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1.

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑

_{bT<αβ}is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_{bT<αβ}is the ordinal of the set*M*of 2-element subsets, where*M*is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all*n*-element subsets for each natural number*n*≥ 2. Moreover, series ∑_{β<α}f(β) are investigated and evaluated, where α is a limit ordinal and the function*f*belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic. 相似文献2.

P.A. Ramachandran 《Numerical Methods for Partial Differential Equations》2006,22(4):831-846

Time‐dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi‐implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 相似文献

3.

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures. 相似文献

4.

Deborah M. Greaves 《国际流体数值方法杂志》2006,50(6):693-711

An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at

*Re*=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献5.

We consider the plane-strain buckling of a cylindrical shellof arbitrary thickness which is made of a Varga material andis subjected to an external hydrostatic pressure on its outersurface. The WKB method is used to solve the eigenvalue problemthat results from the linear bifurcation analysis. We show thatthe circular cross-section buckles into a non-circular shapeat a value of µ

_{1}which depends on*A*_{1}/*A*_{2}and a mode number,where*A*_{1}and*A*_{2}are the undeformed inner and outer radii, andµ_{1}is the ratio of the deformed inner radius to*A*_{1}. Inthe large mode number limit, we find that the dependence ofµ_{1}on*A*_{1}/*A*_{2}has a boundary layer structure: it is constantover almost the entire region of 0 <*A*_{1}/*A*_{2}< 1 and decreasessharply from this constant value to unity as*A*_{1}/*A*_{2}tends tounity. Our asymptotic results for*A*_{1}– 1 = O(1) and*A*_{1}– 1 = O(1/*n*) are shown to agree with the numerical resultsobtained by using the compound matrix method. 相似文献6.

We treat here of the question of absorbing boundary conditionsfor nonlinear diffusion equations. We use the conditions designedfor the linear equation, we prove them to be well posed forthe nonlinear problem, and through numerical experiments thatthey are well suited for reaction–diffusion equations. 相似文献

7.

Anuj Mittal Swaminathan Sivaram 《Journal of polymer science. Part A, Polymer chemistry》2005,43(21):4996-5008

A tridentate ligand, BPIEP: 2,6‐bis[1‐(2,6‐diisopropyl phenylimino) ethyl] pyridine, having central pyridine unit and two peripheral imine coordination sites was effectively employed in controlled/“living” radical polymerization of MMA at 90°C in toluene as solvent, Cu

^{I}Br as catalyst, and ethyl‐2‐bromoisobutyrate (EB*i*B) as initiator resulting in well‐defined polymers with polydispersities*M*_{w}/*M*_{n}≤ 1.23. The rate of polymerization follows first‐order kinetics,*k*_{app}= 3.4 × 10^{?5}s^{?1}, indicating the presence of low radical concentration ([P*] ≤ 10^{?8}) throughout the reaction. The polymerization rate attains a maximum at a ligand‐to‐metal ratio of 2:1 in toluene at 90°C. The solvent concentration (v/v, with respect to monomer) has a significant effect on the polymerization kinetics. The polymerization is faster in polar solvents like, diphenylether, and anisole, as compared to toluene. Increasing the monomer concentration in toluene resulted in a better control of polymerization. The molecular weights (*M*_{n,SEC}) increased linearly with conversion and were found to be higher than predicted molecular (*M*_{n,Cal}). However, the polydispersity remained narrow,*i.e*., ≤1.23. The initiator efficiency at lower monomer concentration approaches a value of 0.7 in 110 min as compared to 0.5 in 330 min at higher monomer concentration. The aging of the copper salt complexed with BPIEP had a beneficial effect and resulted in polymers with narrow polydispersitities and higher conversion. PMMA obtained at room temperature in toluene (33%, v/v) gave PDI of 1.22 (*M*_{n}= 8500) in 48 h whereas, at 50°C the PDI is 1.18 (*M*_{n}= 10,300), which is achieved in 23 h. The plot of ln*k*_{app}versus 1/*T*gave an apparent activation energy of polymerization as (Δ*E*^{≠}_{app}) 58.29 KJ/mol and enthalpy of equilibrium (Δ*H*^{0}_{eq}) to 28.8 KJ/mol. Reverse ATRP of MMA was successfully performed using AIBN in bulk as well as solution. The controlled nature of the polymerization reaction was established through kinetic studies and chain extension experiments. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 4996–5008, 2005 相似文献8.

