A quadratic Bolza-type problem in a Riemannian manifold

A classical nonlinear equation on a complete Riemannian manifold is considered. The existence of solutions connecting any two points is studied, i.e., for T>0 the critical points of the functional with x(0)=x₀,x(T)=x₁. When the potential V has a subquadratic growth with respect to x, J_T admits a minimum critical point for any T>0 (infinitely many critical points if the topology of is not trivial). When V has an at most quadratic growth, i.e., , this property does not hold, but an optimal arrival time T(λ)>0 exists such that, if 0<T<T(λ), any pair of points in can be joined by a critical point of the corresponding functional. For the existence and multiplicity results, variational methods and Ljusternik-Schnirelman theory are used. The optimal value is fulfilled by the harmonic oscillator. These ideas work for other related problems.     

Generic embeddings associated to an indestructibly weakly compact cardinal

I use generic embeddings induced by generic normal measures on P_κ(λ) that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower Q_<κ works much like it does when κ is a Woodin limit of Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness.     

On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.