We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation ∂xi u∂xj u∂ 2 xixj u = f in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions property, of the visc...

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For the theoretical study of real-world networks, we propose a model of scale-free randomgraphs with underlying geometry that we call geometric inhomogeneous random graphs (GIRGs).GIRGs generalize hyperbolic random graphs, which are a popular model to test algorithms forsocial and technological networks. Our generalization overcomes some limitations of hyperbolicrandom graphs, w...

In this thesis, we study a class of control problems which involves controlling a large number of dynamical systems with different values of parameters governing the system dynamics by using the same control signal. We call such problems control of inhomogeneous ensembles. The motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in Nuclear Magne...

Single-particle diffusion, also known as Brownian motion, is the random motion of a particle in space due to collisions with its surrounding medium. The theory of diffusion in a homogeneous medium, one with a constant diffusion coefficient, is well established and recognized. The theoretical basis of diffusion in heterogeneous environments, where the diffusion coefficient varies in space, is fa...

We introduce a multi-parameter generalization of the Lambda-determinant of Robbins and Rumsey, based on the cluster algebra with coefficients attached to a T -system recurrence. We express the result as a weighted sum over alternating sign matrices.

We present exact inhomogeneous and anisotropic cosmological solutions of low-energy string theory containing dilaton and axion fields. The spacetime metric possesses cylindrical symmetry. The solutions describe ever-expanding universes with an initial curvature singularity and contain known homogeneous solutions as subcases. The asymptotic form of the solution near the initial singularity has a...

The sudden injection of energy in a strongly coupled conformal field theory and its subsequent thermalization can be holographically modeled by a shell falling into anti-de Sitter space and forming a black brane. For a homogeneous shell, Bhattacharyya and Minwalla were able to study this process analytically using a weak field approximation. Motivated by event-by-event fluctuations in heavy ion...

Some exact solutions for the Einstein eld equations corresponding to inhomogeneous G2 cosmologies with an exponential-potential scalar eld which generalize solutions obtained previously are considered. Several particular cases are studied and the properties related to generalized in ation and asymptotic behaviour of the models are discussed.

The ‘classical’ random graphs, introduced by Erdős and Rényi half a century ago, are homogeneous in the sense that all their vertices play the same role and the degrees of the vertices tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been much recent interest in de...