Line of fixed points

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. 

The asymptotics discussed above force both the concentrations of the substrate a = c- r - s - 2[RS] and the product rs to vanish ultimately. These two conditions define a line of fixed points in the three-dimensional r - s - [RS] phase space. If the initial state has a prejudice to R enantiomer such as ro > s(l, then the system ends up on a fixed line r + 2[RS] = c on a s = 0 plane, as shown in Fig. 5a. Otherwise with ro < so> the system flows to another fixed... 

Each of these systems has a fixed point x — 0 with f x ) = 0. However the stability is different in each case. Figure 2.4.1 shows that (a) is stable and (b) is unstable. Case (c) is a hybrid case we II call half-stable, since the fixed point is attracting from the left and repelling from the right. We therefore indicate this type of fixed point by a half-filled circle. Case (d) is a whole line of fixed points perturbations neither grow nor decay. 

Something dramatic happens when a = 0 (Figure 5.1.5d). Now (la) becomes x(z) = Xq and so there s an entire line of fixed points along the x-axis. All trajectories approach these fixed points along vertical 1 ines. 

If A = 0, at least one of the eigenvalues is zero. Then the origin is not an isolated fixed point. There is either a whole line of fixed points, as in Figure 5.1.5d, or a plane of fixed points, if 4 = 0. 

Fig. 9.10. Drift velocity field determined from Eqs.(9.43), (9.42), (9.48) for (a) dp/A = 0.45, (b) dp/X = 1.0. Thin solid lines represent lines of fixed points that satisfy Eq. (9.49) (compare text). Thick solid lines depict trajectories of the spiral center computed for the Oregonator model (9.1) with fc/ , = 0.02 and t = 0 [53].

Actually, we must observe that this principle of stability and universality of the critical exponents is very general and applies to a great number of critical phenomena, as was shown by Griffiths.31 As we shall see later, the critical exponents are related to the existence of fixed points for the Hamiltonians describing the systems under study. But, in general, these fixed points are isolated, and it is only in very special circumstances that there exist lines of fixed points, along which the critical exponents may vary in a continuous manner. 

Figure 3. a) The lines of fixed points of the SAWs on a lattice with point-like uncorrelated disorder (see the text). 

As far as stationary points of trajectory bundles of distillation at finite reflux lay on trajectories of reversible distillation, these trajectories were also called the lines of stationarity (pinch lines, lines of fixed points) (Serafimov, Timofeev, Balashov, 1973a, 1973b). These lines were used to deal with important applied tasks connected with ordinary and extractive distillation under the condition of finite... 

On the other hand, for the simpler system with a propagating front without periodic pattern [lO] it turns out, that the fixed point at r coincides with the end-point r of the line of fixed point r < r, rj=0. In this case the marginal stability criterion gives exactly the operating point of the system. 

The KTTS depends upon an absolute 2ero and one fixed point through which a straight line is projected. Because they are not ideally linear, practicable interpolation thermometers require additional fixed points to describe their individual characteristics. Thus a suitable number of fixed points, ie, temperatures at which pure substances in nature can exist in two- or three-phase equiUbrium, together with specification of an interpolation instmment and appropriate algorithms, define a temperature scale. The temperature values of the fixed points are assigned values based on adjustments of data obtained by thermodynamic measurements such as gas thermometry. 

Fig. 6.3 Schematic phase diagram for lamellar PS-PB diblocks in PS homopolymer (volume fraction 0h). where the homopolymer Mv is comparable to that of the PS block (Jeon and Roe 1994). L is a lamellar phase, I, and I2 are disordered phases, M may correspond to microphase-separated copolymer micelles in a homopolymer matrix. Point A is the order-disorder transition.The horizontal lines BCD and EFG are lines where three phases coexist at a fixed temperature and are lines of peritectic points. The lines BE and EH denote the limit of solubility of the PS in the copolymer as a function of temperature.

Low (<8.5) pH Solutions. The Np solution concentrations are plotted as a function of measured pe in Figure 2 for various equilibration periods. The data points for the low pH (<8.5) suspensions were fitted by a linear least squares method to a line of fixed unit slope. When the slopes were allowed to vary, their values ranged from 0.8 to 1.1. 

One of the main goals of this book is to help you develop a solid and practical understanding of bifurcations. This chapter introduces the simplest examples bifurcations of fixed points for flows on the line. We ll use these bifurcations to model such dramatic phenomena as the onset of coherent radiation in a laser and the outbreak of an insect population. (In later chapters, when we step up to two-and three-dimensional phase spaces, we ll explore additional types of bifurcations and their scientific applications.)... 

Fig. 15.15. Phase differences 4>i (left) and phase profile Bi — 6 (b,d) in units of 27t for a chain of 500 phase oscillators (15.16) with uniformly distributed frequencies wi [—0.1,0. ] and open boundaries. (a,b) Antisymmetric coupling (15.14) with 7 = 0. (c,d) unidirectional coupling (15.17) with F(< ) = Q(). (e) System (15.14) with 7 = 2. (f) Transfer map TQ[i) (15.20) for Eqs. (15.14) with 7 = 2 (solid line) and fixed points 4> (filled circles).

The ensemble of fixed points (points, lines or planed) of a symmetry operation are called symmetry elements. To the fixed point of l or S , we must add the fixed line corresponding to the operation A or the plane which is perpendicular to it. Rotation axes correspond to the operations A , centers and rotoinversion axes to the operations l , and mirror planes and rotoreflection axes to the... 

In the quaternary system water-dodecane-pentanol-SDS, the critical behavior is much more complex. As mentioned in Sec. II. B, at fixed temperature T = 21°C, this quaternary mixture presents a line of critical points that extends in the phase diagram between a critical point belonging to the ternary mixture water-SDS-pentanol and a critical endpoint P . located in the oil-rich part of the diagram. The X values (expressed in weight) corresponding to P p and P are 0.95 and 6.6, respectively. 

Now, consider the case that the restriction of fixed points is withdrawn, and we are able to estimate the integral from the area under a straight line that joins any two points on the curve. By choosing these points in proper positions, a straight line that balances the positive and negative errors can be drawn, as illustrated in Fig. 4.5. Asa result, we obtain an improved estimate of the integral. 

Suppose that, near some fixed point G F, dT, the graph F, is a straight line segment parallel to the x axis. Let G (0, T) be an arbitrary fixed point and let Re C denote the ball of a sufficiently small radius with centre (a °,t°). First, we examine the smoothness of the function X = (IF, w). Let D stand for a first-order derivative and let (p denote an arbitrary smooth function in i 2s such that p = 0 outside Rzeji 0 < (> < 1, and dpidy = 0 on F. ... 

Pressure. Standard atmospheric pressure is defined to be the force exerted by a column of mercury 760-mm high at 0°C. This corresponds to 0.101325 MPa (14.695 psi). Reference or fixed points for pressure caUbration exist and are analogous to the temperature standards cited (23). These points are based on phase changes or resistance jumps in selected materials. For the highest pressures, the most rehable technique is the correlation of the wavelength shift, /SX with pressure of the mby, R, fluorescence line and is determined by simultaneous specific volume measurements on cubic metals... 

Fig. 3. Profitabihty diagram for Venture A. (a) Simple diagram. NRR is net return rate IRR, the internal rate of return, is a given fixed point, (b) Three NRR cutoff lines for Venture A where B, C, and D represent NRR values of 15, 10, and 5%/yr, respectively. For example, at a discount rate of 10% per year, the NRR cutoff for Venture A could be as high as 10.74% per year for marginal acceptance (point X). Acceptable levels are to the left of NRR cutoff...

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

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