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991.
In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R. In this case Ei will denote the primitive idempotent of A associated with thetai(0?i?d). We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ∈R such that ΔAΔ-1 is symmetric. Let Γ(A) denote the directed graph with vertex set {0,1,…,d}, where i→j whenever i≠j and Aij≠0.Theorem.Assume that each entry ofAis nonnegative. Then the following are equivalent for0≤s,t≤d.
- (i)
- The graphΓ(A)is a bidirected path with endpointss,t:s↔*↔*↔?↔*↔t.
- (ii)
- The matrixAis symmetrizable and multiplicity-free. Moreover the(s,t)-entry ofEitimes(θi-θ0)?(θi-θi-1)(θi-θi+1)?(θi-θd)is independent of i for0≤i≤d, and this common value is nonzero.
992.
Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆Vi-1+Vi+Vi+1 for 0?i?d, where V-1=0 and Vd+1=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A∗ is called sharp whenever . It is known that if F is algebraically closed then A,A∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture. 相似文献
993.
In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur. 相似文献
994.
Ádám Besenyei 《Linear algebra and its applications》2011,435(5):984-997
Some functions f:R+→R+ induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define a linear mapping on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of in the case of a few concrete functions f. This problem has been motivated by applications in quantum information. 相似文献
995.
We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. Strongly regular graphs and complete bipartite graphs are examples of such graphs, but we also construct more exotic families of examples from conference graphs, projective planes, and certain quasi-symmetric designs. 相似文献
996.
We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? KrN, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space Lp(?)(X, μ) into a certain Musielak‐Orlicz space Lp(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
997.
We consider some functional Banach algebras with multiplications as the usual convolution product * and the so‐called Duhamel product ?. We study the structure of generators of the Banach algebras (C(n)[0, 1], *) and (C(n)[0, 1], ?). We also use the Banach algebra techniques in the calculation of spectral multiplicities and extended eigenvectors of some operators. Moreover, we give in terms of extended eigenvectors a new characterization of a special class of composition operators acting in the Lebesgue space Lp[0, 1] by the formula (Cφf)(x) = f(φ(x)). 相似文献
998.
We consider the robust estimation of regression parameters in linear models with long memory and heavy-tailed errors. Asymptotic Bahadur-type representations of robust estimates are developed and their limiting distributions are obtained. It is shown that the limiting distributions are very different from those obtained under short memory. A simulation study is carried out to compare the performance of various asymptotic representations. 相似文献
999.
Peng Zeng 《Journal of multivariate analysis》2011,102(3):550-562
One important step in regression analysis is to identify significant predictors from a pool of candidates so that a parsimonious model can be obtained using these significant predictors only. However, most of the existing methods assume linear relationships between response and predictors, which may be inappropriate in some applications. In this article, we discuss a link-free method that avoids specifying how the response depends on the predictors. Therefore, this method has no problem of model misspecification, and it is suitable for selecting significant predictors at the preliminary stage of data analysis. A test statistic is suggested and its asymptotic distribution is derived. Examples are used to demonstrate the proposed method. 相似文献
1000.
A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist. 相似文献