Recent zbMATH articles in MSC 30https://www.zbmath.org/atom/cc/302021-02-12T15:23:00+00:00WerkzeugBoundedness of \(l\)-index and completely regular growth of entire functions.https://www.zbmath.org/1452.300142021-02-12T15:23:00+00:00"Bandura, A. I."https://www.zbmath.org/authors/?q=ai:bandura.a-i"Skaskiv, O. B."https://www.zbmath.org/authors/?q=ai:skaskiv.oleh-bohdanovych|skaskiv.oleg-bSummary: We study the relationship between the class of entire functions of completely regular growth of order \(\rho\) and the class of entire functions with bounded \(l\)-index, where \(l(z) = |z|^{ \rho -1} + 1\) for \(|z| \geq 1\). Possible applications of these functions in the analytic theory of differential equations are considered. We formulate three new problems on the existence of functions with given properties that belong to the differences of these classes. For the fourth problem, we obtain an affirmative answer, namely, we present sufficient conditions for an infinite product to be an entire function of completely regular growth of order \(\rho\) with unbounded \(l_\rho \)-index whose zeros do not satisfy the well-known Levin conditions (C) and (C'). We also construct an entire function of completely regular growth of order \(\rho\) with unbounded \(l_\rho \)-index whose zeros do not satisfy the Levin conditions (C) and (C').Refinement of Pellet radii for matrix polynomials.https://www.zbmath.org/1452.300072021-02-12T15:23:00+00:00"Melman, A."https://www.zbmath.org/authors/?q=ai:melman.aaronSummary: We derive a new family of polynomial multipliers that improve the Pellet radii of matrix polynomials, which determine an annulus in the complex plane devoid of eigenvalues, and that generalize and improve previously obtained results, while relaxing the conditions under which these hold. As a special limit case we also obtain an improvement of the Cauchy radius, which is an upper bound on the moduli of the eigenvalues. Results for the special case of scalar polynomials are presented.A note on the rigidity of convergence of Möbius transformations.https://www.zbmath.org/1452.300222021-02-12T15:23:00+00:00"Lascurain, Antonio"https://www.zbmath.org/authors/?q=ai:lascurain.antonio"Nicolás, Francisco"https://www.zbmath.org/authors/?q=ai:nicolas.franciscoSummary: It is proved using elementary techniques that if a given sequence of Möbius transformations acting in \(\widehat{\mathbb{R}}^n\) converges pointwise, and for three different points in \(\widehat{\mathbb{R}}^n\) the sequence converges to three different points in \(\widehat{\mathbb{R}}^n\), one has that the sequence converges to a Möbius transformation (Theorem 3.1). Moreover, it is proved that the number three is the best lower bound.
For the entire collection see [Zbl 1429.53001].Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree.https://www.zbmath.org/1452.320342021-02-12T15:23:00+00:00"Brotbek, Damian"https://www.zbmath.org/authors/?q=ai:brotbek.damian"Deng, Ya"https://www.zbmath.org/authors/?q=ai:deng.yaIn [Hyperbolic manifolds and holomorphic mappings. New York, NY: Marcel Dekker, Inc. (1970; Zbl 0207.37902)] \textit{S. Kobayashi} made the following famous conjecture, which is often called the logarithmic Kobayashi conjecture in the literature.
Conjecture (Kobayashi). The complement \(\mathbb{P}^n \backslash D\) of a general hypersurface \(D \subset \mathbb{P}^n\) of sufficiently large degree \(d \geq d_n\) is Kobayashi hyperbolic.
In this paper, the authors prove that in any projective manifold, the complements of general hypersurfaces of sufficiently
large degree, are Kobayashi hyperbolic. The proof also provides an effective lower bound on the degree. The naturality of the results allows us to enunciate the following quite clear version.
Main Theorem. Let \(Y\) be a smooth complex projective variety having dimension at least 2. Fix any very ample line bundle \(A\) on \(Y\). Then for a general smooth hypersurface \(D \in |A^d |\) with
\[
d \geq (n + 2)^{n+3} (n + 1)^{n+3}\sim_{ n \to \infty}e^3 n^{2n+6},
\]
the following assertions hold.
(i) The complement \(Y \backslash D\) is hyperbolically embedded into \(Y\). In particular, \(Y \backslash D\) is Kobayashi hyperbolic.
(ii) For any holomorphic entire curve (possibly algebraically degenerate) \(f : \mathbb{C} \longrightarrow Y\) which is not contained in \(D\), one has
\[
T_f (r, A)\leq N_f^{(1)} (r, D) + C (\log T_f (r, A) + \log r) \vert \vert
.\]
Here \(T_f (r, A)\) is the Nevanlinna order function, \(N_f^{(1)} (r, D)\) is the truncated counting function, and the symbol \(\vert \vert\) means that the inequality holds outside a Borel subset of \((1, + \infty)\) of finite Lebesgue measure.
(iii) The (Campana) orbifold \((Y, (1 - \frac{1}{d})D)\) is orbifold hyperbolic, i.e., there exists no entire curve \(f : \mathbb{C} \longrightarrow Y\) so that
\[
f(\mathbb{C})\not\subset D\quad\text{with}\quad\text{mult}_t (f^* D) \ge d\quad \text{for all}\quad t \in f^{- 1} (D).
\]
(iv) Let \(\pi : X \longrightarrow Y\) be the cyclic cover of \(Y\) obtained by taking the \(d\)-th root along \(D\). Then \(X\) is Kobayashi hyperbolic.
The proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies on the construction of higher order logarithmic connections,
allowing the construction logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials, that the authors are able to construct. See also [\textit{D. Brotbek}, Publ. Math., Inst. Hautes Étud. Sci. 126, 1--34 (2017; Zbl 06827883)] and [\textit{Y. Deng}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 3, 787--814
(2020; Zbl 1447.32042)].
Compare the result (iii), about the orbifold hyperbolicity for generic geometric orbifolds, with [\textit{F. Campana} et al., ``Orbifold hyperbolicity'', Preprint, \url{arXiv:1803.10716}].
A result related with (iv) is due to [\textit{X. Roulleau} et al., J. Lond. Math. Soc., II. Ser. 87, No. 2, 453--477 (2013; Zbl 1276.14053)].
As far as we known, the optimatility of the bound in (i) remains as an open problem. Certainly, the main theorem in the present article is a genuine constribution to the study of the Kobayashi hyperbolicity property.
Reviewer: Jesus Muciño Raymundo (Morelia)A Dirichlet problem in noncommutative potential theory.https://www.zbmath.org/1452.300212021-02-12T15:23:00+00:00"Wu, Kuang-Ru"https://www.zbmath.org/authors/?q=ai:wu.kuang-ruSummary: We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.On the exceptional set of transcendental functions with integer coefficients in a prescribed set: the problems A and C of Mahler.https://www.zbmath.org/1452.110862021-02-12T15:23:00+00:00"Marques, Diego"https://www.zbmath.org/authors/?q=ai:marques.diego"Moreira, Carlos Gustavo"https://www.zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-aA \textit{transcendental function} is a function \( f(x) \) such that the only complex polynomial \( P \) satisfying \( P(x, f(x)) =0 \), for all \( x \) in its domain, is the zero polynomial. Trigonometric functions, the exponential function, and their inverses are some of the examples of transcendental functions. Denote by \( \bar{\mathbb{Q}} \) the field of algebraic numbers. For a function \( f \) analytic in the complex domain \( \mathcal{D} \), define the exceptional set \( S_f \) of \( f \) as
\(\displaystyle{S_f=\left\{ \alpha \in \bar{\mathbb{Q}}\cap \mathcal{D}: f(\alpha) \in \bar{\mathbb{Q}} \right\}}\). For example, the exceptional sets of the functions \( 2^{z} \) and \( e^{z\pi+1} \) are \( \mathbb{Q} \) and \( \emptyset \), respectively, as shown by the Gelfond-Schneider theorem and Baker's theorem.
