Bisymmetric matrices have wide range of applications in statistics, engineering problems, information theory and computer science including coding theory and cryptography. In cryptography, a rhotrix being a couple matrix doubles the security of the cryptosystem. Here, we construct maximum distance separable (MDS) bisymmetric rhotrices using self-dual bases and conjugate elements of finite fields. MDS rhotrices are very crucial for the designing of block ciphers and hash functions in cryptography.
The coronavirus caused havoc around the world. There was a terrible situation in villages and cities, and no one knew how to deal with it. Although the governments of each country tried their best to save the common people, vaccination programs and testing centers were built everywhere. However, people were not utilizing it due to fear. Because of this, the infection spread rapidly. The qualitative study of the mathematical model here is in context with the situation when bivalent vaccination and testing are available for an epidemic. The mathematical model combines the exposed period and influenza model with vaccination included under the peer effect in rural India under vital dynamics. The boundedness and positivity of the solution for the proposed model and its unique disease-free equilibrium point with its local and global stability are established. Threshold and sensitivity analysis of the model are also taken in context to study the role of parameters. Finally, simulation supports the established theoretical results.
This research work considers a single server queueing model with differentiated vacations. In addition there is a possibility of two types of failures when the server is in a busy period; namely hard failure and soft failure. In the time of soft failure server may work with a slow service rate. We analyzed as a Quasi-Birth-and-Death (QBD) process, using the matrix geometric method, the steady state probability vector of the number of customers in the queue and the stability conditions are produced. Busy period analysis of the proposed model in given. The effects of various parameters on the system performance measures are illustrated numerically.
We examine CUSUM-type test for detecting changes in unconditional variance within Bilinear GARCH models. We derive the asymptotic distribution of the test statistic under both null and alternative hypotheses and assess test effectiveness in identifying single structural breaks. Simulation studies support our theoretical results and demonstrate the practical utility of the test.
The elliptic restricted three-body problem investigates the motion behaviour of the variable mass infinitesimal body under the gravitational forces of the radiated oblate primary and dipole secondary. The equations of motion of the infinitesimal body are determined using Jeans law and Meshcherskii space time transformations. Using the Lindstedt-Poincar & eacute; method, we perform the solutions of the equations of motion. With the use of these solutions and the equations of motion, we numerically illustrate the time series, phase spaces, projections and the Halo orbits.
This study presents a semi-analytical shooting method for solving nonlinear higher-order boundary value problems by integrating the Adomian Decomposition Method into the shooting technique, enabling series-form solutions. To enhance convergence, new higher-order shooting slopes and their corresponding supplementary equation formulas were introduced. Three numerical examples demonstrated the method's accuracy: for the first two, absolute errors were computed using available exact solutions, while the third was compared with reference literature due to the absence of an exact solution. The method achieved very small absolute errors in the first two cases, and results from the third closely matched the literature. Tolerance values-defined as differences between successive shooting slopes-were also evaluated for all cases, showing a consistent decrease with increasing slopes. These findings confirm the effectiveness and reliability of the proposed approach in producing accurate results and ensuring convergence for higher-order boundary value problems.
In this article, we consider a multi server Markovian queueing system with working vacation. During busy period, the arrival and service completion are generated by K distinct randomly varying environments. At a service completion epoch, if no customer in the system, the servers take vacation, the vacation policy is multiple vacation policy and the vacation period follows negative exponential distribution. In addition, during vacation period the servers serve customers if they arrive. Based on the vacation termination point we define two Models. For the two models, the steady state probability vector of number of customers in the queue, the stability condition and some performance measures are derived. Some illustrative examples are also provided.
Without killing vector fields, the Szekeres metric is an explicit example of an anisotropic and inhomogeneous solution to the Einstein equations. The inhomogeneous Szekeres cosmological models (ISCM) within the Big Bang singularity (BBS) are obtained. This indicates that the Szekeres solution represents a more general class of exact solutions. It is known to exhibit axial symmetry. We investigate a Big Bang theory of the cosmos, which unexpectedly predicts that the universe started at the so-called BBS a finite length of time ago. A growing number of astrophysical researchers are using inhomogeneous extensions of the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution and, by extension, Lemaitre-Tolman-Bondi (LTB) solution to investigate cosmic events. In this particular scenario, the Scwarzschild-Kruskal-Szekeres metrics, dust Robertson Walker, and LTB models with zero pressure are all contained in the Szekeres metric. In this paper, we first provide the Szekeres model solutions, and then we expand on recent discussions on the BBS in anisotropic and inhomogeneous szekeres cosmological models. The BBS for the Szekeres inhomogeneous model and some new solutions are presented.
