PUBLICATION PLANNED FOR: June 2023
FORMAT: PaperbackISBN: 9781009243902
Since its inception in the early 20th century, Functional Analysis has become a core part of modern mathematics. This accessible and lucid textbook will guide students through the basics of Functional Analysis and the theory of Operator Algebras. The text begins with a review of Linear Algebra and Measure Theory. It progresses to concepts like Banach spaces, Hilbert spaces, Dual spaces and Weak Topologies. Subsequent chapters introduce the theory of operator algebras as a guide to study linear operators on a Hilbert space and cover topics such as Spectral Theory and C*-algebras. Theorems have been introduced and explained through proofs and examples, and historical background to the mathematical concepts have been provided wherever appropriate. At the end of chapters, practice exercises have been segregated in a topic-wise manner for targeted practice, making the book ideal both for classroom teaching as well as self-study.
Lucid and classroom-style language promotes ease of understanding
More than 450 practice exercises for sharpening problem-solving skills
'Aside' sections provide a deeper understanding of concepts
'Additional reading' sections provide further information for interested students
Preface
Notation
1. Preliminaries
2. Normed Linear Spaces
3. Hilbert Spaces
4. Dual Spaces
5. Operators on Banach Spaces
6. Weak Topologies
7. Spectral Theory
8. C*-Algebras
9. Measure and Integration
10. Normal Operators on Hilbert Spaces
Appendices
A.1 The Stone?Weierstrass Theorem
A.2 The Radon?Nikodym Theorem
Bibliography
Index.
PUBLICATION PLANNED FOR: March 2023
FORMAT: Hardback ISBN: 9781009260282
FORMAT: Paperback ISBN: 9781009260305
Networks surround us, from social networks to protein?protein interaction networks within the cells of our bodies. The theory of random graphs provides a necessary framework for understanding their structure and development. This text provides an accessible introduction to this rapidly expanding subject. It covers all the basic features of random graphs ? component structure, matchings and Hamilton cycles, connectivity and chromatic number ? before discussing models of real-world networks, including intersection graphs, preferential attachment graphs and small-world models. Based on the authors' own teaching experience, it can be used as a textbook for a one-semester course on random graphs and networks at advanced undergraduate or graduate level. The text includes numerous exercises, with a particular focus on developing students' skills in asymptotic analysis. More challenging problems are accompanied by hints or suggestions for further reading.
Contains more than 250 exercises, with a focus on asymptotic analysis
Covers all the basic properties of random graphs, providing a foundation on which to build
Introduces network models and explains some of the properties of real-world networks
Conventions/Notation
Part I. Preliminaries:
1. Introduction
2. Basic tools
Part II. Erdos?Renyi?Gilbert Model:
3. Uniform and binomial random graphs
4. Evolution
5. Vertex degrees
6. Connectivity
7. Small subgraphs
8. Large subgraphs
9. Extreme characteristics
Part III. Modeling Complex Networks:
10. Inhomogeneous graphs
11. Small world
12. Network processes
13. Intersection graphs
14. Weighted graphs
References
Author index
Main index.
Part of Cambridge Tracts in Mathematics
PUBLICATION PLANNED FOR: April 2023
FORMAT: HardbackISBN: 9781009346108
This book establishes the moduli theory of stable varieties, giving the optimal approach to understanding families of varieties of general type. Starting from the Deligne?Mumford theory of the moduli of curves and using Mori's program as a main tool, the book develops the techniques necessary for a theory in all dimensions. The main results give all the expected general properties, including a projective coarse moduli space. A wealth of previously unpublished material is also featured, including Chapter 5 on numerical flatness criteria, Chapter 7 on K-flatness, and Chapter 9 on hulls and husks.
Gradually builds from the simplest theories to the most general one, allowing readers to see the development of the ideas and reach important special cases quickly
Provides many worked-out examples, demonstrating the natural limits of all concepts and theorems
Includes a wealth of previously unpublished material, providing a basis and solid references for future work
Introduction
Notation
1. History of moduli problems
2. One-parameter families
3. Families of stable varieties
4. Stable pairs over reduced base schemes
5. Numerical flatness and stability criteria
6. Moduli problems with flat divisorial part
7. Cayley flatness
8. Moduli of stable pairs
9. Hulls and husks
10. Ancillary results
11. Minimal models and their singularities
References
Index.
