18,083 research outputs found
Convergence of nonlinear semigroups under nonpositive curvature
The present paper is devoted to semigroups of nonexpansive mappings on metric
spaces of nonpositive curvature. We show that the Mosco convergence of a
sequence of convex lsc functions implies convergence of the corresponding
resolvents and convergence of the gradient flow semigroups. This extends the
classical results of Attouch, Brezis and Pazy into spaces with no linear
structure. The same method can be further used to show the convergence of
semigroups on a sequence of spaces, which solves a problem of [Kuwae and
Shioya, Trans. Amer. Math. Soc., 2008].Comment: Accepted for publication in Trans. Amer. Math. So
Ultrafilter extensions of linear orders
It was recently shown that arbitrary first-order models canonically extend to
models (of the same language) consisting of ultrafilters. The main precursor of
this construction was the extension of semigroups to semigroups of
ultrafilters, a technique allowing to obtain significant results in algebra and
dynamics. Here we consider another particular case where the models are
linearly ordered sets. We explicitly calculate the extensions of a given linear
order and the corresponding operations of minimum and maximum on a set. We show
that the extended relation is not more an order however is close to the natural
linear ordering of nonempty half-cuts of the set and that the two extended
operations define a skew lattice structure on the set of ultrafilters
Nonlinear Markov semigroups and interacting Lévy type processes
Semigroups of positivity preserving linear operators on measures of a measurable space describe the evolutions of probability distributions of Markov processes on . Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions on describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such semigroups is given in case when is the Euclidean space (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation model
Maximal subgroups of free idempotent generated semigroups over the full linear monoid
We show that the rank r component of the free idempotent generated semigroup
of the biordered set of the full linear monoid of n x n matrices over a
division ring Q has maximal subgroup isomorphic to the general linear group
GL_r(Q), where n and r are positive integers with r < n/3.Comment: 37 pages; Transactions of the American Mathematical Society (to
appear). arXiv admin note: text overlap with arXiv:1009.5683 by other author
Effective dimension of finite semigroups
In this paper we discuss various aspects of the problem of determining the
minimal dimension of an injective linear representation of a finite semigroup
over a field. We outline some general techniques and results, and apply them to
numerous examples.Comment: To appear in J. Pure Appl. Al
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