Dimensional dual hyperovals in projective spaces and c.AGast-geometries
The two subjects of the title are studied, first independently and then by making them interact. Many questions arise from this interaction.In our intrinsic study of the first subject, we construct a new family of examples and define the notions of quotient and universality. For the second subject, we restrict the class c.AG* of circular extensions of dual affine spaces under the two geometrical conditions (LL) and (T) by showing that, apart from some extreme cases, every such rank three geometry can be erected into a rank 4 geometry. In particular, c.AG*n(q)-geometries satisfying (LL) and (T) with n ≥ 3 must have q even. We then deduce similar results for the first subject.
A Construction of Extended Generalized Quadrangles Using the Veronesean
A family ofc.C2-geometries with point residues isomorphic to the dual of the Tits quadrangles.T*2(O) for the regular hyperovalOcan be constructed using the quadratic Veronesean.
General expression for the density effect for the ionization loss of charged particles
The equations for the density-effect correction $\ensuremath{\delta}(X)$ for the ionization loss as given in a previous paper have been extended to the case where $\ensuremath{\delta}({X}_{0})\ensuremath{\equiv}{\ensuremath{\delta}}_{0}g0$. Here ${X}_{0}$ is the lower limit of the range ${X}_{0}lXl{X}_{1}$, where a particularly simple expression for $\ensuremath{\delta}(X)$ applies; $X={log}_{10}(\frac{p}{{m}_{0}c})$, where $p$ is the momentum and ${m}_{0}$ is the mass of the incident particle. The case ${\ensuremath{\delta}}_{0}g0$ applies directly to the calculation of $\ensuremath{\delta}(X)$ for conductors (metals).