In the paper under review, the authors consider Problem A and Problem C in the book of \textit{K. Mahler} [Lectures of transcendental numbers (1976; Zbl 0332.10019)], who suggested three problems, which he named Problem A, B and C, on the arithmetic behaviour of transcendental functions. Problems B and C have been completely solved by the authors in [Math. Ann. 368, No. 3--4, 1059--1062 (2017; Zbl 1387.11056); Bull. Aust. Math. Soc. 98, No. 1, 60--63 (2018; Zbl 1422.11162)], but Problem A remains open in general. Recall that, as usual, \( \mathbb{Z}{\{z\}} \) denotes the set of the power series analytic in the unit ball \( B(0,1) \) and with integer coefficients. Problems A and C are stated as follows.
\begin{itemize}
\item[A.] Does there exist a transcendental function \( f\in \mathbb{Z}{\{z\}} \) with bounded coefficients and such that \( f(\bar{\mathbb{Q}}\cap B(0,1)) \subseteq \bar{\mathbb{Q}} \)?
\item[C.] Does there exist for every choice of \( S \) (closed under complex conjugation and such that \( 0\in S \)) a transcendental entire function with rational coefficients for which \( S_f=S \)?
\end{itemize}
In this paper, the authors generalize the main result of \textit{J. Haung}, et al. [Bull. Aust. Math. Soc. 82, No. 2, 322--327 (2010; Zbl 1204.11113)]. As a consequence, the authors improve their main result in [Acta Arith. 192, No. 4, 313--327 (2020; Zbl 1450.11078)] as well as providing a variant version of Problem A (for coefficients belonging to some zero asymptotic density sets). Recall that an \( n \)-smooth integer is an integer (possibly negative) whose prime factors are all less than or equal to \( n \). The main result in this paper is the following.
Theorem. Let \( A \) be a countable subset of \( B(0,1) \) which is closed under complex conjugation. For each \( \alpha \in A \), fix a dense subset \( E_\alpha \subseteq \mathbb{C} \) (such that \( 0\in A \) , then \( 1\in E_0 \), \( E_\alpha \) is dense in \( \mathbb{R} \) whenever \( \alpha \in \mathbb{R} \), and such that \(\bar{E_\alpha} = E_{\bar{\alpha}}\), for all \( \alpha\in A \)). Then there exist uncountably many transcendental functions \( \displaystyle{f(z)=\sum_{n\ge 0}a_nz^{n} \in \mathbb{Z}\{z\}} \), such that \( a_n \) is a \( 3 \)-smooth number (for all \( n\ge 0 \)) and \( f(\alpha) \in E_\alpha \), for all \( \alpha \in A \).
Reviewer: Mahadi Ddamulira (Saarbrücken)A panorama of positivity. II: Fixed dimension.https://www.zbmath.org/1452.150212021-02-12T15:23:00+00:00"Belton, Alexander"https://www.zbmath.org/authors/?q=ai:belton.alexander-c-r"Guillot, Dominique"https://www.zbmath.org/authors/?q=ai:guillot.dominique"Khare, Apoorva"https://www.zbmath.org/authors/?q=ai:khare.apoorva"Putinar, Mihai"https://www.zbmath.org/authors/?q=ai:putinar.mihaiThis second part of the survey on positivity presents ``a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions).'' A special emphasis is given to operations which preserve positivity. Many techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are used. With comments and full bibliographical references some partially forgotten classical roots in metric geometry and distance transforms are reminded. Some modern applications to high-dimensional covariance estimation and regularisation are added.
For Part I see [the authors, in: Analysis of operators on function spaces. The Serguei Shimorin memorial volume. Including extended versions of lectures of the conference in honor of the memory of Serguei Shimorin at the Mittag-Leffler Institute, Stockholm, Sweden in the summer of 2018. Cham: Springer. 117--165 (2019; Zbl 07121791)].
For the entire collection see [Zbl 1444.30001].
Reviewer: Mihail Voicu (Iaşi)Conservation of module and the product of modules of foliations.https://www.zbmath.org/1452.580012021-02-12T15:23:00+00:00"Kaźmierczak, Anna"https://www.zbmath.org/authors/?q=ai:kazmierczak.annaThe author considers a diffeomorphism \(G\) between two Riemannian manifolds and examines the link between the structure of its Jacobian and the question whether \(G\) conserves the module of a single foliation or the product of the modules of a system of foliations. Sufficient conditions for the product of the modules of more than two foliations to be equal to \(1\) are given in Section 4. Other author paper directly connected to this subject is [\textit{A. Kaźmierczak} and \textit{A. Pierzchalski}, Balkan J. Geom. Appl. 21, No. 1, 51--57 (2016; Zbl 1354.53039)].
Reviewer: Dorin Andrica (Riyadh)Variational integral and some inequalities of a class of quasilinear elliptic system.https://www.zbmath.org/1452.300242021-02-12T15:23:00+00:00"Lu, Yueming"https://www.zbmath.org/authors/?q=ai:lu.yu"Lian, Pan"https://www.zbmath.org/authors/?q=ai:lian.panSummary: This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincaré inequalities for the case \(q<1\).A criterion for the sequence of roots of holomorphic function with restrictions on its growth.https://www.zbmath.org/1452.300052021-02-12T15:23:00+00:00"Menshikova, E. B."https://www.zbmath.org/authors/?q=ai:menshikova.e-b"Khabibullin, B. N."https://www.zbmath.org/authors/?q=ai:khabibullin.b-nSummary: The main result of the paper is a criterion for a sequence of points in a domain of the complex plane, giving necessary and sufficient conditions under which this sequence of points is an exact sequence of zeros of some holomorphic function whose logarithm of modulus is majored by a given subharmonic function in the domain under consideration. Our criterion for the distribution of zeros of holomorphic functions with a given majorant is formulated in terms of special integral estimates and uses a new notion we recently introduced of affine balayage of measures. In one of our previous joint communications this criterion was announced without any proof. Here we fill this gap and give a criterion with exact definitions and a complete proof.Jacobi-type continued fractions and congruences for binomial coefficients.https://www.zbmath.org/1452.110112021-02-12T15:23:00+00:00"Schmidt, Maxie D."https://www.zbmath.org/authors/?q=ai:schmidt.maxie-dNew properties and congruence relations satisfied by the integer-order binomial coefficients through two specifics, and new Jacobi-type continued fraction expansions of a formal power series in z studied in the article. New continued fraction results lead to new exact formulas and finite difference equations for binomial coefficient variants, and new congruences for the binomial coefficients modulo any (prime or composite) integers \(h \geq 2 \). Main theorems of the article include new exact formulas for the binomial coefficients and new congruence properties for binomial coefficient variants.
Reviewer: Michael M. Pahirya (Mukachevo)Divergence and parallelism of cylindrical stretch lines.https://www.zbmath.org/1452.300232021-02-12T15:23:00+00:00"Théret, Guillaume"https://www.zbmath.org/authors/?q=ai:theret.guillaumeSummary: A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.The Hornich space and a subspace of BMOA.https://www.zbmath.org/1452.300312021-02-12T15:23:00+00:00"Cima, Joseph A."https://www.zbmath.org/authors/?q=ai:cima.joseph-aThe paper introduces a linear isomorphism between the so-called Hornich space and a subspace of BMOA on the unit disc. As a consequence of this isomorphism, some classes of functions in BMOA are identified that have not been studied before.
Reviewer: Kehe Zhu (Albany)Some aspects of summability.https://www.zbmath.org/1452.400012021-02-12T15:23:00+00:00"Mozo Fernández, Jorge"https://www.zbmath.org/authors/?q=ai:mozo-fernandez.jorgeSummary: This text contains a review of classical summability theory for holomorphic functions in one variable, and a short introduction to the theory of monomial summability in two complex variables, with some applications.
