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Expander families and Cayley graphs : a beginner's guide
\"The theory of expander graphs is a rapidly developing topic in mathematics and computer science, with applications to communication networks, error-correcting codes, cryptography, complexity theory, and much more. Expander Families and Cayley Graphs: A Beginner's Guide is a comprehensive introduction to expander graphs, designed to act as a bridge between classroom study and active research in the field of expanders. It equips those with little or no prior knowledge with the skills necessary to both comprehend current research articles and begin their own research. Central to this book are four invariants that measure the quality of a Cayley graph as a communications network-the isoperimetric constant, the second-largest eigenvalue, the diameter, and the Kazhdan constant. The book poses and answers three core questions: How do these invariants relate to one another? How do they relate to subgroups and quotients? What are their optimal values/growth rates? Chapters cover topics such as: ℗ʺ Graph spectra ℗ʺ A Cheeger-Buser-type inequality for regular graphs ℗ʺ Group quotients and graph coverings ℗ʺ Subgroups and Schreier generators ℗ʺ Ramanujan graphs and the Alon-Boppana theorem ℗ʺ The zig-zag product and its relation to semidirect products of groups ℗ʺ Representation theory and eigenvalues of Cayley graphs ℗ʺ Kazhdan constants The only introductory text on this topic suitable for both undergraduate and graduate students, Expander Families and Cayley Graphs requires only one course in linear algebra and one in group theory. No background in graph theory or representation theory is assumed. Examples and practice problems with varying complexity are included, along with detailed notes on research articles that have appeared in the literature. Many chapters end with suggested research topics that are ideal for student projects\"-- Provided by publisher.
Book
Large Order Behavior Near the AD Point: The Case of ð'© = 2, su(2), Nf = 2
A non-perturbative effect in κ (renormalized string coupling) obtained from the large order behavior in the vicinity of the prototypical Argyres–Douglas critical point of su(2), Nf = 2, $\\mathcal {N} =2$ supersymmetric gauge theory can be studied in the Gross–Witten–Wadia unitary matrix model with the log term: one as the work done against the barrier of the effective potential by a single eigenvalue lifted from the sea and the other as a non-perturbative function contained in the solutions of the nonlinear differential Painlevé II equation that goes beyond the asymptotic series. The leading behaviors are of the form $\\exp \\{[-({4}/{3})({1}/{\\kappa })] \\, [1, \\left({s}/{K}\\right)^{{3}/{2}}]\\}$. We make comments on their agreement.
Journal Article
Multiparameter eigenvalue problems : Sturm-Liouville theory
\"With special attention to the Sturm-Liouville theory, this book discusses the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of these multiparameter problems in detail for both the regular and singular cases. The text covers eignencurves, the essential spectrum, eigenfunctions, oscillation theorems, the distribution of eigencurves, the limit point, limit circle theory, and more. This text is the culmination of more than two decades of research by F.V. Atkinson, one of the masters in the field, and his successors, who continued his work after he passed away in 2002\"-- Provided by publisher.
Book
Simple zeros of automorphic -functions
We prove that the complete
$L$
-function associated to any cuspidal automorphic representation of
$\\operatorname{GL}_{2}(\\mathbb{A}_{\\mathbb{Q}})$
has infinitely many simple zeros.
Journal Article
Hidden symmetries in the HFB norm overlap functions
A brief consideration about hidden symmetries in the HFB norm overlap functions is presented, in particular, in association with the presence of pair-wise degeneracy of the eigenvalue spectrum of a product of two antisymmetric matrices.
Conference Proceeding
Research Spotlights
The analysis of generalized eigenvalue problems is central to understanding a number of complex phenomena, including the stability of nonlinear waves. One generally seeks a characterization of a linearized spectrum in relation to the complex plane (e.g., eigenvalues strictly in the right half plane) from which one can deduce the stability of the system and the presence of features such as bifurcations of Hamiltonian--Hopf type. Two fundamentally different but useful tools for analyzing spectral stability include the Krein signature and the Evans function. The Krein signature is helpful in investigating the stability of purely imaginary eigenvalues (i.e., whether eigenvalues will move toward the right half plane under perturbations), while the Evans function can be used to detect eigenvalue locations. The paper \"Graphical Krein Signature Theory and Evans--Krein Functions,\" by Richard Kollar and Peter Miller, highlights a graphical interpretation of the Krein signature and more specifically stresses the utility of this graphical interpretation. On the computational side, the graphic interpretation is used to adapt the notion of an Evans function to an Evans--Krein function. The new generalization allows one to calculate the Krein signature in a way that is easy to incorporate into existing simulation capabilities that are already capable of evaluating an Evans function. This is in contrast to the traditional Evans function which cannot generally be used to directly deduce the Krein signature. In addition to this computational utility, the graphical interpretation of the Krein signature has nice theoretical properties as demonstrated by a set of proofs associated with index theorems for linearized Hamiltonians and includes relations to the well-known Vakhitov--Kolokolov criterion. [PUBLICATION ABSTRACT]
Journal Article
Shape Invariance, Exactly Solvable Systems, and Semi-Classical Quantization
In the 1980s, Comtet et al. found that a modified version of the WKB quantization condition yields exact eigenvalues for all exactly solvable potentials that were known at the time. This intriguing property prompted investigations into the underlying reasons for such exact solvability. In this paper, we trace the journey that reveals shape invariance as the fundamental cause of this exactness and identifies the set of potentials for which it holds. We demonstrate that while shape invariance ensures this exactness in conjunction with an additional condition, it alone is not sufficient.
Journal Article
A Sequence on the Maximum Eigenvalue of a Nonnegative Matrix
For an irreducible matrix with nonnegative entries, a sequence is constructed on the matrix trace. And we prove the convergence of the sequence in two cases when the irreducible matrix with nonnegative entries is a primitive matrix and a non-primitive matrix.
Journal Article
