This paper is concerned with determining the shortest path for a pursuer aiming to intercept a moving target travelling at a constant speed. To address this challenge, we introduce an efficient mathematical model outlined as an optimal control problem. The proposed model is based on Dubin's path, where we concatenate two possible paths: a left-circular curve or a right-circular curve followed by a straight line. We develop and explore this model, providing a comprehensive geometric interpretation, and design an algorithm tailored to implement the proposed mathematical approach efficiently. Extensive numerical experiments involving diverse target positions highlight the strength of the model. The method exhibits a remarkably high convergence rate in finding solutions. We compare the proposed model and demonstrate its advantages through examples. For experiment purposes, we utilized the modelling software AMPL, employing a range of solvers to solve the problem. Subsequently, we simulated the obtained solutions using MATLAB, demonstrating the efficiency of the model in intercepting a moving target. The proposed model distinguishes itself by employing fewer parameters and making fewer assumptions, setting the model simplifies the complexities, and thus, makes it easier for experts to design optimal path plans.

This paper presents a comprehensive mathematical model designed to determine the most efficient trajectory for intercepting a stationary target, with a primary focus on minimizing interception time and resource utilization. The proposed model incorporates key parameters such as target location, interception speed, and environmental constraints to formulate a set of differential equations that describe the motion dynamics of both the pursuer and the target. The objective function is defined to optimize the interception path. The study employs advanced mathematical techniques, including optimal control theory and numerical optimization algorithms, to solve the formulated equations and derive the optimal path. The model is adaptable to various scenarios, accounting for different pursuit vehicle dynamics and target characteristics. Simulation results are presented to validate the effectiveness of the proposed mathematical model in practical applications. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 049- 064

Asymptomatic transmission of the coronavirus disease and the infected individual prediction has become very important in the COVID-19 outbreak study. The asymptomatic and symptomatic transmission studies are still ongoing to assess their impacts on disease monitoring and burden. However, there has been limited research on how asymptomatic and symptomatic transmissions together can affect the coronavirus disease outbreak. A mathematical model is therefore needed to be developed in order to assess the effect of these transmissions on the coronavirus disease dynamics. This paper develops a mathematical model concerning asymptomatic and symptomatic disease transmission processes in the COVID-19 outbreak. The model sensitivity has been analysed in terms of the variance of each parameter, and the local stability at two equilibrium points have been discussed in terms of the basic reproduction number (R0). It is found that the disease-free equilibrium gets stable for R0 < 1 whereas the endemic equilibrium becomes stable for R0 > 1 and unstable otherwise. The proportion of the effect of asymptomatic and symptomatic transmission rates on R0 is calculated to be approximately between 1 and 3. The results demonstrate that asymptomatic transmission has a significant impact compared to symptomatic transmission in the disease outbreak. Outcomes of this study will contribute to setting an effective control strategy for the COVID-19 outbreak.

We propose new mathematical models of inventory management in a reverse logistics system. The proposed models extend the model introduced by Nahmias and Rivera with the assumption that the demand for newly produced and repaired (remanufacturing) items are not the same. We derive two mathematical models and formulate unconstrained and constrained optimization problems to optimize the holding cost. We also introduce the solution procedures of the proposed problems. The exactness of the proposed solutions has been tested by numerical experiments. Nowadays, it is an essential commitment for industries to reduce greenhouse gas (GHG) emissions as well as energy consumption during the production and remanufacturing processes. This paper also extends along this line of research, and therewith develops a three-objective mathematical model and provides an algorithm to obtain the Pareto solution.

The path planning problem for unmanned aerial vehicles (UAVs) is important for scheduling the UAV missions. This paper presents an optimal path planning model for UAV to control its direction during target touring, where UAV and target are at the same altitude. Geometric interpretation of the given model is provided when the vehicles consider connecting an initial position to the destination position with specific target touring. We develop a nonlinear constrained model based on an arc parameterization approach to determine the UAV’s optimal path touring a target. The model is then extended to touring finite numbers of targets and optimizing the routes. The model is found reliable through several simulations. Numerical experiments are conducted and we have shown that the UAV’s generated path satisfies vehicle dynamics constraints, tours the targets, and arrives at its destination.

Generalized orthogonal designs have many useful applications in statistics, web engineering, and coding theory along with signal processing. In this paper, for every prime number p > 3, we develop a method of constructing a set of four circulant amicable matrices {V-i} satisfying ViViT = q(i)I(n) (I-n is the identity matrix of order n) over the prime field F(p) Using these matrices with T-matrices, we suggest some approaches to construct infinite classes of generalized orthogonal designs (GODs) over the prime fields F(p).

Quantum field theory can be understood through gauge theories. It is already established that the gauge theories can be studied either perturbatively or non-perturbatively. Perturbative means using Feynman diagrams and non-perturbative means using Path-integral method. Operator regularization (OR) is one of the exceptional methods to study gauge theories because of its two-fold prescriptions. That means in OR two types of prescriptions have been introduced, which gives us the opportunity to check the result in self consistent way. In an earlier paper, we have evaluated basic QED loop diagrams in (3 + 1) dimensions using the both methods of OR and Dimensional regularization (DR). Then all three results have been compared. It is seen that the finite part of the result is almost same. In this paper, we are interested to evaluate the same basic loop diagrams in (2 + 1) space-time dimensions, because of two reasons: the main reason in (2 + 1) space-time dimensions, these loops diagrams are finite, on other hand, there are divergences in (3 + 1) space-time dimensions and the other reason is to see validity of using OR to evaluate Feynman loop diagrams in all dimensions. Here we have used both prescriptions of OR and DR to evaluate the basic loop diagrams and compared the results. Interestingly the results are almost same in all cases.