Smooth Frame-Path Termination for Higher Order Sigma-Delta Quantization

We study the performance of finite frames for the encoding of vectors by applying standard higher-order sigma-delta quantization to the frame coefficients. Our results are valid for any quantizer with accuracy ε > 0 operating in the no-overload regime. The frames under consideration are obtained from regular sampling of a path in a Hilbert space. In order to achieve error bounds that are comparable to results on higher-order sigma-delta for the quantization of oversampled bandlimited functions, we construct frame paths that terminate smoothly in the zero vector, that is, with an appropriate number of vanishing derivatives at the endpoint.     

Bohr's inequality for uniform algebras

We prove a uniform algebra analogue of a classical inequality of Bohr's concerning Fourier coefficients of bounded holomorphic functions. The classical inequality follows trivially.

     

Solid products from thermal decomposition of polyethylene terephthalate: Investigation by CP/MAS 13C-NMR and fourier transform-IR spectroscopy

The solid reaction products from pyrolysis of polyethylene terephthalate in the presence and absence of red phosphorus were characterized by CP/MAS ¹³C-NMR, FR-IR, and MAS ³¹P-NMR spectroscopy. Over the temperature range of 300–400°C, polyethylene terephthalate was converted in a sealed vial to a highly crosslinked polymer of terephthalic acid. Pyrolysis in the presence of red phosphorus, which functions as a flame retardant by increasing the amount of char, yielded an intractible polyaromatic phosphate ester. After thermal cleavage of polyethylene terephthalate with formation of free carboxyl and vinyl ester groups, there are two competing reaction pathways. The smaller molecular weight fragments may enter the vapor phase where they undergo further degradation primarily to CO₂, CO, and acetaldehyde, as described by others. However, if volatilization of the oligomeric fragments is inhibited, an alternate reaction pathway gives rise to the formation of highly crosslinked char. Red phosphorus decreases the volatility of the oligomeric fragments by converting them to phosphates and thereby enhances char formation.     

A synchrotron radiation study of the one‐dimensional complex of sodium with (1S)‐N‐carboxylato‐1‐(9‐deazaadenin‐9‐yl)‐1,4‐dideoxy‐1,4‐imino‐d‐ribitol,a member of the 'immucillin' family

The sodium salt of [immucillin‐A–CO₂H]⁻ (Imm‐A), namely catena‐poly[[[triaquadisodium(I)](μ‐aqua)[μ‐(1S)‐N‐carboxylato‐1‐(9‐deazaadenin‐9‐yl)‐1,4‐dideoxy‐1,4‐imino‐d ‐ribitol][triaquadisodium(I)][μ‐(1S)‐N‐carboxylato‐1‐(9‐deazaadenin‐9‐yl)‐1,4‐dideoxy‐1,4‐imino‐d ‐ribitol]] tetrahydrate], {[Na₂(C₁₂H₁₃N₄O₆)₂(H₂O)₇]·4H₂O}_n, (I), forms a polymeric chain via Na⁺—O interactions involving the carboxylate and keto O atoms of two independent Imm‐A molecules. Extensive N,O—H...O hydrogen bonding utilizing all water H atoms, including four waters of crystallization, provides crystal packing. The structural definition of this novel compound was made possible through the use of synchrotron radiation utilizing a minute fragment (volume ∼2.4 × 10⁻⁵ mm⁻³) on a beamline optimized for protein data collection. A summary of intra‐ring conformations for immucillin structures indicates considerable flexibility while retaining similar intra‐ring orientations.     

Mass-dependent bond vibrational dynamics influence catalysis by HIV-1 protease

Protein motions that occur on the microsecond to millisecond time scale have been linked to enzymatic rates observed for catalytic turnovers, but not to transition-state barrier crossing. It has been hypothesized that enzyme motions on the femtosecond time scale of bond vibrations play a role in transition state formation. Here, we perturb femtosecond motion by substituting all nonexchangeable carbon, nitrogen, and hydrogen atoms with (13)C, (15)N, and (2)H and observe the catalytic effects in HIV-1 protease. According to the Born-Oppenheimer approximation, isotopic substitution alters vibrational frequency with unchanged electrostatic properties. With the use of a fluorescent peptide to report on multiple steps in the reaction, we observe significantly reduced rates in the heavy enzyme relative to the light enzyme. A possible interpretation of our results is that there exists a dynamic link between mass-dependent bond vibrations of the enzyme and events in the reaction coordinate.     

A classification theorem for essentially binormal operators

An essentially binormal operator on Hilbert space is an operator which is unitarily equivalent to a 2 × 2 matrix of essentially commuting, essentially normal operators. A natural invariant of essentially binormal operators up to unitary equivalence in the Calkin Algebra is the reducing essential 2 × 2 matricial spectrum. A nonempty compact subset X of the set of 2 × 2 matrices is called hypoconvex, if it is the reducing essential 2 × 2 matricial spectrum of an operator on Hilbert space. The set EN₂(X) is then defined to be the set of all equivalence classes (up to unitary equivalence in the Calkin algebra) of essentially binormal operators whose reducing essential 2 × 2 matricial spectrum coincides with X. The aim of this paper is to prove a result that enables one to compute EN₂(X) in terms of the topological structure of the space $\text{X}\text{?}$ of unitary orbits of X. Indeed, it is shown that for every hypoconvex subset X of the set of 2 × 2 matrices, there exists a natural homomorphism from $\text{Ext}\text{(}\text{X}\text{?}\text{)}$ onto EN₂(X). Also, a six term cyclic exact sequence is obtained, which produces a characterization of the kernel of the above-mentioned homomorphism.     

Equivariant maps and bimodule projections

We construct a counterexample to Solel's [B. Solel, Contractive projections onto bimodules of von Neumann algebras, J. London Math. Soc. 45 (2) (1992) 169–179] conjecture that the range of any contractive, idempotent, MASA bimodule map on is necessarily a ternary subalgebra. Our construction reduces this problem to an analogous problem about the ranges of idempotent maps that are equivariant with respect to a group action. Such maps are important to understand Hamana's theory [M. Hamana, Injective envelopes of C^*-dynamical systems, Tohoku Math. J. 37 (1985) 463–487] of G-injective operator spaces and G-injective envelopes.     

Transition state analogue discrimination by related purine nucleoside phosphorylases

Transition state analogues of PNP, the Immucillins and DADMe-Immucillins, were designed to match transition state features of bovine and human PNPs, respectively. The inhibitors with or without the hydroxyl and hydroxymethyl groups of the substrate demonstrate that inhibitor geometry mimicking that of the transition state confers binding affinity discrimination. This finding is remarkable since crystallographic analysis indicates complete conservation of active site residues and contacts to ligands in human and bovine PNPs.