DIRK Schemes with High Weak Stage Order

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.Comment: 10 pages, 5 figure

An algebraic Birkhoff decomposition for the continuous renormalization group

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited

Linear and nonlinear susceptibilities of a decoherent two-level system

The linear and nonlinear dynamical susceptibilities of a two level system are calculated as it undergoes a transition to a decoherent state. Analogously to the Glover-Tinkham-Ferrell sum rule of superconductivity, spectral weight in the linear susceptibility is continuously transferred from a finite frequency resonance to nearly zero frequency, corresponding to a broken symmetry in the thermodynamic limit. For this reason, the behavior of the present model (the Mermin model) differs significantly from the spin-boson model. The third order nonlinear susceptibility, corresponding to two-photon absorption, has an unexpected non-monotonic behavior as a function of the environmental coupling, reaching a maximum within the decoherent phase of the model. Both linear and nonlinear susceptibilities may be expressed in a universal form.Comment: 10 pages, 9 figure

t-3-Pentyl-r-2,c-6-diphenyl­piperidin-4-one

In the title mol­ecule, C22H27NO, the piperidine ring adopts a chair conformation, with all substituents equatorial. The dihedral angle between the two phenyl rings is 56.90 (5)°. In the crystal, mol­ecules are linked by weak C—H⋯O hydrogen bonds. A C—H⋯π inter­action involving the phenyl ring at the 6-position is also found in the crystal structure

Photometric Properties of Galaxy Population in the Cluster EIS 0048-2942 at z~0.64

~Deep photometric data in the V-, R-, I-, z- and K-bands for the cluster of galaxies EIS 0048-2942 are used to investigate the properties of the galaxy populations at z~0.64 in a field of 2.5x2.5 Mpc^2. The sample of candidate cluster members (N = 171) is selected by the photometric redshift technique and is complete up to I=22.5. Galaxies were classified as spheroids and disks according to the shape of the light profile in the I-band, as parametrized by the Sersic index. In both optical and NIR, spheroids define a sharp colour-magnitude sequence, whose slope and zero points are consistent with a high formation redshift (z_f > 2). The disk population occupies a different region in the colour-magnitude diagram, having bluer colours with respect to the red sequence. Interestingly, we find some level of mixing between the properties of the two classes: some disks lie on the colour-magnitude sequence or are redder, while some spheroids turn out to be bluer. The spatial distribution of cluster galaxies show a clumpy structure, with a main over-density of radius ~0.5 Mpc, and at least two other clumps distant ~1 Mpc from the center. The various sub-structures are mostly populated by the red galaxies, while the blue population has an almost uniform distribution. The fraction of blue galaxies in EIS 0048-2942 is f_B=0.11 +/-0.07. This is much lower than what expected on the basis of the Butcher-Oemler effect at lower redshifts.Comment: 22 pages, 12 figures. A&A in pres

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

Evolution since z = 0.5 of the Morphology-Density relation for Clusters of Galaxies

Using traditional morphological classifications of galaxies in 10 intermediate-redshift (z~0.5) clusters observed with WFPC-2 on the Hubble Space Telescope, we derive relations between morphology and local galaxy density similar to that found by Dressler for low-redshift clusters. Taken collectively, the `morphology-density' relationship, M-D, for these more distant, presumably younger clusters is qualitatively similar to that found for the local sample, but a detailed comparison shows two substantial differences: (1) For the clusters in our sample, the M-D relation is strong in centrally concentrated ``regular'' clusters, those with a strong correlation of radius and surface density, but nearly absent for clusters that are less concentrated and irregular, in contrast to the situation for low redshift clusters where a strong relation has been found for both. (2) In every cluster the fraction of elliptical galaxies is as large or larger than in low-redshift clusters, but the S0 fraction is 2-3 times smaller, with a proportional increase of the spiral fraction. Straightforward, though probably not unique, interpretations of these observations are (1) morphological segregation proceeds hierarchically, affecting richer, denser groups of galaxies earlier, and (2) the formation of elliptical galaxies predates the formation of rich clusters, and occurs instead in the loose-group phase or even earlier, but S0's are generated in large numbers only after cluster virialization.Comment: 35 pages, 19 figures, uses psfig. Accepted for publication in Ap

Is the Butcher-Oemler effect a function of the cluster redshift ?

Using PSPC {\it Rosat} data, we measure x-ray surface brightness profiles, size and luminosity of the Butcher-Oemler (BO) sample of clusters of galaxies. The cluster x-ray size, as measured by the Petrosian r_{\eta=2} radius, does not change with redshift and is independent from x-ray luminosity. On the other hand, the x-ray luminosity increases with redshift. Considering that fair samples show no-evolution, or negative luminosity evolution, we conclude that the BO sample is not formed from the same class of objects observed at different look-back times. This is in conflict with the usual interpretation of the Butcher-Oemler as an evolutionary (or redshift-dependent) effect, based on the assumption that we are comparing the same class of objects at different redshifts. Other trends present in the BO sample reflect selection criteria rather than differences in look-back time, as independently confirmed by the fact that trends loose strength when we enlarge the sample with x-ray selected sample of clusters. The variety of optical sizes and shapes of the clusters in the Butcher-Oemler sample, and the Malmquist-like bias, are the reasons for these selection effects that mimic the trends usually interpreted as changes due to evolution.Comment: ApJ, in press, scheduled on May, 10 issue. 17 pages & 11 figure