共查询到20条相似文献,搜索用时 46 毫秒
1.
Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let . It is shown that if , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved. 相似文献
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David L Russell 《Journal of Mathematical Analysis and Applications》1982,87(2):528-550
We suppose that K is a countable index set and that is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L2[a, b], a < b. Let Σ = {σ1, σ2,…, σm} consist of m complex numbers not in Λ. Then, with p(λ) = Πk = 1m (λ ? σk), forms a Riesz (strong) bas Sobolev space Hm[a, b]. If we take σ1, σ2,…, σm to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tneλkt. 相似文献
4.
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
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Let be a Dirichlet form in , where Ω is an open subset of n, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let k be a Dirichlet form on some k-dimensional submanifold of Ω. The paper is devoted to the study of the closability of the forms E with domain and defined by: ki where 1 ? kp < ? < n, and where , gki denote restrictions of ?, g in to . Conditions are given for E to be closable if, for each i = 1,…, p, one has ki = n ? i. Other conditions are given for E to be nonclosable if, for some i, ki < n ? i. 相似文献
6.
Allen J. Schwenk 《Discrete Mathematics》1977,18(1):71-78
Let denote the polynomial obtained from the cycle index of the symmetric group Z(Sn) by replacing each variable si by f(x1). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=xk + ak+1xk+1 + ak+2xk+2+? with every a1?0. Finally, let 0<x<1 and let x??. We prove that This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of .In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented. 相似文献
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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
9.
Let and be polynomials with real zeros satisfying An?1 = Bn?1 = 0, and let Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials. 相似文献
10.
Ludwig Arnold 《Linear algebra and its applications》1976,13(3):185-199
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
11.
Peter Lancaster 《Linear algebra and its applications》1977,18(3):213-222
The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix where the Ai are n×n complex matrices and detAl ≠ 0. In this paper general solutions are formulated for difference equations of the form . The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of for matrix S. 相似文献
12.
A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, Im(x, y) denotes , and Am(x, y) denotes an approximation of Im(x, y) of the form , where ak and yik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and xik = x + (y ? x)yik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F(1) exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, , then Further, if F(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when Aj is substituted for Ij. 相似文献
13.
Let π = (a1, a2, …, an), ? = (b1, b2, …, bn) be two permutations of . A rise of π is pair ai, ai+1 with ai < ai+1; a fall is a pair ai, ai+1 with ai > ai+1. Thus, for i = 1, 2, …, n ? 1, the two pairs ai, ai+1; bi, bi+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ωn denotes the number of pairs with RR forbidden, it is proved that , . More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then . 相似文献
14.
Austin J. Lemoine 《Stochastic Processes and their Applications》1973,1(3):251-268
A delayed random walk is defined here as a partial sum process of independent random variables in which the first N summands (N optional) are distributed F1,…,FN, respectively, while all remaining summands are distributed F0, where {Fk, k ≥ 0} is a sequence of proper distribution functions on the real line. Delayed random walks arise naturally in the study of certain generalized single server queues. This paper examines optional times of the process such as . Conditions insuring the finiteness of E {π} and E {π2} are obtained, generating functions calculated, and illustrative examples given. The bivariate functions and are studied for the case where N ≡ 1. 相似文献
15.
Stanisław Lewanowicz 《Journal of Computational and Applied Mathematics》1979,5(3):193-206
In this paper we are constructing a recurrence relation of the form for integrals (called modified moments) in which Ck(λ) is the k-th Gegenbauer polynomial of order , and f is a function satisfying the differential equation of order n, where p0, p1, …, pn ? 0 are polynomials, and mk〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense. 相似文献
16.
D Zwick 《Journal of Mathematical Analysis and Applications》1984,104(2):435-436
For a(1) ? a(2) ? ··· ? a(n) ? 0, b(1) ? b(2) ? ··· ? b(n) ? 0, the ordered values of ai, bi, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that where is the integer such that and . The inequality is shown to be sharp. When p < 1 and a(i)'s are in increasing order then the inequality is reversed. 相似文献
17.
Tom M. Apostol 《Journal of Number Theory》1982,15(1):14-24
An elementary proof is given of the author's transformation formula for the Lambert series relating Gp(e2πiτ) to Gp(e2πiAτ), where p > 1 is an odd integer and is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function , and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions. 相似文献
18.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
19.
Let Ω = {1, 0} and for each integer n ≥ 1 let (n-tuple) and for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that . For each A in , let TA be the first occurrence time of A with respect to the stochastic process {Ym}m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in , there is an element B in such that the probability that TB is less than TA is greater than . This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in , there is an element B also in such that the probability that TB is less than TA is greater than ; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a1, a2,…,an) in , there is an element C also in such that the probability that TA is less than TC is greater than if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process. 相似文献
20.
Robert Donaghey 《Journal of Combinatorial Theory, Series A》1976,21(2):155-163
This paper treats the class of sequences {an} that satisfy the recurrence relation between the odd and even terms of {an} that involves the coefficients of tan(t), namely A combinatorial setting is then provided to elucidate the appearance of the tangent coefficients in this equation. 相似文献