A Perron theorem for matrices with negative entries and applications to Coxeter groups

Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a matrix $A$ is a simple root of the characteristic polynomial and is strictly greater than the modulus of any other root, then $A$ is conjugate to a matrix $Z$ some power of which is positive. In this article, we provide an explicit conjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple and dominant eigenvalue of $A$ if and only if $Z$ is eventually positive. For $n\times n$ real matrices with each row-sum equal to $1$, this criterion can be declined into checking that each entry of some power is strictly larger than the average of the entries of the same column minus $\frac{1}{n}$. We apply the criterion to elements of irreducible infinite nonaffine Coxeter groups to provide evidences for the dominance of the spectral radius, which is still unknown.Comment: 14 page

33-dimensional Continued Fraction Algorithms Cheat Sheets

Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone $\mathbb{R}^d_+$ for $d=3$. We include well-known and old ones (Poincar\'e, Brun, Selmer, Fully Subtractive) and new ones (Arnoux-Rauzy-Poincar\'e, Reverse, Cassaigne). For each algorithm, one page (called cheat sheet) gathers a handful of informations most of them generated with the open source software Sage with the optional Sage package \texttt{slabbe-0.2.spkg}. The information includes the $n$-cylinders, density function of an absolutely continuous invariant measure, domain of the natural extension, lyapunov exponents as well as data regarding combinatorics on words, symbolic dynamics and digital geometry, that is, associated substitutions, generated $S$-adic systems, factor complexity, discrepancy, dual substitutions and generation of digital planes. The document ends with a table of comparison of Lyapunov exponents and gives the code allowing to reproduce any of the results or figures appearing in these cheat sheets.Comment: 9 pages, 66 figures, landscape orientatio

A note on matrices mapping a positive vector onto its element-wise inverse

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.Comment: 7 pages, 2 figure

Evolution of Galaxy Stellar Mass Functions, Mass Densities, and Mass to Light Ratios from z 7 to z 4

We derive stellar masses from SED fitting to rest-frame optical and UV fluxes for 401 star-forming galaxies at z 4, 5, and 6 from Hubble-WFC3/IR observations of the ERS combined with the deep GOODS-S Spitzer/IRAC data (and include a previously-published z 7 sample). A mass-luminosity relation with strongly luminosity-dependent M/Luv ratios is found for the largest sample (299 galaxies) at z 4. The relation M \propto L_{UV,1500}^(1.7+/-0.2) has a well-determined intrinsic sample variance of 0.5 dex. This relation is also consistent with the more limited samples at z 5-7. This z 4 mass-luminosity relation, and the well-established faint UV luminosity functions at z 4-7, are used to derive galaxy mass functions (MF) to masses M~10^8 at z 4-7. A bootstap approach is used to derive the MFs to account for the large scatter in the M--Luv relation and the luminosity function uncertainties, along with an analytical crosscheck. The MFs are also corrected for the effects of incompleteness. The incompleteness-corrected MFs are steeper than previously found, with slopes \alpha_M-1.4 to -1.6 at low masses. These slopes are, however, still substantially flatter than the MFs obtained from recent hydrodynamical simulations. We use these MFs to estimate the stellar mass density (SMD) of the universe to a fixed M_{UV,AB}<-18 as a function of redshift and find a SMD growth \propto(1+z)^{-3.4 +/-0.8} from z 7 to z 4. We also derive the SMD from the completeness-corrected MFs to a mass limit M~10^{8} Msun. Such completeness-corrected MFs and the derived SMDs will be particularly important for model comparisons as future MFs reach to lower masses.Comment: 7 pages, 4 figures, 1 table. Submitted to ApJL, version after comments from refere