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91.
Gregory A. Freiman Boris L. Granovsky 《Transactions of the American Mathematical Society》2005,357(6):2483-2507
We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.
92.
We study the asymptotic behavior of the maximal multiplicity μn = μn(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμn/(6n)1/2 converges weakly to max jXj/j as n→∞, where X1, X2, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05 相似文献
93.
快速造P(n,k)大表的左肩法则和斜线法则 总被引:10,自引:0,他引:10
设P(n,k)为整数n分为k部的无序分拆的个数,每个分部≥1,它为大师欧拉所建立(1707-1783).它是组合图论和数论里最重要的数据之一.然而,它却十分难于计数和造表.本文,由公式P(n,k)=P(n-1,k-1)+P(n-k,k)定义了P(n,k)的左肩数和锐角数,并由此得到求P(n,k)的左肩法则(第一法则).还根据本文作者[5]的一些重要定理得到求 P(n,k)的斜线法则(第二法则).使用这些法则得到造P(n,k)大表的有趣原理.为方便计,我们仅用第一法则设计了计算机程序,用此程序即可快速造出任意大的P(n,k)表. 相似文献
94.
We consider sequences of integers (1,..., k) defined by a system of linear inequalities i j>iaijj with integer coefficients. We show that when the constraints are strong enough to guarantee that all i are nonnegative, the generating function for the integer solutions of weight n has a finite product form
, where the bi are positive integers that can be computed from the coefficients of the inequalities. The results are proved bijectively and are used to give several examples of interesting identities for integer partitions and compositions. The method can be adapted to accommodate equalities along with inequalities and can be used to obtain multivariate forms of the generating function. We show how to extend the technique to obtain the generating function when the coefficients ai,i+1 are allowed to be rational, generalizing the case of lecture hall partitions. Our initial results were conjectured thanks to the Omega package (G.E. Andrews, P. Paule, and A. Riese, European J. Comb. 22(7) (2001), 887–904).Research supported by NSA grants MDA 904-00-1-0059 and MDA 904-01-0-0083. 相似文献
95.
The object of this paper is to propose and prove a new generalization of the Andrews-Gordon Identities, extending a recent result of Garrett, Ismail and Stanton. We also give a combinatorial discussion of the finite form of their result which appeared in the work of Andrews, Knopfmacher, and Paule. 相似文献
96.
We use elementary methods to prove formulas that represent sums of restricted classes of Schur functions as ratios of determinants. This includes recent formulas for sums over bounded partitions with even parts and sums over bounded partitions whose conjugates have only even parts. All of these formulas imply plane partition generating functions. 相似文献
97.
Pietro Caputo Fabio Martinelli Fabio Lucio Toninelli 《Random Structures and Algorithms》2011,39(1):83-114
We study a single‐flip dynamics for the monotone surface in (2 + 1) dimensions obtained from a boxed plane partition. The surface is analyzed as a system of non‐intersecting simple paths. When the flips have a non‐zero bias we prove that there is a positive spectral gap uniformly in the boundary conditions and in the size of the system. Under the same assumptions, for a system of size M, the mixing time is shown to be of order M up to logarithmic corrections. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 83–114, 2011 相似文献
98.
The combinatorial tool of generating functions for restricted partitions is used to generalize a quantum physics theorem relating distinct multiplets of different angular momenta in the composite Fermion model of the fractional quantum Hall effect. Specifically, if gℓ(N,M) denotes the number of distinct multiplets of angular momentum ℓ and total angular momentum M, we prove thatwhere the sum is taken over all positive divisors of N and L(k)=kℓ-kN/2+3k/2-N+N/(2k)-1/2. The original Quinn–Wójs theorem results when k=1 and it appears that this generalization will be useful in further investigations of nuclear shells modeling elementary particle interactions when the particles are clustered together. 相似文献
99.
100.
Eisenkölbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon. 相似文献