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Ｓtatic ['sｔætɪｋ]静力Dｙｎａmic ［daɪ＇ｎæmɪｋ］动力Expｌｉcｉｔ[ɪｋ'ｓplɪsɪｔ]•显示ａdｊ. 明确的;清楚的;直率的;详述的•Tｉme pｅrｉoｄ~ ['pɪrɪəｄ] n. 周期,期间;时期;月经；课时;（语法学)句点，句号时间长度Nlｇeｏm 几何非线性Include ａdiaｂａtic heaｔinｇ effects•~(adｉａｂatic ［，ædɪə'bætɪk] ａｄj．［物]绝热的；隔热的）［'hｉtɪŋ]~包括绝热效应Incrｅmeｎｔａtiｏｎ•(inｃrｅment [＇ɪŋｋrɪm(ə)nt] n. [数] 增量;增加;增额;盈余)增量Automatic ['ɔtə＇mætɪｋ］自动Ｓtablｅincremｅｎt estｉmatoｒ•['steɪb(ə）l] 稳定~['estɪmeɪtə］ｎ． [统计］估计量；评价者稳定增量步数Ｕnlimiｔed [ʌｎ'lɪmɪtɪｄ]无限制的Uｓe ｓcａled mass and “ｔhroughｏutstep”dｅfｉnitions•（thｒｏuｇhout ［θrʊ＇aʊt] 整个，adv．自始至终,到处；全部；pｒep. 贯穿，遍及•dｅfinitiｏｎ[deｆɪ＇nɪʃ(ə)ｎ］n. 定义;[物] 清晰度;解说) 使用前一分析步的缩放系数和“整个分析步”定义Cｏｎtacｔ［＇kɑnｔækｔ]接触Tangeｎｔial behaｖior［tæn'dʒenʃ(ə)l] ［bɪ'hｅｖjɚ]切向行为Normａl beｈａvior ['nɔrｍl]~法向行为Frictｉonlｅss [f'rɪkʃnlｅs]•（fｒｉction [＇ｆrɪkʃəｎ] n. 摩擦,[力］摩擦力)无摩擦•Penａｌty ［'pen(ə)ltɪ] ｎ．罚款,罚金；处罚罚Frictｉｏn formｕlａtｉon•（fｏｒmuｌation ［fɔːmjʊ'leɪʃn] n．构想，规划；公式化；简洁陈述)摩擦公式Directioｎaliｔy ［daɪrɛkʃə'ｎæｌəｔi]方向性Iｓotｒｏｐiｃ[,ａɪｓə'trɑpɪk]各向同性Ａnisｏｔｒoｐｉc [，ænaɪsə＇trɑｐɪk]各向异性Ｕｓe sｌｉｐ-rａte-dｅpｅndent ｄatａ•(rate [reɪt］ｎ．比率,率；速度；价格;等级•Depenｄeｎt [ｄɪ'ｐɛndəｎt] adj．依靠的；从属使用基于滑动率的数据的;取决于…的）使用依赖接触压力的数据Ｕse conｔact-prｅssurｅ-ｄepｅndｅntdaｔａ摩擦系数Frｉctioｎcoｅｆf•(coefｆ[kəｕf］ｎ．多项式系数)Ｓhｅar sｔresｓ[ʃɪə]~剪应力压力过盈Pｒessure-overｃlosure ~ [əʊvək'ｌəʊʒər]•（closure [＇kləʊʒə] n. 关闭;终止,结束•vt. 使终止）约束执行方法Ｃonstｒaｉnｔenfoｒｃeｍeｎt mｅthod•（ｅnfｏrcemeｎｔ[ɪｎ＇ｆorsｍənｔ] n. 执行，实施；强制•Ｍethod [＇mｅθəd] n. 方法；条理;类函数•adｊ. 使用体验派表演方法的）Ａllow separatiｏn ａfteｒ cｏnｔact允许接触后分离•（Ａlｌow [ə'laʊ］ｖｔ. 允许;给予；认可vi. 容许；考虑•Ｓepａratiｏn [sepə'ｒeɪʃ(ə)ｎ］n. 分离，分开;间隔,距离;[法] 分居;缺口）Coｕｐliｎｇ[＇kʌｐｌɪŋ］耦合接触领域Contact domain•（dｏmaiｎ[də(ʊ)'ｍeɪｎ]n. 领域；域名;产业;地产)Excluded sｕrfａce pａirｓ排除的表面对•（Exclude ［ɪk'skluːd; ek－] vt. 排除;排斥；拒绝接纳;逐出)Ｅxterior ［ɪk'ｓtɪərɪə； eｋ-]•adj. 外部的；表面的;外在的•n. 外部；表面；外型;外貌Ｓeｇｍenｔｓ［'ｓegm(ə)nt］•vi．分割 n．段；部分vt. 分割Aｔtriｂute aｓsiｇｎment属性指派•(Attｒibute [ə'ｔrɪbjuｔ] ｎ. 属性；特质•vｔ. 归属；把…归于•ａｓsiｇnmｅnt [ə'saɪｎməｎt］ｎ. 分配;任务;作业;功课)Macｈaｎicａl ｃonsｔrａiｎt ｆormｕ力学约束公式化lation(Macｈanical[ｍəˈkænɪkl]adj. 机械的，机械学的; 呆板的; 体力的; 手工操作的;Kineｍatｉc coｎtact method运动摩擦法•(Kｉnｅｍａｔｉc [,ｋɪnə'mæｔɪk] ａｄｊ．［力] 运动学上的,［力]运动学的)有限滑移Finite sｌｉdinｇ•(Ｆinｉｔe ['faɪnaɪt］ａｄj.有限的；限定的n.有限之物•Ｓlidｉnｇ[＇ｓlａɪdɪŋ]•Clearaｎce ［'ｋlɪrəｎｓ］n．清除；空隙过盈量•Ｐｒocｅduｒe ［prə'sidʒɚ] n. 程序，手续;步骤步骤•Freqｕｅnｃy [＇frikwəｎsｉ］ｎ. 频率;频繁频率•Inｔerval ['ɪntəv（ə)ｌ] n．间隔;间距;幕间休息间隔场输出:S，应力分量和不便量S, Sｔrｅss comｐoｎentｓ and invariａｎts•(componｅnt ［kəm＇ｐonəｎｔ］adj．组成的,构成的•n.成分;组件；［电子]元件•ｉｎvariant ［ɪn＇ｖｅərɪənt] adj. 不变的•n． [数]不变量;［计] 不变式PＥ,塑性应变分量PＥ, Plasｔic ｓtrａｉｎ cｏmｐｏnｅnts•（Pｌaｓtｉc ['plæsｔɪk]adｊ. 塑料的;(外科)造型的;可塑的;ｎ. 塑料制品；整形;可塑体PEEQ，等效塑性应变ＰEEQ, Ｅqｕｉvａlｅnt plastic sｔraｉn•（Eqｕivalｅnt [ɪ'ｋｗɪvələnt] aｄj. 等价的,相等的；同意义的•n. 等价物，相等物PEMAG，塑性应变PEMＡG，Ｐlａstｉc strａin magnｉtuｄe•(mａgｎituｄｅ［'mæɡnɪtud]ｎ．大小;量级；［地震] 震级;重要;光度LE,对数应变分量LE, Ｌogａrithmic stｒain ｃomｐｏnents•(Logarｉthmｉc ［lɔɡə'ｒɪðmɪk]ａｄｊ. 对数的位移、速度、加速度Dｉspｌａcement／Ｖelocity／Ａcｃeleration•（Velocity ［və'lɑｓəti]n. [力］速率；迅速;周转率•Ａcｃeｌeｒation [əｋsｅlə'ｒeɪʃ(ə)n] n. 加速,促进;[物] 加速度U,位移和转动Ｕ,Translａｔion and ｒotaｔiｏns•（rotation [rə(ʊ）'teɪʃ(ə)n]ｎ．旋转;循环,轮流UT,Trａｎｓlations UT,平移UＲ,Ｒotａtｉoｎs UR，旋转Foｒces/Reactioｎs作用力/反作用力•（Reａｃtｉoｎ［ｒɪ'æｋʃəｎ]n．反应，感应;反动，复古;反作用RF,Reａction ｆoｒｃes and mｏments RＦ，反作用力和力矩CF,Ｃoｎｃentrａted forces aｎｄｍｏCF，集中力和弯矩menｔs•（Cｏｎceｎtrａte ['kɑnｓn'ｔｒeｔ]vi. 集中;浓缩;全神贯注;聚集•vｔ.集中;浓缩•n. 浓缩,精选;浓缩液CＳTＲESＳ, Coｎtact strｅss CSTRESS,接触应力CDISP,接触位移CDISP, Contａcｔ displacemｅnts•（Coｎtaｃt ['ｋɑntæｋt］n. 接触，联系•ｖt.使接触•vi. 联系，接触均匀时间间隔Eveｎly spacｅd ｔiｍe intervals•(Ｅｖenlｙ['ivnlｉ] adv．均匀地;平衡地；平坦地;平等地•Sｐaced [spｅst]adｊ．隔开的；以规定距离排列的;间隔排列的•v. 隔开;空出（space的过去分词）每x个时间单位Eｖｅｒy x uｎｉtｓof tiｍe•（unit ['juｎɪt] n. 单位,单元;装置;[军］部队；部件Concenｔrateｄ force 集中力Mｏｍent 弯矩Pressure 压力表面载荷Surfaｃｅtraction•(tractｉon [＇træｋʃən] ｎ. 牵引；[机][车辆] 牵引力重力•Graviｔy ['ɡræｖəｔｉ] ｎ．重力，地心引力；严重性;庄严Lineａr ｐerturｂatioｎ线性摄动•(pｅrturbaｔion [,pɜːtə＇beɪʃ(ə)n］n. [数]［天］摄动；不安；扰乱逐个•Ｉndｉｖidｕally [ˌɪndɪˈｖɪdʒuəli］aｄv．个别地,单独地对称/反对称/完全固定Symmetｒy/Ａntisymmetry/Ｅnｃastｒｅ•(Symｍetry ['sɪｍətrｉ] n. 对称(性)；整齐,匀称Antisymmetry ['æntɪsɪmɪtrɪ] n.反对称性Encastre [en'kɑ:stər]n. 端部固定•Maintaiｎ[men'ten]•ｖｔ．维持;继续；维修；主张;供养Suｂｓequent ['sʌｂsɪkｗənｔ]•ａdｊ．后来的，随后的Actｉvｅ［'æktɪv］•adj. 积极的;活跃的;主动的；有效的;现役的Ａpplicａble ['æplɪkəbl]•adj. 可适用的;可应用的；合适的Allocatｉoｎ[，ælə'keʃən]•n. 分配,配置;安置esｔｉｍatｅ['esｔɪmeɪt]估计•vｉ. 估计，估价•ｎ. 估计,估价；判断，看法•vt. 估计,估量;判断，评价Ｐaｒalleｌｉｚａtion [pærəlɪｌaɪ'zeɪʃn]并行Use multiｐle pｒｏcessｏrｓ运行多个处理器•(ｍultiple ［'mʌltəｐl]adj. 多重的;多样的;许多的•n．倍数;[电] 并联•pｒocessor ['ｐｒɑｓɛsɚ] n. [计］处理器;处理程序；加工者精度•Precisioｎ[prɪ＇sɪʒ(ə)ｎ] n. 精度,［数］精密度;精确•adj. 精密的,精确的提交•Submit [ｓəｂ＇mɪｔ]vt. 使服从；主张；呈递•vi. 提交；服从监控•Ｍonitoｒ［＇mɒnɪtə] n．监视器;监听器；监控器;班长•vt. 监控边界条件：速度／角速度Ｖeｌociｔy/Ａngulaｒ velociｔy•(Velocitｙ［və'ｌɑｓəti］n. [力] 速率；迅速;周转率•Aｎgｕlar ['æŋgｊʊlə] aｄj.[生物］有角的；生硬的,笨拙的;瘦削的加速度／角加速度Acｃeｌｅration/Angulaｒａcceｌｅrａｔion可视化：Reｎder ['renｄə］渲染风格Wirefｒame ['ｗaiəｆreim］线框变形缩放系数Dｅfoｒmatiｏn scale ｆａctor•（Ｄｅforｍatiｏｎ[,diːfɔː＇ｍｅɪʃ(ə)n]n. 变形Uniｆorm ［'ｊunɪｆɔrｍ]一致Noｎｕmiforｍ不一致Ｖiｓible eｄges可见边•（Visｉble ［'vɪｚəbｌ] ａdj．明显的；看得见的;现有的;可得到的•n. 可见物;进出口贸易中的有形项目Exteｒior edges外部边•(Exｔerｉor [ɪk'ｓtɪərɪə; ｅk-] adj.外部的;表面的；外在的•n. 外部；表面;外型;外貌ODＢ dispｌａy oｐtｉons:细化精度Ｒeｆｉnement lｅvel•(Refinemeｎt [rɪ＇faɪｎｍənt] n. 精制;文雅；[化工]［油气][冶] 提纯•Lｅveｌ[＇lev(ə）l] n．水平；标准；水平面•ａdｊ.水平的；平坦的;同高的•ｖi.瞄准；拉平;变得平坦•vt. 使同等;对准;弄平•Ｃoａrse [kɔrs] aｄj. 粗糙的;粗俗的;下等的稍粗极粗Exｔｒa coarｓe•（Eｘtra [＇eｋｓｔrə］ａdv．特别地,非常;另外•n. 临时演员;号外；额外的事物；上等产品•ａdj. 额外的，另外收费的;特大的中•Ｍedium ['midɪəm］adj．中间的，中等的；半生熟的•ｎ. 方法；媒体;媒介;中间物Fiｎｅ稍细Extra finｅ极细辅助显示•Ideaｌiｚatｉonｓ[aɪ＇dɪəlɪ＇zｅʃəｎ] n. 理想化；理想化的事物考虑未激活的单元Ａcｃｏunｔ fｏｒ deactivated elemeｎts•(Aｃcount [ə'kaʊnt]ｎ. 账户；解释;账目,账单;理由 vi.解释;导致；报账•ｖｔ. 认为;把…视为•ｄeactivate [dｉ'æktə,ｖet]vt. 使无效；使不活动;遣散;复员Mirｒｏr/Patｔerｎ镜像／图样•(Ｐａtterｎ['pæt(ə)n] n．模式;图案;样品•ｖt.模仿；以图案装饰•vi. 形成图案Pattｅrn 模式PatteｒｎＣSＹS 阵列坐标系直角•Rｅcｔａnｇｕｌar ［rek'tæŋgjʊlə] adj. 矩形的;成直角的Ciｒcular 圆形云图绘制选项:Conｔour tｙpe 云图类型Isoｓｕrfａce 等值表面离散•Disｃｒete [dɪ'sｋriｔ] ａdj.离散的,不连续的•n.分立元件；独立部件Iｎterval type 间隔类型Ｌoｇ对数用户定义Usｅr-defined•(dｅｆｉned [dɪ'faɪnｄ] ａdj. 有定义的,确定的；清晰的,轮廓分明的•v．使明确；给．．．下定义；使..．的轮廓分明（definｅ的过去分词）显示位置Sｈｏw location•(loｃａｔion [ｌə（ʊ）'keɪʃ(ə)n] ｎ. 位置（形容词locatｉonal);地点;外景拍摄场地动画Animatｉon [ænɪ'meɪʃ（ə)n]•n. 活泼，生气；激励；卡通片绘制使用所有桢的最大最小值Usｅlimits frｏm all fｒames•(frame ［freɪｍ]n. 框架;结构;[电影］画面•vｔ. 设计；建造；陷害;使…适合•vi. 有成功希望•adｊ. 有木架的；有构架的Ａｎimａte scale ｆａctor 动画缩放系数动画:谐振Aniｍate harmoｎic•(ｈarｍonｉc ［hɑr＇mɑｎɪk] ａdj. 和声的；谐和的;音乐般的•n. [物] 谐波；和声动画选项：Fｒａme ｒate 帧频率显示帧计数Ｓhoｗ frａme coｕｎter•(counｔeｒ['kａʊnｔə］n．柜台;对立面；计数器;(某些棋盘游戏的）筹码•ｖｔ. 反击，还击;反向移动，对着干;反驳，回答•vi. 逆向移动,对着干；反驳•ａdj. 相反的•adv．反方向地;背道而驰地积分点Integraｔion point•(Ｉntegrａｔiｏn [ɪntɪ'ｇreɪʃ(ə)n]n. 集成；综合质心Centrｏid ［＇sentrɒɪd］•n. 图心Eｌｅｍent ｎodal单元结点•（ｎodal ［'ｎodl］ａdj. 节的;结的；节似的唯一结点的Unique nodal•（Uｎique [ｊʊ'nik] aｄj.独特的，稀罕的；[数]唯一的,独一无二的•n．独一无二的人或物网格：近似全局尺寸Apｐroximaｔe gｌoｂal ｓize•(Aｐｐroximate ［ə'prɑksɪｍət] vt. 近似;使…接近;粗略估计•vi. 接近于;近似于•adj.［数] 近似的；大概的曲率控制Curvaｔｕre cｏntrol•(Ｃurvatｕre ['kɜːvətʃə］ｎ.弯曲,[数] 曲率Mａxiｍuｍ deｖiation fａctoｒ最大偏离因子•（deviaｔion [diːvɪ'ｅɪʃ（ə）n] n．偏差；误差;背离按占全局尺寸的比例Bｙ fraction of ｇlobal size•(fraｃtｉon ['frækʃ(ə)ｎ］n. 分数;部分；小部分；稍微按绝对值Bｙ absolutｅ vａlue•（aｂｓｏlute ['æbｓəlut]adj. 绝对的;完全的;专制的•ｎ. 绝对；绝对事Hex 六面体六面体为主Heｘ-domｉnated•（doｍinaｔed [＇dɔmiｎｅitｉd］ａｄj. 占主导地位的；强势的;占统治地位的；[数] 受控的•v. 控制,支配;处于支配地位(dominatｅ的过去式Ｔet 四面体楔形Wedge [ｗｅｄʒ］•ｖt．楔入；挤进;楔住•vi. 楔入;挤进•n．楔子；楔形物；导致分裂的东西Tｅｃhｎique ［tek＇ｎiːk］技术•ｎ. 技巧，技术;手法Sｔｒuctureｄ['ｓｔrʌｋtʃəd］结构•adｊ．有结构的；有组织的•v.组织;构成（strucｔuｒe的过去分词）;建造Ｍediａl aｘiｓ中性轴算法Aｐｐropriatｅ[ə'ｐrəʊｐrɪət]•ａdj．适当的•vt．占用；拨出Lｉnear 线性二次Ｑuadratic [kwɑ＇dｒæｔɪk］•ａdj. [数]二次的•n. 二次方程式杂交公式Hybrid foｒmulａtiｏn•(Hybriｄ['hａɪbｒɪd] n．杂种,混血儿;混合物•ａdｊ. 混合的；杂种的•ｆoｒmulation [ｆɔːmjʊ'leɪʃn］ｎ. 构想，规划；公式化;简洁陈述Reｄuced inteｇrａtｉon 减缩积分Iｎcompａtｉbｌe modes 非协调模式Hｏurｇｌass stiffｎｅss 沙漏刚度Viｓcosiｔy 粘性Kinｅmaｔic sｐlit 运动裂纹Sｅｃonｄ-ordeｒ accuracｙ二阶精度Disｔｏrｔioｎｃontｒｏl 扭曲控制Hｏurｇｌass ｃontｒoｌ沙漏控制Enｈanced 增强Relａx stｉfｆｎess 松弛刚度Sｔiｆfｎeｓs 刚度Ｖiscous 粘性Ｃoｍbｉｎed 组合Eleｍeｎt deletiｏｎ单元删除Maｘ Dｅgrａｄaｔion 最大下降Sｃaling ｆacｔors 缩放系数Ｌiｎeａr bulk viscosiｔｙ线性体积粘性Qｕaｄratic bｕlk vｉscosｉty 二次体积粘度优化：Frｅeze ｌoad rｅｇｉons 冻结载荷区域冻结边界条件区域Freeze bouｎdaｒy cｏnditioｎregｉｏnｓDｅnsity ｕpdate sｔraｔegｙ密度更新策略Ｃｏnｓervaｔiｖe 保守Ａggresｓｉｖe 激进Ｉｎｉtiａl densｉｔy 初始密度Maxｉｍｕn change per deｓign cycle 每个设计循环的最大改变Ｃoｎverｇenｃe cｒiteriａ收敛准则Firｓt desiｇn ｃyclｅｕｓed ｔｏ首次设计循环作为评价标准evaluatｅcriｔｅｒｉaＣriteria ｔo bｅｆｕｌfｉlled 要适应的准则目标函数deltａ准则Objｅctｉve fuｎctiｏn dｅlta ｃｒｉterionＡlgoｒithm 算法Seｎｓｉｔiviｔy－bａsed 基于敏度Ｃoｎditiｏn-ｂaseｄoptimization 基于条件的优化Mｅthod 方法Matｅrial interpolation tｅｃhniqｕe 材料内插技术Vｏlume 体积Ｆrozｅn arｅa 冻结区域Ｍｅmber size 构件尺寸Mｅmolｄ cｏntrｏl 脱模控制Rotatiｏnal symmetrｙ轴对称Cyｃlｉc ｓymmeｔrｙ循环对称Opｅratｏr 运操作符Coｍｐaｒisｏn oｐｅrａtion 对比操作Previous iｔerａｔion 前一次迭代Modifｙ rｅfeｒence ｖalｕe by ｆacto按因子修改参考值ｒ有材料增加引起的位移Displaceｍent ｂy addｉｔiｏn of ｍaｔｅriaｌRｅducｔｉoｎ减少Tｏｔaｌａbsolute displaｃement 总觉得位移Equｉvalent stｒesｓ等效应力Eｑuivalｅnｔ stress of restriｃted poｉ任务区域内约束点的等效应力nｔｓiｎ tasｋ rｅgiｏn载荷里的速度。
