Edge singularities and structure of the 3-D Williams expansion

The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202]. To cite this article: T. Apel et al., C. R. Mecanique 336 (2008).     

Functional polymers. XVI. Synthesis and polymerization of 2(2-hydroxy-5-isopropenylphenyl)-2H-benzotriazole and a new synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole

2(2-Hydroxy-5-isopropenylphenyl)2H-benzotriazole was synthesized in 40% overall yield starting from o-nitroaniline. Diazotization in aqueous hydrochloric acid gave o-nitrophenyl diazonium chloride which was condensed with p-hydroxyacetophenone; the azo compound was reduced to 2(2-hydroxy-5-acetylphenyl) 2H-benzotriazole with zinc powder in sodium hydroxide solution and the 2-hydroxy group of the compound was acetylated. Treatment of the acetyl compound with methyl Grignard reagent resulted in the methylation of the 5-acetyl group to 2[2-acetoxy-5(2-hydroxy-2-propyl)phenyl]2H-benzotriazole which was then dehydrated with potassium hydrogen sulfate to the desired 2(2-hydroxy-5-isopropenylphenyl)2H-benzotriazole. This monomer did not homopolymerize, but was copolymerized readily with styrene, methyl methacrylate, and n-butyl acrylate with azobisisobutyronitrile as the initiator. 2(2-Acetoxy-5-acetylphenyl)2H-benzotriazole was also reduced with sodium borohydride to form 2[2-acetoxy-5-(1-hydroxyethyl)phenyl]2H-benzotriazole which was dehydrated and hydrolyzed to the known 2(2-hydroxy-5-vinylphenyl)-2H-benzotriazole. This route provides a novel and simpler synthesis of 2(2-hydroxy-5-vinylphenyl)2H-benzotriazole.     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

Sequent Systems for Negative Modalities

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.     

Realistic Experiments for Measuring the Wave Function of a Single Particle

We suggest scattering experiments which implement the concept of protective measurements allowing the measurement of the complete wave function even when only one quantum system (rather than an ensemble) is available. Such scattering experiments require massive, slow, projectiles with kinetic energies lower than the first excitation of the system in question. The results of such experiments can have a (probabilistic) distribution (as is the case when the Born approximation for the scattering is valid) or be deterministic (in a semiclassical limit).