Abelian functions satisfy an Algebraic Addition Theorem

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Non-Abelian L Functions for Function Fields

In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves deﬁned over ﬁnite ﬁelds. It consists of two chapters. More precisely, in Chapter I, we ﬁrst introduce new yet genuine non-abelian zeta functions for curves deﬁned over ﬁnite ﬁelds. This is achieved by a ‘weighted count’ on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. We justify our construction by establishing basic properties for these new zetas such as functional equation and rationality, and show that if only line bundles are involved, our newly deﬁned zetas coincide with Artin’s Zeta. All this, in particular, the rationality, then leads naturally to our deﬁnition of (global) non-abelian zeta functions (for curves deﬁned over number ﬁelds), which themselves are justiﬁed by a convergence result. We end this chapter with a detailed study on rank two non-abelian zeta functions for genus two curves, based on what we call inﬁnitesimal structures of Brill-Noether loci (and Weierstrass points). In Chapter II, we begin with a similar construction for the ﬁeld of rationals to motivate what follows. In particular, we show that there is an intrinsic relation between our non-abelian zeta functions and Eisenstein series. Due to this, instead of introducing general non-abelian L-functions for curves deﬁned over ﬁnite ﬁelds with more general test functions (as what Tate did in his Thesis for abelian L-functions), we then deﬁne non-abelian L-functions for curves over ﬁnite ﬁelds as integrations of Eisenstein series associated to L2 -automorphic forms over certain generalized moduli spaces. Here geometric truncations play a key role. Basic properties for these non-abelian L-functions, such as meromorphic continuation, functional equations and singularities, are established as well, based on the theory of Eisenstain series of Langlands and Morris. We end this chapter by establishing a closed formula for what we call the abelian parts of non-abelian Lfunctions associated with Eisenstein series for cusp forms, via the Rankin-Selberg method, motivated by a formula of Arthur and Langlands. This work is an integrated part of our vast yet still under developing Program for Geometric Arithmetic [We1], and is motivated by our new non-abelian L-functions for number ﬁelds [We2] in connection with non-abelian arithmetic aspects of global ﬁelds.

数学专业英语词汇（H）_数学物理英语词汇

h closed h闭的haar condition 哈尔条件haar measure 哈尔测度hadamard criterion 阿达玛判别准则hadamard gap condition 阿达玛间断条件hadamard matrix 阿达玛矩阵hadamard method of descent 阿达玛下降法hadamard multiplication theorem 阿达玛乘法定理hadamard three circles theorem 阿达玛三圆定理half 半half angle 半角half angle formulas 半角公式half axis 半轴half closed interval 半闭区间half exact 半正合的half line 半直线half neighborhood 半邻域half plane 半无限平面half plane of absolute convergence 绝对收敛半平面half plane of convergence 收敛半平面half round 半圆的half side formulas 半边公式half space 半空间halve 对分halving method 二等分法hamilton characteristic function 哈密顿特寨数hamilton formula 哈密顿公式hamilton function 哈密顿函数hamilton jacobi equation 哈密顿雅可比方程hamilton jacobi theory 哈密顿雅可比理论hamilton principle 哈密顿原理hamiltonian 哈密顿函数hamiltonian circuit 哈密顿回路hamiltonian group 哈密顿群hamiltonian operator 哈密顿算子hamiltonian path 合密顿道路hand 边handle 