Integral convolution

In the case of our linear, stationary and causal device, input and output are linked by the convolution integral ... 

The deconvolution is the numerical solution of this convolution integral. The theory of the inverse problem that we exposed in the previous paragraph shows an idealistic character because it doesn t integrate the frequency restrictions introduced by the electro-acoustic set-up and the mechanical system. To attenuate the effect of filtering, we must deconvolve the emitted signal and received signal. 

The second tenu in the Omstein-Zemike equation is a convolution integral. Substituting for h r) in the integrand, followed by repeated iteration, shows that h(r) is the sum of convolutions of c-fiinctions or bonds containing one or more c-fiinctions in series. Representing this graphically with c(r) = o-o, we see that... 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

Suppose X and Y are process components such that the failure of either one fails the train. The probaility of failing the train is the probability that one or the other or both fail, i.e., z = x + y with failure rates distributed as pjx), Py(y). Their combined distribution is expressed by the convolution integral (equation 2.7-1),... 

The integrals in Eqs. (17) and (18) are called convolution integrals. In Fourier space they are products of the Fourier transforms of c r). Thus, Eq. (18) is a geometric series in Fourier space, which can be summed. Performing this summation and returning to direct space, we have the OZ equation... 

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

A transformation of the peak voltammogram to the sigmoidal shape shown in the preceding section, Fig. 5.13, is achieved by the convolution analysis method proposed by K. Oldham. The experimental function j(t) = j[T(E — Ej)/v] is transformed by convolution integration... 

When you come across the term convolution integral later in Eq. (4-10) and wonder how it may come about, take a look at the form of Eq. (2-3) again and think about it. If you wonder about where (2-3) comes from, review your old ODE text on integrating factors. We skip this detail since we will not be using the time domain solution in Eq. (2-3). 

This expression is the discrete form of the convolution integral defined in Eq. (11-13). 

The remaining terms in Eq. (4-24) are the nth-order corrections to approximate the real system, in which the expectation value ( c is called cumulant, which can be written in terms of the standard expectation value ( by cumulant expansion in terms of Gaussian smearing convolution integrals ... 

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 ... 

The values of C(t) can be evaluated from the current-time history using the convolution integral... 

To develop the curve of growth, we shall roughly approximate the convolution integral in Eq. (3.33) by treating the two broadenings separately, i.e. we take... 

The convolution integral and the Exponential Piston Flow Model (EPM) were used to relate measured tracer concentrations to historical tracer input. The tritium input function is based on tritium concentrations measured monthly since the 1960s near Wellington, New Zealand. CFC and SF6 input functions are based on measured and reconstructed data from southern hemisphere sites. The EPM was applied consistently in this study because statistical justification for selection of some other response function requires a substantial record of time-series tracer data which is not yet available for the majority of NGMP sites, and for those NGMP sites with the required time-series data, the EPM and other response functions yield similar results for groundwater age. 

Where is a convolution integral and FT is the Fourier transform. The phase-contrast imaging performance of an HRTEM is controlled by sin y, which contains the basic phase-contrast sinusoidal terms modified by an envelope... 

The general case of non-uniform initial conditions and time-dependent boundary conditions can also be expressed analytically, and leads to convolution integrals in space and time. Nevertheless, for most practical applications, the assumptions of uniform initial conditions and constant boundary conditions are adequate. 

The function Q cannot be an operator that introduces multi-point terms such as gradients or space/time-convolution integrals. 

The CD model can also be written in terms of a convolution integral with respect to the joint composition PDF. However, its properties are more easily understood in terms of a mechanistic model for a mixing event. 

A somewhat more complex case is that of an integral equation that cannot be formulated in any closed form. This is a frequently encountered situation, the integral in the equation often being a convolution integral involving the linear diffusion function 1 / fwz, while the other side contains a function, F, of the function sought, ij/ 9... 

Calculation of the convolution integral in equation (6.5) may be performed as depicted in Section 2.2.8, leading to the results displayed in Figure 1.12. 

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