共查询到20条相似文献,搜索用时 562 毫秒
1.
Li Delang 《Journal of Number Theory》1981,13(4):485-494
It is proved that for any given positive integers N and k the number of integers n < N for which the equation is unsolvable in positive integers x, y, z is not greater than , where c is a constant depending only on k. 相似文献
2.
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
3.
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
4.
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
5.
The system is investigated, where x and y are scalar functions of time (t ? 0), and n space variables , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution , which is bounded for r ? 0 and satisfies x(0) >x0, y(0) > y0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique. 相似文献
6.
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
7.
Patrick J Browne 《Journal of Differential Equations》1977,23(2):285-292
In this paper we study the linked nonlinear multiparameter system , where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ . 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 . Here [a, b] = [a1, b1 × … × [ak, bk]. 相似文献
8.
A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, Im(x, y) denotes , and Am(x, y) denotes an approximation of Im(x, y) of the form , 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, , then 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
9.
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
10.
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
11.
As an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, as x → ∞, holds for each ε > 0. Here integers a ≥ 0 and r ≥ 1 are given. Br(x) denotes the rth Bernoulli polynomial and {x} denotes the fractional part of x. The special case a = 0, r = 2 of this conjecture was also mentioned by S. Chowla. In this paper we prove this conjecture for all and r ≥ 2 with ε = 0 (with xε replaced by log x in case ). 相似文献
12.
13.
Jacob A Brandler 《Journal of Number Theory》1973,5(4):271-286
Necessary and sufficient conditions for representing certain classes of primes by given quadratic forms are found by generalizing techniques of rational number theory. The main result is that if m = 5 or 13, and if p is a rational prime such that , then a necessary and sufficient condition that x2 + 4my2 = p for some rational integers x and y is that , where ?m denotes the fundamental unit of the field and [/] denotes the Legendre symbol of the ring of integers of with p a prime ideal divisor of p in . 相似文献
14.
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
15.
Let A be an n × n matrix; write A = H+iK, where i2=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if , where g has degree 2, then for some unitary matrix U, the matrix U1AU is the direct sum of copies of a 2×2 matrix A1, where A1 is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)r[g(x,y,z)]s, where g(x,y,z) is irreducible of degree 2. 相似文献
16.
R.A MacLeod 《Journal of Number Theory》1982,14(2):185-227
Elementary methods are used to study sums of the form for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form for integers p and t, t > 0, where Pt(x) = Bt({x}) and Bt(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σn≤xntσa(n) and Σn≤xEt(n) for integers t and a, a ≥ 0, where σa(n) is the sum of the ath powers of the divisors of n and Et(x) is the error term in the sum Σn≤xntσa(n). 相似文献
17.
Brother Joseph Heisler 《Journal of Number Theory》1974,6(1):50-51
We shall establish for all finite fields GF(pn) the following result of Chowla: given a positive integer m greater than one and the finite field GF(p), p a prime, such that xm = ?1 is solvable in GF(p), then there exists an absolute positive constant c, , such that for each set of s nonzero elements ai of GF(p), has a non-trivial zero in GF(p) if s ≥ c ln m. 相似文献
18.
A function f(z) = z ? ∑∞n = 2anzn, an ? 0, analytic and univalent in the unit disk, is said to be in the family , a real and b ? 0, if for all z in the unit disk. A complete characterization is found for when a ? 1. Also, sharp coefficient bounds are determined for certain subclasses of when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family. 相似文献
19.
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
20.
Leslie E Shader 《Journal of Combinatorial Theory, Series A》1974,16(2):221-229
The problem of determining the number of finite central groupoids (an algebraic system satisfying the identity (x · y) · (y ? z) = y) is equivalent to the problem of determining the number of solutions of the matrix equation A2 = J, where A is a 0, 1 matrix and J is a matrix of 1's.The existence of solutions of A2 = J of all ranks r, where , and A is n2 × n2, is proven. Since these are the only possible values, the question of existence solutions of all possible ranks is completely answered. The techniques and proofs are of a constructive nature. 相似文献