Search

Multifunctions are discrete functions defined on a finite set and returning as their values all subsets of the considered set. The paper considers the classification of multifunctions defined on a two-element set with respect to the E-closure operator. E-closed sets of multifunctions are sets that are closed under superposition, the closure operator with branching by the equality predicate, the identification of variables, and the addition of dummy variables. The concept of separating sets was introduced using a greedy algorithm for the set covering problem, and 22 classes of separating sets were obtained. It is shown that the classification under consideration leads to a finite set of closed classes. The work describes all 359 E-closed classes of multifunctions, among which there are 138 pairs of dual classes and 83 self-dual classes. For each class consisting only of multifunctions, its generating system is indicated.

In recent years, the direction associated with the study of maps from a finite set A to the set of all subsets of the set A, including the empty one, has been intensively developing. Such mappings are called multifunctions on A, as well as hyperfunctions on A, if an empty subset is excluded from the subsets under consideration. It is not difficult to see that the so-called undefined or undefined functions, which are studied in many works, are most directly related to this field of research. The power of the set A is called the rank of multifunction or hyperfunction. Obviously, multifunctions and hyperfunctions generalize well-known functions of k-valued logic, however, it should be noted that the usual superposition of functions of k-valued logic is not suitable for multifunctions and hyperfunctions. Two types of superpositions are most often considered here, one of them leads to sets closed relative to the superposition, which are called multiclones and hyperclones, and for the second type of superposition, closed sets are called ultraclones and partial ultraclones. In this article, the elements of the rank 2 ultraclone lattice are considered. By now, all the maximum and minimum elements of this lattice are known. For example, Panteleev V.I. described all maximal ultraclones in the predicate language, which allowed us to prove the completeness criterion of an arbitrary system of hyperfunctions of rank 2. We managed to prove the completeness criterion in the maximal ultraclone of linear hyperfunctions of rank 2. Thus, all submaximal ultraclones of linear hyperfunctions are described.

In this research, we define a novel concept termed neutrosophic soft metric space as well as in vestigate its fundamental characteristics. Additionally, in neutrosophic soft metric spaces, we present various topological features of this newly developed space, such as the soft open sphere and the soft closed sphere. The neutrosophic soft metric space has already been studied in depth, and its topological as well as structural features have been mapped out.

In this research, we define a novel concept termed neutrosophic soft metric space as well as investigate its fundamental characteristics. Additionally, in neutrosophic soft metric spaces, we present various topological features of this newly developed space, such as the soft open sphere and the soft closed sphere. The neutrosophic soft metric space has already been studied in depth, and its topological as well as structural features have been mapped out. [ABSTRACT FROM AUTHOR]

The concept of ⋆-metric, based on the relaxation of triangle inequality of metric axioms by using a t-definer, was introduced by Khatami and Mirzavaziri. This paper extends and generalizes some well-known results of classical metric space. Considering the definition of ⋆-metric space, it studies the notion of a closed ball. The paper proves some results related to closed sets, convergent sequences, Cauchy sequences, and the diameter of a set. This paper contains the study on the metrizability of ⋆-metric space and provides an alternative approach to the proof of metrizability for ⋆-metric space using the famous 'Niemytski and Wilson's metrization theorem'. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new classes of sets called E-closed sets and Eα-closed sets in topological spaces and we study some of its basic properties. This class lies between the class of closed sets and the class of g-closed sets. [ABSTRACT FROM AUTHOR]

Visual counting, a task that aims to estimate the number of objects from an image/video, is an open-set problem by nature as the number of population can vary in [ 0 , + ∞) in theory. However, collected data are limited in reality, which means that only a closed set is observed. Existing methods typically model this task through regression, while they are prone to suffer from unseen scenes with counts out of the scope of the closed set. In fact, counting has an interesting and exclusive property—spatially decomposable. A dense region can always be divided until sub-region counts are within the previously observed closed set. We therefore introduce the idea of spatial divide-and-conquer (S-DC) that transforms open-set counting into a closed set problem. This idea is implemented by a novel Supervised Spatial Divide-and-Conquer Network (SS-DCNet). It can learn from a closed set but generalize to open-set scenarios via S-DC. We provide mathematical analyses and a controlled experiment on synthetic data, demonstrating why closed-set modeling works well. Experiments show that SS-DCNet achieves state-of-the-art performance in crowd counting, vehicle counting and plant counting. SS-DCNet also demonstrates superior transferablity under the cross-dataset setting. Code and models are available at: https://git.io/SS-DCNet. [ABSTRACT FROM AUTHOR]

