Sample Size for Testing Homogeneity of Two a Priori Dependent Binomial Populations Using the Bayesian Approach

This paper establishes new methodology for calculating the optimal sample size when a hypothesis test between two binomial populations is performed. The problem is addressed from the Bayesian point of view, with prior information expressed through a Dirichlet distribution. The approach of this paper sets an upper bound for the posterior risk and then chooses as “optimum ”the combined sample size for which the likelihood of the data does not satisfy this bound. The combined sample size is divided equally between the two binomials. Numerical examples are discussed for which the two proportions are equal to either a fixed or to a random value.     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

Substituent effects and the mechanism of alkene metathesis catalyzed by ruthenium dichloride catalysts

Density functional theory calculations are reported concerning the dissociative mechanism for alkene metathesis by ruthenium dichloride catalysts, including both bisphosphine and diaminocarbene/phosphine complexes. The calculations use a hierarchy of models, ranging from [(L)(PH(3))Ru(Cl)(2)(CH(2))](L=PH(3) or diaminocarbene) through the larger [(L)(PMe(3))Ru(Cl)(2)(CHPh)] to the "real"[(L)(PCy(3))Ru(Cl)(2)(CHPh)]. Calculations show that the rate-limiting step for metathesis is either ring closing from an alkene complex to form a ruthena-cyclobutane, or ring-opening of the latter intermediate to form an isomeric alkene complex. The higher efficiency of the diaminocarbene based catalysts is due to the stabilization of the formal +iv oxidation state of the ruthenium centre in the metallacycle. This effect is partly masked in the smaller model systems due to a previously unnoticed stereoelectronic effect. The calculations do not reproduce the experimental observation whereby the initiation step, phosphine dissociation, is more energetically demanding and hence slower for the diaminocarbene-containing catalyst system than for the bisphosphine. Further calculations on the corresponding bond energies using a variety of DFT and hybrid DFT/molecular mechanics methods all find instead a larger phosphine dissociation energy for the bisphosphine catalyst. This reversed order of binding energies would in fact be the one expected based on the stronger trans influence of the diaminocarbene ligand. The discrepancy with experiment is small and could have a number of causes which are discussed here.     

Small molecules as inhibitors of cyclin-dependent kinases

Cell division (mitosis) is one of the basic requirements for multicellular oranisms. The capability of a cell to replicate enables a complex assembly to be created. Faulty regulation of the control mechanism in the cell cycle leads to an excessive cell proliferation and is the cause of cancer. The key position of the cyclin-dependent kinases (CDKs) and their direct partners, as well as the fact that the majority of malign illnesses show defects in at least one of these key players of the cell cycle, is of great interest for the development of low-molecular-weight CDK inhibitors. In this Review an overview of the different structural classes of ATP-competitive inhibitors of CDKs are given, whose devlopment was aimed at battling cancer. The Review shows how far the development of selective CDK inhibitors has progressed and to what extent the expectations for such drugs have so far been fulfilled.     

Halogen derivatives of m-phenylene(carbeno)nitrene: a switch in ground-state multiplicity

m-Phenylene-coupled carbenonitrenes [(3-nitrenophenyl)methylene (2-H), (3-nitrenophenyl)fluoromethylene (2-F), (3-nitrenophenyl)chloromethylene (2-Cl), (3-nitrenophenyl)bromomethylene (2-Br)] have been investigated computationally (with B3LYP, MCSCF, CASPT2, ROMP2, and QCISD(T) methods) and experimentally (with IR, UV, and ESR spectroscopy). For each species, five electronic states were considered. At the highest level of theory explored, the parent compound (2-H) has a quintet ground state, but its halogen derivatives (2-X, X = F, Cl, and Br) have triplet ground states. A linear relationship between the Q[bond]T energy gap of 2-X and the T-S gap of the corresponding phenylcarbenes 8-X is found, which can be helpful in rationalizing and predicting ground-state multiplicities in m-phenylene-linked carbenonitrenes and similar species. Precursors for the photochemical generation of 2-X (X = H, F, Cl, and Br) were synthesized and photolyzed in matrixes (Ar, triacetin) at low temperatures. IR (Ar, 13 K) and ESR (triacetin, 77 K) data are compatible with the generation of triplet halocarbenonitrenes 2-X, (X = F, Cl, and Br). All four compounds upon further irradiation undergo isomerization to substituted cyclopropenes 5-X (X = H, F, Cl, and Br), as suggested by their IR spectra.