This chapter provides a brief literature review together with detailed descriptions of the authors’ work on the stability and control of systems represented by linear time-periodic delay-differential equations using the Chebyshev and temporal finite element analysis (TFEA) techniques. Here, the theory and examples assume that there is a single fixed discrete delay which is equal to the principa...

We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all poly...

It is shown that the tensor product JH⊗̃ǫJH fails Pe lczńyski’s property (u). The proof uses a result of Kwapień and Pe lczńyski on the main triangle projection in matrix spaces. The Banach space JH constructed by Hagler [1] has a number of interesting properties. For instance, it is known that JH contains no isomorph of l, and has property (S): every normalized weakly null sequence has a subseq...

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to c...

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

It was proved in [4] that the ultraweakly closed algebras generated by certain contractions on Hubert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to Hx, was used in [3] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with Hx, and a...

The method called Arnoldi is currently a very popular method to solve largescale eigenvalue problems. The general purpose of this paper is to generalize Arnoldi to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem. The DDE can equivalently be expressed with a linear infinite dimensional operator which eigenvalues are the solutions to the ...

Chemical reaction networks (CRNs) are directed graphs with reactant or product complexes as vertices, and reactions arcs. A CRN is weakly reversible if each of its connected components strongly connected. Weakly can be considered the most important class networks. Now, stoichiometric subspace a network linear span vectors (i.e., difference between complexes). decomposition independent (incidenc...

‎It is commonly accepted that fractional differential equations play‎ ‎an important role in the explanation of many physical phenomena‎. ‎For‎ ‎this reason we need a reliable and efficient technique for the‎ ‎solution of fractional differential equations‎. ‎This paper deals with‎ ‎the numerical solution of a class of fractional differential‎ ‎equation‎. ‎The fractional derivatives are described...

A Banach space E has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space P(kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric k-fold projective tensor product of E (i.e., the predual of P...