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class="ant-col ThreeColumnLayout_leftColumn__yqNgV"> <div> <div> <div id="article-table-of-contents" data-ismobile="false"> <div class="TableOfContents_tableOfContents__0jAuv"> <div class="TableOfContents_tableOfContents__title__CH9Lz"> 在这一页上</d我v> <a class="TableOfContents_tableOfContents__item__JJil1" href="#abstract">文摘</gydF4y2Baa> <a class="TableOfContents_tableOfContents__item__JJil1" href="#introduction">介绍</gydF4y2Baa> <a class="TableOfContents_tableOfContents__item__JJil1" href="#preliminaries">预赛</gydF4y2Baa> <a class="TableOfContents_tableOfContents__item__JJil1" href="#references">引用</gydF4y2Baa> <a class="TableOfContents_tableOfContents__item__JJil1" href="#copyright">版权</gydF4y2Baa> <a class="TableOfContents_tableOfContents__item__JJil1" href="#related-articles">相关文章</gydF4y2Baa> </div> </div> </div> </div> </div> <div class="ant-col ThreeColumnLayout_middleColumn__sbd41"> <div class="ArticleMiddleSection_articleMiddleSection__gAv9z"> <div class="articleHeader"> <strong>研究文章|<!gydF4y2Ba-- -->开放获取</年代trong><d我v class="articleHeader__meta"> <span>体积<!gydF4y2Ba-- -->2011年<!gydF4y2Ba-- -->|</年代pan> <span>文章的ID<!gydF4y2Ba-- -->321903年<!gydF4y2Ba-- -->|</年代pan> <span class="articleHeader__meta_doiLink"><a href="https://doi.org/10.1155/2011/321903" aria-label="Doi-link" target="_blank" rel="noreferrer">https://doi.org/10.1155/2011/321903</gydF4y2Baa></span> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示引用</gydF4y2Babutton> </div> <h1 class="articleHeader__title">解的存在性和唯一性与罗宾Time-Fractional扩散方程的边界条件</gydF4y2Bah1> <div class="articleHeader__authors"> <span class="articleHeader__authors_author"><strong>Jukka Kemppainen</年代trong><a href="//www.newsama.com/cdn-cgi/l/email-protection" aria-label="Mail Option"><span role="img" class="anticon"> <svg version="1.0" xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 1401 1101"> <path fill="#6ba439" d="M132.5 2.1C101.4 6 71.1 20.5 48.3 42.4c-22.7 21.7-37.5 48.3-44.6 80l-2.2 10.1v837l2.2 10.1c6.7 30 19.3 53.9 39.8 75.3 23.3 24.3 56.4 40.9 90 45 12.8 1.6 1122.2 1.6 1135 0 38.9-4.8 75.9-25.6 100-56.4 15-19.3 24.2-39 29.8-63.9l2.2-10.1v-837l-2.2-10.1c-7.1-31.8-21.9-58.3-44.6-80.1-20.2-19.2-43.9-31.7-73.6-38.5l-9.6-2.3L705 1.4c-321.9-.1-568.5.2-572.5.7zm1123.4 99.5c1.7.4 3.1 1 3.1 1.4 0 .5-125.6 87.7-279 194L700.9 490.2 422 297C268.5 190.7 143 103.4 143 103c0-.5 1-1.1 2.3-1.3 3.5-.7 1107.1-.8 1110.6-.1zM401 404c164.4 113.8 299.4 207 300 207 .6 0 135.6-93.2 300.1-207 164.5-113.9 299.3-207 299.5-207 .2 0 .4 170.9.4 379.7 0 417.9.5 386-6.1 398.3-6.2 11.6-19 21.4-31.9 24.4-5.8 1.4-67.4 1.6-562 1.6s-556.2-.2-562-1.6c-12.9-3-25.7-12.8-31.9-24.4-6.6-12.3-6.1 19.6-6.1-398.3 0-208.8.2-379.7.5-379.7S236.6 290.1 401 404z"></path> </svg></span></a><sup>1</年代up></年代pan> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示更多</gydF4y2Babutton> </div> <div class="articleHeader__academicEditor"> <strong>学术编辑器:</年代trong><年代pan>w·a·柯克</年代pan> </div> <div class="articleHeader__timeline"> <div class="articleHeader__timeline_item "> <strong>收到了</年代trong><年代pan>2011年1月10</年代pan> </div> <div class="articleHeader__timeline_item "> <strong>修改后的</年代trong><年代pan>2011年3月01</年代pan> </div> <div class="articleHeader__timeline_item "> <strong>接受</年代trong><年代pan>2011年3月08</年代pan> </div> <div class="articleHeader__timeline_item articleHeader__timeline_item_sticky"> <strong>发表</年代trong><年代pan>2011年5月08</年代pan> </div> </div> </div> <div class="articleBody"> <div class="xml-content"> <h4 class="header" id="abstract">文摘</gydF4y2Bah4> <p>解的存在性和唯一性与罗宾time-fractional扩散方程边界条件与李雅普诺夫有限域边界证明连续函数空间的边界。因为绿色问题的矩阵是已知的,我们可以寻求解决方法的线性组合单层潜力,体积势,泊松积分。那最初的问题可能会减少沃尔泰拉积分方程的第二种关联到一个紧凑的运营商。古典分析可以用来显示相应的边界积分方程有唯一解,如果数据是连续的,初始数据是连续可微的,空间中的源项是持有人连续变量。这反过来证明了原问题有一个独特的解决方案。</p><年代pan class="end-abs"></span> <h4 id="introduction">1。介绍</gydF4y2Bah4> <p>在本文中,我们研究time-fractional扩散方程的可解性(TFDE)<年代pan class="equation" id="EEq1"> <svg height="93.349998" id="M1" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 93.349998" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,93.35)"> <g transform="translate(72,2.68)"> <text transform="matrix(1,0,0,-1,67.16,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,73.55,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,78.99,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Φ</t年代pan> <tspan style="font-size: 12.50px; " x="11.615287" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.778786" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.655436" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.869186" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " 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transform="matrix(1,0,0,-1,67.01,24.87)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.3765302" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.678522" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.842022" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.718672" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="40.09" y="-8.75"> +</t年代pan> <tspan style="font-size: 12.50px; " x="51.430222" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.490002" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="64.653496" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="71.530151" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="76.7314" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.644836" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="86.896339" y="-8.7600002"> Φ</t年代pan> <tspan style="font-size: 12.50px; " x="98.524124" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="102.68762" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="109.56428" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="114.76553" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="118.67896" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="126.3183" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="138.35869" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="147.09828" y="-8.7600002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="151.26178" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="158.13843" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="163.33968" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="167.25311" y="-8.7600002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="173.50462" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="188.67076" y="-8.7600002"> o</t年代pan> <tspan style="font-size: 12.50px; " x="194.92226" y="-8.7600002"> n</t年代pan> <tspan style="font-size: 12.50px; " x="205.34976" y="-8.7600002"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,280.17,30.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,290.35,33.62)"> <tspan style="font-size: 12.50px; " x="59.52" y="0.029999999"> ]</t年代pan> <tspan style="font-size: 12.50px; " x="65.769997" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="22.155315" y="0"> ×</t年代pan> <tspan style="font-size: 12.50px; " x="35.02" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="39.183498" y="-0.0099999998"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="45.434998" y="-0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.636246" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="-151.37" y="32.5"> Φ</t年代pan> <tspan style="font-size: 12.50px; " x="-139.75471" y="32.5"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-135.59122" y="32.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="-128.71457" y="32.5"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-123.51331" y="32.5"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="-117.26182" y="32.5"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-109.62248" y="32.5"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-97.582092" y="32.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="-86.10434" y="32.5"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-81.940842" y="32.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="-75.064194" y="32.5"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-68.825195" y="32.5"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-53.646553" y="32.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="-43.294067" y="32.5"> ∈</t年代pan> </text> <g> <path d="m 259.1,12.46 9.3,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,259.1,1.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="205.64999" y="-30.41"> (</t年代pan> <tspan style="font-size: 12.50px; " x="209.81349" y="-30.41"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="216.065" y="-30.41"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="219.19075" y="-30.41"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="225.44225" y="-30.41"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="13.4875" id="M2" style="vertical-align:-2.34499pt;width:51.662498px;" version="1.1" viewbox="0 0 51.662498 13.4875" width="51.662498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.303911" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.455387" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="31.832638" y="0"> </t年代pan> </text> </g> </g> </svg>是任何给定的函数,<年代vg height="11.5875" id="M3" style="vertical-align:-0.3135pt;width:48.875px;" version="1.1" viewbox="0 0 48.875 11.5875" width="48.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.5875)"> <g transform="translate(72,-62.73)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="12.765563" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="24.805952" y="0"> ℝ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.8,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="12.3" id="M4" style="vertical-align:-1.29163pt;width:35.099998px;" version="1.1" viewbox="0 0 35.099998 12.3" width="35.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 2</t年代pan> </text> </g> </g> </svg>是一个有界与李雅普诺夫域边界<年代vg height="14.575" id="M5" style="vertical-align:-0.33858pt;width:60.650002px;" version="1.1" viewbox="0 0 60.650002 14.575" width="60.650002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.575)"> <g transform="translate(72,-60.34)"> <text transform="matrix(1,0,0,-1,-71.95,60.73)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="10.802592" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.842981" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-39.14,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="11.0625" id="M6" style="vertical-align:-0.30096pt;width:62.987499px;" version="1.1" viewbox="0 0 62.987499 11.0625" width="62.987499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.995176" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="44.035564" y="0"> 1</t年代pan> </text> </g> </g> </svg>,<年代pan class="equation" id="EEq2"> <svg height="39.174999" id="M7" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 39.174999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,39.175)"> <g transform="translate(72,-40.66)"> <text transform="matrix(1,0,0,-1,69.84,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,76.23,58.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,81.67,52.96)"> <tspan style="font-size: 12.50px; " x="56.419998" y="-7.25"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0019197" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.165419" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.078857" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="23.718191" y="0"> =</t年代pan> </text> <g> <path d="m 118.63,56.09 45.17,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,118.63,44.21)"> <tspan style="font-size: 12.50px; " x="48.450001" y="-1.42"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.98" y="-16"> d</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="22.605425" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="33.945644" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.997337" y="0"> )</t年代pan> </text> <g> <path d="m 177.65,56.09 10.17,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,177.65,44.21)"> <tspan style="font-size: 12.50px; " x="13.45" y="-1.05"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,198.1,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,203.81,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.849105" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.189325" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.590862" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,236.57,58.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,248,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0019197" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.165419" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.566956" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.818455" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="31.069956" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.247158" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="51.587379" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,303.51,57.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,314.94,52.96)"> <tspan style="font-size: 12.50px; " x="22.59" y="7.3299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0019197" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.165419" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="18.41692" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="149.81" y="-0.23999999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="153.9735" y="-0.23999999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="160.22501" y="-0.23999999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="163.35075" y="-0.23999999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="169.60225" y="-0.23999999"> )</t年代pan> </text> </g> </g> </svg></span>是分数卡普托时间导数的秩序<年代vg height="11.0625" id="M8" style="vertical-align:-0.30096pt;width:63.349998px;" version="1.1" viewbox="0 0 63.349998 11.0625" width="63.349998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.282745" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="44.323135" y="0"> 1</t年代pan> </text> </g> </g> </svg>。身体部分扩散方程描述反常扩散等复杂系统某些非晶态半导体或强烈多孔材料(见[<gydF4y2Baa href="#B15">1</gydF4y2Baa>)和引用)。</p><p>gydF4y2Ba分数阶扩散方程的数学理论,只有第一个步骤了。在文献中,主要是这些方程的柯西问题被认为是到目前为止(见[<gydF4y2Baa href="#B1">2</gydF4y2Baa>- - - - - -<gydF4y2Baa href="#B13">5</gydF4y2Baa>)和引用)。广义解的存在性和唯一性的证明了广义time-fractional扩散方程初边值问题(<gydF4y2Baa href="#B14">6</gydF4y2Baa>]。然而,古典解的唯一性和存在性是由于只有在一个特殊的维情况。</p><p>gydF4y2Ba我们的模型问题是比治疗更简单的例子,在<gydF4y2Baa href="#B2">3</gydF4y2Baa>,<gydF4y2Baa href="#B13">5</gydF4y2Baa>]。然而,本文使用边界积分方法可以用在更一般的情况。我们决定专注于一个简单的模型,而不是更一般的澄清的基本思想。边界积分方法还允许我们研究(TFDE)或其概括等较弱的空间<年代vg height="11.2" id="M9" style="vertical-align:-0.0pt;width:17.1625px;" version="1.1" viewbox="0 0 17.1625 11.2" width="17.1625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.2)"> <g transform="translate(72,-63.04)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.1,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>空间或各向异性索伯列夫空间的规模。</p><p>gydF4y2Ba本文组织如下。在预赛中,我们回忆起势的定义和泊松积分。我们介绍著名的性能从理论pde的抛物线型,这需要证明解的存在性和唯一性。我们回忆的边界行为单层的潜力。我们表明,体积潜在解决nonhomogeneus TFDE零初始条件。此外,我们证明了泊松积分解决了均匀TFDE与给定初始基准。最后一部分是致力于解决方案的存在性和唯一性的证明。</p><gydF4y2Bah4 id="preliminaries">2。预赛</gydF4y2Bah4> <p>在这里我们回忆起势和泊松积分及其基本性质。结果我们将假定函数出现在定义足够光滑,这样相应的积分存在。</p><p>gydF4y2Ba可以被定义为单层潜力<年代pan class="equation" id="EEq3"> <svg height="38.387501" id="M10" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.387501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.3875)"> <g transform="translate(72,-41.29)"> <text transform="matrix(1,0,0,-1,45.07,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="66.699997" y="7.71"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.28695" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.613949" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.852945" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.016443" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.893093" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="43.106846" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.020283" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="54.659618" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,118.77,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,124.48,45.26)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,130.23,41.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,137.95,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,144.34,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="11.605152" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,159.36,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="54.763142" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="66.103363" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.504898" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.756393" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.171394" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="93.334892" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.536385" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="104.73763" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="111.13917" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="117.39067" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="123.64217" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="133.06943" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="137.23293" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="143.43442" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="149.67342" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="155.92491" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="162.32645" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="177.5051" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="187.85757" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="199.88547" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="209.1877" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="305.39001" y="-0.56"> (</t年代pan> <tspan style="font-size: 12.50px; " x="309.5535" y="-0.56"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="315.80499" y="-0.56"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="318.93076" y="-0.56"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="325.18225" y="-0.56"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="13.55" id="M11" style="vertical-align:-2.29482pt;width:26.049999px;" version="1.1" viewbox="0 0 26.049999 13.55" width="26.049999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.2139912" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.37749" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.578978" y="0"> )</t年代pan> </text> </g> </g> </svg>表示向外单位法<年代vg height="13.325" id="M12" style="vertical-align:-2.29482pt;width:36.4375px;" version="1.1" viewbox="0 0 36.4375 13.325" width="36.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.325)"> <g transform="translate(72,-61.34)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6648188" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.705208" y="0"> Γ</t年代pan> </text> </g> </g> </svg>和<年代pan class="equation" id="eq1"> <svg height="66.9375" id="M13" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 66.9375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,66.9375)"> <g transform="translate(72,-18.45)"> <text transform="matrix(1,0,0,-1,15.04,42.1)"> <tspan style="font-size: 12.50px; " x="51.349998" y="-14.79"> ⎧</t年代pan> <tspan style="font-size: 12.50px; " x="51.349998" y="-7.52"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="51.349998" y="3.1900001"> ⎨</t年代pan> <tspan style="font-size: 12.50px; " x="51.349998" y="13.9"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="51.349998" y="21.110001"> ⎩</t年代pan> <tspan style="font-size: 12.50px; " x="59.360001" y="-7.5"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.515263" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.678762" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.555412" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.75666" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.670099" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="39.309433" y="0"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,82,54.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,99.65,49.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,103.56,54.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9361281" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,119.37,49.61)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,131.25,54.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.09,49.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,154.13,54.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.12" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.256" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,165.46,39.87)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.46" y="-16.98"> 1</t年代pan> </text> <g> <path d="m 174.92,52.73 6.26,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,174.92,40.85)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="7.4499998" y="-8.7600002"> |</t年代pan> <tspan style="font-size: 12.50px; " x="9.9505997" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.82725" y="-8.7600002"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.25,54.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,199.12,49.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,203.04,54.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,217.94,49.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∣</t年代pan> <tspan style="font-size: 12.50px; " x="81.839996" y="9.7299995"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.5" y="-10.12"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.663498" y="-10.12"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.702686" y="-10.12"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="46.916439" y="-10.12"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.968132" y="-10.12"> )</t年代pan> <tspan style="font-size: 12.50px; " x="9.79" y="9.8599997"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.953499" y="9.8599997"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.16749" y="9.8599997"> /</t年代pan> <tspan style="font-size: 12.50px; " x="23.643324" y="9.8599997"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="29.894823" y="9.8599997"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.096073" y="9.8599997"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="41.347572" y="9.8599997"> )</t年代pan> <tspan style="font-size: 12.50px; " x="47.599072" y="9.8599997"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="52.80032" y="9.8599997"> (</t年代pan> <tspan style="font-size: 12.50px; " x="56.963818" y="9.8599997"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="63.215321" y="9.8599997"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="68.42907" y="9.8599997"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="74.680573" y="9.8599997"> )</t年代pan> <tspan style="font-size: 12.50px; " x="92.190002" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="107.80624" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="118.15873" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="130.19913" y="0"> ℝ</t年代pan> </text> <text transform="matrix(1,0,0,-1,357.48,54.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,362.33,49.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.754021" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="28.794409" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-287.92999" y="26.6"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="-281.6785" y="26.6"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-36.57" y="26.6"> </t年代pan> <tspan style="font-size: 12.50px; " x="-26.230019" y="26.6"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="-14.18963" y="26.6"> ℝ</t年代pan> </text> <text transform="matrix(1,0,0,-1,357.48,28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,362.33,23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.754021" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="28.794409" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="102.42" y="-19.1"> (</t年代pan> <tspan style="font-size: 12.50px; " x="106.5835" y="-19.1"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="112.835" y="-19.1"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="115.96075" y="-19.1"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="122.21225" y="-19.1"> )</t年代pan> </text> </g> </g> </svg></span>的基本解决方案(TFDE) (<gydF4y2Baa href="#B2">3</gydF4y2Baa>,<gydF4y2Baa href="#B4">7</gydF4y2Baa>- - - - - -<gydF4y2Baa href="#B17">9</gydF4y2Baa>]。在这里<年代vg height="19.200001" id="M14" style="vertical-align:-4.35121pt;width:26.737499px;" version="1.1" viewbox="0 0 26.737499 19.200001" width="26.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.12" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.256" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>是狐狸<年代vg height="10.325" id="M15" style="vertical-align:-0.0pt;width:15.175px;" version="1.1" viewbox="0 0 15.175 10.325" width="15.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>定义的函数,通过Mellin-Barnes积分表示<年代pan class="equation" id="eq2"> <svg height="38.575001" id="M16" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.575001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.575)"> <g transform="translate(72,-41.14)"> <text transform="matrix(1,0,0,-1,19.81,53.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,31.85,58.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.2460001" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.18,53.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.029325" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="24.41836" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="36.458748" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,91.69,58.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.2460001" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,103.02,46.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="97.910004" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="107.73" y="-7.3299999"> =</t年代pan> <tspan style="font-size: 12.50px; " x="126.5" y="-14.58"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="6.3499999" y="-7.3299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.064831" y="-7.3299999"> ∣</t年代pan> <tspan style="font-size: 12.50px; " x="46.560001" y="-16.700001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="50.723499" y="-16.700001"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.775192" y="-16.700001"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="62.988941" y="-16.700001"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.040634" y="-16.700001"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.85" y="0.79000002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.013498" y="0.79000002"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.227489" y="0.79000002"> /</t年代pan> <tspan style="font-size: 12.50px; " x="39.703323" y="0.79000002"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="45.954823" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="51.156071" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="57.407574" y="0.79000002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="63.659073" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="68.872826" y="0.79000002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="73.036324" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="79.287819" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="84.489067" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="90.74057" y="0.79000002"> )</t年代pan> </text> <g> <path d="m 223.98,56.57 17.33,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,223.98,44.69)"> <tspan style="font-size: 12.50px; " x="20.610001" y="-1.0599999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.853324" y="0"> 我</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.34,41.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,261.1,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="23.268084" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="32.29525" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="43.635471" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.136791" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="55.38829" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="64.815552" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="68.97905" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="78.006218" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="89.346436" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.847755" y="0"> )</t年代pan> </text> <g> <path d="m 261.1,56.57 99.02,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,284.87,44.69)"> <tspan style="font-size: 12.50px; " x="76.440002" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.405617" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="34.758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.810032" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.311352" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,367.55,58.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,382.07,53.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="82.68" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="86.843498" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="93.095001" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="96.220749" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="102.47225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="11.1" id="M17" style="vertical-align:-0.1881pt;width:12.575px;" version="1.1" viewbox="0 0 12.575 11.1" width="12.575" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1)"> <g transform="translate(72,-63.12)"> <text transform="matrix(1,0,0,-1,-71.95,63.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>复平面上无限轮廓负实轴逆时针方向传播。</p><p>gydF4y2Ba潜在的被定义为量<年代pan class="equation" id="EEq4"> <svg height="38.387501" id="M18" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.387501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.3875)"> <g transform="translate(72,-41.29)"> <text transform="matrix(1,0,0,-1,65.99,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="65.760002" y="7.71"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.349225" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.676224" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.91522" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.078718" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.955368" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="42.169121" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.082558" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="53.721893" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,138.76,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.26" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,144.46,45.26)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,150.21,41.