Symbolic Mathematics Research Articles

Devito (v3.1.0): an embedded domain-specific language for finite differences and geophysical exploration

Abstract. We introduce Devito, a new domain-specific language for implementing high-performance finite-difference partial differential equation solvers. The motivating application is exploration seismology for which methods such as full-waveform inversion and reverse-time migration are used to invert terabytes of seismic data to create images of the Earth's subsurface. Even using modern supercomputers, it can take weeks to process a single seismic survey and create a useful subsurface image. The computational cost is dominated by the numerical solution of wave equations and their corresponding adjoints. Therefore, a great deal of effort is invested in aggressively optimizing the performance of these wave-equation propagators for different computer architectures. Additionally, the actual set of partial differential equations being solved and their numerical discretization is under constant innovation as increasingly realistic representations of the physics are developed, further ratcheting up the cost of practical solvers. By embedding a domain-specific language within Python and making heavy use of SymPy, a symbolic mathematics library, we make it possible to develop finite-difference simulators quickly using a syntax that strongly resembles the mathematics. The Devito compiler reads this code and applies a wide range of analysis to generate highly optimized and parallel code. This approach can reduce the development time of a verified and optimized solver from months to days.

Recurrence scheme for FDTD-compatible discrete Green's function derived based on properties of Gauss hypergeometric function

ABSTRACTIn this paper, the formulation of one-dimensional FDTD (Finite-difference time-domain)-compatible discrete Green's function (DGF) is derived based on the Gauss hypergeometric function (GHF). The properties of GHF make it possible to derive the recurrence scheme only in the time domain for the DGF generation. Furthermore, this recurrence scheme is valid for any stable time-step size and can be implemented using standard numerical precision of computations. The proposed derivation is obtained without processing in any symbolic mathematics software and relies on the application of known properties of GHF. The difference between the developed recurrence scheme and the direct FDTD simulation is approximately at the level of numerical noise, which confirms the correctness of our derivation. The results obtained should be useful for the development of computational techniques employing FDTD and the diakoptic approach.

The Maple ® symbolic mathematics system in the method of projections for discrete optimization problems

The Maple ® symbolic mathematics system in the method of projections for discrete optimization problems

"Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns

Solving linear systems with a relatively large number of equationsand unknowns can be achieved using an approximate method to obtain a solution with specified accuracy within numerical mathematics. Obtaining theexact solution using the computer today is only possible within the frameworkof symbolic mathematics. It is possible to define an algorithm that does notsolve the system of equations in the usual mathematical way, but still findsits exact solution in the exact number of steps already defined. The methodconsists of simple computations that are not cumulative. At the same time,the number of operations is acceptable even for a relatively large number ofequations and unknowns. In addition, the algorithm allows the process to startfrom an arbitrary initial n-tuple and always leads to the exact solution if itexists.

Development of symbolic mathematics language literacy perception scale for junior high school students and analysis of literacy perceptions based on certain variables

ABSTRACTThe present study aimed to develop a valid and reliable instrument to determine the symbolic mathematics language literacy perceptions of junior high school students, and to investigate literacy perceptions based on gender, final mathematics achievement scores, significance of using symbols, and school grade variables. The study group included 312 students in the pilot scheme and 123 students in the main application during the development of the scale. Exploratory factor analysis demonstrated that a 5-point Likert-type scale with a Cronbach Alpha internal consistency coefficient of 0.85 that included 4 factors in 18 items was developed. The factor structure of the scale was confirmed by confirmatory factor analysis. Furthermore, it was determined that the relayed literacy perceptions of the students did not differ based on gender and grade variables, however there was a significant difference between literacy perceptions based on the final mathematics achievement scores and significance of using mathematical symbols, and it was found that literacy perceptions of the successful students who found it important to use mathematical symbols were higher when compared to other students. Based on the results of multiple regression analysis, the combined gender, mathematical achievement, significance of symbol use, and grade variables explained about 20% of symbolic mathematical language literacy. Based on the findings, it was suggested that future studies could be conducted in schools of different types and utilizing different variables.

