Computable Carathéodory Theory

The conformal Riemann mapping of the unit disk onto a simply-connected domain W is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carathéodory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds.     

Carathéodory selections for multivalued mappings

We prove the existence of a Carathéodory selection for a set-valued mapping satisfying standard conditions. Also, an application to random fixed point for random operators is established.     

Controlled extensions of Carathéodory functions

In this paper we present two theorems on controlled extensions of Carathéodory functions. We obtain our results from corresponding ones for continuous functions by means of the Marczewski function.     

Polyhedra with the Integer Carathéodory Property

A polyhedron P has the Integer Carathéodory Property if the following holds. For any positive integer k and any integer vector w∈kP, there exist affinely independent integer vectors x₁,…,xt∈P and positive integers n₁,…,nt such that n₁+?+nt=k and w=n₁x₁+?+ntxt.In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a totally unimodular matrix, then P and projections of P have the Integer Carathéodory Property. For the matroid base polytope this answers a question by Cunningham from 1984.     

A random Euler scheme for Carathéodory differential equations

We study a random Euler scheme for the approximation of Carathéodory differential equations and give a precise error analysis. In particular, we show that under weak assumptions, this approximation scheme obtains the same rate of convergence as the classical Monte–Carlo method for integration problems.     

Invariant subspaces for operators whose spectra are Carathéodory regions

In this paper it is shown that if an operator T satisfies ‖p(T)‖?‖p_‖σ(T) for every polynomial p and the polynomially convex hull of σ(T) is a Carathéodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of σ(T) has at most finitely many prime ends corresponding to singular points on ∂D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace.     

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.     

On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

In the paper we study the existence and uniqueness of bounded solutions for differential equations of the form: x^′−Ax=f(t,x), x^″−Ax=f(t,x), where A∈L(Rm), is a Carathéodory function and the homogeneous equations x^′−Ax=0, x^″−Ax=0 have nontrivial solutions bounded on R. Using a perturbation of the equations, the Leray-Schauder Topological Degree and Fixed Point Theory, we overcome the difficulty that the linear problems are non-Fredholm in any reasonable Banach space.     

Carathéodory bounds for integer cones

We provide analogues of Carathéodory's theorem for integer cones and apply our bounds to integer programming and to the cutting stock problem. In particular, we provide an NP certificate for the latter, whose existence has not been known so far.     

On existence of integrable solutions of a functional integral equation under Carathéodory conditions

We study the solvability of a functional integral equation in the space of Lebesgue integrable functions on an unbounded interval. Using the conjunction of the technique of measures of weak noncompactness with the classical Schauder fixed point principle we show that the equation in question is solvable in the mentioned function space. Our existence result is obtained under the assumption that functions involved in the investigated functional integral equation satisfy Carathéodory conditions. Moreover, that result generalizes several ones obtained earlier in many research papers and monographs.     

Sandwich-type theorem for Carathéodory multifunctions

In this paper we present a sandwich-type theorem for Carathéodory multifunctions. We obtain our result by means of the Marczewski function.     

A higher order analogue of the Carathéodory-Julia theorem

A higher order analogue of the classical Carathéodory-Julia theorem on boundary derivatives is proved.     

Carathéodory's method for a class of dynamic games

The method of equivalent variational methods, originally due to Carathéodory for free problems in the calculus of variations is extended to investigate feedback Nash equilibria for a class of n-person differential games. Both the finite-horizon and infinite-horizon cases are considered. Examples are given to illustrate the presented results.     

Carathéodory's method for a class of second order differential equations on the half-line

The method of equivalent variational methods, originally due to Carathéodory for free problems in the calculus of variations is extended to investigate boundary value problems for a class of second order differential equations on the half-line. Some applications are presented to illustrate the potential of this method.     

The Hilbert–Carathéodory Form for Parametric Multiple Integral Problems in the Calculus of Variations

We define, and discuss some properties of, a form which bears a relation to the Carathéodory form for multiple integral problems in the calculus of variations similar to the relation that the Hilbert 1-form of Finsler geometry bears to the Poincaré–Cartan form for one-dimensional variational problems.     

A new linear spectral transformation associated with derivatives of Dirac linear functionals

In this contribution, we analyze the regularity conditions of a perturbation on a quasi-definite linear functional by the addition of Dirac delta functionals supported on N points of the unit circle or on its complement. We also deal with a new example of linear spectral transformation. We introduce a perturbation of a quasi-definite linear functional by the addition of the first derivative of the Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. Necessary and sufficient conditions for the quasi-definiteness of the new linear functional are obtained. Outer relative asymptotics for the new sequence of monic orthogonal polynomials in terms of the original ones are obtained. Finally, we prove that this linear spectral transform can be decomposed as an iteration of Christoffel and Geronimus linear transformations.     

Hermitian调和映照的Schwarz引理

由Jost和Yau引进的Hermitian调和映照是Riemannian流形上通常的调和映照在Hermitian流形上的一种自然的类比.本文证明了复分析中经典的Schwarz引理对一大类Hermitian调和映照仍然成立.作为推论,我们得到了半共形Hermitian调和映照的Liouville性质.     

A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields

In this note we prove a Fefferman-Poincaré type inequality in spaces with metric induced by Carnot-Carathéodory vector fields.

     

On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)