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The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forrey's transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.

This article is from PLoS ONE , volume 9 . Abstract MYH9 encodes non-muscle myosin heavy chain IIA (NMMHCIIA), the predominant force-generating ATPase in non-muscle cells. Several lines of evidence implicate a role for MYH9 in podocytopathies. However, NMMHCIIA‘s function in podocytes remains unknown. To better understand this function, we performed immuno-precipitation followed by mass-spectrometry proteomics to identify proteins interacting with the NMMHCIIA-enriched actin-myosin complexes. Computational analyses revealed that these proteins belong to functional networks including regulators of cytoskeletal organization, metabolism and networks regulated by the HIV-1 gene nef. We further characterized the subcellular localization of NMMHCIIA within podocytes in vivo, and found it to be present within the podocyte major foot processes. Finally, we tested the effect of loss of MYH9 expression in podocytes in vitro, and found that it was necessary for cytoskeletal organization. Our results provide the first survey of NMMHCIIA-enriched actin-myosin-interacting proteins within the podocyte, demonstrating the important role of NMMHCIIA in organizing the elaborate cytoskeleton structure of podocytes. Our characterization of NMMHCIIA’s functions goes beyond the podocyte, providing important insights into its general molecular role.

This article is from PLoS ONE , volume 9 . Abstract MYH9 encodes non-muscle myosin heavy chain IIA (NMMHCIIA), the predominant force-generating ATPase in non-muscle cells. Several lines of evidence implicate a role for MYH9 in podocytopathies. However, NMMHCIIA‘s function in podocytes remains unknown. To better understand this function, we performed immuno-precipitation followed by mass-spectrometry proteomics to identify proteins interacting with the NMMHCIIA-enriched actin-myosin complexes. Computational analyses revealed that these proteins belong to functional networks including regulators of cytoskeletal organization, metabolism and networks regulated by the HIV-1 gene nef. We further characterized the subcellular localization of NMMHCIIA within podocytes in vivo, and found it to be present within the podocyte major foot processes. Finally, we tested the effect of loss of MYH9 expression in podocytes in vitro, and found that it was necessary for cytoskeletal organization. Our results provide the first survey of NMMHCIIA-enriched actin-myosin-interacting proteins within the podocyte, demonstrating the important role of NMMHCIIA in organizing the elaborate cytoskeleton structure of podocytes. Our characterization of NMMHCIIA’s functions goes beyond the podocyte, providing important insights into its general molecular role.

This article is from PLoS ONE , volume 9 . Abstract MYH9 encodes non-muscle myosin heavy chain IIA (NMMHCIIA), the predominant force-generating ATPase in non-muscle cells. Several lines of evidence implicate a role for MYH9 in podocytopathies. However, NMMHCIIA‘s function in podocytes remains unknown. To better understand this function, we performed immuno-precipitation followed by mass-spectrometry proteomics to identify proteins interacting with the NMMHCIIA-enriched actin-myosin complexes. Computational analyses revealed that these proteins belong to functional networks including regulators of cytoskeletal organization, metabolism and networks regulated by the HIV-1 gene nef. We further characterized the subcellular localization of NMMHCIIA within podocytes in vivo, and found it to be present within the podocyte major foot processes. Finally, we tested the effect of loss of MYH9 expression in podocytes in vitro, and found that it was necessary for cytoskeletal organization. Our results provide the first survey of NMMHCIIA-enriched actin-myosin-interacting proteins within the podocyte, demonstrating the important role of NMMHCIIA in organizing the elaborate cytoskeleton structure of podocytes. Our characterization of NMMHCIIA’s functions goes beyond the podocyte, providing important insights into its general molecular role.

This article is from Nucleus , volume 4 . Abstract Linker of nucleoskeleton and cytoskeleton (LINC) complexes span the double membrane of the nuclear envelope (NE) and physically connect nuclear structures to cytoskeletal elements. LINC complexes are envisioned as force transducers in the NE, which facilitate processes like nuclear anchorage and migration, or chromosome movements. The complexes are built from members of two evolutionary conserved families of transmembrane (TM) proteins, the SUN (Sad1/UNC-84) domain proteins in the inner nuclear membrane (INM) and the KASH (Klarsicht/ANC-1/SYNE homology) domain proteins in the outer nuclear membrane (ONM). In the lumen of the NE, the SUN and KASH domains engage in an intimate assembly to jointly form a NE bridge. Detailed insights into the molecular architecture and atomic structure of LINC complexes have recently revealed the molecular basis of nucleo-cytoskeletal coupling. They bear important implications for LINC complex function and suggest new potential and as yet unexplored roles, which the complexes may play in the cell.

"Unabridged and unaltered republication of the third edition, published ... [in one volume] in 1918."

