Complex analysis, complex variables Books

Book SynopsisThe first part of this book is mainly intended as a textbook for students at the Sophomore-Junior level, majoring in mathematics, engineering, or the sciences in general. The book includes the basic topics in Ordinary Differential Equations, normally taught at the undergraduate level, such as linear and nonlinear equations and systems, Bessel functions, Laplace transform, stability, etc. It is written with ample flexibility to make it appropriate either as a course stressing application, or a course stressing rigor and analytical thinking. It also offers sufficient material for a one-semester graduate course, covering topics such as phase plane analysis, oscillation, Sturm-Liouville equations, Euler-Lagrange equations in Calculus of Variations, first and second order linear PDE in 2D. There are substantial lists of exercises at the ends of the chapters. In this edition complete solutions to all even number problems are given in the back of the book.The 2nd edition also includes some new problems and examples. An effort has been made to make the material more suitable and self-contained for undergraduate students with minimal knowledge of Calculus. For example, a detailed review of matrices and determinants has been added to the chapter on systems of equations. The second edition also contains corrections of some misprints and errors in the first edition.

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Book SynopsisThis book may serve as a basis for students and teachers. The text should provide the reader with a quick overview of the basics for Optimal Control and the link with some important conceptes of applied mathematical, where an agent controls underlying dynamics to find the strategy optimizing some quantity. There are broad applications for optimal control across the natural and social sciences, and the finale to this text is an invitation to read current research on one such application. The balance of the text will prepare the reader to gain a solid understanding of the current research they read.

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Book SynopsisWe introduce mixed twistor D-modules and establish their fundamental functorial properties. We also prove that they can be described as the gluing of admissible variations of mixed twistor structures. In a sense, mixed twistor D-modules can be regarded as a twistor version of M. Saito's mixed Hodge modules. Alternatively, they can be viewed as a mixed version of the pure twistor D-modules studied by C. Sabbah and the author. The theory of mixed twistor D-modules is one of the ultimate goals in the study suggested by Simpson's Meta Theorem and it would form a foundation for the Hodge theory of holonomic D-modules which are not necessarily regular singular.Table of ContentsIntroduction.- Preliminary.- Canonical prolongations.- Gluing and specialization of r-triples.- Gluing of good-KMS r-triples.- Preliminary for relative monodromy filtrations.- Mixed twistor D-modules.- Infinitesimal mixed twistor modules.- Admissible mixed twistor structure and variants.- Good mixed twistor D-modules.- Some basic property.- Dual and real structure of mixed twistor D-modules.- Derived category of algebraic mixed twistor D-modules.- Good systems of ramified irregular values.

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Book SynopsisThis monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point. Inside, readers will find a detailed introduction into the theory of parabolic bifurcation, Fatou coordinates, Écalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization. The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both experts in the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.Trade Review“The book under review is devoted to the study of parabolic renormalization. … The book is very well written and self-contained … and most results are stated together with their proofs.” (Jasmin Raissy, zbMATH 1342.37051, 2016)Table of Contents​1 Introduction.- 2 Local dynamics of a parabolic germ.- 3 Global theory.- 4 Numerical results.- 5 For dessert: several amusing examples.- Index.

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Book SynopsisThis text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called “arithmetic hyperbolic surfaces”, the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them. After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss. The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics.Trade Review“The French book under review gives an introduction to hyperbolic surfaces with an emphasis on the Selberg conjecture. … it is intended for advanced graduate students but is also well suited for all those who want to acquaint themselves with harmonic analysis on hyperbolic surfaces and automorphic forms.” (Frank Monheim, zbMATH, August, 2017)“This book gives a very nice introduction to the spectral theory of the Laplace-Beltrami operator on hyperbolic surfaces of constant negative curvature. … mainly intended for students with a knowledge of basic differential geometry and functional analysis but also for people doing research in other domains of mathematics or mathematical physics and interested in the present day problems in this very active field of research. … book gives one of the best introductions to this fascinating field of interdisciplinary research.” (Dieter H. Mayer, Mathematical Reviews, August, 2017)Table of ContentsPreface.- Introduction.- Arithmetic Hyperbolic Surfaces.- Spectral Decomposition.- Maass Forms.- The Trace Formula.- Multiplicity of lambda1 and the Selberg Conjecture.- L-Functions and the Selberg Conjecture.- Jacquet-Langlands Correspondence.- Arithmetic Quantum Unique Ergodicity.- Appendices.- References.- Index of notation.- Index.- Index of names.

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Book SynopsisThis book offers a modern introduction to Nevanlinna theory and its intricate relation to the theory of normal families, algebraic functions, asymptotic series, and algebraic differential equations.Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value distribution of differential polynomials and illustrates how value- and pair-sharing problems are linked to algebraic curves and Briot–Bouquet differential equations. In addition to discussing classical applications of Nevanlinna theory, the book outlines state-of-the-art research, such as the effect of the Yosida and Zalcman–Pang method of re-scaling to algebraic differential equations, and presents the Painlevé–Yosida theorem, which relates Painlevé transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes of meromorphic functions.Aimed at graduate students interested in recent developments in the field and researchers working on related problems, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations will also be of interest to complex analysts looking for an introduction to various topics in the subject area. With examples, exercises and proofs seamlessly intertwined with the body of the text, this book is particularly suitable for the more advanced reader.Trade Review“The book by Steinmetz is clearly written, including a substantial number of exercises related to and complementing the actual text.” (Ilpo Laine, Mathematical Reviews, June, 2018)“The list of references contains more than 200 items including very recent results of the author and other. … I recommend this book to any person who is interested in complex analysis, in particular, in value distribution theory and complex differential equations.” (Igor Chyzhykov, zbMATH, 2018)Table of ContentsIntroduction and preface.- Selected Topics in Complex Analysis.- Nevanlinna Theory.- Selected Applications of Nevanlinna Theory.- Normal Families.- Algebraic Differential Equations.- Higher-Order Algebraic Differential Equations.- Index.

