Classical Linear Methods Research Articles

Wind tunnel wall interference correction for transonic airfoils with data-reduced ensemble Kalman filter

The uncertain factors in wind tunnel experiments, especially the interference effects, will cause the flow around the test model to differ from that under the free flow condition. The wind tunnel wall interference effects are challenging to be eliminated, especially in transonic flow regions where classical linear correction methods are ineffective. To address this issue, an efficient wind tunnel wall interference correction framework based on data assimilation is proposed for steady and shock buffet flows around the airfoils in the transonic region in this work. The ensemble Kalman filter (EnKF) is used to find the combination of inflow conditions, which minimizes the differences between the pressure coefficients measured in the wind tunnel experiment and those estimated by computational fluid dynamics (CFD). The polynomial chaos expansion method is used to propagate uncertainty in the forecast step of the EnKF in order to improve efficiency. Typical wind tunnel wall interference correction problems are studied, including two steady cases: the transonic flow around the Royal Aircraft Establishment 2822 (RAE2822) and the National Aerospace Laboratory 7301 (NLR7301) airfoils, and one unsteady case: the transonic shock buffet on the National Aeronautics and Space Administration supercritical (phase 2)-0714 (NASA SC(2)-0714) airfoil. It is demonstrated that with the corrected Mach number and angle of attack, the estimated CFD results are in better agreement with the measured experimental data than traditional linear methods and the computational efficiency is improved compared to the original EnKF.

A novel optimal design method for tuned mass dampers with elastic motion‐limiting stoppers

AbstractTuned Mass Dampers (TMDs) are commonly used passive control devices in practical engineering applications. However, motion‐limiting stoppers are usually installed to control the excessive TMD displacement due to the building space limitation, resulting in piecewise nonlinearity and detuning of TMD. This paper studies the influence of elastic motion‐limiting stoppers on the optimal design of TMDs through a piecewise stiffness TMD (PSTMD) model. Performance of a PSTMD with classical design is first investigated and proven to be ineffective. To optimize the PSTMD parameters, the motion of PSTMD is decoupled from the controlled structure, and the frequency response equation of PSTMD is obtained analytically through the averaging method. Subsequently, the solution of the optimal design frequency for PSTMD is transformed into the solution of the jump frequency in the frequency response equation. With the optimal frequency of PSTMDs, the optimal damping and control performance of PSTMDs are discussed and analyzed compared with classical linear design, which fully showcases the effectiveness of the novel design method. Finally, the effectiveness of the novel design method is verified using a nine‐story benchmark frame structure, and the results demonstrate that the control performance of the optimal PSTMD can be improved by nearly under specific seismic excitation, compared to the PSTMD with classical linear method. In summary, the novel design method can effectively take into account the influence of piecewise nonlinearity caused by elastic motion‐limiting stoppers and improve the optimal control performance of TMD in a more realistic engineering environment.

Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

AbstractA nonlinear correction technique for finite element methods of advection‐diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well‐known restrictions on diffusion coefficients and geometry of mesh‐cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non‐physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non‐physical oscillations around sharp layers in advection‐dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion‐dominate and advection‐dominate problems.

Novel efficient iterative schemes for linear finite element approximations of elliptic optimal control problems with integral constraint on the state

Centering on a distributed optimal control problem governed by elliptic equations with a unilateral integral constraint on the state, we recall the first-order optimality conditions to explore variational formulations. The classical linear finite element method is employed to approximate the equivalent formulae. Taking into account the distinctive structural characteristics of discretized schemes, we divide the discretized optimal conditions into two equivalent matrix schemes. Subsequently, we introduce an efficient iterative algorithm designed to solve the approximation schemes. We demonstrate the convergence of our iterative algorithm and investigate its robustness and uniform optimality under a specified constraint, particularly focusing on small regularization parameters. Finally, numerical tests are conducted using various h and α to illustrate the efficiency of our proposed algorithm. These tests also validate a tremendous reduction in the number of required iterations.

Degrees of Freedom: Search Cost and Self-consistency

Model degrees of freedom ( df ) is a fundamental concept in statistics because it quantifies the flexibility of a fitting procedure and is indispensable in model selection. To investigate the gap between df and the number of independent variables in the fitting procedure, Tibshirani (2015) introduced the search degrees of freedom ( sdf ) concept to account for the search cost during model selection. However, this definition has two limitations: it does not consider fitting procedures in augmented spaces and does not use the same fitting procedure for sdf and df . We propose a modified search degrees of freedom ( msdf ) to directly account for the cost of searching in either original or augmented spaces. We check this definition for various fitting procedures, including classical linear regressions, spline methods, adaptive regressions (the best subset and the lasso), regression trees, and multivariate adaptive regression splines (MARS). In many scenarios when sdf is applicable, msdf reduces to sdf . However, for certain procedures like the lasso, msdf offers a fresh perspective on search costs. For some complex procedures like MARS, the df has been pre-determined during model fitting, but the df of the final fitted procedure might differ from the pre-determined one. To investigate this discrepancy, we introduce the concepts of nominal df and actual df , and define the property of self-consistency, which occurs when there is no gap between these two df ’s. We propose a correction procedure for MARS to align these two df ’s, demonstrating improved fitting performance through extensive simulations and two real data applications.

