ISBN 9781032260372
June 9, 2023 Forthcoming
240 Pages 199 Color Illustrations
Basics of Ramsey Theory serves as a gentle introduction to Ramsey theory for students interested in becoming familiar with a dynamic segment of contemporary mathematics that combines ideas from number theory and combinatorics. The core of the of the book consists of discussions and proofs of the results now universally known as Ramsey’s theorem, van der Waerden’s theorem, Schur’s theorem, Rado’s theorem, the Hales?Jewett theorem, and the Happy End Problem of Erd?s and Szekeres. The aim is to present these in a manner that will be challenging but enjoyable, and broadly accessible to anyone with a genuine interest in mathematics.
Suitable for any undergraduate student who has successfully completed the standard calculus sequence of courses and a standard first (or second) year linear algebra course.
Filled with visual proofs of fundamental theorems.
Contains numerous exercises (with their solutions) accessible to undergraduate students.
Serves as both a textbook or as a supplementary text in an elective course in combinatorics and aimed at a diverse group of students interested in mathematics.
1. Introduction: Pioneers and Trailblazers. 1.1. Complete Disorder is Impossible. 1.2 Paul Erd?s. 1.3. Frank Plumpton Ramsey. 1.4 Ramsey Theory. 2. Ramsey’s Theorem. 2.1. The Pigeonhole Principle. 2.2. Acquaintances and Strangers. 2.3. Ramsey’s Theorem for Graphs. 2.4. Ramsey’s Theorem: Infinite Case. 2.5. Ramsey’s Theorem: General Case. 2.6. Exercises. 3. van der Waerden’s Theorem. 3.1. Bartel van der Waerden. 3.2. van der Waerden’s Theorem: 3?Term Arithmetic Progressions. 3.3. Proof of van der Waerden’s Theorem. 3.4. van der Waerden’s Theorem: How Far and Where? 3.5. van der Waerden’s Theorem: Some Related Questions. 3.6. Exercises. 4. Schur’s Theorem and Rado’s Theorem. 4.1 Issai Schur. 4.2. Schur’s Theorem. 4.3. Richard Rado. 4.4 Rado’s Theorem. 4.5. Exercises. 5. The Hales?Jewett Theorem. 5.1. Combinatorial Lines. 5.2. Generalized Tic?Tac?Toe Game. 5.3. The Hales?Jewett Theorem. 5.4. Exercises. 6. Happy End Problem. 6.1. The Happy End Problem: Triangles, Quadrilaterals, and Pentagons. 6.2. The Happy End Problem ? General Case. 6.3. Erd?s?Szekeres’ Upper and Lower Bounds. 6.4. Progress on the Conjecture OF Erd?s and Szekeres. 6.5. Exercises. 7. Solutions.
ISBN 9780367208264
July 3, 2023 Forthcoming
328 Pages 8 B/W Illustrations
Model theory is the meta-mathematical study of the concept of mathematical truth. After Afred Tarski coined the term Theory of Models in the early 1950’s, it rapidly became one of the central most active branches of mathematical logic. In the last few decades, ideas that originated within model theory have provided powerful tools to solve problems in a variety of areas of classical mathematics, including algebra, combinatorics, geometry, number theory, and Banach space theory and operator theory.
The two volumes of Beyond First Order Model Theory present the reader with a fairly comprehensive vista, rich in width and depth, of some of the most active areas of contemporary research in model theory beyond the realm of the classical first-order viewpoint. Each chapter is intended to serve both as an introduction to a current direction in model theory and as a presentation of results that are not available elsewhere. All the articles are written so that they can be studied independently of one another.
This second volume contains introductions to real-valued logic and applications, abstract elementary classes and applications, interconnections between model theory and function spaces, nonstucture theory, and model theory of second-order logic.
A coherent introduction to current trends in model theory.
Contains articles by some of the most influential logicians of the last hundred years. No other publication brings these distinguished authors together.
Suitable as a reference for advanced undergraduate, postgraduates, and researchers.
Material presented in the book (e.g, abstract elementary classes, first-order logics with dependent sorts, and applications of infinitary logics in set theory) is not easily accessible in the current literature.
The various chapters in the book can be studied independently.
I. Real-Valued Structures and Applications.
Chapter 1. Metastable Convergence and Logical Compactness.