For the entire collection see [Zbl 1436.35004].The Gauss-Lucas theorem.https://www.zbmath.org/1452.300032021-02-12T15:23:00+00:00"Gasull, Armengol"https://www.zbmath.org/authors/?q=ai:gasull.armengol(no abstract)Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions.https://www.zbmath.org/1452.460572021-02-12T15:23:00+00:00"Chicourrat, Monique"https://www.zbmath.org/authors/?q=ai:chicourrat.monique"Diarra, Bertin"https://www.zbmath.org/authors/?q=ai:diarra.bertin"Escassut, Alain"https://www.zbmath.org/authors/?q=ai:escassut.alainThe article is devoted to maximal ideals $M$ of analytic function algebras over a perfect field~$F$. Semi-admissible and semi-compatible algebras $U$ are studied. Conditions are found under which $M$ is of codimension~$ 1$ over~$ F$.
Reviewer: Sergey Ludkovsky (Moskva)Spectra of algebras of symmetric and subsymmetric analytic functions.https://www.zbmath.org/1452.460412021-02-12T15:23:00+00:00"Zagorodnyuk, A. V."https://www.zbmath.org/authors/?q=ai:zagorodnyuk.andriy-v"Chernega, I. V."https://www.zbmath.org/authors/?q=ai:chernega.i-vSummary: Algebras of symmetric and subsymmetric analytic functions of bounded type on spaces \(L_1[0,\infty)\cap L_{\infty}[0,\infty)\) and \(L_{\infty}[0,1]\) and their spectra are investigated.The geometry of the space of BPS vortex-antivortex pairs.https://www.zbmath.org/1452.300062021-02-12T15:23:00+00:00"Romão, N. M."https://www.zbmath.org/authors/?q=ai:romao.nuno-m"Speight, J. M."https://www.zbmath.org/authors/?q=ai:speight.james-martinSummary: The gauged sigma model with target \({\mathbb{P}}^1\), defined on a Riemann surface \(\Sigma \), supports static solutions in which \(k_+\) vortices coexist in stable equilibrium with \(k_-\) antivortices. Their moduli space is a noncompact complex manifold \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\) of dimension \(k_++k_-\) which inherits a natural Kähler metric \(g_{L^2}\) governing the model's low energy dynamics. This paper presents the first detailed study of \(g_{L^2}\), focussing on the geometry close to the boundary divisor \(D=\partial \, {\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). On \(\Sigma =S^2\), rigorous estimates of \(g_{L^2}\) close to \(D\) are obtained which imply that \({\mathsf{M}}_{(1,1)}(S^2)\) has finite volume and is geodesically incomplete. On \(\Sigma ={\mathbb{R}}^2\), careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for \(g_{L^2}\) in the limits of small and large separation. All these results make use of a localization formula, expressing \(g_{L^2}\) in terms of data at the (anti)vortex positions, which is established for general \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). For arbitrary compact \(\Sigma \), a natural compactification of the space \({{\mathsf{M}}}_{(k_+,k_-)}(\Sigma)\) is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for \(\text{Vol}(\mathsf{M}_{(1,1)}(S^2))\), and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of \(\Sigma \), and that the entropy of mixing is always positive.The strata do not contain complete varieties.https://www.zbmath.org/1452.140232021-02-12T15:23:00+00:00"Gendron, Quentin"https://www.zbmath.org/authors/?q=ai:gendron.quentinThe moduli space of holomorphic differentials on Riemann surfaces with prescribed numbers of zeros and multiplicities is called a stratum. In this paper the author shows that any stratum of holomorphic differentials does not contain complete algebraic curves (i.e. no compact Riemann surfaces can be embedded in the stratum algebraically). The proof applies the maximum modulus principle in a cute way to the shortest saddle connections joining zeros of the differentials under the induced flat metric. It remains an open question to determine whether the projectivized strata (i.e. parameterizing the underlying effective canonical divisors) can contain a complete algebraic curve or not. Note that if one considers the strata of strictly meromorphic differentials or non-effective canonical divisors, then they do not contain any complete algebraic curve [\textit{D. Chen}, J. Inst. Math. Jussieu 18, No. 6, 1331--1340 (2019; Zbl 1423.14184)].
Reviewer: Dawei Chen (Chestnut Hill)Radial distribution of Julia sets of entire solutions to complex difference equations.https://www.zbmath.org/1452.300182021-02-12T15:23:00+00:00"Chen, Jinchao"https://www.zbmath.org/authors/?q=ai:chen.jinchao"Li, Yezhou"https://www.zbmath.org/authors/?q=ai:li.yezhou"Wu, Chengfa"https://www.zbmath.org/authors/?q=ai:wu.chengfaSummary: In this paper, entire solutions \(f\) of a class of nonlinear difference equations are studied. By considering the order and deficiency of the coefficients in the equations, we investigate the properties of the radial distribution of the Julia set of \(f\), and estimate the lower bound of the measure of the set defined by the common limiting directions of Julia sets of shifts of \(f\).On \(p\)-valent analytic functions defined by Noor integral operator.https://www.zbmath.org/1452.300102021-02-12T15:23:00+00:00"Noor, Khalida Inayat"https://www.zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: Let \(A(p)\) denote the class of functions \(f:f(z)=z^p+ \sum^\infty_{m=2}a_{p+m}z^{p+m}\), which are analytic in the unit disc \(E=\{z:|z|<1\}\). For \(n\in N_0=\{0,1,2,\dots\}\), Noor integral operator \(I_{n+p-1}:A(p)\to A(p)\) is defined as \(I_{n+p-1}f=f^{-1}_{n+p-1}*f\) such that \[(f^{-1}_{n+p-1}*f_{n+p-1})=\frac{z^p} {(1-z)^p},\] where \(f_{n+p-1}(z)=\frac{z^p}{(1-z)^{n+p}}\), \(n>-p\), and \(*\) denotes convolution. Using this operator, we define the new classes \(T_p(n,k)\), \(n\ne N_0\) and \(k\ge 2\), which contain the well known Kaplan class of close-to-convex univalent functions \(T_1(1,2)\). Some interesting properties such as inclusion results and invariance under convex convolution are studied for these classes.A convolution property of univalent harmonic right half-plane mappings.https://www.zbmath.org/1452.310012021-02-12T15:23:00+00:00"Ali, Md Firoz"https://www.zbmath.org/authors/?q=ai:ali.md-firoz"Allu, Vasudevarao"https://www.zbmath.org/authors/?q=ai:allu.vasudevarao"Ghosh, Nirupam"https://www.zbmath.org/authors/?q=ai:ghosh.nirupamThe authors discuss the convolution of right half-plane harmonic mappings in the unit disk with respective dilatations \(e^{i\alpha}\frac{z+a}{1+az}\) and \(-z\), where \(-1 < a < 1\) and \(\alpha\in {\mathbb R}\). It is obtained that under certain conditions, such convolutions are locally univalent and convex in the horizontal direction.
Reviewer: Marius Ghergu (Dublin)An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces.https://www.zbmath.org/1452.300202021-02-12T15:23:00+00:00"Falcó, Javier"https://www.zbmath.org/authors/?q=ai:falco.javier"Gauthier, Paul M."https://www.zbmath.org/authors/?q=ai:gauthier.paul-mMotivated by the classical Dirichlet problem, the authors prove that for every open set \(U\) in an arbitrary Riemann surface and every continuous function \(\varphi\) on the boundary \(\partial U\), there exists a holomorphic function \(\widetilde{\varphi}\) on \(U\) such that \(\forall p\in\partial U, \ \widetilde{\varphi}(x)\xrightarrow{} \varphi(p)\), as \(x \to p\) outside a set of density 0 at \(p \) relative to \(U\).
First they slighty extend a result of \textit{A. Boivin} [Math. Ann. 275, 57--70 (1986; Zbl 0583.30035)] which characterizes the sets of tangential approximation in an open Riemann surface and then they prove that every chaplet \(E\) of an open Riemann surface \(M\) is a set of tangential approximation. The latter result plays a central role to obtain their goal.
