Nonempty Set Research Articles

A study on set-cordial labeling of graphs

For a non-empty ground set XX, finite or infinite, the set-valuation or set-labeling of a given graph GG is an injective function f:V(G)→P(X)f:V(G)→P(X), where P(X)P(X) is the power set of the set XX. In this paper, we introduce a new type of set-labeling, called set-cordial labeling and study the characteristics of graphs which admit the set-cordial labeling.

Implementation of Rough Set on A Group Structure

Let be a non-empty set and an equivalence relation on . Then, is called an approximation space. The equivalence relation on forms disjoint equivalence classes. If , then we can form a lower approximation and an upper approximation of . If X⊆U, then we can form a lower approximation and an upper approximation of X. In this research, rough group and rough subgroups are constructed in the approximation space for commutative and non-commutative binary operations.

Safe Sets in Some Graph Families

For a connected simple graph G , a non-empty set \(S \subseteq V(G)\) of vertices is a safe set if, for every component \(A \text { of }\langle S\rangle_{G}\) and every component \(B \text { of }\langle V(G)-S\rangle_{G}\) adjacent to A , it holds that \(|A| \geq|B|\). The safe number denoted by s(G) of G is the minimum cardinality of a safe set G . In this paper, it examines the characterization of a safe set in complete bipartite graph. It also discusses the minimum cardinality of a safe sets of path graph and cycle graph via modulus. Moreover, this study generates the possible exact values of the safe number of the complete graph, complete bipartite graph, and star graph.

Algebraic Hyperstructure of Multi-Fuzzy Soft Sets Related to Polygroups

The combination of two elements in a group structure is an element, while, in a hypergroup, the combination of two elements is a non-empty set. The use of hypergroups appears mainly in certain subclasses. For instance, polygroups, which are a special subcategory of hypergroups, are used in many branches of mathematics and basic sciences. On the other hand, in a multi-fuzzy set, an element of a universal set may occur more than once with possibly the same or different membership values. A soft set over a universal set is a mapping from parameters to the family of subsets of the universal set. If we substitute the set of all fuzzy subsets of the universal set instead of crisp subsets, then we obtain fuzzy soft sets. Similarly, multi-fuzzy soft sets can be obtained. In this paper, we combine the multi-fuzzy soft set and polygroup structure, from which we obtain a new soft structure called the multi-fuzzy soft polygroup. We analyze the relation between multi-fuzzy soft sets and polygroups. Some algebraic properties of fuzzy soft polygroups and soft polygroups are extended to multi-fuzzy soft polygroups. Some new operations on a multi-fuzzy soft set are defined. In addition to this, we investigate normal multi-fuzzy soft polygroups and present some of their algebraic properties.

On closed weakly m-convexsets

In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn 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 Rn 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 Rn 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 Rn 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 Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.

On t-Derivations of BE-algebras

BE-algebra is a non-empty set with a binary operation and a constant 1 which is denoted by , and fulfill certain axioms. In this article, the concept of t-derivation in BE-algebra is defined and some of its properties are investigated. Then, the properties of the fixed set and kernel of a t-derivation in BE-algebra are identified based on their relation to the subalgebra and its elements.

Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded

Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number.

Fixed Point of Contractive Mappings in Cone Metric Space

The concept of a fixed point of a function is closely related to the abstract space of the function it is in, one of which is the cone metric space which is a generalization of the metric space and which was first introduced by Zhang and Huang in 2007. The cone metric space is any non-empty set that with cone metric function. The cone metric function has the domain of any non-empty set and has the codomain of an ordered Banach space. The purpose of this study is to recognize the concept of the cone metric space and its relation to the metric space, as well as to examine the fixed point theorem on the cone metric space.

On r-Ideals and m-k-Ideals in BN-Algebras

A BN-algebra is a non-empty set X with a binary operation “∗” and a constant 0 that satisfies the following axioms: (B1) x∗x=0, (B2) x∗0=x, and (BN) (x∗y)∗z=(0∗z)∗(y∗x) for all x, y, z ∈X. A non-empty subset I of X is called an ideal in BN-algebra X if it satisfies 0∈X and if y∈I and x∗y∈I, then x∈I for all x,y∈X. In this paper, we define several new ideal types in BN-algebras, namely, r-ideal, k-ideal, and m-k-ideal. Furthermore, some of their properties are constructed. Then, the relationships between ideals in BN-algebra with r-ideal, k-ideal, and m-k-ideal properties are investigated. Finally, the concept of r-ideal homomorphisms is discussed in BN-algebra.

Set-valued solutions of an equation of Jensen type

In this paper we provide a characterization theorem for set-valued solutions of the functional equation for the case of compact, convex valued maps. We prove that every solution of the functional equation is the sum of an additive single-valued function and a nonempty compact and convex set.

