Sine map

Plot the graph of the sine map = rsinstA, for 0 < r < I and 0 < x < 1, and compare it to the logistic map. Then plot the orbit diagrams for both maps, and list some similarities and differences. 

Figure 10.6.2 shows the orbit diagrams for the sine map (top panel) and the logistic map (bottom panel). The resemblance is incredible. Note that both diagrams have the same vertical scale, but that the horizontal axis of the sine map diagram is scaled by a factor of 4. This normalization is appropriate because the maximum of rsin nx is r, whereas that of rx(l - x) is r. 

The first two exercises deal with the sine map x , = r sin rx , where 0 < r < 1 and X e [0,1]. The goal is to learn about some of the practical problems that come up when one tries to estimate <5 numerically. 

Superstable cycles to the rescue) The critical slowing down encountered in the previous problem is avoided if we compute instead of r ,. Here denotes the value of r at which the sine map has a superstable cycle of period 2". ... 

Qualitative universality of patterns) The U-sequence dictates the ordering of the windows, but it actually says more it dictates the iteration pattern within each window. (See Exercise 10.4.7 for the definition of iteration patterns.) For instance, consider the large period-6 window for the logistic and sine maps, visible inFigure 10.6.2. 

Sinee the poorest performanee of the assembly distribution would oeeur when shifted, Cpk values rather than Cp values are better design targets. The Cp value ean be used as a target for the assembly based on the severity of applieation and minimum failure eost of 0.01% of the total produet eost as determined by the Conformability Map. If only eapable solutions are to be generated, whieh have a minimum proeess eapability index of Cp =1.33 (or 30 ppm) for both eomponent and assembly distributions, then the number of eomponents in the assembly staek ean be as low as three using the proposed statistieal model (Harry and Stewart, 1988). The overall requirement is... 

Sinee there are no methods whieh are guaranteed to loeate all TSs (short of mapping the whole surfaee, whieh is impossible for more than three or four variables), it is essentially impossible to prove that a TS does not exist. The failure to locate a TS eonnecting two minima may simply be due to the inability to generate a sufficiently good trial structure for NR methods, or interpolation methods converging to a TS not eonneeting the two desired minima. 

Alternative approaches to mapping secondary structure have also been reported. These include alanine-and lysine-scanning mutagenesis, both of which do not rely on chemical modification (Yan et al., 1999 Sine et al., 2002). 

The abnormal T- and B-cell functions in patients with SCID are the result of ADA dehciency. The ADA gene has been mapped to chromosome 20q.l3, and a number of point and deletion mutations have been identihed in SCID patients [5-7]. ADA catalyses the irreversible deamination of adenosine and 2 -deoxyadenosine to inosine and 2 -deoxyi-nosine as a part of purine nucleoside metabolism. Adenosine and deoxyadeno-sine are suicide inachvators of S-adenosyl-homocysteine (SAH) hydrolase, and lead indirectly to intracellular accumulation of SAH, which is a potent inhibitor of methy-lation reactions. Cellular methylation function is essential for detoxihcation of adenosine and deoxyadenosine. As a result ADA dehciency leads to accumulation to... 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

Lithic artifacts were collected from three sites south of the industrial town of Sines along the western coast of the southern Portuguese Alentejo province (see Map I). Two of these sites, Samouqueira and Palheiroes do Alegra, are 30 km apart on cliffs above the modem sea shore the third site, Fiais, is 12... 

The Fourier theorem states that any periodic function may be resolved into cosine and sine terms involving known constants. Since a crystal has a periodically repeating internal structure, this can be represented, in a mathematically useful way, by a three-dimensional Fourier series, to give a three-dimensional Fourier or electron density map. In X-ray diffraction studies the magnitudes of the coefficients may be derived from... 

Fourier synthesis The summation of sine and cosine waves to give a periodic function an example is the computation of an electron density map from waves of known phase, frequency, and amplitude F (. See also the definition in Chapter 1. 

The intensity differences between the Bragg reflections hkl and hkl suggested to Yoshi H. Okaya and Raymond Pepinsky their use in a sine function (rather than the cosine function of the normal Patterson map) to give an analogue of the Patterson map. This sine-Patterson map has some interesting properties (see Figure 14.27). Peaks in the map represent vectors from the anomalously scattering atom, A, to all other atoms. 

X, but now the vectors between atoms have a distinguishable direction. When peaks in the sine-Patterson map are positive they represent A- X vectors, while the holes (troughs) in the map represent X- A vectors. 

FIGURE 14.27. (a) The cosine (normal) and (b) the sine Patterson maps of potassium isocitrate (Ref. 13). Peaks in these maps are marked with filled circles if they are positive and open circles if they are negative. Note that the high K- -K peaks at u = 0.25, V — 0.50 and u = 0.50, v = 0.0 (present in the cosine Patterson map), are missing in the sine Patterson map. Note that each vector in the sine Patterson map has a sign (positive or negative). 

Thus, by comparing peaks in a normal Patterson map with those in the sine-Patterson function, a sign for each peak can be found that directly gives the absolute structure of the crystal. This is shown in the example in Figure 14.27. 

To connect the two markedly different scenarios observed in the static and the well-mixed environments, it is natural to analyze the role of increasing mobility (Reichenbach et al., 2007). Karolyi et al. (2005) studied the above competition model combined with dispersion by a chaotic map that represents advection of fluid elements in the alternating sine-flow. By continuously changing the frequency of the chaotic dispersion as a control parameter, it is possible to follow the transitions between the two limiting situations. When the chaotic mixing is much faster than the local population dynamics, the killer and resistant cells gradually disappear from the population and only the sensitive cells survive. This is because the killer cells... 

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