Complexity Estimates for Two Uncoupling Algorithms

Uncoupling algorithms transform a linear differential system of first order into one or several scalar differential equations. We examine two approaches to uncoupling: the cyclic-vector method (CVM) and the Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the scalar equations produced by CVM, and design a fast variant of CVM whose complexity is quasi-optimal with respect to the output size. We exhibit a strong structural link between CVM and DBZ enabling to show that, in the generic case, DBZ has polynomial complexity and that it produces a single equation, strongly related to the output of CVM. We prove that algorithm CVM is faster than DBZ by almost two orders of magnitude, and provide experimental results that validate the theoretical complexity analyses.Comment: To appear in Proceedings of ISSAC'13 (21/01/2013

Computer algebra tools for Feynman integrals and related multi-sums

In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite nested integrals or sums. Furthermore, one seeks for solutions of coupled systems of linear differential equations, that can be represented in terms of indefinite nested sums (or integrals). In this article we elaborate the main tools and the corresponding packages, that we have developed and intensively used within the last 10 years in the course of our QCD-calculations

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.Comment: 8 page

Refined Holonomic Summation Algorithms in Particle Physics

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle indefinite nested sums and products in the setting of difference rings and holonomic sequences. Relevant mathematics is reviewed and the underlying advanced difference ring machinery is elaborated upon. The flexibility of this new toolbox contributed substantially to evaluating complicated multi-sums coming from particle physics. Illustrative examples of the functionality of the new software package RhoSum are given.Comment: Modified Proposition 2.1 and Corollary 2.

Partitioned methods for coupled fluid flow problems

Many flow problems in engineering and technology are coupled in their nature. Plenty of turbulent flows are solved by legacy codes or by ones written by a team of programmers with great complexity. As knowledge of turbulent flows expands and new models are introduced, implementation of modern approaches in legacy codes and increasing their accuracy are of great concern. On the other hand, industrial flow models normally involve multi-physical process or multi-domains. Given the different nature of the physical processes of each subproblem, they may require different meshes, time steps and methods. There is a natural desire to uncouple and solve such systems by solving each subphysics problem, to reduce the technical complexity and allow the use of optimized legacy sub-problems' codes. The objective of this work is the development, analysis and validation of new modular, uncoupling algorithms for some coupled flow models, addressing both of the above problems. Particularly, this thesis studies: i) explicitly uncoupling algorithm for implementation of variational multiscale approach in legacy turbulence codes, ii) partitioned time stepping methods for magnetohydrodynamics flows, and iii) partitioned time stepping methods for groundwater-surface water flows. For each direction, we give comprehensive analysis of stability and derive optimal error estimates of our proposed methods. We discuss the advantages and limitations of uncoupling methods compared with monolithic methods, where the globally coupled problems are assembled and solved in one step. Numerical experiments are performed to verify the theoretical results

The evolutionary trajectory of mitochondrial carrier family during metazoan evolution

BACKGROUND: Exploring metabolic evolution is a way to understand metabolic complexity. The substrate transport of mitochondrial carrier family (MCF) influences direct metabolic activities, making it possible to understand indirectly metabolic evolution from the evolution of substrate transport of MCF. However, the evolutionary study of substrate transport of MCF does not mean that all the concrete structures of mitochondrial carriers (MCs) must first be gained. RESULTS: Here we studied the alternation of MCF structure and potential correlated functions of MCF during metazoan evolution. The data analysis indicates that the types of substrates transported by MCF as a whole were maintained during metazoan evolution. However, the size of the substrates transported by members of MCs continuously diminished during the evolutionary process. We have found that the ratio of hydrophobic amino acids at specific helix-helix interfaces increases significantly during vertebrate evolution. Amino acid's spatial positioning and the calculating of packing values both indicate the increase in the number of hydrophobic amino acids would lead to a more "tight" structure of the TR domain, which is in agreement with the trend of diminishing size of substrates transported by MCs. In addition, there was a significant increase in the number of carriers of MCF during vertebrate evolution. CONCLUSIONS: We propose that the more "tight" TR structure generated by the increase of the hydrophobic amino acids at specific helix-helix interfaces during vertebrate evolution enhances the substrate selectivity of MCF, reflecting the evolutionary trajectory of MCF during metazoan evolution