The cubic symmetry S 4 contains A 4 and S 3 , both of which have been used to study neutrino mass matrices. Using S 4 as the family symmetry of a complete super-symmetric theory of leptons, it is shown how the requirement of breaking S 4 at the seesaw scale without breaking supersymmetry enforces a special form of the neutrino mass matrix which exhibits maximal ν µ − ν τ mixing as well as zero ...

Motivated by the hypothesis that financial crashes are macroscopic examples of critical phenomena associated with a discrete scaling symmetry, we reconsider the evidence of log-periodic precursors to financial crashes and test the prediction that log-periodic oscillations in a financial index are embedded in the conditional expectation function of this index. In particular, we examine the first...

We consider the problem of calculating the weak and strong topological indices in noncentrosymmetric time-reversal (T ) invariant insulators. In 2D we use a gauge corresponding to hybrid Wannier functions that are maximally localized in one dimension. Although this gauge is not smoothly defined on the two torus, it respects the T symmetry of the system and allows for a definition of the Z2 inva...