Koszul homology and extremal properties of Gin and Lex

In a polynomial ring $R$ with $n$ variables, for every homogeneous ideal $I$ and for every $p\leq n$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms and define the Koszul-Betti number $\beta_{ijp}(R/I)$ of $R/I$ to be the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic 0, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of any gin of $I$ and also by those of the Lex-segment of $I$. We also investigate the set $Gins(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $Gins(I)$ are bounded below by those of the gin-revlex of $I$ and present examples showing that in general there is no $J$ is $Gins(I)$ such that the Koszul-Betti numbers of any ideal in $Gins(I)$ are bounded above by those of $J$.Comment: 21 pages, preprint 200

Self-Energy of the Lorentzian EPRL-FK Spin Foam Model of Quantum Gravity

We calculate the most divergent contribution to the non-degenerate sector of the self-energy (or "melonic") graph in the context of the Lorentzian EPRL-FK Spin Foam model of Quantum Gravity. We find that such a contribution is logarithmically divergent in the cut-off over the SU(2)-representation spins when one chooses the face amplitude guaranteeing the face-splitting invariance of the foam.We also find that the dependence on the boundary data is different from that of the bare propagator. This fact has its origin in the non-commutativity of the EPRL-FK Y-map with the projector onto SL(2,C)-invariant states. In the course of the paper, we discuss in detail the approximations used during the calculations, its geometrical interpretation as well as the physical consequences of our result.Comment: 58 pages, 8 figures. Corrections about the degenerate sector (suppression of a paragraph, introduction of a new appendix), plus minor corrections and improvements in the introduction and in the conclusion

Charged Higgs in Models with Singlet Neutrino in Large Extra Dimensions

The charged Higgs decay in the channel H^- ->tau_L^- nu in models with a singlet neutrino can provide a test of large extra dimensions models with TeV scale quantum gravity since in the standard two Higgs doublet model type II, H^- -> tau_L^- nu is suppressed. In the present study, we focus on the decay H^- -> tau_L^- nu at the LHC for Higgs masses larger than the top-quark mass.Comment: 4 pages, LaTeX2e, 2 figures, 2 tables, Talk given at XXXVIIth Rencontres de Moriond, ElectroWeak Interactions and Unified Theories, March 9th - 16th 2002, Les Arcs, Franc

Groebner bases for spaces of quadrics of codimension 3

Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gr\"obner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.Comment: Minor changes, to appear in the J. Pure Applied Algebr

Little Higgs and precision electroweak tests

I consider the low energy limit of Little Higgs models. The method consists in eliminating the heavy fields using their classical equations of motion in the infinite mass limit. After the elimination of the heavy degrees of freedom one can directly read off deviations from the precision electroweak data. I also examine the effects on the low energy precision experiments.Comment: Talk given at 39th Rencontres de Moriond on Electroweak Interactions and Unified Theories, La Thuile, Italy, 21-28 Mar 200

Ginzburg-Landau Approach to Holographic Superconductivity

We construct a family of minimal phenomenological models for holographic superconductors in d=4+1 AdS spacetime and study the effect of scalar and gauge field fluctuations. By making a Ginzburg-Landau interpretation of the dual field theory, we determine through holographic techniques a phenomenological Ginzburg-Landau Lagrangian and the temperature dependence of physical quantities in the superconducting phase. We obtain insight on the behaviour of the Ginzburg-Landau parameter and whether the systems behaves as a Type I or Type II superconductor. Finally, we apply a constant external magnetic field in a perturbative approach following previous work by D'Hoker and Kraus, and obtain droplet solutions which signal the appearance of the Meissner effect.Comment: 41 pages, 31 figures, calculations adde

Reduction numbers and initial ideals

The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k=m^{k+1}. Vasconcelos conjectured that the reduction number of A=R/I can only increase by passing to the initial ideal, i.e r(R/I)\leq r(R/in(I)). The goal of this note is to prove the conjecture.Comment: 6 page

Notes on two interesting specimens of Hippotion celerio (Linnaeus) (Lepidoptera, Sphingidae)

Moths of the genus Hippotion Hübner, 1822 (Family: Sphingidae) are distributed in Africa, Central and southern Asia and as migrants in southern Europe and Australia. Currently, around 30 species are accommodated in this genus. Hippotion celerio is a regular migrant to the Maltese Islands. On the onset of warmer weather conditions, colonies establish themselves in North Africa and Europe. It is a resident species in the Azores and the Atlantic coast of Morocco and most probably also in the Levant, the Arabian Peninsula and Egyptpeer-reviewe