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Book SynopsisThis book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds.Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory.Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.Table of ContentsPart I Stein Manifolds.- 1 Preliminaries.- 2 Stein Manifolds.- 3 Stein Neighborhoods and Approximation.- 4 Automorphisms of Complex Euclidean Spaces.- Part II Oka Theory.- 5 Oka Manifolds.- 6 Elliptic Complex Geometry and Oka Theory.- 7 Flexibility Properties of Complex Manifolds and Holomorphic Maps.- Part III Applications.- 8 Applications of Oka Theory and its Methods.- 9 Embeddings, Immersions and Submersions.- 10 Topological Methods in Stein Geometry.- References.- Index.

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Book SynopsisThe central theme of this reference book is the metric geometry of complex analysis in several variables. Bridging a gap in the current literature, the text focuses on the fine behavior of the Kobayashi metric of complex manifolds and its relationships to dynamical systems, hyperbolicity in the sense of Gromov and operator theory, all very active areas of research. The modern points of view expressed in these notes, collected here for the first time, will be of interest to academics working in the fields of several complex variables and metric geometry. The different topics are treated coherently and include expository presentations of the relevant tools, techniques and objects, which will be particularly useful for graduate and PhD students specializing in the area.Table of Contents1. Invariant Distances.- 2. Dynamics in Several Complex Variables.- 3. Gromov Hyperbolic Spaces and Applications to Complex Analysis.- 4. Gromov Hyperbolicity of Bounded Convex Domains.- 5. Quasi-conformal Mappings.- 6. Carleson Measures and Toeplitz Operators. References.

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Book SynopsisThe focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved. The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps. Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.Table of Contents1. Introduction.- 2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces.- 3. Symbol Escape Rates and the Survivor Set K(Un).- 4. Escape Rates for Conformal GDMSs and IFSs.- 5. Applications: Escape Rates for Multimodal Mapsand One-Dimensional Complex Dynamics.

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Book SynopsisThis text provides a comprehensive introduction to Berezin–Toeplitz operators on compact Kähler manifolds. The heart of the book is devoted to a proof of the main properties of these operators which have been playing a significant role in various areas of mathematics such as complex geometry, topological quantum field theory, integrable systems, and the study of links between symplectic topology and quantum mechanics. The book is carefully designed to supply graduate students with a unique accessibility to the subject. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation; prerequisites have been kept to a minimum. Readers are encouraged to enhance their understanding of the material by working through the many straightforward exercises. Trade Review“The book … represents an essential prerequisite for anyone who wants to work in the field. The author have managed to make it readable by non-specialists.” (Béchir Dali, zbMATH 1452.32002, 2021)Table of ContentsPreface.- 1. Introduction.- 2. A short introduction to Kähler manifolds.- 3. Complex line bundles with connections.- 4. Quantization of compact Kähler manifolds.- 5. Berezin–Toeplitz operators.- 6. Schwartz kernels.- 7. Asymptotics of the projector Pi_k.- 8. Proof of product and commutator estimates.- 9. Coherent states and norm correspondence.- A. The circle bundle point of view.- Bibliography.

