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We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon's identity in the one variable case, which seems to not appear in the literature.

 The present volume is closely related in its contents to the author’s book Theory of analytic functions of several complex variables, published in English translation by the American Mathematical Society in 1963. These two volumes together constitute the second edition, considerably revised and enlarged, of the monograph Theory of analytic functions of several complex variables published in 1948. In the second edition, as well as in the first, the author does not aim to cover, even to any extent, all the material which has accumulated in the theory of analytic functions of several complex variables.   The present volume is divided into five chapters: approximation of functions and domains, coherent analytic sheaves and the solution of fundamental problems, domains analytically convex in the "sense of Hartogs, holomorphic extension of domains, biholomorphic mappings. In order to understand these chapters, the reader should be familiar with the concepts contained in the first two chapters of Theory of analytic functions of several complex variables. In addition, for § 2 of Chapter II the reader must be acquainted with Weil’s integral representations (§ 22, Chapter IV, (1)) and Cousin’s first theorem (§ 25, Chapter V, (1)), for §§8—12 of Chapter II with properties of holomorphically complete complex manifolds (§ 14.1 and §18. 3—4, Chapter III, (1)), for §l4 of Chapter III with Weil’s integral representations and the methods of solving Cousin’s first problem (throughout §25, Chapter V, (I) and §7, Chapter :11) and with properties of sequences of domains (§ 6, Chapter I), for §§ 17 and 18 of Chapter IV with the theory of plurisubharmonic functions (§13, Chapter III) and for Chapter V with properties of the kernel function in a domain (§§ 4 and 5, Chapter 1). The remaining parts Of the present volume, except for certain cross-references, are independent of one another and of Chapters III, IV and V of the first part of the book. To shorten the volume and simplify the text the proofs of several propositions are not developed in the most general form. For example, Theorems (-A) and (B) of H. Cartan are proved for complex manifolds, but not for spaces; the theorem of K. Oka on the domains convex in the sense of Hartogs is proved for the case of spaces of only two complex variables. The "edge of the wedge” theorem of N.N. Bogoljubov is also proved in a simplified form by introducing some hypotheses. The actual text of the book itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   At the request of the author, the first draft of the text of §§8—12, dealing with the theory of coherent analytic sheaves and its application to the solution of the fundamental problem, was written by D.B. Fuks, §19, dealing with the "edge of the wedge” theorem, by V.S. Vladimirov, and §24, dealing with homogeneous bounded domains, by S.G. Gindikin. The latter two sections contain new results which are due to the above-mentioned persons and are introduced here, as a rule, without reference to the original articles. The author is indebted for advice and a number of valuable remarks to L.A. Ai’zenberg, who looked over the entire text while it was being prepared for the press, and to V.S. Vladimirov, who looked over the text of Chapters III and V. To all the above persons I wish to express my profound gratitude. I also wish to take this opportunity of thanking other mathematicians who looked over various parts of the book and sent me their suggestions. Results belonging to many mathematicians are presented in this book. It should be noted, however, that the greatest influence on its contents is due to works of S. Bergman, concerning the kernel function in a domain and its applications, of G. Bremermann, concerning domains convex in the sense of Hartogs and the holomorphic extension. of domains, of H. Cartan who established, in collaboration with J.P. Sam and others, the theory of coherent analytic sheaves and its applications to many important problems of the theory of functions, and of K. Oka, concerning approximation of functions, Cousin’s problems and the solution of the inverse problem of Hartogs . Translated from : СПЕЦИАЛЬНЫЕ ГЛАВЫ ТЕОРИИ АНАЛИТИЧЕСКИХ ФУНКЦИИ МНОГИХ.КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ  Б.А. ФУКС  Государственное Издательство Физике—Математической Литературы  Москва 1963 

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $ \chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $ \chi^{(3)}$ and $ \chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.

We define a normal form (called the canonical image) of an arbitrary measurable function of several variables with respect to a natural group of transformations; describe a new complete system of invariants of such a function (the system of joint distributions); and relate these notions to the matrix distribution, another invariant of measurable functions found earlier, which is a random matrix. Bibliography: 7 titles.