A single server retrial queueing-inventory model is investigated in this study. In Bernoulli vacation, after providing service to the customer, the server may opt to avail vacation or start the service to subsequent customer. During the busy period, breakdown and balking may occur. The inventory is replenished to an (s, S) policy and the replenishing time is assumed to adopt an exponential distribution. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. We integrate the emergency replenishment into the system to ensure customer satisfaction. For our system, a stability criterion was developed and the stationary probability vector was evaluated by utilizing the matrix analytical approach. This model also examined the study of busy time and performance measures. Using two and three dimensional graphs, the numerical illustrations are shown.
There are many uses of queues, where services are provided in groups; these types of queues are widely studied in the literature. In this paper we examine a particular queueing model wherein the services are provided in groups and the group size may be less than or equal to the size initially fixed. The arrival follows a Markovian arrival process. The service time of each individual customer follows phase type distribution. The maximum of each customer's individual service time within a group is defined as the group's service time. At the service completion moment if there are fewer customers than the initially fixed size, the server won't begin the subsequent service until the system's customer size reaches the initially fixed size or a randomly assigned admission period expires, whichever happens first. The phase type representation of the service times depends on the group's size. If there is no customer block in the waiting line after the server finishes serving, the server will leave for vacation. If any customer block arrives within the designated vacation period, the server immediately will start serving them at a slower pace than the regular pace. After a regular service each customer group has an option of receiving the optional service from the server. The Markov chain's stability condition is determined and stationary probability vector is computed. Formulas for the primary system performance measures are given. Waiting time distribution of the model is derived. Numerical and graphical representations of the proposed model are illustrated.
This paper provides a considerably efficient numerical approach to acquire the solutions of a biomathematical model administrating oral and intravenous distribution of pharmaceuticals in the human body. The proposed numerical approach based on an artificial neural network is employed to extract numerical solutions for a detailed set of ordinary differential equations and analyze the change in concentration of drug diffusion via the compartments of blood and tissue medium. We primarily focus on analyzing three different models established on the diffusion process, exercising laws of mass action and Fick's principle. In this work, the existing model is reformulated as an optimization problem by investigating the drug distribution impacted by multiple factors related to the human body. Based on the drug efficacy, the rate constants (governing the law of mass action) are applied at different interfaces. The posed optimization problem is then solved by minimizing the concerned loss function. Also, all the attached numerical parameters have been considered while computing the drug concentration within distinct compartments. In addition, with the aid of Python programming, the presented plots show how the medication concentration changes over time. The obtained graphical results signify that the rate of change in the concentration of drugs rises gradually in other compartments while decreasing in the first. Compared to the traditional methods, the experimental results demonstrate the accuracy and efficacy of the proposed methodology evidently.
In this study, the ruled surfaces generated by Smarandache curves are expressed according to the modified orthogonal frame defined according to both curvature and torsion of the given unit speed curve in 3-dimensional Euclidean space; some special characterizations of these surfaces such as developability, striction curves, distribution parameters are given. In addition, some examples of these surfaces are given in graphical form. We use Maple to draw the graphs of the examples.
In this paper we introduce a design, called "geodetic-design" arising from geodetic sets in a graph. A geodetic-design, over a regular graph is an ordered pair D = (V, B), where V = V(G) and B, the set of all geodetic sets, called blocks, containing vertices belonging to geodetic sets, such that every pair of non-adjacent vertices appears in exactly mu blocks. We first find governing results of a geodetic-design, if it exists, and then get such PBIB-designs for different products of graphs. It is common to have geodetic sets of graphs to be independent sets, hence we extend the designs obtained from maximum independent sets of graphs to geodetic-designs for products of graphs.
The COVID-19 pandemic significantly disrupted various sectors, with higher education being one of the most severely affected. Students in higher education faced numerous challenges transitioning to online learning, leading to a surge in mental health issues. The abrupt shift in the mode of education and the inability of many students to adapt exacerbated their mental health struggles. This, in turn, contributed to a notable rise in student suicide rates in India during the pandemic-induced isolation period. Addressing this critical socio-psychological issue requires effective strategies for stress detection and management. The proposed study employed the Online Education Stress Scale (Online ESS) to collect data from students enrolled in colleges affiliated with Dr. Bhimrao Ambedkar University, Agra. The research introduced a methodology to analyse stress levels by categorizing data into three primary stress factors. For each factor, a fuzzy inference system was developed. By applying fuzzy logic, the study tackled the imprecision and vagueness inherent in psychological data, providing a more reliable system for understanding the relationships between psychological variables and stress levels.