环柄handle of the second kind 交叉套handlebody 环柄体hankel transformation 汉克尔变换harmonic analysis 低分析harmonic analyzer 傅里叶分析仪harmonic conjugate 低共轭点harmonic constant 低常数harmonic curve 低曲线harmonic differential equation 低微分方程harmonic division 低分割harmonic function 低函数harmonic integral 低积分harmonic mapping 低映射harmonic mean 低平均harmonic measure 低测度harmonic motion 低运动harmonic oscillation 谐振动harmonic progression 低级数harmonic ratio 低比harmonic series 低级数harmonic synthesis 傅里叶综合法harmonicity 低性hasse diagram 哈塞图hausdorff group 豪斯道夫群hausdorff measure 豪斯道夫测度hausdorff metric 豪斯道夫度量hausdorff separation axiom 廉斯道夫分离公理hausdorff space 分离空间haversine 半正矢heat 热heat conduction 热传导hecke character 黑克特贞hecke operator 黑克算子hectoliter 百升hectometer 百米helicograph 螺旋规helicoid 螺旋面helicoidal surface 螺旋面helix 螺旋线hemi continuous 半连续的hemihedry 半对称hemipyramid 半棱锥体hemisphere 半球hemispherical 半球面的hemispherical shape 半球形hendecagon 十一边形henselization 享泽莱化heptagon 七边形heptahedron 七面体hereditarily enumerable set 遗传可数集hereditarily generating system 遗传的生成系hereditarily indecomposable continuum 遗传不可分解的连续统hereditarily normal space 遗传正规空间hereditary class 遗传类hereditary property 遗传性质hereditary set 遗传集hereditary system of sets 集的遗传系heredity 遗传性hermite function 埃尔米特函数hermite interpolation formula 埃尔米特插值公式hermite interpolation polynomial 埃尔米特插值多项式hermite normal form 埃尔米特正规形式hermite polynomial 埃尔米特多项式hermite reciprocity law 埃尔米待互反律hermitian bilinear functional 埃尔米特双线性泛函hermitian conjugate 埃尔米特共轭阵hermitian form 埃尔米特形式hermitian inner product module 埃尔米特内积模hermitian inner product space 埃尔米特空间hermitian kernel 埃尔米特核hermitian matrix 埃尔米特矩阵hermitian metric 埃尔米特度量hermitian operator 埃尔米特算子hermitian polynomiat 埃尔米特多项式hermitian transformation 埃尔米特变换hero formula 海伦公式hesse normal form 海赛正规形式hessian 海赛形式hessian group 海赛群hessian matrix 海赛矩阵hexagon 六边形hexagonal 六边形的hexagonal net 六边形网格hexagonal system 六角系hexahedral 六面体的hexahedron 六面体hexakistetrahedron 六四面体hierarchical classification 谱系分类hierarchy 分层high speed computer 高速计算机higher algebra 高等代数higher commutator 广义换位子higher derivative 高阶导数higher mathematics 高等数学higher order term 高阶项higher plane curve 高次平面曲线higher singularity 高次奇异性highest common divisor 最大公约highest common factor 最大公因子highest derivative 最高阶导数highest order 最高位highest weight 最高权hilbert basis theorem 希耳伯特基定理hilbert cube 希耳伯特超平行体hilbert inequality 希耳伯特不等式hilbert integral 希耳伯特积分hilbert matrix 希耳伯特矩阵hilbert modular form 希耳伯特模形式hilbert modular function 希耳伯特模函数hilbert modular group 希耳伯特模群hilbert nullstellensatz 希耳伯特零点定理hilbert parallelotope 希耳伯特超平行体hilbert problems 希耳伯特问题hilbert space 希耳伯特空间hill differential equation 希耳微分方程histogram 直方图history 履历hodograph 速端曲线hodograph transformation 速端曲线变换hodometer 路程表holding domain 解域holomorph convex manifold 全形凸廖holomorph separable manifold 全形可分廖holomorphic 正则的holomorphic completeness 全纯完全性holomorphic convexity 正则凸性holomorphic differential 全纯微分holomorphic differential form 全纯微分形式holomorphic divisor 全纯除子holomorphic function 全纯函数holomorphic manifold 复解析廖holomorphic mapping 全纯映射holomorphic part 全纯部分holomorphy 正则holonomic condition 完全性条件holonomic reference system 完整参考系holonomic system 完整系holonomy 完整holonomy group 完整群homeomorph 同胚象homeomorphic 同胚的homogeneity 齐性homogeneity formula 齐性公式homogeneity of variances 同方差性homogeneity relation 齐性关系homogeneous 均匀的homogeneous cartesian co ordinates 齐次笛卡儿坐标homogeneous coordinates 