ترغب بنشر مسار تعليمي؟ اضغط هنا

A proof of a Dodecahedron conjecture for distance sets

175   0   0.0 ( 0 )
 نشر من قبل Hiroshi Nozaki
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

A finite subset of a Euclidean space is called an $s$-distance set if there exist exactly $s$ values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in $mathbb{R}^3$ is 20, and every $5$-distance set in $mathbb{R}^3$ with $20$ points is similar to the vertex set of a regular dodecahedron.



قيم البحث

اقرأ أيضاً

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products ${alpha, -alpha}$, $alphain[0,1)$, are called equ iangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $mathbb{R}^n$, $ngeq 7$, is $frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k in mathbb{N}$. We also prove the universal upper bound $sim frac 2 3 n a^2$ for equiangular sets with $alpha=frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.
We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.
We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operato rs on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $tilde{AA}$ acting on this space, and interpret the right generalization of the $ abla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)mapsto (q^{-1},t^{-1})$.
347 - Masashi Shinohara 2013
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the only 12-poi nt three-distance set in $mathbb{R}^3$ up to isomorphism. In this paper, we prove the uniqueness of 12-point three-distance sets in $mathbb{R}^3$.
In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $eta(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $eta(G)$. This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا