Algebraic Manipulation Research Articles

Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{q}$$\\end{document}-rational Functions and Interpolation with Complete Nevanlinna–Pick Kernels

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproducing kernel Hilbert spaces and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna–Pick kernels.

Constraining parameters for the accelerating universe in [formula omitted] gravity

In the paper, we present an accelerating cosmological model in f(R,Lm) gravity with the parameter constrained through the cosmological data sets. At the beginning, we have employed a functional form of f(R,Lm)=R2+αR2+Lmβ, where α and β are model parameters. This model is well motivated from the Starobinsky model in f(R) gravity and the power law form of f(Lm). The Hubble parameter has been derived with some algebraic manipulation and constrained by Hubble data and Pantheon+ data. With the constraint parameters, present value of deceleration parameter has been obtained to as q0≈−0.63 with the transition at zt≈0.7. It shows the early deceleration and late time acceleration behaviour. The present value of other geometric parameters such as the jerk and snap parameter are obtained to be j0≈0.78 and s0≈0.1 respectively. The state finder diagnostic test gives the quintessence behaviour at present and converging to ΛCDM at late times. Moreover the Om(z) diagnostics gives negative slope which shows that the model favours the state finder diagnostic result. Also the current age of Universe has been obtained as, t0=13.64Gyrs. The equation of state parameter also shows the quintessence behaviour. Based on the present analysis, it indicates that the f(R,Lm) gravitational theory may be another alternative to study the dark energy models.

On computation of solution for [formula omitted] dimensional fractional order general wave equation

In this research article, we handle a class of (2+1) dimensional wave equations under fractional-order derivatives using an iterative integral transform due to Laplace. The concerned derivative is taken in the Caputo’s sense. The proposed method is a purely algebraic manipulation approach to compute solutions without a priori knowledge of geometry and physical meaning related to the proposed problem. In fact, in this procedure, we combine two novel techniques, Laplace transforms (LT) and iterative procedures to form a hybrid technique for computation of the solution to the proposed problem. The method is rapidly convergent. Here, we give various examples for the validation of our proposed method. The superiority of the method over the existing numerical method is that it does not require any prior discretization or collocation of functions. Also, the method is independent of axillary parameters as needed in the homotopy methods, because such auxiliary parameters control the efficiency of the mentioned methods. The proposed procedure is simple and straightforward. In addition, some comparison between exact and approximate solutions is also given. The proposed method is compared with the homotopy perturbation method (HPM) which shows that the proposed technique is more efficient and easy to implement.

Center Conditions for Nilpotent Singularities in the Plane Using Invariant Solutions

Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particular curve allows to use it to determine necessary conditions to have a center for nilpotent singularities in the plane. Several involve methods to solve the center problem are known all based in the existence of a change of variables and a scaling transformation of time bringing any differential system with a nilpotent center into a time-reversible system. Here we present a new algebraic method based on the existence of such particular solution curve not passing through the singular point and the involution associated to the nilpotent system with a center. The algebraic method needs the computation of this particular curve up to certain order, which can be done with the help of an algebraic manipulator. Finally a new algebraic method is derived computing the vanishing of a unique function which really gives a scalar method for computing the necessary conditions.

FlexibleSUSY extended to automatically compute physical quantities in any beyond the standard model theory: Charged lepton flavor violation processes, Higgs decays, and user-defined observables

FlexibleSUSY is a framework for the automated computation of physical quantities (observables) in models beyond the Standard Model (BSM). This paper describes an extension of FlexibleSUSY which allows to define and add new observables that can be enabled and computed in applicable user-defined BSM models. The extension has already been used to include Charged Lepton Flavor Violation (CLFV) observables, but further observables can now be added straightforwardly. The paper is split into two parts. The first part is non-technical and describes from the user's perspective how to enable the calculation of predefined observables, in particular CLFV observables. The second part of the paper explains how to define new observables such that their automatic computation in any applicable BSM model becomes possible. A key ingredient is the new NPointFunctions extension which allows to use tree-level and loop calculations in the model-independent setup of observables. Three examples of increasing complexity are fully worked out. This illustrates the features and provides code snippets that may be used as a starting point for implementation of further observables. Program summaryProgram title:NPointFunctionsCPC Library link to program files:https://doi.org/10.17632/kf7m8gn8vp.2Developer's repository link:https://github.com/FlexibleSUSY/FlexibleSUSYLicensing provisions: GPLv3Programming language:C++, Wolfram Language, Fortran, Bourne shellJournal reference of previous version:: Comput. Phys. Commun. 230 (2018) 145–217; PoS CompTools2021 (2022) 036Does the new version supersede the previous version?: YesReasons for the new version: Program extension including new observables and file structuresNature of problem: Determining observables for an arbitrary extension of the Standard Model supported by FlexibleSUSY, input by the user.Solution method: Generation of the code from automated algebraic manipulations. Automatic filling and compiling of predefined template files.Additional comments including restrictions and unusual features: Vertices with a direct product of Lorentz and color structures are supported. Settings of the advanced NPointFunctions mode rely on explicit specification of topologies.

