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We prove that, for any positive integer $m$, a segment may be partitioned into $m$ possibly degenerate or empty segments with equal values of a continuous function $f$ of a segment, assuming that $f$ may take positive and negative values, but its value on degenerate or empty segments is zero.
We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $mge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible higher-dimensional generali
We show that for any compact convex set $K$ in $mathbb{R}^d$ and any finite family $mathcal{F}$ of convex sets in $mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $mathcal{F}$ contains an isometric copy of $K$ of volume $1$
Given a torus $E = S^{1} times S^{1}$, let $E^{times}$ be the open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th singular Betti number $h^{i}(mathrm{Conf}^{n}(E^{times}))$ of the unordered configuration space of $
To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X({ n_x : x in X })$. This is obtained from the blowup $X[{bf n}]$ - which
We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $Lambda$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sim