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.

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.

The Root of the Problem

2015 ◽  
Vol 109 (2) ◽  
pp. 98-102 ◽  
Author(s):  
Dana L. Grosser-Clarkson
Keyword(s):  
Value Function ◽  
Absolute Value ◽  
Square Roots ◽  
The Absolute

The absolute value function opens the door to a better way to solve problems involving square roots.

scholarly journals Estimation of Non-smooth Functionals in Hilbert Sample Space Using the Edgeworth Expansions

2019 ◽  
pp. 1-15
Author(s):  
M. M. Kololi ◽  
G. O. Orwa ◽  
J. K. Mung’atu ◽  
R. O. Odhiambo
Keyword(s):  
Value Function ◽  
Hermite Polynomials ◽  
Taylor Series Expansion ◽  
Accurate Estimate ◽  
Sample Space ◽  
Edgeworth Expansions ◽  
Absolute Value ◽  
Optimal Estimator ◽  
Gaussian Density ◽  
The Absolute

An arbitrary non-smooth functional is estimated using a nonparametric set-up. Exploratory data analysis methods are relied on to come up with the functional form for the sample to allow both robustness and optimality to be achieved. An infinite number of parameters are involved and thus the Hilbert sample space is a natural choice. An important step in understanding this problem is the normal means problem, . The basic difficulty of estimating  as defined can be traced back to the non differentiability of the absolute value function, at the origin. Accordingly, constructing an optimal estimator is not easy partly due to the nonexistence of an unbiased estimate of the absolute value function. Therefore, best polynomial approximation was used to smooth the singularity at the origin and then an unbiased estimator for every term in the expansion constructed by use of Hermite polynomials when the averages are bounded by a given constant M > 0 say. The expansion of the Gaussian density function in terms of Hermite polynomials gives a clear and almost accurate estimate that admits cumulant generating function; the Saddle point approximation. Additional precision is obtained by using a higher order Taylor series expansion about the mean resulting in Edgeworth expansion techniques.

scholarly journals Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: two theoretical perspectives

2017 ◽  
Vol 11 (2) ◽  
pp. 97-124
Author(s):  
Luis Roberto Pino-Fan ◽  
Ismenia Guzmán ◽  
Vicenç Font ◽  
Raymond Duval
Keyword(s):  
Detailed Analysis ◽  
Value Function ◽  
Cognitive Activity ◽  
Mathematical Cognition ◽  
Absolute Value ◽  
Theoretical Perspectives ◽  
Semiotic Approach ◽  
The Absolute

This paper presents a study of networking of theories between the theory of registers of semiotic representation (TRSR) and the onto-semiotic approach of mathematical cognition and instruction (OSA). The results obtained show complementarities between these two theoretical perspectives, which might allow more detailed analysis of the students’ performance.Análisis de la actividad cognitiva subyacente en la resolución de una tarea sobre la derivabilidad de la función valor absoluto: dos perspectivas teóricas En este artículo se presenta un estudio de networking of theories, entre la teoría de los registros de representación semióticos (TRRS) y el enfoque onto-semiótico de la cognición e instrucción matemáticos (OSA). Los resultados obtenidos revelan complementariedades entre estas dos perspectivas teóricas cuya aplicación simultánea permitiría hacer análisis más pormenorizados de las producciones de los estudiantes.Handle: http://hdl.handle.net/10481/44148WOS-ESCIScopus record and citations 

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