Recent zbMATH articles in MSC 30https://zbmath.org/atom/cc/302024-04-15T15:10:58.286558ZWerkzeugDirichlet series associated to sum-of-digits functionshttps://zbmath.org/1530.110702024-04-15T15:10:58.286558Z"Everlove, Corey"https://zbmath.org/authors/?q=ai:everlove.coreyFix \(\beta > 1\). The author considers \(d_\beta(n)\), the sum of base-\(\beta\) digits of \(n\), a summatory function
\[
S_\beta(n) = \sum_{m<n}d_\beta(n),
\]
and the Dirichlet series associated with these functions
\[
F_\beta(s) = \sum_n d_\beta(n)n^{-s},\qquad \sigma>1,
\]
and
\[
G_\beta(s) = \sum_n S_\beta(n)n^{-s},\qquad \sigma>2.
\]
In the case when \(\beta = b\) is an integer, the author obtains meromorphic continuations of order \(\geq 2\) of \(F_b\) and \(G_b\) to the entire complex plane, determines their poles and gives closed-form formulas for the residues. A similar goal is achieved for \(F_\beta\) and \(G_\beta\) with general \(\beta>1\) in the half-planes \(\sigma>0\) and \(\sigma >1\) respectively.
Reviewer: Maciej Radziejewski (Poznań)Sublinear quasiconformality and the large-scale geometry of Heintze groupshttps://zbmath.org/1530.201462024-04-15T15:10:58.286558Z"Pallier, Gabriel"https://zbmath.org/authors/?q=ai:pallier.gabrielSummary: This article analyzes sublinearly quasisymmetric homeomorphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly bi-Lipschitz equivalence (generalized quasi-isometry).Large-scale sublinearly Lipschitz geometry of hyperbolic spaceshttps://zbmath.org/1530.201472024-04-15T15:10:58.286558Z"Pallier, Gabriel"https://zbmath.org/authors/?q=ai:pallier.gabrielSummary: Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly bi-Lipschitz Equivalences (SBE) are a weak variant of quasi-isometries, with the only requirement of still inducing bi-Lipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of boundary extensions of SBEs, reminiscent of quasi-Möbius (or quasisymmetric) mappings. We give a dimensional invariant of the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druţu.Convexity properties of area integral means over the annulihttps://zbmath.org/1530.300012024-04-15T15:10:58.286558Z"Duan, Yucong"https://zbmath.org/authors/?q=ai:duan.yucong"Wang, Chunjie"https://zbmath.org/authors/?q=ai:wang.chunjieSummary: For positive numbers \(t\), \(p\), \(q\), \(c\) and an analytic function \(f(z)\) in an annulus \(R_1<|z| < R_2\) let \(M_{t, \varphi,q, c} (f,r)\) be the area integral means of \(f\) with respect to the weighted area measure \(\varphi' (|z|^q) |z|^{q-2} dA(z)\), where \(R_1\leqslant c < R_2\). We show that \(M_{t,\varphi,q,c} (f,r)^{\frac{1}{p}}\) is a convex function of \(r\) if \(f\) and \(\varphi\) satisfy certain conditions. The convexities of \(\log M_{t,\varphi,q,c}(f,r)\) in \(r\) and \(\log r\) can be obtained as special cases.A note on a theorem of T. J. Rivlinhttps://zbmath.org/1530.300022024-04-15T15:10:58.286558Z"Mir, Abdullah"https://zbmath.org/authors/?q=ai:mir.abdullah"Wani, Ajaz"https://zbmath.org/authors/?q=ai:wani.ajaz-ahmad"Hussain, Imtiaz"https://zbmath.org/authors/?q=ai:hussain.imtiazSummary: We obtain a result that improves the results of Govil and Nwaeze, Qazi and the classical result of Rivlin.The Bohr radius and its modifications for linearly invariant families of analytic functionshttps://zbmath.org/1530.300032024-04-15T15:10:58.286558Z"Ponnusamy, S."https://zbmath.org/authors/?q=ai:ponnusamy.saminathan"Shmidt, E. S."https://zbmath.org/authors/?q=ai:shmidt.e-s"Starkov, V. V."https://zbmath.org/authors/?q=ai:starkov.victor-vSummary: In 1914, \textit{H. Bohr} [Proc. Lond. Math. Soc. (2) 13, 1--5 (1914; JFM 44.0289.01)] published an article in which he considered the class \(\mathcal{B}\) of all analytic functions in the unit disk \(| z | < 1\), bounded in absolute value by 1. The article contains proofs that for any function \(f(z) = \sum_{n = 0}^\infty a_n z^n\) from \(\mathcal{B}\), the inequality \(\sum_{n = 0}^\infty | a_n z^n | \leq 1\) holds in the disk of radius 1/3 centered at the origin, and the value 1/3 is optimal. Also, Bohr himself in this article proved the corresponding result in a circle of radius 1/6 and adds, following the request of Wiener, Wiener's later proof for the disk of radius 1/3. Since then, the constant 1/3 in this problem has been called the Bohr radius. This was followed by a series of papers dealing with analogues of the Bohr radius or its estimates in other classes of functions. In this article, we give estimates for the Bohr radius in some classes of analytic functions in \(\mathbb{D}\), associated with linearly invariant families of finite order.On a result of Bao-Qin Li concerning Dirichlet series and shared valueshttps://zbmath.org/1530.300042024-04-15T15:10:58.286558Z"Fan, Xiao-Yan"https://zbmath.org/authors/?q=ai:fan.xiaoyan"Li, Xiao-Min"https://zbmath.org/authors/?q=ai:li.xiaomin"Yi, Hong-Xun"https://zbmath.org/authors/?q=ai:yi.hongxunSummary: \textit{B. Q. Li} [Adv. Math. 226, No. 5, 4198--4211 (2011; Zbl 1221.11192)] proved that if two \(\mathrm{L}\)-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) satisfy the same functional equation with \(a(1)=1\) and \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), then \(L_1=L_2\). Later on, \textit{S. M. Gonek} et al. [Math. Z. 278, No. 3--4, 995--1004 (2014; Zbl 1306.11069)] proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c)=L_2^{-1}(c)\) for a finite non-zero complex number \(c\), then \(L_1=L_2\). This implies that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), then \(L_1=L_2\). In this paper, we prove that if two \(\mathrm{L}\)-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the zero degrees and satisfy \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), and if \(a_1(1)=a_2(1)\) or \(\lim_{r\rightarrow+\infty}\frac{T(r,L_2)}{T(r,L_1)}=1\), then \(L_1= L_2 \). The main results obtained in this paper improve Theorem 1 of [Li; loc. cit.] when the \(\mathrm{L}\)-functions in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the zero degrees. Some examples are provided to show that the results obtained in this paper, in a sense, are best possible.On complex modified Bernstein-Stancu operatorshttps://zbmath.org/1530.300052024-04-15T15:10:58.286558Z"Çetin, Nursel"https://zbmath.org/authors/?q=ai:cetin.nurselSummary: The present paper deals with complex form of a generalization of perturbed Bernstein-type operators. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation by these operators attached to functions analytic in a disk centered at the origin with radius greater than 1 are obtained in this study.On the \(s\mathrm{th}\) derivative of a polynomialhttps://zbmath.org/1530.300062024-04-15T15:10:58.286558Z"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchand"Krishnadas, Kshetrimayum"https://zbmath.org/authors/?q=ai:krishnadas.kshetrimayumSummary: We extend and improve some well-known Bernstein-type inequalities for the \(s\mathrm{th}\) derivative of a \(q^\mathrm{th}\) degree polynomial \(f(\omega )\) not vanishing in the interior of a disk, \(|\omega |<\tau , \tau >0\).
For the entire collection see [Zbl 1515.00024].Improvement and generalization of polynomial inequality due to Rivlinhttps://zbmath.org/1530.300072024-04-15T15:10:58.286558Z"Singha, Nirmal Kumar"https://zbmath.org/authors/?q=ai:singha.nirmal-kumar"N., Reingachan"https://zbmath.org/authors/?q=ai:n.reingachan"Devi, Maisnam Triveni"https://zbmath.org/authors/?q=ai:devi.maisnam-triveni"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z| < 1\). In this paper, by involving some coefficients of the polynomial, we prove an inequality that not only improves as well as generalizes the well-known result proved by Rivlin but also has some interesting consequences.Improved bounds of polynomial inequalities with restricted zerohttps://zbmath.org/1530.300082024-04-15T15:10:58.286558Z"Soraisam, Robinson"https://zbmath.org/authors/?q=ai:soraisam.robinson"Singha, Nirmal Kumar"https://zbmath.org/authors/?q=ai:singha.nirmal-kumar"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchandSummary: Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z| < k\), \(k \geq 1\). Then \textit{M. A. Malik} [J. Lond. Math. Soc., II. Ser. 1, 57--60 (1969; Zbl 0179.37901)] obtained the following inequality:
\[
\max_{|z| = 1}|p^\prime(z)| \leq \frac{n}{1+k}\max_{|z| = 1}|p(z)|.
\]
In this paper, we shall first improve as well as generalize the above inequality. Further, we also improve the bounds of two known inequalities obtained by \textit{N. K. Govil} et al. [Ill. J. Math. 23, 319--329 (1979; Zbl 0408.30003)].Turán-type \(L^r\)-inequalities for polar derivative of a polynomialhttps://zbmath.org/1530.300092024-04-15T15:10:58.286558Z"Soraisam, Robinson"https://zbmath.org/authors/?q=ai:soraisam.robinson"Singh, Mayanglambam Singhajit"https://zbmath.org/authors/?q=ai:singh.mayanglambam-singhajit"Chanam, Barchand"https://zbmath.org/authors/?q=ai:chanam.barchandSummary: If \(p(z)\) is a polynomial of degree \(n\) having all its zeros in \(\vert z\vert\le k\), \(k\ge1\), then for
any complex number \(\alpha\) with \(\vert\alpha\vert\ge k\) and \(r\ge 1\), \textit{A. Aziz} [J. Approximation Theory 55, No. 2, 232--239 (1988; Zbl 0675.30002)] proved
\[
\left\{\int_0^{2\pi}\left\vert 1+k^ne^{i\theta}\right\vert^r d\theta\right\}^{\frac{1}{r}}\max_{\vert z\vert=1}\left\vert p'(z)\right\vert\ge
n\left\{\int_0^{2\pi}\left\vert p\left(e^{i\theta}\right)\right\vert^r d\theta\right\}^{\frac{1}{r}}.
