Non-isomorphism

S Doubhe, S Xiang, Cl Gilmore, G Bricogne, GW Carter Ir. Overcoming non-isomorphism by phase permutation and likelihood scoring Solution of the TrpRS crystal structure. Acta Crystallogr A 50 164-182, 1994. 

Rudenko, G., et al. (2003). MADTy phasing the extracellular domain of the LDL receptor a medium-sized protein, large timgsten clusters and multiple non-isomorphous crystals. Acta Crystallogr. D 59,1978-1986. 

CO ligand by a laser (Fig. 7). As mentioned earlier Lane methods have been used which avoid the effects of non-isomorphism in a kinetic sequence. 

The (3, Sf-boundary sequence (34345)252(34345)252 admits two different (3,5)-fillings (by 34 3-gons and 30 vertices) shown on Figure 5.4. Those fillings are non-isomorphic and have the same symmetry as the boundary, i.e. Cz- This (3,5)-boundary sequence might be minimal for the number of3-gons. 

Figure 5.4 Two different non-isomorphic (3,5)-fillings (with symmetry C2) of the same boundary with symmetry C2...

Figure 5.5 Several non-isomorphic (6,3)-fillings of the same (6,3>boundary are obtained by different (6,3)-fiIlings of above 3 components...

Theorem 7.2.1 For parabolic and hyperbolic parameters (r,q), there exists a continuum of non-isomorphic elementary (r, q)-polycycles. 

We then see that non-isomorphic line bundles on a curve can embed it with isomorphic normal bundles. 

If all phase trajectories by means of continuous deformation can be tranformed into one another, then the summation in Eq. (11) can be extended to all possible paths. However, if the phase space consists of topological non-isomorphic classes C, the summation in Eq. (11) extends only to paths from a given class and the problem of determination of the topological invariant arises. 

The opening lines of the chapter on enumeration from the monograph of Gutman and Cyvin [22] read By enumeration of benzenoid systems, the counting of all possible non-isomorphic members within a class of benzenoids is understood. Usually, but not always, the number of hexagons (h) is the leading parameter. Thus the enumeration for h — 1,2, 3, 4, etc. is to be executed. ... 

When the expressions from (32)-(34) and (36)—(38) are added appropriately the total number of (non-isomorphic) generalized hexagon-shaped benzenoids, say Nb(n2), is obtained. The result was rendered into the form ... 

In Fig. 13 all the non-isomorphic generalized hexagon-shaped benzenoids with n2 up to 20 are depicted. 

Tosic et al. [79] defined the title systems in the course of a particularly efficient algorithm for enumerations of branched catacondensed benzenoids with regular trigonal (D3h) symmetry. The SCS s are (unbranched and branched) catacondensed benzenoids defined in such a way that isomorphic systems of this kind may be reckoned as different , depending on their orientation with respect to an axis. The counting of different SCS s is therefore not a single counting of non-isomorphic... 

Let N be the number of all the non-isomorphic catacondensed benzenoids (C) with h + 2 hexagons each, which can be constructed according to (iii). Then N is taken by definition to be the number of different SCS s with h hexagons each, as obtained by the last step (iv). 

The problem of enumerating benzenoid isomers (C HS) consists of finding the number of non-isomorphic benzenoids for a given pair of the invariants n and s. It is also of interest to classify the set of isomers into Kekulean and non-Kekulean systems and occasionally further into more subclasses. 

In most of the enumerations of benzenoid systems the number of hexagons, h, has been used as a leading parameter. This is to say that the numbers of non-isomorphic benzenoids with a given h have been determined, and this set has occasionally been subdivided into different classes see e.g. a consolidated report [30] with supplements [31]. Also when special classes of benzenoids have been generated specifically the numbers of benzenoids were produced as a function of h. 

The second term in Eq. (1.36) accounts for the non-isomorphic change of cell shape in the process of its transformation into a polyhedron. The value of this term does not exceed 2.8% of the main equation term and equals zero when volume fraction in close-packed spheres). 

The set of all transformed reaction centres is generated by a straightforward implementation of the usual combinatorial algorithm (cf. Sect. 4.2.3), and the transformed reaction centers are checked to fulfill several restrictive conditions which allow one to suppress the possibility of combinatorial explosion at the very beginning of the SPS generation. The generation of transformed reaction centers is very fast (it usually takes no more then 10-15% of the overall execution time when the reaction center has no more then 4 atoms in a molecule with more then 10-15 atoms). Only non-isomorphic transformed reaction centers are processed by the second step. 

The only calculation in this direction we could make, is the following. Let p= 3i and let C be a hyperelliptic curve (i.e. a curve of genus 2), with Jacobian V2iriety J. Then o(J) <2 implies that J is isogeneous to a product of two elliptic curves (non--isomorphic curves have non-isomorphic polarized Jacobian varieties, by Torelli s theorem, but their Jacobians may be isomorphic). We use notations of IGUSA, cf. C20], page 6kk/Sk C can be of type n, with l[Pg.77]

Non-isomorphic graphs having the same - characteristic polynomial. 

Quintas, L.V. and Slater, P.J. (1981). Pairs of Non-Isomorphic Graphs Having the Same Path Degree Sequence. MATCH (Comm.Math.Comp.Chem.), 12, 75-86. 

The greatest problem in scaling multiple crystals, however, is non-isomorphism, and this may arise from several sources. If the crystals were not grown from the same batch of protein, grown under the same crystallization conditions, or if they were exposed to different... 

---