Finally, they also present the following interesting approximation result, where the approximation function has interpolation and Picard-type properties: Let \(M\) be an open Riemann surface, let \(E\) be a chaplet in an open subset \(U\subset M\) and let \((x_n)_n\) be a sequence of distinct points in \( U \smallsetminus E\) with no accumulation points in \(U\). Then for every function \( f\in A(E)\), every sequence \((y_n)_n\subset\mathbb{C}\) and every positive and continuous function \(\varepsilon\) on \(E\), there exists a function \(g\) holomorphic in \(U\) such that \( g(x_n) = y_n, \ \forall n\in\mathbb{N}\) and \(|g(x)-f(x)| <\varepsilon (x), \ \forall x\in E\). Furthermore, g can be chosen so that for every \(p\in\partial U\) and for every complex number \(\beta\) there exists a sequence \((a_{j})_j\) in \(U\smallsetminus E\) with \(a_j\to p\), when \(j\) goes to infinity and \(g(a_j) = \beta\) for each natural number \(j\).
Reviewer: Vagia Vlachou (Patras)The measure transition problem for meromorphic polar functions.https://www.zbmath.org/1452.300162021-02-12T15:23:00+00:00"Buescu, J."https://www.zbmath.org/authors/?q=ai:buescu.jorge"Paixão, A. C."https://www.zbmath.org/authors/?q=ai:paixao.a-cSummary: In very general conditions, meromorphic polar functions (i.e. functions exhibiting some kind of positive or co-positive definiteness) separate the complex plane into horizontal or vertical strips of holomophy and polarity, in each of which they are characterized as integral transforms of exponentially finite measures. These measures characterize both the function and the strip. We study the problem of transition between different holomorphy strips, proving a transition formula which relates the measures on neighbouring strips of polarity. The general transition problem is further complicated by the fact that a function may lose polarity upon strip crossing and in general we cannot expect polarity, or even some specific related form of integral representation, to exist. We show that, even in these cases, a relevant analytical role will be played by exponentially finite signed measures, which we construct and study. Applications to especially significant examples like the \(\Gamma\), \(\zeta\) or Bessel functions are performed.Rational approximation and Sobolev-type orthogonality.https://www.zbmath.org/1452.410062021-02-12T15:23:00+00:00"Díaz-González, Abel"https://www.zbmath.org/authors/?q=ai:diaz-gonzalez.abel"Pijeira-Cabrera, Héctor"https://www.zbmath.org/authors/?q=ai:pijeira-cabrera.hector"Pérez-Yzquierdo, Ignacio"https://www.zbmath.org/authors/?q=ai:perez-yzquierdo.ignacioIn this contribution the authors deal with sequences \(\{S_{n}(x)\}_{n\geq0}\) of polynomials orthogonal with respect to a Sobolev-type inner product
\[\langle f, g \rangle = \int_{-1}^{1} f(x) g(x) d\mu(x) + \sum_{j=1}^{N} \sum _{i=0}^{d_{j}} M_{j,i} f^{(i)} (c_{j}) g^{(i)} (c_{j}).\]
Here, \(\mu\) is a measure in the Nevai class \(M(0,1)\) (see [\textit{P. G. Nevai}, Orthogonal polynomials. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0405.33009)]), \(M_{j,i} \geq 0, i=0, 1, \cdots, d_{j}-1, M_{j,d_{j}}>0,\) and \(c_{j}, j=1, 2, \cdots, N\) are real numbers with \(|c_{j}| >1, j=1, 2, \cdots, N.\)
If we assume the above Sobolev-type inner product is sequentially-ordered, i. e. \(M_{j, i}=0, i=0, 1, \cdots, d_{j}-1, j= 1, 2, \cdots, N,\) then the polynomial \(S_{n}(z)\) has at least \(n-N\) changes of sign in \((-1,1).\) Moreover, for \(n\) large enough all its zeros are real, simple and each sufficiently small neighborhood of \(c_{j}, j= 1, 2, \cdots,N\) contains exactly one zero of \(S_{n}\) and the remaining zeros lie on \((-1,1).\) This last result is a direct consequence of the outer relative asymptotics for Sobolev-type orthogonal polynomials given in [\textit{G. López Lagomasino} et al., Constr. Approx. 11, No. 1, 107--137 (1995; Zbl 0840.42017)].
Let denote by \(\{Q_{n}(x)\}_{n\geq0}\) the sequence of orthogonal polynomials with respect to the measure \(\rho(x) d \mu(x),\) where \(\rho(x)= \prod_{c_{j} <-1} (z-c_{j})^{d_{j} +1} \prod_{c_{j} >1} (c_{j}-z)^{d_{j} +1}\)is a positive polynomial on \((-1,1).\) The authors introduce the sequence of polynomials \(\{S_{n}^{[k]} (z) \} _{n\geq0}\) defined as
\[
S_{n}^{[k]} (z)= \int_{-1}^{1} \frac{S_{n+k}(z)- S_{n+k}(x)}{z-x} Q_{k-1}(x) \rho(x) d\mu(x).\]
Notice that they constitute a natural extension of the so called associated polynomials of \(k\)th kind in the theory of standard orthogonal polynomials (see [\textit{W. Van Assche}, J. Comput. Appl. Math. 37, No. 1--3, 237--249 (1991; Zbl 0744.42012)]).
The following extended Markov's theorem is proved. Let consider a sequentially ordered discrete Sobolev inner product and let \(\mu\) be a measure in the Nevai class \(M(0,1).\) Then, for \(k \in \mathbb{N}\) the sequence of rational functions \(\{\frac{S_{n}^{[k]}(z)} {S_{n+k} (z)}\}_{n\geq0}\) uniformly converges to the function \(\int_{-1}^{1} \frac {Q_{k-1}(x)}{z-x} \rho(x) d\mu(x)\) in every compact subset of the exterior of \([-1,1] \bigcup \{c_{1}, c_{2}, \cdots, c_{N}\}\) in the complex plane.
The function on the right hand side is said to be the \(k\)th Markov type function associated with the measure \(\rho(x) d\mu(x).\) On the other hand, an estimate of the degree of convergence for the sequence of rational functions in the left hand side is deduced.
Reviewer: Francisco Marcellán (Leganes)Asymptotic mean value Laplacian in metric measure spaces.https://www.zbmath.org/1452.300342021-02-12T15:23:00+00:00"Minne, Andreas"https://www.zbmath.org/authors/?q=ai:minne.andreas"Tewodrose, David"https://www.zbmath.org/authors/?q=ai:tewodrose.davidSummary: We use the mean value property in an asymptotic way to provide a notion of a pointwise Laplacian, called AMV Laplacian, that we study in several contexts including the Heisenberg group and weighted Lebesgue measures. We focus especially on a class of metric measure spaces including intersecting submanifolds of \(\mathbb{R}^n\), a context in which our notion brings new insights; the Kirchhoff law appears as a special case. In the general case, we also prove a maximum and comparison principle, as well as a Green-type identity for a related operator.On Dubrovin's Frobenius structures on Hurwitz spaces.https://www.zbmath.org/1452.140302021-02-12T15:23:00+00:00"Lando, Sergei K."https://www.zbmath.org/authors/?q=ai:lando.sergei-kSummary: Hurwitz spaces are spaces of meromorphic functions on algebraic curves. B. Dubrovin introduced Frobenius structures on Hurwitz spaces, which serve as one of the most spectacular examples of such structures. On the other hand, Hurwitz spaces admit natural compactification by stable maps. The main goal of the present paper is to formulate a question concerning the behavior of Dubrovin's Frobenius structures on Hurwitz spaces on the boundary of the compactification. Sample computations justifying the question are given.
For the entire collection see [Zbl 1446.53004].Algebraic structure of the range of a trigonometric polynomial.https://www.zbmath.org/1452.300042021-02-12T15:23:00+00:00"Kovalev, Leonid V."https://www.zbmath.org/authors/?q=ai:kovalev.leonid-v"Yang, Xuerui"https://www.zbmath.org/authors/?q=ai:yang.xueruiLet \(p(z) =\sum_{k=-m}^na_kz^k\), \(z\in \mathbb{C}\), be a Laurent polynomial with complex coefficients. The main result of the paper states that the image of the unit circle \(p(\mathbb{T})\) is contained in the real algebraic set \(V=\{(x,y)\in \mathbb{R}^2:q(x,y)=0\} \), where \(q(x,y)\) is a polynomial of degree \(2 \max(m, n)\). The difference \(V\setminus p(\mathbb{T})\) is finite when \(m\ne n\) but may be infinite when \(m = n\).
