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Unicity Of Meromorphic Mappings

Author: Pei-Chu Hu
Publisher: Springer Science & Business Media
ISBN: 1475737750
Size: 12.16 MB
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For a given meromorphic function I(z) and an arbitrary value a, Nevanlinna's value distribution theory, which can be derived from the well known Poisson-Jensen for mula, deals with relationships between the growth of the function and quantitative estimations of the roots of the equation: 1 (z) - a = O. In the 1920s as an application of the celebrated Nevanlinna's value distribution theory of meromorphic functions, R. Nevanlinna [188] himself proved that for two nonconstant meromorphic func tions I, 9 and five distinctive values ai (i = 1,2,3,4,5) in the extended plane, if 1 1- (ai) = g-l(ai) 1M (ignoring multiplicities) for i = 1,2,3,4,5, then 1 = g. Fur 1 thermore, if 1- (ai) = g-l(ai) CM (counting multiplicities) for i = 1,2,3 and 4, then 1 = L(g), where L denotes a suitable Mobius transformation. Then in the 19708, F. Gross and C. C. Yang started to study the similar but more general questions of two functions that share sets of values. For instance, they proved that if 1 and 9 are two nonconstant entire functions and 8 , 82 and 83 are three distinctive finite sets such 1 1 that 1- (8 ) = g-1(8 ) CM for i = 1,2,3, then 1 = g.

Distribution Theory Of Algebraic Numbers

Author: Pei-Chu Hu
Publisher: Walter de Gruyter
ISBN: 3110208261
Size: 67.46 MB
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The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions • Algebraic numbers • Algebraic geometry • Height functions • The abc-conjecture • Roth's theorem • Subspace theorems • Vojta's conjectures • L-functions.

Value Distribution Theory Related To Number Theory

Author: Pei-Chu Hu
Publisher: Springer Science & Business Media
ISBN: 3764375698
Size: 77.30 MB
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The subject of the book is Diophantine approximation and Nevanlinna theory. This book proves not just some new results and directions but challenging open problems in Diophantine approximation and Nevanlinna theory. The authors’ newest research activities on these subjects over the past eight years are collected here. Some of the significant findings are the proof of Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, generalized abc-conjecture, and more.

Value Distribution Theory And Related Topics

Author: Grigor A. Barsegian
Publisher: Springer Science & Business Media
ISBN: 1402079516
Size: 11.85 MB
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The Nevanlinna theory of value distribution of meromorphic functions, one of the milestones of complex analysis during the last century, was c- ated to extend the classical results concerning the distribution of of entire functions to the more general setting of meromorphic functions. Later on, a similar reasoning has been applied to algebroid functions, subharmonic functions and meromorphic functions on Riemann surfaces as well as to - alytic functions of several complex variables, holomorphic and meromorphic mappings and to the theory of minimal surfaces. Moreover, several appli- tions of the theory have been exploited, including complex differential and functional equations, complex dynamics and Diophantine equations. The main emphasis of this collection is to direct attention to a number of recently developed novel ideas and generalizations that relate to the - velopment of value distribution theory and its applications. In particular, we mean a recent theory that replaces the conventional consideration of counting within a disc by an analysis of their geometric locations. Another such example is presented by the generalizations of the second main theorem to higher dimensional cases by using the jet theory. Moreover, s- ilar ideas apparently may be applied to several related areas as well, such as to partial differential equations and to differential geometry. Indeed, most of these applications go back to the problem of analyzing zeros of certain complex or real functions, meaning in fact to investigate level sets or level surfaces.

Current Topics In Pure And Computational Complex Analysis

Author: Santosh Joshi
Publisher: Springer
ISBN: 8132221133
Size: 42.65 MB
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The book contains 13 articles, some of which are survey articles and others research papers. Written by eminent mathematicians, these articles were presented at the International Workshop on Complex Analysis and Its Applications held at Walchand College of Engineering, Sangli. All the contributing authors are actively engaged in research fields related to the topic of the book. The workshop offered a comprehensive exposition of the recent developments in geometric functions theory, planar harmonic mappings, entire and meromorphic functions and their applications, both theoretical and computational. The recent developments in complex analysis and its applications play a crucial role in research in many disciplines.