齐次笛卡儿坐标homogeneous distribution 均匀分布homogeneous element 齐次元素homogeneous equation 齐次方程homogeneous function 齐次函数homogeneous function of order k k阶齐次函数homogeneous ideal 齐次理想homogeneous integral equation 齐次积分方程homogeneous linear boundary value problem 齐次线性边值问题homogeneous linear differential equation 齐次线性微分方程homogeneous linear transformation 齐次线性变换homogeneous lineare transformation 齐次线性变换homogeneous markov chain 齐次马尔可夫链homogeneous markov process 齐次马尔可夫过程homogeneous operator 齐次算子homogeneous polynomial 齐次多项式homogeneous space 商空间homogeneous system of differential equations 齐次微分方程组homogeneous system of linear equations 齐次线性方程组homogeneous variational problem 齐次变分问题homographic function 单应函数homological algebra 同碟数homological dimension 同惮数homological invariant 同祷变量homologous mappings 同党射homologous simplicial map 同单形映射homologous to zero 同第零homology 同调homology algebra 同碟数homology class 同掂homology equivalence 同等价homology equivalent complex 同等价复形homology functor 同弹子homology group 同岛homology manifold 同滴homology module 同担homology operation 同邓算homology sequence 同凋列homology simplex 同单形homology spectral sequence 同底序列homology sphere 同凋homology theory 同帝homology type 同低homomorphic group 同态群homomorphic image 同态象homomorphism 同态homomorphism theorem 同态定理homoscedastic 同方差的homoscedasticity 同方差性homothetic transformation 相似扩大homothety 相似扩大homotopic 同伦的homotopic invariant 同伦不变量homotopic map 同伦映射homotopic path 同伦道路homotopically equivalent space 同伦等价空间homotopy associativity 同伦结合性homotopy category of topological spaces 拓扑空间同伦范畴homotopy chain 同伦链homotopy class 同伦类homotopy classification 同伦分类homotopy equivalence 同伦等价homotopy excision theorem 同伦分割定理homotopy extension 同伦扩张homotopy group 同伦群homotopy group functor 同伦群函子homotopy inverse 同伦逆的homotopy operator 同伦算子homotopy sequence 同伦序列homotopy set 同伦集homotopy sphere 同伦球面homotopy theorem 同伦定理homotopy theory 同伦论homotopy type 同伦型homotopyassociative 同伦结合的horizon 水平线horizontal axis 水平轴horizontal component 水平分量horizontal coordinates 水平坐标horizontal plane 水平面horizontal projection 水平射影horned sphere 角形球面horocycle 极限圆horosphere 极限球面horse power 马力hungarian method 匈牙利法hurewicz isomorphism theorem 胡列维茨同构定理hydrodynamics 铃动力学hydromechanics 铃力学hydrostatics 铃静力学hyper graeco latin square 超格勒科拉丁方格hyper octahedral group 超八面体群hyperabelian function 超阿贝耳函数hyperalgebraic manifold 超代数廖hyperarithmetical 超算术的hyperarithmetical relation 超算术关系hyperbola 双曲线hyperbolic 双曲线的hyperbolic automorphism 双曲代换hyperbolic catenary 双曲悬链线hyperbolic cosecant 双曲余割hyperbolic cosine 双曲余弦hyperbolic cotangent 双曲余切hyperbolic cylinder 双曲柱hyperbolic elliptic motion 双曲椭圆运动hyperbolic equation 双曲型方程hyperbolic function 双曲函数hyperbolic geometry 双曲几何学hyperbolic inverse point 双曲逆点hyperbolic involution 双曲对合hyperbolic line 双曲线hyperbolic motion 双曲运动hyperbolic orbit 双曲线轨道hyperbolic paraboloid 双曲抛物面hyperbolic plane 双曲平面hyperbolic point 双曲点hyperbolic riemann surface 双曲型黎曼曲面hyperbolic rotation 双曲旋转hyperbolic secant 双曲正割hyperbolic sine 双曲正弦hyperbolic space 双曲空间hyperbolic spiral 双曲螺线hyperbolic substitution 双曲代换hyperbolic system 双曲型组hyperbolic tangent 双曲正切hyperbolic tangent function 双曲正切hyperbolic type 双曲型hyperbolicity 双曲性hyperboloid 双曲面hyperboloid of one sheet 单叶双曲面hyperboloid of revolution 旋转双曲面hyperboloid of two sheets 双叶双曲面hypercohomology 超上同调hypercomplex 超复数hypercomplex number 超复数hypercone 超锥hyperconjugation 超共轭hypercyclic group 