Calculus and mathematical analysis Books

Book SynopsisThe experimental achievement of Bose-Einstein condensation (1995) and of Fermi degeneracy (1999) in ultra-cold, dilute gases has opened a new field in atomic physics and condensed matter physics. This thesis presents an overview of theoretical and experimental facts on ultra-cold atomic gases. A Green's function scheme is examined, and the book also applies a novel spin-density-functional approach to the study of Fermi gases inside one-dimensional optical lattices.Table of ContentsPreface.- 1. Reviews on Ultra-cold quantum gases.- 2. Theory of matter transport in quasi-1D arrays.- 3. Density-functional theory of 1D Fermi gases.- 4. Ultra-cold attractive fermions in 1D optical lattices.

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Book SynopsisThese are the proceedings of a one-week international conference centered on asymptotic analysis and its applications. They contain major contributions dealing with: mathematical physics: PT symmetry, perturbative quantum field theory, WKB analysis, local dynamics: parabolic systems, small denominator questions, new aspects in mould calculus, with related combinatorial Hopf algebras and application to multizeta values, a new family of resurgent functions related to knot theory.

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Book SynopsisIt isn't that they can't see Approach your problems from the solution. the right end and begin with It is that they can't see the the answers. Then one day, perhaps you will find the problem. final question. G. K. Chesterton. The Scandal 'The Hermit Clad in Crane of Father Brown 'The Point of a Pin'. Feathers' in R. van Gulik's The Chinese Maze l1urders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.Table of Contents1. Random Operators in Hilbert Space.- 1. Basic Definitions.- 1.1 Strong Random Operator.- 1.2 Weak Random Operator.- 1.3 Product of Random Operators.- 2. Characteristic Functions of Random Operators.- 2.1 Definition.- 2.2 Characteristic Functions of Strong and Bounded Operators.- 2.3 Gaussian Random Operators.- 3. Convergence of Random Operators.- 3.1 Weak Convergence of Random Operators.- 3.2 Strong Convergence of Random Operators.- 3.3 Convergence of Distributions corresponding to Random Operators.- 2. Functions of Random Operators.- 4. Spectral Representation for Symmetric Random Operators.- 4.1 Symmetric Random Operators and Selfadjoint Extensions.- 4.2 Spectral Representation of a Selfadjoint Random Operator.- 4.3 Spectral Representation of a Strong Symmetric Operator.- 5. Equations with Symmetric Random Operators.- 5.1 Evolution Equations.- 5.2 Schrödinger-type Equations.- 5.3 Spectral Moment Functions.- 5.4 Equation of Fredholm Type.- 6. Equations with Semi-Bounded Random Operators.- 6.1 Nonnegative Closed Random Operators.- 6.2 Resolvent of a Nonnegative Operator.- 6.3 Resolvent of a Nonnegative Random Operator.- 6.4 Equations of Fredholm Type.- 6.5 Equations of Evolution Type.- 3. Operator-Valued Martingales.- 7. Operator-Valued Martingale Sequences.- 7.1 Weak Operator-valued Martingale.- 7.2 Strong Operator-valued Martingales.- 7.3 Operator-valued Martingale.- 8. Convergence of Infinite Products of Independent Random Operators.- 8.1 Infinite Products as Martingales.- 8.2 Convergence of Infinite Products given the Existence of Two Moments.- 8.3 Convergence of Infinite Products in Absolute Norm.- 9. Continuous Operator-Valued Martingales.- 9.1 Some Properties of Continuous Real-valued Local Martingales.- 9.2 Continuous Martingales with values in X.- 9.3 Operator-valued Continuous Martingales.- 9.4 Strong Operator-valued Wiener Processes.- 4. Stochastic Integrals and Equations.- 10. Stochastic Integrals with Respect to an X-Valued Martingale.- 10.1 Definition.- 10.2 Integrals for Processes with Regular Characteristics.- 10.3 Stochastic Integral with respect to a Wiener Process.- 11. Stochastic Integral with Respect to an Operator-Valued Martingale.- 11.1 Integrals of X-valued Functions.- 11.2 Integrals of Operator-valued Functions.- 12. Stochastic Operator Equations.- 12.1 Operator-valued Functions of Random Operators.- 12.2 Stochastic Equations Involving I(Z, Y)t.- 12.3 Stochastic Equations Involving I*(Z, Y)t.- 12.4 Some Generalizations.- 5. Linear Stochastic Operator Equations.- 13. Generalization of the Stochastic Operator Integral.- 13.1 General Form of the Linear Equation.- 13.2 A Generalization of the Stochastic Integral.- 14. Linear Differential Operator Equations.- 14.1 Definition of a Linear Equation.- 14.2 Existence of Uniqueness of Solution.- 14.3 Linear Transformations of Solutions.- 14.4 Equations for Moments of the Solution of a Stochastic Equation.- 15. Continuous Stochastic Semigroups.- 15.1 Solutions of Simple Linear Equations -Stochastic Semigroups.- 15.2 Time Reversal in Stochastic Differential Equations.- 15.3 Definition of Stochastic Semigroups.- 15.4 Semigroups which are Martingales.

