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Convex functions. --- Subdifferentials. --- Calculus, Subdifferential --- Subdifferential calculus --- Convex functions --- Functions, Convex --- Functions of real variables
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Lively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression. In studying this book, students will encounter: the interconnections between set theory and mathematical statements and proofs; the fundamental axioms of the natural, integer, and real numbers; rigorous ε-N and ε-δ definitions; convergence and properties of an infinite series, product, or continued fraction; series, product, and continued fraction formulæ for the various elementary functions and constants. Instructors will appreciate this engaging perspective, showcasing the beauty of these fundamental results.
Mathematics. --- Functions of real variables. --- Sequences (Mathematics). --- Sequences, Series, Summability. --- Real Functions. --- Math --- Science --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Mathematical analysis. --- Real variables --- Functions of complex variables
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Based on the bestselling Artech House classic title, Hilbert Transforms Signal Processing, this comprehensive new resource introduces complex and hypercomplex analytic signals and their applications. Professionals find in-depth explanations of the theory of multidimensional complex and hypercomplex signals illustrated with numerous examples and followed by practical applications. The survey of chosen hypercomplex algebras and the orthants of the n-dimensional Cartesian space and single-orthant operators are explored. This book also covers topics including, the polar representation of analytic signals, quasi-analytic signals, the space-frequency of n-D complex and hypercomplex signals as well as the causality of signals.
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This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.
Mathematics. --- Algebra. --- Matrix theory. --- Functions of real variables. --- Real Functions. --- General Algebraic Systems. --- Linear and Multilinear Algebras, Matrix Theory. --- Real variables --- Math --- Mathematics --- Mathematical analysis --- Science --- Functions of complex variables
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This book is about the rise and supposed fall of the mean value theorem. It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus. It was called “the fundamental theorem of the differential calculus” because of its power to provide simple and rigorous proofs of basic results encountered in a first-year course in calculus. In mathematical terms, the book is a thorough treatment of this theorem and some related results in the field; in historical terms, it is not a history of calculus or mathematics, but a case study in both. MVT: A Most Valuable Theorem is aimed at those who teach calculus, especially those setting out to do so for the first time. It is also accessible to anyone who has finished the first semester of the standard course in the subject and will be of interest to undergraduate mathematics majors as well as graduate students. Unlike other books, the present monograph treats the mathematical and historical aspects in equal measure, providing detailed and rigorous proofs of the mathematical results and even including original source material presenting the flavour of the history.
Mathematics. --- Functions of real variables. --- History. --- Real Functions. --- History of Mathematical Sciences. --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Mathematical analysis --- Functions --- Geometry, Infinitesimal --- Math --- Science --- Annals --- Auxiliary sciences of history --- Real variables --- Functions of complex variables
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This book is dedicated to the memory of Mikael Passare, an outstanding Swedish mathematician who devoted his life to developing the theory of analytic functions in several complex variables and exploring geometric ideas first-hand. It includes several papers describing Mikael’s life as well as his contributions to mathematics, written by friends of Mikael’s who share his attitude and passion for science. A major section of the book presents original research articles that further develop Mikael’s ideas and which were written by his former students and co-authors. All these mathematicians work at the interface of analysis and geometry, and Mikael’s impact on their research cannot be underestimated. Most of the contributors were invited speakers at the conference organized at Stockholm University in his honor. This book is an attempt to express our gratitude towards this great mathematician, who left us full of energy and new creative mathematical ideas.
Geometry. --- Mathematics. --- Partial differential equations. --- Functions of complex variables. --- Several Complex Variables and Analytic Spaces. --- Partial Differential Equations. --- Mathematics --- Euclid's Elements --- Differential equations, partial. --- Partial differential equations --- Differential equations, partial --- Complex variables --- Elliptic functions --- Functions of real variables
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This book is devoted to the study of certain integral representations for Neumann, Kapteyn, Schlömilch, Dini and Fourier series of Bessel and other special functions, such as Struve and von Lommel functions. The aim is also to find the coefficients of the Neumann and Kapteyn series, as well as closed-form expressions and summation formulas for the series of Bessel functions considered. Some integral representations are deduced using techniques from the theory of differential equations. The text is aimed at a mathematical audience, including graduate students and those in the scientific community who are interested in a new perspective on Fourier–Bessel series, and their manifold and polyvalent applications, mainly in general classical analysis, applied mathematics and mathematical physics.
