A new algebra of Toeplitz-plus-Hankel matrices and applications

We introduce a new simultaneously diagonalizable real algebra of symmetrical centrosymmetrical matrices having a Toeplitz-plus-Hankel structure. We give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors. An application is the construction of a symmetrical Toeplitz-plus-centrosymmetrical Hankel matrix of equal row sums having a prescribed real spectrum. This matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows. In particular, for the isospectral flow corresponding to the construction of a regular Toeplitz matrix having prescribed eigenvalues. Moreover, if A is a noise representation of an unknown matrix in of rank k then we give a procedure to approximate A by a matrix in of rank k.     

Spectral and structural properties of some pentadiagonal symmetric matrices

By using suitable bordering and modification techniques, spectral properties of some classes of pentadiagonal symmetric matrices are found.     

Spectral portrait for non-hermitian large sparse matrices

The spectral portrait of a matrix is the picture of its ?-spectra forε ∈[ε ₁,ε ₂], where an ?-spectrum ofA is the union of all the eigenvalues of all the matricesA+Δ with ∥Δ∥₂≤ε∥A∥₂. The spectral portrait is, for example, useful to study the stability of various problems, or, as we illustrate in this paper, to visualize the condition number of an eigenvalue. Some methods to estimate the spectral portrait already exist, but only for small matrices. We propose here a new algorithm for non hermitian large sparse matrices.     

Spectral properties of infinite-dimensional closed-loop systems

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator A^K. We give a useful characterization of that part of the spectrum σ(A^K) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of A^K if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σ_e(A), the essential spectrum of A, is invariant, that is, it is equal to σ_e(A^K). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σ_e(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.     

Brownian matrices: properties and extensions

Brownian matrices arise in certain models used in digital signal processing. Fast algorithms for solving brownian systems of linear equations can be derived because any brownian matrix is congruent to a tridiagonal matrix. This, and related properties, are developed and extended to matrices having a more general patterned structure.     

Some properties of inverses of the full matrices

In this paper, we discuss the problem of the number of elements equal to zero included in the inverse matrix, in case when the given matrix is full or it satisfies some specific algebraic-geometrical conditions.     

On recognizing graph properties from adjacency matrices

A conjecture of Aanderaa and Rosenberg [15] motivates this work. We investigate the maximum number C(P) of arguments of P that must be tested in order to compute P, a Boolean function of d Boolean arguments. We present evidence for the general conjecture that C(P) = d whenever P(0^d) ≠ P(1^d) and P is invariant under a transitive permutation group acting on the arguments. A non-constructive argument (not based on the construction of an “oracle”) settles this question for d a prime power. We use this result to prove the Aanderaa-Rosenberg conjecture: at least $\text{v}{}^2\text{16}$ entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property.     

Spectral properties of the SPH Laplacian operator

In order to address the question of the SPH (Smoothed Particle Hydrodynamics) Laplacian conditioning, a spectral analysis of this discrete operator is performed. In the case of periodic Cartesian particle network, the eigenfunctions and eigenvalues of the SPH Laplacian are found on theoretical grounds. The theory agrees well with numerical eigenvalues. The effects of particle disorder and non-periodicity conditions are then investigated from numerical viewpoint. It is found that the matrix condition number is proportional to the square of the particle number per unit length, irrespective of the space dimension and kernel choice.     

Spectral properties of pseudo-resolvents under structured perturbations

The changes in the spectrum caused by structured perturbations of pseudo-resolvents and operators on Banach spaces are considered. In particular, if a point is in the resolvent set of an operator, necessary and sufficient conditions for it to remain in the resolvent set under structured perturbations are given. The structured perturbations of an operator are specified by an operator node that has three generating operators and a characteristic function together with an admissible feedback operator. In addition, the robustness of stability under structured perturbations is analyzed. The results are applied to boundary control systems and impedance passive systems.     

Generation and properties of banded viscous damping matrices

Numerical analysis of discrete vibratory systems subjected to broad band excitation usually requires specification of a viscous damping matrix. A method of determining the entries in such matrices directly is presented in this paper. The modal damping ratios, which can normally be estimated, are used to produce a fully populated damping matrix. To facilitate numerical integration of the equations of motion, a procedure termed stripping has been established for reducing the matrix to banded form. Additionally, using a procedure termed replacement, small element matrices were identified, extracted from the fully populated damping matrix and utilized (much as element mass and stiffness matrices are) to construct the global damping matrix. The effects of stripping or replacement were evaluated for a number of boundary conditions on plates, beams, and shells. Errors caused by variation of the number of system degrees of freedom, the number of diagonal rows remaining in the stripped matrix, and the magnitude and mo dal distribution of damping ratios were studied. Results were, in general, good. Damping ratios, damped natural frequencies, and eigenvectors were closely approximated by both type of systems.     

