西安交通大学2005你数学建模选拔赛试题

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< ><p><FONT face="Times New Roman" size=3> </FONT></p></P>
<H1  align=center>试题<FONT face="Times New Roman">A: </FONT>乐谱识别问题<p></p></H1>
< ><FONT size=3>从获得的信息量角度看，听觉是人类仅次于视觉的第二感觉，因而，在机器人研究领域中，机器听力成为了一个十分热门的话题。特别是由于听力对于人的思想熏陶和感情气质的巨大作用，使人们联想到可以通过听力来训练计算机获得感情，由此必须首先能够让计算机从音乐信号中识别出喜怒哀乐等各色情感。</FONT></P>
< ><FONT size=3>人的情感十分复杂，表现在音乐中更是千差万别，这使得机器识别的困难很大。作为机器感情识别的初步研究，我们首先给出<FONT face="Times New Roman">20</FONT>首风格各异的曲目，分成两类，请从中提取特征，再构造分类方法，并用这些已知的分类组去衡量你的方法是否准确。随后希望能够用你的方法对后<FONT face="Times New Roman">30</FONT>首未分类的曲目进行判别。</FONT>
<FONT face="Times New Roman" size=3>    </FONT><FONT size=3>为方便大家使用计算机进行阅读曲谱，曲谱已经转为文本格式，各符号定义如下：</FONT></P>
<TABLE  cellSpacing=0 cellPadding=0 border=1>

<TR >
<TD  vAlign=top width=114>
<P >符号</P></TD>
<TD  vAlign=top width=158>
<P >含义</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P >符号</P></TD>
<TD  vAlign=top width=165>
<P >含义</P></TD></TR>
<TR >
<TD  vAlign=top width=114>
<P ><FONT face="Times New Roman">1</FONT>－<FONT face="Times New Roman">7</FONT></P></TD>
<TD  vAlign=top width=158>
<P >音高</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P ><FONT face="Times New Roman">_</FONT></P></TD>
<TD  vAlign=top width=165>
<P >节拍减半</P></TD></TR>
<TR >
<TD  vAlign=top width=114>
<P ><FONT face="Times New Roman">^</FONT></P></TD>
<TD  vAlign=top width=158>
<P >升<FONT face="Times New Roman">8</FONT>度</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P >－</P></TD>
<TD  vAlign=top width=165>
<P >延长一拍</P></TD></TR>
<TR >
<TD  vAlign=top width=114>
<P ><FONT face="Times New Roman">#</FONT></P></TD>
<TD  vAlign=top width=158>
<P >降<FONT face="Times New Roman">8</FONT>度</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P >.</P></TD>
<TD  vAlign=top width=165>
<P >浮点音符</P></TD></TR>
<TR >
<TD  vAlign=top width=114>
<P ><FONT face="Times New Roman">0</FONT></P></TD>
<TD  vAlign=top width=158>
<P >空拍</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P >|<p></p></P></TD>
<TD  vAlign=top width=165>
<P >小节分格符</P></TD></TR>
<TR >
<TD  vAlign=top width=114>
<P ><FONT face="Times New Roman">||</FONT></P></TD>
<TD  vAlign=top width=158>
<P >乐曲终止符</P></TD>
<TD  vAlign=top width=24>
<P ><p><FONT face="Times New Roman"> </FONT></p></P></TD>
<TD  vAlign=top width=108>
<P >$<p></p></P></TD>
<TD  vAlign=top width=165>
<P >文件的行结束标志位</P></TD></TR></TABLE>
<P ><FONT size=3>例如<FONT face="Times New Roman"> </FONT>有下面一段曲谱：</FONT>
<v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape><v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape></P>
<P ><v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape>
<v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape>
<v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape></P>
<P ><v:shape><v:imagedata><FONT size=3></FONT></v:imagedata></v:shape>
<FONT face="Times New Roman" size=3> </FONT><FONT size=3>转换成<FONT face="Times New Roman">txt</FONT>文件后为：</FONT></P>
<P ><v:group><lock aspectratio="t" v:ext="edit"></lock><v:shape><FONT size=3><v:fill detectmouseclick="t"></v:fill><v:path connecttype="none" extrusionok="t"></v:path><lock text="t" v:ext="edit"></lock></FONT></v:shape><v:rect><v:textbox style="mso-next-textbox: #_x0000_s1028">
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV>
<P ><FONT face="Times New Roman">6_ 7 6_ 6_ 3_ 5_ 6_| 6 3--| 2_ 3 2_ 2_ 6#_ 1_ 2_ |1_ 2_ 6#--|$</FONT></P>
<P ><FONT face="Times New Roman">1_ 2 3_ 3_ 6# 3_ | 2-- 3_ 5_ | 6_ 7_ 6_ 7_ 6__ 7__ 3_ |5-- 5_ 6_|$</FONT></P>
