Related papers: On a problem by Steklov
We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization…
We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-$N$ and continuum limits, the average Lanczos coefficients and the recursion coefficients become…
A polynomial algorithm is obtained for the NP-complete linear ordering problem.
Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field and $f$ an automorphism of $X$. Suppose that the pullback $f^*|_{\mathsf{N}^1(X)_\mathbf{R}}$ of $f$ on the real N\'eron--Severi space…
In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use…
Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schr\"odinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic…
We establish the Pogorelov type estimates for degenerate prescribed k-curvature equations as well as k-Hessian equations. Furthermore,we investigate the interior C1,1 regularity of the solutions for Dirichlet problems. These techniques also…
We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic…
I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost-every Aleksandrov measure…
The Goldberg-Ostrovskii problem asks whether finite-order solutions of a linear differential equation inherit the property of completely regular growth (c.r.g.) from its coefficients. While Bergweiler's counterexample demonstrated that the…
In this work, a new approach has been developed to obtain numerical solution of linear Volterra type integral equations by obtaining asymptotic approximation to solutions. Using the classical Bernoulli polynomials, a set of orthonormal…
We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see…
We introduce a canonical notion of entropy for polynomials analogue to that of random variables in probability. We prove that entropy increases smoothly with respect to finite free addition. In particular we get the new inequality : $…
In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization.…
The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the…
Markov's inequality for algebraic polynomials on $\left[-1,1\right]$ goes back to more than a century and it is widely used in approximation theory. Its asymptotically sharp form for unions of finitely many intervals has been found only in…
Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The…
This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th…
We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of their spectra on Euclidean annular domains $\mathbb{B}^n_1\setminus \overline{\mathbb{B}^n_\epsilon}$ as…
We investigate a class of estimators of the Markov order for stationary ergodic processes which form a slight modification of the constructions by Merhav, Gutman, and Ziv in 1989 as well as by Ryabko, Astola, and Malyutov in 2006 and 2016.…