Configurations of Points and Lines

Grünbaum, Branko

doi:10.1090/gsm/103

Cited by 152 publications

(281 citation statements)

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“…Here, we shall only be concerned with specific point-line incidence structures called configurations [9]. A (v r , b k )-configuration is a C where: (1) v = |P| and b = |L|; (2) every line has k points and every point is on r lines; and (3) two distinct lines intersect in at most one point and every two distinct points are joined by at most one line; a configuration where v = b and r = k is called symmetric (or balanced), and usually denoted as a…”

Section: Introductionmentioning

confidence: 99%

From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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Abstract:Given a 2 N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6, we first observe that the multiplication table of its imaginary units e a , 1 ≤ a ≤ 2 N − 1, is encoded in the properties of the projective space PG(N − 1, 2) if these imaginary units are regarded as points and distinguished triads of them {e a , e b , e c }, 1 ≤ a < b < c ≤ 2 N − 1 and e a e b = ±e c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b = c. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG (N − 1, 2), the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C N ; in particular, C 3 (octonions) is isomorphic to the Pasch (6 2 , 4 3 )-configuration, C 4 (sedenions) is the famous Desargues (10 3 )-configuration, C 5 (32-nions) coincides with the Cayley-Salmon (15 4 , 20 3 )-configuration found in the well-known Pascal mystic hexagram and C 6 (64-nions) is identical with a particular (21 5 , 35 3 )-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C N leads to a conjecture that C N is isomorphic to a combinatorial Grassmannian of type G 2 (N + 1).

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Section: Introductionmentioning

confidence: 99%

From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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“…For an example of a point-splittable (T2) configuration see Figure 2. The configuration on Figure 2 is isomorphic to a configuration on Figure 5.1.11 from [8]. For an example of a line-splittable (T3) configuration see Figure 3.…”

Section: Splittable and Unsplittable Configurations (And Graphs)mentioning

confidence: 99%

“…The idea of unsplittable configuration was conceived in 2004 and formally introduced in the monograph [8] by Grünbaum. Later, it was also used in [19].…”

Section: Introductionmentioning

confidence: 99%

Splittable and unsplittable graphs and configurations

Bašić¹,

Grošelj²,

Grünbaum³

et al. 2018

Ars Math. Contemp.

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“…Choosing an appropriate basis for the 3-dimensional subspace of Such a matrix can only arise in characteristic 2, a result that is discussed in detail in the projective geometry literature; see, e.g., [Grü09]. Since the matrix problem specified by the Fano plane does not have solutions over all sufficiently large finite fields F q , it cannot 'come from' the F q -vector labelings of a code graph.…”

Section: An Open Question On F Q -Vector Labelings Of Code Graphsmentioning

confidence: 99%

Representations of the Multicast Network Problem

Anderson¹,

Halbawi

Kaplan

et al. 2017

Association for Women in Mathematics Series

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We approach the problem of linear network coding for multicast networks from different perspectives. We introduce the notion of the coding points of a network, which are edges of the network where messages combine and coding occurs. We give an integer linear program that leads to choices of paths through the network that minimize the number of coding points. We introduce the code graph of a network, a simplified directed graph that maintains the information essential to understanding the coding properties of the network. One of the main problems in network coding is to understand when the capacity of a multicast network is achieved with linear network coding over a finite field of size q. We explain how this problem can be interpreted in terms of rational points on certain algebraic varieties.

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Configurations of Points and Lines

Cited by 152 publications

References 0 publications

From Cayley-Dickson Algebras to Combinatorial Grassmannians

From Cayley-Dickson Algebras to Combinatorial Grassmannians

Splittable and unsplittable graphs and configurations

Representations of the Multicast Network Problem

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