172,222 research outputs found
Ultrafilter extensions of linear orders
It was recently shown that arbitrary first-order models canonically extend to
models (of the same language) consisting of ultrafilters. The main precursor of
this construction was the extension of semigroups to semigroups of
ultrafilters, a technique allowing to obtain significant results in algebra and
dynamics. Here we consider another particular case where the models are
linearly ordered sets. We explicitly calculate the extensions of a given linear
order and the corresponding operations of minimum and maximum on a set. We show
that the extended relation is not more an order however is close to the natural
linear ordering of nonempty half-cuts of the set and that the two extended
operations define a skew lattice structure on the set of ultrafilters
Development of the fuzzy sets theory: weak operations and extension principles
The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α -cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α -weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α -cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information
Development of the fuzzy sets theory: weak operations and extension principles
The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α-cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α-weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α-cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information
A linear time algorithm for a variant of the max cut problem in series parallel graphs
Given a graph , a connected sides cut or
is the set of edges of E linking all vertices of U to all vertices
of such that the induced subgraphs and are connected. Given a positive weight function defined on , the
maximum connected sides cut problem (MAX CS CUT) is to find a connected sides
cut such that is maximum. MAX CS CUT is NP-hard. In this
paper, we give a linear time algorithm to solve MAX CS CUT for series parallel
graphs. We deduce a linear time algorithm for the minimum cut problem in the
same class of graphs without computing the maximum flow.Comment: 6 page
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