On the use of variational methods for solving Boltzmann equations involving non-Hermitian operators

Variational principles yielding upper and lower bounds on transport coefficients can readily be applied to the Boltzmann equation, provided it has the form of a linear, inhomogeneous integrodifferential equation with a Hermitian operator acting on the deviation from equilibrium of the distribution function. In transport problems involving a magnetic field or an alternating electric field, this operator is non-Hermitian. By suitably transforming the transport equation, we show how Variational principles may still give upper and lower bounds. The bounds are used for considering the frequency-dependent conductivity associated with a general scattering operator, and the longitudinal magnetoresistivity in the relaxation time approximation for the scattering operator. Explicit results are presented for (1) the frequency-dependent conductivity of a charged Fermi liquid and (2) the longitudinal magnetoresistivity for a weakly anisotropic Fermi surface.     

Invariance principles for renewal processes when only moments of low order exist

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419–1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.     

Stability and extremum principles for post yield analysis of finite plasticity

The post yield behavior of rigid-perfectly plastic solids at the collapse load is studied based on the finite deformation theory. By using the general duality theory developed by Gao-Strang (1989), a global stability criteria is proposed and a pair of dual extremum principles, expressed in terms of displacements, displacement rates and the Kirchhoff stresses are established for plastic collapse analysis. It is proved that under large deformations, the existence of the plastic limit state at the collapse load depends on the directional derivative of a so-called complementary gap function. The application to the nonlinear plastic collapse theory yields a pair of dual bounding theorems for limit loading factor associated with any transient displacement of the deformed body when the global extremum criteria are satisfied. Dedicated to Professor Y.K. Cheung on the occasion of his 60th birthday     

On a class of constrained functional minimization problems and their numerical solution

We investigate a class of functional minimization problems with constraints. By means of variational principles, optimal control theory, and numerical methods for nonlinear equations, numerical methods and the corresponding computer software are established to solve the problems. These tools can be used in fitting curves with arbitrary smoothness, different boundary conditions, and constraints. For special boundary conditions, analytical expressions of the curves are derived. Numerical examples are given to demonstrate the effectiveness of the algorithms by the means of curve fitting.     

The Enskog repeated ring equations for tagged molecule motion and variational solutions

The Enskog repeated ring equations (ERRE) for tagged molecule motion are derived from the BBGKY hierarchy. The hierarchical deviation demonstrates the problem associated with using naive truncations for complex fluid systems. A moment-variational solution is suggested for the ERRE. The moment-variational method is applied to the ERRE for the two- and three-dimensional overlapping Lorentz gas (LG). Both self-consistent and non-self-consistent equations for the diffusion constant are solved. The results compare favorably with previous calculations and molecular dynamics (MD).     

Uncertainty principles for the continuous Hankel Wavelet transform

The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Hankel wavelet transform. We also analyse the concentration of this transform on sets of finite measure. Benedicks-type uncertainty principle is given.     

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly-nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.     

Structural,electronic, and magnetic properties of 3d transition metal doped GaN nanosheet: A first‐principles study

We have performed the first‐principles calculations on the structural, electronic, and magnetic properties of 3d transition‐metal? (Cr, Mn, Fe, Co, and Ni) atoms doped 2D GaN nanosheet. The results show that 3d TM atom substituting one Ga leads to a structural reconstruction around the 3d TM impurity compared to the pristine GaN nanosheet. The doping of TM atom can induce magnetic moments, which are mainly located on the 3d TM atom and its nearest‐neighbor N atoms. It is found that Mn‐ and Ni‐doped GaN nanosheet with 100% spin polarization characters seem to be good candidates for spintronic applications. When two Ga atoms are substituted by two TM dopants, the ferromagnetic (FM) ordering becomes energetically more favorable for Cr‐, Mn‐, and Ni‐doped GaN nanosheet with different distances of two TM atoms. On the contrary, the antiferromagnetic (AFM) ordering is energetically more favorable for Fe‐doped GaN nanosheet. In addition, our GGA + U calculations show the similar results with GGA calculations. © 2016 Wiley Periodicals, Inc.     

The structural,electronic, and magnetic properties of the stoichiometric (001) surface of double perovskite Sr2FeMoO6

The structural, electronic, and magnetic properties of the stoichiometric (001) surface of double perovskite Sr₂FeMoO₆ have been studied by using a 10‐layer FeMoO₄ and SrO terminated (001)‐oriented slab model and the first‐principles projector augmented wave potential within the generalized gradient approximation as well as taking into account the on‐site Coulomb repulsive (U = 2.0 eV for Fe and 1.0 eV for Mo). An outwards relaxation is observed for several layers near surface, and the accompanying layer rumpling has a decrease tend from surface layer to inner layer. Along Fe–O–Mo–O–Fe or Mo–O–Fe–O–Mo chains, the oxygen atom is closer to the adjacent Mo atom than to the adjacent Fe atom. In FeO₆ or MoO₆ octahedra, the two axial TM?O bonds are not equal, and especially, the surface dangling bond makes the remaining one axial TM?O bond slightly shorter than four equally equatorial TM?O bonds. The half‐metallic nature and a complete (100%) spin polarization character ensure the FeMoO₄ and SrO terminated (001)‐oriented slab of double perovskite Sr₂FeMoO₆ a potential application in spintronics devices. The Fe⁺³ and Mo⁺⁵ ions are still in the (3d⁵, S = 5/2) and (4d¹, S = 1/2) states with positive and negative magnetic moments respectively and thus antiferromagnetic coupling via oxygen between them. There is no direct interaction between two nearest Fe–Fe or Mo–Mo pairs, whereas the hybridizations between Fe 3d and 4s, O 2s and 2p, as well as Mo 4d, 5s and 5p orbitals are fairly significant. Copyright © 2016 John Wiley & Sons, Ltd.     

First‐principles calculations of oxygen adsorption on the Ti3Al (0001) surface

Adsorption energies and density of states for O atoms adsorption on the Ti₃Al (0001) surface have been calculated using first‐principles calculations based on density functional theory. It is found that the order of O atom adsorption on the Ti₃Al (0001) surface is associated with the adsorption energy as well as the distance of O atoms because of the interaction. The adsorption energy mainly depends on the bond number and bond strength between O and Ti atoms, and the adsorption site with rich‐Ti surface (H_I and HCP_Al) is first priority. The adsorption energy decreases with the increase of the oxygen coverage because of the characteristics of the valence d‐orbitals of transition metals surface. Furthermore, the density of states indicates that the hybridization peak of O and Ti atoms is mainly from the contribution of Ti 3d‐ and O 2p‐orbitals, and the hybridization peak of O and Al atoms from the contribution of Al 2p‐ and O 2p‐orbitals. Copyright © 2016 John Wiley & Sons, Ltd.