Maximal Open Set Research Articles

New types of connected, component spaces and maximal open set via s-operation

This paper aims to introduce a kind of connected space called λrc-connected. We analyze the characterizations, properties, and relationships of λrc-connected spaces with other types of connected spaces including, λrc-locally connected spaces. Additionally, we define a class of maximal open sets which called maximal λrc-open sets. Some of its basic properties, characterization and relationships are studied and obtained.

Hesitant Fuzzy Maximal and Minimal Clopen Sets

This article is to study the notions of hesitant fuzzy minimal clopen and hesitant fuzzy maximal clopen sets. In this article, we have proved hesitant fuzzy maximal and minimal clopen sets are independent to hesitant fuzzy minimal and maximal open sets(resp. closed sets). Using hesitant fuzzy disconnectedness certain properties of both hesitant fuzzy minimal and maximal clopen sets are discussed.

MINIMAL AND MAXIMAL OPEN SETS IN NANO TOPOLOGICAL SPACE

This paper deals with nano minimal open sets and nano maximal open sets. Thereafter nano maximal, nano minimal open sets, nano minimal continuous and nano maximal continuous are studied. Also we study the interrelationship among these concepts. Finally we obtain a result which is not possible in classical topology.

A study on soft rough minimal continuous and soft rough maximal continuous

The notions of soft rough minimal open set, soft rough maximal open set, soft rough minimal closed set, soft rough maximal closed set, soft rough minimal continuous and soft rough maximal continuous are introduced and studied. We also investigate some related properties of these concepts.

A NOTE ON SOFT ROUGH MINIMAL AND SOFT ROUGH MAXIMAL OPEN SETS.

The notions of some theorems on soft rough minimal open set, soft rough maximal open set and also soft rough minimal irresolute and soft rough maximal irresolute are introduced and studied. We also investigate some related properties of these concepts.

FUZZY MAXIMAL, MINIMAL OPEN AND CLOSED SETS

Adv. Math. Sci. J. Vol. 9, No. 10 Open Access FUZZY MAXIMAL, MINIMAL OPEN AND CLOSED SETS A. Swaminathan and S. Sivaraja DOI: https://doi.org/10.37418/amsj.9.10.5 Full Text Abstract Abstract By means of fuzzy maximal and fuzzy minimal open sets, we obtain few conditions for fuzzy disconnectedness; we obtain some identical results by fuzzy minimal … page AMSJ9-N10-5 Read More »

Some New Properties of Fuzzy Maximal Regular Open Sets

A proper nonempty open subset of a fuzzy topological space is said to be a fuzzy maximal regular open set , if any regular open set which contains is or . The purpose of this paper is to study some new fundamental properties of fuzzy maximal regular open sets. The decomposition theorems for a fuzzy maximal regular open set are investigated. Notion and basic properties of radical of fuzzy maximal regular open sets are established, such as the law of fuzzy radical closure. Some new properties and characterization theorems of fuzzy maximal regular open set are achieved.

On maximal, minimal and mean open sets

We see that the real numbers system with the usual topology contains no minimal open sets. This observation instigates us to study topological spaces having no minimal and maximal open sets. We find that such topological spaces if connected are not cut-point spaces. We also characterize mean open sets in $$T_{1}$$ connected topological spaces.

English

A subset $S$ of vertices in a graph $G$ is an open packing set if no pair of vertices of $S$ has a common neighbor in $G$. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by $\rho ^{\rm o}(G)$. A subset $S$ in a graph $G$ with no isolated vertex is called a total dominating set if any vertex of $G$ is adjacent to some vertex of $S$. The total domination number of $G$, denoted by $\gamma _t(G)$, is the minimum cardinality of a total dominating set of $G$. We characterize graphs of order $n$ and minimium degree at least two with $\rho ^{\rm o}(G)=\gamma _t(G)=\frac 12n$.

