49 research outputs found
The cubic period-distance relation for the Kater reversible pendulum
We describe the correct cubic relation between the mass configuration of a
Kater reversible pendulum and its period of oscillation. From an analysis of
its solutions we conclude that there could be as many as three distinct mass
configurations for which the periods of small oscillations about the two pivots
of the pendulum have the same value. We also discuss a real compound Kater
pendulum that realizes this property.Comment: 25 pages 4figure
The theory of the exponential differential equations of semiabelian varieties
The complete first order theories of the exponential differential equations
of semiabelian varieties are given. It is shown that these theories also arises
from an amalgamation-with-predimension construction in the style of Hrushovski.
The theory includes necessary and sufficient conditions for a system of
equations to have a solution. The necessary condition generalizes Ax's
differential fields version of Schanuel's conjecture to semiabelian varieties.
There is a purely algebraic corollary, the "Weak CIT" for semiabelian
varieties, which concerns the intersections of algebraic subgroups with
algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new
introductio
Evaluation Codes from smooth Quadric Surfaces and Twisted Segre Varieties
We give the parameters of any evaluation code on a smooth quadric surface.
For hyperbolic quadrics the approach uses elementary results on product codes
and the parameters of codes on elliptic quadrics are obtained by detecting a
BCH structure of these codes and using the BCH bound. The elliptic quadric is a
twist of the surface P^1 x P^1 and we detect a similar BCH structure on twists
of the Segre embedding of a product of any d copies of the projective line.Comment: 10 pages. Presented at the conference Workshop on Coding theory and
Cryptography 201
Virtually abelian K\"ahler and projective groups
We characterise the virtually abelian groups which are fundamental groups of
compact K\"ahler manifolds and of smooth projective varieties. We show that a
virtually abelian group is K\"ahler if and only if it is projective. In
particular, this allows to describe the K\"ahler condition for such groups in
terms of integral symplectic representations
Del Pezzo surfaces of degree 1 and jacobians
We construct absolutely simple jacobians of non-hyperelliptic genus 4 curves,
using Del Pezzo surfaces of degree 1. This paper is a natural continuation of
author's paper math.AG/0405156.Comment: 24 page
Singularity, complexity, and quasi--integrability of rational mappings
We investigate global properties of the mappings entering the description of
symmetries of integrable spin and vertex models, by exploiting their nature of
birational transformations of projective spaces. We give an algorithmic
analysis of the structure of invariants of such mappings. We discuss some
characteristic conditions for their (quasi)--integrability, and in particular
its links with their singularities (in the 2--plane). Finally, we describe some
of their properties {\it qua\/} dynamical systems, making contact with
Arnol'd's notion of complexity, and exemplify remarkable behaviours.Comment: Latex file. 17 pages. To appear in CM
Modular Lie algebras and the Gelfand-Kirillov conjecture
Let g be a finite dimensional simple Lie algebra over an algebraically closed
field of characteristic zero. We show that if the Gelfand-Kirillov conjecture
holds for g, then g has type A_n, C_n or G_2.Comment: 20 page