Narrow Results

In this paper, we consider the reconstruction of matrices whose graph is a star with root at the central vertex from given partial eigen data. Three inverse spectral problems are discussed. The necessary and sufficient conditions for the solvability of the problems are studied. Furthermore, the corresponding numerical algorithms are presented. Some numerical examples are also provided to demonstrate the applicability of the results obtained here.

The usual QR algorithm for finding the eigenvalues of a Hessenberg matrix H is based on vector-vector operations, e.g. adding a multiple of one row to another. The opportunities for parallelism in such an algorithm are limited. In this paper, we describe a reorganization of the QR algorithm to permit either matrix-vector or matrix-matrix operations to be performed, both of which yield more efficient implementations on vector and parallel machines. The idea is to chase a k by k bulge rather than a 1 by 1 or 2 by 2 bulge as in the standard QR algorithm. We report our preliminary numerical experiments on the CONVEX C-1 and CYBER 205 vector machines.

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix$L=D-A$ and directed Laplacian energy$LE\left(G\right)=\sum _{i=1}^n{\lambda }_i^2$ using the second spectral moment of $L$ for a digraph $G$ with $n$ vertices, where $D$ is the diagonal out-degree matrix, and $A=\left(a_{ij}\right)$ with $a_{ij}=1$ whenever there is an arc $(i,j)$ from the vertex $i$ to the vertex $j$ and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

We consider the spectral problem associated with the Klein–Gordon equation for unbounded electric potentials such that the spectrum is contained in two disjoint real intervals related to positive and negative energies, respectively. If the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.

In this article, we extend Pólya's legendary inequality for the Dirichlet Laplacian to the fractional Laplacian. Pólya's argument is revealed to be a powerful tool for proving such extensions on tiling domains. As in the Dirichlet Laplacian case, Pólya's inequality for the fractional Laplacian on any bounded domain is still an open problem. Moreover, we also investigate the equivalence of several related inequalites for bounded domains by using the convexity, the Lieb–Aizenman procedure (the Riesz iteration), and some transforms such as the Laplace transform, the Legendre transform, and the Weyl fractional transform.

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is the construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ${\ell }^{\infty }$-space on $Z$. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.

We give a lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the interior radius. The result sharpens earlier estimates.

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra M_k^∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L²(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

In this paper, we consider the boundary value problem

We show that for any two $n\times n$ matrices $X$ and $Y$ we have the inequality $s_j^2(I+XY)\le {\lambda }_j((I+X^{\ast }X)(I+Y^{\ast }Y)),$ where $s_j(T)$ and ${\lambda }_j(T)$ denote the decreasingly ordered singular values and eigenvalues of $T$. As an application, we show that for $2n\times 2n$ real positive definite matrices the symplectic eigenvalues $d_j,$ under some special conditions, satisfy the inequality $d_j(A+B)\ge d_j(A)+d_1(B)$.

We analyze cross-correlation between return fluctuations of stocks of an emerging market by using random matrix theory (RMT). We test the statistics of eigenvalues of cross-correlation (C) between stocks of the Tehran Price Index (TEPIX) as an emerging market and compare these with a mature market (US market). According to the "null hypothesis," a random correlation matrix constructed from mutually uncorrelated time series, the deviation from the Gaussian orthogonal ensemble of RTM is a good criterion. We find that a majority of the eigenvalues of C fall within the bulk (RMT bounds between λ₊ and λ_-) for the eigenvalues of the random correlation matrices. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the largest deviating eigenvalues, display systematic deviations from the RMT prediction. Analyzing the components of the deviating eigenvectors by Inverse Participation Ratio, leads us to know that the largest eigenvalue corresponds to an influence common to the whole market. Our analysis of the other deviating eigenvectors shows distinct industries, whose identities corresponds to the structure of the Iran business environment.

