Cardinality Of Set Research Articles

Irredundance trees of diameter 3

A set D of vertices of a graph G=(V,E) is irredundant if each non-isolated vertex of G[D] has a neighbour in V−D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G).The IR-graph of G has the IR(G)-sets as vertex set, and sets D and D′ are adjacent if and only if D′ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in D′. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K1,2n.

Arens regularity for totally ordered semigroups

Let S be a semigroup. We shall consider the centres of the semigroup (beta ,S, ,Box ,) and of the algebra (M(beta ,S), ,Box ,), where M(beta ,S) is the bidual of the semigroup algebra (ell ^{,1}(S),,star ,), and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S^* and of M(S^*) that are ‘determining for the left topological centre’ (DLTC sets) of beta ,S and M(beta ,S). It is known that, when the semigroup S is cancellative, ell ^{,1}(S) is strongly Arens irregular and that there is a DLTC set consisting of two points of S^*. In contrast, there is little that has been published about the Arens regularity of ell ^{,1}(S) when S is not cancellative. Totally ordered, abelian semigroups, with the map (s,t)rightarrow s wedge t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of beta ,S and of M(beta ,S) for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for beta S or for M(beta S). There was no previously-known example of an abelian semigroup S for which beta ,S or M(beta S) did not have a finite DTC set.

Resolving sets tolerant to failures in three-dimensional grids

An ordered set S of vertices of a graph G is a resolving set for G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any k vertices from the set. This is equivalent to finding (k+1)-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differs in at least k+1 coordinates. This problem is also related with the study of the (k+1)-metric dimension of a graph, defined as the minimum cardinality of a (k+1)-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of kge 1 for which there exists a (k+1)-resolving set and construct such a resolving set of minimum cardinality in almost all cases.

Connecting $k$-Naples Parking Functions and Obstructed Parking Functions via Involutions

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$.
 In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.

ПРАКТИЧЕСКОЕ ПРИМЕНЕНИЕ МЕТОДОВ МАШИННОГО ОБУЧЕНИЯ НА ПРИМЕРЕ ОПРЕДЕЛЕНИЯ АКТИВНОСТИ ТУБЕРКУЛЕЗНОГО ПРОЦЕССА У ЛИЦ С МИНИМАЛЬНЫМИ ТУБЕРКУЛЕЗНЫМИ ИЗМЕНЕНИЯМИ, ВЫЯВЛЕННЫМИ НА РЕНТГЕНОГРАММЕ ОРГАНОВ ГРУДНОЙ КЛЕТКИ

Background: According to the World Health Organization (WHO), over 23% of the human population infected with tuberculosis bacilli – M.tuberculosis. For each infected person, the likelihood of a transition from a state of latent tuberculosis infection (LTBI) to active tuberculosis remains. WHO considers testing and treatment for LTBI in groups at high risk of reactivations as a necessary condition for tuberculosis elimination. Early detection of the transition of LTBI into active tuberculosis presents a certain difficulty due to the absence of clinically and radiographically distinguishable symptoms of the onset of the disease, therefore, immunodiagnostics, including the use of a skin test with a recombinant tuberculosis allergen, and methods of clinical laboratory diagnostics come to the aid of clinicians. The objective of our study was to demonstrate the possibilities of using an artificial intelligence system to identify the level of activity of tuberculosis infection in children with the presence of small tuberculosis changes in the respiratory organs, detected by X-ray. Material and methods The total number of patients registered in anti-tuberculosis institutions enrolled in the study was 489, including: the main group – a training sample consisting of patients with confirmed active tuberculosis (n1 = 369); the control group – a test sample of patients in whom the pathogen was in an inactive form (n2 = 120). As variables for calculations: anamnesis, laboratory parameters and X-ray data, were obtained by routine methods and used in accordance with current national standards of care and clinical guidelines, which did not require additional invasive interventions, equipment and material costs. The above survey results: age, gender, medical history, BCG vaccination, blood biochemical parameters in dynamics, X-ray signs, formalized according to the binary principle (presence / absence), were retrieved from patients files into the study database based on MS Excel spreadsheets for further processing. At the initial stage, the Wolfram Mathematica software package was used for calculations; six classical machine learning (ML) methods were carried out: Logistic Regression, Naive Bayes, Nearest Neighbors, Neural Network and Random Forest. Results: The results of the calculations based on combinations of categorical features did not suit us in terms of the quality of the forecast, and we proceeded to the search for a decision rule based on quantitative features. All spelled above methods predicted the presence of the disease significantly better than its absence. The Random Forest method showed the best results for both categorical and quantitative traits, however, interpretation of its results was not possible for clinical decision making. Convinced of the non-optimality of applying classical ML methods, it was decided to apply the author’s committee machine method with the possibility of minimal correction of conditions for significantly different cardinalities of the separable sets and subsequent geometric interpretation of the results. As a result of the application of the committee machine method, 7 most informative parameters were identified to create a decision rule that makes possible to distinguish patients with inactive pathogen and who do not require treatment in children with suspected tuberculosis. Conclusion: the committee machine method in a geometric formulation lead to localize areas in the feature space that correspond to sick and healthy patients from the training sample. That areas were unambiguously described in the form of a system of inequalities and could be easily explained to clinicians and allow moving from a geometric interpretation to a meaningful description of cause-and-effect relationships between the laboratory parameters in a certain area and the patient’s condition.

