The Continuum

A Constructive Approach to Basic Concepts of Real Analysis

Author: Rudolf Taschner

Publisher: Springer Science & Business Media

ISBN: 332282036X

Category: Mathematics

Page: 136

View: 4221


In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the famous theorems of Brouwer concerning the continuity of all functions that are defined on "whole" intervals, the uniform continuity of all functions that are defined on compact intervals, and the uniform convergence of all pointwise converging sequences of functions defined on compact intervals. The constructive approach is interesting both in itself and as a contrast to, for example, the formal axiomatic one.

Real Analysis

A Constructive Approach

Author: Mark Bridger

Publisher: John Wiley & Sons

ISBN: 1118031563

Category: Mathematics

Page: 320

View: 9603


A unique approach to analysis that lets you apply mathematics across a range of subjects This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences. The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes: Early use of the Completeness Theorem to prove a helpful Inverse Function Theorem Sequences, limits and series, and the careful derivation of formulas and estimates for important functions Emphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsets Construction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integrals Differentiation, emphasizing the derivative as a function rather than a pointwise limit Properties of sequences and series of continuous and differentiable functions Fourier series and an introduction to more advanced ideas in functional analysis Examples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging. This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.

Sets and Extensions in the Twentieth Century

Author: N.A

Publisher: Elsevier

ISBN: 0080930662

Category: Mathematics

Page: 880

View: 2116


Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights

The Higher Infinite

Large Cardinals in Set Theory from Their Beginnings

Author: Akihiro Kanamori

Publisher: Springer Science & Business Media

ISBN: 3540888667

Category: Mathematics

Page: 538

View: 4869


Over the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.

Celebrating Perfection in Education

Dawn of Total Knowledge

Author: N.A

Publisher: N.A


Category: Education

Page: 196

View: 1958


The Field Of Total Knowledge At The Basis Of All Disciplines Is The Foundation For Celebrating The Availability Of Total Knowledge For Every Student.

Encyclopedia of Philosophy: Cabanis - Destutt de Tracy

Author: Donald M. Borchert

Publisher: Thomson Gale/MacMillan Reference USA

ISBN: 9780028657820

Category: Philosophy

Page: 10

View: 1728


Containing material from hundreds of highly distinguished contributors representing the world's top universities and institutions, the second edition has a truly global perspective. It contains more than 2,100 entries -- including more than 450 new articles. Among the many topics covered are African, Islamic, Jewish, Russian, Chinese, and Buddhist philosophies; bioethics and biomedical ethics; art and aesthetics; epistemology; metaphysics; peace and war; social and political philosophy; the Holocaust; feminist thought; and much more. Additionally, the second edition also features 1,000 biographical entries on major figures in philosophical thought throughout history.

Interpreting Godel

Critical Essays

Author: Juliette Kennedy

Publisher: Cambridge University Press

ISBN: 1107002664

Category: Science

Page: 288

View: 7940


In this groundbreaking volume, leading philosophers and mathematicians explore Kurt Gödel's work on the foundations and philosophy of mathematics.

A Primer of Infinitesimal Analysis

Author: John L. Bell

Publisher: Cambridge University Press

ISBN: 0521887186

Category: Mathematics

Page: 124

View: 7547


A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.


A Classified Cumulation : Volumes 1-10, March 1964--February 1974

Author: Richard K. Gardner,Phyllis Grumm

Publisher: N.A


Category: Best books

Page: N.A

View: 9770


Handbook of Analysis and Its Foundations

Author: Eric Schechter

Publisher: Academic Press

ISBN: 9780080532998

Category: Mathematics

Page: 883

View: 1429


Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. Covers some hard-to-find results including: Bessagas and Meyers converses of the Contraction Fixed Point Theorem Redefinition of subnets by Aarnes and Andenaes Ghermans characterization of topological convergences Neumanns nonlinear Closed Graph Theorem van Maarens geometry-free version of Sperners Lemma Includes a few advanced topics in functional analysis Features all areas of the foundations of analysis except geometry Combines material usually found in many different sources, making this unified treatment more convenient for the user Has its own webpage:

What is Mathematics?

An Elementary Approach to Ideas and Methods

Author: Richard Courant,Herbert Robbins,Ian Stewart

Publisher: Oxford University Press, USA

ISBN: 9780195105193

Category: Mathematics

Page: 566

View: 8854


A discussion of fundamental mathematical principles from algebra to elementary calculus designed to promote constructive mathematical reasoning.

Balancing Agility and Discipline

A Guide for the Perplexed, Portable Documents

Author: Barry Boehm,Richard Turner

Publisher: Addison-Wesley Professional

ISBN: 9780321623881

Category: Computers

Page: 304

View: 6379


Agility and discipline: These apparently opposite attributes are, in fact, complementary values in software development. Plan-driven developers must also be agile; nimble developers must also be disciplined. The key to success is finding the right balance between the two, which will vary from project to project according to the circumstances and risks involved. Developers, pulled toward opposite ends by impassioned arguments, ultimately must learn how to give each value its due in their particular situations. Balancing Agility and Discipline sweeps aside the rhetoric, drills down to the operational core concepts, and presents a constructive approach to defining a balanced software development strategy. The authors expose the bureaucracy and stagnation that mark discipline without agility, and liken agility without discipline to unbridled and fruitless enthusiasm. Using a day in the life of two development teams and ground-breaking case studies, they illustrate the differences and similarities between agile and plan-driven methods, and show that the best development strategies have ways to combine both attributes. Their analysis is both objective and grounded, leading finally to clear and practical guidance for all software professionals--showing how to locate the sweet spot on the agility-discipline continuum for any given project.