Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHalf of an antipodal spherical design
70
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$.
rate research
Read More
How can $d+k$ vectors in $mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $theta(d,k):=min_Xmax_{x eq yin X}|langle x,yrangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $Xsubseteqmathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $dtoinfty$. In establishing bounds on $theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1choose 2}$ equiangular lines in $mathbb{R}^k$. Using this connection, we are able to pin down $theta(d,k)$ whenever $kin{1,2,3,7,23}$ and establish asymptotics for general $k$. The main tool is an upper bound on $mathbb{E}_{x,ysimmu}|langle x,yrangle|$ whenever $mu$ is an isotropic probability mass on $mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $mathbb{C}^k$, also known as SIC-POVM in physics literature.
We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position in ${mathbb{R}}^d$, affinely spanning ${mathbb{R}}^d$, is $n + d(d - 1)/2 - 1$. Let ${underline{sa}}^s_d(n)$ be the minimum of the number of strictly antipodal pairs of points among any $n$ points in ${mathbb{R}}^d$, with affine hull ${mathbb{R}}^d$, and in strictly convex position. The value of ${underline{sa}}^s_d(n)$ was known for $d le 3$ and any $n$. Moreover, ${underline{sa}}^s_d(n) = lceil n/2rceil $ was known for $n ge 2d$ even, and $n ge 4d+1$ odd. We show ${underline{sa}}^s_d(n) = 2d$ for $2d+1 le n le 4d-1$ odd, we determine ${underline{sa}}^s_d(n)$ for $d=4$ and any $n$, and prove ${underline{sa}}^s_d(2d -1) = 3(d - 1)$. The cases $d ge 5 $ and $d+2 le n le 2d - 2$ remain open, but we give a lower and an upper bound on ${underline{sa}}^s_d(n)$ for them, which are of the same order of magnitude, namely $Theta left( (d-k)d right) $. We present a simple example of a strictly antipodal set in ${mathbb{R}}^d$, of cardinality const,$cdot 1.5874...^d$. We give simple proofs of the following statements: if $n$ segments in ${mathbb{R}}^3$ are pairwise antipodal, or strictly antipodal, then $n le 4$, or $n le 3$, respectively, and these are sharp. We describe also the cases of equality.
The multicolor Ramsey number problem asks, for each pair of natural numbers $ell$ and $t$, for the largest $ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of ErdH{o}s on graphs with no large cliques and few large independent sets. We also make some basic observations about this problem.
We generalize valuations on polyhedral cones to valuations on fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion-restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion-restriction proof of a result of Klivans-Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.
In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in $X$ on the equator and point-hyperplane frameworks with the vertices in $X$ representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
