2 edition of **Generalizations of a theorem of Carathéodory.** found in the catalog.

Generalizations of a theorem of Carathéodory.

John R. Reay

- 133 Want to read
- 4 Currently reading

Published
**1965**
by American Mathematical Society in Providence, R.I
.

Written in English

**Edition Notes**

Series | Memoirs of the American Mathematical Society -- no.54 |

The Physical Object | |
---|---|

Pagination | 50p. |

Number of Pages | 50 |

ID Numbers | |

Open Library | OL20175864M |

3. GENERALIZATIONS There are many applications of Polya’s and de Bruijn’s theorems, for instance see [2,3,5,7,8, 11, 12, Here we shall apply these theorems to number theory. Our following Theorem l(a) is not much a generalization; one may consider it as another proof of Wilson’s theorem. Generalization of a theorem of Carathéodory. By S. Ciccariello A and A. Cervellino B. Abstract. Carathéodory showed that n complex numbers c1,,cn can uniquely be written in the form cp = ∑ m j=1 ρjǫj p with p = 1,,n, where the ǫjs are different unimodular complex numbers, the ρjs are strictly positive numbers and integer m never.

At the heart of Caratheodory's idea is the observation that given an equilibrium state A, all other states fall into 3 categories: (a) states that are mutually accessible, (b) states that are accessible but from which state A is not accessible, (c) states that are not . Abstract. In this chapter generalizations of the Douady-Oesterlé theorem (Theorem , Chap. 5) are obtained for maps and vector fields on Riemannian proof of the generalized Douady-Oesterlé theorem on manifolds is given in Sect. In Sect. it is shown that the Lyapunov dimension is an upper bound for the Hausdorff dimension.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Carathéodory showed that n complex numbers c1,,cn can uniquely be written in the form cp = ∑ m j=1 ρjǫj p with p = 1,,n, where the ǫjs are different unimodular complex numbers, the ρjs are strictly positive numbers and integer m never exceeds n. We give the conditions to be obeyed for the . this reformulation the following theorem can be seen as a generalization of Carathéodory's theorem. THEOREM 2. Let be a probability measure on R m such that Sq(x) m. Let V be a subset of R m such that supp(g) = cl(V). Then for i = there exists a probability measure v on R m with 7m (x) dp(x) ãT2(x) du(x).

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Carathéodory's Generalizations of a theorem of Carathéodory. book is a theorem in convex states that if a point x of R d lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset P ′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P ′.

Equivalently, x lies in an r-simplex with vertices in P, where ≤. Generalizations of a theorem of Carathéodory. Providence, American Mathematical Society, (OCoLC) Material Type: Internet resource: Document Type: Book, Internet Resource: All Authors / Contributors: John R Reay.

Introduction 1. Standard generalizations of Carathéodory's theorem 2. Positive bases of linear spaces 3. Carathéodory's theorem with connectedness conditions 4. Carathéodory's theorem with at least two points in $\operatorname {con} X$ 5.

Carathéodory's theorem with symmetry conditions. Series Title. However, further choices of c 0 are still possible and the following considerations will show that some of these lead to a non-trivial generalization of Carathéodory's theorem.

First, it is convenient to determine all the distinct Toeplitz matrices that can be obtained from (2) by choosing c 0 equal to the opposite of each eigenvalue of by: 4. In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory.

Carathéodory's theorem (conformal mapping), about the extension of conformal mappings to the boundary Carathéodory's theorem (convex hull), about the convex hulls of sets in Euclidean space Carathéodory's existence theorem, about the existence of solutions to.

One of the basic results ([]) in convexity, with many applications in different principle it states that every point in the convex hull of a set S ⊂ R n can be represented as a convex combination of a finite number (n + 1) of points in the set for example [], [], [], [], [], [].Generalizations of the theorem can be found in [] and [].

Constantin Carathéodory was born in in Berlin to Greek parents and grew up in father Stephanos, a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Carathéodory family, originally from Bosnochori or Vyssa, was well established and respected in.

We mention 'hat using Theorem (or directly ) we can get further `mclniphed' versions of several generalization of Helly's theorem. (For these generalizations see [4] for instance.) We mention further that Theorem can be regarded as a further (though sm a)1 as it may be) step towards characterizing the possible types of intersection.

Caratheodory’s theorem can be generalized so as to avoid the requirement that the sign of´ all the scattering charges be positive. This paper shows how to obtain this generalization according to the following plan.

In the ﬁrst of the three subsections of section 2,wereview the proof of Caratheodory’s theorem given by Grenander and Szeg. Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange. Generalization of a theorem of Carathéodory Article (PDF Available) in Journal of Physics A General Physics 39(48) November with 77 Reads How we measure 'reads'. Reay JR () Generalizations of a theorem of Carathéodory. Memoirs Amer Math Soc Google Scholar.

Rockafellar RT () Convex analysis. Search book. Search within book. Type for suggestions. Table of contents Previous. Page 4. Navigate to page number. of Next. About this reference work.

The Colorful Carathéodory theorem by Bárány () states that given d + 1 sets of points in R d, the convex hull of each containing the origin, there exists a simplex (called a ‘rainbow simplex’) with at most one point from each point set, which also contains the lently, either there is a hyperplane separating one of these d + 1 sets of points.

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence 's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general.

Colette De Coster, Patrick Habets, in Mathematics in Science and Engineering, A Generalization: the Carathéodory Case. Observe that if φ is continuous, any L p-Carathéodory function f that satisfies the Nagumo condition () is L ∞ the next result, we extend the Nagumo condition so as to deal with L p-Carathéodory functions which are not L ∞.

The classical Julia-Wolff-Caratheodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a. In particular, in the case of functions holomorphic in the unit disk in C, this generalization of Carathéodory’s inequality implies the classical inequalities of Carahtéodory and Landau.

As an application, new results on multidimensional analogues of Bohr’s theorem on. There have been many generalizations of Theorem mostly weakening the prerequisites for the existence of a colorful simplex containing the origin; see Section as well as [22,5, 11].Kalai.

A generalization of Carathéodory's existence theorem for ordinary differential equations. By Jan Persson. Cite. BibTex; Full citation; Publisher: Elsevier BV. Year: DOI identifier: /x(75) OAI identifier: Provided by: MUCC. Generalization of a theorem of Carathéodory This generalization is relevant to neutron scattering.

Its proof is made possible by a lemma stating the necessary and sufficient conditions to be obeyed by the coefficients of a polynomial equation for all the roots to lie on the unit circle.

This lemma is an interesting side result of our analysis. Abstract. This note is a short introduction to the Julia-Wolff-Carathéodory theorem, and its generalizations in several complex variables, up to very recent results for infinitesimal generators of .A set in dimension 2 satisfying the condition of Theorem but not the one of Theorem Up to an additional argument to handle degenerate configurations, Theorem .L.

Pogliani, M.N. Berberan-Santos / Constantin Carathéodory but which concerns thermal equilibrium, that is, “if t1, t2 and t3 are equilibrium states of three systems such as t1 is in thermal equilibrium with t2, and t2 is in thermal equilibrium with t3,thent3 is also in thermal equilibrium with t1”.This law strongly resembles the.