288pp Jan 2018
ISBN: 978-981-3231-59-7 (hardcover)
The aim of the book is to give a smooth analytic continuation from basic subjects including linear algebra, group theory, Hilbert space theory, etc. to number theory. With plenty of practical examples and worked-out exercises, and the scope ranging from these basic subjects made applicable to number-theoretic settings to advanced number theory, this book can then be read without tears. It will be of immense help to the reader to acquire basic sound skills in number theory and its applications.
Number theory used to be described as the queen of mathematics, that is, there is no practical use. However, with the development of computers and the security of internet communications, the importance of number theory has been exponentially increasing daily. The raison d'etre of the present book in this situation is that it is extremely reader-friendly while keeping the rigor of serious mathematics and in-depth analysis of practical applications to various subjects including control theory and pseudo-random number generation. The use of operators is prevailing rather abundantly in anticipation of applications to electrical engineering, allowing the reader to master these skills without much difficulty. It also delivers a very smooth bridging between elementary subjects including linear algebra and group theory (and algebraic number theory) for the reader to be well-versed in an efficient and effortless way. One of the main features of the book is that it gives several different approaches to the same topic, helping the reader to gain deeper insight and comprehension. Even just browsing through the materials would be beneficial to the reader.
Linear Algebraic Approach to Algebraic Number Theory
Group-Theoretic Approach to Algebraic Number Theory
Arithmetic Functions and Stieltjes Integrals
Number Theory in the Unit Disc
Hilbert Space and Number Theory
Number Theory and Control Theory
280pp Feb 2018
ISBN: 978-981-3231-06-1 (hardcover)
Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots.
This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications.
Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.
Introduction
Categories, Functors and Natural Transformations
Limits and Colimits
Adjunctions and Monads
Applications in Algebra
Applications in Topology and Algebraic Topology
Applications in Homological Algebra
Hints at Higher Dimensional Category Theory
References
Indices
Graduate students and researchers of mathematics, computer science, physics.
Essential Textbooks in Mathematics
264pp Feb 2018
ISBN: 978-1-78634-471-7 (hardcover)
Introduction to Number Theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers.
The book begins with introductory material, including uniqueness of factorization of integers and polynomials. Subsequent topics explore quadratic reciprocity, Hensel's Lemma, p-adic powers series such as exp(px) and log(1+px), the Euclidean property of some quadratic rings, representation of integers as norms from quadratic rings, and Pell's equation via continued fractions.
Throughout the five chapters and more than 100 exercises and solutions, readers gain the advantage of a number theory book that focuses on doing calculations. This textbook is a valuable resource for undergraduates or those with a background in university level mathematics.
About the Author
Acknowledgments
Introduction
Euclid's Algorithm
Polynomial Rings
Congruences Modulo Prime Numbers
p-Adic Methods in Number Theory
Diophantine Equations and Quadratic Rings
Solutions to Exercises
Bibliography
Index
Students and educators in a university course on number theory.
Series on Knots and Everything: Volume 62
950pp Apr 2018
ISBN: 978-981-3233-55-3 (hardcover)
In this richly illustrated book, the contributors describe the Mereon Matrix, its dynamic geometry and topology. Through the definition of eleven First Principles, it offers a new perspective on dynamic, whole and sustainable systems that may serve as a template information model. This template has been applied to a set of knowledge domains for verification purposes: pre-life-evolution, human molecular genetics and biological evolution, as well as one social application on classroom management.
The importance of the book comes in the following ways:
The dynamics of the geometry unites all Platonic and Kepler Solids into one united structure and creates 11 unique trefoil knots. Its topology is directly related to the dynamics of the polyhedra.
The Mereon Matrix is an approach to the unification of knowledge that relies on whole systems modelling. it is a framework charting the emergence of the Platonic and Kepler solids in a sequential, emergent growth process that describes a non-linear whole system, and includes a process of 'breathing' as well as multiplying ('birthing');
This dynamic/kinematic structure provides insight and a new approach to General Systems Theory and non-linear science, evolving through a new approach to polyhedral geometry. A set of 11 First Principles is derived from the structure, topology and dynamics of the Mereon Matrix, which serve well as a template information system.
The Mereon Matrix is related to a large number of systems, physical, mathematical, and philosophical, and in linking these systems, provides access to new relationships among them by combining geometry with process thinking. The new perspective on systems is hypothesized as universal ? this is, applicable in all areas of science, natural and social. Such applicability has been demonstrated for applications as diverse as pre-life evolution, biological evolution and human molecular genetics, as well as a classroom management system for the educational system.
Care has been taken to use images and languaging that are understandable across domains, connecting diverse disciplines, while making this complex system easily accessible.
