Structure invariants

For z-invariant structures the spatial field evolution is simply ruled by... 

A slightly larger fragment containing five D-mannose and two hexosamine residues per molecule seems to be common to all of the glycopeptides that have been isolated from ovalbumin. A useful step forward would be to ascertain whether this fragment has an invariant structure. 

From the structural point of view, manninotriose has been of critical significance in relationship to its parent tetrasaccharide, stachyose. Almost invariably, structural studies on manninotriose have gone hand-in-glove with complementary studies on stachyose. 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

This behavior reminds us of chaotic itinerancy found in dynamical systems with many degrees of freedom [18,19,21,38]. Chaotic itinerancy is the behavior where orbits repetitively approach and leave invariant structures of the phase space. Such behavior has been found in coupled maps [19], turbulence [18], neural networks [38], and Hamilton systems [21]. The mechanism of chaotic itinerancy is not yet fully understood. The study of NHIMs and how their stable and unstable manifolds intersect could offer some clues in revealing its mechanism [20]. 

The conventional theory of reaction processes relies on equilibrium statistical physics where the equi-energy surface is uniformly covered by orbits as shown in Fig. 34. To the contrary, the phase space in multidimensional chaos has various invariant structures, and orbits wander around these structures as shown in Fig. 35. In these processes, those degrees of freedom that constitute the movement along stable or unstable manifolds vary from NHIM to NHIM. Their variance reveals how reaction coordinates change during successive processes in reaction dynamics. 

To characterize these invariant structures and the changes of reaction coordinates, the concept of finite-time Lyapunov exponents can be useful [44]. The original definition of the Lyapunov exponents needs ergodicity (see, e.g.. Ref. 45) to make sure that the time average of the exponents converges. However, for chaotic itinerancy, the exponents would not converge. Moreover, the finite-time Lyapunov exponents can be more useful to detect whether... 

Nonlinear resonances are important factors in reaction processes of systems with many degrees of freedom. The contributions of Konishi and of Honjo and Kaneko discuss this problem. Konishi analyzes, by elaborate numerical calculations, the so-called Arnold diffusion, a slow movement along a single resonance under the influence of other resonances. Here, he casts doubt on the usage of the term diffusion. In other words, Arnold diffusion is a dynamics completely different from random behavior in fully chaotic regions where most of the invariant structures are lost. Hence, understanding Arnold diffusion is essential when we go beyond the conventional statistical theory of reaction dynamics. The contribution of Honjo and Kaneko discusses dynamics on the network of nonlinear resonances (i.e., the Arnold web), and stresses the importance of resonance intersections since they play the role of the hub there. 

Where Cf is a structural parameter that counts available reactive carbon sites and c, is a coefficient that account for distribution of reactive carbon sites types and catalytic effects and thus a variation in c, may change kinetic parameters. The gas composition vector (F) in J X) may beside the CO2 partial pressure also include such gas partial pressures as KOH since the likelihood that a catalytic site is activated is a function of the partial pressure of the catalyst, the site-catalyst attraction forces and the temperature. Since from Eq. (2) the structural profile invariance SPI assumption is by no means obvious we suggest that a temperature and partial pressure range are always given for the validity of the structural profile invariance assumption. Only if the invariant structural profile assumption is approximately valid a reference profile (/ /) can be used to eliminate the structural profile to form a normalised reactivity (R ) and to determine kinetics up to a constant... 

MD simulations with a constant energy is nothing but Hamiltonian dynamics. Recent accumulation of MD simulations will certainly contribute to our further understanding of Hamiltonian systems, especially in higher dimensions. The purpose of this section is to sketch briefly how the slow relaxation process emerges in the Hamiltonian dynamics, and especially to show that transport properties of phase-space trajectories reflect various underlying invariant structures. 

It is true that the hyperbolic system is an ideal dynamical system to understand from where randomness comes into the completely deterministic law and why the loss of memory is inevitable in the chaotic system, but generic physical and chemical systems do not belong strictly to such ideal systems. They are not uniformly hyperbolic, meaning that invariant structures are heterogeneously distributed in phase space, and there may not exist a lower bound of instability. It is believed that dynamical systems of such classes are certainly to be explored for our understanding of dynamical aspects of all relevant physical and chemical phenomena. 

To summarize, variable structural features of lysins suggest ways to attack the elucidation of the molecular mechanism of species-specific sperm-egg recognition in abalones. The invariant structural features of lysins suggest ways to explore the molecular mechanism lysin uses to destroy nonenzymatically the integrity of the VE to allow the sperm to pass through this protective envelope and contact the egg cell membrane. 

The subtle question has been addressed in some detail as to whether the cone conformers of calix[4]arenes exist in solution as time-invariant structures with C4 symmetry or as rapidly interconverting structures with C2 symmetry, as illustrated in Figure 4.7. 

Other aspects of structure - Although the nucleosome itself is a nearly invariant structure in eukaryotes, the length of DNA between nucleosomes may vary from about 20 bp to over 100 bp. Exactly what determines the arrangement of nucleosomes along the DNA is still not understood completely. However, it is now clear that at least some nucleosomes occupy defined positions. For... 

The heme group has such a characteristic and largely invariant structure that its properties are only slightly modulated by the proteins into which it is incorporated. That is, the g value is smaller when the external magnetic field is in the heme plane than when the field lies along the z... 

Our method makes it possible to use simple linear rules for exploring complicated nonlinear systems. A simple application is the study of connectivity among various chemical species in complicated reaction networks. In the simple case of homogeneous systems with time-invariant structure, the susceptibility matrix x = [Xhh ] = X depends only on the transit time and not on time itself. The matrix elements Xuu ( ) are proportional to the elements (t) of a Green function matrix G (t) = [G / (t)], which is the exponential of a connectivity matrix K, that is, G (t) = exp [tK]. It follows that from a response experiment involving a system with time-invariant structure, it is possible to evaluate the connectivity matrix, K, which contains information about the relations among the different chemical species involved in the reaction mechanism. The nondiagonal elements of the matrix K = Kuu I show whether in the reaction mechanism there is a direct connection between two species in particular, if Kuu 0, there is a connection from the species u to the species u the reverse connection, from u io u, exists if Ku u 0. 

Mandelbrot [2, 3] systematized and organized mathematical ideas concerning complex structures such as trees, coastlines and non-equilibrium growth processes. He pointed out that such patterns share a central property and symmetry which may be called scale invariance. These objects are invariant under a transformation, which replaces a small part with bigger part that is under a change in a scale of the picture. Scale-invariant structures are called fractals [7]. More recently the relevance of natural and mathematical structure has become clearer with the help of computer simulation. Self-similarity turns out to be a general invariance principle of these structures. 

Another difficulty in the correct estimation of the kinetic parameters of deformation is the need to maintain the invariable structure of polymers during measurement, whereas structural changes are inevitable in the process of considerable changes in deformation values for polymers. Therefore, it is impossible to learn deformation kinetics as a function of the magnitude of deformation using the generally accepted techniques for such experiments. 

Eor time-invariant structural reliability, the problem is generally described by a performance function G(X), in which X is a vector of random variables of system parameters, and the probability of failure (limit state being reached) can be formulated by... 

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