Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in $\mathbb{R}^{d}$ , d≥3, in general position, that is, no (d+1) points of S lie in a common (d?1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3≤k≤d we provide a lower bound of $\varOmega(n^{d-k+1+2^{-d}})$ and strengthen this to Ω(n ^d?2/3) for k=2. On the way we provide various results on triangulations of point sets in  $\mathbb{R}^{d}$ . In particular, for any constant dimension d≥3, we prove that every set of n points (n sufficiently large), in general position in $\mathbb{R}^{d}$ , admits a triangulation with at least dn+Ω(logn) simplices.     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.      

Inverse semigroups and combinatorial C*-algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case. *Partially supported by CNPq.     

Identification of the Active Principle Conferring Anti-Inflammatory and Antinociceptive Properties in Bamboo Plant

Early plants began colonizing earth about 450 million years ago. During the process of coevolution, their metabolic cellular pathways produced a myriad of natural chemicals, many of which remain uncharacterized biologically. Popular preparations containing some of these molecules have been used medicinally for thousands of years. In Brazilian folk medicine, plant extracts from the bamboo plant Guadua paniculata Munro have been used for the treatment of infections and pain. However, the chemical basis of these therapeutic effects has not yet been identified. Here, we performed protein biochemistry and downstream pharmacological assays to determine the mechanisms underlying the anti-inflammatory and antinociceptive effects of an aqueous extract of the G. paniculata rhizome, which we termed AqGP. The anti-inflammatory and antinociceptive effects of AqGP were assessed in mice. We identified and purified a protein (AgGP), with an amino acid sequence similar to that of thaumatins (~20 kDa), capable of repressing inflammation through downregulation of neutrophil recruitment and of decreasing hyperalgesia in mice. In conclusion, we have identified the molecule and the molecular mechanism responsible for the anti-inflammatory and antinociceptive properties of a plant commonly used in Brazilian folk medicine.     

Lipozyme 435-Mediated Synthesis of Xylose Oleate in Methyl Ethyl Ketone

In this paper, we have performed the Lipozyme 435-catalyzed synthesis of xylose oleate in methyl ethyl ketone (MEK) from xylose and oleic acid. The effects of substrates’ molar ratios, reaction temperature, reaction time on esterification rates, and Lipozyme 435 reuse were studied. Results showed that an excess of oleic acid (xylose: oleic acid molar ratio of 1:5) significantly favored the reaction, yielding 98% of xylose conversion and 31% oleic acid conversion after 24 h-reaction (mainly to xylose mono- and dioleate, as confirmed by mass spectrometry). The highest Lipozyme 435 activities occurred between 55 and 70 °C. The predicted Ping Pong Bi Bi kinetic model fitted very well to the experimental data and there was no evidence of inhibitions in the range assessed. The reaction product was purified and presented an emulsion capacity close to that of a commercial sugar ester detergent. Finally, the repeated use of Lipozyme 435 showed a reduction in the reaction yields (by 48 and 19% in the xylose and oleic acid conversions, respectively), after ten 12 h-cycles.     

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.     

Blaschke’s problem for hypersurfaces

We solve Blaschke’s problem for hypersurfaces of dimension . Namely, we determine all pairs of Euclidean hypersurfaces that induce conformal metrics on M ⁿ and envelop a common sphere congruence in .     

Conformal immersions of warped products

We prove a decomposition theorem for conformal immersions \(f\colon\;M^n\to {\mathbb{R}}^{N}\) into Euclidean space of a warped product of Riemannian manifolds \(M^n:=M_0\times_\rho\Pi_{i=1}^k M_i\) of dimension n ≥ 3 under the assumption that the second fundamental form \(\alpha \colon TM \times TM\to T^\perp M\) of f satisfies \(\alpha|_{TM_i\times TM_j}=0\) for i ≠ j. It generalizes the corresponding theorem of Nölker for isometric immersions as well as our previous result on conformal immersions of Riemannian products. In particular, we determine all conformal representations of Euclidean space of dimension n ≥ 3 as a warped product of Riemannian manifolds. As a consequence, we classify the conformally flat warped products.