Fabrication of grating structures with variable period

Photoresist gratings with variable period have been fabricated by recording the interference pattern of a collimated laser beam with a converging one. The period variation as a function of position, as well as the wavelength dependent focal properties of the gratings, has also been analyzed.     

From a theoretical concept to biochemical reactions: strain-induced bond localization (SIBL) in oxidation of vitamin E

The regioselectivity of the oxidation of alpha-tocopherol (the main component of vitamin E) to an ortho-quinone methide (oQM) has been explained in the literature mostly by the ill-defined term "Mills-Nixon effect". In this paper we describe the preparation of eleven alpha-tocopherol derivatives, different from each other by the sum of annulation angles, but keeping the electronic factors unchanged. These compounds underwent Ag(2)O oxidation, forming two isomeric oQMs that were trapped by vinylmethyl ether. It was found that the isomeric product ratio changes smoothly as a function of the annulation angles, not abruptly from one regioisomer to the other on going from five- to six- and seven-membered rings, as predicted by the Mills-Nixon effect. The relative amounts of the products were determined at four different temperatures, and assuming that the product ratio represents the relative rates ratio, the relative enthalpy of activations could be obtained. Theoretically (at B3LYP/6-31G* theoretical level) four different intermediates were considered. Each of these underwent angular angles deformations to model the two regioisomers. At each deformation angle the energy difference between the two intermediates models was correlated to the experimental data for each of the four intermediates. It was found that the angle-deformed lithium (6-methyl-2-benzylium)phenolate correlated best (R>0.994) to the experimental data. This study confirms that the regioselectivity of the two isomeric oQMs in the oxidation of alpha-tocopherol and related compounds is simply a function of angular strain, best explained by the SIBL (strain-induced bond localization) model. In addition, this study provides evidence that the highest energy intermediate in the oxidation of vitamin E is a phenolate-benzyl cation.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Angular gaps in radial diffusion-limited aggregation: two fractal dimensions and nontransient deviations from linear self-similarity

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial diffusion-limited aggregation is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D = 1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than M = 10(6) particles. Intermediate size gaps exhibit an effective dimension around 1.67, even for M--> infinity. They dominate the distribution for clusters with M<10(6). The largest gap approaches a finite limit extremely slowly, with a correction of order M(-0.17).     

Nonlinear dispersion in a coupled-resonator optical waveguide

The propagation of an optical pulse in a coupled-resonator optical waveguide may be calculated nonperturbatively to all orders of dispersion, in the conventional tight-binding approximation, even though the dispersion relationship is nonlinear. Working in this framework, we discuss limits of the physical parameters and approximations to the exact formulation that highlight the conditions under which pulse distortion can be minimized. The results are fundamental to the design of coupled-resonator optical waveguides and are also relevant to other applications of the tight-binding method.     

Failure of Brown representability in derived categories

Let be a triangulated category with coproducts, the full subcategory of compact objects in . If is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors are the restrictions of representable functors on , and all natural transformations are the restrictions of morphisms in . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories of rings, and homological functors which are not restrictions of representables.     

Additivity for derivator K-theory

We prove the additivity theorem for the K-theory of triangulated derivators. This solves one of the conjectures made by Maltsiniotis in [G. Maltsiniotis, La K-théorie d'un dérivateur triangulé, in: Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack, Amnon Neeman (Eds.), Categories in Algebra, Geometry and Physics, Conference and Workshop in honor of Ross Street's 60th Birthday, in: Contemp. Math., vol. 431, Amer. Math. Soc., 2007, pp. 341-368]. We also review some basic definitions and results in the theory of derivators in the sense of Grothendieck.