443,795 research outputs found
On the regularity over positively graded algebras
We study the relationship between the Tor-regularity and the local-regularity
over a positively graded algebra defined over a field which coincide if the
algebra is a standard graded polynomial ring. In this case both are
characterizations of the so-called Castelnuovo--Mumford regularity. Moreover,
we can characterize a standard graded polynomial ring as an algebra with
extremal properties with respect to the Tor- and the local-regularity. For
modules of finite projective dimension we get a nice formula relating the two
regularity notions. Interesting examples are given to help to understand the
relationship between the Tor- and the local-regularity in general.Comment: 13 pages; Revised version of the pape
About [q]-regularity properties of collections of sets
We examine three primal space local Hoelder type regularity properties of
finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and
uniform [q]-regularity as well as their quantitative characterizations.
Equivalent metric characterizations of the three mentioned regularity
properties as well as a sufficient condition of [q]-subregularity in terms of
Frechet normals are established. The relationships between [q]-regularity
properties of collections of sets and the corresponding regularity properties
of set-valued mappings are discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1309.700
Szemer\'edi's Regularity Lemma for matrices and sparse graphs
Szemer\'edi's Regularity Lemma is an important tool for analyzing the
structure of dense graphs. There are versions of the Regularity Lemma for
sparse graphs, but these only apply when the graph satisfies some local density
condition. In this paper, we prove a sparse Regularity Lemma that holds for all
graphs. More generally, we give a Regularity Lemma that holds for arbitrary
real matrices
Regularity of Stochastic Kinetic Equations
We consider regularity properties of stochastic kinetic equations with
multiplicative noise and drift term which belongs to a space of mixed
regularity (-regularity in the velocity-variable and Sobolev regularity in
the space-variable). We prove that, in contrast with the deterministic case,
the SPDE admits a unique weakly differentiable solution which preserves a
certain degree of Sobolev regularity of the initial condition without
developing discontinuities. To prove the result we also study the related
degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a
suitable stochastic flow
On almost distance-regular graphs
Distance-regular graphs are a key concept in Algebraic Combinatorics and have
given rise to several generalizations, such as association schemes. Motivated
by spectral and other algebraic characterizations of distance-regular graphs,
we study `almost distance-regular graphs'. We use this name informally for
graphs that share some regularity properties that are related to distance in
the graph. For example, a known characterization of a distance-regular graph is
the invariance of the number of walks of given length between vertices at a
given distance, while a graph is called walk-regular if the number of closed
walks of given length rooted at any given vertex is a constant. One of the
concepts studied here is a generalization of both distance-regularity and
walk-regularity called -walk-regularity. Another studied concept is that of
-partial distance-regularity or, informally, distance-regularity up to
distance . Using eigenvalues of graphs and the predistance polynomials, we
discuss and relate these and other concepts of almost distance-regularity, such
as their common generalization of -walk-regularity. We introduce the
concepts of punctual distance-regularity and punctual walk-regularity as a
fundament upon which almost distance-regular graphs are built. We provide
examples that are mostly taken from the Foster census, a collection of
symmetric cubic graphs. Two problems are posed that are related to the question
of when almost distance-regular becomes whole distance-regular. We also give
several characterizations of punctually distance-regular graphs that are
generalizations of the spectral excess theorem
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