Wiener

English Culture and the Decline of the Industrial Spirit Martin J. Wiener... 

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transfonns. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization teclmique. This method has been generalized to charged hard spheres. 

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.

R. A. E, C. Paley and N. Wiener, Fourier Transforms in the Complex Domam, American Physical Society, New York, 1934. 

Hence, they take account of only the structure constitution (topology). One of the first and most frequently used topological indices is the Wiener index. It has the form of Eq. (3), where D,y are all the routes from atom i to atom j. 

Z eb index, Wiener index. Balaban J index, connectivity indices chi (x), kappa (k) shape indices, molecular walk counts, BCUT descriptors, 2D autocorrelation vector... 

For historical reasons the Wiener index, W, is introduced in this section. It was defined in 1947 and is still a starting point for the invention of new topological indices. 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

Because of the symmetry of the distance matrix, the Wiener index can be expressed as Eq. (16). 

With Eq. (16) the Wiener index of compound 2 can be calculated from the distance matrix as shown in Eq, (17)... 

Duduchava R., Wendland W. (1995) The Wiener-Hopf method for system of pseudodifferential equations with applications to crack problems. Integr. Eqs. and Oper. Theory 23, 294-335. 

Wiener, N. (1949) The Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley, New York. 

Wiener, E. L. (1985). Beyond the Sterile Cockpit. Human Factors 27(1), 75-90. 

Wiener, wienerisch, a. Vienna, Viennese. — Wiener Xtzpulver, Vienna paste, Vienna caustic. — Wiener Griin, Vienna green. — Wiener Kalk, Vienna hme, Vienna white. — Wiener Lack, Vienna lake. — Wiener Metall, Vienna metal (a white copper-antimony alloy). 

It is a well known fact, called the Wiener-Khintchine Theorem [gardi85], that the correlation function and power spectrum are Fourier Transforms of one another ... 

From the Wiener-Kinchine theorem, the integrated PSD yields the r.m.s. roughness vs the lenglh/scale / ... 

N. Wiener, Generalized Harmonic Analysis, Acta Mathematical 55, 117-258 (1930) ... 

As a further application of the Wiener-Khinchine theorem, we shall now calculate the power density spectrum of the shot noise process. The autocorrelation function for such a process is given by Campbell s theorem, Eq. (3-262), repeated below... 

N. Wiener, Cybernetics, The Technology Press and John Wiley A Sons, Inc., New York, 1948 Extrapolation, Interpolation, and Smoothing of Stationary Time Series, The Technology Press and John Wiley A Sons, Inc., New York, 1948. 

This is the autocorrelation and by the Wiener-Khintchine theorem the power spectrum of the disturbance is given by its Fourier transform,... 

Figure 2b. Profiles of the modulation transfer function (MTF), its inverse and Wiener inverse-filter.

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