Algebraic Equations Research Articles

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

Modeling the Efficiency of Biogas Plants by Using an Interval Data Analysis Method

This article considers the task of developing mathematical models and their computer implementation that would establish the dependence of pH (acidity of the environment) on the volume and structure of raw materials for daily loading, as well as on the operating parameters of temperature and humidity based on the interval analysis of experimental data obtained during BGP research of a given type. In the process of research, based on the developed interval models, it was established that this indicator depends on the volume and structure of raw materials, as well as on the temperature and humidity of the substrate in the bioreactor. To build this mathematical model, it is proposed to use the method of interval data analysis and the method of identification of model parameters based on multidimensional optimization. The results of experimental studies for a specific type of biogas plant are given, and interval models with guaranteed prognostic properties that characterize the pH of the environment depending on the specific type of bio-raw material of solid and liquid fractions, temperature, and humidity are obtained. Based on the use of different types of raw materials, the developed models, based on experimental data, describe different configurations of the structure and volumes of raw materials for daily loading. The obtained mathematical models are an algebraic nonlinear equation that can be applied to control the level of pH of the environment in the bioreactor by determining the optimal volumes of raw materials of each type during the loading period depending on the temperature and humidity of the substrate in the bioreactor.

Rationality and parametrizations of algebraic curves under specializations

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications.

Dynamical analysis of a stochastic maize streak virus epidemic model with logarithmic Ornstein-Uhlenbeck process.

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein-Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus.

Electrodynamic modeling of slot gratings by the generalized scattering matrix method – spherical wave decomposition

An algorithm is presented for the electrodynamic analysis of a two-dimensional waveguide slot array of finite dimensions. To solve the boundary value problem, the generalized scattering matrix method is used. The complex problem for a structure with large electrical dimensions is divided into two subproblems: wave scattering on one lattice element and the interaction of waves within the lattice. In accordance with this method, the electromagnetic field of a solitary lattice element is represented in the form of an expansion in incident and scattered spherical waves. The solution to the first subproblem is given by the scattering operator, which relates the amplitudes of the incident and scattered waves. The solution to the second subproblem yields an interaction matrix that relates the amplitudes of waves incident on the mth array element with the amplitudes of waves scattered by the nth element. Application of the scattering operator and interaction matrix to the analysed lattice leads to a system of linear algebraic equations for the amplitudes of the scattered waves. A non-periodic slot grating, focused in the Fresnel zone, containing up to a thousand elements is analysed. The obtained numerical results are in good agreement with the known behaviour of focused leaky wave gratings. Possible areas of application of the method are discussed.

Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.

Haar Wavelet Approach for the Mathematical Model on Hepatitis B Virus

Abstract The Haar wavelet collocation method, a wavelet technique, is discussed in this article to examine the mathematical model of Hepatitis B virus infection. We took into account the HB virus, cytotoxic T lymphocytes (CTL) immune response, birth rate, death rate, and infected and uninfected hepatocytes to identify the dynamics of the hepatitis B virus infection. An ordinary differential equation (ODE) system that is nonlinear makes up this model. Using this method, the Hepatitis B Virus model can be solved by expressing each dependent variable as a Haar wavelet and then converting the system of ordinary differential equations into a system of nonlinear algebraic equations. The unknown coefficient values are thought to be extracted using the collocation procedure and the Newton–Raphson method. Tables and graphs are used to illustrate the characteristics of the Hepatitis B virus. The obtained results show that the current approach outperforms other approaches found in the literature in terms of accuracy. Mathematica software is utilized to obtain numerical results and nature.

