Constant values

In statistical terms, a perceptual improvement is therefore obtained if the amplitude distribution in the filtered signal (image) is more concentrated around zero than in the raw data (contrast enhancement). A more concentrated amplitude distribution generally means smaller entropy. Thus, from an operator perception point of view, interesting results should be obtained if the raw data can be filtered to yield low entropy amplitude distributions. However, one should note that the entropy can be minimized by means of a (pathological) filter which always outputs zero or another constant value. Thus, appropriate restrictions must be imposed on the filter construction process. 

In order to perform this procedure on all elements of matrix A a margin with constant values aj and a ,j and a length of X (left/right) or % (top/bottom) is added on each side of matrix A. The two new matrices A and Ai are combined to form a resulting matrix Ax for the filtering in x-direction ... 

F/B due to the changing defect area s ratio as follows is examined, when f/ X is a constant value. 

The pulse shape in each point in Fig.7 (a) are shown in Fig.8 respectively. The direction of the intial part of pulse shape and pulse amplitude has changed by the defect position when the defect area s ratio Sr/So is a constant value. 

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

In practice, 7s 7sv is negligible as is dys/dT for systems having large contact angles. Also, low energy surfaces have a relatively constant value of dyst/dT = 0.07 0.02 erg cm K , leaving... 

Figure A3.8.2 The correlation fimction k( ) for particular case of the reaction of methyl vinyl ketone with cyclopentadiene in water. The leveling-off of this function to reach a constant value at the plateau time tp is clearly seen.

As an alternative to sampling the canonical distribution, it is possible to devise equations of motion for which the iiiechanicaT temperature is constrained to a constant value [84, 85, 86]. The equations of motion are... 

The bond lengths appearing on the right-hand side of Eq. (46) are assumed to take their equilibrium (constant) values. 

The dipole moments of the hydrogen halides decrease with increasing atomic number of the hydrogen, the largest difference occurring between HF and HCl, and association of molecules is not an important factor in the properties of FICl, HBr and HI. This change in dipole moment is reflected in the diminishing permittivity (dielectric constant) values from HF to HI. 

In fact, the pressures will never settle completely to constant values until diffusion is complete and the composition is the same in both chambers, but by making the chambers sufficiently large this rate of drift of the pressures can be reduced below any desired level. 

They represent the generalization of our earlier equations (11.12) and include an additional equation for the pressure, since this can no longer be assumed to take the constant value p throughout the pellet. The differential equation relating X to f is derived exactly as before and has the same form, except that p cannot be replaced by p In the present case. Thus we have... 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

For nitrations in sulphuric and perchloric acids an increase in the reactivity of the aromatic compound being nitrated beyond the level of about 38 times the reactivity of benzene cannot be detected. At this level, and with compounds which might be expected to surpass it, a roughly constant value of the second-order rate constant is found (table 2.6) because aromatic molecules and nitronium ions are reacting upon encounter. The encounter rate is measurable, and recognisable, because the concentration of the effective electrophile is so small. 

The term on the left-hand side is a constant and depends only on the constituent values provided by the reference laboratory and does not depend in any way upon the calibration. The two terms on the right-hand side of the equation show how this constant value is apportioned between the two quantities that are themselves summations, and are referred to as the sum of squares due to regression and the sum of squares due to error. The latter will be the smallest possible value that it can possibly be for the given data. 

The idea of a constant value for the molecular area of argon on different solids would seem reasonable in view of the non-specific nature of argon... 

In this section we consider a different experimental situation the case of creep. In a creep experiment a is maintained at a constant value and the time dependence of the strain is measured. Thus it is the exact inverse of the relaxation... 

One of the most sensitive tests of the dependence of chemical reactivity on the size of the reacting molecules is the comparison of the rates of reaction for compounds which are members of a homologous series with different chain lengths. Studies by Flory and others on the rates of esterification and saponification of esters were the first investigations conducted to clarify the dependence of reactivity on molecular size. The rate constants for these reactions are observed to converge quite rapidly to a constant value which is independent of molecular size, after an initial dependence on molecular size for small molecules. The effect is reminiscent of the discussion on the uniqueness of end groups in connection with Example 1.1. In the esterification of carboxylic acids, for example, the rate constants are different for acetic, propionic, and butyric acids, but constant for carboxyUc acids with 4-18 carbon atoms. This observation on nonpolymeric compounds has been generalized to apply to polymerization reactions as well. The latter are subject to several complications which are not involved in the study of simple model compounds, but when these complications are properly considered, the independence of reactivity on molecular size has been repeatedly verified. 

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