Translation planes constructed by multiple hyper-regulus replacement

New classes of mutually disjoint hyper-reguli of order q ⁿ , for n > 2, are determined, which are not André hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q?1) = 1, there are (q?1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2^(q-1)/2 translation planes constructed is not André or generalized André. There are also new constructions of mixed subgeometry partitions.     

Flag-Transitive affine planes of order at most 125

We complete the determination of all the flag-transitive affine planes of order at most 125. In particular, all such planes of order 81 are determined. We show that those with a solvable collineation group are obtained by constructions due to Kantor and Suetake.     

Net replacements in the Hughes-Kleinfeld semifield planes

The bilinear flocks of Cherowitzo are generalized to a large variety of bilinear flocks of the cone ${\mathcal{C}_q}$ . Using these ideas, net replacements of certain Hughes-Kleinfeld planes of order q ⁴ are obtained that construct every André plane in PG(3, q).     

A new class of square order planes

Only a few classes of square order planes are known. These are generalized André planes (including Hall planes), flag transitive planes, Hering's planes and Walker's planes. A new class of planes of order 5^2r, where r is an odd natural number, is constructed and the translation complements of the corresponding planes have been determined. The translation complements divide the sets of distinguished points into 4 orbits of lengths 1, 1, 5^r ? 1, and 5^2r ? 5^r and it is of order 5^r(5^r ? 1)² if r ≠ 1. In the particular case of r = 1, the translation complement divides the set of distinguished points into two orbits of lengths 6 and 20 and it is of order 480.     

Sur l'infini

Sans résumé Traduit par André Weil à Paris     

The André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q 2)

The André/Bruck and Bose representation ([1], [5,6]) of PG(2,q ²) in PG(4,q) is a tool used by many authors in the proof of recent results. In this paper the André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q ²) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2,q ²) is a normal rational curve of order 2, 3, or 4 in the André/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class. Received 1 September 1999; revised 17 July 2000.     

Homotopy and homology of simplicial abelian Hopf algebras

Let A be a simplicial bicommutative Hopf algebra over the field with the property that . We show that is a functor of the André-Quillen homology of A, where A is regarded as an algebra. Then we give a method for calculating that André-Quillen homology independent of knowledge of . Received November 15, 1996 ; in final form March 15, 1997     

On quasiaffine spaces and coherent configurations

In [2] André deduced a (1?1) correspondence between the class of homogeneous coherent configurations and the class of certain noncommutative spaces which he called quasiaffine. In this note we establish a (1?1) correspondence between (not necessarily homogeneous) coherent configurations and weakly quasiaffine spaces which generalizes André's. Furthermore we consider some applications of this correspondence to quasiaffine spaces; especially we characterize such spaces with maximal diameter with respect to one direction (compare with [5]).     

Book review

B. Andráspai:Introductory Graph Theory. Akadémiai Kiadó, Budapest 1977, 266 S., Ft 170.     

Asymptotic sine laws arising from alternating random permutations and sequences

In the last century, Désiré André obtained many remarkable properties of the numbers of alternating permutations, linking them to trigonometric functions among other things. By considering the probability that a random permutation is alternating and that a random sequence (from a uniform distribution) is alternating, and by conditioning on the first element of the sequence, his results are extended and illuminated. In particular, several “asymptotic sine laws” are obtained, some with exponential rates of convergence. © 1996 John Wiley & Sons, Inc.     

Strutture Di Andre Eds-Spazi Con Traslazioni

In this note we consider finite André structures possessing a sufficiently large number of translations. Results are obtained on the structure of the dilatation group and the group generated by central translations. The case of Sperner space is examined in detail.

     

An abelian analogue of Schanuel’s conjecture and applications

The Ramanujan Journal - In this article we study an abelian analogue of Schanuel’s conjecture. This conjecture falls in the realm of the generalised period conjecture of André. As shown...     

Topological and equidistributional refinement of the André–Pink–Zannier conjecture at finitely many places

We present some applications of recent results in homogeneous dynamics to an unlikely intersections problem in Shimura varieties (the André–Pink–Zannier conjecture) and its refinements.     

Blocking sets of Rédei type in projective Hjelmslev planes

In this paper, we introduce Rédei type blocking sets in projective Hjelmslev planes over finite chain rings. We construct, in Hjelmslev planes over chain rings of nilpotency index 2 that contain the residue field as a proper subring, the Baer subplanes associated with this subring as Rédei type blocking sets. Two further examples of Rédei type blocking sets are given for planes over Galois rings generalizing familiar constructions in projective planes over finite fields.     

Non-commutative André–Quillen cohomology for differential graded categories

This article is the companion of [G. Tabuada, Postnikov towers, k-invariants and obstruction theory for DG categories, J. Algebra, in press]. By inspiring ourselves in André–Quillen's work [D. Quillen, On the (co)-homology of commutative rings, in: Proc. Sympos. Pure Math., vol. 17, Amer. Math. Soc., 1970, pp. 65–87], we develop a non-commutative André–Quillen cohomology theory for differential graded categories. As in the classical case of commutative rings, there are derivations, square-zero extension, (non-commutative) cotangent complexes … . We prove that our cohomology theory satisfies transitivity, a Mayer–Vietoris's property, and is the natural algebraic setting for the k-invariants and obstruction classes constructed in [G. Tabuada, Postnikov towers, k-invariants and obstruction theory for DG categories, J. Algebra, in press].     

DEGENERATION OF SEMISIMPLE ALGEBRAS

Some observations and conjectures concerning finite dimensional associative commutative algebras that are degenerations of semisimple algebras.

“…on sait qu'on ne sait rien sur cette sorte d'algèbre.” André Weil     

The motive of the moduli stack of <Emphasis Type="Italic">G</Emphasis>-bundles over the universal curve

We define relative motives in the sense of André. After associating a complex in the derived category of motives to an algebraic stack we study this complex in the case of the moduli of G-bundles varying over the moduli of curves.     

Algebraic and analytic parallel transport on p-adic curves

We relate two different partial p-adic analogues of the classical Riemann-Hilbert correspondence on curves. The first one comes from Deninger-Werner and Faltings and is of algebraic nature. The second one comes from André and Berkovich and is defined on Berkovich analytic spaces.     

The algebra of marriage: an episode in applied group theory

In 1949, André Weil contributed a mathematical appendix to Claude Lévi-Strauss's landmark book, The elementary structures of kinship. In this appendix, Weil (one of the Bourbaki mathematicians) used group-theoretic techniques to model Australian marriage systems. Weil's paper marked the beginning of mathematical anthropology. This essay describes Weil's analysis of marriage systems and traces the uneasy history of the application of group theory to kinship studies.     

An unconditional proof of the Andr���COort conjecture for Hilbert modular surfaces

We give an unconditional proof of the André?COort conjecture for Hilbert modular surfaces asserting that an algebraic curve contained in such a surface and containing an infinite set of special points, is special. The proof relies on a combination of Galois-theoretic techniques and results from the theory of o-minimal structures.