Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

Suppose O is an alternative division algebra that is quadratic over some subfield K of its center Z(O). Then with (O, K), there is associated a dual polar space. We provide an explicit representation of this dual polar space into a (6n + 7)-dimensional projective space over K, where n D dim(K)(O). We prove that this embedding is the universal one, provided vertical bar K vertical bar > 2. When O is not an inseparable field extension of K, we show that this universal embedding is the unique polarized one. When O is an inseparable field extension of K, then we determine the minimal full polarized embedding, and show that all homogeneous embeddings are either universal or minimal. We also provide explicit generators of the corresponding projective representations of the little projective group associated with the ( dual) polar space

Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.Comment: 20 Pages, 3 Figure

Fiber waveguide sensors for intelligent materials

This report, an addendum to the six month report submitted to NASA Langley Research Center in December 1987, covers research performed by the Fiber and Electro-Optics Research Center (FEORC) at Virginia Tech for the NASA Langley Research Center, Grant NAG1-780, for the period from December 1987 to June 1988. This final report discusses the research performed in the following four areas as described in the proposal: Fabrication of Sensor Fibers Optimized for Embedding in Advanced Composites; Fabrication of Sensor Fiber with In-Line Splices and Evaluation via OTR methods; Modal Domain Optical Fiber Sensor Analysis; and Acoustic Fiber Waveguide Implementation

The RARE model: a generalized approach to random relaxation processes in disordered systems

This paper introduces and analyses a general statistical model, termed the RARE model, of random relaxation processes in disordered systems. The model considers excitations, that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance. The framework of the RARE model is robust, and a detailed stochastic analysis of the random relaxation processes is established. Analytic results regarding the duration and the range of the random relaxation processes, as well as the model's thermodynamic limit, are obtained in closed form. In particular, the case of power-law inputs, which turn out to yield stretched exponential relaxation patterns and asymptotically Paretian relaxation ranges, is addressed in detail.Comment: 10 pages, REVTeX

The ideal center of the dual of a Banach lattice

Let $E$ be a Banach lattice. Its ideal center $Z(E)$ is embedded naturally in the ideal center $Z(E')$ of its dual. The embedding may be extended to a contractive algebra and lattice homomorphism of $Z(E)"$ into $Z(E')$. We show that the extension is onto $Z(E')$ if and only if $E$ has a topologically full center. (That is, for each $x\in E$, the closure of $Z(E)x$ is the closed ideal generated by $x$.) The result can be generalized to the ideal center of the order dual of an Archimedean Riesz space and in a modified form to the orthomorphisms on the order dual of an Archimedean Riesz space