In this paper the output regulation of a linear distributed parameter system with a nonautonomous periodic exosystem is considered. It is shown that the solvability of the output regulation problem can be characterized by the solvability of a certain constrained infinite-dimensional Sylvester differential equation. Conditions are given for the existence of feedforward and feedback controllers s...

p/k + 4b + c. Our proof is inspired by the so called Stepanov method, which involves bounding the size of the solution set of a system of equations by constructing a nonzero low degree auxiliary polynomial that vanishes with high multiplicity on the solution set. The proof uses basic algebra and number theory along with a determinant identity that generalizes both the Sylvester and the Vandermo...

We prove that if p ≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then both of the equations x + y = p and x + y = p have a solution with x, y ∈ Q.

The theorem in question asserts that, if n > k, then, in the set of integers n, n+1, n-J-2, . . ., n +k-1, there is a number containing a prime divisor greater than k . If n = k+ 1, we obtain the well-known theorem of Chebyshev . The theorem was first asserted and proved by Sylvester t about forty-five years ago . Recently Schurt has rediscovered and again proved the theorem . The following pro...

A new method for constructing Sylvester-type resultant matrices for multivariate elimination is proposed. Unlike sparse resultant constructions discussed recently in the literature or the Macaulay resultant construction, the proposed method does not explicitly use the support of a polynomial system in the construction. Instead, a multiplier set for each polynomial is obtained from the Dixon for...

Based on the higher-order restricted flows, first type of integrable deformed fourth-order matrix NLS equations, that is, equations with self-consistent sources (FMNLSSCS), is derived. By virtue $${\bar{\partial }}$$ -dressing method, second called NLS–Maxwell–Bloch system (FMNLS-MB) presented. We prove equivalence FMNLSSCS and FMNLS-MB successfully. Furthermore, N-soliton solutions are explici...

We present new iterative methods based on refinement process for solving large sparse Sylvesterobserver equations applied in state estimation of a continuous-time system. These methods use projection methods to produce low-dimensional Sylvester-observer matrix equations that are solved by the direct methods. Moreover, the refinement process described in this paper has the capability of improvin...

We consider numerical computation of Taylor expansions of invariant manifolds around equilibria of maps and flows. These are obtained by writing the corresponding functional equation in a number of points, setting up a nonlinear system of equations and solving that using a simplified Newton’s method. This approach will avoid symbolic or explicit numerical differentiation. The linear algebra iss...