1,679 research outputs found
The quaternion core inverse and its generalizations
In this paper we extend notions of the core inverse, core EP inverse, DMP
inverse, and CMP inverse over the quaternion skew-field and get
their determinantal representations within the framework of the theory of
column-row determinants previously introduced by the author. Since the
Moore-Penrose inverse and the Drazin inverse are necessary tools to represent
these generalized inverses, we use their determinantal representations
previously obtained by using row-column determinants. As the special case, we
give their determinantal representations for matrices with complex entries as
well. A numerical example to illustrate the main result is given.Comment: 34 page
The inverse along an element in rings with an involution, Banach algebras and -algebras
Properties of the inverse along an element in rings with an involution,
Banach algebras and -alegbras will be studied unifying known expressions
concerning generalized inverses.Comment: 18 pages, original research articl
Three limit representations of the core-EP inverse
In this paper, we present three limit representations of the core-EP inverse.
The first approach is based on the full-rank decomposition of a given matrix.
The second and third approaches, which depend on the explicit expression of the
core-EP inverse, are established. The corresponding limit representations of
the dual core-EP inverse are also given. In particular, limit representations
of the core and dual core inverse are derive
A universal invariant of four-dimensional 2-handlebodies and three-manifolds
This paper is the second part of our work on 4-dimensional 2-handlebodies. In
the first part (arXiv:math.GT/0407032) it is shown that up to certain set of
local moves, connected simple coverings of B^4 branched over ribbon surfaces,
bijectively represent connected orientable 4-dimensional 2-handlebodies up to
2-deformations (handle slides and creations/cancellations of handles of index
<= 2). We factor this bijective correspondence through a map onto the closed
morphisms in a universal braided category freely generated by a Hopf algebra
object H. In this way we obtain a complete algebraic description of
4-dimensional 2-handlebodies. This result is then used to obtain an analogous
description of the boundaries of such handlebodies, i.e. 3-dimensional
manifolds, which resolves for closed manifolds the problem posed by Kerler in
"Towards an algebraic characterization of 3-dimensional cobordisms",
Contemporary Mathematics 318 (2003). (cf. Problem 8-16 (1) in T. Ohtsuki,
"Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4
(2002).Comment: 86 pages, 146 postscript figures, 29 references. LaTeX 2.09 file.
Uses: geom.sty epsf.st
Further results on the -inverse, the outer inverse and the Moore-Penrose inverse in the Banach context
In this article properties of the -inverse, the inverse along an
element, the outer inverse with prescribed range and null space and the Moore-Penrose inverse will be studied in the contexts of Banach
spaces operators, Banach algebras and -algebras. The main properties to be
considered are the continuity, the differentiability and the openness of the
sets of all invertible elements defined by all the aforementioned outer
inverses but the Moore-Penrose inverse. The relationship between the -inverse and the outer inverse will be also characterized.Comment: 27 pages, original research articl
Hyperplane Arrangements and Diagonal Harmonics
In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial
interpretation of the -Catalan numbers and the Hilbert series of diagonal
harmonics. In this paper we propose a new combinatorial interpretation in terms
of the affine Weyl group of type . In particular, we define two statistics
on affine permutations; one in terms of the Shi hyperplane arrangement, and one
in terms of a new arrangement - which we call the Ish arrangement. We prove
that our statistics are equivalent to the {\sf area'} and {\sf bounce}
statistics of Haglund and Loehr. In this setting, we observe that {\sf bounce}
is naturally expressed as a statistic on the root lattice. We extend our
statistics in two directions: to "extended" Shi arrangements and to the bounded
chambers of these arrangements. This leads to a (conjectural) combinatorial
interpretation for all integral powers of the Bergeron-Garsia nabla operator
applied to the elementary symmetric functions.Comment: 27 pages, 12 figure
Anyons and Deformed Lie Algebras
We discuss the connection between anyons (particles with fractional
statistics) and deformed Lie algebras (quantum groups). After a brief review of
the main properties of anyons, we present the details of the anyonic
realization of all deformed classical Lie algebras in terms of anyonic
oscillators. The deformation parameter of the quantum groups is directly
related to the statistics parameter of the anyons. Such a realization is a
direct generalization of the Schwinger construction in terms of fermions and is
based on a sort of bosonization formula which yields the generators of the
deformed algebra in terms of the undeformed ones. The entire procedure is well
defined on two-dimensional lattices, but it can be consistently reduced also to
one-dimensional chains.Comment: Lectures given at the Varenna School on ``Quantum Groups and Their
Applications in Physics'' (June 1994), DFTT 33/94 and DFT-US 2/94, 31 pp. (5
figures and 1 table on request) LaTex file (the macro subeqn.sty is appended
at the end the LaTex file
Anyons from Three-Body Hard-Core Interactions in One Dimension
Traditional anyons in two dimensions have generalized exchange statistics
governed by the braid group. By analyzing the topology of configuration space,
we discover that an alternate generalization of the symmetric group governs
particle exchanges when there are hard-core three-body interactions in
one-dimension. We call this new exchange symmetry the traid group and
demonstrate that it has abelian and non-abelian representations that are
neither bosonic nor fermionic, and which also transform differently under
particle exchanges than braid group anyons. We show that generalized exchange
statistics occur because, like hard-core two-body interactions in two
dimensions, hard-core three-body interactions in one dimension create defects
with co-dimension two that make configuration space no longer simply-connected.
Ultracold atoms in effectively one-dimensional optical traps provide a possible
implementation for this alternate manifestation of anyonic physics.Comment: 34 pages double spaced, 8 figures, 85 refs. In v4, Sects. 1, 2, and
references are revise
Integral bases for TQFT modules and unimodular representations of mapping class groups
We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus
one and two at roots of unity of prime order and show that the corresponding
mapping class group representations preserve a unimodular Hermitian form over a
ring of algebraic integers. For higher genus surfaces the Hermitian form
sometimes must be non-unimodular. In one such case, genus 3 and p=5, we still
give an explicit basis
Moduli space for generic unfolded differential linear systems
In this paper, we identify the moduli space for germs of generic unfoldings
of nonresonant linear differential systems with an irregular singularity of
Poincar\'e rank at the origin, under analytic equivalence. The modulus of a
given family was determined in \cite{HLR}: it comprises a formal part depending
analytically on the parameters, and an analytic part given by unfoldings of the
Stokes matrices. These unfoldings are given on "Douady-Sentenac" (DS) domains
in the parameter space covering the generic values of the parameters
corresponding to Fuchsian singular points. Here we identify exactly which
moduli can be realized. A necessary condition on the analytic part, called
compatibility condition, is saying that the unfoldings define the same
monodromy group (up to conjugacy) for the different presentations of the
modulus on the intersections of DS domains. With the additional requirement
that the corresponding cocycle is trivial and good limit behavior at some
boundary points of the DS domains, this condition becomes sufficient. In
particular we show that any modulus can be realized by a -parameter family
of systems of rational linear differential equations over with , or singular points (with multiplicities). Under
the generic condition of irreducibility, there are precisely singular
points which are Fuchsian as soon as simple. This in turn implies that any
unfolding of an irregular singularity of Poincar\'e rank is analytically
equivalent to a rational system of the form , with polynomial of degree at most and is the
generic unfolding of the polynomial .Comment: 45 pages, 19 figure
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