12. Comparative Statics of Optimal Solutions

广义连续统力学generalized continuum mechanics简单物质simple material纯力学物质purely mechanical material微分型物质material of differentialtype积分型物质material of integral type混合物组份constituents of a mixture非协调理论incompatibility theory微极理论micropolar theory决定性原理principle of determinism等存在原理principle of equipresence局部作用原理principle of objectivity客观性原理principle of objectivity电磁连续统理论theory of electromagnetic conti-nuum内时理论endochronic theory非局部理论nonlocal theory混合物理论theory of mixtures里夫林-矣里克森张量Rivlin-Ericksen tensor 声张量acoustic tensor半向同性张量hemitropic tensor各向同性张量isotropic tensor应变张量strain tensor伸缩张量stretch tensor连续旋错continuous dislination连续位错continuous dislocation动量矩平衡angular momentum balance余本构关系complementary constitutiverela-tions共旋导数co-rotational derivative, Jaumann derivative非完整分量anholonomic component爬升效应climbing effect协调条件compatibility condition错综度complexity当时构形current configuration能量平衡energy balance变形梯度deformation gradient有限弹性finite elasticity熵增entropy production标架无差异性frame indifference弹性势elastic potential熵不等式entropy inequality极分解polar decomposition低弹性hypoelasticity参考构形reference configuration响应泛函response functional动量平衡momentum balance奇异面singular surface贮能函数stored-energy function内部约束internal constraint物理分量physical components本原元primitive element普适变形universal deformation速度梯度velocity gradient测粘流动viscometric flow当地导数local derivative岩石力学rock mechanics原始岩体应力virgin rock stress构造应力tectonic stress三轴压缩试验three-axial compression test三轴拉伸试验three-axial tensile test 三轴试验triaxial test岩层静态应力lithostatic stress吕荣lugeon地压强geostatic pressure水力劈裂hydraulic fracture咬合[作用] interlocking内禀抗剪强度intrinsic shear strength 循环抗剪强度cyclic shear strength残余抗剪强度residual shear strength 土力学soil mechanics孔隙比void ratio内磨擦角angle of internal friction休止角angle of repose孔隙率porosity围压ambient pressure渗透系数coefficient of permeability [抗]剪切角angle of shear resistance 渗流力seepage force表观粘聚力apparent cohesion粘聚力cohesion稠度consistency固结consolidation主固结primary consolidation次固结secondary consolidation固结仪consolidometer浮升力uplift扩容dilatancy有效应力effective stress絮凝[作用] flocculation主动土压力active earth pressure 被动土压力passive earth pressure 土动力学soil dynamics应力解除stress relief次时间效应secondary time effect 贯入阻力penetration resistance 沙土液化liquefaction of sand泥流mud flow多相流multiphase flow马格努斯效应Magnus effect韦伯数Weber number环状流annular flow泡状流bubble flow层状流stratified flow平衡流equilibrium flow二组份流two-component flow冻结流frozen flow均质流homogeneous flow二相流two-phase flow气-液流gas-liquid flow气-固流gas-solid flow液-气流liquid-gas flow液-固流liquid-solid flow液体-蒸气流liquid-vapor flow浓相dense phase稀相dilute phase连续相continuous phase离散相dispersed phase悬浮suspension气力输运pneumatic transport气泡形成bubble formation体密度bulk density壅塞choking微滴droplet挟带entrainment流型flow pattern流[态]化fluidization界面interface跃动速度saltation velocity非牛顿流体力学non-Newtonian fluid mechanics非牛顿流体non-Newtonian fluid幂律流体power law fluid拟塑性流体pseudoplastic fluid触稠流体rheopectic fluid触变流体thixotropic fluid粘弹性流体viscoelastic fluid流变测量学rheometry震凝性rheopexy体[积]粘性bulk viscosity魏森贝格效应Weissenberg effect流变仪rheometer稀薄气体动力学rarefied gas dynamics物理化学流体力学physico-chemical hydrodynamics空气热化学aerothermochemistry绝对压强absolute pressure绝对反应速率absolute reaction rate绝对温度absolute temperature吸收系数absorption coefficient活化分子activated molecule活化能activation energy绝热压缩adiabatic compression绝热膨胀adiabatic expansion绝热火焰温度adiabatic flame temperature 电弧风洞arc tunnel原子热atomic heat雾化atomization自燃auto-ignition自动氧化auto-oxidation可用能量available energy缓冲作用buffer action松密度bulk density燃烧率burning rate燃烧速度burning velocity接触面contact surface烧蚀ablation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 boundarylayer大气-海洋相互作用atmosphere-ocean interaction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 pressure tech-nique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V oigt-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, bloodrheology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magnetohydrodynamic stability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V oigt 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尘暴dust storm计尘仪koniscope盛行风prevailing wind输沙率rate of sand transporting重演距离repetition distance跃移[运动] saltation跃移质saltation load沙波纹sand ripple沙影sand shadow沙暴sand storm流沙shifting sand翻滚tumble植物固沙vegetative sand-control流速线velocity line泥石流debris flow连续泥石流continuous debris flow 泥石铺床bed-predeposit of mud泥石流地声geosound of debris flow 气浪airsurge冻胀力frost heaving pressure冻土强度frozen soil strength雪崩avalanche冰崩iceslide冰压力ice pressure重力侵蚀gravity erosion分凝势segregation potential滑波landslide山洪torrent爆发blow up雪暴snowstorm火爆fire storm闪点flash point闪耀flare up阴燃smolder轰燃flashover飞火spotting, firebrand地表火surface fire地下火ground fire树冠火crown fire烛炬火candling fire狂燃火running fire火焰强度flame intensity火焰辐射flame radiation火龙卷fire tornado火旋涡fire whirl火蔓延fire spread对流柱convection column隔火带fire line隔火带强度fireline intensity非线性动力学nonlinear dynamics动态系统dynamical system原象preimage控制参量control parameter霍普夫分岔Hopf bifurcation倒倍周期分岔inverse period- doubling bifurca-tion全局分岔global bifurcation魔[鬼楼]梯devil's staircase非线性振动nonlinear vibration侵入物invader锁相phase- locking猎食模型predator- prey model[状]态空间state space[状]态变量state variable吕埃勒-塔肯斯道路Ruelle- Takens route 斯梅尔马蹄Smale horseshoe混沌chaos李-约克定理Li-Yorke theorem李-约克混沌Li-Yorke chaos洛伦茨吸引子Lorenz attractor混沌吸引子chaotic attractorKAM环面KAM torus费根鲍姆数Feigenbaum number费根鲍姆标度律Feigenbaum scaling KAM定理Kolmogorov-Arnol'd Moser theorem, KAM theorem勒斯勒尔方程Rossler equation混沌运动chaotic motion费根鲍姆函数方程Feigenbaum functional equation蝴蝶效应butterfly effect同宿点homoclinic point异宿点heteroclinic point同宿轨道homoclinic orbit异宿轨道heteroclinic orbit排斥子repellor超混沌hyperchaos阵发混沌intermittency chaos内禀随机性intrinsic stochasticity含混吸引子vague attractor [of Kolmogorov]VAK奇怪吸引子strange attractorFPU问题Fermi-Pasta- Ulam problem,FPU problem初态敏感性sensitivity to initial state反应扩散方程reaction-diffusion equation 非线性薛定谔方程nonlinear Schrodinger equation逆散射法inverse scattering method孤[立]波solitary wave奇异摄动singular perturbation正弦戈登方程sine-Gorden equation科赫岛Koch island豪斯多夫维数Hausdorff dimensionKS[动态]熵Kolmogorov-Sinai entropy, KS entropy卡普兰-约克猜想Kaplan -Yorke conjecture 康托尔集[合] Cantor set欧几里得维数Euclidian dimension茹利亚集[合] Julia set科赫曲线Koch curve谢尔平斯基海绵Sierpinski sponge李雅普诺夫指数Lyapunov exponent芒德布罗集[合] Mandelbrot set李雅普诺夫维数Lyapunov dimension谢尔平斯基镂垫Sierpinski gasket雷尼熵Renyi entropy雷尼信息Renyi information分形fractal分形维数fractal dimension分形体fractal胖分形fat fractal退守物defender覆盖维数covering dimension信息维数information dimension度规熵metric entropy多重分形multi-fractal关联维数correlation dimension拓扑熵topological entropy拓扑维数topological dimension拉格朗日湍流Lagrange turbulence布鲁塞尔模型Brusselator贝纳尔对流Benard convection瑞利-贝纳尔不稳定性Rayleigh-Benard instability闭锁键blocked bond元胞自动机cellular automaton浸渐消去法adiabatic elimination连通键connected bond, unblocked bond自旋玻璃spin glass窘组frustration窘组嵌板frustration plaquette窘组函数frustration function窘组网络frustration network窘组位形frustrating configuration逾渗通路percolation path逾渗阈[值] percolation threshold入侵逾渗invasion percolation扩程逾渗extend range percolation多色逾渗polychromatic percolation快变量fast variable慢变量slow variable卷筒图型roll pattern六角[形]图形hexagon pattern主[宰]方程master equation役使原理slaving principle耗散结构dissipation structure离散流体[模型] discrete fluid自相似解self-similar solution协同学synergetics自组织self-organization跨越集团spanning cluster奇点singularity多重奇点multiple singularity多重定态multiple steady state不动点fixed point吸引子attractor自治系统autonomous system结点node焦点focus简单奇点simple singularity单切结点one-tangent node极限环limit cycle中心点center鞍点saddle [point]映射map[ping]逻辑斯谛映射logistic map[ping]沙尔科夫斯基序列Sharkovskii sequence 面包师变换baker's transformation吸引盆basin of attraction生灭过程birth-and death process台球问题biliard ball problem庞加莱映射Poincar'e map庞加莱截面Poincar'e section猫脸映射cat map[of Arnosov][映]象image揉面变换kneading transformation倍周期分岔period doubling bifurcation单峰映射single hump map[ping]圆[周]映射circle map[ping]埃农吸引子Henon attractor分岔bifurcation分岔集bifurcation set余维[数] co-dimension叉式分岔pitchfork bifurcation鞍结分岔saddle-node bifurcation次级分岔secondary bifurcation跨临界分岔transcritical bifurcation开折unfolding切分岔tangent bifurcation普适性universality突变catastrophe突变论catastrophe theory折叠[型突变] fold [catastrophe]尖拐[型突变] cusp [catastrophe]燕尾[型突变] swallow tail抛物脐[型突变] parabolic umbilic双曲脐[型突变] hyperbolic umbilic椭圆脐[型突变] elliptic umbilic蝴蝶[型突变] butterfly阿诺德舌[头] Arnol'd tongueBZ反应Belousov-Zhabotinskireaction, BZ reaction法里序列Farey sequence法里树Farey tree洛特卡-沃尔泰拉方程Lotka-V olterra equation 梅利尼科夫积分Mel'nikov integral锁频frequency-locking滞后[效应] hysteresis突跳jump准周期振动quasi-oscillation。

