Department of Mathematical Sciences
http://hdl.handle.net/1805/4424
2019-06-26T04:54:02ZA new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators
http://hdl.handle.net/1805/18546
A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators
Cowen, Carl C.; Gallardo-Gutiérrez, Eva A.
A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D acting on the classical Hardy space H² is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe’s Theorem based on analytic Toeplitz operators.
2017-01-01T00:00:00ZLower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus
http://hdl.handle.net/1805/17874
Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus
Lu, Kang
The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin model associated to the Lie algebras of types B and C. In this paper, we give lower bounds for the numbers of real self-dual spaces in intersections of Schubert varieties related to osculating flags in the Grassmannian. The higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of this form.
2018-01-01T00:00:00ZSzeg\H{o}-type asymptotics for ray sequences of Frobenius-Pad\’e approximants
http://hdl.handle.net/1805/17778
Szeg\H{o}-type asymptotics for ray sequences of Frobenius-Pad\’e approximants
Aptekarev, Alexander I.; Bogolubsky, Alexey I.; Yattselev, Maxim I.
Let $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $\mu$, $\mathrm{supp}(\mu)\cap\mathrm{supp}(\sigma)=\varnothing$. An $(m,n)$-th Frobenius-Pad\'e approximant to $\widehat\sigma$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Pad\'e approximants to $\widehat\sigma$ along ray sequences $\frac n{n+m+1}\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.
2016-01-01T00:00:00ZFarey–Lorenz Permutations for Interval Maps
http://hdl.handle.net/1805/17693
Farey–Lorenz Permutations for Interval Maps
Geller, William; Misiurewicz, Michał
Lorenz-like maps arise in models of neuron activity, among other places. Motivated by questions about the pattern of neuron firing in such a model, we study periodic orbits and their itineraries for Lorenz-like maps with nondegenerate rotation intervals. We characterize such orbits for the simplest such case and gain substantial information about the general case.
2018-02-01T00:00:00Z