Compact representation of knowledge bases in inductive logic programming

空间域和变换域When it comes to the concept of space domain and transform domain, it can be quite overwhelming for many individuals who are not familiar with the field of signal processing. The space domain refers to the physical space or location where a signal is originally acquired or processed. On the other hand, the transform domain involves the process of converting a signal from its original form in the space domain to a new representation in a different domain, such as the frequency domain.谈到空间域和变换域的概念，对于不熟悉信号处理领域的许多人来说，这可能会令人感到不知所措。

空间域是指信号最初获取或处理的物理空间或位置。

另一方面，变换域涉及将信号从其在空间域的原始形式转换为不同域中的新表示，例如频率域。

In signal processing, the choice between working in the space domain or the transform domain depends on the specific objectivesof the analysis. While the space domain is more intuitive and easierto understand for beginners, the transform domain often provides amore efficient and compact representation of the signal, making it easier to analyze and process.在信号处理中，选择在空间域还是变换域中工作取决于分析的具体目标。

t distribution 学生t分布t number t数t statistic t统计量t test t检验t1topological space t1拓扑空间t2topological space t2拓扑空间t3topological space 分离空间t4topological space 正则拓扑空间t5 topological space 正规空间t6topological space 遗传正规空间table 表table of random numbers 随机数表table of sines 正弦表table of square roots 平方根表table of values 值表tabular 表的tabular value 表值tabulate 制表tabulation 造表tabulator 制表机tacnode 互切点tag 标签tame 驯顺嵌入tame distribution 缓增广义函数tamely imbedded 驯顺嵌入tangency 接触tangent 正切tangent bundle 切丛tangent cone 切线锥面tangent curve 正切曲线tangent function 正切tangent line 切线tangent of an angle 角的正切tangent plane 切平面tangent plane method 切面法tangent surface 切曲面tangent vector 切向量tangent vector field 切向量场tangent vector space 切向量空间tangential approximation 切线逼近tangential component 切线分量tangential curve 正切曲线tangential equation 切线方程tangential stress 切向应力tangents method 切线法tape 纸带tape inscription 纸带记录tariff 税tautology 重言taylor circle 泰勒圆taylor expansion 泰勒展开taylor formula 泰勒公式taylor series 泰勒级数technics 技术technique 技术telegraph equation 电报方程teleparallelism 绝对平行性temperature 温度tempered distribution 缓增广义函数tend 倾向tendency 瞧tension 张力tensor 张量tensor algebra 张量代数tensor analysis 张量分析tensor bundle 张量丛tensor calculus 张量演算法tensor density 张量密度tensor differential equation 张量微分方程tensor field 张量场tensor form 张量形式tensor form of the first kind 第一张量形式tensor function 张量函数tensor of torsion 挠率张量tensor product 张量乘积tensor product functor 张量乘积函子tensor representation 张量表示tensor space 张量空间tensor subspace 张量子空间tensor surface 张量曲面tensorial multiplication 张量乘法term 项term of higher degree 高次项term of higher order 高次项term of series 级数的项terminability 有限性terminable 有限的terminal decision 最后判决terminal edge 终结边terminal point 终点terminal unit 级端设备terminal vertex 悬挂点terminate 终止terminating chain 可终止的链terminating continued fraction 有尽连分数terminating decimal 有尽小数termination 终止terminology 专门名词termwise 逐项的termwise addition 逐项加法termwise differentiation 逐项微分termwise integration 逐项积分ternary 三元的ternary connective 三元联结ternary form 三元形式ternary notation 三进制记数法ternary number system 三进制数系ternary operation 三项运算ternary relation 三项关系ternary representation og numbers 三进制记数法tertiary obstruction 第三障碍tesseral harmonic 田形函数tesseral legendre function 田形函数test 检验test for additivity 加性检验test for uniform convergence 一致收敛检验test function 测试函数test of dispersion 色散检验test of goodness of fit 拟合优度检验test of hypothesis 假设检验test of independence 独立性检验test of linearity 线性检验test of normality 正规性检验test point 测试点test routine 检验程序test statistic 检验统计量tetracyclic coordinates 四圆坐标tetrad 四元组tetragon 四角形tetragonal 正方的tetrahedral 四面角tetrahedral angle 四面角tetrahedral co ordinates 四面坐标tetrahedral group 四面体群tetrahedral surface 四面曲面tetrahedroid 四面体tetrahedron 四面形tetrahedron equation 四面体方程theorem 定理theorem for damping 阻尼定理theorem of alternative 择一定理theorem of identity for power series 幂级数的一致定理theorem of implicit functions 隐函数定理theorem of mean value 平均值定理theorem of principal axes 轴定理theorem of residues 残数定理theorem of riemann roch type 黎曼洛赫型定理theorem on embedding 嵌入定理theorems for limits 极限定理theoretical curve 理论曲线theoretical model 理论模型theory of automata 自动机理论theory of cardinals 基数论theory of complex multiplication 复数乘法论theory of complexity of computations 计算的复杂性理论theory of correlation 相关论theory of differential equations 微分方程论theory of dimensions 维数论theory of elementary divisors 初等因子理论theory of elementary particles 基本粒子论theory of equations 方程论theory of errors 误差论theory of estimation 估计论theory of functions 函数论theory of games 对策论theory of hyperbolic functions 双曲函数论theory of judgment 判断论theory of numbers 数论theory of ordinals 序数论theory of perturbations 摄动理论theory of probability 概率论theory of proportions 比例论theory of relativity 相对论theory of reliability 可靠性理论theory of representations 表示论theory of sets 集论theory of sheaves 层理论theory of singularities 奇点理论theory of testing 检验论theory of time series 时间序列论theory of transversals 横断线论theory of types 类型论thermal 热的thermodynamic 热力学的thermodynamics 热力学theta function 函数theta series 级数thick 厚的thickness 厚度thin 薄的thin set 薄集third boundary condition 第三边界条件third boundary value problem 第三边界值问题third fundamental form 第三基本形式third isomorphism theorem 第三同构定理third proportional 比例第三项third root 立方根thom class 汤姆类thom complex 汤姆复形three body problem 三体问题three dimensional 三维的three dimensional space 三维空间three dimensional torus 三维环面three eighths rule 八分之三法three faced 三面的three figur 三位的three place 三位的three point problem 三点问题three series theorem 三级数定理three sheeted 三叶的three sided 三面的three sigma rule 三规则three termed 三项的three valued 三值的three valued logic 三值逻辑three valued logic calculus 三值逻辑学threshold logic 阈逻辑time interval 时程time lag 时滞time series analysis 时序分析timesharing 分时toeplitz matrix 托普利兹矩阵tolerance 容许tolerance distribution 容许分布tolerance estimation 容许估计tolerance factor 容许因子tolerance level 耐受水平tolerance limit 容许界限tolerance region 容许区域top digit 最高位数字topological 拓扑的topological abelian group 拓扑阿贝耳群topological algebra 拓扑代数topological cell 拓扑胞腔topological circle 拓扑圆topological completeness 拓扑完备性topological complex 拓扑复形topological convergence 拓扑收敛topological dimension 拓扑维topological direct sum 拓扑直和topological dynamics 拓扑动力学topological embedding 拓扑嵌入topological field 拓扑域topological group 拓扑群topological homeomorphism 拓扑同胚topological index 拓扑指数topological invariant 拓扑不变量topological limit 拓扑极限topological linear space 拓扑线性空间topological manifold 拓扑廖topological mapping 拓扑同胚topological pair 拓扑偶topological polyhedron 曲多面体topological product 拓扑积topological residue class ring 拓扑剩余类环topological ring 拓扑环topological simplex 拓扑单形topological skew field 拓扑非交换域topological space 拓扑空间topological sphere 拓扑球面topological structure 拓扑结构topological sum 拓扑和topological type 拓扑型topologically complete set 拓扑完备集topologically complete space 拓扑完备空间topologically equivalent space 拓扑等价空间topologically nilpotent element 拓扑幂零元topologically ringed space 拓扑环式空间topologically solvable group 拓扑可解群topologico differential invariant 拓扑微分不变式topologize 拓扑化topology 拓扑topology of bounded convergence 有界收敛拓扑topology of compact convergence 紧收敛拓扑topology of uniform convergence 一致收敛拓扑toroid 超环面toroidal coordinates 圆环坐标toroidal function 圆环函数torque 转矩torsion 