Format: Hardback, 490 pages, height x width: 235x155 mm, 61 Tables, color; 64 Illustrations, color;
63 Illustrations, black and white; X, 490 p. 127 illus., 64 illus. in color.
Series: Operator Theory: Advances and Applications 293
Pub. Date: 19-Oct-2023
ISBN-13: 9783662678701
This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schroedinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.
Very personal introduction.- How to define differential operators on metric graphs.- Vertex conditions.- Elementary spectral properties of quantum graphs.- The characteristic equation.- Standard Laplacians and secular polynomials.- Reducibility of secular polynomials.- The trace formula.- Trace formula and inverse problems.- Arithmetic structure of the spectrum and crystalline measures.- Quadratic forms and spectral estimates.- Spectral gap and Dirichlet ground state.- Higher eigenvalues and topological perturbations.- Ambartsumian type theorems.- Further theorems inspired by Ambartsumian.- Magnetic fluxes.- M-functions: definitions and examples.- M-functions: properties and first applications.- Boundary control: BC-method.- Inverse problems for trees.- Boundary Control for graphs with cycles: dismantling graphs.- Magnetic Boundary Control I: graphs with several cycles.- Magnetic Boundary Control II: graphs on one cycle and dependent subtrees.- Discrete graphs.
https://doi.org/10.1142/13366 | July 2023
Pages: 364
ISBN: 978-981-127-475-6 (hardcover)
This book deals with the role played by symmetry in the understanding of the physical world, beginning with the notion of geometric symmetries of the ancient Greek philosophers and mathematicians. The recognition of the existence of symmetries led to the notion of transformations, which led from one state of the system to another. It was then realized that such transformations, under the operation of multiplication, constitute an interesting set, whose study led to the branch of mathematics known as Group Theory. With the emergence of quantum mechanics, this theory became much more interesting and led to some additional applications. The theory got another boost with the need for of the internal degrees of freedom in describing physical systems. This way the notion of symmetry is no longer purely geometric and evolved into a useful tool in the study of all physical sciences.
For practical reasons as well as pedagogical reasons, group theory is usually split into two parts. The first deals with discrete groups, with the group elements being countable, usually finite in number, while the second deals with continuous groups, whose elements depend on continuous parameters. This volumefocuses the discussion on discrete groups.
Given that group theory should be presented from a unified perspective, involving not only the mathematical rigor and beauty of symmetries, but also the ability to use it as a tool for applications, either currently popular or expected to become so in the future, this approach will surely be more beneficial to the dedicated reader. It is not intended for those who would like to just look up a formula or use the results of a table, without understanding their derivation.
The Role of Symmetries in Physics ? A Prelude
Introduction to Discrete Groups
Discrete Groups ? Basic Theorems
Elements of Representation Theory
Representation Reduction ? Schur's Lemmas ? Character Tables
Simple Applications in Quantum Mechanics
Symmetries and Normal Modes of Oscillation
Space Groups
Irreducible Representations of Space Groups
Normal Modes in Crystals
The Symmetric Group Sn
Further Applications in Molecular Physics and Crystal Structure ? Crystal Harmonics
Applications of Discrete Groups in Particle Physics
Exotic Discrete Groups for Quantum Mechanics ? Field Theory
Appendices:
Proofs of Various Theorems
Representation Reduction Via a Chain of Group Operators
Generators and Character Tables of Point Groups
Undergraduate and graduate students in related field for their course and
researchers in group theory.
Pages: 636
ISBN: 978-981-127-784-9 (hardcover)
The first half of the book walks the reader through methods of counting, both direct elementary methods and the more advanced method of generating functions. Then, in the second half of the book, the reader learns how to apply these methods to fascinating objects, such as graphs, designs, random variables, partially ordered sets, and algorithms. In short, the first half emphasizes depth by discussing counting methods at length; the second half aims for breadth, by showing how numerous the applications of our methods are.
New to this fifth edition of A Walk Through Combinatorics is the addition of Instant Check exercises ? more than a hundred in total ? which are located at the end of most subsections. As was the case for all previous editions, the exercises sometimes contain new material that was not discussed in the text, allowing instructors to spend more time on a given topic if they wish to do so. With a thorough introduction into enumeration and graph theory, as well as a chapter on permutation patterns (not often covered in other textbooks), this book is well suited for any undergraduate introductory combinatorics class.
Foreword
Preface
Acknowledgments
Basic Methods:
Seven is More Than Six. The Pigeon-Hole Principle
One Step at a Time. The Method of Mathematical Induction
Enumerative Combinatorics:
There are a Lot of Them. Elementary Counting Problems
No Matter How You Slice It. The Binomial Theorem and Related Identities
Divide and Conquer. Partitions
Not So Vicious Cycles. Cycles in Permutations
You Shall Not Overcount. The Sieve
A Function is Worth Many Numbers. Generating Functions
Graph Theory:
Dots and Lines. The Origins of Graph Theory
Staying Connected. Trees
Finding a Good Match. Coloring and Matching
Do Not Cross. Planar Graphs
Horizons:
Does It Clique? Ramsey Theory
So Hard to Avoid. Subsequence Conditions on Permutations
Who Knows What It Looks Like, But It Exists. The Probabilistic Method
At Least Some Order. Partial Orders and Lattices
As Evenly as Possible. Block Designs and Error Correcting Codes
Are They Really Different? Counting Unlabeled Structures
The Sooner the Better. Combinatorial Algorithms
Does Many Mean More Than One? Computational Complexity
Bibliography
Index
This book is primarily suitable for advanced undergraduate students and instructors teaching an introductory combinatorics class. However, it would also be applicable to graduate students in fields other than combinatorics who need an introduction to Combinatorics.