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Book SynopsisIn his Retiring Presidential address, delivered before the Annual Meeting of The American Mathematical Society on December, 1948, the late Professor Einar Hille spoke on his recent results on the Lie theory of semigroups of linear transformations, . . • "So far only commutative operators have been considered and the product law . . . is the simplest possible. The non-commutative case has resisted numerous attacks in the past and it is only a few months ago that any headway was made with this problem. I shall have the pleasure of outlining the new theory here; it is a blend of the classical theory of Lie groups with the recent theory of one-parameter semigroups. " The list of references in the subsequent publication of Hille's address (Bull. Amer. Math •. Soc. 56 (1950)) includes pioneering papers of I. E. Segal, I. M. Gelfand, and K. Yosida. In the following three decades the subject grew tremendously in vitality, incorporating a number of different fields of mathematical analysis. Early papers of V. Bargmann, I. E. Segal, L. G~ding, Harish-Chandra, I. M. Singer, R. Langlands, B. Konstant, and E. Nelson developed the theoretical basis for later work in a variety of different applications: Mathematical physics, astronomy, partial differential equations, operator algebras, dynamical systems, geometry, and, most recently, stochastic filtering theory. As it turned out, of course, the Lie groups, rather than the semigroups, provided the focus of attention.Trade Review`...the reader obtains the impression that there remains much to discover in commutation theory, and this monograph provides both motivation and a guide to the current state of knowledge.' Mathematical Reviews (1986) Table of ContentsI: Some Main Results on Commutator Identities.- 1. Introduction and Survey.- 1A General Objectives of the Monograph.- 1B Contact with Prior Literature.- 1C The Main Results in Commutation Theory.- 1D The Main Results in Exponentiation Theory.- 1E Results on (Semi) Group-invariant C?-domains.- 1F Typical Applications of Commutation Theory.- 1G Typical Applications of Exponentiation Theory.- 2. The Finite-Dimensional Commutation Condition.- 2A Implications of Finite-Dimensionality in Commutation Theory.- 2B Examples involving Differential Operators.- 2C Examples from Universal and Operator Enveloping Algebras.- 2D Relaxing the Finite-Dimensionality Condition.- II: Commutation Relations and Regularity Properties for Operators in the Enveloping Algebra of Representations of Lie Groups.- 3. Domain Regularity and Semigroup Commutation Relations.- 3A Lie Algebras of Continuous Operators.- 3B Semigroups and Ad-Orbits.- 3C Variations upon the Regularity Condition.- 3D Infinite-Dimensional OA(B).- 4. Invariant-Domain Commutation Theory applied to the Mass-Splitting Principle.- 4A Global Invariance/Regularity for Heat-Type Semigroups.- 4B Formulation of the Generalized Mass-Splitting Theorem.- 4C The Mass-Operator as a Commuting Difference of Sub-Laplacians.- 4D Remarks on General Minkowskian Observables.- 4E Fourier Transform Calculus and Centrality of Isolated Projections.- III: Conditions for a System of Unbounded Operators to Satisfy a given Commutation Relation.