Maximum norm

Fig. 15.8 Example problem illustrating the iteration path for the hybrid Newton and time-integration approach (solid lines) and the time-marching approach alone (dashed line). The contours are for the maximum norm of residuals of the steady-state equations.

I. The maximum norm X, = max, jCj, which means that the norm is equal to the absolute value of that element x, whose absolute value is equal to or greater than that of any other element of X. 

Similarly, the minimization of the maximum norm of r, lll oo= maxi[Pg.180]

We convert every sample voice to the frequency series using fast Fourier transform (FFT) with 500 Hz as the initial frame rate [18] to work out the correlation coefficient between all possible combinations of the frequency series. Grubbs test (maximum normed residual test) indicates the states of correlation coefficients between a particular person s voice class and others as outlier. Therefore, we use frame rates to convert human voices to the frequency series by filtering and repeating the process while increasing 500 Hz windows until the frame rate is greater than 20 kFlz. 

J. J. H. Miller, E. O Riordan, and G. 1. Shishkin, Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimension. World Scientific Publishing, Singapore, 1996. 

A variant of the DGR method was suggested by Luo et al. (2010a) for the reduction of reaction mechanisms containing many isomers. Luo et al. recommended the application of the maximum norm instead of the... 

Because RPSA is appHed to gain maximum product rate from minimum adsorbent, single beds are the norm. In such cycles where the steps take only a few seconds, flows to and from the bed are discontinuous. Therefore, surge vessels are usuaHy used on feed and product streams to provide unintermpted flow. Some RPSA cycles incorporate delay steps unique to these processes. During these steps, the adsorbent bed is completely isolated and any pressure gradient is aHowed to dissipate (68). The UOP Polybed PSA system uses five to ten beds to maximize the recovery of the less selectively adsorbed component and to extend the process to larger capacities (69). 

In equation (9.131), sup is short for supremum, which means the final result is the least upper bound. Thus the //qo-optimal controller minimizes the maximum magnitude of the weighted sensitivity function over frequency range uj, or in mathematical terms, minimizes the oo-norm of the sensitivity function weighted by fE(jtj). 

Maximum rate of change alert The second alert (i.e., maximum rate of change alert) is used to automatically notify the operator that based on statistical data the rate of degradation has increased above the pre-selected norm. Since the vibration amplitudes of all machine-trains increase as normal wear occurs, the statistical rate of this normal increase should be trended. A drastic change in this rate is a major indication that a problem is developing. 

A typical feed rate is 1 to 3 ppm of active amine. Most filmer products contain between 3 and 10% amine (3-5% being the norm), so that product feed rates are around 10 to 30 ppm, dosed continuously to achieve 0.1 to 0.5 ppm amine (as ODA) in the returned condensate. However, products generally are diluted to 2 to 20% strength before application. Solutions are best prepared by using warm to hot condensate (150-160 °F/66-71 °C maximum) and agitated with a mechanical mixer until a uniform solution is reached. The solution should always be kept over 60 °F/16 °C. 

Stability of lOS. The main goal of stability consideration is to establish that the uniform convergence with the rate 0 t + /ip) follows from a summarized approximation obtained. This can be done using the maximum principle and a priori estimates in the grid norm of the space C for a solution of problem (21)-(23) expressing the stability of the scheme concerned with respect to the initial data, the right-hand side and boundary conditions. 

As known, the norm of the operator Tn is ecjual to the greatest eigenvalue max Aj,(Tn). Further replacements of Q/. and in (11) by continuous variables a and / lead to increased maximum of the right-hand side of (11) meaning... 

For H at T in Ge, Pickett et al. (1979) carried out empirical-pseudopotential supercell calculations. Their band structures showed a H-induced deep donor state more than 6 eV below the valence-band maximum in a non-self-consistent calculation. This binding energy was substantially reduced in a self-consistent calculation. However, lack of convergence and the use of empirical pseudopotentials cast doubt on the quantitative accuracy. More recent calculations (Denteneer et al., 1989b) using ab initio norm-conserving pseudopotentials have shown that H at T in Ge induces a level just below the valence-band maximum, very similar to the situation in Si. The arguments by Pickett et al. that a spin-polarized treatment would be essential (which would introduce a shift in the defect level of up to 0.5 Ry), have already been refuted in Section II.2.d. 

The period-specific throughput share tpPt Vr e R, t e T is the capacity volume normed on the available period time deducted by shut-down hours in relation to the maximum throughput basis. 

Model-free methods do neither supply absolute information about the concentrations or about the spectra. Essentially they only deliver the shapes for the profiles. In this and future examples, we normalise the concentration profiles in C to a maximum of 1 and adjust the species spectra of A in such a way that the product CA is correct. This is done in the function norm max, m. 

The subsequent computation of the absorption spectra A from C and Y is a simple linear regression. This is followed by the normalisation of the concentration profiles to a maximum of one, as has been outlined already in the preceding chapter Iterative Refinement of the Concentration Profiles. The normalisation is done using the routine norm max. m (p.275). 

Table Vlll lists the norms of internal, semi-internal, and external excitation amplitudes obtained in the CASSCF(6e, 5o, NO)/L-CTSD calculations with the 6-3IG basis set. The percentage of retained amplitudes for internal and semi-internal excitations are also shown in the table. The norm of the external amplitudes, which primarily contribute to dynamic correlation, does not fluctuate much across the potential curve. The maximums of the internal and semi-internal amplitudes are found at the intermediate bond region. More internal and semi-internal excitation operators are retained as the bond length ron is increased.

Normalizing so that the maximum intensiw is equal to 1 is accomplished by dividing each dement in the vector by the tnfinirj norm, defined as the maximum (in absolute value) of the vector. 

PROBLEM SHE STARTING POINT THRESHOLD ON FUNCTION NORM THRESHOLD ON STEP LEN6TH MAXIMUM NUMBER OF ITERATIONS... 

As illustrated, here a single variable (the maximum temperature) is chosen as a characteristic function of the solution. For the premixed twin flame, this is a good choice. However, in other circumstances, like an opposed-flow diffusion flame, the choice of a characteristic scalar is less clear. Vlachos avoids the need for a choice by using a norm of the full-solution vector to characterize the solution in the arc length [415,416], The Nish-... 

On the whole, as a result of the coupled effect of climatic and anthropogenic impacts, the salinity maximum in the Sea of Azov in 1975-1977 exceeded the natural norm by up to 3.0 psu and, in Taganrog Bay, even more. Beginning from 1978, the regime of the sea reached its new water-rich phase, the mean annual salinity of the sea acquired a tendency to fall, and, in 1980s, it comprised 11-12 psu. By 2000 salinity lowered even to 10-11 psu. 

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