Unions of three starshaped sets in R2

LetS be a compact, simply connected set inR ². If every boundary point ofS is clearly visible viaS from at least one of the three pointsa, b, c, thenS is a union of three starshaped sets whose kernels containa, b, c, respectively. The result fails when the number three is replaced by four.As a partial converse, ifS is a union of three starshaped sets whose kernels containa, b, c, respectively, then the set of points in the boundary ofS clearly visible from at least one ofa, b, orc is dense in the boundary ofS.Supported in part by NSF grant DMS-8705336.     

Preparation and characterization of dealuminated metakaolin and its use in the transformation of waste plastics to aromatic hydrocarbons

Acid activated metakaolins (AAMKs) have been prepared by calcination of the natural clay at 600 degrees C to provide a metakolin which was then leached at 80 degrees C for 3 h using 1M, 2M, 3M, and 6M HCl. These materials were characterized and their ability to transform the off gases from HDPE decomposition into useful aromatic species was evaluated. The amount of adsorbed water and the number of acid sites increased with the severity of acid treatment. Variable temperature DRIFTS spectroscopy of pyridine treated samples revealed that both Br?nsted and Lewis acid centers were present until 425 degrees C. Pyridine bonded to the Lewis acid centers was more thermally stable. The AAMKs were all selective to the production of toluene with respectable, but lesser, amounts of xylenes and trimethylbenzenes. This selectivity contrasts with that of acid leached and pillared smectites which are selective toward trimethylbenzene.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

An intersection property for starshaped sets in the plane

Let be a family of compact starshaped sets in the plane. If every three and every two members of have a union which is connected and simply connected, then {F:F in } is simply connected and nonempty. Of course, if every three and every two members of have a starshaped union, the same result holds.Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma.     

A Krasnosel'skii-type theorem for nonclosed sets in the plane

This work will be concerned with a Krasnosel'skii-type theorem for nonclosed bounded sets in R², and the following result will be obtained: Let S be a nonempty bounded set in R² whose complement consists of a finite number of locally compact components. Assume that every 4 or fewer points in S see a common point via S. Then for some point p in cl S, the set A {s s in S and (p,s] S} is nowhere dense in S. The number 4 is best possible.     

Dual Illumination Enhances Transformation of an Engineered Green-to-Red Photoconvertible Fluorescent Protein

The molecular mechanisms for the photoconversion of fluorescent proteins remain elusive owing to the challenges of monitoring chromophore structural dynamics during the light-induced processes. We implemented time-resolved electronic and stimulated Raman spectroscopies to reveal two hidden species of an engineered ancestral GFP-like protein LEA, involving semi-trapped protonated and trapped deprotonated chromophores en route to photoconversion in pH 7.9 buffer. A new dual-illumination approach was examined, using 400 and 505 nm light simultaneously to achieve faster conversion and higher color contrast. Substitution of UV irradiation with visible light benefits bioimaging, while the spectral benchmark of a trapped chromophore with characteristic ring twisting and bridge-H bending motions enables rational design of functional proteins. With the improved H-bonding network and structural motions, the photoexcited chromophore could increase the photoswitching-aided photoconversion while reducing trapped species.     

Monoidal Categories and Multiextensions

We associate to a group-like monoidal grupoid a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of . The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.