“Approximate number system” training: A perceptual learning approach

Recent research suggests that humans perceive quantity using a non-symbolic "number sense." This sense is then thought to provide a foundation for understanding symbolic numbers in formal education. Given this link, there has been interest in the extent to which the approximate number system (ANS) can be improved via dedicated training, as this could provide a route to improving performance in symbolic mathematics. However, current evidence regarding the trainability of the ANS comes largely from studies that have used short training durations, leaving open the question of whether improvements occur over a longer time span. To address this limitation, we utilized a perceptual learning approach to investigate the extent to which long-term (8,000+ trials) training modifies the ANS. Consistent with the general methodological approach common in the domain of perceptual learning (where learning specificity is commonly observed), we also examined whether ANS training generalizes to: (a) untrained locations in the visual field; (b) an enumeration task; (c) a higher-level ratio comparison task; and (d) arithmetic ability. In contrast to previous short-term training studies showing that ANS learning quickly asymptotes, our long-term training approach revealed that performance continued to improve even after thousands of trials. We further found that the training generalized to untrained visual locations. At post-test there was non-significant evidence for generalization to a low-level enumeration task, but not to our high-level tasks, including ratio comparison, multi-object tracking, and arithmetic performance. These results demonstrate the potential utility of long-term psychophysical training, but also suggest that ANS training alone (even long-duration training) may be insufficient to modify higher-level math skills.

Влияние внутрискважинного компенсатора на падение напряжения в элементах электротехнического комплекса добывающей скважины

Актуальность исследования обусловлена необходимостью определения требуемого напряжения на выходе питающего трансформатора с учетом компенсации реактивной мощности внутри скважины, с целью предотвращения аварийных ситуаций в процессе эксплуатации установок электроцентробежных насосов. Цель исследования: оценить изменение величины падения напряжения в токопроводящих элементах электротехнического комплекса установки электроцентробежного насоса для добычи нефти при использовании внутрискважинного компенсатора реактивной мощности. Объект исследования: электротехнический комплекс установок электроцентробежных насосов для добычи нефти. Методы. При разработке математической модели электротехнического комплекса погружной установки для добычи нефти использованы положения теоретических основ электротехники, корректные допущения при составлении математических моделей и компьютерное моделирование в программном пакете символьной математики Wolfram Mathematica. Результаты. Разработана схема замещения и математическая модель электротехнического комплекса установки электроцентробежных насосов с внутрискважинным компенсатором реактивной мощности. С помощью качественной векторной диаграммы напряжений и токов в ветвях и узлах схемы замещения показано распределение величин падений напряжений на элементах комплекса. Анализ векторной диаграммы показал, что внутрискважинный компенсатор уменьшает продольную и поперечную составляющие падения напряжения, что положительно влияет на режим работы установки для добычи нефти. Получены поверхности требуемого напряжения на выходе питающего трансформатора в зависимости от мощности погружного электродвигателя для различных длин и сечений кабельной линии. Установлено, что использование внутрискважинного компенсатора позволяет уменьшить падение напряжения в токопроводящих элементах электротехнического комплекса установки для добычи нефти до 50 В.

THE INFLUENCE OF STATIC ARCHES ON THE EFFICIENCY OF TECHNOLOGICAL PROCESSES IN A HOPPER

Objectives. In practice, the process of loose material flowing from hoppers in different technological processes is often disrupted, resulting in reduced processing efficiency of contained loose materials. One of the most common reasons for the interruption of flows is the formation of dynamic arches in the hopper. From this perspective, the aim of the study is to assess the effect of static arches on the efficiency of technological processes in the hopper. Methods. The study was based on a system of differential equations describing the stationary and equilibrium state of the loose medium, which is integrated in the boundary conditions corresponding to round silos and hoppers. In this case, the analytical expressions for the pressure components of loose material in different directions are derived. Results. The obtained solutions are checked for logical consistency and illustrated with the help of graphs constructed in the area of symbolic mathematics for modeled and real parameters of silos and hoppers. The results of mathematical modeling are compared with experimental data for typical silos. The scatter of the experimental values is related to the processes of compacting the loose material, the time of its storage in the silo, as well as the microvibrational directions of a seismic or technogenic nature. The loose material in the silo resides in an intermediate position between stationary and equilibrium states at all times. The parameters of the loose material (bulk density, external and internal friction angles) represent some generalised values that do not correspond to a specific loose material, which, in addition, can vary over time. Conclusion. It is confirmed that arches formed in the hopper during the static state acquire a dynamic character during material flow, hampering the flow process and consequently reducing the efficiency of the hoppers.

