Submanifolds of the Cayley projective plane with bounded second fundamental form

We consider totally complex submanifolds of the Cayley projective plane with estimates on the length squared of the second fundamental form. We determine those bounds for which the second fundamental form is parallel and for which the submanifold is totally geodesic. The case of totally real submanifolds is also included.     

Geometric Implications of Linearizability

Linearizability for hyperbolic fixed points is shown to be equivalent to the existence of certain invariant foliations. The analogue for partial linearizability of nonhyperbolic fixed points is considered.     

Flat affine spheres in R 3

Nondegenerate affine surfaces in R ³ which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x ²+y ²)z=1.Partially supported by NSF Grant DMS 8802664.Partially supported by NSERC Operating Grant A2501.     

Crystal Structure of 1-nitro-2-phenylcycloprop-2-enecarbonitrile

The synthesis and x-ray crystal structure of 1 -nitro-2-phenylcycloprop-2-enecarbonitrile [1] is reported. The C-NO₂ bond length in1 is similar to but slightly shorter than that reported for 1, 2-diphenyl-3-nitrocyclopropene (2). Unlike2, 1 shows no tendency to ionize to the corresponding cyclopropenium ion in solution. This result is further supported by ab initio calculations.     

Structure and shear orientation of a side chain liquid crystal polymer studied by small angle X-ray scattering

We have studied the structure and shear flow behaviour of a side chain liquid crystal polymer, namely P₁₂, in the smectic (SmA_d) phase. Simultaneous oscillatory shear and small angle X-ray scattering experiments showed that, as expected, the shearing of P₁₂ at high frequency and large strain amplitudes leads to the alignment of the smectic layer normals parallel to the neutral direction. It was also found that, on erasing the memory of the prior shear alignment through melting, P₁₂ can unexpectedly orient with the smectic layers perpendicular to the shear flow direction (b-configuration). We speculate that the b-configuration is adopted due to a counter-balance between the rather long spacer in P₁₂ which decouples mesogens from the backbone, and the oscillatory character of the shear applied to P₁₂.     

Progress in liquid crystalline dispersions: Cubosomes

Dispersed particles of bicontinuous cubic liquid crystalline phase, cubosomes, are self-assembled nanostructured particles that can be formed in aqueous lipid and surfactant systems. Contributions to cubosome research have come from the fields of biology, material science, medicine, and mathematics and much is known about their formation and properties. At the center of much of the discovery and innovation is the technique of cryo-transmission electron microscopy. Most of the research into cubosomes is motivated by potential applications in drug delivery and material synthesis although no commercialized product based on cubosomes is known. Recent advances in understanding and use of cubosomes are discussed in the context of some of the more promising application areas and the opportunities for microscopy techniques to make unique contributions to these areas.