rongduo 发表于 2006-12-21 17:47:15

[原创]魔方角块色向问题的群论模型

<h3 style="MARGIN: 13pt 0cm;"><span style="FONT-WEIGHT: normal; FONT-SIZE: 12pt; LINE-HEIGHT: 173%; FONT-FAMILY: 楷体_GB2312; mso-bidi-font-weight: bold;">《魔方组合原理》新增附录<span lang="EN-US"><p></p></span></span></h3><h3 align="center" style="MARGIN: 13pt 0cm; TEXT-ALIGN: center;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"><font size="5">魔方角块色向问题的群论模型</font></span></h3><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">本书正文在网上挂出后,有读者问:既然在第八章的结尾引用了置换群的知识来简化方块置换组合数的计算,那么群论能够处理方块的色向问题吗?本文将解答这一疑问,不过这要求读者拥有群论和向量代数的初步知识。</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"></span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt;"><p><font face="Times New Roman"></font></p></span></p><p><font face="Times New Roman"></font></p><p></p><p><font face="Times New Roman"></font></p><p></p><p><font face="Times New Roman"></font></p><p></p><p><font face="Times New Roman"></font></p><p></p><p><font face="Times New Roman"></font></p><p></p><p><font face="Times New Roman"></font></p><p></p><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">又以下论述凡提及魔方,皆指全部角块已经归位的鲁毕克魔方。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p></span><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">在正文第四章(二)中,为处理角块的色向组合问题,我们在集合</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">A</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">={</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">, </font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">T, -T}</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">上定义了一个运算“<span lang="EN-US">+</span>”<span lang="EN-US">(</span>直接称为“加”、“加运算”<span lang="EN-US">)</span>。容易看出,<span lang="EN-US">A</span>中元素对于这一运算构成群,且此群是一个交换群和循环群。为便于进一步讨论,我们称此群为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">角的精细群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">或直接称为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">精细群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,并将它记为<span lang="EN-US">E,</span>按群论的常例表示为:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">E = {A, +}<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">又称此群中的每一个元素</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">t</span><font face="Times New Roman"><sub><span lang="EN-US" style="FONT-SIZE: 12pt;">i </span></sub><span lang="EN-US" style="FONT-SIZE: 12pt;">(</span></font><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">注意</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">t<sub>i</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">A</span></span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">)</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">为一个</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">精细值</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">,精细值的运算仍沿用正文中状态值的“</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">+</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">”运算。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">记定义于集合<span lang="EN-US">A</span>上的任一<span lang="EN-US">n</span>维<span lang="EN-US">(</span>魔方的角块有<span lang="EN-US">8</span>个,所以是<span lang="EN-US">8</span>维<span lang="EN-US">)</span>向量为:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt;<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">并记所有这样的向量的集合为<span lang="EN-US">B</span>:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">B = { b|b</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">是定义在集合<span lang="EN-US">A</span>上的<span lang="EN-US">n</span>维向量<span lang="EN-US"> }<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">从魔方直观看,任意一个<span lang="EN-US">b =</span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt;</span>表示由组装或转动而得到的所有魔方角块的一种色向组合图案,其中编号为<span lang="EN-US">i</span>的那个角块的状态为<span lang="EN-US">t<sub>i </sub></span>,<span lang="EN-US">t<sub>i</sub></span>取值的范围为集合</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">A={</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">, </font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">T, -T}</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">任取<span lang="EN-US">B</span>中的两个向量元素<span lang="EN-US">b<sub>1</sub>, b<sub>2</sub></span>:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1 </sub>= &lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt;<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>2 </sub>= &lt;t</span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">1</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,t</span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">2</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t</span></span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">n </span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&gt;<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">现给出这两个元素之间加运算的定义如下。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定义<span lang="EN-US">1&nbsp; </span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1 </sub>+ b<sub>2</sub> = &lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt; + &lt;t</span></span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">1</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,t</span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">2</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t</span></span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">n </span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&gt;<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 4;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>= &lt;t<sub>1</sub>+t</span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">1</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">, t<sub>2</sub>+t</span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">2</span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n</sub>+t</span></span><sub><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;">i</span></sub><sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">n </span></sub><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&gt;</span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-FAMILY: 宋体;"><p></p></span></b></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">此定义的魔方直观意义是:<span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">取两个色向图案未必相同但角块编号相同的魔方</span>,把第一个魔方与第二个魔方中所有序号对应相等的角块的精细值(在正文中叫“状态值”)分别相加,其结果是唯一确定的一个魔方图案,这种图案的魔方也可以通过组装或转动而得到。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">显见定义<span lang="EN-US">1</span>与普通代数中向量加法的定义完全一致。我们知道在一般向量加法的定义中,等号两端加号的意义并不一样。定义<span lang="EN-US">1</span>也是如此,其前两个加号表示两个魔方的角块按一定规则的虚拟叠合,后边尖括号中的加号则表示叠合的规则<span lang="EN-US">:</span>相同编号的角块状态值(或精细值)相加。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><span style="FONT-SIZE: 12pt; COLOR: red; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;"><span lang="EN-US"><p></p></span></span><p></p><p></p><p></p>
[此贴子已经被作者于2006-12-23 19:28:27编辑过]

