Linearity Property

One of the most important properties of Laplace transformation is that it is linear. 

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Axioms (2) and (3) are met by virtue of the linearity property. The validity of (4) is stipulated by the fact that the operator A is positive. The meaning of the self-adjointness of the operator A is that we should have... 

That is to say, the meaning of stability of scheme (21) is that a solution (21) depends continuously on the right-hand side and this dependence is uniform in the parameter h. This implies that a small change of the right-hand side results in a small change of the solution. If the scheme is solvable and stable, it is correct. Note that the uniqueness of the scheme (21) solution is a consequence of its solvability and stability and, hence, we might get rid of the uniqueness requirement in condition (1). Indeed, assume to the contrary that there were two solutions to equation (21), say and By the linearity property of the operator A, their difference = y — yf should satisfy the homogeneous equation... 

When the MLF is used for classification its non-linear properties are also important. In Fig. 44.12c the contour map of the output of a neural network with two hidden units is shown. It shows clearly that non-linear boundaries are obtained. Totally different boundaries are obtained by varying the weights, as shown in Fig. 44.12d. For modelling as well as for classification tasks, the appropriate number of transfer functions (i.e. the number of hidden units) thus depends essentially on the complexity of the relationship to be modelled and must be determined empirically for each problem. Other functions, such as the tangens hyperbolicus function (Fig. 44.13a) are also sometimes used. In Ref. [19] the authors came to the conclusion that in most cases a sigmoidal function describes non-linearities sufficiently well. Only in the presence of periodicities in the data... 

The linear property is one very important reason why we can do partial fractions and inverse transform using a look-up table. This is also how we analyze more complex, but linearized, systems. Even though a text may not state this property explicitly, we rely heavily on it in classical control. 

This approach works because of the linear property of Laplace transform. 

Before being able to study the nonlinear optical properties of any material, it is necessary to have a complete understanding of its linear optical properties. Therefore, we start this section with a brief discussion of the techniques used to measure some of the most important linear properties, e.g., linear absorption, fluorescence, anisotropy, and fluorescence quantum yield. 

In general, molecular crystals are too soft for them to be of interest as structural materials. Also, they fracture readily. Because of their transparencies and non-linear properties some of them are of interest for optical applications, but most of them suffer from optical damage at low intensities of light. 

As discussed in Subramaniam and Pope (1998), the EMST model does not satisfy the independence and linearity properties proposed by Pope (1983). 

Applying the linearity property of the expectation to the change of variable given by equation (4.2.29), we get... 

The connection between noise in the time and in the frequency domain is given by the linearity property of the Fourier transform (see, e.g., Ref. 31). Let s(t) and n(t) be the signal and noise in the time domain and S(a>) and N((u) their corresponding Fourier transforms, then... 

The linearity property of the Fourier transform does not hold for phase delay values. Let a, be the intrinsic, single measurement, time domain standard deviation of the filtered signal (in units of amplitude). Also let o>(n) be the standard deviation of the phase of the nth harmonic averaged j times. The phase noise as a function of harmonic (frequency) is given by... 

As shown by Marechal et al. (1999), the linearity property of the isotopic array formed in log-log plots by all the measurements of a same solution can be used to obtain the isotopic ratio of a sample with respect to the same ratio in a standard solution of known isotopic properties of a different element. Combining either Equations (21) or (22) for both the sample and the standard, we obtain ... 

A. LINEARITY. The linearity property is easily proved from the definition of z transformation. 

Konno and Yamazaki (1991) proposed a large-scale portfolio optimization model based on mean-absolute deviation (MAD). This serves as an alternative measure of risk to the standard Markowitz s MV approach, which models risk by the variance of the rate of return of a portfolio, leading to a nonlinear convex quadratic programming (QP) problem. Although both measures are almost equivalent from a mathematical point-of-view, they are substantially different computationally in a few perspectives, as highlighted by Konno and Wijayanayake (2002) and Konno and Koshizuka (2005). In practice, MAD is used due to its computationally-attractive linear property. 

