Download or read book Extension Spaces of Oriented Matroids written by Bernd Sturmfels and published by . This book was released on 1991 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Extension Spaces of Oriented Matriods written by Bernd Sturmfels and published by . This book was released on 1991 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We study the space of all extensions of a real hyperplance arrangement by a new pseudo-hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the 'Generalized Baues Problem' of Billera, Kapranov & Sturmfels, via the Bohne-Dress Theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not even known whether the extension space is connected. We show that the subspace of realizable extensions is always connected but not necessarily spherical."
Download or read book Two Constructions of Oriented Matroids with Disconnected Extension Space written by Nicolai E. Mnëv and published by . This book was released on 1992 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Triangulations of Oriented Matroids written by Francisco Santos and published by American Mathematical Soc.. This book was released on 2002 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera-Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroid $\mathcal{M}$ and extensions of its dual $\mathcal{M}^*$, via the so-called lifting triangulations. We show that this duality behaves particularly well in the class of Lawrence matroid polytopes. In particular, that the extension space conjecture for realizable oriented matroids is equivalent to the restriction to Lawrence polytopes of the Generalized Baues problem for subdivisions of polytopes. We finish by showing examples and a characterization of lifting triangulations.
Download or read book Oriented Matroids written by Anders Björner and published by Cambridge University Press. This book was released on 1999-11-18 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: First comprehensive, accessible account; second edition has expanded bibliography and a new appendix surveying recent research.
Download or read book Computational Oriented Matroids written by Jürgen Bokowski and published by Cambridge University Press. This book was released on 2006-05-08 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: Oriented matroids play the role of matrices in discrete geometry, when metrical properties, such as angles or distances, are neither required nor available. Thus they are of great use in such areas as graph theory, combinatorial optimization and convex geometry. The variety of applications corresponds to the variety of ways they can be defined. Each of these definitions corresponds to a differing data structure for an oriented matroid, and handling them requires computational support, best realised through a functional language. Haskell is used here, and, for the benefit of readers, the book includes a primer on it. The combination of concrete applications and computation, the profusion of illustrations, many in colour, and the large number of examples and exercises make this an ideal introductory text on the subject. It will also be valuable for self-study for mathematicians and computer scientists working in discrete and computational geometry.
Download or read book Purity and Separation for Oriented Matroids written by Pavel Galashin and published by American Mathematical Society. This book was released on 2023-09-27 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Download or read book New Perspectives in Algebraic Combinatorics written by Louis J. Billera and published by Cambridge University Press. This book was released on 1999-09-28 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text contains expository contributions by respected researchers on the connections between algebraic geometry, topology, commutative algebra, representation theory, and convex geometry.
Download or read book Extension equivalence of oriented matroids written by Achim Bachem and published by . This book was released on 1984 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Jerusalem Combinatorics 93 written by Hélène Barcelo and published by American Mathematical Soc.. This book was released on 1994 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains twenty-two papers presented at the International Conference in Combinatorics, held in Jerusalem in May 1993. The papers describe some of the latest developments in algebraic combinatorics, enumeration, graph and hypergraph theory, combinatorial geometry, and geometry of polytopes and arrangements. The papers are accessible to specialists as well as nonspecialists.
Download or read book Handbook of Convex Geometry written by Bozzano G Luisa and published by Elsevier. This book was released on 2014-06-28 with total page 803 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.
Download or read book Face Lattices of Oriented Matroids written by Beth Spellman Munson and published by . This book was released on 1981 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Oriented Matroids and the Linear Complementarity Problem written by Walter Duff Morris and published by . This book was released on 1986 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Purity and Separation for Oriented Matroids written by Pavel Galashin and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in [n] has the same cardinality. In this paper, we extend these notions and define M-separated collections for any oriented matroid M. We show that maximal by size M-separated collections are in bijection with fine zonotopal tilings (if M is a realizable oriented matroid), or with one-element liftings of M in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid M is pure if M-separated collections form a pure simplicial complex, i.e., any maximal by inclusion M-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank 3, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank 3 is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an n-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank 3, graphical, uniform).
Download or read book Realization Spaces of Polytopes written by Jürgen Richter-Gebert and published by Springer. This book was released on 2006-11-13 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book collects results about realization spaces of polytopes. It gives a presentation of the author's "Universality Theorem for 4-polytopes". It is a comprehensive survey of the important results that have been obtained in that direction. The approaches chosen are direct and very geometric in nature. The book is addressed to researchers and to graduate students. The former will find a comprehensive source for the above mentioned results. The latter will find a readable introduction to the field. The reader is assumed to be familiar with basic concepts of linear algebra.
Download or read book The Topology of Baues Complexes and Flip Graphs written by Xue Liu (Mathematician) and published by . This book was released on 2017 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis contains several results on the connectedness of Baues complexes and flip graphs, which are topological spaces modeling certain sets coming from geometric combinatorics. These sets include the triangulations of a polytope, the tilings of a zonotope, and the extensions of an oriented matroid. Some long-standing conjectures are resolved, including the connectedness of triangulations of a product of two simplices and the sphericity of extension spaces of realizable oriented matroids. This thesis covers the main construction which is common to these proofs, but defers the details specific to each problem to other papers.
Download or read book On intersection properties of oriented matroids written by Achim Bachem and published by . This book was released on 1985 with total page 3 pages. Available in PDF, EPUB and Kindle. Book excerpt:
