Abstract. We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. Generalizing Gel’fond-Mahler inequality for the unit disk and Kneser-Borwein inequality for the segment [−1, 1], we prove an asymptotically sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set in plane. Applying similar techniques, we prod...

For a monomial x in d variables, the problem of best approximation to x by polynomials of lower degrees is studied on the unit sphere, the unit ball and the standard simplex. For the uniform norm we discuss what is known in d = 2, for which complete solutions are known, and in d ≥ 3 for which only a few cases have been successfully solved. For the L norm, we present the complete solution, inclu...

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. Generalizing Gel’fond-Mahler inequality for the unit disk and Kneser-Borwein inequality for the segment [−1, 1], we prove an asymptotically sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set in plane. Applying similar techniques, we produce a rela...

The primary goal of this paper is the study of polynomials with integer coefficients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform nor...

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations with a gravitational force and degenerate viscosity coefficients. Under certain assumptions that imposed on the initial data, we obtain the global existence and uniqueness of the weak solution and give some uniform bounds (with respect to time) of the solution. Moreover, we obtain some...

We propose a simple and intuitive method to derive the exact convergence rate of global L2-norm error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and Müller-Gronbach (2004). We conclude that any strong numerical scheme of order γ > 1/2 has the same optimal convergence rate for this error. The method clearly reveals the...

As explained by Axelrod in his seminal work An Evolutionary Approach to Norms, punishment is a key mechanism to achieve the necessary social control and to impose social norms in a self-regulated society. In this paper, we distinguish between two enforcing mechanisms. i.e. punishment and sanction, focusing on the specific ways in which they favor the emergence and maintenance of cooperation. Th...

• We consider the reaction diffusion problem for singular perturbed case when term dominates equation. Using optimal test norm and SP reformulation, we provide efficient discretization processes. present a preconditioning strategy that works large range of perturbation parameter. introduce simplified Bramble-Pasciak-Vassilevski type preconditioner. ways to discretize precondition in concepts sa...