Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003     

Julia lines of random Dirichlet series

For the random entire functions defined by some random Dirichlet series almost surely every horizontal line is a Julia line without exceptional values. These results contain and generalize partially a theorem of P.L. Davies. The proofs are different.     

Zeros of the partial sums of $\cos(z)$ and $\sin(z)$ II

Summary.   We study here in detail the location of the real and complex zeros of the partial sums of and , which extends results of Szeg? (1924) and Kappert (1996). Received November 9, 2000 / Published online August 17, 2001     

Covering relations, cone conditions and the stable manifold theorem

We show how to effectively link covering relations with cone conditions. We give a new, ‘geometric,’ proof of the stable manifold theorem for hyperbolic fixed point of a map.     

Uniform definability of the Weierstrass ?\wp functions and generalized tori of dimension one

No Abstract.  .     

A Frobenius Theorem for Locally Convex Global Analysis

 We prove a Frobenius theorem for Banach space complemented subbundles of the tangent bundle of a manifold modelled on locally convex spaces. The proof is based on an implicit function theorem for maps from locally convex spaces to Banach spaces proved in a recent paper of the author. (Received 15 March 1999; in revised form 2 June 1999)     

Bloch’s theorem for mappings of bounded and finite distortion

We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form. Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.     

Principal points of a multivariate mixture distribution

A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.     

Conformal mapping by the method of alternating projections

Summary The functional analytic principle of alternating projections is used to construct an iterative method for numerical conformal mapping of the unit disc onto regions with smooth boundaries. The result is a simple method which requires in each iterative step only two complex Fourier transforms. Local convergence can be proved using a theorem of Ostrowski. Convergence is linear. The asymptotic convergence factor is equal to the spectral radius of a certain operator. A version with overrelaxation as well as a discretized version are discussed along the same lines. For regions which are close to the unit disc convergence is fast. For some familiar regions the convergence factors can be calculated explicitly. Finally, the method is compared with Theodorsen's.Dedicated to the memory of Peter Henrici     

Problème du bord dans les variétés $q$-convexes et phénomène de Hartogs-Bochner

Résumé.   Soient E un fibré vectoriel holomorphe au dessus d'une variétéq-complète X et M une sous-variété réelle de dimension 2p-1 () de E. Nous montrons que le problème du bord pour M est résoluble si et seulement si M est maximalement complexe. Dans le cas où nous retrouvons le théorème de Harvey et Lawson [17]. Comme conséquence nous obtenons une généralisation du théorème de Hartogs-Bochner pour les applications CR-méromorphesà valeurs dans une variétéq-convexe et une généralisation au cas CR-méromorphe du théorème d'extension de Chazal [5].

---

Let E be a holomorphic vector bundle over a -complete manifold X and M be a real submanifold of dimension 2p-1 () of E. We prove that M has a solution to the boundary problem in X if and only if M is maximally complex. In the case , this is a result of Harvey and Lawson [17]. As a consequence, we obtain a generalization of the Hartogs-Bochner theorem for CR-meromorphic maps taking their values in -convex manifolds and a generalization to the CR-meromorphic case of a theorem of Chazal [5].

Received: 27 April 2000 / Published online: 23 July 2001     

The combinatorial encoding of disjoint convex sets in the plane

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2n-2 combinatorially distinct ways. The encoding generalizes the authors’ encoding of point configurations by “allowable sequences” of permutations. Since it applies as well to a collection of compact connected sets with a specified pseudoline arrangement of separators and double tangents, the result extends the Edelsbrunner-Sharir theorem to the case of geometric permutations induced by pseudoline transversals compatible with . Supported in part by NSA grant MDA904-03-I-0087 and PSC-CUNY grant 65440-0034. Supported in part by NSF grant CCR-9732101.     

Radially symmetric systems with a singularity and asymptotically linear growth

We prove the existence of infinitely many periodic solutions for radially symmetric systems with a singularity of repulsive type. The nonlinearity is assumed to have a linear growth at infinity, being controlled by two constants which have a precise interpretation in terms of the Dancer-Fu?ik spectrum. Our result generalizes an existence theorem by Del Pino et al. (1992) [4], obtained in the case of a scalar second order differential equation.     

Univalent holomorphic mappings on a complex manifold with a C1 exhaustion function

Let M be a complex manifold with a exhaustion function ρ. Assume that ρ attains its minimum at a single point o and dρ≠ 0 on M\{o}. In this case, we will give a necessary condition and a sufficient condition for locally biholomorphic mappings on M to be biholomorphic and to have Φ-like images. Received: 15 May 1998 / Revised version: 8 March 1999     

Finite gap Jacobi matrices: An announcement

We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szeg?’s theorem, Jost solutions, and Szeg? asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007.     

Minimizing measures of risk by saddle point conditions

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

A Local Expansion Theorem for the Löwner Equation

If W(z) is a power series with complex coefficients which represents an injective function bounded by one in the unit disk and which vanishes at the origin then a Grunsky space $\mathcal{G}(W)$ exists. It is contained contractively in the Dirichlet space for the unit disk. In this paper an admissible family of weighted Dirichlet spaces is used as in the proof of Bieberbach conjecture to construct a Local Grunsky space. An expansion theorem is presented for such a Local Grunsky space. The proof relies on the reproducing kernel function for coefficients of powers of z and Löwner differential equation.