87,229 research outputs found
Overview of (pro-)Lie group structures on Hopf algebra character groups
Character groups of Hopf algebras appear in a variety of mathematical and
physical contexts. To name just a few, they arise in non-commutative geometry,
renormalisation of quantum field theory, and numerical analysis. In the present
article we review recent results on the structure of character groups of Hopf
algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild
assumptions on the Hopf algebra or the target algebra the character groups
possess strong structural properties. Moreover, these properties are of
interest in applications of these groups outside of Lie theory. We emphasise
this point in the context of two main examples: The Butcher group from
numerical analysis and character groups which arise from the Connes--Kreimer
theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on
"New Developments in Discrete Mechanics, Geometric Integration and
Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai
Matrix Shanks Transformations
Shanks' transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix -algorithm of Wynn. Then, the transformation is related to matrix Pad\ue9-type and Pad\ue9 approximants. Numerical experiments showing the interest of this transformation end the paper
Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints
We examine two multidimensional optimization problems that are formulated in
terms of tropical mathematics. The problems are to minimize nonlinear objective
functions, which are defined through the multiplicative conjugate vector
transposition on vectors of a finite-dimensional semimodule over an idempotent
semifield, and subject to boundary constraints. The solution approach is
implemented, which involves the derivation of the sharp bounds on the objective
functions, followed by determination of vectors that yield the bound. Based on
the approach, direct solutions to the problems are obtained in a compact vector
form. To illustrate, we apply the results to solving constrained Chebyshev
approximation and location problems, and give numerical examples.Comment: Mathematical Methods and Optimization Techniques in Engineering:
Proc. 1st Intern. Conf. on Optimization Techniques in Engineering (OTENG
'13), Antalya, Turkey, October 8-10, 2013, WSEAS Press, 2013, pp. 86-91. ISBN
978-960-474-339-
Modernizing PHCpack through phcpy
PHCpack is a large software package for solving systems of polynomial
equations. The executable phc is menu driven and file oriented. This paper
describes the development of phcpy, a Python interface to PHCpack. Instead of
navigating through menus, users of phcpy solve systems in the Python shell or
via scripts. Persistent objects replace intermediate files.Comment: Part of the Proceedings of the 6th European Conference on Python in
Science (EuroSciPy 2013), Pierre de Buyl and Nelle Varoquaux editors, (2014
- …