Reviewer: Alexander Ulanovskii (Stavanger)On ELSV-type formulae, Hurwitz numbers and topological recursion.https://www.zbmath.org/1452.140522021-02-12T15:23:00+00:00"Lewanski, Danilo"https://www.zbmath.org/authors/?q=ai:lewanski.daniloSummary: We present several recent developments on ELSV-type formulae and topological recursion concerning Chiodo classes and several kind of Hurwitz numbers.
For the entire collection see [Zbl 1404.14006].Higher order commutators and multi-parameter BMO.https://www.zbmath.org/1452.420142021-02-12T15:23:00+00:00"Petermichl, Stefanie"https://www.zbmath.org/authors/?q=ai:petermichl.stefanieThe text under review is the author's invited lecture at the ICM 2018. She highlights the interplay
of multi-parameter BMO spaces and boundedness of corresponding commutators. This goes back to a classical result by \textit{Z. Nehari} [Ann. Math. (2) 65, 153--162 (1957; Zbl 0077.10605)]. One of its equivalent formulations says that the \(L^2\to L^2\) norm of the commutator of
the multiplication operator with symbol function and the Hilbert transform is equivalent to the norm of
the symbol in BMO. It turns out that such a phenomenon is typical in a variety of settings. The author gives
a comprehensive picture of the known results where boundedness of certain commutators involving Hilbert or Riesz transforms are a testing condition for the symbol to belong to one of the BMO spaces.
For the entire collection see [Zbl 1437.00045].
Reviewer: Elijah Liflyand (Ramat-Gan)Nearly invariant subspaces with applications to truncated Toeplitz operators.https://www.zbmath.org/1452.300282021-02-12T15:23:00+00:00"O'Loughlin, Ryan"https://www.zbmath.org/authors/?q=ai:oloughlin.ryanSummary: In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.Bernstein-type integral inequalities for a certain class of polynomials. II.https://www.zbmath.org/1452.300012021-02-12T15:23:00+00:00"Mir, Abdullah"https://www.zbmath.org/authors/?q=ai:mir.abdullah"Ahmad, Abrar"https://www.zbmath.org/authors/?q=ai:ahmad.abrarSummary: In this paper, we establish some new inequalities in the plane that are inspired by some classical Bernstein-type inequalities that relate the sup-norm of a polynomial to that of its derivative on the unit circle. The obtained results relate the \(L^\gamma\)-norm of the polar derivative and the polynomial. Our results besides derive polar derivative analogues of some classical Bernstein-type inequalities also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials, as well.Connecting the \(q\)-multiplicative convolution and the finite difference convolution.https://www.zbmath.org/1452.260132021-02-12T15:23:00+00:00"Leake, Jonathan"https://www.zbmath.org/authors/?q=ai:leake.jonathan-d"Ryder, Nick"https://www.zbmath.org/authors/?q=ai:ryder.nick-rThe Walsh additive and Grace-Szegö multiplicative polynomial convolution on \(f,g\in \mathbb{C}_{n}[x]\), denoted by \(\boxplus^{n}\) and \(\boxtimes^{n}\) respectively, are defined by
\[
f\boxplus^{n} g = \frac{1}{n!} \sum^{n}_{k=0}\partial^{k}_{x}f\cdot (\partial_{x}^{n-k}g)(0) \quad \text{and} \quad
f\boxtimes^{n} g = \sum^{n}_{k=0}\binom{n}{k}^{-1} (-1)^{k} f_{k}g_{k}x^{k},
\]
where \(f_{k}\) and \(g_{k}\) denote the coefficient of \(x^{k}\) of \(f\) and \(g\), respectively.
It is known that the additive convolution can only increase root mesh, which is defined as the minimum absolute difference between any pair of roots of a given polynomial. As well, for polynomials with only non-negative roots, the multiplicative convolution can only increase logarithmic root mesh.
Properties of logarithmic mesh preservation are also known for the \(q\)-multiplicative convolution. This convolution is defined as follows, where \(p_{k}\) and \(r_{k}\) are the coefficients of \(x^{k}\) of \(p\) and \(r\),
\[
p\boxtimes_{q}^{n} r=\sum^{n}_{k=0}\binom{n}{k}^{-1}_{q}q^{-\binom{k}{2}}(-1)^{k}p_{k}r_{k}x^{k},
\]
with \((x)_{q}\) meaning \(\frac{1-q^{x}}{1-q}\).
Writing \(\operatorname{lmesh}(p)\) for the minimum ratio between any pair of non-zero roots of \(p\), the result is that, for \(p\), \(r\) polynomials with real roots, one has \(\operatorname{lmesh}(p\boxtimes_{q}^{n}r)\ge q\) whenever \(\operatorname{lmesh}(p)\ge q\) and \(\operatorname{lmesh}(r)\ge q\) for some \(q\in (1,\infty)\).
In this paper, the authors show that the \(b\)-additive convolution preserves the space of polynomials with root mesh at least \(b\). This gives a characterization of finite difference operators which preserve root mesh at least \(b\). The \(b\)-additive convolution of \(p\) and \(r\) is
\[
p\boxplus_{b}^{n} r=\frac{1}{n!}\sum^{n}_{k=0}\Delta_{b}^{k} p\cdot (\Delta_{b}^{n-k}r)(0),
\]
where \(\Delta_{b}\) is a finite \(b\)-difference operator defined by
\[
\Delta_{b}: p\mapsto \frac{p(x)-p(x-b)}{b}.
\]
A real-rooted polynomial \(p\) is \(b\)-mesh if the minimum non-negative difference of any pair of roots of \(p\) is at least \(b\). In this situation, one writes \(\operatorname{mesh}(p)\ge b\).
The following result answers an open question:
\begin{itemize} \item
Let \(p\) and \(r\) be polynomials of degree at most \(n\) such that \(\operatorname{mesh}(p) \ge b\) and \(\operatorname{mesh}(r) \ge b\), for some \(b\in (0,\infty)\). Then, \(\operatorname{mesh}(p\boxplus_{b}^{n} r) \ge b\).
\end{itemize}
The proof is based in a way to pass root properties of the \(q\)-multiplicative convolution to the \(b\)-additive convolution. To formulate it one needs some notation.
For \(b\ge 0\) and \(q\ge 0\) consider the following bases of \(\mathbb{C}[x]\):
\begin{align*}
v_{q,b}^{k}&= \frac{(1-x)(1-q^{b}x)\dotsm (1-q^{(k-1)b}x)}{(1-q)^{k}},\\
\nu_{b}^{k}&=x(x+b)(x+2b)\dotsm (x+(k-1)b),
\end{align*}
and define the following generalized ``exponential map'':
\[
E_{q,b}: \nu_{b}^{k}\mapsto v_{q,b}^{k}.
\]
The main result, which gives an analytic link between the \(b\)-additive and \(q\)-multiplicative convolutions is the following:
\begin{itemize} \item
Fix \(b\ge 0\) and let \(p\), \(r\) be polynomials of degree \(n\). We have the following, where convergence is uniform on compact sets:
\[
\lim_{q\to 1}(1-q)^{n} \Bigl[E_{q,b}(p)\boxtimes_{q^{b}}^{n} E_{q,b}(r)\Bigr] (q^{x})=p\boxplus_{b}^{n}r.
\]
\end{itemize}
The corresponding result in the classical case, which gives an analytic connection between the additive and multiplicative convolutions is
\begin{itemize} \item
Let \(p,r\in\mathbb{C}[x]\) be of degree at most \(n\). We have the following:
\[
\lim_{q\to 1}(1-q)^{n} [E_{q,0}(p)\boxtimes^{n} E_{q,0}(r)] (q^{x})=p\boxplus^{n}r.
