Format: Paperback / softback, 201 pages, height x width: 235x155 mm, 77 Illustrations, color; 3 Illustrations, black and white; X, 190 p. 60 illus. in color., 1 Paperback / softback
Series: Lecture Notes in Physics 1032
Pub. Date: 06-Nov-2024
ISBN-13: 9783031701993
This textbook introduces topological defects and solitons at a level suitable for advanced undergraduates and beginning graduate students in physics and materials science. It avoids the formal mathematics of topology, and instead concentrates on the physical properties of these topological structures.
The first half of the book concentrates on fundamental principles of defects and solitons, and illustrates these principles with a single example?the xy model for 2D magnetic order. It begins by defining the concept of a winding number, and uses this concept to describe the topology of defects (vortices or disclinations) and solitons (domain walls), carefully identifying the similarities and differences between these two types of topological structures. It then goes on to discuss physical properties of defects and solitons, including free energy, dynamics, statistical mechanics, and coupling with curvature. It shows how these concepts emerge from a theory with variable magnitude of order, and hence how topology can be viewed as an approximation to physics.
The second half goes on to explore a wider range of topological defects and solitons. First, it considers more complex types of order?2D nematic liquid crystals, 3D magnetic or liquid-crystal order, 2D or 3D crystalline solids?and shows how each type of order leads to specific topological structures. Next, it discusses defects and solitons that are characterized by 2D or 3D measuring surfaces, not just 1D loops, including hedgehogs, skyrmions, and hopfions. These structures are more complex, but they can still be understood using the same fundamental principles. A final chapter describes the formation of phases with regular arrays of defects or solitons.
Chapter 1. Introduction to Defects.
Chapter 2. Introduction to Solitons.
Chapter 3. Free Energy.
Chapter 4. Dynamics and Statistical Mechanics.
Chapter 5. Prequel to Defects: Variable Magnitude of Order.
Chapter 6. Further Issues: Defect Phase/Orientation, Charge Density, Curvature.
Chapter 7. 2D Nematic Order, Active Liquid Crystals.
Chapter 8. 3D Polar or Nematic Order.
Chapter 9. Defects in Crystals.
Chapter 10. 2D Measuring Surface: Hedgehogs, Skyrmions.
Chapter 11. 3D Measuring Surface: Hopfions.
Chapter 12. Phases With Regular Arrays of Defects or Solitons.
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Format: Paperback / softback, 571 pages, height x width: 235x155 mm, 101 Illustrations, black and white; Approx. 535 p., 1 Paperback / softback
Series: Mathematics Study Resources 15
Pub. Date: 17-Nov-2024
ISBN-13: 9783662696965
This book provides essential foundations of mathematics, covering set theory, algebra, the theory of real and complex numbers, and topology. It serves as the basis for further exploration of mathematics. Not only are the necessary concepts introduced, but also essential ? including deeper ? statements are proven. The material is illustrated and supplemented by unusual examples and diverse exercises. The book is suitable for self-study, but primarily designed as companion reading from the beginning for a study in mathematics, physics, and computer science. Its coherent approach makes it highly readable and comparatively easy to understand.
This book is a translation of the original German edition eGrundkonzepte der Mathematikf by Uwe Storch and Hartmut Wiebe, published by Springer-Verlag GmbH, DE in 2017. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content (by the author Hartmut Wiebe only, as Uwe Storch is no longer with us).
Foreword.- Introduction.- List of Symbols.- Fundamentals of Set Theory.- Algebraic Foundations.- Real and Complex Numbers.- Topological Foundations.- Subject Index.- Bibliography.
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Format: Paperback / softback, 138 pages, height x width: 235x155 mm, 2 Illustrations, color;
7 Illustrations, black and white; VIII, 138 p. 9 illus., 2 illus. in color., 1 Paperback / softback
Series: SpringerBriefs on PDEs and Data Science
Pub. Date: 06-Sep-2024
ISBN-13: 9789819759477
This book is the result of various master and summer school courses the author has taught. The objective is to provide the reader with an introduction to control theory and to the main tools allowing to treat general control systems. The author hopes this book will serve as motivation to go deeper into the theory or numerical aspects that are not covered in this book.
This book might be helpful for graduate students and researchers in the field of control theory.
Chapter 1 Controllability.
Chapter 2. Optimal control.
Chapter 3 Stabilization.
Chapter 4 Semigroup theory.
Chapter 5 Linear control systems in Banach spaces.
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Format: Paperback / softback, 453 pages, height x width: 235x155 mm, 38 Illustrations, black and white; X, 457 p. 19 illus., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2360
Pub. Date: 23-Nov-2024
ISBN-13: 9783031672873
This book studies the relation between conformal invariants and dynamical invariants and their applications, taking the reader on an excursion through a wide range of topics. The conformal invariants, called here the conformal modules of conjugacy classes of elements of the fundamental group, were proposed by Gromov in the case of the twice punctured complex plane. They provide obstructions to Gromov's Oka Principle. The invariants of the space of monic polynomials of degree n appeared earlier in relation to Hilbert's 13th Problem, and are called the conformal modules of conjugacy classes of braids.