Bibliography: p. 319-320

In the spaces $S^p$ of functions of several variables, $2\pi$-periodic in each variable, we study the approximative properties of operators $A^\vartriangle_{\varrho,r}$ and $P^\vartriangle_{\varrho,s}$, which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes $S^pH_\omega$.

We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $\theta (z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $\theta (z), \theta_1(z)$ and $\theta_2(z)$ to be Agler--Schur-class functions on the polydisk $\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametric linear system $\alpha $--a conservative realization of $\theta (z)$, and give the criterion for its solvability in terms of common invariant subspaces for the $N$-tuple of main operators of $\alpha $.

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^{\psi}$ in the integral metric.

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^{\psi}$ in the integral metric.

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^{\psi}$ in the integral metric.

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^{\psi}$ in the integral metric.

We find the exact order estimates of the approximations of the classes ${\cal F}_{q,r}^{\psi}$ of functions of several variables by greedy approximants in the integral metric. We also obtain the exact order estimates of the best $n$-term orthogonal trigonometric approximations of the classes ${\cal F}_{q,r}^{\psi}$ in the integral metric.

Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of $\partial_1, ..., \partial_n$ with $\partial_i = d$ $1 \leq i \leq n$. As in the single variable case, Lax equations can be constructed using such pseudodifferential operators, whose solutions can be provided by Baker functions. We extend the usual notion of tau functions to the case of pseudodifferential operators of several variables so that each Baker function can be expressed in terms of the corresponding tau function.

We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic.

 The present volume has five chapters. The first chapter deals with the fundamental properties of analytic funcrions in the space of several complex variables, and the second chapter with the properties of analytic funCtions in covering regions over a suitable space. These two chapters may be considered as a textbook for readers who are looking for basic information, in as elementary a form as possible, about the theory of functions of several complex variables. The next three chapters deal respectively with complex spaces, inten representations of functions of several complex variables, and functions meromorphic of one another in content but each of in the whole space. They are independent them makes a great deal of use of the material of the first two chapters. In contradistinction to the first two chapters, the lasr three are to a great extent in the nature of a survey. These chapters may serve as an introduction to the current technical literature on the various branches of the theory of functions. The actual exposition itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   The present book constitutes the first part of a second edition, considerably revised and enlarged, of the author’s book Theory of analytic functions of several complex variables published in 1948. The second part, which is to appear soon after the first, will discuss a number of special chapters in the theory of functions. At the request of the author the firsr draft of the text of subsections 1—3, § 23, dealing with integral representations in n-circular regions, was written by L.A. Aizenberg, subsections 4—6, §23, dealing with integral representations in tubular regions, by S.G. Gindikin, and section 26, dealing with methods of characterizing the growth of entire functions, by L.I. Ronkin. These sections contain a number of new results, which are due to the above mentioned persons and are introduced here, as a rule, without reference to the original articles. An exposition of a number of original results referring to integral representations was kindly placed at my disposal by A.A. Temljakov. I am also indebted to L.A. Aizenberg and D.B. Fuks, who looked over the entire text while it was being prepared for the press and gave me valuable advice. To all the above persons I wish to express my profound gratitude. Many sections of this book were first presented to the seminar on the theory of analytic functions at the University of Moscow. I wish to take this opportunity of thanking the members of the seminar, and several other mathematicians, who looked over various parts of the book and sent me their suggestions.  Translated From : ВВЕДЕНИЕ В TEOPИЮ АНАЛИТИЧЕСКИХ ФУНК МНОГИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ - Б.А. ФУКС  Государственное Издательство Физике—Матсматхічсской Литературы Москва - 1962 VVEDENIYe V TEOPIYU ANALITICHESKIKH FUNK MNOGIKH KOMPLEKSNYKH PEREMENNYKH - B.A. FUKS  Gosudarstvennoye Izdatel'stvo Fizike—Matsmatkhíchsskoy Literatury Moskva - 1962

