The Mobius Function

Hence this approach is feasible, provided that the lattice of subgroups of G is known. For the point groups used in chemistry it is easy to obtain the subgroup lattice with the help of a suitable software package, for example GAP or MAGMA, which also deliver the Mobius function and the other matrices that we introduce now. 

First, we recall the classical facts about the Mobius function for posets. As mentioned in Section 10.4, for an arbitrary poset P, let I P) denote the set of the finite intervals of P. Define the function p I P) —> C as follows ... 

The function /x is called the Mobius function. An alternative way to define y, is via the incidence algebra of P. All functions / I P) — C form a C-algebra, with pointwise addition, pointwise multiplication by complex numbers, and the following convolution product for /, I P) — C we define... 

The Mobius function for posets is in fact a special case of the notion of a Mobius function for acyclic categories. 

Proposition 10.28. Let C he an arbitrary finite acyclic category. The equation jj,o = e has a unique solution, which is also called the Mobius function. This solution will automatically satisfy the equation fj, = e as well. 

Originally, the Mobius function was introduced for integers. The textbook [Sta97] contains a great deal of material about the Mobius function for posets, including Theorem 10.24, which is also called Hall s Theorem. 

When additionally the complex A is pure, it follows from Theorem 12.3(2) that the reduced Betti number is nonzero only in the top dimension. Therefore, by the Euler-Poincare formula, in this case the cohomology groups can be computed simply by computing the Euler characteristic. In the even more special case that A is an order complex of a poset A = A P), by Hall s theorem, it suffices to compute the value of the Mobius function pp 0,1). 

When an acyclic category is shown to be shellable, and aU of its maximal chains have the same number of morphisms, the number of spheres can be found by computing the Mobius function. The latter can often be done by some explicit combinatorial counting procedure. 

In the next subsection we will see that one can in general relate the Mobius function in an acyclic category to the values of the tableau entries in a spectral sequence computing the cohomology groups of the associated nerve. Here, Proposition 16.5 specializes to... 

Rota GC (1964) On the Foundations of Combinatorial Theory. I Theory of Mobius Functions. Zeit. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 340-368... 

Walba et al. extended their conception to the synthesis of belt-shaped crown ethers by lengthening the open-chained precursor 32 by one unit [29]. The cyclization of the diol-ditosylate 36 was carried out analogously to the method cited above and gave two products with the same molecular weight as determined by Cl mass spectrometry in 50% overall yield. As CPK models suggest the four functional groups of 36 can react with each other not only parallel but also crossed . This leads to the cylinder 37 and the first molecular Mobius strip 38. A Mobius strip as an isomer of a cylinder has only one surface and one edge. 

Figure 5.7 The Periodic Function mapped to a Mobius hand...

The graphical representation of the way in which chemical periodicity varies continuously as a function of the limiting ratio (Figure 5.3), 1 < Z/N < 0, appears strangely unsymmetrical, despite perfect symmetry at the extreme values. By adding an element of mirror symmetry a fully symmetrical closed function, that now represents matter and antimatter, is obtained. To avoid self overlap the graphical representation of the periodic function is transferred to the double cover of a Mobius band, which in closed form defines a projective plane. 

Kuhn [24, 26, 34], Mobius [28, 35], and others [36, 37] have studied the distance dependence of the rate of photoinduced electron transfer in LB films. Their observed dependence agrees in a qualitahve sense with the other experimental [38-48] and theorehcal [29] results described by Eq. (11) and Eq. (12). As they did, the distances between the three functional moieties, that is. A, S, and D, within LB monolayers can be closely controlled at known values. 

Some examples of this Mobius function are shown in Figure 10.20. When X is an object our previously used shorthand notation from Definition 10.25 translates to /x(x) = /x(m), where m is the unique morphism m x —> t. 

When specializing to a spectral sequence for the cohomology of the nerve of an acyclic category, we immediately observe that its Mobius function can be read off from the P -tableau, for any nonnegative integer n. 

As we have seen earlier, formula (16.17) specializes to several well-known formulas for Mobius function computations, once the spectral sequence is specified. 

Gordon et al. ° have obtained a discrete form of the partition function of Fixman for an isolated coil. Using some elegant mathematics involving the Mobius inversion, they have shown that the so called excluded volume can be re-included in an almost Markovian theory. The result is a convergent series for the expansion factor of a polymer coil and a clean formulation for the partition function for an isolated coil which, for all but very short chains, may be better than any previously available. 

Fig. 11.7 The Mobius strip representation of a spin = function. There is a node on each side of the strip at the point indicated by an arrow (this is the point of contact of the two ends of the top diagram). In the bottom diagram only the signs of the function are given. There is no attempt to indicate amplitude. Note that two different Mobius strip representations exist—the twist can be made in either of two senses.

Fig. 11.8 The Mobius strip constructions for angular momentum = and f functions.

Generalization of either the frontier orbital, the orbital symmetry, or the transition-state aromaticity analysis leads to the same conclusion about the preferred stereochemistry for concerted thermal electrocyclic reactions The stereochemistry is a function of the number of electrons involved. Processes involving 4n +2 electrons will be disrotatory and involve a Hiickel-type transition state, whereas those involving 4n electrons will be conrotatory and the orbital array will be of the Mobius type. 

By using the determinant form of the electronic wave functions, it is readily shown that a phase-inverting reaction is one in which an even number of electron pairs are exchanged, while in a phase-preserving reaction, an odd number of electron pairs are exchanged. This holds for Hiickel-type reactions, and is demonstrated in Appendix A. For a definition of Hiickel and Mobius-type reactions, see Section III. 

Day21 has given a careful account of the relationship between the Woodward-Hoffmann rules and Mobius/Hiickel aromaticity, and has defined the terms supra-facial and antarafacial in terms of the nodal structure of the atomic basis functions. His approach makes quite explicit the assumption that the transition state involves a cyclic array of basis functions. Thus the interconversion of prismane (10) and benzene, apparently an allowed (n2s+ 2S+ 2S) process, is in fact forbidden because there are additional unfavourable overlaps across the ring.2... 

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