Let Rij be a given set of μi× μj matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[Fij], i,j=1,…, n, such that Fij=Rij for |i?j|?m, F admits either a left or right block triangular factorization, and (F?1)ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given Rij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied. 相似文献
We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? m, v?n, find an (m,n) matrix X ? 0 with prescribed row sums Σnj=1Xij = ui (i = 1,…,m) and prescribed column sums Σmi=1Xij = vj (j = 1,…,n) which fits the relations Xij = Aij + λiWij + Wij + Wijμj for all i,j and some λ?m, μ?n. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2]. 相似文献
If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.Pn will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. Pm × Pn will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(Pm × Pn) = min(m, n). 相似文献
It is shown that, whenever m1, m2,…, mn are natural numbers such that the pairwise greatest common divisors, dij=(mi, mj), i≠j are distinct and different from 1, then there exist integers a1, a2,…,an such that the solution sets of the congruences x≡i (modmi), i= 1,2,…,n are disjoint. 相似文献
Consider a standard row-column-exchangeable array X = (Xij : i,j ≥ 1), i.e., Xij = f(a, ξi, ηj, λij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′1, ξ′2,…, η′1, η′2,…, are conditionally independent given ∩n ≥ 1σ(X′ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous. 相似文献
In connection with the problem of finding the best projections of k-dimensional spaces embedded in n-dimensional spaces Hermann König asked: Given m∈ and n∈, are there n×n matrices C=(cij), i, j=1,…,n, such that cii=m for all i, |cij|=1 for i≠j, and C2=(m2+n?1)In? König was especially interested in symmetric C, and we find some families of matrices satisfying this condition. We also find some families of matrices satisfying the less restrictive condition CCT=(m2+n?1)In. 相似文献
It is well known that the ideal classes of an order Z[μ], generated over Z by the integral algebraic number μ, are in a bijective correspondence with certain matrix classes, that is, classes of unimodularly equivalent matrices with rational integer coefficients. If the degree of μ is ?3, we construct explicitly a particularly simple ideal matrix for an ideal which is a product of different prime ideals of degree 1. We obtain the following special n×n matrix (cij) in the matrix class corresponding to the ideal class of our ideal: ci+1,i=1(i=1,…,n?2); cij=0(?i?n, 1?j?n? 2, and i≠j+1); cnj=0(j)=2,…,n?1). The remaining coefficients are given as explicit polynomials in an integer z which depends on the ideal. It is shown that the matrix class of every regular ideal class of Z[μ] contains a special matrix of this kind. 相似文献
Let P=[pij] be a m×n matrix and let C be the coefficient matrix of Σj=1npijxij=ui, 1≤i≤m, Σi=1mpijxij=vj, 1≤j≤n. The relation between the reducibility of P and the rank of C is investigated. An application to martingale extension is given. 相似文献
Let A = (aij) be an n × m matrix with aij ∈ K, a field of characteristic not 2, where Σi=1naij2 = e, 1 ≤ j ≤ m, and Σi=1naijaij′ = 0 for j ≠ j′. The question then is when is it possible to extend A, by adding columns, to obtain a matrix with orthogonal columns of the same norm. The question is answered for n ? 7 ≤ m ≤ n as well as for more general cases. Complete solutions are given for global and local fields, the answer depending on what congruence class modulo 4 n belongs to and how few squares are needed to sum to e. 相似文献
Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G. 相似文献
We consider matrices M with entries mij = m(λi, λj) where λ1, … ,λn are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries wij = 1/m (λi, λj) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means. 相似文献