 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

Source: Digital Library of India Scanning Centre: Allama Iqbal Library, University of Kashmir Source Library: Women College M.A Road Srinagar Date Accessioned: 7/14/2015 21:44 The Digital Library of India was a project under the auspices of the Government of India.

Book Source: Digital Library of India Item 2015.164336 dc.contributor.author: Dr.h.durege dc.contributor.other: Carnegie Mellon University dc.date.accessioned: 2015-07-06T23:23:55Z dc.date.available: 2015-07-06T23:23:55Z dc.date.citation: 1896 dc.identifier.barcode: 1990040033780 dc.identifier.origpath: /data/upload/0033/785 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/164336 dc.description.scanningcentre: SASTRA University, Thanjavur dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 301 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: N.srinivasan dc.publisher: G.e.fisher And I.j.schwatt,philadelphia dc.rights: Out_of_copyright dc.title: Elements Of The Theory Of Functions Of A Complex Variable

Book Source: Digital Library of India Item 2015.153706 dc.contributor.author: A. R. Forsyth dc.date.accessioned: 2015-07-06T19:21:34Z dc.date.available: 2015-07-06T19:21:34Z dc.date.digitalpublicationdate: 2004-06-29 dc.date.citation: 1918 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 2999990018958 dc.identifier.origpath: /data/upload/0018/963 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/153706 dc.description.numberedpages: 885 dc.description.numberedpages: 37 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 922 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Cambridge University Press dc.rights: Public Domain dc.title: Theory Of Functions Of A Complex Variable

Source: Digital Library of India Scanning Centre: C-DAC, Noida Source Library: Central Library, Bits- Pilani Date Accessioned: 6/24/2015 2:01 The Digital Library of India was a project under the auspices of the Government of India.

Source: Tamil Digital Library Book ID: jZY9lup2kZl6TuXGlZQdjZMdkJly

Source: Digital Library of India Scanning Centre: C-DAC, Noida Source Library: Central Library, Bits- Pilani Date Accessioned: 6/24/2015 2:05 The Digital Library of India was a project under the auspices of the Government of India.

Source: Digital Library of India Scanning Centre: C-DAC, Noida Source Library: Central Library, Bits- Pilani Date Accessioned: 6/24/2015 2:05 The Digital Library of India was a project under the auspices of the Government of India.

The purpose of this paper is to contribute to development a general theory of dual-complex numbers. We start by define the notion of dual-complex and their algebraic properties. In addition, we develop a simple mathematical method based on matrices, simplifying manipulation of dual-complex numbers. Inspired from complex analysis, we generalize the concept of holomorphicity to dual-complex functions. Moreover, a general representation of holomorphic dual-complex functions has been obtained. Finally, and as concrete examples, some usual complex functions have been generalized to the algebra of dual-complex numbers.

The purpose of this paper is to contribute to development a general theory of dual-complex numbers. We start by define the notion of dual-complex and their algebraic properties. In addition, we develop a simple mathematical method based on matrices, simplifying manipulation of dual-complex numbers. Inspired from complex analysis, we generalize the concept of holomorphicity to dual-complex functions. Moreover, a general representation of holomorphic dual-complex functions has been obtained. Finally, and as concrete examples, some usual complex functions have been generalized to the algebra of dual-complex numbers.

14

We present an ongoing development of an existing code for calculating ground-state, steady-state, and transient properties of many-particle systems. The development involves the addition of the full four-index two electron integrals, which allows for the calculation of transport systems, as well as the extension to multi-level electronic systems, such as atomic and molecular systems and other applications. The necessary derivations are shown, along with some preliminary results and a summary of future plans for the code.

The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators.

The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators.

The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators.

The microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators.

Journal of Research of the National Bureau of Standards

This paper discusses some common iterations of complex functions. The presentation is such that similar processes can easily be implemented and understood by undergraduate students. The aim is to illustrate some of the beauty of complex dynamics in an informal setting, while providing a couple of results that are not otherwise readily available in the literature. (Contains 6 figures.)

"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

"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 give a simple uniqueness criterion (and some derived criteria) for holomorphic Abel functions and show that Kneser's real analytic Abel function of the exponential is subject to this criterion.

We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests of a connected graph G completely determines G.

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ of $r$-partitions and a sign $+, -$. It is expected that there exists a close connection between those Kostka functions and the intersection cohomology associated to the enhanced variety $X$ of level $r$. In this paper, we study combinatorial properties of Kostka functions by making use of the geometry of $X$. In particular, we show that if $\mu$ is of the form $\mu = (-,\dots, -, \xi)$ and $\lambda$ is arbitrary, $K^-_{\lambda, \mu}(t)$ has a Lascoux-Sch\"utzenberger type combinatorial description.