超循环群hypercyclide 超四次圆纹曲面hyperelliptic 超椭圆的hyperelliptic function 超椭圆函数hyperelliptic integral 超椭圆积分hyperelliptic theta function 超椭圆函数hyperfinite c* algebra 超有限c*代数hypergeometric differential equation 超几何微分方程hypergeometric distribution 超几何分布hypergeometric function 超几何函数hypergeometric function of the second kind 第二类超几何函数hypergeometric series 超几何级数hypergeometry 超几何学hypergroup 超群hypermatrix 超矩阵hypernormal dispersion 超正态方差hyperplane 超平面hyperplane coordinates 超平面坐标hyperplane of support 支撑超平面hyperplane section 超平面截面hyperquadric 超二次曲面hyperreal numbers 超实数hyperspace 超空间hypersphere 超球面hyperstonian space 超斯通空间hypersurface 超曲面hypocycloid 内摆线hypocycloidal 圆内旋轮线的hypoelliptic operator 次椭圆型算子hypoellipticity 次椭圆性hypotenuse 斜边hypothesis 假设hypothetical population 假言总体hypotrochoid 长短辐圆内旋轮线。

数学专业课中英文词汇

数学专业英语词汇大全Aa priori bound 先验界限a priori distribution 先验分布a priori probability 先验概率a summable a可和的abacus 算盘/ˈæbəkəs/abbreviate 略abbreviation 简化abel equation 阿贝耳方程abel identity 阿贝耳恒等式abel inequality 阿贝耳不等式abel summation method 阿贝耳求和法abelian algebra 阿贝耳代数/ˈældʒəbrə/abelian differential 阿贝耳微分/ˌdɪfəˈrenʃəl◂/abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分/ˈɪntəɡrəl/abelian summation 阿贝耳求和法abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号/əˈbrɪdʒd/abscissa 横坐标[æbˈsɪsə]abscissa of absolute convergence 绝对收敛坐标[kənˈvɜːdʒəns] abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute 绝对形absolute address 绝对地址absolute class field 绝对类域absolute coding 绝对编码absolute cohomology 绝对上同调absolute conic 绝对二次曲线absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率absolute geometry 绝对几何[dʒiˈɒmətri]absolute homology group 绝对同岛absolute homotopy group 绝对同伦群absolute inequality 绝对不等式absolute instability 绝对不稳定性absolute maximum 绝对极大值absolute minimum 绝对极小值absolute moment 绝对矩absolute neighborhood 绝对邻域absolute neighborhood retract 绝对邻域收缩核absolute norm 绝对范数absolute number 不名数absolute parallelism 绝对平行性absolute quadric 绝对二次曲面absolute ramification index 绝对分歧指数[ˌræmɪfɪˈkeɪʃn] absolute rotation 绝对旋转absolute singular homology group 绝对奇异同岛absolute space 绝对空间absolute space time 绝对时空absolute stability 绝对稳定性absolute term 常数项absolute unit 绝对单位absolute value 绝对值absolute value sign 绝对值符号absolute velocity 绝对速度[vəˈlɒsəti]absolutely additive measure 完全加性测度absolutely compact set 绝对紧集absolutely complete system 绝对完备系absolutely continuous 绝对连续的[kənˈtɪnjuəs]absolutely continuous distribution 绝对连续分布absolutely continuous function 绝对连续函数absolutely continuous measure 绝对连续测度absolutely continuous part 绝对连续部分absolutely continuous transformation 绝对连续变换absolutely convergent 绝对收敛的absolutely convergent integral 绝对收敛积分absolutely convergent series 绝对收敛级数absolutely convex hull 绝对凸包[ˈkɒnveks]absolutely discontinuous function 绝对不连续函数absolutely integrable 绝对可积的absolutely irreducible character 绝对不可约特征[ˌɪrɪˈdjuːsəbl] absolutely irreducible representation 绝对不可约表示absolutely irreducible variety 绝对不可约簇absolutely normal number 绝对范数absolutely prime ideal 绝对素理想absolutely semisimple algebra 绝对半单代数absolutely simple group 绝对单群absolutely summable sequence 绝对可和序列absolutely unbiased estimator 绝对无偏估计量[ˈɛstɪmeɪtə] absolutely unramified extension 绝对非分歧扩张absorbing barrier 吸收障碍absorbing medium 吸收媒体absorbing set 吸收集absorbing state 吸收态absorption 吸收[əbˈzɔːpʃn]absorption coefficient 吸收系数[ˌkəʊɪˈfɪʃnt]absorption curve 吸收曲线absorption factor 吸收因数absorption index 吸收指数absorption law 