Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

AbstractWe present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results

Reaction-diffusion processes and their interdisciplinary applications

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-12-201

Holographic Brownian Motion in 1+1 Dimensions

We study the motion of a stochastic string in the background of a BTZ black hole. In the 1+1 dimensional boundary theory this corresponds to a very heavy external particle (e.g, a quark), interacting with the fields of a CFT at finite temperature, and describing Brownian motion. The equations of motion for a string in the BTZ background can be solved exactly. Thus we can use holographic techniques to obtain the Schwinger-Keldysh Green function for the boundary theory for the force acting on the quark. We write down the generalized Langevin equation describing the motion of the external particle and calculate the drag and the thermal mass shift. Interestingly we obtain dissipation even at zero temperature for this 1+1 system. Even so, this does not violate boost (Lorentz) invariance because the drag force on a constant velocity quark continues to be zero. Furthermore since the Green function is exact, it is possible to write down an effective membrane action, and thus a Langevin equation, located at a "stretched horizon" at an arbitrary finite distance from the horizon.Comment: 39 pages, 4 figures; New discussion on holographic RG in subsection 5.1 and new section 6 on different time scales. Modified discussion on zero temperature dissipation in section 3. Typos corrected , references added. Revised version to appear in Nucl. Phys.