Arithmetic purity of strong approximation for homogeneous spaces

We prove that any open subset $U$ of a semi-simple simply connected quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$ over a number field satisfies strong approximation by establishing a fibration of $G$ over a toric variety. We also prove a similar result of strong approximation with Brauer-Manin obstruction for a partial equivariant smooth compactification of a homogeneous space where all invertible functions are constant and the semi-simple part of the linear algebraic group is quasi-split. Some semi-abelian varieties of any given dimension where the complements of a rational point do not satisfy strong approximation with Brauer-Manin obstruction are given

Heat kernel transform for nilmanifolds associated to the Heisenberg group

We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.Comment: Revised version; to appear in Revista Mathematica Iberoamericana, 28

The asymmetric structure of the Galactic halo

Using the stellar photometry catalogue based on the latest data release (DR4) of the Sloan Digital Sky Survey (SDSS), a study of the Galactic structure using star counts is carried out for selected areas of the sky. The sample areas are selected along a circle at a Galactic latitude of +60$^\circ$, and 10 strips of high Galactic latitude along different longitudes. Direct statistics of the data show that the surface densities of $\ell$ from $180^{\circ}$ to $360^{\circ}$ are systematically higher than those of $\ell$ from $0^{\circ}$ to $180^{\circ}$, defining a region of overdensity (in the direction of Virgo) and another one of underdensity (in the direction of Ursa Major) with respect to an axisymmetric model. It is shown by comparing the results from star counts in the $(g-r)$ colour that the density deviations are due to an asymmetry of the stellar density in the halo. Theoretical models for the surface density profile are built and star counts are performed using a triaxial halo of which the parameters are constrained by observational data. Two possible reasons for the asymmetric structure are discussed.Comment: 17 pages, 7 figures, 5 tables, MNRAS accepte

Design of Photonic Crystal Klystrons

2D Pho­ton­ic crys­tals (PC) with de­fects can act as stand­ing-wave res­onators, which offer ben­e­fit of high mode se­lec­tiv­i­ty for build­ing novel RF sources. We in­tro­duce our work on de­sign­ing two-cav­i­ty sin­gle-beam and mul­ti-beam klystrons using tri­an­gu­lar lat­tice metal­lic PCs. We pre­sent the cold test re­sults of the stub-cou­pled sin­gle-beam struc­ture, which show that at res­o­nance a very low re­flec­tion can be ob­tained, and the waves are well con­fined. We also pre­sent bead-pull mea­sure­ment re­sults of field strengths in the de­fect, using mod­i­fied per­tur­ba­tion equa­tion for small unit di­elec­tric cylin­der, which are in very good agree­ment to nu­mer­i­cal re­sults. A 6-beam klystron cav­i­ty is de­signed as a 6-cou­pled-de­fect struc­ture with a cen­tral stub, which only cou­ples to the in-phase mode at the low­est fre­quen­cy. Fi­nal­ly, we pre­sent a fea­si­bil­i­ty dis­cus­sion of using this mul­ti-de­fect PC struc­ture to con­struct an in­te­grat­ed klystron-ac­cel­er­a­tor cav­i­ty, along with nu­mer­i­cal re­sults show­ing a peak ac­cel­er­a­tion field of 22MV/m can be achieved