Analysis
Department of Mathematics and Statistics
Texas Tech University
N/A
In this talk we will study mappings $ f $ preserving harmonic measures of boundary sets. We will show that every homeomorphism $ f $ : $ \overline{D} $ → $ \overline{Ω} $ between Greenian domains $ D $ and $ Ω $ in $ R^n $, $ n $ ≥ 2, preserving harmonic measures, is a harmonic morphism. We will also study problems on conformality of mappings preserving harmonic measures of some specific sets on the boundaries of planar domains. Joint work with S. Pouliasis.In 1967 Bombieri conjectured in very precise terms the behavior of the coefficients of normalized univalent functions close to the Koebe function $K(z)=\frac{z}{(1-z)^2}$. Namely, he proposed that the two real numbers
$$
\sigma_{mn}:=\liminf_{f\to K} \frac{n-{\rm Re\,}a_n}{m-{\rm Re\,}a_m} \qquad \text{and} \qquad B_{mn} := \min_{t\in\mathbb{R}} \, \frac{n\sin t -\sin(nt)}{m\sin t -\sin(mt)}
$$
should be equal for all $m,n\geq2$. Although it is known that $0\leq \sigma_{mn} \leq B_{mn}$ and that $\sigma_{mn} = B_{mn}$ when $K$ is approached only by functions with real coefficients, the Bombieri conjecture has been disproved by Greiner and Roth (2001) in the case $(m,n) = (3,2)$, while disproofs for the points $(2,4), (3,4)$ and $(4,2)$ were then furnished by Prokhorov and Vasil'ev (2005).
Recently, Leung used a second variation formula for the Koebe function to prove that the conjecture is false at the points $(m,2)$ for every $m\geq3$, and, also, at $(m,3)$ for every odd $m\geq5$. Complementing his work we prove that the conjecture is false in many more points $(m,n)$ that lie in some sectors.
We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed. This is joint work with Emine Celik (Sakarya University) and Thinh Kieu (University of North Georgia, Gainesville Campus).We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed. This is joint work with Emine Celik (Sakarya University) and Thinh Kieu (University of North Georgia, Gainesville Campus).In this talk, we shall introduce the multiple zeta values (Euler-Zagier sums) from an analytic perspective. Mostly, we are interested
in exploring the $\mathbb{Q}$-linear relations among these values (MZV's). Also, we present the proof of some of the evaluations
which go back to Hoffman and Zagier such as $\zeta(2, 2, \ldots, 2)$ or $\zeta(1, 3, \ldots, 1,3)$. This talk should be accessible to
graduate students and non-specialists.
In this second talk, we explain in more details the techniques that appear in Zagier's famous paper
(https://annals.math.princeton.edu/
wp-content/uploads/annals-v175-n2-p11-p.pdf)
on the evaluation of the multiple zeta values
$$H(r, s)=\zeta(\underbrace{2, 2, \ldots, 2}_{r}, 3, \underbrace{2, 2, \ldots, 2}_{s} )$$
as a $\mathbb{Q}$-linear combination of $\pi^{2m}\zeta(2n+1)$ with $m+n=r+s+1$.
The formula is derived indirectly by considering two generating functions which at first, can be expressed in two completely different quantities, but in the
end, it turns out that they are equal!
Zagier's proof uses a combination of techniques involving
${}_{3}F_{2} $-hypergeometric function as well as products of sine and digamma
functions. Moreover, we also give an account into a direct proof using rational zeta series involving the coefficient $\zeta(2n)$.First we shall present some basic facts about condenser capacity,
analytic capacity and holomorphic motions.
Then we will discuss superharmonicity properties of those
capacities under holomorphic motions and, given a condenser, we will
describe the holomorphic motions for which harmonicity occurs. Finally, we will examine a
special holomorphic motion and characterize the condensers for which
harmonicity occurs.
We establish the existence of global pathwise solutions for the stochastic non-Newtonian incompressible fluid equations in two space dimensions. Moreover, we show that said solutions converge in probability to solutions of the stochastic Navier-Stokes equations in the appropriate limit.We are interested in classifying various types of canonical domains which are conformally equivalent to any domain arbitrarily given in the complex plane. Among the techniques of conformal uniformization, extremum problems play a significant role. The central object of the theory is that univalent analytic functions in a given domain form a normal family which guarantees the existence of an extremal element for a reasonable extremum problem. We give an extremal problem which uniformized a given finitely connected subdomain $\Omega$ in $z$-sphere with $\infty \in \Omega $ onto a domain bounded by rectangles with sides parallel to real and imaginary axis having the conformal module $m$.The purpose of this talk is to give one example of a regularizing property of noise in a partial differential equation. The Hall-magnetohydrodynamic system forced by Levy noise is just one example.
Global well-posedness of a partial differential equation in a certain Banach space informally implies particularly that given any initial data in that Banach space, there exists a unique solution in that Banach space for all time. When a partial differential equation is not known to be globally well-posed, the second best result may be the ``small initial data result;'' i.e. there exists a positive constant such that for any initial data in that Banach space with its norm less than this constant, the unique solution exists for all time. It is well known that for non-diffusive equations in fluid mechanics, such a ``small initial data'' is very difficult to obtain. In particular, obtaining such a ``small initial data'' result for deterministic Euler equations is a completely open problem, to the best of the speaker's knowledge. Nevertheless, if you force with a certain noise, then for such non-diffusive equations, including the Euler equations, the ``small initial data results'' are attainable. Therefore, a noise (random forcing term) can possibly display a regularizing property for its solution. In this talk I will explain the idea behind this phenomenon.
Stationary surfaces with boundaries frequently arise as the critical points of energy functionals which depend on the curvature. For example, the area functional gives rise to minimal surfaces while the Willmore functional corresponds to Willmore surfaces. In this talk, we study various boundary-value problems and characterize free-boundary stationary surfaces with rotational symmetry in case the functional is scaling invariant. We also discuss interesting boundary-to-interior effects in case the functional is not scaling invariant. This is a joint work with Anthony Gruber and Magdalena Toda.