¢ 10¡6

1:13 ¢ 10¡6

0:6 ¢ 10¡6

3

3

1:9 ¢ 10¡9

2:7 ¢ 10¡9

1:12 ¢ 10¡9

cos ¼4 x

0

3

0:28 ¢ 10¡7

0:39 ¢ 10¡7

0:32 ¢ 10¡7

cos ¼4 x

1

2

0:24 ¢ 10¡7

0:34 ¢ 10¡7

0:29 ¢ 10¡7

cos ¼4 x

2

2

0:69 ¢ 10¡10

0:94 ¢ 10¡10

0:79 ¢ 10¡10

cos ¼4 x

0

5

0:57 ¢ 10¡13

0:79 ¢ 10¡13

0:66 ¢ 10¡13

cos ¼4 x

2

3

0:4 ¢ 10¡13

0:55 ¢ 10¡13

0:46 ¢ 10¡13

sin ¼4 x

0

4

0:34 ¢ 10¡11

0:48 ¢ 10¡11

0:47 ¢ 10¡11

sin ¼4 x

2

2

0:32 ¢ 10¡11

0:45 ¢ 10¡11

0:44 ¢ 10¡11

sin ¼4 x

0

5

0:36 ¢ 10¡14

0:55 ¢ 10¡14

0:45 ¢ 10¡14

sin ¼2 x

1

1

0:14 ¢ 10¡3

0:14 ¢ 10¡3

0:12 ¢ 10¡3

sin ¼2 x

0

4

0:67 ¢ 10¡8

0:67 ¢ 10¡8

0:54 ¢ 10¡8

sin ¼2 x

2

2

0:63 ¢ 10¡8

0:63 ¢ 10¡8

0:53 ¢ 10¡8

sin ¼2 x

3

3

0:63 ¢ 10¡13

0:63 ¢ 10¡13

0:5 ¢ 10¡13

tan ¼4 x

1

1

0:64 ¢ 10¡5

0:64 ¢ 10¡5

0:57 ¢ 10¡5

tan ¼4 x

2

1

0:16 ¢ 10¡7

0:16 ¢ 10¡7

0:14 ¢ 10¡7

tan ¼4 x

2

2

0:25 ¢ 10¡10

0:25 ¢ 10¡10

0:22 ¢ 10¡10

arctan x

0

7

0:75 ¢ 10¡7

10¡7

10¡7

arctan x

2

3

0:16 ¢ 10¡7

0:51 ¢ 10¡7

0:27 ¢ 10¡7

arctan x

0

9

0:15 ¢ 10¡8

0:28 ¢ 10¡8

0:23 ¢ 10¡8

arctan x

3

3

0:54 ¢ 10¡9

1:9 ¢ 10¡9

0:87 ¢ 10¡9

arctan x

4

4

0:12 ¢ 10¡11

0:48 ¢ 10¡11

0:17 ¢ 10¡11

arctan x

5

4

0:75 ¢ 10¡13

3:7 ¢ 10¡13

0:71 ¢ 10¡13

min

Table 1

to another (because of the calculation errors) gives rise to large perturbations in the coe±cients of the approximant. Fortunately, the e®ect of error autocorrection improves the situation, and the errors in the approximant undergo no substantial changes under this perturbation. This fact is described in the Introduction, where concrete examples are also given. Consider a few more examples connected with passing from one computer to another (see [6], [7] for details). The branch of the algorithm which corresponds to the normalization condition (14) (i.e., to b0 = 1) is considered. For arctan x, the calculation of an

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

47

approximant of the form (18) on the segment [¡ 1; 1] for m = n = 5 gave an approximant with the absolute error ¢ = 0:35 ¢ 10¡12 and the relative error ¯ = 0:16 ¢ 10¡11 . The corresponding system of linear algebraic equations has the condition number of order 1030 ! Passing to another computer we obtain the following: ¢ = 0:5 ¢ 10¡14 , ¯ = 0:16 ¢ 10¡12 , the condition number is of order 1014 , and the errors ¢a1 and ¢b1 in the coe±cients a1 and b1 in (18) are greater in absolute value than 1! This example shows that the problem of computing the condition number of an ill-conditioned system is, in its turn, ill-conditioned. Indeed, the condition number is, roughly speaking, determined by values of coe±cients of the inverse matrix, every column of the inverse matrix being the solution of the system of equations with the initial matrix of coe±cients, i.e., of an ill-conditioned system. Consider in detail the e®ect of error autocorrection for the approximant of the form (17) to the function cos ¼4 x on the segment [¡ 1; 1]for m = 2, n = 3. For our two di®erent computers, two di®erent approximants were obtained with the coe±cients ai ,bi and a~i ,~bi respectively. In both cases, the condition number is of order 109 , and the absolute error is ¯ = 0:55 ¢ 10¡13 . These errors are close to best possible. The coe±cients of the approximants obtained on the computers mentioned above and the coe±cients of the error approximant (see Section 3 above) are as follows: a~0 = 0:9999999999999600;

a0 = 0:9999999999999610;

¢a0 = ¡ 10¡15 ; ~a1 = ¡ 0:2925310453579570;

a1 = ¡ 0:2925311264716216;

¢a1 = 10¡7 ¢ 0:811136646; a~2 = 10¡1 ¢ 0:1105254254716866;

a2 = 10¡1 ¢ 0:1105256585556549;

¢a2 = ¡ 10¡7 ¢ 0:2330839683; a~3 = 10¡3 ¢ 0:1049474500904401;

a3 = 10¡3 ¢ 0:1049482094850086;

¢a3 = 10¡9 ¢ 0:7593947685; b0 = 1;

~b0 = 1;

¢b0 = 0; ~b1 = 10¡1 ¢ 0:1589409217324021;

b1 = 10¡1 ¢ 0:1589401105960337;

¢b1 = 10¡7 ¢ 0:8111363684; ~b2 = 10¡3 ¢ 0:1003359011092697;

b2 = 10¡3 ¢ 0:1003341918083529;

¢b2 = 10¡8 ¢ 0:17093009168: Thus, the error approximant has the form ¢P ¢a0 + ¢a1 x2 + ¢a2 x4 + ¢a3 x6 = : ¢Q ¢b1 x2 + ¢b2 x4

(21)

48

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

If the relatively small quantity ¢a0 = ¡ 10¡15 in (21) is omitted, then, as testing by means of a computer shows (2000 checkpoints), this expression is an approximant to the function cos ¼4 x on the segment [¡ 1; 1] with the absolute and relative errors ¢ = ¯ = 0:22 ¢ 10¡6 . But the polynomial ¢Q is zero at x = 0, and the polynomial ¢P takes a small, but nonzero value at x = 0. Fortunately, relation (3) can be rewritten in the following way: Pe ¡ e Q

P ¢P = e ¡ Q Q

¢Q P ¢ : e Q Q

(22)

Thus, as ¢Q ! 0, the e®ect of error autocorrection arises because the quantity ¢P is close to zero, and the error of the approximant P =Q is determined by the error in the coe±cient a0 . The same situation also takes place when the polynomial ¢Q vanishes at an arbitrary point x0 belonging to the segment [A; B] where the function is approximated. It is clear that if one chooses the standard normalization (b0 = 1), then the error approximant actually has two coe±cients less than the initial one. It is clear that in the general case the normalization conditions an = 1 or bm = 1 result in the following: the coe±cients of the error approximant form an approximate solution of the homogeneous system of linear algebraic equations whose exact solution determines the Pad¶e{Chebyshev approximant having one coe±cient less than the initial one. The e®ect of error autocorrection improves the accuracy of this error approximant as well. Thus,\the snake bites its own tail". A situation also arises in the case when the approximant to an even function of the form (17) is constructed by solving the system of equations (20). Sometimes it is possible to decrease the error of the approximant by choosing a good normalization condition. As an example, consider the approximation of the function ex on the segment [¡ 1; 1] by rational functions of the form (1) for m = 15, n = 0. For the traditionally accepted normalization b0 = 1, the PADE program yields an approximant with the absolute error ¢ = 1:4 ¢ 10¡14 and the relative error ¯ = 0:53 ¢ 10¡14 . After passing to the normalization condition b15 = 1, the errors are reduced nearly one half: ¢ = 0:73 ¢ 10¡14 , ¯ = 0:27 ¢ 10¡14 . Note that the condition number increases: in the ¯rst case it is 2 ¢ 106 , and in the second case it is 6 ¢ 1016 . Thus the error decreases notwithstanding the fact that the system of equations becomes drastically ill-conditioned. This example shows that the increase in the accuracy of the error approximant can be accompanied by the increase of the condition number, and, as experiments show, by the increase of errors of the numerator and the denominator of the approximant. The best choice of the normalization condition depends on the particular situation. A speci¯c situation arises when the degree of the numerator (or of the denominator) of the approximant is equal to zero. In this case a bad choice of the normalization condition results in the following: the error approximant becomes zero or is not well-de¯ned. For n = 0 it is expedient to choose condition (15), as it was done in the example given above. For m = 0 (the case of the polynomial approximation) it is usually expedient to choose condition (16). One could search for the numerator and the denominator of the approximant in the

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

form P =

n X

ai T i ;

Q=

i=0

m X

bj Tj ;

49

(23)

j=0

where T i are the Chebyshev polynomials. In this case the system of linear equations determining the coe±cients would be better conditioned. But the calculation of the polynomials of the form (23) by, for example, the Chenshaw method, results in lengthening the computation time, although it has a favorable e®ect upon the error. The transformation of the polynomials P and Q from the form (23) into the standard form also requires additional e®orts. In practice it is more convenient to use approximants represented in the form (1), (17), or (18), and calculate the fraction’s numerator and denominator according to the Horner scheme. In this case the normalization an = 1 or bm = 1 allows to reduce the number of multiplications. The use of the algorithm does not require that the approximated function be expanded into a series or a continued fraction beforehand. Equations (12) or (13) and the quadrature formula (19) show that the algorithm uses only the values of the approximated function f (x) at the interpolation points of the quadrature formula (which are the zeros of some Chebyshev polynomial). On the segment [¡ 1; 1] the linear Pad¶e{Chebyshev approximants give a considerably smaller error than the classical Pad¶e approximants. For example, the Pad¶e approximant of the form (1) to the function ex for m = n = 2 has the absolute error ¢(1) = 4 ¢ 10¡3 at the point x = 1, but the PADE program gives an approximant of the same form with the absolute error ¢ = 1:9 ¢ 10¡4 (on the entire the segment), i.e., the latter is 20 times smaller than the former. The absolute error of the best approximant is 0:87 ¢ 10¡4 .

6

The \cross{multiplied" linear Pad¶e{Chebyshev approximation

As a rule, linear Pad¶e{Chebyshev approximants are constructed according to the following scheme, see, e.g., [21], [11], [16]. Let the approximated function be decomposed into the series in Chebyshev polynomials f (x) =

1 X 0 i=0

where the notation

m P 0

i=1

1 ci Ti (x) = c0 + c1 T1 (x) + c2 T 2 (x) + : : : ; 2

(24)

means that the ¯rst term u0 in the sum is replaced by u0 =2. The

rational approximant of the form

R(x) =

n P 0

i=0 m P 0

j=0

ai Ti (x) ; bj Tj (x)

(25)

50

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

is desired, where the coe±cients bj satisfy the following system of linear algebraic equations m X0 j=0

bj (ci+j + cji¡jj) = 0;

i = n + 1; : : : ; n + m;

(26)

and the coe±cients ai are determined by the equalities m

1 X0 ai = bj (ci+j + cji¡jj ) = 0; 2 j=0

i = 0; 1; : : : ; n:

(27)

It is not di±cult to verify that this algorithm must lead to the same results as the algorithm described in Section 5 if the calculation errors are not taken into account. The coe±cients ck for k = 0; 1; : : : ; n + 2m, are present in (26) and (27), i.e., it is necessary to have the ¯rst n + 2m + 1 terms of series (24). The coe±cients ck are known, as a rule, only approximately. To determine them one can take the truncated expansion of f (x) into the series in powers of x (the Taylor series) and by means of the well-known economization procedure transform it into the form n+2m X

c~i Ti (x):

(28)

i=0

7

Nonlinear Pad¶e{Chebyshev approximations

A rational function R(x) of the form (1) or (25) is called a nonlinear Pad¶ e{Chebyshev approximant to the function f (x) on the segment [¡ 1; 1], if Z1

(f (x) ¡

R(x))T k (x)w(x) dx = 0;

k = 0; 1; : : : ; m + n;

(29)

¡1

p where Tk (x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 . The paper [22] describes the following algorithm of computing the coe±cients of (29) is given. Let the approximated function f (x) be expanded into series (24) in Chebyshev polynomials. Determine the auxiliary quantities ® i from the system of linear algebraic equations m X

j=0

® j cjk¡jj = 0;

k = n + 1; n + 2; : : : ; n + m;

(30)

assuming that ® 0 = 1. The coe±cients of the denominator in expression (25) are determined by the relations bj = · P

m¡j X

® i®

i+j ;

i=0

where · ¡1 = 1=2 ni=1 ® i2 ; this implies b0 = 2. Finally, the coe±cients of the numerator are determined by formula (27). It is possible to solve system (30) explicitly and to indicate the formulas for computing the quantities ® i . One can also estimate explicitly the absolute error of the approximant. This algorithm is described in detail in the book [20]; see also [16].

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

51

In contrast to the linear Pad¶e{Chebyshev approximants, the nonlinear approximants of this type do not always exist, but it is possible to indicate explicitly veri¯able conditions guaranteeing the existence of such approximants [20]. The nonlinear Pad¶e{Chebyshev approximants (in comparison with the linear ones) have, as a rule, somewhat smaller absolute errors, but can have larger relative errors. Consider, as an example, the approximant of the form (1) or (25) to the function ex on the segment [¡ 1; 1] for m = n = 3. In this case the absolute error for a nonlinear Pad¶e{Chebyshev approximant is ¢ = 0:258 ¢ 10¡6 , and the relative error, ¯ = 0:252 ¢ 10¡6 ; for the linear Pad¶e{Chebyshev approximant ¢ = 0:33 ¢ 10¡6 and ¯ = 0:20 ¢ 10¡6 .

8

Applications of the computer algebra system REDUCE to the construction of rational approximants

The computer algebra system REDUCE [23] allows us to handle formulas at the symbolic level and is a convenient tool for the implementation of algorithms of computing rational approximants. The use of this system allows us to bypass the procedure of working out the algorithm for computing the approximated function if this function is presented in analytical form or when the Taylor series coe±cients are known or are determined analytically from a di®erential equation. The round-o® errors can be eliminated by using the exact arithmetic of rational numbers represented in the form of ratios of integers. Within the framework of the REDUCE system, the code package for enhanced precision computations and construction of rational approximants is implemented; see, for example [4]. In particular, the algorithms from Sections 6 and 7 (which have similar structure) are implemented, the approximated function being ¯rst expanded into the P (k) power (Taylor) series, f = 1 (0)xk =k!, and then the truncated series k=0 f N X

k=0

f (k) (0)

xk ; k!

(31)

consisting of the ¯rst N + 1 terms of the Taylor series (the value N is determined by the user) are transformed into a polynomial of the form (28) by means of the economization procedure. The algorithms implemented by means of the REDUCE system allow us to obtain approximants in the form (1) or (25), estimates of the absolute and the relative error, and the error curves. The output includes the Fortran code for computing the corresponding approximant. The constants of rational arithmetic are transformed into the standard °oating point form. When computing the values of the obtained approximant, this approximant can be transformed into the form most convenient for the user. For example, one can calculate values of the numerator and the denominator of the fraction of the form (1) according to the Horner scheme, and for the fraction of the form (25), according to Clenshaw’s scheme, and transform the rational expression into a continued fraction or a Jacobi fraction as well.

52

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

The ALGOL-like input language of the REDUCE system and convenient tools for solving problems of linear algebra guarantee the simplicity and compactness of the code. For example, the length of the program for computing linear Pad¶e{Chebyshev approximants is 62 lines.

9

Error approximants for linear and nonlinear Pad¶e{Chebyshev approximations

Relations (29) can be regarded as a system of equations for the coe±cients of the ape e proximant. Let the approximants R(x) = P (x)=Q(x) and R(x) = Pe (x)= Q(x), where e e P (x), P (x) are polynomials of degree n, and Q(x), Q(x) are polynomials of degree m, be obtained by approximate solving the indicated system of equations. Consider the error e ¡ Q(x). Subapproximant ¢P (x)=¢Q(x), where ¢P (x) = Pe (x) ¡ P (x), ¢Q(x) = Q(x) e stituting R(x) and R(x) into (29) and subtracting one of the obtained expressions from the other, we see that the following approximate equality holds: Z1 µ ~ P (x)

¡1

~ Q(x)

¡

¶

P (x) T k (x)w(x) dx º 0; Q(x)

~ ¡ which directly implies that R(x) the approximate equality Z1 µ

¡1

¢P (x) ¡ ¢Q(x)

k = 0; 1; : : : ; m + n;

R(x) is close to zero. This and the equality (3) imply ¶

P (x) ¢Q T (x)w(x) dx º 1; ~ k Q(x) Q

(32)

p where k = 0; 1; : : : ; m + n, w(x) = 1= 1 ¡ x2 . If the quantity ¢Q is relatively large (this is connected with the fact that the system of equations (30) is ill-conditioned), then, as follows from equality (32), we can naturally expect that the error approximant is close to P =Q and, consequently, to the approximated function f (x). Due to the fact that the arithmetic system of rational numbers is used, the software described in Section 7 allows us to eliminate the round-o® errors and to estimate the \pure" in°uence of errors in the approximated function on the coe±cients of the nonlinear Pad¶e{Chebyshev approximant. In this case the e®ect of error autocorrection can be substantiated by a more accurate reasoning which is valid for both the linear and the nonlinear Pad¶e{Chebyshev approximants, and even for the linear generalized Pad¶e approximants connected with di®erent systems of orthogonal polynomials. This reasoning is analogous to Y. L. Luke’s considerations [13] for the case of classical Pad¶e approximants. Assume that the function f (x) is expanded into series (24) and that the rational approximant R(x) = P (x)=Q(x) of the form (25) is desired. Let ¢bj be the errors in the coe±cients of the approximant’s denominator Q. In the linear case these errors arise when solving the system of equations (26), and in the nonlinear case, when solving the system of equations (27). In both cases the coe±cients

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

53

in the approximant’s numerator are determined by equations (27), whence we have m

¢ai =

1 X0 ¢bj (ci+j + cji¡jj); 2 j=0

i = 0; 1; : : : ; n:

(33)

This implies the following fact: the error approximant ¢P =¢Q satis¯es the relations Z1

(f (x)¢Q(x) ¡

¢P (x))T i (x)w(x) dx = 0;

i = 0; 1; : : : ; n;

(34)

¡1

which are analogous to relations (12) de¯ning the linear Pad¶e{Chebyshev approximants. Indeed, let us use the well-known multiplication formula for Chebyshev polynomials: 1 T i (x)Tj (x) = [Ti+j (x) + Tji¡jj (x)]; 2

(35)

where i, j are arbitrary indices; see, for example [16], [20]. Taking (35) into account, the quantity f ¢Q ¡ ¢P can be rewritten in the following way: f ¢Q ¡

¢P = =

µX m 0

¢bj Tj j=0 1 · m 1 X0 X0 2 i=0

j=0

¶µX 1

0

¶

ci T i ¡

i=0

n X 0

¢ai T i

i=0

¸

¢bj (ci+j + cji¡jj) Ti ¡

n X 0

¢ai T i :

i=0

This formula and (33) imply that f ¢Q ¡

¢P = O(T n+1);

(36)

i.e., in the expansion of the function f ¢Q ¡ ¢P into the series in Chebyshev polynomials, the ¯rst n + 1 terms are absent, and the latter is equivalent to relations (34) by virtue of the fact that the Chebyshev polynomials form an orthogonal system. Consider an arbitrary rational function of the form (1) or (8) Rm;n (x) =

a0 + a1 x + ¢ ¢ ¢ + a n x n P n (x) = : m b0 + b1 x + ¢ ¢ ¢ + bm x Qm (x)

We shall say that Rm;n (x) is a generalized linear Pad¶ e-Chebyshev approximant of order N to the function f (x) if Z 1

¡1

©(x)T k (x)w(x)dx = 0;

k = 0; 1; ¢ ¢ ¢ ; N;

p where Tk (x) = cos(n arccos x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 , ©(x) = f (x)Q m (x) ¡ Pn (x). This means that the ¯rst N +1 terms in the expansion of the function ©(x) into the series in Chebyshev polynomials (\the Fourier-Chebyshev series") are absent, i.e. f (x)Q m (x) ¡ Pn (x) = O(T N+1): If N = m + n, then we have the usual linear Pad¶e-Chebyshev approximant discussed above in Section 4. Formula (36) means that the following result is valid.

54

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

Theorem 9.1. Let ¢P be the error approximant to f (x) generated by the approxi¢Q mant (25) for the case of linear or nonlinear Pad¶e{Chebyshev approximation and algo¢P rithms described in Sections 6 and 7. Then this error approximant ¢Q is a generalized linear Pad¶e-Chebyshev approximant of order n to the function f (x). An equivalent result was discussed in [5], [7]. When carrying out actual computations, the coe±cients ci are known only approximately, and thus the equalities (33), (34) and (35) are also satis¯ed approximately.

10

Computer experiments for the nonlinear Pad¶e{Chebyshev approximation

Consider the results of computer experiments that were performed by means of the software implemented in the framework of the REDUCE system and brie°y described in Section 7 above. At the author’s request, computer calculations were carried out by A. Ya. Rodionov. We begin with the example considered in Section 5 above, where the linear Pad¶e{Chebyshev approximant of the form (17) to the function cos ¼4 x was constructed on the segment [¡ 1; 1] for m = 2, n = 3. To construct the corresponding nonlinear Pad¶e{Chebyshev approximant, it is necessary to specify the value of the parameter N determining the number of terms in the truncated Taylor series (31) of the approximated function. In this case the calculation error is determined, in fact, by the parameter N . The coe±cients in approximants of the form (17) which are obtained for N = 15 and N = 20 (the nonlinear case) and the coe±cients in the error approximant are as follows: a~0 = 0:4960471034987563;

a0 = 0:4960471027757504;

¢a0 = 10¡8 ¢ 0:07230059; ~a1 = ¡ 0:1451091945278387;

a1 = ¡ 0:1451091928755736;

¢a1 = ¡ 10¡8 ¢ 0:16522651; ~a2 = 10¡2 ¢ 0:5482586543334515;

a2 = 10¡2 ¢ 0:548258121085953;

¢a2 = ¡ 10¡9 ¢ 0:42224856; ~a3 = ¡ 10¡4 ¢ 0:5205903601778259;

a3 = ¡ 10¡4 ¢ 0:5205902238186334;

¢a3 = ¡ 10¡10 ¢ 0:13635919; ~b0 = 0:4960471034987759;

b0 = 0:4960471027757698;

¢b0 = 10¡8 ¢ 0:07230061; ~b1 = 10¡2 ¢ 0:7884201590727615;

b1 = 10¡2 ¢ 0:7884203019999351;

¢b1 = ¡ 10¡10 ¢ 0:1429272;

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

~b2 = 10¡4 ¢ 0:4977097973870693;

55

b2 = 10¡4 ¢ 0:4977100977750249;

¢b2 = ¡ 10¡10 ¢ 0:300388: Both approximants have absolute errors ¢ equal to 0:4 ¢ 10¡13 and relative errors ¯ equal to 0:6 ¢ 10¡13 . These values are close to the best possible. The condition number of the system of equations (30) in both cases is 0:4 ¢ 108 . The denominator ¢Q of the error approximant is zero for x = x0 º 0:70752 : : :; the point x0 is also close to the root of the numerator ¢P which for x = x0 is of order 10¡8 . Such a situation was considered in Section 5 above. Outside a small neighborhood of the point x0 the absolute and the relative errors have the same order as in the \linear case" considered in Section 5. Now consider the nonlinear Pad¶e{Chebyshev approximant to the function tan ¼4 x on the segment [¡ 1; 1] of the form (17) for m = n = 3. In this case the Taylor series converges very slowly, and, as the parameter N increases, the values of the coe±cients of the rational approximant undergo substantial (even in the ¯rst decimal digits) and intricate changes. The situation is illustrated in Table 2, where the following values are given: the absolute errors ¢, the absolute errors ¢0 of error approximants (there the approximants are compared for N = 15 and N = 20, for N = 25 and N = 35, for N = 40 and N = 50), and also the values of the condition number cond of the system of linear algebraic equations (30). In this case the relative errors coincide with the absolute ones. The best possible error is ¢min = 0:83 ¢ 10¡17 . A small neighborhood of the root of the polynomial ¢Q is eliminated as before. N

15

20

25

35

40

50

cond

0:76 ¢ 107

0:95 ¢ 108

0:36 ¢ 1010

0:12 ¢ 1012

0:11 ¢ 1012

0:11 ¢ 1012

¢

0:13 ¢ 10¡4

0:81 ¢ 10¡6

0:13 ¢ 10¡7

0:12 ¢ 10¡10

0:75 ¢ 10¡12

0:73 ¢ 10¡15

¢0

0:7 ¢ 10¡4

0:7 ¢ 10¡8

0:2 ¢ 10¡9

Table 2

11

Small deformations of approximated functions and acceleration of convergence of series

Let a function f (x) be expanded into the series in Chebyshev polynomials, i.e., suppose P that f (x) = 1 i=0 ci T i . Consider a partial sum f^N (x) =

N X

ci Ti

(37)

i=o

of this series. Using formula (35), it is easy to verify that the linear Pad¶e{Chebyshev approximant of the form (1) or (25) to the function f (x) coincides with the linear Pad¶e{ Chebyshev approximant to polynomial (37) for N = n + 2m, i.e., it depends only on the ¯rst n + 2m + 1 terms of the Fourier{Chebyshev series of the function f (x). A similar

56

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

result is valid for the approximant of the form (17) or (18) to even or odd functions, respectively. Note that for N = n + 2m the polynomial f^N is the result of application of the algorithm of linear (or nonlinear) Pad¶e{Chebyshev approximation to f (x), where the exponents m and n are replaced by 0 and 2m + n. The interesting e®ect mentioned in [2] consists of the fact that the error of the polynomial approximant f^n+2m depending on n + 2m + 1 parameters can exceed the error of the corresponding Pad¶e{Chebyshev approximant of the form (1) which depends only on n + m + 1 parameters. For example, consider an approximant of the form (18) to the function tan ¼4 x on the segment [¡ 1; 1]. For m = n = 3 the linear Pad¶e{Chebyshev approximant has error of order 10¡17 , and the corresponding polynomial approximant of the form (37) has the error of order 10¡11 . This polynomial of degree 19 (odd functions are involved, and hence m = n = 3 in (18) corresponds to m = 6, n = 7 in (1)) can be regarded as a result of a deformation of the approximated function tan ¼4 x. This deformation does not a®ect the ¯rst twenty terms in the expansion of this function in Chebyshev polynomials and, consequently, does not a®ect the coe±cients in the corresponding rational Pad¶e{Chebyshev approximant, but leads to an increase of several orders in its error. Thus, a small deformation of the approximated function can result in a sharp change in the order of the error of a rational approximant. Moreover the e®ect just mentioned means that the algorithm extracts additional information concerning the following components of the Fourier{Chebyshev series from the polynomial (37). In other words, in this case the transition from the Fourier{Chebyshev series to the corresponding Pad¶e{Chebyshev approximant accelerates the convergence of the series. A similar e®ect of acceleration of convergence of power series by passing to the classical Pad¶e approximant is known (see, e.g., [16]). It is easy to see that the nonlinear Pad¶e{Chebyshev approximant of the form (1) to the function f (x) depends only on the ¯rst m + n + 1 terms of the Fourier{Chebyshev series for f (x), so that for such approximants a more pronounced e®ect of the type indicated above takes place. Since one can change the \tail" of the Fourier{Chebyshev series in a quite arbitrary way without a®ecting the rational Pad¶e{Chebyshev approximant, the e®ect of acceleration of convergence can take place only for the series with an especially regular behavior (and for the corresponding \well-behaved" functions). See [5], [7] for some details.

12

Error autocorrection and Interval Analysis

Undoubtedly one of the most relevant problems in Interval Analysis in the sense of [8], [9], [10] is getting realistic interval estimates for calculation errors, i.e. to get e±cient estimates close to the virtual calculation errors. Di±culties arise where intermediate errors cancel out each other. For the sake of simplicity let us suppose that we calculate values of a real smooth function z = F (y1 ; : : : ; yN ) of real variables y1 ; : : : ; yN , and suppose that all round-o® errors are negligible with respect to input data errors. This situation has been examined in detail in the framework of Ju.V. Matijasevich’s \a posteriori interval

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

57

analysis", see, e.g., [10]. In this case the error ¢F of F (y1 ; : : : ; yN ) can be evaluated by the formula N X @F ¢ ¢yi + r; ¢F = (38) i=1 @yi P

where r is negligible. The sum N i=1 (@F =@yi ) ¢ ¢yi in (38) can be treated as a scalar product of the gradient vector f@F=@yi g and the vector of errors f¢yi g. The e®ect of error autocorrection corresponds to the case, where the gradient f@F =@yi g is large but the scalar product is relatively small. In this case these vectors are almost orthogonal and the following approximate equation holds: N X @F i=1

@yi

¢ ¢yi º 0

(39)

This e®ect is typical for some ill-posed problems. For all the standard interval methods, the best possible estimation for ¢F is given by the formula N X @F j¢F j µ j j ¢ j¢yi j: (40) i=1 @yi This estimate is good if the errors ¢yi are \independent" but it is not realistic in the case discussed in this paper (calculation of values of rational approximants when the error autocorrection e®ect is at work). In this case F (y1 ; : : : ; yN ) = R(x; a0 ; : : : ; an ; b0 ; : : : ; bm ) =

a0 + a 1 x + a 2 x 2 + : : : + a n x n ; b0 + b1 x + b2 x2 + : : : + bm xm

where N = m + n+ 3, fy1 ; : : : ; yN g = fx; a0 ; : : : ; an ; b0 ; : : : ; bm g. For the sake of simplicity let us suppose that ¢x = 0. In this case we can use the equality (22) to transform the formula (38) into the formula ¢R º

¢P ¡ Q

n m X P ¢Q X xi P (x)xj = ¢a + ¢bj : i 2 Q2 i=0 Q(x) j=0 Q (x)

(41)

So the estimation (39) transforms into the estimation n X

m X jxi j jP (x)xj j ¢ j¢ai j + ¢ j¢bj j: ¢R µ 2 jQ(x)j i=0 j=0 Q (x)

(42)

It is easy to check that estimations of this type are not realistic. Consider the following example discussed in the Introduction: f (x) = arctan x on the segment [¡ 1; 1], R(x) has the form (1) for m = 8, n = 9. In this case the estimation (41) is of order 10¡4 but in fact ¢R is of order 10¡11 . This situation is typical for examples examined in this paper. In fact we have an approximate equation ¢R º

n X i=0

m X xi P (x)xj ¢ai + ¢bj º " º 0; 2 Q(x) j=0 Q (x)

(43)

58

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

where " is the absolute error of the approximation method used. Of course, this approximate equation holds only if our approximation method is good and the uncertainty relation (discussed above in Section 2) is valid. Then the approximate equation (43) corresponds to the approximate equation (39). The error autocorrection e®ect appears not only in rational approximation but in many problems. Other examples (where this e®ect occurs in a weaker form) are the method of least squares and some popular methods for the numerical integration of ordinary and partial di®erential equations, see, e.g., [24], [25], [26], [27], [28], [29]. In principle, the error autocorrection e®ect appears if input data form an (approximate) solution (or solutions) of an equation (or equations or systems of equations). Then the corresponding errors could form an approximate solution (or solutions) for another equation (or equations or systems of equations). As a result this could lead to corrections for standard interval estimates. Of course, theoretically we can include all the preliminary numerical problems to our concrete problem and to use, e.g., a posteriori interval analysis for the \united" problem. However, in practice this is not convenient. In practice, situations of this kind often appear if we use approximate solutions to ill conditioned systems of linear algebraic equations. If the condition number of the system is great and the residual of the solution (with respect to the system) is small, then our software must send us a \warning". This means that an additional investigation for error estimates is needed. In the theory of interval analysis this corresponds to a further development of \a posteriori interval methods" in the spirit of [10], [27], [28], [29], [30]. Remark 12.1. We have discussed \smooth" computations. Note that for many \nonsmooth" optimization problems all the interval estimates could be good and absolutely exact. A situation of this kind (related to solving systems of linear algebraic equations over idempotent semirings) is described in [31].

Acknowledgments Partly supported by the Fields Institute for Research in Mathematical Sciences (Toronto, Canada).

References [1] G.L. Litvinov et al., Mathematical Algorithms and Programs for Small Computers, Finansy i Statistika Publ., Moscow, 1981 (in Russian). [2] G.L. Litvinov and V.N. Fridman, Approximate construction of rational approximants, C. R. Acad. Bulgare Sci., 36, No. 1 (1981), 49{52 (in Russian). [3] I.A. Andreeva, G. L. Litvinov, A. Ya. Rodionov and V. N. Fridman, The PADEprogram for Calculation of Rational Approximants. The Program Speci¯cation and its Code, Fond Algoritmov i Programm NIVTs AN SSSR, Puschino, 1985 (in Russian).

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

59

[4] A.P. Kryukov, G.L. Litvinov and A.Ya. Rodionov, Construction of rational approximations by means of REDUCE, in: Proceeding of the ACM{SIGSAM Symposium on Symbolic and Algebraic Computation (SYMSAC 86), Univ. of Waterloo, Canada, 1986, pp. 31{33. [5] G.L. Litvinov, Approximate construction of rational approximations and an e®ect of error autocorrection, in: Mathematics and Modeling, NIVTs AN SSSR, Puschino, 1990, 99{141 (in Russian). [6] G.L. Litvinov, Error auto-correction Computations, 4 (6), (1992), 14{18.

in

rational

approximation,

Interval

[7] G.L. Litvinov, Approximate construction of rational approximations and the e®ect of error autocorrection. Applications, Russian J. Math. Phys., 1, No. 3 (1994), 14{18. [8] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York{London, 1983. [9] R.B. Kearfott, Interval computations { introduction, uses, and resources, Euromath. Bulletin, 2(1), (1996), 95{112. [10] Y. Matijasevich, A posteriori version of interval analysis, in: Proc. Fourth Hung. Computer Sci. Conf., Topics in the Theoretical Basis and Applications of Computer Science, eds. M. Arato, I. Katai, L. Varga, Acad. Kiado, Budapest, 1986, pp. 339{349. [11] J.F. Hart et al., Computer Approximations, Wiley, New York, 1968. [12] V.V. Voevodin, Numerical Principles of Linear Algebra, Nauka Publ., Moscow, 1977 (in Russian). [13] Y.L. Luke, Computations of coe±cients in the polynomials of Pad¶e approximations by solving systems of linear equations, J. Comp. and Appl. Math., 6, No. 3 (1980), 213{218. [14] Y.L. Luke, A note on evaluation of coe±cients in the polynomials of Pad¶e approximants by solving systems of linear equations, J. Comp. and Appl. Math., 8, No. 2 (1982), 93{99. [15] G.E. Forsythe, M. Malcolm and C. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cli®s, N. J., 1977. [16] G.A.Baker and P. Graves-Morris, Pad¶ e Approximants. Part I: Basic Theory. Part II: Extensions and Applications, Encyclopaedia of Mathematics and its Applications 13, 14, Addison-Wesley, Reading, Mass., 1981. [17] W.J. Cody, W. Fraser and J.F. Hart, Rational Chebyshev approximation using linear equations, Numer.Math., 12 (1968) 242{251. [18] E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966. [19] D.S. Lubinsky and A. Sidi, Convergence of linear and nonlinear Pad¶e approximants from series of orthogonal polynomials, Trans. Amer. Math. Soc., 278, No. 1, (1983) 333{345. [20] S. Paszkowski, Zastosowania Numeryczne Wielomian¶ow i Szereg¶ow Czebyszewa, Panstwowe Wydawnictwo Naukowe, Warszawa, 1975 (in Polish). [21] H.J. Maehly, Rational approximations for transcendental functions, in: Proceedings of the International Conference on Information Processing, UNESCO, Butterworths, London, 1960, pp. 57{62.

60

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

[22] C.K. Clenshaw and K. Lord, Rational approximations from Chebyshev series, in: Studies in Numerical Analysis, ed. B.K.P. Scaife, Academic Press, London and New York, 1974, pp. 95{113. [23] A.C. Hearn, REDUCE User’s Manual, Rand. Publ., 1982. [24] E.A. Volkov, Two-sided di®erence methods for solving linear boundary-value problems for ordinary di®erential equations, Proc. Steklov Inst. Math. 128 (1972), 131{152 (translated from Russian by AMS in 1974). [25] E.A. Volkov, Pointwise estimates of the accuracy of a di®erence solution of a boundary-value problem for an ordinary di®erential equation, Di®erential Equations, vol. 9, No 4 (1973), 717{726 (in Russian; translated into English by Plenum Publ. Co. in 1975, 545{552). [26] B.S. Dobronets and V.V. Shaydurov, Two-sided numerical methods, Nauka Publ., Novosibirsk, 1990 (in Russian). [27] B.S. Dobronets, On some two-sided methods for solving systems of ordinary di®erential equations, Interval Computations, No 1(3) (1992), 6{21. [28] B.S. Dobronets, Interval methods based on a posteriory estimates, Interval Computations, No 3(5) (1992), 50{55. [29] L.F. Shampine, Ill-conditioned matrices and the integration of sti® ODEs, J. of Computational and Applied Mathematics 48 (1993), 279{292. [30] Yu.V. Matijasevich, Real numbers and computers. { In: Kibernetika i Vychislitelnaya Tekhnika, vol. 2 (1986), 104{133 (in Russian). [31] G.L. Litvinov and A.S. Sobolevskii, Idempotent interval analysis and optimization problems, Reliable Computing 7 (2001), 353{377.