\]
In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by \textit{N. A. Rather, I. Dar} and \textit{A. Iqbal} [`` Some Lower Bound Estimates for the Polar Derivatives of Polynomals with Restricted Zeros'', J. Anal. Num. Theory, 9(1) (2021), 1--5] into \(L^r\)-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.On zeros of holomorphic functionshttps://zbmath.org/1530.300102024-04-15T15:10:58.286558Z"Khodos, Olga V."https://zbmath.org/authors/?q=ai:khodos.olga-vSummary: The aim of the article is to find conditions on the coefficients of the Taylor expansion of a holomorphic function in \(\mathbb{C}\) that guarantee a absence of zeros.On continuous parameter dependence of roots of analytic functionshttps://zbmath.org/1530.300112024-04-15T15:10:58.286558Z"Möller, Manfred"https://zbmath.org/authors/?q=ai:moller.manfredSummary: Let \({\mathcal{A}}_n(\Omega)\) be the set of analytic functions on a domain \(\Omega\) of the complex plane which have \(n\) roots in \(\Omega\), counted with multiplicity. In this note we consider functions in \({\mathcal{A}}_n(\Omega)\) which depend continuously on a parameter. A simple short proof shows that the set of roots in the Hausdorff metric depends continuously on the parameter. If the parameter space is connected and all roots are known to lie in one of two disjoint open subsets \(\Omega_1\), \(\Omega_2\) of the complex plane, then the number of the roots, counted with multiplicity, in \(\Omega_1\) and \(\Omega_2\), respectively, is independent of the parameter. Each set of roots generates a unique monic polynomial. It is shown that the map which associates with each function in \({\mathcal{A}}_n(\Omega)\) the corresponding monic polynomial is continuous when \({\mathcal{A}}_n(\Omega)\) is equipped with the topology of uniform convergence on compact subsets of \(\Omega\). Possible applications are indicated.On extrema of the Mityuk radius for doubly connected domainshttps://zbmath.org/1530.300122024-04-15T15:10:58.286558Z"Kazantsev, A. V."https://zbmath.org/authors/?q=ai:kazantsev.andrei-vitalevich"Kinder, M. I."https://zbmath.org/authors/?q=ai:kinder.m-i.1Summary: We study extrema of the Mityuk radius depending on the choice of the canonical domain. Turning to the doubly connected case allows us to use the explicit form of mapping functions onto canonical domains. We obtain the results on the localization of critical points of the Mityuk radius for two types of such domains.Computing \(h\)-functions of some planar simply connected two-dimensional regionshttps://zbmath.org/1530.300132024-04-15T15:10:58.286558Z"Mahenthiram, Arunmaran"https://zbmath.org/authors/?q=ai:mahenthiram.arunmaranSummary: The \(h\)-function \(h(r)\) represents the probability that a Brownian traveller released from a fixed point \(z_0\) in a region \(\Omega\), first exits the region \(\Omega\) somewhere within distance \(r\) of \(z_0\). We compute the \(h\)-functions of some planar simply connected regions, and investigate the asymptotics of these \(h\)-functions at certain values of \(r\). Also, we analyse the derivatives of \(h(r)\) at a value of \(r\) where two regimes meet.\(\Phi\)-like analytic functions associated with a vertical domainhttps://zbmath.org/1530.300142024-04-15T15:10:58.286558Z"Araci, Serkan"https://zbmath.org/authors/?q=ai:araci.serkan"Karthikeyan, K. R."https://zbmath.org/authors/?q=ai:karthikeyan.kadhavoor-ragavan"Murugusundaramoorthy, G."https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Khan, Bilal"https://zbmath.org/authors/?q=ai:khan.bilalSummary: In this article, using the principle of subordination we introduce a new class of \(\Phi\)-like functions associated with a vertical strip domain and provided some interesting deviations or adaptation which are helpful in unification and extension of various studies of analytic functions. Furthermore, we illustrated the impact of vertical strip domain on various conic region. Inclusion relations, geometrical interpretation, coefficient estimates, inverse function coefficient estimates and solution to the Fekete-Szegő problem of the de fined class are discussed. Applications of our main results are given as corollaries.Three families of functions of complexity onehttps://zbmath.org/1530.300152024-04-15T15:10:58.286558Z"Beloshapka, Valery K."https://zbmath.org/authors/?q=ai:beloshapka.valerii-kSummary: Three rare families of functions of analytic complexity one were studied. Main results are the description of linear differential equations with solutions of complexity one (Theorem 2), the description of \(L\)-pairs of complexity one (Theorem 5), the description of \(O(2)\)-simple functions (Theorem 7).Some results on a starlike class with respect to \((j; m)\)-symmetric functionshttps://zbmath.org/1530.300162024-04-15T15:10:58.286558Z"Devi, K. Renuka"https://zbmath.org/authors/?q=ai:devi.k-renuka"Sivasubramanian, S."https://zbmath.org/authors/?q=ai:sivasubramanian.srikandan|sivasubramanian.sivaramakrishnan"Shamsan, Hamid"https://zbmath.org/authors/?q=ai:shamsan.hamid"Latha, S."https://zbmath.org/authors/?q=ai:latha.satyanarayana|latha.s-r|latha.sridar|latha.s-k|latha.sridharSummary: In the present article, the class \(S_s^{*(j,m)}(\mu ,\eta )\) is introduced using \((j, m)\)-symmetrical functions. Further, we discuss certain interesting properties for the functions of this class.
For the entire collection see [Zbl 1515.00024].The second- and third-order Hermitian Toeplitz determinants for some subclasses of analytic functions associated with exponential functionhttps://zbmath.org/1530.300172024-04-15T15:10:58.286558Z"Gurusamy, P."https://zbmath.org/authors/?q=ai:gurusamy.palpandy"Jayasankar, R."https://zbmath.org/authors/?q=ai:jayasankar.r"Sivasubramanian, S."https://zbmath.org/authors/?q=ai:sivasubramanian.srikandanSummary: In the current investigation, estimates for the second- and third-order Hermitian Toeplitz determinants for few subclasses of analytic functions associated with exponential function are obtained.
For the entire collection see [Zbl 1515.00024].Sharp coefficients bounds for starlike functions associated with Gregory coefficientshttps://zbmath.org/1530.300182024-04-15T15:10:58.286558Z"Kazımoğlu, Sercan"https://zbmath.org/authors/?q=ai:kazimoglu.sercan"Deniz, Erhan"https://zbmath.org/authors/?q=ai:deniz.erhan"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: In this paper we introduced the class \(\mathcal{S}_G^\ast\) of analytic functions which is related with starlike functions and generating function of Gregory coefficients. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain several sharp coefficient bounds on the first six coefficients and also further bounds on the corresponding Hankel determinants for functions in the class \(\mathcal{S}_G^\ast\). Additionally, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the considered class \(\mathcal{S}_G^\ast\) were estimated.Radius of \(\gamma\)-spirallikeness of order \(\alpha\) of some special functionshttps://zbmath.org/1530.300192024-04-15T15:10:58.286558Z"Kazımoğlu, Sercan"https://zbmath.org/authors/?q=ai:kazimoglu.sercan"Gangania, Kamaljeet"https://zbmath.org/authors/?q=ai:gangania.kamaljeetSummary: In light of the Alexander transformation, the class of spirallike functions is significant. The characteristics of special functions also appear very frequently in Geometric function theory. In this paper, we find the radii of \(\gamma\)-spirallike and convex \(\gamma\)-spirallike of order \(\alpha\) of certain special functions.Some properties for meromorphic functions associated with integral operatorshttps://zbmath.org/1530.300202024-04-15T15:10:58.286558Z"Kota, W. Y."https://zbmath.org/authors/?q=ai:kota.wafaa-y"El-Ashwah, R. M."https://zbmath.org/authors/?q=ai:el-ashwah.rabha-mohamedSummary: In the present paper we aim at proving some subordinations properties for meromorphic functions analytic in the punctured unit disc \(\Delta^* =\{ z:0<|z|<1\}\) with a simple pole at the origin. The functions under investigation are associated with two integral operators \(\mathcal{P}_{\sigma}^{\gamma}\) and \(\mathcal{Q}_{\sigma}^{\gamma}\) (see \textit{A. Y. Lashin} in [Comput. Math. Appl. 59, No. 1, 524--531 (2010; Zbl 1189.30025)]). Several other results and numerical examples are also obtained.Study of the analytic function related to the Le-Roy-type Mittag-Leffler functionhttps://zbmath.org/1530.300212024-04-15T15:10:58.286558Z"Mehrez, K."https://zbmath.org/authors/?q=ai:mehrez.khaledSummary: We study some geometric properties (such as univalence, starlikeness, convexity, and close-to-convexity) of Le-Roy-type Mittag-Leffler function. In order to solve the posed problem, we use new two-sided inequalities for the digamma function. Some examples are also provided to illustrate the obtained results. Interesting consequences are deduced to show that these results improve several results available in the literature for the two-parameter Mittag-Leffler function.Two applications of Grunsky coefficients in the theory of univalent functionshttps://zbmath.org/1530.300222024-04-15T15:10:58.286558Z"Obradović, Milutin"https://zbmath.org/authors/?q=ai:obradovic.milutin"Tuneski, Nikola"https://zbmath.org/authors/?q=ai:tuneski.nikolaSummary: Let \(\mathcal{S}\) denote the class of functions \(f\) which are analytic and univalent in the unit disk \(\mathbb{D} = \{z : |z| < 1\}\) and normalized with \(f(z) = z + \sum^\infty_{n=2} a_n z^n\). Using a method based on Grusky coefficients we study two problems over the class \(\mathcal{S}\): estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference \(|\alpha_5| - |\alpha_4|\).Starlike functions associated with \(\tanh z\) and Bernardi integral operatorhttps://zbmath.org/1530.300232024-04-15T15:10:58.286558Z"Rai, Pratima"https://zbmath.org/authors/?q=ai:rai.pratima"Çetinkaya, Asena"https://zbmath.org/authors/?q=ai:cetinkaya.asena"Kumar, Sushil"https://zbmath.org/authors/?q=ai:kumar.sushilSummary: We determine the necessary and sufficient convolution conditions for the starlike functions on the open unit disk and related to some geometric aspects of the function \(\tanh z\). We also determine sharp bounds on second and third order Hermitian-Toeplitz determinants for such functions. Further, we compute estimates on some initial coefficients and the Hankel determinants of third and fourth order. In addition, using the concept of Briot-Bouquet type differential subordination, we establish a subordination inclusion involving Bernardi integral operator.Coefficient estimates for starlike and convex functions related to sigmoid functionshttps://zbmath.org/1530.300242024-04-15T15:10:58.286558Z"Raza, M."https://zbmath.org/authors/?q=ai:raza.mohsan"Thomas, D. K."https://zbmath.org/authors/?q=ai:thomas.derek-keith"Riaz, A."https://zbmath.org/authors/?q=ai:riaz.aminaSummary: We give sharp coefficient bounds for starlike and convex functions related to modified sigmoid functions. We also provide some sharp coefficient bounds for the inverse functions and sharp bounds for the initial logarithmic coefficients and some coefficient differences.On a class of \(A\)-analytic functionshttps://zbmath.org/1530.300252024-04-15T15:10:58.286558Z"Sadullaev, Azimbai"https://zbmath.org/authors/?q=ai:sadullaev.azimbai-sadullaevich|sadullaev.azimbai-s"Jabborov, Nasridin M."https://zbmath.org/authors/?q=ai:jabborov.nasridin-mSummary: We consider \(A\)-analytic functions in case when \(A\) is anti-holomorphic function. In paper for \(A\)-analytic functions the integral theorem of Cauchy, integral formula of Cauchy, expansion to Taylor series, expansion to Loran series, Picard's big theorem and Montel's theorem are proved.Univalence criteria for locally univalent analytic functionshttps://zbmath.org/1530.300262024-04-15T15:10:58.286558Z"Hu, Zhenyong"https://zbmath.org/authors/?q=ai:hu.zhenyong"Fan, Jinhua"https://zbmath.org/authors/?q=ai:fan.jinhua"Wang, Xiaoyuan"https://zbmath.org/authors/?q=ai:wang.xiaoyuanSummary: Suppose that \(p(z) = 1 + z \varphi '' (z) / \varphi ' (z)\), where \(\varphi (z)\) is a locally univalent analytic function in the unit disk \(\mathbf{D}\) with \(\varphi (0) = \varphi ' (1) - 1 = 0\). We establish the lower and upper bounds for the best constants \(\sigma_0\) and \(\sigma_1\) such that \({e}^{{-\sigma }_0/2}<\left|p\left(z\right)\right|<{e}^{{\sigma }_0/2}\) and \(|p(w)/p(z)| < {e}^{{\sigma }_1}\) for \(z, w \in \mathbf{D} \), respectively, imply the univalence of \(\varphi (z)\) in \(\mathbf{D}\).A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence formhttps://zbmath.org/1530.300272024-04-15T15:10:58.286558Z"Guarnotta, Umberto"https://zbmath.org/authors/?q=ai:guarnotta.umberto"Mosconi, Sunra"https://zbmath.org/authors/?q=ai:mosconi.sunra-j-nSummary: For solutions of \(\operatorname{Div}(DF(Du))=f\) we show that the quasiconformality of \(z\mapsto DF(z)\) is the key property leading to the Sobolev regularity of the stress field \(DF(Du)\), in relation with the summability of \(f\). This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present two applications: a nonlinear Cordes condition for equations in divergence form and some partial results on the \(C^{p'}\) conjecture.On distortions of the transfinite diameter of disk imagehttps://zbmath.org/1530.300282024-04-15T15:10:58.286558Z"Salimov, R."https://zbmath.org/authors/?q=ai:salimov.ruslan-radikovich"Vyhivska, L."https://zbmath.org/authors/?q=ai:vyhivska.liudmyla-vyacheslavivna"Klishchuk, B."https://zbmath.org/authors/?q=ai:klishchuk.bogdan-anatolevichThis paper deals with the so-called ring \(Q\)-homeomorphisms with respect to the \(p\)-modulus for \(p>2\) in the complex plane. This important theory in complex analysis has been developed by several authors, e.g., [\textit{V. I. Ryazanov} and \textit{E. A. Sevost'yanov}, Sib. Mat. Zh. 48, No. 6, 1361--1376 (2007); translation in Sib. Math. J. 48, No. 6, 1093--1105 (2007; Zbl 1164.30364); \textit{O. Martio} et al., Contemp. Math. 364, 193--203 (2004; Zbl 1069.30039); Ann. Acad. Sci. Fenn., Math. 30, No. 1, 49--69 (2005; Zbl 1071.30019); Moduli in modern mapping theory. New York, NY: Springer (2009; Zbl 1175.30020); the first author, Izv. Math. 72, No. 5, 977--984 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 141--148 (2008; Zbl 1175.31006)], among others. Some introductory definitions are necessary in order to present the results of the paper.
A Borel function \(\rho:\mathbb C\to[0,\infty]\) is called \textit{admissible} for a family \(\Gamma\) of curves in \(\mathbb C\) iff
\[\int_\gamma \rho(z)\,|dz|\geq1\] for each \(\gamma\in\Gamma\). The set of all admissible Borel functions \(\rho\) for \(\Gamma\) is denoted by \(\mathrm{adm\,}\Gamma\).
For \(p\in(1,\infty)\) the \textit{\(p\)-modulus} of the family \(\Gamma\) is defined by
\[ M_p(\Gamma):=\inf_{\rho\in\mathrm{adm\,}\Gamma}\int_{\mathbb C}\rho^p(z)\,dx\,dy.\]
For arbitrary \(E,F,G\subset\mathbb C\), the set of all curves \(\gamma:[a,b]\to\mathbb C\) such that
\(\gamma(a)\in E\), \(\gamma(b)\in F\) and \(\gamma(t)\in G\) for \(a<t<b\), is denoted by
\(\Delta(E,F,G)\).