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,159.31,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="54.763142" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="66.115868" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.517403" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.756393" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.171394" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="93.334892" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.536385" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="104.75014" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="111.15167" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="117.39067" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="123.64217" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="131.93166" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="138.18315" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="144.58469" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="159.75082" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="170.10332" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="182.14371" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="305.44" y="-0.56"> (</t年代pan> <tspan style="font-size: 12.50px; " x="309.60349" y="-0.56"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="315.85501" y="-0.56"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="318.98074" y="-0.56"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="325.23224" y="-0.56"> )</t年代pan> </text> </g> </g> </svg></span>为<年代vg height="9.875" id="M19" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>这样<年代vg height="14.075" id="M20" style="vertical-align:-2.72118pt;width:107.775px;" version="1.1" viewbox="0 0 107.775 14.075" width="107.775" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.075)"> <g transform="translate(72,-60.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="4.863667" y="0"> u</t年代pan> <tspan style="font-size: 12.50px; " x="11.115167" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="17.366667" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="31.945166" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.272163" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="44.435661" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="48.01152" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="53.225269" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.13871" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="64.778046" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="76.805931" y="0"> Ω</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.45" id="M21" style="vertical-align:-2.21957pt;width:57.737499px;" version="1.1" viewbox="0 0 57.737499 13.45" width="57.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.922558" y="0"> ]</t年代pan> </text> </g> </g> </svg>。</p><p>gydF4y2Ba泊松积分的定义是<年代pan class="equation" id="EEq5"> <svg height="41.075001" id="M22" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 41.075001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,41.075)"> <g transform="translate(72,-39.14)"> <text transform="matrix(1,0,0,-1,85.81,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="66" y="7.71"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.586782" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.91378" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.152777" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.316277" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.192924" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="42.406677" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.320114" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="53.95945" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,158.81,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,164.52,45.26)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,170.27,41.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,176.78,39.27)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,182.99,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.908855" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.110344" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.237534" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.891533" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="73.055031" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="79.256516" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="85.508018" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="91.759514" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="97.961006" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="113.12714" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="123.47963" y="0"> ∈</t年代pan> </text> <g> <path d="m 318.52,64.3 9.3,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,318.52,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="9.3022318" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="146.23" y="0.51999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="150.39349" y="0.51999998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="156.645" y="0.51999998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="159.77075" y="0.51999998"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="166.02225" y="0.51999998"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="14.75" id="M23" style="vertical-align:-3.25793pt;width:17.85px;" version="1.1" viewbox="0 0 17.85 14.75" width="17.85" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>一些社区的<年代vg height="15.85" id="M24" style="vertical-align:-0.0pt;width:11.75px;" version="1.1" viewbox="0 0 11.75 15.85" width="11.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.85)"> <g transform="translate(72,-59.32)"> <g> <path d="m -71.95,70.71 9.3,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> </g> </g> </svg>和<年代pan class="equation" id="EEq6"> <svg height="30.700001" id="M25" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 30.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.7)"> <g transform="translate(72,-47.44)"> <text transform="matrix(1,0,0,-1,119.05,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.792969" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="28.00672" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="39.55949" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="51.587379" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,178.25,61.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,195.9,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,207.78,61.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,221.14,58.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.53"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,230.66,64.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,242,51.33)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.5600004" y="-12.17"> 1</t年代pan> </text> <g> <path d="m 250.56,59.37 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,250.56,47.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="7.4499998" y="-8.7600002"> |</t年代pan> <tspan style="font-size: 12.50px; " x="9.9505997" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.82725" y="-8.7600002"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,269.88,61.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,274.76,56.25)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,278.67,61.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,290.1,51.33)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="174.64999" y="-5.27"> (</t年代pan> <tspan style="font-size: 12.50px; " x="178.81349" y="-5.27"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="185.065" y="-5.27"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="188.19075" y="-5.27"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="194.44225" y="-5.27"> )</t年代pan> </text> </g> </g> </svg></span>与<年代pan class="equation" id="EEq7"> <svg height="38.575001" id="M26" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.575001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.575)"> <g transform="translate(72,-41.14)"> <text transform="matrix(1,0,0,-1,29.17,55.98)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.53"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,38.69,62.08)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,50.02,53.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.029325" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.069715" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,92.14,58.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.2460001" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,103.47,46.12)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="97.910004" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="107.73" y="-7.3299999"> =</t年代pan> <tspan style="font-size: 12.50px; " x="126.5" y="-14.58"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="6.3600001" y="-7.3299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.062328" y="-7.3299999"> ∣</t年代pan> <tspan style="font-size: 12.50px; " x="46.970001" y="-16.700001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.133499" y="-16.700001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="57.384998" y="-16.700001"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="62.586246" y="-16.700001"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.637939" y="-16.700001"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.860001" y="0.79000002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="30.023499" y="0.79000002"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.237492" y="0.79000002"> /</t年代pan> <tspan style="font-size: 12.50px; " x="39.713326" y="0.79000002"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="45.964825" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="51.166073" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="57.417572" y="0.79000002"> )</t年代pan> <tspan style="font-size: 12.50px; " x="63.65657" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="68.870323" y="0.79000002"> (</t年代pan> <tspan style="font-size: 12.50px; " x="73.033821" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="79.285316" y="0.79000002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="84.499069" y="0.79000002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="90.750572" y="0.79000002"> )</t年代pan> </text> <g> <path d="m 224.44,56.57 17.32,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,224.44,44.69)"> <tspan style="font-size: 12.50px; " x="20.6" y="-1.0599999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.840821" y="0"> 我</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.8,41.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,261.55,61.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="23.268084" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="32.29525" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="43.635471" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.136791" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="55.38829" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="64.815552" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="68.97905" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="78.006218" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="89.346436" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.847755" y="0"> )</t年代pan> </text> <g> <path d="m 261.55,56.57 99.02,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,285.72,44.69)"> <tspan style="font-size: 12.50px; " x="76.050003" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="22.617928" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="33.958149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.009838" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="46.511162" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,368.01,58.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,378.35,53.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="86.400002" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="90.563499" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="96.815002" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="99.94075" y="0"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="106.19225" y="0"> )</t年代pan> </text> </g> </g> </svg></span>和<年代vg height="11.1" id="M27" style="vertical-align:-0.1881pt;width:12.575px;" version="1.1" viewbox="0 0 12.575 11.1" width="12.575" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1)"> <g transform="translate(72,-63.12)"> <text transform="matrix(1,0,0,-1,-71.95,63.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>的定义<年代vg height="19.200001" id="M28" style="vertical-align:-4.35121pt;width:26.737499px;" version="1.1" viewbox="0 0 26.737499 19.200001" width="26.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.2)"> <g transform="translate(72,-56.64)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.12" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.256" y="9.3400002"> 2</t年代pan> </text> </g> </g> </svg>。</p><p>gydF4y2Ba注意,在经典抛物型偏微分方程相比,我们有一个绿色的矩阵<年代vg height="13.575" id="M29" style="vertical-align:-2.26974pt;width:105.0875px;" version="1.1" viewbox="0 0 105.0875 13.575" width="105.0875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> {</t年代pan> <tspan style="font-size: 12.50px; " x="6.00144" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.928582" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="19.092081" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.968731" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="31.169979" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.08342" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="39.246918" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="44.460667" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="53.637871" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="57.801369" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.678017" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="69.879265" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="73.792709" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="77.956207" y="0"> }</t年代pan> </text> </g> </g> </svg>而不是一个根本的解决方案。我们也强调,格林函数奇点在时间和空间变量与经典抛物型pd,奇点只发生在时间。</p><p>gydF4y2Ba现在让我们国家上述数量的基本属性。由于证据是狐狸强烈详细分析的基础上<年代vg height="10.325" id="M30" style="vertical-align:-0.0pt;width:15.175px;" version="1.1" viewbox="0 0 15.175 10.325" width="15.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>功能,我们回忆起他们的基本属性。这些功能的详情,我们指的是(<gydF4y2Baa href="#B2">3</gydF4y2Baa>,<gydF4y2Baa href="#B9">10</gydF4y2Baa>,<gydF4y2Baa href="#B16">11</gydF4y2Baa>]。</p><p>gydF4y2Ba为了简化符号,我们介绍下面的函数定义<年代vg height="11.0625" id="M31" style="vertical-align:-0.30096pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 11.0625" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 0</t年代pan> </text> </g> </g> </svg>:<年代pan class="equation" id="EEq8"> <svg height="38.087502" id="M32" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.087502" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.0875)"> <g transform="translate(72,-41.53)"> <text transform="matrix(1,0,0,-1,67.94,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,78.85,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,91.54,54.65)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="36.810001" y="-7.2399998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.408997" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.035828" y="0.0099999998"> =</t年代pan> </text> <g> <path d="m 122.82,57.77 17.33,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,122.82,45.89)"> <tspan style="font-size: 12.50px; " x="20.6" y="-1.05"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.840821" y="0"> 我</t年代pan> </text> <text transform="matrix(1,0,0,-1,149.18,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159.93,62.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4147587" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.578258" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="23.268084" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="32.29525" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="43.647972" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.149292" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="55.38829" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="62.727551" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="66.891052" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="75.918213" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="87.270943" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="92.772263" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,256.87,68.04)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 159.93,57.77 101.72,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,165.85,43.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4200001" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.583499" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.410856" y="-0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="34.75108" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.802769" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.304089" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="53.555592" y="-0.0099999998"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="60.894852" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="65.05835" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="70.559669" y="-0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,240.58,48.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.2797279" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.274848" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,262.84,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,269.08,59.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,279.43,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="26.918959" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.496258" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="48.536648" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="54.788147" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="60.001896" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="66.253395" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="185.32001" y="1"> (</t年代pan> <tspan style="font-size: 12.50px; " x="189.48351" y="1"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="195.735" y="1"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="198.86075" y="1"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="205.11224" y="1"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>注意,特别是,<年代vg height="19.612499" id="M33" style="vertical-align:-4.68874pt;width:109.5625px;" version="1.1" viewbox="0 0 109.5625 19.612499" width="109.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.6125)"> <g transform="translate(72,-56.31)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-50.33,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.029325" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.069715" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.21,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.3400002"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="3.2460001" y="9.3400002"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,1.04,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> </text> </g> </g> </svg>。以下的属性<年代vg height="16.237499" id="M34" style="vertical-align:-4.74141pt;width:27.025px;" version="1.1" viewbox="0 0 27.025 16.237499" width="27.025" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.2375)"> <g transform="translate(72,-59.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> </g> </g> </svg>是必要的。</p><p><年代pan class="statement" id="lem1"><span class="statement-title">引理2.1。</年代pan><i>的功能<年代vg height="16.237499" id="M35" style="vertical-align:-4.74141pt;width:27.025px;" version="1.1" viewbox="0 0 27.025 16.237499" width="27.025" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.2375)"> <g transform="translate(72,-59.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> </g> </g> </svg>,拥有</我><年代pan class="list"><span class="list-item"><span class="list-label">(我)</年代pan><span class="list-content"><i>微分公式<年代vg height="19.612499" id="M36" style="vertical-align:-4.68874pt;width:179.3875px;" version="1.1" viewbox="0 0 179.3875 19.612499" width="179.3875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.6125)"> <g transform="translate(72,-56.31)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="13.890833" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="20.142332" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.38133" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.544828" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-30.49,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.79,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="18.041828" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="30.069715" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="38.63427" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,25.09,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,35.96,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,46.88,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,57.58,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> z</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> )</t年代pan> </text> </g> </g> </svg>,</我></年代pan></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>在无穷远处的渐近行为,<年代pan class="equation" id="EEq9"> <svg height="19.674999" id="M37" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.674999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.675)"> <g transform="translate(72,-56.26)"> <text transform="matrix(1,0,0,-1,59.36,62.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.48"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,72.77,57.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,85.46,61.04)"> <tspan style="font-size: 12.50px; " x="14.57" y="-1.74"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.57" y="1.74"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="20.540001" y="0.0099999998"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="32.567886" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.957397" y="0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,132.67,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.7828002" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,146.41,61.03)"> <tspan style="font-size: 12.50px; " x="22.219999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> e</t年代pan> <tspan style="font-size: 12.50px; " x="5.551332" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="28.07" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.634556" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.973816" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,196.63,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.7234716" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="14.099472" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.094591" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.021969" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,225.07,58.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="29.810249" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.625343" y="-2.51"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="47.014378" y="-2.51"> =</t年代pan> <tspan style="font-size: 12.50px; " x="59.042263" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,291.18,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9273758" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="7.3604321" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="11.736432" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="17.731552" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,316.43,61.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="13.190665" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="24.530886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.582579" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="37.83408" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="148.32001" y="0.039999999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="152.48351" y="0.039999999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="158.735" y="0.039999999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="161.86075" y="0.039999999"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="168.11224" y="0.039999999"> )</t年代pan> </text> </g> </g> </svg></span>为<年代vg height="13.55" id="M38" style="vertical-align:-2.29482pt;width:49.299999px;" version="1.1" viewbox="0 0 49.299999 13.55" width="49.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.5772982" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.617687" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="27.869186" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="33.082939" y="0"> 2</t年代pan> </text> </g> </g> </svg>和<年代vg height="12.3" id="M39" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,</我></年代pan></span><span class="list-item"><span class="list-label">(3)</年代pan><span class="list-content"><i>渐近行为接近于零<年代pan class="equation" id="EEq10"> <svg height="70.400002" id="M40" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 70.400002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,70.4)"> <g transform="translate(72,-15.68)"> <text transform="matrix(1,0,0,-1,96.72,42.46)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.48"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,110.14,37.59)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,122.83,40.72)"> <tspan style="font-size: 12.50px; " x="14.56" y="-1.74"> |</t年代pan> <tspan style="font-size: 12.50px; " x="14.56" y="1.74"> |</t年代pan> <tspan style="font-size: 12.50px; " x="44.049999" y="-16.17"> ⎧</t年代pan> <tspan style="font-size: 12.50px; " x="44.049999" y="-7.8200002"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="44.049999" y="3.1900001"> ⎨</t年代pan> <tspan style="font-size: 12.50px; " x="44.049999" y="14.2"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="44.049999" y="22.5"> ⎩</t年代pan> <tspan style="font-size: 12.50px; " x="52.049999" y="-20.309999"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="20.540001" y="0.0099999998"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="32.567886" y="0.0099999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,181.12,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.7828002" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,229.2,61.03)"> <tspan style="font-size: 12.50px; " x="-54.32" y="20.4"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> f</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.480154" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="33.520542" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="43.948044" y="0"> o</t年代pan> <tspan style="font-size: 12.50px; " x="50.199547" y="0"> r</t年代pan> <tspan style="font-size: 12.50px; " x="58.526543" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.21637" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="80.244255" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="86.49575" y="0"> ,</t年代pan> </text> <text transform="matrix(1,0,0,-1,181.12,45.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,188.08,42.22)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="26.110001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="26.110001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="-13.2" y="21.99"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> l</t年代pan> <tspan style="font-size: 12.50px; " x="5.9758339" y="1.59"> o</t年代pan> <tspan style="font-size: 12.50px; " x="12.227334" y="1.59"> g</t年代pan> <tspan style="font-size: 12.50px; " x="20.554333" y="1.59"> z</t年代pan> <tspan style="font-size: 12.50px; " x="41.119999" y="1.59"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="44.595833" y="1.59"> f</t年代pan> <tspan style="font-size: 12.50px; " x="52.922832" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.600155" y="1.59"> =</t年代pan> <tspan style="font-size: 12.50px; " x="74.640541" y="1.59"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="80.892044" y="1.59"> ,</t年代pan> </text> <text transform="matrix(1,0,0,-1,181.12,25.23)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,229.2,20.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> f</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.480154" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="33.520542" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="39.772041" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="229.28999" y="-20.48"> (</t年代pan> <tspan style="font-size: 12.50px; " x="233.45351" y="-20.48"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="239.705" y="-20.48"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="242.83075" y="-20.48"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="249.08224" y="-20.48"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="255.33376" y="-20.48"> )</t年代pan> </text> </g> </g> </svg></span>为<年代vg height="13.55" id="M41" style="vertical-align:-2.29482pt;width:49.299999px;" version="1.1" viewbox="0 0 49.299999 13.55" width="49.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.5772982" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.617687" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="27.869186" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="33.082939" y="0"> 2</t年代pan> </text> </g> </g> </svg>和<年代vg height="12.3" id="M42" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>。</我></年代pan></span></span><i>的常量<年代vg height="13.45" id="M43" style="vertical-align:-2.21957pt;width:20.25px;" version="1.1" viewbox="0 0 20.25 13.45" width="20.25" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0519323" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.940365" y="0"> )</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.45" id="M44" style="vertical-align:-2.21957pt;width:25.112499px;" version="1.1" viewbox="0 0 25.112499 13.45" width="25.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0519323" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.940365" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.828798" y="0"> )</t年代pan> </text> </g> </g> </svg>可以依赖<年代vg height="7.1374998" id="M45" style="vertical-align:-0.10033pt;width:7.8874998px;" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1375)"> <g transform="translate(72,-66.29)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="9.875" id="M46" style="vertical-align:-2.29482pt;width:7.7624998px;" version="1.1" viewbox="0 0 7.7624998 9.875" width="7.7624998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="7.1750002" id="M47" style="vertical-align:-0.1254pt;width:8.9375px;" version="1.1" viewbox="0 0 8.9375 7.1750002" width="8.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.175)"> <g transform="translate(72,-66.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</我></年代pan></p> <p><span class="statement" id="proof1"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>证明遵循Mellin-Barnes积分表示和分析性的功能<年代vg height="16.237499" id="M48" style="vertical-align:-4.74141pt;width:27.025px;" version="1.1" viewbox="0 0 27.025 16.237499" width="27.025" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.2375)"> <g transform="translate(72,-59.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> </g> </g> </svg>(<gydF4y2Baa href="#B2">3</gydF4y2Baa>,<gydF4y2Baa href="#B9">10</gydF4y2Baa>,<gydF4y2Baa href="#B5">12</gydF4y2Baa>]。</年代pan></p> <p><span class="statement" id="rem1"><span class="statement-title"><i class="statement-title">2.2的话。</我></年代pan>上面的续集,<年代vg height="10.6125" id="M49" style="vertical-align:-0.15048pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 10.6125" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6125)"> <g transform="translate(72,-63.51)"> <text transform="matrix(1,0,0,-1,-71.95,63.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>表示一个通用的常数,这可能取决于不同的数量。唯一重要的是,在我们的计算<年代vg height="10.6125" id="M50" style="vertical-align:-0.15048pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 10.6125" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6125)"> <g transform="translate(72,-63.51)"> <text transform="matrix(1,0,0,-1,-71.95,63.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>将独立的<年代vg height="7.1624999" id="M51" style="vertical-align:-0.11285pt;width:8.7250004px;" version="1.1" viewbox="0 0 8.7250004 7.1624999" width="8.7250004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="9.125" id="M52" style="vertical-align:-0.11285pt;width:5.0124998px;" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.125)"> <g transform="translate(72,-64.7)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</年代pan></p> <p>现在让我们专注于势的属性。我们开始与单层的潜力<年代vg height="13.425" id="M53" style="vertical-align:-2.29482pt;width:20.6875px;" version="1.1" viewbox="0 0 20.6875 13.425" width="20.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1269503" y="0"> </t年代pan> </text> </g> </g> </svg>。首先,计算表明,标准<年代vg height="13.425" id="M54" style="vertical-align:-2.29482pt;width:20.6875px;" version="1.1" viewbox="0 0 20.6875 13.425" width="20.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1269503" y="0"> </t年代pan> </text> </g> </g> </svg>解决了方程<年代vg height="15.4875" id="M55" style="vertical-align:-3.36943pt;width:95.5px;" version="1.1" viewbox="0 0 95.5 15.4875" width="95.5" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.4875)"> <g transform="translate(72,-59.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.4,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-53.18,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-32.81,59.89)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.5,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.553252" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> 0</t年代pan> </text> </g> </g> </svg>。此外,我们需要知道的边界行为单层的潜力,在下面的结果。</p><p><年代pan class="statement" id="thm1"><span class="statement-title">定理2.3。</年代pan><i>让<年代vg height="19.5375" id="M56" style="vertical-align:-3.13504pt;width:70.925003px;" version="1.1" viewbox="0 0 70.925003 19.5375" width="70.925003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.790329" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="23.830717" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.795609" y="0"> (</t年代pan> </text> <g> <path d="m -33.98,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-33.98,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.18,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.47,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>。定义的单层势(<gydF4y2Baa href="#EEq3">2.1</gydF4y2Baa>)是连续的<年代vg height="19.75" id="M57" style="vertical-align:-3.13504pt;width:20.137501px;" version="1.1" viewbox="0 0 20.137501 19.75" width="20.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.75)"> <g transform="translate(72,-56.2)"> <g> <path d="m -71.95,70.71 16.01,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.65,56.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>零初始值。此外,对于<年代vg height="11.1125" id="M58" style="vertical-align:-0.33858pt;width:39.737499px;" version="1.1" viewbox="0 0 39.737499 11.1125" width="39.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1125)"> <g transform="translate(72,-63.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> Ω</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.5375" id="M59" style="vertical-align:-3.25793pt;width:43.375px;" version="1.1" viewbox="0 0 43.375 14.5375" width="43.375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.5375)"> <g transform="translate(72,-60.37)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.07,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,-56.73,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="12.040389" y="0"> Γ</t年代pan> </text> </g> </g> </svg>,<年代vg height="14.75" id="M60" style="vertical-align:-3.25793pt;width:123.85px;" version="1.1" viewbox="0 0 123.85 14.75" width="123.85" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∇</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.81,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-57.5,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.290449" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.617447" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.780947" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="28.944445" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.821095" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="41.022343" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.935783" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="51.887451" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="58.238976" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.452965" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="68.616463" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,18,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,22.87,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>价值有以下限制:</我><年代pan class="equation" id="eq3"> <svg height="78.425003" id="M61" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 78.425003" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,78.425)"> <g transform="translate(72,-9.26)"> <text transform="matrix(1,0,0,-1,18.3,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,24.69,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,36.76,55.82)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,40.39,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.8,61.13)"> <tspan style="font-size: 12.50px; " x="18.540001" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.389999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1269503" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,75.07,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,79.94,61.13)"> <tspan style="font-size: 12.50px; " x="9.1199999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="18.450001" y="0"> ∶</t年代pan> <tspan style="font-size: 12.50px; " x="24.839033" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.61758" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="43.093414" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="46.569248" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,116.81,53.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.2751842" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="15.832368" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,137.46,51.45)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,141.09,61.13)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ∇</t年代pan> </text> <text transform="matrix(1,0,0,-1,150.23,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,157.62,61.14)"> <tspan style="font-size: 12.50px; " x="68.629997" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="74.480003" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.290449" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.617447" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.856443" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="31.019943" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.896591" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="43.110344" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.023785" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="53.962948" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="60.326977" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.97,58.01)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,243.85,58.62)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-145.46001" y="37.700001"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-132.23" y="30.440001"> 1</t年代pan> </text> <g> <path d="m 111.62,24.05 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,111.62,12.17)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="7.4499998" y="-8.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.860001" y="-6.2399998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.709999" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,142.21,17.8)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,147.08,20.92)"> <tspan style="font-size: 12.50px; " x="9.1199999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.75" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="29.1" y="7.6900001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="5.2012482" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,183.18,34.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.52"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,188.89,13.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,194.64,9.43)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,202.36,20.92)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,208.75,17.8)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,220.83,15.61)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,224.45,17.8)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,227.86,20.92)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.51" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.360001" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,252.1,17.8)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,259.75,20.92)"> <tspan style="font-size: 12.50px; " x="47.189999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.130001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.54171" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="22.755461" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.444565" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="40.784786" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="65.540001" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="69.703499" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.904984" y="-0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="81.106232" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="87.507774" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="93.75927" y="-0.0099999998"> d</t年代pan> <tspan style="font-size: 12.50px; " x="100.01077" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="109.43803" y="-0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="113.60153" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="119.80302" y="-0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="126.04202" y="-0.0099999998"> d</t年代pan> <tspan style="font-size: 12.50px; " x="132.29352" y="-0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="198.74001" y="-16.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="202.9035" y="-16.58"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="209.155" y="-16.58"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="212.28075" y="-16.58"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="218.53224" y="-16.58"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="224.78375" y="-16.58"> )</t年代pan> </text> </g> </g> </svg></span><i>作为<年代vg height="7.1624999" id="M62" style="vertical-align:-0.11285pt;width:8.7250004px;" version="1.1" viewbox="0 0 8.7250004 7.1624999" width="8.7250004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>倾向于<年代vg height="11.075" id="M63" style="vertical-align:-3.25793pt;width:14.8125px;" version="1.1" viewbox="0 0 14.8125 11.075" width="14.8125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.075)"> <g transform="translate(72,-63.14)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.07,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>nontangentially。</我></年代pan></p> <p><span class="statement" id="proof2"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>证明遵循相同的行像单层潜在的热方程(<gydF4y2Baa href="#B3">13</gydF4y2Baa>,5.2章)和基于内核的详细分析<年代vg height="10.75" id="M64" style="vertical-align:-0.15048pt;width:11.9125px;" version="1.1" viewbox="0 0 11.9125 10.75" width="11.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。由于证据是相当漫长的,我们只给出文献[<gydF4y2Baa href="#B5">12</gydF4y2Baa>定理1和2)。</年代pan></p> <p>体积的潜力,我们有下面的结果。</p><p><年代pan class="statement" id="thm2"><span class="statement-title">定理2.4。</年代pan><i>让<年代vg height="19.5375" id="M65" style="vertical-align:-3.13504pt;width:69.199997px;" version="1.1" viewbox="0 0 69.199997 19.5375" width="69.199997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.407776" y="0"> (</t年代pan> </text> <g> <path d="m -35.37,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-35.37,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.57,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.86,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>这样<年代vg height="13.6125" id="M66" style="vertical-align:-2.34499pt;width:35.075001px;" version="1.1" viewbox="0 0 35.075001 13.6125" width="35.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="14.678522" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="19.87977" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.793209" y="0"> )</t年代pan> </text> </g> </g> </svg>持有人持续一致吗<年代vg height="13.125" id="M67" style="vertical-align:-1.95624pt;width:57.737499px;" version="1.1" viewbox="0 0 57.737499 13.125" width="57.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.125)"> <g transform="translate(72,-61.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.922558" y="0"> ]</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.075" id="M68" style="vertical-align:-2.72118pt;width:100.825px;" version="1.1" viewbox="0 0 100.825 14.075" width="100.825" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.075)"> <g transform="translate(72,-60.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="4.863667" y="0"> u</t年代pan> <tspan style="font-size: 12.50px; " x="11.115167" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="17.366667" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="27.781666" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.720833" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="38.884331" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="42.46019" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="47.661434" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.574875" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="59.214207" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="71.254601" y="0"> Ω</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.125" id="M69" style="vertical-align:-1.95624pt;width:57.737499px;" version="1.1" viewbox="0 0 57.737499 13.125" width="57.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.125)"> <g transform="translate(72,-61.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.922558" y="0"> ]</t年代pan> </text> </g> </g> </svg>。然后体积势<年代vg height="13.4875" id="M70" style="vertical-align:-2.34499pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 13.4875" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1892252" y="0"> </t年代pan> </text> </g> </g> </svg>与<年代vg height="10.575" id="M71" style="vertical-align:-0.20064pt;width:12.625px;" version="1.1" viewbox="0 0 12.625 10.575" width="12.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.575)"> <g transform="translate(72,-63.54)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>定义为(<gydF4y2Baa href="#EEq4">2.4</gydF4y2Baa>)解决<年代vg height="15.4875" id="M72" style="vertical-align:-3.36943pt;width:95.237503px;" version="1.1" viewbox="0 0 95.237503 15.4875" width="95.237503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.4875)"> <g transform="translate(72,-59.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.56,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.13,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7020884" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.05,59.89)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-25.74,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> </t年代pan> </text> </g> </g> </svg>零初始条件。</我></年代pan></p> <p><span class="statement" id="proof3"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>零初始条件遵循<年代vg height="10.75" id="M73" style="vertical-align:-0.15048pt;width:11.9125px;" version="1.1" viewbox="0 0 11.9125 10.75" width="11.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>局部可积。事实上,我们把积分<年代vg height="13.4875" id="M74" style="vertical-align:-2.34499pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 13.4875" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1892252" y="0"> </t年代pan> </text> </g> </g> </svg>分成两部分<年代vg height="14.2375" id="M75" style="vertical-align:-3.13504pt;width:44.962502px;" version="1.1" viewbox="0 0 44.962502 14.2375" width="44.962502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-58.3,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>这取决于<年代vg height="16.8375" id="M76" style="vertical-align:-2.29482pt;width:186.4875px;" version="1.1" viewbox="0 0 186.4875 16.8375" width="186.4875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="50.212048" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.901154" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="68.241371" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="74.642906" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,6.87,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,18.3,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152916" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.493137" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.694626" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,50.5,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,58.85,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>或<年代vg height="12.3" id="M77" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>如果<年代vg height="12.3" id="M78" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,我们使用这一事实<年代vg height="17.1625" id="M79" style="vertical-align:-2.72118pt;width:120.7375px;" version="1.1" viewbox="0 0 120.7375 17.1625" width="120.7375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.1625)"> <g transform="translate(72,-58.27)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.71,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-58.68,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> e</t年代pan> <tspan style="font-size: 12.50px; " x="5.551332" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="24.293329" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="28.456827" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="37.021381" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.360645" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-8.07,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.7234716" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="14.099472" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.094591" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.021969" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,20.37,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>是一致有界的吗<年代vg height="13.6125" id="M80" style="vertical-align:-2.34499pt;width:50.950001px;" version="1.1" viewbox="0 0 50.950001 13.6125" width="50.950001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.3390212" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="11.552772" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.367867" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="34.408257" y="0"> 0</t年代pan> </text> </g> </g> </svg>。然后引理<gydF4y2Baa href="#lem1">2.1</gydF4y2Baa>一起的定义<年代vg height="10.75" id="M81" style="vertical-align:-0.15048pt;width:11.9125px;" version="1.1" viewbox="0 0 11.9125 10.75" width="11.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>收益率<年代pan class="equation" id="EEq11"> <svg height="20.700001" id="M82" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 20.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.7)"> <g transform="translate(72,-55.44)"> <text transform="matrix(1,0,0,-1,86.74,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.190002" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.190002" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.00276" y="1.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.16626" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.831078" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.171299" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.372787" y="1.58"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.586536" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.275642" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="68.615868" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.017403" y="1.58"> )</t年代pan> <tspan style="font-size: 12.50px; " x="85.160004" y="1.58"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="97.187889" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="106.57764" y="1.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="110.74113" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="117.44275" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="128.78297" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.18451" y="1.58"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,226.09,64.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9361281" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.867376" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="20.313393" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.308512" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="30.684511" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="36.67963" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="41.615761" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="45.965504" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="48.39856" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,279.37,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.139813" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.58"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,311.57,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,326.88,59.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="131.61" y="-1.3099999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="135.7735" y="-1.3099999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="142.02499" y="-1.3099999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="145.15074" y="-1.3099999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="151.40225" y="-1.3099999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="157.65375" y="-1.3099999"> )</t年代pan> </text> </g> </g> </svg></span>如果我们选择<年代vg height="13.6125" id="M83" style="vertical-align:-2.34499pt;width:112.1625px;" version="1.1" viewbox="0 0 112.1625 13.6125" width="112.1625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.2139912" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="9.6898251" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="18.716991" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="30.057213" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="39.784546" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="51.812431" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="61.63979" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="73.680176" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="79.894173" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="83.370003" y="0"> 2</t年代pan> </text> </g> </g> </svg>,我们看到,<年代vg height="15.15" id="M84" style="vertical-align:-3.49493pt;width:128.175px;" version="1.1" viewbox="0 0 128.175 15.15" width="128.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.15)"> <g transform="translate(72,-59.88)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.1,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.1922078" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="13.749392" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.47,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.47,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-15.6,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.040149" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.241398" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154837" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.794168" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.834557" y="0"> 0</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>另一方面,如果<年代vg height="12.3" id="M85" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,然后引理<gydF4y2Baa href="#lem1">2.1</gydF4y2Baa>收益率<年代pan class="equation" id="EEq12"> <svg height="78.387497" id="M86" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 78.387497" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,78.3875)"> <g transform="translate(72,-9.29)"> <text transform="matrix(1,0,0,-1,12.62,39.11)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.18" y="-7.1054274e-015"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.18" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="108.67" y="-17.780001"> ⎧</t年代pan> <tspan style="font-size: 12.50px; " x="108.67" y="-6.9200001"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="108.67" y="4.7800002"> ⎨</t年代pan> <tspan style="font-size: 12.50px; " x="108.67" y="16.469999"> ⎪</t年代pan> <tspan style="font-size: 12.50px; " x="108.67" y="27.27"> ⎩</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.00276" y="1.59"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.16626" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.818575" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.171299" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.372787" y="1.59"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.574036" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.263142" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="68.615868" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.017403" y="1.59"> )</t年代pan> <tspan style="font-size: 12.50px; " x="85.160004" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="97.187889" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="116.67" y="-21.75"> (</t年代pan> <tspan style="font-size: 12.50px; " x="120.8335" y="-21.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="127.52261" y="-21.75"> −</t年代pan> <tspan style="font-size: 12.50px; " x="138.86282" y="-21.75"> </t年代pan> <tspan style="font-size: 12.50px; " x="145.26436" y="-21.75"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,162.05,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="13.285536" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="17.635281" y="0"> )</t年代pan> <tspan style="font-size: 8.75px; " x="20.549696" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.485825" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="27.91888" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.85,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="165.52" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="175.20982" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="187.23772" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="193.48921" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="198.70296" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="204.95447" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-65.559998" y="24.85"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-61.3965" y="24.85"> </t年代pan> <tspan style="font-size: 12.50px; " x="-54.707397" y="24.85"> −</t年代pan> <tspan style="font-size: 12.50px; " x="-43.367176" y="24.85"> </t年代pan> <tspan style="font-size: 12.50px; " x="-36.965637" y="24.85"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,162.05,41.18)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="16.926369" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,185.94,31.08)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.3600001" y="-11.84"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.3600001" y="-1.1799999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.3600001" y="1.99"> |</t年代pan> <tspan style="font-size: 12.50px; " x="30.01" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.380001" y="-6.5100002"> |</t年代pan> <tspan style="font-size: 12.50px; " x="37.380001" y="-3.3499999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="67.080002" y="-6.5100002"> |</t年代pan> <tspan style="font-size: 12.50px; " x="67.080002" y="-3.3499999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="9.8599997" y="-4.9200001"> l</t年代pan> <tspan style="font-size: 12.50px; " x="13.335834" y="-4.9200001"> o</t年代pan> <tspan style="font-size: 12.50px; " x="19.587334" y="-4.9200001"> g</t年代pan> <tspan style="font-size: 12.50px; " x="39.880001" y="-4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.519814" y="-4.9200001"> −</t年代pan> <tspan style="font-size: 12.50px; " x="60.872536" y="-4.9200001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,255.52,42.76)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.39,36.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.192825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.59436" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.15,41.18)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,304.58,31.08)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.6100001" y="-11.84"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.6100001" y="-1.1799999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.6100001" y="1.99"> |</t年代pan> <tspan style="font-size: 12.50px; " x="30.48" y="7.1054274e-015"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.88" y="-4.9200001"> +</t年代pan> <tspan style="font-size: 12.50px; " x="24.220221" y="-4.9200001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="40.16" y="-4.9200001"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="55.78875" y="-4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="65.478577" y="-4.9200001"> =</t年代pan> <tspan style="font-size: 12.50px; " x="77.506462" y="-4.9200001"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="83.757957" y="-4.9200001"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-175.28999" y="17.23"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-171.1265" y="17.23"> </t年代pan> <tspan style="font-size: 12.50px; " x="-164.43739" y="17.23"> −</t年代pan> <tspan style="font-size: 12.50px; " x="-153.09717" y="17.23"> </t年代pan> <tspan style="font-size: 12.50px; " x="-146.69563" y="17.23"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,162.05,19.02)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="16.926369" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,183.86,15.43)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.139813" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.58"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,216.06,20.6)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 4</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,231.28,13.85)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="129.09" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="138.77983" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="150.80771" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="157.0592" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="227.21001" y="-23.67"> (</t年代pan> <tspan style="font-size: 12.50px; " x="231.3735" y="-23.67"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="237.625" y="-23.67"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="240.75075" y="-23.67"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="247.00224" y="-23.67"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="253.25375" y="-23.67"> )</t年代pan> </text> </g> </g> </svg></span><br>然后<年代vg height="15.15" id="M87" style="vertical-align:-3.49493pt;width:128.175px;" version="1.1" viewbox="0 0 128.175 15.15" width="128.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.15)"> <g transform="translate(72,-59.88)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.1,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.1922078" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="13.749392" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.47,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.47,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-15.6,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.040149" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.241398" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154837" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.794168" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.834557" y="0"> 0</t年代pan> </text> </g> </g> </svg>之后立即<年代vg height="12.8875" id="M88" style="vertical-align:-1.76814pt;width:49.424999px;" version="1.1" viewbox="0 0 49.424999 12.8875" width="49.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.8875)"> <g transform="translate(72,-61.69)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="27.969212" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="33.182961" y="0"> 3</t年代pan> </text> </g> </g> </svg>。如果<年代vg height="10.8125" id="M89" style="vertical-align:-0.10033pt;width:35.099998px;" version="1.1" viewbox="0 0 35.099998 10.8125" width="35.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8125)"> <g transform="translate(72,-63.35)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 4</t年代pan> </text> </g> </g> </svg>我们可以使用这一事实<年代vg height="14.6125" id="M90" style="vertical-align:-2.73372pt;width:55.9375px;" version="1.1" viewbox="0 0 55.9375 14.6125" width="55.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6125)"> <g transform="translate(72,-60.31)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.71,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.77,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="4.576098" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="8.0519323" y="0"> o</t年代pan> <tspan style="font-size: 12.50px; " x="14.303432" y="0"> g</t年代pan> <tspan style="font-size: 12.50px; " x="24.730934" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.969931" y="0"> |</t年代pan> </text> </g> </g> </svg>是有界的<年代vg height="13.45" id="M91" style="vertical-align:-2.21957pt;width:32.674999px;" version="1.1" viewbox="0 0 32.674999 13.45" width="32.674999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.616247" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="21.867746" y="0"> ]</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.6125" id="M92" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>。如果<年代vg height="11.0625" id="M93" style="vertical-align:-0.30096pt;width:35.099998px;" version="1.1" viewbox="0 0 35.099998 11.0625" width="35.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 4</t年代pan> </text> </g> </g> </svg>,我们使用<年代vg height="12.8125" id="M94" style="vertical-align:-1.29163pt;width:48.799999px;" version="1.1" viewbox="0 0 48.799999 12.8125" width="48.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.8125)"> <g transform="translate(72,-61.75)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.71,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.3,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.6125" id="M95" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>。在前面的两个例子,我们获得<年代pan class="equation" id="eq4"> <svg height="20.700001" id="M96" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 20.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.7)"> <g transform="translate(72,-55.44)"> <text transform="matrix(1,0,0,-1,90.43,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.18" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="79.18" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.00276" y="1.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.16626" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.818575" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="39.158794" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.360283" y="1.58"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="50.574036" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.263142" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="68.615868" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.017403" y="1.58"> )</t年代pan> <tspan style="font-size: 12.50px; " x="85.160004" y="1.58"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="97.187889" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="106.57764" y="1.58"> (</t年代pan> <tspan style="font-size: 12.50px; " x="110.74113" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="117.43024" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="128.77046" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.172" y="1.58"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,229.78,64.45)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9273758" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="9.3733921" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="15.368512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="20.30464" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.299761" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.96,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.58"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.58"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.58"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,293.16,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 4</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="14.720864" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.715984" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="25.091984" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,323.2,59.28)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="135.28999" y="-1.3099999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="139.45351" y="-1.3099999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="145.705" y="-1.3099999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="148.83075" y="-1.3099999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="155.08224" y="-1.3099999"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="161.33376" y="-1.3099999"> )</t年代pan> </text> </g> </g> </svg></span>如果我们选择<年代vg height="13.6125" id="M97" style="vertical-align:-2.34499pt;width:62.474998px;" version="1.1" viewbox="0 0 62.474998 13.6125" width="62.474998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.582579" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.622967" y="0"> 2</t年代pan> </text> </g> </g> </svg>,我们看到,<年代vg height="15.15" id="M98" style="vertical-align:-3.49493pt;width:128.175px;" version="1.1" viewbox="0 0 128.175 15.15" width="128.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.15)"> <g transform="translate(72,-59.88)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.1,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.1922078" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="13.749392" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.47,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.47,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-15.6,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.040149" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.241398" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.154837" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="27.794168" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="39.834557" y="0"> 0</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>证明的第一个要求,我们指的是(<gydF4y2Baa href="#B2">3</gydF4y2Baa>章节5.2和5.3),证明在一个更一般情况下的time-fractional扩散方程。</年代pan></p> <p>最后,对于泊松积分有持有以下定理。</p><p><年代pan class="statement" id="thm3"><span class="statement-title">定理2.5。</年代pan><i>让<年代vg height="9.875" id="M99" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续函数<年代vg height="14.75" id="M100" style="vertical-align:-3.25793pt;width:17.85px;" version="1.1" viewbox="0 0 17.85 14.75" width="17.85" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>。然后,泊松积分<年代vg height="13.1875" id="M101" style="vertical-align:-2.29482pt;width:21.15px;" version="1.1" viewbox="0 0 21.15 13.1875" width="21.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.1875)"> <g transform="translate(72,-61.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0"> </t年代pan> </text> </g> </g> </svg>与<年代vg height="10.325" id="M102" style="vertical-align:-0.0pt;width:11.75px;" version="1.1" viewbox="0 0 11.75 10.325" width="11.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>定义为(<gydF4y2Baa href="#EEq5">2.5</gydF4y2Baa>)解决<年代vg height="15.4875" id="M103" style="vertical-align:-3.36943pt;width:92.5px;" version="1.1" viewbox="0 0 92.5 15.4875" width="92.5" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.4875)"> <g transform="translate(72,-59.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.56,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.13,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7020884" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.05,59.89)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-25.74,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> 0</t年代pan> </text> </g> </g> </svg>与<年代vg height="13.55" id="M104" style="vertical-align:-2.29482pt;width:91px;" version="1.1" viewbox="0 0 91 13.55" width="91" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.9264221" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.966572" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="22.167818" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="28.419319" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="36.058651" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="48.099041" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.488792" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.652294" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.528946" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="16.275" id="M105" style="vertical-align:-0.33858pt;width:39.737499px;" version="1.1" viewbox="0 0 39.737499 16.275" width="39.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.275)"> <g transform="translate(72,-58.98)"> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ∈</t年代pan> </text> <g> <path d="m -49.56,70.71 9.3,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-49.56,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> </g> </g> </svg>。</我></年代pan></p> <p><span class="statement" id="proof4"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>这一事实<年代vg height="13.1875" id="M106" style="vertical-align:-2.29482pt;width:21.15px;" version="1.1" viewbox="0 0 21.15 13.1875" width="21.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.1875)"> <g transform="translate(72,-61.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0"> </t年代pan> </text> </g> </g> </svg>解决了<年代vg height="15.4875" id="M107" style="vertical-align:-3.36943pt;width:92.5px;" version="1.1" viewbox="0 0 92.5 15.4875" width="92.5" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.4875)"> <g transform="translate(72,-59.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.56,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="0" y="8.3500004"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-60.13,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7020884" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="20.042309" y="0"> Δ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-31.05,59.89)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-25.74,63.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3897533" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.430141" y="0"> 0</t年代pan> </text> </g> </g> </svg>遵循从给定的计算(<gydF4y2Baa href="#B11">8</gydF4y2Baa>]。注意,内部分化的积分是允许的,因为没有奇点<年代vg height="9.125" id="M108" style="vertical-align:-0.11285pt;width:5.0124998px;" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.125)"> <g transform="translate(72,-64.7)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。因此,它仍然证明初始条件。<gydF4y2Babr>我们继续在<gydF4y2Baa href="#B3">13</gydF4y2Baa>定理的证明<年代vg height="10.8625" id="M109" style="vertical-align:-0.13794pt;width:31.387501px;" version="1.1" viewbox="0 0 31.387501 10.8625" width="31.387501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8625)"> <g transform="translate(72,-63.31)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="15.62875" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="18.754499" y="0"> 1</t年代pan> </text> </g> </g> </svg>),考虑第一个常数的情况下<年代vg height="9.875" id="M110" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。整体分为两部分<年代vg height="16.3375" id="M111" style="vertical-align:-4.12299pt;width:131.45px;" version="1.1" viewbox="0 0 131.45 16.3375" width="131.45" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.3375)"> <g transform="translate(72,-58.93)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.590281" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.967531" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.131029" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="29.294529" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.171181" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="41.38493" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.29837" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="52.937702" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="64.978088" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.97,59.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,8.79,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,27.26,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.1099997"> </t年代pan> </text> </g> </g> </svg>,在那里<年代vg height="14.2375" id="M112" style="vertical-align:-3.13504pt;width:16.35px;" version="1.1" viewbox="0 0 16.35 14.2375" width="16.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>积分球吗<年代vg height="13.45" id="M113" style="vertical-align:-2.21957pt;width:48.087502px;" version="1.1" viewbox="0 0 48.087502 13.45" width="48.087502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7020884" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.865587" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.742237" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="24.943485" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.195705" y="0"> )</t年代pan> </text> </g> </g> </svg>与<年代vg height="10.325" id="M114" style="vertical-align:-0.0pt;width:11.6875px;" version="1.1" viewbox="0 0 11.6875 10.325" width="11.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>这么小,<年代vg height="13.45" id="M115" style="vertical-align:-2.21957pt;width:48.087502px;" version="1.1" viewbox="0 0 48.087502 13.45" width="48.087502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.7020884" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.865587" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.742237" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="24.943485" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.195705" y="0"> )</t年代pan> </text> </g> </g> </svg>包含在<年代vg height="14.75" id="M116" style="vertical-align:-3.25793pt;width:17.85px;" version="1.1" viewbox="0 0 17.85 14.75" width="17.85" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.75)"> <g transform="translate(72,-60.2)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.65,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>和<年代vg height="16.3375" id="M117" style="vertical-align:-4.12299pt;width:16.35px;" version="1.1" viewbox="0 0 16.35 16.3375" width="16.35" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.3375)"> <g transform="translate(72,-58.93)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.82,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.13" y="9.1099997"> </t年代pan> </text> </g> </g> </svg>表示它的互补部分。自<年代vg height="10.325" id="M118" style="vertical-align:-0.0pt;width:11.6875px;" version="1.1" viewbox="0 0 11.6875 10.325" width="11.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是固定的,没有奇点在空间变量的渐近行为<年代vg height="10.75" id="M119" style="vertical-align:-0.15048pt;width:11.9125px;" version="1.1" viewbox="0 0 11.9125 10.75" width="11.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>显示,<年代vg height="16.3375" id="M120" style="vertical-align:-4.12299pt;width:100.4px;" version="1.1" viewbox="0 0 100.4 16.3375" width="100.4" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.3375)"> <g transform="translate(72,-58.93)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.1,59.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.1922078" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="13.749392" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.47,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-19.35,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.12" y="9.1099997"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-10.02,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 0</t年代pan> </text> </g> </g> </svg>。我们需要证明<年代vg height="15.15" id="M121" style="vertical-align:-3.49493pt;width:104.3125px;" version="1.1" viewbox="0 0 104.3125 15.15" width="104.3125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.15)"> <g transform="translate(72,-59.88)"> <text transform="matrix(1,0,0,-1,-71.95,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> l</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="6.9516678" y="0"> 米</t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.1,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.1922078" y="0"> →</t年代pan> <tspan style="font-size: 8.75px; " x="13.749392" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="18.125393" y="0"> +</t年代pan> </text> <text transform="matrix(1,0,0,-1,-26.47,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.47,60.28)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-10.02,63.41)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>介绍球坐标,得到<年代pan class="equation" id="eq5"> <svg height="140.22501" id="M122" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 140.22501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,140.225)"> <g transform="translate(72,40.18)"> <text transform="matrix(1,0,0,-1,85.63,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,91.63,49.08)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,102.08,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,121.72,57.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,139.37,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,147.88,49.08)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,152.73,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.48" y="7.6999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,171.21,66.24)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,180.27,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,185.37,57.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.76,54.74)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.53"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,208.28,60.84)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,219.61,47.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.5600004" y="-12.17"> 1</t年代pan> </text> <g> <path d="m 228.17,55.33 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,228.17,43.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="7.4499998" y="-8.7600002"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,239.53,57.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,250.96,52.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,256.06,57.21)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.94,47.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6800003" y="-4.9200001"> d</t年代pan> <tspan style="font-size: 12.50px; " x="15.9315" y="-4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="-158.86" y="37.419998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="-146.83211" y="37.419998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,121.72,14.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,139.37,9.87)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,147.88,6.74)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,152.73,9.87)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.48" y="7.6999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,171.21,23.89)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.47648" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="8.9095364" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="13.285536" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,187.23,27.4)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 6.25px; " x="3.5261281" y="0"> /</t年代pan> <tspan style="font-size: 6.25px; " x="5.264184" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,169.96,-1.63)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.7,9.87)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,203.8,14.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,217.19,12.4)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.53"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,226.71,18.5)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,238.05,7.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8499999" y="-2.52"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,249,14.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,253.87,7.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.52"> d</t年代pan> <tspan style="font-size: 12.50px; " x="14.1915" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="-151.78999" y="35.860001"> ⟶</t年代pan> <tspan style="font-size: 12.50px; " x="-128.64694" y="35.860001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,132.83,-23.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,150.48,-28.51)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159,-31.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,163.85,-28.51)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.47" y="7.6999998"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,182.32,-14.48)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,193.01,-28.51)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,198.11,-23.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,211.5,-25.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.54"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,221.02,-19.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,232.35,-31.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.8600001" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,243.31,-23.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,248.18,-31.02)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> d</t年代pan> <tspan style="font-size: 12.50px; " x="14.1915" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.292725" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="34.458862" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.835632" y="-2.51"> ⟶</t年代pan> <tspan style="font-size: 12.50px; " x="64.991188" y="-2.51"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="71.242691" y="-2.51"> +</t年代pan> <tspan style="font-size: 12.50px; " x="79.807243" y="-2.51"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="210.31" y="-43.799999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="214.4735" y="-43.799999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="220.72501" y="-43.799999"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="223.85075" y="-43.799999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="230.10225" y="-43.799999"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="236.35374" y="-43.799999"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="11.0125" id="M123" style="vertical-align:-3.20526pt;width:16.825001px;" version="1.1" viewbox="0 0 16.825001 11.0125" width="16.825001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0125)"> <g transform="translate(72,-63.19)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-63.44,63.31)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>表示单位球体的表面积<年代vg height="11.2" id="M124" style="vertical-align:-0.0pt;width:17.862499px;" version="1.1" viewbox="0 0 17.862499 11.2" width="17.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.2)"> <g transform="translate(72,-63.04)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℝ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.61,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>评估最后用积分<年代vg height="10.325" id="M125" style="vertical-align:-0.0pt;width:9.0375004px;" version="1.1" viewbox="0 0 9.0375004 10.325" width="9.0375004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,我们注意到被积函数的渐近行为保证绝对可积性。通过改变变量<年代vg height="13.9125" id="M126" style="vertical-align:-0.11285pt;width:36.875px;" version="1.1" viewbox="0 0 36.875 13.9125" width="36.875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.9125)"> <g transform="translate(72,-60.87)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-66.85,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-58.5,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> </text> </g> </g> </svg>,我们看到积分只是梅林变换的一半<年代vg height="22.225" id="M127" style="vertical-align:-3.13504pt;width:26.737499px;" version="1.1" viewbox="0 0 26.737499 22.225" width="26.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,22.225)"> <g transform="translate(72,-54.22)"> <text transform="matrix(1,0,0,-1,-69.43,59.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.52" y="2.54"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-59.91,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.76"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.76"> 2</t年代pan> </text> </g> </g> </svg>,<年代pan class="equation" id="eq6"> <svg height="34.674999" id="M128" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 34.674999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,34.675)"> <g transform="translate(72,-44.26)"> <text transform="matrix(1,0,0,-1,99.44,55.27)"> <tspan style="font-size: 12.50px; " x="22.709999" y="-7.25"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.4772739" y="0"> =</t年代pan> </text> <g> <path d="m 122.15,58.4 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,122.15,46.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="7.4499998" y="-8.75"> ℳ</t年代pan> <tspan style="font-size: 12.50px; " x="22.940001" y="-3.8299999"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.82" y="-11.29"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.299999" y="-8.75"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,164.49,63.9)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="0" y="11.75"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="11.75"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,173.74,50.35)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6899996" y="-4.9299998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.853499" y="-4.9299998"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.354818" y="-4.9299998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="26.981649" y="-4.9299998"> =</t年代pan> <tspan style="font-size: 12.50px; " x="40.220001" y="-13.14"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="49.634758" y="-13.14"> (</t年代pan> <tspan style="font-size: 12.50px; " x="53.79826" y="-13.14"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.012249" y="-13.14"> /</t年代pan> <tspan style="font-size: 12.50px; " x="63.488083" y="-13.14"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="72.527756" y="-13.14"> +</t年代pan> <tspan style="font-size: 12.50px; " x="83.867973" y="-13.14"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.369293" y="-13.14"> )</t年代pan> <tspan style="font-size: 12.50px; " x="95.608292" y="-13.14"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="105.03555" y="-13.14"> (</t年代pan> <tspan style="font-size: 12.50px; " x="109.19905" y="-13.14"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="118.22622" y="-13.14"> +</t年代pan> <tspan style="font-size: 12.50px; " x="129.57893" y="-13.14"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.08026" y="-13.14"> )</t年代pan> </text> <g> <path d="m 213.96,58.4 99.03,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,235.01,46.52)"> <tspan style="font-size: 12.50px; " x="79.169998" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="19.829758" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="28.856924" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="40.209648" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.261341" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.762661" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="223.48" y="-8.4899998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="227.64349" y="-8.4899998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="233.895" y="-8.4899998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="237.02075" y="-8.4899998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="243.27225" y="-8.4899998"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="249.52374" y="-8.4899998"> )</t年代pan> </text> </g> </g> </svg></span>评估在点<年代vg height="10.9125" id="M129" style="vertical-align:-0.17555pt;width:34.200001px;" version="1.1" viewbox="0 0 34.200001 10.9125" width="34.200001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9646511" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.005039" y="0"> 0</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>因此,我们可以得出这样的结论:索赔<年代vg height="9.875" id="M130" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是恒定的。