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ODEs and the calculus of variations
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Next, we consider the following one-sided limit: δJ(y;h):=limα→0+J(y+αh)−J(y)α, for y ∈ V, h∈B∖{0}. If this limit exists, it is called the directional derivative or first (right) variation of J in y in the direction h. This concept of directional derivative in functional spaces was made mathematically rigorous thanks to the work of René Eugène Gâteaux.
Analytics Toolsets
Published in Ali Soofastaei, Data Analytics Applied to the Mining Industry, 2020
Russell Molaei, Ali Soofastaei
Two-sided or unilateral are the two ways of expressing trust limits. Two-sided links are added to suggest that a particular trust is kept in the amount of interest. The one-sided limit is used to demonstrate that the amount of interest above the lower bound is higher. Besides, with exact confidence, a lower bond is seen. The correct bound type depends on the entry. For example, the analyst should apply unilateral lower reliability and unilateral higher reliability for percentage failure below guarantee and a bidirectional boundary of distribution variables. For example, the 95% smaller two-sided band is the 95% lowest single bond, and the 95% most massive double-sided band is the 95% larger unilateral bond.
An Algebraic Approach to Hankel Norm Approximation Problems
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
In this section we develop a duality theory in the context of Hankel norm approximation problems. There are three operations applied to a given, antistable, transfer function. Namely, inversion of the restricted Hankel operator, taking the adjoint map and finally one sided multiplication by unitary operators. The last two operations do not change the singular values, whereas the first operation inverts them.
Design of variance control charts with estimated parameters: A head to head comparison between two perspectives
Published in Journal of Quality Technology, 2022
Martin G. C. Sarmiento, Felipe S. Jardim, Subhabrata Chakraborti, Eugenio K. Epprecht
With the resulting value of the adjusted upper one-sided limit factor () and limit () can be found using Eq. [1]. These exact values of were provided by Goedhart et al. (2017) and Faraz, Heuchenne, and Saniga (2018) for various settings. Faraz, Woodall, and Heuchenne (2015) also obtained the using the bootstrap procedure by Jones and Steiner (2012) and Gandy and Kvaløy (2013). However, since we assume normality of the original data, it is possible to find analytically.
Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions
Published in Optimization Methods and Software, 2018
H. Martin Bücker, Johannes Willkomm
This article is concerned with a hierarchical AD approach for the special functions called the associated Legendre functions. These functions frequently arise, for instance, from quantum physics during the solutions of the Schrödinger equation in spherical polar coordinates [7]. MATLAB, Octave, and Scilab offer a rich set of high-level functions including the associated Legendre functions. So, from a conceptual point of view, associated Legendre functions are treated by AD implementations for these languages as ‘elemental functions’. The associated Legendre functions are defined on the interval , where the ordinary limit defining derivatives does not exist at the boundaries . Thus, it is difficult for a black-box AD approach to mechanically generate AD code that is capable of evaluating the derivatives at these boundaries. However, using a hierarchical AD approach, it is possible to treat these boundaries rigorously, by replacing the ordinary limit by a one-sided limit. The main contributions of this article are as follows. First, we derive the analytic formulæ for the first-order one-sided derivatives of the associated Legendre functions at the boundaries. Second, this hierarchical AD approach is implemented in the AD tool ADiMat [3,20] that implements AD for programs written in MATLAB. Third, we demonstrate the feasibility of this AD approach by estimating the expansion coefficients of a geomagnetic field model and quantify the differences when the same parameter estimation problem is solved via divided differences.
Non-iterative two-step method for solving scalar inverse 3D diffraction problem
Published in Inverse Problems in Science and Engineering, 2020
M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak
We consider inhomogeneities of the domain Q of two types. Inhomogeneities of the first type are presented by continuous functions whereas inhomogeneities of the second type are described by piecewise-continuous functions In the latter case, we set where are arbitrary Hölder continuous functions. At the points of the boundaries , the function can be defined via the one-sided limit from any side of the surface.