System Maple Research Articles

Computational Research of the Efficiency of Using a Three-Layer Panel Made of Highly Porous Polystyrene Concrete.

This paper presents linking computational research of the multilayer structure of the cladding of a three-layer panel made of highly porous polystyrene concrete developed using a new technology in comparison with traditional ones. The calculation of the thermal efficiency of the exterior fence was carried out in three stages, where the thermal regime was calculated from the values of temperature fields in the ELCUT 6.6 system, and the humidity and air modes were determined by the analytical method in the Maple system. The territory of central Kazakhstan (Karaganda) was selected as the research region, where the research showed that equating the thickness by the values of the actual and required heat transfer resistances of traditional multilayer structures to the developed one, the thickness of traditional structures increases from 3.09% to 27.83%. Moisture accumulation relative to the developed one occurs in all the studied structures. Thus, if in some cases of traditional structures moisture is collected by 2.61% and 9.48% less, in others moisture is collected by 27.94% and 119% more. However, the value of evaporated moisture during the drying period showed that all the moisture will evaporate during the specified period. Thus, all the structures meet the conditions for the inadmissibility of moisture for the annual period and the period of moisture accumulation. Moreover, the values of the actual and required permeabilities to air satisfy the condition, which affected the values of the temperature fields taking into account air filtration; the developed structure showed a positive effect for this value, and in traditional structures, the value of τint decreased to 1.35 °C depending on the option. The analytical results of the thermal inertia values of the developed and traditional multilayer structures showed that the developed structure exceeds traditional ones by up to 30.04% depending on the option, which is positive in the cold period. It was also found that the market prices of all traditional structures exceed the developed one by 1.2-2.5 times, depending on the design, which also emphasizes the positive aspects of the new design. Thus, the findings of this research will positively complement the catalog of products of external multilayer cladding structures made of effective materials and can be used by research communities and design organizations in the design of residential buildings.

Calculation of symmetric polynomials in the maple computer mathematics system

The article studies the theory of symmetric polynomials. As a result of the study, it was shown that one of the methods for solving systems of higher degree equations is the method using symmetric polynomials. The practical implementation of the solution method has a computer implementation in the Maple system.

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.

Nonlinear Singularly Perturbed Cauchy Problem With Inner Layer

In the article, the asymptotics of the solution of the Cauchy problem for a nonlinear autonomous ordinary differential equation of the first order, which is a simple mathematical model with an inner layer, is constructed. For all theoretical calculations, evidence-based calculations in the Maple system are given.

Computational Technology for the Basis and Coefficients of Geodynamo Spectral Models in the Maple System

Spectral models are often used in the study of geodynamo problems. Physical fields in these models are presented as stationary basic modes combinations with time-dependent amplitudes. To construct a model it is necessary to calculate the modes parameters, and to calculate the model coefficients (the Galerkin coefficients). These coefficients are integrals of complex multiplicative combinations of modes and differential operators. The paper proposes computing technology for the calculation of parameters, the derivation of integrands and the calculation of the integrals themselves. The technology is based on computer algebra methods. The main elements for implementation of technology in the Maple system are described. The proposed computational technology makes it possible to quickly and accurately construct fairly wide classes of new geodynamo spectral models.

Изгиб секторальной пластины: использование систем компьютерной алгебры

Purpose: To investigate the stress-strain state of a thin homogeneous isotropic plate in the form of a sector by numerical-analytical method. To consider the possibility of using computer algebra systems (CAS) to calculate sectoral plates operating under bending due to a transverse load. To demonstrate the effectiveness of applying one of these CASs through the example of the Maple system for calculations using the Ritz method — performing analytical transformations when calculating the integral that determines the total potential energy functional, forming and solving the main resolving system of linear algebraic equations with respect to unknown numerical coefficients in the formula approximating the deflection of the plate, visualization of the obtained solution. Methods: A direct method is used to solve the variational problem of minimizing the functional of the total potential energy of deformation of a thin homogeneous isotropic plate in the form of a sector — the Ritz method. The solution is constructed in the form of a series in terms of basis functions. As basis functions, polynomial functions are chosen that exactly satisfy all boundary conditions. Results: An approximate numerical-analytical solution has been obtained for the problem of bending a sectoral plate in the form of a quarter of a circle, clamped along the contour and loaded with a uniformly distributed load. The effectiveness of using the Maple analytical computing system for solving the problem of bending a sectoral plate by the variational Ritz method is demonstrated. It is shown that the resulting solution quickly converges both for deflection and for bending moments and stresses. Practical significance: The algorithm proposed in the article for solving the problems of bending sectoral plates can be recommended for practical use in determining the stress-strain state of such plates.

