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数学专业英语-How to Organize a paper (For Beginers)?

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发表于 2004-5-6 09:56:31 | 显示全部楼层 |阅读模式
< ><FONT face="Times New Roman" size=3>The usual journal article is aimed at experts and near-experts, who are the people most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There are many good ways of organizing a paper which can be learned by studying papers of the better expositors. The following suggestions describe a standard acceptable style.</FONT></P>
< ><FONT face="Times New Roman" size=3>Choose a title which helps the reader place in the body of mathematics. A useless title: Concerning some applications of a theorem of J. Doe. A. good title contains several well-known key words, e. g. Algebraic solutions of linear partial differential equations. Make the title as informative as possible; but avoid redundancy, and eschew the medieval practice of letting the title serve as an inflated advertisement. A title of more than ten or twelve words is likely to be miscopied, misquoted, distorted, and cursed.</FONT></P>
< ><FONT face="Times New Roman" size=3>The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it is feasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.</FONT></P>
<P ><FONT face="Times New Roman" size=3>It is sometimes useful to follow the introduction with a brief section that establishes notation and refers to standard sources for basic concepts and results. Normally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.</FONT></P>
<P ><FONT face="Times New Roman" size=3>The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a major theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.</FONT></P>
<P ><FONT face="Times New Roman" size=3>Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed. To achieve conceptual proofs, it is often helpful for an author to adopt an initial attitude such as one would take when communicating mathematics orally (as when walking with a friend). Decide how to state results with a minimum of symbols and how to express the ideas of the proof without computations. Then add to this framework the details needed to clinch the results.</FONT></P>
<P ><FONT face="Times New Roman" size=3>Omit any computation which is routine (i.e. does not depend on unexpected tricks). Merely indicate the starting point, describe the procedure, and state the outcome.</FONT></P>
<P ><FONT face="Times New Roman" size=3>It is good research practice to analyze an argument by breaking it into a succession of lemmas, each stated with maximum generality. It is usually bad practice to try to publish such an analysis, since it is likely to be long and uninteresting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice later on.</FONT></P>
<P ><FONT face="Times New Roman" size=3>The rudiments of grammar are important. The few lines written on the blackboard during an hour’s lecture are augmented by spoken commentary, and aat the end of the day they are washed away by a merciful janitor. Since the published paper will forever speak for its author without benefit of the cleansing sponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless appropriate.</FONT></P>
<P ><FONT face="Times New Roman" size=3>The barbarism called the dangling participle has recently become more prevalent, but not less loathsome. “Differentiating both sides with respect to x, the equation becomes---”is wrong, because “the equation” cannot be the subject that does the differentiation. Write instead “differentiating both sides with respect to x, we get the equation---,” or “Differentiation of both sides with respect to x leads to the equation---”</FONT></P>
<P ><FONT face="Times New Roman" size=3>Although the notion has gained some currency, it is absurd to claim that informal “we” has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed be banished from those parts of a paper which will be printed in italics. But fifteen consecutive pages of formality are altogether foreign to the spirit of the twentieth century, and nearly all authors who try to sustain an impersonal dignified text of such length succeed merely in erecting elaborate monuments to slumsiness.</FONT></P>
<P ><FONT face="Times New Roman" size=3>A sentence of the form “if P,Q” can be understood. However “if P,Q,R,S,T” is not so good, even if it can be deduced from the context that the third comma is the one that serves the role of “then.” The reader is looking at the paper to learn something, not with a desire for mental calisthenics.</FONT></P>                 
 楼主| 发表于 2004-5-6 09:56:44 | 显示全部楼层
<DIV class=Section1 style="LAYOUT-GRID:  15.6pt none">< 0cm 0cm 0pt; TEXT-INDENT: 21.75pt"><B><FONT face="Times New Roman">  Vocabulary<p></p></FONT></B></P></DIV><BR auto; mso-break-type: section-break" clear=all><DIV class=Section2 style="LAYOUT-GRID:  15.6pt none">< 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">preliminary </FONT>序<FONT face="Times New Roman">,</FONT>小引<FONT face="Times New Roman">(</FONT>名<FONT face="Times New Roman">)</FONT>开端的<FONT face="Times New Roman">,</FONT>最初的<FONT face="Times New Roman">(</FONT>形<FONT face="Times New Roman">)</FONT></P>< 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">eschew   </FONT>避免<FONT face="Times New Roman">                                                                                                                                                                                                                                                                                                                          </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">medieval  </FONT>中古的,中世纪的<FONT face="Times New Roman">                           </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">inflated  </FONT>夸张的<FONT face="Times