 The present volume has five chapters. The first chapter deals with the fundamental properties of analytic funcrions in the space of several complex variables, and the second chapter with the properties of analytic funCtions in covering regions over a suitable space. These two chapters may be considered as a textbook for readers who are looking for basic information, in as elementary a form as possible, about the theory of functions of several complex variables. The next three chapters deal respectively with complex spaces, inten representations of functions of several complex variables, and functions meromorphic of one another in content but each of in the whole space. They are independent them makes a great deal of use of the material of the first two chapters. In contradistinction to the first two chapters, the lasr three are to a great extent in the nature of a survey. These chapters may serve as an introduction to the current technical literature on the various branches of the theory of functions. The actual exposition itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   The present book constitutes the first part of a second edition, considerably revised and enlarged, of the author’s book Theory of analytic functions of several complex variables published in 1948. The second part, which is to appear soon after the first, will discuss a number of special chapters in the theory of functions. At the request of the author the firsr draft of the text of subsections 1—3, § 23, dealing with integral representations in n-circular regions, was written by L.A. Aizenberg, subsections 4—6, §23, dealing with integral representations in tubular regions, by S.G. Gindikin, and section 26, dealing with methods of characterizing the growth of entire functions, by L.I. Ronkin. These sections contain a number of new results, which are due to the above mentioned persons and are introduced here, as a rule, without reference to the original articles. An exposition of a number of original results referring to integral representations was kindly placed at my disposal by A.A. Temljakov. I am also indebted to L.A. Aizenberg and D.B. Fuks, who looked over the entire text while it was being prepared for the press and gave me valuable advice. To all the above persons I wish to express my profound gratitude. Many sections of this book were first presented to the seminar on the theory of analytic functions at the University of Moscow. I wish to take this opportunity of thanking the members of the seminar, and several other mathematicians, who looked over various parts of the book and sent me their suggestions.  Translated From : ВВЕДЕНИЕ В TEOPИЮ АНАЛИТИЧЕСКИХ ФУНК МНОГИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ - Б.А. ФУКС  Государственное Издательство Физике—Матсматхічсской Литературы Москва - 1962 VVEDENIYe V TEOPIYU ANALITICHESKIKH FUNK MNOGIKH KOMPLEKSNYKH PEREMENNYKH - B.A. FUKS  Gosudarstvennoye Izdatel'stvo Fizike—Matsmatkhíchsskoy Literatury Moskva - 1962

 The present volume has five chapters. The first chapter deals with the fundamental properties of analytic funcrions in the space of several complex variables, and the second chapter with the properties of analytic funCtions in covering regions over a suitable space. These two chapters may be considered as a textbook for readers who are looking for basic information, in as elementary a form as possible, about the theory of functions of several complex variables. The next three chapters deal respectively with complex spaces, inten representations of functions of several complex variables, and functions meromorphic of one another in content but each of in the whole space. They are independent them makes a great deal of use of the material of the first two chapters. In contradistinction to the first two chapters, the lasr three are to a great extent in the nature of a survey. These chapters may serve as an introduction to the current technical literature on the various branches of the theory of functions. The actual exposition itself is preceded by an introductory essay giving the most frequently used information from closely related mathematical disciplines. It is recommended that the reader refer to this essay whenever he finds it necessary.   The present book constitutes the first part of a second edition, considerably revised and enlarged, of the author’s book Theory of analytic functions of several complex variables published in 1948. The second part, which is to appear soon after the first, will discuss a number of special chapters in the theory of functions. At the request of the author the firsr draft of the text of subsections 1—3, § 23, dealing with integral representations in n-circular regions, was written by L.A. Aizenberg, subsections 4—6, §23, dealing with integral representations in tubular regions, by S.G. Gindikin, and section 26, dealing with methods of characterizing the growth of entire functions, by L.I. Ronkin. These sections contain a number of new results, which are due to the above mentioned persons and are introduced here, as a rule, without reference to the original articles. An exposition of a number of original results referring to integral representations was kindly placed at my disposal by A.A. Temljakov. I am also indebted to L.A. Aizenberg and D.B. Fuks, who looked over the entire text while it was being prepared for the press and gave me valuable advice. To all the above persons I wish to express my profound gratitude. Many sections of this book were first presented to the seminar on the theory of analytic functions at the University of Moscow. I wish to take this opportunity of thanking the members of the seminar, and several other mathematicians, who looked over various parts of the book and sent me their suggestions.  Translated From : ВВЕДЕНИЕ В TEOPИЮ АНАЛИТИЧЕСКИХ ФУНК МНОГИХ КОМПЛЕКСНЫХ ПЕРЕМЕННЫХ - Б.А. ФУКС  Государственное Издательство Физике—Матсматхічсской Литературы Москва - 1962 VVEDENIYe V TEOPIYU ANALITICHESKIKH FUNK MNOGIKH KOMPLEKSNYKH PEREMENNYKH - B.A. FUKS  Gosudarstvennoye Izdatel'stvo Fizike—Matsmatkhíchsskoy Literatury Moskva - 1962