Interestingly, the conformal module of a conjugacy class of braids is inversely proportional to a popular dynamical invariant, the entropy, which was studied in connection with Thurston's celebrated theory of surface homeomorphisms. This result, proved here for the first time, is another instance of the numerous manifestations of the unity of mathematics, and it has applications.
After prerequisites on Riemann surfaces, braids, mapping classes and elements of Teichmuller theory, a detailed introduction to the entropy of braids and mapping
classes is given, with thorough, sometimes new proofs.
Estimates are provided of Gromov's conformal invariants of the twice punctured complex plane and it is shown that the upper and lower bounds differ by universal multiplicative constants. These imply estimates of the entropy of any pure three-braid, and yield quantitative statements on the limitations of Gromov's Oka Principle in the sense of finiteness theorems, using conformal invariants which are related to elements of the fundamental group (not merely to conjugacy classes). Further applications of the concept of conformal module are discussed. Aimed at graduate students and researchers, the book proposes several research problems.
1. Introduction.-
2. Riemann Surfaces, Braids, Mapping Classes, and Teichmueller Theory.-
3. The entropy of surface homeomorphisms.-
4. Conformal invariants of homotopy classes of curves. The Main theorem.-
5. Reducible pure braids. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem.-
6. The general case. Irreducible nodal components, irreducible braid components, and the proof of the Main Theorem.-
7. The conformal module and holomorphic families of polynomials.-
8. Gromovs Oka Principle and conformal module.-
9. Gromovs Oka Principle for (g, m)-fiber bundles.-
10. Fundamental groups and bounds for the extremal length.-
11. Counting functions.-
12. Riemann surfaces of second kind and finiteness theorems.- A. Several complex variables.- B. A Lemma on Conjugation.- C. Koebes Theorem.- Index.- References.
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Format: Paperback / softback, 439 pages, height x width: 235x155 mm, XII, 439 p., 1 Paperback / softback
Series: Universitext
Pub. Date: 05-Nov-2024
ISBN-13: 9783031709081
This textbook provides a modern introduction to advanced concepts and methods of mathematical analysis.
The first three parts of the book cover functional analysis, harmonic analysis, and microlocal analysis. Each chapter is designed to provide readers with a solid understanding of fundamental concepts while guiding them through detailed proofs of significant theorems. These include the universal approximation property for artificial neural networks, Brouwer's domain invariance theorem, Nash's implicit function theorem, Calderon's reconstruction formula and wavelets, Wiener's Tauberian theorem, Hormander's theorem of propagation of singularities, and proofs of many inequalities centered around the works of Hardy, Littlewood, and Sobolev. The final part of the book offers an overview of the analysis of partial differential equations. This vast subject is approached through a selection of major theorems such as the solution to Calderon's problem, De Giorgi's regularity theorem for elliptic equations, and the proof of a StrichartzBourgain estimate. Several renowned results are included in the numerous examples.
Based on courses given successively at the Ecole Normale Superieure in France (ENS Paris and ENS Paris-Saclay) and at Tsinghua University, the book is ideally suited for graduate courses in analysis and PDE. The prerequisites in topology and real analysis are conveniently recalled in the appendix
Part I Functional Analysis.
- 1 Topological Vector Spaces.
- 2 Fixed Point Theorems.
- 3 Hilbertian Analysis, Duality and Convexity.-
Part II Harmonic
Analysis.
- 4 Fourier Series.
- 5 Fourier Transform.
- 6 Convolution.
- 7 Sobolev Spaces.
- 8 Harmonic Functions.-
Part III Microlocal Analysis.
- 9 Pseudo-Differential Operators.
- 10 Symbolic Calculus.
- 11 Hyperbolic Equations.
- 12 Microlocal Singularities.-
Part IV Analysis of Partial Differential Equations.
- 13 The Calderon Problem.
- 14 De Giorgis Theorem.
-15 Schauders Theorem.
- 16 Dispersive Estimates.-
Part V Recap and Solutions to the Exercises.
- 17 Recap on General Topology.
- 18 Inequalities in Lebesgue Spaces.
- 19 Solutions.
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Format: Hardback, 342 pages, height x width: 279x210 mm, 120 Illustrations, color; 21 Illustrations, black and white; XX, 330 p., 1 Hardback
Pub. Date: 01-Nov-2024
ISBN-13: 9783031672019
This book investigates sequences and series with a clear and focused approach, presenting key theoretical concepts alongside a diverse range of examples and proposed problems, complete with solutions. It is designed to be largely self-contained, offering formal proofs when they enhance understanding. Solutions are provided separately, encouraging students to develop their problem-solving skills.
