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1.
A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence. Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000  相似文献
2.
A Simple Proof of the Restricted Isometry Property for Random Matrices   总被引:12,自引:0,他引:12  
We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.   相似文献
3.
Some remarks on greedy algorithms   总被引:10,自引:0,他引:10  
Estimates are given for the rate of approximation of a function by means of greedy algorithms. The estimates apply to approximation from an arbitrary dictionary of functions. Three greedy algorithms are discussed: the Pure Greedy Algorithm, an Orthogonal Greedy Algorithm, and a Relaxed Greedy Algorithm.This research was supported by the Office of Naval Research Contract N0014-91-J1343.  相似文献
4.
Adaptive wavelet methods for elliptic operator equations: Convergence rates   总被引:9,自引:0,他引:9  

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution of the equation by a linear combination of wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to by an arbitrary linear combination of wavelets (so called -term approximation), which would be obtained by keeping the largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to with error in the energy norm, whenever such a rate is possible by -term approximation. The range of 0$"> for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to . The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.

  相似文献

5.
Triangulated categories and Kac-Moody algebras   总被引:7,自引:0,他引:7  
By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T 2=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000  相似文献
6.
The best m-term approximation and greedy algorithms   总被引:5,自引:0,他引:5  
Two theorems on nonlinear ‐term approximation in , are proved in this paper. The first one (theorem 2.1) says that if a basis is ‐equivalent to the Haar basis then a near best >‐term approximation to any can be realized by the following simple greedy type algorithm. Take the expansion and form a sum of terms with the largest out of this expansion. The second one (theorem 3.3) states that nonlinear ‐term approximations with regard to two dictionaries: the Haar basis and the set of all characteristic functions of intervals are equivalent in a very strong sense. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献
7.
Adaptive Finite Element Methods with convergence rates   总被引:5,自引:5,他引:0  
Summary. Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H1 norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n1, it can be approximated to accuracy O(n–s) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.Mathematics Subject Clasification (2000): 65N30, 65Y20, 65N12, 65N50, 68W40, 68W25.This work has been supported by the Office of Naval Research Contract Nr. N00014-03-10051, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation Grants DMS 0221642, DMS 9872890 the Deutsche Forschungsgemeinschaft grant SFB 401, the European Communitys Human Potential Programme under Contract HPRN-CT-2002-00286, Breaking Complexity.  相似文献
8.
The best m-term approximation and greedy algorithms   总被引:5,自引:0,他引:5  
Two theorems on nonlinear ‐term approximation in , are proved in this paper. The first one (theorem 2.1) says that if a basis is ‐equivalent to the Haar basis then a near best >‐term approximation to any can be realized by the following simple greedy type algorithm. Take the expansion and form a sum of terms with the largest out of this expansion. The second one (theorem 3.3) states that nonlinear ‐term approximations with regard to two dictionaries: the Haar basis and the set of all characteristic functions of intervals are equivalent in a very strong sense. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献
9.
Continuous Function Characterizations of Stratifiable Spaces   总被引:3,自引:0,他引:3  
The insertion of a continuous function between pairs of semicontinuous functions in a monotone manner is investigated. In particular, it is established that a topological space is stratifiable if and only if for every pair of semicontinuous functions......  相似文献
10.
关于高维Willmore问题   总被引:3,自引:0,他引:3       下载免费PDF全文
马志圣 《数学学报》1999,42(6):0-1046
本文考虑高维欧氏空间中子流形M的一组有较好意义的共形不变的泛函.给出这些泛函通过M的Betti数的下界估计;给出对于管状超曲面的下界和对于双球环的下界以及达到这些下界的相应的子流形,并且证明对于管状超曲面所得的有关Betti数的下界是不精确的,方法是不适当的.给出类似Willmore猜测的一些猜测.  相似文献
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