A Formula Omitted-ary function Formula Omitted in Formula Omitted variables is an Formula Omitted- form if Formula Omitted for any nonzero Formula Omitted in Formula Omitted and Formula Omitted in ...Formula Omitted. Let Formula Omitted be a positive even integer, Formula Omitted an odd prime, and Formula Omitted an element of Formula Omitted provided that Formula Omitted if Formula Omitted. Let Formula Omitted be a Formula Omitted-ary bent function in Formula Omitted variables of Formula Omitted-form with Formula Omitted and Formula Omitted, and let Formula Omitted. We denote by Formula Omitted the Cayley graph Formula Omitted. Our main results are as follows: 1) if there is weakly regular Formula Omitted-ary bent Formula Omitted which is not regular, then Formula Omitted is 2; 2) if Formula Omitted, then Formula Omitted is weakly regular Formula Omitted-ary bent if and only if the Cayley graph Formula Omitted is strongly regular; 3) if Formula Omitted, then Formula Omitted is regular Formula Omitted-ary bent if and only if the Cayley graph Formula Omitted is strongly regular; 4) Formula Omitted can be replaced by Formula Omitted in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of Formula Omitted.
We elucidate the band structures and scattering properties of the simplest one-dimensional parity–time (PT)-symmetric photonic crystal. Its unit cell comprises one gain layer and one balanced loss ...layer. Herein, the analytic expressions of the band structures and scattering properties are derived, and based on these relations, we reveal and explain the following phenomena: Exceptional point pairs appear from Brillouin boundaries at a nonzero non-Hermiticity. With an increase in non-Hermiticity, each of these pairs moves toward the Brillouin center, finally coalescing into a single point at the Brillouin center at a critical non-Hermiticity value. Near the exceptional point, singular scattering is observed and explained. This refers to the phenomenon whereby transmittances and reflectances for left and right incidences reach exceptionally large values simultaneously. Moreover, these are infinite at some discrete points at which poles and zeros of the scattering matrix are attained. In forbidden gaps, unidirectional weak visibility, where transmittances are zero, is disclosed and analyzed: specifically, the reflectance for incidence from one side is very large, whereas that for incidence from the other side is very small. In this phenomenon, the eigenstates of the scattering matrix are the incident waves from the left and right sides, and their eigenvalues are the corresponding reflectances. Our results are important as new functional optical devices can potentially be developed by utilizing these novel phenomena.
Research Spotlights Tuminaro, Ray
SIAM review,
03/2014, Volume:
56, Issue:
1
Journal Article
Peer reviewed
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
Let Formula omitted. and Formula omitted. be Formula omitted. real numbers that satisfy the strict second-order Cauchy interlacing inequalities Formula omitted. for Formula omitted. and the ...nondegeneracy conditions Formula omitted. for Formula omitted. . Given a connected graph Formula omitted. on Formula omitted. vertices with adjacent vertices Formula omitted. and Formula omitted. , it is proven that there is a real symmetric matrix Formula omitted. whose graph is Formula omitted. such that Formula omitted. has eigenvalues Formula omitted. and Formula omitted. has eigenvalues Formula omitted. , provided some necessary combinatorial conditions on Formula omitted. are satisfied. We also provide generalizations when the two deleted vertices are not adjacent, as well as interpretation of the results in terms of perturbing one or two diagonal entries.
Abstract 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.
In this thesis, we study the spectrum of Schrödinger operators with complex potentials and Dirichlet Laplace operators on domains with rough boundaries. The focus is on spectral approximation ...results and a-priori bounds for the location and distribution of eigenvalues. Chapter 1 provides an overview of our main results and Chapters 2 - 5 are based on the papers 130, 129, 79, 121 respectively. In Chapter 2, spectral inclusion and pollution results are proved for sequences of linear operators of the form T0+iγsn on a Hilbert space, where sn is strongly convergent to the identity operator and γ > 0. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps. In Chapter 3, we consider Schrödinger operators of the form HR = −d 2/dx 2 +q+iγχ0,R for large R > 0, where q ∈ L 1 (0,∞) and γ > 0. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this operator, for sufficiently large R. In Chapter 4, we prove upper and lower bounds for sums of eigenvalues of Lieb- Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases. In Chapter 5, we prove a Mosco convergence theorem for H 1 0 spaces of bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.
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