Quadratic

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

More accurately, as the inverse problem process computes a quadratic error with every point of a local area around a flaw, we shall limit the sensor surface so that the quadratic error induced by the integration lets us separate two close flaws and remains negligible in comparison with other noises or errors. An inevitable noise is the electronic noise due to the coil resistance, that we can estimate from geometrical and physical properties of the sensor. Here are the main conclusions ... 

We try to estimate the function H(u), noted H, by minimization of the quadratic residual error... 

More correctly, the regression problem involves means instead of averages in (1). Furthermore, when the criterion function is quadratic, the general (usually nonlinear) optimal solution is given by y = [p u ], i.e., the conditional mean of y given the observation u . 

Often, it is a fair approximation to tnmcate the series at the quadratic tenn with The energy levels are then approximated as... 

The first tenn is known as the hannonic contribution and the second tenn as the quadratic anhannonic correction. 

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

There are many physical systems which are modelled by Hamiltonians, which can be transfonned tln-ough a canonical transfomiation to a quadratic fomi ... 

This leads to a quadratic equation whose solutions are... 

Van der Waals (1890) extended his theory to mixtures of components A and B by introducing mole-fraction-dependent parameters a and b defined as quadratic averages... 

As we have seen, all the analytic coexistence curves are quadratic in the limit, so for all these analytic theories, tire exponent (3=1/2. 

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

In figure A3.3.9 the early-time results of the interface fonnation are shown for = 0.48. The classical spinodal corresponds to 0.58. Interface motion can be simply monitored by defining the domain boundary as the location where i = 0. Surface tension smooths the domain boundaries as time increases. Large interconnected clusters begin to break apart into small circular droplets around t = 160. This is because the quadratic nonlinearity eventually outpaces the cubic one when off-criticality is large, as is the case here. 

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

The bromate-ferroin reaction has a quadratic autocatalytic sequence, but in this case the induction period is detennined primarily by the time required for the concentration of the hiliibitor bromide ion to fall to a critical low value tlirough the reactions... 

Hizhnyakov V and Tehver I 1988 Transform method in resonance Raman scattering with quadratic Franck-Condon and Herzberg-Teller interactions J. Raman Spectrosc. 19 383-8... 

Since there is a definite phase relation between the fiindamental pump radiation and the nonlinear source tenn, coherent SH radiation is emitted in well-defined directions. From the quadratic variation of P(2cii) with (m), we expect that the SH intensity 12 will also vary quadratically with the pump intensity 1 ... 

In this approach [51], the expectation value ( T // T ) / ( T ) is treated variationally and made stationary with respect to variations in the C and. coeflScients. The energy fiinctional is a quadratic function of the Cj coefficients, and so one can express the stationary conditions for these variables in the secular fonu... 

Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke).

The simplest smooth fiuictioii which has a local miiiimum is a quadratic. Such a function has only one, easily detemiinable stationary point. It is thus not surprising that most optimization methods try to model the unknown fiuictioii with a local quadratic approximation, in the fomi of equation (B3.5.1). 

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

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