哲学术语英汉对照-CAL-FENGHAI.-(YICAI)-Company One1a prioria posteriori distinction 先验-后验的区分 abstract ideas 抽象理念abstract objects 抽象客体ad hominem argument 谬误论证 alienation/estrangement 异化，疏离altruism 利他主义analysis 分析analyticsynthetic distinction 分析-综合的区分aporia 困惑argument from design 来自设计的论证artificial intelligence (AI) 人工智能association of ideas 理念的联想autonomy 自律axioms 公理Categorical Imperative 绝对命令categories 范畴Category mistake 范畴错误causal theory of reference 指称的因果论causation 因果关系certainty 确定性chaos theory 混沌理论class 总纲、类clearness and distinctness 清楚与明晰cogito ergo sum 我思故我在concept 概念consciousness 意识consent 同意consequentialism 效果论conservative 保守的consistency 一致性，相容性constructivism 建构主义contents of consciousness 意识的内容 contingentnecessary distinction 偶然-必然的区分 continuum 连续体continuum hypothesis 连续性假说contradiction 矛盾（律）conventionalism 约定论counterfactual conditional 反事实的条件句criterion 准则，标准critique 批判，批评Dasein 此在，定在deconstruction 解构主义defeasible 可以废除的definite description 限定摹状词deontology 义务论dialectic 辩证法didactic 说教的dualism 二元论egoism 自我主义、利己主义eliminative materialism 消除性的唯物主义empiricism 经验主义Enlightenment 启蒙运动（思想）entailment 蕴含essence 本质ethical intuition 伦理直观ethical naturalism 伦理的自然主义eudaimonia 幸福主义event 事件、事变evolutionary epistemology 进化认识论expert system 专门体系explanation 解释fallibilism 谬误论family resemblance 家族相似fictional entities 虚构的实体first philosophy 第一哲学form of life 生活形式formal 形式的foundationalism 基础主义free will and determinism 自由意志和决定论function 函项（功能）function explanation 功能解释good 善happiness 幸福hedonism 享乐主义hermeneutics 解释学（诠释学，释义学）historicism 历史论（历史主义）holism 整体论iconographic 绘画idealism 理念论ideas 理念identity 同一性illocutionary act 以言行事的行为imagination 想象力immaterical substance 非物质实体immutable 不变的、永恒的individualism 个人主义（个体主义）induction 归纳inference 推断infinite regress 无限回归intensionality 内涵性intentionality 意向性irreducible 不可还原的Leibnizs Law 莱布尼茨法则logical atomism 逻辑原子主义logical positivism 逻辑实证主义logomachy 玩弄词藻的争论material biconditional 物质的双向制约materialism 唯物论（唯物主义）maxim 箴言，格言method 方法methodologica 方法论的model 样式modern 现代的modus ponens and modus tollens 肯定前件和否定后件 natural selection 自然选择necessary 必然的neutral monism 中立一无论nominalism 唯名论nonEuclidean geometry 非欧几里德几何nonmonotonic logics 非单一逻辑OckhamRazor 奥卡姆剃刀omnipotence and omniscience 全能和全知ontology 本体论（存有学）operator 算符(或算子)paradox 悖论perception 知觉phenomenology 现象学picture theory of meaning 意义的图像说pluralism 多元论polis 城邦possible world 可能世界postmodernism 后现代主义prescriptive statement 规定性陈述presupposition 预设primary and secondary qualities 第一性的质和第二性质 principle of noncontradiction 不矛盾律proposition 命题quantifier 量词quantum mechanics 量子力学rational numbers 有理数real number 实数realism 实在论reason 理性，理智recursive function 循环函数reflective equilibrium 反思的均衡relativity (theory of) 相对（论）rights 权利rigid designator 严格的指称词Rorschach test 相对性（相对论）rule 规则rule utilitarianism 功利主义规则Russells paradox 罗素悖论sanctions 制发scope 范围，限界semantics 语义学sense data 感觉材料，感觉资料set 集solipsism 唯我论social contract 社会契约subjectiveobjective distinction 主客区分sublation 扬弃substance 实体，本体sui generis 特殊的，独特性supervenience 偶然性syllogism 三段论thingsinthemselves 物自体thought 思想thought experiment 思想实验threevalued logic 三值逻辑transcendental 先验的truth 真理truth function 真值函项understanding 理解universals 共相，一般，普遍verfication principle 证实原则versimilitude 逼真性vicious regress 恶性回归Vienna Circle 维也纳学派virtue 美德。