挠率torsion coefficient 挠系数torsion form 挠率形式torsion free group 非挠群torsion group 挠群torsion module 挠模torsion of a curve 曲线的挠率torsion product 挠积torsion subgroup 挠子群torsion tensor 挠率张量torsion vector 挠向量torsionfree connection 非挠联络torsionfree module 无挠模torsionfree ring 无挠环torus 环面torus function 圆环函数torus group 环面群torusknot 环面纽结total 总和total correlation 全相关total curvature 全曲率total degree 全次数total differential 全微分total differential equation 全微分方程total error 全误差total graph 全图total image 全象total inspection 全检查total instability 全不稳定性total inverse image 全逆象total matrix algebra 全阵环total matrix ring 全阵环total order 全序total predicate 全谓词total probability 总概率total probability formula 总概率公式total regression 总回归total relation 通用关系total space 全空间total stability 全稳定性total step iteration 整步迭代法total step method 整步迭代法total stiefel whitney class 全斯蒂费尔惠特尼类total subset 全子集total sum 总和total variation 全变差totally bounded set 准紧集totally bounded space 准紧空间totally differentiable 完全可微分的totally differentiable function 完全可微函数totally disconnected 完全不连通的totally disconnected graph 完全不连通图totally disconnected groupoid 完全不连通广群totally disconnected set 完全不连通集totally disconnected space 完全不连通空间totally geodesic 全测地的totally nonnegative matrix 全非负矩阵totally ordered group 全有序群totally ordered set 线性有序集totally positive 全正的totally positive matrix 全正矩阵totally quasi ordered set 完全拟有序集totally real field 全实域totally reflexive relation 完全自反关系totally regular matrix method 完全正则矩阵法totally singular subspace 全奇异子空间totally symmetric loop 完全对称圈totally symmetric quasigroup 完全对称拟群touch 相切tournament 竞赛图trace 迹trace form 迹型trace function 迹函数trace of dyadic 并向量的迹trace of matrix 矩阵的迹trace of tensor 张量的迹tracing point 追迹点track 轨迹tractrix 曳物线trajectory 轨道transcendence 超越性transcendence basis 超越基transcendence degree 超越次数transcendency 超越性transcendental element 超越元素transcendental equation 超越方程transcendental function 超越函数transcendental integral function 超越整函数transcendental number 超越数transcendental singularity 超越奇点transcendental surface 超越曲面transfer 转移transfer function 转移函数transfinite 超限的transfinite diameter 超限直径transfinite induction 超限归纳法transfinite number 超限序数transfinite ordinal 超限序数transform 变换transformation 变换transformation equation 变换方程transformation factor 变换因子transformation formulas of the coordinates 坐标的变换公式transformation function 变换函数transformation group 变换群transformation of air mass 气团变性transformation of coordinates 坐标的变换transformation of parameter 参数变换transformation of state 状态变换transformation of the variable 变量的更换transformation rules 变换规则transformation theory 变换论transformation to principal axes 轴变换transgression 超渡transient response 瞬态响应transient stability 瞬态稳定性transient state 瞬态transient time 过渡时间transition function 转移函数transition graph 转换图transition matrix 转移矩阵transition probability 转移函数transitive closure 传递闭包transitive graph 传递图transitive group of motions 可迁运动群transitive law 可迁律transitive permutation group 可迁置换群transitive relation 传递关系transitive set 可递集transitivity 可递性transitivity laws 可迁律translatable design 可旋转试验设计translate 转移translation 平移translation curve 平移曲线translation group 平移群translation invariant 平移不变的translation invariant metric 平移不变度量translation number 殆周期translation of axes 坐标轴的平移translation operator 平移算子translation surface 平移曲面translation symmetry 平移对称translation theorem 平移定理transmission channel 传输通道transmission ratio 传输比transport problem 运输问题transportation algorithm 运输算法transportation matrix 运输矩阵transportation network 运输网络transportation problem 运输问题transpose 转置transposed inverse matrix 转置逆矩阵transposed kernel 转置核transposed map 转置映射transposed matrix 转置阵transposition 对换transversal 横截矩阵胚transversal curve 横截曲线transversal field 模截场transversal lines 截线transversality 横截性transversality condition 横截条件transverse axis 横截轴transverse surface 横截曲面trapezium 不规则四边形trapezoid 不规则四边形trapezoid formula 梯形公式trapezoid method 梯形公式traveling salesman problem 转播塞尔斯曼问题tree 树trefoil 三叶形trefoil knot 三叶形纽结trend 瞧trend line 瞧直线triad 三元组trial 试验triangle 三角形triangle axiom 三角形公理triangle condition 三角形公理triangle inequality 三角形公理triangulable 可三角剖分的triangular decomposition 三角分解triangular form 三角型triangular matrix 三角形矩阵triangular number 三角数triangular prism 三棱柱triangular pyramid 四面形triangular surface 三角曲面triangulate 分成三角形triangulation 三角剖分triaxial 三轴的triaxial ellipsoid 三维椭面trichotomy 三分法trident of newton 牛顿三叉线tridiagonal matrix 三对角线矩阵tridimensional 三维的trigammafunction 三函数trigonometric 三角的trigonometric approximation polynomial 三角近似多项式trigonometric equation 三角方程trigonometric function 三角函数trigonometric moment problem 三角矩问题trigonometric polynomial 三角多项式trigonometric series 三角级数trigonometrical interpolation 三角内插法trigonometry 三角学trihedral 三面形的trihedral angle 三面角trihedron 三面体trilateral 三边的trilinear 三线的trilinear coordinates 三线坐标trilinear form 三线性形式trinomial 三项式;三项式的trinomial equation 三项方程triplanar point 三切面重点 ?triple 三元组triple curve 三重曲线triple integral 三重积分triple point 三重点triple product 纯量三重积triple product of vectors 向量三重积triple root 三重根triple series 三重级数triple tangent 三重切线triply orthogonal system 三重正交系triply tangent 三重切线的trirectangular spherical triangle 三直角球面三角形trisecant 三度割线trisect 把...三等分trisection 三等分trisection of an angle 角的三等分trisectrix 三等分角线trivalent map 三价地图trivector 三向量trivial 平凡的trivial character 单位特贞trivial cohomology functor 平凡上同弹子trivial extension 平凡扩张trivial fibre bundle 平凡纤维丛trivial graph 平凡图trivial homogeneous ideal 平凡齐次理想trivial knot 平凡纽结trivial solution 平凡解trivial subset 平凡子集trivial topology 密着拓扑trivial valuation 平凡赋值triviality 平凡性trivialization 平凡化trochoid 摆线trochoidal 余摆线的trochoidal curve 摆线true error 真误差true formula 真公式true proposition 真命题true sign 直符号true value 真值truncated cone 截锥truncated cylinder 截柱truncated distribution 截尾分布truncated pyramid 截棱锥truncated sample 截样本truncated sequence 截序列truncation 舍位truncation error 舍位误差truncation point 舍位点truth 真值truth function 真值函项truth matrix 真值表truth set 真值集合truth symbol 真符号truth table 真值表truthvalue 真值tube 管tubular knot 管状纽结tubular neighborhood 管状邻域tubular surface 管状曲面turbulence 湍流turbulent 湍聊turing computability 图灵机可计算性turing computable 图灵机可计算的turing machine 图录机turn 转向turning point 转向点twice 再次twice differentiable function 二次可微函数twin primes 素数对twisted curve 空间曲线twisted torus 挠环面two address 二地址的two address code 二地址代码two address instruction 二地址指令two body problem 二体问题two decision problem 二判定问题two digit 二位的two dimensional 二维的two dimensional laplace transformation 二重拉普拉斯变换two dimensional normal distribution 二元正态分布two dimensional quadric 二维二次曲面two dimensional vector space 二维向量空间two fold transitive group 双重可迁群two person game 两人对策two person zero sum game 二人零和对策two phase sampling 二相抽样法two place 二位的two point distribution 二点分布two point form 两点式two sample method 二样本法two sample problem 二样本问题two sample test 双样本检验two sheet 双叶的two sided condition 双边条件two sided decomposition 双边分解two sided divisor 双边因子two sided ideal 双边理想two sided inverse 双边逆元two sided module 双边模two sided neighborhood 双侧邻域two sided surface 双侧曲面two sided test 双侧检定two stage sampling 两阶段抽样法two termed expression 二项式two valued logic 二值逻辑two valued measure 二值测度two variable matrix 双变量矩阵two way array 二向分类two way classification 二向分类twopoint boundary value problem 两点边值问题type 型type problem 类型问题typenumber 型数typical mean 典型平均。

