Complexity increases with increasing system size in everything from organisms
to organizations. The nonlinear dependence of a system’s functionality on its
size, by means of an allometry relation, is argued to be a consequence of their
joint dependency on complexity (information). In turn, complexity is proven to
be the source of allometry and to provide a new kind of force entailed by a
system‘s information gradient. Based on first principles, the scaling behavior of
the probability density function is determined by the exact solution to a set of
fractional differential equations. The resulting lowest order moments in system
size and functionality gives rise to the empirical allometry relations. Taking
examples from various topics in nature, the book is of interest to researchers in
applied mathematics, as well as, investigators in the natural, social, physical and
life sciences.
Complexity
Empirical allometry
Statistics, scaling and simulation
Allometry theories
Strange kinetics
Fractional probability calculus
* First ever mathematical treatment of allometry independent of disciplines.
* Employs fractional calculus to capture complexity in allometry.
* Discusses the role of scaling in the control of complex systems in allometry relations.
Bruce J. West, US Army Research Office, Cary, US
The philosophy of the book, which makes it quite distinct from many existing
texts on the subject, is based on treating the concepts of measure and integration
starting with the most general abstract setting and then introducing and studying
the Lebesgue measure and integration on the real line as an important particular
case.
The book consists of nine chapters and appendix, with the material flowing from
the basic set classes, throughmeasures,
outer measures and the general
procedure of measure extension, through measurable functionsandvarious
types of convergence of sequences of such based on the idea of measure, to the
fundamentals of the abstract Lebesgue integration, the basic limit theorems, and
the comparison of the Lebesgue and Riemann integrals. Also, studied are Lp
spaces, the basics of normed vector spaces, and signed measures. The novel
approach based on the Lebesgue measure and integration theory is applied to
develop a better understanding of differentiation and extend the classical total
change formula linking differentiation with integration to a substantially wider
class of functions.
Being designed as a text to be used in a classroom, the book constantly calls for
the student's actively mastering the knowledge of the subject matter. There are
problems at the end of each chapter, starting with Chapter 2 and totaling at 125.
Many important statements are given as problems and frequently referred to in
the main body. There are also 358 Exercises throughout the text, including
Chapter 1 and the Appendix, which require of the student to prove or verify a
statement or an example, fill in certain details in a proof, or provide an
intermediate step or a counterexample. They are also an inherent part of the
material. More difficult problems are marked with an asterisk, many problems
and exercises are supplied with ``existential'' hints.
The book is generous on Examples and contains numerous Remarks
accompanying definitions, examples, and statements to discuss certain
subtleties, raise questions on whether the converse assertions are true, whenever
appropriate, or whether the conditions are essential.
Withplenty of examples, problems, and exercises, this well-designed text is
ideal for a one-semester Master's level graduate course on real analysis with
emphasis on the measure and integration theory for students majoring in
mathematics, physics, computer science, and engineering.
* A concise but profound and detailed presentation of the basics of real analysis
with emphasis on the measure and integration theory.
*Designed for a one-semester graduate course, with plethora of examples,
problems, and exercises.
* Is of interest to students and instructors in mathematics, physics, computer
science, and engineering.
* Prepares the students for more advanced courses in functional analysis and
operator theory.
Preliminaries
Basic Set Classes
Measures
Extension of Measures
Measurable Functions
Abstract Lebesgue Integral
Lp Spaces
Differentiation and Integration
Signed Measures
The Axiom of Choice and Equivalents
* A lucid, pedagogically-written book on real analysis
* The presented material is exactly designed for a one-semester course, with
plenty of examples and exercises
* Of interest to students and lecturers in mathematics, physics, and engineering
Marat V. Markin, California State University, Fresno, USA
Periodic differential equations appear in many contexts such as in the theory of
nonlinear oscillators, in celestial mechanics, or in population dynamics with
seasonal effects. The most traditional approach to study these equations is based
on the introduction of small parameters, but the search of nonlocal results leads
to the application of several topological tools. Examples are fixed point
theorems, degree theory, or bifurcation theory. These well-known methods are
valid for equations of arbitrary dimension and they are mainly employed to
prove the existence of periodic solutions.