中文英文英文中文艾利应力函数Airy stress function Airy stress function艾利应力函数板plate anti-sysmetric tensor反对称张量板边bounday of plate applied elasticity应用弹性力学板的抗弯强度flexural rigidity of plate axisymmetry轴对称板的内力internal force of plate base vector基矢量板的中面middle plane of plate basic assumptions ofelasticity弹性力学基本假定贝尔特拉米-米歇尔方程Beltrami-Michellequationbasic equation for thebending of thin plate薄板弯曲的基本方程贝蒂互换定理Betti reciprocal theorem Beltrami consistencyequation贝尔特拉米相容方程变温temperature change Beltrami-Michellequation 贝尔特拉米-米歇尔方程表层波surface wave Betti reciprocal theorem贝蒂互换定理半逆解法semi-inverse method body force体力薄板thin plate boundary condition边界条件薄板弯曲的基本方程basic equation for thebending of thin platebounday of plate板边薄膜比拟membrage analogy Boussinesq problem布西内斯克问题布西内斯克问题Boussinesq problem Boussinesq solution布西内斯克解答布西内斯克解答Boussinesq solution Boussinesq solution布西内斯克解答布西内斯克-伽辽金通解Boussinesq-Galerkingeneral solutionBoussinesq-Galerkingeneral solution布西内斯克-伽辽金通解半空间体semi-infinite body bulk modulus体积模量半平面体semi-infinite plane Castigliano formula卡斯蒂利亚诺公式贝尔特拉米相容方程Beltrami consistencyequationCauchy equation柯西方程边界条件boundary condition Cerruti problem塞路蒂问题变分法（能量法）variationalmethod,energy method characteristic equationof stress state应力状态特征方程薄板内力internal forces of thinplate coefficient of lateralpressure侧压力系数薄板弹性曲面elatic surface of thinplate complex potential复位势薄板弹性曲面微分方程differential equation ofelastic surface of thinplatecondition of single-value displacement位移单值条件薄板弯曲刚度flexural rigidity of thinplateconsistency equation相容方程布西内斯克解答Boussinesq solution contact problem接触问题产熵entropy prodction continuity连续性沉陷settlement continuous hypothesis连续性假设侧压力系数coefficient of lateralpressure coordinate curves坐标曲线ELASTICITY(弹性力学)常用专业名词中英文对照差分法finite-differencemethord coordinate surface坐标曲面差分公式finite-differencefromulate coupling耦合重三角级数double triangle series curvilinear coordinates曲线坐标大挠度问题large deflection problem deflection挠度单位张量unit tensor deformation形变单元分析element analysis density of comlementarystrain energy应变余能密度单元刚度矩阵element stiffness matrix density of internalenergy 内能密度等容波equivoluminal wave diaplacement位移等容的位移场equivoluminaldisplacement field diaplacementcomponents位移分量叠加原理superposition principle diaplacement method位移解法度量张量metric tensor diaplacement method位移法对称张量symmetric tensor diaplacement shapefunction位移的形函数单连体simply connected body diaplacement variationalequation位移变分方程单三角级数解single triangle series differential equation ofelastic surface弹性曲面的微分方程单元节点载荷列阵elemental nodal loadmatrix differential equation ofelastic surface of thinplate薄板弹性曲面微分方程单元劲度矩阵elemental stiffnessmatrix differential equation ofequilibrium平衡微分方程多连体multiply connected body differential equation ofequilibrium in terms ofdisplacement 以位移表示的平衡微分方程二阶张量second order tensor dilatation wave膨胀波反对称张量anti-sysmetric tensor discretization离散化符拉芒解答Flamant soluton discretization structure离散化结构反射reflection displacement boundarycondition位移边界条件傅里叶变换Fourier transform displacement model位移模式傅里叶积分Fourier integral distrotion wave畸变波复位势complex potential double triangle series重三角级数格林公式Green formula dummy index哑指标各向同性假设isotropic hypothesis elastic body弹性体供熵entropy supply elastic constants弹性常数广义变分原理generanized variatianalprincipleelastic matrix弹性矩阵广义胡克定律generanized Hooke law elastic principledirection 弹性主方向刚体位移rigid body displacement elastic symmetric plane弹性对称面各向同性isotropy elastic wave弹性波哈密顿变分原理Hamiton varitionalprincipleelasticity弹性哈密顿作用量Hamiton action elasticity弹性力学赫林格-赖斯纳变分原理Hellinger-Reissnervariational principleelatic surface of thinplate薄板弹性曲面亥姆霍兹定理Helmholtz theorem element analysis单元分析横观各向同性弹性体transverse isotropicelastic bodyelement stiffness matrix单元刚度矩阵横波transverse wave elemental nodal loadmatrix单元节点载荷列阵厚板thick plate elemental stiffnessmatrix 单元劲度矩阵胡海昌-鹫津久一郎变分原理Hu Haichang-Washizuvariational principleenergy method能量法混合边值问题mixed boundary-valueproblementropy prodction产熵胡克定律Hooke law entropy supply供熵混合边界条件mixed boundarycondition equation of stresscompatibility应力协调方程畸变波distrotion wave equivalent shear forcetorsional moment扭矩等效剪力基尔霍夫假设Kirchhoff hypothesis equivoluminaldisplacement field等容的位移场基矢量base vector equivoluminal wave等容波几何方程geometrical equation Euler method欧拉法几何可能的位移geometrically possibledisplacementEuler strain components欧拉应变分量几何可能的应变geometrically possiblestriainexternal force外力几何线性的假设geometrically linearhypothesisfinite element有限元伽辽金法Galerkin method finite element method有限单元法伽辽金矢量Galerkin vector finite-differencefromulate 差分公式结点node finite-differencemethord 差分法结点荷载nodal load first law ofthermodynamics热力学第一定律结点力nodal force first(second,third)kindboundary-value problemof elasticity 弹性力学的第一（第二、第三）类边值条件结点位移nodal displacement Flamant soluton符拉芒解答解的唯一性定理theorem of uniquenesssolutionflexural rigidity of plate板的抗弯强度静力可能的应力statically possible stress flexural rigidity of thinplate薄板弯曲刚度均匀性假设homogeneoushypothesis Fourier integral傅里叶积分局部编码local coding Fourier transform傅里叶变换基尔斯解答Kirsch solution free energy density自由能密度极小势能原理princile of minimumpotential energyfree index自由指标接触问题contact problem Galerkin method伽辽金法均匀性homogeneity Galerkin vector伽辽金矢量卡斯蒂利亚诺公式Castigliano formula generanized Hooke law广义胡克定律开尔文问题Kelvin problem generanized variatianalprinciple广义变分原理扭转刚度torsional rigidity geometrical equation几何方程柯西方程Cauchy equation geometrically linearhypothesis几何线性的假设克罗内克δ符号Kroneckerdelta symbol geometrically possibledisplacement几何可能的位移空间轴对称问题spatial axisymmetryproblem geometrically possiblestriain几何可能的应变孔口应力集中stress concentration ofholesglobal analysis整体分析拉梅解答Lame slution global analysis整体分析离散化结构discretization structure global coding总体编码理想弹性体perfect elastic body global equivalent nodalload vector整体等效结点荷载列阵连续性continuity global nodaldisplacement vector整体结点位移列阵拉格朗日法Lagrange method global stiffness matrix总刚度矩阵拉格朗日函数Lagrange function global stiffness matrix整体劲度矩阵拉格朗日应变函数Lagrange straincomponentsGreen formula格林公式拉梅常数Lamé constants Hamiton action哈密顿作用量拉梅系数Lamé coefficient Hamiton varitionalprinciple哈密顿变分原理拉梅方程Lamé equation heat-conductionequation 热传导方程拉梅应变势Lamé strain potential Hellinger-Reissnervariational principle 赫林格-赖斯纳变分原理莱维方程Lévy equation Helmholtz theorem亥姆霍兹定理勒夫应变函数Love strain function homogeneity均匀性离散化discretization homogeneoushypothesis 均匀性假设连续性假设continuous hypothesis Hooke law胡克定律梁的纯弯曲pure bending of beam Hooke's law of volume体应变胡克定律莱维解Lévy solution Hu Haichang-Washizuvariational principle 胡海昌-鹫津久一郎变分原理面力surface force infinitesimaldeformation hypothesis小变形假设膜板membrane plate internal force内力米歇尔相容方程Michell consistencyequationinternal force of plate板的内力挠度deflection internal forces of thinplate 薄板内力内力internal force inverse method逆解法能量法energy method irrotationaldisplacement field无旋的位移场逆解法inverse method irrotational wave无旋波扭矩等效剪力equivalent shear forcetorsional momentisotropic hypothesis各向同性假设扭转torsion isotropy各向同性纳维解Navier solution Kelvin problem开尔文问题内能密度density of internalenergy Kirchhoff hypothesis基尔霍夫假设纽勃-巴博考维奇通解Neuber-Papkovichgeneral solutionKirsch solution基尔斯解答欧拉法Euler method Kroneckerdelta symbol克罗内克δ符号欧拉应变分量Euler strain components Lagrange function拉格朗日函数耦合coupling Lagrange method拉格朗日法膨胀波dilatation wave Lagrange straincomponents拉格朗日应变函数平衡微分方程differential equation ofequilibriumLamé coefficient拉梅系数平面波plane wave Lamé constants拉梅常数平面应力问题plane stress problem Lamé equation拉梅方程平面应变问题plane strain problem Lame slution拉梅解答泊松比Poisson ratio Lamé strain potential拉梅应变势普朗特比拟Prandtl analogy large deflection problem大挠度问题普朗特应力函数Prandtl stress function Lévy equation莱维方程切变模量shear modulus Lévy solution莱维解切应变shear strain linear elasticity线性弹性力学切应力shear stress linear expansioncoefficient线膨胀系数切应力互等定理reciprocal theorem ofshear stresslinear thermal elasticity线性热弹性力学切应力线shear stress lines local coding局部编码求和约定summation convention longitudinal wave纵波球面波spherical wave Love strain function勒夫应变函数曲线坐标curvilinear coordinates mathematical elasticity数学弹性力学热力学第一定律first law ofthermodynamicsmembrage analogy薄膜比拟热力学第二定律second law ofthermodynamicsmembrane plate膜板热弹性应变势thermal elastic strainpotentialmetric tensor度量张量热应力thermal stress Michell consistencyequation米歇尔相容方程热传导方程heat-conductionequation middle plane of plate板的中面瑞利波Rayleigh wave mixed boundarycondition 混合边界条件瑞利-里茨法Rayleigh-Ritz method mixed boundary-valueproblem混合边值问题三阶张量third order tensor multiply connected body多连体塞路蒂问题Cerruti problem Navier solution纳维解圣维南扭转函数Saint-Venant torsionfunction Neuber-Papkovichgeneral solution纽勃-巴博考维奇通解圣维南方程Saint-Venant equation no initial stresshypothesis 无初始应力的假设圣维南原理Saint-Venant principle nodal displacement结点位移数学弹性力学mathematical elasticity nodal force结点力弹性elasticity nodal load结点荷载弹性波elastic wave node结点弹性常数elastic constants normal strain线应变弹性对称面elastic symmetric plane normal strain正应变弹性力学的平面问题plane problem ofelasticitynormal stress正应力弹性力学的第一（第二、第三）类边值条件first(second,third)kindboundary-value problemof elasticityorthotropic elastic body正交各向异性弹性体弹性曲面的微分方程differential equation ofelastic surfaceperfect elastic body理想弹性体弹性体elastic body perfect elasticity完全弹性弹性体的虚功原理principle of virtual workfor elastic solidperfectly elastic body完全弹性体弹性主方向elastic principledirection perfectly elastichypothesis完全弹性的假设弹性矩阵elastic matrix permulation tensor置换张量体力body force physical equation物理方程体应变胡克定律Hooke's law of volume physically linerhypothesis 物理线性的假设弹性力学elasticity plane problem ofelasticity 弹性力学的平面问题弹性力学基本假定basic assumptions ofelasticityplane strain problem平面应变问题体积模量bulk modulus plane stress problem平面应力问题体积应力volumetric strain plane wave平面波体应变volumetric strain plate板完全弹性的假设perfectly elastichypothesisPoisson ratio泊松比完全弹性体perfectly elastic body potential energy ofexternal force外力势能位移边界条件displacement boundarycondition potential functiondecomposition ofdisplacement field位移场的势函数分解式位移变分方程diaplacement variationalequationPrandtl analogy普朗特比拟位移场的势函数分解式potential functiondecomposition ofdisplacement fieldPrandtl stress function普朗特应力函数位移分量diaplacementcomponentspressure tunnel压力隧道位移解法diaplacement method princile of minimumpotential energy极小势能原理位移的形函数diaplacement shapefunctionprincipal plane主平面无初始应力的假设no initial stresshypothesisprincipal shear stress主切应力无旋波irrotational wave principal strain主应变无旋的位移场irrotationaldisplacement fieldprincipal stress主应力物理线性的假设physically linerhypothesis principle direction ofstrain应变主方向外力external force principle direction ofstress应力主方向外力功work of external force principle of least work最小功原理外力势能potential energy ofexternal force principle of minimum complementary energy最小余能原理完全弹性perfect elasticity principle of minimumpotential energy最小势能原理位移diaplacement principle of virtual workfor elastic solid弹性体的虚功原理位移单值条件condition of single-value displacementprinciple plane of stress应力主面位移法diaplacement method pure bending of beam梁的纯弯曲位移模式displacement model quadratic surface ofstrain 应变二次曲面物理方程physical equation quadratic surface ofstress 应力二次曲面线膨胀系数linear expansioncoefficientRayleigh wave瑞利波线性弹性力学linear elasticity Rayleigh-Ritz method瑞利-里茨法线性热弹性力学linear thermal elasticity reciprocal theorem ofshear stress切应力互等定理相对位移张量relative displacementtensorreflection反射小变形假设infinitesimaldeformation hypothesisrefraction折射小挠度问题small deflection matrix relative displacementtensor相对位移张量形函数矩阵shape function matrix rigid body displacement刚体位移虚位移virtual displacement rotation components转动分量虚位移方程virtual displacementequationrotation vector转动矢量虚应变virtual strain Saint-Venant equation圣维南方程虚应力virtual stress Saint-Venant principle圣维南原理虚应力方程virtual stress equation Saint-Venant torsionfunction圣维南扭转函数线应变normal strain second law ofthermodynamics热力学第二定律相容方程consistency equation second order tensor二阶张量形变deformation semi-infinite body半空间体形变势能strain erergy semi-infinite plane半平面体形函数shape function semi-inverse method半逆解法虚功方程virtual work equation settlement沉陷哑指标dummy index shape function形函数杨氏模量Young modulus shape function matrix形函数矩阵一点的应变状态state of strain at a point shear modulus切变模量一点的应力状态state of stress at a point shear strain切应变以位移表示的平衡微分方程differential equation ofequilibrium in terms ofdisplacementshear stress切应力应变二次曲面quadratic surface ofstrain shear stress lines切应力线应变分量strain components simply connected body单连体应变能密度strain energy density single triangle series单三角级数解应变矩阵strain matrix small deflection matrix小挠度问题应变协调方程strain compatibilityequation spatial axisymmetryproblem空间轴对称问题应变余能密度density of comlementarystrain energyspherical wave球面波应变张量strain tensor state of strain at a point一点的应变状态应变张量不变量strain tensor invariant state of stress at a point一点的应力状态应变主方向principle direction ofstrain statically possible stress静力可能的应力应力变分方程stress variationalequation strain compatibilityequation应变协调方程应力边界条件stress boundarycondition strain components应变分量应力二次曲面quadratic surface ofstress strain energy density应变能密度应力分量stress components strain erergy形变势能应力环量stress circulation strain matrix应变矩阵应力解法stress method strain tensor应变张量应力矩阵stress matrix strain tensor invariant应变张量不变量应力协调方程equation of stresscompatibility stress boundarycondition应力边界条件应力张量stress tensor stress circulation应力环量应力张量不变量stress tensor invariant stress components应力分量应力主方向principle direction ofstress stress concentration ofholes孔口应力集中应力状态特征方程characteristic equationof stress statestress matrix应力矩阵应用弹性力学applied elasticity stress method应力解法有限元finite element stress method应力法圆柱体扭转torsion of circular bar stress tensor应力张量压力隧道pressure tunnel stress tensor invariant应力张量不变量应力法stress method stress variationalequation 应力变分方程应力主面principle plane of stress summation convention求和约定有限单元法finite element method superposition principle叠加原理折射refraction surface force面力整体等效结点荷载列阵global equivalent nodalload vectorsurface wave表层波整体结点位移列阵global nodaldisplacement vectorsymmetric tensor对称张量整体分析global analysis temperature change变温正应变normal strain theorem of uniquenesssolution解的唯一性定理正应力normal stress thermal elastic strainpotential热弹性应变势正交各向异性弹性体orthotropic elastic body thermal stress热应力置换张量permulation tensor thick plate厚板主应变principal strain thin plate薄板主应力principal stress third order tensor三阶张量主平面principal plane torsion扭转主切应力principal shear stress torsion of circular bar圆柱体扭转转动矢量rotation vector torsional rigidity扭转刚度转动分量rotation components total complementaryenergy总余能自由能密度free energy density total potential energy总势能自由指标free index transverse isotropicelastic body横观各向同性弹性体纵波longitudinal wave transverse wave横波总刚度矩阵global stiffness matrix unit tensor单位张量总势能total potential energy variationalmethod,energy method变分法（能量法）总余能total complementaryenergyvirtual displacement虚位移总体编码global coding virtual displacementequation虚位移方程最小功原理principle of least work virtual strain虚应变最小势能原理principle of minimumpotential energyvirtual stress虚应力最小余能原理principle of minimumcomplementary energyvirtual stress equation虚应力方程坐标曲面coordinate surface virtual work equation虚功方程坐标曲线coordinate curves volumetric strain体积应力整体分析global analysis volumetric strain体应变整体劲度矩阵global stiffness matrix work of external force外力功轴对称axisymmetry Young modulus杨氏模量。

