Roughness/structural stability

The notion of roughness/structural stability can be extended to the highdimensional case without any problem. However, some other problems do arise here when we need to find out explicitly the necessary and sufficient conditions for roughness. We have remarked that Andronov and Pontryagin, as well as Peixoto, had used the classification of proper two-dimensional systems in an essential way. So, we must stop here to get acquainted with some basic notions and facts from the general theory of dynamical systems. 

The structural stability of mixed-metal hemoglobin hybrids also has allowed us to study low-temperature electron transfer in this system. We first reported the temperature dependence of triplet-state quenching within the [ (ZnP), Fe (H20)P] hybrids, which we attributed to the ZnP Fe P ET reaction [7d]. The rate constant dropped smoothly as the temperature was lowered from room temperature to 200 K. Below this temperature the rate constant remained roughly constant with a tunnelling rate constant of kt 9 s (Fig. 7). 

Note that in this proof, we use essentially the roughness (versus structural stability) of the system, i.e. we assume that the homeomorphism establishing the topological equivalence of sufficiently closed systems is close to identity. However, without this assumption the claim is still true, though the proof becomes more involved. 

Resuming our consideration, we may make a preliminary conclusion typical dynamical systems are divided into two basic classes depending on whether the system has a finite number of periodic orbits in a boimded sub-region of its phase space or the niunber of periodic orbits is infinite. In the first case such systems are usually called systems with simple dynamics. The second class is composed of systems with complex dynamics. The notion of roughness or structural stability is easily applied to the systems with simple dynamics. The situation for systems with complex dynamics is more uncertain. 

The modern theory of bifurcations of dynamical system is directly linked to the notion of non-roughness, or structural instability of a system. The main motivation is that the analysis of a system will be rather incomplete if we restrict our consideration to only the regions of structural stability of the system. Indeed, by changing parameters we can move from one structurally stable system to another, a qualitatively different system, upon crossing some boundaries in the parameter space that correspond to non-rough systems. 

In the two-dimensional case, rough systems compose an open and dense set in the space of all systems on a plane. The non-rough systems fill the boundaries between different regions of structural stability in this space. This nice structure allows for a mathematical description for transformations of... 

Consider some finite-parameter family of smooth systems Xg, where e = ( 1,..., 6p) assumes its values from some region V e W. If is non-rough, then q is said to be a bifurcation parameter value. The set of all such values in V is called a bifurcation set. It is obvious that once we know the bifurcation set, we can identify all regions of structural stability in the parameter space. Hence, the first step in the study of a model is identifying its bifurcation set. This emphasizes a special role of the theory of bifurcations among all tools of nonlinear dynamics. 

An explicit mathematical formulation to the finite-parameter approach to the local bifurcations was given by Arnold [19], based on the notion of versal families. Roughly speaking, versality is a kind of structural stability of the family in the space of families of dynamical systems. Different versions of such stability are discussed in detail in [97]. 

Note that in many special cases attention is restricted to the study of the smaller spaces of systems, e.g. systems with some specified symmetries, Hamiltonian systems, etc. In view of that, the notion of structural stability in, say, Hamiltonian systems with one-degree-of-freedom becomes completely meaningful. So, for example, equilibrium states such as centers and saddles of such systems, become structurally stable. Moreover, if there are no heteroclinic cycles containing different saddles, we can naturally distinguish such systems as rough in the set of all systems of the given class. 

Obviously, any statement concerning a real system derived from an analysis of its theoretical idealization, i.e. from its model, must not be too sensible to small uncontrolled variations of the parameters. Hence, it is a standard requirement that one must consider not only a stand-alone system but must also understand what happens with all neighboring systems. This works well for rough (structurally stable) equilibrium states and periodic orbits in this case the qualitative structure is not modified by small perturbations of the right-hand side of the system. In contrast, for systems on the stability boundary, the analysis of close systems may become a real problem. 

A study of the lithium-ammonia reduction of 14-en-16-ones would extend our understanding of the configuration favored at C-14 in metal-ammonia reductions. Although several simple 14-en-16-ones are known, their reduction by lithium and ammonia apparently has not been described in the literature. Lithium-ammonia reduction of A-nortestosterone, a compound that structurally is somewhat analogous to a 14-en-16-one, affords roughly equal amounts of the 5a- and 5 -dihydro-A-nortestosterones. " This finding was interpreted as indicating that there is little difference in thermodynamic stability between the two stereoisomeric products. 

The current-voltage and luminance-voltage characteristics of a state of the art polymer LED [3] are shown in Figure 11-2. The luminance of this device is roughly 650 cd/m2 at 4 V and the luminous efficiency can reach 2 lm/W. This luminance is more than adequate for display purposes. For comparison, the luminance of the white display on a color cathode ray tube is about 500 cd/m2l5J. The luminous efficiency, 2 lm/W, is comparable to other emissive electronic display technologies [5], The device structure of this state of the art LED is similar to the first device although a modified polymer and different metallic contacts are used to improve the efficiency and stability of the diode. Reference [2] provides a review of the history of the development of polymer LEDs. 

Many complex ions, such as NH4+, N(CH3)4+, PtCle", Cr(H20)3+++, etc., are roughly spherical in shape, so that they may be treated as a first approximation as spherical. Crystal radii can then be derived for them from measured inter-atomic distances although, in general, on account of the lack of complete spherical symmetry radii obtained for a given ion from crystals with different structures may show some variation. Moreover, our treatment of the relative stabilities of different structures may also be applied to complex ion crystals thus the compounds K2SnCle, Ni(NH3)3Cl2 and [N(CH3)4]2PtCl3, for example, have the fluorite structure, with the monatomic ions replaced by complex ions and, as shown in Table XVII, their radius ratios fulfil the fluorite requirement. Doubtless in many cases, however, the crystal structure is determined by the shapes of the complex ions. 

C13-0124. Design a protein containing ten amino acids whose tertiary structure would be roughly spherical with a hydrophobic interior and a hydrophilic exterior. Include one S—S bridge that would help stabilize the structure. 

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