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.

scholarly journals Combining Predictors

2000 ◽  
Vol 29 (550) ◽  
Author(s):  
Jakob Vogdrup Hansen
Keyword(s):  
Machine Learning ◽  
Cross Validation ◽  
Error Function ◽  
Variance Decomposition ◽  
Ensemble Method ◽  
Error Functions ◽  
The Family ◽  
Bias Variance ◽  
The Cross ◽  
Opinion Pool

The most important theoretical tool in connection with machine learning is the bias/variance decomposition of error functions. Together with Tom Heskes, I have found the family of error functions with a natural bias/variance decomposition that has target independent variance. It is shown that no other group of error functions can be decomposed in the same way. An open problem in the machine learning community is thereby solved. The error functions are derived from the deviance measure on distributions in the one-parameter exponential family. It is therefore called the deviance error family.<br /> <br /> A bias/variance decomposition can also be viewed as an ambiguity decomposition for an ensemble method. The family of error functions with a natural bias/variance decomposition that has target independent variance can therefore be of use in connection with ensemble methods.<br /> <br /> The logarithmic opinion pool ensemble method has been developed together with Anders Krogh. It is based on the logarithmic opinion pool ambiguity decomposition using the Kullback-Leibler error function. It has been extended to the cross-validation logarithmic opinion pool ensemble method. The advantage of the cross-validation logarithmic opinion pool ensemble method is that it can use unlabeled data to estimate the generalization error, while it still uses the entire labeled example set for training.<br /> <br /> The cross-validation logarithmic opinion pool ensemble method is easily reformulated for another error function, as long as the error function has an ambiguity decomposition with target independent ambiguity. It is therefore possible to use the cross-validation ensemble method on all error functions in the deviance error family.

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 Some Discussions about the Error Functions on SO(3) and SE(3) for the Guidance of a UAV Using the Screw Algebra Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Yi Zhu ◽  
Xin Chen ◽  
Chuntao Li
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Error Function ◽  
Elliptic Cylinder ◽  
General Situation ◽  
3 Dimensional ◽  
Error Functions ◽  
Aerial Vehicle ◽  
Algebra Theory ◽  
In The Beginning

In this paper a new error function designed on 3-dimensional special Euclidean group SE(3) is proposed for the guidance of a UAV (Unmanned Aerial Vehicle). In the beginning, a detailed 6-DOF (Degree of Freedom) aircraft model is formulated including 12 nonlinear differential equations. Secondly the definitions of the adjoint representations are presented to establish the relationships of the Lie groups SO(3) and SE(3) and their Lie algebras so(3) and se(3). After that the general situation of the differential equations with matrices belonging to SO(3) and SE(3) is presented. According to these equations the features of the error function on SO(3) are discussed. Then an error function on SE(3) is devised which creates a new way of error functions constructing. In the simulation a trajectory tracking example is given with a target trajectory being a curve of elliptic cylinder helix. The result shows that a better tracking performance is obtained with the new devised error function.

Fekete Szegö theorem for a close-to-convex error function

2019 ◽  
Vol 69 (2) ◽  
pp. 391-398
Author(s):  
C. Ramachandran ◽  
D. Kavitha ◽  
Wasim Ul-Haq
Keyword(s):  
Error Function ◽  
Error Functions ◽  
Sharp Estimates

Abstract By motivating the result of Ramachandran et al. [Certain results on q-starlike and q-convex error functions, Math. Slovaca, 68(2) (2018), 361–368], in this present investigation we derive the classical Fekete Szegö theorem for a close-to-convex error function of order β and the sharp estimates also obtained for real μ.

"An Overview of Using Error function in Adsorption Isotherm Modeling "

2020 ◽  
Vol 8 (1) ◽  
pp. 22-30
Author(s):  
Hawraa K. Hami ◽  
Ruba F. Abbas ◽  
Amel S. Mahdi ◽  
Asma A. Maryoosh
Keyword(s):  
Adsorption Equilibrium ◽  
Error Function ◽  
Common Error ◽  
Analysis And Design ◽  
Error Functions ◽  
The Past ◽  
Adsorption Data ◽  
Best Fitting ◽  
Definition Of ◽  
The Right

"Over the past years, a large number of statistical expressions used as a measure of accuracy, collectively referred to as error functions. These functions use to determine the best fitting data. Since accurate adsorption equilibrium information are necessary for the analysis and design of adsorption, error functions are used to valuation the validity of the adsorption mathematical models with experimental results by finding the most appropriate isotherm. Therefore, this overall review provides a definition of a number of common error function and explains the use of these functions in determine optimal adsorption data and chose the right isotherm.

scholarly journals Comparison of Neural Network Error Measures for Simulation of Slender Marine Structures