Because of the limitations of both electronics and photonics a hybrid technology, opto-electronics, has been a major area of research since the 1980s, especially into the non-linear properties of molecules and their potential applications in communications, data storage and information processing. ... 

The analysis of these 25 compounds confirms the flexibility of dpg/dpg+. It can adopt a propeller like conformation and induce the formation of chiral crystals. It establishes predominantly two-dimensional H-bonding networks with the counter ion. From the perspective of the crystal engineer, an anticlinal conformation would maximize the non-linear properties of the molecule. However the non-linear optical response depends greatly on the molecular alignment. Non-centrosymmetric dispositions have not yet been achieved for dpg with an anti-anti conformation. The best strategy to attain NLO samples is by inducing a syn-anti conformation where a propeller structure might induce a chiral disposition of molecules and thus a non-centrosymmetric order. Another successful approach is to force crystallization with a chiral counterion. 

Quantitative evidence regarding chain entanglements comes from three principal sources, each solidly based in continuum mechanics linear viscoelastic properties, the non-linear properties associated with steady shearing flows, and the equilibrium moduli of crosslinked networks. Data on the effects of molecular structure are most extensive in the case of linear viscoelasticity. The phenomena attributed to chain entanglement are very prominent here, and the linear viscoelastic properties lend themselves most readily to molecular modeling since the configuration of the system is displaced for equilibrium only slightly by the measurement. 

Viscoelastic behavior is classified as linear or non-linear according to the manner by which the stress depends upon the imposed deformation history (SO). Insteady shear flows, for example, the shear rate dependence of viscosity and the normal stress functions are non-linear properties. Linear viscoelastic behavior is obtained for simple fluids if the deformation is sufficiently small for all past times (infinitesimal deformations) or if it is imposed sufficiently slowly (infinitesimal rate of deformation) (80,83). In shear flow under these circumstances, the normal stress differences are small compared to the shear stress, and the expression for the shear stress reduces to a statement of the Boltzmann superposition principle (15,81) ... 

How can (10.9) make sense as a geometrical scalar product From the chain-rule linearity property (10.4) of partial derivatives, one can see that the (R/ R7) values defined by (10.9) will automatically satisfy the distributive property (9.27a) ... 

We have seen that the limitations of the time characteristics of electronic devices requires the use of optical delays between the pump and probe pulses in ps flash photolysis. There are also indirect ways of using optical properties to measure the kinetics of laser pulses and of fluorescence, known as autocorrelation and up-conversion . These rely on the non-linear properties of certain materials or chemical systems, i.e. they are based on fast biphotonic processes. 

In order to generate the second harmonic of an electromagnetic wave, one needs to make use of some device which has a non-linear property. In the case we are considering, the non-linear relationship made use of is that between applied electric field and electric polarisation. One can write... 

In addition to photoconductivity, polysilanes have been found to exhibit marked nonlinear optical properties,95-97 suggesting that they may eventually be useful in laser and other optical technology. The third-order non-linear susceptibility, X3, is a measure of the strength of this effect. The non-linear properties of polysilanes, like the absorption spectra, seem to be dependent on chain conformation and are enhanced for polymers having an extended, near anti conformation (Table 5.5). The value of 11 x 10 12 esu observed for (n-Hex2Si) below its transition temperature is the largest ever observed for a polymer which is transparent in the visible region. 

When deciding to study the dynamics of electronic excitation energy transfer in molecular systems by conventional spectroscopic techniques (in contrast to those based on non-linear properties such as photon echo spectroscopy) one has the choice between time-resolved fluorescence and transient absorption. This choice is not inconsequential because the two techniques do not necessarily monitor the same populations. Fluorescence is a very sensitive technique, in the sense that single photons can be detected. In contrast to transient absorption, it monitors solely excited state populations this is the reason for our choice. But, when dealing with DNA components whose quantum yield is as low as 10-4, [3,30] such experiments are far from trivial. 

The discrete Sis calculated with Eqs. (24) or (25) give a set for molecules A and B. Petke proved that the discrete MEP-SI defined by Eq. (25) has general linear properties independent of any functional relation between VAi and VBi [116]. 

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