共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
Gideon Nettler 《Journal of Number Theory》1981,13(4):456-462
In a previous paper it was proven that given the continued fractions where the a's and b's are positive integers, then A, B, A ± B, and AB are irrational numbers if for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, and AB are transcendental numbers if an > bn > an?1(n?1)2 for all n sufficiently large. 相似文献
4.
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
5.
John Riordan 《Discrete Mathematics》1973,4(4):383-387
The level code representation of the simplest ballot problem (weak lead lattice paths from (0, 0) to (n, n) is the set of sequences (b1,…, bn) defined by b1 = 1, bi ?1 ≤ bi ≤ i, 2 ≤ i ≤n. Each sequence is monotone non-decreasing, has a specification (c1, c2,…, cn) with ci the number of sequence elements equal to i (hence c1 + c2…+ cn = n), and may be permuted in ways. The set of permuted sequences, as noted in [4], is the set of parking functions, introduced by Konheim and Weiss in [1]. To count parking functions by number of fixed points, associate the rook polynomial for matching a deck of cards of specification (c1,…,cn), ci cards marked i, with a deck of n distinct cards. The hit polynomial Hn(x) corresponding to the sum of such rook polynomials over all sequences (I am using the terminology of [2]) is the required enumerator and turns out to be simply . 相似文献
6.
P. Révész 《Stochastic Processes and their Applications》1983,15(2):169-179
Let U1, U2,… be a sequence of independent, uniform (0, 1) r.v.'s and let R1, R2,… be the lengths of increasing runs of {Ui}, i.e., X1=R1=inf{i:Ui+1<Ui},…, Xn=R1+R2+?+Rn=inf{i:i>Xn?1,Ui+1<Ui}. The first theorem states that the sequence can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R1+R2+?+Rτ(n)?n, and . Here Mn is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of Mn.The limit distribution of the lengths of increasing runs is our third problem. 相似文献
7.
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
8.
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
9.
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
10.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,41(2):300-312
We study certain functionals and obtain an inverse Hölder inequality for n functions f1a1,…,fnan (fk concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f1,…,fn, where .Finally we give an inverse Minkowski inequality for concave functions. 相似文献
11.
A set {b1,b2,…,bi} ? {1,2,…,N} is said to be a difference intersector set if {a1,a2,…,as} ? {1,2,…,N}, j > ?N imply the solvability of the equation ax ? ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,…,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (, , , (where () denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets. 相似文献
12.
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
13.
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
14.
Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, , where z is the center of mass . 相似文献
15.
This paper deals with sequences a1a2a3 ··· of symbols 0 and 1 with the property that they contain no arbitrary long blocks of the form ai+1 ? ai+k = ww. The behaviour of this class of sequences with respect to some operations is examined. Especially the following is shown: Let be , then there exists a sequence without arbitrary long adjacent identical blocks such that no limk→∞a(n)k exists. Let be α? (0, 1), then there exists such a sequence with limk→∞a(1)k = α. Furthermore a class of sequences appearing in computer graphics is considered. 相似文献
16.
R.J Cook 《Journal of Combinatorial Theory, Series B》1979,26(3):337-345
Let G be a planar graph having n vertices with vertex degrees d1, d2,…,dn. It is shown that . The main term in this upper bound is best possible. 相似文献
17.
Let ρ21,…,ρ2p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ1,…,δp, where , i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ1,…,δp, given a random matrix F whose eigenvalues have the same distribution as , i = 1,…,p, where r1,…,rp are the sample canonical correlation coefficients. 相似文献
18.
László Babai 《Journal of Combinatorial Theory, Series B》1979,27(2):180-189
By a result of L. Lovász, the determination of the spectrum of any graph with transitive automorphism group easily reduces to that of some Cayley graph.We derive an expression for the spectrum of the Cayley graph X(G,H) in terms of irreducible characters of the group G: for any natural number t, where ξi is an irreducible character (over ), of degree ni , and λi,1 ,…, λi,ni are eigenvalues of X(G, H), each one ni times. (σni2 = n = | G | is the total'number of eigenvalues.) Using this formula for t = 1,…, ni one can obtain a polynomial of degree ni whose roots are λi,1,…,λi,ni. The results are formulated for directed graphs with colored edges. We apply the results to dihedral groups and prove the existence of k nonisomorphic Cayley graphs of Dp with the same spectrum provided p > 64k, prime. 相似文献
19.
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
20.
Recently (see De Vylder & Goovaerts (1984), this issue) so called credibility matrices have been introduced and studied in the framework of general properties of matrices, such as non-negativity, total positivity etc. In the present note we characterize a class of credibility matrices generated by the normed sequence of functions (pl, pl,…, pn) on K = [0, b] where , i=0, …, n, θ ? K, and where ?, g, h are nonnegative (eventually depending on n, n may be finite or infinite). For simplicity we suppose h to be monotonic and continuous. 相似文献