Self-intersections

Let the mid-surface of the Kirchhoff-Love plate occupy a domain flc = fl Tc, where C is a bounded domain with the smooth boundary T, and Tc is the smooth curve without self-intersections recumbent in fl (see Fig.3.4). The mid-surface of the plate is in the plane z = 0. Coordinate system (xi,X2,z) is assumed to be Descartes and orthogonal, x = xi,X2)-... 

Let C be a bounded domain with the smooth boundary L, which has an inside smooth curve Lc without self-intersections. We denote flc = fl Tc. Let n = (ni,ri2) be a unit normal vector at L, and n = ( 1,1 2) be a unit normal vector at Lc, which defines a positive and a negative surface of the crack. We assume that there exists a closed continuation S of Lc dividing fl into two domains the domain fl with the outside normal n at S, and the domain 12+ with the outside normal —n at S (see Section 1.4). By doing so, for a smooth function w in flc, we define the traces of w at boundaries 912+ and, in particular, the traces w+ and the jump [w] = w+ — w at Lc. Let us consider the bilinear form... 

A. H. Schoen. Inifinite periodic minimal surfaces without self-intersections. Technical Report TN D-5541, NASA, May 1970. 

A. H. Schoen, Infinite Periodic Minimal Surfaces without Self-Intersections, NASA Technical Note No. D-5541 (1970). 

Hie linear macromolecule as simulated bv the chain of M sites on the volume-centered lattice allowing the self-intersection with minimum loop of 4 chain units. 

The procedure of the simulation included the following steps the random conformation was built in the computer, the number of reactive contacts /I.e. noncross-linked self-intersections 7 was calculated and then one of the contacts was cross-linked with a. probability... 

We suggest here an approximation based on one of the results obtained in computer experiment. It was Jound that the mean square of the radius of gyration P2 and the total number of self-intersections of the partially crcss-linJed chain g /"it consists of the reactive contacts and "dead" contacts, cross-links / are related bv the fol low in cr rel at ion s h in ... 

As it was shwn above the Nineties of intramolecular cross-lin inq rs cor.r -Letoly cetsrTinrtcl bTr tbe number reactive contscts,. /i.i.C-1 is eouai to tbe number of self-intersections minus the number cro -linta s /w do not consider the multiols 9-Uf-JnteSect W Then this relationship turns tne kinetic pro s lap to the average dimensions calculation. 

Imagine your frustration (or perhaps delight) if you tried to paint just the outside of a Klein bottle. You start on the bulbous outside and work your way down the slim neck. The real 4-D object does not self-intersect, allowing you to continue to follow the neck that is now inside the bottle. As the neck opens up to rejoin the bulbous surface, you find you are now painting inside the bulb. 

NON-SELF-INTERSECTING RANDOM WALKS IN LATTICES WITH NEAREST-NEIGHBOR INTERACTIONS... 

The partition function, Z (x), for the non-self-intersecting chain with nearest-neighbor interactions is10... 

So far, the effects of the chain ends were neglected in our stochastic model for the restricted chain. Therefore, n must be much larger than the number of steps needed to form the largest excluded polygon. The partition function, which incorporates the chain-end effects and which could be also employed for exact statistical description of short non-self-intersecting chains can be obtained as follows Assume, as before, that we eliminate only lowest-order polygons of t steps. Therefore, the first t — 1 steps in the chain are described as a sequence of independent events. Eq (9), then, will be replaced by... 

Interactions, Near-Neighbor, Non-Self Intersecting Random Walks... 

Non-Self Intersecting Random Walks on Lattices with Near-Neighbor... 

We can also use link polynomials to prove that certain unoriented links are topologically chiral. For example, let L denote the (4,2)-torus link which is illustrated on the left in Figure 12. This is called a torus link because it can be embedded on a torus (i.e. the surface of a doughnut) without any self-intersections. It is a (4,2)-torus link, because, when it lies on the torus, it twists four times around the torus in one direction, while wrapping two times around the torus the other way. Let L denote the oriented link that we get by putting an arbitrary orientation on each component of the (4,2)-torus link, for example, as we have done in Figure 12. Now the P-polynomial of L is P(L ) = r5m l - r3m x + ml 5 -m3r + 3m r3. 

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