CEJM 1 (2003) 36{60

Error Autocorrection in Rational Approximation and Interval Estimates. [A survey of results.] Grigori L. Litvinov¤ Independent University of Moscow B. Vlasievskii per., 11, Moscow, 121002 Moscow, Russia

Received 21 August 2002; revised 15 September 2002 Abstract: The error autocorrection e¬ect means that in a calculation all the intermediate errors compensate each other, so the ­ nal result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the e¬ect which appears in rational approximations to functions. The error autocorrection e¬ect occurs in all e¯ cient methods of rational approximation (e.g., best approximations, Pad´e approximations, multipoint Pad´e approximations, linear and nonlinear Pad´e-Chebyshev approximations, etc.), where very signi­ cant errors in the approximant coe¯ cients do not a¬ect the accuracy of this approximant. The reason is that the errors in the coe¯ cients of the rational approximant are not distributed in an arbitrary way, but form a collection of coe¯ cients for a new rational approximant to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The e¬ect of error autocorrection indicates that variations of an approximated function under some deformations of rather a general type may have little e¬ect on the corresponding rational approximant viewed as a function (whereas the coe¯ cients of the approximant can have very signi­ cant changes). Accordingly, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation is not stable under small deformations of the approximated functions. This property is \individual", in the sense that it holds for speci­ c functions. ® c Central European Science Journals. All rights reserved.

¤

Keywords: error autocorrection, rational approximation, algorithms, interval estimates. MSC (2000): Primary 41A20; Secondary 65Gxx

E-mail: [email protected] and [email protected]

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

1

37

Introduction

The paper contains a brief survey description of some, but not all, results published in [1], [2], [3], [4], [5], [6], [7] and some new material on the general form of the error autocorrection e®ect and its relation to interval analysis in the spirit of [8], [9], [10]. Some old results are presented in a new form. The treatment is partly heuristic and based on computer experiments. The error autocorrection e®ect means that in a calculation all the intermediate calculating errors compensate each other, so the ¯nal result is much more accurate than the intermediate results. In this case standard interval estimates (e.g., in the framework of Yu. Matijasevich’s a posteriori interval analysis) are too pessimistic. The error autocorrection e®ect appears in some popular numerical methods, e.g., in the least squares method. In principle this e®ect is already known to experts in interval computations. We shall discuss a very strong form of the e®ect. For the sake of simplicity let us suppose that we calculate values of a real smooth function z = F (y1 ; ¢ ¢ ¢ ; yn ) where y1 ; ¢ ¢ ¢ ; yn are real, and suppose that all round-o® errors are negligible with respect to input data errors. In this case the error ¢F of the function F can be evaluated by the formula ¢F =

N X @F i=1

@yi

¢ ¢yi + r;

P

@F where r is negligible. The sum N i=1 @yi ¢ ¢yi in this formula can be treated as a scalar @F g and the vector of errors f¢yi g. For all the standard product of the gradient vector f @y i interval methods the best possible estimate for ¢F is given by the formula

j¢F j µ

N X @F

j

i=1

@yi

j ¢ j¢yi j:

@F g and f¢yi g are large but the scalar This estimate is too pessimistic if both vectors f @y i product is small enough for these vectors to be almost orthogonal. That is the case of the error autocorrection e®ect. To calculate the values of a function we usually use four arithmetic operations (applying to arguments and constants) and this leads to a rational approximation to the calculated function. The author came across the phenomenon of error autocorrection in the late 1970ies while developing nonstandard algorithms for computing elementary functions on small computers. It was desired to construct rational approximants of the form

R(x) =

a0 + a 1 x + a 2 x2 + : : : + a n xn b0 + b1 x + b2 x2 + : : : + bm xm

(1)

to certain functions of one variable x de¯ned on ¯nite segments of the real line. For this purpose a simple method (described in [1] and below) was used: the method allows to determine the family of coe±cients ai , bj of the approximant (1) as a solution of a certain system of linear algebraic equations. These systems turned out to be ill conditioned, i.e.,

38

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

the problem of determining the coe±cients of the approximant is, generally speaking, ill-posed and small perturbations of the approximated function f (x) or calculation errors lead to very signi¯cant errors in the values of coe±cients. Nevertheless, the method ensures a paradoxically high quality of the obtained approximants. In fact the errors are close to the best possible [1], [2] For example, for the function cos x the approximant of the form (1) on the segment [¡ º =4; º =4] obtained by the method mentioned above for m = 4, n = 6 has the relative error equal to 0:55 ¢ 10¡13 , and the best possible relative error is 0:46 ¢ 10¡13 [11]. The corresponding system of linear algebraic equations has the condition number of order 109 . Thus we risk to lose 9 accurate decimal digits in the solution because of calculation errors. Computer experiments show that this is a serious risk. The method mentioned above was implemented in Fortran code. The calculations were carried out with double precision (16 decimal places) on two di®erent computers. These computers were very similar in architecture, but when passing from one computer to another the system of linear equations and the computational process are perturbed because of calculation errors, including round-o® errors. As a result, the coe±cients of the approximant mentioned above to the function cos x experience a perturbation at the sixth{ninth decimal digits. But the error in the rational approximant itself remains invariant and is 0:4 ¢ 10¡13 for the absolute error and 0:55 ¢ 10¡13 for the relative error. The same thing happens for approximants of the form (1) to the function arctan x on the segment [¡ 1; 1] obtained by the method mentioned above for m = 8, n = 9. The relative error is 0:5 ¢ 10¡11 and does not change while passing from one computer to another although the corresponding system of linear equations has the condition number of order 1011 , and the coe±cients of the approximant experience a perturbation with a relative error of order 10¡4 . Thus the errors in the numerator and the denominator of a rational approximant compensate each other. The e®ect of error autocorrection is connected with the fact that the errors in the coe±cients of a rational approximant are not distributed in an arbitrary way, but form the coe±cients of a new approximant to the approximated function. It can be easily understood that all standard methods of interval arithmetic (see, for example, [6], [8], [9], [10]) do not allow us to take into account this e®ect and, as a result, to estimate the error in the rational approximant accurately (see section 12 below). Note that the application of standard procedures known in the theory of ill-posed problems results in this case in a loss of accuracy. For example, if one applies the regularization method, then two thirds of the accurate ¯gures are lost [12]; in addition, the number of calculations required increases rapidly. The matter of import is that the exact solution of the system of equations in the present case is not the ultimate goal; the aim is to construct an approximant that is precise enough. This approach allows to \rehabilitate" (i.e., to justify) and to simplify a number of algorithms intended for the construction of the approximant, and to obtain (without additional transformations) approximants in a form that is convenient for applications. Professor Yudell L. Luke kindly drew the author’s attention to his papers [13], [14] where the e®ect of error autocorrection for the classical Pad¶e approximants was described

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

39

and explained at a heuristic level. The method mentioned above leads to the linear Pad¶e{ Chebyshev approximants if the calculation errors are ignored. In the present paper the error autocorrection mechanism is considered for a very general situation (including linear methods for the construction of rational approximants and nonlinear generalized Pad¶e approximations). The e±ciency of the construction algorithms used for rational approximants is due to the error autocorrection e®ect (at least in the case when the number of coe±cients is large enough). Our new understanding of the error autocorrection mechanism allows us, to some extent, to control calculation errors by changing the construction procedure depending on the form of the approximant. It is shown that the use of a control parameter allowing to take into account the error autocorrection mechanism ensures the decrease of the calculation errors in some cases. Construction methods for linear and nonlinear Pad¶e{Chebyshev approximants involving the computer algebra system REDUCE (see [4]) are also brie°y described. Computation results characterizing the comparative precision of these methods are given. We analyze the e®ect described in [2] ewith regard to the error autocorrection phenomenon. This e®ect is connected with the fact that a small variation of an approximated function can lead to a sharp decrease in accuracy of the Pad¶e{Chebyshev approximants. The error autocorrection e®ect occurs not only in rational approximation but appears in some other cases - approximate solutions of linear di®erential equations, the method of least squares, etc. In this more general situation relations between the error autocorrection e®ect and standard methods of interval analysis are also discussed in this paper. The author is grateful to Y.L. Luke, B.S. Dobronets, and S.P Shary for stimulating discussions and comments. The author wishes to express his thanks to I. A. Andreeva, A. Ya. Rodionov and V. N. Fridman who participated in the programming and organization of computer experiments.

2

Error autocorrection in rational approximation

Let f’0 ; ’1 ; : : : ; ’n g and fÃ0 ; Ã 1 ; : : : ; Ãm g be two sets of linearly independent functions of the argument x belonging to some (possibly multidimensional) set X . Consider the problem of constructing an approximant of the form R(x) =

a 0 ’ 0 + a 1 ’ 1 + : : : + an ’ n b0 à 0 + b1 à 1 + : : : + bm à m

(2)

to a given function f (x) de¯ned on X . If X coincides with a real line segment [A; B] and if ’k = xk and Ãk = xk for all k, then the expression (2) turns out to be a rational function of the form (1) (see the Introduction). It is clear that expression (2) also gives a rational function in the case when we take Chebyshev polynomials T k or, for example, Legendre, Laguerre, Hermite, etc. polynomials as ’k and Ãk . Fix an abstract construction method for an approximant of the form (2) and consider the problem of computing the coe±cients ai , bj . Quite often this problem is ill-conditioned

40

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

(ill-posed). For example, the problem of computing coe±cients for best rational approximants (including polynomial approximants) for high degrees of the numerator or the denominator is ill-conditioned. The instability with respect to the calculation error can be related both to the abstract construction method of approximation (i.e., with the formulation of the problem) and to the particular algorithm implementing the method. The fact that the problem of computing coe±cients for the best approximant is ill-conditioned is related to the formulation of this problem. This is also valid for other construction methods for rational approximants with a su±ciently large number of coe±cients. But an unfortunate choice of the algorithm implementing a certain method can aggravate troubles connected with ill-conditioning. Let the coe±cients ai , bj give an exact or an approximate solution of this problem, and let the ~ai , ~bj give another approximate solution obtained in the same way. Denote by ¢ai , ¢bj the absolute errors of the coe±cients, i.e., ¢ai = ~ai ¡ ai , ¢bj = ~bj ¡ bj . These errors arise due to perturbations of the approximated function f (x) or due to calculation errors. Set P (x) = ¢P (x) =

Pn

i=0

Pn

i=0

ai ’i ;

Q(x) =

¢ai ’i ;

¢Q(x) =

Pe (x) = P + ¢P;

Pm

j=0 bj à j ;

Pm

j=0 ¢bj à j ;

e Q(x) = Q + ¢Q:

It is easy to verify that the following exact equality is valid: P + ¢P ¡ Q + ¢Q

Ã

P ¢Q ¢P ¡ = Q Q ¢Q

!

P : Q

(3)

As mentioned in the Introduction, the fact that the problem of calculating coe±cients is ill-conditioned can nevertheless be accompanied by high accuracy of the approximants e are close to the approxiobtained. This means that the approximants P=Q and Pe =Q mated function and, therefore, are close to each other, although the coe±cients of these e = ¢Q=(Q + ¢Q) of approximants di®er greatly. In this case the relative error ¢Q= Q the denominator considerably exceeds in absolute value the left-hand side of equality (3). This is possible only in the case when the di®erence ¢P =¢Q ¡ P=Q is small, i.e., the function ¢P =¢Q is close to P =Q, and, hence, to the approximated function. Thus the function ¢P =¢Q will be called the error approximant. For a special case, this concept was actually introduced in [13]. For "e±cient" methods, the error approximant provides e indeed a good approximation for the approximated function and, thus, P =Q and Pe = Q di®er from each other by a product of small quantities in the right-hand side of (3). Usually the following \uncertainty relation" is valid: Ã

P ¢ Q

!

=¯ Q¢

Ã

¢P ¡ ¢Q

P Q

!

º¯ Q¢

Ã

¢P ¡ ¢Q

f

!

¹

";

¢Q ¡ f is where ¯ Q = Q+¢Q is the relative error of the denominator Q, the di®erence ¢P ¢Q ¢P the absolute error of the error approximant ¢Q to the function f , and " is the absolute \theoretical" error of our method; the argument x can be treated as ¯xed.

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

41

The function f (x) is usually treated as an element of a Banach space with a norm P k ¢ k. The absolute error ¢ of the approximant (2) is de¯ned as ¢ = kf ¡ Q k. Its relative P P P error ¯ is the de¯ned as ¯ = k(f ¡ Q )=f k or ¯ = k(f ¡ Q )= Q k. In what follows, we shall consider the space C [A; B] of all continuous functions de¯ned on the real line segment [A; B] with the norm kf (x)k = max jf (x)j: A·x·B

Below we discuss examples of the error autocorrection e®ect for linear and nonlinear methods of rational approximation.

3

Error autocorrection for linear methods in rational approximation

Several construction methods for approximants of the form (2) are connected with solving systems of linear algebraic equations. This procedure can lead to a large error if the corresponding matrix is ill-conditioned. Consider an arbitrary system of linear algebraic equations Ay = h; (4) where A is a given square matrix of order N with components aij (i; j = 1; : : : ; N ), h is a given column vector with components hi , and y is an unknown column vector with components yi . De¯ne the vector norm by the equality kxk =

N X

jxi j

i=1

(this norm is more convenient for calculations than norm is determined by the equality kAk = max kAyk = max kyk=1

1·j·N

q

x21 + : : : + x2N ). Then the matrix

N X

kaij k:

i=1

If a matrix A is nonsingular, then the quantity cond(A) = kAk ¢ kA¡1 k

(5)

is called the condition number of the matrix A (see, for example, [15]). Since y = A¡1 h, we see that the absolute error ¢y of the vector y is connected with the absolute error of the vector h by the relation ¢y = A¡1 ¢h, whence k¢yk µ kA ¡1 k ¢ k¢hk and k¢yk=kyk µ kA ¡1 k ¢ (khk=kyk)(k¢hk=khk): Taking into account the fact that khk µ kAk ¢ kyk, we ¯nally obtain k¢yk=kyk µ kAk ¢ kA ¡1 k ¢ k¢hk=khk;

(6)

42

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

i.e., the relative error of the solution y is estimated via the relative error of the vector h by means of the condition number. It is clear that (6) can turn into an equality. Thus, if the condition number is of order 10k , then, because of round{o® errors in h, we can lose k decimal digits of y. The contribution of the error of the matrix A is evaluated similarly. Finally, the dependence of cond(A) on the choice of a norm is weak. A method of rapid estimation of the condition number is described in [15], Section 3.2. Let an abstract construction method for the approximant of the form (2) be linear in the sense that the coe±cients of the approximant can be determined from a homogeneous system of linear algebraic equations. The homogeneity condition is connected to the fact that, when multiplying the numerator and the denominator of fraction (2) by the same nonzero number, the approximant (2) does not change. Denote by y the vector whose components are the coe±cients a0 ; a1 ; : : : ; an , b0 ; b1 ; : : : ; bm . Assume that the coe±cients can be obtained from the homogeneous system of equations H y = 0;

(7)

where H is a matrix of dimension (m + n + 2) £ (m + n + 1). The vector y~ is an approximate solution of system (7) if the quantity kH y~k is small. If y and y~ are approximate solutions of system (7), then the vector ¢y = y~ ¡ y is also an approximate solution of this system since kH ¢yk = kH y~ ¡ H yk µ kH y~k + kH yk. Thus it is natural to assume that the function ¢P =¢Q corresponding to the solution ¢y is an approximant to f (x). It is clear that the order of the residual of the approximate solution ¢y of system (7), i.e., of the quantity kH ¢yk, coincides with the order of the largest of the residuals of the approximate solutions y and y~. For a ¯xed order of the residual the increase in the error ¢y is compensated by the fact that ¢y satis¯es the system of equations (7) with greater \relative" accuracy, and the latter, generally speaking, leads to the increase in the accuracy of the error approximant. To obtain a particular solution of system (7), one usually adds to this system a normalization condition of the form n X

¶ i ai +

i=0

m X

· j bj = 1;

(8)

j=0

where ¶ i , · j are numerical coe±cients. As a rule, the relation b0 = 1 is taken as the normalization condition (but this is not always successful with respect to minimizing the calculation errors). Adding equation (8) to system (7), we obtain a nonhomogeneous system of m + n + 2 linear algebraic equations of type (4). If the approximate solutions y and y~ of system (7) satisfy condition (8), then the vector ¢y satis¯es the condition n X i=0

¶ i ¢ai +

m X

· j ¢bj = 0:

(9)

j=0

Of course, the above reasoning is not very rigorous; for each speci¯c construction method for approximations it is necessary to carry out some additional analysis. More

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

43

accurate arguments are given below for the linear and nonlinear Pad¶e{Chebyshev approximants. The presence of the error autocorrection mechanism described above is also veri¯ed by numerical experiments (see below). It is clear that classical Pad¶e approximations, multipoint Pad¶e approximations, linear generalized Pad¶e approximations in the sense of [16] (e.g., linear Pad¶e{Chebyshev approximations) give us good examples of linear methods in rational approximation. From our point of view, the methods for obtaining the best approximations can be treated as linear. Indeed the coe±cients of the best Chebyshev approximant satisfy a system of linear algebraic equations and are computed as approximate solutions of this system on the last step of the iteration process in algorithms of Remez’s type (see [7], [17] for details). Thus, the construction methods for the best rational approximants can be regarded as linear. At least for some functions (say, for cos((º =4)x), ¡ 1 µ x µ 1) the linear and the nonlinear Pad¶e{Chebyshev approximants are very close to the best ones in relative and absolute errors, respectively. The results that arise when applying calculation algorithms for Pad¶e{Chebyshev approximants can be regarded as approximate solutions of the system which determines the best approximants. Thus the presence of the e®ect of error autocorrection for Pad¶e{Chebyshev approximants gives an additional argument in favor of the conjecture that this e®ect also takes place for the best approximants. Finally, note that the basic relation (3) becomes meaningless if one seeks an approximant in the form a0 ’0 + a1 ’1 + : : : + an ’n , i.e., the denominator in (2) is reduced to 1. However, in this case the e®ect of error autocorrection (although much weakened) is also possible. This is connected to the fact that the errors ¢ai approximately satisfy certain relations. Such a situation can arise when using the least squares method.

4

Linear Pad¶e{Chebyshev approximations and the PADE program

Let us begin to discuss a series of examples. Consider the approximant of the form (1) Rm;n (x) =

a0 + a 1 x + a 2 x2 + : : : + an xn b0 + b1 x + b2 x2 + : : : + bm xm

(10)

to a function f (x) de¯ned on the segment [¡ 1; 1]. The absolute error function ¢(x) = f (x) ¡

Rm;n (x)

obviously has the following form: ¢(x) = ©(x)=Qm (x); where ©(x) = f (x)Q m (x) ¡

Pn (x):

(11)

44

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

The function Rm;n (x) = P n (x)=Q m (x) is called the linear Pad¶ e{Chebyshev approximant to the function f (x) if Z1

©(x)Tk (x)w(x) dx = 0;

k = 0; 1; : : : ; m + n;

(12)

¡1

p where Tk (x) = cos(n arccos x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 . This concept allows a generalization to the case of other orthogonal polynomials (see, e.g., [16], [18], [19], [20]). Approximants of this kind always exist [18]. The system of equations (12) is equivalent to the following system of linear algebraic equations with respect to the coe±cients ai , bj : m X

j=0

bj

Z1

¡1

xj T k (x)f (x) p dx ¡ 1 ¡ x2

n X i=0

ai

Z1

¡1

xi T (x) p k dx = 0: 1 ¡ x2

(13)

The homogeneous system (12) can be transformed into a nonhomogeneous one by adding a normalization condition. In particular, any of the following relations can be taken as this condition: b0 = 1;

(14)

bm = 1;

(15)

am = 1:

(16)

The sources [1], [2] brie°y describe the program PADE, written in Fortran, with double precision) which constructs rational approximants by solving the system of equations of type (13). The complete text of a certain version of this code and its detailed description can be found in the Collection of Algorithms and Codes of the Research Computer Center of the Russian Academy of Sciences [3]. For even functions the program looks for an approximant of the form R(x) =

a0 + a1 x2 + : : : + an (x2 )n ; b0 + b1 x2 + : : : + bm (x2 )m

(17)

a0 + a1 x2 + : : : + an (x2 )n ; b0 + b1 x2 + : : : + bm (x2 )m

(18)

and for odd functions it is R(x) = x

The program computes the values of coe±cients of the approximant, the absolute and the relative errors ¢ = maxA·x·B j¢(x)j and ¯ = maxA·x·B j¢(x)=f (x)j, and gives the information which allows us to estimate the quality of the approximation (see [7] and [3] for details). Using a subroutine, the user introduces the function de¯ned by means of any algorithm on an arbitrary segment [A; B], introduces the boundary points of this segment, the numbers m and n, and the number of control parameters. In particular, one can choose the normalization condition of type (14){(16), look for an approximant in

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

45

the form (17) or (18) and so on. The change of the variable reduces the approximation on any segment [A; B] to the approximation on the segment [¡ 1; 1]. Therefore, we shall consider the case when A = ¡ 1, B = 1 in the sequel unless otherwise stated. For the calculation of integrals, the Gauss{Hermite{Chebyshev quadrature formula is used: Z1 s ³ ’(x) º X 2i ¡ 1 ´ p dx = ’ cos º ; (19) 2 s i=1 2s ¡ 1 x ¡1 where s is the number of interpolation points. For polynomials of degree 2s¡ 1 this formula is exact, so the precision of formula (19) increases rapidly as the parameter s increases and depends on the quality of the approximation of the function ’(s) by polynomials. To calculate the values of Chebyshev polynomials, the well-known recurrence relation is applied. If the function f (x) is even and of the form (17) is desired, then the system (13) is transformed into the following system of equations: n X i=0

ai

Z1

¡1

x2i T 2k (x) p dx ¡ 1 ¡ x2

m X

j=0

bj

Z1

¡1

x2j T2k (x)f (x) p dx = 0; 1 ¡ x2

(20)

where k = 0; 1; : : : ; m + n. If f (x) is an odd function and an approximant of the form (18) is desired, then one ¯rst determines an approximant of the form (17) to the even function f (x)=x by solving the system (20) complemented by one of the normalization conditions. Then the obtained approximant is multiplied by x. This procedure allows us to avoid a large relative error for x = 0. This algorithm is rather simple; for its implementation only two standard subroutines are needed (for solving systems of linear algebraic equations and for numerical integration). However, the algorithm is e±cient. The capabilities of the PADE code are demonstrated in Table 1. This table contains errors for certain approximants obtained by means of this program. For every approximant, the absolute error ¢, the relative error ¯ , and (for comparison) p the best possible relative error ¯ min given in [11] are indicated. The function x is approximated on the segment [1=2; 1] by an expression of the form (1), the function cos ¼4 x is approximated on the segment [¡ 1; 1] by an expression of the form (17), and all the others are approximated on the same segment by an expression of the form (18).

5

Error autocorrection for the PADE program

The condition numbers of systems of equations that arise while calculating, by means of the PADE program, the approximants considered above are very large. For example, for calculating the approximant of the form (18) on the segment [¡ 1; 1] to sin ¼2 x for m = n = 3, the corresponding condition number is of order 1013 . As a result, the coe±cients of the approximant are determined with a large error. In particular, a small perturbation of the system of linear equations arising when passing from one computer

46

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

Function p x p x

m

n

¢

¯

¯

2

2

0:8 ¢ 10¡6

1:13 ¢ 10¡6

0:6 ¢ 10¡6

3

3

1:9 ¢ 10¡9

2:7 ¢ 10¡9

1:12 ¢ 10¡9

cos ¼4 x

0

3

0:28 ¢ 10¡7

0:39 ¢ 10¡7

0:32 ¢ 10¡7

cos ¼4 x

1

2

0:24 ¢ 10¡7

0:34 ¢ 10¡7

0:29 ¢ 10¡7

cos ¼4 x

2

2

0:69 ¢ 10¡10

0:94 ¢ 10¡10

0:79 ¢ 10¡10

cos ¼4 x

0

5

0:57 ¢ 10¡13

0:79 ¢ 10¡13

0:66 ¢ 10¡13

cos ¼4 x

2

3

0:4 ¢ 10¡13

0:55 ¢ 10¡13

0:46 ¢ 10¡13

sin ¼4 x

0

4

0:34 ¢ 10¡11

0:48 ¢ 10¡11

0:47 ¢ 10¡11

sin ¼4 x

2

2

0:32 ¢ 10¡11

0:45 ¢ 10¡11

0:44 ¢ 10¡11

sin ¼4 x

0

5

0:36 ¢ 10¡14

0:55 ¢ 10¡14

0:45 ¢ 10¡14

sin ¼2 x

1

1

0:14 ¢ 10¡3

0:14 ¢ 10¡3

0:12 ¢ 10¡3

sin ¼2 x

0

4

0:67 ¢ 10¡8

0:67 ¢ 10¡8

0:54 ¢ 10¡8

sin ¼2 x

2

2

0:63 ¢ 10¡8

0:63 ¢ 10¡8

0:53 ¢ 10¡8

sin ¼2 x

3

3

0:63 ¢ 10¡13

0:63 ¢ 10¡13

0:5 ¢ 10¡13

tan ¼4 x

1

1

0:64 ¢ 10¡5

0:64 ¢ 10¡5

0:57 ¢ 10¡5

tan ¼4 x

2

1

0:16 ¢ 10¡7

0:16 ¢ 10¡7

0:14 ¢ 10¡7

tan ¼4 x

2

2

0:25 ¢ 10¡10

0:25 ¢ 10¡10

0:22 ¢ 10¡10

arctan x

0

7

0:75 ¢ 10¡7

10¡7

10¡7

arctan x

2

3

0:16 ¢ 10¡7

0:51 ¢ 10¡7

0:27 ¢ 10¡7

arctan x

0

9

0:15 ¢ 10¡8

0:28 ¢ 10¡8

0:23 ¢ 10¡8

arctan x

3

3

0:54 ¢ 10¡9

1:9 ¢ 10¡9

0:87 ¢ 10¡9

arctan x

4

4

0:12 ¢ 10¡11

0:48 ¢ 10¡11

0:17 ¢ 10¡11

arctan x

5

4

0:75 ¢ 10¡13

3:7 ¢ 10¡13

0:71 ¢ 10¡13

min

Table 1

to another (because of the calculation errors) gives rise to large perturbations in the coe±cients of the approximant. Fortunately, the e®ect of error autocorrection improves the situation, and the errors in the approximant undergo no substantial changes under this perturbation. This fact is described in the Introduction, where concrete examples are also given. Consider a few more examples connected with passing from one computer to another (see [6], [7] for details). The branch of the algorithm which corresponds to the normalization condition (14) (i.e., to b0 = 1) is considered. For arctan x, the calculation of an

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

47

approximant of the form (18) on the segment [¡ 1; 1] for m = n = 5 gave an approximant with the absolute error ¢ = 0:35 ¢ 10¡12 and the relative error ¯ = 0:16 ¢ 10¡11 . The corresponding system of linear algebraic equations has the condition number of order 1030 ! Passing to another computer we obtain the following: ¢ = 0:5 ¢ 10¡14 , ¯ = 0:16 ¢ 10¡12 , the condition number is of order 1014 , and the errors ¢a1 and ¢b1 in the coe±cients a1 and b1 in (18) are greater in absolute value than 1! This example shows that the problem of computing the condition number of an ill-conditioned system is, in its turn, ill-conditioned. Indeed, the condition number is, roughly speaking, determined by values of coe±cients of the inverse matrix, every column of the inverse matrix being the solution of the system of equations with the initial matrix of coe±cients, i.e., of an ill-conditioned system. Consider in detail the e®ect of error autocorrection for the approximant of the form (17) to the function cos ¼4 x on the segment [¡ 1; 1]for m = 2, n = 3. For our two di®erent computers, two di®erent approximants were obtained with the coe±cients ai ,bi and a~i ,~bi respectively. In both cases, the condition number is of order 109 , and the absolute error is ¯ = 0:55 ¢ 10¡13 . These errors are close to best possible. The coe±cients of the approximants obtained on the computers mentioned above and the coe±cients of the error approximant (see Section 3 above) are as follows: a~0 = 0:9999999999999600;

a0 = 0:9999999999999610;

¢a0 = ¡ 10¡15 ; ~a1 = ¡ 0:2925310453579570;

a1 = ¡ 0:2925311264716216;

¢a1 = 10¡7 ¢ 0:811136646; a~2 = 10¡1 ¢ 0:1105254254716866;

a2 = 10¡1 ¢ 0:1105256585556549;

¢a2 = ¡ 10¡7 ¢ 0:2330839683; a~3 = 10¡3 ¢ 0:1049474500904401;

a3 = 10¡3 ¢ 0:1049482094850086;

¢a3 = 10¡9 ¢ 0:7593947685; b0 = 1;

~b0 = 1;

¢b0 = 0; ~b1 = 10¡1 ¢ 0:1589409217324021;

b1 = 10¡1 ¢ 0:1589401105960337;

¢b1 = 10¡7 ¢ 0:8111363684; ~b2 = 10¡3 ¢ 0:1003359011092697;

b2 = 10¡3 ¢ 0:1003341918083529;

¢b2 = 10¡8 ¢ 0:17093009168: Thus, the error approximant has the form ¢P ¢a0 + ¢a1 x2 + ¢a2 x4 + ¢a3 x6 = : ¢Q ¢b1 x2 + ¢b2 x4

(21)

48

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

If the relatively small quantity ¢a0 = ¡ 10¡15 in (21) is omitted, then, as testing by means of a computer shows (2000 checkpoints), this expression is an approximant to the function cos ¼4 x on the segment [¡ 1; 1] with the absolute and relative errors ¢ = ¯ = 0:22 ¢ 10¡6 . But the polynomial ¢Q is zero at x = 0, and the polynomial ¢P takes a small, but nonzero value at x = 0. Fortunately, relation (3) can be rewritten in the following way: Pe ¡ e Q

P ¢P = e ¡ Q Q

¢Q P ¢ : e Q Q

(22)

Thus, as ¢Q ! 0, the e®ect of error autocorrection arises because the quantity ¢P is close to zero, and the error of the approximant P =Q is determined by the error in the coe±cient a0 . The same situation also takes place when the polynomial ¢Q vanishes at an arbitrary point x0 belonging to the segment [A; B] where the function is approximated. It is clear that if one chooses the standard normalization (b0 = 1), then the error approximant actually has two coe±cients less than the initial one. It is clear that in the general case the normalization conditions an = 1 or bm = 1 result in the following: the coe±cients of the error approximant form an approximate solution of the homogeneous system of linear algebraic equations whose exact solution determines the Pad¶e{Chebyshev approximant having one coe±cient less than the initial one. The e®ect of error autocorrection improves the accuracy of this error approximant as well. Thus,\the snake bites its own tail". A situation also arises in the case when the approximant to an even function of the form (17) is constructed by solving the system of equations (20). Sometimes it is possible to decrease the error of the approximant by choosing a good normalization condition. As an example, consider the approximation of the function ex on the segment [¡ 1; 1] by rational functions of the form (1) for m = 15, n = 0. For the traditionally accepted normalization b0 = 1, the PADE program yields an approximant with the absolute error ¢ = 1:4 ¢ 10¡14 and the relative error ¯ = 0:53 ¢ 10¡14 . After passing to the normalization condition b15 = 1, the errors are reduced nearly one half: ¢ = 0:73 ¢ 10¡14 , ¯ = 0:27 ¢ 10¡14 . Note that the condition number increases: in the ¯rst case it is 2 ¢ 106 , and in the second case it is 6 ¢ 1016 . Thus the error decreases notwithstanding the fact that the system of equations becomes drastically ill-conditioned. This example shows that the increase in the accuracy of the error approximant can be accompanied by the increase of the condition number, and, as experiments show, by the increase of errors of the numerator and the denominator of the approximant. The best choice of the normalization condition depends on the particular situation. A speci¯c situation arises when the degree of the numerator (or of the denominator) of the approximant is equal to zero. In this case a bad choice of the normalization condition results in the following: the error approximant becomes zero or is not well-de¯ned. For n = 0 it is expedient to choose condition (15), as it was done in the example given above. For m = 0 (the case of the polynomial approximation) it is usually expedient to choose condition (16). One could search for the numerator and the denominator of the approximant in the

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

form P =

n X

ai T i ;

Q=

i=0

m X

bj Tj ;

49

(23)

j=0

where T i are the Chebyshev polynomials. In this case the system of linear equations determining the coe±cients would be better conditioned. But the calculation of the polynomials of the form (23) by, for example, the Chenshaw method, results in lengthening the computation time, although it has a favorable e®ect upon the error. The transformation of the polynomials P and Q from the form (23) into the standard form also requires additional e®orts. In practice it is more convenient to use approximants represented in the form (1), (17), or (18), and calculate the fraction’s numerator and denominator according to the Horner scheme. In this case the normalization an = 1 or bm = 1 allows to reduce the number of multiplications. The use of the algorithm does not require that the approximated function be expanded into a series or a continued fraction beforehand. Equations (12) or (13) and the quadrature formula (19) show that the algorithm uses only the values of the approximated function f (x) at the interpolation points of the quadrature formula (which are the zeros of some Chebyshev polynomial). On the segment [¡ 1; 1] the linear Pad¶e{Chebyshev approximants give a considerably smaller error than the classical Pad¶e approximants. For example, the Pad¶e approximant of the form (1) to the function ex for m = n = 2 has the absolute error ¢(1) = 4 ¢ 10¡3 at the point x = 1, but the PADE program gives an approximant of the same form with the absolute error ¢ = 1:9 ¢ 10¡4 (on the entire the segment), i.e., the latter is 20 times smaller than the former. The absolute error of the best approximant is 0:87 ¢ 10¡4 .

6

The \cross{multiplied" linear Pad¶e{Chebyshev approximation

As a rule, linear Pad¶e{Chebyshev approximants are constructed according to the following scheme, see, e.g., [21], [11], [16]. Let the approximated function be decomposed into the series in Chebyshev polynomials f (x) =

1 X 0 i=0

where the notation

m P 0

i=1

1 ci Ti (x) = c0 + c1 T1 (x) + c2 T 2 (x) + : : : ; 2

(24)

means that the ¯rst term u0 in the sum is replaced by u0 =2. The

rational approximant of the form

R(x) =

n P 0

i=0 m P 0

j=0

ai Ti (x) ; bj Tj (x)

(25)

50

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

is desired, where the coe±cients bj satisfy the following system of linear algebraic equations m X0 j=0

bj (ci+j + cji¡jj) = 0;

i = n + 1; : : : ; n + m;

(26)

and the coe±cients ai are determined by the equalities m

1 X0 ai = bj (ci+j + cji¡jj ) = 0; 2 j=0

i = 0; 1; : : : ; n:

(27)

It is not di±cult to verify that this algorithm must lead to the same results as the algorithm described in Section 5 if the calculation errors are not taken into account. The coe±cients ck for k = 0; 1; : : : ; n + 2m, are present in (26) and (27), i.e., it is necessary to have the ¯rst n + 2m + 1 terms of series (24). The coe±cients ck are known, as a rule, only approximately. To determine them one can take the truncated expansion of f (x) into the series in powers of x (the Taylor series) and by means of the well-known economization procedure transform it into the form n+2m X

c~i Ti (x):

(28)

i=0

7

Nonlinear Pad¶e{Chebyshev approximations

A rational function R(x) of the form (1) or (25) is called a nonlinear Pad¶ e{Chebyshev approximant to the function f (x) on the segment [¡ 1; 1], if Z1

(f (x) ¡

R(x))T k (x)w(x) dx = 0;

k = 0; 1; : : : ; m + n;

(29)

¡1

p where Tk (x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 . The paper [22] describes the following algorithm of computing the coe±cients of (29) is given. Let the approximated function f (x) be expanded into series (24) in Chebyshev polynomials. Determine the auxiliary quantities ® i from the system of linear algebraic equations m X

j=0

® j cjk¡jj = 0;

k = n + 1; n + 2; : : : ; n + m;

(30)

assuming that ® 0 = 1. The coe±cients of the denominator in expression (25) are determined by the relations bj = · P

m¡j X

® i®

i+j ;

i=0

where · ¡1 = 1=2 ni=1 ® i2 ; this implies b0 = 2. Finally, the coe±cients of the numerator are determined by formula (27). It is possible to solve system (30) explicitly and to indicate the formulas for computing the quantities ® i . One can also estimate explicitly the absolute error of the approximant. This algorithm is described in detail in the book [20]; see also [16].

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

51

In contrast to the linear Pad¶e{Chebyshev approximants, the nonlinear approximants of this type do not always exist, but it is possible to indicate explicitly veri¯able conditions guaranteeing the existence of such approximants [20]. The nonlinear Pad¶e{Chebyshev approximants (in comparison with the linear ones) have, as a rule, somewhat smaller absolute errors, but can have larger relative errors. Consider, as an example, the approximant of the form (1) or (25) to the function ex on the segment [¡ 1; 1] for m = n = 3. In this case the absolute error for a nonlinear Pad¶e{Chebyshev approximant is ¢ = 0:258 ¢ 10¡6 , and the relative error, ¯ = 0:252 ¢ 10¡6 ; for the linear Pad¶e{Chebyshev approximant ¢ = 0:33 ¢ 10¡6 and ¯ = 0:20 ¢ 10¡6 .

8

Applications of the computer algebra system REDUCE to the construction of rational approximants

The computer algebra system REDUCE [23] allows us to handle formulas at the symbolic level and is a convenient tool for the implementation of algorithms of computing rational approximants. The use of this system allows us to bypass the procedure of working out the algorithm for computing the approximated function if this function is presented in analytical form or when the Taylor series coe±cients are known or are determined analytically from a di®erential equation. The round-o® errors can be eliminated by using the exact arithmetic of rational numbers represented in the form of ratios of integers. Within the framework of the REDUCE system, the code package for enhanced precision computations and construction of rational approximants is implemented; see, for example [4]. In particular, the algorithms from Sections 6 and 7 (which have similar structure) are implemented, the approximated function being ¯rst expanded into the P (k) power (Taylor) series, f = 1 (0)xk =k!, and then the truncated series k=0 f N X

k=0

f (k) (0)

xk ; k!

(31)

consisting of the ¯rst N + 1 terms of the Taylor series (the value N is determined by the user) are transformed into a polynomial of the form (28) by means of the economization procedure. The algorithms implemented by means of the REDUCE system allow us to obtain approximants in the form (1) or (25), estimates of the absolute and the relative error, and the error curves. The output includes the Fortran code for computing the corresponding approximant. The constants of rational arithmetic are transformed into the standard °oating point form. When computing the values of the obtained approximant, this approximant can be transformed into the form most convenient for the user. For example, one can calculate values of the numerator and the denominator of the fraction of the form (1) according to the Horner scheme, and for the fraction of the form (25), according to Clenshaw’s scheme, and transform the rational expression into a continued fraction or a Jacobi fraction as well.

52

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

The ALGOL-like input language of the REDUCE system and convenient tools for solving problems of linear algebra guarantee the simplicity and compactness of the code. For example, the length of the program for computing linear Pad¶e{Chebyshev approximants is 62 lines.

9

Error approximants for linear and nonlinear Pad¶e{Chebyshev approximations

Relations (29) can be regarded as a system of equations for the coe±cients of the ape e proximant. Let the approximants R(x) = P (x)=Q(x) and R(x) = Pe (x)= Q(x), where e e P (x), P (x) are polynomials of degree n, and Q(x), Q(x) are polynomials of degree m, be obtained by approximate solving the indicated system of equations. Consider the error e ¡ Q(x). Subapproximant ¢P (x)=¢Q(x), where ¢P (x) = Pe (x) ¡ P (x), ¢Q(x) = Q(x) e stituting R(x) and R(x) into (29) and subtracting one of the obtained expressions from the other, we see that the following approximate equality holds: Z1 µ ~ P (x)

¡1

~ Q(x)

¡

¶

P (x) T k (x)w(x) dx º 0; Q(x)

~ ¡ which directly implies that R(x) the approximate equality Z1 µ

¡1

¢P (x) ¡ ¢Q(x)

k = 0; 1; : : : ; m + n;

R(x) is close to zero. This and the equality (3) imply ¶

P (x) ¢Q T (x)w(x) dx º 1; ~ k Q(x) Q

(32)

p where k = 0; 1; : : : ; m + n, w(x) = 1= 1 ¡ x2 . If the quantity ¢Q is relatively large (this is connected with the fact that the system of equations (30) is ill-conditioned), then, as follows from equality (32), we can naturally expect that the error approximant is close to P =Q and, consequently, to the approximated function f (x). Due to the fact that the arithmetic system of rational numbers is used, the software described in Section 7 allows us to eliminate the round-o® errors and to estimate the \pure" in°uence of errors in the approximated function on the coe±cients of the nonlinear Pad¶e{Chebyshev approximant. In this case the e®ect of error autocorrection can be substantiated by a more accurate reasoning which is valid for both the linear and the nonlinear Pad¶e{Chebyshev approximants, and even for the linear generalized Pad¶e approximants connected with di®erent systems of orthogonal polynomials. This reasoning is analogous to Y. L. Luke’s considerations [13] for the case of classical Pad¶e approximants. Assume that the function f (x) is expanded into series (24) and that the rational approximant R(x) = P (x)=Q(x) of the form (25) is desired. Let ¢bj be the errors in the coe±cients of the approximant’s denominator Q. In the linear case these errors arise when solving the system of equations (26), and in the nonlinear case, when solving the system of equations (27). In both cases the coe±cients

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

53

in the approximant’s numerator are determined by equations (27), whence we have m

¢ai =

1 X0 ¢bj (ci+j + cji¡jj); 2 j=0

i = 0; 1; : : : ; n:

(33)

This implies the following fact: the error approximant ¢P =¢Q satis¯es the relations Z1

(f (x)¢Q(x) ¡

¢P (x))T i (x)w(x) dx = 0;

i = 0; 1; : : : ; n;

(34)

¡1

which are analogous to relations (12) de¯ning the linear Pad¶e{Chebyshev approximants. Indeed, let us use the well-known multiplication formula for Chebyshev polynomials: 1 T i (x)Tj (x) = [Ti+j (x) + Tji¡jj (x)]; 2

(35)

where i, j are arbitrary indices; see, for example [16], [20]. Taking (35) into account, the quantity f ¢Q ¡ ¢P can be rewritten in the following way: f ¢Q ¡

¢P = =

µX m 0

¢bj Tj j=0 1 · m 1 X0 X0 2 i=0

j=0

¶µX 1

0

¶

ci T i ¡

i=0

n X 0

¢ai T i

i=0

¸

¢bj (ci+j + cji¡jj) Ti ¡

n X 0

¢ai T i :

i=0

This formula and (33) imply that f ¢Q ¡

¢P = O(T n+1);

(36)

i.e., in the expansion of the function f ¢Q ¡ ¢P into the series in Chebyshev polynomials, the ¯rst n + 1 terms are absent, and the latter is equivalent to relations (34) by virtue of the fact that the Chebyshev polynomials form an orthogonal system. Consider an arbitrary rational function of the form (1) or (8) Rm;n (x) =

a0 + a1 x + ¢ ¢ ¢ + a n x n P n (x) = : m b0 + b1 x + ¢ ¢ ¢ + bm x Qm (x)

We shall say that Rm;n (x) is a generalized linear Pad¶ e-Chebyshev approximant of order N to the function f (x) if Z 1

¡1

©(x)T k (x)w(x)dx = 0;

k = 0; 1; ¢ ¢ ¢ ; N;

p where Tk (x) = cos(n arccos x) are the Chebyshev polynomials, w(x) = 1= 1 ¡ x2 , ©(x) = f (x)Q m (x) ¡ Pn (x). This means that the ¯rst N +1 terms in the expansion of the function ©(x) into the series in Chebyshev polynomials (\the Fourier-Chebyshev series") are absent, i.e. f (x)Q m (x) ¡ Pn (x) = O(T N+1): If N = m + n, then we have the usual linear Pad¶e-Chebyshev approximant discussed above in Section 4. Formula (36) means that the following result is valid.

54

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

Theorem 9.1. Let ¢P be the error approximant to f (x) generated by the approxi¢Q mant (25) for the case of linear or nonlinear Pad¶e{Chebyshev approximation and algo¢P rithms described in Sections 6 and 7. Then this error approximant ¢Q is a generalized linear Pad¶e-Chebyshev approximant of order n to the function f (x). An equivalent result was discussed in [5], [7]. When carrying out actual computations, the coe±cients ci are known only approximately, and thus the equalities (33), (34) and (35) are also satis¯ed approximately.

10

Computer experiments for the nonlinear Pad¶e{Chebyshev approximation

Consider the results of computer experiments that were performed by means of the software implemented in the framework of the REDUCE system and brie°y described in Section 7 above. At the author’s request, computer calculations were carried out by A. Ya. Rodionov. We begin with the example considered in Section 5 above, where the linear Pad¶e{Chebyshev approximant of the form (17) to the function cos ¼4 x was constructed on the segment [¡ 1; 1] for m = 2, n = 3. To construct the corresponding nonlinear Pad¶e{Chebyshev approximant, it is necessary to specify the value of the parameter N determining the number of terms in the truncated Taylor series (31) of the approximated function. In this case the calculation error is determined, in fact, by the parameter N . The coe±cients in approximants of the form (17) which are obtained for N = 15 and N = 20 (the nonlinear case) and the coe±cients in the error approximant are as follows: a~0 = 0:4960471034987563;

a0 = 0:4960471027757504;

¢a0 = 10¡8 ¢ 0:07230059; ~a1 = ¡ 0:1451091945278387;

a1 = ¡ 0:1451091928755736;

¢a1 = ¡ 10¡8 ¢ 0:16522651; ~a2 = 10¡2 ¢ 0:5482586543334515;

a2 = 10¡2 ¢ 0:548258121085953;

¢a2 = ¡ 10¡9 ¢ 0:42224856; ~a3 = ¡ 10¡4 ¢ 0:5205903601778259;

a3 = ¡ 10¡4 ¢ 0:5205902238186334;

¢a3 = ¡ 10¡10 ¢ 0:13635919; ~b0 = 0:4960471034987759;

b0 = 0:4960471027757698;

¢b0 = 10¡8 ¢ 0:07230061; ~b1 = 10¡2 ¢ 0:7884201590727615;

b1 = 10¡2 ¢ 0:7884203019999351;

¢b1 = ¡ 10¡10 ¢ 0:1429272;

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

~b2 = 10¡4 ¢ 0:4977097973870693;

55

b2 = 10¡4 ¢ 0:4977100977750249;

¢b2 = ¡ 10¡10 ¢ 0:300388: Both approximants have absolute errors ¢ equal to 0:4 ¢ 10¡13 and relative errors ¯ equal to 0:6 ¢ 10¡13 . These values are close to the best possible. The condition number of the system of equations (30) in both cases is 0:4 ¢ 108 . The denominator ¢Q of the error approximant is zero for x = x0 º 0:70752 : : :; the point x0 is also close to the root of the numerator ¢P which for x = x0 is of order 10¡8 . Such a situation was considered in Section 5 above. Outside a small neighborhood of the point x0 the absolute and the relative errors have the same order as in the \linear case" considered in Section 5. Now consider the nonlinear Pad¶e{Chebyshev approximant to the function tan ¼4 x on the segment [¡ 1; 1] of the form (17) for m = n = 3. In this case the Taylor series converges very slowly, and, as the parameter N increases, the values of the coe±cients of the rational approximant undergo substantial (even in the ¯rst decimal digits) and intricate changes. The situation is illustrated in Table 2, where the following values are given: the absolute errors ¢, the absolute errors ¢0 of error approximants (there the approximants are compared for N = 15 and N = 20, for N = 25 and N = 35, for N = 40 and N = 50), and also the values of the condition number cond of the system of linear algebraic equations (30). In this case the relative errors coincide with the absolute ones. The best possible error is ¢min = 0:83 ¢ 10¡17 . A small neighborhood of the root of the polynomial ¢Q is eliminated as before. N

15

20

25

35

40

50

cond

0:76 ¢ 107

0:95 ¢ 108

0:36 ¢ 1010

0:12 ¢ 1012

0:11 ¢ 1012

0:11 ¢ 1012

¢

0:13 ¢ 10¡4

0:81 ¢ 10¡6

0:13 ¢ 10¡7

0:12 ¢ 10¡10

0:75 ¢ 10¡12

0:73 ¢ 10¡15

¢0

0:7 ¢ 10¡4

0:7 ¢ 10¡8

0:2 ¢ 10¡9

Table 2

11

Small deformations of approximated functions and acceleration of convergence of series

Let a function f (x) be expanded into the series in Chebyshev polynomials, i.e., suppose P that f (x) = 1 i=0 ci T i . Consider a partial sum f^N (x) =

N X

ci Ti

(37)

i=o

of this series. Using formula (35), it is easy to verify that the linear Pad¶e{Chebyshev approximant of the form (1) or (25) to the function f (x) coincides with the linear Pad¶e{ Chebyshev approximant to polynomial (37) for N = n + 2m, i.e., it depends only on the ¯rst n + 2m + 1 terms of the Fourier{Chebyshev series of the function f (x). A similar

56

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

result is valid for the approximant of the form (17) or (18) to even or odd functions, respectively. Note that for N = n + 2m the polynomial f^N is the result of application of the algorithm of linear (or nonlinear) Pad¶e{Chebyshev approximation to f (x), where the exponents m and n are replaced by 0 and 2m + n. The interesting e®ect mentioned in [2] consists of the fact that the error of the polynomial approximant f^n+2m depending on n + 2m + 1 parameters can exceed the error of the corresponding Pad¶e{Chebyshev approximant of the form (1) which depends only on n + m + 1 parameters. For example, consider an approximant of the form (18) to the function tan ¼4 x on the segment [¡ 1; 1]. For m = n = 3 the linear Pad¶e{Chebyshev approximant has error of order 10¡17 , and the corresponding polynomial approximant of the form (37) has the error of order 10¡11 . This polynomial of degree 19 (odd functions are involved, and hence m = n = 3 in (18) corresponds to m = 6, n = 7 in (1)) can be regarded as a result of a deformation of the approximated function tan ¼4 x. This deformation does not a®ect the ¯rst twenty terms in the expansion of this function in Chebyshev polynomials and, consequently, does not a®ect the coe±cients in the corresponding rational Pad¶e{Chebyshev approximant, but leads to an increase of several orders in its error. Thus, a small deformation of the approximated function can result in a sharp change in the order of the error of a rational approximant. Moreover the e®ect just mentioned means that the algorithm extracts additional information concerning the following components of the Fourier{Chebyshev series from the polynomial (37). In other words, in this case the transition from the Fourier{Chebyshev series to the corresponding Pad¶e{Chebyshev approximant accelerates the convergence of the series. A similar e®ect of acceleration of convergence of power series by passing to the classical Pad¶e approximant is known (see, e.g., [16]). It is easy to see that the nonlinear Pad¶e{Chebyshev approximant of the form (1) to the function f (x) depends only on the ¯rst m + n + 1 terms of the Fourier{Chebyshev series for f (x), so that for such approximants a more pronounced e®ect of the type indicated above takes place. Since one can change the \tail" of the Fourier{Chebyshev series in a quite arbitrary way without a®ecting the rational Pad¶e{Chebyshev approximant, the e®ect of acceleration of convergence can take place only for the series with an especially regular behavior (and for the corresponding \well-behaved" functions). See [5], [7] for some details.

12

Error autocorrection and Interval Analysis

Undoubtedly one of the most relevant problems in Interval Analysis in the sense of [8], [9], [10] is getting realistic interval estimates for calculation errors, i.e. to get e±cient estimates close to the virtual calculation errors. Di±culties arise where intermediate errors cancel out each other. For the sake of simplicity let us suppose that we calculate values of a real smooth function z = F (y1 ; : : : ; yN ) of real variables y1 ; : : : ; yN , and suppose that all round-o® errors are negligible with respect to input data errors. This situation has been examined in detail in the framework of Ju.V. Matijasevich’s \a posteriori interval

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

57

analysis", see, e.g., [10]. In this case the error ¢F of F (y1 ; : : : ; yN ) can be evaluated by the formula N X @F ¢ ¢yi + r; ¢F = (38) i=1 @yi P

where r is negligible. The sum N i=1 (@F =@yi ) ¢ ¢yi in (38) can be treated as a scalar product of the gradient vector f@F=@yi g and the vector of errors f¢yi g. The e®ect of error autocorrection corresponds to the case, where the gradient f@F =@yi g is large but the scalar product is relatively small. In this case these vectors are almost orthogonal and the following approximate equation holds: N X @F i=1

@yi

¢ ¢yi º 0

(39)

This e®ect is typical for some ill-posed problems. For all the standard interval methods, the best possible estimation for ¢F is given by the formula N X @F j¢F j µ j j ¢ j¢yi j: (40) i=1 @yi This estimate is good if the errors ¢yi are \independent" but it is not realistic in the case discussed in this paper (calculation of values of rational approximants when the error autocorrection e®ect is at work). In this case F (y1 ; : : : ; yN ) = R(x; a0 ; : : : ; an ; b0 ; : : : ; bm ) =

a0 + a 1 x + a 2 x 2 + : : : + a n x n ; b0 + b1 x + b2 x2 + : : : + bm xm

where N = m + n+ 3, fy1 ; : : : ; yN g = fx; a0 ; : : : ; an ; b0 ; : : : ; bm g. For the sake of simplicity let us suppose that ¢x = 0. In this case we can use the equality (22) to transform the formula (38) into the formula ¢R º

¢P ¡ Q

n m X P ¢Q X xi P (x)xj = ¢a + ¢bj : i 2 Q2 i=0 Q(x) j=0 Q (x)

(41)

So the estimation (39) transforms into the estimation n X

m X jxi j jP (x)xj j ¢ j¢ai j + ¢ j¢bj j: ¢R µ 2 jQ(x)j i=0 j=0 Q (x)

(42)

It is easy to check that estimations of this type are not realistic. Consider the following example discussed in the Introduction: f (x) = arctan x on the segment [¡ 1; 1], R(x) has the form (1) for m = 8, n = 9. In this case the estimation (41) is of order 10¡4 but in fact ¢R is of order 10¡11 . This situation is typical for examples examined in this paper. In fact we have an approximate equation ¢R º

n X i=0

m X xi P (x)xj ¢ai + ¢bj º " º 0; 2 Q(x) j=0 Q (x)

(43)

58

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

where " is the absolute error of the approximation method used. Of course, this approximate equation holds only if our approximation method is good and the uncertainty relation (discussed above in Section 2) is valid. Then the approximate equation (43) corresponds to the approximate equation (39). The error autocorrection e®ect appears not only in rational approximation but in many problems. Other examples (where this e®ect occurs in a weaker form) are the method of least squares and some popular methods for the numerical integration of ordinary and partial di®erential equations, see, e.g., [24], [25], [26], [27], [28], [29]. In principle, the error autocorrection e®ect appears if input data form an (approximate) solution (or solutions) of an equation (or equations or systems of equations). Then the corresponding errors could form an approximate solution (or solutions) for another equation (or equations or systems of equations). As a result this could lead to corrections for standard interval estimates. Of course, theoretically we can include all the preliminary numerical problems to our concrete problem and to use, e.g., a posteriori interval analysis for the \united" problem. However, in practice this is not convenient. In practice, situations of this kind often appear if we use approximate solutions to ill conditioned systems of linear algebraic equations. If the condition number of the system is great and the residual of the solution (with respect to the system) is small, then our software must send us a \warning". This means that an additional investigation for error estimates is needed. In the theory of interval analysis this corresponds to a further development of \a posteriori interval methods" in the spirit of [10], [27], [28], [29], [30]. Remark 12.1. We have discussed \smooth" computations. Note that for many \nonsmooth" optimization problems all the interval estimates could be good and absolutely exact. A situation of this kind (related to solving systems of linear algebraic equations over idempotent semirings) is described in [31].

Acknowledgments Partly supported by the Fields Institute for Research in Mathematical Sciences (Toronto, Canada).

References [1] G.L. Litvinov et al., Mathematical Algorithms and Programs for Small Computers, Finansy i Statistika Publ., Moscow, 1981 (in Russian). [2] G.L. Litvinov and V.N. Fridman, Approximate construction of rational approximants, C. R. Acad. Bulgare Sci., 36, No. 1 (1981), 49{52 (in Russian). [3] I.A. Andreeva, G. L. Litvinov, A. Ya. Rodionov and V. N. Fridman, The PADEprogram for Calculation of Rational Approximants. The Program Speci¯cation and its Code, Fond Algoritmov i Programm NIVTs AN SSSR, Puschino, 1985 (in Russian).

G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

59

[4] A.P. Kryukov, G.L. Litvinov and A.Ya. Rodionov, Construction of rational approximations by means of REDUCE, in: Proceeding of the ACM{SIGSAM Symposium on Symbolic and Algebraic Computation (SYMSAC 86), Univ. of Waterloo, Canada, 1986, pp. 31{33. [5] G.L. Litvinov, Approximate construction of rational approximations and an e®ect of error autocorrection, in: Mathematics and Modeling, NIVTs AN SSSR, Puschino, 1990, 99{141 (in Russian). [6] G.L. Litvinov, Error auto-correction Computations, 4 (6), (1992), 14{18.

in

rational

approximation,

Interval

[7] G.L. Litvinov, Approximate construction of rational approximations and the e®ect of error autocorrection. Applications, Russian J. Math. Phys., 1, No. 3 (1994), 14{18. [8] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York{London, 1983. [9] R.B. Kearfott, Interval computations { introduction, uses, and resources, Euromath. Bulletin, 2(1), (1996), 95{112. [10] Y. Matijasevich, A posteriori version of interval analysis, in: Proc. Fourth Hung. Computer Sci. Conf., Topics in the Theoretical Basis and Applications of Computer Science, eds. M. Arato, I. Katai, L. Varga, Acad. Kiado, Budapest, 1986, pp. 339{349. [11] J.F. Hart et al., Computer Approximations, Wiley, New York, 1968. [12] V.V. Voevodin, Numerical Principles of Linear Algebra, Nauka Publ., Moscow, 1977 (in Russian). [13] Y.L. Luke, Computations of coe±cients in the polynomials of Pad¶e approximations by solving systems of linear equations, J. Comp. and Appl. Math., 6, No. 3 (1980), 213{218. [14] Y.L. Luke, A note on evaluation of coe±cients in the polynomials of Pad¶e approximants by solving systems of linear equations, J. Comp. and Appl. Math., 8, No. 2 (1982), 93{99. [15] G.E. Forsythe, M. Malcolm and C. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cli®s, N. J., 1977. [16] G.A.Baker and P. Graves-Morris, Pad¶ e Approximants. Part I: Basic Theory. Part II: Extensions and Applications, Encyclopaedia of Mathematics and its Applications 13, 14, Addison-Wesley, Reading, Mass., 1981. [17] W.J. Cody, W. Fraser and J.F. Hart, Rational Chebyshev approximation using linear equations, Numer.Math., 12 (1968) 242{251. [18] E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966. [19] D.S. Lubinsky and A. Sidi, Convergence of linear and nonlinear Pad¶e approximants from series of orthogonal polynomials, Trans. Amer. Math. Soc., 278, No. 1, (1983) 333{345. [20] S. Paszkowski, Zastosowania Numeryczne Wielomian¶ow i Szereg¶ow Czebyszewa, Panstwowe Wydawnictwo Naukowe, Warszawa, 1975 (in Polish). [21] H.J. Maehly, Rational approximations for transcendental functions, in: Proceedings of the International Conference on Information Processing, UNESCO, Butterworths, London, 1960, pp. 57{62.

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G. L. Litvinov / Central European Journal of Mathematics 1 (2003) 36{60

[22] C.K. Clenshaw and K. Lord, Rational approximations from Chebyshev series, in: Studies in Numerical Analysis, ed. B.K.P. Scaife, Academic Press, London and New York, 1974, pp. 95{113. [23] A.C. Hearn, REDUCE User’s Manual, Rand. Publ., 1982. [24] E.A. Volkov, Two-sided di®erence methods for solving linear boundary-value problems for ordinary di®erential equations, Proc. Steklov Inst. Math. 128 (1972), 131{152 (translated from Russian by AMS in 1974). [25] E.A. Volkov, Pointwise estimates of the accuracy of a di®erence solution of a boundary-value problem for an ordinary di®erential equation, Di®erential Equations, vol. 9, No 4 (1973), 717{726 (in Russian; translated into English by Plenum Publ. Co. in 1975, 545{552). [26] B.S. Dobronets and V.V. Shaydurov, Two-sided numerical methods, Nauka Publ., Novosibirsk, 1990 (in Russian). [27] B.S. Dobronets, On some two-sided methods for solving systems of ordinary di®erential equations, Interval Computations, No 1(3) (1992), 6{21. [28] B.S. Dobronets, Interval methods based on a posteriory estimates, Interval Computations, No 3(5) (1992), 50{55. [29] L.F. Shampine, Ill-conditioned matrices and the integration of sti® ODEs, J. of Computational and Applied Mathematics 48 (1993), 279{292. [30] Yu.V. Matijasevich, Real numbers and computers. { In: Kibernetika i Vychislitelnaya Tekhnika, vol. 2 (1986), 104{133 (in Russian). [31] G.L. Litvinov and A.S. Sobolevskii, Idempotent interval analysis and optimization problems, Reliable Computing 7 (2001), 353{377.

CEJM 1 (2003) 61{78

Existence and Nonexistence Results for Reaction-Di® usion Equations in Product of Cones Abdallah El Hamidi1¤ , Gennady G. Laptev2y 1

2

Universit¶e de La Rochelle, Laboratoire de Math¶ematiques Avenue Michel Cr¶epeau 17000 La Rochelle Department of Function Theory, Steklov Mathematical Institute Gubkina 8, 119991 Moscow, Russia

Received 22 July 2002; revised 11 October 2002 Abstract: Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution di¬erential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and V´eron. The existence result is established using the method of supersolutions. c Central European Science Journals. All rights reserved. ® Keywords: nonexistence, blow-up, evolution di® erential inequalities, cone MSC (2000): Primary 35G25; Secondary 35R45, 35K55, 35L70

1

Introduction

The paper is devoted to establishing the conditions for the nonexistence of global nontrivial solutions of semilinear di®erential inequalities of parabolic type in a product of conical domains. Such a formulation implies that the domain is unbounded and the corresponding problem has a nontrivial local solution. Complications occur when one attempts to extend this solution to a global one, that is, to ¯nd a solution of the Cauchy problem that is de¯ned in the entire domain under consideration. Here, even for semilinear problem of the form ( @u ¡ ¢u = uq in RN £ (0; 1); @t y

¤

u(x; 0) = u0 (x) ¶ 0

E-mail: [email protected] E-mail: [email protected]

in RN

62

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

there existes a so-called critical nonlinearity exponent (equal in the present case to the Fujita q ¤ = 1 + 2=N ) such that for 1 < q µ q ¤ no local (in t) solution can be extended to a global one (by an `extension’ we mean here, of course, one that keeps the solution in some local function space). Results of this sort can be formulated also as theorems on the nonexistence of global solutions. Surprising here is the fact that we make no assumptions about the growth of the global solution at in¯nity. For a more detailed setting of the problem and a survey of the literature see [22]. In place of the above equation one can consider the following di®erential inequality without initial conditions in the local space of functions with continuous second derivatives: ( @u ¡ ¢u ¶ uq in RN £ (0; 1); @t u(x; 0) ¶ 0

in RN :

A critical exponent also arises here. It is equal to the similar-type exponent for equations. An interesting feature of inequalities is the fact that it is sometimes possible to explicitly construct solutions for a supercritical nonlinearity, easily establishing in this way the de¯nitive character of the results obtained. In the present paper we consider the di®erential inequality @ku ¡ @tk

¡ ¢ ¢ jujm¡1 u ¶ jujq in K 1 £ K 2 £ (0; +1);

and its various generalizations. That is, in place of the entire space RN one considers a product of cones K 1 and K 2 , m ¶ 1, and k is a positive integer. Then the critical exponent depends on certain characteristics of the cones expressible in terms of the ¯rst eigenvalue of the corresponding elliptic problem on the unit sphere. The ¯rst results for this problem (with k = 1, only for positive solutions) were obtained in [23]. In the present paper the initial and boundary data may change sign but we have to verify some integral nonnegativity. The theory of linear elliptic boundary value problems on a cone goes back to Kondratiev. The nonexistence of solutions of the corresponding semilinear and nonlinear elliptic and parabolic problems is mainly studied by means of a reduction to an integral equation, using results similar to comparison theorems and the maximum principle. For the case of parabolic equations we point out the papers [2, 16] and the already classical book [29]. The state of the art is re°ected in the surveys [14] and [4]. In the present paper we prove nonexistence of solutions using the test-function method and do not use comparison principles which are characteristic for the theory of secondorder equations. The method enables us to demonstrate virtually immediately, using the techniques of Mitidieri-Pohozaev [22], the nonexistence of solutions for the critical nonlinearity exponent (which previously presented serious di±culties) and to consider systems of di®erential inequalities and other classes of problems for which the maximum principle does not hold. The central problem is now the choice of a test function and the estimation of the corresponding integrals involving this function. The nonexistence of solutions for the single cone has been investigated using this technique in [8, 10, 12].

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63

In order to show that our result is sharp in the parabolic case (m = k = 1), we show that the problem @u ¡ ¢u = jujq in K1 £ K 2 £ (0; 1); @t admits global nonnegative solutions for q greater that the critical exponent. The proof of this result is based on the method of supersolutions.

2

Notations and preliminary results

Pn Let n ¶ 1 be an integer, Ni ¶ 3, i 2 f1; 2; :::; ng, and N = i=1 Ni . The polar coordinates in RNi will be denoted by (ri ; !i ). Let S Ni¡1 be the unit sphere in RNi and ­ i a domain of S Ni¡1 with a su±ciently smooth boundary @­ i . We will denote by Ki the cone Ki = fxi = (ri ; !i) 2 RNi ; ri > 0 and !i 2 ­ i g and K = K1 £K 2 £:::£K n . The boundary of Ki (resp. K ) is designed by @Ki (resp. @K ). The outward normal vector to @­ i (resp. @K ) is denoted by ¸ i (resp. ¸ ). Let ¶ i > 0 be the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on ­ i and let ©i be the associated eigenfunction such that 0 < ©i µ 1, and the function © : (!1 ; !2 ; :::; !n ) 7¡ !

n Y

©i (!i ):

i=1

Qn

Let us introduce also the functions ©(i) = j=1; j6=i ©j (!j ). Throughout this paper, the letter C denotes a constant which may vary from line to line but is independent of the terms which will take part in any limit process. For any real number q > 1, we de¯ne the real q 0 such that 1=q + 1=q 0 = 1. We will use the set ­ =­

1

£­

2

£ ::: £ ­ n ;

whose elements are denoted by ! = (!1 ; !2 ; :::; !n ), and ¯nally d! = d!1 d!2 ::: d!n . We shall construct the test function which will be used in our proofs. Let ± 2 C 01 (R+ ) be the standard cut-o® function 8 > > 1 if 0 µ y µ 1; > > < ± (y) = and 8 y 2 R; 0 µ ± (y) µ 1: > > > > : 0 if y ¶ 2 Let p0 ¶ k + 1 and ² the function de¯ned by

² (y) = [± (y)]p0 : Explicit computation shows that there is a positive constant C(² ) > 0 such that, for any y ¶ 0 and any p, 1 < p µ p0 , the estimates ¯ i ¯p ¯d ² ¯ p¡1 ¯ ¯ (y) (y); 1 µ i µ k; (1) ¯ dy i ¯ µ C(² )²

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hold true. Consider the function t 7! ² (t=» µ ), where » and ³ are positive parameters. We have ¯ ¯ supp ¯² (t=» µ )¯ = ft 2 R+ ; 0 µ t µ 2» µ g and

¯ k ¯ ¯d ² ¯ supp ¯¯ k (t=» µ )¯¯ = ft 2 R+ ; » µ µ t µ 2» µ g; dt where "supp" denotes the support. It follows that ¯ k ¯p ¯d ´ µ ¯ Z ¯ dtk (t=» )¯ dt µ c´ » ¡µ(kp¡1) : ¯ k ¯ ¯d ² ¯ ² p¡1 (t=» µ ) ³ supp ¯ dtk (t=½ )¯

(2)

Now, consider the functions

¹ 1 (r1 ; ::; rn ) =

n Y

risi ;

i=1

¹ 2 (r1 ; ::; rn ) =

n Y

² (ri =» );

i=1

and ¹ = ¹

1

£ ¹ 2 , where si > 0, i 2 f1; 2; :::; ng. Let us set, for any i in f1; 2; :::; ng, ¹

(i) 1 (r1 ; :::; rn )

=

n Y

s

rj j

j=1;j6=i

and ¹

(i) 2 (r1 ; :::; rn )

n Y

=

² (rj =» ):

j=1;j6=i

We shall give estimates concerning @¹ =@ri and @ 2 ¹ =@ri2 . Since @¹ = si risi¡1 ¹ @ri there is a constant cp > 0 such that ¯ ¯ h ¯ @¹ ¯p ¯ ¯ µ cpsp r p(si¡1) ¹ i i ¯ @r ¯ i

(i) 1 ¹ 2

(i) 1

ip

¹

1 + ² 0 (ri =» )¹ 1 ¹ »

p 2

+

(i) 2 ;

»

cp

h

p 1

j² 0 (ri =» )jp¹ p

¹

(i) 2

ip

:

Then, there exists C > 0, independent of » and rj , j 2 f1; 2; :::; ng, such that ¯ ¯ µ p¶ h ip ¯ @¹ ¯p r ¡1) p(s (i) p¡1 i i ¯ ¯ ¹ 1 ¹ 2 1+ p : ¯ @r ¯ µ Cri » i

(3)

Similarly,

@2¹ = si (si ¡ @ri2 =

risi¡2

1)risi¡2 ¹ ½

si (si ¡

(i) 1 ¹ 2

1)¹

+

(i) 1 ¹ 2

2si si¡1 0 r ² (ri =» )¹ » i ri + 2si ² 0 (ri =» )¹ »

(i) (i) 1 ¹ 2

(i) (i) 1 ¹ 2

+

+

1 2

»

»

ri2 2

² 00 (ri =» )¹ 1 ¹ 00

² (ri =» )¹

(i) 2

(i) (i) 1 ¹ 2

¾

:

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

65

There exists C > 0, independent of » and rj , j 2 f1; 2; :::; ng, such that ¯ 2 ¯p µ p 2p ¶ h ip ¯@ ¹ ¯ r r ¡2) p(s (i) p¡1 i i i ¯ ¯ ¹ 1 ¹ 2 1 + p + 2p : ¯ @r2 ¯ µ C ri » » i

(4)

On the other hand, we have ¢(¹ 1 ©) =

n X

¢i (¹ 1 ©) =

i=1

n n X

risi¡2 [si (si ¡

1) + si (Ni ¡

1) ¡

¶ i] ¹

(i) 1

i=1

Let Ni ¡ 2 s¤i = ¡ + 2 be the positive root of si (si ¡

1) + si (Ni ¡

sµ

Ni ¡ 2 2

1) ¡ ¶

¹ ¤ (r1 ; r2 ; :::; rn ) =

i

¶2

+¶

o

©:

i

= 0 and

n Y

s¤

ri i :

i=1

Then we have ¢(¹ ¤ ©) = 0: Finally, we introduce the test function (independent of t) ý (x) = ¹ ¤ (r1 ; r2 ; :::; rn ) ¹ 2 (r1 ; r2 ; :::; rn ) ©(!1 ; !2 ; :::; !n ): If ¸

i

denotes the outward normal vector to ­ i , then @ý @©i (!i ) = ¹ ¤ ¹ 2 ©(i) : @¸ i @¸ i

The Hopf lemma implies that @©i (!i ) µ 0; @¸ i

and

@ý µ 0: @¸ i

Consequently, we conclude that

@Ã ½ j@K µ 0: (5) @¸ Let ¢i be the Laplacian operator with respect to the variable xi = (ri ; !i) 2 RNi , i 2 f1; 2; :::; ng. Then ¢(Ã ½ )(x) =

n X

¢i (ý )(x)

i=1 n ½ X

@ 2 (ý ) Ni ¡ 1 @(ý ) = (x) + (x) + 2 @r r @r i i i i=1 ¾ n ½ 2 X @ Ni ¡ 1 @ ¶ i ¡ = + ý (x) @ri2 ri @ri ri2 i=1 n ½ 2 X @ Ni ¡ 1 @ ¡ = ©(!1 ; ::; !n ) + 2 @r r @r i i i i=1

1 ¢! (Ã ½ )(x) ri2 i

¶

i ri2

¾

¾

(¹ ¤ ¹ 2 )(r1 ; ::; rn );

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since ¢!i (à ½ )(x) = ¡ ¶ i (à ½ )(x). Hence, ¯½ 2 ¯p ¾ ¯ @ ¯ Ni ¡ 1 @ ¶ i ¯ j¢i (à ½ )j = © ¯ ¡ + (¹ ¤ ¹ 2 )¯¯ 2 2 @ri ri @ri ri µ ¶ £ (i) ¤p p¡1 rip ri2p p p(si¡2) µ C© ri ¹ ¤ ¹ 2 1 + p + 2p » » µ p 2p ¶ ¹ ¤ r r µ Cà ½p¡1 2p 1 + ip + i2p : » » ri p

p

Since ² (ri =» ) = 1 for ri µ » and ² (ri =» ) = 0 for ri ¶ 2» , if we set N then N i » fx 2 K ; » < ri < 2» g, and the expression 1+ is bounded for any x 2 N

i.

