Representation of a function

More often, however, s.a. methods are applied at the outset, with more or less arbitrary initial approximations, and without previous application of a direct method. Usually this is done if the matrix is extremely large and extremely sparse, which is most often true of the matrices that arise in the algebraic representation of a functional equation, especially a partial differential equation. The advantage is that storage requirements are much more modest for s.a. methods. 

Graphical representation of a function of two variables reduced to a function of one variable by direct substitution. The unconstrained minimum is at (0,0), the center of the contours. 

Cormack AM. Representation of a function by its line integrals, with some radiological applications. J Appl Phys 1963 34(9) 2722-2727. 

Wavelet analysis is the representation of a function by wavelets that are mathematical functions used to divide a given function into different frequency components and study each component with a resolution that matches its scale [Depczynski, Jetter et al, 1997, 1999b Alsberg, 2000 Jetter, Depczynski et al, 2000 Walczak, 2000]. 

There are two important features associated with the generation of power series representations of functions. First, a value of jc lying in the domain of the function must be chosen for the expansion point, a second, the function must be infinitely differentiable at the chosen point in its domain. In other words, differentiation of the function must never yield a constant function because subsequent derivatives will be zero, and the series will be truncated to a polynomial of finite degree. The question as to whether the power series representation of a function has the same domain as the function itself is discussed in a later subsection. The next subsection is concerned with determining the coefficients, c for the two kinds of power series used to represent some of the functions introduced in Chapter 2 of Volume 1. 

A very important property of an asymptotic expansion is the manner in which it converges to the function that it is intended to represent. Two facts can be stated that relate intimately to the nature of the convergence of an asymptotic expansion. First, if a function such as T(x e) has an asymptotic expansion for small e (either for all x or at least in some subdomain of x), then this expansion is unique (at least in the subdomain). However, second, more than one function T may have the same asymptotic representation through any finite number of terms. The second of these statements implies that one cannot sum an asymptotic expansion to find a unique function T(x e) as would be possible (in the domain of convergence) with a normal power-series representation of a function. This distinction between an asymptotic and infinite-series representation is reflected in a more formal statement of the convergence properties of both an infinite series and an asymptotic expansion. In the case of an infinite-series representation of some function T(x e), namely,... 

Although a list of several temperatures and the corresponding several values of the vapor pressure qualifies as a mathematical function, such a function is not useful for other temperatures. We need a representation of the function that will provide a value of the vapor pressure for other temperatures. Although mathematicians frequently have exact representations of their functions, approximate representations of a function must generally be used in physical chemistry. These representations include interpolation between values in a table, use of an approximate curve in a graph, and approximate mathematical formulas. In all these approximate representations of a function we hope to make our rule for generating new values give nearly the same values as the correct function. 

A Fourier transform is a representation of a function as an integral instead of a sum. Many modem instruments use Fourier transforms to produce spectra from raw data in another form. 

A Laplace transform is a representation of a function that is similar to a Fourier... 

In general, it is useful to regard 5 as a complex number s = a jw in these analyses (41, 42). Then one can calculate real-axis and imaginary-axis frequency domain representations of a function. For example, the real-axis transform of E t) is... 

Wavelet transforms inspect signals at different scales or resolutions. At the coarse level, only the most prominent large features can be seen, whereas at the higher levels finer details are captured. Multiresolution develops representations of a function f(t) at various levels of resolution where each level is expanded in terms of translated scaling functions (t)(2jt — k). As mentioned earlier in this book a sequence of embedded subspaces Vj is created... 

Transforming functions between different coordinate systems can often simplify the description. In some cases, it may also be advantageous to switch between different representations of a function. A function in real space, for example, can be transformed into a reciprocal space, where the coordinate axes have units of inverse length. 

This is rather like an orthogonality condition, for any positive values Aj and A2, in the interval 0 a 00. We are guided to the conclusion that a representation of a function fix) in such semi-infinite intervals must involve all possible functions of the type in Eq. C15, where A is not restricted to discrete values, but can take on any positive number, as a continuiun in A > 0. Previously, we represented fix) by the infinite series in the region Q <,x L ... 

Fourier analysis The representation of a function f(x), which is periodic in as an infinite series of sine and cosine functions,... 

Other interesting consequence is the coordinate representation of a function of coordinate operator f acting on a state vector m) ... 

In this example the exact calculation could be made more easily than the approximation. However, in physical chemistry it is frequently the case that a representation of a function is not known, but values for the partial derivatives are available, so that approximation can be made while the exact calculation cannot. For example, there is usually no simple formula giving the thermodynamic energy as a function of its independent variables. However, we can write... 

Because a functional element represents an activity, it is natural to consider its set of variables as containing two subsets Iht functional parameters that describe the output or result of the activity, and the variables describing interactions required in order to provide the output, which we may call dependencies. The latter are related to the functional parameters through the set of relationships that forms part of the definition of a functional element and describes the behaviour of the element. But there is a third subset, consisting of those variables that, while they do not describe intended interactions, describe necessary external interactions. Typical examples might be ambient conditions and interest rate. We shall call these variables influences. Consequently, a symbolic representation of a functional element takes the form shown in Fig. C3.1. 

Before considering in more detail the approximation methods used in quantum mechanics, we review some of the mathematical tools available 4 One of the most powerful of these concerns the representation of a function as a mixture, with carefully chosen coefficients, of a number of more elementary functions ... 

Chemical structure of the Silicalite-1 zeolite (a), schematic representation of a functionalized silica nanotube (b), chemical formula of silane functional group used in the silica nanotube (c). 

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