The Lyapunov second method is an eigenvalue-based technique which consist of finding a Lyapunov function candidate for studying the stability of dynamical systems which is hard to deal with especially in most of nonlinear cases. Studying the stability and some of other qualitative behaviors based on the Lyapunov second method is research topic of actuality because of the wide range of applications of the differential equations. Many authors in the literature have used the second method of Lyapunov in the study of some qualitative behaviors from which the stability, the boundedness and the square integrability of solutions for various kinds of differential equations that are different in terms of the order, the linearity, the autonomy, the delay or the neutral case. The present paper contains two main results. The first part is dedicated to establish sufficient conditions that guarantee the boundedness and the square integrability of solutions for a given third order neutral differential equation with delay. The second part of this work is devoted to the purpose of breaking the barrier of reaching exponential stability for some cases of the previous third order differential equation using the direct method of Lyapunov. In the end of the paper, a concrete example is given to illustrate the obtained results.
In this paper, we study the dynamical analysis of a stochastic Leslie-Gower biological predator-prey model. Earlier, the Leslie-Gower model was studied in the context of biological systems, including cases involving cannibalism. In our model, we investigate the dynamic properties of a stochastic Leslie-Gower predator-prey ecological system using the stability of invariant measures on invariant sets, where the invariant measures are shown to be ergodic. We also conduct a threshold analysis to study the stochastic persistence and extinction of species. Stochastic bifurcation is also examined. The theoretical results are supported by numerical simulations and examples. Intra-species competition is considered and described through theoretical analysis. The numerical results provide insights for modeling new stochastic ecological systems. Simulations are carried out to verify all the results.
Analysis of the effects of heat and mass transfer on the chemically responding boundary layer flow of a Casson fluid across a porous stretched sheet in the existence of a crosswise magnetic field is the aim of the existing work. The partial differential equations that control the current flow problems can be converted into similarity equations by applying the right similarity technique. Our goal is to use the DTM-Pade approximation to analytically solve the derived similarity equations. The transformed similarity equations are a group of nonlinear ordinary differential equations for the present flow problem. The analytical approach of the differential transform method (DTM) with Pade approximant has been used to solve the group of nonlinear ordinary differential equations. For the velocity, temperature, and concentration, graphical approximate analytical solutions are presented. It is invigilated that increasing the Casson parameter decreases the velocity field, and the temperature and concentration increase while enhancing the Casson parameter.
The study aims to investigate the effect of magneto-hydrodynamic on a non-Newtonian unsteady blood flow with internal heat energy in the presence of blood ironic properties characterized by stenosis. The formulated mathematical equations resulted in differential forms and were solved analytically by Differential Transform Method. The obtained solutions were displayed by graphs showing different flow physiognomies like blood velocity, temperature profile, Nusselt number, wall shear stress and stream function. The results indicated that velocity profile increases as magnetic field, Darcy number and aneurysmal artery rise, while it decreases as heat radiation, Reynold number, and Casson parameter speedup. The temperature profile increases as magnetic field, and Reynold number rise. Furthermore, wall shear stress increases as heat radiation increases but drops as Reynold number accelerates. This happens due to the fact that magnetic field on blood flow increases the viscosity of the blood flow that bring rise in the Lorentz force and Reynolds number indicates the significant dominance of viscous forces over inertial forces, which keeps the flow in the laminar path.
This manuscript deals with an infinite-capacity queueing system under multiple differentiated working vacations and customers' impatience. The first vacation is assumed to be a working vacation where the server, instead of being idle, serves the customers at a lower rate. In contrast, the second one is considered a non-working vacation of a different duration. The customers may leave the system at any time due to long delays in service during vacations but, via some convincing mechanisms, they are retained in the system. The operating characteristics of the system are obtained in a steady state. The results obtained are illustrated numerically and graphically with the help of MATLAB software. The cost model is formulated for the proposed system, and the optimal cost is obtained relative to the service rate.
This research introduces a new two-parameter Marshall-Olkin Garima distribution model. The novel model has many sub-models that are useful in modeling real-life data, such as the extended Garima distribution, exponentiated Garima distribution, exponential distribution, Lindley distribution, Kumaraswamy Garima distribution, and normal distribution. The proposed model demonstrates a high level of suitability in modeling both reliability and survival data. It is flexible in accommodating various failures. The quantile function, density shapes, hazard rate functions, and order statistics are a few of the statistical features that have been explored. Maximum likelihood estimation methods were employed to estimate the parameters. Using five data sets, the suggested distribution's flexibility was shown with nuclear reactions that are important systems in the field of nuclear physics. The proposed distribution was compared with its sub-models and other existing models. The findings demonstrated that, compared to the other competing distributions, the suggested distribution offered a superior fit to the data sets.