Pages: 248
ISBN: 978-981-127-669-9 (hardcover)
This book presents a novel development of fundamental and fascinating aspects of algebraic topology and mathematical physics: "extra-ordinary" and further generalized cohomology theories enhanced to "twisted" and differential-geometric form, with focus on, firstly, their rational approximation by generalized Chern character maps, and then, the resulting charge quantization laws in higher n-form gauge field theories appearing in string theory and the classification of topological quantum materials.
Although crucial for understanding famously elusive effects in strongly interacting physics, the relevant higher non-abelian cohomology theory ("higher gerbes") has had an esoteric reputation and remains underdeveloped.
Devoted to this end, this book's theme is that various generalized cohomology theories are best viewed through their classifying spaces (or moduli stacks) ? not necessarily infinite-loop spaces ? from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby not only uniformly subsuming the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applicable in the hitherto elusive generality of (twisted, differential, and) non-abelian cohomology.
In laying out this result with plenty of examples, this book provides a modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories; 2. generalized cohomology in its homotopical incarnation; 3. rational homotopy theory seen via homotopy Lie theory, whose fundamental theorem we recast as a (twisted) non-abelian de Rham theorem, which naturally induces the (twisted) non-abelian character map.
Preface
Introduction
Non-abelian cohomology:
Model Category Theory
Non-abelian Cohomology Theories
Twisted Non-abelian Cohomology
Non-abelian de Rham Cohomology:
Dgc-algebras and L∞-algebras
?-rational Homotopy Theory
Non-abelian de Rham Theorem
The (Differential) Non-abelian Character Map:
Chern-Dold Character
Chern-Weil Homomorphism
Cheeger-Simons Homomorphism
The Twisted (Differential) Non-abelian Character Map:
Twisted Chern Character on Higher K-theory
Twisted Differential Non-abelian Character
Twisted Character on Twisted Differential Cohomotopy
Bibliography
Index
Graduate students, researchers in differential geometry, algebraic topology, and their applications to physics. Advanced undergraduate in mathematics and physics, novice researchers interested in a modern introduction to homotopy theory and techniques.
Pages: 280
ISBN: 978-981-127-766-5 (hardcover)
ISBN: 978-981-127-870-9 (softcover)
This book is translated from the Chinese version published by Science Press, Beijing, China, in 2017. It was written for the Chern class in mathematics of Nankai University and has been used as the textbook for the course Abstract Algebra for this class for more than five years. It has also been adapted in abstract algebra courses in several other distinguished universities across China.
The aim of this book is to introduce the fundamental theories of groups, rings, modules, and fields, and help readers set up a solid foundation for algebra theory. The topics of this book are carefully selected and clearly presented. This is an excellent mathematical exposition, well-suited as an advanced undergraduate textbook or for independent study. The book includes many new and concise proofs of classical theorems, along with plenty of basic as well as challenging exercises.
Textbook for an advanced undergraduate course on abstract algebra. Reference book for graduate students in physics, engineering, and computer science. Any students interested in abstract algebra.
Pages: 292
ISBN: 978-981-127-809-9 (hardcover)
Stirling numbers are one of the most known classes of special numbers in Mathematics, especially in Combinatorics and Algebra. They were introduced by Scottish mathematician James Stirling (1692?1770) in his most important work, Differential Method with a Tract on Summation and Interpolation of Infinite Series (1730). Stirling numbers have rich history; many arithmetic, number-theoretical, analytical and combinatorial connections; numerous classical properties; as well as many modern applications.
This book collects together much of the scattered material on the two subclasses of Stirling numbers to provide a holistic overview of the topic. From the combinatorial point of view, Stirling numbers of the second kind S(n,k) count the number of ways to partition a set of n different objects (i.e., a given n-set) into k non-empty subsets. Stirling numbers of the first kind s(n, k) give the number of permutations of n elements with k disjoint cycles. Both subclasses of Stirling numbers play an important role in Algebra: they form the coefficients, connecting well-known sets of polynomials.
This book is suitable for students and professionals, providing a broad perspective of the theory of this class of special numbers, and many generalizations and relatives of Stirling numbers, including Bell numbers and Lah numbers. Throughout the book, readers are presented with exercises to test and cement their understanding.
Notations
Preface
Preliminaries
Combinatorics of Partitions
Stirling Numbers of the Second Kind
Stirling Numbers of the First Kind
Generalizations and Relatives of Stirling Numbers
Zoo of Numbers
Mini Dictionary
Exercises
Teachers and students (esp. at university) interested in Combinatorics, Number Theory, General Algebra, Cryptography and related fields, as well as general audience of amateurs of Mathematics.
Pages: 440
ISBN: 978-981-128-008-5 (hardcover)
Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar?Moh?Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata.
Readers will find the book covers vast basic material on an extremely rigorous level:
It begins with an introduction to algebraic geometry which comprises almost all results in commutative algebra and algebraic geometry.
Arguments frequently used in affine algebraic geometry are elucidated by treating affine lines embedded in the affine plane and automorphism theorem of the affine plane. There is also a detailed explanation on affine algebraic surfaces which resemble the affine plane in the ring-theoretic nature and for actions of algebraic groups.
The Jacobian conjecture for these surfaces is also considered by making use of the results and tools already presented in this book. The conjecture has been thought as one of the most unattackable problems even in dimension two.
Advanced results are collected in appendices of chapters so that readers can understand the main streams of arguments.
There are abundant problems in the first three chapters which come with hints and ideas for proof.
Preface
Introduction to Algebraic Geometry
Geometry on Affine Surfaces
Geometry and Topology of Polynomial Rings
Postscript
Mathematics students, both undergraduate and graduate, where knowledge of group, ring and linear algebra is required, and researchers.
If the book is used as a textbook, it is for students in the beginning class of algebraic geometry and commutative algebra.