Accuracy order in L2 grammatical morphemes: Corpus evidence from different proficiency levels of Turkish learners of English

The present study empirically scrutinizes the fixed natural order of grammatical morphemes relying on a manual analysis of an EFL learner corpus. Specifically, we test whether the accuracy order of L2 grammatical morphemes in the case of L1 Turkish speakers of English deviates from Krashen’s (1977) natural order and whether proficiency levels play a role in the order of acquisition of these morphemes. With this in mind, we focus on the (in)accuracy of nine English grammatical morphemes with 2883 cases manually tagged by the UAM Corpus Tool in the written exam scripts of Turkish learners of English. The results based on target-like use scores provide evidence for deviation from what is widely believed to be a set order of acquisition of these grammatical morphemes by second language learners. In light of such findings, we challenge the view that the internally driven processes of mastering grammatical morphemes in English for interlanguage users are largely independent of their L1. Regardless of L2 grammar proficiency in our data, the observed accuracy of some morphemes ranked low in comparison with the so-called natural order. These grammatical morphemes were almost exclusively non-existent features in participants’ mother tongue (e.g., third person singular –s, articles and the irregular past tense forms), thus suggesting the influence of L1 in this respect.

Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

Simulation of heat transfer with considering permafrost thawing in 3D media

An approach to mathematical modeling of heat transfer with a permafrost algorithm in 3D media based on the idea of localizing the phase transition area is considered. The paper presents a problem statement for a non-stationary heat transfer and a description of a numerical method based on a predictor-corrector scheme. For a better understanding of the proposed splitting method, the accuracy order of approximation considering inhomogeneous right-hand side was studied. The phase changes in the numerical implementation of permafrost thawing is considered in the temperature range and requires recalculation of coefficients values of the heat equation at each iteration step with respect to time. A brief description of the parallel algorithm based on a 3D decomposition method and the parallel sweep method is presented. A study of the parallel algorithm implementations using a high-performance computing system of the Siberian Supercomputer Center of the SB RAS was performed. The results of the permafrost algorithm on models with wellbores are also presented.

HIGHER ACCURACY ORDER IN DIFFERENTIATION-BY-INTEGRATION

In this text explicit forms of several higher precision order kernel functions (to be used in the differentiation-by-integration procedure) are given for several derivative orders. Also, a system of linear equations is formulated which allows to construct kernels with an arbitrary precision for an arbitrary derivative order. A computer study is realized and it is shown that numerical differentiation based on higher precision order kernels performs much better (w.r.t. errors) than the same procedure based on the usual Legendre-polynomial kernels. Presented results may have implications for numerical implementations of the differentiation-by-integration method.

The Variational Iterations Method for the Three-dimensional Equations Analysis of Mathematical Physics and the Solution Visualization with its Help

The aim of the work is to use the variational iterations method to study the three-dimensional equations of mathematical physics and visualize the solutions obtained on its basis and the 3DsMAX software package. An analytical solution of the three-dimensional Poisson equations is obtained for the first time. The method is based on the Fourier idea of variables separation with the subsequent application of the Bubnov-Galerkin method for reducing partial differential equations to ordinary differential equations, which in the Western scientific literature has become known as the generalized Kantorovich method, and in the Eastern European literature has known as the variational iterations method. This solution is compared with the numerical solution of the three-dimensional Poisson equation by the finite differences method of the second accuracy order and the finite element method for two finite element types: tetrahedra and cubic elements, which is a generalized Kantorovich method, based on the solution of the three-dimensional stationary differential heat equation. As the method study, a set of numerical methods was used. For the results reliability, the convergence of the solutions by the partition step is checked. The results comparison with a change in the geometric parameters of the heat equation is made and a conclusion is drawn on the solutions reliability obtained. Solutions visualization using the 3Ds max program is also implemented.

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

КЭЭ БИР ЭЛЕМЕНТАРДЫК ФУНКЦИЯЛАРДЫ ДАРАЖАЛУУ КАТАРГА АЖЫРАТУУДА СТУДЕНТТЕРДИН ЧЫГАРМАЧЫЛЫК ОЙ ЖҮГҮРТҮҮСҮН ЖОГОРУЛАТУУНУН ПЕДАГОГИКАЛЫК ЫКМАЛАРЫ