Ideas of defining non-manifold topologies on flat and curved spacetime taking into account the causal structure of spacetime, paved the way for the use of topology in general relativity. The A, f, s, fine, t and space topologies are some of the topologies defined on Makowski space and the path topology, space topology, separating topology, Zeeman topology and geodesic topology are some of the non-manifold topologies defined on Lorentz manifolds with a reasonable homeomorphism group making these topologies physically significant and interesting for their mathematical study. New line In the present paper, it is obtained that the Makowski space with the A-topology is Harsdorf, path connected, separable, non-first countable, non-regular, non-compact and non-simply connected. It is also found that a set is compact in Makowski space with the A-topology if and only if it is compact in the Euclidean space and does not contain the image of a Zeno sequence. The A-topology on the Makowski space has been generalized to Lorentz manifolds and studied its compact sets and topological properties. New line Further, it is obtained that the homeomorphism group of Makowski space with the f-topology is the group generated by Lorentz transformations together with translations and dilatations. [ABSTRACT FROM AUTHOR]

The main intention of this paper is to develop the idea of Neutrosophic Semi-generalized pre-closed set in neutrosophic topological space. We also study relations and some properties between the existing Neutrosophic closed set. The examples are provided wherever necessary. Besides, we discuss some applications of Neutrosophic Semi-generalized pre closed set.

The problem of instability of a closed positively invariant set M of a semi-dynamical system on an arbitrary metric space X is considered. The Lyapunov’s direct method for such problems has been developed quite completely in the case when M is compact and X is locally compact. In this article, we obtain sufficient conditions for instability in terms of Lyapunov functions in two situations: M has a neighborhood of positive Lagrange stable semi-trajectories; the space X is asymptotically compact in some neighborhood of M.

This paper studies the regularity of the extensions of a double fuzzy topological space. The interrelations among certain families of closed sets in a double fuzzy topological space and its extensions are also investigated. [ABSTRACT FROM AUTHOR]

In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space R n, n ą 1, known as weakly mconvex, m = 1, . . ., n ´ 1. An open set of R n is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of R n is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of R n to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n ´ 1)-convex set in R n with non-empty set of (n ´ 1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in R n with non-empty set of mnonconvexity points are constructed for any n ą 2 and any m = 1, . . ., n´2. [ABSTRACT FROM AUTHOR]

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is 129. All types of bases are described and it is proved that the maximum cardinality of a basis is 4.

Attribute reduction and function selection have end up one of the essential steps for pattern recognition and machine getting to know responsibilities. The purpose of attribute reduction is to discover a minimal attribute subset that satisfies a few particular criteria, even as the minimal attribute subset is referred to as attribute reduce. In this paper a pretopological space in an information system is to be constructed, as well as defining some essential operations for the pretopological space and a brief study of its properties will be presented. Moreover, we use similarity method by using pretopological space' matrix to reduce the attribute. We will compare the pretopologies resulting after the removing of the attributes with the original pretopology resulting from the information system. [ABSTRACT FROM AUTHOR]

Multioperations are operations from a finite set A to set of all subsets of A. The usual composition operator leads to a continuum of closed sets. Therefore, the research of closure operators, which contain composition and other operations becomes necessary. In the paper, the closure of multioperations that can be obtained using the operations of adding dummy variables, identifying variables, composition operator, and operator with the equality predicate branching is studied. We obtain eleven precomplete closed classes of multioperations of rank 2 and prove the completeness criterion. The diagram of inclusions for one of the precomplete class is presented.

