Computer Algebra System Maple Research Articles

Information and logical transformations in Schaeffer and Pierce Bases in Maple

One of the tasks of synthesis and analysis of the functioning of combinational devices is the representation of a Boolean function in various bases. The absolute completeness of the use and implementation of various transformations of Boolean functions is impossible without the implementation of transformations into universal bases of Schaeffer (AND-NOT) or Webb (OR-NOT). The Logic library of the Maple computer algebra system allows you to use only the operations &nand (Schaeffer basis) and &nor (Webb basis) to build a model of a digital device. Therefore, there is a need to expand the functionality of the Logic library of the Maple computer algebra system by representing Boolean functions in Schaeffer and/or Pierce (Webb) bases. In this regard, the authors of the article have developed procedures that allow the representation of disjunctive or conjunctive normal forms in the Schaeffer basis or in the Webb basis. The development was carried out taking into account the existing data processing functions and procedures. The article describes the technology of extending the Logic library of the Maple computer algebra system by developed procedures that implement the transformation of a given or received Boolean function in Schaeffer or Webb bases.

Information modeling of the normal forms of logical functions in Maple system

The article discusses possible ways to use the Maple computer algebra system to obtain perfect conjunctive and disjunctive normal forms. Computer algebra systems make it possible to expand the possibilities for creating, applying and using mathematical models in the daily activities of engineers, researchers, and also contribute to improving the education of students of various training profiles. In addition, the use of computer technology can help to apply various methods and expand approaches to the representation of the studied objects (Boolean functions) of discrete form, including in education. Computer algebra systems such as Maple, Mathematica, etc. are the most adapted for studying various sections of discrete mathematics and mathematics in general. The possibilities of the Maple program are shown for use at the initial stages of mastering the discrete mathematics section of the university course - Boolean functions, in particular, for work on the construction and study of perfect conjunctive and disjunctive normal forms according to the truth table.

Computer assisted investigation of alienness of linear functional equations

AbstractIn this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations.

Revealing patterns of thermophysical parameters in the designed energy-saving structures for external fencing with air channels

This study reports the design of new models of energy-saving enclosing structures with air channels. To calculate the thermophysical parameters of external fences, the Maple computer algebra system was used; the value of thermal resistance of structures was determined on the basis of a finite-element method in ANSYS. The result of the structure analysis showed that the value of thermal inertia of the traditional design and the average value of the thermal inertia of the developed structures were equal. However, the vibration amplitude of the designed enclosing structures was up to 20.72 % more efficient than the traditional one. At the same time, it was revealed that the air gaps did not affect the thermal inertia of the strucure, and its parameters depended only on the total thickness of the material. The analysis showed that the vapor permeability of the inner wall of the designed structures was equal to the traditional one. However, the value of resistance to vapor permeation of the fence of the developed structures was 3.21 % more effective. At the same time, the use of a closed air layer with a heat-reflecting screen makes it possible to shift the possible condensation zone towards the outer surface of the fence. An analysis of the check for the non-condensation of condensate in the ventilated air gap showed that condensate did not fall out in the ventilated air gap in all the considered schemes, and the results of the analysis by the air permeability value showed that all fencing schemes met the requirements for air permeability. Solving the problems of energy saving in construction through the development of new energy-efficient designs of enclosing structures help reduce the cost of thermal energy of buildings, which is an urgent task all over the world today

Bibliographic Study on the Computer Algebra System Maple

The authors were surprised by the number of bibliographic citations to the computer algebra system DERIVE long after it was discontinued and developed a small bibliographic study about its evolution. They later carried out a similar one about the successful dynamic geometry system GeoGebra. Finally, they presented at Maple Conference 2022 a comparative small bibliographic study about Maple and other computer algebra systems, which can be found here.

