Homogeneous Linear Equations Research Articles

Existence and Uniqueness of Homogeneous Linear Equation Solutions in Supertropical Algebra

This research investigates the existence and uniqueness of solutions to homogeneous linear equations in supertropical algebra. We analyze the structure of supertropical matrices to identify the conditions in which nontrivial solutions exist for the system of equations A⊗𝒙 ⊨ 𝜀, where A is a matrix over a supertropical semiring and x is a vector. By applying determinant-based criteria, we demonstrate how tropical and supertropical values influence the solution space. The research applies theorems that determine the presence of trivial and nontrivial solutions and uses examples to illustrate practical methods for solving homogeneous matrix systems. This highlights the distinct characteristics of supertropical algebra compared to classical linear algebra. Our findings provide a deeper insight into solution behaviors in supertropical systems, paving the way for further research in tropical mathematics.

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it.The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. The general solution of the homogeneous linear equation can be found using the concept of generalized functions expressed as a series expansion in terms of coordinate eigenfunctions dependent on p parameter. The p parameter is related with the FE mass and at some certain values results in bifurcation points of the boundary value problem for eigenfunctions. The partial solution can be found in several ways: using the Fourier transform, the method of Green's function, and in terms of series expansion by eigenfunctions.Eigenfunction expansion coefficients in an explicit form have been obtained and the numerical solution accuracy has been estimated for some important particular cases. The results of the obtained exact solutions appear to be very close to the results of the ANSYS numerical estimations. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 400 FE. Therefore, the advantage of the exact solution in practical use is that its calculation is much faster as compared with the finite element method done with ANSYS.

On a nonhomogeneous heat equation on the complex plane

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form [Formula: see text], where [Formula: see text], the function [Formula: see text] is smooth and subharmonic on [Formula: see text], and [Formula: see text] is the formal adjoint of [Formula: see text]. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in [Formula: see text], J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Solution forms for generalized hyperbolic cotangent type systems of p-difference equations

Due to the recent increasing interest in hyperbolic-cotangent types of scalar-or two-dimensional systems of difference equations and treatment of some particular states. This paper presents a natural extension of the p-dimensional of four-systems of this generalized type and treats general states. Which is an extension of Stevic's work (J. Inequal. Appl., 2021, 184 (2021)). We also show these systems are solvable by using appropriate variable transformations and obtaining systems of homogeneous linear difference equations with constant coefficients. Some numerical examples of these systems are presented.

On Some Exact Formulas for \(2\)-Color Off-diagonal Rado Numbers

Let \(\varepsilon_{0}\), \(\varepsilon_{1}\) be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the \(2\)-color off-diagonal Rado number \(R_2(\varepsilon_{0},\varepsilon_{1})\) to be the smallest \(N\) such that for any 2-coloring of \([1,N]\), it must admit a monochromatic solution to \(\varepsilon_{0}\) of the first color or a monochromatic solution to \(\varepsilon_{1}\) of the second color. Mayers and Robertson gave the exact \(2\)-color off-diagonal Rado numbers \(R_2(x+qy=z,x+sy=z). \) Xia and Yao established the formulas for \(R_2(3x+3y=z,3x+qy=z) \) and \(R_2(2x+3y=z,2x+2qy=z) \). In this paper, we determine the exact numbers \(R_2(2x+qy=2z,2x+sy=2z)\), where \(q, s\) are odd integers with \(q>s\geq1\).

Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.

Application of First Order differential Equations in R L Circuits

Abstract: The application of first order differential equation in Growth and Decay problems will study the method of variable separable and the model of Malthus (Malthusian population model), where we use the methods to find the solution to the population problems which are of use in mathematics, physics and biology especially in dealing with problems involving growth and decay problems that requires the use of Malthus model. Differential equations are fundamental importance in engineering mathematics because any physical laws and relations appear mathematically in the form of such equations. In this paper we discussed about first order linear homogeneous equations, first order linear non homogeneous equations and the application of first order differential equation in electrical circuits

A note on Laplace Adomian decomposition method on a linear problem

In last two decades, Laplace Adomian Decomposition Method (LADM) is vastly used to solve non-linear (or even fractional order) differential equations. The method approaches the solution with the partial sums of function series. However, it is not easy to show that the limit of the function series is the exact solution of the problem. In this article, we consider a simple problem such as homogenous second-order linear ordinary differential equation with constant coefficients. We prove analytically that the LADM gives the right exact solution to the considered problem.

Solving systems of homogeneous linear algebraic equations in pricing problems of an alternative financial model

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Global existence and decay rates of solutions to the Oldroyd-B model with stress tensor diffusion

In this paper, we consider the Oldroyd-B model with stress tensor diffusion in Rd with d≥2. We first establish the global well-posedness of solutions to this model for small initial data (u0,τ0)∈(B˙p,1dp−1∩B˙p,1dp+1)d×(B˙p,1dp)d×d with 1≤p<∞. Furthermore, under some additional L2 type conditions on (u0,τ0), but without any more smallness restrictions, we get the L2 decay rates of all derivatives of (u,τ). It is shown that the velocity u decays as fast as the solution to the corresponding homogeneous linear heat equation, and the symmetry τ decays as fast as ∇u. In particular, when d=3 the velocity u admits the decay rate faster than (1+t)−34 in L2.

