Nonempty Subset Research Articles

Optimizing HX-Group Compositions Using C++: A Computational Approach to Dihedral Group Hyperstructures

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group (G,·), they constructed another group (G,∗)⊂P∗(G), where P∗(G) is the set of non-empty subsets of G. The hypercomposition “∗” is thus defined for any A, B from G, A∗B={a·b|a∈A,b∈B}. In this article, we consider a particular group, G, to be the dihedral group Dn,n is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group Dn as a support. The HX-groups were studied algebraically, but the novelty of this article is that it is a computer analysis of the HX-groups by creating a program in C++. This code aims to improve the calculation time regarding the composition of the HX-groups. In the first part of the paper, we present some results from the hypergroup theory and HX-groups. We create another hyperstructure formed by reuniting all the HX-groups associated with a dihedral group Dn as a support for a natural fixed number n. In the second part, we present the C++ code created in the Microsoft Visual Studio program, and we provide concrete examples of the program’s application. We created this program because the code aims to improve the calculation time regarding the composition of HX-groups.

Geodetic numbers of tensor product and lexicographic product of graphs

A shortest u - v path between two vertices u and v of a graph G is a u - v geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some u - v geodesic of G. For a nonempty subset S of V ( G ) , let I ( S ) = ∪ u , v ∈ S I [ u , v ] . If I ( S ) = V ( G ) , then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by g ( G ) . In this paper, the exact geodetic numbers of the product graphs T × K m and T ° K ¯ m are obtained, where T is a tree, K ¯ m denotes the complement of the complete graph K m and, × and ° denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

The monoid of all partial transformations with restricted range

Let [Formula: see text] be a nonempty set and let [Formula: see text] be the monoid of all partial transformations of [Formula: see text]. Given a nonempty subset [Formula: see text] of [Formula: see text], we denote by [Formula: see text] the subsemigroup of [Formula: see text] of all partial transformations with range contained in [Formula: see text]. In 2014, Fernandes and Sanwong [On the ranks of semigroups of transformations on a finite set with restricted range, Algebra Colloquium. 21(3) (2014) 497–510] proved that the semigroup [Formula: see text] is the largest regular subsemigroup of [Formula: see text] and showed that this subsemigroup determines Green’s relations on [Formula: see text]. In this paper, we classify completely the maximal subsemigroups of [Formula: see text] when [Formula: see text]. Moreover, we prove that the rank of [Formula: see text] is equal to [Formula: see text] when [Formula: see text] and [Formula: see text].

On algebraic properties of power monoids of numerical monoids

AbstractLet S ⊂ ℕ0 be a numerical monoid and let $$\cal{P}_{\text{fin}}(S)$$ P fin ( S ) , resp. $$\cal{P}_{\text{fin},0}(S)$$ P fin , 0 ( S ) , denote the power monoid, resp. the restricted power monoid, of S, that is the set of all finite nonempty subsets of S, resp. the set of all finite nonempty subsets of S containing 0, with set addition as operation. The arithmetic of power monoids received some attention in recent literature. We complement these investigations by studying algebraic properties of power monoids, such as their prime spectrum. Moreover, we show that almost all elements of $$\cal{P}_{\text{fin},0}(S)$$ P fin , 0 ( S ) are irreducible (i.e., they are not proper sumsets).

HOW MANY SUBSETS WHICH THEIR OPERATIONS WITH A FIXED SUBSET CONTAIN AT LEAST ONE ELEMENT OF A GIVEN COLLECTION?

We pose the following problem related to binary set operations on finite sets. Given a finite set . Let a binary set operation and be a non-empty collection of non-empty subsets of . For a fixed subset of , where , how many subsets of which their operation with contains at least one element of ?. In this paper, we give the solution of this problem, especially for the subsets of size , using the inclusion-exclusion principle, Corrádi’s lemma, and Bonferroni’s inequality. In this context, the problem is related to determining the degree of nodes in certain graphs, such as graphs constructed with the adjacency rule depends on and the node set is a hypergraph.

