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We give an elementary proof of a theorem that characterizes quasisymmetric maps of the unit circle in terms of shear coordinates on the Farey tesselation. The proof only uses the normal family argument for quasisymmetric maps and some elementary hyperbolic geometry.
We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $φ\colon \partial \mathbb D \to \partial \mathbb Y$ as a Sobolev homeomorphism of the complex plane. Investigating such a Sobolev variant of the classical Jordan-Schönflies theorem is motivated by the well-posedness of the related pure displacement variational questions in the theory of Nonlinear Elasticity (NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for the boundary mapping $φ$ to admit a $W^{1,p}$-Sobolev homeomorphic extension is that it first admits a continuous $W^{1,p}$-Sobolev extension. For an arbitrary target domain $\mathbb Y$ this, however, is not sufficent.
The aim of this Note is to specify the links between the three kinds of Lisbon integrals, trace functions and trace forms with the corresponding D--modules.
Starting with a question of Yuan-Li-Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lithuanian Math. J., 58(2)(2018), 249-262] we have studied the uniqueness of a meromorphic function f and an L-function L sharing two finite sets. At the time of execution of our work, we have pointed out a serious lacuna in the proof of a recent result of a of Sahoo-Halder [ Some results on L-functions related to sharing two finite sets, Comput. Methods Funct. Theo., 19(2019), 601-612] which makes most of the part of the Sahoo-Halder's paper under question. In context of our choice of sets, we have rectified Sahoo-Halder's result in a convenient manner.
The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an L-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results in {W. J. Hao and J. F. Chen, Uniqueness of L-functions concerning certain differential polynomials, Discrete Dyn. Nat. Soc., 2018, DOI. 10.1155/2018/4673165}, {F. Liu, X. M. Li and H. X. Yi, Value distribution of L-functions concerning shared values and certain differential polynomials, Proc. Japan. Acad. Ser. A, 93 (2017), 41-46} and {P. Sahoo and S. Haldar, Uniqueness results related to L-functions and certain differential polynomials, Tbilisi Math. J., 11(4) (2018), 67-78}.
The aim of this paper is to study some features of slice semi-regular functions $\mathcal{RM}(Ω)$ on a circular domain $Ω$ contained in the skew-symmetric algebra of quaternions $\mathbb{H}$ via the analysis of a family of linear operators built from left and right $*$-multiplication on $\mathcal{RM}(Ω)$; this class of operators includes the family of Sylvester-type operators $\mathcal{S}_{f,g}$. Our strategy is to give a matrix interpretation of these operators as we show that $\mathcal{RM}(Ω)$ can be seen as a $4$-dimensional vector space on the field $\mathcal{RM}_{\mathbb{R}}(Ω)$. We then study the rank of $\mathcal{S}_{f,g}$ and describe its kernel and image when it is not invertible. By using these results, we are able to characterize when the functions $f$ and $g$ are either equivalent under $*$-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. We also provide a complete classification of idempotents and zero divisors on product domains of $\mathbb{H}$.
Bounded and compact generalized weighted composition operators acting from the weighted Bergman space $A^p_ω$, where $0<p<\infty$ and $ω$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling condition, to a Lebesgue space $L^q_ν$ are characterized. On the way to the proofs a new embedding theorem on weighted Bergman spaces $A^p_ω$ is established. This last-mentioned result generalizes the well-known characterization of the boundedness of the differentiation operator $D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_α$ to the Lebesgue space $L^q_μ$, induced by a positive Borel measure $μ$, to the setting of doubling weights.
In this work, we investigate the positivity of logarithmic and orbifold cotangent bundles along hyperplane arrangements in projective spaces. We show that a very interesting example given by Noguchi (as early as in 1986) can be pushed further to a very great extent. Key ingredients of our approach are the use of Fermat covers and the production of explicit global symmetric differentials. This allows us to obtain some new results in the vein of several classical results of the literature on hyperplane arrangements. These seem very natural using the modern point of view of augmented base loci, and working in Campana's orbifold category. As an application of our results, we derive two new orbifold hyperbolicity results, going beyond some classical results of value distribution theory.
We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can have rotational, translational, or mirror symmetry according to our chosen mapping functions. An appealing feature of these designs is how they reveal important properties of Euclidean and non-Euclidean geometries. We can also generate animations of our designs. Our goal is to create designs and animations having significant artistic content.
We introduce the notion of {\it quasi-inner} function and show that the product $u=ρ_\infty\prod ρ_v$ of $m+1$ ratios of local {$L$-}factors {$ρ_v(z)=γ_v(z)/γ_v(1-z)$} over a finite set $F$ of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line $\Re(z)= \frac 12$ in the following sense. The off diagonal part $u_{21}$ of the matrix of the multiplication by $u$ in the orthogonal decomposition of the Hilbert space $L^2$ of square integrable functions on the critical line into the Hardy space $H^2$ and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios $ρ_v$ is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio $ρ_\infty$ into a product of $m$ quasi-inner functions whose product with each $ρ_v$ retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part $u_{22}$ for the quasi-inner function $u=ρ_\infty$, and when $u(
Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) $f''+P(z)f=~0$, where $P(z)$ is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E.~Hille's 1969 book \emph{Lectures on Ordinary Differential Equations} are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille's book.
