Krull dimension,overrings and semistar operations of an integral domain

In this paper, we will present new developments in the study of the links between the cardinality of the sets $O(R)$ of all overrings of R, $\mathit{SSF}c(R)$ of all semistar operations of finite character when finite to the Krull dimension of an integral domain R. In particular, we prove that if $|\mathit{SSF}c(R)|=n+\dim R$, then R has at most $n?1$ distinct maximal ideals. Moreover, R has exactly $n?1$ maximal ideals if and only if $n=3$. In this case R is a Prüfer domain with exactly two maximal ideals and Y-graph spectrum. We also give a complete characterizations for local domains R such that $|\mathit{SSF}c(R)|=3+\dim R$, and nonlocal domains R with $|\mathit{SSF}c(R)|=|O(R)|=n+\dim R$ for $n=4$, $n=5$, $n=6$ and $n=7$. Examples to illustrate the scopes and limits of the results are constructed.     

Small-amplitude nonlinear waves on a black hole background

Let $G(x)$ be a $C^0$ function such that $|G(x)|⩽K{|x|}^p$ for $|x|⩽c$, for constants $K,c>0$. We consider spherically symmetric solutions of ${\square }_g\varphi =G(\varphi )$ where g is a Schwarzschild or more generally a Reissner–Nordström metric, and such that ϕ and ∇ϕ are compactly supported on a complete Cauchy surface. It is proven that for $p>4$, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, $|\varphi |⩽C{(\max \{v,1\})}^{\mathrm{-1}}$, where v denotes an Eddington–Finkelstein advanced time coordinate.     

On mixed codes with covering radius 1 and minimum distance 2: (extended abstract)

Let R, S and T be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when r divides s, when $r=s-1$, when s is large, relative to r, when r is large, relative to s, as well as $K(3r,2r,2r;2)$. Finally, a table with bounds on $K(r,s,s;2)$ is given.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.