Accurate Exponential Representations for the Ground State Wave Functions of the Collinear Two-Electron Atomic Systems

In the framework of the study of helium-like atomic systems possessing the collinear configuration, we propose a simple method for computing compact but very accurate wave functions describing the relevant S-state. It is worth noting that the considered states include the well-known states of the electron–nucleus and electron–electron coalescences as a particular case. The simplicity and compactness imply that the considered wave functions represent linear combinations of a few single exponentials. We have calculated such model wave functions for the ground state of helium and the two-electron ions with nucleus charge 1≤Z≤5. The parameters and the accompanying characteristics of these functions are presented in tables for number of exponential from 3 to 6. The accuracy of the resulting wave functions are confirmed graphically. The specific properties of the relevant codes by Wolfram Mathematica are discussed. An example of application of the compact wave functions under consideration is reported.

Dynamic and Interactive Tools to Support Teaching and Learning

The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the theoretical results and often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. The main goal of this work is to show how some new dynamic and interactive tools, created with Mathematica and available in the Computable Document Format (CDF), can be used as active learning tools to promote better student activity and engagement in the learning process. The CDF format allows anyone with a computer to use them, at no cost, even without an active Wolfram Mathematica license. Besides that, the presented tools are very intuitive to use which makes it suitable for less experienced users. Some tools applicable to several mathematics concepts taught in higher education will be presented. This kind of tools can be used either in a remote or classroom learning environment. The corresponding CDF files are made available as supplement of the online edition of this article.

Исследование поведения классических критериев множественных сравнений, на ненормальных неоднородных распределениях, методом Монте-Карло

Традиционные аналитические методы исследования применимости методов множественных сравнений эффективны только при весьма жёстких ограничениях на соответствующие генеральные совокупности. В то же время для решения этого вопроса с успехом можно применять компьютерные симуляции и метод Монте-Карло. Методом Монте-Карло мы симулируем проведение тестов, выполняемых при множественных сравнениях на выборках малого объёма из искаженных (по сравнению с нормальным) распределений. Исследуется возможность применения классических критериев дисперсионного анализа (ANOVA) и непараметрического теста Краскела — Уоллиса для выборок малого объёма с ненормальным распределением и/или неоднородных по дисперсии. В качестве критерия однородности выборок по дисперсиям используется тест Левене. Нормальность (Гауссовость) выборок проверяется с помощью теста Шапиро –Уилка. Для искажения нормальности выборок используются генеральные совокупности, распределенные по хи-квадрат и t-распределению Стьюдента с малым числом степеней свободы. Также ненормальность распределений отслеживается с помощью параметров: эксцесс (коэффициент островершинности) и асимметрия. В качестве генератора псевдослучайных чисел применяется так называемый вихрь Мерсенна реализованный в рамках пакета программ Wolfram Mathematica. Число испытаний для каждого набора параметров доведено до миллиона. Вычисляются эффективные вероятности ошибок 1-го рода и делаются выводы о влиянии негомогенности дисперсий, «ненормальности» эксцесса и асимметрии на эффективность исследуемых критериев. В результате можно сказать, что зачастую нет оснований использовать непараметрические методы вместо параметрических в ущерб мощности соответствующих критериев.

Tonality analysis in Wolfram Mathematica 12 computer algebra system

Sentiment analysis is a class of methods that are used to automatically determine the tonality of statements in the text. There are many solutions and technologies to define of text tonality at the moment and the choice depends on task. With the release of Wolfram Mathematica 12 versions, it became possible to use included nlp methods for determination of text tonality. This article is about definition of text tonality in the wolfram Mathematica 12 using nlp and compare nlp methods with standard definition function.

Modelación de andamios porosos basados en las estructuras triplemente periódicas P y G

En este trabajo se estudia la modelación de estructuras híbridas de andamios porosos para regeneración de tejido óseo basadas en las superficies minimales triplemente periódicas Giroide (G) y primitiva de Schwarz (P). El diseño de las probetas prismáticas híbridas, con dimensiones según la norma ASTM D695_15, se logra a partir de las ecuaciones que definen a cada estructura utilizando la función de enlace sigmoidea con valor k=0.5 mediante el software CAS Wolfram Mathematica v11.2. Los aspectos relacionados con el uso de Mathematica como herramienta para el diseño de las probetas son discutidos en detalle. Las constantes de la ecuación de cada estructura son utilizadas como variables en un diseño factorial 32 para estudiar su efecto en la porosidad y tamaño de poros. A partir de regresión multilineal se obtienen las ecuaciones que relacionan los factores con las variables dependientes y se discuten los modelos obtenidos. Se concluye que el modelo bilineal es adecuado para la descripción de las variables de respuesta lo que justifica la elección del diseño experimental utilizado.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Dynamical Behaviors of Sinusoidal Nanocomposite Plates Subjected to Thermomechanical Loads