吸收律absorption probability 吸收概率abstract 抽象的abstract algebra 抽象代数abstract algebraic geometry 抽象代数几何[ˌældʒɪˈbreɪɪk] abstract automaton 抽象自动机[ɔːˈtɒmətən]abstract category 抽象范畴abstract code 理想码abstract complex 抽象复形abstract group 抽象群abstract interval function 抽象区间函数abstract mathematics 抽象数学abstract number 不名数abstract ordered simplicial complex 抽象有序单纯复形abstract simplex 抽象单形abstract simplicial subcomplex 抽象单子复形abstract space 抽象空间abstraction 抽象abstraction operator 抽象算子absurd 谬论的[əbˈsɜːd]absurdity 谬论[əbˈsɜːdɪti]abundance ratio 丰度比abundant number 过剩数accelerated motion 加速运动acceleration 加速度acceleration of convergence 收敛性的加速acceleration of gravity 重力加速度acceptable quality level 合格质量水平acceptance 肯定acceptance inspection 接受检查acceptance limit 接受界限acceptance line 接受线acceptance number 接受数acceptance probability 接受概率acceptance region 接受区域acceptance zone 接受带access 存取access speed 存取速度access time 存取时间accessibility 可达性accessible boundary point 可达边界点accessible ordinal number 可达序数[ˈɔːdɪnl] accessible point 可达点accessible set 可达集accessible vertex 可达顶点[ˈvɜːteks]accessory extremal 配连极值[əkˈsesəri]accidental 偶然的accidental coincidence 偶然符合accidental error 随机误差accommodation 第accumulated error 累积误差accumulating point 聚点accumulation 累积accumulation point 聚点accumulator 累加器存储器accuracy 准确性accuracy grade 准确度accuracy of measurement 测量精确度accuracy rating 准确度acnod 孤点acount 计算action integral 酌积分action variable 酌变量active restriction 有效限制actual 真实的actual infinity 实无穷[ɪnˈfɪnəti]acute 尖锐的[əˈkjuːt] adj. 十分严重的;(疾病)急性的;灵敏的acute angle 锐角acute angled triangle 锐角三角形acute triangle 锐角三角形acuteness 锐度acyclic 非循环的[ˌeɪˈsaɪklɪk]acyclic complex 非循环复形acyclic graph 环道自由图acyclic model theorem 非循环模型定理ad infinitum 无穷地[ɪnfɪˈnaɪtəm]adams circle 阿达姆斯圆adams extrapolation method 阿达姆斯外插法adaptability 适应性adaptation 适应adapted basis 适应基add 加added circuit 加法电路addend 加数adder 加法器addition 加法addition formulas 加法公式addition sign 加号addition system 加法系addition table 加法表addition theorem 加法定理addition theorem of probability 概率的加法定理additional 加法的additional code 附加代码additional condition 附加条件additional error 附加误差additive 加法的additive category 加性范畴additive class 加性类additive functional 加性泛函数additive functional transformation 加性泛函变换additive group 加法群additive interval function 加性区间函数additive inverse element 加性逆元素additive number theory 堆垒数论additive operator 加性算子additive process 加性过程additive relation 加性关系additive separable 加法可分的additive theory of numbers 堆垒数论additive valuation 加法赋值additively commutative ordinal numbers 加性交换序数additivity 加法性address 地址address part 地址部分address register 地址寄存器addressing 指定箱位adele 阿代尔adele group 阿代尔群adequate 适合的adherent point 触点[ədˈhɪərənt]adhesion 附着[ədˈhiːʒn]adjacency 邻接adjacency matrix 邻接矩阵[ˈmeɪtrɪks]adjacent angles 邻角adjacent edge 邻棱adjacent side 邻边adjacent supplementary angles 邻角adjacent vertex 邻顶adjoint boundary value problem 伴随边值问题adjoint determinant 伴随行列式[dɪˈtɜːmɪnənt] n. 决定因素;决定条件adjoint difference equation 伴随差分方程adjoint differential equation 伴随微分方程adjoint differential expression 伴随微分式adjoint form 伴随形式adjoint function 伴随函数adjoint functor 伴随函子adjoint graph 导出图adjoint group 伴随群adjoint hilbert problem 伴随希耳伯特问题adjoint integral equation 伴随积分方程adjoint kernel 伴随核[ˈkɜːnl]adjoint linear map 伴随线性映射adjoint matrix 伴随阵adjoint operator 伴随算子adjoint process 伴随过程adjoint representation 伴随表示adjoint space 伴随空间adjoint surface 伴随曲面adjoint system 伴随系adjoint system of differential equations 微分方程的伴随系adjoint transformation 伴随算子adjoint vector 伴随向量adjunct 代数余子式[ˈædʒʌŋkt]adjunction 附加adjunction of an identity element 单位元的附加adjustable point 可胆点adjustment 蝶？