The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.

The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of $2^{\omega }$.

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[a,b\right]$ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $\left[a,b\right]$ is Denjoy-Riemann integrable on $\left[a,b\right]$, that a Denjoy-Riemann integrable function on $\left[a,b\right]$ is Denjoy-McShane integrable on $\left[a,b\right]$ and that a Denjoy-McShane integrable function on $\left[a,b\right]$ is Denjoy-Pettis integrable on $\left[a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_0$, a measurable Denjoy-McShane integrable...

We construct a differentiable function $f:{\mathbf{R}}^n\to \mathbf{R}$ ($n\ge 2$) such that the set ${\left(\nabla f\right)}^{-1}\left(B(0,1)\right)$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.

Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.

This is an expository paper dealing with Jan Marik's results concerning perimeter and the divergence theorem of Gauss-Green-Ostrogradski.

For any $x\in (0,1]$, let $$x=\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots +\frac{1}{d_1(d_1-1)\cdots d_{n-1}(d_{n-1}-1)d_n}+\cdots$$ be its Lüroth expansion. Denote by $P_n\left(x\right)/Q_n\left(x\right)$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${\{P_n\left(x\right)/Q_n\left(x\right)\}}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of...