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 $V(\phi, \chi)= {m^{2}\phi^{2}\over 2}+{g^{2}\phi^{2}\chi^{2}\over 2}$. We discuss two possible sources of such perturbations: a) due to the coupling between the perturbation of the matter field $\delta \chi$ and the background part of the matter field $\chi_{0}(t)$, b) due to non-linear fluctuations in a condensate of ``particles'' of the field $\chi$. 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 ℓ\ell-adic valuation of certain Jacobi sums

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $\chi : \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(\chi^{\ell}, \chi^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.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 $\mathfrak{a} \subset k[{\chi}\_1$,..., ${\chi}\_n]=k[\chi]$ is of the {\it principal class} and thus {\it unmixed} if it has rank $r$ and is generated by $r$ polynomials. Having in mind this definition, a primary ideal $\mathfrak{q}$ with associated prime ideal $\mathfrak{p} = rad(\mathfrak{q})$ is such that any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a} \subset \mathfrak{q}$ determines a primary ideal of greater {\it multiplicity} over $k$. In particular, we have $dim\_k(k[\chi]/({\chi}\_1$,...,${\chi}\_n)^2)=n+1$ because, passing to a system of PD equations for one unknown $y$, the parametric jets are \{${y,y\_1, ...,y\_n}$\} but any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a}\subset ({\chi}\_1,{\^a},{\chi}\_n)^2$ is contained into a {\it simple} ideal, that is a primary ideal $\mathfrak{q}$ such that $rad(\mathfrak{q})=\mathfrak{m}\in max(k[\chi])$ is a maximal and thus prime ideal with $dim\_k(M)=dim\_k(k[\chi]/\mathfrak{q})=2^n$ at least. Accordingly, any primary ideal $\mathfrak{q}$ may not be a member of the primary decomposition of an unmixed ideal $\mathfrak{a} \subseteq \mathfrak{q}$ of the principal class. Otherwise, $\mathfrak{q}$ 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 $n=4$. The importance of such an example is that it allows for the first time to exhibit symbols which are $2,3,4$-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 $\chi$ of pristine and brominated arc-produced sample of multiwall carbon nanotubes was measured from 4.2 to 400 K. An additional contribution $\Delta \chi(T)$ to diamagnetic susceptibility $\chi(T)$ of carbon nanotubes was found at T $<$ 50 K for both samples. It is shown that $\Delta \chi(T)$ are dominated by quantum correction to $\chi$ for interaction electrons (interaction effects-IE). The IE shows a crossover from two-dimensional to three-dimensional at $B$ = 5.5 T. The effective interaction between electrons for interior layers of nanotubes are repulsion and the electron-electron interaction $\lambda$$_c$ was estimated to be $\lambda_c\sim$ 0.26.Comment: 10 pages, 7 figure