A quantization procedure based on completely positive maps and Markov operators

We describe $\omega$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases

Internet-delivered therapy for alcohol misuse: engagement, satisfaction, and outcomes when patients select their preference for therapist- or self-guided treatment

Background Alcohol misuse is common and causes substantial harm. Internet-delivered cognitive behaviour therapy (ICBT) is effective in reducing alcohol misuse; however, the literature investigating how treatment outcomes are impacted by patients’ preferences for therapist- versus self-guided ICBT for alcohol misuse is sparse. Methods In this preference trial, 74 eligible patients (who reported ≥ 14 drinks in the previous week and obtained scores suggesting hazardous or harmful drinking) chose between enrolling in either therapist- or self-guided ICBT for alcohol misuse. We investigated whether those who chose therapist- versus self-guided ICBT differed in their (a) drinking outcomes—as measured by Timeline Follow-Back (TLFB) and heavy drinking days (HDD) at post-treatment and 3-month follow-up—and (b) post-treatment ICBT engagement and satisfaction. Results The majority (81.1%) of eligible patients chose therapist-guided ICBT. These patients reported higher psychotropic medication use, drinking difficulties, and anxiety symptoms. For both the therapist- and self-guided patients, a modified intention-to-treat analysis revealed large within-group treatment effects for TLFB (β =  − 2.64, SE 0.66; p < 0.001) and HDD (β =  − 0.34, SE 0.07; p < 0.001), with large pre-to-post-treatment Cohen’s effect sizes of d = 0.97 (95% CI [0.49, 1.45]) for TLFB and d = 1.19 (95% CI [0.69, 1.68]) for HDD. The interaction comparing the effects of therapist- to self-guided ICBT over time was not significant for TLFB (p = 0.34) or HDD (p = 0.06). With treatment, for both therapist- and self-guided patients, there was a significant improvement in drinking difficulties, cravings, and confidence with controlling cravings, as well as in anxiety, depression, and functional impairment. Further, the majority (75.7%) of patients completed five or more lessons, as well as reported overall satisfaction with the treatment (88.9%) and increased confidence in managing their symptoms (86.7%); these outcomes also did not differ between therapist- and self-guided patients. Conclusions The current study shows that ICBT for alcohol misuse is associated with reduced drinking and comorbid mental health difficulties over time, irrespective of whether patients chose to complete the course on their own or with therapist guidance. Trial registration number: NCT04611854 (https://clinicaltrials.gov/ct2/show/NCT04611854)

QUENCHING OF REVERSE SMOLDER

A simple model of reverse smolder in a porous medium is analyzed using asymptotic methods. When the only chemical reaction is exothermic oxidation, the burning rate is a single-valued function of the blowing rate, increasing from zero to a maximum, and then returning to zero. When endothermic pyrolysis is added to the description, the burning rate is double-valued for blowing rates less than some maximum. Beyond this maximum there are no solutions. The upper branch of the double-valued solution is the physically relevant one. On it, for certain choices of parameters, the burning rate increases from zero to a maximum, and then decreases until quenching occurs at the maximum blowing rate. This behavior mimics experimental observations by Torero, Fernandez-Pello, and Kitano [15]