Synthesis of readily cleavable immobilized 1,10-phenanthroline resins

1,10-Phenanthroline is derivatized and ultimately immobilized on two different polystyrene/divinylbenzene solid supports using convenient methodology. All syntheses are amenable to semiautomatic processing and are scalable for high-throughput screening. A domino copper-catalyzed coupling-cyclization reaction is used to illustrate applicability in catalytic studies.     

All frame-spun knots are slice

Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of . We show that all frame-spun knots are slice (null-cobordant).

     

Experimental implementation of an adiabatic quantum optimization algorithm

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.     

To fold or to assemble?

We introduce a new class of foldable oligomers consisting of alternating rigid and flexible regions. The rigid segments overlap to give pi-stacked folded conformers whose formation is driven mostly by pi-pi molecular orbital overlaps. As the oligomer concentration increases, the folded molecular structures further self-assemble into larger nanostructures. The dynamic processes of folding and self-organization are monitored with absorption, fluorescence, and NMR spectroscopies. Our results show that folding dominates at low concentrations (< approximately 1 mM) and precedes self-assembly, which occurs over the initial concentration range of approximately 1-100 mM.     

Integral kašin splittings

Forx ∈ ℝ ⁿ andp≥1 put ‖x‖ _p :=(n ⁻¹Σ|x _i| ^p )^1/p . An orthogonal direct sum decomposition ℝ^2k =E⊕E ^⊥ where dimE=k and ‖x‖₂/‖x‖₁≤C is called here a (k, C)-splitting. By a theorem of Kašin there existsC>0 such that (k, C)-splittings exist for allk, and by the volume ratio method of Szarek one can takeC=32eπ. All proofs of existence of (k, C)-splittings heretofore given are nonconstructive. Here we investigate the representation of (k, C)-splittings by matrices with integral entries. For everyC>8e _1/2 π _−1/2 and positive integerk we specify a positive integerN(k, C) such that in the set ofk by 2k matrices with integral entries of absolute value not exceedingN(k, C) there exists a matrix with row span a summand in a (k, C)-splitting. We haveN(k, C)≤2^18k fork large enough depending onC. We explain in detail how to test a matrix for the property of representing a (k, C)-splitting. Taken together our results yield an explicit (if impractical) construction of (k, C)-splittings.     

A generalization of Filliman duality

Filliman duality expresses (the characteristic measure of) a convex polytope containing the origin as an alternating sum of simplices that share supporting hyperplanes with . The terms in the alternating sum are given by a triangulation of the polar body . The duality can lead to useful formulas for the volume of . A limiting case called Lawrence's algorithm can be used to compute the Fourier transform of .

In this note we extend Filliman duality to an involution on the space of polytopal measures on a finite-dimensional vector space, excluding polytopes that have a supporting hyperplane coplanar with the origin. As a special case, if is a convex polytope containing the origin, any realization of as a linear combination of simplices leads to a dual realization of .

     

Non-treeability for product group actions

A product of two lcsc non-compact groups G ₁ × G ₂ acting freely by measure preserving transformation on a standard Borel probability space gives rise to a non-treeable equivalence relation unless both groups are amenable. Partially supported by NSF grant DMS 0140503     

Displacement affinity chromatography of protein phosphatase one (PP1) complexes

Background  

Protein phosphatase one (PP1) is a ubiquitously expressed, highly conserved protein phosphatase that dephosphorylates target protein serine and threonine residues. PP1 is localized to its site of action by interacting with targeting or regulatory proteins, a majority of which contains a primary docking site referred to as the RVXF/W motif.