Hyperbola method

Other variants are the rise between two parallel plane surfaces of glass, and the so-called hyperbola method, which measures the rise between two vertical plates inclined at a small angle. If the parallel plates are at a distance b apart, the radius of curvature is infinite in one direction and b in the other. The height of rise is therefore (neglecting any deviation of the meniscus from a... 

The hyperbola method has two vertical plates inclined at a small angle radians to each other, so that their distance apart increases with increasing... 

In seismology a 2-D source location with no information about the depth of the event is called epicenter. At least 3 sensors are needed for a 2-D localization. Assuming constant velocity and three measured arrival times ti, t2 and t of the compressional wave, at three different sensors, the epicenter can be calculated by the hyperbola method (Bath 1979 Pujol 2004). The epicenter must be located on a curve for which the arrival time difference between two sensors e.g. G - ti is constant (Fig. 6.6). Such a curve is a hyperbola with the corresponding sensor coordinates of sensor 1 and sensor 2 as foci. 

Fig. 6.6. 2-dimensional localization using the hyperbola method. t2 and t are the arrival times of the compressional wave at the corresponding sensors. 

Due to the fact that generally one arrival time is greater than the other e.g. 6 > the epicenter loeation is limited to one branch of the hyperbola. The hyperbolas of the other station pairs ti, ts and t2, ts) are caleulated in a similar way. The epicenter is the interseetion point of the 3 h erbolas (Fig. 6.5). Due to measurement errors the three h erbolas may not intersect at one point. For such a case, using more than 3 sensors should improve the localization accuracy and statistical methods must be applied. For example, Joswig [2004] uses a jackknife test to improve the results of the hyperbola method in an overdetermined case. 

Besides the hyperbola method, a circle method for 2-D manual localization that uses only the arrival times of the compressional wave can be applied (Bath 1979). Another circle method that requires the arrival times of both the compressional and the shear waves (Havskov et al. 2002) can also be used. 

We should nevertheless be aware that this method does not guarantee that all points will fall onto the same branch of a hyperbola, o... 

Fitting velocity data directly to a hyperbohc curve has several advantages over linear methods, transformed or otherwise. The major advantages are that no transformation of data is necessary, curves are fitted easily with currently available graphing software, and variations in behavior from a simple Michaelis-Menten one-substrate equation usually result in an equation which still describes a hyperbola, thus requiring no change in the analytical approach. 

Hyperbola segments described by eqn. (51) for two sets of values of and h are shown in Fig. 5. Values of g and h for a specific reaction process can be obtained using methods described by Waterman [29]. For real mixtures, values of x and y must both lie between 0 and 1 also, x + y < 1. Therefore, only those segments of the curve which lie within the triangle shown in Fig. 5 have physical meaning. Graphical procedures similar to these have been used to describe a wide variety of chemical processes [29]. The values of g and h in eqn. (51) apply to a particular type of reactor. 

For the case where is not very small, Ferguson and Vogel1 have calculated the corrections to (12) and Griinmach2 has constructed an instrument with a series of hyperbolas ruled on the plates the angle is altered until the surface of liquid coincides with one of them, when the surface tension can be calculated from (12). The method is ingenious but not capable of very high accuracy. 

If we set up the same enzyme assay with a fixed amount of enzyme but vary the substrate concentration we will observe that initial velocity (va) will steadily increase as we increase substrate concentration ([S]) but at very high [S] the va will asymptote towards a maximal value referred to as the Vmax (or maximal velocity). A plot of va versus [S] will yield a hyperbola, that is, v0 will increase until it approaches a maximal value. The initial velocity va is directly proportional to the amount of enzyme—substrate complex (E—S) and accordingly when all the available enzyme (total enzyme or E j) has substrate bound (i.e. E—S = E i -S and the enzyme is completely saturated ) we will observe a maximal initial velocity (Pmax)- The substrate concentration for half-maximal velocity (i.e. the [S] when v0 = Vmax/2) is termed the Km (or the Michaelis—Menten constant). However because va merely asymptotes towards fT ax as we increase [S] it is difficult to accurately determine Vmax or Am by this graphical method. However such accurate determinations can be made based on the Michaelis-Menten equation that describes the relationship between v() and [S],... 

Some definitions are contradictory, meaningless, without benefit or will cause much expenditure of personnel and measurement capacity, e.g. Limit of determination. This is the smallest analyte content for which the method has been vahdated with specific accuracy and precision . Apart from the fact that precision is included in the explanation of accuracy the definition manifests a fundamental inability to give a definition which is fit for practice. A useful definition of the detection and quantification limit is based on a statistical approach to the confidence hyperbola of a methods calibration curve, elaborated by the Deutsche Forschungsgemeinschaft [12]-... 

Fig. 3-7 Location of Laue spots (a) on ellipses in transmission method and (b) on hyperbolas in back-reflection method. (C = crystal, F = film, Z.A. = zone axis.)...

In either Laue method, the diffraction spots on the film, due to the planes of a single zone in the crystal, always lie on a curve which is some kind of conic section. When the film is in the transmission position, this curve is a complete ellipse for sufficiently small values of 0, the angle between the zone axis and the transmitted beam (Fig. 8-12). For somewhat larger values of 0, the ellipse is incomplete because of the finite size of the film. When 0 = 45°, the curve becomes a parabola when 0 exceeds 45°, a hyperbola and when 0 = 90°, a straight line. In all cases, the curve passes through the central spot formed by the transmitted beam. 

