The Average-Case Area of Heilbronn-Type Triangles

From among ${n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < \mu_n < C/n^3$ for all large enough values of $n$, where $\mu_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a roll, {\em New Scientist}, November 6, 1999, 44--4

Pade-resummed high-order perturbation theory for nuclear structure calculations

We apply high-order many-body perturbation theory for the calculation of ground-state energies of closed-shell nuclei using realistic nuclear interactions. Using a simple recursive formulation, we compute the perturbative energy contributions up to 30th order and compare to exact no-core shell model calculations for the same model space and Hamiltonian. Generally, finite partial sums of this perturbation series do not show convergence with increasing order, but tend to diverge exponentially. Nevertheless, through a simple resummation via Pade approximants it is possible to extract rapidly converging and highly accurate results for the ground state energy.Comment: 6 pages, 2 figures, 1 tabl

Follow-up of referrals to the Social Service Department from the Out-Patient Department at the Boston State Hospital during 1950.

Thesis (M.S.)--Boston Universit

Pinching of the First Eigenvalue of the Laplacian and almost-Einstein Hypersurfaces of the Euclidean Space

In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost Einstein is diffeomorpic and almost isometric to a standard sphere.Comment: 18 page