Symbolic Mathematics Research Articles

Mathematical model for predicting the yield of apple trees on rootstock 62-396

Traditionally, forecasting apple tree yields was carried out on the basis of statistics of already obtained long-term yield dynamics or based on generative formations formed on the tree. The purpose of the work is to identify the most significant soil parameters for the formation of the apple tree varieties yield on the dwarf rootstock 62-396 and to build a mathematical model for predicting fruit yield in the conditions of the Central Chernozem region of the Russian Federation. Field research was carried out in 2004-2022 in industrial apple tree plantations at the age of full fruiting in the Lipetsk (two farms) and Tambov (eight farms) regions. Seven horticultural soil types have been studied. Based on statistical analysis, a mathematical model of apple tree productivity was obtained in the form of an analytical dependence on three factors. A geometric interpretation of the regression model is given using the Mathcad symbolic mathematics system in the form of response surfaces and the corresponding level lines. Practical recommendations are given for choosing rational parameters within selected intervals of factor variation.

Typical Mathematical Education Cannot Predict Non‐Verbal Number Sense

ABSTRACTFormal education would promote symbolic number processing ability, but the relationship between non‐symbolic number sense and mathematical education remains controversial. The current investigation hypothesized that non‐symbolic number sense is independent from the limited human experience of mathematical education, while symbolic number processing could be influenced by related closely to mathematical education. Experiment 1 compared both non‐symbolic number sense and arithmetic computation of undergraduates majoring in STEM fields and humanities. Experiment 2 compared homologous undergraduates from China and Kyrgyzstan in non‐symbolic number sense and mathematical cognitive processing. Both of two experiments found that mathematical education was significantly associated with symbolic mathematics, but not with non‐symbolic number sense. The results suggest that while mathematical education might improve symbolic mathematics, it does not alter non‐symbolic number sense.

Large language models for automatic equation discovery of nonlinear dynamics

Equation discovery aims to directly extract physical laws from data and has emerged as a pivotal research domain in nonlinear systems. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require handcrafted representation rules and complex optimization algorithms. In this paper, we introduce a novel framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically extracting governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse candidate equations in string form and then evaluate the generated equations based on observations. The best equations are preserved and further refined iteratively using the reasoning capacity of LLMs. We propose two alternately iterated strategies to collaboratively optimize the generated equations. The first strategy uses LLMs as a black-box optimizer to achieve equation self-improvement based on historical samples and their performance. The second strategy instructs LLMs to perform evolutionary operations for a global search. Experiments are conducted on various nonlinear systems described by partial differential equations, including the Burgers equation, the Chafee–Infante equation, and the Navier–Stokes equation. The results demonstrate that our framework can discover correct equations that reveal the underlying physical laws. Further comparisons with state-of-the-art models on extensive ordinary differential equations showcase that the equations discovered by our framework possess physical meaning and better generalization capability on unseen data.

Computer Algebra in R Bridges a Gap Between Symbolic Mathematics and Data in the Teaching of Statistics and Data Science

Computer Algebra in R Bridges a Gap Between Symbolic Mathematics and Data in the Teaching of Statistics and Data Science

Differential cognitive correlates in processing symbolic and situational mathematics

AbstractSymbolic and situational mathematics are the two major representations of mathematical knowledge. Although previous literature has studied the relationship between the two from the perspective of teaching practice, learning effectiveness and behavioural performance, there is still a lack of empirical psychological research on cognitive mechanisms to explore the psychological processes of the two. The current study investigated the relationship between symbolic and situational mathematics by determining the differences in cognitive correlates between the two in fourth‐grade children. Their symbolic and situational mathematics abilities were assessed using symbolic and situational enumeration tests under the same conditions. Several types of general cognitive abilities, language processing and academic achievements were also examined. Results showed that both situational and symbolic mathematics are crucial for mathematical achievement. Arithmetic computation is closely correlated with symbolic mathematics, whereas spatial processing and inductive reasoning ability are uniquely correlated with situational mathematics. The results suggest that situational and symbolic mathematics have separate cognitive correlates, which means the two are distinct in terms of psychological processing.

