Elementary Solutions Research Articles

An analog of the pendulum/rotor system with elementary solutions

Abstract We explore a novel one-dimensional model potential, V(x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly solvable, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 < E < ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).

Fluid flow between two parallel active plates

This paper investigates the fluid flow phenomenon arising from the combined action of two parallel plates, which can expand/squeeze, absorb/inject, and stretch/shrink at different rates. These physical mechanisms are incorporated into the governing unsteady Navier–Stokes equations, which are then reduced to a fourth-order nonlinear differential equation with boundary conditions reflecting the imposed wall constraints. By letting the permeable Reynolds number (controlling the nonlinear convective terms) limit to zero, we demonstrate the existence of exact solutions expressed in terms of advanced mathematical functions. Additionally, in the absence of wall expansion/contraction, elementary exponential solutions are obtained under particular relationships between the stretching/shrinking and permeability parameters. A shear-like exact solution with broader applicability across various physical parameters is also identified. For moderate values of the expansion/squeezing parameters and permeable Reynolds numbers, we propose an efficient double-expansion perturbation analysis to approximate the flow behavior. Otherwise, for general physical parameters, a comprehensive mathematical analysis is provided and numerical simulations are employed to extract insights into the complex fluid motion between the parallel plates.

An analytic method to investigate hemodynamics of the cardiovascular system: Biventricular system.

Previously, we found analytic solutions for single ventricular system based on the lumped parameter model (LPM). In this study, we generalized the method to biventricular system and derived its analytic solutions. LPM is just a set of differential equations, but it is difficult to solve due to time-varying ventricular elastance and high order. Mathematically, there exist no elementary solutions for time-varying equations. It turns out that instead of differential equations, according to volume conservation, a set of algebraic equations can be carried out. The solutions of the set of equations are just physiological states at end of systolic and diastolic phases such as end systolic/diastolic pressure/volume of left ventricle. As a preliminary application, the method is utilized to deduce the hemodynamic effects of VA ECMO. Left ventricular (LV) distension, a serious complication of VA ECMO, is usually attributed to factors such as increased afterload, inadequate LV unloading, reduced myocardial contractility or aortic valve regurgitation (AR), bronchial and Thebesian return in the absence of aortic valve (AoV) opening. Among these, reduced contractility and AR are strongly associated with LV distension. However, in the absence of reduced contractility or AR, it is less clear whether increased afterload or inadequate LV unloading alone can cause LV distension. This leads to the critical question: under what conditions does LV distension occur in the absence of reduced contractility or AR? The analytic formulas derived in this study give conditions for LV distension. Furthermore, the results show that the analytic hemodynamics are coincident with simulated results.

Analytical calculation of Kerr and Kerr-Ads black holes in f(R) theory

In this paper, we extend Chandrasekhar’s method of calculating rotating black holes into f(R) theory. We show that the solution with constant Ricci scalar always exists in a general f(R) gravity and derive the Kerr and Kerr-Ads metric by using the analytical mathematical method. Suppose that the spacetime is a 4-dimensional Riemannian manifold with a general stationary axisymmetric metric, we calculate Cartan’s equation of structure and derive the Einstein tensor. In order to reduce the solving difficulty, we fix the gauge freedom to transform the metric into a more symmetric form. We solve the field equations in the two cases of the Ricci scalar R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document} and R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}. In the case of R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document}, the Ernst’s equations are derived. We give the elementary solution of Ernst’s equations and show the way to obtain more solutions including Kerr metric. In the case of R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}, we reasonably assume that the solution to the equations consists of two parts: the first is Kerr part and the second is introduced by the Ricci scalar. Giving solution to the second part and combining the two parts, we obtain the Kerr-Ads metric. The calculations are carried out in a general f(R) theory. Furthermore, the whole solving process can be treated as a standard calculation procedure to obtain rotating black holes, which can be applied to other modified gravities.

Causes of Deterioration of Elementary Education and Potential Solutions

This research study aimed to find out the causes of the deterioration of elementary education and potential solutions perceived by teachers. The major objective of this study was to examine the causes of deterioration in elementary education and potential solutions perceived by teachers. All male and female elementary teachers in both public and private schools in the district Muzaffargarh were the population. The sample consisted of 274 school teachers (150 male and 124 female), who were selected through a stratified sampling technique. For data collection, a 40-item self-structured questionnaire was designed for teachers. Findings showed that causes of deterioration of Elementary education as perceived by teachers include insufficient resources, inadequate funding, large class sizes, lack of teacher training, outdated curriculum, and lack of parental involvement. The solutions included increased funding for education, reducing class sizes, providing opportunities for professional development for teachers, updating the curriculum to be more relevant and engaging, improving access to resources and technology, parental involvement, and fostering stronger collaboration between teachers, parents, and administration.

