Long Time Dynamics of Resonant Systems

This thesis studies the long time dynamics of resonant systems in the weakly nonlinear regime. It is divided into two main parts. In the first one, we consider the resonant equation, which captures the energy transfer between normal modes of the system. Different tools to extract analytic information from the resonant equation are developed. After that, we apply them to a large number of resonant models. Some of them consist of a scalar field in different geometries as well as the Gross-Pitaevskii equation. In the second part of this thesis, asymptotically anti-de Sitter geometries subject to time-periodic boundary conditions are studied. The phenomenology allowed by these conditions is explored through the environment of time-periodic geometries. In particular, we construct their phase-space and delimit the regions of linear stability. We also present a protocol to dynamically construct time-periodic geometries

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Holographic thermalization in finite-size systems

The AdS/CFT correspondence has provided a fascinating window into the real-time dynamics of strongly-coupled QFTs. On the other hand, studies of gravitational collapse in asymptotically global AdS spacetimes have unraveled a surprisingly rich landscape of possible routes to final black hole formation or its absence thereof, that depend both on the precise dynamics and the initial state considered. In the light of the equivalence between gravitational collapse and thermalization implied by the duality, it is natural to wonder which universal lessons these results entail on the out-of-equilibrium physics of isolated, macroscopic quantum systems at strong coupling. This thesis aims to be a modest step in elucidating this question

Modified Theories of Gravity and Cosmological Applications

This reprint focuses on recent aspects of gravitational theory and cosmology. It contains subjects of particular interest for modified gravity theories and applications to cosmology, special attention is given to Einstein–Gauss–Bonnet, f(R)-gravity, anisotropic inflation, extra dimension theories of gravity, black holes, dark energy, Palatini gravity, anisotropic spacetime, Einstein–Finsler gravity, off-diagonal cosmological solutions, Hawking-temperature and scalar-tensor-vector theories

Mathematics & Statistics 2017 APR Self-Study & Documents

UNM Mathematics & Statistics APR self-study report, review team report, response report, and initial action plan for Spring 2017, fulfilling requirements of the Higher Learning Commission

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described