This book describes the theory of profinite groups, from the basics of the theory to topics which are active areas of current research. It is the first textbook on profinite groups to make their use in studying residually finite groups via their profinite completions a central focus.
The first part of the book gives the subject a firm theoretical underpinning from category theory and introduces profinite groups as objects worthy of study in their own right. The reader is not expected to have a background in category theory. The connection of a residually finite group to its profinite completion is explored in detail, with emphasis on various separability properties and profinite rigidity.
The study of group cohomology is a key tool in the exploration of profinite groups. The central portion of this book gives a standalone first course in group cohomology before showing the modification of this theory for use with profinite groups. There is special emphasis on the unique features of profinite group cohomology such as Pontryagin duality and Sylow theory.
Later chapters of the book collect together for the first time important results concerning the relation of the cohomology of a group to that of its profinite completion, and introduce the concept of an action of a profinite group on a profinite tree. This material aims to be a useful reference for researchers as well as a learning resource.
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The present book is a new, substantially enlarged, version of a previous set of lecture notes on diophantine analysis, published first in 2008, and then in revised form a few years later, by the Edizioni della Scuola Normale.
The content mixes a number of rather classical results on diophantine equations and diophantine approximation, with the basic theory of heights and a few more recent results and applications of it.
The exposition has been generally kept at an elementary and essentially self-contained level, focusing on some main ideas rather than finer technical results which can be obtained by similar methods. In fact, the book is addressed also to readers outside the relevant fields, with the hope that also more expert readers might find something relevant to them.
The present second edition contains substantial new material, in the form of new sections, supplements, remarks, and exercises. The arguments for the exercises are developed in full by means of ghintsh, which in fact are much more than scattered suggestions, and practically contain complete details. Occasionally the remarks and the exercises contain miscellaneous results which have not been explicitly published elsewhere.
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Format: Hardback, 484 pages, height x width: 254x178 mm, 140 Illustrations, black and white; XIV, 483 p. 67 illus., 1 Hardback
Series: Springer Undergraduate Texts in Mathematics and Technology
Pub. Date: 10-Nov-2024
ISBN-13: 9783031647949
This text lays the foundation for understanding the beauty and power of discrete-time models. It covers rich mathematical modeling landscapes, each offering deep insights into the dynamics of biological systems. A harmonious balance is achieved between theoretical principles, mathematical rigor, and practical applications. Illustrative examples, numerical simulations, and empirical case studies are provided to enhance mastery of the subject and facilitate the translation of discrete-time mathematical biology into real-world challenges. Mainly geared to upper undergraduates, the text may also be used in graduate courses focusing on discrete-time modeling.
Chapters 14 constitute the core of the text. Instructors will find the dependence chart quite useful when designing their particular course. This invaluable resource begins with an exploration of single-species models where frameworks for discrete-time modeling are established. Competition models and Predator-prey interactions are examined next followed by evolutionary models, structured population models, and models of infectious diseases. The consequences of periodic variations, seasonal changes, and cyclic environmental factors on population dynamics and ecological interactions are investigated within the realm of periodically forced biological models.
This indispensable resource is structured to support educational settings:
A first course in biomathematics, introducing students to the fundamental mathematical techniques essential for biological research. A modeling course with a concentration on developing and analyzing mathematical models that encapsulate biological phenomena. An advanced mathematical biology course that offers an in-depth exploration of complex models and sophisticated mathematical frameworks designed to tackle advanced problems in biology.
With its clear exposition and methodical approach, this text educates and inspires students and professionals to apply mathematical biology to real-world situations. While minimal knowledge of calculus is required, the reader should have a solid mathematical background in linear algebra.
Preface.- 1.Scalar Population Models.-
2. Linear Structured Population Models.-
3. Linear and Nonlinear Systems.-
4. Infectious Disease Models I.-
5. Models with Multiple Attractors.-
6. Nonlinear Structured Population Models.-
7. Infectious Disease Models II.-
8. Evolutionary Models.-
9. Autonomous Models.- Bibliography.- Index.
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Format: Paperback / softback, 136 pages, height x width: 235x155 mm, X, 136 p., 1 Paperback / softback
Series: SpringerBriefs in Mathematics
Pub. Date: 30-Aug-2024
ISBN-13: 9783031695858
Introducing a groundbreaking framework for stochastic partial differential equations (SPDEs), this work presents three significant advancements over the traditional variational approach.
Firstly, Stratonovich SPDEs are explicitly addressed. Widely used in physics, Stratonovich SPDEs have typically been converted to Ito form for mathematical treatment. While this conversion is understood heuristically, a comprehensive treatment in infinite dimensions has been lacking, primarily due to insufficient rigorous results on martingale properties.
Secondly, the framework incorporates differential noise, assuming the noise operator is only bounded from a smaller Hilbert space into a larger one, rather than within the same space. This necessitates additional regularity in the Ito form to solve the original Stratonovich SPDE. This aspect has been largely overlooked, despite the increasing popularity of gradient-dependent Stratonovich noise in fluid dynamics and regularisation by noise studies.
