arXiv subjects

Michal Stanislaw Wojcik

Publications and source records attributed to Michal Stanislaw Wojcik.

3 recordsLinked to original sources

arXiv2013

Random variables with an invariant random shift in compact metrisable abelian groups

The main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that $P(Y\in A)=1$ and $X + a$ has the same distribution as $X$ for all $a\in A$. As a conclusion from the above it is shown that for independent random variables $X, Y$ such that $X+Y$ has the same distribution as $X$, $X+Y$ and $Y$ are also independent. It becomes also apparent that the distribution of $X$ is the Haar measure (uniform distribution) if for each open set $U$, $P(Y\in U) > 0$.

math.PR↗

arXiv2013

The closed graph function from plane with the network of discontinuity points

The main result of this paper states, that if a function $f:\R^2\to [0, +\infty)$ has a closed graph and the set of discontinuity points is a network (as defined by Kuratowski in Topology II, 61.IV), then the graph of $f$ is disconnected. It is also proven that this result can be easily generalised to a function $f:\R^2\to Y$ where $Y$ is a $σ$-locally compact metrisable space.

math.GN↗

arXiv2013

Discontinuity points of a function with a closed and connected graph

The main result of this paper states that for a function $f:\R^2\to Y$ with a closed, connected and locally connected graph, where $Y$ is a locally compact, second-countable metrisable space, the graph over discontinuity points remains locally connected.

math.GN↗