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 ${\Bbb C}^3$. 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.