ISBN 9781138361409
March 17, 2021 Forthcoming
437 Pages 521 B/W Illustrations
Format : Hardback
Graph theory is the study of interactions, conflicts, and connections. The relationship between collections of discrete objects can inform us about the overall network in which they reside, and graph theory can provide an avenue for analysis.
This text, for the first undergraduate course, will explore major topics in graph theory from both a theoretical and applied viewpoint. Topics will progress from understanding basic terminology, to addressing computational questions, and finally ending with broad theoretical results. Examples and exercises will guide the reader through this progression, with particular care in strengthening proof techniques and written mathematical explanations.
Current applications and exploratory exercises are provided to further the readerfs mathematical reasoning and understanding of the relevance of graph theory to the modern world.
The first chapter introduces graph terminology, mathematical modeling using graphs, and a review of proof techniques featured throughout the book
The second chapter investigates three major route problems: eulerian circuits, hamiltonian cycles, and shortest paths.
The third chapter focuses entirely on trees ? terminology, applications, and theory.
Four additional chapters focus around a major graph concept: connectivity, matching, coloring, and planarity. Each chapter brings in a modern application or approach.
Hints and Solutions to selected exercises provided at the back of the book.
Author
Karin R. Saoub is an Associate Professor of Mathematics at Roanoke College in Salem, Virginia. She earned her PhD in mathematics from Arizona State University and BA from Wellesley College. Her research focuses on graph coloring and on-line algorithms applied to tolerance graphs. She is also the author of A Tour Through Graph Theory, published by CRC Press.
Chapter 1: Graph Models, Terminology, and Proofs
Chapter 2: Graph Routes
Chapter 3: Trees
Chapter 4: Connectivity and Flow
Chapter 5: Matching and Factors
Chapter 6: Graph Coloring
Chapter 7: Planarity
Appendix
Selected Hints and Solutions
Dr. Karin R. Saoub is an Associate Professor of Mathematics at Roanoke College in Salem, Virginia. She received her PhD in Mathematics from Arizona State University and a Bachelor of Arts degree from Wellesley College. Her research focuses on graph coloring and on-line algorithms applied to tolerance graphs. She is also the author of A Tour Through Graph Theory, published by CRC Press.
ISBN 9780367483883
March 31, 2021 Forthcoming
276 Pages
Format : Hardback
Outside of randomized experiments, association does not imply causation, and yet there is nothing defective about our knowledge that smoking causes lung cancer, a conclusion reached in the absence of randomized experimentation with humans. How is that possible? If observed associations do not identify causal effects in observational studies, how can a sequence of such associations become decisive?
Two or more associations may each be susceptible to unmeasured biases, yet not susceptible to the same biases. An observational study has two evidence factors if it provides two comparisons susceptible to different biases that may be combined as if from independent studies of different data by different investigators, despite using the same data twice. If the two factors concur, then they may exhibit greater insensitivity to unmeasured biases than either factor exhibits on its own.
Replication and Evidence Factors in Observational Studies includes four parts:
A concise introduction to causal inference, making the book self-contained
Practical examples of evidence factors from the health and social sciences with analyses in R
The theory of evidence factors
Study design with evidence factors
A companion R package evident is available from CRAN.
ISBN 9780367740290
June 14, 2021 Forthcoming
456 Pages 56 B/W Illustrations
Format : Hardback
Level-Crossing Problems and Inverse Gaussian Distributions: Closed-Form Results and Approximations focusses on inverse Gaussian approximation for the distribution of the first level-crossing time in a shifted compound renewal process framework. This approximation, whose name was coined by the author, is a successful competitor of the normal (or Cramer's), diffusion, and Teugels' approximations, being a breakthrough in its conditions and accuracy.
Since such approximations underlie numerous applications in risk theory, queueing theory, reliability theory, mathematical theory of dams and inventories, this book is of interest not only to professional mathematicians, but also to physicists, engineers, and economists.
People from industry, with theoretical background in level-crossing problem, e.g., from the insurance industry, can benefit from reading this book.
