614 research outputs found
Toric moment mappings and Riemannian structures
Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in
six dimensions, and we use this correspondence to interpret symplectic
fibrations between these orbits, and to analyse moment polytopes associated to
the standard Hamiltonian torus action on the coadjoint orbits. The theory is
then applied to describe so-called intrinsic torsion varieties of Riemannian
structures on the Iwasawa manifold.Comment: 25 pages, 14 figures; Geometriae Dedicata 2012, Toric moment mappings
and Riemannian structures, available at
http://www.springerlink.com/content/yn86k22mv18p8ku2
The extension problem for partial Boolean structures in Quantum Mechanics
Alternative partial Boolean structures, implicit in the discussion of
classical representability of sets of quantum mechanical predictions, are
characterized, with definite general conclusions on the equivalence of the
approaches going back to Bell and Kochen-Specker. An algebraic approach is
presented, allowing for a discussion of partial classical extension, amounting
to reduction of the number of contexts, classical representability arising as a
special case. As a result, known techniques are generalized and some of the
associated computational difficulties overcome. The implications on the
discussion of Boole-Bell inequalities are indicated.Comment: A number of misprints have been corrected and some terminology
changed in order to avoid possible ambiguitie
Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes
We study the exactness of certain combinatorially defined complexes which
generalize the Orlik-Solomon algebra of a geometric lattice. The main results
pertain to complex reflection arrangements and their restrictions. In
particular, we consider the corresponding relation complexes and give a simple
proof of the -formality of these hyperplane arrangements. As an application,
we are able to bound the Castelnouvo-Mumford regularity of certain modules over
polynomial rings associated to Coxeter arrangements (real reflection
arrangements) and their restrictions. The modules in question are defined using
the relation complex of the Coxeter arrangement and fiber polytopes of the dual
Coxeter zonotope. They generalize the algebra of piecewise polynomial functions
on the original arrangement
Construction and Analysis of Projected Deformed Products
We introduce a deformed product construction for simple polytopes in terms of
lower-triangular block matrix representations. We further show how Gale duality
can be employed for the construction and for the analysis of deformed products
such that specified faces (e.g. all the k-faces) are ``strictly preserved''
under projection. Thus, starting from an arbitrary neighborly simplicial
(d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose
projection to the last dcoordinates yields a neighborly cubical d-polytope. As
an extension of thecubical case, we construct matrix representations of
deformed products of(even) polygons (DPPs), which have a projection to d-space
that retains the complete (\lfloor \tfrac{d}{2} \rfloor - 1)-skeleton. In both
cases the combinatorial structure of the images under projection is completely
determined by the neighborly polytope Q: Our analysis provides explicit
combinatorial descriptions. This yields a multitude of combinatorially
different neighborly cubical polytopes and DPPs. As a special case, we obtain
simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler
(2000) as well as of the ``projected deformed products of polygons'' that were
announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets
arbitrarily close to 9.Comment: 20 pages, 5 figure
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