Symbolic Algebra Software Research Articles

Handy formulas for binomial moments

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.

FeynMG: A FeynRules extension for scalar-tensor theories of gravity

The ability to represent perturbative expansions of interacting quantum field theories in terms of simple diagrammatic rules has revolutionized calculations in particle physics (and elsewhere). Moreover, these rules are readily automated, a process that has catalyzed the rise of symbolic algebra packages. However, in the case of extended theories of gravity, such as scalar-tensor theories, it is necessary to precondition the Lagrangian to apply this automation or, at the very least, to take advantage of existing software pipelines. We present a Mathematica code FeynMG, which works in conjunction with the well-known package FeynRules, to do just that: FeynMG takes as inputs the FeynRules model file for a non-gravitational theory and a user-supplied gravitational Lagrangian. FeynMG provides functionality that inserts the minimal gravitational couplings of the degrees of freedom specified in the model file, determines the couplings of the additional tensor and scalar degrees of freedom (the metric and the scalar field from the gravitational sector), and preconditions the resulting Lagrangian so that it can be passed to FeynRules, either directly or by outputting an updated FeynRules model file. The Feynman rules can then be determined and output through FeynRules, using existing universal output formats and interfaces to other analysis packages. Program summaryProgram title: FeynMGCPC Library link to program files:https://doi.org/10.17632/bjpfmtt55k.1Developer's repository link:https://gitlab.com/feynmg/FeynMGLicensing provisions: MIT licenseProgramming language:Mathematica 13.2Nature of problem: Determining the additional interactions that scalar-tensor theories of gravity induce between the fields of the Standard Model of particle physics and its extensions is a tedious and time-consuming task that is ripe for automation. FeynMG is a package that provides this automation. It can expand the spacetime metric around a flat background and perform the necessary field redefinitions for a given theory in order to provide a Lagrangian that is in a form that can be processed by FeynRules[1]. The Feynman rules for the theory can then be determined using the existing functionality of FeynRules and output in formats that can be read into other particle-physics analysis packages.Solution method: (1) Load both FeynRules and FeynMG into Mathematica. (2) Load a model file describing a given matter sector and Lagrangian. (3) Use the FeynMG implementation to: (3a) append the gravitational sector and incorporate curvature-dependent objects, such as the metric, curvature scalars and tensors, and covariant derivatives; (3b) effect a Weyl transformation or expand the spacetime metric up to second order in graviton interactions with the matter fields; and (3c) diagonalize mass and kinetic mixings, and canonically normalize dynamical fields. (4) Pass the output directly to FeynRules or output a new model file from FeynMG.

Bit-twiddling hacks for gamma matrices

For some research questions that involve Spin(p, q) representation theory, using symbolic algebra based techniques might be an attractive option for simplifying and manipulating expressions. Yet, for some such problems, especially as they arise in the study of various limits of M-theory (such as dimensional reductions), the complexity of the resulting expressions can become computationally challenging when using popular symbolic algebra packages in a straightforward manner.This work discusses some general properties of Gamma matrices that are computationally useful, down to the level of what one would call “bit-twiddling hacks” in computer science. It is presented in a self-contained way that should be accessible to both physicists and computer scientists. Code is available alongside the TeX source of the preprint version of this article on arXiv.

Automated handling of complex chemical structures in Z-matrix coordinates-The chemcoord library.

In this work, we present a fully automated method for the construction of chemically meaningful sets of hierarchical nonredundant internal coordinates (ICs; also commonly denoted as Z-matrices) from the Cartesian coordinates of a molecular system. Particular focus is placed on avoiding ill-definitions of angles and dihedrals due to linear arrangements of atoms, to consistently guarantee a well-defined transformation to Cartesian coordinates, even after structural changes. The representations thus obtained are particularly well suited for pathway construction in double-ended methods for transition state search and optimizations with nonlinear constraints. Analytical gradients for the transformation between the coordinate systems were derived for analytical geometry optimizations purely in Z-matrix coordinates. The geometry optimization was coupled with a Symbolic Algebra package to support arbitrary nonlinear constraints in Z-matrix coordinates, while retaining analytical energy gradient conversion. The difference to the commonly used nonhierarchical IC transformations is discussed. Sample applications are provided for a number of common chemical reactions and illustrative examples.