A Method to Obtain the Transducers Impulse Response (TIR) in Photoacoustic Imaging

Photoacoustic tomography (PAT) is an emerging imaging technique with great potential for a wide range of biomedical imaging applications. The transducers impulse response (TIR) is a key factor affecting the performance of photoacoustic imaging (PAI). It is customary in PAI to assume that TIR is known or obtain it from experiments. In this paper, we investigate the possibility of obtaining TIR in another way. A new method is proposed to extract TIR from observed optoacoustic signal (OPAS) data, without prior knowledge, as a known condition. It is based on the relation between the OPAS data and the photoacoustic pressure signal (PAPS) at transducer positions. The relation can be expressed as a homogeneous linear equation. The TIR is solved by solving the homogeneous equation. The numerical test verifies the effectiveness of the presented method. This article also discusses the effect of calculation parameters on the extracting precision of TIR.

Validating an algebraic approach to characterizing resonator networks

Resonator networks are ubiquitous in natural and engineered systems, such as solid-state materials, electrical circuits, quantum processors, and even neural tissue. To understand and manipulate these networks it is essential to characterize their building blocks, which include the mechanical analogs of mass, elasticity, damping, and coupling of each resonator element. While these mechanical parameters are typically obtained from response spectra using least-squares fitting, this approach requires a priori knowledge of all parameters and is susceptible to large error due to convergence to local minima. Here we validate an alternative algebraic means to characterize resonator networks with no or minimal a priori knowledge. Our approach recasts the equations of motion of the network into a linear homogeneous algebraic equation and solves the equation with a set of discrete measured network response vectors. For validation, we employ our approach on noisy simulated data from a single resonator and a coupled resonator pair, and we characterize the accuracy of the recovered parameters using high-dimension factorial simulations. Generally, we find that the error is inversely proportional to the signal-to-noise ratio, that measurements at two frequencies are sufficient to recover all parameters, and that sampling near the resonant peaks is optimal. Our simple, powerful tool will enable future efforts to ascertain network properties and control resonator networks in diverse physical domains.

NATURAL ELECTROMAGNETIC MODES OF A COMPOSITE OPEN STRUCTURE INVOLVING A PERFECTLY CONDUCTING STRIP GRATING, AN INHOMOGENEOUS FERRITE LAYER, AND A MONOLAYER OF GRAPHENE

Subject and Purpose. Сonsidered are the natural modes and their correspondent eigenfrequencies of a composite structure which is nonuniform along one of the coordinates and consists of a lossy ferromagnetic layer placed in a static magnetic field. The layer involves a perfectly conducting strip grating at one of its boundaries and a graphene monolayer at the other. Methods and Methodology. The above stated problem can be approached within the analytical regularization procedure developed for dual series equations. The latter concern a broad class of diffraction problems which include, in particular, the diffraction of monochromatic plane waves on strip gratings placed at the boundary of a gyromagnetic medium. The amplitudes of the electromagnetic eigenmodes can be obtained from the infinite set of homogeneous linear algebraic equations solvable within a truncation technique. The roots of the system’s determinant represent complex-valued eigenfrequencies of the system under investigation. The material parameters adopted in our computations for the ferromagnetic layer correspond to such of yttrium iron garnet. Results. A number of numerical programs have been developed which permit analyzing the dependences of wave field eigenfunctions and complex eigenfrequencies upon geometrical parameters of the structure (such as grating slot width and period, and thickness of the lossy layer), as well as on electrodynamic parameters of the ferromagnet and graphene characteristics, specifically the chemical potential and relaxation energy of electrons. A number of behavioral regularities have been established, as well as the effect of non-uniformity of ferrite layer parameters upon the structure’s eigenfrequencies and wave field eigenfunctions. Conclusions. The structure under study has been shown to be is an open oscillatory system with a set of complex-valued natural frequencies demonstrating finite points of accumulation. The real parts of these eigenfrequencies lie in a certain interval determined by characteristic frequencies of the ferrite layer, while the imaginary parts are negative, such that the correspondent natural modes show an exponential decay with time. The grating edges represent the mirrors which the natural surface oscillations are reflected from, being supported at that by the ferromagnetic medium. The results obtained in this paper can be useful for creating the elemental base for microwave devices and the devices themselves.

О дифференцированиях линейных алгебр специального типа

In this work, Lie algebras of differentiation of linear algebra, the op­eration of multiplication in which is defined using a linear form and two fixed elements of the main field are studied. In the first part of the work, a definition of differentiation of linear algebra is given, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differentiation. An embedding of the Lie algebra of differenti­ations into the Lie algebra of square matrices of order n over the field P is constructed. This made it possible to give an upper bound for the dimen­sion of the Lie algebra of derivations. It has been proven that the dimen­sion of the algebra of differentiation of the algebras under study is equal to n2 – n, where n is the dimension of the algebra. Next we give a result on the maximum dimension of the Lie algebra of derivations of a linear alge­bra with identity. Based on the above facts, it is proven that the algebras under study cannot have a unit.