rongduo 发表于 2006-12-21 17:56:24

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; COLOR: red; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">(接上帖)<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">1&nbsp; </span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">向量集合<span lang="EN-US">B</span>与定义<span lang="EN-US">1</span>所确定的加运算构成一个群。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">证明 </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">1</span>)运算的封闭性显然。又<span lang="EN-US">B</span>中必有这样一个元素</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: Batang; mso-hansi-font-family: Batang;">:</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 102pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><shapetype id="_x0000_t202" ospt="202" coordsize="21600,21600" path="m,l,21600r21600,l21600,xe"></shapetype><shapetype></shapetype><stroke joinstyle="miter"></stroke><stroke></stroke><stroke></stroke><stroke></stroke><path oconnecttype="rect" gradientshapeok="t"></path><path></path><path></path><path></path><shapetype></shapetype><shapetype></shapetype><shape id="_x0000_s1026" type="#_x0000_t202" stroked="f" style="MARGIN-TOP: 19.7pt; Z-INDEX: 1; LEFT: 0px; MARGIN-LEFT: 148.3pt; WIDTH: 1in; TEXT-INDENT: 0px; POSITION: absolute; HEIGHT: 23.4pt; TEXT-ALIGN: left;"></shape><shape></shape><fill opacity="0"></fill><fill></fill><fill></fill><fill></fill><textbox style="LAYOUT-FLOW: vertical-ideographic; mso-next-textbox: #_x0000_s1026;"></textbox><textbox></textbox><textbox></textbox><textbox></textbox><shape></shape><shape></shape><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">=&lt;</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&gt;&nbsp;&nbsp;&nbsp; (尖括号中有n个<span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ)</span><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><shape id="_x0000_s1027" type="#_x0000_t202" stroked="f" style="MARGIN-TOP: -0.1pt; Z-INDEX: 2; LEFT: 0px; MARGIN-LEFT: 147.5pt; WIDTH: 63.1pt; POSITION: absolute; HEIGHT: 23.4pt; TEXT-ALIGN: left;"></shape><shape></shape><fill opacity="0"></fill><fill></fill><fill></fill><fill></fill><textbox style="mso-next-textbox: #_x0000_s1027;"></textbox><textbox></textbox><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">此元素显然符合群单位元的要求。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">2</span>)对于任意的<span lang="EN-US">b=&lt;t<sub>1</sub>,t<sub>2</sub>,</span>…<span lang="EN-US">,t<sub>n </sub>&gt;</span>∈<span lang="EN-US">B</span>,总有另一个属于<span lang="EN-US">B</span>的元素<span lang="EN-US">b’=&lt;-t<sub>1</sub>,-t<sub>2</sub>,</span>…<span lang="EN-US">,-t<sub>n </sub>&gt;</span>,使得<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b + b’= &lt;t<sub>1</sub>+(-t<sub>1</sub>),t<sub>2</sub>+(-t<sub>2</sub>),</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n</sub>+(-t<sub>n</sub>)&gt; </span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">= &lt;</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">,</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&gt; = </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang; mso-fareast-font-family: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">所以任一元素的逆元存在。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">3</span>)按定义,集合<span lang="EN-US">B</span>中向量的加运算,其实仅仅是对两个向量的对应分量分别相加,而每一个分量又是一个精细值,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在正文第四章(二)中已证明精细值的加运算满足结合律,故而</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">B</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">中向量的加运算也一定满足结合律。至此,定理</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">1</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">完全</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">得证。</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">我们称定理<span lang="EN-US">1</span>所说的向量群为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">角色向组合群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,或简称为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">角色向群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">、</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">色向群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">。记此群为<span lang="EN-US">Cd</span>:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">Cd={B</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,定义<span lang="EN-US">1</span>所确定的运算“<span lang="EN-US">+</span>”<span lang="EN-US">}<p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">自然,群<span lang="EN-US">Cd</span>是一向量加群。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">下来讨论<span lang="EN-US">Cd</span>的性质。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">2 </span></span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-spacerun: yes;">&nbsp;</span><span style="mso-spacerun: yes;">&nbsp;</span></span></b><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">Cd</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">是交换群。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">欲证此定理,可比照定理<span lang="EN-US">1</span>证明结合律的思路,此处从略。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><span lang="EN-US" style="FONT-SIZE: 12pt; COLOR: red; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">(</span><span style="FONT-SIZE: 12pt; COLOR: red; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">未完<span lang="EN-US">)<p></p></span></span><p></p>
[此贴子已经被作者于2006-12-22 18:36:34编辑过]

