CC BY
37
Вестник Челябинского государственного университета. 2015. № 3 (358).
Математика. Механика. Информатика. Вып. 17. С. 50-61.
УДК 512.7
ББК В144.5
ALGEBRAIC G-FUNCTIONS ASSOCIATED TO MATRICES OVER A GROUP-RING*
S. Garoufalidis, J. Bellissard
Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic G-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman's theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a G-function. We ask whether the latter holds for general hyperbolic groups.
Keywords: rational function, algebraic function, holonomic function, G-function, generating series, non-commuting variables, moment, hamiltonian, resolvant, regular language, context-free language, Hadamard product, group-ring, free probability, Schur complement method, free group, von Neumann algebra, polynomial Hamiltonian,spectral theory,norm.
1. Introduction
1.1. Algebricity of the Green's function for the free group
Given a group G, consider the group-algebra Q[G], and define a trace map:
Tr : Q[G] ^ C, Tr(P) = constant term of P, where the constant term is the coefficient of the identity element of G. Let MN (R) denote the set of N by N matrices with entries in a ring R. We can extend the trace to the algebra Mn (Q[G]) by:
N
Tr : Mn (<Q>[G] ) ^ C, Tr(P) = ).
j=1
Definitiom 1. Given P e MN(Q[G]), consider the sequence (aPn), aPn = Tr(Pn), and the
generating series RP (z) = ^aPnzn.
n=0
Let Fr denote the free group of rank r.
Theorem 1. The Green's function RP(z) of every element P of MN(Q[Fr]) is algebraic. Theorem 1 appears in the cross-roads of several areas of research:
(a) operator algebras;
(b) free probability;
(c) linguistics and context-free languages;
(d) non-commutative combinatorics;
(e) mathematical physics.
In fact, Woess proves Theorem1 when N = 1 using linguistics and context-free languages; see [1; 2]. In [3] Sauer also gives a proof using linguistics, with emphasis the rationality of the Novikov-Shubin invariants. Voiculescu proves Theorem 1 using the R and S transforms of free probability; see [4; 5]. For additional results using free probability, see [6; 7] and also [8 — 10].
* Work is supported in part by the National Science Foundation.
It is well-known that Theorem 1 provides an exact calculation of the norm of P e MN (Q[Fr ]) c MN (L(Fr)) , where L(Fr) denotes the reduced C*-algebra completion of the group-algebra C[Fr]. For a detailed discussion, see the above references.
Our proof of Theorem 1 uses the notion of an algebraic function in non-commuting variables and a theorem of Haiman, which itself is based on a theorem of Chomsky - Schutzenberger on context-free languages. A by-product of our proof is the fact that the moment generating series is a matrix of algebraic power series in non-commuting variables (see Proposition 1), which is a statement a priori stronger than Theorem 1.
An alternative proof of Theorem 1 uses methods from functional analysis, and most notably the Schur complement method (see below). We will discuss in detail the first proof and postpone the third proof to a later publication. Either proof explains the close relation between the differential properties of the generating function RP (z) and the word problem in G.
Our aim is to give a proof of algebraicity in the case of the free group, discuss holonomicity in the case of the free abelian group and formulate a question regarding holonomicity for hyperbolic groups. As it turns out, algebricity is well-studied in the above mentioned literature whereas holonomicity is largely absent.
1.2. Related work
Our paper was completed in the summer of 2007, and posted on the arxiv arXiv:0708.4234. In the fall of 2008, M. Kontsevich brought to the attention of the second author a related earlier paper of Sauer [3] from 2003 that gives a proof of Theorem 1 with emphasis on the Novikov-Shubin invariants. Sauer's and our work has been cited by M. Kontsevich in the Arbeitstagung talk Bonn 2011, and (from what we have heard) in other talks too. Theorem 1 keeps attracting attention in diverse areas of mathematics. In the summer 2013, C. Kassel informed the second author of related article of Kassel-Reutenauer [11] around the theme of Theorem 1. Kassel was unaware of Sauer's work and of our work. In view of the interest of Theorem 1 and its connections to several branches of mathematics, we were encouraged to submit our article for publication.
