Recent zbMATH articles in MSC 30 https://zbmath.org/atom/cc/30 2021-11-25T18:46:10.358925Z Werkzeug Book review of: U. Daepp et al., Finding ellipses. What Blaschke products, Poncelet's theorem, and the numerical range know about each other. https://zbmath.org/1472.00042 2021-11-25T18:46:10.358925Z "Zeytuncu, Yunus E." https://zbmath.org/authors/?q=ai:zeytuncu.yunus-ergyn Review of [Zbl 1419.51001]. Favorite mathematics topics from the 12-th century to the 21-st century https://zbmath.org/1472.01006 2021-11-25T18:46:10.358925Z "Ghusayni, Badih" https://zbmath.org/authors/?q=ai:ghusayni.badih The article describes historical mathematical examples from various centuries. The author chooses one example, more precisely one favourite mathematician, per century, except for the 20th and 21st century, where he gives two each. Most of the content is fairly well known and it is obvious that the term favourite'' refers to the author's favourites. The chosen favourites are mostly, but not exclusively, from number theory. In particular, two famous old formulas (the explicit formula for Fibonacci numbers, Fibonacci being chosen to represent the 13th century; and the Wallis product, Wallis being chosen to represent the 17th century) are proven with a modern mathematical treatment. On the nature of four models of symmetric walks avoiding a quadrant https://zbmath.org/1472.05006 2021-11-25T18:46:10.358925Z "Dreyfus, Thomas" https://zbmath.org/authors/?q=ai:dreyfus.thomas "Trotignon, Amélie" https://zbmath.org/authors/?q=ai:trotignon.amelie Summary: We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series. Applications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbers https://zbmath.org/1472.05010 2021-11-25T18:46:10.358925Z "Simsek, Yilmaz" https://zbmath.org/authors/?q=ai:simsek.yilmaz Summary: The aim of this article is to construct some new families of generating-type functions interpolating a certain class of higher order Bernoulli-type, Euler-type, Apostol-type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol-type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented. Hankel continued fractions and Hankel determinants of the Euler numbers https://zbmath.org/1472.11074 2021-11-25T18:46:10.358925Z "Han, Guo-Niu" https://zbmath.org/authors/?q=ai:han.guo-niu The author presents the Hankel continued fraction expansion and Hankel determinants of the Euler numbers, thus bringing together known results for the even (secant numbers) and odd Euler numbers (tangent numbers). The proofs are based on Flajolet's continued fraction for permutation statistics and combinatorial interpretations of the Euler number. The continued fraction expansion also leads to a $$q$$-analogue of the Euler numbers. Effective sup-norm bounds on average for cusp forms of even weight https://zbmath.org/1472.11119 2021-11-25T18:46:10.358925Z "Friedman, J. S." https://zbmath.org/authors/?q=ai:friedman.joshua-s "Jorgenson, J." https://zbmath.org/authors/?q=ai:jorgenson.jay-a "Kramer, J." https://zbmath.org/authors/?q=ai:kramer.jurg Summary: Let $$\Gamma \subset \operatorname{PSL}_2(\mathbb{R})$$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $$\mathbb{H}$$. Consider the $$d_{2k}$$-dimensional space of cusp forms $$\mathcal{S}_{2k}^{\Gamma }$$ of weight $$2k$$ for $$\Gamma$$, and let $$\{f_1,\ldots ,f_{d_{2k}}\}$$ be an orthonormal basis of $$\mathcal{S}_{2k}^{\Gamma }$$ with respect to the Petersson inner product. In this paper, we will give effective upper and lower bounds for the supremum of the quantity $$S_{2k}^{\Gamma }(z):=\sum _{j=1}^{d_{2k}}\vert f_j(z)\vert ^2\operatorname{Im}(z)^{2k}$$ as $$z$$ ranges through $$\mathbb{H}$$. On good universality and the Riemann hypothesis https://zbmath.org/1472.11232 2021-11-25T18:46:10.358925Z "Nair, Radhakrishnan" https://zbmath.org/authors/?q=ai:nair.radhakrishnan-b "Verger-Gaugry, Jean-Louis" https://zbmath.org/authors/?q=ai:verger-gaugry.jean-louis "Weber, Michel" https://zbmath.org/authors/?q=ai:weber.michel-j-g In the paper, the authors deal with a sequence $$(k_n)_{n \geq 1}\in \mathbb{N}$$ which is $$L^p$$-good universal (later, for brevity, LPGU; if for each dynamical system $$(X,\beta,\mu,T)$$ and for each $$g \in L^p(X,\beta,\mu)$$ the limit $$l_{T,g}=\lim_{N \to \infty}\frac{1}{N}\sum_{n=0}^{N-1}g(T^{k_n}x)$$ exists $$\mu$$ almost everywhere), uniformly distributed modulo one sequence $$(x_n)_{n=1}^N$$ (if $$\lim_{N\to \infty}\frac{1}{N}\#\{1 \leq n \leq N:\{x_n\}\in I\}=|I|$$ for every interval $$I \subseteq [O,1)$$), and a sequence of natural numbers $$(k_n)_{n \geq 1}$$ which is Hartman uniformly distributed on $$\mathbb{Z}$$ (later, for brevity, HUD; if it is uniformly distributed among the residue classes modulo $$m$$ for each natural number $$m>1$$, and for each irrational number $$\alpha$$, the sequence $$(\{k_n \alpha\})_{n \geq 1}$$ is uniformly distributed modulo 1; here $$\{ a\}$$ denotes the fractional part of number $$a$$). First several average ergodic type theorems are established. For example, if the meromorphic function $$f$$ on the half-plane $$\{z \in \mathbb{C}: \Re(z)>c\}$$, $$c \in \mathbb{R}$$, satisfying certain additional conditions, and $$(k_n)_{n\geq 1}$$ is LPGU and HUD, for any $$\{z \in \mathbb{C}: \Re(z)>c\}\setminus \{z \in \mathbb{C}: \Re(z)=c\}$$, it holds an equality $\lim_{N \to \infty}\frac{1}{N}\sum_{n=0}^{N-1}f(s+iT^{k_n}_{\alpha,\beta}(x))=\frac{\alpha}{\pi}\int_{\mathbb{R}}\frac{f(s+i\tau)}{\alpha^2+(\tau-\beta)^2}d\tau$ for almost all $$x \in \mathbb{R}$$. Later, using this theorem, new characterisations of the extended Lindelöf hypothesis are obtained for the Riemann zeta-function, Dirichlet $$L$$-functions, Dedekind zeta-functions over number field, Hurwitz zeta-functions. Later, replacing the LPGU and HUD conditions for sequence $$(k_n)_{n \geq 1}$$ to be Stoltz sequence, similar limit theorems are proved. Also some examples of LPGU sequences and HUD sequences are given. The determinant inner product and the Heisenberg product of $$\mathrm{Sym}(2)$$ https://zbmath.org/1472.15005 2021-11-25T18:46:10.358925Z "Crasmareanu, Mircea" https://zbmath.org/authors/?q=ai:crasmareanu.mircea Let $$A$$ be a finite subset of a field and denote by $$D^{n(A)}$$ the set of all possible determinants of matrices with entries in $$A$$. In this paper, the following problem, typical in additive combinatorics, is investigated: how big is the image set of the determinant function compared to the set $$A$$? Interesting results are obtained, that remain also true also for the set of permanents. On effective criterion of stability of partial indices for matrix polynomials https://zbmath.org/1472.15028 2021-11-25T18:46:10.358925Z "Adukova, N. V." https://zbmath.org/authors/?q=ai:adukova.natalya-viktorovna "Adukov, V. M." https://zbmath.org/authors/?q=ai:adukov.viktor-mikhailovich|adukov.viktor-michailovich Summary: In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg-Krein-Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg-Krein-Bojarsky theorem. A proof of Carleson's $$\varepsilon^2$$-conjecture https://zbmath.org/1472.28005 2021-11-25T18:46:10.358925Z "Jaye, Benjamin" https://zbmath.org/authors/?q=ai:jaye.benjamin-j "Tolsa, Xavier" https://zbmath.org/authors/?q=ai:tolsa.xavier "Villa, Michele" https://zbmath.org/authors/?q=ai:villa.michele Summary: In this paper we provide a proof of the Carleson $$\varepsilon^2$$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $$\varepsilon^2$$-square function. Asymptotics of Chebyshev polynomials. IV: Comments on the complex case https://zbmath.org/1472.30001 2021-11-25T18:46:10.358925Z "Christiansen, Jacob S." https://zbmath.org/authors/?q=ai:christiansen.jacob-stordal "Simon, Barry" https://zbmath.org/authors/?q=ai:simon.barry.1 "Zinchenko, Maxim" https://zbmath.org/authors/?q=ai:zinchenko.maxim The Chebyshev polynomial $$T_n$$ of a compact infinite set $$E\subset{\mathbb C}$$ is that monic polynomial of degree-$$n$$ which minimizes $${\|P_n\|}_E$$ over all degree $$n$$ monic polynomials $$P_n$$, where $${\|\cdot\|}_E$$ denotes the supremum norm on $$E$$. In this paper, which is the fourth part of a series of papers (the second joint with Yuditskii), all of them devoted to Chebyshev polynomials and related problems, the authors present some results for rather general sets $$E$$ in the complex plane. On the one hand, they prove some interesting results concerning the asymptotics of the zeros of $$T_n$$, and on the other hand, they give explicit Totik-Widom upper bounds for certain complex sets $$E$$. For Part III, see [the authors, Oper. Theory: Adv. Appl. 276, 231--246 (2020; Zbl 1448.41026)]. Critical points, critical values, and a determinant identity for complex polynomials https://zbmath.org/1472.30002 2021-11-25T18:46:10.358925Z "Dougherty, Michael" https://zbmath.org/authors/?q=ai:dougherty.michael-r|dougherty.michael-m "McCammond, Jon" https://zbmath.org/authors/?q=ai:mccammond.jon Let $$\theta: \mathbb C^n \to \mathbb C^n$$ be the map defined by $\theta(z) := (p_z(z_1), \ldots, p_z(z_n)),\qquad p_z(\zeta) := \int_0^\zeta (w - z_1)\cdot \ldots \cdot (w - z_n) d w.$ Note that, for each $$z \in \mathbb C^n$$, the expression $$p_z(\zeta)$$ is a polynomial in $$\zeta$$, it has the entries $$z_i$$ of $$z$$ as critical points and the value $$n$$-tuple $$\theta(z) := (p_z(z_1), \ldots, p_z(z_n))$$ coincides with the set of the critical values of the polynomial $$p_z(\zeta)$$. The map $$\theta$$ is known to be surjective [\textit{A. F. Beardon} et al., Constr. Approx. 18, No. 3, 343--354 (2002; Zbl 1018.30003)], proving that any point of $$\mathbb C^n$$ arises as the set of critical values of some polynomial. It is also known that the Jacobian $$\mathbb J(z)$$ of the map $$\theta$$ at a point $$z$$ is invertible if and only if the entries of $$z$$ are all distinct. In this paper, the authors establish an analogue of this property for the Jacobians of the maps of the following class. For any choice of $$m$$ positive integers $$\mathbf{a} = (a_1, \ldots, a_m)$$ with sum $$\sum_{j = 1}^m a_j = n$$, let us denote by $$\theta_{\mathbf{a}} : \mathbb C^m \to \mathbb C^m$$ the map defined by $\theta_{\mathbf{a}} (z) := (p_{\mathbf{a}, z}(z_1), \ldots, p_{\mathbf{a} , z}(z_m)),\qquad p_{\mathbf{a} , z}(\zeta) := \int_0^\zeta (w - z^1)^{a_1}\cdot \ldots \cdot (w - z_m)^{a_m} d w.$ The main result consists of an explicit formula for the Jacobian $$\mathbb J_{\mathbf{a}}(z)$$ of $$\theta_{\mathbf{a}}$$ at any point $$z$$. Such a formula implies that \textit{$$\mathbb J_{\mathbf{a}}(z)$$ is invertible if and only if the entries of $$z$$ are all distinct and non-zero}. As a corollary, the authors obtain that \textit{for any choice of a partition $$\lambda$$ of $$[n]$$ and for the corresponding stratum $$\mathbb C^n_{(\lambda)}$$ of $$\mathbb C^n$$ given by the points whose entries are such that $$z_i = z_j$$ for any $$i, j$$ belonging to the same block in $$\lambda$$, the restricted map $$\theta|_{\mathbb C^n_{(\lambda)} \setminus\{0\}}: \mathbb C^n_{(\lambda)}\setminus\{0\} \to \overline{ \mathbb C^n_{(\lambda)}}$$ is a local homeomorphism.} On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle https://zbmath.org/1472.30003 2021-11-25T18:46:10.358925Z "Erdélyi, Tamás" https://zbmath.org/authors/?q=ai:erdelyi.tamas The so-called Rudin-Shapiro polynomials are defined recursively by $P_{k+1}(z)=P_k(z)+z^{2^k}Q_k(z), \quad Q_{k+1}(z)=P_k(z)-z^{2^k}Q_k(z),$ for $$k=0,1,2,\ldots$$, and $$P_0(z)=Q_0(z)=1$$. Both, $$P_k(z),Q_k(z)$$ are polynomials of degree $$n-1$$ with $$n=2^k$$ and all coefficients are in $$\{-1,1\}$$. Since they have good autocorrelation properties and their values on the unit circle are small, these polynomials have applications in signal processing. In this paper, the author proves a series of theorems concerning the asymptotic (as $$n\to\infty$$) behaviour of the number of zeros of $$P_k(z),Q_k(z)$$. For instance, it is proved that $$P_k(z),Q_k(z)$$ have $$o(n)$$ zeros on the unit circle. On the number of non-real zeroes of a homogeneous differential polynomial and a generalisation of the Laguerre inequalities https://zbmath.org/1472.30004 2021-11-25T18:46:10.358925Z "Tyaglov, Mikhail" https://zbmath.org/authors/?q=ai:tyaglov.mikhail "Atia, Mohamed J." https://zbmath.org/authors/?q=ai:atia.mohamed-jalel Summary: Given a real polynomial $$p$$ with only real zeroes, we find upper and lower bounds for the number of non-real zeroes of the differential polynomial $F_\varkappa [p](z) \overset{d e f}{=} p(z) p''(z) - \varkappa [ p^\prime ( z ) ]^2,$ where $$\varkappa$$ is a real number. We also construct a counterexample to a conjecture by \textit{B.