TB近似参数对称性摘要：利用3种近似对称性方法(近似Lie对称性法、近似Noether对称性法和近似Mei对称性法)研究典型微扰力学系统的一阶近似对称性和近似守恒量。
结果表明, 利用近似Lie 对称性法找到的6个一阶近似对称性和近似守恒量与利用近似Noether对称性法找到的相同, 而利用近似Mei对称性法只能找到其中5个一阶近似对称性和近似守恒量。
Approximate Lie Symmetries, Approximate Noether Symmetries and Approximate Mei Symmetries of Typical Perturbed Mechanical SystemAbstract: Three methods, which are approximate Lie symmetry method, approximate Noether symmetry method and approximate Mei symmetry method, are adopted to study the first order approximate symmetries and approximate conserved quantities of a typical perturbed mechanical system. Six identical first order approximate symmetries and approximate conserved quantities of the typical perturbed mechanical system are obtained by approximate Lie symmetry method and approximate Noether symmetry method, but only five of them can be obtained by approximate Mei symmetry method.Key words: perturbed mechanicalsystem approximate Liesymmetry approximate Noethersymmetry approximate Meisymmetry approximate conservedquantity分析力学中研究力学系统的对称性与守恒量有3种对称性方法[1-2]: Lie对称性法、Noether对称性法和Mei对称性法。
V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partition ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Normal line ：法线Normal vector ：法向量Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Laplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyperboloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauchy’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
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V alue of function ：函数值V ariable ：变数V ector ：向量V elocity ：速度V ertical asymptote ：垂直渐近线V olume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean V alue Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected V alued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme V alue Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differentia l equation ：微分方程partial ：偏微分方程Differentia tion ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean V alue Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
SPSS术语中英文对照【常用软件】SPSS术语中英文对照Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interac tion Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 一般线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。
abbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 抛物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲; 圆锥曲conic section 二次曲; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer's rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre's theorem 棣美弗定理De Morgan's law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法)determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns(variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler's formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of… 以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat's last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron's formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder's Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke's law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分)invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分)kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L'Hospital's rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami's law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直线line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度measurement 量度mechanical energy 机械能median (1)中位数；(2)中线meet 相交；相遇mensuration 计量；求积法method 方法method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法；换元法method of successive substitution 逐次代换法; 逐次调替法method of superposition 迭合法metric unit 十进制单位mid-point 中点mid-point formula 中点公式mid-point theorem 中点定理million 百万minimize 极小minimum point 极小点minimum value 极小值Minkowski Inequality 闵可夫斯基不等式minor (1)子行列式；(2)劣；较小的minor arc 劣弧；小弧minor axis 短轴minor of a determinant 子行列式minor sector 劣扇形；小扇形minor segment 劣弓形；小弓形minus 减minute 分mixed number(fraction) 带分数modal class 众数组mode 众数model 模型modulo (1)模; 模数; (2)同余modulo arithmetic 同余算术modulus 模; 模数modulus of a complex number 复数的模modulus of elasticity 弹性模(数)moment arm (1)矩臂; (2)力臂moment of a force 力矩moment of inertia 贯性矩momentum 动量monomial 单项式monotone 单调monotonic convergence 单调收敛性monotonic decreasing 单调递减monotonic decreasing function 单调递减函数monotonic function 单调函数monotonic increasing 单调递增monotonic increasing function 单调递增函数motion 运动movable pulley 动滑轮multinomial 多项式multiple 倍数multiple angle 倍角multiple-angle formula 倍角公式multiple root 多重根multiplicand 被乘数multiplication 乘法multiplication law (of probability) (概率)乘法定律multiplicative inverse 乘法逆元multiplicative property 可乘性multiplicity 重数multiplier 乘数；乘式multiply 乘multi-value 多值的mulually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根；n次方根n the root of unity 单位的n次根Napierian logarithm 纳皮尔对数; 自然对数natural logarithm 自然对数natural number 自然数natural surjection 自然满射necessary and sufficient condition 充要条件necessary condition 必要条件negation 否定式negative 负negative angle 负角negative binomial distribution 负二项式分布negative index 负指数negative integer 负整数negative number 负数negative vector 负向量; 负矢量neighborhood 邻域net 净(值)net force 净力Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's law of motion 牛顿运动定律Newton's method 牛顿方法n-gon n边形nonagon 九边形non-collinear 不共线non-commutative 非交换的non-linear 非线性non-linear equation 非线性方程non-negative 非负的non-reflexive 非自反的。
SYMMETRY STUDIESMARLOS VIANAEYE RESEARCH INSTITUTE,UNIVERSITY OF ILLINOIS AT CHICAGO1.IntroductionSymmetry studies are aimed at the statistical/probabilistic assessment of experimental hypotheses related to generic processing of related stimuli,ω,by a sensory mechanisms,H,when the relationship among these stimuli is characterized by certain algebraic structures.This leads to the notion of representing and quan-tifying the processing of symmetry by the interfacing mechanism H in response to symmetrically related stimuli.The mechanism or interface looks at symmetrically displaced copies ofω,represented here by gω, processes or accounts for their symmetries,and returns a measurable quantity H(gω).If the mechanism correctly processes the symmetry displacement represented by the transformation g,then,the association ω→g′H(gω),where g′stands for the inverse of the transformation represented by g,should in principle be a constant operator.Here is an example:letωrepresent the set object△and gωthe object▽,the result of a half-circle rotation.If H processes g(the symmetry operation)accordingly and without statistical error, then the stable operation is represented by−−−→△.−−−→▽g′−−−→▽H△gThe probabilistic aspect comes from the natural fact that H processesωcorrectly only up to a probabil-ity law-that is,the response H(gω)follows a probability law,in addition to the randomness inωalone. In one(linear)example we then have experimental data on Y=U′HUω,U follows a probability law in G,ωa probability law in its natural space and we want to deconvolute the law of H,based on observables from Y.In this formulation,the quantification of processing and representation of symmetry is inclusive of bio-logical,sensory,or cognitive responses to properly defined sets(G)of algebraic operations(g)acting on the inputωto a given process(or directly on a given data structure).Technically,as detailed elsewhere,ωis a set and G is an algebraic group acting on it,the action of which is represented here by gω.The amount of symmetry submitted to the interface H may be characterized by(indexes related to)the size of the class of operations in G acting on the objectω(e.g.,Rosen(1995)),or by defining a suitable distance between the symmetrically displaced copies gωof the objectωand the original objectω,or by investigating the proper-ties of the operatorω→g′H(gω).This formulation leads,in particular,to the notion of uniformly random permutations and to its usefulness as a method of generating the stimuli data gωfor experimental symmetry studies.Preliminary applications of random permutation matrices include the analysis of multivariate data with circular coordinates,such as those obtained from corneal topography maps.The following are selected references.2MARLOS VIANA EYE RESEARCH INSTITUTE,UNIVERSITY OF ILLINOIS AT CHICAGOReferencesAasen,T.(1993),‘Chaos theory applied to caloric response of the vestibular system’,Computer and Bio-medical Research26,556–567.Crosby,A.W.(1997),The Measure of Reality-Quantification and Western Society1250-1600,Cambridge, New York.Detrelova,D.(1998),‘Bilateral retinal detachment in marfan’s syndrome’,European Journal of Ophthalmolgy 8(2),102–105.Diaconis,P.(1988),Group Representation in Probability and Statistics,IMS,Hayward,California. Eaton,M.L.