Faced with both identity theft and the theft of means of authentication, users of digital services are starting to look rather suspiciously at online systems. To increase access security it is necessary to introduce some new factor of implicit authentication such as user behavior analysis. A behavior is made up of a series of observable actions and taken as a whole, the most frequent of these actions amount to habit. The challenge is to detect identity theft as quickly as possible and, reciprocally, to validate a legitimate identity for as long as possible. To take up this challenge, we introduce in this paper a closed set-based learning classifier. This classifier is inspired by classification in concept lattices from positive and negative examples and several works on emerging patterns. We also rely on the tf-idf parameter used in the context of information retrieval. We propose three heuristics named H t f − i d f c , H s u p c and H s u p M i n c to select closed patterns for each class to be described. To compute performance of our models we have followed an experimental protocol described in a previous study which had the same purpose. Then, we compared the results from our own dataset of web navigation connection logs of 3,000 users over a six-month period with the heuristic H s u p introduced in this study. Moreover, to strengthen our analysis, we have designed and set up one model based on the naive Bayes classifier to be used as a reference statistical tool. [ABSTRACT FROM AUTHOR]

In this paper , we introduces a new definition in bi-supra topological space , called M-θ- ii- open and via this definition , we introduce a new types of functions called quasi M-θ-ii-continuous functions which unifies some weak forms of quasi θ-ii-continuous functions in bi-supra topological spaces and investigate their properties.

The research of multifunctions is a one of the directions in discrete function’s investigations. Multifunction is a discrete function from a finite set A to all subsets of A. The set of hyperfunctions is a subset of set of multifunctions. Hyperfunction is a discrete function from a finite set A to all nonempty subsets of A. The subset relation of hyperfunctions to maximal multiclones is a relation of equivalence and according it all hyperfunctions are divided into equivalence classes. Based on this equivalence it is possible to estimate the cardinality of all possible bases, calculate the number of different types of bases of the same cardinality, and construct some intervals of the clone lattice. Knowing the number of maximal clones, we can obtain upper-bound estimate of the number of such classes as the cardinality of the set of all subsets of the set of maximal clones. It possible to lower this estimate by using properties of the hyperfunctions . A lower-bound estimate for the number of classes can be obtained by constructing the corresponding classes. It is noteworthy that the complexity of the problem of describing all equivalence classes increases essentially with the cardinality of the set of maximal clones increasing. This paper considers multifunctions on a two-element set. The number of maximal multiclones equals to 15 (Panteleyev V.I., 2009). The primary purpose of this research is describing the classification of the set of hyperfunctions into equivalence classes. As Kazimirov A.S. and Panteleyev V.I. showed there are 18 equivalence classes on the set of Boolean functions, which is a subset of the set of hyperfunctions. In this paper were obtained special properties of hyperfunctions and were described all the equivalence classes generated by functions of 3 arguments having applied a computer experiment. The obtained results made it possible to show that the subset relation of hyperfunctions to maximal multiclones splits the set of all hyperfunctions into 67 equivalence classes.

Abstract This note is a somewhat personal account of a paper that L.E.J. Brouwer published in 1908 and that dealt with the possible cardinalities of subsets of the continuum. That paper is of interest because it represents the first time that Brouwer presented his ideas on foundations in an international forum. I found Brouwer's notions and arguments at times hard to grasp if not occasionally perplexing. I hope that this note contributes to a further discussion of the definitions and reasonings as presented in Brouwer's paper. [ABSTRACT FROM AUTHOR]

Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi(x) = xi. [ABSTRACT FROM AUTHOR]

In this work we consider the closure operator with the equality predicate branching (E-operator) on the set of hyperfunctions on two-element set. With respect to this operator closed classes of hyperfunctions are generated. We show that there are four submaximal classes and prove the criterion of functional completeness. The relation of equivalence is considered on the set of hyperfunctions obtained by their membership in E-submaximal classes. All hyperfunctions are divided into 14 equivalence classes. In closed sets the minimal closed subsets named basis are derived. We show that basis of hyperfunctions can have cardinality from 1 to 3 and there is no basis with cardinality more than 3. There is only one kind of basis with cardinality 1. The function from that basis does not belong to any of four E-submaximal classes. We obtain 23 kind of basis with cardinality 2 and 11 with cardinality 3.

In this article, we introduce the notion of generalized closed sets and generalized open sets in multiset topological spaces. We investigate their different properties. We have introduced the notion of some separation axioms and discussed some examples. [ABSTRACT FROM AUTHOR]

Pushpalatha and K. Anitha [1] introduce a new concept of functions which is g*s-closed function in topological spaces. In this paper, we modify this concept and study the properties of some types of these functions.