"The Linear Dependence of Functions of Several Variables, and Completely Integrable Systems of Homogeneous Linear Partial Differential Equations" is an article from Transactions of the American Mathematical Society, Volume 17 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1988834

xvi, 593-927, A1-A28 pages : 27 cm

We prove a analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.

The paper deals with (multidimensional and one-dimensional) Bochner-Phillips functional calculus. Bounded perturbations of Bernstein functions of (one or several commuting) semigroup generators on Banach spaces are considered, conditions for Lipschitzness and Frechet-differentiability of such functions are obtained, estimates for the norm of commutators are proved, and a generalization of Livschits-Kre\u{\i}n trace formula derived.

"A Note Concerning Maxima and Minima of Functions of Several Variables" is an article from The American Mathematical Monthly, Volume 14 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2968092

"A Note Concerning Maxima and Minima of Functions of Several Variables" is an article from The American Mathematical Monthly, Volume 14 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2968092

"Factorization of Analytic Functions of Several Variables." is an article from The Annals of Mathematics, Volume 19 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967766

For a class of functions of several variables, which contains the continuous functions, we show that there exists a sum of functions of one variable that minimizes the distance from the given function to the space of such sums. For functions of two variables we show that such a minimizing sum may be constructed by an iterative scheme. (Author)

For a class of functions of several variables, which contains the continuous functions, we show that there exists a sum of functions of one variable that minimizes the distance from the given function to the space of such sums. For functions of two variables we show that such a minimizing sum may be constructed by an iterative scheme. (Author)

"Fourier's Constants of Functions of Several Variables" is an article from American Journal of Mathematics, Volume 39 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370530

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of several variables. By using these polynomials, recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function and interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given.

This work was motivated by the problem of obtaining a smooth density function over a geographical region from data aggregated over irregular subregions. Minimization of a family of roughness criteria given 'volume' data lead to smooth multivariate functions - Laplacian histosplines, having a certain order of the iterated Laplacian of constant value in each of the subregions and satisfying natural boundary conditions on the boundary of the region. For inexact data, e.g., in case of estimating an underlying density given counts of events by subregions, Laplacian smoothing histosplines are constructed, analogous to smoothing splines in the univariate case, and a method for choosing the smoothing parameter is presented. For both cases of exact and inexact data, modified roughness criteria, independent of the region, are discussed, and results known for point-evaluation data are extended to the case of aggregated data. (Author)

This work was motivated by the problem of obtaining a smooth density function over a geographical region from data aggregated over irregular subregions. Minimization of a family of roughness criteria given 'volume' data lead to smooth multivariate functions - Laplacian histosplines, having a certain order of the iterated Laplacian of constant value in each of the subregions and satisfying natural boundary conditions on the boundary of the region. For inexact data, e.g., in case of estimating an underlying density given counts of events by subregions, Laplacian smoothing histosplines are constructed, analogous to smoothing splines in the univariate case, and a method for choosing the smoothing parameter is presented. For both cases of exact and inexact data, modified roughness criteria, independent of the region, are discussed, and results known for point-evaluation data are extended to the case of aggregated data. (Author)

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f.We then characterize meromorphic functions admitting a harmonic star function.

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f.We then characterize meromorphic functions admitting a harmonic star function.

