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9084 results in Discrete Mathematics, Information Theory and Coding

Page 422 of 455

  • Kinematic measures for sets of support figures

  • Part of
  • William J. Firey
  • Journal:
    Mathematika / Volume 21 / Issue 2 / December 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. 270-281
    Print publication:
    December 1974

  • 1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

    see [4; p. 214]. Blaschke's formula [4; p. 243] gives

    as the measure, to within a normalization, of overlapping positions of K1 relative to K0.

  • MTK volume 21 issue 2 Front matter

  • Journal:
    Mathematika / Volume 21 / Issue 2 / December 1974
    Published online by Cambridge University Press:
    26 February 2010, pp. f1-f2
    Print publication:
    December 1974
    • Article
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  • Cut-set lattices of graphs

  • Edited by T. P. McDonough, V. C. Mavron
  • Book:
    Combinatorics
    Published online:
    05 April 2013
    Print publication:
    21 November 1974, pp 35-36

  • Summary

    Let G be a connected, undirected, simple graph. Let V(G) and E(G) denote the vertex and edge sets respectively. Consider two different vertices a, b of G. A subset T of the vertex (edge) set is called an a, b-vertex (edge) cut if T separates a and b but no proper subset T' of T does, i. e. if in G - T the vertices a and b belong to different connected components but in G - T' there is always an a, b path. Cuts with the smallest cardinality are called minimal.

    Let now S, T be two a, b-vertex (edge) cuts and order them in such a way that S ≦ T if and only if no a, b-path meets T ‘before’ S. Then it can be shown that ≦ is a partial order; indeed we have (for proofs see for example [1]):

    Theorem 1.Let Γ1and Γ2be the sets of all a, b-vertex cuts and a, b-edge cuts, respectively, of a graph G with respect to two different, fixed vertices a, b of G. Then (Γ1, ≦) and (Γ2, ≦) constitute complete lattices.

    Theorem 2.Let Δ1and Δ2be the sets of all minimal a, b-vertex cuts and minimal a, b-edge cuts, respectively, of a graph G with respect to two different, fixed vertices a, b of G. If the cardinality of the cuts is finite, then (Δ1, ≦) and (Δ2, ≦) constitiite distributive lattices.

  • On a theorem of R. A. Liebler

  • Edited by T. P. McDonough, V. C. Mavron
  • Book:
    Combinatorics
    Published online:
    05 April 2013
    Print publication:
    21 November 1974, pp 53-56

  • Summary

    The conjecture that ‘any affine plane A admitting a collineation group which is rank 3 on points is a translation plane’ was stated by D. G. Higman [3]; M. J. Kallaher [4] proved a slightly weaker statement namely that A is either a translation plane or the dual of a translation plane, and R. A. Liebler [5] overcame the possible ambiguity to prove the conjecture. We offer a proof which, though similar to [4] and [5] in the first stages, completes the final steps more smoothly. Thus we prove

    Theorem (Liebler). An affine plane A admitting a collineation group H which is rank 3 on points is a translation plane, (and H contains all translations).

    Let A be an affine plane of order k, with projective completion P having line at infinity l(∞). P(∞), P(l), …, P(k) are the points of l(∞) and l(l), …, l(k) the affine lines through P(∞). For a point P and line l of P, (P) denotes the set of lines through P and (l) the set of points on l. If G is a group acting on the set Ω, then GΩ is the natural permutation group induced by G on Ω.

Page 422 of 455