276,000 research outputs found
On the equivalence of linear sets
Let be a linear set of pseudoregulus type in a line in
, or . We provide examples of
-order canonical subgeometries such
that there is a -space with the property that for , is the projection
of from center and there exists no collineation of
such that and .
Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes
Cryptogr. 56:89-104, 2010) states the existence of a collineation between the
projecting configurations (each of them consisting of a center and a
subgeometry), which give rise by means of projections to two linear sets. It
follows from our examples that this condition is not necessary for the
equivalence of two linear sets as stated there. We characterize the linear sets
for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented.
The final version is to appear in Designs, Codes and Cryptograph
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
Rough sets and matroidal contraction
Rough sets are efficient for data pre-processing in data mining. As a
generalization of the linear independence in vector spaces, matroids provide
well-established platforms for greedy algorithms. In this paper, we apply rough
sets to matroids and study the contraction of the dual of the corresponding
matroid. First, for an equivalence relation on a universe, a matroidal
structure of the rough set is established through the lower approximation
operator. Second, the dual of the matroid and its properties such as
independent sets, bases and rank function are investigated. Finally, the
relationships between the contraction of the dual matroid to the complement of
a single point set and the contraction of the dual matroid to the complement of
the equivalence class of this point are studied.Comment: 11 page
Estimating Maximally Probable Constrained Relations by Mathematical Programming
Estimating a constrained relation is a fundamental problem in machine
learning. Special cases are classification (the problem of estimating a map
from a set of to-be-classified elements to a set of labels), clustering (the
problem of estimating an equivalence relation on a set) and ranking (the
problem of estimating a linear order on a set). We contribute a family of
probability measures on the set of all relations between two finite, non-empty
sets, which offers a joint abstraction of multi-label classification,
correlation clustering and ranking by linear ordering. Estimating (learning) a
maximally probable measure, given (a training set of) related and unrelated
pairs, is a convex optimization problem. Estimating (inferring) a maximally
probable relation, given a measure, is a 01-linear program. It is solved in
linear time for maps. It is NP-hard for equivalence relations and linear
orders. Practical solutions for all three cases are shown in experiments with
real data. Finally, estimating a maximally probable measure and relation
jointly is posed as a mixed-integer nonlinear program. This formulation
suggests a mathematical programming approach to semi-supervised learning.Comment: 16 page
On the Number of Membranes in Unary P Systems
We consider P systems with a linear membrane structure working on objects
over a unary alphabet using sets of rules resembling homomorphisms. Such a
restricted variant of P systems allows for a unique minimal representation of
the generated unary language and in that way for an effective solution of the
equivalence problem. Moreover, we examine the descriptional complexity of unary
P systems with respect to the number of membranes
Hierarchy of Conservation Laws of Diffusion--Convection Equations
We introduce notions of equivalence of conservation laws with respect to Lie
symmetry groups for fixed systems of differential equations and with respect to
equivalence groups or sets of admissible transformations for classes of such
systems. We also revise the notion of linear dependence of conservation laws
and define the notion of local dependence of potentials. To construct
conservation laws, we develop and apply the most direct method which is
effective to use in the case of two independent variables. Admitting
possibility of dependence of conserved vectors on a number of potentials, we
generalize the iteration procedure proposed by Bluman and Doran-Wu for finding
nonlocal (potential) conservation laws. As an example, we completely classify
potential conservation laws (including arbitrary order local ones) of
diffusion--convection equations with respect to the equivalence group and
construct an exhaustive list of locally inequivalent potential systems
corresponding to these equations.Comment: 24 page
Subgeometries and linear sets on a projective line
We define the splash of a subgeometry on a projective line, extending the
definition of \cite{BaJa13} to general dimension and prove that a splash is
always a linear set. We also prove the converse: each linear set on a
projective line is the splash of some subgeometry. Therefore an alternative
description of linear sets on a projective line is obtained. We introduce the
notion of a club of rank , generalizing the definition from \cite{FaSz2006},
and show that clubs correspond to tangent splashes. We determine the condition
for a splash to be a scattered linear set and give a characterization of clubs,
or equivalently of tangent splashes. We also investigate the equivalence
problem for tangent splashes and determine a necessary and sufficient condition
for two tangent splashes to be (projectively) equivalent
Basis Functions for Linear-Scaling First-Principles Calculations
In the framework of a recently reported linear-scaling method for
density-functional-pseudopotential calculations, we investigate the use of
localized basis functions for such work. We propose a basis set in which each
local orbital is represented in terms of an array of `blip functions'' on the
points of a grid. We analyze the relation between blip-function basis sets and
the plane-wave basis used in standard pseudopotential methods, derive criteria
for the approximate equivalence of the two, and describe practical tests of
these criteria. Techniques are presented for using blip-function basis sets in
linear-scaling calculations, and numerical tests of these techniques are
reported for Si crystal using both local and non-local pseudopotentials. We
find rapid convergence of the total energy to the values given by standard
plane-wave calculations as the radius of the linear-scaling localized orbitals
is increased.Comment: revtex file, with two encapsulated postscript figures, uses epsf.sty,
submitted to Phys. Rev.
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
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