223 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
Parametric matroid of rough set
Rough set is mainly concerned with the approximations of objects through an
equivalence relation on a universe. Matroid is a combinatorial generalization
of linear independence in vector spaces. In this paper, we define a parametric
set family, with any subset of a universe as its parameter, to connect rough
sets and matroids. On the one hand, for a universe and an equivalence relation
on the universe, a parametric set family is defined through the lower
approximation operator. This parametric set family is proved to satisfy the
independent set axiom of matroids, therefore it can generate a matroid, called
a parametric matroid of the rough set. Three equivalent representations of the
parametric set family are obtained. Moreover, the parametric matroid of the
rough set is proved to be the direct sum of a partition-circuit matroid and a
free matroid. On the other hand, since partition-circuit matroids were well
studied through the lower approximation number, we use it to investigate the
parametric matroid of the rough set. Several characteristics of the parametric
matroid of the rough set, such as independent sets, bases, circuits, the rank
function and the closure operator, are expressed by the lower approximation
number.Comment: 15 page
Covering matroid
In this paper, we propose a new type of matroids, namely covering matroids,
and investigate the connections with the second type of covering-based rough
sets and some existing special matroids. Firstly, as an extension of
partitions, coverings are more natural combinatorial objects and can sometimes
be more efficient to deal with problems in the real world. Through extending
partitions to coverings, we propose a new type of matroids called covering
matroids and prove them to be an extension of partition matroids. Secondly,
since some researchers have successfully applied partition matroids to
classical rough sets, we study the relationships between covering matroids and
covering-based rough sets which are an extension of classical rough sets.
Thirdly, in matroid theory, there are many special matroids, such as
transversal matroids, partition matroids, 2-circuit matroid and
partition-circuit matroids. The relationships among several special matroids
and covering matroids are studied.Comment: 15 page
Some characteristics of matroids through rough sets
At present, practical application and theoretical discussion of rough sets
are two hot problems in computer science. The core concepts of rough set theory
are upper and lower approximation operators based on equivalence relations.
Matroid, as a branch of mathematics, is a structure that generalizes linear
independence in vector spaces. Further, matroid theory borrows extensively from
the terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
Matroid connectivity and singularities of configuration hypersurfaces
Consider a linear realization of a matroid over a field. One associates with
it a configuration polynomial and a symmetric bilinear form with linear
homogeneous coefficients. The corresponding configuration hypersurface and its
non-smooth locus support the respective first and second degeneracy scheme of
the bilinear form. We show that these schemes are reduced and describe the
effect of matroid connectivity: for (2-)connected matroids, the configuration
hypersurface is integral, and the second degeneracy scheme is reduced
Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the
second degeneracy scheme is integral. In the process, we describe the behavior
of configuration polynomials, forms and schemes with respect to various matroid
constructions.Comment: 64 pages, 4 figure
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