Do quality criteria of instructional explanatory videos in algebra influence student choice? A feasibility study using conjoint analysis to assess implicit reasoning in educational contexts

ABSTRACT Many students struggle with algebra and often turn to instructional explanatory videos (IEVs) as learning resources. This feasibility study explored whether students implicitly recognize quality indicators in algebra videos and how these influence their choices. Using Conjoint analysis in a 2×2 design, we examined the focus on subject-matter didactical and multimedia features among 34 students aged 14–15 years. We presented authentic IEVs on solving algebraic equations varying in didactical and multimedia features. Bayesian analysis revealed that both didactical and multimedia features influenced video choices with no notable differences in impact strength. Student characteristics (mathematics achievement, interest, and self-concept) affected preferences only for the didactical features. The study concludes that students can identify quality indicators in math videos, although this recognition is limited and may be further enhanced through targeted guidance. Methodologically, conjoint analysis demonstrated its efficacy in mathematics education research, effectively addressing social desirability bias and implicit behaviors.

Viscous Current-Induced Forces.

We study the motion (translational, vibrational, and rotational) of a diatomic impurity immersed in an electron liquid and exposed to electronic current. An approach based on the linear response time-dependent density functional theory combined with the Ehrenfest dynamics leads to a system of linear algebraic equations, which account for the competing and counteracting effects of the current-induced force (electron wind) and the electronic friction. We find and emphasize the coupling between the center of mass motion and that of the nuclei relative to each other, the feature due to the mediation of the two-body interaction by the environment. The current-induced forces, by means of the dynamic exchange-correlation (xc) kernel f_{xc}(r,r^{'},ω), include the electronic viscosity contribution. Starting from the ground state at the equilibrium internuclear distance and applying a current pulse, we observe three phases of the motion: (i)acceleration due to the prevalence of the current-induced force, (ii)stabilization upon balancing of the two forces, and (iii)deceleration due to the friction after the end of the pulse. At lower, but still metallic, electron densities, the dynamic xc contribution to the force significantly affects the acceleration (deceleration) at the first (third) phase of the process. For the Cs density (r_{s}≈6 a.u.), this correction amounts up to 40% in the rotation regime.

On G2 approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

Massless spin 2 field in 50-component approach: exact solutions with cylindrical symmetry, eliminating the guage degrees of freedom

We begin with some known results of the 50-component theory for a spin-2 field described in cylindrical coordinates. This theory is based on the use of a 2nd-rank symmetric tensor and a 3rd-rank tensor symmetric in two indices. In the massive case, this theory describes a spin-2 particle with an anomalous magnetic moment. According to the Fedorov – Gronskiy method, which is based on projective operators, all 50 functions involved in the description of the spin-2 field for the case of the free particle can be expressed in terms of only 7 different functions constructed from Bessel functions. This leads to a homogeneous system of linear algebraic equations for 50 numerical parameters. We have found 6 independent solutions to these equations. Additionally, we have obtained explicit expressions for 4 guage solutions defined in accordance with the Pauli – Fierz approach. These solutions are exact and correspond to non-physical states that do not affect observable quantities, such as the energy-momentum tensor. Finally, we have constructed two classes of solutions that represent physically observable states.

Observer-based bipartite output containment control of heterogeneous multi-agent systems over structurally unbalanced signed graphs

In this paper, we propose state feedback and output feedback observer-based protocols that solve the bipartite output containment (BOC) problem for heterogeneous multi-agent systems over structurally unbalanced signed graphs. The proposed protocols rely solely on the exchange of output information among neighboring agents, resulting in a significant reduction or complete elimination of communication costs. Two design methods for determining controller parameters based on algebraic Riccati equations and linear matrix inequalities (LMIs) are proposed. Each method offers distinct advantages while ensuring that the closed-loop system poles are placed within a predefined region in the complex plane. Furthermore, an extensive solvability analysis is conducted, demonstrating that the solvability of the BOC problem using the proposed approach can be pre-guaranteed for a broad range of agent dynamics.

Finite-horizon and infinite-horizon linear quadratic optimal control problems: A data-driven Euler scheme

In this paper, we devote our focus to studying finite-horizon and infinite-horizon linear quadratic optimal control (LQOC) problems in continuous time. We focus on algorithms for approximating optimal controls of these two problems without the knowledge of all system coefficients. Firstly, we transform the backward differential Riccati equation (DRE) corresponding to the finite-horizon LQOC problem into a forward ordinary differential equation. Then, we discretize the forward equation by the Euler method. Subsequently, we use the collected state and input data to calculate all discretized points of the forward equation. At the same time, we obtain the optimal control of the finite-horizon LQOC problem. Moreover, we approximate the optimal control of the infinite-horizon LQOC problem by virtue of an asymptotic behavior between the DRE and an algebraic Riccati equation (ARE) arising in the infinite-horizon LQOC problem. Finally, we validate the obtained results via two practically motivated examples.

