Recent zbMATH articles in MSC 30Chttps://www.zbmath.org/atom/cc/30C2021-11-25T18:46:10.358925ZWerkzeugApplications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbershttps://www.zbmath.org/1472.050102021-11-25T18:46:10.358925Z"Simsek, Yilmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmazSummary: 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.The determinant inner product and the Heisenberg product of \(\mathrm{Sym}(2)\)https://www.zbmath.org/1472.150052021-11-25T18:46:10.358925Z"Crasmareanu, Mircea"https://www.zbmath.org/authors/?q=ai:crasmareanu.mirceaLet \(A\) 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.A proof of Carleson's \(\varepsilon^2\)-conjecturehttps://www.zbmath.org/1472.280052021-11-25T18:46:10.358925Z"Jaye, Benjamin"https://www.zbmath.org/authors/?q=ai:jaye.benjamin-j"Tolsa, Xavier"https://www.zbmath.org/authors/?q=ai:tolsa.xavier"Villa, Michele"https://www.zbmath.org/authors/?q=ai:villa.micheleSummary: 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 casehttps://www.zbmath.org/1472.300012021-11-25T18:46:10.358925Z"Christiansen, Jacob S."https://www.zbmath.org/authors/?q=ai:christiansen.jacob-stordal"Simon, Barry"https://www.zbmath.org/authors/?q=ai:simon.barry.1"Zinchenko, Maxim"https://www.zbmath.org/authors/?q=ai:zinchenko.maximThe 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 polynomialshttps://www.zbmath.org/1472.300022021-11-25T18:46:10.358925Z"Dougherty, Michael"https://www.zbmath.org/authors/?q=ai:dougherty.michael-r|dougherty.michael-m"McCammond, Jon"https://www.zbmath.org/authors/?q=ai:mccammond.jonLet \(\theta: \mathbb C^n \to \mathbb C^n\) 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 circlehttps://www.zbmath.org/1472.300032021-11-25T18:46:10.358925Z"Erdélyi, Tamás"https://www.zbmath.org/authors/?q=ai:erdelyi.tamasThe 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 inequalitieshttps://www.zbmath.org/1472.300042021-11-25T18:46:10.358925Z"Tyaglov, Mikhail"https://www.zbmath.org/authors/?q=ai:tyaglov.mikhail"Atia, Mohamed J."https://www.zbmath.org/authors/?q=ai:atia.mohamed-jalelSummary: 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 polynomialshttps://www.zbmath.org/1472.300052021-11-25T18:46:10.358925Z"Totik, Vilmos"https://www.zbmath.org/authors/?q=ai:totik.vilmosThe 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 periodhttps://www.zbmath.org/1472.300062021-11-25T18:46:10.358925Z"Baddoo, Peter J."https://www.zbmath.org/authors/?q=ai:baddoo.peter-jonathan"Crowdy, Darren G."https://www.zbmath.org/authors/?q=ai:crowdy.darren-gSummary: 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 functionshttps://www.zbmath.org/1472.300072021-11-25T18:46:10.358925Z"Fournier, Richard"https://www.zbmath.org/authors/?q=ai:fournier.richardThe 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 operatorshttps://www.zbmath.org/1472.300082021-11-25T18:46:10.358925Z"Mohammed, Aabed"https://www.zbmath.org/authors/?q=ai:mohammed.aabed"Darus, Maslina"https://www.zbmath.org/authors/?q=ai:darus.maslinaSummary: 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://www.zbmath.org/1472.300092021-11-25T18:46:10.358925Z"Barnard, Roger W."https://www.zbmath.org/authors/?q=ai:barnard.roger-w"Richards, Kendall C."https://www.zbmath.org/authors/?q=ai:richards.kendall-cFor \(z,\omega\in \mathbb{C}\) 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 setshttps://www.zbmath.org/1472.300102021-11-25T18:46:10.358925Z"Ntalampekos, Dimitrios"https://www.zbmath.org/authors/?q=ai:ntalampekos.dimitriosRemovability 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 lemmahttps://www.zbmath.org/1472.300112021-11-25T18:46:10.358925Z"Mashreghi, Javad"https://www.zbmath.org/authors/?q=ai:mashreghi.javadThe 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 