Following the approach initiated by Massera, this book presents some more
delicate techniques whose validity is restricted to two dimensions. These
typically produce additional dynamical information such as the instability of
periodic solutions, the convergence of all solutions to periodic solutions, or
connections between the number of harmonic and subharmonic solutions.
The qualitative study of periodic planar equations leads naturally to a class of
discrete dynamical systems generated by homeomorphisms or embeddings of the
plane. To study these maps, Brouwer introduced the notion of a translation arc,
somehow mimicking the notion of an orbit in continuous dynamical systems.
The study of the properties of these translation arcs is full of intuition and often
leads to "non-rigorous proofs". In the book, complete proofs following ideas
developed by Brown are presented and the final conclusion is the Arc
Translation Lemma, a counterpart of the Poincare?Bendixson theorem for
discrete dynamical systems.
Applications to differential equations and discussions on the topology of the
plane are the two themes that alternate throughout the five chapters of the book.
* A detailed study of periodic differential equations, planar topology, and
discrete dynamics
* Presents refined qualitative and topological results
* Of interest to researchers and graduate students in differential equations and
dynamical systems
De Gruyter Series in Nonlinear Analysis and Applications 29
Approx. xii, 200 pages, 91 Figures (bw)
Hardcover:
ISBN 978-3-11-055040-5
March 2019
https://doi.org/10.1142/11217 | March 2019
Pages: 232
Since its birth in Poincare's seminal 1894 "Analysis Situs", topology has become a cornerstone of mathematics. As with all beautiful mathematical concepts, topology inevitably ? resonating with that Wignerian principle of the effectiveness of mathematics in the natural sciences ? finds its prominent role in physics. From Chern?Simons theory to topological quantum field theory, from knot invariants to Calabi?Yau compactification in string theory, from spacetime topology in cosmology to the recent Nobel Prize winning work on topological insulators, the interactions between topology and physics have been a triumph over the past few decades.
In this eponymous volume, we are honoured to have contributions from an assembly of grand masters of the field, guiding us with their world-renowned expertise on the subject of the interplay between "Topology" and "Physics". Beginning with a preface by Chen Ning Yang on his recollections of the early days, we proceed to a novel view of nuclei from the perspective of complex geometry by Sir Michael Atiyah and Nick Manton, followed by an entree toward recent developments in two-dimensional gravity and intersection theory on the moduli space of Riemann surfaces by Robbert Dijkgraaf and Edward Witten; a study of Majorana fermions and relations to the Braid group by Louis H Kauffman; a pioneering investigation on arithmetic gauge theory by Minhyong Kim; an anecdote-enriched review of singularity theorems in black-hole physics by Sir Roger Penrose; an adventure beyond anyons by Zhenghan Wang; an apercu on topological insulators from first-principle calculations by Haijun Zhang and Shou-Cheng Zhang; finishing with synopsis on quantum information theory as one of the four revolutions in physics and the second quantum revolution by Xiao-Gang Wen. We hope that this book will serve to inspire the research community.
Preface Early Examples of Topological Concepts in Physics (C N Yang)
Complex Geometry of Nuclei and Atoms (M F Atiyah and N S Manton)
Developments in Topological Gravity (Robbert Dijkgraaf and Edward Witten)
Majorana Fermions and Representations of the Braid Group (Louis H Kauffman)
Arithmetic Gauge Theory: A Brief Introduction (Minhyong Kim)
Singularity Theorems (Roger Penrose)
Beyond Anyons (Zhenghan Wang)
Four Revolutions in Physics and the Second Quantum Revolution A Unification of Force and Matter by Quantum Information (Xiao-Gang Wen)
Topological Insulators from the Perspective of First-principles Calculations (Haijun Zhang and Shou-Cheng Zhang)
Appendix SO symmetry in a Hubbard Model (C N Yang and Shou-Cheng Zhang)
Readership: Senior undergraduate student; graduate students; researchers.