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.

Frugal LMs Trained to Invoke Symbolic Solvers Achieve Parameter-Efficient Arithmetic Reasoning

Large Language Models (LLM) exhibit zero-shot mathematical reasoning capacity as a behavior emergent with scale, commonly manifesting as chain-of-thoughts (CoT) reasoning. However, multiple empirical findings suggest that this prowess is exclusive to LLMs that have exorbitant sizes (beyond 50 billion parameters). Meanwhile, educational neuroscientists suggest that symbolic algebraic manipulation be introduced around the same time as arithmetic word problems so as to modularize language-to-formulation, symbolic manipulation of the formulation, and endgame arithmetic. In this paper, we start with the hypothesis that much smaller LMs, which are weak at multi-step reasoning, can achieve reasonable arithmetic reasoning if arithmetic word problems are posed as a formalize-then-solve task. In our architecture, which we call SyReLM, the LM serves the role of a translator to map natural language arithmetic questions into a formal language (FL) description. A symbolic solver then evaluates the FL expression to obtain the answer. A small frozen LM, equipped with an efficient low-rank adapter, is capable of generating FL expressions that incorporate natural language descriptions of the arithmetic problem (e.g., variable names and their purposes, formal expressions combining variables, etc.). We adopt policy-gradient reinforcement learning to train the adapted LM, informed by the non-differentiable symbolic solver. This marks a sharp departure from the recent development in tool-augmented LLMs, in which the external tools (e.g., calculator, Web search, etc.) are essentially detached from the learning phase of the LM. SyReLM shows massive improvements (e.g., +30.65 absolute point improvement in accuracy on the SVAMP dataset using GPT-J 6B model) over base LMs, while keeping our testbed easy to diagnose and interpret, and within the reach of most researchers.

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Examining the Contributions of Various Source Layers and Transport Pathways to Dose Rates Exterior to a Low- and Intermediate-Level Radioactive Waste Storage Facility Using ADVANTG/MCNP

An in-depth shielding analysis of a large indoor facility designed for storing low- and intermediate-level radioactive waste generated from nuclear power plant decommissioning was performed. The facility has a storage capacity of 1260 box-type containers, which are stacked in five layers. The characteristics of the radiation field around the facility were investigated comprehensively through efficient ADVANTG/MCNP simulations. The contribution of each source layer to the dose rates exterior to the facility was determined. Furthermore, dose components from various transport pathways (e.g., direct, streaming, and skyshine) were isolated using algebraic manipulation of the outputs obtained from additional simulations. The combined results provided valuable information for shielding optimization during the design phase and could facilitate the future operation of the storage facility.

Mathematical Model of Impact Projectile Flight Dynamics as an Element of its Digital Twin

This paper gives the analysis of the structure and characteristics of the mathematical model of the impact projectile’s flight dynamics. The model is designed for being used as an element of the projectile’s digital twin. The model is based on differential motion equations of a gyro-stabilized solid with an axisymmetric mass distribution. Different types of angle variables were chosen for describing aerodynamics and formulating equations of motion. Non-linear (considering the nutation angle) dependences for aerodynamic coefficients are proposed. They are created by applying proven scientific concepts and research methods in aerodynamics of axisymmetric body and by comparing with known numerical and experimental results obtained in exterior ballistics of gyro-stabilized aviation and artillery projectiles. Special aspects of initial conditions for angles and angular velocity were also studied. Since the impact projectile is considered as an axially symmetric body, its self-rotation angle is not of practical inte rest. Using algebraic manipulations, the differential equation for this angle was eliminated from the set of equations. This has made it possible to significantly reduce stiff of the remaining system of differential equations. The Dormand-Prince method is recommended as a method of numerical integration. The method of the eight-order (with seventh-order uncertainty estimate) allows getting the high accurate solution of the differential equations set under relatively small computing costs. The model allows computing the projectile trajectory under various initial conditions, including the flight with high nutation angles up to 87°—89°. As a result, there is a possibility to determine the nature of the interaction between impact projectiles and typical targets (ricochet, surface effect, after-penetration effect) within a wide range of approach angles to the target’s surface (skin) unattainable during the full-scale tests. The possibility of solving similar problems allows to recommend the designed model as an element of the impact projectile’s digital twin intended for testing its exterior ballistics on the digital (virtual) test range. All testing calculations and final modeling were made by using the "GNU Octave" computational software package.

Investigation of more solitary waves solutions of the stochastics Benjamin-Bona-Mahony equation under beta operator

<p>This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}} $-expansion and the modified $ \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} $-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.</p>

Slope conception as rate of change starter kit: Malaysian pre-service secondary mathematics teachers’ subject matter knowledge in rate of change

Past research on calculus has shown that students often struggle to understand derivatives due to an overemphasis on algebraic manipulation and procedures rather than grasping the underlying concept. A key factor contributing to this challenge is the lack of a solid understanding of slope as a rate of change. Conceptualizing slope as a rate of change is essential, as it serves as the basis for comprehending derivative concepts. To address this gap, our research explored subject matter knowledge of rate of change, specifically focusing on the slope concept, among Malaysian pre-service mathematics teachers. Our research followed a qualitative methodology, conducting task-based interviews with two pre-service mathematics teachers, Zheng and Amitha, who are majoring in mathematics. The interviews included seven tasks related to slope and ratio concepts. The findings revealed inconsistencies in their notion of rate of change, as they have a loose connection between rate of change and slope concepts, along with its multiplicative property. While they recognized that equivalent values of rate of change indicate a constant rate, they did not grasp that slope also represents a rate of change. Their knowledge appeared limited to viewing derivatives as a method to calculate rate of change, without conceiving the changes occurring between quantities and their multiplicative relationship.

Semantic embedding for quantum algorithms

The study of classical algorithms is supported by an immense understructure, founded in logic, type, and category theory, that allows an algorithmist to reason about the sequential manipulation of data irrespective of a computation’s realizing dynamics. As quantum computing matures, a similar need has developed for an assurance of the correctness of high-level quantum algorithmic reasoning. Parallel to this need, many quantum algorithms have been unified and improved using quantum signal processing (QSP) and quantum singular value transformation (QSVT), which characterize the ability, by alternating circuit ansätze, to transform the singular values of sub-blocks of unitary matrices by polynomial functions. However, while the algebraic manipulation of polynomials is simple (e.g., compositions and products), the QSP/QSVT circuits realizing analogous manipulations of their embedded polynomials are non-obvious. This work constructs and characterizes the runtime and expressivity of QSP/QSVT protocols where circuit manipulation maps naturally to the algebraic manipulation of functional transforms (termed semantic embedding). In this way, QSP/QSVT can be treated and combined modularly, purely in terms of the functional transforms they embed, with key guarantees on the computability and modularity of the realizing circuits. We also identify existing quantum algorithms whose use of semantic embedding is implicit, spanning from distributed search to proofs of soundness in quantum cryptography. The methods used, based in category theory, establish a theory of semantically embeddable quantum algorithms, and provide a new role for QSP/QSVT in reducing sophisticated algorithmic problems to simpler algebraic ones.

The impact of teachers' knowledge on the connection between technology supported exploration and mathematical proof

Technology is recognized for its potential to implement exploration tasks. The ease and speed with which it becomes possible to observe many cases of a situation, allows the development of conjectures and brings conviction about their veracity. Mathematical proof, assumed as the essence of Mathematics, tends to appear to the students as something dispensable. Based on KTMT – <i>Knowledge for Teaching Mathematics with Technology </i>model, this study intends to understand the impact of the teachers’ knowledge on mathematical proof in a context of technology integration. The study adopts a qualitative and interpretative methodology, based on case study, analyzing the practice of one teacher. The conclusions emphasize the relevance of the teacher’s MTK – <i>Mathematics and Technology Knowledge</i>, and TLTK – <i>Teaching and Learning and Technology Knowledge</i>. The teacher's MTK guides her decisions, leading her to focus on helping students understand the meaning of conjecture and proof, valuing, at the same time, the relevance of algebraic manipulations. However, the teacher’s TLTK guides her practice, where the knowledge about the students is determinant. The study provides evidence about the difficulty of articulating proof and technology, but it also clarifies the relevance of this articulation and of how the teacher’s KTMT can impact the teacher’s decisions.

Absolute nodal coordinate formulation – Multilevel finite element framework for the nonlinear multi-scale multibody dynamic analysis of composite structures

In this paper, a multi-scale modeling approach for the dynamic response of flexible multibody system made of composite material is suggested using the absolute nodal coordinate formulation (ANCF) – multilevel finite element (FE2) method. On many occasions, the analysis of composite structures requires a multi-scale modeling approach. FE2 method is one of the famous multi-scale modeling methods. And it is a versatile computational homogenization approach that is commonly applicable to many different materials. Because the present FE2 is capable of computing the mechanical response at two different scales simultaneously, ANCF element will be used at the macroscopic scale. And the microscopic RVE at each macroscopic integration point will be obtained by the finite element method. Since ANCF element uses only the absolute nodes and gradients, the mass matrix will be constant. Due to such characteristics, when combined with FE2, the linearization process will become easier compared against the existing formulation. And Coriolis or centrifugal force will not need to be considered in the dynamic formulation. Therefore, computational cost will be reduced, and the relevant algebraic manipulation will become simplified. Several numerical examples are presented to demonstrate the improved accuracy and diminished computational cost of the present suggestion.

