In this paper we highlight how the Fonseca and Müller blow-up technique is particularly well suited for homogenization problems. As examples we give a simple proof of the nonlinear homogenization theorem for integral functionals and we prove a homogenization theorem for sets of finite perimeter.

The Regularity Lemma [16] is a powerful tool in Graph Theory and its applications. It basically says that every graph can be well approximated by the union of a constant number of random-looking bipartite graphs called regular pairs (see the definitions below). These bipartite graphs share many local properties with random bipartite graphs, e.g. most degrees are about the same, most pairs of ve...

The equation ut = ∆u + u with homegeneous Dirichlet boundary conditions has solutions with blow-up if p > 1. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.

Bershadsky, Cecotti, Ooguri and Vafa constructed a real-valued invariant for Calabi–Yau manifolds, which is now called the BCOV invariant. In this paper, we extend to such pairs $(X,D)$ , where $X$ compact Kähler manifold $D$ pluricanonical divisor on with simple normal crossing support. We also study behavior of extended under blow-ups. The results in paper lead joint work Fu proving that bira...

The HIV/AIDS epidemic has many serious consequences for children. These consequences are, however, rarely inevitable. Families can provide a protective barrier that deflects blows, or minimises their impact and a supportive nurturing environment that can help children recover from harm. If strong enough, and with sufficient access to quality services and support from communities, families can r...