~Shapiro} [Arnold Math. J. 1, No.~1, 91--99 (2015; Zbl 1321.26032)] on the number of real zeroes of the polynomial $$F_{\frac{ n - 1}{ n}} [p](z)$$ in the case when the real polynomial $$p$$ of degree $$n$$ has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture. Multiplicity of zeros of polynomials https://zbmath.org/1472.30005 2021-11-25T18:46:10.358925Z "Totik, Vilmos" https://zbmath.org/authors/?q=ai:totik.vilmos The paper grew out of the known result of \textit{P. Erdős} and \textit{P. Túran} [Ann. Math. (2) 41, 162--173 (1940; Zbl 0023.02201)] on zero distributions and bounds for their multiplicities of monic polynomials with all their zeros in $$[-1,1]$$. Theorem 1.1. Let $$K$$ be a compact set consisting of pairwise disjoint $$C^{1+\alpha}$$-smooth Jordan curves or arcs lying exterior to each other. Given a monic polynomial $$P_n$$ of degree at most $$n$$ with a zero $$a\in K$$ of multiplicity $$m=m(a)$$, the following lower bound holds $\|P_n\|_k\ge e^{c\frac{m^2}{n}} (\mathrm{cap}\,K)^n, \qquad c>0,$ where $$\mathrm{cap}\,K$$ is the logarithmic capacity of $$K$$, $$\|\cdot\|_K$$ the supremum norm on~$$K$$. In the case when $$K$$ is an analytic Jordan curve or arc, the result turns out to be sharp. Periodic Schwarz-Christoffel mappings with multiple boundaries per period https://zbmath.org/1472.30006 2021-11-25T18:46:10.358925Z "Baddoo, Peter J." https://zbmath.org/authors/?q=ai:baddoo.peter-jonathan "Crowdy, Darren G." https://zbmath.org/authors/?q=ai:crowdy.darren-g Summary: We present an extension to the theory of Schwarz-Christoffel (S-C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the $$x$$-direction in an $$(x, y)$$-plane, three cases are considered; these differ in whether the period window extends off to infinity as $$y \rightarrow \pm \infty$$, or extends off to infinity in only one direction $$(y \rightarrow + \infty$$ or $$y \rightarrow - \infty )$$, or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S-C mapping formulae are shown to be expressible in terms of the Schottky-Klein prime function associated with the circular preimage domains. As usual for an S-C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results. One more note on neighborhoods of univalent functions https://zbmath.org/1472.30007 2021-11-25T18:46:10.358925Z "Fournier, Richard" https://zbmath.org/authors/?q=ai:fournier.richard The paper under review is aimed to pay tribute to the late German mathematician Stephan Ruscheweyh by raising some open questions on the concept of neighbourhood of a univalent function. Let $$\mathcal{A}_0$$ denote the class of analytic functions $$f$$ in the unit disk $$\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$$ that satisfy the conditions $f(0)=f^\prime (0)-1=0.$ Let $$f(z)=z+\sum_{n=2}^\infty a_nz^n$$ be an element of $$\mathcal{A}_0$$. It is well known that the condition $\sum_{n=2}^\infty |a_n|\le 1$ implies that $$f$$ is one-to-one (univalent) in $$\mathbb{D}$$, and $$f(\mathbb{D})$$ is star-like with respect to the origin. Let $$\mathcal{S}$$ denote the subclass of $$\mathcal{A}_0$$ of star-like functions (functions $$f$$ with the property that $$f(\mathbb{D})$$ is star-like with respect to the origin. In [Proc. Am. Math. Soc. 81, 521-527 (1981; Zbl 0458.30008)], \textit{S. Ruschweyh} introduced the following notion of $$\delta$$-neighbourhood for a given function $$f(z)=z+\sum_{n=2}^\infty a_nz^n$$: $N_\delta(f)=\left \{g(z)=z+\sum_{n=2}^\infty b_nz^n\in \mathcal{A}_0: \sum_{n=2}^\infty n|a_n - b_n|\le \delta\right \}.$ It is clear that if $$I$$ is the identity function, then $$N_1(I)$$ coincides with the class of star-like functions. It is also clear that the class of convex functions $$K$$ (those functions $$f\in\mathcal{A}_0$$ with the property that $$f(\mathbb{D})$$ is convex) is a subclass of $$\mathcal{A}_0$$. A result of Ruscheweyh states that: If $$f\in\mathcal{A}_0,\, \delta>0$$, and $\frac{f(z)+\epsilon z}{1+\epsilon}\in\mathcal{S},\quad -\delta <\epsilon <\delta,$ then $$N_\delta(f)\subseteq \mathcal{S}$$. Ruscheweyh asked in [loc. cit.] if this result is valid if $$\mathcal{S}$$ is replaced by the class $$C$$ of close-to-convex functions. The author discusses a partial answer to this question, and states that the question is still unsolved. On new $$p$$-valent meromorphic function involving certain differential and integral operators https://zbmath.org/1472.30008 2021-11-25T18:46:10.358925Z "Mohammed, Aabed" https://zbmath.org/authors/?q=ai:mohammed.aabed "Darus, Maslina" https://zbmath.org/authors/?q=ai:darus.maslina Summary: We define new subclasses of meromorphic $$p$$-valent functions by using certain differential operator. Combining the differential operator and certain integral operator, we introduce a general $$p$$-valent meromorphic function. Then we prove the sufficient conditions for the function in order to be in the new subclasses. A direct proof of Brannan's conjecture for $$\beta = 1$$ https://zbmath.org/1472.30009 2021-11-25T18:46:10.358925Z "Barnard, Roger W." https://zbmath.org/authors/?q=ai:barnard.roger-w "Richards, Kendall C." https://zbmath.org/authors/?q=ai:richards.kendall-c For $$z,\omega\in \mathbb{C}$$ with $$|z| < 1 = |\omega|$$ write $\frac{(1+\omega z)^\alpha}{(1-z)^\beta}=\sum_{n=0}^{\infty}\mathcal{A}_n(\alpha,\beta,\omega)z^n.$The coefficients $$\mathcal{A}_n$$ can be written as $\mathcal{A}_n(\alpha,\beta,\omega) = \frac{(\beta)_n}{n!}\,_2F_1(-n, -\alpha; 1-\beta - n; -\omega),$ where $$\,_2F_1$$ is the Gaussian hypergeometric function. Brannan's conjecture states that the inequality $\mathcal{A}_n(\alpha,\beta,e^{i\theta})\leq \mathcal{A}_n(\alpha,\beta,1)$holds for all odd $$n$$, $$\alpha,\beta\in(0,1]$$ and $$\theta\in (-\pi,\pi].$$ Recent results of several authors provide a proof of Brannan's conjecture for the case that $$\beta=1$$, which relies on a computer-assisted argument. The present paper presents a direct analytical proof of this result. A removability theorem for Sobolev functions and detour sets https://zbmath.org/1472.30010 2021-11-25T18:46:10.358925Z "Ntalampekos, Dimitrios" https://zbmath.org/authors/?q=ai:ntalampekos.dimitrios Removability of compact sets for continuous Sobolev functions is studied, i.e., if $$K$$ be a compact set in $$\mathbb{R}^n$$ and $$f \in C(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n\setminus K)$$, is $$f \in W^{1,p}(\mathbb{R}^n)$$? The problem is intimately connected to the removability problem for quasiconformal maps: If $$f: U \rightarrow\mathbb{R}^n$$ is a homeomorphism and $$f|U \setminus K$$ is quasiconformal, is $$f$$ quasiconformal in $$U$$? The stronger removability, without the continuity assumption, for Sobolev functions has been studied by [\textit{P. Koskela}, Ark. Mat. 37, No. 2, 291--304 (1999; Zbl 1070.46502)]. In the plane it has been shown that boundaries of domains $$\Omega$$ satisfying the quasihyperbolic boundary condition and, in particular, boundaries of John domains are removable [\textit{P. Jones} and \textit{S. Smirnov}, Ark. Math. 38, No. 2, 363--379 (2000)]. The author concentrates on sets $$K$$ which have infinitely many complementary components. A typical such set is the standard $$1/3$$-Sierpinski carpet $$S$$ in the plane which is not $$W^{1,p}$$-removable for any $$p \geq 1$$ and so the author focuses on the Sierpinski and Apollonian gaskets. The first is constructed using triangles and the latter using disks. It is shown that the planar Sierpinski and Apollonia gaskets are removable for $$p > 2$$. The proof, which holds in $$\mathbb{R}^n$$, is based on the result concerning detour sets. This means, roughly speaking, that the set $$K$$ has the property that for almost every line $$L$$ intersecting $$K$$ there is a path which intersects only finitely many complementary components of $$K$$ and still remains arbitrarily close to $$L$$. In addition to this the complementary components $$D$$ of $$K$$ need to be uniformly Hölder. It is shown that for a Hölder domain each point $$x \in \partial D$$ can be reached from the base point by a quasihyperbolic geodesic, see also [\textit{O. Martio} and \textit{J. Väisälä}, Pure Appl. Math. Q. 7, No. 2, 395--409 (2011; Zbl 1246.30041)]. The results are used to obtain equivalent conditions for a homeomorphism $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$ that is quasiconformal in $$\mathbb{R}^2 \setminus K$$ where $$K$$ is a Sierpinki gasket to be quasiconformal in $$\mathbb{R}^2$$. The paper also contains a nice overview of removability results for Sobolev functions outside different Sierpinski-type sets. A note on Schwarz's lemma https://zbmath.org/1472.30011 2021-11-25T18:46:10.358925Z "Mashreghi, Javad" https://zbmath.org/authors/?q=ai:mashreghi.javad The author uses the classical ideas related to the Alhlfors-Schwarz lemma and involving conformal metrics $$\rho$$ and their Gaussian curvature $$k_\rho$$ to prove the following result: Suppose that $$\rho_1,\rho_2$$ are two metrics with strictly negative curvature on a planar domain $$\Omega$$. Assume that $$\rho_2$$ is strictly positive and continuous on $$\Omega$$, and that $$\rho_1/\rho_2$$ attains its maximum inside $$\Omega$$. Then for every $$z\in\Omega$$, $\frac{\rho_1(z)}{\rho_2(z)}\leq \left \|\frac{k_{\rho_1}(z)}{k_{\rho_2}(z)}\right \|_\infty^{1/2}.$ The author gives an application of this result for extremal metrics on star-shaped domains. Differential subordinations for nonanalytic functions https://zbmath.org/1472.30012 2021-11-25T18:46:10.358925Z "Oros, Georgia Irina" https://zbmath.org/authors/?q=ai:oros.georgia-irina "Oros, Gheorghe" https://zbmath.org/authors/?q=ai:oros.gheorghe Summary: In the paper [Math. Rev. Anal. Numér. Théor. Approximation, Math. 22(45), 77--83 (1980; Zbl 0457.30038)], \textit{P. T. Mocanu} has obtained sufficient conditions for a function in the classes $$C^1(U)$$, respectively, and $$C^2(U)$$ to be univalent and to map $$U$$ onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper [Math. Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 10, 75--79 (1981; Zbl 0481.30014)], \textit{P. T. Mocanu} has obtained sufficient conditions of univalency for complex functions in the class $$C^1$$ which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes $$C^1$$ and $$C^2$$ following the classical theory of differential subordination for analytic functions introduced by \textit{S. S. Miller} and \textit{P. T. Mocanu} in their papers [J. Math. Anal. Appl. 65, 289--305 (1978; Zbl 0367.34005); Mich. Math. J. 28, 157--171 (1981; Zbl 0439.30015)] and developed in their book [Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)]. Let $$\Omega$$ be any set in the complex plane $$\mathbb{C}$$, let $$p$$ be a nonanalytic function in the unit disc $$U$$, $$p \in C^2(U)$$, and let $$\psi(r, s, t; z) : \mathbb{C}^3 \times U \rightarrow \mathbb{C}$$. In this paper, we consider the problem of determining properties of the function $$p$$, nonanalytic in the unit disc $$U$$, such that $$p$$ satisfies the differential subordination $$\psi(p(z), D p(z), D^2 p(z) - D p(z); z) \subset \Omega \Rightarrow p(U) \subset \Delta$$. Correction to: Interrelation between Nikolskii-Bernstein constants for trigonometric polynomials and entire functions of exponential type'' https://zbmath.org/1472.30013 2021-11-25T18:46:10.358925Z "Gorbachev, Dmitriĭ Viktorovich" https://zbmath.org/authors/?q=ai:gorbachev.dmitrii-viktorovich "Mart'yanov, Ivan Anatol'evich" https://zbmath.org/authors/?q=ai:martyanov.ivan-anatolevich Correction to the authors' paper [ibid. 20, No. 3(71), 143--153 (2019; Zbl 1439.30050)]. Entire functions with separated zeros and 1-points https://zbmath.org/1472.30014 2021-11-25T18:46:10.358925Z "Bergweiler, Walter" https://zbmath.org/authors/?q=ai:bergweiler.walter "Eremenko, Alexandre" https://zbmath.org/authors/?q=ai:eremenko.alexandre-e The paper concerns transcendental entire functions of finite order with zeros and $$1$$-points in disjoint sectors. By a theorem of Biernacki, it is not possible for the set of all zeros of a transcendental entire function of finite order to accumulate in one direction and the set of all $$1$$-points to accumulate in another. Here by the statement that the set of $$a$$-points is accumulating in a direction, it is understood that there exists a ray such that for every open sector bisected by this ray all but possibly finite number of $$a$$-points lie in the sector. The authors present three generalizations of Biernacki's theorem. The results concern possible existence and form of entire functions with zeros and $$1$$-points in disjoint sectors. One of the results is the following. Theorem 1.1. Let $$S_0$$ and $$S_1$$ be closed sectors in $$\mathbb{C}$$ satisfying $$S_0\cap S_1=\{0\}.