An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n of P ²=P _k ² ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position. This paper was written while the author was member of C.N.R., Sez. 3 of G.N.S.A.G.A. and was supported by M.P.I. funds     

Carathéodory Convergence and Harmonic Measure

We give several new characterizations of Carathéodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

     

On 4-flattening theorems and the curves of Carathéodory, Barner and Segre

We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below) Barner, Segre and Carathéodory, we prove that Barner $\subset$ (Segre $\cap$ Carathéodory). We also prove that (Segre)\ (Segre $\cap$ Carathéodory) and (Carathéodory)\(Segre $\cap$ Carathéodory) are open sets in the space of closed smooth curves with the C-topology. Finally, we define a class of closed curves containing the class of Segre curves and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any curve of our class has at least 4 flattenings.     

Non-Commutative Carathéodory Interpolation

We prove a Carathéodory–Fejér type interpolation theorem for certain matrix convex sets in mathbbC^d{mathbb{C}^d} using the Blecher–Ruan–Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ for the d-dimensional non-commutative polydisc.     

Effective formulas for the Carathéodory distance

For a class of Reinhardt domains we give formulas for the Carathéodory distance. As an application we discuss the product property of the Carathéodory distance when one factor domain is a Reinhardt domain of special type.     

Nonnegative Polynomials and Their Carathéodory Number

In 1888 Hilbert showed that every nonnegative homogeneous polynomial with real coefficients of degree $2d$ in $n$ variables is a sum of squares if and only if $d=1$ (quadratic forms), $n=2$ (binary forms) or $(n,d)=(3,2)$ (ternary quartics). In these cases, it is interesting to compute canonical expressions for these decompositions. Starting from Carathéodory’s Theorem, we compute the Carathéodory number of Hilbert cones of quadratic forms and binary forms.     

A topological principle for retarded functional differential equations of Carathéodory type

The paper presents a Razumikhin-type version of Wa?ewski's principle for RFDEs of Carathéodory type, thus generalizing the author's previous results (J. Differential Equations36, 117–138 (1980)). A few facts from the theory of RFDEs are reviewed, and the concept of a regular polyfacial set with respect to an RFDE of Carathéodory type is introduced and the results are state and proved. Most of the examples given in the above work can easily be adapted to the more general setting of Carathéodory systems. An indication how to do it is given in only one case. Throughout the paper, standard notation is used.     

Borel–Carathéodory Type Theorem for Monogenic Functions

In this paper we give a generalization of the classical Borel–Carathéodory theorem in complex analysis to higher dimensions in the framework of Quaternionic Analysis. Submitted: November 14, 2007., Revised: February 25, 2008.     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

The Carathéodory topology for multiply connected domains I

We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.     

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Nonlinear evolutions with Carathéodory forcing

Let A be an m-accretive operator in a real Banach space X and f : J x X X a function of Carathéodory type, where $ J = [0, a] \subset \mathbb{R} $. This paper investigates the existence of mild solutions of the evolution system satisfying additional time-dependent constraints $ u(t) \in K(t) $ on J for a given tube K(·). Main emphasis is on existence results that are valid under minimal assumptions on f, K and X.AMS Subject Classification: Primary: 47J35, 35K90, Secondary: 34G25, 47H20.     

Il corpo convesso di Carathéodory

Sunto Si da una topologizzazione di S_∞ che permette di parlare di corpo convesso diCarathéodory in modo sintetico e tale che l'interno corrisponda alle funzioni positive. A Mauro Picone nel suo 70^mo compleanno.     

A remark on Carathéodory balls

Member of URA CNRS D 1322 Groupes de Lie et Géométrie     

Pseudo-Taylor expansions and the Carathéodory–Fejér problem

We give a new solvability criterion for the boundary Carathéodory–Fejér problem: given a point $x\in \mathbb{R}$ and, a finite set of target values $a^0,a^1,\dots ,a^n\in \mathbb{C}$, to construct a function f in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z\to x$ nontangentially in the upper half-plane is $a^k$ for $k=0,1,\dots ,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.     

Notes About the Carathéodory Number

In this paper we give sufficient conditions for a compactum in ? ⁿ to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.     

Carathéodory’s Theorem and moduli of local connectivity

We give a quantitative proof of the Carathéodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus.