Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P³${\mathbb{P}}^3$ and derived category tools. The method allows one to prove the existence of new examples of size 10×10$10\times 10$ and 14×14$14\times 14$ via instanton bundles of charge 2 and 4 respectively, and it provides an explanation for what used to be the only known example (Westwick 1996 [25]). We also give an algorithm to construct explicitly a matrix of size 14 of this type.     

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

In this work we propose an original implementation of a large family of lowest-order methods for diffusive problems including standard and hybrid finite volume methods, mimetic finite difference-type schemes, and cell centered Galerkin methods. The key idea is to regard the method at hand as a (Petrov–)Galerkin scheme based on possibly incomplete, broken affine spaces defined from a gradient reconstruction and a point value. The resulting unified framework serves as a basis for the development of a FreeFEM-like domain specific language targeted at defining discrete linear and bilinear forms. Both the back-end and the front-end of the language are extensively discussed, and several examples of applications are provided. The overhead of the language is evaluated with respect to a more traditional implementation. A benchmark including the comparison with more classical finite element methods on standard meshes is also proposed.     

Yttrium segregation and intergranular defects in alumina

Intergranular segregation of yttrium is investigated in a rhombohedral twin grain boundary of alumina. The deviation from the twin orientation is compensated by a periodic arrangement of intergranular dislocations. TEM tools (CTEM and HREM, EDXS, EFTEM and EELS/ELNES) have been used to characterize changes in chemical and electronic environments along this twin. Within the experimental limits, no Y is detected in the perfect twin parts. On the other hand, Y segregation occurs very locally in the dislocation cores on about four atomic planes perpendicularly to the GB, which in fact corresponds to the step height associated with the interfacial dislocations. By comparison with an undoped bicrystal, both the dislocation distribution and the Y segregation localization confirm the influence of Y on the dislocation mobility. Furthermore, in the Y-rich defects, high spatial resolution analysis of the energy loss near edge structures (ELNES) of the Al-L₂₃ absorption edge brings some enlightenments about the Al³⁺ cation environment, provided the effects of local radiation damage are fully considered and under control.     

Weil Representation and Metaplectic Groups over an Integral Domain

Given F a locally compact, nondiscrete, non-archimedean field of characteristic ≠ 2 and R an integral domain such that a non-trivial smooth character χ: F → R ^× exists, we construct the (reduced) metaplectic group attached to χ and R. We show that it is in the expected cases a double cover of the symplectic group over F. Finally, we define a faithful infinite dimensional R-representation of the metaplectic group analogue to the Weil representation in the complex case.     

Analysis of a discontinuous galerkin method for heterogeneous diffusion problems with low‐regularity solutions

We study the convergence of the Symmetric Weighted Interior Penalty discontinuous Galerkin method for heterogeneous diffusion problems with low‐regularity solutions only belonging to W^{2, p} with p ∈ (1,2]. In 2d, we infer an optimal algebraic convergence rate. In any space dimension d, we achieve the same result for p > 2 d/( d + 2). We also prove convergence without algebraic rates for exact solutions only belonging to the energy space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Bounded enumeration reducibility and its degree structure

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤_be, which is a natural extension of s-reducibility ≤_s. We show that ≤_s, ≤_be, and enumeration reducibility do not coincide on the ${\Pi^0_1}$ –sets, and the structure ${\boldsymbol{\mathcal{D}_{\rm be}}}$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of ${\boldsymbol{\mathcal{D}_{\rm be}}}$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987).     

Bent polyynes: ring geometry studied by Raman and IR spectroscopy

Infrared (IR) and Raman spectroscopy have been used to examine the vibrational characteristics of a series of three macrocyclic tetraynes in comparison with an acyclic analog. By changing the length of the alkyl tether of the macrocycles, varying degrees of bending of the tetrayne moiety can be achieved, and the joint use of IR and Raman spectroscopies provides an avenue to probe the impact of bending on the sp‐chain. The spectroscopic data show a general trend toward decreasing activation of Raman bands in the IR spectra, and vice versa, as bending of the polyyne chain is decreased. Density functional theory calculations provide a detailed rationalization of the experimental observations. Copyright © 2011 John Wiley & Sons, Ltd.     

Raman spectroscopy of polyconjugated molecules with electronic and mechanical confinement: the spectrum of Corallium rubrum

The chemical nature of the red pigment of Corallium rubrum (CR) has not yet been clearly identified. We have recorded the Raman spectra of CR, canthaxantin, and parrot feather, and we propose an interpretation with the help of quantum chemistry and of the effective conjugation coordinate theory, which accounts for the main Raman lines (ν₁, ν₂, ν₃) as originating from the polyene backbone. In this study, two additional lines in the 1000 cm⁻¹ range (ν₄ and ν₅) are considered as well as a series of overtones and combinations in the second order Raman spectrum (2000–3000 cm⁻¹). Density Functional Theory calculations predict that, moving with successive methylation from a simple unsubstituted polyene chain with 9 C═C bonds (as psyttacofulvins) to a 9, 9′, 13, 13′ tetramethylated chain (as carotenoids), the geometric structures of the molecules and their Raman spectra show changes that can be rationalized if the effective conjugation coordinate theory is extended to account for mechanical confinement. This turns out to be a new concept that helps the interpretation of the Raman spectra of partially methylated conjugated oligoenes. Evidence is found that the main component of the pigment of CR does not posess a fully demethylated polyene chain (psyttacofulfine) nor a tetramethylated chain (carotenoid), thus indicating that the polyene chain is partially methylated. Because we consider resonance Raman spectra, the target of this study is the structure of the polyene backbone; no direct spectroscopic information can be obtained in this way on the chemical nature of the groups attached to both chain ends. Copyright © 2012 John Wiley & Sons, Ltd.