Piecewise continuous functions

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

The a priori information involved by this modified Beta law (5) does not consider the local correlation between pixels, however, the image f is mainly constituted from locally constant patches. Therefore, this a priori knowledge can be introduced by means of a piecewise continuous function, the weak membrane [2]. The energy related to this a priori model is ... 

A similar obstacle occurs in trying to construct difference approximations of the operator Lu = ku ), where k x) is a piecewise continuous function (see Chapter 3). 

Pattern functionals are defined in the class of piecewise continuous functions ... 

We assume that the a-cuts G (a), Gg(a), and G(-(a) of three fuzzy sets A, B, and C, respectively, depend at least piecewise continuously on the a parameter from the unit interval [0,1], where continuity is understood within the metric topology of the underlying space X. On the closed interval [0,1], the scaled Hausdorff distance ah GJ a),Ggia)) is an at least piecewise continuous function of the level set value a. This function either attains its maximum ft(G ( a ), Gg a )) at some value a within [0,1] or it converges to its supremum value... 

In this averaging procedure we imagine that at any point r in a two-phase flow, phase k passes intermittently so a function ipk associated with phase k will be a piecewise continuous function. However, the interfaces are not stationary so they do not occupy a fixed location for finite time intervals. For this reason the average macroscopic variables are expected to be continuous functions (but this hypothesis has been questioned as it can be shown that the first order time derivative might be discontinuous which is not physical, hence an amended double time averaging operator was later proposed as a way of dealing with this problem [43, 47]). Since T is the overall time period over which the time averaging is performed, phase k is observed within a subset of residence time intervals so that T = for all the phases in the... 

There are two approaches to such interpolation, a local and a global one. The local approach is based on a set of piecewise-continuous functions, usually polynomials. These functions are designed to interpolate locally between the grid points. The interpolation... 

For a piecewise continuous function F x) over a finite interval 0 < x < it the finite Fourier cosine transform of F(x) is... 

I am assuming here that a ie) has the properties Indicated by Fig. 8. The statement of uniqueness applies to the class of piecewise continuous functions [a(X),e(X)] having values In the Indicated domains. 

Selmic, R.R. Ixvis, F.L. 2002. Neural-network approximation of piecewise continuous functions application to function compensation. IEEE Transactions on Neural Networks. 13(3) 745-751. 

Kinetic Models in the Form of Equations Containing Piecewise Continuous Functions... 

We assume that the functions that represent the behavior of physical systems are piecewise continuous with respect to each variable. If we temporarily keep all but one of the independent variables fixed, the function behaves as a piecewise continuous function of that variable. We also assume that the functions are piecewise single-valued. That is, except at isolated points, there is one and only one value of the dependent variable for a given set of values of the independent variables. 

In Ouypomprasert et al. (1989), it has been pointed out that for reliability analysis, it is most important to obtain support points for the response surface very close to or exactly at the limit state g(x) = 0. This finding has been further extended in Kim and Na (1997) and Zheng and Das (2000). In Brenner and Bucher (1995), the response surface concept has been applied to problems involving random fields and nonlinear structural dynamics. Besides polynomials of different orders, piecewise continuous functions such as hyperplanes or simplexes can also be utilized as response surface models. 

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