rongduo 发表于 2006-12-21 18:00:56

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定义<span lang="EN-US">2&nbsp; </span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">对<span lang="EN-US">Cd</span>中的任一元素<span lang="EN-US">b=&lt;t<sub>1</sub>,t<sub>2</sub>,</span>…<span lang="EN-US">,t<sub>n </sub>&gt;</span>,称<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">S(b)=t<sub>1</sub>+t<sub>2</sub>+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US"> +t<sub>n<p></p></sub></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">为<span lang="EN-US">b</span>的</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">精细值的和</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,或简称为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">精细和</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">。它也就是正文所说的状态和。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">3&nbsp; </span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">Cd</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">中任意两个元素<span lang="EN-US">:<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1</sub>=&lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt;<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>2</sub>=&lt;t<sub>21</sub>,t<sub>22</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>2n </sub>&gt;<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">相加结果的精细和,等于<span lang="EN-US">b<sub>1</sub></span>的精细和与<span lang="EN-US">b<sub>2</sub></span>的精细和的和。即:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">S(b<sub>1</sub>+b<sub>2</sub>) = S(b<sub>1</sub>) + S(b<sub>2</sub>)<p></p></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">证明&nbsp; </span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1</sub>+b<sub>2 </sub><b style="mso-bidi-font-weight: normal;">= </b>&lt;t<sub>1</sub>,t<sub>2</sub>,</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">,t<sub>n </sub>&gt; + &lt;t<sub>21</sub>,t<sub>22</sub>,</span>…<span lang="EN-US">,t<sub>2n </sub>&gt;<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 2;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span><span style="mso-spacerun: yes;">&nbsp;&nbsp;&nbsp;&nbsp; </span>= &lt;t<sub>1</sub>+ t<sub>21 </sub>, t<sub>2</sub>+ t<sub>22 </sub>, </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">, <state wst="on"></state><place wst="on"></place>t<sub>n</sub>+ <place></place><state></state>t<sub>2n</sub> &gt;<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">故而,<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">S(b<sub>1</sub>+b<sub>2</sub>) = S(&lt;t<sub>1 </sub>+ t<sub>21 </sub>, t<sub>2 </sub>+ t<sub>22 </sub>, </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">, <state wst="on"></state><place wst="on"></place>t<sub>n</sub>+ <place></place><state></state>t<sub>2n</sub> &gt;)<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 108pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">=(t<sub>1 </sub>+ t<sub>21 </sub>)+(t<sub>2 </sub>+ t<sub>22 </sub>)+</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US">+(<state wst="on"></state><place wst="on"></place>t<sub>n</sub>+ <place></place><state></state>t<sub>2n</sub>)<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 126pt; TEXT-INDENT: 6pt; mso-char-indent-count: .5; mso-para-margin-left: 12.0gd;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">=(t<sub>1</sub>+t<sub>2 </sub>+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US"> + t<sub>n</sub>) + (t<sub>21</sub> + t<sub>22</sub> +</span>…<span lang="EN-US"> + t<sub>2n</sub>)<p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 126pt; TEXT-INDENT: 6pt; mso-char-indent-count: .5; mso-para-margin-left: 12.0gd;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">= S(b<sub>1</sub>) + S(b<sub>2</sub>) <p></p></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">证毕。</span></p><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span lang="EN-US"><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">易知,<span lang="EN-US">Cd</span>中任一元素<span lang="EN-US">b</span>的精细和的可能的取值,仅为集合<span lang="EN-US">A</span>中的所有的三个元素:</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">, </font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">T, -T</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">。据此可以把</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">Cd</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">的集合<span lang="EN-US">B</span>划分为互不相交的三个子集:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">B<sub>0</sub> = { b|S(b)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">}<p></p></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">B<sub>1</sub> = { b|S(b)=T }<p></p></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">B<sub>2</sub> = { b|S(b)=-T}<p></p></span></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&shy;&shy;<span style="font-emphasize: dot;">B<sub>0</sub></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; font-emphasize: dot;">为正文所说的组装正确、符合跷跷板原理的魔方角块的图案集合</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">;<span lang="EN-US" style="font-emphasize: dot;">B<sub>1</sub></span><span style="font-emphasize: dot;">和<span lang="EN-US">B&shy;&shy;<sub>2</sub></span>为组装错误从而不符合跷跷板原理的魔方角块的图案集合</span>。 <span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p></span></span>
[此贴子已经被作者于2006-12-22 18:34:53编辑过]

