Other (2002)

The class of simplicial complexes representing triangulations and subdivisions of
Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for
oriented matroid duality.

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## Scholarly Works (8 results)

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Other (2002)

The class of simplicial complexes representing triangulations and subdivisions of
Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for
oriented matroid duality.

The Hirsch conjecture was posed in 1957 in a question from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n−d. The number n of facets is the minimum number of closed half-spaces needed to form the polytope and the conjecture asserts that one can go from any vertex to any other vertex using at most n−d edges.
Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound n−d is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.

We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the \(C^{d-2}_{d-1}\)-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.

Mathematics Subject Classifications: 52C25, 52B40

Keywords: Rigidity, hyperconnectivity, moment curve, cofactor rigidity

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Faculty (2000)

A dissection of a convex d-polytope is a partition of the polytope into d-simplices
whose vertices are among the vertices of the polytope. Triangulations are dissections that
have the additional property that the set of all its simplices forms a simplicial complex.
The size of a dissection is the number of d-simplices it contains. This paper compares
triangulations of maximal size with dissections of maximal size. We also exhibit lower and
upper bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean
solids, and combinatorial d-cubes.

Faculty (2007)

This paper discusses properties of the graphs of 2-way and 3-way transportation
polytopes, in particular, their possible numbers of vertices and their diameters. Our main
results include a quadratic bound on the diameter of axial 3-way transportation polytopes
and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue
disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et
al. (1984). It also allowed us to discover some new results. For example, we prove that the
number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest
common divisor of $m$ and $n$.

Collective signaling for a quorum is found in a wide range of organisms that face collective action problems whose successful solution requires the participation of some quorum of the individuals present. These range from humans, to social insects, to bacteria. The mechanisms involved, the quorum required, and the size of the group may vary. Here we address the general question of the evolution of collective signaling at a high level of abstraction. We investigate the evolutionary dynamics of a population engaging in a signaling N-person game theoretic model. Parameter settings allow for loners and cheaters, and for costly or costless signals. We find a rich dynamics, showing how natural selection, operating on a population of individuals endowed with the simplest strategies, is able to evolve a costly signaling system that allows individuals to respond appropriately to different states of Nature. Signaling robustly promotes cooperative collective action, in particular when coordinated action is most needed and difficult to achieve. Two different signaling systems may emerge depending on Nature's most prevalent states.

The 17th International HLA and Immunogenetics Workshop (IHIW) organizers conducted a Pilot Study (PS) in which 13 laboratories (15 groups) participated to assess the performance of the various sequencing library preparation protocols, NGS platforms and software in use prior to the workshop. The organizers sent 50 cell lines to each of the 15 groups, scored the 15 independently generated sets of NGS HLA genotyping data, and generated "consensus" HLA genotypes for each of the 50 cell lines. Proficiency Testing (PT) was subsequently organized using four sets of 24 cell lines, selected from 48 of 50 PS cell lines, to validate the quality of NGS HLA typing data from the 34 participating IHIW laboratories. Completion of the PT program with a minimum score of 95% concordance at the HLA-A, HLA-B, HLA-C, HLA-DRB1 and HLA-DQB1 loci satisfied the requirements to submit NGS HLA typing data for the 17th IHIW projects. Together, these PS and PT efforts constituted the 17th IHIW Quality Control project. Overall PT concordance rates for HLA-A, HLA-B, HLA-C, HLA-DPA1, HLA-DPB1, HLA-DQA1, HLA-DQB1, HLA-DRB1, HLA-DRB3, HLA-DRB4 and HLA-DRB5 were 98.1%, 97.0% and 98.1%, 99.0%, 98.6%, 98.8%, 97.6%, 96.0%, 99.1%, 90.0% and 91.7%, respectively. Across all loci, the majority of the discordance was due to allele dropout. The high cost of NGS HLA genotyping per experiment likely prevented the retyping of initially failed HLA loci. Despite the high HLA genotype concordance rates of the software, there remains room for improvement in the assembly of more accurate consensus DNA sequences by NGS HLA genotyping software.