Let Lj (j = 1, …, n + 1) be real linear functions on the convex set of probability distributions. We consider the problem of maximization of Ln+1(F) under the constraint F ? and the equality constraints L1(F) = z1 (i = 1, …, n). Incorporating some of the equality constraints into the basic set , the problem is equivalent to a problem with less equality constraints. We also show how the dual problems can be eliminated from the statement of the main theorems and we give a new illuminating proof of the existence of particular solutions.The linearity of the functions Lj(j = 1, …, n + 1) can be dropped in several results. 相似文献
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
Let Fn(x) be the empirical distribution function based on n independent random variables X1,…,Xn from a common distribution function F(x), and let be the sample mean. We derive the rate of convergence of to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for . 相似文献
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
Let B(Z1, z2) be a power series and C(z1, z2) be the least mean-square inverse approximation of B among polynomials of a fixed order. This paper discusses conditions under which , the least mean-square inverse polynomial approximation of C, does not vanish on the unit bicylinder. 相似文献
Let x1,…,xs be a form of degree d with integer coefficients. How large must s be to ensure that the congruence (x1,…,xs) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if has coefficients in a finite additive group G, how large must s be in order that the equation (x1,…,xs) = 0 has a solution of this type? We deal with these questions as well as related problems in the group of integers modulo 1 and in the group of reals. 相似文献
Let R1,…,Rm be rectangles on the plane with sides parallel to the coordinate axes. An algorithm is described for finding the contour of F = R1 ∪ … ∪ Rm in time, where p is the number of edges in the contour. This is O(m2) in the general case, and O(m log m) when F is without holes (then p ≤ 8m ? 4); both of these performances are optimal. 相似文献
Let F1(x, y),…, F2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of Fi(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field ((?q)1/2). 相似文献
Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P0(z) and P0(z)fi(z) ? Pi(z), where Pi(z) are polynomials of degree σ ? ρi (i=0,1,…,n), P0(z)fi(z) ? Pi(z) are power series in which the terms with zk, 0?k?σ, vanish (i=1,2,…,n), (ρ0,ρ1,…,ρn) is an (n+1)-tuple of nonnegative integers, σ=ρ0+ρ1+?+ρn, and {fi}ni=1 is the set of hypergeometric functions {1F1(1;ci;z)}ni=1 or {2F0(ai,1;z)}ni=1 under the condition ρ0?ρi ? 1 (i=1,2,…,n). 相似文献
Let (Xn)n? be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions FXn, and Sn = Σi=1nXi. The authors present limit theorems together with convergence rates for the normalized sums ?(n)Sn, where ?: → +, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝f(x) d[F?(n)Sn(x) ? FX(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors. 相似文献
Let be a Schrödinger operator in Rn. Here is an “exploding” radially symmetric potential which is at least C2 monotone nonincreasing and O(r2) as r → ∞. V is a general potential which is short range with respect to VE. In particular, VE 0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = limr → ∞VE(r) and R(z) = (H ? z)?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset ?(Λ, ∞), in a suitable operator topology . And L ? L2(Rn) is a weighted L2-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing. 相似文献
Let F be a field, F1 be its multiplicative group, and = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let Dn be the dihedral group of degree n, H be a nontrivial group in , and τn(H) = {α = (α1, α2,…, αn):αi?H}. For σ?Dn and α?τn(H), let P(σ, α) be the matrix whose (i,j) entry is αiδiσ(j) (i.e., a generalized permutation matrix), and . Let Mn(F) be the vector space of all n×n matrices over F and P(Dn, H) = {T:T is a linear transformation on Mn (F) to itself and T(P(Dn, H)) = P(Dn, H)}. In this paper we classify all T in P(Dn, H) and determine the structure of the group P(Dn, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper. 相似文献
Let C = (V,E) be a digraph with n vertices. Let f be a function from E into the real numbers, associating with each edge e ∈ E a weight. Given any sequence of edges σ = e1,e2,…,ep define w(σ), the weight of σ, as , and define m(σ), the mean weight of σ, as w(σ)?p. Let where C ranges over all directed cycles in G; λ1 is called the minimum cycle mean. We give a simple characterization of λ1, as well as an algorithm for computing it efficiently. 相似文献
Let n be a positive integer, L a subset of {0, 1,…,n}. We discuss the existence of partitions (or tilings) of the n-dimensional binary vector space Fn into L-spheres. By a L-sphere around an x in Fn we mean {y ? Fn, d(x, y) ? L}, d(x, y) being the Hamming distance betwe en x and y. These tilings are generalizations of perfect error correcting codes. We show that very few such tilings exist (Theorem 2) and characterize them all for any L ? {0, 1,…,[n]}. 相似文献
Let Z = {Z0, Z1, Z2,…} be a martingale, with difference sequence X0 = Z0, Xi = Zi ? Zi ? 1, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(supi |Xi| ≥ λ) ≤ C supiE |Zi|, for λ > 0, is C = (log 2)?1. If Z is finite of length n, it is proved that the best constant is . The analogous best constant Cn(z) when Z0 ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity Gn(z, E) = supzP(maxi=1,…,n |Xi| ≥ 1), the supremum being taken over all martingales Z with Z0 ≡ z and E|Zn| = E. The expression for Gn(z,E) is derived by induction, using methods from the theory of moments. 相似文献
Let A be a subalgebra of the full matrix algebra Mn(F), and suppose J∈A, where J is the Jordan block corresponding to xn. Let be a set of generators of A. It is shown that the graph of determines whether A is the full matrix algebra Mn(F). 相似文献
Let be a family of connected graphs. With each element α ∈ , we can associate a weight wα. Let G be a graph. An F-cover of G is a spanning subgraph of G in which every component belongs to . With every F-cover we can associate a monomial π(C) = Παwα, where the product is taken over all components of the cover. The F-polynomial of G is Σπ(C), where the sum is taken over all F-covers in G. We obtain general results for the complete graph and complete bipartite graphs, and we show that many of the well-known graph polynomials are special cases of more general F-polynomials. 相似文献
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
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