- 5. Graph-Density applied to Resolvent Commutation, and Operational Calculus.- 5A Augmented Spectra and Resolvent Commutation Relations.- 5B Commutation Relations on D1.- 5C Analytic Continuation of Commutation Relations.- 5D Commutation Relations for the Holomorphic Operational Calculus.- 6. Graph-Density Applied to Semigroup Commutation Relations.- 6A Semigroup Commutation Relations with a Closable Basis.- 6B Variants of Sections 5B and 6A for General M.- 6C Automatic Availability of a Closable Basis.- 6D Remarks on Operational Calculi.- 7. Construction of Globally Semigroup-invariant C?-domains.- 7A Fréchet C?-domains in Banach Spaces.- 7B The Extrinsic Two-Operator Case.- 7C The Lie Algebra Case.- 7D C?-action of Resolvents, Projections, and Operational Calculus.- IV: Conditions for a Lie Algebra of Unbounded Operators to Generate a Strongly Continuous Representation of the Lie Group.- 8. Integration of Smooth Operator Lie Algebras.- 8A Smooth Lie Algebras and Differentiable Representations.- 8B Applications in C?-vector spaces.- 9. Exponentiation and Bounded Perturbation of Operator Lie Algebras.- 9A Discussion of Exponentiation Theorems and Applications.- 9B Proofs of the Theorems.- 9C Phillips Perturbations of Operator Lie Algebras and Analytic Continuation of Group Representations.- 9D Semidirect Product Perturbations.- Appendix to Part IV.- V: Lie Algebras of Vector Fields on Manifolds.- 10. Applications of Commutation Theory to Vector-Field Lie Algebras and Sub- Laplacians on Manifolds.- 10A Exponentials versus Geometric Integrals of Vector-Field Lie Algebras.- 10B Exponentiation on Lp spaces.- 10C Sub-Laplacians on Manifolds.- 10D Solution Kernels on Manifolds.- VI: Derivations on Modules of Unbounded Operators with Applications to Partial Differential Operators on Riemann Surfaces.- 11. Rigorous Analysis of Some Commutator Identities for Physical Observables.- 11A Variations upon the Graph-Density and Kato Conditions.- 11B Various forms of Strong Commutativity.- 11C Nilpotent Commutation Relations of Generalized Heisenberg-Weyl Type.- Appendix to Part VI.- VII: Lie Algebras of Unbounded Operators: Perturbation Theory, and Analytic Continuation of s?(2, ?)-Modules.- 12. Exponentiation and Analytic Continuation of Heisenberg-Matrix Representations for s?(2, ?).- 12A Connections to the Theory of TCI Representations of Semisimple Groups on Banach Spaces.- 12B The Graph-Density Condition and Base-Point Exponentials.- 12C C?-integrals and Smeared Exponentials on ?p.- 12D The Operators A0, A1 and A2.- 12E Compact and Phillips Perturbations.- 12F Perturbations and Analytic Continuation of Smeared Representations.- 12G Irreducibility, Equivalences, Unitarity, and Single-Valuedness.- 12H Perturbation and Reduction Properties of Other Analytic Series.- 12I A Counter-Theorem on Group-Invariant Domains.- Appendix to Part VII.- General Appendices.- Appendix A. The Product Rule for Differentiable Operator Valued Mappings.- Appendix B. A Review of Semigroup Folklore, and Integration in Locally Convex Spaces.- Appendix C. The Square of an Infinitesimal Group Generator.- Appendix E. Compact Perturbations of Semigroups.- Appendix G. Bounded Elements in Operator Lie Algebras.- References.- References to Ouotations.- List of Symbols.