Prediction of Refractive Index of Petroleum Fluids by Empirical Correlations and ANN

The refractive index is an important physical property that is used to estimate the structural characteristics, thermodynamic, and transport properties of petroleum fluids, and to determine the onset of asphaltene flocculation. Unfortunately, the refractive index of opaque petroleum fluids cannot be measured unless special experimental techniques or dilution is used. For that reason, empirical correlations, and metaheuristic models were developed to predict the refractive index of petroleum fluids based on density, boiling point, and SARA fraction composition. The capability of these methods to accurately predict refractive index is discussed in this research with the aim of contrasting the empirical correlations with the artificial neural network modelling approach. Three data sets consisting of specific gravity and boiling point of 254 petroleum fractions, individual hydrocarbons, and hetero-compounds (Set 1); specific gravity and molecular weight of 136 crude oils (Set 2); and specific gravity, molecular weight, and SARA composition data of 102 crude oils (Set 3) were used to test eight empirical correlations available in the literature to predict the refractive index. Additionally, three new empirical correlations and three artificial neural network (ANN) models were developed for the three data sets using computer algebra system Maple, NLPSolve with Modified Newton Iterative Method, and Matlab. For Set 1, the most accurate refractive index prediction was achieved by the ANN model, with %AAD of 0.26% followed by the new developed correlation for Set 1 with %AAD of 0.37%. The best literature empirical correlation found for Set 1 was that of Riazi and Daubert (1987), which had %AAD of 0.40%. For Set 2, the best performers were the models of ANN, and the new developed correlation of Set 2 with %AAD of refractive index prediction was 0.21%, and 0.22%, respectively. For Set 3, the ANN model exhibited %AAD of refractive index prediction of 0.156% followed by the newly developed correlation for Set 3 with %AAD of 0.163%, while the empirical correlations of Fan et al. (2002) and Chamkalani (2012) displayed %AAD of 0.584 and 0.552%, respectively.

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

AbstractThe main idea of this work is an application of probabilistic entropy and also relative entropy in the numerical analysis of uncertainty propagation in the homogenization of some composite materials. The homogenization method is based on the determination of deformation energy for the representative volume elements computed with the use of some specific finite element method experiments. Uncertainty propagation concerns material and geometrical design parameters of particulate composites and is performed thanks to an application of polynomial responses; probabilistic moments of the effective tensor are computed via the iterative generalized stochastic perturbation technique and the semi‐analytical probabilistic method. Probabilistic entropy is determined according to Shannon theory, while relative entropies equations employed here follow mathematical models created by Kullback & Leibler, Jeffreys, Hellinger, and Bhattacharyya. Deterministic analyses have been performed with the use of the system ABAQUS, while all the remaining procedures have been programmed in the computer algebra system MAPLE.

An Application of Relative Entropy in Structural Safety Analysis of Elastoplastic Beam under Fire Conditions

The main aim of this work is to propose a new algorithm of reliability assessment for steel civil engineering structures subjected to fire temperatures. This new algorithm is based upon the relative probabilistic entropy concept elaborated by Bhattacharyya, and this probabilistic distance is sought in-between extreme and admissible deformations of some structural beam subjected to higher temperatures. Similar to the First Order Reliability Method, this approach uses the first two probabilistic characteristics of the structural response, when structural output may be modelled with the use of Gaussian distribution. The probabilistic structural response is found here using hybrid computational technique–the Finite Element Method system ABAQUS with its fully coupled thermo-elastic analysis with 3D solid elements and probabilistic modules implemented in the computer algebra system MAPLE. The probabilistic response is determined via a triple stochastic analysis, which is based on the classical Monte-Carlo simulation, iterative generalized stochastic perturbation technique, and also using semi-analytical probabilistic calculus. Final determination of the relative entropy in the Serviceability Limit State of the given structure and its comparison with the results obtained using the FORM analysis enables to calibrate this new technique to numerical values proposed in the engineering designing codes. Hence, a more accurate probabilistic method may use some experimental-based admissible values included in the existing design of legal provisions.

Study of Bulk Properties Relation to SARA Composition Data of Various Vacuum Residues Employing Intercriteria Analysis

Twenty-two straight run vacuum residues extracted from extra light, light, medium, heavy, and extra heavy crude oils and nine different hydrocracked vacuum residues were characterized for their bulk properties and SARA composition using four and eight fractions (SAR-ADTM) methods. Intercriteria analysis was employed to determine the statistically meaningful relations between the SARA composition data and the bulk properties. The determined strong relations were modeled using the computer algebra system Maple and NLPSolve with the Modified Newton Iterative Method. It was found that the SAR-ADTM saturates, and the sum of the contents of saturates and ARO-1 can be predicted from vacuum residue density, while the SAR-ADTM asphaltene fraction content, and the sum of asphaltenes, and resins contents correlate with the softening point of the straight run vacuum residues. The softening point of hydrocracked vacuum residues was found to strongly negatively correlates with SAR-ADTM Aro-1 fraction, and strongly positively correlates with SAR-ADTM Aro-3 fraction. While in the straight run vacuum residues, the softening point is controlled by the content of SAR-ADTM asphaltene fraction in the H-Oil hydrocracked vacuum residues, the softening point is controlled by the content of SAR-ADTM Aro-3 fraction content. During high severity H-Oil operation, resulting in higher conversion, hydrocracked vacuum residue with higher SAR-ADTM Aro-3 fraction content is obtained, which makes it harder and more brittle.