On Shannon entropy computations in selected plasticity problems

AbstractThis paper considers the problem of determining probabilistic entropy fluctuations, which are important for understanding uncertainty propagation in mechanical systems in the elasto‐plastic regime. Probabilistic entropy is conceptualized based on an initial definition by Shannon, which demands discrete representation of the uncertainty source. Numerical analysis is performed using the Response Function Method with polynomial bases. Coefficients are found and order optimization is completed using polynomial interpolations or the Least Squares Method. Approximations are based on the Finite Element Method. Local polynomial bases enable nonlinear increment analysis, and allow for a given degree of freedom in the FEM model to be described as a function of a random input parameter. Academic FEM software and the ABAQUS system were used for numerical experiments. Polynomial approximations, probabilistic moment computations, and statistical entropy estimations were programmed in the symbolic algebra package MAPLE. Transformation of the input probability density into the output function was performed using the Monte‐Carlo simulation algorithm for statistically optimized polynomial bases of extreme displacement functions. Two computational examples are given to demonstrate probabilistic entropy fluctuations for a small statically indeterminate aluminum truss structure and also for practical engineering case study of the steel round bar under uniform tensile stress. In these examples, some material and geometrical uncertainties distributed according to Gaussian, triangular, uniform as well as lognormal distributions were analyzed. The presented approach could be used for constitutive models of solids, computational fluid dynamics, and in other discrete numerical methods.

Numerical implementation of drop spin and tilt method for five-axis tool positioning for tensor product surfaces

This paper presents an extension of multipoint machining technique, called the Drop Spin and Tilt (DST) method, that spins the tool on two axes, allowing for the generation of multiple contact points at varying distances around the first point of contact. The multiple DST second points of contact were used to manually generate a toolpath with uniform spacing between the two points of contact. The original DST method used a symbolic algebra package to position the tool on a bi-quadratic surface; our extension is a numerical solution that allows positioning a toroidal tool on a tensor product Bézier surface, which we tested on bi-degrees 2, 3, 4, and 5. On the three bicubic surfaces we tested, the average number of initial seeds for our numerical method was less than 5 across all three surfaces, and the average number of Newton iterations was less than 20 except when the curvature of the tool closely matches the local surface curvature (in which case more iterations were required). Furthermore, we investigate the spread of possible second points of contact as the tool is spun around these two axes, demonstrating the feasibility of using the method to control the machining strip width.

Plain correlation asymptotes to predict the center and mean temperatures and total heat transfer in simple solid objects with uniform surface temperature at limiting small-time conditions

Within the platform of unsteady, one-dimensional heat conduction in simple solid objects (large plate, long cylinder and sphere) cooled/heated with uniform surface temperature, the three important thermal quantities that need to be quantified are: the center temperature, the mean temperature and the total heat transfer. One objective of the study is the accurate determination of new dimensionless critical times for the mean and center temperatures that describe the large time sub-domain using a symbolic algebra software. Another objective of the study is to use regression analysis to construct plain correlation asymptotes for the center temperature, the mean temperature and the total heat transfer in the small time sub-domain using as input data the exact, analytical temperature distributions for the plate, cylinder and sphere expressed by infinite series for all time. Excellent agreement for the center temperature, mean temperature and total heat transfer are achieved in the small time sub-domain using the two different computational procedures.

Probabilistic and stochastic aspects of rubber hyperelasticity

The main aim of this work is to compare various models of rubber elasticity, i.e. neo-Hookean, Mooney-Rivlin, Yeoh, Gent, Arruda-Boyce as well as the Extended Tube model in terms of their application to the probabilistic analysis. Some discussions concerning failure analysis of the rubbers according to these models is provided also. Constitutive relations following these theories are tested for the case of uniaxial tension of the incompressible material, where deformation of a rubber specimen is treated as Gaussian random variable having a priori given expectation and standard deviations varying in some interval with bounds driven by various experimentation techniques. Probabilistic analysis is provided here in two alternative ways—via traditional Monte-Carlo technique as well as using higher order stochastic perturbation method implemented both in the symbolic computer algebra software. An application of non-Gaussian distributions relevant to the considered deformation, like lognormal one for instance has been also considered. This analysis includes computational determination of the first four basic probabilistic characteristics, i.e. expectation, coefficient of variation, skewness and kurtosis, and is provided to verify the resulting probabilistic distribution of the induced stress and its entropy. Some conclusions are drawn for the generalization of this method to other stress softening materials.

