1,717 research outputs found
Road Systems and Betweenness
A road system is a collection of subsets of a set—the roads—such that every singleton subset is a road in the system and every doubleton subset is contained in a road. The induced ternary (betweenness) relation is defined by saying that a point c lies between points a and b if c is an element of every road that contains both a and b . Traditionally, betweenness relations have arisen from a plethora of other structures on a given set, reflecting intuitions that range from the order-theoretic to the geometric and topological. In this paper we initiate a study of road systems as a simple mechanism by means of which a large majority of the classical interpretations of betweenness are induced in a uniform way
On the apparent failure of the topological theory of phase transitions
The topological theory of phase transitions has its strong point in two
theorems proving that, for a wide class of physical systems, phase transitions
necessarily stem from topological changes of some submanifolds of configuration
space. It has been recently argued that the lattice -model
provides a counterexample that falsifies this theory. It is here shown that
this is not the case: the phase transition of this model stems from an
asymptotic () change of topology of the energy level sets, in spite
of the absence of critical points of the potential in correspondence of the
transition energy.Comment: 5 pages, 4 figure
The densest lattices in PGL3(Q2)
We find the smallest possible covolume for lattices in PGL3(Q2), show that
there are exactly two lattices with this covolume, and describe them
explicitly. They are commensurable, and one of them appeared in Mumford's
construction of his fake projective plane. We also discuss a new 2-adic
uniformization of another fake projective plane.Comment: Minor error correcte
Apparent contours of nonsingular real cubic surfaces
We give a complete deformation classification of real Zariski sextics, that
is of generic apparent contours of nonsingular real cubic surfaces. As a
by-product, we observe a certain "reversion" duality in the set of deformation
classes of these sextics.Comment: 61 pages, 8 figures, Revised version to be published in Transactions
AMS: some minor corrections, a missing lemma is include
Arithmetic lattices and weak spectral geometry
This note is an expansion of three lectures given at the workshop "Topology,
Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University
in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and
Arithmetic of Hyperbolic Spaces". Comments welcom
A THEORY OF QUALITATIVE SIMILARITY
The central result of this paper establishes an isomorphism between two types of mathematical structures: ""ternary preorders"" and ""convex topologies."" The former are characterized by reflexivity, symmetry and transitivity conditions, and can be interpreted geometrically as ordered betweenness relations; the latter are defined as intersection-closed families of sets satisfying an ""abstract convexity"" property. A large range of examples is given. As corollaries of the main result we obtain a version of Birkhoff''s representation theorem for finite distributive lattices, and a qualitative version of the representation of ultrametric distances by indexed taxonomic hierarchies.
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