第13课。
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14FiniteGeometriesAndLatinSquares.tex
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14FiniteGeometriesAndLatinSquares.tex
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\chapter{有限几何与拉丁方}
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\section{``小世界''：有限几何}
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\subsection{The Fano Plane}
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下面的这个平面被称作Fano Plane：
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a) at (0,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b) at (30:1) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c) at (150:1) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (d) at (270:1) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (e) at (-30:2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (f) at (90:2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (g) at (210:2) {};
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\draw (a)--(b) (a)--(c) (a)--(d) (a)--(e) (a)--(f) (a)--(g) (e)--(b)--(f)--(c)--(g)--(d)--(e) (a) circle (1);
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\end{tikzpicture}
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\caption{The Fano plane}
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\label{The Fano plane}
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\end{figure}
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与我们之前所学的欧氏几何不同，这个平面是由七个点与连接它们的线组成的。注意，图中一些未涂黑的线之间的交点不是图中的点。
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Fano Plane有很多很好的性质。
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\begin{enumerate}[label=(\alph{*})]
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\item 任何两个点可以唯一确定一条线。
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\item 任何两条线都有唯一的交点。
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\item 任何一条线上都有至少三个点。
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\end{enumerate}
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有了这三条性质的条件，我们可以推出Fano Plane和其它满足这些条件的图还有如下的性质:
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\begin{enumerate}[label=(\alph{*}), resume]
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\item 所有的线上都有相同数量的点。
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\item 所有的点都有相同数量的线穿过。
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\end{enumerate}
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同时，如果我们将某两个点的名字交换，我们总有方法可以对应改变其它所有点的名字，使得变换后图中点的相互关系与变换前完全相同，即
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\begin{enumerate}[label=(\alph{*}), resume]
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\item 所有的点是相似的，所有的边也是相似的。
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\end{enumerate}
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\begin{definition}[有限投影平面]
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对一个含有有限元素（称为点）的集合，它的一些子集称为边。如果它满足：
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\begin{enumerate}[label=(\alph{*})]
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\item 任何两个点可以唯一确定一条线；
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\item 任何两条线都有唯一的交点；
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\item 任何一条线上都有至少三个点，
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\end{enumerate}
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则称这个平面为一个\newnoun{有限投影面}{finite projective plane}。
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\end{definition}
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下面让我们来看另外一种有限的图形。
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\subsection{The Tictactoe Plane}
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\begin{figure}[H]
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\centering
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\subfloat{
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\begin{tikzpicture}
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a1) at (0,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a2) at (2,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a3) at (4,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b1) at (0,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b2) at (2,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b3) at (4,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c1) at (0,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c2) at (2,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c3) at (4,4) {};
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\draw (a1)--(a3) (b1)--(b3) (c1)--(c3) (a1)--(c1) (a2)--(c2) (a3)--(c3);
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\draw (c1)--(a3) (c2) ..controls (2.4,3.8) and (3.8,3)..(b3).. controls (3.8,1) and (2,0.6)..(a1);
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\draw (c3) ..controls (2,3.4) and (0.2,3).. (b1) .. controls(0.2,1) and (1.6,0.2) ..(a2);
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\draw[dashed] (c1)..controls (2,3.4) and (3.8,3) .. (b3) .. controls (3.8,1) and (2.4,0.2) .. (a2);
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\draw[dashed] (c2)..controls (1.6, 3.8) and (0.2,3) .. (b1) .. controls (0.2,1) and (2,0.6) ..(a3);
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\draw[dashed] (a1)--(c3);
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\end{tikzpicture}
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}
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\hspace{2cm}
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\subfloat{
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\begin{tikzpicture}
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a1) at (0,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a2) at (2,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a3) at (4,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (a4) at (6,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (a5) at (8,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b1) at (0,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b2) at (2,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (b3) at (4,2) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b4) at (6,2) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b5) at (8,2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c1) at (0,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c2) at (2,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c3) at (4,4) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (c4) at (6,4) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (c5) at (8,4) {};
