共查询到20条相似文献,搜索用时 31 毫秒
1.
Bakir Farhi 《Journal of Number Theory》2007,125(2):393-411
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z3, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{12+1,22+1,…,n2+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (k∈N, n∈N∗). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k). 相似文献
2.
We consider the linear nonautonomous system of difference equations xn+1−xn+P(n)xn−k=0, n=0,1,2,… , where k∈Z, P(n)∈Rrxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system. 相似文献
3.
Wanzhou Ye 《Discrete Mathematics》2011,(21):2437
Let Fk be a mapping from RZ to RZ, satisfying that for x∈RZ and n∈Z, Fk(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for k∈{2,3}, if there exists n0∈Z such that x is locally finitely convergent with respect to Fk on {n0,…,n0+k−1}, then x is finitely convergent with respect to Fk. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true. 相似文献
4.
Victor Tkachenko 《Journal of Mathematical Analysis and Applications》2005,303(1):173-187
We present several conditions sufficient for global stability of the zero solution of nonautonomous difference equation xn+1=qxn+fn(xn,…,xn−k), n∈Z, when the nonlinearities fn satisfy a sort of negative feedback condition. Moreover, for every k∈N, we indicate qk such that one of our stability conditions is sharp if q∈(0,qk]. We apply our results to discrete versions of Nicholson's blowflies equation, the Mackey-Glass equations, and the Wazewska and Lasota equation. 相似文献
5.
Vladimir Varlamov 《Journal of Mathematical Analysis and Applications》2010,370(2):687-1648
A new integral representation of the Hankel transform type is deduced for the function Fn(x,Z)=Zn−1Ai(x−Z)Ai(x+Z) with x∈R, Z>0 and n∈N. This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function 2|Ai(z)| with z∈C. 相似文献
6.
The uniformly optimal graph problem with node failures consists of finding the most reliable graph in the class Ω(n,m) of all graphs with n nodes and m edges in which nodes fail independently and edges never fail. The graph G is called uniformly optimal in Ω(n,m) if, for all node-failure probabilities q∈(0,1), the graph G is the most reliable graph in the class of graphs Ω(n,m). This paper proves that the multipartite graphs K(b,b+1,…,b+1,b+2) are uniformly optimal in their classes Ω((k+2)(b+1),(k2+3k+2)(b+1)2/2−1), where k is the number of partite sets of size (b+1), while for i>2, the multipartite graphs K(b,b+1,…,b+1,b+i) are not uniformly optimal in their classes Ω((k+2)b+k+i,(k+2)(k+1)b2/2+(k+1)(k+i)b+k(k+2i−1)/2). 相似文献
7.
In this paper it is shown that if T∈L(H) satisfies
- (i)
- T is a pure hyponormal operator;
- (ii)
- [T∗,T] is of rank two; and
- (iii)
- ker[T∗,T] is invariant for T,
8.
We say that a matrix R∈Cn×n is k-involutary if its minimal polynomial is xk-1 for some k?2, so Rk-1=R-1 and the eigenvalues of R are 1,ζ,ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m, A∈Cm×n, S∈Cn×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS-1=ζμA, and A is (R,S,α,μ)-symmetric if RAS-α=ζμA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈Cn×t and B∈Cm×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈Cm×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A. 相似文献
9.
Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
10.
Let X be a topological space and let F be a filter on N, recall that a sequence (xn)n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?1 and in Banach spaces, we characterize the P-filters, the P-filters+, the weak P-filters, the Q-filters, the Q-filters+, the weak Q-filters, the selective filters and the selective+ filters. 相似文献
11.
Oleg Okunev 《Topology and its Applications》2011,158(16):2158-2164
We prove that if X and Y are t-equivalent spaces (that is, if Cp(X) and Cp(Y) are homeomorphic), then there are spaces Zn, locally closed subspaces Bn of Zn, and locally closed subspaces Yn of Y, n∈N+, such that each Zn admits a perfect finite-to-one mapping onto a closed subspace of Xn, Yn is an image under a perfect mapping of Bn, and Y=?{Yn:n∈N+}. It is deduced that some classes of spaces, which for metric spaces coincide with absolute Borelian classes, are preserved by t-equivalence. Also some limitations on the complexity of spaces t-equivalent to “nice” spaces are obtained. 相似文献
12.
Grzegorz Bobiński 《Journal of Pure and Applied Algebra》2011,215(4):642-654
Throughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N+, we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i,j∈Z, [i,j]:={l∈Z∣i≤l≤j} (in particular, [i,j]=∅ if i>j). 相似文献
13.
Chun-Gang Ji 《Discrete Mathematics》2008,308(23):5860-5863
Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have
{a(k,pln)∣n,k∈N}=Z. 相似文献
14.
In this paper we generalize the Prouhet-Tarry-Escott problem (PTE) to any dimension. The one-dimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n,k∈N, for two different multi-sets {(x1,y1),…,(xn,yn)}, of points from Z2 such that for all d,j∈{0,…,k} with j?d. We present parametric solutions for n∈{2,3,4,6} with optimal size, i.e., with k=n−1. We show that these solutions come from convex 2n-gons with all vertices in Z2 such that every line parallel to a side contains an even number of vertices and prove that such convex 2n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers. 相似文献
15.
Rigoberto Medina 《Journal of Mathematical Analysis and Applications》2007,335(1):615-625
We present sufficient conditions for the stability of the nonautonomous difference system , k∈Z+, with m?1, when the (n×n)-matrices Aj(⋅) are slowly varying coefficients. The proposed approach is based on the generalization of the “freezing” method for ordinary differential equations. The stability conditions are formulated in terms of the corresponding Cauchy's function. 相似文献
16.
For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn=Zn/E[Zn|ξ] to its limit W∞: we show a central limit theorem for W∞−Wn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for Wn+k−Wn for each fixed k∈N∗. 相似文献
17.
Arno van den Essen Andrzej Nowicki Andrzej Tyc 《Journal of Pure and Applied Algebra》2003,177(1):43-47
Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X1,…,Xn]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1. 相似文献
18.
Yukihide Takayama 《Journal of Pure and Applied Algebra》2010,214(7):1110-1120
We consider a family of slightly extended version of Raynaud’s surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H1(X,Z−n)≠0 for all 1≤n≤N with some N≥1. The surfaces are at least normal but smooth under a special condition. We also give a fairly large family of non-Mumford-Szpiro type polarizations Za,b with Kodaira non-vanishing. 相似文献
19.
For all non-negative integers n1,n2,n3,j1,j2 and j3 with nk+jk>1 for k=1,2,3, (nk,jk)≠(nl,jl) if k≠l, j3=n3−1 and jk≠nk−1 for k=1,2, we study the center variety of the 6-parameter family of real planar polynomial vector given, in complex notation, by , where z=x+iy and A,B,C∈C\{0}. 相似文献
20.
Thierry Huillet 《Journal of Computational and Applied Mathematics》2010,233(10):2449-2467
We consider the random walk on Z+={0,1,…}, with up and down transition probabilities given the chain is in state x∈{1,2,…}:
(1) 相似文献