47 research outputs found
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Detailed analysis of the continuum limit of a supersymmetric lattice model in 1D
We present a full identification of lattice model properties with their field
theoretical counter parts in the continuum limit for a supersymmetric model for
itinerant spinless fermions on a one dimensional chain. The continuum limit of
this model is described by an superconformal field theory
(SCFT) with central charge c=1. We identify states and operators in the lattice
model with fields in the SCFT and we relate boundary conditions on the lattice
to sectors in the field theory. We use the dictionary we develop in this paper,
to give a pedagogical explanation of a powerful tool to study supersymmetric
models based on spectral flow. Finally, we employ the developed machinery to
explain numerically observed properties of the particle density on the open
chain presented in Beccaria et al. PRL 94:100401 (2005).Comment: 28 pages, 7 figures, 3 tables, 1 appendix, this work is based on
chapter 4 of the authors PhD Thesis: L. Huijse, A supersymmetric model for
lattice fermions, University of Amsterdam (2010
Approximations from Anywhere and General Rough Sets
Not all approximations arise from information systems. The problem of fitting
approximations, subjected to some rules (and related data), to information
systems in a rough scheme of things is known as the \emph{inverse problem}. The
inverse problem is more general than the duality (or abstract representation)
problems and was introduced by the present author in her earlier papers. From
the practical perspective, a few (as opposed to one) theoretical frameworks may
be suitable for formulating the problem itself. \emph{Granular operator spaces}
have been recently introduced and investigated by the present author in her
recent work in the context of antichain based and dialectical semantics for
general rough sets. The nature of the inverse problem is examined from
number-theoretic and combinatorial perspectives in a higher order variant of
granular operator spaces and some necessary conditions are proved. The results
and the novel approach would be useful in a number of unsupervised and semi
supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings,
Springe
Economical adjunction of square roots to groups
How large must an overgroup of a given group be in order to contain a square
root of any element of the initial group? We give an almost exact answer to
this question (the obtained estimate is at most twice worse than the best
possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2:
minor correction
The structure of quotients of the Onsager algebra by closed ideals
We study the Onsager algebra from the ideal theoretic point of view. A
complete classification of closed ideals and the structure of quotient algebras
are obtained. We also discuss the solvable algebra aspect of the Onsager
algebra through the use of formal Lie algebras.Comment: 33 pages, Latex, small topos corrected-Journal versio
Invariants of solvable rigid Lie algebras up to dimension 8
The invariants of all complex solvable rigid Lie algebras up to dimension
eight are computed. Moreover we show, for rank one solvable algebras, some
criteria to deduce to non-existence of non-trivial invariants or the existence
of fundamental sets of invariants formed by rational functions of the Casimir
invariants of the associated nilradical.Comment: 16 pages, 7 table
The Seven-sphere and its Kac-Moody Algebra
We investigate the seven-sphere as a group-like manifold and its extension to
a Kac-Moody-like algebra. Covariance properties and tensorial composition of
spinors under are defined. The relation to Malcev algebras is
established. The consequences for octonionic projective spaces are examined.
Current algebras are formulated and their anomalies are derived, and shown to
be unique (even regarding numerical coefficients) up to redefinitions of the
currents. Nilpotency of the BRST operator is consistent with one particular
expression in the class of (field-dependent) anomalies. A Sugawara construction
is given.Comment: 22 pages. Macropackages used: phyzzx, epsf. Three epsf figure files
appende