The Kohn Sham Equation For Deformed Crystals

Author : Weinan E,Jianfeng Lu
Publisher : Unknown
Page : 97 pages
File Size : 42,7 Mb
Release : 2012
Category : Deformations (Mechanics)
ISBN : 0821894668

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The Kohn-Sham Equation for Deformed Crystals by Weinan E,Jianfeng Lu Pdf

The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, we also establish a number of fundamental properties of the Kohn-Sham map.

Author : Weinan E,Jianfeng Lu
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 49,8 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821875605

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The Kohn-Sham Equation for Deformed Crystals by Weinan E,Jianfeng Lu Pdf

The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, the authors also establish a number of fundamental properties of the Kohn-Sham map.

Author : Robert J. Buckingham,Peter D. Miller
Publisher : American Mathematical Soc.
Page : 136 pages
File Size : 40,9 Mb
Release : 2013-08-23
Category : Mathematics
ISBN : 9780821885451

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The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates by Robert J. Buckingham,Peter D. Miller Pdf

The authors study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. They show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.

Author : Florica C. Cîrstea
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 47,6 Mb
Release : 2014-01-08
Category : Mathematics
ISBN : 9780821890226

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A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials by Florica C. Cîrstea Pdf

Author : Joachim Krieger,Jacob Sterbenz
Publisher : American Mathematical Soc.
Page : 99 pages
File Size : 51,9 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821844892

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Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space by Joachim Krieger,Jacob Sterbenz Pdf

This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on $(6+1)$ and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space $\dot{H}_A^{(n-4)/{2}}$. Regularity is obtained through a certain ``microlocal geometric renormalization'' of the equations which is implemented via a family of approximate null Cronstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic $L^p$ spaces, and also proving some bilinear estimates in specially constructed square-function spaces.

Author : Kening Lu,Qiudong Wang,Lai-Sang Young
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 46,7 Mb
Release : 2013-06-28
Category : Mathematics
ISBN : 9780821884843

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Strange Attractors for Periodically Forced Parabolic Equations by Kening Lu,Qiudong Wang,Lai-Sang Young Pdf

The authors prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

Author : Ariel Barton
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 44,8 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821887400

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Elliptic Partial Differential Equations with Almost-Real Coefficients by Ariel Barton Pdf

In this monograph the author investigates divergence-form elliptic partial differential equations in two-dimensional Lipschitz domains whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. He shows that for such operators, the Dirichlet problem with boundary data in $L^q$ can be solved for $q1$ small enough, and provide an endpoint result at $p=1$.

Author : Andrew Knightly,C. Li
Publisher : American Mathematical Soc.
Page : 132 pages
File Size : 52,9 Mb
Release : 2013-06-28
Category : Mathematics
ISBN : 9780821887448

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Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms by Andrew Knightly,C. Li Pdf

The authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $\operatorname{GL}(2)$ over $\mathbf{Q}$. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. The authors include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, they show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.

Author : Jose Angel Pelaez,Jouni Rattya
Publisher : American Mathematical Soc.
Page : 124 pages
File Size : 47,7 Mb
Release : 2014-01-08
Category : Mathematics
ISBN : 9780821888025

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Weighted Bergman Spaces Induced by Rapidly Increasing Weights by Jose Angel Pelaez,Jouni Rattya Pdf

Author : Florin Diacu
Publisher : American Mathematical Soc.
Page : 80 pages
File Size : 51,6 Mb
Release : 2014-03-05
Category : Mathematics
ISBN : 9780821891360

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Relative Equilibria in the 3-Dimensional Curved n-Body Problem by Florin Diacu Pdf

Author : Hajime Koba
Publisher : American Mathematical Soc.
Page : 127 pages
File Size : 51,6 Mb
Release : 2014-03-05
Category : Mathematics
ISBN : 9780821891339

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Nonlinear Stability of Ekman Boundary Layers in Rotating Stratified Fluids by Hajime Koba Pdf

A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This book constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. The author calls such stationary solutions Ekman layers. This book shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, the author discusses the uniqueness of weak solutions and computes the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. The author also shows that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.

Author : Emmanuel Schertzer ,Rongfeng Sun,Jan M. Swart
Publisher : American Mathematical Soc.
Page : 160 pages
File Size : 50,7 Mb
Release : 2014-01-08
Category : Mathematics
ISBN : 9780821890882

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Stochastic Flows in the Brownian Web and Net by Emmanuel Schertzer ,Rongfeng Sun,Jan M. Swart Pdf

It is known that certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a `stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is characterized by its -point motions. The authors' work focuses on a class of stochastic flows of kernels with Brownian -point motions which, after their inventors, will be called Howitt-Warren flows. The authors' main result gives a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called "erosion flow", can be constructed from two coupled "sticky Brownian webs". The authors' construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, the authors show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart. Using these constructions, the authors prove some new results for the Howitt-Warren flows.

Author : Alejandro D. de Acosta,Peter Ney
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 51,9 Mb
Release : 2014-03-05
Category : Mathematics
ISBN : 9780821890899

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Operator Theory, Operator Algebras, and Applications by Alejandro D. de Acosta,Peter Ney Pdf

Author : Ioan Bejenaru,Daniel Tataru
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 51,6 Mb
Release : 2014-03-05
Category : Mathematics
ISBN : 9780821892152

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Near Soliton Evolution for Equivariant Schrödinger Maps in Two Spatial Dimensions Ioan Bejenaru, University of California, San Diego, La Jolla, CA, and Daniel Tataru, University of California, Berkeley, Berkeley, CA by Ioan Bejenaru,Daniel Tataru Pdf

The authors consider the Schrödinger Map equation in 2+1 dimensions, with values into \mathbb{S}^2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that Q is unstable in the energy space \dot H^1. However, in the process of proving this they also show that within the equivariant class Q is stable in a stronger topology X \subset \dot H^1.

Author : Victor Reiner,Franco Saliola,Volkmar Welker
Publisher : American Mathematical Soc.
Page : 109 pages
File Size : 53,8 Mb
Release : 2014-03-05
Category : Mathematics
ISBN : 9780821890950

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Spectra of Symmetrized Shuffling Operators by Victor Reiner,Franco Saliola,Volkmar Welker Pdf

For a finite real reflection group $W$ and a $W$-orbit $\mathcal{O}$ of flats in its reflection arrangement--or equivalently a conjugacy class of its parabolic subgroups--the authors introduce a statistic $\operatorname{noninv}_\mathcal{O}(w)$ on $w$ in $W$ that counts the number of ``$\mathcal{O}$-noninversions'' of $w$. This generalizes the classical (non-)inversion statistic for permutations $w$ in the symmetric group $\mathfrak{S}_n$. The authors then study the operator $\nu_\mathcal{O}$ of right-multiplication within the group algebra $\mathbb{C} W$ by the element that has $\operatorname{noninv}_\mathcal{O}(w)$ as its coefficient on $w$.