\]
\end{itemize}
Finally, the authors prove that the \(b\)-additive convolution preserves a root-interlacing property for \(b\)-mesh polynomials.
Let \(f,g\in \mathbb{R}[x]\) two real-rooted polynomials with positive leading coefficients and roots \(\alpha_{1}\ge\dotsb\ge\alpha_{n}\) and \(\beta_{1}\ge\dotsb\ge \beta_{m}\), respectively. If \(n-m\in \{0,1\}\), then one says that the roots of \(g\) interlaces the roots of \(f\) whenever
\[
\alpha_{1}\ge \beta_{1}\ge\alpha_{2}\ge \beta_{2}\ge\alpha_{3}\ge\dotsb
\]
Then one has
\begin{itemize} \item
Let \(f,g\in \mathbb{R}_{n}[x]\) be \(b\)-mesh polynomials of degree \(n\). Let \(T_{g}: \mathbb{R}_{n}[x]\to
\mathbb{R}_{n}[x]\) be the real linear operator defined by \(T_{g}: r\mapsto r\boxplus_{b}^{n}g\).
Then, \(T_{g}\) preserves the set of polynomials whose roots interlace the roots of \(f\).
\end{itemize}
Reviewer: Julià Cufí (Bellaterra)On some Chebyshev type inequalities for the complex integral.https://www.zbmath.org/1452.260192021-02-12T15:23:00+00:00"Dragomir, Silvestru Sever"https://www.zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Assume that \(f\) and \(g\) are continuous on \(\gamma\), \(\gamma\subset\mathbb{C}\) is a piecewise smooth path parametrized by \(z(t)\), \(t\in[a,b]\) from \(z(a)=u\) to \(z(b)=w\) with \(w\ne u\), and the complex Chebyshev functional is defined by \[\mathcal{D}_\gamma(f,g): =\frac{1}{w-u}\int_\gamma f(z)g(z)dz-\frac{1}{w-u}\int_\gamma f(z) dz\frac{1}{w-u}\int_\gamma g(z)dz.\] In this paper we establish some bounds for the magnitude of the functional \(\mathcal{D}_\gamma(f,g)\) under Lipschitzian assumptions for the functions \(f\) and \(g\), and provide a complex version for the well known Chebyshev inequality.On the structure of the zero sets of functions of the \(F\)-algebras \(M^q\), \(0<q<1\), in the unit disc.https://www.zbmath.org/1452.300292021-02-12T15:23:00+00:00"Gavrilov, V. I."https://www.zbmath.org/authors/?q=ai:gavrilov.valerian-ivanovich"Subbotin, A. V."https://www.zbmath.org/authors/?q=ai:subbotin.a-vSummary: The paper deals with the \(F\)-algebras \(M^q\), \(0<q<1\), of analytic functions \(f\) in the unit disc \( D: |z|<1\) on the complex \(z\)-plane such that the functions \(\displaystyle\log_+\biggl(\sup_{0\le r<1} |f(re^{i\theta})|\biggr)\) belong to the Lebesgue class \(L^q[0,2\pi]\), \(0<q<1\). A necessary condition is proved for a sequence of points in \(D\) to be the sequence of zeros for a function in \(M^q\), \(0<q<1\).On the coefficients of \(\mathcal{B}_1(\alpha)\) Bazilevič functions.https://www.zbmath.org/1452.300082021-02-12T15:23:00+00:00"Bano, Khadija"https://www.zbmath.org/authors/?q=ai:bano.khadija"Raza, Mohsan"https://www.zbmath.org/authors/?q=ai:raza.mohsan"Thomas, Derek K."https://www.zbmath.org/authors/?q=ai:thomas.derek-keithSummary: Denote by \(\mathcal{A}\), the class of functions \(f\), analytic in \(\mathbb{D} =\{z:|z|<1\}\) and given by \(f(z)=z+\sum_{n=2}^\infty a_nz^n\) for \(z\in\mathbb{D}\), and by \(\mathcal{S}\) the subset of \(\mathcal{A}\) whose elements are univalent in \(\mathbb{D}\). The class \(\mathcal{B}_1(\alpha) \subset \mathcal{S}\), of Bazilevič functions is defined by \(Re\frac{zf'(z)}{f(z)} \left(\frac{f(z)}{z}\right)^\alpha >0\), for \(\alpha \geq 0\) and \(z\in\mathbb{D}\). We give sharp bounds for \(|\gamma_n|\), where \(\log \frac{f(z)}{z}=2\sum_{n=1}^\infty \gamma_nz^n\), when \(n=1,2,3\), and \(\alpha \geq 0\), and obtain the sharp bound for \(|\gamma_4|\) when \(0\leq \alpha \leq \alpha^*\) (\(\alpha^*\approx 1.5464\)), together with another bound for \(|\gamma_4|\) when \(\alpha \geq 0\). Sharp bounds for some initial coefficients of the inverse function when \(f\in \mathcal{B}_1(\alpha)\) are also found, which augment known results.On some limit sets of real-valued functions.https://www.zbmath.org/1452.300192021-02-12T15:23:00+00:00"Berberyan, S. L."https://www.zbmath.org/authors/?q=ai:berberyan.samvel-l(no abstract)On the initial coefficients for certain class of functions analytic in the unit disc.https://www.zbmath.org/1452.300112021-02-12T15:23:00+00:00"Obradović, Milutin"https://www.zbmath.org/authors/?q=ai:obradovic.milutin"Tuneski, Nikola"https://www.zbmath.org/authors/?q=ai:tuneski.nikolaSummary: Let function \(f\) be analytic in the unit disk \(\mathbb{D}\) and be normalized so that \(f (z) = z + a_2 z^2 + a_3 z^3 + \cdots\) In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if \(f\) satisfies \[\bigg|\arg\bigg[\bigg(\frac{z}{f(z)}\bigg)^{1+\alpha} f'(z)\bigg] \bigg|< \gamma \frac{1}{2} \pi \quad (z \in \mathbb{D})\] for \(0 < \alpha < 1\) and \(0 < \gamma \leq 1\).Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials.https://www.zbmath.org/1452.140262021-02-12T15:23:00+00:00"Aggarwal, Amol"https://www.zbmath.org/authors/?q=ai:aggarwal.amol"Delecroix, Vincent"https://www.zbmath.org/authors/?q=ai:delecroix.vincent"Goujard, Élise"https://www.zbmath.org/authors/?q=ai:goujard.elise"Zograf, Peter"https://www.zbmath.org/authors/?q=ai:zograf.peter"Zorich, Anton"https://www.zbmath.org/authors/?q=ai:zorich.antonThe moduli space of quadratic differentials (with at worst simple poles) on Riemann surfaces can be stratified according to the number and multiplicities of singularities of the differentials. For each stratum of quadratic differentials, period coordinates with respect to the induced flat metric give a natural volume form, leading to a finite volume on the hyperboloid of unit-area differentials, called the Masur-Veech volume. Computing Masur-Veech volumes and analyzing their asymptotic behavior has motivated a number of interesting works in the fields of Teichmüller dynamics, intersection theory, representation theory, and combinatorics. For the case of abelian differentials (whose global squares give quadratic differentials), large genus asymptotics of Masur-Veech volumes were conjectured by \textit{A. Eskin} and \textit{A. Zorich} [Arnold Math. J. 1, No. 4, 481--488 (2015; Zbl 1342.32012)], and were settled respectively by \textit{A. Aggarwal} [J. Am. Math. Soc. 33, No. 4, 941--989 (2020; Zbl 1452.14025)] via a combinatorial method and by \textit{D. Chen} et al. [Invent. Math. 222, No. 1, 283--373 (2020; Zbl 1446.14015)] via intersection theory. In this paper the authors make conjectural descriptions on the large genus asymptotic behavior of Masur-Veech volumes of the strata of (primitive) quadratic differentials as well as their (area) Siegel-Veech constants. Numerical evidence is also provided for these conjectures.
Reviewer: Dawei Chen (Chestnut Hill)Properties of \(\beta\)-Cesàro operators on \(\alpha\)-Bloch space.https://www.zbmath.org/1452.300302021-02-12T15:23:00+00:00"Kumar, Shankey"https://www.zbmath.org/authors/?q=ai:kumar.shankey"Sahoo, Swadesh Kumar"https://www.zbmath.org/authors/?q=ai:sahoo.swadesh-kumarSummary: For each \(\alpha > 0\), the \(\alpha\)-Bloch space consists of all analytic functions \(f\) on the unit disk satisfying \(\operatorname{sup}_{|z|<1} (1-|z|^2)^\alpha | f'(z)|<+\infty\). We consider the following complex integral operators, namely the \(\beta\)-Cesàro operator \[C_\beta(f)(z) = \int_0^z \frac{f(w)}{w(1-w)^{\beta}} d w \] and its generalization, acting from the \(\alpha\)-Bloch space to itself, where \(f(0) = 0\) and \(\beta \in \mathbb{R}\). We investigate the boundedness and compactness of the \(\beta\)-Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.On the support of the bifurcation measure of cubic polynomials.https://www.zbmath.org/1452.370512021-02-12T15:23:00+00:00"Inou, Hiroyuki"https://www.zbmath.org/authors/?q=ai:inou.hiroyuki"Mukherjee, Sabyasachi"https://www.zbmath.org/authors/?q=ai:mukherjee.sabyasachiAuthors' abstract: We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.
Reviewer: Klaus Schiefermayr (Wels)Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in \(L^1\) over metric measure spaces.https://www.zbmath.org/1452.300332021-02-12T15:23:00+00:00"Matsuoka, Katsuo"https://www.zbmath.org/authors/?q=ai:matsuoka.katsuo"Mizuta, Yoshihiro"https://www.zbmath.org/authors/?q=ai:mizuta.yoshihiro"Shimomura, Tetsu"https://www.zbmath.org/authors/?q=ai:shimomura.tetsuThe authors provide weak \(M^{1,q,a}(X)\)-estimates for the maximal and Riesz potential operators. Under certain assumptions on the growth of the measures of balls in \(X\) (which hold in the Euclidean setting), it is shown in Theorems 3.3 and 4.6 that the maximal operator and the Riesz potential operator are bounded from \(M^{1,q,a}(X)\) to \(WM^{\varphi,q,a}(X)\). Moreover, quantitative estimates for the boundedness of these operators are provided in Theorems 3.5 and 4.10 for functions \(f\) satisfying
\[
\Vert f \Vert_{N^{p,q,a}(X)}
:=
\Vert f \Vert_{L^p(B(x_0,2))}
+
\left(
\int_1^{\infty} \left( r^a \Vert f \Vert_{L^p(X \setminus B(x_0,r))}\right)^q \frac{dr}{r}
\right)^{1/q}
< 1
\]
for \(p = 1\).
The paper concludes with a discussion of the duality between \(M^{1,q,a}(X)\) and the space \(N^{\infty,q',a}(X)\) consisting of those measurable functions on \(X\) which satisfy
\(\Vert f \Vert_{N^{\infty,q',a}(X)} < \infty\).
In particular, \(N^{\infty,q',a}(X)\) is precisely the associate space of \(M^{1,q,a}(X)\), i.e.,
\[
\Vert f \Vert_{N^{\infty,q',a}(X)}
=
\sup_{g \in M^{1,q,a}(X) \, : \, \Vert g \Vert_{M^{1,q,a}(X)}\leq 1}
\int_X |f(x)g(x)| \, d \mu (x).
\]
Reviewer: Scott Zimmerman (Marion)A uniqueness theorem for the two-dimensional sigma function.https://www.zbmath.org/1452.300152021-02-12T15:23:00+00:00"Domrin, A. V."https://www.zbmath.org/authors/?q=ai:domrin.andrei-victorovichSummary: We prove that the sigma functions of Weierstrass \((g = 1)\) and Klein \((g = 2)\) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of \(2g\) linear differential heat equations in a nonholonomic frame (for a function of \(3g\) variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For \(g =1\), we give a complete description of the envelopes of holomorphy of such solutions.The study of univalence criteria linked with Pescar's-typed univalence criteria.https://www.zbmath.org/1452.300092021-02-12T15:23:00+00:00"Faisal, M. I."https://www.zbmath.org/authors/?q=ai:faisal.muhammad-imranSummary: In this article, I establish univalence criteria concerning the class of analytic functions to be univalent in the open unit disk. Besides, I review its application in the open unit disk of univalent functions. It is observed that for a special case, the criteria developed here reduces to Pescar's-typed univalence criteria.Quasi-Möbius invariant quaternionic spaces.https://www.zbmath.org/1452.300252021-02-12T15:23:00+00:00"Pérez Hernández, Jorge"https://www.zbmath.org/authors/?q=ai:perez-hernandez.jorge"Reséndis O., Lino F."https://www.zbmath.org/authors/?q=ai:resendis-ocampo.lino-feliciano"Tovar S., Luis M."https://www.zbmath.org/authors/?q=ai:tovar.luis-manuelSummary: With the help of some embedding of quaternions in the Clifford Algebra \(Cl(4)\) and by using some bounded operators, we prove that certain weighted harmonic Bergman spaces of quaternionic valued functions, are complete and quasi Möbius invariant.
For the entire collection see [Zbl 1429.35006].Masur-Veech volumes of quadratic differentials and their asymptotics.https://www.zbmath.org/1452.140342021-02-12T15:23:00+00:00"Yang, Di"https://www.zbmath.org/authors/?q=ai:yang.di"Zagier, Don"https://www.zbmath.org/authors/?q=ai:zagier.don-bernard"Zhang, Youjin"https://www.zbmath.org/authors/?q=ai:zhang.youjinLet \(\mathcal Q_{g,n}\) be the moduli space of quadratic differentials on genus \(g\) Riemann surfaces with at worst simple poles at the \(n\) marked points. It carries a natural volume form given by period coordinates with respect to the induced flat metric, and the associate volume \(\mathrm{Vol}~\mathcal Q_{g,n}\) (after suitable normalization) is called the Masur-Veech volume of the principal strata of quadratic differentials. In [``Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials'', Preprint, \url{arXiv:1912.02267}] \textit{D. Chen} et al. gave an expression of \(\mathrm{Vol}~\mathcal Q_{g,n}\) via certain linear Hodge integrals on the Deligen-Mumford moduli space of curves. Based on this formula, the authors apply the theory of integrable systems to derive a number of relations for the generating series of \(\mathrm{Vol}~\mathcal Q_{g,n}\). Moreover, they provide refinements of the conjectural formulas given in [\textit{A. Aggarwal} et al., Arnold Math. J. 6, No. 2, 149--161 (2020; Zbl 1452.14026)] for the large genus asymptotics of the \(\mathrm{Vol}~\mathcal Q_{g,n}\) as well as the associated area Siegel-Veech constants. As a remark, the original version of the large genus asymptotic conjecture for \(\mathrm{Vol}~\mathcal Q_{g,n}\) has been recently setted by \textit{A. Aggarwal} [``Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials'', Preprint, \url{arXiv:2004.05042}].
