1,721,128 research outputs found
On affine planes with 3-regular group of projectivities
No full textThe author looks at affine planes from von Staudt's point of view, by investigating the consequences of regularity assumptions for the group Πa of affine projectivities. This group Πa, which consists of all products of parallel projections, is always doubly transitive; it is 2-regular only in Desarguesian affine planes, and it is 4-regular in free affine planes [A. Barlotti et al., Rend. Sem. Mat. Univ. Padova 60 (1978), 183--200; MR0555963 (81g:51005)]. Concerning 3-regularity, the author proves the following theorem: Let A be an affine plane, and assume that Πa is 3-regular. Then A is a translation plane, and if the kernel of A is not GF(2) (or if A is finite), then A is in fact Desarguesian
Vier-reguläre Möbius-Ebenen
No full textConsider the following regularity condition (Pn): every projectivity fixing n points is the identity. It was known that if the subgroup Π of all proper projectivities satisfies (P3) the plane must be Miquelian. The author shows that this is true even if Π satisfies (P4)
Der Satz von v. Staudt-Schleiermacher in Minkowski-Ebenen
No full textLet M be a Minkowski (incidence) plane and let Π(M) be the group of so-called`free' projectivities of M. Then M is Miquelian if Π(M) satisfies condition (P_5), i.e., every free projectivity with 5 fixed points is the indentity. But first a lemma is proved, which holds also in Möbius and Laguerre (incidence) planes: If Π(M) satisfies (P_5), then every affine derivation of M is Pappian
Cyclic Difference Sets of Positive Deficiency
No full textA subset is called a {\it cyclic difference set modulo}
{\it of order} {\it and deficiency} if, for with , the
differences s_i - s_j \; (\makebox{mod} \, n) are pairwise
distinct. Planar cyclic difference sets provide instances having
deficiency , whereas --mark Golomb rulers produce infinitely
many examples of cyclic difference sets modulo of order
and positive deficiency, for all where
denotes the length of an optimal --mark Golomb ruler. We
present two constructions which yield deficient difference sets
modulo with . As an application, these results
fill some gaps in the spectrum of cyclic configurations of type
On configurations of type n_κ with constant degree of irreducibility
No full textFor each symmetric configuration, a criterion of Martinetti irreducibility is obtained in terms of weights associated with every anti-flag of the configuration. With the aid of this concept (finite) projective planes and generalized quadrangles are characterized in the class of all symmetric configurations
On a Class of Amply Regular Graphs
No full textAmply regular graphs with parameters can be characterized as regular graphs of order and valency such that every edge lies in some 3-cycle, but in no 4-cycle. Such graphs can exist only if (mod ) and (mod ). Extremal graph theory provides some further restriction: we conjecture that a lower bound is given by and show that eventually this bound is sharp for even prime powers . For , there exist connected instances for all feasible orders (mod ) with , e.g. line graphs of generalized Petersen graphs with girth at least . For , the existence of instances remains an open problem for many values of . We show that connected amply regular graphs with parameters can be amalgamated to a connected amply regular graph with parameters
On the existence of little Desargues configurations in planes satisfying certain weak configurational conditions
No full textIn a k-net, relationships between the structure of be the group of all projectivities on one hand and the validity of certain closure conditions such as the conditions of Desargues, Pappos, Reidemeister and Thomsen on the other are investigated
On configurations in translation planes of positive characteristic
No full textA projective plane is a characteristic 2 Moufang plane if and only if it satisfies the generalized Desargues condition (d2)
Octagonality conditions in projective and affine planes
No full textA projectie confined configuration C in terms of a non-degenerate octagon gives rise to a configurational condition, whose affine specialization is equivalent to the affine Pappos condition, whereas the little specialization is equivalent to the little Desdargues condition. In Pappian projective planes the configuration C can be completed to a configuration of type
Regularität in freien Benz-Ebenen
No full textIn a free Möbius, Laguerre, or Minkowski plane, the group of projectivities is 6-regular, i.e. each projectivity with 6 fixed points is the identity
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