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

A note on fibers and Vietoris topologies of paratopological groups

If f: G → Y is an irreducible closed continuous mapping defined on a regular weakly collectionwise normal first-countable meta-Lindelöf locally σ paratopological group G onto a T 1-space Y which has a neighborhood ωω -base at a point y ∈ Y, then f −1(y) is σ-compact in G. We prove that if a Fréchet-Urysohn space X has strong α 4 -property and a weakly countably complete base , then X is first-countable, where M is a separable and metrizable space and = {F: F is a non-empty compact subset of M } and with the Vietoris topology. By this result we can get the first-countability of certain paratopological groups.

Minimal solutions of fuzzy relation equations via maximal independent elements

Fuzzy relation equations (FRE) are a useful formalism with a broad number of applications in different computer science areas. Testing if a solution exists and, if so, computing the unique greatest solution is straightforward. In contrast, the computation of minimal solutions is more complex. In particular, even in FRE with a very simple structure, the number of minimal solutions can increase exponentially. However, minimal solutions are immensely useful since, under mild conditions, they (together with the greatest solution) allow one to describe the entire space of solutions to an FRE. The main result of this work is a new method for enumerating the set of minimal solutions. It works by establishing a relationship between coverings of FRE and maximal independent elements of (hyper-)boxes. We can thus make efficient enumeration methods for maximal independent elements of (hyper-)boxes applicable also to our setting of FRE, where the operator considered in the composition of fuzzy relations only needs to preserve suprema of arbitrary subsets and infima of non-empty subsets. More specifically, we thus show that the enumeration of the minimal solutions of an FRE can be done with incremental quasi-polynomial delay.

The Monophonic-triangular Number of a Graph

A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.

Sublinear message bounds of authenticated implicit Byzantine agreement

This paper studies the message complexity of authenticated Byzantine agreement (BA) in synchronous, fully-connected distributed networks under an honest majority. We focus on the so-called implicit Byzantine agreement problem where each node starts with an input value and at the end a non-empty subset of the honest nodes should agree on a common input value by satisfying the BA properties (i.e., there can be undecided nodes)3. We show that a sublinear (in n, number of nodes) message complexity BA protocol under honest majority is possible in the standard PKI model when the nodes have access to an unbiased global coin and hash function. In particular, we present a randomized Byzantine agreement algorithm which, with high probability achieves implicit agreement, uses O˜(n) messages, and runs in O˜(1) rounds while tolerating (1/2−ϵ)n Byzantine nodes for any fixed ϵ>0, the notation O˜ hides a O(polylogn) factor4. The algorithm requires standard cryptographic setup PKI and hash function with a static Byzantine adversary. The algorithm works in the CONGEST model and each node does not need to know the identity of its neighbors, i.e., works in the KT0 model. The message complexity (and also the time complexity) of our algorithm is optimal up to a polylog n factor, as we show a Ω(n) lower bound on the message complexity. We further extend the result to Byzantine subset agreement, where a non-empty subset of nodes should agree on a common value. Lastly, we analyze several relevant results that follow from the construction of the main result.To the best of our knowledge, this is the first sublinear message complexity result of Byzantine agreement. A quadratic message lower bound is known for any deterministic BA protocol (due to Dolev-Reischuk [JACM 1985]). The existing randomized BA protocols have at least quadratic message complexity in the honest majority setting. Our result shows the power of a global coin in achieving significant improvement over the existing results. It can be viewed as a step towards understanding the message complexity of randomized Byzantine agreement in distributed networks with PKI.

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Carathéodory-Hahn-Kluvánek extension theorem in locally convex spaces with application

In this paper, we present a Carathéodory-Hahn-Kluvánek-type theorem for a multimeasure H:A→cwk(X) on A, where A is an algebra of subsets of a set Ω and cwk(X) is the family of all convex weakly compact nonempty subsets of a non-separable locally convex topological vector space X. Applying extension results for additive multimeasures, it will be proved a Radon-Nikodým theorem for an additive multimeasure in terms of “nonempty weakly density”.

