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An Introduction to Chaotic Dynamical Systems
An Introduction to Chaotic Dynamical Systems by
Call Number: QA614.8 .D481 2022
There is an explosion of interest in dynamical systems in the mathematical community, as well as in many areas of science. Throughout this text, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, rather than the many and diverse applications of this theory. The field of dynamical systems (and particularly the study of chaotic systems) has been hailed as one of the important breakthroughs in science in the past century, and its importance continues to expand.
A Mathematical Introduction to General Relativity by This book aims to give a mathematical presentation of the theory of general relativity (that is, spacetime-geometry-based gravitation theory) to advanced undergraduate mathematics students. Mathematicians will find spacetime physics presented in the definition-theorem-proof format familiar to them. The given precise mathematical definitions of physical notions help avoiding pitfalls, especially in the context of spacetime physics describing phenomena that are counter-intuitive to everyday experiences. In the first part, the differential geometry of smooth manifolds, which is needed to present the spacetime-based gravitation theory, is developed from scratch. Here, many of the illustrating examples are the Lorentzian manifolds which later serve as spacetime models. This has the twofold purpose of making the physics forthcoming in the second part relatable, and the mathematics learnt in the first part less dry. The book uses the modern coordinate-free language of semi-Riemannian geometry. Nevertheless, to familiarize the reader with the useful tool of coordinates for computations, and to bridge the gap with the physics literature, the link to coordinates is made through exercises, and via frequent remarks on how the two languages are related. In the second part, the focus is on physics, covering essential material of the 20th century spacetime-based view of gravity: energy-momentum tensor field of matter, field equation, spacetime examples, Newtonian approximation, geodesics, tests of the theory, black holes, and cosmological models of the universe. Prior knowledge of differential geometry or physics is not assumed. The book is intended for self-study, and the solutions to the (over 200) exercises are included.
Call Number: QC173.6 .S27 2022
Deformation Theory of Discontinuous Groups by This book contains the latest developments of the theory of discontinuous groups acting on homogenous spaces, from basic concepts to a comprehensive exposition. It develops the newest approaches and methods in the deformation theory of topological modules and unitary representations and focuses on the geometry of discontinuous groups of solvable Lie groups and their compact extensions. It also presents proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and experienced researchers in Lie theory, discontinuous groups, and deformation (and moduli) spaces.
Call Number: QA174.2 .B35 2022
Nonlinear Higher Order Differential and Integral Coupled Systems by Boundary value problems on bounded or unbounded intervals, involving two or more coupled systems of nonlinear differential and integral equations with full nonlinearities, are scarce in the literature. The present work by the authors desires to fill this gap. The systems covered here include differential and integral equations of Hammerstein-type with boundary constraints, on bounded or unbounded intervals. These are presented in several forms and conditions (three points, mixed, with functional dependence, homoclinic and heteroclinic, amongst others). This would be the first time that differential and integral coupled systems are studied systematically. The existence, and in some cases, the localization of the solutions are carried out in Banach space, following several types of arguments and approaches such as Schauders fixed-point theorem or Guo-Krasnoselskii fixed-point theorem in cones, allied to Greens function or its estimates, lower and upper solutions, convenient truncatures, the Nagumo condition presented in different forms, the concept of equiconvergence, Carathéodory functions, and sequences. Moreover, the final part in the volume features some techniques on how to relate differential coupled systems to integral ones, which require less regularity. Parallel to the theoretical explanation of this work, there is a range of practical examples and applications involving real phenomena, focusing on physics, mechanics, biology, forestry, and dynamical systems, which researchers and students will find useful.
Call Number: QA379 .M555 2022
An Introduction to Probability : With Mathematica by The main objective of this text is to facilitate a students smooth learning transition from a course on probability to its applications in various areas. To achieve this goal, students are encouraged to experiment numerically with problems requiring computer solutions.
Call Number: QA273.A5 K36 2022
Amplitudes, Hodge Theory and Ramification : From Periods and Motives to Feynman Amplitudes by This is the first volume of the lectures presented at the Clay Mathematics Institute 2014 Summer School, "Periods and Motives: Feynman amplitudes in the 21st century", which took place at the Instituto de Ciencias Matemáticas-ICMAT (Institute of Mathematical Sciences) in Madrid, Spain. It covers the presentations by S. Bloch, by M. Marcolli and by L. Kindler and K. Rülling. The main topics of these lectures are Feynman integrals and ramification theory. On the Feynman integrals side, their relation with Hodge structures and heights as well as their monodromy are explained in Bloch's lectures. Two constructions of Feynman integrals on configuration spaces are presented in Ceyhan and Marcolli's notes. On the ramification theory side an introduction to the theory of l -adic sheaves with emphasis on their ramification theory is given. These notes will equip the reader with the necessary background knowledge to read current literature on these subjects.
Call Number: QC19.2 .C586 2020
Finding eBooks & eResources
Differential Geometry of Plane Curves by This book features plane curves-the simplest objects in differential geometry-to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem. The last chapter connects the classical with the modern by giving an introduction to the curve-shortening flow that is based on original articles but requires a minimum of previous knowledge. Over 200 figures and more than 100 exercises illustrate the beauty of plane curves and test the reader's skills. Prerequisites are courses in standard one variable calculus and analytic geometry on the plane.
Call Number: QA567 .A437 2022
What are Tensors Exactly? by Tensors have numerous applications in physics and engineering. There is often a fuzzy haze surrounding the concept of tensor that puzzles many students. The old-fashioned definition is difficult to understand because it is not rigorous; the modern definitions are difficult to understand because they are rigorous but at a cost of being more abstract and less intuitive. The goal of this book is to elucidate the concepts in an intuitive way but without loss of rigor, to help students gain deeper understanding. As a result, they will not need to recite those definitions in a parrot-like manner any more.
Call Number: QA200 .G86 2021