'This book dismantles the final, most daunting barriers to learning about moduli of higher dimensional varieties, from the point of view of the Minimal Model Program. The first chapter draws the reader in with a compelling history; a discussion of the main ideas; a visitor's trail through the subject, complete with guardrails around the most dangerous traps; and a rundown of the issues that one must overcome. The text that follows is the outcome of Kollar's monumental three-decades-long effort, with the final stones laid just in the last few years.' Dan Abramovich, Brown University
'This is a fantastic book from Janos Kollar, one of the godfathers of the compact moduli theory of higher dimensional varieties. The book contains the definition of the moduli functor, the prerequisites required for the definition, and also the proof of the existence of the projective coarse moduli space. This is a stunning achievement, completing the story of 35 years of research. I expect this to become the main reference book, and also the principal place to learn about the theory for graduate students and others interested.' Zsolt Patakfalvi, EPFL
'This excellent book provides a wealth of examples and technical details for those studying birational geometry and moduli spaces. It completely addresses several state-of-the-art topics in the field, including different stability notions, K-flatness, and subtleties in defining families of stable pairs over an arbitrary base. It will be an essential resource for both those first learning the subject and experts as it moves through history and examples before settling many of the (previously unknown) technicalities needed to define the correct moduli functor.' Kristin DeVleming, University of Massachusetts Amherst
Copyright Year 2023
ISBN 9781032332055
ISBN 9781032017204
February 23, 2023 Forthcoming
160 Pages
Introduction to Number Theory covers the essential content of an introductory number theory course including divisibility and prime factorization, congruences, and quadratic reciprocity. The instructor may also choose from a collection of additional topics.
Aligning with the trend toward smaller, essential texts in mathematics, the author strives for clarity of exposition. Proof techniques and proofs are presented slowly and clearly.
The book employs a versatile approach to the use of algebraic ideas. Instructors who wish to put this material into a broader context may do so, though the author introduces these concepts in a non-essential way.
A final chapter discusses algebraic systems (like the Gaussian integers) presuming no previous exposure to abstract algebra. Studying general systems urges students realize unique factorization into primes is a more subtle idea than may at first appear; students will find this chapter interesting, fun and quite accessible.
Applications of number theory include several sections on cryptography and other applications to further interest instructors and students alike.
Introduction. What is Number Theory?
Divisibility
Congruences and Modular Arithmetic
Cryptography: An Introduction
Perfect Numbers
Primitive Roots
Quadratic Reciprocity
Arithmetic Beyond the Integers
Author(s)
Biography
Mark Hunacek has advanced degrees in both mathematics (Ph.D., Rutgers University) and law (J.D., Drake University Law School). He is now a Teaching Professor Emeritus at Iowa State University, and before entering academia he was an Assistant Attorney General for the state of Iowa.
Copyright Year 2023
ISBN 9780367458393
March 29, 2023 Forthcoming by Chapman & Hall
366 Pages 130 B/W Illustrations
This book takes a deep dive into several key linear algebra subjects as they apply to data analytics and data mining. The book offers a case study approach where each case will be grounded in a real-world application.
This text is meant to be used for a second course in applications of Linear Algebra to Data Analytics, with a supplemental chapter on Decision Trees and their applications in regression analysis. The text can be considered in two different but overlapping general data analytics categories, clustering and interpolation.
Knowledge of mathematical techniques related to data analytics, and exposure to interpretation of results within a data analytics context, are particularly valuable for students studying undergraduate mathematics. Each chapter of this text takes the reader through several relevant and case studies using real world data.
All data sets, as well as Python and R syntax are provided to the reader through links to Github documentation. Following each chapter is a short exercise set in which students are encouraged to use technology to apply their expanding knowledge of linear algebra as it is applied to data analytics.
A basic knowledge of the concepts in a first Linear Algebra course are assumed; however, an overview of key concepts are presented in the Introduction and as needed throughout the text.