矛盾contradiction – resolve contradiction 解决矛盾contradictory立论make a point 对立面立论antithesis驳论refutation, rebuttal论证过程process of reasoning逻辑体系logic systemarchitecture logical政策policy, strategy客观结果objective results主观愿望subjective desire理想主义idealism唯心的idealistic立足点(立脚点)foothold(立场) standpoint出发点jump-off placeout of… consideration三段论法syllogism前提premise 大前提major premise小前提minor premise推理(by) inference解决难题put the axe in the helve激化矛盾intensification of contradictions 激化intensify 四书four books 五经five classics文艺复兴Revival of Learning宗教改革Reformation罗马法复兴Reception of Roman Law社会进步social progress经济繁荣economic boom经济衰退economic recession经济萧条economic depression虚假繁荣borrowed prosperity人民币升值压力upward pressure on the RMB exchange rate 次贷危机subprime mortgage crisis信用额度credit line消费信心consumer confidence人均收入per capita income股市stock market赤字deficit 财政赤字financial deficit减少预算budget reduction财政finance货币政策monetary policy收支平衡break even降息reduction of interest退休金pension, superannuation养老保险endowment insurance医疗改革healthcare reformreforms to the public healthcare system民生问题bread-and-butter issueslivelihood issuespocketbook issues以人为本putting people firstpeople orientedpeople foremost不动摇不懈怠不折腾We don’t sway back and forth, relax our effort社会责任public responsibilitiesSocial accountability发挥道德模范榜样作用let paragons of virtue serve as role models of society宇航员美国宇航员astronaut苏联宇航员cosmonaut中国宇航员taikonaut教育改革educative reformeducation reform入学率enrollment rate辍学率dropout rate提高人口素质quality of population科技创新technological innovation就业问题problem of employment解决劳动力就业问题tackle the problem of labor force utilization产业结构调整adjustment of industrial structure学历academic qualification学分制the credit system专业课specialized courses选修课elective/optional courses必修课required/compulsory courses基础课basic courses通识课general knowledge courses四有新人new generation of people with lofty ideals, moral integrity, good education and strong sense of discipline国家科技进步奖national prize of progress in science and technological大国superpower国家形象image of a nation国际关系international relation(履行)国际义务(fulfill) international obligations台海和平稳定peace and stability in the Taiwan straits记者招待会press conference亚太问题the Asia-Pacific region中东问题Middle East affairs中东问题四方Quartet for Middle East peace化干戈为玉帛beat your sward into plowshares国家主权national sovereignty国家主权和领土完整national sovereignty and territorial integrity国家利益national interests和平崛起peaceful rising战略资源strategic resources战略资源室（英国贸工部）Strategy Unit沙文主义Chauvinism军国主义Kultur, Jingoism霸权Hegemony, Hegemonism帝国主义Imperialism强权政治power politics不平等条约treaty of inequalityunequal treaty经济封锁economic blockade禁运embargo实行禁运lay embargo on 解除禁运lift/remove the embargo on 第一岛链First Island Chain意识形态ideology关税保护tariff protection贸易壁垒trade barrierBarrier to trade贸易保护主义trade protectionism朝核问题the nuclear issue on Korea Peninsular六方会谈the Six-Party Talks反恐战争Anti-Terrorism War侵略战争aggressive warwar of aggressive战略威胁strategic threats中国威胁论the “China Threat” theory国家导弹防御National Missile Defense同盟alliance, league经济一体化economic integration欧元区Euro Area北约NATO北约东扩the eastward expansion of NATO一国两制One Country, Two System人权human rights生命权right to life自由权liberty财产权property rights1自由转让权rights to transfer2收入享受权exclusive right to derive income3私人使用权exclusive use right尊严权right to dignity获助权rights to getting help公正权equitable interests（现代社会的衍生）发展权rights to development民族自决权right of national self-determination物权法property law民主进程democratization加快民主进程speed up the pace of democracy指导思想guiding ideology民权运动Civil Rights Movement启蒙运动the Enlightenment低保basic living allowance资本主义民主Capitalist Democracy论法的精神the Spirit of the Law暴力机关organ of violence国家机器the state machinerythe state apparatus立法legislation立法机关legislative body全国人大National People’s Congress全国人大代表deputy to the National People’s Congress全国人大常委会Standing Committee of the National People’s Congress试点pilot project地方法规local lawsLocal Regulation城管urban management force执法law enforcement自卫self-defense , self-protection合理合法的legitimate合理性legitimacy法律面前人人平等Equality Before the Law加强执政能力建设特警special armed policeregimental police便衣plainclothes police突发性群体事件谈判negotiation, 讨价还价bargain缓和局势缓和矛盾mitigate a contradictionalleviate a contradiction群众监督supervision by the people加强党的执政能力建设strengthen the construction of governing capacity of CPC 执法犯法inside job行政透明Administrative Transparency 行政透明化Transparency in Procedure 维权safeguard legal rights上访appeal to the higher authorities for help社会矛盾social contradiction贫富分化polarization between the rich and the poor失业问题unemployment problem农民工migrant workers再就业reemployment社会保障social security医疗保险hospitalization insurancemedicare养老保险endowment insurance失业保险unemployment insurance救济金alms, relief fund血汗工厂sweatshop犯罪心理criminal mentalityguilty mind犯罪情节circumstance of a crime犯罪动机crime motive有组织犯罪Organized Crime网络犯罪Cyber Crime刑罚penalty刑法criminal law罪刑法定原则principal of a legally prescribed punishment for a specified crime 死刑death penalty在十字架上钉死crucifixion电椅hot seat被处死刑be put to death严刑torture 酷刑excruciation徒刑imprisonment无期徒刑life imprisonment 有期徒刑specified sentence劳教reeducation through labor劳改reform through labor罚款amerce, forfeiture危害公共安全endangered public security危害公共安全罪crime against public security危害国家安全罪crime threatening state security宽恕condone, mercy, forgiveness愿上帝宽恕你的罪过May God grant your pardon for your sins悔过resipiscence故意杀人voluntary manslaughter过失杀人manslaughter防卫过当excessive defense怯懦white feather。