<P ><FONT face="Times New Roman">1^-- 7__ 1^__ 7__ 6__|6-- 5| 6_ 1 3_ 2_ 6#__ 1__ 7#_ 6#_| 6#_ 5#. 5#_ 6#_|$</FONT></P>
<P ><FONT face="Times New Roman">1. 6#_ 3. 2__ 3__| 2-- 3| 7#_ 5#_ 3__ 2__ 7#_ 3_ 2__ 3__ 5#_|1_ 6#. 6#-|$</FONT></P>
<P ><FONT face="Times New Roman">1^_ 6. 6-|1^_ 2^. 1^_ 2^_ 1^__ 2^__ 6| 1^_ 6. 6-| 6---||$</FONT></P>
<P ><FONT face="Times New Roman">80</FONT></P></DIV></TD></TR></TABLE></v:textbox></v:rect><w:wrap type="none"></w:wrap><w:anchorlock></w:anchorlock></v:group></P>
<P ><FONT size=3>注：最后的数字为曲目演奏速度（每分钟的节拍数）</FONT></P>
<P ><FONT size=3>全部乐谱文件在<FONT face="Times New Roman">”</FONT>试题<FONT face="Times New Roman">A</FONT>附件<FONT face="Times New Roman">.txt”</FONT>中给出，其中<FONT face="Times New Roman">1</FONT>－<FONT face="Times New Roman">10</FONT>为<FONT face="Times New Roman">I</FONT>类，<FONT face="Times New Roman">11</FONT>－<FONT face="Times New Roman">20</FONT>为<FONT face="Times New Roman">II</FONT>类，<FONT face="Times New Roman">21</FONT>－<FONT face="Times New Roman">50</FONT>为待分类的乐曲。</FONT></P>
<P ><p><FONT face="Times New Roman" size=3> </FONT></p></P>
<P ><p><FONT face="Times New Roman" size=3> </FONT></p></P>
<P ><p><FONT face="Times New Roman" size=3> </FONT></p></P>
<P ><p><FONT face="Times New Roman" size=3> </FONT></p></P>
<P  align=center>试题B：最佳泄洪方案<p></p></P>
<P >在洪水肆虐时，从全局出发有必要采取破堤泄洪，但从何处破堤分洪要考虑破堤的最小损失。现在选定在河岸一边完全封闭的某一区域破堤泄洪，根据区域内地形以及当前地面财产总数的不同，可将该区域分成<FONT face="Times New Roman">17</FONT>个小区域，各个相邻小区之间有相对高度为<st1:chmetcnv w:st="on" UnitName="米" SourceValue="1.2" HasSpace="False" Negative="False" NumberType="1" TCSC="0"><FONT face="Times New Roman">1.2</FONT>米</st1:chmetcnv>的小堤互相间隔。如下图所示：<p></p></P>
<P ><p><FONT face="Times New Roman"> </FONT></p></P>
<P ><v:line><FONT face="Times New Roman"></FONT></v:line><FONT face="Times New Roman">           ----------------</FONT>河<FONT face="Times New Roman">----------------------------</FONT>流<FONT face="Times New Roman">----------------------------<p></p></FONT></P>
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<TR >
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">H4.0<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">S8.4<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">K3.0<p></p></FONT></P></TD>
<TD  vAlign=top width=127 colSpan=2>
<P  align=center><FONT face="Times New Roman">3.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">6.1<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">2.4<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">4.7<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">7.0<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">5.8<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">3.8<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">4.8<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">2.0<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">4.4<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">9.3<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">3.3<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><v:line><FONT face="Times New Roman"></FONT></v:line><FONT face="Times New Roman">5.2<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">3.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">5.2<p></p></FONT></P></TD></TR>
<TR >
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">3.3<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">3.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">9.4<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">3.2<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">0.9<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.8<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">2.5<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">8.5<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">5.0<p></p></FONT></P></TD>
