scholarly journals Sigmoid functions for the smooth approximation to $ \vert{x}\vert $

2019 ◽  
Author(s):  
Yogesh J. Bagul ◽  
Christophe Chesneau
Keyword(s):  
Error Function ◽  
Smooth Approximation ◽  
Sigmoid Functions ◽  
Smooth Approximations

We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.

scholarly journals Sigmoid functions for the smooth approximation to the absolute value function

2021 ◽  
Vol 7 (1) ◽  
pp. 12-19
Author(s):  
Yogesh J. Bagul ◽  
Christophe Chesneau
Keyword(s):  
Value Function ◽  
Error Function ◽  
Smooth Approximation ◽  
Absolute Value ◽  
The Absolute ◽  
Sigmoid Functions ◽  
Smooth Approximations

AbstractWe present smooth approximations to the absolute value function |x| using sigmoid functions. In particular, x erf(x/μ) is proved to be a better smooth approximation for |x| than x tanh(x/μ) and \sqrt {{x^2} + \mu } with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.

The Pricing Error Function

2008 ◽  
Author(s):  
Huntley Schaller ◽  
Lynda Khalaf
Keyword(s):  
Error Function ◽  
Pricing Error

scholarly journals Searching for optimum adsorption curve for metal sorption on soils: comparison of various isotherm models fitted by different error functions

2021 ◽  
Vol 3 (3) ◽  
Author(s):  
Péter Sipos
Keyword(s):  
Experimental Data ◽  
Error Function ◽  
Isotherm Models ◽  
Metal Sorption ◽  
Error Functions ◽  
Sorption System ◽  
Geochemical Properties ◽  
Function Combination ◽  
Soil Metal ◽  
Sorption Curve

AbstractStudies comparing numerous sorption curve models and different error functions are lacking completely for soil-metal adsorption systems. We aimed to fill this gap by studying several isotherm models and error functions on soil-metal systems with different sorption curve types. The combination of fifteen sorption curve models and seven error functions were studied for Cd, Cu, Pb, and Zn in competitive systems in four soils with different geochemical properties. Statistical calculations were carried out to compare the results of the minimizing procedures and the fit of the sorption curve models. Although different sorption models and error functions may provide some variation in fitting the models to the experimental data, these differences are mostly not significant statistically. Several sorption models showed very good performances (Brouers-Sotolongo, Sips, Hill, Langmuir-Freundlich) for varying sorption curve types in the studied soil-metal systems, and further models can be suggested for certain sorption curve types. The ERRSQ error function exhibited the lowest error distribution between the experimental data and predicted sorption curves for almost each studied cases. Consequently, their combined use could be suggested for the study of metal sorption in the studied soils. Besides testing more than one sorption isotherm model and error function combination, evaluating the shape of the sorption curve and excluding non-adsorption processes could be advised for reliable data evaluation in soil-metal sorption system.

scholarly journals From Stress to Shape: Equilibrium of Cloister and Cross Vaults

2021 ◽  
Vol 11 (9) ◽  
pp. 3846
Author(s):  
Andrea Montanino ◽  
Carlo Olivieri ◽  
Giulio Zuccaro ◽  
Maurizio Angelillo
Keyword(s):  
Error Function ◽  
Membrane Surface ◽  
Equilibrium Analysis ◽  
Test Cases ◽  
Order Partial Differential Equation ◽  
Negative Semidefinite ◽  
Historical Heritage ◽  
Masonry Vaults ◽  
Membrane Shape ◽  
The Given

The assessment of the equilibrium and the safety of masonry vaults is of high relevance for the conservation and restoration of historical heritage. In the literature many approaches have been proposed for this tasks, starting from the 17th century. In this work we focus on the Membrane Equilibrium Analysis, developed under the Heyman’s theory of Limit Analysis. Within this theory, the equilibrium of a vault is assessed if it is possible to find at least one membrane surface, between the volume of the vaults, being in equilibrium under the given loads through a purely compressive stress field. The equilibrium of membranes is described by a second order partial differential equation, which is definitely elliptic only when a negative semidefinite stress is assigned, and the shape is the unknown of the problem. The proposed algorithm aims at finding membrane shapes, entirely comprised between the geometry of the vault, in equilibrium with admissible stress fields, through the minimization of an error function with respect to shape parameters of the stress potential, and then, with respect to the boundary values of the membrane shape. The application to two test cases shows the viability of this tool for the assessment of the equilibrium of existing masonry vaults.

