弯曲衬底上应变异质外延薄膜形态失稳的相场模拟

论文旨在研究弯曲衬底上应变异质外延薄膜的表面非线性演化行为,研究采用了基于Eshelby等效方法的相场微弹性模型来模拟二维应变Si1-xGex/Si薄膜/衬底系统的形态失稳.建立了关于等效特征应变和长程序参量的自由能泛函,数值求解了时间相关的Ginzburg-Landau动力学方程.系统自由能包括化学能,弹性应变能以及薄膜、衬底与真空相两两之界面能.跟踪了全时的形态演化过程,给出了指定时刻的量子点形态轮廓图.结果表明,量子点倾向于沉积在弯曲衬底的波谷处,波谷处是能量有利位置,量子点在此处比在波峰处更加稳定.论文所做的相场模拟可以用来预测量子点形成的轮廓、尺寸和位置,可以为控制和生成周期性自组装表面纳米结构提供理论指导.     

The study of wave motion in the Talbot interferometer with a lens

We present the study of the wave motion in the Talbot interferometer with an additional element such as a lens for all related audiences. Our solutions are in the analytic form. A general principle of the Talbot effect, which is the optical near-field effect, is the Fresnel diffraction. The Fresnel integral is rather complicated. We therefore introduce an alternative method which is based on the wave propagation through the transmission functions of the grating and the lens. Our method has been proved by a simple experimental setup.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR GENERALIZED QUANTUM MHD EQUATION

We prove the existence of a weak solution for a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation,energy estimates, and weak convergence for the adiabatic exponent γ 1.     

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor‐corrector (P‐C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1043–1069, 2017     

Asymptotic spectral analysis in semiconductor nanowire heterostructures

Mathematical settings in which heterogeneous structures affect electron transport through a tube-shaped quantum waveguide are studied, highlighting the interaction between material composition and geometric parameters like curvature and torsion. First, the macroscopic behaviour of a nanowire made of composite fibres with microscopic periodic texture is analysed, which amounts to determining the asymptotic behaviour of the spectrum of an elliptic Dirichlet eigenvalue problem with finely oscillating coefficients in a tube with shrinking cross-section. A suitable formal expansion suggests that the effective one-dimensional limit problem is of Sturm–Liouville type and yields the explicit formula for the underlying potential. In the torsion-free case, these findings are made rigorous by performing homogenization and 3d–1d dimension reduction for the two-scale problem in a variational framework by means of Γ-convergence. Second, waveguides with non-oscillating inhomogeneities in the cross-section are investigated. This leads to explicit criteria for propagation and localization of eigenmodes.     

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Rank 2

We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component B ₀(λ) containing the unit element for arbitrary rank 2 cases. We also present the explicit form of B ₀(λ) containing the highest (or lowest) weight vector by the polyhedral realization method.     

New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups.     

Global existence and large time behavior of solutions for the bipolar quantum hydrodynamic models in the quarter plane

In this paper, we discuss a bipolar transient quantum hydrodynamic model for charge density, current density, and electric field in the quarter plane. This model takes the form of a classical Euler–Poisson system with the additional dispersion terms caused by the quantum (Bohn) potential. We show global existence of smooth solutions for the initial boundary value problem when the initial data are near the nonlinear diffusive waves, which are different from the steady state. We also show the asymptotical behavior of the global smooth solution towards the nonlinear diffusive waves and obtain the algebraic decay rates. These results are proved by elaborate energy methods. Finally, using the Fourier analysis, we obtain the optimal convergence rates of the solutions towards the nonlinear diffusion waves. As far as we known, this is the first result about the initial boundary value problem of the one‐dimensional bipolar quantum hydrodynamic model in the quarter plane. Copyright © 2013 John Wiley & Sons, Ltd.     

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, ) onto (U, *, ) the set L of Lyndon words on A and their images { exp(w) w L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, ). We define a deformation *_q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.