hmmmmmmmmmm. you are an interesting person.These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase 'complete Archimedean field' instead of 'complete ordered field'. Since it can be proved that any uniformly complete Archimedean field must also be Dedekind-complete (and vice versa, of course), this justifies using 'the' in the phrase 'the complete Archimedean field'. This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.
(,
Wed 3 Jun 2009, 13:37,
archived)
fucking hell, sir!*is impressed - genuinely so*
(
J Peasemould Gruntfuttock Alpha Mike Foxtrot,
Wed 3 Jun 2009, 13:47,
archived)
the fact that you understand *any*is cause enough, in my book*.
* This slim volume here, entitled 'My Knowledge of Stuff'
(* This slim volume here, entitled 'My Knowledge of Stuff'
J Peasemould Gruntfuttock Alpha Mike Foxtrot,
Wed 3 Jun 2009, 13:53,
archived)
well, cauchy sequences are easy enough to get your head around.add up a bunch of stuff that is getting increasingly smaller and eventually you get a final number.
Here's something I writted myself:
The presence of a shell dramatically changes the response to an acoustic driving pressure pulse. To successfully model the behaviour of modern contrast agents the shell layer must be considered. The differences in the acoustic response between and a bubble without a shell and a bubble with a shell were shown in Section (\ref{sec:church}). In general the behaviour of contrast agents is also greatly affected by the shell properties. The highly nonlinear stable cavitation bubble oscillations are reduced by the shell. The presence of the shell will make the bubble more viscous, hence, more of the energy from the pulse will be absorbed and converted to heat, rather than being scattered or radiated by the bubble. Several models exist for the shell layer, such as the viscoelastic model studied here. It is possible to use different rheological models, and no one model is considered to be completely encompassing of all the contrast agents.
(,
Wed 3 Jun 2009, 13:57,
archived)
Here's something I writted myself:
The presence of a shell dramatically changes the response to an acoustic driving pressure pulse. To successfully model the behaviour of modern contrast agents the shell layer must be considered. The differences in the acoustic response between and a bubble without a shell and a bubble with a shell were shown in Section (\ref{sec:church}). In general the behaviour of contrast agents is also greatly affected by the shell properties. The highly nonlinear stable cavitation bubble oscillations are reduced by the shell. The presence of the shell will make the bubble more viscous, hence, more of the energy from the pulse will be absorbed and converted to heat, rather than being scattered or radiated by the bubble. Several models exist for the shell layer, such as the viscoelastic model studied here. It is possible to use different rheological models, and no one model is considered to be completely encompassing of all the contrast agents.
you're starting to scare me now...;-)
( J Peasemould Gruntfuttock Alpha Mike Foxtrot,
Wed 3 Jun 2009, 14:05,
archived)
everyone's doing it...www.google.com.au/search?hl=en&q=The+Portuguese+Legion+(Portuguese%3A+Legião+Portuguesa)+was+a+Portuguese+paramilitary+state+organization+founded+in+1936+during+dictator+António+de+Oliveira+Salazar's+right-wing+regime%2C+the+Estado+Novo.+It+was+dissolved+in+1974.&btnG=Google+Search&meta=&aq=f&oq=
although, I can copy stuff I've written.
(,
Wed 3 Jun 2009, 13:53,
archived)
although, I can copy stuff I've written.
