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Abstract: We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense \(G_{\delta }\) -subset of meromorphic functions that are common universal for certain uncountable families of these mappings. While a corresponding result for translations exists for entire functions, our result for dilations has no holomorphic counterpart. We further obtain an analogue of Ansari’s Theorem for the mappings we consider, which is used as a key tool in the proofs of our main results. PubDate: 2022-01-17
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Abstract: The theory of multidimensional quasiconformal mappings employs three main approaches: analytic, geometric (modulus) and metric ones. In this paper, we use the last approach and establish the relationship between homeomorphisms of finite metric distortion (FMD-homeomorphisms), finitely bi-Lipschitz, quasisymmetric and quasiconformal mappings on Riemannian manifolds. One of the main results shows that FMD-homeomorphisms are lower Q-homeomorphisms. As an application, there are obtained some sufficient conditions for boundary extensions of FMD-homeomorphisms. These conditions are illustrated by several examples of FMD-homeomorphisms. PubDate: 2022-01-15
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Abstract: The famous Jones–Ruscheweyh theorem states that n distinct points on the unit circle can be mapped to n arbitrary points on the unit circle by a Blaschke product of degree at most \(n-1\) . In this paper, we provide a new proof using real algebraic techniques. First, the interpolation conditions are rewritten into complex equations. These complex equations are transformed into a system of polynomial equations with real coefficients. This step leads to a “geometric representation” of Blaschke products. Then another set of transformations is applied to reveal some structure of the equations. Finally, the following two fundamental tools are used: a Positivstellensatz by Prestel and Delzell describing positive polynomials on compact semialgebraic sets using Archimedean module of length N. The other tool is a representation of positive polynomials in a specific form due to Berr and Wörmann. This, combined with a careful calculation of leading terms of occurring polynomials finishes the proof. PubDate: 2021-12-14
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Abstract: We develop a numerical solver for three-dimensional poroelastic wave propagation, based on a high-order discontinuous Galerkin (DG) method, with the Biot poroelastic wave equation formulated as a first order conservative velocity/strain hyperbolic system. To derive an upwind numerical flux, we find an exact solution to the Riemann problem; we also consider attenuation mechanisms both in Biot’s low- and high-frequency regimes. Using either a low-storage explicit or implicit–explicit (IMEX) Runge–Kutta scheme, according to the stiffness of the problem, we study the convergence properties of the proposed DG scheme and verify its numerical accuracy. In the Biot low frequency case, the wave can be highly dissipative for small permeabilities; here, numerical errors associated with the dissipation terms appear to dominate those arising from discretisation of the main hyperbolic system. We then implement the adjoint method for this formulation of Biot’s equation. In contrast with the usual second order formulation of the Biot equation, we are not dealing with a self-adjoint system but, with an appropriate inner product, the adjoint may be identified with a non-conservative velocity/stress formulation of the Biot equation. We derive dual fluxes for the adjoint and present a simple but illuminating example of the application of the adjoint method. PubDate: 2021-12-01
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Abstract: The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four. PubDate: 2021-12-01
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Abstract: For a quasiregular mapping \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\) , with \(n\ge 2\) , a Julia limiting direction \(\theta \in S^{n-1}\) arises from a sequence \((x_n)_{n=1}^{\infty }\) contained in the Julia set of f, with \( x_n \rightarrow \infty \) and \(x_n/ x_n \rightarrow \theta \) . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in \({\mathbb {R}}^3\) for a set \(E\subset S^2\) to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in \({\mathbb {R}}^3\) are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball \({\mathbb {B}}^3\) under an ambient quasiconformal mapping of \({\mathbb {R}}^3\) onto itself. PubDate: 2021-12-01
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Abstract: In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let \(E\not \subset {\hat{\mathbb {R}}}\) be an arc or a circle. If a map \(f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}\) preserves cross-ratios in E, then f is a Möbius transformation when \({\bar{E}}\not =E\) and f is a Möbius or conjugate Möbius transformation when \({\bar{E}}=E\) , where \({\bar{E}}=\{{\bar{z}} z\in E\}\) . (II) Let \(E\subset {\hat{\mathbb {R}}}\) be an arc satisfying the condition that the cardinal number \(\#(E\cap \{0,1,\infty \})<2\) . If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption \(\#(E\cap \{0,1,\infty \})<2\) is necessary. PubDate: 2021-12-01
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Abstract: We consider minimal graphs \(u = u(x,y) > 0\) over unbounded domains \(D \subset R^2\) bounded by a Jordan arc \(\gamma \) on which \(u = 0\) . We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector $$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$ then the rate of growth is at most \(r^{\pi /\lambda }\) . PubDate: 2021-12-01
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Abstract: For a transcendental entire function f, the property that there exists \(r>0\) such that \(m^n(r)\rightarrow \infty \) as \(n\rightarrow \infty \) , where \(m(r)=\min \{ f(z) : z =r\}\) , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r). PubDate: 2021-12-01
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Abstract: If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path \(\gamma \) from 0 to infinity such that \(f(z) - a\) tends to 0 as z tends to infinity along \(\gamma \) . The Denjoy–Carleman–Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2, mean type. A long-standing problem asks whether this conclusion holds for entire functions having n distinct asymptotic (entire) functions, each of growth at most order 1/2, minimal type. In this paper conditions on the function f and associated asymptotic paths are obtained that are sufficient to guarantee that f satisfies the conclusion of the Denjoy–Carleman–Ahlfors Theorem. In addition, for each positive integer n, an example is given of an entire function of order n having n distinct, prescribed asymptotic functions, each of order less than 1/2. PubDate: 2021-12-01
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Abstract: We find explicit forms for all the entire solutions of a certain type of non-linear binomial differential equation. This has connections to results of Hayman, Mues, Langley and Bergweiler. Observations about entire solutions of two similar types of binomial differential equations are discussed, and open questions about all three types of equations are posed. PubDate: 2021-12-01
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Abstract: We propose a system of non-linear equations equivalent to the fifth Painlevé equation, which enables us to examine the general singular solution given by Andreev and Kitaev along the positive real axis. We present a two-parameter family of asymptotic solutions corresponding to this general singular solution, and pose a conjecture. PubDate: 2021-12-01
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Abstract: In this note we present an alternative to Ford’s construction of the isometric circle of a Möbius map. This construction is based on the double coset decomposition of a group, together with the action of Möbius maps on spherical and hyperbolic spaces. PubDate: 2021-12-01
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Abstract: The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions. PubDate: 2021-12-01
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Abstract: Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in \(L^2(D)\) , and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale. PubDate: 2021-09-08 DOI: 10.1007/s40315-021-00402-8
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Abstract: Let \(\Omega \) be a smooth, bounded, convex domain in \({\mathbb {R}}^n\) and let \(\Lambda _k\) be a finite subset of \(\Omega \) . We find necessary geometric conditions for \(\Lambda _k\) to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k. PubDate: 2021-08-30 DOI: 10.1007/s40315-021-00410-8
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