Écriture bicamérale | EPFL Graph Search

Une écriture bicamérale est une écriture comprenant des lettres minuscules et des lettres capitales. Plus précisément, elle oppose deux œils de format (ou « casse ») — et parfois de tracé — différents pour chaque caractère. Par opposition, une écriture dans laquelle il n’existe pas une telle opposition est dite monocamérale ou unicamérale. On appelle les lettres des minuscules, tandis que les lettres d’un format plus grand, utilisées dans certains cas régis par la grammaire et l’orthotypographie, sont les majuscules (à ne pas confondre avec capitales). On compte, entre autres, au nombre de ces écritures :

l’alphabet latin ;
l’alphabet grec ;
l’alphabet cyrillique ;
l’alphabet arménien ;
l’alphabet copte.

Voici, à titre d'illustration, les sept premières lettres dans les deux casses (d’abord capitales puis minuscules) pour les cinq alphabets cités : Kana japonais Dans l’écriture japonaise, trois types de caractères sont utilisés : les sinogrammes, app

MATHICSE Technical Report: Block Krylov subspace methods for functions of matrices II: Modified block FOM

Kathryn Dianne Lund

We analyze an expansion of the generalized block Krylov subspace framework of [Electron.\ Trans.\ Numer.\ Anal., 47 (2017), pp. 100-126]. This expansion allows the use of low-rank modifications of the matrix projected onto the block Krylov subspace and contains, as special cases, the block GMRES method and the new block Radau-Arnoldi method. Within this general setting, we present results that extend the interpolation property from the non-block case to a matrix polynomial interpolation property for the block case, and we relate the eigenvalues of the projected matrix to the latent roots of these matrix polynomials. Some convergence results for these modified block FOM methods for solving linear system are presented. We then show how {\em cospatial} residuals can be preserved in the case of families of shifted linear block systems. This result is used to derive computationally practical restarted algorithms for block Krylov approximations that compute the action of a matrix function on a set of several vectors simultaneously. We prove some convergence results and present numerical results showing that two modifications of FOM, the block harmonic and the block Radau-Arnoldi methods for matrix functions, can significantly improve the convergence behavior.

MATHICSE2019

Looping probabilities of elastic chains: A path integral approach

Ludovica Cotta-Ramusino, John Maddocks

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e. g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

2010

Can a Helmholtz solver run on a cluster?

Marc Anthony David Habisreutinger, Vincent Keller, Emmanuel Leriche

Our vectorized Helmholtz solver runs at 85% efficiency on a NEC SX-5. The most time-consuming parts have been ported on SMP, NUMA, and cluster architectures. It is shown that an OpenMP version can deliver similar performance when running it on a 16 processor SGI Altix. A partial parallelisation using MPI is made to validate the Gamma model to predict application behaviours on parallel machines. This model is then applied to simulate the behaviour of a hypothetical full MPI version on different distributed memory machines. It is found that only the Cray XT3 with its very fast internode communication network will be able to deliver the performance of a NEC SX-8 with the advantage that bigger case could be handled

IEEE2006

Looping probabilities of elastic chains: A path integral approach

Ludovica Cotta-Ramusino, John Maddocks

2010