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Carathéodory's work on outer measures found many applications in measure-theoretic set theory ( outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem ), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension.
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Carathéodory's and work
The history of statements of the law for closed systems has two main periods, before and after the work of Bryan ( 1907 ), of Carathéodory ( 1909 ), and the approval of Carathéodory's work given by Born ( 1921 ).
Carathéodory's paper asserts that its statement of the first law corresponds exactly to Joule's experimental arrangement, regarded as an instance of adiabatic work.
A respected text disregards the Carathéodory's exclusion of mention of heat from the statement of the first law for closed systems, and admits heat calorimetrically defined along with work and internal energy.
Optics: Carathéodory's work in optics is closely related to his method in the calculus of variations.
Šilahvý ( 1997 ) notes that Carathéodory's approach does not work for the description of irreversible processes that involve both heat conduction and conversion of kinetic energy into internal energy by viscosity ( which is another prime example of irreversibility ), because " the mechanical power and the rate of heating are not expressible as differential forms in the ' external parameters '".
Carathéodory's and on
This fact is part of the basic idea of heat, and is related also to the so-called zeroth law, though the textbooks ' statements of the zeroth law are usually reticent about that, because they have been influenced by Carathéodory's basing his axiomatics on the law of conservation of energy and trying to make heat seem a theoretically derivative concept instead of an axiomatically accepted one.
Carathéodory's and outer
* Carathéodory's theorem ( measure theory ), about outer measures in measure theory
Carathéodory's and many
Academic contacts in Germany: Carathéodory's contacts in Germany were many and included such famous names as: Minkowski, Hilbert, Klein, Einstein, Schwarz, Fejér.
Carathéodory's and set
Any set which has a well-defined Lebesgue measure is said to be " measurable ", but the construction of the Lebesgue measure ( for instance using Carathéodory's extension theorem ) does not make it obvious whether there exist non-measurable sets.
* Every Borel set B ⊆ R < sup > n </ sup > is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ R < sup > n </ sup >,
In convex geometry Carathéodory's theorem states that if a point x of R < sup > d </ sup > lies in the convex hull of a set P, there is a subset P ′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P ′.
Carathéodory's theorem states that any point in the convex hull of some set of points is also within the convex hull of a subset of at most d + 1 of the points ; that is, that the given point is part of a Radon partition in which it is a singleton.
Carathéodory's and theory
According to Münster ( 1970 ), " A somewhat unsatisfactory aspect of Carathéodory's theory is that a consequence of the Second Law must be considered at this point the statement of the first law, i. e. that it is not always possible to reach any state 2 from any other state 1 by means of an adiabatic process.
Carathéodory's and are
In fact, according to Carathéodory's theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N + 1 points are sufficient in the definition above.
Carathéodory's celebrated presentation of equilibrium thermodynamics refers to closed systems, which are allowed to contain several phases connected by internal walls of various kinds of impermeability and permeability, explicitly including walls that are permeable only to heat.
Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.
Carathéodory's and for
An illustration of Carathéodory's theorem ( convex hull ) for a square in R < sup > 2 </ sup >.
An illustration of Carathéodory's theorem for a square in R < sup > 2 </ sup >
: See also Carathéodory's theorem for other meanings
In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets P in R < sup > d </ sup >.
Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ < sub > 1 </ sub > and Γ < sub > 2 </ sub >, a conformal map
Carathéodory's and proof
One proof of Carathéodory's theorem uses a technique of examining solutions to systems of linear equations, similar to the proof of Radon's theorem, to eliminate one point at a time until at most d + 1 remain.
Carathéodory's and extension
* Carathéodory's extension theorem
* Carathéodory's theorem ( conformal mapping ), about the extension of conformal mappings to the boundary
* Carathéodory's extension theorem, about the extension of a measure
Carathéodory's and theorem
* Carathéodory's theorem
* Carathéodory's theorem ( convex hull )
* Carathéodory's theorem ( disambiguation )
* Carathéodory's theorem ( convex hull )
** Carathéodory's theorem ( conformal mapping )
* Carathéodory's theorem ( convex hull )
In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory:
* Carathéodory's theorem ( convex hull ), about the convex hulls of sets in Euclidean space
* Carathéodory's existence theorem, about the existence of solutions to ordinary differential equations
# REDIRECT Carathéodory's theorem
# redirect Carathéodory's theorem
Carathéodory's and was
Carathéodory's version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred.
Carathéodory's " first axiomatically rigid foundation of thermodynamics " was acclaimed by Max Planck and Max Born.
Carathéodory's and by
In mathematical complex analysis, Carathéodory's theorem, proved by, states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map
Carathéodory's and called
Thus, in Bailyn's survey of thermodynamics, Carathéodory's approach is called " mechanical ", as distinct from " thermodynamic ".
Carathéodory's and .
A more general version of Carathéodory's theorem is the following.
Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis.
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