Radiative Contributions to the Effective Action of Self-Interacting Scalar Field on a Manifold with Boundary

The effect of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The analysis is most easily performed in a space of constant curvature the boundary of which is characterised by constant extrinsic curvature. An extension of the spherical formulation in the presence of a boundary is attained through use of the method of images. Contrary to the consolidated vanishing effect in maximally symmetric space-times the contribution of the massless "tadpole" diagram no longer vanishes in dimensional regularisation. As a result, conformal invariance is broken due to boundary-related vacuum contributions. The evaluation of one-loop contributions to the two-point function suggests an extension, in the presence of matter couplings, of the simultaneous volume and boundary renormalisation in the effective action.Comment: 14 pages, 1 figure. Additional references and minor elucidating remarks added. To appear in Classical and Quantum Gravit

Learning to Use Visualizations (an example with elevation and temperature)

The purpose of this activity is to introduce students to visualizations as a tool for scientific problem-solving, using elevation and temperature as an example. Students color in visualizations of elevation and temperature so that important patterns in the data become evident. The relationship between the two quantities is studied by using them to compute the lapse rate, the rate at which temperature falls with increasing elevation. Intended outcomes are that students can identify and communicate important patterns in a dataset by drawing a visualization, can begin to interpret those patterns, and can analyze the correlation between two variables using visualization as a tool. Educational levels: Middle school, High school

Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces

We model a problem motivated by road design as a feasibility problem. Projections onto the constraint sets are obtained, and projection methods for solving the feasibility problem are studied. We present results of numerical experiments which demonstrate the efficacy of projection methods even for challenging nonconvex problems

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

On the nontrivial projection problem

The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension greater than one) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true "up to a logarithmic factor."Comment: 17 page

Existence and non existence results for the singular Nirenberg problem

In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793–821, 1991), of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities. Besides, we find a class of functions on S^2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp