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Aaesthetic 美观alveolar 牙槽骨anchorage resistance to unwanted tooth movement caused by the reactive component of an orthodontic force; refers also to the intra- and extraoral structures that supply the resistanceAngle's classification of occlusion a definition of malocclusion based on the relationships of the first permanent molarsangulation the tilt of the long axis of a tooth in a mesial or distal direction; see inclinationAnkylosed PrimaryTeeth乳牙粘连anterior crossbite one or more teeth in the maxillary anterior segment is lingual to one or more of the opposing teeth in the mandibular anterior segment in maximum intercuspationanterior segment all of the canine and incisor teeth in a given dental archarch circumference or arch perimeter the distance from the mesial contact of one first permanent molar to its antimere as measured through the contact points or buccal cusp tips of all of the intervening teeth, ignoring those teeth that are malpositioned or blocked out so that the measurement represents an ideal arch form; see arch lengtharch depth the perpendicular distance from a point between the central incisors to a line connecting the mesial contacts of the first permanent molars; see arch lengtharch length same as arch depth; but note that 'arch length' is often used as a synonym for arch circumference or arch perimeterarch wire a wire applied to two or more teeth through fixed attachments to cause or guide orthodontic tooth movementattached lingualfrenum舌系带过短Bband a circular metal strip that is adapted to fit closely around a tooth; various components of an orthodontic appliance may be welded or soldered to itbase, bracket the part of a bracket that is attached either to a metal band or bonding padBegg appliance Begg矫治器bimaxillary both the upper and lower jawblocked out a tooth that is positioned away from its proper position in the dental arch due to insufficient spacebodily movement 整体移动Bolton analysis Bolton指数分析bonded lingual 舌侧粘结式保持retainerbonding pad the retentive portion of a fixed orthodontic attachment which locks it mechanically to the bonding material; the pad usually has a fine mesh surfacebrachycephalic a short skull, with a cranial index of 80 or more; see cranial index This term, or a variant "brachyfacial," is sometimes used to describe a short, wide face, properly referred to as euryprosopic .bracket a metal, plastic, or ceramic fixed attachment which holds an arch wire buccal segment all of the premolar and molar teeth in a given quadrantbuccal tube a fixed attachment which is open only at each end. Tubes may be round or rectangular in cross section; round tubes are usually .045 inches in diameter to receive auxiliaries such as a facebow or lip bumper, rectangular tubes are either .018 x .025 or .022 x .028 inches in order to receive arch wires and generally are placed on the most distal molar tooth in the appliance.Ccamouflagedorthodontic treatment掩饰性正畸治疗Cast Analysis modelanalysis模型分析center of resistance 阻抗中心center of rotation 旋转中心Cephalometrics X线头影测量cervical headgear 颈牵引chief complaint 主诉chin-cup 颏兜Cleft jaw and face 颌、面裂Cleft lip and palate 唇、腭裂Clinical Examination 临床检查combination headgear 联合（水平）牵引condylar 髁突Continuous force 持续力correctiveorthodontics一般性矫治craniofacial growthand development颅面生长发育crowding or spacing 拥挤和间隙curve of Spee Spee曲线DDelayed IncisorEruption切牙迟萌dental age 牙龄dental history 牙科病史dentofacial牙合面矫形orthopedicDiagnosis and诊断和治疗计划treatment planningdifferential tooth差动牙移动movementEEctopic Eruption 异位萌出edgewise appliance 方丝弓矫治器Enamel釉质脱矿demineralizationextraoral orthodontic口外正畸力forceextraoral orthopedic口外矫形力forceextrusion 伸长Ffixed appliance 固定矫治器fixed retaining固定保持器applianceGGingivitis 牙龈炎Growth Modification 生长改良HHand wrist x-ray 手腕骨X片harmony 平衡Hawley retainer Hawley保持器Headcap 头帽Heredity and遗传和环境environmenthigh-pull headgear 高位牵引IIdeal normal occlusion 理想正常合Idiopathic Root特发性吸收resorptionimplant anchorage 种植支抗impression taking 取模Individual normal个别正常合occlusionInterceptive阻断性矫治orthodonticsintermaxillary颌间支抗anchorageIntermittent force 间歇力intramaxillary颌内支抗anchorageintrusion 压低Invisalign 无托槽隐形矫治器Jjob model 工作模型LLeeway space 替牙间隙Mmalocclusion 错合畸形Mandibular Deficiency 上颌发育不足Maxillary Excess 下颌发育过度Maxillary Midline上中切牙间隙Diastemamechanical retention 机械保持medical history 医学病史Minimal Root微小性吸收resorptionminor tooth小范围牙移动movement(MTM)Mixed Dentition 混合牙列，替牙合molar relationship 磨牙关系Nnature retetion 自然保持Oobstructive sleepapnea hypopnea阻塞性睡眠呼吸暂停低通气综合征syndrome OSAHSopenbite 开合optimal orthodontic最适矫治力forceoral examination 口腔检查Oral health education 口腔健康教育oral hygiene口腔卫生指导introductionorthodontic force 正畸力Orthodontics 口腔正畸学Orthognathic Surgery 正颌手术orthopedic force 矫形力overcorrection 过度矫正PPalatal Expansion 腭开展Pattern of growth 生长型Periodontitis 牙周炎Permanent Dentition 恒牙合permanent tooth 恒牙Plaque 菌斑positioner 正位器Post-operative术后正畸orthodonticsPreoperative术前正畸orthodonticsPreventive预防性矫治orthodonticsPrimary Dentition 乳牙合primary tooth 乳牙Primate space 灵长间隙Progressive Root进行性吸收resorptionRrelapse 复发Remodeling and改建和位移translationremovable retaining活动保持器applianceRetetion 保持rotation 旋转Sself-ligation bracket 自锁托槽serial extraction 序列拔牙Six keys to normal正常合六项标准occlusionskeletal age 骨龄skeletal malocclusion 骨性错合畸形Space analysis 间隙分析space available 可用间隙space required 必需间隙stable 稳定straight wire直丝弓矫治器appliancestudy model 记存模型、研究模型Supernumerary Teeth 多生牙Surgical orthodontics 正颌外科TTemporarymalocclusion in mixed暂时性错合dentitionTemporomandibular颞下颌关节jointsthe incidence of错合畸形的患病率malocclusionthree-dimensionalcomplex of颅面三维结构复合体craniofacialstructurethree-dimensional三维数字化重建digitalreconstructionthumb sucking 吮指习惯Time and sequence of萌出时间和顺序eruptiontip 轴倾角tipping movement 倾斜移动torque 转矩角Transverse Maxillary上颌横向狭窄ConstrictionWWire Bending 弓丝弯制。
字号：大中小力学mechanics牛顿力学Newtonian mechanics经典力学classical mechanics静力学statics运动学kinematics动力学dynamics动理学kinetics宏观力学macroscopicmechanics,macromechanics细观力学mesomechanics微观力学microscopicmechanics,micromechanics一般力学general mechanics固体力学solid mechanics流体力学fluid mechanics 理论力学theoretical mechanics应用力学applied mechanics工程力学engineering mechanics 实验力学experimental mechanics 计算力学computational mechanics 理性力学rational mechanics物理力学physical mechanics地球动力学geodynamics力force作用点point of action作用线line of action力系system of forces力系的简化reduction of force system 等效力系equivalent force system刚体rigid body力的可传性transmissibility of force 平行四边形定则parallelogram rule 力三角形force triangle力多边形force polygon零力系null-force system平衡equilibrium力的平衡equilibrium of forces平衡条件equilibrium condition平衡位置equilibrium position平衡态equilibrium state分析力学analytical mechanics拉格朗日乘子Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号Lagrange bracket循环坐标cyclic coordinate循环积分cyclic integral哈密顿[量] Hamiltonian哈密顿函数Hamiltonian function正则方程canonical equation正则摄动canonical perturbation正则变换canonical transformation正则变量canonical variable哈密顿原理Hamilton principle作用量积分action integral哈密顿--雅可比方程Hamilton-Jacobi equation 作用--角度变量action-angle variables阿佩尔方程Appell equation劳斯方程Routh equation拉格朗日函数Lagrangian function诺特定理Noether theorem泊松括号poisson bracket边界积分法boundary integral method并矢dyad运动稳定性stability of motion轨道稳定性orbital stability李雅普诺夫函数Lyapunov function渐近稳定性asymptotic stability结构稳定性structural stability久期不稳定性secular instability弗洛凯定理Floquet theorem倾覆力矩capsizing moment自由振动free vibration固有振动natural vibration暂态transient state环境振动ambient vibration反共振anti-resonance衰减attenuation库仑阻尼Coulomb damping同相分量in-phase component非同相分量out-of-phase component超调量overshoot参量[激励]振动parametric vibration模糊振动fuzzy vibration临界转速critical speed of rotation阻尼器damper半峰宽度half-peak width集总参量系统lumped parameter system 相平面法phase plane method相轨迹phase trajectory等倾线法isocline method跳跃现象jump phenomenon负阻尼negative damping达芬方程Duffing equation希尔方程Hill equationKBM方法KBM method,Krylov-Bogoliu-bov-Mitropol'skii method 马蒂厄方程Mathieu equation平均法averaging method组合音调combination tone解谐detuning耗散函数dissipative function硬激励hard excitation硬弹簧hard spring, hardening spring谐波平衡法harmonic balance method久期项secular term自激振动self-excited vibration分界线separatrix亚谐波subharmonic软弹簧soft spring ,softening spring软激励soft excitation邓克利公式Dunkerley formula瑞利定理Rayleigh theorem分布参量系统distributed parameter system 优势频率dominant frequency模态分析modal analysis固有模态natural mode of vibration同步synchronization超谐波ultraharmonic范德波尔方程van der pol equation频谱frequency spectrum基频fundamental frequencyWKB方法WKB method,Wentzel-Kramers-Brillouin method缓冲器buffer风激振动aeolian vibration嗡鸣buzz倒谱cepstrum颤动chatter蛇行hunting阻抗匹配impedance matching机械导纳mechanical admittance机械效率mechanical efficiency机械阻抗mechanical impedance随机振动stochastic vibration, random vibration隔振vibration isolation减振vibration reduction应力过冲stress overshoot喘振surge摆振shimmy起伏运动phugoid motion起伏振荡phugoid oscillation驰振galloping陀螺动力学gyrodynamics陀螺摆gyropendulum陀螺平台gyroplatform陀螺力矩gyroscoopic torque陀螺稳定器gyrostabilizer陀螺体gyrostat惯性导航inertial guidance姿态角attitude angle方位角azimuthal angle舒勒周期Schuler period机器人动力学robot dynamics多体系统multibody system多刚体系统multi-rigid-body system机动性maneuverability凯恩方法Kane method转子[系统]动力学rotor dynamics转子[一支承一基础]系统rotor-support-foundationsystem静平衡static