2. The case of the free abelian group
2.1. Holonomic, algebraic and G-functions
A priori, RP(z) is only a formal power series. However, it is easy to see that (aPn) is bounded exponentially by n, which implies that RP(z) defines an analytic function in a neighborhood of z = 0. The paper is concerned with differential / algebraic properties of the function RP(z). Algebraic and holonomic functions are well-studied objects. Let us recall their definition here.
Definition 2. (a) A holonomic function f(z) is one that satisfies a linear differential equation with polynomial coefficients. In other words, we have:
cd (z)f (d)(z) + ... + c,(z)f (z) = 0
where cj(z) e <Q>[z] for all j = 0,., d and f(j)(z) = (dj/ dzj)f(z).
(b) An algebraic function f(z) is one that satisfies a polynomial equation Q(f(z), z) = 0 where Q(y, z) e Q[y, z].
Lesser known to the combinatorics community are G-functions, which originated in the work Siegel on arithmetic problems in elliptic integrals, and transcendence problems in number theory; see [12]. G-functions originate naturally in:
(a) algebraic geometry, related to the regularity properties of the Gauss-Manin connection, see for example [13 — 15];
(b) arithmetic, see for example [16—18];
(c) enumerative combinatorics, as was recently shown in [19].
œ
Definition 3. A G-function f(z) = yanzn is one which satisfies the following conditions:
n=0
(a) for every n e N , we have an e Q ;
(b) there exist a constant Cf >0 such that for every n e N we have: | an | ^Cj (for every conjugate of an) and the common denominator of a0,..., an is less than or equal to Cf ;
(c) f(z) is holonomic.
The next theorem summarizes the analytic continuation and the shape of the singularities of algebraic functions and G-functions. Part (a) follows from the general theory of differential equations (see eg. [20]), parts (b) and (d) follow from [21. Lem. 2.2] (see also [18] and [22]) and (c) follows from a combination of Katz's theorem, Chudnovsky's theorem and André's theorem; see [16. P. 706] and also [23].
Theorem 2. (a) A holonomic function f(z) can be analytically continued as a multivalued function in C \ §f where §f c Q is the finite set of singular points of f(z).
(b) Every algebraic function f(z) is a G-function.
(c) In a neighborhood of a singular point Xe §f, a G-function f(z) can be written as a finite sum of germs of the form:
(z -X)ax (log(z -X))px h (z -X) (2.1)
where aX e Q, PX e N ,and hX a holonomic G-function.
(d) In addition, pX =0 if f(z) is algebraic.
Remark 1. Local expansions of the form (2.1) are known in the literature as Nilsson series (see [24]), and minimal order linear differential equations that they satisfy are known to be regular singular, with rational exponents {aX} and quasi-unipotent monodromy. For a discussion, see [14; 15; 19] and references therein.
It is classical and easy to show that the existence of analytic continuation of a function implies the existence of asymptotic expansion of its Taylor series; see for example [12; 25] and also [26. Sec. 7] and [19].
œ
Lemma 1. If f (z) = yanzn is holonomic and analytic at z = 0, then the nth Taylor coeffi-
n=0
cient an has an asymptotic expansion in the sense of Poincaré
œ c
an - yX-nn-ax-1(log n)Px£
Xe§ s=0 n
where §f is the set of singularities of f, aX, PX e Q ,and cXs e C .
2.2. The case of the free abelian group
In this section we will summarize what is known about the generating functions RP (z) when G = Zr is the free abelian group or rank r. The next theorem is shown in [19], using André main theorems from [16]. An alternative proof uses the regular holonomicity of the Gauss - Manin connection and the rationality of its exponents. This was kindly communicated to us by C. Sabbah (see also [27]). Holonomicity of RP (z) also follows from a fundamental result of Wilf-Zeilberger, explained in [19].
Theorem 3. [19]. For every P e MN(Q[Zr ]), RP(z) is a G-function.
2.3. A complexity remark
Given P e MN(Q[Fr]) (resp. P e MN(Q[Fr])), one may ask for the complexity of a minimal polynomial Q(y, z) e Q[y, z] (resp. minimal degree differential operator D(z, dz) e Q(z, dz)) so that Q(RP (z), z) = 0 (resp. D(z, d z )RP (z) = 0). One expects that the y-degree of Q(y, z) and
the dz-degree of D(z, dz) is exponential in the complexity of P, where the latter can be defined to be the degree of P and the maximum of the absolute values of the coefficients of the enrties of P. This prohibits explicit calculations in general.
