26 research outputs found
Key Polynomials
The notion of key polynomials was first introduced in 1936 by S. Maclane in
the case of discrete rank 1 valuations. . Let K -> L be a field extension and
{\nu} a valuation of K. The original motivation for introducing key polynomials
was the problem of describing all the extensions {\mu} of {\nu} to L. Take a
valuation {\mu} of L extending the valuation {\nu}. In the case when {\nu} is
discrete of rank 1 and L is a simple algebraic extension of K Maclane
introduced the notions of key polynomials for {\mu} and augmented valuations
and proved that {\mu} is obtained as a limit of a family of augmented
valuations on the polynomial ring K[x].
In a series of papers, M. Vaqui\'e generalized MacLane's notion of key
polynomials to the case of arbitrary valuations {\nu} (that is, valuations
which are not necessarily discrete of rank 1).
In the paper Valuations in algebraic field extensions, published in the
Journal of Algebra in 2007, F.J. Herrera Govantes, M.A. Olalla Acosta and M.
Spivakovsky develop their own notion of key polynomials for extensions (K,
{\nu}) -> (L, {\mu}) of valued fields, where {\nu} is of archimedian rank 1
(not necessarily discrete) and give an explicit description of the limit key
polynomials.
Our purpose in this paper is to clarify the relationship between the two
notions of key polynomials already developed by vaqui\'e and by F.J. Herrera
Govantes, M.A. Olalla Acosta and M. Spivakovsky.Comment: arXiv admin note: text overlap with arXiv:math/0605193 by different
author
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Simple but Stronger UO, Double but Weaker UNMe Bonds: The Tale Told by Cp2UO and Cp2UNR
The free energies of reaction and the activation energies are calculated, with DFT (B3PW91) and small RECP (relativistic core potential) for uranium, for the reaction of Cp2UNMe and Cp2UO with MeCCMe and H3Si-Cl that yields the corresponding addition products. CAS(2,7) and DFT calculations on Cp2UO and Cp2UNMe give similar results, which validates the use of DFT calculations in these cases. The calculated results mirror the experimental reaction of [1,2,4-(CMe3)3C5H2]2UNMe with dimethylacetylene and [1,2,4-(CMe3)3C5H2]2UO with Me3SiCl. The net reactions are controlled by the change in free energy between the products and reactants, not by the activation energies, and therefore by the nature of the UO and UNMe bonds in the initial and final states. A NBO analysis indicates that the U-O interaction in Cp2UO is composed of a single U-O bond with three lone pairs of electrons localized on oxygen, leading to a polarized U-O fragment. In contrast, the U-NMe interaction in Cp2UNMe is composed of a and component and a lone pairof electrons localized on the nitrogen, resulting in a less polarized UNMe fragment, in accord with the lower electronegativity of NMe relative to O. The strongly polarized U(+)-O(-) bond is calculated to be about 70 kcal mol-1 stronger than the less polarized U=NMe bond
A Qualitative Theory of Motion Based on Spatio-Temporal Primitives
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