Method of solution to fuzzy relation equations in a complete Brouwerian lattice

Up to now, how to solve a fuzzy relation equation in a complete Brouwerian lattice is still an open problem as Di Nola et al. point out. To this problem, the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable. In this paper, we first show that there is a minimal element in the solution set of a fuzzy relation equation AX=b (where A=(a₁,a₂,…,a_n) and b are known, and X=(x₁,x₂,…,x_n)^T is unknown) when its solution set is nonempty, and b has an irredundant finite join-decomposition. Further, we give the method to solve AX=b in a complete Brouwerian lattice under the same conditions. Finally, a method to solve a more general fuzzy relation equation in a complete Brouwerian lattice when its solution set is nonempty is also given under similar conditions.     

Solutions to Operator Equations on Hilbert C*-Modules II

In this paper, we study the solvability of the operator equations A*X + X*A = C and A*XB + B*X*A = C for general adjointable operators on Hilbert C*-modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B*(X* + X)B ≥ 0, and of a solution X with B*XB ≥ 0. Furthermore in the special case that R(B) í [`(R(A*))]{R(B)\subseteq\overline{R(A*)}} we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges.     

A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results.     

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A?BXB*?CYC* and then solve two conjectures on the maximal and minimal possible ranks of A?BXB*?CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles

We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if X is a compact connected manifold, then for a C^r (r ≥ 1) open and dense set of maps A, either (T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of (T, A) vanish for the generic measurable T. For the continuous case, we obtain that if X is of dimension greater than 2, then for a C^r (r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either (T, A) is uniformly hyperbolic or the Lyapunov exponents of (T, A) vanish. *Partially supported by CNPq-Profix and Franco-Brazilian cooperation program in Mathematics.     

Approximate fixed-rank closures of covering problems

Consider a 0/1 integer program min{c^Tx :Ax≥b, x ∈ {0,1}ⁿ} where A is nonnegative. We show that if the number of minimal covers of Ax≥b is polynomially bounded, then for any ε>0 and any fixed q, there is a polynomially large lift-and-project relaxation whose value is at least (1−ε) times the value of the rank ≤q relaxation. A special case of this result is that given by set-covering problems, or, generally, problems where the coefficients in A and b are bounded. This research was partially funded by NSF awards ITR:CCR-0213848 and DMI-0200221 formerly: Set covering problems and Chvátal-Gomory cuts     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

一四元数矩阵方程组的广义(反)反射解

该文给出了四元数矩阵方程组X_1B_1=C_1,X_2B_2=C2,A_1X_1B_3+A_2X_2B_4=C_b可解的充要条件及其通解的表达式,利用此结果建立了四元数矩阵方程组XB_a=C_a,A_bXB_b=C_b有广义(反)反射解的充要条件及其有此种解时通解的表达式.     

Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L ¹(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L ^p (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L ¹(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C ₀-semigroup. Günter Lumer in memoriam     

Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X ₀ and X ₁ in solution X = X ₀ + iX ₁ to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)^- = C + iD of a complex matrix A + iB are also examined.     

An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C

An iteration method is constructed to solve the linear matrix equation AXB=C over symmetric X. By this iteration method, the solvability of the equation AXB=C over symmetric X can be determined automatically, when the equation AXB=C is consistent over symmetric X, its solution can be obtained within finite iteration steps, and its least-norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new matrix equation . Finally, numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation AXB=C.     

On one class of topological *-algebras with standard identities

Let A be a unital semisimple topological nuclear *-algebra over C and let Z be its center. The algebra A is topologically isomorphic to M _n (Z) if and only if A satisfies the standard identity and the maximality condition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 140–143, January, 2007.     

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X ^T B = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.     

On Optimal Feedback Control for Stationary Linear Systems

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems A X+B U=Z with output Y=C X+D U from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.     

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

An example of L-fuzzy join space

On a generalized deMorgan lattice (X, ≤, ∨, ∧,′) we introduce a family of join hyperoperations * _p , parametrized by a parameterp εX. As a result we obtain a family of join spaces (X, * _p ). We show that: for everya,b εX the family {a*_pb} _pεX can be considered as thep-cuts of aL-fuzzy seta*b; in this manner we synthesize aL-fuzzy hyperoperation * which takes pairs fromX toL-fuzzy subsets ofX. We then show that (X, * _p ) is aL-fuzzy hypergroup (in the sense of Corsini) and can be considered as aL-fuzzy join space. Furthermore,a*b is aL-fuzzy interval for alla,b εX.     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Inertias and ranks of some Hermitian matrix functions with applications

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of