Acknowledgments
Introduction
1 Graph Theory
1.1 Basic Terminology
1.2 The Power of the Adjacency Matrix
1.3 Eigenvalues and Eigenvectors as Key Players
1.4 CASE STUDY: Applications in Sport Ranking
1.5 CASE STUDY: Gerrymandering
1.6 Exercises
2. Stochastic Processes
2.1 Markov Chain Basics
2.2 Hidden Markov Models
2.2.1 The Likelihood Problem
2.2.2 The Decoding Problem
2.2.3 The Learning Problem
2.3 CASE STUDY: Spread of Infectious Disease
2.4 CASE STUDY: Text Analysis and Autocorrect
2.5 CASE STUDY: Tweets and Time Series
2.6 Exercises
3. SVD and PCA
3.1 Vector and Inner Product Spaces
3.2 Singular Values
3.3 Singular Value Decomposition
3.4 Compression of Data Using Principal Component Analysis (PCA)
3.5 PCA, Covariance, and Correlation
3.6 Linear Discriminant Analysis
3.7 CASE STUDY: Digital Humanities
3.8 CASE STUDY: Facial Recognition Using PCA and LDA
3.9 Exercises
4. Interpolation
4.1 Lagrange Interpolation
4.2 Orthogonal Families of Polynomials
4.3 Newtonfs Divided Difference
4.3.1 Newtonfs interpolation via divided difference
4.3.2 Newtonfs interpolation via the Vandermonde matrix
4.4 Chebyshev interpolation
4.5 Hermite interpolation
4.6 Least Squares Regression
4.7 CASE STUDY : Chebyshev Polynomials and Cryptography
4.8 CASE STUDY: Racial Disparities in Marijuana Arrests
4.9 CASE STUDY : Interpolation in Higher Education Data
4.10 Exercises
5. Optimization and Learning Techniques for Regression
5.1 Basics of Probability Theory
5.2 Introduction to Matrix Calculus
5.2.1 Matrix Differentiation
5.2.2 Matrix Integration
5.3 Maximum Likelihood Estimation
5.4 Gradient Descent Method
5.5 Introduction to Neural Networks
5.5.1 The Learning Process
5.5.2 Sigmoid Activation Functions
5.5.3 Radial Activation Functions
5.6 CASE STUDY: Handwriting Digit Recognition
5.7 CASE STUDY: Poisson Regression and COVID Counts
5.8 Exercises
6 Decision Trees and Random Forests
6.1 Decision Trees
6.1.1 Decision Trees Regression
6.2 Regression Trees
6.3 Random Decision Trees and Forests
6.4 CASE STUDY: Entropy of Wordle
6.5 CASE STUDY : Bird Call Identification
6.6 Exercises
7. Random Matrices and Covariance Estimate
7.1 Introduction to Random Matrices
7.2 Stability
7.3 Gaussian Orthogonal Ensemble
7.4 Gaussian Unitary Ensemble
7.5 Gaussian Symplectic Ensemble
7.6 Random Matrices and the Relationship to the Covariance
7.7 CASE STUDY: Finance and Brownian Motion
7.8 CASE STUDY: Random Matrices in Gene Interaction
7.9 Exercises
8. Sample Solutions to Exercises
8.1 Chapter 1
8.2 Chapter 2
8.3 Chapter 3
8.4 Chapter 4
8.5 Chapter 5
8.6 Chapter 6
8.7 Chapter 7
Github Links 349
Bibliography 351
Index 355
...
Dr. Crista Arangala is Professor of Mathematics and Chair of the Department of Mathematics and Statistics at Elon University in North Carolina. She and has been teaching and researching in a variety of fields including in inverse problems, applied partial differential equations, applied linear algebra, mathematical modeling and service learning education. She runs a traveling science museum with her Elon University students in Kerala, India. Dr. Arangala was chosen to be a Fulbright Scholar in 2014 as a visiting lecturer at the University of Colombo where she continued her projects in inquiry learning in Linear Algebra and began working with a modeling team focusing on Dengue fever research. Dr. Arangala has published several textbooks that implores inquiry learning techniques including Exploring Linear Algebra: Labs and Projects with Matlab and Mathematical Modeling: Branching Beyond Calculus.