Methodologies] Symbolic and Algebraic Manipulation —Algorithms G.1.2 [Mathematics of Compu

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a-level进阶纯数学英文版教材a-level further pure mathematics

a-level进阶纯数学英文版教材a-level further pure mathematicsTitle: ALevel Further Pure Mathematics TextbookAs an esteemed university professor specializing in English language and adept at composing various types of essays, I have taken on the task of crafting a comprehensive essay based on the given title: "ALevel Further Pure Mathematics Textbook."Introduction:The realm of mathematical understanding knows no bounds, and the pursuit of knowledge in the purest form is a quest that requires dedication, precision, and a keen analytical mind. This essay delves into the development and significance of the ALevel Further Pure Mathematics textbook, exploring its contents, structure, and the value it offers to students and educators alike.Content:The ALevel Further Pure Mathematics textbook is a seminal resource designed to amplify the understanding and application of advanced mathematical concepts for students at the ALevel stage. It navigates through a plethora of topics, including complex numbers, matrices, proof by induction, and further calculus, among others. Each chapter delves into the intricacies of mathematical theory, providing detailed explanations, examples, and exercises to bolster comprehension and problemsolving skills.One of the key strengths of this textbook lies in its comprehensive coverage of topics that are essential for students aspiring to delve deeper into the realms of theoretical mathematics. From algebraic structures to rigorous proofs, the textbook serves as a gateway to unlocking the mysteries of pure mathematics, equipping learners with the tools needed to tackle challenging problems with confidence and precision.Furthermore, the ALevel Further Pure Mathematics textbook adopts a holistic approach to learning, fostering critical thinking, logical reasoning, and creativity in its readers. By encouraging students to engage in problemsolving methodologies and explore diverse mathematical techniques, the textbook nurtures a deeprooted appreciation for the elegance and beauty of pure mathematics.Conclusion:In conclusion, the ALevel Further Pure Mathematics textbook stands as a beacon of excellence in the realm of advanced mathematical education. Its comprehensive coverage, pedagogical approach, and emphasis on deepening mathematical understanding make it an indispensable companion for students embarking on the journey of furthering their knowledge in pure mathematics. As educators, it is our responsibility to instill in our students a love formathematical inquiry and equip them with the necessary tools to excel in the realm of higher mathematics. The ALevel Further Pure Mathematics textbook serves as a guiding light in this endeavor, paving the way for future mathematicians to explore, discover, and innovate in the endlessly fascinating world of pure mathematics.。

托福物理学专业词汇：方法论Methodology

托福物理学专业词汇：方法论Methodology托福物理学学科分类词汇：方法论Methodology方法论,Methodology英语短句,例句大全方法论,Methodology1)Methodology[英][,Meθ?'D?L?D?I][美]['M?Θ?'Dɑl?D??]方法论1.Theoretical System And Methodology Of Coal Structural Chemistry;煤结构化学的理论体系与方法论2.Discussion On Research Methodology Of Law Of Evidence.;证据法学研究的方法论问题3.Thinking The System Biology And Its Methodology;对统生物学及其方法论的思考英文短句/例句1.On Innovation From Methodology Of Law Economics To Traditional Law Methodology;论法经济学方法论对传统法学方法论的创新2.On The Methods And Methodology Of Feminist Research In Education;论女性主义教育研究的方法和方法论3.One Methodology Suspicion--The Evolutionary Theories Of Gene On "Rule By Law";方法论猜想——“法治”基因进化论4.The Scientific Development Theory And The Ecological Trend Of Jurisprudence Methodology;论科学发展观与法学方法论的生态化5.Methodology Of Law And Economics：A General Review;法和经济学方法论：一个综述性的评论6.On Litigation Object论诉讼证明对象——以法律方法论为启示7.On System Analytical And Synthetic Method And Its Methodology试论系统分析综合法及其方法论启示8.The Object Of Existentialism,Theory Of Existence And Generative Methodology;存有论、生存论与生成性方法论旨趣9.Considering Poetry Is Just Like Considering Chan--An Analysis Of Yan Yu S Methodology Of "Considering Poetry As Chan";论诗如论禅——严羽“以禅喻诗”方法论辨析10.Epistemological And Methodological Significances Of Pound S Theory Of Translation;庞德翻译理论的理解论、方法论意义11.The Elaboration Of Marx S Influences In Logic And Reality;论马克思的跨越理论及其方法论意义12.On Jiang Zemin S Theory Of Innovation And Its Methodological Meaning;试论江泽民创新理论及其方法论意义13.On The Epistemological And Methodological Problems In The Probability Theory浅谈概率论中的理解论及方法论问题14.Housing Price Forecasting Method Based On TEI@I Methodology;基于TEI@I方法论的房价预测方法15.An Analysis Of The Necessity Of The Artistic Methodology Used In Education Research Methodology;解析教育研究方法论也需要艺术方法16.Exploration Of Teaching Way Of Practice Of "Middle School Mathematics Methodology;《中学数学方法论》教学实践方法探讨17.The Criminal Rebuilding Ways,Theory And Methodology In USA美国犯罪重建的方法、原理与方法论18.On Source Of Law--From The Perspective Of Legal And Jurisprudence Methodologies;论法律渊源——以法学方法和法律方法为视角相关短句/例句Method[英]['Meθ?D][美]['M?Θ?D]方法论1.A Research On The Methods Of Key Technology;关键技术选择与评价的方法论研究2.On The Method Characteristics Of Jiang Ze - Min Thought And Theory;论江泽民思想理论的方法论特征3.The Article Elaborates The Four Levels And Meaning Of The Industry Design Method.阐述了工业设计方法论内容的四个层次及意义,后结合实际,重点用创新设计法、形态组构法、设计管理法等理论对宝马这个世界汽车品牌及其旗下第五代新产品宝马5系轿车的设计研发过程实行分析。

