8,945 research outputs found
Hindman's finite sums theorem and its application to topologizations of algebras
The first part of the paper is a brief overview of Hindman's finite sums
theorem, its prehistory and a few of its further generalizations, and a modern
technique used in proving these and similar results, which is based on
idempotent ultrafilters in ultrafilter extensions of semigroups. The second,
main part of the paper is devoted to the topologizability problem of a wide
class of algebraic structures called polyrings; this class includes Abelian
groups, rings, modules, algebras over a ring, differential rings, and others.
We show that the Zariski topology on such an algebra is always non-discrete.
Actually, a much stronger fact holds: if is an infinite polyring, a
natural number, and a map of into is defined by a term in
variables, then is a closed nowhere dense subset of the space
with its Zariski topology. In particular, is a closed nowhere dense
subset of . The proof essentially uses a multidimensional version of
Hindman's finite sums theorem established by Bergelson and Hindman. The third
part of the paper lists several problems concerning topologization of various
algebraic structures, their Zariski topologies, and some related questions.
This paper is an extended version of the lecture at Journ\'ees sur les
Arithm\'etiques Faibles 36: \`a l'occasion du 70\`eme anniversaire de Yuri
Matiyasevich, delivered on 7th July, 2017, in Saint Petersburg.Comment: The main result of the paper, Theorem 2.4.1, was proved around 2010
but not published until 2017 though presented at several seminars and
conferences, e.g. Colloquium Logicum 2012 in Paderborn, and included in
author's course lectured at the Steklov Mathematical Institute in 201
Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective
Phylogenetic trees are the fundamental mathematical representation of
evolutionary processes in biology. As data objects, they are characterized by
the challenges associated with "big data," as well as the complication that
their discrete geometric structure results in a non-Euclidean phylogenetic tree
space, which poses computational and statistical limitations. We propose and
study a novel framework to study sets of phylogenetic trees based on tropical
geometry. In particular, we focus on characterizing our framework for
statistical analyses of evolutionary biological processes represented by
phylogenetic trees. Our setting exhibits analytic, geometric, and topological
properties that are desirable for theoretical studies in probability and
statistics, as well as increased computational efficiency over the current
state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl
Algebraic description of spacetime foam
A mathematical formalism for treating spacetime topology as a quantum
observable is provided. We describe spacetime foam entirely in algebraic terms.
To implement the correspondence principle we express the classical spacetime
manifold of general relativity and the commutative coordinates of its events by
means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the
limit essentially correcte
Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology
We consider two relatively natural topologizations of the set of all cellular
automata on a fixed alphabet. The first turns out to be rather pathological, in
that the countable space becomes neither first-countable nor sequential. Also,
reversible automata form a closed set, while surjective ones are dense. The
second topology, which is induced by a metric, is studied in more detail.
Continuity of composition (under certain restrictions) and inversion, as well
as closedness of the set of surjective automata, are proved, and some
counterexamples are given. We then generalize this space, in the sense that
every shift-invariant measure on the configuration space induces a pseudometric
on cellular automata, and study the properties of these spaces. We also
characterize the pseudometric spaces using the Besicovitch distance, and show a
connection to the first (pathological) space.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
- …