On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V−AVF=BW and Sylvester matrix equations AV−EVF=BW,AV+BW=EVF and AV−VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.     

Inducing <Emphasis Type="Italic">W</Emphasis>-graphs

 Let ℋ be the Hecke algebra associated with a Coxeter group W. Many interesting ℋ-modules can be described using the concept of a W-graph, as introduced in the influential paper [4] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of ℋ has an associated W-graph. The purpose of this note is to show that if W _J is a parabolic subgroup of W and V is a module for the corresponding Hecke algebra ℋ _J , then a W _J -graph structure for V gives rise to a W-graph structure for the induced module ℋ⊗ _ℋJ V. In the case that W _J is the identity subgroup and V has dimension 1, our construction coincides with that given by Kazhdan and Lusztig for the regular representation. For arbitrary J and V of dimension 1 we recover the constructions of Couillens [1] and Deodhar [3]. Received: 14 June 2002; in final form: 13 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 20C08     

Energy bounds in dynamical problems for a semi-infinite magnetoelastic beam

The aim of this paper is to investigate the behaviour of the total energy of a magnetoelastic conductor occupying a semi-infinite prismatic cylinder in dynamical conditions. Precisely, we deduce some estimates for the energyW(x ₃,t) of the portion of the medium at distance greater thanx ₃ from the base in terms of the data. First of all, we prove that the total energyW(0,t) is finite for allt > 0 providedW(0, 0) is finite. Then, using the first Korn inequality, we obtain that the estimate forW(x ₃,t) depends only on the initial data ift<x ₃/V (V=computable positive material constant); ift>x ₃/_V then the bound forW(x ₃,t) depends on all the data of the problem.     

Vector Lattices Associated with Ordered Vector Spaces

Let V be a real, Archimedian ordered, vector space, whose positive cone V ⁺ satisfies V =  V ⁺ – V ⁺. To V we associate a Dedekind complete vector lattice W containing V (by abuse of notation). In the case when V has an order unit the determination of W is already known. Let W₀ ì W{W_0 \subset W} be the vector lattice generated by V. We study W ₀ in the case when the cone C of all positive linear forms on V separates the elements of V. The determination of W ₀ involves the extreme rays of C. We determine the cone of positive linear forms on W ₀ in terms of conical measures on C.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

Precoloring extension for K4‐minor‐free graphs

Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K₄‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009     

On the Weight Hierarchy of Product Codes

Let V and W be codes and let C = V W be the product code of V and W. In [6] Wei and Yang, conjectured a formula for the generalized Hamming weights of V W in terms of those of V and W provided that both V and W satisfy the chain condition. Recently the conjecture has been proved by Schaathun [4]. In this paper we generalize the formula to a product code with more than two components.     

A generalization of Borel's theorem and microlocal Gevrey regularity in involutive structures

In this paper, we use Borel's procedure to construct Gevrey approximate solutions of an initial value problem for involutive systems of Gevrey complex vector fields. As an application, we describe the Gevrey wave-front set of the boundary values of approximate solutions in wedges W of Gevrey involutive structures (M,V). We prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone ΓT(W) contained in RV∩TX where X is a maximally real edge of W. We also prove a partial converse.     

Regularized minimization under weaker hypotheses

LetV andW be two Banach spaces, withV reflexive, a bounded convex set ofV, A a linear mapping fromV intoW, and letF be a convex functional onW. We minimizeJ(v)=F(Av) on using hypotheses about particular sequences in IfV is uniformly convex, we prove existence and uniqueness of a solution of minimal norm minimizingJ. In the Hilbert space case, withF defined byF(w)=w–f ²,f given inW, we get existence and uniqueness of the projection off on A(), which generalizes the case where A() is a closed set ofW (taking closed andA continuous). Finally, we give examples, and we study an unbounded operator case.     

