Functionality mathematically ambiguous

The mathematical representation of the ambiguity function for this second method is now different. The matched filtering is performed for each echo and the final output will be a summation of the bistatic ambiguity functions of the various bistatic pairs. The following equation outlines this ... 

Here Fq and F are the frequencies of the non-coated or coated quartz in the (100) mode of the fundamental wave. Because of the ambiguity of the mathematical functions used, the Z value calculated in this way is not always a positively defined variable. This has no consequences of any significance because M is determined in another way by assessing Z and measuring the frequency shift. Therefore, the thickness and rate of the coating are calculated one after the other from the known M. 

Spectroscopists also generally fit their data to potential functions of the general form of Eqs. (15) and (16) or some other convenient mathematical expression, and the potential coefficients (the a —s or V —s) are thereby determined. Not all of these coefficients need to be of importance. Some may even vanish for reasons of symmetry. Naturally, it would be desirable to be able to determine those coefficients on which the potential function primarily depends. Particularly for the methyl groups, accurate barriers are now available. The situation is not this fortunate in other cases. Often, rather limited experimental data are accessible. The quality of some of this material may be poor or even ambiguous, and rather arbitrary assumptions are now and then made. As a result, only the first few potential coefficients, often of low precision, are obtained. In fact, even today cases where more than four potential coefficients have been deduced are exceptional. 

At this point the complex behaviors of these ambiguous figures have been related to plausible mathematical form. When the behaviors of interest can be described by the behavior of the observable x, a potential mechanism can be found by looking for the identity and system descriptions of the control variables u and v. One of the advantages of this particular treatment is that the manifold is a potential surface of the function of interest. Thus, this mathematical form allows a potential energy cost to be related to a particular behavior. It should be appreciated that other manifolds and equations might be found for other behaviors. 

Here we have two possible signs for p, because we have an arccosine function, which gives two possible solntions for a single argument. Thus, the phase ambiguity is actually a mathematical problem. 

The mentioned uniqueness of the Fade approximant for the given input Maclaurin series (4) represents a critical feature of this method. In other words, the ambiguities encountered in other mathematical modelings are eliminated from the outset already at the level of the definition of the PA. Moreover, this definition contains its "figure of merit" by revealing how well the PA can really describe the function G(z ) to be approximated. More precisely, given the infinite sum G(2 ) via Eq. (4), the key question to raise is about the best agreement between from Eq. (2) and G(z ) from... 

Pore Size Distribution. The pore structure is sometimes interpreted as a characteristic pore size, which is sometimes ambiguously called porosity. More generally, pore structure is characterized by a pore size distribution, characteristic of the sample of the porous medium. The pore size distribution/ ) is usually defined as the probability density function of the pore volume distribution with a corresponding characteristic pore size 6. More specifically, the pore size distribution function at 5 is the fraction of the total pore volume that has a characteristic pore size in the range of 5 and 5 + dd. Mathematically, the pore size distribution function can be expressed as... 

As mentioned in the introduction, preferences may be represented by one-dimensional comparison, which we discussed in the previous two sections, or in terms of multidimensional comparison, which we will discuss in this and the following section. Note that in one-dimensional comparison, we implicitly or explicitly assume that = 0 and that no ambiguity exists in preference. Once the value function or proper regret function is determined, MCDM becomes a one-dimensional comparison or a mathematical programming problem. In this section we shall tackle the problems with 0. 

The software reqrrirements shoirld be analysed for ambiguities, inconsistencies and undefined conditions deficiencies shonld be identified, clarified and corrected. The software requirements shoirld be verifiable and consistent. The functional requirements should be stated in mathematical terms where possible. Non-fimctional software requirements, such as minimum precision and accnracy, time behavionr, independence of execution threads and fail-safe behavionr, shonld also be stated exphcitly and, whenever required, in quantitative terms. 

In order to keep its practical advantages and avoid these mentioned ambiguities, one can use a well-studied mathematical tool (the topological analysis) to characterize the Fukui function [44]. This has been applied before to analyze both the... 

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