Algebraic Solution Research Articles

Finite orbits of the braid group actions

We study the finite orbits of the braid group Bn action on the space of n×n upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on 3×3 matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic n×n matrices to be useful to finding solutions of higher order Painleve type differential equations.

Theoretical analysis of water-droplet condensation and evaporation in prescribed environment with radiation

Abstract A quasi-explicit algebraic solution of the problem of quasi-steady water droplet heat- and mass-transfer with condensation or evaporation in a prescribed environment including radiation has been obtained. The solution is based on the mass-fraction form of Fick’s law of species diffusive mass transport, which allows accounting for variation in gas density through the diffusion layer surrounding the droplet. A new term, the average vapor supersaturation through the diffusion layer, has been included that allows the model to be extended to environmental conditions significantly outside those for which it is theoretically derived (dilute or near-saturation), even to conditions of relatively high temperature, low relative humidity, and low pressure. The latter two have atmospheric significance because evaporative cooling can induce significant decrease in droplet temperature and possibly induce freezing. Basic modeling assumptions are confirmed by comparison with time-dependent temperature and evaporation-rate measurements of a laboratory laser-heated droplet. Accuracy of the new model is assessed by comparison with exact equations. For temperatures between −30 and 0°C and negligible radiation, relative-error magnitudes in droplet temperature and evaporation rate are less than 5%, even for relative humidity down to 10% and pressure down to 60 kPa. This represents an improvement in accuracy over the conventional model, in which property variations of density and supersaturation through the diffusion layer are not taken into account. Differences in predicted evaporation rates between the mass-fraction and the more common partial-density Fick’s laws, which can be significant (>10% above 70% relative humidity) in coupled, implicit equations, are shown to be made negligibly small (<0.5%) by linearizing about the saturation state.

Teaching data‐driven decision making for inventory analysis with Monte Carlo simulation

AbstractInventory management and the ability to make data‐driven decisions under uncertainty are two critical components of supply chain management. This brief describes how an Excel‐based Monte Carlo simulation of fuel and lottery purchases can be used to teach students to analyze a system with randomness. This exercise can serve as a group or individual assignment to demonstrate the ability of Monte Carlo simulation to estimate solutions involving uncertainty and to teach undergraduate business students how to implement a basic multiperiod fixed‐interval inventory policy with order‐up‐to‐levels. The important concepts of cycle service levels and confidence intervals are emphasized as are the proper implementation of one‐tailed and two‐tailed critical values. Students are encouraged to compare algebraic solutions to their simulation results.

Algebraic solutions of linear differential equations: An arithmetic approach

Given a linear differential equation with coefficients in Q ( x ) \mathbb {Q}(x) , an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck–Katz p p -curvature conjecture.

The impact of various teaching methods on the knowledge of students of different genders: The case of mathematics word problems abstract

Despite significant progress in recent years, there is still evidence of gender differences and underrepresentation of women in leadership positions in mathematics. To promote equal opportunities for all students, it is essential to comprehend gender differences in math learning at school and identify potential barriers to educational equality. In this regard, our study examined the gender differences in solving word problems, and whether they relate to the solution strategy the students were taught.The study involved 100 10th-grade students divided into two groups and taught different solving approaches. One group was taught an algebraic approach using linear equations, while the other group was taught a combined approach, which incorporates the algebraic solution with a functional-graphic solution using a graph of a linear function.The research findings reveal gender differences in various aspects: Girls who learned the algebraic approach had better attainments than boys; Boys who learned the combined approach opted to solve problems with functions and graphs, whereas girls in the same group preferred an algebraic solution; Boys who learned the combined approach achieved significantly higher than boys who learned the algebraic approach.This study illuminates the importance of teaching word problems using diverse solution strategies because different populations of students respond differently to different strategies.

How is the elementary students' creative thinking process in solving fraction problems using geometric and algebraic solutions?

