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61.
Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for
the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration
phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper
bounds on spectral gaps.
Received: 10 June 1996 / In revised form: 9 August 1996 相似文献
62.
Arthur G. Werschulz 《Numerical Functional Analysis & Optimization》2013,34(9-10):945-967
This paper deals with the optimal solution of ill-posed linear problems, i.e..linear problems for which the solution operator is unbounded. We consider worst-case ar,and averagecase settings. Our main result is that algorithms having finite error (for a given setting) exist if and only if the solution operator is bounded (in that setting). In the worst-case setting, this means that there is no algorithm for solving ill-posed problems having finite error. In the average-case setting, this means that algorithms having finite error exist if and only lf the solution operator is bounded on the average. If the solution operator is bounded on the average, we find average-case optimal information of cardinality n and optimal algorithms using this information, and show that the average error of these algorithms tends to zero as n→∞. These results are then used to determine the [euro]-complexity, i.e., the minimal costof finding an [euro]-accurate approximation. In the worst-case setting, the [euro]comp1exity of an illposed problem is infinite for all [euro]>0; that is, we cannot find an approximation having finite error and finite cost. In the average-case setting, the [euro]-complexity of an ill-posed problem is infinite for all [euro]>0 iff the solution operator is not bounded on the average, moreover, if the the solutionoperator is bounded on the average, then the [euro]-complexity is finite for all [euro]>0. 相似文献
63.
Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model. 相似文献
64.
Lech Drewnowski 《Indagationes Mathematicae》2009,20(1):43-47
A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(Uj Aj ) = supj η(Aj) for all countable disjoint families of sets (Aj) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L∞-spaces. 相似文献
65.
Let be a nontrivial involution, i.e., R=R−1≠±In. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition. 相似文献
66.
We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev
constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces
(X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x
1, …, x
n
} ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x
j
) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number”
of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede
and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers. 相似文献
67.
We study two-stage, finite-scenario stochastic versions of several combinatorial optimization problems, and provide nearly
tight approximation algorithms for them. Our problems range from the graph-theoretic (shortest path, vertex cover, facility
location) to set-theoretic (set cover, bin packing), and contain representatives with different approximation ratios.
The approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its
deterministic counterpart. Furthermore, we show that common techniques for designing approximation algorithms such as LP rounding,
the primal-dual method, and the greedy algorithm, can be adapted to obtain these results. 相似文献
68.
Sergio Albeverio Volodymyr Koshmanenko Mykola Pratsiovytyi Grygoriy Torbin 《Positivity》2006,10(1):39-49
We introduce the conflict interaction with two positions between a couple of image probability measures and consider the associated
dynamical system. We prove the existence of invariant limiting measures and find the criteria for these measures to be a pure
point, absolutely continuous, or singular cotinuous as well as to have any topological type and arbitary Hausdorff dimension. 相似文献
69.
Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s
pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this
Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x0, δ) denote the set of points x in M such that the distance from x0 to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that
if τ = 1 and
if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient
generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case. 相似文献
70.
We consider a new family of convex weakly compact valued integrable random sets which is called an adapted array of convex weakly compact valued integrable random variables of type p (1?p?2). By this concept, more general laws of large numbers will be established. Some illustrative examples are provided. 相似文献