Chern–Simons,Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g., in the Chern–Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as \(p= \langle F, F\rangle \) where F is the curvature 2-form and \(\langle \cdot , \cdot \rangle \) is an invariant scalar product on the corresponding Lie algebra \(\mathfrak g\). The descent for p gives rise to an element \(\omega =\omega _3+\omega _2+\omega _1+\omega _0\) of mixed degree. The 3-form part \(\omega _3\) is the Chern–Simons form. The 2-form part \(\omega _2\) is known as the Wess–Zumino action in physics. The 1-form component \(\omega _1\) is related to the canonical central extension of the loop group LG. In this paper, we give a new interpretation of the low degree components \(\omega _1\) and \(\omega _0\). Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara–Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class p. In more detail, we define a 1-cocycle C which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara–Vergne equation F is mapped to \(\omega _1=C(F)\). Furthermore, the component \(\omega _0\) is related to the associator \(\Phi \) corresponding to F. It is surprising that while F and \(\Phi \) satisfy the highly nonlinear twist and pentagon equations, the elements \(\omega _1\) and \(\omega _0\) solve the linear descent equation.     

Recurrence of Ancestral Lines and Offspring Trees in Time Stationary Branching Populations

For time stationary Galton- Watson-branching populations on a general type space, the structure of the “individually positive recurrent part” of the system is described: its building blocks consist of finitely many “clans” with positive recurrent trunks. Conditions are given when this nubsystem is void, and when it equals the full system. In addition, positive recurrence on the clan level is characterized. Whereas individual positive recurrence turns out to be a symmetric concept with respect to forward and backward time direction (i. e., with respect to anceatral lines and offspring trees), with individual null recurrence this symmetry can fail even in the absence of branching, i.e., for independently migrating particle systems (Example 13.1). For discrete type spaces a classification of types as to the various individual recurrence concepts (positive, null, forward and backward in time) is proposed and illustrated by a couple of results and examples. For finite type spaces conditiom on the branching dynamics and its mean matrix for the existence of nontrivial equilibria are given.     

Towards experiments with highly charged ions at HESR

The atomic physics collaboration SPARC is a part of the APPA pillar at the future Facility for Antiproton and Ion Research. It aims at atomic-physics research across virtually the full range of atomic matter. An emphasis of this contribution are the atomic physics experiments addressing the collision dynamics in strong electro-magnetic fields as well as the fundamental interactions between electrons and heavy nuclei at the HESR. Here we give a short overview about the central instruments for SPARC experiments at this storage ring.     

Organoboron Derivatives of Stereoregular Phenylcyclosilsesquioxanes

This study presents the synthesis of organoboron derivatives of stereoregular 4-, 6-, and 12-unit phenylcyclosilsesquioxanes. All compounds obtained were isolated in good yields (70–80 %) and were fully characterized by ¹H, ¹³C, ²⁹Si, ¹¹B NMR, IR spectroscopy, HRMS ESI, and elemental microanalysis. The structure of the key modifier, obtained for the first time, 4-(tetramethyl-1,3,2-dioxaborolan-2-yl)phenyl) dimethylvinylsilane, was also confirmed by single-crystal XRD.     

Aliphatic hydroxylation and epoxidation of capsaicin by cytochrome P450 CYP505X

Microbial cytochrome P450 enzymes (CYPs) are able to mimic the metabolism of human CYPs. One challenge is to identify the respective drug metabolites and to compare substrate specificities to those of the human enzymes. In this study, a class VIII self-sufficient CYP from Aspergillus fumigatus (CYP505X) and variants of this enzyme were heterologously expressed in E. coli. The substrate scope of the variants was determined using active pharmaceutical ingredients (APIs) and (hetero)cyclic compounds. Capsaicin – the active compound in chili peppers – was oxidized most efficiently (4.36?μM/min) in a whole cell mediated biotransformation. The products were isolated, purified and their structures elucidated by 1D and 2D NMR. The two major metabolites showed modifications on the lipophilic side chain. Specifically, capsaicin was hydroxylated at position 8 to give (E)-8-hydroxy-N-(4-hydroxy-3-methoxybenzyl)-8-methylnon-6-enamide and epoxidized at the double bond to give N-(4-hydroxy-3-methoxybenzyl)-5-(3-isopropyloxiran-2-yl)-pentanamide.     

Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ ^γ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln ^α (1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.     

Monte Carlo simulation of brain sensing by optical diffuse spectroscopy

We report on development of adapted Monte Carlo based algorithm and code for simulation of light propagation in turbid media with complex geometry aimed for simulation of optical diffuse spectroscopy signal in noninvasive brain sensing. Simulation will allow to determine optimal characteristics of a prototype device for optical diffuse brain sensing. The developed Monte Carlo code can be efficiently parallelized both for SMP and distributed memory systems. We show that the speed-up of the developed algorithm almost linearly depends on the number of nodes/threads in a utilized system.     

Global estimates of fundamental solutions for higher-order Schr?dinger equations

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr?dinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L ^p ?CL ^q estimates on the Schr?dinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).