balancing动平衡dynamic balancing静不平衡static unbalance动不平衡dynamic unbalance现场平衡field balancing不平衡unbalance不平衡量unbalance互耦力cross force挠性转子flexible rotor分频进动fractional frequency precession 半频进动half frequency precession油膜振荡oil whip转子临界转速rotor critical speed自动定心self-alignment亚临界转速subcritical speed涡动whirl连续过程continuous process碰撞截面collision cross section 通用气体常数conventional gas constant 燃烧不稳定性combustion instability稀释度dilution完全离解complete dissociation火焰传播flame propagation组份constituent碰撞反应速率collision reaction rate燃烧理论combustion theory浓度梯度concentration gradient阴极腐蚀cathodic corrosion火焰速度flame speed火焰驻定flame stabilization火焰结构flame structure着火ignition湍流火焰turbulent flame层流火焰laminar flame燃烧带burning zone渗流flow in porous media, seepage达西定律Darcy law赫尔-肖流Hele-Shaw flow毛[细]管流capillary flow过滤filtration爪进fingering不互溶驱替immiscible displacement不互溶流体immiscible fluid互溶驱替miscible displacement互溶流体miscible fluid迁移率mobility流度比mobility ratio渗透率permeability孔隙度porosity多孔介质porous medium比面specific surface迂曲度tortuosity空隙void空隙分数void fraction注水water flooding可湿性wettability地球物理流体动力学geophysical fluid dynamics 物理海洋学physical oceanography大气环流atmospheric circulation海洋环流ocean circulation海洋流ocean current旋转流rotating flow平流advection埃克曼流Ekman flow埃克曼边界层Ekman boundary layer大气边界层atmospheric boundary layer大气-海洋相互作用atmosphere-oceaninteraction埃克曼数Ekman number罗斯贝数Rossby unmber罗斯贝波Rossby wave斜压性baroclinicity正压性barotropy内磨擦internal friction海洋波ocean wave盐度salinity环境流体力学environmental fluid mechanics斯托克斯流Stokes flow羽流plume理查森数Richardson number污染源pollutant source污染物扩散pollutant diffusion噪声noise噪声级noise level噪声污染noise pollution排放物effulent工业流体力学industrical fluid mechanics流控技术fluidics轴向流axial flow并向流co-current flow对向流counter current flow横向流cross flow螺旋流spiral flow旋拧流swirling flow滞后流after flow混合层mixing layer抖振buffeting风压wind pressure附壁效应wall attachment effect, Coanda effect 简约频率reduced frequency爆炸力学mechanics of explosion终点弹道学terminal ballistics动态超高压技术dynamic ultrahigh pressuretechnique流体弹塑性体hydro-elastoplastic medium 热塑不稳定性thermoplastic instability空中爆炸explosion in air地下爆炸underground explosion水下爆炸underwater explosion 电爆炸discharge-induced explosion激光爆炸laser-induced explosion核爆炸nuclear explosion点爆炸point-source explosion殉爆sympathatic detonation强爆炸intense explosion粒子束爆炸explosion by beam radiation聚爆implosion起爆initiation of explosion爆破blasting霍普金森杆Hopkinson bar电炮electric gun电磁炮electromagnetic gun爆炸洞explosion chamber轻气炮light gas gun马赫反射Mach reflection基浪base surge成坑cratering能量沉积energy deposition爆心explosion center爆炸当量explosion equivalent火球fire ball爆高height of burst蘑菇云mushroom侵彻penetration规则反射regular reflection崩落spallation应变率史strain rate history流变学rheology聚合物减阻drag reduction by polymers挤出[物]胀大extrusion swell, die swell无管虹吸tubeless siphon剪胀效应dilatancy effect孔压[误差]效应hole-pressure[error]effect 剪切致稠shear thickening剪切致稀shear thinning触变性thixotropy反触变性anti-thixotropy超塑性superplasticity粘弹塑性材料viscoelasto-plastic material滞弹性材料anelastic material本构关系constitutive relation麦克斯韦模型Maxwell model沃伊特-开尔文模型Voigt-Kelvin model宾厄姆模型Bingham model奥伊洛特模型Oldroyd model幂律模型power law model应力松驰stress relaxation应变史strain history应力史stress history记忆函数memory function衰退记忆fading memory应力增长stress growing粘度函数voscosity function相对粘度relative viscosity复态粘度complex viscosity拉伸粘度elongational viscosity拉伸流动elongational flow第一法向应力差first normal-stress difference 第二法向应力差second normal-stress difference 德博拉数Deborah number魏森贝格数Weissenberg number动态模量dynamic modulus振荡剪切流oscillatory shear flow宇宙气体动力学cosmic gas dynamics等离[子]体动力学plasma dynamics电离气体ionized gas行星边界层planetary boundary layer阿尔文波Alfven wave泊肃叶-哈特曼流] Poiseuille-Hartman flow哈特曼数Hartman number生物流变学biorheology生物流体biofluid生物屈服点bioyield point生物屈服应力bioyield stress电气体力学electro-gas dynamics铁流体力学ferro-hydrodynamics 血液流变学hemorheology, blood rheology血液动力学hemodynamics磁流体力学magneto fluid mechanics磁流体动力学magnetohydrodynamics, MHD磁流体动力波magnetohydrodynamic wave 磁流体流magnetohydrodynamic flow磁流体动力稳定性magnetohydrodynamicstability生物力学biomechanics生物流体力学biological fluid mechanics生物固体力学biological solid mechanics 宾厄姆塑性流Bingham plastic flow开尔文体Kelvin body沃伊特体Voigt body可贴变形applicable deformation可贴曲面applicable surface边界润滑boundary lubrication液膜润滑fluid film lubrication向心收缩功concentric work离心收缩功eccentric work关节反作用力joint reaction force微循环力学microcyclic mechanics微纤维microfibril渗透性permeability生理横截面积physiological cross-sectional area 农业生物力学agrobiomechanics纤维度fibrousness硬皮度rustiness胶粘度gumminess粘稠度stickiness嫩度tenderness渗透流osmotic flow易位流translocation flow蒸腾流transpirational flow过滤阻力filtration resistance压扁wafering风雪流snow-driving wind停滞堆积accretion遇阻堆积encroachment沙漠地面desert floor流沙固定fixation of shifting sand流动阈值fluid threshold。
弹性力学专业英语英汉互译词汇弹性力学elasticity弹性理论theory of elasticity均匀应力状态homogeneous state of stress应力不变量stress invariant应变不变量strain invariant应变椭球strain ellipsoid均匀应变状态homogeneous state ofstrain应变协调方程equation of straincompatibility拉梅常量Lame constants各向同性弹性isotropic elasticity旋转圆盘rotating circular disk楔wedge开尔文问题Kelvin problem布西内斯克问题Boussinesq problem艾里应力函数Airy stress function克罗索夫--穆斯赫利Kolosoff- 什维利法Muskhelishvili method基尔霍夫假设Kirchhoff hypothesis板Plate矩形板Rectangular plate圆板Circular plate环板Annular plate波纹板Corrugated plate加劲板Stiffened plate,reinforcedPlate中厚板Plate of moderate thickness 弯[曲]应力函数Stress function of bending 壳Shell扁壳Shallow shell旋转壳Revolutionary shell球壳Spherical shell [圆]柱壳Cylindrical shell锥壳Conical shell环壳Toroidal shell封闭壳Closed shell波纹壳Corrugated shell扭[转]应力函数Stress function of torsion 翘曲函数Warping function半逆解法semi-inverse method瑞利--里茨法Rayleigh-Ritz method 松弛法Relaxation method莱维法Levy method松弛Relaxation量纲分析Dimensional analysis自相似[性] self-similarity影响面Influence surface接触应力Contact stress赫兹理论Hertz theory协调接触Conforming contact滑动接触Sliding contact滚动接触Rolling contact压入Indentation各向异性弹性Anisotropic elasticity颗粒材料Granular material散体力学Mechanics of granular media 热弹性Thermoelasticity超弹性Hyperelasticity粘弹性Viscoelasticity对应原理Correspondence principle褶皱Wrinkle塑性全量理论Total theory of plasticity 滑动Sliding微滑Microslip粗糙度Roughness非线性弹性Nonlinear elasticity大挠度Large deflection突弹跳变snap-through有限变形Finite deformation格林应变Green strain阿尔曼西应变Almansi strain弹性动力学Dynamic elasticity运动方程Equation of motion准静态的Quasi-static气动弹性Aeroelasticity水弹性Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波Cylindrical wave水平剪切波Horizontal shear wave竖直剪切波Vertical shear wave 体波body wave无旋波Irrotational wave畸变波Distortion wave膨胀波Dilatation wave瑞利波Rayleigh wave等容波Equivoluminal wave勒夫波Love wave界面波Interfacial wave边缘效应edge effect塑性力学Plasticity可成形性Formability金属成形Metal forming耐撞性Crashworthiness结构抗撞毁性Structural crashworthiness 拉拔Drawing破坏机构Collapse mechanism回弹Springback挤压Extrusion冲压Stamping穿透Perforation层裂Spalling塑性理论Theory of plasticity安定[性]理论Shake-down theory运动安定定理kinematic shake-down theorem静力安定定理Static shake-down theorem 率相关理论rate dependent theorem 载荷因子load factor加载准则Loading criterion加载函数Loading function加载面Loading surface塑性加载Plastic loading塑性加载波Plastic loading wave简单加载Simple loading比例加载Proportional loading卸载Unloading卸载波Unloading wave冲击载荷Impulsive load阶跃载荷step load脉冲载荷pulse load极限载荷limit load中性变载nentral loading拉抻失稳instability in tension 加速度波acceleration wave本构方程constitutive equation 完全解complete solution名义应力nominal stress过应力over-stress真应力true stress等效应力equivalent stress流动应力flow stress应力间断stress discontinuity应力空间stress space主应力空间principal stress space静水应力状态hydrostatic state of stress 对数应变logarithmic strain工程应变engineering strain等效应变equivalent strain应变局部化strain localization应变率strain rate应变率敏感性strain rate sensitivity 应变空间strain space有限应变finite strain塑性应变增量plastic strain increment 累积塑性应变accumulated plastic strain 永久变形permanent deformation内变量internal variable应变软化strain-softening理想刚塑性材料rigid-perfectly plasticMaterial刚塑性材料rigid-plastic material理想塑性材料perfectl plastic material 材料稳定性stability of material应变偏张量deviatoric tensor of strain 应力偏张量deviatori tensor of stress 应变球张量spherical tensor of strain 应力球张量spherical tensor of stress 路径相关性path-dependency线性强化linear strain-hardening应变强化strain-hardening随动强化kinematic hardening各向同性强化isotropic hardening强化模量strain-hardening modulus幂强化power hardening塑性极限弯矩plastic limit bendingMoment塑性极限扭矩plastic limit torque弹塑性弯曲elastic-plastic bending弹塑性交界面elastic-plastic interface 弹塑性扭转elastic-plastic torsion粘塑性Viscoplasticity非弹性Inelasticity理想弹塑性材料elastic-perfectly plasticMaterial极限分析limit analysis极限设计limit design极限面limit surface上限定理upper bound theorem上屈服点upper yield point下限定理lower bound theorem下屈服点lower yield point界限定理bound theorem初始屈服面initial yield surface后继屈服面subsequent yield surface屈服面[的]外凸性convexity of yield surface 截面形状因子shape factor of cross-section沙堆比拟sand heap analogy屈服Yield屈服条件yield condition屈服准则yield criterion屈服函数yield function屈服面yield surface塑性势plastic potential 能量吸收装置energy absorbing device 能量耗散率energy absorbing device 塑性动力学dynamic plasticity塑性动力屈曲dynamic plastic buckling 塑性动力响应dynamic plastic response 塑性波plastic wave运动容许场kinematically admissibleField 静力容许场statically admissibleField流动法则flow rule速度间断velocity discontinuity滑移线slip-lines滑移线场slip-lines field移行塑性铰travelling plastic hinge 塑性增量理论incremental theory ofPlasticity米泽斯屈服准则Mises yield criterion 普朗特--罗伊斯关系prandtl- Reuss relation 特雷斯卡屈服准则Tresca yield criterion洛德应力参数Lode stress parameter莱维--米泽斯关系Levy-Mises