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\draw (a1)--(a3) (b1)--(b3) (c1)--(c3) (a1)--(c1) (a2)--(c2) (a3)--(c3) (c1)--(a3) (c2)--(a4) (c3)--(b4)--(a5);
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\draw[dashed] (a1)--(c3) (a2)--(b3)--(c4) (a3)--(b4)--(c5);
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\end{tikzpicture}
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}
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\caption{The Tictactoe plane}
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\label{The Tictactoe plane}
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\end{figure}
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将左图中的第一、第二列在右侧重复一遍后，可以把图示画得更为美观。可以看到，在Tictactoe plane中，过线外一点有且仅有一条线与已知的线平行，即欧氏几何中的平行公理成立。
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\begin{definition}
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对一个含有有限元素（称为点）的集合，它的一些子集称为边。如果它满足：
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\begin{enumerate}[label=(\alph{*})]
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\item 任何两个点可以唯一确定一条线；
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\item 过线外一点有且仅有一条线与已知的线平行，
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\end{enumerate}
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则称这个平面为一个\newnoun{有限仿射面}{finite affine plane}。
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\end{definition}
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\begin{theorem}
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任一有限仿射面可通过添加新点与新线的方式扩充至有限投影面；删去有限投影面中任意一条边及边上所有的点可以得到一个有限投影面。
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\end{theorem}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (a1) at (0,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (a2) at (2,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (a3) at (4,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b1) at (0,2) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b2) at (2,2) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b3) at (4,2) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (c1) at (0,4) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (c2) at (2,4) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (c3) at (4,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (d1) at (-2,-2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (d2) at (2, -2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (d3) at (6,-2) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (d4) at (10,-2) {};
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\draw (a1)--(a2)--(a3) (b1)--(b2)--(b3) (c1)--(c2)--(c3) (a1)--(b1)--(c1) (a2)--(b2)--(c2) (a3)--(b3)--(c3);
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\draw (c1)--(b2)--(a3) (c2) ..controls (2.4,3.8) and (3.8,3)..(b3).. controls (3.8,1) and (2,0.6)..(a1);
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\draw (c3) ..controls (2,3.4) and (0.2,3).. (b1) .. controls(0.2,1) and (1.6,0.2) ..(a2);
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\draw (a1)--(d2) (a2)--(d2) (a3)--(d2);
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\draw (a1)--(d3) (a2)--(d3) (a3)--(d3);
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\draw (a3)--(d4) (b3)--(d4) (c3)--(d4);
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\draw (d1)--(d2)--(d3)--(d4);
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\draw[dashed] (a1)--(d1) (a2)--(d1) (a3)--(d1);
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\draw[dashed] (c1)..controls (2,3.4) and (3.8,3) .. (b3) .. controls (3.8,1) and (2.4,0.2) .. (a2);
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\draw[dashed] (c2)..controls (1.6, 3.8) and (0.2,3) .. (b1) .. controls (0.2,1) and (2,0.6) ..(a3);
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\draw[dashed] (a1)--(b2)--(c3);
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\end{tikzpicture}
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\caption{为图中加上四个点和一条线，并延长原有的线得到有限投影面}
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\end{figure}
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\subsection{The Cube Space}
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (a) at (0,0) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (b) at (3,0) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (c) at (4,1) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (d) at (1,1) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (e) at (0,3) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (f) at (3,3) {};
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\node[draw=black, circle, inner sep=0, minimum size=8pt] (g) at (4,4) {};
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\node[draw=black, fill=black, circle, inner sep=0, minimum size=8pt] (h) at (1,4) {};
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\draw (a)--(b)--(c)--(f)--(b) (a)--(e) (c)--(g)--(h)--(e)--(f)--(g);
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\draw[dashed] (a)--(d)--(c) (e)--(d)--(h);
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\end{tikzpicture}
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\caption{The Cube space}
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\label{The Cube space}
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\end{figure}
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在图中，我们定义``面''为：
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\begin{enumerate}[label=（\arabic{*}）]
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\item 四个点形成了图中正方体的一个面（欧氏几何面），共六个；
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\item 四个点分别是一对相对的边的边的顶点，共六个；
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\item 图中四个黑色或四个白色顶点。
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\end{enumerate}
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这些面有很多好性质：
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\begin{enumerate}[label=(\Alph{*})]
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\item 任意三点唯一确定一个面；
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\item 两平面要么平行，要么相相交于一条线；
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\item 过平面外一点有且仅有一个平面与已知平面平行；
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\item 任意两点相似；
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\item 任意两面相似。
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\end{enumerate}
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\section{区组设计}
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\subsection{Kirkman 15女生问题}
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1847年Kirkman提出了下面这样一个有趣的组合数学问题：有一个女教师，每天要带领一个班的15名女生去散步。问能否设计一种连续一周的散步方案，使得每天散步的时候，15名女生都能分成5组，每组由3名女生组成，而且在7天里每一个女生和其她女生都要在同一组中同时出现一次而且只出现一次?