functionshttps://www.zbmath.org/1472.300122021-11-25T18:46:10.358925Z"Oros, Georgia Irina"https://www.zbmath.org/authors/?q=ai:oros.georgia-irina"Oros, Gheorghe"https://www.zbmath.org/authors/?q=ai:oros.gheorgheSummary: 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://www.zbmath.org/1472.300132021-11-25T18:46:10.358925Z"Gorbachev, Dmitriĭ Viktorovich"https://www.zbmath.org/authors/?q=ai:gorbachev.dmitrii-viktorovich"Mart'yanov, Ivan Anatol'evich"https://www.zbmath.org/authors/?q=ai:martyanov.ivan-anatolevichCorrection to the authors' paper [ibid. 20, No. 3(71), 143--153 (2019; Zbl 1439.30050)].Blow-up solutions of Liouville's equation and quasi-normalityhttps://www.zbmath.org/1472.300152021-11-25T18:46:10.358925Z"Grahl, Jürgen"https://www.zbmath.org/authors/?q=ai:grahl.jurgen"Kraus, Daniela"https://www.zbmath.org/authors/?q=ai:kraus.daniela"Roth, Oliver"https://www.zbmath.org/authors/?q=ai:roth.oliverLet \(D\) 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.Zeros of slice functions and polynomials over dual quaternionshttps://www.zbmath.org/1472.300212021-11-25T18:46:10.358925Z"Gentili, Graziano"https://www.zbmath.org/authors/?q=ai:gentili.graziano"Stoppato, Caterina"https://www.zbmath.org/authors/?q=ai:stoppato.caterina"Trinci, Tomaso"https://www.zbmath.org/authors/?q=ai:trinci.tomasoStarting 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.Continuity of condenser capacity under holomorphic motionshttps://www.zbmath.org/1472.310052021-11-25T18:46:10.358925Z"Pouliasis, Stamatis"https://www.zbmath.org/authors/?q=ai:pouliasis.stamatisA 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: ``Maximal open radius for Strebel point''https://www.zbmath.org/1472.320082021-11-25T18:46:10.358925Z"Yao, Guowu"https://www.zbmath.org/authors/?q=ai:yao.guowuA 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.Spectral stability estimates of Dirichlet divergence form elliptic operatorshttps://www.zbmath.org/1472.352542021-11-25T18:46:10.358925Z"Gol'dshtein, Vladimir"https://www.zbmath.org/authors/?q=ai:goldshtein.vladimir"Pchelintsev, Valerii"https://www.zbmath.org/authors/?q=ai:pchelintsev.valerii"Ukhlov, Alexander"https://www.zbmath.org/authors/?q=ai:ukhlov.alexanderThe 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
\[
A_{\frac{4\beta}{\beta -1},2}(\Omega) \leq \inf\limits_{p\in \left(\frac{4\beta}{3\beta -1},2\right)} \left(\frac{p-1}{2-p}\right)^{\frac{p-1}{p}} \frac{\left(\sqrt{\pi}\cdot\sqrt[p]{2}\right)^{-1}|\Omega|^{\frac{\beta-1}{4\beta}}}{\sqrt{\Gamma(2/p) \Gamma(3-2/p)}}~~.
\]Difference equations related to number theoryhttps://www.zbmath.org/1472.390322021-11-25T18:46:10.358925Z"Heim, Bernhard"https://www.zbmath.org/authors/?q=ai:heim.bernhard-ernst"Neuhauser, Markus"https://www.zbmath.org/authors/?q=ai:neuhauser.markusAs 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].Scaling limits and fluctuations for random growth under capacity rescalinghttps://www.zbmath.org/1472.600452021-11-25T18:46:10.358925Z"Liddle, George"https://www.zbmath.org/authors/?q=ai:liddle.george"Turner, Amanda"https://www.zbmath.org/authors/?q=ai:turner.amanda-gSummary: 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 functionshttps://www.zbmath.org/1472.600462021-11-25T18:46:10.358925Z"Michelen, Marcus"https://www.zbmath.org/authors/?q=ai:michelen.marcus"Sahasrabudhe, Julian"https://www.zbmath.org/authors/?q=ai:sahasrabudhe.julianSummary: 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 \(X_n\).Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couplehttps://www.zbmath.org/1472.742092021-11-25T18:46:10.358925Z"Wang, Xu"https://www.zbmath.org/authors/?q=ai:wang.xu"Schiavone, Peter"https://www.zbmath.org/authors/?q=ai:schiavone.peterSummary: 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.