Formulas for Calculating Deformations and Natural Frequency of Free Vibrations of a Hexagonal Tower

Statement of the problem. A scheme of a six-sided prismatic statically determinate spatial girder is proposed. The task is set by induction to derive formulas for the dependence of the deflection of the structure and the lower limit of the main frequency of natural vibrations on the number of panels along the height of the prism. Materials and methods. The forces in the rods along with the reactions of the supports are found in an analytical form by means of the method of cutting nodes in the Maple symbolic mathematics system. One of the nodes at the base of the truss has a spherical support, one has a cylindrical support, the remaining four supports are racks. The top deflection is determined by the Maxwell-Mohr formula. From the analysis of the sequences of coefficients in the formulas for individual structures with a different number of panels, their common members are determined, which are included in the desired calculation formula. The Dunkerley method is used to estimate the first frequency of free oscillations. Results. For various types of loads, formulas for the dependence of girderf deflections on the number of panels are obtained. The coefficients in the solution are polynomial in the number of panels. The derived analytical dependence of the first frequency on the number of panels in comparison with the numerical solution has a small error, which decreases with increasing number of panels. Conclusions. A design of an axisymmetric statically determinate tower-type girder has been developed, which allows analytical solutions to the problem of deflection and the problem of the first natural frequency for an arbitrary number of panels. The resulting formulas can be used to assess the accuracy of numerical solutions and for preliminary calculations of models of structures of this type.

PARMESAN: Meteorological Timeseries and Turbulence Analysis Backed by Symbolic Mathematics

PARMESAN: Meteorological Timeseries and Turbulence Analysis Backed by Symbolic Mathematics

Mental Comparison of Students Learning Abacus-Arithmetic and Not Learning Abacus-Arithmetic on Mathematics Material

Students' mental mastery in elementary school mathematics lessons in Indonesia is weak, slow, inaccurate, and declining. Mastery problems among elementary school students who have studied mental abacus arithmetic were found to be low. This is an urgent matter to research because there is a gap between theory, expectations, and reality. The purpose of this research was to compare the ability to solve mathematical problems between students who studied abacus mental arithmetic and students who did not study abacus mental arithmetic. This research involved 70 students. Data collection techniques using instruments, the instruments used were the first-semester mathematics exam and mental arithmetic exam. Data analysis techniques using SPSS Version 25.0 statistics, namely the t-test, were used to compare the ability to solve mathematical problems between students who studied mental abacus-arithmetic and students who did not study mental abacus-arithmetic. Pearson correlation was used to determine the relationship between students' mental arithmetic learning achievement and their ability to solve mathematical problems. The results of the research showed that there was a significant difference (p<0.05) in learning achievement on symbolic mathematics questions and mental arithmetic achievement between students who studied mental abacus calculation and students who did not study mental abacus calculation. The minimum score of the group that studied mental abacus calculation was higher compared to the group that did not study mental abacus calculation. However, there was no significant difference (p<0.05) in mathematics learning achievement between students who studied mental abacus-arithmetic and students who did not study mental abacus-arithmetic.

Simulation of distributed network elements in CAS MAXIMA and server online calculations in the educational process

Pressure pipelines are an important part of the water management infrastructure. Water supply networks pump the liquid to the consumer with the help of pumps (or supply it from a water tower) to specified points. The functioning of the entire urban infrastructure depends on the correct calculation and reliable operation of these areas. This paper shows the implementation of a computer calculation of one of the branched network options using numerical methods in the CAS MAXIMA computer algebra system, its graphic subsystem of symbolic mathematics, and the use of the author's online calculation program with a client-server implementation - https://www.k123.org.ua/jgIHjob1.html .
 The implementation of elements of computer calculations in the educational process allows the generation of an unlimited number of individual variants of initial data. The implementation of computer calculation requires the availability of automatic online generation of answers for each stage of iterative calculations. The task of generating online answers is the basis of the program development.
 An educational online platform is offered for researching the optimality of numerical, graphic and audio formats for setting the problem. The formats of the web form for entering output data have been studied. Online calculations reduce the time spent on typical monotonous operations. Additional possibilities of instant online calculations and visual graphic visualization of calculation results expand the possibilities of modeling individual elements of a branched network.
 The online program allows you to use computer calculations, expand the culture of using applied mathematical methods for engineering calculations, introduce algorithms and data structures, and solve the main task of training future engineers - to ensure a modern level of education and readiness to solve new problems.

