Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

Diffusion in the Lorentz Gas

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.     

Isolation and Connectivity in Random Geometric Graphs with Self-similar Intensity Measures

Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show that nonuniformity can break the Poisson distribution property, but it strengthens the link between isolation and connectivity. It also stretches out the connectivity transition. Finite ramification is another mechanism for lack of connectivity. The same considerations apply to fractal distributions as smooth, with some technical differences in evaluation of the integrals and analytical arguments.     

Prearranged Glycosides. Part 8. Intramolecular α-Galactosylation Via Succinoyl Tethered Glycosides

ABSTRACT

Benzyl protected phenyl 1-thio-galactopyranoside donors which were tethered by a succinoyl linker at their positions 2 and 6, respectively, to position 3 of a blocked benzyl glucopyranoside acceptor with a 4-OH group solely afforded the corresponding α-(1→4)-linked disaccharides upon intramolecular glycosylation. 4,6-Siloxane protected mannosides react with rearrangement of the siloxane group under similar conditions.     

Characterization and quantification of polyphenols in Amazon grape (Pourouma cecropiifolia Martius)

The phenolic profile of Amazon grape fruit (Pourouma cecropiifolia Martius) was investigated by high-performance liquid chromatography-electrospray ionization mass spectrometry (HPLC-ESI-MS/MS). For this purpose, suitable extraction and liquid chromatographic methods were developed. Anthocyanins, flavonols and chlorogenic acids were found mainly in the peel. Besides the main anthocyanins, i.e. delphinidin 3-glucoside, cyanidin 3-glucoside and cyanidin 3-(6"-malonyl)glucoside, several minor anthocyanins were identified in the peel. Among these, cyanidin 3,5-diglucoside, delphinidin 3-galactoside, cyanidin 3-rutinoside, cyanidin 3-(3"-malonyl)glucoside, malvidin 3-glucoside, pelargonidin 3-glucoside, peonidin 3-glucoside and petunidin 3-glucoside were characterized on the basis of their fragmentation patterns in MS/MS experiments. The total anthocyanin content in the peel was 420.26±3.07 mg kg(-1) fresh weight. The pulp contained mainly 5-O-caffeoylquinic acid (210.39±3.43 mg kg(-1) fresh weight). Rutin was the predominant flavonol found in Amazon grape (peel 155.45 ± 2.06 mg kg(-1) fresh weight and pulp 2.64±1.21 mg kg(-1) fresh weight). Total polyphenols content was higher in the peel than in the pulp.     

New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor \(\sqrt{t}\) are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by \(\sqrt {t\ln t}\), with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.     

Note on Chaos and Diffusion

Using standard definitions of chaos (as positive Kolmogorov–Sinai entropy) and diffusion (that multiple time distribution functions are Gaussian), we show numerically that both chaotic and nonchaotic systems exhibit diffusion, and hence that there is no direct logical connection between the two properties. This extends a previous result for two time distribution functions.     

Lyapunov Spectra of Periodic Orbits for a Many-Particle System

The Lyapunov spectrum corresponding to a periodic orbit for a two-dimensional many-particle system with hard core interactions is discussed. Noting that the matrix to describe the tangent space dynamics has the block cyclic structure, the calculation of the Lyapunov spectrum is attributed to the eigenvalue problem of 16×16 reduced matrices regardless of the number of particles. We show that there is the thermodynamic limit of the Lyapunov spectrum in this periodic orbit. The Lyapunov spectrum has a step structure, which is explained by using symmetries of the reduced matrices.