Reviewer: Dawei Chen (Chestnut Hill)Fourier expansion of the Riemann zeta function and applications.https://www.zbmath.org/1452.111022021-02-12T15:23:00+00:00"Elaissaoui, Lahoucine"https://www.zbmath.org/authors/?q=ai:elaissaoui.lahoucine"Guennoun, Zine El Abidine"https://www.zbmath.org/authors/?q=ai:el-abidine-guennoun.zineSummary: We study the distribution of values of the Riemann zeta function \(\zeta(s)\) on vertical lines \(\Re s + i \mathbb{R}\), by using the theory of Hilbert space. We show among other things, that, \(\zeta(s)\) has a Fourier expansion in the half-plane \(\Re s \geq 1 / 2\) and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of \(\zeta(s) - s /(s - 1)\). Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients. For a video summary of this paper, please visit \url{https://youtu.be/wI5fIJMeqp4}.Brück conjecture and certain solution of some differential equation.https://www.zbmath.org/1452.300172021-02-12T15:23:00+00:00"Ahamed, Molla Basir"https://www.zbmath.org/authors/?q=ai:ahamed.molla-basir"Linkha, Santosh"https://www.zbmath.org/authors/?q=ai:linkha.santoshSummary: We investigate on the famous Brück conjecture, and improved some of the existing results by extending them up to a differential monomial \(M[f]\) sharing small function with certain power of \(f^{d_M}\) of a meromorphic function. The class of all meromorphic solutions of the differential equation \(f^{d_M}\equiv M[f]\) has been explored. For the generalization of our main result, some relevant questions finally have been posed for further study in this direction.A note on Eneström-Kakeya theorem for a polynomial with quaternionic variable.https://www.zbmath.org/1452.300272021-02-12T15:23:00+00:00"Tripathi, Dinesh"https://www.zbmath.org/authors/?q=ai:tripathi.dineshSummary: In this paper, we present certain results concerning the location of the zeros of polynomials with quaternionic variable which generalize and refine some known Eneström-Kakeya type bounds for the zeros of polynomials.Bicomplex Bergman and Bloch spaces.https://www.zbmath.org/1452.300262021-02-12T15:23:00+00:00"Reséndis O., L. F."https://www.zbmath.org/authors/?q=ai:resendis-ocampo.lino-feliciano"Tovar S., L. M."https://www.zbmath.org/authors/?q=ai:tovar.luis-manuelSummary: In this article, we define the bicomplex weighted Bergman spaces on the bidisk and their associated weighted Bergman projections, where the respective Bergman kernels are determined. We study also the bicomplex Bergman projection onto the bicomplex Bloch space.Fekete-Szegö problem for certain class of bi-stralike functions involving \(q\)-differential operator.https://www.zbmath.org/1452.300122021-02-12T15:23:00+00:00"Shrigan, M. G."https://www.zbmath.org/authors/?q=ai:shrigan.m-g"Kamble, P. N."https://www.zbmath.org/authors/?q=ai:kamble.prakash-namdeoSummary: Making use of \(q\)-derivative operator, in this paper, we introduce new subclasses of the function class \(\Sigma\) of normalized analytic and bi-starlike functions defined in the open disk \(\mathbb{U}\). Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients \(|a_2|\) and \(|a_3|\). Moreover, we obtain Fekete-Szegö inequalities for the new function classes.Some examples of residual Julia set for a class of meromorphic functions with no wandering domains.https://www.zbmath.org/1452.300132021-02-12T15:23:00+00:00"Hernández, Iván"https://www.zbmath.org/authors/?q=ai:hernandez.ivanSummary: In this paper, we define the residual Julia set of functions which belong to a class of meromorphic functions, denoted by \(\mathcal{K}\). We provide one example of a function in class \(\mathcal{K}\) where the residual Julia set is not empty by using approximation theory.
For the entire collection see [Zbl 1429.92002].Mass equidistribution for random polynomials.https://www.zbmath.org/1452.320042021-02-12T15:23:00+00:00"Bayraktar, Turgay"https://www.zbmath.org/authors/?q=ai:bayraktar.turgaySummary: The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.Blaschke products and zero sets in weighted Dirichlet spaces.https://www.zbmath.org/1452.300322021-02-12T15:23:00+00:00"Idrissi, H. Bahajji-El"https://www.zbmath.org/authors/?q=ai:idrissi.hafid-bahajji-el"El-Fallah, O."https://www.zbmath.org/authors/?q=ai:el-fallah.omarSummary: In this paper, we deal with superharmonically weighted Dirichlet spaces \(\mathcal{D}_{\omega } \). First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for \(\mathcal{D}_{\omega } \). Our conditions improve Shapiro-Shields condition for \(\mathcal{D}_{\alpha } \), when \(\alpha \in (0,1)\).Classes of meromorphic harmonic functions and duality principle.https://www.zbmath.org/1452.310022021-02-12T15:23:00+00:00"Dziok, Jacek"https://www.zbmath.org/authors/?q=ai:dziok.jacekSummary: We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions.A note on the removability of totally disconnected sets for analytic functions.https://www.zbmath.org/1452.300022021-02-12T15:23:00+00:00"Pokrovskii, A. V."https://www.zbmath.org/authors/?q=ai:pokrovskii.andrei-vladimirovichSummary: We prove that each totally disconnected closed subset \(E\) of a domain \(G\) in the complex plane is removable for analytic functions \(f(z)\) defined in \(G\setminus E\) and such that, for any point \(z_0 \in E\), the real or imaginary part of \(f(z)\) vanishes at \(z_0\).Bounds on dimension reduction in the nuclear norm.https://www.zbmath.org/1452.460172021-02-12T15:23:00+00:00"Regev, Oded"https://www.zbmath.org/authors/?q=ai:regev.oded"Vidick, Thomas"https://www.zbmath.org/authors/?q=ai:vidick.thomasLet \(\mathsf{S}_1\) denote the Schatten-von Neumann trace class, that is, the Banach space of all nuclear linear operators \(T:\ell_2\to \ell_2\) with its nuclear norm. Let \(\mathsf{S}_1^m\) be the normed linear space of all operators \(T:\ell_2^m\to \ell_2^m\) with its nuclear norm.
The authors show that the geometry of \(\mathsf{S}_1^m\) is quite different from the geometry of the normed spaces \(\ell_p^m\), \(1\le p\le \infty\), in the following sense: there exists a set of \(n\) elements in \(\mathsf{S}_1^m\) such that embedding of this set into \(\mathsf{S}_1^d\) with ``very'' low distortion requires \(d\) to be exponential in \(n\).
More detailed statement (from the abstract): ``For all \(n \ge 1\), we give an explicit construction of \(m \times m\) matrices \(A_1,\ldots,A_n\) with \(m = 2^{\lfloor n/2 \rfloor}\) such that for any \(d\) and \(d \times d\) matrices \(A'_1,\ldots,A'_n\) that satisfy
\[
\|A'_i-A'_j\|_{\mathsf{S}_1} \,\leq\, \|A_i-A_j\|_{\mathsf{S}_1}\,\leq\, (1+\delta) \|A'_i-A'_j\|_{\mathsf{S}_1}
\]
for all \(i,j\in\{1,\ldots,n\}\) and small enough \(\delta = O(n^{-c})\), where \(c>0\) is a universal constant, it must be the case that \(d \ge 2^{\lfloor n/2\rfloor -1}\).''
``Our proof is based on matrices derived from a representation of the Clifford algebra generated by \(n\) anti-commuting Hermitian matrices that square to identity, and borrows ideas from the analysis of nonlocal games in quantum information theory.''
This result contrasts with the well-known results of \textit{K. Ball} [Eur. J. Comb. 11, No. 4, 305--311 (1990; Zbl 0712.46008)] on existence of isometric embeddings of \(n\)-element subsets of \(\ell_p\), \(1\le p\le \infty\), into \(\ell_p^m\) with \(m=\frac12n(n-1)\).
In Lemma 13.19 the authors show that for any \(0<\delta<1\) the metric space constructed to prove the result above can be embedded with the distortion \(1+\delta\) into \(\mathsf{S}_1^d\) for \(d=n^{O(1/\delta^2)}\)
The results of this paper are closely related to the results of \textit{A. Naor} et al. [Discrete Comput. Geom. 63, No. 2, 319--345 (2020; Zbl 1442.46017)]. However the results are incomparable because Naor et al. [loc. cit.] consider embeddings into \textbf{any} subspace of \(\mathsf{S}_1\). The mentioned paper of Naor et al. also contains (see page 322) information on the importance of \(\mathsf{S}_1\) and its metric properties in many areas.
For the entire collection see [Zbl 1446.00030].
Reviewer: Mikhail Ostrovskii (New York)