Symmetrizing the Dai–Liao (DL) search direction matrix by a rank-one modification, we propose a one-parameter class of nonlinear conjugate gradient (CG) methods which includes the memoryless Broyden–Fletcher–Goldfarb–Shanno (MLBFGS) quasi-Newton updating formula. Then, conducting an eigenvalue analysis, we suggest two choices for the parameter of the proposed class of CG methods which simultaneously guarantee the descent property and well-conditioning of the search direction matrix. A global convergence analysis is made for uniformly convex objective functions. Computational experiments are done on a set of unconstrained optimization test problems of the CUTEr collection. Results of numerical comparisons made by the Dolan–Moré performance profile show that proper choices for the mentioned parameter may lead to promising computational performances.

In the present paper, the problem of two-dimensional static deformation of an isotropic elastic half-space of irregular thickness has been studied using the eigenvalue approach, following a Fourier transform. The irregularity is expressed by a rectangular shape. As an application, the normal line-load acting inside an irregular isotropic half-space has been considered. Further, the results for the displacements have been derived in the closed form. To examine the effect of irregularity, variations of the displacements with horizontal distance have been shown graphically for different values of irregularity size, and they are compared with those for the medium with uniform shape. It is found that irregularity affects the deformation significantly.

The paper discusses the problem of a two-dimensional static deformation as the result of normal line-load acting inside an irregular initially stressed isotropic half-space. The eigenvalue approach method has been used. The irregularity is expressed by a rectangle shape. Further, the results for the displacements and stresses have been derived in the closed form. The effect of initial stress and irregularity are shown graphically. It was found that the initial stresses as well as irregularity have a notable effect on this deformation.

Analysis of flows in crowd videos is a remarkable topic with practical implementations in many different areas. In this paper, we present a wide overview of this topic along with our own approach to this problem. Our approach treats the difficulty of crowd flow analysis by distinguishing single versus multiple flows in a scene. Spatiotemporal features of two consecutive frames are extracted by optical flows to create a three-dimensional tensor, which retains appearance and velocity information. Tensor’s upper left minor matrix captures intensity structure. A normalized continuous rank-increase measure for each frame is calculated by a generalized interlacing property of the eigenvalues of these matrices. In essence, measure values put through the knowledge of existing flows. Yet they do not go into effect desirably due to optical flow estimation error and some other factors. A proper set of the degree of polynomial fitting functions decodes their existence. But how can we estimate that set? Its detailed study is performed. Zero flow, single flow, multiple flows, and interesting events are detected as frame basis using thresholds on the polynomial fitting measure values. Plausible mean outputs of recall rate (88.9%), precision rate (86.7%), area under the receiver operating characteristic curve (98.9%), and accuracy (92.9%) reported from conducted experiments on PETS2009 and UMN benchmark datasets make clear and visible that our method gains high-quality results to detect flows and events in crowd videos in terms of both robustness and potency.

Over the last 40 years, the design of $n$-dimensional hyperchaotic systems with a maximum number ($n-2$) of positive Lyapunov exponents has been an open problem for research. Nowadays it is not difficult to design $n$-dimensional hyperchaotic systems with less than ($n-2$) positive Lyapunov exponents, but it is still extremely difficult to design an $n$-dimensional hyperchaotic system with the maximum number ($n-2$) of positive Lyapunov exponents. This paper aims to resolve this challenging problem by developing a chaotification approach using average eigenvalue criteria. The approach consists of four steps: (i) a globally bounded controlled system is designed based on an asymptotically stable nominal system with a uniformly bounded controller; (ii) a closed-loop pole assignment technique is utilized to ensure that the numbers of eigenvalues with positive real parts of the controlled system be equal to ($n-1$) and ($n-2$), respectively, at two saddle-focus equilibrium points; (iii) the number of average eigenvalues with positive real parts is ensured to be equal to ($n-2$) for the controlled system over a given control period; (iv) the smallest value of the positive real parts of the average eigenvalues is ensured to be greater than a given threshold value. Finally, the paper is closed with some typical examples which illustrate the feasibility and performance of the proposed design methodology.

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.