On the algebraic structure of the unitary group

We consider the unitary group \U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever \U acts by isometries on a metric space, every orbit is bounded. Equivalently, \U is not the union of a countable chain of proper subgroups, and whenever \E\subseteq \U generates \U, it does so by words of a fixed finite length

On the Granular Stress-Geometry Equation

Using discrete calculus, we derive the missing stress-geometry equation for rigid granular materials in two dimensions, in the mean-field approximation. We show that (i) the equation imposes that the voids cannot carry stress, (ii) stress transmission is generically elliptic and has a quantitative relation to anisotropic elasticity, and (iii) the packing fabric plays an essential role.Comment: 6 page

Does the Nominal Exchange Rate Regime Affect the Long Run Properties of Real Exchange Rates?

This paper examines whether the behaviour of the real exchange rate is associated with a particular regime for the nominal exchange rate, like fixed and flexible exchange rate arrangements. The analysis is based on 16 annual real exchange rates and covers a long time span, 1870-2006. Four subperiods are distinguished and linked to exchange rate regimes: the Gold Standard, the interwar float, the Bretton Woods system and the managed float thereafter. Panel integration techniques are applied to increase the power of the tests. Cross section correlation is embedded via common factor structures. The evidence shows that real exchange rates properties are affected by the exchange rate regime, although the impact is not very strong. A unit root is rejected in both fixed and flexible exchange rate systems. Regarding fixed-rate systems, mean reversion of real exchange rates is more visible for the Gold Standard. The half lives of shocks have increased after WWII, probably due to a higher stickiness of prices and a lower weight of international trade in the determination of exchange rates. Both for the periods before and after WWII, half lives are lower in flexible regimes. This suggests that the nominal exchange rate plays some role in the adjustment process towards PPP.Real exchange rate persistence, exchange rate regime, panel unit roots

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of no-photon Quantum Electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of $N$ electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant $\alpha$ is small whereas $\alpha Z$ and the particle number $N$ are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree-Fock ground state.Comment: Final version, to appear in Arch. Rat. Mech. Ana

Lucas congruences for the Ap\'ery numbers modulo p2p^2

The sequence $A(n)_{n \geq 0}$ of Ap\'ery numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.Comment: 13 pages, 1 figure; significantly shorter proof of Theorem

Circular Coloring of Random Graphs: Statistical Physics Investigation

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table