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Mehmet Celik

Publications and source records attributed to Mehmet Celik.

10 recordsLinked to original sources
arXiv2024

Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them

In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik's conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these products.

math.CV↗
arXiv2021

A Sufficient condition for compactness of Hankel operators

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$. We show that if $φ\in C^{1}(\overlineΩ)$ is holomorphic along analytic varieties in $bΩ$, then $H^{q}_φ$, the Hankel operator with symbol $φ$, is compact. We have shown the converse earlier, so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these symbols are Fredholm (of index zero).

math.CV↗
arXiv2020

Compactness of Hankel operators with continuous symbols on convex domains

Let $Ω$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $ϕ\in C(\barΩ)$. If the Hankel operator $H^{q-1}_ϕ$ on $(0,q-1)$--forms with symbol $ϕ$ is compact, then $ϕ$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $ϕ\in C(\barΩ)$ is analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_ϕ$ is compact.

math.CV↗
arXiv2019

Convex domains, Hankel operators, and maximal estimates

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $(n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $q$ if and only if they hold at all levels.

math.CV↗
arXiv2016

Analysis on the Intersection of Pseudoconvex Domains

In this note, we discuss the preservation of certain analytic properties of the $\overline{\partial}$-Neumann operator, Bergman projection and Hankel operators on the intersection of pseudoconvex domains.

math.CV↗
arXiv2013

Compactness of the dbar-Neumann operator and commutators of the Bergman projection with continuous functions

Let D be a bounded pseudoconvex domain in $C^n, n\geq 2, 0\leq p\leq n,$ and $1\leq q\leq n-1.$ We show that compactness of the dbar-Neumann operator, $N_{p,q+1},$ on square integrable (p,q+1)-forms is equivalent to compactness of the commutators $[P_{p,q}, \bar{z}_j]$ on square integrable dbar-closed (p,q)-forms for $1\leq j\leq n$ where $P_{p,q}$ is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with functions continuous on the closure percolates up in the dbar-complex on dbar-closed forms and square integrable holomorphic forms.

math.CV↗
arXiv2012

Coherent Population Trapping resonances on lower atomic levels of Doppler broadened optical lines

We have detected and analysed narrow high-contrast coherent population trapping (CPT) resonances, which are induced in absorption of the weak probe light beam by the counterpropagating two-frequency pumping radiation. Our experimental investigations have been carried out on example of nonclosed three level Lambda systems formed by spectral components of the Doppler broadened D2 line of cesium atoms. We have established that CPT resonances in transmission of the probe beam (in the cesium vapor), at definite conditions, may have not only more contrast but also much lesser width in comparison with well- known CPT resonances in transmission of the corresponding two-frequency pumping radiation. Thus CPT resonances, detected by the elaborated method, may be used in atomic frequency standards and sensitive magnetometers (based on the CPT phenomenon) and also in ultahigh resolution spectroscopy of atoms and molecules.

physics.atom-ph↗
arXiv2011

On compactness of the dbar-Neumann problem and Hankel operators

Let $\D=\D_1\setminus \Dc_2$, where $\D_1$ and $\D_2$ are two smooth bounded pseudoconvex domains in $\C^n, n\geq 3,$ such that $\Dc_2\subset \D_1.$ Assume that the $\dbar$-Neumann operator of $\D_1$ is compact and the interior of the Levi-flat points in the boundary of $\D_2$ is not empty (in the relative topology). Then we show that the Hankel operator on $\D$ with symbol $ϕ, H^{\D}_ϕ,$ is compact for every $ϕ\in C(\Dc)$ but the $\dbar$-Neumann operator on $\D$ is not compact.

math.CV↗