Some Solutions for a Class of Singular Equations

In this paper we obtain all solutions which depend only on r for a class of partial differential equations of higher order with singular coefficients.     

Finite Lifetime Eigenfunctions of Coupled Systems of Harmonic Oscillators

We consider a vector-valued Hermite-type basis for which the eigenvalue problem associated to the operator H _A,B :=B(; _x ² )+Ax ² acting on becomes a three-terms recurrence. Here A and B are 2 × 2 constant positive definite matrices. Our main result provides an explicit characterization of the eigenvectors of H _A,B that lie in the span of the first four elements of this basis when AB BA.This revised version was published online in March 2005 with corrections to the cover date.     

Re-based heterogeneous catalysts for olefin metathesis prepared by surface organometallic chemistry: reactivity and selectivity

Herein we describe the catalytic activity of 1, a well-defined Re alkylidene complex supported silica, in the reaction of olefin metathesis. This system is highly active for terminal and internal olefins with initial rates up to 0.7 mol per mol Re per s. It also catalyses the self-metathesis of methyl oleate (MO) without the need of co-catalysts. The turnover numbers can reach up to 900 for MO, which is unprecedented for a heterogeneous Re-based catalyst. Moreover the use of silica as a support can bring major advantages, such as the possibility to use branched olefins like isobutene, which are usually incompatible with alumina-based supports; therefore, the formation of isoamylene from the cross-metathesis of propene and isobutene can be performed. All these results are in sharp contrast to what has been found for other silica- or alumina-supported rhenium oxide systems, which are either completely inactive (silica system) or typically need co-catalysts when functionalised olefins are used. Finally the initiation step corresponds to a cross-metathesis reaction to give a 3:1 mixture of 3,3-dimethylbutene and trans-4,4-dimethylpent-2-ene, and make this catalyst the first generation of well-defined Re-based heterogeneous catalysts.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

Meson-baryon s-wave resonances with strangeness -3

Starting from a consistent SU(6) extension of the Weinberg-Tomozawa (WT) meson-baryon chiral Lagrangian ( Phys. Rev. D 74, 034025 (2006)), we study the s-wave meson-baryon resonances in the strangeness S = - 3 and negative-parity sectors. Those resonances are generated by solving the Bethe-Salpeter equation with the WT interaction used as kernel. The considered mesons are those of the 35-SU(6)-plet, which includes the pseudoscalar (PS) octet of pions and the vector (V) nonet of the rho-meson. For baryons we consider the 56-SU(6)-plet, made of the 1/2⁺ octet of the nucleon and the 3/2⁺ decuplet of the Delta. Quantum numbers I(J ^P) = 0(3/2^-) are suggested for the experimental resonances Ω ^*(2250)^- and Ω ^*(2380)^-. Among other, resonances with I = 1 are found, which minimal quark content is sssˉl', being s the strange quark and l, l' any of the the light up or down quarks. A clear signal for such a pentaquark would be a baryonic resonance with strangeness -3 and electric charge -2 or 0, in proton charge units. We suggest looking for K ^- Ξ ^- resonances with masses around 2100 and 2240MeV in the sector 1(1/2^-), and for π ^± Ω ^- and K ^- Ξ ^*- resonances with masses around 2260MeV in the sector 1(3/2^-).     

Carbon nanotubes-supported palladium nanoparticles for the Suzuki reaction in supercritical carbon dioxide: A facile method for the synthesis of tetrasubstituted olefins

A facile and efficient method for the synthesis of tetrasubstituted olefins in supercritical carbon dioxide was developed by using carbon nanotubes-supported palladium nanoparticles (Pd/CNTs) as the catalyst. Compared with common Pd/C, Pd/CNTs could more effectively catalyze the reaction of dibromo-substituted olefins with boronic acids, affording the corresponding tetrasubstituted olefins with moderate to good yields. This environmentally benign route with an easy-to-handle catalyst provides an appealing alternative to the currently available methods.     

Symmetries of nonlinear hyperbolic systems of the Toda chain type

We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example of models of this type is given by two-dimensional open Toda chains. For systems that have integrals, we construct a differential operator that takes integrals into symmetries. For systems of the chosen type, this proves the existence of higher symmetries dependent on arbitrary functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 344–355, May, 2008.     

Reaction and surface characterization study of Zn/Cr-based higher-alcohol synthesis catalysts X: Effects of excess promoter loading on surface chemistry

Promoter loading level significantly affects the catalytic behavior of Zn/Cr spinel higher alcohol synthesis (HAS) catalysts. In this study a 5.9 wt.% Pd-promoted Zn/Cr spinel with a 5 and 7 wt.% K promoter have been tested and compared. At 1500 psig and 400°C, the catalyst containing 5 wt.% K results in the production of slightly more butanol, less methanol and more hydrocarbons. Using ion scattering spectroscopy and X-ray photoelectron spectroscopy, differences are found between the two catalyst surfaces which must be responsible for the differences in catalytic behavior.