Wiener series

They differ from the kernels it (ti, ..., r ) of the Volterra series only by a faster signal decay with increasing time arguments [Bliil]. For coinciding time arguments the crosscorrelation function is the sum of the n-dimensional impulse-response function h with the impulse-response functions hm of lower orders m < n. The stochastic impulse-response functions h are the kernels of an expansion of the system response y(t) similar to the Volterra series (4.2.4) but with functionals orthogonalized for white-noise excitation x t) [Bliil, Marl, Leel, Schl], This expansion is known by the name Wiener series, and the h are referred to as Wiener kernels. 

Schl M. Schetzen, The Volterra and Wiener Series of Nonlinear Systems, Wiley, New York, 1980. 

The inability to estimate the Volterra kernels in the general case of an infinite series prompted Wiener to suggest the orthogonalization of the Volterra series when a GWN test input is used. The functional terms of the Wiener series are constructed on the basis of a Gram-Schmidt orthogonalization procedure requiring that the covariance between any two Wiener functionals be zero. The resulting Wiener series expansion takes the form ... 

The orthogonality of the Wiener series allows decoupling of the various Wiener functionals and the estimation of the respective Wiener kernels from input-output data through cross-correlation [Lee and Schetzen, 1965]... 

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

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. 

An original formalism for the treatment of many-particle effects in the A + B — B reaction was developed in a series of papers by Berezhkovskii, Machnovskii and Suris [54-59]. It is based on the so-called Wiener trajectories and related the Wiener sausages concept (the spatial region visited by a spherical Brownian particle during its random walks) [55, 60, 61]. It was shown that the convential survival probability for a walker among traps, which could be presented in a form [47]... 

Fig. 4. Plot of relaxivity versus molecular weight for a series of gadolinium based MRI contrast agents (circles Wiener et al. [32] diamonds Aime et al. [35] triangles Martin et al. [4] squares Ranganathan et al. [36])...

Hanson, Rouvray 1987 Topological indices Balaban, Wiener, and carbon number Homologous series... 

Relation to a Noise Current. The fluctuating polarization results from a movement of charge in the dielectric medium. In a short-drcuited condenser, the polarization changes induce a corresponding current /(/) in the external drcuit. The mean square current G y) per unit frequency bandwidth at frequency v can be related to a series of observations of /(/) by the important Wiener-Khinchin theorem... 

The topological substituent indices 5a, 5b, 5ab have been used to model the biological activity of series of congeneric compounds in the models, the Wiener index Wm of the parent structure can be neglected as it represents the constant term due to the main bulk of the molecules. These substituent indices, unlike the - substituent constants, can be calculated easily moreover, they depend on the site at which the substitution takes place and the interaction index 5ab overcomes the additivity scheme of the substituent constants. 

The factor in front of the sum term ensures that bond paths are only counted once. The Wiener index is particularly useful to describe quantitative structure-property relationships (QSPR). An example is the correlation of saturation vapor pressure with the chemical structure. Since the Wiener index does not distinguish atom types, those correlations can only be achieved for homologues series of compounds. 

The Wiener index was the first proposed index of molecular branching [Bonchev and Trinajstic, 1977] it is a function, inversely related to branching, of the number, length, and position of branches as well as of the number of atoms. For an isomeric series, it can be considered mainly dependent on molecular branching. Other specific molecular descriptors... 

The 1950s saw a change in emphasis from the analysis of bio-chemistry-as-kinetics to that of biochemistry-as-information. The theoretical rationale for this transition was provided by the growth of the new sciences associated with the development of computers. Theories of control , feedback , and information transfer were collated in 1948 by the American engineer and mathematician Norbert Wiener under the name of cybernetics. As more and more became known about the mechanisms of individual enzymic reactions, about their energy-requirements, and about the workings of series of enzymes in the harmony of metabolic pathways, biochemists seized on these new concepts in order to probe the ways in which the cell controlled and regulated its own metabolism how, so to speak, it decided at any one time... 

A. S. Mikhailov. Modeling pattern formation in excitable media The legacy of Norbert Wiener. In J. Milton, P. Jung (eds.), Epilepsy as a dynamic disease. Biological and Medical Physics Series. Springer, Berlin Heidelberg, 125-163, 2003. 

Wiener, N. (1950) Extrapolation, interpolation and smoothing of stationary time series, Wiley Sons, New York. 

Gugliotta, L. Alba, D. Meira, G. Correction for instrumental broadening in SEC through a stochastic matrix approach based on Wiener filtering theory. In Detection and Data Analysis in Size Exclusion Chromatography ACS Symposium Series No. 352 Provder, T., Ed. American Chemical Society Washington, 1987 287-298. 

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