Strictly positive

In this case it is required that the original data in X are strictly positive. The effect of the transformation appears from Table 31.6. Column-means are zero, while column-standard deviations tend to be more homogeneous than in the case of simple column-centering in Table 31.4 as can be seen by inspecting the corresponding values for Na and Cl. 

It is assumed that the original data in X are strictly positive. As is evident from Table 31.7 both the row-means m and the column-means of the transformed table Z are equal to zero. 

The conviction that C should be strictly positive was so compelling that it has been typically believed to also apply under cr-control conditions (see Refs. 7 and 36 for a review). In some other cases, confusion was caused by an opposite extreme, dropping the restrictions on C < 0 even for -control. 

Noting that U < Ne, the temperature is observed to be a strictly positive quantity. Solving the equation of state for the energy gives... 

Regulation of Flavonoid Synthesis in C. americanum. Biosynthesis of methylated flavonol glucosides seems to be under tight regulation, not only by the substrate specificity of the enzymes involved, but also by other factors, among which are (a) the strict position specificity of these enzymes towards their hydroxylated or partially methylated substrates (b) the apparent difference in microenvironment of the different methyl-transferases, whereby those earlier in the pathway utilized aglycones whereas later enzymes accepted only glucosides as substrates (c) the subtle characteristic differences in methyl-transferases with respect to their pH optima, pi values and requirement for Mg ions, despite their similar molecular size ... 

In the paired t-test setting it is the normality of the differences (response on A — response on B) that is required for the validity of the test. The log transformation on the original data can sometimes be effective in this case in recovering normality for these differences. In other settings, such as ANOVA, ANCOVA and regression, log transforming the outcome variable is always worth trying, where this is a strictly positive quantity, as an initial attempt to recover normality. 

Figure 3.6. The interval [—1. 1] is closed, since every point x outside the interval lies in an open ball (i.e., open interval of strictly positive length e) outside the interval.

Exercise 3.1 (Used in Section 3.5) In this exercise we show how to make sense of inequalities on Lebesgue equivalence classes of functions. Suppose S is a set with an integral defined on it and f is a real-valued functions on S. Let [] denote the Lebesgue equivalence class of f. We say that [] is strictly positive (0 < []) if for every function fi such that 0 < lA (x) for all x e S, we have... 

Exercise 3.2 (For students of Lebesgue measure) Show thatQ < [] if and only iff is strictly positive on the complement of a set of measure zero. 

Exercise 3.25 Suppose M is an nyconjugate transpose of M. Suppose every eigenvalue of M is strictly positive. Define... 

Exercise 3.30 (Used in Proposition 7.5) Suppose R is a strictly positive real number. Show that the set... 

We now proceed to the determination of all possible parameters. Before that, a few remarks are in order. Equation (9.6) is useful only for the relations between k, a, b, i, and j. The difficulty is that an equation, say, pa(T) = 0 does not imply Pa(p, P) = 0, since a function can be strictly positive and have a zero limit. The global equation (9.6) does not give local information. For local reasoning, our only possibility is to use corona arguments, i.e. those based on possible corona of faces. 

To remove this weakness of the Fritz John necessary conditions, we need to determine the required restrictions under which fi0 is strictly positive (fi0 > 0). These restrictions are called first-order constraint qualifications and will be discussed in the following section. 

Consequently, condition (4.23) requires that, for very large 12, the difference between the coolant and the initial temperature must be strictly positive. 

A single-variable function is convex if the second derivative is strictly positive over the range of the dependent variable, as shown in Fig. 4(1). As show in Fig. 4(2), a function is concave when the second derivative is negative over the dependent variable range. For a multivariable function, the matrix of second derivatives (termed the Hessian, H(x)) is used to check the convexity (or concavity) of the function ... 

Subtracting E0 from E and assuming that the ground state is nondegenerate we get the following inequality where all excitation energies AEt are strictly positive ... 

The use of an exponential model has several attractive features. It will ensure that all parameters are strictly positive and avoids so the estimation of negative non-physiological individual values. Also square root of the estimate is approximately the coefficient of variation in the model parameters. 

Thus, the fundamental difference between N-NDR and HN-NDR systems is that the former s stationary polarization curve exhibits a range of negative real impedance, whereas for the latter the zero-frequency impedance is strictly positive in the potential region of interest. From this observation one might get the impression that the mechanisms of electrode reactions are fundamentally different for systems in the two groups. But in fact it is only a small step, or more precisely, one additional potential-dependent process, that transforms an N-NDR system into an HN-NDR system. Formally, any HN-NDR system is composed of a subsystem with an N-shaped stationary polarization curve whose NDR is hidden by at least one further slow and potential-dependent step of the interfacial kinetics of the total system. This step dominates the faradaic impedance at low perturbation frequencies, whereas at higher... 

It can also be shown that in the interval where Rp is greater than 0 and less than M2// i, F2(RP) is a strictly positive non-decreasing function. This is done by showing that the ratio F2(RP + e)/F2(Rp) for any positive e less than ( [M2// i] — Rp) is strictly greater than 1. If we write this ratio ... 

This first step would involve the displacement of a pyridine ligand by the aromatic amine. Some evidence points to this being the rate-determining step. A strict positive correlation between the basicity of the diamine relative to pyridine and the oxidative coupling rate (as measured by the oxygen absorption) was established (3). 

A symmetric matrix A is said to be positive-definite if the quadratic form uTAu > 0 for all nonzero vectors u. Similarly, the symmetric matrix A is positive-semidefinite if uTAu 2 0 for all nonzero vectors u. Positive-definite matrices have strictly positive eigenvalues. We classify A as negative-definite if u Au < 0 for all nonzero vectors u. A is indefinite if uTAu is positive for some u and negative for others. 

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