rongduo 发表于 2006-12-21 18:02:24

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">4</span></span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">
                                <span style="mso-spacerun: yes;">&nbsp;</span></span></b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">集合<span lang="EN-US">B<sub>0</sub></span>对向量的加运算构成群。<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">证明</span><b style="mso-bidi-font-weight: normal;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">
                                <span style="mso-spacerun: yes;">&nbsp;</span></span></b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">因为<span lang="EN-US">B<sub>0</sub></span>是群<span lang="EN-US">Cd</span>的集合<span lang="EN-US">B</span>的子集,其运算的结合性是自然的。我们只须证明单位元和逆元的存在以及加运算的封闭性。<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">1</span>)∵<span lang="EN-US"> S(</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">)= </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><span style="mso-tab-count: 2;"><font face="Times New Roman">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">B<sub>0<p></p></sub></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">即集合<span lang="EN-US">B<sub>0</sub></span>包含单位元。<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">2</span>)若<span lang="EN-US">b=&lt;t<sub>1</sub>,t<sub>2</sub>,</span>…<span lang="EN-US">,t<sub>n </sub>&gt;</span>∈<span lang="EN-US">B<sub>0</sub></span>,则<span lang="EN-US">S(b)=</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ。</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">其逆元<span lang="EN-US">b’</span>∈<span lang="EN-US">B</span>为<span lang="EN-US">&lt;-t<sub>1</sub>,-t<sub>2</sub>,</span>…<span lang="EN-US">,-t<sub>n </sub>&gt;</span>,<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∵ </span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">b’= b’</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang; mso-fareast-font-family: 宋体;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴<span lang="EN-US"> S(b’) = S(b’+ </span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang; mso-fareast-font-family: 宋体;">)<p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang; mso-fareast-font-family: 宋体;"><span style="mso-tab-count: 2;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span style="mso-spacerun: yes;">&nbsp; </span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-spacerun: yes;">&nbsp;</span>= S(b’) + S(</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;">Φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">)<p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 2;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span style="mso-spacerun: yes;">&nbsp;&nbsp; </span>= [(-t<sub>1</sub>)+(-t<sub>2</sub>)+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US"> +(-t<sub>n</sub>)]+</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">已知</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">S(b)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ,所以又有:</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US" style="FONT-SIZE: 12pt;"><span style="mso-tab-count: 4;"><font face="Times New Roman">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </font></span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">S(b’) = [(-t<sub>1</sub>)+(-t<sub>2</sub>)+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US"> +(-t<sub>n</sub>)]+ S(b)<p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 6;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>= [(-t<sub>1</sub>)+(-t<sub>2</sub>)+ </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">…<span lang="EN-US"> +(-t<sub>n</sub>)]+<p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 6;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>= </span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">所以</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">的逆元</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b’</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">也属于<span lang="EN-US">B<sub>0</sub></span>。<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(<span lang="EN-US">3</span>)对属于<span lang="EN-US">B<sub>0&shy;</sub></span>的任意两个元素<span lang="EN-US">b<sub>1</sub></span>和<span lang="EN-US">b<sub>2</sub></span>有<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>S(b<sub>1</sub>)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ,</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 2;">&nbsp;&nbsp;&nbsp;&nbsp; </span>S(b<sub>2</sub>)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">按照定理<span lang="EN-US">3</span>有<span lang="EN-US"><p></p></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>S(b<sub>1</sub>+b<sub>2</sub>)= S(b<sub>1</sub>)+ S(b<sub>2</sub>)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">+</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">=</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">所以</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1</sub>+b<sub>2</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">也属于<span lang="EN-US">B&shy;<sub>0</sub></span>,即<span lang="EN-US">B&shy;<sub>0</sub></span>对于向量加运算保持封闭性。至此定理完全得证。<span lang="EN-US"><p></p></span></span></p>