We introduce à la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called disc polynomials that form an orthogonal basis of the whole weighted Hilbert space.
We analyze the elliptic function ${\rm dn}_3$ introduced by Li-Chien Shen, contributing to the Ramanujan theory of elliptic functions in signature three. A famous hypergeometric identity emerges from our analysis.
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics' in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow. This leads to global convergence results, and applications to complex geometry. A purely mathematical introduction to the physical idea of T-duality is given, and a discussion of its relationship to generalized Ricci flow.
Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L_p(0, 2π)$, where $\dot{F}(e^{it})=\frac{d}{dt} F(e^{it})$ and $p\geq 1$. Recently, the author in \cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\leq p< 2$, then $f_{z}$ and $\overline{f_{\overline{z}}}\in \mathcal{B}^{p}(\mathbb{D}),$ the classical Bergman spaces of $\mathbb{D}$ \cite[Theorem 1.2]{Zhu}; $(2)$ if $f$ is a harmonic quasiregular mapping and $1\leq p\leq \infty$, then $f_{z},$ $\overline{f_{\overline{z}}}\in \mathcal{H}^{p}(\mathbb{D}),$ the classical Hardy spaces of $\mathbb{D}$ \cite[Theorem 1.3]{Zhu}. These are the main results in \cite{Zhu}. The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, \cite[Theorem 1.2]{Zhu} is true when $1\leq p< \infty$. Also, we show that \cite[Theorem 1.2]{Zhu} is not true when $p=\infty$. Second, we demonstrate that \cite[Theorem 1.3]{Zhu} still holds true when the assumption $f$ being a harmonic quasiregular mapping is replaced by the weaker one $f$ being a harmonic elliptic mapping.
Denote by $C^α(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are Hölder continuous with the exponent $α$, and denote by $C^{1, α}(\mathbb{D})$ the space which consists of differentiable functions $f$ such that their derivatives are in the space $C^α(\mathbb{D})$. Let $\mathcal{C}$ be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of $\|\mathcal{C}\|_{L^p\to L^q}$, where $3/2<p<2$ and $q=p/(p-1)$. As an application, we show that if $3/2<p<2$, then $u\in C^μ(\mathbb{D})$, where $μ=2/p-1$. We also show that if $2<p<\infty$, then $u\in C^{1, ν}(\mathbb{D})$, where $ν=1-2/p$. Finally, for the case $p=\infty$, we show that $u$ is not necessarily in $C^{1, 1}(\mathbb{D})$, but its gradient, i.e., $|\nabla u|$ is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by Chapter 4 of [Astala, Iwaniec, Martin: Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, Vol. 48, Princeton University Press, Princeton, NJ, 2009, p. xviii+677] and [Kalaj, Cauchy transform and Poisson's equation, Adv. Math. \textbf{231} (2012), 213-
Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $φ^* E$ is trivial for some surjective holomorphic map $φ$, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.
In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two and three dimensional setting.
The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati s method. The main concepts are the Picard Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $φ: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_φ$ on $H^2$ is defined by \[ (C_φ f)(z)=f(φ(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}). \] Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is \[ \mathcal{S}(\mathbb{D}) = \{ψ\in H^\infty(\mathbb{D}): \|ψ\|_{\infty} := \sup_{z \in \mathbb{D}} |ψ(z)| \leq 1\}. \] The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: Characterize $φ$, holomorphic self maps of $\mathbb{D}$, and inner functions $θ\in H^\infty(\mathbb{D})$ such that the Beurling type invariant subspace $θH^2$ is an invariant subspace for $C_φ$. We prove the following result: $C_φ (θH^2) \subseteq θH^2$ if and only if \[ \frac{θ\circ φ}θ \in \mathcal{S}(\mathbb{D}). \] This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of co
We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies - sequences of scalar Riemann-Hilbert problems - which allows us to express all higher order correction terms in closed form. In fact, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.
Let $f$ be a transcendental meromorphic function defined in the complex plane $\mathbb{C}$, and $φ(\not\equiv 0,\infty)$ be a small function of $f$. In this paper, We give a quantitative estimation of the characteristic function $T(r, f)$ in terms of $N\left(r,\frac{1}{M[f]-φ(z)}\right)$ as well as $\ol{N}\left(r,\frac{1}{M[f]-φ(z)}\right)$, where $M[f]$ is the differential monomial, generated by $f$.\par Moreover, we prove one normality criterion: Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k(\geq1)$, $q_{0}(\geq 3)$, $q_{i}(\geq0)$ $(i=1,2,\ldots,k-1)$, $q_{k}(\geq1)$ be positive integers. If for each $f\in \mathscr{F}$, $f$ has only zeros of multiplicity at least $k$, and $f^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}\not=1$, then $\mathscr{F}$ is normal on domain $D$.
Given a pseudoconvex domain U with C^1-boundary in P^n, n>2, we show that if H^{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the \dbar-equation in X=P^n\ U, for data smooth (0,1) forms on X. We also discuss Levi-flat domains in surfaces. If Z is a real algebraic hypersurface in P^2, (resp a real-analytic hypersurface with a point of strict pseudoconvexity), then there is a strictly psh function in a neighborhood of Z.
In this paper, we study a special one dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal-Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb's uncertainty principle. We provide also the reproducing kernel associated to the Gabor space considered in this setting.
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