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Book SynopsisThis book is an introduction to basic topics in inequalities and their applications. These include the arithmetic mean-geometric mean inequality, Cauchy-Schwarz inequality, Chebyshev inequality, rearrangement inequality, convex and concave functions and Muirhead's theorem. More than 400 problems are included in the book and their solutions are explained. A chapter on geometric inequalities is a special feature of this book. Most of these problems are from International Mathematical Olympiads and from many national mathematical olympiads.The book is intended to help students who are preparing for various mathematical competitions. It is also a good source book for graduate students in consolidating their knowledge of inequalities and their applications. The author has been involved in training Indian students for mathematical olympiads for more than 25 years.Table of Contents 1 Some basic inequalities 2 Techniques for proving inequalities 3 Geometric inequalities 4 Applications involving inequalities 5 Problems on inequalities 6 Solutions to problems 7 Bibliography 8 Abbreviation 9 Index

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Book SynopsisThis book represents material suitable for a two semester graduate course in analysis. Based on courses given by the author since 2007, it is targeted towards graduate students preparing for a research career in mathematics. After reviewing what a typical student would have learned in single variable and multivariable calculus courses, the first chapter presents familiar material from a mature mathematical perspective. The second chapter surveys measure theory and highlights the essential theorems that have now become ubiquitous in mathematics. The third chapter on Fourier transforms derives the inversion theorem and gives diverse applications ranging from probability theory to mathematical physics. The fourth chapter reviews complex analysis giving a synthetic, rigorous development of the calculus of residues and its applications to a wide array of problems. The final chapter is an introduction to algebraic topology and is included to show the symbiosis between algebra and analysis.

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Book SynopsisThis means that semigroup theory may be applied directly to the study of the equation I'!.f = h on M. In [45] Yau proves that, for h ~ 0, there are no nonconstant, nonnegative solutions f in [j' for 1 < p < 00. From this, Yau gets the geometric fact that complete noncom pact Riemannian manifolds with nonnegative Ricci curvature must have infinite volume, a result which was announced earlier by Calabi [4]. 6. Concluding Remarks In several of the above results, positivity of the semigroup plays an important role. This was also true, although only implicitly, for the early work of Hille and Yosida on the Fokker-Planck equation, i.e., Equation (4) with c = O. But it was Phillips [41], and Lumer and Phillips [37] who first called attention to the importance of dissipative and dispersive properties of the generator in the context of linear operators in a Banach space. The generation theorems in the Batty-Robinson paper appear to be the most definitive ones, so far, for this class of operators. The fundamental role played by the infinitesimal operator, also for the understanding of order properties, in the commutative as well as the noncommutative setting, are highlighted in a number of examples and applications in the different papers, and it is hoped that this publication will be of interest to researchers in a broad spectrum of the mathematical sub-divisions.Table of ContentsPositive Semigroups of Operators, and Applications: Editors’ Introduction.- Positive One-Parameter Semigroups on Ordered Banach Spaces.- Asymptotic Behavior of One-Parameter Semigroups of Positive Operators.- Positivity in Time Dependent Linear Transport Theory.- Quantum Dynamical Semigroups, Symmetry Groups, and Locality.- Stochastic Dilations of Uniformly Continuous Completely Positive Semigroups.- Order Properties of Attractive Spin Systems.- Book Reviews:.- E. B. Davies: One-Parameter Semigroups (William G. Faris).- L. Asimow and A. J. Ellis: Convexity Theory and its Applications in Functional Analysis (E. B. Davies).- Publications Received.- Announcement.

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Book SynopsisOne SCI\'ice mathematics bas rendered the 'Et moi, ...si j'avait su comment en revcnir. je n'y serais point aile: human race. It bas put common sc:nsc back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.Table of Contents0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.

