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The purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.

This study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.

The stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.

In this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

We consider, as discretization in space, the nonconforming mesh developed in SUSHI (Scheme Using Stabilization and Hybrid Interfaces) developed in Eymard et al. (2010) for a semi-linear heat equation. The time discretization is performed using a uniform mesh. We are concerned with a nonlinear scheme that has been studied in Bradji (2016) in the context of the general framework GDM (Gradient Discretization Method) (Droniou et al., 2018) which includes SUSHI. We provide sufficient conditions on the size of the spatial mesh and the time step which allow to prove a W1,∞(L2)-error estimate. This error estimate can be viewed as an improvement for the W1,2(L2)-error estimate proved in Bradji (2016). The W1,∞(L2)-error estimate we want to prove in this note was stated without proof in Bradji (2016, Remark 7.2, Page 1302). Its proof is based on a comparison with an appropriately chosen auxiliary finite volume scheme along with the derivation of some new estimates on its solution.

The purpose of this paper is to propose a semiparametric estimator for the tail index of Pareto-type random truncated data that improves the existing ones in terms of mean square error. Moreover, we establish its consistency and asymptotic normality.

To construct a root mean squared error (RMSE)-reduced estimator of the tail index, the authors used the semiparametric estimator of the underlying distribution function given by Wang (1989). This allows us to define the corresponding tail process and provide a weak approximation to this one. By means of a functional representation of the given estimator of the tail index and by using this weak approximation, the authors establish the asymptotic normality of the aforementioned RMSE-reduced estimator.

In basis on a semiparametric estimator of the underlying distribution function, the authors proposed a new estimation method to the tail index of Pareto-type distributions for randomly right-truncated data. Compared with the existing ones, this estimator behaves well both in terms of bias and RMSE. A useful weak approximation of the corresponding tail empirical process allowed us to establish both the consistency and asymptotic normality of the proposed estimator.

A new tail semiparametric (empirical) process for truncated data is introduced, a new estimator for the tail index of Pareto-type truncated data is introduced and asymptotic normality of the proposed estimator is established.

The main purpose of this paper is to introduce the gradient discretisation method (GDM) to a system of reaction diffusion equations subject to non-homogeneous Dirichlet boundary conditions. Then, the authors show that the GDM provides a comprehensive convergence analysis of several numerical methods for the considered model. The convergence is established without non-physical regularity assumptions on the solutions.

In this paper, the authors use the GDM to discretise a system of reaction diffusion equations with non-homogeneous Dirichlet boundary conditions.

The authors provide a generic convergence analysis of a system of reaction diffusion equations. The authors introduce a specific example of numerical scheme that fits in the gradient discretisation method. The authors conduct a numerical test to measure the efficiency of the proposed method.

This work provides a unified convergence analysis of several numerical methods for a system of reaction diffusion equations. The generic convergence is proved under the classical assumptions on the solutions.

In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.

To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.

Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.

In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

We consider the kernel-based classifier proposed by Younso (2017). This nonparametric classifier allows for the classification of missing spatially dependent data. The weak consistency of the classifier has been studied by Younso (2017). The purpose of this paper is to establish strong consistency of this classifier under mild conditions. The classifier is discussed in a multi-class case. The results are illustrated with simulation studies and real applications.

In this research, the main purpose is to use a suitable structure to predict the trading signals of the stock market with high accuracy. For this purpose, two models for the analysis of technical adaptation were used in this study.

It can be seen that support vector machine (SVM) is used with particle swarm optimization (PSO) where PSO is used as a fast and accurate classification to search the problem-solving space and finally the results are compared with the neural network performance.

Based on the result, the authors can say that both new models are trustworthy in 6 days, however, SVM-PSO is better than basic research. The hit rate of SVM-PSO is 77.5%, but the hit rate of neural networks (basic research) is 74.2.

In this research, two approaches (raw-based and signal-based) have been developed to generate input data for the model: raw-based and signal-based. For comparison, the hit rate is considered the percentage of correct predictions for 16 days.

While there exist many surveys on the use stochastic frontier analysis (SFA), many important issues and techniques in SFA were not well elaborated in the previous surveys, namely, regular models, copula modeling, nonparametric estimation by Grenander’s method of sieves, empirical likelihood and causality issues in SFA using regression discontinuity design (RDD) (sharp and fuzzy RDD). The purpose of this paper is to encourage more research in these directions.

A literature survey.

While there are many useful applications of SFA to econometrics, there are also many important open problems.

This is the first survey of SFA in econometrics that emphasizes important issues and techniques such as copulas.

In this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true.

In this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space.

The existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space.

The purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.

The purpose of this paper is to search for the critical-scenarios of autonomous vehicles (AVs) quickly and comprehensively, which is essential for verification and validation (V&V).

The author adopted the index F1 to quantitative critical-scenarios' coverage of the search space and proposed the improved particle swarm optimization (IPSO) to enhance exploration ability for higher coverage. Compared with the particle swarm optimization (PSO), there were three improvements. In the initial phase, the Latin hypercube sampling method was introduced for a uniform distribution of particles. In the iteration phase, the neighborhood operator was adapted to explore more modals with the particles divided into groups. In the convergence phase, the convergence judgment and restart strategy were used to explore the search space by avoiding local convergence. Compared with the Monte Carlo method (MC) and PSO, experiments on the artificial function and critical-scenarios search were carried out to verify the efficiency and the application effect of the method.

Results show that IPSO can search for multimodal critical-scenarios comprehensively, with a stricter threshold and fewer samples in the experiment on critical-scenario search, the coverage of IPSO is 14% higher than PSO and 40% higher than MC.

The critical-scenarios' coverage of the search space is firstly quantified by the index F1, and the proposed method has higher search efficiency and coverage for the critical-scenarios search of AVs, which shows application potential for V&V.