247,470 research outputs found
On generation of metric perturbations during preheating
We consider the generation of the scalar mode of the metric perturbations
during preheating stage in a two field model with the potential . We discuss two possible
sources of such perturbations: a) due to the coupling between the perturbation
of the matter field and the background part of the matter field
, b) due to non-linear fluctuations in a condensate of
``particles'' of the field . Both types of the metric perturbations are
assumed to be small, and estimated using the linear theory of the metric
perturbations. We estimate analytically the upper limit of the amplitude of the
metric perturbations for all scales in the limit of so-called broad resonance,
and show that the large scale metric perturbations are very small, and taking
them into account does not influence the standard picture of the production of
the metric perturbations in inflationary scenario.Comment: This version is to be published in PRD, new references added and
typos correcte
On the -adic valuation of certain Jacobi sums
Fix distinct primes and , a finite field such that
, and a character of exact order . We present a new -adic
congruence for the Jacobi sum . These Jacobi sums are
Frobenius eigenvalues of the curve .Comment: 18 pages; improved notation, reorganized sections accordingly,
clarified a small subtlety with ell=2 cas
Pure Differential Modules and a Result of Macaulay on Unmixed Polynomial Ideals
The first purpose of this paper is to point out a curious result announced by
Macaulay on the Hilbert function of a differential module in his famous book
The Algebraic Theory of Modular Systems published in 1916. Indeed, on page
78/79 of this book, Macaulay is saying the following: " A polynomial ideal
,..., is of the {\it
principal class} and thus {\it unmixed} if it has rank and is generated by
polynomials. Having in mind this definition, a primary ideal
with associated prime ideal is such that any
ideal of the principal class with determines a primary ideal of greater {\it multiplicity} over
. In particular, we have ,...,
because, passing to a system of PD equations for one unknown , the
parametric jets are \{\} but any ideal of
the principal class with is
contained into a {\it simple} ideal, that is a primary ideal
such that is a maximal and
thus prime ideal with at least.
Accordingly, any primary ideal may not be a member of the
primary decomposition of an unmixed ideal
of the principal class. Otherwise, is said to be of the {\it
principal noetherian class} ". Our aim is to explain this result in a modern
language and to illustrate it by providing a similar example for . The
importance of such an example is that it allows for the first time to exhibit
symbols which are -acyclic without being involutive. Another interest of
this example is that it has properties quite similar to the ones held by the
system of conformal Killing equations which are still not known. For this
reason, we have put all the examples at the end of the paper and each one is
presented in a rather independent way though a few among them are quite tricky.
Meanwhile, the second purpose is to prove that the methods developped by
Macaulay in order to study {\it unmixed polynomial ideals} are only particular
examples of new formal differential geometric techniques that have been
introduced recently in order to study {\it pure differential modules}. However
these procedures are based on the formal theory of systems of ordinary
differential (OD) or partial differential (PD) equations, in particular on a
systematic use of the Spencer operator, and are still not acknowledged by the
algebraic community
Electron-electron interaction in multiwall carbon nanotubes
Magnetic susceptibility of pristine and brominated arc-produced sample
of multiwall carbon nanotubes was measured from 4.2 to 400 K. An additional
contribution to diamagnetic susceptibility of carbon
nanotubes was found at T 50 K for both samples. It is shown that are dominated by quantum correction to for interaction
electrons (interaction effects-IE). The IE shows a crossover from
two-dimensional to three-dimensional at = 5.5 T. The effective interaction
between electrons for interior layers of nanotubes are repulsion and the
electron-electron interaction was estimated to be 0.26.Comment: 10 pages, 7 figure
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