In this article, we study an optimal control problem associated with a nonlinear hyperbolic equation, including cubic polynomial nonlinearity and homogeneous Neumann conditions. Starting from an integral functional dependent on the initial data and the right-hand side of the equation, we establish the necessary conditions for optimality using a variational approach. The analysis is based on the differentiability of a functional and on the solution of the adjoint problem, allowing the formulation of the Hamiltonian and the first-order conditions. The results obtained apply to a wide class of controlled problems governed by nonlinear hyperbolic partial differential equations.
For a map $g: X \rightarrow Y$ between connected CW-complexes, we introduce the relative self-length $L \mathcal{E}(g)$ as the number of strict inclusions in the descending chain of relative monoids $\mathcal{A}_{\sharp}^k(g)$. We prove that $L \mathcal{E}(g) \leq N \mathcal{E}(g)$, where $N \mathcal{E}(g)$ is the relative selfcloseness number. For the projection $p_Y: X \times Y \rightarrow Y$, we prove $$N \mathcal{E}\left(p_Y\right)=N \mathcal{E}(X) \quad \text { and } \quad L \mathcal{E}\left(p_Y\right)=L \mathcal{E}(X) .$$ As an application, we construct two projections with the same relative self-closeness number but different relative self-lengths.
This paper introduces two prefix replacement operators acting on real sequences together with convolution. These operators replace finitely many initial entries of a sequence with either zeros or ones while preserving the remaining entries. Their fundamental algebraic properties are established, including idempotency and composition rules. Explicit formulas describing their actions on ordinary generating functions are derived, together with their interactions with the convolution property. The obtained results provide a direct operator framework for symbolic manipulations of ordinary generating functions without repeated coefficient expansions.
Cancer is the uncontrolled growth of abnormal cells in the body. There are many perspectives one can look at cancer and we focus on a computational aspect of how cancer grows under conditions. A kinetic model for describing cancer development and growth which takes into account the effects of Titanium Dioxide nanoparticles is proposed and investigated. The model has five parameters that control the growth of the cancer tumor; the mitosis of cancer cells, the binding with nanoparticles, the unbinding with nanoparticles, the toxicity to cancer cells by nanoparticles, and the necrosis of dead cancer cells. The model is then implemented by stochastic cellular automata simulations in silico. A sensitivity analysis of some parameters in the Monte Carlo simulation has been carried out. The simulated results are presented and discussed.
In this paper, we introduce and investigate a Ma-Minda-type subclass of bi-univalent functions of complex order governed by a $q$-analytic balloon mapping that generates a symmetric balloon-shaped image domain. The class is defined via suitable subordination conditions imposed on both the function and its inverse, thereby incorporating the geometric influence of $q$-calculus into the analytic structure of the associated mappings. We derive explicit coefficient estimates for the initial Taylor-Maclaurin coefficients $\left|\vartheta_2\right|$ and $\left|\vartheta_3\right|$, expressed in terms of $q$-integers and the governing parameters of the class. Furthermore, we establish a piecewise Fekete-Szegö inequality for the functional $\left|\vartheta_3-\lambda \vartheta_2^2\right|$, revealing its precise dependence on the deformation parameter $\lambda$ and the balloon-shaped geometry. The limiting behavior as $q \rightarrow 1^{-}$is examined, showing that several known results from the classical Ma-Minda theory are recovered as special cases.
This paper extends Yitzhaki’s Extended Gini Coefficient (EGC) to the evaluation of multidimensional welfare distributions. The proposed Multidimensional Extended Gini Coefficient (MEGC) is constructed by applying the EGC framework to an aggregate welfare index combining several dimensions of well-being, including income, education, and health. The approach preserves the normative interpretation of the inequality-aversion parameter while allowing inequality to be decomposed into dimension-specific contributions. We establish the main properties inherited from the underlying extended Gini framework and derive an additive decomposition under linear aggregation. The empirical behavior of the proposed measure is illustrated through simulation experiments based on synthetic multidimensional populations. Results show that the MEGC increases with inequality aversion and provides useful information on the relative contribution of each dimension to overall inequality. The proposed framework offers a flexible tool for multidimensional welfare analysis and policy evaluation.
In this paper, we introduce the concept of $\ast$-tri-multiplier on a ring $\mathcal{R}$, where $\ast$ represents involution on $\mathcal{R}$, and obtain some commutativity theorems for a prime ring by utilizing the role of $\ast$.