Reinforcement Learning Control of Hypersonic Vehicles and Performance Evaluations

This work presents a new framework for model-based continuous-time reinforcement learning (CT-RL) control of hypersonic vehicles (HSVs). The predominant classes of CT-RL methods for general nonlinear systems in adaptive dynamic programming (ADP) and deep RL tend to either present substantial theoretical results but lack practical synthesis capability (ADP) or show empirical promise without offering theoretical guarantees (deep RL). Meanwhile, RL control frameworks developed directly for HSVs tend to require a simplified model and complicated control structure, and they lack the substantial numerical evaluations essential for real-world flight implementation. To directly address these challenges, we propose a new decentralized excitable integral reinforcement learning framework within which the reference input-based exploration improves persistence of excitation. Together with new insights on prescaling and established decentralized control structure for HSVs, we demonstrate the resulting controller for significant performance improvement over classical Linear Quadratic Regulator (LQR) and feedback linearization methods. Additionally, we provide convergence, optimality, and closed-loop stability guarantees of the proposed method. We demonstrate these performance guarantees over a set of substantial and systematic numerical evaluations on an unstable, nonminimum phase HSV model subject to varying modeling errors and initial conditions.

Using machine learning to identify environmental factors that collectively determine microbial community structure of activated sludge

Activated sludge (AS) microbial communities are influenced by various environmental variables. However, a comprehensive analysis of how these variables jointly and nonlinearly shape the AS microbial community remains challenging. In this study, we employed advanced machine learning techniques to elucidate the collective effects of environmental variables on the structure and function of AS microbial communities. Applying Dirichlet multinomial mixtures analysis to 311 global AS samples, we identified four distinct microbial community types (AS-types), each characterized by unique microbial compositions and metabolic profiles. We used 14 classical linear and nonlinear machine learning methods to select a baseline model. The extremely randomized trees demonstrated optimal performance in learning the relationship between environmental factors and AS types (with an accuracy of 71.43%). Feature selection identified critical environmental factors and their importance rankings, including latitude (Lat), longitude (Long), precipitation during sampling (Precip), solids retention time (SRT), effluent total nitrogen (Effluent TN), average temperature during sampling month (Avg Temp), mixed liquor temperature (Mixed Temp), influent biochemical oxygen demand (Influent BOD), and annual precipitation (Annual Precip). Significantly, Lat, Long, Precip, Avg Temp, and Annual Precip, influenced metabolic variations among AS types. These findings emphasize the pivotal role of environmental variables in shaping microbial community structures and enhancing metabolic pathways within activated sludge. Our study encourages the application of machine learning techniques to design artificial activated sludge microbial communities for specific environmental purposes.

Towards a Unified Approach for Continuously-Variable Impedance Control of Powered Prosthetic Legs over Walking Speeds and Inclines.

Research in powered prosthesis control has explored the use of impedance-based control algorithms due to their biomimetic capabilities and intuitive structure. Modern impedance controllers feature parameters that smoothly vary over gait phase and task according to a data-driven model. However, these recent efforts only use continuous impedance control during stance and instead utilize discrete transition logic to switch to kinematic control during swing, necessitating two separate models for the different parts of the stride. In contrast, this paper presents a controller that uses smooth impedance parameter trajectories throughout the gait, unifying the stance and swing periods under a single, continuous model. Furthermore, this paper proposes a basis model to represent intertask relationships in the impedance parameters-a strategy that has previously been shown to improve model accuracy over classic linear interpolation methods. In the proposed controller, a weighted sum of Fourier series is used to model the impedance parameters of each joint as continuous functions of gait cycle progression and task. Fourier series coefficients are determined via convex optimization such that the controller best reproduces the joint torques and kinematics in a reference able-bodied dataset. Experiments with a powered knee-ankle prosthesis show that this simpler, unified model produces competitive results when compared to a more complex hybrid impedance-kinematic model over varying walking speeds and inclines.

