Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
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DOI：
10.1137/050626636
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发表时间：
2006-11
期刊：
影响因子：
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通讯作者：
Xiaojun Chen;R. Womersley
中科院分区：
文献类型：
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作者：
Xiaojun Chen;R. Womersley
本文涉及证明欠定方程组解的存在性，以及将 $R^3$ 中的单位球面 $S^2$ 上的 $(t+1)^2$ 点应用于球面 $t$ 设计的存在性。我们证明球形设计的构造等效于欠定方程的解。利用布劳威尔不动点定理导出了一种新的欠定方程验证方法。该方法的应用提供了接近极值（最大行列式）点的球形 $t$ 设计，并且点的数量具有最佳阶数 $O(t^2)$。提供了计算的球形设计的误差范围。
This paper is concerned with proving the existence of solutions to an underdetermined system of equations and with the application to existence of spherical $t$-designs with $(t+1)^2$ points on the unit sphere $S^2$ in $R^3$. We show that the construction of spherical designs is equivalent to solution of underdetermined equations. A new verification method for underdetermined equations is derived using Brouwer’s fixed point theorem. Application of the method provides spherical $t$-designs which are close to extremal (maximum determinant) points and have the optimal order $O(t^2)$ for the number of points. An error bound for the computed spherical designs is provided.