Computations in Large N Matrix Mechanics

The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on the planar connected parts X_w, namely that they should decrease rapidly with increasing order. This leads to algebraic/variational schemes of computation which show remarkably rapid convergence in numerical tests on some many- matrix models. The method allows the calculation of all moments of the ground state, in a sequence of approximations, and excited states can be determined as well. There are two unexpected findings: a large d expansion and a new selection rule for certain types of interaction.Comment: 27 page

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions

A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements

A parallel fast direct solver for rank-compressible block tridiagonal linear systems is presented. Algorithmic synergies between Cyclic Reduction and Hierarchical matrix arithmetic operations result in a solver with $O(N \log^2 N)$ arithmetic complexity and $O(N \log N)$ memory footprint. We provide a baseline for performance and applicability by comparing with well known implementations of the $\mathcal{H}$-LU factorization and algebraic multigrid with a parallel implementation that leverages the concurrency features of the method. Numerical experiments reveal that this method is comparable with other fast direct solvers based on Hierarchical Matrices such as $\mathcal{H}$-LU and that it can tackle problems where algebraic multigrid fails to converge

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002