Climate change and maple syrup: Producer observations, perceptions, knowledge, and adaptation strategies

IntroductionClimate change is impacting forest-based agricultural systems with implications for producer decision-making and livelihoods. This article presents a case study on the observations, perceptions, knowledge, and adaptation strategies of maple syrup producers in the United States to climate change.MethodsWe carried out two semi-structured surveys with maple producers on: (1) climate change and its impacts on the maple system (n = 106 participants); and (2) responses to climate adaptation scenarios (n = 98 participants). Additionally, we carried out two focus groups and key informant interviews (n = 70+) to understand barriers and opportunities for climate adaptation. One of these focus groups and follow up key informant interviews was with tribally affiliated community members with the intention to acknowledge Indigenous Peoples’ voices, history, and relationships to the land.ResultsFindings highlight that most of the surveyed producers (89%) have experienced the negative impacts of climate on maple syrup production. While 40% of participants feel concerned regarding the future of the maple system, 39% feel hopeful, with significant differences based on the age of the surveyed producers. The majority of producers have adapted their harvesting practices to climate effects. Producers shared knowledge of multiple adaptation strategies in response to climate scenarios comprised of: (1) stand management practices such as diversification of sap species tapped; (2) harvesting practices such as changing the type and number of taps; (3) sap processing practices focused on the integration of technology such as the use of an evaporator and reverse osmosis; and (4) marketing practices such as innovation of products and marketing different maple syrup characteristics. Responses shared by tribally affiliated producers highlight knowledge of multiple adaptation strategies that focus on long-term ecological management of forests rather than technological solutions.DiscussionOverall, findings emphasize the importance of cooperation and diversification at every level and dimension of the maple system for its long-term resilience.

Self-Learning: Application of differential equations of hyperbolic type to some engineering problems using information and communication technologies

This article analyzes the issue of organizing students' self-study, which is one of the urgent problems of the credit-modular system. The student independently discusses in detail what to pay attention to in order to solve an engineering problem using information and communication technologies using differential equations with special derivatives of hyperbolic type. It is stated that the initial-boundary value problem of the process is solved by the Fourier method. The construction of a decision graph using the Maple system is explained. The student's attention is focused on the analysis of the results based on the nature of the considered engineering problem.

The use of Maple system in solving the problems of balance analysis

The article describes the current state of the use of computer mathematics systems and the advantages of using the Maple computer mathematics system. The possibility and relevance of using the Maple computer mathematics system in solving problems of balance analysis of the course of further mathematics for economic specialties is proved, and the promising aspects of further research are determined.

Kinematic analysis and estimation of the frequency of natural oscillations of a planar lattice

Introduction. A statically determined regular symmetrical rectangular lattice is fixed on two supports. The mass of the lattice is distributed over its nodes. A kinematic analysis is presented that reveals the variability of the lattice for a certain number of panels. An algorithm for searching for the distribution of virtual velocities of lattice nodes in this case is described. A formula is derived for the dependence of the lower estimate of the first frequency of natural oscillations of the structure on the number of panels from the range of permissible values.
 
 Materials and methods. Calculation of forces in the bars of a statically determinate lattice is performed by the method of cutting out nodes using Maple computer mathematics system operators. To determine the rigidity of the structure, the Maxwel – Mohr formula is used on the assumption that the elastic moduli and cross-sectional areas of all lattice rods are the same.
 
 Results. It is shown that when the number of panels is a multiple of three, the determinant of the matrix of the system of equilibrium equations degenerates, and the system becomes an instantly variable mechanism. A corresponding picture of the distribution of knot velocities is given. Using the approximate Dunkerley estimate, a formula is derived for the dependence of the lower limit of the first oscillation frequency of the truss on the number of panels. The generalization of a series of particular solutions to an arbitrary number of panels is performed in the Maple system by induction.
 
 Conclusions. The proposed lattice model admits an analytical solution for the lower estimate of the first natural frequency. Comparison of the result with the numerical solution obtained for the lowest frequency of the entire frequency spectrum of natural lattice vibrations shows its high accuracy. It is shown that the error of the found analytical estimate decreases with an increase in the number of panels. The solution can be used in optimization problems and for a preliminary assessment of the designed structure.

