Analytical solution of a class of nonlinear differential equations of the third order in the complex area

В работе приводится доказательство теоремы существования и единственности аналитического решения класса нелинейных дифференциальных уравнений третьего порядка, правая часть которого представлена полиномом шестой степени, в комплексной области. Расширен класс рассматриваемых уравнений за счет новой замены переменных. Получена априорная оценка аналитического приближенного решения. Представлен вариант численного эксперимента оптимизации априорных оценок с помощью апостериорных. The article presents a proof of the theorem of the existence and uniqueness of the analytical solution of the class of nonlinear differential equations of the third order, with a polynomial right-hand side of the sixth degree, in the complex domain. The class of the considered equations has been extended by means of a new change of variables. An a priori estimate of the analytical approximate solution is obtained. A variant of the numerical experiment of optimizing a priori estimates using a posteriori estimates is presented.

A Neural Network-Based Approach to A Priori Assessment of the Student Academic Performance in the Mirera Digital Learning Platform

В статье рассматривается опыт авторов по построению априорной оценки финальных результатов успеваемости студентов в цифровой образовательной платформе Мирера. Оценка строится по результатам промежуточной проверки успеваемости, полученным из промежуточных проверок на семинарах, при выполнении домашних заданий и проверочных работ. При этом учитываются как непосредственные результаты проверок, так и поведение студента при их выполнении. В предлагаемом подходе студенты условно разделены на три категории: отстающие студенты с неудовлетворительным финальным результатом, удовлетворительно успевающие студенты со средним результатом и студенты с высоким результатом. Для каждой категории студентов можно определить характер и целесообразность автоматизации корректирующих действий преподавателя для «подтягивания» отстающих. Оценка строится с использованием искусственных нейронных сетей. Полученная априорная оценка может быть использована для раннего обнаружения студентов, которые могут быть отчислены за неуспеваемость и которым необходима помощь преподавателя, а также для построения адаптивных треков обучения средне и хорошо успевающих студентов. Предлагаемый подход может быть применен только при условии цифровой трансформации учебного процесса. The paper presents our approach to an a priori assessment of the final student performance in the Mirera digital learning platform. The assessment is based on interim tests at seminars, homework evaluations, and individual tests. In this case, both the test results and the student behavior during the tests are considered. In the proposed approach, students are conditionally divided into three categories: underperforming students with unsatisfactory final results, satisfactory performing students with average results, and high performing students. For each category, the type and feasibility of automating the teacher’s corrective actions to improve the student’s final scores can be identified. The score is generated using artificial neural networks. The a priori estimate can be used for early detection of underperforming students who need help, as well as for building adaptive learning tracks for average and high performing students. The proposed approach can be applied only to digitally transformed academic process. The authors are implementing adaptive learning technologies in the Mirera digital learning platform.  

A Time-Continuation Solver for Hydraulic Fracture Propagation

We present an efficient time-continuation scheme for fluid-driven fracture propagation problems in the frame-work of the extended finite element method (XFEM). The fully coupled, fully implicit hydro-mechanical system is solved in conjunction with the linear elastic fracture propagation criterion by the Newton-Raphson method. Therefore, at the end of each time-step solve, the model ensures the energy release rate of weakest fracture tips within the equilibrium propagation regime. Besides, an initialization procedure for newly created fracture space as well as a priori estimate of stress intensity factor (SIF) growth rates are also developed to further improve the solver performance. We validate the model by the analytical solution and extend the problem to the multiple fracture propagation where stress shadow phenomenon occur.

