Proof of the Boltzmann-Sinai Ergodic Hypothesis for Typical Hard Disk Systems

We consider the system of $N$ ($\ge2$) hard disks of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^2$. We prove the ergodicity (actually, the B-mixing property) of such systems for almost every selection $(m_1,...,m_N;r)$ of the outer geometric parameters.Comment: 58 page

Net worth and housing equity in retirement

This paper documents the trends in the life-cycle profiles of net worth and housing equity between 1983 and 2004. The net worth of older households significantly increased during the housing boom of recent years. However, net worth grew by more than housing equity, in part because other assets also appreciated at the same time. Moreover, the younger elderly offset rising house prices by increasing their housing debt, and used some of the proceeds to invest in other assets. We also consider how much of their housing equity older households can actually tap, using reverse mortgages. This fraction is lower at younger ages, such that young retirees can consume less than half of their housing equity. These results imply that ‘consumable’ net worth is smaller than standard calculations of net worth. JEL Classification: G11, E2

Bounds for solid angles of lattices of rank three

We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in $\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical configurations come up in connection with the kissing number problem.Comment: 12 pages; to appear in the Journal of Combinatorial Theory

Frobenius problem and the covering radius of a lattice

Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. The condition that $gcd(a_1,...,a_N)=1$ implies that such number exists. The general problem of determining the Frobenius number given $N$ and $a_1,...,a_N$ is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating Frobenius number to the covering radius of the null-lattice of this $N$-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.Comment: 12 pages; minor revisions; to appear in Discrete and Computational Geometr

On a Problem in Diophantine Approximation

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.Comment: 16 page