Let H be a subset of the set Sn of all permutations C=6cij6 a real n?n matrix Lc(s)=c1s(1)+c2s(2)+???+cns(n) for s ? H. A pair (H, C) is the existencee of reals a1,b1,a2,b2,…an,bn, for which cij=a1+bj if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described. 相似文献
Let A be an n × n normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i1,…, ik, ik+1,…, im, such that α(ij) = β(ij), i = 1,…, k, and {α(ik+1),…, α(im) } ∩ {β(ik+1),…, β(im) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let n be the group of n × n unitary matrices. Define the nonnegative number where | α ∩ β| = k. It is proved that Let A be semidefinite hermitian. We conjecture that ρ0(A) ? ρ1(A) ? ··· ? ρm(A). These inequalities have been tested by machine calculations. 相似文献
Let Kn denote the set of all n X n nonnegative matrices whose entries have sum n, and let φ be a real valued function defined on Kn by φ(X) = πin=1 n, + πj=1cj−n per X for X E Kn with row sum vector (r1,…, rn) and column sum vector (cl,hellip;, cn). For the same X, let φij(X)= πk≠irk + π1≠jc1 - per X(i| j). AϵKn is called a φ-maximizing matrix if φ(A) > φ(X) for all X ϵ Kn. Dittert's conjecture asserts that Jn = [1/n]n×n is the unique (φ-maximizing matrix on Kn. In this paper, the following are proved: (i) If A = [aij] is a φ-maximizing matrix on Kn then φij(A) = φ (A) if aij > 0, and φij (A) ⩽ φ (A)if aij = 0. (ii) The conjecture is true for n = 3. 相似文献
Given a sequence of random variables taking values in {?v,…,0,…,+u}, let's define the local score of the sequence by Hn=max1?i?j?n(∑k=ijXk). The local score is used to analyze biological sequences pointing out regions of the sequences with interesting biological properties. In order to separate randomly events from really interesting segments, we establish here the distribution of the local score of Hn when the sequence is a Markov chain of order 1. To cite this article: S. Mercier, C. Hassenforder, C. R. Acad. Sci. Paris, Ser. I 336 (2003).相似文献
Let Xj (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y1,…,Ym) and X′ = (X1,…,Xn) be independently distributed, and A = (ajk) be an n × n random coefficient matrix with ajk = ajk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ~ N(O,I) if and only if X ~ N(O,I). provided that certain conditions defined in terms of the ajk are satisfied. The task of this paper is to delete the identical assumption on X1,…,Xn and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered. 相似文献
A sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (aij) such that at least one of aij and aij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and aij=0 iffaji=0. We also show that for any n×nL-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects. 相似文献
Let Kn denote the set of all n × n nonnegative matrices whose entries have sum n, and let ϕ be a real function on Kn defined by ϕ (X) = Πni=1Σnj=1xij + Πnj=1Σni=1xij − per X for X = [xij] ϵ Kn. A matrix A ϵ Kn is called a ϕ -maximizing matrix on Kn if ϕ (A) ⩾ ϕ (X) for all X ϵ Kn. It is conjectured that Jn = [1/n]n × n is the unique ϕ-maximizing matrix on Kn. In this note, the following are proved: (i) If A is a positive ϕ-maximizing matrix, then A = Jn. (ii) If A is a row stochastic ϕ-maximizing matrix, then A = Jn. (iii) Every row sum and every column sum of a ϕ-maximizing matrix lies between 1 − √2·n!/nn and 1 + (n − 1)√2·n!/nn. (iv) For any p.s.d. symmetric A ϵ Kn, ϕ (A) ⩽ 2 − n!/nn with equality iff A = Jn. (v) ϕ attains a strict local maximum on Kn at Jn. 相似文献
Let X1, X2, …, Xm be finite sets. The present paper is concerned with the m2 ? m intersection numbers |Xi ∩ Xj| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T1, T2, …, Tm be finite sets and let m ? 3. We assume that each of the elements in the set union T1 ∪ T2 ∪ … ∪ Tm occurs in at least two of the subsets T1, T2, …, Tm. We further assume that every pair of sets Ti and Tj (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set Tk (k ≠ i, k ≠ j) such that Tk intersects both Ti and Tj. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}. 相似文献