We investigate the transmission dynamics of antimicrobial resistance in a hospital setting using a deterministic compartmental model with uncolonized and colonized patients and environmental bacterial load, incorporating external importation mediated by hospital visitors. The model is shown to be well-posed, and its equilibrium and stability are analyzed. The basic reproduction number is derived using the next-generation matrix method, and sensitivity analysis is performed. Numerical simulations are conducted to illustrate the analytical results. The findings show that external importation alters system behavior, eliminating classical threshold dynamics leading to persistent infection characterized by a unique positive equilibrium.
A vertex cover $S\subseteq V(G)$ is called a total vertex cover of $G$ if the graph $\langle S \rangle$ induced by set $S$ does not contain isolated vertices, i.e., $ | N_G(v) \cap S | \ge 1$ for every $v \in S$. The total vertex cover number of $G$, denoted by $\beta_t(G)$, is the minimum cardinality of a total vertex covering of $G$. In this paper, we show that given two positive integers $a$ and $b$ such that $2 \le a \le b$, there exist a connected graph $G$ such that $\gamma _t(G) = a$ and $\beta_t(G) = b$, where $\gamma _t(G)$ is the total domination number of $G$. We also characterize the total vertex covers of the join, corona, edge corona, and lexicographic product of two graphs. From these characterizations, we determine a bound or the exact value of the total vertex cover number of each of these graphs.
In this paper, we introduce and investigate a novel Ma-Minda type family of $q$-Yamakawa-type bi-univalent functions associated with a $q$-deformation of a symmetric balloon-shaped domain within the framework of $q$-calculus for $0
In this paper, we introduce and investigate a new family of special polynomials called the degenerate bivariate poly-Fubini polynomials through an appropriate generating function involving the polylogarithm function and the degenerate exponential function. Several fundamental properties of these polynomials are derived, including addition formulas, derivative identities, and explicit representations. We further establish relationships between the degenerate bivariate poly-Fubini polynomials and the degenerate poly-Bernoulli polynomials. In addition, identities involving the Stirling numbers of the second kind and the degenerate Stirling numbers of the second kind are obtained. Several special cases and limiting forms are also discussed, showing that the introduced polynomials generalize the classical poly-Fubini polynomials and the degenerate Fubini polynomials.
In this paper, we investigate dispersion properties of bivariate count models through their stochastic generation procedures. We develop cross-tables to systematically identify overdispersion, equidispersion and underdispersion in pairs of count random variables. We review several dispersion indices, including the scaled generalized variance, the generalized dispersion index, the multiple marginal dispersion index and the Minkova-Balakrishnan index. We also establish relationship between them. After several authors had constructed the bivariate extended Poisson laws of type 1, type 2, and type 3, in this paper, we have introduced the type 4. At the end, we study real data and simulate the bivariate extended Poisson laws of type 1, type 2, type 3, and type 4. It results that the type 1 model is undetermined while the others are underdispersed.
In this article, the Elzaki homotopy perturbation method (EHPM) is applied to solve fractional-order linear and nonlinear partial differential equations. This hybrid approach combines the Elzaki transform with the homotopy perturbation method (HPM). The introduction of the Elzaki transform helps overcome certain limitations related to convergence conditions in classical semi-analytical methods, such as the homotopy perturbation method (HPM), the variational iteration method, and the Adomian decomposition method. The solutions obtained using the EHPM are compared with those provided by the finite difference method. The results show satisfactory agreement between the exact solutions and the numerical approximations while highlighting the practical advantages of EHPM: reduced computational effort, ease of implementation, and user-friendliness. Thus, this method emerges as a promising alternative for applications in engineering and other scientific disciplines.
Let $G$ be a simple undirected graph with vertex and edge sets $V(G)$ and $E(G)$, respectively. Then a set $S\subseteq V(G)$ is a pointwise non-dominating set if for each $v \in V(G)\setminus S$, there exists $w \in S$ such that $vw \notin E(G)$. A pointwise non-dominating set $S$ is 2-step movable pointwise non-dominating if for each $x \in S$, $S \setminus \{x\}$ or $[S \setminus \{x\}] \cup \{y\}$ for some $y \in (V(G) \setminus S) \setminus N_G(x)$, is pointwise non-dominating. The minimum cardinality of a 2-step movable pointwise non-dominating set, denoted $[pnd]_m^2 (G)$, is called the 2-step movable pointwise non-domination number of $G$. In this paper, we characterize those graphs which admit a 2-step movable pointwise non-dominating set and give bounds on the 2-step movable pointwise non-domination number of a graph. We also determine the 2-step movable pointwise non-domination number of some classes of graphs. Moreover, we use the newly defined concept to characterize the 2-step movable hop dominating sets in the join of graphs and determine the value of the corresponding parameter on this graph.
