Mobius transformation

Taking advantage of this fact, we first map the v-sphere to an auxiliary u-sphere by the Mobius transformation... 

For our purposes the best way to study the convergence of the continued fractions in Eq. (3.47) is by reducing them to a succession of Mobius transformations (Wall (1948)). Let t be the first continued fraction in Eq. (3.46) Then... 

Stable solutions require the amplification factor to be on or within the unit circle in the complex plane. The Mobius transformation maps the unit circle on the left complex half-plane, and thus in general stability can be analyzed in terms of the variable z by the Routh-Hurwitz criterion. 

Equation 35 is a Mobius transformation (Needham, 2007), and using this fact, Eqn. 34 can be expressed analytically. Mobius transformations, denoted by p, are mappings that rotate, stretch, shrink, or invert curves on the complex plane, and take the following form (Needham, 2007) p z) = flz-E b/cz-f d, wherein fl,i),c,d, and 2 are complex numbers. Remarkably, fxmctional compositions of p can also be calculated by multiplying matrices derived from it, termed Mobius matrices, drastically reducing the complexity of the fxmctiona 1-composition problem. 

Experts give values of the lower and upper bound of (i.e. BlefF) and Pl F)) or m N). From plausibility and belief measures, the basic mass assignment is computed by the Mobius transform. Here we have... 

Electrocyclic reactions are examples of cases where n-electron bonds transform to sigma ones [32,49,55]. A prototype is the cyclization of butadiene to cyclobutene (Fig. 8, lower panel). In this four electron system, phase inversion occurs if no new nodes are formed along the reaction coordinate. Therefore, when the ring closure is disrotatory, the system is Hiickel type, and the reaction a phase-inverting one. If, however, the motion is conrotatory, a new node is formed along the reaction coordinate just as in the HC1 + H system. The reaction is now Mobius type, and phase preserving. This result, which is in line with the Woodward-Hoffmann rules and with Zimmerman s Mobius-Hiickel model [20], was obtained without consideration of nuclear symmetry. This conclusion was previously reached by Goddard [22,39]. 

If A transforms to B by an antara-type process (a Mobius four electron reaction), the phase would be preserved in the reaction and in the complete loop (An i2p loop), and no conical intersection is possible for this case. In that case, the only way to equalize the energies of the ground and excited states, is along a trajectory that increases the separation between atoms in the molecule. Indeed, the two are computed to meet only at infinite interatomic distances, that is, upon dissociation [89]. 

When a cyclic polyene is large enough, it can exist in both cis- and iraws-forms. Our approach to polyene cyclization has tacitly assumed an all cis -n chain in the form of a band or ribbon that would slip smoothly on to the surface of a cylinder of appropriate diameter. Should the orbitals of the two polyenes in (36) have a mismatch in their orbital symmetries, a single twist in the tt band of one of them could remedy this (Fig. 10c). Cycloaddition would now be allowed and the reaction would proceed, provided other factors were favorable. Such cases of Mobius (Zimmerman, 1966), anti (Fukui and Fujimoto, 1966b) or axisymmetric (Lemal and McGregor, 1966), as opposed to Hiickel, syn, or sigma-symmetric ring closure are unknown (or, at least, rare). A Mobius form has, however, been proposed as the key intermediate in the photochemical transformations of benzene (Farenhorst, 1966) in (48) in place of the disrotatory cyclization proposed by van Tamelen (1965). 

We now turn to a different type of photochemistry and different mechanistic questions, namely photochemical pericyclic reactions and the utility of the Mobius-Hiickel treatment of these transformations. 

Experimental data about the properties of Mobius strip-like structures as the ladder 129a or knots like 127 are rather scarce. However, a theoretical analysis of the peculiarities of these constructions (see refs. 18a-d, 21b, 21c and literature cited therein) led to some conclusions of general importance. Thus it was established that a new phenomenon of topological chirality should be observed for compounds having the shape of trefoil knots or Mobius strips. Normally, chemists deal with chiral objects which can be (in principle) transformed into their mirror image by a continuous deformation. For... 

The transformation from P to P is therefore not an identity transformation, but rather an involution, with P and F as conjugate points. The identity transformation corresponds to a double rotation of 27t along the Mobius surface. The two sides of the paper corresponds to a double covering of the non-orientable topological Mobius surface. 

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