Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation

Nonabelian Discrete Symmetries, Fermion Mass Textures and Large Neutrino Mixing

Nonabelian discrete groups are an attractive tool to describe fermion masses and mixings. They have nonsinglet representations which seem particularly suitable for distinguishing the lighter generations from the heavier ones. Also, they do not suffer from the extra constraints a continuous group must obey, e.g. limits on extra particles. Some of the simplest groups are the nonabelian discrete subgroups of SO(3) and SU(2), the so called dihedral groups D_n and dicyclic groups Q_2n, which both have only singlet and doublet representations. After studying which vacuum expectation value (VEV) directions of representations of dihedral and dicyclic groups preserve which subgroups, we construct a simple model based on the group Q_6 \times Q_6. The model reproduces the masses and mixings of all quarks and leptons, including neutrinos. It has a large mixing angle in the mu - tau neutrino sector, in accordance with the recent SuperKamiokande results, while keeping a small quark mixing in the bottom - charm sector. The reason is similar to the one found in the literature based on the SU(5) group: the large_left_ handed mixing angle in the lepton sector corresponds to the large unphysical_right_ handed in the down quark sector. The large mixing is also responsible for the different hierarchies of the two heaviest families in the up and down sector, and can be summarized as the order of magnitude relation: m_s/m_b \sim tan(theta_\mu\tau) \sqrt{m_c/m_t}Comment: 15 pages, revtex, no figures. Shortened version, dirac neutrino mass and typos corrected, reference adde

The effective temperature

This review presents the effective temperature notion as defined from the deviations from the equilibrium fluctuation-dissipation theorem in out of equilibrium systems with slow dynamics. The thermodynamic meaning of this quantity is discussed in detail. Analytic, numeric and experimental measurements are surveyed. Open issues are mentioned.Comment: 58 page

Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations and initial conservation laws, follow from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for generation of symmetries is developed and applied to discrete KdV. It can also be applied to all the other ABS equations

Many Higgs doublet supersymmetric model, flavor changing interactions and spontaneous CP violation

I study many higgs doublet supersymmetric model with spontaneous CP violation. The damaging flavor changing interactions and large CP violation are brought under control simultaneously by an approximate PQ symmetry. I relate the smallness of the CP violating parameter in the kaon system to the small {m_b \over m_t} ratio