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Book SynopsisMean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.Table of Contents

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Book SynopsisThis book systematically introduces the theory of nonlinear analysis, providing an overview of topics such as geometry of Banach spaces, differential calculus in Banach spaces, monotone operators, and fixed point theorems. It also discusses degree theory, nonlinear matrix equations, control theory, differential and integral equations, and inclusions. The book presents surjectivity theorems, variational inequalities, stochastic game theory and mathematical biology, along with a large number of applications of these theories in various other disciplines. Nonlinear analysis is characterised by its applications in numerous interdisciplinary fields, ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. Organised into ten chapters, the book shows the elegance of the subject and its deep-rooted concepts and techniques, which provide the tools for developing more realistic and accurate models for a variety of phenomena encountered in diverse applied fields. It is intended for graduate and undergraduate students of mathematics and engineering who are familiar with discrete mathematical structures, differential and integral equations, operator theory, measure theory, Banach and Hilbert spaces, locally convex topological vector spaces, and linear functional analysis.Trade Review“This book cover many important fundamental concepts and various techniques. It shall be very useful for both students and researchers to understand and to prepare themselves for conducting research in these two areas.” (Satit Saejung, zbMATH 1447.47002, 2020)Table of Contents​Chapter 1. Fundamentals.- Chapter 2. Geometry in Banach Spaces and Duality Mappings.- Chapter 3. Differential Calculus in Banach Spaces.- Chapter 4. Monotone Operators, Phi-accretive Operators and Their Generalizations.- Chapter 5. Fixed Point Theorems.- Chapter 6. Degree Theory, K-Set Contractions and Condensing Operators.- Chapter 7. Random Fixed Point Theory and Monotone Operators.- Chapter 8. Applications of Monotone Operator Theory to Differential and Integral Equations.- Chapter 9. Applications of Fixed Point Theorems.- Chapter 10. Applications of Fixed Point Theorems for Multifunction to Integral Inclusions.

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Book SynopsisThis book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 ≤ k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

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Book SynopsisThis book discusses all the major topics of complex analysis, beginning with the properties of complex numbers and ending with the proofs of the fundamental principles of conformal mappings. Topics covered in the book include the study of holomorphic and analytic functions, classification of singular points and the Laurent series expansion, theory of residues and their application to evaluation of integrals, systematic study of elementary functions, analysis of conformal mappings and their applications—making this book self-sufficient and the reader independent of any other texts on complex variables. The book is aimed at the advanced undergraduate students of mathematics and engineering, as well as those interested in studying complex analysis with a good working knowledge of advanced calculus. The mathematical level of the exposition corresponds to advanced undergraduate courses of mathematical analysis and first graduate introduction to the discipline. The book contains a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic skills and test the understanding of concepts. Other problems are more theoretically oriented and illustrate intricate points of the theory. Many additional problems are proposed as homework tasks whose level ranges from straightforward, but not overly simple, exercises to problems of considerable difficulty but of comparable interest.Table of Contents1. Introduction.- 2. Analytic Functions and Their Properties.- 3. Singular Points. Laurent Series. Residues.- 4. Conformal Mappings. Elementary Functions.- 5. Fundamental Principles of Conformal Mappings. Transformations of Polygons.

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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 presents original papers reflecting topics featured at the international symposium entitled “Fusion of Mathematics and Biology” and organized by the editor of the book. The symposium, held in October 2020 at Osaka University in Japan, was the core event for the final year of the research project entitled “Establishing International Research Networks of Mathematical Oncology.” The project had been carried out since April 2015 as part of the Core-to-Core Program of Japan Society for the Promotion of Science (JSPS). In this book, the editor presents collaborative research from prestigious organizations in France, the UK, and the USA. By utilizing their individual strengths and realizing the fusion of life science and mathematical science, the project achieved a combination of mathematical analysis, verification by biomedical experiments, and statistical analysis of chemical databases.Mathematics is sometimes regarded as a universal language. It is a valuable property that everyone can understand beyond the boundaries of culture, religion, and language. This unifying force of mathematics also applies to the various fields of science. Mathematical oncology has two aspects, i.e., data science and mathematical modeling, and definitely helps in the prediction and control of biological phenomena observed in cancer evolution.The topics addressed in this book represent several methods of applying mathematical modeling to scientific problems in the natural sciences. Furthermore, novel reviews are included that may motivate many mathematicians to become interested in biological research.Table of ContentsPART 1: Mathematical Modeling: D. Guan, X. Luo, and H. Gao, Constitutive Modelling of Soft Biological Tissue from Ex Vivo to In Vivo: Myocardium as an Example.- T. Colin, T. Michel, and C. Poignard, Mathematical Modeling of Gastro-intestinal Metastasis Resistance to Tyrosine Kinase Inhibitors.- Y. Tanaka and T. Yasugi, Mathematical Modeling and Experimental Verification of the Proneural Wave.- D. Kumakura and S. Nakaoka, Exploring Similarity between Embedding Dimension of Time-series Data and Flows of an Ecological Population Model.- T. Hayashi, Mathematical Modeling for Angiogenesis.- S. Collin, Corridore and C. Poignard, Floating Potential Boundary Condition in Smooth Domains in an Electroporation Context.- N. L. Othman and T. Suzuki, Free Boundary Problem of Cell Deformation and Invasion.- L. Preziosi and M. Scianna, Multi-level Mathematical Models for Cell Migration in Confined Environments.- S. Magi, Mathematical Modeling of Cancer Signaling Addressing Tumor Heterogeneity.- N. Sfakianakis and Mark A.J. Chaplain, Mathematical Modelling of Cancer Invasion: A Review.- T. Williams, A. Wilson, and N. Sfakianakis, The First Step towards the Mathematical Understanding of the Role of Matrix Metalloproteinase-8 in Cancer Invasion.- PART II: Biological Prediction: T. Ito, T. Suzuki, and Y. Murakami, Mathematical Modeling of the Dimerization of EGFR and ErbB3 in Lung Adenocarcinoma.- H. Kubota, Selective Regulation of the Insulin-Akt Pathway by Simultaneous Processing of Blood Insulin Pattern in the Liver.- D. Oikawa, N. Hatanaka, T. Suzuki, and F. Tokunaga, Mathematical Simulation of Linear Ubiquitination in T Cell Receptor-mediated NF-κB Activation Pathway.- Y. Ito, D. Minerva, S. Tasaki, M. Yoshida, T. Suzuki, and A. Goto, Time Changes in the VEGF-A Concentration Gradient Lead Neovasculature to Engage in Stair-like Growth.- N. Hatanaka, M. Futakuchi, and T. Suzuki, Mathematical Modeling of Tumor Malignancy in Bone Microenvironment.- M. Yamamoto and Jun-ichiro Inoue, Signaling Networks Involved in the Malignant Transformation of Breast Cancer.- PART III: Data Science: R. Morishita, H. Takahashi, and T. Sawasaki, Cell-free Based Protein Array Technology.- Y. Nojima and Y. Takeda, Omics Data Analysis Tools for Biomarker Discovery and the Tutorial.- M. Oyama and H. Kozuka-Hata, Integrative Network Analysis of Cancer Cell Signaling by High-resolution Proteomics.- N. Nakamura and R. Yamada, Distance-matrix-based Extraction of Motility Features from Functionally Heterogeneous Cell Populations.- S. Kawasaki, H. Hayashi, and Y. Tominaga, Data Analytic Study of Genetic Mechanism of Ovarian Carcinoma from Single Cell RNA-seq Data.

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Book SynopsisThis first of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It studies metric spaces and general topology. It starts with the concept of the metric which is an abstraction of distance in the Euclidean space. The special structure of a metric space induces a topology that leads to many applications of topology in modern analysis and modern algebra, as shown in this volume. This volume also studies topological properties such as compactness and connectedness. Considering the importance of compactness in mathematics, this study covers the Stone–Cech compactification and Alexandroff one-point compactification. This volume also includes the Urysohn lemma, Urysohn metrization theorem, Tietz extension theorem, and Gelfand–Kolmogoroff theorem. The content of this volume is spread into eight chapters of which the last chapter conveys the history of metric spaces and the history of the emergence of the concepts leading to the development of topology as a subject with their motivations with an emphasis on general topology. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power, and active learning of the subject, all the while covering a wide range of theories and applications in a balanced unified way.Trade Review“With an encyclopedic range of topics and terse exposition, Basic Topology 1 may make a reasonable reference for self-motivated learners … .” (Timothy Clark, MAA Reviews, March 20, 2023)Table of Contents1. Prerequisites: Sets, Algebraic Systems, and Classical Analysis.- 2. Metric Spaces and Normed Linear Spaces.- 3. Topological Spaces and Continuous Maps.- 4. Separation Axioms.- 5. Compactness and Connectedness.- 6. Real-valued Continuous Functions.- 7. Countability, Separability and Embedding.- 8. Brief History of General Topology.

