spherical designs
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2021 ◽  
Vol 54 ◽  
pp. 137-144
Author(s):  
Matt Sourisseau ◽  
Yu Guang Wang ◽  
Robert S. Womersley ◽  
Hau-Tieng Wu ◽  
Wei-Hsuan Yu

10.37236/9891 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Zhiqiang Xu ◽  
Zili Xu ◽  
Wei-Hsuan Yu

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X, \mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.


2020 ◽  
Author(s):  
Matt Sourisseau ◽  
Yu Guang Wang ◽  
Robert S. Womersley ◽  
Hau-Tieng Wu ◽  
Wei-Hsuan Yu

AbstractConcentration of frequency and time (ConceFT) is a generalized multitaper algorithm introduced to analyze complicated non-stationary time series. To avoid the randomness in the original ConceFT algorithm, we apply the novel complex spherical design technique to standardize ConceFT, which we coin CQU-ConceFT. The proposed CQU-ConceFT is applied to visualize the spindle structure in the electroencephalogram signal during the N2 sleep stage and other physiological time series.


2020 ◽  
Vol 127 (3) ◽  
pp. 238-248
Author(s):  
Richard Evan Schwartz ◽  
Serge Tabachnikov

2019 ◽  
Vol 65 (1) ◽  
pp. 244-260
Author(s):  
Peter Boyvalenkov ◽  
Konstantin Delchev ◽  
Matthieu Jourdain

Author(s):  
Stefan Steinerberger

Abstract We study manifolds $M$ equipped with a quadrature rule \begin{equation*} \int_{M}{\phi(x)\,\mathrm{d}x} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.\end{equation*}We show that $n$-point quadrature rules with nonnegative weights on a compact $d$-dimensional manifold cannot integrate more than at most the 1st $c_{d}n + o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result is new even on $\mathbb{S}^2$ where it generalizes results on spherical designs.


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