Local variational study of 2d lattice energies and application to Lennard–Jones type interactions

In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furthermore, we apply our result to Lennard–Jones type interacting potentials that appear to be accurate in many physical and biological models. The goal of this investigation is to understand how the minimum of the Lennard–Jones lattice energy varies with respect to the density of the points. Considering the lattices of fixed area A, we find the maximal open set to which A must belong so that the triangular lattice is a minimizer (resp. a maximizer) among lattices of area A. Similarly, we find the maximal open set to which A must belong so that the square lattice is a minimizer (resp. a saddle point). Finally, we present a complete conjecture, based on numerical investigations and rigorous results among rhombic and rectangular lattices, for the minimality of the classical Lennard–Jones energy per point with respect to its area. In particular, we prove that the minimizer is a rectangular lattice if the area is sufficiently large.

On Minimal and Maximal Regular Open Sets

The purpose of this paper is to investigate the concepts of minimal and maximal regular open sets and their relations with minimal and maximal open sets. We study several properties of such concepts in a semi-regular space. It is mainly shown that if X is a semi-regular space, then miO(X) = miRO(X). We introduce and study new type of sets called minimal regular generalized closed. A special interest type of topological space called rTmin space is studied and obtain some of its basic properties.

Minimal and Maximal Beta Open Sets

The present paper deals and discusses new types of sets all of these concepts completely depended on the concept of Beta open set. The importance concepts which introduced in this paper are minimal -open and maximal -open sets. Besides, new types of topological spaces introduced which called and spaces. Also we present new two maps of continuity which called minimal -continuous and maximal -continuous. Additionally we investigated some fundamental properties of the concepts which presented in this paper.

Some properties of f max. o. s. and f. min. o. s. in fuzzy topological spaces defined on a fuzzy set

In this work we introduce (Fuzzy Maximal Open Set) and (Fuzzy Minimal Open Set). We introduce these fuzzy set in a fuzzy topological spaces defined on a fuzzy set Iƒ instead of crisp set . We studied some properties of these fuzzy sets and some interrelationships between them.

From resolvent estimates to unique continuation for the Schrödinger equation

In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation ( i ∂ t + Δ ) u = V ( x ) u (i\partial _t+\Delta )u=V(x)u . In general the problem is to find a class of potentials V V which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted L 2 L^2 resolvent estimates ‖ ( − Δ − z ) − 1 f ‖ L 2 ( | V | ) ≤ C ‖ f ‖ L 2 ( | V | − 1 ) \|(-\Delta -z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})} . We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato H H -smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution u u . In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.

Asymmetric maximal and minimal open sets

We introduce the notions of maximal and minimal open sets in bitopological spaces and obtain some properties of them. In contrary to maximal and minimal open sets in topological spaces, we observe that maximal and minimal open sets in bitopological spaces behave differently. The maximal and minimal open sets in a bitopological space under the operations of union and intersection respectively sometimes become slightly different types of maximal and minimal open sets in that bitopological space. We also obtain results concerning an asymmetric minimal open set on a subspace of a bitopological space.

Minimal and maximal soft open sets

In this paper, we introduce new types of minimal and maximal sets via soft topological spaces namely minimal and maximal soft open sets and their complements. These sets are depended on the soft open sets. Many interested result are presented to reveal some properties of these new sets.

ON MAXIMAL, MINIMAL OPEN AND CLOSED SETS

We obtain some conditions for disconnectedness of a topolog- ical space in terms of maximal and minimal open sets, and some similar results in terms of maximal and minimal closed sets along with interre- lations between them. In particular, we show that if a space has a set which is both maximal and minimal open, then either this set is the only nontrivial open set in the space or the space is disconnected. We also obtain a result concerning a minimal open set on a subspace.

Packing parameters in graphs

In a graph G = (V,E), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The minimum and maximum cardinalities of a maximal open packing set are respectively called the lower open packing

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.

Sweeping out sectional curvature

We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of rigidity statements in comparison geometry.