Translation interchangeability

Theorem — Let G be a bipartite regular polymer graph G with cyclic boundary conditions and with a Kekule structure K, which has edge set (K). Then the difference 8(G,e,K) between the orders of e n (K) and of es+n (K) for translationally equivalent boundary sets e varies with a period of no more than 2. The period can only be 2 if the primitive translation interchanges starred unstarred sites. 

It has been realized for many years that interchange of electronic and vibrational energy is more probable than interchange of electronic and translational energy. However, it still seems very inefficient except for a near coincidence of vibrational and electronic energy levels. The ultra-simplified theory seems inadequate (Ref 34)... 

In this section we restrict considerations to an nxn nansingular matrix A. As shown in Section 1.1, the Gauss-Jordan elimination translates A into the identity matrix I. Selecting off-diagonal pivots we interchange some rows of I, and obtain a permutation matrix P instead, with exactly one element 1 in each row and in each column, all the other entries beeing zero. Matrix P is called permutation matrix, since the operation PA will interchange some rows of A. ... 

There are two important points to note about the list of special positions and their symmetries. The point i, 5, z is identical to 0, 0, z, because they are related by the centering translation. The point at i, 0, z may not appear equivalent to the listed one, 0, z since the a and b directions in the lattice are not equivalent. However, it is because the two twofold axes at the centers of the a and b edges are interchanged by the twofold rotations about the axes at a = b = i, and so on. 

For the crystallographer, one of the most useful ways to describe unit-cell symmetry is by equivalent positions, positions in the unit cell that are superimposed on each other by the symmetry operations. In a P2, cell with an atom located at (x,y,z), an identical atom can be found at (-x, -y, /2 + z), because the operation of a 2j screw axis interchanges these positions. So a P2t cell has the equivalent positions (x, y, z) and (-x, -y, V2 + z). (The V2 means one-half of a unit translation along c, or a distance c/2 along the z-axis.)... 

These conclusions are based on the assumption that there is no interchange of translational and vibrational energy prior to the crossing of the barrier. The validity of the assumption depends on the exact form of the potential energy surface as well as on the masses of the particles. In order to get a quick estimate of the mass effects, a mass-weighted coordinate system can be used, see Appendix D. 

The structure of cubic zinc sulfide (zinc blende, sphalerite) may be described as a ccp of S atoms, in which half of the tetrahedral sites are filled with Zn atoms the arrangement of the filled sites is such that the coordination numbers of S and Zn are both four, as shown in Fig. 10.1.7. The crystal belongs to space group 7 2 — / 43m. Note that the roles of the Zn and S atoms can be interchanged by a simple translation of the origin. 

There is a wealth of theory on the process of evolution that is largely overlooked in the experimental literature. Part of the difficulty is the abundance of jargon that is specific to relatively small clusters of theoretical literature. Theoretical studies on protein evolution, RNA evolution, DNA evolution, and algorithms in computer science often have interchangeable results. For instance, motifs that have important implications for experiments with proteins emerge from RNA secondary structure studies. However, using the information requires a substantial translation of the language. 

Note that for a fixed operation time, t in Equation 9.1, the profit will increase with the increase in the distillate amount and a maximum profit optimisation problem will translate into a maximum distillate optimisation problem (Mujtaba and Macchietto, 1993 Diwekar, 1992). However, for any reaction scheme (some presented in Table 9.1) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for reversible or irreversible reaction scheme and is already explained in the introduction section. Note for batch reactive distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For non-reactive distillation system, of course, the maximum distillate problem has to be solved. 

Interchangeability of translational components 5.1. Interchangeability of ribosomal subunits... 

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