АЛГОРИТМИ ПОБУДОВИ МІНІМАЛЬНОЇ ПОРОДЖУВАЛЬНОЇ МНОЖИНИ РОЗВ’ЯЗКІВ СИСТЕМ ЛІНІЙНИХ РІВНЯНЬ

We consider optimizing transformations of the algorithm for construction of minimal generating sets of solutions of systems of linear homogeneous equations (SLHE) over the set of natural numbers. The features of such SLHEs are described, optimization transforms are substantiated, and examples of algorithm operation before and after optimization transforms are given. The application of the algorithm is illustrated by examples of the analysis of the properties of Petri nets and the construction of a set of basic solutions in the fields of complex, real, and rational numbers and over finite fields. Keywords: systems of linear equations, algorithms, solutions, optimization, complexity.

О размерности алгебр Ли инфинитезимальных аффинных преобразований прямых произведений более двух пространств аффинной связности первого типа

The theory of motions in generalized spaces is one of the directions in modern differential geometry. Such scientists as E. Cartan, P. K. Rashev­sky, P. A. Shirokov, I. P. Egorov, A.Ya. Sultanov and other scientists were engaged in the study of movements in various spaces of affine connec­tions. The question of movements in direct products of two spaces of af­fine connection was considered in M. V. Morgun’s work. In the case of a direct product of more than two spaces of affine con­nection, the question of the dimension of Lie algebras of infinitesimal affine transformations of a given space remained open. In this article, an estimate of the upper bound of the dimension of the Lie algebra of infinitesimal affine transformations of affine connection spaces, representing a direct reproduction of at least three non-projective Euclidean spaces of a certain type, is obtained. To solve this problem, a system of linear homogeneous equations is obtained, which is satisfied by the components of an arbitrary infinitesi­mal affine transformation. This system is found using the properties of the Lie derivative applied to the tensor field of curvature of the spaces under consideration. The evaluation of the rank of this system allows us to ob­tain an estimate from below of the rank of the matrix of the system under consideration.

Properties of the Wrońskian determinant of a system of solutions to a linear homogeneous equation: The case when the number of solutions is less than the order of the equation

Objectives. The work sets out to study the properties of the Wrońskian determinant of the system of solutions to a linear homogeneous equation in cases when the number of solutions is less than the order of the equation, comparing them with the known properties of the same determinant when the number of solutions is equal to the order of the equation.Methods. The work uses the methods of linear algebra according to the theory of ordinary differential equations, as well as mathematical and complex analysis.Results. It is shown that the vanishing of a considered determinant on an arbitrarily small interval implies its vanishing on the entire domain of definition; the solutions turn out to be linearly dependent. A stronger result is obtained in three cases: (1) if the coefficients of the equation are analytic functions; (2) if the number of solutions is equal to one; (3) if the number of solutions is one less than the order of the equation. Namely, if the set of zeros of the considered Wrońskian has a limit point belonging to the domain of definition of solutions, then the determinant is identically equal to zero and the solutions are linearly dependent.Conclusions. According to the obtained results, the Wrońskian of a system of solutions of a linear homogeneous equation can serve as an indicator of the linear dependence or independence of this system in cases where the number of solutions is lower than the order of the equation; here, the solutions are linearly dependent if and only if their Wrońskian is identically equal to zero. In this case, there is no need to check whether the determinant vanishes over the entire domain of definition, since it is sufficient to do this on an arbitrarily chosen interval or even (in the special cases listed above) on an arbitrarily chosen set having a limit point.

A Note on Lebesgue Solvability of Elliptic Homogeneous Linear Equations with Measure Data

A Note on Lebesgue Solvability of Elliptic Homogeneous Linear Equations with Measure Data

ЗАВИСИМОСТЬ ОСНОВНОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ ПЛОСКОЙ ШПРЕНГЕЛЬНОЙ ФЕРМЫ ОТ ЧИСЛА ПАНЕЛЕЙ

A scheme of a statically determined planar truss with downward braces and truss reinforcement is proposed. The problems of the lower and upper boundaries of the main natural frequency of free oscillations of the structure are solved. A model with masses concentrated at the nodes is adopted. To find the formula for the dependence of the oscillation frequency limits on the number of panels, the Dunkerley and Rayleigh methods are used. The expressions for the forces in the truss rods are found from the solution of the system of linear equilibrium equations for all nodes. To obtain the matrix of the system of equations, Maple computer mathematics operators are used. The rigidity of the structure is calculated using the Maxwell – Mohr formula. The sequence of solutions obtained for trusses with different numbers of panels is generalized by induction to an arbitrary case. For the coefficients of the formula in the desired solution, homogeneous linear recurrent equations are compiled and solved. The obtained solution is compared with the numerical solution of the problem of the oscillation spectrum of a structure with many degrees of freedom. It is shown that the first frequency of the spectrum is close to its analytical estimate. With an increase in the number of panels in the truss, the accuracy of the derived formulas increases.

On Local Unique Solvability for a Class of Nonlinear Identification Problems

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders.