This book gives a systematic presentation of real algebraic varieties.
Real algebraic varieties are ubiquitous.They are the first objects encountered when learning of coordinates, then equations, but the systematic study of these objects, however elementary they may be, is formidable.
This book is intended for two kinds of audiences: it accompanies the reader, familiar with algebra and geometry at the masters level, in learning the basics of this rich theory, as much as it brings to the most advanced reader many fundamental results often missing from the available literature, the gfolkloreh. In particular, the introduction of topological methods of the theory to non-specialists is one of the original features of the book.
The first three chapters introduce the basis and classical methods of real and complex algebraic geometry. The last three chapters each focus on one more specific aspect of real algebraic varieties. A panorama of classical knowledge is presented, as well as major developments of the last twenty years in the topology and geometry of varieties of dimension two and three, without forgetting curves, the central subject of Hilbert's famous sixteenth problem.
Various levels of exercises are given, and the solutions of many of them are provided at the end of each chapter.
Related Subjects:
Mathematics and Computing, Mathematics, Algebra, Algebraic Geometry, Topology, Manifolds and Cell Complexes, Analysis, Several Complex Variables and Analytic Spaces
Delivery Status: Available
Subject: Mathematics and Statistics
Copyright Year: 2020
ISBN: 978-3-030-43106-8
Format:Book: Generic (Soft cover)
Product Category:Monographs
Paris of the year 1900 left two landmarks: the Tour Eiffel, and David Hilbert's celebrated list of twenty-four mathematical problems presented at a conference opening the new century. Kurt Godel, a logical icon of that time, showed Hilbert's ideal of complete axiomatization of mathematics to be unattainable. The result, of 1931, is called Godel's incompleteness theorem. Godel then went on to attack Hilbert's first and second Paris problems, namely Cantor's continuum problem about the type of infinity of the real numbers, and the freedom from contradiction of the theory of real numbers. By 1963, it became clear that Hilbert's first question could not be answered by any known means, half of the credit of this seeming faux pas going to Godel. The second is a problem still wide open. Godel worked on it for years, with no definitive results; The best he could offer was a start with the arithmetic of the entire numbers.
This book, Godel's lectures at the famous Princeton Institute for Advanced Study in 1941, shows how far he had come with Hilbert's second problem, namely to a theory of computable functionals of finite type and a proof of the consistency of ordinary arithmetic. It offers indispensable reading for logicians, mathematicians, and computer scientists interested in foundational questions. It will form a basis for further investigations into Godel's vast Nachlass of unpublished notes on how to extend the results of his lectures to the theory of real numbers. The book also gives insights into the conceptual and formal work that is needed for the solution of profound scientific questions, by one of the central figures of 20th century science and philosophy.
Related Subjects:
Mathematics and Computing, Mathematics, History of Mathematical Sciences, Humanities and Social Sciences, Philosophy, Logic, History of Philosophy
Delivery Status: In production
Subject: Mathematics and Statistics
Copyright Year:2021
ISBN: 978-3-030-87295-3
Format: Book: Generic (Hard cover)
Product Category: Monographs
This first of the three-volume book is targeted as a basic course in topology for undergraduate and graduate students of mathematics. It studies metric spaces and general topology. It starts with the concept of the metric which is an abstraction of distance in the Euclidean space. The special structure of a metric space induces a topology that leads to many applications of topology in modern analysis and modern algebra, as shown in this volume. This volume also studies topological properties such as compactness and connectedness. Considering the importance of compactness in mathematics, this study covers the Stone?Cech compactification and Alexandroff one-point compactification. This volume also includes the Urysohn lemma, Urysohn metrization theorem, Tietz extension theorem, and Gelfand?Kolmogoroff theorem.
The content of this volume is spread into eight chapters of which the last chapter conveys the history of metric spaces and the history of the emergence of the concepts leading to the development of topology as a subject with their motivations with an emphasis on general topology. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power, and active learning of the subject, all the while covering a wide range of theories and applications in a balanced unified way.
Related Subjects:
Mathematics and Computing, Mathematics, Topology, Analysis
Subject: Mathematics and Statistics
Copyright Year: 2021
ISBN: 978-981-16-6508-0
Format: Book: Generic (Hard cover)
Product Category: Textbooks
This book gives an overview of research on graphs associated with commutative rings. The study of the connections between algebraic structures and certain graphs, especially finite groups and their Cayley graphs, is a classical subject which has attracted a lot of interest. More recently, attention has focused on graphs constructed from commutative rings, a field of study which has generated an extensive amount of research over the last three decades. The aim of this text is to consolidate this large body of work into a single volume, with the intention of encouraging interdisciplinary research between algebraists and graph theorists, using the tools of one subject to solve the problems of the other. The topics covered include the graphical and topological properties of zero-divisor graphs, total graphs and their transformations, and other graphs associated with rings.
The book will be of interest to researchers in commutative algebra and graph theory and anyone interested in learning about the connections between these two subjects.
Related Subjects:
Mathematics and Computing, Mathematics, Algebra, Commutative Rings and Algebras, Discrete Mathematics, Graph Theory
Delivery Status: Available
Subject: Mathematics and Statistics
Copyright Year: 2021
ISBN: 978-3-030-88409-3
Format: Book: Generic (Hard cover)
Product Category: Monographs
This book presents a thorough discussion of the theory of abstract inverse linear problems on Hilbert space. Given an unknown vector f in a Hilbert space H, a linear operator A acting on H, and a vector g in H satisfying Af=g, one is interested in approximating f by finite linear combinations of g, Ag, A2g, A3g, c The closed subspace generated by the latter vectors is called the Krylov subspace of H generated by g and A. The possibility of solving this inverse problem by means of projection methods on the Krylov subspace is the main focus of this text.
After giving a broad introduction to the subject, examples and counterexamples of Krylov-solvable and non-solvable inverse problems are provided, together with results on uniqueness of solutions, classes of operators inducing Krylov-solvable inverse problems, and the behaviour of Krylov subspaces under small perturbations. An appendix collects material on weaker convergence phenomena in general projection methods.
This subject of this book lies at the boundary of functional analysis/operator theory and numerical analysis/approximation theory and will be of interest to graduate students and researchers in any of these fields.
Related Subjects:
Mathematics and Computing, Mathematics, Analysis, Differential Equations, Functional Analysis, Computational Mathematics and Numerical Analysis, Numerical Analysis, Operator Theory
Subject: Mathematics and Statistics
Copyright Year: 2022
ISBN: 978-3-030-88158-0
Format: Book: Generic (Hard cover)
Product Category: Monographs