Copyright 2024
Hardback
ISBN 9781032738000
Paperback
ISBN 9781032706108
256 Pages 126 B/W Illustrations
April 22, 2024
This is a book about infinity ? specifically the infinity of numbers and sequences. Amazing properties arise, for instance, some kinds of infinity are argued to be greater than others. Along the way the author will demonstrate how infinity can be made to create beautiful eartf, guided by the development of underlying mathematics. This book will provide a fascinating read for anyone interested in number theory, infinity, math art, and/or generative art, and could be used a valuable supplement to any course on these topics.
Beautiful examples of generative art
Accessible to anyone with a reasonable high school level of mathematics
Full of challenges and puzzles to engage readers.
1. What is this book about? 1.1. Infinitely many. 1.2. Infinite sequences and turtle figures. 1.3. Morphic sequences and symmetry. 1.4. Fractal turtle figures. 1.5. Mathematical challenges. 1.6. Who is this book for, how is it organized and how to read it? Challenge: the paint pot problem. 2. Numbers of the simplest kind. 2.1. Natural numbers. 2.2. Induction Strong induction. 2.3. Addition. 2.4. Multiplication. 2.5. Divisors and prime numbers. Challenge: number of divisors. 3. More complicated numbers. 3.1. Integer numbers. 3.2. Rational numbers. 3.3. Real numbers. 3.4. Complex numbers. Challenge: ten questions. 4. Flavors of infinity. 4.1. The Hilbert Hotel. 4.2. Smaller than? 4.3. Equal size? 4.4. Countable sets. 4.5. Uncountable sets. 4.6. Computable numbers. 4.7. Cardinal numbers. Challenge: monotone functions. 5 Infinite sequences. 5.1. Operations on sequences Morphisms. 5.2. Periodic and ultimately periodic sequences. 5.3. Decimal notation of numbers. 5.4. Frequency of symbols. 5.5. Challenge: the marble box. 6. Turtle figures. 6.1. Turtle figures of words and sequences. 6.2. Turtle figures of periodic sequences Some theory. 6.3. Finite turtle figures of periodic sequences. 6.4. Infinite turtle figures of periodic sequences. 6.5. Ultimately periodic sequences. Challenge: subword with zero angle. 7. Programming. 7.1. Turtle programming in Python. 7.2. Turtle programming in Lazarus. 7.3. Some theory. Challenge: knight moves. 8. More complicated sequences. 8.1. Random sequences. 8.2. The spiral sequence. 8.3. Pure morphic sequences. 8.4. Morphic sequences. 8.5. Programming morphic sequences. Challenge: a variation on the spiral sequence. 9. The Thue-Morse sequence. 9.1 A fair distribution. 9.2. The Thue-Morse sequence as a morphic sequence 9.3. Alternative characterizations. 9.4. Finite turtle figures of more general sequences. 9.5. Finite turtle figures of the Thue-Morse sequence. 9.6. Thue-Morse is cube free. 9.7. Stuttering variants of Thue-Morse. Challenge: finiteness in the spiral sequence. 10. More finite turtle figures. 10.1 A new theorem. 10.2. Two equal consecutive symbols. 10.3. Rosettes. 10.4. More Symbols. 10.5. Adding a tail. Challenge: one more finite turtle figure. 11. Fractal turtle figures. 11.1. Mandelbrot sets. 11.2. Fractal turtle figures. 11.3. The main theorem. 11.4. Examples with rotation. 11.5. Examples with u = u. 11.6. Other examples. Challenge: googol. 12. Variations on Koch. 12.1. Koch curve. 12.2. Koch curve as a turtle figure. 12.3. Other scaling factors. 12.4. The period-doubling sequence. 12.5. Fractal turtle figures of variants of p. Challenge: googolplex. 13. Simple morphic sequences. 13.1. Koch-like turtle figures of Thue-Morse. 13.2. Relating t and p. 13.3. Finite turtle figures. 13.4. Other simple morphic sequences. 13.5. The binary Fibonacci sequence. 13.6. Turtle figures of the binary Fibonacci sequence. 13.7. Frequency of symbols in morphic sequences. Challenge: frequency of 1 percent. 14. Looking back. 14.1. Turtle figures of morphic sequences. 14.2. Other types of turtle figures. 14.3. More exciting pictures: cellular automata. 14.4. Mathematical challenges. 14.5. Almost infinite. Challenge: the greatest value.
Softcover ISBN: 978-1-4704-7577-2
Product Code: SURV/222.S
Mathematical Surveys and Monographs Volume: 222;
2017; 344 pp
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices.
One of the main conclusions of the theory is that the spectrum of the integers is gintrinsically curvedh; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
Chapters
Introduction
Algebraic background
Classical differential geometry revisited
Arithmetic differential geometry: Generalities
Arithmetic differential geometry: The case of GLn
Curvature and Galois groups of Ehresmann connections
Curvature of Chern connections
Curvature of Levi-Civita connections
Curvature of Lax connections
Open problems
Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.
Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonne quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them.
A pro-Lie group is a complete topological group G in which every identity neighborhood U of G contains a normal subgroup N such that the quotient G/N is a Lie group. Every locally compact connected topological group and every compact group is a pro-Lie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of pro-Lie groups is.
For half a century, locally compact pro-Lie groups have drifted through the literature, yet this is the first book which systematically treats the Lie theory and the structure theory of pro-Lie groups irrespective of local compactness. So it fits very well into that current trend which addresses infinite dimensional Lie groups. The results of this text are based on a theory of pro-Lie algebras which parallels the structure theory of finite dimensional real Lie algebras to an astonishing degree even though it has to overcome technical obstacles.
A topological group is said to be almost connected if the quotient group of its connected components is compact. This book exposes a Lie theory of almost connected pro-Lie groups (and hence of almost connected locally compact groups) and illuminates the variety of ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is therefore a continuation of the authors' monograph on the structure of compact groups (1998, 2006, 2014, 2020, 2023) and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of research, which has so many fruitful interactions with other fields of mathematics.
Softcover ISBN: 978-1-4704-7104-0
Product Code: CONM/793
Contemporary Mathematics Volume: 793;
2024; 358 pp
This book consists of a series of papers focusing on the mathematical and computational modeling and analysis of some real-life phenomena in the natural and engineering sciences. The book emphasizes three main themes: (i) the design and analysis of robust and dynamically-consistent nonstandard finite-difference methods for discretizing continuous-time dynamical systems arising in the natural and engineering sciences, (ii) the mathematical study of nonlinear oscillations, and (iii) the design and analysis of models for the spread and control of emerging and re-emerging infectious diseases.
Specifically, some of the topics covered in the book include advances and challenges on the design, analysis and implementation of nonstandard finite-difference methods for approximating the solutions of continuous-time dynamical systems, the design and analysis of models for the spread and control of the COVID-19 pandemic, modeling the effect of prescribed fire and temperature on the dynamics of tick-borne disease, and the design of a novel genetic-epidemiology framework for malaria transmission dynamics and control.
The book also covers the impact of environmental factors on diseases and microbial populations, Monod kinetics in a chemostat setting, structure and evolution of poroacoustic solitary waves, mathematics of special (periodic) functions and the numerical discretization of a phase-lagging equation with heat source.
Roumen Anguelov and Jean M.-S. Lubuma ? A second-order nonstandard finite difference scheme and application to a model of biological and chemical processes
Manh Tuan Hoang ? A generalized nonstandard finite difference method for a class of autonomous dynamical systems and its applications
Fawaz K. Alalhareth, Madhu Gupta, Hristo V. Kojouharov and Souvik Roy ? Higher-order modified nonstandard finite difference methods for autonomous dynamical systems
Hermann J. Eberl ? A simple NSFD inspired method for Monod kinetics with small half saturation constants in the chemostat setting
Michael Chapwanya and Phindile Dumani ? Dynamics preserving nonstandard finite difference scheme for a microbial population model incorporating environmental stress
Treena S. Basu, Ron Buckmire, Zaheer Coovadia, Mayra Diaz, David A. Iniguez and Alexandra Scott ? Using unity approximations to construct nonstandard finite difference schemes for Bernoulli differential equations
C. L. Shimp, J. V. Lambers and P. M. Jordan ? On the structure and evolution of poroacoustic solitary waves: Finite-time gradient catastrophe under the Darcy?Jordan model
Cui-Cui Ji and Weizhong Dai ? A fractional-order equation and its finite difference scheme for approximating a delay equation
Laura F. Strube and Lauren M. Childs ? Multistability in a discrete-time SI epidemic model with Ricker growth: Infection-induced changes in population dynamics
Jemal Mohammed-Awel and Abba B. Gumel ? A genetic-epidemiology modeling framework for malaria mosquitoes and disease
M. Adrian Acuna-Zegarra, Mario Santana-Cibrian, Carlos E. Rodriguez Hernandez-Vela, Ramses H. Mena and Jorge X. Velasco-Hernandez ? A retrospective analysis of COVID-19 dynamics in Mexico and Peru: Studying hypothetical changes in the contact rate
Gustavo Morais Rodrigues Costa, Marcelo Lobosco, Matthias Ehrhardt and Ruy Freitas Reis ? Mathematical analysis and a nonstandard scheme for a model of the immune response against COVID-19
Alexander Fulk and Folashade B. Agusto ? Effects of rising temperature and prescribed fire on Amblyomma Americanum with ehrlichiosis
Elena N. Naumova, Maryam B. Yassai, Jack Gorski and Yuri N. Naumov ? Modeling T-cell repertoire response to a viral infection with short immunity
Sandra A. Rucker ? Geometric approach to the construction of Leah-type periodic functions: Basic and analytic properties
Isom H. Herron ? Discursion on a paper of R. E. Mickens and J. E. Wilkins, Jr.
