Functions linear

The simplest functional relation between two variables has the general form 

FIGURE 5.1 Plot of functional relation /(j , = 0 in Cartesian coordinates or the 

FIGURE 5.2 Standard form for equation of straight line, a, i , and m are the X intercept, y intercept, and slope, respectively. The alternative intercept form of the equation can be written as Eq. (5.7). 

The parameter m in Eq. (5.6) is called the slope. It measures how steeply y rises or falls with x. Differential calculus, which we begin in the next chapter, is at its most rudimentary level, the computation of slopes of functions at different points. The slope tells how many units of y you go up or down when you travel along one unit of x. Symbolically, 

Slope might be used to describe the the degree of inclination of a road. If a mountain road rises 5 meters for every 100 meters of horizontal distance on the map (which would be called a 5% grade ), the slope equals 5/100 = 0.05. In the following chapter on differential calculus, we will identify the slope with the first derivative, using the notation 

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In Section 5.2.8 we shall look at pressure-depth relationships, and will see that the relationship is a linear function of the density of the fluid. Since water is the one fluid which is always associated with a petroleum reservoir, an understanding of what controls formation water density is required. Additionally, reservoir engineers need to know the fluid properties of the formation water to predict its expansion and movement, which can contribute significantly to the drive mechanism in a reservoir, especially if the volume of water surrounding the hydrocarbon accumulation is large. 

Derive the equation of state, that is, the relationship between t and a, of the adsorbed film for the case of a surface active electrolyte. Assume that the activity coefficient for the electrolyte is unity, that the solution is dilute enough so that surface tension is a linear function of the concentration of the electrolyte, and that the electrolyte itself (and not some hydrolyzed form) is the surface-adsorbed species. Do this for the case of a strong 1 1 electrolyte and a strong 1 3 electrolyte. 

The heat of immersion is measured calorimetrically with finely divided powders as described by several authors [9,11-14] and also in Section XVI-4. Some hi data are given in Table X-1. Polar solids show large heats of immersion in polar liquids and smaller ones in nonpolar liquids. Zetdemoyer [15] noted that for a given solid, hi was essentially a linear function of the dipole moment of the wetting liquid. 

For a quadratic surface, the gradient vector is a linear function of the coordinates. An alternative way of using... 

When two electronie states are degenerate at a particular point in configuration space, the elements of the diabatie potential energy matiix can be modeled as a linear function of the coordinates in the following fonn ... 

Additionally, as in all Tl-based approaches, the free energy differences are linear functions of the potential. Thus non-rigorous decompositions may be made into contributions from the different potential energy terms, parts of system and individual coordinates, providing valuable insight into the molecular mechanisms of studied processes [8, 9, 10). 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

We shall begin with the simplest case of a linear function passing through the origin to intr oduce the method and set up the ground rules. The more complicated... 

If the linear function through the origin y = mx were obeyed with perfect precision by an experimental data set [xi, yi], we would have... 

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

An illustrative example generates a 2 x 2 calibration matrix from which we can determine the concentrations xi and X2 of dichromate and permanganate ions simultaneously by making spectrophotometric measurements yi and j2 at different wavelengths on an aqueous mixture of the unknowns. The advantage of this simple two-component analytical problem in 3-space is that one can envision the plane representing absorbance A as a linear function of two concentration variables A =f xuX2). 

In the case of drug design, it may be desirable to use parabolic functions in place of linear functions. The descriptor for an ideal drug candidate often has an optimum value. Drug activity will decrease when the value is either larger or smaller than optimum. This functional form is described by a parabola, not a linear relationship. 

The first detector for optical spectroscopy was the human eye, which, of course, is limited both by its accuracy and its limited sensitivity to electromagnetic radiation. Modern detectors use a sensitive transducer to convert a signal consisting of photons into an easily measured electrical signal. Ideally the detector s signal, S, should be a linear function of the electromagnetic radiation s power, P,... 

Absorbance is the more common unit for expressing the attenuation of radiation because, as shown in the next section, it is a linear function of the analyte s concentration. 

Brown and Lin reported a quantitative method for methanol based on its effect on the visible spectrum of methylene blue. In the absence of methanol, the visible spectrum for methylene blue shows two prominent absorption bands centered at approximately 610 nm and 660 nm, corresponding to the monomer and dimer, respectively. In the presence of methanol, the intensity of the dimer s absorption band decreases, and that of the monomer increases. For concentrations of methanol between 0 and 30% v/v, the ratio of the absorbance at 663 nm, Asss, to that at 610 nm, Asio, is a linear function of the amount of methanol. Using the following standardization data, determine the %v/v methanol in a sample for which Agio is 0.75 and Ag63 is 1.07. 

Thus, the limiting current, is a linear function of the concentration of O in bulk solution, and a quantitative analysis is possible using any of the standardization methods discussed in Chapter 5. Equations similar to equation 11.35 can be developed for other forms of voltammetry, in which peak currents are related to the analyte s concentration in bulk solution. 

Peak currents in anodic stripping voltammetry are a linear function of concentration... 

Peak currents in differential pulse polarography are a linear function of the concentration of analyte thus... 

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

If all anharmonic constants except cOgXg are neglected, is a linear function of v... 

The linear functional F obviously possesses the needful property of the weak continuity. Thus we obtain the weak lower semicontinuity of the functional J. 

Density and refractive index are nearly linear functions of formaldehyde and methanol concentration. Based on available data (16—19), the density may be expressed ia g/cm by the following approximation ... 

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