Real Constant Coefficients Research Articles

Data-based stability analysis of strongly autonomous discrete nD systems

The aim is to provide data-based conditions for checking stability of strongly autonomous nD systems. It is known that because of the lack of natural ordering of the domain, the problem of defining and studying stability of nD systems is not straightforward. We focus on the following two notions of stability: (i) stability with respect to a given direction, and more generally, (ii) conic stability, that is, stability with respect to subsets of the domain having the structure of a cone. For a strongly autonomous nD system, these two notions of stability have been studied in the literature from a model based perspective. Here, we analyze their data-based counterparts.We assume that the data is noise-free, and that it is measured from a linear system of partial difference equations, with constant real coefficients, having n independent variables. We further assume that the data-generating system is a strongly autonomous nD system, that is, the solution trajectories can be uniquely determined using finite “initial conditions”. Given a finite set of data, that is, a finite set of multi-indexed sequences, we first define a notion of informativity of nD trajectories. Using informative nD trajectories, we provide data-based tests for checking stability of nD systems with respect to a given direction, and, more generally, for conic stability.

Planar Polynomial Differential Systems of Degree One: Full Characterization of Its First Integrals

In this work, we classify the first integrals of all planar polynomial differential systems of degree one with real constant coefficients. Additionally, we characterize when these first integrals are either polynomial, or rational, or nonalgebraic.

Bisimulation of strongly autonomous discrete nD systems

The notion of bisimulation is an important tool for analysis and synthesis of complex dynamical systems. While the notion of bisimulation for systems described by ordinary differential and difference equations is well studied in the literature, a notion of bisimulation for systems of partial differential and difference equations is practically absent. In this paper, we consider the problem of formulating a notion of bisimulation for a class of dynamical systems described by linear partial difference equations with real constant coefficients. We define two notions of bisimulation, namely, S-bisimulation and Π-induced bisimulation, and provide algebraic characterizations for the same.

Polynomial structures in generalized geometry

On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphisms satisfying an arbitrary polynomial equation with real constant coefficients. We investigate the compatibility of these structures with the de Rham operator and the Dorfman bracket. In particular, we isolate several conditions that when restricted to the motivating example of generalized almost complex structure are equivalent to the notion of integrability.

Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences

Let (Fn)n=1∞ be the classical Fibonacci sequence. It is well known that the limFn+1/Fn exists and equals the Golden Mean. If, more generally, (Fn)n=1∞ is an order-k linear recurrence with real constant coefficients, i.e., Fn=∑j=1kλk+1−jFn−j with n>k, λj∈R, j=1,…,k, then the existence of the limit of ratios of consecutive terms may fail. In this paper, we show that the limit exists if the first k elements F1,F2,…,Fk of (Fn)n=1∞ are positive, λ1,…,λk−1 are all nonnegative, at least one being positive, and max(λ1,…,λk)=λk≥k. The limit is characterized as fixed point, bounded below by λk and bounded above by λ1+λ2+⋯+λk.

Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classes

Let m ∈ N $m\in \mathbb {N}$ , P ( D ) : = ∑ | α | = 2 m ( − 1 ) m a α D α $P(D):=\sum _{|\alpha |=2m}(-1)^m a_\alpha D^\alpha$ be a 2 m $2m$ -order homogeneous elliptic operator with real constant coefficients on R n $\mathbb {R}^n$ , and V $V$ a real-valued measurable function on R n $\mathbb {R}^n$ . In this article, the authors introduce a new generalized Schechter class concerning V $V$ and show that the higher order Schrödinger operator L : = P ( D ) + V $\mathcal {L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when V $V$ belongs to this new class. The Davies–Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of L $\mathcal {L}$ .

On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj-1){(k_j - 1)}th-order normal derivatives where 1≤k1...kl≤2l{1\le k_1 < ... < k_l \le 2l}; for kj=j{k_j = j} it turns into the Dirichlet problem, and for kj=j+1{k_j = j +1} into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.

Некоторые аналитические и геометрические свойства решений кососимметрических эллиптических систем

We study the properties of complex-valued functions of a complex variable, whose real and imaginary parts satisfy a second-order skew-symmetric strongly elliptic system with constant real coefficients in the plane. The behavior of such functions and their dilatations near singular points is investigated and the dependence of the type of the singularity on the form of the Laurent expansion of the function under consideration is established. The principle of the argument is established for the functions with poles under study, analogs of the Ruschet and Hurwitz theorems are proved

On the stability of equilibrium states of the dynamical systems in critical cases

The stability of equilibrium states described by an autonomous system of linear differential equations with constant real coefficients is studied. The cases when, among the eigenvalues of the matrix of coefficients of the system, there are simple or multiple zero eigenvalues or complex eigenvalues with zero real part (critical cases of the Lyapunov’s general theory of stability) are discussed in detail. Stability criteria for the equilibrium states in critical cases are formulated using only information about the order of the matrix of coefficients, rank of the corresponding λ-matrix, and the multiplicity of the corresponding eigenvalues without reducing to the Jordan canonical form. Particular examples of the use of the proposed stability criteria are given. The dynamics of phase transitions due to thermal fluctuations for systems described by the Landau-type kinetic potential is discussed separately. An identity is given for matrices with a zero determinant and a given rank, which can be used to analyze the stability of solutions of linear dynamical systems.