New Roman">                                      </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">comprehensible  </FONT>可领悟的,可了解的<FONT face="Times New Roman">                    </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">pinpoint  </FONT>准确指出(位置)<FONT face="Times New Roman">                           </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">weave   </FONT>插入,嵌入<FONT face="Times New Roman">                                  </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">unobtrusivcly  </FONT>无妨碍地<FONT face="Times New Roman">                               </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">triviality   </FONT>平凡琐事<FONT face="Times New Roman">                                  </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">barbarism  </FONT>野蛮,未开化<FONT face="Times New Roman">                              </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">portray   </FONT>写真,描写<FONT face="Times New Roman">                                  </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">clinch   </FONT>使终结<FONT face="Times New Roman">                                         </FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">rudiment  </FONT>初步,基础</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">commentary  </FONT>注解,说明</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">janitor   </FONT>看守房屋者</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">sponge  </FONT>海绵</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">dangling participle </FONT>不连结分词</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">prevalent </FONT>流行的,盛行</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">loathsome </FONT>可恶地</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">absurd   </FONT>荒谬的</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">banish  </FONT>排除</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">sustain  </FONT>维持,继续</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">slumsiness  </FONT>粗俗,笨拙</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">monument    </FONT>纪念碑</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">calisthenics   </FONT>柔软体操,健美体操</P></DIV><BR always; mso-break-type: section-break" clear=all><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman"> <p></p></FONT></P>
 楼主| 发表于 2004-5-6 09:56:55 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B><FONT face="Times New Roman">notes<p></p></FONT></B></P>< 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 18.0pt; mso-list: l37 level1 lfo35; mso-line-height-rule: exactly"><FONT face="Times New Roman">1.       </FONT>本课文选自美国数学会出版的小册子<FONT face="Times New Roman">A mamual for authors of mathematical paper</FONT>的一节,本文对准备投寄英文稿件的读者值得一读。</P>< 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 18.0pt; mso-list: l37 level1 lfo35; mso-line-height-rule: exactly"><FONT face="Times New Roman">2.       Choose a title which helps the reader place in the body of mathematics.</FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">         </FONT>意思是:选择一个可帮助读者进入数学核心的标题。</P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">3</FONT>.<FONT face="Times New Roman"> For an example of the difference, </FONT>……<FONT face="Times New Roman">in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed.</FONT></P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">        </FONT>意思是:作为这种差别的一个例子,可参看<FONT face="Times New Roman">J.E.Littlewood A mathematician’s Miscellany</FONT>一文,在那里,他把野蛮的(令人讨厌的)证明与文明的证明这两者之间的对比很漂亮地和有趣地给予描绘出来,这里“差别”是指<FONT face="Times New Roman">conceptual proof </FONT>与<FONT face="Times New Roman">computation proof  </FONT>差别。<FONT face="Times New Roman">Portray</FONT>的意思是:“人像”,这里作动词用,作“描绘”解。</P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">4</FONT>.<FONT face="Times New Roman"> The reader wants to see the path--------not to examine it with a microscope.</FONT></P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">        </FONT>意思是:读者想知道的是有关论证的途径——而不想使用显微镜去观察。这里作者所要表达的意思是:写文章的人只需把论证的要点写出即可,无需把论证的整个分析过程写得过于冗长。</P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">5</FONT>.<FONT face="Times New Roman"> The barbarism called the dangling participles had recently become prevalent, but not less loathsome.</FONT></P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">        </FONT>意思是:一种称为“不连结分词”的句子,最近变得盛行起来,但这类句子毕竟是令人讨厌的。关于“不连结分词”,请参看Ⅱ<FONT face="Times New Roman">A</FONT>第三课注<FONT face="Times New Roman">2</FONT>。</P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">6</FONT>.<FONT face="Times New Roman"> The reader is looking at the paper to learn something, not with a desire for mental calisthenics.</FONT></P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">       </FONT>意思是:读者阅读文章是为了学到一点东西,而不是抱着一种做智力体操的愿望去阅读的。这里作者是在批评有些写文章的人使用了一些令读者摸不着头脑,而要读者去猜其真实意思的句子(例如用<FONT face="Times New Roman">”if P,Q,R,S,T”</FONT>表达<FONT face="Times New Roman">”if P,Q,R then S,T</FONT>这样的句子。)</P><P 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; LINE-HEIGHT: 24pt; mso-char-indent-count: -2.0; mso-char-indent-size: 10.5pt; mso-line-height-rule: exactly"><FONT face="Times New Roman"> <p></p></FONT></P>