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Book SynopsisThis third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.Table of Contents1. Prerequisite Concepts of Topology, Algebra and Category Theory.- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups.- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences.- 4. Topology of Fiber Bundles.- 5. Homotopy Theory of Bundles.- 6. Some Applications of Algebraic Topology.- 7. Brief History on Algebraic Topology and Fiber Bundles.

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Book SynopsisThis second of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, algebra, and the theory of numbers. Offering a proper background on topology, analysis, and algebra, this volume discusses the topological groups and topological vector spaces that provide many interesting geometrical objects which relate algebra with geometry and analysis. This volume follows a systematic and comprehensive elementary approach to the topology related to manifolds, emphasizing differential topology. It further communicates the history of the emergence of the concepts leading to the development of topological groups, manifolds, and also Lie groups as mathematical topics with their motivations. This book will promote the scope, power, and active learning of the subject while covering a wide range of theories and applications in a balanced unified way.Table of Contents1. Background on Topology, Analysis and Algebra.- 2. Topological Groups.- 3. Topology of Manifolds.- 4. Lie Groups and Lie Algebra.- 5. Brief History of Topological Groups, Manifold and Lie Groups.

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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 SynopsisThis unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.Table of ContentsTangents and Double Points; Derivatives by Algebra; Exponential Functions; Completeness of Real Numbers; The Base of the Natural Exponential and Logarithm Functions; Continuity of Functions; Differentiability; Chain Rule and Other Rules for Derivatives; Derivatives of Trigonometric Functions; Mean Value Inequality and Theorem; Basic Differential Equations; Motion with Constant Acceleration; Linear and Higher Order Approximations; The Antiderivative Problem; Definite Integrals; Fundamental Theorem of Calculus; Integrability of Monotonic Functions; Integrability of Functions with Bounded Derivative; Substitution; Integration by Parts; Taylor's Theorem; Analytic Functions; The Euler Identity;

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Book SynopsisBackground in Potential Theory.- Fundamentals of Fine Potential Theory.- Further Developments.- Fine Complex Potential Theory.

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Book Synopsis

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Book SynopsisChapter 1 The System of Real Numbers.- Chapter 2 Real Sequences.- Chapter 3 Infinite Series of Numbers.- Chapter 4 Limits, Continuity and Differentiability.- Chapter 5 Metric Spaces.

