Unimodularity criteria for Poisson structures on foliated manifolds

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.     

Rotational and Translational Mobility of Nitroxide Spin Probes in Ionic Liquids and Molecular Solvents

Rotational and translational movements of 1-oxyl-2,2,6,6-tetramethyl-4-oxypiperidine (TEMPOL) spin probe in the room temperature ionic liquid (RTIL) 1-octyl-3-methylimidazolium tetrafluoroborate (omimBF₄) and in two molecular solvents, 1-propanol and isopropyl benzene (cumene), have been studied by X-band electron paramagnetic resonance (EPR) spectroscopy. Rotational correlation times τ _c of spin probes and the intermolecular spin exchange rate constants k _e were measured from EPR spectra at different temperatures and TEMPOL concentrations, and compared with the published data. The τ _c values were calculated both by known equations and from the EPR spectra simulation. Rotation movements of TEMPOL in omimBF₄ cannot be described by the model of the isotropic Brownian diffusion, which is valid for conventional solvents. The correct modeling of EPR spectra in RTIL can be achieved with the assumption of different rotational mobility of the spin probe around different molecular axes. The rotational, D _rot, and translational, D _tr, diffusion coefficients were calculated from τ _c and k _e values. The Debye–Stokes–Einstein law is valid in all three solvents while the dependence of D _tr on T/η is not linear in Stokes–Einstein coordinates. The effective activation energy E _rot^a of the rotational movements in omimBF₄ is noticeably higher than the corresponding values for conventional solvents, while the effective activation energies E _tr^a of the translational movements are comparable in all solvents studied.     

DISTINGUISHED PRE-NICHOLS ALGEBRAS

We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity.     

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements.Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site.The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane.The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.     

Azobenzene derivatives carrying a nitroxide radical

Several trans-azobenzene derivatives carrying a nitroxide (aminoxyl) radical (2a, 6a-12a) were prepared, and their photoisomerization reactions to the corresponding cis-isomers were investigated. Although no fruitful results could be obtained for the photoisomerizations of the derivatives with para-subsituents (9a-12a), the unsubstututed derivatives at the para-position (2a, 6a, 7a, 8a) were found to show photoisomerizations by irradiation to give the corresponding cis-isomers (2b, 6b, 7b, 8b), being isolated as relatively stable solid materials, and the change of the intermolecular magnetic interactions was apparently observed by the structural change for each photochromic couple.     

Free radicals in photolysis of mixed phosphonium–iodonium ylides and in their reactions with acetylenes

UV irradiation of mixed phosphonium–iodonium ylide in CH₂Cl₂ leads to formation of free radicals with lifetimes of a few minutes detected by EPR. In mixtures of ylides with acetylenes, the structure of radicals changes, and their concentration and stability increase. In the presence of acetylenes, the radicals contain ylide and acetylene residues, and their EPR spectra have hyperfine coupling constants typical for ³¹P nuclei in C-radicals and for ¹H nuclei, depending on the acetylene structure. It has been demonstrated that the observed radical products are formed from short-lived primary radicals.     

Preparative synthesis of 2,2-dimethyl-5-(trifluoroacetyl)-1,3-dioxane-4,6-dione (2-trifluoroacetyl Meldrum's acid) and 2,2-dimethyl-6-(trifluoromethyl)-4H-1,3-dioxin-4-one and their synthetic usefulness as (trifluoroacetyl)ketene precursors

A simple and reliable method for the preparation of 2,2-dimethyl-5-(trifluoroacetyl)-1,3-dioxane-4,6-dione and 2,2-dimethyl-6-(trifluoromethyl)-4H-1,3-dioxin-4-one on a multigram scale was developed. These (trifluoroacetyl)ketene precursors were used in the hetero-Diels-Alder reaction with dialkylcyanamides and 1-ethoxyprop-1-yne, as well as in some reactions with nucleophiles.     

Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher. Received: August 2006, Accepted: January 2007, Revision: February 2007