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Book SynopsisNunmehr kann ich auch den zweiten Teil meiner Lehre von den Kettenbrüchen, der den analytischen Kettenbrüchen gewidmet ist, als Band 11 in neuer Be­ arbeitung den Fachgenossen vorlegen. Ebenso wie bei dem im Jahr 1954 er­ schienenen Band I ging mein Bemühen dah~, den heutigen Stand der Wissen­ schaft in möglichst leicht verständlicher Weise darzustellen. Die leichte Ver­ ständlichkeit kann natürlich nicht bedeuten, daß der Leser das Buch wie einen Roman durcheilen kann. Wenn er aber die Technik der Differential-und Integral­ rechnung beherrscht, wenn er schon etwas von der Gammafunktion und von linearen Differentialgleichungen gehört hat und ein klein wenig Funktionen­ theorie weiß, kann er unschwer folgen; nur darf er, um in Einzelheiten ein­ zudringen, nicht die Mühe scheuen, gelegentlich Papier und Bleistift zur Hand zu nehmen und einfache Rechnungen nach gegebener Anweisung selbst durch­ zuführen. Es geht alles nach geläufigen Methoden. Der allgemeine Rahmen des Buches ist der alte geblieben; doch sind die sechs Kapitel mit weitgehend verändertem Inhalt gefüllt. Namentlich die ersten drei und auch die zweite Hälfte des vierten sind mannigfach umgestaltet und er­ weitert, während in den letzten zwei nur geringere Änderungen nötig und sogar Kürzungen möglich waren, um Raum für den neuen Stoff der früheren zu ge­ winnen. Überall in der Welt, besonders in der Neuen, ist in den letzten Dezennien ein reiches Material von neuen Kettenbruchtypen und neuen Erkenntnissen, vor allem in bezug auf Konvergenz, gewonnen worden, das gesichtet, geordnet und systematisch eingearbeitet werden mußte.Table of ContentsI. Transformation von Kettenbrüchen..- § 1. Rekapitulation.- § 2. Null als Teilzähler. — Äquivalente Kettenbrüche.- § 3. Kettenbrüche mit vorgegebenen Näherungsbrüchen.- § 4. Kontraktion und Extension.- § 5. Äquivalenz von Kettenbrüchen und Reihen.- § 6. Äquivalenz von Kettenbrüchen und Produkten.- § 7. Die Transformation von Bauer und Muir.- § 8. Weitere Anwendungen. Haupformel von Ramanujan.- II. Kriterien für Konvergenz und Divergenz..- § 9. Bedingte und unbedingte Konvergenz.- § 10. Allgemeine Kriterien von Broman, Stern und Scott-Wall.- § 11. Konvergenz bei positiven Elementen.- § 12. Konvergenz bei reellen Elementen.- § 13. Irrationalität gewisser Kettenbrüche.- § 14. Die Konvergenzkriterien von Pringsheim.- § 15. Die Konvergenzkriterien von van Vleck-Jensen und Hamburger-Mall-Wall.- § 16. Anwendung: Geltungsbereich der Ramanujan-Formel.- § 17. Einige neuere Kriterien. — Das Parabeltheorem.- § 18. Periodische Kettenbrüche.- § 19. Limitärperiodische Kettenbrüche.- § 20. Die Gleichung % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WG4bWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamiEamaaBaaaleaacaaI % XaaabeaaaaGccqGH9aqpcaWGIbWaaSbaaSqaaiaaicdaaeqaaOGaey % 4kaSYaaSaaaeaadaabcaqaaiaadggadaWgaaWcbaGaaGymaaqabaaa % kiaawIa7aaqaamaaeeaabaGaamOyamaaBaaaleaacaaIXaaabeaaaO % Gaay5bSdaaaiabgUcaRmaalaaabaWaaqGaaeaacaWGHbWaaSbaaSqa % aiaaikdaaeqaaaGccaGLiWoaaeaadaabbaqaaiaadkgadaWgaaWcba % GaaGOmaaqabaaakiaawEa7aaaacqGHRaWkcqWIVlctaaa!4F24! $$ \frac{{{x_0}}}{{{x_1}}} = {b_0} + \frac{{\left. {{a_1}} \right|}}{{\left| {{b_1}} \right.}} + \frac{{\left. {{a_2}} \right|}}{{\left| {{b_2}} \right.}} + \cdots $$als Folge des Rekursionssystems % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa % aaleaacaWG2baabeaakiabg2da9iaadkgadaWgaaWcbaGaamODaaqa % baGccaWG4bWaaSbaaSqaaiaadAhacqGHRaWkcaaIXaaabeaakiabgU % caRiaadggadaWgaaWcbaGaamODaiabgUcaRiaaigdaaeqaaOGaamiE % amaaBaaaleaacaWG2bGaey4kaSIaaGOmaaqabaaaaa!4763! $$ {x_v} = {b_v}{x_{v + 1}} + {a_{v + 1}}{x_{v + 2}} $$.- III. Verschiedene Zuordnungen von Potenzreihen zu Kettenbrüchen..- § 21. Allgemeine C-Kettenbrüche.- § 22. Quadratwurzeln.- § 23. Regelmäßige C-Kettenbrüche.- § 24. Die Kettenbrüche von Gauß, Heine und damit verwandte.- § 25. Der assoziierte Kettenbruch.- § 26. Zusammenhang zwischen dem korrespondierenden und assoziierten Kettenbruch. — Einige Transformationen des korrespondierenden Kettenbruches.- § 27. Konvergenz und Divergenz.- § 28. Konvergenz der Kettenbrüche von Gauß, Heine usw.- § 29. Ein bemerkenswertes Divergenzphänomen.- § 30. J-Kettenbrüche und ihre Anwendung auf Polynome, deren Wurzeln negative reelle Teile haben.- § 31. Weitere Typen von Kettenbrüchen, denen man Potenzreihen zuordnen kann.- IV. Die Kettenbrüche von Stieltjes..- § 32. Der Integralbegriff von Stieltjes.- § 33. Der korrespondierende und assoziierte Kettenbruch eines Stieltjessehen Integrals.- § 34. Der Satz von Markoff.- § 35. Die Wurzeln der Näherungsnenner von G-, H- und S-Kettenbrüchen.- § 36. Das Grommersche Auswahltheorem.- § 37. Konvergenz und analytischer Charakter der S- und H-Kettenbrüche.- § 38. Die vollständige Konvergenz der G-Kettenbrüche.- § 39. Das Momentenproblem.- V. Die P adésehe Tafel..- § 40. Begriff der Padéschen Tafel.- § 41. Normale und anormale Tafel.- § 42. Die Exponentialfunktion.- § 43. Die Laguerresche Differentialgleichung.- § 44. Die Kettenbrüche der Padéschen Tafel.- § 45. Die Konvergenzfrage.- VI. Kettenbrüche, deren Elemente a, und b, rationale Funktionen von v sind..- § 46. Die Konvergenz dieser Kettenbrüche.- § 47. Zusammenhang mit Differentialgleichungen.- § 48. Die Kettenbrüche mit dem allgemeinen Glied % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada % abcaqaaiaadggadaWgaaWcbaGaamODaaqabaaakiaawIa7aaqaamaa % eeaabaGaamOyamaaBaaaleaacaWG2baabeaaaOGaay5bSdaaaiabg2 % da9maalaaabaWaaqGaaeaacaWGHbGaey4kaSIaamOyamaaBaaaleaa % caWG2baabeaaaOGaayjcSdaabaWaaqqaaeaacaWGJbGaey4kaSIaam % izamaaBaaaleaacaWG2baabeaaaOGaay5bSdaaaaaa!4961! $$ \frac{{\left. {{a_v}} \right|}}{{\left| {{b_v}} \right.}} = \frac{{\left. {a + {b_v}} \right|}}{{\left| {c + {d_v}} \right.}} $$.- § 49. Die Kettenbrüche mit dem allgemeinen Glied % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada % abcaqaaiaadggadaWgaaWcbaGaamODaaqabaaakiaawIa7aaqaamaa % eeaabaGaamOyamaaBaaaleaacaWG2baabeaaaOGaay5bSdaaaiabg2 % da9maalaaabaWaaqGaaeaacaWGHbGaey4kaSIaamOyamaaBaaaleaa % caWG2baabeaakiabgUcaRiaadogacaWG2bWaaWbaaSqabeaacaaIYa % aaaaGccaGLiWoaaeaadaabbaqaaiaadsgacqGHRaWkcaWGLbGaamOD % aaGaay5bSdaaaaaa!4CE5! $$ \frac{{\left. {{a_v}} \right|}}{{\left| {{b_v}} \right.}} = \frac{{\left. {a + {b_v} + c{v^2}} \right|}}{{\left| {d + ev} \right.}} $$.- § 50. Die Methode von Cesàro.- § 51. Die Formel von Pincherle.- Literatur.- Verzeichnis der bemerkenswerten Formeln.

