Normalized variables

The normalized concentration in a mixture of n end-members is defined in a similar way 

This situation is entirely analogous to the cases of elemental or isotopic ratios which have been discussed above, except that we have artificially created a new species by summing the concentrations of m different species. Actually, any sort of linear combination of elemental concentrations would show the same properties. 

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Very clean sands are rare and normally variable amounts of c/ay will be contained in the reservoir pore system, the clays being the weathering products of rock constituents such as feldspars. The quantity of clay and its distribution within the reservoir exerts a major control on permeability and porosity. Figure 5.2 shows several types of clay distribution. 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

These are generally called Second Order Reliability Methods, where the use of independent, near-Normal variables in reliability prediction generally come under the title First Order Reliability Methods (Kjerengtroen and Comer, 1996). For economy and speed in the calculation, however, the use of First Order Reliability Methods still dominates presently. 

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

Table 20.5.2 also can be used to determine probabilities concerning normal random variables tliat are not standard normal variables. The required probability is first converted to tm equivalent probability about a standard normal variable. For example if T, the time to failure, is normally distributed with mean p = 100 and stanchird deviation a = 2 tlien (T - 100)/2 is a standard normal variable and... 

From Table 20.6.2, z, is found to be -1.28. The corresponding simulated value of Ta is obtained by noting tliat (Ta - 100)/20 is a standard normal variable. Therefore, tlie simulated value of Ta corresponding to z, = -1.28 is... 

Since (In Z - p)/a is a standard normal variable, we can refer to a table tliat gives areas under tlie standard nonnal curve (see Table 21.5.2) to find tliat... 

We shall now consider the properties of systems the state of which is determined by the values of the absolute temperature T, and n other independent variables x , 2, x3i. . . xn. If the latter are chosen in such a way that no external work is done when the temperature changes provided all the s are maintained constant, they, along with T, are called the normal variables, and the state so defined is said to be normally defined (Duhem Mecanique chimiqne, I., 83). 

The normal variables may, according to the nature of the system considered, be the temperature and either geometrical... 

If the temperature is changed as well as the normal variables, the free and bound energies alter as follows ... 

Let the temperature now be raised from T to T + ST whilst all the normal variables remain unchanged with the values (b). This is a change of temperature at constant configuration, and the line traced out would be an adynamic. 

The Suffix Ax shows that the difference between the initial and final sets of normal variables is to be maintained constant. Thus, if the only normal configuration variable is the volume, x — v and Ax — i — n = Ar, i.e., the expansion is to be kept the same whilst the temperature changes. Ax may be said to define the... 

In this equation, Y is the catalyst performance, the variables X and ni are normalized variables representing the reaction conditions and catalyst s metal weight loadings, respectively. The model coefficients C, a , and (3 , are functions of the catalyst composition, as shown in Eqns (6) and (7), where m.j refers to the nominal weight loading of Pt, Ba, or Fe. The equation for (3 takes the same form as Eqns (6) and (7). 

In this equation, R is the catalyst performance, determined by average NO, concentration. CT and LF are normalized variables representing the operating conditions, i.e., cycle time and lean fraction. The operating condition variables were normalized according to Eqn (5) by assigning the low value listed in Table 11.10 as —1 and the high value as +1. 

While this qualifies as a small population-effect, the normal variability of individual susceptibility means that there may be a larger IQ deficit in particularly vulnerable individuals. 

Generally, porous materials have the porosity of 0.2-0.95. The porosity means the fraction of pore volume to the total volume. Let us consider a sample of total volume V. The volume of the solid phase is Vs, and the volume of the pore phase (the holes) is Vp, where V=VS +VP. The volume fraction is normalized variable that is generally more useful. The volume fraction of the pore phase is commonly called the porosity, and is denoted as ( )= Vp/V. The solid volume fraction is then described as 1-( )29. 

A small bowel transit study can be used to evaluate intestinal propulsion and clearance, and the presence of Enterobacteriacea (Gram-negative bacilli) in the small bowel indicates delayed transit [111]. The wide normal variability, however, makes transit tests rather insensitive, and thus less useful clinically [126, 127], It is also a problem that accelerated and delayed transit may coexist in neuropathies and confuse the interpretation. Finally, as nutrients are mostly absorbed in the proximal small bowel, and the rate and pattern of transit vary along the intestine, segmental failure of transit is easily missed by global... 

A random variable X is distributed as a log-normal distribution if In X is distributed as a normal distribution. If U is a random normal variable, a log-normal distribution is the distribution of a variable X such as... 

The physical and conceptual importance of the normal distribution rests on one unique property the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n- oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo, or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. 

The ratio of two normal random variables with zero mean is distributed as a Cauchy variable. Isotopic ratios such as 206Pb/204Pb and 207Pb/204Pb therefore should not be described as normal variables since ratios of ratios (e.g., 207Pb/206Pb) should be distributed with a consistent distribution. A consistent distribution for isotopic ratios is the log-normal distribution. 

Consequently, the distribution of the sum Z of two normal variables X and Y with respective moment generating functions... 

The sum Z of two normal variables X and Y is a normal variable. Its mean is the sum of individual means, its variance the sum of individual variances. If X is normal variable with mean n and variance a2, 2X is normal with mean 2/i and variance 2a2. This property is easily extended to the sum of any number of normal variables. An important consequence on sampling distributions is that, from equation (4.1.40), the mean x of m observations Xj from the same normal distribution with mean n and variance a2 is a normal variable with mean mfi/m=fi and variance ma2/m2 - a2/ /m. 

Find the distribution of Y = ex where X is a normal variable with mean g and variance a2. 

Make a table of 20 crustal values of eNd(0) which is assumed to be a normal variable with mean fi= —12 and standard deviation [Pg.199]

Adding variances on different variables at the denominator, e.g. pH and temperature in solutions, does not make much sense and is certainly not invariant upon rescaling. Proportions of explained total variance do not survive a simple change of units For this reason, PCA is commonly carried out instead on normalized variables such as... 

It is useful to introduce normalized variables, functions, and parameters so as to obtain a dimensionless formulation, resolve the problem at this level, and finally, come back to the real experimental quantities. This strategy allows one to find out, here and in more complicated cases, the minimal number of parameters that are actually governing the electrochemical responses. We thus introduce the following normalized variables ... 

We introduce the same normalized variables and parameters as in Section 6.1.2 and, in addition,... 

The main difference with the EC mechanism (Section 6.2.1) is that C is reduced as soon as it reaches the electrode hence the replacement of the boundary condition (QCc/Qx)x=0 = 0 by the condition (Cc)x=0 = 0. A second difference is the contribution to the current provided by the reduction of C. Introduction of the same normalized variables and parameters as in Sections 6.1.2 and 6.2.1 leads to... 

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