Determinants of Estimate Difference between Geometric Measure and Standard Deviation

Measures of variation are statistical measures which assist in describing the distribution of data set. These measures are either used separately or together to give a wide variety of ways of measuring variability of data. Researchers and mathematicians found out that these measures were not perfect, they violated the algebraic laws and they possessed some weakness that they could not ignore. As a result of these facts, a new measure of variation known as geometric measure of variation was formulated. The new measure of variation was able to overcome all the weaknesses of the already existing measures. It obeyed all the algebraic laws, allowed further algebraic manipulation and was not affected by outliers or skewed data sets. Researchers were also able to determine that geometric measure was more efficient than standard deviation and that its estimates were always smaller than those of standard deviation but they did not determine their main relationship and how the sample characteristics affect the minimum difference between geometric measure and standard deviation. The main aim of this study was to empirically determine the ratio factor between standard deviation and geometric measure and specifically how certain variable such as sample size, outliers and geometric measure affects the minimum difference between geometric measure and standard deviation. Data simulation was the concept that was used to achieve the studies objectives. The samples were simulated individually under four different types of distributions which were normal, Poisson, Chi-square and Bernoulli distribution. A Hierarchical linear regression model was fitted on the normal, skewed, binary and countable data sets and results were obtained. Based on the results obtained, there is always a positive significant ratio factor between the geometric measure and standard deviation in all types of data sets. The ratio factor was influenced by the existence of outliers and sample size. The existence of outliers increased the difference between the geometric measure and standard deviation in skewed and countable data sets while in binary it decreased the difference between the standard deviation and geometric measure. For normal and binary data sets, increase in sample size did not have any significant effect on the difference between geometric measure and standard deviation but for skewed and countable data sets the increase in sample size decreased the difference between geometric measure and standard deviation.

Analytical solution for Fredlund–Hasan unsaturated consolidation using mode superposition method

AbstractThis note presents an exact analytical solution of Fredlund–Hasan consolidation for unsaturated soils under an arbitrary loading using the mode superposition method. Air pressure and water pressure are expressed in a series of products of Eigen functions with respect to depth and normal coordinates with respect to time, respectively. A pressure function substitutes for the related homogeneous problem. Eigen functions can be first obtained based on two pairs of drainage condition. Then, the nonhomogeneous governing equations of unsaturated soils under constant loading, exponential loading, and sine loading are considered. After matrix algebraic manipulation, the normal coordinates can be obtained based on the initial conditions that compose the final solutions of the Fredlund–Hasan unsaturated consolidation. This solution can avoid inconvenient Laplace transform and inverse Laplace transform. Through comparisons with a classic example, the validation of the proposed analytical solution is verified against other analytical solutions and the finite difference method solutions.

Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

This study analyses an approach to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by an improper discretisation of the required source terms. The current work focuses on advection-diffusion models, in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used in heat and mass transfer problems or multiphase flows. The investigated scheme is applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. Further application of this work can be found in the method of manufactured solutions or in the immersed boundary method. This paper is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. Firstly, it identifies the differences observed between the ways in which source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the observed differences, and to identify their origin. Secondly, it analyses in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is valid for any source term discretization scheme. It is a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model and the Allen-Cahn equation. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scalings are identified and discussed. Furthermore, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

Collision-Avoiding Model Predictive Rendezvous Strategy to Tumbling Launcher Stages

This paper considers the situation where a small satellite shall autonomously rendezvous with a tumbling object in a circular low Earth orbit (LEO) and derives a path-based model predictive controller that uses the docking point state and position of the chaser to guide it to a safe docking autonomously. The strategy embeds collision avoidance elements and reduces the computational effort for calculating the pulses to be provided by the thrusters through opportune algebraic manipulations, a Runge–Kutta 4 propagation method using linearized state transition matrices, and implicit embedding of dynamically equivalent thrust models, leading to constant state propagation matrices. Furthermore, the inputs design optimization problem and the embedded collision avoidance scheme are modeled and explicitly crafted as convex problems, contributing positively to low computational requirements. The docking and collision avoidance capabilities of the proposed scheme are extensively tested in an environment that accounts for all the perturbations relevant to LEO frameworks, for realistic thrust schemes, and for uncertainties in the measurement. Numerical results assess which tumbling objects can be docked or not by means of the proposed schemes as a function of the tumbling rates versus the thrust capabilities and hardware uncertainty of the docker.