a r X i v :h e p -t h /0112244v 1 28 D e c 2001TIT/HEP–473UT-984hep-th/yymmnnDecember,2001SUSY Breaking by stable non-BPS configurations Nobuhito Maru a ∗,Norisuke Sakai b †Yutaka Sakamura b ‡and Ryo Sugisaka b §a Department of Physics,University of Tokyo 113-0033,JAPAN and b Department of Physics,Tokyo Institute of Technology Tokyo 152-8551,JAPAN Abstract A simple mechanism for SUSY breaking is proposed due to the coexistence of BPS domain walls.It requires no messenger fields nor complicated SUSY breaking sector on any of the walls.We assumed that our world is on a BPS domain wall and that the other BPS wall breaks the SUSY preserved by our wall.We obtain an N =1model in four dimensions which admits anexact solution of a stable non-BPS configuration of two walls.The stabilityis assured by a topological quantum number associated with the winding onthe field space of the topology of S 1.We propose that the overlap of thewave functions of the Nambu-Goldstone fermion and those of physical fieldsprovides a practical method to evaluate SUSY breaking mass splitting on ourwall thanks to a low-energy theorem.This is based on our recent works hep-th/0009023and hep-th/0107204.1IntroductionSupersymmetry(SUSY)provides the most realistic models to solve the hierarchy problem in unified theories[1].One of the most important issues in model building of SUSY unified theories has been for some years how to understand the SUSY breaking in our observable world.Many models of SUSY breaking uses some kind of mediation of the SUSY breaking from the hidden sector to our observable sector.Recently the“Brane World”scenario has become quite popular where our four-dimensional spacetime is realized on the wall embedded in a higher dimensional spacetime[2,3].In order to discuss the stability of such a wall,it is often useful to consider SUSY theories as the fundamental theory.Moreover,SUSY theories in higher dimensions are a natural possibility in string theories. These SUSY theories in higher dimensions have8or more supercharges,which should be broken partially if we want to have a phenomenologically viable SUSY unified model in four dimensions. Such a partial breaking of SUSY is nicely obtained by topological defects[4].Domain walls or other topological defects preserving part of the original SUSY in the fundamental theory are called the BPS states in SUSY theories.Walls have co-dimension one and typically preserve half of the original SUSY,which are called1/2BPS states[5,6,7].Junctions of walls have co-dimension two and typically preserve a quarter of the original SUSY[8,9].The new possibility offered by the brane world scenario stimulated studies of SUSY break-ing.Recently we have proposed a simple mechanism of SUSY breaking due to the coexistence of different kinds of BPS domain walls and proposed an efficient method to evaluate the SUSY breaking parameters such as the boson-fermion mass-splitting by means of overlap of wave func-tions involving the Nambu-Goldstone(NG)fermion[10]–[12].We have exemplified these points by taking a toy model in four dimensions,which allows an exact solution of coexisting walls with a three-dimensional effective theory[10].Although thefirst model is only meta-stable,we were able to show approximate evaluation of the overlap allows us to determine the mass-splitting reliably.More recently,we have constructed a stable non-BPS configuration of two walls in an N=1supersymmetric model in four dimensions to demonstrate our idea of SUSY breaking due to the coexistence of BPS walls.We have also extended our analysis to more realistic case offour-dimensional effective theories and examined the consequences of our mechanism in detail [11].Our proposal for a SUSY breaking mechanism requires no messengerfields,nor complicated SUSY breaking sector on any of the walls.We assume that our world is on a wall and SUSY is broken only by the coexistence of another wall with some distance from our wall.The NG fermion is localized on the distant wall and its overlap with the wave functions of physicalfieldson our wall gives the boson-fermion mass-splitting of physicalfields on our wall thanks to a low-energy theorem[13].The purpose of this paper is to illustrate our idea of SUSY breaking due to the coexistence of BPS walls by taking a simple soluble model with a stable non-BPS configuration of two walls and to extend our analysis to more realistic case of four-dimensional effective theories.We work out how various soft SUSY breaking terms can arise in our framework.Phenomenological implications are briefly discussed.We alsofind that effective SUSY breaking scale observed on our wall becomes exponentially small as the distance between two walls grows.The NG fermion is localized on the distant wall and its overlap with the wave functions of physicalfields on our wall gives the boson-fermion mass-splitting of physicalfields on our wall thanks to a low-energy theorem.We have proposed that this overlap provides a practical method to evaluate the mass-splitting in models with SUSY breaking due to the coexisting walls.2BPS equation and topological quantum numberLet us illustrate the BPS equation and topological quantum number for1Φ′3.3 After eliminating the auxiliaryfield F the bosonic part of the Lagrangian becomesL=−∂µA′∗∂µA′− 1−A′2 .(2.1) We have absorbed possible constants into the normalization offield and coordinates for simplicity. The model has two SUSY vacua at A′=±1.The supertransformation of the fermionψ′is givenby√2ǫF′.(2.2)δψ′=iIf we choose A′to depend only on one coordinate,say,x2=y,and chooseǫ=iσ2¯ǫ,the half of supersymmetry is conserved by the configuration satisfying the BPS equationdA′=−(1−A′2),(2.5)dywhich admits a wall solution connecting the SUSY vacuum+1at y=−∞to another SUSY vacuum−1at y=∞A′(2)cl(y)=−tanh(y−y2),(2.6)where y2denotes the position of the wall.If we combine these two solutions,we obtain a wall anti-wall configuration.In fact we have found exact solution of the equation of motion which is a non-BPS state and gives an example of the SUSY breaking due to the coexistence of BPS and anti-BPS walls[10].The wall anti-wall configuration is unstable due to the annihilation into vacuum.It is desirable to have a model with stable but non-BPS two wall configuration.We have found a way to give the topological quantum number.We shall give a topology of S1tofield space so that we can have a notion of winding from a compactified base space which is also S1.To achieve that goal,we changefield variable A′into a periodic variable AA′=sin A,Φ′=sinΦ.(2.7) Then the SUSY vacua occurs at A=π n+1=cos A.(2.8)dyThe BPS solution(2.4)is mapped into a solution of this transformed BPS Eq.(2.8)sin A(1)cl(y)=tanh(y−y1)(2.9) connecting the SUSY vacuum A=−π/2at y=−∞to A=π/2at y=∞.The solution of the anti-BPS equation connecting the SUSY vacuum A=π/2at y=−∞to A=3π/2at y=∞. can also be obtainedsin A(2)cl(y)=−tanh(y−y2).(2.10) Now we can see these solutions can be smoothly connected in thefield space since thefield A=π/2at right end point of the BPS wall is the same as thefield at the left end point of the anti-BPS wall.This suggests that we may have a non-BPS solution of two wall configuration which is non-BPS.Indeed we found that a simple model with minimal kinetic term provides the BPS equation(2.3)and that there is an exact solution for the non-BPS configuration of two walls which winds around thefield space A once[11]1.3Stable non-BPS configuration of two wallsIn order to illustrate our basic ideas,we consider three dimensional domain walls in four-dimensional spacetime.Our model readsL=¯ΦΦ|θ2¯θ2+W(Φ)|θ2+h.c.,W(Φ)=Λ3ΛΦ .(3.1) We have introduced a scale parameterΛwith a mass-dimension one and a dimensionless coupling constant g,both of which are real positive.Choosing y=X2as the extra dimension and compactify it on S1of radius R.Other coordinates are denoted as x m(m=0,1,3),i.e., Xµ=(x m,y).The bosonic part of the model isL bosonic=−∂µA∗∂µA−Λ4ΛA 2.(3.2) The target space of the scalarfield A has a topology of a cylinder.This model has two vacua at A=±πΛ/(2g),both lie on the real axis.In the limit R→∞,we have a BPS domain wall solutionsingΛA(2)cl(y)=−tanh(Λ(y−y2)),(3.4) which interpolates the vacuum at A=πΛ/(2g)to that at A=3πΛ/(2g)=−πΛ/(2g)and preserves another real two component supercharge Q(2)α.Here y1and y2are integration constants and represent the location of the walls along the extra dimension.The four-dimensional super-charge Qαis a sum of these two supercharges Qα=12(Q(1)α+iQ(2)α).Each wall breaks a half of the bulk supersymmetry and all of the bulk supersymmetry will be broken if these walls coexist.For this model,we have found an exact solution of the non-BPS two wall configuration which is stable due to the winding number:π(S1)=Z.Such a configuration should be a solution ofthe equation of motion,∂µ∂µA+Λ3ΛA∗ cos g gam ΛQ Q (1)(2),Q Q (1)(2),Figure 1:The profile of the classical solution A cl (y ).The dotted lines A =−πΛ/(2g )and A =3πΛ/(2g )are identified.where k and y 0are real parameters and the function am(u,k )denotes the amplitude function,which is defined as an inverse function of u (ϕ)= ϕ0d θ1−k 2sin 2θ.If k <1,the solution A cl (y )is a monotonically increasing function withA cl y +4kK (k )g .(3.7)This is the solution that we want.Since the field A is an angular variable A =A +2πΛ/g ,we can choose the compactified radius 2πR =4kK (k )/Λso that the classical field configuration A cl (y )contains two walls and becomes periodic modulo 2πΛ/g .We shall take y 0=0to locate one of the walls at y =0.Then we find that the other wall is located at the anti-podal point y =πR ofthe compactified circle.We have computed the energy of a superposition of the first wall A (1)cl (y )located at y =y 1in Eq.(3.3)and the second wall A (2)cl (y )located at y =y 2in Eq.(3.4).Thisenergy can be regarded as a potential between two walls in the adiabatic approximation and has a peak at |y 1−y 2|=0implying that two walls experience a repulsion.This is in contrast to a BPS configuration of two walls which should exert no force between them.Thus we can explain that the second wall is settled at the anti-podal point y =πR in our stable non-BPS configuration because of the repulsive force between two walls.Since the repulsive force forces the other wall to oscillate around the anti-podal point when a small fluctuation is added,we have a physical reason to obtain a stable spectrum without any tachyon.Figure2:The mode functions for the bosonic modes a R,0and a R,1.The solid line represents the profile of b R,0(y)and the dashed line is that of b R,1(y).In the limit of R→∞,i.e.,k→1,A cl(y)approaches near y=0to the BPS configuration A(1)cl(y)with y1=0which preserves Q(1),and near y=πR to A(2)cl(y)with y2=πR which preserves Q(2).The profile of the classical solution A cl(y)is shown in Fig.1.We will refer to the wall at y=0as“our wall”and the wall at y=πR as“the other wall”.4Mode expansion and effective LagrangianThefluctuationfields around the background A cl(y)can be expanded into modesA(X)=A cl(y)+12(A R(X)+iA I(X)),Ψα(X)=12(Ψ(1)α(X)+iΨ(2)α(X)).(4.1)The four-dimensionalfluctuationfields can be expanded asA R(X)= p b R,p(y)a R,p(x),A I(X)= p b I,p(y)a I,p(x),(4.2)Ψ(1)(X)= p f(1)p(y)ψ(1)p(x),Ψ(2)(X)= p f(2)p(y)ψ(2)p(x).(4.3) Exact mode functions and mass-eigenvalues can be found for several light modes of b R,p(y),b R,0(y)=C R,0dn ΛyFigure3:The mode functions for fermionic zero-modesψ(1)0andψ(2)0.The solid line represents the profile of f(1)0(y)and the dashed line is that of f(2)0(y).b R,1(y)=C R,1cn Λy k2Λ2,b R,2(y)=C R,2sn Λy k2,(4.4) where functions dn(u,k),cn(u,k),sn(u,k)are the Jacobi’s elliptic functions and C R,p are nor-malization factors.For b I,p(y),we canfind all the eigenmodesb I,p(y)=12πRe i pR2,(p∈Z).(4.5)The masslessfield a R,0(x)is the Nambu-Goldstone(NG)boson for the breaking of the trans-lational invariance in the extra dimension.Thefirst massivefield a R,1(x)corresponds to the oscillation of the background wall around the anti-podal equilibrium point and hence becomes massless in the limit of R→∞.All the other bosonicfields remain massive in that limit.For fermions,only zero modes are known explicitly,f(1)0(y)=C0 dn Λy k,k ,f(2)0(y)=C0 dn Λy k,k ,(4.6)where C0is a normalization factor.These fermionic zero modes are the NG fermions for the breaking of Q(1)-SUSY and Q(2)-SUSY,respectively.Thus there are fourfields which are massless or become massless in the limit of R→∞: a R,0(x),a R,1(x),ψ(1)0(x)andψ(2)0(x).The profiles of their mode functions are shown in Fig.2and Fig.3.Otherfields are heavier and have masses of the order ofΛ.We will concentrate ourselves on the breaking of the Q(1)-SUSY,which is approximately preserved by our wall at y=0.So we call thefieldψ(2)0(x)the NG fermion.We can obtain a three-dimensional effective Lagrangian by substituting the mode-expanded fields into the Lagrangian(3.1),and carrying out an integration over yL(3)=−V0−12∂m a R,1∂m a R,1−i2ψ(2)0∂/ψ(2)0−1√ΛA cl(y) b R,1(y)f(1)0(y)f(2)0(y)=g2C20[4]E.Witten and D.Olive,Phys.Lett.B78(1978)97.[5]M.Cvetic,S.Griffies and S.Rey,Nucl.Phys.B381(1992)301[hep-th/9201007].[6]G.Dvali and M.Shifman,Phys.Lett.B396(1997)64[hep-th/9612128];Nucl.Phys.B504(1997)127[hep-th/9611213].[7]A.Kovner,M.Shifman,and A.Smilga,Phys.Rev.D56(1997)7978[hep-th/9706089];A.Smilga and A.Veselov,Phys.Rev.Lett.79(1997)4529[hep-th/9706217];V.Kaplunovsky,J.Sonnenschein,and S.Yankielowicz,Nucl.Phys.B552(1999)209 [hep-th/9811195];B.de Carlos and J.M.Moreno,Phys.Rev.Lett.83(1999)2120[hep-th/9905165];J.Edelstein,M.L.Trobo,F.Brito and D.Bazeia,Phys.Rev.D57(1998)7561 [hep-th/9707016].[8]E.R.C.Abraham and P.K.Townsend,Nucl.Phys.B351(1991)313;M.Cvetic,F.Quevedo and S.Rey,Phys.Rev.Lett.67(1991)1836.[9]G.Gibbons and P.Townsend,Phys.Rev.Lett.83(1999)1727[hep-th/9905196];S.M.Car-roll,S.Hellerman and M.Trodden,Phys.Rev.D61(2000)065001[hep-th/9905217];A.Gorsky and M.Shifman,Phys.Rev.D61(2000)085001[hep-th/9909015];H.Oda,K.Ito, M.Naganuma and N.Sakai,Phys.Lett.B471(1999)140[hep-th/9910095];Nucl.Phys.B586(2000)231[hep-th/0004188];D.Binosi and T.ter Veldhuis,Phys.Lett.B476(2000) 124[hep-th/9912081];S.Nam and K.Olsen,JHEP0008(2000)001.[10]N.Maru,N.Sakai,Y.Sakamura,and R.Sugisaka,Phys.Lett.B496(2000)98[hep-th/0009023].[11]N.Maru,N.Sakai,Y.Sakamura,and R.Sugisaka,Nucl.Phys.B616(2001)47[hep-th/0107204].[12]N.Maru,N.Sakai,Y.Sakamura,and R.Sugisaka,“SUSY Breaking by Stable non-BPSWalls”,to appear in the Proceedings of the10th Tohwa international symposium on string theory[hep-th/0109087].[13]T.Lee and G.H.Wu,Phys.Lett.B447(1999)83[hep-ph/9805512];Mod.Phys.Lett.A13(1998)2999[hep-ph/9811458];T.E.Clark and S.T.Love,Phys.Rev.D54(1996)5723[hep-ph/9608243].[14]M.Naganuma,M.Nitta,and N.Sakai,to appear in Phys.Rev.[hep-th/0108179].。

