We study differential-difference equation of the form d dx t(n+ 1, x) = f(t(n, x), t(n + 1, x), d dx t(n, x)) with unknown t(n, x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n ± k, x)}k=−∞, { d dxk t(n, x) }∞ k=1 , such that DxF = 0 and DI = I, where Dx is ...

3 Analytic classification of admissible q-difference modules 10 3.1 Generalities on q-difference modules . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Simple objects in the category of admissible q-difference modules . . . . . . . 11 3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Analytic vs formal classification . . . . . . . . . . . . ...

In this work we give sufficient conditions in order that the difference equation with advanced arguments x(n + 1) = A(n)x(n) + B(n)x(σ1(n)) + f(n, x(n), x(σ2(n))), σi(n) ≥ n + 1 has Φ−bounded solutions in the space `Φ of Φ−bounded sequences, are given.

Denote by x = (x0, x1, ..., xn) = (x0, x′) the coordinates in R × R. Sometimes we refer to the first coordinate x0 as “time” and to x′ as “space” coordinates. The Strichartz estimates for solutions to the homogeneous wave equation in R× R, 2u = 0, u(0) = u0, ut(0) = u1, have the form ‖|D|u‖Lp(Lq) . ‖∇u0‖L2 + ‖u1‖L2. (1.1) Such an estimate holds for all pairs (ρ, p, q) satisfying the relations 2...

Children have concepts of the physical world long before they enter school, and in the course of further development these rudimentary concepts typically change in everyday settings, even without formal instruction. This kind of knowledge is often termed “intuitive physics.” Understanding children's concepts about the physical world is valuable for both practical and theoretical reasons. Practi...

The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space around these fractal trees. We present the canonical and degenerate form of this fractal space and extend the fractal geometrical space to R explicitly and R ...

This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.

We introduce a group theoretical tool on which one can derive a family of identities from sequences that are defined by a recursive relation. As an illustration it is shown that n−1 ∑ i=1 Fn−iF 2 i = 1 2 n ∑ i=1 (−1)(F2i − Fi) = ( Fn+1 2 )

The primary aim of this paper is to (provide tools in order to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of omnipresent differential equations in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.