Nodar Kandelaki

An attempt is made to give a more or less systematic approach to the study of some basic probability concepts related to random vectors with values in a complex Hilbert space. In a natural way, one can associate with such a random vector a pair of random vectors with values in a relevant real Hilbert space, and this yields two possible approaches to describe complex random vectors. These two approaches are actually the same with respect to the measurability problem and the concept of mathematical expectation. The situation is different, however, with regard to problems connected with a covariance operator. The main concept of the paper is a proper random vector the one for which these two approaches give the same results for problems related to covariance operator as well. We study the circle of problems connected with the concept of proper random vectors.

New oscillation and nonoscillation criteria are established for the equation

u″ + p(t)|u|α|u′|1–α sgn u = 0,

where α ∈]0, 1] and the function p :]0, +∞[→] – ∞, +∞[ is locally integrable.

The present investigation is stimulated by the works T. Chantladze, N. Kandelaki, and A. Lomtatidze, On zeros of solutions of a second order singular semilinear equation. Mem. Differential Equations Math. Phys. 17(1999), 127-155. 2. T. Chantladze, N. Kandelaki, and A. Lomtatidze, Oscillation and nonoscillation criteria for a second order equation. Georgian Math. J. 6(1999), No. 5, 401-414. 3. N. Kandelaki, A. Lomtatidze, and D. Ugulava, On oscillation and nonoscillation of a second order half-linear equation. Georgian Math. J. 7(2000), No. 2 in which the authors study oscillatory properties of half-linear ordinary differential equations, of the so-called P -Laplacian. Here we consider its two-dimensional version. Moreover, 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 will 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 the analysis, the Legendre transformation to the investigation of the P -Laplacian.

Using the theory of spline functions, we investigate the problem of minimization of a generalized Dirichlet integral. Key words and phrases:: Dirichlet integral; spline functions; Sobolev space; generalized derivative; l-Laplacian

A sequence of matrices is constructed, these are used to construct representations of a Clifford algebra for special quadratic forms.

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 \[ \int ^{+\infty }\frac{u^2(s)}{s^i}ds<+\infty \,. \]

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