Photochemistry of trifluromethyl substituted 1‐methylpyrazoles

Trifluromethyl substitution on the pyrazole ring was found to enhance photoreactivity via the P₄ pathway which involves interchange of the N2‐C3 ring atoms. Thus, 1‐methyl‐3‐(trifluromethyl)pyrazole (1) and 1‐methyl‐5‐(trifluromethyl)pyrazole (3) transposed with a P₄/P₆ or P₄/P₆+P₇ ratio of 2.1:1. 1‐methyl‐4‐(trifluromethyl)pyrazole (2) transposed regiospecifically by the P₄ transposition pathway. Compounds 2 and 3 were also observed to undergo photocleavage to yield enaminonitrile and enaminoisocyanide products.     

The structure of CeAlO3 by Rietveld refinement of X-ray powder diffraction data

The room temperature structure of perovskite CeAlO₃ has been reinvestigated by X-ray powder diffraction. The Rietveld refinement has confirmed the tetragonal symmetry; but revealed a super cell, a=5.32489(6) Å and c=7.58976(10) Å, with the space group I4/mcm. In CeAlO₃, the distortion from the ideal cubic perovskite is caused by the cooperative tilting of the AlO₆ octahedra around the primitive cubic [001]_p-axis.     

Poincaré-Hopf inequalities

In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.

     

Criteria for right disfocality of linear difference equations

Right disfocality and criteria for right disfocality of linear difference equations are considered. The criteria require the positivity of appropriate minors of a Wronskian determinant and are analogues to criteria for the disconjugacy of linear ordinary differential equations and linear difference equations.     

Investigation into the absolute stereochemistry of the marine sponge alkaloid pyrinodemin A

The absolute configuration of the marine sponge alkaloid pyrinodemin A is established by organic synthesis.