D. Pröpper, A. N. Yaresko, T. I. Larkin, T. N. Stanislavchuk, A. A. Sirenko, T. Takayama, A. Matsumoto, H. Takagi, B. Keimer, and A. V. Boris Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany Department of Physics, New Jersey Institute of Technology, Newark, New Jersey 07102, USA Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan (...

There are exactly 16 non-isomorphic Steiner systems S(2,4, 25) with nontrivial automorphism group. It is interesting to note that each of these designs has an automorphism of order 3. These 16 designs are presented along with their groups and other invariants. In particular, we determine and tabulate substructures for each of the sixteen designs inciuding Fano subplanes, ovals, complete 5-arcs,...

Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n+1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 ...

We consider the following Turán problem. How many edges can there be in a (q + 1)-uniform hypergraph on n vertices that does not contain a copy of the projective geometry PGm(q)? The case q = m = 2 (the Fano plane) was recently solved independently and simultaneously by Keevash and Sudakov (The Turán number of the Fano plane, Combinatorica, to appear) and Füredi and Simonovits (Triple systems n...

In this paper we study embeddings of the Fano plane as a bipartite graph. We classify all possible embeddings especially focusing on those with non-trivial automorphism group. We study them in terms of rotation systems, isomorphism classes and chirality. We construct quotients and show how to obtain information about face structure and genus of the covering embedding. As a by-product of the cla...

In this work, we consider matroid theory. After presenting three different (but equivalent) definitions of matroids, we mention some of the most important theorems of such theory. In particular, we note that every matroid has a dual matroid and that a matroid is regular if and only if it is binary and includes no Fano matroid or its dual. We show a connection between this last theorem and octon...

The existence of a connected 12-regular {K4, K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 i...

We investigate the effect of decoherence on Fano resonances in wave transmission through resonant scattering structures. We show that the Fano asymmetry parameter q follows, as a function of the strength of decoherence, trajectories in the complex plane that reveal detailed information on the underlying decoherence process. Dissipation and unitary dephasing give rise to manifestly different tra...

During the past few years we have obtained several new computer classification results on association schemes and in particular distance regular and strongly regular graphs. Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents Ei of these association schemes : the fact that they are positive semidefinite and that they have known rank. Inc...