34 research outputs found
On finite matroids with two more hyperplanes than points
No full textAbstractOne of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of Basterfield, Kelly and Green (1968/1970) (J.G. Basterfield, L.M. Kelly, A characterization of sets of n points which determine n hyperplanes, in: Proceedings of the Cambridge Philosophical Society, vol. 64, 1968, pp. 585–588; C. Greene, A rank inequality for finite geometric lattices, J. Combin Theory 9 (1970) 357–364) affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented
Le Simmetrie dei Fregi Ornamentali
No full textI fregi ornamentali sono presenti in molti luoghi diversi dai fregi degli edifici. Senza tenere conto della scala e del motivo di base, ma considerando solo le simmetrie in cui tali schemi sono lasciati invariati, vedremo che ci sono solo sette possibili tipi di modelli di fregi ornamentali. La geometria delle isometrie del piano euclideo e alcuni elementi della teoria dei gruppi saranno strumenti necessari per classificare i gruppi delle simmetrie dei fregi. In questo modo, speriamo che il fascino di riconoscere le simmetrie nelle architetture e nei fregi possa motivare l'acquisizione dei concetti necessari di geometria e algebr
Bose-Burton type theorems for finite Grassmannians
No full textIn this paper both blocking sets with respect to the s-subspaces and covers with t-subspaces in a finite Grassmannian are investigated, especially focusing on geometric descriptions of blocking sets and covers of minimum size. When such a description exists, it is called a Bose–Burton type theorem. The canonical example of a blocking set with respect to the s-subspaces is the intersection of s linear complexes. In some cases such an intersection is a blocking set of minimum size, that can occasionally be characterized by a Bose–Burton type theorem. In particular, this happens for the 1-blocking sets of the Grassmannian of planes of PG( 5 , q )
A Characterization of Grassmann Spaces of Index h of a Projective Space
No full textAbstractIn 1982/83 Bichara and Tallini characterized Grassmann spaces Gr(h,P ) of a projective space P involving intersection properties of the two disjoint families of maximal subspaces of Gr(h, P). In 1984 Melone and Olanda characterized Gr(1,P ) using only one family of maximal subspaces. In this paper the generalization of the result of Melone and Olanda to general index is given. More precisely, I prove that the natural extension of the axiom of Melone and Olanda is not sufficient to characterize Gr(h, P) when h> 1, since the affine Grassmann spaceGr (1, A) of the lines of an affine space A satisfies the axioms, too. Thus, an additional axiom is given and the characterization follows
Classification of Veronesean caps
No full textAbstractIn this paper all Veronesean caps of projective spaces of finite dimension over skewfields are classified. More precisely, if PG(M,K), K a skewfield, contains a Veronesean cap X, then K is a field and X is either a Veronese variety or a projection of a Veronese variety. This result extends analogous theorems of Mazzocca and Melone [Caps and Veronese varieties in projective Galois spaces. Discrete Math. 48 (1984) 243–252] and Thas and Van Maldeghem [Classification of finite Veronesean caps, European J. Combin. 25(2) (2004) 275–285] for finite projective spaces
