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  Volume 6, Issue 3, Article 80
 
A Minimum Energy Condition of 1-Dimensional Periodic Sphere Packing
    Authors: Kanya Ishizaka,  
    Keywords: Packing, Energy, One-dimension.  
    Date Received: 13/04/05  
    Date Accepted: 21/07/05  
    Subject Codes:

05B40, 74G65.

  
    Editors: Sever S. Dragomir,  
 
    Abstract:

Let $ X !subset! mathbf{R}/mathbf{Z}$ be a non-empty finite set and $ f(x) $ be a real-valued function on $ [0,frac{1}{2}]$. Let an energy of $ X$ be the average value of $ f ( Vert x-y Vert )$ for $ x,y in X$ where $ Vert ! cdot ! Vert$ is the Euclidean distance on $ mathbf{R}/mathbf{Z}$. Let $ X_n subset! mathbf{R}/mathbf{Z}$ be an equally spaced $ n$-point set. It is shown that if $ f$ is monotone decreasing and convex, then among all $ n$ -point sets, the energy is minimized by $ X_n$. Moreover, by giving a variant of a result of Bennett and Jameson, it is shown that if $ f$ is convex, $ f^{prime}(x^frac{1}{2})$ is concave and $ lim_{x to frac{1}{2}} f^{prime}(x) = 0$, then the energy of $ X_n$ decreases with $ n$.

         
       
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