Acknowledgement
The authors wish to thank K. Dykema, R. Gilman, F. Flajolet, L. Mosher, C. Sabbah and D. Zeilberger for stimulating conversations and F. Lehner and D. Voiculescu for bringing to our attention relevant literature.
3. A theorem of Haiman and a proof of Theorem 1
In [28] Haiman proves the following theorem.
Theorem 4. [28]. Let K be a field with a rank 1 discrete valuation v; Kv its completion with respect to the metric induced by v. Let f(xv...,xr,yv...,yr) be a rational power series over K in non-commuting indeterminants. Any coefficient of f(x^... , xr, xt ,..., xr ing over Kv is algebraic over K.
Letting K = <(z), and Kv = Q((z)) the ring of formal Laurent series in z, and considering the element (1 - zP)-1, where P e MN (Q[Fr ]) , gives an immediate proof of Theorem 1.
In the next section we will give a detailed description of Haiman's argument which exhibits a close relation to linguistics, as well as an obstruction to generalizing Theorem 1 to groups other than the free group.
4. Algebraic and rational functions in noncommuting variables
4.1. Rational, algebraic and holonomic functions in one variable
In this section all functions will be analytic in a neighborhood of z = 0. Let Q0at(z), Qjfs(z) and Q|jol(z) denote respectively the set of rational, algebraic and holonomic functions, analytic at z = 0. Let <[[z]] denote the set of formal power series in z. Using the injective Taylor series map around z = 0, we
will consider Q0at(z), Q0'g(z) and Q0hol(z) <[[z]]
Q0at(z) c Q0lg(z) c Q0ol(z) c Q[[z]]. <[[z]] has two multiplications:
• the usual multiplication of formal power series
With respect to the usual multiplication, <[[z]] is an algebra and Q0at(z), Q0's(z) and Q°Jol(z) are subalgebras. In case two power series are convergent in a neighborhood of zero, so is their Hadamard product. Hadamard, Borel and Jungen studied the analytic continuation and the singularities of the Hadamard product of two functions; see [25; 29]. Their method used an integral representation of the Hadamard product, and a deformation of the contour of integration; see [25. Fig. 2, p. 303]. Let us summarize these classical results.
Theorem 5. (a) If f and g are rational,so is f © g.
(b) If f is rational and g is algebraic, then f © g is algebraic.
(c) If f and g are holonomic (resp. regular holonomic with rational exponents),so is f © g.
(d) If f and g are algebraic,then f © g is not necessarily algebraic.
For a proof, see Thm. 7, 8, E and the example of p. 298 from [25].
• the Hadamard product
4.2. Rational and algebraic functions in noncommuting variables
In this section we discuss a generalization of the previous section to non-commuting variables. Let X be a finite set, and let X* denote the free monoid on X. In other words, X consists of the set of all words in X, including the empty word e. Let Q<X) (resp. Q<<X))) denote the algebra of polynomials (resp. formal power series) in non-commuting variables. In [30], Schutzenberger defines the notion of a rational and an algebraic power series in non-commuting variables. Let Qrat < X) and Qalg < X) denote the sets of rational (resp. algebraic) power series. Then, we have an inclusion: Qrat<X) c Qalg<X) c Q<<X)). Q<<X)) has two multiplications: • the usual multiplication of formal power series in non-commuting variables:
( A ( A ( A
Zaww • Zbww = Z Z awb
J V weX
weX
w w
V w',w":w'w"=w
w;
V weX
• the Hadamard product:
( AT A
Zaww ® Zbww = Zawbww.
V weX J V weX J weX
With respect to the usual multiplication, Q<<X)) is a non-commutative algebra and Qrat < X) and Qalg <X) are subalgebras. We have the following analogue of Theorem 5. Theorem 6. [30. Pro. 2.2]. (a) If f e Qrat<X) and g e Qrat<X), then f © g e Qrat<X). (b) If f e Qrat<X) and g e Qalg<X), then f © g e Qalg<X).
