112 research outputs found
Gaussian cubature arising from hybrid characters of simple Lie groups
Lie groups with two different root lengths allow two mixed sign homomorphisms
on their corresponding Weyl groups, which in turn give rise to two families of
hybrid Weyl group orbit functions and characters. In this paper we extend the
ideas leading to the Gaussian cubature formulas for families of polynomials
arising from the characters of irreducible representations of any simple Lie
group, to new cubature formulas based on the corresponding hybrid characters.
These formulas are new forms of Gaussian cubature in the short root length case
and new forms of Radau cubature in the long root case. The nodes for the
cubature arise quite naturally from the (computationally efficient) elements of
finite order of the Lie group.Comment: 23 pages, 3 figure
Magic numbers in the discrete tomography of cyclotomic model sets
We report recent progress in the problem of distinguishing convex subsets of
cyclotomic model sets by (discrete parallel) X-rays in prescribed
-directions. It turns out that for any of these model sets
there exists a `magic number' such that any two
convex subsets of can be distinguished by their X-rays in any set
of prescribed -directions. In particular, for
pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible
numbers are in that very order 11, 9, 11 and 13.Comment: 6 pages, 1 figure; based on the results of arXiv:1101.4149 [math.MG];
presented at Aperiodic 2012 (Cairns, Australia
Recurrence in 2D Inviscid Channel Flow
I will prove a recurrence theorem which says that any () solution
to the 2D inviscid channel flow returns repeatedly to an arbitrarily small
neighborhood. Periodic boundary condition is imposed along the
stream-wise direction. The result is an extension of an early result of the
author [Li, 09] on 2D Euler equation under periodic boundary conditions along
both directions
Representations of Double Affine Lie algebras
We study representations of the double affine Lie algebra associated to a
simple Lie algebra. We construct a family of indecomposable integrable
representations and identify their irreducible quotients. We also give a
condition for the indecomposable modules to be irreducible, this is analogous
to a result in the representation theory of quantum affine algebras. Finally,
in the last section of the paper, we show, by using the notion of fusion
product, that our modules are generically reducible
Using Insights from Cognitive Neuroscience to Investigate the Effects of Event-Driven Process Chains on Process Model Comprehension
Business process models have been adopted by enterprises for more than a decade. Especially for domain experts, the comprehension of process models constitutes a challenging task that needs to be mastered when creating or reading these models. This paper presents the results we obtained from an eye tracking experiment on process model comprehension. In detail, individuals with either no or advanced expertise in process modeling were confronted with models expressed in terms of Event-driven Process Chains (EPCs), reflecting different levels of difficulty. The first results of this experiment confirm recent findings from one of our previous experiments on the reading and comprehension of process models. On one hand, independent from their level of exper-tise, all individuals face similar patterns, when being confronted with process models exceeding a certain level of difficulty. On the other, it appears that process models expressed in terms of EPCs are perceived differently compared to process models specified in the Business Process Model and Notation (BPMN). In the end, their generalization needs to be confirmed by additional empirical experiments. The presented expe-riment continues a series of experiments that aim to unravel the factors fostering the comprehension of business process models by using methods and theories stemming from the field of cognitive neuroscience and psychology
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
Clifford algebras are naturally associated with quadratic forms. These
algebras are Z_2-graded by construction. However, only a Z_n-gradation induced
by a choice of a basis, or even better, by a Chevalley vector space isomorphism
Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition
into scalars, vectors, tensors, and so on, mandatory in physics. We show that
the Chevalley isomorphism theorem cannot be generalized to algebras if the
Z_n-grading or other structures are added, e.g., a linear form. We work with
pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which
we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford
algebras'. It turns out, that in this sense, all multi-vector Clifford algebras
of the same quadratic but different bilinear forms are non-isomorphic. The
usefulness of such algebras in quantum field theory and superconductivity was
shown elsewhere. Allowing for arbitrary bilinear forms however spoils their
diagonalizability which has a considerable effect on the tensor decomposition
of the Clifford algebras governed by the periodicity theorems, including the
Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which
can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1}
\otimes Cl_{1,1}. The general case used in quantum field theory lacks this
feature. Theories with non-symmetric bilinear forms are however needed in the
analysis of multi-particle states in interacting theories. A connection to
q-deformed structures through nontrivial vacuum states in quantum theories is
outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International
Conference on Clifford Algebras and their Applications in Mathematical
Physics, Ixtapa, Mexico, June 27 - July 4, 199
Aperiodic order and pure point diffraction
We give a leisurely introduction into mathematical diffraction theory with a
focus on pure point diffraction. In particular, we discuss various
characterisations of pure point diffraction and common models arising from cut
and project schemes. We finish with a list of open problems.Comment: 14 page
Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals
This list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field
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