Improving the Learning and Teaching of Mathematical Logic Elements Using Maple

The study of the elements of mathematical logic and its sections, especially in the process of teaching in high school in general and specialized courses of mathematics or higher educational institutions, is a rather tricky task. The paper demonstrates how one can design and develop a package of interactive applications to help learn, understand, and apply the essential elements of mathematical logic. The research aims to develop and describe an interactive application for studying the order of constructing truth tables and studying and demonstrating the production of principal normal disjunctive and conjunctive forms. The development is based on the methods of the Maple computer algebra system of the Maplet library. In addition, other methods are used to solve the research issues – theoretical and accurate analytical solutions to mathematical logic issues. All constructed material (Maplet) will serve as a good simulator for students, allowing one to perform an infinite number of attempts to analyze, synthesize, and build truth tables.

The Sum of Reduced Harmonic Series Generated by Any Number of Positive Integer Factors

This paper is a free continuation of the author’s previous papers dealing with the sums of the reduced harmonic series generated of reciprocals of all products generated by all prime divisors of the numbers 2002 and 2022, that were inspired by one task on the sum of a special infinite series on the Berkeley Math Circle. We determined the sums of these series, i.e. the series of all the unit fractions that have denominators with only factors consisting of all prime divisors of the numbers 2022 and 2002, analytically and also by calculation in computer algebra system Maple. In this paper, we generalize our considerations and derive two formulas that obviously hold not only to series generated by n prime numbers, but also to series generated by n positive integers.

Optimal formulas for the approximate-analytical solution of the general Abel integral equation in the Sobolev space

This article discusses the development of a new algorithm, which is based on optimal quadrature formulas for obtaining solutions to the generalized Abel integral equations. Based on the study, it was found that the proposed scheme is very effective, extremely accurate and can be extended to other special tasks. The quadrature formulas presented in this paper are optimal in the Sobolev space of functions that have square integrable derivatives of order m. Using the quadrature formula and the Maple computer algebra system, exact and approximate values of the Abel integral equations are found, illustrating the effectiveness of the proposed approach.

Analysis of Some Properties of an Activated Sludge Wastewater Treatment Model

The aim of the paper is to study some properties of a model describing the wastewater treatment process with activated sludge. The process configuration consists of an aeration tank and of a secondary settler. The mathematical model is described by three nonlinear ordinary differential equations. The paper presents important results on existence and uniqueness of positive model solutions. Numerical computations establish the stability properties of the equilibrium points with respect to two important control model parameters and present the corresponding input-output static characteristics. The dynamic behaviour of the model solutions is illustrated through computer experiments carried out in the computer algebra system Maple. The results can be useful in designing real control strategies of the wastewater treatment processes with activated sludge.

Analysis of Longitudinal Forest Data on Individual-Tree and Whole-Stand Attributes Using a Stochastic Differential Equation Model

This paper focuses on individual-tree and whole-stand growth models for uneven-aged and mixed-species stands in Lithuania. All the growth models were derived using a single trivariate diffusion process defined by a mixed-effect parameters trivariate stochastic differential equation describing the tree diameter, potentially available area, and height. The mixed-effect parameters of the newly developed trivariate transition probability density function were estimated using an approximate maximum likelihood procedure. Using the relationship between the multivariate probability density and univariate marginal (conditional) densities, the growth equations were derived to predict or forecast the individual-tree and whole-stand variables, such as diameter, potentially available area, height, basal area, and stand density. All the results are illustrated using an observed dataset from 53 permanent experimental plots remeasured from 1 to 7 times. The computed statistical measures showed high predictive and forecast accuracy compared with validation data that were not used to find parameter estimates. All the results were implemented in the Maple computer algebra system.

