Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations

The influence of fractional diffusion in Fisher-KPP equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable L\'evy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model

A note on the convergence of parametrised non-resonant invariant manifolds

Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds

Electrostática y EAO : una experiencia de simulación

In this work the inclusion of educational software in the teaching of physics is analyzed. The program used corresponds to the computer simulation of electrostatic fields, and it was applied to students of first year of universitary level in a Faculty of Chemistry

On the use of fast blue, fluoro-gold and diamidino yellow for retrograde tracing after peripheral nerve injury: uptake, fading, dye interactions, and toxicity

The usefulness of three retrograde fluorescent dyes for tracing injured peripheral axons was investigated. The rat sciatic was transected bilaterally and the proximal end briefly exposed to either Fast Blue (FB), Fluoro-Gold (FG) or to Diamidino Yellow (DY) on the right side, and to saline on the left side, respectively. The nerves were then resutured and allowed to regenerate. Electrophysiological tests 3 months later showed similar latencies and amplitudes of evoked muscle and nerve action potentials between tracer groups. The nerves were then cut distal to the original injury and exposed to a second (different) dye. Five days later, retrogradely labelled neurones were counted in the dorsal root ganglia (DRGs) and spinal cord ventral horn, The number of neurones labelled by the first tracer was similar for all three dyes in the DRG and ventral horn except for FG, which labelled fewer motoneurones. When used as second tracer, DY labelled fewer neurones than FG and FB in some experimental situations. The total number of neurotics labelled by the first and/or second tracer was reduced by about 30% compared with controls. The contributions of cell death as well as different optional tracer combinations for studies of nerve regeneration are discussed

H^s versus C^0-weighted minimizers

We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H^s-topology.Comment: 15 page

Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s \in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.Comment: 42 page