Paperback ISBN: 9780323990202
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and how they work. The book offers techniques for both reading and writing proofs, discusses techniques in proving if/then statements by contrapositive and proofing by contradiction, includes the negation statement, and/or, examines various theorems, such as the if and only-if, equivalence theorems, existence theorems, and the uniqueness theorems. In addition, the use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are also covered. The book also provides mathematical topics for practicing proof techniques.
Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book accessible as well as invaluable.
1. Getting Started
2. Basic Techniques to Prove If/Then Statements
3. Special Kinds of Theorems
4. Some Mathematical Topics on Which to Practice Proof Techniques
5. Review Exercises
ISBN 9780367549664
Published January 9, 2023 by Chapman & Hall
281 Pages 25 B/W Illustrations
Typically, undergraduates see real analysis as one of the most di?cult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to ?nd a way to bridge the gap between a studentfs preparation and the mathematical skills that are required to be successful in such a course.
Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies o?ered in this textbook will be appreciated by all readers.
Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis
Suitable for junior or senior undergraduates majoring in mathematics.
1. Proofs, Sets, Functions, and Induction. 1.1. Proofs. 1.2. Sets. 1.3. Functions. 1.4. Mathematical Induction. 2. The Real Numbers. 2.1. Introduction. 2.2. R is an Ordered Field. 2.3 The Completeness Axiom. 2.4. The Archimedean Property. 2.5. Nested Intervals Theorem. 3. Sequences. 3.1 Convergence. 3.2 Limit Theorems for Sequences. 3.3. Subsequences. 3.4. Monotone Sequences. 3.5. Bolzano?Weierstrass Theorems. 3.6. Cauchy Sequences. 3.7. Infinite Limits. 3.8. Limit Superior and Limit Inferior. 4. Continuity. 4.1. Continuous Functions. 4.2. Continuity and Sequences. 4.3. Limits 0f Functions. 4.4. Consequences 0f Continuity. 4.5 Uniform Continuity. 5. Differentiation. 5.1. The Derivative. 5.2. The Mean Value Theorem. 5.3. Taylorfs Theorem. 6. _ Riemann Integration. 6.1. The Riemann Integral. 6.2. Properties of The Riemann Integral. 6.3. Families of Integrable Functions. 6.4. The Fundamental Theorem of Calculus. 7. Infinite Series. 7.1. Convergence and Divergence. 7.2 Convergence Tests. 7.3. Regrouping and Rearranging Terms of a Series. 8. Sequences and Series of Functions. 8.1 Pointwise and Uniform Convergence. 8.2. Preservation Theorems. 8.3. Power Series. 8.4. Taylor Series. Appendix A: Proof of the Composition Theorem. Appendix B: Topology on the Real Numbers. Appendix C: Review of Proof and Logic.
In the series De Gruyter Textbook
The definitive guide to the game-theoretic and probabilistic underpinning for Bitcoinfs security model. The book begins with an overview of probability and game theory. Nakamoto Consensus is discussed in both practical and theoretical terms.
Describes attacks and exploits with mathematical justifications, including selfish mining.
Identifies common assumptions such as the Market Fragility Hypothesis, establishing a framework for analyzing incentives to attack.
Outlines the block reward schedule and economics of ASIC mining.
Discusses how adoption by institutions would fundamentally change the security model.
Analyzes incentives for double-spend and sabotage attacks via stock-flow models.
Overviews coalitional game theory with applications to majority takeover attacks
Presents Nash bargaining with application to unregulated environments
This book is intended for students or researchers wanting to engage in a serious conversation about the future viability of Bitcoin as a decentralized, censorship-resistant, peer-to-peer electronic cash system. @
Explains the game theoretic underpinning of Bitcoin from inception to large-scale adoption. Combines basics of game theory, probability and the Proof-of-Work protocol. Includes end of chapter exercises, and python code snippets.
Analysis
Applied Mathematics
Business and Economics
Mathematics
Mathematics
Mathematics and Statistics for Economists
Probability and Statistics
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Volume 41 in the series De Gruyter Series in Nonlinear Analysis and Applications
This work is devoted to fixed point theory as well as the theory of accretive operators in Banach spaces. The goal is to develop, in self-contained way, the main results in both theories. Special emphasis is given to the study how both theories can be used to study the existence and uniqueness of solution of several types of partial differential equations and integral equations.
Clear and detailed writing style
Blends theory and applications
Discusses the latest techniques and applications pertinent to evolution equations and boundary value problems; accretive operators and fixed-point theory
Jesus Garcia-Falset, University of Valencia, ES; Khalid Latrach, Universite Clermont Auvergne, FR.
Analysis
Differential Equations and Dynamical Systems
Mathematics
Volume 91 in the series De Gruyter Studies in Mathematics
This is an indispensable reference for those mathematicians that conduct research activity in applications of fixed-point theory to boundary value problems for nonlinear functional equations. Coverage includes second-order finite difference equations and systems of difference equations subject to multi-point boundary conditions, various methods to study the existence of positive solutions for difference equations, and Green functions.
Presents second-order finite difference equations and systems of difference equations subject to multi-point boundary conditions.
Covers Green functions and the construction used in positive solutions.
Authors are recognized experts.
Johnny Henderson, Baylor University, USA; Rodica Luca, Technical University of Iasi, RO.
Differential Equations and Dynamical Systems
Mathematics