The Identity Correspondence Problem and its Applications

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the published journal version of this article, see footnote 3 on page 1

Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise

Future neutrino oscillation facilities

The recent discovery that neutrinos have masses opens a wide new field of experimentation. Accelerator-made neutrinos are essential in this program. Ideas for future facilities include high intensity muon neutrino beams from pion decay (`SuperBeam'), electron neutrino beams from nuclei decays (`Beta Beam'), or muon and electron neutrino beams from muon decay (`Neutrino Factory'), each associated with one or several options for detector systems. Each option offers synergetic possibilities, e.g. some of the detectors can be used for proton decay searches, while the Neutrino Factory is a first step towards muon colliders. A summary of the perceived virtues and shortcomings of the various options, and a number of open questions are presented.Comment: Originally written for the CERN Strategy Grou

On in situ Determination of Earth Matter Density in Neutrino Factory

We point out that an accurate in situ determination of the earth matter density \rho is possible in neutrino factory by placing a detector at the magic baseline, L = \sqrt{2} \pi / G_{F} N_{e} where N_{e} denotes electron number density. The accuracy of matter density determination is excellent in a region of relatively large theta_{13} with fractional uncertainty \delta \rho / \rho of about 0.43%, 1.3%, and \lsim 3% at 1 sigma CL at sin^2 2theta_{13}=0.1, 10^{-2}, and 3 x 10^{-3}, respectively. At smaller theta_{13} the uncertainty depends upon the CP phase delta, but it remains small, 3%-7% in more than 3/4 of the entire region of delta at sin^2 2theta_{13} = 10^{-4}. The results would allow us to solve the problem of obscured CP violation due to the uncertainty of earth matter density in a wide range of theta_{13} and delta. It may provide a test for the geophysical model of the earth, or it may serve as a method for stringent test of the MSW theory of neutrino propagation in matter once an accurate geophysical estimation of the matter density is available.Comment: 21 pages, 4 figures, version to appear in PR

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

TLEP, first step in a long-term vision for HEP

The discovery of H(126) has renewed interest in circular e+e- colliders that can operate as Higgs factories, which benefit from three unique characteristics: i) high luminosity and reliability, ii) the availability of several interaction points, iii) superior beam energy accuracy. TLEP is an e+e- storage ring of 80-km circumference that can operate with very high luminosity from the Z peak (90 GeV) to the top quark pair threshold (350 GeV). It can achieve transverse beam polarization at the Z peak and WW threshold, giving it unparalleled accuracy on the beam energy. A preliminary study indicates that an 80 km tunnel could be constructed around CERN. Such a tunnel would allow a 100 TeV proton-proton collider to be established in the same ring (VHE-LHC), offering a long term vision.Comment: This is a contribution to the the Snowmass process 2013: Frontier Capabilitie

La Croatie dans l’Union Européenne : une insertion différenciée

International audienc

Comments on "Wall-plug (AC) power consumption of a very high energy e+/e- storage ring collider" by Marc Ross

The paper arXiv:1308.0735 questions some of the technical assumptions made by the TLEP Steering Group when estimating in arXiv:1305.6498 the power requirement for the very high energy e+e- storage ring collider TLEP. We show that our assumptions are based solidly on CERN experience with LEP and the LHC, as well accelerators elsewhere, and confirm our earlier baseline estimate of the TLEP power consumption.Comment: 6 page

On the improvement of the low energy neutrino factory

The low energy neutrino factory has been proposed as a very sensitive setup for future searches for CP violation and matter effects. Here we study how its performance is affected when the experimental specifications of the setup are varied. Most notably, we have considered the addition of the 'platinum' nu_{mu} -> nu_{e} channel. We find that, whilst theoretically the extra channel provides very useful complementary information and helps to lift degeneracies, its practical usefulness is lost when considering realistic background levels. Conversely, an increase in statistics in the 'golden' nu_{e} -> nu_{mu} channel and, to some extent, an improvement in the energy resolution, lead to an important increase in the performance of the facility, given the rich energy dependence of the 'golden' channel at these energies. We show that a low energy neutrino factory with a baseline of 1300 km, muon energy of 4.5 GeV, and either a 20 kton totally active scintillating detector or 100 kton liquid argon detector, can have outstanding sensitivity to the neutrino oscillation parameters theta13, delta and the mass hierarchy. For our estimated exposure of 2.8 x 10^{23} kton x decays per muon polarity, the low energy neutrino factory has sensitivity to theta13 and delta for sin^{2}(2theta13) > 10^{-4} and to the mass hierarchy for sin^{2}(2theta13) > 10^{-3}.Comment: 13 pages, 8 eps figures. Version published in PRD - experimental section with preliminary results removed, abstract and conclusions re-written accordingly, title changed, author list amended

Four-path interference and uncertainty principle in photodetachment microscopy

We study the quantal motion of electrons emitted by a pointlike monochromatic isotropic source into parallel uniform electric and magnetic fields. The two-path interference pattern in the emerging electron wave due to the electric force is modified by the magnetic lens effect which periodically focuses the beam into narrow filaments along the symmetry axis. There, four classical paths interfere. With increasing electron energy, the current distribution changes from a quantum regime governed by the uncertainty principle, to an intricate spatial pattern that yields to a semiclassical analysis.Comment: submitted to Europhysics Letter