Necklace problem

By this time Polya s Theorem had become a familiar combinatorial tool, and it was no longer necessary to explain it whenever it was used. Despite that, expositions of the theorem have continued to proliferate, to the extent that it would be futile to attempt to trace them any further. I take space, however, to mention the unusual exposition by Merris [MerRSl], who analyzes in detail the 4-bead 3-color necklace problem, and interprets it in terms of symmetry classes of tensors — an interpretation that he has used to good effect elsewhere (see [MerRSO, 80a]). 

Let us consider again, and now solve, the necklace problem that was mentioned in the discussion of De Bruijn s Theorem, namely, to enumerate necklaces of six beads of two colors, red and green, where the two colors can be interchanged. We shall ask only for the total number, and can therefore use a simpler (unweighted) form of the power group enumeration theorem which gives as the required number the expression... 

The light shed by Redfield s paper on the close connection between Polya theory and symmetric function theory is well illustrated by a particularly simple way of looking at Polya s Theorem -- one that shows the way to further developments. Suppose the store of figures consists of n distinct figures, as for example with necklace problems using n kinds of beads. The figure generating function is then... 

Consider the following necklace problem. How many necklaces can be made from six beads, with three colors of beads available, subject to the restriction that no two adjacent beads are to be the same color Without the restriction the problem is a straight Polya-type problem of a kind that we have already met, and the solution is Z(Dg 3,3,. ..) 92. With the resifiction, however, the problem no... 

This is similar to a limiting reactant problem. We determine how many necklaces can be made from each quantity of beads. 

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

The materials most commonly used as gems and ornamental stones are listed in Table 2.9. This is by no means a complete listing of all materials ever used in jewelry or for decorative purposes. There are many worked specimens that are one-of-a-kind, made from unexpected materials that were opportunistically obtained. These often pose problems of identification and consequently of conservation, since once a stone has been worked it loses its natural luster and form. Stones have been altered with dyes and heat for thousands of years, so it does not hold true that just because something is in an old artifact or Grandma s necklace that it cannot be dyed or otherwise not natural. Synthetics are relatively new, but imitations are as old as the stones themselves. If someone wanted a red gem, and there were no rubies available, then a garnet or spinel could be used instead. No emeralds Use an olivine (peridot) or green sapphire. A synthetic must have the same composition and internal structure as the natural material, but an imitation just has to look like the natural stone. 

This rule Is surprisingly strong even polymers which can bind up to 300 micelles do not join to form a 3-d network. Horeover, these necklaces will not bind to each other even If salt Is added to the solution to screen the repulsion between micelles (precipitation occurs first). Because of this problem we must return to the mechanisms through which a system of particles with adsorbed polymers may build a 3-d network (4), and examine why these mechanisms do not operate when the particles are small. 

Some forms of hematite, a mineral composed of iron(III) oxide, can be used to make jewelry. Because of its iron content, hematite jewelry has a unique problem among stone jewelry. It shows signs of rusting. How many moles of iron are there in a necklace that contains 78.435 g of Fe203 ... 

An intriguing question is, however, whether the pearl necklace, or some other types of multidomain intramolecular structures of low symmetry, may correspond to the equilibrium conformation of a partially collapsed star-branched PE. This problem has been recently addressed by Kosovan et al. [158] by means of MD simulations. 

The conclusions that we formulated in early 1990s are, in principle, in agreement with up-to-date knowledge. Water is a poor (but not too bad) solvent for non-ionized PMA and, hence, the conformational transition is expected to proceed not as a sharp transition, but via a cascade of pearl necklace structures. In Sect. 4.1, we will show that our recent MD simulations support the above description of the behavior. We included this almost-forgotten experimental study in the present feature article mainly because we returned to this problem and studied it theoretically within the POLYAMPHI network. 

In contrast, the AFM pictures as well as cryo-TEM of sample CB-PLL55 prepared from 5 mM NaBr aqueous solution (Fig. 20) show extended cylinders. The cylinders exhibit undulations of the cross-section that are reminiscent of the pinned clusters [74, 91] postulated by scaling arguments. The occurrence of pearl-necklace-type structures, where pinned clusters of side chains alternate with regimes that are almost free of side chains, has also been seen in simulations of bottlebrushes, provided one has poor solvent conditions. These clusters are formed by collective collapse of several neighboring side chains [92]. We return to this problem in Sect. 3.5. 

A similar problem occurs if we start with < 0. Then we can again deflne blobs, of size and monomer numter g. Each blob is still quasi-ideal, but the necklace of blob is a collapsed structure— the blobs fill the available space, with a certain filling density. 

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