写一篇你最喜欢的一件学习用品的英文作文全文共3篇示例，供读者参考篇1One of my favorite study tools is my trusty notebook. It may seem simple, but my notebook is an essential part of my learning routine and has been a constant companion throughout my academic journey.My notebook is a sleek, black Moleskine with lined pages that are perfect for jotting down notes, ideas, and thoughts. I have always been a fan of writing things down by hand, as I find that it helps me retain information better and allows me to make connections between different concepts. I carry my notebook with me everywhere I go, whether I'm in class, studying at the library, or even just sitting in a café brainstorming ideas.One of the things I love most about my notebook is its versatility. I use it for everything from taking notes in lectures and seminars, to creating to-do lists and setting goals for myself. Its compact size makes it easy to carry around, and thehigh-quality paper ensures that my thoughts and musings are preserved for posterity.In addition to its practical uses, my notebook also holds sentimental value for me. It is a record of my academic achievements, a repository of my ideas and inspirations, and a reflection of my personal growth and development. Flipping through its pages, I can see how far I have come and how much I have learned over the years.Overall, my notebook is more than just a study tool – it is a trusted companion, a source of inspiration, and a tangible representation of my intellectual curiosity and passion for learning. I am grateful for its presence in my life and the role it plays in helping me navigate the challenges and joys of the academic world. Here's to many more pages filled with knowledge, insights, and revelations!篇2One of my favorite learning tools is my trusty notebook. It may seem like a simple item, but for me, my notebook is an essential part of my daily learning routine.I have always been a firm believer in the power of writing things down. Whether it's taking notes during a lecture, jotting down key points from a book, or simply brainstorming ideas, my notebook is always by my side. I love the feeling of pen on paper,the act of physically writing something down helps me to internalize and remember information better than typing on a computer.My notebook is not just a place for taking notes, it's also a reflection of my personality and creativity. I often doodle in the margins or decorate my pages with colorful washi tape and stickers. This personal touch makes my notebook feel like a part of me, and I find that I am more inclined to engage with the material when it's presented in a visually appealing way.Another reason why I love my notebook is its portability. I can take it with me wherever I go, whether it's to a coffee shop, the library, or a park. This flexibility allows me to study and learn in different environments, which helps to keep me motivated and engaged.In conclusion, my notebook is not just a tool for learning, it's a companion that has been with me through thick and thin. It helps me to organize my thoughts, express my creativity, and ultimately, become a better student. I can't imagine my life without it, and I will continue to cherish and use it for many years to come.篇3One of my favorite items for learning is my trusty notebook. As a student, I have found that a good notebook is a crucial tool to help me organize my thoughts, take notes in class, and keep track of my assignments.My notebook is always with me, whether I’m in class, studying at the library, or doing homework at home. I have filled its pages with countless notes, diagrams, and ideas over the years. It has become like a second brain to me, holding all of the important information I need to succeed in my studies.One of the things I love most about my notebook is its versatility. I can use it to take notes during a lecture, jot down ideas for a paper, create to-do lists for assignments, and even sketch out rough drafts of projects. It’s the perfect tool for keeping all of my thoughts and ideas in one place.Another reason I love my notebook is that it allows me to personalize my learning experience. I can choose the color, style, and layout of the notebook that best suits my needs and preferences. This customization helps me stay motivated and engaged in my studies.Overall, my notebook is an essential tool that helps me stay organized, focused, and productive in my academic pursuits. Ican’t imagine my life as a student without it. It is truly my favorite learning accessory.。

