"Weak yet strong'' restrictions of Hindman's Finite Sums Theorem

We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman’s Theorem with analogous propertie

"Weak yet strong" restrictions of Hindman's Finite Sums Theorem

We present a natural restriction of Hindman's Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman's Theorem with analogous properties

Cuspidal divisor class groups of non-split Cartan modular curves

I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,\chi}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.Comment: 35 page

First implications of LHCb data on models beyond the Standard Model

We discuss the theoretical and experimental details of two of the main results obtained by LHCb with the 2011 data, namely the measurement of the mixing-induced CP-violation in the decay B_s -> J/psi phi and the upper limits on the decays B_(s) -> mu+ mu-. Then we describe the possible strategies to obtain new constraints on two different New Physics models in the light of these results.Comment: 5 pages, Proceedings of "QCD@Work 2012" - June 18-21, 2012 - Lecce (Italy

On the large order behaviour of the Potts model

Following the work by Houghton, Reeve and Wallace about an alternative formulation of the $n \to 0$ limit of the $(n+1)$ state Potts model in field theory for the large order behaviour of the perturbative expansion, we generalise their technique to all $n$ by establishing an equivalence in perturbation theory order by order with another bosonic field theory. Restricting ourselves to a cubic interaction, we obtain an explicit expression (in terms of~$n$) for the large order behaviour of the partition function.Comment: 16 pages, require TeXdra