Progress In Analysis

Author: Heinrich G W Begehr
Publisher: World Scientific
ISBN: 9814485233
Size: 38.92 MB
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The biannual ISAAC congresses provide information about recent progress in the whole area of analysis including applications and computation. This book constitutes the proceedings of the third meeting. Contents: Volume 1: Function Spaces and Fractional Calculus (V I Burenkov & S Samko)Asymptotic Decomposition (Methods of Small Parameters, Averaging Theory) (J A Dubinski)Integral Transforms and Applications (S Saitoh et al.)Analytic Functionals, Hyperfunctions and Generalized Functions (M Morimoto & H Komatsu)Geometric Function Theory (G Kohr & M Kohr)omplex Function Spaces (R Aulaskari & I Laine)Value Distribution Theory and Complex Dynamics (C C Yang)Clifford Analysis (K Gürlebeck et al.)Octonions (T Dray & C Monogue)Nonlinear Potential Theory (O Martio)Classical and Fine Potential Theory, Holomorphic and Finely Holomorphic Functions (P Tamrazov)Differential Geometry and Control Theory for PDEs (B Gulliver et al.)Differential Geometry and Quantum Physics (-)Dynamical Systems (B Fiedler)Attractors for Partial Differential Equations (G Raugel)Spectral Theory of Differential Operators (B Vainberg)Pseudodifferential Operators, Quantization and Signal Analysis (M W Wong)Microlocal Analysis (B-W Schulze & M Korey) Volume 2: Complex and Functional Analytic Methods in PDEs (A Cialdea et al.)Geometric Properties of Solutions of PDEs (R Magnanini)Qualitative Properties of Solutions of Hyperbolic and Schrödinger Equations (M Reissig & K Yagdjian)Homogenization Moving Boundaries and Porous Media (A Bourgeat & R P Gilbert)Constructive Methods in Applied Problems (P Krutitskii)Waves in Complex Media (R P Gilbert & A Wirgin)Nonlinear Waves (I Lasiecka & H Koch)Mathematical Analysis of Problems in Solid Mechanics (K Hackl & X Li)Direct and Inverse Scattering (L Fishman)Inverse Problems (G N Makrakis et al.)Mathematical Methods in Non-Destructive Evaluation and Non-Destructive Testing (A Wirgin)Numerical Methods for PDEs, Systems and Optimization (A Ben-Israel & I Herrera) Readership: Graduate students and researchers in real, complex, numerical analysis, as well as mathematical physics.Keywords:Fractional Calculus;Function Theory;Potential Theory;Dynamical Systems;Differential Operators;Pseudodifferntial Operators;Partial Differential Equations;Inverse Problems

Proceedings Of The Second Isaac Congress

Author: Heinrich G.W. Begehr
Publisher: Springer Science & Business Media
ISBN: 9780792365976
Size: 22.34 MB
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This book is the Proceedings of the Second ISAAC Congress. ISAAC is the acronym of the International Society for Analysis, its Applications and Computation. The president of ISAAC is Professor Robert P. Gilbert, the second named editor of this book, e-mail: [email protected] The Congress is world-wide valued so highly that an application for a grant has been selected and this project has been executed with Grant No. 11-56 from *the Commemorative Association for the Japan World Exposition (1970). The finance of the publication of this book is exclusively the said Grant No. 11-56 from *. Thus, a pair of each one copy of two volumes of this book will be sent to all contributors, who registered at the Second ISAAC Congress in Fukuoka, free of charge by the Kluwer Academic Publishers. Analysis is understood here in the broad sense of the word, includ ing differential equations, integral equations, functional analysis, and function theory. It is the purpose of ISAAC to promote analysis, its applications, and its interaction with computation. With this objective, ISAAC organizes international Congresses for the presentation and dis cussion of research on analysis. ISAAC welcomes new members and those interested in joining ISAAC are encouraged to look at the web site http://www .math. udel.edu/ gilbert/isaac/index.html vi and http://www.math.fu-berlin.de/ rd/ ag/isaac/newton/index.html.

The Beltrami Equation

Author: Vladimir Gutlyanskii
Publisher: Springer Science & Business Media
ISBN: 1461431913
Size: 47.21 MB
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This book is devoted to the Beltrami equations that play a significant role in Geometry, Analysis and Physics and, in particular, in the study of quasiconformal mappings and their generalizations, Riemann surfaces, Kleinian groups, Teichmuller spaces, Clifford analysis, meromorphic functions, low dimensional topology, holomorphic motions, complex dynamics, potential theory, electrostatics, magnetostatics, hydrodynamics and magneto-hydrodynamics. The purpose of this book is to present the recent developments in the theory of Beltrami equations; especially those concerning degenerate and alternating Beltrami equations. The authors study a wide circle of problems like convergence, existence, uniqueness, representation, removal of singularities, local distortion estimates and boundary behavior of solutions to the Beltrami equations. The monograph contains a number of new types of criteria in the given problems, particularly new integral conditions for the existence of regular solutions to the Beltrami equations that turned out to be not only sufficient but also necessary. The most important feature of this book concerns the unified geometric approach based on the modulus method that is effectively applied to solving the mentioned problems. Moreover, it is characteristic for the book application of many new concepts as strong ring solutions, tangent dilatations, weakly flat and strongly accessible boundaries, functions of finite mean oscillations and new integral conditions that make possible to realize a more deep and refined analysis of problems related to the Beltrami equations. Mastering and using these new tools also gives essential advantages for the reader in the research of modern problems in many other domains. Every mathematics graduate library should have a copy of this book.​