Polynomial series

Since A x,y) can really be any function bounded by the aperture of the system it is best to use as general description as possible. One such description of this function is to expand A x, y) = A p, 0) about (p, 0) in an infinite polynomial series. One set of polynomials that are frequently used are Zernike polynomials. Thus one can write A(p, 0) = Y.m,n CmnFmn p, 0). 

Approximating a function with an orthogonal polynomial series means that it is not necessary to solve N equations simultaneously. 

The virial equation of state represents the pressure as a polynomial series in the inverse molar volume as... 

Table 1. Coefficients a, in the polynomial series expansions (equation (8)) of Eg (ra), e(r ), N(Ta), (Ta) and E+(ra) along the a normal mode distortion path for AsF3 and AsBr3 values of la,-1 with i > 6 are smaller than 0.015...

As for AsF3 and AsBr3 a rather good fit of the curves in Fig. 5 to a polynomial series in Tb as in equation (8) is possible, with parameters listed in Table 4. Again N( rb) is approximated reasonably well by a linear function, while for Zsg(rb), Zse(rb) and E (rb) at least one additional an harmonic term a4rb is needed in addition to a2. 7h, in contrast to the assumptions in equations (10) and (11). 

Table 4. Coefficients at in the polynomial series expansions of the various energies along the Tb distortion path Td — C2v for the s2-lone pair molecule SeF4...

Where 8 and d2 are, respectively, the intrinsic 7j and an experimental factor accounting for a spatially dependent imperfect spin inversion. In some cases, generic parametric models (e.g., polynomial series) can be used to identify complicated responses, such as the free induction decay (FID) signal dephasing induced by the presence of magnetic field inhomogeneities.3... 

This equation is based on theoretical arguments and is a closed expression for an infinite polynomial series. It contains three parameters a, b, and c however. 

The absorption spectrum consists of sequences of transitions from v" = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

The time evolution operator can be expanded in any classical orthogonal polynomial such as Qiebychev [93,97], Newtonian [98], Faber [99], Hermite [100,101] and others. In general, the evolution operator can be written in polynomial series... 

While relatively benign in appearance, this transform is difficult to invert analytically. The inversion scheme of Kubin (22) and Kucera (30), which uses a Her-mite polynomial series expansion, is too cumbersome to be of any practical use. The coefficients are difficult to obtain algebraically and the series converges slowly. 

High and Danner found that the Koningsveld-Kleintjens expression (Equation 16.39) is superior to the polynomial series in correlating and extrapolating concentration-dependent FH interaction parameters. 

We now see that there are physical reasons why we are trying to fit a section of a polynomial series and not a section of a power series. 

Once we turn least squares-or any other fitting rule-loose to fit what we may think are sections of power series, it will fit what is best thought of as a section of another polynomial series. The nature of this polynomial series is fixed by the data locations, the assumed weights, and the selected fitting rule. No physics enters. There is no reason why this polynomial series should be the same as the physically meaningful polynomial series, and it probably never has been. 

A very simple and widely used method, known as the method of cumulants, was described by D. Koppel [45], This method allows one to obtain mean value of T(0) and the width of distribution, characterized by the polydispersity index, PI. In the method of cumulants the natural logarithm of g(l)(xcorr, 0) is expanded into polynomial series, in which usually only the first two terms are retained, i.e. ... 

The first major contribution was made in 1901 by Kamerlingh Onnes, who described the isothermal volumetric behavior of a gas by the finite polynomial series... 

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 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-... 

The Zernike polynomials are functions of the polar coordinates (r, 0) of positions r and angle 6 in the exit pupil. Because the same point in the exit pupil is specified by 6 and (6 -1-360 deg), it is convenient for the polynomials to be expressed as functions of sin(u0) and cos(u0), where n is an integer." Although the Zernike polynomials comprise an orthogonoal polynomial series with an... 

Hirata (47) also presented an equation for the calculation of activity coefficients for this quaternary system. The equation was based on the Margules equation and consisted of 64 constants rearranged as a polynomial series in terms of mole fractions. It was tested by Pilavakis and the results found inconsistent even for binary mixtures. 

These functions are in fact the even members of a Legendre polynomial series. Any surface which encloses a volume of rotation about the axis and has a central symmetry plane normal to this axis may be described by a summation of the component functions each with an appropriate amplitude, written (PjnCcos cr)). Loss of the central symmetry plane would then require the odd components of the... 

The Guth-Gold model (29) is an extension of the Einstein model, whereby interactions of neighboring particles are incorporated. Normalized initial moduli, normalized against the polymer modulus, of filled polymers are related to the filler volume fraction by a polynomial series as follows ... 

The director determines only the direction of the preferred orientation of the molecules, and indicates nothing about the degree of orientational order in the mesophase. The order parameter, 5, which is the first moment P2 in the expansion of the cylindrical distribution function of molecules in the Legendre polynomial series, provides just such a measure of the long-range orientational order... 

Physical theories often require mathematical approximations. When functions are expressed as polynomial series, approximations can be systematically improved by keeping terms of increasingly higher order. One of the most important expansions is the Taylor series, an expression of a function in terms of its derivatives. These methods show that a Gaussian distribution function is a second-order approximation to a binomial distribution near its peak. We will hnd this useful for random walks, which are used to interpret diffusion, thermal conduction, and polymer conformations. In the next chapter we develop additional mathematical tools. 

As for the pure gas case, the transport coefficients are related to individual expansion coefficients of the polynomial series. Thus, for the partial thermal conductivity of a gas mixture, one has... 

The general geometry is illustrated in Fig. 2. We begin by considering the configuration where the molecular dipole is oriented perpendicular to the surface, i.e. y/ = 0. The potential is expanded in an infinite Legendre polynomial series as... 

Chebyshev orthogonal polynomial series along the nonperiodic (shear) direction ... 

The hierarchy of reactions is labeled with the basic reaction graph as the most abstract, the reaction graph with shell bonds, and the fully defined reaction center graph (RCG) with the atom types specified. Fujita elaborates mathematically, in two ways, the enumeration of RCGs by Polya s method, using a polynomial series as generating function to afford cycle indexes for the symmetry of each basic group. All... 

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