在有限元分析中的非弹性响应模型中,通过分离变形到恢复(弹性)和不可恢复的(非弹性)来区分弹性和非弹性的反应。

这种分离是基于假设有一种应变率之间的附加的关系:是总应变率，是弹性应变的变化率，是非弹性应变的变化率。

一个更一般的假设是总变形，，是由非弹性变形和纯粹的弹性变形组成的(伴随着加在任何阶段流程中的刚体旋转):在《附加应变率分解》章节1.4.4,方程4.1.1-1和 4.1.1-2是近似方程。

我们认为,如果1.方程4.1.1-1中使用的总应变速率是测量变形的速率:是速度，是当前空间质点的位置；2.弹性压力很小。

并且这种近似是一致的。

有限元分析使用变形率作为在限制变形问题中测量应变率的原因。

(只有当压力忽略不计时，不同的应变测量之间的区别很重要,也就是限制变形问题。

)实际中许多材料的弹性压力始终保持很小，例如,一个金属的屈服应力通常是小于它的弹性模量三个数量级,这意味着弹性应变应乘以。

然而,一些材料(例如，聚合物)也可以接受大的弹性应变并且伸缩自如,在这种情况下,附加应变率分解不再是一个近似值。

有限元分析提供了各种弹性响应模型。

其中最简单的是线性弹性:是可能取决于温度的一个矩阵,但并不依赖于变形(除非这种依赖是由损伤模型引入的)。

这种弹性模型用于小变形问题或模拟在弹性压力总是小的弹塑性模型中的弹性。

一个扩展的弹性行为的类型是次弹性模型:可能取决于变形的地方。

在这种情况下,弹性可能不再是非线性,但有限元分析仍然基于假设弹性压力总是很小。

在多孔和散体介质,弹性性质主要依赖于体积应变;多孔介质的弹性行为在《多孔弹性》的4.1.1节中讲解。

最通用的一种非线性弹性行为是超弹性的模型,我们定义假设有一个的由压力产生U为潜在的能量密度。

和是共轭应力和应变措施。

这种形式的弹性模型一般用来模拟弹性体:长期反应是完全可恢复的(弹性)的大变形材料。

有限元分析中提供的超弹性建模在《大应变超弹性平面分析程序》章节4.6中叙述过。

On Solving The Partial MAX-SAT ProblemZhaohui Fu and Sharad MalikDepartment of Electrical EngineeringPrinceton UniversityPrinceton,NJ08544,USA{zfu,sharad}@Abstract.Boolean Satisfiability(SAT)has seen many successful applications invariousfields such as Electronic Design Automation and Artificial Intelligence.However,in some cases,it may be required/preferable to use variations of thegeneral SAT problem.In this paper,we consider one important variation,the Par-tial MAX-SAT problem.Unlike SAT,Partial MAX-SAT has certain constraints(clauses)that are marked as relaxable and the rest are hard,i.e.non-relaxable.Theobjective is tofind a variable assignment that satisfies all non-relaxable clausestogether with the maximum number of relaxable ones.We have implementedtwo solvers for the Partial MAX-SAT problem using a contemporary SAT solver,zChaff.Thefirst approach is a novel diagnosis based algorithm;it iteratively ana-lyzes the UNSAT core of the current SAT instance and eliminates the core througha modification of the problem instance by adding relaxation variables.The secondapproach is encoding based;it constructs an efficient auxiliary counter that con-strains the number of relaxed clauses and supports binary search or linear scan forthe optimal solution.Both solvers are complete as they guarantee the optimalityof the solution.We discuss the relative strengths and thus applicability of the twosolvers for different solution scenarios.Further,we show how both techniquesbenefit from the persistent learning techniques of incremental SAT.Experimentsusing practical instances of this problem show significant improvements over thebest known solvers.1IntroductionIn the last decade Boolean Satisfiability(SAT)has seen many great advances,includ-ing non-chronological backtracking,conflict driven clause learning,efficient Boolean Constraint Propagation(BCP)and UNSAT core generation.As a consequence,many applications have been able to successfully use SAT as a decision procedure to deter-mine if a specific instance is SAT or UNSAT.However,there are many other variations of the SAT problem that go beyond this decision procedure use of SAT solvers.For example,the MAX-SAT Problem[8]seeks the maximum number of clauses that can be satisfied.This paper examines a generalization of this problem referred to as Partial MAX-SAT[2,15].2Partial MAX-SAT[2,15](PM-SAT)sits between the classic SAT problem and MAX-SAT.While the classic SAT problem requires all clauses to be satisfied,PM-SAT relaxes this requirement by having certain clauses marked as relaxable or soft and others to be non-relaxable or hard.Given n relaxable clauses,the objective is tofind an assignment that satisfies all non-relaxable clauses together with the maximum number of relaxable clauses(i.e.a minimum number k of these clauses get relaxed).PM-SAT can thus be used in various optimization tasks,e.g.multiple property checking,FPGA routing,uni-versity course scheduling,etc.In these scenarios,simply determining that an instance is UNSAT is not enough.We are interested in obtaining a best way to make the instance satisfiable allowing for some clauses to be unsatisfied.The difference between PM-SAT and MAX-SAT[8]is that every clause in MAX-SAT can be relaxed,which clearly makes MAX-SAT a special case of PM-SAT.Though decision versions of both problems are NP-Complete[5],PM-SAT is clearly more ver-satile.1.1Previous WorkPM-SAT wasfirst defined by Miyazaki et al.[15]during their work on optimization of database queries in1996.In the same year,Kautz et al.[10]proposed thefirst heuristic algorithm based on local search for solving this ter in1997Cha et al.[2] proposed another local search technique to solve the PM-SAT problem in the the context of university course scheduling.In2005,Li used two MinCostSat solvers,eclipse-stoc[12]and wpack[12],for the transformed PM-SAT problem in FPGA routing.MinCostSat is a SAT problem which minimizes the cost of the satisfying assignment.For example,assigning a variable to be true usually incurs a positive cost while assigning it to be false incurs no cost.The objective is tofind a satisfying assignment with minimum total cost.By inserting a slack variable[12]to each of the relaxable clauses,Li transforms the PM-SAT problem into a MinCostSat problem with each slack variable having a unit cost.eclipse-stoc is a general purpose MinCostSat solver and wpack is specialized for FPGA routing bench-marks.Li demonstrated some impressive results using wpack in his thesis.However, both eclipse-stoc and wpack are based on local search techniques and hence are not complete solvers,i.e.the solver provides no guarantee on the optimality of the solution.Argelich and Many`a uses a branch and bound approach for the over constrained MAX-SAT problems[1].However,as we will show in Section4,branch and bound based algorithms,including bsolo,do not work well on the PM-SAT problem.1.2Our ContributionIn this paper,we propose two practically efficient approaches to solve the PM-SAT problem optimally.Both approaches use the state-of-the-art SAT solver zChaff[16] with certain extensions.1.Diagnosis Based.Thefirst approach is based on the ability of SAT solvers to pro-vide an UNSAT core[21]for unsatisfiable instances.This core is a subset of origi-nal clauses that are unsatisfiable by themselves and in some sense can be considered3 to be the“cause”of the unsatisfiability.This core is generated as a byproduct of the proof of the unsatisfiability.The UNSAT core is analyzed and each relaxable clause appearing in the core is augmented with a distinct relaxation variable.Additional clauses are added to the original SAT instance to ensure the one-hot property of these relaxation variables.This augmentation essentially eliminates this core from the SAT instance.The procedure continues until the SAT instance is satisfiable.We give a proof of the optimality of thefinal solution using these relaxation variables and the one-hot property.2.Encoding Based.The second approach constructs an efficient auxiliary logic counter,i.e.an adder and comparator,to constrain the number of clauses that can be relaxedsimultaneously.It then uses either binary search or linear scan techniques tofind the minimum number of k(out of n)clauses that need to be relaxed.The logic counter is carefully designed such that maximum amount of learned information can be re-used across different invocations of the decision procedure.2Diagnosis Based Approach:Iterative UNSAT Core Elimination Being the best solver in the Certified UNSAT Track of SAT2005Competition,zChaff is very efficient in generating UNSAT cores.Our diagnosis based approach takes full advantage of this feature.It iteratively identifies the reason of the unsatisfiability of the instance,i.e.the UNSAT core[21],and uses relaxation variables to eliminate these UNSAT cores one by one until the instance becomes satisfiable.Definition1.An unsatisfiable core is a subset of the original CNF clauses that are unsatisfiable by themselves.Modern SAT solvers provide the UNSAT core as a byproduct of the proof of unsat-isfiability[21].2.1The Optimal Algorithm with ProofThe diagnosis based approach is illustrated in Algorithm1.We use CNF to represent the original SAT instance and V(CNF)is the set of all Boolean variables and C(CNF) is the set of all clauses.An UNSAT core UC is a set of clauses,i.e.UC⊆C(CNF).A clause c∈C(CNF)consists of a set of literals.A literal l is just a Boolean variable, v∈V(CNF),with positive or negative phase,i.e.l=v or l=v .Given an UNSAT core UC,for each relaxable clause c∈UC,a distinct relaxation variable is added to this clause,i.e.c is replaced by c∪{v}.Setting this variable to true makes the associated clause satisfied(and hence relaxed).An UNSAT core is said to be eliminated when at least one of its clauses is satisfied(relaxed)by a relaxation variable setting to be true.Let S be the set of relaxation variables from UNSAT core UC,the one-hot constraint over a set S of Boolean variables requires that one and only one of the variables in S is assigned to be true and the other|S|−1variables must be false.The number of clauses added due to the one-hot constraint is|S|×(|S|−1)41:S:=/02:while SAT solver returns UNSATISFIABLE do3:Let UC be the UNSAT core provided by the SAT solver4:S:=/05:for all Clause c∈UC do6:if c is relaxable then7:Allocate a new relaxation variable v8:c:=c∪{v}9:S:=S∪{v}10:end if11:end for12:if S=/0then13:Return CNF UNSATISFIABLE14:else15:Add clauses enforcing the One-Hot constraint for S to the SAT solver 16:S:=S∪S17:end if18:end while19:R:={v|v∈S,v=1};k:=|R|20:Return Satisfying Assignment,k,R.5 It is worth mentioning that the UNSAT core extraction is not compulsory.One could add a relaxation variable to each relaxable clause and require this batch of relaxation variables to be one-hot for every iteration in which the problem remains unsatisfiable. This naive approach is still capable offinding the minimum number of clauses to be relaxed.However,recall that the one-hot constraint requires O(|S|2)additional clauses where S is the set of relaxable clauses in the UNSAT core.Therefore it is impractical to enforce the one-hot constraint on the relaxation variables for all relaxable clauses.For example,the PM-SAT instance might have100relaxation clauses while only3appear in the UNSAT core.The naive approach adds100×992+1=4clauses.The diagnosis based approachexploits the availability of the UNSAT core to keep the number of relaxation variables and one-hot constraint clauses small.2.2An Illustrative ExampleIt is worth mentioning that Algorithm1does not require the UNSAT core UC to be min-imal.Furthermore,the UNSAT cores encountered by Algorithm1need not be disjoint. The following example shows a simple CNF formula that contains two overlapping cores.Suppose we have four Boolean variables x1,x2,x3and x4.Relaxable clauses are shown with square brackets and denotes the resolution operator.(x 1+x 2)(x 1+x3)(x 1+x 3)(x 2+x4)(x 2+x 4)[x1][x2]This CNF formula is unsatisfiable since(x 1+x 2)[x1][x2]form an UNSAT core because(x 1+x 2) [x1] [x2]=(x 2) [x2]=()Note that whether a clause is relaxable or not does not affect the resolution.Recall that a UNSAT core is a set of original clauses that are unsatisfiable and they resolve to an empty clause(),which can never be satisfied.The only relaxable clauses in this core are[x1][x2].So in thefirst iteration we add two distinct relaxation variables r1and r2 to each of them respectively and enforce r1and r2to be one-hot.The resulting CNF formula is(x 1+x 2)(x 1+x3)(x 1+x 3)(x 2+x4)(x 2+x 4)[x1+r1][x2+r2](r 1+r 2)(r1+r2)Note that clauses due to the one-hot constraint are not relaxable.However,the relaxable clauses are still marked as relaxable even after inserting relaxation variables.This is be-cause,as we will show,one relaxation variable may not be enough to make the instance satisfiable.The current CNF formula is still unsatisfiable as(x 1+x3) (x 1+x 3) [x1+r1] (r 1+r 2) [x2+r2]=(x2)(x2) (x 2+x 4)=(x 4)(x2) (x 2+x4)=(x4)(x 4) (x4)=()6So in the second iteration,we add another two relaxation variables r3and r4to the relaxable clauses[x1+r1][x2+r2]in the core.Together with clauses due to the one-hot constraint of r3and r4,the CNF formula becomes(x 1+x 2)(x 1+x3)(x 1+x 3)(x 2+x4)(x 2+x 4)[x1+r1+r3][x2+r2+r4](r 1+r 2)(r1+r2)(r 3+r 4)(r3+r4)This formula is satisfiable with the following assignmentx1=0,x2=0,x3=1,x4=1,r1=1,r2=0,r3=0,r4=1.Based on the above satisfying assignment,both[x1][x2]should be relaxed to make the problem satisfiable,i.e.k=n=2.Note that there is no constraint among the relaxation variables added in different iterations and the one-hot constraint only applies to all relaxation variables added due to the same UNSAT core in one iteration.The numberof the relaxation variables needed only depends on the number of relaxation clauses in the current UNSAT core1,and not the total number of relaxation clauses in the entire SAT instance.In Section4we will see some cases where the total number of relaxable clauses is large and our diagnosis based approach still performs well on these cases.The iterative core elimination requires the SAT solver to be able to provide the UN-SAT core(or proof)as part of answering UNSAT.This feature does incur some over-head.For example,the SAT solver needs to record the resolution trace for each learned clause2.Even when a learned clause is deleted,which happens very frequently in most state-of-the-art SAT solvers,the resolution trace for that particular learned clause can-not be deleted because it might be used to resolve other learned clauses that are not yet deleted.In case of an unsatisfiable instance,we need all the resolution information so that we could trace back from the conflict to the original clauses,which then form the UNSAT core.Recording the resolution trace not only slows down the search speed,but also uses a large amount of memory,which could otherwise be used for learned clauses.3Encoding Based Approach:Constructing An Auxiliary Counter With highly optimized state-of-the-art SAT solvers like zChaff[16],Berkmin[6],Siege[17] and MiniSat[4],the most straightforward way is to translate the PM-SAT problem di-rectly into a SAT instance.Such an implementation is likely to be efficient since the translated SAT instance takes advantages of all the sophisticated