Some applications of optimization routing problems with additional constraints

The paper deals with an extremal routing problem with constraints. In the general formulation, it is assumed that the objects of visiting are any non-empty finite sets — megalopolises. The main applied problem considered in this study is the tool path optimization problem for CNC sheet-cutting machines, known as the Cutting Path Problem. This problem arises at the stage of developing control programs for CNC machines. Other applications are also possible. In particular, the results obtained in the chapter can be used in the problem of minimizing the radiation dose when dismantling a system of radiation-hazardous elements after accidents at nuclear power plants and in transport problems. Among tasks constraints, the precedence constraints are investigated. These constraints can be used to reduce computational complexity. As the main method, the study used broadly understood dynamic programming. The offered realization of the method takes into account the precedence constraints and the dependence of the objective functions on the task list. This dependence belongs to the class of very complex conditions that determine the route admissibility at each routing step, depending on the tasks already completed or, on the contrary, not yet completed. As applied to the Cutting Path Problem, the dependence of the objective function on the task list makes it possible to reduce thermal deformations of the material during cutting. The chapter provides a mathematical formalization of an extremal routing problem with additional constraints, a description of the method, and the exact algorithm obtained with its help. The order of task execution, the specific trajectory of the process, and the starting point are optimized.

Convex invariants of M-fuzzifying convex spaces

The numbers of Helly, Carathéodory and Radon are a central theme in classic convex spaces and each of them is invariant under isomorphism. The theory of M-fuzzifying convex spaces is an active research field since it is presented. This paper focus on convex invariants of M-fuzzifying convex spaces. The degrees of Helly independent, Carathéodory independent and Radon independent of non-empty set are defined in the framework of M-fuzzifying convex spaces. By those definitions, we introduce the Helly number, Carathéodory number and Radon number of M-fuzzifying convex spaces. Finally, we inspect M-fuzzifying topology and M-fuzzifying JHC which are characterized by convex invariants of M-fuzzifying convex spaces.

On some module amenability-like properties of Banach algebras with applications

In this paper, we introduce a new notion of module strong pseudo-amenability for Banach algebras. We study the relation between this new concept to other various notions at this issue, module pseudo-contractibility, module pseudo-amenability and module approximate amenability. For an inverse semigroup [Formula: see text] with the set of all idempotents [Formula: see text], we show that [Formula: see text] is module strong pseudo-amenable as an [Formula: see text]-module if and only if [Formula: see text] is amenable. For specific types of semigroups such as Brandt semigroups and bicyclic semigroups, we investigate the module strong pseudo-amenability of [Formula: see text]. We show that for every non-empty set [Formula: see text], [Formula: see text] under this new notion is forced to have a finite index as an [Formula: see text]-module, where [Formula: see text].

Geometric algebra for sets with betweenness relations

Given a betweenness relation on a nonempty set E, a certain abelian group {mathbb {T}}= {{mathbb {T}}}_E given in terms of generators and relations is investigated. This group controls the given betweenness relation in an algebraic form. That is, the group structure algebraically unfolds geometric relations, and in turn allows us to read off geometric properties from algebraic relations emerging from them. The most important examples for betweenness relations arise from ordered sets on the one side and from intervals in metric spaces on the other side. The structure of {mathbb {T}} will be determined completely in case of totally ordered sets as well as for several classes of metric spaces.

Channel-based coherence of quantum states

Quantum coherence is one of the most fundamental and striking features in quantum physics. Considered the standard coherence (SC), the partial coherence (PC) and the block coherence (BC) as variance of quantum states under some quantum channels (QCs) [Formula: see text], we propose the concept of channel-based coherence of quantum states, called [Formula: see text]-coherence for short, which contains the SC, PC and BC, but does not contain the positive operator-valued measure (POVM)-based coherence. By our definition, a state [Formula: see text] is said to be [Formula: see text]-incoherent if it is a fixed point of a QC [Formula: see text], otherwise, it is said to be [Formula: see text]-coherent. First, we find the set [Formula: see text] of all [Formula: see text]-incoherent states for some given channels [Formula: see text] and prove that the set [Formula: see text] forms a nonempty compact convex set for any channel [Formula: see text]. Second, we define [Formula: see text]-incoherent operations ([Formula: see text]-IOs) and prove that the set of all [Formula: see text]-IOs is a nonempty convex set. We also establish some characterizations of a [Formula: see text]-IO in terms of its Kraus operators. Lastly, we discuss the problem of quantifying [Formula: see text]-coherence and prove some related properties.

Clique and Independence Number of Rough Co-Zero Divisor Graph and Its Applications in Analyzing a Twitter Data

In view of this paper, we have defined Rough Co-zero divisor graph [Formula: see text] of a rough semiring [Formula: see text] for a given approximation space [Formula: see text] where [Formula: see text] is nonempty finite set of objects and [Formula: see text] is an arbitrary equivalence relation on [Formula: see text]. It is proved that the Rough Co-zero divisor graph and the subgraph of Rough Ideal-based rough edge Cayley graph [Formula: see text] are complement to each other. Also the Clique number and Independence number of [Formula: see text] and [Formula: see text] are obtained. The developed concepts are illustrated with suitable examples. A sentiment analysis is also made using [Formula: see text] for a Twitter data.

