Dimer problem

EXAMPLE Consider an m x n lattice, where rn is even. Prom the explicit solution to the dimer problem [boll79], given by... 

The dimer problem effectively consists of exactly enumerating the number of ways an arbitrary lattice can be decomposed into non-intersecting edges, without any leftover links covering an n X n chessboard with n /2 dominoes, for example, so that the entire board is covered without overlap or gaps. 

It is well known that delocalized stable radicals may have potential for the construction of solid state conducting materials. The phenylalenyl radical 40 has been considered a good candidate with its spin density delocalized over 13 carbons in its jt-conjugated system. Unfortunately, 40 exists in equilibrium with its dimer and it decomposes at modest temperatures. To overcome the dimerization problem, Goto et al. and Koutentis et al. synthesized substituted radicals 41 and 42. 

Very thorough theoretical work on the dimer problem in photosynthetic RCs has been published in a series of papers by Reimers and Hush, see references 64,75. Very recently the authors gave a unified description of the electrochemical, charge distribution and spectroscopic properties of P,+ in bRCs.76... 

Dow Chemicals (2001) gives the following storage life data regarding Isonate 125M, which is a pure MDI (Table 3.1). A prepolymer prepared with pure MDI does not have the dimerization problems of the diisocyanate. 

The dimer problem and the self-avoiding random walk problems are related to other statistical mechanical problems. However, since our standpoint in this article is to treat macromolecular solutions in terms of the general theory of solutions, we shall not discuss the details of these problems. 

In this chapter this simple model is used to illustrate all these effects. We follow the treatment of Berne and Giniger (1973). In fact, with linearization and a suitable redefinition of parameters, the results for the above reaction can be applied to the dimerization problem (see Section 6.5)... 

In the introduction it was stated that transition metal-containing systems have often been considered as unusually difficult to treat by theoretical methods. Even though respect for these systems has been exaggerated in many ways, there are definitely cases for which it is still very difficult to obtain results of even qualitative accuracy, and this does not refer to obvious cases of very large molecules. The classically difficult case is the chromium dimer, which has challenged method-oriented theoreticians for at least a decade. However, as solution to the chromium dimer problem is now in sight, and some of the more recent development are described briefly below. 

Temperley HNV, Fisher ME (1961) Dimer Problem in Statistical Mechanics — An Exact Result. Phil Mag (Ser 8) 6 1061... 

J.K. Percus, One more technique for the dimer problem, J. Math. Phys. 10 (1969) 1881-1884. J.K. Percus, Combinatorial methods, Springer, Berlin, 1971. 

E. H. Lieb, Solution of the dimer problem by the transfer matrix method, J. Math. Phys. 8... 

The polymer described in the last problem is commercially called poly (phenylene oxide), which is not a proper name for a molecule with this structure. Propose a more correct name. Use the results of the last problem to criticize or defend the following proposition The experimental data for dimer polymerization can be understood if it is assumed that one molecule of water and one molecule of monomer may split out in the condensation step. Steps involving incorporation of the monomer itself (with only water split out) also occur. 

To an experienced operator trained in the handling of industrial chemicals, the dimers present Httle cause for concern in handling or storage. The finished polymer coating presents even less of a health problem contact with the reactive monomer is unlikely. In the ancillary operations, such as cleaning or adhesion promotion, the operator must observe suitable precautions. Before using the process chemicals, operators must read and understand the current Material Safety Data Sheets, which are available from the manufacturers. 

EMPl, selected by phage display from random peptide libraries, demonstrates that a dimer of a 20-residue peptide can mimic the function of a monomeric 166-residue protein. In contrast to the minimized Z domain, this selected peptide shares neither the sequence nor the structure of the natural hormone. Thus, there can be a number of ways to solve a molecular recognition problem, and combinatorial methods such as phage display allow us to sort through a multitude of structural scaffolds to discover novel solutions. 

The inaccuracy seems not to prohibit study of the structural properties of associating fluids, at least at low values of the association energy. However, what is most important is that this difficulty results in the violation of the mass action law, see Refs. 62-64 for detailed discussion. To overcome the problem, one can apply thermodynamical correspondence between a dimerizing fluid and a mixture of free monomers of density p o = P/30 = Po/2 and dimer species [12]. The equation of state of the corresponding mixture... 

The main conclusion which can be drawn from the results presented above is that dimerization of particles in a Lennard-Jones fluid leads to a stronger depletion of the proflles close to the wall, compared to a nonassociating fluid. On the basis of the calculations performed so far, it is difficult to conclude whether the second-order theory provides a correct description of the drying transition. An unequivocal solution of this problem would require massive calculations, including computer simulations. Also, it would be necessary to obtain an accurate equation of state for the bulk fluid. These problems are the subject of our studies at present. 

Each of the membranes acts like a hard wall for dimer molecules. Consequently, in parts I and III we observe accumulation of dimer particles at the membrane. The presence of this layer can prohibit translation of particles through the membrane. Moreover, in parts II and IV of the box, at the membranes, we observe a depletion of the local density. This phenomenon can artificially enhance diffusion in the system. In order to avoid the problem, a double translation step has been applied. In one step the maximum displacement allows a particle to jump through the surface layer in the second step the maximum translation is small, to keep the total acceptance ratio as desired. 

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