Аннотация: Биздин заманда билим алууга болгон көз караш өзгөрдү: мурун маалымат алуу абдан маанилүү болсо, азыр маалыматтарды колдонууну билиш керек. Себеби, азыркы турмушта Google сыяктуу маалымат булактары бар. Биз биргелешкен математика курсу синергияны пайда кылып, алгебра менен геометриянын элементтерин өздөштүрүүгө жардам берет деп ишенебиз. Алгебралык, дифференциалдык жана интегралдык теӊдемелердин жакындаштырылган чыгарылыштарын тургузууда жана ошондой эле ар кандай интегралдарды баалоодо параметрдин же көз карандысыз өзгөрүлмөнүн даражасы бар катарлар менен иштөөгө туура келет. Негизинен даражалуу катарга ажыратуу Ньютондун биномунун формуласынын жардамы менен же Тейлордун катарын колдонуу жолу аркылуу тургузулат. Бул илимий макалада ошол тууралуу сөз болот. Түйүндүү сөздөр: Тейлордун катары, Маклорендин катары, катарга ажыратуу, көрсөткүчтүү функция, тригонометриялык функциялар, сумма, интервал, бардык чыныгы сандардын огу, жыйналуучу катар, Коши-Адамардын формуласы, Лагранж формуласындагы калдык мүчө, көрсөткүчү бар биномдук катар, логарифмикалык функция, барабардык, касиеттер, аргументтин мааниси, даража, тактык, тартип, баалоо. Аннотация: В области математики знание точных формулировок определений, теорем и т.п. теперь не столь важно, как умение их использовать для решения задач, связанных с окружающей действительностью. Мы убеждены в том, что курс математики, объединяющий элементы алгебры и геометрии поможет повысить уровень усвоения материала за счет эффекта синергии, возникающего при этом. При построении приближенных решений алгебраических, дифференциальных и интегральных уравнений, а также при оценке различных интегралом нам приходится иметь дело с рядами по степеням параметра или независимой переменной. Такие разложения в степенные ряды строятся обычно либо с помощью формулы бинома Ньютона, либо путем использования рядов Тейлора. О них и пойдет речь ниже. Ключевые слова: Ряд Тейлора, ряд Маклорена, разложения в ряд, Показательная функция, тригонометрические функции, сумма, интервал, на всей действительной оси, сходящийся ряд, формула Коши-Адамара, остаточный член в формуле Лагранжа, биноминальный ряд с показателем , логарифмическая функция, равенства, свойства, значение аргумента, степень, точность, порядок, оценка. Аnnotation: Nowadays, getting general information is easy an ditisim portant to beable to correctly interpretand use existing data. In the field of mathematics, knowledge of exact formulations of definitions, theorems, etc. now it is not so important as the ability to use them for solving problems related to the surround dingreality. We are convinced that the course of mathematics, combining the elements of Algebra and Geometry, will help to in crease the level of mastering matterdueto the synergy effect thatarises. In constructing approximate solutions of algebraic differential, and integral equations, as well as in estimating various integrals, we have to deal with series in powers of a parameter or an independent variable. Such power series expansions are usually constructed either using the Newton binomial formula, or by using the Taylor series. About them find it below. Keywords: Taylor series, Maclaurin series, series expansions, Exponential function, trigonometric functions, sum, interval, on the whole real axis, convergent series, Cauchy-Hadamard formula, residual term in Lagrange formula, binomial series with exponent μ, logarithm function, equalities, properties, argument value, degree, accuracy, order, evaluation.

Using IIR filters to build high-order finite difference schemes for the unsteady Schrödinger equation

High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.

DISCRETIZATION EFFECTS DURING NUMERICAL INVESTIGATION OF HODGKIN-HUXLEY NEURON MODEL

Computer design is a valuable tool in the course of designing neuro-morphic systems. In particular it allows investigating basic mechanisms of neuron pulse activities in networks. For computer modeling it is necessary to digitize a continuous model of the system by means of the application of discrete operators able to keep basic properties of a prototype. But the accuracy of discrete models may decrease because of negative effects caused by the type of the method used, by a discretization pitch and errors in rounding off. This fact is significant for the analysis of non-linear systems to which belong the models of biological neurons. As a possible solution of the problem may be the development of specialized tools for the analysis of dynamic systems with the focus upon numerical methods used. In this paper by the example of the modeling of the neuron described by Hodgkin-Huxley classical equations there is considered a set of widespread methods for ODU solution. In the course of investigations there are shown possible negative consequences of incorrect use of some discrete operators. In the paper the results of two sets of computer experiments are presented. The first ones determine the limitations for the practical use of the methods of the first accuracy order during modeling neurons in the mode of resonance generation of action potentials. The second ones show discretization effects connected with chaotic modes of neurons functioning: incorrect behavior of discrete models which is manifested in the emergence of chaotic transition processes. The investigation results may be used at the formation of modeling tool packages both, non-linear dynamic systems in the whole, and neuro-morphic systems in particular.