Reviving the shear-free perfect fluid conjecture in general relativity

Employing a Mathematica symbolic computer algebra package called xTensor, we present -covariant special case proofs of the shear-free perfect fluid conjecture in general relativity. We first present the case where the pressure is constant, and where the acceleration is parallel to the vorticity vector. These cases were first presented in their covariant form by Senovilla et al. We then provide a covariant proof for the case where the acceleration and vorticity vectors are orthogonal, which leads to the existence of a Killing vector along the vorticity. This Killing vector satisfies the new constraint equations resulting from the vanishing of the shear. Furthermore, it is shown that in order for the conjecture to be true, this Killing vector must have a vanishing spatially projected directional covariant derivative along the velocity vector field. This in turn implies the existence of another basic vector field along the direction of the vorticity for the conjecture to hold. Finally, we show that in general, there exists a basic vector field parallel to the acceleration for which the conjecture is true.

Minimum time optimal control simulation of a GP2 race car

In this work, optimal control theory is applied to minimum lap time simulation of a GP2 car, using a multibody car model with enhanced load transfer dynamics. The mathematical multibody model is formulated with use of the symbolic algebra software MBSymba and it comprises 14 degrees of freedom, including full chassis motion, suspension travels and wheel spins. The kinematics of the suspension is exhaustively analysed and the impact of tyre longitudinal and lateral forces in determining vehicle trim is demonstrated. An indirect optimal control method is then used to solve the minimum lap time problem. Simulation outcomes are compared with experimental data acquired during a qualifying lap at Montmeló circuit (Barcelona) in the 2012 GP2 season. Results demonstrate the reliability of the model, suggesting it can be used to optimise car settings (such as gearing and aerodynamic setup) before executing track tests.

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂xJ(x,ϵ)=A(x,ϵ)J(x,ϵ) finds a basis transformation T(x,ϵ), i.e., J(x,ϵ)=T(x,ϵ)J′(x,ϵ), such that the system turns into the epsilon form : ∂xJ′(x,ϵ)=ϵS(x)J′(x,ϵ), where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ϵ. That makes the construction of the transformation T(x,ϵ) crucial for obtaining solutions of the initial system.In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals. Program summaryProgram Title:FuchsiaProgram Files doi:http://dx.doi.org/10.17632/zj6zn9vfkh.1Licensing provisions: MITProgramming language:Python 2.7Nature of problem: Feynman master integrals may be calculated from solutions of a linear system of differential equations with rational coefficients. Such a system can be easily solved as an ϵ-series when its epsilon form is known. Hence, a tool which is able to find the epsilon form transformations can be used to evaluate Feynman master integrals.Solution method: The solution method is based on the Lee algorithm (Lee, 2015) which consists of three main steps: fuchsification, normalization, and factorization. During the fuchsification step a given system of differential equations is transformed into the Fuchsian form with the help of the Moser method (Moser, 1959). Next, during the normalization step the system is transformed to the form where eigenvalues of all residues are proportional to the dimensional regulator ϵ. Finally, the system is factorized to the epsilon form by finding an unknown transformation which satisfies a system of linear equations.Additional comments including Restrictions and Unusual features: Systems of single-variable differential equations are considered. A system needs to be reducible to Fuchsian form and eigenvalues of its residues must be of the form n+mϵ, where n is integer. Performance depends upon the input matrix, its size, number of singular points and their degrees. It takes around an hour to reduce an example 74 × 74 matrix with 20 singular points on a PC with a 1.7 GHz Intel Core i5 CPU. An additional slowdown is to be expected for matrices with complex and/or irrational singular point locations, as these are particularly difficult for symbolic algebra software to handle.

(Invited) Numerical Simulations of Mass Transport Using Separation of Variables: an Old Method Rejuvenated with Symbolic Algebra Software

