Infinite Linear Equations Research Articles

Tight infinite matrices

We give a simple proof of a recent result of Gollin and Joó (Matching variables to equations in infinite linear equation systems. Linear Algebra Appl. 2023;660:40–46. doi: 10.1016/j.laa.2022.12.002): if a possibly infinite system of homogeneous linear equations A x → = 0 → , where A = ( a i , j ) is an I × J matrix, has only the trivial solution, then there exists an injection ϕ : J → I , such that a ϕ ( j ) , j ≠ 0 for all j ∈ J .

Matching variables to equations in infinite linear equation systems

A fundamental result in linear algebra states that if a homogenous linear equation system has only the trivial solution, then there are at most as many variables as equations. We prove the following generalisation of this phenomenon. If a possibly infinite homogenous linear equation system with finitely many variables in each equation has only the trivial solution, then there exists an injection from the variables to the equations that maps each variable to an equation in which it appears.

Dynamic analysis of non‐confocal elliptical lined tunnel under the action of SH wave

The scattering characteristics of non‐confocal elliptical lined tunnels under SH‐wave disturbance are investigated. The problem is essentially a boundary value problem and its theoretical treatment is based on the wave function expansion method, which is an exact treatment of the problem. Multi‐elliptic co‐ordinate systems are employed and wave function translations between multi‐elliptic co‐ordinate systems are performed using the extra‐domain addition theorem of the Mathieu function. The method can be used to fit and handle scattering problems with a wide range of geometries by varying the minor‐and‐major axis ratio, lining thickness and relative position of the tunnel. Calculations have been performed on the truncated infinite linear equation established by considering the boundary conditions. Based on the calculation results a parametric study is carried out to analyze the effect of the dimensionless eccentricity distance, the minor‐and‐major axis ratio and the dimensionless wave number on the dynamic stresses around the tunnel. Two typical application examples of concrete lining in granite and steel lining in granite are presented.

Increasable doubly substochastic matrices with application to infinite linear equations

It has been proved that every doubly substochastic matrix can be “increased” into a doubly stochastic matrix in the sense that there is a doubly stochastic matrix with ([Marshall AW, Olkin I, Arnold BC. Inequalities; theory of majorization and its applications. 2nd ed. New York (NY): Springer; 2011], Theorem C.1). In this paper, we show this assertion does not necessarily hold for all infinite doubly substochastic matrices. Then we characterize all such matrices which are called increasable doubly substochastic matrices. We also provide an application of this result to the existence of a non-negative solution for some infinite linear equations.

Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

Doubly (sub)stochastic operators on ℓp spaces

A square matrix is said to be doubly stochastic if its elements are non-negative and all row sums and column sums are equal one. An important tool in the study of majorization for infinite dimensional spaces ℓp(I) are doubly stochastic operators. These operators are a generalization of doubly stochastic matrices. In this paper, we compare some properties of doubly stochastic operators in finite and infinite dimensions. We will see that if D:ℓp(I)→ℓp(I) is a doubly stochastic operator then ‖D‖≤1. Moreover, the existence of such an operator with ‖D‖<1 is equivalent to 1<p<∞ and I is an infinite set. We discuss some other properties of doubly stochastic operators such as compactness and closedness and also, provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.

Linearization of seismic response of structures equipped with nonlinear viscous dampers using perturbation technique

Seismic analysis of structures equipped with nonlinear viscous dampers is commonly difficult and time-consuming due to the strong nonlinearity existing in such devices. This paper aims to develop a linearization method to convert the governing nonlinear differential equations to a series of infinite linear equations using perturbation technique. The order of accuracy depends on the number of successive equations solved. The accuracy of developed method was verified through the analysis of a variety of SDOF and MDOF structures. The results showed a good agreement with those obtained by the existing direct methods for nonlinear dynamic analyses.

Analytical and numerical results of fractional differential-difference equations

Abstract In this paper, we examine the fractional differential-difference equation (FDDE) by employing the proposed sensitivity approach (SA) and Adomian transformation method (ADTM). In SA the nonlinear differential-difference equation is converted to infinite linear equations which have a wide criterion to solve for the analytical solution. By ADTM, the FDDE is converted into ordinary differential-difference equation that can be solved. We test both the techniques through some test problems which are arising in nonlinear dynamical systems and found that ADTM is equivalently appropriate and simpler method to handle than SA.

