Let the space $\mathbb R^n$ be endowed with a Minkowski structure $M$ (that is, $M\colon \mathbb R^n \to [0,+\infty )$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$ ), and let $d^M(x,y)$ be the (asymmetric) distance associated to $M$ . Given an open domain $\Omega \subset \mathbb R^n$ of class $C^2$ , let $d_{\Omega }(x) := \inf \{d^M(x,y);\ y\in \partial \Omega \}$ be the Minkowski distance of a point $x\in \Omega$ from the boundary of $\Omega$ . We prove that a suitable extension of $d_{\Omega }$ to $\mathbb R^n$ (which plays the rôle of a signed Minkowski distance to $\partial \Omega$ ) is of class $C^2$ in a tubular neighborhood of $\partial \Omega$ , and that $d_{\Omega }$ is of class $C^2$ outside the cut locus of $\partial \Omega$ (that is, the closure of the set of points of nondifferentiability of $d_{\Omega }$ in $\Omega$ ). In addition, we prove that the cut locus of $\partial \Omega$ has Lebesgue measure zero, and that $\Omega$ can be decomposed, up to this set of vanishing measure, into geodesics starting from $\partial \Omega$ and going into $\Omega$ ...

Introgressive hybridization represents one of the long-lasting debated genetic consequences of species range expansion. Mitochondrial DNA has been shown to heavily introgress between interbreeding animal species that meet in new sympatric areas and, often, asymmetric introgression from local to the colonizing populations has been observed. Disentangling among the evolutionary and ecological processes that might shape this pattern remains difficult, because they continuously act across time and space. In this context, long-term studies can be of paramount importance. Here, we investigated the dynamics of mitochondrial introgression between two mosquito species (Aedes mariaeandAe. zammitii ) during a colonization event that started in 1986 after a translocation experiment. By analyzing 1,659 individuals across 25 years, we showed that introgression occurred earlier and at a higher frequency in the introduced than in the local species, showing a pattern of asymmetric introgression. Throughout time, introgression increased slowly in the local species, becoming reciprocal at most sites. The rare opportunity to investigate the pattern of introgression across time during a range expansion along with the characteristics of our study-system allowed us to support a role of demographic dynamics in determining the observed introgression pattern.

This paper is concerned with the Cauchy problem $(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\overline{u}(x),$ for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear. Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\overline{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique "viscosity solution", defined for $t\in [0,T]$, depending continuously on the initial data.