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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.

This paper aims to consider a multiscale electromagnetic wave problem for a housing with a ventilation grill. Using the standard finite element method to discretise the apertures leads to an unduly large number of unknowns. An efficient approach to simulate the multiple scales is introduced. The aim is to significantly reduce the computational costs.

A domain decomposition technique with upscaling is applied to cope with the different scales. The idea is to split the domain of computation into an exterior domain and multiple non-overlapping sub-domains. Each sub-domain represents a single aperture and uses the same finite element mesh. The identical mesh of the sub-domains is efficiently exploited by the hybrid discontinuous Galerkin method and a Schur complement which facilitates the transition from fine meshes in the sub-domains to a coarse mesh in the exterior domain. A coarse skeleton grid is used on the interface between the exterior domain and the individual sub-domains to avoid large dense blocks in the finite element discretisation matrix.

Applying a Schur complement to the identical discretisation of the sub-domains leads to a method that scales very well with respect to the number of apertures.

The error compared to the standard finite element method is negligible and the computational costs are significantly reduced.

The purpose of this paper is to investigate the design method of partial observer canonical form (POCF), which is one of the important research tools for industrial plants.

Motivated by the two-steps method proposed in Xu et al. (2020), this paper extends this method to the case of Multi-Input Multi-Output (MIMO) nonlinear system. It decomposes the original system into two subsystems by observable decomposition theorem first and then transforms the observable subsystem into OCF. Furthermore, the necessary and sufficient conditions for the existing of POCF are proved.

The proposed method has a wide range of applications including completely observable nonlinear system, noncompletely observable nonlinear system, autonomous nonlinear system and forced nonlinear system. Besides, comparing to the existing results (Saadi et al., 2016), the method requires less verified conditions.

The new method concerning design POCF has better plants compatibility and less validation conditions.

Since the last few decades, Artificial Neural Networks have been the center of attraction of a large number of researchers for solving diversified problem domains. Due to its distinguishing features such as generalization ability, robustness and strong ability to tackle nonlinear problems, it appears to be more popular in financial time series modeling and prediction. In this paper, a Pi-Sigma Neural Network is designed for foretelling the future currency exchange rates in different prediction horizon. The unrevealed parameters of the network are interpreted by a hybrid learning algorithm termed as Shuffled Differential Evolution (SDE). The main motivation of this study is to integrate the partitioning and random shuffling scheme of Shuffled Frog Leaping algorithm with evolutionary steps of a Differential Evolution technique to obtain an optimal solution with an accelerated convergence rate. The efficiency of the proposed predictor model is actualized by predicting the exchange rate price of a US dollar against Swiss France (CHF) and Japanese Yen (JPY) accumulated within the same period of time.