rongduo 发表于 2006-12-21 18:04:37

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">出于显见的理由,我们把<span lang="EN-US">B<sub>0</sub></span>与向量加法所构成的这个群称为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">角色向的跷跷板群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">或直接称为</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">跷跷板群</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">,并记为<span lang="EN-US">Ss:<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">Ss = { B<sub>0 </sub>, + }<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">自然,<span lang="EN-US">Ss</span>是<span lang="EN-US"> Cd={B</span>,<span lang="EN-US">+}</span>的一个子群。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">5</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"> 若<span lang="EN-US">b<sub>0</sub></span>∈<span lang="EN-US">B<sub>0</sub></span>,<span lang="EN-US">b<sub>1</sub></span>∈<span lang="EN-US">B<sub>1</sub>, b<sub>2</sub></span>∈<span lang="EN-US">B<sub>2</sub></span>,则<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(i) b<sub>0</sub>+b<sub>1</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">B<sub>1</sub></span>,<span lang="EN-US"><span style="mso-tab-count: 1;">&nbsp;&nbsp;&nbsp; </span>b<sub>0</sub>+b<sub>2</sub></span>∈<span lang="EN-US">B<sub>2</sub></span><sub> </sub>;<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">(ii) b<sub>1</sub>+b<sub>2</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">B<sub>0 </sub></span>。</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: Batang;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">证明</span><b style="mso-bidi-font-weight: normal;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;"> </span></b><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">(1)</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">&nbsp;</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∵</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;"> S(</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>0</sub>+b<sub>1</sub></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">)=S(</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>0</sub></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">)+S(</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1</sub></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">)=</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">+</font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">T</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">=</font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;">T<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;"><span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴ <span lang="EN-US">b<sub>0</sub>+b<sub>1</sub></span>∈<span lang="EN-US">B<sub>1</sub></span></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span lang="EN-US"><span style="mso-tab-count: 3;"><font face="Times New Roman">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<font size="3">&nbsp;</font></font></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;"><font size="3">同理, </font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>0</sub>+b<sub>2</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">B<sub>2</sub></span><sub> </sub></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">。</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt;"><span lang="EN-US"><font face="Times New Roman"><font size="4">(2)</font>&nbsp; &nbsp;</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∵ <span lang="EN-US">S(b<sub>1</sub>+b<sub>2</sub>)</span>=<span lang="EN-US">S(b<sub>1</sub>)+S(b<sub>2</sub>)=T+(-T)=</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 24pt; TEXT-INDENT: 18pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴<span lang="EN-US"><span style="mso-spacerun: yes;">&nbsp; </span>b<sub>1</sub>+b<sub>2</sub></span>∈<span lang="EN-US">B<sub>0</sub></span>。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">定理<span lang="EN-US">5</span>证毕。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">现记任意集合<span lang="EN-US">R</span>的元素个数为<span lang="EN-US">|R|</span>,我们有如下的定理:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">定理<span lang="EN-US">6</span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"> 集合<span lang="EN-US">B<sub>0</sub></span>,<span lang="EN-US">B<sub>1</sub></span>,<span lang="EN-US">B<sub>2</sub></span>的元素个数都相等。即:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 60pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">|B<sub>0</sub>|=|B<sub>1</sub>|=|B<sub>2</sub>|</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">证明</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: 宋体;"> 取任一元素</span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">b<sub>1</sub></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∈<span lang="EN-US">B<sub>1</sub></span>,分别与<span lang="EN-US">B<sub>0</sub></span>中所有的元素相加,则得到<span lang="EN-US">|B<sub>0</sub>|</span>个互异的元素,按定理<span lang="EN-US">5</span>,这些互异的元素都属于<span lang="EN-US">B<sub>1</sub></span>,<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴ <span lang="EN-US">|B<sub>0</sub>|</span>≤<span lang="EN-US">|B<sub>1</sub>|<span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span>①<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">又任取一元素<span lang="EN-US">b<sub>2</sub></span>∈<span lang="EN-US">B<sub>2</sub></span>,分别与<span lang="EN-US">B<sub>1</sub></span>的所有元素相加,则得到<span lang="EN-US">|B<sub>1</sub>|</span>个互异的元素,仍按定理<span lang="EN-US">5</span>,这些互异的元素都属于<span lang="EN-US">B<sub>0</sub></span>,<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴<span lang="EN-US"> |B<sub>1</sub>|</span>≤<span lang="EN-US">|B<sub>0</sub>|<span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span>②<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">综合①,②可知<span lang="EN-US">|B<sub>0</sub>|=|B<sub>1</sub>|</span>;同理可知<span lang="EN-US">|B<sub>0</sub>|=|B<sub>2</sub>|</span>。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴ <span lang="EN-US">|B<sub>0</sub>|=|B<sub>1</sub>|=|B<sub>2</sub>|<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-font-kerning: 1.0pt; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">证毕。</span>
[此贴子已经被作者于2006-12-21 18:35:45编辑过]

