Concentrates on separation axioms in localic topology
Provides an in-depth discussion on axioms of Hausdorff's type
Written in a pleasant style
This book is the first systematic treatment of this area so far scattered in a vast number of
articles. As in classical topology, concrete problems require restricting the (generalized pointfree) spaces
by various conditions playing the roles of classical separation axioms. These are
typically formulated in the language of points; but in the point-free context one has either
suitable translations, parallels, or satisfactory replacements. The interrelations of separation
type conditions, their merits, advantages and disadvantages, and consequences are discussed.
Highlights of the book include a treatment of the merits and consequences of subfitness,
various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of
the separation conditions put them in a new perspective, and, a.o., gave some of them
unexpected importance. The text contains a lot of quite recent results; the reader will see the
directions the area is taking, and may find inspiration for her/his further work. The book will be
of use for researchers already active in the area, but also for those interested in this growing
field (sometimes even penetrating into some parts of theoretical computer science), for
graduate and PhD students, and others. For the reader's convenience, the text is supplemented
with an Appendix containing necessary background on posets, frames and locales.
Mathematics : Order, Lattices, Ordered Algebraic Structures
Due 2020-12-15
1st ed. 2020, X, 330 p. 13 illus.
Hardcover
ISBN 978-3-030-53478-3
Product category : Monograph
Provides a comprehensive, self-contained introduction to decidability of firstorder theories,
using detailed proofs and examples to illustrate and clarify complex concepts
Incorporates computability theory and reduction techniques to determine the
decidability of theories
Illustrates a variety of ways to deduce logical consequences from a theory,
including the use of Gentzen calculus for first-order logic
This textbook provides a self-contained introduction to decidability of first-order theories and
their combination. The technical material is presented in a systematic and universal way and
illustrated with plenty of examples and a range of proposed exercises. After an overview of
basic first-order logic concepts, the authors discuss some model-theoretic notions like
embeddings, diagrams, and elementary substructures. The text then goes on to explore an
applicable way to deduce logical consequences from a given theory and presents sufficient
conditions for a theory to be decidable. The chapters that follow focus on quantifier
elimination, decidability of the combination of first-order theories and the basics of
computability theory. The inclusion of a chapter on Gentzen calculus, cut elimination, and Craig
interpolation, as well as a chapter on combination of theories and preservation of decidability,
help to set this volume apart from similar books in the field. Decidability of Logical Theories
and their Combinationis ideal for graduate students of Mathematics and is equally suitable for
Computer Science, Philosophy and Physics students who are interested in gaining a deeper
understanding of the subject. The book is also directed to researchers that intend to get
acquainted with first-order theories and their combination.
Mathematics : Mathematical Logic and Foundations
Due 2020-11-01
1st ed. 2020, XIII, 178 p. 5 illus.
Hardcover
ISBN 978-3-030-56553-4
Product category : Graduate/advanced undergraduate textbook
Series : Studies in Universal Logic
This book presents 30 articles on the topic areas discussed at the 30th gInternational
Workshop on Operator Theory and its Applicationsh, held in Lisbon in July 2019. The
contributions include both expository essays and original research papers reflecting recent
advances in the traditional IWOTA areas and emerging adjacent fields, as well as the
applications of Operator Theory and Functional Analysis. The topics range from C*?algebras
and Banach *?algebras, Sturm-Liouville theory, integrable systems, dilation theory, frame
theory, Toeplitz, Hankel, and singular integral operators, to questions from lattice, group and
matrix theories, complex analysis, harmonic analysis, and function spaces. Given its scope, the
book is chiefly intended for researchers and graduate students in the areas of Operator Theory,
Functional Analysis, their applications and adjacent fields.
Mathematics : Operator Theory
Due 2021-01-04
1st ed. 2020, X, 490 p. 23 illus., 10 illus. in color.
Hardcover
ISBN 978-3-030-51944-5
Product category : Proceedings
Series : Operator Theory: Advances and Applications
https://doi.org/10.1142/11870 | September 2020
Pages: 320
ISBN: 978-981-122-183-5 (hardcover)
This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses.
An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer.
Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work.
The first three chapters develop the basic theory of elliptic curves.
For this edition, the text has been completely revised and updated.
Algebraic Curves
Basic Theory of Elliptic Curves
Elliptic Curves over the Complex Numbers
The Arithmetic of Elliptic Curves
Elliptic Curves and Modular Forms
Graduate students and lecturers in mathematics, specifically number theory
courses.