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Niels H. Christiansen ◽  
Per Erlend Torbergsen Voie ◽  
Ole Winther ◽  
Jan Høgsberg
Keyword(s):  
Neural Network ◽  
Error Function ◽  
Error Measure ◽  
Error Functions ◽  
Error Measures ◽  
Specific Objective ◽  
The Mean ◽  
Stress Cycles ◽  
Fatigue Life Analysis ◽  
Artificial Neural Network Ann

Training of an artificial neural network (ANN) adjusts the internal weights of the network in order to minimize a predefined error measure. This error measure is given by an error function. Several different error functions are suggested in the literature. However, the far most common measure for regression is the mean square error. This paper looks into the possibility of improving the performance of neural networks by selecting or defining error functions that are tailor-made for a specific objective. A neural network trained to simulate tension forces in an anchor chain on a floating offshore platform is designed and tested. The purpose of setting up the network is to reduce calculation time in a fatigue life analysis. Therefore, the networks trained on different error functions are compared with respect to accuracy of rain flow counts of stress cycles over a number of time series simulations. It is shown that adjusting the error function to perform significantly better on a specific problem is possible. On the other hand. it is also shown that weighted error functions actually can impair the performance of an ANN.

scholarly journals A Study of Some Families of Multivalent q-Starlike Functions Involving Higher-Order q-Derivatives

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1470
Author(s):  
Bilal Khan ◽  
Zhi-Guo Liu ◽  
Hari M. Srivastava ◽  
Nazar Khan ◽  
Maslina Darus ◽  
...  
Keyword(s):  
Starlike Functions ◽  
Higher Order ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Function Classes ◽  
The Family ◽  
Radius Problems ◽  
Coefficient Inequalities ◽  
Distortion Theorems

In the present investigation, by using certain higher-order q-derivatives, the authors introduce and investigate several new subclasses of the family of multivalent q-starlike functions in the open unit disk. For each of these newly-defined function classes, several interesting properties and characteristics are systematically derived. These properties and characteristics include (for example) distortion theorems and radius problems. A number of coefficient inequalities and a sufficient condition for functions belonging to the subclasses studied here are also discussed. Relevant connections of the various results presented in this investigation with those in earlier works on this subject are also pointed out.

Constant Modulus Algorithm Based on Different Error Functions Weighted by Variable Coefficient

2011 ◽  
Vol 66-68 ◽  
pp. 1579-1585
Author(s):  
Qiao Xi Zhou ◽  
Ye Cai Guo
Keyword(s):  
Theoretical Analysis ◽  
Convergence Rate ◽  
Variable Coefficient ◽  
Gradient Descent ◽  
Error Function ◽  
Descent Method ◽  
Gradient Descent Method ◽  
Constant Modulus ◽  
Constant Modulus Algorithm ◽  
Error Functions

To improve equalization performance of the constant modulus algorithm (CMA), we study that error functions have an influence on the performance of the algorithm in this paper. Aiming at the character of different error functions, a new style of error function weighted by a variable coefficient is proposed. And a new CMA based on the new error function (VCMA) is proposed too. Because of variable-coefficient adjustability, the value of this new error function can become larger at the beginning of iteration and smaller at the end of iteration in the new algorithm. From gradient descent method, VCMA can have faster convergence rate and lower residual error than the CMA. Both theoretical analysis and experimental results have shown the effectiveness of the proposed algorithm.

scholarly journals A note on the theorem of Jarník-Besicovitch

1997 ◽  
Vol 39 (2) ◽  
pp. 233-236 ◽  
Author(s):  
H. Dickinson
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Error Function ◽  
Positive Function ◽  
Real Numbers ◽  
Linear Forms ◽  
Error Functions ◽  
General Error

This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function | sinq|/qτ rather than the strictly decreasing function qτ of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.

scholarly journals On approximately monotone and approximately Hölder functions

2020 ◽  
Vol 81 (1) ◽  
pp. 65-87
Author(s):  
Angshuman R. Goswami ◽  
Zsolt Páles
Keyword(s):  
Structural Properties ◽  
Total Variation ◽  
Error Function ◽  
Decomposition Theorem ◽  
Open Interval ◽  
Jordan Decomposition ◽  
Optimal Error ◽  
Hölder Functions ◽  
Error Functions ◽  
Function Classes

Abstract A real valued function f defined on a real open interval I is called $$\Phi $$ Φ -monotone if, for all $$x,y\in I$$ x , y ∈ I with $$x\le y$$ x ≤ y it satisfies $$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$ f ( x ) ≤ f ( y ) + Φ ( y - x ) , where $$ \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+$$ Φ : [ 0 , ℓ ( I ) [ → R + is a given nonnegative error function, where $$\ell (I)$$ ℓ ( I ) denotes the length of the interval I. If f and $$-f$$ - f are simultaneously $$\Phi $$ Φ -monotone, then f is said to be a $$\Phi $$ Φ -Hölder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for $$\Phi $$ Φ -monotonicity and $$\Phi $$ Φ -Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper $$\Phi $$ Φ -monotone and $$\Phi $$ Φ -Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.

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