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We study some nil-ai-semiring varieties. We establish a model for the free object in the variety FC generated by all commutative flat semirings. Also, we provide two sufficient conditions under which a finite ai-semiring is nonfinitely based. As a consequence, we show that the power semiring P S ˙ c (W) of the finite nil-semigroup S ˙ c (W) is nonfinitely based, where W is a finite set of words in the free commutative semigroup X c + over an alphabet X, whenever the maximum of lengths of words in W is k ≥ 3 and W does not contain the kth power of a letter. This partially answers a problem raised by Jackson et al. (J Algebr 611: 211–245, 2022). [ABSTRACT FROM AUTHOR]

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]

We develop a statistical approach to model handwriting that accommodates all styles of writing (cursive, print, connected print). The goal is to compute a posterior probability of writership of a questioned document given a closed set of candidate writers. Such probabilistic statements can support examiner conclusions and enable a quantitative forensic evaluation of handwritten documents. Writing is treated as a sequence of disjoint graphical structures, which are extracted using an automated and open‐source process. The graphs are grouped based on the similarity of their shapes through a K‐means clustering template. A person's writing pattern can be characterized by the rate at which graphs are emitted to each cluster. The cluster memberships serve as data for a Bayesian hierarchical model with a mixture component. The rate of mixing between two parameters in the hierarchy indicates writing style. [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]

In this paper, we study a robust optimization problem whose constraints include nonsmooth and nonconvex functions and the intersection of closed sets. Using advanced variational analysis tools, we first provide necessary conditions for the optimality of the robust optimization problem. We then establish sufficient conditions for the optimality of the considered problem under the assumption of generalized convexity. In addition, we present a dual problem to the primal robust optimization problem and examine duality relations. [ABSTRACT FROM AUTHOR]

The this paper, we introduce a new class of closed sets called Beta generalized closed set in topological spaces is to describe and analyze the basics of the set of closed sets, especially Beta generalized closed set in topological spaces. We also study their properties with few existing examples of closed sets, along with their characterizations. [ABSTRACT FROM AUTHOR]

The present work considers properties of generally convex sets in the n-dimensional real Euclidean space ℝn, n > 1, known as weakly m-semiconvex, m = 1, 2, ... , n − 1. For all that, the subclass of not m-semiconvex sets is distinguished from the class of weakly m-semiconvex sets. A set of the space ℝn is called m-semiconvex if, for any point of the complement of the set to the whole space, there is an m-dimensional half-plane passing through this point and not intersecting the set. An open set of ℝn is called weaklym-semiconvex if, for any point of the boundary of the set, there exists an m-dimensional half-plane passing through this point and not intersecting the given set. A closed set of ℝn is called weaklym-semiconvex if it is approximated from the outside by a family of open weakly m-semiconvex sets. An example of a closed set with three connected components of the subclass of weakly 1-semiconvex but not 1-semiconvex sets in the plane is constructed. It is proved that this number of components is minimal for any closed set of the subclass. An example of a closed set of the subclass with a smooth boundary and four components is constructed. It is proved that this number of components is minimal for any closed, bounded set of the subclass having a smooth boundary and a not 1-semiconvex interior. It is also proved that the interior of a closed, weakly 1-semiconvex set with a finite number of components in the plane is weakly 1-semiconvex. Weakly m-semiconvex but not m-semiconvex domains and closed connected sets in ℝn are constructed for any n ≥ 3 and any m = 1, 2, ... , n − 2. [ABSTRACT FROM AUTHOR]

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]

We prove completeness theorems for the set of all d-closed d-bounded sets in a (q1, q2)-quasimetric space (X, d) equipped with suitable analogs of the Hausdorff distance. [ABSTRACT FROM AUTHOR]

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 make further investigation on I- convergence of filters and introduce I- closed sets in a topological space. We have used I- closed sets to determine topology on an arbitrary set. [ABSTRACT FROM AUTHOR]

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]

This paper generalizes the notion of antimatroids from the case that the ground set X is a finite set to the case that X is an arbitrary set. It not only deals with the relationships between geometries and independence spaces, but also demonstrates the relationships between convex geometries and arbitrary antimatroids. [ABSTRACT FROM AUTHOR]