relation亨基应力方程Hencky stress equation赫艾--韦斯特加德应Haigh-Westergaard 力空间stress space洛德应变参数Lode strain parameter德鲁克公设Drucker postulate盖林格速度方程Geiringer velocityEquation结构力学structural mechanics结构分析structural analysis结构动力学structural dynamics拱Arch三铰拱three-hinged arch抛物线拱parabolic arch圆拱circular arch穹顶Dome空间结构space structure空间桁架space truss雪载[荷] snow load风载[荷] wind load土压力earth pressure地震载荷earthquake loading弹簧支座spring support支座位移support displacement支座沉降support settlement超静定次数degree of indeterminacy机动分析kinematic analysis结点法method of joints截面法method of sections结点力joint forces共轭位移conjugate displacement影响线influence line三弯矩方程three-moment equation单位虚力unit virtual force刚度系数stiffness coefficient柔度系数flexibility coefficient力矩分配moment distribution力矩分配法moment distribution method 力矩再分配moment redistribution分配系数distribution factor矩阵位移法matri displacement method 单元刚度矩阵element stiffness matrix 单元应变矩阵element strain matrix总体坐标global coordinates贝蒂定理Betti theorem高斯--若尔当消去法Gauss-Jordan eliminationMethod屈曲模态buckling mode复合材料力学mechanics of composites复合材料composite material纤维复合材料fibrous composite单向复合材料unidirectional composite泡沫复合材料foamed composite颗粒复合材料particulate composite 层板Laminate夹层板sandwich panel正交层板cross-ply laminate斜交层板angle-ply laminate层片Ply多胞固体cellular solid膨胀Expansion压实Debulk劣化Degradation脱层Delamination脱粘Debond纤维应力fiber stress层应力ply stress层应变ply strain层间应力interlaminar stress比强度specific strength强度折减系数strength reduction factor 强度应力比strength -stress ratio 横向剪切模量transverse shear modulus 横观各向同性transverse isotropy正交各向异Orthotropy剪滞分析shear lag analysis短纤维chopped fiber长纤维continuous fiber纤维方向fiber direction纤维断裂fiber break纤维拔脱fiber pull-out纤维增强fiber reinforcement致密化Densification最小重量设计optimum weight design网格分析法netting analysis混合律rule of mixture失效准则failure criterion蔡--吴失效准则Tsai-W u failure criterion 达格代尔模型Dugdale model断裂力学fracture mechanics概率断裂力学probabilistic fractureMechanics格里菲思理论Griffith theory线弹性断裂力学linear elastic fracturemechanics, LEFM弹塑性断裂力学elastic-plastic fracturemecha-nics, EPFM 断裂Fracture脆性断裂brittle fracture解理断裂cleavage fracture蠕变断裂creep fracture延性断裂ductile fracture晶间断裂inter-granular fracture 准解理断裂quasi-cleavage fracture 穿晶断裂trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹elliptical crack深埋裂纹embedded crack[钱]币状裂纹penny-shape crack预制裂纹Precrack短裂纹short crack表面裂纹surface crack裂纹钝化crack blunting裂纹分叉crack branching裂纹闭合crack closure裂纹前缘crack front裂纹嘴crack mouth裂纹张开角crack opening angle,COA 裂纹张开位移crack opening displacement,COD裂纹阻力crack resistance裂纹面crack surface裂纹尖端crack tip裂尖张角crack tip opening angle,CTOA裂尖张开位移crack tip openingdisplacement, CTOD裂尖奇异场crack tip singularityField裂纹扩展速率crack growth rate稳定裂纹扩展stable crack growth定常裂纹扩展steady crack growth亚临界裂纹扩展subcritical crack growth 裂纹[扩展]减速crack retardation 止裂crack arrest止裂韧度arrest toughness断裂类型fracture mode滑开型sliding mode张开型opening mode撕开型tearing mode复合型mixed mode撕裂Tearing撕裂模量tearing modulus断裂准则fracture criterionJ积分J-integral J阻力曲线J-resistance curve断裂韧度fracture toughness应力强度因子stress intensity factor HRR场Hutchinson-Rice-RosengrenField 守恒积分conservation integral 有效应力张量effective stress tensor 应变能密度strain energy density 能量释放率energy release rate内聚区cohesive zone塑性区plastic zone张拉区stretched zone热影响区heat affected zone, HAZ 延脆转变温度brittle-ductile transitiontempe- rature 剪切带shear band剪切唇shear lip无损检测non-destructive inspection 双边缺口试件double edge notchedspecimen, DEN specimen 单边缺口试件single edge notchedspecimen, SEN specimen 三点弯曲试件three point bendingspecimen, TPB specimen 中心裂纹拉伸试件center cracked tensionspecimen, CCT specimen 中心裂纹板试件center cracked panelspecimen, CCP specimen 紧凑拉伸试件compact tension specimen,CT specimen 大范围屈服large scale yielding小范围攻屈服small scale yielding韦布尔分布Weibull distribution帕里斯公式paris formula空穴化Cavitation应力腐蚀stress corrosion概率风险判定probabilistic riskassessment, PRA 损伤力学damage mechanics损伤Damage连续介质损伤力学continuum damage mechanics 细观损伤力学microscopic damage mechanics 累积损伤accumulated damage脆性损伤brittle damage延性损伤ductile damage宏观损伤macroscopic damage细观损伤microscopic damage微观损伤microscopic damage损伤准则damage criterion损伤演化方程damage evolution equation 损伤软化damage softening损伤强化damage strengthening损伤张量damage tensor损伤阈值damage threshold损伤变量damage variable损伤矢量damage vector损伤区damage zone疲劳Fatigue低周疲劳low cycle fatigue应力疲劳stress fatigue随机疲劳random fatigue蠕变疲劳creep fatigue腐蚀疲劳corrosion fatigue疲劳损伤fatigue damage疲劳失效fatigue failure疲劳断裂fatigue fracture 疲劳裂纹fatigue crack疲劳寿命fatigue life疲劳破坏fatigue rupture疲劳强度fatigue strength 疲劳辉纹fatigue striations 疲劳阈值fatigue threshold 交变载荷alternating load 交变应力alternating stress 应力幅值stress amplitude 应变疲劳strain fatigue应力循环stress cycle应力比stress ratio安全寿命safe life过载效应overloading effect 循环硬化cyclic hardening 循环软化cyclic softening 环境效应environmental effect 裂纹片crack gage裂纹扩展crack growth, crackPropagation裂纹萌生crack initiation 循环比cycle ratio实验应力分析experimental stressAnalysis工作[应变]片active[strain] gage基底材料backing material应力计stress gage零[点]飘移zero shift, zero drift 应变测量strain measurement应变计strain gage应变指示器strain indicator应变花strain rosette应变灵敏度strain sensitivity机械式应变仪mechanical strain gage 直角应变花rectangular rosette引伸仪Extensometer应变遥测telemetering of strain 横向灵敏系数transverse gage factor 横向灵敏度transverse sensitivity 焊接式应变计weldable strain gage平衡电桥balanced bridge粘贴式应变计bonded strain gage粘贴箔式应变计bonded foiled gage粘贴丝式应变计bonded wire gage 桥路平衡bridge balancing电容应变计capacitance strain gage 补偿片compensation technique 补偿技术compensation technique 基准电桥reference bridge电阻应变计resistance strain gage 温度自补偿应变计self-temperaturecompensating gage半导体应变计semiconductor strainGage 集流器slip ring应变放大镜strain amplifier疲劳寿命计fatigue life gage电感应变计inductance [strain] gage 光[测]力学Photomechanics光弹性Photoelasticity光塑性Photoplasticity杨氏条纹Young fringe双折射效应birefrigent effect等位移线contour of equalDisplacement 暗条纹dark fringe条纹倍增fringe multiplication 干涉条纹interference fringe等差线Isochromatic等倾线Isoclinic等和线isopachic应力光学定律stress- optic law主应力迹线Isostatic亮条纹light fringe光程差optical path difference 热光弹性photo-thermo -elasticity 光弹性贴片法photoelastic coatingMethod光弹性夹片法photoelastic sandwichMethod动态光弹性dynamic photo-elasticity 空间滤波spatial filtering空间频率spatial frequency起偏镜Polarizer反射式光弹性仪reflection polariscope残余双折射效应residual birefringentEffect应变条纹值strain fringe value应变光学灵敏度strain-optic sensitivity 应力冻结效应stress freezing effect应力条纹值stress fringe value应力光图stress-optic pattern暂时双折射效应temporary birefringentEffect脉冲全息法pulsed holography透射式光弹性仪transmission polariscope 实时全息干涉法real-time holographicinterfero - metry 网格法grid method全息光弹性法holo-photoelasticity 全息图Hologram全息照相Holograph全息干涉法holographic interferometry 全息云纹法holographic moire technique 全息术Holography全场分析法whole-field analysis散斑干涉法speckle interferometry 散斑Speckle错位散斑干涉法speckle-shearinginterferometry, shearography 散斑图Specklegram白光散斑法white-light speckle method 云纹干涉法moire interferometry [叠栅]云纹moire fringe[叠栅]云纹法moire method 云纹图moire pattern离面云纹法off-plane moire method参考栅reference grating试件栅specimen grating分析栅analyzer grating面内云纹法in-plane moire method脆性涂层法brittle-coating method条带法strip coating method坐标变换transformation ofCoordinates计算结构力学computational structuralmecha-nics加权残量法weighted residual method 有限差分法finite difference method 有限[单]元法finite element method 配点法point collocation里茨法Ritz method广义变分原理generalized variationalPrinciple 最小二乘法least square method胡[海昌]一鹫津原理Hu-Washizu principle赫林格-赖斯纳原理Hellinger-ReissnerPrinciple 修正变分原理modified variationalPrinciple 约束变分原理constrained variationalPrinciple混合法mixed method杂交法hybrid method边界解法boundary solution method有限条法finite strip method半解析法semi-analytical method协调元conforming element非协调元non-conforming element混合元mixed element杂交元hybrid element边界元boundary element 强迫边界条件forced boundary condition 自然边界条件natural boundary condition 离散化Discretization离散系统discrete system连续问题continuous problem广义位移generalized displacement广义载荷generalized load广义应变generalized strain广义应力generalized stress界面变量interface variable节点node, nodal point [单]元Element角节点corner node边节点mid-side node内节点internal node无节点变量nodeless variable 杆元bar element桁架杆元truss element梁元beam element二维元two-dimensional element 一维元one-dimensional element 三维元three-dimensional element 轴对称元axisymmetric element板元plate element壳元shell element厚板元thick plate element三角形元triangular element四边形元quadrilateral element 四面体元tetrahedral element曲线元curved element二次元quadratic element线性元linear element三次元cubic element四次元quartic element等参[数]元isoparametric element超参数元super-parametric element 亚参数元sub-parametric element节点数可变元variable-number-node element 拉格朗日元Lagrange element拉格朗日族Lagrange family巧凑边点元serendipity element巧凑边点族serendipity family无限元infinite element单元分析element analysis单元特性element characteristics 刚度矩阵stiffness matrix几何矩阵geometric matrix等效节点力equivalent nodal force节点位移nodal displacement节点载荷nodal load位移矢量displacement vector载荷矢量load vector质量矩阵mass matrix集总质量矩阵lumped mass matrix相容质量矩阵consistent mass matrix阻尼矩阵damping matrix瑞利阻尼Rayleigh damping刚度矩阵的组集assembly of stiffnessMatrices载荷矢量的组集consistent mass matrix质量矩阵的组集assembly of mass matrices 单元的组集assembly of elements局部坐标系local coordinate system局部坐标local coordinate面积坐标area coordinates体积坐标volume coordinates曲线坐标curvilinear coordinates静凝聚static condensation合同变换contragradient transformation 形状函数shape function试探函数trial function检验函数test function权函数weight function样条函数spline function代用函数substitute function降阶积分reduced integration零能模式zero-energy modeP收敛p-convergenceH收敛h-convergence掺混插值blended interpolation等参数映射isoparametric mapping双线性插值bilinear interpolation小块检验patch test非协调模式incompatible mode节点号node number单元号element number带宽band width带状矩阵banded matrix变带状矩阵profile matrix带宽最小化minimization of band width 波前法frontal method子空间迭代法subspace iteration method 行列式搜索法determinant search method 逐步法step-by-step method纽马克法Newmark威尔逊法Wilson拟牛顿法quasi-Newton method牛顿-拉弗森法Newton-Raphson method 增量法incremental method初应变initial strain初应力initial stress切线刚度矩阵tangent stiffness matrix 割线刚度矩阵secant stiffness matrix 模态叠加法mode superposition method 平衡迭代equilibrium iteration子结构Substructure子结构法substructure technique超单元super-element网格生成mesh generation结构分析程序structural analysis program 前处理pre-processing后处理post-processing网格细化mesh refinement应力光顺stress smoothing组合结构composite structure。
