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Kilmer S: An Introduction to Point-Set
Topology

An Introduction to Point-Set Topology

Developments in Mathematics

A large mathematical community throughout the world actively works in functional analysis and uses profound techniques from topology.&nbsp;Written by experts in the field, this book is a treasure trove for researchers and graduate students studying the interplay among the areas of point-set and descriptive topology, modern analysis, set theory, topological vector spaces, including Banach spaces, and continuous function spaces. This second edition continues in the same spirit of the acclaimed first edition, providing new insights into the connections between the topological properties of linear function spaces and their applications in functional analysis. It has been expanded by adding completely new Chapters 17-21, presenting results concerning, but not limited to, topological spaces and groups with G-bases, various concepts related to networks and their applications in topology and functional analysis, and those that develop topological and analytic methods related to Grothendieck Banach spaces and Boolean algebras with the Nikodym property.

The book will continue to serve as a reference for present and future work done in this area and could serve as a valuable supplement to advanced graduate courses in functional analysis, set-theoretic topology, or the theory of function spaces.

Updated and Expanded

Descriptive Topology in Selected Topics of Functional Analysis

Princeton Mathematical Series

Princeton Legacy Library

An authoritative introduction to the essential features of étale cohomology A. Grothendieck’s work on algebraic geometry is one of the most important mathematical achievements of the twentieth century. In the early 1960s, he and M. Artin introduced étale cohomology to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry but also in several different branches of number theory and in the representation theory of finite and p-adic groups. In this classic book, James Milne provides an invaluable introduction to étale cohomology, covering the essential features of the theory. Milne begins with a review of the basic properties of flat and étale morphisms and the algebraic fundamental group. He then turns to the basic theory of étale sheaves and elementary étale cohomology, followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Milne proves the fundamental theorems in étale cohomology—those of base change, purity, Poincaré duality, and the Lefschetz trace formula—and applies these theorems to show the rationality of some very general L-series.

Étale Cohomology

Series on Knots & Everything

A classic knot is an embedded simple loop in 3-dimensional space. It can be described as a 4-valent planar graph or network in the horizontal plane, with the vertices or crossings corresponding to double points of a projection. At this stage we have the shadow of the knot defined by the projection. We can reconstruct the knot by lifting the crossings into two points in space, one above the other. This information is preserved at the vertices by cutting the arc which appears to go under the over crossing arc. We can then act on this diagram of the knot using the famous Reidemeister moves to mimic the motion of the knot in space. The result is classic combinatorial knot theory. In recent years, many different types of knot theories have been considered where the information stored at the crossings determines how the Reidemeister moves are used, if at all.

In this book, we look at all these new theories systematically in a way which any third-year undergraduate mathematics student would understand. This book can form the basis of an undergraduate course or as an entry point for a postgraduate studying topology.

Combinatorial Knot Theory

Problem Solving in Mathematics and Beyond

Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The circle, one of the basic aspects of geometry, has a plethora of unexpected curiosities, which the authors present in an easily understandable fashion requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.

The book is intended to be widely appreciated by a general readership, whose love for geometry should be greatly enhanced through exploring these many unexpected relationships. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating circle properties.

Geometric Gems: An Appreciation For Geometric Curiosities - Volume Iii: The Wonders Of Circles

Get the whole story on fractions Fractions are one of the toughest concepts to master for young mathematicians. Fractions For Dummies makes it easier, with clear explanations, step-by-step solutions, and plenty of example problems-including word problems and tips on how to solve them. You'll also explore the world of decimals and percentages, and you'll learn how all these concepts are related. Students who understand fractions can cook, budget, tip at restaurants, and succeed in their future math learning. Make fractions a piece of pie with this Dummies guide! Inside... <ul><li>Basic math operations with fractions</li> <li>Clear, easy explanations</li> <li>Tips for solving word problems</li> <li>Convert between fractions, decimals, and percents</li> <li>Working with mixed numbers</li> <li>New fraction strategies being taught in schools</li></ul>

Fractions For Dummies

This book covers computational group theory (CGT), an active, rapidly expanding field at the interface of mathematics and computer science. It is the first systematic effort to establish computing with infinite matrix groups on the same footing as the rest of CGT. The book emphasizes practical methods based on results of linear group theory, solutions of fundamental algorithmic problems, and provision of software. Additionally, the book sets out a research program to make further progress in the area.

This book covers computational group theory (CGT), an active, rapidly expanding field at the interface of mathematics and computer science. It is the first systematic effort to establish computing with infinite matrix groups on the same footing as the rest of CGT. The book emphasizes practical methods based on results of linear group theory, solutions of fundamental algorithmic problems, and provision of software. Additionally, the book sets out a research program to make further progress in the area.

Linear Groups and Computation

Textbooks in Mathematics

This text focuses on the primary topics in a first course in Linear Algebra including additional advanced topics related to data analysis, singular value decomposition and connections to differential equations. This is a lab text that would lead a class through Linear Algebra using Mathematica demonstrations and Mathematica coding

This text focuses on the primary topics in a first course in Linear Algebra including additional advanced topics related to data analysis, singular value decomposition and connections to differential equations. This is a lab text that would lead a class through Linear Algebra using Mathematica demonstrations and Mathematica coding

Labs and Projects with Mathematica ®

Exploring Linear Algebra

This book deals with the problem of bounding numerical invariants of nonsingular projective algebraic varieties from a unified point of view. Starting with a new insightful proof of the Castelnuovo theorem, the author proceeds with giving (sharp) bounds for basic numerical invariants (such as Betti, Hodge, and Chern numbers) of algebraic varieties of arbitrary dimension and classifying the varieties on the boundary. Many important results are proved in several different ways underlining different aspects of the topic. However, studying numerical invariants is just a pretext for the exploration of the rich and versatile interplay between geometry and topology of projective algebraic varieties which forms the core of the book. A special role in this study is played by the dual varieties and their degrees classically called classes. While making an extensive use of Lefschetz theory (which is now also available for varieties over an algebraically closed field of positive characteristic), the author also develops a new variant of Morse theory which yields statements (including bounds) on the level of CW-complexes and provides bounds for the homology of real algebraic varieties. In this way one also gets stronger forms of some classical results on the topology of projective varieties, such as weak and hard Lefschetz theorems, Hopf type formula etc. A detailed outline of the exciting centennial history of this topic and a list of intriguing open problems contribute to the value of the book. The exposition is accessible to students with only basic knowledge of algebraic geometry and topology.

Bounding Numerical Invariants Of Algebraic Varieties

This presents a new topology algorithm for structural optimization called Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT). It presents the basic theory of SEMDOT, explains its connections with the corresponding optimizers, and uses it to address the jagged edge problem facing classical topology optimization algorithms.

Smooth Topological Design of Continuum Structures focuses on the use of a newly-proposed topology algorithm for structural optimization called Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT).The book presents the basic theory of SEMDOT and explains connections between this method and the corresponding optimizers. The method is used to address the long-standing jagged edge problem facing classical topology optimization algorithms and is presented as a perfect tool for combining additive manufacturing and topology optimization, which are increasingly coupled together to produce new part designs. A range of representative case studies are also included to illustrate applications.This serves as a textbook and reference for graduate and senior undergraduate students in the area, as well as engineers in the structural optimization field. Full modifiable MATLAB codes for each chapter are available online.

Smooth Topological Design of Continuum Structures

Annals of Mathematics Studies

A comprehensive new theory of adelic line bundles on quasi-projective varieties over finitely generated fieldsThis book introduces a comprehensive theory of adelic line bundles on quasi-projective varieties over finitely generated fields, developed in both geometric and arithmetic contexts. In the geometric setting, adelic line bundles are defined as limits of line bundles on projective compactifications under the boundary topology. In the arithmetic setting, they are defined as limits of Hermitian line bundles on projective arithmetic compactifications, also under the boundary topology. After establishing these foundational definitions, the book uses the theory to explore key concepts such as intersection theory, effective sections, volumes, and positivity of adelic line bundles. It also applies these results to study height functions of algebraic points and prove an equidistribution theorem on quasi-projective varieties. This theory has broad applications in the study of numerical, dynamical, and Diophantine properties of moduli spaces, quasi-projective varieties, and varieties over finitely generated fields.

Adelic Line Bundles on Quasi-projective Varieties

An Introduction to Point-Set Topology

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