This Letter generalises frequency‐domain oriented design guidelines for optimised single‐resonant three‐parameter AC current regulators for grid‐connected converters. Unlike in previously established set of design guidelines, possible practical restrictions imposed by relatively low‐switching frequency are taken into account in the proposed methodology. In addition, resonant term bandwidth is selected according to the desired robustness to fundamental frequency variations. As a result, a closed set of three non‐linear equations is created. The solution maximises the loop gain crossover frequency for a given phase margin while respecting frequency robustness and resonant gain restrictions, set a priori. [ABSTRACT FROM AUTHOR]

In this work, we present a notion of an (H, F)-closed set and prove the existence of a coupled coincidence point theorem for a pair {F,H} of mappings F,H : X × X → X with ϕ-contraction mappings in partially ordered metric spaces without H-increasing property of F and mixed monotone property of H. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by H using the mixed monotone property and H-increasing property of F. We also show the uniqueness of a coupled coincidence point of the given mappings. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mappings in partially ordered G-metric spaces with H-increasing property of F and mixed monotone property of H. These results generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest that it is complete coanalytic, but which turns out to be a rather simple Borel set. [ABSTRACT FROM AUTHOR]

The last years witnessed an explosive progress in networking, storage, and processing technologies resulting in an unprecedented amount of digitalization of data. Hence, there has been a considerable need for tools or techniques to delve and efficiently discover valuable, non-obvious information from large databases. In this situation, data mining is an important research field which offers efficient solutions for such an extraction. Much research in data mining from large databases have focused on the discovery of frequent patterns which are then used to identify relationships between sets of items in a database, through for example association rule derivation. In practice, however, the number of frequently occurring patterns is very large, hampering their effective exploitation by the end-users. In this situation, many works have been interested in defining manageably-sized sets of patterns, called concise representations, from which redundant patterns can be regenerated. In this paper, we concentrate on exact concise representations of frequent patterns. Thus, we describe their close relation with important concepts like the framework of ℇ-adequate representation and the minimum description length principle. Based on the mathematical settings of Formal Concept Analysis, we also show the complementarity between minimal generators and closed itemsets. Then, we focus on the key role played by these patterns for solving several problem associated to various pattern classes. In this respect, we classify concise representations of frequent itemsets according to their common characteristics. Then, we analyze a representative of each class and show its close link with minimal generators. Finally, we carry out a critical study of concise representations with respect to several aspects and comparative criteria which proves the importance of considering closed sets and minimal generators. [ABSTRACT FROM AUTHOR]

Abstract: In relational databases the original definition of a multivalued dependency is dependent on the underlying relation schema. In this context, the implication of multivalued dependencies has been characterised from multiple perspectives. Logically, it is equivalent to the logical implication of certain material implications in Boolean propositional logic. Proof-theoretically, the Chase procedure offers a convenient tool to decide implication. And algebraically, the implication can be characterised by the notion of closed attribute sets with respect to multivalued dependencies. The assumption of having a fixed underlying relation schema is not always feasible in practice, and also distinguishes multivalued dependencies from other classes of data dependencies. In this paper, we establish logical, proof-theoretical and algebraic characterisations for Biskupʼs notion of multivalued dependency implication over undetermined universes. That is, we unburden the current theory of the assumption of having a fixed underlying relation schema. From the perspective of probability theory this means that is unnecessary to fix the set of discrete probabilistic variables in order to utilise conditional independencies. [Copyright &y& Elsevier]

We continue our study of topological X-rays begun in Lynch [Topological X-rays and MRI's, iJMEST 33(3) (2002), pp. 389–392]. We modify our definition of a topological magnetic resonance imaging and give an affirmative answer to the question posed there: Can we identify a closed set in a box by defining X-rays to probe the interior and without looking inside the box? [ABSTRACT FROM PUBLISHER]

With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: 'If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed'. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the definition of a closure operator for a finite matroid to a pi-space and obtain the corresponding closure axioms. Then we discuss some properties of pi-spaces using the closure axioms and prove the non-existence for the dual of a pi-space. We also present some results on the automorphism group of a pi-space. [ABSTRACT FROM AUTHOR]

