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A Generalization Method of Partitioned Activation Function for Complex NumberFeb 08 2018A method to convert real number partitioned activation function into complex number one is provided. The method has 4em variations; 1 has potential to get holomorphic activation, 2 has potential to conserve complex angle, and the last 1 guarantees interaction ... More

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

Nevanlinna theory and value distribution in the unicritical polynomials familyFeb 08 2018In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicritical polynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed measure) $T_f$ ... 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

Complete complex hypersurfaces in the ball come in nonsingular foliationsFeb 06 2018In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. ... More

The Loewner energy is the $L^2$-norm of the pre-Schwarzian of the uniformizing mapFeb 06 2018Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function. In the present paper, we show that the Dirichlet energy of this driving function (also known as the ... More

Extensions to the boundary of Riemann maps on varying domains in the complex planeFeb 06 2018We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.

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

On the Lichnerowicz conjecture for CR manifolds with mixed signatureJan 31 2018We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold ... More

Extremal and stationary discs for the Kobayashi $k$-metricJan 30 2018We generalize Lempert's and Poletsky's works on the description of extremal discs for the Kobayashi metric to a higher order setting.

Characterizations for inner functions in certain function spacesJan 30 2018For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\Theta$ for which $$\|\Theta'\|_{A^{p,q}_\omega}^q=\int_0^1 \left(\int_0^{2\pi} |\Theta'(re^{i\theta})|^p d\theta\right)^{q/p} ... More

Positive-definiteness and integral representations for special functionsJan 29 2018We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on $\mathbb{R}$. The classical theorems of Bochner on positive definite ... More

Fekete-Szego Inequality for Analytic and Bi-univalent Functions Subordinate to Chebyshev PolynomialsJan 29 2018In the present paper, a new subclass of analytic and bi-univalent functions by means of Chebyshev polynomials is introduced. Certain coefficient bounds for functions belong to this subclass are obtained. Furthermore, the Fekete-Szego problem in this subclass ... 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.

Computing polynomial conformal models for low-degree Blaschke productsJan 23 2018Jan 29 2018For any finite Blaschke product $B$, there is an injective analytic map $\varphi:\mathbb{D}\to\mathbb{C}$ and a polynomial $p$ of the same degree as $B$ such that $B=p\circ\varphi$ on $\mathbb{D}$. Several proofs of this result have been given over the ... More

Almost Periodic Functions in terms of Bohr's Equivalence RelationJan 22 2018In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner's result that characterizes these spaces of functions. In fact, with respect to the topology of ... More

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

Ramanujan's Master Theorem and two formulas for zero-order Hankel transformJan 19 2018Using Ramanujan's Master Theorem, two formulas are derived which define the Hankel transforms of order zero with even functions by inverse Mellin transforms, provided these functions and their derivatives obey special conditions. Their validity is illustrated ... More

On connected preimages of simply-connected domains under entire functionsJan 19 2018Let $f$ be a transcendental entire function, and let $U,V\subset\mathbb{C}$ be disjoint simply-connected domains. Must one of $f^{-1}(U)$ and $f^{-1}(V)$ be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove ... 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

Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functionsJan 06 2018We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ has a factor of genus zero or one. In particular, we show that if $A$ is not a "generalized ... More

The Leray transform on model hypersurfaces in $\mathbb{C}\mathbb{P}^2$Dec 25 2017We investigate the Leray transform on a family $\mathcal{S}_{\beta}$ of unbounded hypersurfaces in two complex dimensions. The $\mathcal{S}_{\beta}$ can be used to approximate any smooth, strongly $\mathbb{C}$-convex hypersurface to two orders of tangency. ... 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

Gradient Descent Learns One-hidden-layer CNN: Don't be Afraid of Spurious Local MinimaDec 03 2017We consider the problem of learning a one-hidden-layer neural network with non-overlapping convolutional layer and ReLU activation function, i.e., $f(\mathbf{Z}; \mathbf{w}, \mathbf{a}) = \sum_j a_j\sigma(\mathbf{w}^\top\mathbf{Z}_j)$, in which both the ... More

Tighter Lifting-Free Convex Relaxations for Quadratic Matching ProblemsNov 29 2017In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they lift the original ... 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

On strongly convex projectively flat and dually flat complex Finsler metricsOct 03 2017In this paper, we prove that a strongly convex complex Finsler metric $F$ on a domain $D\subset\mathbb{C}^n$ is projectively flat (resp. dually flat) if and only if $F$ comes from a strongly convex complex Minkowski metric.

Approximation by random complex polynomials and rational functionsSep 24 2017We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

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

A properly embedded holomorphic disc in the ball with finite area and dense boundary curveSep 04 2017In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function ... 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

Model based learning for accelerated, limited-view 3D photoacoustic tomographyAug 31 2017Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed-up. In this work we present a deep neural network that is specifically designed to provide ... More

Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler GeometryAug 12 2017Jan 18 2018It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out ... More

On the truncated matricial Stieltjes moment problem $\mathsf{M}[[α,\infty);(s_j)_{j=0}^m,\leq]$Jul 19 2017This paper gives via Stieltjes transform a complete description of the solution set of a matricial truncated Stieltjes-type power moment problem in the non-degenerate and degenerate cases. The approach is based on the Schur type algorithm which was worked ... 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

Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical SystemsJun 12 2017Nov 01 2017Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation ... 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

Jet determination of smooth CR automorphisms and generalized stationary discsMay 03 2017We prove finite jet determination for (finitely) smooth CR diffeomorphisms of (finitely) smooth Levi degenerate hypersurfaces in $\mathbb{C}^{n+1}$ by constructing generalized stationary discs glued to such hypersurfaces.

