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 problem As in the case of Mumford's geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. A counterexample to Hilbert's Fourteenth Problem in dimension six The kernel of a certain derivation of the polynomial ringk[6] is shown to be nonfinitely generated overk (a field of charactersitic zero), thus giving a new counterexample to Hilbert's Fourteenth Problem. Real and Rational Forms of Certain O2(?)-actions, and a Solution to the Weak Complexification Problem On Freudenburg's counterexample to the Fourteenth Problem of Hilbert In this paper we study Freudenburg's counterexample to the fourteenth problem of Hilbert and counterexamples derived from it. We also study the problem of uniqueness of a direct sum decomposition of objects in ${\mathcal M}(G,R).$ We prove that the Krull--Schmidt theorem holds in many cases. This is done by reducing the problem to the case of bipartite quivers of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. Tight Frames of Polynomials and the Truncated Trigonometric Moment Problem A simple parametrization is given for the set of positive measures with finite support on the circle group T that are solutions of the truncated trigonometric moment problem: Finally, the problem of inversion of a multiplier will be analyzed for smooth functions that have a specified structure near their zeros. On the Uniqueness in the Inverse Conductivity Problem The inverse conductivity problem to the the elliptic equation ${\rm div}((1+(k-1)\chi_D)\nabla u)=0\ {\rm in }\ \Omega$ is considered. Prediction for Two Processes and the Nehari Problem We exploit an analogy between the trigonometric moment problem and prediction theory for a stationary stochastic process. Prediction for two processes and the nehari problem

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