共查询到20条相似文献,搜索用时 390 毫秒
1.
Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)c on a set of positive density. Letting Fk(x) denote the number of n ≤ x for which G(n) = k, it is shown that , where logrx denotes the r-fold iterated logarithm. 相似文献
2.
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
3.
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
4.
Let {Xn} be a stationary Gaussian sequence with E{X0} = 0, {X20} = 1 and E{X0Xn} = rnn Let cn = (2ln n), bn = cn? c-1n ln(4π ln n), and set Mn = max0 ?k?nXk. A classical result for independent normal random variables is that Berman has shown that (1) applies as well to dependent sequences provided rnlnn = o(1). Suppose now that {rn} is a convex correlation sequence satisfying rn = o(1), (rnlnn)-1 is monotone for large n and o(1). Then for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {Mn}, there are others. In particular, the limit distribution is given below when rn is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when rn decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2). 相似文献
5.
Douglas Hensley 《Journal of Number Theory》1984,18(2):206-212
For a > 0 let , the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about () that , where pa(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of pa(u) is worked out. 相似文献
6.
Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min1 ? t ? n(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, is rational, and g(n) = δnI. (2) α + β > 1, min(α, β) > 1, is rational, and (a) g(n) = δn1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases , where γ is defined by α?γ + β?γ = 1, and that does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max1 ? t ? n(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even. 相似文献
7.
Satendra K Vaish 《Journal of Mathematical Analysis and Applications》1984,101(1):23-29
In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then , where n(r) is the number of zeros of f(z) in and . 相似文献
8.
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
9.
Tom M. Apostol 《Journal of Number Theory》1982,15(1):14-24
An elementary proof is given of the author's transformation formula for the Lambert series relating Gp(e2πiτ) to Gp(e2πiAτ), where p > 1 is an odd integer and is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function , and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions. 相似文献
10.
V.B Headley 《Journal of Mathematical Analysis and Applications》1985,108(1):283-292
Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim on an arc A of ?Δ with length . It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = supz?Δ, where C1 = limn→∞. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that almost everywhere. It is proved that inff?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C1. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥fp∥ ? p∥f∥, for any positive integer p. 相似文献
11.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
12.
Matania Ben-Artzi 《Journal of Differential Equations》1980,38(1):51-60
Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = VS(r) + VL. Let λ = lim supr→∞VL(r) < ∞ (we allow λ = ? ∞) and set λ+ = max(λ, 0). Assume that for some r0, VL(r) ?C2k(r0, ∞) and that there exists δ > 0 such that . Assume further that and that 2kδ > 1. It is shown that: (a) The restriction of H to C∞(Rn) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous. 相似文献
13.
Charles J Monlezun 《Journal of Mathematical Analysis and Applications》1974,47(1):133-152
A theory of scattering for the time dependent evolution equations (1) is developed. The wave operators are defined in terms of the evolution operators Uj(t, s), which govern (1). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H0(t) = ?Δ, Δ denoting the Laplacian, and H1(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable (). 相似文献
14.
A.M Odlyzko 《Advances in Mathematics》1982,44(2):180-205
It is shown that the coefficients an of the power series f(z) = ∑∞n=1anzn which satisfies the functional equation display periodic oscillations; as n → ∞, where and u(x) is a positive, nonconstant, continuous function which is periodic with period log(4 ? ø). Similar results are obtained for a wide class of power series that satisfy similar functional equations. Power series of these types are of interest in combinatorics and computer science since they often represent generating functions. For example, the nth coefficient of the power series satisfying (1) enumerates 2, 3-trees with n leaves. 相似文献
15.
J.S. Hwang 《Journal of Mathematical Analysis and Applications》1983,91(2):434-443
For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) , for all π lying on some arc Af ? ?Δ with arclength . We show that for each 0 < ε < 1, there is a π0 > 0 such that for any f?D(π) with π < π0, the Bloch and Doob norm respectively satisfy These two estimates do not hold with ε = 0. 相似文献
16.
The probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in n (1 ≤ r ≤ n ? 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 . 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 , 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 n invariant under the group of rotations in n, 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 ≤ r ≤ n. 相似文献
17.
Jeffrey Rauch 《Journal of Functional Analysis》1980,35(3):304-315
Suppose that e2?|x|V ∈ ReLP(R3) for some p > 2 and for g ∈ R, H(g) = ? Δ + gV, H(g) = ?Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is . We find a series expansion in powers of whose values for g < g0 correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues. 相似文献
18.
Tom Brylawski 《Discrete Mathematics》1977,18(3):243-252
In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if X(G) = Σμ(0, x)λr ? r(x) = Σ (?1)r ? kWkλk is the characteristic polynomial of G, then Thus γ(G) ? 2r ? 1 (n+2), where γ(G) = Σwk. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then |μ| ? (r? 1)n; and γ ? (2r ? 1 ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, , and then . Further, if β is the Crapo invariant, then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network. 相似文献
19.
Harry Dym 《Journal of Functional Analysis》1978,28(1):33-57
Let PT denote the orthogonal projection of L2(R1, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure , and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if is summable, if is locally summable, and if belongs to the span in L∞ of e?iyTH∞:T ? 0, in which h is chosen to be an outer function and h#(γ) agrees with the complex conjugate of h(γ) on the line, then exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections. 相似文献
20.
Wolfgang Wasow 《Linear algebra and its applications》1977,18(2):163-170
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 , as ε→0+. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion , as ε→0+, is constructed, such that the transformation y = P(x,ε)z takes the differential equation a positive integer, into , 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