Chapters 1 and 2 focus on sequences and numerical series, drawing primarily on knowledge acquired in high school. Calculus concepts become important from the end of Chapter 2, extending into Chapter 3, which is entirely dedicated to function series. This includes in-depth discussions of Taylor, Maclaurin, and Fourier series. Many of the exercises have been rigorously tested in actual classes and exams.
The book is enriched by historical facts about mathematicians who have contributed to the subject, fostering students' motivation. It is valuable reading for undergraduates in mathematics, engineering, and other STEM-related fields, as well as for any student with a specific interest in the matter.
Sequences.- Sequences of Real Numbers.- Solved Exercises.- Proposed Exercises.- Numerical Series.- Generalization of the Addition Operation.- Series Definition. Convergence. General Properties.- Alternating Series.- Absolute Convergence.- Series of Non-Negative Terms.- Multiplication of Series.- Solved Exercises.- Proposed Exercises.- Series of Functions.- Introduction. Sequences of Functions - Pointwise Convergence and Uniform Convergence of Series of Functions - Power Series - Taylor Series and Maclaurin Series - Introduction to Fourier Series - Solved Exercises - Proposed Exercises.- Additional Proofs.- Sequences of Real Numbers.- Solutions to Proposed Exercises.- Bibliography.- Index.
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Format: Hardback, 470 pages, height x width: 235x155 mm, 13 Illustrations, color; 20 Illustrations, black and white; Approx. 480 p., 1 Hardback
Series: Contributions to Statistics
Pub. Date: 11-Nov-2024
ISBN-13: 9783031680076
This book of selected works exhibits a significant part of Carlos A. Coelhofs research activity in multivariate statistics, focusing on topics related to likelihood ratio tests. After two introductory chapters, including a biography and an overview of the contributions and related research, the articles are presented in their chronological order of publication, thus also showing the research path followed by the author. The volume shows how likelihood ratio tests useful in multivariate analysis can be developed, and how in most cases exact distributions can be obtained for their statistics in a manageable form. Furthermore, it shows how extremely sharp asymptotic (near-exact) approximations can be found in the remaining cases. Circular distributions are also addressed and, more generally, distributions of products of random variables. The book will be useful for advanced students as well as researchers and practitioners who deal with likelihood ratio tests in multivariate analysis.
Preface.- Biography of Carlos A. Coelho.- Overview of Carlos A. Coelhos
contributions included in the book and related research.- C. A. Coelho: The
Generalized Integer Gamma Distribution a Basis for Distributions in
Multivariate Statistics. Journal of Multivariate Analysis.- C. A. Coelho:
Addendum to the paper The Generalized Integer Gamma Distribution - a Basis
for Distributions in Multivariate Analysis.- C. A. Coelho: The Generalized
Near-Integer Gamma distribution a basis for near-exact approximations to
the distributions of statistics which are the product of an odd number of
particular independent Beta random variables.- C. A. Coelho: The wrapped
Gamma distribution and wrapped sums and linear combinations of independent
Gamma and Laplace distributions.- Coelho and J. T. Mexia: On the distribution
of the product and ratio of independent generalized Gamma-ratio random
variables.- C. A. Coelho and F. J. Marques: The advantage of decomposing
elaborate hypotheses on covariance matrices into conditionally independent
hypotheses in building near-exact distributions for the test statistics.- C.
A. Coelho, B. C. Arnold, and F. J. Marques: Near-exact distributions for
certain likelihood ratio test statistics.- F. J. Marques, C. A. Coelho, and
B. C. Arnold: A general near-exact distribution theory for the most common
likelihood ratio test statistics used in Multivariate Analysis.- B. C.
Arnold, C. A. Coelho, and F. J. Marques: The distribution of the product of
powers of independent Uniform random variables a simple but useful tool to
address and better understand the structure of some distributions.- C. A.
Coelho, F. J. Marques, and B. C. Arnold: The exact and near-exact
distributions of the main likelihood ratio test statistics used in the
complex multivariate normal setting.- C. A. Coelho and A. Roy: Testing the
hypothesis of a doubly exchangeable covariance matrix.- C. A. Coelho: On the
Distribution of Linear Combinations of Chi-Square Random Variables.- C. A.
Coelho and R. P. Alberto: On the Distribution of the Product of Independent
Beta Random Variables Applications.- C. A. Coelho: Testing equality of mean
vectors with block-circular and block compound-symmetric covariance
matrices.- C. A. Coelho and J. Pielaszkiewicz: The Likelihood Ratio Test of
Equality of Mean Vectors with a Doubly Exchangeable Covariance Matrix.
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