在一般的情况下<年代vg height="9.875" id="M131" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>我们可以进行如(<gydF4y2Baa href="#B3">13</gydF4y2Baa>定理的证明<年代vg height="10.8625" id="M132" style="vertical-align:-0.13794pt;width:31.387501px;" version="1.1" viewbox="0 0 31.387501 10.8625" width="31.387501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8625)"> <g transform="translate(72,-63.31)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="15.62875" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="18.754499" y="0"> 1</t年代pan> </text> </g> </g> </svg>。]</年代pan></p> <h4 id="existence-and-uniqueness-of-the-solution">3所示。解决方案的存在性和唯一性</gydF4y2Bah4> <p>介绍中提到,我们寻求解决方案的形式<年代pan class="equation" id="EEq13"> <svg height="16.5375" id="M133" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 16.5375" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,91.17,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.0019197" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.165419" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="19.042068" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="24.255819" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.169258" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="35.808594" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="47.836479" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="51.999977" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.126926" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.453926" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="74.705421" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="78.868927" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="85.745575" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="90.959328" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="94.872765" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="101.81193" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="113.15215" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="117.31565" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="124.74243" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="134.13219" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="140.38368" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="144.54718" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="151.42383" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="156.62508" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="160.53851" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="167.49019" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="178.83041" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="182.99391" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="190.18314" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="198.62267" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="204.87416" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="209.03766" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="215.91431" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="221.11555" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="225.02899" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="231.28049" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="373.57999" y="0.63"> (</t年代pan> <tspan style="font-size: 12.50px; " x="377.7435" y="0.63"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="383.995" y="0.63"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="387.12076" y="0.63"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="393.37225" y="0.63"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="9.875" id="M134" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>还有待确定。密度<年代vg height="9.875" id="M135" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是由减少最初的问题相应的积分方程。</p><p>gydF4y2Ba我们假设<年代vg height="13.425" id="M136" style="vertical-align:-2.29482pt;width:8.8500004px;" version="1.1" viewbox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续函数<年代vg height="19.5375" id="M137" style="vertical-align:-3.13504pt;width:18.262501px;" version="1.1" viewbox="0 0 18.262501 19.5375" width="18.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <g> <path d="m -71.95,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-64.15,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。我们需要计算的法向导数<年代vg height="13.425" id="M138" style="vertical-align:-2.29482pt;width:20.6875px;" version="1.1" viewbox="0 0 20.6875 13.425" width="20.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1269503" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="13.4875" id="M139" style="vertical-align:-2.34499pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 13.4875" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1892252" y="0"> </t年代pan> </text> </g> </g> </svg>,<年代vg height="13.1875" id="M140" style="vertical-align:-2.29482pt;width:21.15px;" version="1.1" viewbox="0 0 21.15 13.1875" width="21.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.1875)"> <g transform="translate(72,-61.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0"> </t年代pan> </text> </g> </g> </svg>。为<年代vg height="13.1875" id="M141" style="vertical-align:-2.29482pt;width:21.15px;" version="1.1" viewbox="0 0 21.15 13.1875" width="21.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.1875)"> <g transform="translate(72,-61.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0"> </t年代pan> </text> </g> </g> </svg>在积分定义,我们观察到分化<年代vg height="10.325" id="M142" style="vertical-align:-0.0pt;width:11.75px;" version="1.1" viewbox="0 0 11.75 10.325" width="11.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>允许由于没有奇点<年代vg height="9.125" id="M143" style="vertical-align:-0.11285pt;width:5.0124998px;" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.125)"> <g transform="translate(72,-64.7)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。为<年代vg height="13.4875" id="M144" style="vertical-align:-2.34499pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 13.4875" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1892252" y="0"> </t年代pan> </text> </g> </g> </svg>内部的分化积分是合理的,计算了(<gydF4y2Baa href="#B11">8</gydF4y2Baa>]。最后,定理<gydF4y2Baa href="#thm1">2.3</gydF4y2Baa>给出了边界的法向导数值<年代vg height="13.425" id="M145" style="vertical-align:-2.29482pt;width:20.6875px;" version="1.1" viewbox="0 0 20.6875 13.425" width="20.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.1269503" y="0"> </t年代pan> </text> </g> </g> </svg>。然后罗宾边界条件是等价的<年代pan class="equation" id="EEq14"> <svg height="30.700001" id="M146" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 30.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,30.7)"> <g transform="translate(72,-47.44)"> <text transform="matrix(1,0,0,-1,147.35,51.33)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.5600004" y="-12.17"> 1</t年代pan> </text> <g> <path d="m 155.91,59.37 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,155.91,47.49)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="69.120003" y="-3.8399999"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4400001" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.217106" y="-8.7600002"> +</t年代pan> <tspan style="font-size: 12.50px; " x="27.557327" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.78574" y="-8.7600002"> +</t年代pan> <tspan style="font-size: 12.50px; " x="53.138466" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.115139" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="78.800003" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="90.602829" y="-8.7600002"> =</t年代pan> <tspan style="font-size: 12.50px; " x="102.63071" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="110.35757" y="-8.7600002"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="308.84" y="-9.1099997"> (</t年代pan> <tspan style="font-size: 12.50px; " x="313.00351" y="-9.1099997"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="319.255" y="-9.1099997"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="322.38074" y="-9.1099997"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="328.63226" y="-9.1099997"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代pan class="equation" id="eq7"> <svg height="38.387501" id="M147" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.387501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.3875)"> <g transform="translate(72,-41.29)"> <text transform="matrix(1,0,0,-1,68.16,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="70.029999" y="7.71"> </t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.612748" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.939747" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="30.178743" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="34.342243" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.218891" y="0.0099999998"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="46.432644" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="50.346081" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="57.985416" y="0.0099999998"> =</t年代pan> </text> <text transform="matrix(1,0,0,-1,145.19,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,150.9,45.26)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,156.65,41.46)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Γ</t年代pan> </text> <text transform="matrix(1,0,0,-1,164.37,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,170.76,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,186.25,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="54.763142" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="66.115868" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.517403" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.756393" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.171394" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="93.334892" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.536385" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="104.75014" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="111.15167" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="117.39067" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="123.64217" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="133.06943" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="137.23293" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="143.43442" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="149.68591" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="155.93741" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="278.5" y="-0.56"> (</t年代pan> <tspan style="font-size: 12.50px; " x="282.66351" y="-0.56"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="288.91501" y="-0.56"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="292.04074" y="-0.56"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="298.29224" y="-0.56"> )</t年代pan> </text> </g> </g> </svg></span>定理的积分<gydF4y2Baa href="#thm1">2.3</gydF4y2Baa>和<年代pan class="equation" id="EEq15"> <svg height="115.2875" id="M148" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 115.2875" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,115.2875)"> <g transform="translate(72,20.23)"> <text transform="matrix(1,0,0,-1,81.29,54.64)"> <tspan style="font-size: 12.50px; " x="98.639999" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.615287" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.778786" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.655436" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="27.869186" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.782625" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="39.409454" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="51.449844" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.189442" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="64.352943" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="71.229591" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="76.443344" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.356781" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="87.295944" y="0"> −</t年代pan> </text> <text transform="matrix(1,0,0,-1,185.69,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.2,40.95)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.4,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,204.79,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,220.28,54.64)"> <tspan style="font-size: 12.50px; " x="-90.300003" y="41.73"> −</t年代pan> <tspan style="font-size: 12.50px; " x="-78.959999" y="49.43"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.581139" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.92136" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.122849" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.237534" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.966789" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="74.130287" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.331772" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="86.57077" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="92.822273" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,148.33,26.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.26" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,154.03,5.21)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159.78,1.41)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,168.88,12.91)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,175.27,9.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,190.76,12.91)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="54.763142" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="66.115868" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="72.517403" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="78.756393" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.28392" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="93.447418" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.648911" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="104.86266" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="111.2642" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="117.5032" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="123.75469" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="132.04419" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="138.29568" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="-60.779999" y="29.34"> −</t年代pan> <tspan style="font-size: 12.50px; " x="-49.439777" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-40.387608" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-36.22411" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-29.347458" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="-24.133707" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-20.220268" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="-13.968768" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="-9.8052692" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="-2.3784871" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.0112662" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="13.250263" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.413761" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.290413" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="29.504164" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.417603" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="40.356766" y="29.34"> −</t年代pan> <tspan style="font-size: 12.50px; " x="51.709492" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.761662" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="64.925163" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="71.801811" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="77.015564" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.929001" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="87.180504" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="91.344002" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="98.533226" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="106.97275" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="113.21175" y="29.34"> (</t年代pan> <tspan style="font-size: 12.50px; " x="117.37524" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="124.2519" y="29.34"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="129.46565" y="29.34"> </t年代pan> <tspan style="font-size: 12.50px; " x="133.37909" y="29.34"> )</t年代pan> <tspan style="font-size: 12.50px; " x="139.63058" y="29.34"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="273.98999" y="-9.8500004"> (</t年代pan> <tspan style="font-size: 12.50px; " x="278.1535" y="-9.8500004"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="284.405" y="-9.8500004"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="287.53076" y="-9.8500004"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="293.78226" y="-9.8500004"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>我们将证明(<gydF4y2Baa href="#EEq14">3.2</gydF4y2Baa>为任何有界函数)承认一个独特的解决方案<年代vg height="10.325" id="M149" style="vertical-align:-0.0pt;width:12.05px;" version="1.1" viewbox="0 0 12.05 10.325" width="12.05" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。因此,它是需要确定的条件,保证有界性。</p><p>gydF4y2Ba右边第二个积分(<gydF4y2Baa href="#EEq15">3.4</gydF4y2Baa>),我们使用下面的结果。</p><p><年代pan class="statement" id="lem2"><span class="statement-title">引理3.1。</年代pan><i>让<年代vg height="16.8375" id="M150" style="vertical-align:-2.29482pt;width:123.2125px;" version="1.1" viewbox="0 0 123.2125 16.8375" width="123.2125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="48.549149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.201466" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="69.541687" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.743172" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,6.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.18,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,15.09,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> </g> </g> </svg>与<年代vg height="10.8875" id="M151" style="vertical-align:-0.33858pt;width:37.287498px;" version="1.1" viewbox="0 0 37.287498 10.8875" width="37.287498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.8875)"> <g transform="translate(72,-63.29)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> Γ</t年代pan> </text> </g> </g> </svg>和<年代vg height="13.55" id="M152" style="vertical-align:-2.29482pt;width:38.887501px;" version="1.1" viewbox="0 0 38.887501 13.55" width="38.887501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6648188" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.705208" y="0"> Ω</t年代pan> </text> </g> </g> </svg>。下面的法向导数的估计<年代vg height="10.75" id="M153" style="vertical-align:-0.15048pt;width:11.9125px;" version="1.1" viewbox="0 0 11.9125 10.75" width="11.9125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.75)"> <g transform="translate(72,-63.4)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>持有:</我><年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content"><i>如果<年代vg height="12.3" id="M154" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,然后<年代pan class="equation" id="eq8"> <svg height="28.9625" id="M155" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 28.9625" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,28.9625)"> <g transform="translate(72,-48.83)"> <text transform="matrix(1,0,0,-1,37.7,58.99)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.3900001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.7"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,46.59,54.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,62.08,57.29)"> <tspan style="font-size: 12.50px; " x="56.16" y="-1.7"> |</t年代pan> <tspan style="font-size: 12.50px; " x="56.16" y="1.6900001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.987473" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.130001" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="74.157883" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="82.547401" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,148.55,62.29)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="15.280992" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="17.714048" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="22.090048" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="28.085169" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,183.6,58.88)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="56.5" y="6.5100002"> </t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.279999" y="1.59"> e</t年代pan> <tspan style="font-size: 12.50px; " x="39.831333" y="1.59"> x</t年代pan> <tspan style="font-size: 12.50px; " x="46.082832" y="1.59"> p</t年代pan> <tspan style="font-size: 12.50px; " x="65.470001" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="74.034554" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.361313" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,268.89,62.29)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="13.355552" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="16.269968" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="20.645967" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.641088" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="31.577215" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,303.89,58.88)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.139813" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,336.09,64.05)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.7234716" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="14.099472" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.094591" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.021969" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,364.53,52.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.05" y="-4.9200001"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="100.22" y="-4.9200001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="104.3835" y="-4.9200001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="110.635" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="113.76075" y="-4.9200001"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="120.01225" y="-4.9200001"> )</t年代pan> </text> </g> </g> </svg></span></i></span></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>如果<年代vg height="12.3" id="M156" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,然后<年代pan class="equation" id="eq9"> <svg height="52.662498" id="M157" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 52.662498" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,52.6625)"> <g transform="translate(72,-29.87)"> <text transform="matrix(1,0,0,-1,41.01,49.51)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.3900001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.7"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,49.89,44.68)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,65.39,47.81)"> <tspan style="font-size: 12.50px; " x="56.16" y="-1.7"> |</t年代pan> <tspan style="font-size: 12.50px; " x="56.16" y="1.6900001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="85.639999" y="9.7299995"> </t年代pan> <tspan style="font-size: 12.50px; " x="99.099998" y="-8.7700005"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.987473" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="62.130001" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="74.157883" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,168.4,61.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="16.917616" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.28,58.17)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="34.290001" y="-5.3299999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="34.290001" y="5.3400002"> |</t年代pan> <tspan style="font-size: 12.50px; " x="34.290001" y="8.5"> |</t年代pan> <tspan style="font-size: 12.50px; " x="56.939999" y="6.5100002"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.300003" y="7.1054274e-015"> |</t年代pan> <tspan style="font-size: 12.50px; " x="64.300003" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="94" y="7.1054274e-015"> |</t年代pan> <tspan style="font-size: 12.50px; " x="94" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.790001" y="1.59"> l</t年代pan> <tspan style="font-size: 12.50px; " x="40.265835" y="1.59"> o</t年代pan> <tspan style="font-size: 12.50px; " x="46.517334" y="1.59"> g</t年代pan> <tspan style="font-size: 12.50px; " x="66.800003" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="76.452316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="87.792534" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,288.78,63.34)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.66,56.58)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,297.57,61.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,309,51.66)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.5999999" y="-11.84"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.5999999" y="-1.17"> |</t年代pan> <tspan style="font-size: 12.50px; " x="7.5999999" y="1.99"> |</t年代pan> <tspan style="font-size: 12.50px; " x="-144.50999" y="17.24"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.6" y="-4.9200001"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="26.075834" y="-4.9200001"> f</t年代pan> <tspan style="font-size: 12.50px; " x="34.402832" y="-4.9200001"> n</t年代pan> <tspan style="font-size: 12.50px; " x="44.130165" y="-4.9200001"> =</t年代pan> <tspan style="font-size: 12.50px; " x="56.170555" y="-4.9200001"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="62.422054" y="-4.9200001"> ,</t年代pan> </text> <text transform="matrix(1,0,0,-1,168.4,39.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="16.917616" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,190.2,36.01)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="-7.1054274e-015"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,222.4,41.18)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 3</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,331.6,34.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="3.4758339" y="0"> f</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> n</t年代pan> <tspan style="font-size: 12.50px; " x="21.530167" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="33.570557" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="39.822056" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="133.14999" y="-13.39"> (</t年代pan> <tspan style="font-size: 12.50px; " x="137.31349" y="-13.39"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="143.565" y="-13.39"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="146.69075" y="-13.39"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="152.94225" y="-13.39"> )</t年代pan> </text> </g> </g> </svg></span></i></span></span></span><i></i></span></p> <p><span class="statement" id="proof5"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>应用微分公式引理<gydF4y2Baa href="#lem1">2.1</gydF4y2Baa>,我们得到<年代pan class="equation" id="EEq16"> <svg height="42.400002" id="M158" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 42.400002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,42.4)"> <g transform="translate(72,-38.08)"> <text transform="matrix(1,0,0,-1,37.24,54.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,43.63,51.23)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,59.12,54.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.666259" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.318575" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.658794" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.860283" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.074036" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.987473" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="59.626808" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="71.667198" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="80.23175" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,146.96,59.36)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="12.77792" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,165.8,63.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⟨</t年代pan> <tspan style="font-size: 12.50px; " x="5.0012002" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.653516" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="25.993736" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.195225" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="37.408978" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.710968" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="49.874466" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="56.751118" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="60.914616" y="0"> ⟩</t年代pan> </text> <g> <path d="m 165.8,57.48 68,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,176.09,43.51)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1600001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,208.29,48.68)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,235,54.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,238.91,59.36)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9361281" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,256.8,51.85)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1900001" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="13.403991" y="-2.51"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,281.12,51.23)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,293.91,54.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="17.329159" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="28.681883" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="34.93338" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,339.76,51.23)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,352.55,54.36)"> <tspan style="font-size: 12.50px; " x="14.56" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.84" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="112.2" y="2.4400001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="116.3635" y="2.4400001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="122.615" y="2.4400001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="125.74075" y="2.4400001"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="131.99225" y="2.4400001"> )</t年代pan> </text> </g> </g> </svg></span>在哪里<年代vg height="16.8375" id="M159" style="vertical-align:-2.29482pt;width:123.2125px;" version="1.1" viewbox="0 0 123.2125 16.8375" width="123.2125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="48.549149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.201466" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="69.541687" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.743172" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,6.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.18,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,15.09,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="14.6" id="M160" style="vertical-align:-2.67102pt;width:28.075001px;" version="1.1" viewbox="0 0 28.075001 14.6" width="28.