On the optimality of the neighbor-joining algorithm

The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map. From this point of view, NJ is ``optimal'' when the algorithm outputs the tree which minimizes the balanced minimum evolution criterion. We use the fact that the NJ tree topology and the BME tree topology are determined by polyhedral subdivisions of the spaces of dissimilarity maps ${\R}_{+}^{n \choose 2}$ to study the optimality of the neighbor-joining algorithm. In particular, we investigate and compare the polyhedral subdivisions for $n \leq 8$. A key requirement is the measurement of volumes of spherical polytopes in high dimension, which we obtain using a combination of Monte Carlo methods and polyhedral algorithms. We show that highly unrelated trees can be co-optimal in BME reconstruction, and that NJ regions are not convex. We obtain the $l_2$ radius for neighbor-joining for $n=5$ and we conjecture that the ability of the neighbor-joining algorithm to recover the BME tree depends on the diameter of the BME tree

On Quantization of Black Holes

A simple argument is presented in favour of the equidistant spectrum in semiclassical limit for the horizon area of a black hole. The following quantization rules for the mass $M_N$ and horizon area $A_{Nj}$ are proposed: M_N = m_p [N(N+1)]^{1/4}; A_{Nj} = 8\pi l_p^2 [\sqrt{N(N+1)} + \sqrt{N(N+1) - j(j+1)} ]. Here both $N$ and $j$ are nonnegative integers or half-integers.Comment: 4 pages, late

The Christian Sacraments of Initiation: Baptism, Confirmation, Eucharist

Reviewed Book: Osborne, Kenan B. The Christian Sacraments of Initiation: Baptism, Confirmation, Eucharist. Mahwah, NJ: Paulist Press, 1987

Book Review: Thinking About Art

Book review for Thinking About Art, Edmund Burke Feldman, Prentice Hall, Englewood Cliffs, NJ, 1985

Imaginal preaching: an archetypal perspective

Reviewed Book: Wallace, James A. Imaginal preaching: an archetypal perspective. Mahwah, NJ: Paulist Pr, 1995

The Covenant Never Revoked: Biblical Reflections on a Christian-Jewish Dialogue

Reviewed Book: Lohfink, Norbert. The Covenant Never Revoked: Biblical Reflections on a Christian-Jewish Dialogue. Mahwah, NJ: Paulist Press, 1991