On modified Runge–Kutta trees and methods

AbstractModified Runge–Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge–Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested

Local invariants of stabilizer codes

In [Phys. Rev. A 58, 1833 (1998)] a family of polynomial invariants which separate the orbits of multi-qubit density operators $\rho$ under the action of the local unitary group was presented. We consider this family of invariants for the class of those $\rho$ which are the projection operators describing stabilizer codes and give a complete translation of these invariants into the binary framework in which stabilizer codes are usually described. Such an investigation of local invariants of quantum codes is of natural importance in quantum coding theory, since locally equivalent codes have the same error-correcting capabilities and local invariants are powerful tools to explore their structure. Moreover, the present result is relevant in the context of multipartite entanglement and the development of the measurement-based model of quantum computation known as the one-way quantum computer.Comment: 10 pages, 1 figure. Minor changes. Accepted in Phys. Rev.

Poisson-Vlasov : Stochastic representation and numerical codes

A stochastic representation for the solutions of the Poisson-Vlasov equation, with several charged species, is obtained. The representation involves both an exponential and a branching process and it provides an intuitive characterization of the nature of the solutions and its fluctuations. Here, the stochastic representation is also proposed as a tool for the numerical evaluation of the solutionsComment: 17 pages, 5 figure

On the limiting distribution of the metric dimension for random forests

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure

A New Parallel N-body Gravity Solver: TPM

We have developed a gravity solver based on combining the well developed Particle-Mesh (PM) method and TREE methods. It is designed for and has been implemented on parallel computer architectures. The new code can deal with tens of millions of particles on current computers, with the calculation done on a parallel supercomputer or a group of workstations. Typically, the spatial resolution is enhanced by more than a factor of 20 over the pure PM code with mass resolution retained at nearly the PM level. This code runs much faster than a pure TREE code with the same number of particles and maintains almost the same resolution in high density regions. Multiple time step integration has also been implemented with the code, with second order time accuracy. The performance of the code has been checked in several kinds of parallel computer configuration, including IBM SP1, SGI Challenge and a group of workstations, with the speedup of the parallel code on a 32 processor IBM SP2 supercomputer nearly linear (efficiency $\approx 80\%$) in the number of processors. The computation/communication ratio is also very high ($\sim 50$), which means the code spends $95\%$ of its CPU time in computation.Comment: 21 Pages Latex file Figures available from anonymous ftp to astro.princeton.edu under /xu/tpm.ps, POP-57