$$ Let $$\theta_j$$ denote the opening angle of $$S_j$$ and suppose that $\min\{\theta_0,\theta_1\}<\frac{\pi}{2}\mbox{ and } \max\{\theta_0,\theta_1\}<\pi.$ Then there is no transcendental entire function of finite order for which all but finitely many zeros are in $$S_0$$ while all but finitely many $$1$$-points are in $$S_1.$$ The other results also involve hypotheses concerning upper bounds for the opening angles of the sectors in question. The results are illustrated with examples showing exactness of the hypotheses. Blow-up solutions of Liouville's equation and quasi-normality https://zbmath.org/1472.30015 2021-11-25T18:46:10.358925Z "Grahl, Jürgen" https://zbmath.org/authors/?q=ai:grahl.jurgen "Kraus, Daniela" https://zbmath.org/authors/?q=ai:kraus.daniela "Roth, Oliver" https://zbmath.org/authors/?q=ai:roth.oliver Let $$D$$ be a domain in the complex plane and $$C > 0$$. Let $$\mathcal{F}_C$$ be the set of all functions $$f$$ meromorphic in $$D$$ for which the spherical area of $$f(D)$$ on the Riemann sphere is at most $$C \pi$$. Then it is shown that $$\mathcal{F}_C$$ is quasi-normal of order at most $$C$$. In particular, for every sequence $$\{ f_m \}$$ in $$\mathcal{F}_C$$ (after taking a subsequence), there is an $$f$$ in $$\mathcal{F}_C$$ such that (1) or (2) below holds. (1) $$\{ f_m \}$$ converges locally uniformly in $$D$$ to $$f$$; (2) There exists a finite nonempty set $$S \subset D$$ with at most $$C$$ points for which (2a) $$\{ f_m \}$$ converges locally uniformly in $$D \backslash S$$ to $$f$$, and for each $$p$$ in $$S$$ there exists a sequence $$\{ z_m \}$$ in $$D$$ such that $$\{z_m \}$$ converges to $$p$$ and $$\{ f_m^{\#} (z_m) \}$$ converges to $$+\infty$$; and (2b) for each $$p$$ in $$S$$ there exists a real number $$\alpha_p \geq 1$$ such that in the measure theoretic sense $\frac{1}{\pi}(f_m^{\#})^2\text{ converges to } \sum_{p\in S}\alpha_p\delta_p+\frac{1}{\pi}(f^{\#})^2.$ The authors note that the above may be viewed as extending to all meromorphic functions in $$\mathcal{F}_C$$ some well-known work of \textit{H. Brézis} and \textit{F. Merle} [Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006)] on solutions of $$-\Delta u =4e^{2u}$$ for locally univalent meromorphic functions. In the comparison (2a) may be seen to correspond with Bubbling'', while (2b) corresponds with Mass Concentration'' in the Brezis-Merle work. Section 2 of the current manuscript contains a lengthy set of remarks and questions (including open questions) regarding the above comparison, while Section 3 on quasi-normality observes a criterion of Montel and Valiron may be applied to obtain $$\mathcal{F}_C$$ quasi-normal. Also introduced in Section 3 is an extension of the Montel-Valiron criterion for quasi-normality where exceptional values are replaced by exceptional functions allowed to depend on the individual members of the family. Erratum to: The local universality of Muttalib-Borodin ensembles when the parameter $$\theta$$ is the reciprocal of an integer'' https://zbmath.org/1472.30016 2021-11-25T18:46:10.358925Z "Molag, L. D." https://zbmath.org/authors/?q=ai:molag.leslie-d Erratum to the author's paper [ibid. 34, No. 5, 3485--3564 (2021; Zbl 1468.30066)]. Maximally stretched laminations on geometrically finite hyperbolic manifolds https://zbmath.org/1472.30017 2021-11-25T18:46:10.358925Z "Guéritaud, François" https://zbmath.org/authors/?q=ai:gueritaud.francois "Kassel, Fanny" https://zbmath.org/authors/?q=ai:kassel.fanny Summary: Let $$\Gamma_0$$ be a discrete group. For a pair $$(j,\rho)$$ of representations of $$\Gamma_0$$ into $$\operatorname{PO}(n,1)=\operatorname{Isom}(\mathbb{H}^n)$$ with $$j$$ geometrically finite, we study the set of $$(j,\rho)$$-equivariant Lipschitz maps from the real hyperbolic space $$\mathbb{H}^n$$ to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is maximally stretched'' by all such maps when the minimal constant is at least $$1$$. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups $$\Gamma$$ of $$\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)$$ on $$\operatorname{PO}(n,1)$$ by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups $$\Gamma$$ the action remains properly discontinuous after any small deformation of $$\Gamma$$ inside $$\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)$$. Pluripotential theory on Teichmüller space. I: Pluricomplex Green function https://zbmath.org/1472.30018 2021-11-25T18:46:10.358925Z "Miyachi, Hideki" https://zbmath.org/authors/?q=ai:miyachi.hideki In the paper [Bull. Am. Math. Soc., New Ser. 27, No. 1, 143--147 (1992; Zbl 0766.30016)], \textit{S. L. Krushkal} announced the following result on the pluricomplex Green function on Teichmüller space: Theorem. Let $$T_{g, m}$$ be the Teichmüller space of Riemann surfaces of analytically finite-type $$(g,m)$$, and let $$d_T$$ be the Teichmüller distance on $$T_{g, m}$$. Then, the pluricomplex Green function $$g_{T_{g, m}}$$ on $$T_{g, m}$$ satisfies $g_{T_{g, m}}(x, y)\log \tanh d_T(x, y)$ for $$x, y \in g_{T_{g, m}}$$. The author has a programm to investigate of the pluripotential theory on Teichmüller space. In this first one of a series of works, the author gave an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. In comparison with the original approach by Krushkal, the strategy is more direct here. He first shows that the Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. Then he gives a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas' symplectic structure on the space of holomorphic quadratic differentials [\textit{D. Dumas}, Acta Math. 215, No. 1, 55--126 (2015; Zbl 1334.57020)]. Counting one-sided simple closed geodesics on Fuchsian thrice punctured projective planes https://zbmath.org/1472.30019 2021-11-25T18:46:10.358925Z "Magee, Michael" https://zbmath.org/authors/?q=ai:magee.michael Since the work of \textit{M. Mirzakhani} [Ann. Math. (2) 168, No. 1, 97--125 (2008; Zbl 1177.37036)], it is known that for an orientable surface $$S$$ with a hyperbolic metric $$J$$ of finite area, the number of $$n_{J}(L)$$ of isotopy classes of simple closed curves of length at most $$L$$ satisfies the asymptotic formula $$n_{J}(L)=cL^{d}+o(L^{d})$$ where $$c$$ is a constant depending on $$J$$ and $$d$$ is the (integer) dimension of a space of measured laminations on $$S$$ ($$d$$ only depends on the topology of the surface). This paper provides an analogous asymptotic formula $$cL^{\beta}+o(L^{\beta})$$ in the case of the (nonorientable) real projective plane $$\Sigma$$ with three punctures. In contrast with the case of orientable hyperbolic surfaces, exponent $$\beta$$ is not an integer and is estimated to be in the range $$2.430< \beta < 2.477$$. The proof is obtained by an identification of the Teichmüller space of $$\Sigma$$ with an algebraic variety $$V$$ defined by a Markoff-Hurwitz equation, drawing on [\textit{Y. Huang} and \textit{P. Norbury}, Geom. Dedicata 186, 113--148 (2017; Zbl 1360.30040)]. Then, the action of the mapping class group on the curve complex of $$\Sigma$$ is related to the action of the combinatorial symmetries of $$V$$ (the so-called Markoff moves). Finally, the counting of integer points of the variety provides the estimate of the number of isotopy classes of simple closed curves. On linearly independent solutions of the homogeneous Schwarz problem https://zbmath.org/1472.30020 2021-11-25T18:46:10.358925Z "Nikolaev, V. G." https://zbmath.org/authors/?q=ai:nikolaev.vladimir-gennadevich Summary: We study the homogeneous Schwarz problem for Douglis analytic functions. We consider two-dimensional matrices $$J$$ with a multiple eigenvalue and a eigenvector, which is not proportional to a real vector. We obtain a sufficient condition for the matrix $$J$$ under which there exist two linearly independent solutions of the problem defined in a certain domain $$D$$. We present an example. Zeros of slice functions and polynomials over dual quaternions https://zbmath.org/1472.30021 2021-11-25T18:46:10.358925Z "Gentili, Graziano" https://zbmath.org/authors/?q=ai:gentili.graziano "Stoppato, Caterina" https://zbmath.org/authors/?q=ai:stoppato.caterina "Trinci, Tomaso" https://zbmath.org/authors/?q=ai:trinci.tomaso Starting with the class of slice functions over a real alternative *-algebra introduced by \textit{R. Ghiloni} and \textit{A. Perotti} [Adv. Math. 226, No. 2, 1662--1691 (2011; Zbl 1217.30044)], the authors extend the class of slice functions formely introduced by the first author and \textit{D. C. Struppa} [C. R., Math., Acad. Sci. Paris 342, No. 10, 741--744 (2006; Zbl 1105.30037)] to the algebra of dual quaternions. The main goal is a faithful classification of the zeros of slice functions in interplay with the problem of factorizing motion polynomials. To this end, the authors construct the \textit{primal part} of slice functions to exploit some well-known results over the algebra of [commutative] polynomials that seemlessly comprises the algebraic and geometric nature of the dual quaternions. In particular, the authors establish with the proof of Theorem 5.2. and Proposition 5.3. a tangible interplay between the zero set of a slice function with the zero set of its primal part. With the proof of Theorem 6.4. -- depicted on Table 1. (see p. 5532) -- the authors classify in detail the zero set of slice products. Finally, with the proof of Theorems 7.2. and 7.3. they study the discreteness of the zeros of slice regular functions. Finally, the results obtained are applied in Section 8 to enrich the classification obtained [\textit{G. Hegedüs} et al. [Factorization of rational curves in the study quadric'', Mech. Mach. Theory 69, 142--152 (2013; \url{doi:10.1016/j.mechmachtheory.2013.05.010})]. As a whole, this approach proved to be fruitful much beyond the traditional study of rings of noncommutative polynomials. For example, the group of rigid body transformations $$\operatorname{SE}(3)$$, obtained through the $$1-1$$ identification of a vector $$(x_1,x_2,x_3)$$ of $$\mathbb{R}^3$$ as a dual quaternion $$1+\epsilon (x_1 i+x_2j+x_3k)$$ as well as the corresponding translation and rotation invariants (see Subsection 8.1.) lies on the spirit of Cartan's generalization of Klein's Erlangen program. Extension and tangential CRF conditions in quaternionic analysis https://zbmath.org/1472.30022 2021-11-25T18:46:10.358925Z "Maggesi, Marco" https://zbmath.org/authors/?q=ai:maggesi.marco "Pertici, Donato" https://zbmath.org/authors/?q=ai:pertici.donato "Tomassini, Giuseppe" https://zbmath.org/authors/?q=ai:tomassini.giuseppe The paper deals with some extension results for the class of quaternionic functions of several quaternionic variables that are regular in the sense of Fueter. After a general introduction on Fueter regularity in one and several variables, the authors define the notion of \textit{domain splitting} for an open domain $$\Omega\subset\mathbb{H}^2$$, given by a triple $$(S,U^+,U^-)$$, where $$S$$ is a smooth closed hypersurface of $$\Omega$$ and $$U^+$$, $$U^-$$ are open and such that $$\Omega\setminus S=U^+\cup U^-$$ and their boundary is $$S$$. A smooth function $$f:S\to\mathbb{H}$$ is called