313 research outputs found
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
Prodsimplicial-Neighborly Polytopes
Simultaneously generalizing both neighborly and neighborly cubical polytopes,
we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to
that of a product of r simplices. We construct PSN polytopes by three different
methods, the most versatile of which is an extension of Sanyal and Ziegler's
"projecting deformed products" construction to products of arbitrary simple
polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1.
Using topological obstructions similar to those introduced by Sanyal to bound
the number of vertices of Minkowski sums, we show that this dimension is
minimal if we additionally require that the PSN polytope is obtained as a
projection of a polytope that is combinatorially equivalent to the product of r
simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction
The CONEstrip algorithm
Uncertainty models such as sets of desirable gambles and (conditional) lower previsions can be represented as convex cones. Checking the consistency of and drawing inferences from such models requires solving feasibility and optimization problems. We consider finitely generated such models. For closed cones, we can use linear programming; for conditional lower prevision-based cones, there is an efficient algorithm using an iteration of linear programs. We present an efficient algorithm for general cones that also uses an iteration of linear programs
Quantum Computing with NMR
A review of progress in NMR quantum computing and a brief survey of the
literatureComment: Commissioned by Progress in NMR Spectroscopy (95 pages, no figures
Algorithms for Highly Symmetric Linear and Integer Programs
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title
slightly change
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
Complementary vertices and adjacency testing in polytopes
Our main theoretical result is that, if a simple polytope has a pair of
complementary vertices (i.e., two vertices with no facets in common), then it
has at least two such pairs, which can be chosen to be disjoint. Using this
result, we improve adjacency testing for vertices in both simple and non-simple
polytopes: given a polytope in the standard form {x \in R^n | Ax = b and x \geq
0} and a list of its V vertices, we describe an O(n) test to identify whether
any two given vertices are adjacent. For simple polytopes this test is perfect;
for non-simple polytopes it may be indeterminate, and instead acts as a filter
to identify non-adjacent pairs. Our test requires an O(n^2 V + n V^2)
precomputation, which is acceptable in settings such as all-pairs adjacency
testing. These results improve upon the more general O(nV) combinatorial and
O(n^3) algebraic adjacency tests from the literature.Comment: 14 pages, 5 figures. v1: published in COCOON 2012. v2: full journal
version, which strengthens and extends the results in Section 2 (see p1 of
the paper for details
Exponents of 2-multiarrangements and multiplicity lattices
We introduce a concept of multiplicity lattices of 2-multiarrangements,
determine the combinatorics and geometry of that lattice, and give a criterion
and method to construct a basis for derivation modules effectively.Comment: 14 page
On the existence of star products on quotient spaces of linear Hamiltonian torus actions
We discuss BFV deformation quantization of singular symplectic quotient
spaces in the special case of linear Hamiltonian torus actions. In particular,
we show that the Koszul complex on the moment map of an effective linear
Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of
Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that
reduced spaces of such actions admit continuous star products.Comment: 9 pages, 4 figures, uses psfra
Proto-Plasm: parallel language for adaptive and scalable modelling of biosystems
This paper discusses the design goals and the first developments of
Proto-Plasm, a novel computational environment to produce libraries
of executable, combinable and customizable computer models of natural and
synthetic biosystems, aiming to provide a supporting framework for predictive
understanding of structure and behaviour through multiscale geometric modelling
and multiphysics simulations. Admittedly, the Proto-Plasm platform is
still in its infancy. Its computational framework—language, model library,
integrated development environment and parallel engine—intends to provide
patient-specific computational modelling and simulation of organs and biosystem,
exploiting novel functionalities resulting from the symbolic combination of
parametrized models of parts at various scales. Proto-Plasm may define
the model equations, but it is currently focused on the symbolic description of
model geometry and on the parallel support of simulations. Conversely, CellML
and SBML could be viewed as defining the behavioural functions (the model
equations) to be used within a Proto-Plasm program. Here we exemplify
the basic functionalities of Proto-Plasm, by constructing a schematic
heart model. We also discuss multiscale issues with reference to the geometric
and physical modelling of neuromuscular junctions
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