Analysis Seminar in Crete (2018-19)

Σεμιναριο Αναλυσης

    Department of Mathematics and Applied Math / Fourier and Functional Analysis / Previous years: 2017-18/ 2016-17/ 2015-16/ 2014-15/ 2013-14/ 2012-13/ 2011-12/ 2010-11/ 2009-10/ 2008-09/ 2007-08/ 2006-07/ 2005-06/ 2004-05 / Summer 2004 / 2003-04 / 2002-03 / 2001-02 / 2000-01 / 1999-00

    Analysis Seminars in the World / Analysis seminars in Greece

In chronological ordering

9/7/2018 11:15 B201Agelos GeorgakopoulosUniversity of Warwick Analytic functions in bond percolation
Abstract: We consider Bernoulli bond percolation on various graphs/groups and prove that certain functions are analytic in the percolation parameter $p$. This will be an overview talk; hardly any background will be assumed.
Joint work with C. Panagiotis.
9/19/2018 9:00 B201Speakers Fourier Bases 2018
9/20/2018 9:00 B201Speakers Fourier Bases 2018
9/21/2018 9:00 B201Speakers Fourier Bases 2018
10/30/2018 10:00 A303George CostakisUniversity of Crete Unions of numerical ranges
11/13/2018 10:00 A303Themis MitsisUniversity of Crete Projection theorems for antichains
1/24/2019 11:00 A303Nikos TsirivasUniversity of Ioannina Computation of the acceleration in the uniform circular motion

In physics circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

Examples of circular motion include: an artificial satellite orbiting the Earth at a constant height, a ceiling fan's blades rotating around a hub, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. In this talk we derive the formula of acceleration of the uniform circular motion using direct computation (the argument seems to be not widely known).

The talk will be targeted towards undergraduate and graduate students.

2/12/2019 10:00 A303Vasiliki EvdoridouThe Open University Singularities of inner functions and entire maps of finite order

Let $f$ be a transcendental entire function of finite order and $U$ an unbounded, invariant connected component of the Fatou set of $f$. We can associate an inner function, $g$ say, to the restriction of $f$ to $U$. We show that for two classes of entire functions whose set of singular values is bounded, the number of singularities of $g$ on the unit circle is at most twice the order of $f$.

This is joint work with N. Fagella, X. Jarque and D. Sixsmith.

2/19/2019 10:00 A303Vassili NestoridisUniversity of Athens Mergelyan approximations in several complex variables and a new algebra of functions
3/13/2019 10:00 A303Georgios CostakisUniversity of Crete Dynamics of linear operators and similarity transformations
4/3/2019 10:00 A303Rachel GreenfeldBar Ilan University Fuglede's spectral set conjecture for convex polytopes
Abstract: A set $\Omega \subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. In 1974, B. Fuglede conjectured that spectral sets could be characterized geometrically as sets which can tile the space by translations. Although this conjecture inspired extensive research, the precise connection between the notions of spectrality and tiling is still not clear. In the talk I will survey the subject, and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.
4/17/2019 10:00 A303Yurii LyubarskiiNTNU Norway Exponential Basis in Convex Polygons

Let $D$ be a convex polygon in the plane which is symmetric with respect to the origin. We offer a construction of a Riesz basis in $L^2(D)$ of the form $\{\exp(i\langle x, \lambda \rangle)\}_{\lambda \in \Lambda}$, where $\Lambda$ is a sequence of points in $\mathbb R ^2$. The construction is based on solution of interpolation problem in the corresponding space of entire functions.

This is a joint work with A. Rashkovskii.

5/28/2019 11:00 A303Christos PelekisInstitute of Mathematics, Czech Academy of Sciences A continuous analogue of Erdos' $k$-Sperner theorem
Abstract: A chain in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We show that the $1$-dimensional Hausdorff measure of a chain in the unit $n$-cube is at most $n$, and that the bound is sharp. Given this result, we consider the problem of maximising the $n$-dimensional Lebesgue measure of a measurable set $A\subset [0,1]^n$ subject to the constraint that it satisfies $H^1(A\cap C) \leq \kappa$ for all chains $C\subset [0,1]^n$, where $\kappa$ is a fixed real number from the interval $(0,n]$. We show that the measure of $A$ is not larger than the measure of the following optimal set: \[ A^{\ast}_{\kappa} = \left\{ (x_1,\ldots,x_n)\in [0,1]^n : \frac{n-\kappa}{2}\le \sum_{i=1}^{n}x_i \le \frac{n+ \kappa}{2} \right\} \, . \] Our result may be seen as a continuous counterpart to a theorem of Erdos, regarding $k$-Sperner families of finite sets. Joint work with Themis Mitsis and Vaclav Vlasak
6/12/2019 10:00 A303George AndroulakisUniv. of South Carolina The role of entropy in quantum communications
Abstract: We will review the properties of quantum entropy and indicate its uses in quantum communications.
7/3/2019 11:15 A303Mate MatolcsiRenyi Institute Fuglede's conjecture for convex bodies
Abstract: We prove that a convex body $K$ tiles $\mathbb{R}^n$ if and only if it is spectral, i.e. the space $L^2(K)$ has an orthogonal basis of exponentials. The proof relies on a construction of a so-called "diffraction measure" and a corresponding weak tiling property for $K$. Joint work with Nir Lev.

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