S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsIndividual2:30–5:00pm,July 9,2011(Please select 5problems to solve)1.a)Compute the integral: ∞−∞x cos xdx (x 2+1)(x 2+2),b)Show that there is a continuous function f :[0,+∞)→(−∞,+∞)such that f ≡0and f (4x )=f (2x )+f (x ).2.Solve the following problem: d 2u dx 2−u (x )=4e −x ,x ∈(0,1),u (0)=0,dudx(0)=0.3.Find an explicit conformal transformation of an open set U ={|z |>1}\(−∞,−1]to the unit disc.4.Assume f ∈C 2[a,b ]satisfying |f (x )|≤A,|f(x )|≤B for each x ∈[a,b ]and there exists x 0∈[a,b ]such that |f (x 0)|≤D ,then |f (x )|≤2√AB +D,∀x ∈[a,b ].5.Let C ([0,1])denote the Banach space of real valued continuous functions on [0,1]with the sup norm,and suppose that X ⊂C ([0,1])is a dense linear subspace.Suppose l :X →R is a linear map (not assumed to be continuous in any sense)such that l (f )≥0if f ∈X and f ≥0.Show that there is a unique Borel measure µon [0,1]such that l (f )= fdµfor all f ∈X .6.For s ≥0,let H s (T )be the space of L 2functions f on the circle T =R /(2πZ )whose Fourier coefficients ˆf n = 2π0e−inx f (x )dx satisfy Σ(1+n 2)s ||ˆf n |2<∞,with norm ||f ||2s =(2π)−1Σ(1+n 2)s |ˆf n |2.a.Show that for r >s ≥0,the inclusion map i :H r (T )→H s (T )is compact.b.Show that if s >1/2,then H s (T )includes continuously into C (T ),the space of continuous functions on T ,and the inclusion map is compact.1S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyIndividual9:30–12:00am,July10,2011(Please select5problems to solve)1.Suppose M is a closed smooth n-manifold.a)Does there always exist a smooth map f:M→S n from M into the n-sphere,such that f is essential(i.e.f is not homotopic to a constant map)?Justify your answer.b)Same question,replacing S n by the n-torus T n.2.Suppose(X,d)is a compact metric space and f:X→X is a map so that d(f(x),f(y))=d(x,y)for all x,y in X.Show that f is an onto map.3.Let C1,C2be two linked circles in R3.Show that C1cannot be homotopic to a point in R3\C2.4.Let M=R2/Z2be the two dimensional torus,L the line3x=7y in R2,and S=π(L)⊂M whereπ:R2→M is the projection map. Find a differential form on M which represents the Poincar´e dual of S.5.A regular curve C in R3is called a Bertrand Curve,if there existsa diffeomorphism f:C→D from C onto a different regular curve D in R3such that N x C=N f(x)D for any x∈C.Here N x C denotes the principal normal line of the curve C passing through x,and T x C will denote the tangent line of C at x.Prove that:a)The distance|x−f(x)|is constant for x∈C;and the angle made between the directions of the two tangent lines T x C and T f(x)D is also constant.b)If the curvature k and torsionτof C are nowhere zero,then there must be constantsλandµsuch thatλk+µτ=16.Let M be the closed surface generated by carrying a small circle with radius r around a closed curve C embedded in R3such that the center moves along C and the circle is in the normal plane to C at each point.Prove thatMH2dσ≥2π2,and the equality holds if and only if C is a circle with radius √2r.HereH is the mean curvature of M and dσis the area element of M.1S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsIndividual2:30–5:00pm,July 10,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let K =Q (√−3),an imaginary quadratic field.(a)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=S 3?(Here S 3is the symmetric group in 3letters.)(b)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Z /4Z ?(c)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Q ?Here Q is the quaternion group with 8elements {±1,±i,±j,±k },a finite subgroup of the group of units H ×of the ring H of all Hamiltonian quaternions.2.Let f be a two-dimensional (complex)representation of a finite group G such that 1is an eigenvalue of f (σ)for every σ∈G .Prove that f is a direct sum of two one-dimensional representations of G3.Let F ⊂R be the subset of all real numbers that are roots of monic polynomials f (X )∈Q [X ].(1)Show that F is a field.(2)Show that the only field automorphisms of F are the identityautomorphism α(x )=x for all x ∈F .4.Let V be a finite-dimensional vector space over R and T :V →V be a linear transformation such that(1)the minimal polynomial of T is irreducible;(2)there exists a vector v ∈V such that {T i v |i ≥0}spans V .Show that V contains no non-trivial proper T -invariant subspace.5.Given a commutative diagramA →B →C →D →E↓↓↓↓↓A →B →C →D →E1Algebra,Number Theory and Combinatorics,2011-Individual2 of Abelian groups,such that(i)both rows are exact sequences and(ii) every vertical map,except the middle one,is an isomorphism.Show that the middle map C→C is also an isomorphism.6.Prove that a group of order150is not simple.S.-T.Yau College Student Mathematics Contests 2011Applied Math.,Computational Math.,Probability and StatisticsIndividual6:30–9:00pm,July 9,2011(Please select 5problems to solve)1.Given a weight function ρ(x )>0,let the inner-product correspond-ing to ρ(x )be defined as follows:(f,g ):= baρ(x )f (x )g (x )d x,and let f :=(f,f ).(1)Define a sequence of polynomials as follows:p 0(x )=1,p 1(x )=x −a 1,p n (x )=(x −a n )p n −1(x )−b n p n −2(x ),n =2,3,···wherea n =(xp n −1,p n −1)(p n −1,p n −1),n =1,2,···b n =(xp n −1,p n −2)(p n −2,p n −2),n =2,3,···.Show that {p n (x )}is an orthogonal sequence of monic polyno-mials.(2)Let {q n (x )}be an orthogonal sequence of monic polynomialscorresponding to the ρinner product.(A polynomial is called monic if its leading coefficient is 1.)Show that {q n (x )}is unique and it minimizes q n amongst all monic polynomials of degree n .(3)Hence or otherwise,show that if ρ(x )=1/√1−x 2and [a,b ]=[−1,1],then the corresponding orthogonal sequence is the Cheby-shev polynomials:T n (x )=cos(n arccos x ),n =0,1,2,···.and the following recurrent formula holds:T n +1(x )=2xT n (x )−T n −1(x ),n =1,2,···.(4)Find the best quadratic approximation to f (x )=x 3on [−1,1]using ρ(x )=1/√1−x 2.1Applied Math.Prob.Stat.,2011-Individual 22.If two polynomials p (x )and q (x ),both of fifth degree,satisfyp (i )=q (i )=1i,i =2,3,4,5,6,andp (1)=1,q (1)=2,find p (0)−q (0)y aside m black balls and n red balls in a jug.Supposes 1≤r ≤k ≤n .Each time one draws a ball from the jug at random.1)If each time one draws a ball without return,what is the prob-ability that in the k -th time of drawing one obtains exactly the r -th red ball?2)If each time one draws a ball with return,what is the probability that in the first k times of drawings one obtained totally an odd number of red balls?4.Let X and Y be independent and identically distributed random variables.Show thatE [|X +Y |]≥E [|X |].Hint:Consider separately two cases:E [X +]≥E [X −]and E [X +]<E [X −].5.Suppose that X 1,···,X n are a random sample from the Bernoulli distribution with probability of success p 1and Y 1,···,Y n be an inde-pendent random sample from the Bernoulli distribution with probabil-ity of success p 2.(a)Give a minimum sufficient statistic and the UMVU (uniformlyminimum variance unbiased)estimator for θ=p 1−p 2.(b)Give the Cramer-Rao bound for the variance of the unbiasedestimators for v (p 1)=p 1(1−p 1)or the UMVU estimator for v (p 1).(c)Compute the asymptotic power of the test with critical region |√n (ˆp 1−ˆp 2)/ 2ˆp ˆq |≥z 1−αwhen p 1=p and p 2=p +n −1/2∆,where ˆp =0.5ˆp 1+0.5ˆp 2.6.Suppose that an experiment is conducted to measure a constant θ.Independent unbiased measurements y of θcan be made with either of two instruments,both of which measure with normal errors:fori =1,2,instrument i produces independent errors with a N (0,σ2i )distribution.The two error variances σ21and σ22are known.When ameasurement y is made,a record is kept of the instrument used so that after n measurements the data is (a 1,y 1),...,(a n ,y n ),where a m =i if y m is obtained using instrument i .The choice between instruments is made independently for each observation in such a way thatP (a m =1)=P (a m =2)=0.5,1≤m ≤n.Applied Math.Prob.Stat.,2011-Individual 3Let x denote the entire set of data available to the statistician,in this case (a 1,y 1),...,(a n ,y n ),and let l θ(x )denote the corresponding log likelihood function for θ.Let a =n m =1(2−a m ).