7,240 research outputs found
A Unified Algebraic Framework for Fuzzy Image Compression and Mathematical Morphology
In this paper we show how certain techniques of image processing, having
different scopes, can be joined together under a common "algebraic roof"
Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
We investigate properties of the most general PT-symmetric non-Hermitian
Hamiltonian of cubic order in the annihilation and creation operators as a ten
parameter family. For various choices of the parameters we systematically
construct an exact expression for a metric operator and an isospectral
Hermitian counterpart in the same similarity class by exploiting the
isomorphism between operator and Moyal products. We elaborate on the subtleties
of this approach. For special choices of the ten parameters the Hamiltonian
reduces to various models previously studied, such as to the complex cubic
potential, the so-called Swanson Hamiltonian or the transformed version of the
from below unbounded quartic -x^4-potential. In addition, it also reduces to
various models not considered in the present context, namely the single site
lattice Reggeon model and a transformed version of the massive sextic
x^6-potential, which plays an important role as a toy modelto identify theories
with vanishing cosmological constant.Comment: 21 page
Baxter Q-Operators and Representations of Yangians
We develop a new approach to Baxter Q-operators by relating them to the
theory of Yangians, which are the simplest examples for quantum groups. Here we
open up a new chapter in this theory and study certain degenerate solutions of
the Yang-Baxter equation connected with harmonic oscillator algebras. These
infinite-state solutions of the Yang-Baxter equation serve as elementary,
"partonic" building blocks for other solutions via the standard fusion
procedure. As a first example of the method we consider sl(n) compact spin
chains and derive the full hierarchy of operatorial functional equations for
all related commuting transfer matrices and Q-operators. This leads to a
systematic and transparent solution of these chains, where the nested Bethe
equations are derived in an entirely algebraic fashion, without any reference
to the traditional Bethe ansatz techniques.Comment: 27 pages, 5 figures; v2: typos fixed, references updated and adde
Erlangen Programme at Large 3.2: Ladder Operators in Hypercomplex Mechanics
We revise the construction of creation/annihilation operators in quantum
mechanics based on the representation theory of the Heisenberg and symplectic
groups. Besides the standard harmonic oscillator (the elliptic case) we
similarly treat the repulsive oscillator (hyperbolic case) and the free
particle (the parabolic case). The respective hypercomplex numbers turn to be
handy on this occasion, this provides a further illustration to Similarity and
Correspondence Principle.
Keywords: Heisenberg group, Kirillov's method of orbits, geometric
quantisation, quantum mechanics, classical mechanics, Planck constant, dual
numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics,
interference, Fock--Segal--Bargmann representation, Schr\"odinger
representation, dynamics equation, harmonic and unharmonic oscillator,
contextual probability, symplectic group, metaplectic representation,
Shale--Weil representationComment: LaTeX2e, 12 pages, 3 EPS pictures in one figures; v2: the
illustration is added, several small improvements; v3: minor corrections,
several references are added; v4: minor correction
Geometric Aspects of Frame Representations of Abelian Groups
We consider frames arising from the action of a unitary representation of a
discrete countable abelian group. We show that the range of the analysis
operator can be determined by computing which characters appear in the
representation. This allows one to compare the ranges of two such frames, which
is useful for determining similarity and also for multiplexing schemes. Our
results then partially extend to Bessel sequences arising from the action of
the group. We apply the results to sampling on bandlimited functions and to
wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two
sampling transforms to have orthogonal ranges, and two analysis operators for
wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient
condition is easy to compute in terms of the periodization of the Fourier
transform of the frame generators.Comment: 20 pages; contact author: Eric Webe
Quantale Modules and their Operators, with Applications
The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology
Simplicial gauge theory on spacetime
We define a discrete gauge-invariant Yang-Mills-Higgs action on spacetime
simplicial meshes. The formulation is a generalization of classical lattice
gauge theory, and we prove consistency of the action in the sense of
approximation theory. In addition, we perform numerical tests of convergence
towards exact continuum results for several choices of gauge fields in pure
gauge theory.Comment: 18 pages, 2 figure
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