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Book SynopsisZ. Bartosiewicz, Systems on time scales: theory and applications in modeling.- R. Deb, A. K. Das, Complementarity problem with Nekrasov Z tensor.- S. Debnath, Conformal ?-Einstein soliton on (LCS)n-manifold.- A. Gunn and S. Smith, How Much Variability Can an Impulsive Threshold Maintain Before the Outcome Switches?.- T. Ghosh, S. Ghosh, and E. Schwartz, P. Kumar Roy, Mathematical Overview of Bio-Chemical and Physical Interactions Considering Reverse Transcriptase and Integrase Enzymes to Explore HIV Progression: A Control Theoretic Approach.- T. Mondal, S. Ghosh, S. Mukherjee and A. Dénes, Mathematical Modeling of Leprosy Examining the Effect of Awareness Programs in the Disease Transmission: An optimal Control Approach.- S. Kushary, P. Kumar Roy, X. Cao, Introducing MSC Therapy to Inhibit ?Th?_1 and Th?_17 Mediated Cytokines: A Mathematical Study to Regulate Psoriasis.- H. Bose, S. Mukherjee, Xue- Zhi Li, Ratio-Dependent Vector-Host Model on Cutaneous Leishmaniasis - A Mathematical Study Based on Caputo-Fabrizio Fractional Derivative.- X. Cao, T. Ghosh, S. Ghosh.- Mathematical Modelling to Exhibit the Influence of Latently Infected Schwann cells in Leprosy: An Optimal Control-based Study.- Mst Sebi Khatun, Pritha Das, Dynamical behaviour of an SIR epidemic model with Holling type III treatment function and media control.- S. Majee1, S. Jana, and T. Kumar Kar, Dynamical study of the spread of monkeypox including optimal vaccination and treatment with cost-effectiveness.- S. Mukherjee, T. Ghosh, S. Ghosh, Impact of Stem Cell Therapy in Leprosy Pathogenesis through Mathematical Study: An Impulsive Control based Treatment Design.- P. Samui and Jayanta Mondal, Determining pathogenic behaviors of chronic HCV Infection in presence of incubation delay.- P. Das, M. S. Khatun, and K. B. Mahato, Dynamics of a stochastic prey-predator model with Allee effect and fear-dependent prey refuge function.- Amit Kumar Bag, and Amar Nath Chatterjee, The role of screening programs in cervical cancer prevention: A mathematical study.- I. Schreiber and  J. Cerveny, Reaction network based model for carbon-nitrogen metabolism and ultradian rhythms in cyanobacteria.- Sanju Sardar, Priti Kumar Roy, Subhankar Kushary, and David Greenhalgh, Mathematical insight of Methanol Toxicity Management using Charcoal and Folinic Acid.- Sk Mosaraf Ahammed1, Priti Kumar Roy1, and Shu Wang, Effect of Ultrasound Technique for Production of Biodiesel using Enzyme as Catalyst: A Mathematical Study.- Ujjwal Kumar Deb, Temperature effects on microalgae cell growth in an outdoor HLTP.- A. Kumar Paul and M. H. A. Biswas, A Mathematical Theory for Studying and Controlling the Disinformation System Dynamics.- S. Aktar, F. Rahaman, Thin accretion disks around traversable wormholes of Teo class.- P. Baidya1, S.K. Panja and S.C. Mandal, Stress Intensity factor of two co-linear rigid strips under impact load.- Amrita Bhattacharya , Discussion on strong field lensing by wormholes as black hole foils.- P. Mukherjee and S. Kumar Sardar, On a canonical multiplicative semigroup related with the C^*-algebra of a digraph.-  Narendranath Mitra, Generalization of Ramanujan's Famous Nested Radicals to the nth Root and their Evaluation.- Shipan Chandra Deb Nath1, Rajib Karmaker2*, Ujjwal Kumar Deb, Comparison of Iron and Steel Helical Suspension Springs for Mechanical Load Management using Finite Element Method.- Marina Lallallawmzuali1, Sundararajan Muniyan, Analysis of Rainfall Trend and Its Pattern in Uttarakhand Using Appropriate Probability Distributive Functions.- Manojit Das, Shariful Alam, Strategic product launch and pricing: Maximizing profitability in a dynamic consumer market.- Amit K. Ghosh*, Lopamudra Roy and Stuti Saxena, Mathematical attributes applicable on the evolution of life forms and their adaptation with changing climate in the geologic past.- K. Lalnunsiami1, M. Sundararajan and J.K. Pandey, Dispersion Pattern of PM10, PM2.5 and PM1.0 in and around an Opencast Coal Mine Using Statistical and Gaussian Emission Models.- Divya Sharma1, Deepti Kaur and R. K. Mohanty, Bi-parametric compact sixth-order two-stage BGE iteration method for two-dimensional Helmholtz equation.- B.S. Mahapatra , S. Golui , M.K. Mondal, M.B. Bera, and G.S. Mahapatra, Fuzzy TOPSIS method for resource planning selection  incorporating correlation measure.- S. Naskar,  Shariful Alam,  A. Garai, A study on closed-loop dual-channel supply chain with advertising level of online products.

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