a \textit{smooth jump} relative to a domain splitting $$(S,U^+,U^-)$$ if can be extended by two Fueter regular functions $$F^+$$ and $$F^-$$ on $$U^+$$ and $$U^-$$, respectively. Then, the authors introduce the notion of \textit{admissible function} and, after a deep and detailed analysis of this concept, they are able to prove their first main theorem stating the following: given a convex domain $$\Omega$$ and a splitting $$(S,U^+,U^-)$$, then if $$f:S\to\mathbb{H}$$ is a smooth admissible function, it is in fact a smooth jump. As applications of this result, the authors prove two extension results. After this neat analysis, the paper continues with a section deepening weak conditions in order to obtain the same mentioned results and with an appendix concerning the octonionic context. Quaternionic inner and outer functions https://zbmath.org/1472.30023 2021-11-25T18:46:10.358925Z "Monguzzi, Alessandro" https://zbmath.org/authors/?q=ai:monguzzi.alessandro "Sarfatti, Giulia" https://zbmath.org/authors/?q=ai:sarfatti.giulia "Seco, Daniel" https://zbmath.org/authors/?q=ai:seco.daniel The paper contributes to a deeper understanding and characterization of the outer and inner functions in the non-commutative quaternionic context. These functions have proven to be useful in a factorization theorem for the Hardy space of the so-called slice regular functions in the quaternionic unit ball. Higher order Dirichlet-type problems in 2D complex quaternionic analysis https://zbmath.org/1472.30024 2021-11-25T18:46:10.358925Z "Schneider, B." https://zbmath.org/authors/?q=ai:schneider.berthold.1|schneider.bruce|schneier.bruce|schneider.barry-i|schneider.brandon|schneider.bernd|schneider.baruch|schneider.brit|schneider.bernhard|schneider.barbara The author studies a higher-order Dirichlet boundary value problem for solutions of the two-dimensional Helmoltz equation in $$\mathbb R^2$$. In Section 1, Introduction, the author poses the problem to be solved. In Section 2, Preliminaries, the author recalls the following: \begin{enumerate} \item[(1)] The Dirac operator in $$\mathbb R^2$$: $\displaystyle \mathcal D = \frac{\partial }{\partial x } e_1 + \frac{\partial }{\partial y } e_2 ;$ \item[(2)] The Helmholtz operator in $$\mathbb R^2$$: $$\displaystyle \Delta_{\mathbb R^2} + \lambda \mathcal I$$, where $$\mathcal I$$ denotes the identity operator and $$\lambda \in \mathbb C \setminus \{ 0 \}$$; \item[(3)] The factorization $\displaystyle - \left( \mathcal D + \alpha \mathcal I \right) \left( \mathcal D - \alpha \mathcal I \right) = \Delta_{\mathbb R^2} + \lambda \mathcal I ,$ where $$\alpha$$ is a complex number such that $$\alpha^2 = \lambda$$; \item[(4)] The perturbed Dirac operator: $$\mathcal D_\alpha := \mathcal D + \alpha \mathcal I$$. \item[(5)] The set of $$\alpha$$-hyperholomorphic functions on $$\Omega \subset \mathbb R^2$$, i.e., the set of functions defined on $$\Omega$$ such that $$\mathcal D_\alpha f =0$$. \item[(6)] The fundamental solution $$\Theta_\alpha^{(M)} (z)$$, of the operator $$\displaystyle \left( \Delta_{\mathbb R^2} + \alpha^2 \right)^M$$ in $$\mathbb R^2$$, where $$M \in \mathbb N$$ and the Cauchy Kernel $$\displaystyle \mathcal K_\alpha (z) := - \mathcal D_{- \alpha} \left[ \Theta_\alpha^{(1)} (z) \right]$$; \end{enumerate} In Section 3.1, the author reminds us the definitions and main properties of the Teodorescu integral operator, the Cauchy integral operator and the singular integral operator. In Section 3.2 the author examines an orthogonal decomposition of the Sobolev space $$\displaystyle \mathbf W^k_2 \left( \Omega , \, \mathbb H ( \mathbb C) \right)$$, $$k \in \mathbb N \cup \{ 0 \}$$ with respect to the high-order Dirac operator $$\mathcal D_\alpha^k$$. Using the formula $\displaystyle \mathcal D \Theta_\alpha^{ (M)} (z) = \frac{1}{ 2 (M-1) } \Theta_\alpha^{ (M - 1)} (z) , \quad M \in \mathbb N, \; \; M \not= 1, \; \; \forall z \in \mathbb R^2 \setminus \{ 0 \} ,$ a formula for $$\mathcal D_\alpha \Theta_\alpha^{(M)}$$ is obtained. Hence two orthogonal decompositions of the Sobolev space $$\displaystyle \mathbf W^k_2 \left( \Omega , \, \mathbb H ( \mathbb C) \right)$$ are given in Theorem 3.8. In Theorem 3.11, it is proved that given $$f \in \mathbf W_2^k (\Omega, \mathbb H ( \mathbb C))$$, $$g_0 \in \mathbf W_2^{k + \frac{7}{2}} (\Gamma, \mathbb H ( \mathbb C))$$, and $$g_1 \in \mathbf W_2^{k + \frac{3}{2} } (\Gamma, \mathbb H ( \mathbb C))$$, $$k \geq 0$$, then the following boundary value problem has a unique solution $$h \in \mathbf W_2^{k +4} (\Omega, \mathbb H ( \mathbb C))$$: $\left\{ \begin{array}{l} \left( - \Delta_{\mathbb R^2} + 2 \alpha \mathcal D + \alpha^2 \right)^2 h = f \quad \text{in} \quad \Omega , \\ h= g_0 \quad \text{on} \quad \Gamma , \\ \left( - \Delta_{\mathbb R^2} + 2 \alpha \mathcal D + \alpha^2 \right) h = g_1 \quad \text{on} \quad \Gamma. \end{array} \right.$ At the end of the paper, in Theorem 3.13, it is proved that, given $$f \in \mathbf W_2^k (\Omega, \mathbb H ( \mathbb C))$$ and $$g_p \in \mathbf W_2^{k + 2M \frac{4p +1 }{2}} (\Gamma, \mathbb H ( \mathbb C))$$, then the following boundary value problem of higher order has a unique solution $$h \in \mathbf W_2^{k +2M} (\Omega, \mathbb H ( \mathbb C))$$: $\left\{ \begin{array}{l} \left( - \Delta_{\mathbb R^2} + \lambda \right)^M h = f \quad \text{in} \quad \Omega , \\ h= g_0 \quad \text{on} \quad \Gamma , \\ \left( - \Delta_{\mathbb R^2} + \lambda \right) h = g_1 \quad \text{on} \quad \Gamma , \\ \vdots \\ \left( - \Delta_{\mathbb R^2} + \lambda \right)^{M-1} h = g_{M - 1} \quad \text{on} \quad \Gamma . \end{array} \right.$ Generalized integration operators on Hardy spaces https://zbmath.org/1472.30025 2021-11-25T18:46:10.358925Z "Chalmoukis, Nikolaos" https://zbmath.org/authors/?q=ai:chalmoukis.nikolaos Let $H^p$, with a positive exponent $p$, be the Hardy space of analytic function in the unit disc, i.e., functions $f$ holomorphic in $\mathbb D$ such that $$\|f\|_p^p:=\frac1{2\pi}\sup_{0<r<1}\int_0^{2\pi}|f(re^{i\theta})|^p\,d\theta<\infty.$$ Let $I$ be the integration operator, i.e., $If(z):=\int_0^zf(t)\,dt$. Fixing now an analytic symbol $g$ and a sequence of coefficients $a=(a_1,\dots,a_{n-1})\in \mathbb C^{n-1}$, define the generalized integration operator $T_{g,a}$ by $$T_{g,a}f(z):=I^n\left(fg^{(n)}+a_1f'g^{(n-1)}+\cdots+a_{n_1}f^{(n-1)}g'\right)(z).$$ The paper gives sufficient and necessary conditions on $g$ for the boundedness and compactness of the operator $T_{g,a}$. Indeed, the author obtains \begin{enumerate} \item $T_{g,a}$ is bounded from $H^p$ to itself if and only if $g\in \operatorname{BMOA}$; \item $T_{g,a}$ is compact from $H^p$ to itself if and only if $g\in \operatorname{VMOA}$. \end{enumerate} Moreover, the author also proves that: Let $0<p<q<\infty$ and $a\in\mathbb C^{n-1}$. If $g\in H^s$, where $\frac1s=\frac1q-\frac1p$, then $T_{g,a}$ is bounded from $H^p$ to $H^q$. In the special case that $n=2$ and $a=0$, the boundedness of $T_{g,a}$ from $H^p$ to $H^q$ implies $g\in H^s$. A characterization of one-component inner functions https://zbmath.org/1472.30026 2021-11-25T18:46:10.358925Z "Nicolau, Artur" https://zbmath.org/authors/?q=ai:nicolau.artur "Reijonen, Atte" https://zbmath.org/authors/?q=ai:reijonen.atte In this interesting paper the authors study the class $$\mathcal I_c$$ of one-component inner functions. This class comprises all inner functions $$\Theta$$ whose level set $$\{z\in\mathbb D:|\Theta(z)|<\epsilon\}$$ is connected for some $$\epsilon\in ]0,1[$$ (hence for all bigger $$\epsilon$$). Their goal is to answer several questions by \textit{J. Cima} and the rewiever [Complex Anal. Synerg. 3, Paper No. 2, 15 p. (2017; Zbl 1364.30064); in: Advancements in complex analysis. From theory to practice. Cham: Springer. 39--49 (2020; Zbl 1442.30064)]. Using Carleson squares, the authors first give a characterization of the class $$\mathcal I_c$$ in terms of the location of the zeros and mass distribution of the singular measures associated with $$\Theta$$. Let us give some sample results: if $$B$$ is a Blaschke product with infinitely many zeros contained in a Stolz angle with vertex at $$e^{i\theta}$$, then $$B\in\mathcal I_c$$ if and only if $$\limsup_{r\to 1} |B(re^{i\theta})|<1$$. Also, if $$\Theta$$ is any inner function whose boundary singular set has Lebesgue measure zero, then there is an interpolating Blaschke product $$B$$ such that $$B\Theta\in\mathcal I_c$$. It is also shown that singular inner functions associated with a symmetric Cantor measure belong to $$\mathcal I_c$$. Finally, if $$\sigma$$ is a positive singular measure supported on a closed countable set $$E\subset \mathbb T$$ with $$\sigma(\lambda)>0$$ for all $$\lambda \in E$$, then the associated singular inner function $$I_\sigma$$ belongs to $$\mathcal I_c$$. An explicit example of a discrete singular inner function is constructed which does not belong to $$\mathcal I_c$$. Potential theory on minimal hypersurfaces. I: Singularities as Martin boundaries https://zbmath.org/1472.30027 2021-11-25T18:46:10.358925Z "Lohkamp, Joachim" https://zbmath.org/authors/?q=ai:lohkamp.joachim The author develops a detailed potential theory on (almost) minimizing hypersurfaces applicable to large classes of linear elliptic second-order operators. Let $$H$$ be an (almost) minimizing hypersurface containing the singularity set $$\Sigma \subset H$$. By $$\mathcal S$$-uniformity, we can regard $$H \setminus \Sigma$$ as a generalized convex set and $$\Sigma$$ as its boundary. Then the author proves a generalized boundary Harnack inequality, and use it to deduce other interesing results concerning Martin theory, the Dirichlet problem and Hardy inequalities. Properties of normal harmonic mappings https://zbmath.org/1472.31002 2021-11-25T18:46:10.358925Z "Deng, Hua" https://zbmath.org/authors/?q=ai:deng.hua "Ponnusamy, Saminathan" https://zbmath.org/authors/?q=ai:ponnusamy.saminathan "Qiao, Jinjing" https://zbmath.org/authors/?q=ai:qiao.jinjing Let $$\mathbb{D}$$ denote the open unit disk in the complex plane and let $$\rho$$ denote the hyperbolic distance on $$\mathbb{D}$$. A harmonic mapping $$f:\mathbb{D}\to \mathbb{C}$$ is said to be normal if $$f$$ is Lipschitz as mapping from the hyperbolic disk to the extended plane endowed with the chordal distance $$\chi$$, that is, $$\sup_{z\not=w} \chi(f(z), f(w))/\rho(z,w)<\infty$$. The condition turns out to be equivalent to $\alpha:=\sup_{z \in \mathbb{D}} (1-|z|)^2 f^\#(z)<\infty,$ where $$f^\#=(|h'|+|g'|)/(1+|f|^2)$$ if $$f=h+\overline{g}$$ with $$f, g$$ holomorphic in $$\mathbb{D}$$, and in the case of meromorphic functions $$f$$ equivalent to the normality of the family $$\{f \circ \varphi: \varphi \in\mathrm{Aut}(\mathbb{D})\}$$. In this paper the authors present necessary, sufficient and also necessary and sufficient conditions for a harmonic $$f$$ to be normal, and they discuss a maximum principle in terms of $$\alpha$$. Moreover, they show that the existence of an asymptotic value at a point $$\xi\in \partial \mathbb{D}$$ already implies the existence of the non-tangential limit at $$\xi$$ and they investigate sequences of normal harmonic functions. Finally, a five-point theorem for sense-preserving harmonic functions is proven. Continuity of condenser capacity under holomorphic motions https://zbmath.org/1472.31005 2021-11-25T18:46:10.358925Z "Pouliasis, Stamatis" https://zbmath.org/authors/?q=ai:pouliasis.stamatis A condenser in the complex plane $$\mathbb{C}$$ is a pair $$(E,F)$$ where $$E$$ and $$F$$ are non-empty disjoint compact subsets of $$\mathbb{C}$$. A holomorphic motion of a set $$A \subset \mathbb{C}$$, parameterized by a domain $$D \subset \mathbb{C}$$ containing $$0$$, is a map $$f:D \times A \mapsto \mathbb{C}$$ such that $$f(\cdot,z )$$ is holomorphic in $$D$$ for any fixed $$z\in A$$, $$f(\lambda,\cdot ):=f_\lambda (\cdot)$$ is an injection for any fixed $$\lambda \in D$$ and $$f(0,\cdot )$$ is the identity on $$A$$. If $$(E,F)$$ is a condenser with positive capacity and $$f$$ is a holomorphic motion of $$E\cup F$$ parameterized by a domain $$D$$ containing $$0$$, then $$(f_\lambda (E),f_\lambda (F))$$ is also a condenser. In the