This study aimed to explore students’ creative thinking process in fifth- grade elementary school in fraction problem solving based on gender differences. Subjects consisted of one male and one female with high mathematical ability. Data were collected by task-based interviews and analyzed through the sequence: data classification, data reduction, data presentation, interpretation, and conclusions. The creative thinking process in this research is a series of stages that students go through in creative thinking, including: exploring facts, identifying problems, generating ideas and completing ideas to produce solutions. By these stages, the subjects produced solutions in geometric and algebraic that showed their creative thinking that fulfilled fluency, flexibility and novelty. The subjects used algorithm solutions by repeating the completion steps. In this research, both subjects related their algebraic solutions with geometric solutions that has been produced. It was found that geometric solutions supported the subjects to deliver algebraic solutions and they related their understanding of fractions concepts to solve the problem. The more students’ knowledge or experiences associated with the problem, the more ideas for solutions they found. In this research, it was also found that there were some similarities and differences between the creative thinking processes of both subjects.

Algebraic solution and thermodynamic properties for the one- and two-dimensional Dirac oscillator with minimal length uncertainty relations

We study the quantum characteristics of the Dirac oscillator within the framework of Heisenberg's generalized uncertainty principle. This principle leads to the appearance of a minimal length of the order of the Planck length. Hidden symmetries are identified to solve the model algebraically. The presence of the minimal length leads to a quadratic dependence of the energy spectrum on the quantum number n, implying the hard confinement property of the system. Thermodynamic properties are calculated using the canonical partition function. The latter is well determined by the method based on Epstein's zeta function. The results reveal that the minimal length has a significant effect on the thermodynamic properties.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).

An analytical formula evolved from thermal-structural coupling model for life prediction and structural design of reusable thrust chambers

The thermal-structural analytic model is essential for the life prediction of reusable rocket thrust chambers. The common numerical model based on the finite element method is computationally intensive. In order to simplify the life prediction method and improve the calculation efficiency, a thermal-structural analytical formula is proposed. The one-dimensional analytical heat transfer algorithm for accurately calculating the temperature of the thrust chamber is established, and Porowski’s beam model for calculating the structural deformation is adopted. Finally, the analytic formula for life prediction is directly established by reasonable simplification. This is an algebraic solution describing the relationship between life and the cooling channel parameters, and the computational efficiency is improved by about 12,000 times. The calculated temperature and deformation of the inner wall are verified by the finite element method with an error of less than 10 %, indicating the proposed analytical formula is accurate. Further, the analytical formula is applied to design the cooling channel parameters according to the given target life. The structural design method is validated by numerical simulation and experimental data. The analytical formula and the structural design method provide new ideas for the analysis and design of reusable rocket engines, which have broad application prospects.

BPS Spectra and Algebraic Solutions of Discrete Integrable Systems

This paper extends the correspondence between discrete Cluster Integrable Systems and BPS spectra of five-dimensional N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}=1$$\\end{document} QFTs on R4×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^4\ imes S^1$$\\end{document} by proving that algebraic solutions of the integrable systems are exact solutions for the system of TBA equations arising from the BPS spectral problem. This statement is exemplified in the case of M-theory compactifications on local del Pezzo Calabi–Yau threefolds, corresponding to q-Painlevé equations and SU(2) gauge theories with matter. A degeneration scheme is introduced, allowing to obtain closed-form expression for the BPS spectrum also in systems without algebraic solutions. By studying the example of local del Pezzo 3, it is shown that when the region in moduli space associated to an algebraic solution is a “wall of marginal stability”, the BPS spectrum contains states of arbitrarily high spin, and corresponds to a 5d uplift of a four-dimensional nonlagrangian theory.

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

Topology and Dynamics of Transcriptome (Dys)Regulation.

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a "healthy" sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt-1, it must be aperiodic, and fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a "wild" dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases.