268区域治理ON THE W AY作者简介：林海都，生于1993年，研究生，研究方向为发展与教育心理学。
尺寸一致性效应及理论解释江西师范大学 林海都摘要：尺寸作为个体知觉辨认最初的最基本的维度，其与数字加工的关系密不可分，研究者把数字和尺寸的关系称之为尺寸一致性效应(Henik& Tzelgov, 1982)。
该效应通常采用数字Stroop范式加以研究。
本文阐述了尺寸一致性效应的发现、含义和理论解释，有助于研究者了解尺寸一致性效应，并进一步探讨该效应背后的机制。
关键词：尺寸一致性效应；数字Stroop；数字加工中图分类号：S781.62文献标识码：A文章编号：2096-4595（2020）34-0268-0001一、尺寸一致性效应的发现数字数量与物理尺寸存在交互，被称之为尺寸一致性效应(size congruity effect)，或称数字Stroop 效应(numerical stroop effect)。
在数字Stroop 任务中，被试需要对同时呈现的不同物理尺寸大小的数字对进行数量比较（或尺寸比较），并忽略物理尺寸（数量）的影响。
当判断哪个数字数量更大时，会受到与任务无关的尺寸维度信息的影响。
具体而言，被试对两个数字进行数量比较时，数量较大的数字刺激同时具有较大的物理尺寸时反应更快，称为一致条件（如2 7）；当数量较大的数字刺激的物理尺寸较小时，则为不一致条件（如2 7），且反应变慢。
同样的，在进行尺寸比较任务中，与任务无关的数量信息也会影响到尺寸的比较(Besner& Coltheart, 1979; Henik& Tzelgov,1982)。
由于这种具有双向影响的尺寸一致性效应是快速且自动发生的，它可能意味着数量和物理尺寸两种不同的信息存在于同个数量加工机制中(Kadosh, Lammertyn& Izard, 2008)，至少并非是完全独立的两个维度(Santens& Verguts,2011)。
斯仑贝谢所有测井曲线英文名称解释OCEAN DRILLING PROGRAMACRONYMS USED FOR WIRELINE SCHLUMBERGER TOOLS ACT Aluminum Clay ToolAMS Auxiliary Measurement SondeAPS Accelerator Porosity SondeARI Azimuthal Resistivity ImagerASI Array Sonic ImagerBGKT Vertical Seismic Profile ToolBHC Borehole Compensated Sonic ToolBHTV Borehole TeleviewerCBL Casing Bond LogCNT Compensated Neutron ToolDIT Dual Induction ToolDLL Dual LaterologDSI Dipole Sonic ImagerFMS Formation MicroScannerGHMT Geologic High Resolution Magnetic ToolGPIT General Purpose Inclinometer ToolGR Natural Gamma RayGST Induced Gamma Ray Spectrometry ToolHLDS Hostile Environment Lithodensity SondeHLDT Hostile Environment Lithodensity ToolHNGS Hostile Environment Gamma Ray SondeLDT Lithodensity ToolLSS Long Spacing Sonic ToolMCD Mechanical Caliper DeviceNGT Natural Gamma Ray Spectrometry ToolNMRT Nuclear Resonance Magnetic ToolQSST Inline Checkshot ToolSDT Digital Sonic ToolSGT Scintillation Gamma Ray ToolSUMT Susceptibility Magnetic ToolUBI Ultrasonic Borehole ImagerVSI Vertical Seismic ImagerWST Well Seismic ToolWST-3 3-Components Well Seismic ToolOCEAN DRILLING PROGRAMACRONYMS USED FOR LWD SCHLUMBERGER TOOLSADN Azimuthal Density-NeutronCDN Compensated Density-NeutronCDR Compensated Dual ResistivityISONIC Ideal Sonic-While-DrillingNMR Nuclear Magnetic ResonanceRAB Resistivity-at-the-BitOCEAN DRILLING PROGRAMACRONYMS USED FOR NON-SCHLUMBERGER SPECIALTY TOOLSMCS Multichannel Sonic ToolMGT Multisensor Gamma ToolSST Shear Sonic ToolTAP Temperature-Acceleration-Pressure ToolTLT Temperature Logging ToolOCEAN DRILLING PROGRAMACRONYMS AND UNITS USED FOR WIRELINE SCHLUMBERGER LOGSAFEC APS Far Detector Counts (cps)ANEC APS Near Detector Counts (cps)AX Acceleration X Axis (ft/s2)AY Acceleration Y Axis (ft/s2)AZ Acceleration Z Axis (ft/s2)AZIM Constant Azimuth for Deviation Correction (deg)APLC APS Near/Array Limestone Porosity Corrected (%)C1 FMS Caliper 1 (in)C2 FMS Caliper 2 (in)CALI Caliper (in)CFEC Corrected Far Epithermal Counts (cps)CFTC Corrected Far Thermal Counts (cps)CGR Computed (Th+K) Gamma Ray (API units)CHR2 Peak Coherence, Receiver Array, Upper DipoleCHRP Compressional Peak Coherence, Receiver Array, P&SCHRS Shear Peak Coherence, Receiver Array, P&SCHTP Compressional Peak Coherence, Transmitter Array, P&SCHTS Shear Peak Coherence, Transmitter Array, P&SCNEC Corrected Near Epithermal Counts (cps)CNTC Corrected Near Thermal Counts (cps)CS Cable Speed (m/hr)CVEL Compressional Velocity (km/s)DATN Discriminated Attenuation (db/m)DBI Discriminated Bond IndexDEVI Hole Deviation (degrees)DF Drilling Force (lbf)DIFF Difference Between MEAN and MEDIAN in Delta-Time Proc. (microsec/ft) DRH HLDS Bulk Density Correction (g/cm3)DRHO Bulk Density Correction (g/cm3)DT Short Spacing Delta-Time (10'-8' spacing; microsec/ft)DT1 Delta-Time Shear, Lower Dipole (microsec/ft)DT2 Delta-Time Shear, Upper Dipole (microsec/ft)DT4P Delta- Time Compressional, P&S (microsec/ft)DT4S Delta- Time Shear, P&S (microsec/ft))DT1R Delta- Time Shear, Receiver Array, Lower Dipole (microsec/ft)DT2R Delta- Time Shear, Receiver Array, Upper Dipole (microsec/ft)DT1T Delta-Time Shear, Transmitter Array, Lower Dipole (microsec/ft)DT2T Delta-Time Shear, Transmitter Array, Upper Dipole (microsec/ft)DTCO Delta- Time Compressional (microsec/ft)DTL Long Spacing Delta-Time (12'-10' spacing; microsec/ft)DTLF Long Spacing Delta-Time (12'-10' spacing; microsec/ft)DTLN Short Spacing Delta-Time (10'-8' spacing; microsec/ftDTRP Delta-Time Compressional, Receiver Array, P&S (microsec/ft)DTRS Delta-Time Shear, Receiver Array, P&S (microsec/ft)DTSM Delta-Time Shear (microsec/ft)DTST Delta-Time Stoneley (microsec/ft)DTTP Delta-Time Compressional, Transmitter Array, P&S (microsec/ft)DTTS Delta-Time Shear, Transmitter Array, P&S (microsec/ft)ECGR Environmentally Corrected Gamma Ray (API units)EHGR Environmentally Corrected High Resolution Gamma Ray (API units) ENPH Epithermal Neutron Porosity (%)ENRA Epithermal Neutron RatioETIM Elapsed Time (sec)FINC Magnetic Field Inclination (degrees)FNOR Magnetic Field Total Moment (oersted)FX Magnetic Field on X Axis (oersted)FY Magnetic Field on Y Axis (oersted)FZ Magnetic Field on Z Axis (oersted)GR Natural Gamma Ray (API units)HALC High Res. Near/Array Limestone Porosity Corrected (%)HAZI Hole Azimuth (degrees)HBDC High Res. Bulk Density Correction (g/cm3)HBHK HNGS Borehole Potassium (%)HCFT High Resolution Corrected Far Thermal Counts (cps)HCGR HNGS Computed Gamma Ray (API units)HCNT High Resolution Corrected Near Thermal Counts (cps)HDEB High Res. Enhanced Bulk Density (g/cm3)HDRH High Resolution Density Correction (g/cm3)HFEC High Res. Far Detector Counts (cps)HFK HNGS Formation Potassium (%)HFLC High Res. Near/Far Limestone Porosity Corrected (%)HEGR Environmentally Corrected High Resolution Natural Gamma Ray (API units) HGR High Resolution Natural Gamma Ray (API units)HLCA High Res. Caliper (inHLEF High Res. Long-spaced Photoelectric Effect (barns/e-)HNEC High Res. Near Detector Counts (cps)HNPO High Resolution Enhanced Thermal Nutron Porosity (%)HNRH High Resolution Bulk Density (g/cm3)HPEF High Resolution Photoelectric Effect (barns/e-)HRHO High Resolution Bulk Density (g/cm3)HROM High Res. Corrected Bulk Density (g/cm3)HSGR HNGS Standard (total) Gamma Ray (API units)HSIG High Res. Formation Capture Cross Section (capture units) HSTO High Res. Computed Standoff (in)HTHO HNGS Thorium (ppm)HTNP High Resolution Thermal Neutron Porosity (%)HURA HNGS Uranium (ppm)IDPH Phasor Deep Induction (ohmm)IIR Iron Indicator Ratio [CFE/(CCA+CSI)]ILD Deep Resistivity (ohmm)ILM Medium Resistivity (ohmm)IMPH Phasor Medium Induction (ohmm)ITT Integrated Transit Time (s)LCAL HLDS Caliper (in)LIR Lithology Indicator Ratio [CSI/(CCA+CSI)]LLD Laterolog Deep (ohmm)LLS Laterolog Shallow (ohmm)LTT1 Transit Time (10'; microsec)LTT2 Transit Time (8'; microsec)LTT3 Transit Time (12'; microsec)LTT4 Transit Time (10'; microsec)MAGB Earth's Magnetic Field (nTes)MAGC Earth Conductivity (ppm)MAGS Magnetic Susceptibility (ppm)MEDIAN Median Delta-T Recomputed (microsec/ft)MEAN Mean Delta-T Recomputed (microsec/ft)NATN Near Pseudo-Attenuation (db/m)NMST Magnetometer Temperature (degC)NMSV Magnetometer Signal Level (V)NPHI Neutron Porosity (%)NRHB LDS Bulk Density (g/cm3)P1AZ Pad 1 Azimuth (degrees)PEF Photoelectric Effect (barns/e-)PEFL LDS Long-spaced Photoelectric Effect (barns/e-)PIR Porosity Indicator Ratio [CHY/(CCA+CSI)]POTA Potassium (%)RB Pad 1 Relative Bearing (degrees)RHL LDS Long-spaced Bulk Density (g/cm3)RHOB Bulk Density (g/cm3)RHOM HLDS Corrected Bulk Density (g/cm3)RMGS Low Resolution Susceptibility (ppm)SFLU Spherically Focused Log (ohmm)SGR Total Gamma Ray (API units)SIGF APS Formation Capture Cross Section (capture units)SP Spontaneous Potential (mV)STOF APS Computed Standoff (in)SURT Receiver Coil Temperature (degC)SVEL Shear Velocity (km/s)SXRT NMRS differential Temperature (degC)TENS Tension (lb)THOR Thorium (ppm)TNRA Thermal Neutron RatioTT1 Transit Time (10' spacing; microsec)TT2 Transit Time (8' spacing; microsec)TT3 Transit Time (12' spacing; microsec)TT4 Transit Time (10' spacing; microsec)URAN Uranium (ppm)V4P Compressional Velocity, from DT4P (P&S; km/s)V4S Shear Velocity, from DT4S (P&S; km/s)VELP Compressional Velocity (processed from waveforms; km/s)VELS Shear Velocity (processed from waveforms; km/s)VP1 