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6)"> <g transform="translate(72,-60.32)"> <text transform="matrix(1,0,0,-1,-71.95,63.04)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ⟨</t年代pan> <tspan style="font-size: 12.50px; " x="5.0012002" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="8.5770578" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="13.778306" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="17.354164" y="0"> ⟩</t年代pan> </text> </g> </g> </svg>表示的内积<年代vg height="11.2" id="M161" style="vertical-align:-0.0pt;width:17.862499px;" version="1.1" viewbox="0 0 17.862499 11.2" width="17.862499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.2)"> <g transform="translate(72,-63.04)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℝ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.61,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>使用的定义<年代vg height="16.237499" id="M162" style="vertical-align:-4.74141pt;width:27.025px;" version="1.1" viewbox="0 0 27.025 16.237499" width="27.025" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.2375)"> <g transform="translate(72,-59.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="7.1941438" y="0"> )</t年代pan> </text> </g> </g> </svg>和财产<年代vg height="13.45" id="M163" style="vertical-align:-2.21957pt;width:107.5375px;" version="1.1" viewbox="0 0 107.5375 13.45" width="107.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="7.3392611" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.50276" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="20.517424" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="31.857643" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="38.109142" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="45.748478" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="57.776363" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.015358" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="71.354622" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="75.51812" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.757118" y="0"> )</t年代pan> </text> </g> </g> </svg>γ函数,它遵循的梅林变换<年代vg height="16.5375" id="M164" style="vertical-align:-4.68874pt;width:87.400002px;" version="1.1" viewbox="0 0 87.400002 16.5375" width="87.400002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.5375)"> <g transform="translate(72,-58.77)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.2139912" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-54.82,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-41.34,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="17.591721" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-12.83,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> </g> </g> </svg>是<年代pan class="equation" id="eq10"> <svg height="34.674999" id="M165" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 34.674999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,34.675)"> <g transform="translate(72,-44.26)"> <text transform="matrix(1,0,0,-1,72.17,55.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ℳ</t年代pan> <tspan style="font-size: 12.50px; " x="15.49" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.34" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.553991" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,110.64,52.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,124.12,55.27)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="17.591721" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,152.63,52.15)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,163.34,52.76)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.470001" y="-9.7600002"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="7.9299998" y="-2.52"> (</t年代pan> <tspan style="font-size: 12.50px; " x="12.093499" y="-2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="17.594818" y="-2.52"> )</t年代pan> <tspan style="font-size: 12.50px; " x="25.221649" y="-2.52"> =</t年代pan> </text> <g> <path d="m 201.81,58.4 6.25,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,201.81,46.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="8.6400003" y="-16.969999"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="18.054758" y="-16.969999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="22.218258" y="-16.969999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="26.381758" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.371414" y="-16.969999"> +</t年代pan> <tspan style="font-size: 12.50px; " x="46.724136" y="-16.969999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="52.975639" y="-16.969999"> )</t年代pan> <tspan style="font-size: 12.50px; " x="59.214634" y="-16.969999"> /</t年代pan> <tspan style="font-size: 12.50px; " x="62.690468" y="-16.969999"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="71.717636" y="-16.969999"> +</t年代pan> <tspan style="font-size: 12.50px; " x="83.070358" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="88.571678" y="-16.969999"> )</t年代pan> <tspan style="font-size: 12.50px; " x="94.810677" y="-16.969999"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="104.23794" y="-16.969999"> (</t年代pan> <tspan style="font-size: 12.50px; " x="108.40144" y="-16.969999"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="117.4286" y="-16.969999"> +</t年代pan> <tspan style="font-size: 12.50px; " x="128.76883" y="-16.969999"> </t年代pan> <tspan style="font-size: 12.50px; " x="134.27014" y="-16.969999"> )</t年代pan> </text> <g> <path d="m 210.45,58.4 129.81,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,249.61,46.52)"> <tspan style="font-size: 12.50px; " x="91.839996" y="-8.75"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="9.4272623" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="13.590761" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.418119" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="34.758339" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.810032" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="47.311352" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="215.14" y="-8.4899998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="219.3035" y="-8.4899998"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="225.55499" y="-8.4899998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="228.68076" y="-8.4899998"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="234.93225" y="-8.4899998"> )</t年代pan> </text> </g> </g> </svg></span>这只不过是梅林变换的一半吗<年代vg height="16.174999" id="M166" style="vertical-align:-4.68874pt;width:27.15px;" version="1.1" viewbox="0 0 27.15 16.174999" width="27.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.175)"> <g transform="translate(72,-59.06)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-61.03,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="7.2904158" y="0"> )</t年代pan> </text> </g> </g> </svg>与<年代vg height="7.1374998" id="M167" style="vertical-align:-0.10033pt;width:7.8874998px;" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1375)"> <g transform="translate(72,-66.29)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>取而代之的是<年代vg height="11.325" id="M168" style="vertical-align:-0.51414pt;width:33.362499px;" version="1.1" viewbox="0 0 33.362499 11.325" width="33.362499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9896574" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.329878" y="0"> 2</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>使用估计(<gydF4y2Baa href="#EEq9">2.9</gydF4y2Baa>)的引理<gydF4y2Baa href="#lem1">2.1</gydF4y2Baa>与<年代vg height="11.325" id="M169" style="vertical-align:-0.51414pt;width:33.362499px;" version="1.1" viewbox="0 0 33.362499 11.325" width="33.362499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9896574" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.329878" y="0"> 2</t年代pan> </text> </g> </g> </svg>而不是<年代vg height="7.1374998" id="M170" style="vertical-align:-0.10033pt;width:7.8874998px;" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1375)"> <g transform="translate(72,-66.29)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>,我们获得的第一个估计<年代vg height="12.3" id="M171" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>。<gydF4y2Babr>如果<年代vg height="12.3" id="M172" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,我们使用估计(<gydF4y2Baa href="#EEq10">2.10</gydF4y2Baa>)的引理<gydF4y2Baa href="#lem1">2.1</gydF4y2Baa>与<年代vg height="11.325" id="M173" style="vertical-align:-0.51414pt;width:60.5625px;" version="1.1" viewbox="0 0 60.5625 11.325" width="60.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9896574" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.329878" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="30.057213" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="42.085098" y="0"> 4</t年代pan> </text> </g> </g> </svg>和<年代vg height="11.325" id="M174" style="vertical-align:-0.51414pt;width:60.5625px;" version="1.1" viewbox="0 0 60.5625 11.325" width="60.5625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.9896574" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="20.329878" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="30.057213" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="42.085098" y="0"> 4</t年代pan> </text> </g> </g> </svg>获得第二个估计。</年代pan></p> <p>让我们回到r.h.s.第二积分的估计(<gydF4y2Baa href="#EEq15">3.4</gydF4y2Baa>)。我们再次把整体分割成两部分<年代vg height="14.2375" id="M175" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.2375" id="M176" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>这取决于<年代vg height="12.3" id="M177" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>或<年代vg height="12.3" id="M178" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>与<年代vg height="16.8375" id="M179" style="vertical-align:-2.29482pt;width:159.27499px;" version="1.1" viewbox="0 0 159.27499 16.8375" width="159.27499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="48.549149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.201466" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="69.541687" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.743172" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,6.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.18,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.852604" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.192825" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.59436" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,43.94,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> </g> </g> </svg>。如果<年代vg height="13.4875" id="M180" style="vertical-align:-2.34499pt;width:10.675px;" version="1.1" viewbox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个有界函数,<年代vg height="12.3" id="M181" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,拥有<年代pan class="equation" id="eq11"> <svg height="38.387501" id="M182" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.387501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.3875)"> <g transform="translate(72,-41.29)"> <text transform="matrix(1,0,0,-1,17.98,54.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,26.48,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,31.35,54.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="18.007887" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.397638" y="1.59"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="33.936707" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.875874" y="1.59"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,78.77,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,84.97,53.2)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,91.26,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,100.68,47.65)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,105.62,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,111.11,45.26)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,118.12,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,123.82,52.97)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1700001" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.859105" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.199326" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.600863" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,156.59,58.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9273758" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="7.3604321" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="11.736432" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="17.731552" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,179.2,52.96)"> <tspan style="font-size: 12.50px; " x="14.74" y="7.6999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,200.94,66.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> <tspan style="font-size: 8.75px; " x="-1.25" y="25.530001"> 4</t年代pan> </text> <text transform="matrix(1,0,0,-1,211.63,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,216.73,57.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,223.66,52.96)"> <tspan style="font-size: 12.50px; " x="22.219999" y="2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> e</t年代pan> <tspan style="font-size: 12.50px; " x="5.551332" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="28.07" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.634556" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.973816" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,272.74,57.96)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.7234716" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="14.099472" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.094591" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.021969" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,301.18,50.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> d</t年代pan> <tspan style="font-size: 12.50px; " x="14.1915" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.756056" y="-2.51"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="34.796444" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.186195" y="-2.51"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="50.725266" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.664433" y="-2.51"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,365.39,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,371.59,53.2)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,377.87,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,387.3,47.65)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,392.24,49.83)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,395.65,52.96)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="69.099998" y="-0.56"> (</t年代pan> <tspan style="font-size: 12.50px; " x="73.263496" y="-0.56"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="79.514999" y="-0.56"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="82.640747" y="-0.56"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="88.89225" y="-0.56"> )</t年代pan> </text> </g> </g> </svg></span>我们使用的球坐标在哪里<年代vg height="16.8375" id="M183" style="vertical-align:-2.29482pt;width:129.89999px;" version="1.1" viewbox="0 0 129.89999 16.8375" width="129.89999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="20.604944" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="24.768442" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.457548" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="42.797768" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.199306" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.57,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="13.355552" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-0.34,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152916" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.493137" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.694626" y="0"> |</t年代pan> </text> </g> </g> </svg>。</p><p>gydF4y2Ba如果<年代vg height="12.3" id="M184" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,我们必须考虑不同情况下的<年代vg height="7.1374998" id="M185" style="vertical-align:-0.10033pt;width:7.8874998px;" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1375)"> <g transform="translate(72,-66.29)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>单独的。例如,让我们考虑的情况<年代vg height="10.9125" id="M186" style="vertical-align:-0.17555pt;width:35.099998px;" version="1.1" viewbox="0 0 35.099998 10.9125" width="35.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9125)"> <g transform="translate(72,-63.27)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6773224" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.71771" y="0"> 3</t年代pan> </text> </g> </g> </svg>。我们有<年代pan class="equation" id="eq12"> <svg height="38.112499" id="M187" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.112499" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.1125)"> <g transform="translate(72,-41.51)"> <text transform="matrix(1,0,0,-1,59.08,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.51" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,67.59,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,72.46,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.9699998" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="18.010389" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.400143" y="1.59"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="33.939213" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.878376" y="1.59"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,119.88,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,126.08,54.88)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,132.36,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,141.79,49.33)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,146.73,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,152.22,46.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,157.97,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> <tspan style="font-size: 8.75px; " x="6.511488" y="0"> ×</t年代pan> <tspan style="font-size: 8.75px; " x="12.112768" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="15.027184" y="0"> 0</t年代pan> <tspan style="font-size: 8.75px; " x="19.403185" y="0"> ,</t年代pan> <tspan style="font-size: 8.75px; " x="21.591185" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="24.330561" y="0"> )</t年代pan> <tspan style="font-size: 8.75px; " x="27.244976" y="0"> ∩</t年代pan> <tspan style="font-size: 8.75px; " x="32.671215" y="0"> {</t年代pan> </text> <text transform="matrix(1,0,0,-1,194.85,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> z</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.73,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> }</t年代pan> </text> <text transform="matrix(1,0,0,-1,215.89,54.65)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.849105" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="22.189325" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="28.590862" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,248.65,59.82)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.9361281" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="9.382144" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="15.377264" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="20.313393" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.308512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,279.83,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,312.04,61.4)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="10.37112" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,327.35,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.528486" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="20.779985" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.181522" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="131.14" y="1.01"> (</t年代pan> <tspan style="font-size: 12.50px; " x="135.3035" y="1.01"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="141.55499" y="1.01"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="144.68076" y="1.01"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="150.93225" y="1.01"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="157.18375" y="1.01"> )</t年代pan> </text> </g> </g> </svg></span>在这样一个事实<年代vg height="12.8125" id="M188" style="vertical-align:-1.29163pt;width:48.799999px;" version="1.1" viewbox="0 0 48.799999 12.8125" width="48.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.8125)"> <g transform="translate(72,-61.75)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.71,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.3,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.6125" id="M189" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>使用。选择<年代vg height="13.6125" id="M190" style="vertical-align:-2.34499pt;width:74.625px;" version="1.1" viewbox="0 0 74.625 13.6125" width="74.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.75522" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.582579" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="43.622967" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="49.874466" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="53.3503" y="0"> 2</t年代pan> </text> </g> </g> </svg>,我们看到,<年代vg height="14.2375" id="M191" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>是有界的。</p><p>gydF4y2Ba使用估计(<gydF4y2Baa href="#EEq11">2.12</gydF4y2Baa>)和(<gydF4y2Baa href="#EEq12">2.13</gydF4y2Baa>在定理的证明<gydF4y2Baa href="#thm2">2.4</gydF4y2Baa>,我们看到,<年代vg height="13.4875" id="M192" style="vertical-align:-2.34499pt;width:19.6625px;" version="1.1" viewbox="0 0 19.6625 13.4875" width="19.6625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.1892252" y="0"> </t年代pan> </text> </g> </g> </svg>是有界的。</p><p>gydF4y2Ba右边第一积分,我们使用以下结果<gydF4y2Baa href="#B2">3</gydF4y2Baa>命题1]。</p><p><年代pan class="statement" id="lem3"><span class="statement-title">引理3.2。</年代pan><i>让<年代vg height="16.75" id="M193" style="vertical-align:-2.21957pt;width:97.8125px;" version="1.1" viewbox="0 0 97.8125 16.75" width="97.8125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.75)"> <g transform="translate(72,-58.6)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="48.549149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.4258" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,-14.02,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,-9.14,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-5.23,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> </g> </g> </svg>。为<年代vg height="10.325" id="M194" style="vertical-align:-0.0pt;width:12.225px;" version="1.1" viewbox="0 0 12.225 10.325" width="12.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>持有以下:</我><年代pan class="list"><span class="list-item"><span class="list-label">(1)</年代pan><span class="list-content"><i>如果<年代vg height="12.3" id="M195" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,然后<年代pan class="equation" id="eq13"> <svg height="19.700001" id="M196" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.700001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.7)"> <g transform="translate(72,-56.24)"> <text transform="matrix(1,0,0,-1,77.61,62.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> ∇</t年代pan> </text> <text transform="matrix(1,0,0,-1,89.25,57.73)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,94.56,60.86)"> <tspan style="font-size: 12.50px; " x="36.09" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="36.09" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.765323" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.928822" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.805471" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="28.00672" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="42.07" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="54.097885" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.4874" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,160.97,65.86)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="13.836912" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="18.186657" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="24.181776" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="28.557775" y="0"> )</t年代pan> <tspan style="font-size: 8.75px; " x="31.472193" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="33.905247" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,201.84,60.86)"> <tspan style="font-size: 12.50px; " x="22.219999" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> e</t年代pan> <tspan style="font-size: 12.50px; " x="5.551332" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="29.41" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="37.974556" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="45.313816" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,251.07,65.86)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="13.355552" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="16.269968" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="20.645967" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.641088" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="31.577215" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,286.07,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,297.95,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0"> /</t年代pan> <tspan style="font-size: 8.75px; " x="6.8090558" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.7234716" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="14.099472" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="20.094591" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="25.021969" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,326.39,58.34)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.2700005" y="-2.52"> ;</t年代pan> <tspan style="font-size: 12.50px; " x="132.10001" y="-2.6600001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="136.2635" y="-2.6600001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="142.515" y="-2.6600001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="145.64075" y="-2.6600001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="151.89224" y="-2.6600001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="158.14375" y="-2.6600001"> )</t年代pan> </text> </g> </g> </svg></span></i></span></span><span class="list-item"><span class="list-label">(2)</年代pan><span class="list-content"><i>如果<年代vg height="12.3" id="M197" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>,然后<年代pan class="equation" id="eq14"> <svg height="18.725" id="M198" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 18.725" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,18.725)"> <g transform="translate(72,-57.02)"> <text transform="matrix(1,0,0,-1,142.87,62.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.51" y="1.59"> ∇</t年代pan> </text> <text transform="matrix(1,0,0,-1,154.51,57.73)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,159.83,60.86)"> <tspan style="font-size: 12.50px; " x="36.080002" y="-1.59"> |</t年代pan> <tspan style="font-size: 12.50px; " x="36.080002" y="1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.792969" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="28.00672" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.920158" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="42.060001" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="54.087887" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.477398" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,226.23,65.86)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,237.66,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.3772497" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,249.54,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="16.339985" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,270.75,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="187.74001" y="-0.51999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="191.9035" y="-0.51999998"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="198.155" y="-0.51999998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="201.28075" y="-0.51999998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="207.53224" y="-0.51999998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="213.78375" y="-0.51999998"> )</t年代pan> </text> </g> </g> </svg></span></i></span></span></span><i></i></span></p> <p>我们把右边的第一积分(<gydF4y2Baa href="#EEq15">3.4</gydF4y2Baa>)分为两部分<年代vg height="14.2375" id="M199" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.2375" id="M200" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>这取决于<年代vg height="16.8375" id="M201" style="vertical-align:-2.29482pt;width:150.425px;" version="1.1" viewbox="0 0 150.425 16.8375" width="150.425" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.8375)"> <g transform="translate(72,-58.53)"> <text transform="matrix(1,0,0,-1,-71.95,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="25.906216" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="32.157715" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="35.633549" y="0"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="41.885052" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="46.04855" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="48.549149" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.201466" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="69.541687" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="75.743172" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,6.31,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,11.18,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,15.09,65.87)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,30,60.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>或<年代vg height="12.3" id="M202" style="vertical-align:-1.29163pt;width:35.125px;" version="1.1" viewbox="0 0 35.125 12.3" width="35.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="21.742718" y="0"> 1</t年代pan> </text> </g> </g> </svg>。如果<年代vg height="9.875" id="M203" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个有界函数,然后利用引理吗<gydF4y2Baa href="#lem3">3.2</gydF4y2Baa>我们有<年代pan class="equation" id="eq15"> <svg height="19.424999" id="M204" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.424999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.425)"> <g transform="translate(72,-56.46)"> <text transform="matrix(1,0,0,-1,147.42,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,155.92,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,160.8,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.9699998" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="17.997887" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.3874" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,191.11,66.