A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

Local orbitals by minimizing powers of the orbital variance

It is demonstrated that a set of local orthonormal Hartree-Fock (HF) molecular orbitals can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller maximum orbital spreads are encountered. Calculations on superbenzene, C(60), and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. In fact, when increasing the cardinal number for Dunning's correlation consistent basis sets, it is seen that for larger penalties, the virtual orbitals become more local than the occupied ones. We also show that the local virtual HF orbitals are significantly more local than the redundant projected atomic orbitals, which often have been used to span the virtual orbital space in local correlated wave function calculations. Our local molecular orbitals thus appear to be a good candidate for local correlation methods.     

Complex quantum enveloping algebras as twisted tensor products

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.     

Fabrication and characterization of stable ultrathin film micropatterns containing DNA and photosensitive polymer diazoresin

Stable, ultrathin DNA micropatterns were fabricated from photosensitive polymer diazoresin (DR) through a self-assembly technique. The micropatterns were achieved on LBL ultrathin film after UV exposure through a photomask. The patterns were characterized systematically with scanning electron microscopy, atomic force microscopy, X-ray photoelectron spectroscopy and fluorescence microscopy. All of the results indicate that the combined LBL self-assembly and photolithography technique is a promising method for constructing stable, well-defined micropatterns with a nanoscale thickness.     

First Passage Time Distribution for Linear Functions of a Random Walk

In this paper, theorems about asymptotic behavior of the local probabilities of crossing the linear boundaries by a perturbed random walk are proved.     

The Existence Of Nonhomogeneous,Viscous And Incompressible Flow In Unbounded Domains

The existence of global (in time) solution {ρu,p} (density, velocity field and pressure) of a partial differential problem modelling the behaviour of a viscous, nonhomogenous and incompressible flow has been recently obtained viscous, nonhomogenous and incompressible flow has been recently obtained by J. Simon. In his most general result, he finds a solution which satifies an initial condition in an appropriate weak sense. In this paper, we present an extension of this result to the case of arbitrary unbounded three–dimensional fluid domains.

1. Introduction.

2. The main result.

3. Some technical Lemmas.

4. Proof of Theorem 1.

4.1 The existence and properties of {ρR, U^R PR}

4.2 Some estimates for ρR,U^R and their time derivatives.

4.3 Further estimates.

4.4 The choice of a convergent sequence and conclusions.

References.     

Strict Monotonicity of Eigenvalues and Unique Continuation

In this paper, we show that the strict mononicity of the eigen-values of an uniformly elliptic operator of second order is equivalent to a unique continuation property.     

On normality of the Wijsman topology

Let (X, ρ) be a metric space and (CL(X), W _ρ ) be the hyperspace of all nonempty closed subsets of X equipped with the Wijsman topology. The Wijsman topology is one of the most important classical hyperspace topologies. We give a partial answer to a question posed in Di Maio (Quaderni di Matematica, 3:55–92, 1998) whether the normality of (CL(X), W _ρ ) is equivalent to its metrizability. If (X, ρ) is a linear metric space, then (CL(X), W _ρ ) is normal if and only if (CL(X), W _ρ ) is metrizable. Some further results concerning normality of the Wijsman topology on CL(X) are also proved.