Convolution Property

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

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One of the most useful relations in signal processing is the convolution property. The convolution of two discrete (sampled) functions x(t) and y(t) is defined as... 

As already noted, the properties of convolution and correlation are the same, whether or not a continuous or discrete transformation is used, but because of the cyclic nature of sampled sequences discussed previously, the mechanics of calculating correlation and convolution of functions are somewhat different. The discrete convolution property is applied to a periodic signal 5 and a finite, but periodic, sequence r. The period of 5 is N, so that 5 is completely determined by the N samples s0, Sj,. .., %. The duration of the finite sequence r is assumed to be the same as the period of the data N samples. Then, the convolution of 5 and r is... 

Note that direct inversion of (8.2.8) and (8.2.9) using the convolution property leads to... 

These boundary gradients can be accounted for with only a slight modification to Eq. (6.17) [39]. Using the convolution property of Fourier Transforms, it can be shown that the scattering intensity (at high q) for an isotropic system can then be approximated as ... 

The essential property that we use, is the transformation of the product of convolution in a sum. 

The last relation in equation (Al.6.107) follows from the Fourier convolution theorem and tlie property of the Fourier transfonn of a derivative we have also assumed that E(a) = (-w). The absorption spectmm is defined as the total energy absorbed at frequency to, nonnalized by the energy of the incident field at that frequency. Identifying the integrand on the right-hand side of equation (Al.6.107) with the total energy absorbed at frequency oi, we have... 

Patent documents differ from journal Hterature in several ways. First of all, they are legal documents whose disclosures support one or more claims that define an area of property rights. The language in patent documents can therefore be quite convoluted "patentese" as the appHcant strives to achieve the broadest possible scope of coverage. Examples provided in patents may never have happened. Based on the appHcant s understanding of the technical... 

The thermoplastic rubbers have properties similar to those of the cast polyurethane rubbers but, because of the absence of covalent cross-links, have rather higher values for compression set, a common problem with thermoplastic rubbers. Their main uses are for seals, bushes, convoluted bellows and bearings. 

It is not easy to determine detailed properties of the tube terminations using STM or AFM. These microscopes cannot image undercut surfaces and the tip shape is convoluted with the cap shape of the nanotube. How ever, the tips may have very sharp edges... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

For the correlation bands obtained by a convolution of many shake-up lines, the size-consistency and size-intensivity requirements imply a convergence towards some asymptotic profile, when going to the polymer limit. This must ideally be achieved through a balance of the scaling properties of the individual shake-up lines, and the dispersion of the intensity of Ih lines over a rapidly increasing number of excited states with increasing system size. 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

The convolution and general properties of the Fourier transform, as presented in Section 11.1, are equally applicable to the Laplace transform. Thus,... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

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