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Book SynopsisFrom the reviews: "Theory of Stein Spaces provides a rich variety of methods, results, and motivations - a book with masterful mathematical care and judgement. It is a pleasure to have this fundamental material now readily accessible to any serious mathematician." --J. Eells in Bulletin of the London Mathematical Society (1980)Trade Review"Theory of Stein Spaces provides a rich variety of methods, results, and motivations - a book with masterful mathematical care and judgement. It is a pleasure to have this fundamental material now readily accessible to any serious mathematician."J. Eells in Bulletin of the London Mathematical Society (1980) "Written by two mathematicians who played a crucial role in the development of the modern theory of several complex variables, this is an important book."J.B. Cooper in Internationale Mathematische Nachrichten (1979)Table of ContentsA. Sheaf Theory.- B. Cohomology Theory.- I. Coherence Theory for Finite Holomorphic Maps.- II. Differential Forms and Dolbeault Theory.- III. Theorems A and B for Compact Blocks ?m.- IV. Stein Spaces.- V. Applications of Theorems A and B.- VI. The Finiteness Theorem.- VII. Compact Riemann Surfaces.- Table of Symbols.- Addendum.- Errors and Misprints.

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Book SynopsisEasily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)Trade ReviewFrom the reviews: "The book under review provides an introduction to the contemporary theory of compact complex manifolds, with a particular emphasis on Kähler manifolds in their various aspects and applications. As the author points out in the preface, the text is based on a two-semester course taught in 2001/2002 at the University of Cologne, Germany. Having been designed for third-year students, the aim of the course was to acquaint beginners in the field with some basic concepts, fundamental techniques, and important results in the theory of compact complex manifolds, without being neither too basic nor too sketchy. Also, as complex geometry has undergone tremendous developments during the past five decades, and become an indispensable framework in modern mathematical physics, the author has tried to teach the subject in such a way that would enable the students to understand the more recent developments in the field, too, up to some of the fascinating aspects of the stunning interplay between complex geometry and quantum field theory in theoretical physics. The present text, as an outgrowth of this special course in complex geometry, does evidently reflect these emphatic intentions of the author's in a masterly manner. Keeping the prerequisites from complex analysis and differential geometry to an absolute minimum, he provides a streamlined introduction to the theory of compact complex manifolds and Kählerian geometry, with many outlooks and applications, but without trying to be encyclopedic or panoramic. without trying to be encyclopedic or panoramic. As to the precise contents, the text consists of six chapters and two appendices. [...] The author has added two general appendices at the end of the book. Those aremeant to help the unexperienced reader to recall a few basic concepts and facts from differential geometry, Hodge theory on differentiable manifolds, sheaf theory, and sheaf cohomology. This very user-friendly service makes the entire introductory text more comfortable for less seasoned students, perhaps also for interested and mathematically less experienced physicists, although the author does not claim absolute self-containedness of the book. The entire text comes with a wealth of enlightening examples, historical remarks, comments and hints for further reading, outlooks to other directions of research, and numerous exercises after each section. The exercises are far from being bland and often quite demanding, but they should be mastered by ambitious and attentive readers, in the last resort after additional reading. Finally, there is a very rich bibliography of 118 references, also from the very recent research literature, which the author profusely refers to throughout the entire text. The whole exposition captivates by its clarity, profundity, versality, and didactical strategy, which lead the reader right to the more advanced literature in complex geometry as well as to the forefront of research in geometry and its applications to mathematical physics. No doubt, this book is an outstanding introduction to modern complex geometry." KIeinert (Berlin), Zentralblatt für Mathematik 1055 (2005) This is a very interesting and nice book. It provides a clear and deep introduction about complex geometry, namely the study of complex manifolds. These are differentiable manifolds endowed with the additional datum of a complex structure that is more rigid than the geometrical structures used in differential geometry. Complex geometry is on the crossroad of algebraic and differential geometry. Complex geometry is also becoming a stimulating and useful tool for theoretical physicists working in string theory and conformal field theory. The physicist, will be very glad to discover the interplay between complex geometry and supersymmetry and mirror symmetry. The book begins by explaining the local theory and all you need to understand the global structure of complex manifolds. Then we get an introduction to the complex manifolds as such, where the reader can progressively perceive the difference between real manifolds and complex ones. Then he gets an account of the theory of Kälher manifolds. And the physicist will be glad to find therein a first step on the road going from complex geometry to conformal field theory and supersymmetry. One chapter is dedicated to the study of holomorphic vector bundles (connections, curvature, Chern classes). In this context, the reader will clarify the relations between Riemannian and Kälher geometries. With all this stuff it is then possible to focus on some applications of cohomology. This leads to a nice introduction to the famous Hirzebruch-Riemann-Roch theorem and to Kodaira vanishing and embedding theorems. The last chapter of the book tackles the very important topics of deformations of complex structures. This chapter will be interesting especially for readers that are studying Calabi-Yau manifolds and mirror symmetries. The main text of the book is completed by two pedagogical appendices. One about Hodge theory and the other about sheaf cohomology. Thus this beautiful textbook will be very interesting for both pure mathematicians and theoretical physicists working in recent domains of field theory. It can be used by students or scientists for a first introduction in this field. It is always very accessible and the reader will find a detailed account of the basic concepts and many well-chosen exercises that illustrate the theory. Many illuminating examples help the reader in the understanding of all fundamental notions. I could certainly recommend this textbook to my students attending my lectures on differential geometry. Professor Dominique LAMBERT, University of Namur; Department « sciences, philosophies et sociétés » Rue de Bruxelles 61 B-5000 Namur Belgium "As complex geometry has undergone tremendous developments … the author has tried to teach the subject in such a way that would enable the students to understand the more recent developments in the field … . This very user-friendly … more comfortable for less seasoned students … . The entire text comes with a wealth of enlightening examples, historical remarks, comments and hints … . Finally, there is a very rich bibliography … . The whole exposition captivates by its clarity, profundity, versality, and didactical strategy … . an outstanding introduction to modern complex geometry." (Werner Kleinert, Zentralblatt Math, Vol. 1055, 2005) "The book contains detailed accounts of the basic concepts and the many exercises illustrate the theory. Appendices to various chapters allow an outlook to recent research directions." (L’Enseignment Mathematique, Vol. 50 (3-4), 2004) "This is the book that a generation of complex geometers will wish had existed when they first learned the subject, and that the next generation of geometers will surely use. … Inserted into the standard material are some excellent appendices to stimulate interest and further reading … . the reader learning the basic material is brought quickly and often to some fascinating areas of current research. Exercises introduce many examples … . The result is an excellent course in complex geometry." (Richard P. Thomas, Mathematical Reviews, 2005h) "The book is based on a year course on complex geometry and its interaction with Riemannian geometry. It prepares a basic ground for a study of complex geometry as well as for understanding ideas coming recently from string theory. … The book is a very good introduction to the subject and can be very useful both for mathematicians and mathematical physicists." (EMS Newsletter, June, 2005) "The book under review is a textbook, based on a 2-semester course to third year undergraduates in the University of Cologne. … In the UK I think the book would be regarded as more suitable for a masters’ level course for students well versed in standard complex analysis and differential geometry." (Peter Giblin, The Mathematical Gazette, Vol. 91 (520), 2007)Table of ContentsLocal Theory.- Complex Manifolds.- Kähler Manifolds.- Vector Bundles.- Applications of Cohomology.- Deformations of Complex Structures.

