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Construction of Mendelsohn designs by using quasigroups of ( 2 , q ) -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

Let q be a positive integer. An algebra is said to have the property ( 2 , q ) if all of its subalgebras generated by two distinct elements have exactly q elements. A variety 𝒱 of algebras is a variety with the property ( 2 , q ) if every member of 𝒱 has the property ( 2 , q ) . Such varieties exist only in the case of q prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property ( 2 , q ) , 2 < q , blocks of Steiner system of type ( 2 , q ) are obtained. The stated correspondence between Steiner...

Construction, properties and applications of finite neofields

Anthony Donald Keedwell (2000)

Commentationes Mathematicae Universitatis Carolinae

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

Contraction of excess fibres between the McKay correspondences in dimensions two and three

Samuel Boissière, Alessandra Sarti (2007)

Annales de l’institut Fourier

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions...

Contraction par Frobenius de G -modules

Michel Gros, Masaharu Kaneda (2011)

Annales de l’institut Fourier

Soit G un groupe algébrique semi-simple simplement connexe défini sur un corps algébriquement clos 𝕜 de caractéristique positive. Nous donnons une nouvelle preuve de l’existence d’un scindage de Frobenius de la variété des drapeaux de G ainsi que de la nature G -équivariante de celui-ci. L’outil principal est un scindage de l’endomorphisme de Frobenius défini sur toute l’algèbre des distributions de G qui permet de « détordre » la structure des G -modules.

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