The automorphism group of a free-by-cyclic group in rank 2

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Stable rank of leavitt path algebra

We characterize the values of the stable rank for Leavitt path algebras, by giving concrete criteria in terms of properties of the underlying graph.

Real rank zero algebras have the corona factorization property

The purpose of this short note is to prove that a stable separable C*-algebra with real rank zero has the so-called corona factorization property, that is, all the full multiplier projections are properly in finite. Enroute to our result, we consider conditions under which a real rank zero C*-algebra admits an injection of the compact operators (a question already considered in [21]).

Validation of order rank scales based on compositional data analysis: a proposal

Usually, psychometricians apply classical factorial analysis to evaluate construct validity of order rankscales. Nevertheless, these scales have particular characteristics that must be taken into account: totalscores and rank are highly relevant

Rank estimation of trajectory matrix in motion segmentation

A novel technique for estimating the rank of the trajectory matrix in the local subspace affinity (LSA) motion segmentation framework is presented. This new rank estimation is based on the relationship between the estimated rank of the trajectory matrix and the affinity matrix built with LSA. The result is an enhanced model selection technique for trajectory matrix rank estimation by which it is possible to automate LSA, without requiring any a priori knowledge, and to improve the final segmentation

Reliable detection of directional couplings using rank statistics

To detect directional couplings from time series various measures based on distances in reconstructed state spaces were introduced. These measures can, however, be biased by asymmetries in the dynamics' structure, noise color, or noise level, which are ubiquitous in experimental signals. Using theoretical reasoning and results from model systems we identify the various sources of bias and show that most of them can be eliminated by an appropriate normalization. We furthermore diminish the remaining biases by introducing a measure based on ranks of distances. This rank-based measure outperforms existing distance-based measures concerning both sensitivity and specificity for directional couplings. Therefore, our findings are relevant for a reliable detection of directional couplings from experimental signals.

Testing the descriptive performance of the rank-dependent utility in the domain of health profiles

Expected utility theory (EUT) has been challenged as a descriptive theoryin many contexts. The medical decision analysis context is not an exception.Several researchers have suggested that rank dependent utility theory (RDUT)may accurately describe how people evaluate alternative medical treatments.Recent research in this domain has addressed a relevant feature of RDU models-probability weighting-but to date no direct test of this theoryhas been made. This paper provides a test of the main axiomatic differencebetween EUT and RDUT when health profiles are used as outcomes of riskytreatments. Overall, EU best described the data. However, evidence on theediting and cancellation operation hypothesized in Prospect Theory andCumulative Prospect Theory was apparent in our study. we found that RDUoutperformed EU in the presentation of the risky treatment pairs in whichthe common outcome was not obvious. The influence of framing effects onthe performance of RDU and their importance as a topic for future researchis discussed.

A general theory of rank testing

This paper develops an approach to rank testing that nests all existing rank tests andsimplifies their asymptotics. The approach is based on the fact that implicit in every ranktest there are estimators of the null spaces of the matrix in question. The approach yieldsmany new insights about the behavior of rank testing statistics under the null as well as localand global alternatives in both the standard and the cointegration setting. The approach alsosuggests many new rank tests based on alternative estimates of the null spaces as well as thenew fixed-b theory. A brief Monte Carlo study illustrates the results.

Examples of retracts in a free group that are not the fixed subgroup of any group of automorphisms

Let F be a free group of rank at least three. We show that some retracts of F previously studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where, by the Galois monoid of a subgroup H of F, we mean the monoid consisting of all endomorphisms of F that fix H.

Free triples, large indifference classes and the majority rule

We present a new domain of preferences under which the majority relation is always quasi-transitive and thus Condorcet winners always exist. We model situations where a set of individuals must choose one individual in the group. Agents are connected through some relationship that can be interpreted as expressing neighborhood, and which is formalized by a graph. Our restriction on preferences is as follows: each agent can freely rank his immediate neighbors, but then he is indifferent between each neighbor and all other agents that this neighbor "leads to". Hence, agents can be highly perceptive regarding their neighbors, while being insensitive to the differences between these and other agents which are further removed from them. We show quasi-transitivity of the majority relation when the graph expressing the neighborhood relation is a tree. We also discuss a further restriction allowing to extend the result for more general graphs. Finally, we compare the proposed restriction with others in the literature, to conclude that it is independent of any previously discussed domain restriction.

Stochastic dominance and absolute risk aversion

In this paper we propose the infimum of the Arrow-Pratt index of absolute risk aversion as a measure of global risk aversion of a utility function. We then show that, for any given arbitrary pair of distributions, there exists a threshold level of global risk aversion such that all increasing concave utility functions with at least as much global risk aversion would rank the two distributions in the same way. Furthermore, this threshold level is sharp in the sense that, for any lower level of global risk aversion, we can find two utility functions in this class yielding opposite preference relations for the two distributions.

An Atkinson-Gini family of social evaluation functions

We investigate the properties of a family of social evaluation functions and inequality indices which merge the features of the family of Atkinson (1970) and S-Gini (Donaldson and Weymark (1980, 1983), Yitzhaki (1983) and Kakwani (1980)) indices. Income inequality aversion is captured by decreasing marginal utilities, and aversion to rank inequality is captured by rank-dependent ethical weights, thus providing an ethically-flexible dual basis for the assessment of inequality and equity. These ocial evaluation functions can be interpreted as average utility corrected for the illfare of relative deprivation. They can alternatively be understood as averages of altruistic well-being in a population. They moreover have a simple graphical interpretation.

Infinite index subgroups and finiteness properties of intersections of geometrically finite groups

We explore which types of finiteness properties are possible for intersections of geometrically finite groups of isometries in negatively curved symmetric rank one spaces. Our main tool is a twist construction which takes as input a geometrically finite group containing a normal subgroup of infinite index with given finiteness properties and infinite Abelian quotient, and produces a pair of geometrically finite groups whose intersection is isomorphic to the normal subgroup.

Cat(0) and Cat (-1) dimensions of torsion free

We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimensional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.