Polynomials Hermitian

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

The symbol He (y) denotes the Hermitian polynomials, which are known to be orthogonal to each other when weighted with the function exp[ — y /2], that is. 

In consequence, the Hermitian polynomials H (y) can also be identified with the eigenstate f (y) of Note that... 

Well-known recursion formulas concerning the form H (x) of the Hermitian polynomials, after some algebra, lead us to... 

This difference equation is satisfied by the Hermitian polynomials. Thus, we obtain... 

A Gauss pulse and a polynomial multiplied by a Gauss pulse satisfy the requirements of rapid fall-off in both the time and the frequency domains. A pulse the shape of which is determined by a product of a Hermitian polynomial and a Gauss function is a Hermite pulse [Sill, War2]. Its width can be more narrow than that of a Gauss pulse. A fairly uniform rotation of z magnetization can be generated if the sum of a zeroth- and a... 

The next higher interpolation step are extended Hermitian polynomials in which not only the wave-function and its derivative but in addition its second derivative are taken into account. Hence we get... 

The stationary amplitude distribution is obtained from the solutions. For the confocal resonator it can be represented by the product of Hermitian polynomials, a Gaussian function, and a phase factor ... 

Here, C is a normalization factor. The function Hm is the Hermitian polynomial of mth order. The last factor gives the phase r) in the plane z = zo at a distance r = (x + y ) / from the resonator axis. The arguments x and y depend on the mirror separation d and are related to the coordinates x, y by X = flxlvo and y = Vlylw, where... 

From the definition of the Hermitian polynomials [5.31], one can see that the indices m and n give the number of nodes for the amplitude A(x, y) in the X- (or the y-) direction. Figures 5.9,5.10 illustrate some of these transverse electromagnetic standing waves, which are called TEM, modes. The diffraction effects do not essentially influence the transverse character of the waves. While Fig. 5.9a shows the one-dimensional amplitude distribution A(x) for some modes. Fig. 5.9b depicts the two-dimensional field amplitude A(x, y) in Cartesian coordinates and A(r, d) in polar coordinates. Modes with m = n = 0 are called fundamental modes or axial modes (often zero-order transverse modes as well), while configurations with m > 0 or n > 0 are transverse modes of higher order. The intensity distribution of the fundamental mode /qq oc Aqo qo derived from (5.30). With... 

Recent years have seen great advances in nonlinear analysis of frame structures. These advances were led by the development and implementation of force-based elements (Spacone et al. 1996), which are superior to classical displacement-based elements in tracing material nonlinearities such as those encountered in reinforced concrete beams and columns. In the classical displacement-based frame element, the cubic and linear Hermitian polynomials used to interpolate the transverse and axial displacement fields, respectively, are only approximations of the actual displacement fields in the presence of non-uniform beam cross-section and/or nonhnear material behaviour. On the other hand, force-based frame element formulations stem from equilibrium between section and nodal forces, which can be enforced exactly in the case of a frame element. The exact flexibiUty matrix can be computed for an arbitrary (geometric) variation of the cross-section and for any section/material constitutive law. Thus, force-based elements enable, at no significant additional computational costs, a drastic reduction in the number of elements required for a given level of accuracy in the simulated response of a EE model of a frame structure. 

The cylindrical tank wall is modeled using axisymmetric shell finite elements (Fig. 2). Shape functions of the 2nd order polynomials are used for the axial and circumferential displacements z(f) and Uff(i). However, those of the 4th order Hermitian polynomials... 

Handy, C. R., Khan, D. and Wang, Xiao-Qian, (2000) Multiscale Reference Function Quantization of the —(ix) Non-Hermitian Polynomial Potentials , preprint Clark Atlanta University. 

The function H is the Hermitian polynomial of m " order. The last factor... 

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