Principal simple linear regression

In this paper, we propose a new method called principal simple linear regression for predicting a continuous response variable, $Y$ using a single continuous predictor, $X$ by utilizing multiple regression lines. This method is based on the theory of principal points and offers several advantages over classical simple linear regression methods, such as the ability to predict central, dispersion, and distributional tendencies of $Y$ on $X$, and simultaneous estimation. We provide the main properties, inferences, and limiting behavior of the estimators. Additionally, we conduct a comprehensive simulation study to validate our theoretical results. The model is also applied to real datasets to demonstrate its effectiveness.

Understanding complex processes better-A case study on increasing patient safety and efficiency in a central operating room

Various professional groups are involved in the daily work of the central operating room with the aim of providing the best possible treatment for each individual using modern medical technology (sociotechnical system) in acost-effective manner. Ensuring perioperative patient safety is of particular importance. At the same time, the efficient use of the central operating room is essential for the economic success of ahospital. Preoperative preparation is acomplex process with many substeps that are often difficult to manage. Historically, the focus has been on retrospective learning from errors and incidents. More recent approaches take asystemic view. Acentral idea is to consider the mostly positive course of treatment and the adjustments to daily work that are currently required by the people involved (Safety-II). By taking greater account of how the many components of the system interact, processes can be better understood and specific measures derived. This strengthens the system's ability to adapt to changes and disturbances, thus ensuring that goals are achieved. The functional resonance analysis method (FRAM) is an internationally recognized method for modelling work as done compared to work as imagined. This paper presents the application of FRAM to preoperative preparation in a major regional hospital. Is FRAM suitable for improving process understanding in preoperative preparation? An interdisciplinary project team identified relevant functions of preoperative preparation through document analysis and walkthroughs. Based on this, more than 30guided interviews were conducted with functionaries. The results were presented graphically and specific information, such as safety-related statements or reasons for the variability of functions, were also presented textually. In the next phase, statements were evaluated and compared with the target model and the job descriptions. The FRAM revealed the process as acomplex network of relationships. During the modelling process, avarying degree of centrality and variability of certain functions became apparent. From the observations, the project team selected those with high relevance for patient safety and for the efficiency of the overall process in order to prioritize starting points for deriving measures to increase resilience. These starting points relate either to single functions, such as surgical site marking or to multiple functions that are variable in their execution, such as delays due to nonsynchronized duty times. The FRAM conducted provides valuable new insights into the functioning of complex sociotechnical systems that go far beyond classical linear methods. The awareness of operational processes gained and the resulting dynamic view of interactions within the system enable specific measures to be derived that promote resilient behavior and reduce critical variability, thus contributing to increased patient safety and efficiency.

Koopman-Based MPC With Learned Dynamics: Hierarchical Neural Network Approach.

This article presents a data-driven control strategy for nonlinear dynamical systems, enabling the construction of a Koopman-based linear system associated with nonlinear dynamics. The primary idea is to apply the deep learning technique to the Koopman framework for globally linearizing nonlinear dynamics and impose a Koopman-based model predictive control (MPC) approach to stabilize the nonlinear dynamical systems. In this work, we first generalize the Koopman framework to nonlinear control systems, enabling comprehensive linear analysis and control methods to be effective for nonlinear systems. We next present a hierarchical neural network (HNN) approach to deal with the crucial challenge of the finite-dimensional Koopman representation approximation. In particular, a scale-invariant constrained network in the HNN includes four modules, in which a predictor module and a linear module can accurately approximate the finite Koopman eigenfunctions and Koopman operator, respectively, thus forming the lifted linear system. Then, we design the Koopman-based MPC scheme for controlling nonlinear systems with constraints by adopting the modified MPC with a saturation-like function on the lifted linear system. Importantly, the Koopman-based MPC enjoys higher computational efficiency compared to the classical linear MPC and nonlinear MPC methods. Finally, a physical experiment on an overhead crane system is provided to demonstrate the effectiveness of the proposed data-driven control framework.

Fusing physics-based and machine learning models for rapid ground-motion-adaptative probabilistic seismic fragility assessment