The continued fraction expansion of the real number $x=a_0+x_0, a_0\in {\ZZ},$ is given by $0\leq x_n

Book Source: Digital Library of India Item 2015.212410 dc.contributor.other: Hedrick, Earle Raymond dc.date.accessioned: 2015-07-09T17:58:30Z dc.date.available: 2015-07-09T17:58:30Z dc.date.digitalpublicationdate: 2004-01-05 dc.date.citation: 1916 dc.identifier.barcode: 99999990081881 dc.identifier.origpath: /data_copy/upload/0081/886 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/212410 dc.description.scanningcentre: IIAp, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 291 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Dover Publications Inc, New York dc.rights: In Public Domain dc.source.library: Indian Institute Of Astrophysics dc.title: Functions Of A Complex Variable

In this paper we show that a methodology based on a sampling with the Gaussian function of kind $h\,{e^{ - {{\left( {t/c} \right)}^2}}}/\left( {{c}\sqrt \pi } \right)$, where ${c}$ and $h$ are some constants, leads to the Fourier transform that can be represented as a weighted sum of the complex error functions. Due to remarkable property of the complex error function, the Fourier transform based on the weighted sum can be significantly simplified and expressed in terms of a damping harmonic series. In contrast to the conventional discrete Fourier transform, this methodology results in a non-periodic wavelet approximation. Consequently, the proposed approach may be useful and convenient in algorithmic implementation.

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.

xiii, 288 p. 25 cm

xiii, 288 p. 25 cm

Starting from Feynman's Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynman's and Vernon's influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmonic oscillators. As an example we calculate the propagator for the reduced Wigner function associated with the Caldeira-Legett model.

Title: Lectures on the Theory of Functions of a Complex Variable: Volume II: Geometric Theory Author(s): G. Sansone; J. Gerretsen Series: Lectures on the Theory of Functions of a Complex Variable 2 Language: English Year: 1968 Publisher: Springer ISBN: 9028609040, 9789028609044 Pages: 700 (bibliographic) / 709 (file) Format: PDF File size: 21.32 Mb (22354934 bytes) Time added / time modified: 2023-03-24 18:03:05 Commentary: OCR'd with ABBYY Finereader (not proofread). The covers is blank so I generated one.

While integrals like _ f(z) dz and _ M dx+ N dy have been de�ned in the text (p.101), di_erential forms like dx, dy and dz = dx + i dy have not been de�ned (and the de�nition is more subtle), we shall develop the theory of harmonic functions (p.162-170) without di_erential forms.

14

We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, $k \ge 1$) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as $e^{-cn^\alpha}$ strictly inside E, where c and $\alpha$ are positive constants independent of the degree n of the approximating polynomial.

We show how the space of complex spin structures of a closed oriented three-manifold embeds naturally into a space of quadratic functions associated to its linking pairing. Besides, we extend the Goussarov-Habiro theory of finite type invariants to the realm of compact oriented three-manifolds equipped with a complex spin structure. Our main result states that two closed oriented three-manifolds endowed with a complex spin structure are undistinguishable by complex spin invariants of degree zero if, and only if, their associated quadratic functions are isomorphic.

The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions ${}_{p+1}F_p$.

The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions ${}_{p+1}F_p$.

Assumption of certain hierarchy of soft ferromagnet energy terms, realized in small enough flat nano-elements, allows to obtain explicit expressions for their magnetization distributions. By minimising the energy terms sequentially, from most to the least important, magnetization distributions are expressed as solutions of Riemann-Hilbert boundary value problem for a function of complex variable. A number of free parameters, corresponding to positions of vortices and anti-vortices, still remain in the expression. These parameters can be found by computing and minimizing the total magnetic energy of the particle with no approximations. Thus, the presented approach is a factory of realistic Ritz functions for analytical micromagnetic calculations. These functions are so versatile, that they may even find applications on their own (e.g. for fitting magnetic microscopy images). Examples are given for multi-vortex magnetization distributions in circular cylinder, and for 2-dimensional domain walls in thin magnetic strips.

A thorough analysis of values of the function $m\mapsto\mbox{sn}(K(m)u\mid m)$ for complex parameter $m$ and $u\in (0,1)$ is given. First, it is proved that the absolute value of this function never exceeds 1 if $m$ does not belong to the region in $\mathbb{C}$ determined by inequalities $|z-1| 1$. The global maximum of the function under investigation is shown to be always located in this region. More precisely, it is proved that, if $u\leq1/2$, then the global maxim is located at $m=1$ with the value equal to $1$. While if $u>1/2$, then the global maximum is located in the interval $(1,2)$ and its value exceeds $1$. In addition, more subtle extremal properties are studied numerically. Finally, applications in a Laplace-type integral and spectral analysis of some complex Jacobi matrices are presented.