(1989),Group Invariance Applications in Statistics,IMS-ASA,Hayward,California. Enquist,M.&Arak,A.(1994),‘Symmetry,beaty and evolution’,Nature371,169–72.Hannan,E.J.(1965),‘Group representations and applied probability’,Journal of Applied Probability2,1–68. Hoffman,W.(1966),‘The lie algebra of visual perception’,Journal of Mathematical Psychology3,65–98. James,A.T.(1954),‘Normal multivariate analysis and the orthogonal group’,Annals of Mathematical Statistics25,40–75.Rosen,J.(1995),Symmetry in Science,An Introduction to the General Theory,Springer-Verlag,New York. Smith,S.(1998),‘Unilateral alpha reactivity:an electroencephalographicfinding in albinism’,Journal of Clinical Neurophysiology15(2),146–149.Swaddle,J.P.(1999),‘Visual signalling by assymetry:a review of perceptual processes’,Philos Trans R Soc Lond B Biol Science.Viana,M.A.G.,Olkin,I.&McMahon,T.(1993),‘Multivariate assessment of computer analyzed corneal topographers’,Journal of the Optical Society of America-A10(8),1826–1834.Wagemans,J.(1999),‘Parallel visual processes in symmetry perception:Normality and pathology’,Docu-menta Ophthalmologica95,359–70.Weyl,H.(1952),Symmetry,Princeton University Press,Princeton.Wijsman,R.A.(1990),Invariant Measures on Groups and Their Use in Statistics,Vol.14,Institute of Mathematical Statistics,Hayward,California.Wood,J.M.&Bullimore,M.A.(1996),‘Interocular differences in visual function in normal subjects’, Ophthalmic Physiol Opt16(6),507–512.。
(完整版)微积分术语中英文对照微积分术语中英文对照A、B:Absolute convergence :绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：原函数,反导数Approximate integration ：近似积分（法）Approximation ：逼近法by differentials :用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant :任意常数Arc length ：弧长Area ：面积under a curve :曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积 of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal :水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate :坐标轴Axes of ellipse ：椭圆之对称轴Binomial series ：二项式级数Binomial theorem:二项式定理C：Calculus ：微积分differential :微分学integral ：积分学Cartesian coordinates :笛卡儿坐标一般指直角坐标Cartesian coordinates system :笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西中值定理Chain Rule ：链式法则Circle ：圆Circular cylinder ：圆柱体，圆筒Closed interval ：闭区间Coefficient ：系数Composition of function :复合函数Compound interest ：复利Concavity :凹性Conchoid :蚌线Conditionally convergent：条件收敛Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point :在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function :连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence :收敛数列series ：收敛级数Coordinates：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes :坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point :临界点Cubic function :三次函数Curve ：曲线Cylinder：圆筒，圆柱体, 柱面Cylindrical Coordinates :圆柱坐标D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial :多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变化率 right—hand :右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation :微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial :偏微分法term by term ：逐项求导法(完整版)微积分术语中英文对照Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance :距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral :二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分E、F、G：Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid :外摆线Equation :方程式Even function ：偶函数Expected Valued ：期望值Exponential Function :指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem :极值定理Factorial :阶乘First Derivative Test :一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions :分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series :几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式H：Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative :高阶导数Higher mathematics 高等数学Horizontal asymptote :水平渐近线Horizontal line ：水平线Hyperbola :双曲线Hyperboloid ：双曲面I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：反常积分, 广义积分Increasing，Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变量Indeterminate from ：不定型Inequality ：不等式Infinite point :无穷极限点Infinite series ：无穷级数Inflection point :反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积函数Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中值定理Interval :区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral :逐次积分L:Laplace transform ：Laplace 变换Law of sines：正弦定理Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative :左导数Left—hand limit :左极限Lemniscate ：双钮线Length ：长度Level curve :等高线L’Hospital’s rule : 洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation :线性方程式Linear function :线性函数Linearity :线性Linearization :线性化Line in the plane :平面上之直线Line in space :空间之直线Local extreme ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm :对数Logarithmic function ：对数函数M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals :重积分Multiplier ：乘子Natural exponential function :自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector :法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval :开区间Optimization problems :最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal :正交的(完整版)微积分术语中英文对照P、Q：Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：平行线Parameter :参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions :部分分式Partial integration :部分积分Partition ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form :点斜式Power function ：幂函数Product :积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number :实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum :黎曼和Right-hand derivative ：右导数Right—hand limit ：右极限Root ：根S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative :二阶偏导数Sector ：扇形Sequence ：数列Series :级数Set :集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant, Oblique asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface :平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing :严格递减Strictly increasing ：严格递增Substitution rule ：替代法则Sum ：和Surface ：曲面Surface integral :面积分Surface of revolution ：旋转曲面Symmetry ：对称T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Taylor’s formula ：泰勒公式Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals :三角积分Trigonometric substitutions :三角代换法Tripe integrals ：三重积分V、X、Z:Value of function ：函数值Variable ：变量Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume :体积X-axis :x轴X -coordinate :x坐标X -intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial :多项式的零点。
武汉大学生命科学学院2012－2013学年度第一学期期末考试《生物化学》试卷（A卷）正确答案C o l l e g e o f L i f e S c i e n c e s,W u h a n U n i v e r s i t yB i o c h e m i s t r y F i n a l E x a m(2013.1.9)Name__________Student No.Major__________Score__________ 1．Multiple choice (1 point each, 25 points total)__1)According to the fluid mosaic model of cell membranes, which type of molecule spans the membrane from its inner to outer surface?A.CholesterolB.PhospholipidsC.ProteinD.triacylglycerol__2)If a mutant enzyme is found to have the same K m but a lower V max, it is most likely that energy of the transition stateA.has remained the sameB.has decreasedC.has increasedD.has decreased substrate binding__3)In an erythrocyte undergoing glycolysis, a sudden increase in the concentration of the following would lead to the increase of glycolysis EXCEPTA.[AMP]B.[Citrate]C.[fructose-1,6-bisphosphate]D.[fructose-2,6-bisphosphate]__4). Acetyl groups (e.g. for lipid biosynthesis) are transported out of the mitochondria indirectly usingA.acetyl carnitine.B.acetyl phosphate.C.citrate.D.malate.__5). The source of oxygen for O2 production during photosynthesis by higher plants isA. CO2B. HCO3 -C. H2OD. ATP__6). HMG-CoA reductase is the rate limiting enzyme in synthesis of:A.low molecular weight triglyceridesB.cholesterolC.High molecular weight triglyceridesD.Chylomicrons__7). The final electron acceptor in the electron transport chain isA.FADB.NAD+C.CoQD.Oxygen__8). Absorbed lipid is transported away from the gut mucosal cells in the form ofA.VLDLB.LDLC.HDLD.Chylomycrons__9). During DNA replication the negative supercoiling of the bacterial chromosome is maintained by the action ofA.DNA gyrase.B.DnaB helicase.C.DNA ligase.D.Primase.__10). Which of the following is true?A.Liquid and solid water have the same number of hydrogen bonds per molecule ofwaterB.Liquid water has more hydrogen bonds per molecule than does solid waterC.Solid water has more hydrogen bonds per molecule than does liquid waterD.Hydrogen bonding is a major factor in the structure of gaseous water__11). Electrons from complex I are transferred directly to which of thefollowing electron transport components?plex IIplex IIIC.cytochrome cD.ubiquinone12). Which diagram best depicts the state immediately following peptide bond formation between the initiating met and the second amino acid (aa2) of the peptide prior to the first translocation step (movement of the ribosome by one codon toward the 3' end)?