共查询到20条相似文献,搜索用时 15 毫秒
1.
Noga Alon 《Journal of Combinatorial Theory, Series A》1985,40(1):82-89
Let X1, …, Xn be n disjoint sets. For 1 ? i ? n and 1 ? j ? h let Aij and Bij be subsets of Xi that satisfy |Aij| ? ri and |Bij| ? si for 1 ? i ? n, 1 ? j ? h, for 1 ? j ? h, for 1 ? j < l ? h. We prove that . This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra). 相似文献
2.
Rudolf Wegmann 《Journal of Mathematical Analysis and Applications》1976,56(1):113-132
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, , p) with the same distribution Fa. Suppose that all moments | a | k, k = 1, 2, … are finite, a=0 and | a | 2. Let with complex numbers θσ and finite products Pσ of factors A and (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples , , , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n. 相似文献
3.
Let A be an n-square 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} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i1,…,ik, ik+1,…,im, such that α(ij)=β(ij), j=1,…,k, and {α(ik+1),…,α(im)};∩{β(ik+1),…,β(im)}=ø. Let be the group of n-square unitary matrices. Define the nonnegative number , where |α∩β|=k. Theorem 1 establishes a bound for ?k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that . 相似文献
4.
M. Neumann 《Linear algebra and its applications》1976,14(1):41-51
In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?Cm × nr. The splitting A = M ? N is called subproper if R(A) ? R(M) and . Consider the iteration . We characterize the convergence of this scheme to a solution of the linear system. When A?Rm×nr, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution. 相似文献
5.
An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λi, singular values σi, and inverse B = (bij). We are interested in the four closely related problems of finding λ(n) = minA, i|λi|, σ(n) = minA, iσi, χ(n) = maxA, i, j |bij|, and μ(n) = maxAΣijb2ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between and c√n (2.274)?n, where c is a constant, , and . We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry. 相似文献
6.
Zsolt Baranyai 《Journal of Combinatorial Theory, Series B》1979,26(3):276-294
This paper contains and generalizes the solution of the following classical problem:If h | n then the h-element subsets of an n-element set can be partitioned into (h?1n?1) classes so that every class contains disjoint h-element sets and every h-element set appears in exactly one class. A short formulation of this statement is: If h | n then the hypergraph is 1-factorizable. In this paper we study the factorization and edge-coloring problems of the hypergraph Krxmh (which is the complete, regular, h-uniform, r-partite hypergraph with m vertices in each of the r classes of vertices). 相似文献
7.
Gérald Tenenbaum 《Journal of Number Theory》1982,15(3):331-346
The condition , where Ω(n) stands for the number of prime factors, counted according to multiplicity, of the positive integer n, is shown to be necessary and sufficient for the integer sequence with characteristic function χ to have divisor density z, i.e., Σd|nχ(d) = (z + o(1)) Σd|n 1 when n → ∞ if one neglects a sequence of asymptotic density zero. Among the applications, the following result, first conjectured by R. R. Hall, is proved: given any positive α, we have, for almost all n's, and uniformly with respect to z in |0, 1|, 相似文献
8.
K.B. Athreya 《Statistics & probability letters》1983,1(3):147-150
Let X1, X2, X3, … be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,…) and integers n and m, construct Yn,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution for 1 ? j ? n. ( denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981). 相似文献
9.
Ming-Po Chen Cheh-Chih Yeh Cheng-Shu Yu 《Journal of Mathematical Analysis and Applications》1977,59(2):211-215
For nonlinear retarded differential equations and the sufficient conditions are given on fi, pi, Fi, and h under which every bounded nonoscillatory solution of () or () tends to zero as t → ∞. 相似文献
10.
