Format: Hardback, 514 pages, height x width: 235x155 mm, 6 Illustrations, color;
141 Illustrations, black and white; XVI, 514 p. 147 illus., 6 illus. in color
Series: Texts in Applied Mathematics 75
Pub. Date: 13-Mar-2023
ISBN-13: 9783031219153
This book is about the dynamics of neural systems and should be suitable for those with a background in mathematics, physics, or engineering who want to see how their knowledge and skill sets can be applied in a neurobiological context. No prior knowledge of neuroscience is assumed, nor is advanced understanding of all aspects of applied mathematics! Rather, models and methods are introduced in the context of a typical neural phenomenon and a narrative developed that will allow the reader to test their understanding by tackling a set of mathematical problems at the end of each chapter. The emphasis is on mathematical- as opposed to computational-neuroscience, though stresses calculation above theorem and proof. The book presents necessary mathematical material in a digestible and compact form when required for specific topics. The book has nine chapters, progressing from the cell to the tissue, and an extensive set of references. It includes Markov chain models for ions, differential equations for single neuron models, idealised phenomenological models, phase oscillator networks, spiking networks, and integro-differential equations for large scale brain activity, with delays and stochasticity thrown in for good measure. One common methodological element that arises throughout the book is the use of techniques from nonsmooth dynamical systems to form tractable models and make explicit progress in calculating solutions for rhythmic neural behaviour, synchrony, waves, patterns, and their stability. This book was written for those with an interest in applied mathematics seeking to expand their horizons to cover the dynamics of neural systems. It is suitable for a Masters level course or for postgraduate researchers starting in the field of mathematical neuroscience.
Overview.- Single neuron models-. Phenomenological models and their analysis.- Axons, dendrites, and synapses.- Response properties of single neurons.- Weakly coupled oscillator networks.- Strongly coupled spiking networks.- Population models.- Firing rate tissue models.- Stochastic calculus.- Model Details.- References.
Format: Hardback, 273 pages, height x width: 235x155 mm, 1 Illustrations, color;
3 Illustrations, black and white; XIII, 273 p. 4 illus., 1 illus. in color
Series: Texts in Applied Mathematics 76
Pub. Date: 28-Feb-2023
ISBN-13: 9783031211386
This book introduces optimal control problems for large families of deterministic and stochastic systems with discrete or continuous time parameter. These families include most of the systems studied in many disciplines, including Economics, Engineering, Operations Research, and Management Science, among many others. The main objective is to give a concise, systematic, and reasonably self contained presentation of some key topics in optimal control theory. To this end, most of the analyses are based on the dynamic programming (DP) technique. This technique is applicable to almost all control problems that appear in theory and applications. They include, for instance, finite and infinite horizon control problems in which the underlying dynamic system follows either a deterministic or stochastic difference or differential equation. In the infinite horizon case, it also uses DP to study undiscounted problems, such as the ergodic or long-run average cost. After a general introduction to control problems, the book covers the topic dividing into four parts with different dynamical systems: control of discrete-time deterministic systems, discrete-time stochastic systems, ordinary differential equations, and finally a general continuous-time MCP with applications for stochastic differential equations. The first and second part should be accessible to undergraduate students with some knowledge of elementary calculus, linear algebra, and some concepts from probability theory (random variables, expectations, and so forth). Whereas the third and fourth part would be appropriate for advanced undergraduates or graduate students who have a working knowledge of mathematical analysis (derivatives, integrals, ...) and stochastic processes.
Introduction: optimal control problems-. Discrete-time deterministic systems.- Discrete-time stochastic control systems.- Continuous-time deterministic systems.- Continuous-time Markov control processes.- Controlled diffusion processes.- Appendices.- Bibliography.- Index.
Format: Paperback / softback, 253 pages, height x width: 235x155 mm,
43 Illustrations, black and white; X, 253 p. 43 illus.
Series: Lecture Notes in Mathematics 2326
Pub. Date: 03-Mar-2023
ISBN-13: 9783031219511
The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings. This book is based on Steven G. Krantz's notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974. This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.
Introductory material.- More on Hardy Spaces.- Background on H^p Spaces.- Hardy Spaces on D.- Hardy Spaces on R^n.- Developments Since 1974.- Concluding Remarks.- Bibliography.- Index.
Format: Hardback, 440 pages, height x width: 235x155 mm, 24 Illustrations, color;
5 Illustrations, black and white; X, 440 p. 29 illus., 24 illus. in color.
Series: MS&A 21
Pub. Date: 20-Mar-2023
ISBN-13: 9783031218323
The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.
1 In dimension "zero".- 2 Homogenization in dimension 1.- 3 Dimension 2: The "simple" cases: abstract or periodic settings.- 4 Dimension 2: Some explicit cases beyond the periodic setting.- 5 Numerical Approaches.- 6 Beyond the diffusion equation and miscellaneous topics.- Appendix A: Some basic elements of PDE analysis.
Format: Hardback, 444 pages, height x width: 235x155 mm, 1 Illustrations, color;
113 Illustrations, black and white; XXIV, 444 p. 114 illus., 1 illus. in color
Series: Grundlehren der mathematischen Wissenschaften 360
Pub. Date: 29-Apr-2023
ISBN-13: 9783031217999
The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by Laszlo Fejes Toth in his Notes to the 2nd edition. The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.
Part I. Lagerungen - Arrangements in the Plane, on the Sphere, and in Space.-
1. Some Theorems from Elementary Geometry.-
2. Theorems from the Theory of Convex Bodies.-
3. Problems on Packing and Covering in the Plane.-
4. Efficiency of Packings and Coverings with a Sequence of Convex Disks.-
5. Extremal Properties of Regular Polyhedra.-
6. Irregular Packing on the Sphere.-
5. Packing in Space.- Part II. Notes and Additional
Chapters to the English Edition.-
8. Notes.-
9. Finite Variations on the Isoperimetric Problem.-
10. Higher Dimensions.-
11. Ball Packings in Hyperbolic Space.-
12. Mutliple Arrangements.-
13. Neighbors.-
14. Packing and Covering Properties fo Sequences of Convex Bodies.-
15. Four Classic Problems.-
16. Miscellaneous Problems about Packing and Covering.- References for Part I.- References for Part II.- Name Index.- Subject Index.