Characteristic functions, definition

As an example of these techniques, we shall calculate the characteristic function of the gaussian distribution with zero mean and unit variance and then use it to calculate moments. Starting from the definition of the characteristic function, we obtain18 ... 

We conclude this section with a derivation of the characteristic function of the Poisson distribution. Starting from the definition, Eq. (3-77), we obtain... 

Now from the definition of a characteristic function of a random variable x, i.e. 

Exercise. There is no reason why the characteristic function should be positive for all k. Why does that not restrict the validity of the definition (2.8) of the cumulants ... 

Consequently R(t) is positive definite. It follows from Bochner s theorem that the normalized memory function can be regarded as the characteristic function of the probability distribution, P(coi), such that... 

A more general definition is given by Feller [12]. We denote a Levy stable pdf dF(x) as La(x) and call a the Levy (stable) index. It can be shown that a stable law has a characteristic function of the form... 

The characteristic function of this diffusion process is by definition the Fourier transform of p(x, t), namely... 

The characteristic function for a distribution law of a random variable is the Laplace transform of the expression of the distribution law. For the analysis of the properties of the distribution of a random variable, the characteristic function is good for the rapid calculation of the centred or not, momentum of various orders. Here below, we have the definition of the characteristic function cp (s) and its particularization with the case under discussion ... 

Allahverdyan and Nieuwenhuizen " consider Thomson s formulation of the second law and argue that the variable of the quantum FRs in previous studies should not be considered to be the work. They discuss different definitions of the work in quantum systems and argue for introduction of a new definition. Esposito and Mukamel present transient and steady state FRs as well as JE by developing a quantum mechanical trajectory, and then carrying out a derivation in a similar way to that used for stochastic dynamics. Talkner and Flanggi recently derived a quantum version of the Crooks FR using a characteristic function for the work obtained by Talkner et al ... 

Characteristic function of a system, 91.—76. Characteristic function of a perfect gas, 92.-77. The characteristic function considered as available energy, 98.-78. Definite form of the equilibrium condition of a system kept at a given constant temperature, 94. —79. Internal thermodynamic potential, 94.—80. Total thermodynamic potential under constant pressure or at constant volume,... 

Since the probability density is normalizable at all times, the real part of g(k) must be negative definite. In order for the characteristic function to retain the product form at all spatial scales, it must be infinitely divisible. If we scale the Fourier variable k by a constant factor b, then in order for the probability density to be infinitely divisible, g(k) must be homogeneous ... 

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

In this way, the probabilities w(k) can be obtained from Eq. (12.22). The definition of the Fourier transform is not unique, but it is historically grown from the different fields of science. Various conventions are in use. In Table 12.7 we show some pairs of Fourier transforms. The last definition in Table 12.7 involves two parameters a, b) and is the most versatile one. In the integral form, the characteristic function uses (1, 1). 

If the cell jumps are normally distributed, the characteristic function of in(x) is u)(k) = exp (iaiki - aijkik. The positive definite matrix can be written in terms of the first two moments ... 

The definition of set operations on extended characteristic functions is done in a two step manner. In the first step the operator is applied to the set elements without considering attributes. In the second step the operator is applied to the attribute values. The semantics of the second step is dependent on the attribute values. 

The definition of the characteristic function is extended to sets of states with the following definitions ... 

One functional definition of flocculation is that it describes the ability of yeast strains to clump together and fall out of solution. Near the end of fermentation, single cells aggregate into clumps of thousands of cells. Different strains of yeast have different flocculation characteristics. Some strains flocculate earlier during fermentation and subsequently do not attenuate (i.e. finish the fermentation) normally. Flocculating too early results in a beer that is under attenuated and sweet however, when yeast fails to flocculate entirely, it results in a beer that is cloudy with a yeasty flavour (Speers, 2012). 

The definition of system is quite varied, but a common element is that it focuses on the whole entity, for example A system is a construct or collection of different elements that together produce results not obtainable by the elements alone. The elements, or parts, can include people, hardware, software, facihties, poHcies, and documents that is, all things required to produce systems-level results. The results include system-level qualities, properties, characteristics, functions, behavior and performance. The value added by the system as a whole, beyond that contributed independently by the parts, is primarily created by the relationship among the parts that is, how they are interconnected. Using this definition, one can identify a variety of systems within the clinical setting. For example, an MRI device is... 

This is another function of force ratio and the coupling pwameter, so we do not exp>ect the extremes in the two functions at the same value of the force ratio. This is one of the reasons some experts discuss, that not all biological systems maximize their efficacy, instead they try to optimize other "cost-function" depending its complexity and its connection on the surroundings [55]. In the following we shall try to summarize these fxmctions and their roles. Additionally to the above mentioned two characteristic functions, the isotherm Power output function (Pout) and the Ecological function (Ec) are used. Their definitions are [55]. 

Thus different parts of the molecule may carry hydration shells that have significantly different properties. This leads to mutually destructive intramolecular overlap of water shells and the resulting overall hydration characteristics will definitely represent these complexities. This makes it difficult to define hydrophobic parameters for organic functional groups because they will depend on their position in the molecule. Only long alkyl functionalities do not suffer from these overlap effects. Recent kinetic studies have demonstrated these effects and rough estimates have been made of the extent of the overlap region in the total hydration shell vide supra). 

We have seen that the method, involving changing the variables to express the internal energy, quickly results in complex expressions, which is why we prefer to use a better adapted characteristic function T for each variable set, which has the same properties as the internal energy, i.e. it rerrtains a characteristic function whose knowledge will allow, as for U in Sq variables, a complete definition of the phase studied in the thermodynamic plane, and this remains a state fimction. However, this function will be expressed in a set of variables to which it will be adapted, called its canonical variables. 

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