admissibility limit 容许界限admissible 容许的[ədˈmɪsəbl]admissible category 容许范畴admissible chart 容许图admissible control 可行控制admissible decision function 容许判决函数admissible decision rule 容许判决函数admissible deformation 容许形变admissible domain 容许区域admissible function 容许函数admissible homomorphism 容许同态admissible hypothesis 容许假设admissible lifting 容许提升admissible map 容许映射admissible sequence 容许序列admissible space 容许空间admissible strategy 容许策略admissible subgroup 容许子群admissible test 容许检定admissible value 容许值affine algebraic set 仿射代数集[əˈfaɪn]affine collineation 仿射直射变换affine connection 仿射联络affine coordinates 平行坐标affine curvature 仿射曲率affine differential geometry 仿射微分几何学affine distance 仿射距离affine figure 仿射图形affine function 仿射函数affine geometry 仿射几何学affine group 仿射群affine group scheme 仿射群概型[skiːm] affine isothermal net 仿射等温网[ˌaɪsəʊˈθɜːməl] affine length 仿射长度affine line 仿射直线affine normal 仿射法线affine parameter 仿射参数[pəˈræmɪtə(r)] 决定因素;规范;范围affine principal curvature 仿射助率affine rational transformation 仿射有理变换affine space 仿射空间affine sphere 仿射球面affine surface 仿射曲面affine transformation 仿射变换affine variety 仿射簇affinely connected manifold 仿射连通廖[ˈmænɪfəʊld]affinely connected space 仿射连通空间affinity 仿射变换[əˈfɪnəti] n. 喜好;喜爱;密切的关系;类同affirmation 肯定affirmative proposition 肯定命题affix 附标[əˈfɪks , ˈæfɪks]after effect 后效酌aggregate 集aggregation 聚合agreement 一致air coordinates 空间坐标airy function 亚里函数airy integral 亚里积分aitken interpolation 艾特肯插值[ɪnˌtɜːpəˈleɪʃn]aitken interpolation formula 艾特肯插值公式albanese variety 阿尔巴内斯簇aleph 阿列夫aleph zero 阿列夫零alexander cohomology module 亚历山大上同担alexander cohomology theory 亚历山大上同帝alexander matrix 亚历山大阵alexander polynomial 亚历山大多项式[ˌpɒli'nəʊmiəl]algebra 代数学algebra of events 事件场algebra of logic 逻辑代数algebra of tensors 张量代数algebra over k 环k上的代数algebraic 代数的algebraic adjunction 代数的附加algebraic affine variety 仿射代数集algebraic algebra 代数的代数algebraic branch point 代数分歧点algebraic calculus 代数计算algebraic closure 代数闭包algebraic closureoperator 代数闭包算子algebraic complement 代数余子式algebraic cone 代数锥[kəʊn]algebraic correspondence 代数对应algebraic curve 代数曲线algebraic equation 代数方程algebraic expression 代数式algebraic extension 代数扩张algebraic form 代数形式algebraic fraction 代数分式algebraic function 代数函数algebraic function field 代数函数域algebraic geometry 代数几何学algebraic group 代数群algebraic hull 代数包algebraic hypersurface of the seconed order 二阶代数超曲面algebraic integer 代数整数algebraic irrational number 代数无理数algebraic lie algebra 代数的李代数algebraic logic of pocket calculator 袖珍计算机的代数逻辑algebraic multiplicity 代数重度algebraic number 代数数algebraic number field 代数数域algebraic number theory 代数数论algebraic operation 代数运算algebraic polynomial 代数多项式algebraic singularity 代数奇点algebraic space 代数空间algebraic spiral 代数螺线algebraic structure 代数结构algebraic sum 代数和algebraic surface 代数曲面algebraic system 代数系algebraic variety 代数簇algebraically closed field 代数闭域algebraically dependent elements 代数相关元algebraically equivalent 代数等价的algebraically independent elements 代数无关元algebraization 代数化algebro geometric 代数几何的algebroid function 代数体函数algebroidal function 代数体函数algorithm 算法[ˈælɡərɪðəm]algorithm of division 辗转相除法algorithm of euclid 欧几里得算法[ˈjuːklɪd]algorithm theory 算法论algorithmic language 算法语言algorithmization 算法化aligned