Piecewise differentiable functions

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator from the known values of the field p(r, u>) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise real-analytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Wcidelt s and Gusarov s uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). 

It is advantageous to replace Eq. (11) by an equivalent expression with lower differentiability requirements on the basis functions, in which the boundary conditions are automatically satisfied. The appropriate integration by parts formula is the Surface Divergence Theorem (SDT Weatherburn 1927), which is an integral relation for a piecewise-differentiable vector-value function F defined on a surface ... 

In the original implementation of the PME algorithm [109,111], Lagrangian weight functions W2p(x) [116] were used for interpolation of the complex exponents, by using values at 2p points in the interval x < p. Unfortunately, these functions W2p(x) are only piecewise differentiable, so the approximate reciprocal lattice sum caimot be differentiated to arrive at the reciprocal part of the Coulomb forces. Thus, the forces were interpolated as well. 

For unphysical energies, this means that the expression o(b) should become unbounded, as 6 - oo. Alternatively, it is also possible that o(b) is a piecewise differentiable solution to the Schrodinger equation. In this case, o b) could be bounded (lim i, >oo oip) = 0), but have discontinuous (first order) derivatives (i.e. at the turning points). The second derivative, for such a spline type configuration, could still be zero, at the turning points thereby still satisfying the TPQ conditions. As an example, the functions 1 + c x + a (for a < 0) and 1 + c+a + (for x > 0) have continuous zeroth and second order derivatives, at the origin however, they have discontinuous first order derivatives (i.e. c c+). 

In mathematics a spline is a piecewise polynomial function, made up of individual polynomial sections or segments that are joined together at (user-selected) points known as knot points. Splines used in term structure modeling are generally made up of cubic polynomials. The reason they are often cubic polynomials, as opposed to polynomials of order, say, two or five, is explained in straightforward fashion by de la Grandville (2001). A cubic spline is a function of order three and a piecewise cubic polynomial that is twice differentiable at each knot point. At each knot point the slope and curvature of the curve on either side must match. The cubic spline approach is employed to fit a smooth curve to bond prices (yields) given by the term discount factors. 

When the second stage decisions are real-valued variables, the value function Qu(x) is piecewise-linear and convex in x. However, when some of the second stage variables are integer-valued, the convexity property is lost. The value function Qafx) is in general non-convex and non-differentiable in x. The latter property prohibits the use of gradient-based search methods for solving (MASTER). 

For present purposes, the functions of time, f(f), which will be encountered will be piecewise continuous, of less than exponential order and defined for all positive values of time this ensures that the transforms defined by eqn. (A.l) do actually exist. Table 9 presents functional and graphical forms of f(t) together with corresponding Laplace transforms. The simpler of these forms can be readily verified using eqn. (A.l), but as extensive tables of functions and their transforms are available, derivation is seldom necessary, (see, for instance, ref. 75). A simple introduction to the Laplace transform, to some of its properties and to its use in solving linear differential equations, is presented in Chaps. 2—4 of ref. 76, whilst a more complete coverage is available in ref. 77. 

The technical conditions on f are quite reasonable if a physical situation has a discontinuity, we might look for solutions with discontinuities in the function f and its derivatives. In this case, we might have to consider, e.g., piecewise-defined combinations of smooth solutions to the differential equation. These solutions might not be linear combinations of spherical harmonics. 

In actual computations, the numerical differentiation can introduce a large error and should be avoided. A simple solution to this would be to fit spline functions, or a piecewise polynomial and overall smooth function of E, to the numerically calculated eigenphase sum 5(E), and then to differentiate the spline functions analytically [52, 53]. In the E-matrix method [44], the analytic E dependence of the R matrix and associated matrices can be taken advantage of in the direct differentiation of these quantities. This technique was found to be useful for automatic and fast analysis of the results of E-matrix method calculations [54-56]. 

For many applications, interpolations of functions of two or three variables defined in two-and three-dimensional domains must be considered. For example, global interpolations in two- and three-dimensional systems are analogous to polynomial interpolation in onedimensional systems however, global interpolants do not exist in 2- and 3D. This is a big drawback in numerical analysis because a basic tool available for one variable is not available for multivariable approximation [21], The best developed aspect of this theory is that of piecewise polynomial approximation, associated with finite element and finite volume approximations for partial differential equations, which will be examined in detail in Chapters 9 and 10. 

The use of piecewise bicubic Hermite functions in collocation schemes for the solution of elliptic partial differential equations has been described by Prenter (13,17) a short outline is presented here. 

In the finite element method (Courant 1943 Turner et al. 1956), the basis functions are of compact support—that is, each (j)j is nonzero only on a small rectangle in D. Since the quantities that we want to calculate from the solutions, such as area, volume, and other surface or volume integrals such as the scattering function, do not call for explicit knowledge of the second or high partial derivatives, the most economical choice is bilinear basis functions, which are piecewise first differentiable. A method for estimating the Gaussian curvature from the solution is described later in this section. 

Let us observe that in both examples 1 and 2 the function G(% >) is piecewise linear for any realization of >, and hence is not everywhere differentiable. The same holds for the optimal value function Q(-, p) of the second-stage problem (15). If the distribution of the corresponding random variables is discrete, then the resulting expected value function is also piecewise hnear and hence is not everywhere differentiable. 

Here is a chosen pdf, z is a random vector having pdf , and L x, z) = f Z + Tx)/likelihood ratio function. It can be shown by duality arguments of linear programming that G(-) is a piecewise Unear convex function. Therefore, Q(x, h) is piecewise constant and discontinuous, and hence second order derivatives of tf[Q(x, h)] cannot be taken inside the expected vrilue. On the other hand, the likelihood ratio function is as smooth as the pdf /( ). Therefore, if /( ) is twice differentiable, then the second order derivatives can be taken inside the expected vrilue in the right-hand side of (30), and consequently the second order derivatives of tf[Q(x, h)] can be consistently estimated by a sample average. 

If the function g(x) is twice differentiable, then the above sample path method produces estimators that converge to an optimal solution of the true problem at the same asymptotic rate as the stochastic approximation method, provided that the stochastic approximation method is applied with the asymptotically optimal step sizes (Shapiro 1996). On the other hand, if the underlying probability distribution is discrete and g(x) is piecewise linear and convex, then w.p.l the sample path method provides an exact optimal solution of the true problem for N large enough, and moreover the probability of that event approaches one exponentially fast as A — (Shapiro and Homem-de-MeUo 1999). 

This fimction is continuous and varies monotonically from a lower bound of 0 to an upper bound of 1 and has a continuous derivative. The transfer function in the output layer can be different from than that used in the rest of the network. Often, it is linear, f(NET)=NET, since this speeds up the training process. On the other hand, a sigmoid function has a high level of noise immunity, a feature that can be very useful. Currently, the majority of current CNNs use a nonlinear transfer function such as a sigmoid since it provides a number of advantages. In theory, however, any nonpolynomial function that is bounded and differentiable (at least piecewise) can be used as a transfer... 

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