АНАЛИТИЧЕСКАЯ ОЦЕНКА ПЕРВОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ МНОГОПРОЛЕТНОЙ ФЕРМЫ

The task is to obtain the dependence of the lower limit of the fundamental frequency of natural oscillations of a statically determinate truss on the number of spans. The upper belt of the truss is straight, the lower belt of spans is arched. The inertial properties of the truss are modeled by the same concentrated masses in the nodes. Each mass has two degrees of freedom. The forces in the rods and the reactions of the supports are found by cutting out the nodes in symbolic form from the solution of the general system of equations for the equilibrium of the truss nodes. All transformations are made in the Maple computer mathematics system. To find the structural stiffness matrix, the Maxwell – Mohr formula is used. The first natural frequency of the structure is found using the approximate Dunkerley method. The generalization of a series of individual solutions found for trusses with a different number of spans to an arbitrary number of spans is performed by induction using Maple system operators. For comparison, the lowest oscillation frequency of a system with many degrees of freedom obtained numerically is used. A formula is derived for the lower estimate of the first frequency depending on the size of the truss and the number of spans. Comparison of the analytical solution with the numerical one shows a good degree of approximation of the formula for the first frequency. It is shown that the error significantly depends on the height of the truss. Cases of kinematic variability of the structure with an even number of spans are revealed.

Welcome to Maple Transactions

Maple was conceived over forty years ago as a general purpose system for mathematical calculations. Its strength, however, has always been its community. The work of hundreds of researchers from around the world has produced a mathematical engine unique in its depth, breath and efficiency. Forward thinking educators have used Maple to transform the way mathematics is taught, all the way supporting each other with advice, examples and myriads of Maple worksheets. Scientists and engineers have been taking advantage of the power and ease of use of the Maple system to help them in their discovery and the development of new products. Together we have tackled environmental issues, taken on disease and reached for the stars.
 
 At Maplesoft, we are firm believers that Math Matters and our mission is to provide technology to explore, derive, capture, solve and disseminate mathematical problems and their applications, and to make math easier to learn, understand, and use. This mission, we share with hundreds of thousands of Maple users from all over the world and indeed we rely on that community’s constant stream of feedback and support.
 
 With Maple Transactions, our community is gaining a new place to come together. A place to exchange ideas, share experiences and discoveries. A place to welcome newcomers and discuss possibilities. The drive, vision and energy of editor in chief Prof Rob Corless together with the fantastic editorial board that he assembled, have given me a glimpse into a bright future for the journal and this first issue bears witness to the high quality of contributions we can expect.

The analysis of dependence of the vibration frequency of a space cantilever truss on the number of panels

Introduction. The first (lowest) frequency of natural vibrations of a structure is one of its most important dynamic characteristics. Analytical solutions supplement numerical ones; they can be efficiently used to perform a rapid assessment of properties of structures, to analyze and optimize constructions and to test numerical results. A space cantilever truss consisting of three planar trusses with a rectangular grid is considered in the article. The objective is to find the analytical dependence between the frequency of natural vibrations of a structure and the number of panels. It is assumed that the truss mass is distributed among the joints. Only the vertical mass displacement is taken into account.
 Materials and methods. Forces, arising in cantilever rods, are calculated by the Maple software as symbolic expressions, and the method of joint isolation is used here. The stiffness matrix is identified using the Mohr integral. Rods are assumed to be elastic, they have identical stiffness. The lower value of the vibration frequency is determined using the Dunkerley method. The final calculation formula used to identify the value of the vibration frequency is derived using the method of induction applied to a series of analytical solutions developed for trusses with a consistently increasing number of panels. When common members of sequences are found, genfunc operators of the Maple system are used. The analytical solution is compared with the numerical solution in terms of the first frequency using the analysis of the system spectrum featuring many degrees of freedom. The eigenvalues of the characteristic matrix are identified using the Eigenvalues operator from the Linear Algebra package.
 Results. The comparison between the analytical values and the numerical solution shows that the Dunkerley method ensures the accuracy varying from 20 % for a small number of panels to 3 % if the number of panels exceeds ten. The size of the structure, the weight and stiffness of rods have little effect on the accuracy of the obtained values.
 Conclusions. The lowest value obtained using the Dunkerley method in the form of a fairly compact formula has good accuracy, its application to a space structure with an arbitrary number of panels has a polynomial form equal to the number of panels, and it can be used in practical calculations.

Generation of equations for shell models of turbulence in the Maple system

The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.

