1st ed. 2016, XXXVIII, 643 p. 51 illus., 8 illus. in color.
Hardcover
ISBN 978-3-319-44704-9
Series: Springer Monographs in Mathematics
* The book builds on the one-dimensional theory of complex
dimensions (the case of fractal strings)
* The content is self-contained and relatively easily accessible to
a wide variety of readers with different levels of mathematical
maturity, beginning at the advanced graduate level
The exposition is gentle with several instructive examples and illustrations
This monograph gives a state-of-the-art and accessible treatment of a new general
higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets
of Euclidean spaces, as well as for their natural generalization, relative fractal drums.
It provides a significant extension of the existing theory of zeta functions for fractal
strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension.
Two new classes of fractal zeta functions are introduced, namely, the distance and tube
zeta functions of bounded sets, and their key properties are investigated. The theory is
developed step-by-step at a slow pace, and every step is well motivated by numerous
examples, historical remarks and comments, relating the objects under investigation
to other concepts. Special emphasis is placed on the study of complex dimensions
of bounded sets and their connections with the notions of Minkowski content and
Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time
that essential singularities of fractal zeta functions can naturally emerge for various classes
of fractal sets and have a significant geometric effect. The theory developed in this book
leads naturally to a new definition of fractality, expressed in terms of the existence of
underlying geometric oscillations or, equivalently, in terms of the existence of nonreal
complex dimensions.
The connections to previous extensive work of the first author and his collaborators
on geometric zeta functions of fractal strings are clearly explained. Many concepts are
discussed for the first time, making the book a rich source of new thoughts and ideas to
be developed further. The book contains a large number of open problems and describes
many possible directions for further research. The beginning chapters may be used as a
part of a course on fractal geometry.
1st ed. 2016, XII, 436 p.
Softcover
ISBN 978-3-319-45631-7
Series: Universitext
* Tailored to novice mathematicians and non-specialists who wish to
learn functional analysis with minimal prerequisites
* Contains many interesting examples and challenging exercises
* Covers a selection of topics that, while decades old, are of lasting
value to both pure and applied mathematics
* Presents an outstanding treatment of Banach spaces and operator
theory, being written by a specialist in these areas
This book provides a unique path for graduate or advanced undergraduate students to
begin studying the rich subject of functional analysis with fewer prerequisites than is
normally required. The text begins with a self-contained and highly efficient introduction
to topology and measure theory, which focuses on the essential notions required for the
study of functional analysis, and which are often buried within full-length overviews of
the subjects. This is particularly useful for those in applied mathematics, engineering, or
physics who need to have a firm grasp of functional analysis, but not necessarily some of
the more abstruse aspects of topology and measure theory normally encountered. The
reader is assumed to only have knowledge of basic real analysis, complex analysis, and
algebra.
The latter part of the text provides an outstanding treatment of Banach space theory and
operator theory, covering topics not usually found together in other books on functional
analysis. Written in a clear, concise manner, and equipped with a rich array of interesting
and important exercises and examples, this book can be read for an independent
study, used as a text for a two-semester course, or as a self-contained reference for the
researcher.
2nd ed. 2016, XXIV, 200 p. 6 illus., 3 illus. in color.
Hardcover
ISBN 978-3-319-43372-1
Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 89
* Covers the latest advances in the study of the Monge-Ampere
equation and its applications
* Includes new chapters on the Harnack inequality for the linearized
Monge-Ampere equation and on interior Holder estimates for second
derivatives
* Bibliographic notes provided at the end of each chapter for further
exploration of topics
Now in its second edition, this monograph explores the Monge-Ampere equation and
the latest advances in its study and applications. It provides an essentially self-contained
systematic exposition of the theory of weak solutions, including regularity results by L.
A. Caffarelli. The geometric aspects of this theory are stressed using techniques from
harmonic analysis, such as covering lemmas and set decompositions. An effort is made
to present complete proofs of all theorems, and examples and exercises are offered to
further illustrate important concepts. Some of the topics considered include generalized
solutions, non-divergence equations, cross sections, and convex solutions. New to this
edition is a chapter on the linearized Monge-Ampere equation and a chapter on interior
Holder estimates for second derivatives. Bibliographic notes, updated and expanded
from the first edition, are included at the end of every chapter for further reading on
Monge-Ampere-type equations and their diverse applications in the areas of differential
geometry, the calculus of variations, optimization problems, optimal mass transport,
and geometric optics. Both researchers and graduate students working on nonlinear
differential equations and their applications will find this to be a useful and concise
resource.
Proceedings of the Conference on 60 Years of Yang-Mills Gauge Field Theories:
C N Yang's Contributions to Physics
Nanyang Technological University, Singapore, 25- 28 May 2015
During the last six decades, Yang-Mills theory has increasingly become
the cornerstone of theoretical physics. It is seemingly the only fully
consistent relativistic quantum many-body theory in four space-time dimensions.
As such it is the underlying theoretical framework for the Standard Model
of Particle Physics, which has been shown to be the correct theory at the
energies we now can measure. It has been investigated also from many other
perspectives, and many new and unexpected features have been uncovered
from this theory. In recent decades, apart from high energy physics, the
theory has been actively applied in other branches of physics, such as
statistical physics, condensed matter physics, nonlinear systems, etc.
This makes the theory an indispensable topic for all who are involved in
physics.
The conference celebrated the exceptional achievements using Yang?Mills
theory over the years but also many other truly remarkable contributions
to different branches of physics from Prof C N Yang. This volume collects
the invaluable talks by Prof C N Yang and the invited speakers reviewing
these remarkable contributions and their importance for the future of physics.
