Computer Algebra Program Research Articles

Relative Entropy Application to Study the Elastoplastic Behavior of S235JR Structural Steel.

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback-Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies-Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions.

Examples of Expansions in Fractional Powers, and Applications

We approximate the solution of a generalized form of the Bagley–Torvik equation using Taylor’s expansions in fractional powers. Then, we study the fractional Laguerre-type logistic equation by considering the fractional exponential function and its Laguerre-type form. To verify our findings, we conduct numerical tests using the computer algebra program Mathematica©.

F. S. Macaulay: From plane curves to Gorenstein rings

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

Necessary and Sufficient Conditions for Difference Flatness

We show that the flatness of a nonlinear discrete-time system can be checked by computing a unique sequence of involutive distributions. The well-known test for static feedback linearizability is included as a special case. Since the computation of the sequence of distributions requires only the solution of algebraic equations, it allows an efficient implementation in a computer algebra program. In case of a positive result, a flat output can be obtained by straightening out the involutive distributions with the Frobenius theorem.

A symbolic framework to obtain mid-fidelity models of flexible multibody systems with application to horizontal-axis wind turbines

Abstract. The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.

Mathematical modeling of typical electrical circuits, oscillators and pendulums

The work is devoted to mathematical and simulation modeling of electromechanical systems, in which oscillations in electrical circuits, oscillators and pendulums are studied under the action of forces of various physical nature. To compile the equations of electromechanical systems, the apparatus of analytical mechanics was used, in which the electromagnetic and mechanical quantities characterizing the system appear as formally equal. The equations of motion are based on the Lagrange-Maxwell equations. The simulation was performed using computer algebra programs.

LOCAL PARAMETRIZATION OF AN ALGEBRAIC VARIETY NEAR SOME ITS SINGULARITIES

In theoretical physics, when studying invariant Einstein metrics, there was a need to study an algebraic variety, which is described by an equation of the twelfth degree in three variables. 4 singular points and 2 curves of singular points of the variety were found early. Using algorithms of power geometry and computer algebra programs, local parametrizations of a manifold near 2 its singular points of the third order and near one curve of singular points are obtained.

Logarithm of a Non-Singular Complex Matrix via the Dunford–Taylor Integral

Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.

WDVV equations and invariant bi-Hamiltonian formalism

The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.

Limit relations involving 2-orthogonal polynomials

ABSTRACT As an extension of the Askey-scheme, recently, it has been introduced the so-called d-Askey-scheme, which is a table describing families of hypergeometric d-orthogonal polynomials. In this paper, we derive limit relations linking between all levels of the 2-Askey-scheme (Figure 1). To this end, we use three methods. The first one is based on a recurrence relation. The second one uses difference and differential equations, and the third one deals with the hypergeometric expression of the considered polynomials. The recurrence relations, differential and difference equations will be determined by the use of Computer Algebra program Maple. That allows us to get, explicitly, the 2-Askey-scheme with various properties of their polynomials, which appears to be new.

Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Investigation into Class E/F3 with Parallel Network

The system of equations for processes in the amplifier output network is analytically formulated. This system of equations considers parameters of resonant networks at higher harmonics. To calculate amplifier output network, the system of five equations was built for five unknowns, to which the condition of positive second voltage derivative at extremum of drain voltage was added. Two equations correspond to class E conditions, another two — quadrature waveforms at load and at additional resonant network. The last equation is the condition of extremum at the point near middle of drain voltage pulse. This system was solved using computer algebra program. The circuit elements and waveforms were calculated using the derived parameters. By choosing different parameters, it is possible to obtain various amplifier realizations, which will demonstrate features of different class F variants. The obtained amplifier parameters drain voltage and current waveforms were verified with calculated ones using the harmonic balance simulating software. The variant, which is closer to class E/F3 mode, was chosen to build an experimental amplifier prototype on frequency 2MHz using IRF530 MOSFET as a switch. The prototype was tested in the range of supply dc voltage up to 24V with the output power greater than 6W, while the amplifier efficiency was >80%. In the experiment, the ratio of peak drain voltage to dc supply voltage was measured to be 3.3 at the duty ratio 50%, unlike class E amplifier, where this value is around 3.65, and on practice, considering non-linear drain to source capacitance, it may achieve 4. The experimental second harmonic level amounted to be -20 dB relatively to fundamental, and the third one — 28.5 dB, which is due to an additional second harmonic filter. The paper results are useful for introduction of such circuits to practice.

Integration of the Kenzo System within SageMath for New Algebraic Topology Computations

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before.

