Polynomials Chebychev

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

Following Tal-Ezer and Kosloff (1984) the time evolution-operator is expanded in terms of Chebychev polynomials ipk according to... 

The fulfil the same recursion relation as the Chebychev polynomials, namely... 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

If the expansion functions are derived from orthogonal polynomials, the matrix d can be obtained from the recursion relation for the orthogonal polynomials (32). If there is a fast transform for G (which is true for the Chebychev polynomial expansion), then applying Eq. (38) will scale as 0(Ng log Ng). 

Another widely used propagation scheme is the Chebychev polynomial expansion method introduced by Kosloff and Kosloff (8). This is a global propagator in the sense that it expands the propagator e u h)H in the interval [0, /]. The method is based on the Chebychev expansion relation for the function exp(iRX) (X E [—1, 1]) (13),... 

Here, Tn is the nth Chebychev polynomial satisfying the recursion relation... 

When the Chebychev polynomial series is developed until the order k equals n - 1, i.e. until there are as many polynomial coefficients as there are observations, the matrix X can be considered an orthogonal basis for n-di-mensional space. The coefficients Pj are the co-ordinates in this alternative system of axes, this other domain, as it is often called. We could speak of the Chebychev domain in this case. Eq. (10) describes the basis transformation, i.e. the projection of the signal onto the alternative basis. The transform has only changed our perspective on the data nothing has been changed or lost. So we could also transform back, using the model at the start of all this ... 

The Chebychev polynomial is just one possibility to construct a fixed orthogonal basis for n-dimensional space. There are many others. Interesting members of the family are the Hermite polynomial. Fourier and Wavelets. As there are many, the question arises how to choose between them. Before we are able to answer that question, we need to deal with another, more fundamental one why do a basis transformation in the first place ... 

The Fourier polynomial series is not a sequence of increasing powers of x. like the Chebychev polynomial, but a series of sines and cosines of increasing frequency. In fact, there is no longer a notion of x and y, as in the initial example of polynomial approximation, but just y. a series of num-... 

Resistance versus temperature characteristics for the blends show a trend similar to that of a germanium thermometer (GRT) in the temperature range between 0.35 K and 10 K. The data were fitted using Chebychev polynomials [105]. Sensitivity of the blends is high (better than 0.1 mK) at temperatures below about 1 K, because of the higher resistance. In the temperature range between 2 K and 50 K, the sensitivity is about 1.0 mK. Monotonic trend in MR, where... 

The non-symmetric Lanczos algorithm is an effident tool to investigate the stability of flexible mechanisms. It permits to take into account the influence of mechanical effects introduced by modeling of the gyroscopic effect and dry friction. Improvements such as an acceleration technique by use of Chebychev polynomials [3] would still render it more attractive. The algorithm is still under test on more general examples. 

The numerical methods commonly used to carry out either TDW or TIW calculations are the symmetric split operator (SSO) approximation to the evolution operator, exp(-i//r//i), and the Chebychev polynomial expansions of either exp(-i//r/fi) or the causal Green s function, E — H 4- ie)". These methods are described in great detail in the literature, but are too technically involved to go into in this article. Of course, there are other propagation methods as well which are being used or are under development. One which is very interesting and which has the attractive feature of being readily applied even in the case of an explicit time-dependence in the Hamiltonian is the modified Cayley method . ... 

A similar measure to the Gabor filter bank, Subsection 3.5, may be extracted using the volume reflection spectral (VRS) decomposition [16]. In this approach, the seismic traces are decomposed by a Chebychev polynomial, and the polynomial coefficients are used as spectral components, representing an attribute volume. An example for this is shown in Figure 11. 

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