The conclusion of these comments is that the trial functions and the results reported in [3] do not provide a reliable description of the two-dimensional hydrogen atom ground-state energy for confinements by an angle and by a hyperbola. The solutions of the Schrodinger equation for the hydrogen atom confined by a hyperbola can also be constructed transparently and accurately using standard methods. 

In this and subsequent spreadsheet exercises, we will use the method of least-squares to fit data to a function rather than to repeat measurements. This is based on several assumptions (1) that, except for the effect of random fluctuations, the experimental data can indeed be described by a particular function (say, a straight line, a hyperbola, a circle, etc.), that (2) the random fluctuations are predominantly in the dependent parameter, which we will here call y, so that random fluctuations in the independent parameterxcanbeneglected, and (3) that those random fluctuations can be described by a single Gaussian distribution. 

When a freezing curve is obtained, the values may vary at times, due to the nonlinearity of the temperature in the entire system, defects in the temperature detection, and so on. The determination of an actual curve to fit the experimental data presents a difficult problem. Various techniques, such as the use of a flexible spline, have been employed to draw this curve. An optical method, using a lantern projector, has been employed successfully by Saylor (34). Another method that has been suggested is that given by Kienitz (35). A hyperbola is constructed from certain values of time and temperature which best represent the measured curve. In this way, the freezing point of a sample is better obtained than with the analytical or geometrical methods of evaluation via three points on the equilibrium curve. 

But since in the rectangular hyperbola KN = PM, the isothermal elasticity of a gas is equal to the pressure (2). The adiabatic elasticity of a gas may be obtained by a similar method to that used for equation (1). If the gas be subject to an adiabatic change of pressure and volume it is known that... 

The curve will have a point of inflexion when the fraction 3a(v - b)s/v = BT. By the methods already described you can show that 3a v - b)slv will be zero when v — b and that it will attain the maximum value 34a/44fe when v = 4b. Every value of v which makes (5) zero will correspond with a point of inflexion. BT may be equal to, greater, or less than 34a/446. For all values of BT between 2sa/3sb and 34a/446, there will be two points of inflexion, as shown at F and G (Fig. 89). When BT exceeds the value 34a/446, we have a branch of the rectangular hyperbola as shown for 91°. 

The rectification of a great number of curves furnishes expressions. which can only be integrated by approximation methods— say, in series. The lemniscate and the hyperbola furnish elliptic integrals of the first class which can only be evaluated in series. In the ellipse, the ratio oFJoP2 (Fig. 22, page 100) is called the eccentricity of the ellipse, the e of Ex. (3), page 115. Therefore, c = ae ) but, c2 = a2 - b2,. . b2/a2 = 1 - e2. 

If becomes unity (log a = 0) a separation of the two components by distillation is not possible. Only selective methods of separation (section 6.2) can then lead to a solution of the problem. The larger the value of a, the higher does the hyperbola of the Ideal equilibrium curve lie and the easier is the separation. 

This method was introduced by Chin Fung Kee in 1970, and had been widely used in Malaysia. The method assumes that the shape of the load-settlement curve is in the shape of a hyperbola that can be expressed as... 

Before the advent of computer technology and computational methods, the linear transformations of the Michaelis-Menten equation were extensively used for the calculation of kinetic parameters (Allison Puiich, 1979) with the aid of a linear transformation of rectangular hyperbola, one can calculate with precision the asymptotes (Kmax and Kjd by linear regression (Fig. 5). The merits of various transformations were estimated with respect to the statistical bias inherent in most linear transformations of the Michaelis-Menten equation (Wilkinson, 1961 Johanson Lumry, 1961 Johanson Faunt, 1992 Straume Johnson, 1992 Ritchie Prvan, 1996). The detailed statistical treatment of initial rate data, however, is presented in Chapter 18. 

Since the replots of slopes or intercepts versus / are nonlinear, it is not possible to determine directly the values of kinetic constants from the data in Fig. 2 instead, it is necessary to apply a differential method to rate equations, in order to obtain a graphical solution (Cleland, 1967, 1979). By using the differential method, we are raising the horizontal axis in Fig. 2 ensuring that curves become hyperbola that start at the origin ... 

An IR spectroscopic method was developed for the determination of vinyl acetate content in thick (about 100 micrometres) films of EVAs. The A3460/A3610 ratio was used for the quantitative analysis, the function of this ratio plotted against vinyl acetate content of the EVA films being a convex hyperbola. The technique was simple and rapid and did not require a complicated sample preparation procedure. 11 refs. 

Two general classes of methods are considered. The first is semlempirical models in which equations that describe the data on either critical or safe arrays in terms of some selected set of array parameters are written. Most of these methods (as, for example, the density analog technique) are directly based on comparing arrays to single units, to addition to the density analog, these models include the surface density and equilateral hyperbola techniques. The second Is array unit interaction models, which are based on equations written to describe the neutron balance for each unit. The solution... 

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