Encounters with degeneracy: A personal view

We give a brief history of degeneracy and cycling and discuss aspects of these topics that have been of interest to the author.     

Blocks and the nonorientable genus of graphs

Examples are given to show that the nonorientable genus of a graph is not additive over its blocks. A nonorientable analog for the Battle, Harary, Kodama, and Youngs Theorem is proved; this completely determines the nonorientable genus of a graph in terms of its blocks. It is also shown that one of the above counterexamples has the minimum possible order.     

Possibility of a doubly-charged structure at theS-meson mass

The various symmetry breaking patterns that are possible with a 24-plet and a 5-plet of Higgs fields in SU(5) are classified. Spontaneous breakdown of SU(5) to SU(3)_colour×U(1) _em can be achieved (excluding the gauge hierarchy problem) without imposing unnatural constraints on the parameters of the Higgs potential.     

Production of Z (W) in association with a jet at unusually large transverse momentum

We point out an unusual energy-independent component of the elementary two-body cross section for quark-antiquark annihilation into an intermediate vector boson in association with a jet which arises from a hypothetical spin-zero quantum. This leads to an unexpected extension of the transverse momentum distribution, with possible relevance to some of the novel events seen at the CERN Sp$\text{p}$S collider.     

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.     

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 .