Recent zbMATH articles in MSC 30Dhttps://zbmath.org/atom/cc/30D2022-09-13T20:28:31.338867ZWerkzeugAn open problem of Lü, Li and Yanghttps://zbmath.org/1491.300112022-09-13T20:28:31.338867Z"Majumder, Sujoy"https://zbmath.org/authors/?q=ai:majumder.sujoySummary: In this paper, we use the idea of normal family to investigate the problem of entire functions that share two entire functions with one of their derivatives. In particular, we solve an open problem posed in the last section of [\textit{W. Lü} et al., Bull. Korean Math. Soc. 51, No. 5, 1281--1289 (2014; Zbl 1300.30063)]. Some examples have been exhibited to show that the conditions used in the paper are sharp.A note on the solvability of homogeneous Riemann boundary problem with infinity indexhttps://zbmath.org/1491.300122022-09-13T20:28:31.338867Z"Bory-Reyes, Juan"https://zbmath.org/authors/?q=ai:moreno-garcia.tania|bory-reyes.juanAuthors' abstract: In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the number of linearly independent admissible solutions to linear differential and linear difference equationshttps://zbmath.org/1491.340962022-09-13T20:28:31.338867Z"Heittokangas, Janne"https://zbmath.org/authors/?q=ai:heittokangas.janne-m"Yu, Hui"https://zbmath.org/authors/?q=ai:yu.hui"Zemirni, Mohamed Amine"https://zbmath.org/authors/?q=ai:zemirni.mohamed-amineAuthors' abstract: A classical theorem of Frei states that if \(A_p\) is the last transcendental function in the sequence \(A_0,\dots,A_{n-1}\) of entire functions, then each solution base of the differential equation
\[f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_1f'+A_0f=0\]
contains at least \(n-p\) entire functions of infinite order. Here, the transcendental coefficient \(A_p\) dominates the growth of the polynomial coefficients \(A_{p+1},\dots,A_{n-1}\). By expressing the dominance of \(A_p\) in different ways and allowing the coefficients \(A_{p+1},\dots,A_{n-1}\) to be transcendental, we show that the conclusion of Frei's theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that \(0\) is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich's theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex \(q\)-difference equations.
Reviewer: Shamil Makhmutov (Muscat)On zeros and growth of solutions of second order linear differential equationshttps://zbmath.org/1491.340972022-09-13T20:28:31.338867Z"Kumar, Sanjay"https://zbmath.org/authors/?q=ai:kumar.sanjay.2|kumar.sanjay-v|kumar.sanjay.1"Saini, Manisha"https://zbmath.org/authors/?q=ai:saini.manishaIn this paper under review, the authors investigate the growth and the oscillation of solutions of the homogeneous second order linear differential equation \[ f^{\prime \prime }+A\left( z\right) f^{\prime }+B\left( z\right) f=0, \tag{1} \] where \(A\left( z\right) \) and \(B\left( z\right) \) are entire functions. In order to give the main results of this paper, we recall some definitions.
Let \(f\) be a meromorphic function. Then the order \(\rho \left( f\right) \) of \(f\) is defined by
\[
\rho \left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup }\frac{\log T\left( r,f\right) }{\log r},
\]
where \(T\left( r,f\right) \) is the Nevanlinna characteristic function of \(f\) . If \(f\) is an entire function, then the order \(\rho \left( f\right) \) of \(f\) is defined by
\[
\rho \left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup }\frac{\log T\left( r,f\right) }{\log r}=\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log \log M\left( r,f\right) }{\log r},
\]
where \(M\left( r,f\right) =\max_{\left\vert z\right\vert =r}\left\vert f\left( z\right) \right\vert \). The exponent of convergence of the sequence of zeros of \(f\) is defined by
\[
\lambda \left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup }\frac{ \log N\left( r,\frac{1}{f}\right) }{\log r},
\]
where \(N\left( r,\frac{1}{f}\right) \) is the integrated counting function of zeros of \(f\) in \(\left\{ z:\left\vert z\right\vert \leq r\right\} \).\newline We say that an entire function \(g\left( z\right) =\sum_{n=0}^{+\infty }a_{\lambda _{n}}z^{\lambda _{n}}\) satisfies the Fabry gap if the sequence of exponents \(\left\{ \lambda _{n}\right\} \) satisfies the condition
\[
\frac{\lambda _{n}}{n}\rightarrow +\infty \text{ as \ }n\rightarrow +\infty .
