Representations of the discrete inhomogeneous Lorentz group and Dirac wave equation on the lattice

We propose the fundamental and two dimensional representation of the Lorentz groups on a (3+1)-dimensional hypercubic lattice, from which representations of higher dimensions can be constructed. For the unitary representation of the discrete translation group we use the kernel of the Fourier transform. From the Dirac representation of the Lorentz group (including reflections) we derive in a natural way the wave equation on the lattice for spin 1/2 particles. Finally the induced representation of the discrete inhomogeneous Lorentz group is constructed by standard methods and its connection with the continuous case is discussed.Comment: LaTeX, 20 pages, 1 eps figure, uses iopconf.sty (late submission

Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice

We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on non-homogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org

Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission

The double torus as a 2D cosmos: groups, geometry and closed geodesics

The double torus provides a relativistic model for a closed 2D cosmos with topology of genus 2 and constant negative curvature. Its unfolding into an octagon extends to an octagonal tessellation of its universal covering, the hyperbolic space H^2. The tessellation is analysed with tools from hyperbolic crystallography. Actions on H^2 of groups/subgroups are identified for SU(1, 1), for a hyperbolic Coxeter group acting also on SU(1, 1), and for the homotopy group \Phi_2 whose extension is normal in the Coxeter group. Closed geodesics arise from links on H^2 between octagon centres. The direction and length of the shortest closed geodesics is computed.Comment: Latex, 27 pages, 5 figures (late submission to arxiv.org

Effects of Including a Penetration Test in Motorcyclist Helmet Standards: Influence on Helmet Stiffness and Impact Performance

Regulation ECE-22.05/06 does not require a helmet penetration test. Penetration testing is controversial since it has been shown that it may cause the helmet to behave in a non-desirable stiff way in real-world crashes. This study aimed to assess the effect of the penetration test in the impact performance of helmets. Twenty full-face motorcycle helmets were penetration tested at multiple locations of the helmet shell. Then, 10 helmets were selected and split into two groups (hard shell and soft shell) depending on the results of the penetration tests. These 10 helmets were then drop tested at front, lateral, and top areas at two different impact speeds (5 m/s and 8.2 m/s) to assess their impact performance against head injuries. The statistical analyses did not show any significant difference between the two groups (hard/soft shell) at 5 m/s. Similar results were observed at 8.2 m/s, except for the top area of the helmet in which the peak linear acceleration was significantly higher for the soft shell group than for the hard shell group (230 ± 12 g vs. 211 ± 11 g; p-value = 0.038). The results of this study suggest that a stiffer shell does not necessarily cause helmets to behave in a stiffer way when striking rigid flat surfaces. These experiments also showed that hard shell helmets can provide better protection at higher impact speeds without damaging helmet performance at lower impact speeds. © 2022 by the authors. Licensee MDPI, Basel, Switzerland

Physical Properties of Asteroid (308635) 2005 YU55 derived from multi-instrument infrared observations during a very close Earth-Approach

The near-Earth asteroid (308635) 2005 YU55 is a potentially hazardous asteroid which was discovered in 2005 and passed Earth on November 8th 2011 at 0.85 lunar distances. This was the closest known approach by an asteroid of several hundred metre diameter since 1976 when a similar size object passed at 0.5 lunar distances. We observed 2005 YU55 from ground with a recently developed mid-IR camera (miniTAO/MAX38) in N- and Q-band and with the Submillimeter Array (SMA) at 1.3 mm. In addition, we obtained space observations with Herschel/PACS at 70, 100, and 160 micron. Our thermal measurements cover a wide range of wavelengths from 8.9 micron to 1.3 mm and were taken after opposition at phase angles between -97 deg and -18 deg. We performed a radiometric analysis via a thermophysical model and combined our derived properties with results from radar, adaptive optics, lightcurve observations, speckle and auxiliary thermal data. We find that (308635) 2005 YU55 has an almost spherical shape with an effective diameter of 300 to 312 m and a geometric albedo pV of 0.055 to 0.075. Its spin-axis is oriented towards celestial directions (lam_ecl, beta_ecl) = (60 deg +/- 30deg, -60 deg +/- 15 deg), which means it has a retrograde sense of rotation. The analysis of all available data combined revealed a discrepancy with the radar-derived size. Our radiometric analysis of the thermal data together with the problem to find a unique rotation period might be connected to a non-principal axis rotation. A low to intermediate level of surface roughness (r.m.s. of surface slopes in the range 0.1 - 0.3) is required to explain the available thermal measurements. We found a thermal inertia in the range 350-800 Jm^-2s^-0.5K^-1, very similar to the rubble-pile asteroid (25143) Itokawa and indicating a mixture of low conductivity fine regolith with larger rocks and boulders of high thermal inertia on the surface.Comment: Accepted for publication in Astronomy & Astrophysics, 12 pages, 10 figure

Confined Motion: Motility of Active Microparticles in Cell-Sized Lipid Vesicles

[EN] Active materials can transduce external energy into kinetic energy at the nano and micron length scales. This unique feature has sparked much research, which ranges from achieving fundamental understanding of their motility to the assessment of potential applications. Traditionally, motility is studied as a function of internal features such as particle topology, while external parameters such as energy source are assessed mainly in bulk. However, in real-life applications, confinement plays a crucial role in determining the type of motion active particles can adapt. This feature has been however surprisingly underexplored experimentally. Here, we showcase a tunable experimental platform to gain an insight into the dynamics of active particles in environments with restricted 3D topology. Particularly, we examined the autonomous motion of coacervate micromotors confined in giant unilamellar vesicles (GUVs) spanning 10¿50 ¿m in diameter and varied parameters including fuel and micromotor concentration. We observed anomalous diffusion upon confinement, leading to decreased motility, which was more pronounced in smaller compartments. The results indicate that the theoretically predicted hydrodynamic effect dominates the motion mechanism within this platform. Our study provides a versatile approach to understand the behavior of active matter under controlled, compartmentalized conditions.The authors would like to acknowledge the support from the Dutch Ministry of Education, Culture and Science (Gravitation program 024.001.035 and Spinoza premium) and the ERC Advanced Grant (Artisym 694120) . A.L.-L. acknowledges the support from the MSCA Cofund Project of Life, which has received funding from the European Union's Horizon 2020 research and innovation program under the grant agreement 847675, and the Maria Zambrano Program from the Spanish Government funded by NextGenerationEU from the European Union. Dr. Bastiaan Buddingh is thanked for useful discussions regarding GUV preparation and handling. Dr. Shoupeng Cao is thanked for providing the azido-functionalized block polymer. We specially thank Prof. Samuel Sanchez for the tailor-made particle-tracking software based on Python.Song, S.; Llopis-Lorente, A.; Mason, AF.; Abdelmohsen, LK.; Van Hest, JCM. (2022). Confined Motion: Motility of Active Microparticles in Cell-Sized Lipid Vesicles. Journal of the American Chemical Society. 144:13831-13838. https://doi.org/10.1021/jacs.2c05232138311383814