Three-neutrino mixing in matter is studied through a set of evolution equations which are based on a rephasing invariant parametrization. Making use of the known properties of measured neutrino parameters, analytic, approximate, solutions are obtained. Their accuracy is confirmed by comparison with numerical integration of these equations. The results, when expressed in the elements squared of the mixing matrix, exhibit striking patterns as the matter density varies.Comment: Revised, 5 pages, 3 figures, references added

The elements (squared) of the neutrino mixing matrix are found to satisfy, as functions of the induced mass, a set of differential equations. They show clearly the dominance of pole terms when the neutrino masses "cross." Using the known vacuum mixing parameters as initial conditions, it is found that these equations have very good approximate solutions, for all values of the induced mass. The results are applicable to long baseline experiments.

In the seesaw model with hierarchical Dirac masses, the neutrino mixing angle exhibits the behavior of a narrow resonance. In general, the angle is strongly suppressed, but it can be maximal for special parameter values. We delineate the small regions in which this happens, for the two flavor problem. On the other hand, the physical neutrino masses are hierarchical, in general, except in a large part of the region in which the mixing angle is sizable, where they are nearly degenerate. It is also shown that, if one mixing angle is small, the three neutrino problem reduces to two, two flavor problems.

We obtain explicitly the renormalization group equations for the quark mass matrices in terms of a set of rephasing invariant parameters. For a range of assumed high energy values for the mass ratios and mixing parameters, they are found to evolve rapidly and develop hierarchies as the energy scale decreases. To achieve the experimentally observed high degree of hierarchy, however, the introduction of new models with specific properties becomes necessary.

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