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881.
We present the design, analysis, and implementation of an algorithm for the computation of any number of digits of the roots of a polynomial with complex coefficients. The real and the imaginary parts of the coefficients may be integer, rational, or floating point numbers represented with an arbitrary number of digits. The algorithm has been designed to deal also with numerically hard polynomials like those arising from the symbolic preprocessing of systems of polynomial equations, where the degree and the size of the coefficients are typically huge.The algorithm is based on an adaptive strategy which automatically exploits any specific feature of the input polynomial, like its sparsity or the conditioning of its roots, in order to speed up the computation. We introduce different concepts and tools suitably designed to arrive at an adaptive implementation, such as the concepts of root neighborhood, inclusion discs and some inclusion and conditioning theorems for their determination. The main engine for shrinking the inclusion discs is the simultaneous iteration method of Ehrlich–Aberth, complemented with a suitable technique for cluster analysis that is used for getting rid of the slow convergence in case of clustered or multiple roots.The algorithm, implemented in C, relies on the GNU multiprecision package GMP and allows many options. Counting, isolating and approximating all roots in a given set
are the main goals that the algorithm provides. Automatic determination of multiplicities and the detection of real or imaginary roots can be selected as well. Polynomials having coefficients with a bounded precision may be processed too. Comparisons with the polynomial rootfinders of the packages Mathematica, Mapleand Pari, performed on a wide class of test polynomials show that our algorithm is generally much faster: in most cases the speedup factor is greater than 10 and, for certain polynomials, it is greater than 1000. The MPSolve package can be downloaded from the numeralgo library of netlib. 相似文献
882.
Chang Min Oh Joon Yong Choi In Ah Bae Hong Taek Kim Seong Su Hong Jay Kyun Noah Yong Chool Boo 《Molecules (Basel, Switzerland)》2022,27(9)
Hemp (Cannabis sativa L.) contains a variety of secondary metabolites, including cannabinoids, such as psychoactive (−)-trans-Δ⁹-tetrahydrocannabinol. The present study was conducted to identify the major phenolic components contained in hemp root, which has been relatively under-researched compared to other parts of hemp. The aqueous ethanol extract of hemp roots was fractionated into methylene chloride (MC), ethyl acetate (EA), and water (WT) fractions, and high-performance liquid chromatography with photodiode array detection (HPLC-DAD) analysis was performed. The main ultraviolet (UV)-absorbing phenolic compound contained in the EA fraction was identified as p-coumaric acid by comparing the retention time and UV absorption spectrum with a standard. Silica gel column chromatography was performed to isolate a hydrophobic derivative of p-coumaric acid contained in the MC fraction. Nuclear magnetic resonance (NMR) analysis identified the isolated compound as ethyl p-coumarate. For comparative purposes, ethyl p-coumarate was also chemically synthesized by the esterification reaction of p-coumaric acid. The content of p-coumaric acid and ethyl p-coumarate in the total extract of hemp root was estimated to be 2.61 mg g−1 and 6.47 mg g−1, respectively, by HPLC-DAD analysis. These values correspond to 84 mg Kg−1 dry root and 216 mg Kg−1 dry root, respectively. In conclusion, this study identified p-coumaric acid and ethyl p-coumarate as the main phenolic compounds contained in the hemp roots. 相似文献
883.
884.
885.
In this paper we prove that the groupws SL(n,q), q=pm, are factors of the modular groups PSL(2,Z) When n=5,6,7 and P≠2, q≠9 相似文献
886.
Alexandr?JegorovEmail author Ladislav?Cvak Jan??ejka Bohumil?Kratochvíl Petr?Sedmera Vladimír?Havlí?ek 《Journal of chemical crystallography》2005,35(8):621-627
Mollugin was isolated from the extract of Rubia tinctorum roots. Lucidin was obtained by semisynthesis from xanthopurpurin. Mollugin crystallises in space group P-1 (No. 2), a = 8.5857(7), b = 11.473(4), c = 15.024(1) Å, α = 77.64(2)∘, β = 89.36(1)∘, γ = 89.71(2)∘, and V = 1445.5(5) Å3, lucidin crystallises in space group P21/c (No.14), a = 16.800(6), b = 9.637(2), c = 7.073(7) Å, β = 98.01(5)∘, and V = 1134(1) Å3. 相似文献
887.
We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension 2 or Gorenstein of dimension =1.In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.Every derivation of the Lie algebra is inner. 相似文献
888.
We study multiple series extensions of basic hypergeometric series related to the root system Dn. We make a small change in the notation used for Cn and Dn series to bring them closer to An series. This allows us to combine the three types of series, and get Dn extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' 6 5 sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define An and Cn extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system An (U(n + 1) basic hypergeometric series). 相似文献
889.
研究了一个带有强迫项的多时滞差分方程△x_n+sum from j=1 to m p_jx_(n-k_j)=r_n解的某种渐近性态.当这个方程的系数以及强迫项满足一定的约束条,其中p_j,r_n是实数,n,r_j是非负整数,λ_0是上述差分方程的特征方程的唯一实特征根,则表达式■λ_0~(-n)·x_n存在,并且给出了它的极限值. 相似文献
890.
设φ(x)=∑Nn=0anxE(x-n).当E是z -tile且a是一正有理数时,我们证明了{e2ximxφ(x-na):m,n∈z}成为框架的充要条件是多项式p(z)=∑Nn=0anzn无单位根.此结果推广了Casazza和Kalton的结果[3],并且给出了Weyl-Heisenberg框架与多项式根的相互关系. 相似文献