In this paper, a new plate structure has been found with the change of profile according to the sine function which we temporarily call as the sinusoidal plate. The classical plate theory and Galerkin’s technique have been utilized in estimating the nonlinear vibration behavior of the new non-rectangular plates reinforced by functionally graded (FG) graphene nanoplatelets (GNPs) resting on the Kerr foundation. The FG-GNP plates were assumed to have two horizontal variable edges according to the sine function. Four different configurations of the FG-GNP plates based on the number of cycles of sine function were analyzed. The material characteristics of the GNPs were evaluated in terms of two models called the Halpin–Tsai micromechanical model and the rule of mixtures. First, to verify this method, the natural frequencies of new non-rectangular plates made of metal were compared with those obtained by the Finite Element Method (FEM). Then, the numerical outcomes are validated by comparing with the previous papers for rectangular FGM/GNP plates — a special case of this structure. Furthermore, the impacts of the thermal environment, geometrical parameters, and the elastic foundation on the dynamical responses are scrutinized by the 2D/3D graphical results and coded in Wolfram-Mathematica. The results of this work proved that the introduced approach has the advantages of being fast, having high accuracy, and involving uncomplicated calculation.

Some New Solutions of Non Linear Evolution Equations With Mutable Coefficients

We construct the traveling wave solutions of some NonLinear Evolution Equations (NLEEs) with mutable coefficients arising in different branches of physics and mathematics. we apply a novel (G′G)-formalism to construct more general solitary traveling wave solutions of NLEEs such as Sharma-Tasso-Olver with mutable coefficients and Zakharov Kuznetsov equation. Interesting solutions of NLEEs are investigated by traveling wave solutions which are in form of trigonometric, rational, and hyperbolic functions. This may build more unified new solutions for different kinds of such NLEEs with mutable coefficients arising in mathematics and physics. Wolfram Mathematica 11 is used to perform the computation work and their corresponding plots and counter graphs are plotted. This method is found to be more useful and efficient for searching the exact solutions of NLEEs.

МЕХАНИКАЛЫҚ ЖӘНЕ ЭЛЕКТРОМАГНИТТІК ТЕРБЕЛІСТЕРДІ ҮЙРЕТУДЕ КОМПЬЮТЕРЛІК МОДЕЛЬДЕУДІ МЕКТЕПТЕ ҚОЛДАНУ – ЗАМАН ТАЛАБЫ

Мақалада қазіргі оқыту жүйесінде физика сабақтарында ақпараттық технологияларды, соның ішінде компьютерлік модельдеу технологиясы арқылы оқытудың алғышарттары, ерекшеліктері қарастырылған. Физиканың механика тарауындағы маңызды бөлім «Электромагниттік және механикалық тербелістерді» барынша түсінікті оқыту үшін тербелістердің графиктері мен үш өлшемді (3D) нобайларын көрнекі сипатта түсіндірудің жолдарының кең мүмкіндіктері талданды. Осы арқылы оқушыларды жаңа технологиялармен жұмыс жасауға бейімдеуге және ғылыми зерттеулердің жаңа әрі басым бағыты компьютерлік модельдеу жұмыстарының алғашқы командалық жұмыстары қарастырылды. Оқытудың технологиялық ресурсы ретінде – Wolfram Mathematica атты компьютерлік бағдарламасы қолданылып, осы бағдарлама негізінде тербелістерді оқытудың кезеңдері жүйеленіп көрсетілді. Сонымен қатар, физика сабағында оқушылар ақпараттық технологиялар жұмыс жасап, бір жағынан физикалық құбылыстардың сипатына идеал модель негізінде зер салып, олардың түрлі-түсті көрінісіне ерекше назар аударып, тербелісті сипаттайтын физикалық шамалардың арасындағы тәуелділік қатынастарын да ажыратуына болады. Мұндай заман ағымына дес бермес технология көмегімен информатика және қазіргі ғылым тілі – ағылшын тілімен кіріктіре оқыту әдісі де қатар жүреді. Компьютерлік модельдеуді механикалық және электромагниттік тербелістерді оқытуда қолдану физикалық құбылыстарды да тереңінен түсіндіруге және оқушылардың танымдық қабілетін дамытуға мүмкіндік бар.

An Efficient Non-Parametric Statistical Test for Assessing Some Treatment Methods of Clinical Data

The recent rapid spread of deadly epidemics have precipitated an urgent need to speed up the development of different treatments, as well as methods of evaluating their efficacy. The first step towards this is the collection of data relating to the cure rate in groups of patients who have had different treatments applied to them. As most of the available data in these cases is random, it is now the role of statisticians to provide efficient statistical tests to assess the treatment methods through the data. This research aims to provide a new statistical test with high efficiency to reach the right decision with accurate results as quickly as possible using parallel computing algorithms through Wolfram Mathematica software.