Mutigrid preconditioner for nonconforming discretization of elliptic problems with jump coefficients

In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming Crouzeix-Raviart discretization of second order elliptic problems with jump coe fficients. The preconditioner uses the standard conforming subspaces as coarse spaces. Numerical tests show both robustness with respect to the jump in the coe fficient and near-optimality with respect to the number of degrees of freedom.

Schwarz methods for a preconditioned WOPSIP method for elliptic problems

We construct and analyze non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) method for elliptic problems.

An electronic voting platform with elliptic curve cryptography

The future of elections seems to be electronic voting systems du to its advantatges over the traditional voting. Nowadays, there are some different paradigms to ensure the security and reliability of e-voting. This document is part of a wider project which presents an e-Voting platform based on elliptic curve cryptography. It uses an hybrid combination of two of the main e-Voting paradigms to guarantee privacy and security in the counting phase, these are precisely, the mixnets and the homomorphic protocols. This document is focused in the description of the system and the maths and programming needed to solve the homomorphic part of it. In later chapters, there is a comparison between a simple mixing system and our system proposal.

Photometric Bgr survey of the distant clusters of galaxies CL 1613+3104 and CL 1600+4109

We present a set of photometric data concerning two distant clusters of galaxies: Cl 1613+3104 (z=0.415) and Cl 1600+4109 (z=0.540). The photometric survey extends to a field of about 4' x 3'. It was performed in 3 filters: Johnson B, and Thuan-Gunn g and r. The sample includes 679 objects in the field of Cl 1613+3104 and 334 objects in Cl 1600+4109.

New measurements of radial velocities in clusters of galaxies

Redshifts for 100 galaxies in 10 clusters of galaxies are presented based on data obtained between March 1984 and March 1985 from Calar Alto, La Palma, and ESO, and on data from Mauna Kea. Data for individual galaxies are given, and the accuracy of the velocities of the four instruments is discussed. Comparison with published data shows the present velocities to be shifted by + 4.0 km/s on average, with a standard deviation in the difference of 89.7 km/s, consistent with the rms of redshift measurements which range from 50-100 km/s.

Photometric and Spectroscopic Observations of the Cluster of Galaxies Abell 2390

Spectroscopic and photometric observations in a 6 arcmin x 6 arcmin field centered on the rich cluster of galaxies Abell 2390 are presented. The photometry concerns 700 objects and the spectroscopy 72 objects. The redshift survey shows that the mean redshift of the cluster is 0.232. An original method for automatic determination of the spectral type of galaxies is presented.

Photometric and spectroscopic survey of the cluster of galaxies Abell A2218

We present new photometric and spectroscopic observations of objects in the field of the cluster of galaxies Abell 2218. The photometric survey, centered on the cluster core, extends to a field of about 4 x 4 arcmin. It was performed in 5 bands (B,g,r,i and z filters). This sample, which includes 729 objects, is about three times larger than the survey made by Butcher and collaborators (Butcher et al., 1983, Butcher and Oemler, 1984) in the same central region of the field. Only 228 objects appear in both catalogues since our survey covers a smaller region. The spectral range covered by our filters is wider and the photometry is much deeper, up to magnitude 27 in r. The spectroscopic survey concerns 66 objects, on a field comparable to that of Butcher and collaborators. From our observations we calculate the mean redshift of the cluster, 0.1756, and its velocity dispersion, 1370 km/s. The spectral types are determined for many galaxies in the sample by comparing their spectra with synthetic ones from Rocca-Volmerange and Guiderdoni (1988).

On Ito's formula for elliptic diffusion processes

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83-109] prove an extension of Ito's formula for F(Xt, t), where F(x, t) has a locally square-integrable derivative in x that satisfies a mild continuity condition in t and X is a one-dimensional diffusion process such that the law of Xt has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303-328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function F has a locally integrable derivative in t, we can avoid the mild continuity condition in t for the derivative of F in x.

J2 effect and elliptic inclined periodic orbits in the collision three-body problem

The existence of a new class of inclined periodic orbits of the collision restricted three-body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet

Algorithms and cryptographic protocols using elliptic curves

En els darrers anys, la criptografia amb corbes el.líptiques ha adquirit una importància creixent, fins a arribar a formar part en la actualitat de diferents estàndards industrials. Tot i que s'han dissenyat variants amb corbes el.líptiques de criptosistemes clàssics, com el RSA, el seu màxim interès rau en la seva aplicació en criptosistemes basats en el Problema del Logaritme Discret, com els de tipus ElGamal. En aquest cas, els criptosistemes el.líptics garanteixen la mateixa seguretat que els construïts sobre el grup multiplicatiu d'un cos finit primer, però amb longituds de clau molt menor. Mostrarem, doncs, les bones propietats d'aquests criptosistemes, així com els requeriments bàsics per a que una corba sigui criptogràficament útil, estretament relacionat amb la seva cardinalitat. Revisarem alguns mètodes que permetin descartar corbes no criptogràficament útils, així com altres que permetin obtenir corbes bones a partir d'una de donada. Finalment, descriurem algunes aplicacions, com són el seu ús en Targes Intel.ligents i sistemes RFID, per concloure amb alguns avenços recents en aquest camp.

Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.

Volcanoes of l-isogenies of elliptic curves over finite fields: the case l=3

This paper is devoted to the study of the volcanoes of l-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the l-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case l = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results are also provided.

Evolution and Stellar Content of AGN Host Galaxies

In this dissertation, Active Galactic Nuclei (AGN) and their host galaxies are discussed. Together with transitional events, such as supernovae and gamma-ray bursts, AGN are the most energetic phenomena in the Universe. The dominant fraction of their luminosity originates from the center of a galaxy, where accreting gas falls into a supermassive black hole, converting gravitational energy to radiation. AGN have a wide range of observed properties: e.g. in their emission lines, radio emission, and variability. Most likely, these properties depend significantly on their orientation to our line-of-sight, and to unify AGN into physical classes it is crucial to observe their orientation-independent properties, such as the host galaxies. Furthermore, host galaxy studies are essential to understand the formation and co-evolution of galactic bulges and supermassive black holes. In this thesis, the main focus is on observationally characterizing AGN host galaxies using optical and near-infrared imaging and spectroscopy. BL Lac objects are a class of AGN characterized by rapidly variable and polarized continuum emission across the electromagnetic spectrum, and coredominated radio emission. The near-infrared properties of intermediate redshift BL Lac host galaxies are studied in Paper I. They are found to be large elliptical galaxies that are more luminous than their low redshift counterparts suggesting a strong luminosity evolution, and a contribution from a recent star formation episode. To analyze the stellar content of galaxies in more detail multicolor data, especially observations at blue wavelengths, are essential. In Paper III, optical - near-infrared colors and color gradients are derived for low redshift BL Lac host galaxies. They show bluer colors and steeper color gradients than inactive ellipticals which, most likely, are caused by a relatively young stellar population indicating a different evolutionary stage between AGN hosts and inactive ellipticals. In Paper II, near-infrared imaging of intermediate redshift radio-quiet quasar hosts is used to study their luminosity evolution. The hosts are large elliptical galaxies, but they are systematically fainter than the hosts of radio-loud quasars at similar redshifts, suggesting a link between the luminosity of the host galaxies and the radio properties of AGN. In Paper IV, the characteristics of near-infrared stellar absorption features of low redshift radio galaxies are compared with those of inactive early-type galaxies. The comparison suggests that early-type galaxies with AGN are in a different evolutionary stage than their inactive counterparts. Moreover, radio galaxies are found to contain stellar populations containing both old and intermediate age components.