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As applications, the wave equation on?+ × ?+ and the heat equation in a semiinfinite rod are considered in detail.


In a much cited article, Yau [5] proved that when the Ricci curvature is bounded uniformly below, then the only bounded solution to the heat equation ?tμ=Δμ on [0, ∞) × M which vanishes at t=0 is the one which vanishes evarywhere.


Wellposedness of a semilinear heat equation with weak initial data


In the first part the initial value problem (IVP) of the semilinear heat equation with initial data in is studied.


For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms.

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 The absorption wavemeter investigated consists of a single loop with a variable gap which is formed by a pair of parallel plates. A calibration formula for the meter was derived on the assumption that the stray capacity could be considered constant and the stray inductance negligible. This simple and useful formula was ex perimentally checked on four different meters, covering the range from about oneandquarter meter to onehalf meter. These melors were carefully constructed and accurately calibrated with... The absorption wavemeter investigated consists of a single loop with a variable gap which is formed by a pair of parallel plates. A calibration formula for the meter was derived on the assumption that the stray capacity could be considered constant and the stray inductance negligible. This simple and useful formula was ex perimentally checked on four different meters, covering the range from about oneandquarter meter to onehalf meter. These melors were carefully constructed and accurately calibrated with reference to a quartz oscillator by the doubleheating method through the use of an auxiliary oscillator of a continuous range of wavelengths from six to ten meters.  在吸收式波长计之理论上,以其杂储电量为常数而略其杂磁感量,得一简便之公式虽在波长一公尺左右亦可用。所研究之吸收式波长计之构造,系一黄铜制成之单圈,用二黄铜片制成之一缝隙。其缝隙之宽度,可由一螺旋调节。此种波长计,曾 在国立中央研究院物理研究所制造室制造四具,可量波长一有四分之一公尺至半公尺亦曾在此研究所之实验室内,与水晶振荡器校准先制一振荡器,可在六至十公尺之间,发生无论任何波长之振荡。并将此振荡器与一水晶振荡器用拍谐法校准。再制一有四分之一公尺至半公尺之振荡器,与已与水晶校准之振荡器,再用拍谐法校准,而同时以所制之吸收式波长计量之其实验与理论所得之各结果极为符合。  In the statistical theory of superlattices in binary alloys, the dependence of the interaction energy upon atomic arrangements is taken into account by regarding the interaction energy in Bethe's theory as an average quantity depending on the degree of order and the composition of the alloy. Two simple assumptions concerning the functional relationship of the interaction energy with order and composition are made. The first is a linear function of order and composition. The second is a linear function of the... In the statistical theory of superlattices in binary alloys, the dependence of the interaction energy upon atomic arrangements is taken into account by regarding the interaction energy in Bethe's theory as an average quantity depending on the degree of order and the composition of the alloy. Two simple assumptions concerning the functional relationship of the interaction energy with order and composition are made. The first is a linear function of order and composition. The second is a linear function of the average numbers of pairs of atoms. The result of applying these assumptions to superlattices of the type AB is that the critical temperature as a function of the composition is a maximum for equal sumber of A and B atoms only when a certain relation between the coefficients in the assumed function is satisfied. In the cass of superlattices of type AB3 the theory of Bragg and Williams is used for simplicity. It is shown that when the composition varies, the maximum of the critical temperature may occur at any desired composition by a suitables adjustment of the coefficients in the assumed functions. There is thus a hope of removing the discrepancy between theory and experiment on this line. The anomalous specific heat at the critical temperature is also calculated for different compositions. In the case of the AB type of superlattices, Bethe's formula for the energy is no longer valid, and in order to calculate the specific heat, an approximate formula for the energy is obtained by analogy with the theory of Bragg and Williams. Finally, the problem of separation into more than one phase is briefly discussed.  在二元合金超格之统计力学理论中,原子间互作用能量,因原子之排列不同而异,其所生之影响,吾人擬於此篇中讨论之。吾人认为有Bethe氏理论中之相互作用能量,实为一平均值,其值因合金之秩序程度及其成分而异。吾人作二简单假设:一设相互作用能量为秩序及成分之线性函数,另一设其与原子对偶之数成线性函数。将此等假设应用於AB类之合金,则必须在所设函数中之系数间,有适当关系,合金之临界温度,始在成分为1:1时,有极大值。在AB_3类之合金,吾人乃应用Bragg及Williams二氏之理论以求简便。於此可证明若所设函数中之系数,可任意调整则所计算出之临界温度之极大值可在任何成分发生。故关於此点理论与实验不合之处,可望解决。又合金之反常比热,亦经算出。在AB类之合金,Bethe氏原来之能量公式不復可用,故另用与Bragg及Williams理论比较而得之公式计算。又关於合金可分为二相或多相之问题,此篇亦大略论及。  General formulas for the. critical temperature and the discontinuity of the ipecific heat of a, superlattice are obtained by the generalized quasichemical method, The results may be applied to any approximation.  本文应用推广准化学方法求得二元超格结晶在任何几近计算中之临界温度及比热突变公式。   << 更多相关文摘 
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