Moving point method

C.L. Farmer (1985) A moving point method for arbitrary Peclet number multidimensional convection-diffusion equations. IMA Journal of Numerical Analysis 5, 465-480. 

C.L. Farmer (1987) Moving point methods. Advances in Transport Phenomena in Porous Media, J. Bear and M.Y. Corapcioglu (eds.), NATO ASI Series E Applied Sciences no. 128, Martinus Nijhoff, Dordrecht, 953-1004. 

Transport numbers are intended to measure the fraction of the total ionic current carried by an ion in an electrolyte as it migrates under the influence of an applied electric field. In essence, transport numbers are an indication of the relative ability of an ion to carry charge. The classical way to measure transport numbers is to pass a current between two electrodes contained in separate compartments of a two-compartment cell These two compartments are separated by a barrier that only allows the passage of ions. After a known amount of charge has passed, the composition and/or mass of the electrolytes in the two compartments are analyzed. Erom these data the fraction of the charge transported by the cation and the anion can be calculated. Transport numbers obtained by this method are measured with respect to an external reference point (i.e., the separator), and, therefore, are often referred to as external transport numbers. Two variations of the above method, the Moving Boundary method [66] and the Eiittorff method [66-69], have been used to measure cation (tR+) and anion (tx ) transport numbers in ionic liquids, and these data are listed in Table 3.6-7. 

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

Product appearance was unremarkable. The pH was examined using a control chart. Because this is a single point observation, the moving range method was employed. The chart disclosed that the process operates within the calculated control limits. No trends were apparent. Individual batch results all met specification, and the process average (4.07) is close to the target value of 4.10. (See Fig. 11.)... 

In the practical application of the moving boundary method one boundary only is observed, and so the necessity of finding two indicator solutions is obviated the method of calculation is as follows. If one faraday of electricity passes through the system, equiv. of the cation must pass any given point in one direction if c equiv. per unit volume is the concentration of the solution in the vicinity of the boundary formed by the M ions, this boundary must sweep through a volume t /c while one faraday is passing. The volume swept out by the cations for the pa.ssage of Q coulombs is thus... 

The principal numerical problem associated with the solution of (7) is that lengthy calculations are required to integrate several coupled nonlinear equations in three dimensions. However, models based on a fixed coordinate approach may be used to predict pollutant concentrations at all points of interest in the airshed at any time. This is in contrast to moving cell methods, wherein predictions are confined to the paths along which concentration histories are computed. 

In the development of the moving boundary method it was pointed out by early workers that the electrochemical reactions which occur at the electrodes on the passage of current are accompanied by volume changes which may affect the observed displacement of the boundary. Denison and Steele considered the effect to be negligible, though their conclusion was subsequently shown to be incorrect by Lewis20 whose discussion of the matter is essentially that given below. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

The moving edge method provides an opportunity widely used. The sample under investigation has to be substituted by an edge, normally a razor blade (Fig. 5). The razor blade is scanned through the focus area in the x-direction and the transmitted energy for a fixed position of the razor blade (x=l, one measuring point) is determined with a detector. The coordinate system (x,y) is depicted in Fig. 6. 

Most of the Monte Carlo moves discussed up to this point involve the localized rearrangement of one or at most all sites of a single molecule. In contrast to such moves, collective methods involve simultaneous rearrangement of several or all molecules in the system. 

First-order analysis by use of the jSxed observation point method is precluded in such a case, and — d[R]/dt must be found by measurement of [R] along the axis of the flow tube at a series of points from the origin of the reaction zone. This may be accomplished for absorption spectrophotometry by moving the detection system on a railway parallel to the tube axis (Fig. 4.4) or, also, in some cases, by using a moveable inlet (or a series of fixed inlets) to enable the reaction origin to be shifted relative to a fixed detector, as in e.p.r. spectrometry (Fig. 4.3). 

Although two pairs of data points from a TSC spectrum are required for AEM-u, at least three points satisfy the requirement by using one point in common for both pairs. The convenient method to select data by its temperature coordinate, is shown in Fig. 3. In the method, named "moving coordinates method", two pair a-b and b-c) of points at a same temperature separation (AT), as shown in Fig.3, is selected, holding a point at temperature (Tb) in common and inputting their coordinates into the calculation. The Et and v values at... 

Electrofocusing makes it possible to measure the pi of a protein with great accuracy. This fundamental characteristic of a protein can be demined with better controlled and more stable conditions in the surroundings than in early experiments, where isoelectric points were measured with the help of the moving boundary method. 

When working by visual information, it is necessary to grasp what you have in the outside world at sight, and a result of recognition must be reflected by an action. A two-dimensional moving point with one camera mounted on a robot was chased [1-3]. But, two cameras are necessary in the case of three dimensions. Binocular disparity of a base-line stereo that both optical axes are parallel is usually used [4]. Gaze control is a method to determine an object on a intersection point of optical axes of both cameras and makes it possible to always cateh an observation objeet at the center of a field of vision [5, 6]. This method has coneentration of gaze... 

Simplex methods are based on the idea of moving from one vertex to the adjacent one, while Interior Point methods are based on the idea of generating a barrier that prevents constraints being approached. 

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