Numerical simulation of 3D vorticity dynamics with the Diffused Vortex Hydrodynamics method

In this paper the three-dimensional extension of the Diffused Vortex Hydrodynamics (DVH) is discussed along with free vorticity dynamics simulations. DVH is a vortex particle method developed in-house and widely validated in a 2D framework. The DVH approach has been embedded in a new frontend to the open-source code PEPC, the Pretty Efficient Parallel Coulomb solver. Within this parallel Barnes–Hut tree code, a superposition of elementary heat equation solutions in a cubic support is performed during the diffusion step. This redistribution avoids excessive clustering or rarefaction of the vortex particles, providing robustness and high accuracy of the method. An ascending vortex dipole at various resolutions is selected as a test-case and heuristic convergence measurements are performed, taking into account the conservation of prime integrals and the energy–enstrophy balance.

Exact solutions of the Landau–Ginzburg–Higgs equation utilizing the Jacobi elliptic functions

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations.

Elementary Problems and Solutions

Elementary Problems and Solutions

Modeling ocean eddies using exact solutions of the Charney–Obukhov equation

New exact solutions of the Charney–Obukhov equation for the ocean are obtained in the form of a partial superposition of elementary solutions with different wave numbers. The boundary conditions for the ocean are satisfied due to the presence of a carrier zonal flow in the solution. The existing arbitrariness in the choice of wave numbers and other solution parameters makes it possible to simulate an arbitrary stream function profile at a fixed ocean depth on an interval of a fixed length using a Fourier series or in a circle of a fixed radius using a Fourier–Bessel series. An example of modeling a Gaussian stream function profile on the ocean surface in the presence of circular symmetry is considered.

From Domain Decomposition to Model Reduction for Large Nonlinear Structures

The numerical simulation of multiscale and multiphysics problems requires efficient tools for spatial localization and model reduction. A general strategy combining Domain Decomposition and Nonuniform Transformation Field Analysis (NTFA) is proposed herein for the simulation of nuclear fuel assemblies at the scale of a full nuclear reactor. The model at subdomain level solves the full elastic problem but with a reduced nonlinear loading, based on simplified boundary conditions, reduced creep flow rules, projected sign preserving contact conditions, and a NTFA like reduced friction law to get the evolution of each slipping mode. With this loading reduction, the local solution can be explicitly obtained from a small set of precomputed elementary elastic solutions. The numerical tests indicate that considerable cost reduction (a factor of 50 to 1000) can be achieved while preserving engineering accuracy.

Science teachers’ implementation of science and engineering practices in different instructional settings

ABSTRACT This article explores science teachers’ implementation of science and engineering practices (SEPs) under different instructional settings. We compared the number of SEPs science teachers reported using in face-to-face instruction (traditional), online-only instruction (virtual), or HyFlex instruction (synchronously online and in-person) from August 2020 to May 2021. Records and artefacts of the teachers’ instructional practices were collected over three one-week periods. Interview data were used to validate teachers’ instructional activities, the context of SEP implementation, and their challenges when navigating the different instructional settings. Through a lens of consequential transition perspective, our findings revealed that science teachers implemented significantly more SEPs in a HyFlex or traditional setting than in a virtual setting. The results also showed that regardless of the instructional setting, elementary and secondary teachers generally implemented few investigating SEPs. Among elementary teachers, developing explanations and solutions were the most frequently used SEPs across all instructional settings. Among secondary teachers, the developing explanations and solutions SEPs and evaluating SEPs were prevalent but varied across the different instructional settings. Our findings suggest that science teachers need to continue to build their knowledge and practice of the SEPs, and have different supports to facilitate their SEP implementation in different instructional environments.

Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Abstract There are several problems in physics, such as kinetic energy equation, wave equation, anomalous diffusion process, and viscoelasticity that are described well in the fractional differential equation form. Therefore, the solutions with elementary solution method cannot be solved and described deliberately with detailed physics of the problems, so these problems are solved with the help of special operators such as Mittag–Leffler (M–L) functions equipped with Riemann–Liouville (R–L) fractional operators. Hence, keeping in view the above-mentioned problems in physics in the current study, the generalized properties are derived M–L functions connected with R–L fractional operators that are investigated in the generalized form. These extended special operators will be used for the solutions of generalized kinetic energy equation. The M–L function is a fundamental special function with a wide range of applications in mathematics, physics, engineering, and various scientific disciplines. Ayub et al. gave the definition of newly extended M–L ( p , s , k ) \left(p,s,k) function. Also, they gave its convergence condition and found several results relevant to that. The purpose of this study is to investigate newly extended M–L function and study its elementary properties and integral transforms such as Whittaker transform and fractional Fourier transform. The R–L fractional operator is a fundamental concept in fractional calculus, a branch of mathematics that generalizes differentiation and integration to non-integer orders. In this study, we discuss the relation of M–L ( p , s , k ) \left(p,s,k) -function and R–L fractional operators. In some cases, fractional calculus is used to describe kinetic energy equations, particularly in systems where fractional derivatives are more appropriate than classical integer-order derivatives. The M–L function can appear as a solution or as a part of the solution to these fractional kinetic energy equations. Also, we gave the generalization of kinetic energy equation and its solution in terms of newly extended M–L function.

Does a language model “understand” high school math? A survey of deep learning based word problem solvers

AbstractFrom the latter half of the last decade, there has been a growing interest in developing algorithms for automatically solving mathematical word problems (MWP). It is a challenging and unique task that demands blending surface level text pattern recognition with mathematical reasoning. In spite of extensive research, we still have a lot to explore for building robust representations of elementary math word problems and effective solutions for the general task. In this paper, we critically examine the various models that have been developed for solving word problems, their pros and cons and the challenges ahead. In the last 2 years, a lot of deep learning models have recorded competing results on benchmark datasets, making a critical and conceptual analysis of literature highly useful at this juncture. We take a step back and analyze why, in spite of this abundance in scholarly interest, the predominantly used experiment and dataset designs continue to be a stumbling block. From the vantage point of having analyzed the literature closely, we also endeavor to provide a road‐map for future math word problem research.This article is categorized under: Technologies > Machine Learning Technologies > Artificial Intelligence Fundamental Concepts of Data and Knowledge > Knowledge Representation

The Shortest Distance Problem: An Elementary Solution

Summary We offer an elementary solution to the shortest distance problem. In addition to being accessible to freshman Calculus students, our approach also provides a bit of extra information, namely a way quantify the “length excess” in terms of how much the path deviates from being a straight line.

ON THE MINIMALITY OF A PART OF EIGEN- AND ASSOCIATED VECTORS FOR A CLASS OF THIRD-ORDER QUASI-ELLIPTIC OPE

In the paper, we derive sufficient conditions on the coefficients of a thirdorder quasi-elliptic operator pencil, ensuring the minimality of its system of eigen- and associated vectors corresponding to eigenvalues from the left half-plane. Additionally, we prove a theorem on the minimality of decreasing elementary solutions to a homogeneous equation in a Sobolev-type space.

Elementary Solution to the Fair Division Problem

Elementary Solution to the Fair Division Problem

Elementary Problems and Solutions

Elementary Problems and Solutions

Fast Analytic–Numerical Algorithms for Calculating Mutual and Self-Inductances of Air Coils

This paper deals with a method of calculating the mutual and self-inductances of various air coils located arbitrarily in space. Known elementary solutions (the Biot–Savart formulas) were used to determine the magnetic field of infinitely thin current loops and infinitely thin wires of finite length magnetically linking other coils. Unlike commonly used algorithms, these elementary solutions were not extensively transformed analytically but were used to perform calculations via direct numerical integration. This enabled the very quick and accurate obtaining of the self-inductance values, as well as determining the dependence of mutual inductances on the positions of both coils. This method allows for the analysis of different coil configurations (misaligned coils, inclined to each other, etc.) that other methods do not cover. It also enables the determination of the forces acting on the coils, as well as the calculation of the magnetic field distribution from any coil configuration. The obtained results were compared with those presented by other authors (both computational and measurement results).

The distance from a point to a line or a plane in coordinate geometry: A revisit with three elementary solutions

The distance from a point to a line or a plane in coordinate geometry: A revisit with three elementary solutions

Representation of the Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

We define the elementary solution of the polyharmonic equation, with the help of which an explicit representation of the Green’s function of the Dirichlet problem for the polyharmonic equation in the unit ball is given for all space dimensions except for some finite set. On the basis of the obtained Green’s function, the solution of the homogeneous Dirichlet problem in the unit ball is constructed. As an example, an explicit form of the solution of the homogeneous Dirichlet problem for the inhomogeneous polyharmonic equation with the simplest polynomial right-hand side is found.