11,927 research outputs found
Modeling the electron with Cosserat elasticity
We suggest an alternative mathematical model for the electron in dimension
1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose
material points can experience no displacements, only rotations. This framework
is a special case of the Cosserat theory of elasticity. Rotations of material
points are described mathematically by attaching to each geometric point an
orthonormal basis which gives a field of orthonormal bases called the coframe.
As the dynamical variables (unknowns) of our theory we choose a coframe and a
density. We then add an extra (third) spatial dimension, extend our coframe and
density into dimension 1+3, choose a conformally invariant Lagrangian
proportional to axial torsion squared, roll up the extra dimension into a
circle so as to incorporate mass and return to our original (1+2)-dimensional
spacetime by separating out the extra coordinate. The main result of our paper
is the theorem stating that our model is equivalent to the Dirac equation in
dimension 1+2. In the process of analyzing our model we also establish an
abstract result, identifying a class of nonlinear second order partial
differential equations which reduce to pairs of linear first order equations
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Ermakov Systems with Multiplicative Noise
Using the Euler-Maruyama numerical method, we present calculations of the
Ermakov-Lewis invariant and the dynamic, geometric, and total phases for
several cases of stochastic parametric oscillators, including the simplest case
of the stochastic harmonic oscillator. The results are compared with the
corresponding numerical noiseless cases to evaluate the effect of the noise.
Besides, the noiseless cases are analytic and their analytic solutions are
briefly presented. The Ermakov-Lewis invariant is not affected by the
multiplicative noise in the three particular examples presented in this work,
whereas there is a shift effect in the case of the phasesComment: 12 pages, 4 figures, 22 reference
Shadow world evaluation of the Yang-Mills measure
A new state-sum formula for the evaluation of the Yang-Mills measure in the
Kauffman bracket skein algebra of a closed surface is derived. The formula
extends the Kauffman bracket to diagrams that lie in surfaces other than the
plane. It also extends Turaev's shadow world invariant of links in a circle
bundle over a surface away from roots of unity. The limiting behavior of the
Yang-Mills measure when the complex parameter approaches -1 is studied. The
formula is applied to compute integrals of simple closed curves over the
character variety of the surface against Goldman's symplectic measure.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-17.abs.htm
Geometric gradient-flow dynamics with singular solutions
The gradient-flow dynamics of an arbitrary geometric quantity is derived
using a generalization of Darcy's Law. We consider flows in both Lagrangian and
Eulerian formulations. The Lagrangian formulation includes a dissipative
modification of fluid mechanics. Eulerian equations for self-organization of
scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic
equations. We identify singular solutions of these equations corresponding to
collapsed (clumped) states and discuss their evolution.Comment: 28 pages, 1 figure, to appear on Physica
Formation and Evolution of Singularities in Anisotropic Geometric Continua
Evolutionary PDEs for geometric order parameters that admit propagating
singular solutions are introduced and discussed. These singular solutions arise
as a result of the competition between nonlinear and nonlocal processes in
various familiar vector spaces. Several examples are given. The motivating
example is the directed self assembly of a large number of particles for
technological purposes such as nano-science processes, in which the particle
interactions are anisotropic. This application leads to the derivation and
analysis of gradient flow equations on Lie algebras. The Riemann structure of
these gradient flow equations is also discussed.Comment: 38 pages, 4 figures. Physica D, submitte
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