Prologues:
Sustainability: Mathematical Elegance, Solid Science and Social Grace (L Dennis and L H Kauffman)
Lynnclaire Dennis & R Buckminster Fuller Investigation (R W Gray)
The Matrix We Call Mereon (L H Kauffman)
First Things First :
Building on the Known: A Quintessential Jitterbug (L Dennis, J Brender McNair, N J Woolf and L H Kauffman)
Methodology (J Brender McNair and L Dennis)
Philosophical Thoughts and Thinking Aloud Allowed (L Dennis)
Belonging ? Education as Transformation (L Dennis)
Meme, Pattern and Perspective (L Dennis, N J Woolf and L H Kauffman)
Including and Beyond the Point:
The Context ? Form Informing Function (L Dennis, J Brender McNair, N J Woolf and L H Kauffman)
Flow and Scale (L Dennis and L H Kauffman)
The Core ? Sharp Distinctions to Elegant Curves (L Dennis and L H Kauffman)
Connections, Ligatures and Knots:
Mereon Thoughts ? Knots and Beyond (L H Kauffman)
The Mereon Trefoil ? Asymmetrical with Perfect Symmetry (L Dennis)
Applying Mereon to Knowledge Domains:
Exploring the Mereon Matrix (and Beyond) with the CymaScope Technology (L Dennis and P McNair)
The Origin of Matter: Life, Learning and Survival (N J Woolf and L Dennis)
ATCG ? An Applied Theory for Human MoleCular Genetics (J Brender McNair, P McNair, L Dennis and Z Tumer)
A Unifying Theory of Biological Evolution (J Brender McNair)
Epilogue: Divine Architecture & the Art of Becoming Human (L Dennis)
Reference Material:
Definition of Key Concepts (L Dennis and J Brender McNair)
The Emergent Science of Cymatics (J S Reid)
Bibliography
Researchers in knot theory, geometry and systems theory.
550pp Jun 2018
ISBN: 978-981-3231-52-8 (hardcover)
This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder?Narasimhan and Atiyah?Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE.
This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research.
Non-Abelian Zeta Functions
Rank Two Zeta Functions
Eisenstein Periods and Multiple Zeta Functions
Zeta Functions for Reductive Groups
Symmetries and Riemann Hypothesis
Stability and Riemann Hypothesis
Arithmetic Cohomology in Dimension Two
Graduate students and researchers in the theory of zeta functions.
400pp Nov 2018
ISBN: 978-981-3230-79-8 (hardcover)
Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics.
Originally, M L Lapidus and M van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M L Lapidus and H Maier on inverse spectral problems for fractal strings and the Riemann hypothesis.
One of the main themes of the book is to provide a rigorous framework within which the corresponding question "Can one hear the shape of a fractal string?" or, equivalently, "Can one obtain information about the geometry of a fractal string, given its spectrum?" can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator.
The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space.
It is shown that the quasi-invertibilty of the spectral operator is intimately connected to the existence of critical zeros of the Riemann zeta function, leading to a new spectral and operator-theoretic reformulation of the Riemann hypothesis. Accordingly, the spectral operator is quasi-invertible for all values of the dimensional parameter c in the critical interval (0,1) (other than in the midfractal case when c =1/2) if and only if the Riemann hypothesis (RH) is true. A related, but seemingly quite different, reformulation of RH, due to the second author and referred to as an 'asymmetric criterion for RH', is also discussed in some detail: namely, the spectral operator is invertible for all values of c in the left-critical interval (0,1/2) if and only if RH is true.
These spectral reformulations of RH also led to the discovery of several 'mathematical phase transitions' in this context, for the shape of the spectrum, the invertibilty, the boundedness or the unboundedness of the spectral operator, and occurring either in the midfractal case or in the most fractal case when the underlying fractal dimension is equal to ? or 1, respectively. In particular, the midfractal dimension c=1/2 is playing the role of a critical parameter in quantum statistical physics and the theory of phase transitions and critical phenomena.
Furthermore, the authors provide a 'quantum analog' of Voronin's classical theorem about the universality of the Riemann zeta function. Moreover, they obtain and study quantized counterparts of the Dirichlet series and of the Euler product for the Riemann zeta function, which are shown to converge (in a suitable sense) even inside the critical strip.
For pedagogical reasons, most of the book is devoted to the study of the quantized Riemann zeta function. However, the results obtained in this monograph are expected to lead to a quantization of most classic arithmetic zeta functions, hence, further 'naturally quantizing' various aspects of analytic number theory and arithmetic geometry.
The book should be accessible to experts and non-experts alike, including mathematics and physics graduate students and postdoctoral researchers, interested in fractal geometry, number theory, operator theory and functional analysis, differential equations, complex analysis, spectral theory, as well as mathematical and theoretical physics. Whenever necessary, suitable background about the different subjects involved is provided and the new work is placed in its proper historical context. Several appendices supplementing the main text are also included.
Overview
Preface
List of Figures
List of Tables
Introduction
Generalized Fractal Strings and Complex Dimensions
Direct and Inverse Spectral Problems for Fractal Strings
The Heuristic Spectral Operator ac
The Infinitesimal Shift ∂c
The Spectrum of the Infinitesimal Shift ∂c
The Spectral Operator ac = ζ(∂c): Quantized Dirichlet Series, Euler Product and Analytic Continuation
A Spectral Reformulation of the Riemann Hypothesis
Zeta Values, Riemann Zeros and Phase Transitions for ac = ζ(∂c)
A Quantum Analog of the Universality of ζ(s)
Concluding Comments and Future Research Directions
Riemann's Explicit Formula
Natural Boundary Conditions for ∂c
The Momentum Operator and Normality of ∂c
Finiteness of Some Useful Integrals
The Spectral Mapping Theorem
The Range and Growth of ζ(s) on Vertical Lines
Some Extensions of the Universality of ζ(s)
Acknowledgements
Bibliography
Author Index
Subject Index
Index of Symbols
Conventions
Researchers in fractal geometry, analysis and differential equations, complex analysis, harmonic analysis, operator theory, spectral theory, mathematical physics, algebra and number theory.