Adaptive mesh reconstruction: Total Variation Bound

We consider 3-point numerical schemes for scalar Conservation Laws, that are oscillatory either to their dispersive or anti-diffusive nature. Oscillations are responsible for the increase of the Total Variation (TV); a bound on which is crucial for the stability of the numerical scheme. It has been noticed (\cite{Arvanitis.2001}, \cite{Arvanitis.2004}, \cite{Sfakianakis.2008}) that the use of non-uniform adaptively redefined meshes, that take into account the geometry of the numerical solution itself, is capable of taming oscillations; hence improving the stability properties of the numerical schemes. In this work we provide a model for studying the evolution of the extremes over non-uniform adaptively redefined meshes. Based on this model we prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We moreover prove under more strict assumptions that the increase of the TV -due to the oscillatory behaviour of the numerical schemes- decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D)

Higgsed Gauge-flation

We study a variant of Gauge-flation where the gauge symmetry is spontaneously broken by a Higgs sector. We work in the Stueckelberg limit and demonstrate that the dynamics remain (catastrophically) unstable for cases where the gauge field masses satisfy $\gamma < 2$, where $\gamma = g^2\psi^2/H^2$, $g$ is the gauge coupling, $\psi$ is the gauge field vacuum expectation value, and $H$ is the Hubble rate. We compute the spectrum of density fluctuations and gravitational waves, and show that the model can produce observationally viable spectra. The background gauge field texture violates parity, resulting in a chiral gravitational wave spectrum. This arises due to an exponential enhancement of one polarization of the spin-2 fluctuation of the gauge field. Higgsed Gauge-flation can produce observable gravitational waves at inflationary energy scales well below the GUT scale.Comment: 52 pages, 14 figure

A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix

Current biological knowledge supports the existence of a secondary group of cancer cells within the body of the tumour that exhibits stem cell-like properties. These cells are termed Cancer Stem Cells (CSCs}, and as opposed to the more usual Differentiated Cancer Cells (DCCs), they exhibit higher motility, they are more resilient to therapy, and are able to metastasize to secondary locations within the organism and produce new tumours. The origin of the CSCs is not completely clear; they seem to stem from the DCCs via a transition process related to the Epithelial-Mesenchymal Transition (EMT) that can also be found in normal tissue. In the current work we model and numerically study the transition between these two types of cancer cells, and the resulting "ensemble" invasion of the extracellular matrix. This leads to the derivation and numerical simulation of two systems: an algebraic-elliptic system for the transition and an advection-reaction-diffusion system of Keller-Segel taxis type for the invasion

Tensor Spectra Templates for Axion-Gauge Fields Dynamics during Inflation

$SU(2)$ gauge fields can generate large gravitational waves during inflation, if they are coupled to an axion which can be either the inflaton or a spectator field. The shape of the produced tensor power spectrum $\mathcal{P}_h$ depends on the form of the axion potential. We derive analytic expressions and provide general templates for $\mathcal{P}_h$ for various types of the spectator axion potential. Furthermore, we explore the detectability of the oscillatory feature, which is present in $\mathcal{P}_h$ in the case of an axion monodromy model, by possible future CMB B-mode polarization observations.Comment: 31 pages, 11 figure