Mathematics. --- Functions of complex variables. --- Differential equations. --- Functions of real variables. --- Sequences (Mathematics). --- Special functions. --- Astronomy. --- Astrophysics. --- Cosmology. --- Special Functions. --- Sequences, Series, Summability. --- Real Functions. --- Functions of a Complex Variable. --- Ordinary Differential Equations. --- Astronomy, Astrophysics and Cosmology. --- Astronomy --- Deism --- Metaphysics --- Astronomical physics --- Cosmic physics --- Physics --- Physical sciences --- Space sciences --- Special functions --- Mathematical analysis --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Real variables --- Functions of complex variables --- 517.91 Differential equations --- Differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Functions, special. --- Differential Equations. --- Bessel functions. --- Cylindrical harmonics --- Transcendental functions --- Bessel polynomials --- Harmonic analysis --- Harmonic functions
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The book presents an introduction to the geometry of Hilbert spaces and operator theory, targeting graduate and senior undergraduate students of mathematics. Major topics discussed in the book are inner product spaces, linear operators, spectral theory and special classes of operators, and Banach spaces. On vector spaces, the structure of inner product is imposed. After discussing geometry of Hilbert spaces, its applications to diverse branches of mathematics have been studied. Along the way are introduced orthogonal polynomials and their use in Fourier series and approximations. Spectrum of an operator is the key to the understanding of the operator. Properties of the spectrum of different classes of operators, such as normal operators, self-adjoint operators, unitaries, isometries and compact operators have been discussed. A large number of examples of operators, along with their spectrum and its splitting into point spectrum, continuous spectrum, residual spectrum, approximate point spectrum and compression spectrum, have been worked out. Spectral theorems for self-adjoint operators, and normal operators, follow the spectral theorem for compact normal operators. The book also discusses invariant subspaces with special attention to the Volterra operator and unbounded operators. In order to make the text as accessible as possible, motivation for the topics is introduced and a greater amount of explanation than is usually found in standard texts on the subject is provided. The abstract theory in the book is supplemented with concrete examples. It is expected that these features will help the reader get a good grasp of the topics discussed. Hints and solutions to all the problems are collected at the end of the book. Additional features are introduced in the book when it becomes imperative. This spirit is kept alive throughout the book.
Mathematics. --- Functional analysis. --- Operator theory. --- Functions of complex variables. --- Operator Theory. --- Functional Analysis. --- Several Complex Variables and Analytic Spaces. --- Hilbert space. --- Banach spaces --- Hyperspace --- Inner product spaces --- Functional analysis --- Differential equations, partial. --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Complex variables --- Elliptic functions --- Functions of real variables
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The main purpose of this book is to give a detailed and complete survey of recent progress related to the real-variable theory of Musielak–Orlicz Hardy-type function spaces, and to lay the foundations for further applications. The real-variable theory of function spaces has always been at the core of harmonic analysis. Recently, motivated by certain questions in analysis, some more general Musielak–Orlicz Hardy-type function spaces were introduced. These spaces are defined via growth functions which may vary in both the spatial variable and the growth variable. By selecting special growth functions, the resulting spaces may have subtler and finer structures, which are necessary in order to solve various endpoint or sharp problems. This book is written for graduate students and researchers interested in function spaces and, in particular, Hardy-type spaces.
Mathematics. --- Fourier analysis. --- Functional analysis. --- Operator theory. --- Functions of real variables. --- Fourier Analysis. --- Functional Analysis. --- Operator Theory. --- Real Functions. --- Real variables --- Functions of complex variables --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Function spaces. --- Spaces, Function
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The book presents major topics in semigroups, such as operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics, and applications. Along with a systematic development of the subject, the book emphasises on the explorations of the contact areas and interfaces, supported by the presentations of explicit computations, wherever feasible. Designed into seven chapters and three appendixes, the book targets to the graduate and senior undergraduate students of mathematics, as well as researchers in the respective areas. The book envisages the pre-requisites of a good understanding of real analysis with elements of the theory of measures and integration, and a first course in functional analysis and in the theory of operators. Chapters 4 through 6 contain advanced topics, which have many interesting applications such as the Feynman–Kac formula, the central limit theorem and the construction of Markov semigroups. Many examples have been given in each chapter, partly to initiate and motivate the theory developed and partly to underscore the applications. The choice of topics in this vastly developed book is a difficult one, and the authors have made an effort to stay closer to applications instead of bringing in too many abstract concepts.
Mathematics. --- Functional analysis. --- Differential equations. --- Partial differential equations. --- Functions of real variables. --- Functional Analysis. --- Real Functions. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Differential Equations. --- Differential equations, partial. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Real variables --- Functions of complex variables
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