Equations line breaks

Equation (16) stops to be valid when x reaches zero, i.e., the two lines collide. Beyond this point, the line breaks up into a number of segments, each segment ending at a pair of pinching points. The velocity of the pinching points should be of the order, of... 

When an equation is too long to fit on one line, break it after an operator that is not within an enclosing mark (parentheses, brackets, or braces) or break it between sets of enclosing marks. Do not break equations after integral, product, and summation signs after trigonometric and other functions set in roman type or before derivatives. 

Moreover, the sediment balance that leads to the one-line governing equation, only accounts for longitudinal transport induced by wave breaking. To include the effect of other sources of sediment (e.g., artificial nom-ishment or river discharge) a nonhomogeneous equation has to be solved. 

One of the most striking features of Fig. 2.10 is the change of slope. Let us begin our discussion of this feature by examining the slopes of the linear portions of these lines on either side of the break. Representing the equation of a straight line by y = ax -i- b, we can write an empirical equation for the phenomena shown in Fig. 2.10 as... 

Another line of study was started by Quincke3 It has been found, experimentally, that the breaking load Ww of wires having different radii r could be expressed by the equation... 

EXAMPLE 7.4 Determination of Surface Excess Concentration from Surface Tension Data. The slope of the 25°C line in Figure 7.15 on the low-concentration side of the break is about -16.7 mN m 1. Calculate the surface excess and the area per molecule for the range of concentrations shown. How would Figure 7.15 be different if accurate measurements could be made over several more decades of concentration in the direction of higher dilution Could the data still be interpreted by Equation (49) in this case ... 

At a certain critical load, the crack spontaneously breaks through to the free surface (the cut-through ), usually with additional crack depth increase D, and changes into a well developed semicircle (solid-line semicircle). This transformation may also be induced in another, stable, manner, e.g., by the action of rosette type stresses around the strain region while loading the indenter before the cut-through . The model given in Fig. 6.2.8 is described by the equation... 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

We expected that the adsorption isotherms could be described with help of Dubinin s equation, but analysis of adsorption isotherms was shown that ones have a breaking of line in region of low pressure and this single-parameter equation isn t able to describe them exactly. 

The assumptions and approximations used in deriving Eq. (3.42) break down at low temperatures or at high-applied voltages. At room temperature and at low applied voltage the values of J(V) obtained using this equation agree closely with the numerical solutions. Therefore Eq. (3.42) should also yield Arrhenius straight lines if log I values are plotted as a function of l/T. Recent work has shown that this is indeed the case [49],... 

Figure 5 Pure dephasing widths, l/inT]), of the asymmetrical CO-stretching mode of Rh(CO)2acac in DBP versus temperature on a log plot. The solid line through the data is a tit to Equation (4), the sum of a power law and an exponentially activated process. The inset is an Arrhenius plot at higher temperatures showing that the process is activated. Note that there is no break at the experimental glass transition temperature, 169 K. The best fit has the power law exponent, a = 1.0, and the activation energy, AE = 385 cm-1.

Figure 32. The a-relaxation times for the glass formers studied in the present work (cf. Fig. 27). In addition data of diglycidyl ether of bisphenol A (DGEBA) and phenyl glycidyl ether (PGE) are included time constants as obtained from DS data sets of m-TCP and 2-picoline were combined with xrl from conductivity and light scattering measurements, respectively, (a) Relaxation times as a function of T Ts. The systems differ by the slope of Ta at Tg. (b) By plotting xr, as a function of the rescaled temperature z = m(T/Tg — 1) the effect of an individual fragility is removed and a master curve is obtained for systems with similar To. Solid line represents Eq. (41) with Kf) — 17. (c) Upper part master curve for xa according to Eq. (42). Deviations of the data from Eq. (42) (solid line) indicate break-down of the VFT equation. Lower part The ratio lg(ra/rvft) shows deviations from a VFT behavior most clearly. Dashed vertical lines indicate shortest and fastest tx, respectively, observed. All the figures taken from Ref. [275].

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