This paper deals with two analytic questions on a connected compact Lie group G. i) Let a ∈ G and denote by γ the diffeomorphism of G given by γ (x) = ax (left translation by a). We give necessary and sufficient conditions for the existence of solutions of the cohomological equation f - f ∘ γ = g on the Fréchet space C^∞ (G) of complex C^∞ functions on G. ii) When G is the torus ${\Bbb T}^n$, we compute explicitly the distributions on ${\Bbb T}^n$ invariant by an affine automorphism γ, that is, γ (x) = A (x + a) with A ∈ GL(n, ℤ) and a ∈ ${\Bbb T}^n$. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations.

We show that certain holomorphic loop algebra-valued 1-forms over Riemann surfaces yield minimal Lagrangian immersions into the complex 2-dimensional projective space via the Weierstrass type representation, hence 3-dimensional special Lagrangian submanifolds of ℂ³. A particular family of 1-forms on ℂ corresponds to solutions of the third Painlevé equation which are smooth on (0, +∞).

The Navier-Stokes equations with bounded initial data admit unique local-in-time smooth mild solutions. It is shown that the solution can be extended globally-in-time, if the initial velocity has a special structure. Thanks to the structure, the annihilation of the pressure occurs, and then the mild solution is a solution to the viscous Burgers equations. By the maximum principle, it is derived an a priori bound for velocity, uniformly in time and space.

We compute the K-theory groups for the group C^*-algebras of certain solvable discrete groups. The solvable discrete groups considered are the discrete elementary ax + b group and the generalized discrete elementary ax + b groups and their proper versions, and also the generalized discrete elementary Mautner groups and products of the generalized discrete elementary ax + b groups and their proper versions.

This paper puts forward two applications of matrix volume. First, we present a new method to determine whether a matrix is orthogonal; second, a new way is given to indicate whether a linear system is consistent.