Коэффициент Пуассона TiNi

A review of the literature data and methods for calculating the Poisson coefficient of the TiNi intermetallic compound in the poly- and single-crystal state is performed. The results of our own research are also presented. Significant variability of the presented data is noted, due to differences in the thermomechanical processing of the alloys and the measurement and calculation methods used. By averaging the matrices of elastic constants and compliance coefficients using the Voigt, Reuss, and Hill approximations, we obtained the values of the parameters of the effective elastic properties of TiNi polycrystals and calculated the Poisson’s ratio. Using analytical expressions to calculate the values of the extreme values, the extrema of the Poisson’s ratio of cubic TiNi crystals are determined for standard orientations. Based on a number of data, TiNi crystals are auxetics (materials having negative Poisson’s ratio values), on the basis of others they are not. We found that TiNi crystals belong to the so-called partial auxetics, in this case the signs of the inequalities (s12 < 0, s = s11 + s12 − s44/2 > 0 or s12 > 0, s = s11 + s12 − s44/2 < 0) are opposite. The values of the Poisson’s ratio TiNi averaged over the transverse directions of deformation are analyzed. Isosurfaces of the Poisson’s ratio and their sections are presented using the ELATE computational graphic package and the MATHCAD computer algebra program. Aspects of TiNi elastic anisotropy, its parameters, and their relationship with martensitic transformations in TiNi and alloys based on it are discussed.

Distinguishing crystallographic groups by their finite quotients

Using the computer algebra program GAP, we show that all crystallographic groups in dimensions at most 4 are distinguished from each other by their sets of finite quotients.

A Kenzo interface for algebraic topology computations in SageMath

In this work we present an interface between SageMath and Kenzo, together with an optional Kenzo package. Our work makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, dealing in particular with simplicial objects of infinite nature.

Analytical description of a luminescent solar concentrator

An analytical solution for the optical efficiency of a luminescent solar concentrator is presented. Due to a large number of input parameters and their complex effect on the device efficiency numerical simulations have been previously used for this purpose. The formulas, provided here, derived using a probabilistic approach, significantly reduce the complexity of the problem. The equations were validated by simulations and the obtained explicit expressions provide a clear common ground for the theoretical description of such devices. Implementation by a computer algebra program yields instant results for any set of input parameters. It allows a higher level of analysis, where an inverse task of finding parameters for a given efficiency can be readily solved.

Algebraic properties of the monopole formula

The monopole formula provides the Hilbert series of the Coulomb branch for a 3-dimensional mathcal{N}=4 gauge theory. Employing the concept of a fan defined by the matter content, and summing over the corresponding collection of monoids, allows the following: firstly, we provide explicit expressions for the Hilbert series for any gauge group. Secondly, we prove that the order of the pole at t = 1 and t → ∞ equals the complex or quaternionic dimension of the moduli space, respectively. Thirdly, we determine all bare and dressed BPS monopole operators that are sufficient to generate the entire chiral ring. As an application, we demonstrate the implementation of our approach to computer algebra programs and the applicability to higher rank gauge theories.

Higher order bell polynomials and the relevant integer sequences

The recurrence relation for the coefficients of higher order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a multiple composite function, is proved. Therefore, starting from this recurrence relation and by using the computer algebra program Mathematica?, some tables for complete higher order Bell polynomials and the relevant numbers are derived.

Product-quotient surfaces: new invariants and algorithms

In this article we suggest a new approach to the systematic, computer-aided construction and to the classification of product-quotient surfaces, introducing a new invariant, the integer \gamma , which depends only on the singularities of the quotient model X = (C_1 \times C_2)/G . It turns out that \gamma is related to the codimension of the subspace of H^{1,1} generated by algebraic curves coming from the construction (i.e., the classes of the two fibers and the Hirzebruch-Jung strings arising from the minimal resolution of singularities of X ). Profiting from this new insight we developed and implemented an algorithm in the computer algebra program MAGMA which constructs all regular product-quotient surfaces with given values of \gamma and geometric genus. Being far better than the previous algorithms, we are able to construct a substantial number of new regular product-quotient surfaces of geometric genus zero. We prove that only two of these are of general type, raising the number of known families of product-quotient surfaces of general type with genus zero to 75. This gives evidence to the conjecture that there is an effective bound \Gamma (p_g,q) \geq \gamma (cf. Conjecture 4.5). Finally we introduce a duality among product-quotient surfaces and prove that the dual surface of a surface of geometric genus zero has maximal Picard number, thus providing several new examples of surfaces with maximal Picard number.