 楼主| 发表于 2004-5-6 09:57:10 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><FONT face="Times New Roman"><B>Exercise</B><B><p></p></B></FONT></P>< 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">                        (Miscellaneous exercises (continued))<p></p></FONT></P>< 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly">Ⅰ<FONT face="Times New Roman">.Translate the following sentensces into English:</FONT></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">1.       </FONT>若行列式中有两行成比例,则行列式为零。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">2.       </FONT><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>矩阵和<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>矩阵是两种常用的矩阵</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">3.       </FONT>两个<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>次多项式<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>上的值相等,则<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">4.       </FONT>对任意两个多项式<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>一定存在有多项式<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>或者相等于零或者它的次数小于<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的次数。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">5.       </FONT>如果取双曲线的渐近线做为坐标轴,则双曲线方程将得到特别简单的形式。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">6.       </FONT>抛物线与椭圆和双曲线不同,它没有中心,它的另一个特殊性是它仅有一个焦点。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">7.       </FONT>通过平面上任何<FONT face="Times New Roman">5</FONT>个不同的点,其中没有<FONT face="Times New Roman">4</FONT>点同在一直线上(共线),有一条仅有一条二级曲线。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">8.       </FONT>当椭圆的长轴等于它的短轴时,它化为一圆。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">9.       </FONT>显然,无界序列不收敛。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">10.   </FONT>设<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>在<FONT face="Times New Roman">[a,b]</FONT>上连续,若存在两个点<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>,则必有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">11.   </FONT>若我们能证明连续函数级数在紧集<FONT face="Times New Roman">D</FONT>上一致收敛,则在此集上可对级数进行逐项积分。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">12.   </FONT>此定理给出了用<FONT face="Times New Roman">n</FONT>次泰勒多项式来近似代替<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>时余项太小的一种估计。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">13.   </FONT>用同样的方法,我们还可以证明定理<FONT face="Times New Roman">A</FONT>。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">14.   </FONT>定理中的条件是缺一不可的。</P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l20 level1 lfo36; mso-line-height-rule: exactly"><FONT face="Times New Roman">15.   </FONT>最后,我们再举出两个能说明问题的例子来结束文章。</P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly">Ⅱ<FONT face="Times New Roman">.Translate the following sentences into Chinese(Pay attention to the words underlined):</FONT></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l18 level1 lfo37; mso-line-height-rule: exactly"><FONT face="Times New Roman">1.       The compact and Fredholm operators lately have been receiving renewed attention because of the applications to<U> integral</U> operators and partial differential elliptic operators.</FONT></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l18 level1 lfo37; mso-line-height-rule: exactly"><FONT face="Times New Roman">2.       We dcnote by [x] the greatest <U>integral</U> part which is less than or equal to x.</FONT></P><P 0cm 0cm 0pt 34.5pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 34.5pt; mso-list: l18 level1 lfo37; mso-line-height-rule: exactly"><FONT face="Times New Roman">3.       Global analysis on manifolds has come into its owm, both in its integral and differential aspects. It is therefore desirable to integrate manifolds in analysis courses.</FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly">Ⅲ<FONT face="Times New Roman">.Translate the following passages into Chinese:</FONT></P><P 0cm 0cm 0pt 39.75pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 39.75pt; mso-list: l19 level1 lfo38; mso-line-height-rule: exactly"><FONT face="Times New Roman">1.       The concept of ordering was abstracted form various relations, such as the inequality relation between real numbers and the inclusion relation between sets. Suppose that we are given a set X={x, y, z ,</FONT>…<FONT face="Times New Roman">}</FONT>,<FONT face="Times New Roman">the relation between the elements of X, denoted by&lt;or other symbols, is called an ordering (partially ordering, semi—ordering, order relation or simply order),if the following three laws hold (i) the reflexive law, x&lt;x; (ii) the anti—symmetric law, x&lt;y and y&lt;x imply x=y and (iii) the transitive law, x&lt;y and y&lt;z imply x&lt;z.</FONT></P><P 0cm 0cm 0pt 39.75pt; TEXT-INDENT: -18pt; LINE-HEIGHT: 24pt; tab-stops: list 39.75pt; mso-list: l19 level1 lfo38; mso-line-height-rule: exactly"><FONT face="Times New Roman">2.       </FONT><FONT face="Times New Roman">Suppose we are given a relation R (usually denoted by the symbol</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">) between elements of a set X such that for any elements x and y of X, either xRy orits negation holds The relation is called an equivalence relation (on X)if it satisfies the following three conditions:</FONT></P><P 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; LINE-HEIGHT: 24pt; tab-stops: list 78.75pt; mso-list: l19 level2 lfo38; mso-line-height-rule: exactly"><FONT face="Times New Roman">(i)                   xRx</FONT></P><P 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; LINE-HEIGHT: 24pt; tab-stops: list 78.75pt; mso-list: l19 level2 lfo38; mso-line-height-rule: exactly"><FONT face="Times New Roman">(ii)                xRy implies yRx,</FONT></P><P 0cm 0cm 0pt 78.75pt; TEXT-INDENT: -36pt; LINE-HEIGHT: 24pt; tab-stops: list 78.75pt; mso-list: l19 level2 lfo38; mso-line-height-rule: exactly"><FONT face="Times New Roman">(iii)               xRy and yRz imply xRz.</FONT></P><P 0cm 0cm 0pt 42.75pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">Conditions (i),(ii),and(iii)are called the reflexive, symmetric and transitive laws respectively. Together they are called equivalence properties. The relations of congruence and similarity between figures are equivalence relations.</FONT></P><P 0cm 0cm 0pt; LINE-HEIGHT: 24pt; mso-line-height-rule: exactly"><FONT face="Times New Roman">                  </FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>
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