1,932 research outputs found
Refractors in anisotropic media associated with norms
We show existence of interfaces between two anisotropic materials so that
light is refracted in accordance with a given pattern of energy. To do this we
formulate a vector Snell law for anisotropic media when the wave fronts are
given by norms for which the corresponding unit spheres are strictly convex.Comment: 26 pages, 2 figure
Extremal Problems in Minkowski Space related to Minimal Networks
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan
[FLM]: Is there an upper bound polynomial in for the largest cardinality of
a set S of unit vectors in an n-dimensional Minkowski space (or Banach space)
such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n
and that equality holds iff the space is linearly isometric to \ell^n_\infty,
the space with an n-cube as unit ball. We also remark on similar questions
raised in [FLM] that arose out of the study of singularities in
length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence
has been answered in Discrete & Computational Geometry 21 (1999) 437-44
Parametric polynomial minimal surfaces of arbitrary degree
Weierstrass representation is a classical parameterization of minimal
surfaces. However, two functions should be specified to construct the
parametric form in Weierestrass representation. In this paper, we propose an
explicit parametric form for a class of parametric polynomial minimal surfaces
of arbitrary degree. It includes the classical Enneper surface for cubic case.
The proposed minimal surfaces also have some interesting properties such as
symmetry, containing straight lines and self-intersections. According to the
shape properties, the proposed minimal surface can be classified into four
categories with respect to , and . The explicit
parametric form of corresponding conjugate minimal surfaces is given and the
isometric deformation is also implemented
Uniqueness of planar tangent maps in the modified Ericksen model
We prove the uniqueness of homogeneous blow-up limits of maps minimizing the
modified Ericksen energy for nematic liquid crystals in a planar domain. The
proof is based on the Weiss monotonicity formula, and a blow-up argument,
originally due to Allard and Almgren \cite{AA} for minimal surfaces, and L.
Simon \cite{SL} for energy-minimizing maps into analytic targets, which
exploits the integrability of certain Jacobi fields.Comment: 17 page
Should we solve Plateau's problem again?
After a short description of various classical solutions of Plateau's
problem, we discuss other ways to model soap films, and some of the related
questions that are left open. A little more attention is payed to a more
specific model, with deformations and sliding boundary conditions.Comment: Lecture for the conference in Honor of E. Stein, 201
Machine Learning, Quantum Mechanics, and Chemical Compound Space
We review recent studies dealing with the generation of machine learning
models of molecular and solid properties. The models are trained and validated
using standard quantum chemistry results obtained for organic molecules and
materials selected from chemical space at random
Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in
In this paper, we investigate an optimal boundary control problem for a two
dimensional simplified Ericksen--Leslie system modelling the incompressible
nematic liquid crystal flows. The hydrodynamic system consists of the
Navier--Stokes equations for the fluid velocity coupled with a convective
Ginzburg--Landau type equation for the averaged molecular orientation. The
fluid velocity is assumed to satisfy a no-slip boundary condition, while the
molecular orientation is subject to a time-dependent Dirichlet boundary
condition that corresponds to the strong anchoring condition for liquid
crystals. We first establish the existence of optimal boundary controls. Then
we show that the control-to-state operator is Fr\'echet differentiable between
appropriate Banach spaces and derive first-order necessary optimality
conditions in terms of a variational inequality involving the adjoint state
variables
Partielle Differentialgleichungen
The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric problems, regularity of free boundaries, conformal invariance and the Willmore functional
- …