Central limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ( $H<\frac{1}{2})$ and regular ( $H>\frac{1}{2})$ types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter $H<\frac{3}{4}$) is the so-called 4th moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

An advanced statistical analysis of patients’ faces after specific surgical procedures that temporarily negatively affect the patient’s mimetic muscles is presented. For effective planning of rehabilitation, which typically lasts several months, it is crucial to correctly evaluate the improvement of the mimetic muscle function. The current way of describing the development of rehabilitation depends on the subjective opinion and expertise of the clinician and is not very precise concerning when the most common classification (House–Brackmann scale) is used. Our system is based on a stereovision Kinect camera and an advanced mathematical approach that objectively quantifies the mimetic muscle function independently of the clinician’s opinion. To effectively deal with the complexity of the 3D camera input data and uncertainty of the evaluation process, we designed a three-stage data-analytic procedure combining the calculation of indicators determined by clinicians with advanced statistical methods including functional data analysis and ordinal (multiple) logistic regression. We worked with a dataset of 93 distinct patients and 122 sets of measurements. In comparison to the classification with the House–Brackmann scale the developed system is able to automatically monitor reinnervation of mimetic muscles giving us opportunity to discriminate even small improvements during the course of rehabilitation.

Metadynamics method has been widely used to enhance sampling in molecular simulations. Its original form suffers two major drawbacks, poor convergence in complex (especially biomolecular) systems and its serial nature. The first drawback has been addressed by introduction of a convergent variant known as well-tempered metadynamics. The second was addressed by introduction of a parallel multisystem metadynamics referred to as altruistic metadynamics. Here, we combine both approaches into well-tempered altruistic metadynamics. We provide mathematical arguments and trial simulations to show that it accurately predicts free energy surfaces.

This paper focuses on the statistical analysis of mimetic muscle rehabilitation after head and neck surgery causing facial paresis in patients after head and neck surgery. Our work deals with an evaluation problem of mimetic muscle rehabilitation that is observed by a Kinect stereo-vision camera. After a specific brain surgery, patients are often affected by face palsy, and rehabilitation to renew mimetic muscle innervation takes several months. It is important to be able to observe the rehabilitation process in an objective way. The most commonly used House–Brackmann (HB) scale is based on the clinician’s subjective opinion. This paper compares different methods of supervised learning classification that should be independent of the clinician’s opinion. We compare a parametric model (based on logistic regression), non-parametric model (based on random forests), and neural networks. The classification problem that we have studied combines a limited dataset (it contains only 122 measurements of 93 patients) of complex observations (each measurement consists of a collection of time curves) with an ordinal response variable. To balance the frequencies of the considered classes in our data set, we reclassified the samples from HB4 to HB3 and HB5 to HB6—it means that only four HB grades are used for classification algorithm. The parametric statistical model was found to be the most suitable thanks to its stability, tractability, and reasonable performance in terms of both accuracy and precision.

Membrane performance in gas separation is quantified by its selectivity, determined as a ratio of measured gas permeabilities of given gases at fixed pressure difference. In this manuscript a nonlinear dependence of gas permeability on pressure difference observed in the measurements of gas permeability of graphene oxide membrane on a manometric integral permeameter is reported. We show that after reasoned assumptions and simplifications in the mathematical description of the experiment, only static properties of any proposed governing equation can be studied, in order to analyze the permeation rate for different pressure differences. Porous Medium Equation is proposed as a suitable governing equation for the gas permeation, as it manages to predict a nonlinear behavior which is consistent with the measured data. A coefficient responsible for the nonlinearity, the polytropic exponent, is determined to be gas-specific—implications on selectivity are discussed, alongside possible hints to a deeper physical interpretation of its actual value.