Comment on " Gain coefficient method for amplified spontaneous emission in thin waveguided film of a conjugated polymer " [APL 93, 163307 (2008)]

Comment on " Gain coefficient method for amplified spontaneous emission in thin waveguided film of a conjugated polymer " [APL 93, 163307 (2008)

Saddle towers in Heisenberg space

We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to ensure the convergence of a family of bounded minimal disks. Such a barrier is actually a periodic deformation of a minimal plane with prescribed asymptotic behavior. A consequence of the barrier construction is that the number of disjoint minimal graphs suppoerted on domains is not bounded in Heisenberg space.Comment: 20 pages. V2: addition of a result. V3: minor correction

Anisotropic oracle inequalities in noisy quantization

The effect of errors in variables in quantization is investigated. We prove general exact and non-exact oracle inequalities with fast rates for an empirical minimization based on a noisy sample $Z_i=X_i+\epsilon_i,i=1,\ldots,n$, where $X_i$ are i.i.d. with density $f$ and $\epsilon_i$ are i.i.d. with density $\eta$. These rates depend on the geometry of the density $f$ and the asymptotic behaviour of the characteristic function of $\eta$. This general study can be applied to the problem of $k$-means clustering with noisy data. For this purpose, we introduce a deconvolution $k$-means stochastic minimization which reaches fast rates of convergence under standard Pollard's regularity assumptions.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1205.141

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