Fault detection and isolation in uncertain dynamic systems using composite optimization and inferential sensing

Inferential sensing is a cost-effective and reliable approach to replace expensive and impractical hardware sensing, while yielding robust fault detection and isolation (FDI) during system maintenance or operation. Yet, their high computational footprint necessitates the employment of advanced and efficient estimation techniques, along with symbolic mathematics and automatic differentiation. Herein, we attempt to combine model-driven sensor selection methods with inferential sensing to accurately identify faults in the presence of system noise and uncertainty. During sensor selection, we employ criteria from information theory to maximize the estimability of system faults from available system measurements. The most informative sensor set stems from a comparison of all possible sensor suites via information theoretic criteria expressed as functions of the Fisher Information Matrix. Optimal inferential sensors are calculated through a novel fusion of symbolic regression and information theory. For built-in test deployment, we utilize k-Nearest Neighbors classification to assess the capability of each sensor suite for all plausible instantiations of uncertainty and system noise, and evaluate FDI performance benefits from the inclusion of inferential sensor(s). The proposed computational framework is deployed with steady-state and dynamic models of a cross-flow plate-fin heat exchanger, at varying levels of measurement noise and uncertainty. As shown, the augmented system of composite sensors (i.e., inferential and hardware) provides accurate information on system fault(s) and reduces the evidence of uncertainty. Compared to traditional sensor approaches, the proposed techniques are shown to be more precise while simultaneously granting increased insight into details otherwise hidden by noise. Thus, they form a robust and reliable tool when conducting fault diagnosis and possibly toward system safety.

Wittgenstein and Stenlund on Mathematical Symbolism

In recent work, Sören Stenlund (2015) contextualizes Wittgenstein’s philosophy of mathematics as aligned with the tradition of symbolic mathematics. In the early modern era, mathematicians began using purely formal methods disconnected from any obvious empirical applications, transforming their subject into a symbolic discipline. With this, Stenlund argues, they were freeing themselves of ancient ontological presuppositions and discovering the ultimately autonomous nature of mathematical symbolism, which eventually formed the basis for Wittgenstein’s thinking. A crucial premise of Wittgenstein’s philosophy of mathematics, on this view, is that the development of mathematical concepts is independent of any ontological implications and occurs in principle without normative connections to empirical applicability. This paper critically examines this narrative and arrives at the conclusion that Stenlund’s view of mathematical progress is in stark contrast to the later Wittgenstein’s writing, which emphasizes links between symbolisms and their applications.

Responsefun: Fun with Response Functions in the Algebraic Diagrammatic Construction Framework.

We present the open-source responsefun package, which implements a universally applicable procedure for computing molecular response properties within the algebraic diagrammatic construction (ADC) framework, exploiting the intermediate state representation (ISR) approach. With symbolic mathematics, the user can simply enter textbook sum-over-states (SOS) expressions from time-dependent perturbation theory, which are then automatically translated into the corresponding symbolic ADC/ISR formulations. Using the data structures provided by the hybrid Python/C++ module adcc for calculating excited states with ADC, the specified response property is directly evaluated, and the result is returned to the user. Employing the novel responsefun package, we present the first ADC/ISR calculations of second-order hyperpolarizability tensors and three-photon-absorption matrix elements.

Verbalized arithmetic principles correlate with mathematics achievement.

When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.