On a periodic Schrödinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part –+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension N2.Mathematics Subject Classification (2000):35Q55, 35Q40, 35J10, 35J20, 35J60, 46N50, 49J35, 81V70in final form: 14 November 2003     

Infinite dimensional irreducible Lie algebras containing transformations of finite rank

Let be a field of characteristic zero and let V be an infinite dimensional vector space over . A linear transformation x of V is called finitary if . The aim of this paper is to describe irreducible Lie subalgebras of containing nonzero finitary transformations. It turns out that any such algebra is a semidirect product of a finite dimensional Lie algebra and a “dense” Lie subalgebra of for some vector space W. Received January 4, 2000 / Published online March 12, 2001     

Perturbation analysis for the trace quotient problem

Given a symmetric matrix B?∈?? ^m×m and a symmetric and positive-definite matrix W?∈?? ^m×m , maximizing the ratio trace(V ^? BV)/trace(V ^? WV) with respect to V?∈?? ^m×? (??≤?m) subject to the orthogonal constraint V ^? V?=?I _? is called the trace quotient problem or the trace ratio problem (TRP). TRP arises originally from the linear discriminant analysis (LDA), which is a popular approach for feature extraction and dimension reduction. It has been known that TRP is equivalent to a nonlinear extreme eigenvalue problem and very efficient method has been proposed to find a global optimal solution successfully. The matrices B and W arising in LDA are constructed from samples, and thereby are contaminated by noises and errors. In this article, we perform a perturbation analysis for TRP assuming the original B and W are perturbed. The upper perturbation bounds of both the global optimal value and the set of global optimal solutions are derived, and numerical investigation is carried out to illustrate these perturbation estimates.     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

The eigenvalues of four-dimensional rotations

Let X be an orthosymplectie Lie superalgebra of type B or D. The weight structure of the tensor product moduleW=? ^M V, of M-copies of the natural representation. V, of X is studied from a partition point of view. A combinatorial characterization of the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules is given. This partition point of view allows us to prove that the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules stabilize as the rank of X gets large.     

4-Webs on hypersurfaces of 4-axial space

V. B. Lazareva investigated 3-webs formed by shadow lines on a surface embedded in 3-dimensional projective space and assumed that the lighting sources are situated on 3 straight lines. The results were used, in particular, for the solution of the Blaschke problem of classification of regular 3-webs formed by pencils of circles in a plane. In the present paper, we consider a 4-web W formed by shadow surfaces on a hypersurface V embedded in 4-dimensional projective space assuming that the lighting sources are situated on 4 straight lines. We call the projective 4-space with 4 fixed straight lines a 4-axial space. Structure equations of 4-axial space and of the surface V , asymptotic tensor of V , torsions and curvatures of 4-web W, and connection form of invariant affine connection associated with 4-web W are found.     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

0-Minimal Quasi-ideals of Generalized Linear Transformation Semigroups

Abstract

For vector spaces V and W over a field F, L _F (V, W) denotes the set of all linear transformations α : V → W, and for a cardinal number k > 0, let L _F (V, W, k) be the set of all α ∈ L _F (V, W) of rank less than k. For θ ∈ L _F (W, V), let (L _F (V, W, k), θ) denote the semigroup L _F (V, W, k) under the operation ? defined by α ? β = αθβ for all α, β ∈ L _F (V, W, k). In this paper, all 0-minimal quasi-ideals of the semigroup (L _F (V, W, k), θ) are completely characterized. It is also shown from this characterization that every nonzero semigroup (L _F (V, W, k), θ) always has a 0-minimal quasi-ideal.     

Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C^1,α boundary to ℝ³ which belongs to the critical Sobolev space W^2,2 is C¹ up to the boundary. More generally C¹ regularity up to the boundary holds for all scalar functions V ∈ W^2,2(S) which satisfy det ∇²V=0. If S has only Lipschitz boundary we show such V can be approximated in W^2,2 by functions V_k ∈ W^1,∞∩W^2,2 with det ∇²V_k=0.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

An Elementary Approach to Exponential Spaces

In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.