Transverse family

An ideal case (and this is possible sometimes) would be one where all close transverse families are qualitatively equal (such families were called versal due to Arnold [20]). Having studied bifurcations in one transverse family, we would have a complete description for all neighboring ones. 

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

Moreover, in more complex cases the problem of presenting a complete description, or of proving that a family imder consideration is versal, is not even set up. However, the general approach remains the same a bifurcating system is considered as a point on some smooth bifurcation surface of a finite codimension. Then, a transverse family is constructed and the qualitative... 

Consider first the case where the first Lyapunov value I2 is non-zero. Following the scheme outlined in the preceding section, we first derive the equation of the boundary of the stability region near e = 0. Next we will find the conditions under which is a smooth surface of codimension one. Finally, we will select the governing parameter and investigate the transverse families. 

Suppose next that Z2 = Z3 = 0 and Z4 0. Then, the transverse family assumes the form... 

Consequently, the right-hand side of the system on the center manifold will contain no terms with even power of x. The associated transverse family can then be represented in the form... 

The other mechanism frequently encountered in applications is when it is known a priori that the equilibrium state does not disappear in the bifurcation. If it resides at the origin, then the transverse family has the form... 

I2 = 0, Z3 0. The transverse family in this case assumes the form... 

Then, in the case of general position the surface (which corresponds to equilibrium states with a pair of purely imaginary eigenvalues and with the first k — 1) zero Lyapunov values equal to zero) is a -smooth surface of codimension k passing through the point e = 0 in the parameter space. All transverse families in this case depend on k governing parameters /io,..., /ifc-i and may be written in polar coordinates as follows ... 

Remark 6. It follows from the above analysis that all properties of the Poincare map of the transverse family are exhibited by the simplified map... 

Note that she did not use finite-parameter families. Nevertheless, it is understood that an appropriate transverse family can be found such that the governing parameters are /i, e,, crj,..., (Tn-i in the first case of Theorem 13.5, or /i, e, si, 0 1,..., in the second case. 

In the case of a saddle-focus, all possibilities allowed by Theorems 13.11 and 13.12 are encountered in any transverse family The bifurcation... 

Figure 10.2 A family of short, transverse corrosion-fatigue cracks originating on the external surface.

As stated in Section H.A, the moment free energy does not give exact results beyond the onset of phase coexistence—that is, in the regime where the coexisting phases occupy comparable fractions of the total system volume. As shown in Section III.A, the calculated phases will still be in exact thermal equilibrium but the lever rule will now be violated for the transverse degrees of freedom of the density distributions. This is clear from Eq. (11) In general, no linear combination of distributions from this family can match the parent p (a) exactly. 

Figure 16. Extended dislocations. The thickness of the sample increases from right to left perpendicular to the first family, and from bottom to top past the unique transversal line.

Figure 17. Schematic fitting of regions of different thickness in Figure 16. Horizontal first family of lines vertical unique transversal line.

There are some transport pathways for mRNA export which do not require an importin/3 family member. Mex67/TAP is a factor with a direct role in mRNA export (Strasser and Hurt, 2000 Stutz et al., 2000 Nakielny and Dreyfuss, 1999). Mex67/TAP binds RNA and Nups and has the ability to undergo nucleocytoplasmic shuttling. Another factor might be involved (Hodge et al., 1999). How the actions of these factors are coordinated to mediate mRNA export is presently unknown. Once recognized by the export machinery, RNAs must transverse the nuclear pore. 

D-Ala-deltorphin-I and -II transverse the blood brain barrier in vivo and in vitro [51]. Recently, D-Ala-deltorphin-II was identified as a transport substrate of organic anion transporting polypeptides (Oatp/OATP), a family of polyspecific membrane transporters, strongly expressed at the rat and human blood brain barrier [52]. Modified analogues of these peptides were synthesized to improve their transit across the blood brain barrier [48,49,53]. Because they resist enzyme degradation and can cross endothelial barriers into the CNS, the deltorphins meet the criteria for peptides with potential for systemic administration. 

TTF type. The optical anisotropy of such two-dimensional conductors and their electron parameters may also be deduced from reflectance studies. As an example, from the (TMTSF)2X family we present the polarized reflectance of (TMTSF)2PF6 at three temperatures (Fig. 7). It is evident that optical anisotropy decreases at low temperature, and a reasonably well-defined plasma edge appears in the b direction at 25 K. The transverse reflectance edge appears at the frequency about 10 times lower than that of the stacking axis edge (tb< = 22 meV, about 10 times smaller than ta) [46]. Drude parameters for typical (TMTSF)2X salt are eq = 3.5, 1500 cm-1 < cop < 2000 cm-1, 250 cm-1 < y < 500 cm-1, and tb = 0.02 eV. 

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