Miscellaneous Books – serie

Number Theory Revealed: A Masterclass acquaints enthusiastic students with the ""Queen of Mathematics''. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod $p$ and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves.This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the $abc$-conjecture including its proof for polynomials.

Linear algebra permeates mathematics, as well as physics and engineering. In this text for junior and senior undergraduates, Sadun treats diagonalization as a central tool in solving complicated problems in these subjects by reducing coupled linear evolution problems to a sequence of simpler decoupled problems. This is the Decoupling Principle. Traditionally, difference equations, Markov chains, coupled oscillators, Fourier series, the wave equation, the Schrodinger equation, and Fourier transforms are treated separately, often in different courses. Here, they are treated as particular instances of the decoupling principle, and their solutions are remarkably similar. By understanding this general principle and the many applications given in the book, students will be able to recognize it and to apply it in many other settings. Sadun includes some topics relating to infinite-dimensional spaces. He does not present a general theory, but enough so as to apply the decoupling principle to the wave equation, leading to Fourier series and the Fourier transform. The second edition contains a series of Explorations. Most are numerical labs in which the reader is asked to use standard computer software to look deeper into the subject. Some explorations are theoretical, for instance, relating linear algebra to quantum mechanics. There is also an appendix reviewing basic matrix operations and another with solutions to a third of the exercises.

This book presents a rich, insightful exploration of the interplay between randomness, the Riemann Hypothesis, and analytic number theory. Written for readers with a moderate background in mathematics (preferably having some familiarity with probability theory and complex analysis), it offers a deeper and more comprehensive account of the Riemann Hypothesis and its far-reaching consequences than is typically available in books aimed at general audiences. The book takes a relaxed and engaging approach, highlighting the beauty and intuition behind the key ideas. Technical proofs are kept to a minimum, with accessible arguments presented fully, while deeper results are described in context and supported by historical background and comprehensive references. This combination allows readers to appreciate the elegance of the subject and to navigate their own path toward more advanced study. Bringing together a wealth of material that is otherwise widely scattered across the literature, the book serves as an indispensable reference for researchers and enthusiasts alike. Throughout, it illuminates the crucial role that randomness plays in modern number theory-inspiring conjectures, guiding proofs, and opening new directions of inquiry. Six valuable appendices round out the volume, providing opportunities to review-or learn-essential background material across a range of mathematical topics.