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* -biregular rings

Christopher J. Duckenfield (1972)

Commentationes Mathematicae Universitatis Carolinae

( S 3 , S 6 ) -Amalgams IV

Wolfgang Lempken, Christopher Parker, Peter Rowley (2005)

Rendiconti del Seminario Matematico della Università di Padova

( S 3 , S 6 ) -Amalgams V

Wolfgang Lempken, Christopher Parker, Peter Rowley (2007)

Rendiconti del Seminario Matematico della Università di Padova

( S 3 , S 6 ) -Amalgams VI

Wolfgang Lempken, Christopher Parker, Peter Rowley (2007)

Rendiconti del Seminario Matematico della Università di Padova

( S 3 , S 6 ) -Amalgams VII

Wolfgang Lempken, Christopher Parker, Peter Rowley (2008)

Rendiconti del Seminario Matematico della Università di Padova

2 - ( n 2 , 2 n , 2 n - 1 ) designs obtained from affine planes

Andrea Caggegi (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The simple incidence structure 𝒟 ( 𝒜 , 2 ) formed by points and unordered pairs of distinct parallel lines of a finite affine plane 𝒜 = ( 𝒫 , ) of order n > 2 is a 2 - ( n 2 , 2 n , 2 n - 1 ) design. If n = 3 , 𝒟 ( 𝒜 , 2 ) is the complementary design of 𝒜 . If n = 4 , 𝒟 ( 𝒜 , 2 ) is isomorphic to the geometric design A G 3 ( 4 , 2 ) (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a 2 - ( n 2 , 2 n , 2 n - 1 ) design to be of the form 𝒟 ( 𝒜 , 2 ) for some finite affine plane 𝒜 of order n > 4 . As a consequence we obtain a characterization of small designs 𝒟 ( 𝒜 , 2 ) .

2-frieze patterns and the cluster structure of the space of polygons

Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov (2012)

Annales de l’institut Fourier

We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n -gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.

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