On the solutions of infinite systems of linear equations

New theorems about the existence of solution for a system of infinite linear equations with a Vandermonde type matrix of coefficients are proved. Some examples and applications of these results are shown. In particular, a kind of these systems is solved and applied in the field of the General Relativity Theory of Gravitation. The solution of the system is used to construct a relevant physical representation of certain static and axisymmetric solution of the Einstein vacuum equations. In addition, a newtonian representation of these relativistic solutions is recovered. It is shown as well that there exists a relation between this application and the classical Haussdorff moment problem.

A new approach to (s, S) inventory systems

In this paper, we consider period review (s, S) inventory systems with independent and identically distributed continuous demands and full backlogging. Using an approach recently proposed by Gong and Hu (1992), we derive an infinite system of linear equations for all moments of inventory level. Based on this infinite system, we develop two algorithms to calculate the moments of the inventory level. In the first one, we solve a finite system of linear equations whose solution converges to the moments as its dimension goes to infinity. In the second one, we in fact obtain the power series of the moments with respect to s and S. Both algorithms are based on some very simple recursive procedures. To show their efficiency and speed, we provide some numerical examples for the first algorithm.(s, S) INVENTORY SYSTEMS; DYNAMIC RECURSIVE EQUATIONS; INFINITE LINEAR EQUATIONS; MACLAURIN SERIES

A new approach to (s, S) inventory systems

In this paper, we consider period review (s, S) inventory systems with independent and identically distributed continuous demands and full backlogging. Using an approach recently proposed by Gong and Hu (1992), we derive an infinite system of linear equations for all moments of inventory level. Based on this infinite system, we develop two algorithms to calculate the moments of the inventory level. In the first one, we solve a finite system of linear equations whose solution converges to the moments as its dimension goes to infinity. In the second one, we in fact obtain the power series of the moments with respect to s and S. Both algorithms are based on some very simple recursive procedures. To show their efficiency and speed, we provide some numerical examples for the first algorithm. (s, S) INVENTORY SYSTEMS; DYNAMIC RECURSIVE EQUATIONS; INFINITE LINEAR EQUATIONS; MACLAURIN SERIES

A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness

The T-matrix formalism is used to calculate scattering of a plane wave from a doubly infinite fluid–solid interface with doubly periodic surface roughness. The Helmholtz–Kirchhoff integral equations are used to represent the scattered pressure field in the fluid and the displacement field in the solid. The boundary conditions are applied and a system of four coupled integral equations is obtained. The incident and scattered pressure fields in the fluid, as well as the surface pressure field, are represented by infinite series of scalar Floquet plane waves, while the scattered displacement field in the solid and the surface displacement field are represented by infinite series of rectangular vector basis functions constructed from Floquet plane waves. This process discretizes the integral equations and transforms them into a system of four coupled doubly infinite linear equations. The extended boundary condition is applied and the T matrix that relates the spectral amplitudes of the incident field to the spectral amplitudes of the scattered fields is constructed. An exact analytic solution and numerical results are obtained for plane wave scattering from doubly periodic sinusoidal and triangular surface roughness.

A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness

The T-matrix formalism is used to calculate scattering of a pressure wave from a doubly infinite fluid–solid interface with doubly periodic surface roughness. The Helmholtz–Kirchhoff integral equations are used to represent the scattered pressure field in the fluid and the displacement field in the solid. The boundary conditions are applied and a system of four coupled integral equations is obtained. The incident field, the surface fields, and scattered pressure field in the fluid and displacement field in the solid, are represented by infinite series of Floquet plane waves. This process discretizes the integral equations and transforms them into a system of four coupled doubly infinite linear equations. The extended boundary condition is applied and the T matrix that relates the spectral amplitudes of the incident field to the spectral amplitudes of the scattered fields is constructed. An exact analytic solution and numerical results are obtained for scattering from a doubly periodic rough surface constructed by superposing two sinusoids each of which depends on a single but different orthogonal coordinate.

