Specifying attracting cycles for Newton maps of polynomials

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result in Plaza and Romero [6], which shows how to find such a polynomial of degree 2n. Moreover, we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n ≥ 2 and degree d ≥ 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.     

Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h^1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

New open modified Newton Cotes type formulae as multilayer symplectic integrators

New modified open Newton Cotes integrators are introduced in this paper. For the new proposed integrators the connection between these new algorithms, differential methods and symplectic integrators is studied. Much research has been done on one step symplectic integrators and several of them have obtained based on symplectic geometry. However, the research on multistep symplectic integrators is very poor. Zhu et al. [1] studied the well known open Newton Cotes differential methods and they presented them as multilayer symplectic integrators. Chiou and Wu [2] studied the development of multistep symplectic integrators based on the open Newton Cotes integration methods. In this paper we introduce a new open modified numerical method of Newton Cotes type and we present it as symplectic multilayer structure. The new obtained symplectic schemes are applied for the solution of Hamilton’s equations of motion which are linear in position and momentum. An important remark is that the Hamiltonian energy of the system remains almost constant as integration proceeds. We have applied also efficiently the new proposed method to a nonlinear orbital problem and an almost periodic orbital problem.     

A reduction model for eigensolutions of damped viscoelastic sandwich structures

The aim of this paper is to develop a reduction method to determine the modal characteristics of viscoelastic sandwich structures. The method is based on the high order Newton algorithm and reduction techniques. Numerical tests have been performed in the case of sandwich beams and cylindrical shells. The comparison of the results obtained by the reduction method with those given by direct simulation shows both a good agreement and a significant reduction in computational cost.     

Force-field parametrization based on radial and energy distribution functions

We propose a novel force-field-parametrization procedure that fits the parameters of potential functions in a manner that the pair distribution function (DF) of molecules derived from candidate parameters can reproduce the given target DF. Conventionally, approaches to minimize the difference between the candidate and target DFs employ radial DFs (RDF). RDF itself has been reported to be insufficient for uniquely identifying the parameters of a molecule. To overcome the weakness, we introduce energy DF (EDF) as a target DF, which describes the distribution of the pairwise energy of molecules. We found that the EDF responds more sensitively to a small perturbation in the pairwise potential parameters and provides better fitting accuracy compared to that of RDF. These findings provide valuable insights into a wide range of coarse graining methods, which determine parameters using information obtained from a higher-level calculation than that of the developed force field. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.     

A simple,fast, and accurate thermodynamic‐based approach for transfer and prediction of gas chromatography retention times between columns and instruments Part I: Estimation of reference column geometry and thermodynamic parameters

The transfer of retention times based on thermodynamic models between columns can aid in separation optimization and compound identification in gas chromatography. Although earlier investigations have been reported, this problem remains unsuccessfully addressed. One barrier is poor predictive accuracy when moving from a reference column or system to a new target column or system. This is attributed to challenges associated with the accurate determination of the effective geometric parameters of the columns. To overcome this, we designed least squares‐based models that account for geometric parameters of the columns and thermodynamic parameters of compounds as they partition between mobile and stationary phases. Quasi‐Newton‐based algorithms were then used to perform the numerical optimization. In this first of three parts, the model used to determine the geometric parameters of the reference column and the thermodynamic parameters of compounds subjected to separation is introduced. As will be shown, the overall approach significantly improves the predictive accuracy and transferability of thermodynamic data (and retention times) between columns of the same stationary phase chemistry. The data required for the determination of the thermodynamic parameters and retention time prediction are obtained from fast and simple experiments. The proposed model and optimization algorithms were tested and validated using simulated and experimental data.     

A simple,fast, and accurate thermodynamic‐based approach for transfer and prediction of gas chromatography retention times between columns and instruments Part III: Retention time prediction on target column

This is the third part of a three‐part series of papers. In Part I, we presented a method for determining the actual effective geometry of a reference column as well as the thermodynamic‐based parameters of a set of probe compounds in an in‐house mixture. Part II introduced an approach for estimating the actual effective geometry of a target column by collecting retention data of the same mixture of probe compounds on the target column and using their thermodynamic parameters, acquired on the reference column, as a bridge between both systems. Part III, presented here, demonstrates the retention time transfer and prediction from the reference column to the target column using experimental data for a separate mixture of compounds. To predict the retention time of a new compound, we first estimate its thermodynamic‐based parameters on the reference column (using geometric parameters determined previously). The compound's retention time on a second column (of previously determined geometry) is then predicted. The models and the associated optimization algorithms were tested using simulated and experimental data. The accuracy of predicted retention times shows that the proposed approach is simple, fast, and accurate for retention time transfer and prediction between gas chromatography columns.     

Numerical Experiences with New Truncated Newton Methods in Large Scale Unconstrained Optimization

Recently, in [12] a very general class oftruncated Newton methods has been proposed for solving large scale unconstrained optimization problems. In this work we present the results of an extensive numericalexperience obtained by different algorithms which belong to the preceding class. This numerical study, besides investigating which arethe best algorithmic choices of the proposed approach, clarifies some significant points which underlies every truncated Newton based algorithm.