Positive definite functions on the Heisenberg group

Work supported in part by the NSF     

Dimorphism in (2Z)‐2‐benzyl­idene‐N,7‐dimethyl‐3‐oxo‐5‐phenyl‐2,3‐dihydro‐5H‐1,3‐thia­zolo[3,2‐a]pyrimidine‐6‐carboxamide

The title compound, C₂₂H₁₉N₃O₂S, crystallizes in two polymorphic forms having the same space group, viz. P, with Z′ = 2 and Z′ = 1. In both polymorphs, the planar thia­zole ring is fused cis with the dihydro­pyrimidine ring, the carbamoyl group is in an extended conformation with an anti­clinal orientation with respect to the pyrimidine ring, and the phenyl ring is attached to the pyrimidine ring approximately at a right angle. The two polymorphs have different inter­planar angles between the phenyl and thia­zole rings. The mol­ecules are linked by N—H⋯O and C—H⋯O hydrogen bonds.     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

Unbounded derivations tangential to compact groups of automorphisms

We consider unbounded derivations in C¹-algebras commuting with compact groups of ¹-automorphisms. A closed ¹-derivation δ in a C¹-algebra $\text{U}$ is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈$\text{R}$→τ(t)? Aut($\text{U}$) such that $\text{δ =}\text{d}\text{dt}\text{τ(t)¦}{}_{\mathrm{t\; =\; 0}}$. If δ is known to commute with a compact abelian action α:G→Aut($\text{U}$), and if δ(a) = 0 for all a in the fixed point algebra $\text{U}$^α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that $\text{v(γ)}{}^1\text{= ?v(γ), v(γ)}$ is possibly unbounded but affiliated with the center of {$\text{U}$^α}″, and e^tδ(x) = xe^tv(γ) for all x in the Arveson spectral subspace $\text{U}$^α(γ). In particular, if $\text{U}$ is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group.     

N-Aryl-N-Hydroxy Urethanes as Peroxidase Substrates

 N-Aryl-N-hydroxy urethanes (AHUs), which are promising N*OH mediators for oxidoreductase catalysis, are electrochemically active redox compounds with a redox potential of 0.35–0.40 V vs. SCE (0.59–0.64 V vs. NHE). Alkyl substituents in the carbamic acid residue have, as expected, only little influence on the redox potential. The highest potential was noticed for N-hydroxy-N-(4-cyanophenyl)-carbamic acid methyl ester. Recombinant Coprinus cinereus peroxidase (rCiP) catalyzes oxidation of the AHUs with apparent bimolecular constants k _ox of 2.5ċ10⁴–7.5ċ10⁴ M ⁻¹s⁻¹ at pH = 8.5 and 25°C. Structure-function connectivities of the substrates were analysed within the framework of the Marcus cross relationship and by using ab initio quantum chemical calculations. An excellent correlation of the redox potentials and the HOMO energies could be found. However, no correlation of log(k _ox) with redox potential and HOMO energy was indicated as predicted by theory. This was explained by specific docking of the substrates in the active center of rCiP.     

Moment computations for graphs with fractal property

In this paper, we use explicit formulas for the moments of a self-adjoint operator (radial operator), induced by a certain discrete structure, in Hilbert space. Our main theorem shows that the structures are classified by the moment computations producing an equivalence relation. Our motivation in turn derives from a groupoid-theoretic approach to spectral problems as they arise in quantum mechanics. While it is typically difficult to obtain explicit formulas for spectra, we demonstrate that our moment formulas serve as a substitute. The discrete structures $\Bbb{G}$ we study have a built in fractal feature: any portion of $\Bbb{G}$ is similar to the whole. And this fact (like in renormalization groups) serves to facilitate computations.