Exterior controllability properties for a fractional Moore–Gibson–Thompson equation

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that \(b>0\) and \(\alpha -\frac{\tau c^2}{b}>0\), we show that if \(0<s<1\) and \(\varOmega \subset {\mathbb {R}}^N\) (\(N\ge 1\)) is a bounded domain with a Lipschitz continuous boundary \(\partial \varOmega \), then there is no control function g such that the following system $$\begin{aligned} {\left\{ \begin{array}{ll} \tau u_{ttt} + \alpha u_{tt}+c^2(-\varDelta )^{s} u + b(-\varDelta )^{s} u_{t}=0 &{} \text{ in } \; \varOmega \times (0,T),\\ u=g\chi _{{\mathcal {O}}} &{} \text{ in } \; ({\mathbb {R}}^N\setminus \varOmega )\times (0,T) ,\\ u(\cdot ,0) = u_0, u_t(\cdot ,0) = u_1, u_{tt}(\cdot ,0)=u_2 &{} \text{ in } \; \varOmega , \end{array}\right. } \end{aligned}$$is exactly or null controllable in time \(T>0\). However, we prove that for \(0<s<1\), the system is approximately controllable for every \(g\in H^1((0,T);L^{2}({\mathcal {O}}))\), where \(\mathcal O\subset {\mathbb {R}}^N\setminus {\overline{\varOmega }}\) is an arbitrary non-empty open set.

Homomorphisms of Lattice-Valued Intuitionistic Fuzzy Subgroup Type-3.

The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the L-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval [0,1]. In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

A Parameterized Approximation Scheme for Min $k$-Cut

In the Min $k$-Cut problem, input is an edge weighted graph $G$ and an integer $k$, and the task is to partition the vertex set into $k$ non-empty sets, such that the total weight of the edges with endpoints in different parts is minimized. When $k$ is part of the input, the problem is NP-complete and hard to approximate within any factor less than $2$. Recently, the problem has received significant attention from the perspective of parameterized approximation. Gupta et al.~[SODA 2018] initiated the study of FPT-approximation for the Min $k$-Cut problem and gave an $1.9997$-approximation algorithm running in time $2^{\mathcal{O}(k^6)}n^{\mathcal{O}(1)}$. Later, the same set of authors~[FOCS 2018] designed an $(1 +\epsilon)$-approximation algorithm that runs in time $(k/\epsilon)^{\mathcal{O}(k)}n^{k+\mathcal{O}(1)}$, and a $1.81$-approximation algorithm running in time $2^{\mathcal{O}(k^2)}n^{\mathcal{O}(1)}$. More, recently, Kawarabayashi and Lin~[SODA 2020] gave a $(5/3 + \epsilon)$-approximation for Min $k$-Cut running in time $2^{\mathcal{O}(k^2 \log k)}n^{\mathcal{O}(1)}$. In this paper we give a parameterized approximation algorithm with best possible approximation guarantee, and best possible running time dependence on said guarantee (up to Exponential Time Hypothesis (ETH) and constants in the exponent). In particular, for every $\epsilon > 0$, the algorithm obtains a $(1 +\epsilon)$-approximate solution in time $(k/\epsilon)^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$. The main ingredients of our algorithm are: a simple sparsification procedure, a new polynomial time algorithm for decomposing a graph into highly connected parts, and a new exact algorithm with running time $s^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ on unweighted (multi-) graphs. Here, $s$ denotes the number of edges in a minimum $k$-cut. The latter two are of independent interest.

On coloring of graphs of girth $2\l+1$ without longer odd holes

A hole is an induced cycle of length at least 4. Let $\l\ge~2$ be an integer and $r=\min\{\l-1,~\lfloor{l\over~2}\rfloor+1\}$, let ${\cal~G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in~{\cal~G}_{\l}$. For a vertex $u\in~V(G)$ and a nonempty set $S\subseteq~V(G)$, let $d(u,~S)=\min\{d(u,~v):v\in~S\}$, and let $L_i(S)=\{u\in~V(G)~\mbox{~and~}d(u,~S)=i\}$ for any integer $i\ge~0$. We show that if $G[S]$ is connected and $G[L_i(S)]$ is bipartite for each $i\in\{1,~\ldots,~r\}$, then $G[L_i(S)]$ is bipartite for each $i&gt;0$, and consequently $\chi(G)\le~4$, where $G[S]$ denotes the subgraph induced by $S$.Let $\theta^-$ be the graph obtained from the Petersen graph by deleting three vertices which induce a path, let $\theta^+$ be the graph obtained from the Petersen graph by deleting two adjacent vertices, and let $\theta$ be the graph obtained from $\theta^+$ by removing an edge incident with two vertices of degree 3. For a non-Petersen graph $G\in{\cal~G}_2$, we show that if $G$ is 3-connected and has no unstable 3-cutset then $G$ must induce either $\theta$ or $\theta^-$ but does not induce $\theta^+$. As corollaries, $\chi(G)\le~3$ for every graph $G$ of ${\cal~G}_2$ that induces neither $\theta$ nor $\theta^-$, and minimal non-3-colorable graphs of ${\cal~G}_2$ induce no $\theta^+$.