A Note on One Less Known Class of Generated Residual Implications

This paper builds on our contribution [Havlena and Hlinena, 2016] which studied modelling of the conjunction in human language. We have discussed three different ways of constructing conjunction. We have dealt with generated t-norms, generated means and Choquet integral. In this paper we construct the residual operators based on the above conjunctions. The only operator based on a t-norm is an implication. We show that this implication belongs to the class of generated implications I^g_N which was introduced in [Smutna, 1999] and studied in [Biba and Hlinena, 2012]. We study its properties. More, we investigate this class of generated implications. Some important properties, including relations between some classes of implications, are given

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT-results to compare SRPT with FIFO from a large-deviations point of view.Comment: 22 page

More "normal" than normal: scaling distributions and complex systems

One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems

Generic IRS in free groups, after Bowen

Let $E$ be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space $(X,B,\mu)$. Let $([E],d_{u})$ be the the (Polish) full group endowed with the uniform metric. If $F_r = \langle s_1, \ldots, s_r \rangle$ is a free group on $r$-generators and $\alpha \in \operatorname{Hom}(F_r,[E])$ then the stabilizer of a $\mu$-random point $\alpha(F_r)_x$ is a random subgroup of $F_r$ whose distribution is conjugation invariant. Such an object is known as an "invariant random subgroup" or an IRS for short. Bowen's generic model for IRS in $F_r$ is obtained by taking $\alpha$ to be a Baire generic element in the Polish space $\operatorname{Hom}(F_r, [E])$. The "lean aperiodic model" is a similar model where one forces $\alpha(F_r)$ to have infinite orbits by imposing that $\alpha(s_1)$ be aperiodic. In this setting we show that for $r < \infty$ the generic IRS $\alpha(F_r)_x$ is of finite index in $F_r$ a.s. if and only if $E = E_0$ is the hyperfinite equivalence relation. For any ergodic equivalence relation we show that a generic IRS coming from the lean aperiodic model is co-amenable and core free. Finally, we consider the situation where $\alpha(F_r)$ is highly transitive on almost every orbit and in particular the corresponding IRS is supported on maximal subgroups. Using a result of Le-Ma\^{i}tre we show that such examples exist for any aperiodic ergodic $E$ of finite cost. For the hyperfinite equivalence relation $E_0$ we show that high transitivity is generic in the lean aperiodic model.Comment: 15 pages, 1 figur