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Title: On convergence of Padé and Hermite — Padé approximants for functions with branch points, Part II Speaker: E. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130
Title: On convergence of Padé and Hermite — Padé approximants for functions with branch points Speaker: E. Rakhmanov Time: 4:00pm–5:00pm Place: CMC 130
I will start with some historic remarks on the above-mentioned approximations. In particular, it is interesting to ask what happen in the theory for a century between Markov-Stieltjes and Stahl's theorems on Padé approximants. Then I plan to discuss chances of obtain a Hermite-Padé analogue of Stahl theorem.
Title: An affirmative answer to a problem of Cater Speaker: Arthur Danielyan Time: 4:00pm–5:00pm Place: CMC 130
The talk is dedicated to the solution of the following problem from the classical real analysis. Does there exist an increasing absolutely continuous function \(f\) on \([0,1]\) such that \(\{x: f'(x)=0\}\) is both countable and dense? This problem was proposed by F. Cater about two decades ago. We give an affirmative answer.
Title: L'Hôpital's Rule for Infinite Determinants, Part II Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
Title: L'Hôpital's Rule for Infinite Determinants Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
Consider the (formal) infinite determinants associated with the spectra of an elliptic differential operator on a compact space and of a perturbation, respectively. Their ratio is a generalization of the perturbation determinant and plays an important role in computing the analytic index of the operator. We explore extensions of this result to the case of the complex spectra.
Title: Dirichlet energy and focusing NLS soliton condensates of minimal intensity Speaker: Alex Tovbis, UCF Time: 4:00pm–5:00pm Place: CMC 130
We consider the family of (poly)continua \(K\) in the upper complex half-plane that contain a preassigned finite anchor set \(E\). For a given harmonic external field, we define a Dirichlet energy functional \(I(K)\) and show that within each “connectivity class” of the family, there exists a minimizing compact \(K^*\) consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential dp associated with the finite gap solutions of the focusing Nonlinear Schröedinger equation (fNLS) defined by a hyperelliptic Riemann surface \(R\) branched at the points \(E\) and \(E^*\).
An fNLS soliton condensate is defined by a compact spectral support \(K\), whereas the average intensity of the condensate is proportional to \(I(K)\) with the external field \(-2\operatorname{Im} z\). The motivation for this work lies in the problem of soliton condensate of least average intensity where the anchor set \(E\) belongs to \(K\). We prove that spectral support \(K^*\) indeed provides the fNLS soliton condensate of the least average intensity within a given “connectivity class”.
Title: On the weak closure of the invertible elements of the disc algebra Speaker: Arthur Danielyan Time: 4:00pm–5:00pm Place: CMC 130
Let \(A\) be the disc algebra and let \(G(A)\) be the group of invertible elements of \(A\). It is an elementary fact that the norm closure of \(G(A)\) coincides with the subset of \(A\) each member of which has no zeros in the open unit disc. In this talk we investigate the weak closure of \(G(A)\).
No seminar this week.
Title: Taboos, smash sums, and ideals: the other side of Anderson localization Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
(Very) recently, Hannah Cairns has proven that the Diaconis-Fulton “smash sum” operation on open sets generating a commutative algebra (graded by cardinality) is essentially unique. The smash sum is known to generate the DLA model in the countable case and quadrature domains otherwise. The proof of uniqueness now provides a new way to characterize the support of superharmonic functions associated to the eigenvectors in Anderson localization.
(No prior knowledge of Anderson localization is assumed.)
Title: Around Bourgain's theorem on the Anderson-Bernoulli localization Speaker: Razvan Teodorescu Time: 4:00pm–5:00pm Place: CMC 130
In 2004, Jean Bourgain proved that Anderson localization (Phil Anderson, 1958) holds in dimensions \(n\) equal to \(1\) and higher for a Bernoulli distribution of disorder and potentials with compact support. This remains the strongest result yet for discrete distributions of disorder, as opposed to the continuous case (Goldsh'ein-Molchanov-Pastur for \(n=1\) and Frohlich-Spencer, Altshuler, and Efetov for \(n>1\)). We consider a deformation of the Anderson-Bernoulli model which allows to “go around” Bourgain's theorem and prove that dimension \(n=2\) is critical for the discrete distribution of disorder, as well.