Xavier Caicedo, Eduardo Duenez, and Jose Iovino
Chapter 2. Model Theory for Real-Valued Structures
H. Jerome Keisler
Chapter 3. Spectral Gap and Definability
Isaac Goldbring
II. Abstract Elementary Classes and Applications
Chapter 4. Lf Groups, AEC Amalgamation, Few Automorphisms
Saharon Shelah
III. Model Theory and Topology of Spaces of Functions
Chapter 5. Cp-Theory for Model Theorists
Clovis Hamel and Franklin D. Tall
IV. Constructing Many Models
Chapter 6. General Non-Structure Theory.
Saharon Shelah
V. Model Theory of Second Order Logic
Chapter 7. Model Theory of Second Order Logic
Jouko Vaananen
This book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.
He has published more than 80 papers in computer and intelligent systems (see my DBLP record https://dblp.org/pid/66/4092.html ), and two books in Computer Science: 1. Soft Computing Applications in Industry, Copyright c 2008, Springer-Verlag , 386p. ISBN 978-3-540-77465-5, 2. Big Data Integration Theory, Copyright c 2014, XX, Springer-Verlag , 516p. ISBN 978-3-319-04155-1 February 10, 2014 This second book obtained in 2021 four achievements 1. Best book in Database Theory of all times, 2. Best Big Data book of all times 3. Best Computer Science e-Book of all times 4. Best Database Schema book of all times More about my research and academic activities and publishing you can find at my Homepage https://zoranmajkic.webs.com/
Artificial Intelligence
Computer Sciences
Information Technology
Logic and Set Theory
Mathematics
Theoretical Computer Sciences
In the series De Gruyter Proceedings in Mathematics
This proceedings volume documents the contributions presented at the CONIAPS XXVII international Conference on Recent Advances in Pure and Applied Algebra. The entries focus on modern trends and techniques in various branches of pure and applied Algebra and highlight their applications in coding theory, cryptography, graph theory, and fuzzy theory.
A selection of up-to-date contributions in pure and applied algebra and applicable applications in coding theory, cryptography, graph theory, and fuzzy theory.
Of interest to mathematicians working in pure and applied algebra and related areas
Author information
Manoj Kumar Patel, NIT, Nagaland, India; Ratnesh Kumar Mishra, NIT, Jamshedpur, India; Shiv Datt Kumar, NIT, Allahabad, India.
Algebra and Number Theory
Analysis
Applied Mathematics
Mathematics
First unified study of the relationship between syllogistic logic and mathematical proof from antiquity to the middle of the nineteenth century
Advances the scholarly debate in an important area in the history of logic and mathematics
Provides a systematic analysis of attempts to put mathematical proof in syllogistic form
Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics.
Introduction
1:Aristotelian Syllogism and Mathematics in Antiquity and the Medieval Period
2:Extensions of the Syllogism in Medieval Logic
3:Syllogistic and Mathematics: The Case of Piccolomini
4:Obliquities and Mathematics in the 17th and 18th Centuries: From Jungius to Wolff
5:The Extent of Syllogistic Reasoning: From Rudiger to Wolff
6:Lambert and Kant
7:Bernard Bolzano on Non-Syllogistic Reasoning
8:Thomas Reid, William Hamilton and Augustus De Morgan
Conclusion
For centuries, mathematicians the world over have tried, and failed, to solve the zeta-3 problem. Math genius Leonhard Euler attempted it in the 1700s and came up short. The straightforward puzzle considers if there exists a simple symbolic formula for the following: 1+(1/2)^3+(1/3)^3+(1/4)^3+… . But why is this issue?the sum of the reciprocals of the positive integers cubed?so important? With In Pursuit of Zeta-3, popular math writer Paul Nahin investigates the history and significance of this mathematical conundrum.
Drawing on detailed examples, historical anecdotes, and even occasionally poetry, Nahin sheds light on the richness of the nature of zeta-3. He shows its intimate connections to the Riemann hypothesis, another mathematical mystery that has stumped mathematicians for nearly two centuries. He looks at its links with Euler’s achievements and explores the modern research area of Euler sums, where zeta-3 occurs frequently. An exact solution to the zeta-3 question wouldn’t simply satisfy pure mathematical interest: it would have critical ramifications for applications in physics and engineering, such as quantum electrodynamics. Challenge problems with detailed solutions and MATLAB code are included at the end of each of the book’s sections.
Detailing the trials and tribulations of mathematicians who have approached one of the field’s great unsolved riddles, In Pursuit of Zeta-3 will tantalize curious math enthusiasts everywhere.