paper under review the author proves that the capacity of $$(f_\lambda (E),f_\lambda (F))$$ is a continuous subharmonic function on $$D$$. Moreover, he shows that the equilibrium measure of $$(f_\lambda (E),f_\lambda (F))$$ is continuous with respect to weak-star convergence. A condenser $$(E,F)$$ is called a ring if both $$E$$ and $$F$$ are connected and $$\mathbb{C} \backslash (E\cup F)$$ is a doubly connected domain. One way to characterize uniform perfectness is the following. A compact set $$K\subset \mathbb{C}$$ is uniformly perfect if and only if the supremum of the equilibrium energy of all the rings that separate $$K$$ is finite. Let $$P(K)$$ denote this supremum. If $$K$$ is a uniformly perfect compact set and $$f$$ is a holomorphic motion parameterized by a bounded domain $$D$$ containing $$0$$, then the author finds an upper and a lower estimate for $$P(f_\lambda (K))$$ involving the Harnack distance. The paper is well organized and helps the reader to follow it. Corrigendum to: Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces'' https://zbmath.org/1472.31015 2021-11-25T18:46:10.358925Z "Björn, Anders" https://zbmath.org/authors/?q=ai:bjorn.anders "Björn, Jana" https://zbmath.org/authors/?q=ai:bjorn.jana Corrigendum to the authors' paper [ibid. 266, No. 1, 44--69 (2019; Zbl 1420.31002)]. Moduli space of meromorphic differentials with marked horizontal separatrices https://zbmath.org/1472.32005 2021-11-25T18:46:10.358925Z "Boissy, Corentin" https://zbmath.org/authors/?q=ai:boissy.corentin A (compact) translation surface is a pair $$(X, \omega)$$, where $$X$$ is a (compact) Riemann surface and $$\omega$$ is a holomorphic 1-form on the surface. Locally integrating the form defines a flat metric on the surface, with conical singularities. If one only asks the form $$\omega$$ to be \emph{meromorphic}, obtaining a non-compact translation surface with infinite area (if the poles of $$\omega$$ are not all simple). The study of such objects is motivated by the fact that they appear naturally when dealing with compactifications of the moduli space of translation surfaces. More precisely, if a sequence $$(X_n,\omega_n)$$ converges to the boundary of the moduli space in the Deligne-Mumford compactification, then the thick components of $$X_n$$, appropriately rescaled, converge to meromorphic differentials, see, e.g., [\textit{A. Eskin} et al., Publ. Math., Inst. Hautes Étud. Sci. 120, 207--333 (2014; Zbl 1305.32007)]. In this article, the author studies the topology of the moduli spaces of translation surfaces with poles equipped with an extra combinatorial data: the choice, for each singularity of (an equivalence class of) horizontal separatrix, denoted $$\mathcal{H}^{\text{hor}}$$. Here, by horizontal separatrix we mean either an horizontal geodesic line ending (or beginning) at a conical singularity or an horizontal geodesic going to infinity in the flat metric if the singularity is a non simple pole. The main result of the paper is a complete characterization of the connected components of $$\mathcal{H}^{\text{hor}}$$. Similarly to the case of compact translation surfaces, proven in [\textit{M. Kontsevich} and \textit{A. Zorich}, Invent. Math. 153, No. 3, 631--678 (2003; Zbl 1087.32010)], in the general case of genus greater than 1 and underlining surfaces not belonging to the hyperelliptic component, there are at most 2 connected components, classified but an invariant $$\operatorname{Sp}$$ which is a generalization of the classical $$\operatorname{Arf}$$. In the hyperelliptic case the extra symmetry of the underlining surface yields more components. Finally, the genus-0 case is the most complicated one, depending also on the combinatorics of the singularities. The main result is obtained by reducing the problem to the study of some cyclic group coming for the forgetful map from $$\mathcal{H}^{\text{hor}}$$ to $$\mathcal{H}^{\text{sing}}$$, which is the space of translation surfaces with poles in which singularities have given names. The latter space has the same connected components of the moduli space of translation with poles, which were classified by the author in [Comment. Math. Helv. 90, No. 2, 255--286 (2015; Zbl 1323.30060)]. Generalizing two local surgeries introduced in the article by Kontsevich and Zorich [loc. cit.], called \emph{breaking up a zero} and \emph{bubbling a handle}, so that they can be performed also on poles, one constructs some important elements in the cyclic groups coming from the covering. Using some topological analysis, one then proceeds to show that these elements always exist. If the above elements generate the whole cyclic group, then the corresponding moduli space is connected. If they generate a finite index subgroup, then the index gives the number of connected components. Depending on the genus and on whether or not we are in the hyperelliptic case, components are distinguished by some topological invariant, related to the classical $$\operatorname{Arf}$$ invariant and to the parity of the spin structure. Topics on Teichmüller spaces https://zbmath.org/1472.32006 2021-11-25T18:46:10.358925Z "Seppälä, Mika" https://zbmath.org/authors/?q=ai:seppala.mika (no abstract) Growth of the Weil-Petersson inradius of moduli space https://zbmath.org/1472.32007 2021-11-25T18:46:10.358925Z "Wu, Yunhui" https://zbmath.org/authors/?q=ai:wu.yunhui For a surface $$S_{g,n}$$ of genus $$g$$ with $$n$$ punctures satisfying $$3g-3+n>0$$, let Teich($$S_{g,n}$$) be the Teichmüller space of $$S_{g,n}$$ with the Weil-Petersson metric and let $$\mathcal{M}_{g,n}$$ be the moduli space of $$S_{g,n}$$ defined as the quotient of Teich$$(S_{g,n})$$ by the mapping class group Mod($$S_{g,n}$$). The author studies the asymptotic behavior of the inradius of $$\mathcal{M}_{g,n}$$ either as $$g\to \infty$$ or $$n\to\infty$$. It is proved that for all $$n\ge 0$$ and $$g\ge 2$$, the inradius of $$\mathcal{M}_{g,n}$$ is comparable to $$\sqrt{\ln g}$$ by a constant independent of $$g$$; for all $$g\geq 0$$ and $$n\geq 4$$, it is comparable to 1 by a constant independent of $$n$$. Here, the inradius is defined by the maximum of the Weil-Petersson distances dist$${}_{wp}(X, \partial\overline{\mathcal{M}}_{g,n})$$ among all $$X\in \mathcal{M}_{g,n}$$. To prove these results, the author considers the systole function $$\ell_{sys}$$ on Teich($$S_{g,n}$$) and gives a key theorem which establishes Lipschitz continuity of the square root of $$\ell_{sys}$$ with respect to the Weil-Petersson distance, where the Lipschitz constant can be taken independently of $$g$$ and $$n$$. For the proof of Lipschitz continuity the author uses a thin-thick decomposition of the Weil-Petersson geodesics connecting two points in Teich($$S_{g,n}$$) and estimates the norm of the gradient of the square root of geodesic length functions. \par Let $$\mathcal{M}_{g,n}^{\geqslant \epsilon}$$ denote the $$\epsilon$$-thick part of of $$\mathcal{M}_{g,n}$$. The moduli space $$\mathcal{M}_{g,n}$$ is foliated by $$\partial\mathcal{M}_{g,n}^{\geqslant \epsilon}$$ for all $$\epsilon$$. The author shows that for any $$s>t\geq 0$$ the Weil-Petersson distance between $$\partial\mathcal{M}_{g,n}^{\geqslant s}$$ and $$\partial\mathcal{M}_{g,n}^{\geqslant t}$$ is comparable to $$\sqrt{s}-\sqrt{t}$$ by a constant independent of $$g$$ and $$n$$. \par Another interesting result in this paper is that for a closed surface $$S_g$$ the author shows the asymptotic behavior of the Weil-Petersson volume of geodesic balls as $$g\to \infty$$, where the geodesic balls have a finite radius and are away from the boundary of the completion of Teich($$S_g$$). Corrigendum to: Maximal open radius for Strebel point'' https://zbmath.org/1472.32008 2021-11-25T18:46:10.358925Z "Yao, Guowu" https://zbmath.org/authors/?q=ai:yao.guowu A correction of a minor error to the proof of Theorem 2 in [ibid. 178, No. 2, 311--324 (2015; Zbl 1329.32006), lines 17--20 on page 323] is given. Meromorphic solutions of certain types of non-linear differential equations https://zbmath.org/1472.34153 2021-11-25T18:46:10.358925Z "Liu, Huifang" https://zbmath.org/authors/?q=ai:liu.huifang "Mao, Zhiqiang" https://zbmath.org/authors/?q=ai:mao.zhiqiang In this paper under review, the authors study the following non-linear differential equation $f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z},$ where $$n\geq 2$$ is an integer, $$p_{1},$$ $$p_{2}$$ and $$\alpha _{1},$$ $$\alpha _{2}$$ are non-zero constants and $$P_{d}\left( z,f\right)$$ is a differential polynomial in $$f$$ of degree $$d$$. The authors find the forms of meromorphic solutions with few poles of the above equation when $$d=n-1$$ under some restrictions on $$\alpha _{1},$$ $$\alpha _{2},$$ The Theorems 1.1 and 1.2 obtained extend the result established by \textit{P. Li} [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; Zbl 1206.30046)] provided $$\alpha _{1}\neq$$ $$\alpha _{2}$$ and $$d\leq n-2$$. Some examples are given to illustrate the results. On Petrenko's deviations and second order differential equations https://zbmath.org/1472.34154 2021-11-25T18:46:10.358925Z "Heittokangas, Janne" https://zbmath.org/authors/?q=ai:heittokangas.janne-m "Zemirni, Mohamed Amine" https://zbmath.org/authors/?q=ai:zemirni.mohamed-amine The authors consider the oscillation of solutions of $f''+A(z)f=0\tag{1.1}$ and the growth of solutions of $f''+A(z)f'+B(z)f=0\tag{1.2}$ were $$A$$ and $$B$$ are entire functions.'' For the first equation, an improved (see [\textit{I. Laine} and \textit{P. Wu}, Rev. Roum. Math. Pures Appl. 44, No. 4, 609--615 (1999; Zbl 1004.34079)]) estimate of the value of $$\Lambda (E)$$ is obtained, where $$E$$ is a product of two linearly independent solutions of the equation (1.1) and $$\Lambda (E)$$ denotes the exponent of convergence of zeros of $$E$$. For the equation (1.2), conditions are considered under which all its nontrivial solutions have infinite order. A new result is obtained in this direction and, as a consequence, the following statement Corollary. Let $$A$$ and $$B$$ be entire functions. Suppose there exists a sector where $$\log^+|A(z)|=O(\log(|z|))$$, and suppose that $$B$$ is transcedental with Fabry gaps. Then every non-trivial solution of (1.2) is of infinite order. Order and hyper-order of solutions of second order linear differential equations https://zbmath.org/1472.34155 2021-11-25T18:46:10.358925Z "Kumar, Sanjay" https://zbmath.org/authors/?q=ai:kumar.sanjay-v|kumar.sanjay.2|kumar.sanjay.1 "Saini, Manisha" https://zbmath.org/authors/?q=ai:saini.manisha In this paper, the authors investigate the growth of solutions of second order linear differentilal equations. They determine new conditions on the coefficients in order that every non-trivial solution has an infinite order by introducing the notions of function extremal to Yang's inequality and extremal to Denjoy's conjecture. They also study the hyper-order of these solutions. Moreover, they extend these results to higher order linear differential equations. So this work is interesting. Infinite growth of solutions of second order complex differential equations with entire coefficient having dynamical property https://zbmath.org/1472.34156 2021-11-25T18:46:10.358925Z "Zhang, Guowei" https://zbmath.org/authors/?q=ai:zhang.guowei "Yang, Lianzhong" https://zbmath.org/authors/?q=ai:yang.lianzhong This paper is concerned with the classical problem on the growth of solutions of second order linear complex differential equations $$f''+A(z)f'+B(z)f=0$$, where $$A(z)$$ and $$B(z)$$ are two entire functions as coefficients of this equation. The authors give some properties for these two coefficients to obtain the solutions of the equation having infinite growth order. This topic is a research hotspot and many literatures focused on this research areas. The authors introduce some new methods and technology to get the infinity order of the solutions. In the main results of this paper the authors suppose that the coefficient $$B(z)$$ has a dynamical property, that is, has a multiple Fatou component. This property ensures that the maximum and minimum modules of $$B(z)$$ satisfy an important inequality as $$|z|=r$$ is in an