Tamaz Chantladze

The present investigation is stimulated by earlier works of the authors where they studied oscillatory properties of half-linear ordinary differential equations, the so-called p-Laplacian. Here we consider its two-dimensional version. It turns out that the fundamentals of the p-Laplacian theory can be successfully involved in a more general scheme which we call the theory of connected pairs. We see that it contains, on the one hand, the notion of analyticity and, on the other hand, it allows one to apply a very important transformation of analysis, the Legendre transformation, to the investigation of the

p-Laplacian.

Conjugacy and disconjugacy criteria are established for the equation u00 + p(t)u = 0 , where p :] − ∞, +∞[→] − ∞, +∞[ is a locally summable function.

Integral criteria are established for $\dim V_i(p)=0$ and $\dim V_i(p)=1, i\in \lbrace 0,1\rbrace $, where $V_i(p)$ is the space of solutions $u$ of the equation \[ u^{\prime \prime }+p(t)u=0 \] satisfying the condition \[ \int ^{+\infty }\frac{u^2(s)}{s^i}ds<+\infty \,. \]

Using the theory of spline functions the authors investigate the problem of minimization of a generalized Dirichlet integral Fλ(u) = ∫ Ω ( λ 2 + ∑n i=1 u 2 xi )p/2 dσ, 1 < p < ∞, where Ω is a bounded domain of an n-dimensional Euclidean space R n, λ ≥ 0 is a fixed number, and uxi is a generalized according to Sobolev with respect to xi derivative of the function u defined on Ω. Minimization is realized with respect to the functions u whose boundary values on Γ form a preassigned function, and for them Fλ(u) is finite.

A relation between Gaussian complex random vectors and Dirichlet L-series is established.

The authors construct a particular real matrix representation of real Clifford algebras. Shortcomings are that the relation to other realizations is not discussed, that not all real Clifford algebras obey real matrix representations, and that the claim that their realizations generalize to infinite dimensional Clifford algebras is not proven.

Problems of approximative nature are studied for some spaces of real or complex valued functions, and also measures defined on a locally compact Abelian group. 

Points of infinity order for elliptic curves related to θ-congruent numbers are found. A cryptosystem created by reduction of such curves over finite fields is considered. An example illustrating the cryptosystem is given.

The problem of summability of Fourier series of continuous almost periodic on locally compact Abelian groups functions with values in Banach spaces is considered.