On splitting the area under curves

Two area-splitting problems involving real-valued functions of a real variable are investigated. The second of these is essentially equivalent to finding all functions a?C¹((0, r)) with 0 < a(x) < x which satisfy the functional differential equation $\text{a′ (a(x)) =}\text{a(x)}\text{x}\text{for}\text{x ? (0, r)}$. All solutions analytic at x = 0 (and many which are not) are exhibited in closed form.     

SU(5) Higgs problem with adjoint + vector representations

Recently Kim has given a general method, using group-invariant orbit parameters, for determining the energy and residual symmetry of the Higgs potential minimum. In this paper we illustrate the method by working out the case of a quartic SU(5) Higgs potential with $\text{5}$ and $\text{24}$ higgsons. In this method the Gell-Mann-Slansky conjecture concerning possible little groups of the potential minimum takes a geometric form, which is verified for our case. The results are used to discuss the hierarchical symmetry breaking of SU(5) grand unification theory. We generalize our results to the SU(N) adjoint + vector models, which are all closely related.     

Thermodynamic properties of a fluid of hard spheres with water-like dipole and quadrupole moments

The thermodynamic properties of a fluid of hard spheres with dipole and tetrahedral quadrupole moments are investigated using thermodynamic perturbation theory. The drpole-quadrupole coupling is found to be very important at values of the multipole moments that are close to those of water.