Lastly, the framework departs from the explicit duality structure (Gelfand Triple), which is typically expected in the study of analytically strong solutions. This extension builds on the classical variational framework established by Rockner and Pardoux, advancing it in all three key aspects.
Explore this innovative approach that not only addresses existing challenges but also opens new avenues for research and application in SPDEs.
1 Introduction.- 2 Stochastic Calculus in Infinite Dimensions.- 3
Stochastic Differential Equations in Infinite Dimensions.- 4 A Toolbox for
Nonlinear SPDEs.- 5 Existence Theory for Nonlinear SPDEs and the
Stochastic Navier-Stokes Equations.- A Appendix.- References .- Index .
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Format: Hardback, 642 pages, height x width: 235x155 mm, 59 Illustrations, color; 169 Illustrations, black and white; Approx. 650 p., 1 Hardback
Pub. Date: 13-Nov-2024
ISBN-13: 9783031649981
This textbook has been designed to support the initial study of Complex Analysis, progressing to Complex Dynamics. It focuses on the fundamental aspects of one-variable complex functions, covering the geometric theory and dynamics of iterations of rational mappings. Following the standard material, the book delves into an extensive range of advanced topics, encompassing the requirements for a one-year graduate-level course or a preliminary exam.
In this work, the reader will discover three distinctive characteristics: it simplifies and unifies ideas and concepts that might appear disparate or complicated in real analysis; it contributes to the development of other areas in mathematics; and it showcases relevance for applications in Science and Engineering, with many exercises. Historical notes throughout the text help to contextualize the theory.
With its flexible structure, this textbook provides a solid foundation for a first course in Complex Analysis and for a second more advanced course, establishing a robust basis for subsequent studies.
Complex plane.- Functions.- Derivative.- Integral.- Analyticity.- Uni cation of holomorphy cauchy theorem and analyticity.- Global cauchy theorem.- Meromorphic functions.- Harmonic functions.- Conformal regions.- Analytic continuation and riemann surfaces.- Uniformization of riemann surfaces.- Complex dynamics.
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Format: Paperback / softback, 241 pages, height x width: 235x155 mm, 35 Illustrations, color; 8 Illustrations, black and white; Approx. 300 p., 1 Paperback / softback
Pub. Date: 11-Nov-2024
ISBN-13: 9783031703256
This textbook proposes an informal access to the most important issues of multidimensional differential and integral calculus. The traditional style?characterized by listing definitions, theorems, and proofs?is replaced by a conversational approach, primarily oriented to applications. The topics covered, developing along the usual path of a textbook for undergraduate courses, are always introduced by thoroughly carried out examples. This drives the reader in building the capacity of properly use the theoretical tools to model and solve practical problems. To situate the contents within a historical perspective, the book is accompanied by a number of links to the biographies of all scientists mentioned as leading actors in the development of the theory.
Chapter 1. Basic concepts and parametrisation of curves.- Chapter
2. Differential and geometric properties of curves.- Chapter 3. Curves in
space: the Frenet frame.- Chapter 4. Functions of a vector variable.- Chapter
5. Continuity and differentiability of functions of a vector
variable.- Chapter 6. Partial derivatives.- Chapter 7. Sequences of
functions.- Chapter 8. Series of functions.- Chapter 9. Taylor series for
functions of several variables.- Chapter 10. Applications of the Taylor
series.- Chapter 11. Integration of functions of two variables.- Chapter
12. Samples of two-dimensional integration and change of variables.- Chapter
13. Two-dimensional integration and area of a surface.- Chapter 14. Vector
functions of vector variables.- Chapter 15. Line integral and flux of vector
functions.- Chapter 16. Triple integrals and coordinate changes.- Chapter
17. Greens formulae for the integral calculus.-Chapter 18. Application of
Greens formulae.- Chapter 19. Gauss and Stokes theorems.- Chapter
20. Partial differential equations.- Etc...
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Format: Paperback / softback, 130 pages, height x width: 235x155 mm, 1 Illustrations, black and white; X, 130 p. 1 illus., 1 Paperback / softback
Series: SpringerBriefs in Physics
Pub. Date: 23-Sep-2024
ISBN-13: 9783031679148
This book offers a non-standard introduction to quantum mechanics and quantum field theory, approaching these topics from algebraic and geometric perspectives. Beginning with fundamental notions of quantum theory and the derivation of quantum probabilities from decoherence, it proceeds to prove the expression for the scattering matrix in terms of Green functions (LSZ formula), along with a similar expression for the inclusive scattering matrix. The exposition relies on recent findings by the author that provide a deeper understanding of the structure of quantum theory and extend beyond its traditional boundaries. The book is suitable for graduate students and young researchers in mathematics and theoretical physics seeking to delve into innovative concepts within quantum theory. The book contains many recent results therefore it should be interesting also to accomplished physicists and mathematicians.
1. Quantum Theory in Algebraic and Geometric Approaches.-
2. Scattering Theory.-
3. Deterministic Physical Theories.-
4. Appendix.