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="16.926369" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="21.862495" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,217.92,61.91)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,240.39,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,246.58,62.16)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,252.87,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,262.3,56.6)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,265.92,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,458.49,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="13.540749" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="26.043749" y="0"> )</t年代pan> </text> </g> </g> </svg></span>对于任何<年代vg height="13.6125" id="M205" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>,因为<年代vg height="17.075001" id="M206" style="vertical-align:-2.72118pt;width:120.25px;" version="1.1" viewbox="0 0 120.25 17.075001" width="120.25" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.075)"> <g transform="translate(72,-58.34)"> <text transform="matrix(1,0,0,-1,-71.95,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.7023277" y="0"> ↦</t年代pan> <tspan style="font-size: 12.50px; " x="24.755939" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.95,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-33.92,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> e</t年代pan> <tspan style="font-size: 12.50px; " x="5.551332" y="0"> x</t年代pan> <tspan style="font-size: 12.50px; " x="11.802832" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="22.217831" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="26.38133" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="34.945885" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.285145" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,14.61,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,19.99,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>是一致有界的<年代vg height="13.45" id="M207" style="vertical-align:-2.21957pt;width:39.337502px;" version="1.1" viewbox="0 0 39.337502 13.45" width="39.337502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.45)"> <g transform="translate(72,-61.24)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="15.616247" y="0"> ∞</t年代pan> <tspan style="font-size: 12.50px; " x="27.194025" y="0"> )</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.6125" id="M208" style="vertical-align:-2.34499pt;width:76.599998px;" version="1.1" viewbox="0 0 76.599998 13.6125" width="76.599998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9641709" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.341421" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="22.692945" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="32.070194" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="42.885288" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="54.925678" y="0"> 0</t年代pan> </text> </g> </g> </svg>。同样的,对<年代vg height="14.2375" id="M209" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>那里拥有<年代pan class="equation" id="eq16"> <svg height="38.974998" id="M210" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 38.974998" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,38.975)"> <g transform="translate(72,-40.82)"> <text transform="matrix(1,0,0,-1,82.02,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,90.52,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,95.39,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="18.007887" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.397638" y="1.59"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="33.936707" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.326462" y="1.59"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,145.26,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,151.46,54.88)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,157.75,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,167.17,49.33)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,170.8,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,176.29,46.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,182.04,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,188.56,40.95)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.18,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∩</t年代pan> <tspan style="font-size: 8.75px; " x="5.42624" y="0"> {</t年代pan> <tspan style="font-size: 8.75px; " x="9.6272001" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="13.994448" y="0"> ≤</t年代pan> <tspan style="font-size: 8.75px; " x="19.989569" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="24.365568" y="0"> }</t年代pan> </text> <text transform="matrix(1,0,0,-1,223.33,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,227.24,59.64)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.931248" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="16.926369" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="21.862495" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,254.05,56.23)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="29.700001" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152316" y="1.59"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.492537" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,286.25,61.4)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="16.339985" y="0"> 1</t年代pan> <tspan style="font-size: 8.75px; " x="20.715984" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="26.711103" y="0"> 2</t年代pan> <tspan style="font-size: 8.75px; " x="31.087105" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,322.28,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="6.2515001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="136.21001" y="1.36"> (</t年代pan> <tspan style="font-size: 12.50px; " x="140.3735" y="1.36"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="146.625" y="1.36"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="149.75075" y="1.36"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="156.00224" y="1.36"> 4</t年代pan> <tspan style="font-size: 12.50px; " x="162.25375" y="1.36"> )</t年代pan> </text> </g> </g> </svg></span>对于任何<年代vg height="13.6125" id="M211" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>,因为<年代vg height="12.8125" id="M212" style="vertical-align:-1.29163pt;width:48.799999px;" version="1.1" viewbox="0 0 48.799999 12.8125" width="48.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.8125)"> <g transform="translate(72,-61.75)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.71,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-51.3,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≥</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 1</t年代pan> </text> </g> </g> </svg>对于任何<年代vg height="13.6125" id="M213" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> ></t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 0</t年代pan> </text> </g> </g> </svg>。选择<年代vg height="13.6125" id="M214" style="vertical-align:-2.34499pt;width:47.424999px;" version="1.1" viewbox="0 0 47.424999 13.6125" width="47.424999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="28.106745" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="31.582579" y="0"> 2</t年代pan> </text> </g> </g> </svg>,我们有<年代pan class="equation" id="eq17"> <svg height="19.424999" id="M215" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 19.424999" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.425)"> <g transform="translate(72,-56.46)"> <text transform="matrix(1,0,0,-1,145.86,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,154.36,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,159.23,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="3.1700001"> |</t年代pan> <tspan style="font-size: 12.50px; " x="5.98" y="1.59"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="18.007887" y="1.59"> </t年代pan> <tspan style="font-size: 12.50px; " x="26.3974" y="1.59"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,189.55,66.91)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.922496" y="0"> +</t年代pan> <tspan style="font-size: 8.75px; " x="16.917616" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="21.853745" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,216.35,61.91)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="15.928822" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,238.82,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,245.02,62.16)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,251.31,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,260.73,56.6)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,264.36,58.78)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,267.77,61.91)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="190.72" y="0.81"> (</t年代pan> <tspan style="font-size: 12.50px; " x="194.8835" y="0.81"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="201.13499" y="0.81"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="204.26074" y="0.81"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="210.51225" y="0.81"> 5</t年代pan> <tspan style="font-size: 12.50px; " x="216.76375" y="0.81"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>我们可以看到,<年代vg height="14.2375" id="M216" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>吹成<年代vg height="11.325" id="M217" style="vertical-align:-0.51414pt;width:51.900002px;" version="1.1" viewbox="0 0 51.900002 11.325" width="51.900002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.325)"> <g transform="translate(72,-62.94)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.4647713" y="0"> →</t年代pan> <tspan style="font-size: 12.50px; " x="26.593882" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="32.845383" y="0"> +</t年代pan> </text> </g> </g> </svg>。因此,我们必须承担更多的平滑度<年代vg height="9.875" id="M218" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>保证有界性。假设<年代vg height="9.875" id="M219" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>是一个连续可微的函数<年代vg height="19.9125" id="M220" style="vertical-align:-3.25793pt;width:17.85px;" version="1.1" viewbox="0 0 17.85 19.9125" width="17.85" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.9125)"> <g transform="translate(72,-56.07)"> <g> <path d="m -71.95,70.71 14.18,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-62.65,56.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>。然后分部积分收益更好的内核<年代vg height="10.325" id="M221" style="vertical-align:-0.0pt;width:12.225px;" version="1.1" viewbox="0 0 12.225 10.325" width="12.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。的渐近行为<年代vg height="10.325" id="M222" style="vertical-align:-0.0pt;width:12.225px;" version="1.1" viewbox="0 0 12.225 10.325" width="12.225" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>保证产生的积分是一致有界<年代vg height="11.1875" id="M223" style="vertical-align:-1.76814pt;width:25.3375px;" version="1.1" viewbox="0 0 25.3375 11.1875" width="25.3375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1875)"> <g transform="translate(72,-63.05)"> <text transform="matrix(1,0,0,-1,-71.95,64.86)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.8766499" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="16.241398" y="0"> </t年代pan> </text> </g> </g> </svg>(见[<gydF4y2Baa href="#B2">3</gydF4y2Baa>命题1])。我们有<年代pan class="equation" id="eq18"> <svg height="44.075001" id="M224" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 44.075001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,44.075)"> <g transform="translate(72,-36.74)"> <text transform="matrix(1,0,0,-1,49.56,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="10.67"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="21.33"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="24.5"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5" y="19.950001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,57.81,39.75)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,64.32,37.56)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,70.53,51.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,76.92,48.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,92.41,51.25)"> <tspan style="font-size: 12.50px; " x="99.029999" y="-12.25"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="9.0799999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="12.25"> |</t年代pan> <tspan style="font-size: 12.50px; " x="128.52" y="4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.908855" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.110344" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.237534" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.954285" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="74.117783" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.319275" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="86.57077" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="92.822273" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="105.01" y="0.0099999998"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="117.03789" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="135.88" y="0.0099999998"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="142.41907" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="151.80882" y="0.0099999998"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.76,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,256.96,50.31)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,263.24,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> </text> <g> <path d="m 266.16,54.88 10.13,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,266.16,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,272.67,44.75)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,276.29,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,282.48,51.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="17.87929" y="0"> ∇</t年代pan> <tspan style="font-size: 12.50px; " x="27.018984" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.408737" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,325.44,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,331.63,50.31)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,337.92,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> </text> <g> <path d="m 340.83,54.88 10.14,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,340.83,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,347.34,44.75)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,350.97,46.94)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,354.38,46.33)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6899996" y="-4.9099998"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="104.11" y="-4.9200001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="108.2735" y="-4.9200001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="114.525" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="117.65075" y="-4.9200001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="123.90225" y="-4.9200001"> 6</t年代pan> <tspan style="font-size: 12.50px; " x="130.15375" y="-4.9200001"> )</t年代pan> </text> </g> </g> </svg></span></p> <p>和上面同样的理由说明<年代vg height="13.1875" id="M225" style="vertical-align:-2.29482pt;width:21.15px;" version="1.1" viewbox="0 0 21.15 13.1875" width="21.15" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.1875)"> <g transform="translate(72,-61.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4267821" y="0"> </t年代pan> </text> </g> </g> </svg>是有界的。</p><p>gydF4y2Ba现在我们已经准备好证明(<gydF4y2Baa href="#EEq14">3.2</gydF4y2Baa>)有一个独特的解决方案。</p><p><年代pan class="statement" id="thm4"><span class="statement-title">定理3.3。</年代pan><i>让<年代vg height="14.9875" id="M226" style="vertical-align:-3.13504pt;width:80.4375px;" version="1.1" viewbox="0 0 80.4375 14.9875" width="80.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.9875)"> <g transform="translate(72,-60.01)"> <text transform="matrix(1,0,0,-1,-71.95,63.19)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-40.65,68.19)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,-32.05,63.19)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.58,60.06)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-11.87,63.19)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="19.5375" id="M227" style="vertical-align:-3.13504pt;width:68.212502px;" version="1.1" viewbox="0 0 68.212502 19.5375" width="68.212502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6148071" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.620087" y="0"> (</t年代pan> </text> <g> <path d="m -36.16,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-36.16,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.36,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-21.65,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="19.9125" id="M228" style="vertical-align:-3.25793pt;width:77.925003px;" version="1.1" viewbox="0 0 77.925003 19.9125" width="77.925003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.9125)"> <g transform="translate(72,-56.07)"> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.853084" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="24.893473" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.08,64.68)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-32.21,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> </text> <g> <path d="m -28.05,70.71 14.18,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-28.05,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.74,56.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,-13.87,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>。然后边界积分方程(<gydF4y2Baa href="#EEq14">3.2</gydF4y2Baa>)承认一个独特的有界、持续的解决方案<年代vg height="9.875" id="M229" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</我></年代pan></p> <p><span class="statement" id="proof6"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>使用类似的估计如引理<gydF4y2Baa href="#lem2">3.1</gydF4y2Baa>在定理的证明<gydF4y2Baa href="#thm2">2.4</gydF4y2Baa>,我们看到,<年代vg height="13.425" id="M230" style="vertical-align:-2.29482pt;width:52.0625px;" version="1.1" viewbox="0 0 52.0625 13.425" width="52.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.545307" y="0"> </t年代pan> </text> </g> </g> </svg>是一个弱奇异积分算子与内核。注意,估计的法向导数的引理<gydF4y2Baa href="#lem2">3.1</gydF4y2Baa>可以乘<年代vg height="16.75" id="M231" style="vertical-align:-2.29482pt;width:46.912498px;" version="1.1" viewbox="0 0 46.912498 16.75" width="46.912498" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.75)"> <g transform="translate(72,-58.6)"> <text transform="matrix(1,0,0,-1,-71.95,60.93)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.152916" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="23.493137" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.694626" y="0"> |</t年代pan> </text> <text transform="matrix(1,0,0,-1,-39.75,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>在估计<年代vg height="10.575" id="M232" style="vertical-align:-0.20064pt;width:17.9625px;" version="1.1" viewbox="0 0 17.9625 10.575" width="17.9625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.575)"> <g transform="translate(72,-63.54)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>由于李雅普诺夫平滑的边界<年代vg height="10.475" id="M233" style="vertical-align:-0.0pt;width:9.3000002px;" version="1.1" viewbox="0 0 9.3000002 10.475" width="9.3000002" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.475)"> <g transform="translate(72,-63.62)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> </text> </g> </g> </svg>。详情我们指<gydF4y2Baa href="#B5">12</gydF4y2Baa>]。<gydF4y2Babr>我们得出这样的结论:<年代vg height="13.425" id="M234" style="vertical-align:-2.29482pt;width:52.0625px;" version="1.1" viewbox="0 0 52.0625 13.425" width="52.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.425)"> <g transform="translate(72,-61.26)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.228414" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.545307" y="0"> </t年代pan> </text> </g> </g> </svg>是一个紧凑的运营商<年代vg height="19.5375" id="M235" style="vertical-align:-3.13504pt;width:41.137501px;" version="1.1" viewbox="0 0 41.137501 19.5375" width="41.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.9648914" y="0"> (</t年代pan> </text> <g> <path d="m -57.82,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-57.82,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-50.02,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-43.31,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>(<gydF4y2Baa href="#B12">14</gydF4y2Baa>定理2.22)。此外,同样在(<gydF4y2Baa href="#B8">15</gydF4y2Baa>我们可以证明存在一个整数<年代vg height="14.625" id="M236" style="vertical-align:-3.25793pt;width:14.6875px;" version="1.1" viewbox="0 0 14.6875 14.625" width="14.6875" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.625)"> <g transform="translate(72,-60.3)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.17,60.47)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> </g> </g> </svg>这样<年代pan class="equation" id="EEq17"> <svg height="35.275002" id="M237" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 35.275002" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,35.275)"> <g transform="translate(72,-43.78)"> <text transform="matrix(1,0,0,-1,95.09,52.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.702568" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="16.954067" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.182482" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="42.522701" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="48.774204" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="55.750877" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="64.753036" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,164.02,57.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,168.76,55.69)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,172.38,57.88)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,175.84,52.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="8.3269978" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,190.71,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,196.9,52.95)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,203.19,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,211.57,47.39)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,216.5,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,223.39,52.71)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="13.23" y="-8.2200003"> (</t年代pan> <tspan style="font-size: 12.50px; " x="17.393499" y="-8.2200003"> </t年代pan> <tspan style="font-size: 12.50px; " x="29.959015" y="-8.2200003"> </t年代pan> <tspan style="font-size: 12.50px; " x="38.836143" y="-8.2200003"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,266.39,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <g> <path d="m 236.62,55.83 33.23,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,249.17,43.95)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="3.9759541" y="0"> !</t年代pan> <tspan style="font-size: 12.50px; " x="21.870001" y="-8.7600002"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="28.409069" y="-8.7600002"> </t年代pan> <tspan style="font-size: 12.50px; " x="36.736069" y="-8.7600002"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,292.45,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,298.65,52.95)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,304.93,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,313.31,47.39)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,318.24,49.58)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,458.49,54.76)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="13.540749" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> 7</t年代pan> <tspan style="font-size: 12.50px; " x="26.043749" y="0"> )</t年代pan> </text> </g> </g> </svg></span>为一个常数<年代vg height="10.325" id="M238" style="vertical-align:-0.0pt;width:17.237499px;" version="1.1" viewbox="0 0 17.237499 10.325" width="17.237499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和所有<年代vg height="10.9875" id="M239" style="vertical-align:-0.33858pt;width:35.387501px;" version="1.1" viewbox="0 0 35.387501 10.9875" width="35.387501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9875)"> <g transform="translate(72,-63.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4392848" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.479673" y="0"> ℕ</t年代pan> </text> </g> </g> </svg>。尤其是,这意味着齐次方程<年代vg height="13.55" id="M240" style="vertical-align:-2.29482pt;width:154.52499px;" version="1.1" viewbox="0 0 154.52499 13.55" width="154.52499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="8.3269978" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="14.578498" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="18.054333" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="24.305832" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.469332" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.246437" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="48.586658" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.815071" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="74.155296" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.131966" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="89.258919" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="93.422417" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="105.22525" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="117.26564" y="0"> 0</t年代pan> </text> </g> </g> </svg>有一个独特的解决方案。此外,<年代vg height="16.75" id="M241" style="vertical-align:-2.29482pt;width:92.887497px;" version="1.1" viewbox="0 0 92.887497 16.75" width="92.887497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.75)"> <g transform="translate(72,-58.6)"> <text transform="matrix(1,0,0,-1,-71.95,60.93)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.643414" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="35.983635" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="42.235134" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.211807" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.213966" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,-9.57,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-4.82,63.92)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,-1.2,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>一些收缩吗<年代vg height="10.9875" id="M242" style="vertical-align:-0.33858pt;width:35.387501px;" version="1.1" viewbox="0 0 35.387501 10.9875" width="35.387501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.9875)"> <g transform="translate(72,-63.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.4392848" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.479673" y="0"> ℕ</t年代pan> </text> </g> </g> </svg>。因此,<年代vg height="13.55" id="M243" style="vertical-align:-2.29482pt;width:107.6px;" version="1.1" viewbox="0 0 107.6 13.55" width="107.6" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="13.890833" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="20.142332" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="24.305832" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="33.082939" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="44.423161" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.651573" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="69.991791" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="76.968468" y="0"> </t年代pan> </text> </g> </g> </svg>是可逆的,逆的诺伊曼系列<年代pan class="equation" id="eq19"> <svg height="48.262501" id="M244" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 48.262501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,48.2625)"> <g transform="translate(72,-33.39)"> <text transform="matrix(1,0,0,-1,87.26,49.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> (</t年代pan> <tspan style="font-size: 12.50px; " x="8.3234987" y="0.0099999998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="14.574999" y="0.0099999998"> /</t年代pan> <tspan style="font-size: 12.50px; " x="18.050833" y="0.0099999998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="24.302334" y="0.0099999998"> )</t年代pan> <tspan style="font-size: 12.50px; " x="28.465832" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="37.242939" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="48.58316" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="62.811573" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="74.164299" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="81.140968" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="90.143127" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,181.57,54.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,195.91,49.77)"> <tspan style="font-size: 12.50px; " x="0" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,218.23,65.71)"> <tspan style="font-size: 8.75px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,214.2,43.22)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,214.72,36.04)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.7435842" y="0"> =</t年代pan> <tspan style="font-size: 8.75px; " x="10.738704" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,230.36,49.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> −</t年代pan> <tspan style="font-size: 12.50px; " x="12.724555" y="0.0099999998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="18.976055" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,253.5,54.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,258.74,49.78)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1599998" y="0.0099999998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="10.4115" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.639914" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="35.980137" y="0.0099999998"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="42.231636" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="49.208309" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="58.210468" y="0.0099999998"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,321.12,54.95)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,326.37,49.77)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="132.12" y="0.2"> (</t年代pan> <tspan style="font-size: 12.50px; " x="136.28349" y="0.2"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="142.535" y="0.2"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="145.66075" y="0.2"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="151.91225" y="0.2"> 8</t年代pan> <tspan style="font-size: 12.50px; " x="158.16374" y="0.2"> )</t年代pan> </text> </g> </g> </svg></span><br>由于系列是一致收敛,我们有<年代pan class="equation" id="eq20"> <svg height="17.7125" id="M245" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 17.7125" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.7125)"> <g transform="translate(72,-57.83)"> <text transform="matrix(1,0,0,-1,144.05,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.866067" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,165.45,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,171.65,63.44)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,177.94,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,186.31,57.88)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,191.25,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,198.14,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.417639" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="27.956709" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="35.683563" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,240.37,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,246.56,63.44)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,252.85,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,261.23,57.88)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,266.16,60.07)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,269.58,63.2)"> <tspan style="font-size: 12.50px; " x="0" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="188.91" y="1.41"> (</t年代pan> <tspan style="font-size: 12.50px; " x="193.0735" y="1.41"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="199.325" y="1.41"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="202.45074" y="1.41"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="208.70226" y="1.41"> 9</t年代pan> <tspan style="font-size: 12.50px; " x="214.95375" y="1.41"> )</t年代pan> </text> </g> </g> </svg></span>和连续性<年代vg height="9.875" id="M246" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>之前的<年代vg height="10.325" id="M247" style="vertical-align:-0.0pt;width:12.05px;" version="1.1" viewbox="0 0 12.05 10.325" width="12.05" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</年代pan></p> <p>总之,<年代vg height="7.1624999" id="M248" style="vertical-align:-0.11285pt;width:7.5374999px;" version="1.1" viewbox="0 0 7.5374999 7.1624999" width="7.5374999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>定义为(<gydF4y2Baa href="#EEq13">3.1</gydF4y2Baa>)解决(TFDE)提供<年代vg height="9.875" id="M249" style="vertical-align:-2.29482pt;width:10.5375px;" version="1.1" viewbox="0 0 10.5375 9.875" width="10.5375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>解决(<gydF4y2Baa href="#EEq14">3.2</gydF4y2Baa>)。