The non{linear dynamics of self{gravitating collisionless matter with vanishing vorticity is studied in a general relativistic framework, using synchronous and comoving (i.e. Lagrangian) coordinates. Writing the equations in terms of the metric tensor of the spatial sections orthogonal to the uid ow allows an unambiguous expansion in inverse powers of the speed of light. The Newtonian and post{...

It is shown that every set of n integers contains a subset of size Ω(n) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for every > 0 and every m > m( ) the following holds. If A1, . . . , Am are m subsets of cardinality at least m each, then there are a1 ∈ A1, . . . , am ∈ Am so that the sum of every nonempty subset of the set ...

Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. non-vanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Al...

We study long-range interacting systems perturbed by external stochastic forces. Unlike the case of short-range systems, where stochastic forces usually act locally on each particle, here we consider perturbations by external stochastic fields. The system reaches stationary states where external forces balance dissipation on average. These states do not respect detailed balance and support non-...

We express the total equation of state parameter of a spatially flat FriedmanRobertson-Walker universe in terms of derivatives of the red-shift dependent spinweighted angular moments of the two-point correlation function of the three dimensional cosmic shear. We first express the total equation of state parameter in terms of the growing mode of the gauge invariant metric perturbation in the con...

Starting out with new deenitions for the gauge-invariant perturbations in energy density and particle number density, evolution equations for these quantities are derived which are manifestly gauge-invariant. In the case of vanishing pressure, these equations are identical to Poisson's equation. This fact is in contrast to what is found in earlier treatments based on other dee-nitions.

Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus by Kevin Purbhoo Doctor of Philosophy in Mathematics University of California at Berkeley Professor Allen Knutson, Chair We investigate several related vanishing problems in Schubert calculus. First we consider the multiplication problem. For any complex reductive Lie group G, many of the structure constants of the ordinary co...

The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved in [1] under Jensen’s Diamond Principle. In this note, the same is verified using the ZFC axioms alone. In the construction of large indecomposable divisible modules over certain valuation domainsR, the first author used the property thatR satisfied Ext1R(Q,U) 6= 0, where ...

Let f be a newform of even weight k, level M and character ψ and let g be a newform of even weight l, level N and character η. We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet L-functions, in the context of twisted modular L-functions associated to f and g. Using this result, we find a lower bound in terms of Q for the number of primitive Dirichlet cha...