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Book SynopsisThe geometry of real submanifolds in complex manifolds and the analysis of their mappings belong to the most advanced streams of contemporary Mathematics. In this area converge the techniques of various and sophisticated mathematical fields such as P.D.E.s, boundary value problems, induced equations, analytic discs in symplectic spaces, complex dynamics. For the variety of themes and the surprisingly good interplaying of different research tools, these problems attracted the attention of some among the best mathematicians of these latest two decades. They also entered as a refined content of an advanced education. In this sense the five lectures of this volume provide an excellent cultural background while giving very deep insights of current research activity.Table of ContentsPreface.- M. Abate: Angular Derivatives in Several Complex Variables.- J.E. Fornaess: Real Methods in Complex Dynamics.- X. Huang: Local Equivalence Problems for Real Submanifolds in Complex Spaces.- J.-P. Rosay: Introduction to a General Theory of Boundary Values.- A. Tumanov: Extremal Discs and the Geometry of CR Manifolds.

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Book SynopsisAuthor received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equationsTrade ReviewFrom the reviews: "...these volumes are excellently written and make for greatly profitable reading. For years to come they will surely be a main reference for anyone wishing to study partial differential operators."-- MATHEMATICAL REVIEWS "This volume focuses on linear partial differential operators with constant coefficients … . Each chapter ends with notes on the literature, and there is a large bibliography. … The binding of this softcover reprint seems quite good … . Overall, it is great to have this book back at an affordable price. It really does deserve to be described as a classic." (Fernando Q. Gouvêa, MathDL, January, 2005)Table of ContentsExistence and Approximation of Solutions of Differential Equations.- Interior Regularity of Solutions of Differential Equations.- The Cauchy and Mixed Problems.- Differential Operators of Constant Strength.- Scattering Theory.- Analytic Function Theory and Differential Equations.- Convolution Equations.

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Book SynopsisKodaira is a Fields Medal Prize Winner. (In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.) Kodaira is an honorary member of the London Mathematical Society. Affordable softcover edition of 1986 classicTable of ContentsHolomorphic Functions.- Complex Manifolds.- Differential Forms, Vector Bundles, Sheaves.- Infinitesimal Deformation.- Theorem of Existence.- Theorem of Completeness.- Theorem of Stability.

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Book SynopsisDie ersten vier Kapitel dieser Darstellung der klassischen Funktionentheorie vermitteln mit minimalem Begriffsaufwand und auf geringen Vorkenntnissen aufbauend zentrale Ergebnisse und Methoden der komplexen Analysis einer Veränderlichen. Sie gipfeln in einem Beweis des kleinen Riemannschen Abbildungssatzes und einer Charakterisierung einfach zusammenhängender Gebiete. Weitere Themen sind: elliptische Funktionen (Weierstraßscher, Jacobischer Ansatz), die elementare Theorie der Modulformen einer Variablen, Anwendungen der Funktionen- auf die Zahlentheorie (einschl. eines Beweises des Primzahlsatzes). Plus: über 400 Übungsaufgaben mit Lösungen. Trade Review"... Jeder einzelne Abschnitt enthält sorgfältig ausgewählte Übungsaufgaben." Monatshefte für Mathematik "... Positiv hervorzuheben sind die optisch sehr übersichtliche Aufbereitung und der Versuch der Autoren, alle Begriffsbildungen dem Leser gegenüber soweit wie möglich zu motivieren ..." Internationale Mathematische Nachrichten ÖsterreichTable of ContentsDifferentialrechnung im Komplexen.- Integralrechnung im Komplexen.- Folgen und Reihen analytischer Funktionen, Residuensatz.- Konstruktion analytischer Funktionen.- Elliptische Funktionen.- Elliptische Modulformen.- Analytische Zahlentheorie.

