Shannon equation

The Shannon Equation Eq. (1)) [4] enables one to evaluate the information content, I (also known as the Shannon entropy), of the system. 

In order to use the orbital distribution of the graph connections as measure for molecular complexity, Bertz modified the Shannon equation (i.e., the Mowshowitz 80) variant) adding the term N lb N ... 

Information-Theoretical Indices Information theory has been employed to define topological indices based on the Shannon equation [17] ... 

Atom uniqueness or redundancy is certainly a shape attribute that must play some role in the influence of structure on function. One approach to the quantitation of uniqueness is the use of the Shannon equation for information content,which has been studied quite thoroughly by Brillouin and Bon-chev. " Kier has made use of the equation to encode molecular uniqueness, or negentropy, and to relate to biological and physical properties. 

Two interrelated values may be derived from the Shannon equation. The first of these is 1, the information content per atom ... 

Shannon s method, expands a Boolean function of n variables in minterms consisting of all combinations of occurrences and non-occurrences of the events of interest. Consider a function of n Boolean variables XJ which may be expanded about X, as shown in Equation 2.2-3 where f(l, Xj,..., XJ where 1 replaces X,. This says that a function of Boolean variables equals the function with a variable set to I plus the product of NOT the variable limes the function with the variable set to 0. By extending Equation 2.2-3, a Boolean function may be expanded about all of its... 

Shannon s expansion is demonstrated for two, 3-variable functions in Equation 2.2. 5 which equates the first function to its expansion. Equations 2.2-6a-g evaluate the functions (or all combinations of values for X, Y, and Z Equation 2.2-7 equates the second function to its expansion, and equations 2.2-8a-g evaluate the functions for all combinations of values for X, Y, and Z. 

Given an object com[)osed of N interconnected and interacting [)arts, one might at first be te,m[)ted to equate the complexity of an object with its conventional information content, as defined l)y Shannon [.shann49] ... 

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

Figure 24. Lattice strain model applied to zircon-melt partition coefficients from Hinton et al. (written comm.) for a zircon phenocryst in peralkaline rhyolite SMN59 from Kenya. Ionic radii are for Vlll-fold coordination (Shannon 1976). The curves are fits to Equation (1) at an estimated eraption temperature of 700°C (Scaillet and Macdonald 2001). Note the excellent fit of the trivalent lanAanides, with the exception of Ce, whose elevated partition coefficient is due to the presence of both Ce and Ce" in the melt, with the latter having a much higher partition coefficient into zircon. The 4+ parabola cradely fits the data from Dj, and Dy, through Dzi to Dih, but does not reproduce the observed DuIDjh ratio. We speculate that this is due to melt compositional effects on Dzt and (Linnen and Keppler 2002), and possibly other 4+ cations, in very silicic melts. Because of its Vlll-fold ionic radius of 0.91 A (vertical line), Dpa is likely to be at least as high as Dwh, and probably considerably higher.

The first numerical study on the transient flow of a single liquid droplet impinging onto a flat surface, into a shallow or deep pool was performed by Harlow and Shannon)397 In their work, the full Navier-Stokes equations were solved numerically in cylindrical... 

This equation was first defined and used in a non-thermodynamic context by Shannon [1] when working with problems of the capacity of communication channels, and the transmission of signals down noisy lines. Suppose we have m constraints, expressed as expectation values, ),... 

Brown (1970) calculated the following equation, which is valid for olivine compounds (ionic radii of Shannon and Prewitt, 1969) ... 

Table 1.9 Parameters relating covalence to bond strength and bond length in equations 1.48 and 1.50 (adapted from Brown and Shannon, 1973).

Solving these equations for Y3Fe2pe30i2, taken as an example of a garnet with a well-refined structure Shannon and Prewitt s table for bond lengths yields the following parameters (observed values are in parentheses) ... 

One procedure for recovering the continuous (band-limited) function exactly is provided by the Whittaker-Shannon sampling theorem, which is expressed by the equation... 

Boltzmann s tombstone in Vienna bears the famous formula 5 = k log W (W = Wahrscheinlichkeit—probability) that was a signature of his audacious concepts. The alternative formula (13.69) (which reduces to Boltzmann s in the limit of equal a priori probabilities pa) was ultimately developed by Gibbs, Shannon, and others in a more general and productive way (see Sidebar 13.4), but the key step of employing probability to trump Newtonian determinism was his. Boltzmann was long identified with efforts to establish the //-theorem and Boltzmann equation within the context of classical mechanics, but each such effort to justify the second law (or existence of atoms) in the strict framework of Newtonian dynamics proved futile. Boltzmann s deep intuition to elevate probability to a primary physical principle therefore played a key role in efforts to find improved foundation for atomic and molecular concepts in the pre-quantum era. 

Shannon27 analyzed in some detail the theory as it applies to the thermal decomposition of solids. He found that of the 31 reactions for which he compared experimental rate constants with those calculated from the Polanyi-Wigner equation, only a third showed order-of-magnitude agreement. In Shannon s view, the lack of agreement stems from neglecting rotational and other vibrational degrees of freedom. 

Shannon calculated the rate constant for thermal decomposition of a solid from absolute reaction rate theory. The resulting equation is of the same form as Equation 1.27, but v is replaced by a partition function ratio ... 

The catalytic activity decreases with the time and temperature. The change in activity is described according to eq. (5) of Butt and Weng [14]. This is a relation similar to equation (3) in the paper of Shannon and Pask [15]. The activity is given by the following formula ... 

Incidentally, Equation (1.15) is also called the Shannon formula for entropy. Claude Shannon was an engineer who developed his definition of entropy, sometimes called information entropy, as a measure of the level of uncertainty of a random variable. Shannon s formula is central in the discipline of information theory. 

In bioinformatics, the invariance (or conservatism) of the position of amino acid j in a polypeptide chain is typically estimated using the Shannon entropy as an integral characteristic of the probabiftstic (stochastic) process (see also equation (6.2)) ... 

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