rongduo 发表于 2006-12-21 18:10:05

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">定理<span lang="EN-US">6</span>的一个自然的</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 黑体;">推论</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">是:<span style="font-emphasize: dot;">角色向的跷跷板群的阶是角色向组合群的阶的三分之一</span>,也就是:<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><span style="mso-tab-count: 3;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>|B<sub>0</sub>| =(1/3)·|B|<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">熟知定义于<span lang="EN-US">m</span>个元素上的<span lang="EN-US">n</span>维向量共有<span lang="EN-US">m<sup>n</sup></span>个,本例中<span lang="EN-US">,<p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">m =|A|=|{</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">φ</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">, </font></span><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">T, -T }|=3<p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">又因为角色向组合群<span lang="EN-US">Cd</span>的集合<span lang="EN-US">B</span>是定义在<span lang="EN-US">A</span>上的<span lang="EN-US">n</span>维向量的集合,<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴<span lang="EN-US"> |B|=3<sup>n</sup><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 63pt; TEXT-INDENT: 21pt;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">∴ <span lang="EN-US">|B<sub>0</sub>| =(1/3)·|B| =(1/3)</span>·<span lang="EN-US">3<sup>n</sup> = 3<sup>n - 1<p></p></sup></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">又知在鲁毕克魔方中<span lang="EN-US">n=8</span>,故组装正确且已全部归位的魔方角块的图案共有<span lang="EN-US">3</span></span><font size="3"><font size="4"><sup><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;"><span lang="EN-US" style="FONT-SIZE: 13pt; FONT-FAMILY: 宋体; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">8 - 1</span></span></sup></font><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">=3</span></font><sup><span lang="EN-US" style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体;"><span lang="EN-US" style="FONT-SIZE: 13pt; FONT-FAMILY: 宋体; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">7</span></span></sup><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">种。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 25.8pt; mso-char-indent-count: 2.15;"><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><p>&nbsp;</p></span></p><p>&nbsp;</p><p></p><p>&nbsp;</p><p></p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">可以把本文中精细群<span lang="EN-US">E</span>的集合<span lang="EN-US">A={</span>φ<span lang="EN-US">, T, -T }</span>推广到任意<span lang="EN-US">m</span>个元素的集合<span lang="EN-US">A<sub>m</sub>={T<sub>0</sub>,T<sub>1</sub>,T<sub>2</sub>,</span>…<span lang="EN-US">,T<sub>m - 1</sub>}</span>上。这样的推广至少有两个好处:第一,当<span lang="EN-US">m=2</span>时,所得的色向群可用于描述边块的状态,这意味着<font face="细明体" color="#bb4444" size="4" style="BACKGROUND-COLOR: #e3fdbd;"><em>边块和角块的状态在群论中可得到统一的描述</em></font>;第二,如果存在某种异形魔方,其角块不止<span lang="EN-US">8</span>个,角块的状态也不止<span lang="EN-US">3</span>种,仍然可以用推广后的色向群来描述。<span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">不过本文已经不适合进行这样的推广了。</span><span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 黑体; mso-hansi-font-family: 宋体;">最后的结论</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">:可以用置换群和向量加群来完整、统一地描述鲁毕克魔方。<span lang="EN-US"><p></p></span></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; TEXT-ALIGN: center; mso-char-indent-count: 2.0;"><chsdate wst="on" year="2006" month="12" day="14" islunardate="False" isrocdate="False"></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">2006</span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;">年<span lang="EN-US">12</span>月<span lang="EN-US">14</span>日</span><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><chsdate></chsdate><span lang="EN-US" style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><span style="FONT-SIZE: 12pt; COLOR: red; FONT-FAMILY: 宋体; mso-bidi-font-family: &quot;Times New Roman&quot;; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">(全文完)</span>
[此贴子已经被作者于2006-12-23 7:39:27编辑过]