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Book SynopsisTable of ContentsI. Konvergente Potenzreihenalgebren.- § 0. Formale Potenzreihen.- 1. Potenzreihen. Ordnung.- 2. Substitutionshomomorphismen.- 3. Partielle Ableitungen. Kettenregel.- 4. Topologie der koeffizientenweisen Konvergenz.- § 1. Analytische k-Banachalgebren.- 0. Bewertungen.- 1. Definition der Bt.- 2. Partielle Ableitungen.- 3. Topologische Eigenschaften der Bt.- § 2. Weierstraßsche Formel und Weierstraßscher Vorbereitungssatz für Bt.- 1. Weierstraßsche Formel.- 2. Weierstraßscher Vorbereitungssatz.- § 3. Konvergente Potenzreihen.- 1. Definition konvergenter Potenzreihen.- 2. Analytische Homomorphismen.- 3. Partielle Ableitungen.- 4. Schwache Topologie und analytische Konvergenz.- § 4. Weierstraßsche Formel und Weierstraßscher Vorbereitungssatz für Kn.- 1. Weierstraßsche Formel und Vorbereitungssatz.- 2. Scherungen.- 3. Analytische Karten in Kn.- Supplement zu § 4. Der Stickelberger-Siegelsche Beweis des Vorbereitungssatzes.- 1. Der Stickelbergersche Beweis.- 2. Der Siegeische Beweis.- 3. Herleitung der Weierstraßschen Formel aus dem Vorbereitungssatz.- § 5. Algebraische Struktur des Ringes Kn.- 1. Weierstraßhomomorphismen und Weierstraßpolynome.- 2. Noethereigenschaft.- 3. Unbeschränktheit der Corangfunktion.- 4. Cartanscher Abgeschlossenheitssatz.- 5. Primfaktorzerlegung.- 6. Henselsches Lemma.- Supplement zu § 5. Noethersche Banachalgebren über ? und ?.- § 6. Die Folgentopologie des Kn.- 1. Finale Topologien.- 2. Folgentopologie auf Kn.- 3. Stetigkeit analytischer Homomorphismen.- § 7. Folgentopologien bei lokal-kompaktem Grundkörper.- 1. Produkttopologie. Silvasche Topologie.- 2. Produkttopologie von Silvatopologien.- 3. Ausgezeichnete Umgebungen. Charakterisierung konvergenter Folgen.- 4. Folgentopologie auf Kn.- 5. Erstes Abzählbarkeitsaxiom und Folgenabschluß.- § 8. Silvatopologie auf Vektorräumen und Algebren.- 1. Definitionen.- 2. Restklassenräume und Restklassenalgebren.- 3. Beschränkte Mengen.- 4. Silvasche Vektorräume und Silvasche Algebren.- 5. Kompakte Mengen.- 6. Lokale Konvexität.- 7. Ausblick.- II. Analytische k-Stellenalgebren.- § 0. Analytische k-Stellenalgebren und analytische Moduln.- 1. Die Kategorie U.- 2. Die Kategorie MA.- § 1. Topologie auf analytischen Stellenalgebren und analytischen Moduln.- 1. Schwache Topologie auf analytischen Stellenalgebren.- 2. Folgentopologie auf analytischen Stellenalgebren.- 3. Schwache Topologie und Folgentopologie auf analytischen Moduln.- § 2. Quasi-endliche und endliche Homomorphismen.- 1. Quasi-endliche Moduln.- 2. Quasi-endliche und endliche analytische Homomorphismen.- 3. Analytische Epimorphismen und analytische Erzeugendensysteme.- 4. Ganze Elemente und endliche Homomorphismen.- 5. Analytische k-Unterstellenalgebren.- 6. Invarianz der Modultopologie.- 7. Relativtopologie und strikte Homomorphismen.- § 3. Einbettungsdimension. Epimorphismen. Umkehrsatz.- 1. Cotangentialraum. Einbettungsdimension. Ableitung.- 2. Epimorphiekriterium.- 3. Jacobischer Umkehrsatz.- 4. Satz über implizite Funktionen.- 5. Einbettungsdimension und Epimorphismen.- § 4. Dimensionstheorie analytischer k-Stellerialgebren. Aktives Lemma.- 1. Aktive Elemente.- 2. Artinsche Algebren.- 3. Dimension.- 4. Aktives Lemma.- 5. Konstruktion aktiver Elemente.- 6. Konstruktion von Parametersystemen.- 7. Tiefe eines Ideals.- § 5. Dimension und endliche analytische Homomorphismen.- 1. Invarianz der Dimension.- 2. Endliche Monomorphismen. Osgoodsches Beispiel.- 3. Reguläre analytische k-Stellenalgebren.- § 6. Krullsche Dimension. Rein-dimensionale analytische Stellenalgebren.- 1. Primidealketten.- 2. Krullscher Hauptidealsatz.- 3. Rein-dimensionale analytische k-Stellenalgebren.- § 7. Endliche Erweiterungen analytischer Stellenalgebren. Normalisierung.- 1. Endliche Erweiterungen.- 2. Normalisierung reduzierter analytischer Stellenalgebren.- III. Weiterführende Theorie analytischer k-Stellenalgebren und analytischer Moduln.- § 1. Homologische Codimension (Profondeur).- 1. M-Sequenzen.- 2. Homologische Codimension. Maximale M-Sequenzen.- 3. Profondeur und endliche Homomorphismen.- 4. Cohen-Macaulay-Moduln.- 5. Unvermischtheit.- 6. Freie Moduln und Macaulay-Moduln.- 7. Beispiele von Macaulay-Moduln.- 8. Beispiele von nicht-Macaulayschen Ringen.- § 2. Homologische Dimension (Syzygientheorie).- 1. Minimale Epimorphismen.- 2. Minimale-freie Auflösungen.- 3. Syzygienmoduln.- 4. Homologische Dimension.- 5. Homologische Dimension und homologische Codimension. Syzygiensatz.- 6. Konstruktion von Hilbert-Auflösungen.- 7. Koszul-Komplexe.- § 3. Invariante analytische k-Unterstellenalgebren.- 1. Invariante Algebren zu endlichen Automorphismengruppen.- 2. Linearisierung.- 3. Beispiele. Zyklische Gruppen.- § 4. Derivations- und Differentialmoduln.- 1. Derivationen.- 2. Differentialmoduln.- 3. Existenz von Differentialmoduln.- 4. Eigenschaften der Differentialmoduln.- 5. Regularitätskriterium.- 6. Äußere Differentialformen über kn. Poincaré-Sequenz.- 7. Exaktheit der Poincaré-Sequenz.- § 5. Analytische Tensorprodukte.- 1. Definition und Existenz.- 2. Endlichkeit und Freiheit.- 3. Faseralgebren und endliche Homomorphismen.- 4. Das analytische Tensorprodukt analytischer Moduln.- 5. Invarianz unter endlichen Homomorphismen.- 6. Einbettungsdimension und Dimension.- 7. Normalität und Nullteilerfreiheit.- 8. Reduziertheit.- 9. Homologische Codimension.- 10. Differentialmoduln.- Anhang. Algebraische Hilfsmittel.- § 1. Ringe und Moduln.- 1. Idealpotenzen. Nilpotente Ideale.- 2. Primideale.- 3. Radikale. Reduzierte Ringe. Multiplikative Mengen.- 4. Torsionsmoduln. Quotientenmoduln.- 5. Rang und Corang.- 6. Noethersche Moduln.- 8. Zerlegungssatz von Lasker-Noether.- § 2. Endliche Moduln über noetherschen Stellenringen.- 2. Lemma von Nakayama.- 3. Krullscher Durchschnittsatz.- 4. Corang.- 5. Jacobirang.- 6. Einbettungsdimension.- 7. Freie Moduln.- § 3. Normale noethersche Integritätsringe.- 1. Ganze Elemente. Dedekindsches Lemma.- 2. Ganzer Abschluß. Normalisierung.- 3. Charakterisierung ganz-abgeschlossener Ringe.- 4. Hauptidealsatz.- 5. Minimale Primideale.- 6. Teilbarkeitstheorie.- § 4. Reduzierte und noethersche Ringe.- 1. Direkte Summen von Ringen.- 2. Epimorphiesatz.- 3. Reduzierte noethersche Ringe.- 4. Charakterisierung von Torsionsmoduln.- Literatur.