(a)Show that the maximum likelihood estimate of θis given by ˆθ= n m =11/σ2a m −1 n m =1y m /σ2a m.(b)Express the expected Fisher information I θand the observedFisher information I x in terms of n ,σ21,σ22,and a .What hap-pens to the quantity I θ/I x as n →∞?(c)Show that a is an ancillary statistic,and that the conditional variance of ˆθgiven a equals 1/I x .Of the two approximations ˆθ·∼N (θ,1/I θ)and ˆθ·∼N (θ,1/I x ),which (if either)would you use for the purposes of inference,and why?S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsTeam9:00–12:00am,July 9,2011(Please select 5problems to solve)1.Let H 2(∆)be the space of holomorphic functions in the unit disk ∆={|z |<1}such that ∆|f |2|dz |2<∞.Prove that H 2(∆)is a Hilbert space and that for any r <1,the map T :H 2(∆)→H 2(∆)given by T f (z ):=f (rz )is a compact operator.2.For any continuous function f (z )of period 1,show that the equation dϕdt=2πϕ+f (t )has a unique solution of period 1.3.Let h (x )be a C ∞function on the real line R .Find a C ∞function u (x,y )on an open subset of R containing the x -axis such that u x +2u y =u 2and u (x,0)=h (x ).4.Let S ={x ∈R ||x −p |≤c/q 3,for all p,q ∈Z ,q >0,c >0},show that S is uncountable and its measure is zero.5.Let sl (n )denote the set of all n ×n real matrices with trace equal to zero and let SL (n )be the set of all n ×n real matrices with deter-minant equal to one.Let ϕ(z )be a real analytic function defined in a neighborhood of z =0of the complex plane C satisfying the conditions ϕ(0)=1and ϕ (0)=1.(a)If ϕmaps any near zero matrix in sl (n )into SL (n )for some n ≥3,show that ϕ(z )=exp(z ).(b)Is the conclusion of (a)still true in the case n =2?If it is true,prove it.If not,give a counterexample.e mathematical analysis to show that:(a)e and πare irrational numbers;(b)e and πare also transcendental numbers.1S.-T.Yau College Student Mathematics Contests2011Applied Math.,Computational Math.,Probability and StatisticsTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Let A be an N-by-N symmetric positive definite matrix.The con-jugate gradient method can be described as follows:r0=b−A x0,p0=r0,x0=0FOR n=0,1,...αn= r n 22/(p TnA p n)x n+1=x n+αn p n r n+1=r n−αn A p nβn=−r Tk+1A p k/p TkA p kp n+1=r n+1+βn p nEND FORShow(a)αn minimizes f(x n+αp n)for allα∈R wheref(x)≡12x T A x−b T x.(b)p Ti r n=0for i<n and p TiA p j=0if i=j.(c)Span{p0,p1,...,p n−1}=Span{r0,r1,...,r n−1}≡K n.(d)r n is orthogonal to K n.2.We use the following scheme to solve the PDE u t+u x=0:u n+1 j =au nj−2+bu nj−1+cu njwhere a,b,c are constants which may depend on the CFL numberλ=∆t ∆x .Here x j=j∆x,t n=n∆t and u njis the numerical approximationto the exact solution u(x j,t n),with periodic boundary conditions.(i)Find a,b,c so that the scheme is second order accurate.(ii)Verify that the scheme you derived in Part(i)is exact(i.e.u nj =u(x j,t n))ifλ=1orλ=2.Does this imply that the scheme is stable forλ≤2?If not,findλ0such that the scheme is stable forλ≤λ0. Recall that a scheme is stable if there exist constants M and C,which are independent of the mesh sizes∆x and∆t,such thatu n ≤Me CT u0for all∆x,∆t and n such that t n≤T.You can use either the L∞norm or the L2norm to prove stability.1Applied Math.Prob.Stat.,2011-Team2 3.Let X and Y be independent random variables,identically dis-tributed according to the Normal distribution with mean0and variance 1,N(0,1).(a)Find the joint probability density function of(R,),whereR=(X2+Y2)1/2andθ=arctan(Y/X).(b)Are R andθindependent?Why,or why not?(c)Find a function U of R which has the uniform distribution on(0,1),Unif(0,1).(d)Find a function V ofθwhich is distributed as Unif(0,1).(e)Show how to transform two independent observations U and Vfrom Unif(0,1)into two independent observations X,Y fromN(0,1).4.Let X be a random variable such that E[|X|]<∞.Show thatE[|X−a|]=infE[|X−x|],x∈Rif and only if a is a median of X.5.Let Y1,...,Y n be iid observations from the distribution f(x−θ), whereθis unknown and f()is probability density function symmetric about zero.Suppose a priori thatθhas the improper priorθ∼Lebesgue(flat) on(−∞,∞).Write down the posterior distribution ofθ.Provides some arguments to show that thisflat prior is noninforma-tive.Show that with the posterior distribution in(a),a95%probability interval is also a95%confidence interval.6.Suppose we have two independent random samples{Y1,i=1,...,n} from Poisson with(unknown)meanλ1and{Y i,i=n+1,...,2n}from Poisson with(unknown)meanλ2Letθ=λ1/(λ1+λ2).(a)Find an unbiased estimator ofθ(b)Does your estimator have the minimum variance among all un-biased estimators?If yes,prove it.If not,find one that has theminimum variance(and prove it).(c)Does the unbiased minimum variance estimator you found at-tain the Fisher information bound?If yes,show it.If no,whynot?S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Suppose K is afinite connected simplicial complex.True or false:a)Ifπ1(K)isfinite,then the universal cover of K is compact.b)If the universal cover of K is compact thenπ1(K)isfinite.pute all homology groups of the the m-skeleton of an n-simplex, 0≤m≤n.3.Let M be an n-dimensional compact oriented Riemannian manifold with boundary and X a smooth vectorfield on M.If n is the inward unit normal vector of the boundary,show thatM div(X)dV M=∂MX·n dV∂M.4.Let F k(M)be the space of all C∞k-forms on a differentiable man-ifold M.Suppose U and V are open subsets of M.a)Explain carefully how the usual exact sequence0−→F(U∪V)−→F(U)⊕F V)−→F(U∩V)−→0 arises.b)Write down the“long exact sequence”in de Rham cohomology as-sociated to the short exact sequence in part(a)and describe explicitly how the mapH kdeR (U∩V)−→H k+1deR(U∪V)arises.5.Let M be a Riemannian n-manifold.Show that the scalar curvature R(p)at p∈M is given byR(p)=1vol(S n−1)S n−1Ric p(x)dS n−1,where Ric p(x)is the Ricci curvature in direction x∈S n−1⊂T p M, vol(S n−1)is the volume of S n−1and dS n−1is the volume element of S n−1.1Geometry and Topology,2011-Team2 6.Prove the Schur’s Lemma:If on a Riemannian manifold of dimension at least three,the Ricci curvature depends only on the base point but not on the tangent direction,then the Ricci curvature must be constant everywhere,i.e.,the manifold is Einstein.S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsTeam9:00–12:00pm,July 9,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let F be a field and ¯Fthe algebraic closure of F .Let f (x,y )and g (x,y )be polynomials in F [x,y ]such that g .c .d .(f,g )=1in F [x,y ].Show that there are only finitely many (a,b )∈¯F×2such that f (a,b )=g (a,b )=0.Can you generalize this to the cases of more than two-variables?2.Let D be a PID,and D n the free module of rank n over D .Then any submodule of D n is a free module of rank m ≤n .3.Identify pairs of integers n =m ∈Z +such that the quotient rings Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m );and identify pairs of integers n =m ∈Z +such that Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m ).4.Is it possible to find an integer n >1such that the sum1+12+13+14+ (1)is an integer?5.Recall that F 7is the finite field with 7elements,and GL 3(F 7)is the group of all invertible 3×3matrices with entries in F 7.(a)Find a 7-Sylow subgroup P 7of GL 3(F 7).(b)Determine the normalizer subgroup N of the 7-Sylow subgroupyou found in (a).(c)Find a 2-Sylow subgroup of GL 3(F 7).6.For a ring R ,let SL 2(R )denote the group of invertible 2×2matrices.Show that SL 2(Z )is generated by T = 1101 and S = 01−10 .What about SL 2(R )?1。