Construction of compositional geological models using the variational grid method of geomapping and the object-hierarchical approach

The variational grid method of geomapping is based on approximation by bicubic splines, the permissible grid size of which is limited by the amount of computer RAM. The paper describes an approach to solving problems of constructing maps of geological parameters, using this method, when computer resources are insufficient. The approach is based on the sequential mapping for a set of fragments of the entire mapping area. The implementation of the approach is considered for two options. The first, if the data density is sufficiently high, consists of sequential calculations over partially overlapping “bands”. If the data distribution is significantly uneven, the second option is implemented, which consists in calculating a detailed grid for local areas and smoothly pasting it into the general map. Smoothness at the boundaries of bands and insets is ensured through the condition of equality of spline coefficients for coinciding nodes, which is set by adjusting the system of linear algebraic equations when solving the mapping problem. Based on these approaches, a multiscale structural-geological model of the sedimentary cover over the territory of Western Siberia was implemented. The formation of a compositional model is carried out on the basis of an object-hierarchical approach, which ensures consistency in the calculations of all its elements, as well as automation of constructions.

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

Noise intensity of a Markov chain.

Stochastic transitions between discrete microscopic states play an important role in many physical and biological systems. Often these transitions lead to fluctuations on a macroscopic scale. A classic example from neuroscience is the stochastic opening and closing of ion channels and the resulting fluctuations in membrane current. When the microscopic transitions are fast, the macroscopic fluctuations are nearly uncorrelated and can be fully characterized by their mean and noise intensity. We show how, for an arbitrary Markov chain, the noise intensity can be determined from an algebraic equation, based on the transition rate matrix; these results are in agreement with earlier results from the theory of zero-frequency noise in quantum mechanical and classical systems. We demonstrate the validity of the theory using an analytically tractable two-state Markovian dichotomous noise, an eight-state model for a calcium channel subunit (De Young-Keizer model), and Markov models of the voltage-gated sodium and potassium channels as they appear in a stochastic version of the Hodgkin-Huxley model.

Unlocking Synergy: Leveraging SysML and Modelica with Bi‐Directional Transformation and Simulation Integration Standards

AbstractBoth the SysML and Modelica standards are used in the field of Systems Engineering (SE) to model systems from different perspectives, on different abstraction levels. SysML is strong when modeling systems on a functional level because it provides different views. With the capabilities of other simulation specifications, the engineers can simulate the system architecture. On the other hand, an open standard, such as Modelica is a key enabler for representing multi‐physical systems described by differential, algebraic, and discrete equations. With the symbolic manipulation, the dynamics of the systems are represented in state space form, and solved by the numerical integration methods fixed or variable step. However, it is clear that the connection between systems engineering and system simulation, with their respective domain knowledge of the actual equipment in their system, is missing. By seeing these complementary values, the authors demonstrate language interaction to integrate SysML and Modelica to achieve complimentary values through bi‐directional transformation and simulation.

A nonlinear optimal control approach for 3-DOF four-cable driven parallel robots

In this article, a nonlinear optimal control approach is proposed for the dynamic model of 3-DOF four-cable driven parallel robots (CDPR). To solve the associated nonlinear optimal control problem, the dynamic model of the 3-DOF cable-driven parallel robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the 3-DOF cable-driven parallel robot a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs and a minimum dispersion of energy.

Regularization of the Solution to Degenerate Systems of Algebraic Equations Exemplified by Identification of the Virial Equation of State of a Real Gas

Regularization of the Solution to Degenerate Systems of Algebraic Equations Exemplified by Identification of the Virial Equation of State of a Real Gas