On the existence of orthonormal geodesic bases for Lie algebras

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product

BiplotGUI: Interactive Biplots in R

Biplots simultaneously provide information on both the samples and the variables of a data matrix in two- or three-dimensional representations. The BiplotGUI package provides a graphical user interface for the construction of, interaction with, and manipulation of biplots in R. The samples are represented as points, with coordinates determined either by the choice of biplot, principal coordinate analysis or multidimensional scaling. Various transformations and dissimilarity metrics are available. Information on the original variables is incorporated by linear or non-linear calibrated axes. Goodness-of-fit measures are provided. Additional descriptors can be superimposed, including convex hulls, alpha-bags, point densities and classification regions. Amongst the interactive features are dynamic variable value prediction, zooming and point and axis drag-and-drop. Output can easily be exported to the R workspace for further manipulation. Three-dimensional biplots are incorporated via the rgl package. The user requires almost no knowledge of R syntax.

Generalized Spectral Decomposition for Stochastic Non Linear Problems

International audienceWe present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e. below the critical temperature $T_c$, the disorder and thermally averaged correlation function $B(r)$ of the phase field $\theta(x)$, $B(r) = \bar{}$ behaves, for $r \gg a$, as $B(r) \simeq A(\tau) \ln^2 (r/a)$ where $r = |r|$ and $a$ is a microscopic length scale. We derive the RG equations up to cubic order in $\tau = (T_c-T)/T_c$ and predict the universal amplitude ${A}(\tau) = 2\tau^2-2\tau^3 + {\cal O}(\tau^4)$. The universality of $A(\tau)$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute ${A}(\tau)$ numerically and obtain a remarkable agreement with our analytical prediction, up to $\tau \approx 0.5$.Comment: 5 pages, 3 figure