Wiener’s method

To our knowledge, the most comprehensive evidence in support of the IFP mechanism was in Lewis et al. [6] where the differences among three systems were shown (i) an IFP system, (ii) an IFP system with a small-molecule inhibitor added so that only diffusion and no front occurred, and (iii) a system with only monomer solution and no seed so that bulk polymerization occurred. The differences in these systems were shown using Wiener s method [48, 49], a laser sheet deflection technique (LLD) that illuminated changes in RI. The experimental system of this study contained a polymer seed and its monomer solution in a cuvette (Figure 5.8a, side view) [6]. To explain the results of this experiment, the authors first explained how LLD illuminated an IFP sample A low-power laser (about 8mW) passed... 

Figure 5.8 (a) Side and front views of Wiener s method (LLD). (b) Raw photo of an IFP system, (c) Plot of exiting light - incoming light vs... 

Figure 5.9 Wiener s method (LLD) illuminating an IFP sample (a), a sample with no seed that exhibits homogeneous polymerization (b), and an IFP sample with inhibitor added to prevent IFP (c). (Reprinted with permission from Ref. [5]. Copyright 2005 Wiley.)...

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

The wave lengths of electrons are so short that it is not possible to perform experiments like Young s or Wiener s experiments with them. The method by which Davisson established the wave aspect of electrons was similar to that used... 

The calculation method of Papirer and Brendle [178] is based on molecular topology and on calculation of Wiener s indices [179] as modified by Barysz et al. [180]. This method is based on the determination of an index that considers the molecular structure of the probe molecule. The calculation leads to a parameter Xt that represents a non-whole number of carbon atoms of a hypothetical alkane that would interact with a solid surface with the same intensity as the molecule that it represents. 

Wiener, M. C. and White, S. H. (1991b). Fluid bilayer stmcture determination by the combined use of x-ray and neutron diffraction II. Composition space refinement method. Biophys. J. 59 174. 

Cash, G.G., Klavzar, S. and Petrovsek, M. (2002) Three methods for calculation of the hyper-Wiener index of molecular graphs. J. Chem. Inf. Comput. Sci., 42, 571-576. 

Feller, K.L., Dammar and mastic infrared analysis. Science, 120 1069-1070 (1954). Shedrinsky, A.M., D. Grimaldi, T.R Wampler, and N.S. Baer, Amber and copal pyrolysis gas chromatographic (PyGC) studies of prevenance, Wiener Berichte fiber Naturwissenschaft in der Kunst, 6niS (1989/90/91), pp. 37—62. van Aarssen, B.G.K., J.W. de Leeuw, and B. Horsfield, A comparative study of three different pyrolysis methods used to characterize a biopolymer isolated from fossil and extant dammar resins, J. Anal. Appl. PyroL, 20 125-139 (1991). 

Baxter discussed the Wiener-Hopf factorization of the inverse structure factor for disordered fluids [182]. He was able to split OZ2 Eq. (Ill) into a pair of equations, which provide a route to determine c (R. Dixon and Hutchinson complemented Baxter s developments by stating conditions for proper R values (in simulation work with cubic boxes / co T / 2) and proposed a highly accurate minimization method to solve Baxter s equations [183]. These R proper values will be denoted by R and termed zeros hereafter. As proven by the present author in extensive calculations [96,103,155], there is always (at least) one zero when physically significant functions are analyzed. By following this procedure (BDH), except for very low densities, one normally obtains more than one R [Pg.111]

Glick SJ, Penney BJ, King MA (1991) Filtering of SPECT reconstruction made using Bellini s attenuation correction method a comparison of three pre-construction filters and a post reconstruction Wiener filter. IEEE Trans Nucl Sci 38 663-669... 

K. Ito, S. Qoi, Y. Kinura, and R. Hayakawa. New measurement method for the quasielastic light scattering with the sinusoidal electric field by use of an extended Wiener-Khinchin theorem. J. Chem. Phys., 101 (1994), 4463-4465. 

The first method is the use of the chain variables R s) directly, and we remember equation (86) for the probability of finding a conformation. If a crosslink fixes segment s from chain i to segment 5 j on chain j we have the constraint Ri s ) = Rj s)) permanently. The probability distribution of the network is given by the constraint averaged over Gaussian conformations of chains. In terms of the Wiener measure this probability can be written as... 

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