Christoffel-Minkowski flows

We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term.Comment: 25 page

Orlicz-Minkowski flows

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p$-Minkowski problem for $p>1$. In the final section, we use a curvature flow with no global term to solve a class of $L_p$-Christoffel-Minkowski type problems.Comment: 30 page

Housing benefit and financial returns to employment for tenants in the social sector

This paper examines the impact of the UK housing benefit system on the financial returns to employment of people in local authority or Housing Association accommodation. It outlines the current structure of housing benefit and examines its effects on the returns to employment using data from the Family Expenditure Survey. It analyses the consequences of a number of reforms to the current system — lowering social rents, increasing the levels of housing benefit received in work and restricting the amount of rent covered by housing benefit payments. This analysis highlights the trade-offs involved in various strategies available for restructuring the present system.

Recurrent Acceleration in Dilaton-Axion Cosmology

A class of Einstein-dilaton-axion models is found for which almost all flat expanding homogeneous and isotropic universes undergo recurrent periods of acceleration. We also extend recent results on eternally accelerating open universes.Comment: 8 pages, 7 figures. minor changes. Version 4 corrects a figure captio

Dense Subgraphs in Random Graphs

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest to $\frac{2}{\alpha(\gamma,p)}\big(\log n-\log\log n+\log\frac{e\alpha(\gamma,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap