Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights

The asymptotic behavior of thenth root of the leading coefficient of orthogonal polynomials on (–,) and the distribution of their zeros is studied for nonsymmetric weights that behave like exp(–2B|x|) whenx>0 and exp(–2Ax ) whenx<0,>AB. These results generalize previous investigations of Rakhmanov and Mhaskar and Saff who handle the symmetric caseA=B.Communicated by Paul Nevai.     

On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems

The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s>0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented.     

Nomura algebras of nonsymmetric Hadamard models

We show that the Nomura algebra of the nonsymmetric Hadamard model coincides with the Bose–Mesner algebra of the directed Hadamard graph.     

Investigation of the three-dimensional micromechanical behavior of woven-fabric composites

A three-dimensional representative volume-element model is presented to study the micromechanical behavior of woven-fabric composites. The effects of the fiber undulation zone and the fiber braid angle on the elastic modulus of the composites are taken into account in the unit cell. Based on isostrain and isostress assumptions, a standard homogenization procedure is used to calculate the effective elastic properties of woven-fabric composites, and all the final stiffness components are expressed in an explicit form. The results obtained by the model considered agree with published experimental results very well. The relationship between the geometric parameters, such as fiber width, thickness, volume fraction, etc., and the macromechanical behavior of the composites can be obtained by this model. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 2, pp. 209–220, April–May, 2006.     

On cone of nonsymmetric positive semidefinite matrices

In this paper, we analyze and characterize the cone of nonsymmetric positive semidefinite matrices (NS-psd). Firstly, we study basic properties of the geometry of the NS-psd cone and show that it is a hyperbolic but not homogeneous cone. Secondly, we prove that the NS-psd cone is a maximal convex subcone of P₀-matrix cone which is not convex. But the interior of the NS-psd cone is not a maximal convex subcone of P-matrix cone. As the byproducts, some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given. Finally, we present some properties of metric projection onto the NS-psd cone.     

Symmetric and nonsymmetric equilibria of a rod-and-spring model

A simple rod-and-spring model is used to illustrate bifurcationin the presence of symmetry. Our aim is to demonstrate the relevanceof certain general mathematical ideas and computational techniquesin a comparatively simple physical example, a full appreciationof which can be intuitive. The model discussed here has twodistinct reflectional symmetries, i.e. Z₂Z₂ symmetry. The symmetry-grouprepresentations, invariant subspaces, projection matrices, projectedequa-tions, and extended systems used to compute loci of singularitiesare all given explicitly. A closely allied nonsymmetric modelis then developed and its behaviour is examined numerically.Connections between singularities in the symmetric and asymmetricmodels are explored.     

The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form $f(z_1\text{,}{z}_2)=\frac{1}{\overline{P(z_1\text{,}{z}_2)}Q(z_1\text{,}{z}_2)}$ where $P(z_1\text{,}{z}_2)$ and $Q(z_1\text{,}{z}_2)$ are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.     

Regularity for solutions of nonlocal,nonsymmetric equations

We study the regularity for solutions of fully nonlinear integro differential equations with respect to nonsymmetric kernels. More precisely, we assume that our operator is elliptic with respect to a family of integro differential linear operators where the symmetric parts of the kernels have a fixed homogeneity σ and the skew symmetric parts have strictly smaller homogeneity τ  . We prove a weak ABP estimate and C^1,α$C^{1,\alpha }$ regularity. Our estimates remain uniform as we take σ→2$\sigma \to 2$ and τ→1$\tau \to 1$ so that this extends the regularity theory for elliptic differential equations with dependence on the gradient.     

Comparison theorems for some nonsymmetric classes of functions

We prove comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 969–975, July, 2008.