498 research outputs found
Pseudofinite structures and simplicity
We explore a notion of pseudofinite dimension, introduced by Hrushovski and
Wagner, on an infinite ultraproduct of finite structures. Certain conditions on
pseudofinite dimension are identified that guarantee simplicity or
supersimplicity of the underlying theory, and that a drop in pseudofinite
dimension is equivalent to forking. Under a suitable assumption, a
measure-theoretic condition is shown to be equivalent to local stability. Many
examples are explored, including vector spaces over finite fields viewed as
2-sorted finite structures, and homocyclic groups. Connections are made to
products of sets in finite groups, in particular to word maps, and a
generalization of Tao's algebraic regularity lemma is noted
Voting in small networks with cross-pressure
We present a model of participation in elections in small networks, in which citizens su¤er from cross-pressures if voting against the alternative preferred by some of their social contacts. We analyze how the existence of cross-pressures may shape voting decisions, and so, political outcomes; and how candidates may exploit this e¤ect to their interest.Network; Voting; Cross-Cutting.
Non-local energetics of random heterogeneous lattices
In this paper, we study the mechanics of statistically non-uniform two-phase
elastic discrete structures. In particular, following the methodology proposed
in (Luciano and Willis, Journal of the Mechanics and Physics of Solids 53,
1505-1522, 2005), energetic bounds and estimates of the Hashin-Shtrikman-Willis
type are developed for discrete systems with a heterogeneity distribution
quantified by second-order spatial statistics. As illustrated by three
numerical case studies, the resulting expressions for the ensemble average of
the potential energy are fully explicit, computationally feasible and free of
adjustable parameters. Moreover, the comparison with reference Monte-Carlo
simulations confirms a notable improvement in accuracy with respect to
approaches based solely on the first-order statistics.Comment: 32 pages, 8 figure
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
Definable orthogonality classes in accessible categories are small
We lower substantially the strength of the assumptions needed for the
validity of certain results in category theory and homotopy theory which were
known to follow from Vopenka's principle. We prove that the necessary
large-cardinal hypotheses depend on the complexity of the formulas defining the
given classes, in the sense of the Levy hierarchy. For example, the statement
that, for a class S of morphisms in a locally presentable category C of
structures, the orthogonal class of objects is a small-orthogonality class
(hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from
the existence of a proper class of supercompact cardinals if S is \Sigma_2, and
from the existence of a proper class of what we call C(n)-extendible cardinals
if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a
new hierarchy, and we show that Vopenka's principle is equivalent to the
existence of C(n)-extendible cardinals for all n. As a consequence, we prove
that the existence of cohomological localizations of simplicial sets, a
long-standing open problem in algebraic topology, is implied by the existence
of arbitrarily large supercompact cardinals. This result follows from the fact
that cohomology equivalences are \Sigma_2. In contrast with this fact, homology
equivalences are \Sigma_1, from which it follows (as is well known) that the
existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies
have been correcte
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