By E. von Glasersfeld

ISBN-10: 0792312570

ISBN-13: 9780792312574

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**Extra resources for Bernhard Riemann's Gesammelte mathematische Werke und Wissenschaftlicher Nachlass**

**Sample text**

3) was deemed to be mandatory in the approach to differential geometry originally adopted by Einstein (1916). , [N], Chapter 6) - and we shall call it the M-coordinate-dependent formulation of structures associated with a differential manifold. The above definition of smooth functions and curves enables, however, the modem approach to differential geometry [C,I,NT,SC] - to which we shall refer as the M-coordinate-independent formulation of the same structures. In this lauer formulation a vector X tangent to a smooth curve r = {x(t)1 a ~ t ~ b} passing through a point x E M is defined in a coordinate-independent manner by the following procedure.

Hence, in such a context there is no need for assuming the spontaneous compactification of six dimensions in Mlo, that is postulated in the standard approach to string theory (Green et al. 1987; Kaku, 1988). On the other hand, one of the pioneers of modem string theory has pointed out the following: "One ought to formulate string theory in a mueh larger space - something like the space of all possible positions of a string. The reason we use the language of ten dimensions or four dimensions is because we have so far been forced to talk about string theories in an approximate way" (Green, 1989, p.

In addition, it will also have to incorporate all the intemal symmetry groups characteristic of a given model. The well-known standard model of elementary particle physics (HaI zen and Martin, 1984) suggests adopting SU(3)xSU(2)xU(1) as an internal symmetry group, so that photons, gravitons, leptons, quarks and gluons are to be cast in the role of excitons, with the hadrons being viewed as composite objects. However, the still unresolved problem of quark confinement (Nachtmann, 1990), and the possibility of subquark structure, such as a common preon substructure for quarks and leptons (Pati and Salam, 1975, 1984), indicate that the question of "ultimate constituents" is still open for debate, and that other possible alternatives for hadron structure are not to be altogether dismissed within the OS context.

### Bernhard Riemann's Gesammelte mathematische Werke und Wissenschaftlicher Nachlass by E. von Glasersfeld

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