Mathematical modeling of heat treatment processes conserving biological activity of plant bioresources

The aim of this study is to develop a mathematical model of the heat exchange process of LT-processing to estimate the dynamics of temperature field changes and optimize the regime parameters, due to the non-stationarity process, the physicochemical and thermophysical properties of food systems. The application of LT-processing, based on the use of low-temperature modes in thermal culinary processing of raw materials with preliminary vacuum packaging in a polymer heat- resistant film is a promising trend in the development of technics and technology in the catering field. LT-processing application of food raw materials guarantees the preservation of biologically active substances in food environments, which are characterized by a certain thermolability, as well as extend the shelf life and high consumer characteristics of food systems that are capillary-porous bodies. When performing the mathematical modeling of the LT-processing process, the packet of symbolic mathematics “Maple” was used, as well as the mathematical packet flexPDE that uses the finite element method for modeling objects with distributed parameters. The processing of experimental results was evaluated with the help of the developed software in the programming language Python 3.4. To calculate and optimize the parameters of the LT processing process of polycomponent food systems, the differential equation of non-stationary thermal conductivity was used, the solution of which makes it possible to identify the temperature change at any point of the solid at different moments. The present study specifies data on the thermophysical characteristics of the polycomponent food system based on plant raw materials, with the help of which the physico-mathematical model of the LT- processing process has been developed. The obtained mathematical model allows defining of the dynamics of the temperature field in different sections of the LT-processed polycomponent food systems on the basis of calculating the evolution profiles of temperature fields, which enable one to analyze the efficiency of the regime parameters of heat treatment.

CALCULATION OF A RECTANGULAR PLATE ON THE TWO-LAYER BASIS

Operating experience of concrete with vigorous motion of heavy vehicles has shown that, despite the high strength concrete slabs themselves, they need a strong base. When concrete is close to the ground in it, as a result pushes elastically flex when the aisles plates vehicles accumulated residual deformation. Initially, under the plates formed cavity plate lose contact with the ground, and then these cracks. The increase in base strength increases the cost of the pavement. In this paper, a static analysis of rectangular plate on two-layer elastic foundation in the traditional setting. As the base, taken homogeneoussoils, with the upper and lower layer in the form of half-spaces. Considered plate loading variant symmetrical about two axes from the calculated external load wheels. The calculation is performed by B. N. Zhemochkina using symbolic mathematics package. When this plate was divided into 72 rectangular portion in the center of which was placed vertical brace, through which the elastic plate contact with a base. The resulting statically indeterminate system, calculated by the method of mixed structural mechanics, where efforts were taken for unknown in vertical connections, as well as linear and two angular displacements inputted pinching plate along the axis of symmetry. Coefficients of the unknowns related efforts B. N. Zhemochkina determined by the method described previously by one of the authors. The calculation of the vertical plate prepared surface displacements reactive voltage across the contact plate and an elastic base. The proposed approach can be used to find the optimal solutions to (sufficiently) base strength to the estimated vehicle as well as for calculations hinged precast concrete plates.

Methodological approaches of mathematics intersubject training with the use of computer mathematical packages in the training system of future bachelors and masters of non-mathematical disciplines under the conditions of informatization

Quality of mathematical knowledge in the training system of bachelor’s and master’s of non-mathematical disciplines is a problem of current education. Unfortunately, the newly appeared opportunities of improving the mathematical training quality of the above-mentioned bachelors are not used due to origin of the wealthiest tools of computational and symbolic mathematics. We will think as before: if we have a mathematical thinking, we will get the knowledge. But formation of mathematical thinking of the whole professional community representatives is not only feasible, but also economically profitable, and there is no need for this as well. The use of mathematical methods in various areas of professional activity under the conditions of implementation of information technologies into all spheres of human activity requires revision of teaching methods, content of mathematical training, study of computational methods and mathematical simulation tools. This article focuses on innovative approaches to mathematical training of bachelors and masters of non-mathematical areas associated with the use of mathematics as a tool for solving professional problems.