Pn

i=1

»

+

p

ri2p 2p

»

j¢i (Ã ½ )(x)jp µ CÃ ½p¡1 (x)

i;

¹ »

j¢(Ã ½ )j µ c

n X

j¢i (à ½ )jp µ cC ýp¡1 (x)

i=1

Z 2½ Z n X C » 2p ½ ­ i i=1

(Z

¤ : 2p

(6)

¢i (Ã ½ ). Then from (6) we obtain p

Futhermore,

= fx 2 K ; ¢i ý 6= 0g,

Consequently, there is a constant C > 0 such that

8x 2 N Recall that ¢(ý ) =

rip

i

Z

K

1 [0;2½]n¡

»

¤ : 2p

j¢(ý )jp dx µ (p¡1)¾j à p¡1 ½ j=1 jxj j

Qn

Z

Assume that s¤j + Nj ¡ which is equivalent to ¼

¹

j

Q

n Y

l6=i K!j j=1

(p ¡

1)¼

s¤ +Nj ¡1¡(p¡1)¾j rj j

j

Y

drl d!j

l6=i

)

d!i dri :

> 0; 1 µ j µ n;

s¤j + Nj < ; 1 µ j µ n; p¡ 1

then we have the estimate Z j¢(ý )jp dx µ C » Qn (p¡1)¾j à p¡1 jx j ½ K j j=1

P ¤ ¡2p+ n j=1 (sj +Nj ¡(p¡1)¾j )

(7)

:

Finally, we consider the test function, which depends on all the variables, µ ¶ t ’½ (x; t) = ² Ã ½ (x): » µ

(8)

(9)

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67

Using (7) and (8), we obtain the ¯rst estimate concerning ’½ : Z µ

Z

+1 0 2½³ 0

µC»

Z

j¢(’½ )jp (x; t) dx dt (p¡1)¾j ’ p¡1 K ½ j=1 jx j j µ ¶ Z j¢(ý )jp t ² dt dx Q n (p¡1)¾j à p¡1 » µ jx j ½ K j j=1 Qn

P ¤ µ¡2p+ n j=1 (sj +Nj ¡(p¡1)¾j )

:

(10)

Similarly, using (7) and (2), we obtain the second estimate concerning ’½ : ¯ k ¯p ¯ @ ’» ¯ Z Z ¯ @tk (x; t)¯ ¯ ¯ dx dt ¯ k ¯ Qn p¡1 supp ¯¯ @@t’k» ¯¯ j=1 jxj j(p¡1)¾j ’½ (x; t) ¯ k ¯p ¯d ´ µ ¯ Z Z ¯ dtk (t=» )¯ à (x) Qn ½ (p¡1)¾ dx µ dt ¯ ¯ k p¡1 (t=» µ ) j K supp ¯¯ d k² (t=½³ )¯¯ ² j=1 jxj j dt

3

µC»

Pn

µC»

¡µ(kp¡1)+

¤ j=1 (sj +Nj )¡(p¡1)¾j

Pn

»

¡µ(kp¡1)

¤ j=1 (s j +Nj ¡(p¡1)¾j )

:

(11)

Nonexistence Results

Let us consider the nonexistence problem for weak solutions to the problem 8 k n Y ¡ m¡1 ¢ @ u > q > ¡ ¢ juj jxi j¾i ; x 2 K; t > 0; > u ¶ juj > k > @t > i=1 > > > > > > > > Z Z > < @ k¡1 u lim inf ::: (x; 0) ª(x) dx ¶ 0; x 2 K; (E) k¡1 Ri!+1 jx j 1 1 > 1·i·n > > > > > > > > Z R0 Z Z > > @ª(x) > > ju(x; t)jm¡1 u(x; t) ::: dx dt µ 0; x 2 @K; > lim inf : Ri!+1 0 @¸ jx1j
where ¼

i

¶ 0 (i = 1; : : : ; n),

ª(x) = ª(r1 ; :::; rn ; !1 ; :::; !n ) = ¹ ¤ (r1 ; :::; rn )©(!1 ; :::; !n ); for any x 2 K . De¯nition 3.1. A weak solution u of the system (E) on K £]0; +1[ is a continuous j function on K £ [0; +1[ such that the traces @@tuj (x; 0), j 2 f1; ::; k ¡ 1g, are well de¯ned

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A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

and locally integrable on K and the inequality ! Z 1Z Ã n k Y @ ’ jujm¡1 u ¢’ ¡ u (¡ 1)k k + ’jujq jxi j¾i dx dt¡ @t 0 K i=1 Z 1Z Z k¡1 k¡1¡j X @’ @ u @j’ ¡ jujm¡1 u dx dt + (¡ 1)j (x; 0) (x; 0) dx µ 0; k¡1¡j j @¸ @t @t 0 @K K j=0 holds true for any nonnegative test function ’ 2 C such that ’j@K£]0;+1[ = 0.

2;k

(12)

(K £]0; +1[) with compact support

Theorem 3.2. Assume that 0µ¼ and

j

<

q¡ m

m¡

¢ s¤j + Nj ; 1 µ j µ n

Pn (2 + i=1 ¼ i ) (1 + (k ¡ 1)m) 1 < q µ q (k; m) = m + ³ : ¢´ Pn ¡ ¤ k ¡ 2 + j=1 sj + Nj + 2 ¤

Then the problem (E) does not have any nontrivial global weak solutions. We start by proving the following lemmas. Lemma 3.3. The assumption Z lim inf Ri!+1 1·i·n

:::

jx1j
implies that lim inf ½!+1

Z

K

Z

jxnj
@ k¡1 u (x; 0) ª(x) dx ¶ 0 @tk¡1

@ k¡1 u (x; 0)Ã ½ (x) dx ¶ 0: @tk¡1

Proof. Let us set v(r1 ; r2 ; :::; rn ) =

Z

­

@ k¡1 u (x; 0)ª(x) d!: @tk¡1

Then, we have µ ¶ Z Z n Y @ k¡1 u rj (x; 0)ý (x) dx = v(r1 ; r2 ; :::; rn ) ² dr1 dr2 ::: drn : k¡1 » K @t [0;2½]n j=1 Using integration by parts with respect to variable r1 and the fact that ² (r1 =» ) = 1 for 0 µ r1 µ » and ² (r1 =» ) = 0 for r2 ¶ 2» , we have µ ¶ ¶ µ ¶ Z 2½ Z µZ r1 r1 1 2½ r1 v(r1 ; r2 ; :::; rn )² dr1 = ¡ v(s1 ; r2 ; :::; rn ) ds1 ² 0 dr1 » » 0 » 0 0 ¶ µ ¶ Z µZ r1 1 2½ r1 =¡ v(s1 ; r2 ; :::; rn ) ds1 ² 0 dr1 » ½ » 0 µ ¶ Z ½1 » 1 0 =¡ ² v(s1 ; r2 ; :::; rn ) ds1 ; » 0

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

69

where » 1 is such that » µ » 1 µ 2» , according to the intermediate value theorem. Proceeding in the same manner for the other variables, r2 , r3 ,..., rn , we obtain µ ¶ Z ½1 Z ½n Z n Y @ k¡1 u » j 0 n (x; 0)Ã ½ (x) dx = (¡ 1) ² ::: v(s1 ; :::; sn ) ds1 ::: dsn ; k¡1 » K @t 0 0 j=1 where » j are such that » µ » decreasing on [1; 2] then

µ 2» , for j 2 f1; 2; :::; ng. Since the cut-o® function ² is

j

(¡ 1)

n

n Y

j=1

On the other hand, Z ½1 Z ::: 0

²

0

µ »

»

j

¶

½n

v(s1 ; :::; sn ) dsn ::: ds1 =

0

¶ 0:

Z

D»

@ k¡1 u (x; 0)ª(x) dx; @tk¡1

where D½ = fx = (x1 ; :::; xn ) 2 K ; jxj j < » j ; for 1 µ j µ ng. Hence, the assumption Z Z @ k¡1 u lim inf ::: (x; 0) ª(x) dx ¶ 0; k¡1 Ri!+1 jx j
implies that lim inf ½!+1

Z

D»

@ k¡1 u (x; 0)ª(x) dx ¶ 0: @tk¡1

Finally, since the expressions Z Z @ k¡1 u @ k¡1 u (x; 0)ª(x) dx and (x; 0)Ã ½ (x) dx k¡1 k¡1 D» @t K @t have the same sign, it follows that Z @ k¡1 u lim inf (x; 0)Ã ½ (x) dx ¶ 0; ½!+1 K @tk¡1 which completes the proof. Lemma 3.4. The assumption Z R0 Z Z lim inf ::: Ri !+1 0·i·n

0

jx1j
ju(x; t)jm¡1 u(x; t) jxnj
implies that lim inf ½!+1

Z

+1 0

Z

jujm¡1 u @K

@ª(x) dx dt µ 0; x 2 @K @¸

@’½ dx dt µ 0: @¸

Proof. The same arguments as in the last proof give the result. Now, we are able to give the proof of Theorem 3.2:

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Proof. Assume that the problem (E) admits a global weak solution u. In de¯nition 3.1, we choose the test function ’(x; t) = ’½ (x; t), de¯ned in (9). Note that ’½ satis¯es the equalities @ j ’½ (x; 0) = 0; for j 2 f1; 2; :::; k ¡ 1g: @tj Then, the inequality (12) implies that Z 1Z Z 1Z n Y @’½ q ¾j jxj j dx dt ¡ jujm¡1 u ’½ juj dx dt+ @¸ 0 K 0 @K j=1 ¶ Z 1Z µ k @ k¡1 u m¡1 k @ ’½ juj + (x; 0)’½ (x; 0) dx µ u (¡ ¢’½ ) + u (¡ 1) dx dt: k¡1 @tk K @t 0 K Let " be an arbitrary positive real number. Using Lemma 3.3 and Lemma 3.4, there is » 0 > 0 such that for any » ¶ » 0 ,we have ¶ Z 1Z Z 1Z µ n k Y q ¾j m¡1 k @ ’½ juj ’½ jxj j dx dt µ juj u (¡ ¢’½ ) + u (¡ 1) dx dt + " k @t 0 K 0 K j=1 ¯ k ¯¶ Z 1Z µ ¯ @ ’½ ¯ m µ juj j¢’½ j + juj ¯¯ k ¯¯ dx dt + ": (13) @t 0 K Z

Let us set

8 Z 1Z n Y > > q > I½ = juj ’½ jxj j¾j dx dt; > > > 0 K > j=1 > > 0 Z Z > 1 ( mq ) > > j¢(’ )j ½ > < A½ = ³ ´( mq )0¡1 dx dt; 0 K Qn ¾ j > j=1 jxj j ’ ½ (x; t) > > ¯ ¯0 > > ¯ @ k’» ¯q > Z 1Z > ¯ > @tk ¯ > > B > ³Q ´q0¡1 dx dt: ½ = > > n 0 K ¾j ’ (x; t) : jx j j ½ j=1 R1R Applying the HÄolder inequality to 0 K jujm j¢’½ j dx dt, we have Z 1Z 1 m ( q )0 jujm j¢’½ j dx dt µ I½q A½ m : 0

K

Similarly, the estimate Z

1 0

Z

¯ k ¯ 1 1 ¯ @ ’½ ¯ 0 juj ¯¯ k ¯¯ dx dt µ I½q B½q ; @t K

holds true, and we conclude that

m

(

1 q 0

)

1

1 0

I½ µ I½q A½ m + I½q B½q + ":

(14)

Applying the Young inequality to the right-hand side of (14), there is a positive constant C, independent of » , such that I½ µ C (A½ + B½ ) + ":

(15)

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

71

Combining the estimates (10), (11) and (15) we conclude, for » su±ciently large, that I½ µ C (»

1

+»

2

) + ";

(16)

where ¬

1

¬

2

=³ ¡

n µ ³ q ´0 X 2 + s¤j + Nj ¡ m j=1 0

= ¡ ³ (kq ¡

1) +

n X

(s¤j + Nj ¡

µ³ (q 0 ¡

¶

¶ 1 ¼ j ;

q ´0 ¡ m 1)¼ j );

j=1

and C is a positive constant independent of » . At this stage, we choose the parameter ³ to equal the exponents of » in the last estimate. Explicit computation gives à à ! ! P n n X X 1 q¡ 1 1 2(q ¡ 1) + (m ¡ 1) ni=1 ¼ i ³ = 2+ ¼ i ¡ ¼ i = ¢ : k q¡ m k q¡ m i=1 i=1 Since k ¶ 1 and m ¶ 1 it follows that ³ > 0. Now, the estimate (16) can be rewritten I½ µ C » where ¬ =q

( Ã k

¡ 2+

n X

(s¤j + Nj )

j=1

!

+2

) ¡

¬ k(q¡ m)

2¡

+ ";

(m(k ¡

1) + 1)

n X

¼

j

¡

Now, we require

j=1

¬ k(q ¡

m)

km

n X

(s¤j + Nj ):

j=1

µ 0;

which is equivalent to P (2 + ni=1 ¼ i ) (1 + (k ¡ 1)m) ´ q µ q (k; m) = m + ³ : Pn ¤ k ¡ 2 + j=1 (sj + Nj ) + 2 ¤

Whence, I½ is bounded uniformly with respect to the parameter » . Moreover, the function I(» ) is increasing in » . Consequently, the monotone convergence theorem implies that the function n Y q ² ¡ 7 ! ju jxj j¾j (x; t) (r; !; t) (x; t)j ª(x) j=1

is in L1 (K £]0; +1[). Furthermore, note that supp(¢’½ ) » ft 2 R+ ; and supp

µ

@ k ’½ @tk

¶

» ft 2 R+ ;

0 µ t µ 2» µ g £ fx 2 K;

»

µ

µ t µ 2» µ g £ fx 2 K;

» µ jxj j µ 2» ; 1 µ j µ ng

0 µ jxj j µ 2» ; 1 µ j µ ng:

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A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

Whence, instead of (14) we have more precisely m

1 q 0

1

1

0 ( ) I½ µ I~½q A½ m + I~½q B½q + ";

where I~½ =

Z

’½ jujq C»

n Y

jxj j¾j dx dt;

j=1

and C

½

= supp(¢’½ ) [ supp

µ

@ k ’½ @tk

¶

:

Finally, using the dominated convergence theorem, we obtain that there is » that I½ µ "; for any » ¶ » 00 ;

0 0

> 0 such

which is equivalent to lim I½ = 0:

½!+1

This means that u ² 0, which contradicts the fact that u is assumed to be a nontrivial weak solution to (E).

4

Existence Results

In this section, we will limit ourselves to the problem 8 > @u > ¡ ¢u = uq in K £]0; +1[; > > @t > > > > > > > < ~ (E) = u(x; t) = 0 on @K £ [0; +1[; > > > > > > > > > > > : u(x; 0) = u0 (x) ¶ 0 in K; From Theorem 3.2, we know that if

2 ¤ i=1 (N i + si )

1 < q µ q ¤ = 1 + Pn

~ has no global nontrivial solutions. We complete this result with an then the problem (E) existence theorem: Theorem 4.1. If

2 ¤ i=1 (N i + si )

q > q ¤ ² 1 + Pn

~ exist. then nontrivial global solutions of (E)

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

73

Proof. The proof is based on the method of supersolutions [30, 31] and the arguments used are inspired by the ideas of [15, 16, 23]. Let v be a positive solution of 8 > @v > ¡ ¢v = 0 in K £]0; +1[; > > @t > > > > > > > < v(x; t) = 0 on @K £ [0; +1[; > > > > > > > > > > > : v(x; 0) = v0 (x) ¶ 0 in K;

and let the function w be de¯ned on K £]0; +1[ by w(x; t) = ¬ (t)v(x; t), where the function ¬ has to be de¯ned. If ¬ is selected such that · ¸q¡1 0 q ¬ (t) = (¬ (t)) sup v(x; t) ; x2K

~ on its domain. Let us set ¬ to be the solution of the then w is a supersolution of (E) Cauchy problem 8 > > ¬ 0 (t) = (¬ (t))q jjv(:; t)jjq¡1 > L1 (K) ; t > 0; > < (17) > > > > : ¬ (0) = ¬ > 0: 0 It is easy to see that the solution of (17) is global if, and only if, Z and 0<¬

0

<

+1 0

µ

(q ¡

jjv(:; t)jjq¡1 L1 (K) dt < +1

1)

Z

+1

0

(18)

1 ¡ ¶ q¡ 1

jjv(:; t)jjq¡1 L1 (K ) dt

:

At this stage, we will construct the function v on K £ [0; +1[. Consider the function vi de¯ned on K i £]0; +1[ by 1 1 vi (xi ; t) = Iº 1 t + 1 r 2 (Ni ¡2) i i

µ

ri 2(t + 1)

¶

µ

ri2 + 1 exp ¡ 4(t + 1)

¶

©i (!i );

where xi = (ri ; !i), ¸ i = s¤i + Ni2¡2 and Iºi is the modi¯ed Bessel function of order ¸ The function vi is a positive solution of @vi ¡ @t

¢vi = 0 in Ki £]0; +1[

i

[32].

74

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

(see, for example, [16]). Recall that the asymptotic behaviour of Iºi in the neighborhood of 0 and +1 is given respectively by [32] Iºi (z) ¹

z ºi as z ¡ ! 0+ 2ºi ¡(¸ i + 1)

(19)

and

exp(z) p as z ¡ ! +1: (20) 2º z Then, using the fact that ©i = 0 on @­ i and (19), we conclude that vi vanishes on @Ki £ [0; +1[. Let us set now Iºi (z) ¹

v(x; t) =

n Y

vi (xi ; t); for any x = (x1 ; x2 ; :::; xn ) 2 K; t > 0

i=1

and

1 1 Vi (ri ; t) = Iº 1 t + 1 r 2 (Ni¡2) i i

µ

ri 2(t + 1)

¶

µ exp ¡

ri2 + 1 4(t + 1)

¶

:

The function v is a positive solution of @v ¡ @t

¢v = 0 in K £]0; +1[;

which vanishes on @K £ [0; +1[. Moreover, if the following estimate 2 3 n Y 1 lim sup (t + 1) q¤ ¡1 4 sup V i (ri ; t)5 < +1 t!+1

(21)

ri>0 1·i·n i=1

holds then the condition (18) will be satis¯ed for any q > q ¤ . Indeed, it su±ces to remark that 0 < ©i µ 1 for 1 µ i µ n, and Z +1 q¡ 1 (t + 1)¡ q¤ ¡1 dt < +1; for any q > q ¤ : 0

We will show now that the esimate (21) is satis¯ed. First, since lim

n Y

ri!0 1·i·n i=1

V i (ri ; t) = rlim !1 i

n Y

Vi (ri ; t) = 0; 8 t > 0;

1·i·n i=1

then, for any t > 0, there exists 0 < ri¤ (t) < +1, 1 µ i µ n, such that n Y

Vi (ri¤ (t); t)

i=1

= sup

ri>0 1·i·n i=1

Let V (t) = (t + 1)

n Y

1 q¤ ¡ 1

n Y i=1

V i (ri ; t):

Vi (ri¤ (t); t):

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

Using the fact that

n

1 1X = (Ni + s¤i ); q¤ ¡ 1 2 i=1

we can write V (t) =

n Y i=1

(

(t + 1)

s¤i 2

µ

ri¤ (t) t+1

¶¡ Ni2¡2

If we set yi¤ (t) then V (t) = e

n ¡ 4(t+1)

n ½ Y

(t + 1)

s¤i 2

Iºi

µ

ri¤ (t) 2(t + 1)

¶

µ

(ri¤ (t))2 + 1 exp ¡ 4(t + 1)

¶)

:

ri¤ (t) = ; 2(t + 1)

(yi¤ (t))¡

Ni ¡ 2 2

2 ¤ Iºi (yi¤ (t))e¡(t+1)(yi (t))

i=1

Let

75

s¤i

V i (t) = (t + 1) 2 (yi¤ (t))¡

Ni ¡ 2 2

¤

¾

:

2

Iºi (yi¤ (t))e¡(t+1)(yi (t)) :

for 1 µ i µ n. Suppose that there is a sequence (tk )k2N ! +1 such that lim V (tk ) = +1:

tk!+1

Three cases can arise for each i 2 f1; 2; :::; ng: ° Case 1: There exists a subsequence, also denoted by (tk )k2N , such that lim (yi¤ (tk )) = +1:

tk!+1

In this case V i (tk ) ¹

s¤i

const ¢ (tk + 1) 2 (yi¤ (tk ))¡

Ni ¡ 1 2

¤

¤

2

eyi (tk) e¡(tk+1)(yi (tk)) as tk ! +1;

which implies that lim V i (tk ) = 0:

tk!1

° Case 2: There exists a subsequence, also denoted by (tk )k2N , such that lim (yi¤ (tk )) = 0:

tk!+1

In this case V i (tk ) ¹

£

const ¢ (tk +

1)(yi¤ (tk ))2

¤ s2¤i

¤

2

e¡(tk +1)(yi (tk )) as tk ! +1; s¤i

which implies that V i (tk ) is bounded, since the function z 7¡ ! z 2 e¡z is bounded on R+ . ° Case 3: There are two constants A i and Bi such that the sequence (yi¤ (tk ))k2N satis¯es 0 < A i µ yi¤ (tk ) µ Bi < +1:

76

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

In this case, the expression V i (tk ) is clearly bounded. Whence, there is no subsequence of (tk )k2N ! +1 such that lim V i (tk ) = +1;

tk!+1

which implies that there is no sequence (tk )k2N ! +1 such that lim V (tk ) = +1:

tk!+1

This ends the proof.

References [1] C. Bandle, H.A. Levine, On the existence and nonexistence of global solutions of reaction{di®usion equations in sectorial domains, Trans. Amer. Math. Soc., Vol.316 (1989), 595{622. [2] C. Bandle, H.A. Levine, Fujita type results for convective-like reaction-di®usion equations in exterior domains, Z. Angew. Math. Phys., Vol.40 (1989), 665{676. [3] C. Bandle, M. Essen, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., Vol.112 (1990), 319{338. [4] K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., Vol.243 (2000), 85{126. [5] H. Egnell, Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc., Vol.330 (1992), 191{201. [6] N. Igbida, M. Kirane, Blowup for completely coupled Fujita type reaction-di®usion system, Collocuim Mathematicum, Vol.92 (2002), 87{96. [7] R. Labbas, M. Moussaoui, M. Najmi, Singular behavior of the Dirichlet problem in Hlder spaces of the solutions to the Dirichlet problem in a cone, Rend. Ist. Mat. Univ. Trieste, 30 No.1-2 (1998), 155-179. [8] G.G. Laptev, The absence of global positive solutions of systems of semilinear elliptic inequalities in cones, Russian Acad. Sci. Izv. Math., Vol.64 (2000), 108{124. [9] G.G. Laptev, On the absence of solutions to a class of singular semilinear di®erential inequalities, Proc. Steklov Inst. Math., Vol.232 (2001), 223{235. [10] G.G. Laptev, Nonexistence of solutions to semilinear parabolic inequalities in cones, Mat. Sb., Vol.192 (2001), 51{70. [11] G.G. Laptev, Some nonexistence results for higher{order evolution inequalities in cone{like domains, Electron. Res. Announc. Amer. Math. Soc., Vol.7 (2001), 87{93. [12] G.G. Laptev, Nonexistence of solutions for parabolic inequalities in unbounded cone{ like domains via the test function method, J. Evolution Equations. 2002 (in press) [13] G.G. Laptev, Nonexistence results for higher{order evolution partial di®erential inequalities, Proc. Amer. Math. Soc. 2002 (in press). [14] H.A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., Vol.32 (1990), 262{288.

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77

[15] H.A. Levine, P. Meier, A blowup result for the critical exponent in cones, Israel J. Math., Vol.67 (1989), 1{7. [16] H.A. Levine, P. Meier, The value of the critical exponent for reaction-di®usion equations in cones, Arch. Rational Mech. Anal., Vol.109 (1990), 73{80. [17] E. Mitidieri, S.I. Pohozaev, Nonexistence of global positive solutions to quasilinear elliptic inequalities, Dokl. Russ. Acad. Sci., Vol.57 (1998), 250{253. [18] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for a system of quasilinear elliptic equations and inequalities in RN , Dokl. Russ. Acad. Sci., Vol.59 (1999), 1351{1355. [19] E. Mitidieri, S.I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on RN , Proc. Steklov Inst. Math., Vol.227 (1999), 192{222. [20] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on Rn , J. Evolution Equations, Vol.1 (2001), 189{ 220. [21] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degenerate and singular hyperbolic problems on RN , Proc. Steklov Inst. Math., Vol.232 (2001), 240{ 259. [22] E. Mitidieri, S.I. Pohozaev, A priori Estimates and Nonexistence of Solutions to Nonlinear Partial Di®erential Equations and Inequalities, Moscow, Nauka, 2001. (Proc. Steklov Inst. Math., Vol.234). [23] S. Ohta, A. Kaneko, Critical exponent of blowup for semilinear heat equation on a product domain, J. Fac. Sci. Univ. Tokyo. Sect. IA. Math., Vol.40 (1993), 635{650. [24] S.I. Pohozaev, Essential nonlinear capacities induced by di®erential operators, Dokl. Russ. Acad. Sci., Vol.357 (1997), 592{594. [25] S.I. Pohozaev, A. Tesei, Blow-up of nonnegative solutions to quasilinear parabolic inequalities, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., Vol.11 (2000), 99{109. [26] S.I. Pohozaev, A. Tesei, Critical exponents for the absence of solutions for systems of quasilinear parabolic inequalities, Di®er. Uravn., Vol.37 (2001), 521{528. [27] S.I. Pohozaev, L. Veron, Blow-up results for nonlinear hyperbolic inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Vol.29 (2000), 393{420. [28] S.I. Pohozaev, L. Veron, Nonexistence results of solutions of semilinear di®erential inequalities on the Heisenberg group, Manuscripta math., Vol.102 (2000), 85{99. [29] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Nauka, Moscow, 1987 (in Russian). English translation: Walter de Gruyter, Berlin/New York, 1995. [30] D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lect. Notes Math., Vol. 309, Springer, New York, N. Y., 1973. [31] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana U. Mtah. J. 21 (1972), 979-1000. [32] G. N. Watson, A treatise on the Theory of Bessel Functions, Cambridge University Press, London/New York, 1966.

78

A. El Hamidi, G. G. Laptev / Central European Journal of Mathematics 1 (2003) 61{78

[33] Qi Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., Vol.97 (1999), 515{539. [34] Qi Zhang, The quantizing e®ect of potentials on the critial number of reaction{ di®usion equations, J. Di®erential Equations, Vol.170 (2001), 188{214. [35] Qi Zhang, Global lower bound for the heat kernel of ¡ ¢ + Soc., Vol.129 (2001), 1105{1112.

c jxj2 ,

Proc. Amer. Math.

CEJM 1 (2003) 79{85

Optimal cubature formulas in a re° exive Banach space V. L. Vaskevich¤ Sobolev Institute of Mathematics, Siberian Division of Russian Academy of Sciences, pr. Koptyuga 4, 630090, Novosibirsk, Russia

Received 15 July 2002; revised 11 October 2002 Abstract: Sequences of cubature formulas with a joint countable set of nodes are studied. Each cubature formula under consideration has only a ­ nite number of nonzero weights. We call a sequence of such kind a multicubature formula. For a given re®exive Banach space it is shown that there is a unique optimal multicubature formula and the sequence of the norm of optimal error functionals is monotonically decreasing to 0 as the number of the formula nodes tends to in­ nity. c Central European Science Journals. All rights reserved. ® Keywords: optimal cubature formulas, re° exive Banach spaces, best approximation, reproducing mappings, extremal functions of cubature formulas MSC (1991): 65D32, 41A55, 46N05

1

Introduction

¤

Let ­ » Rn be a bounded domain with a su±ciently smooth boundary. The setting is a separable re°exive Banach space X = X (­ ) and a re°exive Banach space Y = X ¤ , dual to X . The members of X are real valued continuous functions with domain ­ . Let X be embedded in the Banach space C (­ ) of functions which are continuous on ­ and let the embedding be linear and bounded. We also assume that for a given ¯nite subset F of ­ there exists a u(x) in X such that the values of u(x) at points of F are prescribed real numbers. It will be true if, for example, every polynomial belongs to X . Furthermore, let X be a strictly normed linear space. This constraint on the norm is easily seen to be equivalent to the geometric condition that the unit ball of X be rotund. E-mail: [email protected]

80

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85

Since X is strictly normed it follows that Y = X ¤ is a smoothly normed re°exive Banach space [1, p. 173]. In particular [1, p. 169{170], 1 ¤;0 (NG (l); m) = lim (N ¤ (l + tm) ¡ t!0 t

N ¤ (l))

¤;0 exists for all m; l 2 Y , kl j Y k = 1, and de¯nes a functional NG (l) in Y ¤ = X . The ¤;0 functional NG (l) is a Gateaux di®erential of the norm N ¤ (¢) at l. Let Y also be a strictly normed linear space. Then there exists a Gateaux di®erential of the norm N (u) = ku j X k at each unit vector u 2 X . Examples of strictly and smoothly normed spaces are Hilbert spaces and the Sobolev spaces W p(m) (­ ); 1 < p < 1, pm > n. Let ¢ = f¢k g1 k=0 be a sequence of ¯nite subsets of ­ . If the union of all ¢ k is dense in (0) ­ , then ¢ = f¢k g1 k=0 is said to be a multigrid in ­ . Let ¢ 0 = fx j j j = 1; 2; : : : ; ¼ (0)g, (k) (k) and for k ¶ 1 ¢k = ¢k¡1 [ fxj j j = 1; 2; : : : ; ¼ (k)g, where xj 2 = ¢k¡1 . ¢k is said to be (k) a k{level of ¢ ; and vectors xj 2 ­ are nodes of ¢ . Given a multigrid ¢ , we introduce (k) (0) the sequence Nodj of subsets of ¢ by putting Nod1 = ; and (k)

Nodj

(k)

(k)

= Nodj¡1 [ fxj¡1 g;

(k)

(k+1)

Nod¾(k)+1 = Nod1

j = 2; 3; : : : ; ¼ (k);

= ¢k :

(k) (k ) If k < k 1 or (k = k1 and j µ j1 ), then ¢k¡1 » Nodj » Nodj1 1 » ¢k1 . (k) (m) (m) (k) The linear hull Lj = span f¯ (x¡ xi ) j 8 xi 2 Nodj g is a closed ¯nite dimensional subspace of Y . We ¯nd it convenient to formulate a general problem as follows. (0)

Problem 1.1. Given an arbitrary nonzero functional l1 2 Y , ¯nd an element of best (0) (k) approximation to l1 from Lj . As we know, every closed convex subset of a re°exive strictly normed Banach space is a Chebyshev set. It means that there exists a unique solution to Problem 1.1. (0) (0) (0) (0) Let l1 2 Y , l1 6= 0. For example, l1 may be the indicator À ­ (x) of ­ . To l1 and (m) (m) (k) every vector fci j xi 2 Nodj g of real numbers, we assign the associated sequence of error functionals by putting (k)

(0)

lj (x) = l1 (x) ¡

X

(m)

xi

(m)

ci

¯ (x ¡

(m)

xi );

(k)

2Nodj

(k) where k = 0; 1; : : : and j = 1; 2; : : : ; ¼ (k). We call the sequence flj g the error multifunctional. The corresponding sequence of cubature formulas is said to be a multicubature formula. (k) (k) (k) (0) (k) (k) Let Y j be a °at parallel to Lj ; Y j = l1 + Lj . Then Y j is an a±ne variety and (k)

dim Y j

= N (k; j) = ¼ (0) + ¼ (1) + ¢ ¢ ¢ + ¼ (k ¡ (k)

(k )

1) + j ¡

1:

If k < k 1 or (k = k1 and j µ j1 ), then Y j » Y j1 1 . For given k and j we use the symbol (k) (0) (k) ¯ j;opt(x) to designate the element of best approximation to l1 from Lj . We denote the

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85

coe±cients of the expansion of ¯ (m) (m) ci;opt = ci;opt (j; k), i.e., ¯

(k) j;opt (x)

(k) j;opt (x)

=

with respect to delta functions ¯ (x ¡ X

(m)

xi (k)

(0)

(m)

ci;opt ¯ (x ¡

(m)

xi

81 (m)

xi

) by

):

(k)

2Nodj

(k)

Let lj;opt = l1 ¡ ¯ j;opt. The corresponding cubature formula is said to be X -optimal on (k) (k) (0) (k) the set Nodj of nodes [2]. The norm klj;opt j Y k equals E(l1 ; Lj ), where E(w; N ) is the distance from w 2 Y to a linear subspace N of Y . Theorem 1.2. Let X be a strictly normed Banach space such that there is a Frechet di®erential of k¢ j X k at each point v 2 X , v 6= 0. Then the sequence of the norm (k) klj;opt j Y k is monotonically decreasing to 0 as k ! 1.

2

Extremal Functions and Reproducing Mappings (0)

(0)

(0)

(0)

Let l1 2 Y and u1 2 X , l1 6= 0, u1 6= 0. If the following equalities hold (0)

(0)

(0)

(0)

kl1 j Y k2 = (l1 ; u1 ) = ku1 j X k2 ; (0)

(0)

(1)

(0)

then u1 is said to be an extremal function for l1 [2] and l1 is said to be a generated (0) extremal function for u1 . By the re°exivity of X and James Theorem (see, e.g., [3]), (0) there exists an extremal function for an arbitrary functional l1 . Since X is a strictly (0) (0) normed space it follows that for a given functional l1 an extremal function u1 2 X (0) is unique. By the same reasons, for a given function u1 2 X there exists a unique (k) generated extremal function. Hereafter we will denote the extremal function for lj;opt by (k) uj;opt . As our next step, we consider the properties of extremal functions. There are proofs of the theorems of this section in [4]. (0)

Theorem 2.1. For given l1 2 Y the extremal function u 2 X is the element of best approximation to zero element of X from (0) (0) V = fv 2 X j (l1 ; v) = kl1 k2 g: (0)

If M is the kernel of l1 and E(w; N ) is the distance from w 2 X to a linear subspace N of X , then (0) E(u; M ) = E(0; V ) = kl1 j Y k = ku j X k: (0) Theorem 2.2. For given u1 2 X the generated extremal function l 2 Y is the element of best approximation to zero element of Y from (0)

(0)

V ¤ = fm 2 Y j (m; u1 ) = ku1 k2 g:

82

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85

(0) If M ¤ = fm 2 Y j (m; u1 ) = 0g and E(l; N ¤ ) is the distance from l 2 Y to a linear subspace N ¤ of Y , then (0)

E(l; M ¤ ) = E(0; V ¤ ) = ku1 j X k = kl j Y k: (0)

(0)

(0)

Let l1 2 Y , u1 2 X , l1 6= 0, and (1) holds. Then we can de¯ne the mapping (0) (0) (0) º : Y ! X by º (l1 ) = u1 . We also assume that º (0) = 0. By the de¯nition, º (l1 ) is (0) the extremal function for l1 2 Y . By Theorem 2.1, º is a single-valued mapping with domain Y . For 8 l 2 Y and 8 ¬ 2 R we have kº (l) j X k2 = (l; º (l)) = kl j Y k2 ;

º (¬ l) = ¬ º (l):

(0) (0) Together with º , we consider a mapping º ¤ : X ! Y , dual to º . Let u1 2 X , u1 6= 0, (0) (0) (0) l1 2 Y , and (1) holds. We assume that º ¤ (u1 ) = l1 and º ¤ (0) = 0. By Theorem 2.2, º ¤ is a single-valued mapping with domain X . For 8 u 2 X and 8 ¬ 2 R we have

kº

¤

(u) j Y k2 = (º

¤

(u); u) = ku j X k2 ;

¤

º

(¬ u) = ¬ º

¤

(u):

Let l 2 Y and u 2 X . By Theorems 2.1 and 2.2, l = º ¤ (u) i® u = º (l). In particular, for 8 l 2 Y l = º ¤ (º (l)) and for 8 u 2 X u = º (º ¤ (u)). Hence º : Y ! X and º ¤ : X ! Y are reciprocal and surjective mappings. If X is a Hilbert space, then º = º ¤ is said to be a reproducing mapping of X [5, p. 23]. We also ¯nd it convenient to use the term \reproducing mapping" in the case of a Banach space. To be more precise, we call º (resp. º ¤ ) the reproducing mapping of the Banach space Y (resp. X ). By assumption, X is a smoothly normed space. In this case let 1 (NG0 (v); w) = lim (N (v + tw) ¡ t!0 t

N (v));

this is de¯ned (by assumption) whenever v 6= 0 and NG0 (v) is a Gateaux di®erential of the norm N (¢) = k¢ j X k at v; NG0 (v) 2 Y . We have (0)

(0)

(0)

(0)

6 0; u 2 X and (l1 ; u) = kl1 k2 . The Theorem 2.3. Let M is the kernel of l1 2 Y ; l1 = (0) function u is extremal for l1 i® (NG0 (u); w) = 0 for 8 w 2 M . Moreover, the extremal (0) function u 2 X for l1 is the solution to the following problem 8 > > 0 > < NG (u) = > > > : (l1(0) ; u) =

1 (0) l ; (0) 1 kl1 k (0) kl1 k2 :

(2)

(0)

Conversely, every solution u 2 X to (2) is the extremal function for l1 . There is a unique solution to (2). (0)

By the hypotheses of Theorem 2.3, the image of l1 the unique solution to (2).