Independent Transversal Domination Number for Some Transformation Graphs $G^{xyz}$ when xyz=+-+

A dominating set of a graph $G$ which intersects every independent set of a maximum cardinality in $G$ is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of $G$ and is denoted by $\gamma_{it}(G)$. In this paper we investigate the independent transversal domination number for the transformation graph of the path graph $P_{n}^{+-+}$, the cycle graph $C_{n}^{+-+}$, the star graph $S_{1,n}^{+-+}$, the wheel graph $W_{1,n}^{+-+}$ and the complete graph $K_{n}^{+-+}$.

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = (V,E,c) , compute a minimum cardinality set of vertices S⊆ V such that no nontrivial automorphism of X fixes all vertices in S . A closely related problem is computing a minimum base S for a permutation group G ≤ Sym( n ) given by generators, i.e., a minimum cardinality subset S ⊆ [ n ] such that no nontrivial permutation in G fixes all elements of S . Our focus is mainly on the parameterized complexity of these problems. We show that when k =| S | is treated as parameter, then both problems are MINI [1]-hard. For the dual problems, where k = n – | S | is the parameter, we give FPT algorithms. (2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.

A method to construct all the paving matroids over a finite set

We give a characterization of paving matroids through their sets of hyperplanes and an algorithm to construct all of them. We also give a simple proof of Rota’s basis conjecture for the case of sparse-paving matroids and for the case of paving matroids of rank $${\varvec{r}}$$ on a set of cardinality $${\varvec{n\le 2r}}$$ , and a counterexample to Oxley’s characterization of paving matroids.

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.

The cardinality of the partitions of a set in the absence of the Axiom of Choice

Abstract In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, $2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.

BILANGAN INVERS DOMINASI TOTAL PADA GRAF BUNGA DAN GRAF TRAMPOLIN

Given a simple, finite, undirected and contains no isolated vertices graph , with is the set of vertices in and is the set of edges in . The set is called the dominating set in if for every vertex of is adjacent to at least one vertex in . The set is called the total dominating set in graph if for every vertex in is adjacent to at least one vertices in . If is the total domination set with minimum cardinality of the graph and contains another total domination set, for example , then is called the inverse set of total domination respect to . The minimum cardinality of an inverse set of total domination is called the inverse of total domination number which is denoted by .The set of domination and total domination is not singular. A graph that has a total domination set does not necessarily have a inverse total domination set. In this study, exact values are found of , and and,, n is even and , where be a flower graph and T<span style='font-size:10.0pt;mso-ansi-font-siz

Bilangan Invers Dominasi Total Pada Triangular Snake Graph, Line Triangular Snake Graph, dan Shadow Triangular Snake Graph

Let G = (V(G), E(G)) be a connected graph, where V(G) is the set of vertices and E(G) is the set of edges. The set Dt(G) is called the total domination set in G if every vertex v 2 V(G) is adjacent to at least one vertex in Dt (G). Furthermore, Dt(G) must satisfy the property N(Dt ) = V(G), where N(Dt) is an open neighbourhood set of Dt(G). Suppose that Dt(G) is the total domination set with minimum cardinality. If V(G) - Dt(G) contains a total domination set Dt-1(G), then Dt-1(G) is the inverse set of total domination relative to the total domination set Dt (G). The inverse’s number of the total domination set denotes the minimum cardinality of the inverse set of total domination. This number is denoted by gt-1 (G). This article discusses the inverse’s number of total domination of the triangular snake graph (Tn), line triangular snake graph (L (Tn)), and shadow triangular snake graph (D2 (Tn)). Graph Tn is a graph obtained from the path graph (Pn) by replacing each side of the path with a cycle graph (C3). Graph L (Tn) is a graph where the vertex set in L(Tn) is the edge set on Tn, or V(L(Tn)) = E(Tn). Graph D2 (Tn) is a graph obtained by combining two copies of a graph Tn, namely Tn0 and T00n. This research shows that the graph Tn does not have an inverse of domination total, gt-1 (L (Tn)) = n for n = 4, 6, 8, gt-1 (L (Tn)) = n - 1 for n = 3, 5, 7, or n ≥ 9 with n 2 N, and gt-1 (D2 (Tn)) = b23nc for n ≥ 3 with n 2 N.

