The singular values of the hadamard product of a positive semidefinite and a skew-symmetric matrix

Let σ₁(X)≤ · ≤ σ_N(X)≤0 denote the ordered singular values ofan n × n matrix X and let α₁ (X) ≤ α₂(X)≤ · ≤ α_n(X) denote its ordered main diagonal entries (assuming that they are real). Let B be any complex n × n skew-symmetric matrix and ||.|| any unitarily invariant norm. It is shown that for any rea positive semidefinite n × n matrix A.     

Generalized Path Spaces,II

A combination of the LIAPUNOV-SCHMIDT procedure, the implicit function theorems and the topological degree theory is used to investigate bifurcation points of equations of the form T(v) = L(λ, v) + M(λ, v), (λ, v) ? A × D?, where A is an open subset in a normed space and for every fixed λ ? A, T, L(λ ·) and M(λ ·) are mappings from the closure D? of a neighborhood D of the origin in a BANACH space X into another BANACH space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let Λ be a characteristic value of the pair (T, L) such that T ? L( λ ,·) is a FREDHOLM mapping with nullity p and index s, p > s ≧ 0. Under suitable hypotheses on T. L and M, (λ , 0) is a bifurcation point of the above equations. This generalizes the results of [4], [6], [8], [13] and [14] etc. An application of the obtained results to the axisymmetric buckling problem of a thin spherical shell will be given.     

Properties of Modules and Rings Relative to Some Matrices

Let R be a ring and  _β×α(R) (?   _β×α(R)) the set of all β × α full (row finite) matrices over R where α and β ≥ 1 are two cardinal numbers. A left R-module M is said to be “injective relative” to a matrix A ? ?   _β×α(R) if every R-homomorphism from R ^(β) A to M extends to one from R ^(α) to M. It is proved that M is injective relative to A if and only if it is A-pure in every module which contains M as a submodule. A right R-module N is called flat relative to a matrix A ?  _β×α(R) if the canonical map μ: N? R ^(β) A → N ^α is a monomorphism. This extends the notion of (m, n)-flat modules so that n-projectivity, finitely projectivity, and τ-flatness can be redefined in terms of flatness relative to certain matrices. R is called left coherent relative to a matrix A ?  _β×α(R) if R ^(β) A is a left R-ML module. Some results on τ-coherent rings and (m, n)-coherent rings are extended.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

CertainC*-algebras with real rank zero and their corona and multiplier algebras: II

We first determine the homotopy classes of nontrivial projections in a purely infinite simpleC*-algebraA, in the associated multiplier algebraM(A) and the corona algebraM A/A in terms ofK _*(A). Then we describe the generalized Fredholm indices as the group of homotopy classes of non-trivial projections ofA; consequently, we determine theK _*-groups of all hereditaryC*-subalgebras of certain corona algebras. Secondly, we consider a group structure of *-isomorphism classes of hereditaryC*-subalgebras of purely infinite simpleC*-algebras. In addition, we prove that ifA is aC*-algebra of real rank zero, then each unitary ofA, in caseA it unital, each unitary ofM(A) and ofM(A)/A, in caseA is nonunital but -unital, can be factored into a product of a unitary homotopic to the identity and a unitary matrix whose entries are all partial isometries (with respect to a decomposition of the identity).Partially supported by a grant from the National Science Foundation.     

Robust stabilization of some class of uncertain systems

Consider an uncertain system $$ \dot x = A( \cdot )x + B( \cdot )u, $$ where A(·) ∈ ? ^{n × n} , B(·) ∈ ? ^{n × m} , and the elements of matrices A(·) and B(·) are arbitrary functionals. It is assumed that all elements are uniformly bounded, and that the first r elements counted from above and situated on a certain fixed upper superdiagonal are alternating. It is also assumed that m = n ? r, and that a matrix formed by the last m rows of matrix B(·) is nonsingular. The control u = S(·)x is synthesized, and conditions on the admissible matrix B(·) ensuring the global asymptotical stability of the system are obtained. We consider the case when modulation of the components of the vector u is realized by means of synchronous amplitude-frequency pulse modulators of the first kind. A lower estimate for the pulse frequency under which the pulse system is globally asymptotically stable is obtained.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Global stabilization of nonlinear systems by quadratic Lyapunov functions

The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t ₀ and x ∈ ℝ ⁿ , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t ₀ and x ∈ ℝ ⁿ . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.     

Necessary and sufficient conditions for quadratic stabilizability of an uncertain system

Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R ⁿ is the state,u(t) ∈R ^m is the control,r(t) ∈ ? ?R ^p represents the model parameter uncertainty, ands(t) ∈L ?R ^l represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ?,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ?,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ?,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.     

On a matrix associated to a matrix pair

For a c–stable matrix pair (E,A), i.e., all finite eigenvalues of (E,A) have negative real part, an associated c–stable matrix is provided. It has all finite eigenvalues of (E,A) as eigenvalues and an additional stable eigenvalue −α, where α may be chosen arbitrarily, which corresponds to the eigenvalue ∞ of (E,A). In the case of positive systems, this associated c–stable matrix is, in addition, shown to be a −M-matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult _m (A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult _a (A, ∥ · ∥) be the subset of the ϕ ∈ Mult _m (A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult _m (A, ∥ · ∥) \ Mult _a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult _a (A, ∥ · ∥) is dense inside Mult _m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult _a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult _m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult _a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult _a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.     

A bound on the scrambling index of a primitive matrix using Boolean rank

The scrambling index of an n×n primitive matrix A is the smallest positive integer k such that A^k(A^t)^k=J, where A^t denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M=AB for some m×b Boolean matrix A and b×n Boolean matrix B. In this paper, we give an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M). Furthermore we characterize all primitive matrices that achieve the upper bound.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

Generalized matrix near-rings

Let R be a right near-ring with identity. The k×k matrix near-ring over R, Mat_k(R R), as defined by Meldrum and van der Walt, regards R as a left mod-ule over R. Let M be any faithful left R-module. Using the action of R on M, a generalized k×k matrix near-ring, Mat_k(R M), is defined. It is seen that Mat_k(R M) has many of the features of Mat_k(R R). Differences be-tween the two classes of near-rings are shown. In spe- cial cases there are relationships between Mat_k(R M) and Mat_k(R R). Generalized matrix near-rings Mat_k(R M) arise as the “right near-ring” of finite centraiizer near-rings of the form M _A{G)> where G is a finite group and A is a fixed point free automorphism group on G.