Formulas for Calculating the Deflections of a Triangular Truss of Spatial Cover

Statement of the problem. The scheme of triangular in terms of dome type cover is proposed. The construction is statically determinate. The formulas for the dependence of the deflection on the number of panels and sizes are derived by generalizing a series of individual solutions by induction. Materials and methods. The forces in the coating rods are performed by cutting nodes in symbolic form using operators of the Maple symbolic mathematics system. The unknown systems of equilibrium equations in projections on the coordinate axis include the reactions of vertical supports located on the sides of the truss. One of the corners of the truss also has a spherical support, one is cylindrical. The Maxwell—Mohr formula is used to calculate the deflection of the vertex. The analysis of sequences of coefficients in solutions for individual trusses with different numbers of panels yields expressions for common terms included in the desired calculation formula. Results. Formulas for the dependence of the deflections of the truss on the number of panels for a vertical load evenly distributed over the nodes of the truss and a horizontal wind load applied to one of the sides of the structure are obtained. The solutions have a simple polynomial form. The curves of the dependence of the horizontal displacement of the dome top on the number of panels reveal a minimum. Asymptotics of the solutions is identified. Conclusions. A scheme of a statically determinate symmetric spatial dome is developed and its mathematical model is constructed, allowing analytical solutions with an arbitrary number of panels. The identified dependencies can be used both to evaluate the accuracy of numerical solutions and to find optimal combinations of structural dimensions in terms of rigidity.

Deformations of a three-dimensional model of a trihedral double lattice rod tower

Introduction. The calculation of deformations of spatial structures, as a rule, is carried out numerically on the basis of the finite element method. With the development of computer mathematical systems for regular systems an opportunity to obtain analytical solutions appeared. Such solutions can be used as test ones to evaluate numerical solutions and for preliminary calculation of a structure at design stage. The task has been set to derive deformation dependences of a spatial truss under various loads from its size and number of panels.
 
 Materials and methods. The truss of the tower structure is statically determined. The calculation of force values in the bars is performed by cutting out nodes in a Maple symbolic mathematics system. The Maxwell – Mohr formula is used to determine the displacement of a node at the top of the frame, considering only the longitudinal deformations of the rods. From the generalization of a series of analytical solutions for trusses with consecutively increasing number of panels onto an arbitrary number of panels the sought formula is derived by the induction method.
 
 Results. An algorithm of derivation of formulas for the deflections of a building along two mutually perpendicular horizontal axes under the action of lateral uniformly distributed nodal loads is presented. Effort distribution patterns are obtained for the truss rods and analytical stress dependences of some rods on the number of panels are found. Linear asymptotics of the solutions for the number of panels and the points of extremum are found and computed.
 
 Conclusions. The considered model of the spatial statically determined tower truss allows deriving exact formulas for deformations under the action of various loads within the limits of the adopted model. It is possible to use derived formulas for a preliminary estimation of designed structure and to apply as test ones for numerical calculations. The extremum points and analytical expressions for asymptotes allow the derived formulas to be used for solving stiffness optimization problems of a structure.

On the training of scientific personnel at post-graduate school

The possibilities of modern packages of symbolic mathematics for the scientific work of novice researchers in mathematics are discussed. Counterexamples for Schwartz theorem on the equality of mixed derivatives are considered. It is shown how you can use the graphical and computational capabilities of symbolic mathematics packages for their analysis so that novice researchers with the traditional engineering background in mathematics understand them. The article focuses on the problems of ambiguous perception of the PC user of three-dimensional graphic images on the monitor screen. This is important for operators who make decisions based on information obtained in a mixed virtuality or mixed reality environment. The analysis of counterexamples for the Schwartz theorem is carried out using both classical mathematics and possibilities of symbolic mathematics packages. It is shown that this assertion is not necessary. During the virtual demonstration of three-dimensional graphic objects, recommendations on their perception in terms of information environments are provided.