Invariant holomorphic discs in some non-convex domainsApr 10 2017We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.

Local approximation of non-holomorphic discs in almost complex manifoldsApr 06 2017Aug 27 2017We provide a local approximation result of non-holomorphic discs with small d-bar by pseudoholomorphic ones. As an application, we provide a certain gluing construction.

Darboux charts around holomorphic Legendrian curves and applicationsFeb 02 2017Jun 16 2017In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann ... More

The star function for meromorphic functions of several complex variablesJan 31 2017We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f.We then characterize meromorphic ... More

Distributed methods for synchronization of orthogonal matrices over graphsJan 25 2017Apr 07 2017This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as a least-squares ... 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

Birational geometry of foliations associated to simple derivationsJan 03 2017Jun 30 2017We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in ... 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

Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degreeDec 08 2016Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition of the postcritical volume. This allows us to generalize to this setting the theory of stability and bifurcation ... More

Fundamental theorem of Algebra - A Nevanlinna theoretic proofDec 08 2016The aim of this short note is to produce a new proof of Fundamental Theorem of Algebra using Nevanlinna Theory.

Cohomological Laplace transform on non-convex cones and Hardy spaces of $\bar{\partial}$-cohomology on non-convex tube domainsDec 08 2016We consider a class of non-convex cones $V$ in $\mathbb{R}^n$ which can be presented as (not unique) union of convex cones of some codimension $q$ which we call the index of non-convexity. This class contains non-convex symmetric homogeneous cones studied ... 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

Criteria of univalence and fully α--accessibility for p--harmonic and p--analytic functionsDec 07 2016In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields new a criterion for a polyharmonic functions to be fully $\alpha$--accessible. Several examples are presented to illustrate the use of these ... 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

Finiteness Theorems for Products and Symmetric Products of Hyperbolic Riemann SurfacesDec 07 2016We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y = \Omega/\Gamma$ is a complex manifold, where $\Omega$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the space of dominant ... More

Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensemblesDec 06 2016We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of ... More

Mostow's Fibration for canonical embeddings of compact homogeneous CR manifoldsDec 06 2016We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy-Riemann ... More

Applications of singular connections in symplectic and almost complex geometryDec 05 2016In this paper, we give two direct applications of the theory of singular connections developped by Harvey-Lawson [10]. The first one is a version of Lelong-Poincar\'e formula for vector bundle over an almost complex manifold. The second is a convergence ... 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

Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on toriDec 05 2016In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that hyperbolic maps ... More

Algebraic isomonodromic deformations of the five punctured sphere arising from quintic plane curvesDec 04 2016In this paper, we classify the algebraic isomonodromic deformations that can be obtained through restriction to generic lines of logarithmic flat connections on the complex projective plane $\mathbb{P}^2_\mathbb{C}$ whose singular locus is a quintic curve. ... 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

Approximation properties of univalent mappings on the unit ball in $\mathbb{C}^n$Dec 04 2016Let $n\geq 2$. In this paper, we obtain approximation properties of various families of normalized univalent mappings $f$ on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ ... More

Linear measure and $K$-quasiconformal harmonic mappingsDec 03 2016In this paper, we investigate the relationships between linear measure and harmonic mappings.

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

Orbits of real forms, Matsuki duality and CR-cohomologyDec 02 2016We discuss the relationship between some groups of tangential CR cohomology of some compact homogeneous CR manifolds and the corresponding Dolbeault cohomology groups of their canonical complex embeddings.

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 ... More

On Lebesgue Constants for Interpolation Points on a Quasiconformal ArcDec 02 2016Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (see IMA Journal of Numerical Analysis 30 (2010) 462--486) on the behavior of the Lebesgue constants for interpolation points on a compact set ... More

On the Christoffel function for the generalized Jacobi measures on a quasidiskDec 01 2016We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted.

Topology and complex structures of leaves of foliations by Riemann surfacesDec 01 2016We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.

Monge's Optimal Transport Distance with Applications for Nearest Neighbour Image ClassificationDec 01 2016This paper focuses on a similarity measure, known as the Wasserstein distance, with which to compare images. The Wasserstein distance results from a partial differential equation (PDE) formulation of Monge's optimal transport problem. We present an efficient ... More

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

A disproof of a conjecture of Al Baernstein IINov 30 2016We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. The conjecture is not true generically.

The sharp upper bounds for the first positive eigenvalue of Kohn-Laplacian on compact strictly pseudoconvex hypersurfacesNov 30 2016We give sharp and explicit upper bounds for the first positive eigenvalue $\lambda_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that ... 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

Level functions of quadratic differentials, signed measures, and the Strebel propertyNov 29 2016In this paper, motivated by the classical notion of a Strebel qua- dratic differential on a compact Riemann surface without boundary, we in- troduce several classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which ... 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 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 functions $z_j$, $j =1, \ldots, n$, and let $T$ be a bounded linear operator on $H^2(\mathbb{D}^n)$. Then: ... 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

Degeneration of endomorphisms of the complex projective space in the hybrid spaceNov 25 2016Consider a meromorphic familly of endomorphims of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated ... 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

On boundary points at which the squeezing function tends to oneNov 25 2016J. E. Fornaess has posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, if the asymptotic limit of the squeezing function is 1. The purpose of this paper is to give an affirmative answer when ... More