We study a generalization of a continued fraction of Ramanujan with random, complex-valued coefficients. A study of the continued fraction is equivalent to an analysis of the convergence of certain stochastic difference equations and the stability of random dynamical systems. We determine the convergence properties of stochastic difference equations and so the divergence of their corresponding continued fractions.

To each closed subset $S$ of a finite-dimensional Euclidean space corresponds a $\sigma$-ideal of sets $𝒥\left(S\right)$ which is $\sigma$-generated over $S$ by the convex subsets of $S$. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self-maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set-theoretic methods for dealing with formal independence as a means of geometric investigations.

It is proved that, in the sense of Baire category, almost every Markov operator acting on Borel measures is asymptotically stable and the Hausdorff dimension of its invariant measure is equal to zero.

We survey recent results on the structure of the range of the derivative of a smooth mapping $f$ between two Banach spaces $X$ and $Y$. We recall some necessary conditions and some sufficient conditions on a subset $A$ of $ℒ\left(X,Y\right)$ for the existence of a Fréchet differentiable mapping $f$ from $X$ into $Y$ so that $f^{\prime }\left(X\right)=A$. Whenever $f$ is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping $f$ from ${\ell }^1\left(\mathbb{N}\right)$ into ${ℝ}^2$, which is bounded, Lipschitz-continuous, and so that for all $x,y\in {\ell }^1\left(\mathbb{N}\right)$, if $x\ne y$, then $\Vert f^{\prime }\left(x\right)-f^{\prime }\left(y\right)\Vert >1$.

The main aim of this survey paper is to give basic information about properties and applications of $\sigma$-porous sets in Banach spaces (and some other infinite-dimensional spaces). This paper can be considered a partial continuation of the author's 1987 survey on porosity and $\sigma$-porosity and therefore only some results, remarks, and references (important for infinite-dimensional spaces) are repeated. However, this paper can be used without any knowledge of the previous survey. Some new results concerning $\sigma$-porosity in finite-dimensional spaces are also briefly mentioned. However, results concerning porosity (but not $\sigma$-porosity) are mentioned only exceptionally.

We show that in every Banach space, there is a $g$-porous set, the complement of which is of ${\mathscr{H}}^1$-measure zero on every $C^1$ curve.

We give a sharp estimate on the cardinality of point preimages of a uniform co-Lipschitz mapping on the plane. We also give a necessary and sufficient condition for a ball noncollapsing Lipschitz function to have a point with infinite preimage.