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Translating between the representations of a ranked convex geometry [article]

Oscar Defrain and Lhouari Nourine and Simon Vilmin
2021 arXiv   pre-print
In this paper, we consider the problem of translating between the two representations in acyclic convex geometries.  ...  In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets.  ...  Hence, the problem of translating between implicational bases and meet-irreducible elements is critical in order to reap the benefits of both representations.  ... 
arXiv:1907.09433v2 fatcat:zshldvesr5d6hjoyvdf4dpxb7m

Matroids from Modules

Nils Anders Danielsson, Michael B. Smyth
2003 Electronical Notes in Theoretical Computer Science  
The trick is to emulate the structure of a vector space within the module, thereby allowing matroid methods to be used as if the module were a vector space.  ...  It is also shown that Hübler's axiomatic discrete geometry can be characterised in terms of modules over the ring of integers.  ...  Acknowledgement We would like to thank the anonymous referees for their helpful comments.  ... 
doi:10.1016/s1571-0661(04)80763-1 fatcat:rd3i4xgtovaj5lyjfs7o4swvi4

Geometric structures and representations of discrete groups [article]

Fanny Kassel
2018 arXiv   pre-print
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie groups.  ...  In Section 6 we discuss a situation in which the links between geometric structures and Anosov representations are particularly tight, in the context of convex projective geometry.  ...  We now discuss a situation, in the setting of convex projective geometry, in which the links between (G, X)-structures and Anosov representations are particularly tight and go in both directions, yielding  ... 
arXiv:1802.07221v1 fatcat:c4m23fglq5dznm5v3nld3l66w4

Rigidity of symmetric spaces

Inkang Kim
1999 Séminaire de théorie spectrale et géométrie  
The author gives special thanks to Gérard Besson for his hospitality during his stay at Institut Fourier and acknowledges the partial support of CNRS-KOSEF grant.  ...  Let &(T) be the space of faithful, discrete, convex cocompact représentations from Tinto the isometry group of rank one symmetrie space X of non-compact type.  ...  Mainly we are concerned about the 132 I.KIM geometrie boundary of a rank one symmetrie space, since the extended action of isometries on the Furstenberg boundary of higher rank symmetrie spaces are not  ... 
doi:10.5802/tsg.211 fatcat:4oy5fkyutfbhbc4klisy7n4eaa

Guest Editors' Foreword

Imre Bárány, Luis Montejano, Deborah Oliveros
2010 Discrete & Computational Geometry  
Bracho show that the lattice of linear partitions in a projective geometry of rank n has a Helly number.  ...  Strausz introduces the notion of hyperseparoids as a generalization of separoids, and proves a representation theorem for hyperseparoids in terms of Tverberg partitions. I. Bárány ( ) Rényi Institute,  ...  Roldan-Pensado present some bounds on λ(K, 4) and λ(K, 3), for families of translated copies of a convex body K in the plane. X. Goaoc, S. König, and S.  ... 
doi:10.1007/s00454-010-9289-5 fatcat:ejgblstegjal7gwjj77tqjeopa

Combinatorial geometries, convex polyhedra, and schubert cells

I.M Gelfand, R.M Goresky, R.D MacPherson, V.V Serganova
1987 Advances in Mathematics  
The equivalence of ( 1) and (2) establishes a one to one correspondence between representable (over C) combinatorial geometries (or matroids) and certain convex polyhedra.  ...  We will explore a remarkable connection between the geometry of the Schubert cells in the Grassmann manifold, the theory of convex polyhedra, and the theory of combinatorial geometries in the sense of  ... 
doi:10.1016/0001-8708(87)90059-4 fatcat:de2pdm2o65edldtvttlsl6baai

Page 9059 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
The class of closed planar sets falls into a hierarchy of order type 52 CONVEX AND DISCRETE GEOMETRY 2001m:52006 @, + 1 when ordered by d-rank.  ...  “The rank 6°(S) of a set S is defined by means of topological complexity of 3-cliques in the set.  ... 

Page 3500 of Mathematical Reviews Vol. , Issue 85h [page]

1985 Mathematical Reviews  
He proves the existence of a smooth solution which also is representable as such a graph.  ...  The work is concerned with the study of the geometry of shortest paths and geodesics on general convex hypersurfaces (in spaces of constant curvature) and with respect to irregular multidimensional convex  ... 

Gravitational and magnetic anomaly inversion using a tree-based geometry representation

Raymond A. Wildman, George A. Gazonas
2009 Geophysics  
Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently  ...  Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services  ...  S.Army Research Laboratory ͑USARL͒ administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U. S. Department of Energy and USARL.  ... 
doi:10.1190/1.3110042 fatcat:mtz7ow2grnap5ntdu4ad2a45yy

Page 1549 of Mathematical Reviews Vol. , Issue 83d [page]

1983 Mathematical Reviews  
Let M be a packing [covering] of [by] translates of a disc G in [of] the Euclidean plane R*.  ...  Linhart, J. 83d:52015 Closest packings and closest coverings by translates of a convex disc. Studia Sci. Math. Hungar. 13 (1978), no. 1-2, 157-162 (1981).  ... 

Page 5134 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
Here A is a dual element of the Lie algebra of G, t is a representation of the isotropy G, and Z is the representation of G induced by tT.  ...  Let G=SN be a connected unimodular Lie group which is a semidirect product of a closed normal nilpotent subgroup N and semisimple Lie group S whose center is finite and whose rank is the rank of a maximal  ... 

Page 1246 of Mathematical Reviews Vol. 49, Issue 3 [page]

1975 Mathematical Reviews  
The general equation of this type is translated into the language of symmetrical traceless Cartesian tensors.  ...  In the left-hand side of Equation (1) all twice repeated Greek indices imply the summation from 0 to 3, P, (p=9, 1, 2, 3) is a 4-momentum which transforms as a covariant tensor of rank 1 in the Lorentz  ... 

Page 612 of Mathematical Reviews Vol. 16, Issue 6 [page]

1955 Mathematical Reviews  
convex sets in L, there is a semispace S at ¢ such that A=f)\ze2(x+5S) and B=(),e4(y+5"*). (4) If C isa convex set in L, then L\C is convex if and only if the family of all translates of L is linearly  ...  Also dis- cussed is a relationship between semispaces and extreme points, and it is shown how the result (3) leads to a type of separation theorem valid for an arbitrary disjoint pair of convex sets.  ... 

Page 318 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
Summary: “We prove that an oriented matroid over a set E can be regarded as a subset W of vertices of a cube [—1,+1]* CR®’, s 52 CONVEX AND DISCRETE GEOMETRY 318 E' C E, symmetric with respect to the origin  ...  Suppose P is as in the above theo- rem and is also centrally symmetric; then any convex body whose projections on i-dimensional subspaces agree with those of P is a translate of P.  ... 

Rank-One Hilbert Geometries [article]

Mitul Islam
2020 arXiv   pre-print
We introduce and study the notion of rank-one Hilbert geometries, or rank-one properly convex domains, in ℙ(ℝ^d+1). This is in the spirit of rank-one non-positively curved Riemannian manifolds.  ...  We define rank-one isometries of a Hilbert geometry Ω and characterize them precisely as the contracting elements in the automorphism group Aut(Ω) of the Hilbert geometry.  ...  Benoist studied divisible Hilbert geometries and established connections between the geometry of Ω and the regularity of its boundary ∂Ω [Ben04] .  ... 
arXiv:1912.13013v2 fatcat:u6vmyj6azrddjdruaeqindojqu
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