Dual rectangular fuzzy complex matrix equations: extended solution, algebraic solution, solution and their calculating1

In this paper, we focus on generalized fuzzy complex numbers and propose a straightforward matrix method to solve the dual rectangular fuzzy complex matrix equations C · Z ˜ + L ˜ = R · Z ˜ + W ˜ , in which C and R are crisp complex matrices and Z ˜ , L ˜ and M ˜ are fuzzy complex number matrices. The existing methods for solving fuzzy complex matrix equations involve separately calculating the extended solution and the corresponding parameters of the real and imaginary parts, whereby we obtain the algebraic solution of the equations. By means of the interval arithmetic and embedding approach, the n × n dual rectangular fuzzy complex linear systems could be converted into 2n × 2n fuzzy linear systems, which are also equivalent to the 4n × 4n real linear systems. By directly solving the 4n × 4n real linear systems, the algebraic solutions can be obtained. The general dual rectangular fuzzy complex matrix equations and dual rectangular fuzzy complex linear systems are investigated by the generalized inverses of matrices. Finally, some examples are given to illustrate the effectiveness of method.

Control co-design optimization of nonlinear wave energy converters

This paper presents a study in which both the control and the shape of a Wave Energy Converter (WEC) are optimized simultaneously. A heaving point absorber WEC is assumed. To optimize the shape of the WEC’s buoy, nonlinear hydrodynamics need to be evaluated. One main contribution of this paper is the integration of nonlinear hydrodynamics and nonlinear control during the optimization of the WEC’s buoy shape. This approach is referred to as Control Co-Design (CCD). In this work, we present a control co-designed nonlinear heaving point absorber WEC that leverages the nonlinear dynamic, static Froude–Krylov (FK) forces to maximize power extraction. The nonlinear FK forces are approximated using a variation of the algebraic solution; the hydrodynamic forces of the body are computed using an analytic formulation leveraging the methods of eigenfunction expansion and separation of variables. The nonlinear geometry of the buoy is modeled as a series of inclined panels; the inclination angles are optimized to arrive at the optimal shape. The performance of the optimized shape is compared to that of a nonlinear spherical WEC. It is found that an average of 20% improvement is achieved by the optimized geometry over the spherical device.

A Novel Numerical Method for Solving Unknown Statistical Quantities in Multivariate Regression Models

Simulation studies are the basic tools of quantitative methodologists used to obtain empirical solutions to statistical problems that may be impossible to derive through direct mathematical computations. The successful execution of many simulation studies relies on the accurate generation of correlated multivariate data that adhere to a particular model with known parameter values. In this article, we use a kernel inspired by path tracing rules to algebraically solve unknown causal effects in the context of a multivariate general linear model. The algebraic solution is the basis of the mathematical extension, which integrates a model solver. Examples are used to illustrate a range of applications, where information regarding parameter values and predictor correlations can be partially specified. Code for examples is provided.

SECOND ORDER DIFFERENTIAL OPERATORS WITH ALGEBRAIC SOLUTIONS

We are surveying recent results that describe second order differential operators having only algebraic solutions in the sense of Galois theory. We call such operators algebraic. For hypergeometric operators, this problem was studied by Schwarz and Klein who also gave results that describe all second order linear differential operators with a full set of algebraic solutions. Starting from their work, we see algebraic operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We are surveying some of the main results describing second order operators with a full set of algebraic solutions, especially those obtained by using the properties of the pull-back functions. Using the Grothendieck correspondence, these properties transfer to properties for their corresponding dessins d’enfants.

An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

A Study for Solving Pseudo-Parabolic Viscous Diffusion, Telegraph, Poisson and Helmholtz PDE using Legendre-Collocation Method

Legendre collocation approach is implemented employing the operative-differential array escorted by the shifted Legendre polynomials. The significance of the proposed approach is reducing the intended partial differential equation problem into an algebraic system of equations by the aid of Legendre polynomials as orthogonal basis. Using mesh collocation points, the domain of the resulting algebraic system solution domain is discretized, and it is then solved to obtain an approximation solution. To demonstrate the accuracy of the procedure, the maximum absolute error and the estimate error are calculated. Also, Illustrative numerical examples are demonstrated to display the veracity of the potential approach in particular pseudo-parabolic viscous diffusion, telegraph, Poisson and Helmholtz partial differential equations. The generated results are contrasted with the exact solutions/other methods to highlight the efficiency of the proposed approach.