An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials

2020 ◽  
Vol 70 (3) ◽  
pp. 599-604
Author(s):  
Şahsene Altinkaya
Keyword(s):  
Error Function ◽  
Trigonometric Polynomials ◽  
Error Functions ◽  
The Family ◽  
Coefficient Inequalities

AbstractIn this present investigation, we will concern with the family of normalized analytic error function which is defined by$$\begin{array}{} \displaystyle E_{r}f(z)=\frac{\sqrt{\pi z}}{2}\text{er} f(\sqrt{z})=z+\overset{\infty }{\underset {n=2}{\sum }}\frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}. \end{array}$$By making the use of the trigonometric polynomials Un(p, q, eiθ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.

Regularized error function-based extended Kalman filter for estimating the cancer chemotherapy dosage: impact of improved grey wolf optimization

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Utkarsha L. Mohite ◽  
Hirenkumar G. Patel
Keyword(s):  
Tumor Cells ◽  
Error Function ◽  
Drug Dosage ◽  
Parameters Estimation ◽  
Gray Wolf ◽  
Robust Performance ◽  
Parameter Uncertainties ◽  
Initial State ◽  
Robust Controller ◽  
Fine Tune

AbstractObjectivesThe main aim of this work is to introduce a robust controller for controlling the drug dosage.MethodsThe presented work establishes a novel robust controller that controls the drug dosage and it also carried out parameters estimation. Along with this, a Regularized Error Function-based EKF (REF-EKF) is introduced for estimating the tumor cells that could be adapted for different conditions. It also assists in solving the overfitting problems, which occur during the drug dosage estimation. Moreover, the performance of the adopted controller is compared over other conventional schemes, and the attained outcomes reveal the appropriate impact of drug dosage injection on immune, normal, and tumor cells. It is also ensured that the presented controller does a robust performance on the parameter uncertainties. Moreover, to enhance the performance of the proposed system and for fast convergence, it is aimed to fine-tune the initial state of EKF optimally using a new Improved Gray Wolf Optimization (GWO) termed as Adaptive GWO (AGWO). Finally, analysis is held to validate the betterment of the presented model.ResultsThe outcomes, the proposed method has accomplished a minimal value of error with an increase in time, when evaluated over the compared models.ConclusionsThus, the improvement of the proposed REF-EKF-AGWO model is proved from the attained results.

scholarly journals A Multi-Feature Fusion Slam System Attaching Semantic Invariant to Points and Lines

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1196
Author(s):  
Gang Li ◽  
Yawen Zeng ◽  
Huilan Huang ◽  
Shaojian Song ◽  
Bin Liu ◽  
...  
Keyword(s):  
Feature Matching ◽  
Feature Fusion ◽  
Error Function ◽  
Line Segments ◽  
Cumulative Error ◽  
Localization And Mapping ◽  
Tracking Process ◽  
Line Features ◽  
Point Line ◽  
Segment Data

The traditional simultaneous localization and mapping (SLAM) system uses static points of the environment as features for real-time localization and mapping. When there are few available point features, the system is difficult to implement. A feasible solution is to introduce line features. In complex scenarios containing rich line segments, the description of line segments is not strongly differentiated, which can lead to incorrect association of line segment data, thus introducing errors into the system and aggravating the cumulative error of the system. To address this problem, a point-line stereo visual SLAM system incorporating semantic invariants is proposed in this paper. This system improves the accuracy of line feature matching by fusing line features with image semantic invariant information. When defining the error function, the semantic invariant is fused with the reprojection error function, and the semantic constraint is applied to reduce the cumulative error of the poses in the long-term tracking process. Experiments on the Office sequence of the TartanAir dataset and the KITTI dataset show that this system improves the matching accuracy of line features and suppresses the cumulative error of the SLAM system to some extent, and the mean relative pose error (RPE) is 1.38 and 0.0593 m, respectively.

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