数学中英语专业名词

数学中英语专业名词Aabelian group：阿贝尔群；absolute geometry：绝对几何；absolute value：绝对值；abstract algebra：抽象代数；addition：加法；algebra：代数；algebraic closure：代数闭包；algebraic geometry：代数几何；algebraic geometry and analytic geometry：代数几何和解析几何；algebraic numbers：代数数；algorithm：算法；almost all：绝大多数；analytic function：解析函数；analytic geometry：解析几何；and：且；angle：角度；anticommutative：反交换律；antisymmetric relation：反对称关系；antisymmetry：反对称性；approximately equal：约等于；Archimedean field：阿基米德域；Archimedean group：阿基米德群；area：面积；arithmetic：算术；associative algebra：结合代数；associativity：结合律；axiom：公理；axiom of constructibility：可构造公理；axiom of empty set：空集公理；axiom of extensionality：外延公理；axiom of foundation：正则公理；axiom of pairing：对集公理；axiom of regularity：正则公理；axiom of replacement：代换公理；axiom of union：并集公理；axiom schema of separation：分离公理；axiom schema of specification：分离公理；axiomatic set theory：公理集合论；axiomatic system：公理系统；BBaire space：贝利空间；basis：基；Bézout's identity：贝祖恒等式；Bernoulli's inequality：伯努利不等式；Big O notation：大O符号；bilinear operator：双线性算子；binary operation：二元运算；binary predicate：二元谓词；binary relation：二元关系；Boolean algebra：布尔代数；Boolean logic：布尔逻辑；Boolean ring：布尔环；boundary：边界；boundary point：边界点；bounded lattice：有界格；Ccalculus：微积分学；Cantor's diagonal argument：康托尔对角线方法；cardinal number：基数；cardinality：势；cardinality of the continuum：连续统的势；Cartesian coordinate system：直角坐标系；Cartesian product：笛卡尔积；category：范畴；Cauchy sequence：柯西序列；Cauchy-Schwarz inequality：柯西不等式；Ceva's Theorem：塞瓦定理；characteristic：特征；characteristic polynomial：特征多项式；circle：圆；class：类；closed：闭集；closure：封闭性或闭包；closure algebra：闭包代数；combinatorial identities：组合恒等式；commutative group：交换群；commutative ring：交换环；commutativity:：交换律；compact：紧致的；compact set：紧致集合；compact space：紧致空间；complement：补集或补运算；complete lattice：完备格；complete metric space：完备的度量空间；complete space：完备空间；complex manifold：复流形；complex plane：复平面；congruence：同余；congruent：全等；connected space：连通空间；constructible universe：可构造全集；constructions of the real numbers：实数的构造；continued fraction：连分数；continuous：连续；continuum hypothesis：连续统假设；contractible space：可缩空间；convergence space：收敛空间；cosine：余弦；countable：可数；countable set：可数集；cross product：叉积；cycle space：圈空间；cyclic group：循环群；Dde Morgan's laws：德·摩根律；Dedekind completion：戴德金完备性；Dedekind cut：戴德金分割；del：微分算子；dense：稠密；densely ordered：稠密排列；derivative：导数；determinant：行列式；diffeomorphism：可微同构；difference：差；differentiable manifold：可微流形；differential calculus：微分学；dimension：维数；directed graph：有向图；discrete space：离散空间；discriminant：判别式；distance：距离；distributivity：分配律；dividend：被除数；dividing：除；divisibility：整除；division：除法；divisor：除数；dot product：点积；Eeigenvalue：特征值；eigenvector：特征向量；element：元素；elementary algebra：初等代数；empty function：空函数；empty set：空集；empty product：空积；equal：等于；equality：等式或等于；equation：方程；equivalence relation：等价关系；Euclidean geometry：欧几里德几何；Euclidean metric：欧几里德度量；Euclidean space：欧几里德空间；Euler's identity：欧拉恒等式；even number：偶数；event：事件；existential quantifier：存在量词；exponential function：指数函数；exponential identities：指数恒等式；expression：表达式；extended real number line：扩展的实数轴；Ffalse：假；field：域；finite：有限；finite field：有限域；finite set：有限集合；first-countable space：第一可数空间；first order logic：一阶逻辑；foundations of mathematics：数学基础；function：函数；functional analysis：泛函分析；functional predicate：函数谓词；fundamental theorem of algebra：代数基本定理；fraction：分数；Ggauge space：规格空间；general linear group：一般线性群；geometry：几何学；gradient：梯度；graph：图；graph of a relation：关系图；graph theory：图论；greatest element：最大元；group：群；group homomorphism：群同态；HHausdorff space：豪斯多夫空间；hereditarily finite set：遗传有限集合；Heron's formula：海伦公式；Hilbert space：希尔伯特空间；Hilbert's axioms：希尔伯特公理系统；Hodge decomposition：霍奇分解；Hodge Laplacian：霍奇拉普拉斯算子；homeomorphism：同胚；horizontal：水平；hyperbolic function identities：双曲线函数恒等式；hypergeometric function identities：超几何函数恒等式；hyperreal number：超实数；Iidentical：同一的；identity：恒等式；identity element：单位元；identity matrix：单位矩阵；idempotent：幂等；if：若；if and only if：当且仅当；iff：当且仅当；imaginary number：虚数；inclusion：包含；index set：索引集合；indiscrete space：非离散空间；inequality：不等式或不等；inequality of arithmetic and geometric means：平均数不等式；infimum：下确界；infinite series：无穷级数；infinite：无穷大；infinitesimal：无穷小；infinity：无穷大；initial object：初始对象；inner angle：内角；inner product：内积；inner product space：内积空间；integer：整数；integer sequence：整数列；integral：积分；integral domain：整数环；interior：内部；interior algebra：内部代数；interior point：内点；intersection：交集；inverse element：逆元；invertible matrix：可逆矩阵；interval：区间；involution：回旋；irrational number：无理数；isolated point：孤点；isomorphism：同构；JJacobi identity：雅可比恒等式；join：并运算；K格式：Kuratowski closure axioms：Kuratowski 闭包公理；Lleast element：最小元；Lebesgue measure：勒贝格测度；Leibniz's law：莱布尼茨律；Lie algebra：李代数；Lie group：李群；limit：极限；limit point：极限点；line：线；line segment：线段；linear：线性；linear algebra：线性代数；linear operator：线性算子；linear space：线性空间；linear transformation：线性变换；linearity：线性性；list of inequalities：不等式列表；list of linear algebra topics：线性代数相关条目；locally compact space：局部紧致空间；logarithmic identities：对数恒等式；logic：逻辑学；logical positivism：逻辑实证主义；law of cosines：余弦定理；L??wenheim-Skolem theorem：L??wenheim-Skolem 定理；lower limit topology：下限拓扑；Mmagnitude：量；manifold：流形；map：映射；mathematical symbols：数学符号；mathematical analysis：数学分析；mathematical proof：数学证明；mathematics：数学；matrix：矩阵；matrix multiplication：矩阵乘法；meaning：语义； measure：测度；meet：交运算；member：元素；metamathematics：元数学；metric：度量；metric space：度量空间；model：模型；model theory：模型论；modular arithmetic：模运算；module：模；monotonic function：单调函数；multilinear algebra：多重线性代数；multiplication：乘法；multiset：多样集；Nnaive set theory：朴素集合论；natural logarithm：自然对数；natural number：自然数；natural science：自然科学；negative number：负数；neighbourhood：邻域；New Foundations：新基础理论；nine point circle：九点圆；non-Euclidean geometry：非欧几里德几何；nonlinearity：非线性；non-singular matrix：非奇异矩阵；nonstandard model：非标准模型；nonstandard analysis：非标准分析；norm：范数；normed vector space：赋范向量空间；n-tuple：n 元组或多元组；nullary：空；nullary intersection：空交集；number：数；number line：数轴；Oobject：对象；octonion：八元数；one-to-one correspondence：一一对应；open：开集；open ball：开球；operation：运算；operator：算子；or：或；order topology：序拓扑；ordered field：有序域；ordered pair：有序对；ordered set：偏序集；ordinal number：序数；ordinary mathematics：一般数学；origin：原点；orthogonal matrix：正交矩阵；Pp-adic number：p进数；paracompact space：仿紧致空间；parallel postulate：平行公理；parallelepiped：平行六面体；parallelogram：平行四边形；partial order：偏序关系；partition：分割；Peano arithmetic：皮亚诺公理；Pedoe's inequality：佩多不等式；perpendicular：垂直；philosopher：哲学家；philosophy：哲学；philosophy journals：哲学类杂志；plane：平面；plural quantification：复数量化；point：点；Point-Line-Plane postulate：点线面假设；polar coordinates：极坐标系；polynomial：多项式；polynomial sequence：多项式列；positive-definite matrix：正定矩阵；positive-semidefinite matrix：半正定矩阵；power set：幂集；predicate：谓词；predicate logic：谓词逻辑；preorder：预序关系；prime number：素数；product：积；proof：证明；proper class：纯类；proper subset：真子集；property：性质；proposition：命题；pseudovector：伪向量；Pythagorean theorem：勾股定理；QQ.E.D.：Q.E.D.；quaternion：四元数；quaternions and spatial rotation：四元数与空间旋转；question：疑问句；quotient field：商域；quotient set：商集；Rradius：半径；ratio：比；rational number：有理数；real analysis：实分析；real closed field：实闭域；real line：实数轴；real number：实数；real number line：实数线；reflexive relation：自反关系；reflexivity：自反性；reification：具体化；relation：关系；relative complement：相对补集；relatively complemented lattice：相对补格；right angle：直角；right-handed rule：右手定则；ring：环；Sscalar：标量；second-countable space：第二可数空间；self-adjoint operator：自伴随算子；sentence：判断；separable space：可分空间；sequence：数列或序列；sequence space：序列空间；series：级数；sesquilinear function：半双线性函数；set：集合；set-theoretic definition of natural numbers：自然数的集合论定义；set theory：集合论；several complex variables：一些复变量；shape：几何形状；sign function：符号函数；singleton：单元素集合；social science：社会科学；solid geometry：立体几何；space：空间；spherical coordinates：球坐标系；square matrix：方块矩阵；square root：平方根；strict：严格；structural recursion：结构递归；subset：子集；subsequence：子序列；subspace：子空间；subspace topology：子空间拓扑；subtraction：减法；sum：和；summation：求和；supremum：上确界；surreal number：超实数；symmetric difference：对称差；symmetric relation：对称关系；system of linear equations：线性方程组；Ttensor：张量；terminal object：终结对象；the algebra of sets：集合代数；theorem：定理；top element：最大元；topological field：拓扑域；topological manifold：拓扑流形；topological space：拓扑空间；topology：拓扑或拓扑学；total order：全序关系；totally disconnected：完全不连贯；totally ordered set：全序集；transcendental number：超越数；transfinite recursion：超限归纳法；transitivity：传递性；transitive relation：传递关系；transpose：转置；triangle inequality：三角不等式；trigonometric identities：三角恒等式；triple product：三重积；trivial topology：密着拓扑；true：真；truth value：真值；Uunary operation：一元运算；uncountable：不可数； uniform space：一致空间；union：并集；unique：唯一；unit interval：单位区间；unit step function：单位阶跃函数；unit vector：单位向量；universal quantification：全称量词；universal set：全集；upper bound：上界；Vvacuously true：??；Vandermonde's identity：Vandermonde 恒等式；variable：变量；vector：向量；vector calculus：向量分析；vector space：向量空间；Venn diagram：文氏图；volume：体积；von Neumann ordinal：冯·诺伊曼序数；von Neumann universe：冯·诺伊曼全集；vulgar fraction：分数；ZZermelo set theory：策梅罗集合论；Zermelo-Fraenkel set theory：策梅罗-弗兰克尔集合论；ZF set theory：ZF 系统；zero：零；zero object：零对象；。

ACM期刊及会议文献列表

ACM期刊及会议文献列表ACM 期刊清单1 ACM Computing Surveys (CSUR)2 ACM Journal of Computer Documentation (JCD)3 ACM Letters on Programming Languages and Systems (LOPLAS)4 Journal of Educational Resources in Computing (JERIC)5 Journal of Experimental Algorithmics (JEA)6 Journal of the ACM (JACM)ACM 杂志清单1 Communications of the ACM2 Crossroads3 StandardView4 Ubiquity5 eLearn6 intelligence7 interactions8 netWorkerACM学报清单1 ACM Transactions on Asian Language Information Processing (TALIP)2 ACM Transactions on Computational Logic (TOCL)3 ACM Transactions on Computer Systems (TOCS)4 ACM Transactions on Computer-Human Interaction (TOCHI)5 ACM Transactions on Database Systems (TODS)6 ACM Transactions on Design Automation of Electronic Systems (TODAES)7 ACM Transactions on Graphics (TOG)8 ACM Transactions on Information Systems (TOIS)9 ACM Transactions on Information and System Security (TISSEC)10 ACM Transactions on Internet Technology (TOIT)11 ACM Transactions on Mathematical Software (TOMS)12 ACM Transactions on Modeling and Computer Simulation (TOMACS)13 ACM Transactions on Programming Languages and Systems (TOPLAS)14 ACM Transactions on Software Engineering and Methodology (TOSEM)15 IEEE/ACM Transactions on Networking (TON)ACM会议清单1 ACM Policy2 ACM Southeast Regional Conference3 ACM/CSC-ER4 Computer graphics, virtual reality, visualisation and interaction in Africa5 International Conference on Autonomous Agents6 ACM International Conference Proceeding Series7 Analysis of Neural Net Applications Conference8 Annual Simulation Symposium9 Aspect-oriented software development10 International Conference on APL11 with EDA Technofair Design Automation Conference Asia and South Pacific12 Architectural Support for Programming Languages and Operating Systems13 ACM SIGACCESS Conference on Assistive Technologies14 AVI15 Creativity and Cognition16 International Conference on Compilers, Architecture and Synthesis for Embedded Systems17 Computer Architecture Workshop18 Conference on Computer and Communications Security19 Consortium for Computing Sciences in Colleges20 Contemporary Computing in Ukraine21 Conference On Computing Frontiers22Computers, Freedom and Privacy23 Code Generation and Optimization24 Conference on Human Factors in Computing Systems25 Conference on Information and Knowledge Management26 Conference On Information Technology Education27 International Conference on Hardware Software Codesign28 Annual Workshop on Computational Learning Theory29 Applications, Technologies, Architectures,and Protocols for Computer Communication30 Special Interest Group on Computer Personnel Research Annual Conference31 Symposium on Computers and the Quality of Life32 ACM Annual Computer Science Conference33 Computer Supported Cooperative Work34 ACM Conference on Universal Usability35 Collaborative Virtual Environments36 Annual ACM IEEE Design Automation Conference37 Designing Augmented Reality Environments38 Design, Automation, and Test in Europe39 Distributed event-based systems40 Workshop on Discrete Algothrithms and Methods for MOBILE Computing and Communications41 Symposium on Designing Interactive Systems41 International Conference on Digital Libraries43 Data Mining And Knowledge Discovery44 Data Warehousing and OLAP45 International Symposium on Databases for Parallel and Distributed Systems46 Designing Pleasurable Products And Interfaces47 ACM Workshop On Digital Rights Management48 DSL49 Designing For User Experiences50 Document Engineering51 Workshop on Dynamic and Adaptive Compilation and Optimization53 Electronic Commerce54 Ethics in the Computer Age55 OOPSLA workshop on eclipse technology eXchange56 European Design and Test Conference57 International Conference On Embedded Software58 Annual ERLANG Workshop59 Eye Tracking Research & Application60 ACM SIGOPS European Workshop61 European Design Automation Conference62 Workshop on Formal Methods in Security Engineering63 Formal Methods in Software Practice64 Formal Ontology in Information Systems65 Functional Programming Languages and Computer Architecture66 International Symposium on Field Programmable Gate Arrays67 Genetic And Evolutionary Computation Conference68 Geographic Information Systems69 Great Lakes Symposium on VLSI70 GRAPHICON71 Computer graphics and interactive techniques in Austalasia and South East Asia72 Conference on Supporting Group Work73 History of Medical Informatics74 History of Programming Languages75 History of Personal Workstations76 History of Scientific and Numeric Computation77 Conference on Hypertext and Hypermedia78 SIGGRAPH/EUROGRAPHICS Workshop On Graphics Hardware79 Haskell80 Hypercube Concurrent Computers and Applications81 International Conference on Artificial Intelligence and Law82 International Conference on Computer Aided Design83 International Conference on Information and Computation Economies84 International Conference on Functional Programming85 International Conference on Information Systems86 ICMI87 International Conference on Supercomputing88 International Conference on Software Engineering89 International Conference On Service Oriented Computing90 Interaction Design And Children91 International conference on Industrial and engineering applications of artificial intelligence and expert systems92 Internet Measurement Conference93 Workshop on I/O in Parallel and Distributed Systems94 Information Processing In Sensor Networks95 Information Quality in Informational Systems96 International Workshop on Information Retrieval with Asia Languages97 International Workshop on Real-time Ada Issues98 International Conference on Computer Architecture 99 International Symposium on Low Power Electronics and Design100 International Symposium on Methodologies for Intelligent Systems101 International Symposium on Memory Management102 International Symposium on Physical Design103 International Software Process Workshop104 International Conference on Symbolic and Algebraic Computation105 International Symposium on Systems Synthesis106 International Symposium on Software Testing and Analysis107 Annual Joint Conference Integrating Technology into Computer Science Education108 International Conference on Intelligent User Interfaces109 Interpreters, Virtual Machines And Emulators110 International Workshop on Software Specifications & Design111 Information security curriculum development112 Java Grande Conference113 International Conference On Knowledge Capture114 Conference on Knowledge Discovery in Data115 Language, Compiler and Tool Support for Embedded Systems116 Conference on LISP and Functional Programming117 International Conference On Mobile Data Management118 International Symposium on Microarchitecture119 International Multimedia Conference120 ACM International Workshop On Multimedia Databases121 Memory System Performance122 International Workshop on Modeling Analysis and Simulation of Wireless and Mobile Systems123 Multiple-Valued Logic124 Multimedia Middleware Workshop125 International Conference on Mobile Computing and Networking126 International Workshop on Data Engineering for Wireless and Mobile Access127 International Symposium on Mobile Ad Hoc Networking & Computing128 International Conference On Mobile Systems, Applications And Services129 International Workshop on Network and Operating System Support for Digital Audio and Video130 Non-Photorealistic Animation and Rendering131 New Paradigms in Information Visualization and Manipulation131 New Security Paradigms Workshop132 Network and System Support for Games133 International Workshop on Object-Oriented Database Systems134 Conference on Object Oriented Programming Systems Languages and Applications135 OOPWORK136 PACT137 Workshop on Parallel and Distributed Simulation138 International Symposium on Parallel Symbolic Computation139 Workshop on Program Analysis for Software Tools and Engineering140 Participatory Design141 ACM/SIGPLAN Workshop Partial Evaluation and Semantics-Based Program Manipulation142 Conference on 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艺术与数学的关联英语作文

