We present numerical and comprehensive analyses of the sixty hybrid textures of neutrinos, which have an equality of matrix elements and one zero. These textures are possibly derived in the models with discrete flavor symmetry. Only six textures among sixty ones are excluded by the present experimental data. Since there are many textures which give similar predictions, the textures are classified based on the numerical results. The neutrinoless double beta decay is also examined in these textures. Our results suggest that there remain still rich structures of the neutrino mass matrix in the phenomenological point of view.

It is shown that the neutrino mass matrices in the flavour basis yielding a vanishing U e3 are characterized by invariance under a class of Z 2 symmetries. A specific Z 2 in this class also leads to a maximal atmospheric mixing angle θ 23 . The breaking of that Z 2 can be parameterized by two dimensionless quantities, ǫ and ǫ ′ ; the effects of ǫ, ǫ ′ = 0 are studied perturbatively and numerically.The induced value of |U e3 | strongly depends on the neutrino mass hierarchy. We find that |U e3 | is less than 0.07 for a normal mass hierarchy, even when ǫ, ǫ ′ ∼ 30%. For an inverted mass hierarchy |U e3 | tends to be around 0.1 but can be as large as 0.17. In the case of quasi-degenerate neutrinos, |U e3 | could be close to its experimental upper bound 0.2. In contrast, |cos 2θ 23 | can always reach its experimental upper bound 0.28. We propose a specific model, based on electroweak radiative corrections in the MSSM, for ǫ and ǫ ′ . In that model, both |U e3 | and |cos 2θ 23 |, could be close to their respective experimental upper bounds if neutrinos are quasi-degenerate. *

Using the seesaw mechanism and a discrete symmetry, we construct a class of models for the neutrino mass matrix where the inverse of that matrix is the sum of a µ-τ antisymmetric background and a perturbation. We consider various possibilities for that perturbation. The simplest possible perturbations lead to fourparameter neutrino mass matrices which are unable to fit the experimental data. More complicated perturbations give rise to viable six-parameter mass matrices; we present detailed predictions of each of them. *

We present an entirely new approach towards a realization of the supersymmetric Yang-Mills theory on the lattice. The action consists of the staggered fermion and the plaquette variables distributed in the Euclidean space with a particular pattern. The system is shown to have fermionic symmetries relating the fermion and the link variables.

We present a new formulation of a staggered fermion on a D-dimensional lattice based on the SO(2D) Clifford algebra, which is naturally contained in the action. The action of a massless staggered fermion is invariant under discrete rotations and SO(2D) chiral and other discrete transformations. From the transformation properties of the fermion, we find two local meson operators (one scalar and one pseudoscalar), in addition to two standard meson operators. * )