For a domain \(D\subset\mathbb C\) and a Lebesgue-measurable function \(Q:D\to[0,\infty]\), a homeomorphism \(f:D\to\mathbb C\) is called a \textit{ring \(Q\)-homeomorphism w.r.t.\ the \(p\)-modulus at \(z_0\in D\)\/} iff
\[
M_p\bigg( \Delta(fS_1,fS_2,fD)\bigg)
\leq \int_{\mathbb{A}} Q(z)\,\eta^p\big( |z-z_0| \big)\,dx\,dy
\]
for all rings \(\mathbb{A}=\mathbb{A}(z_0,r_1,r_2)=\big\{ r_1<|z-z_0|<r_2 \big\}\)
with \(0<r_1<r_2<d_0:=\mathrm{dist}(z_0,\partial D)\) and for each measurable function
\(\eta:(r_1.r_2)\to[0,\infty]\) such that \(\int_{r_1}^{r_2}\eta(r)\,dr=1\).
For a bounded and closed set \(E\subset\mathbb C\) the authors consider the \textit{transfinite diameter} \(d(E)\) of \(E\) defined as \(\lim_{n\to\infty}d_n(E)\), where \[d_n=\max\Big\{ \prod_{1\leq i<j\leq n} |z_i-z_j|^{2/(n(n-1))}\; \Big\vert\; z_i,z_j\in E \Big\}\] is the \textit{\(n\)-th diameter} of \(E\), \(n\in\mathbb N\).
The authors establish then their first main result regarding a lower bound for the
distortion of the transfinite diameter of disk image.
Theorem 1. Suppose that \(D,D'\) are bounded domains in \(\mathbb C\) and \(f:D\to D'\) is a ring
\(Q\)-homeomorphism with respect to the \(p\)-modulus at a point \(z_0\in D\) for \(p>2\).
Let \(0\leq t< d_0:=\mathrm{dist}(z_0,\partial D)\) and for \(t\in(0,d_0)\) let
\[ q_{z_0}(t)=\frac{1}{2\pi t} \int_{|z-z_0|=t} Q(z)\,|dz|.\] Assume that the function \(Q\) satisfies the condition \(q_{z_0}(t)\leq \kappa t^{-\alpha}\) for \(\kappa>0\), \(\alpha\geq0\) and for almost all \(t\in(0,d_0)\). Then, for all \(r\in(0,d_0)\), the following estimate holds:
\[d\big( f \left\{ |z-z_0|\leq r \right\} \big)
\geq \kappa^{-1/(p-2)} \left(\frac{p-2}{p-1} \right)^{(p-1)/(p-2)} r^{(p+\alpha-2)/(p-2)}.\]
Then, for a domain \(D\subset\mathbb C\) and a fixed \(z_0\in D\), and numbers \(p>2\),
\(\kappa=\kappa(z_0)>0\) and \(\alpha\geq0\), the authors consider the set
\(\mathcal{D}=\mathcal{D}(z_0,p,\kappa,\alpha)\) of all ring \(Q\)-homeomorphisms
\(f:D\to\mathbb C\) w.r.t.\ the \(p\)-modulus at a point \(z_0\in D\) with the property
\[q(t):=\frac{1}{2\pi t}\int_{|z-z_0|=t} q(z)\,|dz|\leq\kappa t^{-\alpha}\]
for almost all \(t\in(0,d_0)\). On the set \(\mathcal{D}\) the authors consider the \textit{distorsion functional} \(\mathbf{d}_r(f)=d\big( f \{|z-z_0|\leq r\} \big)\), \(f\in\mathcal{D}\), and obtain their second main result concerning the solution of the problem of minimization of the distorsion functional of the transfinite diameter of a disk
in the \(\mathcal{D}\)-class of ring \(Q\)-homeomorphisms w.r.t.\ the \(p\)-modulus,
namely
Theorem 2. \(\displaystyle \min_{f\in\mathcal{D}} \mathbf{d}_r(f) = \kappa^{-1/(p-1)}
\left( \frac{p-2}{p+\alpha-2} \right)^{(p-1)/(p-2)} r^{(p+\alpha-2)/(p-2)}\),\quad
\(r\in[0,d_0)\).
Reviewer: Luis Salinas (Valparaíso)Rigidity of holomorphic mappings from local and global viewpointshttps://zbmath.org/1530.300292024-04-15T15:10:58.286558Z"Ito, Manabu"https://zbmath.org/authors/?q=ai:ito.manabuSummary: We consider rigidity of holomorphic mappings from local and global viewpoints. For instance, we derive a generalization of the famous multi-point Schwarz-Pick lemma of Alan Frank Beardon and Carl David Minda that contains a number of known variations of the classical Schwarz lemma. Such global conclusions will be reached by using comparison with proper holomorphic mappings.Analytic function that map the unit disk into the inside of the lemniscate of Bernoullihttps://zbmath.org/1530.300302024-04-15T15:10:58.286558Z"Yadav, Shalu"https://zbmath.org/authors/?q=ai:yadav.shalu"Ravichandran, Vaithiyanathan"https://zbmath.org/authors/?q=ai:ravichandran.vSummary: The function \(\varphi_L\) defined by \(\varphi_L(z)=\sqrt{1+z}\) maps the unit disk \(\mathbb{D}\) onto \(\Omega=\{w\in\mathbb{C}: |w^2-1|<1\}\), the region in the right half-plane bounded by the lemniscate of Bernoulli \(|w^2-1|=1\). This paper deals with starlike functions defined on \(\mathbb{D}\) with \(zf'(z)/f(z)\in\Omega\) or equivalently \(zf'(z)/f(z)\) is subordinated to \(\varphi_L(z)\) and these functions are related to the analytic function \(p:\mathbb{D}\to\mathbb{C}\) with \(p(z)\in\Omega\) for all \(z\in\mathbb{D}\) by \(p(z)=zf'(z)/f(z)\). Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions \(p\) to satisfy \(p(z)\in\Omega\). As applications, we give several sufficient conditions for functions \(f\) to satisfy \(zf'(z)/f(z)\in\Omega\).Logan-Hermite extremal problems for entire functions of exponential typehttps://zbmath.org/1530.300312024-04-15T15:10:58.286558Z"Gorbachev, D. V."https://zbmath.org/authors/?q=ai:gorbachev.dmitrii-viktorovich"Ivanov, V. I."https://zbmath.org/authors/?q=ai:ivanov.valerii-ivanovich(no abstract)On inclusion relations between weighted spaces of entire functionshttps://zbmath.org/1530.300322024-04-15T15:10:58.286558Z"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardSummary: We characterize the inclusions of weighted classes of entire functions in terms of the defining weights resp. weight systems. First we treat weights defined in terms of a so-called associated weight function where the weight(system) is based on a given sequence. The abstract weight function case is then reduced to the weight sequence setting by using the so-called associated weight sequence. As an application of the main statements we characterize closedness under point-wise multiplication of these classes.Isometries, direct sum decompositions, analytic type function spaces, and radial operatorshttps://zbmath.org/1530.300332024-04-15T15:10:58.286558Z"Vasilevski, Nikolai"https://zbmath.org/authors/?q=ai:vasilevski.nikolai-lSummary: The paper connects rather general statements about the direct sum decomposition of a Hilbert space generated by isometries with the study of analytic type subspaces in weighted \(L_2\)-spaces over the unit disk \({\mathbb{D}}\) or the complex plane \({\mathbb{C}}\), and with characterization of the so-called radial operators acting on these spaces. We characterize the properties of the so-called \((m, n)\)-\textit{analytic} functions, those that are annihilated by \(\frac{\partial^m }{\partial z^m}\frac{\partial^n }{\partial{{\overline{z}}}^n}\), which as particular cases include analytic, anti-analytic, harmonic, and various related to them functions. We characterize also the so-called radial operators, with a particular attention to Toeplitz ones, that act on different subspaces of our weighted \(L_2\)-spaces.Some approximation theorems in weighted Smirnov-Orlicz spaceshttps://zbmath.org/1530.300342024-04-15T15:10:58.286558Z"Avsar, A. H."https://zbmath.org/authors/?q=ai:avsar.ahmet-hamdi"Yildirir, Y. E."https://zbmath.org/authors/?q=ai:yildirir.yunus-emreSummary: In this study, we investigate the degree of approximation of matrix transforms obtained by Faber series and prove some approximation theorems in weighted Smirnov-Orlicz spaces.Toward a certified greedy Loewner framework with minimal samplinghttps://zbmath.org/1530.300352024-04-15T15:10:58.286558Z"Pradovera, Davide"https://zbmath.org/authors/?q=ai:pradovera.davideSummary: We propose a strategy for greedy sampling in the context of non-intrusive interpolation-based surrogate modeling for frequency-domain problems. We rely on a non-intrusive and cheap error indicator to drive the adaptive selection of the high-fidelity samples on which the surrogate is based. We develop a theoretical framework to support our proposed indicator. We also present several practical approaches for the termination criterion that is used to end the greedy sampling iterations. To showcase our greedy strategy, we numerically test it in combination with the well-known Loewner framework. To this effect, we consider several benchmarks, highlighting the effectiveness of our adaptive approach in approximating the transfer function of complex systems from a few samples.Universal Taylor series on specific compact setshttps://zbmath.org/1530.300362024-04-15T15:10:58.286558Z"Tsirivas, Nikolaos"https://zbmath.org/authors/?q=ai:tsirivas.nikolaosSummary: It is well-known that there exist holomorphic functions \(f\) on the open unit disc \(D\) for which the sequence \((S_n(f))\) of partial sums of its Taylor series with center 0 approximate all the polynomials on all compact subsets \(K\subset\mathbb{C}\setminus D\) with connected complement. This result fails if we consider the sequence of the weighted partial sums \((2^nS_n(f))\) instead of \((S_n(f))\). In the present paper we show that there exist holomorphic functions \(f\) on \(D\) for which the sequence \((2^nS_n(f))\) approximate all the polynomials on \textit{only some specific compact sets} \(K\). The geometry of \(K\) plays a crucial role. We also generalize these results for weights other than \(2^n\) and on arbitrary simply connected domains.Asymptotics of the deformed Fredholm determinant of the confluent hypergeometric kernelhttps://zbmath.org/1530.300372024-04-15T15:10:58.286558Z"Dai, Dan"https://zbmath.org/authors/?q=ai:dai.dan.2|dai.dan|dai.dan.1"Zhai, Yu"https://zbmath.org/authors/?q=ai:zhai.yuSummary: In this paper, we consider the deformed Fredholm determinant of the confluent hypergeometric kernel. This determinant represents the gap probability of the corresponding determinantal point process where each particle is removed independently with probability \(1- \gamma\), \(0 \leq \gamma <1\). We derive asymptotics of the deformed Fredholm determinant when the gap interval tends to infinity, up to and including the constant term. As an application of our results, we establish a central limit theorem for the eigenvalue counting function and a global rigidity upper bound for its maximum deviation.
{{\copyright} 2022 Wiley Periodicals LLC.}On local properties of singular integralshttps://zbmath.org/1530.300382024-04-15T15:10:58.286558Z"Mamedkhanov, J. I."https://zbmath.org/authors/?q=ai:mamedkhanov.jamal-islam-oglu"Jafarov, S. Z."https://zbmath.org/authors/?q=ai:jafarov.sadulla-zSummary: Let \(\gamma\) be a regular curve. We study the local properties of singular integrals in the class of functions \({H}_{\alpha }^{\alpha +\beta }\left({t}_0,\gamma \right).\) We obtain a strengthening of the Plemelj-Privalov theorem for functions from the class \({H}_{\alpha }^{\alpha +\beta }\left({t}_0,\gamma \right).\) It is proved that, at the point \(t_0\) of increased smoothness for \(\alpha+ \beta < 1\), there is only a logarithmic loss.Remarks on a boundary value problem for a matrix valued \(\overline{\partial}\) equationhttps://zbmath.org/1530.300392024-04-15T15:10:58.286558Z"Kenig, Carlos E."https://zbmath.org/authors/?q=ai:kenig.carlos-eTaking a \(2 \times 2\) matrix \(A\) with complex entries in \(\mathbb{R}^2\), with \(\|A\|_\infty \leq M\), and given a matrix \(H\) bounded in \(\partial D\), where \(D = \{z \in\mathbb{C} : |z| < 1\}\), such that \(\|H\|_\infty \leq N_1\), and that the matrix \(H H^\ast\) is strictly positive definite, with \(\|(H H^\ast)^{-1}\|_\infty \leq N_2\), it is concluded that the problem
\[
\left\{ \begin{array}{lll} \overline{\partial} P = AP &\mbox{in}& D\\
PP^\ast = HH^\ast &\mbox{in}& \partial D, \quad \mbox{a.e. (non-tangentially)} \end{array} \right.
