Convolution approximation

The interfacial fluctuations broaden laterally averaged profiles. Within the convolution approximation ( B3.6.15) one obtains a profile with the shape of the erfc fimction [49] ... 

Expression (2.14), referred to as the convolution approximation, is widely applied in crystallographic work. 

With the aid of Eq. (48), we can show that 6ik (o) = (k + l)N(co) for t(co) = 0. The object estimate consists of noise at frequencies that t does not pass. The noise grows with each iteration. This problem can be alleviated if we bandpass-filter the data to the known extent of z to reject frequencies that t is incapable of transmitting. Practical applications of relaxation methods typically employ such filtering. Least-squares polynomial filters, applied by finite discrete convolution, approximate the desired characteristics (Section III.C.5). For k finite and t 0, but nevertheless small,... 

Szabo [170]. Berg assumed that at any moment of dissociation a newborn A-B pair is surrounded by the equilibrium distribution of B particles. As a result, this theory works well near the limit, when the dissociation events are rare and the trap is mostly bound (cKeq 1). At the same time, it correctly reduces to the opposite, geminate limit, when c = 0 and an isolated A-B pair is formed as a result of the decay of C (Section V.C). There are also other authors who started later from the same assumption and obtained the same result [170,177,245]. In Ref. 170 it was called the convolution approximation (CA). 

In practice, it is important, of course, to take into account that the width of the interface is not due to capillary waves alone, but also the intrinsic width needs to be accounted for. This can be done by a convolution approximation [272,273] for the total profile ( >(z) of the order parameter,... 

The importance of assessing the inaccuracies introduced by the "convolution approximation" (CA), defined in Sec. 1.3, was discussed by Chatzidimitrou-Dreismann et al. [Chatzidimitriou-Dreismann 1997 (a)] in the first published measurement of cross section anomalies on VESUVIO in H2O / D2O mixtures. In order to eliminate the possibility that the CA could be responsible for the observed anomalies in this system, two independent checks were made. 

We have also clearly shown by employing experimental tests that recently published criticisms of the data analysis are unjustified. In addition to previous work [Chatzidimitriou-Dreismann 2002 (b) Abdul-Redah 2003] we have shown here that the convolution approximation criticized earlier [Blostein 2001] is correct and justified. We also demonstrated here that the neutron incident energy spectrum as well as all involved Jacobians - which have been considered as possible explanations for the anomalies [Cowley 2003] - are accounted properly in the data analysis procedure on VESUVIO. 

The coupled-channel calculations are used as benchmark results to check simple models of the impact parameter dependence of the electronic energy loss. A detailed description of such models (convolution approximation) may be found elsewhere [25,26]. Here we present only a short outline of the method. The electronic energy loss involves a sum over all final target states for each impact parameter. Usually this demands a computational effort that precludes its direct calculation in... 

Thus, the even orders of an Zp expansion, as included in the unitary convolution approximation (UCA), dominate the non-perturbative efifects. The present UCA results are plotted as a solid curve. This curve lies close to the average of the AO results for particles and antiparticles. Hence, although the present UCA does not include sign-of-charge efifects it perfectly describes the majority of the energy transfer processes (dominated by ionization) of fast heavy particles at small impact parameters. 

The harmonic-convolution approximation, according to which the density units rigidly follow the motion of the center to which they are attached, ensures a closed analytic form for the structure factor ... 

Each E (l,2) in Eq. (17) is an integrand whose structure can be represented by an "elementary diagram with base points 1,2, say. For the precise definition of these diagrams the reader is referred to Reference 30. Because of the form of E r), Eq. (16), while exact, apart from some questions of convergence of the series for (r), hardly provides a practicable method of determining g r), since many if not all of the terms of E(r) have to be computed. E (r) has so far had to be drastically approximated. The simplest procedure is to set E(r) = 0 in Eq. (16) [E[r) contains no term linear in p this yields the so-called hyper-netted chain or convolution approximation. In a second approximation, the E[i) approximation, the first "elementary diagram having two... 

The third approximation conventionally employed is the convolution approximation [39], under which the application of P2 to R k) given in... 

Spectral relaxation rates for nondilute solutions are affected by interchain interactions. Studies of S q. t) of nondilute polymer solutions may be traced back to the seminal study of Bueldt(15), who showed for smaller c and q that Dm is linear in c - a result extensively confirmed below - and independent of At larger c. Dm at large q increased more rapidly than q. Bueldt proposed dividing the / and q dependences of S q. t) into the time dependences of the self- and distinct parts of S(, t). A rationale based on the Vineyard convolution approximation(16) then plausibly explained why both parts of S q, t) are governed by the same diffusion coefficient. Much further work, whether inspired by this study or not, has expanded on these findings. 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

Due to the convoluted mass and depth scales present in an RBS spectrum, it may not be possible to accurately describe an unknown sample using a single RBS spectrum. For example. Figure 4a is an RBS spectrum acquired at a backscattering angle of 160° from a sample implanted with 2.50 x 10 atoms/cm of As at a depth of approximately 140 nm. If this were a totally unknown sample it would not be possible to determine positively the mass and depth of the implanted species from this spectrum alone, since the peak in the RBS spectrum also could have been caused by a heavier element at greater depth, such as Sb at 450 nm, or Mo at 330 nm, or by a... 

It is seen that however sophisticated the software might be, it would be virtually impossible to de-convolute the peak into the three components. The peaks shown in the diagram are discernible because the peaks themselves were assumed and the composite envelope calculated. The envelope, however, would provide no basic data there is no hint of an approximate position for any peak maximum and absolutely no indication of the peak width of any of the components. The use of the diode array detector, monitoring at different wavelengths, might help by identifying uniquely one or more of the... 

In many macroscopic systems, the massive behavior is a convoluted answer to many microscopic features of the system. For example, the catalysis of the electrooxidation of an organic molecule may be generated by some local arrangement of atoms on a catalyst, defined at the atomic level. If some hypotheses are available to explain the enhancement of the reaction, this can be checked by inserting these hypotheses in the model. In a first approximation, a qualitative explanation is often sought. If this is... 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

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 treatments of Kochendorfer, Porod, and Warren-Averbach identify superposition with the mathematical operation of a convolution. While this is true for translational superposition, for dilational superposition it is a coarse approximation that is only valid for small polydispersity. In the latter case the convolution must be replaced by the Mellin convolution (Eq. (8.85), p. 168) governed by a dilation factor distribution and the structure of the reference crystal, the structure of each observed crystal is generated by affine dilation of the reference crystal (Stribeck [2]). 

Therefore we now turn to the noise part of the S/N ratio. As we saw just above, the two-point derivative approximation can be put into the framework of the S-G convolution functions, and we will therefore not treat them as separate methods. 

Thus the spatial rate of energy loss is (q)/A. The stopping power is actually the mean value of the ratio of energy loss to path length, and to this extent the derivation is an approximation. The path length distribution Pjx) in n collisions may be given as a convolution—that is,... 

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