The Homogeneous Linear Equation and Wronskians

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

Rank-one solutions for homogeneous linear matrix equations over the positive semidefinite cone

Exponential polynomials, i.e. solutions of linear homogeneous differential equations with constant coefficients, are characterized as continuous solutions of linear homogeneous difference equations with constant coefficients. Namely, denote by C the \({\Bbb C}\)-linear space of all continuous functions on \({\Bbb R}\), and define for \( \omega \in {\Bbb R}_+\) the shift operator \(z_{\omega }:C\to C\) by \(\eta \mapsto z_{\omega }\eta \) such that \(z_{\omega }\eta (t)=\eta (t+{\omega })\) for all \(t\in {\Bbb R}\). Every ideal \({\mathfrak a}\) of \(R:={\Bbb C}[x,y]\) determines a subspace¶¶\( C_{\frak a}=\{\eta \in C:f(z_{{\omega }_1},z_{{\omega }_2})\eta =0\) for all \(f(x,y)\in {\frak a}\}\,.\)¶¶For incommensurable \({\omega }_1,{\omega }_2\in {\Bbb R}_+\) and ideals \({\frak a}\) in R with a factor ring \(R/{\frak a}\) , which is finite dimensional as a \({\Bbb C}\)-linear space, the set \(C_{\mathfrak a}\) is characterized. The main idea consists in applying the Kronecker approximation theorem to the analytic transition from the explicit representation of the solutions of recurrence sequences to that of the continuous solutions of difference equations. For an alternative proof, the Kronecker approximation theorem is combined with the Jordan decomposition theorem.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

SECTION 6 - Solutions of Linear Equations

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.

Construction of a fundamental set of solutions of an arbitrary homogeneous linear difference equation

Exponential dichotomy and pseudo-almost automorphy for partial neutral functional differential equations

A probabilistic point of view on peak effects in linear difference equations

The one-dimensional linear homogeneous Boltzmann equation is solved for a binary mixture of quasi-Maxwellian particles in the presence of a time-dependent external field. It is assumed that the charged particles move in a bath of neutral scatterers. The neutral scatterers are in thermal equilibrium and the concentration of the charged particles is low enough to neglect collisions between them. Two cases are considered in detail, the constant and the periodic external field. The quantities calculated are the equilibrium and the stationary distribution function, respectively, from which any desired property can be derived. The solution of the Boltzmann equation for Maxwellian particles can be reduced to the solution of the so-called cold gas equation by employing the one-dimensional variant of a convolution theorem due to Wannier. The two limiting cases, the Lorentz gas (mA→0) and the Rayleigh gas (mA→∞) are treated explicitly. Furthermore, by computing the central moments, the deviations from the Gaussian approximation are discussed, and in particular the large-velocity tails are evaluated.

be a linear differential operator with coefficients in F, a finite algebraic extension of Q(x). We shall show that one can find, in a finite number of steps, a basis for the vector space of liouvillian solutions of L ( Y) = 0 (i.e., those solutions which can be built up from the rational functions by algebraic operations, taking exponentials and by integration; see Section 2 for a precise definition). In particular, we show how to decide if all solution of L(Y) = 0 are liouvillian or if there are any solutions which are liouvillian. Our algorithm in conjunction with the algorithm in [8], also allows one to determine the algebraic relationships among the liouvillian solutions of L(Y) = 0, and, in particular, determine if all solutions are algebraic (c.f. [111). For second order linear homogeneous equations over Q(x), the problem of determining if all solutions are algebraic functions was considered by Fuchs, Klein and Schwarz, but none of these mathematicians seems to have presented a complete decision procedure. Building on the work of Klein, Baldassarri and Dwork [21 have given such a procedure. Baldassarri [11 has extended this to consider linear homogeneous equations whose coefficients are algebraic functions. For third order equations, Painleve and Boulanger gave a procedure which, in effect, reduced the problem of finding algebraic solutions of L( Y) = 0 to the problem of effectively bounding the torsion of the jacobian variety of a given curve. A complete procedure for deciding if all solutions of an n-th order homo-

Monoids determined by a homogenous linear diophantine equation and the half-factorial property

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented.

The balancing numbers satisfy the second order linear homogeneous dierence equation Bn+1 = 6Bn - Bn-1, on the other hand the Fibonacci numbers are solution of the second order linear homogeneous dierence equation Fn+1 = Fn + Fn-1; where Bn and Fn denote the nth balancing number and nth Fibonacci number respectively. In a paper, Smith introduce Fibonometry in connection with a dierential equation called Fibonometric dierential equation. In this study, we first introduce the balancometric dierential equation and then obtain the balancometric functions as solutions of this equation.

The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities of these equations and is based on the description of spectral properties of an operator that is induced by the right-hand side of the equation and acts in the scale of spaces of infinite sequences.