Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

A determination of the total width of the ϒ(9.46) meson

Using the DASP detector at the DESY storage ring DORIS we have continued measuring e⁺e^? annihilations near and on the ?(9.46) resonance. From the cross sections for e⁺e^? → μ⁺μ^? and e⁺e^? → hadrons we obtain a μ⁺μ^? branching ratio for the ?(9.46) of (2.9 ± 1.3 ± 0.5) %, a leptonic width $\text{г}{}_{\mathrm{<mtext>ee</mtext>}}\text{= (1.35 ± 0.11 ± 0.22)}\text{keV}$ and a total width of (47_?15⁺³⁷ keV.     

Phase networks of cross-β peptide assemblies

Recent evidence suggests that simple peptides can access diverse amphiphilic phases, and that these structures underlie the robust and widely distributed assemblies implicated in nearly 40 protein misfolding diseases. Here we exploit a minimal nucleating core of the Aβ peptide of Alzheimer's disease to map its morphologically accessible phases that include stable intermolecular molten particles, fibers, twisted and helical ribbons, and nanotubes. Analyses with both fluorescence lifetime imaging microscopy (FLIM) and transmission electron microscopy provide evidence for liquid-liquid phase separations, similar to the coexisting dilute and dense protein-rich liquid phases so critical for the liquid-solid transition in protein crystallization. We show that the observed particles are critical for transitions to the more ordered cross-β peptide phases, which are prevalent in all amyloid assemblies, and identify specific conditions that arrest assembly at the phase boundaries. We have identified a size dependence of the particles in order to transition to the para-crystalline phase and a width of the cross-β assemblies that defines the transition between twisted fibers and helically coiled ribbons. These experimental results reveal an interconnected network of increasing molecularly ordered cross-β transitions, greatly extending the initial computational models for cross-β assemblies.