一到十二月英语缩写形式January - Jan.February - Feb.March - Mar.April - Apr.May - MayJune - Jun.July - Jul.August - Aug.September - Sept.October - Oct.November - Nov.December - Dec.The English abbreviations for the months of the year are widely used in various contexts, from calendars and schedules to formal documents and casual conversations. These concise representations of the month names serve to save space and time, making them a practical and efficient tool in our daily lives.The month of January, often associated with the start of a new year, is abbreviated as "Jan." This shorthand version quickly conveys themeaning without the need for the full word. Similarly, February is abbreviated as "Feb.," allowing for a more compact representation of the second month of the year.March, the third month, is shortened to "Mar.," while April, the fourth month, is represented as "Apr." May remains unchanged, as the abbreviation "May" is the same as the full month name. June, on the other hand, is shortened to "Jun.," maintaining the essence of the month while reducing the number of characters.The seventh month of the year, July, is abbreviated as "Jul.," a concise form that is widely recognized. August is shortened to "Aug.," a familiar abbreviation that is commonly used in various contexts.September, the ninth month, is represented as "Sept.," a form that retains the essence of the full month name. October, the tenth month, is abbreviated as "Oct.," a convenient shorthand that is easily recognizable.November, the eleventh month, is shortened to "Nov.," a succinct representation of the month. Finally, December, the twelfth and final month of the year, is abbreviated as "Dec.," a widely accepted and understood form.These month abbreviations serve as a testament to the efficiency and practicality of the English language. They enable users to convey important information in a concise manner, saving time and space while still maintaining the core meaning of the month names. Whether in formal documents, calendars, or casual conversations, these abbreviations have become an integral part of our communication and record-keeping practices.。

