Can we make a Finsler metric complete by a trivial projective change?

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.Comment: 11 pages, one figure, submitted to the proceedings of VI International Meeting on Lorentzian Geometry (Granada

Finsler geodesics in the presence of a convex function and their applications

We obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime.Comment: 16 pages, AMSLaTex. v2 is a minor revision: title changed, references updated, typos fixed; it matches the published version. This preprint and arXiv:math/0702323v3 [math.DG] substitute arXiv:math/0702323v2 [math.DG

The causal structure of spacetime is a parameterized Randers geometry

There is a by now well-established isomorphism between stationary 4-dimensional spacetimes and 3-dimensional purely spatial Randers geometries - these Randers geometries being a particular case of the more general class of 3-dimensional Finsler geometries. We point out that in stably causal spacetimes, by using the (time-dependent) ADM decomposition, this result can be extended to general non-stationary spacetimes - the causal structure (conformal structure) of the full spacetime is completely encoded in a parameterized (time-dependent) class of Randers spaces, which can then be used to define a Fermat principle, and also to reconstruct the null cones and causal structure.Comment: 8 page

Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold

Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.Comment: 14 pages, AMSLaTex, 2 figures. v2: typos fixed, added one reference and several comments, statement of last proposition correcte

The artificial retina for track reconstruction at the LHC crossing rate

We present the results of an R&D study for a specialized processor capable of precisely reconstructing events with hundreds of charged-particle tracks in pixel and silicon strip detectors at $40\,\rm MHz$, thus suitable for processing LHC events at the full crossing frequency. For this purpose we design and test a massively parallel pattern-recognition algorithm, inspired to the current understanding of the mechanisms adopted by the primary visual cortex of mammals in the early stages of visual-information processing. The detailed geometry and charged-particle's activity of a large tracking detector are simulated and used to assess the performance of the artificial retina algorithm. We find that high-quality tracking in large detectors is possible with sub-microsecond latencies when the algorithm is implemented in modern, high-speed, high-bandwidth FPGA devices.Comment: 3 pages, 3 figures, ICHEP14. arXiv admin note: text overlap with arXiv:1409.089

Simulation and performance of an artificial retina for 40 MHz track reconstruction

We present the results of a detailed simulation of the artificial retina pattern-recognition algorithm, designed to reconstruct events with hundreds of charged-particle tracks in pixel and silicon detectors at LHCb with LHC crossing frequency of $40\,\rm MHz$. Performances of the artificial retina algorithm are assessed using the official Monte Carlo samples of the LHCb experiment. We found performances for the retina pattern-recognition algorithm comparable with the full LHCb reconstruction algorithm.Comment: Final draft of WIT proceedings modified according to JINST referee's comment

On the energy functional on Finsler manifolds and applications to stationary spacetimes

In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows us to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.Comment: 23 pages, AMSLaTeX. v4 matches the published versio

The artificial retina processor for track reconstruction at the LHC crossing rate

We present results of an R&D study for a specialized processor capable of precisely reconstructing, in pixel detectors, hundreds of charged-particle tracks from high-energy collisions at 40 MHz rate. We apply a highly parallel pattern-recognition algorithm, inspired by studies of the processing of visual images by the brain as it happens in nature, and describe in detail an efficient hardware implementation in high-speed, high-bandwidth FPGA devices. This is the first detailed demonstration of reconstruction of offline-quality tracks at 40 MHz and makes the device suitable for processing Large Hadron Collider events at the full crossing frequency.Comment: 4th draft of WIT proceedings modified according to JINST referee's comments. 10 pages, 6 figures, 2 table

A Specialized Processor for Track Reconstruction at the LHC Crossing Rate

We present the results of an R&D study of a specialized processor capable of precisely reconstructing events with hundreds of charged-particle tracks in pixel detectors at 40 MHz, thus suitable for processing LHC events at the full crossing frequency. For this purpose we design and test a massively parallel pattern-recognition algorithm, inspired by studies of the processing of visual images by the brain as it happens in nature. We find that high-quality tracking in large detectors is possible with sub-$\mu$s latencies when this algorithm is implemented in modern, high-speed, high-bandwidth FPGA devices. This opens a possibility of making track reconstruction happen transparently as part of the detector readout.Comment: Presented by G.Punzi at the conference on "Instrumentation for Colliding Beam Physics" (INSTR14), 24 Feb to 1 Mar 2014, Novosibirsk, Russia. Submitted to JINST proceeding

First prototype of a silicon tracker using an artificial retina for fast track finding

We report on the R\&D for a first prototype of a silicon tracker based on an alternative approach for fast track finding. The working principle is inspired from neurobiology, in particular by the processing of visual images by the brain as it happens in nature. It is based on extensive parallelisation of data distribution and pattern recognition. In this work we present the design of a practical device that consists of a telescope based on single-sided silicon detectors; we describe the data acquisition system and the implementation of the track finding algorithms using available digital logic of commercial FPGA devices. Tracking performance and trigger capabilities of the device are discussed along with perspectives for future applications.Comment: 9 pages, 7 figures, Technology and Instrumentation in Particle Physics 2014 (TIPP 2014), conference proceeding