Remark 2. The notion of rational and algebraic functions works for an arbitrary ring % of characteristic zero, instead of Q. Theorem 6 is still valid.
4.3. Proof of Theorem 1
Let Fr denote the free group of rank r with generating set {uu...,ur}, and
X {Xi,..., Xr, Xl,..., X r }.
Consider the monoid map:
n : X* ^ Fr, %(xi ) = ui, n(xi ) = u-1. The kernel Ker(n) of n is the set of those words in X which reduce to the identity under the relations x{xi = Xix{ = e. Let A = Z we Q<<X)). The next proposition is attributed to
weKer(n)
Chomsky-Schutzenberger by Haiman. For a proof, see [28. Sec. 3]. Proposition 1. [31]. A is algebraic.
The map n has a right inverse (that satisfies n ° i = IF ) i: Fr ^ X, defined by mapping a reduced word in u{ to a corresponding word in X. For every f e Q[Fr ] we have a key relation between trace and Hadamard product: Tr(f) = ^(i(f) © A), where ^ is a Q-linear map defined by:
^ : Q<X)^ Q, ^(w) = l for w e X*. Now, fix P e MN(Q[Fr]). Let AN denote the N by N matrix with entries equal to A, and
to
% = Q(z). Let Pz = zi(P) e MN(%<X)) , P* = ZP" e MN (%<<X))). Notice that P* is well-defined since Pz has no z-constant term. "=0 Lemma 2. We have P* e MN (%rat<X)).
Proof. P* satisfies the matrix equation (l - Pz )PZ = I with entries in %<X). □
Lemma 2, together with Propositions l and part (b) of 6 imply the following result, which we can think as a noncommutative analogue of Theorem l.
to
Proposition 2. For every P e MN(Q[Fr]), we have Z*"(i(P))n © AN e MN(%alg<X)).
"=0
Consider the abelianization ring homomorphism y: %<<X)) ^ %[[X]], where %[[X]] is the formal power series ring in commuting variables. Haiman proves the following. Proposition 3. [28. Prop. 3.3]. If f e %alg<X), then y(f) is algebraic over %(X).
It follows that y(Pz* © AN) e MN(nalg(X)). Consider now the subalgebra nconv[[X]] of n[[X]] that contains all elements of the form ^aww where aw e zl(w)Q[[z]], l(w) denotes
weX
the length of w. Then, we can define an algebra map:
4z : nconv[[X]] ^ Q[[z]], *z(w) = 1. for x e X.
Haiman shows that if f e nalg(X) n nconv[[X]], then (f ) e Qalg . To state our final conclusion, we define for 1 < i, j < N, the sequence (ajn) by ajn = Tr((Pn)j) and the matrix of
generating series Ap(z) e Mn(<№]]) by (AP(z))iy = Yaï,nzn.
n=0
Lemma 3. We have: (*z ° y)(Pz* © AN) = AP(z). Thus, Ap(z) e MN(Q0'g(z)).
Proof. The conclusion follows from the above discussion. □
Thus, the entries of Ap(z) are algebraic functions, convergent at z = 0. Since by definition we have
N
RP(z) = AP(z))ii it follows that Rp(z) e Q0'g(z). This completes the proof of Theorem 1. □
i=1
5. Some Linguistics
5.1. Regular and context-free languages
Haiman's proof uses the key Proposition 1 from linguistics. Let us recall some concepts from this field. See for example [32 — 34] and references therein. Given a finite set X (the alphabet), a language L is a collection of words in X. In other words, C c X*. The generating series FL of a language is FC = ^w e Q«X». It follows that for two languages C1 and C2 we
weL
have Fq= Fq © . A language L is called rational (resp. context-free) iff FL e Qrat(X)
(resp. Fl e Qalg(X) ). In this context, Theorem 6 takes the following form. Theorem 7. [31]. (a) If q and C2 are rational languages,so is q<n q. (b) If q is rational and q is (unambiguous) context-free,then q n q is (unambiguous) context-free.
It was pointed out to us independently by D. Zeilberger and F. Flajolet that the above theorem essentially proves Theorem 1.
5.2. Some questions
Let us end this short paper with some questions. Despite the similarity in their statements and the multitude of proofs, Theorems 1 and 3 have different assumptions, different proofs and different conclusions.
Consider a generating set X for a group G such that every element of G can be written as a word in X with nonnegative exponents. Given X and G, let Cx denote the set of all words in X that map to the identity in G. Deciding membership in Cx is the word problem in G.
Definition 4. A group G has context-free word problem if it has a generating set X such that the language Cx is context-free.
The proof of Theorem 1 applies to groups with a context-free word problem. Muller-Schupp classified those groups. In [35] Muller-Schupp prove that G has context-free word problem iff G has a free finite-index subgroup.
On the other hand, if G is the fundamental group of a hyperbolic manifold of dimension not equal to 2, then G does not have a free finite-index subgroup.
Thus, the linguistics proof of Theorem 1 does not apply to the case of hyperbolic groups in dimension three. Neither does it apply to the case of Zr since the latter does not have context-free word problem.
Example 1. If P is a hyperbolic group and P e MN(Q[G]), is it true that RP(z) is a G-function?
The question may be relevant to low dimensional topology, when one tries to compute the I2 -torsion of a hyperbolic manifold using Luecke's theorem; [36]. In that case, the matrix P comes from Fox (free differential) calculus of a presentation of the fundamental group G of the hyperbolic manifold. See also [37].
Example 2. Given P e MN(Q[Fr]), consider the abelianization Pab e MN(Q[Zr]), and the G-functions RP(z) and R b (z) . How are the singularities of RP(z) and R b (z) related?
Example 3. What is a holonomic function in non-commuting variables?
6. A functional analysis interpretation of Theorem 1
The present paper is focusing on results and techniques inspired by algebra, non-commutative algebraic combinatorics. However it is worth mentioning that Theorem 1 has applications to problems coming from functional analysis, spectral theory, and the spectrum of Schroding-er operators. For instance, the Schrodinger equation describing the electron motion in a d-dimensional periodic crystal, can be well approximated by the difference equation on a lattice of same dimension. The corresponding operator can be seen as an element of the group ring of Zd . The function RP(z) defined previously is noting but the diagonal element of the resolvent and is used to compute the spectral measure, through the Charles de la Vallée Poussin theorem. There are instances for which, this operator is better approximated by the free group analog. For instance the retracable path approximation was used by Brinkman and Rice [38] in 1971 to treat the effect of spin-orbit coupling in the Hall effect, while it was used in [39] to compute the electronic Density of States when the electron is submitted to a random magnetic field. The same operator, seen as an element of the free group ring, is used to describe various infinite dimension approximations. The seminal work of Georges and Kotliar [40] used this free group approximation to give the first model known with a Mott-Hubbard transition.
Another domain in which the Theorem 1 may apply is the Voiculescu Theory of Free Probability [5; 41]. The so-called R-transform used to treat the convolution of free random variables, is also based upon the Schur complement formula. In particular the free central limit theorem asserts that a sum of identically distributed free random variable obey the semicircle law, is a special case of the present result.
Besides the two proofs of Theorem 1 discussed in this paper, the algebraic character of RP(z) can also be deduced from the used of the Schur complement method [42]. This is what makes the free group approximation so attractive to theoretical physicists. This method, also known under the name of Feshbach method [43 — 45] is used in many domains of Physics, Quantum Chemistry, Solid State Physics, Nuclear Physics, to reduce the Hilbert space to a finite dimensional one and make the problem amenable to numerical calculations. However, very few Mathematical Physicists have paid attention to the fact that algebraicity or holonomy can give rise to results concerning the explicit computation of the spectral radius, or more generally, to the band edges, of the Hamiltonian they consider. This later problem is known to be notably hard with other methods.
For the benefit of the reader, we include some history of that method. The Schur complement method [42] is widely used in numerical analysis under this name, while Mathematical Physicists prefer the reference to Feshbach [43]. In Quantum Chemistry, the common reference is Feshbach-Fano [46] or Feshbach-Lowdin [47]. This method is used in various algorithms in Quantum Chemistry (ab initio calculations), in Solid State Physics (the muffin tin approximation, LMTO) as well as in Nuclear Physics. The formula used above is found in the original paper of Schur [42. P. 217].
