23,939 research outputs found
On the Lattice of Intervals and Rough Sets
Rough sets, developed by Pawlak [6], are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].Grabowski Adam - Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandJastrzębska Magdalena - Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, PolandGrzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004.Amin Mousavi and Parviz Jabedar-Maralani. Relative sets and rough sets. Int. J. Appl. Math. Comput. Sci., 11(3):637-653, 2001.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Z. Pawlak. Rough sets. International Journal of Parallel Programming, 11:341-356, 1982, doi:10.1007/BF01001956.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Y. Y. Yao. Interval-set algebra for qualitative knowledge representation. Proc. 5-th Int. Conf. Computing and Information, pages 370-375, 1993.Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990
Mean lattice point discrepancy bounds, II: Convex domains in the plane
We consider planar curved strictly convex domains with no or very weak
smoothness assumptions and prove sharp bounds for square-functions associated
to the lattice point discrepancy.Comment: Revised version, to appear in Journal d'Analyse Mathematiqu
Approximations from Anywhere and General Rough Sets
Not all approximations arise from information systems. The problem of fitting
approximations, subjected to some rules (and related data), to information
systems in a rough scheme of things is known as the \emph{inverse problem}. The
inverse problem is more general than the duality (or abstract representation)
problems and was introduced by the present author in her earlier papers. From
the practical perspective, a few (as opposed to one) theoretical frameworks may
be suitable for formulating the problem itself. \emph{Granular operator spaces}
have been recently introduced and investigated by the present author in her
recent work in the context of antichain based and dialectical semantics for
general rough sets. The nature of the inverse problem is examined from
number-theoretic and combinatorial perspectives in a higher order variant of
granular operator spaces and some necessary conditions are proved. The results
and the novel approach would be useful in a number of unsupervised and semi
supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings,
Springe
Stick-slip statistics for two fractal surfaces: A model for earthquakes
Following the observations of the self-similarity in various length scales in
the roughness of the fractured solid surfaces, we propose here a new model for
the earthquake. We demonstrate rigorously that the contact area distribution
between two fractal surfaces follows an unique power law. This is then utilised
to show that the elastic energy releases for slips between two rough fractal
surfaces indeed follow a Guttenberg-Richter like power law.Comment: 9 pages (Latex), 4 figures (postscript
Order-by-disorder in classical oscillator systems
We consider classical nonlinear oscillators on hexagonal lattices. When the
coupling between the elements is repulsive, we observe coexisting states, each
one with its own basin of attraction. These states differ by their degree of
synchronization and by patterns of phase-locked motion. When disorder is
introduced into the system by additive or multiplicative Gaussian noise, we
observe a non-monotonic dependence of the degree of order in the system as a
function of the noise intensity: intervals of noise intensity with low
synchronization between the oscillators alternate with intervals where more
oscillators are synchronized. In the latter case, noise induces a higher degree
of order in the sense of a larger number of nearly coinciding phases. This
order-by-disorder effect is reminiscent to the analogous phenomenon known from
spin systems. Surprisingly, this non-monotonic evolution of the degree of order
is found not only for a single interval of intermediate noise strength, but
repeatedly as a function of increasing noise intensity. We observe noise-driven
migration of oscillator phases in a rough potential landscape.Comment: 12 pages, 13 figures; comments are welcom
Modelling potential movement in constrained travel environments using rough space-time prisms
The widespread adoption of location-aware technologies (LATs) has afforded analysts new opportunities for efficiently collecting trajectory data of moving individuals. These technologies enable measuring trajectories as a finite sample set of time-stamped locations. The uncertainty related to both finite sampling and measurement errors makes it often difficult to reconstruct and represent a trajectory followed by an individual in space-time. Time geography offers an interesting framework to deal with the potential path of an individual in between two sample locations. Although this potential path may be easily delineated for travels along networks, this will be less straightforward for more nonnetwork-constrained environments. Current models, however, have mostly concentrated on network environments on the one hand and do not account for the spatiotemporal uncertainties of input data on the other hand. This article simultaneously addresses both issues by developing a novel methodology to capture potential movement between uncertain space-time points in obstacle-constrained travel environments
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