On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

Extensions of the Hadamard determinant theorem

Denote byH _n the set ofn byn, positive definite hermitian matrices. Hadamard proved thath(A)≧det(A) for allA∈H _n, whereh(A) is the product of the main diagonal elements ofA. Subsequently, M. Marcus showed that per(A)≧h(A) for allA∈H _n. This article contains a result for all generalized matrix functions from which it follows thath(A)≧(per(A^1/n )) ⁿ ,A∈H _n.     

Vector Majorization Via Hessenberg Matrices

In this paper it is proved that, for real n-vectors x and y,x is majorized by y if and only if x = PHQy for some permutationmatrices P, Q, and for some doubly stochastic matrix H whichis a direct sum of doubly stochastic Hessenberg matrices. Thisresult reveals that any n-vector which is majorized by a vectory can be expressed as a convex combination of at most (n² –n + 2)/2 permutations of y.     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

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.     

TC semigroups and inflations

An algebraA satisfiesTC (the term condition) if for any and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS _E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS _Reg (the set of regular elements ofS) andS _Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.     

Scaling of matrices to achieve specified row and column sums

IfA is ann ×n matrix with strictly positive elements, then according to a theorem ofSinkhorn, there exist diagonal matricesD ₁ andD ₂ with strictly positive diagonal elements such thatD ₁ A D ₂ is doubly stochastic. This note offers an alternative proof of a generalization due toBrualdi, Parter andScheider, and independently toSinkhorn andKnopp, who show that A need not be strictly positive, but only fully indecomposable. In addition, we show that the same scaling is possible (withD ₁ =D ₂) whenA is strictly copositive, and also discuss related scaling for rectangular matrices. The proofs given show thatD ₁ andD ₂ can be obtained as the solution of an appropriate extremal problem.The scaled matrixD ₁ A D ₂ is of interest in connection with the problem of estimating the transition matrix of a Markov chain which is known to be doubly stochastic. The scaling may also be of interest as an aid in numerical computations.Research sponsored in part by the Boeing Scientific Research Laboratories.     

p-Minors of a doubly stochastic matrix at which the permanent achieves a minimum

The author proves that if A is a matrix at which the permanent achieves a local minimum relative to the set of n x n doubly stochastic matrices, then for a_ij=0,

$\text{per A (i|j)?per A}$

.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

On non-isotropic harmonic maps of surfaces into complex projective spaces

It is proved by purely algebraic method that weakly conformai, conformai andA _z ³ = 0 are mutually equivalent if ϕ :Ω→ℂP ⁿ is a non-isotropic harmonic map and the harmonic maps with isotropy order ≥3 are uniquely determined by a system of ordinary differential equations. A method is given, by which the isotropy orders of non-isotropic harmonic maps can be computed.     

Solutions of the matrix equation (1):X m+A 1Xm−1+…+A m=0

SupposeX and the coefficientsA ₁, …,A _m aren×n matrices. LetB be anmn×mn matrix as in (7). LetJ be the Jordan canonical matrix ofB andB=PJP ⁻. LetE _i denote thei×i unit matrix.V is defined bydV/dt=JV andV(t=0)=E _mn. ThenZ=PV satisfiesdZ/dt=BZ.P ^* is a matrix which consists of the firstn rows ofP. The author proves: There is a solution of (1) ↔ there are anmn×n matrixC, ann×n matrixQ and ann×n function matrixN such thatP ^*VC=QN, where detQ≠0 andN is defined byN(t=0)=E _n anddN/dt=RN, in whichR is ann×n Jordan canonical matrix. There are three cases regarding the solutions of (1): No solution, finitek solutions, 1<k<C _n ^m , and infinite solutions which containj parameters, 1<-j<-mn ².     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems

A particular class of preconditioners for the conjugate gradient method and other iterative methods is proposed for the solution of linear systemsA _n,mx=b, whereA _n,m is ann×n positive definite block Toeplitz matrix withm×m Toeplitz blocks. In particular we propose a sparse preconditionerP _n,m such that the condition number of the preconditioned matrix turns out to be less than a suitable constant independent of bothn andm, even if the condition number ofA _n,m tends to . This leads to iterative methods which require a number of steps independent ofm andn in order to reduce the error by a given factor.     

Hypersurfaces of prescribed mean curvature enclosing a given body

Given a smooth domain Ω in ℝ ^m+1 with compact closure and a smooth integrable functionh: ℝ ^m+1→ℝ satisfyingh(x)≥H _∂Ω (x) on ∂Ω whereH _∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ ^m+1 with the properties andH _∂A=h on ∂A.     

Universal PI-algebras and algebras of generic matrices

Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM _n (Ω[ξ]) generated by the matricesY _k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X ₁, …,X _m) is a polynomial linear in all theX _i but one (similar to Formanek’s central polynomials for matrix rings) andf ² is central forM _n (Ω), thenf is central forM _n (Ω). (The existence of a polynomial not central forM _n (Ω), but whose square is central forM _n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.     

Non variational basic elliptic systems of second order

Denoting byu a vector in R ^N defined on a bounded open set Ω ⊂ R ⁿ , we setH(u)={Dij u} and consider a basic differential operator of second ordera(H(u)) wherea(ξ) is a vector in R ^N , which is elliptic in the sense that it satisfies the condition (A). After a rapid comparison between this condition (A) and the classical definition of ellipticity, we shall prove that, if seu∈H ² (Ω) is a solution of the elliptic systema(H(u))=0 in Ω thenH(u)∈H _loc ^{2, q} for someq>2. We then deduce from this the so called fundamental internal estimates for the matrixH(u) and for the vectorsDu andu. We shall then present a first risult on h?lder regularity for the solutions of the system withf h?lder continuous in Ω, and a partial h?lder continuity risult for solutionsu∈H ² (Ω) of a differential systema (x, u, Du, H (u))=b(x, u, Du)     

Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution

Let D be any division ring with an involution,Hn （D） be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank（A - B） = 1. It is proved that if φ is a bijective map from Hn（D）（n ≥ 2） to itself such that φ preserves the adjacency, then φ^-1 also preserves the adjacency. Moreover, if Hn（D） ≠J3（F2）, then φ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe-Xian is answered for geometry of symmetric and hermitian matrices.     

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.     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.