This monograph is devoted to the theory of entire functions of several variables. A definition of bounded index was supposed by B. Lepson. We generalised his definition for several variables and obtained criteria of L-index boundedness in direction. In particular we obtained obtain analogues of one-dimensional criterion of boundedness L-index in terms of behaviour the logarithmic derivative outside of zero sets, sufficient conditions of L-index boundedness in direction for some subclass of entire functions with "plane" zeros, sufficient conditions of L-index boundedness of entire solutions for some partial differential equations. Besides differential equations, entire functions of bounded index have application in the theory of value distribution. The monograph is published by "Chyslo": http://chyslo.com.ua/index.php?route=product/category&path=62

This monograph is devoted to the theory of entire functions of several variables. A definition of bounded index was supposed by B. Lepson. We generalised his definition for several variables and obtained criteria of L-index boundedness in direction. In particular we obtained obtain analogues of one-dimensional criterion of boundedness L-index in terms of behaviour the logarithmic derivative outside of zero sets, sufficient conditions of L-index boundedness in direction for some subclass of entire functions with "plane" zeros, sufficient conditions of L-index boundedness of entire solutions for some partial differential equations. Besides differential equations, entire functions of bounded index have application in the theory of value distribution. The monograph is published by "Chyslo": http://chyslo.com.ua/index.php?route=product/category&path=62

The theorem on the existence of three commuting contractions on a Hilbert space and of a linear homogeneous matrix function of three independent variables for which the generalized von Neumann inequality fails is proved.

"Primary Prime Functions in Several Variables and a Generalization of an Important Theorem of Dedekind" is an article from American Journal of Mathematics, Volume 24 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370006

Technical report for period February 1974 - March 1974.

"On Functions of Several Complex Variables" is an article from Transactions of the American Mathematical Society, Volume 17 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1988823

"On Functions of Several Complex Variables" is an article from Transactions of the American Mathematical Society, Volume 17 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1988823

We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship between operator convexity and operator monotonicity for functions of one variable is extended also to functions of several variables.

We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality \tau(x_1... x_n)\le\tau(y_1... y_n) for a trace \tau on a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices.

This paper is devoted to the family $\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients $(\ell_1+...+\ell_n)!/(\ell_1!...\ell_n!)$, where $n\in\ZZ_{\ge 1}$. All these series belong to the family of the general Appell-Lauricella's series. It is shown that each function $G_n$ can be expressed by an integral involving the previous one, $G_{n-1}$. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand-Kapranov-Zelevinsky's theory of $A$-hypergeometric systems or with the Aomoto's theory of hypermeotric functions. The quasi-invariance of each function $G_n$ with regard to the action of a finite number of involutions of $\CC^{*n}$ is also established. Finally, a particular attention is reserved to the study of the functions $G_2$ and $G_3$, each of which is proved to be algebraic or to be expressed by the Legendre's elliptic function of the first kind.

A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Previously subspace collections had been introduced when there was an inner product on the vector (or Hilbert) space, and appropriate subspaces were mutually orthogonal. In that setting certain normalization and reduction operations on subspace collections led to a continued fraction expansion of the associated function, which allowed one to bound the function in terms of a set of weight matrices and normalization matrices that are derived from series expansions. Here we also initiate the theory of normalization and reduction operations, appropriate when there is no inner product on the space.

We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and restrictions on the growth of functions near the boundary of domain.

It is shown that if three distinct values of a meromorphic function f:C^n -> P^1 of hyper-order strictly less than 2/3 have forward invariant pre-images with respect to a translation t:C^n -> C^n, t(z)=z+c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green's Picard-type theorem in the special case where the hyper-order of f is less than 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the lemma on the logarithmic derivative and of the second main theorem of Nevanlinna theory for meromorphic functions C^n -> P^1 are given, and their applications to partial difference equations are discussed.

"On the Linear Dependence of Functions of Several Variables, and Certain Completely Integrable Systems of Partial Differential Equations" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 2 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-83935

"On the Linear Dependence of Functions of Several Variables, and Certain Completely Integrable Systems of Partial Differential Equations" is an article from Proceedings of the National Academy of Sciences of the United States of America, Volume 2 . View more articles from Proceedings of the National Academy of Sciences of the United States of America . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-83935

Algebraic techniques in interpolation theory for functions of several variables

Algebraic techniques in interpolation theory for functions of several variables

Bell System Technical Journal, 43: 2. March 1964 pp 785-804. The ALPAK System for Nonnumerical Algebra on a Digital Computer - II: Rational Functions of Several Variables and Truncated Power Series with Rational-Function Coefficients. (Brown, W.S.; Hyde, J.P.; Tague, B.A.)