We prove development of type I singularities for a suitable concept of weak evolution in the context of the quintic focussing NLW on R3`1, for data near and “above” the ground state static solution.

1 Departamento de Fı́sica, Ingenierı́a de Sistemas y Teorı́a de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain 2 Instituto Universitario de Fı́sica Aplicada a las Ciencias y las Tecnologı́as, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain 3 Departamento de Ciencias Médicas, Facultad de Medicina, Universidad de Castilla-La Mancha, C/Almansa No. 14, 02006 Albacete, ...

In this paper we first characterize all quasi–homogeneous but non–homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefold...

We construct a hypersurface of degree 5 in projective space P(C) which contains exactly 23436 ordinary nodes and no further singularities. This limits the maximum number μ8(5) of ordinary nodes a hyperquintic in P(C) can have to 23436 ≤ μ8(5) ≤ 27876. Our method generalizes the approach by the 3 author for the construction of a quintic threefold with 130 nodes in an earlier paper.

Manin’s conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A2.

Surfaces of general type with positive second Segre number s2 := c 2 1 − c2 > 0 are known by results of Bogomolov to be algebraically quasihyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of g...