Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userClassifying unavoidable Tverberg partitions
263
0
0.0
(
0
)
Added by
Gabriel Nivasch
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $mathbb{R}^d$ cannot be so divided, and in these cases we ask if the set can nonetheless be $P(r,d)$--partitioned, i.e., split into $r$ subsets so that there exist $r$ points, one from each resulting convex hull, which form the vertex set of a prescribed convex $d$--polytope $P(r,d)$. Our main theorem shows that this is the case for any generic $T(r,2)-2$ points in the plane and any $rgeq 3$ when $P(r,2)=P_r$ is a regular $r$--gon, and moreover that $T(r,2)-2$ is tight. For higher dimensional polytopes and $r=r_1cdots r_k$, $r_i geq 3$, this generalizes to $T(r,2k)-2k$ generic points in $mathbb{R}^{2k}$ and orthogonal products $P(r,2k)=P_{r_1}times cdots times P_{r_k}$ of regular polygons, and likewise to $T(2r,2k+1)-(2k+1)$ points in $mathbb{R}^{2k+1}$ and the product polytopes $P(2r,2k+1)=P_{r_1}times cdots times P_{r_k} times P_2$. As with Tverbergs original theorem, our results admit topological generalizations when $r$ is a prime power, and, using the constraint method of Blagojevic, Frick, and Ziegler, allow for dimensionally restrict
The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have non-empty $q$-fold intersection. The affine cases, true for all $q$, constitute Tverbergs famous 1966 generalization of the classical Radons Theorem. Although established for all prime powers in 1987 by Ozaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps textit{below} the tight dimension $N(q,d)$ are nonetheless guaranteed $q$ pairwise disjoint subfaces -- including when $q$ is not a prime power -- which satisfy a variety of average value coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.
In recent work, M. Schneider and the first author studied a curious class of integer partitions called sequentially congruent partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $ngeq 1,$ that $p_{mathcal S}(n) = p_{square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers.
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.
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as $n$ tends to infinity while $m$ is fixed or tends to infinity. In particular, if $m$ goes to infinity not fast enough, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We derive the asymptotic distributions of the parts jointly and that of the largest part by taking limit of $n$ and $m$ in the same manner as that in the first probability measure.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
