Gusein Sh. Guseinov

Integration on time scales

In this paper we study the process of Riemann and Lebesgue integration on time scales. The relationship of the Riemann and Lebesgue integrals is considered and a criterion for Riemann integrability is established.

The Convolution on Time Scales

The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider the q-difference equations case.

Multiple Lebesgue integration on time scales

We study the process of multiple Lebesgue integration on time scales. The relationship of the Riemann and the Lebesgue multiple integrals is investigated.

Integrable equations on time scales

Integrable systems are usually given in terms of functions of continuous variables (on R), in terms of functions of discrete variables (on Z), and recently in terms of functions of q-variables (on Kq). We formulate the Gel’fand-Dikii (GD) formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers (difference equations) and over q-numbers (q-difference equations). We formulate the GD formalism also in terms of shift operators for all regular-discrete time scales. We give a method allowing to construct the recursion operators for integrable systems on time scales. Finally, we give a trace formula on time scales and then construct infinitely many conserved quantities (Casimirs) of the integrable systems on time scales.

Riemann and Lebesgue Integration

In [86, Section 1.4], the concept of integration on time scales is defined by means of an antiderivative (or pre-antiderivative) of a function and is called the Cauchy integral (we remark that in [191, p. 255] such an integral is named as the Newton integral).

Further properties of the Laplace transform on time scales with arbitrary graininess

In this work, we generalize several properties of the usual Laplace transform to the Laplace transform on arbitrary time scales. Among them are translation theorems, transforms of periodic functions, integration of transforms, transforms of derivatives and integrals, and asymptotic values.

Introduction to the Time Scales Calculus

In this chapter we introduce some basic concepts concerning the calculus on time scales that one needs to know to read this book. Most of these results will be stated without proof. Proofs can be found in the book by Bohner and Peterson [86]. A time scale is an arbitrary nonempty closed subset of the real numbers. Thus

Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

\mathbb{R},\mathbb{Z},\mathbb{N},\mathbb{N}_0 ,