共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
We characterize the uniform algebras A on a compact Hausdorff space X which contain a sequence {uj}j = 0∞ of unimodular elements with and closed span in terms of the maximal ideal space of A. Roughly, the essential set of A looks like (at most) countably many copies of the boundary of the unit disk, and A looks like the disk algebra on each. 相似文献
3.
Let Ω be a finite set with k elements and for each integer let (n-tuple) and and aj ≠ aj+1 for some 1 ≦ j ≦ n ? 1}. Let {Ym} be a sequence of independent and identically distributed random variables such that P(Y1 = a) = k?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in and in Ω?n with respect to the stochastic process {Ym}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process. 相似文献
4.
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
5.
The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x1,…,xk: xi?0, i=1,…,k, s?∑k1xi?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes , where now each Vi is a p×p symmetric matrix and A?B means that A?B is positive semidefinite. 相似文献
6.
Let a complex pxn matrix A be partitioned as A′=(A′1,A′2,…,A′k). Denote by ?(A), A′, and A? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A(14) be an inner inverse of A such that A(14)A is a Hermitian matrix. Let B=(A(14)1,A(14)2,…,Ak(14)) and .Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A(14) or for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition is no longer true. 相似文献
7.
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
8.
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
9.
Elliptic boundary value problems for systems of nonlinear partial differential equations of the form , i = 1(1)N, j, k = 1(1)n, pi ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small. 相似文献
10.
J.C. Bermond 《Discrete Mathematics》1974,9(4):313-321
Given k directed graphs G1,…,Gk the Ramsey number R(G1,…, Gk) is the smallest integer n such that for any partition (U1,…,Uk) of the arcs of the complete symmetric directed graph Kn, there exists an integer i such that the partial graph generated by U1 contains G1 as a subgraph. In the article we give a necessary and sufficient condition for the existence of Ramsey numbers, and, when they exist an upper bound function. We also give exact values for some classes of graphs. Our main result is: , where G is a hamltonian directed graph with p vertices and denotes the directed path of length nt 相似文献
11.
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
12.
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
13.
14.
Michael A. Golberg 《Journal of Mathematical Analysis and Applications》1983,91(2):394-409
We consider solving operator equation (1) Hu + Ku = f, where H and K are bounded linear operators between two real Hilbert spaces H1 and H2. Operator H is assumed to have a finite-dimensional nullspace N(H) and a bounded right inverse H1:H2 → H1 and K is compact. It follows that dim(N(H + K)) = dim(N(H)), so that to obtain uniqueness the m additional conditions (2) 〈u,φk〉1 = bk, k=1, 2, h.,dim(N(H)) = m are imposed, where the {φk}k = 1m are an orthonormal basis for N(H). To solve (1) and (2), these equations are converted to an equivalent equation of the second kind to which Galerkin's method is applied using the basis . It was shown that this method is equivalent to the method of weighted residuals when (the adjoint of H). The results are applied to obtain convergence proofs of some numerical methods for solving several classes of Cauchy singular integral equations whose kernels are only square integrable. 相似文献
15.
Given a polynomial , we calculate a subspace Gp of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property (the algebra generated by Gp, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X1,…,Xn) and Q(X1,…,Xn) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as and respectively, where . 相似文献
16.
Patrick J. Browne 《Journal of Mathematical Analysis and Applications》1977,60(1):259-273
In this paper we continue our investigation of multiparameter spectral theory. Let H1,…, Hk be separable Hilbert spaces and H = ?r = 1kHr, be their tensor product. In each space Hr we have densely defined self-adjoint operators Tr and continuous Hermitian operators Vrs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ1,…, λn) ?Rk and eigenvectors such that . We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations. 相似文献
17.
Karen A Ames 《Journal of Mathematical Analysis and Applications》1984,103(1):172-183
Solutions of Cauchy problems for the singular equations (in a Hilbert space setting) and , ω={(x1,…,xM)εm: 0 < xi < ci for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced. 相似文献
18.
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
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.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献