Orthogonally functionalized

The time-independent case corresponds to fixed time t=0. The only constraints on this expansion are that the Hilbert space norm of the orthogonal functions v should be less than or equal to one, if the functions are to be interpreted as... 

Functions that obey equation (2.41) are called orthogonal functions. If the orthogonal functions are also normalized, as in equation (2.42), then they are... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

In both calculations, the boundary conditions are linear with respect to 0 and its first-order derivatives. The solution of the Fourier equation, with respect to the space variables, may be developed in a series of orthogonal functions, winch are exponential with respect to the time variable [for the solution of similar problems, see (45)]- The time-dependance of the temperature distribution along a single space variable r, resulting from a unit pulse, is therefore given by... 

Solution Let us first choose a reference orthonormal set (j>, jp) to be used consistently in displaying the various matrices and vectors under discussion. For simplicity, we choose (pi, (p2 to be the ( Lowdin-orthogonalized ) functions that are closest to xa and Xb >n the mean-squared-deviation sense. The non-orthogonal functions xa and Xb (with(xJXb) = S ) can then be expressed in terms of reference orthonormal functions as... 

The purpose of this Chapter is not to present an exhaustive theory of linear algebra that would take more than a volume by itself to be presented adequately. It is rather to introduce some fundamental aspects of vectors, matrices and orthogonal functions together with the most common difficulties that the reader most probably has encountered in scientific readings, and to provide some simple definitions and examples with geochemical connotations. Many excellent textbooks exist which can complement this introductory chapter, in particular that of Strang (1976). 

A widely used example of orthogonal functions is the set of sines and cosines. For example, given any real number a, and the function sin nx for integer values of n, 3) is equal to [a, a+ 2ji]. We can check that... 

The powers of the variable x (1, x, x2,..., xa,...) are not orthogonal functions over a unique interval. However, particular sets of polynomials present the orthogonality property. A simple and useful example is that of Legendre polynomials. Let us choose over the range [—1, +1] the first two polynomials... 

Its functional versatility enabling the preparation of dendrons with differentiated [6, 7], and usually orthogonal, functionalities located respectively at the... 

Surface Spherical Harmonics. From the two sets of orthogonal functions ITU (cos 0), cos ( up) we can form a third set of functions... 

Peterson used the skill score to evaluate the performance of his empirical statistical model based on orthogonal functions. The skill score equals 1.0 when all calculated and observed concentrations agree, but 0 when the number of correctly predicted results equals that expected by chance occurrences. The statistical technique had a skill score of 0.304. An 89-day, 40-station set of the data was used to check a Gaussian diffusion model, and this technique gave the diffusion model a skill score of only 0.15. (Recall that the statistical empirical model was used for 24-h averaged sulfur dioxide concentrations at 40 sites in St. Louis for the winder of 1964-1965.)... 

Peterson, J. T. The calculation of sulfur dioxide concentrations over a metropolitan area by using empirical orthogonal functions. American Institute of Aeronautics and Astronautics, Paper 70-113, January 1970. 7 pp. 

We may use the concept of orthogonal functions to identify components. In the present case, our component functions are sinusoids, and we find that the Fourier transform of the sum is the sum of the Fourier transforms ... 

Da] Davis, H.F., Fourier Series and Orthogonal Functions, Dover, New York, 1989. (Unabridged republication of tbe edition pubUsbed by Allyn and Bacon, Boston, 1963.)... 

For a two-dimensional representation, we require two orthogonal functions, which jointly form a basis for the representation. We have one, but we require its partner. To obtain it, we recall that any member of a set of functions forming the basis for a representation must be affected by the symmetry operations of the group in one of two ways ... 

Since only two orthogonal functions are needed to provide a basis for the Eu representations, we have clearly, in this simple case, reached the end of our quest. The two functions, in normalized form, that we require are... 

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