\]
has a solution \(P\) which is a \(2\times 2\) complex matrix in \(D\), with \(P\) and \(P^{-1}\) being bounded in \(D\). Moreover, any two solutions of the problem are equivalent up to a constant unitary matrix.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Contribution of \(n\)-cylinder square-tiled surfaces to Masur-Veech volume of \(\mathcal{H}(2g-2)\)https://zbmath.org/1530.300402024-04-15T15:10:58.286558Z"Yakovlev, Ivan"https://zbmath.org/authors/?q=ai:yakovlev.ivanSquare-tiled surfaces are special kinds of quadrangulations of surfaces. Their asymptotic enumeration is tightly linked with the so-called Masur-Veech volumes of the strata of the moduli space of abelian and quadratic differentials as shown by \textit{A. Zorich} [in: Rigidity in dynamics and geometry. Berlin: Springer. 459--471 (2002; Zbl 1038.37015)]. This link has been used to deduce many values and properties of the Masur-Veech volumes.
The simplest stratum of the moduli space of abelian differentials is \(\mathcal{H}(2g-2)\) which consists of pairs \((X, \omega)\) where \(X\) is a Riemann surface and \(\omega\) an abelian differential on \(X\) with a single zero of degree \(2g-2\). Their Masur-Veech volumes have been studied from an algebraic geometry perspective in a work of \textit{A. Sauvaget} [Geom. Funct. Anal. 28, No. 6, 1756--1779 (2018; Zbl 1404.14035)]. In this article, an alternative computation of the Masur-Veech volumes of \(\mathcal{H}(2g-2)\) is provided using square-tiled surface enumeration. The main result shows that a natural refined asymptotic count of square-tiled surfaces in \(\mathcal{H}(2g-2)\) is a one-parameter deformation of the original generating series of A. Sauvaget.
The strategy of the proof follows the initial idea of \textit{J. Athreya} et al. [Geom. Dedicata 170, 195--217 (2014; Zbl 1290.32012)] and the reviewer et al. [Duke Math. J. 170, No. 12, 2633--2718 (2021; Zbl 1471.14066)]. Namely, square-tiled surfaces are counted with an additional parameter taking into account the combinatorics of their cylinders. That required the author to develop a complete framework for the asymptotic enumeration of metric unicellular bipartite maps (i.e., metric graphs embedded in surface with a proper 2-coloring of its vertices and whose complement is homeomorphic to a disk). One important tool turns out to be the so-called Chapuy-Féray-Fusy bijection between unicellular maps and decorated trees [\textit{G. Chapuy} et al., J. Comb. Theory, Ser. A 120, No. 8, 2064--2092 (2013; Zbl 1278.05081)].
Reviewer: Vincent Delecroix (Bordeaux)The infimum of the dual volume of convex cocompact hyperbolic 3-manifoldshttps://zbmath.org/1530.300412024-04-15T15:10:58.286558Z"Mazzoli, Filippo"https://zbmath.org/authors/?q=ai:mazzoli.filippoLet \(M\) be a complete convex cocompact hyperbolic \(3\)-manifold and let \(CM\) be its convex core. Schlenker introduced the notion of dual volume \(V^{*}_{C}(M)\) of the convex core, which can be computed as \(V_{C}(M)-\frac{1}{2}\ell_{m}(\mu)\), where \(V_{C}(M)\) stands for the usual Riemannian volume of the convex core and \(\ell_{m}(\mu)\) denotes the length of the bending measured lamination \(\mu\) with respect to the hyperbolic metric \(m\) of the boundary of the convex core of \(M\).
The paper studies \(V_{C}^{*}(M)\) as a function over the space of quasi-isometric deformations of a given convex cocompact \(3\)-manifold \(M\) with incompressible boundary. The author shows that the infimum of the dual volume coincides with the infimum of the Riemannian volume of the convex core.
This result can be seen as the dual volume counterpart of a similar theorem proved by \textit{M. Bridgeman} et al. [Duke Math. J. 168, No. 5, 867--896 (2019; Zbl 1420.32007)], who showed that the infimum of the renormalized volume over all quasi-isometric deformations of a given convex cocompact \(3\)-manifold \(M\) with incompressible boundary coinced with the infimum of the Riemannian volume of the convex core.
Reviewer: Andrea Tamburelli (Houston)Groups of automorphisms of bordered non-orientable Klein surfaces of topological genus 3https://zbmath.org/1530.300422024-04-15T15:10:58.286558Z"Bujalance, E."https://zbmath.org/authors/?q=ai:bujalance.emilio"Etayo, J. J."https://zbmath.org/authors/?q=ai:etayo-gordejuela.jose-javier"Martínez, E."https://zbmath.org/authors/?q=ai:martinez.ernesto-cSummary: In this paper we study the automorphism groups \(G\) of compact non-orientable Klein surfaces of topological genus 3 with \(k > 0\) boundary components. We obtain that the group \(G\) is one of the following: \( C_2, C_3, C_4, C_6, C_2 \times C_2, D_3, D_4\) and \(D_6\). Besides, for each \(k > 0\) there exists such a Klein surface admitting each of these groups as an automorphism group, excepting that \(C_6\) and \(D_6\) do not act on surfaces with \(k \equiv 1 \bmod 3\). Moreover, if \(G\) is an automorphism group of a bordered non-orientable Klein surface of topological genus 3, with \(k\) boundary components, then it is the full automorphism group of some such surface excepting \(C_3\) for \(k = 1\) and \(2\), \(C_4\) for \(k = 1\), and \(C_6\) for \(k = 2\) and 5.Non-convexity of extremal lengthhttps://zbmath.org/1530.300432024-04-15T15:10:58.286558Z"Sagman, Nathaniel"https://zbmath.org/authors/?q=ai:sagman.nathanielSummary: With respect to every Riemannian metric, the Teichmüller metric, and the Thurston metric on Teichmüller space, we show that there exist measured foliations on surfaces whose extremal length functions are not convex. The construction uses harmonic maps to \(\mathbb{R}\)-trees and minimal surfaces in \(\mathbb{R}^n\).A non-Archimedean second main theorem for small functions and applicationshttps://zbmath.org/1530.300442024-04-15T15:10:58.286558Z"Ta Thi Hoai An"https://zbmath.org/authors/?q=ai:ta-thi-hoai-an."Nguyen Viet Phuong"https://zbmath.org/authors/?q=ai:nguyen-viet-phuong.Summary: We establish a slowly moving target second main theorem for meromorphic functions on a non-Archimedean field, with counting functions truncated to level \(1\). As an application, we show that two meromorphic functions on a non-Archimedean field must coincide if they share \(q\) (\(q \geq 5\)) distinct small functions, ignoring multiplicities. Thus, our work improves the results in [\textit{A. Escassut} and \textit{C. C. Yang},
Rend. Circ. Mat. Palermo (2) 70, No. 2, 623--630 (2021; Zbl 1476.30154)].A right inverse of curl which is divergence-free invariant and some applications to generalized Vekua-type problemshttps://zbmath.org/1530.300452024-04-15T15:10:58.286558Z"Delgado, Briceyda B."https://zbmath.org/authors/?q=ai:delgado.briceyda-b"Macías-Díaz, Jorge E."https://zbmath.org/authors/?q=ai:macias-diaz.jorge-eduardo(no abstract)Cousin's problems I and II: the bicomplex casehttps://zbmath.org/1530.300462024-04-15T15:10:58.286558Z"Bravo Ortega, Yesenia"https://zbmath.org/authors/?q=ai:bravo-ortega.yesenia"Reséndis Ocampo, Lino Feliciano"https://zbmath.org/authors/?q=ai:resendis-ocampo.lino-feliciano"Tovar Sánchez, Luis Manuel"https://zbmath.org/authors/?q=ai:tovar.luis-manuelSummary: We present the bicomplex versions of Cousin's problems I and II as well as their relationship with the bicomplex versions of Weierstrass' and Mittag-Leffler's theorems. We establish relations between these theorems and Cousin's problems, which reveal peculiarities of bicomplex meromorphic function theory.On some quaternionic serieshttps://zbmath.org/1530.300472024-04-15T15:10:58.286558Z"González Cervantes, J. Oscar"https://zbmath.org/authors/?q=ai:gonzalez-cervantes.jose-oscar"Cordero, J. Emilio Paz"https://zbmath.org/authors/?q=ai:cordero.j-emilio-paz"González Campos, Daniel"https://zbmath.org/authors/?q=ai:gonzalez-campos.danielSummary: The aim of this work is to show that given \(u\in\mathbb{H}\setminus\mathbb{R}\), there exists a differential operator \(G^{-u}\) whose solutions expand in quaternionic power series expansion \(\sum_{n=0}^\infty(x-u)^n a_n\) in a neighborhood of \(u\in\mathbb{H}\). This paper also presents Stokes and Borel-Pompeiu formulas induced by \(G^{-u}\) and as consequence we give some versions of Cauchy's Theorem and Cauchy's Formula associated to these kind of regular functions.On a class of automorphisms in \(\mathbf{H}^2\) which resemble the property of preserving volumehttps://zbmath.org/1530.300482024-04-15T15:10:58.286558Z"Prezelj, Jasna"https://zbmath.org/authors/?q=ai:prezelj.jasna"Vlacci, Fabio"https://zbmath.org/authors/?q=ai:vlacci.fabioThe authors give a possible extension for shears and overshears in the case of two non-commutative variables. They investigate what are the minimal conditions to define good generalizations of complex holomorphic shears and overshears in relation with the associated vector fields and flows in the non commutative (mainly quaternionic) setting. The authors restrict to the case of mappings represented by convergent quaternionic power series.
The authors note that the class of quaternionic automorphisms with volume 1 is not defined, since there does not exist a quaternionic volume form on \(\mathbf H^n\), and since automorphisms with convergent power series as components are not necessarily regular in the sense given in a previous paper by
\textit{R. Ghiloni} and \textit{A. Perotti} [in: Progress in analysis. Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Moscow, Russia, August 22--27, 2011. Volume 1. Moscow: Peoples' Friendship University of Russia. 179--186 (2012; Zbl 1298.30044)].
Therefore, the authors present an alternative definition of partial derivative, divergence and rotor in the quaternionic setting, and detemine the subclasses of vector fields with divergence or rotor. They define also automorphisms with volume to be deformations of the identity by vector fields with divergence, and then the authors show that these automorphisms form a proper class to which the Andersen-Lempert theory [\textit{E. Andersén} and \textit{L. Lempert}, Invent. Math. 110, No. 2, 371--388 (1992; Zbl 0770.32015)] applies.
Finally, the authors exhibit an example of a quaternionic automorphism, which is not in the closure of the set of finite compositions of volume-preserving quaternionic shears while its restriction to the complex variables is approximable by a finite composition of (complex) shears.
Reviewer: María Elena Luna Elizarrarás (Holon)Best constants in inequalities involving analytic and co-analytic projections and Riesz's theorem in various function spaceshttps://zbmath.org/1530.300492024-04-15T15:10:58.286558Z"Melentijević, Petar"https://zbmath.org/authors/?q=ai:melentijevic.petar"Marković, Marijan"https://zbmath.org/authors/?q=ai:markovic.marijanSummary: Let \(P_+\) be the Riesz's projection operator and let \(P_- = I - P_+\). We consider estimates of the expression \(\|(|P_+ f |^s + |P_- f |^s)^{\frac{1}{s}} \|_{L^p (\mathbf{T})}\) in terms of Lebesgue \(p\)-norm of the function \(f \in L^p(\mathbf{T})\). We find the accurate estimates for \(p \geq 2\) and \(0 < s \leq p\), thus significantly improving \textit{D. Kalaj}'s result [Trans. Am. Math. Soc. 372, No. 6, 4031--4051 (2019; Zbl 1422.30002)] who treated this problem for \(s = 2\) and \(1<p< \infty \). Interestingly, for this range of \(s\) the appropriate vector-valued inequality holds with the same constant. Additionally, we obtain the right asymptotic of the constants for large \(s\). This proves the conjecture of \textit{B. Hollenbeck} and and \textit{I. E. Verbitsky} [Operator Theory: Advances and Applications 202, 285--295 (2010; Zbl 1193.42064)] on the Riesz projection operator in some cases.Generalized integration operators from weighted Bergman spaces into general function spaceshttps://zbmath.org/1530.300502024-04-15T15:10:58.286558Z"Zhu, Xiangling"https://zbmath.org/authors/?q=ai:zhu.xianglingSummary: This article studies the boundedness of the inclusion mapping from weighted Bergman spaces \(A_\alpha^p\) into a class of tent type space \(T_s^{p,n} (\mu)\). As an application, the boundedness, compactness and essential norm of generalized integral operators \(T_g^{n,k}\) and \(S_g^{n,0}\) from \(A^p_\alpha\) to general function spaces are also investigated.Non-radial weights and polynomial approximation in spaces of analytic functionshttps://zbmath.org/1530.300512024-04-15T15:10:58.286558Z"Abkar, Ali"https://zbmath.org/authors/?q=ai:abkar.aliSummary: We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights.The constant of interpolation in Bloch type spaceshttps://zbmath.org/1530.300522024-04-15T15:10:58.286558Z"Miralles, Alejandro"https://zbmath.org/authors/?q=ai:miralles.alejandro"Maletzki, Mario P."https://zbmath.org/authors/?q=ai:maletzki.mario-pA sequence \((z_n)\) in the unit disc \(\mathbb D\) is said to be an interpolating sequence for \(\mathcal B_v\), if for any \((a_n)\in \ell_\infty\) there exists \(f\in \mathcal B_v\) such that \(v(z_n) f'(z_n)=a_n\) for all \(n\in \mathbb N\). It is said to be \(\delta\)-separated if \(\rho(z_k,z_j)\ge \delta\) for \(k\ne j\) where \(\rho(z,w)\) stands for the pseudohyperbolic distance in \(\mathbb D\). It was shown by \textit{K. R. M. Attele} [Glasg. Math. J. 34, No. 1, 35--41 (1992; Zbl 0751.30032)] and \textit{K. Madigan} and \textit{A. Matheson} [Trans. Am. Math. Soc. 347, No. 7, 2679--2687 (1995; Zbl 0826.47023)] that there exists a universal constant \(\Delta_1\) such that \(\Delta_1\)-separated sequences are interpolating for \(\mathcal B_v\) in the case \(v(z)=1-|z|^2\). The authors show that the same situation happens for \(v(z)=(1-|z|^2)^p\) with \(p\ge 1\) and get lower and upper estimates for the constant \(\Delta_p\) in these cases, improving the known estimates for \(\Delta_1\).
