Bogdan V. Petrenko
On the conjectures of Atiyah and Sutcliffe.
Thu, 02/24/2011 - 16:01 — SomNambulAuthors: Marcin Mazur, Bogdan V. Petrenko
Subjects: Algebraic Geometry
AbstractMotivated by certain questions in physics, Atiyah defined a determinant
function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb
R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and
Sutcliffe stated three intriguing conjectures about this determinant. They
provided compelling numerical evidence for the conjectures and an interesting
physical interpretation of the determinant.On the smallest number of generators and the probability of generating an algebra.
Tue, 01/19/2010 - 16:01 — SomNambulAuthors: Rostyslav V. Kravchenko, Marcin Mazur, Bogdan V. Petrenko
Subjects: Rings and Algebras
AbstractIn this paper we study algebraic and asymptotic properties of generating sets
of algebras over orders in number fields. Let $A$ be an associative algebra
over an order $R$ in an algebraic number field. We assume that $A$ is a free
$R$-module of finite rank. We develop a technique to compute the smallest
number of generators of $A$. For example, we prove that the ring
$\M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. For a
given positive integer $m$, we define the density of the set of all ordered
$m$-tuples of elements of $A$ which generate it as an $R$-algebra.
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