Remarks on infinitesimally desarguesian families of curves

Infinitesimally Desarguesian two-parameter families of curves in the plane which are in a sense close to the family of straight lines are discussed. Their properties, examples, and multidimensional generalizations are considered.     

The horospherical duality

We discuss the horospherical duality as a geometrical background of harmonic analysis on semisimple symmetric spaces.     

Energy landscapes of dynamic ensembles of rolling triplet repeat bulge loops: implications for DNA expansion associated with disease states

DNA repeat domains can form ensembles of canonical and noncanonical states, including stable and metastable DNA secondary structures. Such sequence-induced structural diversity creates complex conformational landscapes for DNA processing pathways, including those triplet expansion events that accompany replication, recombination, and/or repair. Here we demonstrate further levels of conformational complexity within repeat domains. Specifically, we show that bulge loop structures within an extended repeat domain can form dynamic ensembles containing a distribution of loop positions, thereby yielding families of positional loop isomers, which we designate as "rollamers". Our fluorescence, absorbance, and calorimetric data are consistent with loop migration/translocation between sites within the repeat domain ("rollamerization"). We demonstrate that such "rollameric" migration of bulge loops within repeat sequences can invade and disrupt previously formed base-paired domains via an isoenthalpic, entropy-driven process. We further demonstrate that destabilizing abasic lesions alter the loop distributions so as to favor "rollamers" with the lesion positioned at the duplex/loop junction, sites where the flexibility of the abasic "universal hinge" relaxes unfavorable interactions and/or facilitates topological accommodation. Another strategic siting of an abasic site induces directed loop migration toward denaturing domains, a phenomenon that merges destabilizing domains. In the aggregate, our data reveal that dynamic ensembles within repeat domains profoundly impact the overall energetics of such DNA constructs as well as the distribution of states by which they denature/renature. These static and dynamic influences within triplet repeat domains expand the conformational space available for selection and targeting by the DNA processing machinery. We propose that such dynamic ensembles and their associated impact on DNA properties influence pathways that lead to DNA expansion.     

Causal cells: spacetime polytopes with null hyperfaces

We consider polyhedra and 4-polytopes in Minkowski spacetime—in particular, null polyhedra with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces. We present the basic properties of several classes of null-faced 4-polytopes: 4-simplices, “tetrahedral diamonds” and 4-parallelotopes. We propose a “most regular” representative of each class. The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four lightrays in a tetrahedral pattern. We speculate on the relevance of this construct for discretizations of curved spacetime and for quantum gravity.     

Giant light extraction enhancement of medical imaging scintillation materials using biologically inspired integrated nanostructures

We have utilized biologically inspired (bio-inspired), moth-eye nanostructures and further improved this biomimetic structure to enhance the scintillator materials external quantum efficiency significantly. As a proof of concept, we have demonstrated very high light output efficiency enhancement for Lu(2)SiO(5):Ce(3+) (LSO:Ce) film in large area, the X-ray mammographic instrument was employed to demonstrate the light output enhancement of the Lu(2)SiO(5):Ce thin film with biologically inspired (bio-inspired) moth-eye-like nanophotonic structures. Our work could be extended to other thin film scintillator materials and is promising to achieve lower patient dose, higher resolution images of human organs and even smaller scale medical imaging.     

Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Let G/H be a semisimple symmetric space. Then the space L²(G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L²(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_mc²(G/H) of L²(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.Supported in part by NSF-grant DMS-0070816 and the MSRISupported in part by NSF-grant DMS-0097314 and the MSRISupported in part by NSF-grant DMS-0070607 and the MSRI     

Defect-induced whispering-gallery-mode resonances in optical microdisk resonators

In this Letter we present results of theoretical and experimental studies of whispering-gallery modes in optical microdisk resonators interacting with subwavelength dielectric particles. We predict theoretically and confirm by direct observations that, contrary to the generally accepted models, both peaks of the particle-induced doublet of resonances are redshifted with respect to the position of the initial resonance.     

Holomorphic harmonic functions

We consider boundary problems for holomorphic harmonic functions on hyperboloids at ℂ ⁿ using concepts of integral geometry. Memory of Lenya Volevich