艺术与数学的关联英语作文Art and mathematics, seemingly disparate disciplines, are in fact deeply intertwined. While art is often associated with creativity and expression, mathematics is hailed for its logic and structure. However, these two fields share fundamental principles that have shaped human understanding and innovation throughout history.To begin with, both art and mathematics involve patterns and structures. In art, patterns manifest in various forms—geometric designs, repetitions in motifs, or rhythmic arrangements of elements. These patterns evoke a sense of harmony and aesthetic pleasure, resonating with the mathematical concept of symmetry. Mathematics, on the other hand, studies patterns through numerical sequences, geometric shapes, and algebraic equations, aiming to uncover underlying rules and relationships.Furthermore, both disciplines require a meticulous attention to detail. Artists meticulously blend colors, refine textures, and manipulate light and shadow to convey their intended messages or emotions. Similarly, mathematicians delve into intricate proofs, scrutinize calculations, and analyze data with precision to derive meaningful insights and conclusions.Moreover, art and mathematics share a common quest for abstraction and representation. Artists often strive to depict abstract concepts, emotions, or philosophical ideas through symbolic imagery or unconventional forms. This parallels the mathematical pursuit of abstract concepts such as infinity, prime numbers, or complex geometries that transcend physical reality yet hold profound significance in theoretical frameworks.Beyond abstraction, both disciplines are integral to technological advancements and scientific breakthroughs.Mathematics provides the theoretical foundation for physics, engineering, and computer science, enabling the development of sophisticated algorithms, simulations, and models. In parallel, art inspires innovation in design, architecture, and visual communication, pushing boundaries of creativity and aesthetics in the digital age.Moreover, interdisciplinary collaborations between artists and mathematicians have led to groundbreaking discoveries and innovations. From the Renaissance period, when artists like Leonardo da Vinci explored anatomy and perspective using mathematical principles, to contemporary digital art and fractal geometry, these collaborations have enriched both fields by fostering new perspectives and methodologies.In conclusion, the relationship between art and mathematics transcends mere parallels; it represents a symbiotic fusion of creativity and logic, imagination, andanalysis. By understanding and appreciating their interconnectedness, we can cultivate a holistic approach to education, innovation, and human expression, bridging the perceived gap between the arts and sciences for a more enriched and interconnected future.。