An inner product and orthonormalization inl p -spaces and solution of infinite systems of infinite linear equations

Let the rows of an infinite square matrixM be elements ofl p -space (p>1) andX be an infinite column vector of unknowns andC an infinite column vector of real numbers. To our knowledge the solvability ofMX=C has nowhere been satisfactorily studied in the literature. This is also true of Riesz’classical work [2]. A reason for this is that not until recently [1] an appropriate inner product and the corresponding orthonormalization forp≠2 has been introduced. In this paper, based on [1], it is shown thatMX=C has a solution which is an element ofl q if and only if upon our process of orthonormalization of the rows ofM the system yields an equivalent systemAX=K where the rows ofA form an orthonormal sequence (in our sense) of elements ofl p andK becomes an element ofl q withp −1+q −1=1. A solution is then given byX=(A (q) (AA (q) )−1)K whereA (q) is ourq-transpose ofA. This solution is the solution of the minimall q -norm. Otherwise, the obvious dual concept of the best approximating solution inl q -norm is introduced and obtained.

A new solution for the space crack problem from hypersingular integral equation

In this paper, a hypersingular integral equation for the plane crack problem of arbitrary shape in 3-dimensional isotropic elastic medium is studied, the solution for this equation is given. Based on above result, a new method for solving the space crack problem is obtained. Using this method the difficulties of solving the infinite linear equation system is avoidable. Some examples are given including the known closed form solution of the penny-shape crack under constant pressure

Compactness of the product topology and the solvability of infinite systems of infinite linear equations

If every finite subsystem of an infinite system of linear equations (say, over the field of real numbers) each with finitely many unknowns has a solution then the entire system has a solution. The situation is not so if the equations contain infinitely many unknowns. In this case, as shown below, the solvability of every finite subsystem implies the solva. bility of the entire system provided finite subsystems have solution with common upper and lower bounds and the coefficients of ever equation satisfy some boundedness or convergence conditions. The passage from the solvability of finite subsystem to the solvability of the entire system is achieved based on Tychnoff’s theorem stating that any product of compact topological spaces is compact in their product topology.

The effective conductivity of a periodic array of spheres

We consider the problem of determining the effective conductivity k * of a composite material consisting of equal-sized spheres of conductivity α arranged in a cubic array within a homogeneous matrix of unit conduc­tivity. We modify Zuzovski &amp; Brenner’s (1977) method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by McKenzie et al . (1978) using a method originally devised by Rayleigh (1892). On solving these equations we derive expressions for k * to O ( c 9 ) ─ c being the volume fraction of the spheres ─ for simple, body-centred and face-centred cubic arrays, and also obtain numerical values for k * over the whole range of α and c . We show that these results for cubic arrays can be used to estimate k * for random arrays of identical spheres. For arrays of highly conducting and nearly touching spheres, Batchelor &amp; O’Brien (1977) showed that k * ∼ {─ K 1 In (1 ─ χ ) ─ K 2 ( α = ∞, χ = ( c / c max ) ⅓ → 1), 2 K 1 In α ─ K ʹ 2 ( χ = 1, α → ∞), where c max corresponds to the volume fraction when the spheres are actually touching each other, and determined K 1 for the three cubic arrays. Our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants K 2 and Kʹ 2 thereby obtained do not quite satisfy the relation Kʹ 2 = K 1 (3.9 ─ In 2) + K 2 given by Batchelor &amp; O’Brien. We have been unable to find the reason for this slight discrepancy.

Diffraction of electromagnetic waves by two spheres

The solution method consists in the following. With each body we couple a local coordinate system within which the electromagnetic field is represented by the amplitudes of impressed-current waves and by unknown wave amplitudes determined by the presence of another body. We use the boundary conditions at the surface of each body in the local coordinate system to form an incomplete system of infinite linear equations in the unknown wave amplitudes. The wave-amplitude transformations complete this system of equations for transition from one local system to another. Solution of this system by the truncation method provides the solution to our problem.