Unique straight line

This test was successfully carried out (18), and resulting values of qm exhibit a dependance upon T, fitting well with Equation 8 (17). However, such a method is rather long, and in this work, we preferred to look for a and W0 by successive iterations. A unique straight line was achieved in a plot of Equation 7 in the following form ... 

It is easily seen from Figure 2.3 or Figure 2.4 that a unique straight line through the point (1, 1) is tangent to the detonation branch of the Hugoniot curve. This can be proven formally by considering the equation... 

Unfortunately, the basalt flows on the southern nunataks of Vestfjella (Fig. 14.10) cannot be dated by the whole-rock Rb-Sr method. The samples that were analyzed by Fumes et al. (1987) scatter on the Rb-Sr isochron diagram in Fig. 14.14a and do not define a unique straight line. Instead, we have drawn a reference line based on an assumed age of 170 Ma and an initial ratio of 0.70416 0.00021 (2a) which... 

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

The usual assumptions are made here saying that the boundary F is smooth enough and the intersection of the domain G and a straight line passing through any point x (E G and in parallel to the axes Ox, a = 1,2, consists of a unique interval. The latter should not confine generality. 

A unique strength of concordia diagrams is their ability to reveal modes of diagenetic disturbance. If U-Th-Pa isotopic data for numerous co-eval sub-samples that have experienced different degrees of the same diagenetic process are plotted in or vs. space, they should fall on a curve or straight line... 

With the exception of some unique symbols of William Higgins in 1789, generally, straight lines appeared in published chemical formulas only when Archibald Couper introduced them in 1858 to indicate valences (units of atomicity, saturation capacity, or quantivalence).77 Whereas innocent accent marks or superscript dashes had been used at midcentury to indicate valence or value, straight lines now suggested a less abstract meaning, despite disclaimers like Alexander Crum Brown s that the lines indicated the "chemical," not "physical," positions of atoms. 78... 

Because the determinant is equal to zero, the (X X) matrix cannot be inverted, and a unique solution does not exist. An interpretation of the zero determinant is that the slope P, and the response intercept Po are both undefined (see Equations 5.14 and 5.15). This interpretation is consistent with the experimental design used and the model attempted the best straight line through the two points would have infinite slope (a vertical line) and the response intercept would not exist (see Figure 5.8). 

A convenient way of distinguishing between unique and multiple solutions is to use a flow diagram , in which the two rates—net inflow and chemical reaction—are plotted as functions of concentration. Figure 1.12 shows a number of flow diagrams, corresponding to different values for kl,a0,b0, and flow rate k. The straight line L is the net rate of inflow... 

With the exponential approximation (y 0) and the assumption that the inflow and ambient temperatures are equal, we have a stationary-state equation which links ass to tres and which involves two other unfolding parameters, 0ad and tn. Depending on the particular values of the last two parameters the (1 — ass) versus rres locus has one of five possible qualitative forms. These different patterns are shown in Fig. 7.4 as unique, single hysteresis loop, isola, mushroom, and hysteresis loop plus isola. The five corresponding regions in the 0ad-rN parameter plane are shown in Fig. 7.5. This parameter plane is divided into these regions by a straight line and a cusp, which cut each other at two points. 

The concept of a unique hydrodynamic volume for all rodlike polymers was derived from examination of the Mark-Houwink constants, K and a, of the equation [rj ] = KMa. Macromolecules with values of a greater than unity are commonly accepted to be stiff or rigid rods. However, it was also found that such molecules (even for values of a less than unity) obey a relation illustrated by close concordance with the curve in Fig. lb (13) flexible, branched or otherwise irregular polymers, on the other hand, show dispersion around the upper part of the curve. The straight line curve in Fig. lb implies that the constants K and a are not independent parameters for the regular macromolecules to which they apply. Poly (a- and polyQJ-phenylethyl isocyanide) fall on this line the former has a value of a > 1 while the latter has a value a < 1 (14) both polymers give linear concentration dependence of reduced specific viscosity for fractionated samples... 

When data obtained from similar measurements on different gases and at different pressures are plotted, it is found that the straight lines can be extended to the same point (Fig. 4.12). The extension of a graph beyond the data is called extrapolation, and we say that, in this case, the straight lines extrapolate to the same point. For Charles s law, this unique point corresponds to zero volume and —273.15°C. Because a volume cannot be negative, this temperature must be the lowest possible temperature. It is the value corresponding to 0 on the Kelvin scale. It follows that if we use temperature, T, on the Kelvin scale, then we can write Charles s law as... 

Consider a real function y = f(x) of a real variable x. By this we mean a mapping of the real number x to a unique real number y, given by the rule /. Furthermore, let us assume it to be continuous. We will introduce the concept of the derivative of f(x) with respect to x in terms of the slope of the tangent line at the point x,f(x)). In order to do this, we need to consider three simple constructive rules using the slope of a straight line as our starting point, and Leibniz rule as our keystone. The slope is calculated as a ratio of two displacements rise over run . Hence, we define the derivative of y with respect to r as a quotient of the two corresponding differentials, denoted by dy (the rise ) and dx (the run ) ... 

We can draw a unique tangent line (a straight line whose slope matches the curve s slope) at each point on the curve. Recall that the slope of a line is defined as the amount y changes if x is changed by one for example, the line y = 3x + 6 has a slope of three. 

A unique method and result of applying this Fenske-Underwood equation is that, when plotted on log-log graph paper, the components other than key component separations can be determined. Consider Fig. 2.3 notice the straight line drawn through the key component points. The unique thing here is that all other component values are also determined from this straight-line plot. 

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