Cardinality Of Set Research Articles

Infinite multisets: Basic properties and cardinality

This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ0 (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ0 there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.

Regular singular differential equations and free proalgebraic groups

AbstractWe determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere. It is the free proalgebraic group on a set of cardinality .

On the general position number of the k-th power graphs

A set is called a general position set of a graph G if any triple set V 0 of R is non-geodesic, that is, three elements of V 0 cannot lie on the same geodesic of G. The general position number gp(G) of a graph G is the cardinality of a largest general position set of G. The k-th power, Gk , of a graph G is obtained from G by adding a new edge between each pair of vertices with a distance of at most k in G. In this paper we establish the bounds for gp(Gk ) and and give an explicit formula of . Also we consider the relation between gp(G 2) and gp(G). We deduce a formula of and provide the upper bound on gp(G 2)−gp(G) for general block graphs G. Moreover, we determine the gp-number for the square of trees.

Automated Category Tree Construction: Hardness Bounds and Algorithms

Category trees, or taxonomies, are rooted trees where each node, called a category, corresponds to a set of related items. The construction of taxonomies has been studied in various domains, including e-commerce, document management, and question answering. Multiple algorithms for automating construction have been proposed, employing a variety of clustering approaches and crowdsourcing. However, no formal model to capture such categorization problems has been devised, and their complexity has not been studied. To address this, we propose in this work a combinatorial model that captures many practical settings and show that the aforementioned empirical approach has been warranted, as we prove strong inapproximability bounds for various problem variants and special cases when the goal is to produce a categorization of the maximum utility. In our model, the input is a set of n weighted item sets that the tree would ideally contain as categories. Each category, rather than perfectly match the corresponding input set, is allowed to exceed a given threshold for a given similarity function. The goal is to produce a tree that maximizes the total weight of the sets for which it contains a matching category. A key parameter is an upper bound on the number of categories an item may belong to, which produces the hardness of the problem, as initially each item may be contained in an arbitrary number of input sets. For this model, we prove inapproximability bounds, of order \(\tilde{\Theta }(\sqrt {n}) \) or \(\tilde{\Theta }(n) \) , for various problem variants and special cases, loosely justifying the aforementioned heuristic approach. Our work includes reductions based on parameterized randomized constructions that highlight how various problem parameters and properties of the input may affect the hardness. Moreover, for the special case where the category must be identical to the corresponding input set, we devise an algorithm whose approximation guarantee depends solely on a more granular parameter, allowing improved worst-case guarantees, as well as the application of practical exact solvers. We further provide efficient algorithms with much improved approximation guarantees for practical special cases where the cardinalities of the input sets or the number of input sets each items belongs to are not too large. Finally, we also generalize our results to DAG-based and non-hierarchical categorization.

Two-cardinal ideal operators and indescribability

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner, Feng and the first author, from the cardinal setting to the two-cardinal setting, we study hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.

J-Open Independent Sets in Graphs

Let $G\neq \overline{K}_n$ be a graph with vertex and edge-sets $V(G)$ and $E(G)$, respectively. Then\linebreak $O$ $\subseteq$ $V(G)$ is called a J-open independent set of $G$ if for every $a,b \in V(G)$ where $a\neq b$, $d_G(a,b)$ $\neq 1$, and $N_G(a) \backslash N_G(b) \neq \varnothing$ and $N_G(b)\backslash N_G(a) \neq \varnothing$. The maximum cardinality of a J-open independent set of G, denoted by $\alpha_J(G)$, is called the J-open independence number of $G$. In this paper, we introduce new independence parameter called J-open independence. We show that this parameter is always less than or equal to the standard independence (resp. J-total domination) parameter of a graph. In fact, their differences can be made arbitrarily large. In addition, we show that J-open independence parameter is incomparable with hop independence parameter. Moreover, we derive some formulas and bounds of the parameter for some classes of graphs and the join of two graphs.

Special directions on the finite affine plane

AbstractIn this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.

Revisiting drug-protein interaction prediction: a novel global-local perspective.

Accurate inference of potential drug-protein interactions (DPIs) aids in understanding drug mechanisms and developing novel treatments. Existing deep learning models, however, struggle with accurate node representation in DPI prediction, limiting their performance. We propose a new computational framework that integrates global and local features of nodes in the drug-protein bipartite graph for efficient DPI inference. Initially, we employ pre-trained models to acquire fundamental knowledge of drugs and proteins and to determine their initial features. Subsequently, the MinHash and HyperLogLog algorithms are utilized to estimate the similarity and set cardinality between drug and protein subgraphs, serving as their local features. Then, an energy-constrained diffusion mechanism is integrated into the transformer architecture, capturing interdependencies between nodes in the drug-protein bipartite graph and extracting their global features. Finally, we fuse the local and global features of nodes and employ multilayer perceptrons to predict the likelihood of potential DPIs. A comprehensive and precise node representation guarantees efficient prediction of unknown DPIs by the model. Various experiments validate the accuracy and reliability of our model, with molecular docking results revealing its capability to identify potential DPIs not present in existing databases. This approach is expected to offer valuable insights for furthering drug repurposing and personalized medicine research. Our code and data are accessible at: https://github.com/ZZCrazy00/DPI.

Inverse Fair Restrained Domination in the Join of Two Graphs

The inverse fair restrained domination number of G denoted by γ_frd^(-1) (G) is the minimum cardinality of an inverse fair restrained dominating set of G. An inverse fair restrained dominating set of cardinality γ_frd^(-1) (G) is called γ_frd^(-1)-set. In this paper, we investigate the concept and give some important results on inverse fair restrained dominating sets under the join of two graphs.