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简单分析一下，要求解这个问题就是要找到满足下面条件的三元子集组：
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\begin{enumerate}[label=（\arabic{*}）]
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\item 需要构造由15名女生组成的集合的$7\times 5 = 35$个三元子集族，使得每两个女生在三元组里正好出现一次；
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\item 由于每天的散步，每个女生出现而且只能出现在一个三元组中。所以必须能将35个三元组分成7部分，每部分有5个三元组。每一个女生正好在每部分出现一次，即在一个三元组出现。
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\end{enumerate}
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1850年，人们得到了这个问题的解：
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\begin{enumerate}
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\item 星期一：$\{1, 2, 3\}, \{4, 8, 12\}, \{5, 10, 15\}, \{6, 11, 13\}, \{7, 9, 14\}$；
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\item 星期二：$\{1, 4, 5\}, \{2, 8, 10\}, \{3, 13, 14\}, \{6, 9, 15\}, \{7, 11, 12\}$；
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\item 星期三：$\{1, 6, 7\}, \{2, 9, 11\}, \{3, 12, 15\}, \{4, 10, 14\}, \{5, 8, 13\}$；
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\item 星期四：$\{1, 8, 9\}, \{2, 12, 14\}, \{3, 5, 6\}, \{4, 11, 15\}, \{7, 10, 13\}$；
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\item 星期五：$\{1, 10, 11\}, \{2, 13, 15\}, \{3, 4, 7\}, \{5, 9, 12\}, \{6, 8, 14\}$；
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\item 星期六：$\{1, 12, 13\}, \{2, 4, 6\}, \{3, 9, 10\}, \{5, 11, 14\}, \{7, 8, 15\}$；
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\item 星期日：$\{1, 14, 15\}, \{2, 5, 7\}, \{3, 8, 11\}, \{4, 9, 13\}, \{6, 10, 12\}$。
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\end{enumerate}
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\subsection{区组设计概论}
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\begin{definition}
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设$X$是一个有限集合，则称$X$的子集族$B = \{B_1, B_2, \cdots, B_b\}$为$X$的一个\newnoun{区组设计}{block design}，记作$D = (X, B)$。$X$称为此设计的基集，而子集族$B$中的诸子集$B_i (i = 1, 2, \cdots, b)$则称为此设计的\newnoun{区组}{block}。
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基集$X$中元素的个数$\vert X \vert$称为设计的阶。对于$i = 1, 2, \cdots, b$，区组$B_i$的元素个数$\vert B_i \vert$又称为该区组的长度。
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对于$x \in X$，$B$中含有$x$的区组的个数称为元素$x$的重复数，记为$r(x)$。
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对于$x,y \in X, x \neq y$，$B$中以$\{x, y\}$为子集的集合数称为元素$x$和$y$的相遇数，记为$\lambda(x,y)$。
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区组设计可以用区组（关联）矩阵来描述。设$X = \{x_1, x_2, \cdots, x_v\}$，$B = \{B_1, B_2, \cdots, B_b\}$，其关联矩阵定义为
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\(A = \begin{bmatrix}
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a_{ij}
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\end{bmatrix}_{v \times b}\)，其中
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\[a_{ij} =
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\begin{cases}
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1, x_i \in B_j\\
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0, x_i \not \in B_j
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\end{cases}\eqper\]
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\end{definition}
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容易看出，第$i$行出现1的次数即为$r(x_i)$，第$j$列出现1的个数是$B_j$的长度。第$j$列可以看作是$B_j$的特征函数。
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同时可以得到$\lambda (x_i, x_j)$与$A$中第$i$行与第$j$行的内积相等。
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\begin{remark}
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区组族$B$中的区组是可以重复的，因此$B$是一个族而不是一个集合。
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\end{remark}
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\begin{example}
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在Kirkman 15女生问题中，这些参数为：
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\begin{enumerate}
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\item 设计的阶数为15；
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\item 每个区组的长度为3；
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\item 任意一个元素的重复数为7；
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\item 任意两个元素的相遇数为1；
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\item 区组总数为35。
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\end{enumerate}
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需要强调的是，这些参数并不是互相独立的，其中的某些参数可以由其它参数得到。
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\end{example}
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\subsection{平衡不完全的区组设计（BIBD）}
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\begin{definition}
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设$X = \{x_1, x_2, \cdots, x_v\}$，而$k (k < v)$和$\lambda$是确定的飞赴整数。如果区组设计$D = (X, B)$满足如下两个条件：
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\begin{enumerate}
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\item $B$中每个区组的长度都是$k$；
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\item $X$中每两个不同的元素$x, y$在$D$中的相遇数$\lambda_D(x, y)$都是$\lambda$，
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\end{enumerate}
|
||||
则称区组设计$D$为一个$(v,k,\lambda)$-\newnoun{平衡不完全区组设计}{balanced incomplete blockdesign}，简称$(v,k,\lambda)$-BIBD。
|
||||
\end{definition}
|
||||
|
||||
\begin{example}
|
||||
The Fano plane（图\ref{The Fano plane}）是$(7,3,1)$-BIBD。
|
||||
|
||||
The Tictactoe plane（图\ref{The Tictactoe plane}）是$(9,3,1)$-BIBD。
|
||||
|
||||
The Cube（图\ref{The Cube space}）是$(8,4,3)$-BIBD。
|
||||
\end{example}
|
||||
|
||||
为了排除平凡情况，总假设$k > 1$，从而$\lambda > 0$。
|
||||
\begin{theorem}
|
||||
设$k >1$，$D = (X,B)$是一个$(v,k,\lambda)$-BIBD，则有
|
||||
\begin{enumerate}
|
||||
\item $X$中所有元素的重复数都等于
|
||||
\[r = \frac{\lambda(v-1)}{k-1}\eqco\]
|
||||
\item $B$中所含的区组总数
|
||||
\[b = \frac{\lambda v (v-1)}{k(k-1)} = \frac{vr}{k}\eqper\]
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
如果存在$(v,k,\lambda)$-BIBD，其中$k > 1$，则必然有
|
||||
\[\lambda(v-1) \equiv 0 \pmod{(k-1)} \tag{1} \label{BIBD必要条件1}\]
|
||||
与
|
||||
\[\lambda v (v-1) \equiv 0 \pmod{k} \tag{2} \label{BIBD必要条件2}\eqper\]
|
||||
\end{theorem}
|
||||
Reference in New Issue
Block a user