Resolutions via homological perturbation

The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes produced by the method discussed here with those produced by other methods show

Resolutions which split off of the bar construction

AbstractResolutions which split off of the bar construction are quite common, but explicit formulae expressing these splittings are not often encountered. Given explicit splitting data, perturbations of resolutions can be computed and the perturbed resolutions can be used tomake complete effective calculations where previously only partial or indirect results were obtainable.This paper gives a foundation for the perturbation method in homological algebra by providing a symbolic encoding of binomial coefficient functions which is useful in deriving formulae for an infinite class of resolutions. Formulae for perturbations of those resolutions are then derived. Applications to certain infinite families of groups and monoids are given.The research for this theory as well as the calculation of closed formulae within the theory was aided by new methods in symbolic computation using the Axiom (formerly called Scratchpad) system

Introduction to the Quantum Yang-Baxter Equation and Quantum Groups : An Algebraic Approach

XX, 293 tr.; 25 cm

A fixed point approach to homological perturbation theory

We show that the problem handled by classical homological perturbation theory can be reformulated as a fixed point problem leading to new insights into the nature of its solutions. We show, under mild conditions that the solution is essentially unique. x1

Computing Resolutions over Finite p-Groups

. A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite p-group is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented. 1 Introduction In this paper, we present a uniform constructive approach to calculating relatively small resolutions over nite p-groups. The algorithm we use comes from [32, 8.1.8 and the penultimate paragraph of 9.4]. There has been a massive amount of work done on the structure of p-groups since the beginning of group theory. A good introduction is [22]. We combine mathematical and computer methods to construct the uniform resolut..

AIAA #2007-4260 Quasi-Random Monte Carlo Integration for Computing Dissociation Rates

We describe our work towards implementing quasi-random Monte Carlo integration techniques for computing dissociation rates in chemical kinetics. Preliminary results indicated a significant advantage with quasi-random Monte Carlo methods, which yield accurate results at a reduced computational cost as compared to standard pseudo-random monte carlo. The internal energy relaxation processes of vibrational energy transfer, dissociation and recombination were modeled using state-to-state kinetics of diatomic nitrogen. Computational results for this test case indicate that (i) the quasi-random method converges to the same solution as the standard technique and (ii) the variability in the results are correspondingly reduced. Provided that the underlying theory is correct, the new method is an advantageous alternative to pseudo-random Monte Carlo integration. The state-specific rates can be incorporated into a solution of the master kinetic equations coupled to the fluid dynamic equations to describe the thermo-chemical non-equilibrium phenomenon in high temperature hypersonic flowfields. Introduction Shock waves in high speed flow of air present considerable difficulties for accurate numerical simulation of the flow around aerospace vehicles. The shock wave redistributes the high kinetic energy of the oncoming flow into various internal energy modes with varying time scales, leading to significant chemical and thermal non-equilibrium in the stagnation region of the vehicle. In the gas kinetic description, intermolecular collisions change the translational, rotational, vibrational, and electronic energies of the collision partners. 1 Thermal dissociation for these high temperature flows is a common occurrence, which needs to be modeled accurately to predict aerodynamic and heat loads experienced by the vehicle. The statistical aspect in the modeling of the dissociation process is the subject of the present study. Generally, in modern hypersonic codes, the rate-controlling temperature to determine the non-equilibrium rates for dissociation is taken to be the geometric mean of the equilibrated translational-rotational temperature and the single vibrational temperature of the diatomic molecule. This empirical model known as the Park model 2 is widely used in hypersonic codes today due to its simplicity. However, the Park model lacks a physical basis and has been shown to be inaccurate for a wide range of temperatures. 3 The state kinetic modeling approach advanced by Billing 4, 5 using the semiclassical theory to calculate transition rates for atom-diatom and diatom-diatom collisions was extended by Macheret and Adamovich 6 to develop a theory of dissociation of diatomic molecules based on the anharmonicity-corrected and energy-symmetrized forced harmonic oscillator (FHO) quantum scaling 7 in conjunction with free-rotation or impulsive energy-transfer models. The model predicts state specific dissociation rates by accounting for molecular rotation and three-dimensional collisions and has the advantage of being computationally tractable without any adjustable parameters. However, in the modeling of state kinetic dissociation, the evaluation of the multi-dimensional integrals in the dissociation cross section model for atom-molecule and molecule-molecule collisions is currently done using Monte Carlo methods and its use in today's hypersonic codes is not fully realized due to the extremely high computational requirements. In the work of Billing and subsequent work of Macheret and Adamovich 6 the multi-dimensional Monte Carlo integration was implemented by repeated sub routine calls to a pseudo random number generator 8 by using a different seed each time. In their method, each subroutine call returns a random value for use as a given parameter of the phase space and also returns a random seed for the random generation of the next parameter. In this wa