共查询到20条相似文献,搜索用时 873 毫秒
1.
The following results are proved: Let A = (aij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1. 相似文献
2.
Let be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let be the stochastic matrix which is obtained from ? by substituting the Kronecker product of An11,…,Anppfor each term Xn11·?·Xnpp. In this paper, we present necessary and sufficient conditions for the Cesàro limit of the sequence of the powers of to be equal to the Kronecker product of the Cesàro limits associated with each of A1,…,Ap. These conditions show that the equality of these two matrices depends only on the number of ergodic sets under and?or the cyclic structure of the ergodic sets under A1,…,Ap, respectively. As a special case of these results, we obtain necessary and sufficient conditions for the interchangeability of the Kronecker product and the Cesàro limit operator. 相似文献
3.
Frank Markham Brown 《Discrete Applied Mathematics》1982,4(2):87-96
Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation over a finite Boolean algebra B may be expressed as the union of a collection of subsets of Bn, each of the form {(x1, …, xn) | ai≤xi≤bi, ai?B, bi?B, i = 1, …, n}. We refer to such subsets of Bn as segments and to the collection as a segmental cover of K. We show that is consistent if and only if ? can be expressed by one of a class of sum-of-products expressions which we call segmental formulas. The object of this paper is to relate segmental covers of K to segmental formulas for ?. 相似文献
4.
C.J.K Batty 《Journal of Functional Analysis》1984,57(3):233-243
Let (A, G, α) be a C1-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in and a finite constant κ such that whenever xi lies in a spectral subspace , where . This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G. 相似文献
5.
Dudley Paul Johnson 《Stochastic Processes and their Applications》1985,19(1):183-187
We show that under mild conditions the joint densities Px1,…,xn) of the general discrete time stochastic process Xn on can be computed via where ? is in a Hilbert space , and T (x), x ? are linear operators on . We then show how the Central Limit Theorem can easily be derived from such representations. 相似文献
6.
Let (Wt) = (W1t,W2t,…,Wdt), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from + into the space of d × d skew-symmetric matrices and x(t) such a function into d. A class of stochastic processes (LtA,x), a particular example of which is Levy's “stochastic area” , is dealt with.The joint characteristic function of Wt and L1A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (Wt,LtA,x) is given. 相似文献
7.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If n denotes the symmetric group on {1,2,…,n} then we define the projection by the formula , where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If , then x1?x2? … ?xn denotes the member of Tn(V) such that for each y1 ,2,…,yn in V, and x1·x2… xn denotes . If B? Sn(V) and there exists , such that B = x1·x2…xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C. 相似文献
8.
Let Ω be a simply connected domain in the complex plane, and , the space of functions which are defined and analytic on , if K is the operator on elements defined in terms of the kernels ki(t, s, a1, …, an) in by is the identity operator on , then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on defined in terms of a kernel w(t, s, a1, …, an) in by Wu = ∝antw(t, s, a1, …, an) u(s, a1, …, an) ds. ΠW is the operator; ΠWu = ∝an ? 1w(t, s, a1, …, an) u(s, a1, …, an) ds. ΠK is the operator; ΠKu = ∑i = 1n ? 1 ∝aitki(t, s, a1, …, an) ds + ∝an ? 1tkn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where and maps elements of into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator on functions u in , by . A determinant of the operator is defined as an element of . This is mapped into by setting an + 1 = t to give m(t, a1, …, an). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in , explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case. 相似文献
9.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
10.
Palle E.T. Jørgensen 《Advances in Mathematics》1982,44(2):105-120
Let Ω be an arbitrary open subset of n of finite positive measure, and assume the existence of a subset Λ ? n such that the exponential functions eλ = exp i(λ1x1 + … + λnxn), λ = (λ1,…, λn) ∈ Λ, form an orthonormal basis for with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of (n, +) by K = Λ0 = {γ ∈ n:γ·λ ∈ 2π}, A = {a ∈ n:Ua U1a = }, where Ut is the unitary representation of n on given by Ute = eitλeλ, t ∈ n, λ ∈ Λ, and where is the multiplication algebra of on L2. Assume that A is discrete. Then there is a discrete subgroup D ? A of dimension n, a fundamental domain for D, and finite sets of representers RΛ, RΓ, , each containing 0, RΛ for in K0, and for in A such that Ω is disjoint union of translates of : Ω = ∪a∈RΩ (a + ), neglecting null sets, and Λ = RΛ ⊕ D0. If RΓ is a set of representers for in D, then Γ = RΓ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = n, direct sum, (neglecting null sets). The case A = n corresponds to Ω = , Λ = D0 and Γ = K. This last case corresponds in turn to a function theoretic assumption of Forelli. 相似文献
11.
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
12.
Jorge L.C Sanz Thomas S Huang 《Journal of Mathematical Analysis and Applications》1984,104(1):302-308
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? , i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? , i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?, i=1,…, n; and (2) conjugation: . 相似文献
13.
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
14.
Hans G Weidner 《Journal of Number Theory》1976,8(1):99-108
Let g = (g1,…,gr) ≥ 0 and h = (h1,…,hr) ≥ 0, g?, h? ∈ , be two vectors of nonnegative integers and let λ ? , λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. (g1,…,gr). Define It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i? {1,…,r} is such that then holds true for all sufficiently large λ, λ ≡ 0 mod d. 相似文献
15.
Let A be an infinite sequence of positive integers a1 < a2 <… and put , . In Part I, it was proved that . In this paper, this theorem is sharpened by estimating DA(x) in terms of fA(x). It is shown that limx→+∞sup DA(x) exp(?c1(logfA(x))2) = +∞ and that this assertion is not true if c1 is replaced by a large constant c2. 相似文献
16.
R.N. Buttsworth 《Journal of Number Theory》1980,12(4):487-498
The polynomial functions f1, f2,…, fm are found to have highest common factor h for a set of values of the variables x1, x2,…,xm whose asymptotic density is For the special case f1(x) = f2(x) = … = fm(x) = x and h = 1 the above formula reduces to , the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f1, f2,…, fm for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density. 相似文献
17.
Let (m?n) denote the linear space of all m × n complex or real matrices according as = or . Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?m×n is the quantity . where σ1(A)???σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1???dm?0. We consider the linear isometries between the normed spaces and , and prove that they are dual transformations of the linear operators which map (d) onto (c), where . 相似文献
18.
D.C Doss 《Journal of multivariate analysis》1979,9(3):460-464
The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s1,…,sn)]?k where β is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,…,xn) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,…, m, be subsets of {s1,…, sn} with empty ∩t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t. 相似文献
19.
We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B1(t),…,Bp(t)) U-1, t ∈ ?, and if each matrix function Bk has the polynomial form then A itself has the polynomial form , where , dk being the degree of the minimal polynomial of the matrix Ck, for every k ∈ {1,…,p}. 相似文献
20.
With quasicommutative n-square complex matrices A1,…,As and s-square hermitian G=(gij), relationships are given between the image of a linear transformation on n being positive definite and the action of H on generalized inertial decompositions of n. 相似文献