Local knots

Nevertheless, A.N. Semenov in his works [56-59] have used the concept of local quasi-knot for Gaussian coil state, because the end-to-end distance in this state ( N1/2) is still less than the chain length ( N). Within the framework of this approach, the authors of Refs. [56, 57] have shown that the topological constraints do not influence the dynamics of the whole polymer coil suspended in dilute 0-solutions but lead to the essential slowing-down of the internal coil excitation modes. In accordance with Refs. [56-59], the maximum relaxation time of a coil is limited by the diffusion of the local knots along the chain, i.e. the... 

Ferry considers that the temporary crosslinks are of topological nature and they are not obtained due to intermolecular forces [724], The coupling of the entanglements can be realized through 1 - adhesion or temporary network 2 - local knots 3 -superpositions of large loops, Figure 3.228. 

Figure 3.228. Types of entanglement coupling (1) temporary cross-link (2) local knot (3) long-range contour loop.

Proposition 2. Any standard radiation electromagnetic field in empty space with Faraday 2-form. F, regular in a bounded spacetime domain D, coincides locally with a knot around any point P C D in the following sense. There is a knot with 2-form 3Fkn, such that Fst = Fkn around P, except perhaps ifP is in a zero measure set. The same property holds for Fst. 

This means that the difference between the set of the radiation solutions of the Maxwell equations and the set of the electromagnetic knots is not local but global. In other words Radiation fields and knots are locally equal. A proof is the following. 

This shows that there is still some linearity. In particular, there is a subset of knots that form a vector space and is therefore a linear sector of the model. It is the set of the knots with zero helicity or with unlinked lines. Note also that the theory is fully linear from the local point of view, as a consequence of the local equivalence with Maxwell s theory shown in Section V.A. By this we mean that the set of the electromagentic knots contains all the linear combinations of standard solutions around any point. 

Products most likely to be subject to significant inhomogeneity include natural materials such as timber (treated or untreated) and some painted or printed surfaces. Knots in wood and particularly dense areas of color can cause relatively intense, localized sources of emissions. Potential uncertainty due to inhomogeneity is best addressed by using larger samples or by carrying out multiple emission tests on smaller pieces of the same sample. 

For the knot plane projection with defined passages, the following Reidemeister theorem is valid [39] different knots (or links) are topologically isomorphic to each other if they can be transformed continuously into one another by means of a sequence of simple local Reidemeister moves of types 1, 2 and 3 (see Fig. 9). Two knots are called regular isotopic if they are isomorphic with respect to the last two types of moves (2 and 3) if they are isomorphic with respect to all types of Reidemeister moves, they are called ambient isotopic. As can be seen from Fig. 9, a Reidemeister move of type 1 leads to the cusp creation on chain projection. At the same time, it is noteworthy that all real 3D-knots (links) are of ambient isotopy. 

The quasi-knot concept seems to be most obvious in the case when the knot can be considered as a local one, i.e. when it has its own internal scale much smaller than the chain scale as a whole. However, for chains in the coil state the computer experiments do not indicate the existence of such an internal knot scale (for the globular state the detailed experiments are absent). 

In the real polydisperse foam along with coalescence there always acts another process of internal collapse. This is the diffusion decrease in the specific surface which is accompanied by structural rearrangement, i.e. shift of knots and borders, and change in their orientation. This leads to the origination of various local disturbances (Act, Apa, AC, etc.). These local disturbances along with the rupture of individual films cause destruction either of other films and borders or of local volumes or of the whole foam (see Sections 6.5 and 6.6). Finally, various external factors can affect the foam (pressure drop, applied to the liquid phase reduced pressure of the liquid vapour above the foam, leading to evaporation the effect of antifoam droplets a-particle irradiation vibration, etc.). 

Special cases are discussed in some detail in the literature [112,197,198], where the shape representation P is chosen as a space curve representing a protein backbone and the topological descriptors Fj(s) on the local tangent plane projections are either graphs or knots defined by the crossing pattern on the planar projection at each tangent plane T(s) of the sphere S. 

FIGURE 4-14 A wind rose, which graphically portrays the statistical pattern of wind velocities (in knots) at O Hare Airport in Chicago from 1965 to 1969. Note the large fraction of time when winds are from the west to the south, as would be expected in the region of westerlies. At times, however, local and regional effects (e.g., storm systems) override the air movement attributable to global circulation (Boubel et ah, 1994). 

However, the step-independence criterion is still a relevant one, and, if we insist on it, the knot intervals have to be determined from local data at every... 

As one might anticipate, the two key characteristics influencing the strength of clearwood are density and MFA. However, a further effect is that of branches/knots on the local strength in their immediate vicinity. Discussion on the role of knots is deferred to Chapter 10. 

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