smok 发表于 2006-12-21 22:26:34

<p>提几个问题:</p><p>1。中心块如何描述?</p><p>2。边角块/中棱块色向和分别为零早已是众所周知的事实,况且可以用简单很多的方式描述,是不是一定要用群论描述?</p><p>3。三阶的状态太容易计算,作者是不是可以用跷跷板原理计算其它阶的状态数?如果可能,试着推导一下N阶通用的算式,我认为即然做为原理,应该具有晋适性,反正吧中早有现成的通用计算公式,可以相互比对。</p><p>4。状态描述在鲁毕克魔方上早已没有悬念,作者是不是可以将自已的原理导向最小步研究以避免重复劳动。</p><p>5。纵观作者的文章,发现尚不俱备通用自足的状态描述能力。</p><p>6。看不出作者是如何将魔方众所周知的性质导入数学工具中分析,相反,感觉很像是将数学原则硬塞给魔方,恕直言,而已有的能够通用描述魔方状态的理论,都是基于大家所熟悉的魔方性质而建立。</p><p></p>
[此贴子已经被作者于2006-12-21 22:44:25编辑过]

smok 发表于 2006-12-22 13:47:36

<p>跷跷板原理定义本身表达的理念是:一凸必有一凹,有一黑必有一白,显然是一种静态规则。但是,角块上,可以独立存在三个顺转色向或三个逆转色向,显然不满足"一凸必有一凹,有一黑必有一白"这一静态理念。但从变换的角度,显然满足色向和为零这一原则。动态原则显然不是跷跷原理的定义的精神,那么跷跷板原理想表达的理念到底是什么?如果连解决问题的思路都表达不清,还有必要关注所选择的数学工具?这不能不让人联想起以前某个理论,感觉几乎就是数学的垒彻,只可惜在基本提问的要求下,预言了自身的破产。</p>
[此贴子已经被作者于2006-12-22 13:57:16编辑过]

smok 发表于 2006-12-24 07:47:55

半遮半掩有碍讨论的公正性,rongduo何不将自已的理论及其修正版在此或在你认为合适的任何版块发布出来,让大家有一个全面的了解,当初你用组合数质问PW3的正确性,获得了比预期更完整的答复,同样,现在有人质疑你的理论,为何不用公开,公正,有力的答复来消除一切疑云?错就是错,对就是对,也许“发难者”不会得到以前某个大师的说法:“这是科学家关心的事,你只管用就行了”
[此贴子已经被作者于2006-12-24 7:50:14编辑过]

