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A time-dependent singularly perturbed boundary turning point problem is considered. The numerical approximation to the solution of the problem is achieved by using a weighted-$𝜃$ method on uniform mesh in time direction and streamline diffusion finite element method (SDFEM) on layer adapted mesh in space direction. Properties of the discrete Green’s functions are utilized to ensure $𝜀$-uniform stability and optimality of the SDFEM. We perform the convergence analysis for the proposed scheme both in space and time direction. The convergence results are obtained in the maximum norm by the decomposition of the solution into smooth and layer component. It is demonstrated that the proposed numerical scheme is $𝜀$-uniformly convergent of order one for ($𝜃=1$) and of almost order two for ($𝜃=1/2$). Some relevant numerical examples are taken to illustrate the theoretical findings and efficiency of the numerical method.

Accurate predictions for the spread and evolution of epidemics have significant societal and economic impacts. The temporal evolution of infected (or dead) persons has been described as an epidemic wave with an isolated peak and tails. Epidemic waves have been simulated and studied using the classical SIR model that describes the evolution of susceptible (S), infected (I), and recovered (R) individuals. To illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions, we propose a Korteweg–de Vries (KdV)–SIR equation and obtain its analytical solutions. Among classical and simplified SIR models, our KdV–SIR equation represents the simplest system that produces a solution with both exponential and oscillatory components. The KdV–SIR model is mathematically identical to the nondissipative Lorenz 1963 model and the KdV equation in a traveling-wave coordinate. As a result, the dynamics of an epidemic wave and its predictability can be understood by applying approaches used in nonlinear dynamics, and by comparing the aforementioned systems. For example, a typical solitary wave solution is a homoclinic orbit that connects a stable and an unstable manifold at the saddle point within the $I$–$I^{\prime }$ space. The KdV–SIR equation additionally produces two other types of solutions, including oscillatory and unbounded solutions. The analysis of two critical points makes it possible to reveal the features of solutions near a turning point. Using analytical solutions and hypothetical observed data, we derive a simple formula for determining predictability horizons, and propose a method for predicting timing for the peak of an epidemic wave.

The reaction path is an important concept in theoretical chemistry. We discuss different definitions, their merits as well as their drawbacks: IRC (steepest descent from saddle), reduced gradient following (RGF), gradient extremals, and some others. Many properties and problems are explained by two-dimensional figures. This paper is both a review and a pointer to future research. The branching points of RGF curves are valley-ridge inflection (VRI) points of the potential energy surface. These points may serve as indicators for bifurcations of the reaction path. The VRI points are calculated with the help of Branin's method. All the important features of the potential energy surface are independent of the coordinate system. Besides the theoretical definitions, we also discuss the numerical use of the methods.

This paper provides a quantitative analysis of how the changing dual economic structure and urbanization affect inequality in Asia. Focusing on data for four countries—the Peoples’ Republic of China, India, Indonesia, and the Philippines—the paper asks three questions. First, how much of the past increase in inequality can be attributed to urbanization per se—the rising share of urban population, as opposed to other drivers related to the region's dual economic structure, such as the urban–rural income gap, inequality within the urban sector, and inequality within the rural sector? Second, how might urbanization affect these countries’ inequality in the future as its process continues? Third, moving forward, what is the relative importance of each of these drivers in containing rising inequality in Asia? It is hoped that the framework developed and calculations presented in this paper provide more insights into the dynamics of rising inequality in Asia and can help policy makers prioritize policy actions for confronting it.

When a robot navigates in indoor environments must often identify and then execute a turning point. For this purpose our work intends to give a useful tool that allows in real time to individualize and to determine the distance from the end of a hallway and so from a turning point. The implemented system is based on the simple Visual Looming method known in literature to determine the distance from well-defined objects. In this context the innovative idea is the application of the Visual Looming as method to calculate the distance from a turning point, that is a not well-defined object but it is identified by the intersection of three lines in a particular form. Using a mobile robot that performs a turning point in a corridor, experimental results have demonstrated the efficiency of this method to detect the turning point and then calculate the distance from this.

Let {p_n(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e^−υ(x), where υ(x) = x^2m + … is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for p_n(x), as n→∞, which holds uniformly for 0 ≤ x ∞ O(n^½m). As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.