The a-average Degree Edge-Connectivity of Bijective Connection Networks

Abstract The conditional edge-connectivity is an important parameter to evaluate the reliability and fault tolerance of multi-processor systems. The $n$-dimensional bijective connection networks $B_{n}$ contain hypercubes, crossed cubes, Möbius cubes and twisted cubes, etc. The conditional edge-connectivity of a connected graph $G$ is the minimum cardinality of edge sets, whose deletion disconnects $G$ and results in each remaining component satisfying property $\mathscr{P}$. And let $F$ be the edge set as desired. For a positive integer $a$, if $\mathscr{P}$ denotes the property that the average degree of each component of $G-F$ is no less than $a$, then the conditional edge-connectivity can be called the $a$-average degree edge-connectivity $\overline{\lambda }_{a}(G)$. In this paper, we determine that the exact value of the $a$-average degree edge-connectivity of an $n$-dimensional bijective connection network $\overline{\lambda }_{a}(B_{n})$ is $(n-a)2^a$ for each $0\leq a \leq n-1 $ and $n\geq 1$. 1

Truthful ownership transfer with expert advice

When a company undergoes a merger or transfers its ownership, the existing governing body has an opinion on which buyer should take over as the new owner. Similar situations occur while assigning the host of big sports tournaments, like the World Cup or the Olympics. In all these settings, the values of the external bidders are as important as the opinions of the internal experts. Motivated by such scenarios, we consider a social welfare maximizing approach to design and analyze truthful mechanisms in hybrid social choice settings, where payments can be imposed to the bidders, but not to the experts. Since this problem is a combination of mechanism design with and without monetary transfers, classical solutions like VCG cannot be applied, making this a novel mechanism design problem. We consider the simple but fundamental scenario with one expert and two bidders, and provide tight approximation guarantees of the optimal social welfare. We distinguish between mechanisms that use ordinal and cardinal information, as well as between mechanisms that base their decisions on one of the two sides (either the bidders or the expert) or both. Our analysis shows that the cardinal setting is quite rich and admits several non-trivial randomized truthful mechanisms, and also allows for closer-to-optimal welfare guarantees.

A graph theoretic approach to non-anticipativity constraint generation in multistage stochastic programs with incomplete scenario sets

Multistage-stochastic programming (MSSP) problems under both endogenous and exogenous uncertainties with gradual realization have been gaining attention in the literature, particularly, in chemical engineering applications. Recently several approaches have been introduced to generate the minimum cardinality non-anticipativity constraint (NAC) set for stochastic programs with various scenario set structures. However, they have been limited to uncertain parameters where the realizations occur instantaneously or the full set of scenarios is required, and may require reformulating the problem. This paper proposes an algorithm for generating a minimum cardinality set of NACs for MSSP problems under both endogenous and exogenous uncertainties with gradual realization without the need for (often non-trivial) reformulations. In addition, a heuristic version of this algorithm, which does not guarantee minimum NAC set, but allows for an efficient decomposable implementation, is developed. A thorough computational study demonstrates the benefit of reducing the NAC set, and compares the case of minimum vs near-minimum NAC sets, which has not been widely studied before.

The semantics of N-soft sets, their applications, and a coda about three-way decision

This paper presents the first detailed analysis of the semantics of N-soft sets. The two benchmark semantics associated with soft sets are perfect fits for N-soft sets. We argue that N-soft sets allow for an utterly new interpretation in logical terms, whereby N-soft sets can be interpreted as a generalized form of incomplete soft sets. Applications include aggregation strategies for these settings. Finally, three-way decision models are designed with both a qualitative and a quantitative character. The first is based on the concepts of V-kernel, V-core and V-support. The second uses an extended form of cardinality that is reminiscent of the idea of scalar sigma-count as a proxy of the cardinality of a fuzzy set.

Paired domination stability in graphs

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γpr(G), of G is the minimum cardinality of a paired dominating set of G. A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γpr−-stability of G, denoted stγpr−(G). The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γpr(G) ≥ 4, then stγpr−(G) ≤ 2Δ(G) where Δ(G) is the maximum degree in G, and we characterize the infinite family of trees that achieve equality in this upper bound.

A note on simplification of z-ordinal sum construction

We study the properties of z-ordinal sum construction and show that it can always be expressed in the reduced basic form. This means that it is enough to assume only trivial semigroups in the branching set and we can always remove semigroups with duplicate carriers. We also investigate the cardinality of the minimal branching set corresponding to monotone (and non-monotone) functions defined on the unit interval constructed via z-ordinal sum construction.

Deterministic bootstrap percolation on trees

In a graph, k-bootstrap percolation is a process by which an “infection” spreads from an initial set of infected vertices, according to the rule that on each iteration an uninfected vertex with k infected neighbors becomes infected. This process continues until either every vertex is infected or every uninfected vertex has fewer than k infected neighbors. We are particularly interested in the case where every vertex is eventually infected. The cardinality of a smallest set that results in this is the k-bootstrap percolation number of the graph. In this paper, we determine the k-bootstrap percolation number for trees of small diameter, spiders, complete N-ary trees, and caterpillars. For these graph families we also consider the smallest number of iterations needed for any smallest set to spread to the entire graph. Finally, we give an upper bound for the k-bootstrap percolation number for general trees which improves upon previous results.

On crossing-families in planar point sets

A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of n points which do not contain a crossing family of size larger than 8⌈n41⌉. Both results improve the previously best known bounds.