We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.

Compositional data, multivariate observations that hold only relative information, need a special treatment while performing statistical analysis, with respect to the simplex as their sample space ([Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986.], [Aitchison, J., Greenacre, M.: Biplots of compositional data. Applied Statistics 51 (2002), 375–392.], [Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V. (eds): Compositional data analysis in the geosciences:...

Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.

For a $C^{\infty }$ function $\varphi :M\to ℝ$ (where $M$ is a real algebraic manifold) the following problem is studied. If ${\varphi }^{-1}\left(0\right)$ is an algebraic subvariety of $M$, can $\varphi$ be approximated by rational regular functions $f$ such that $f^{-1}\left(0\right)={\varphi }^{-1}\left(0\right)?$We find that this is possible if and only if there exists a rational regular function $g:M\to ℝ$ such that $g^{-1}\left(0\right)={\varphi }^{-1}\left(0\right)$ and g(x)$·\varphi \left(x\right)\ge 0$ for any $x$ in ${ℝ}^n$. Similar results are obtained also in the analytic and in the Nash cases.For non approximable functions the minimal flatness locus is also studied.

We relate the notion of arc-analyticity and the one of analyticity on restriction to polynomial arcs and we prove that in the subanalytic setting, these two notions coincide.

We present a brief overview of the classification of real Enriques surfaces completed recently and make an attempt to systemize the known classification results for other special types of surfaces. Emphasis is also given to the particular tools used and to the general phenomena discovered; in particular, we prove two new congruence type prohibitions on the Euler characteristic of the real part of a real algebraic surface.