In this article, we have discussed the properties of the probability law "T" called functional capacity and other closely related functionals "Q and C" pertaining to random closed sets. We are interested in the most widely used functional in random set theory "T". We have established the belonging of "T" to the interval [0,1], and proven that it is increasing in the sense of inclusion, and its sub-additivity property through probability techniques. Moreover, we have explored the various types of convergences of a sequence of random closed sets, such as weak convergence, strong convergence (almost surely in the sense of Hausdorff), convergence in the sense of Painlevé -Kuratowski and Wijsman-Mosco, as well as convergence in probability. In the second part of our work, we have proven a new corollary which states that the strong convergence in the sense of Hausdorff implies the convergence in probability of a sequence of random closed sets at infinity. Our proof involves the definition of mathematical expectation for a discrete variable and the indicator variable, which is a random variable that takes two possible values, 0 or 1.

Sometimes draws or ties occur in sports.Tiebreakers are the forms of competition that break ties and decide the winner when a draw or a tie occurs.Depending on types of tiebreakers, some take shorter and some take longer to end the competition.In this article, we are interested in calculating the expectation and variance of the number of games that will continue after a draw from types of tiebreakers that require players to win by two points.We focus on three types of win by two points that are used in many popular sports, such as tennis, volleyball and racquetball.By calculating the expected number of games, we can compare the number of games in each type of tiebreakers that will approximately be taken to end the game.In these kinds of sports, the rules to gain each point are usually the same.This means that there are the same finite states that the players or teams can reach in each point and each possible state depends only on the previous state.Since we know that a Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event, we can use an application of Markov chains to solve the problems.

The generalized inverse is crucial in matrix theory. In many applications, such as control systems, robotics, and signal processing, the generalized inverse of matrices is critical. The generalized inverse of a picture fuzzy matrix is critical to solving a variety of real-world problems. Because of their ability to handle uncertain and imprecise medical data, applications of the generalized inverse of picture fuzzy matrix have gained significant attention in the medical field. Numerous researchers have investigated generalized inverses in fuzzy matrices and intuitionistic matrices. The fuzzy picture is an effective mathematical model for dealing with uncertain realworld issues. The picture fuzzy matrix is a generalization of the classical fuzzy matrix and the intuitionistic fuzzy matrix. In this research, a method for determining the generalized inverse (g-inverse) of a picture fuzzy matrix is implemented. In addition, the concept of a standard basis for picture fuzzy vectors is established. A few results related to the g-inverse of a fuzzy picture matrix are premeditated with relevant examples. An algorithm for evaluating the generalized inverse of a fuzzy picture matrix is provided. This study concludes with an appli-cation of the g-inverse of a picture fuzzy matrix.

A Joint chance-constrained programming (JCCP) technique is regarded as one of the most useful applicable techniques of stochastic programming techniques. It is more suitable for solving uncertain real problems, especially in economics and social problems, where some of model parameters are positive dependent random variables and follow well-known probability distributions. In this paper, we take into account a linear JCCP problem where some right-hand side random parameters are dependent and follow the Dagum distributions. So, firstly we derive a bivariate Dagum distribution with seven parameters with marginals following the Dagum distribution with three parameters. This proposed bivariate Dagum distribution is based on the Farlie-Gumbel-Morgensten copula (as presented in theorem (2.1)). Secondly, the proposed bivariate distribution is used in the context of JCCP technique to transform a linear JCCP model into an exact equivalent deterministic nonlinear programming model through theorem (3.1). Thirdly, through theorem (3.2), we prove that the obtained exact equivalent deterministic nonlinear programming model is a convex model, hence any nonlinear programming method can be used to solve it and find the global optimal solution. Finally, in order to demonstrate how to convert a linear JCCP model into an equivalent deterministic nonlinear programming model and solve it using the cutting plane method, a numerical example is included.

In this article, we generalize the Riemann-Liouville fractional differential and integral operators that can be applied to the functions of a bicomplex variable. For this purpose, we consider the bicomplex Cauchy integral formula and some contours in bicomplex space. We elaborate these operators through some examples. Also, we contemplate some significant properties of these operators which include a discussion of bicomplex analytical behavior of generalized bicomplex functions through Pochhammer contours, the law of exponents, generalized Leibniz rule along with a depiction of the region of convergence, and generalized chain rule for Riemann-Liouville fractional operators of bicomplex order. We give an application of our work in the construction of fractional Maxwell's type equations in vacuum and sourcefree domains equipped with the Riemann-Liouville derivative operator. For this, we define bicomplex grad, div, and curl operator with the help of these newly defined operators. The advantage of this fractional construction of Maxwell's equation is that it may be used to build fractional non-local electronics in bicomplex space. By considering bicomplex vector fields for the respective domains, we reduce the number of these fractional Maxwell's type equations by half, which makes it easier to extract electric and magnetic fields from the bicomplex vector fields.