From Tilings of Orientable Surfaces to Topological Interlocking Assemblies

A topological interlocking assembly (TIA) is an assembly of blocks together with a non-empty subset of blocks called the frame such that every non-empty set of blocks is kinematically constrained and can therefore not be removed from the assembly without causing intersections between blocks of the assembly. TIA provides a wide range of real-world applications, from modular construction in architectural design to potential solutions for sound insulation. Various methods to construct TIA have been proposed in the literature. In this paper, the approach of constructing TIA by applying the Escher trick to tilings of orientable surfaces is discussed. First, the strengths of this approach are highlighted for planar tilings, and the Escher trick is then exploited to construct a planar TIA that is based on the truncated square tiling, which is a semi-regular tiling of the Euclidean plane. Next, the Escher-Like approach is modified to construct TIAs that are based on arbitrary orientable surfaces. Finally, the capabilities of this modified construction method are demonstrated by constructing TIAs that are based on the unit sphere, the truncated icosahedron, and the deltoidal hexecontahedron.

Adaptive Multiple Comparisons With the Best.

Subset selection methods aim to choose a nonempty subset of populations including a best population with some prespecified probability. An example application involves location parameters that quantify yields in agriculture to select the best wheat variety. This is quite different from variable selection problems, for instance, inregression. Unfortunately, subset selection methods can become very conservative when the parameter configuration is not least favorable. This will lead to a selection of many non-best populations, making the set of selected populations less informative. To solve this issue, we propose less conservative adaptive approaches based on estimating the number of best populations. We also discuss variants of our adaptive approaches that are applicable when the sample sizes and/or variances differ between populations. Using simulations, we show that our methods yield a desirable performance. As an illustration of potential gains, we apply them to two real datasets, one on the yield of wheat varieties and the other obtained via genome sequencing of repeated samples.

Restrictions on Anosov subgroups of 𝑆𝑝(2𝑛,ℝ)

Let n ∈ N n\in \mathbb {N} and let Θ ⊂ { 1 , … , n } \Theta \subset \{1,\dots ,n\} be a nonempty subset. We prove that if Θ \Theta contains an odd integer, then any P Θ P_\Theta -Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) is virtually isomorphic to a free or surface group. On the other hand, if Θ \Theta does not contain any odd integers, then there exists a P Θ P_\Theta -Anosov subgroup of Sp ⁡ ( 2 n , R ) \operatorname {Sp}(2n,\mathbb {R}) which is not virtually isomorphic to a free or surface group. We also exhibit new examples of maximally antipodal subsets of certain flag manifolds; these arise as limit sets of rank 1 1 subgroups.

Ideals in Bipolar Quantum Linear Algebra

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c∈(0,1] such that I=(−x,y)∈R2|cx≤y≤xcandx,y≥0. This shows that its shape has to be symmetric with the graph y=−x.

On the numerical corroboration of an obstacle problem for linearly elastic flexural shells.

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

On join-complete implication algebras

In this paper, first, we consider an algebra that has a binary operation and a join of arbitrary nonempty subset. A lattice implication algebra is a lattice with a binary operation, which has a join and a meet of finite nonempty subsets. In this work, the notion of join-complete implication algebras L is defined as a join-complete lattice with a binary operation, and some properties of this algebra L are searched. Moreover, we prove that the interval [a, 1] in L is a lattice implication algebra and show that L satisfies the completely distributive law when it has the smallest element 0. Finally, we state the concept of filter and multipliers of L and provide finite and infinite examples of them. In addition, we research some properties of these concepts in detail.

On Strong Regal and Strong Royal Colorings

Two colorings have been introduced recently where an unrestricted coloring c assigns nonempty subsets of [ k ] = { 1 , … , k } to the edges of a (connected) graph G and gives rise to a vertex-distinguishing vertex coloring by means of set operations. If each vertex color is obtained from the union of the incident edge colors, then c is referred to as a strong royal coloring. If each vertex color is obtained from the intersection of the incident edge colors, then c is referred to as a strong regal coloring. The minimum values of k for which a graph G has such colorings are referred to as the strong royal index of G and the strong regal index of G respectively. If the induced vertex coloring is neighbor distinguishing, then we refer to such edge colorings as royal and regal colorings. The royal chromatic number of a graph involves minimizing the number of vertex colors in an induced vertex coloring obtained from a royal coloring. In this paper, we provide new results related to these two coloring concepts and establish a connection between the corresponding chromatic parameters. In addition, we establish the royal chromatic number for paths and cycles.

Nash’s Existence Theorem for Non-Compact Strategy Sets

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.