The formula has been proposed also by an astronomer Tadeusz Banachiewicz in 1937, even though closely related results were obtained in 1923 by Hans Boltz and in 1933 by Ralf Rohan [48]. Applied to the Green function of a selfadjoint operator with finite rank perturbation, it becomes the Kren formula [49].
Let us end this section with a small dictionary that compares our notions with those in physics:
H e Mn(Q [Fr]) Hamiltonian
1 / (z - H) resolvant
1 / zRH (1 / z) trace of the resolvant
Tr(Hn) n th moment of H
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42. Schur J. Der potenzreihen, die im innern des einheitskreises beschrnkt sind. J. für die reine und angewandte Mathematik, 1918, no. 148, pp. 122.
43. Feshbach H. Unified theory of nuclear reactions. Annals of Physics (N. Y.), 1958, vol. 5, no. 4, pp. 357.
44. Herman F. A unified theory of nuclear reactions. II. Annals of Physics, 1962, vol. 19, pp. 287-313.
45. Feshbach H. The unified theory of nuclear reactions. III. Overlapping resonances. Ann. Phys. (N. Y.), 1967, vol. 43, no. 3, pp. 410-420.
46. Fano U. Absorption spectrum of the noble gases near the limit of the arc spectrum. Nuovo Cimento, 1935, vol. 12, pp. 154-161.
47. Lowdin P.-O. Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism. J. of Mathematical Physics, 1962, vol. 3, no. 5, pp. 969-982.
48. Puntanen S., Styan G. P. H. Historical introduction: Issai Schur and the early development of the Schur complementm. Numerical Methods and Algorithms, 2005, vol. 4, pp. 1-16.
49. Krein M. Concerning the resolvents of an Hermitian operator with the deficiency-index (m,m). Doklady Akademii nauk SSSR [Reports of academy of Sciences of the USSR], 1946, vol. 52, pp. 651-654.
About the authors
Stavros Garoufalidis, professor, School of Mathematics of Georgia Institute of Technology, Atlanta, USA. stavros@math.gatech.edu, www.math.gatech.edu/~stavros.
Jean Bellissard, professor, School of Mathematics of Georgia Institute of Technology, Atlanta, USA. jeanbel@math.gatech.edu, www.math.gatech.edu/~jeanbel.
Bulletin of Chelyabinsk State University. 2015. № 3 (358). Mathematics. Mechanics. Informatics. Issue 17. Р. 50-61.
АЛГЕБРАИЧЕСКИЕ G-ФУКНЦИИ, АССОЦИИРОВАННЫЕ С МАТРИЦАМ НАД ГРУППОВЫМ КОЛЬЦОМ
С. Гаруфалидис, Ж. Беллисард
Для каждой матрицы с элементами из группового кольца некоторой группы можно построить последовательность следов (в смысле группового кольца) их степеней. Мы доказываем, что соответствующий производящий ряд является алгебраической G-функцией (в смысле Зигеля) в случае, когда группа является свободной конечного ранга. Следовательно, норма таких элементов является точно вычислимым алгебраическим числом, и их функция Грина является алгебраической. Наше доказательство использует понятия рациональных и алгебраических степенных рядов с некоммутирующими переменными и опирается на теорему Хаймана. В основе этой теоремы лежат результаты о регулярных и контекстно-свободных языках. С другой стороны, когда группа является свободной абелевой конечного ранга, то соответствующий производящий ряд представляет собой G-функцию. Вопрос состоит в том, выполняется ли это для любой гиперболической группы.
Ключевые слова: рациональная функция, алгебраическая функция, голономная функция, G-функция, производящий ряд, некоммутирующие переменные, момент, гамильтониан, резольвенты, регулярный язык, контекстно-свободный язык, произведение Адамара, групповое кольцо,свободная вероятность,метод дополнений Шура,свободная группа,алгебра фон Неймана, полиномиальный Гамильтониан, спектральная теория, норма.
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Сведения об авторах
Гаруфалидис Ставрос, профессор, Школа математики Технологического института Джорджии, Атланта, США. stavros@math.gatech.edu; www.math.gatech.edu/~stavros.
Белиссард Жан, профессор, Школа математики Технологического института Джорджии, Атланта, США. jeanbel@math.gatech.edu; www.math.gatech.edu/~jeanbel.