Performance-based earthquake engineering (PBEE) has asserted probabilistic seismic fragility assessment (PSFA) as the main research content in light of its irreplaceable significance for seismic decision-making in recent decades. Among the multiple approaches of PSFA implementation, the classical linear regression method (LRM) dominates over practice regarded as one of the most widely-used. However, the general LRM adopts quantile regression method (QRM) on the group of fragility curves to approximate a deterministic probability density distribution (PSD) of structural fragility against certain intensity measure (IM) of potentially confronting earthquake. Consequently, the QRM-derived fragility representation might not be credible enough while evaluating a newly-occurred seismic event owing to its neglect of specificity of stochastic ground motion. To address this issue, a fusing physics-based and machine learning models towards rapid ground-motion-adaptative probabilistic seismic fragility assessment (GmaPSFA) is proposed in present study. With sophisticated framework design and novel fragility hyperparameters estimation, the involved design philosophy and mechanism translating are both elaborated. To validate the method, both the LRM and GmaPSFA were conducted on a six-story frame structure, where a novel fully-automatic batch processing approach fusing APDL and coding languages was propounded for structural analysis. The comprehensive validating cases confirmed the powerful capability and significant superiority of proposed GmaPSFA in realizing rapid ground-motion-adaptative probabilistic seismic fragility assessment catering for modern PBEE.

Artificial Neural Network and Multiple Regression Analysis Applied to 2D-QSAR Studies: the Case of Imidazolidine-2,4-dione as PTP1B Inhibitors

In this investigation, we undertook a comprehensive quantitative structure-activity relationship (QSAR) analysis using both modern artificial neural network (ANN) methods and classical multiple linear regression methods (MLR). Our primary focus was on revealing the intricate connection between antidiabetic activity and the molecular structure of thirty-nine imidazolidine- 2,4-dione derivatives. The B3LYP hybrid functional and 6-31G (d) basis set computed electronic properties at the quantum level. Rigorous benchmarking against experimental data validated the reliability of our quantum theory approach. Our statistical model effectively predicted activities closely aligned with experimental antidiabetic activities, quantified by the IC50 values. They also revealed a significant superiority of the ANN architecture (6-4-1) over the MLR method. This study stands as a meaningful contribution to the field, providing valuable insights for designing new antidiabetic drugs, particularly as potential inhibitors of tyrosine phosphate 1B (PTP1B). The explicit articulation of our primary aim and emphasis on the study’s significance underscore its potential impact on advancing drug development within the realm of antidiabetic therapeutics.

The Influence of Service Quality on Train Passenger Satisfaction at Rantau Prapat Station PT KAI Persero Divre I North Sumatra

In order to keep clients using this kind of transportation and to keep the business afloat, PT KAI places a high priority on their satisfaction. Employees of KAI are required to utilize five dimensions in its implementation: tangibles, assurance, responsiveness, empathy, and consistency. It is undoubtedly not ideal in its implementation at each station, as some travelers continue to be dissatisfied with the offers that are made. Examples of this include the fact that some people are still unable to board the train without a ticket and the indifference of staff members. Following basic observations by the researchers, there was a decline in staff responsiveness and overall satisfaction. The goal of this study is to determine how much the independent variable affects the dependent variable. In this quantitative analysis, primary and secondary data sources are consulted. Three techniques were employed to gather data: questionnaire completion, documentation, and observation. Using SPSS (Statistical Product and Service Solution), the instrument test, classical assumption test, hypothesis test, and simple linear regression method are the data analysis techniques employed. Once the researcher had finished using the data that had been collected. With a significance value (Sig) of 0.000, it may be inferred that the quality of the services offered significantly affects passenger happiness. Less than 0.05. The T test decision-making framework indicates that H1 is accepted and H0 is rejected, indicating that service quality has an impact on passengers' pleasure with trains. When the coefficient of determination hypothesis is evaluated using the SPSS version 26 software, a value of 0.906 can be obtained. This indicates that there is a 90.6% link between passenger satisfaction and service quality, with the remaining 9.4% coming from factors that have not been thoroughly investigated.

Control of Telecommunication Network Parameters under Conditions of Uncertainty of the Impact of Destabilizing Factors

The classification of the natural and anthropogenic destabilizing factors of a telecommunications network as a complex system is presented herein. This research shows that to evaluate the parameters of a telecommunications network in the presence of destabilizing factors, it is necessary to modify classical linear methods to reduce their sensitivity to the incompleteness of a priori information. Using generalized linear models of multiple regression, a combined method was developed for assessing and predicting the survivability of a telecommunications network under conditions of uncertainty regarding the influence of destabilizing factors. The method consists of accumulating current information about the parameters and state of the network, the statistical analysis and processing of information, and the extraction of sufficient sample statistics. The basis of the developed method was balancing multiple correlation–regression analysis with the number of regression equations and the observed results. Various methods of estimating the mathematical expectation and correlation matrix of the observed results under the conditions of random loss of part of the observed data (for example, removing incomplete sample elements, substituting the average, pairwise crossing out, and substituting the regression) were analyzed. It was established that a shift in the obtained estimates takes place under the conditions of a priori uncertainty of the statistics of the observed data. Given these circumstances, recommendations are given for the correct removal of sample elements and variables with missing values. It is shown that with significant unsteadiness of the parameters and state of the network under study and a noticeable imbalance in the number of regression equations and observed results, it is advisable to use stepwise regression methods.

