Analytic relationship

The shape of a frequency distribution curve will depend on how the size increments were chosen. With the common methods for specifying increments, the curve will usually take the general form of a skewed probability curve with a single peak. However, it may also have multiple peaks, as in Fig 2, There are various analytical relationships for representing size distribution. One or the other may give a better fit of data in a particular instance. There are times, however, when analytical convenience may justify one. The log-probability relationship is particularly useful in this respect... 

Derive an analytical relationship between openloop oiajumum log modulus and damping coeflident for a second-order underdamped openloop system with a gain of unity. Show that a damping coeflident of 0.4 corresponds to a maximum log modulus of +2J dedbels. 

If the relationship in Eq. (13) cannot be expressed analytically, the necessary double integration may be carried out graphically. Fortunately, this is usually not necessary, as several analytical relationships between shear stress and shear rate have been proposed as generally useful for these purposes ... 

From Eq. (11), and taking into account the expression v = (M F02)-1, valid for ideal networks, it can be seen that the analytical relationship between the equilibrium swelling degree Q and the molecular weight M of the elastic chains is rather complicated. Several additional points have to be considered ... 

These equations can be used to establish additional analytical relationships when dealing with the matrix elements of operators of physical quantities in the case of complex electron configurations. 

The gauge has proved equally accurate for constant pressure and blowdown systems, and can also be adapted to vented systems. A typical curve for normalized krypton concn vs the amt of proplnt remaining in the tank is shown in Fig 1. Also shown is the analytical relationship between tracer concn and proplnt remaining in the tank. Statistical error analyses showed the typical average gauging error thruout the entire range of proplnt expulsion to be less than . 3% with a one sigma deviation of less than . 4%. This illustrates the consistency and reproducibility of this measurement technique... 

Two exceptions are noteworthy. In the case of homogeneous pure fluids, density can usually be determined more accurately by an indirect measurement, namely, the measurement of pressure and temperature which is then coupled with the analytical relationship between these in-... 

When considering multiple analytes in a composite, information regarding analyte relationships in individual samples will be lost. Hence, half of the individual samples should be saved before compositing. 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

Very often in some design problem, we know which variables or system properties are important, but we do not know the precise analytical relationship between them. That is, we know the variables but cannot write an equation. We can determine the relationships experimentally by varying one parameter and measuring the effect on the dependent variable of interest. We will obtain a graph of this relationship when we plot the results. Let us say a particular problem has two variables, x and y. We run five experiments in which we vary x and measure the resultant change in y. Our data look like those in Figure 26.4. 

So we see that for a particular problem or system we can, because we are bright engineers, specify all the pertinent variables, and can determine the dimensions of those variables. We do not know the analytical relationship between these variables, but we do know that they are the pertinent ones. 

There is a simple analytical relationship between the density of states and the partition function ... 

The ratio of two angles symmetric about 90°, commonly known as a disymmetry measurement, has been used for many years to measure polymer and particle sizes (6) however, the two angles need not be symmetric about 90° for equation 6 to apply. Sizes can be calculated from this ratio using an analytical relationships for P 0) for a specific... 

The development of mass transfer models require knowledge of three properties the diffusion coefficient of the solute, the viscosity of the SCF, and the density of the SCF phase. These properties can be used to correlate mass transfer coefficients. At 35 C and pressures lower than the critical pressure (72.83 atm for CO2) we use the diffusivity interpolated from literature diffusivity data (2,3). However, a linear relationship between log Dv and p at constant temperature has been presented by several researchers U>5) who correlated diffusivities in supercritical fluids. For pressures higher than the critical, we determined an analytical relationship using the diffusivity data obtained for the C02 naphthalene system by lomtev and Tsekhanskaya (6), at 35 C. 

The presence of symmetry (Chapter 1) coupled with well-defined analytical relationships determining both the directions and intensities of scattered beams (Chapter 2), in addition to known properties of both the specimen and instrument employed to obtain a powder diffraction pattern (Chapter 3), makes it possible to develop both the general methodology and algorithm(s) suitable for automation. Given the amount of numerical data collected in a typical powder diffraction experiment, their interpretation and processing usually involves a broad use of computers. 

This is a very accurate calculation of thermodynamic functions of Se2(g) from spectroscopic data using the wavenumber AVe = 509.95 cm for the important splitting of ground electronic state into the components XO and Xlg. Calculated values were tabulated at 100 K intervals in the temperature range 100 to 3000 K. The presented analytical relationship of the temperature dependence of the heat capacity does not extend all the way down to 300 K due to the curvature of the heat capacity at low temperatures. Corresponding expressions were evaluated by this review from the tabulated values divided into three different temperature intervals, whereby the whole temperature range... 

The mother is there with the family. We know from contemporary research in child development that the basic trust in life goes back to a good relationship to the mothering figure in the earliest years. So these children presumably have that basic trust in life. In analysis we try to address these needs or their lack through the analytic relationship, which fosters creative introversion."... 

If the equilibrium relationship y = /(jc) is in graphical or tabular form, for which no analytical relationship is available, integration of (9-3) can be performed graphically. 

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