17 research outputs found
An Example of Clifford Algebras Calculations with GiNaC
This example of Clifford algebras calculations uses GiNaC
(http://www.ginac.de/) library, which includes a support for generic Clifford
algebra starting from version~1.3.0. Both symbolic and numeric calculation are
possible and can be blended with other functions of GiNaC. This calculations
was made for the paper math.CV/0410399.
Described features of GiNaC are already available at PyGiNaC
(http://sourceforge.net/projects/pyginac/) and due to course should propagate
into other software like GNU Octave (http://www.octave.org/), gTybalt
(http://www.fis.unipr.it/~stefanw/gtybalt.html), which use GiNaC library as
their back-end.Comment: 20 pages, LaTeX2e, 12 PS graphics in one figure; v3 code
improvements; v4 small code correction for new libraries; v5 comments are
redesined to be more readabl
Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains
This paper lays down a foundation for a systematic treatment of three main
(elliptic, parabolic and hyperbolic) types of analytic function theory based on
the representation theory of SL(2,R) group. We describe here geometries of
corresponding domains. The principal role is played by Clifford algebras of
matching types.
Keywords: analytic function theory, semisimple groups, elliptic, parabolic,
hyperbolic, Clifford algebrasComment: 14 pages, AMS-LaTeX, 31 PS graphcs in 7 Figures; v2--the missing PS
file is included now; v3 minor corrections, reference added; v4 numerous
misprints correcte
Introduction to the GiNaC Framework for Symbolic Computation within the C++ Programming Language
The traditional split-up into a low level language and a high level language
in the design of computer algebra systems may become obsolete with the advent
of more versatile computer languages. We describe GiNaC, a special-purpose
system that deliberately denies the need for such a distinction. It is entirely
written in C++ and the user can interact with it directly in that language. It
was designed to provide efficient handling of multivariate polynomials,
algebras and special functions that are needed for loop calculations in
theoretical quantum field theory. It also bears some potential to become a more
general purpose symbolic package
Schwerdtfeger-Fillmore-Springer-Cnops Construction Implemented in GiNaC
This paper presents an implementation of the
Schwerdtfeger-Fillmore-Springer-Cnops construction (SFSCc) along with
illustrations of its usage. SFSCc linearises the linear-fraction action of the
Moebius group in R^n. This has clear advantages in several theoretical and
applied fields including engineering. Our implementation is based on the
Clifford algebra capacities of the GiNaC computer algebra system
(http://www.ginac.de/), which were described in cs.MS/0410044.
The core of this realisation of SFSCc is done for an arbitrary dimension of
R^n with a metric given by an arbitrary bilinear form. We also present a
subclass for two dimensional cycles (i.e. circles, parabolas and hyperbolas),
which add some 2D specific routines including a visualisation to PostScript
files through the MetaPost (http://www.tug.org/metapost.html) or Asymptote
(http://asymptote.sourceforge.net/) packages.
This software is the backbone of many results published in math.CV/0512416
and we use its applications their for demonstration. The library can be ported
(with various level of required changes) to other CAS with Clifford algebras
capabilities similar to GiNaC.
There is an ISO image of a Live Debian DVD attached to this paper as an
auxiliary file, a copy is stored on Google Drive as well.Comment: LaTeX, 82 p; 11 PS graphics in two figures, the full source files and
ISO image of Live DVD are included; v9: library update for the book on
Moebius transformations; v10: an ISO image of a Live DVD is attached to the
paper; v11: a bug is fixed; v12: Library is uupdated, the reference to a
larger project is adde
Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers
In the search for hypercomplex analytic functions on the half-plane, we
review the construction of induced representations of the group G=SL(2,R).
Firstly we note that G-action on the homogeneous space G/H, where H is any
one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on
hypercomplex numbers. Thus we investigate various hypercomplex characters of
subgroups H. The correspondence between the structure of the group SL(2,R) and
hypercomplex numbers can be illustrated in many other situations as well. We
give examples of induced representations of SL(2,R) on spaces of hypercomplex
valued functions, which are unitary in some sense. Raising/lowering operators
for various subgroup prompt hypercomplex coefficients as well.
The paper contains both English and Russian versions.
Keywords: induced representation, unitary representations, SL(2,R),
semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius
transformations, split-complex numbers, parabolic numbers, hyperbolic numbers,
raising/lowering operators, creation/annihilation operatorsComment: LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two
Figures; v2: minor improvements and corrections; v3: a section on
raising/lowering operators is added; v4: typos are fixed; v5: Introduction is
added, open problems are expanded.v6: Russian translation is added,
references areupdated, NoWeb and C++ source codes are added as ancillary
files. arXiv admin note: substantial text overlap with arXiv:0707.402
Conformal Parametrisation of Loxodromes by Triples of Circles
We provide a parametrisation of a loxodrome by three specially arranged
cycles. The parametrisation is covariant under fractional linear
transformations of the complex plane and naturally encodes conformal properties
of loxodromes. Selected geometrical examples illustrate the usage of
parametrisation. Our work extends the set of objects in Lie sphere
geometry---circle, lines and points---to the natural maximal
conformally-invariant family, which also includes loxodromes.Comment: 14 pages. 9 PDF in four figures, AMS-LaTe