Analytical calculation of the deflection of the lattice truss

An algorithm is given for deriving the dependence of the deflection of a planar statically determinate beam truss on the number of panels, dimensions and load. Three load cases are considered: uniform load on the lower belt, upper belt and vertical force in the middle of the span. By induction, generalizing a series of solutions for trusses with a consecutively increasing number of panels, the desired formula is obtained for the deflection and horizontal displacement of the mobile support of the truss. All transformations are performed in the system of symbolic mathematics Maple. For a sequence of coefficients of the desired formula, using the special Maple operators, homogeneous recurrent equations are constructed and solved. The coefficients found are in the form of polynomials in the number of panels. The asymptotic property of the solution is found. On the graphs of the dependence of the deflection on the number of panels and on the height, extreme points are found. The solution can be used to test the calculations obtained numerically.

Oscillations of a statically indeterminate system with a finite number of degrees of freedom (the experience with the application of mathematical packages in the technical university course of mechanics)

Symbolic mathematics packages give the opportunity to execute the difficult symbolic transformations with use of computer, abandoning graphic methods. The resilient weightless beam fixed by resilient links and carrying two concentrated masses is considered. Instead of building the bending moment diagram and the later use of Vereshchagin's method for disclosure of static indeterminacy, the equation of distribution of the bending and single moments along the beam length is written, and Mohr's integral is calculated.

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

This paper presents an explicit integration scheme to compute the stiffness matrix of a nine node linear convex quadrilateral element of Lagrange family using symbolic mathematics and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique , In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in the bivariates over a nine node 2-square (-1 ) with nodes at the points {(-1,-1),(1,-1),(1,1),(-1,1),(0,-1),(1,0).(0,1),(- 1,0),(0,0)} in local parametric space. In this paper, we have computed these integrals in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all nine node linear convex quadrilaterals for a polygonal domain which provides the nodal coordinates and element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is unknown physical variable and in with given Dirichlet boundary conditions over the given convex polygonal domain.

Approximate number sense correlates with math performance in gifted adolescents

Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical approximation in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the approximate number task. These results suggest that the Approximate Number System is linked to symbolic math performance even at the top levels of math performance.

Does the Approximate Number System Serve as a Foundation for Symbolic Mathematics?

ABSTRACTIn this article we first review evidence for the approximate number system (ANS), an evolutionarily ancient and developmentally conservative cognitive mechanism for representing number without language. We then critically review five different lines of support for the proposal that symbolic representations of number build upon the ANS, and discuss potential causes of conflicting findings in the literature. Finally, we consider potential mechanisms that could drive a relationship between the ANS and symbolic math. We conclude that while there is considerable evidence the relationship between the ANS and symbolic math is meaningful, we are far from understanding the cognitive and neural mechanisms that underlie this relationship.

An All Quadrilateral Automesh Generation Technique and Explicit Integration Scheme for Finite Elements to Solve Some Elliptic Boundary Value Problems in Two Space

This paper proposes the explicit integration scheme for a unique linear convex quadrilateral which can be obtained from an arbitrary linear triangle by joining the centroid to the midpoints of sides of the triangle. The explicit integration scheme proposed for these unique linear convex quadrilaterals is derived by using the standard transformations in two steps. We first map an arbitrary triangle into a standard right isosceles triangle by using an affine linear transformation from global (x, y) space into a local space (u, v). We discretise the standard right isosceles triangle in (u, v) space into three unique linear convex quadrilaterals. We have shown that any unique linear convex quadrilateral in (x, y) space can be mapped into one of the unique quadrilaterals in (u, v) space. We can always map these linear convex quadrilaterals into a 2-aquare in a natural space by an appropriate bilinear transformation. Using these two mappings, we have established an integral derivative product relation between the linear convex quadrilaterals in the (x, y) space interior to the arbitrary triangle and the linear convex quadrilaterals of the local space (u, v) interior to the standard right isosceles triangle. Further , we have shown that the product of global derivative integrals in (x, y) space can be expressed as a matrix triple product P ( X (2 * area of the arbitrary triangle in (x, y) space ) in which P is a geometric properties matrix and is the product of global derivative integrals in (u, v) space.We have shown that the explicit integration of the product of local derivative integrals in (u, v) space over the unique quadrilateral spanning vertices (1/3 ,1/3), (0, ½), (0, 0) , (1/2, 0) is now possible by use of symbolic processing capabilities in MATLAB which are based on Maple – V software package. In a similar manner we have found explicit integrals of ihe shape function and global derivative products as well as the product of shape functions.We use these integral values in computing stiffness matrix for some elliptic equations with constant coefficients.The proposed explicit integration scheme is a useful technique for boundary value problems governed by either a single equation or a system of second order partial differential equations. The physical applications of such problems are numerous in science, and engineering, the examples of single equations are the well known Laplace and Poisson equations with suitable boundary conditions. These problems are already studied in authors recent paper. We have demonstrated the proposed explicit integration scheme to solve the elliptic equations with constant coefficients subject to Dirichlet boundary conditions over simple polygonal domains like 1-square and 2-square.We have demonstrated the propsed scheme and technique to study five typical elliptic boundary value problems.Three FEM models with 2400,5400 and 7776 elements and 2481,5521 and 7921 nodes respectively are used.Tables of fem computed values,theoretical values at selected nodes and the finite element meshes are appended.The findings confirm with surface plots of exact solutions for the considered elliptic boundary value problems. Key words: Explicit Integration, Finite Element Method, Matlab Symbolic Mathematics, All Quadrilateral Mesh Generation Technique,Elliptic equations with constant coefficients ,Dirichlet Boundary Conditions ,Polygonal Domain,2-square,1-square