Compressional Velocity, from DT, DTLN, or MEAN (km/s)VP2 Compressional Velocity, from DTL, DTLF, or MEDIAN (km/s)VCO Compressional Velocity, from DTCO (km/s)VS Shear Velocity, from DTSM (km/s)VST Stonely Velocity, from DTST km/s)VS1 Shear Velocity, from DT1 (Lower Dipole; km/s)VS2 Shear Velocity, from DT2 (Upper Dipole; km/s)VRP Compressional Velocity, from DTRP (Receiver Array, P&S; km/s) VRS Shear Velocity, from DTRS (Receiver Array, P&S; km/s)VS1R Shear Velocity, from DT1R (Receiver Array, Lower Dipole; km/s) VS2R Shear Velocity, from DT2R (Receiver Array, Upper Dipole; km/s) VS1T Shear Velocity, from DT1T (Transmitter Array, Lower Dipole; km/s) VS2T Shear Velocity, from DT2T (Transmitter Array, Upper Dipole; km/s) VTP Compressional Velocity, from DTTP (Transmitter Array, P&S; km/s) VTS Shear Velocity, from DTTS (Transmitter Array, P&S; km/s)#POINTS Number of Transmitter-Receiver Pairs Used in Sonic Processing W1NG NGT Window 1 counts (cps)W2NG NGT Window 2 counts (cps)W3NG NGT Window 3 counts (cps)W4NG NGT Window 4 counts (cps)W5NG NGT Window 5 counts (cps)OCEAN DRILLING PROGRAMACRONYMS AND UNITS USED FOR LWD SCHLUMBERGER LOGSAT1F Attenuation Resistivity (1 ft resolution; ohmm)AT3F Attenuation Resistivity (3 ft resolution; ohmm)AT4F Attenuation Resistivity (4 ft resolution; ohmm)AT5F Attenuation Resistivity (5 ft resolution; ohmm)ATR Attenuation Resistivity (deep; ohmm)BFV Bound Fluid Volume (%)B1TM RAB Shallow Resistivity Time after Bit (s)B2TM RAB Medium Resistivity Time after Bit (s)B3TM RAB Deep Resistivity Time after Bit (s)BDAV Deep Resistivity Average (ohmm)BMAV Medium Resistivity Average (ohmm)BSAV Shallow Resistivity Average (ohmm)CGR Computed (Th+K) Gamma Ray (API units)DCAL Differential Caliper (in)DROR Correction for CDN rotational density (g/cm3).DRRT Correction for ADN rotational density (g/cm3).DTAB AND or CDN Density Time after Bit (hr)FFV Free Fluid Volume (%)GR Gamma Ray (API Units)GR7 Sum Gamma Ray Windows GRW7+GRW8+GRW9-Equivalent to Wireline NGT window 5 (cps) GRW3 Gamma Ray Window 3 counts (cps)-Equivalent to Wireline NGT window 1GRW4 Gamma Ray Window 4 counts (cps)-Equivalent to Wireline NGT window 2GRW5 Gamma Ray Window 5 counts (cps)-Equivalent to Wireline NGT window 3GRW6 Gamma Ray Window 6 counts (cps)-Equivalent to Wireline NGT window 4GRW7 Gamma Ray Window 7 counts (cps)GRW8 Gamma Ray Window 8 counts (cps)GRW9 Gamma Ray Window 9 counts (cps)GTIM CDR Gamma Ray Time after Bit (s)GRTK RAB Gamma Ray Time after Bit (s)HEF1 Far He Bank 1 counts (cps)HEF2 Far He Bank 2 counts (cps)HEF3 Far He Bank 3 counts (cps)HEF4 Far He Bank 4 counts (cps)HEN1 Near He Bank 1 counts (cps)HEN2 Near He Bank 2 counts (cps)HEN3 Near He Bank 3 counts (cps)HEN4 Near He Bank 4 counts (cps)MRP Magnetic Resonance PorosityNTAB ADN or CDN Neutron Time after Bit (hr)PEF Photoelectric Effect (barns/e-)POTA Potassium (%) ROPE Rate of Penetration (ft/hr)PS1F Phase Shift Resistivity (1 ft resolution; ohmm)PS2F Phase Shift Resistivity (2 ft resolution; ohmm)PS3F Phase Shift Resistivity (3 ft resolution; ohmm)PS5F Phase Shift Resistivity (5 ft resolution; ohmm)PSR Phase Shift Resistivity (shallow; ohmm)RBIT Bit Resistivity (ohmm)RBTM RAB Resistivity Time After Bit (s)RING Ring Resistivity (ohmm)ROMT Max. Density Total (g/cm3) from rotational processing ROP Rate of Penetration (m/hr)ROP1 Rate of Penetration, average over last 1 ft (m/hr).ROP5 Rate of Penetration, average over last 5 ft (m/hr)ROPE Rate of Penetration, averaged over last 5 ft (ft/hr)RPM RAB Tool Rotation Speed (rpm)RTIM CDR or RAB Resistivity Time after Bit (hr)SGR Total Gamma Ray (API units)T2 T2 Distribution (%)T2LM T2 Logarithmic Mean (ms)THOR Thorium (ppm)TNPH Thermal Neutron Porosity (%)TNRA Thermal RatioURAN Uranium (ppm)OCEAN DRILLING PROGRAMADDITIONAL ACRONYMS AND UNITS(PROCESSED LOGS FROM GEOCHEMICAL TOOL STRING)AL2O3 Computed Al2O3 (dry weight %)AL2O3MIN Computed Al2O3 Standard Deviation (dry weight %) AL2O3MAX Computed Al2O3 Standard Deviation (dry weight %) CAO Computed CaO (dry weight %)CAOMIN Computed CaO Standard Deviation (dry weight %) CAOMAX Computed CaO Standard Deviation (dry weight %) CACO3 Computed CaCO3 (dry weight %)CACO3MIN Computed CaCO3 Standard Deviation (dry weight %) CACO3MAX Computed CaCO3 Standard Deviation (dry weight %) CCA Calcium Yield (decimal fraction)CCHL Chlorine Yield (decimal fraction)CFE Iron Yield (decimal fraction)CGD Gadolinium Yield (decimal fraction)CHY Hydrogen Yield (decimal fraction)CK Potassium Yield (decimal fraction)CSI Silicon Yield (decimal fraction)CSIG Capture Cross Section (capture units)CSUL Sulfur Yield (decimal fraction)CTB Background Yield (decimal fraction)CTI Titanium Yield (decimal fraction)FACT Quality Control CurveFEO Computed FeO (dry weight %)FEOMIN Computed FeO Standard Deviation (dry weight %) FEOMAX Computed FeO Standard Deviation (dry weight %) FEO* Computed FeO* (dry weight %)FEO*MIN Computed FeO* Standard Deviation (dry weight %) FEO*MAX Computed FeO* Standard Deviation (dry weight %) FE2O3 Computed Fe2O3 (dry weight %)FE2O3MIN Computed Fe2O3 Standard Deviation (dry weight %) FE2O3MAX Computed Fe2O3 Standard Deviation (dry weight %) GD Computed Gadolinium (dry weight %)GDMIN Computed Gadolinium Standard Deviation (dry weight %) GDMAX Computed Gadolinium Standard Deviation (dry weight %) K2O Computed K2O (dry weight %)K2OMIN Computed K2O Standard Deviation (dry weight %)K2OMAX Computed K2O Standard Deviation (dry weight %) MGO Computed MgO (dry weight %)MGOMIN Computed MgO Standard Deviation (dry weight %) MGOMAX Computed MgO Standard Deviation (dry weight %)S Computed Sulfur (dry weight %)SMIN Computed Sulfur Standard Deviation (dry weight %) SMAX Computed Sulfur Standard Deviation (dry weight %)SIO2 Computed SiO2 (dry weight %)SIO2MIN Computed SiO2 Standard Deviation (dry weight %) SIO2MAX Computed SiO2 Standard Deviation (dry weight %) THORMIN Computed Thorium Standard Deviation (ppm) THORMAX Computed Thorium Standard Deviation (ppm)TIO2 Computed TiO2 (dry weight %)TIO2MIN Computed TiO2 Standard Deviation (dry weight %) TIO2MAX Computed TiO2 Standard Deviation (dry weight %) URANMIN Computed Uranium Standard Deviation (ppm) URANMAX Computed Uranium Standard Deviation (ppm) VARCA Variable CaCO3/CaO calcium carbonate/oxide factor。
花样滑冰专业术语花样滑冰 figure skating 花样滑冰鞋 figure skate 冰场 ice arena; rink 起滑脚 starting foot 冰刀套 skate guard 花样 pattern起跳 take off奥运会花样滑冰四个项目男子单人滑 men女子单人滑 ladies双人滑 pairs冰舞 ice dancing(包括:规定舞 compulsory dance创编舞 original dance 和自由舞 freedance)男子单人滑和女子单人滑两个项目自由滑 free skating 短节目 short program花样滑冰技术动作特种圆形 advanced figure 半周、半圆 half-circle 旋转 spin燕式旋转 arabesque spin 倒滑;退滑 back skating 直立滑行 ride冰上表演 ice show艺术印象 artistic impression附加动作 additional move 前一周半跳 axel-paulsen(源自阿克塞尔?保尔森，19世纪挪威花样溜冰选手)旋转轴 axle of revolution 倒滑压步 back cross over 后内刃 back in 后内括弧形 back in bracket 后内变刃形 back in change 后内圆形 back in circle 后内外勾形 back in counter 后内结环形 back in loop后内勾形 back in rocker 后内环绕，后内螺旋形 back in spiral 后辅刃back out后内括弧形 back out bracket 后外变刃形 back out change 后外圆形 back out circle 后外勾形 back out counterback out loop 后外结环形后外环绕，后外螺旋形 back out spiral 环绕图形 spiral figure单脚蹲转 Haines spin单脚直立旋转 one-foot upright spin单脚括弧形 paragraph bracket 单脚结环形 paragraph loop 单个动作individual part 单人旋转 solo spin抱身 hand-to-body grip 连接动作 connecting move 连接步 connecting step 步法 footwork技术水平 technical merit 侧翻举 cartwheel lift变刃 change of edge借跳 partner-assisted jump 旋转动作 spinning movement 滑脚 employed foot;gliding foot;tracting foot滑步 glide滑弧线 curved stroke滑区 skating area编排 combination硬转身 forced turn集体动作 group move外刃 outside;outer edge内刃 inner edge;inside右脚外刃 right outside右脚内刃 right inside用刀刃蹬冰 stroke from the edge 用刀齿蹬冰 stroke from the point of the skate握臂 hand-to-arm grip 携手 hand-to-hand grip 稳健 sureness撑竿式跳 pole vaule jump 横一字 spread eagle镜子滑 mirror skating刀齿旋转 pirouette;toe-scratch 刀齿向下 toe of the skate pointing downward刀齿蹬冰 toe push双人旋转 pair spin分腿 split分腿举 split lift平刃旋转 flat-foot spin 规定图形滑 skating of prescribed movements 同脚变刃的前进跳跃 forward jump 前刃变后刃半周跳 Mohawk评分 assign marks人工冷冻冰场artificially frozen rink “6”字形 figure six“8”字形 curve eight浮脚 free foot双人动作 pair move转体 turn转体半周 half turn转体两周 double turn圆形滑 concentric stroking 同足后内结环一周跳 one-foot Salchow jump 异足后外结环一周跳 half loop jump 后内点冰一周跳 flip jump;toe Salchow 后内点冰“3”字跳 half toe Salchow。