2 Y under the mapping º

is

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85

83

Theorem 2.4. The reproducing mappings º and º ¤ are demicontinuous. If for each ¤;0 nonzero v 2 X (resp. l 2 Y ) there is the Frechet di®erential NG0 (v) (resp. NG (l)) of the norm at v (resp. l), then º ¤ (resp. º ) is continuous. If there is a Frechet di®erential of k¢ j X k at v 2 X , v 6= 0, and a Frechet di®erential of k¢ j Y k at l 2 Y , l 6= 0, then it follows from Theorem 2.4 that º ¤ and º are homeomorphisms of X and Y . (k) Since the a±ne variety Y j of error functionals is an unbounded subset of Y it follows (k) (k) that the image X j of Yj under º is an unbounded subset of X . If k < k 1 or (k = k1 (k) (k ) (k) and j µ j1 ), then Xj » X j1 1 . Let º ¤ be continuous. Since Yj is a closed subset of Y (k) it follows that X j is a closed subset of X . (k)

Theorem 2.5. The norm of the extremal function uj;opt for the optimal error functional (k) (k) lj;opt is less than the norm of an arbitrary element of X j ; (k) (k) uj;opt = arg minfkv j X k j v 2 X j g: (k)

(k)

There is a unique element of Xj with this property. A function u 2 Xj (k) the optimal error functional lj;opt i® u(x(p) s ) = 0

3

is extremal for

(k) 8 x(p) s 2 Nodj :

Convergence of the optimal multicubature formula

Proof 3.1 (of Theorem 1.2). . Let k and j be nonnegative integers such that ¼ (k) ¶ j. (k) (k+1) (k) Since Nodj » Nodj it follows that lj may be considered as the error functional with (k+1) (k+1) (k) n Nodj . Whence and from the nodes in Nodj and zero weights for nodes in Nodj (k+1) (k) de¯nition of best approximation the following inequality holds klj;opt j Y k µ klj;opt j Y k. Consequently, the sequence of norms under consideration is monotone and bounded. We (k) denote the limit of this sequence by d; d = limk!1 klj;opt j Y k, 0 µ d < 1. If d = 0 then Theorem 1.2 is evident. Let d > 0. By the re°exivity of X every ball in X is weakly compact. Hence there is a subse(ks) quence k 1 < ¢ ¢ ¢ < k j < k j+1 < : : : such that lj;opt is weakly convergent to l1 2 Y as (ks) s ! 1. We shall prove that lims!1 klj;opt ¡ l1 j Y k = 0. Given a ’ 2 X we observe that (ks) (ks) j(l1 ; ’)j = j lim (lj;opt j Y k ¢ k’ j X k = dk’ j X k: ; ’)j µ lim klj;opt s!1

s!1

Hence kl1 j Y k µ d. By Mazur’s theorem, for a given ° > 0 there is a convex combination p P

s=1

¬

(ks) s lj;opt ,

¬

s

¶ 0,

p P

¬

s=1

s

= 1, such that kl1 ¡

p P

s=1

¬

(ks) s lj;opt

j Y k µ ° . By the de¯nition of

the optimal error functional it follows from the equality p X

s=1

¬

(ks) s lj;opt (x)

(0)

= l1 (x) ¡

X

(m)

ci

(kp) (m) xi 2Nodj

¯ (x ¡

(m)

xi

)

84

that k

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85 p P

s=1

¬

(ks) s lj;opt

(k )

p j Y k ¶ klj;opt j Y k. Let ° < d=2. Then, by the triangle inequality,

¯ p ¯ X (ks) kl1 j Y k ¶ ¯¯k jYk¡ ¬ s lj;opt

¶

s=1 (kp) klj;opt

jYk¡

kl1 ¡

° ¶d¡

p X

¬

(ks) s lj;opt

s=1

° > 0:

¯ ¯ j Y k¯¯

Taking the limit as ° ! 0 we arrive at the following inequality kl1 j Y k ¶ d. Finally, kl1 j Y k = d. (ks) Let ms (x) = (ks1) lj;opt (x) and let u1 2 X be the extremal function for l1 . By klj;optjY k (ks) lj;opt , the

1 the de¯nition of sequence fms g1 s=1 is weakly convergent to d l1 (x) as s ! 1. Furthermore, kms j Y k = 1 and

1 1 lim (m s ; u1 ) = 2 (l1 ; u1 ) = 1: d d

s!1

By the hypothesis, there is a Frechet di®erential of the norm k¢ j X k at d1 u1 (x) 2 X . Whence and from the Shmulian criterion [6, p. 147] it follows that fm s g1 s=1 converges to m1 2 Y in the norm of Y . The limit functional m1 must be equal to the weak limit (ks) 1 1 l of fms g1 s=1 . Hence flj;opt gs=1 converges to l1 in the norm of Y . d 1 By Theorem 2.4 the reproducing mapping º : Y ! X is demicontinuous. Hence (ks) 1 gs=1 under º is a weakly convergent the image of the strongly convergent sequence flj;opt (ks) (ks) (ks) ¤ sequence uj;opt = º (lj;opt ) in X = Y and the weak limit of uj;opt is u1 = º (l1 ). In (m) particular, for a given xi 2 ¢ the following equalities hold (m)

u1 (xi

) = (¯ (x ¡

(m)

xi

= lim (¯ (x ¡ s!1

(k )

); u1 (x)) (m)

xi

(m)

s = lim uj;opt (xi

s!1

(k )

(k )

s ); uj;opt (x))

):

(m)

s By Theorem 2.5, lims!1 uj;opt (xi ) = 0. Since the continuous function u1 equals 0 at all nodes of the initial multigrid ¢ , which is dense in ­ , it follows that u1 equals 0 everywhere in ­ . Hence d = ku1 j X k = 0, a contradiction.

References [1] Holmes, R., Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics 24, Springer{Verlag. 1975. [2] Sobolev, S.L. and Vaskevich, V.L. The Theory of Cubature Formulas. Kluwer Academic Publishers, Dordrecht, 1997. [3] Kutateladze S.S., Fundamentals of Functional Analysis., Kluwer texts in the Math. Sciences: Volume 12, Kluwer Academic Publishers, Dordrecht, 1996, 229 pp. [4] Vaskevich, V.L., Best approximation and hierarchical bases. Sel»cuk Journal of Applied Mathematics. 2001. V. 2, No. 1. P. 83{106. The full text version of the article is available via http://www5.in.tum.de/selcuk/sjam012207.html.

V.L. Vaskevich / Central European Journal of Mathematics 1 (2003) 79{85

85

[5] Bezhaev, A.Yu. and Vasilenko, V.A. Variational Spline Theory. Bull. of Novosibirsk Computing Center. Series: Numerical Analysis. Special Issue: 3. 1993. [6] Holmes, R., A Course on Optimization and Best Approximation., Lecture Notes in Math. 257, Springer{Verlag. 1972.

CEJM 1 (2003) 86{96

On Asymptotic Independence Of The Exit Moment And Position From A Small Domain For Di® usion Processes Vitalii A. Gasanenko¤ Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivska 3, 01601, Kiev-4, Ukraine

Received 9 August 2002; revised 1 October 2002 Abstract: If ¹ (t) is the solution of homogeneous SDE in Rm , and T² is the ­ rst exit moment of the process from a small domain D² , then the total expansion for the following functional showing independence of the exit time and exit place is E exp(¡ ¶ T² )f (

¹ (T² ) )¡ °

E exp(¡ ¶ T² ) E f (

¹ (T² ) ); ° & 0; ¶ > 0: °

c Central European Science Journals. All rights reserved. ® Keywords: Di® usion process, exit moment and exit place, elliptic boundary problems MSC (1991): 60 J 50

Let’s consider the solution for an m - dimension stochastic di®erential equation d¹ (t) = a(¹ (t))dt +

m X

bk (¹ (t))dwk (t);

¹ (0) = 0:

(1)

k=1

Here functions bk (¢); a(¢) : R m ! Rm are smooth, and W (t) = (wk (t); k = 1; m) is the standard m - dimension Wiener process. Let’s introduce the following designations and agreements: D is some bounded connected domain in R m and f0g 2 D; S , or @D , is the boundary of D; and D represents the closure of D . Furthermore, the domain D ² is de¯ned as the ° contraction of the domain D : D² = ° D. The ¯rst moment of hitting with a process ¹ (t) on the boundary S² is denoted by T² = infft : ¹ (t) 2 = D² g: ¤

E-mail: [email protected]

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

87

In this article we investigate the total expansion in powers of ° of the following functional, which is indepedent of the moment and place of exit from D² , I² (¶ ; f ) = E exp(¡ ¶ T ² )f (

¹ (T² ) )¡ °

E exp(¡ ¶ T² ) E f (

¹ (T ² ) ); ° & 0; ¶ > 0: °

Here f : R m ! R there is a smooth function. The ¯rst principal members of the expansion of I² (¶ ; f ) were obtained in [1] under the condition that ¹ (t) is the Wiener process in an m - dimension Riemann manifold and S² is a small geodesic sphere with radius ° . We shall obtain the total expansion through the total expansions of solutions of relevant elliptic boundary problems. The constructions of algorithms for the total expansion of I² (¶ ; f ) is based on the approach from [2]. In this paper one obtains the total expansion of the following functional H² (¶ ; f ) = E[exp(¡ ¶

T² ¹ (T² ) )f ( )]; 2 ° °

¶ ¶ 0:

Now, we shall introduce some de¯nitions and results from the theory of the elliptic equations [3]. De¯nitions: ­ - bounded connected domain in an m - dimensional Euclidean space Em ; S - boundary of ­ , ­ - closure of ­ ; oscfu(x); ­ g- °uctuation u(x) on ­ : oscfu(x); ­ g = ess sup u(x) ¡

ess inf u(x) ­

­

K½ is an arbitrary open ball with radius » in space Em ; ­ ½ := K ½ \ ­ : Let’s say, a function u(x) satis¯es the HÄ older condition with parameter ¬ 2 (0; 1) , ( ) and ¯niteness HÄ older constant hui­ in domain ­ , if the following equality holds sup »

( ) oscfu; ­ ½ g = hui­ ;

¡

where the sup is calculated over all ­ ½ ; » µ » 0 ; » 0 is de¯ned below. C (­¹ ) is the Banach space of all continuous functions u(x), with x 2 ­ , and the ( ) ¯niteness norm hui­ : The norm in C (­¹ ) is de¯ned as follows juj(­

)

( )

= sup juj + hui­ : ­

Here D k is the symbol of the derivative u(x) with respect to x of order k, C l+ (­ ) is the Banach space of the functions continuous in ­ , which have continuous derivatives in ­ of order l and the following value is ¯niteness juj(l+ ­

)

=

l X X

k=0 (k)

max jD k (x))j + ­

X (l)

hD l ui(­ ):

88

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

Here the symbol

P

denotes the summation over all derivatives of order k.

(k)

It is possible to consider elements from C l+ (­ ) as functions which are continuous and l times continuously di®erentiable in ­ . For this it is necessary to complete a de¯nition for u(x) and its derivatives on a boundary S by continuity. Let’s introduce the classi¯cation of boundaries. We shall suppose that a boundary S of domain ­ satis¯es the folowing condition: There are two positive numbers » 0 and ³ 0 , such that for any ball K ½ with the center on S with radius » µ » 0 , and for the intersection ­ ½ of a ball K½ with ­ , we have the following inequality: ­ ½ µ (1 ¡ ³ 0 )K ½ . Let’s designate by CR;L the cylinder fx :

m¡1 P i=1

x2i < R2 ; ¡ 2LR < xm < 2RLg, and

we shall designate the point x = 0 to be the center of the cylinder. The domain ­ is called strictly Lipschitzian, if for 8x0 2 S it is possible to introduce the coordinates P yk = ckl ((xl ¡ x0l ), where (ckl ) is an orthogonal matrix, with k; l = 1; m, so that the l

intersection of S by the cylinder C¹R;L (cylinder is related to the coordinates y) is de¯ned by 0 0 0 the equation ym = !(ym ), where ym := (y1 ; : : : ; ym¡1 ). The function !(ym ) is Lipschitzian 0 for jym j µ R with a Lipschitz constant not exceeding L, and 0 0 j µ R; !(ym ­¹ \ C¹R;L = fy : jym ) µ ym µ 2LRg:

The numbers R and L for the given domain ­ are ¯xed. Let x0 = (x01 ; : : : ; x0m ) be a point of S, in which the surface S has a tangent plane. Let’s name (y1 ; : : : ; ym) as the local system of coordinates with the beginning of a point P x0 ; y and x are connected by the equality yi = aik (xk ¡ x0k ); i = 1; : : : m, where (aik ) k

is an orthogonal matrix, and the axis ym is directed on the normal to S at an external point x0 with respect to ­ . The domain ­ (surface S) class C l+ ; l ¶ 1, if the domain ­ is strictly Lipschizian, where y, participating in this de¯nition, are local coordinates, and the functions 0 0 j µ R). ym = !(ym ), which de¯ne the equation of surface S, belong to space C l+ (jym Let us consider the equation Au :=

X

ai;j (x)uxixj +

1·i;j·m

X

ai (x)uxi + a(x)u = g(x):

(2)

i

The coe±cients of this equation and free member g(x) are de¯ned in the bounded domain ­ and belong to space C l¡2+ (­ ); l ¶ 2; ¬ 2 (0; 1): Let’s assume, also, that condition aij = aji is satis¯ed and equation (2) is elliptic in ­ : X i;j

ai;j (x)² i ²

j

¶¸

m X

² i2 ;

¸ = const > 0:

(3)

i=1

for any ¯xed vector (² 1 ; ¢ ¢ ¢ ; ² m ) 2 Rm . For function u(x), which satis¯es the equation (2) and condition on the boundary S: ujS = ’(s);

(4)

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

89

the following existence theorem [3](page.157) holds. Theorem 1. If the coe±cients of A belonging to the space C (­ ) satisfy the inequalities of (3) and a(x) µ 0 and S belongs to C 2+ , then the problem (2), (4) has a unique solution in C 2+ (­ ) for all g(x) in C (­ ) and all ’(x) in C 2+ (­ ): De¯ne the following di®erential operator in space C 2 (D). Lu(x) =

m X

ai (x)

i=1

m m @u(x) 1 X @ 2 u(x) X + bki (x)bkj (x) @xi 2 i;j=1 @xi @xj k=1

Here, coe±cients are taken from (1). Now, we shall prove the total expansion for the following functional, which enters in the de¯nition for the functional I² (¶ ; f ): ©² (¶ ; f ) = E exp(¡ ¶ T ² )f (

¹ (T² ) ) °

It’s well-known [4] that ©² (¶ ; f ) = ¿ ² (¶ ; 0) where the function ¿ ² (¶ ; y) is solution to the following boundary problem ¶ ¿ ² (¶ ; y) ¡

y 2 int D² y ¿ ² (¶ ; y) = f ( ); y 2 @D ² : °

L¿ ² (¶ ; y) = 0;

After a change of variables z = y=° , we obtain that the function · ² (¶ ; z) := ¿ ² (¶ ; ° z) is solution to the following problem:

¶ · ² (¶ ; z) ¡

L² · ² (¶ ; z) = 0;

z 2 int D

· ² (¶ ; z) = f (z); Here L² u(z) := °

¡1

m X i=1

ai (° z)

z 2 S:

m m @u(z) ° ¡2 X @ 2 u(z) X + bki (° z)bkj (° z): @zi 2 i;j=1 @zi @zj k=1

The expansion takes the following form: · ² (¶ ; z) =

X

° k·

k (¶

; z):

k¸0

It means that we shall de¯ne the approximamation of · ² (¶ ; z) as the partial sums of this series. Initially, we assume that the functions ai (x) and bki (x), 1 µ k; i µ m, are real analytical functions in the neighbourhood of zero and continuous at zero. Now the following Taylor-series expansion in zero follows: ai (° z) =

X

ail (z)° l ;

l¸0

bkj (° z) =

X l¸0

bkjl (z)° l :

(5)

90

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

From de¯nition we shall obtain X 1 @ l ai (0) jm z j 1 ¢ ¢ ¢ ¢ ¢ zm l! j1+¢¢¢+jm=l @ j1 z1 ¢ ¢ ¢ ¢ ¢ @ jm zm 1

ail (z) =

X 1 @ l bkj (0) z j1 ¢ ¢ ¢ ¢ ¢ zjm l! j1+¢¢¢+jm=l @1j1 ¢ ¢ ¢ ¢ ¢ @ jmzm

bkjl = Put

Akijl (z) :=

l X

bkjs (z)bkil¡s (z)

s=0

L0 u(z) :=

m 1 X @ 2 u(z) X bkj (0)bki (0) 2 1·i;j·m @zi @zi k=1

Thus, if we have the solutions (5) for ai (z) and bki (z) , then operator L² has the following form: L² = ° ¡2 L0 + ° ¡1 L1 + ¢ ¢ ¢ + ° n Ln+2 + ¢ ¢ ¢ ; here Ln =

m X

ain¡1 (z)

i=1

m @ 1 X @2 X + Akijn(z); @zi 2 1·i;j·m @zi @zj k=1

ai;¡1 = 0:

If we expand in series · ² (¶ ; z) by series of the operators, which represents L² and the grouping of coe±cients with identical degrees ° , under the condition that the sum of it should equal zero, we shall obtain the equations for the functions · k (¶ ; z). Since the boundary function f (z) does not depend on ° , then only for · 0 (¶ ; z) do we have a nonzero boundary condition on S : · 0 (¶ ; z) = f (z); · k (¶ ; z) = 0; z 2 S; k ¶ 1. Thus, formally for de¯nition of functions · k (¶ ; z); k ¶ 0, we have a system of elliptic problems k = 0 : L0 · ·

0 (¶

; z) = f (z); z 2 S

0 (¶

k = 1 : L0 · ·

1 (¶

; z) = ¡ L1 ·

0 (¶

; z);

z 2 int D

; z) = 0; z 2 S

1 (¶

k ¶ 2 : L0 ·

; z) = 0; z 2 int D

k (¶

; z) = ¡

k X

Ls ·

(6) k¡s (¶

; z) + ¶ ·

k¡2 (¶

; z); z 2 int D

s=0

· Put ·

²;n (¶

; z) =

; z) = 0; z 2 S:

k (¶ n P

k=0

·

k (¶

; z)° k .

Theorem 2. If the following conditions are carried out 10 The functions ai (x) and bik (x), 1 µ k; i µ m, have n + 2 derivatives in a neighbourhood of zero; P 20 The functions ¼ ij (x) = bki (x)bkj (x) satisfy condition (3); 30 f (z) 2 C 2+ (D);

k

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

40 50

91

@D belongs to the class C 2+ ; The functions · k (¶ ; z) are the solutions of system (6) for k µ n; for the function r²;n;¹ (¶ ; z) = · ² (¶ ; z) ¡ ·

²;n (¶

; z)

the following inequality holds jr²;n;¹(¶ ; z) µ °

sup

¹ ¸>0; z2D

n+1

K 1;

here K1 is a bounded constant. Proof. The condition 10 guarantees the following representations ai (° z) =

n¡1 X

ail (z)° l + ain;² (z)°

n

l=0

bkj (° z) =

n X

bkjl (z)° l + bkjn+1;² (z)°

n+1

;

i; k; j = 1; m;

l=0

where the functions ain;² (z) and bkjn+1;² (z) have the following form: ain;² (z) =

X 1 @ n ai (³ ² (z)) j1 jm j1 jm z1 ¢ ¢ ¢ zm n! j1+¢¢¢+jm=n @z1 ¢ ¢ ¢ @zm

³ ² (z) = (³ 1 ° z1 ; ¢ ¢ ¢ ; ³ m ° zm ); 0 µ ³ i µ 1; i = 1; m X 1 @ n+1 bkj (­ ² (z)) j1 jm bkjn+1;² (z) = jm z1 ¢ ¢ ¢ zm (n + 1)! j1+¢¢¢+jm=n+1 @z1j1 ¢ ¢ ¢ @zm ­ ² (z) = (­ 1 ° z1 ; ¢ ¢ ¢ ; ­ m ° zm );

0 µ ­ i µ 1;

i = 1; m:

Thus the operator L² has the following decomposition L² = ° where Ln+1 (° ; z) =

¡2

L0 + °

m X i=1

¡1

ain;² (z)

L1 + L0 + ¢ ¢ ¢ + °

n¡1

Ln+1 (° ; z);

m X @ 1X @2 + A kijn+1;² (z) : @zi 2 k=1 1·i;j·m @zi @zj

Further from the conditions of theorem 2 and theorem 1, for the ¯rst task from system (6) there is a unique solution · 0 (¶ ; z). Notice, that this solution belongs to the space C 2+ (D). From the last task, condition 10 , and the algorithm of the construction of the free member of the second task from (6), we make a conclusion that the free member of this task belongs to space C (D). Now again using the theorem 1, we shall obtain the unique solution · 1 (¶ ; z) of the second task from system (6), such that u1 2 C 2+ (D). Similarly, we can consistently prove the existence of the unique solutions u2 ; : : : ; un from the relevant tasks from system (6). Thus the function · ²;n (¶ ; z) is correctly de¯ned. Now from the de¯nition of the function · ² (¶ ; z) , the construction of the functions · k (¶ ; z); k = 0; 1; ¢ ¢ ¢ ; n, and the decomposition of the operator L² , it follows that the

92

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

function of an error of approximation r²;n;¹ (¶ ; z) is the solution of the following boundary problem 2

° L² r²;n;¹ ¡

2

° ¶ r²;n;¹

m X m m 1 X @ 2 r²;n;¹ X @r²;n;¹ ¡ = bki (° z)bkj (° z) + ° ai (° z) 2 i;j=1 k=1 @zi @zj @zi i=1

=

Ã

° 2 (°

¡2

L0 + ¢ ¢ ¢ + °

2n+2 X

n¡2

Ln + °

n¡1

Ln+1 ) + 1

l X m X m X

2 ~bkjs (z)~bkil¡s (z) @ )A r²;n;¹ ¡ + ° l( @zi @zj s=0 k=1 i;j=1 l=n+2 m X n X

n+1

= °

+ + := °

ail (z)

i=1 l=0 2n m X l ³X X l

°

l=n+1 3n+2 X

@ · @zi

n¡l (¶

a~il (z)

i=1 s=0

°

l

l=n+1 k=1 i;j=1 s=0 n+1 K²;n (¶ ; z)

here for 1 µ i; j µ m

8 > > <

~ail (z) = > >

´ @ · ~l¡s (¶ ; z) + @zi

8 > >
~bkjl (z) = > >

2 ~bkjr (z)~bkis¡r (z)) @ · ~l¡s (¶ ; z) @zi @zj r=0

if

kjl (z);

: 0;

¶

lµn

ail (z); if

: 0;

° 2 ¶ r²;n;¹

; z) +

µX m X m X l X s

(

° 2 ¶ r²;n;¹

l > n:

if

l µn+1

if

l > n + 1:

and the functions · ~ l (¶ ; z) are de¯ned analogously in the following way: 8 > > <

· ~l (¶ ; z) = > >

· l (z); if

: 0;

if

lµn l > n:

¹ at every ° and It is clear, that at a ¯xed n, the function K²;n (¶ ; z) 2 C0; (B) sup

¹ ²¸0;¸¸0;z2B

From the construction of ·

k (¶

jK ²;n (¶ ; z)j µ K < 1:

; z) the boundary condition for r²;n;¹(¶ ; z) follows:

r²;n;¹(¶ ; z) = 0;

z 2 S:

Following results of the monography [3](page 160), for the function r²;n;¹ we shall obtain the following apriori estimation max jr²;n;¹(¶ ; z)j µ m max e¯jz1j +

¹ D;¸¸0

¹ z2D

max (jr²;n;¹(¶ ; z)j + me¯jz1j ):

z2@D;¸¸0

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

Here m = max f°

n+1

D;¸¸0

jK ² (¶ ; z)e¯jz1j g;

following inequality a^11 ­ 2 + a^1 ­ ¡ a^11 =

min

z2D;²·²0

­ is a positive number which satis¯es the

° 2 ¶ ¶ 1 and under the ¯xed small ° m X

93

b2k1 (° z); ^a1 =

k=1

0

>0

min a1 (° z):

z2D;²·²0

From last the estimation it follows that the max jr²;n;¹ (¶ ; z)j µ °

n+1

¹ D

K1;

K 1 < 1:

and the theorem is proved. Thus ©² (¶ ; f ) =

n X

° k·

k (¶

; 0) + O(°

n+1

):

k=0

In a similar way we shall obtain the expansions of other functional, which de¯ne I² (¶ ; f ) , as corollaries of theorem 2. For the functional ¹ (T² ) ©² (0; f ) = E f ( ): ° the algorithm of the total expansion is de¯ned by expansion of the solution of the following boundary problem L¸ ² (y) = 0; y 2 int D² ;

y ¸ ² (y) = f ( ); y 2 @D² : °

Thus, we have ©² (0; f ) = ¸ ² (0): Further, by the change of variables z = y² we shall remove parameter ° from the boundary conditions. Now the function v² (z) := ¸ (° z) is a solution to the following boundary problem L² v² (z) = 0; z 2 int D; v² (z) = f (z); z 2 S: We expand v² (z) as ° ! 0 , v² (z) =

X

uk (z)° k :

k¸0

Now under conditions of theorem 1, and operating similarly to the proof of this theorem, it is possible to prove the existence of such functions uk (z); k µ n , which are unique solutions to the following system of boundary problems k = 0 : L0 u0 (z) = 0; z 2 int D; uk (z) = f (z); z 2 S: k ¶ 1 : L0 uk (z) = ¡

k X

Ls uk¡s (z); z 2 int D; uk (z) = 0; z 2 S:

k

is the approximation to v² (z). Thus, the following

(7)

s=1

And the partial sum theorem takes place

n P

k=0

uk (z)°

94

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

Corollary 1. If the conditions 10 ¡ 40 of the theorem 2 are carried out and the functions uk (z) are solutions of system (7) for k µ n , then n X

r²;n;v (z) = v² (z) ¡

uk (z)°

k

k=0

satis¯es the following inequality sup jr²;n;v (z)j µ °

n+1

¹ z2 B

K 2;

here K2 is the ¯niteness constant. Thus, Corollary 1 guarantees the following representation ©² (0; f ) =

n X

uk (0)°

k

+ O(°

n+1

):

k=0

Analogously to the proved theorems, we show that the following functional ©² (¶ ; 1) = E exp(¡ ¶ T² ) is de¯ned by equality ©(¶ ; 1) = h² (¶ ; 0) where the function h² (¶ ; z) is the solution to the following boundary problem

¶ h² (¶ ; z) ¡

L² h² (¶ ; z) = 0; z 2 int D h² (¶ ; z) = 1;

z 2 S:

We show the approximation to h² (¶ ; z) to be the following partial sum h²;n (¶ ; z) =

n X

° k hk (¶ ; z):

k=0

Here the functions hk (¶ ; z) are the solutions to the following boundary problems k = 0 : L0 h0 (¶ ; z) = 0; z 2 int D h0 (¶ ; z) = 1; z 2 S k = 1 : L0 h1 (¶ ; z) = ¡ L1 h0 ; z 2 int D

(8)

h1 (¶ ; z) = 0; z 2 S k ¶ 2 : L0 hk (¶ ; z) = ¡

k X

Ls hk¡s (¶ ; z) + ¶ hk¡2 (¶ ; z); z 2 int D

s=0

hk (¶ ; z) = 0; z 2 S: Corollary 2. If the conditions 10 ¡ 40 are carried out and the functions hk (¶ ; z) are solutions to the system of problems (8) for k µ n , then for the function r²;n;h (¶ ; z) = h² (¶ ; z) ¡

h²;n (¶ ; z):

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

95

the following estimation holds sup

¹ ¸>0; z2 D

jr²;n;h(¶ ; z)j µ °

n+1

K3 :

where K3 is a bounded constant. Thus, we have the following asymptotic expansion ©² (¶ ; 1) =

n X

° k hk (¶ ; 0) + O(°

n+1

):

k=0

Finally, the theorem of the expansion of I² (¶ ; f ) follows from theorems 1{4. Theorem 3. If the conditions 10 ¡ 40 are executed, then the following expansion takes place à ! n X

I² (¶ ; f ) =

°

k

·

; 0) ¡

k (¶

uk (0) ¡

k X

hs (¶ ; 0)uk¡s(0) + O(°

n+1

):

s=2

k=2

Proof. If we analyze the solutions of the ¯rst problems from systems (6), (7), (8), which de¯ne the associated functions · k (¶ ; z); uk (z); hk (¶ ; z), then it is easy to establish the following statements: a) The functions · 0 and · 1 do not depend on ¶ and coincide with the associated functions u0 (z) and u1 (z). This is the result of the unique solutions to the identical boundary problems. Thus ·

0 (¶

; z) = ·

0 (z)

= u0 (z);

·

1 (¶

; z) = ·

1 (z)

= u1 (z):

¹ is equal to a unit and the function h1 (¶ ; z) at b) The function h0 (¶ ; z) at z 2 D ¹ is equal to zero. Thus z2D h0 (¶ ; z) ² 1;

h1 (¶ ; z) ² 0;

¹ z 2 D:

¶ > 0;

Now from theorem 2, corollaries 1{2 and statements a) and b) the chain of equalities follows: I² (¶ ; f ) = ©² (¶ ; f ) ¡ =·

; 0) ¡

n² (¶

=

n X

k=0

=·

·

k (¶

n X

°

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k=2

+O(° =

n X

k=2

°

k

Ã

n+1

Ã

n X

uk (0) + O(°

; 0) ¡

k X

·

k (¶

; 0) ¡

1 (¶

; 0) ¡

h0 (¶ ; 0)uk (0) ¡

n+1

)

h0 (¶ ; 0)u1 (0) ¡

h1 (¶ ; 0)uk¡1(0) ¡

h1 (¶ ; 0)u0 (0)) + k X

; 0) ¡

uk (0) ¡

Thus the theorem is proved.

k X

s=2

!

hs (¶ ; 0)uk¡s (0) + O(°

n+1

!

hs (¶ ; 0)uk¡s(0) +

s=2

) k (¶

)

hs (¶ ; 0)uk¡s (0) + O(°

h0 (¶ ; 0)u0 (0) + ° (· ·

n+1

!

s=0

; 0) ¡

0 (¶

+

hn;² (¶ ; 0)

k=0

Ã

k

°

©² (¶ ; 1)©² (0; f )

):

96

Vitalii A. Gasanenko / Central European Journal of Mathematics 1 (2003) 86{96

Acknowledgment This research was supported (in part) by the Ministry of Education and Science of Ukraine, project No 01.07/103.

References [1] M.Liao, Hitting distributions of small geodesic spheres, Ann. Probab., 16 (1988), 1039 - 1050 [2] V.A. Gasanenko, A total expansion functional of exit time from a small ball for di®usion process, International Journal Istatistik, Vol. 3, Issue 3 (2000), 83-91 [3] O.A. Ladyjenskai and N.N. Ural’ceva, Linear and quasilinear equation of elliptic type, \Nauka", Moscow (1973) [4] I.I.Gikhman and A.V. Skorokhod, Introduction in the theory of random, \Nauka", Moscow (1977)

CEJM 1 (2003) 97{107

Representation-¯nite triangular algebras form an open scheme StanisÃlaw Kasjan¤ Faculty of Mathematics and Computer Science Nicholas Copernicus University Chopina 12/18, 87-100 Toru¶ n, Poland

Received 13 November 2002; revised 20 December 2002 Abstract: Let V be a valuation ring in an algebraically closed ­ eld K with the residue ­ eld R. Assume that A is a V -order such that the R-algebra A obtained from A by reduction modulo the radical of V is triangular and representation-­ nite. Then the K-algebra KA ¹= A «V K is again triangular and representation-­ nite. It follows by the van den Dries’s test that triangular representation-­ nite algebras form an open scheme. c Central European Science Journals. All rights reserved. ® Keywords: representation-¯nite algebra, valuation ring, ¯rst-order formula MSC (2000): 16G60, 16G30, 03C60.

1

Introduction

¤

It is proved by Gabriel in [13] that the representation-¯nite algebras induce a Zariski-open subset in the variety of all associative algebras of ¯xed dimension over a ¯xed algebraically closed ¯eld. Moreover, by [16, Corollary 12.57], they form a constructible Z-scheme. The question if they form an open Z-scheme (see Theorem 1.2 for an explanation of this concept) is still open. A positive answer to an analogous question regarding the class of representation-directed algebras is given in [18]. The present paper is devoted to the class of triangular algebras which are representation-¯nite. Before the formulation of the main result we introduce the basic notation. Let : K ¡ !G [ f1g be a valuation of an algebraically closed ¯eld K with values in an ordered group G and V the corresponding valuation ring with the maximal ideal . Recall that, for every nonzero x 2 K , either x 2 V or x¡1 2 V . Every ¯nitely generated E-mail: [email protected]

98

S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

ideal in V is principal. Every torsion-free V -module is °at and every ¯nitely generated torsion-free V -module is free. The reader is referred to [4, Chapter III] for basic facts about valuation rings. Denote by R the residue ¯eld V = of V . Then R is algebraically closed since K is so. The value group is divisible. Assume that A is a V -order, that is, A is a V -algebra (associative, with a unit) which is ¯nitely generated and free as a V -module. Let d be the V -rank of A. We denote by A(K) and A the algebras A «V K and A «V R, respectively. There is a canonical ring homomorphism A¡ !A with the kernel A. The value of this homomorphism on an element a 2 A is denoted by a. For every V -module X , we set X (K) = X «V K and X = X «V R. Let rk V (X ) denote the V -rank of X . A right A-module X is a right lattice over A if X is ¯nitely generated and free as a V -module. Every A-lattice structure on a V -module X induces an A (K )-module structure on X (K ) and an A-module structure on X . The right A-lattices form a full subcategory latt(A) in the category mod(A) of ¯nitely generated right A-modules. Let K be a ¯eld. A ¯nite dimensional K -algebra B is said to be representation-¯nite is there is only ¯nitely many isomorphism classes of indecomposable (¯nite dimensional) B-modules. We say that B is triangular if there is no cycle in the category of projective B-modules, that is, there is no sequence of non-zero non-isomorphisms P0 ¡ ! P1 ¡ ! ::: ¡ ! P m with m ¶ 1, P0 ¹= P m and P i projective indecomposable for i = 0; :::; m. It is well known that this condition is equivalent to the fact that the ordinary quiver of the basic algebra associated to B (or the reduced form of B, according to the terminology in [2]) is directed (see [1], [2]). The reader is referred to [1], [2] for a representation theory terminology and to [9] for a background on lattices over classical orders. Our main results are the following two theorems. Theorem 1.1. Let A be a V -order. Assume that the R-algebra A is triangular and representation-¯nite. Then the K -algebra A (K ) is also triangular and representation-¯nite. Theorem 1.2. Representation-¯nite triangular algebras of ¯xed dimension d over algebraically closed ¯elds form and open Z-scheme, that is, there exist polynomials G1 ; :::; G m 2 Z[Xijk : 1 µ i; j; k µ d] such that if L is an algebraically closed ¯eld and ® = (® ijk )1·i;j;k·d is a system of structure constants of a d-dimensional L-algebra B then B is representation-¯nite and triangular if and only if Gi (® ) 6= 0 for some i = 1; :::; m. Let us formulate now a technical fact we need in the proof of Theorem 1.2. The reader is referred to [16] for the model-theoretical notions appearing in the following lemma. Lemma 1.3. The class of triangular algebras of a ¯xed dimension d over algebraically closed ¯elds is ¯nitely axiomatizable as a subclass of the class of all d-dimensional algebras over algebraically closed ¯elds.

S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

99

Proof. A K -algebra B is triangular if and only if for any complete system e1 ; :::; en of primitive pairwise orthogonal idempotents, a sequence i0 ; :::; im = i0 , 1 µ m µ n of elements of f1; :::; ng and a sequence a1 ; :::; am such that al 2 eil Beil¡1 for l = 1; :::; m one of the conditions hold: (1) al = 0 for some l = 1; :::; m, (2) for some l = 1; :::; m there exist bl 2 eil¡1 Beil such that al bl and bl al are invertible in the local algebra eil Beil and eil¡1 Beil¡1 respectively. It is easy to express the above property in a suitable ¯rst order language. The proof of Theorem 1.1 is given in Section 4. Theorem 1.2 is a direct consequence of Theorem 1.1, ¯nite axiomatizability of the class of representation-¯nite algebras [15], [16, Theorem 12.54], Lemma 1.3 and van den Dries’s test [16, Theorem 12.7], [23]. The arguments are identical as those used in [18], hence we will not repeat them here and we concentrate on the proof of Theorem 1.1. In Section 2 we reduce the problem to the case of V -orders de¯ned by quivers with relations and we ¯nish the proof in Section 4. The idea of the proof is the following: show that A is described by the same combinatorial data as a suitable degeneration of A(K) . The results of this paper were presented at Algebra Conference, Venezia 2002. The author thanks to Daniel Simson for reading the preliminary version of the article and useful remarks.