techniques used in a contemporary SAT solver.Furthermore,this approach requires very little or no modifi-cation to the SAT solver itself and hence could continuously benefit from the advancesin SAT.7 However,conventional SAT solvers do not support integer arithmetic,which is nec-essary in PM-SAT for expressing the constraint of≤k clauses left unsatisfied.We use an auxiliary logic counter[11]to represent this≤k condition,whose output is a Boolean variable and the entire counter could then be translated into CNF in a straightforward way.There are various ways of constructing such an an auxiliary logic counter[11]. Xu considers four types of these logic counters,namely chain counter,hierarchical tree counter,routing counter and sorting counter in her work on subSAT[20].subSAT is a MAX-SAT problem but with the assumption that k n,where n here is the total number of clauses.In other words,the problem becomes satisfiable with very small number(usually k<5)of clauses removed(relaxed).In the subSAT implementation, one mask variable(which is equivalent to our relaxation variable)is added to each of the n clauses and they constrain that only≤k mask variables can be true by using one of the four logic counters as mentioned above.The chain counter method creates a lg k+1 bit adder for each clause and concatenates them together.Thefinal output from the last adder is constrained to be≤k.The hierarchical tree counter creates a tree using lg k+1 bit adders as internal nodes that sum up all n mask variables and gives a lg k+1 bit output at the root of the tree.The routing counter implements k k-to-n decoders with k inputs all set to be1.The sorting counter uses a sorting circuit with k max operators(range from n bit to n−k+1bit)to move the1s to one side of the output and then checks the k th bit of the output.Xu states that thefirst two counters(chain and tree counters)are more efficient than the others in terms of the amount of additional logic.3.1An Efficient Hierarchical Tree AdderThe most significant differences between our proposed encoding based approach and the subSAT approach are that our hierarchical tree adder is independent of k and we do not assume k n.In addition to the linear scan for minimum k,we also use a binary search on[0,n]for the minimum k(subSat only uses linear scan due to their assumption of k n).We design our tree adder to be independent of the value k for two obvious reasons.First,we only need to construct the adder once at the beginning and re-use it during each iteration of the binary search,as compared to constructing the adder lg n times for binary search and k times for linear scan.Second,using the idea of incremental SAT[18],all clauses associated with the adder can be kept intact since they are always consistent with the problem.Maintaining the learned information is very important to the performance of most contemporary SAT solvers.Unfortunately,all the above4types of auxiliary counters proposed by Xu are dependent on k,particularly for the routing and sorting counters.We propose a hierarchical tree adder that is independent of k using elementary adders,e.g.half adder and full adders.Figure1gives an example of such an adder with n=9.It can be shown that the total number of additional2-input logic gates is ≤5n as follows.Consider starting with n≥3bit input,we use a full adder to sum up 3bits while returning a sum bit and a carry bit[9].The sum bit needs to be added with the other n−3bits left and the carry bit will only be used in next level.So each full adder reduces the number of inputs left by2and n8Fig.1.An efficient hierarchical tree adder that sums the number of1s from the n=9bit input (top)and gives a4bit binary value(bottom).Thefirst level uses4full adders(FA);the second level needs1full adder and1half adder(HA);the third level just needs1half adder.S and C are the sum and carry bits of the adder respectively.second level,we consider all the carry bits from the previous level and there are at most n2 +n8+...+1≤nEach full adder requires52-input logic gates(2AND,2XOR and1OR gate),which gives the total number of additional logic gates≤5n.Note that we can sometimes replace a full adder by a half adder due to simplification by constant value(0),as shown in the second and third levels in Figure1.There is an important distinction between our hierarchical tree adder in Figure1 and the one used by Xu[20].Instead of using full/half adders as internal nodes of the tree,Xu uses a lg k+1 bit adder for each of the internal nodes,which introduces a large amount of redundancy with relatively large k.For example,thefirst level inputs to the adder are at most1and in a binary representation of lg k+1 bits,at least lg k+1 −1bits are just0s.Our hierarchical tree adder is free of such redundancy due to the judicious use of full/half adders.The hierarchical tree adder outputs a lg n+1 bit binary value,which is then com-pared against a given value k using a logic comparator that outputs true if and only if the sum is less than or equal to k.Note that this logic comparator is dependent on k for efficiency reasons3.This hierarchical tree adder with comparator provides us an ef-ficient platform for searching the minimum k.Generally binary search has advantages over linear scan on the benchmarks with k>lg n.9 It is worth mentioning that when this logic counter(adder with comparator)are translated into CNF clauses,the hierarchical tree adder generates many more clauses than the comparator does.In general,the number of CNF clauses from the adder is O(n)while from the comparator is O(lg n).For each iteration during the binary search or linear scan,only O(lg n)original CNF clauses with related learned clauses need to be changed.The remaining clauses include both original and learned clauses correspond-ing to the original problem instance and the adder.The learned clauses capture the logic relationship among the Boolean variables used in the problem instance and the adder and they cannot be learned without the adder.One disadvantage of using an auxiliary counter is the introduction of a large number of XOR gates.Each full adder consists of two XOR gates and the entire counter results in2n XOR gates.Though the number of additional logic gates is only linear in n,the situation could get worse when many XOR gates are chained together.For example,the least significant bit of the sum comes from an XOR chain of length lg n .XOR chains are well known to cause poor performance of SAT solvers.One main reason is that unlike AND/OR gates,Boolean constraint propagation over XOR gates is very limited. This complication makes this approach no longer efficient for solving problems with very large n.It is worth mentioning that a large n does not necessarily imply a large k though obviously k≤n.3.2A Discussion on Incremental SATIncremental SAT wasfirst formalized by Strichman[18].It is the process of solving a series of SAT instancesϕ1,ϕ2,...,ϕn.The consecutive SAT instances,ϕi andϕi+1, are similar,i.e.only a small number of clauses(and variables)are different.Given the solution ofϕi,we could solveϕi+1incrementally by only updating the different clauses while keeping most learned clauses inϕi,which are still consistent withϕi+1,intact. Maintaining the maximum amount of the learned clauses,i.e.recording the most visited search space,is a great advantage than starting from scratch each time.A key issue in the implementation of incremental SAT is the efficient updating from instanceϕi toϕi+1,which usually includes both deletion and addition of origi-nal clauses.Addition of new original clauses is trivial.However,deletion of original clauses implies the additional deletion of all learned clauses related to the deleted orig-inal clauses in order to maintain the integrity of the clauses database.This deletion can be performed efficiently with the use of group IDs.A group ID indicates a particular group,to which the clause belongs.The group IDs of a learned clause is the union of all the group IDs from the clauses used to generate this learned clauses(through reso-lution).Deletion according to a particular group ID removes all clauses(both original and learned)having this ID.For the encoding based approach,we utilize the incremental SAT feature of zChaff and group all CNF clauses associated with the comparator using the same group ID. This implementation enable us to only change a very small fraction of all clauses(both original and learned)that are related to the comparator for each different value of k dur-ing binary search or linear scan.All clauses associated with the adder are independent of k and hence remain unchanged throughout the entire incremental SAT.Recall that the adder corresponds to many more clauses than the comparator does.10However,unlike the encoding based approach,the diagnosis based approach re-quires us to update the original clauses by inserting some relaxation variables.This makes it harder to use the incremental SAT algorithm.However,we can still group all the hard constraint clauses in a group and reuse all learned clauses that are generated within this group.In other words,we only delete the learned clauses associated with the relaxable constraint clauses.4Experimental ResultsWe implemented the subSat approach for PM-SAT using the chain counter and hierar-chical tree counter proposed by Xu[20]for comparison.In addition,we translate our PM-SAT benchmarks into MinCostSat instances so that we can have an extensive com-parison using other general purpose solvers.Recall that a MinCostSat problem is a SAT problem with a cost function for each satisfying assignment.We add a unique relax-ation variable to each relaxable clause in PM-SAT and the cost of this relaxation vari-able is1.All other variables have a cost of0.Non-relaxable clauses remain unchanged in the above translation.The resulting problem is now a MinCostSat instance where the minimum cost corresponds to the minimum number of relaxation variables setting to be1,which in turn implies that minimum number of clauses are relaxed.We then use Scherzo[3],bsolo[13]and cplex[7]to solve the translated MinCostSat problem. Scherzo is a well known branch-and-bound solver for Binate/Uniate Covering Problem (BCP/UCP)that incorporates many state-of-the-art techniques,including Maximum In-dependent Set[5]based lower bounding,branch variable selection and various search pruning rules.The BCP problem is essentially a MinCostSat problem[12]with a spe-cific cost function.UCP has the additional restriction that all variables appear in only one phase.But unfortunately,Scherzo is not able to solve any of the benchmarks in the following tables.bsolo is another state-of-the-art branch-and-bound BCP/UCP solver based on the SAT solver GRASP[14].cplex is the cutting edge commercial Linear Programming(LP)solver that is also capable offinding integer solutions efficiently.All the experiments are conducted on a Dell PowerEdge700running Linux Fe-dora core1.0(g++GCC3.3.2)with single Pentium42.8GHz,1MB L2cache CPU on 800MHz main bus.4.1FPGA Routing BenchmarksWe conduct our experiments mainly on industrial benchmarks.Table1shows the results of industrial examples resulting from a SAT based FPGA router.Each relaxable clause corresponds to a net-arc(single source,single destination)in the routing problem.Re-laxation of clauses in the unsatisfiable SAT instance to make it satisfiable represent finding the fewest number of net-arcs which,if re-routed elsewhere,e.g.route-around, would allow the remaining set of net-arcs to be routed simultaneously.The benchmark name in Table1shows the number of net-arcs in the actual FPGA routing problem.For example,thefirst row shows the result of benchmark FPGATable1.Performance comparison on FPGA routing benchmarks.Timeout for all solvers:1hour.∗indicates server times out,the best solution found is reported.Bench-Num.Min.Diagnosis Encoding subSat Gen.Solver Vars.Cls.Core Rmv Linear Tree cplex 27135373 2.13 2.6421.29 FPGA1786931380.83309.75860.1812∗3292023 1.10 1.12 6.56 FPGA90773318.6526.4427.934∗392286547.768.8359.07 FPGA65664410.128.3612.805∗which corresponds to the27net-arcs to be routed.The optimal solution is a relaxation of k=3clauses(out of27)that makes the entire problem satisfiable.The diagnosis based approach takes1.85seconds tofind the optimal solution.The binary search and linear scan of the encoding based approach take2.13and1.65seconds respectively.The subSat approach using linear scan with the chain counter and hierarchical tree counter need2.64and2.85seconds respectively.For the translated MinCostSat problem,bsolo takes21.29seconds.cplex only reports a solution of k=3but it cannot prove its opti-mality.Scherzo could not report any solution found within the1hour time limit for all our benchmarks and hence is omitted from all of our tables.4.2Multiple Property Checking BenchmarksTable2shows the results of multiple property checking using circuits from ISCAS85 and ITC99benchmarks.Relaxable clauses are the properties(assertions)that assume each output signal of the entire circuit to be1or0.The non-relaxable clauses are trans-lated from the circuit structure.The corresponding PM-SAT instance is tofind the max-imum number of outputs that can be1or0(satisfying the property).Benchmarks that are satisfiable without any relaxation are excluded from the tables.All benchmarks start with a c are from the iscas85family and the rests are from the itc99family.4.3Randomized UNSAT BenchmarksTable3shows the results of classic UNSAT benchmarks with randomly chosen relax-able clauses.These benchmarks are from the fvp-unsat-2.0(verification of super-scalar microprocessors)family by Velev[19].Note that all the benchmarks in Table3 have k=1,which makes it inefficient to use binary search.Table1,Table2and Table3clearly show that both approaches constantly outper-form the best known solvers.For benchmarks with a large number of relaxable clauses, e.g.b17,b20and b22in Table2and all benchmarks in Table3,the diagnosis based approach has obvious advantage over the search approach(either binary or linear), which suffers from the large auxiliary adder.With most other benchmarks like c6288 in Table2and FPGA。

