Some triviality results for quasi-Einstein manifolds and Einstein warped products

In this paper we prove a number of triviality results for Einstein warped products and quasi-Einstein manifolds using different techniques and under assumptions of various nature. In particular we obtain and exploit gradient estimates for solutions of weighted Poisson-type equations and adaptations to the weighted setting of some Liouville-type theorems.Comment: 15 pages, fixed minor mistakes in Section

Renormalization in Quantum Mechanics

We implement the concept of Wilson renormalization in the context of simple quantum mechanical systems. The attractive inverse square potential leads to a \b function with a nontrivial ultraviolet stable fixed point and the Hulthen potential exhibits the crossover phenomenon. We also discuss the implementation of the Wilson scheme in the broader context of one dimensional potential problems. The possibility of an analogue of Zamolodchikov's $C$ function in these systems is also discussed.Comment: 16 pages, UR-1310, ER-40685-760. (Additional references included.

Vlasov Equation In Magnetic Field

The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are infinitely degenerate, which causes difficulty solving this initial value problem by using the conventional eigenfunction expansion method. Finally, the Vlasov equation is solved by the resolvent method.Comment: 15 page

A Map of Update Constraints in Inductive Inference

We investigate how different learning restrictions reduce learning power and how the different restrictions relate to one another. We give a complete map for nine different restrictions both for the cases of complete information learning and set-driven learning. This completes the picture for these well-studied \emph{delayable} learning restrictions. A further insight is gained by different characterizations of \emph{conservative} learning in terms of variants of \emph{cautious} learning. Our analyses greatly benefit from general theorems we give, for example showing that learners with exclusively delayable restrictions can always be assumed total.Comment: fixed a mistake in Theorem 21, result is the sam

Scattering Theory for Jacobi Operators with Steplike Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators with steplike quasi-periodic finite-gap background in the same isospectral class. We derive the corresponding Gel'fand-Levitan-Marchenko equation and find minimal scattering data which determine the perturbed operator uniquely. In addition, we show how the transmission coefficients can be reconstructed from the eigenvalues and one of the reflection coefficients.Comment: 14 page

General flux to a trap in one and three dimensions

The problem of the flux to a spherical trap in one and three dimensions, for diffusing particles undergoing discrete-time jumps with a given radial probability distribution, is solved in general, verifying the Smoluchowski-like solution in which the effective trap radius is reduced by an amount proportional to the jump length. This reduction in the effective trap radius corresponds to the Milne extrapolation length.Comment: Accepted for publication, in pres

Dimensional renormalization: ladders to rainbows

Renormalization factors are most easily extracted by going to the massless limit of the quantum field theory and retaining only a single momentum scale. We derive factors and renormalized Green functions to all orders in perturbation theory for rainbow graphs and vertex (or scattering diagrams) at zero momentum transfer, in the context of dimensional renormalization, and we prove that the correct anomalous dimensions for those processes emerge in the limit D -> 4.Comment: RevTeX, no figure

Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data

We develop direct and inverse scattering theory for Jacobi operators (doubly infinite second order difference operators) with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give necessary and sufficient conditions for the scattering data in the case of perturbations with finite second (or higher) moment.Comment: 23 page