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1.
A two-dimensional framework (G,p) is
a graph G = (V,E) together with a map p: V → ℝ2. We view (G,p) as a straight line realization of G in ℝ2. Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length.
A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points
corresponding to u and v is the same
in all pairs of equivalent generic realizations of G. The graph G is globally rigid
if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the
first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs.
As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid
graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given
generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3]. 相似文献
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Let G = (V,E) be a graph or digraph and r : V → Z+. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν1,…,νr(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003 相似文献
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Spikmans V Lane SJ Leavens B Manz A Smith NW 《Rapid communications in mass spectrometry : RCM》2002,16(14):1377-1388
A system is presented that demonstrates the principle of on-line and on-chip post-column derivatisation reactions in micro-high-performance liquid chromatography (micro-HPLC) hyphenated to electrospray time-of-flight mass spectrometry (ESI-TOFMS). In this micro-HPLC-chip-MS set-up, the analytes are separated using gradient micro-HPLC and subsequently derivatised on-chip and detected. One of the major limitations of MS detection is its dependency on the degree of ionisation, which is widely variable and compound-specific. Optimising and controlling the degree of ionisation in a simple manner would allow MS detection to be truly generic. One way of achieving this is by pre-ionisation of analytes using simple derivatisation procedures that are both rapid and quantitative. Performing this in situ on the system described here overcomes issues of sample handling and efficiency losses when time-consuming "bench chemistry" is necessary prior to analysis. The power of the system is demonstrated by the separation of primary and secondary amines, which are subsequently derivatised with a positively charged phosphonium complex and detected in an enhanced manner. Typically, molecular cations (M(+)) are detected showing that the ionisation process is dominated by the phosphonium species, leading to more constant ionisation for a variety of compounds. In addition, stable isotopically labelled ((12)C/(13)C)-phosphonium reagent is used for the reactions, allowing for inherent signal/noise (S/N) improvement and automated data processing using cluster analysis. A similar reaction scheme is used for the derivatisation of ketones and aldehydes, also demonstrating dramatic increases in sensitivity, especially with increasing temperature. Minimal loss in chromatographic fidelity in terms of retention times is observed by the introduction of the micromixer chip into the system. Optimal flow rates in micro-HPLC and ESI-MS are compatible with flow rates for the chip as well as a multitude of in-line optical detectors including UV and fluorescence. In addition, the micromixer chip can be positioned pre-column if preferred. The system is robust, easily fully automated and applicable to a wide variety of reactions. The system has a major advantage in its simple robust connection to the "normal scale" outside world. 相似文献
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In the structure Ba12F19Cl5 [hexagonal space group P6 2m] the two chlorides on the sites Cl(1) and Cl(2) can partially be replaced by bromide ions. Single crystals of the type Ba12F19ClδBr5–δ with a chloride to bromide ratio up to 2 : 3 could be obtained by cooling a flux of 75 mol% BaF2 and 25 mol% BaX2 with X = Cl, Br. The crystal quality decreases with increasing bromide concentration. Structural parameters of five selected single crystals with different chloride/bromide ratio were studied by single crystal X-ray diffraction methods. The refined total Cl?/Br? population ratio in the crystals is close to the one of the flux. The lattice parameters and interatomic distances change in various ways, when the smaller chloride ion is replaced by the bigger bromide ion. The refinements show a statistical disorder on the halide sites with preferential bromide substitution on site Cl(1). 相似文献