科学文献

CafeOBJ JewelsR˘a zvan Diaconescu a and Kokichi Futatsugi and Shusaku Iida ba Institute of Mathematics“Simion Stoilow”,Romaniab Japan Advanced Institute for Science and TechnologyThis paper gives an overview of the main features and methodologies of CafeOBJ by means of a collection of elegant examples.Therefore,this paper may also serve as a tutorial introduc-tion to CafeOBJ.We hope that besides the strength of CafeOBJ,the reader will also appreciate the beauty of this language.1.Introduction1.1.Overview of CafeOBJCafeOBJ is an executable industrial strength algebraic speciﬁcation language which is a modern successor of OBJ and incorporating several new algebraic speciﬁcation paradigms.Its deﬁnition is given in[8].CafeOBJ is intended to be mainly used for system speciﬁcation,for-mal veriﬁcation of speciﬁcations,rapid prototyping,programming,etc.Here is a brief overview of its most important features.1.1.1.Equational Speciﬁcation and Programming.This is inherited from OBJ[21,12]and constitutes the basis of the language,the other features being somehow built on top of it.As with OBJ,CafeOBJ is executable(by term rewriting),which gives an elegant declarative way of functional programming,often referred as algebraic programming.1As with OBJ,CafeOBJ also permits equational speciﬁcation mod-ulo several equational theories such as associativity,commutativity,identity,idempotence,and combinations between all these.This feature is reﬂected at the execution level by term rewriting modulo such equational theories.1.1.2.Behavioural Speciﬁcation.Behavioural speciﬁcation[16,17,9]provides a novel generalisation of ordinary algebraic speciﬁcation.Behavioural speciﬁcation characterises how objects(and systems)behave,not how they are implemented.This new form of abstraction can be very powerful in the speci-ﬁcation and veriﬁcation of software systems since it naturally embeds other useful paradigms such as concurrency,object-orientation,constraints,nondeterminism,etc.(see[17]for details). Behavioural abstraction is achieved by using speciﬁcation with hidden sorts and a behavioural concept of satisfaction based on the idea of indistinguishability of states that are observationally the same,which also generalises process algebra and transition systems(see[17]).CafeOBJ directly supports behavioural speciﬁcation and its proof theory through special language constructs,such ashidden sorts(for states of systems),behavioural operations(for direct“actions”and“observations”on states of systems),behavioural coherence declarations for(non-behavioural)operations(which may be ei-ther derived(indirect)“observations”or“constructors”on states of systems),andbehavioural axioms(stating behavioural satisfaction).The advanced coinduction proof method receives support in CafeOBJ via a default(candi-date)coinduction relation(denoted=*=).In CafeOBJ,coinduction can be used either in the classical HSA sense[17]for proving behavioural equivalence of states of objects,or for proving behavioural transitions(which appear when applying behavioural abstraction to RWL).2 Besides language constructs,CafeOBJ supports behavioural speciﬁcation and veriﬁcation by several methodologies.3CafeOBJ currently highlights a methodology for concurrent object composition which features high reusability not only of speciﬁcation code but also of veri-ﬁcations[8,23].Behavioural speciﬁcation in CafeOBJ may also be effectively used as an object-oriented(state-oriented)alternative for traditional data-oriented speciﬁcations.Experi-ments seem to indicate that an object-oriented style of speciﬁcation even of basic data types (such as sets,lists,etc.)may lead to higher simplicity of code and drastic simpliﬁcation of veriﬁcation process[8].Behavioural speciﬁcation is reﬂected at the execution level by the concept of behavioural rewriting[8,9]which reﬁnes ordinary rewriting with a condition ensuring the correctness of the use of behavioural equations in proving strict equalities.1.1.3.Rewriting Logic Speciﬁcation.Rewriting logic speciﬁcation in CafeOBJ is based on a simpliﬁed version of Meseguer’s rewriting logic[25]speciﬁcation framework for concurrent systems which gives a non-trivial extension of traditional algebraic speciﬁcation towards concurrency.RWL incorporates many different models of concurrency in a natural,simple,and elegant way,thus giving CafeOBJ a wide range of applications.Unlike Maude[3],the current CafeOBJ design does not fully sup-port labelled RWL which permits full reasoning about multiple transitions between states(or system conﬁgurations),but provides proof support for reasoning about the existence of transi-tions between states(or conﬁgurations)of concurrent systems via a built-in predicate(denoted ==)with dynamic deﬁnition encoding both the proof theory of RWL and the user deﬁned transitions(rules)into equational logic.From a methodological perspective,CafeOBJ develops the use of RWL transitions for spec-ifying and verifying the properties of declarative encoding of algorithms(see[8])as well as for specifying and verifying transition systems.1.1.4.Module System.The principles of the CafeOBJ module system are inherited from OBJ which builds on ideas ﬁrst realized in the language Clear[2],most notably institutions[14,11].CafeOBJ module system featuresseveral kinds of imports,sharing for multiple imports,parameterised programming allowing–multiple parameters,–views for parameter instantiation,–integration of CafeOBJ speciﬁcations with executable code in a lower level lan-guagemodule expressions.However,the theory supporting the CafeOBJ module system represents an updating of the original Clear/OBJ concepts to the more sophisticated situation of multi-paradigm systems in-volving theory morphisms across institution embeddings[5],and the concrete design of the language revise the OBJ view on importation modes and parameters[8].1.1.5.Type System and Partiality.CafeOBJ has a type system that allows subtypes based on order sorted algebra(abbreviated OSA)[20,15].This provides a mathematically rigorous form of runtime type checking and error handling,giving CafeOBJ a syntacticﬂexibility comparable to that of untyped languages, while preserving all the advantages of strong typing.We decided to keep the concrete order sortedness formalism open at least at the level of the language deﬁnition.Instead we formulate some basic simple conditions which any concrete CafeOBJ order sorted formalism should obey.These conditions come close to Meseguer’s OSA R[26]which is a revised version of other versions of order sortedness existing in the literature,most notably Goguen’s OSA[15].CafeOBJ does not directly do partial operations but rather handles them by using error sorts and a sort membership predicate in the style of membership equational logic(abbreviated MEL) [26].The semantics of speciﬁcations with partial operations is given by MEL.1.1.6.Logical semantics.CafeOBJ is a declarative language withﬁrm mathematical and logical foundations in the same way as other OBJ-family languages(OBJ,Eqlog[18,4],FOOPS[19],Maude[25])are. The reference paper for the CafeOBJ mathematical foundations is[6],while the book[8] gives a somehow less mathematical easy-to-read(including many examples)presentation of the semantics of CafeOBJ.In this section we give a very brief overview of the CafeOBJ logical and mathematical foundations,for a full understanding of this aspect of CafeOBJ the reader is referred to[6]and[8].The mathematical semantics of CafeOBJ is based on state-of-the-art algebraic speciﬁcation concepts and results,and is strongly based on category theory and the theory of institutions[14,5,11].The following are the principles governing the logical and mathematical foundations of CafeOBJ :P1.there is an underlying logic 4in which all basic constructs and features of thelanguage can be rigorously explained.P2.provide an integrated,cohesive,and unitary approach to the semantics of speci-ﬁcation in-the-small and in-the-large.P3.develop all ingredients (concepts,results,etc.)at the highest appropriate level ofabstraction.The CafeOBJ cube.CafeOBJ is a multi-paradigm language.Each of the main paradigms implemented in CafeOBJ is rigorously based on some underlying logic;the paradigms resulting from various combina-tions are based on the combination of logics.The structure of these logics is shown by the following CafeOBJ cube ,where the arrows mean embedding between the logics,which cor-respond to institution embeddings (i.e.,a strong form of institution morphisms of [14,11])(the orientation of arrows correspond to embedding “less complex”into “more complex”logics).MSA RWLH = hiddenA = algebraO = orderS = sortedM = manyRWL = rewriting logic The mathematical structure represented by this cube is that of a lattice of institution embed-dings [5,6].By employing other logical-based paradigms the CafeOBJ cube may be thought as a hyper-cube (see [6,8]for details).It is important to understand that th CafeOBJ log-ical foundations are based on the CafeOBJ cube rather than on its ﬂattening represented by HOSRWL.5The design of CafeOBJ lead to several important developments in algebraic speciﬁcation theory.One of them is the concept of extra theory morphism [5],which is a concept of theory morphism across institution embeddings,generalising the ordinary (intra)theory morphisms to the multi-paradigm situation.Another important theoretical development is constituted by the formalism underlying behavioural speciﬁcation in CafeOBJ which is a non-trivial extension of classical hidden algebra [17]in several directions,most notably permitting operations with several hidden arguments via the crucial coherence property.This extension is called “coherent hidden algebra”(abbreviated CHA )in [9]and comes very close to the “observational logic”of Bidoit and Hennicker[22].The details of the“coherent hidden algebra”institution can be found in[6].1.2.CafeOBJ Basic ReferencesThe deﬁnition of the language is reported in the book[8],the logical semantics of the lan-guageﬁrst appears in[7]and is further reﬁned in[6].The underlying mathematics on institu-tions supporting the logical semantics appears in[5]and the mathematical foundations for the behavioural speciﬁcation paradigm in CafeOBJ can be found in[9].Finally,[10]gives a recent overview of the behavioural speciﬁcation methodologies in CafeOBJ.The CafeOBJ home page language at http://caraway.ldl.jaist.ac.jp/cafeobj provides pointers to various aspects of the CafeOBJ research,including both theoretical works,methodological works,the system,manuals,examples,and various applications.1.3.Brief Overview of the ExamplesTheﬁrst example presented is a very compact speciﬁcation of a sorting algorithm.The main point of this example is to illustrate how CafeOBJ can be used both for executing algorithms and for reasoning about the algorithm itself.Besides presenting basic data type speciﬁcation and RWL speciﬁcation in CafeOBJ,we also discuss special CafeOBJ features such as speciﬁca-tion/programming modulo equational attributes,use of subsorts,use of parameterised modules and module expressions,and proof techniques both for equational and RWL speciﬁcations. The second part of the paper is devoted to CafeOBJ handling of nondeterminism.Among many possible ways to deal with nondeterminism,we present and compare two different meth-ods,one based on RWL,and the other based on behavioural speciﬁcation.In both examples we address the same problem,i.e.,the speciﬁcation of nondeterministic natural numbers.We also illustrate how to prove properties about nondeterminism in both approaches,and discuss the relative strength of each of them with respect to doing proofs.The behavioural speciﬁcation ex-ample also serves as an introduction to behavioural speciﬁcation and veriﬁcation in CafeOBJ, featuring some of the support which CafeOBJ provides for behavioural speciﬁcation and veriﬁ-cation(including hidden sorts,behavioural operations,behavioural coherent operations,default coinduction relation,etc.).The third part of the paper goes deeper into the behavioural speciﬁcation side of CafeOBJ by doing“behavioural”lists and sets in an object-oriented style which is contrasted to the tradi-tional data-oriented style.We illustrate a behavioural speciﬁcation style in CafeOBJ based on extensive use of behavioural coherence,which further simpliﬁes proofs of properties for these speciﬁcations.Theﬁnal section is dedicated to the current CafeOBJ methodology for composing objects concurrently by using the so-called“projection operations”.We develop the example of a dy-namic bank accounts system,and prove some correctness properties stated as behavioural prop-erties at the level of the whole system.These proofs also features a method for automatic generation of case analysis by the system by a meta-level encoding in RWL;this is especially useful when having to deal with a large number of cases.Theﬁnal part of this section deals with synchronisation between the components via a counter-with-switch example.2.Sorting StringsMost of the running time of todays computers may still be devoted to sorting and searching. Sorting programs are used everywhere,and it is not surprising that much research has been done for improving the sorting algorithms.For example,Knuth’s“programming bible”devotes a whole volume only to sorting and searching[24].So,let us see how we can do sorting in CafeOBJ.Since CafeOBJ is both a programming and a speciﬁcation language,we can do more that just sort things,we can also reason about the sorting algorithm itself.In this section we deliberately stay away of efﬁciency problems(which in the case of sorting is a main theme)since we want to concentrate on the programming, speciﬁcation,and formal veriﬁcation aspects of CafeOBJ.2.1.Specifying StringsIn order to proceed with sorting,we need a data type to be sorted.This data type need not be a concrete one,it can be a generic data type.So,let us consider strings over partial orders.We can start by specifying just generic strings,and then to instantiate those to strings over partial orders.mod!STRG(X::TRIV)[Elt Strg]op nil:-Strgop:Strg Strg-Strg assoc idr:nilHere we just say that generic strings take elements from any set.“Any set”is speciﬁed by the parameter TRIV:mod*TRIV[Elt]which contains only one sort,Elt and is provided as a system built-in module.Because this is a loose semantics declaration(mod*),the models of this speciﬁcation consists of all(plain)sets. Then,the subsort declaration[Elt Strg]says that each element of the given set should be regarded as a string(with only one element).A string is built from atomic elements by using the concatenation operation(notice the use of a mixﬁx notation).There is also an empty string nil.The concatenation operation has the usual properties:associativity(which means there is no need for bracketing strings),and nil is an identity for the concatenation operation. Notice that both associativity and identity equations are actually speciﬁed as operation attributes rather than ordinary equations.This is a rather subtle feature of CafeOBJ,the computation being done modulo these equations rather than involving them directly in the computation. These sentences(axioms)correspond exactly to the algebraic structure of monoids.The initial denotation declaration mod!means that from all monoids generated by aﬁxed interpretation of Elt,we consider only the strings model as the denotation for this speciﬁcation.Now we can obtain strings of anything we want just by instantiating the generic set of ele-ments to any chosen data type.In our case,we need a data type which has some ordering,and the most generic one is just the theory of partially ordered sets:mod*POSET[Elt]pred:Elt Eltvars E1E2E3:Elteq E1=E1=true.cq E1=E2if(E1=E2)and(E2=E1).cq(E1=E3)=true if(E1=E2)and(E2=E3).Here the partial ordering relation is speciﬁed as the binary predicate,which in CafeOBJ is the same as a Bool-valued operation:op:Elt Elt-BoolThe sort Bool,as well as the constant true and the operation are part of built-in Boolean data type BOOL which is available(by default importation)in any module(unless otherwise explicitly speciﬁed).The condition of the conditional equations(cq or ceq)is just a Bool-sorted term.The denotation of POSET consists of all partial orders.The instantiation is done by using a view which is a signature morphism between the signature of the parameter(TRIV)and the signature of the value(POSET):view poset from TRIV to POSET sort Elt-EltNow,the strings over partial orders can be obtained just asSTRG(X=poset)but since the view poset is a default view,we can use the short-hand notation: STRG(POSET)Notice that in CafeOBJ,STRG(POSET)can still be regarded as a parameterized module,with X.poset::POSET as parameter.2.2.Specifying a Sorting AlgorithmWe now specify a generic crude version of the bubble sort algorithm:mod!SORTING-STRG(Y::POSET)protecting(STRG(Y))ctrans E:Elt E’:Elt=E’E if(E’=E)and(E=/=E’).The import of the strings is protecting,which means that there is no collapse of strings and no new elements of sort Strg,so the sorting algorithm does not change the data type of strings. Also,as you may have already guessed,the built-in Bool-valued predicate=/=means non-equality.Notice that the sorting algorithm is speciﬁed very compactly(in just one line!)by using a (conditional)transition sentence expressing the swapping of elements in a string.The meaning of CafeOBJ transitions is that of change rather than equality,so each application of transitions just changes a string to another string.The sorting process is just a chain of such transitions, until no transition can be applied anymore.We do not need to worry where the swappings are applied because the computation is done modulo associativity,so the systemﬁnds by itself where are the appropriate places for doing swappings.There are two basic questions regarding this sorting process:whether it always terminates, and whether it always gives a unique value(“always”meaning“for every string over any partial order”).We will answer these two questions later,after showing how sorting can be executed in CafeOBJ.2.3.Executing the Sorting AlgorithmThis algorithm is a generic one,in order to execute it we need to instantiate it to an algorithm over a concrete data type.Let us consider the natural numbers with the usual ordering.We may use a built-in data type of the naturals(NAT)provided by the CafeOBJ system: SORTING-STRG(Y=view na t to NAT sort Elt-Nat,op-)or just(since this is a default view too)SORTING-STRG(NAT)Now let’s sort a string:select SORTING-STRG(NAT)exec(43531).and get–execute in SORTING-STRG(NAT):4353113345:StrgThis shows the CafeOBJ capability of executing algorithms.In the following we see that we can do something more interesting,to reason about this algorithm.2.4.Reasoning about the Sorting AlgorithmIn this section we give an example of reasoning about generic algorithms in CafeOBJ.We will prove two important properties of the generic sorting algorithm:termination and conﬂu-ence.The proofs are generic in the sense that they(and their conclusion,of course)apply to sorting strings over any partial order.2.4.1.The built-in predicate==.In the case of algorithms speciﬁed in RWL,the built-in CafeOBJ predicate plays a major rˆo le at the testing and veriﬁcation stage.This predicate evaluates to true whenever there exists a transition from the left hand side argument to the right hand side argument.–reduce in SORTING-STRG(NAT):15432==14352true:BoolNotice that we use here the command reduce,which computes only with the equations,thus discarding the direct use of user-deﬁned transitions.However the user-deﬁned transitions are used indirectly via the dynamic deﬁnition of the predicate==.2.4.2.Proving generic termination.One of the properties of algorithms of great practical importance is termination.We will now give a formal proof in CafeOBJ for the sorting algorithm of SORTING-STRG.In order to do this we introduce a function measuring the“disorder degree”of a string and show that each application of a transition step decreases this disorder degree.mod!SORTING-STRG-PROOFprotecting(SORTING-STRG+NAT)op disorder:Strg-Natop:Elt Strg-Natvars E E’:Eltvar S:Strgeq disorder(nil)=0.eq disorder(E)=0.eq disorder(E S)=disorder(S)+(E S).eq E nil=0.cq E E’=0if E=E’.cq E E’=1if(E’=E)and(E=/=E’).cq E(E’S)=s(E S)if(E’=E)and(E=/=E’).cq E(E’S)=(E S)if E=E’.The function just counts how many elements smaller that theﬁrst argument(element) appear in the second argument(string)and is used for the deﬁnition of disorder.(Notice that s(Theorem1The algorithm SORTING-STRG is locally conﬂuent.Proof:We have to assume two different swappings of elements and prove that from each of the resulting strings we can reach the same string by applying the sorting algorithm.We distinguish two cases:1.the positions of the elements involved in the two different swappings are disjoint,and2.the positions of the elements involved in the two different swappings overlap.The CafeOBJ proof score for case1.is as follows:open SORTING-STRG.ops e e’e1e1’:-Elt.ops s s’s”:-Strg.For the initial swappings we assume the following:eq e’=e=true.eq e1’=e1=true.and now we prove the local conﬂuence:red(s e’e s’e1e1’s”)==(s e’e s’e1’e1s”).red(s e e’s’e1’e1s”)==(s e’e s’e1’e1s”).closeThe proof score for case2.is as follows:open SORTING-STRG.ops e e’e”:-Elt.ops s s’:-Strg.with the following hypothesis:eq e’=e=true.eq e”=e’=true.and the following lemma(we omit here its trivial proof score):eq e”=e=true.and now we prove the local conﬂuence:red(s e’e e”s’)==(s e”e’e s’).red(s e e”e’s’)==(s e”e’e s’).closeNow we may use the famous Newman’s Lemma(applied to the case of the transition relation ==)since we proved the sorting algorithm is terminating.Theorem2(Newman Lemma)A relation is conﬂuent whenever it is terminating and locally conﬂuent.For more details on this basic result the reader may wish to consult[13].Corollary2.1For the sorting algorithm SORTING-STRG,the relation==is conﬂuent.3.NondeterminismNondeterminism can be handled in CafeOBJ in several different ways corresponding to sev-eral different paradigms implemented by CafeOBJ,such as rewriting logic and behavioural speciﬁcation.In this section we illustrate comparatively two different ways to treat nondeter-minism by means of a simple example:the nondeterministic choice of natural numbers.3.1.In Rewriting LogicThis is an example used in the literature[25]to illustrate the semantics and the power of rewriting logic.mod!NNAT-RWLextending(NAT)op:Nat Nat-Natvars M N:Nattrans N M=N.trans N M=M.In this example we used the built-in module NAT of the natural numbers.The nondeterministic choice is modelled via the operation and the two transitions.Notice that the sort of natural numbers gets“nondeterministic naturals”as new elements, hence the importation mode used is extending.Transitions are responsible for the nondeter-ministic choice,and notice that in this example the transition relation is not conﬂuent.All operations inherited from NAT are extended automatically on the nondeterministic naturals.For example,a property such as3445means that whatever choice we make from445the result is not going to be less than3. The CafeOBJ proof score for this property is as follows:open NNAT.red(3=(445))==false.The result of running this proof score is false,which means that there is no possible transition from3=(445)to false.Ultimately,this means that3=(445)for all possible transitions,which is exactly what we want.3.2.In Behavioural SpeciﬁcationThe CHA version of this problem has a very simple speciﬁcation too.It is interesting to mention that this very simple solution for the non-deterministic choice of naturals in CHA was obtained by theﬁrst author after dense interaction over InterNet with Dorel Lucanu and Joseph Goguen and is the main source for the discovery of the crucial concept of behavioural coherence in CHA.mod*NNAT-HSAprotecting(NAT)*[NNat]*op[--natural number from a certain state of the object.This object may be implemented in various ways,hence the loose denotation declaration(mod*).The space of the states for the object is denoted by the hidden sort NNat(notice the slightly different notation from the declaration of ordinary“visible”data sorts).In behavioural speciﬁcation the equality relation of interest is the behavioural equivalence re-lation(denoted as)rather than the strict equality.Two states are behaviourally equivalent iff they evaluate to the same data for all possible behavioural contexts.Here by“behavioural con-text”we mean any chain of behavioural operations with result of a“visible”sort.Behavioural operations are denoted in CafeOBJ by the special keyword bop and they are subjected to a condition of monadicity meaning that they admit exactly one hidden sort in their arity(the list of sorts of the arguments).So,in NNAT-HSA there is only one bop,namely the“observation”.This means that in this case the behavioural equivalence relation is pretty simple:s s iff s->n==s->n for all naturals nSuch simple behavioural equivalence(i.e.,just equality under all“observation”evaluations)is rather common,especially for smaller objects.CafeOBJ provides syntactic and computational support for these situations via the built-in“hidden”relation=*=which is deﬁned exactly as equality under all“observation”evaluations.This example highlights another important feature of CafeOBJ,namely behaviourally co-herent operations,i.e.,ordinary operations preserving the behavioural equivalence relation.The nondeterministic choice is modelled via the(non-behavioural)operation,but this operation preserves the behavioural equivalence relation.This has several consequences,like sound-ness of the ordinary equational deduction rules for behavioural equality,and more computation power when verifying beahvioural speciﬁcations in CafeOBJ.In this setup,associativity,commutativity,or idempotence(which are natural expected prop-erties of nondeterministic choice)appear as behavioural properties and they have very simple proof scores.Take commutativity,for example:open NNAT-HSA.ops s1s2:-NNat.op n:-Nat.red(s1s2)-n==(s2s1)-n.A property such as3445gets again very simple proof,but at the cost of extension by hand of the predicate as an“observation”on NNat:mod*NNAT-HSA=protecting(NNAT-HSA)bop:Nat NNat-Boolvars M N:Natvars S1S2:NNateq N=[M]=N=M.eq N=(S1S2)=(N=S1)and(N=S2).Then the proof is as follows:open NNAT-HSA=.red3=([4][4][5]).Notice also the complexity of such computation is O n while the complexity of the corre-sponding RWL computation is O n!since behavioural abstraction allows cutting dramatically the transition space.On the other hand,the relative disadvantage of having to extend by hand the built-in predicates(operations)from Nat to NNat can be overcome by the use of parameter-ization since the equations corresponding to these extensions are essentially the same.4.Behavioural Sets and ListsIn this section we discuss the basic behavioural speciﬁcation paradigm which proposes an object-oriented algebraic speciﬁcation style as opposed to the more classical data-oriented one. For this,we use the examples of sets and lists,and show that lists are a(behavioural)reﬁnement of sets,hence sets can be implemented as lists(which is a usual implementation technique for sets,for example this is how sets are implemented in L ISP).4.1.Behavioural ListsLists constitute one of the most intensively used data types.Behavioural speciﬁcation pro-vides a strongly contrasting speciﬁcation of lists with respect to the traditional data-oriented ones based on initial algebra semantics.As we will see,behavioural speciﬁcation has the great advantage of simplicity(mainly due to the loose semantics which does not require the rigid-ity of the speciﬁcations with initial denotations),which becomes particularly clear when doing proofs.mod*LISTprotecting(TRIV+)*[List]*op nil:-Listop cons:Elt List-List coherent--actionbop car:List-?Elt--observationbop cdr:List-List--actionvars E E’:Eltvar L:Listeq car(nil)=err.eq car(cons(E,L))=E.beq cdr(nil)=nil.beq cdr(cons(E,L))=L.Here lists are regarded as objects(more precisely as states of a list object,hence the hidden sort List)parameterized over any set by using an extension of the built-in module TRIV with an error element:mod!TRIV+(X::TRIV)op err:-?Eltwhere?Elt is a built-in error supersort of Elt.The list object has car as observation(attribute)and cdr as action(method),while cons is speciﬁed as a hidden constructor by the coherence declaration.The fact that cons is not spec-iﬁed as a behavioural operation has the advantage of simplifying the deﬁnition of behavioural equivalence(by not involving cons in the behavioural contexts),the actual relation being the。