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Book SynopsisThis is the proceedings of the meeting entitled “The 12th MSJ International Research Institute of the Mathematical Society of Japan 2003”. The papers cover several important topics in Singularity theory. Especially some of them are survey on motivic integrations, Thom polynomials, complex analytic singularity theory, generic differential geometry etc.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North AmericaTable of ContentsInvariants of combinatorial line arrangements and Rybnikov's example by E. A. Bartolo, J. C. Ruber, J. I. Cogolludo-Agustin, and M. A. Marco-Buzunariz On time averaged optimization of dynamic inequalities on a circle by A. Davydov Thom polynomial computing strategies. A survey by L. M. Feher and R. Rimanyi The complex crystallographic groups and symmetries of $J_{10}$ by V. Goryunov and S. H. Man $tt^*$ geometry and mixed Hodge structures by C. Hertling Thom polynomials by M. Kazarian Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture by K. Kurdyka and A. Parusinski Homotopy groups of complements to ample divisors by A. Libgober Massey products of complex hypersurface complements by D. Matei On degree of mobility for complete metrics by V. S. Matveev Valuations and moduli of Goursat distributions by P. Mormul Semidifferentiabilite et version lisse de la conjecture de fibration de Whitney by C. Murolo and D. Trotman Submanifolds with a nondegenerate parallel normal vector field in euclidean spaces by J. J. Nuno-Ballesteros Weighted homogeneous polynomials and blow-analytic equivalence by O. M. Abderrahmane Characteristic classes of singular varieties by A. Parusinski On the classification of 7th degree real decomposable curves by G. M. Polotovskiy $\mathcal A$-topological triviality of map germs and Newton filtrations by M. J. Saia and L. M. Soares On the topology of symmetry sets of smooth submanifolds in $\mathbb{R}^k$ by V. D. Sedyh An infinitesimal criterion for topological triviality of families of sections of analytic varieties by M. A. S. Ruas and J. N. Tomazella Lines of principal curvature near singular end points of surfaces in $\mathbb{R}^3$ by J. Sotomayor and R. Garcia $r$ does not imply $n$ or $(npf)$ for definable sets in non polynomially bounded o-minimal structures by D. Trotman and L. Wilson Valuations and local uniformization by M. Vaquie Arc spaces, motivic integration and stringy invariants by W. Veys Finite Dehn surgery along A'Campo's divide knots by Y. Yamada.

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Book SynopsisThis book deals with the classical theory of Nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds. The theory was extended to several variables by S. Kobayashi, T. Ochiai, J. Carleson, and P. Griffiths in the early 1970s. K. Kodaira took up this subject in his course at The University of Tokyo in 1973 and gave an introductory account of this development in the context of his final paper, contained in this book. The first three chapters are devoted to holomorphic mappings from C to complex manifolds. In the fourth chapter, holomorphic mappings between higher dimensional manifolds are covered. The book is a valuable treatise on the Nevanlinna theory, of special interests to those who want to understand Kodaira's unique approach to basic questions on complex manifolds.Table of ContentsPreface1. Nevanlinna Theory of One Variable (1)1.1 metrics of compact Rimann surfaces1.2 integral formula1.3 holomorphic maps over compact Riemann surfaces whose genus are greater than 21.4 holomorphic maps over Riemann sphreres1.5 Defect relation2. Schwarz--Kobayashi's Lemma2.1 Schwarz--Kobayashi's Lemma2.2 holomorphic maps over algebraic varieties (general type)2.3 hyperbolic measures3. Nevanlinna Theory of One Variable (2)3.1 holomorphic maps over Riemann shpres3.2 the first main theorem3.3 the second main theorem4. Nevanlinna Theory of Several Variables4.1 Biebelbach's example4.2 the first main theorem4.3 the second main theorem4.4 defect relation4.5 applicationsReferences

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Book SynopsisThe Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kähler cases. In this book, the unsolved cases of the conjecture will be discussed.It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds. Another important tool in our approach is the Chow norm introduced by Zhang. This is closely related to Ding’s functional, and plays a crucial role in our differential geometric study of stability. By discussing the Chow norm from various points of view, we shall make a systematic study of the existence problem of extremal Kähler metrics.Trade Review“The concise style of exposition likely means that this monograph is best suited for experts with background knowledge in canonical Kähler metrics. … It can be recommended also to those who would like a review of important results concerning the generalised Kähler-Einstein metrics, with various examples, and the moduli space of Lp-spaces.” (Yoshinori Hashimoto, Mathematical Reviews, May, 2023)Table of ContentsIntroduction.- The Donaldson-Futaki invariant.- Canonical Kähler metrics.- Norms for test configurations.- Stabilities for polarized algebraic manifolds.- The Yau-Tian-Donaldson conjecture.- Stability theorem.- Existence problem.- Canonical Kähler metrics on Fano manifolds.- Geometry of pseudo-normed graded algebras.- Solutions.