rongduo 发表于 2006-12-24 10:17:23

<div class="msgheader">QUOTE:</div><div class="msgborder"><b>以下是引用<i>smok</i>在2006-12-24 7:47:55的发言:</b><br/>半遮半掩有碍讨论的公正性,rongduo何不将自已的理论及其修正版在此或在你认为合适的任何版块发布出来,让大家有一个全面的了解,当初你用组合数质问PW3的正确性,获得了比预期更完整的答复,同样,现在有人质疑你的理论,为何不用公开,公正,有力的答复来消除一切疑云?错就是错,对就是对,也许“发难者”不会得到以前某个大师的说法:“这是科学家关心的事,你只管用就行了”<br/></div><span style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;"><h1 align="center" style="MARGIN: 17pt 0cm 16.5pt; TEXT-ALIGN: center;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></h1><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 25.2pt; mso-char-indent-count: 2.1;"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">本主题帖原本是准备与那些学习过近世代数的魔友交流的。敢于评论自己知识范围以外的帖子,这需要坦然的参与精神或超人的勇气。出于敬重,我愿意牺牲时间最后一次回答你。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 67.95pt; TEXT-INDENT: -42.75pt; mso-list: l0 level1 lfo1; tab-stops: list 67.95pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; mso-fareast-font-family: &quot;Times New Roman&quot;;"><span style="mso-list: Ignore;"><font face="Times New Roman">1.<span style="FONT: 7pt &quot;Times New Roman&quot;;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">对于</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><font face="Times New Roman">PW3</font></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">,我有过评论,正面和<b style="mso-bidi-font-weight: normal;"><span style="COLOR: fuchsia;">稍微</span></b>负面的都有,但我极为谨慎,决不信口开河。但我好像并未觉察他给出的组合数有什么问题,更谈不上<b style="mso-bidi-font-weight: normal;"><span style="COLOR: fuchsia;">质问</span></b>其正确性。请你拿出<b style="mso-bidi-font-weight: normal;"><span style="COLOR: fuchsia;">质问的</span></b>证据来,一定啊!</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 67.95pt; TEXT-INDENT: -42.75pt; mso-list: l0 level1 lfo1; tab-stops: list 67.95pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; mso-fareast-font-family: &quot;Times New Roman&quot;;"><span style="mso-list: Ignore;"><font face="Times New Roman">2.<span style="FONT: 7pt &quot;Times New Roman&quot;;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">我准备修订《魔方组合原理》,但只是文字上,其基本理论不变,架构和章节不变。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 67.95pt; TEXT-INDENT: -42.75pt; mso-list: l0 level1 lfo1; tab-stops: list 67.95pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; mso-fareast-font-family: &quot;Times New Roman&quot;;"><span style="mso-list: Ignore;"><font face="Times New Roman">3.<span style="FONT: 7pt &quot;Times New Roman&quot;;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">你所有的疑问在原书(尚未修订的)都有答案,坐下来读一读吧,那是一本通俗读物,挺好读的。(不要无谓地把时间过多地花在本主题帖上,这需要一定的近世代数基础)。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 67.95pt; TEXT-INDENT: -42.75pt; mso-list: l0 level1 lfo1; tab-stops: list 67.95pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; mso-fareast-font-family: &quot;Times New Roman&quot;;"><span style="mso-list: Ignore;"><font face="Times New Roman">4.<span style="FONT: 7pt &quot;Times New Roman&quot;;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">非常感谢你对我的原理和小书过人的关注,即使是批评,也让更多的人知道了这本书的存在并产生了阅读的欲望。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 67.95pt; TEXT-INDENT: -42.75pt; mso-list: l0 level1 lfo1; tab-stops: list 67.95pt;"><span lang="EN-US" style="FONT-SIZE: 12pt; mso-fareast-font-family: &quot;Times New Roman&quot;;"><span style="mso-list: Ignore;"><font face="Times New Roman">5.<span style="FONT: 7pt &quot;Times New Roman&quot;;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span><span style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体; mso-ascii-font-family: &quot;Times New Roman&quot;; mso-hansi-font-family: &quot;Times New Roman&quot;;">我今后将不会有时间来回答你更多的质疑,这真不好意思。不过坦率地说,除了涉及群论背景的问题,迄今为止我还没有发现谁的质疑在原书中没有答案。</span><span lang="EN-US" style="FONT-SIZE: 12pt;"><p></p></span></p><p></p><p></p><p></p><p></p><p></p></span>
[此贴子已经被作者于2006-12-24 10:32:57编辑过]
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