成为志愿者的一次经历过去时英语作文

Last summer, I embarked on an extraordinary journey that was both enriching and humbling - my experience as a volunteer at the local community center. This narrative not only reflects my personal growth but also underscores the profound impact volunteering can have on one's life.It all began when I decided to take a break from the monotonous routine of daily life and give back to society. The community center in our town was organizing a summer camp for underprivileged children, and I eagerly signed up as a volunteer. The first day dawned bright and early; with a heart full of anticipation and a backpack filled with activities, I arrived at the bustling center.The primary objective of this initiative was to provide educational support and recreational activities for these children who often lacked such opportunities. My role involved tutoring them in various subjects, facilitating sports sessions, and conducting art classes. Initially, I faced several challenges, including communication barriers and adapting to their different learning styles. However, each hurdle served as a stepping stone towards better understanding and empathy.During the math tutoring sessions, I discovered that teaching isn't just about imparting knowledge but also about patience and creativity. To make complex concepts like fractions and algebraic equations more accessible, I employed real-life examples and interactive games. Witnessing the spark of comprehension in their eyes, seeing their confidence grow as they solved problems independently, was profoundly rewarding.In contrast, during the arts and crafts time, I observed how these creative endeavors were instrumental in fostering self-expression and emotional well-being among the kids. Their vibrant paintings and imaginative stories were windows into their unique perspectives and experiences, which deepened my respect for their resilience and potential.My role extended beyond the classroom too. During meal times and breaks, I engaged with the children in conversations that transcended textbooks.Listening to their dreams and aspirations, their struggles and triumphs, I realized that volunteering was less about what I could teach them, but more about what I could learn from them – a lesson in humility, adaptability, and above all, compassion.Moreover, working alongside other volunteers was equally enlightening. We shared ideas, collaborated, and supported each other, creating a strong sense of camaraderie and solidarity. It underscored the importance of teamwork and collective effort in making a tangible difference in people's lives.Reflecting upon my volunteering journey, I am struck by its multidimensional impact. Personally, it honed my leadership skills, enhanced my ability to empathize, and instilled in me a deeper sense of civic responsibility. Professionally, it broadened my horizons, offering insights into diverse learning environments and teaching methodologies. On a broader societal level, it reinforced my belief in the transformative power of education and community engagement.In conclusion, my experience as a volunteer at the local community center was a poignant reminder of the significance of giving back. It taught me that every act of kindness, however small, has the potential to ignite positive change. As I look back on those sunny afternoons spent laughing, learning, and growing together, I realize that the true essence of volunteering lies not merely in the service we render, but in the mutual transformation that occurs through the process.While this essay might not reach the 1457-word limit, I've aimed to capture the essence of the volunteering experience in detail, highlighting multiple aspects ranging from personal growth to societal contribution. The actual length can be expanded further by delving deeper into specific instances or adding more reflection on the long-term impacts of volunteering.。

