Block-Projections Algorithms with Blocks Containing Mutually Orthogonal Rows and Columns

In this paper we present an algorithm which, for a given (sparse) matrix, constructs a partition of its set of row-indices, such that each subset of this partition (except the last one obtained) contains indices which correspond to mutually orthogonal rows. We then use such decompositions in some classes of block-projections methods, previously extended by the author to general inconsistent linear least-squares problems. Numerical experiments on an inconsistent and rank-deficient least-squares model problem are described in the last section of the paper.     

Optical resonator for high-power transverse flow CO2 lasers

A new design of the U-type resonator is described. In this way, a laser beam with symmetrical intensity profile (regarding to a symmetry plane) can be extracted from an active medium that exhibits gain asymmetry along one of the transverse directions. The whole area of the active medium cross-section can be used, and consequently the laser efficiency will be increased. This resonator structure was applied for efficiency power extraction (as a low order TEM modes laser beam) from a DC excited transverse flow CO₂ laser with cylindrical geometry. Although the cross-section area of the discharge was entirely used (including the cathode fall region), a symmetrical intensity profile of the laser beam (regarding to the two orthogonal symmetry planes) was obtained in the near field as well as in the far field; the gain asymmetry along the flow direction was compensated by the gas circulation fluidodynamical circuit with two counterflowing discharge channels. A double-U optical resonator was introduced in order to provide a laser beam with axial symmetry.For the practical construction of these two types of optical resonators we have developed two new types of 90° deflection elements: the first one, which does not reverse the image (and which has the properties of the pentaprism), and the second one, which rotates the image with 90° angle. Both elements exhibit good focusability if they are equipped with two concave mirrors.     

Strong approximation of quadrics and representations of quadratic forms

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V. Suppose that M is a lattice on V, and that N is a lattice on U which is represented by M locally everywhere. The main result of this paper is a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.     

Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve

Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B. Oblatum 17-XII-1996 & 26 VI-1997     

Regularity and convergence of stochastic convolutions in duals of nuclear Fréchet spaces

Let Φ′ be the strong dual of a nuclear Fréchet space Φ. In this paper we present regularity properties, weak convergence, and convergence in probability and in mean square of Φ′-valued stochastic evolution equations and convolutions with respect to Φ′-valued cadlag martingales.     

Study of collision-induced decomposition of phenothiazine by ion trap mass spectrometry

The collision-induced decomposition pathways of phenothiazine were studied in detail by ion trap mass spectrometry. With this approach, different amounts of energy can be deposited into the fragment ions, thus favouring particular structures. Different relative abundances in the fragmentation patterns of selected ions are observed, depending on whether they are produced by electron impact or multiple mass spectrometry [(MS)ⁿ] methods.     

Onsager's conjecture on the energy conservation for solutions of Euler's equation

We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.     

Graphes et ordonnés démontables, propriété de la clique fixe

This work extends to dismantable graphs many properties of dismantable posets dealing with products, exponentiation, existence of paths in a graph of all homomorphisms, fixed clique property, etc. We show that a poset is dismantlable in the sense of Rival if and only if its comparability graph is dismantlable, thus proving that, for posets, dismantlability is a comparability invariant. We establish the analogues for graphs of results of Duffus, Poguntke and Rival about fixed point sets and cores in posets.