The Rudin-Shapiro Sequence, Ising Chain, and Paperfolding

This is the proceedings of theworkshop on recent developments in ergodic theory and dynamical systemson March 2011and March 2012 at the University of North Carolina at Chapel Hill. Thearticles in this volume cover several aspects of vibrant research in ergodic theory and dynamical systems. It contains contributions to Teichmuller dynamics, interval exchange transformations, continued fractions, return times averages, Furstenberg Fractals, fractal geometry of non-uniformly hyperbolic horseshoes, convergence along the sequence of squares, adic and horocycle flows, and topological flows. These contributions illustrate the connections between ergodic theory and dynamical systems, number theory, harmonic analysis, probability, andalgebra. Two surveys are included which give a nice introduction for interested young or senior researcher to some active research areas. Overall this volume provides a very useful blend of techniques and methods as well as directions of research on general convergence phenomena in ergodic theory and dynamical systems.

The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments. This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

This volume contains selected contributions from a very successful meeting on Number Theory and Dynamical Systems held at the University of York in 1987. There are close and surprising connections between number theory and dynamical systems. One emerged last century from the study of the stability of the solar system where problems of small divisors associated with the near resonance of planetary frequencies arose. Previously the question of the stability of the solar system was answered in more general terms by the celebrated KAM theorem, in which the relationship between near resonance (and so Diophantine approximation) and stability is of central importance. Other examples of the connections involve the work of Szemeredi and Furstenberg, and Sprindzuk. As well as containing results on the relationship between number theory and dynamical systems, the book also includes some more speculative and exploratory work which should stimulate interest in different approaches to old problems.

Fractional Brownian Motions, Fractional Noises and Applications

This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

The study of integer partitions is a significant area in number theory, intertwining arithmetic, algebra, and combinatorics. This theoretical study delves into the arithmetic and algebraic properties of integer partitions, examining their structure, relationships, and applications. An integer partition refers to a way of expressing a positive integer as the sum of other positive integers, where the order of summands does not matter. The paper explores the generating functions that encapsulate the partition function, focusing on their deep connections to modular forms and q-series. We examine the partition identities, such as Euler's pentagonal number theorem and their implications for partition theory. Furthermore, the study investigates algebraic approaches, such as the use of symmetric functions and Young tableaux, to provide insights into the combinatorial nature of partitions. The paper also emphasizes the arithmetic properties of partitions, such as congruences and partition identities modulo certain integers. Through rigorous analysis, we explore the role of integer partitions in solving Diophantine equations, their applications in statistical mechanics, and their contribution to the understanding of the structure of the partition lattice. Finally, the study draws connections to other areas of mathematics, such as q-algebras and representation theory, highlighting the interplay between algebraic structures and partition theory.

Relations between number theory, quantum mechanics and statistical mechanics are of interest to mathematicians and physicists since it was suggested that the zeros of the Riemann zeta function might be related to the spectrum of a self-adjoint quantum mechanical operator related to a one-dimensional Hamiltonian $$ H = xp $$ known as Berry–Keating–Connes Hamiltonian. However, this type of Hamiltonian is integrable and the classical trajectories of particles are not closed leading to a continuum spectrum. Recently, Sierra and Rodriguez-Laguna conjectured that the Hamiltonian $$ H = x(p \,+\, {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ where $$ \xi $$ is a coupling constant with dimensions of momentum square is characterized by a quantum spectrum which coincides with the average Riemann zeros and contains closed periodic orbits. In this paper, we show first that the Sierra–Rodriguez-Laguna Hamiltonian may be obtained by means of non-standard singular Lagrangians and besides the Hamiltonians $$ H = x(p + {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ and $$ H(x,p) = px $$ are not the only semiclassical Hamiltonians connected to the average Riemann zeros. We show the presence of a new Hamiltonian where its quantization revealed a number of interesting properties, in particular, the sign of a trace of the Riemann zeros.

My title seems self contradictory, for surely random happenings are essentially unpredictable—so how can they be regular? And yet this apparent contradiction is the key which justifies the mathematical model of probability theory. For the regularity is the basic reason for our model providing a new mathematical tool with powerful applications to analysis, potential theory, number theory, quantum theory, statistical mechanics.... Let us start by asking, “What is probability?” What do I mean if I say “When you toss a coin there is a probability ½ that it turns up a head” or “When you pick a card from a well shuffled pack there is probability that it will be an ace”?

