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A *-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil *-clean rings are the *-version of nil-clean rings introduced by Diesl.This paper is about the nil *-clean property of rings with emphasis on matrix rings.We show that a *-ring R is nil *-clean if and only if J(R) is nil and R/J(R) is nil*-clean.For a 2-primal *-ring R,with the induced involution given by (aij)* =(a*ij)T,the nil *-clean property of Mn(R) is completely reduced to that of Mn(Z2).Consequently,Mn(R) is not a nil *-clean ring for n =3,4,and M2(R) is a nil *-clean ring if and only if J(R) is nil,R/J(R) is a Boolean ring and a*-a ∈ J(R) for all a ∈ R. 相似文献
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We construct the Grothendieck rings of a class of 2n2dimensional semisimple Hopf Algebras H2n2,which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H2n2-modules are classified.Furthermore,we describe the Grothendieck rings r(H2n2)by generators and relations explicitly. 相似文献
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A detailed study of the geometry, aromatic character, electronic and magnetic properties for a series of positively charged N-doped polycyclic aromatic hydrocarbons (PAHs) was performed. Magnetic properties of the examined molecules were analyzed by means of the magnetically induced current density calculated using the diamagnetic-zero version of the continuous transformation of origin of current density (CTOCD-DZ) method. The comparative study of the local aromaticity of the studied molecules was performed using several different indices: energy effect (ef), harmonic oscillator model of aromaticity (HOMA) index, six centre delocalization index (SCI) and nucleus independent chemical shifts (NICS). The presence of N-atoms in the inner rings was found to cause a planarity distortion in the studied N-doped systems. The geometric changes and charged nature of the studied N-doped systems do not significantly influence the current density and the local aromaticity distribution in comparison with the corresponding parent benzenoid hydrocarbons. The present study demonstrates how quantum chemical calculations can be used for rational design of novel PAHs and for fine tuning of their properties. 相似文献
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We consider the dynamics and formation of vortices from ring dark solitons in a two-dimensional Bose–Einstein condensate with the Rashba spin–orbit coupling based on the time-dependent coupled Gross–Pitaevskii equation. Compared with previous results, the system exhibits complex dynamical behaviors in the presence of the spin–orbit coupling. With the modulation of the spin–orbit coupling, not only the lifetime of ring dark solitons is greatly prolonged, but also their attenuation kinetics is significantly affected. For two shallow ring dark solitons with the equal strength of the spin–orbit coupling, the radius of ring dark solitons increases to a maximum value over time and then shrinks into a minimum value. Due to the effect of the snake instability, ring dark solitons split into a series of ring-like clusters of vortex pairs, which perform complex oscillations. This indicates that the system is strongly dependent on the presence of the spin–orbit coupling. Furthermore, the effect of different initial modulation depths on the dynamics of ring dark solitons is investigated. 相似文献
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Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved. 相似文献
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We designed a reconfigurable dual-interferometer coupled silicon nitride microring resonator.By tuning the integrated heater on interferometer's arms,the"critical coupling"bandwidth of resonant mode is continuously adjustable whose quality factor varies from 7.9×104 to 1.9×105 with the extinction ratio keeping higher than 25 dB.Also a variety of coupling spanning from"under-coupling"to"over-coupling"were achieved,showing the ability to tune the quality factor from 6.0×103 to 2.3×105.Our design can provide an adjustable filtering method on silicon nitride photonic chip and contribute to optimize the nonlinear process for quantum photonics and all-optical signal processing. 相似文献
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Hans‐Jrg Himmel 《Angewandte Chemie (International ed. in English)》2019,58(34):11600-11617
Electron‐deficient small boron rings are unique in their formation of σ‐ and π‐delocalized electron systems as well as the avoidance of “classical” structures with two‐center‐two‐electron (2c,2e) bonds. These rings are tolerant of several skeletal electron numbers, which makes their redox chemistry highly interesting. In the past few decades, a range of stable compounds have been synthesized with various electron numbers in their B3 and B4 cores. The electronic structures were evaluated by quantum‐chemical calculations. On the other hand, the chemistry of these rings is still very much underdeveloped, being generally limited to the protonation and redox reactions of individual systems. The linkage of several B3 and/or B4 ring systems should give compounds with attractive electronic properties, thus leading the way to novel boron‐based materials. By summarizing important experimental and theoretical results, this Review intends to provide the basis for the exploration of the chemistry of these rings and, in particular, their integration into larger molecular architectures. 相似文献