Celmech: A Python Package for Celestial Mechanics

We present celmech, an open-source Python package designed to facilitate a wide variety of celestial mechanics calculations. The package allows users to formulate and integrate equations of motion incorporating user-specified terms from the classical disturbing function expansion of the interaction potential between pairs of planets. The code can be applied, for example, to isolate the contribution of particular resonances to a system’s dynamical evolution and develop simple analytical models with the minimum number of terms required to capture a particular dynamical phenomenon. Equations and expressions can be easily manipulated by leveraging the extensive symbolic mathematics capabilities of the sympy Python package. The celmech package is designed to interface seamlessly with the popular N-body code REBOUND to facilitate comparisons between calculation results and direct N-body integrations. The code is extensively documented, and numerous example Jupyter notebooks illustrating its use are available online.

On technical debt in mathematical programming: An exploratory study

The Technical Debt (TD) metaphor describes development shortcuts taken for expediency that cause the degradation of internal software quality. It has served the discourse between engineers and management regarding how to invest resources in maintenance and extend into scientific software (both the tools, the algorithms and the analysis conducted with it). Mathematical programming has been considered ‘special purpose programming’, meant to program and simulate particular problem types (e.g., symbolic mathematics through Matlab). Likewise, more traditional mathematical programming has been considered ‘modelling programming’ to program models by providing programming structures required for mathematical formulations (e.g., GAMS, AMPL, AIMMS). Because of this, other authors have argued the need to consider mathematical programming as closely related to software development. As a result, this paper presents a novel exploration of TD in mathematical programming by assessing self-reported practices through a survey, which gathered 168 complete responses. This study discovered potential debts manifested through smells and attitudinal causes towards them. Results uncovered a trend to refactor and polish the final mathematical model and use version control and detailed comments. Nonetheless, we uncovered traces of negative practices regarding Code Debt and Documentation Debt, alongside hints indicating that most TD is deliberately introduced (i.e., modellers are aware that their practices are not the best). We aim to discuss the idea that TD is also present in mathematical programming and that it may hamper the reproducibility and maintainability of the models created. The overall goal is to outline future areas of work that can lead to changing current modellers’ habits and assist in extending existing mathematical programming (both practice and research) to eventually manage TD in mathematical programming.

Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)

Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which are essential for deepening our understanding of physical processes and predicting future responses. However, the PDEs of many real-world problems are uncertain, which calls for PDE discovery. We propose the symbolic genetic algorithm (SGA-PDE) to discover open-form PDEs directly from data without prior knowledge about the equation structure. SGA-PDE focuses on the representation and optimization of PDE. Firstly, SGA-PDE uses symbolic mathematics to realize the flexible representation of any given PDE, transforms a PDE into a forest, and converts each function term into a binary tree. Secondly, SGA-PDE adopts a specially designed genetic algorithm to efficiently optimize the binary trees by iteratively updating the tree topology and node attributes. The SGA-PDE is gradient-free, which is a desirable characteristic in PDE discovery since it is difficult to obtain the gradient between the PDE loss and the PDE structure. In the experiment, SGA-PDE not only successfully discovered nonlinear Burgers' equation, Korteweg-de Vries (KdV) equation, and Chafee-Infante equation, but also handled PDEs with fractional structure and compound functions that cannot be solved by conventional PDE discovery methods.

On solutions of one of the second-order nonlinear differential equation: An in-depth look and critical review

A critical review of recent articles by two scientific groups, which have considered a well-known nonlinear differential equation of the second order, is presented. One of these groups is led by G. Akram et. al. from Pakistan (Department of mathematics, University of the Punjab, Lahore). Another group is led by K.-J. Wang from China (School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo). In a number of papers published by these authors there have been presented a lot of solutions of the well-known differential equation. In fact, this differential equation was studied more than 150 years ago in the works of outstanding mathematicians Niels Henrik Abel (1827), Karl Gustav Jacob Jacobi (1829) and Karl Weierstrass (1855, 1862). However, the scientific groups of Akram and Wang, apparently not being familiar with the works of prominent mathematicians and not realizing that this equation has a unique solution on the complex plane, have been trying to rewrite the solution of this equation using symbolic mathematics programs misleading by that the scientific community. Although there are several erroneous works by Akram and Wang, only a few articles are analyzed here. The errors of a few works by these authors are discussed. The correct solutions of this popular equation, which is often encountered in nonlinear optics, are presented.