Separation of variables is a classic method of solving partial differential equations, such as the convective diffusion equations that are ubiquitous in electrochemistry. It leads to infinite series solutions, which are exact analytical solutions. However, various parameters involved in the series have to be numerically evaluated, and the series converge rather slowly, which has meant that this method has not been used for practical numerical calculations of concentration profiles for electrochemical problems. We show here that these difficulties can be overcome, and demonstrate the use of the symbolic algebra program Maple in a practical implementation of this for steady-state convective diffusion past electrodes in a 2-D channel [1]. The key to successful implementation has been the use of asymptotic formulas to reliably locate the eigenvalues, so that the numerical solver does not miss any. The method has the following advantages: 1. It is a mesh-free method, so a separate step of validating the mesh or grid is not necessary. 2. The global error can be estimated by evaluating the series to more terms. 3. Each segment (above an electrode or insulating section) is solved at once, so changing the length of an electrode does not change the difficulty of the calculation. 4. Workup of the concentrations to derived quantities such as currents or collection efficiencies can be done without any degradation of accuracy. The present implementation assumes diffusivities of reactant and product are equal, which allows the generation of both concentration profiles in a single calculation by using a recently developed transformation method [2]. It also neglects axial diffusion. Future work will seek to remove these limitations, and extend the results to 3-D and time-dependent systems.

Lap time optimisation of a racing go-kart

The minimum lap time optimal control problem has been solved for a go-kart model. The symbolic algebra software Maple has been used to derive equations of motion and an indirect method has been adopted to solve the optimal control problem. Simulation has been successfully performed on a full track lap with a multibody model endowed with seven degrees of freedom. Geometrical and mechanical characteristics of a real kart have been measured by a lab test to feed the mathematical model. Telemetry recorded in an entire lap by a professional driver has been compared to simulation results in order to validate the model. After the reliability of the optimal control model was proved, the simulation has been used to study the peculiar dynamics of go-karts and focus to tyre slippage dynamics, which is highly affected by the lack of differential.

A perturbative algorithm for quasi-periodic linear systems close to constant coefficients

A perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi-periodic but close to constant coefficients. The scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter. It is recursive, can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties. This algorithm can be used, in particular, to carry out systematic stability analyses in the parameter space of a given system by considering variational equations.

Heat transfer and fluid flow of a non-Newtonian nanofluid over an unsteady contracting cylinder employing Buongiorno’s model

Purpose – The purpose of this paper is to discuss a combined similarity-numerical approach that is used to study the unsteady two-dimensional flow of a non-Newtonian nanofluid over a contracting cylinder using Buongiorno’s model and the Casson fluid model that is used to characterize the non-Newtonian fluid behavior. Design/methodology/approach – Similarity transformations are employed to transform the unsteady Navier-Stokes partial differential equations into a system of ordinary differential equations. The transformed equations are then solved numerically by means of the very robust symbolic computer algebra software MATLAB employing the routine bvpc45. Findings – The effect of increasing values of the Casson parameter is to suppress the velocity field (in absolute sense), the temperature and concentration decrease as Casson parameter increase. The heat and mass transfer rates decrease with the increase of unsteadiness parameters and Brownian motion parameter. In addition, they increase as the Casson parameter and the thermophoresis parameter increase. Originality/value – The problem is relatively original and represents a very important contribution to the field of non-Newtonian nanofluids.

