Critical phenomena and relativistic gravitational collapse

Recent calculations have shown the existence of critical phenomena in general relativity associated with the collapse of wavepackets of massless fields that are near, in parameter space, the onset of black hole formation (the critical point). Two physically distinct systems have been explored: collapse of spherically-symmetric massless scalar field and collapse of vacuum, axisymmetric gravitational waves. Nonlinear effects dominate near the critical point. Black-hole mass serves as an order parameter and has a power-law dependence on critical separation in the supercritical region of parameter space. Remarkably, the values of the critical exponent of the power law are nearly identical in the two systems. The nonlinearity induces the fields to oscillate. Each successive oscillation is an echo, obeying a spatial and temporal scaling relation.     

Canp-adic numbers be useful to regularize divergent expectation values of quantum observables?

We show howp-adic analysis can be used in some cases to treat divergent series in quantum mechanics. We consider examples in which the usual theory of the Schrödinger equation would give rise to an infinite expectation value of the energy operator. By usingp-adic analysis, we are able to get a convergent expansion and obtain a finite rational value for the energy. We present also the main ideas to interpret a quantum mechanical state by means ofp-adic statistics.     

The definition and calculation of interfacial energies for thin films

We provide a new definition of the interfacial energy which eliminates three physically extraneous contributions from the conventional definition: (1) the strain or stress energy due to lattice mismatch between film and substrate; (2) the surface energy of the film-vacuum interface; and, (3) the substrate surface energy contribution from substrate layers below the film layers. This new interface energy then quantifies the variation in interactions among film/substrate, film/film and substrate/substrate bonding. Using this new definition, we derive the equations for evaluation of the interfacial energy in terms of the interaction energy for any atom in each layer of the film/substrate, film/film and substrate/substrate systems. With this formulation, it is simple to determine the dependence of the interfacial energy on the film thickness using virtually any interaction potential. Using a corrected effective medium theory, we present results for a few pseudomorphic film systems containing Ni/Cu, Ni/Ag, Cu/Ag and Rh/Ag on (111) and (100) surfaces. These systems cover a wide range of lattice mismatch and alloy formation energies. The results demonstrate that the new definition of interfacial energy converges after only 3–4 film layers, regardless of the degree of lattice mismatch. We also show that the interfacial energies at (100) and (111) interfaces differ and that the interfacial energy for a given pair of materials depends on which of the materials is the film.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.

     

Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule

A comparison principle for viscosity sub- and super-solutions of second order elliptic partial differential equations is derived using the “fuzzy sum rule” of non-smooth calculus. This method allows us to weaken the assumptions made on the function F when the equation F(x,u,Du,D²u)=0 is under consideration.