Herbert E. Salzer 《Journal of Computational and Applied Mathematics》1976,2(4):241-248
Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(xi), i = 1(1)2n+1, gives a trigonometric sum of the nth order, S2n+1(x = a0 + ∑jn = 1(ajcos jx + bjsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S2r+1(x) is one that satisfies and two other arbitrarily imposable conditions needed to make S2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S4r?1(x). We have where the nodes xj, j = 1(1)2r, are the zeros of the orthogonal S2r+1(x). It is proven that Aj > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S2r+1(x), xjandAj, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy. 相似文献
11.
Winfried Stute 《Journal of multivariate analysis》1984,14(1):83-93
For a given score function ψ = ψ(x, θ), let θn be Huber's M-estimator for an unknown population parameter θ. Under some mild smoothness assumptions it is known that is asymptotically normal. In this paper the stopping times associated with the sequence of confidence intervals for θ are investigated. A useful representation of M-estimators is derived, which is also appropriate for proving laws of the iterated logarithm and Donskertype invariance principles for (πn)n. 相似文献
12.
It has been conjectured that if A is a doubly stochastic n>× n matrix such that per A(i, j)≥perA for all i, j, then either A = Jn, then n × n matrix with each entry equal to , or, up to permutations of rows and columns, , where Pn is a full cycle permutation matrix. This conjecture is proved. 相似文献
13.
Let be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of . Furthermore, let Mk(Ω;) be the set of -valued functions defined in an open subset Ω of Rm+1 (1 ? m ? n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator and k? N. The aim of this paper is to characterize the dual and bidual of Mk(Ω;). It is proved that, if Mk(Ω;) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits Mk(Ω;) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized. 相似文献
14.
Rainer Kress Hans Ludwig De Vries Rudolf Wegmann 《Linear algebra and its applications》1974,8(2):109-120
For the eigenvalues λi of an n × n matrix A the inequality is proved, where and 6 ● 6; denotes the euclidean norm. Conditions for equality are stated. 相似文献
15.
Let a complex pxn matrix A be partitioned as A′=(A′1,A′2,…,A′k). Denote by ?(A), A′, and A? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A(14) be an inner inverse of A such that A(14)A is a Hermitian matrix. Let B=(A(14)1,A(14)2,…,Ak(14)) and .Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A(14) or for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition is no longer true. 相似文献
16.
A new result on products of matrices is proved in the following theorem: let Mi (i=1,2,…) be a bounded sequence of square matrices, and K be the l.u.b. of the spectral radii ρ(Mi). Then for any positive number ε there is a constant A and an ordering p(j) (j = 1,2,…) of the matrices such that . The ordering is well defined by p(j), a one-to-one mapping on the set of positive integers. In general the inequality does not hold for any ordering p(j) (a counterexample is provided); however, some sufficient conditions are given for the result to remain true irrespective of the order of the matrices. 相似文献
17.
《Journal of Computational and Applied Mathematics》2002,142(1):125-135
We show that in the evolution of the random d-uniform hypergraph the phase transition occurs when M=n/d(d−1)+O(n2/3). We also prove local limit theorems for the distribution of the size of the largest component of in the subcritical and in the early supercritical phase. 相似文献
18.
Moshe Roitman 《Advances in Mathematics》1981,41(3):301-311
For a finite group G and a set I ? {1, 2,…, n} let ,where We prove, among other results, that the positive integers for 1 ? j ? r, Ij1 ∩ Ij2 ∩ Ij3 ∩ Ij4 = Ø for any 1 ? j1 <j2 <j3 <j4 ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups. 相似文献
19.
Elliptic boundary value problems for systems of nonlinear partial differential equations of the form , i = 1(1)N, j, k = 1(1)n, pi ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small. 相似文献
20.
Harald K. Wimmer 《Linear algebra and its applications》1975,10(2):139-146
The matrix equation , W ?0, is studied. In the case A1H+HA = W[H?A1HA = W], the controllability matrix of (A1,W) is used to determine the number of eigenvalues of A on the imaginary axis [on the unit circle]. As an application a result of Pták on the critical exponent of the spectral norm is obtained. Estimates for the eigenvalues of A satisfying fH(A) = M are given. 相似文献