Plain fundamentals of Fundamental Planes: Analytics and algorithms

The coefficients a and b of the Fundamental Plane relation R ~ Sigma^a I^b depend on whether one minimizes the scatter in the R direction or orthogonal to the Plane. We provide explicit expressions for a and b (and confidence limits) in terms of the covariances between logR, logSigma and logI. Our analysis is more generally applicable to any other correlations between three variables: e.g., the color-magnitude-Sigma relation, the L-Sigma-Mbh relation, or the relation between the X-ray luminosity, Sunyaev-Zeldovich decrement and optical richness of a cluster, so we provide IDL code which implements these ideas, and we show how our analysis generalizes further to correlations between more than three variables. We show how to account for correlated errors and selection effects, and quantify the difference between the direct, inverse and orthogonal fit coefficients. We show that the three vectors associated with the Fundamental Plane can all be written as simple combinations of a and b because the distribution of I is much broader than that of Sigma, and Sigma and I are only weakly correlated. Why this should be so for galaxies is a fundamental open question about the physics of early-type galaxy formation. If luminosity evolution is differential, and Rs and Sigmas do not evolve, then this is just an accident: Sigma and I must have been correlated in the past. On the other hand, if the (lack of) correlation is similar to that at the present time, then differential luminosity evolution must have been accompanied by structural evolution. A model in which the luminosities of low-L galaxies evolve more rapidly than do those of higher-L galaxies is able to produce the observed decrease in a (by a factor of 2 at z~1) while having b decrease by only about 20 percent. In such a model, the Mdyn/L ratio is a steeper function of Mdyn at higher z.Comment: 11 pages, 1 figure, associated IDL code, MNRAS accepte

Counting coloured planar maps: differential equations

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial polynomial differential equation withrespect to the edge variable. We give explicitly a differential systemthat characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. Instatistical physics terms, we solvethe q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating functionwas proved to be algebraic for certain values of q,including q=1, 2 and 3. It isknown to be transcendental in general. In contrast, our differential system holds for an indeterminate q.For certain special cases of combinatorial interest (four colours; properq-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.Comment: 43 p

Bijective counting of Kreweras walks and loopless triangulations

We consider lattice walks in the plane starting at the origin, remaining in the first quadrant and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations with a distinguished spanning tree. We obtain simultaneously a bijective way of counting loopless triangulations.Comment: 25 page

A simple model of trees for unicellular maps

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the "recursive part" of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and the Goupil-Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a refinement by degree of the Harer-Zagier formula and more details in some proof

Curvature in the color-magnitude relation but not in color-sigma: Major dry mergers at M* > 2 x 10^11 Msun?

The color-magnitude relation of early-type galaxies differs slightly but significantly from a pure power-law, curving downwards at low and upwards at large luminosities (Mr>-20.5 and Mr<-22.5). This remains true of the color-size relation, and is even more apparent with stellar mass (M* < 3x10^10 Msun and M* > 2x10^11 Msun). The upwards curvature at the massive end does not appear to be due to stellar population effects. In contrast, the color-sigma relation is well-described by a single power law. Since major dry mergers change neither the colors nor sigma, but they do change masses and sizes, the clear features observed in the scaling relations with M*, but not with sigma > 150 km/s, suggest that M* > 2x10^11 Msun is the scale above which major dry mergers dominate the assembly history. We discuss three models of the merger histories since z ~ 1 which are compatible with our measurements. In all three models, dry mergers are responsible for the flattening of the color-M* relation at M* > 3x10^10 Msun - wet mergers only matter at smaller masses. At M* > 2 x 10^11 Msun, the merger histories in one model are dominated by major rather than minor dry mergers, as suggested by the axis ratio and color gradient trends. In another, although both major and minor mergers occur at the high mass end, the minor mergers contribute primarily to the formation of the ICL, rather than to the mass growth of the central massive galaxy. A final model assumes that the reddest objects were assembled by a mix of major and minor dry mergers.Comment: 22 pages, 22 figures and 3 tables. Accepted for publication in MNRA

Physical Origin of the One-Quarter Exact Exchange in Density Functional Theory

Exchange interactions are a manifestation of the quantum mechanical nature of the electrons and play a key role in predicting the properties of materials from first principles. In density functional theory (DFT), a widely used approximation to the exchange energy combines fractions of density-based and Hartree-Fock (exact) exchange. This so-called hybrid DFT scheme is accurate in many materials, for reasons that are not fully understood. Here we show that a 1/4 fraction of exact exchange plus a 3/4 fraction of density-based exchange is compatible with a correct quantum mechanical treatment of the exchange energy of an electron pair in the unpolarized electron gas. We also show that the 1/4 exact-exchange fraction mimics a correlation interaction between doubly-excited electronic configurations. The relation between our results and trends observed in hybrid DFT calculations is discussed, along with other implications