infinitely logarithmic measure set. The coefficient $$A(z)$$ satisfies the extreme of Yang's inequality, which says that for an entire function with nonzero finite lower order the number of its Borel direction is the double of the number of its finite deficient values, or $$A(z)$$ is a nontrivial solution of another differential equation $$w''+P(z)w=0$$, where $$P(z)$$ is a polynomial. The author also discusses some other properties to get the infinity growth of the solutions. In my opinion, this paper is of great significance for future research. Sharp estimate for the critical parameters of $$SU(3)$$ Toda system with arbitrary singularities. I https://zbmath.org/1472.35136 2021-11-25T18:46:10.358925Z "Lin, Chang-Shou" https://zbmath.org/authors/?q=ai:lin.chang-shou "Yang, Wen" https://zbmath.org/authors/?q=ai:yang.wen Summary: To obtain the a priori estimate of Toda system, the first step is to determine all the possible local masses of blow up solutions. In this paper we study this problem and improve the main results in [the first author et al., Anal. PDE 8, No. 4, 807--837 (2015; Zbl 1322.35038)]. Our method is based on a recent work by \textit{A. Eremenko} et al. [Ill. J. Math. 58, No. 3, 739--755 (2014; Zbl 1405.30005)]. Nonexistence of variational minimizers related to a quasilinear singular problem in metric measure spaces https://zbmath.org/1472.35204 2021-11-25T18:46:10.358925Z "Garain, Prashanta" https://zbmath.org/authors/?q=ai:garain.prashanta "Kinnunen, Juha" https://zbmath.org/authors/?q=ai:kinnunen.juha The goal of this article is to show that there are cases where the equation $-\Delta_p u=-\operatorname{div}(|\nabla|^{p-2}\nabla u)=f(u)$ does not have a positive solution. The idea is to consider a singular problem, that is an example where $$f$$ blows up near zero. The novelty of the paper is that the nonexistence is studied in quite general metric measure spaces. Hence, not exactly the above equation is studied, but an integral version of it with solutions in Sobolev spaces. In the metric setting, this makes it possible to use Newtonian spaces as Sobolev spaces. As $$f$$, the authors choose the function given by $$u\mapsto u^{-\delta}$$. The main part of the proof is a lemma that provides an energy estimate for a positive variational minimizer. This is then used to show that such a minimizer must vanish, a contradiction to the existence of a positive minimizer. Spectral stability estimates of Dirichlet divergence form elliptic operators https://zbmath.org/1472.35254 2021-11-25T18:46:10.358925Z "Gol'dshtein, Vladimir" https://zbmath.org/authors/?q=ai:goldshtein.vladimir "Pchelintsev, Valerii" https://zbmath.org/authors/?q=ai:pchelintsev.valerii "Ukhlov, Alexander" https://zbmath.org/authors/?q=ai:ukhlov.alexander The paper is aimed on applying quasiconformal mappings to spectral stability estimates of the Dirichlet eigenvalues of $$A$$-divergent form elliptic operators $L_{A}=-\text{div} [A(w) \nabla g(w)]\in \widetilde{\Omega}, \quad w|_{\partial \widetilde{\Omega}}=0,$ in non-Lipschitz domains $$\widetilde{\Omega} \subset \mathbb{C}$$ with $$2 \times 2$$ symmetric matrix functions $$A(w)=\left\{a_{kl}(w)\right\}$$, $$\textrm{det} A=1$$, with measurable entries satisfying the uniform ellipticity condition. The main results of the article concern to spectral stability estimates in domains that the authors call as $$A$$-quasiconformal $$\beta$$-regularity domains. Namely, a simply connected domain $$\widetilde{\Omega} \subset \mathbb{C}$$ is called an $$A$$-quasiconformal $$\beta$$-regular domain about a simply connected domain $${\Omega} \subset \mathbb{C}$$ if $\iint\limits_{\widetilde{\Omega}} |J(w, \varphi)|^{1-\beta}~dudv < \infty, \,\,\,\beta>1,$ where $$J(w, \varphi)$$ is a Jacobian of an $$A$$-quasiconformal mapping $$\varphi: \widetilde{\Omega}\to\Omega$$. The main result of the article states that, if a domain $$\widetilde{\Omega}$$ is $$A$$-quasiconformal $$\beta$$-regular about $$\Omega$$, then for any $$n\in \mathbb{N}$$ the following spectral stability estimates hold: $|\lambda_n[I, \Omega]-\lambda_n[A, \widetilde{\Omega}]| \leq c_n A^2_{\frac{4\beta}{\beta -1},2}(\Omega) \left(|\Omega|^{\frac{1}{2\beta}} + \|J_{\varphi^{-1}}\,|\,L^{\beta}(\Omega)\|^{\frac{1}{2}} \right) \cdot \|1-J_{\varphi^{-1}}^{\frac{1}{2}}\,|\,L^{2}(\Omega)\|,$ where $$c_n=\max\left\{\lambda_n^2[A, \Omega], \lambda_n^2[A, \widetilde{\Omega}]\right\}$$, $$J_{\varphi^{-1}}$$ is a Jacobian of an $$A^{-1}$$-quasiconformal mapping $$\varphi^{-1}:\Omega\to\widetilde{\Omega}$$, and $A_{\frac{4\beta}{\beta -1},2}(\Omega) \leq \inf\limits_{p\in \left(\frac{4\beta}{3\beta -1},2\right)} \left(\frac{p-1}{2-p}\right)^{\frac{p-1}{p}} \frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\Omega|^{\frac{\beta-1}{4\beta}}}{\sqrt{\Gamma(2/p) \Gamma(3-2/p)}}~~.$ Evolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potential https://zbmath.org/1472.35321 2021-11-25T18:46:10.358925Z "Colombo, F." https://zbmath.org/authors/?q=ai:colombo.fabrizio "Gantner, J." https://zbmath.org/authors/?q=ai:gantner.jonathan "Struppa, D. C." https://zbmath.org/authors/?q=ai:struppa.daniele-carlo Summary: In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift. Local fractional Moisil-Teodorescu operator in quaternionic setting involving Cantor-type coordinate systems https://zbmath.org/1472.35428 2021-11-25T18:46:10.358925Z "Bory-Reyes, Juan" https://zbmath.org/authors/?q=ai:bory-reyes.juan "Pérez-de la Rosa, Marco Antonio" https://zbmath.org/authors/?q=ai:perez-de-la-rosa.marco-antonio Summary: The Moisil-Teodorescu operator is considered to be a good analogue of the usual Cauchy-Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in $$\mathbb{R}^3$$. In the present work, a general quaternionic structure is developed for the local fractional Moisil-Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil-Teodorescu operator. Application of the fractional Sturm-Liouville theory to a fractional Sturm-Liouville telegraph equation https://zbmath.org/1472.35433 2021-11-25T18:46:10.358925Z "Ferreira, M." https://zbmath.org/authors/?q=ai:ferreira.michel|ferreira.max|ferreira.miguel-h|ferreira.marta|ferreira.m-n|ferreira.m-d-c|ferreira.mario-f-s|ferreira.maria-c-f|ferreira.maria-joao.2|ferreira.m-p|ferreira.maria-teodora|ferreira.mauricio-a|ferreira.miguel-jorge-bernabe|ferreira.maria-joao.1|ferreira.marco-s|ferreira.marina-a|ferreira.marcelo-rodrigo-portela|ferreira.marisa|ferreira.mauro-s|ferreira.marco-a-r|ferreira.marizete-a-c|ferreira.marcio-j-r|ferreira.mardson|ferreira.marcos-r-s|ferreira.manoel-m-jun|ferreira.mariana|ferreira.m-margarida-a|ferreira.m-luisa|ferreira.marcio-v|ferreira.milton|ferreira.maria-margarida|ferreira.marcelo-c "Rodrigues, M. M." https://zbmath.org/authors/?q=ai:rodrigues.maria-manuela|rodrigues.maikol-m "Vieira, N." https://zbmath.org/authors/?q=ai:vieira.newton-j|vieira.nelson Summary: In this paper, we consider a non-homogeneous time-space-fractional telegraph equation in $$n$$-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm-Liouville operator defined in terms of right and left fractional Riemann-Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented. Zeros, growth and Taylor coefficients of entire solutions of linear $$q$$-difference equations https://zbmath.org/1472.39011 2021-11-25T18:46:10.358925Z "Bergweiler, Walter" https://zbmath.org/authors/?q=ai:bergweiler.walter The author studies the linear $$q$$-difference equation $\sum_{j=0}^m a_j(z)f(q^jz) = b(z),$ where $$q\in\mathbb{C}$$ with $$0 < |q| < 1$$ and $$b$$ and the $$a_j$$ are polynomials. He determines the asymptotic behavior of the Taylor coefficients of the transcendental entire solutions. He proves that the zeros of the transcendental entire solutions are asymptotic to finitely many geometric progressions under a suitable assumption on the associated Newton-Puiseux diagram. He also sharpens the results on the growth rate of entire solutions due to the author and \textit{W. K. Hayman} [Comput. Methods Funct. Theory 3, No. 1--2, 55--78 (2003; Zbl 1087.39022)]. Difference equations related to number theory https://zbmath.org/1472.39032 2021-11-25T18:46:10.358925Z "Heim, Bernhard" https://zbmath.org/authors/?q=ai:heim.bernhard-ernst "Neuhauser, Markus" https://zbmath.org/authors/?q=ai:neuhauser.markus As part of a collection on difference equations and applications [Zbl 1467.39001], this article requires a high level of knowledge on the subject. The authors recall the Dedekind's $$\eta$$-function they already studied in [Res. Math. Sci. 7, No. 1, Paper No. 3, 8 p. (2020; Zbl 1472.11122)], clarifying how its powers are linked to a polynomial defined recursively as follows: \begin{align*} P_1(x) &= x , \\ P_n(x) &= \frac{x}{n} \left( \sigma(n) + \sum_{k=1}^{n-1} \sigma(k) P_{n-k}(x) \right) , \end{align*} where $$x \in \mathbb{C}$$ and $$\sigma(k)$$ is the sum of the divisors of $$k$$. After remarking the importance of $$P_n(x)$$ in number theory, the authors acknowledge its irreducibility to a recurrence relation of bounded length and they propose to generalize it through the arithmetic functions $$g,h$$: \begin{align*} P_1^{g,h}(x) &= x , \\ P_n^{g,h}(x) &= \frac{x}{h(n)} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g,h}(x) \right) , \end{align*} with $$g: \mathbb{N} \rightarrow \mathbb{C}$$, $$h: \mathbb{N} \rightarrow \mathbb{R}$$, and $$g(1)=h(1)=1$$. Beside a quick mention to the classical orthogonal polynomials as solutions of a specific differential equation, the authors distinguish the following subcases: \begin{align*} P_n^g (x) &= \frac{x}{n} \left( g(n) + \sum_{k=1}^{n-1} g(k) P_{n-k}^{g}(x) \right) , \\ Q_n^g (x) &= x \left( g(n) + \sum_{k=1}^{n-1} g(k) Q_{n-k}^{g}(x) \right) , \end{align*} related, respectively, to the associated Laguerre polynomials and to the Chebyshev polynomials of the second kind. The authors employ some of their previous results, obtained in collaboration with \textit{R. Tröger} [J. Difference Equ. Appl. 26, No. 4, 510--531 (2020; Zbl 1456.30010)], in order to analyze the limiting behavior of the main sequence $$\left( P_n^{g,h}(x) \right)_{n \in \mathbb{N}}$$ and of the subsequence $$\left( Q_n^g (x) \right)_{n \in \mathbb{N}}$$. Then the authors focus on the recurrence relations of $$P_n^g (x)$$ and $$Q_n^g (x)$$ for $$g(n)=1$$: \begin{align*} P_n^1 (x) &= (-1)^n \binom{-x}{n} , \\ Q_n^1 (x) &= (x+1)^{n-1}x , \end{align*} finding a connection between them via a theorem of \textit{H. Poincaré} [Am. J. Math. 7, 203--258 (1885; JFM 17.0290.01)]; they eventually suggest a further investigation under arbitrary conditions. For the entire collection see [Zbl 1467.39001]. On approximation by Kantorovich exponential sampling operators https://zbmath.org/1472.41009 2021-11-25T18:46:10.358925Z "Bajpeyi, Shivam" https://zbmath.org/authors/?q=ai:bajpeyi.shivam "Kumar, A. Sathish" https://zbmath.org/authors/?q=ai:kumar.angamuthu-sathish|sathish-kumar.a In this article, the authors extended their work on Kantorovich type exponential sampling operators which were introduced by Butzer and Stens in $$1993$$. The authors studied the Voronovskaya type theorem and its quantitative estimates for these operators in terms of an appropriate K-functional. Further, they improved the order of approximation by using the convex type linear combinations of these operators. The authors also gave few examples of kernels along with the graphical representations. The saddle-point method in $$\mathbb{C}^N$$ and the generalized Airy functions https://zbmath.org/1472.41018 2021-11-25T18:46:10.358925Z "Pinna, Francesco" https://zbmath.org/authors/?q=ai:pinna.francesco "Viola, Carlo" https://zbmath.org/authors/?q=ai:viola.carlo The authors consider the saddle-point method for multiple integrals $F(\Lambda):=\int_\Gamma e^{\Lambda h(z_1,z_2,\dots,z_N)}g(z_1,z_2,\dots,z_N)dz_1dz_2\cdot\cdot\cdot dz_N,$ when $$\Lambda\to\infty$$; where $$h$$ and $$g$$ are holomorphic functions of all their variables over a suitable manifold $$\Gamma\subset C^N$$. This problem was first studied in [\textit{M. V. Fedoryuk}, The saddle-point method. (Metod perevala) (Russian). Moskow: Nauka, Moscow (1977; Zbl 0463.41020)] by using