<TD  vAlign=top width=127 colSpan=2>
<P  align=center><FONT face="Times New Roman">5.0<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.8<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">7.2<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">4.4<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">4.3<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.6<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">5.2<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">2.3<p></p></FONT></P></TD></TR>
<TR >
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">3.0<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">4.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.8<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">5.4<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">1.5<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">4.1<p></p></FONT></P></TD>
<TD  vAlign=top width=127 colSpan=2>
<P  align=center><FONT face="Times New Roman">3.8<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">2.6<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">3.9<p></p></FONT></P></TD>
<TD  vAlign=top width=63>
<P  align=center><FONT face="Times New Roman">4.2<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">2.3<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">5.1<p></p></FONT></P></TD>
<TD  vAlign=top width=127 colSpan=2>
<P  align=center><FONT face="Times New Roman">3.9<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">6.8<p></p></FONT></P>
<P  align=center><FONT face="Times New Roman">3.2<p></p></FONT></P></TD></TR>
<TR height=0>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD>
<TD  width=63><FONT face="Times New Roman"></FONT></TD></TR></TABLE>
<P ><v:line><FONT face="Times New Roman"></FONT></v:line><v:line><FONT face="Times New Roman"></FONT></v:line><v:line><FONT face="Times New Roman"></FONT></v:line><p></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 ><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><BR  clear=all>
<P >每个小区域内分别标有该区域的海拔高度<FONT face="Times New Roman">h(</FONT>米<FONT face="Times New Roman">)</FONT>、面积<FONT face="Times New Roman">s(</FONT>平方千米<FONT face="Times New Roman">)</FONT>和泄洪后完全淹没时各种损失之和<FONT face="Times New Roman">k</FONT>（百万元）。约定：<p></p></P>
<P ><FONT face="Times New Roman">（1）     </FONT>泄洪后洪水淹没一个小区的损失、小区总资产以及水位的高度<FONT face="Times New Roman">p</FONT>之间的关系如下：<p></p></P>
<P  align=center>损失<FONT face="Times New Roman">=</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>；<p></p></P>
<P >（<FONT face="Times New Roman">2</FONT>）<FONT face="Times New Roman"> </FONT>在大堤和小堤上的决口数不受限制，可在任意地方选择决口，但决口后不可再补合；<p></p></P>
<P >（<FONT face="Times New Roman">3</FONT>）大堤决口后流入小区的洪水量按决口宽度成比例，小区之间一经决口则认为该小堤完全决口；<p></p></P>
<P >（<FONT face="Times New Roman">4</FONT>）在各小区之间，若水位高于某一相邻小堤，则水将向邻近最低的小区自动泄洪，若有多个同高小区，则约定平均泄洪。<p></p></P>
<P >求解：<p></p></P>
<P ><FONT face="Times New Roman">（1）     </FONT>整个区域最大损失的最小洪水量<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>？<p></p></P>
<P ><FONT face="Times New Roman">（2）     </FONT>选取满足条件<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的不同洪水量为<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，制定损失最小的泄洪方案（至少选择<FONT face="Times New Roman">4</FONT>个不同的洪水量）并计算该方案的损失数。<p></p></P>
<P ><p><FONT face="Times New Roman" size=3> </FONT></p></P>