Reviewer: Oscar Blasco (València)Which sets are images of disk-algebra functions?https://zbmath.org/1530.300532024-04-15T15:10:58.286558Z"Mortini, Raymond"https://zbmath.org/authors/?q=ai:mortini.raymond"Pflug, Peter"https://zbmath.org/authors/?q=ai:pflug.peterThe paper gives a characterization of those compact sets in the plane with finitely many holes (i.e., the bounded components of the complement) that are images of disk-algebra functions. The authors also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex (from the abstract).
The paper starts with a useful discussion of topological tools, including a construction of a crosscut \(J\) joining the outer boundary of a locally connected set \(K\) such that \(\overline{K^{\circ}}=K\) with the closure of the unique hole of \(K\) (Lemma 2.6). The construction guarantees that \(K^{\circ}\setminus J\) is simply connected.
Proposition 4.1 gives necessary conditions for the set \(K:=f(\bar{\mathbb{D}})\), where \(f\) belongs to the disk algebra \(A(\bar{\mathbb{D}})\). The set \(K\) must be a locally (path-)connected continuum such that \(K=\overline{K^{\circ}}\) and \(K^{\circ}\) is connected. Such sets are called {\em admissible}.
Theorem 4.2 says that if \(K\) is an admissible set, then the boundary of each complementary component is a Jordan curve. Proposition 5.1 says that if \(K\) is admissible and polynomially convex, then there exists \(f\in A(\overline{\mathbb{D}})\) such that \(f(\overline{\mathbb{D}})=K\). Next the authors turn their attention to sets with holes. Theorem 5.3 says that if \(K\) is admissible with a single hole, then \(K\) is the image of a disk algebra function. A crowning achievement of the paper is Theorem 6.4 which is the same statement for admissible sets with finitely many holes.
In the last section the authors turn their attention to polynomials. It is shown that if \(p\) is a polynomial of degree \(2\) then \(p(\bar{\mathbb{D}})\) has no holes (Proposition 7.1), there exists a polynomial \(p\) of degree \(4\) such that \(p(\bar{\mathbb{D}})\) has a unique hole (Proposition 7.2), a similar statement is also proved for the case of degree \(3\). The section concludes with natural questions concerning polynomials. In particular, is the maximal number of holes of \(p(\bar{\mathbb{D}})\) equal to \(n-2\) whenever a polynomial \(p\) has degree \(n\geq 2\)?
Lastly, the authors discuss difficulties with compacta having infinitely many holes.
Reviewer: Michal Jasiczak (Poznań)Quasisymmetrically co-Hopfian Menger curves and Sierpiński spaceshttps://zbmath.org/1530.300542024-04-15T15:10:58.286558Z"Hakobyan, Hrant"https://zbmath.org/authors/?q=ai:hakobyan.hrantThe paper studies quasisymmetrically co-Hopfian metric spaces, which are those metric spaces \(X\) having the following property: every quasisymmetric embedding of \(X\) into itself is surjective. The main result states that there exists a quasisymmetrically co-Hopfian metric space that is homeomorphic to the Menger curve \(\mathscr M\); this answers a question raised by \textit{S. Merenkov} [Invent. Math. (2) 180, 361--388 (2010; Zbl 1194.37044)]. An explicit example of such a space is constructed, namely the ``double'' \(D\mathfrak M\) of a slit Menger curve \(\mathfrak M\). The author also proves a similar result for higher-dimensional spaces, namely showing that for every natural number \(n\geq 1\) the double of a suitably-chosen slit Sierpiński \(n\)-space is quasisymmetrically co-Hopfian. Both slit Menger curves and slit Sierpiński spaces are objects that are introduced and studied for the first time in this paper. Among other properties of these spaces, the author proves that the set of quasisymmetric equivalence classes of slit Menger curves is uncountable. Some applications to the study of boundaries of Gromov hyperbolic spaces are also obtained.
Reviewer: Enrico Pasqualetto (Jyväskylä)Uniform domains in real Banach spaceshttps://zbmath.org/1530.300552024-04-15T15:10:58.286558Z"Ouyang, Zhengyong"https://zbmath.org/authors/?q=ai:ouyang.zhengyong"Jiao, Bo"https://zbmath.org/authors/?q=ai:jiao.bo"Guan, Tiantian"https://zbmath.org/authors/?q=ai:guan.tiantianSummary: In this paper, we show that certain types of domains in real Banach spaces \(E\) with dimension at least two are uniform domains. Our examples incude annulus domains, bounded convex domains \(C\) and their complements \(E\setminus \overline{C}\), and \(C\setminus \alpha \overline{C}\) for all \(0<\alpha <1\) when the zero vector \(o\in C\).Zero distribution of random Bernoulli polynomial mappingshttps://zbmath.org/1530.320032024-04-15T15:10:58.286558Z"Bayraktar, Turgay"https://zbmath.org/authors/?q=ai:bayraktar.turgay"Çelik, Çiğdem"https://zbmath.org/authors/?q=ai:celik.cigdemSummary: In this note, we study asymptotic zero distribution of multivariable full system of random polynomials with independent Bernoulli coefficients. We prove that with overwhelming probability their simultaneous zeros sets are discrete and the associated normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.On traces in Hardy type analytic spaces in bounded strictly pseudoconvex domains and in tubular domains over symmetric coneshttps://zbmath.org/1530.320052024-04-15T15:10:58.286558Z"Shamoyan, Romi F."https://zbmath.org/authors/?q=ai:shamoyan.romi-f"Kurilenko, Sergey M."https://zbmath.org/authors/?q=ai:kurilenko.sergey-mSummary: We provide some new estimates on traces in new mixed norm Hardy type spaces and related new results on Bergman type intergal operators in Hardy type spaces in tubular domains over symmetric cones and bounded striclty pseudoconvex domains with smooth boundary. We generalize a well-known one dimensional result concerning traces of Hardy spaces obtained previously in the unit disk by various authors.On bisectors in quaternionic hyperbolic spacehttps://zbmath.org/1530.320142024-04-15T15:10:58.286558Z"Almeida, Igor A. R."https://zbmath.org/authors/?q=ai:almeida.igor-a-r"Chamorro, Jaime L. O."https://zbmath.org/authors/?q=ai:chamorro.jaime-leonardo-orjuela"Gusevskii, Nikolay"https://zbmath.org/authors/?q=ai:gusevskii.n-aUnlike the real hyperbolic space, the complex hyperbolic space \(\mathbb{H}^n_{\mathbb{C}}\) of complex dimension \(n\ge 2\) contains no totally geodesic real hypersurfaces. In \(\mathbb{H}^n_{\mathbb{C}}\), bisectors are a particular class of real hypersurfaces which are a good substitute for totally geodesic ones (a bisector is the locus of points equidistant from a particular pair of points). Although not totally geodesic, they are foliated by totally geodesic real and complex submanifolds.
In this paper the authors show that some basic results about bisectors from complex hyperbolic geometry carry over to the quaternionic hyperbolic case. But in the quaternionic hyperbolic space \(\mathbb{H}^n_{\mathbb{Q}}\) the geometry of bisectors is richer: any bisector is a union of totally geodesic submanifolds of \(\mathbb{H}^n_{\mathbb{Q}}\) isometric to \(\mathbb{H}^n_{\mathbb{C}}\), intersecting in a common point (\textit{fan} decomposition) and such decomposition is not unique.
Generalizing a construction of \textit{J. R. Parker} and \textit{I. D. Platis} [J. Differ. Geom. 73, No. 2, 319--350 (2006; Zbl 1100.30037)], they also introduce \textit{complex hyperbolic packs} in \(\mathbb{H}^n_{\mathbb{Q}}\), a new class of hypersurfaces which could be useful in the construction of fundamental polyhedra for discrete groups of isometries of quaternionic hyperbolic case.
Reviewer: Laura Geatti (Roma)Jensen polynomials for holomorphic functionshttps://zbmath.org/1530.330082024-04-15T15:10:58.286558Z"Griffin, Michael"https://zbmath.org/authors/?q=ai:griffin.michael-j"South, Daniel"https://zbmath.org/authors/?q=ai:south.danielSummary: Recent work of \textit{M. Griffin} et al. [Proc. Natl. Acad. Sci. USA 116, No. 23, 11103--11110 (2019; Zbl 1431.11105)]
shows that the Jensen polynomials for the Riemann xi function converge to the Hermite polynomials under a suitable normalization. We generalize this result, proving that the normalized Jensen polynomials for a large class of genus zero or one entire functions converge either to the Hermite polynomials, or to a class of polynomials which can be written as a confluent hypergeometric function.Asymptotic properties of Hermite-Padé polynomials and Katz pointshttps://zbmath.org/1530.330092024-04-15T15:10:58.286558Z"Suetin, Sergey P."https://zbmath.org/authors/?q=ai:suetin.sergei-p(no abstract)Orthogonal polynomials associated with a continued fraction of Hirschhornhttps://zbmath.org/1530.330182024-04-15T15:10:58.286558Z"Bhatnagar, Gaurav"https://zbmath.org/authors/?q=ai:bhatnagar.gaurav"Ismail, Mourad E. H."https://zbmath.org/authors/?q=ai:ismail.mourad-el-houssienySummary: We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn's continued fraction contains as special cases the famous Rogers-Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan's continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.On a generalized homogeneous Hahn polynomialhttps://zbmath.org/1530.330192024-04-15T15:10:58.286558Z"He, Bing"https://zbmath.org/authors/?q=ai:he.bing.2|he.bing.3|he.bing|he.bing.1|he.bing.4Summary: We investigate a generalized homogeneous Hahn polynomial in some detail. This polynomial includes as special cases the homogeneous Hahn polynomial and the homogeneous Rogers-Szegő polynomial. A generating function, which contains a known generating function as a special case, is given. We also give a finite series generating function. Some results on the asymptotic expansion for this polynomial are derived. Certain results on zeros are also obtained. We deduce several results on zeros of certain entire functions involving this generalized Hahn polynomial. As results, one of \textit{R. Zhang}'s results [Proc. Am. Math. Soc. 145, No. 1, 241--250 (2017; Zbl 1368.30012)] as well as others is obtained. Finally, we derive several general results on \(q\)-congruences of the generalized \(q\)-Apéry polynomials, from which two \(q\)-congruences involving the generalized homogeneous Hahn polynomial are deduced.On the transcendental entire functions satisfying some Fermat-type differential-difference equationshttps://zbmath.org/1530.340672024-04-15T15:10:58.286558Z"Mandal, Rajib"https://zbmath.org/authors/?q=ai:mandal.rajib"Biswas, Raju"https://zbmath.org/authors/?q=ai:biswas.rajuSummary: This paper explores about the existence and forms of the finite order transcendental entire function \(f(z)\) which satisfies the following Fermat-type differential-difference equations \[\begin{cases} (Af'(z))^2+(Bf(z+c)+Cf(z))^2=c^{\alpha(z)},\\ (Af'(z))^2+R^2(z)(Bf(z+c)+Cf(z))^2=Q(z),\end{cases}\] where \(A,B,C,c\in\mathbb{C}\setminus\{0\}\), \(\alpha(z)\) is a polynomial with constant coefficients, and \(R(z)\), \(Q(z)\) are non-zero polynomials with constant coefficients. The core part of value distribution theory is used as the key tool to establish these results. These results significantly improve the earlier results due to \textit{K. Liu} et al. [Arch. Math. 99, No. 2, 147--155 (2012; Zbl 1270.34170)]. Moreover, We exhibit some examples to fortify our results.On meromorphic solutions of non-linear differential equations of Tumura-Clunie typehttps://zbmath.org/1530.340772024-04-15T15:10:58.286558Z"Heittokangas, J."https://zbmath.org/authors/?q=ai:heittokangas.janne-m"Latreuch, Z."https://zbmath.org/authors/?q=ai:latreuch.zinelaabidine"Wang, J."https://zbmath.org/authors/?q=ai:wang.jun.2"Zemirni, M. A."https://zbmath.org/authors/?q=ai:zemirni.mohamed-amineA classical problem in Nevanlinna theory is, given a meromorphic function \(h(z)\), to characterize particular meromorphic solutions \(f(z)\) of the ordinary differential equation (ODE) \(f^n + P(z,f) =h\), with \(P\) a differential polynomial of \(f\) and its derivatives.
The main idea of this article is taken from a classical result of Painlevé (1902): requiring some strong property of a given ODE (movable singlevaluedness in Painlevé, meromorphy in Nevanlinna) determine its coefficients as the solutions of ODEs, possibly the same ones.
Instead of giving \(h(z)\) by some explicit expression, the authors only require \(h\) to obey a given second order linear inhomogeneous ODE with rational coefficients. This allows them to derive two important results (theorems 1.3 and 1.4), thus extrapolating all previous results.
Numerous examples illustrate all possible situations, making this article a pleasure to read.
Reviewer: Robert Conte (Gif-sur-Yvette)On Fatou sets containing Baker omitted valuehttps://zbmath.org/1530.370672024-04-15T15:10:58.286558Z"Ghora, Subhasis"https://zbmath.org/authors/?q=ai:ghora.subhasis"Nayak, Tarakanta"https://zbmath.org/authors/?q=ai:nayak.tarakanta"Sahoo, Satyajit"https://zbmath.org/authors/?q=ai:sahoo.satyajitThis paper focuses on dynamics of meromorphic functions with Baker omitted values. An omitted value \(b\), say \textit{bov}, of a transcendental meromorphic function \(f\) is called a Baker omitted
value, if there is a disk \(D\) centered at the \textit{bov} such that each component of the
boundary of \(f^{-1}(D)\) is bounded. Note that if a map has a \textit{bov}, then it must be the limit of critical values, therefore the map has infinitely singular values.
The first part of the paper concerns the connectivity of Fatou components. If \(f\) has a \textit{bov} \(b\) and \(b\) is in the Fatou component \(U\), then a Fatou component of \(f\) is infinitely connected if and only if it lands on \(U\). Further, every other Fatou component is either simply connected or lands on a Herman ring. As the \textit{bov} \(b\) is the limit of infinitely many critical point, the grand orbit of \(U\) contains infinitely critical points. Under further assumption that the number of critical points not in the grand orbit of \(U\) is finite, the connectivity of Fatou components landing on a Herman ring is bounded, and the bound only depends the multiplicity of the critical points not in the grand orbit of \(U\).
If a connected component of Fatou set is multiple connected, then the Julia set must be disconnected. Then the next part of the paper concerns the case when the Julia set is totally connected. That is, if \(b\in U\), where is \(U\) is an invariant attracting or parabolic basin, and all critical value are compactly contained in it then the Julia set is totally disconnected.
The last part of the paper focuses on the existence of a Baker domain and a Herman ring. It is found that:
(1) If the box \(b\) is in the Fatou set, there is no Baker domain;
(2) If \(f\) has a \(2\)-periodic Fatou component which is either a Baker domain or an attracting domain or a parabolic domain containing \(b\), then there is no Herman ring.
Reviewer: Tao Chen (New York)Tame rational functions: decompositions of iterates and orbit intersectionshttps://zbmath.org/1530.370682024-04-15T15:10:58.286558Z"Pakovich, Fedor"https://zbmath.org/authors/?q=ai:pakovich.fedorSummary: Let \(A\) be a rational function of degree at least 2 on the Riemann sphere. We say that \(A\) is tame if the algebraic curve \(A(x)-A(y)=0\) has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions \(A\) and \(B\) have some orbits with infinite intersection, then \(A\) and \(B\) have a common iterate. We also show that for a tame rational function \(A\) decompositions of its iterates \(A^{\circ d}\), \(d\geq 1,\) into compositions of rational functions can be obtained from decompositions of a single iterate \(A^{\circ N}\) for \(N\) large enough.Dimensions of Kleinian orbital setshttps://zbmath.org/1530.370712024-04-15T15:10:58.286558Z"Bartlett, Thomas"https://zbmath.org/authors/?q=ai:bartlett.thomas-e|bartlett.thomas-michael"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-mSummary: Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the \textit{orbital set} is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set, the Poincaré exponent of the Kleinian group, and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that our assumption that the given set is bounded (in the hyperbolic metric) cannot be removed in general.Three lectures on square-tiled surfaceshttps://zbmath.org/1530.370722024-04-15T15:10:58.286558Z"Matheus, Carlos"https://zbmath.org/authors/?q=ai:matheus.carlosSummary: This text corresponds to a minicourse delivered on June 11, 12 and 13, 2018 during the summer school ``Teichmüller dynamics, mapping class groups and applications'' at Institut Fourier, Grenoble, France. \par In this article, we cover the same topics from our minicourse, namely, origamis, Veech groups, affine homeomorphisms, and the Kontsevich-Zorich cocycle.
For the entire collection see [Zbl 1508.37014].Equidistribution of the zeros of higher order derivatives in polynomial dynamicshttps://zbmath.org/1530.390162024-04-15T15:10:58.286558Z"Okuyama, Yûsuke"https://zbmath.org/authors/?q=ai:okuyama.yusukeThe author studies the convergence of averaged distributions of the zeros of the \(m\)-th order derivatives \((f^{n})^{(m)}\) of iterated polynomials \(f^{n}\) of a polynomial \(f\in \mathbb{C}[z]\) of degree bigger than 1 towards the harmonic measure of the filled-in Julia set of \(f\) with pole at \(\infty\) as \(n \rightarrow +\infty\) for every \(m\in \mathbb{N}\), when \(f\) has no exceptional points in \(\mathbb{C}\). This is the main result:
Theorem. If the exceptional set is \(E(f)=\{\infty\}\), then
\[
\lim_{n \rightarrow +\infty} \frac{((f^{n})^{(m)})^{*}\delta_{0}}{d^{n}-m} =\mu_{f} \qquad \text{weakly on} \; \mathbb{P}^{1}.
\]
To prove his results, the author gives some preliminary computations on the higher order derivatives of iterations of a meromorphic function on \(\mathbb{C}\) restricted to an either attracting or parabolic basin. Solutions of Schröder's or Abel's functional equations with a locally uniform non-trivial error estimate are used. The following result is used to prove the above theorem:
Theorem. If \(E(f)=\{\infty\}\), then for every \(m\in \mathbb{N}\),
\[
\lim_{n \rightarrow +\infty} \frac{\log |(f^{n})^{(m)}|}{d^{n}-m} =g_{f} \qquad \text{in} \; L^{1}_{\text{loc}}(\mathbb{C}, m_{2}),
\]
where \(m_{2}\) is the real 2-dimensional Lebesgue measure on \(\mathbb{C}\).
Reviewer: Mohammad Sajid (Buraydah)On Bernstein inequality via Chebyshev polynomialhttps://zbmath.org/1530.410082024-04-15T15:10:58.286558Z"Huang, Yi C."https://zbmath.org/authors/?q=ai:huang.yichi|huang.yicheng|huang.yichun|huang.yichao|huang.yichen|huang.yi-c|huang.yicong|huang.yi-chiehSummary: Motivated by applications to the Carleson embedding theorem with matrix weights, Culiuc and Treil proved a Bernstein-type inequality for complex polynomials in the plane which are positive and satisfy a polynomial growth condition on the positive real axis. A sharp form of this Bernstein inequality, with Chebyshev polynomial of the first kind as an extremizer, was later found by \textit{D. Kraus} et al. [Anal. Math. Phys. 12, No. 1, Paper No. 40, 6 p. (2022; Zbl 1482.42063)]. In this note we show that the Chebyshev polynomial of the first kind is indeed the only extremal polynomial for this sharp Bernstein inequality.Estimates for bilinear \(\theta \)-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaceshttps://zbmath.org/1530.420432024-04-15T15:10:58.286558Z"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghui"Wang, Miaomiao"https://zbmath.org/authors/?q=ai:wang.miaomiao"Tao, Shuangping"https://zbmath.org/authors/?q=ai:tao.shuangpingSummary: Let \((\mathcal{X},d,\mu )\) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space \({M}_p^u(\mu )\), where \(1\le p< \infty\) and \(u(x,r):{\mathcal{X}}\times (0,\infty )\to (0,\infty )\) is a Lebesgue measurable function. Furthermore, under assumption that the measurable functions \(u_1\), \(u_2\), and \(u\) belong to \(\mathbb{W}_\tau\) with \(\tau \in (0,2)\), we prove that the bilinear \(\theta\)-type generalized fractional integral \(\widetilde{T}_{\theta ,\alpha}\) is bounded from the product of spaces \(M_{p_1}^{u_1}(\mu )\times M_{p_2}^{u_2}(\mu )\) into spaces \(M_q^u(\mu )\), where \({u}_1{u}_2=u\), \(\alpha \in (0,1)\), and \(\frac{1}{q}=\frac{1}{p_1}+\frac{1}{p_2}-2\alpha\) with \(p_1,p_2\in (1,\frac{1}{\alpha})\), and also show that the \(\widetilde{T}_{\theta,\alpha}\) is bounded from the product of spaces \(M_{p_1}^{u_1}(\mu ) \times M_{p_2}^{u_2}(\mu )\) into spaces \(M_1^u(\mu )\), where \(1=\frac{1}{p_1}+\frac{1}{p_2}-2\alpha\). Meanwhile, we prove that the commutator \(\widetilde{T}_{\theta,\alpha,b_1,b_2}\) formed by \(b_1,b_2\in \widetilde{\mathrm{RBMO}}(\mu )\) and \(\widetilde{T}_{\theta,\alpha}\) is bounded from the product of spaces \(M_{p_1}^{u_1}(\mu ) \times M_{p_2}^{u_2} (\mu )\) into spaces \(M_q^u(\mu )\), and it is also bounded from the product of spaces \(M_{p_1}^{u_1}(\mu ) \times M_{p_2}^{u_2}(\mu )\) into spaces \(M_1^u(\mu )\).Spectral transformation associated with a perturbed \(R_I\) type recurrence relationhttps://zbmath.org/1530.420492024-04-15T15:10:58.286558Z"Shukla, Vinay"https://zbmath.org/authors/?q=ai:shukla.vinay"Swaminathan, A."https://zbmath.org/authors/?q=ai:swaminathan.anbhuIt is very well known that Orthogonal Polynomials on the Real Line (OPRL) satisfy a three-term recurrence relation that characterizes them. The study of perturbations of the coefficients of this recurrence relation has been considered by several authors from different points of view such as the so-called co-recursive case, adding a constant to the first coefficient, or the general case perturbing the coefficients at any level.
The fact of constructing new sequences by modifying the original ones is a powerful tool with many applications to theoretical and physical problems.
In this paper, the authors study properties of OPRL that satisfy a three-term recurrence relation with new coefficients obtained by modifying the original ones in a general way. They analyze structural relations between the perturbed polynomials and the original ones via two methods: the classical matrix method and the transfer matrix method. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed polynomials. The authors also investigate the behavior of zeros of OPRL and their modified version.
Unlike OPRL, a sequence of monic Orthogonal Polynomials on the Unit Circle (OPUC) satisfies a first-order recurrence relation involving the so-called reciprocal polynomial. The coefficients of this recurrence relation, the values of the polynomials at zero, lie on the unit disk, are called reflection coefficients, Schur parameters, or Verblunsky coefficients and determine completely the OPUC sequence. This fact motivated several authors to study polynomials associated with perturbations of Schur parameters. In the paper under review, the authors also analyze the effects of co-recursion and co-dilation in the sequences of OPUC, in its reflection coefficients as well as in the corresponding orthogonality measures.
Finally, the authors illustrate the preceding results with some examples.
Reviewer: María-José Cantero (Zaragoza)Closed-form solutions for several classes of singular integral equations with convolution and Cauchy operatorhttps://zbmath.org/1530.450032024-04-15T15:10:58.286558Z"Bai, Songwei"https://zbmath.org/authors/?q=ai:bai.songwei"Li, Pingrun"https://zbmath.org/authors/?q=ai:li.pingrun"Sun, Meng"https://zbmath.org/authors/?q=ai:sun.mengThe authors discuss the existence of solutions to certain types of singular integral equations on the real line. The equations are assumed to contain a combination of integrals with Cauchy kernels and integrals with certain convolution kernels. Solutions are sought in the class \(L_2(\mathbb R) \cap \hat H\), where \(\hat H\) is the class of functions \(f\) that satisfy a Hölder condition of arbitrary order \(\nu \in (0,1]\) on some finite interval \([-A, A]\) and a corresponding condition \(|f(x) - f(y)| \le C |x^{-1} - y^{-1}|^\nu\) for \(x, y \notin [-A,A]\). The proofs are based on constructing certain Riemann boundary value problems related to the integral equations in question and on analyzing these boundary value problems.