结合我们的研究结果与结果(<gydF4y2Baa href="#B2">3</gydF4y2Baa>,<gydF4y2Baa href="#B11">8</gydF4y2Baa>),我们已经证明了我们的主要结果,陈述如下。</p><p><年代pan class="statement" id="thm5"><span class="statement-title">定理3.4。</年代pan><i>让<年代vg height="19.5375" id="M250" style="vertical-align:-3.13504pt;width:68.212502px;" version="1.1" viewbox="0 0 68.212502 19.5375" width="68.212502" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6148071" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="21.655195" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="31.620087" y="0"> (</t年代pan> </text> <g> <path d="m -36.16,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-36.16,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-28.36,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-21.65,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="19.9125" id="M251" style="vertical-align:-3.25793pt;width:77.925003px;" version="1.1" viewbox="0 0 77.925003 19.9125" width="77.925003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.9125)"> <g transform="translate(72,-56.07)"> <text transform="matrix(1,0,0,-1,-71.95,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.853084" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="24.893473" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-37.08,64.68)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,-32.21,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> </text> <g> <path d="m -28.05,70.71 14.18,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-28.05,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,-18.74,56.25)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,-13.87,59.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="19.5375" id="M252" style="vertical-align:-3.13504pt;width:69.199997px;" version="1.1" viewbox="0 0 69.199997 19.5375" width="69.199997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,19.5375)"> <g transform="translate(72,-56.37)"> <text transform="matrix(1,0,0,-1,-71.95,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.402496" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.442884" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="32.407776" y="0"> (</t年代pan> </text> <g> <path d="m -35.37,70.71 14.51,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,-35.37,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,-27.57,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-20.86,59.55)"> <tspan style="font-size: 12.50px; " x="0" y="0"> )</t年代pan> </text> </g> </g> </svg>这样<年代vg height="13.6125" id="M253" style="vertical-align:-2.34499pt;width:35.075001px;" version="1.1" viewbox="0 0 35.075001 13.6125" width="35.075001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="6.9391651" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="11.102664" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="14.678522" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="19.87977" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.793209" y="0"> )</t年代pan> </text> </g> </g> </svg>持有人持续一致吗<年代vg height="13.125" id="M254" style="vertical-align:-1.95624pt;width:57.737499px;" version="1.1" viewbox="0 0 57.737499 13.125" width="57.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.125)"> <g transform="translate(72,-61.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.922558" y="0"> ]</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.075" id="M255" style="vertical-align:-2.72118pt;width:100.825px;" version="1.1" viewbox="0 0 100.825 14.075" width="100.825" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.075)"> <g transform="translate(72,-60.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="4.863667" y="0"> u</t年代pan> <tspan style="font-size: 12.50px; " x="11.115167" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="17.366667" y="0"> p</t年代pan> <tspan style="font-size: 12.50px; " x="27.781666" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.720833" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="38.884331" y="0"> ⋅</t年代pan> <tspan style="font-size: 12.50px; " x="42.46019" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="47.661434" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="51.574875" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="59.214207" y="0"> ⊂</t年代pan> <tspan style="font-size: 12.50px; " x="71.254601" y="0"> Ω</t年代pan> </text> </g> </g> </svg>,<年代vg height="13.125" id="M256" style="vertical-align:-1.95624pt;width:57.737499px;" version="1.1" viewbox="0 0 57.737499 13.125" width="57.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.125)"> <g transform="translate(72,-61.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.37677" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="19.417158" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="23.580658" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="29.832157" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="35.04591" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="41.922558" y="0"> ]</t年代pan> </text> </g> </g> </svg>。然后(TFDE)承认一个独特的经典解决方案和解决方案不断取决于数据以下感觉:</我><年代pan class="equation" id="eq21"> <svg height="28.9625" id="M257" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 28.9625" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,28.9625)"> <g transform="translate(72,-48.83)"> <text transform="matrix(1,0,0,-1,65.93,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.553492" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.716991" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.593641" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="30.794889" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.708328" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="40.959827" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,113.43,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> </text> <g> <path d="m 123.32,60.92 11.45,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,123.32,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,129.84,50.8)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,134.77,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,141.66,57.29)"> <tspan style="font-size: 12.50px; " x="23.51" y="4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.870001" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="37.409069" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.348232" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,192.55,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> </text> <g> <path d="m 202.44,60.92 11.45,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,202.44,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,208.95,50.8)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,213.89,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,220.08,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="17.87929" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.030766" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,250.65,54.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.88976" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,266,51.98)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,270.94,54.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,277.13,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="17.87929" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.269043" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,310.94,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,317.92,56.66)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,321.54,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> </text> <g> <path d="m 324.46,60.92 10.13,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,324.46,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,330.97,50.8)"> <tspan style="font-size: 6.25px; " x="0" y="0"> 0</t年代pan> </text> <text transform="matrix(1,0,0,-1,334.59,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,338.01,52.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6800003" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="120.48" y="-4.9200001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="124.6435" y="-4.9200001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="130.895" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="134.02075" y="-4.9200001"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="140.27225" y="-4.9200001"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="146.52374" y="-4.9200001"> )</t年代pan> </text> </g> </g> </svg></span><i></i></span></p> <p>如果<年代vg height="9.875" id="M258" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>有紧凑的支持<年代vg height="10.6875" id="M259" style="vertical-align:-0.0pt;width:11.75px;" version="1.1" viewbox="0 0 11.75 10.6875" width="11.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6875)"> <g transform="translate(72,-63.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Ω</t年代pan> </text> </g> </g> </svg>,我们可以放松的平滑的假设<年代vg height="9.875" id="M260" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和定理的证明<gydF4y2Baa href="#thm5">3.4</gydF4y2Baa>意味着以下。</p><p><年代pan class="statement" id="coro1"><span class="statement-title">推论3.5。</年代pan><i>让<年代vg height="9.9375" id="M261" style="vertical-align:-2.34499pt;width:8.4375px;" version="1.1" viewbox="0 0 8.4375 9.9375" width="8.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.9375)"> <g transform="translate(72,-64.05)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>和<年代vg height="13.4875" id="M262" style="vertical-align:-2.34499pt;width:10.675px;" version="1.1" viewbox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>满足定理的假设<gydF4y2Baa href="#thm5">3.4</gydF4y2Baa>,让<年代vg height="13.725" id="M263" style="vertical-align:-2.29482pt;width:65.737503px;" version="1.1" viewbox="0 0 65.737503 13.725" width="65.737503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.725)"> <g transform="translate(72,-61.02)"> <text transform="matrix(1,0,0,-1,-71.95,63.36)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="12.853084" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="24.893473" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.858364" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="39.021862" y="0"> Ω</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> )</t年代pan> </text> </g> </g> </svg>紧凑的支持。然后(TFDE)承认一个独特的经典解决方案和解决方案不断取决于数据以下感觉:</我><年代pan class="equation" id="eq22"> <svg height="28.9625" id="M264" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 28.9625" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,28.9625)"> <g transform="translate(72,-48.83)"> <text transform="matrix(1,0,0,-1,69.56,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="6.5390692" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="14.540989" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="18.704489" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.581139" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="30.794889" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="34.708328" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="40.947327" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,117.06,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> </text> <g> <path d="m 126.95,60.92 11.45,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,126.95,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,133.46,50.8)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,138.4,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,145.28,57.29)"> <tspan style="font-size: 12.50px; " x="23.51" y="4.9200001"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="12.027886" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.879999" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="37.419067" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="44.358234" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,196.17,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> </text> <g> <path d="m 206.06,60.92 11.45,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,206.06,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,212.58,50.8)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,217.51,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,223.7,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="17.87929" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="24.030766" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,254.28,54.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="9.88976" y="0"> Σ</t年代pan> </text> <text transform="matrix(1,0,0,-1,269.63,51.98)"> <tspan style="font-size: 6.25px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,274.57,54.16)"> <tspan style="font-size: 8.75px; " x="0" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,280.76,57.29)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.327718" y="0"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="17.866787" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="27.25654" y="0"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,314.57,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="6.9753442" y="0"> (</t年代pan> </text> <g> <path d="m 324.46,60.92 6.51,0" style="fill:none;stroke:#000000;stroke-width:0.57800001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path> </g> <text transform="matrix(1,0,0,-1,324.46,52.99)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> <tspan style="font-size: 8.75px; " x="6.511488" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,334.38,52.37)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.6899996" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="124.11" y="-4.9200001"> (</t年代pan> <tspan style="font-size: 12.50px; " x="128.2735" y="-4.9200001"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="134.52499" y="-4.9200001"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="137.65074" y="-4.9200001"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="143.90225" y="-4.9200001"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="150.15375" y="-4.9200001"> )</t年代pan> </text> </g> </g> </svg></span><i></i></span></p> <p><span class="statement" id="proof7"><span class="statement-title"><i class="statement-title">证明。</我></年代pan>所有的参数都是相同的如定理<gydF4y2Baa href="#thm5">3.4</gydF4y2Baa>现在我们可以选择除外<年代vg height="13.6125" id="M265" style="vertical-align:-2.34499pt;width:35.262501px;" version="1.1" viewbox="0 0 35.262501 13.6125" width="35.262501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="9.8148546" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="21.855244" y="0"> 1</t年代pan> </text> </g> </g> </svg>在估计<年代vg height="14.2375" id="M266" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> </g> </g> </svg>和<年代vg height="14.2375" id="M267" style="vertical-align:-3.13504pt;width:13.725px;" version="1.1" viewbox="0 0 13.725 14.2375" width="13.725" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.2375)"> <g transform="translate(72,-60.61)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-65.95,60.66)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> </g> </g> </svg>的<年代pan class="equation" id="eq23"> <svg height="36.137501" id="M268" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 36.137501" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,36.1375)"> <g transform="translate(72,-43.09)"> <text transform="matrix(1,0,0,-1,114.8,46.94)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,120.56,43.14)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,129.65,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,136.04,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,151.53,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.92136" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.122849" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.237534" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.966789" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="74.130287" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.331772" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="86.57077" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="92.822273" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="102.4996" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="114.53999" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,272.08,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,279.73,54.64)"> <tspan style="font-size: 12.50px; " x="0" y="0"> +</t年代pan> <tspan style="font-size: 12.50px; " x="11.340221" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,297.07,51.51)"> <tspan style="font-size: 8.75px; " x="0" y="0"> 2</t年代pan> </text> <text transform="matrix(1,0,0,-1,458.49,54.42)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="13.540749" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="26.043749" y="0"> )</t年代pan> </text> </g> </g> </svg></span>在定理<gydF4y2Baa href="#thm5">3.4</gydF4y2Baa>。因此,我们获得<年代pan class="equation" id="eq24"> <svg height="44.075001" id="M269" preserveaspectratio="xMaxYMin" style="vertical-align:-0pt;width:700.9375px;" version="1.1" viewbox="0 0 700.9375 44.075001" width="700.9375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,44.075)"> <g transform="translate(72,-36.74)"> <text transform="matrix(1,0,0,-1,33.44,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="10.67"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="21.33"> |</t年代pan> <tspan style="font-size: 12.50px; " x="0" y="24.5"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.51" y="19.950001"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,41.7,39.75)"> <tspan style="font-size: 8.75px; " x="0" y="0"> Ω</t年代pan> </text> <text transform="matrix(1,0,0,-1,50.79,51.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,57.18,48.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="4.3497438" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="7.2641602" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="12.07776" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,72.67,51.25)"> <tspan style="font-size: 12.50px; " x="99.029999" y="-12.25"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="-1.58"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="9.0799999"> |</t年代pan> <tspan style="font-size: 12.50px; " x="99.029999" y="12.25"> |</t年代pan> <tspan style="font-size: 12.50px; " x="128.52" y="2.52"> </t年代pan> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="11.75282" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="15.916319" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="25.568636" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="36.92136" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="43.122849" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="48.324097" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="52.237534" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="58.489033" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="69.966789" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="74.130287" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="80.331772" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="86.57077" y="0"> d</t年代pan> <tspan style="font-size: 12.50px; " x="92.822273" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="105.01" y="0.0099999998"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="117.03789" y="0.0099999998"> </t年代pan> <tspan style="font-size: 12.50px; " x="134.37" y="0.0099999998"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="143.39717" y="0.0099999998"> +</t年代pan> <tspan style="font-size: 12.50px; " x="154.73738" y="0.0099999998"> d</t年代pan> <tspan style="font-size: 12.50px; " x="160.98889" y="0.0099999998"> 我</t年代pan> <tspan style="font-size: 12.50px; " x="164.46472" y="0.0099999998"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="169.32838" y="0.0099999998"> t</t年代pan> <tspan style="font-size: 12.50px; " x="175.64" y="1.4210855e-014"> (</t年代pan> <tspan style="font-size: 12.50px; " x="179.8035" y="1.4210855e-014"> 年代</t年代pan> <tspan style="font-size: 12.50px; " x="184.66716" y="1.4210855e-014"> u</t年代pan> <tspan style="font-size: 12.50px; " x="190.91867" y="1.4210855e-014"> p</t年代pan> <tspan style="font-size: 12.50px; " x="197.17017" y="1.4210855e-014"> p</t年代pan> <tspan style="font-size: 12.50px; " x="207.58516" y="1.4210855e-014"> </t年代pan> <tspan style="font-size: 12.50px; " x="216.97491" y="1.4210855e-014"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="222.18867" y="1.4210855e-014"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="229.52792" y="1.4210855e-014"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,306.37,56.42)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> <tspan style="font-size: 8.75px; " x="10.344864" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="16.339985" y="0"> 1</t年代pan> </text> <text transform="matrix(1,0,0,-1,327.58,48.73)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="7.9400001" y="-2.51"> 为</t年代pan> <tspan style="font-size: 12.50px; " x="14.479069" y="-2.51"> </t年代pan> <tspan style="font-size: 12.50px; " x="23.856319" y="-2.51"> 为</t年代pan> </text> <text transform="matrix(1,0,0,-1,357.98,48.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,364.18,51.49)"> <tspan style="font-size: 6.25px; " x="0" y="0"> ∞</t年代pan> </text> <text transform="matrix(1,0,0,-1,370.47,48.12)"> <tspan style="font-size: 8.75px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 8.75px; " x="2.9144161" y="0"> Ω</t年代pan> <tspan style="font-size: 8.75px; " x="9.4259043" y="0"> )</t年代pan> </text> <text transform="matrix(1,0,0,-1,458.49,51.24)"> <tspan style="font-size: 12.50px; " x="0" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="10.414999" y="0"> 。</t年代pan> <tspan style="font-size: 12.50px; " x="13.540749" y="0"> 2</t年代pan> <tspan style="font-size: 12.50px; " x="19.79225" y="0"> 3</t年代pan> <tspan style="font-size: 12.50px; " x="26.043749" y="0"> )</t年代pan> </text> </g> </g> </svg></span>和索赔。</年代pan></p> <p><span class="statement" id="rem2"><span class="statement-title"><i class="statement-title">3.6的话。</我></年代pan>估计给<年代vg height="10.325" id="M270" style="vertical-align:-0.0pt;width:12.05px;" version="1.1" viewbox="0 0 12.05 10.325" width="12.05" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>显示,如果<年代vg height="9.875" id="M271" style="vertical-align:-2.29482pt;width:11.8625px;" version="1.1" viewbox="0 0 11.8625 9.875" width="11.8625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.875)"> <g transform="translate(72,-64.1)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>仅仅是连续的,我们有什么<年代vg height="16.450001" id="M272" style="vertical-align:-2.21957pt;width:95.4375px;" version="1.1" viewbox="0 0 95.4375 16.450001" width="95.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.45)"> <g transform="translate(72,-58.84)"> <text transform="matrix(1,0,0,-1,-71.95,61.1)"> <tspan style="font-size: 12.50px; " x="0" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.227454" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="14.390953" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="21.267603" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="26.468851" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="30.38229" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="34.545788" y="0"> |</t年代pan> <tspan style="font-size: 12.50px; " x="40.522224" y="0"> ≤</t年代pan> <tspan style="font-size: 12.50px; " x="52.562611" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="60.952126" y="0"> </t年代pan> </text> <text transform="matrix(1,0,0,-1,-7.08,66.1)"> <tspan style="font-size: 8.75px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 8.75px; " x="5.99512" y="0"> </t年代pan> </text> </g> </g> </svg>对于一些<年代vg height="13.55" id="M273" style="vertical-align:-2.29482pt;width:97.824997px;" version="1.1" viewbox="0 0 97.824997 13.55" width="97.824997" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0"> 1</t年代pan> <tspan style="font-size: 12.50px; " x="18.279387" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="30.319775" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="40.772282" y="0"> <</t年代pan> <tspan style="font-size: 12.50px; " x="52.812672" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="61.377228" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.428917" y="0"> /</t年代pan> <tspan style="font-size: 12.50px; " x="71.904755" y="0"> 2</t年代pan> </text> </g> </g> </svg>。然后同样的技术在<gydF4y2Baa href="#B3">13</gydF4y2Baa>,5.3节)可以用来证明(TFDE)初始条件所取代<年代vg height="13.55" id="M274" style="vertical-align:-2.29482pt;width:91px;" version="1.1" viewbox="0 0 91 13.55" width="91" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="5.9264221" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="10.089921" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="16.966572" y="0"> ,</t年代pan> <tspan style="font-size: 12.50px; " x="22.167818" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="28.419319" y="0"> )</t年代pan> <tspan style="font-size: 12.50px; " x="36.058651" y="0"> =</t年代pan> <tspan style="font-size: 12.50px; " x="48.099041" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="57.488792" y="0"> (</t年代pan> <tspan style="font-size: 12.50px; " x="61.652294" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="68.528946" y="0"> )</t年代pan> </text> </g> </g> </svg>,<年代vg height="11.1125" id="M275" style="vertical-align:-0.33858pt;width:39.737499px;" version="1.1" viewbox="0 0 39.737499 11.1125" width="39.737499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.1125)"> <g transform="translate(72,-63.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> <tspan style="font-size: 12.50px; " x="10.339981" y="0"> ∈</t年代pan> <tspan style="font-size: 12.50px; " x="22.380369" y="0"> Ω</t年代pan> </text> </g> </g> </svg>有一个独特的解决方案,甚至可能不是连续的,可以无限接近<年代vg height="13.575" id="M276" style="vertical-align:-2.26974pt;width:49.0625px;" version="1.1" viewbox="0 0 49.0625 13.575" width="49.0625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.575)"> <g transform="translate(72,-61.14)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0"> Γ</t年代pan> <tspan style="font-size: 12.50px; " x="10.114927" y="0"> ×</t年代pan> <tspan style="font-size: 12.50px; " x="20.892513" y="0"> {</t年代pan> <tspan style="font-size: 12.50px; " x="26.893953" y="0"> 0</t年代pan> <tspan style="font-size: 12.50px; " x="33.145454" y="0"> }</t年代pan> </text> </g> </g> </svg>因此不是一个经典的解决方案。</年代pan></p> <p><span class="statement" id="rem3"><span class="statement-title"><i class="statement-title">3.7的话。</我></年代pan>与上面相同的方法可用于一般time-fractional扩散方程,在那里<年代vg height="10.6625" id="M277" style="vertical-align:-0.0pt;width:22.112499px;" version="1.1" viewbox="0 0 22.112499 10.6625" width="22.112499" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6625)"> <g transform="translate(72,-63.47)"> <text transform="matrix(1,0,0,-1,-71.95,63.52)"> <tspan style="font-size: 12.50px; " x="0" y="0"> −</t年代pan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0"> Δ</t年代pan> </text> </g> </g> </svg>被均匀椭圆nondivergence形式的二阶微分算子的有界连续实值系数取决于<年代vg height="7.1624999" id="M278" style="vertical-align:-0.11285pt;width:8.7250004px;" version="1.1" viewbox="0 0 8.7250004 7.1624999" width="8.7250004" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>。</年代pan></p> <p><span class="statement" id="rem4"><span class="statement-title"><i class="statement-title">3.8的话。</我></年代pan>在[<gydF4y2Baa href="#B8">15</gydF4y2Baa>),我们已经证明解的存在性和唯一性TFDE零初始条件和零源项狄利克雷边界条件。使用与上面相同的方法,我们也可以考虑非零初始条件和重要的源项。事实上,使用双层拟设导致沃尔泰拉积分方程的第二种。然后,使用相同的参数如上所述,我们可以证明古典解的存在唯一性及其没有任何限制<年代vg height="7.1374998" id="M279" style="vertical-align:-0.10033pt;width:7.8874998px;" version="1.1" viewbox="0 0 7.8874998 7.1374998" width="7.8874998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1375)"> <g transform="translate(72,-66.29)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0"> </t年代pan> </text> </g> </g> </svg>或等边界条件(<gydF4y2Baa href="#B14">6</gydF4y2Baa>]。</年代pan></p> <h4 id="acknowledgment">承认</gydF4y2Bah4> <p>作者要感谢裁判为本文的改进的建议。</p></d我v> </div> <div class="ArticleReferences_xmlContent__p_40j"> <h4 class="ArticleReferences_references__GH2t_" id="references">引用</gydF4y2Bah4> <ol class="ArticleReferences_orderedReferences__mJr9M"> <li class="ArticleReferences_articleReference__ouEuh" id="B15"> <div class="referenceContent"> <p class="referenceText">r·麦茨勒和j . 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14"> <defs> <clippath id="a"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)"></path> </clippath> </defs> <g transform="translate(-3 -5)"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)" fill="currentColor"></path> </g> </svg></span></a> <div class="articleToolbarStatistics_articleToolbarStatistics__8_woN"> <div class="articleToolbarStatistics_column__R2GqD"> <div> <div> <span>的观点</年代pan> <div class="articleToolbarStatistics_amount__vJIDp"> 1787年</d我v> </div> <div> <span>下载</年代pan> <div class="articleToolbarStatistics_amount__vJIDp"> 1233年</d我v> </div> </div> </div> <div class="articleToolbarStatistics_column__R2GqD articleToolbarStatistics_dimensions__pe7pO"> <div> 引用</d我v> <span class="__dimensions_badge_embed__ articleToolbarStatistics_circle__p2dSc" data-doi="10.1155/2011/321903" data-style="small_circle"></span> </div> </div> <ul class="articleToolbarSocials_articleToolbarSocials__g02HH"> <li><a href="https://twitter.com/share?url=https://doi.org/10.1155/2011/321903&via=Hindawi" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="126.444 2.281 589 589"> <circle 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