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311.
It is shown that if two submeasures on that are lim sup's of sequences of measures have the same zero sets, and one is nonatomic, so is the other, and they are (ε-δ)-equivalent. Moreover, if a submeasure η is the lim sup of a sequence of lower semi-continuous (lsc) submeasures and is 0-dominated by the so-called core γ• of an lsc submeasure γ, then η is also (ε-δ)-dominated by γ•. And if a submeasure η of this type is 0-dominated by a nonatomic submeasure, then it is nonatomic as well. 相似文献
312.
Let (X, G) be an association scheme. We say that (X, G) is flat if it is homogeneous and if any two distinct points have at most one common g-neighbor for each g ∈ G. In this paper we prove that any nondiscrete equitable partition of (X, G) has at most one singleton if (X, G) is flat, and {X} is the unique equitable partition without any singleton if (X, G) is flat and |X| is a prime.
This work was supported for two years by Korea Research Foundation Grant (KRF-2006-003-C00010) and Pusan National University
Research Grant.
Received: January 31, 2007. Final version received: Novmeber 14, 2007. 相似文献
313.
Hugh Thomas 《Discrete Mathematics》2006,306(21):2711-2723
The usual, or type An, Tamari lattice is a partial order on , the triangulations of an (n+3)-gon. We define a partial order on , the set of centrally symmetric triangulations of a (2n+2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the An Tamari lattice, and therefore that it deserves to be considered the Bn Tamari lattice. We also define a bijection between and the noncrossing partitions of type Bn defined by Reiner. 相似文献
314.
A vector space partition of a finite dimensional vector space V=V(n,q) of dimension n over a finite field with q elements, is a collection of subspaces U1,U2,…,Ut with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of consists of the subspaces of least dimension d1 in , and the length n1 of the tail is the number of subspaces in the tail. Let d2 denote the second least dimension in .Two cases are considered: the integer qd2−d1 does not divide respective divides n1. In the first case it is proved that if 2d1>d2 then n1≥qd1+1 and if 2d1≤d2 then either n1=(qd2−1)/(qd1−1) or n1>2qd2−d1. These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d1 respectively d2.In case qd2−d1 divides n1 it is shown that if d2<2d1 then n1≥qd2−qd1+qd2−d1 and if 2d1≤d2 then n1≥qd2. The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail. 相似文献
315.
Brian Drake 《Discrete Mathematics》2009,309(12):3936-3953
We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays. 相似文献
316.
We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from a×b×c to (a−1)×(b+1)×c or (a+1)×(b−1)×c. Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions.Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties. 相似文献
317.
Anisse Kasraoui 《Journal of Combinatorial Theory, Series A》2009,116(3):539-563
[E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the p,q-Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon's theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result. 相似文献
318.
One of MacMahon's partition theorems says that the number of partitions of n into parts divisible by 2 or 3 equals the number of partitions of n into parts with multiplicity larger than 1. Recently, Holroyd has obtained a generalization. In this short note, we provide a bijective proof of his theorem. 相似文献
319.