techniques from algebraic topology based on homology groups, but Fedoryuk's method is difficult to apply in concrete examples. It was also studied in [\textit{M. Hata}, Trans. Am. Math. Soc. 352, No. 10, 4557--4583 (2000; Zbl 0996.11048)] for the particular case $$N=2$$. In this paper the authors circumvent the difficulties involved in Fedoryuk's method by introducing a more flexible analytic method to find the nondegenerate saddle-points of $$h(z_1,z_2,\dots,z_N)$$ that are the essential ingredient in the analysis of the asymptotic behavior of $$F(\Lambda)$$. The analysis is based in writing the above multiple integral as an $$N-$$times iterated integral and applying the one-dimensional steepest-descent method to each variable successively. The main result of the paper is that, under certain assumptions over the successive restrictions of the function $$h$$ to the successive complex hypersufaces defined by the iterated integrals, $F(\Lambda)=(2\pi)^{N/2}e^{i(\nu_1+\cdot\cdot\cdot +\nu_N)}g(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)}){\vert e^{h(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})}\vert^{N/2}\sqrt{H(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})}}{e^{\Lambda h(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})} \Lambda^{N/2}}(1+o(1)),$ where $$(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})$$ is the relevant admissible saddle point where $$\nabla h=0$$, and, for $$k=1,2,\dots.,N$$, $\nu_k:=\pi h_k-\frac{1}{2}\mathrm{arg}\left(-e^{-h(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})}H_k(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)})H_{k-1}(z_1^{(0)},z_2^{(0)},\dots,z_N^{(0)}\right),$ for certain $$h_k\in Z$$ and certain Hessian determinants $$H$$ and $$H_k$$ of the function $$h$$ defined in the paper. As an application of the theory, the authors derive asymptotic approximations of the generalized Airy functions $\left(\frac{\Lambda^{1/3}}{2\pi i}\right)^N\int_{\Gamma_1}dz_1\dots.\int_{\Gamma_N}dz_N\, e^{\Lambda h(z_1,\dots,z_N)},$ where $h(z_1,\dots,z_N)=\sum_{k=1}^Nz_k+\sum_{1\le k<n\le N}z_kz_n-\frac{1}{3}\sum_{k=1}^Nz_k^3$ and $$\Gamma_k$$ are the standard integration paths in the integral definition of the classical Airy function. A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem https://zbmath.org/1472.44005 2021-11-25T18:46:10.358925Z "Fritzsche, Bernd" https://zbmath.org/authors/?q=ai:fritzsche.bernd "Kirstein, Bernd" https://zbmath.org/authors/?q=ai:kirstein.bernd "Mädler, Conrad" https://zbmath.org/authors/?q=ai:madler.conrad Summary: The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situations. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper [Linear Algebra Appl. 590, 133--209 (2020; Zbl 1447.44005)], is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail. Sequence space representations for spaces of entire functions with rapid decay on strips https://zbmath.org/1472.46022 2021-11-25T18:46:10.358925Z "Debrouwere, Andreas" https://zbmath.org/authors/?q=ai:debrouwere.andreas Let $$\omega:[0, \infty)\to [0, \infty]$$ be an increasing continuous function with $$\omega(0) = 0$$ and $$\log t = o(\omega(t))$$ as $$t$$ goes to $$\infty.$$ The author considers the Fréchet space $${\mathcal U}_\omega({\mathbb C})$$ consisting of those entire functions $$\varphi$$ such that $\sup_{|{\Im z}| < n}|\varphi(z)| e^{n\omega\left(|{\Re z}|\right)} < \infty \ \quad \text{for all }n\in {\mathbb N}.$ The main results of the paper can be summarized as follows. \par \begin{itemize} \item[(a)] If $$\omega$$ satisfies $$\omega(t+1) = O(\omega(t))$$, then $${\mathcal U}_\omega({\mathbb C})$$ is isomorphic to some power series space of infinite type. \item[(b)] If $$\omega$$ satisfies $$\omega(2t) = O(\omega(t))$$, then $${\mathcal U}_\omega({\mathbb C})$$ is isomorphic to $$\Lambda_\infty\left(\omega^\ast(n)\right),$$ where $$\omega^\ast(n) = \left(n\omega^{-1}(n)\right)^{-1}$$. \end{itemize} \par To prove (a), the author shows that $${\mathcal U}_\omega({\mathbb C})$$ has properties (DN) and ($$\Omega$$). To conclude (b), the diametral dimension of $${\mathcal U}_\omega({\mathbb C})$$ is calculated. This is done by combining a result due to \textit{M. Langenbruch} [Stud. Math. 233, No. 1, 85--100 (2016; Zbl 1356.46007)] with properties of the short-time Fourier transform. As a consequence, a sequence space representation for projective Gelfand-Shilov spaces is obtained. A weaker Gleason-Kahane-Żelazko theorem for modules and applications to Hardy spaces https://zbmath.org/1472.46049 2021-11-25T18:46:10.358925Z "Sebastian, Geethika" https://zbmath.org/authors/?q=ai:sebastian.geethika "Daniel, Sukumar" https://zbmath.org/authors/?q=ai:daniel.sukumar Let $$A$$ be a complex unital Banach algebra. The paper deals with the question of finding sufficient conditions under which a map $$\Lambda:M\rightarrow\mathbb C$$ between a left $$A$$-module $$M$$ and the field $$\mathbb C$$ is linear.\par Let $$A$$ be a complex unital Banach algebra, $$M$$ a left $$A$$-module, $$\Lambda:M\rightarrow\mathbb C$$ a non-zero map and $$S$$ a non-empty subset of $$M$$, satisfying the following conditions: \begin{itemize} \item[(S1)] $$\theta_M\not\in S$$ and $$S$$ generates $$M$$ as an $$A$$-module; \item[(S2)] if $$a$$ is invertible in $$A$$ and $$s\in S$$, then $$as\in S$$; \item[(S3)] for all $$s_1,s_2\in S$$, there exist $$a_1,a_2\in A$$ such that $$a_1s_1=a_2s_2\in S$$. \end{itemize} The main results of the paper are the following: \begin{itemize} \item[(1)] If $$\Lambda$$ is an $$\mathbb R$$-linear functional on $$M$$ such that $$\Lambda(m)s-\Lambda(s)m\not\in S$$ for all $$m\in M$$ and $$s\in S$$, then $$\Lambda$$ is $$\mathbb C$$-linear and there exists a unique character $$\chi:A\rightarrow\mathbb C$$ such that $$\Lambda(am)=\chi(a)\Lambda(m)$$ for all $$a\in A$$ and $$m\in M$$. \item[(2)] If $$\Lambda(\theta_M)=0$$ and $$(\Lambda(m_1)-\Lambda(m_2))s-(m_1-m_2)\Lambda(s)\not\in S$$ for all $$m_1,m_2\in M$$ and $$s\in S$$, then there exists a unique character $$\chi:A\rightarrow\mathbb C$$ such that $$\Lambda(as)=\chi(a)\Lambda(s)$$ for all $$a\in A$$ and $$s\in S$$. \item[(3)] If, in addition, $\sum_{j=1}^n\Lambda(s_j)=\Lambda\Biggl(\sum_{j=1}^ns_j\Biggr)$ for all $$n\in {\mathbb Z}^+$$ and $$s_1,\dots,s_n\in S$$, then $$\Lambda$$ is linear. \item[(4)] If $$\chi:A\rightarrow\mathbb C$$ is a map such that $$\Lambda(am)=\chi(a)\Lambda(m)$$ and $$\Lambda(m)s-\Lambda(s)m\not\in S$$ for all $$a\in A, m\in M$$ and $$s\in S$$, then $$\chi$$ is linear if and only if the map $$\tau_a:{\mathbb C}\rightarrow\mathbb C$$, defined as $$\tau_a(\lambda)=\chi(\lambda e_A-a)$$ for each $$\lambda\in\mathbb C$$, is an entire function on $$\mathbb C$$ for every $$a\in A$$. \end{itemize} In the final chapter, the results of this type are also applied to study the properties of some Hardy spaces. On matrix-valued Stieltjes functions with an emphasis on particular subclasses https://zbmath.org/1472.47011 2021-11-25T18:46:10.358925Z "Fritzsche, Bernd" https://zbmath.org/authors/?q=ai:fritzsche.bernd "Kirstein, Bernd" https://zbmath.org/authors/?q=ai:kirstein.bernd "Mädler, Conrad" https://zbmath.org/authors/?q=ai:madler.conrad Summary: The paper deals with particular classes of $$q{\times}q$$ matrix-valued functions which are holomorphic in $$\mathbb{C}\setminus [\alpha, +\infty)$$, where $$\alpha$$ is an arbitrary real number. These classes are generalizations of classes of holomorphic complex-valued functions studied by \textit{I. S. Kac} and \textit{M. G. Krein} [Transl., Ser. 2, Am. Math. Soc. 103, 1--18 (1974; Zbl 0291.34016)] and by \textit{M. G. Krein} and \textit{A. A. Nudel'man} [The Markov moment problem and extremal problems. Providence, RI: American Mathematical Society (AMS) (1977; Zbl 0361.42014)]. The functions are closely related to truncated matricial Stieltjes problems on the interval $$[\alpha+\infty)$$. Characterizations of these classes via integral representations are presented. Particular emphasis is placed on the discussion of the Moore-Penrose inverse of these matrix-valued functions. For the entire collection see [Zbl 1367.47005]. Composition operators in hyperbolic Bloch-type and $$F \left(p, q, s\right)$$ spaces https://zbmath.org/1472.47019 2021-11-25T18:46:10.358925Z "Kotilainen, Marko" https://zbmath.org/authors/?q=ai:kotilainen.marko "Pérez-González, Fernando" https://zbmath.org/authors/?q=ai:perez-gonzalez.fernando Summary: Composition operators $$C_\varphi$$ from Bloch-type $$\mathcal{B}_\alpha$$ spaces to $$F \left(p, q, s\right)$$ classes, from $$F \left(p, q, s\right)$$ to $$\mathcal{B}_\alpha$$, and from $$F \left(p_1, q_1, 0\right)$$ to $$F \left(p_2, q_2, s_2\right)$$ are considered. The criteria for these operators to be bounded or compact are given. Our study also includes the corresponding hyperbolic spaces. Natural boundary for a sum involving Toeplitz determinants https://zbmath.org/1472.47023 2021-11-25T18:46:10.358925Z "Tracy, Craig A." https://zbmath.org/authors/?q=ai:tracy.craig-a "Widom, Harold" https://zbmath.org/authors/?q=ai:widom.harold Summary: In the theory of the two-dimensional Ising model, the \textit{diagonal susceptibility} is equal to a sum involving Toeplitz determinants. In terms of a parameter $$k$$, the diagonal susceptibility is analytic for $$| k | < 1$$, and the authors proved in [J. Math. Phys. 54, No. 12, 123302, 9 p. (2013; Zbl 1288.82018)] the conjecture that this function has the unit circle as a natural boundary. The symbol of the Toeplitz determinants is a $$k$$-deformation of one with a single singularity on the unit circle. Here we extend the result, first, to deformations of a larger class of symbols with a single singularity on the unit circle, and then to deformations of (almost) general Fisher-Hartwig symbols. For the entire collection see [Zbl 1367.47005]. New characterizations for the products of differentiation and composition operators between Bloch-type spaces https://zbmath.org/1472.47025 2021-11-25T18:46:10.358925Z "Liang, Yu-Xia" https://zbmath.org/authors/?q=ai:liang.yuxia "Dong, Xing-Tang" https://zbmath.org/authors/?q=ai:dong.xingtang Summary: We use a brief way to give various equivalent characterizations for the boundedness and the essential norm of the operator $$C_\varphi D^m$$ acting on Bloch-type spaces. At the same time, we use this method to easily get a known characterization for the operator $$DC_\varphi$$ on Bloch-type spaces. Correction to: The total intrinsic curvature of curves in Riemannian surfaces'' https://zbmath.org/1472.53063 2021-11-25T18:46:10.358925Z "Mucci, Domenico" https://zbmath.org/authors/?q=ai:mucci.domenico "Saracco, Alberto" https://zbmath.org/authors/?q=ai:saracco.alberto From the text: In the authors' paper [ibid. 70, No. 1, 521--557 (2021; Zbl 1466.53064)], in the statements of the main results, Theorems 1--9 and Proposition 3, one has to assume in addition that the curve $$\mathbf{c}$$ is rectifiable. A new proof of a conjecture on nonpositive Ricci curved compact Kähler-Einstein surfaces https://zbmath.org/1472.53083 2021-11-25T18:46:10.358925Z "Guan, Zhuang-Dan Daniel" https://zbmath.org/authors/?q=ai:guan.zhuang-dan-daniel Summary: In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of \textit{Y. Hong} et al. [Acta Math. Sin. 31, No. 5, 595--602 (1988; Zbl 0678.53060); Sci. China, Math. 54, No. 12, 2627--2634 (2011; Zbl 1259.53067)]. Moreover, we proved that any compact Kähler-Einstein surface $$M$$ is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) $$M$$ has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction. The dual Bonahon-Schläfli formula https://zbmath.org/1472.53086 2021-11-25T18:46:10.358925Z "Mazzoli, Filippo" https://zbmath.org/authors/?q=ai:mazzoli.filippo In the paper, for a differentiable deformation of geometrically finite 3-manifold the Bonahon-Schläfli Formula gives the derivative of the volume of the convex cones in terms of the variation of the geometry of their boundaries (the classical Schläfli Formula is applicable for determining the volume of the hyperbolic polyhedra). Here the author studies the analogous problem for the dual volume and gives a self-contained proof of the dual Bonahon-Schläfli Formula (without using Bonahon's results). A Thurston boundary for infinite-dimensional Teichmüller spaces https://zbmath.org/1472.57016 2021-11-25T18:46:10.358925Z "Bonahon, Francis" https://zbmath.org/authors/?q=ai:bonahon.francis "Šarić, Dragomir" https://zbmath.org/authors/?q=ai:saric.dragomir The Teichmüller space of a Riemann surface $$X_0$$ is the space of quasiconformal deformations of the complex structure of $$X_0$$. Thurston compactified the Teichmüller space of a compact Riemann surface $$X_0$$, of genus at least 2, by adding a boundary at infinity consisting of projective measured foliations or, equivalently, projective measured geodesic laminations. In the present paper, the authors introduce a similar construction of a boundary for the Teichmüller space of a noncompact surface $$X_0$$, using the technical tool of geodesic currents. In addition to the fact that Teichmüller spaces of noncompact Riemann surfaces are fundamental objects in complex analysis, our motivation here is to put in evidence the hidden features that underlie Thurston's construction, by tying it more closely to the quasiconformal geometry of $$X_0$$ and less to the purely topological considerations that suffice for compact surfaces.'' Scaling limits and fluctuations for random growth under capacity rescaling https://zbmath.org/1472.60045 2021-11-25T18:46:10.358925Z "Liddle, George" https://zbmath.org/authors/?q=ai:liddle.george "Turner, Amanda" https://zbmath.org/authors/?q=ai:turner.amanda-g Summary: We evaluate a strongly regularised version of the Hastings-Levitov model $$\mathrm{HL}(\alpha)$$ for $$0\leq\alpha<2$$. Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where $$\alpha=0$$ and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where $$0<\alpha<2$$ and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on $$\alpha$$. Furthermore, this field becomes degenerate as $$\alpha$$ approaches 0 and 2, suggesting the existence of phase transitions at these values. Central limit theorems from the roots of probability generating functions https://zbmath.org/1472.60046 2021-11-25T18:46:10.358925Z "Michelen, Marcus" https://zbmath.org/authors/?q=ai:michelen.marcus "Sahasrabudhe, Julian" https://zbmath.org/authors/?q=ai:sahasrabudhe.julian Summary: For each $$n$$, let $$X_n \in \{0, \ldots, n \}$$ be a random variable with mean $$\mu_n$$, standard deviation $$\sigma_n$$, and let $P_n(z) = \sum_{k = 0}^n \mathbb{P}(X_n = k) z^k,$ be its probability generating function. We show that if none of the complex zeros of the polynomials $$\{P_n(z) \}$$ is contained in a neighborhood of $$1 \in \mathbb{C}$$ and $$\sigma_n > n^\varepsilon$$ for some $$\varepsilon > 0$$, then $$X_n^\ast = (X_n - \mu_n) \sigma_n^{- 1}$$ is asymptotically normal as $$n \to \infty$$: that is, it tends in distribution to a random variable $$Z \sim \mathcal{N}(0, 1)$$. On the other hand, we show that there exist sequences of random variables $$\{X_n \}$$ with $$\sigma_n > C \log n$$ for which $$P_n(z)$$ has no roots near 1 and $$X_n^\ast$$ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of $$P_n(z)$$ and the distribution of the random variable $$X_n$$. On the finiteness of moments of the exit time of planar Brownian motion from comb domains https://zbmath.org/1472.60139 2021-11-25T18:46:10.358925Z "Boudabra, Maher" https://zbmath.org/authors/?q=ai:boudabra.maher "Markowsky, Greg" https://zbmath.org/authors/?q=ai:markowsky.greg-t Summary: A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite $$p$$-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the $$p$$-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite $$p$$-th moment for all $$p \geq 1/2$$. Numerical solution of scattering problems using a Riemann-Hilbert formulation https://zbmath.org/1472.65176 2021-11-25T18:46:10.358925Z "Smith, Stefan G. Llewellyn" https://zbmath.org/authors/?q=ai:llewellyn-smith.stefan-g "Luca, Elena" https://zbmath.org/authors/?q=ai:luca.elena Summary: A fast and accurate numerical method for the solution of scalar and matrix Wiener-Hopf (WH) problems is presented. The WH problems are formulated as Riemann-Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix WH problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach. Annular and circular rigid inclusions planted into a penny-shaped crack and factorization of triangular matrices https://zbmath.org/1472.74187 2021-11-25T18:46:10.358925Z "Antipov, Y. A." https://zbmath.org/authors/?q=ai:antipov.yuri-a "Mkhitaryan, S. M." https://zbmath.org/authors/?q=ai:mkhitaryan.s-m Summary: Analytical solutions to two axisymmetric problems of a penny-shaped crack when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of arbitrary profile are embedded into the crack are derived. The problems are governed by integral equations with the Weber-Sonine kernel on two segments. By the Mellin convolution theorem, the integral equations associated with models 1 and 2 reduce to vector Riemann-Hilbert problems with $$3 \times 3$$ and $$2 \times 2$$ triangular matrix coefficients whose entries consist of meromorphic and plus or minus infinite indices exponential functions. Canonical matrices of factorization are derived and the partial indices are computed. Exact representation formulae for the normal stress, the stress intensity factors (SIFs) at the crack and inclusion edges, and the normal displacement are obtained and the results of numerical tests are reported. In addition, simple asymptotic formulae for the SIFs are derived. Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple https://zbmath.org/1472.74209 2021-11-25T18:46:10.358925Z "Wang, Xu" https://zbmath.org/authors/?q=ai:wang.xu "Schiavone, Peter" https://zbmath.org/authors/?q=ai:schiavone.peter Summary: Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes. On the role of Hermite-like polynomials in the Fock representations of Gaussian states https://zbmath.org/1472.81021 2021-11-25T18:46:10.358925Z "Pierobon, Gianfranco" https://zbmath.org/authors/?q=ai:pierobon.gianfranco-l "Cariolaro, Gianfranco" https://zbmath.org/authors/?q=ai:cariolaro.gianfranco-l "Dattoli, Giuseppe" https://zbmath.org/authors/?q=ai:dattoli.giuseppe Summary: The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper. {\copyright 2021 American Institute of Physics} Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform https://zbmath.org/1472.81065 2021-11-25T18:46:10.358925Z "Almalki, Fadhel" https://zbmath.org/authors/?q=ai:almalki.fadhel "Kisil, Vladimir V." https://zbmath.org/authors/?q=ai:kisil.vladimir-v Summary: The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems. A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities https://zbmath.org/1472.81090 2021-11-25T18:46:10.358925Z "Antunes, Pedro R. S." https://zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao "Benguria, Rafael D." https://zbmath.org/authors/?q=ai:benguria.rafael-d "Lotoreichik, Vladimir" https://zbmath.org/authors/?q=ai:lotoreichik.vladimir "Ourmières-Bonafos, Thomas" https://zbmath.org/authors/?q=ai:ourmieres-bonafos.thomas let $$\Omega \subset {\mathbb R}^2$$ be a $$C^\infty$$ simply connected domain and let $$n = (n_1,n_2)^\top$$ be the outward pointing normal field on $$\partial\Omega$$. The Dirac operator with infinite mass boundary conditions in $$L^2(\Omega,{\mathbb C}^2)$$ is defined as $D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\ -2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix},$ with domain $$\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},$$ where $$\mathbf{n} := n_1 + \mathrm{i} n_2$$ and $$\partial_z, \partial_{\bar{z}}$$ are the Wirtinger operators. The spectrum of $$D^\Omega$$ is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity $\cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.$ The authors prove the following estimate $E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D})$ with equality if and only if $$\Omega$$ is a disk, where $$r_i$$ is the inradius of $$\Omega$$ and $$\mathbb D$$ is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of $$E_1(\Omega)$$. $$E > 0$$ is the first non-negative eigenvalue of $$D^\Omega$$ if and only if $$\mu^\Omega(E) = 0$$, where $\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.$ \par The authors propose the following conjecture $\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0$ and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality $$E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})$$ (it is still an open question). Hyperspace fermions, Möbius transformations, Krein space, fermion doubling, dark matter https://zbmath.org/1472.81110 2021-11-25T18:46:10.358925Z "Jaroszkiewicz, George" https://zbmath.org/authors/?q=ai:jaroszkiewicz.george Summary: We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter $$T$$ and equations of motion converted into difference equations. These equations are solved and the physical limit $$T \rightarrow 0$$ then taken. In principle, this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projective geometry. Quantization via Schwinger's action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter. Comparing light-front quantization with instant-time quantization https://zbmath.org/1472.81338 2021-11-25T18:46:10.358925Z "Mannheim, Philip D." https://zbmath.org/authors/?q=ai:mannheim.philip-d "Lowdon, Peter" https://zbmath.org/authors/?q=ai:lowdon.peter "Brodsky, Stanley J." https://zbmath.org/authors/?q=ai:brodsky.stanley-j Summary: In this paper we compare light-front quantization and instant-time quantization both at the level of operators and at the level of their Feynman diagram matrix elements. At the level of operators light-front quantization and instant-time quantization lead to equal light-front time commutation (or anticommutation) relations that appear to be quite different from equal instant-time commutation (or anticommutation) relations. Despite this we show that at unequal times instant-time and light-front commutation (or anticommutation) relations actually can be transformed into each other, with it only being the restriction to equal times that makes the commutation (or anticommutation) relations appear to be so different. While our results are valid for both bosons and fermions, for fermions there are subtleties associated with tip of the light cone contributions that need to be taken care of. At the level of Feynman diagrams we show for non-vacuum Feynman diagrams that the pole terms in four-dimensional light-front Feynman diagrams reproduce the widely used three-dimensional light-front on-shell Hamiltonian Fock space formulation in which the light-front energy and light-front momentum are on shell. Moreover, we show that the contributions of pole terms in non-vacuum instant-time and non-vacuum light-front Feynman diagrams are equal. However, because of circle at infinity contributions we show that this equivalence of pole terms fails for four-dimensional light-front vacuum tadpole diagrams. Then, and precisely because of these circle at infinity contributions, we show that light-front vacuum tadpole diagrams are not only nonzero, they quite remarkably are actually equal to the pure pole term instant-time vacuum tadpole diagrams. Light-front vacuum diagrams are not correctly describable by the on-shell Hamiltonian formalism, and thus not by the closely related infinite momentum frame prescription either. Thus for the light-front vacuum sector we must use the off-shell Feynman formalism as it contains information that is not accessible in the on-shell Hamiltonian Fock space approach. We show that light-front quantization is intrinsically nonlocal, and that for fermions this nonlocality is present in Ward identities. One can project fermion spinors into so-called good and bad components, and both of these components contribute in Ward identities. Central to our analysis is that the transformation from instant-time coordinates and fields to light-front coordinates and fields is a unitary, spacetime-dependent translation. Consequently, not only are instant-time quantization and light-front quantization equivalent, because of general coordinate invariance they are unitarily equivalent.