25,682 research outputs found
Lie algebra configuration pairing
We give an algebraic construction of the topological graph-tree configuration
pairing of Sinha and Walter beginning with the classical presentation of Lie
coalgebras via coefficients of words in the associative Lie polynomial. Our
work moves from associative algebras to preLie algebras to graph complexes,
justifying the use of graph generators for Lie coalgebras by iteratively
expanding the set of generators until the set of relations collapses to two
simple local expressions. Our focus is on new computational methods allowed by
this framework and the efficiency of the graph presentation in proofs and
calculus involving free Lie algebras and coalgebras. This outlines a new way of
understanding and calculating with Lie algebras arising from the graph
presentation of Lie coalgebras.Comment: 21 pages; uses xypic; ver 4. added subsection 3.4 outlining another
computational algorithm arising from configuration pairing with graph
Structure-preserving mesh coupling based on the Buffa-Christiansen complex
The state of the art for mesh coupling at nonconforming interfaces is
presented and reviewed. Mesh coupling is frequently applied to the modeling and
simulation of motion in electromagnetic actuators and machines. The paper
exploits Whitney elements to present the main ideas. Both interpolation- and
projection-based methods are considered. In addition to accuracy and
efficiency, we emphasize the question whether the schemes preserve the
structure of the de Rham complex, which underlies Maxwell's equations. As a new
contribution, a structure-preserving projection method is presented, in which
Lagrange multiplier spaces are chosen from the Buffa-Christiansen complex. Its
performance is compared with a straightforward interpolation based on Whitney
and de Rham maps, and with Galerkin projection.Comment: 17 pages, 7 figures. Some figures are omitted due to a restricted
copyright. Full paper to appear in Mathematics of Computatio
Expanders and right-angled Artin groups
The purpose of this article is to give a characterization of families of
expander graphs via right-angled Artin groups. We prove that a sequence of
simplicial graphs forms a family of expander
graphs if and only if a certain natural mini-max invariant arising from the cup
product in the cohomology rings of the groups
agrees with the Cheeger constant of the
sequence of graphs, thus allowing us to characterize expander graphs via
cohomology. This result is proved in the more general framework of \emph{vector
space expanders}, a novel structure consisting of sequences of vector spaces
equipped with vector-space-valued bilinear pairings which satisfy a certain
mini-max condition. These objects can be considered to be analogues of expander
graphs in the realm of linear algebra, with a dictionary being given by the cup
product in cohomology, and in this context represent a different approach to
expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.Comment: 21 pages. Accepted version. To appear in J. Topol. Ana
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