Simon Rubinstein-Salzedo – författare

Ergodic theory is concerned with the measure-theoretic or statistical properties of a dynamical system. This book provides a conversational introduction to the topic, guiding the reader from the classical questions of measure theory to modern results such as the polynomial recurrence theorem. Applications to number theory and combinatorics enhance the exposition, while also presenting the utility of ergodic theory in other areas of research. The book begins with an introduction to measure theory and the Lebesgue integral. After this, the key concepts of the subject are covered: measure-preserving transformations, ergodicity, and invariant measures. These chapters also cover classical results such as Poincare's recurrence theorem and Birkhoff's ergodic theorem. The book ends with more advanced topics, such as mixing, entropy, and an appendix on the weak* topology. Each chapter ends with numerous exercises with a range of difficulty levels, including a handful of open problems. An excellent resource for anyone wishing to learn about ergodic theory, the book only assumes prior exposure to proof-based mathematics. Familiarity with real analysis would be ideal but is not required.

Algebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years. Each chapter, or lecture, corresponds to one day of class at SUMaC. The book begins with the preliminaries needed for the formal definition of a surface. Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text. This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class. The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.

This text introduces cryptography, from its earliest roots to cryptosystems used today for secure online communication.Beginning with classical ciphers and their cryptanalysis, this book proceeds to focus on modern public key cryptosystems such as Diffie-Hellman, ElGamal, RSA, and elliptic curve cryptography with an analysis of vulnerabilities of these systems and underlying mathematical issues such as factorization algorithms. Specialized topics such as zero knowledge proofs, cryptographic voting, coding theory, and new research are covered in the final section of this book.Aimed at undergraduate students, this book contains a large selection of problems, ranging from straightforward to difficult, and can be used as a textbook for classes as well as self-study. Requiring only a solid grounding in basic mathematics, this book will also appeal to advanced high school students and amateur mathematicians interested in this fascinating and topical subject.

This text introduces cryptography, from its earliest roots to cryptosystems used today for secure online communication.Beginning with classical ciphers and their cryptanalysis, this book proceeds to focus on modern public key cryptosystems such as Diffie-Hellman, ElGamal, RSA, and elliptic curve cryptography with an analysis of vulnerabilities of these systems and underlying mathematical issues such as factorization algorithms. Specialized topics such as zero knowledge proofs, cryptographic voting, coding theory, and new research are covered in the final section of this book.Aimed at undergraduate students, this book contains a large selection of problems, ranging from straightforward to difficult, and can be used as a textbook for classes as well as self-study. Requiring only a solid grounding in basic mathematics, this book will also appeal to advanced high school students and amateur mathematicians interested in this fascinating and topical subject.

This textbook is aimed at transitioning high-school students who have already developed proficiency in mathematical problem solving from numerical-answer problems to proof-based mathematics. It serves to guide students on how to write and understand mathematical proofs. It covers proof techniques that are commonly used in several areas of mathematics, especially number theory, combinatorics, and analysis. In addition to just teaching the mechanics of proofs, this book showcases key materials in these areas, thus introducing readers to interesting mathematics along with proof techniques.