380 research outputs found
Retracts of vertex sets of trees and the almost stability theorem
Let G be a group, let T be an (oriented) G-tree with finite edge stabilizers,
and let VT denote the vertex set of T. We show that, for each G-retract V' of
the G-set VT, there exists a G-tree whose edge stabilizers are finite and whose
vertex set is V'. This fact leads to various new consequences of the almost
stability theorem. We also give an example of a group G, a G-tree T and a
G-retract V' of VT such that no G-tree has vertex set V'.Comment: 15 pages, 0 figures. Formerly titled "Some refinements of the almost
stability theorem". Version
Small cancelation rings
The theory of small cancellation groups is well known. In this paper we
introduce the notion of Group-like Small Cancellation Ring. This is the main
result of the paper. We define this ring axiomatically, by generators and
defining relations. The relations must satisfy three types of axioms. The major
one among them is called the Small Cancellation Axiom. We show that the
obtained ring is non-trivial. Moreover, we show that this ring enjoys a global
filtration that agrees with relations, find a basis of the ring as a vector
space and establish the corresponding structure theorems. It turns out that the
defined ring possesses a kind of Gr\"obner basis and a greedy algorithm.
Finally, this ring can be used as a first step towards the iterated small
cancellation theory which hopefully plays a similar role in constructing
examples of rings with exotic properties as small cancellation groups do in
group theory. This is a short version of paper arXiv:2010.02836Comment: arXiv admin note: substantial text overlap with arXiv:2010.0283
Accidental parabolics and relatively hyperbolic groups
By constructing, in the relative case, objects analoguous to Rips and Sela's
canonical representatives, we prove that the set of images by morphisms without
accidental parabolic, of a finitely presented group in a relatively hyperbolic
group, is finite, up to conjugacy.Comment: Revision, 24 pages, 4 figure
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
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Effects of classification context on categorization in natural categories
The patterns of classification of borderline instances of eight common taxonomic categories were examined under three different instructional conditions to test two predictions: first, that lack of a specified context contributes to vagueness in categorization, and second, that altering the purpose of classification can lead to greater or lesser dependence on similarity in classification. The instructional conditions contrasted purely pragmatic with more technical/quasi-legal contexts as purposes for classification, and these were compared with a no-context control. The measures of category vagueness were between-subjects disagreement and within-subjects consistency, and the measures of similarity based categorization were category breadth and the correlation of instance categorization probability with mean rated typicality, independently measured in a neutral context. Contrary to predictions, none of the measures of vagueness, reliability, category breadth, or correlation with typicality were generally affected by the instructional setting as a function of pragmatic versus technical purposes. Only one subcondition, in which a situational context was implied in addition to a purposive context, produced a significant change in categorization. Further experiments demonstrated that the effect of context was not increased when participants talked their way through the task, and that a technical context did not elicit more all-or-none categorization than did a pragmatic context. These findings place an important boundary condition on the effects of instructional context on conceptual categorization
Raising argument strength using negative evidence: A constraint on models of induction
Both intuitively, and according to similarity-based theories of induction, relevant evidence raises argument strength when it is positive and lowers it when it is negative. In three experiments, we tested the hypothesis that argument strength can actually increase when negative evidence is introduced. Two kinds of argument were compared through forced choice or sequential evaluation: single positive arguments (e.g., “Shostakovich’s music causes alpha waves in the brain; therefore, Bach’s music causes alpha waves in the brain”) and double mixed arguments (e.g., “Shostakovich’s music causes alpha waves in the brain, X’s music DOES NOT; therefore, Bach’s music causes alpha waves in the brain”). Negative evidence in the second premise lowered credence when it applied to an item X from the same subcategory (e.g., Haydn) and raised it when it applied to a different subcategory (e.g., AC/DC). The results constitute a new constraint on models of induction
Assessing Semantic Similarities among Geospatial Feature Class Definitions
The assessment of semantic similarity among objects is a basic requirement for semantic interoperability. This paper presents an innovative approach to semantic similarity assessment by combining the advantages of two different strategies: featurematching process and semantic distance calculation. The model involves a knowledge base of spatial concepts that consists of semantic relations (is-a and part-whole) and distinguishing features (functions, parts, and attributes). By taking into consideration cognitive properties of similarity assessments, this model expects to represent a cognitively plausible and computationally achievable method for measuring the degree of interoperability
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