Fifty years ago, when Claude Shannon was developing the Mathematical Theory of Communications, for reliable data transmission, which evolved into the subject of information theory, another discipline was developing dealing with Feedback Control of Dynamical System, which evolved into a scientific subject dealing with decision, stability, and optimization. More recently, a separate discipline dealing with robustness of uncertain systems was born in response to the codification of high performance and reliability in the presence of modeling uncertainties. In principle, robustness in dynamical systems is captured through power dissipation via induced norms and dynamic games, while reliable data transmission is captured through measures of information via entropy, relative entropy, and certain laws of Large Deviations theory. The main ingredient in Large Deviations is the rate functional (or action functional in the classical mechanics terminology), often identified through the Cramer or Legendre-Fenchel Transform. On the other hand, robustness of stochastic uncertain systems is currently under development, using information theoretic as well as statistical mechanics concepts, such as, partition functions, free energy, relative entropy, and entropy rate functional. This lecture will summarize certain connections between fundamental concepts of robustness, information theory, and statistical mechanics, and possibly make future projections into the convergence of these disciplines.

I remember vividly my first course on dynamical systems when the professor was discussing ergodicity and claimed that if initially all the molecules started out on one side of the room that eventually they would all return to that side, suffocating half his students in the process. While this provocative statement is in some sense true, we know both practically that a fear of suffocation is irrational and intellectually that the recurrence time is so astronomically long as to make the age of the universe seem like the blink of an eye. The resolution of this classical paradox is the foundation of statistical mechanics. Microscopic time-reversible behavior gives rise to irreversible behavior at the macroscale, a truth that many of us tell our students early in their education, but for which they must usually wait until an advanced undergraduate course in statistical mechanics to see explored in any detail. This issue's Education section offering, Boltzmann's Dilemma: An Introduction to Statistical Mechanics via the Kac Ring, by Georg A. Gottwald and Marcel Oliver, revives a classic model in this field and brings an understanding of this paradox within the reach of undergraduates with only a modest familiarity with the concept of a limit from calculus. The paper begins with an engaging historical review; introduces the Kac ring, which is an explicitly solvable, microscopically time-reversible model; and then introduces macroscopic concepts such as ensemble averages and entropy. The authors should be commended for making the fundamentals of statistical mechanics accessible to students at the introductory calculus level. The material could be incorporated as a series of lectures in a first seminar on applied mathematics or mechanics. It also stands on its on as supplemental reading and would nicely complement the introduction of the continuum approximation in fluid mechanics. I am hopeful it will also alleviate my irrational fear of suffocation.

Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.

Since the time of the Ancient Greeks, much has been written about the relation between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student, or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad and involves physics, biology, psychoacoustics, the history of science, and digital technology as well as, of course, mathematics and music. Starting with the structure of the human ear and its relationship with Fourier analysis, the story proceeds via the mathematics of musical instruments to the ideas of consonance and dissonance, and then to scales and temperaments. This is a must-have book if you want to know about the music of the spheres, digital music, and many things in between.

By the use of two examples, we discuss the techniques of Fourier analysis in the study of problems in irregularities of point distribution. Such techniques include classical Fourier series and transforms as well as Fourier-Walsh analysis and wavelet analysis. We also show that often the Fourier analysis can be combined with ideas and techniques in number theory, geometry, probability theory, group theory, characters and duality.

In previous chapters we have met Fourier series and the Fourier transform. These are both incarnations of Fourier analysis on a commutative group, namely the unit circle and the real line under addition. In this chapter we study Fourier analysis in the even simpler setting of the additive group of integers modulo a fixed positive integer n [6, 12]. Here, for obvious reasons, the Fourier transform is called the finite Fourier transform. Although the finite Fourier transform has many interesting applications in abstract algebra, combinatorics, number theory, and complex variables [8], we view it mainly as a tool for approximating Fourier series. Computation of finite Fourier transforms is done efficiently by an algorithm known as the fast Fourier transform [1, 3, 5, 9, 13, 15]. Although it was discovered by Gauss, the fast Fourier transform has come into prominence only with the advent of modern computing. As an indication of its critical role in many scientific and engineering applications, it is often implemented in hardware rather than software.