Finite Element Analysis Of Linear Elastic Torsion For Regular Polygons

This paper presents an explicit finite element integration scheme to compute the stiffness matrices for linear convex quadrilaterals. Finite element formulationals express stiffness matrices as double integrals of the products of global derivatives. These integrals can be shown to depend on triple products of the geometric properties matrix and the matrix of integrals containing the rational functions with polynomial numerators and linear denominator in bivariates as integrands over a 2-square. These integrals are computed explicitely by using symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains.We have also developed an automatic all quadrilateral mesh generation technique for convex polygonal domain which provides the nodal coordinates and element connectivity.We have demonstrated the proposed explicit integration scheme to solve the Poisson Boundary Value Problem for a linear elastic torsion of a non-circular bar with cross sections having profiles of equilateral triangle, a square and regular polygons (pentagon(5-gon)to icosagon(20gon)) which are inscribed in a circle of unit radius. Monotonic convergence from below is observed with known analytical solutions for the Prandtl stress function and torsional constant .We have shown the solutions in Tables which list both the FEM and exact solutions. The graphical solutions of contour level curves and the corresponding finite element meshes are also displayed.

Kindergartners’ fluent processing of symbolic numerical magnitude is predicted by their cardinal knowledge and implicit understanding of arithmetic 2 years earlier

Fluency in first graders’ processing of the magnitudes associated with Arabic numerals, collections of objects, and mixtures of objects and numerals predicts current and future mathematics achievement. The quantitative competencies that support the development of fluent processing of magnitude, however, are not fully understood. At the beginning and end of preschool (M=3years 9months at first assessment, range=3years 3months to 4years 3months), 112 children (51 boys) completed tasks measuring numeral recognition and comparison, acuity of the approximate number system, and knowledge of counting principles, cardinality, and implicit arithmetic and also completed a magnitude processing task (number sets test) in kindergarten. Use of Bayesian and linear regression techniques revealed that two measures of preschoolers’ cardinal knowledge and their competence at implicit arithmetic predicted later fluency of magnitude processing, controlling domain-general factors, preliteracy skills, and parental education. The results help to narrow the search for the early foundation of children’s emerging competence with symbolic mathematics and provide direction for early interventions.

Spontaneous focusing on numerosity and the arithmetic advantage

Children show individual differences in their tendency to focus on the numerical aspects of their environment. These individual differences in ‘Spontaneous Focusing on Numerosity’ (SFON) have been shown to predict both current numerical skills and later mathematics success. Here we investigated possible factors which may explain the positive relationship between SFON and symbolic number development. Children aged 4–5 years (N = 130) completed a battery of tasks designed to assess SFON and a range of mathematical skills. Results showed that SFON was positively associated with children's symbolic numerical processing skills and their performance on a standardised test of arithmetic. Hierarchical regression analyses demonstrated that the relationship between SFON and symbolic mathematics achievement can be explained, in part, by individual differences in children's nonsymbolic numerical processing skills and their ability to map between nonsymbolic and symbolic representations of number.