Application of quasi-rotation surface segments in architectural prototyping

The present paper details an approach to modeling of quasi-rotation surface segments with pre-defined conditions. The desired surfaces and their segments are formed parametrically through Maple computer algebra system. Various analytical approaches to trimming of the excessive parts of quasi-rotation surfaces are indicated. Geometric modeling is performed with use of a previously developed algorithm. An example illustrating the initial stage of design of an overpass between two separate buildings is provided. The design objective contains the basic geometric parameters of the desired architectural element. Three control models were created with the same initial parameters through basic CAD software methods. All of the constructed models are thin-walled. Mechanical properties of the scale 3D models of the target element were compared in equal conditions by means of CAD software. The prototype created through the use of a quasi-rotation surface segment was the most resistant to load. The paper also features a rendered concept of a complex of three towers connected with three overpasses. The acquired results allow us to conclude that quasi-rotation is an effective method of formation of models of architectural elements.

The efficiency of multimedia technologies utilized in studying Mathematics course in the fields of Engineering

The learning model represented in the paper implies the use of specifically designed workbooks and dynamic presentations, original (self-developed) interactive computer games and training systems, LearningApps.org web service, mobile applications in Glide, Kahoot! in-game learning platform, Desmos graphing calculator and the Maple computer algebra system in learning. The applied educational tools give students an opportunity not only to get to know basic mathematical concepts, work out and test their computation skills individually but also to model processes that change over time, changing the process variables interactively. The article provides several examples of such assignments for students. In this model of arranging to learn, an educator is becoming free from a lot of routine work (manual check of tasks and tests). The educational experiment took place in the 2019-2020 and 2020-2021 academic years. 167 Engineering students from Sholom-Aleichem Priamursky State University, Moscow State University of Civil Engineering, and Osh State University participated in it. The conducted experiment proved the effectiveness of the learning model suggested: competent academic performance of students has increased by 15%. Therefore, supported multimedia technologies, involving most of the sensory experience of a student in understanding learning materials, make learning more productive.

SOME APPLICATIONS OF EXPONENTIAL AND LOGISTIC GROWTH MODELS IN BUSINESS AND ECONOMICS

The use of mathematics for predicting and analyzing the future has long been a subject of study due to the fact that it is a measurable source. Correct estimation is vital for the business world and its economic society. In this study, by employing the exponential growth model and the logistic and generalized logistic models, we estimate the size of the population of Turkey, the international investment position and the level of the national income per person in 2025. Furthermore, we examine the changes in sales of houses and the number of mobile phone subscribers in Turkey. We use GeoGebra program to determine the constants and use the computer algebraic system Maple 15 for other computations and graphs

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

Symmetric and Asymmetric Diffusions through Age-Varying Mixed-Species Stand Parameters

(1) Background: This paper deals with unevenly aged, whole-stand models from mixed-effect parameters diffusion processes and Voronoi diagram points of view and concentrates on the mixed-species stands in Lithuania. We focus on the Voronoi diagram of potentially available areas to tree positions as the measure of the competition effect of individual trees and the tree diameter at breast height to relate their evolution through time. (2) Methods: We consider a bivariate hybrid mixed-effect parameters stochastic differential equation for the parameterization of the diameter and available polygon area at age to ensure a proper description of the link between them during the age (time) span of a forest stand. In this study, the Voronoi diagram was used as a mathematical tool for the quantitative characterization of inter-tree competition. (3) Results: The newly derived model considers bivariate correlated observations, tree diameter, and polygon area arising from a particular stand and enables defining equations for calculating diameter, polygon-area, and stand-density predictions and forecasts. (4) Conclusions: From a statistical point of view, the newly developed models produced acceptable statistical measures of predictions and forecasts. All the results were implemented in the Maple computer algebra system.

Some properties of 2-orthogonal polynomials associated with inverse Gaussian distribution

In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated polynomials. A particularly important property is that the associated polynomials satisfy a four-term recurrence relation. Then we state and calculate the famous creation and annihilation operators of the Heisenberg-Weyl Lie algebra that acting on IG polynomials Pn as the multiplicative operator X and the derivative one D on the monomials xn . Finally, I used the Maple computer algebra system to program and Calculate the six first IG polynomials and their roots numerically.