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The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as the testing ground for methods from equivariant algebraic topology. In 2018, Barany and Soberon presented a new variation, the Tverberg plus minus theorem. In this paper, we give a new proof of the Tverberg plus minus theorem, by using a projective transformation. The same tool allows us to derive plus minus analogues of all known affine Tverberg type results. In particular, we prove a plus minus analogue of the optimal colored Tverberg theorem.
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Recent theoretical advances applied to metamaterials have opened new avenues to design a coating that hides objects from electromagnetic radiation and even the sight. Here, we propose a new design of cloaking devices that creates perfect invisibility in isotropic media. A combination of positive and negative refractive indices, called plus-minus construction, is essential to achieve perfect invisibility (i.e., no time delay and total absence of reflection). Contrary to the common understanding that between two isotropic materials having different refractive indices the electromagnetic reflection is unavoidable, our method shows that surprisingly the reflection phenomena can be completely eliminated. The invented method, different from the classical impedance matching, may also find electromagnetic applications outside of cloaking devices, wherever distortions are present arising from reflections.
Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverbergs theorem, and call a partition $mathcal I$ of ${1,2,ldots,T(d,r)}$ into $r$ parts a Tverberg type. We say that $mathcal I$ occurs in an ordered point sequence $P$ if $P$ contains a subsequence $P$ of $T(d,r)$ points such that the partition of $P$ that is order-isomorphic to $mathcal I$ is a Tverberg partition. We say that $mathcal I$ is unavoidable if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $dle 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksmas conjecture on the number of Tverberg partitions.
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 $1$, then the intersection of the whole family contains an isometric copy of $K$ scaled by a factor of $(1-varepsilon)$, where $varepsilon$ is positive and fixed in advance. Unless $K$ is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of $K$. We show how our results imply the existence of randomized algorithms that approximate the largest copy of $K$ that fits inside a given polytope $P$ whose expected runtime is linear on the number of facets of $P$.
We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible.
In basketball and hockey, state-of-the-art player value statistics are often variants of Adjusted Plus-Minus (APM). But APM hasnt had the same impact in soccer, since soccer games are low scoring with a low number of substitutions. In soccer, perhaps the most comprehensive player value statistics come from video games, and in particular FIFA. FIFA ratings combine the subjective evaluations of over 9000 scouts, coaches, and season-ticket holders into ratings for over 18,000 players. This paper combines FIFA ratings and APM into a single metric, which we call Augmented APM. The key idea is recasting APM into a Bayesian framework, and incorporating FIFA ratings into the prior distribution. We show that Augmented APM predicts better than both standard APM and a model using only FIFA ratings. We also show that Augmented APM decorrelates players that are highly collinear.
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