Analysis
Department of Mathematics and Statistics
Texas Tech University
See the colloquium at this link.Let $\varphi$ be an automorphism of the unit disc $\mathbb{D}$ and let $W_{\varphi}^{\alpha}$ be the weighted composition operator, acting on a Banach space of analytic functions in the unit disc, defined by $W_{\varphi}^{\alpha} f= (f \circ \varphi)(\varphi')^{\alpha}$ with $\alpha>0$. We observe that many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces are invariant under these operators. Thus, the main goal of this talk will be to present a general approach to the spaces that satisfy this weighted conformal invariance property. Among other things, we will identify the largest and the smallest as well as the ``unique'' Hilbert space satisfying this property for a given $\alpha >0$. These results are part of a joint work together with Professor Alexandru Aleman.
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Mixed-norm spaces $H(p, q, a)$ are a family of spaces of analytic functions which generalize standard weighted Bergman spaces and, as a limit case, Hardy spaces. They were introduced by Flett, but some of their properties were deduced previously by Hardy and Littlewood. The possible inclusions between them are completely characterized, but the norms of these inclusions are unknown in general. These norms, in some cases, could be of interest for other fields of Mathematics as Mathematical Physics or Number Theory.
In this talk, based on a joint work with D. Vukotic, we will show some conditions under which we are able to prove the contractivity of the inclusion (that is, when the norm of the inclusion operator is exactly 1).
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We study, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.
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We consider the degenerate Einstein’s Brownian motion model for the case when the time interval (τ) of particle Jumps before collision (free jumps) reciprocal to the number of particles per unit volume u(x,t) >0 at the point of observation x at time t. The parameter 0 < τ ≤ C < ∞ , controls characteristic of the fluid "almost decreases" to 0 when u→ ∞. This degeneration leads to the localization of the spread of particle propagation in the media. In our report we will present a structural condition of the time interval of free jumps τ and the frequency of these free jumps φ as functions of u which guarantees the finite speed of propagation of u.
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We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor's conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system. Based on joint work with Tristan Buckmaster and Vlad Vicol.
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We consider sewing machinery between finite difference and analytical solutions defined
at different scale: far away and near source of the perturbation of the flow.
One of the essences of the approach is that coarse problem and boundary value problem
in the proxy of the source model two different flows. We are proposing method to glue
solution via total fluxes, which is predefined on coarse grid. It is important to mention
that the coarse solution "does not see" boundary.
From industrial point of view our report can be considered as a mathematical shirt on
famous Peaceman well-block radius formulae and can be applied in much more general scenario.
This report presents ongoing project with Daniil Anikeev, Ilya Indrupunski and Ernest Zakirov - group of
applied mathematicians from Russian Academy of Science and student Jared Cullingford from Texas Tech university.
We will also formulate several topics/problems, some time in form of conjecture in harmonic analysis relating to Green function and capacity.
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The Boussinesq equations concerned here model buoyancy-driven fluids such
as various atmospheric and oceanographic flows, and Rayleigh-Benard convection.
This talk presents recent stability results on the Boussinesq equations with partial
dissipation near two physically significant steady-states: Couette flow and the
hydrostatic balance. In the case of perturbation near a Couette flow, it is the enhanced
dissipation created by a linear non-self-adjoint operator that makes the nonlinear stability
possible. This is a joint work with Wen Deng and Ping Zhang.
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We extend recently obtained sharp bounds for ratios of zero-balanced hypergeometric
functions to the general $k$-balanced case, $k\in\mathbb{N}$. We also discuss the absolute monotonicity
of generalizations of previously studied functions involving generalized complete elliptic integrals.
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We will discuss several isoperimetric problems
for geometric and functional characteristics of polygons with
$n\ge 3$ sides. I will demonstrate several methods, which can be
used to attack these problems, explain remaining difficulties,
and mention several challenging questions, which remain open for
long time.
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This talk is about a classical problem in complex analysis and geometric function theory: finding geometric conditions for functions to belong in spaces of holomorphic functions. In particular, we will talk about necessary and sufficient conditions for a conformal mapping of the unit disk to belong to Hardy or weighted Bergman spaces by studying the harmonic measure and the hyperbolic metric in the image region. Moreover, we will describe the Hardy number of conformal mappings in terms of the harmonic measure and the hyperbolic distance and give some applications in comb domains.
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