The Review - Fall 2001

IN THIS ISSUE 1 - Message From The Dean 2 - A Special Welcome for Alumni Babies 2 - A Very Special Offer for Our Alumni 3 - Farewell to Joann Ludwig 4 - The Admissions-Alumni Partnership 5 - JAVA is Brewing at Jefferson! 6 - What A Year! 8 - Alumni Update 10 - Alumni News Form 11 - Visiting Scholar 2001: A Nurse Alumna Sets the Agenda 12 - Michael Hartman Elected New CHP Alumni President 13 - Commencement 200

Efficient dot product over word-size finite fields

We want to achieve efficiency for the exact computation of the dot product of two vectors over word-size finite fields. We therefore compare the practical behaviors of a wide range of implementation techniques using different representations. The techniques used include oating point representations, discrete logarithms, tabulations, Montgomery reduction, delayed modulus

Bounds on the coefficients of the characteristic and minimal polynomials

This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise input-dependant bounds on these coefficients. Such bounds are e.g. useful to perform deterministic chinese remaindering of the characteristic or minimal polynomial of an integer matrix

Two-Party Competition with Persistent Policies

This paper studies the Markov perfect equilibrium outcomes of a dynamic game of electoral competition between two policy-motivated parties. I model incumbent policy persistence: parties commit to implement a policy for their full tenure in office, and hence in any election only the opposition party renews its platform. In equilibrium, parties alternate in power and policies converge to symmetric alternations about the median voter's ideal policy. Parties' disutility from opponents' policies leads to alterna- tions that display bounded extremism; alternations far from the median are never limits of equilibrium dynamics. Under a natural restriction on strategies, I find that robust long-run outcomes display bounded moderation; alternations close to the median are reached in equilibrium only if policy dynamics start there. I show that these results are robust to voters being forward-looking, the introduction of term limits, costly policy adjustments for incumbents, and office benefits.

Keeping Your Options Open

In standard models of experimentation, the costs of project development consist of (i) the direct cost of running trials as well as (ii) the implicit opportunity cost of leaving alternative projects idle. Another natural type of experimentation cost, the cost of holding on to the option of developing a currently inactive project, has not been studied. In a (multi-armed bandit) model of experimentation in which inactive projects have explicit maintenance costs and can be irreversibly discarded, I fully characterise the optimal experimentation policy and show that the decision-maker's incentive to actively manage its options has important implications for the order of project development. In the model, an experimenter searches for a success among a number of projects by choosing both those to develop now and those to maintain for (potential) future development. In the absence of maintenance costs, the optimal experimentation policy has a 'stay-with-the-winner' property: the projects that are more likely to succeed are developed first. Maintenance costs provide incentives to bring the option value of less promising projects forward, and under the optimal experimentation policy, projects that are less likely to succeed are sometimes developed first. A project development strategy of 'going-with-the-loser' strikes a balance between the cost of discarding possibly valuable options and the cost of leaving them open.

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Hammer et al. and Merris, the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure

An introspective algorithm for the integer determinant

We present an algorithm computing the determinant of an integer matrix A. The algorithm is introspective in the sense that it uses several distinct algorithms that run in a concurrent manner. During the course of the algorithm partial results coming from distinct methods can be combined. Then, depending on the current running time of each method, the algorithm can emphasize a particular variant. With the use of very fast modular routines for linear algebra, our implementation is an order of magnitude faster than other existing implementations. Moreover, we prove that the expected complexity of our algorithm is only O(n^3 log^{2.5}(n ||A||)) bit operations in the dense case and O(Omega n^{1.5} log^2(n ||A||) + n^{2.5}log^3(n||A||)) in the sparse case, where ||A|| is the largest entry in absolute value of the matrix and Omega is the cost of matrix-vector multiplication in the case of a sparse matrix.Comment: Published in Transgressive Computing 2006, Grenade : Espagne (2006

Symmetric indefinite triangular factorization revealing the rank profile matrix

We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization $\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{L}\mathbf{D}\mathbf{L}^T$ where $\mathbf{P}$ is a permutation matrix, $\mathbf{L}$ is lower triangular with a unit diagonal and $\mathbf{D}$ is symmetric block diagonal with $1{\times}1$ and $2{\times}2$ antidiagonal blocks. The novel algorithm requires $O(n^2r^{\omega-2})$ arithmetic operations. Furthermore, experimental results demonstrate that our algorithm can even be slightly more than twice as fast as the state of the art unsymmetric Gaussian elimination in most cases, that is it achieves approximately the same computational speed. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. There is an obstruction in characteristic $2$ for revealing the rank profile matrix which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard $\mathbf{P}\mathbf{L}\mathbf{D}\mathbf{L}^T\mathbf{P}^T$ decomposition at a negligible cost