\]
In the paper [Comput. Methods Funct. Theory 17, No. 2, 195--209 (2017; Zbl 1369.30029)] the author asked the following question: If \(\lambda \left( A\right) <\rho \left( A\right) <\infty \) and \(\rho \left( B\right) \) is a nonconstant polynomial, then does every nontrivial solution \( f\) of \(\left( 1\right) \) satisfy \(\rho \left( f\right) =+\infty ?\) \newline In their paper, \textit{J. Long} et al. [Ann. Acad. Sci. Fenn., Math. 43, No. 1, 337--348 (2018; Zbl 1391.34139)] gave a partial answer to the above question by proving the following result: \ Let \(A\left( z\right) =C\left( z\right) e^{P\left( z\right) }\), where \(C\not\equiv 0\) is an entire function and \( P\left( z\right) =a_{n}z^{n}+\cdots +a_{1}z+a_{0}\) is a polynomial of degree \(n\) such that \(\rho \left( C\right) <n\). Let \(B\left( z\right) =b_{m}z^{m}+\cdots +b_{1}z+b_{0}\) be a nonconstant polynomial of degree \(m\). Then every solution \(f\not\equiv 0\) of the equation \(\left( 1\right) \) is of infinite order if one of the following condition holds:\newline (i) \(m+2<2n;\)\newline (ii) \(m+2>2n\) and \(m+2\neq 2kn\) for all integers \(k\).\newline (iii) \(m+2=2n\) and \(\frac{a_{n}^{2}}{b_{m}}\) is not real and negative. \newline
In this paper under review, \(B\left( z\right) \) is assumed to be a transcendental entire function. The main results of this paper state as follows:
Theorem. Let \(A\left( z\right) \) be an entire function with \( \lambda \left( A\right) <\rho \left( A\right) .\) Suppose that\newline (i) \(B\left( z\right) \) is a transcendental entire function satisfying \(\rho \left( B\right) \neq \rho \left( A\right) \) or\newline (ii) \(B\left( z\right) \) is a transcendental entire function with Fabry gap. \newline Then every solution \(f\not\equiv 0\) of the equation \(\left( 1\right) \) is of infinite order.
Corollary. Let \(f\left( z\right) =h\left( z\right) e^{Q\left( z\right) }\) be a nontrivial solution of the equation \(\left( 1\right) \), where \(h\left( z\right) \) is a canonical product of zeros of \(f\left( z\right) \) and \(Q\left( z\right) \) is an entire function. If \(\rho \left( B\right) >\max \left\{ \rho \left( A\right) ,\rho \left( Q\right) \right\} ,\) then \(\lambda \left( f\right) =+\infty .\)
Further, the authors extend the main theorem for higher order linear differential equation
\[
f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_{1}(z)f^{\prime }+A_{0}(z)f=0,
\]
where \(k\geq 2\) and \(A_{0}(z)\), \(A_{1}(z)\),\(...\), \(A_{k-1}(z)\) are entire functions.
Reviewer: Benharrat Belaidi (Mostaganem)Topological characterisation of rational maps with Siegel diskshttps://zbmath.org/1491.370472022-09-13T20:28:31.338867Z"Zhang, Gaofei"https://zbmath.org/authors/?q=ai:zhang.gaofeiA famous result of W. Thurston (please refer to [\textit{A. Douady} and \textit{J. H. Hubbard}, Acta Math. 171, No. 2, 263--297 (1993; Zbl 0806.30027)]) characterizes the postcritically finite branched coverings of the Riemann sphere which are equivalent to a rational map. In this paper the author extends this result to certain rational maps with a fixed Siegel disk. Note that rational maps with a Siegel disk cannot be postcritically finite.