Abstract: Two topologies on the Levi-Civita field will be studied: the valuation topology induced by the order on the field, and another weaker topology induced by a family of seminorms, which we will call weak topology. We show that each of the two topologies results from a metric on , that the valuation topology is not a vector topology while the weak topology is, and that is complete in the valuation topology while it is not in the weak topology. Then the properties of both topologies will be studied in details; in particular, we give simple characterizations of open, closed, and compact sets in both topologies. [Copyright &y& Elsevier]

Classes of functions of several variables on arbitrary nonempty domains that are closed under permutation of variables and addition of dummy variables are characterized by generalized constraints, and hereby Hellerstein's Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary nonempty domains that are closed under permutation of variables, addition of dummy variables, and composition are characterized by clusters, and a Galois connection is established between operations and clusters. [ABSTRACT FROM AUTHOR]

For a classifier, besides classification capability, its size is another vital aspect. In pursuit of high performance, many classifiers do not take into consideration their sizes and contain numerous both essential and insignificant rules. This, however, may bring adverse situation to classifier, for its efficiency will been put down greatly by redundant rules. Hence, it is necessary to eliminate those unwanted rules. In this paper, we propose a fast post-processing approach to remove insignificant rules. The basis of this method is the dependent relation between rules regarding to data objects, from which closed sets can be derived. The experimental evaluation on UCI benchmark datasets using two typical classifiers shows that the proposed method is competent for discarding lots of superfluous rules without degrading classification capability greatly. In particular, the computational cost of our approach is extremely lower than the Apriori-like method. [Copyright &y& Elsevier]

Abstract: General topology has its roots in real and complex analysis, which made important uses of the interrelated concepts of open set, of closed set, and of a limit point of a set. This article examines how those three concepts emerged and evolved during the late 19th and early 20th centuries, thanks especially to Weierstrass, Cantor, and Lebesgue. Particular attention is paid to the different forms of the Bolzano–Weierstrass Theorem found in the latter''s unpublished lectures. An abortive early, unpublished introduction of open sets by Dedekind is examined, as well as how Peano and Jordan almost introduced that concept. At the same time we study the interplay of those three concepts (together with those of the closure of a set and of the derived set of a set) in the struggle to determine the ultimate foundations on which general topology was built, during the first half of the 20th century. [Copyright &y& Elsevier]

This work concerns Markov chains with finite state space. Given a real-valued cost function on the state space, the corresponding Varadhan's function, measuring the exponential growth rate of the aggregated costs, is characterized as the unique limit of the fixed points of a family of contraction operators, a conclusion that does not involve any condition on the transition law. [ABSTRACT FROM AUTHOR]

For further studying the theories and applications of rough sets (RS), this paper proposes a new theory on RS, which mainly includes topological space, topological properties, homeomorphism, and its properties on RS by some new definitions and theorems given. The relationship between partition and countable open covering is discussed, and some applications based on the topological rough space and its topological properties are introduced. Moreover, some perspectives for future research are given. Throughout this paper, the advancements of the new theory on RS and topological algebra not only represent an important theoretical value but also exhibit significant applications of RS and topology. [ABSTRACT FROM AUTHOR]

Abstract: Let . For or , has been determined by Hanani. In this paper, we investigate the case of . It is easy to see that if , then . It is known that by Hanani and that by a previous paper of the author. We shall focus on the case of . It is proved that and . [Copyright &y& Elsevier]

At the Hospital for Sick Children, a Graded Profile Analysis (GPA) is used to quantify candidacy evaluations and steer decisions regarding candidacy for cochlear implantation. Over the first 4 years of use, the GPA score was reported to influence the decision to implant and correlate with speech perception outcomes. With the evolution of candidacy criteria, the usefulness of the GPA required reexamination. The GPA was completed on the basis of a multidisciplinary evaluation of cochlear implant candidacy in 172 children assessed over 2 years. GPA scores were analyzed with respect to the decision to implant and behavioural speech perception outcomes in these children. The GPA score was significantly higher for children who were provided with implants compared to those who were not. In addition, higher GPA scores were associated with higher scores on postimplant speech perception tests and lower ages at implantation. The GPA provides an objective measure for candidacy evaluation, reflects changes in candidacy criteria, and provides a prediction of potential speech perception outcomes after implantation. The GPA could be helpful in counselling children/parents prior to implantation. [Copyright &y& Elsevier]

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = (U ∪ V) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed. [ABSTRACT FROM AUTHOR]