26,395 research outputs found

    On the intersection of free subgroups in free products of groups

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    Let (G_i | i in I) be a family of groups, let F be a free group, and let G = F *(*I G_i), the free product of F and all the G_i. Let FF denote the set of all finitely generated subgroups H of G which have the property that, for each g in G and each i in I, H \cap G_i^{g} = {1}. By the Kurosh Subgroup Theorem, every element of FF is a free group. For each free group H, the reduced rank of H is defined as r(H) = max{rank(H) -1, 0} in \naturals \cup {\infty} \subseteq [0,\infty]. To avoid the vacuous case, we make the additional assumption that FF contains a non-cyclic group, and we define sigma := sup{r(H\cap K)/(r(H)r(K)) : H, K in FF and r(H)r(K) \ne 0}, sigma in [1,\infty]. We are interested in precise bounds for sigma. In the special case where I is empty, Hanna Neumann proved that sigma in [1,2], and conjectured that sigma = 1; almost fifty years later, this interval has not been reduced. With the understanding that \infty/(\infty -2) = 1, we define theta := max{|L|/(|L|-2) : L is a subgroup of G and |L| > 2}, theta in [1,3]. Generalizing Hanna Neumann's theorem, we prove that sigma in [theta, 2 theta], and, moreover, sigma = 2 theta if G has 2-torsion. Since sigma is finite, FF is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that sigma = theta whenever G does not have 2-torsion.Comment: 28 pages, no figure

    The notion of ψ\psi-weak dependence and its applications to bootstrapping time series

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    We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.Comment: Published in at http://dx.doi.org/10.1214/06-PS086 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Fixed subgroups are compressed in surface groups

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    For a compact surface Σ\Sigma (orientable or not, and with boundary or not) we show that the fixed subgroup, Fix⁡B\operatorname{Fix} B, of any family BB of endomorphisms of π1(Σ)\pi_1(\Sigma) is compressed in π1(Σ)\pi_1(\Sigma) i.e., rk⁡((Fix⁡B)∩H)≤rk⁡(H)\operatorname{rk}((\operatorname{Fix} B)\cap H)\leq \operatorname{rk}(H) for any subgroup Fix⁡B≤H≤π1(Σ)\operatorname{Fix} B \leq H \leq \pi_1(\Sigma). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, GG, of finitely many free and surface groups, and give a characterization of when GG satisfies that rk⁡(Fix⁡ϕ)≤rk⁡(G)\operatorname{rk}(\operatorname{Fix} \phi) \leq \operatorname{rk}(G) for every ϕ∈Aut(G)\phi \in Aut(G)

    A robust bootstrap approach to the Hausman test in stationary panel data models

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    In panel data econometrics the Hausman test is of central importance to select an e?cient estimator of the models' slope parameters. When testing the null hypothesis of no correlation between unobserved heterogeneity and observable explanatory variables by means of the Hausman test model disturbances are typically assumed to be independent and identically distributed over the time and the cross section dimension. The test statistic lacks pivotalness in case the iid assumption is violated. GLS based variants of the test statistic are suitable to overcome the impact of nuisance parameters on the asymptotic distribution of the Hausman statistic. Such test statistics, however, also build upon strong homogeneity restrictions that might not be met by empirical data. We propose a bootstrap approach to specification testing in panel data models which is robust under cross sectional or time heteroskedasticity and inhomogeneous patterns of serial correlation. A Monte Carlo study shows that in small samples the bootstrap approach outperforms inference based on critical values that are taken from a X?-distribution. --Hausman test,random effects model,wild bootstrap,heteroskedasticity

    No-cloning theorem in thermofield dynamics

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    We discuss the relation between the no-cloning theorem from quantum information and the doubling procedure used in the formalism of thermofield dynamics (TFD). We also discuss how to apply the no-cloning theorem in the context of thermofield states defined in TFD. Consequences associated to mixed states, von Neumann entropy and thermofield vacuum are also addressed.Comment: 16 pages, 3 figure
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