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Second Order Spiral Splines

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 Added by Lyle Noakes
 Publication date 2018
  fields
and research's language is English
 Authors Lyle Noakes




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Second order spiral splines are $C^2$ unit-speed planar curves that can be used to interpolate a list $Y$ of $n+1$ points in $R ^2$ at times specified in some list $T$, where $ngeq 2$. Asymptotic methods are used to develop a fast algorithm, based on a pair of tridiagonal linear systems and standard software. The algorithm constructs a second order spiral spline interpolant for data that is convex and sufficiently finely sampled.

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121 - Wanyue Xu , Bin Wu , Zuobai Zhang 2021
A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In generally, they are simultaneously sparse, scale-free, small-world, and loopy. In this paper, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherence $H_{rm SO}$ characterized in terms of the $mathcal{H}_2$-norm of the vertex systems, which measures the mean deviation of vertex states from their average value. We first study numerically the coherence of some representative real-world networks. We find that their coherence $H_{rm SO}$ scales sublinearly with the vertex number $N$. We then study analytically $H_{rm SO}$ for a class of iteratively growing networks -- pseudofractal scale-free webs (PSFWs), and obtain an exact solution to $H_{rm SO}$, which also increases sublinearly in $N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study $H_{rm SO}$ for Sierpinski gaskets, for which $H_{rm SO}$ grows superlinearly in $N$, with a power exponent much larger than 1. Sierpinski gaskets have the same number of vertices and edges as the PSFWs, but do not display the scale-free and small-world properties. We thus conclude that the scale-free and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of $H_{rm SO}$.
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