Optimism is a powerful force that can significantly influence ones life.It is the tendency to look at the brighter side of life,to expect positive outcomes,and to believe in the inherent goodness of people and situations.Here are some key points to consider when writing an essay on optimism:1.Definition of Optimism:Begin by defining what optimism is and how it contrasts with pessimism.Optimism is a positive mental attitude that involves expecting the best outcome in any situation.2.Benefits of Optimism:Discuss the various benefits of being optimistic.These can include improved mental health,increased resilience in the face of adversity,better physical health,and enhanced social relationships.3.Personal Stories:Share anecdotes or case studies of individuals who have overcome challenges due to their optimistic outlook.This can help to illustrate the power of optimism in reallife situations.4.Cognitive Effects:Explain how optimism can influence cognitive processes. Optimistic people tend to have a more positive interpretation of events,which can lead to more effective problemsolving and decisionmaking.5.Psychological Impact:Delve into the psychological aspects of optimism.It can reduce stress and anxiety,foster a sense of wellbeing,and promote a more positive selfimage.6.Physical Health Correlations:Discuss the correlation between optimism and physical health.Studies have shown that optimistic individuals may have stronger immune systems and are more likely to engage in healthy behaviors.7.Optimism vs.Realism:Address the balance between optimism and realism.While being optimistic is beneficial,its also important to be realistic and not ignore potential risks or challenges.8.Cultivating Optimism:Offer practical advice on how individuals can cultivate a more optimistic mindset.This might include practicing gratitude,reframing negative thoughts, setting achievable goals,and surrounding oneself with positive influences.9.Challenges of Optimism:Acknowledge that there can be challenges associated with optimism,such as the potential for unrealistic expectations or denial of problems.Discuss how to maintain a balanced perspective.10.Conclusion:Summarize the importance of optimism and its impact on personal and professional success.Encourage readers to adopt an optimistic outlook as a means to enhance their overall quality of life.Remember to use clear and concise language,provide evidence to support your points, and structure your essay with a clear introduction,body,and conclusion.By exploring these aspects,your essay on optimism will provide a comprehensive view of its significance and practical applications in everyday life.。