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1.
Let Fn denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=ANxN+ ?+ A1x+A0?Fn[x]. A function ?:Fn→Fn is called a right polynomial function iff there exists an A(x)?Fn[x] such that ?(B)=ANBN+?+A1B+ A0 for every B?Fn. This paper obtains unique representations for and determines the number of right polynomial functions.  相似文献   

2.
A topological system (X,f) is F-transitive if for each pair of opene subsets U and V of X, Nf(U,V)={n∈Z+:fnU∩V≠∅}∈F, where F is a collection of subsets of Z+ which is hereditary upward. (X,f) is F-mixing if (X×X,f×f) is F-transitive. In this paper F-mixing systems are characterized in terms of the chaoticity of the systems. Moreover, weak disjointness is studied via family. We will give conditions such that a dual theorem of the Weiss–Akin–Glasner theorem holds. Examples with this dual theorem fails for some “good” families are obtained.  相似文献   

3.
Let p, q be arbitrary parameter sets, and let H be a Hilbert space. We say that x = (xi)i?q, xi ? H, is a bounded operator-forming vector (?HFq) if the Gram matrixx, x〉 = [(xi, xj)]i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on lq2, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from lq2 to lp2. Then exists a linear operator ǎ from (the Banach space) HFq to HFp on D(A) = {x:x ? HFq, A〈x, x〉12 is p × q bounded on lq2} such that y = ǎx satisfies yj?σ(x) = {space spanned by the xi}, 〈y, x〉 = Ax, x〉 and 〈y, y〉 = A〈x, x〉12(A〈x, x〉12)1. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.  相似文献   

4.
Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, 1N)21λN?(x,Hx)(x,H?1x)?1. Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products M(y)1H and M(y)1H?1, where M(y) is a matrix defined by M(y)=(δij+(1?δij)yiyj. Subsequently, these results are extended to improve the spectral bounds of M(y)1H.  相似文献   

5.
A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, Im(x, y) denotes
χyχv2?χv2i=1mF(νi)dν12?dνm
, and Am(x, y) denotes an approximation of Im(x, y) of the form
(y?x)mk=1naki=1mF(χik)
, 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, (b ? a)n = h < 1, xi = a + hi for i = 0, 1,…, n and 0 < r ? s ? n, then
χr?χs(I+F dχ)?i=rsI+j=1pIji?1i)
=hpH(χr?1s)+O(hp+1)
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   

6.
Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C functions f: C × HR having Fredholm second derivative with respect to x at each (c, x) ?C × H for which D?c(x) = 0; here we write ?c(x) for ?(c, x). Say ? is of standard type if at all critical points of ?c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function fo?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in Rn.  相似文献   

7.
Let H1 = ?∑i = 1Ni + V(xi)) + ∑1 ? i <j ? N¦xi ? xj¦?1, V(xi) = N ∝ ¦x ? y¦?1 ?(y)dy, with ? a normalized Gaussian. Suppose E ≠ 0 and that H = H1 + E · (∑i = 1Nxi) has no eigenfunctions in L2(R3N. If H1ψ = μψ with μ < infσess(H1), then (ψ, e?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.  相似文献   

8.
This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gi(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when φ(x) = max{f(x, y) ¦ y ? Ωy}, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form miny ? Ωxmaxy ? Ωyf(x, y) under the assumption that Ωmξ={x|gi(x)?0, j=1,…,s} ∩Rn, Ωy={y|ζi(y)?0, i-1,…,t} ∩ Rm, with f, gj, ζi continuously differentiable, f(x, ·) concave, ζi convex for i = 1,…, t, and Ωx, Ωy compact.  相似文献   

9.
Let X1, …, Xp have p.d.f. g(x12 + … + xp2). It is shown that (a) X1, …, Xp are positively lower orthant dependent or positively upper orthant dependent if, and only if, X1,…, Xp are i.i.d. N(0, σ2); and (b) the p.d.f. of |X1|,…, |Xp| is TP2 in pairs if, and only if, In g(u) is convex. Let X1, X2 have p.d.f. f(x1, x2) = |Σ|?12 g((x1, x2) Σ?1(x1, x2)′). Necessary and sufficient conditions are given for f(x1, x2) to be TP2 for fixed correlation ?. It is shown that if f is TP2 for all ? >0. then (X1, X2)′ ~ N(0, Σ). Related positive dependence results and applications are also considered.  相似文献   

10.
Let?(x1,…,xp) be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let ??(A1,…,Ap) 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 ??(A1,…,Ap) 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??(A1,…,Ap) 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   

