We consider a surjective morphism f : X → Y from a smooth projective variety X onto a smooth projective variety Y with connected fibers, henceforth called a fibration for short. Typically, if a certain geometric object on X like a cohomology class has a certain property, then its restriction to a smooth fiber of f trivially satisfies the same property. The converse question is of more interest:...

We discuss the radiation problem of total reflection for a time-harmonic generalized Maxwell system in a nonsmooth exterior domain Ω ⊂ RN , N ≥ 3 , with nonsmooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r−τ , τ > 1 , towards the identity. By means of the limiting absorption principle a Fredholm alternative holds true and the eigensolutions decay polynomially...

We consider simple random walk on the family tree T of a nondegenerate supercritical Galton-Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary of T . Concretely, this implies that an exponentially small fraction of the nth level of T carries most of the harmonic measure. First order asymptotics for ...

This article studies the numerical approximation of harmonic maps into surfaces, i.e., critical points for the Dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold. An iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its g...

We discuss the harmonicity of horizontally conformal maps and their relations with the spectrum of the Laplacian. We prove that if φ : M → N is a horizontally conformal map such that the tension field is divergence free, then φ is harmonic. Furthermore, if N is noncompact, then φ must be constant. Also we show that the projection of a warped product manifold onto the first component is harmonic...

Coded distributed computing introduced by Li et al. in 2015 is an efficient approach to trade computing power to reduce the communication load in general distributed computing frameworks such as MapReduce. In particular, Li et al. show that increasing the computation load in the Map phase by a factor of r can create coded multicasting opportunities to reduce the communication load in the Reduce...

In this paper we represent harmonic moments in the language of transfinite functions, that is projective limits of polynomials in infinitely many variables. We obtain also an explicit formula for the Jacobian of a generalized harmonic moment map. Mathematics Subject Classification (2000). 13B35; 30E05; 47A57.

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute n...

An ODE variational calculation shows that an image principle curvature ratio factor, (√ ρ1 ρ2 + √ ρ2 ρ1 ) , can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into R. In certain situations, including all radially symmetry harmonic maps, equality is achieved.