18,183 research outputs found
Characteristic of partition-circuit matroid through approximation number
Rough set theory is a useful tool to deal with uncertain, granular and
incomplete knowledge in information systems. And it is based on equivalence
relations or partitions. Matroid theory is a structure that generalizes linear
independence in vector spaces, and has a variety of applications in many
fields. In this paper, we propose a new type of matroids, namely,
partition-circuit matroids, which are induced by partitions. Firstly, a
partition satisfies circuit axioms in matroid theory, then it can induce a
matroid which is called a partition-circuit matroid. A partition and an
equivalence relation on the same universe are one-to-one corresponding, then
some characteristics of partition-circuit matroids are studied through rough
sets. Secondly, similar to the upper approximation number which is proposed by
Wang and Zhu, we define the lower approximation number. Some characteristics of
partition-circuit matroids and the dual matroids of them are investigated
through the lower approximation number and the upper approximation number.Comment: 12 page
Bitangents of tropical plane quartic curves
We study smooth tropical plane quartic curves and show that they satisfy
certain properties analogous to (but also different from) smooth plane quartics
in algebraic geometry. For example, we show that every such curve admits either
infinitely many or exactly 7 bitangent lines. We also prove that a smooth
tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in
Mathematische Zeitschrif
Diffusivity of a walk on fracture loops of a discrete torus
In this paper we study functions on the discrete torus which have a
crystalline structure. This means that if we fix such a function and walk
around the torus in a positive direction, the function increases on almost
every step, except at a small number of steps where it must go down in order to
meet the periodicity of the torus. It turns out that the down steps are
organised into a small number of closed simple disjoint paths, the fracture
lines of the crystal. We define a random walk on the resulting functions, the
law of which is Brownian in the diffusive limit. We show that in the limit of
the crystal structure becoming microscopic, the diffusivity is given by
, where
and are the number of fractures in the horizontal
and vertical direction respectively. This is the main result of this paper. The
diffusivity of the corresponding one-dimensional model has already been studied
by Espinasse, Guillotin-Plantard and Nadeau, and this paper generalises that
model to two dimensions. However, the methodology involving an analysis of the
fracture lines that we use to calculate the diffusivity is completely novel
Invariant measure for the stochastic Cauchy problem driven by a cylindrical L\'evy process
In this work, we present sufficient conditions for the existence of a
stationary solution of an abstract stochastic Cauchy problem driven by an
arbitrary cylindrical L\'evy process, and show that these conditions are also
necessary if the semigroup is stable, in which case the invariant measure is
unique. For typical situations such as the heat equation, we significantly
simplify these conditions without assuming any further restrictions on the
driving cylindrical L\'evy process and demonstrate their application in some
examples
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A weighty theorem of the heart for the algebraic K-theory of higher categories
We introduce the notion of a bounded weight structure on a stable [infinity symbol]-category and prove a generalization of Waldhausen’s sphere theorem for the algebraic K-theory of higher categories. The algebraic K-theory of a stable [infinity symbol]-category with a bounded non-degenerate weight structure is proven to be equivalent to the algebraic K-theory of the heart of the weight structure. We relate this theorem to previous results as well as new applications.Mathematic
Representation Homology, Lie Algebra Cohomology and Derived Harish-Chandra Homomorphism
We study the derived representation scheme DRep_n(A) parametrizing the
n-dimensional representations of an associative algebra A over a field of
characteristic zero. We show that the homology of DRep_n(A) is isomorphic to
the Chevalley-Eilenberg homology of the current Lie coalgebra gl_n^*(C) defined
over a Koszul dual coalgebra of A. We extend this isomorphism to representation
schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we
define the derived affine scheme DRep_g(a) parametrizing the representations
(in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic
to the Chevalley-Eilenberg homology of the Lie coalgebra g^*(C), where C is a
cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a
canonical DG algebra map \Phi_g(a) : DRep_g(a)^G -> DRep_h(a)^W, which is a
homological extension of the classical restriction homomorphism. We call
\Phi_g(a) a derived Harish-Chandra homomorphism. We conjecture that, for a
two-dimensional abelian Lie algebra a, the derived Harish-Chandra homomorphism
is a quasi-isomorphism, and provide some evidence for this conjecture. For any
complex Lie algebra g, we compute the Euler characteristic of DRep_g(a)^G in
terms of matrix integrals over G and compare it to the Euler characteristic of
DRep_h(a)^W.This yields an interesting combinatorial identity, which we prove
for gl_n and sl_n (for all n). Our identity is analogous to the classical
Macdonald identity, and our quasi-isomorphism conjecture is analogous to the
strong Macdonald conjecture proved by S.Fishel, I.Grojnowski and C.Teleman. We
explain this analogy by giving a new homological interpretation of Macdonald's
conjectures in terms of derived representation schemes, parallel to our
Harish-Chandra quasi-isomorphism conjecture.Comment: 61 pages; minor correction
Grothendieck Rings of Theories of Modules
The model-theoretic Grothendieck ring of a first order structure, as defined
by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the
definable subsets of finite powers of the structure. In this paper we compute
the Grothendieck ring, , of a right -module , where
is any unital ring. As a corollary we prove a conjecture of Prest
that is non-trivial, whenever is non-zero. The main proof uses
various techniques from the homology theory of simplicial complexes.Comment: 42 Page
Canonical matrices for linear matrix problems
We consider a large class of matrix problems, which includes the problem of
classifying arbitrary systems of linear mappings. For every matrix problem from
this class, we construct Belitskii's algorithm for reducing a matrix to a
canonical form, which is the generalization of the Jordan normal form, and
study the set C(m,n) of indecomposable canonical m-by-n matrices. Considering
C(m,n) as a subset in the affine space of m-by-n matrices, we prove that either
C(m,n) consists of a finite number of points and straight lines for every
(m,n), or C(m,n) contains a 2-dimensional plane for a certain (m,n).Comment: 59 page
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