Specifically, the result covers orientation-preserving branched coverings \(f\) of the Riemann sphere such that the restriction of \(f\) to the unit disk is a rigid rotation with an irrational rotation number of bounded type, the unit circle contains no critical point of \(f\), and the forward orbit of each critical point of \(f\) either intersects the closed unit disk, or is eventually periodic, or converges to some holomorphic attracting cycle.
Reviewer: Walter Bergweiler (Kiel)Picard-Borel algebrashttps://zbmath.org/1491.460442022-09-13T20:28:31.338867Z"Esterle, Jean"https://zbmath.org/authors/?q=ai:esterle.jeanSummary: A Picard-Borel algebra is a commutative, unital, complex algebra \(A\) such that every family \((u_\lambda)_{\lambda\in\Lambda}\) of invertible elements of \(A\), which are pairwise linearly independent, is linearly independent. A Picard-Borel algebra is said to be nontrivial, if \(u\notin\mathbb{C}1\) for some invertible element \(u\in A\). \par The algebra \(\mathbb{C}[X]\) of complex polynomials is an obvious example of trivial Picard-Borel algebra, and results from the celebrated 1897 paper ``Sur les zéros des fonctions entières'' by
\textit{É.~Borel} [Acta Math. 20, 357--396 (1897; JFM 28.0360.01)]
show that the algebra \(\mathcal{H}(\mathbb{C})\) of entire functions on \(\mathbb{C}\) is a Picard-Borel algebra. The main result of the paper shows that those Picard-Borel algebras, which are Fréchet algebras, are integral domains.
For the entire collection see [Zbl 1477.46002].Beyond limber: efficient computation of angular power spectra for galaxy clustering and weak lensinghttps://zbmath.org/1491.830202022-09-13T20:28:31.338867Z"Fang, Xiao"https://zbmath.org/authors/?q=ai:fang.xiao.1|fang.xiao"Krause, Elisabeth"https://zbmath.org/authors/?q=ai:krause.elisabeth"Eifler, Tim"https://zbmath.org/authors/?q=ai:eifler.tim"MacCrann, Niall"https://zbmath.org/authors/?q=ai:maccrann.niall(no abstract)Perturbations in tachyon dark energy and their effect on matter clusteringhttps://zbmath.org/1491.830252022-09-13T20:28:31.338867Z"Singh, Avinash"https://zbmath.org/authors/?q=ai:singh.avinash-c|singh.avinash-k"Jassal, H. K."https://zbmath.org/authors/?q=ai:jassal.h-k"Sharma, Manabendra"https://zbmath.org/authors/?q=ai:sharma.manabendra(no abstract)Capturing non-Gaussianity of the large-scale structure with weighted skew-spectrahttps://zbmath.org/1491.830602022-09-13T20:28:31.338867Z"Dizgah, Azadeh Moradinezhad"https://zbmath.org/authors/?q=ai:dizgah.azadeh-moradinezhad"Lee, Hayden"https://zbmath.org/authors/?q=ai:lee.hayden"Schmittfull, Marcel"https://zbmath.org/authors/?q=ai:schmittfull.marcel"Dvorkin, Cora"https://zbmath.org/authors/?q=ai:dvorkin.cora(no abstract)The EFT likelihood for large-scale structurehttps://zbmath.org/1491.850032022-09-13T20:28:31.338867Z"Cabass, Giovanni"https://zbmath.org/authors/?q=ai:cabass.giovanni"Schmidt, Fabian"https://zbmath.org/authors/?q=ai:schmidt.fabian(no abstract)