Unconditional Decompositions in Subspaces of l<Subscript>2</Subscript>(X)

If a Banach space X is not isomorphic to a Hilbert space then l₂X contains a subspace which has a UFDD, but does not admit a UFDD with a uniform bound for the dimensions of the decomposition.     

Fiber fracture and strength degradation in unidirectional graphite/epoxy composite materials

An experimental study was conducted to determine the influence of load factor on fiber fracture development and residual strength of fatigue loaded unidirectional graphite/epoxy composite laminated. 8-phy composite laminates with a layer of release cloth imbedded at the middle ply were fatigued at different load levels and were examined for fiber fracture and residual strength at several stages of life based on the average number of cycles to failure (according to S-N data). From the experimental results, it is evident that the number of fiber fractures is nearly constant after the first few percent of the life. It is also suggested that the load level is much more important than the number of cycles of loading in the determination of the state of fiber fracture. This behavior was interrupted at high load levels (S60% S_u where the final fracture was highly affected by the longitudinal matrix splittings. Residual strength is found to be independent of the global fiber fracture density, and to be controlled by local behavior such as matrix cracking, local clustering of fiber fractures, and other local stress concentrations.     

Around a theorem of F. Dyson and A. Lenard: Energy equilibria for point charge distributions in classical electrostatics

We discuss several results in electrostatics: Onsager’s inequality, an extension of Earnshaw’s theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references.     

The 1/N Expansion of Colored Tensor Models

In this paper, we perform the 1/N expansion of the colored three-dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S ³ contribute to the leading order in the large N limit.     

Catalytic and stoichiometric reactivity of β-silylamido agostic complex of Mo: intermediacy of a silanimine complex and applications to multicomponent coupling

The reaction of complex (ArN═)(2)Mo(PMe(3))(3) (Ar = 2,6-diisopropylphenyl) with PhSiH(3) gives the β-agostic NSi-H···M silyamido complex (ArN═)Mo(SiH(2)Ph)(PMe(3))(η(3)-ArN-SiHPh-H) (3) as the first product. 3 decomposes in the mother liquor to a mixture of hydride compounds, including complex {η(3)-SiH(Ph)-N(Ar)-SiHPh-H···}MoH(3)(PMe(3))(3) characterized by NMR. Compound 3 was obtained on preparative scale by reacting (ArN═)(2)Mo(PMe(3))(3) with 2 equiv of PhSiH(3) under N(2) purging and characterized by multinuclear NMR, IR, and X-ray diffraction. Analogous reaction of (Ar'N═)(2)Mo(PMe(3))(3) (Ar' = 2,6-dimethylphenyl) with PhSiH(3) affords the nonagostic silylamido derivative (Ar'N═)Mo(SiH(2)Ph)(PMe(3))(2)(NAr'{SiH(2)Ph}) (5) as the first product. 5 decomposes in the mother liquor to a mixture of {η(3)-PhHSi-N(Ar')-SiHPh-H···}MoH(3)(PMe(3))(3), (Ar'N═)Mo(H)(2)(PMe(3))(2)(η(2)-Ar'N═SiHPh), and other hydride species. Catalytic and stoichiometric reactivity of 3 was studied. Complex 3 undergoes exchange with its minor diastereomer 3' by an agostic bond-opening/closing mechanism. It also exchanges the classical silyl group with free silane by an associative mechanism which most likely includes dissociation of the Si-H agostic bond followed by the rate-determining silane σ-bond metathesis. However, labeling experiments suggest the possibility of an alternative (minor) pathway in this exchange including a silanimine intermediate. 3 was found to catalyze dehydrogenative coupling of silane, hydrosilylation of carbonyls and nitriles, and dehydrogenative silylation of alcohols and amines. Stoichiometric reactions of 3 with nitriles proceed via intermediate formation of η(2)-adducts (ArN═)Mo(PMe(3))(η(2)-ArN═SiHPh)(η(2)-N≡CR), followed by an unusual Si-N coupling to give (ArN═)Mo(PMe(3))(κ(2)-NAr-SiHPh-C(R)═N-). Reactions of 3 with carbonyls lead to η(2)-carbonyl adducts (ArN═)(2)Mo(O═CRR')(PMe(3)) which were independently prepared by reactions of (ArN═)(2)Mo(PMe(3))(3) with the corresponding carbonyl O═CRR'. In the case of reaction with benzaldehyde, the silanimine adduct (ArN═)Mo(PMe(3))(η(2)-ArN═SiHPh)(η(2)-O═CHPh) was observed by NMR. Reactions of complex 3 with olefins lead to products of Si(ag)-C coupling, (ArN═)Mo(Et)(PMe(3))(η(3)-NAr-SiHPh-CH═CH(2)) (17) and (ArN═)Mo(H)(PMe(3))(η(3)-NAr-SiHPh-CH═CHPh), for ethylene and styrene, respectively. The hydride complex (ArN═)Mo(H)(PMe(3))(η(3)-NAr-SiHPh-CH═CH(2)) was obtained from 17 by hydrogenation and reaction with PhSiH(3). Mechanistic studies of the latter process revealed an unusual dependence of the rate constant on phosphine concentration, which was explained by competition of two reaction pathways. Reaction of 17 with PhSiH(3) in the presence of BPh(3) leads to agostic complex (ArN═)Mo(SiH(2)Ph)(η(3)-NAr-Si(Et)Ph-H)(η(2)-CH(2)═CH(2)) (24) having the Et substituent at the agostic silicon. Mechanistic studies show that the Et group stems from hydrogenation of the vinyl substituent by silane. Reaction of 24 with PMe(3) gives the agostic complex (ArN═)Mo(SiH(2)Ph)(PMe(3))(η(3)-NAr-Si(Et)Ph-H), which slowly reacts with PhSiH(3) to furnish silylamide 3 and the hydrosilylation product PhEtSiH(2). A mechanism involving silane attack on the imido ligand was proposed to explain this transformation.     

On the relation between the A-polynomial and the Jones polynomial

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.