Weighted sum

According to some remarks concerning the physical interaction between the incident ultrasonic wavelet and the defects [4-6], we consider that an Ascan signal, may be described as a weighted sum of few delayed and phase-shifted replicas of the ultrasonic incident wavelet j(r). We can express this mathematically as ... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

How is it that the modulus can be less Suppose we had loaded the composite in the opposite way, at right angles to the fibres (as in Fig. 6.3(b)) It now becomes much more reasonable to assume that the stresses, not the strains, in the two components are equal. If this is so, then the total nominal strain e is the weighted sum of the individual strains ... 

The modulus of the polymer is an average of the stiffnesses of its bonds. But it obviously is not an arithmetic mean even if the stiff bonds were completely rigid, the polymer would deform because the weak bonds would stretch. Instead, we calculate the modulus by summing the deformation in each type of bond using the methods of composite theory (Chapter 25). A stress d produces a strain which is the weighted sum of the strains in each sort of bond... 

The activation function/(.v) (where. v is the weighted sum) can take many forms, some of which are shown in Figure 10.21. From Figure 10.21 it can be seen that the bias bj in equations (10.54) and (10.55) will move the curve along the. v axis, i.e. effectively... 

It is also possible to determine A and B that minimize the weighted sum of squares. The weighting funetion is the square of the independent variable, and the funetion to be minimized is... 

Unconditional Probability (weighted sum) that Quality of Training is high vs. low is . 254. 746 ... 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

For the paramagnetic case the expre.ssion for the photo current in Eq. (2) can be simplified to a concentration weighted sum over the products of the K-resolved partial density of states (DOS) ri (F) and a corresponding matrix element that smoothly varies with energy [13]. This simple interpretation of the XPS-spectra essentially also holds for the more complex spin-resolved ca e in the presence of spin-orbit coupling as studied here. 

Figure 2 Orbital magnetic moments in bcc-Fe Coi-a . The triangles pointing up-and downwards represent the theoretical moments of Fe and Co, respectively, while the concentration weighted sum is given by circles. Full and open symbols stand for results obtained with and without the OP-term included (SOPR- and SPR-KKR-CPA, resp.). Experimental data [15] for the average magnetic moment (bottom) stemming from magneto mechanical and spectroscopic g-factors are given by full squares and diamonds.

A weighted sum of C,E curves for the faces was found to be similar to the C,E curve for a pc electrode. According to Valette and Hamelin,67 all main Ag faces [(111), (100), and (110)] are exposed on the surface, their fractions 0j on the surface being 0.31, 0.23, and 0.46, respectively. These authors demonstrated that the diffuse-layer capacitance minimum potential E a of a pc-Ag electrode was only slightly less negative (30 mV) than the pzc of the Ag(110) face, i.e., for the face with the more negative value of EamQ. The diffuse-layer capacitance minimum for pc-Ag was wider and less deep than for the Ag faces. 

It must be stressed that every Un(t,J) brings dq(0) to a different coordinate system. Consequently, the averaged operator (A7.13) is actually a weighted sum of the quantities in differently oriented reference systems. It can nevertheless be used to find the scalar product (A7.7), that is the orientational correlation function. 

The concept of a well-stirred segregated reactor which also has an exponential residence time distribution function was introduced by Dankwerts (16, 17) and was elaborated upon by Zweitering (18). In a totally segregated, stirred tank reactor, the feed stream is envisioned to enter the reactor in the form of macro-molecular capsules which do not exchange their contents with other capsules in the feed stream or in the reactor volume. The capsules act as batch reactors with reaction times equal to their residence time in the reactor. The reactor product is thus found by calculating the weighted sum of a series of batch reactor products with reaction times from zero to infinity. The weighting factor is determined by the residence time distribution function of the constant flow stirred tank reactor. 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

In a steady state, there can be no net isofractionation in the excreted components (by definition) the mass-weighted sum of all the 5 values for the excreted material must equal that of the diet. It so happens that what is excreted very often is dominated by one component. 

Intermediate liquid 8 values are obtained by mixing liquids of known solubility parameter SPS makes use of this. The 8 value of the mixture is equal to the volume-weighted sum of the individual component liquid 8 values. Thus, the mass uptake of a miscible liquid mixture by an elastomer may be very much greater than the swelling which would occur in the presence of either one of the constituent liquids alone. The mixture could of course comprise more than two liquid components, and an analogous situation would apply MERL have applied this approach for the offshore oil-production industry to allow realistic hydrocarbon model oils to be developed,basically by mixing one simple aliphatic (paraffinic) hydrocarbon, one naphthenic, and one aromatic to proportions that meet two criteria, namely, that... 

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