共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
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
3.
Z.A Karian 《Journal of Number Theory》1976,8(2):233-244
Let k be a positive square free integer, the ring of algebraic integers in and S the unit sphere in Cn, complex n-space. If A1,…, An are n linearly independent points of Cn then L = {u1Au + … + unAn} with is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3. 相似文献
4.
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
5.
Rhonda J Hughes 《Journal of Functional Analysis》1982,49(3):293-314
Perturbations of and suitable m, by distributions V for which , are shown to correspond to self-adjoint operators Hv, in such a way that Hv depends continuously on V, and agrees with H + V when V is sufficiently regular. These results extend joint work with Irving E. Segal [J. Functional Analysis38 (1980), 71–98], in which perturbations of by distributions V with bounded Fourier transforms in L2(R1) were considered. 相似文献
6.
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
7.
Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = akb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Qk,r(x) denotes the number of (k, r)-integers ≤ x, then , where , B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δk,r(x) and prove that , according as or , where ω(x) = exp [B log x(log log x)?1]. 相似文献
8.
R.J Cook 《Journal of Number Theory》1983,17(1):80-92
Let k be an odd positive integer. Davenport and Lewis have shown that the equations with integer coefficients, have a nontrivial solution in integers x1,…, xN provided that Here it is shown that for any ? > 0 and k > k0(?) the equations have a nontrivial solution provided that 相似文献
9.
D de Caen 《Journal of Combinatorial Theory, Series B》1983,34(3):340-349
The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let Xs(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(Xl), given that min(Xl) ≥ 1. The following lower bound on the variance of Xs is proved: , where m = |E(H)| and . This implies the following: putting t(k, l) = limn→∞T(n, l, k)(kn)?1 then t(k, l) ≥ T(s, l, k)((ks) ? 1)?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned. 相似文献
10.
William Alexandre 《Comptes Rendus Mathematique》2003,336(7):555-558
Ck estimates for convex domains of finite type in are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of , of finite type m and m′ such that , for all q=1,…,n?2, there exists a linear operator from to such that for all and all (0,q)-form f, -closed of regularity Ck up to the boundary, is of regularity Ck+1/max(m,m′) up to the boundary and . We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
11.
Let x?Sn, the symmetric group on n symbols. Let θ? Aut(Sn) and let the automorphim order of x with respect to θ be defined by where xθ is the image of x under θ. Let αg? Aut(Sn) denote conjugation by the element g?Sn. Let where s and k are positive integers and denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?Sn let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let gs denote the product of all cycles in g whose lengths do not divide s. Then gs moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z+ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in gss of any given length is smaller than the smallest prime dividing s iff b(gs; s, 1 : c) = 1. If g = (12 … pm)t and then . 相似文献
12.
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
13.
Tosio Kato 《Journal of Functional Analysis》1973,12(4):415-417
It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C0∞(Rm) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C∞(Rm) and if . 相似文献
14.
In two party elections with popular vote ratio , a theoretical model suggests replacing the so-called MacMahon cube law approximation , for the ratio of candidates elected, by the ratio of the two half sums in the binomial expansion of (p+q)2k+1 for some k. This ratio is nearly when k = 6. The success probability for the power law is shown to so closely approximate , if we choose , that for . Computationally, we avoid large binomial coefficients in computing for k>22 by expressing as the sum , whose terms decrease by the factors . Setting K = 4k+3, we compute ak for the large k using a continued fraction derived from the ratio of π to the finite Wallis product approximation. 相似文献
15.
It is shown that if satisfies , where σk(A) denotes the sum of all kth order subpermanent of A, then Per[λJn+(1?λ)A] is strictly decreasing in the interval 0<λ<1. 相似文献
16.
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
17.
Alladi Sitaram 《Journal of Functional Analysis》1978,27(2):179-184
Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space 1(G). We show that the corresponding property is no longer true for the space of double cosets . If G is of real-rank 1, we construct liner functionals for each p, 0 < p ? 2, such that but Tp does not extend to a continuous functional on . In particular, if p ? 1, Tv does not extend to a continuous functional on . We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that . The main tool used is a theorem of Trombi-Varadarajan. 相似文献
18.
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
19.
20.
Adalbert Kerber 《Discrete Mathematics》1978,23(3):319-320
A matrix T=(tik) is introduced, the coefficients of which are defined by , where ai(x) denotes the s the number of i cycles in the element x of the symmetric group Sn. It is shown that these numbers are natural numbers, that they are easy to evaluate, and that they serve very well in order to formulate an infinite number of characterizations of multiply transitive subgroups of symmetric groups in terms of the cycle structure of their elements. 相似文献