An observation on certain spectral properties of Toeplitz matrices

Consider the generalized eigevalue problemAx=λBx,x=(x_i), where A, B are Hermitian Toeplitz matrices. We give lower bounds to the number of the zeros of the «eigenpolynomial» U(z)=Σx_izⁱ, lying on the unit circle in the complex plane.     

Spectral characterization of mixing properties of annular MHD micromixers

We develop a quantitative analysis of mixing regimes in an annular MHD-driven micromixer recently proposed by Gleeson et al. as a prototype for biomolecular applications. The analysis is based on the spectral properties of the advection–diffusion operator, with specific focus on the dependence of the dominant eigenvalue–eigenfunction on the Peclet number and on the system geometry. A theoretical prediction for the dominant eigenvalue encompassing all mixing regimes is developed and validated by comparison with numerical simulations. The theoretical prediction is extended to an open inflow–outflow version of the reactor, which shows the occurrence of new regimes associated with the existence of a nonuniform axial flow.     

Minimal parameterization of fundamental matrices using motion and camera properties

This paper addresses the optimal recovery of the displacement and projection parameters from uncalibrated monocular video sequences. We study the particular cases of camera and object displacements and camera projection in order to extract an optimized parameterization of the problem of parameter recovery for each case.

This work follows previous studies on particular cases of displacement, scene geometry and camera analysis and focuses on the particular forms of fundamental matrices. This paper introduces the idea of using not all particular cases as individual cases but grouping these cases into a tractable number of sets, using properties on fundamental matrices.

Some experiments were performed in order to demonstrate that if several models are correct, the model with the least parameters gives the best estimate, corresponding to the true case.     

A Semiring-based study of judgment matrices: properties and models

In decision making and group decision making, multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information. In this paper, semirings are applied to the discussion of judgment matrix properties. First, two special semirings are constructed. Second, the properties of the consistent judgment matrices are given as a set of equations (all in the semiring sense), which include idempotency equations and fixed point equations. We find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix. Third, optimization models for inconsistent judgment matrices are presented. Finally, we offer some simple illustrative examples.     

Formulation of time variant stiffness matrices due to changing joint properties

Time variant stiffness matrices due to changing joint properties occur in a wide range of structural dynamic response problems. An example is a structure with element forces which can be reacted at a joint in only one direction. Another is one with structural elements which can be alternately attached and detached to an adjacent structure due to joint clearance. A general treatment of stiffness matrices for structures having joints with time dependent properties is presented. Various joints are examined and the approach is demonstrated by deriving time variant stiffness matrices for simple structures which contain each type of joint.     

M/M/1算子的谱特征

本文讨论动态M/M/1排队模型.运用泛函分析中的算子的谱理论及C0-半群理论,证明了该模型非负解的存在唯一性,研究了相应算子的谱特征.     

Spectral absorption properties of dissolved and particulate matter in Lake Erie

Spectral absorption properties of particulate and dissolved matter were determined for Lake Erie waters in order to investigate the natural variability of the absorption coefficients required as inputs to optical models for converting satellite observations of water colour into water quality information. Particulate absorption measured using the quantitative filter technique yielded absorption spectra containing a fraction that could not be attributed to phytoplankton pigments, living heterotrophs, mineral sediments, or organic detritus but were indicative of additional absorption by a fraction of dissolved organic matter present in colloidal and/or particle-bound form. Erroneously high phytoplankton absorption coefficients measured at short wavelengths using the filter technique suggested that this particle-bound DOM is removed along with phytoplankton pigments during bleaching by sodium hypochlorite and as such is mistakenly incorporated into the phytoplankton absorption signal. Observations suggest that the selective sorption of fractions of DOM onto suspended particles may be responsible for significant variability in the absorption coefficients of particulate and dissolved matter and may be an important contributor to the total spectral absorption signals in Lake Erie waters. This reservoir of coloured organic matter, which to date has not been seriously considered in the optical properties of coastal and inland waters, may produce significant uncertainties in the parameterization of optical models and the interpretation of in situ and remotely sensed aquatic colour signals.     

Fast parallel and sequential computations and spectral properties concerning band Toeplitz matrices

We exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a (2k+1)-diagonal Toeplitz matrix with elements in the complex field, either for sequential or for parallel computations. A fast algorithm, to achieve one step of Newton's method, is shown to be suitable to compute the eigenvalues of such a matrix. Bounds to the eigenvalues and necessary and sufficient conditions for positive definiteness, which are easy to check, are given either for matrices with scalar elements or for matrices with blocks. In the case in which the blocks are themselves band Toeplitz matrices such conditions assume a very simple form. Dedicated to Prof. Aldo Ghizzetti on his 75 th birthday