systematic sampling 列系统抽样[ˈsɑːmplɪŋ] alignment chart 列线图aliquot part 整除部分[ˈælɪkwɒt]alligation 混合法allocation problem 配置问题allowable 容许的allowable defects 容许靠allowable error 容许误差allowance 允许almost all 几乎处处almost bounded function 殆有界函数almost certain convergence 几乎必然收敛almost complex manifold 殆复廖[ˈmænɪfəʊld]almost convergent sequence 殆收敛序列almost equivalent 殆等价almost everywhere 几乎处处almost impossible event 殆不可能事件almost invariant set 殆不变集almost periodic function 殆周期函数almost periodicity 殆周期性almost significant 殆显著的alpha capacity 容量alpha limit set 极限集alphabetical 字母的alphanumeric 字母数字式[ˌælfənjuːˈmerɪk]alphanumeric representation of information 信息的字母数字表示alternate angles 错角[ɔːlˈtɜːnət , ˈɔːltəneɪt , ˈɔːltɜːnət] alternating chain 交错链alternating differential form 外微分形式alternating differential of differential form 微分形式的交错微分alternating direction method 交替方向法alternating form 交错形式alternating function 反对称函数alternating group 交错群alternating harmonic series 莱布尼兹级数alternating knot 交错纽结alternating matrix 交错矩阵alternating method 交错法alternating product 外积alternating sequence 交错序列alternating series 交错级数alternating series test 交错级数检验alternating sum 交错和alternating tensor 交错张量alternating tensor density 交错张量密度alternating tree 交错树alternation 交错alternative 交错;择一alternative algebra 交错代数alternative field 交错域alternative hypothesis 择一假设alternative normal form 析取范式alternative proposition 选言命题altitude 高度altitude theorem 高度定理amalgamated product 融合积[əˈmælɡəmeɪtɪd] amalgamated subcategory 融和子范畴amalgamation 合并[əˌmælɡəˈmeɪʃn]ambient space 环绕空间[ˈæmbiənt]ambiguous point 歧点amicable numbers 亲和数amount 量amphicheiral knot 双向纽结ample divisor 丰富除子amplitude 振幅;角amplitude of a complex number 复数角analog computer 模拟计算机analogous 类似的analogue display 相似表示analogue method 相似法analogy 类似analysis 数学分析analysis of time series 时间序列分析analysis of variance 方差分析[ˈveəriəns] analytic 分析的analytic arc 解析弧analytic completion 解析开拓analytic continuation 解析开拓analytic curve 解析曲线analytic dynamics 分析动力学analytic expression 解析式analytic function 分析函数analytic function of several variables 多元解析函数analytic geometry 分析几何学analytic index 解析指数analytic manifold 解析廖analytic method 解析法analytic prime number theory 解析素数论analytic proof 解析证明analytic proposition 解析命题analytic set 解析集analytic space 解析空间analytic transformation 解析变换analytical differential 解析微分analytical geometry 分析几何学analytical hierarchy 解析分层[ˈhaɪərɑːki] analytical mapping 全纯映射analytical transformation 全纯映射analytically continuable 可解析开拓的analytically independent 解析无关的analytically irreducible variety 解析不可约簇analytically representable function 解析可表示的函数anastigmatic 去象散的anchor ring 环面ancillary statistic 辅助统计量[ænˈsɪləri]and circuit 与电路andre permutation 安得列置换[ˌpɜːmjuˈteɪʃn] andre polynomial 安得列多项式anger function 安格尔函数angle 角angle at center 圆心角angle between chord and tangent 弦和切线的角angle function 角函数angle of 落后角angle of advance 超前角angle of attack 迎角angle of contact 接触角angle of contingence 切线角angle of declination 俯角angle of diffraction 衍射角[dəˈfrækʃən]angle of incidence 入射角angle of inclination 斜角angle of intersection 相交角angle of lead 超前角angle of reflection 反射角angle of refraction 折射角angle of rotation 旋转角angle of torsion 挠率角[ˈtɔːʃn]angle preserving 保角的angle preserving map 保角映象angular 角的angular acceleration 角加速度angular coefficient 角系数angular coordinates 角坐标angular correlation 角相关angular derivative 角微商[dɪˈrɪvətɪv]angular dispersion 