Detection of Hopf points by counting sectors in the complex plane

Summary. Let ( real) be a family of real by matrices. A value of is called a Hopf value if has a conjugate pair of purely imaginary eigenvalues , . We describe a technique for detecting Hopf values based on the evolution of the Schur complement of in a bordered extension of where varies along the positive imaginary axis of the complex plane. We compare the efficiency of this method with more obvious methods such as the use of the QR algorithm and of the determinant function of as well as with recent work on the Cayley transform. In particular, we show the advantages of the Schur complement method in the case of large sparse matrices arising in dynamical problems by discretizing boundary value problems. The Hopf values of the Jacobian matrices are important in this setting because they are related to the Hopf bifurcation phenomenon where steady state solutions bifurcate into periodic solutions. Received September 15, 1994 / Revised version received July 7, 1995     

Continuous bordering of matrices and continuous matrix decompositions

Summary. Let be the set of all real -matrices of rank . We prove that for there are no continuous vector fields such that the bordered matrix is regular for all . This result has some relevance for the numerical analysis of steady state bifurcation. As a by-product we show that there is no nonvanishing continuous vector field with for all , where is the set of all matrices of rank deficiency one. This implies that there is no singular value decomposition of depending continuously on in any matrix set which contains . As another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable. Received October 6, 1993 / Revised version received July 18, 1994     

Closed bosonic strings in background gauge fields

The coupling to abelian background gauge fields of closed bosonic strings compactified on a torus is considered. There exist specific field configurations for which the theory cannot be defined consistently, in spite of the fact that the field equations of motion are then solved exactly. This critical behaviour is related to a breakdown of local and global world-sheet symmetries, as well as of spacetime unitarity. A physical interpretation of this phenomenon is proposed. The issue of spacetime duality is also discussed. This symmetry is shown to be partially broken for generic background field configurations.     

Towards supersymmetric subconstituent models

't Hooft's consistency equations for subconstituent models are generalized to supersymmetric theories and a few solutions are outlined.     

Alcoylation des enolates de l'acetylacetate d'ethyle dans le dimethoxyethane. Reactivite des formes associee et dissociee

The alkylation of alkali metal and N⁺Bu₄ ethylacetoacetate énolates by means of ethyl iodide, bromide and tosylate has been studied in diméthoxyéthane, in order to measure the reactivity of the associated forms. C/O alkylation ratios, structures of O-alkylated products and rates of alkylation, have been determined. With ethyl tosylate, the amount of O-alkylated products does not increase in the generally observed order Li⁺ < Na⁺ < K⁺ < Cs⁺ but in the order Na⁺ < K⁺ < Li⁺ ≈ Cs⁺. Some aspects of the alkali ethylacetoacetate énolates reactivity in weakly polar medium such as DME are discussed.     

Multi-vector Spherical Monogenics, Spherical Means and Distributions in Clifford Analysis

New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.     

Representations of the rotation reflection symmetry group of the four-dimensional cubic lattice

The structure and representations of the rotation reflection symmetry group of the four-dimensional cubic lattice are described. Their connections with the representations of the three-dimensional lattice rotation reflection group, and with the representations of the continuous O(3) and O(4) groups are given.     

Selective C(sp3)−H Aerobic Oxidation Enabled by Decatungstate Photocatalysis in Flow

A mild and selective C(sp³)?H aerobic oxidation enabled by decatungstate photocatalysis has been developed. The reaction can be significantly improved in a microflow reactor enabling the safe use of oxygen and enhanced irradiation of the reaction mixture. Our method allows for the oxidation of both activated and unactivated C?H bonds (30 examples). The ability to selectively oxidize natural scaffolds, such as (?)‐ambroxide, pregnenolone acetate, (+)‐sclareolide, and artemisinin, exemplifies the utility of this new method.