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

For an undirected connected graph G = G(V, E) with vertex set V(G) and edge set E(G), a subset R of V is said to be a resolving in G, if each pair of vertices (say a and b; a ≠ b) in G satisfy the relation d(a, k) ≠ d(b, k), for at least one member k in R. The minimum set R with this resolving property is said to be a metric basis for G, and the cardinality of such set R, is referred to as the metric dimension of G, denoted by dim v (G). In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by HCN s ) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of HCN s . Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.

Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG(3,q) $(3,q)$

AbstractLet be the graph whose vertices are the chambers of the finite projective 3‐space , with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of contains , or , or at most elements. For the structure of the largest maximal independent sets is described. For the structure of the maximal independent sets of the three largest cardinalities is described. Using the cardinality of the second largest maximal independent sets, we show that the chromatic number of is .

Secure Inverse Domination in the Corona and Lexicographic Product of Two Graphs

As secure domination and inverse domination garnered attention from various researchers, the combination of the two also raised a certain amount of curiosity. This paper aimed to investigate the secure inverse domination in graphs which is defined as follows. Let G be a connected simple graph and let D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is an inverse dominating set of G with respect to D. The set S is called a secure inverse dominating set of G if for every u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪{u} is a dominating set of G. The secure inverse domination number of G, denoted by γ_s^((-1) ) (G), is the minimum cardinality of a secure inverse dominating set of G. A secure inverse dominating set of cardinality γ_s^((-1) ) (G) is called γ_s^((-1) )-set. Particularly, the researchers examined and provided the characterization of secure inverse dominating set in the corona and lexicographic product of two graphs in this study. Moreover, the secure inverse domination number of graphs under the binary operations corona and lexicographic product were determined.

Outer-restrained Domination in the Lexicographic Product of Two Graphs

Let G be a connected simple graph. A set S⊆V(G) is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in (G)∖S . A set S of vertices of a graph G is an outer-restrained dominating set if every vertex not in S is adjacent to some vertex in S and V(G)∖S is a restrained set. The outer-restrained domination number of G, denoted by (γ_r ) ̃(G) is the minimum cardinality of an outer-restrained dominating set of G. An outer-restrained set of cardinality (γ_r ) ̃(G) will be called a (γ_r ) ̃(G) -set. This study is an extension of an existing research on outer-restrained domination in graphs. In this paper, we characterized the outer-restrained domination in graphs under the lexicographic product of two graphs.

Inverse Doubly Connected Domination in the Lexicographic Product of Two Graphs

Let G be a nontrivial connected graph. A dominating set D⊆V(G) is called a doubly connected dominating set of G if both 〈D〉 and 〈V(G)\D〉 are connected. Let D be a minimum connected dominating set of G. If S⊆V(G)\D is a connected dominating set of G, then S is called an inverse doubly connected dominating set of G with respect to D. Furthermore, the inverse doubly connected domination number, denoted by γ_cc^(-1) (G) is the minimum cardinality of an inverse doubly connected dominating set of G. An inverse doubly connected dominating set of cardinalities γ_cc^(-1) (G) is called γ_cc^(-1)-set. In this paper, we characterized the inverse doubly connected domination in the lexicographic product of two graphs and give some important results.

What Are the Different Infinities?

This article explores the concept of infinity in mathematics, inspired by the work of Georg Cantor. It delves into Cantor's theorem on the cardinality of power sets, the absence of a “largest” set, and the uncountable nature of real numbers.

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

A bound on relative lengths of triangle-free graphs

For a 2-connected graph G, the relative length of G, denoted by diff(G), is the difference between the orders of a longest path and a longest cycle in G. This parameter is used as a measure to estimate how close a given graph is to a hamiltonian graph. Let σk(G) be the least value of the sums of degrees of vertices in independent sets of cardinality k. In 2008, Paulusma and Yoshimoto proved that a 2-connected triangle-free graph G of order n with σ4(G)≥n+2 satisfies diff(G)≤1 unless G is isomorphic to one exceptional graph G0. In this paper, we extend their result and prove that for an integer s with 23(n+4)<s≤n+2, a 2-connected triangle-free graph of order n with σ4(G)≥s satisfies diff(G)≤n+3−s unless s=σ4(G)=n+2 and G=G0.

On the cardinality of unique range set for two minimum weights over a non-Archimedean field

On the cardinality of unique range set for two minimum weights over a non-Archimedean field

On the Minkowski dimension of some invariant sets of dynamical systems

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.

Restrained Inverse Domination in the Lexicographic and Cartesian Products of Two Graphs

Let G be a connected simple graph and D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is called an inverse dominating set of G with respect to D. An inverse dominating set S is called a restrained inverse dominating set of G if every vertex not in S is adjacent to a vertex in S and to a vertex in V(G)∖S. The restrained inverse domination number of G, denoted by, γ_r^((-1) ) (G), is the minimum cardinality of a restrained inverse dominating set of G. A restrained inverse dominating set of cardinality γ_r^((-1) ) (G) is called γ_r^((-1) ) (G) is called γ_r^((-1) )-set. This study is an extension of an existing research on restrained inverse domination in graphs. In this paper, we characterized the restrained inverse domination in graphs under the lexicographic and Cartesian products of two graphs.