a rX iv:mat h /69598v2[mat h.CA]12D ec26A BSTRACT .–We prove that in finite time a trajectory of a Lipschitz vector field in R n can not have infinite rotation around a given point.This result extends to the mutual rotation of two trajectories of a field in R 3:this rotation is bounded from above on any finite time interval.The bounds we give are only in terms of the Lipschitz constant of the field and the length of the time interval.1.Introduction In this paper we investigate the tameness of a geometric behavior of trajectories of vector fields:the rotation of such a trajectory around a point or the mutual rotation of two trajectories.Of course a lot of results have been published on the geometry of solutions of differential equations and trajectories of vector fields,and we simply cannot give an extensive review or even a bibliography on the subject.Nevertheless,we have at our disposal only few global theories and general results concerning the tameness of trajectories.In this very short introduction we just want to focus on two of them.In Gabrielov-Hovanskii’s theory of Pfaffian sets,a Pfaffian function f on an open set U ⊂R n ,is defined as a function that can be written in the following way:f (x )=P (x,c 1(x ),···,c m (x )),where P is a polynomial and the c i ’s are analytic solutions of thepolynomial triangular differential system:Dc i (x )=mj =1P i,j (x,c 1(x ),···,c i (x ))dx j ,i ∈{1,···,m }.(∗)Then if we consider the Pfaffian structure,that is to say the smallest structure containing the semi-Pfaffian sets,it has been proved in [Wi],using a Bezout type theorem of Hovanskii([Ho1],[Ho2]),that this structure is o-minimal(see[Co],[Dr],[Dr-Mi],[Sh]):for short the number of connected components of sets in such a structure isfinite,and consequently no Pfaffian curve(nor set)may infinitely oscillate or spiral around a point and two such curves have bounded mutual rotation.This theory presents a large class of tame objects coming from differential equations, but when we deal with vectorfields,the only way to a priori be sure that the trajectories belong to this category is to assume that the trajectory is a Pfaffian function itself, satisfying equation(∗).We deduce from this assumption that thefield depends only on one variable.A too restrictive hypothesis that obviously not allows a wild behavior for trajectories.On the other hand,starting with a given vectorfield and studying the local geometry of its trajectories in a neighborhood of one of its singularities,we know that this geometry is tame,following[Ku-Mo]and[Ku-Mo-Pa],provided that thefield is an analytic gradient vectorfield:the rotation of a trajectory of an analytic gradient vectorfield around one of its singularities isfinite.As a consequence the limit of the secant lines to the trajectory exists (Thom’s Gradient Conjecture).Let us notice that we do not know wether a trajectory of an analytic gradient vectorfield lies in some o-minimal structure,although we know that trajectories of the gradient of a function definable in a given o-minimal structure have finite length(see[Ku]).On the local behavior of trajectories we would like to refer to the number of deep results produced by the Spanish-Dijon School,and specially the most recent ones:[Ca-Mo-Ro],[Ca-Mo-Sa1,2,3],[Bl-Mo-Ro].The aim of this paper is to give general results about rotation of trajectories of a vectorfield,with no restrictive assumptions on the nature of thefield,besides the Lipschitz property which is a minimal hypothesis for the existence of trajectories.Of course,in this direction,we cannot hope to treat infinite time phenomena,as it is done in[Ku-Mo-Pa]and the Pfaffian theory,because it is well-known that complete trajectories of polynomial vectorfields in the plane may spiral around a singularity of the field(see for instance Remark of Section3.1).We thus obtain our bounds for trajectories defined on afinite time interval.For bounds obtained in the same spirit,the reader may report to[Gr-Yo],[Ho-Ya],[No-Ya1,2],[Ya].The paper is organized as follows.In Section2we introduce and compare some notions of absolute and topological rotation for trajectories around an affine subspace in R n or for two trajectories around each others in R3.In Section3wefirst notice that the rotation of any trajectory of a Lipschitz vectorfield around its stationary points is bounded in terms of the Lipschitz constant(and the time elapsed)only(Proposition3.1).The same is true for the rotation of any trajectory around a linear invariant subspace of thefield(Proposition3.2).Moreover,while the rotation velocity of a trajectory around a non-stationary point of thefield may tend to infinity,we prove ourfirst main result:the“total”rotation around such a point is still bounded in terms of the Lipschitz constant and the time interval(Theorem3.4).Our second main result is a consequent of thefirst one:we give a uniform bound for the rotation of any two trajectories of a given Lipschitz vectorfield,in terms of the time interval and the Lipschitz constant(Theorem3.8).In contrast,we provide an easy example showing that the rotation of a trajectory of C∞vectorfield around a non-invariant subspace may beinfinite infinite time(Example3.3).2.Definition of signed and absolute rotation2.1.Rotation around an affine subspaceFirst of all,we define an absolute rotation of the curveγin R n around the origin 0∈R n.Assuming thatγdoes not pass through the origin,we can define the spherical image ofγas the curveσin the unit sphere S n−1,in the following way:σ(t)=γ(t)dt be the velocity vector ofγ:I→R n,and letγ′r(t)andγ′s(t)be the radial and the spherical components of this velocity vector.Then the velocity of the spherical blowing-upσofγis:σ′(t)=γ′s(t)γ(t)dt.Remark.The absolute rotation R abs(γ,0)is invariant with respect to the monotone reparametrizations of the curveγand in the same time we want our results to be about the geometry of the curve and not depending of one of its parametrizations.This is why in what follows we implicitly consider the geometric trajectoryγ(I)of the injective curve γ:I→γ(I)as the class ofγ:I→R n modulo its injective reparametrizations.For plane curvesγin R2we can define their“signed rotation”R(γ,0)around the origin,which is,of course,the usual rotation index.Indeed,in this case the unit sphere is a circle.Assuming that the orientation of this circle(and of the curveγ)has been chosen, we can define the signed rotation,essentially,by the same expression as above: Definition.The signed rotation R(γ,0)ofγaround the origin is defined by:R(γ,0)=1γ(t)dt.Here the sign under the integral is chosen according to the direction of the tangent to the unit circle vectorγ′s(t).Remark.Notice the normalization by2π,the unit circle length,which appears here to make the linking number,defined below,an integer.In codimension greater or equal to three we do not normalize the length of the spherical curves.An absolute rotation of the curveγin R n around a linear k-dimensional subspace L⊂R n is defined as follows:let L⊥denote the orthogonal subspace to L.Let˜γbe the projection ofγon L⊥.Assuming thatγdoes not touch L,we get˜γnot passing through the origin in L⊥.Definition.The absolute rotation R abs(γ,L)is defined as the absolute rotation of the curve˜γin L⊥around the origin.In the case of a linear subspace L of codimension 2in R n,a signed rotation R(γ,L)ofγaround L is defined as the signed rotation of the curve˜γin the plane L⊥around the origin.Of course the absolute rotation always bounds from above the absolute value of the signed one.For a closed curveγand for a subspace L of codimension2the signed rotation R(γ,L) is an integer,and it is a topological invariant.Forγnon-closed this signed rotation R(γ,L)may take non-integer values.However,it is still an invariant of deformations of γ,preserving the end points,and not touching L.2.2.Rotation of two curves in R3It is well known that the linking number of two closed curves in R3can be defined via an integral expression,the so-called Gauss integral(see,for example[Ar-Kh],[Du-Fo-No]). This gives us a natural way to define also an absolute and a signed rotation of two curves (closed or non-closed)one around the other.For non-closed curves the rotation defined in this(or any other)way cannot be metri-cally or topologically invariant.But on the other hand,the Gauss integral representation provides a powerful analytic tool for its investigation.Our presentation in the next sub-section follows very closely the one given in[Du-Fo-No].2.2.1.The Linking CoefficientConsider a pair of smooth,closed,regular directed curves in R3,which do not intersect.We may assume them to be parametrized in the following way:γi:I i→R3, with I1,I2two compact intervals.We denote our geometric curves byγ1andγ2(instead ofγ1(I1),γ2(I2)).Definition.The linking coefficient of the two curvesγ1,γ2is defined,in terms of the “Gauss integral”,by:{γ1,γ2}=1γ12(t1,t2) 3dt1dt2,whereγ12(t1,t2)=γ2(t2)−γ1(t1).Remark.In this definition the normalization by4πhas to be seen as the normal-ization by the volume of the unit sphere of R3(see the Remark that follows the proof of Theorem2.2).Let us stress that this definition immediately shows that{γ1,γ2}does not depend on the parametrizaton of the curves nor on their rigid transformations.Intuitively speaking the linking coefficient gives the algebraic(i.e.signed)number of loops of one contour around the other.This interpretation is justified by the following result.Theorem 2.2.—Letγ1andγ2be two closed curves in R3and assume that I1=[0,2π].(i)The linking coefficient{γ1,γ2}is an integer,and is unchanged by deformations ofγ1andγ2,involving no intersection of one curve with the other.