Differential techniques

Variational methods - theoretically the variational approach offers the most powerful procedure for the generation of a computational grid subject to a multiplicity of constraints such as smoothness, uniformity, adaptivity, etc. which cannot be achieved using the simpler algebraic or differential techniques. However, the development of practical variational mesh generation techniques is complicated and a universally applicable procedure is not yet available. 

Flow injection analysis (FIA) (Ruzicka and Hansen), since 1975 In continuous flow, stopped flow or with merging zones (FIA scanning or intermittent pumping) Adapted voltammetric electrodes Membranes for Partial dialysis Membrane amperometry (Clark) Differential techniques (Donnan) Computerization, including microprocessors Special measuring requirements in plant control (to avoid voltage leakage, etc., Section 5.5)... 

The differential technique described under (a) has an advantage in removal of the liquid-junction potential and of mechanical faults often encountered in work with reference electrodes of the second kind. The procedure described under (b) suppresses the potential fluctuations, but difficulties can arise from the very high resistance of a cell containing two ISEs. A differential amplifier was designed for this prupose [15]. The two ISEs used can also exhibit different slopes electrode membranes were therefore prepared by cutting a single crystal into two halves, where each half contains a chaimel for passage of the solution and functions as an ISE [163]. 

J. McAndless Does it require use of a differential technique computer based program to subtract out the C-13 NMR rubber... 

A differential technique in mathematics used to obtain information about the composite function f(x)/g(x) where/(x) and g(x) are functions of x. One version of the method requires that both functions are differentiable, symbolized by f (x) and g (x), at every point x greater than a certain fixed M > 0 and that g (x) = 0. If limit (f (x)/g (x)) exists and has a certain value, then... 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

Ziemke, J.R, S. Chandra and P.K. Bhartia (1998) Two new methods for deriving tropospheric column ozone from TOMS measurements Assimilated UARS MLS/HALOE and convective-cloud differential techniques Journal of Geophysical Research 103 22115-22127. 

When the characteristic dimension of the electrode fulfills that Q -C fnDt (with qG being equal to rd, rs, w, or rc for discs, spheres, bands, or cylinders, respectively), stationary or pseudo-stationary conditions are attained. Under these conditions, the expressions for the signal obtained in SWV as well as those corresponding to differential techniques DSCVC and DMPV previously discussed simplify greatly. In the case of discs or spheres, a true steady-state response can be obtained (see Eqs. (7.20) and Table 7.1) ... 

Gas-phase photoacoustic detection of propane in N2 was attempted on a Si chip. In this method, a modulated light beam is incident on the sample. If the wavelength of the modulated light couples to an energy transition in a gas, the gas absorbs the modulated light resulting in periodic gas expansions and contractions, which are manifested as an acoustic wave. This wave can be detected by a microphone. In contrast to conventional absorption spectroscopy, the sensitivity of photoacoustic spectroscopy scales inversely with dimension, and hence this method is favored in the microscale. This is because photoacoustic spectroscopy is a differential technique in which the absorption is measured as the intensity per unit surface area [796],... 

Square-wave voltammetry is a large-amplitude differential technique in which a waveform composed of a symmetric square wave, superimposed on a base staircase potential, is applied to the working electrode (8) (Fig. 3.9). The current is sampled twice during each square-wave cycle, once at the end of the forward pulse (at h) and once at the end of the reverse pulse (at t2). Since the square-wave modulation amplitude is very large, the reverse pulses cause the reverse reaction of the product (of the forward pulse). The difference between the two measurements is plotted versus the base staircase potential. 

Hwang D, Byun DW, Odman MT (1997) An automatic differentiation technique for sensitivity analysis of numerical advection schemes in air quality models. Atmospheric Environment, 31(6) 879-888. 

Two experimental runs were performed. The H2S- and CO2 mole fluxes were obtained from the measured concentration curves by numerical differentiation and are plotted in figure 8a,b together with penetration and film model calculations. It is evident that forced desorption can be realized under practical conditions and can be predicted by the model. In general, measured H2S mole fluxes are between the values predicted by the models, whereas the CO2 forced desorption flux is larger than calculated by the models. The CO2 absorption flux, on the other hand, can correctly be calculated by the models. This probably implies that the rate of the reverse reaction, incorporated in equation (5), is underestimated. Moreover, it should be kept in mind that especially the results of the calculations in the forced desorption range are very sensitive to indirectly obtained parameters (diffusion, equilibrium constants and mass transfer coefficients) and the numerical differentiation technique applied. 

Since DPV is a differential technique, the response is similar to the first derivative of a conventional voltammogram, that is a peak. The peak potential, Ep, can be approximately identified with Em. With increasing irreversibility Ep moves away from Em (reversible system), at the same time as peak width increases and its height diminishes. The degree of reversibility of an electrode reaction is similar to that observed in NPV, since the timescale is the same. 

At the dropping mercury electrode the fact that DPV is better than NPV is due to the residual capacitive current contribution, which is subtracted out in the differential technique. It is relatively easy to demonstrate that the diminution factor, f, is given by... 

At solid electrodes the better response from DPV than from NPV is clear in many cases, especially involving organic compounds. As these often lead to adsorption on the electrode, it is possible that a differential technique discriminates against effects that are moreless constant before and after pulse application. 

L. B. Rail, Automatic Differentiation—Techniques and Applications, Lecture Notes in Computer Science 120, Springer-Verlag, Berlin/New York, 1981. 

The heat o( vaporization at a temperature T may be estimated from vapor pressure data by plotting In p versus l/T, determining [rf(ln p )/d l/T) at the temperature of interest as the slope of the tangent to the curve, and solving Equation 8.4-7 for AHy. The slope may be determined graphically or by any of several numerical differentiation techniques outlined in texts on numerical analysis. 

Xiao H, Levine SP. 1993. Application of computerized differentiation technique to remote-sensing Fourier transform infrared spectrometry for analysis of toxic vapors. Anal Chem 65(17) 2262- 2269. 

The precision of spectrophotometric methods [27-29] depends on the range of the values determined. Under optimum conditions, it ranges usually from 0.5-2%. In the differential technique, precisions ranging from 0.2 to 0.5% are attained. 

Zirconium has been determined by the Alizarin S method in uranium alloys [45], titanium alloys [92], magnesium alloys [27], and rutile [3]. Determination of Zr by the differential technique has also been applied [44]. 

Calculate the exeess molar volume for the solution and partial molar volumes of the two eomponents at mole fractions equal to 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Use the appropriate numerical interpolation and differentiation techniques (see appendix C). 

These are very difficult experiments because the pressures in the G-LE region are very low and great care must therefore be taken to avoid even minute amounts of surface-active impurities, which can cause large drifts in the pressure and erroneously high pressure readings. Kim and Cannell employed a differential technique in which the pressure was measured with respect to that of a monolayer maintained at a fixed area in the two-phase region their results are shown in Fig. 4. 

Fig. 17. Diagrams a-c illustrate different experimental methods for measuring the dichroism needed to calculate the orientation function - see text for discussion of (a) two spectra technique, (b) differential technique and (c) waveplate technique...

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