Analytical assessment of the frequency of natural vibrations of a truss with an arbitrary number of panels

The aim of the work is to derive a formula for the dependence of the first frequency of the natural oscillations of a planar statically determinate beam truss with parallel belts on the number of panels, sizes and masses concentrated in the nodes of the lower truss belt. Truss has a triangular lattice with vertical racks. The solution uses Maple computer math system operators. Methods. The basis for the upper estimate of the desired oscillation frequency of a regular truss is the energy method. As a form of deflection of the truss taken deflection from the action of a uniformly distributed load. Only vertical mass movements are assumed. The amplitude values of the deflection of the truss is calculated by the Maxwell - Mohrs formula. The forces in the rods are determined in symbolic form by the method of cutting nodes. The dependence of the solution on the number of panels is obtained by an inductive generalization of a series of solutions for trusses with a successively increasing number of panels. For sequences of coefficients of the desired formula, fourth-order homogeneous linear recurrence equations are compiled and solved. Results. The solution is compared with the numerical one, obtained from the analysis of the entire spectrum of natural frequencies of oscillations of the mass system located at the nodes of the truss. The frequency equation is compiled and solved using Eigenvalue search operators in the Maple system. It is shown that the obtained analytical estimate differs from the numerical solution by a fraction of a percent. Moreover, with an increase in the number of panels, the error of the energy method decreases monotonically. A simpler lower bound for the oscillation frequency according to the Dunkerley method is presented. The accuracy of the lower estimate is much lower than the upper estimate, depending on the size and number of panels.

Calculation of the normal modes of closed waveguides

The aim of the work is the development of numerical methods for solving waveguiding problems of the theory of waveguides, as well as their implementation in the form of software packages focused on a wide range of practical problems from the classical issues of microwave transmission to the design of optical waveguides and sensors. At the same time, we strive for ease of implementation of the developed methods in computer algebra systems (Maple, Sage) or in software oriented to the finite element method (FreeFem++). The work uses the representation of electromagnetic fields in a waveguide using four potentials. These potentials do not reduce the number of sought functions, but even in the case when the dielectric permittivity and magnetic permeability are described by discontinuous functions, they turn out to be quite smooth functions. A simple check of the operability of programs by calculating the normal modes of a hollow waveguide is made. It is shown that the relative error in the calculation of the first 10 normal modes does not exceed 4%. These results indicate the efficiency of the method proposed in this article.

Calculation of the normal modes of closed waveguides

The aim of the work is the development of numerical methods for solving waveguiding problems of the theory of waveguides, as well as their implementation in the form of software packages focused on a wide range of practical problems from the classical issues of microwave transmission to the design of optical waveguides and sensors. At the same time, we strive for ease of implementation of the developed methods in computer algebra systems (Maple, Sage) or in software oriented to the finite element method (FreeFem++). The work uses the representation of electromagnetic fields in a waveguide using four potentials. These potentials do not reduce the number of sought functions, but even in the case when the dielectric permittivity and magnetic permeability are described by discontinuous functions, they turn out to be quite smooth functions. A simple check of the operability of programs by calculating the normal modes of a hollow waveguide is made. It is shown that the relative error in the calculation of the first 10 normal modes does not exceed 4%. These results indicate the efficiency of the method proposed in this article.

Calculation of the normal modes of closed waveguides

The aim of the work is the development of numerical methods for solving waveguiding problems of the theory of waveguides, as well as their implementation in the form of software packages focused on a wide range of practical problems from the classical issues of microwave transmission to the design of optical waveguides and sensors. At the same time, we strive for ease of implementation of the developed methods in computer algebra systems (Maple, Sage) or in software oriented to the finite element method (FreeFem++). The work uses the representation of electromagnetic fields in a waveguide using four potentials. These potentials do not reduce the number of sought functions, but even in the case when the dielectric permittivity and magnetic permeability are described by discontinuous functions, they turn out to be quite smooth functions. A simple check of the operability of programs by calculating the normal modes of a hollow waveguide is made. It is shown that the relative error in the calculation of the first 10 normal modes does not exceed 4%. These results indicate the efficiency of the method proposed in this article.

Calculation of the deflection of an arched truss with suspended elements depending on the number of panels

The aim of the work - to propose a scheme and analytical calculation of a statically definable planar truss with a suspended lower belt. Methods. The formula for the dependence of the deflection of the truss under the action of a uniform load on the lower belt on its size and the number of panels is derived in the computer mathematics system Maple. The forces in the rods are found from the solution of the general system of equilibrium equations of all nodes in symbolic form. The deflection is calculated using the Maxwell - Mohr's formula. Generalization of a number of formulas for deflection obtained by increasing the number of panels sequentially to an arbitrary number is performed by double induction using two independent parameters. In this case, special operators of the Maple system are used, allowing for a sequence of coefficients in the desired formula to create and solve recurrent equations that satisfy the elements of the sequences. Results. The obtained solutions have a polynomial form for the number of panels. Curves of deflection dependence on the number of panels are constructed and analyzed. Asymptotic properties of solutions are found in the case of a fixed span length of the structure and a given total load. The proposed scheme is a statically determinate structure with two independent parameters of regularity allows for the finding of a fairly simple analytical solution. The resulting formula is most effective in calculating systems with a large number of elements, where numerical methods tend to accumulate rounding errors.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.