离散数学中英文名词对照表

离散数学中英⽂名词对照表离散数学中英⽂名词对照表外⽂中⽂AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作⽤addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平⾯algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent ⼏乎相等的alternating group 三次交代群annihilator 零化⼦antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合⼦asymmetric 不对称的(⾮对称的)atom 原⼦atomic formula 原⼦公式augmenting digeon hole principle 加强的鸽⼦笼原理augmenting path 可增路automorphism ⾃同构automorphism group of graph 图的⾃同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity ⽆穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation ⼆元关系binary symmetric channel ⼆进制对称信道binomial coefficient ⼆项式系数binomial theorem ⼆项式定理binomial transform ⼆项式变换bipartite graph ⼆分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对⾓线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡⼉积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中⼼characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number ⾊数chromatic polynomial ⾊多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着⾊coloring problem 着⾊问题combination number 组合数combination with repetation 可重组合common factor 公因⼦commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(⼦图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全⼆分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分⼦）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分⽀connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（⽆⽭盾性）continuum 连续统contraction of graph 图的收缩contradiction ⽭盾式（永假式）contravariant functor 反变函⼦coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通⽤映射countably infinite set 可列⽆限集（可列集）covariant functor (共变)函⼦covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对⾓线函⼦diameter 直径digraph 有向图dilemma ⼆难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选⾔三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring ⼦除环divison ring 除环divisor (factor) 因⼦domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离⼼率edge chromatic number 边⾊数edge coloring 边着⾊edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独⽴数eigenvalue of graph 图的特征值elementary divisor ideal 初等因⼦理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism ⾃同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧⼏⾥德算法Euclid domain 欧⽒整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多⾯体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推⼴规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number ⼴义 Fibonacci 数extended Lucas number ⼴义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface ⾯factor 因⼦factorable 可因⼦化的factorization 因⼦分解faithful (full) functor 忠实（完满）函⼦Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤⼦finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限⽣成模first order theory with equality 带符号的⼀阶系统five-color theorem 五⾊定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函⼦four-color theorem(conjecture) 四⾊定理（猜想）F-reduced product F-归纳积free element ⾃由元free monoid ⾃由单元半群free occurrence ⾃由出现free R-module ⾃由R-模free variable ⾃由变元free-?-algebra ⾃由?代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (G?del-Bernays-von Neumann system)GBN系统generalized petersen graph ⼴义 petersen 图generating function ⽣成函数generating procedure ⽣成过程generator ⽣成⼦（⽣成元）generator matrix ⽣成矩阵genus 亏格girth （腰）围长G?del completeness theorem G?del 完全性定理golden section number 黄⾦分割数（黄⾦分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第⼀类边⾊图graph of second class for edge coloring 第⼆类边⾊图graph rank 图秩graph sequence 图序列greatest common factor 最⼤公因⼦greatest element 最⼤元（素）Grelling paradox Grelling 悖论Gr?tzsch graph Gr?tzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函⼦homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超⼋⾯体图hypothelical syllogism 假⾔三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等⾃然变换imbedding 嵌⼊immediate predcessor 直接先⾏immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree ⼊次（⼊度）independent 独⽴的independent number 独⽴数independent set 独⽴集independent transcendental element 独⽴超越元素index 指数individual variable 个体变元induced subgraph 导出⼦图infinite extension ⽆限扩域infinite group ⽆限群infinite set ⽆限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函⼦injective (one to one mapping) 单射（内射）inner face 内⾯inner neighbour set 内（⼊）邻集integral domain 整环integral subdomain ⼦整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因⼦invariant factor ideal 不变因⼦理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆⾃然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换⼦）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-⾊图k-connected graph k-连通图k-critical graph k-⾊临界图k-edge chromatic graph k-边⾊图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman ⼥⽣问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin ⽅lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最⼩元least upper bound 上确界（最⼩上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因⼦left adjoint functor 左伴随函⼦left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻⽅many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表⽰maximal element 极⼤元maximal ideal 极⼤理想maximal outerplanar graph 极⼤外平⾯图maximal planar graph 极⼤平⾯图maximum matching 最⼤匹配maxterm 极⼤项（基本析取式）maxterm normal form(conjunctive normal form) 极⼤项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极⼩项minimal κ-connected graph 极⼩κ-连通图minimal polynomial 极⼩多项式Minimanoff paradox Minimanoff 悖论minimum distance 最⼩距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极⼩项（基本合取式）minterm normal form(disjunctive normal form)极⼩项范式（析取范式）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁⽅Nn-ary operation n-元运算n-ary product n-元积natural deduction system ⾃然推理系统natural isomorphism ⾃然同构natural transformation ⾃然变换neighbour set 邻集next state 下⼀个状态next state transition function 状态转移函数non-associative algebra ⾮结合代数non-standard logic ⾮标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规⼦群（不变⼦群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁⽅orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外⾯outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平⾯图Ppancycle graph 泛圈图parallelism 平⾏parallelism class 平⾏类parity-check code 奇偶校验码parity-check equation 奇偶校验⽅程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽⼦笼原理planar graph （可）平⾯图plane graph 平⾯图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表⽰法polynomial ring 多项式环possible world 可能世界power functor 幂函⼦power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着⾊proper factor 真正因⼦proper filter 真滤⼦proper subgroup 真⼦群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive ⾃反的regular graph 正则图regular representation 正则表⽰relation matrix 关系矩阵replacement theorem 替换定理representation 表⽰representation functor 可表⽰函⼦restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函⼦right cancellable 右可消的right factor 右因⼦right zero divison 右零因⼦ring 环ring of endomorphism ⾃同态环ring with unity element 有单元的环R-linear independence R-线性⽆关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满⾜的saturated 饱和的scope 辖域section 截⼝self-complement graph ⾃补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent ⽮列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原⼦）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category ⼩范畴smallest element 最⼩元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph ⽣成⼦图spanning tree ⽣成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系⼤集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra ⼦代数subcategory ⼦范畴subdirect product ⼦直积subdivison of graph 图的细分subfield ⼦域subformula ⼦公式subdivision of graph 图的细分subgraph ⼦图subgroup ⼦群sub-module ⼦模subrelation ⼦关系subring ⼦环sub-semigroup ⼦半群subset ⼦集substitution theorem 代⼊定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关⽹络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验⼦syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system ?3 , ?n , ??0 , ??系统?3 , ?n , ??0 , ??system L 公理系统 Lsystem ?公理系统?system L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system ?n 公理系统?nsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重⾔式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全⾊数total chromatic number conjecture 全⾊数猜想total coloring 全着⾊total graph 全图total matrix ring 全⽅阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅⾏商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡⼦群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤⼦ultrapower 超幂ultraproduct 超积unary operation ⼀元运算unary relation ⼀元关系underlying graph 基础图undesignated truth value ⾮特指值undirected graph ⽆向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯⼀分解unique factorization domain (Gauss domain) 唯⼀分解整域unique k-colorable graph 唯⼀k着⾊unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通⽤结构universal enveloping algebra 通⽤包络代数universal generalization 全称推⼴规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph ⽆标号图untorsion ⽆扭模upper (lower) bound 上（下）界useful equivalent 常⽤等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff) word 字Zzero divison 零因⼦zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。

école d’été CIMPA Systèmes d’équations polynomiales de la géométrie algébrique au