(ii)Let F:D2→R3be a map of the disc D2which agrees withγ1:t→γ1(t),0≤t≤2π, on the boundary∂D2≃S1≃[0,2π]γ1(t1)−γ2(t2)is well defined.An easy geometric consideration shows that the integrand in the Gauss integral is just the Jacobian of the mapϕ.Therefore the Gauss integral above is equal to the degree of the mapϕ.Hence the linking coefficient is indeed an integer.Under deformations of the curves{γ1,γ2}involving no intersection one with the other,the map ϕundergoes a homotopy,so that its degree,and therefore also the linking coefficient,are preserved.Let us stress that in the process of these deformations each of the curvesγ1,γ2 separately may cross itself in an arbitrary way.Of course,the topological linking number is also preserved by such deformations.We now prove(ii).If the curves are not linked(i.e.if by means of a homotopy respecting non-intersection they can be brought to opposite sides of a2-dimensional plane in R3)then it can be verified directly that{γ1,γ2}=degϕ=0.In a general case we can “push”γ1alongγ2in such a way that after this deformation it comes close toγ2only in a neighborhood of exactly one point.Then by applying another deformation(remind that self-intersections of the curves are allowed)we reduce the general case of the problem of calculating the linking coefficient essentially to the following simple situation:the curve γ2is a straight line,whileγ1is a circle,orthogonal toγ1and passed several times in the positive or negative direction.Thus we supposeγ1andγ2to be given respectively by γ1(t1)=(cos t1,sin t1,0),0≤t1≤2πandγ2(t2)=(0,0,t2),−∞<t2<∞.The linking coefficient for these two curves is:{γ1,γ2}=1(1+t22)3/2=1(1+t22)3/2=1ch2(u)=14πt1∈I1t2∈I2<γ′1∧γ′2,γ12> 4πt1∈I1t2∈I2|<γ′1∧γ′2,γ12>|segmentγ1,and then contract it to the point.The rotation of the deformed curves is zero, so it was not preserved in the process of the deformation.However,for one of the curves,sayγ1,closed,we have the following result:Proposition2.3.—Let the curveγ1be closed.Then the signed rotation R(γ1,γ2) is invariant under the deformations of the curveγ2(without crossingγ1)if the end-points ofγ2remainfixed.Proof.Consider a closed curve˜γ2obtained fromγ2by passing it twice in the opposite directions.We have R(γ1,˜γ2)=0,since the rotation of these two closed curves is invariant under deformation,while˜γ2can be deformed into the point without crossingγ1.Now consider another deformation of˜γ2,where one copy ofγ2remainsfixed,while another copy,ˆγ2,undergoes a deformation without crossingγ1and with the end pointsfixed.The signed rotation remain zero in this deformation,so R(γ1,˜γ2)=R(γ1,γ2)−R(γ1,ˆγ2)=0. Hence R(γ1,ˆγ2)remains the same in the deformation.2Another property which can be obtained by a rather straightforward computation,is the following:Proposition2.4.—Forγ1=L a straight line,the signed rotation R(γ1,γ2)(resp. the absolute rotation R abs(γ1,γ2))given by the Gauss integral coincides with the signed rotation R(γ2,L)(resp.the absolute rotation R abs(γ2,L))of the curveγ2around the straight line L,as defined by projection on L⊥in Section2.1above.The absolute rotation R abs(γ1,γ2)given by the expression(2.7)coincides with the absolute rotation of the curveγ2around the straight lineγ1,as defined in Section2.1.Proof.Let us start with the case of the signed rotation.We have,passing to the length parametrization,R(γ1,γ2)=1γ12 3ds1ds2,where n1,n2are the unit tangent vectors to the curvesγ1,γ2,and s1,s2are the length parameters on the curvesγ1,γ2,respectively.Letη2(s2)denote the vector joining the point s2∈γ2with the projection of s2onto the straight lineγ1(see Figure1).We have<n1∧n2,γ12>=<n1∧n2,η2>.Hence:R(γ1,γ2)=1γ12 3ds1ds2.The vector n1,being the tangent vector to the straight lineγ1,is constant.Therefore,the triple product under the above integral depends only on s2and we have:R(γ1,γ2)=1γ12(s1,s2) 3ds2.Now the integral:I1ds1over the straight line γ1has been computed (up to a scaling by the distance η2(s 2) to the line)at the end of Section 2.2.1above.It is equal to22π I 2<n 1∧n 2(s 2),η2(s 2)>η2 ,with the sign defined by the orientation,and we obtain:R (γ1,γ2)=1η2 ds 2.(1)But η2is just the radius-vector of the projection of the curve γ2onto the plane orthogonal to the line γ1,and ˆn 2is the orthogonal component of the velocity vector of this projection.Therefore,according to Lemma 2.1and up to a sign ǫ,the integral (1)is the signed rotation R (γ2,L )of γ2around the straight line L =γ1,as defined in Section 2.1(the sign ǫis +when in the computation of R (γ2,L )we have oriented the plane L ⊥in such a way that if ( u , v )is an oriented orthonormal basis of L ⊥, u ∧ v =−n 1(see Figure 1)).This completes the proof of the Proposition for the signed rotation.γn 12n 1η2η2^γ2γ1Figure 1The proof for the absolute rotation is exactly the same:we just take the absolute value of the triple product in each step.Since the proof above consisted of a chain of point-wise equalities of the integrands,it remains valid also for the absolute values.23.Rotation of trajectories of Lipschitz vectorfieldsFor a Lipschitz vectorfield in R n,the very simple and basic fact is that the angular velocity of its trajectories with respect to any stationary point is bounded by the Lipschitz constant K.As an immediate consequence,the rotation of a trajectory around a stationary point is bounded(up to some universal constant C)by the Lipschitz constant K and the time interval T.The bound has this form:C·K·T.Below we remind the proof of this fact.In fact,we show that the same is true for the rotation speed of the trajectories around any linear subspace L⊂R n,which is an invariant submanifold of ourfield.The main result of this section(Theorem3.4)is that the“rotation”of a trajectory around any point(stationary or not)is bounded in terms of K and T.As the consequence we prove(Theorem3.8)that the mutual rotation of any two trajectories of a Lipschitz vectorfield is essentially bounded by the Lipschitz constant and the length of the time interval.More accurately,the absolute rotation of any two trajectories of a Lipschitz vectorfield on the time intervals T1and T2,respectively,is bounded from above by a linear combination of the expressions K·min(T1,T2)and C·K2·T1·T2.Easy examples, given in the end,show that this bound is sharp.At this point,let us remind that the rotation of a curveω:I→R n around a point, an affine space or another curve,as defined above,only depends on the trajectoryω(I), provided we only admit injective(parametrization of)curves.Consequently our bounds not really depend on thefield,but rather on the geometry of the trajectories of thefields. This is obvious when we look at the type of bounds we obtain:(up to some universal constant)they are combinations of K·T and K2·T1·T2,expressions that are unchanged with respect to any transformation of thefield that preserve the trajectories.For Lipschiz vectorfields,we can summarize the situation by the following slogan:“two trajectories havefinite mutual rotation infinite time”.We cannot expect bounds of this sort to be true for a rotation around a non-invariant subspace of thefield.Indeed,the easy-to-construct Example3.3below shows that a trajectory of a C∞-vectorfield in R3can make infinite time an infinite number of turns around a straight line.Let us remind that it was shown(in some special cases)in[Gr-Yo]that for a trajectory of a polynomial vectorfield(trajectory which is not in general in some o-minimal structure),its rotation rate around any algebraic submanifold is bounded in terms of the degree of the submanifold and the degree and size of the vectorfield.As a consequence, we obtain also a linear in time bound on the number of intersections of the trajectory with any algebraic hypersurface.On the other hand,our bounds on the rotation rate of the trajectories of a polynomial vectorfield imply upper bounds on the multiplicities of the local intersections of such trajectories with algebraic submanifolds in terms of the degree only.As the Example3.3shows,nothing of this sort can be expected even for C∞(and of course,for Lipschitz)vectorfields.Still,the results of this section show that Lipschitzvectorfields exhibit rather strong non-oscillation patterns.As far as the rotation of the trajectories of suchfields around non-invariant submanifolds is concerned,our current understanding is far from being sufficient.In particular,there is a serious gap between the result of Theorem3.4below that a“global”rotation rate of a trajectory of a Lipschitz vectorfield around a non-stationary point is still bounded,and the Example3.3, demonstrating an infinite rotation around a non-invariant straight line.3.1.Rotation of a trajectory around a stationary pointLet v be a vectorfield defined in a certain domain U in R n.We shall always assume v to satisfy a Lipschitz condition with the constant K:v(x)−v(y) ≤K x−y ,for any two points x,y∈ULet x0∈U be a stationary or a singular point of v,ie v(x0)=0,so that the constant curve c(t)=x0,t∈R,is the integral curve of v passing through x0.Then for any x∈U,x=x0, the angular velocity of the trajectory of v,passing through x,with respect to x0,is equal to ˆv(x) / x−x0 ,whereˆv(x)is the projection of the vector v(x)to the