Fractal Time Series: Background, Estimation Methods, and Performances.

Over the past 40years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

Lookahead data-gathering strategies for online adaptive model reduction of transport-dominated problems

Online adaptive model reduction efficiently reduces numerical models of transport-dominated problems by updating reduced spaces over time, which leads to nonlinear approximations on latent manifolds that can achieve a faster error decay than classical linear model reduction methods that keep reduced spaces fixed. Critical for online adaptive model reduction is coupling the full and reduced model to judiciously gather data from the full model for adapting the reduced spaces so that accurate approximations of the evolving full-model solution fields can be maintained. In this work, we introduce lookahead data-gathering strategies that predict the next state of the full model for adapting reduced spaces toward dynamics that are likely to be seen in the immediate future. Numerical experiments demonstrate that the proposed lookahead strategies lead to accurate reduced models even for problems where previously introduced data-gathering strategies that look back in time fail to provide predictive models. The proposed lookahead strategies also improve the robustness and stability of online adaptive reduced models.

Cost-efficient bathymetric mapping method based on massive active–passive remote sensing data

Accurate bathymetric mapping is critical for island reef research, coastal ecosystem monitoring, and nearshore engineering. Increasing amounts of remote sensing data have promoted the development of optical bathymetric remote sensing. However, large-scale and efficient bathymetric mapping of shallow oceanic islands is difficult due to the limited availability of high-quality optical remote sensing images. In this study, we propose a cost-efficient method for bathymetric mapping based on a quadratic polynomial ratio model (QPRM) of massive active–passive remote sensing data. Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) data represent active data and a priori data, and Sentinel-2 data from the Google Earth Engine represent passive data. The key step is to build a QPRM based on the median value of the blue-green band logarithmic ratio from massive Sentinel-2 data. The method is applied to six typical areas located in the Pacific Ocean, Indian Ocean, and South China Sea, and the results showed that the root mean square error (RMSE) and mean absolute error (MAE) of bathymetric mapping in the water depth range of 0–20 m were 0.49–0.71 m and 0.29–0.49 m, respectively. In addition, the RMSE decreased with an increase in the number of Sentinel-2 images, and the results were relatively stable when the number reached approximately 150. The QPRM results were also compared with that of the classical linear ratio model (CLRM) and multiple fitting methods to calculate the median water depth, and the findings showed that the QPRM method has a slight advantage in terms of accuracy and efficiency. The proposed method, which does not depend on in situ data, effectively calculates precise water depths and removes the effects of complex underwater optical paths and surface instabilities, such as waves, boats, and clouds. Therefore, it has great potential for bathymetric mapping of oceanic islands and reefs worldwide.

Bayesian model averaging for predicting factors associated with length of COVID-19 hospitalization

IntroductionThe length of hospital stay (LOHS) caused by COVID-19 has imposed a financial burden, and cost on the healthcare service system and a high psychological burden on patients and health workers. The purpose of this study is to adopt the Bayesian model averaging (BMA) based on linear regression models and to determine the predictors of the LOHS of COVID-19.MethodsIn this historical cohort study, from 5100 COVID-19 patients who had registered in the hospital database, 4996 patients were eligible to enter the study. The data included demographic, clinical, biomarkers, and LOHS. Factors affecting the LOHS were fitted in six models, including the stepwise method, AIC, BIC in classical linear regression models, two BMA using Occam's Window and Markov Chain Monte Carlo (MCMC) methods, and GBDT algorithm, a new method of machine learning.ResultsThe average length of hospitalization was 6.7 ± 5.7 days. In fitting classical linear models, both stepwise and AIC methods (R2 = 0.168 and adjusted R2 = 0.165) performed better than BIC (R2 = 0.160 and adjusted = 0.158). In fitting the BMA, Occam's Window model has performed better than MCMC with R2 = 0.174. The GBDT method with the value of R2 = 0.64, has performed worse than the BMA in the testing dataset but not in the training dataset. Based on the six fitted models, hospitalized in ICU, respiratory distress, age, diabetes, CRP, PO2, WBC, AST, BUN, and NLR were associated significantly with predicting LOHS of COVID-19.ConclusionThe BMA with Occam's Window method has a better fit and better performance in predicting affecting factors on the LOHS in the testing dataset than other models.

Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity

Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $\Lambda=\{\lambda_{\delta}(k)\}$ of functions of the natural argument depending on the real parameter $\delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{\omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.