107 research outputs found
One-basesness and reductions of elliptic curves over real closed fields
Building on the positive solution of Pillay’s conjecture we present a notion of “intrinsic” reduction for elliptic curves over a real closed field K. We compare such a notion with the traditional algebro-geometric reduction and produce a classification of the group of K-points of an elliptic curve E with three “real” roots according to the way E reduces (algebro-geometrically) and the geometric complexity of the “intrinsically” reduced curve
Algebraic closures and their variations
We study possibilities for algebraic closures, differences between definable
and algebraic closures in first-order structures, and variations of these
closures with respect to the bounds of cardinalities of definable sets and
given sets of formulae. Characteristics for these possibilities and differences
are introduced and described. These characteristics are studied for some
natural classes of theories. Besides algebraic closure operators with respect
to sets of formulae are introduced and studied. Semilattices and lattices for
families of these operators are introduced and characteristics of these
structures are described
Diffusivity of a walk on fracture loops of a discrete torus
In this paper we study functions on the discrete torus which have a
crystalline structure. This means that if we fix such a function and walk
around the torus in a positive direction, the function increases on almost
every step, except at a small number of steps where it must go down in order to
meet the periodicity of the torus. It turns out that the down steps are
organised into a small number of closed simple disjoint paths, the fracture
lines of the crystal. We define a random walk on the resulting functions, the
law of which is Brownian in the diffusive limit. We show that in the limit of
the crystal structure becoming microscopic, the diffusivity is given by
, where
and are the number of fractures in the horizontal
and vertical direction respectively. This is the main result of this paper. The
diffusivity of the corresponding one-dimensional model has already been studied
by Espinasse, Guillotin-Plantard and Nadeau, and this paper generalises that
model to two dimensions. However, the methodology involving an analysis of the
fracture lines that we use to calculate the diffusivity is completely novel
Group orderings, dynamics, and rigidity
Let G be a countable group. We show there is a topological relationship
between the space CO(G) of circular orders on G and the moduli space of actions
of G on the circle; as well as an analogous relationship for spaces of left
orders and actions on the line. In particular, we give a complete
characterization of isolated left and circular orders in terms of strong
rigidity of their induced actions of G on and R.
As an application of our techniques, we give an explicit construction of
infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of
circular orders on free groups disproving a conjecture from Baik--Samperton,
and infinitely many nonconjugate isolated points in the space of left orders on
the pure braid group P_3, answering a question of Navas. We also give a
detailed analysis of circular orders on free groups, characterizing isolated
orders
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Piecewise Linear Models for the Quasiperiodic Transition to Chaos
We formulate and study analytically and computationally two families of
piecewise linear degree one circle maps. These families offer the rare
advantage of being non-trivial but essentially solvable models for the
phenomenon of mode-locking and the quasi-periodic transition to chaos. For
instance, for these families, we obtain complete solutions to several questions
still largely unanswered for families of smooth circle maps. Our main results
describe (1) the sets of maps in these families having some prescribed rotation
interval; (2) the boundaries between zero and positive topological entropy and
between zero length and non-zero length rotation interval; and (3) the
structure and bifurcations of the attractors in one of these families. We
discuss the interpretation of these maps as low-order spline approximations to
the classic ``sine-circle'' map and examine more generally the implications of
our results for the case of smooth circle maps. We also mention a possible
connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request
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