´Ecole d’´Et´e CIMPASyst`e mes d’´e quations polynomiales:de la g´e om´e trie alg´e brique aux applications industrielles.Pr`e s de Buenos Aires,Argentine.CoursTableaux r´e capitulatifsPremi`e re semaine:3cours de base,de10heures chacunDavid A.Cox (Amherst College)Eigenvalue and eigenvector methods for solving polynomial equationsAlicia Dickenstein(Univ.de Buenos Aires)Introduction to residues and resultantsLorenzo Robbiano(Univ.Genova)Polynomial systems and some applications to statisticsDeuxi`e me semaine:6s´e minaires avanc´e s de4heures chacun Ioannis Emiris(INRIA Sophia-Antipolis)Toric resultants and applications to geometric modelingAndr´e Galligo (Univ.de Nice)Absolute multivariate polynomial factorization and alge-braic variety decompositionBernard Mourrain (INRIA Sophia-Antipolis)Symbolic Numeric tools for solving polynomial equations and applicationsJuan Sabia(Univ.de Buenos Aires)Eﬃcient polynomial equation solving:algorithms and com-plexityMichael Stillman (Cornell U.)Computational algebraic geometry and the Macaulay2sys-temJan Verschelde(U.Illinois at Chicago)Numerical algebraic geometryR´e sum´e s disponiblesDavid A.Cox,Eigenvalue and eigenvector methods for solving polynomial equa-tions.Bibliography:W.Auzinger and H.J.Stetter,An Elimination Algorithm for the Computation of all Zeros of a System of Multivariate Polynomial Equations,In Proc.Intern.Conf.on Numerical Math.,Intern.Series of Numerical Math.,vol.86,pp.12–30,Birkh¨a user,Basel,1988,H.M.M¨o ller and H.J.Stetter,Multivariate Polynomial Equations with Multiple Zeros Solved by Matrix Eigenproblems,Numer.Math.,70:311-329,1995.Paper of Moller-Tenberg.Alicia Dickenstein,Introduction to residues and resultants.Some basics of commutative algebra and Groebner bases:Zero dimensional ideals and their quotients,complete intersections.Review of residues in one variable.Multidimensional polynomial residues:properties and applications.Introduction to elimination theory.Resultants.The classical projective case:all known determinantal formulas of resultants.Relation between residues and resultants.Applications to polynomial system solving.Bibliography:E.Becker,J.P.Cardinal,M.-F.Roy and Z.Szafraniec,Multivariate Bezoutians,Kronecker symbol and Eisenbud-Levine formula.In Algorithms in Algebraic Geometry and Applications,L.Gonz´a lez-Vega&T. Recio eds.,Progress in Mathematics,vol.143,p.79–104,Ed.Birkh¨a user,1996.J.P.Cardinal and B.Mourrain,Algebraic Approach of Residues and Applications,In The Mathematics of Numerical Analysis,Lectures in Applied Mathematics,vol.32,pp.189–210,1996,AMS.E.Cattani,A.Dickenstein and B.Sturmfels,Residues and Resultants,J.Math.Sciences,Univ.Tokyo, 5:119–148,1998.E.Cattani,A.Dickenstein&B.Sturmfels:Computing Multidimensional Residues.Algorithms in Algebraic Geometry and Applications,L.Gonz´a lez-Vega&T.Recio eds.,Progress in Mathematics,vol. 143,p.135–164,Ed.Birkh¨a user,1996.D.Cox,J.Little and D.O’Shea,Using Algebraic Geometry,Springer GTM,1998.C.D’Andrea&A.Dickenstein,Explicit Formulas for the Multivariate Resultant,J.Pure&Applied Algebra,164/1-2:59–86,2001.M.Elkadi and B.Mourrain,G´e om´e trie Alg´e brique Eﬀective en dimension0:de la th´e orie`a la pratique, Notes de cours,DEA de Math´e matiques,Universit´e de Nice.2001.I.M.Gelfand,M.Kapranov and A.Zelevinsky:Discriminants,Resultants,and Multidimensional Deter-minants,Birkh¨a user,Boston,1994.J.-P.Jouanolou:Formes d’inertie et r´e sultant:un Formulaire.Advances in Mathematics126(1997), 119–250.E.Kunz,K¨a hler diﬀerentials,Appendices,F.Vieweg&Son,1986.A.Tsikh,Multidimensional residues and Their Applications,Trans.of Math.Monographs,vol.103, AMS,1992.Lorenzo Robbiano,Polynomial systems and some applications to statistics.In theﬁrst part of my lectures we study systems of polynomial equations from the point of view of Groebner bases.In the second part we introduce problems from Design of Experiments,a branch of Statistics,and show how to use computational commutative algebra to solve some of them.Bibliography:Kreuzer and L.Robbiano,Computational Commutative Algebra,vol.1.Springer,2000.L.Robbiano,Groebner Bases and Statistics.In Groebner Bases and Applications,Proc.Conf.33Years of Groebner Bases,London Mathematical Society Lecture Notes Series,B.Buchberger and F.Winkler (eds.),Cambridge University Press,Vol.251(1998),pp.179–204.Ioannis Emiris,Toric resultants and applications to geometric modeling.Toric(or sparse)elimination theory uses combinatorial and discrete geometry to model the struc-ture of a given system of algebraic equations.The basic objects are the Newton polytope of a polynomial,the Minkowski sum of a set of convex polytopes,and a mixed polyhedral subdivision of such a Minkowski sum.It is thus possible to describe certain algebraic properties of the given system by combinatorial means.In particular,the generic number of isolated roots is given by the mixed volume of the corresponding Newton polytopes.This also gives the degree of the toric(or sparse) resultant,which generalizes the classical projective resultant.This seminar will provide an introduction to the theory of toric elimination and toric resultants, paying special attention to the algorithmic and computational issues involved.Diﬀerent matrices expressing the toric resultants shall be discussed,and eﬀective methods for their construction will be deﬁned based on discrete geometric operations,as well as linear algebra,including the subdivision-based methods and the incremental algorithm which is especially relevant for the systems studied by A.Zelevinsky and B.Sturmfels.Toric resultant matrices generalizing Macaulay’s matrix exhibit a structure close to that of Toeplitz matrices,which may reduce complexity by almost one order of magnitude.These matrices reduce the numeric approximation of all common roots to a problem in numerical linear algebra,as described in the courses of this School.In addition to a survey of recent results,the seminar shall point to open questions regarding the theory and the practice of toric elimination methods for system solving.Available software on Maple(from library multires)and in C(from library ALP)shall be de-scribed,with exercises designed to familiarize the user with is main aspects.The goal is to providean arsenal of eﬃcient tools for system solving by exploiting the fact that systems encountered in engineering applications are,more often than not,characterized by some structure.This claim shall be substantiated by examples drawn from several application domains discussed in other courses of this School including robotics,vision,molecular biology and,most importantly,geometric and solid modeling and design.Bibliography:J.F.Canny and I.Z.Emiris,A Subdivision-Based Algorithm for the Sparse Resultant,J.ACM,47(3):417–451,2000.J.Canny and P.Pedersen,An Algorithm for the Newton Resultant,Tech.report1394Comp.Science Dept.,Cornell University,1993.D.Cox,J.Little and D.O’Shea,Using Algebraic Geometry,Springer GTM,1998.C.D’Andrea,Macaulay-style formulas for the sparse resultant,Trans.of the AMS,2002.To appear.C.D’Andrea and I.Z.Emiris,Solving Degenerate Polynomial Systems,In Proc.AMS-IMS-SIAM Conf. on Symbolic Manipulation,Mt.Holyoke,Massachusetts,pp.121–139,AMS Contemporary Mathematics, 2001.I.Emiris,Notes creuses sur l’´e limination creuse,Notes au DEA de Maths,U.Nice,2001,ftp://ftp-sop.inria.fr/galaad/emiris/publis/NOTcreuxDEA.ps.gz.I.Z.Emiris,A General Solver Based on Sparse Resultants:Numerical Issues and Kinematic Applications, INRIA Tech.report3110,1997.I.Z.Emiris and J.F.Canny.Eﬃcient incremental algorithms for the sparse resultant and the mixed volume.J.Symbolic Computation,20(2):117–149,August1995.I.Z.Emiris and B.Mourrain,Matrices in elimination theory,put.,28:3–44,1999.I.M.Gelfand,M.Kapranov and A.Zelevinsky,Discriminants,Resultants,and Multidimensional Deter-minants,Birkh¨a user,Boston,1994.B.Mourrain and V.Y.Pan,Multivariate Polynomials,Duality and Structured Matrices,plexity, 16(1):110–180,2000.B.Sturmfels,On the Newton Polytope of the Resultant,J.of binatorics,3:207–236,1994.B.Sturmfels and A.Zelevinsky,Multigraded Resultants of Sylvester Type,J.of Algebra,163(1):115–127, 1994.Bernard Mourrain,Symbolic Numeric tools for solving polynomial equations and applications.This course will be divided into a tutorial part and a problem solving part.In theﬁrst part, we will gives an introductive presentation of symbolic and numeric methods for solving equations. We will brieﬂy recall well-known analytic methods and less known subdivision methods and will move to algebraic methods.Such methods are based on the study of the quotient algebra A of the polynomial ring modulo the ideal I=(f1,...,f m).We show how to deduce the geometry of the solutions,from the structure of A and in particular,how solving polynomial equations reduces to eigencomputations on these multiplication operators.We will mention a new method for computingthe normal of elements in A,used to obtain a representation of the multiplication operators.based on these formulations.We will describe iterative methods exploiting the properties of A,and which can be applied to select a root(among the other roots),which maximize or minimize some criterion, or to count or isolate the roots in a given domain.A major operation in eﬀective algebraic geometry is the projection,which is closely related to the theory of resultants.We present diﬀerent notions and constructions of resultants and diﬀerent geometric methods for solving systems of polynomial equationsIn a second part,we will consider problems from diﬀerent areas such CAD,robotics,computer vision,computational biology,...and show how to apply the methods that we have presented be-fore.Practical experimentations in maple with the package multires and with the library ALP (environment for symbolic and numeric computations)will illustrate these developments.Bibliography:David Cox,John Little and Donal O’Shea:Ideals,Varieties,and Algorithms,second edition,Under-graduate Texts in Mathematics,Springer,1997.D.Eisenbud,Commutative Algebra with a view toward Algebraic Geometry,Berlin,Springer-Verlag, Graduate Texts in Math.150,1994.M.Elkadi and B.Mourrain.G´e om´e trie Alg´e brique Eﬀective en dimension0:de la th´e orie`a la pratique, Notes de cours,DEA de Math´e matiques,Universit´e de Nice.2001.B.Mourrain,An introduction to algebraic methods for solving polynomial equations,Tutorial in Work-shop on Constructive Algebra and Systems Theory,Acad.Art and Science,Amsterdam,2000,2001,sub-mitted.http://www-sop.inria.fr/galaad/mourrain/Cours/2001tutorial.ps.gz.B.Mourrain and H.Prieto,A framework for Symbolic and Numeric Computations,Rapport de Recherche 4013,INRIA,2000.L.Bus´e,M.Elkadi and B.Mourrain,Residual Resultant of Complete Intersection,J.Pure&Applied Algebra,2001.I.Z.Emiris and B.Mourrain,Matrices in elimination theory,put.,28:3–44,1999.Juan Sabia,Eﬃcient polynomial equation solving:algorithms and complexity.This course intends to familiarize the assistants with the notion of algebraic complexity when solving polynomial equation systems.First,it will deal with the notion of dense representation of multivariate polynomials.Some results about the algebraic complexities of the eﬀective Nullstel-lensatz,of quantiﬁer elimination processes and of decomposition of varieties when using this model will be exposed.Then it will be shown how these complexities are essentially optimal in the dense representation model.This leads to a change of encoding of polynomials to get lower bounds for the complexity:the sparse representation and the straight-line program representation will be discussed. Finally,some complexity results in the straight-line program representation model will be shown (eﬀective Nullstellensatz,quantiﬁer elimination procedures,deformation techniques,for example).Bibliography:D.Brownawell,Bounds for the degrees in the Nullstellensatz,Ann.Math.2nd Series,126(3)(1987) 577-591.L.Caniglia,A.Galligo and J.Heintz,Some new eﬀective bounds in computational geometry,Lecture Notes in Computer Science357,Springer,Berlin(1989),131-151.A.L.Chistov and D.Y.Grigor’ev,Subexponential time solving systems of algebraic equations,LOMI preprint E-9-83,Steklov Institute,Leningrad(1983).A.L.Chistov and D.Y.Grigor’ev,Complexity of quantiﬁer elimination in the theory of algebraically closedﬁelds,Lecture Notes in Computer Science176,Springer,Berlin(1984),17-31.M.Elkadi and B.Mourrain,A new algorithm for the geometric decomposition of a variety,Proceedings of the1999International Symposium on Symbolic and Algebraic Computation(1999).N.Fitchas,A.Galligo and J.Morgenstern,Precise sequential and parallel complexity bounds for quan-tiﬁer elimination over algebraically closedﬁelds,J.Pure Appl.Algebra67(1990)1-14.M.Giusti and J.Heintz,Algorithmes-disons rapides-pour la d´e composition d’une vari´e t´e alg´e brique en composantes irr´e ductibles et´e quidimensionnelles,Progress in Mathematics94,Birkhauser(1991)169-193.M.Giusti,J.Heintz and J.Sabia,On the eﬃciency of eﬀective Nullstellensatz,plexity3, (1993)56-95.J.Heintz and C.P.Schorr,Testing polynomials which are easy to compute,Monographie30de l’Enseignement Math´e matique(1982)237-254.G.Jeronimo and J.Sabia,Eﬀective equidimensional decomposition of aﬃne varieties,to appear in J. Pure Appl.Algebra(2001).G.Lecerf,Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions,Proceedings of the ISSAC2000Conference(ACM)(2000).J.Kollar,Sharp eﬀective Nullstellensatz,J.AMS1(1988),963-975.S.Puddu and J.Sabia,An eﬀective algorithm for quantiﬁer elimination over algebraically closedﬁelds using straight-line programs,J.Pure Appl.Algebra129(1998),173-200.Jan Verschelde,Numerical algebraic geometry.In a1996paper,Andrew Sommese and Charles Wampler began developing new area,”Numerical Algebraic Geometry”,which would bear the same relation to”Algebraic Geometry”that”Numerical Linear Algebra”bears to”Linear Algebra”.To approximate all isolated solutions of polynomial systems,numerical path following techniques have been proven reliable and eﬃcient during the past two decades.In the nineties,homotopy methods were developed to exploit special structures of the polynomial system,in particular its sparsity.For sparse systems,the roots are counted by the mixed volume of the Newton polytopes and computed by means of polyhedral homotopies.In Numerical Algebraic Geometry we apply and integrate homotopy continuation methods to describe solution components of polynomial systems.One special,but important problem in Symbolic Computation concerns the approximate factorization of multivariate polynomials with approximate complex coeﬃcients.Our algorithms to decompose positive dimensional solution sets of polynomial systems into irreducible components can be considered as symbolic-numeric,or perhaps rather as numeric-symbolic,since numerical interpolation methods are applied to produce symbolic results in the form of equations describing the irreducible components.Applications from mechanical engineering motivated the development of Numerical Algebraic Geometry.The performance of our software on several test problems illustrate the eﬀectiveness of the new methods.Bibliography:J.Verschelde,Polynomial Homotopies for Dense,Sparse and Determinantal Systems,MSRI Preprint Number1999-041.Sophia-Antipolis,18th of March,2002.。