__13). For the first five steps of glycolysis, the appropriate sequence of enzymes is:A.phosphofructokinase (PFK).B.hexokinase / glucokinase.C.fructose bisphosphate aldolase.D.Phosphoglucoisomerase.E.triose phosphate isomeraseA. A, C, B, E, DB. B, D, C, E, AC. B, D, A, C, ED. D, B, A, C, E__14). DNA replication originates at the indicated origin of replication. Which segment(s) serve as template(s) for lagging strand synthesis?A. B and DB. A and CC. A and BD. A and D__15). Which of the following is meant by the statement that glucose and mannose are epimers?A. One is an aldose and the other is a ketoseB. One is a pyranose and the other is a furanoseC. They are mirror images of each otherD. They differ only in the configuration about one carbon atom16).The first three reactions of β-oxidation of saturated fatty acids are analogous to which sequence of metabolic reactions discussed in the TCA cycle?A. succinate → fumarate → malate → oxaloacetateB. isocitrate →α-ketoglutarate → succinate → fumarateC. oxaloacetate → citrate → isocitrate →α-ketoglutarateD. phosphoenolpyruvate → pyruvate → acetyl-CoA → citrate__17). If glucose was labeled with 14C at carbon #6, after going through glycolysis, which carbon of pyruvate would have the 14C?A. carboxyl carbonB. ketone carbonC. methyl carbonD. the 14C would be lost as CO2__18). The main source(s) of NADPH for fatty acid biosynthesis is (are):A. TCA cycle.B. oxidative phosphorylation.C. malic enzyme and the pentose phosphate pathway.D. glycolysis.__19). For an enzyme-catalyzed reaction, the initial velocity was determined at two different concentrations of the substrate. Which of the following would be closest to the value of K m?[S] (mM)Vo(mM/min)1.02.04.0 2.8A.0.17 mMB.2.7 mMC.0.60 mMD.1.7 mM20). Which amino acids are most likely to be found in the transmembrane spanning domain of a protein such as glycophorin?A. Serine and threonineB. Glutamine and asparagineC. Lysine and arginineE.Leucine and isoleucine21). An obese individual is brought to the emergency room by a concerned friend. The patient has been on a self-imposed "starvation diet" for four months, and has lost 60 pounds while consuming only water and vitamin pills. If extensive blood studies were performed, which of the following would be expected to be elevated?A. Acetoacetic acidB. AlanineC. BicarbonateD. Glucose22) What is the importance of the purine nucleoside cycle?A. It participates in the conversion of purines into pyrimidines.B. It generates fumarate, which inhibits the formation of AMP from IMP.C. It provides fumarate as an anaplerotic enhancement of the citric acid cycle.D. It is used in energy metabolism in the liver.23) Fatty acids are mobilized from adipose cells in response to all of the hormones EXCEPT:A. ACTH (adrenocorticotropic hormone).B. glucagon.C. insulin.D. epinephrine (adrenaline).__24). The complete oxidation of a molecule of Palmitoyl-CoAin respiring mitochondria to CO2 and H2O would cause a net gain of how many ATP?A.130B.128C.108D.102__25) The following compounds represent intermediates in β-oxidation:The order of generation of these intermediates in one cycle of β-oxidation is:A. III, IV, I, IIB. III, I, IV, IIC. III, IV, II, ID. III, II, I, IV2. Match with the appropriate subcellular locations (0.5 point each, 5 points total)1) Glyoxylate cycle ___F____2) Pentose phosphate pathway ___A____3) Glycolysis ___A____4) -oxidation ___G____5) The electron transport chain ___C____6) Pyruvate dehydrogenase ___G____7) Ornithine transcarbamoylase ___G____8) Final step of urea formation ___A____9) Fatty acid biosynthesis __A____10) Insulin receptor ___D____A. CytoplasmB. LysosomeC. Inner mitochondrial membraneD. Cell membraneE. Endoplasmic reticulumF. GlyoxysomeG Mitochondrial matrixH. Nuclear area or nucleoidI. Cell wall3. Fill-in (0.5 point for each blank, 15 points total)1)Animal cells have evolved two important shuttles to transfer NADH across the innermitochondrial membrane:_the glycerol phosphate shuttle __ and_ the malate-aspartate shuttle2)The two nitrogen and one carbon atoms in urea synthesis (urea cycle) are from _ NH4+,CO2, and aspartate __.3)____ Glutamate ___________ and ___ glutamine ____ are the primary nitrogen donormolecules in a transamination reaction.4)Isoprenoids are a large group of biomolecules with repeating units derived fromisopentenyl pyrophosphate.There are two types of isoprenoids, _ terpene_ and_ cholestrol_.5)Nitrogen fixation is the formation of NH4+ from N2.6)The first law of thermodynamics states that the total energy of an isolated system isc onserved /unchanged/constant. The second law states that the universe tends to becomemore disorganized/disordered .7)In photosynthesis, electrons flow from H2O to NADP+, driven by light energy absorbedat the reaction centers.8)Most aspartic proteases have a tertiary structure consisting of two lobes (N-terminal andC-terminal) with approximate two-fold symmetry. But HIV-1 protease is a homodimer, with each subunit contributing a catalytic Asp residue.9)The thousands of proteins in cells can be separated and purified on the basis of size andelectrical charge . Proteins tend to be least soluble at a pH value close to their isoelectric point .10)Increasing ionic strength at first increases the solubility of proteins, then decreases it.Protein subunit interactions depend on weak forces. Separation can be achieved with treatment of (name two conditions):_______, ________.- extreme pH- 8M urea- 6M guanidine HCl- high salt concentration (usually ammonium sulfate)11) The major fuel depots in animals are glycogen in liver and muscle; triacylglycerols in adipose tissue; and protein, mostly in skeletal muscle. The usual order of preference for use of these is glycogen > triacylglycerol > protein12) In ETC, c omplex II does not directly contribute to the proton concentration gradient across the inner mitochondrial membrane. Instead, it supplies electrons from the oxidation of succinate midway along the ETC.4. Definition (2.5 points each, 15 points total)1)Prosthetic group (as in enzymology): tightly bound co-enzyme2)Transition state analogs:Transition state analogs (TSAs) are stable molecules that arechemically and structurally similar to the transition state3)Proteomics: large scale study of proteins4)The Cori Cycle: the metabolic pathway in which lactate produced by anaerobic glycolysisin the muscles moves to the liver and is converted to glucose, which then returns to the muscles and is converted back to lactate5)Allosteric regulation is the regulation of an enzyme or other protein by binding aneffector molecule at the protein's allosteric site (that is, a site other than the protein'sactive site).6)Cell signalling is part of a complex system of communication that governs basic cellularactivities and coordinates cell actions.5. Short questions (4 points each, 20 points total)1) The biosynthesis and degradation pathways are different. Please discuss the major differences between fatty acid breakdown and biosynthesis●Intermediates in synthesis are linked to-SH groups of acyl carrier proteins (as compared to -SH groups of CoA)●Synthesis in cytosol; breakdown in mitochondria●Enzymes of synthesis are one polypeptide, the fatty acid synthase●Biosynthesis uses NADPH/NADP+; breakdown uses NADH/NAD+2) Name at least three different modes of enzyme regulation you can think of. Which type of modification of the protein is required in each case?Is the modification reversible or irreversible?(1) Allosteric control. Binding (at a regulatory site) of a regulatory molecule in addition to the substrate. Reversible.(2) Proteolytic activation. Peptide bonds in the protein are cleaved (hydrolysed) by a protease. Irreversible.(3) Reversible covalent modification, e.g. phosphorylation. Phosphorylation is catalysed by protein kinases, which transfers a phosphate group from ATP to a Ser, Thr or Tyr residue on the enzyme. The reverse, i.e. removal of the phosphate group is performed by protein phosphatases (or protein phosphorylases).(4) Binding of regulatory protein. Reversible.(5) Gene regulation. Does not affect the properties of the enzyme, but the amount.3) What would be the consequences of a carnitine deficiency for fatty acid oxidation?A carniting deficiency would presumably result in defective or limited transport of fatty acids into the mitochondrial matrix and reduce rates of fatty acid oxidation.4) What are the anaplerotic, or “Filling Up” reactions for TCA cycle? Please give anexample.Anaplerotic reactions replenish the TCA cycle intermediates. Examples include PEP carboxylase and pyruvate carboxylase, both of which synthesize oxaloacetate frompyruvate.5) Make a list of the ways that DNA replication in eukaryotes differs from transcriptionin prokaryotes.1.Timing is limited to a specific period referred to as the S phase (vs throughout lifecycle).2.Replication rate is slower, approximately 50 nucleotides/sec per rep fork (vs1000).3.Multiple replicons (vs one) speed overall replication to just a few hours.4.Okazaki fragments are 100-200 nucleotides long. (vs 1000-2000)6. Questions (10 points each, 20 points total)1)Demonstrate your knowledge of biochemistry by showing the sequence(with structures) of reactions u sing pyruvate (labeled with 14C in its keto group) via the pyruvate dehydrogenase reaction and the TCA cycle. Where would the carbon label be at the end of one turn of the TCA cycle? Where would the carbon label be at the end of the second turn of the cycle?The carbonyl carbon of acetyl-CoA is fully retained through on turn of the cycle, but is lost completely in a second turn of the cycle.2) The sequence of reactions from glucose to pyruvate is similar in most organisms and most types of cells. In contrast, the fate of pyruvate is variable. Discuss the fate of pyruvate under different metabolic conditions. Please show the biochemical reactions involved. 1) in yeast and several other microorganisms; 2) in muscle during intense activity; 3) inside mitochondria by the oxidative decarboxylation of pyruvate; 4) converstion to oxaloacetate; 5) transaminated to form Alanine.Answer-1. Ethanol is formed from pyruvate in yeast and several other microorganisms. The first step is the decarboxylation of pyruvate. This reaction is catalyzed by pyruvate decarboxylase, which requires the coenzyme thiamine pyrophosphate.+.2. Lactate is formed from pyruvate in a variety of microorganisms in a process called lactic acid fermentation. The reaction also takes place in the cells of higher organisms when the amount of oxygen is limiting, as in muscle during intense activity. The reduction of pyruvateby NADH to form lactate is catalyzed by lactate dehydrogenase.Figure-2- showing the conversion of pyruvate to Lactate3. Acetyl co A – Only a fraction of the energy of glucose is released in its anaerobic conversion into ethanol or lactate. Much more energy can be extracted aerobically by means of the citric acid cycle and the electron-transport chain. The entry point to this oxidative pathway is acetyl coenzyme A (acetyl CoA), which is formed inside mitochondria by the oxidative decarboxylation of pyruvate.4. Oxaloacetate- Pyruvate can be converted to oxaloacetate. Mitochondrial pyruvate carboxylase catalyzes the carboxylation of pyruvate to oxaloacetate, an ATP-requiring reaction in which the vitamin biotin is the coenzyme. Biotin binds CO2 from bicarbonate as carboxybiotin prior to the addition of the CO2 to pyruvate.Figure –3- showing the conversion of pyruvate to oxaloacetateThe Oxaloacetate can be subsequently used for the synthesis of Aspartate, phosphoenol pyruvate or be utilized in the TCA cycle depending upon the need of the cell.5. Alanine- Pyruvate can be transaminated to form Alanine as per need.Figure-4- showing the conversion of Pyruvate to Alanine by transamination 课程答案网课刷课flyingjgh。
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It is negative if the center of curvature is to the left of the vertex.如果曲率中心在最高点的右边,曲率值为正,如果曲率中心在最高点的左边,则曲率为负image plane像平面Ray Aberration光线相差tangential direction切线方向sagittal direction径向paraxial focus旁轴的Marginal边缘的spherical aberration球面像差Optimization Setup最优化调整variable变量mathematical sense数学角度MFE= Merit Function Editor, Adding constraints增加约束focal length焦矩长度operand操作数the effective focal length有效焦矩primary wavelength主波长initiate开始spot diagram位图表Airy disk艾里斑axial chromatic aberration轴向色差with respect to关于至于exit pupil出射光瞳OPD=optical path difference光学路径差diffraction limited衍射极限chromatic aberration色差chromatic focal shift色焦距变换paraxial focus傍轴焦点axial spherical aberration轴向球差(longitudinal spherical aberration 纵向球差:沿光轴方向度量的球差) lateral spherical aberration垂轴球差(在过近轴光线像点A‵的垂轴平面内度量的球差)coma、comatic aberration彗差meridional coma子午彗差sagittal coma弧矢彗差astigmatism像散local coordinate system本地坐标系统meridional curvature of field子午场曲sagittal curvature of field弧矢场曲decentered lens偏轴透镜orthogonal直角的垂直的conic section圆锥截面account for说明,占有,得分stigmatic optical system无散光的光学系统Newtonian telescope牛顿望远镜parabolic reflector抛物面镜foci焦距chromatic aberration,色差superpose重迭parabola抛物线spherical mirror球面镜RMS=Root Mean Square均方根wavefront波阵面spot size光点直径Gaussian quadrature高斯积分rectangular array矩阵列grid size磨粒度PSF=Point Spread Function点扩散函数FFT=Fast Fourier Transform Algorithm快速傅里叶变换Cross Section横截面Obscurations昏暗local coordinates局部坐标系统vignette把…印为虚光照Arrow key键盘上的箭头键refractive折射reflective反射in phase同相的协调的Ray tracing光线追迹diffraction principles衍射原理order effect式样提出的顺序效果energy distribution能量分配Constructive interference相长干涉dispersive色散的Binary optics二元光学phase advance相位提前achromatic single消色差单透镜diffractive parameter衍射参数Zoom lenses变焦透镜Athermalized lenses绝热透镜Interferometers干涉计Beam splitter分束器Switchable component systems可开关组件系统common application通用symmetry对称boundary constraint边界约束multi-configuration (MC) MC Editor (MCE) perturbation动乱,动摇index accuracy折射率准确性index homogeneity折射率同种性index distribution折射率分配abbe number离差数Residual剩余的Establishing tolerances建立容差figure of merit质量因子tolerance criteria公差标准Modulation Transfer Function (MTF)调制传递函数boresight视轴，瞄准线Monte Carlo蒙特卡洛Tolerance operands误差操作数conic constant ]MC1"{_qT 圆锥常数astigmatic aberration像散误差Mechanical tilt机械倾斜，机械倾角Tolerance Data Editor (TDE)公差资料编辑器compensator补偿棱镜estimated system performance预估了的系统性能iteratively反复的，重迭的statistical dependence统计相关性sequential ray trace model连续光线追迹模型imbed埋葬，埋入multiple多样的，多重的，若干的Non-Sequential Components不连续的组件Corner cube角隅棱镜，三面直角透镜Sensitivity Analysis灵敏度分析Faceted reflector有小面的反射镜emit发射，发出nest嵌套overlap交迭outer lens外透镜brute force强力seidel像差系数aspect ratio长宽比MRA边缘光线角MRH边缘光线高度asynchronous不同时的,异步Apodization factor变迹因子hexapolar六角形dithered高频脉冲衍射调制传递函数（DMTF），衍射实部传递函数（DRTF），衍射虚部传递函数（DITF），衍射相位传递函数（DPTF），方波传递函数（DSWM）logarithmic对数的parity奇偶% Uc,I e longitudinal aberrations 纵向像差赛得系数:球差（SPHA，SI）彗差（COMA，S2），像散（ASTI，S3），场曲（FCUR，S4），畸变（DIST，S5），轴向色差（CLA，CL）和横向色差（CTR，CT）.横向像差系数:横向球差（TSPH），横向弧矢彗差（TSCO），横向子午彗差（TTCO），横向弧矢场曲（TSFC），横向子午场曲（TTFC），横向畸变（TDIS）横向轴上色差（TLAC）。
Partial and Approximate Symmetry Detection for 3D GeometryNiloy J.Mitra Stanford UniversityLeonidas J.Guibas Stanford UniversityMark Pauly ETH Z¨u rich0.02-0.02Figure 1:Symmetry detection on a sculpted model.From left to right:Original model,detected partial and approximate symmetries,color-coded deviations from perfect symmetry as a fraction of the bounding box diagonal.Abstract“Symmetry is a complexity-reducing concept [...];seek it every-where.”-Alan J.PerlisMany natural and man-made objects exhibit signi ficant symmetries or contain repeated substructures.This paper presents a new al-gorithm that processes geometric models and ef ficiently discovers and extracts a compact representation of their Euclidean symme-tries.These symmetries can be partial,approximate,or both.The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space.A clustering stage extracts potential signi ficant symmetries of the object,followed by a veri-fication step.Based on a statistical sampling analysis,we provide theoretical guarantees on the success rate of our algorithm.The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations,including shape compression,segmentation,consistent editing,symmetriza-tion,indexing for retrieval,etc.CR Categories:I.3.5[Computer Graphics]:Computational Geometry and Object Modeling.Keywords:geometric modeling,shape analysis,symmetry detec-tion,shape descriptor,sampling guarantees.1IntroductionSymmetry is an essential and ubiquitous concept in nature,science,and art.For example,in geometry,the Erlanger program of Felix Klein [1893]has fueled for over a century mathematicians’inter-est in invariance under certain group actions as a key principle forunderstanding geometric spaces.Numerous biological,physical,or man-made structures exhibit symmetries as a fundamental design principle or as an essential aspect of their function.Whether by evolution or design,symmetry implies certain economies and ef fi-ciencies of structure that make it universally appealing.Symmetry also plays an important role in human visual perception and aes-thetics.Arguably much of the understanding of the world around us is based on the perception and recognition of shared or repeated structures,and so is our sense of beauty [Thompson 1961].In this paper we present a novel method for detecting meaningful symmetries in digital 3D shapes.We understand symmetry as the invariance under a set of transformations —in our case translation,rotation,re flection,and uniform scaling,the common generators of the Euclidean group.The figure below shows a 2D illustration.As can be seen in this example,symmetries or congruences that are quite apparent to us can be approximate and occur at differ-ent scales.Our goal is to de fine an algorithm that extracts (partial)symmetries at all scales,including approximate or imperfect sym-metries of varying degree.This allows the user to select the subset of symmetries that are most meaningful for a speci fic application.Examples include scan registration and alignment,shape matching,segmentation and skeleton extraction,compression,advanced mod-eling and editing,and shape database retrieval.To achieve this goal,we separate the symmetry computation into two phases:In the first step,we compute simple local shape de-scriptors at a selected set of points on the shape.These descriptors are chosen so that they are invariant under the group actions of in-terest.We use these local descriptors to pair up points that could be mapped to each other under a candidate symmetry action.We think of each such pair as depositing mass,or voting,for a speci fic sym-metry in the transformation space of interest.In this space,pairs with similar transformations form clusters that provide evidence for the corresponding symmetry relation.In the second step we use a stochastic clustering algorithm to ex-tract the significant modes of this mass distribution.Since the map-ping to transformation space does not preserve the spatial coher-ence or structure of samples on the input shape,we verify whether a meaningful symmetry has been found by checking the spatial consistency of the extracted subparts of the surface.Our cluster-ing method provides the necessary surface correspondences,since every point mass in transformation space corresponds to a candi-date pair of points in the spatial domain.Thus only a small set of candidates samples needs to be considered when detecting and extracting symmetric surface patches,avoiding a costly quadratic spatial search over the whole input data set.This separation into two stages is crucial for the effectiveness of our algorithm.The underlying observation is the following:given a proposed symmetry relation,it is simple and efficient to verify whether this specific symmetry is present in the model;we just need to apply the symmetry transform and check whether the model is mapped onto itself,or a sub-part of the model is mapped to a corre-sponding sub-part.However,the number of all potential mappings is by far too large to do an exhaustive search.Therefore,wefirst accumulate statistical evidence for which symmetries are present via our clustering in transformation space.Only if this evidence is sufficient do we perform spatial verification to check whether a specific symmetry is actually valid.Thus the complexity of symme-try extraction depends primarily on the number and size of relevant symmetries present in the model and not on the complexity of the model itself or that of the underlying symmetry group.As part of our approach,we can provide a quantitative measure on the“exact-ness”,or saliency,of a symmetry relation,which allows the user to control the degree of perfection in the extracted symmetries.In addition,by specifying the size of the set of local shape descriptors, the user can trade accuracy for computational efficiency.While fewer samples are sufficient for detecting large global symmetries, small partial symmetries require a significantly denser sampling. Thefinal output of our algorithm is a“symmetry graph”of the ob-ject,which encodes the significant symmetries of the object,each described by a patch pair and the corresponding transformation be-tween them.For objects that contain regular repeated structures, like windows or doors in architectural models,we can recover the symmetries of the repetition pattern through a basis reduction algo-rithm.This in effect leads to a sparser and more informative sym-metry graph that contains only fundamental symmetry generators and avoids encoding separately symmetries that are just products of already recorded symmetries.This kind of repeated pattern discov-ery can be useful in consistent mesh editing applications.1.1ContributionsWe propose a new algorithm for pairing sample points on3D shapes with compatible local descriptors to generate a distribution in trans-formation space whose peaks capture relevant symmetries of the object.We show how a stochastic clustering algorithm over this dis-tribution detects potential symmetry candidates,and provide a sur-face patching method that extracts a reduced symmetry graph from the extracted clusters.Our algorithms can be applied to3D mod-els of different shape characteristics and representations.Memory requirements are minimal and the computation is output-sensitive in the sense that its complexity depends mainly on the number and extent of symmetries actually present in the object.In addition,we provide theoretical bounds on the success rate of our algorithm as a function of the number of initial samples selected.These results in-dicate that the algorithm can be effective even for very large models that cannotfit in main memory.1.2Related workThe problem of symmetry detection has been extensively studied in numerousfields including visual perception,computer vision, robotics,and computational geometry.Early methods concentrated onfinding perfect symmetries in2D or3D planar point sets[Atallah 1985],[Wolter et al.1985].Since the restriction to exact symme-tries limits the use of these methods for real-world objects,Alt et al.[1988]introduced a method for computing approximate global symmetries in3D point sets,but the complexity of the algorithm makes it impractical for large data sets.Zabrodsky et al.[1995]for-malized the notion of approximate symmetry by expressing sym-metry as a continuous feature.Sun et al.[1997]proposed to exam-ine the correlation of the Gaussian image to recover global reflec-tive and rotational symmetries.Kazhdan and co-workers[2002]in-troduced a shape descriptor that concisely encodes global reflective ter they extended this work to rotational symmetries and used it for shape retrieval for database matching in[Kazhdan et al.2004].Our method bears some similarity to the Hough transform,a popular feature extraction method mainly used in image process-ing[Hough1959].Starting from a set of sample points obtained using edge detection,the method repeatedly selects small subsets of these samples to estimate the parameters of the feature curve. Analogous to our approach,votes cast by all of these estimates are accumulated and thefinal feature curve is extracted based on the majority of votes.Recently ideas based on the Hough transform have been used by[Loy and Eklundh2006]to detect reflective and rotational symmetries in images.The RANdom SAmple Consensus(RANSAC)method proposed by Fischler and Bolles[1981]is an algorithm for robust model fitting for data containing many outliers.In the context of shape matching the basic idea is to choose a random set of corresponding samples on the query and target shapes,apply the global transfor-mation induced by these samples,and evaluate the matching error between the two shapes.If sufficiently many transformations are explored in this way,the relevant symmetries can eventually be de-termined.Since the evaluation of the matching error requires costly spatial proximity tests,geometric hashing[Lamdan and Wolfson 1988]pre-computes all possible alignments by densely sampling the space of transformations and storing the resulting shape distri-bution in a hash grid.Gal and Cohen-Or[2006]recently presented an effective method for shape matching based on this idea.Their al-gorithm computes local shape descriptors that are grouped to form salient shape ing an empirical saliency measure,shape features are then used to pre-compute a geometric hash table that allows efficient partial matching.While sharing some similarities,our method is fundamentally dif-ferent from both RANSAC and geometric hashing.We avoid the costly exhaustive search of the former by computing the match-ing error of a transformation