Marcinkiewicz–Zygmund inequalities

We study a generalization of the classical Marcinkiewicz–Zygmund inequalities. We relate this problem to the sampling sequences in the Paley–Wiener space and by using this analogy we give sharp necessary and sufficient computable conditions for a family of points to satisfy the Marcinkiewicz–Zygmund inequalities.     

关于Gerber不等式的一个猜想

本文证明了陈计-单墫关于Ｇｅｒｂｅｒ不等式的一个猜想，作为其应用，导出了单形内一点到顶点的距离与到面的距离的两个不等式．     

The stress driven instability in elastic crystals: Mathematical models and physical manifestations

Summary Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms. The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body (of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals, and discuss possible manifestations of the above-mentioned instability in the problems of crystal growth, materials science, fracture, physical chemistry, and low-temperature physics.     

Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

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and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.     

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Generalized classical theory of magnetism

We consider an Ising model with Kac potential ^dK(¦x¦) which may have arbitrary sign, and show, following Gates and Penrose, that the free energy in the classical limit0+ can be obtained from a variational principle. When the Fourier transform of the potential has its maximum atp=0 one recovers the usual mean-field theory of magnetism. When the maximum occurs forp ₀0, however, one obtains an oscillatory or helicoidal phase in which the magnetization near the critical point oscillates with period 2/¦p ₀¦. An example with a potential possessing parameter-dependent oscillations is shown to exhibit crossover phenomena and a multicritical Lifshitz point in the classical limit.     

Consistency of a linear system of inequalities

We show that a consistency check of a linear system of inequalitiesAxb reduces to check whetherQb0 for a certain matrixQ. It is a direct consequence of the Farkas-Minkowski theorem. Thus, when one has to check consistency for different values ofb, one need not run a numerical algorithm for eachb.On leave at the Electronics Research Laboratory of the University of California at Berkeley in a CNRS/NSF Exchange Program.     

Error in an approximate wave function and an error minimization scheme

An estimate of error in an approximate wave function for a stationary state is put forward that does not require any information about the exact state. The measure is sensitive and direct. Parameters embedded in a trial wave function can be varied to minimize this error as well, leading to a variational principle. Such a scheme works nicely for bound states and the more so for Siegert states, for which the standard energy minimization principle does not apply. Pilot calculations on the anharmonic oscillator system and the radial Stark effect in the hydrogen atom reveal the worth of the endeavor. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Theoretical study and rate constant calculation for the reactions of SH (SD) with Cl2, Br2, and BrCl

The mechanisms of the SH (SD) radicals with Cl2 (R1), Br2 (R2), and BrCl (R3) are investigated theoretically, and the rate constants are calculated using a dual-level direct dynamics method. The optimized geometries and frequencies of the stationary points are calculated at the MP2/6-311G(d,p) and MPW1K/6-311G(d,p) levels. Higher-level energies are obtained at the approximate QCISD(T)/6-311++G(3df, 2pd) level using the MP2 geometries as well as by the multicoefficient correlation method based on QCISD (MC-QCISD) using the MPW1K geometries. Complexes with energies less than those of the reactants or products are located at the entrance or the exit channels of these reactions, which indicate that the reactions may proceed via an indirect mechanism. The enthalpies of formation for the species XSH/XSD (X = Cl and Br) are evaluated using hydrogenation working reactions method. By canonical variational transition-state theory (CVT), the rate constants of SH and SD radicals with Cl2, Br2, and BrCl are calculated over a wide temperature range of 200-2000 K at the a-QCISD(T)/6-311++G(3df, 2pd)//MP2/6-311G(d, p) level. Good agreement between the calculated and experimental rate constants is obtained in the measured temperature range. Our calculations show that for SH (SD) + BrCl reaction bromine abstraction (R3a or R3a') leading to the formation of BrSH (BrSD) + Cl in a barrierless process dominants the reaction with the branching ratios for channels 3a and 3a' of 99% at 298 K, which is quite different from the experimental result of k3a'/k3' = 54 +/- 10%. Negative activation energies are found at the higher level for the SH + Br2 and SH + BrCl (Br-abstraction) reactions; as a result, the rate constants show a slightly negative temperature dependence, which is consistent with the determination in the literature. The kinetic isotope effects for the three reactions are "inverse". The values of kH/kD are 0.88, 0.91, and 0.69 at room temperature, respectively, and they increase as the temperature increases.     

考虑订单取件时间和柔性时间窗的取送货车辆路径问题

研究了同城配送中考虑订单取货时间和柔性时间窗的取送货车辆路径问题,考虑同城配送中订单起终点,订单取货时间和订单配送的柔性时间窗,车容量限制等因素。首先构建以配送成本与超时惩罚成本之和最小化为目标的混合整数线性模型。其次,设计了含多种有效不等式及其对应分离算法的改进分支切割算法对该模型进行精确求解。最后通过实验测试分析了不等式的性能,验证了算法的有效性,实验表明适当的减少车辆数和增大装载能力能够有效的减少成本。