专利名称：RESPONDING TO SITUATIONS USING KNOWLEDGE REPRESENTATION ANDINFERENCE发明人：GUPTA, RAKESH,VASCO CALAIS, PEDRO 申请号：US2006002204申请日：20060119公开号：WO2006083596A3公开日：20080410专利内容由知识产权出版社提供摘要：A system, apparatus and application for providing robots with the ability to intelligently respond to perceived situations are described. A knowledge database is assembled automatically, based on distributed knowledge capture. The knowledge base embodies the "common sense," that is, the consensus, of the subjects who contribute the knowledge. Systems are provided to automatically preprocess, or "clean" the information to make it more useful. The knowledge thus refined is utilized to construct a multidimensional semantic network, or MSN. The MSN provides a compact and efficient semantic representation suitable for extraction of knowledge for inference purposes and serves as the basis for task and response selection. When the robot perceives a situation that warrants a response, an appropriate subset of the MSN is extracted into a Bayes network. The resultant network is refined, and used to derive a set of response probabilities, which the robot uses to formulate a response.申请人：HONDA MOTOR CO., LTD.,GUPTA, RAKESH,VASCO CALAIS, PEDRO更多信息请下载全文后查看。

《计算机英语（第4版）》练习参考答案欧阳歌谷（2021.02.01）Unit One: Computer and Computer ScienceUnit One/Section AI.Fill in the blanks with the information given in the text:1.Charles Babbage; Augusta Ada Byron2.input; output3.VLSI4.workstations; mainframes5.vacuum; transistors6.instructions; software7.digit; eight; byte8.microminiaturization; chipII.Translate the following terms or phrases from English into Chinese and vice versa:1.artificial intelligence 人工智能2.paper-tape reader 纸带阅读器3.optical computer 光计算机4.neural network 神经网络5.instruction set 指令集6.parallel processing 并行处理7.difference engine 差分机8.versatile logical element 通用逻辑元件9.silicon substrate 硅衬底10.vacuum tube 真空管11.数据的存储与处理the storage and handling ofdata12.超大规模集成电路very large-scale integratedcircuit13.中央处理器 central processing unit14.个人计算机 personal computer15.模拟计算机 analogue computer16.数字计算机 digital computer17.通用计算机 general-purpose computer18.处理器芯片 processor chip19.操作指令 operating instructions20.输入设备 input deviceIII.Fill in each of the blanks with one of the words given in the following list, making changes if necessary: We can define a computer as a device that accepts input, processes data, stores data, and produces output. According to the mode of processing, computers are either analog or digital.They can also be classified as mainframes, minicomputers, workstations, or microcomputers. All else (for example, the age of the machine) being equal, this categorization provides some indication of the computer’s speed, siz e, cost, and abilities.Ever since the advent of computers, there have been constant changes. First-generation computers of historic significance, such as UNIV AC (通用自动计算机), introduced in the early 1950s, were based onvacuum tubes. Second-generation computers, appearing in the early 1960s, were those in whichtransistors replaced vacuum tubes. In third-generation computers, dating from the1960s,integrated circuits replaced transistors. In fourth-generation computers such asmicrocomputers, which first appeared in the mid-1970s, large-scale integration enabled thousands of circuitsto be incorporated on onechip. Fifth-generation computers are expected to combine very-large-scale integration with sophisticated approaches to computing, including artificial intelligence and true distributed processing.IV.Translate the following passage from English into Chinese:计算机将变得更加先进，也将变得更加容易使用。