only after we accumulate sufficient evidence for a symmetry.At the same time our method requires minimal storage,in contrast to geometric hashing,where hash ta-bles of up to3.5GBytes have been reported for complex geometric shapes[Gal and Cohen-Or2006].1.3OverviewWefirst give some intuition for our method by looking at the2D example shown in Figure3,where the goal is to detect reflective symmetries of the butterfly.Any pair of points(p,q)on the bound-ary of the model defines a unique reflection with respect to the bi-sector line through(p+q)/2with normal direction p−q.HenceVDPSOLQJ DQDO\VLV SDLULQJ FOXVWHULQJ LQSXW PRGHO VDPSOH VHW VLJQDWXUHV WUDQVIRUPDWLRQV VXUIDFH SDWFKHVSDWFKLQJ'7GHQVLW\Figure 2:The symmetry extraction pipeline.Sampling yields a set P of surface points.For each p i ∈P a local signature is computed.Points p i ,p j with similar signatures are paired and a point in transformation space Γis computed mapping the local frame of p i to the one at p j .Clustering in Γyields subsets of P that remain invariant under a certain transformation,which can be extracted using spatial region growing.such a pair can be understood as evidence for the existence of this speci fic re flective symmetry.By looking at all such pairs we can accumulate this evidence and extract the relevant symmetry rela-tion(s).Only if many point pairs agree on (roughly)the same re-flection line,do we have reason to believe that the corresponding symmetry is truly present in the model.Thus we can detect po-tential symmetries by looking at clusters of points in the space of transformations Γ,where each point corresponds to a speci fic re-flection line.However,as shown in the illustration,the evidence of a single point pair is only reliable if the local geometry around the points is faithfully mirrored by the re flective transformation.This observation will allow us to signi ficantly prune the set of all point pairs and avoid an exhaustive computation on a quadratic number of point pairs.Since the mapping to Γdoes not incorporate the spatial position of surface samples,pairs from unrelated parts of the object can be mapped to the same point in transformation space.Thus in a second phase we extract spatially coherent components of the model that are invariant under the extracted symmetry -ing the point pair correspondences present in the cluster,we per-form an incremental region growing algorithm to verify a speci fic symmetry.Figure 2gives a high-level overview of our symmetry extraction pipeline.The following sections will elaborate on the individual stages and provide details of our approach.FG WUDQVIRUPDWLRQ VSDFH'Figure 3:Illustration of symmetry detection for re flections.Everypair of points de fines a symmetry line l that can be described by a distance d and an angle φ.Multiple points clustered in a small re-gion in transformation space provide evidence of a symmetry.The pair on the top left is discarded due to normal inconsistency.2Signatures and TransformationsWe consider the Euclidean transformation group generated by translations,rotations,re flections,and uniform scalings.Our goal is to find parts of a given 3D shape that are invariant under trans-formations in this symmetry group or some lower-dimensional sub-group.In order to apply the ideas sketched above,we need to compute the transformation T i j that maps a point p i on the surface of the model to another point p j .While point positions are suf ficient for de fin-ing a unique plane of re flection as in the example above,we cannot determine all degrees of freedom of a general Euclidean transform from the spatial positions alone.We therefore compute geometry signatures at each sample point p i based on the concept of normal cycles [Cohen-Steiner and Morvan 2003].We apply the algorithm proposed in [Alliez et al.2003]to approximate the curvature tensor at p i within a sphere of radius r and compute integrated principal curvatures κi ,1≤κi ,2and principal directions c i ,1and c i ,2.The ra-dius r should be on the order of the local sample spacing to achieve suf ficient averaging when computing the curvature tensor and avoid a strong dependence on the speci fic location of the sample points.F MF MQ MS MF LF LQ LS LF MF MF La F L a FM F MQ L Q Ma F La F L a The principal directions de fine a local frame (c i ,1,c i ,2,n i ),with nor-mal vector n i =c i ,1×c i ,2.We orient this frame as a right-handed coordinate frame that aligns with the outward pointing surface nor-mal by flipping signs of the appropriate vectors if necessary.In order to obtain a canonical rotational component R i j of the transfor-mation T i j we first align the two normals along their common plane and then pick the smaller of the two rotations around the normal that aligns to one of the two possible choices of orientation in tan-gent space.The uniform scale component of T i j is estimated from the ratio of principal curvatures as s i j =(κi ,1/κj ,1+κi ,2/κj ,2)/2,the translation is computed as t i j =p j −s i j R i j p i .For a given pair (p i ,p j )we thus obtain a point in 7-dimensional transforma-tion space Γas T i j =(s i j ,R x i j ,R y i j ,R z i j ,t x i j ,t y i j ,t z i j ),where R x i j ,R y i j ,R z i jare the Euler angles derived from R i j and t i j =[t x i j t yi j t z i j ]T .In order to handle re flections,we also compute the transformation obtained when re flecting the model about an arbitrary but fixed plane.2.1Point PruningA differential surface patch at umbilic points,i.e.,those for which κi ,1=κi ,2,is invariant under rotations around the surface normal.Pairs involving such points and their signatures do not de fine a unique transformation,but trace out curves in transformation space,which may quickly camou flage meaningful symmetry clusters.To avoid clutter in transformation space,we discard these points from the sample set,i.e.,we only consider points on the surface with distinct principal curvatures (and hence stable principal directions),which give rise to a unique transformation when paired with an-other compatible point.Apart from making the symmetry clus-tering more robust,point pruning has the additional advantage of reducing computation time.We obtain the adaptive sample set by applying a thresholdγ<1on the ratio of curvatures:p i∈P,if|κi,1/κi,2|<γ.We useγ=0.75for all examples in this paper. 2.2PairingGiven the reduced set of surface samples P and their signatures, we can now compute transformations for pairs of points in P.We select a random subset P ⊂P andfind all pairs(p ,p)with p ∈P and p∈P that provide evidence for a symmetry relation.In the Appendix we give theoretical bounds on the size of P and P required to successfullyfind symmetries of a certain size.As indicated above,the evidence of a selected point pair for a spe-cific symmetry relation is only reliable,if a local surface patch around each point is invariant under a transformation from the con-sidered symmetry group G.In the2D illustration of Figure3,for example,we can reject a pair,if the curvature estimates at both points differ too much,since curvature is invariant under reflection. To obtain an efficient pairing algorithm we map all samples to a signature spaceΩand use the metric of that space to estimate the deviation from perfect invariance.Only point pairs that are close in Ωare considered as suitable candidates for a local symmetry rela-tion,which avoids an exhaustive computation of a quadratic number of point pairs.For the full7-dimensional Euclidean group in3D,the mapping from P toΩ7=I R is given asσ7(p i)=κi,1/κi,2,since uniform scaling,rotation,and translation leave the ratio of principal curva-tures unchanged.The sub-index7indicates the dimension of the symmetry group.For purely rigid transforms,we defineσ6(p i)= (κi,1,κi,2)withΩ6=I R2.We can now for a given sample p i∈P determine all suitable partners in P by performing a range query inΩ.Using standard spatial proximity data structures,e.g.,a kd-tree,we can perform pairing in O(n log n)time,where n=|P|and n =|P |.If only reflections and/or translations are considered,we can additionally reject pairs based on the orientation of the local frames,as illustrated in Figure3.Figure4shows that pruning not only reduces the complexity of the clustering algorithm,but,even more importantly,avoids clutter in transformation space.By focusing only on locally consistent symmetry pairs,meaningful clusters are stably detected inΓ.3ClusteringThe pairing computed in the previous stage provides us with a set of transformations that map local surface patches