Polynomials

The enthalpy of pure hydrocarbons In the ideal gas state has been fitted to a fifth order polynomial equation of temperature. The corresponding is a polynomial of the fourth order ... 

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

In case if amplitude-frequency characteristics of TF can be expressed through polynomial characteristic equation, stability of the system is determined by value of roots of characteristic equation. There are two rules that can be used in this case ... 

For each frequency 100 points were taken along a line running from the surface of the conductor into a depth of 30 mm in that region below the coil, where the maximum eddy currents are located (dashed vertical lines in the sketch). These data are fitted by appropriate polynomials to obtain an analytical expression for s (to, z) in the frequency and depth interval mentioned above. 

Lane improved on these tables with accurate polynomial fits to numerical solutions of Eq. 11-17 [16]. Two equations result the first is applicable when rja 2... 

Functional fonns used for the repulsion include the simple exponential multiplied by a linear combmation of powers (possibly non-mteger) of r, a generalized exponential function exp(-h(r)), where b r) is typically a polynomial in r, and a combination of these two ideas. 

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

Deuflhard P and Wulkow M 1989 Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable Impact of Computing in Science and Engineering vol 1... 

It is convenient to expand7 (0) in a basis of Legendre polynomials / (cos 0) (as these define the natural... 

The first tenn is zero if j due to the orthogonality of the Hemiite polynomials. The recursion relation in equation (B 1,2.4 ) is rearranged... 

Figure Bl.12.11. Angular variation of the second- and fourth-rank Legendre polynomials.

Wlien expanded as a series of Legendre polynomials /Jj (cos 0), tire differential cross section has the following form... 

The Taylor series by itself is not numerically stable, since the individual temis can be very large even if the result is small, but other polynomials which are highly convergent can be found, e.g. Chebyshev [M, M and M] or Lancosz polynomials [, 68]. 

Mandelshtam V A and Taylor H S 1995 A simple recursion polynomial expansion of the Green s function with absorbing boundary conditions. Application to the reactive scattering J. Chem. Phys. 102... 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

This is written in terms of zt-, the moduli of the roots Zk-, since the roots are either real or come in mutually complex conjugate pairs. In any case, this constant term can be absorbed in the polynomial P t) in Eq. (15).]... 

This algorithm was improved by Chen et al. [78] to take into account the surface anhannonicity. After taking a step from Rq to R[ using the harmonic approximation, the true surface information at R) is then used to fit a (fifth-order) polynomial to fomi a better model of the surface. This polynomial model is then used in a coirector step to give the new R,. 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, hamonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the bajectoiy, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

We shall expand the polynomial of z. But recalling that only terms of the even power of z do not vanish, we can write the expansion in the following form ... 

Next, we shall consider the integral basically corresponding to the Rys polynomial problem [15,16]. Letting... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

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