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Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

This monograph is devoted to the proof of two related results. The first one asserts that if The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the For each Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

We study a class of measure-theoretic objects that we call

Much more than a technical book. Erik’s work is a well documented journey into the multiple interactions between safety, work and human nature. A timely contribution to vindicate human beings and their variability from the one sided focus on the evils of human error. A groundbreaking look at ‘the other story’ that will certainly contribute to safer and more productive workplaces. Dr Alejandro Morales, Mutual Seguridad, Chile Safety needs a new maturity. We can no longer improve by simply doing what we have been doing, even by doing it better. DR Hollnagel brings forth new distinctions, interpretations, and narratives that will allow safety to progress to new unforeseen levels. Safety-II is more than just incident and accident prevention. A must read for every safety professional. Tom McDaniel, Global Manager Zero Harm and Human Performance, Siemens Energy, Inc., USA

The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.

We extend the proof of Reifenberg’s Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set On généralise la démonstration du théorème du disque topologique de Reifenberg pour inclure le cas d’ensembles ayant des trous, et on donne des conditions suffisantes sur l’ensemble

Background Implementation outcome measures are essential for monitoring and evaluating the success of implementation efforts. Yet, currently available measures lack conceptual clarity and have largely unknown reliability and validity. This study developed and psychometrically assessed three new measures: the Acceptability of Intervention Measure (AIM), Intervention Appropriateness Measure (IAM), and Feasibility of Intervention Measure (FIM). Methods Thirty-six implementation scientists and 27 mental health professionals assigned 31 items to the constructs and rated their confidence in their assignments. The Wilcoxon one-sample signed rank test was used to assess substantive and discriminant content validity. Exploratory and confirmatory factor analysis (EFA and CFA) and Cronbach alphas were used to assess the validity of the conceptual model. Three hundred twenty-six mental health counselors read one of six randomly assigned vignettes depicting a therapist contemplating adopting an evidence-based practice (EBP). Participants used 15 items to rate the therapist’s perceptions of the acceptability, appropriateness, and feasibility of adopting the EBP. CFA and Cronbach alphas were used to refine the scales, assess structural validity, and assess reliability. Analysis of variance (ANOVA) was used to assess known-groups validity. Finally, half of the counselors were randomly assigned to receive the same vignette and the other half the opposite vignette; and all were asked to re-rate acceptability, appropriateness, and feasibility. Pearson correlation coefficients were used to assess test-retest reliability and linear regression to assess sensitivity to change. Results All but five items exhibited substantive and discriminant content validity. A trimmed CFA with five items per construct exhibited acceptable model fit (CFI = 0.98, RMSEA = 0.08) and high factor loadings (0.79 to 0.94). The alphas for 5-item scales were between 0.87 and 0.89. Scale refinement based on measure-specific CFAs and Cronbach alphas using vignette data produced 4-item scales (α’s from 0.85 to 0.91). A three-factor CFA exhibited acceptable fit (CFI = 0.96, RMSEA = 0.08) and high factor loadings (0.75 to 0.89), indicating structural validity. ANOVA showed significant main effects, indicating known-groups validity. Test-retest reliability coefficients ranged from 0.73 to 0.88. Regression analysis indicated each measure was sensitive to change in both directions. Conclusions The AIM, IAM, and FIM demonstrate promising psychometric properties. Predictive validity assessment is planned.

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than