An algorithm for numerical determination of the structure of a general matrix

An algorithm, proposed by V. N. Kublanovskaya, for solving the complete eigenvalue problem of a degenerate (that is defective and/or derogatory) matrix, is studied theoretically and numerically. It uses successiveQR-factorizations to determine annihilated subspaces.An adaptation of the algorithm is developed which, applied to a matrix with a very ill-conditioned eigenproblem, computes a degenerate matrix. The difference between these matrices is small, measured in the spectral norm. The degenerate matrix will appear in a standard form, whose eigenvalues and principal vectors can be computed in a numerically stable manner.Numerical examples are given.     

Metalation of 4-oxazolinyloxazole derivatives. A convenient route to 2,4-bifunctionalized oxazoles

The synthesis of an array of 5-phenyloxazole derivatives bearing a variety of hydroxyalkyl groups at the C-2 position of the heterocyclic nucleus and possessing a formyl or a carboxyl function at C-4 is reported. These bifunctionalized compounds have been efficiently prepared by addition of carbonylated electrophiles to the 2-lithio derivative of 5-phenyloxazole preliminarily equipped with an oxazoline unit at the 4-position of the oxazole nucleus. It is demonstrated that this protocol offers a double advantage since it suppresses the troublesome electrocyclic ring-opening reaction and allows access to the target compounds by simple chemical transformation of the oxazoline ring system.     

Design,synthesis and biological activity of rigid cannabinoid CB1 receptor antagonists

The design, synthesis and biological activities of potent pyrazole-based tricyclic CB1 receptor antagonists (2) are described. The key synthetic step involves the ring closure of the lithiated alpha, gamma-keto ester adduct (4). The optimal nitroderivative (28) in this series exhibits a high CB1 receptor affinity (pKi=7.2) as well as very potent antagonistic activity (pA2=8.8) in vitro. The regioselectivity of the pyrazole ring closure is shown to depend strongly on the aromatic substitution pattern of the applied arylhydrazine.     

Ion-pairing molecular recognition in water: aggregation at low concentrations that is entropy-driven

Investigations into the thermodynamic parameters that characterize the binding of citrate to tris-guanidinium host 1 in water are reported. The parameters K(a), DeltaH degrees, DeltaS degrees, and DeltaG degrees for the binding event were quantified using isothermal titration calorimetry (ITC) techniques. The 1:1 binding stoichiometry was verified by a Job plot derived from NMR data, and the microcalorimetry data was collected for solutions of 1 and citrate ranging from 1 to 100 mM using phosphate buffer concentrations of 5 and 103 mM. At low buffer concentrations (low ionic strength) complexes with greater than 1:1 stoichiometries were observed by ITC, and K(1) was determined to range from 2.0 x 10(3) to 3.0 x 10(3) M(-1). At higher buffer concentrations (high ionic strength) the higher-order complexes were not detected, and K(1) was determined to be 409 M(-1). The 1:1 association of host 1 and citrate is characterized by a large favorable entropy component and negative enthalpy. However, the complexes with higher-order stoichiometry arise from desolvation processes that result from the association of polyions in aqueous media and is entirely entropy driven. This leads to an unusual observation: the dilution of one component of the host/guest complex leads to the formation of the higher-order complexes. The reason for this observation is discussed.     

Laplace operator and hodge decomposition for quantum groups and quantum spaces

LetΓ=Γ _±,z be one of theN ²-dimensional bicovariant first order differential calculi for the quantum groups GL _q (N), SL _q (N), SO _q (N), or Sp _q (N), whereq is a transcendental complex number andz is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’s external algebraΓ ^{^} coincides with the de Rham cohomologies of its leftinvariant, its right-invariant and its biinvariant subcomplexes. In the cases GL _q (N) and SL _q (N) the cohomology ring is isomorphic to the biinvariant external algebraΓ _inv ^{^} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. It is also applicable for quantum Euclidean spheres. The eigenvalues of the Laplace-Beltrami operator in cases of the general linear quantum group, the orthogonal quantum group, and the quantum Euclidean spheres are given.