Reviewer: Kai Diethelm (Schweinfurt)Conductive homogeneity of compact metric spaces and construction of \(p\)-energyhttps://zbmath.org/1530.460012024-04-15T15:10:58.286558Z"Kigami, Jun"https://zbmath.org/authors/?q=ai:kigami.junStarting in the second half of the 1990s, the concept of Sobolev spaces on metric spaces has emerged. The book [\textit{J. Heinonen} et al., Sobolev spaces on metric measure spaces. An approach based on upper gradients. Cambridge: Cambridge University Press (2015; Zbl 1332.46001)] provides a panoramic view on the topic. The Sobolev spaces described therein mainly consist of a function and an associated function being in some sense a generalization of the weak derivative. Having the concept of a derivative asks for the introduction of \(p\)-energies akin the situation in the Euclidean setting.
Barlow and Bass (e.g., [\textit{M.~T. Barlow} and \textit{R.~F. Bass}, Ann. Inst. Henri Poincaré, Probab. Stat. 25, No.~3, 225--257 (1989; Zbl 0691.60070)]) have studied Brownian motions in the metric setting. The inconvenience is that there are some metric spaces that do not satisfy the Gaussian heat kernel estimate.
The book under review offers a different approach to Sobolev spaces on metric spaces as a remedy to this deficiency by defining Sobolev spaces in such a way that the estimate becomes available in a larger class of metric spaces. The starting point is the following fact mentioned at the very beginning of the book. Namely, if \(I=[0,1]\) and
\[
\mathcal{E}_p^n(f)=\sum_{i=1}^{2^n}\left|f\left(\frac{i-1}{2^n}\right)-f\left(\frac{i}{2^n}\right)\right|^p \tag{1}
\]
for \(n\geq 1\) and \(f\in W^{1,p}(I)\), then, denoting by \(\nabla f\) the weak derivative of \(f\), it follows that
\[
\lim_{n \to \infty}(2^{p-1})^n\mathcal{E}_p^n(f)\to \int_{0}^{1}|\nabla f|^p\, dx. \tag{2}
\]
By looking only at the left-hand side, we obtain a way to talk about an energy without having to refer to the derivative of \(f\). The idea is to understand the points \(\frac{i}{2^n}\) as nodes. More precisely, the author is approximating metric spaces by graphs and then evaluating the function (or rather an approximation) at the nodes. The process to obtain the graphs is by using finer and finer partitions of the metric space into compact sets. Given one such approximation, two of the nodes are connected by an edge if and only if the corresponding sets meet. This leads to a version of (1) for metric spaces. Denoting by \(P_n f\) an approximation of \(f\) urges us to find a proper scaling constant \(\sigma\) such that \(\sigma^n \mathcal{E}_p^n(P_n f)\) converges. This then gives rise to a Sobolev space.
After the introduction, the author carefully introduces the partitions and their descriptions via trees. Furthermore, we get acquainted with the standard assumptions. One of them is more restrictive but contains all others.
A large part of the book deals with generalizing the energy and finding a proper constant \(\sigma\) to obtain an analogue of (2).
The author gives a definition for the energy and its corresponding Sobolev space. The obtained results depend on the parameter \(p\). The situation is much better understood for large enough \(p\). However, the author also shares what is known for small \(p\). In particular, the author proves an existence theorem concerning the heat kernel.
Now that the foundations are laid and the theory is developed, in the later part of the book, the author details many examples for the reader to get a feeling for the introduced concepts.
The book ends with a nice touch: a section concerning open problems and an appendix containing many useful facts and results.
Reviewer: Thomas Zürcher (Katowice)Multipliers on spaces of holomorphic functionshttps://zbmath.org/1530.460212024-04-15T15:10:58.286558Z"Trybuła, Maria"https://zbmath.org/authors/?q=ai:trybula.mariaSummary: We consider multipliers on the space of holomorphic functions of one variable \(H(\Omega)\), \(\Omega\subset\mathbb{C}\) open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to \(\Omega\) and \(\Omega\) is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on \(H(\Omega)\). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when \(\Omega\) is convex.Bi-Lipschitz invariance of planar \(BV\)- and \(W^{1,1}\)-extension domainshttps://zbmath.org/1530.460272024-04-15T15:10:58.286558Z"García-Bravo, Miguel"https://zbmath.org/authors/?q=ai:garcia-bravo.miguel"Rajala, Tapio"https://zbmath.org/authors/?q=ai:rajala.tapio"Zhu, Zheng"https://zbmath.org/authors/?q=ai:zhu.zhengThe authors consider the problem of extending Sobolev functions defined on a particular domain \(\Omega \subset \mathbb{R}^{d}\) to Sobolev functions (of the same regularity) defined on the entire space \(\mathbb{R}^{d}\). A domain \(\Omega \subset \mathbb{R}^{d}\) for which such an extension is possible is called an \textit{extension domain}. More precisely, given a parameter \(1\leq p\leq \infty \), \(\Omega \subset \mathbb{ R}^{d}\) is a \(W^{1,p}\)-extension domain if, for any \(u\in W^{1,p}\left( \Omega \right) \), there exists \(\tilde{u}\in W^{1,p}\left( \mathbb{R} ^{d}\right) \) such that \(\tilde{u}=u\) on \(\Omega \) and
\[
\left\Vert \tilde{u}\right\Vert _{W^{1,p}(\mathbb{R}^{d})}\leq C\left\Vert u\right\Vert _{W^{1,p}(\Omega )},
\]
where \(C>0\) is a constant depending only on \(\Omega \). Some classical examples of \(W^{1,p}\)-extension domains (for any \(1\leq p\leq \infty \)) are the Lipschitz domains (as it was proved by \textit{A. P. Calderón} [Proc. Sympos. Pure Math. 4, 33--49 (1961; Zbl 0195.41103)] and \textit{E. M. Stein} [Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press (1970; Zbl 0207.13501)]) or more generally the \((\varepsilon ,\delta )\)-domains introduced by \textit{P. W. Jones} [Acta Math. 147, 71--88 (1981; Zbl 0489.30017)].
The authors are motivated by the following result of \textit{P.~Hajłasz} et al. [Rev. Mat. Iberoam. 24, No.~2, 645--669 (2008; Zbl 1226.46029)]:
{Theorem 1.1.} If \(\Omega\) and \(\Omega ^{\prime }\) are bi-Lipschitz equivalent, for \(1<p\leq \infty \), then \(\Omega\) is a \(W^{1,p}\)-extension domain if and only if \(\Omega'\) is a \(W^{1,p}\)-extension domain.
It is natural to ask if this statement remains true in the case \(p=1\). Under some additional hypotheses on the domain \(\Omega\), \textit{P.~Koskela} et al. [in: Around the research of Vladimir Maz'ya. I. Function spaces. Dordrecht: Springer; Novosibirsk: Tamara Rozhkovskaya Publisher. 255--272 (2010; Zbl 1196.46025)] proved that Theorem~1.1 can be extended to cover the situation when \(p=1\). The authors of the present paper are able to remove those additional hypotheses on \(\Omega\). They prove the following:
{Theorem 1.3 (or Corollary 1.4).} Let \(\Omega\subset \mathbb{R}^{2}\) be a bounded \(BV\) (or \(W^{1,1}\))-extension domain and \(f:\Omega\rightarrow \Omega^{\prime }\) a bi-Lipschitz map. Then \(\Omega^{\prime }\) is also a \(BV\) (or \(W^{1,1}\))-extension domain.
The proof relies on the results of \textit{J.~Väisälä} [Conform. Geom. Dyn. 12, 58--66 (2008; Zbl 1192.30005)] and on quasiconvexity arguments.
Reviewer: Eduard Curca (Lyon)Hölder continuity of the traces of Sobolev functions to hypersurfaces in Carnot groups and the \(\mathcal{P} \)-differentiability of Sobolev mappingshttps://zbmath.org/1530.460322024-04-15T15:10:58.286558Z"Basalaev, S. G."https://zbmath.org/authors/?q=ai:basalaev.sergey-g"Vodopyanov, S. K."https://zbmath.org/authors/?q=ai:vodopyanov.serguei-kSummary: We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot-Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class \(W^{1,\nu}\) of Carnot groups is continuous, \( \mathcal{P} \)-differentiable almost everywhere, and has the \(\mathcal{N} \)-Luzin property.Metric space mappings connected with Sobolev-type function classeshttps://zbmath.org/1530.460342024-04-15T15:10:58.286558Z"Romanov, A. S."https://zbmath.org/authors/?q=ai:romanov.alexandr-sergeevich|romanov.aleksandr-sergeevichSummary: We study some properties of the metric space mappings connected with the Sobolev-type function classes \(M^1_p(X,d,\mu) \).The fine structure of the spectral theory on the \(S\)-spectrum in dimension fivehttps://zbmath.org/1530.470042024-04-15T15:10:58.286558Z"Colombo, Fabrizio"https://zbmath.org/authors/?q=ai:colombo.fabrizio"De Martino, Antonino"https://zbmath.org/authors/?q=ai:de-martino.antonino"Pinton, Stefano"https://zbmath.org/authors/?q=ai:pinton.stefano"Sabadini, Irene"https://zbmath.org/authors/?q=ai:sabadini.ireneSummary: Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter-Sce-Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called \(S\)-functional calculus for noncommuting operators based on the \(S\)-spectrum. In the second step, this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper, we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the \(S\)-spectrum. The function spaces and the associated functional calculi define the so-called \textit{fine structure of the spectral theories on the \(S\)-spectrum}. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.Bohr operator on operator-valued polyanalytic functions on simply connected domainshttps://zbmath.org/1530.470142024-04-15T15:10:58.286558Z"Allu, Vasudevarao"https://zbmath.org/authors/?q=ai:allu.vasudevarao"Halder, Himadri"https://zbmath.org/authors/?q=ai:halder.himadriSummary: In this article, we study the Bohr operator for the operator-valued subordination class \(S(f)\) consisting of holomorphic functions subordinate to \(f\) in the unit disk \(\mathbb{D} := \{z \in \mathbb{C} : |z|<1\}\), where \(f : \mathbb{D} \rightarrow \mathcal{B}(\mathcal{H})\) is holomorphic and \(\mathcal{B}(\mathcal{H})\) is the algebra of bounded linear operators on a complex Hilbert space \(\mathcal{H}\). We establish several subordination results, which can be viewed as the analogs of a couple of interesting subordination results from scalar-valued settings. We also obtain a von Neumann-type inequality for the class of analytic self-mappings of the unit disk \(\mathbb{D}\) which fix the origin. Furthermore, we extensively study Bohr inequalities for operator-valued polyanalytic functions in certain proper simply connected domains in \(\mathbb{C}\). We obtain Bohr radius for the operator-valued polyanalytic functions of the form \(F(z)= \sum_{l=0}^{p-1} \overline{z}^l f_l(z)\), where \(f_0\) is subordinate to an operator-valued convex biholomorphic function, and operator-valued starlike biholomorphic function in the unit disk \(\mathbb{D}\).Compact linear combination of composition operators on Bergman spaceshttps://zbmath.org/1530.470312024-04-15T15:10:58.286558Z"Kang, Qijian"https://zbmath.org/authors/?q=ai:kang.qijian"Wang, Maofa"https://zbmath.org/authors/?q=ai:wang.maofaSummary: It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang [3] proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each operator is compact on the weighted Bergman space when the sum of the coefficients is not equal to zero. Motivated by that work, in this paper, we discuss which operator is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC).Supercyclicity and resolvent condition for weighted composition operatorshttps://zbmath.org/1530.470322024-04-15T15:10:58.286558Z"Mengestie, Tesfa"https://zbmath.org/authors/?q=ai:mengestie.tesfa-y"Seyoum, Werkaferahu"https://zbmath.org/authors/?q=ai:seyoum.werkaferahuLet \(u\) and \(\psi\) be entire functions. This article presents a few results about weighted composition operators \(W_{u,\psi} f := u (f \circ \psi)\) acting on the Fock space \(\mathcal{F}_p, \ 1 \leq p < \infty\). It is proved that no weighted composition operator on Fock spaces is supercyclic. Conditions under which the operators satisfy Ritt's resolvent growth condition are also identified. In particular, it is shown that a non-trivial composition operator \(C_{\psi}\) on a Fock space satisfies such a growth condition if and only if it is compact.
