We study di erent notions of blow-up of a scheme X along a subscheme Y , depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to the embedding of X into a nonsingular variety. We prove that this latter blow-up is intrinsic of Y and X , and is u...

Weobtain a hypergraph generalisation of the graph blow-up lemma proved byKomlós, Sarközy and Szemerédi, showing that hypergraphs with sufficient regularity and no atypical vertices behave as if they were complete for the purpose of embedding bounded degree hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 39, 275–376, 2011

In this paper we consider the slightly L-supercritical gKdV equations ∂tu + (uxx + u|u|)x = 0, with the nonlinearity 5 < p < 5 + ε and 0 < ε ≪ 1 . In the previous paper [10] we know that there exists an stable selfsimilar blow-up dynamics for slightly L-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of ...