The Set Partition Problem (SPP) is a classical NP-hard combinatorial optimization problem with applications in scheduling, resource allocation, cryptography, and operations research. In this work, a semi-analytical framework based on the Homotopy Analysis Method (HAM) is developed for approximating solutions of the SPP. The discrete partitioning problem is first reformulated as a constrained continuous optimization problem using Lagrange multipliers. A homotopy is then constructed between an initial guess and the full nonlinear system, and a Padé $[1, 1]$ series representation is employed to obtain approximate analytical solutions. The resulting algebraic equations are solved up to second order in the homotopy parameter, and the explicit Taylor-series derivation of these equations is given, together with a discussion of the conditions under which the resulting linear system for the Padé coefficients becomes degenerate. A numerical example involving a six-element set is investigated, and the influence of the convergence-control parameter on a constraint-violation residual is analyzed statistically over multiple random initial guesses, using a statistically motivated outlier criterion. The results show that the residual varies non-monotonically with the convergence-control parameter, with no clearly defined convergence region, and that the rounded partition assignments obtained from the second-order Padé approximation fail to satisfy the exact partition condition for the tested samples. The study illustrates the potential and, in its present second-order form, the significant limitations of semi-analytical homotopy techniques when applied to NP-hard combinatorial optimization problems. Furthermore, it identifies higher-order approximations and pole-avoidance strategies as necessary directions for improvement.
In this study, the HK transformation is employed to solve ordinary differential equations (ODEs) with variable coefficients. A generalized theorem is derived for handling such equations, and the HK transform formulas for the functions and are established. ODEs with variable coefficients play a significant role in analyzing real-life systems arising in various scientific disciplines. Many differential equation models of practical relevance involve variable coefficients due to inherent variations in time, space, or environmental factors. The HK transformation, combined with other analytical techniques, offers a direct and efficient method for converting these ODEs into simpler algebraic equations, facilitating the derivation of exact solutions. Several illustrative examples are presented to confirm the effectiveness of the proposed approach and to demonstrate its applicability in solving complex variable-coefficient ODEs.
In this paper, we investigate the effect of the limiting extensibility parameter constant $J_m$ and shear modulus on the pure azimuthal shear deformation of a cylindrical material whose strain energy is modeled as Gent. Here we first analyze the deformation equation and solve the second order nonlinear ordinary differential equation which results from the analysis to obtain the angular displacement and shear stresses associated with the deformation at every component of the cylinder and, with the help of appropriate boundary conditions, we obtain the constants involved in the solution. Different values of the limiting extensibility material constant and shear modulus are considered in the solution with constant boundary conditions. From the available tables of values, we plot the graphs and establish the effect of limiting extensibility material constant and shear modulus on the deformation of cylindrical material whose strain energy function is Gent. Investigation and results show that increasing the values of the material constant and shear modulus of a given material increases the shear stresses and shear strains of the Gent materials.
We introduce the bicentric edge join of two graphs and investigate its adjacency spectrum. The operation is constructed from two central graphs by joining the subdivision vertices of one central graph to the subdivision vertices of the other. We first obtain a block representation for the adjacency matrix of a general bicentric edge join in terms of the 0-1 incidence matrices of the factor graphs. We then obtain a general characteristic-polynomial formula when the two factors are connected regular graphs of degree at least two. This formula yields compact spectral descriptions for $K_m \divideontimes K_n$, $C_m \divideontimes C_n$, and $C_m \divideontimes K_n$. We also give exact determinant-root descriptions for the path-involving cases $P_m \divideontimes K_n$ and $C_m \divideontimes P_n$. Finally, we show that the operation preserves adjacency cospectrality among regular factors and exhibit an integral bicentric edge join.
Let $G$ be a real compact Lie group and $M$ a Hilbert $C^{\ast}$-module over a $C^{\ast}$-algebra $A$. In this work, we study the Fourier multipliers (operators) on $M$-valued integrable functions on $G$, with symbols by bounded $A$-valued functions on the dual object of $G$. We investigate the boundedness and the adjointability of these operators on Bochner-Lebesgue $L^p$-spaces and on their corresponding $p$-Fourier spaces in the framework of analysis of Hilbert $C^{\ast}$-module-valued maps.
In this paper, we establish results on the existence of fixed points within the framework of perturbed metric spaces for mappings that satisfy an extension of the $\mathbf{F}$-contraction, thereby generalizing the $\mathbf{F}$-contraction concept in metric spaces. Several corollaries are derived from the main theorems to illustrate their connections with more recent results in the literature. As an application, we obtain the existence and uniqueness of solutions to a non-linear Fredholm integral equation.
