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The topology of the set of non-escaping endpointsFeb 08 2018There are several classes of transcendental entire functions for which the Julia set consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Many authors have studied the topological properties of this set ... More On Zero-Sector Reducing OperatorsFeb 07 2018We prove a Jensen-disc type theorem for polynomials $p\in\mathbb{R}[z]$ having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators $T\colon\mathbb{R}[z]\to\mathbb{R}[z]$ ... More Shift invariant subspaces of slice $L^2$ functionsFeb 07 2018In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the ... More A remark on symbolic powersFeb 06 2018Let an ideal $I$ of holomorphic functions on an $n$-dimensional Stein manifold $X$ be generated by $r$ functions holomorphic on a neighbourhood of $\overline X$. By an easy application of Skoda's theorem on ideal generation, we show that symbolic powers ... More Conformal Parametrisation of Loxodromes by Triples of CirclesFeb 06 2018We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical ... More A Parameter Version of Forstnerič's Splitting LemmaFeb 05 2018We construct solution operators to the $\overline{\partial}$-equation that depend continuously on the domain. This is applied to derive a parameter version of Forstneri\v{c}'s splitting lemma: If both the maps and the domains they are defined on vary ... More Multi-parameter extensions of a theorem of PichoridesFeb 05 2018Extending work of Pichorides and Zygmund to the $d$-dimensional setting, we show that the supremum of $L^p$-norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces $H^p_A(\mathbb{T}^d)$ blows up like $(p-1)^{-d}$ ... More Square Sierpiński carpets and Lattès mapsJan 29 2018Jan 30 2018We prove that every quasisymmetric homeomorphism of a standard square Sierpi\'nski carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and completes earlier work by the authors. We also show that a similar conclusion holds for quasisymmetries ... More Sets of values of equivalent almost periodic functionsJan 25 2018In this paper we establish a new equivalence relation on the spaces of almost periodic functions which allows us to prove a result like Bohr's equivalence theorem extended to the case of all these functions. Linear space properties of $H^p$ spaces of Dirichlet seriesJan 19 2018We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More ... More Stably irrational hypersurfaces of small slopesJan 16 2018We show that a very general complex projective hypersurface of dimension N and degree at least $ \lceil \log_2N \rceil+2$ is not stably rational. The same statement holds over any uncountable field of characteristic p>>N. This significantly improves earlier ... More Lagrangian potential theory in symplectic geometryDec 10 2017The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${\bf C}^n$. However, ... More The Local Structure of Generalized Contact BundlesNov 22 2017Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting ... More Holomorphic Jacobi Manifolds and Complex Contact GroupoidsOct 09 2017This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. ... More On Newton Diagrams of Plurisubharmonic PolynomialsOct 03 2017Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do the same. We ... More When is a Convolutional Filter Easy To Learn?Sep 18 2017We analyze the convergence of (stochastic) gradient descent algorithm for learning a convolutional filter with Rectified Linear Unit (ReLU) activation function. Our analysis does not rely on any specific form of the input distribution and our proofs only ... More Online Convolutional Dictionary LearningAug 31 2017Convolutional sparse representations are a form of sparse representation with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in batch mode, requiring simultaneous access to all training ... More Changing Views on Curves and SurfacesJul 06 2017Nov 11 2017Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in ... More Localization of Bott-Chern classes and Hermitian residuesMay 26 2017We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies ... More Dynamics of transcendental Hénon mapsMay 25 2017The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, ... More On the structure of Hausdorff moment sequences of complex matricesJan 16 2017Jan 25 2017The paper treats several aspects of the truncated matricial $[\alpha,\beta]$-Hausdorff type moment problems. It is shown that each $[\alpha,\beta]$-Hausdorff moment sequence has a particular intrinsic structure. More precisely, each element of this sequence ... More A direct approach to quaternionic manifoldsDec 12 2016The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on $\mathbb{H}^n$, in a slice regular ... More Quaternionic toric manifoldsDec 12 2016In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted ... More On the Bargmann-Radon transform in the monogenic settingDec 07 2016In this paper we introduce and study a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane ... More Kähler geometry on Hurwitz spacesDec 07 2016We study the K\"ahler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently ... More A flat Higgs bundle structure on the complexified Kähler coneDec 07 2016We shall construct a natural Higgs bundle structure on the complexified K\"ahler cone of a compact K\"ahler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of ... More On generalized Lattès mapsDec 05 2016We introduce a class of rational functions $A:\,\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ which can be considered as a natural extension of the class of Latt\`es maps and establish basic properties of functions from this class. Entire holomorphic curves on a Fermat surface of low degreeDec 05 2016The purpose of the paper is to study some problems raised by Hayman and Gundersen about the existence of non-trivial entire and meromorphic solutions for the Fermat type functional equation $f^n+g^n+h^n=1$. Hayman showed that no non-trivial meromorphic ... More Conformal mapping of rectangular heptagons IIDec 04 2016A new analytical method for the conformal mapping of a rectangular heptagon with a straight angle at infinity to a half plane and back is proposed. The method is based on the observation that SC integral in this case is an abelian integral on a genus ... More Twisted Hodge filtration: Curvature of the determinantDec 02 2016Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula ... More On the Bohr inequalityDec 02 2016The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex ... Moremath.CV30A10, 30B10, 30B50, 30C35, 30C45, 30C80, 30H05, 32A05, 32A07,
32A10, 46L06, 47A56, 47A13 One sided conformal collars and the reflection principleDec 01 2016If a Jordan curve {\sigma} has a one-sided conformal collar with "good" properties, then, using the Reflection principle, we show that any other conformal collar of {\sigma} from the same side has the same "good" properties. A particular use of this fact ... More Geometric function theory for certain slice regular functionsNov 30 2016Dec 01 2016In this paper, we shall study the geometric function theory for slice regular functions of a quaternionic variable. Specially, we give some coefficient estimates for slice regular functions among which a version of the Bieberbach theorem and the Fekete-Szeg\"{o} ... More Geometric function theory for certain slice regular functionsNov 30 2016In this paper, we shall study the geometric function theory for slice regular functions of a quaternionic variable. Specially, we give some coefficient estimates for slice regular functions among which a version of the Bieberbach theorem and the Fekete-Szeg\"{o} ... More Bloch type spaces on the unit ball of a Hilbert spaceNov 30 2016In this article, we initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to ... More Hedgehogs in higher dimensions and their applicationsNov 29 2016In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate ... More GSV-index for holomorphic Pffaf SystemsNov 28 2016In this work we introduce the GSV-index for varieties invariant by a holomorphic Pffaf system on complex manifolds. We work with Pfaff systems not necessarily locally decomposable. We prove a non-negativity property for the index. As an application, we ... More GSV-index for holomorphic Pfaff SystemsNov 28 2016Nov 30 2016In this work we introduce the GSV-index for varieties invariant by a holomorphic Pfaff system on complex manifolds. We work with Pfaff systems not necessarily locally decomposable. We prove a non-negativity property for the index. As an application, we ... More Frederick William Gehring, Life and MathematicsNov 28 2016Frederick William Gehring was a hugely influential mathematician who spent most of his career at the University of Michigan. Gehring's major research contributions were to Geometric Function Theory, particularly in higher dimensions $\IR^n$, $n\geq 3$. ... More Curvature of higher direct imagesNov 28 2016Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We ... More Regularity of infinitesimal CR automorphismsNov 28 2016We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if $X$ is such an infinitesimal ... More Equivariant bundles and connectionsNov 27 2016Let $X$ be a connected complex manifold equipped with a holomorphic action of a complex Lie group $G$. We investigate conditions under which a principal bundle on $X$ admits a $G$--equivariance structure. Ideals of regular functions of a quaternionic variableNov 27 2016In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the non-commutative ... More Toeplitz and Asymptotic Toeplitz operators on $H^2(\mathbb{D}^n)$Nov 25 2016Dec 05 2016Let $M_{z_j}$ denote the multiplication operator on the Hardy space $H^2(\mathbb{D}^n)$ (over the unit polydisc $\mathbb{D}^n$) by the coordinate function $z_j$, $j =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then: ... More Multipliers between Toeplitz kernelsNov 25 2016Multipliers between kernels of Toeplitz operators are characterised in terms of test functions (so-called maximal vectors for the kernels); these maximal vectors may easily be parametrised in terms of inner and outer factorizations. Immediate applications ... More