International Journal of Mathematics for Industry

In this paper, we consider a model for prognosis of the effect of self-heating of a p–n-heterojunction on its properties. An analytical approach is given for analyzing the processes in the framework of the model.

Iterated stochastic integrals of nonrandom integrands are constructed in the two-dimensional case. They are applied to the velocity fluctuations in a two-dimensional flow and mean kinetic energy of the velocity fluctuations is discussed.

The degree-based topological indices are numerical graph invariants that are used to link a molecule’s structural characteristics to its physical, and chemical characteristics. In the investigation, and study of the structural features of a chemical network, it has emerged as one of the most potent mathematical techniques. In this paper, we study the degree-based topological invariants, called K-Banhatti indices, of the line graphs of some rooted product graphs namely, $C_n\{P_{k+1}\}$, $C_n\{S_{m+1}\}$, and ith vertex rooted product graph $C_{i,r}\{P_{k+1}\}$ which are derived by the concept of subdivision.

In this paper, we introduce a model of transport of a medicinal product in an organism. The model is based on the estimation of spatio-temporal distribution of concentration of product. We introduce an analytical approach for analysis of the considered transport taking into account the changing conditions. We consider the possibility to accelerate and decelerate transport of the above medicinal product.

Let ${𝒰}$ be the three-dimensional uniform spanning tree, whose probability law is denoted by $P$. For $P$-a.s. realization of ${𝒰}$, the recurrence of the simple random walk on ${𝒰}$ is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on ${𝒰}$ collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on ${𝒰}$, which provides another proof of the infinite collision property of ${𝒰}$.

Results of clustering are qualitatively evaluated by quantities called clustering indices. While many clustering indices are proposed, in B. Desgraupes [Clustering Indices (2016)], Desgraupes reviewed 27 indices, most of which are applicable only to crisp clustering and only one of which is applicable to fuzzy clustering. In the previous article [D. R. Sope and M. Fujio, On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors gazed on analytic 3 indices among them and modified them to fit fuzzy clustering by regarding the membership degrees as distribution functions of objects over clusters. In this paper, this method called fuzzification, is applied to all the indices of Desgraupes. This investigation also includes optimization of indices. A significant benefit of fuzzy clustering is that the membership degrees allow the optimization problems to be treated as continuous, whereas the ones in the crisp case are discrete, making it generally easier to solve. After giving a precise description of fuzzification which is briefly given in D. R. Sope and M. Fujio [On a fuzzification of clustering indices, in Proc. Fuzzy System Symposium of the Japan Intelligent Information Fuzzy Society, Vol. 35 (2002), pp. 443–448], the authors fuzzify all of 27 indices of Desgraupes and then solve the fuzzy clustering problems having 13 analytic indices among them as the objective functions by the gradient method.

There are many problems in graph theory, where labeling of graphs is the only alternative to solve it. Graph labeling widely appeared in frequency assignment, communication network addressing, circuit design, X-ray crystallography, coding theory, missile guidance, signal processing, etc. For any graph $G=(V,E)$, where V is the node set and $d(p,q)$ is the distance between nodes p and q, the $L(3,2,1)$-labeling of G is a mapping $\Re :V\to \{0,1,2,\dots \}$ such that $|\Re (p)-\Re (q)|\ge 3$ if $d(p,q)=1$, $|\Re (p)-\Re (q)|\ge 2$ if $d(p,q)=2$ and $|\Re (p)-\Re (q)|\ge 1$ if $d(p,q)=3$. This paper is devoted to $L(3,2,1)$-labeling of squares of path (SOP) $(P_n^2)$ and we obtained unique results. This is the first result about $L(3,2,1)$-labeling of SOP. To the best of our knowledge, no result is available related to $L(3,2,1)$-labeling of square of any classes of graphs.

Graph theory has various applications in computer science, such as image segmentation, clustering, data mining, image capturing, and networking. Fuzzy graph (FG) theory has been widely adopted to handle uncertainty in graph-related problems. Interval-valued picture fuzzy graphs (IVPFGs) are a generalization of FGs, interval-valued FGs, intuitionistic fuzzy graphs (IFGs), and interval-valued IFGs. This paper introduces the concept of interval-valued picture fuzzy sets to graph theory and presents a new type of graph called the IVPFG. Within this framework, we define the degree, order, and size of IVPFGs. The paper further explores various operations on IVPFGs, including the Cartesian product, composition, join, and union. The paper delves into the properties of these operations, providing proofs and examples to support the findings. By studying the operations on IVPFGs, we can gain insights into their behavior and leverage this knowledge for solving graph-based problems in the presence of uncertainty. Also, an application regarding merging of community is provided.

Fractional kinetic equations are of immense importance in describing and solving numerous intriguing problems of physics and astrophysics. Inequalities are important topics in special functions. In this paper, we studied the monotonicity of the extended Gauss and confluent hypergeometric function that are derived by using the inequalities on generalized beta function involving Appell series and Lauricella function. We also establish generalized fractional kinetic equation involving extended hypergeometric and confluent hypergeometric functions. The solutions of generalized fractional kinetic equation is derived and studied as an application of extended hypergeometric and confluent hypergeometric function using the General integral transform. The results obtained here are general and can be used to derive many new solutions of fractional kinetic equations involving various types of special functions.

This paper studies the Gerber–Shiu function for the insurance surplus process with additional investment under the Bachelier model. The Gerber–Shiu function allows us to study the moments of the time of ruin, which is the first time that the surplus is negative. First, we use the martingale theory in deriving the integro-differential equation of the Gerber–Shiu function. Then, we give the exact solution of the ruin probability in case the amount of claims follows the exponential distribution. Under a general distribution case, we propose a numerical method of the Gerber–Shiu function using the finite differential method based on the integro-differential equation. Then, numeric illustrations are provided to study the effect of the parameters on the Gerber–Shiu function.