Reviewer: José Bonet (València)The product of small Hankel operators on the Bergman spacehttps://zbmath.org/1530.470392024-04-15T15:10:58.286558Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.1|zhang.jie.52|zhang.jie.8|zhang.jie.16|zhang.jie.6|zhang.jie.13|zhang.jie.21|zhang.jie.12|zhang.jie.14|zhang.jie.7|zhang.jie.3|zhang.jie|zhang.jie.15|zhang.jie.4"Zhao, Xianfeng"https://zbmath.org/authors/?q=ai:zhao.xianfengSummary: In this paper, we establish necessary conditions and sufficient conditions for the boundedness and compactness of the small Hankel product \(\Gamma_\varphi\Gamma_\psi^\ast\) on the Bergman space, where \(\phi\) and \(\psi\) are square integrable on the open unit disk.On the localization of measure induced Toeplitz operators on the Bergman spacehttps://zbmath.org/1530.470402024-04-15T15:10:58.286558Z"Zorboska, Nina"https://zbmath.org/authors/?q=ai:zorboska.ninaLet \(L_{2,h}(\mathbb D)\) be the Hilbert space of holomorphic functions on the complex unit disk \(\mathbb D\) and let \(\nu\) be a complex Borel measure on \(\mathbb D\) such that its total variation is a Carleson measure, and \(T_\nu\) be a Toeplitz operator with symbol \(\nu\). The author states that \(T_\nu\) is strongly localized on \(L_{2,h}(\mathbb D)\) and pertains to the Toeplitz algebra (Theorem 2.1).
Reviewer: Mohammed El Aïdi (Bogotá)Volterra-type operators on the minimal Möbius-invariant spacehttps://zbmath.org/1530.470542024-04-15T15:10:58.286558Z"Xie, Huayou"https://zbmath.org/authors/?q=ai:xie.huayou"Liu, Junming"https://zbmath.org/authors/?q=ai:liu.junming"Ponnusamy, Saminathan"https://zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: In this note, we mainly study operator-theoretic properties on the Besov space \(B_1\) on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.A Sylvester-Gallai result for concurrent lines in the complex planehttps://zbmath.org/1530.520112024-04-15T15:10:58.286558Z"Cohen, Alex"https://zbmath.org/authors/?q=ai:cohen.alexThe Sylvester-Gallai theorem says that if a finite set of points in the real plane is not contained in a line, then there exists a line containing exactly two points of the set. In the complex plane, the analogous theorem does not hold. The author studies conditions which ensure that the conclusion of the Sylvester-Gallai theorem also holds in the complex plane. The main result says that if a finite set \(S\) of non-collinear points in the complex plane is contained in the union of \(m\) concurrent lines, and if one of those lines contains more than \(m-2\) points of \(S\) (not including the point of concurrency), then there exists a line containing exactly two points of \(S\). The bound is optimal because it is achieved in the Fermat configurations. The new idea in the proof consists in ordering complex numbers by their real part.
Reviewer: Norbert Knarr (Stuttgart)A projection from filling currents to Teichmüller spacehttps://zbmath.org/1530.570112024-04-15T15:10:58.286558Z"Hensel, Sebastian"https://zbmath.org/authors/?q=ai:hensel.sebastian|hensel.sebastian-c|hensel.sebastian-wolfgang"Sapir, Jenya"https://zbmath.org/authors/?q=ai:sapir.jenyaOn a closed surface, the space of geodesic currents, defined by \textit{F. Bonahon} [Invent. Math. 92, No. 1, 139--162 (1988; Zbl 0653.32022)], is the space of measures on the unit tangent bundle of that surface that are invariant by the geodesic flow. This space contains, among other subsets, the set of closed curves up to homotopy as well as the Teichmüller space of that surface. The authors define a geodesic current \(\mu\) to be filling if its intersection with any geodesic current is nonzero. This notion of filling generalizes the fact that when \(\mu\) is a closed geodesic on the surface, it is filling if it cuts the surface into simply connected regions.
In the paper under review, the authors show that there exists a mapping class group equivariant length minimizing projection from the set of filling geodesic currents onto the Teichmüller space of that surface. They prove some basic properties of this projection that show that it is well-behaved. Relations with work of \textit{R. Díaz} and \textit{C. Series} [Algebr. Geom. Topol. 3, 207--234 (2003; Zbl 1066.32020)] on Kerckhoff's lines of minima are highlighted, and applications to questions in higher Teichmüller theory are mentioned.
Reviewer: Athanase Papadopoulos (Strasbourg)Ideally, all infinite-type surfaces can be triangulatedhttps://zbmath.org/1530.570142024-04-15T15:10:58.286558Z"McLeay, Alan"https://zbmath.org/authors/?q=ai:mcleay.alan"Parlier, Hugo"https://zbmath.org/authors/?q=ai:parlier.hugoThe authors prove that any orientable surface of infinite type admits an ideal triangulation. The idea of the proof consists in using a pair-of-pants decomposition (which is known to exist), presenting a surface as a union of flute-type surfaces, and then triangulating these flute-type surfaces (by the so-called perforated Farey triangulation).
This construction also implies that given a collection of non-intersecting arcs such that any arc intersects any simple closed curve only a finite number of times, one may complete this collection to an ideal triangulation.
Reviewer: Nikita Kalinin (Guangdong)Hyperbolic limits of Cantor set complements in the spherehttps://zbmath.org/1530.570192024-04-15T15:10:58.286558Z"Cremaschi, Tommaso"https://zbmath.org/authors/?q=ai:cremaschi.tommaso"Vargas Pallete, Franco"https://zbmath.org/authors/?q=ai:vargas-pallete.francoA manifold is called of infinite type if its fundamental group is not finitely generated. Although infinite type hyperbolic 3-manifolds are very wild objects, there have been several results for them in recent years, including those by the first author. For example, many Cantor set complements in the 3-sphere are hyperbolizable.
In this paper, the authors consider the geometric convergence of infinite type hyperbolic 3-manifolds. Let \(M\) be a hyperbolic 3-manifold with no rank two cusps. Suppose that \(M\) admits an embedding in the 3-sphere and an exhaustion by \(\pi_{1}\)-injective sub-manifolds. Then there exist hyperbolic Cantor set complements in the 3-sphere which converge geometrically to \(M\).
The proof is reduced to the case that \(M\) is convex co-compact. For a convex co-compact hyperbolic 3-manifold \(M\), the desired Cantor set complements are obtained by gluing infinitely many thickened bounded surfaces to \(M\).
Reviewer: Ken Ichi Yoshida (Tōkyō)Flat forms in metric spaceshttps://zbmath.org/1530.580022024-04-15T15:10:58.286558Z"Pfeffer, Washek F."https://zbmath.org/authors/?q=ai:pfeffer.washek-fSummary: In a metric space, we define flat forms by means of tuples of Lipschitz functions multiplied by Borel measurable functions, and use them to represent flat cochains. The representation, which extends Wolfe's theorem to metric spaces, is functorial on the category of metric spaces and Lipschitz maps. It provides flat cochains and flat chains with well-behaved cup and cap products. On compact Lipschitz manifolds, the cohomology of flat cochains is naturally isomorphic to the Čech cohomology with real coefficients -- a version of De Rham's theorem.Law of the SLE tiphttps://zbmath.org/1530.600732024-04-15T15:10:58.286558Z"Butkovsky, Oleg"https://zbmath.org/authors/?q=ai:butkovskii.oleg-yaroslavovich|butkovsky.o-a"Margarint, Vlad"https://zbmath.org/authors/?q=ai:margarint.vlad"Yuan, Yizheng"https://zbmath.org/authors/?q=ai:yuan.yizhengSummary: We analyse the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is the unique solution (up to a multiplicative constant) of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions.Recovering from accuracy deterioration in the contour integral-based eigensolverhttps://zbmath.org/1530.650422024-04-15T15:10:58.286558Z"Hasegawa, Tetsuya"https://zbmath.org/authors/?q=ai:hasegawa.tetsuya"Imakura, Akira"https://zbmath.org/authors/?q=ai:imakura.akira"Sakurai, Tetsuya"https://zbmath.org/authors/?q=ai:sakurai.tetsuyaThe generalized eigenvalue problem involves finding
eigenvalues \(\lambda \in \mathbb{C}\) and the corresponding eigenvectors
\(x\in \mathbb{C}^n\setminus {0}\) that satisfy
\[
Ax = \lambda Bx, \quad A,B \in \mathbb{C}^{n\times n}.
\]
The authors assume that the matrix pencil \(zB-A\) is diagonalizable.
They apply the Sakurai-Sugiura (SS) method that can find eigenvalues located inside a given domain \(\Omega\). Then the
corresponding eigenvectors are computed by using contour integrals along a Jordan curve surrounding the given domain \(\Omega\). These contour integrals are approximated by using \(N-\)point numerical integration. When some eigenvalues exist near a quadrature point, the accuracy of other eigenpairs is deteriorated.
The approximate eigenvalues in \(\Omega\) and the corresponding eigenvectors are extracted from scaled quadrature points by the Rayleigh-Ritz procedure.
The quadrature points are usually placed along a circle or an ellipse. If the matrices \(A, B\) are real and \(zB-A\)
has only real eigenvalues, using real quadrature points can reduce the memory requirements and the computational cost because only real arithmetic is required. The authors suggest to use the Chebyshev points.
When some eigenvalues are placed near the quadrature point \(z_j\), the matrix \(z_j B-A\) becomes ill-conditioned and other eigenvalues not close to \(z_j\) are obtained with low accuracy.
The authors introduce a filter function. They show that calculating the scaled quadrature points is equivalent to multiplying each eigen-component by the filter function \(f_k(\lambda_i)\).
The accuracy deterioration is caused by the oscillation of the filter function.
The authors propose a method to recover the accuracy by suppressing the oscillations.
After obtaining the approximate eigenpairs by the SS method, the authors apply the SS method again with quadrature
points set away from the approximate eigenvalues. As a result, improved approximate eigenpairs are obtained.
However, this approach requires solution of linear equations again. But the computation time for solving the
linear equations is the most of the total computation time. The authors propose a method that avoids solving linear equations again, thus the computational cost is low.
The paper includes two algorithms. The first one is the Sakurai-Sugiura (SS) method. The second algorithm contains a detailed description of the algorithm suggested by the authors. There are also figures that illustrate particular examples.
Reviewer: Drahoslava Janovská (Praha)The quantum mechanics canonically associated to free probability i: free momentum and associated kinetic energyhttps://zbmath.org/1530.810052024-04-15T15:10:58.286558Z"Accardi, Luigi"https://zbmath.org/authors/?q=ai:accardi.luigi"Hamdi, Tarek"https://zbmath.org/authors/?q=ai:hamdi.tarek"Lu, Yun Gang"https://zbmath.org/authors/?q=ai:lu.yun-gangSummary: After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable \(X\), characterized by the fact that its probability distribution is the semi-circle law \(\mu\) on \([-2,2]\). We prove that, in the identification of \(L^2([-2,2],\mu)\) with the \(1\)-mode interacting Fock space \(\Gamma_\mu\), defined by the orthogonal polynomial gradation of \(\mu,X\) is mapped into position operator and its canonically associated momentum operator \(P\) into \(i\) times the \(\mu\)-Hilbert transform \(H_\mu\) on \(L^2([-2,2],\mu)\). In the first part of the present paper, after briefly describing the simpler case of the \(\mu\)-harmonic oscillator, we find an explicit expression for the action, on the \(\mu\)-orthogonal polynomials, of the semi-circle analogue of the translation group \(e^{itP}\) and of the semi-circle analogue of the free evolution \(e^{itP^2/2}\), respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of \(e^{-tH_\mu}\) and \(e^{-itH_\mu^2/2}\) on the \(\mu\)-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.Shor-Laflamme distributions of graph states and noise robustness of entanglementhttps://zbmath.org/1530.810292024-04-15T15:10:58.286558Z"Miller, Daniel"https://zbmath.org/authors/?q=ai:miller.daniel-n|miller.daniel-j|miller.daniel-e|miller.daniel-a"Loss, Daniel"https://zbmath.org/authors/?q=ai:loss.daniel"Tavernelli, Ivano"https://zbmath.org/authors/?q=ai:tavernelli.ivano"Kampermann, Hermann"https://zbmath.org/authors/?q=ai:kampermann.hermann"Bruß, Dagmar"https://zbmath.org/authors/?q=ai:bruss.dagmar"Wyderka, Nikolai"https://zbmath.org/authors/?q=ai:wyderka.nikolaiSummary: The Shor-Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify \(k\)-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of SLDs plays an important role, and of the geometry of quantum states, where SLDs are known as sector length distributions.Weyl-ambient geometrieshttps://zbmath.org/1530.811112024-04-15T15:10:58.286558Z"Jia, Weizhen"https://zbmath.org/authors/?q=ai:jia.weizhen"Karydas, Manthos"https://zbmath.org/authors/?q=ai:karydas.manthos"Leigh, Robert G."https://zbmath.org/authors/?q=ai:leigh.robert-gSummary: Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal \(\mathbb{R}_+\)-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.Free products and AQFThttps://zbmath.org/1530.811182024-04-15T15:10:58.286558Z"Tanimoto, Yoh"https://zbmath.org/authors/?q=ai:tanimoto.yohSummary: We review the free product construction of von Neumann algebras, its application to a question in Algebraic Quantum Field Theory (AQFT) and an application of AQFT techniques to a question of free products. We show the existence of half-sided modular inclusions with trivial relative commutant and nontriviality of relative commutant for an inclusion of free product von Neumann algebras.
For the entire collection see [Zbl 1492.47001].