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Irreducible collineation groups with two orbits forming an oval
No full textLet G be a collineation group of a finite projective plane P of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω_0 and Ω_1 on Ω, and the action of G on Ω_0 is primitive.We show that if G is irreducible, then P has a G-invariant desarguesian subplane P_0 and Ω_0 is a conic of P_0
Sharply transitive decompositions of complete graphs into generalized Petersen graphs
No full textA decomposition of the complete graph into copies of a subgraph Γ is called a sharply transitive Γ-decomposition if it is left invariant by an automorphism group acting sharply transitively on the vertex-set of . For suitable values of v we construct examples of sharply transitive Γ-decompositions when Γ is either a Petersen graph, a generalized Petersen graph or a prism
Neuronal alignment and outgrowth on microwrinkled conducting polymer substrates
No full textWe report on the results of culturing SH-SY5Y neuron-like cells on PEDOT:PSS wrinkled surfaces fabricated by thermally-induced shrinking of commercial polystyrene sheets. Such smart biointerfaces combine the functional properties of conducting polymers with the topographic patterning at the micro-and sub-microscale, as a result of surface wrinkling. By imposing mechanical constraints during shrinking, anisotropic topographic features are formed, with a spatial periodicity in the range 0.7-1.2 um, tunable by varying the thickness of the PEDOT:PSS thin film. The effectiveness of wrinkled surfaces in enhancing and orientating the outgrowth of neurites is demonstrated by a 42% increase in length and by the 85% of neuntes aligned along wrinkles direction (angle 0 < 9< 15°), after 5 days of differentiation. Furthermore, the conductive properties of the PEDOT:PSS film are retained after the surface wrinkling, opening the way for the exploitation of these smart biointerfaces for the electrical stimulation of cells
A Characterization of the sharply 3-transitive finite permutation groups
No full textGeneralizing a result by N. Percsy, we prove a sufficient conditionfor a sharply 3-transitive finite permutation set with identity to be a group; the proof makes use of M. J. Kallaher's theorem on finite Bol quasifield
Intransitive collineation groups of ovals fixing a triangle
Full text linkWe investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let P be a finite projective plane of odd order n containing an oval O. If a collineation group G of P satisfies the properties: (a) G fixes O and the action of G on O yields precisely two orbits O_1 and O_2, (b) G has even order and a faithful primitive action on O_2, (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval O, then n is an element of {7, 9, 27}, the orbit O_2 has length 4 and G acts naturally on O_2 as A_4 or S_4. Each order n in {7, 9, 27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n = 7, 9; the determination of the planes is still incomplete for n = 27
On two-transitive ovals in projective planes of even order
No full textAn oval Ω in a finite projective plane is said to be 2-transitive if the plane admits a collineation group G fixing Ω and acting 2-transitively on its points. In the order n of the plane is assumed to be even then the following result is proved.Theorem. If G fixes an external line and acts 2-transitively on Ω then either n ∈ {2, 4} or n ≡ 0 mod 8 and the Sylow 2-subgroups of G are generalized quaternion groups.The result is obtained by examining the action of G on a G-invariant family of pairwise disjoint ovals (including Ω) with a common knot
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