Improved formulas for the two optimum VFA flip‐angles

The variable flip-angle (VFA) experiment is an efficient way to measure T1 using a spoiled steady-state sequence compared to the gold standard inversion recovery experiment 1-3. The VFA signal curves can be easily linearized and hence only two acquisitions are required 4, although in practice the number of flip-angles used varies 5-8. Deoni et al. showed that the same methods could be used to measure T2 using a balanced steady-state free precession (bSSFP) sequence 9. This method has been less widely used, likely due to off-resonance artifacts at higher field strengths 10-12. If two flip-angles are used, they must be selected carefully. An obvious criterion is that the measurement variance of T1 or T2 should be minimized. Wang et al. derived an expression for the variance in T1 that contains a sum over all flip-angles 13, and found the optimal two by numerical optimization. In contrast, Deoni et al. hypothesized that the optimal flip-angles will generate equal signal intensity but lie either side of the maximum, and derived a formula for the flip-angles in terms of the measured fractional signal f compared to the Ernst signal. Deoni et al. then numerically optimized the product of signal intensity and the distance between points on the regression line and found f = 0.71 for both T1 and T2 (9). This hypothesis, while intuitive and well grounded, was only checked by comparison to a pair of flip-angles calculated by Wang. Instead, if Wang and Deoni's formulas are combined f can be found analytically to be and this value shown to minimize the experimental variance. The following analysis used the Sympy symbolic algebra package (version 0.7.5, www.sympy.org), provided as part of the Canopy distribution (www.enthought.com). The code used is available as Supporting Information. These values also make the derivatives of (5) equal to zero, confirming that the assumption of equal signal intensity is correct. Using values of , given previously in (9,13), yields identical values of and . More practical values of yields and , which differ from those given in (9) by only 1°. This result is more concise than equations [A14–17] in (9). Deoni does not give an example flip-angle for a T2 experiment. For , Eq. 8 gives and . Figure 1 plots Eqs. 5 and 11, expressed as a noise factor (13), across a range of T1 and T2 values with flip-angles chosen with (8) or (13) for T1 or T2, respectively. TR was 5 ms and for the T2 simulations. These plots give an indication of how far from the respective target T1 or T2 the relative accuracy of the two flip-angle method is maintained. For T1, a short target value produces the best accuracy but only over a very narrow range, whereas for T2 a long target gives the best accuracy. For T1 and T2, the curves are broader for long target values, implying more equal accuracy over a range of values. However, the accuracy for values lower than the target becomes compromised, particularly in the case of T2. Plots of Eq. 5 (a) and Eq. 11 (b) using angles optimized for T1 or T2 values given in the legends. Tobias Charles Wood Department of Neuroimaging King's College London London, UK The author would like to thank Professor Gareth Barker for reading the article and Doctor Anna Völker for helpful suggestions. In accordance with the procedures described in our Authors' Guidelines, Drs. Deoni, Peters and Rutt were sent a copy of this Letter to the Editor after it was accepted, along with an invitation to respond. While they did not submit a formal Response manuscript, they pointed out the following reference, which they think may be useful to readers who are interested in this topic: Schabel MC, Morrell GR. Uncertainty in T(1) mapping using the variable flip angle method with two flip angles. Phys Med Biol. 2009;54 N1-8. doi: 10.1088/0031-9155/54/1/N01. Additional Supporting Information may be found in the online version of this article. The derivation of equations [5] to [8] using the Sympy symbolic algebra package. The derivation of equations [11] to [13] using the Sympy symbolic algebra package. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Weyl-Euler-Lagrange Equations of Motion on Flat Manifold

This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.

Sensitivity and uncertainty in homogenization of the CFRP composites via the Response Function Method

The main issue in this work is computational analysis of material parameters variability, deterministic sensitivity as well as of the basic probabilistic characteristics of the homogenized elasticity tensor for Carbon Fiber Reinforced Polymers (CFRP). The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and its components are treated as isotropic and statistically homogeneous media defined uniquely by two independent Gaussian random material parameters – Young modulus and, independently, Poisson ratio. Probabilistic approach is based on the Response Function Method allowing for large random dispersions of the input random variables and is implemented using the polynomial response functions recovered using the Nonlinear Least Squares Method. The same response functions of the homogenized elasticity tensor components related to both material parameters of the fiber and matrix in deterministic analysis support semi-analytical determination of the sensitivity coefficients of this tensor. Homogenization technique proposed in this study consists of equating the deformation energies of the real and homogenized composites under the same uniform deformations of the entire RVE. The cell problem is solved numerically using the Finite Element Method system ABAQUS® and probabilistic computer analysis is carried out entirely in the symbolic algebra package MAPLE®. Computer analysis shows the most influential material parameter in the CFRP composite, random fluctuations of the effective tensor following an input uncertainty as well as some perspective of effective parameters of these composites resulting from permanently increasing Young modulus of the carbon itself.

The Impartial, Anonymous, and Neutral Culture Model: A Probability Model for Sampling Public Preference Structures

We introduce a new probability model, namely the Impartial, Anonymous, and Neutral Culture (IANC) Model, for sampling public preferences concerning a given set of alternatives. The IANC Model treats public preferences through a class of preference profiles named roots, where both names of the voters and the alternatives are immaterial. The general framework along with the theoretical formulation through group actions, an exact formula for the number of roots, and the description of a symbolic algebra package that allows for the generation of roots uniformly are presented. In order to be able to obtain uniform distribution of roots for large electorate size and high number of alternatives which lead to combinatorial explosions, the machinery we developed involves elements of symmetric functions and an application of the Dixon-Wilf algorithm. Using Monte Carlo methods, the model we develop allows for a testbed that can be used to answer various questions about the properties and behaviors of anonymous and neutral social choice rules for large parameters. As applications of the method, the results of two Monte Carlo experiments are presented: the likelihood of the existence of Condorcet winners and the probability of Condorcet and plurality rules to choose the same winner.

Identifying Sets of Constraint Forces by Inspection

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.