We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type.

We compute the Łojasiewicz exponent of $f=(f_1,\ldots,f_n)\colon \Bbb R^2\to\Bbb R^n$ via the real approximation of Puiseux"s expansions at infinity of the curve $f_1\ldots f_n=0$. As a consequence we construct a collection of real meromorphic curves which provide a testing set for properness of $f$ as well as a condition, which is very easy to check, for a local diffeomorphism to be a global one.

We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional unimodular Lie group with left invariant metric, that is, it is one of the 3-dimensional commutative Lie group, the 3-dimensional Heisenberg group, the groups of rigid motions on the Euclidean or the Minkowski planes, the special unitary group $SU(2)$, and the special real linear group $SL(2,\mathbb R)$.

Saeki extended the F. and M. Riesz theorem to $\mathbb{R}^N$ $(N \geqq 2)$, and Takahashi extended Saeki"s result to a LCA group. We give a result, which is relevant to theirs. We also give a strong version of Bochner"s generalization of the F. and M. Riesz theorem.

Let $P^{k}(n,2)$ be the set of all real polynomial map germs $f=(f_1,f_2):(\mathbb R^n,0) \rightarrow (\mathbb R^2,0)$ with degree of $f_1,f_2 \leq k$. The main result of this paper shows that the set of equivalence classes of $P^{k}(n,2)$, with respect to topological contact equivalence, is finite.

A generalized version of weighted Hardy spaces on a Euclidean domain is introduced and it is proved that the spaces are transformed to the same kind of spaces by certain smooth change of variables. Some related properties of the spaces, including a modified form of atomic decomposition, and some examples are also given.

We prove that the group-measure-space von Neumann algebra $L^\infty(\mathbb T^2) \rtimes \mathrm{SL}(2,\mathbb Z)$ is solid. The proof uses topological amenability of the action of $\mathrm{SL}L(2,\mathbb Z)$ on the Higson corona of $\mathbb Z^2$.

From Goursat"s transformation formulas for the hypergeometric function $F(\alpha,\beta,\gamma;z)$, we derive several double sequences given by mean iterations and express their common limits by the hypergeometric function. Our results are analogies of the fact that the arithmetic-geometric mean of 1 and $x\in (0,1)$ can be expressed as the reciprocal of $F \big( {1\over2},{1\over2},1;1-x^2 \big)$.

We will formulate the Monster Goursat manifolds for multi independent variables cases. We will classify the singularities which appears in the prolongation of 2-jet space of 2 independent and 1 dependent variables.