Recent zbMATH articles in MSC 30D15https://zbmath.org/atom/cc/30D152023-05-31T16:32:50.898670ZWerkzeugMöbius disjointness for a class of exponential functionshttps://zbmath.org/1508.300552023-05-31T16:32:50.898670Z"Gu, Weichen"https://zbmath.org/authors/?q=ai:gu.weichen"Wei, Fei"https://zbmath.org/authors/?q=ai:wei.feiSummary: A vast class of exponential functions is shown to be deterministic. This class includes functions whose exponents are polynomial-like or `piece-wise' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the Möbius function.Integral representation of one class of entire functionshttps://zbmath.org/1508.300562023-05-31T16:32:50.898670Z"Khats', R. V."https://zbmath.org/authors/?q=ai:khats.r-vSummary: In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class.Analytic functions of infinite order in half-planehttps://zbmath.org/1508.300712023-05-31T16:32:50.898670Z"Malyutin, K. G."https://zbmath.org/authors/?q=ai:malyutin.konstantin-gennadevich"Kabanko, M. V."https://zbmath.org/authors/?q=ai:kabanko.mikhail-vladimirovich"Shevtsova, T. V."https://zbmath.org/authors/?q=ai:shevtsova.tatyana-vasilevna\textit{J. Miles} [Pac. J. Math. 81, 131--157 (1979; Zbl 0371.30024)] proved that if \(f\) is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, a similar result in the space of functions analytic in the upper half-plane is proved. The analytic function \(f\) in \(\mathbb{C}_+=\{z:\Im z>0\}\) is called proper analytic if \(\limsup_{z\to t}\ln|f(z)|\leq 0\) for all real numbers \(t\in\mathbb{R}\). The space of the proper analytic functions is denoted by \(JA\). The full measure of a function \(f\in JA\) is a positive measure, which justifies the term ``proper analytic function''. The order of a function \(f\in JA\) is defined as \(\displaystyle\rho=\limsup\limits_{r\to\infty}\frac{\ln(r m(r,f))}{\ln r}\), where \(\displaystyle m(r,f):=\frac 1r\int\limits_0^{\pi}\ln^+|f(re^{i\varphi})|\sin\varphi\,d\varphi\). If this limit equals infinity, then the order of the function equals infinity. In this case, the function \(f(z)\) is called functions of infinite order. Otherwise, the function \(f(z)\) is called functions of finite order. Accordingly, the lower order of a function \(f\in JA\) is defined as \(\displaystyle\underline{\rho}=\liminf\limits_{r\to\infty}\frac{\ln(r m(r,f))}{\ln r}\).
The main result is the following theorem.
Theorem. Suppose \(f\) is the proper analytic function in half-plane \(\mathbb{C}_+\) of infinite order with zeros restricted to a finite number of rays \(\mathbb{L}_k\) through the origin\,\(:\) \[\displaystyle\mathbb{L}_k=\left\{z:\arg z=e^ {i\theta_k},\quad0<\theta_k<\pi,\>k\in\overline{1,N_0},\>N_0\in\mathbb{N}\right\}\,.\] Then its lower order equals infinity.
Reviewer: Konstantin Malyutin (Kursk)On closed finite gap curves in spaceforms. IIhttps://zbmath.org/1508.530102023-05-31T16:32:50.898670Z"Klein, Sebastian"https://zbmath.org/authors/?q=ai:klein.sebastian"Kilian, Martin"https://zbmath.org/authors/?q=ai:kilian.martinSummary: We prove that the set of closed finite gap curves in hyperbolic 3-space \(\mathbb{H}^3\) is \(W^{2,2}\)-dense in the Sobolev space of all closed \(W^{2,2}\)-curves in \(\mathbb{H}^3\). We also show that the set of closed finite gap curves in any two-dimensional space form \(\mathbb{E}^2\) is \(W^{2,2}\)-dense in the Sobolev space of all closed \(W^{2,2}\)-curves in \(\mathbb{E}^2\).
For Part I, see [the authors, SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 011, 29 p. (2020; Zbl 1475.53025)].The random normal matrix model: insertion of a point chargehttps://zbmath.org/1508.820462023-05-31T16:32:50.898670Z"Ameur, Yacin"https://zbmath.org/authors/?q=ai:ameur.yacin"Kang, Nam-Gyu"https://zbmath.org/authors/?q=ai:kang.nam-gyu"Seo, Seong-Mi"https://zbmath.org/authors/?q=ai:seo.seong-miSummary: In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (``Mittag-Leffler fields'') and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for \(\log |p_n(\zeta )|\) where \(p_n\) is the characteristic polynomial of an \(n\):th order random normal matrix.