Sharp pointwise estimates for the gradients of solutions to linear parabolic second-order equation in the layer

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second-order equations with real constant coefficients in the layer , where and . The homogeneous equation is considered with initial data in , . For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to , p>n + 2 and . Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.

On constructing boundaries for boundary value problems defined by continuous 2-D autonomous systems

Abstract In this paper, we provide a constructive way of specifying initial/boundary data for a given continuous 2-D autonomous system described by a set of linear partial differential equations (PDEs) with real constant coefficients. One of the ways of specifying initial/boundary data is by specifying the values of various derivatives of the solution trajectories at the origin; the derivatives correspond to a standard monomial set obtained using Grobner basis. However, such an initial/boundary data often lacks physical interpretation. In this paper, we consider subsets of the domain having some algebraic structure (in the form of subspaces and strips of finite width around such subspaces) such that trajectories restricted to these subsets, often called characteristic sets, serve as initial/boundary conditions for the given autonomous system of linear PDEs. We provide a systematic way to construct such characteristic sets with the help of Grobner bases and Oberst-Riquier algorithm. Thus we bridge the gap between initial/boundary conditions involving standard monomials and more conventional initial/boundary conditions in the form of restrictions on characteristic sets. We also show that every scalar system of PDEs admits such a characteristic set given by a rectangular strip of finite width around a subspace whose dimension equals the Krull dimension of the system’s quotient ring.

Critical length: An alternative approach

We provide a numerical method to determine the critical lengths of linear differential operators with constant real coefficients. The need for such a procedure arises when the orders increase. The interest of this article is clearly on the practical side since knowing the critical lengths permits an optimal use of the associated kernels. The efficiency of the procedure is due to its being based on crucial features of Extended Chebyshev spaces on closed bounded intervals.

Best Constant for Hyers–Ulam Stability of Second-Order h-Difference Equations with Constant Coefficients

First, we prove that the best (minimum) constant for Hyers–Ulam stability of first-order linear h-difference equations with a complex constant coefficient is the reciprocal of the absolute value of the Hilger real part of that coefficient; if the coefficient lies on the Hilger imaginary circle, then the equation is unstable in the Hyers–Ulam sense. Second, using this best constant from the first-order complex coefficient case, we determine the best constant for Hyers–Ulam stability of second-order linear h-difference equations with constant real coefficients. The second-order equation also is Hyers–Ulam stable if and only if the values of the characteristic equation do not intersect the Hilger imaginary circle.

The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Holder boundary, the differential properties of the solution are studied.

Linear combinations of the telegraph random processes driven by partial differential equations

Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.

Fundamental Solution of Elliptic Equation with Positive Definite Matrix Coefficient

In this paper, we study the fundamental solution of elliptic equations with real constant coefficients where is a positive definite matrix. We obtained by searching the radial solution so that we solved the equation into ordinary differential equations.

Lebesgue Constants for Some Interpolating L-Splines

We find exact values for the uniform Lebesgue constants of interpolating L-splines that are bounded on the real axis, have equidistant knots, and correspond to the linear thirdorder differential operator L3(D) = D(D2 + α2) with constant real coefficients, where α > 0. We compare the obtained result with the Lebesgue constants of other L-splines.

On characteristic cones of discrete nD autonomous systems: theory and an algorithm

In this paper, we provide a complete answer to the question of characteristic cones for discrete autonomous nD systems, with arbitrary \(n\geqslant 2\), described by linear partial difference equations with real constant coefficients. A characteristic cone is a special subset (having the structure of a cone) of the domain (here \(\mathbb {Z}^n\)) such that the knowledge of the trajectories on this set uniquely determines them over the whole domain. Despite its importance in numerous system-theoretic issues, the question of characteristic sets for multidimensional systems has not been answered in its full generality except for Valcher’s seminal work for the special case of 2D systems (Valcher in IEEE Trans Circuits and Syst Part I Fundam Theory Appl 47(3):290–302, 2000). This apparent lack of progress is perhaps due to inapplicability of a crucial intermediate result by Valcher to cases with \(n\geqslant 3\). We illustrate this inapplicability of the above-mentioned result in Sect. 3 with the help of an example. We then provide an answer to this open problem of characterizing characteristic cones for discrete nD autonomous systems with general n; we prove an algebraic condition that is necessary and sufficient for a given cone to be a characteristic cone for a given system of linear partial difference equations with real constant coefficients. In the second part of the paper, we convert this necessary and sufficient condition to another equivalent algebraic condition, which is more suited from algorithmic perspective. Using this result, we provide an algorithm, based on Grobner bases, that is implementable using standard computer algebra packages, for testing whether a given cone is a characteristic cone for a given discrete autonomous nD system.

On Pointwise Feedback Invariants of Linear Parameter-varying Systems

Linear systems with constant real coefficients are completely described in terms of feedback actions. In this paper the problem is studied in the framework of linear systems where coefficients depending continuously on a set of parameters. Some invariants are given as well as criteria to find a complete classification in low dimension.

On General boundary value problem for an elliptic higher order equation in the plane with constant real coefficients

By means of a new approach, the general boundary value problem for a higher order elliptic equation with two independent variables, and a normal set of boundary conditions and simple complex characteristics is reduced to the Fredholm system of integral equations in a bounded region with a smooth boundary.