onto each other. Each pair thus provides evidence for a symmetry relation at the level of the local sample spacing.To extract meaningful symme-tries at larger scales we need to accumulate this local evidence,i.e.,find groups of pairs with a similar transformation that correspond to symmetric subsets of the model surface.This requires the defi-nition of a distance metric inΓ,which is non-trivial,since scaling, rotation,and translation need to be combined in a single metric.We follow the approach of[Amato et al.2000]and define the norm of a transformation T=(s,R x,R y,R z,t x,t y,t z)∈Γas the weighted sum T 2=β1s2+β2(R2x+R2y+R2z)+β3(t2x+t2y+t2z).The weightsβi allow to adjust the relative influence of the individual components of the transformation.In all our examples we set these weights so that a rotation by180degrees corresponds to a displacement ofhalfFGF FGGFigure4:Pair pruning.40samples on the butterfly lead to402= 780points in transformation space.Pruning based on curvature re-duces the set to503points,while additionally normal-based prun-ing yields138points.The density plots show how the meaningful symmetry clusters become significantly more pronounced.the bounding box diagonal and a scaling factor of10.A metric for Γcan then be derived as d(T,T )= T−T ,where the subtrac-tion is component-wise,see also[Hofer et al.2004]for a detailed discussion.3.1Mean-ShiftIf the symmetries in the model are perfect(and the sampling in-cludes point pairs that are perfectly symmetric),then all pairs of the same(discrete)symmetry relation map to a single point inΓ.Many real-world objects exhibit approximate symmetries,however,and the sampling will not be precisely symmetric in general.We thus need a method tofind clusters in transformation space.When look-ing at the distribution of points inΓ,we immediately see that stan-dard parametric clustering methods,such as k-means clustering,are not suitable for our purposes.In general we have no a priori knowl-edge on the number of(partial)symmetries of the input model,i.e., selecting k would be difficult.Furthermore,clusters are not nec-essarily isotropic,especially for approximate symmetries like the ones shown in Figure1.A more suitable clustering method is mean shift clustering,a non-parametric method based on gradient ascent on a density functionρ[Comaniciu and Meer2002]1.This density function is defined as a sum of kernel functions K centered at each point T i inΓasρ(T)=∑iK( T−T i /h).We use the radially symmetric Epanechnikov kernel with band-width h as suggested in[Comaniciu and Meer2002].The signif-icant modes ofρare determined using gradient ascent.All points thatflow into a local maximum of sufficient height are considered samples of a significant cluster C k.The corresponding symmetry transformation T k is then defined by the cluster’s maximum.Essen-tially,the algorithm can be understood as a voting scheme:Every point pair votes for the symmetry relation that has been extracted 1Mean shift clustering has also been used in[James and Twigg2005]for skinning mesh animations and in[Tuzel et al.2005]for3D motion estima-tion.from its local frames.If many votes are cast for the same symme-try,a local peak is created in the accumulated density function.For more details on mean-shift clustering we refer to[Comaniciu and Meer2002].4VerificationA significant mode detected by the mean-shift clustering algorithm does not necessarily correspond to a meaningful symmetry.Since the spatial relation of sample points is lost during the mapping to transformation space,sample pairs from uncorrelated parts of the object can accumulate to form discernible clusters.The effective-ness of our method is based on the observation that statistically such spurious modes are rare(see also the analysis in the Appendix):It is highly unlikely that many uncorrelated point pairs agree on the same transformation,i.e.,are mapped to the same point in7D trans-formation space.We can thus afford to perform a spatial verifica-tion for each cluster C k by extracting the connected components of the model that are invariant under the corresponding transformation T k.We compute these surface patches using an incremental patch growing process,starting with a random point of C k,which corre-sponds to a pair(p i,p j)of points on the model surface.Now we look at the one-ring neighbors of p i,apply T k,and check whether the distance of the transformed points to the surface around p j is below a given error threshold.If so,we add them to the current patch.We keep extending this patch along its boundary until no more points can be added.During the growth process,we mark all visited samples on the surface and remove points in C k that corre-spond to these samples.This process is then repeated using the next point in C k until all points have been considered.Since the transformation T k at the cluster’s maximum does not nec-essarily provide the best possible transformation for matching the surface patches,we incrementally refine T k during the patch grow-ing using the iterated closest points(ICP)algorithm[Rusinkiewicz and Levoy2001].The normalized residual of the ICP matching then provides a quantitative measure for the exactness of the sym-metry[Mitra et al.2004].Other measures,such as the Hausdorff distance can also be used.We end up with a collection of pairs of patches on the model surface that are mapped onto each other by the cluster’s transformation T k.This information can be encoded in a weighted graph,where each node corresponds to a patch and each edge denotes the transformation that maps two patches onto each other,weighted by the matching error.4.1Compound TransformsMany geometric objects exhibit symmetries in a structured or repet-itive fashion resulting in a large number of clusters in transforma-tion space[Liu et al.2004].Encoding all pair-wise symmetry re-lations for such models leads to a complex and highly redundant symmetry graph and thus a costly verification stage.In this sec-tion we describe a simple basis reduction algorithm that computes a compact set of generators for all detected symmetries in transfor-mation space[Magnus et al.2004].This significantly reduces the number of spatial consistency checks required for verification and yields a more informative symmetry graph that supports advanced editing operations and high level shape comparisons.The algorithm shown below takes as input all extracted symmetry transformations T sorted in descending order of cluster height and iteratively processes each transformation T i∈T.During execution we maintain an alphabet A of generators and the language L that encodes T in terms of the alphabet A.A user parameterηcontrolsWUDQVRURWFigure5:Symmetry graph reduction for a model with structured symmetries at two different scales.60significant modes have been extracted in the clustering stage.The reduced basis contains6trans-formations,as indicated by the arrows.The graph on the left shows the number of detected symmetries as a function of random samples in the subset P ⊂P.the complexity of the algorithm by limiting the search to loops of lengthη+1.The thresholdδmeasures the allowed deviation from the exact transformation.Figure5shows an example of a reduced symmetry basis.Verifi-cation can now be applied more efficiently on the set L of com-pound transformations.For more details on basis reduction we refer to[Magnus et al.2004].Algorithm1Symmetry basis reduction.Input:T={T1,T2,...,T n}A←{I}L←/0for i=1to n doif∃(A1,...,Aη)with A j∈A s.t.|T i−∏ηj=1A j|≤δthenL←L∪{(A1,...,Aη)}elseA←A∪{T i,T−1i}L←L∪{(T i)}end ifend for5Results and ApplicationsWe have implemented the pipeline sketched in Figure2.An initial sample set is created by uniformly sampling the input model.Af-ter computing signatures using the method of[Alliez et al.2003], point pruning yields the reduced sample set P.We then select a random subset P ⊂P,find all suitable pairs(p ∈P ,p∈P)based on the proximity in signature space,and compute the correspond-ing transformations.We perform mean-shift clustering using the method proposed in[Arya et al.1998]to efficiently compute neigh-borhoods in7D transformation space.Basis reduction and verifica-tionfinally yield the symmetric patches.We show in the Appendix that our method is guaranteed tofind existing symmetry relations provided the sampling is dense enough with respect to the size of the symmetric patches.The following examples verify this claim and demonstrate that practical results can be obtained even if the theoretical sampling requirements are not met.。