Primarily aimed at researchers and postgraduates, but may be of interest to some professionals working in related fields, such as the insurance industry
Suitable for advanced courses in Applied Probability and, as a supplementary reading, for basic courses in Applied Probability
1. Introduction: Level-Crossing Problem and Related Fields. 1.1. Sums of independent Random Variables, Gaussian and Inverse Gaussian Distributions. 1.2. Random Walks and Renewal Processes. 1.3. Level-Crossing by a Compound Renewal Process. 1.4. Closed-form Results and Limit Theorems in Level-Crossing. 1.5. Message, Agenda, and Target Audience. 2. Inverse Gaussian and Generalized Inverse Gaussian Distributions. 2.1. Inverse Gaussian Distribution. 2.2. Generalized Inverse Gaussian Distribution. 3. Integral Expressions. 3.1. Elementary Integral Expressions. 3.2. Core Integral Expression. 3.3. Composite Integral Expressions. 4. Distribution of Compound Renewal Process at a Fixed Time Point. 4.1. Compound Renewal Process in Continuous Time. 4.2. Closed-form Results. 4.3. Aspects of Renewal Theory. 4.4. Origin of The Method based on Limit Theorems for Sums. 4.5. Approximation for the Mean. 4.6. Approximation for the Distribution. 4.7. Extensions from Renewal to more General Models. 5. Closed-form Results for the Distribution of First Level-Crossing Time. 5.1. Representations of the Distribution of First Level-Crossing Time. 5.2. Closed-form Results in Exponential Case. 5.3. Closed-form Expression for Conditional Probability. 5.4. Type II Formula and Random Walk with Random Displacements. 5.5. Closed-form Results, When T Is Non-Exponentially Distributed. 5.6. A Result, When Y Is Mixed Exponential. 6. The Inverse Gaussian Approximation. 6.1. Agenda for This Chapter. 6.2. Statement of Main Results. 6.3. Shorthand Notation, Structure Lemmas, Identities Specific to a choice of Arguments, and Centering At Z = 0. 6.4. Expressions of The First Kind. 6.5. Expressions of The Second Kind. 6.6. Expressions of the third kind 6.7. Proof of Theorem 6.1. 6.8. Numerical Illustrations. 6.9. Conclusions. 7. Refinement of the Inverse Gaussian Approximation. 7.1 Asymptotic Expansions: Rigorous Vs. Heuristic. 7.2. Expansion for the Distribution of the First Level-Crossing Time. 7.3. Proof of Theorem 7.1. 7.4. Numerical Illustrations. 8. Derivatives of The First Level-Crossing Time Distribution. 8.1. The Problem and Its Rationale. 8.2. Approximations for Derivatives. 8.3. Fundamental Identities for Derivatives with Respect to C and U. 8.4. Proof of Theorem 8.1. 8.5. Proof of Theorem 8.2. 9. A Breakthrough in the Level-Crossing Problem. 9.1. Neyman's Cycles in Level-Crossing Problem. 9.2. Normal Approximation Versus Inverse Gaussian Approximation. 9.3. Diffusion Approximation Versus inverse Gaussian Approximation: Is there A Mix-Up in This Collation? 9.4. Teugels' Approximation Versus Inverse Gaussian Approximation. 9.5. Conclusions. Appendices. Index.
ISBN 9780367544829
June 16, 2021 Forthcoming
144 Pages 149 B/W Illustrations
Format : Hardback
ISBN 9780367544713
Format : Paperback
Introduction to Math Olympiad Problems aims to introduce high school students to all the necessary topics that frequently emerge in international Math Olympiad competitions. In addition to introducing the topics, the book will also provide several repetitive?type guided problems to help develop vital techniques in solving problems correctly and efficiently. The techniques employed in the book will help prepare students for the topics they will typically face in an Olympiad-style event, but also for future college mathematics courses in Discrete Mathematics, Graph Theory, Differential Equations, Number Theory and Abstract Algebra.
Numerous problems designed to embed good practice in readers, and build underlying reasoning, analysis, and problem-solving skills.
Suitable for advanced high school students preparing for Math Olympiad competitions.
1. Introduction. 1.1. Patterns and Sequences. 1.2. Integers. 1.3. Geometry. 1.4. Venn Diagrams. 1.5. Factorial and Pascal's Triangle. 1.6. Graph Theory. 1.7. Piecewise Sequences. 1.8. Chapter 1 Exercises. 2. Sequences and Summations. 2.1. Linear and Quadratic Sequences. 2.2 Geometric Sequences. 2.3. Factorial and Factorial-Type Sequences. 2.4. Alternating and Piecewise Sequences. 2.5. Formulating Recursive Sequences. 2.6. Solving Recursive Sequences. 2.7. Summations. 2.8. Chapter 2 Exercises. 3. Proofs. 3.1. Algebraic Proofs. 3.2. Proof By Inductions. 3.3. Chapter 3 Exercises. 4 Integers' Characteristics. 4.1. Consecutive Integers. 4.2. Prime Factorization and Divisors. 4.3. Perfect Squares. 4.4. Integers' Ending Digits. 4.5. Chapter 4 Exercises. 5. Pascal's Triangle Identities. 5.1 Horizontally-Oriented Identities. 5.2 Diagonally-Oriented Identities. 5.3. Binomial Expansion. 5.4. Chapter 5 Exercises. 6. Geometry. 6.1. Triangular Geometry. 6.2 Area and Perimeter Geometry. 6.3. Geometry and Proportions. 6.4. Chapter 6 Exercises. 7. Graph Theory. 7.1. Degrees of Vertices and Cycles. 7.2 Regular Graphs. 7.3. Semi-Regular Graphs. 7.4 Hamiltonian Cycles. 7.5. Chapter 7 Exercises. 8. Answers to Chapter Exercises. 9. Appendices. 10. Index. 11. Bibliography.
ISBN 9781138364332
June 18, 2021 Forthcoming
264 Pages 29 B/W Illustrations
Format : Hardback
Ramsey theory is a fascinating topic. The author shares his view of the topic in this contemporary overview of Ramsey theory. He presents from several points of view, adding intuition and detailed proofs, in an accessible manner unique among most books on the topic. This book covers all of the main results in Ramsey theory along with results that have not appeared in a book before.
The presentation is comprehensive and reader friendly. The book covers integer, graph, and Euclidean Ramsey theory with many proofs being combinatorial in nature. The author motivates topics and discussion, rather than just a list of theorems and proofs. In order to engage the reader, each chapter has a section of exercises.
This up-to-date book introduces the field of Ramsey theory from several different viewpoints so that the reader can decide which flavor of Ramsey theory best suits them.
Additionally, the book offers:
A chapter providing different approaches to Ramsey theory, e.g., using topological dynamics, ergodic systems, and algebra in the Stone-?ech compactification of the integers.
A chapter on the probabilistic method since it is quite central to Ramsey-type numbers.
A unique chapter presenting some applications of Ramsey theory.
Exercises in every chapter
The intended audience consists of students and mathematicians desiring to learn about Ramsey theory. An undergraduate degree in mathematics (or its equivalent for advanced undergraduates) and a combinatorics course is assumed.
1. Introduction. 1.1. What is Ramsey Theory? 1.2. Notations and Conventions. 1.3. Prerequisites. 1.4. Compactness Principle. 1.5. Set Theoretic Considerations. 1.6. Exercises. 2. Integer Ramsey Theory. 2.1. Van der Waerden's Theorem. 2.2. Equations. 2.3. Hales-Jewett Theorem. 2.4. Finite Sums. 2.5. Density Results. 2.6. Exercises. 3. Graph Ramsey Theory. 3.1. Complete Graphs. 3.2. Other Graphs. 3.3. Hypergraphs. 3.4. Infinite Graphs. 3.5. Comparing Ramsey and van der Waerden Results. 3.6. Exercises. 4. Euclidean Ramsey Theory. 4.1. Polygons. 4.2. Chromatic Number of the Plane. 4.3. Four Color Map Theorem. 4.4. Exercises. 5. Other Approaches to Ramsey Theory. 5.1. Topological Approaches. 5.2. Ergodic Theory. 5.3. Stone-?ech Compactification. 5.4. Additive Combinatorics Methods. 5.5. Exercises. 6. The Probabilistic Method. 6.1. Lower Bounds on Ramsey, van der Waerden, and Hales-Jewett Numbers. 6.2. Turan's Theorem. 6.3. Almost-surely van der Waerden and Ramsey Numbers. 6.4. Lovasz Local Lemma. 6.5. Exercises. 7. Applications. 7.1. Fermat's Last Theorem. 7.2. Encoding Information. 7.3. Data Mining. 7.4. Exercises. Bibliography. Index.
ISBN 9780367716226
July 15, 2021 Forthcoming
368 Pages 28 B/W Illustrations
Format : Hardback
Introduction to High-Dimensional Statistics, Second Edition preserves the philosophy of the first edition: to be a concise guide for students and researchers discovering the area and interested in the mathematics involved. The main concepts and ideas are presented in simple settings, avoiding thereby unessential technicalities. High-dimensional statistics is a fast-evolving field and much progress has been made on a large variety of topics, providing new insights and methods. Offering a succinct presentation of the mathematical foundations of high-dimensional statistics, this new edition features:
Revised chapters from the previous edition, with the inclusion of many additional materials on some important topics, including compress sensing, estimation with convex constraints, the slope estimator, simultaneously low rank and row sparse linear regression, or aggregation of a continuous set of estimators.
Three new chapters on iterative algorithms, clustering and minimax lower bounds.
Enhanced appendices,minimax lower-bounds mainly with the addition of Davis-Kahan perturbation bound and of two simple versions of Hanson-Wright concentration inequality.
Covers cutting-edge statistical methods including model selection, sparsity and the lasso, iterative hard thresholding, aggregation, support vector machines and learning theory
Provides detailed exercises at the end of every chapter with collaborative solutions on a wikisite.
Illustrates concepts with simple but clear practical examples.
1. Introduction. 2. Model Selection. 3. Minimax Lower Bounds. 4. Aggregation of Estimators. 5. Convex Criteria. 6. Iterative Algorithms. 7. Estimator Selection. 8. Multivariate Regression. 9. Graphical Models. 10. Multiple Testing. 11. Supervised Classification. 12. Clustering.
Christophe Giraud was a student of the Ecole Normale Superieure de Paris, and he received a Ph.D in probability theory from the University Paris 6. He was assistant professor at the University of Nice from 2002 to 2008. He has been associate professor at the Ecole Polytechnique since 2008 and professor at Paris Sud University (Orsay) since 2012. His current research focuses mainly on the statistical theory of high-dimensional data analysis and its applications to life sciences.