We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider their difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are nonstandard notions of regularity of higher order, and the formula of Faà di Bruno for higher order derivatives and a discrete version of it. As an application ...

We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space M0,n of stable genus zero n-pointed cu...

In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has Φsectional curvature −3. I also briefly discuss some relations with the well-know sub-Riemannian geomet...

The G-values of various games, by R. K. Guy and C. A. B. Smith, introduced a broad class of impartial games and gave complete normalplay solutions for many of them. In this paper, we announce misère-play solutions to many of the games considered by Guy and Smith. They are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games....

In earlier work we gave a characterisation of pregeometries which are ‘basic’ (that is, admit no ‘non-degenerate’ quotients) relative to two different kinds of quotient operation, namely taking imprimitive quotients and normal quotients. Each basic geometry was shown to involve a faithful group action, which is primitive or quasiprimitive, respectively, on the set of elements of each type. For ...

The G-values of various games, by R. K. Guy and C. A. B. Smith, introduced a broad class of impartial games and gave complete normalplay solutions for many of them. In this paper, we announce misère-play solutions to many of the games considered by Guy and Smith. They are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games....

We extend previous computations of Calabi-Yau metrics on projective hypersurfaces to free quotients, complete intersections, and free quotients of complete intersections. In particular, we construct these metrics on generic quintics, four-generation quotients of the quintic, Schoen CalabiYau complete intersections and the quotient of a Schoen manifold with Z3 × Z3 fundamental group that was pre...

Symmetric graphs of diameter 2 with complete normal quotients Carmen Amarra University of Western Australia, Australia [email protected] A graph has diameter 2 if it is not a complete graph and if every pair of nonadjacent vertices is joined by a path of length 2. Our general problem is to examine the overall structure of graphs which are both arc-transitive and of diameter 2 using norm...

Ah, la recherche! Du temps perdu. Soit G le quotient du groupe libre a 26 g en era-teurs a; b; c; : : : ; z par les relations A = B, pour tout couple de mots (A; B) pouvant avoir la m^ eme prononciation en anglais. (Un groupe d eeni d'une mani ere analogue a et e consid er e dans Landsburg 1986].) Notre but est de d eterminer la structure du groupe G. Théorème. G est trivial. Démonstration. (No...