PERIMETER CENTROIDS AND CIRCUMSCRIBED QUADRANGLES

Vertical perimeter versus horizontal perimeter

Given k ∈ N, the k’th discrete Heisenberg group, denoted H Z , is the group generated by the elements a1, b1, . . . , ak, bk, c, subject to the commutator relations [a1, b1] = . . . = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ {1, . . . , k} we have [ai, aj ] = [bi, bj ] = [ai, bj ] = [ai, c] = [bi, c] = ...

Centroids and Comparison of Volumes

For −1 < p < 1 we introduce the concept of a polar p-centroid body Γ * p K of a star body K. We consider the question of whether Γ * p K ⊂ Γ * p L implies vol(L) ≤ vol(K). Our results extend the studies by Lutwak in the case p = 1 and Grinberg, Zhang in the case p > 1.

Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in [6] that if p ≥ 5 and e = 23 for integ...

Preferences in quadrangles reconsidered.

Through about 130 years of the history of experimental aesthetics, preferences of figures have been summarized in representative values such as the golden ratio. Researches especially in the golden-section hypothesis overshadowed the basic and profound problem of how people's individual preference would be decided. In the present study I returned to simple quadrangles, and investigated each per...

Finite Generalized Quadrangles