In this paper, graphs called overtrees are introduced and studied. These are connected graphs that contain a single simple cycle. Such graphs are connected graphs following the trees in terms of the number of edges. An overtree can be obtained from a tree by adding an edge to connect two non-adjacent vertices of a tree. The same class of graphs can also be defined as a class of graphs obtained from trees by replacing one vertex of the tree with a simple cycle. The main characteristics of overtrees are <img src=image/13433172_02.gif>, which is the number of vertices, and <img src=image/13433172_03.gif>, which is the number of vertices of a simple cycle (<img src=image/13433172_01.gif>). A formula for the chromatic polynomial of an overtree is obtained, which is determined by the characteristics <img src=image/13433172_02.gif> and <img src=image/13433172_03.gif> only. As a consequence, it is obtained the formula for the chromatic function of a graph which is built from a tree by replacing some of its vertices (possibly all) with simple cycles of arbitrary length. It follows from these formulas that any overtree with an even-length cycle is two-colored, and with an odd-length cycle is three-colored. The same is true for graphs obtained from trees by replacing some vertices with simple cycles.

The concept of partition dimension in graph theory was first introduced by Chartrand et al. [1] as a variation of metric dimension. Since then, numerous studies have attempted to determine the partition dimensions of various types of graphs. However, for many types of graphs, their partition dimensions remain unknown as determining a general graph's partition dimension is an NP-complete problem. In this study, we aim to determine the partition dimension of a specific graph, namely the comb product of a wheel and a tree. One approach to finding the partition dimension of a graph is to determine its upper and lower bounds. In this article, we propose an upper bound for the partition dimension of the comb product using number representation for certain bases. We divide the problem into two cases based on the path graph. For the first case, which is the comb product with a path of a single vertex, Tomescu et al. [2] have already provided an upper bound. In the other case, we utilize the bijection property of a number system on the number copy of the tree to find an upper bound. Our results show that the partition dimension of the second case has a smaller upper bound compared to the general upper bound proposed by Chartrand et al. [1] .

Molecular descriptors act as an important part in mathematical chemistry, in investigating quantitative structure-property relationship and quantitative structureactivity relationship. A topological descriptors, also called a molecular descriptor, is a mathematical formula applied to any graph which produces new molecular structure. In medicine mathematical model, the chemical compound is represented as an undirected graph, where each vertex represents an atom and each edge indicates a chemical bond between these atoms. The Wiener index is the first topological index to be used in chemistry introduced by Harold Wiener [1947]. It is used to compare the boiling points of some alkane isomers. There are various topological indices which are applied in chemistry. Among them, our interest is on Forgotten index which is degree based topological index introduced by Furtula and Gutman in 2015[2], defined as u where d u is the degree of vertex u in G. The mathematicians and chemists have studied several general properties of Forgotten index which may help the chemical and pharmaceutical industry to achieve the significance details by quantitative methods than by experiments. Vaidya et al (2009) proposed the concept of duplication of a vertex by an edge and duplication of an edge by a vertex of graphs. Shobana et al. proposed the double duplication of graphs (2017) [6] . Only connected, simple, undirected and finite graphs are considered throughout this article. Also, some inequalities are obtained by comparing the duplication and double duplication of graphs using Forgotten index which can also be used by chemists to generate new antidrug in future.

Univariate time series forecasting is a crucial machine learning issue across many fields notably sentiment analysis, economy, medicine, agriculture, and finance. In this working paper, we tackled comparing the Support Vector Regression (SVR) to the traditional Autoregressive Integrated Moving Average (ARIMA) algorithms in terms of forecasting through a real case study. In fact, the data set used in this investigation has been extracted from the World Bank. The target time series is the American Foreign direct investment, net outflows (% of GDP) which includes the data for 50 years from 1972 to 2021. For analytical and comparison purposes, all the compilations have been done using the R programming language for Windows 10. The statistical findings revealed that, in short-term prediction, the forecast accuracy of both algorithms reduces in terms of error accuracy, significantly. Comparatively, the analysis conducted in this investigation demonstrates that the machine learning algorithms, especially the SVM one perform better than the ARIMA in short-term forecasting since its accuracy functions are the lowest. Thus, we highly recommend future research to compare the advanced machine learning algorithms especially the recurrent neural network algorithms with the classical algorithms, especially with the ARIMA approach in order to choose the best algorithm in terms of results and predictive performance.

, who introduced the first simple graph G, defined it as a quadratic residue graph modulo n if its vertex set is reduced, a residue system modulo n such that two different vertices a and b are nearby, and a 2 ≡b 2 (mod n). This initiates to study the present article, here we introduce a cartesian product of quadratic residue graphs F = G m •H n , where m and n are either prime or composite, and G m and H n are quadratic residue graphs, respectively. The aforementioned work suggests and evaluates the regular graphs that are produced from graph F and its adjacency matrix. In addition, we define and examine their generating matrices with the help of adjacency matrix of F. Also, in this article we define three linear codes that are taken from the graph F and the parameters of codes denotes [N, k, d], where N denotes length, k denotes the dimension which is taken from the number of vertices and d denotes the distance which is taken from the minimum degree. Moreover, we also introduce an encoding and decoding algorithm for the graph using binary bits which is illustrated with a suitable example. Finally, we test the error correction capability of the code by using sphere packing bounds.