Some properties of the world are fixed by physics derived frommathematical symmetries, while others are selected from an ensemble of possibilities. Several successes and failures of “anthropic” reasoning in this context are reviewed in the light of recent developments in astrobiology, cosmology and unification physics. Specific issues raised include our spacetime location (including the reason ...

Some properties of the universe are fixed by physics derived from mathematical symmetries, others may have been selected from an ensemble of possibilities. Some successes and failures of anthropic reasoning in this context are reviewed in the light of recent developments in astrobiology, cosmology and unification physics. Specific issues raised include our spacetime location (including the reas...

In the framework of the theory of scale relativity, we suggest a solution to the cosmological problem of the formation and evolution of gravitational structures on many scales. This approach is based on the giving up of the hypothesis of differentiability of space-time coordinates. As a consequence of this generalization, space-time is not only curved, but also fractal. In analogy with Einstein...

The pion structure function is investigated in a simple model, where the pion and its constituent quark fields are coupled through the simplest pseudoscalar coupling. The imaginary part of the forward γπ → γπ scattering amplitude is evaluated and related to the structure functions. It is shown that the introduction of nonperturbative effects, linked to the size of the pion, allows a connection ...

In a recent study (Nottale 1996b), it was found that the distribution of the semi-major axes of the firstly discovered exoplanets was clustered around quantized values according to the law a/GM = (n/w0), in the same manner and in terms of the same constant w0 = 144 km/s as in our own inner Solar System. The ratio αg = w0/c actually stands out as a gravitational coupling constant. The number of ...

Until 1980, there was no such subgroup as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I s...

Mansfield showed how to induce representations of crossed products of C∗algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable w...

We consider the problem of testing graph expansion in the bounded degree model. We give a property tester that given a graph with degree bound d, an expansion bound α, and a parameter ε > 0, accepts the graph with high probability if its expansion is more than α, and rejects it with high probability if it is ε-far from any graph (with degree bound 2d) with expansion Ω(α). The algorithm runs in ...

We review the development of renormalizable SO(10) Susy GUTs based on the 210 ⊕⊕126 ⊕ 126 ⊕ 10 Higgs system. These GUTs are minimal by parameter counting. Using the SO(10) → SU(4) × SU(2)L × SU(2)R label decomposition developed by us we calculated the complete GUT scale spectra and couplings and the threshold effects therefrom. The corrections to αG, Sin θW and MX are sensitive functions of the...

We review the development of renormalizable SO(10) Susy GUTs based on the 210 ⊕⊕126 ⊕ 126 ⊕ 10 Higgs system. These GUTs are minimal by parameter counting. Using the SO(10) → SU(4) × SU(2)L × SU(2)R label decomposition developed by us we calculated the complete GUT scale spectra and couplings and the threshold effects therefrom. The corrections to αG, Sin θW and MX are sensitive functions of the...