The determinant of a class of skew-symmetric toeplitz matrices

The determinant of the n×n matrix associated with the finite-difference operator μδ^2r+1 is obtained explicitly for all n and all integral r?0. An interesting combinatorial identity results.     

Perturbation analysis of the variation method

It is pointed out that an assumption common to all general discussions of the perturbation analysis of the variation method is sometimes unjustified, and some consequences are discussed.     

Variable basis sets in perturbation theory: Numerical finite perturbation versus analytic approach

We consider the problem of determining variational, external-field-dependent corrections to nonoptimal zero-field nonlinear parameters. Both a direct analytic perturbation analysis and finite perturbation methods are described in a general way and in detail for the SCF approximation. The abstract theory is illustrated by reference to the results of several explicit calculations. Also, the sensitivity of the results to the choice of zero-field values is discussed.     

Perturbation theory within the variation method

Some of Moccia's formulas (Int. J. Quant. Chem. 8 , 293 (1974)) for corrections to “not completely optimized variational parameters” are compared with those resulting from a perturbation theory analysis. It is not possible to choose the zero-order Hamiltonian (at least not in any simple way) so as to get exact agreement, but one choice does come close.     

Selective Ir-catalysed borylation of polycyclic aromatic hydrocarbons: structures of naphthalene-2,6-bis(boronate), pyrene-2,7-bis(boronate) and perylene-2,5,8,11-tetra(boronate) esters

The selectivity of the iridium boryl complex-catalysed reaction of pyrene and perylene with B2(pin)2 (pin = OCMe2CMe2O) has been employed in the synthesis of D(2h)-symmetry pyrene-2,7-(Bpin)2 and D(4h)-symmetry perylene-2,5,8,11-(Bpin)4 providing direct borylation of positions which are otherwise difficult to derivatise.     

Influence of hydrogen bonding on coordination polymer assembly

An extended dipyridyl ligand (L1) capable of hydrogen bonding with guest species via urea functionalities has been designed and synthesised. Assembly of a silver(I) coordination polymer of L1 is dependent on the nature of the hydrogen bond acceptor in a logical extension of the monopyridyl analogue.     

A version of Sharkovskii's theorem for differential equations

We present a version of the Sharkovskii cycle coexistence theorem for differential equations. Our earlier applicable version is extended here to hold with the exception of at most two orbits. This result, which (because of counter-examples) cannot be improved, is then applied to ordinary differential equations and inclusions. In particular, if a time-periodic differential equation has -periodic solutions with , for all , then infinitely many subharmonics coexist.

     

Finitary coding of Markov random fields

Summary There exists a finitary code from any stationary ergodic Markov random field to any i.i.d. random field of strictly lower entropy.     

Constraints and the Hellmann-Feynman theorem

It is pointed out that Hurley's condition applies even in the presence of Lagrange multipliers. Also a useful formula for E is derived which allows for the possibility that both the trial functions and the constraints may depend on .     

How to study a gluon jet produced by antiproton—proton colliding beams

We show that at the energies and luminosities soon to be attained with antiproton—proton colliding beams, the cross section for antiquark—quark annihilation, leading to production of a gluon jet in association with a Z⁰ at large angles, is measurable and has a magnitude, energy dependence, and angular dependence which can serve to establish the existence of the gluon and to fix its spin.