Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

Asymptotic analysis of time dependent solutions for the coagulation equation with source and efflux

This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.

Numerical Valuation of American Basket Options via Partial Differential Complementarity Problems

We study the principal component analysis based approach introduced by Reisinger and Wittum (2007) and the comonotonic approach considered by Hanbali and Linders (2019) for the approximation of American basket option values via multidimensional partial differential complementarity problems (PDCPs). Both approximation approaches require the solution of just a limited number of low-dimensional PDCPs. It is demonstrated by ample numerical experiments that they define approximations that lie close to each other. Next, an efficient discretisation of the pertinent PDCPs is presented that leads to a favourable convergence behaviour.

Convergence of SP-iteration for generalized nonexpansive mapping in Banach spaces

UDC 517.9 Phuengrattana and Suantai [J. Comput. and Appl. Math., <strong>235</strong>, 3006 – 3014 (2011)] introduced an iteration scheme and they named this iteration as SP-iteration. In this paper, we study the convergence behaviour of SP-iteration scheme for the class of generalized nonexpansive mappings. One weak convergence theorem and two strong convergence theorems in uniformly convex Banach spaces are obtained. We also furnish a numerical example in support of our main result. In process, our results generalize and improve many existing results in the literature.

Dynamics of convergence behaviour in social media crisis communication – a complexity perspective

PurposeThe purpose of this study is to investigate communication on Twitter during two unpredicted crises (the Manchester bombings and the Munich shooting) and one natural disaster (Hurricane Harvey). The study contributes to understanding the dynamics of convergence behaviour archetypes during crises.Design/methodology/approachThe authors collected Twitter data and analysed approximately 7.5 million relevant cases. The communication was examined using social network analysis techniques and manual content analysis to identify convergence behaviour archetypes (CBAs). The dynamics and development of CBAs over time in crisis communication were also investigated.FindingsThe results revealed the dynamics of influential CBAs emerging in specific stages of a crisis situation. The authors derived a conceptual visualisation of convergence behaviour in social media crisis communication and introduced the terms hidden and visible network-layer to further understanding of the complexity of crisis communication.Research limitations/implicationsThe results emphasise the importance of well-prepared emergency management agencies and support the following recommendations: (1) continuous and (2) transparent communication during the crisis event as well as (3) informing the public about central information distributors from the start of the crisis are vital.Originality/valueThe study uncovered the dynamics of crisis-affected behaviour on social media during three cases. It provides a novel perspective that broadens our understanding of complex crisis communication on social media and contributes to existing knowledge of the complexity of crisis communication as well as convergence behaviour.

Convergence behaviour of solvation shells in simulated liquids

Energy and structural properties of locally described solvation shells are shown to not converge to bulk values moving away from the reference point. Reasons for this behaviour and methods to alleviate it are explored.

GMRES Solver for MLPG Method Applied to Heat Conduction

In recent years, meshless local Petrov-Galerkin (MLPG) method has emerged as the promising choice for solving variety of scientific and engineering problems. MLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods (such as GMRES and BiCGSTAB methods) are more competent than the direct solvers for solving a general linear system of larger size (order of millions or billions). This paper presents the use of GMRES solver with MLPG method for the very first time. The restarted version of the GMRES method is applied in connection with the interpolating MLPG method, to solve steady-state heat conduction in three-dimensional regular geometry. The performance of GMRES solver (with and without preconditioner) has been compared with the preconditioned BiCGSTAB method in terms of computation time and convergence behaviour. Jacobi and successive over-relaxation methods have been used as preconditioners in both the solvers. The results show that GMRES solver takes about 18 to 20% less CPU time than the BiCGSTAB solver along with better convergence behaviour.

When Will a Sequence of Points in a Riemannian Submanifold Converge?

Let X be a Riemannian manifold and xn a sequence of points in X. Assume that we know a priori some properties of the set A of cluster points of xn. The question is under what conditions that xn will converge. An answer to this question serves to understand the convergence behaviour for iterative algorithms for (constrained) optimisation problems, with many applications such as in Deep Learning. We will explore this question, and show by some examples that having X a submanifold (more generally, a metric subspace) of a good Riemannian manifold (even in infinite dimensions) can greatly help.