Tangent series

Figure A2.5.16 shows the ooexistenoe ourve obtained from equation (A2.5.16). The logaritluns (or the hyperbolio tangent) oan be expanded in a power series, yielding...

Asymptotes The hmiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that... 

Figure 16 Loss tangent (tanS) at 3 Hz, as a function of temperature for a series of main chain thermotropic polyesters having oxyethylene spacers.

First-order kinetics. Consider a first-order reaction studied with a series of initial concentrations, as depicted here. Show that the initial rate tangents come to a common intercept t" on the x axis, and find an expression for what time this is. 

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

Figure 6 Loss tangent (<5) plotted as a function of complex modulus G for a series of syndiotactic and isotactic polypropylene. Reproduced with permission from Rojo et al. [32]. Copyright Wiley-VCH Verlag GmbH Co. KgaA. iPP, isotactic PP sPP, syndiotactic PP Met-iPP, metallocene-catalyzed PP (see Ref. [32] for full details).

Thus, by drawing the tangent at a series of points on the curve BDC and measuring the corresponding slope — uc and intercept OT, it is possible to establish the solids flux f for any concentration ( ((, < C < Cmax). 

The tangent formula [3] is a key formula in direct methods that lets us refine phase values and determine new ones. Consider the situation in which we have a series of triplets with a common reflection They can be written... 

The orbits from Venus to Ceres are represented by the unimodular series 4. In the outer system the Ford circles of only Uranus and Neptune are tangent, but the likeness to Farey sequences in atomic systems is sufficient to support the self-similarity conjecture. 

When an a.c. voltage is applied to a perfect capacitor, no energy is dissipated. However, a real capacitor dissipates energy because of lead and electrode resistances, d.c. leakage resistance and, most importantly, dielectric losses. These account for the capacitor s dissipation factor or loss tangent tan 3. It is sometimes convenient to regard the lossy capacitor as an ideal capacitor shunted by a resistance Rp or in series with a resistance rs, as shown in Fig. 5.5. 

In many investigations dynamic-mechanical properties have been determined not so much to correlate mechanical properties as to study the influence of polymer structure on thermo-mechanical behaviour. For this purpose, complex moduli are determined as a function of temperature at a constant frequency. In every transition region (see Chap. 2) there is a certain fall of the moduli, in many cases accompanied by a definite peak of the loss tangent (Fig. 13.22). These phenomena are called dynamic transitions. The spectrum of these damping peaks is a characteristic fingerprint of a polymer. Fig. 13.23 shows this for a series of polymers. 

The substituent effects in various unsymmetrical series of [3C (X,Y,Z)] are schematically shown in Fig. 15, where pXr values for various series are plotted against the sum of a with an r value of 0.76, in reference to the linear a plot for the symmetrical (trisaryl) series [3(X = Y = Z)]. All the unsym-metrically substituted series show significantly concave (quadratic) correlations, which intersect with the linear plot of the [3C (X = Y = Z)] series as tangent at the point of X = Y = Z. This implies that the p value of any series based on the tangent at the point (X = Y), where the varying X substituent is the same as the fixed Y substituent, should be identical to the p value for the symmetrical series [3C (X = Y = Z)]. 

A more selective approach consists in trying to influence the kinetics of formation of at least one network in this case, the two networks are formed more or less simultaneously, and the resulting morphology and properties can be expected to vary to some extent without changing the overall composition. The same system as previously studied, PUR/PAc, has been utilized in order to prepare a series of in situ simultaneous IPNs (SIM IPNs), by acting essentially on two synthesis parameters the temperature of the reaction medium and the amount of the polyurethane catalyst. Note that the term simultaneous refers to the onset of the reactions and not necessarily to the process. The kinetics of the two reactions are followed by Fourier transform infra-red (FTIR) spectroscopy as described earlier (7,8). In this contribution, the dynamic mechanical properties, especially the loss tangent behavior, have been examined with the aim to correlate the preceding synthesis parameters to the shape and temperature of the transitions of the IPNs. 

Table I. Storage modulus and loss tangent values at room temperature and at 100 C for a series of 25/75 SIM PUR/PAc IPNs obtained with various amounts of stannous octoate...

As noted previously by Senich et al. (8), the usefulness of the DSA technique in following the cure of epoxy curing systems stems from the fact that a relative maximum Is produced in the composite loss tangent when the viscosity of the resin reaches a predetermined value. In a previous study (23) which considered a series of polystyrenes of varying molecular weights characterized... 

For low potentials several tens of mV for z = 1. the hyperbolic tangent may be replaced by the first, linear, term of its series expansion [A2.9), to give... 

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