数学难的原因英语作文

Mathematics, often perceived as an intricate and challenging subject by many, has long been a topic of discussion among educators, students, and researchers alike. Its complexity stems from various aspects that contribute to the overall perception of difficulty. This essay aims to delve into these multifaceted dimensions, elucidating why math is often considered hard and thereby providing a comprehensive understanding of this intriguing conundrum.Firstly, one key reason for the perceived difficulty of mathematics lies in its abstract nature. Unlike other subjects that deal with tangible entities and real-life scenarios, mathematics operates largely in the realm of abstraction. Concepts like algebraic expressions, geometric shapes, or calculus functions are not immediately connected to everyday experiences. This detachment can lead to cognitive strain on individuals who find it difficult to visualize or conceptualize such abstract ideas. For instance, solving an equation requires understanding the symbolic representation rather than merely recognizing patterns visually, which can be daunting for some learners due to its non-intuitive characteristics.Secondly, mathematics is cumulative in its learning process. Each new concept builds upon previous ones, creating a hierarchical structure where mastery at each level is crucial to progress further. A lack of comprehension or retention of foundational concepts can lead to a domino effect, making subsequent topics appear more complex and challenging. For example, without a solid grasp of basic arithmetic, a student may struggle with advanced algebra or calculus. This sequential learning structure means that any gaps in understanding can significantly amplify the difficulty as one moves along the mathematical spectrum.The language of mathematics also adds to its complexity. It uses a unique vocabulary and notation system that can be alienating to newcomers. Mathematical symbols, equations, and formulas require precise interpretation and execution, leaving little room for ambiguity. Misunderstanding a single symbol could lead to an incorrect solution, which can be frustrating and intimidating for students.Moreover, mathematical reasoning and proof-writing necessitate logical precision and rigor that might not be expected in other disciplines, further contributing to its perceived difficulty.Another aspect to consider is the high demand for logical and analytical thinking skills required in mathematics. Problem-solving in math often involves breaking down complex situations into simpler components and then synthesizing them logically – a process that is mentally demanding and requires sustained concentration. Many individuals find this type of deep, systematic thinking inherently challenging, especially if they have developed habits of surface-level learning or rely heavily on memorization.Furthermore, psychological factors play a significant role. Fear and anxiety associated with mathematics, commonly known as 'math phobia', can hinder performance and deepen the sense of difficulty. This fear can stem from negative early experiences, societal stereotypes, or pressure to perform well, leading to a self-fulfilling prophecy where individuals perceive themselves as incapable before even attempting to solve a problem.Lastly, teaching methodologies and resources can either alleviate or exacerbate the difficulty of mathematics. Inadequate teaching strategies, overemphasis on rote learning, or lack of practical applications can make math seem more abstract and disconnected from reality. Conversely, effective pedagogical practices, including visualization, hands-on activities, and real-world connections, can help demystify mathematics and foster a deeper understanding.In conclusion, the perceived difficulty of mathematics is a product of its abstract nature, cumulative learning process, unique language, requirement for rigorous logic, psychological barriers, and teaching methods. To address this challenge, it's essential to adopt a holistic approach that acknowledges these factors and tailors instruction accordingly. By doing so, we can potentially transform the way mathematics is taught and learned, ultimately reducing its perceived complexity and opening doors to a world of intellectual discovery forall learners.Word Count: 794 words (excluding title)。

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Hybrid Symbolic-Numeric Computation*Erich KaltofenDepartment of MathematicsMassachusetts Institute of T echnology Cambridge,Massachusetts02139-4307,USA kaltofen@Lihong ZhiKey Laboratory of Mathematics Mechanization Academy of Mathematics and Systems ScienceBeijing100080,Chinalzhi@/~lzhiCategories and Subject Descriptors:I.2.1[Comput-ing Methodologies]:Symbolic and Algebraic Manipulation —Algorithms;G.1.2[Mathematics of Computing]:Numeri-cal Analysis—ApproximationGeneral Terms:algorithms,experimentation Keywords:symbolic/numeric hybrid methodsTUTORIAL ABSTRACTSeveral standard problems in symbolic computation,such as greatest common divisor and factorization of polynomials, sparse interpolation,or computing solutions to overdeter-mined systems of polynomial equations have non-trivial so-lutions only if the input coeﬃcients satisfy certain algebraic constraints.Errors in the coeﬃcients due toﬂoating point round-oﬀor through phsical measurement thus render the exact symbolic algorithms unusable.By symbolic-numeric methods one computes minimal deformations of the coeﬃ-cients that yield non-trivial results.We will present hybrid algorithms and benchmark computations based on Gauss-Newton optimization,singular value decomposition(SVD) and structure-preserving total least squares(STLS)ﬁtting for several of the above problems.A signiﬁcant body of results to solve those“approximate computer algebra”problems has been discovered in the past 10years.In the Computer Algebra Handbook the section on “Hybrid Methods”concludes as follows[2]:“The challenge of hybrid symbolic-numeric algorithms is to explore the ef-fects of imprecision,discontinuity,and algorithmic complex-ity by applying mathematical optimization,perturbation theory,and inexact arithmetic and other tools in order to solve mathematical problems that today are not solvable by numerical or symbolic methods alone.”The focus of our tu-torial is on how to formulate several approximate symbolic computation problems as numerical problems in linear al-gebra and optimization and on software that realizes their solutions.∗This research was supported in part by the National Science Foun-dation of the USA under Grants CCR-0305314and CCF-0514585 (Kaltofen)and OISE-0456285(Kaltofen,Yang and Zhi).This research was partially supported by NKBRPC(2004CB318000) and the Chinese National Natural Science Foundation under Grant 10401035(Yang and Zhi).Kaltofen’s permanent address:Dept.of Mathematics,North Car-olina State University,Raleigh,North Carolina27695-8205,USA, kaltofen@.Copyright is held by the author/owner(s).ISSAC’06,July9–12,2006,Genova,Italy.ACM1-59593-276-3/06/0007.Approximate Greatest Common Divisors[3].Our pa-per at this conference presents a solution to the approximate GCD problem for several multivariate polynomials with real or complex coeﬃcients.In addition,the coeﬃcients of the minimally deformed input coeﬃcients can be linearly con-strained.In our tutorial we will give a precise deﬁnition of the approximate polynomial GCD problem and we will present techniques based on parametric optimization(slow) and STLS or Gauss/Newton iteration(fast)for its numeri-cal solution.The fast methods can compute globally optimal solutions,but they cannot verify global optimality.We show how to apply the constrained approximate GCD problem to computing the nearest singular polynomial with a root of multiplicity at least k≥2.Approximate Factorization of Multivariate Polyno-mials[1].Our solution and implementation of the approx-imate factorization problem follows our approach for the approximate GCD problem.Our algorithms are based ona generalization of the diﬀerential forms introduced by W.Ruppert and S.Gao to many variables,and use SVD or STLS and Gauss/Newton optimization to numerically com-pute the approximate multivariate factors.Solutions of Zero-dimensional Polynomial Systems[4].We translate a system of polynomials into a systemof linear partial diﬀerential equations(PDEs)with constant coeﬃcients.The PDEs are brought to an involutive form by symbolic prolongations and numeric projections via SVD.The solutions of the polynomial system are obtained by solv-ing an eigen-problem constructed from the null spaces of the involutive system and its geometric projections.1.REFERENCES[1]Gao,S.,Kaltofen,E.,May,J.P.,Yang,Z.,andZhi,L.Approximate factorization of multivariatepolynomials via diﬀerential equations.In Gutierrez,J.,Ed.ISSAC2004Proc.2004Internat.Symp.Symbolic Algebraic Comput.,pp.167–174.[2]Grabmeier,J.,Kaltofen,E.,and Weispfenning,puter Algebra Handbook Springer Verlag,2003,pp.109–124.[3]Kaltofen,E.,Yang,Z.,and Zhi,L.Approximategreatest common divisors of several polynomials withlinearly constrained coeﬃcients and singularpolynomials.In Dumas,J-G.,Ed.ISSAC2006Proc.2006Internat.Symp.Symbolic Algebraic Comput..[4]Reid,G.,Tang,J.,and Zhi,L.A completesymbolic-numeric linear method for camera posedetermination.In Sendra,J.,Ed.ISSAC2003Proc.2003Internat.Symp.Symbolic Algebraic Comput.,pp.215–223.7。

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