F. Robert | Searcharxiv

arXiv2020

Iterative Retina for high track multiplicity in a barrel-shape tracker and high magnetic field

Real-time track tracking in high energy physics experiments at colliders running at high luminosity is very challenging for trigger systems. To perform pattern-recognition and track fitting in online trigger system, the artificial Retina algorithm has been introduced in the field. Retina can be implemented in the state of the art FPGA devices. Our developments use Retina in an iterative way to identify track for barrel-shape tracker embedded in a high magnetic field and with high track multiplicity. As a benchmark we simulate LHC t-tbar events, with a pile-up of 200 and a GEANT-4 based simulation of a 6-layers barrel tracker detector made of silicon modules. With this sample the performance of the hardware design (resource usage, latency) is evaluated. Both efficiency and purity of the Retina fitting are over 90%. Moreover we have also added a Kalman filter after the Retina fit to improve the resolution on the track parameters. Our simulation results show that the Kalman filter can work well together with the Retina algorithm to find track through t-tbar event and provides high resolutions of the reconstructed parameters.

physics.ins-det↗

arXiv2005

The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev Inequalities

We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain $Ω$ of \mathbb{R}^n: $$ μ_s (Ω) := \inf \{\int_Ω| \nabla u|^2 dx; u \in {H_{1,0}^2(Ω)} \hbox{and} \int_Ω \frac {|u|^{2^{\star}}}{|x|^s} dx =1\}$$ when 0 = 4, the negativity of the mean curvature of $\partial Ω$ at 0 is sufficient to ensure the attainability of $μ_{s}(Ω)$. Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions correrresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [17].

math.AP↗