Projector technique

Projector technique (PT), one early application of optical techniques to membrane fouling visualization, utilizes a projector and screen to monitor fouling deposition. This technique was applied to measure biofilm thickness on the silicone rubber tubular membrane during wastewater filtration [63]. The thickness was then measured and photographed from direct visual observation on the projector screen. 

From this direct visualization, the biofilm layer was shown not to be uniform, but featuring filaments and extensions [63]. The resolution of the PT technique is 1 mm therefore it is unable to give information at the level of individual bacterial cell interactions. However, the PT technique can provide useful aggregate-level information about biofilm growing onto a tubular silicone membrane. 

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A. Rockenbauer and P. Simon, Perturbation solution of the spin Hamiltonian of low symmetry using the projector technique. Mol. Phys., 1974, 28(5), 1113-1126. 

The symmetry eigenvectors corresponding to the G36 acetone-like group are not so easily deduced by tryings and errors, from the character table, as in the case of the one-fold rotor molecules. The projector technique has to be used for [21,34] ... 

The quantitative determination of the magnitude of the tensor component is based on the explicit knowledge of the spatial dependence of the electronic wavefunctions. As pointed out in Section 2 the Z , w, II) functions transform as the basis functions of the irreps of the symmetry group G and can be obtained with the modified projector technique method [23] in tight binding approximation. The so obtained symmetrized wavefunctions preserve the transformation properties as dictated by the irreps of the symmetry group G. 

For example, the wavefunction o o), corresponding to the one dimensional irreducible representation of G and obtained with the projector technique in tight binding approximation, results to be fully symmetric under the symmetry transformations of G. 

Examples of pictures illustrating these techniques can be found in magazines or may be demonstrated on a chalkboard or with the use of an overhead projector. 

The wetting angle can be measured using simple techniques such as a projector, as shown schematically in Fig. 2.54. This technique, originally developed by Zisman [73], can be used in the ASTM D2578 standard test. Here, droplets of known surface tension, at are applied to a film. The measured values of cos are plotted as a function of surface tension, at, as shown in Fig. 2.55, and extrapolated to find the critical surface tension, ac, required for wetting. 

Wen and Fan [6] have provided a comprehensive listing of various tracers and experimental techniques for determining the RTD in flow systems. Recent studies [10,11,12] have been performed employing an impulse tracer to determine the RTD in bubble columns and an oscillatory flow electrochemical reactor. The author [13,14] has employed both step-change and an impulse to determine the RTD of nozzle type reactors analysis of the RTD involves an atomic absorption spectrophotometer (AAS), a cine-projector, and a chart recorder. Figures 8-7 and 8-8 show the nozzle-type reactors and the AAS, respectively. Figure 8-9 gives a typical response curve from the AAS. 

The Partitioning Technique.—Let P denote the projector onto some zero-order model wave function cP0> and Q its complement. The electronic Schrodinger equation... 

These experiments are based on techniques described by Dorfner and arc particularly suited to demonstration using an overhead projector. 

The appropriate technique is the following. First, one encodes the original information a 10) + 6 1) on the two logical states 0/,) = 000) and 1 l) = 111) of a three qubit system. This is simply achieved by adding two physical qubits, initially prepared in the state 0), and by applying some well-chosen unitary transformation C to the compound system of three qubits this operation yields the state a 0/,) I 6 1/,) which is then submitted to the action of the noisy channel. Each of the three physical qubits of the system is likely to independently undergo a bit flip (with probability p). At the end of the channel, one performs the measurement associated with the four projectors... 

Molecular optical potentials for non-reactive processes may be rigorously defined by means of partitioning techniques (see e.g. Feshbach, 1962), which are based on the classification of scattering channels in two groups the first one includes states which are asymptotically selected or detected, and is characterized by a projection operator P the second one includes all other states (in practice those to which flux is lost) and is characterized by the projector Q. An optical potential operator VH may then be constructed as... 

When a freezing curve is obtained, the values may vary at times, due to the nonlinearity of the temperature in the entire system, defects in the temperature detection, and so on. The determination of an actual curve to fit the experimental data presents a difficult problem. Various techniques, such as the use of a flexible spline, have been employed to draw this curve. An optical method, using a lantern projector, has been employed successfully by Saylor (34). Another method that has been suggested is that given by Kienitz (35). A hyperbola is constructed from certain values of time and temperature which best represent the measured curve. In this way, the freezing point of a sample is better obtained than with the analytical or geometrical methods of evaluation via three points on the equilibrium curve. 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT techniques [81]. PAW, however, provides all-electron one-particle waveflmctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in terms of a pseudo-wavefunction (easily expanded in plane waves) term that describes interstitial contributions well, and one-centre corrections expanded in terms of atom-centred functions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential formalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [82. 83]. PAW is also formulated to carry out DFT dynamics, where the forces on nuclei and wavefunctions are calculated from the PAW wavefunctions. (Another all-electron DFT molecular dynamics technique using a mixed-basis approach is applied in [84]. )... 

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