2

Reduction to the bound quiver algebra case

Proposition 2.1. Assume that K ³ L is an extension of algebraically closed ¯elds and B is a ¯nite dimensional K -algebra. Denote by B (L) the L-algebra B «K L. The algebra B is representation-¯nite (resp. triangular) if and only if B (L) is representation-¯nite (resp. triangular). Proof. The assertion about representation-¯niteness follows from [15, Theorem 3.3], [16, Corollary 12.39]. Since K is algebraically closed, for any indecomposable ¯nite dimensional B-module M the induced B (L) -module M «K L is indecomposable. Hence the map P 7! P «K L induces a bijection between the sets of the isomorphism classes of indecomposable projective B-modules and indecomposable projective B (L) -modules. Now the remaining assertion follows easily. Recall that a valuation of a ¯eld K (or the corresponding valuation ring) is maximally complete if cannot be extended to a ¯eld extension L of K with the same value group and the residue ¯eld. If A is an order over a maximally complete valuation ring then idempotents can be lifted modulo the Jacobson radical rad(A) of A and modulo A, [16, Theorem 12.28]. e ; e) of (K; ) which is maxiProposition 2.2. There exists a valued ¯eld extension (K e induces an isomorphism of the residue ¯elds. mally complete and the embedding K ,! K e The ¯eld K is algebraically closed.

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S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

Proof. Follows by applying [21, Corollary 6, Propositon 6]. e ; e) is a maximally complete valued ¯eld extension of (K; ). Let Ve Assume that (K be the valuation ring corresponding to the valuation e and e - its maximal ideal. Denote e e Ae ¹= A as R-algebras. In what follows given by Ae the Ve -order A «V Ve . Observe that A= e e ¹= B (K ) «K K e. a V -order B we set B (K) = B «V K Let Q = (Q0 ; Q1 ) be a ¯nite quiver with the set of vertices (resp. arrows) Q0 (resp. Q1 ). Given an arrow ¬ 2 Q1 let s(¬ ) and t(¬ ) denote the beginning and the end of ¬ , respectively. Recall that the quiver Q is called directed if it has no oriented cycle. If T is a commutative ring then we denote by T Qk the two-sided ideal generated by all paths of length at least k in the path T -algebra T Q of Q, see [2, Chapter 3]. Given a vertex i of Q let ei denote the idempotent of T Q corresponding to i. If I is a two-sided ideal in T Q contained in T Q 1 then the coset ei + I in the the quotient algebra T Q=I will be denoted by ei as well. Lemma 2.3. Let A be a V -order. Assume that A is triangular. There exists an idempotent e 2 Ae := A «V Ve such that e is isomorphic to Ve Q=J for some directed quiver Q and a two (i) The Ve -order B := eAe sided ideal J of Ve Q contained in Ve Q1 . (ii) The R-algebra RQ=J ¹= B is basic and Morita equivalent to A. e e -algebra B (Ke ) ¹= K e Q=J (Ke ) is basic and Morita equivalent to A (Ke ) and A (K) (iii) The K is triangular. e e , respectively. In the above lemma J and J (K) denote J= e J and J «Ve K

Proof. We recall the arguments from [18]. Since Ve is maximally complete there exist e e pairwise orthogonal idempotents "1 ; :::; "n of Ae such that "1 A©:::©" n A and " 1 A©:::©" n A are decompositions of Ae and A respectively into a direct sum of indecomposable modules [16, Corollary 12.26]. Without loss of generality we can assume that there exists a number e e e¹ e m such that "1 A,...," m A are pairwise non-isomorphic and if m < j µ n then " j A = " i A for some i such that 1 µ i µ m. It follows, by [16, Theorem 12.28], that also "1 A,...,"m A are pairwise non-isomorphic and if m < j µ n then "j A ¹= "i A for some i such that e Then the algebra B ¹= "A" is basic and 1 µ i µ m. Let " = "1 + ::: + "m and B = "A". Morita equivalent to A. Note that B is ¯nitely generated and free as a V -module. Let Q be the ordinary quiver of B (see [12], [1], [2, III, Theorem 1.9]) with vertices 1; :::; m. Let º : RQ¡ !B be the canonical surjection such that º (ei ) = "i for i = 1; :::; m with a kernel I satisfying I ³ RQ 2 . Since B is triangular, the quiver Q is directed. For every arrow ¬ of Q let b0 be an element of B such that b0 = º (¬ ) and b = "s( ) b0 "t( ) . Then there exists a Ve -algebra homomorphism ºe : Ve Q¡ !B

S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

101

de¯ned by ºe (ei ) = "i and ºe(¬ ) = b for every vertex i and arrow ¬ of Q. Observe that Im(e º ) + e B = B and ºe is surjective, by Nakayama Lemma. Let J be the kernel of ºe . e -algebra homomorphism This homomorphism induces a surjective K e e Q¡ !B (Ke ) ºe (K) : K

e whose kernel is J (K ), since B is a torsion-free Ve -module. Now we prove that J ³ Ve Q 1 . Otherwise, since Q has no oriented cycles, » ei 2 Ker ºe for some » 2 Ve and i, 1 µ i µ m. Since "i 6= 0 in B it follows that "i 2 = Ker ºe which contradicts the fact that B is torsion-free. Since the quiver Q is directed, the endomorphism ring of any B-module "i B, e e i = 1; :::; n, is isomorphic to Ve . i = 1; : : : ; m, and consequently of any A-module "i A, e e Therefore "iA (K ) are indecomposable A (K )-modules for i = 1; :::; n. Moreover, the ordie e nary quiver of B (K ) is obtained from Q by deleting some arrows hence the B (K) -modules e e "1 B (K ); :::; "m B (K) are pairwise non-isomorphic. e e Now it is clear that B (K ) is basic and Morita equivalent to A(K) and both algebras are triangular.

Thanks to Proposition 2.1 and Lemma 2.3, in order to prove Theorem 1.1, it is enough to prove the assertion under the additional assumption that A ¹= V Q=J for some directed quiver Q and a two-sided ideal J of V Q contained in V Q 1 . The triangularity of A(K) follows already from Lemma 2.3 and Proposition 2.1. Thus we concentrate on the proof that A (K ) is representation-¯nite provided A is so.

3

The variety of algebras 3

Fix an algebraically closed ¯eld K . Let ® = (® ijk )di;j;k=1 2 K d and let ¢ : K d £ K d ¡ ! K d be the multiplication given by the structure constants ® , that is, ei ¢ e j =

d X

®

ijk e k

k=1

for every i; j, 1 µ i; j µ d, if we denote the i-th standard basis vector of K d by ei . It is known (see [13], [19]) that the tuples ® such that this multiplication is associative and has a unit element form an a±ne algebraic variety denoted by AlgK (d). The following theorem is due to Gabriel [13]. Theorem 3.1. For every algebraically closed ¯eld K and a number d the subset of AlgK (d) consisting of points corresponding to representation-¯nite algebras is Zariskiopen.

4

The proof of Theorem 1.1

Let A = V Q=J be a V -order and J ³ V Q1 . Let A(x; y) = ex Aey for x; y 2 Q 0 . We say that A is schurian if rk V (A(x; y)) µ 1 for every x; y 2 Q0 . We identify A with RQ=J and A(K) with K Q=J (K ) . Without loss of generality we can assume that Q is connected.

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S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

Lemma 4.1. If A is representation-¯nite and tringular then A is schurian. Proof. It is well-known that representation-¯nite triangular basic algebra over an algebraically closed ¯eld is schurian (see e.g. [20, Lemma 2.2]). Thus the lemma follows from the formula rkV (ex Aey ) = dimR (ex Aey ) for all x; y 2 Q 0 . Let B be a basic algebra over a ¯eld K with a complete system (ex )x2S0 of primitive pairwise orthogonal idempotents, where S0 is a (¯nite) set. It is convenient to associate with B the K -category [7]: it has S0 as the set of objects and the set B(x; y) of morphisms from x to y equals ey Bex for all x; y 2 S0 . The composition is induced by the multiplication in B. We denote this category with the same letter B. Recall that generally a K -category is a category in which the morphism sets have K -vector space structure such that the composition is K -bilinear. For the purposes of this paper it is convenient to generalize the concept of a K -category to the case when K is a commutative ring, not necessarily a ¯eld. We restrict the attention to locally bounded schurian K -categories. By the de¯nition a locally bounded schurian T -category, where T is a commutative ring, is a category B in which: (1) the morphism sets have structure of T -modules and the multiplication is T -bilinear, (2) B(x; y) ¹= T or B(x; y) = 0 for any objects x; y of B, (3) di®erent objects are non-isomorphic and for any object x of B we have B(x; y) = 0 = B(y; x) for all but ¯nitely many objects y. Note that when T is a ¯eld the above de¯nition coincides with the usual concept of a locally bounded schurian T -category, see [8, 1.3]. Such a category is directed provided for any sequence x0 ; :::; xm¡1 , m ¶ 1, of pairwise di®erent objects of B we have B(xi ; xi+1 ) = 0 for some i = 0; :::; m ¡ 1, if we put xm = x0 . If A is a schurian V -order of the form V Q=J, where J is a two-sided ideal of V Q contained in V Q1 then we can associate with A, in a standard way, a locally bounded schurian V -category, whose set of objects is identi¯ed with the set of vertices of Q. As above we denote this category by A again. Now recall from [8] the construction of the singular complex associated with a schurian algebra or a T -category. Let B be a schurian locally bounded T -category, where T is a commutative ring. A sequence x = (x0 ; :::; xp ) of objects of B induces a map ¿

: B(x0 ; x1 ) £ ::: £ B(xp¡1 ; xp ) ! B(x0 ; xp )

de¯ned by the composition in B. Let Seqp (B) be the set of all sequences of p + 1 pairwise distinct objects of B and Cp (B) be the free abelian group with basis Sp (B) = fx 2 Seqp (B) : ¿

6= 0g:

S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

103

The set of objects of B is identi¯ed with S0 (B). De¯ne also Z-linear maps @p : Cp(B)¡ !Cp¡1 (B) by the formula @p (x0 ; :::; xp ) =

p X

(¡ 1)i (x0 ; :::; xi¡1 ; xi+1 ; :::; xp ):

i=0

We obtain a chain complex S¤ (B) :

@

@

@

3 2 1 :::¡ !C2 (B)¡ !C1 (B)¡ !C0 (B):

Now let us come back to our V -order A (which is schurian by Lemma 4.1). Let S0 (A) be the set of objects of A treated as a V -category. For every (x; y) 2 S1 (A) choose a basic element axy of A(x; y). Then the system (ex ; ayz )x2S0(A);(y;z)2S1(A) is a V -basis of A. The multiplication in A is determined by the system c = (cxyz )(x;y;z)2S2 (A) , cxyz 2 V , such that (+)

axy ayz = cxyz axz

for every (x; y; z) 2 S2 (A). The associativity of the multiplication is equivalent to the system of equations (¤xyzt )

cxyz cxzt = cyztcxyt ;

(x; y; z; t) 2 S3 (A), that is, c a 2-cocycle in the complex Hom(S¤ (A); K ¤ ), see [8]. On the other hand, every system c0 = (c0xyz )(x;y;z)2S2(A) of elements of K satisfying the equations (¤xyzt) for all (x; y; z; t) 2 S3 (A) (in particular every 2-cocycle in Hom(S¤ (A); K ¤ )) determines a K -algebra B(c0 ) with the basis (ex ; ayz )x2S0(A);(y;z)2S1 (A) and multiplication de¯ned by the formula (+) with c interchanged with c0 . By [8, 2.2] if c0 and c00 are 2-cocycles in Hom(S¤ (A); K ¤ ) the K -categories B(c0 ) and B(c00 ) are isomorphic if and only if c0 and c00 induce the same element in the cohomology group. If all the coe±cients c0xyz belong to V we can de¯ne also a V -order A(c0 ) such that A(c0 )(K) ¹= B(c0 ). Note that A(c) = A. Observe that (¤xyzt ) yields: (cxyz ) + (cxzt) = (cyzt ) + (cxyt) for (x; y; z; t) 2 S3 (A). Thanks to Proposition 5.3 below we can ¯nd nonnegative integers vxyz , (x; y; z) 2 S2 (A), satisfying vxyz + vxzt = vyzt + vxyt for (x; y; z; t) 2 S3 (A) and such that vxyz > 0 if and only if (cxyz ) > 0 for every (x; y; z) 2 S2 (A). Given ¶ 2 K , set ¶ v c = (¶ vxyz cxyz )(x;y;z)2S2(A) . Observe that ¶ v c satis¯es (¤xyzt ) for all (x; y; z; t) 2 S3 (A) and this is a 2-cocycle in Hom(S¤ (A); K ¤ ) if ¶ 2 K ¤ . Introduce the following notation: B¸ = B(¶ v c). Observe that B1 ¹= A(K) .

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S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

Lemma 4.2. Let K and L be algebraically closed ¯elds. Assume that B and C are triangular, basic and schurian algebras over K and L, respectively. Let us view B as a K -category and C as a L-category. Assume that Si (B) = Si (C ) for i = 0; 1; 2. Then B is representation-¯nite if and only if C is representation-¯nite. e (resp. C) e be the universal Galois Proof. Assume that B is representation-¯nite. Let B covering of B (resp. C), see [14], comp. with [20]. By [20, Theorem 4.1] B is standard in the sense of [7], see also [6] for an equivalent de¯nition of standard algebra. Then e is simply connected ([7], [6]), and therefore Ce is also simply connected. Indeed, our B e and C e can be identi¯ed in a natural assumptions imply that the sets of objects of B e = Si (C) e for i = 0; 1; 2. Thus the way and under this identi¯cation we have Si (B) e of the complex S¤ (C) e is zero since H 1 (S¤ (B)) e = 0. The ¯rst homology group H1 (S¤ (C)) remaining conditions in the de¯nition of a simply connected L-category in [6] are obviously e (See [3] for a discussion of the concept of simply connected algebra.) Now satis¯ed in C. C is standard and Bongartz’s criterion for representation-¯niteness is applied: it is clear that the conditions a)-c) and d’) in the statement 2) of the main result in [6] (see also e if and only if they are satis¯ed in C, e since the following remark there) are satis¯ed in B they are of a combinatorial nature. We conclude that C is representation-¯nite because B is so. Lemma 4.3. The algebra B0 is representation-¯nite. Proof. Clearly, S0 (B0 ) and S0 (A) can be identi¯ed with the set of vertices of Q. Moreover, under this identi¯cation, S1 (B0 ) = S1 (A) and S2 (B0 ) = f(x; y; z) 2 S2 (A) : cxyz 2 =

g = S2 (A):

Now the assertion follows directly from Lemma 4.2. Proposition 4.4. All the algebras Bt , t 2 K ¤ , are isomorphic to A(K) and representation¯nite. Proof. Any system c0 of elements of K satisfying the equations (¤xyz ), (x; y; z) 2 S2 (A), determines in a canonical way a point in the variety AlgK (d) corresponding to the algebra B(c0 ). It is clear that the point corresponding to B0 belongs to the Zariski-closure of the set of points corresponding to B¸ , ¶ 2 K ¤ . Therefore, by Gabriel’s result (Theorem 3.1), it follows that B¸ is representation-¯nite for all but a ¯nite number of elements ¶ 2 K . Note that the complexes S¤ (A) and S¤ (B¸ ) are isomorphic if ¶ 2 K ¤ . Then we conclude, by [8, Theorem 2.6], that the homology groups H n (S¤ (A)) vanish for n > 1 and H1 (S¤ (A)) is free abelian. Thus the 2nd cohomology group H 2 (S¤ (A); K ¤ ) is trivial and hence all the algebras B¸ are isomorphic [8, 2.2] and therefore representation-¯nite. Proof 4.5 (of Theorem 1.1.). The algebra A(K) is representation-¯nite, by Proposition 4.4. The triangularity of A(K) follows from Lemma 2.3 and Proposition 2.1.

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105

Remark 4.6. It is possible to prove Proposition 4.4 and Theorem 1.1 without referring to Gabriel’s theorem and using only the fact that a degeneration of a representationin¯nite algebra is representation-in¯nite. This proof is rather long and technical, it can be performed in the spirit of [5]. Although in order to prove Theorem 1.2 we need ¯nite axiomatizability of the class of representation-¯nite algebras, which is closely related with the Gabriel’s theorem, (comp. results of [17] concerning the class of tame algebras). Remark 4.7. If B and C are as in Lemma 4.2 then S¤ (B) = S¤ (C).

5

Appendix: on elementary properties of linearly ordered groups

The aim of this section is Proposition 5.3 which is used in Section 4. Let (G; µ) be an ordered group. Then G is torsion-free and if in addition G is divisible then it has a natural Q-module structure. In such situation given g 1 ; :::; gn 2 G P let hg1 ; :::; g n i = ni=1 gi Q. Let W be a ¯nite dimensional Q-vector space. Given elements w1 ; :::; ws 2 W we denote by C (w1 ; :::; ws ) the convex cone generated by w1 ; :::; ws , that is, the set f· 1 w1 + ::: + · s ws : · i 2 Q; · i ¶ 0; i = 1; :::; sg. Lemma 5.1. Suppose that w1 ; :::; ws are nonzero and C(w1 ; :::; ws ) does not contain any nonzero linear subspace of W . Then there exists a Q-linear homomorphism h : W ¡ ! Q such that h(wi ) > 0 for i = 1; :::; s. Proof. We proceed by induction on dimQ W . If dimQ W = 1 the assertion is trivial. Assume that dimQ W > 1. By [22, Theorem 7.1], [24] there exists a nonzero linear functional f : W ¡ ! Q such that f (wi ) ¶ 0 for i = 1; :::; s. If f (wi ) > 0 for all i we put h = f . Otherwise let U = Ker f and assume that f (w1 ) = ::: = f (wr ) = 0 and f (wr+1); :::; f (ws ) > 0 for some r µ s. Clearly C (w1 ; :::; wr ) does not contain any nonzero subspace of U and by inductive hypothesis there exists g : U ¡ ! Q such that g(w1 ); :::; g(wr ) > 0. Let e g : W ¡ ! Q be any linear extension of g to W . Let " > 0 be a rational number such that "je g (wj )j < jf (wj )j for j = r + 1; :::; s. It is easy to observe that h = f + " e g satis¯es the required condition. Lemma 5.2. Assume that G is a divisible ordered group. Let g 1 ; :::; gn 2 G. (a) There exists a Q-linear map h0 : hg 1 ; :::; gn i¡ ! Q such that h0 (gi ) µ 0 (resp. h0 (gi ) ¶ 0) if and only if g i µ 0 (resp. g i ¶ 0) for i = 1; :::; n. (b) There exists a Q-linear map h : hg 1 ; :::; gn i¡ ! Q such that h(gi ) µ h(gj ) if and only if g i µ g j for i; j = 1; :::; n.

Proof. (a) Replacing gi by ¡ g i , if necessary, we can assume that all gi ’s are nonnegative. Let W = hg1 ; :::; g n i. Suppose that the cone C (g1 ; :::; g n ) contains a nonzero subspace of

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S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

W . Then there exist nonnegative integers ai ; bi , i = 1; :::; n such that 0 6=

n X

ai g i = ¡

i=1

n X

bi gi

i=1

and this leads to a contradiction since G is an ordered group. Now the existence of h0 follows from Lemma 5.1. The assertion (b) follows from (a) applied to the elements gi ¡ g j , i; j = 1; :::; n. Proposition 5.3. Assume that g1 ; :::; g n are elements of an ordered group G satisfying the system of linear equations n X

aij g i = 0; j = 1; :::; m

i=1

with aij 2 Z for i = 1; :::; n, j = 1; :::; m. There exist integers ¸ 1 ; :::; ¸ n X

aij ¸

i

n

such that

= 0; j = 1; :::; m

i=1

and gi ¶ 0 (resp. g i µ 0) if and only if ¸

i

¶ 0 (resp. ¸

i

µ 0).

Proof. The order in G induces a structure of an ordered group on G «Z Q which is divisible. Let h0 : hg 1 ; :::; gn i¡ !Q be as in Lemma 5.2 (a). Put ¸ i = N ¢ h0 (g i ), i = 1; :::; n, where N is a positive common multiple of the denominators of h0 (g i). Remark 5.4. Observe that Proposition 5.3 is a consequence of a stronger assertion that every two divisible ordered groups are elementarily equivalent. The latter can be proved using the Ehrenfeucht - Fraijss¶e method (see [11], [10]).

Acknowledgments Supported by Polish KBN Grant 5 P03A 015 21.

References [1] I. Assem, D. Simson and A. Skowro¶nski, "Elements of Representation Theory of Associative Algebras", Vol I: Techniques of Representation Theory, London Math. Soc. Student Texts, Cambridge University Press, Cambridge, to appear. [2] M. Auslander, I. Reiten and S. Smal¿, "Representation theory of Artin algebras", Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. [3] I. Assem and A. Skowro¶nski, On some classes of simply connected algebras, Proc. London Math. Soc., 56 (1988), 417-450. [4] S. Balcerzyk and T. J¶oze¯ak, "Pier¶scienie przemienne", Warszawa: Pa¶ nstwowe Wydawnictwo Naukowe (1985), English translation of the Chapters I-IV:

S. Kasjan / Central European Journal of Mathematics 1 (2003) 97{107

107

"Commutative Noetherian and Krull rings". PWN-Polish Scienti¯c Publishers, Warsaw. Chichester: Ellis Harwood Limited; New York etc.: Halsted Press. (1989). [5] K. Bongartz, Zykellose Algebren sind nicht zÄ ugellos. Representation theory II, Proc. 2nd Int. Conf., Ottawa 1979, Lect. Notes Math. 832, (1980) 97-102. [6] K. Bongartz, A criterion for ¯nite representation type, Math. Ann. 269 (1984), 1-12. [7] K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982), 331-378. [8] O. Bretcher and P. Gabriel, The standard form of a representation-¯nite algebra, Bull. Soc. Math. France 111 (1983), 21-40. [9] Ch. W. Curtis and I. Reiner, "Methods of Representation Theory", Vol. I, Wiley Classics Library Edition, New York, 1990. [10] H. Ebbinghaus and J. Flum, "Finite Model Theory", Perspectives in Mathematical Logic. Berlin: Springer-Verlag 1995. [11] A. Ehrenfeucht, An application of games to the completness problem for formalized theories, Fund. Math. 49, 1961, 129-141. [12] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-102. [13] P. Gabriel, Finite representation type is open, in "Representations of algebras", Lecture Notes in Math. 488, Springer-Verlag, Berlin, Heidelberg and New-York (1975) 132-155. [14] P. Gabriel, The universal cover of a representation-¯nite algebra, in: Lecture Notes in Math. 903, Springer-Verlag, Berlin, Heidelberg and New-York (1981), 68-105. [15] Ch. Jensen and H. Lenzing, Homological dimension and representation type of algebras under base ¯eld extension, Manuscripta Math., 39, 1-13 (1982). [16] Ch. Jensen and H. Lenzing, "Model Theoretic Algebra: with particular emphasis on ¯elds, rings, modules", Algebra, Logic and Applications, 2. Gordon and Breach Science Publishers, New York, 1989. [17] S. Kasjan, On the problem of axiomatization of tame representation type, Fundamenta Mathematicae 171 (2002), 53-67 [18] S. Kasjan, Representation-directed algebras form an open scheme, Colloq. Math. 93 (2002), 237-250. [19] H. Kraft, Geometric methods in representation theory, in: Representations of algebras, 3rd int. Conf., Puebla/Mexico 1980, Lect. Notes Math. 944, 180-258 (1982). [20] R. Martinez-Villa and J.A. de la Pe~ na, The universal cover of a quiver with relations, J. Pure. Appl. Algebra 30 (1983), 277-292. [21] B. Poonen, Maximally complete ¯elds, Enseign. Math. 39 (1993), 87-106. [22] A. Schrijver, "Theory of Linear And Integer Programming", Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. Chichester: John Wiley & Sons Ltd. 1986. [23] L. van den Dries, Some applications of a model theoretic fact to (semi-) algebraic geometry, Nederl. Akad. Indag. Math., 44 (1982), 397-401. [24] H. Weyl, The elementary theory of convex polyhedra. in: Contrib. Theory of Games, Ann. Math. Studies 24, (1950) 3-18.

CEJM 1 (2003) 108{122

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one Andrzej Skowro¶ nski

¤

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toru¶ n, Poland

Received 15 December 2002; revised 23 December 2002 Abstract: Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable ­ nitely generated right A-modules. Denote L A to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by RA the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with L A [ RA co-­ nite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which L A [ RA is co-­ nite in ind A, and derive some consequences. ® c Central European Science Journals. All rights reserved. Keywords: artin algebra, quasi-tilted algebra, generalized double tilted algebra, Auslander-Reiten quiver, projective dimension, injective dimension MSC (2000): Primary 16G70, 18G20; Secondary 16G10. DEDICATED TO STANIS LAW BALCERZYK ON THE OCCASION OF HIS 70TH BIRTHDAY

1

Introduction

¤

Throughout this paper, A will denote a ¯xed artin algebra over a commutative artin ring R. We denote by mod A the category of all ¯nitely generated right A-modules, by ind A the full subcategory of mod A formed by the indecomposable modules, and by D the standard duality HomR (¡ ; I ) on mod A, where I is the injective envelope of R= rad R in E-mail: [email protected]

Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

109

mod R. Further, we denote by ¡A the Auslander-Reiten quiver of A, and by ½ A and ½ A¡ the Auslander-Reiten translations D Tr and Tr D, respectively. We shall not distinguish between an indecomposable A-module and the corresponding vertex of ¡A . By a path in ind A we mean a ¯nite sequence M = X 0 ¡ ! X1 ¡ ! ¢ ¢ ¢ ¡ ! Xr = N , r ¶ 1, of nonzero non-isomorphisms in ind A, and then call M a proper predecessor of N in ind A, and N a proper successor of M in ind A. Every module M in ind A is also called its (trivial) predecessor and successor. Finally, following [15] a module M in ind A is called directing provided M is not its own proper predecessor. It has been shown independently in [12] and [19] that ¡A admits at most ¯nitely many ½ A -orbits containing directing modules. In the representation theory of artin algebras, algebras with small homological dimensions play an important role. Following [2], an artin algebra A is said to be a shod (for small homological dimension) if every module X in ind A has projective dimension pdA X µ 1 or injective dimension idA X µ 1. It is known [6] that if A is shod then gl: dim A µ 3. Moreover, it has been shown in [6] that A is shod with gl: dim A µ 2 if and only if A is quasi-tilted, that is A = End (T ) for a tilting object T in an abelian hereditary R-category . Further, following [6], denote by L A the full subcategory of ind A formed by all modules X such that pdA Y µ 1 for every predecessor Y of X in ind A, and by RA the full subcategory of ind A formed by all modules X such that idA Z µ 1 for every successor Z of X in ind A. It has been shown in [2] that A is shod if and only if ind A = L A [ RA . An important class of quasi-tilted algebras is formed by the tilted algebras, that is algebras of the form EndH (T ), where T is a tilting object in the module category mod H of a hereditary artin algebra H [7]. Recently, it has been shown by D. Happel and I. Reiten [5] that the remaining class of quasi-tilted artin algebras is formed by the quasi-tilted algebras of canonical type (see also [6, 10] for structural results on this class of algebras). Finally, in the joint work with I. Reiten [13], we have proved that A is shod with gl: dim A = 3 if and only if A is strictly double tilted. This completes the classi¯cation of artin algebras A with small homological dimensions (equivalently, with ind A = L A [ RA ). In this paper we are concerned with the structure of artin algebras A for which L A [RA is co-¯nite in ind A. This class of algebras contains all artin algebras of ¯nite representation type, all shod algebras, and a large class of algebras of in¯nite global dimension. We also note that if gl: dim A = 1 then ind A contains a module X with pdA X = 1 and indA X = 1 (see [22]). Recently, in the joint work with I. Reiten [14], we introduced and investigated the class of generalized double tilted algebras (see Section 2 for details). A special class of generalized double tilted algebras, called weakly shod algebras, has been also investigated by F. U. Coelho and M. A. Lanzilotta [3]. The aim of this paper is to prove the following theorem and derive some consequences. Theorem. Let A be a connected artin algebra. The following conditions are equivalent: (i) A is generalized double tilted or quasi-tilted.

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(ii) L

A

[ RA is co-¯nite in ind A.

(iii) There exists a ¯nite set of modules in ind A such that every path in ind A from an injective module to a projective module consists entirely of modules from . We have the following consequences of the theorem and results proved in [13, 14, 21]. Corollary 1.1. Let A be a connected artin algebra. The following conditions are equivalent: (i) A is generalized double tilted.

¡ (ii) ind A n (L A [ RA ) is ¯nite nonempty or L directing module.

A

[½

¡ AL

A

¢

\ (½

A RA

[ RA ) contains a

Corollary 1.2. Let A be a connected artin algebra. The following conditions are equivalent: (i) A is generalized double tilted and ¡A admits a directed connecting component. ¡ ¢ (ii) ind A n (L A [ RA ) consists of directing modules or L A [ ½ A¡ L A \ (½ A RA [ RA ) contains a directing module. Corollary 1.3. Let A be a connected artin algebra. The following conditions are equivalent: (i) ind A n (L

A

[ RA ) is ¯nite and nonempty.

(ii) A is generalized double tilted and ¡A admits a path from an injective module to a projective module containing at least three hooks. We would like to propose the following problem. Problem. Let A be an artin algebra such that for all but ¯nitely many isomorphism classes of modules X in ind A we have pdA X µ 1 or idA X µ 1. Is then L A [ RA co-¯nite in ind A? We note that this proves to be the case if we have pdA X µ 1 (respectively, idA X µ 1) [20] for all but ¯nitely many isomorphism classes of modules X in ind A. For a basic background of the representation theory of artin algebras applied here we refer to [1, 6, 15, 16].

2

Generalized double tilted algebras

In this section we recall basic concepts and results which play a fundamental role in our investigations. Let A be an artin algebra and be a component of ¡A . Following [14], a full valued

Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

connected subquiver ¢ of satis¯ed:

is called a multisection in

111

if the following conditions are

(1) ¢ is almost directed. (2) ¢ is convex in (3) For each ½ (4) j

.

A -orbit

in

we have 1 µ j

\ ¢j = 1 for all but ¯nitely many ½

\ ¢j < 1.

A -orbits

in

.

(5) No proper full convex valued subquiver of ¢ satis¯es the conditions (1){(4). We note that a multisection ¢ in is a section [11, (3.1)] (respectively, double section [13, Section 7]) when j \ j = 1 (respectively, 1 µ j \ j µ 2). Moreover, a full subquiver § of ¡A is called directed (respectively, almost directed ) if all modules from § (respectively, all but ¯nitely many modules from §) do not lie on oriented cycles in ¡A . It has been shown in [14, Theorem 2.5] that a component of ¡A admits a multisection ¢ if and only if is almost directed. For a multisection ¢ in a component of ¡A , the following full valued subquivers of are de¯ned in [14]: 0 ¢l = f X 2 ¢; there is a nonsectional path X ¡ ! ¢ ¢ ¢ ¡ ! P in with projective P g, 0 ¢r = f X 2 ¢; there is a nonsectional path I ¡ ! ¢ ¢ ¢ ¡ ! X in with injective I g, 00 ¢l = f X 2 ¢0l ; ½ A¡ X 2 = ¢0l g, ¢00r = f X 2 ¢0r ; ½ A X 2 = ¢0r g, ¢l = (¢ n ¢0r ) [ ½ A ¢00r , ¢r = (¢ n ¢0l ) [ ½ A¡ ¢00l , ¢c = ¢0l \ ¢0r (the core of ¢). We note that ¢l and ¢r can be empty, for example if is ¯nite and cyclic. Following [14], a connected artin algebra A is said to be a generalized double tilted algebra if the following conditions are satis¯ed: (1) ¡A admits a component

with a faithful multisection ¢.

(2) There exists a tilted factor algebra A l of A (not necessarily connected) such that ¢l is a disjoint union of sections of connecting components of the connected parts of A l , and the category of all predecessors of ¢l in ind A coincides with the category of all predecessors of ¢l in ind A l . (3) There exists a tilted factor algebra Ar of A (not necessarily connected) such that ¢r is a disjoint union of sections of connecting components of the connected parts of A r , and the category of all successors of ¢r in ind A coincides with the category of all successors of ¢r in ind Ar . In the above notation, Al and A r are called the left tilted algebra and the right tilted algebra of A, respectively. Moreover, is called a connecting component of ¡¤ . If ¢

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is a section of , then Al = A = A r , and hence A is a tilted algebra. Moreover, if A is not tilted, then is the unique connecting component of ¡A . We note that both algebras Al and Ar can be empty, for example, if A is a self-injective algebra of ¯nite representation type. Moreover, every connected artin algebra of ¯nite representation type is a generalized double tilted algebra. The following characterization of generalized double tilted algebras has been established in [14, Theorem 3.1] as an extension of the characterizations of tilted and double tilted algebras proved respectively in [17, Theorem 2] and [13, Theorem 7.3]. Theorem 2.1. Let A be a connected artin algebra. The following conditions are equivalent: (i) A is generalized double tilted. (ii) ¡A admits a faithful generalized standard component which is almost directed. (iii) ¡A admits a component with a faithful multisection ¢ such that Hom¤ (X; ½ ¤ Y ) = 0 for all modules X from ¢r and Y from ¢l . Recall that following [18], a component of ¡A is called generalized standard if for any modules X and Y in , the in¯nite radical rad1 A (X; Y ) is zero. Moreover, we have the following consequence of [14, Proposition 2.4]. Proposition 2.2. Let A be a generalized double tilted algebra and a connecting component of ¡A with a multisection ¢. Then ¢c is ¯nite, every oriented cycle of lies in ¢c , and ind A = L A [ ¢c [ RA . We may also reformulate Theorems 3.1 and 3.2 in [20] as follows: Theorem 2.3. Let A be a connected artin algebra. Then the following conditions are equivalent: (i) A is a generalized double tilted algebra and ¡A admits a connecting component containing all indecomposable projective modules. (ii) rad1 (¡ ; A) = 0. (iii) idA X µ 1 for all but ¯nitely many iso-classes of indecomposable A-modules X . Theorem 2.4. Let A be a connected artin algebra. Then the following conditions are equivalent: (i) A is a generalized double tilted algebra and ¡A admits a connecting component containing all indecomposable injective modules. (ii) rad1 (D(A); ¡ ) = 0. (iii) pdA X µ 1 for all but ¯nitely many iso-classes of indecomposable A-modules X .

Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

113

The following fact proved in [21, Corollary B] gives a homological criterion for an algebra A to be tilted. Theorem 2.5. Let A be a connected artin algebra. Then A is tilted if and only if gl: dim A µ 2, ind A = L A [ RA , and L A \ RA contains a directing module. We need also the following characterizations of quasi-tilted algebras established in [6, Theorem II.1.14]. Theorem 2.6. Let A be a connected artin algebra. The following conditions are equivalent: (i) A is quasi-tilted. (ii) L

A

contains all indecomposable projective A-modules.

(iii) RA contains all indecomposable injective A-modules.

3

Preparatory lemmas

Let A be an artin algebra. Denote by add L A (respectively, add RA ) the full subcategory of mod A formed by modules whose all indecomposable direct summands belong to L A (respectively, RA ). We note that L A (respectively, RA ) is nonempty if and only if mod A admits at least one simple projective (respectively, simple injective) module. Assume L A is nonempty and consider a decomposition AA = P ©P 0 where P 2 add L A and P 0 has no indecomposable direct summand from L A . Then A(l) = End A (P ) is a factor algebra of A, and mod A (l) is the full subcategory of mod A consisting of all modules annihilated by the corresponding ideal of A. The following extension of [6, Proposition II.1.15] (see also [9, Theorem 1.2]) holds. Lemma 3.1. The algebra A (l) is quasi-tilted and L

A

³L

A(l) .