The interaction between multiple incompressible air jets has been studied numerically and experimentally. The numerical predictions have been first validated using experimental data for a single jet configuration. The spreading features of five unequal jets in the configuration of one larger central jet surrounded by four smaller equi‐distant jets, have been studied, for different lateral spacing ratios of 1.5, 2.0 and 2.5 and a central jet Reynolds number of 1.24×105 (corresponding to a Mach number of 0.16). Flow of five equal jets has also been simulated, for the sake of comparison. The jet interactions commence at an axial distance of about 3–4 diameters and complete by an axial distance of about 10 diameters for the lowest spacing ratio of 1.5. For larger spacing ratios, the length required for the start and completion of jet interaction increase. Peripheral jets bend more towards the central jet and merge at a smaller distance, when their sizes are smaller than that of the central jet. The entrainment ratio for multiple jets is higher than that for a single jet. Excellent agreement is observed between the experimental data and theoretical predictions for both mean flow field and turbulent quantities, at regions away from the jet inlet. The potential core length and initial jet development, however, are not predicted very accurately due to differences in the assumed and actual velocity profiles at the jet inlet. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献

9.

Peixiang Xing Gilles P. Robertson Michael D. Guiver Serguei D. Mikhailenko Serge Kaliaguine 《Journal of polymer science. Part A, Polymer chemistry》2004,42(12):2866-2876

A series of sulfonated poly(aryl ether ketone)s (SPAEKs) were prepared by aromatic nucleophilic polycondensation of 2,6‐dihydroxynaphthalene with 5,5′‐carbonyl‐bis(2‐fluorobenzenesulfonate) and 4,4′‐difluorobenzophenone. The structure and degree of sulfonation (DS) of the SPAEKs were characterized using

^{1}H NMR spectroscopy. The experimentally observed DS values were close to the expected values derived from the starting material ratios. The thermal stabilities of the SPAEKs were characterized by thermogravimetric analysis, which showed that in acid and sodium salt forms they were thermally stable in air up to about 240 and 380 °C, respectively. Transparent membranes cast from the directly polymerized SPAEKs exhibited good mechanical properties in both dry and hydrated states. The dependence of water uptake and of membrane swelling on the DS at different temperatures was studied. SPAEK membranes with a DS from 0.72 to 1.60 maintained adequate mechanical properties after immersion in water at 80 °C for 24 h. The proton conductivity of SPAEK membranes with different degrees of sulfonation was measured as a function of temperature. The proton conductivity of the SPAEK films increased with increased DS, and the highest room temperature conductivity (4.2 × 10^{?2}S/cm) was recorded for a SPAEK membrane with a DS of 1.60, which further increased to 1.1 × 10^{?1}S/cm at 80 °C. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2866–2876, 2004 相似文献10.

Alberto Mariani Simone Bidali Stefano Fiori Marco Sangermano Giulio Malucelli Roberta Bongiovanni Aldo Priola 《Journal of polymer science. Part A, Polymer chemistry》2004,42(9):2066-2072

By combining frontal polymerization and radical‐induced cationic polymerization, it was possible to cure thick samples of an epoxy monomer bleached by UV light. The effect of the relative amounts of cationic photoinitiator and radical initiator was thoroughly investigated and was related to the front's velocity and its maximum temperature. The materials obtained were characterized by quantitative conversion also in the deeper layers, not reached by UV light. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2066–2072, 2004 相似文献