Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where $\xi_i$ are i.i.d.\ non-degenerate complex variables, and $\{p_i\}$ is a sequence orthonormal with respect to regular measure $\tau$ supported on compact set $K$. We show that zero $G_n$ converges weakly almost surely equilibrium $K$ if only $\mathbb{E} \log(1 + |\xi_0|) < \infty$. This generalizes corresponding resul...

We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with rigid inclusion. Our analysis is based on boundary integral formulation of elastostatic problem and geometric function theory. Specifically, we use coordinate system provided by exterior conformal mapping inclusion to define density basis functions inclusion, Faber polynomials associated ...

A symbolic computation scheme, based on the Lanczos-method, is proposed for obtaining exact polynomial solution to some perturbed diierential equations with suitable boundary conditions. The automated-method uses symbolic Faber polynomials as the perturbation terms for arbitrary circular section of the complex plane and has advantages of avoiding rounding error and easy manipulation over the nu...

The idea of the basic sets of polynomials of one complex variable appeared in 1930’s by Whittaker [33,34,35] who laid down the definition of a basic sets and their effectiveness. The study of the basic sets of polynomials of several complex variables was initiated by Mursi and Makar [25,26], Nassif [27], Kishka and others [13,14,17,18,19], where the representation in polycylindrical and hypersp...

The long-standing Erd\H{o}s-Faber-Lov\'asz conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using colors. In this paper we propose an algebraic framework to the problem and formulate corresponding stronger conjecture. Using Combinatorial Nullstellensatz, reduce existence of non-zero coefficients in certain polynomials. These are turn related ...

For a compactly supported function <p in Rd we study quasiinterpolants based on point evaluations at the integer lattice. We restrict ourselves to the case where the coefficient sequence Xf, for given data /, is computed by applying a univariate polynomial q to the sequence <p\Zd , and then convolving with the data f\Z(¡ . Such operators appear in the well-known Neumann series formulation of qu...