hyperplane orthogonal to x−x0.Hence,this angular velocity does not exceed K:ˆv(x)x−x0 =v(x)−v(x0)ω(t)−x0between t1and t2,does not exceed K·(t2−t1).Remark.Of course on a non-finite time interval,a trajectory may have infinite local rotation around a stationary point,as shown by the example given below(we avoid the obvious example of a cyclic trajectory,since we aim to work with injective trajectories).Let us consider the following algebraicfield in R2with singular point O=(0,0):v(x,y)= (x2+y2−1)x−y,(x2+y2−1)y+x ,and introduce the following notations:r2=x2+y2, τ=(x,y)/r and n=(−y,x)/r.We have:v(x,y)=r(r2−1)· τ+r· n.A trajectory passing at a point p with r(p)=1has to be the unit circle.Now consider ω=(α,β)an integral curve of v passing through a point q with r(q)<1.This trajectory cannot go outside the open unit disc and as we have:d[r2◦ω]dt =2(r2−1)(α2+β2)<0.This proves that the trajectoryωhas the singular pointO as limit(while the unit circle has to be a limit cycle of this trajectory).Furthermore we know that from the point q,the limit O is approached on an infinite time interval I.On the other hand,the velocity v of the spherical blowing-upσofωis:1x−x0 ≤˜v(x)x−x0≤v(x)−v(x0)Here we use the fact that L is invariant for v and hence˜v(x0)=0.By Definition we obtain that the absolute rotation of any trajectory w(t)of thefield v around L in time T does not exceed K·T.This completes the proof of Proposition3.2.2 If x0is not a stationary point of v,the angular velocity of v(x)with respect to x0 tends to infinity as x approaches x0in any direction transverse to v(x0).Indeed,as x approaches x0,v(x)tends to v(x0)=0.By Lemma2.1,the angular velocity of v(x)with respect to x0is equal to ˆv(x) / x−x0 ,whereˆv(x)is the component of v(x)orthogonal to x−x0.So if x approaches x0in a direction transverse to v(x0),ˆv(x)tends toˆv0=0, while x−x0tends to zero.See Figure3.v(x )xx v(x)v(x)^Figure3However,one can show(see Theorem3.4below)that for any trajectoryω(t)of v,and for t2−t1big enough,the length of the spherical curve s(t)=ω(t)−x0/ ω(t)−x0 (i.e. the absolute rotation ofω(t)around x0)is still bounded by C·K·(t2−t1).The intuitive explanation is as follows:as the trajectoryω(t)passes very close to x0,its rotation around x0in the time interval[t1,t2]comes to approximately1/2(see Figure4).x 0v(x )v(x)Figure4However,continuing to move in the same direction,the trajectoryω(t)cannot gain more rotation.So for the total rotation of the trajectoryω(t)around x0to grow,its velocity vector v(ω(t))must change its direction.On the other side,if we know a priori that the directions of v(x)and v(x0)are strongly different with respect to K· x−x0 ,12then the rotation speed satisfies the same Lipschitz upper bound as above.Consequently, the time interval[t1,t2]has to be large in order to gain a large total rotation.Although the rotation infinite time of a trajectory of a Lipschitzfield around a point is bounded,the rotation of a trajectory of a Lipschitz vectorfield v around a straight line, which is not invariant under v,can be unbounded during afinite interval of time,and this phenomena may occur even for v a C∞-vectorfield.Consider for instance the following field:Example3.3.LetΦ:R3→R3be a diffeomorphism,defined by:Φ(x1,x2,x3)=(x1,x2,x3),x1≤0,Φ(x1,x2,x3)=(x1,x2+ω1(x1),x3+ω2(x1)),x1≥0,whereω1(x1)=e−1/x21cos(1/x1),andω2(x1)=e−1/x21sin(1/x1).One can easily check thatΦis a C∞-diffeomorphism of a neighborhood of0∈R3.Now the image of the positive x1-semiaxis underΦis a line w,which makes an infinite number of turns around Ox1in any neighborhood of the origin.Consider the vectorfield v in R3,which is an image underΦof the constant vector field e1=(1,0,0).Clearly,w is a trajectory of the C∞-vectorfield v,and it makes an infinite number of turns around Ox1infinite time.In coordinates,v(x1,x2,x3)=DΦ(x1,x2,x3)(e1)=∂ΦProof.To prove Theorem3.4we need a general geometric lemma,which expresses accurately(in one of many possible ways)the intuitively clear fact that a long curve inside afixed sphere must oscillate.Letω(t)be a C1-curve(or piecewise C1)in the unit sphere S n−1⊂R n.For any two time moments t1and t2denote by s(t1,t2)the length of the curveωbetween t1and t2.For each“equator circle”λin S n−1consider the“longitude”projectionπ:S n−1→λof the sphere(without the poles)on the circleλ.A circleλis given as the intersection of S n−1with a2-dimensional vector planeΛof R n.Thefiber ofπover a point t∈λlies ina(n−2)-dimensional sphere S n−2λ,t ⊂S n−1which is normal toλand characterized as theintersection H∩S n−1,where H is the hyperplane of R n passing through t and containing Λ⊥.The projection of a point x∈S n−1is defined as soon as x is not in the poles,that is to say x∈Λ⊥∩S n−1≃S n−3(see Figure5).Sinceωis(piecewise)C1,by Sard’s theorem, it does not intersect a generic sub-sphere S n−3⊂S n−1.Hence,the projectionπ:ω→λis defined and regular for generic circlesλ.For the computations below it is convenient tofix a real parameterθ>4.Lemma3.5.—Let R>0and S n−1R be the sphere of R n of radius R.If the lengths(t1,t2)of a curveω⊂S n−1R between t1and t2satisfies s(t1,t2)>2πR·θthen there existan“equator circle”λin S n−1and time momentsτ1andτ2,with t1<τ1<τ2<t2,such that the following is true:(i)The“longitude”projectionsπ(ω(τ1))andπ(ω(τ2))of the pointsω(τ1)andω(τ2)onthe circleλcoincide or are antipodal points onλ.(ii)Denoting byℓthe tangent line toλat this common projection point or antipodal points,the norm of the projections of the velocity vectors v(τi)=dω4θ·s(t1,t2)4θ·s(t1,t2)w1and w2,we have:σ1=w1−x0w2−x0.Let v(t)denote,as above,thevelocity vector of the trajectoryω(t),while˜v(t)denotes the velocity vector of the spherical curveσ.The vector˜v(t)is the orthogonal toω(t)−x0component ofv(t)w1−x0 andv24θ·s(t1,t2)w1−x0and V2=v2−v(x0)w1−x0 andv24θ·s(t1,t2)4θ·s(t2,t1)θ−4·K·(t2−t1).In any case we have proved:R abs(ω,x0)=s(t2,t1)≤max(θ,4θBut this bound is true for anyθ>4,and forθ=4+K·(t2−t1),we have both expressions under the maximum sign equal one another.Therefore,for this specific choice ofθ>4 we get:max(θ,4θπR·s(t1,t2)= g∈O(n)#(ω∩S n−2g·S,g·t)dθn(g)Now by Fubini’s theorem:2·s(t1,t2)= t∈S g∈O(n)#(ω∩S n−2g·S,g·t)dθn(g)dt= g∈O(n) t∈S#(ω∩S n−2g·S,g·t)dt dθn(g)We conclude that for some g∈O(n),t∈S #(ω∩S n−2g·S,g·t)dt≥2·s(t1,t2)But this integral is exactly twice the length of the projection ofωonto the circleλ=g·S.So we have shown that for any curveω⊂R n there is a circleλ∈S n−1R such that theprojection ofωontoλhas its length≥s(t1,t2).But by our assumption,we also have:s(t1,t2)>2πR·θConsequently,to prove Lemma3.5it is enough to prove the following Lemma3.6,which is Lemma3.5in the case of a curveωcontained in a circle S1R:16Lemma3.6.—Letθbe any real number>4andωbe a C1(possibly,piecewise C1)curve in S1R,given with an orientation.Let us assume that its length s(t1,t2) between the time moments t1and t2is>2πR·θ.Then there exist time momentsτ1 andτ2,t1<τ1<τ2<t2,such that the following is true:(i)The pointsω(τ1)andω(τ2)coincide or are antipodal points on S1R.(ii)The velocity vectors v(τi)=dω4s(t1,t2)4θ·s(t1,t2)4θ≤14θ·s(t1,t2)θ−4and T=t2−t1the time interval.First of all,we can always reduce the situation to the case of a closed curveω.Indeed, ifωis not closed,but is contained in S1R,we just“close up”this curve with a circular curve joining the endpoints,passed with the velocity2πR4˜T ≥s(t1,t2)4(1+α)T=θ−4t2−t1.On the other hand,by assumptions we haves(t1,t2)4(1+α)T =2πRthe same length both in the positive and in the negative directions.The same is true over any subset A⊂R.Therefore,over each subset A⊂R the“positive length”ofωis equal to its“negative length”.Now let us assume that the statement of Lemma3.6is not true for the curveω. Denote its length s(t1,t2)by L.We thus assume there is no point in w over which the velocities in the opposite directions are both≥L/4T.Consequently,over each point s∈S1R either all the positive velocities ofωare<L/4T,or all the negative velocities of ωare<L/4T,or both.Denote by A1(respectively,A2,A3)the sets of points in S1R where thefirst (respectively,the second or the third)alternative holds.Let us take those of the sets A1,A2over which the total length ofωis larger,say,A1,and let A=A1∪A3.The total length ofωover A is≥L/2.By the remark above,the“positive length”ofωover A is equal to its“negative length”,and hence each is≥L/4.But by the construction,at each positive point ofωover A the velocity is<L/4T.Therefore,the total time required forωto cover A in the positive direction is>L/42)Γ(12),and V n=2Γn(12),whereΓis the Eulerfunction.With these notations,let us state the Euclidean version of Lemma3.5.Proposition3.7.—Letωbe a(piecewise)C1curve in a ball of radius R of R n and let,θ>8be a real number.If the length s(t1,t2)of the curveωbetween t1and t2 satisfies s(t1,t2)>θ·C n·R,then there exist a straight lineλand time momentsτ1and τ2,t1<τ1<τ2<t2,such that the following is true:(i)The orthogonal projectionsπ(ω(τ1))andπ(ω(τ2))of the pointsω(τ1)andω(τ2)onthe lineλcoincide.(ii)The norms of the orthogonal projections of the velocity vectors v(τi)=dω4θ·s(t1,t2)。
