Piecewise-defined functions

The electrostatic potential is a continuous, but piecewise defined function ... 

Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

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

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. 

The piecewise linear function defined by these three relations is represented in fig. 4.25. Three parameters control this map the value M corresponds to the maximum level of substrate reached at the beginning of the active, bursting phase, i.e. at the top of the first peak in )3 parameter a measures the quantity of substrate consumed by the synthesis of each peak of /3 finally, the slope b of the median portion of the linear map determines the value of the minimum m that corresponds to the substrate level reached in the last peak of the volley, at the end of the active phase of bursting. Determining the intersection of the segments defined by eqns (4.5a,b) yields ... 

Because of the implicit use of the first derivative of the coefficient functions, the Simpson-type discretization method may be problematic if piecewise-defined potential terms are present (for some finite-nucleus models such as the homogeneous charge distribution of Eq. (6.151)). For such special cases, it would be more advantageous to use a scheme which employs a diagonal matrix representation of the effective potential (cf. Ref. [492] for details). 

Up to now all the computational procedures have been precisely defined, with the exception of the basis set. In the atomic case all the equations are of course simplified to only the radial coordinate, the angular part being analytical. The radial functions are expanded in a basis set of B-splines over a selected interval [0, RmaxI- These are piecewise polynomial functions, completely defined in terms of... 

The German railroad company (Deutsche Bahn AG) uses wheel and rail models for simulation which are defined by piecewise smooth functions consisting of polynomials and circles. They are given in Figs. 5.20 and 5.21. When using these... 

Next, a finite dimensional subspace consisting of piecewise linear functions must be consuiicted. The sub-intervals of length Az = Zj+i — zj,j = 1, 2,..., AT are defined. As parameters to describe how the function change over the sub-intervals, the basis functions are chosen as the set of ttiangular functions defined as ... 

S = P /m)" exp m) that extended the analytical formula proposed by Hsu and Bernard (1978) defined for m = 1. Furthermore Jennings et al. (1969) proposed a piecewise modulating function given by the following formula ... 

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 simplest function defined in terms of a prescription (such functions are sometimes termed piecewise functions) is the modulus function, fix) = x defined as follows ... 

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. 

Figure 6 The binding free energy landscape for SB203386 generated with the piecewise linear energy function and the ensembles of 6 protein conformations (a) and 32 protein conformations (b). For each two-dimensional temperature slice, the reference energy F R = 0,T) is defined to be zero.

In the finite element and spectral element methods the entire domain f2 is divided into sub-domains or in a number of elements. Piecewise continuous trail (expansion) functions are defined for each element instead of using only one global trail function. Different continuity requirements for connecting the elements can be used. 

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 ... 

Simulation of confinement by penetrable boxes represents a more realistic physical model. A very simple approach was proposed by Marin and Cruz [18], where they used the Rayleigh-Ritz variational method via a trial wave function for the ground state, which consists of two piecewise functions, one for the inner region (r < ro), and the other for the outer region (r > ro). The trial wave function is defined as follows ... 

This equation is only approximately true, because we have multiplied the probability that X is in the subinterval (x, x, - - Ax) by x,-, which is only one of the values of X in the subinterval. However, Eq. (5.68) can be made more and more nearly exact by making n larger and larger and Ax smaller and smaller in such a way that n Ax is constant. In this limit, f becomes independent of Ax. We replace the symbol fi by f(Xi) and assume that /(x,) is an integrable function of x,. It must be at least piecewise continuous. Our formula for the mean value of x now becomes an integral as defined in Eq. (5.21) ... 

If the function /(0 is piecewise continuous and defined for every value of time from t = 0 to t = oo, the rigorous definition (7.1a) reduces to that of (7.1). For almost all the problems that we will be concerned with in this book, the simpler definition given by (7.1) will suffice. 

The function Bj,fc(r) is a piecewise polynomial of degree k-1 inside the interval U[Pg.141]

Fig. 4.28. Self-similarity of sequences of bursting patterns as a function of parameter a in the piecewise linear map defined by eqns (4.5). For a given value of a, the number of values of x equals the number of distinct peaks in the pattern of bursting. The variation of parameter a from 1.76 to 1.82 in (a) illustrates the passage from the pattern tt(6, 2) to the pattern tt(6, 3). In (b), the variation of a over the narrower range between 1.778 and 1.785 shows the transition between the more complex patterns it(6, 2, 5) and tt(6, 2, 4). Each time, more complex patterns alternate with relatively simpler patterns of bursting. The results are obtained by iteration of eqns (4.5) for b = 7 and Af = 11 (Decroly Goldbeter, 1987).

The piecewise linear map defined by eqns (4.5) thus allows us to explain the transitions between different simple or complex patterns of bursting as a function of the variation of parameters whose significance can be related to the properties of the biochemical model from which the map originates. Parameter a, for example, is linked to the quantity of substrate consumed by the production of a peak of product P,. An increase in the maximum rate of the reaction catalysed by enzyme Ej should therefore correspond to an increase in parameter a of the piece-wise linear map. Likewise, a rise in the rate constant results in a decrease in the amount of product Pj within the system enzyme Ej, activated by P2, should therefore become less active as increases. The amount of Pj, the substrate for enzyme Ej, will then tend to increase, owing to its diminished consumption in the second enzyme reaction. As enzyme Ei is activated by Pj, the increased level of this product raises the rate of enzyme Ei, which results in an increased amount of substrate consumed during synthesis of a peak of Pj. Thus we can see how an increase in the rate constant k in the enzyme model can also be associated with a larger value of parameter a in the one-dimensional map studied for bursting. 

This problem may be solved by use of integral calculus in connection with Eq. (16). However, it may be instructive to use Laplace transform methodology to evaluate and piecewise continuous, then the Laplace transform of the function, written /(x)), is defined as a function F(s) of the variable r by the integral... 

Changes in thermod)mamic properties are computed by performing integrations in some situations, the integration amounts to the evaluation of the mean for a continuous function. This mean is defined by the mean value theorem. If f x) is piecewise continuous on [a, b], then there is some value of/, designate it by/, such that... 

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