Proof. Denote by mod P the full subcategory of mod A consisting of all modules X which have a projective presentation P 1 ¡ ! P 0 ¡ ! X ¡ ! 0 with P 0 and P 1 from the additive category add P of P . Then the functor HomA (P; ¡ )jmod P : mod P ¡ ! mod A (l) is an equivalence of categories with add P corresponding to the category of projective A(l) -modules. In order to prove that A (l) is quasi-tilted, it is enough to show that the module A(l) belongs to add L A(l) (Theorem 2.6). Let Q be an indecomposable projective A(l) -module and Y 2 ind A(l) a predecessor of Q in ind A (l) . Then Q = HomA (P; P¹ ) for an indecomposable direct summand P¹ of P and Y = HomA (P; X ) for some indecomposable module X from mod P . Hence X admits a minimal projective presentation P1 ¡ ! P0 ¡ ! X ¡ ! 0 with P0 ; P1 2 add P . Moreover, X is a predecessor of P¹ in ind A. Since P¹ 2 L A , we have an exact sequence 0 ¡ ! P 1 ¡ ! P0 ¡ ! X ¡ ! 0. Applying

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Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

HomA (P; ¡ ) we obtain the exact sequence 0 ¡ ! HomA (P; P1 ) ¡ ! HomA (P; P 0 ) ¡ ! HomA (P; X ) ¡ ! 0 in mod A(l) with Q 1 = HomA (P; P 1 ), Q 0 = HomA (P; P 0 ) projective A(l) -modules and Y = HomA (P; X ), and so pdA(l) Y µ 1. Therefore, A (l) 2 add L A(l) , and hence A (l) is quasi-tilted. For L A ³ L A(l) , take X 2 L A . Then there exists an exact sequence 0 ¡ ! P 1 ¡ ! P 0 ¡ ! X ¡ ! 0 in mod A with P 0 , P 1 projective, and clearly P0 ; P 1 2 L A . But then X 2 mod P = mod A(l) , P0 = HomA (P; P 0 ) and P1 = Hom(P; P 1 ) are projective A(l) -modules, and so pd A(l) X µ 1. Finally, if Z 2 ind A (l) is a predecessor of X in ind A (l) then Z is a predecessor of X in ind A, and hence Z 2 L A . Consequently, pdA(l) Z µ 1. This shows that L A ³ L A(l) . Dually, if RA is nonempty, we may consider a decomposition D(A)A = Q © Q 0 of the injective cogenerator D(A) of mod A such that Q 2 add RA and Q0 has no indecomposable direct summand from RA . Then A (r) = End A (Q)op is a factor algebra of A and the following lemma holds. Lemma 3.2. The algebra A (r) is quasi-tilted and RA ³ RA(r) . It follows from the above considerations that, if an artin algebra A admits at least one simple projective (respectively, simple injective) module then we have a left quasi-tilted factor algebra A (l) (respectively, right quasi-tilted factor algebra A(r) ) of A. But in general mod A(l) (respectively, mod A(r) ) forms a rather small part of the category mod A. Let A be an artin algebra, e a nontrivial idempotent of A, f = 1 ¡ e, ¤ = eAe, ¡ = f Af , ¡ M¤ = f Ae, and assume that eAf = 0. Then A is isomorphic to the matrix algebra of the form 2 3 6¡ M 7 4 5 0 ¤

and we may identify mod A with the category whose objects are triples (V; X; ’), where X 2 mod ¤, V 2 mod ¡, and ’ : V ¡ ! Hom¤ (M; X ) is a ¡-homomorphism. A morphism h : (V; X; ’) ¡ ! (W; Y; Ã) of such triples is given by a pair (f; g), where f : V ¡ ! W is a morphism in mod ¡, g : X ¡ ! Y is a morphism in mod ¤, and Ãf = Hom¤ (M; g)’. Observe that then mod ¤ is identi¯ed with the category of all triples (0; X; 0), X 2 mod ¤. We refer to [15, 16] for matrix extensions of algebras and related vector space category methods. Lemma 3.3. Let X be an indecomposable nonprojective ¤-module, N an indecomposable direct summand of the ¤-module M , and assume that Hom¤ (N; ½ ¤ X ) 6= 0. Then there exists an indecomposable direct summand P of the projective A-module f A, which is a predecessor of X in ind A.

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115

Proof. Consider an Auslander-Reiten sequence f

g

0¡ !Y ¡ !E¡ !X¡ !0 in mod ¤. Then we obtain the following exact sequence in mod A ¯

0 ! (Hom ¤ (M; Y ); Y; id) ! (Hom(M; Y ); E; Hom¤ (M; f )) ! (0; X; 0) ! 0; where ¬ = (id; Hom¤ (M; f )) and ­ = (0; Hom¤ (M; g)). Since Hom¤ (N; Y ) = Hom¤ (N; ½ ¤ X ) 6= 0 and N is a direct summand of M , we have Hom¤ (M; Y ) 6= 0 and Hom¤ (M; f ) 6= 0. Therefore, there exists an indecomposable direct summand Z = (V; U; h) of the A-module (Hom ¤ (M; Y ); E; Hom¤ (M; f )) with h 6= 0, where V 2 mod ¡, U 2 mod ¤, and h : V ¡ ! Hom¤ (M; U ) is a morphism in mod ¡. Hence, HomA (P; Z) 6= 0 for an indecomposable direct summand P of f A. Further, U is a direct summand of E and the restriction of g to U is nonzero, and so HomA (Z; X ) 6= 0. Hence P is a predecessor of X in ind A. In the proofs of our main results we shall use also the following well-known facts (see [1] and [15, (2.4)]). Lemma 3.4. Let A be an artin algebra, I an ideal in A, B = A=I, and X a B-module. Then ½ B X is a submodule of ½ A X . Lemma 3.5. Let A be an artin algebra and X an A-module. Then (i) pdA X µ 1 if and only if HomA (D(A); ½ (ii) idA X µ 1 if and only if HomA (½

4

AX )

¡ A X; A)

= 0.

= 0.

Proof of theorem

Let A be a connected artin algebra. Assume A satis¯es (i). It follows from [6, Corollary II.3.4] that if A is quasi-tilted but not tilted, then there is no path in ind A from an injective module to a projective module. Let A be generalized double tilted. Invoking the properties of module categories of generalized double tilted algebras presented in Section 2 (see also [13, 14]), we infer that every path in ind A from an injective module to a projective module can be re¯ned to a path of irreducible morphisms between modules lying in a ¯nite multisection ¢ of a connecting component of ¡A . Therefore, (i) implies (iii). For (iii) ) (ii), observe that, by Lemma 3.5, for any indecomposable A-module X from ind A n (L A [ RA ) there exists a path in ind A of the form I !½

AY

! U ! Y ! ¢¢¢ ! X ! ¢¢¢ ! Z ! V ! ½

¡ AZ

!P

with I injective and P projective. Therefore, it remains to show that (ii) implies (i). Assume that L A [ RA is co-¯nite in ind A. If ind A = L A [ RA then A is shod [2], and consequently is double tilted or quasi-tilted, by [13, Theorem 8.2]. If A is of ¯nite

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representation type, then A is clearly generalized double tilted. Further, if L A or RA is ¯nite, then, by Theorems 2.3 and 2.4, A is again generalized double tilted. Therefore, we may assume that ind A n (L A [ RA ) is nonempty and the both categories L A and RA are in¯nite. We shall prove that A is generalized double tilted. Since L A and RA are nonempty, we may consider, by Lemmas 3.1 and 3.2, the left quasi-tilted algebra A (l) and the right quasi-tilted algebra A(r) of A. Moreover, since by our assumption A is not quasi-tilted, we have A (l) 6= A 6= A (r) . We also note that A (l) and A(r) are not necessarily connected. We shall prove now that the both algebras A (l) and A(r) are in fact tilted algebras. By duality of considerations, it is enough to prove that A(l) is tilted. Let ¤ = A(l) . It follows from our assumptions that A A admits a decomposition A A = P © P 0 with ¤ = End A (P ) and P 0 6= 0. Put ¡ = End A (P 0 ) and M = HomA (P; P 0 ). Since L A is closed under predecessors in ind A, we have HomA (P 0 ; P ) = 0. Therefore, A is isomorphic to the matrix algebra of the form 2 3 6¡ M 7 4 5 0 ¤

where M = HomA (P; P 0 ) has the induced ¡-¤-bimodule structure. Let ¤ = ¤1 £ ¢ ¢ ¢ £ ¤r and M = M1 ©¢ ¢ ¢© Mr , where ¤1 ; : : : ; ¤r are connected algebras and Mi are ¤i -modules, 1 µ i µ r. Since A is connected, we have Mi 6= 0 for any i 2 f1; : : : ; rg. Moreover, the algebras ¤1 ; : : : ; ¤r are quasi-tilted [6, Proposition II.1.15]. We claim that every ¤i is a tilted algebra and the indecomposable direct summands of Mi belong to one connecting component i of ¡¤i . Fix i 2 f1; : : : ; rg and put B = ¤i . Let N be an indecomposable direct summand of Mi and the connected component of ¡B containing N . Assume ¯rst that ³ RB n L B . Since B is quasi-tilted, all indecomposable projective B-modules belong to L B (Theorem 2.6). Then HomB (B; N ) 6= 0 implies rad1 B (B; N ) 6= 0, and hence there exists an in¯nite path ft

ft¡ 1

f1

f0

¢ ¢ ¢ ¡ ! Xt+1 ¡ ! X t ¡ ! ¢ ¢ ¢ ¡ ! X2 ¡ ! X1 ¡ ! X0 = N 6 0 of irreducible morphisms in mod B consisting of modules from such that ft : : : f1 f0 = for all t ¶ 0. Observe that the B-modules X t ; t ¶ 0, are not projective A-modules. Moreover, the family X t ; t ¶ 0, has in¯nitely many pairwise non-isomorphic modules, because the radicals of the algebras EndB (Xt ) are nilpotent. Therefore, we obtain an in¯nite family ½ A Xt ; t ¶ 0, of indecomposable A-modules with HomA (½ A¡ (½ A Xt ); A) = HomA (X t; A) 6= 0, because N ³ A A , and hence with idA ½ A Xt ¶ 2 (Lemma 3.5). Since the family ½ A X t ; t ¶ 0, contains in¯nitely many pairwise nonisomorphic modules, it follows from our assumption on A that ½ A Xm 2 L A for some m ¶ 0. Further, by Lemma 3.4, ½ B X m is a submodule of ½ A X m . Hence, invoking L A ³ L A(l) = L ¤ (Lemma 3.1), we obtain ½ B Xm 2 L B . This contradicts our assumption ³ RB n L B . Assume now that ³ L B . It follows from [4, Theorem D] that if B is not tilted then does not contain injective modules. Therefore assume that does not contain injective

Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

117

B-modules. Since rad1 B (N; D(B)) = Hom B (N; D(B)) 6= 0, there exists an in¯nite path g0

g1

gs

N = Y0 ¡ ! Y1 ¡ ! Y2 ¡ ! ¢ ¢ ¢ ¡ ! Ys ¡ ! Y s+1 ¡ ! ¢ ¢ ¢ 6 0 of irreducible morphisms in mod B consisting of modules from such that g s : : : g1 g0 = for all s ¶ 0. Observe that the B-modules Ys ; s ¶ 0, are not injective A-modules. Moreover, the family Ys ; s ¶ 0, contains in¯nitely many pairwise non-isomorphic modules, because the radicals of the algebras End B (Y s ) are nilpotent. Therefore, we obtain an in¯nite path ½

¡ BN

=½

¡ B Y0

¡ B Y1

¡ !½

¡ !½

¡ B Y2

¡ ! ¢¢¢ ¡ ! ½

¡ B Ys

¡ !½

¡ B Ys+1

¡ ! ¢¢¢

of irreducible morphisms in mod B consisting of modules from , and containing in¯nitely many pairwise non-isomorphic modules. Further, since Hom¤ (N; ½ ¤ (½ B¡ N )) = Hom¤ (N; ½ ¤ (½ ¤¡ N )) = Hom¤ (N; N ) 6= 0, applying Lemma 3.3, we conclude that there is an indecomposable direct summand P 00 of P 0 such that P 00 is a predecessor of ½ B¡ N , and hence predecessor of all modules ½ B¡ Y s ; s ¶ 0, in ind A. Then the modules ½ B¡ Y s ; s ¶ 0, do not belong to L A . Invoking now our assumption on A, we conclude that there is a natural number n such that the modules ½ B¡ Y s ; s ¶ n, belong to RA . Our next aim is to prove that is directed with ¯nitely many ½ B -orbits. Observe that is a right stable in¯nite component of ¡B . If contains an oriented cycle then, by the dual of [11, Theorem 2.5], we know that is a ray tube. Further, if is directed then, by the dual of [11, Theorem 3.4], we know that there exists a valued connected directed quiver § such that is isomorphic to a full translation subquiver of Z§ which is closed under successors. Suppose is not directed with ¯nitely many ½ B -orbits. Then by applying [4, Theorem 1.5] (see also [19, Lemma 4]), we may conclude that there is in ind B a path ½

¡ B Yn

¡ ! ¢¢¢ ¡ ! ½

BV

¡ !U ¡ !V ¡ !W

where W = N (if has no projective modules) or W is an indecomposable projective B-module lying in . Since HomA (½ A¡ (½ A V ); A) = HomA (V; A) 6= 0, applying Lemma 3.5, we obtain idA ½ A V ¶ 2. On the other hand, by Lemma 3.4, ½ B V is a submodule of ½ A V and ½ B¡ Y n 2 RA , and hence ½ A V 2 RA , a contradiction. Therefore, is a directed component of ¡B with ¯nitely many ½ B -orbits. Observe that the above arguments show also that ½ B¡ Yn 2 RB . Finally, we conclude that all modules in are directing. Indeed, if it is not the case, then there is an oriented cycle in ind B f

g

Z ¡ ! Y ¡ ! ¢¢¢ ¡ ! X ¡ ! L ¡ ! ¢¢¢ ¡ ! Z with L; Z 2 and X; Y 2 = . Then f; g 2 rad1 (mod B), and hence there exist an in¯nite number of paths in of the forms Z = Z 0 ¡ ! Z 1 ¡ ! Z 2 ¡ ! ¢ ¢ ¢ and ¢ ¢ ¢ ¡ ! L2 ¡ ! L1 ¡ ! L0 = L: But then again we have in ind B a path of the form ½

¡ B Yn

¡ ! ¢¢¢ ¡ ! ½

BV

0

¡ ! U0 ¡ ! V 0 ¡ ! W0

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where W 0 = N or W 0 is an indecomposable projective B-module lying in , a contradiction. Consequently, all modules in are directing. In particular, the module ½ B¡ Yn is directing and belongs to L B \ RB . Since B is quasi-tilted, we have also gl: dim B µ 2 and ind B = L B [ RB . Applying now Theorem 2.5 we conclude that B is tilted. Moreover, is a connecting component of ¡B , because consists of directing modules and contains a module from L B \ RB . Assume ¯nally that ¡B admits a connecting component 0 di®erent from and contains an indecomposable direct summand N 0 of Mi . Then one of the components , 0 is a preprojective component without injective modules and the second one is a preinjective component without projective modules. We may assume ³ L B , and as above we conclude that is preprojective and 0 is preinjective. Then ¡ that admits a module Z = ½ B Y n from RA . On the other hand, we have in ind B a path Z ¡ ! ¢ ¢ ¢ ¡ ! ½ B V ¡ ! U ¡ ! V ¡ ! N 0 , and consequently Z is a predecessor of ½ A V in ind A with idA ½ A V ¶ 2, again a contradiction. Therefore, we proved that ¤i = B is a tilted algebra and i = is a connecting component of ¡¤i containing all indecomposable direct summands of Mi . Dually, if D(A) = Q © Q0 , with Q 2 add RA and Q0 without direct summands from RA , ¤0 = A (r) = End A (Q)op , ¡0 = EndA (Q 0 )op , and M 0 = D HomA (Q 0 ; Q), then A is isomorphic to the matrix algebra 2 3 0 0 6¤ M 7 4 5: 0 ¡0

Moreover, if ¤0 = ¤01 £ ¢ ¢ ¢ £ ¤0s and M 0 = M10 © ¢ ¢ ¢ © Ms0 , where ¤01 ; : : : ; ¤0s are connected algebras and Mj0 are (nonzero) ¤0j -modules (1 µ j µ s), then every ¤0j is a tilted algebra and the indecomposable direct summands of Mj0 belong to one connecting component j0 of ¡¤0j . In the ¯nal step of the proof we shall show that ¡A admits a faithful generalized standard component which is almost directed, and consequently is generalized double tilted, by Theorem 2.1. We know that ¤ = A (l) and ¤0 = A(r) are tilted factor algebras of A, and the inclusions L A ³ L ¤ and RA ³ R¤0 hold. In particular, the intersections L A \ (R¤ n L ¤ ) and RA \ (L ¤0 n R¤0) are empty. Moreover, it follows from Lemma 3.3 and its dual that L A does not contain any indecomposable ¤-module U which is a successor in ind ¤ of a module X with Hom¤ (M; ½ ¤ X ) 6= 0, and RA does not contain any indecomposable ¤0 -module V which is a predecessor in ind ¤0 of a module Y with Hom¤0(½ ¤¡0Y; M 0 ) 6= 0. Further, every indecomposable A-module L with HomA (P 0 ; L) 6= 0 does not belong to L A and every indecomposable A-module W with HomA (W; Q0 ) 6= 0 does not belong to RA . Denote by the disjoint union of the connecting components 0 the disjoint union of the connecting components j0 ; 1 µ j µ s. i ; 1 µ i µ r, and by \ L A consists of all modules Z in Then which are not successors of modules X in with Hom¤ (M; ½ ¤ X ) 6= 0 or Hom¤ (D(¤); ½ ¤ X ) 6= 0, and the remaining modules from L A are exactly the indecomposable torsion-free ¤-modules not lying in . Dually, 0 \ RA consists of all modules W in 0 which are not predecessors of modules Y in 0 with Hom¤0(½ ¤¡0Y; M 0 ) 6= 0 or Hom¤0(½ ¤¡0Y; ¤0 ) 6= 0, and the remaining modules

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119

from RA are exactly the torsion ¤0 -modules not lying in 0 . Since ind A n (L A [ RA ) is ¯nite and A is connected, we conclude that ¡A admits a connected component 00 \ L A is a fully translated subquiver of 00 which is closed under predecessuch that sors, 0 \ RA is a fully translated subquiver of 00 which is closed under successors, and 00 n ( \ L A [ 0 \ RA ) = ind A n (L A [ RA ). Observe that 00 is almost directed, \ L A and 0 \ RA are directed, and ind A n (L A [ RA ) is ¯nite. Finally, because 00 suppose that rad1 . Then, since the endomorphism A (X; Y ) 6= 0 for some X and Y in 00 n ( \ L A [ 0 \ RA ) is an artin algebra, algebra of the direct sum of modules in f0

f1

ft

there exist in¯nite paths X = X 0 ¡ ! X1 ¡ ! ¢ ¢ ¢ ¡ ! Xt ¡ ! X t+1 ¡ ! ¢ ¢ ¢ and gs g1 g0 ¢ ¢ ¢ ¡ ! Y s+1 ¡ ! Xs ¡ ! ¢ ¢ ¢ ¡ ! Y 1 ¡ ! Y0 = Y such that ft : : : f1 f0 6= 0, g 0 g 1 : : : g s 6= 0, and HomA (X t ; Ys ) 6= 0 for all s; t ¶ 0. In particular, there are modules Xt 2 0 \ RA \ L A with HomA (X t ; Ys ) 6= 0, a contradiction because 0 \ RA \ L A and and Y s 2 \ L A \ RA are empty. Therefore, 00 is a generalized standard component of ¡A which is almost directed. This completes the proof.

5

Proofs of corollaries 1.1, 1.2, and 1.3

Let A be a connected artin algebra. We shall prove that Corollaries 1.1, 1.2, 1.3 are consequences of the main results of [13, 14, 21], our main theorem and its proof. Proof (of Corollary 1.1). It follows from [21, Theorem A] that A is double tilted if and only if (L A [ ½ A¡ L A ) \ (½ A RA [ RA ) contains a directing module. Further, by [14, Proposition 3.3], if A is generalized double tilted, then ind A n (L A [ RA ) is nonempty if and only if A is not double tilted. Moreover, we know from [5] that if A is quasi-tilted but not tilted (equivalently, quasi-tilted, but not generalized double tilted) then A is of the canonical type, and then by applying results on the structure of module categories of such algebras proved in [10, Section 4], we may conclude that (L A [ ½ A¡ L A ) \ (½ A RA [ RA ) consists of nondirecting modules. Finally, for A quasi-tilted, the equality ind A = L A [RA holds. Therefore, by applying our main result, we obtain that A is generalized double tilted if and only if ind A n (L A [ RA ) is ¯nite nonempty or (L A [ ½ A¡ L A ) \ (½ A RA [ RA ) contains a directing module. Proof (of Corollary 1.2). Assume A is generalized double tilted and ¡A admits a directed connecting component . Then obviously is generalized standard and consists of directing modules. If A is double tilted (equivalently, ind A = L A [ RA ) then, by [21, Theorem A], (L A [ ½ A¡ L A ) \ (½ A RA [ RA ) contains a directing module. Assume ind A n (L A [ RA ) is nonempty. Then every module X from ind A n (L A [ RA ) lies on a path in ind A of the form I !½

AY

! U ! Y ! ¢¢¢ ! X ! ¢¢¢ ! Z ! V ! ½

¡ AZ

!P

with I injective and P projective, and such a path consists entirely of modules from . Hence ind A n (L A [ RA ) consists of directing modules. Therefore (i) implies (ii).

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Andrzej Skowro´nski / Central European Journal of Mathematics 1 (2003) 108{122

Conversely, assume that (ii) holds. If (L A [ ½ A¡ L A ) \ (½ A RA [ RA ) contains a directing module, then by applying [21, Theorem A], we may conclude that A is a double tilted algebra, and ¡A admits a directed connecting component [7, 13]. Therefore, assume that ind A n (L A [ RA ) consists of directing modules. If ind A n (L A [ RA ) is ¯nite, then by applying our main theorem we may conclude that A is generalized double tilted. Moreover, in this case ¡A admits a generalized standard connecting component with a ¯nite multisection ¢ such that ind A = L A [ ¢c [ RA and all oriented cycles in lie entirely in ¢c (see Proposition 2.2). Since ind A n (L A [ RA ) consists of directing modules, we obtain that ¢c , and hence also , is directed. Therefore, assume that ind A n (L A [ RA ) consists of in¯nitely many directing modules. We know from [19, Corollary 2] (see also [12]) that ¡A admits at most ¯nitely many ½ A -orbits containing directing modules. Hence, by duality, we may assume that there exists a directing A-module X such that all modules ½ Ar X; r ¶ 0, are contained in ind A n (L A [ RA ). Therefore, all but ¯nitely many modules ½ Ar X belong to a connected component of the left stable part l ¡A of ¡A . By now invoking the shapes of left stable components [11, Sections 2 and 3] and [19, Lemma 3] (or [4, Lemma 1.5]), we may conclude that is directed with ¯nitely many ½ A -orbits. Then there exists a fully translated subquiver of which is a fully translated subquiver of ¡A closed under predecessors; it consists of all predecessors of a module ½ Am X , for some m ¶ 0, and contains neither a projective module nor an injective module. By now applying [18, Theorem 3.2] (see also [17, Section 3]), we may infer that B = A= ann is a tilted algebra of the form B = End H (T ) for some connected hereditary artin algebra H and a tilting H -module T without preprojective direct summands. Obviously, there are no injective predecessors of modules from in ind B. Further, the Gabriel (valued) quiver of B is a convex subquiver of the Gabriel quiver of A (see [21, Proposition 4]). Since consists of modules from ind A n L A , we conclude that there is an injective A-module such that I = soc I has an indecomposable directed summand M which is a B-module, predecessor of all modules from , and not lying in . Thus M is a preprojective or regular module from the torsion-free part (T ) of mod B. Recall also that all components of (T ) di®erent from the preprojective component are obtained from components of the form ZA1 or ZA1 =(½ r ) by a ¯nite number of ray insertions (see [8] and [15]). By again applying [4, Lemma 1.5] (or [19, Lemma 3]), we may conclude that there is in ind A a path I ¡ !M ¡ !U ¡ !½

¡ BM

¡ ! ¢¢¢ ¡ ! Z ¡ ! ¢¢¢ ¡ ! ½

m AX

¡ ! ¢¢¢ ¡ ! X

with Z a nondirecting B-module. Therefore, Z is a nondirecting module from ind A n (L A [ RA ), which contradicts our assumption. Hence (ii) implies (i), and this completes the proof. Proof (of Corollary 1.3). Observe that, by our main result and [2, 13], ind An(L A [RA ) is ¯nite and nonempty if and only if A is generalized double tilted, but not double tilted. Further, by [14, Proposition 3.3], this is equivalent to the fact that A is generalized double

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121

tilted and ¡A admits a connecting component with a multisection ¢ having the nonempty core ¢c . Finally, we note that ¢c 6= ; if and only if there is in ¢ (equivalently in ¡A ) a path from an injective module to a projective module containing at least three hooks. Therefore, (i) is equivalent to (ii).

Acknowledgments The research described in this paper was initiated during a visit to the Department of Mathematics of Syracuse University (October 2000) and was completed during the author’s stay at the Centre for Advanced Study of Norwegian Academy of Science and Letters in Oslo (August/September 2001). The main results of the paper were presented during the Algebra Conference held in Venice (June 2002). The author also acknowledges support from the Foundation for Polish Science and Polish Scienti¯c KBN Grant No. 5 P03A 008 21.

References [1] M. Auslander, I. Reiten and S. O. Smal¿, \Representation Theory of Artin Algebras," Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge University Press, 1995. [2] F. U. Coelho and M. A. Lanzilotta, Algebras with small homological dimension, Manuscripta Math. 100 (1999), 1{11. [3] F. U. Coelho and M. A. Lanzilotta, Weakly shod algebras, Preprint, Sao Paulo 2001. [4] F. U. Coelho and A. Skowro¶nski, On Auslander-Reiten components of quasi-tilted algebras, Fund. Math. 143 (1996), 67{82. [5] D. Happel and I. Reiten, Hereditary categories with tilting object over arbitrary base ¯elds, J. Algebra, in press. [6] D. Happel and I. Reiten and S. O. Smal¿, Tilting in abelian categories and quasi-tilted algebras, Memoirs Amer. Math. Soc., 575 (1996). [7] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399{443. [8] O. Kerner, Stable components of tilted algebras, J. Algebra 162 (1991), 37{57. [9] M. Kleiner, A. Skowro¶nski and D. Zacharia, On endomorphism algebras with small homological dimensions, J. Math. Soc. Japan 54 (2002), 621{648. [10] H. Lenzing and A. Skowro¶nski, Quasi-tilted algebras of canonical type, Colloq. Math. 71 (1996), 161{181. [11] S. Liu, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. 47 (1993), 405{416. [12] L. Peng and J. Xiao, On the number of D Tr-orbits containing directing modules, Proc. Amer. Math. Soc. 118 (1993), 753{756. [13] I. Reiten and A. Skowro¶nski, Characterizations of algebras with small homological dimensions, Advances Math., in press.

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[14] I. Reiten and A. Skowro¶nski, Generalized double tilted algebras, J. Math. Soc. Japan, in press. [15] C. M. Ringel, \Tame Algebras and Integral Quadratic Forms," Lecture Notes in Math., Vol. 1099, Springer, 1984. [16] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. 4 (Gordon and Breach Science Publishers, Amsterdam 1992). [17] A. Skowro¶nski, Generalized standard components without oriented cycles, Osaka J. Math. 30 (1993), 515{527. [18] A. Skowro¶nski, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), 517{543. [19] A. Skowro¶nski, Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc. 120 (1994), 19{26. [20] A. Skowro¶nski, Minimal representation-in¯nite artin algebras, Math. Proc. Camb. Phil. Soc. 116 (1994), 229{243. [21] A. Skowro¶nski, Directing modules and double tilted algebras, Bull. Polish. Acad. Sci., Ser. Math. 50 (2002), 77{87. [22] A. Skowro¶nski, S. O. Smal¿ and D. Zacharia, On the ¯niteness of the global dimension of Artin rings, J. Algebra 251 (2002), 475{478.

CEJM 1 (2003) 123{140

On characterization of Poisson and Jacobi structures Janusz Grabowski¤1, PaweÃl Urba¶ nski

y2

1

Institute of Mathematics, Polish Academy of Sciences ¶ ul. Sniadeckich 8, P.O.Box 21, 00-956 Warszawa, Poland 2 Division of Mathematical Methods in Physics University of Warsaw Ho_za 74, 00-682 Warszawa, Poland

Received 23 October 2002; revised 20 December 2002 Abstract: We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids. c Central European Science Journals. All rights reserved. ® Keywords: Jacobi structures; Poisson structures; Lie algebroids; tangent lifts MSC (2000): 17B62 17B66 53D10 53D17

1

Introduction

Jacobi brackets are local Lie brackets on the algebra C 1 (M ) of smooth functions on a manifold M . This goes back to the well-known observation by Kirillov [Ki] that in the case of A = C 1 (M ) every local Lie bracket on A is of ¯rst order (an algebraic version of this fact for arbitrary commutative associative algebra A has been proved in [Gr]). Since every skew-symmetric ¯rst-order bidi®erential operator J on C 1 (M ) is of the form J = ¤ + I ^ ¡, where ¤ is a bivector ¯eld, ¡ is a vector ¯eld and I is identity, the corresponding bracket of functions reads ff; ggJ = ¤(f; g) + f ¡(g) ¡

g¡(f ):

(1)

y

¤

The Jacobi identity for this bracket is usually written in terms of the Schouten-Nijenhuis E-mail: [email protected] E-mail: [email protected]

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J. Grabowski, P. Urba´nski / Central European Journal of Mathematics 1 (2003) 123{140

bracket [[¢; ¢]] as follows: [[¡; ¤]] = 0;

[[¤; ¤]] = ¡ 2¡ ^ ¤:

(2)

Hence, every Jacobi bracket on C 1 (M ) can be identi¯ed with the pair J = (¤; ¡) satisfying the above conditions, i.e. with a Jacobi structure on M (cf. [Li]). Note that we use the version of the Schouten-Nijenhuis bracket which gives a graded Lie algebra structure on multivector ¯elds and which di®ers from the classical one by signs. The Jacobi bracket (1) has he following properties: (1) fa; bg = ¡ fb; ag (anticommutativity), (2) fa; bcg = fa; bgc + bfa; cg ¡ fa; 1gbc (generalized Leibniz rule), (3) ffa; bg; cg = fa; fb; cgg ¡ fb; fa; cgg (Jacobi identity), The generalized Leibniz rule tells that the bracket is a bidi®erential operator on C 1 (M ) of ¯rst order. In the case when ¡ = 0 (or, equivalently, when the constant function 1 is a central element), we deal with a Poisson bracket associated with the bivector ¯eld ¤ satisfying [[¤; ¤]] = 0. For a smooth manifold M we denote by ¤M the canonical Poisson tensor on T ¤ M , which in local Darboux coordinates (xl ; pj ) has the form ¤M = @pj ^ @xj . In [GU] the following characterization of Poisson tensors, in terms of the complete (tangent) lift of contravariant tensors X 7! X c from the manifold M to T M , is proved. Theorem 1.1. A bivector ¯eld ¤ on a manifold M is Poisson if and only if the tensors ¤M and ¡ ¤c on T ¤ M and T M , respectively, are ]¤ -related, where ]¤ : T ¤ M ! T M;

]¤ (!x ) = i!x ¤(x):

So, for a Poisson tensor ¤ the map ]¤ : (T ¤ M; ¤M ) ! (T M; ¡ ¤c ) is a Poisson map. The aim of this note is to generalize the above characterization including Jacobi brackets and canonical structures associated with Lie algebroids and Jacobi algebroids.

2

Lie and Jacobi algebroids

A Lie algebroid is a vector bundle ½ : E ! M , together with a bracket [[¢; ¢]] on the C 1 (M )-module Sec(E) of smooth sections of E, and a bundle morphism » : E ! T M over the identity on M , called the anchor of the Lie algebroid, such that (i) the bracket [[¢; ¢]] is R-bilinear, alternating, and satis¯es the Jacobi identity; (ii) [[X; f Y ]] = f [[X; Y ]] + » (X )(f )Y for all X; Y 2 Sec(E) and all f 2 C 1 (M ). From (i) and (ii) it follows easily (iii) » ([[X; Y ]]) = [» (X ); » (Y )] for all X; Y 2 Sec(E). We will often identify sections · of the dual bundle E ¤ with linear (along ¯bres) functions ´ ¹ on the vector bundle E: ´ ¹ (X p ) =< · (p); Xp >. If ¤ is a homogeneous (linear) 2-contravariant tensor ¯eld on E, i.e. ¤ is homogeneous of degree -1 with respect to the Liouville vector ¯eld ¢E , then < ¤; d´ ¹ « d´ º >= f´ ¹ ; ´ º g¤ is again a linear function associated with an element [· ; ¸ ]¤ . The operation [· ; ¸ ]¤ on sections of E ¤ we call the

J. Grabowski, P. Urba´nski / Central European Journal of Mathematics 1 (2003) 123{140

125

bracket induced by ¤. This is the way in which homogeneous Poisson brackets are related to Lie algebroids. Theorem 2.1. There is a one-one correspondence between Lie algebroid brackets [[¢; ¢]]¤ on the vector bundle E and homogeneous (linear) Poisson structures ¤ on the dual bundle E ¤ determined by ´ [[X;Y ]]¤ = f´ X ; ´ Y g¤ = ¤(d´ X ; d´ Y ): (3) For a vector bundle E over the base manifold M , let A(E) = ©k2Z Ak (E), V Ak (E) = Sec( k E), be the exterior algebra of multisections of E. This is a basic geometric model for a graded associative commutative algebra with unity. We will refer to elements of ­ k (E) = Ak (E ¤ ) as to k-forms on E. Here, we identify A0 (E) = ­ 0 (E) with the algebra C 1 (M ) of smooth functions on the base and Ak (E) = f0g for k < 0. Denote by jX j the Grassmann degree of the multisection X 2 A(E). A Lie algebroid structure on E can be identi¯ed with a graded Poisson bracket on A(E) of degree -1 (linear). Such brackets we call Schouten-Nijenhuis brackets on A(E). Recall that a graded Poisson bracket of degree k on a Z-graded associative commutative algebra A = ©i2Z Ai is a graded bilinear map f¢; ¢g : A £ A ! A of degree k (i.e. jfa; bgj = jaj + jbj + k) such that (1) fa; bg = ¡ (¡ 1)(jaj+k)(jbj+k) fb; ag (graded anticommutativity), (2) fa; bcg = fa; bgc + (¡ 1)(jaj+k)jbj bfa; cg (graded Leibniz rule), (3) ffa; bg; cg = fa; fb; cgg ¡ (¡ 1)(jaj+k)(jbj+k)fb; fa; cgg (graded Jacobi identity). It is obvious that this notion extends naturally to more general gradings in the algebra. For a graded commutative algebra with unity 1, a natural generalization of a graded Poisson bracket is graded Jacobi bracket. The only di®erence is that we replace the Leibniz rule by the generalized Leibniz rule fa; bcg = fa; bgc + (¡ 1)(jaj+k)jbj bfa; cg ¡

fa; 1gbc:

(4)

Graded Jacobi brackets on A(E) of degree -1 (linear) we call Schouten-Jacobi brackets. An element X 2 A2 (E) is called a canonical structure for a Schouten-Nijenhuis or Schouten-Jacobi bracket [[¢; ¢]] if [[X; X ]] = 0. As it was already indicated in [KS], Schouten-Nijenhuis brackets are in one-one correspondence with Lie algebroids: Theorem 2.2. Any Schouten-Nijenhuis bracket [[¢; ¢]] on A(E) induces a Lie algebroid bracket on A1 (E) = Sec(E) with the anchor de¯ned by » (X )(f ) = [[X; f ]]. Conversely, any Lie algebroid structure on Sec(E) gives rise to a Schouten-Nijenhuis bracket on A(E) for which A1 (E) = Sec(E) is a Lie subalgebra and » (X )(f ) = [[X; f ]]. We have the following expression for the Schouten-Nijenhuis bracket: [[X 1 ^ : : : ^ Xm ; Y1 ^ ¢ ¢ ¢ ^ Y n ]] =

(5)

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J. Grabowski, P. Urba´nski / Central European Journal of Mathematics 1 (2003) 123{140

X k;l

ck ^ : : : ^ X m ^ Y 1 ^ : : : ^ Ybl ^ : : : ^ Y n ; (¡ 1)k+l [[X k ; Yl ]] ^ : : : ^ X

where Xi ; Y j 2 Sec(E) and the hat over a symbol means that this is to be omitted. A Schouten-Nijenhuis bracket induces the well-known generalization of the standard Cartan calculus of di®erential forms and vector ¯elds [Ma, MX]. The exterior derivative d : ­ k (E) ! ­ k+1 (E) is de¯ned by the standard formula X bi ; : : : ; Xk+1)]] d· (X 1 ; : : : ; Xk+1) = (¡ 1)i+1 [[X i ; · (X1 ; : : : ; X i

+

X i<j

b i; : : : ; X bj ; : : : ; Xk+1 ); (¡ 1)i+j · ([[Xi ; X j ]]; X 1 ; : : : ; X

(6)

where X i 2 Sec(E). For X 2 Sec(E), the contraction iX : ­ p (E) ! ­ p¡1 (E) is de¯ned in the standard way and the Lie di®erential operator $X is de¯ned by the graded commutator $X = iX ¯ d + d ¯ iX : (7) Since Schouten-Nijenhuis brackets on A(E) are just Lie algebroid structures on E, by Jacobi algebroid structure on E we mean a Schouten-Jacobi bracket on A(E) (see [GM]). An analogous concept has been introduced in [IM1] under the name of a generalized Lie algebroid. Every Schouten-Jacobi bracket on the graded algebra A(E) of multisections of E turns out to be uniquely determined by a Lie algebroid bracket on a vector bundle E over M and a 1-cocycle © 2 ­ 1 (E), d© = 0, relative to the Lie algebroid exterior derivative d, namely it is of the form [IM1] [[X; Y ]]© = [[X; Y ]] + xX ^ i© Y ¡

(¡ 1)x yi© X ^ Y;

(8)

where [[; ¢; ¢]] is the Schouten bracket associated with this Lie algebroid and where we use the convention that x = jX j ¡ 1 is the shifted degree of X in the graded algebra A(E). Note that © is determined by the Schouten-Jacobi bracket by i© X = (¡ 1)x [[X; 1]]© , so that (4) is satis¯ed: [[X; Y ^ Z]]© = [[X; Y ]]© ^ Z + (¡ 1)x(y+1) Y ^ [[X; Z]]© ¡

[[X; 1]]© ^ Y ^ Z:

(9)

We already know that there is one-one correspondence between Lie algebroid structures ¤ on E and linear Poisson tensors ¤E on E ¤ . To Jacobi algebroids correspond Jacobi ¤ structures J©E on E ¤ which are homogeneous of degree -1 with respect to the Liouville vector ¯eld ¢E ¤ , namely ¤

¤

J©E = ¤E + ¢E ¤ ^ ©v ¡

I ^ ©v ;

where ©v is the vertical lift of © to a vector ¯eld on E ¤ . The above structure generates a Jacobi bracket which coincides on linear functions with the Poisson bracket associated ¤ with ¤E . One can develop a Cartan calculus for Jacobi algebroids similarly to the Lie algebroid case (cf. [IM1]). For a Schouten-Jacobi bracket associated with a 1-cocycle © the

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127

de¯nitions of the exterior di®erential d© and Lie di®erential $© = d© ¯ i + i ¯ d© are formally the same as (6) and (7), respectively. Since, for X 2 Sec(E), f 2 C 1 (M ), we have [[X; f ]]© = [[X; f ]] + (i© X )f , one obtains d© · = d· + © ^ · . Here [[¢; ¢]] and d are, respectively, the Schouten-Nijenhuis bracket and the exterior derivative associated with the Lie algebroid. Example 2.3. A canonical example of a Lie algebroid over M is the tangent bundle T M with the bracket of vector ¯elds. The corresponding complex (­ (T M ); d) is in this case the standard de Rham complex. A canonical structure for the corresponding SchoutenNijenhuis bracket is just a standard Poisson tensor. Example 2.4. A canonical example of a Jacobi algebroid is (T1 M = T M © R; (0; 1)), where T1 M is the Lie algebroid of ¯rst-order di®erential operators on C 1 (M ) with the bracket [(X; f ); (Y; g)]1 = ([X; Y ]; X (g) ¡

Y (f ));

X; Y 2 Sec(T M );

f; g 2 C 1 (M );

and the 1-cocycle © = (0; 1) is ©((X; f )) = f . A canonical structure with respect to the corresponding Schouten-Jacobi bracket on the Grassmann algebra A(T1 M ) of ¯rst-order polydi®erential operators on C 1 (M ) turns out to be a standard Jacobi structure. Indeed, it is easy to see that the Schouten-Jacobi bracket of A = A 1 + I ^ A 2 2 Aa+1 (T1 M ) and B = B1 + I ^ B2 2 Ab+1(T 1 M ) reads [[A 1 + I ^ A2 ; B1 + I ^ B2 ]]1 = [[A1 ; B1 ]] + (¡ 1)a I ^ [[A1 ; B2 ]] + I ^ [[A 2 ; B1 ]] +aA1 ^ B2 ¡

(¡ 1)a bA 2 ^ B1 + (a ¡

b)I ^ A 2 ^ B2 :

(10)

Hence, the bracket f¢; ¢g on C 1 (M ) de¯ned by a bilinear di®erential operator ¤ + I ^ ¡ 2 A(T1 M ) is a Lie bracket (Jacobi bracket on C 1 (M )) if and only if [[¤ + I ^ ¡; ¤ + I ^ ¡]]1 = [[¤; ¤]] + 2I ^ [[¡; ¤]] + 2¤ ^ ¡ = 0: We recognize the conditions (2) de¯ning a Jacobi structure on M . There is another approach to Lie algebroids. As it has been shown in [GU1, GU2], a Lie algebroid structure (or the corresponding Schouten-Nijenhuis bracket) is determined by the Lie algebroid lift X 7! X c which associates with X 2 T (E) a contravariant tensor ¯eld X c on E. The complete Lie algebroid and Jacobi algebroid lifts are described as follows. Theorem 2.5. ([GU1]) For a given Lie algebroid structure on a vector bundle E over M there is a unique complete lift of elements X 2 Sec(E ­k ) of the tensor algebra T (E) = ©k Sec(E ­k ) to linear contravariant tensors X c 2 Sec((T E)­k ) on E, such that (a) f c = ´ df for f 2 C 1 (M ); (b) X c (´ ¹ ) = ´ X ¹ for X 2 Sec(E); · 2 Sec(E ¤ ); (c) (X « Y )c = X c « Y v + X v « Y c , where X 7! X v is the standard vertical lift of tensors from T (E) to tensors from T (T E), i.e. the complete lift is a derivation with respect to the vertical lift.