Graduate students and research mathematicians interested in theoretical
and computational analysis and modelling of processes arising in ecology,
epidemiology, and population biology.
Softcover ISBN: 978-1-4704-7254-2
Product Code: CONM/794
Expected availability date: April 16, 2024
Contemporary Mathematics Volume: 794;
2024; 258 pp
MSC: Primary 14; 37; 46; 49; 53; 57; 58; 81;
This volume contains the proceedings of the AMS-EMS-SMF Special Session on Recent Advances in Diffeologies and Their Applications, held from July 18?20, 2022, at the Universite de Grenoble-Alpes, Grenoble, France.
The articles present some developments of the theory of diffeologies applied in a broad range of topics, ranging from algebraic topology and higher homotopy theory to integrable systems and optimization in PDE.
The geometric framework proposed by diffeologies is known to be one of the most general approaches to problems arising in several areas of mathematics. It can adapt to many contexts without major technical difficulties and produce examples inaccessible by other means, in particular when studying singularities or geometry in infinite dimension. Thanks to this adaptability, diffeologies appear to have become an interesting and useful language for a growing number of mathematicians working in many different fields. Some articles in the volume also illustrate some recent developments of the theory, which makes it even more deep and useful.
Nico Goldammer, Jean-Pierre Magnot and Kathrin Welker ? On diffeologies from infinite dimensional geometry to PDE constrained optimization
Christian Blohmann ? Elastic diffeological spaces
Alireza Ahmadi ? A remark on stability and the D-topology of mapping spaces
Yael Karshon and Jordan Watts ? Smooth maps on convex sets
Enxin Wu ? A survey on diffeological vector spaces and applications
Ekaterina Pervova ? Finite-dimensional diffeological vector spaces being and not being coproducts
David Miyamoto ? Singular foliations through diffeology
Jordan Watts and Seth Wolbert ? Diffeological coarse moduli spaces of stacks over manifolds
Fiammetta Battaglia and Elisa Prato ? Generalized Laurent monomials in nonrational toric geometry
Iakovos Androulidakis ? On a remark by Alan Weinstein
Anahita Eslami-Rad, Jean-Pierre Magnot, Enrique G. Reyes and Vladimir Rubtsov ? Diffeologies and generalized Kadomtsev-Petviashvili hierarchies
Norio Iwase ? Smooth A
-form on a diffeological loop space
Hiroshi Kihara ? Smooth homotopy of diffeological spaces: theory and applications to infinite-dimensional C
-manifolds
Graduate students and research mathematicians interested in topology, differential geometry, and functional analysis.
Softcover ISBN: 978-1-4704-7334-1
Product Code: STML/106
Expected availability date: March 27, 2024
Student Mathematical Library Volume: 106
MSC: Primary 92; 39; Secondary 37; 15;
This book offers an introduction to the use of matrix theory and linear algebra in modeling the dynamics of biological populations. Matrix algebra has been used in population biology since the 1940s and continues to play a major role in theoretical and applied dynamics for populations structured by age, body size or weight, disease states, physiological and behavioral characteristics, life cycle stages, or any of many other possible classification schemes. With a focus on matrix models, the book requires only first courses in multi-variable calculus and matrix theory or linear algebra as prerequisites.
The reader will learn the basics of modeling methodology (i.e., how to set up a matrix model from biological underpinnings) and the fundamentals of the analysis of discrete time dynamical systems (equilibria, stability, bifurcations, etc.). A recurrent theme in all chapters concerns the problem of extinction versus survival of a population. In addition to numerous examples that illustrate these fundamentals, several applications appear at the end of each chapter that illustrate the full cycle of model setup, mathematical analysis, and interpretation. The author has used the material over many decades in a variety of teaching and mentoring settings, including special topics courses and seminars in mathematical modeling, mathematical biology, and dynamical systems.
Undergraduate and graduate students interested in discrete time models in population, epidemic, and evolutionary dynamics.
Softcover ISBN: 978-1-4704-7033-3
Product Code: STML/107
Expected availability date: April 17, 2024
Student Mathematical Library Volume: 107;
MSC: Primary 51; 11; 32; 52;
This book offers a gentle introduction to the geometry of numbers from a modern Fourier-analytic point of view. One of the main themes is the transfer of geometric knowledge of a polytope to analytic knowledge of its Fourier transform. The Fourier transform preserves all of the information of a polytope, and turns its geometry into analysis. The approach is unique, and streamlines this emerging field by presenting new simple proofs of some basic results of the field. In addition, each chapter is fitted with many exercises, some of which have solutions and hints in an appendix. Thus, an individual learner will have an easier time absorbing the material on their own, or as part of a class.
Overall, this book provides an introduction appropriate for an advanced undergraduate, a beginning graduate student, or researcher interesting in exploring this important expanding field.
Undergraduate and graduate students and researchers interested in analysis and periodical structures.