The Future of Physics ? Revisited (C N Yang)
Quantum Chromodynamics The Perfect Yang-Mills Gauge Field Theory (David Gross)
Maximally Supersymmetric Yang-Mills Theory: The Story of N = 4 Yang-Mills Theory (Lars Brink)
The Lattice and Quantized Yang-Mills Theory (Michael Creutz)
Yang-Mills Theories at High Energy Accelerators (George Sterman)
Yang-Mills Theory at 60: Milestones, Landmarks and Interesting Questions (Ling-Lie Chau)
Discovery of the First Yang-Mills Gauge Particle The Gluon (Sau Lan Wu)
Yang-Mills Gauge Theory and Higgs Particle (Tai Tsun Wu & Sau Lan Wu)
Scenario for the Renormalization in the 4D Yang-Mills Theory (L D Faddeev)
Statistical Physics in the Oeuvre of Chen Ning Yang (Michael E Fisher)
Quantum Vorticity in Nature (Kerson Huang)
Yang-Mills Theory and Fermionic Path Integrals (Kazuo Fujikawa)
Yang-Mills Gauge Theory and the Higgs Boson Family (Ngee-Pong Chang)
On the Physics of the Minimal Length: The Questions of Gauge Invariance (Lay Nam Chang, Djordje Minic, Ahmed Roman, Chen Sun & Tatsu Takeuchi)
Generalization of the Yang-Mills Theory (G Savvidy)
Some Thoughts about Yang-Mills Theory (A Zee)
Gauging Quantum Groups: Yang?Baxter Joining Yang-Mills (Yong-Shi Wu)
The Framed Standard Model (I) A Physics Case for Framing the Yang-Mills Theory? (Chan Hong-Mo & Tsou Sheung Tsun)
The Framed Standard Model (II)? A First Test Against Experiment (Chan Hong-Mo & Tsou Sheung Tsun)
On the Study of the Higgs Properties at a Muon Collider (Mario Greco)
Aharonov?Bohm Types of Phases in Maxwell and Yang-Mills Field Theories (Bruce H J McKellar)
Yang-Mills for Historians and Philosophers (R P Crease)
Gauge Concepts in Theoretical Applied Physics (Seng Ghee Tan & Mansoor B A Jalil)
Yang-Yang Equilibrium Statistical Mechanics: A Brilliant Method (Xi-Wen Guan & Yang-Yang Chen)
Chern-Simons Theory, Vassiliev Invariants, Loop Quantum Gravity and Functional Integration Without Integration (Louis H Kauffman)
The Scattering Equations and Their Off-Shell Extension (York-Peng Yao)
Feynman Geometries (Sen Hu & Andrey Losev)
Particle Accelerator Development: Selected Examples (Jie Wei)
A New Storage-Ring Light Source (Alex Chao)
New Contributions to Physics by Prof C N Yang: 2009-2011 (Zhong-Qi Ma)
Brief Overview of C N Yang's 13 Important Contributions to Physics (Yu Shi)
Readership: Graduate students and scientists working in high energy physics, statistical physics and condensed matter physics.
A First Course in Linear Algebra is written by two experts from algebra
who have more than 20 years of experience in algebra, linear algebra and
number theory. It prepares students with no background in Linear Algebra.
Students, after mastering the materials in this textbook, can already understand
any Linear Algebra used in more advanced books and research papers in Mathematics
or in other scientific disciplines.
This book provides a solid foundation for the theory dealing with finite
dimensional vector spaces. It explains in details the relation between
linear transformations and matrices. One may thus use different viewpoints
to manipulate a matrix instead of a one-sided approach. Although most of
the examples are for real and complex matrices, a vector space over a general
field is briefly discussed. Several optional sections are devoted to applications
to demonstrate the power of Linear Algebra.
Preface
Vector Spaces
Bases and Dimension
Linear Transformations and Matrices
Elementary Matrix Operations
Diagonalization
Canonical Forms
Inner Product Spaces
Readership: Undergraduates who are interested in learning linear algebra and its applications.
Natural numbers are the oldest human inventions. This volume describes
their nature, laws, history and current status. The first five chapters
contain not only the basics of elementary number theory for the convenience
of teaching and continuity of reading, but also many latest research results.
For the first time in history, the Chinese Remainder Theorem is renamed
the Qin Jiushao Theorem to give him the full credit for his establishment
of this famous theorem in number theory. Chapter 6 is about the fascinating
congruence modulo an integer power, and Chapter 7 introduces a new problem
extracted by the author from the classical problems of number theory, which
is out of the combination of additive number theory and multiplicative
number theory.In this volume, there is supplementary material after each
section to broaden the reader's knowledge and imagination. It either discusses
the rudiments of some aspects or introduces new topics, such as the perfect
number problem, Goldbach's conjecture, the twin prime conjecture, the 3x
+ 1 problem, Waring's problem, Catalan's conjecture, Euler's conjecture,
Fermat's Last Theorem, etc.
Originally published in Chinese as in 2014, The Book of Numbers is written
for anyone who loves natural numbers. The author is not only a mathematician,
but also a literary and science writer, with more than 20 books published,
many of which were translated into 20 languages.
The Mystery of Natural Numbers
The Concept of Congruence
Congruence
Quadratic Residue
The nth Power Residues
Congruence Modulo Integer Power
Additive and Multiplicative Number Theory
Readership: Researchers and students on number theory and general mathematics.