Microscopic calculation of the magnetic field of neutron stars

Based on the Dirac equation describing an electron moving in a uniform and cylindrically symmetric magnetic field which may be the result of the self-consistent mean field of the electrons themselves in a neutron star, we have obtained the eigen solutions and the orbital magnetic moments of electrons in which each eigen orbital can be calculated. From the eigen energy spectrum we find that the lowest energy level is the highly degenerate orbitals with the quantum numbers pZ=0, n=0, and m≥0. At the ground state, the electrons fill the lowest eigen states to form many Landau magnetic cells and each cell is a circular disk with the radius λfree and the thickness λe, where λfree is the electron mean free path determined by Coulomb cross section and electron density and λe is the electron Compton wavelength. The magnetic moment of each cell and the number of cells in the neutron star are calculated, from which the total magnetic moment and magnetic field of the neutron star can be calculated. The results are compared with the observational data and the agreement is reasonable.     

On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in \mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+\frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Fatness and thinness of uniform Cantor sets for doubling measures

Let E = E({nk},{ck}) be a fat uniform Cantor set. We prove that E is a minimally fat set for doubling measures if and only if (nkck)p = ∞ for all p < 1 and that E is a fairly fat set for doubling measures if and only if there are constants 0 < p < q < 1 such that (nkck)q < ∞ and (nkck)p = ∞. The classes of minimally thin uniform Cantor sets and of fairly thin uniform Cantor sets are also characterized.     

二元函数连续性条件之探析

对二元函数连续性判定条件给出了详细分析，强调有关问题的关键点，纠正了常见的模糊认识，得到一系列连续性充分条件及其推广形式．     

CFD simulation of oil–water separation characteristics in a compact flotation unit by population balance modeling

Huge amounts of produced water are generated in offshore oil production. The Compact Flotation Unit (CFU) is an excellent pretreatment technology of produced water with high separation efficiency, low residence, and small split ratio. The Computational Fluid Dynamics-population balance model (CFD-PBM) method is used in the present work to study the oil–water separation characteristics in the self-developed Beijing Institute of Petrochemical Technology Compact Flotation Unit (BIPTCFU) at both micro-scale and macro-scale, which would help us gain more insights into the mechanism and the influence of flow field on the oil–water separation process such as the oil droplets’ diameter distribution and separation efficiency. The effects of the inlet diameter, the height of the preliminary separation zone, and the width of the annular space on the oil–water separation characteristics of CFU were discussed systematically. It is illustrated that the appropriate increase of inlet velocity, decrease of annular gap width, and increase of the height in the preliminary separation zone can effectively promote the collision and coalescence process of oil droplets. However, the overlarge height of the preliminary separation zone and the too narrow width of the annular space will both have a significant negative effect on the migration and separation of oil and water and lead to the decrease of separation efficiency.     

Three-dimensional analytical solution for a transversely isotropic functionally graded piezoelectric circular plate subject to a uniform electric potential difference

This paper studies the problem of a functionally graded piezoelectric circular plate subjected to a uniform electric potential difference between the upper and lower surfaces. By assuming the generalized displacements in appropriate forms, five differential equations governing the generalized displacement functions are derived from the equilibrium equations. These displacement functions are then obtained in an explicit form, which still involve four undetermined integral constants, through a step-by-step integration which properly incorporates the boundary conditions at the upper and lower surfaces. The boundary conditions at the cylindrical surface are then used to determine the integral constants. Hence, three-dimensional analytical solutions for electrically loaded functionally graded piezoelectric circular plates with free or simply-supported edge are completely determined. These solutions can account for an arbitrary material variation along the thickness, and thus can be readily degenerated into those for a homogenous plate. A numerical example is finally given to show the validity of the analysis, and the effect of material inhomogeneity on the elastic and electric fields is discussed. Supported by the National Natural Science Foundation of China (Grant Nos. 10472102 and 10432030) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335107)     

The Index of the Complex Eigenvalues of a Parity Progressive Population Operator

In this paper we discuss the index of the complex eigenvalue of a parity progressive population operator. Under certain conditions, we first prove that all the complex eigenvalues of this operator, except at most finitely many ones, are of index 1, and then, as an application of this result, we obtain the asymptotic expansion of the solution of the corresponding parity progressive population equation.     

A Population Dynamics Model with Parental Care

A model for an agestructured unlimited population dynamics with parental care of offspring is presented (migration of individuals is not taken into account). The model consists of six partial integrodifferential equations for single males, single females, pairs with offspring under parental care, pairs without offspring under parental care, and offspring of the male and female sex. A class of separable solutions is constructed.     

Dynamical Localization II with an Application to the Almost Mathieu Operator

Several recent works have established dynamical localization for Schrödinger operators, starting from control on the localization length of their eigenfunctions, in terms of their centers of localization. We provide an alternative way to obtain dynamical localization, without resorting to such a strong condition on the exponential decay of the eigenfunctions. Furthermore, we illustrate our purpose with the almost Mathieu operator, H _{, ,} =–+ cos(2(+x)), 15 and with good Diophantine properties. More precisely, for almost all , for all q>0, and for all functions ²( ) of compact support, we show that The proof applies equally well to discrete and continuous random Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes, supplied in the random case by the multi-scale analysis.