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This complete lift restricted to skew-symmetric tensors is a homomorphism of the corresponding Schouten-Nijenhuis brackets: [[X; Y ]]c = [[X c ; Y c ]]:

(11)

[[X; Y ]]v = [[X c ; Y v ]]:

(12)

Moreover,

Corollary 2.6. If P 2 A2 (E) is a canonical structure for the Schouten bracket, i.e. [[P; P ]] = 0, then P c is a homogeneous Poisson structure on E. The corresponding Poisson bracket determines the Lie algebroid bracket [[¬ ; ­ ]]P = i#P ( ) d­ ¡

i#P (¯) d¬ + d(P (¬ ; ­ ))

(13)

on E ¤ . Remark. For the canonical Lie algebroid E = T M , the above complete lift gives the better-known tangent lift of multivector ¯elds on M to multivector ¯elds on T M (cf. [IY, GU]). In this case the complete lift is an injective operator, so ¤ is a Poisson tensor on M if and only if ¤c is a Poisson tensor on T M . The complete Lie algebroid lift of just sections of E, i.e. the formula (b), was already indicated in [MX1]. Let us see how these lifts look like in local coordinates. Let (xa ) be a local coordinate system on M and let e1 ; : : : ; en be a basis of local sections of E. We denote by e¤1 ; : : : ; e¤n the dual basis of local sections of E ¤ and by (xa ; y i) (resp. (xa ; ¹ i )) the corresponding coordinate system on E (resp. E ¤ ), i.e., ´ ei = ¹ i and ´ e¤ i = y i. The vertical lift is given by (ci1;:::;ik ei1 « ¢ ¢ ¢ « eik )v = ci1;:::;ik @y i1 « ¢ ¢ ¢ « @yik : If for the Lie algebroid bracket we have [ei ; ej ] = ckij ek and if the anchor sends ei to dai @xa , then 1 ¤ ¤E = ckij ¹ k @»i ^ @»j + d ai @»i ^ @xa : (14) 2 Moreover, @f a j fc = d y (15) @xa j and @X k a j (X i ei )c = X i d ai@xa + (X i ckji + d )y @yk : (16) @xa j It follows that, for P = 12 P ij ei ^ ej , we have P c = P ij daj @y i ^ @xa + (P kj cilk +

1 @P ij a l d )y @yi ^ @yj : 2 @xa l

(17)

There is an analog of the Lie algebroid complete lift for Jacobi algebroids which will represent the Schouten-Jacobi bracket on A(E) in the Schouten-Jacobi bracket of ¯rstorder polydi®erential operators on E. Here by polydi®erential operators we understand

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129

skew-symmetric multidi®erential operators. Let [[¢; ¢]]© be the Schouten-Jacobi bracket on A(E) associated with a Lie algebroid structure on E and a 1-cocycle ©. De¯nition. ([GM]) The complete Jacobi lift of an element X 2 T k (E) is the multidifferential operator of ¯rst order on E, i.e. an element of Sec((T1 E)­k ), de¯ned by b© = X c ¡ X

(k ¡

1)´ © X v + iI­d(¶© ) X v ;

(18)

where X c is the complete Lie algebroid lift, X v is the vertical lift and iI­d(¶© ) is the derivation acting on the tensor algebra of contravariant tensor ¯elds which vanishes on functions and satis¯es iI­d(¶© ) X = X (´ © )I on vector ¯elds. The derivation property yields X v v hXi ; ©iX1v « ¢ ¢ ¢ « X i¡1 « I « X i+1 « ¢ ¢ ¢ « Xkv : iI­d(¶© ) (X 1v « ¢ ¢ ¢ « Xkv ) = i

for X 1 : : : ; X k 2 Sec(E). Theorem 2.7. ([GM]) The complete Jacobi lift has the following properties: (a) fb© = ´ d© f for f 2 C 1 (M ); b© = X c + (i© X )v I for X 2 Sec(E); (b) X \ b© « Y v + X v « Yb© ¡ ´ © (X v « Y v ); « Y )© = X (c) (X (d) For skew-symmetric tensors X and Y , b© ; Yb© ]]1 = ([[X; Y ]]© )^ ; [[X ©

where [[¢; ¢]]1 is the Schouten-Jacobi bracket of ¯rst-order polydi®erential operators; (e) For skew-symmetric X and Y b© ; Y v ]]1 = ([[X; Y ]]© )v ; [[X

We remark that in [GM] only skew-symmetric tensors have been considered, but the extension to arbitrary tensors is straightforward. De¯nition. ([GM]) The complete Poisson lift of an element X 2 T k (E) is the contravarib c 2 Sec((T E)­k ), de¯ned by ant tensor X © bc = Xc ¡ X ©

(k ¡

1)´ © X v + i¢E­d(¶© ) X v ;

(19)

where ¢E is the Liouville vector ¯eld on the vector bundle E and X c is the complete Lie algebroid lift, X v is the vertical lift and i¢E­d(¶© ) is the derivation acting on the tensor algebra of contravariant tensor ¯elds which vanishes on functions and satis¯es i¢­d(¶© ) X = X (´ © )¢E on vector ¯elds. Theorem 2.8. ([GM]) The Poisson lift has the following properties: (a) fb©c = ´ d© f for f 2 C 1 (M ); b c (´ ¹ ) = ´ © for X 2 Sec(E), · 2 Sec(E ¤ ); (b) X © X¹

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J. Grabowski, P. Urba´nski / Central European Journal of Mathematics 1 (2003) 123{140 c

\ b c « Y v + X v « Yb c ¡ « Y )© = X (c) (X © © (d) For skew-symmetric X and Y

´ © (X v « Y v );

b c ; Yb c ]] = ([[X; Y ]]© )^c : [[X © © ©

Corollary 2.9. If P 2 A2 (E) is a canonical structure for the Schouten-Jacobi bracket, i.e. [[P; P ]]© = [[P; P ]] + 2P ^ i© P = 0, then Pb© (resp. Pb©c ) is a homogeneous Jacobi (resp. homogeneous Poisson) structure on E. The corresponding Jacobi and Poisson brackets coincide on linear functions and determine the Lie algebroid bracket [[¬ ; ­ ]]P = i#P ( ) d© ­ ¡

i#P (¯) d© ¬ + d© (P (¬ ; ­ ))

(20)

on E ¤ .

3

Characterization of Poisson tensors.

Theorem 1.1 of the Introduction can be generalized in the following way. Let us remark ¯rst that any two-contravariant tensor ¤ (which is not assumed to be skew-symmetric) de¯nes a bracket [¢; ¢]¤ on 1-forms on M by [· ; ¸ ]¤ = i]¤

(¹) d¸

¡

i]¤

(º) d·

+ d < ¤; · « ¸ >;

(21)

where < ¢; ¢ > is the canonical pairing between contravariant and covariant tensors. Theorem 3.1. For a two-contravariant tensor ¤ on a manifold M the following are equivalent: (i) ¤ is a Poisson tensor; (ii) ]¤ induces a homomorphism of [¢; ¢]¤ into the bracket of vector ¯elds: ]¤ ([· ; ¸ ]¤ ) = []¤ (· ); ]¤ (¸ )];

(22)

(iii) The canonical Poisson tensor ¤M and the negative of the complete lift ¡ ¤c are ]¤ -related; (iv) There is a vector bundle morphism F : T ¤ M ! T M over the identity on M such that the canonical Poisson tensor ¤M and the negative of the complete lift ¡ ¤c are F -related; (v) The morphism ]¤ relates ¤M with the complete lift of a 2-contravariant tensor ¤1 . (vi) There is a vector bundle morphism F : T ¤ M ! T M over the identity on M such that F ([· ; ¸ ]¤ ) = [F (· ); F (¸ )]: (23) (vii) There is a 2-contravariant tensor ¤1 on M such that ]¤ ([· ; ¸ ]¤1 ) = []¤ (· ); ]¤ (¸ )];

(24)

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131

Proof. The implication (i) ) (ii) is a well-known fact (cf. e.g. [KSM]). Assume now (ii). To show (iii) one has to prove that the brackets on functions f¢; ¢g¤M and f¢; ¢g¤c induced by tensors ¤M and ¤c by contractions with di®erentials of functions are ]¤ -related, i.e. ¡ ff; gg¤c ¯ ]¤ = ff ¯ ]¤ ; g ¯ ]¤ g¤M

(25)

for all f; g 2 C 1 (T M ). Due to the Leibniz rule, it is su±cient to check (25) for linear functions, i.e. for functions of the form ´ ¹ , where · is a 1-form and ´ ¹ (vx ) =< · (x); vx >. It is well known (see [Co, GU]) that the brackets induced by ¤ and its complete lift are related by f´ ¹ ; ´ º g¤c = ´ [¹;º]¤ : (26) It is also known (cf. (3)) that ´ for vector ¯elds X; Y on M . Since ´

= f´

[X;Y ] ¹

X ; ´ Y g¤ M

¯ ]¤ = ¡ ´

¡ f´ ¹ ; ´ º g¤c ¯ ]¤ = ¡ ´ =´

[¹;º]¤

]¤ (¹) ,

¯ ]¤ = ´

[]¤ (¹);]¤ (º)]

= f´

¹

we get

¯ ]¤ ; ´

º

= f´

]¤ ([¹;º]¤ )

(27)

]¤ (¹) ; ´ ]¤ (º) g¤M

¯ ] ¤ g¤ M

which proves (ii) ) (iii). In fact, (27) proves equivalence of (ii) and (iii). Replacing in (27) the mapping ]¤ by a vector bundle morphism F : T ¤ M ! T M , we get equivalence of (iv) and (vi). Similarly, (v) is equivalent to (vii). The implication (iii) ) (iv) is obvious, so let us show (iv) ) (i). Assume that F relates ¤M and ¤c . We will show that this implies that ¤ is skew-symmetric and F = ]¤ . Since the assertion is local over M we can use coordinates (xa ) in M and the adapted coordinate systems (xa ; pi ) in T ¤ M and (xa ; x_ j ) in T M . Writing ¤ = ¤ij @xi « @xj and F (xa ; pi) = (xa ; F ij pi ), we get @F sk ij ^ i F ¤ (@pi @x ) = F ps @x_ j ^ @x_ k ¡ F ij @xi ^ @x_ j : (28) i @x Since @¤ij k ¤c = x_ @x_ i « @x_ j + ¤ij (@xi « @x_ j + @x_ i « @xj ); (29) @xk comparing the vertical-horizontal parts we get ¤ij = F ij = ¡ F ji , i.e. ¤ is skew-symmetric and F = ]¤ . Going backwards with (27) we get (ii). But for skew tensors we have (cf. [KSM]) 1 [[¤; ¤]](· ; ¸ ; ® ) =< ]¤ ([· ; ¸ ]¤ ) ¡ []¤ (· ); ]¤ (¸ )]; ® >; (30) 2 where [[¢; ¢]] is the Schouten-Nijenhuis bracket, so that [[¤; ¤]] = 0, i.e. ¤ is a Poisson tensor. Finally, (v) is equivalent to (iii), since (iii) ) (v) trivially and exchanging the role of F and ¤ in (28) and (29) we see that, as above, ¤ij = F ij , so that any tensor whose complete lift is ]¤ -related to ¤M equals ¡ ¤.

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A similar characterization is valid for any Lie algebroid. Let us consider a vector bundle E over M with a Lie algebroid bracket [[¢; ¢]] instead of the canonical Lie algebroid T M of vector ¯elds (cf. [Ma, KSM, GU0]). The multivector ¯elds are now replaced by Vk multisections A(E) = ©k Ak (E), Ak (E) = Sec( E), of E and the standard SchoutenNijenhuis bracket with its Lie algebroid counterpart. A Lie algebroid Poisson tensor (canonical structure) is then a skew-symmetric ¤ 2 A2 (E) satisfying [[¤; ¤]] = 0. Such a structure gives a triangular Lie bialgebroid in the sense of [MX]. We have the exterior derivative d on multisections of the dual bundle E ¤ (we will refer to them as to "exterior forms"). For any ¤ 2 Sec(E « E) the formula (21) de¯nes a bracket on "1-forms". We have an analog of the complete lift (cf. [GU1, GU2]) Sec(E ­k ) 3 ¤ 7! ¤c 2 Sec((T E)­k ) of the tensor algebra of sections of E into contravariant tensors on the total space E. The ¤ Lie algebroid bracket corresponds to a linear Poisson tensor ¤E on E ¤ (which is just ¤M ¤ in the case E = T M ) by (3). Since the tensor ¤E may be strongly degenerate, linear maps F : E ¤ ! E do not determine the related tensors uniquely, so we cannot have the full analog of Theorem 3.1. However, since for skew-symmetric tensors the formula (30) remains valid [KSM], a part of Theorem 3.1 can be proved in the same way, mutatis mutandis, in the general Lie algebroid case. Thus we get the Lie algebroid version of Theorem 1.1 (cf. [GU1]). Theorem 3.2. For any bisection ¤ 2 A2 (E) of a Lie algebroid E the following are equivalent: (i) ¤ is a canonical structure, i.e. [[¤; ¤]] = 0; (ii) ]¤ induces a homomorphism of [¢; ¢]¤ into the Lie algebroid bracket: ]¤ ([· ; ¸ ]¤ ) = []¤ (· ); ]¤ (¸ )];

(31)

(iii) The canonical Poisson tensor ¤M and the negative of the complete lift ¡ ¤c are ]¤ -related.

4

Jacobi algebroids and characterization of Jacobi structures

We have introduced in Section 2 Jacobi and Poisson complete lifts related to Jacobi algebroids. For a standard Jacobi structure J = (¤; ¡) on M we will denote these lifts of J by Jb and Jbc , respectively. The Jacobi lift Jb is the Jacobi structure on E = T M © R given by [GM] Jb = (¤c ¡ t¤v + @t ^ (¡c ¡ t¡v ); ¡v ); (32)

where ¤v and ¡v are the vertical tangent lifts of ¤ and ¡, respectively, and t is the standard linear coordinate in R. We consider here tangent lifts as tensors on T M © R = T M £ R instead on T M . The linear Jacobi structure (32) has been already considered by Iglesias and Marrero [IM].

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133

Similarly, the Poisson lift Jbc is the linear Poisson tensor on T M © R given by [GM] Jbc = ¤c ¡

t¤v + @t ^ ¡c + ¢T M ^ ¡v ;

(33)

where ¢T M is the Liouville (Euler) vector ¯eld on the vector bundle T M . This is exactly the linear Poisson tensor corresponding to the Lie algebroid structure on T ¤ M ©R induced by J and discovered ¯rst in [KSB]: [(¬ ; f ); (­ ; g)]J = (L

¡

]¤ ( ) ­

L

¡

]¤ (¯) ¬

d < ¤; ¬ ^ ­ > +f L

< ¤; ­ ^ ¬ > +]¤ (¬ )(g) ¡

¡

¡­

gL

]¤ (­ )(f ) + f ¡(g) ¡

¡¬

¡

i¡ (¬ ^ ­ );

g¡(f ));

(34)

Of course, these lifts and an analog of the bracket (34) are well-de¯ned for any ¯rst-order bidi®erential operator J = ¤ + I « ¡1 + ¡ 2 « I + ¬ I « I ; (35) where ¤ is a 2-contravariant tensor, ¡1 ; ¡2 are vector ¯elds, and ¬ The associated bracket acts on functions on M by

is a function on M .

ff; ggJ =< ¤; df « dg > +f ¡1 (g) + g¡2 (f ) + ¬ f g; The Jacobi lift of J is the ¯rst-order bidi®erential operator on T M © R given by Jb = ¤c ¡

t¤v + @t « (¡c1 ¡

+I «

(¡v1

+¬

v

@t ) +

t¡v1 ) + (¡c2 ¡ (¡v2

+¬

v

t¡v2 ) « @t + (¬

c

¡

t¬

v

t¬

v

)@t « @t

@t ) « I

and the Poisson lift is the 2-contravariant tensor ¯eld Jbc = ¤c ¡ =

t¤v + @t « (¡c1 ¡

+¢E « (¡v1 c v

t¡v1 ) + (¡c2 ¡

t¡v2 ) « @t + (¬

+ ¬ @t ) + (¡v2 + ¬ v @t ) « ¢E ¤ ¡ t¤ + @t « ¡c1 + ¡c2 « @t + (¬ c + t¬ v )@t +¢T M « ¡v1 + ¡v2 « ¢T M :

c

¡

)@t « @t

v

« @t

The mapping ]J : E ¤ = T ¤ M © R ! E = T M © R reads ]J (!x; ¶ ) = (]¤ (!x ) + ¶ ¡1 (x); ¡2 (x)(!x ) + ¬ (x)¶ ): Note that any morphism from the vector bundle E ¤ = T ¤ M © R into E = T M © R over the identity on M is of this form. The bidi®erential operators Jb and Jbc de¯ne brackets f¢; ¢gJb and f¢; ¢gJbc , respectively, on functions on T M © R. These brackets coincide on linear functions which close on a subalgebra with respect to them, so that they de¯ne the bracket [¢; ¢]J on sections of T ¤ M © R (which coincides with the bracket (34) for skew-symmetric operators) by f´

(¹;f ) ; ´ (º;g) gJb

= f´

(¹;f ) ; ´ (º;g) gJ c

=´

[(¹;f );(º;g)]J ;

where · ; ¸ are 1-forms, f; g are functions on M , and ´ (¹;f ) = ´ ¹ + tf v . Here we identify T ¤ M ©R with T ¤ M £R and use the linear coordinate ¶ in R. For the similar identi¯cation of T M © R we use the coordinate t of R in T M £ R, since both R’s play dual roles.

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We have two canonical structures on the vector bundle E ¤ = T ¤ M © R ’ T ¤ M £ R. One is the Jacobi structure (bracket) JM = ¤M + ¢ T ¤

M

^ @ ¸ + @¸ ^ I

(36)

and the other is the Poisson structure ¤M regarded as the product of ¤M on T ¤ M with the trivial structure on R. These brackets coincide on linear functions which close on a subalgebra with respect to both brackets, so that they de¯ne a Lie algebroid structure on the dual bundle E = T M © R. This is the Lie algebroid of ¯rst-order di®erential operators with the bracket [(X; f ); (Y; g)]1 = ([X; Y ]; (X (g) ¡

Y (f ));

where X; Y are vector ¯elds and f; g are functions on M . Theorem 4.1. For a ¯rst-order bidi®erential operator J the following are equivalent: (J1) J is a Jacobi bracket; (J2) The canonical Jacobi bracket JM and ¡ Jb are ]J -related; (J3) There is a ¯rst-order bidi®erential operator J1 such that JM and ¡ Jb are ]J1 -related; (J4) There is a ¯rst-order bidi®erential operator J1 such that JM and ¡ Jb1 are ]J -related; (J5) The contravariant tensors ¤M and ¡ Jbc are ]J -related; (J6) There is a ¯rst-order bidi®erential operator J1 such that ¤M and ¡ Jbc are ]J1 -related; (J7) There is a ¯rst-order bidi®erential operator J1 such that ¤M and ¡ Jb1c are ]J -related; (J8) For any 1-forms · ; ¸ and functions f; g on M ]J ([(· ; f ); (¸ ; g)]J ) = []J (· ; f ); ]J (¸ ; g)]1 :

(J9) There is a ¯rst-order bidi®erential operator J1 such that ]J1 ([(· ; f ); (¸ ; g)]J ) = []J1 (· ; f ); ]J1 (¸ ; g)]1 : (J10) There is a ¯rst-order bidi®erential operator J1 such that ]J ([(· ; f ); (¸ ; g)]J1 ) = []J (· ; f ); ]J (¸ ; g)]1 : Before proving this theorem we introduce some notation and prove a lemma. For a ¯rst-order bidi®erential operator J as in (35), the poissonization of J is the tensor ¯eld on M £ R of the form PJ = e¡s (¤ + @s « ¡1 + ¡2 « @s + ¬ @s « @s );

(37)

where s is the coordinate on R. Identifying T ¤ (M £R) with T ¤ M £T ¤ R (with coordinates (s; ¶ ) in T ¤ R) and T (M £ R) with T M £ T R (with coordinates (s; t) in T R) we can write ¤M £R = ¤M + @¸ ^ @s ; PJc = e¡s (¤c ¡

t¤v + @t « (¡c1 ¡

+@s « (¡v1 + ¬

v

t¡v1 ) + (¡c2 ¡

@t ) + (¡v2 + ¬

v

t¡v2 ) « @t

@t ) « @s + (¬

c

]PJ (!x ; ¶ s ) = e¡s (]¤ (!x ) + ¶ s ¡1 (x); ¡2 (x)(!x ) + ¶ s ¬ (x)):

¡

t¬

v

)@t « @t ;

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135

In local coordinates x = (xl ) on M and adapted local coordinates (x; p) on T ¤ M and (x; x) _ on T M we have (xl ; s; x_ i ; t) ¯ ]PJ = (xl ; s; e¡s (¤kipk + ¶ ¡i1 ); e¡s (¡k2 pk + ¶ ¬ )) for ¤ = ¤ij @xi « @xj , ¡u = ¡ku @xk , u = 1; 2. It is well known that P J is a Poisson tensor if and only if J is a Jacobi structure [Li, GL]. In view of Theorem 3.2, we can conclude that J is a Jacobi structure if and only if ¤M £R and P Jc are related by the map ]PJ : T ¤ (M £ R) ! T (M £ R). Since T (M £ R) ’ E £ R and T ¤ (M £ R) ’ E ¤ £ R, we can consider the bundles E = T M © R and E ¤ = T ¤ M © R as submanifolds of T (M £ R) and T ¤ (M £ R), respectively, given by the equation s = 0. For any function ¿ 2 C 1 (E) we denote by ¿ ¸ the function on T (M £ R) = E £ R given by ¿ ¸(vx ; s) = es ¿ (vx ). Similarly, for any function ’ 2 C 1 (E ¤ ) we denote by ’~ the function on T ¤ (M £ R) = E ¤ £ R given by ’(u ~ x ; s) = es ’(e¡s ux). It is a matter of easy calculations to prove the following. Lemma 4.2. (a) The maps ¿ 7! ¿ ¸ and ¿ 7! ¿ ~ are injective. (b) For any ¯rst-order bidi®erential operator J, » ¿ ¸ ¯ ]PJ = (¿ ¯ ]J ) :

(c) For any ¿ ; Ã 2 C 1 (E), (d) For any ¿ ; Ã 2 C 1 (E ¤ ),

¸ P c = (f¿ ; Ãg b)^ : f¿ ¸; Ãg J J » ~ ¤ f¿ ~; Ãg = (f¿ ; ÃgJM ) : M£ R

(e) For linear ¿ ; Ã 2 C 1 (E), ¸ P c = (f¿ ; ÃgJ c )^ : f¿ ¸; Ãg J (f) For linear ¿ ; Ã 2 C 1 (E ¤ ), » ~ ¤ f¿ ~; Ãg = (f¿ ; Ãg¤M ) : M£ R

Proof of Theorem 4.1. Due to the above Lemma the following identities are valid for arbitrary ¿ ; à 2 C 1 (E) and arbitrary ¯rst-order bidi®erential operators J; J1 : » ¸ P c ¯ ]P (f¿ ; ÃgJb ¯ ]J1 ) = (f¿ ; ÃgJb)^ ¯ ]PJ1 = f¿ ¸; Ãg J1 J » » » ¸ (f¿ ¯ ]J1 ; à ¯ ]J1 gJM ) = f(¿ ¯ ]J1 ) ; (à ¯ ]J1 ) g¤M£R = f¿ ¯ ]PJ1 ; ø ¯ ]PJ1 g¤M£R :

Thus if and only if

¡ f¿ ; ÃgJb ¯ ]J1 = f¿ ¯ ]J1 ; à ¯ ]J1 gJM ¸ P c ¯ ]P = f¿ ¸ ¯ ]P ; ø ¯ ]P g¤ ¡ f¿ ¸; Ãg ; J1 J1 J1 M£ R J

(38)

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which means that JM and ¡ Jb are ]J1 -related if and only if ¤M £R and the complete lift of the poissonization ¡ P Jc are ]PJ1 -related. Due to Theorem 3.2, we get that P J1 = P J and the poissonization PJ is a Poisson tensor which, in turn, is equivalent to the fact that J is a Jacobi bracket. Thus we get (J 1) , (J 2) , (J3) , (J 4): Using now linear functions ¿ ; Ã, we get in a similar way that (38) is equivalent to ¡ f¿ ; ÃgJbc ¯ ]J1 = f¿ ¯ ]J1 ; Ã ¯ ]J1 g¤M

which, due to Theorem 3.2, gives

(J 1) , (J 5) , (J6) , (J 7): Finally, completely analogously to (27) we get (J5) , (J8) , (J 9) , (J10). Remark. In the above proof we get the lifts Jb, J c , and the map ]J in a natural way by using the poissonization and its tangent lift. This is a geometric version of the methods in [Va] for obtaining J c . Note also that JM is the canonical Jacobi structure on T ¤ M £ R regarded as a contact manifold in a natural way and that the equivalence (J 1) , (J 8) is a version of the characterization in [MMP]. The above theorem characterizing Jacobi structures one can generalize to canonical structures associated with Jacobi algebroids as follows. Consider now a Jacobi algebroid, i.e. a vector bundle E over M equipped with a Lie algebroid bracket [¢; ¢] and a `closed 1-form’ © 2 ­ 1 (E). We denote by [[; ¢; ¢]] the SchoutenNijenhuis bracket of the Lie algebroid and by T (E) 3 X 7! X c 2 T (T E) the complete lift from the tensor algebra of E into the tensor algebra of T E. The corresponding SchoutenJacobi bracket we denote by [[¢; ¢]]© and the corresponding complete Jacobi and Poisson b© 2 T (T E) and T (E) 3 X 7! X c 2 T (T E), respectively. lifts by T (E) 3 X 7! X © If the 1-cocycle © is exact, © = ds, we can obtain the bracket [[¢; ¢]]© from [[¢; ¢]] using the linear automorphism of A(E) de¯ned by Ak (E) 3 X 7! e¡(k¡1)s X (cf. [GM1]). This is a version of the Witten’s trick [Wi] to obtain the deformed exterior di®erential d© · = d· + © ^ · via the automorphism of the cotangent bundle given by multiplication by es . Even if the 1-cocycle © is not exact, there is a nice construction [IM1] which allows one b = E £ R over to view © as being exact but for an extended Lie algebroid in the bundle E M £R. The sections of this bundle may be viewed as parameter-dependent (s-dependent) sections of E. The sections of E form a Lie subalgebra of s-independent sections in the b which generate the C 1 (M £ R)-module of sections of E b and the whole Lie algebroid E structure is uniquely determined by putting the anchor » b(X ) of a s-independent section X to be » b(X ) = » (X ) + h©; X i@s , where s is the standard coordinate function in R and » is the anchor in E. All this is consistent (thanks to the fact that © is a cocycle) and b with the exterior derivative d satisfying ds = ©. de¯nes a Lie algebroid structure on E

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137

b be the natural embedding of the tensor algebra of E Let now U : T (E) ! T (E) b It is obvious that on skewinto the tensor subalgebra of s-independent sections of E. symmetric tensors U is a homomorphism of the corresponding Schouten brackets: [[U (X ); U (Y )]]^= U ([[X; Y ]]); b respecwhere we use the notation [[¢; ¢]] and [[¢; ¢]]^ for the Schouten brackets in E and E, b by putting tively. Let us now gauge T (E) inside T (E) P © (X ) = e¡ks U (X )

for any element X 2 Sec(E ­(k+1) ). Note that X 7! P © (X ) preserves the grading but not the tensor product. It can be easily proved (cf. [GM1]) that the Schouten-Jacobi bracket [[¢; ¢]]© can be obtained by this gauging from the Lie algebroid bracket. Theorem 4.3. ([GM1]) For any X 2 A(E); Y 2 A( E) we have [[P © (X ); P © (Y )]]^= P © ([[X; Y ]]© ):

(39)

We will usually skip the symbol U and write simply P © (X ) = e¡ks X , regarding T (E) b b will be denoted by as embedded in T (E). The complete lift for the Lie algebroid E X 7! X bc to distinguish from the lift for E. It is easy to see that (P © (X ))bc = (e¡ks X )bc = e¡ks (X c ¡

k´ © X v + @s ^ (i© X )v ):

b = E £ R in obvious way. Note that Here we understand tensors on E as tensors on E ¤ b d ¤ ) = (E) b ¤ and the linear Poisson tensor ¤E reads (E b¤

¤

¤ E = ¤ E + ©v ^ @ s ;

¤

where ¤E is the Poisson tensor corresponding to the Lie algebroid E and ©v is the vertical lift of ©. Recall that on E ¤ we have also a canonical Jacobi structure ¤

¤

J©E = ¤E + ¢E ¤ ^ ©v ¡

I ^ ©v ¤

which generates a Jacobi bracket which coincides with the Poisson bracket of ¤E on linear functions. Let us remark that the map P © plays the role of a generalized poissonization. Indeed, for the Jacobi algebroid of ¯rst-order di®erential operators E = T M © R the extended b £ R is canonically isomorphic with T (M £ R), U ((X; f )) = X + f @s , and Lie algebroid E for J 2 Sec(E ­2 ) the tensor ¯eld P © (J) coincides with (37). Let now J 2 A2 (E). The tensor J is a canonical structure for the Jacobi algebroid (E; ©), i.e. [[J; J ]]© = 0, if and only if P © (J ) is a canonical structure for the Lie algebroid b i.e. [[P © (J ); P © (J )]]^= 0. Moreover, E, ]P ©

(J) (ux ; s)

= (e¡s ]J (ux); s):

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J. Grabowski, P. Urba´nski / Central European Journal of Mathematics 1 (2003) 123{140

b = E £ R given Like above, for any function ¿ 2 C 1 (E) we denote by ¿ ¸ the function on E 1 ¤ s by ¿ ¸(vx ; s) = e ¿ (vx ) and for any function ’ 2 C (E ) we denote by ’~ the function on b ¤ = E ¤ £ R given by ’(u E ~ x ; s) = es ’(e¡s ux ). Recall that (cf. Section 2) Jb© = J c ¡

and

Jb©c = J c ¡

´ © J v + I ^ (i© J)v

´ © J v + ¢E ^ (i© J)v :

The corresponding brackets on functions on E coincide on linear functions and de¯ne a bracket [¢; ¢]J on sections of E ¤ in the standard way: ´

[¹;º]J

= f´ ¹ ; ´ º gJb© = f´ ¹ ; ´ º gJbc ©

Completely analogously to Lemma 4.2 we get the following. Lemma 4.4. (a) The maps ¿ 7! ¿ ¸ and ’ 7! ’~ are injective. (b) For any J 2 A2 (E) » ¿ ¸ ¯ ]P © (J) = (¿ ¯ ]J ) : (c) For any ¿ ; Ã 2 C 1 (E) ¸ (P © f¿ ¸; Ãg (d) For any ¿ ; Ã 2 C 1 (E ¤ )

(e) For linear ¿ ; Ã 2 C 1 (E)

= (f¿ ; ÃgJb© )^ :

~ Eb ¤ = (f¿ ; Ãg E¤ )» : f¿ ~; Ãg J© ¤ ¸ (P © f¿ ¸; Ãg

(f) For linear ¿ ; Ã 2 C 1 (E ¤ )

(J))cb

(J))cb

= (f¿ ; ÃgJbc )^ :

~ Eb ¤ = (f¿ ; Ãg¤E¤ )» : f¿ ~; Ãg ¤ Now, repeating the arguments from the classical case, one easily derives the following. Theorem 4.5. For any bisection J 2 A2 (E) of the vector bundle E of a Jacobi algebroid (E; ©) the following are equivalent: (1) J is a canonical structure, i.e. [[J; J ]]© = 0; ¤ (2) The canonical Jacobi bracket J©E and ¡ Jb© are ]J -related; ¤ (3) The bivector ¯elds ¤E and ¡ Jb©c are ]J -related; (4) For any `1-forms’ · ; ¸ 2 ­ 1 (E), ]J ([· ; ¸ ]J ) = []J (· ); ]J (¸ )];

where the bracket on the right-hand-side is the Lie algebroid bracket on E. Note that a canonical structure for a Jacobi algebroid gives rise to a triangular Jacobi bialgebroid [GM] (or a triangular generalized Lie bialgebroid in the terminology of [IM1]).

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139

Acknowledgments Supported by KBN, grant No 2 P03A 041 18.

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