First derivatives

Grangeat P. Mathematical framework of cone beam three-dimensional reconstruction via the first derivative of the Radon transform.. Math. Methods in Tomography, V.1947 of Springer Lecturre Notes in Math-cs, Springer-Verlag, Berlin, 1991, p.66-97. 

To find now the optimum excitation frequency, we calculate the first derivative of Equ. (3.1) to find the maximum value of the response field of cracks in different depths (Fig. 3,1). For example a crack (20 x 0.6 x 0,2 mm) in a depth of 9 mm in an aluminium sample (a = 20 MS/m) could be found with highest SNR when using a frequency of 260 Hz Here a double-D... 

Finally, the band pass filters corresponding to the Morlet wavelet have a "quicker" decrease towards null frequencies than filters obtained with the first derivative of gaussian wavelet (fig. 9). As a result, they... 

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

The filter according equation (1) allows a practical application of a second derivative, because it has only the noise amplification like a first derivative. This is shown in fig. 3 on a experimental data set. The SNR of the true second derivative is too low for correct edge detection, whereas the CT filter gives reliable results. 

N. B. a has the inverse role of a in the first derivative of a Gaussian. Deriche proposes the following recursive implementation of the filter/in two dimensions. Deriche retains the same solution as Canny, that is ... 

CO = coq, has a discontinuity in tire first derivative. In figure A1.3.18 the characteristic structure of the joint density of states is presented for each type of critical point. 

This equation was first derived by Zwanzig [72]. Note that p and X always occur together. Expanding about X = 0 at constant p (or equivalently about p = 0 at constant X) one finds... 

Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

Finally the concept of fields penults clarification of the definition of the order of transitions [22]. If one considers a space of all fields (e.g. Figure A2.5.1 but not figure A2.5.3, a first-order transition occurs where there is a discontinuity in the first derivative of one of the fields with respect to anotlier (e.g. (Sp/S 7) = -S... 

The exact quantum expression for the activated rate constant was first derived by Yamamoto [6]. The resulting quantum reactive flux correlation fiinction expression is given by... 

After cancelling out a factor p and regrouping, we obtain a new version of the Sclirodinger equation in which the first derivative temi has been eliminated. 

The Franck-Condon principle says that the intensities of die various vibrational bands of an electronic transition are proportional to these Franck-Condon factors. (Of course, the frequency factor must be included for accurate treatments.) The idea was first derived qualitatively by Franck through the picture that the rearrangement of the light electrons in die electronic transition would occur quickly relative to the period of motion of the heavy nuclei, so die position and iiioiiientiim of the nuclei would not change much during the transition [9]. The quaiitum mechanical picture was given shortly afterwards by Condon, more or less as outlined above [10]. 

Figure Bl.15.6. The EPR spectrum of tire perinaphthenyl radical in mineral oil taken at room temperature. (A) First derivative of the EPR absorption x with respect to the external magnetic field, Bq. (B) Integrated EPR spectrum.

The electronic energy W in the Bom-Oppenlieimer approxunation can be written as W= fV(q, p), where q is the vector of nuclear coordinates and the vector p contains the parameters of the electronic wavefimction. The latter are usually orbital coefficients, configuration amplitudes and occasionally nonlinear basis fiinction parameters, e.g., atomic orbital positions and exponents. The electronic coordinates have been integrated out and do not appear in W. Optimizing the electronic parameters leaves a function depending on the nuclear coordinates only, E = (q). We will assume that both W q, p) and (q) and their first derivatives are continuous fimctions of the variables q- and py... 

Note that there is no first derivative term in Eq. (87), because the first line of Eq. (81) ensures that (x S/04) x) = 0. 

This decomposition into a longitudinal and a hansverse part, as will be discussed in Section III, plays a crucial role in going to a diabatic representation in which this singularity is completely removed. In addition, the presence of the first derivative gradient term W l Rx) Vr x (Rx) in Eq. (15), even for a nonsingular Wi i (Rx) (e.g., for avoided intersections), introduces numerical inefficiencies in the solution of that equation. 

W (Rj.) is an n X n diabatic first-derivative coupling matrix with elements defined using the diabatic electronic basis set as... 

In the -electronic-state adiabatic representation involving real electronic wave functions, the skew-symmetiic first-derivative coupling vector mahix... 

As an example, in a four-electronic-state problem (n = 4) consider the electronic states i = 2 and f = 4 along with the first-derivative coupling vector element Wj4 (Rl) that couples those two states. The ADT matrix ui.4(qx) can... 

We want to choose the ADT matiix U(qx) that either makes the diabatic first-derivative coupling vector matrix W (Rx) zero if possible or that minimizes its magnitude in such a way that the gradient term Vr. x (Rx) in... 

The ADT matrix U(q ) obtained in this way makes the diabatic first-derivative coupling matrix that appears in the diabatic Schrodinger... 

Nevertheless, the residual first-derivative coupling term Vr does not... 

A perfect diabatic basis would be one for which the first-derivative coupling in Eq. (31) vanishes [10]. From the above mentioned... 

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