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Book SynopsisThis book collects original peer-reviewed contributions presented at the "International Conference on Mathematical Analysis and Applications (MAA 2020)" organized by the Department of Mathematics, National Institute of Technology Jamshedpur, India, from 2–4 November 2020. This book presents peer-reviewed research and survey papers in mathematical analysis that cover a broad range of areas including approximation theory, operator theory, fixed-point theory, function spaces, complex analysis, geometric and univalent function theory, control theory, fractional calculus, special functions, operation research, theory of inequalities, equilibrium problem, Fourier and wavelet analysis, mathematical physics, graph theory, stochastic orders and numerical analysis. Some chapters of the book discuss the applications to real-life situations. This book will be of value to researchers and students associated with the field of pure and applied mathematics.Table of ContentsG. K. Srinivasan, A note on isolated removable singularities of harmonic functions.- O. Chadli, Ram N. Mohapatra, B. K. Sahu, Equilibrium Problems and Variational Inequalities: a Survey of Existence Results.- O. Chadli, Ram N. Mohapatra, G. Pany, Nonlinear evolution equations by a Ky Fan minimax inequality approach.- L. A. Wani and A. Swaminathan, Sufficient Conditions Concerning the Unified Class of Starlike and Convex Functions.- S. Menchavez and I. Mae Antabo, One Dimensional Parametrized Test Functions Space of Entire Functions.- D. Raghavan and S. Nagarajan, Extremal mild solutions of Hilfer fractional Impulsive systems.- B. Roy and S. N. Bora, On existence of integral solutions for a class of mixed Volterra-Fredholm integro fractional differential equations.- P. Kumar, A. Kumar, R. Kumar Vats and A. Kumar, Trajectory Controllability of Integro-differential Systems of Fractional Order γ ∈ (1, 2] in a Banach Space with Deviated Argument.- A. S. Kelil and A. Rao Appadu, Shehu-Adomian Decomposition Method for dispersive KdV-type Equations.- A. S. Kelil, A. R. Appadu and S. Arjika, On certain properties of perturbed Freud-type weight: a revisit.- A. K. Singh, Complex chaotic systems and its complexity.- M. Incesu, S. Y. Evren and O. Gursoy, On the bertrand pairs of open non uniform B-spline curves.- M. Verma, P. Sharma and N. Gupta, Convergence analysis of a sixth-order method under weak continuity condition with First-order Frechet derivative.- B. Kour and S. Ram, (m, n)-paranormal composition operators.- T. Yaying, On the domain of q-Euler matrix in c0 and c.- N. Sarkar and M. Sen, Study on some particular class of non linear integral equation with a hybridized approach.- D. Saha, M. Sen and S. Roy, Investigation of the existence criteria for the solution of the functional integral equation in the Lp space.- S. Das and K. Mehrez, Functional Inequalities for the Generalized Wright Functions.- S. Dutta and P. Guha, An Information Theoretic Entropy Related to Ihara $\zeta$ Function and Billiard Dynamics.- S. Baskaran, G. Saravanan and K. Muthunagai, On a new subclass of Sakaguchi type functions using (p;q)- derivative operator.- N. K. Jangid, S. Joshi and S. D. Purohit, Some Double integral Formulae Associated with Q Function and Galue Type Struve Function.- M. Jain, M. Singh and R. K. Meena, Time-dependent analytical and computational study of an M/M/1 queue with disaster failure and multiple working vacations.- M. Datta and N. Gupta, Usual stochastic ordering results for series and parallel systems with components having Exponentiated Chen distribution.

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Book SynopsisThe focus of this book is on the further development of the classical achievements in analysis of several complex variables, the analytic continuation and the analytic structure of sets, to settings in which the q-pseudoconvexity in the sense of Rothstein and the q-convexity in the sense of Grauert play a crucial role. After giving a brief survey of notions of generalized convexity and their most important results, the authors present recent statements on analytic continuation related to them. Rothstein (1955) first introduced q-pseudoconvexity using generalized Hartogs figures. Słodkowski (1986) defined q-pseudoconvex sets by means of the existence of exhaustion functions which are q-plurisubharmonic in the sense of Hunt and Murray (1978). Examples of q-pseudoconvex sets appear as complements of analytic sets. Here, the relation of the analytic structure of graphs of continuous surfaces whose complements are q-pseudoconvex is investigated. As an outcome, the authors generalize results by Hartogs (1909), Shcherbina (1993), and Chirka (2001) on the existence of foliations of pseudoconcave continuous real hypersurfaces by smooth complex ones. A similar generalization is obtained by a completely different approach using L²-methods in the setting of q-convex spaces. The notion of q-convexity was developed by Rothstein (1955) and Grauert (1959) and extended to q-convex spaces by Andreotti and Grauert (1962). Andreotti–Grauert's finiteness theorem was applied by Andreotti and Norguet (1966–1971) to extend Grauert's solution of the Levi problem to q-convex spaces. A consequence is that the sets of (q-1)-cycles of q-convex domains with smooth boundaries in projective algebraic manifolds, which are equipped with complex structures as open subsets of Chow varieties, are in fact holomorphically convex. Complements of analytic curves are studied, and the relation of q-convexity and cycle spaces is explained. Finally, results for q-convex domains in projective spaces are shown and the q-convexity in analytic families is investigated.Table of Contents1. Analytic Continuation and Pseudoconvexity.- 2. q-Plurisubharmonicity.- 3. q-Pseudoconvexity.- 4. q-Convexity and q-Completeness.- References.- Index.

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