⾳频格式介绍⾳频格式介绍电脑上常见的⾳频通常分为合成声⾳（midi）和波形声⾳（pcm）两⼤类。

其中，合成声⾳是⼀种⾳乐演奏指令的序列，就像乐谱⼀样，可以利⽤声⾳输出设备或与电脑相连的电⼦乐器进⾏演奏，本⾝不包含具体声⾳数据，依靠具体的声⾳合成器。

⽽波形声⾳则是通过录⾳设备录制的原始声⾳波形，直接记录了真实声⾳的⼆进制采样数据。

就像是图像中的⽮量图和点阵图⼀样。

相应的⾳频格式有三类：合成声⾳格式：（如铃声），主要有：midi，sp-midi，mmf，rtttl, i-melody，e-melody等。

有⼀些软件可以进⾏相互转换，如：Mobile Music Pro，Quick Ringtone等。

波形声⾳格式：主要有：wav，mp3，mp3pro, ra, rma, wma, ogg, vqf, aiff, au, voc, vox, cda，ac-3,aac，mp4，pcm,adpcm,a-law,u-law, g711, g721, g722, g723.1, g726, g728, g729, amr, amr-wb等。

有很多软件可以进⾏相互转换，如：cool edit，winamp等。

混合声⾳格式：mod、dls等，类似midi⽂件，但还包含波表，不依赖于硬件合成器。

下⾯简单介绍⼀下各种⾳频格式：下⾯有⼀篇⽂章《⾳频⽂件格式全介绍》对⾳频格式介绍的已经⽐较全⾯了。

这⾥只对他没有介绍的⾳频格式进⾏⼀些介绍。

⼀、 sp-midi最近推出的Scalable Polyphony MIDI Specification (简称SP-MIDI)，⼤概可以译成“可升级的MIDI复⾳”。

已经成为3gpp标准⾳频标准的⼀部分。

SP-MIDI的功能实现借助于⼀种新的MIDI消息，这种消息被称为最⼤同时和弦（Maximum Instantaneous Polyphony - MIP）消息。

这种MIP消息根据所需的和弦等级来定义SP-MIDI内容的⾳乐编配。