How to Think About Analysis
How to Think About Analysis

Author: Lara Alcock

Publisher: OUP Oxford

Published: 2014-09-25

Total Pages: 272

ISBN-13: 0191035378

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Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student's existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.

Simulating Hamiltonian Dynamics
Simulating Hamiltonian Dynamics

Author: Benedict Leimkuhler

Publisher: Cambridge University Press

Published: 2004

Total Pages: 464

ISBN-13: 9780521772907

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Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.

Interdisciplinarity
Interdisciplinarity

Author: Julie Thompson Klein

Publisher: Wayne State University Press

Published: 1990

Total Pages: 340

ISBN-13: 9780814320884

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In this volume, Julie Klein provides the first comprehensive study of the modern concept of interdisciplinarity, supplementing her discussion with the most complete bibliography yet compiled on the subject. In this volume, Julie Klein provides the first comprehensive study of the modern concept of interdisciplinarity, supplementing her discussion with the most complete bibliography yet compiled on the subject. Spanning the social sciences, natural sciences, humanities, and professions, her study is a synthesis of existing scholarship on interdisciplinary research, education and health care. Klein argues that any interdisciplinary activity embodies a complex network of historical, social, psychological, political, economic, philosophical, and intellectual factors. Whether the context is a short-ranged instrumentality or a long-range reconceptualization of the way we know and learn, the concept of interdisciplinarity is an important means of solving problems and answering questions that cannot be satisfactorily addressed using singular methods or approaches.

Mathematical Analysis in Interdisciplinary Research
Mathematical Analysis in Interdisciplinary Research

Author: Ioannis N. Parasidis

Publisher: Springer Nature

Published: 2022-03-10

Total Pages: 1050

ISBN-13: 3030847217

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This contributed volume provides an extensive account of research and expository papers in a broad domain of mathematical analysis and its various applications to a multitude of fields. Presenting the state-of-the-art knowledge in a wide range of topics, the book will be useful to graduate students and researchers in theoretical and applicable interdisciplinary research. The focus is on several subjects including: optimal control problems, optimal maintenance of communication networks, optimal emergency evacuation with uncertainty, cooperative and noncooperative partial differential systems, variational inequalities and general equilibrium models, anisotropic elasticity and harmonic functions, nonlinear stochastic differential equations, operator equations, max-product operators of Kantorovich type, perturbations of operators, integral operators, dynamical systems involving maximal monotone operators, the three-body problem, deceptive systems, hyperbolic equations, strongly generalized preinvex functions, Dirichlet characters, probability distribution functions, applied statistics, integral inequalities, generalized convexity, global hyperbolicity of spacetimes, Douglas-Rachford methods, fixed point problems, the general Rodrigues problem, Banach algebras, affine group, Gibbs semigroup, relator spaces, sparse data representation, Meier-Keeler sequential contractions, hybrid contractions, and polynomial equations. Some of the works published within this volume provide as well guidelines for further research and proposals for new directions and open problems.

Variational Methods For Strongly Indefinite Problems
Variational Methods For Strongly Indefinite Problems

Author: Yanheng Ding

Publisher: World Scientific

Published: 2007-07-30

Total Pages: 177

ISBN-13: 9814474509

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This unique book focuses on critical point theory for strongly indefinite functionals in order to deal with nonlinear variational problems in areas such as physics, mechanics and economics. With the original ingredients of Lipschitz partitions of unity of gage spaces (nonmetrizable spaces), Lipschitz normality, and sufficient conditions for the normality, as well as existence-uniqueness of flow of ODE on gage spaces, the book presents for the first time a deformation theory in locally convex topological vector spaces. It also offers satisfying variational settings for homoclinic-type solutions to Hamiltonian systems, Schrödinger equations, Dirac equations and diffusion systems, and describes recent developments in studying these problems. The concepts and methods used open up new topics worthy of in-depth exploration, and link the subject with other branches of mathematics, such as topology and geometry, providing a perspective for further studies in these areas. The analytical framework can be used to handle more infinite-dimensional Hamiltonian systems.