A numerical model for tool–chip friction in intermittent orthogonal machining

This article presents a novel model to study the influence of surface textured cutting tools in near-micromachining conditions. The model utilizes the Challen and Oxley’s asperity deformation model (Van Luttervelt et al., CIRP Ann Manuf Technol, 1998, vol. 47, pp. 587–626; Arrazola et al., CIRP Ann Manuf Technol, 2013, vol. 62, pp. 695–718) paired with an approach to a priori estimate of the interfacial film formation at the tool–chip interface. The procedure considers the chemical effect of the environment, along with the mechanical aspects of the surface texture of the cutting tool’s rake surface. Model performance, in terms of predicting machining forces and coefficient of friction, was validated with existing experimental data (Anand et al., Proceedings of the international conference on advancements and futuristic trends in mechanical and materials engineering, 5–7 October 2012, pp. 661–666). The outcome trend of the proposed model approximately matches with the experimental results. Further, the model tries to explain the impact of cutting tool’s surface roughness on overall tool–chip friction while performing intermittent cutting in the near-micromachining regime.

Estimation of the observation standard deviation error formula thanks to the a posteriori diagnosis

<p><em>Within the UERRA project, a daily precipitation reanalysis at a 5,5km resolution has been realized from 1961 to 2015. The reanalysis was obtained by the MESCAN analysis system which combines an a priori estimate of the atmosphere – called background – and observations using an optimum interpolation (OI) scheme. Such method requires the specification of observations and background errors. In general, constant standard deviation errors are used but more errors are made when high precipitation are observed. Then, to take this effect into account and to avoid a model over-estimation in case of light precipitation, a variable formula of the observation standard deviation error was purposed with a small value for null precipitation and greater values when precipitation are higher, following a linear equation.</em></p><p><em> Desroziers et al proposed a method to determine observations and background errors called a posteriori diagnosis. To use this iterative method, the analysis has to be ran several times until it converged. In this study, the a posteriori diagnosis is used per precipitation class to determine the observation standard deviation error formula. MESCAN was tested using the French operational model AROME at 1,3km resolution and the atmopsheric UERRA analysis downscaled to 5,5km background and combined to the French observational network over the 2016-2018 period. The observation standard deviation error formula obtained by the a posteriori diagnosis is then used in the MESCAN analysis system to produce precipitation analysis over the 2016-2018 period. Results are compared to UERRA precipitation reanalysis over independant observations by comparing bias, RMSE and scores per precipitation class.</em></p>

Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types ∫01vx,tdx and ∫01xnvx,tdx. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results.

On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. Then by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.

Well posedness of a nonlinear mixed problem for a parabolic equation with integral condition

The aim of this work is to prove the well posedness of some posed linear and nonlinear mixed problems with integral conditions. First, an a priori estimate is established for the associated linear problem and the density of the operator range generated by the considered problem is proved by using the functional analysis method. Subsequently, by applying an iterative process based on the obtained results for the linear problem, the existence, uniqueness of the weak solution of the nonlinear problems is established.

Bayesian updating: increasing sample size during the course of a study

Background A priori sample size calculation requires an a priori estimate of the size of the effect. An incorrect estimate may result in a sample size that is too low to detect effects or that is unnecessarily high. An alternative to a priori sample size calculation is Bayesian updating, a procedure that allows increasing sample size during the course of a study until sufficient support for a hypothesis is achieved. This procedure does not require and a priori estimate of the effect size. This paper introduces Bayesian updating to researchers in the biomedical field and presents a simulation study that gives insight in sample sizes that may be expected for two-group comparisons. Methods Bayesian updating uses the Bayes factor, which quantifies the degree of support for a hypothesis versus another one given the data. It can be re-calculated each time new subjects are added, without the need to correct for multiple interim analyses. A simulation study was conducted to study what sample size may be expected and how large the error rate is, that is, how often the Bayes factor shows most support for the hypothesis that was not used to generate the data. Results The results of the simulation study are presented in a Shiny app and summarized in this paper. Lower sample size is expected when the effect size is larger and the required degree of support is lower. However, larger error rates may be observed when a low degree of support is required and/or when the sample size at the start of the study is small. Furthermore, it may occur sufficient support for neither hypothesis is achieved when the sample size is bounded by a maximum. Conclusions Bayesian updating is a useful alternative to a priori sample size calculation, especially so in studies where additional subjects can be recruited easily and data become available in a limited amount of time. The results of the simulation study show how large a sample size can be expected and how large the error rate is.