11.
The probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in Rn (1 ≤ rn ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of Y1 = (y01,…, yr1). Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω with ‘initial’ point at the origin [ω is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and yi1 = (xi ? z)α(p2), where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in Rn invariant under the group of rotations in Rn, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x0,…, xr} for 1 ≤ rn.  相似文献   

12.
Suppose x and y are two points in the upper half-plane H+, and suppose Γ is a discontinuous group of conformal automorphisms of H+ having compact fundamental domain S. Denote by NT(x, y) the number of points of the form γy (γ?Γ) in the closed disc of hyperbolic radius T centered about x, and set QT(x, y) = NT(x, y) ? V(T)A, where V(T) is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ?LxL(QT(x,y))2 is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.  相似文献   

13.
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 X = Σi=1nXin be the sample mean. We derive the rate of convergence of Fn(X) to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for Fn(X).  相似文献   

14.
In this paper we study the linked nonlinear multiparameter system
yrn(Xr) + MrYr + s=1k λs(ars(Xr) + Prs) Yr(Xr) = 0, r = l,…, k
, where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ A ¦ (x) = detars(xr) > 0. Conditions on the polynomial operators Mr, Prs are produced which guarantee a sequence of eigenfunctions for this problem yn(x) = Πr=1kyrn(xr), n ? 1, which form a basis in L2([a, b], ¦ A ¦). Here [a, b] = [a1, b1 × … × [ak, bk].  相似文献   

15.
Let O = limnZ/pnZ, let A = O[g2, g3]Δ, where g2 and g3 are coefficients of the elliptic curve: Y2 = 4X3 ? g2X ? g3 over a finite field and Δ = g23 ? 27g32 and let B = A[X, Y](Y2 ? 4X3 + g2X + g3). Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free A2?ZQ-module H1(X, A2?ZQ). Main results are; Theorem 1.1: X2dY and YdX are basis elements for H1(X, ΓA1(X)2?ZQ); Theorem 1.2: YdX, X2dY, Y?1dX, Y?2dX and XY?2dX are basis elements for H1(X ? (Y = 0), ΓA1(X)2?ZQ), where X is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.  相似文献   

16.
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 Ln denotes the symmetric group on {1,2,…,n} then we define the projection PL : Tn(V) → Sn(V) by the formula (n!)?1Σσ ? Ln Pσ, 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 xi ? V1, 1 ? i ? n, then x1?x2? … ?xn denotes the member of Tn(V) such that (x1?x2· ? ? ?xn)(y1,y2,…,yn) = Пni=1xi(yi) for each y1 ,2,…,yn in V, and x1·x2xn denotes PL(x1?x2? … ?xn). If B? Sn(V) and there exists x i ? V1, 1 ? i ? n, such that B = x1·x2xn, 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   

17.
Let A be an infinite sequence of positive integers a1 < a2 <… and put fA(x) = Σa∈A, a≤x(1a), DA(x) = max1≤n≤xΣa∈A,an1. In Part I, it was proved that limx→+∞supDA(x)fA(x) = +∞. 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   

18.
For (x,y,t)∈Rn × Rn × R, denote Xj = ??xj + 2yj??t, yj = ??yj ? 2xj??t and Lα=?14j=1nXj2 + Yj2 + ??t. When α = n ? 2q, La represents the action of the Kohn Laplacian □b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X1,…, Xn, Y1,…, Yn is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.  相似文献   

19.
For a class C of subsets of a set X, let V(C) be the smallest n such that no n-element set F?X has all its subsets of the form AF, AC. The condition V(C) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both AC = Ø and A ? D for some C, DC, then V(C)=2 if and only if C is linearly ordered by inclusion. If C is of the form C={∩ni=1 Ci:CiCi, i=1,2,…,n}, where each Ci is linearly ordered by inclusion, then V(C)?n+1. If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and C is the collection of all sets {x: f(x)>0} for f in H, then V(C)=n.  相似文献   

20.
Let A(x,ε) be an n×n matrix function holomorphic for |x|?x0, 0<ε?ε0, and possessing, uniformly in x, an asymptotic expansion A(x,ε)?Σr=0Ar(x) εr, as ε→0+. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion P(x,ε)?Σr=0Pr(x)εr, as ε→0+, is constructed, such that the transformation y = P(x,ε)z takes the differential equation εhdydx = A(x,ε)y,h a positive integer, into εhdzdx = B(x,ε)z, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.  相似文献   

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