角色散angular displacement 角位移angular distance 角距angular distribution 角分布angular domain 角域angular excess 角盈angular frequency 角频率angular magnification 角放大率angular measure 角测度angular metric 角度量angular momentum 角动量[məˈmentəm] angular momentum conservation law 角动量守恒律angular motion 角运动angular neighborhood 角邻域angular transformation 角变换angular unit 角的单位angular velocity 角速度anharmonic oscillation 非低振动[ˌɒsɪˈleɪʃn] anharmonic ratio 交比anisotropic 各向异性的anisotropic body 蛤异性体anisotropy 蛤异性annihilator 零化子[əˈnaɪəleɪtə]annular 环[ˈænjələ(r)]annulator 零化子annulus 圆环anomalous magnetic moment 反常磁矩[əˈnɒmələs] anomalous propagation 反常传播anomalous scattering 反常散射anomaly 近点角antecedent 前项anti automorphism 反自同构anti hermitian form 反埃尔米特形式anti isomorphic lattice 反同构格[ˈlætɪs]anti isomorphism 反同构anti position 反位置anti reflexiveness 反自反性anti semiinvariant 反半不变量antianalytic function 反解析函数antichain 反链anticlockwise 逆时针的anticlockwise revolution 逆时针回转anticlockwise rotation 逆时针回转anticoincidence 反重合anticoincidence method 反重法anticommutation 反交换anticommutative 反交换的anticommutativity 反交换性anticommutator 反换位子antiderivative 不定积分的antiholomorphic 反全纯的antihomomorphism 反同态antiisomorphy 反同构antilinear 反线性的antilinear mapping 反线性映射antilinear transformation 反线性变换antilogarithm 反对数antimode 反方式antimodule 反模antinode 反结点antinomy 二律背反antiorder homomorphism operator 反序同态算子antiordered set 反有序集合antiparallel 逆平行的antiplane 反平面antipodal map 对映映射antipodal point 对映点antipodal set 对映集antipode 对映点antipoints 反点antiradical 反根式antiresonance 反共振antisymmetric 反对称的antisymmetric function 反对称函数antisymmetric matrix 反对称矩阵antisymmetric relation 反对称关系antisymmetric tensor 反对称张量antisymmetrical state 反对称态antithesis 反题antitone mapping 反序映射antitone sequence 反序列antitonic function 反序函数antitonicity 反序性antitony 反序性antitrigonometric function 反三角函数antiunitary 反酉的aperiodic damping 非周期衰减aperiodic motion 非周期运动aperiodicity 非周期性aperture ratio 口径比apex 顶点apex angle 顶角apical angle 顶角apolar 非极性的apolarity 从配极性apothem 边心距apparent error 貌似误差apparent force 表观力apparent motion 表观运动apparent orbit 表观轨迹apparent singularity 貌似奇性application 应用applied mathematics 应用数学applied mechanics 应用力学approach 接近approach infinity 接近无穷大approximability 可逼近性approximable 可逼近的approximate 近似的;使近似approximate calculation 近似计算approximate construction 近似准approximate continuity 近似连续性approximate derivative 近似导数[dɪˈrɪvətɪv] approximate differentiability 近似可微性approximate differential 近似微分approximate formula 近似公式approximate integration 近似积分approximate limit 近似极限approximate lower semi continuity 近似下半连续性approximate method 近似法approximate number 近似数approximate partial derivative 近似偏导函数approximate partial derived function 近似偏导函数approximate partial differential 近似偏微分approximate solution 近似解approximate total differentiability 近似全可微性approximate total differential 近似全微分approximate upper semi continuity 近似上半连续性approximate value 近似值approximate value in excess 过剩近似值approximately equal 近似等于approximately semicontinuous 近似半连续的approximation 逼近approximation by excess 过剩逼近approximation calculus 近似计算approximation error 近似误差approximation function 逼近函数approximation in excess 过剩逼近approximation method 近似法approximation methods in physics 物理学中的逼近法approximation theorem 逼近定理approximation theory 逼近理论arbitrarily small 任意小[ˈɑːbɪtrərɪli]arbitrary 任意的arbitrary constant 任意常数arbitrary element 任意元素arbitrary parameter 任意参数arbitrary small number 任意小数arc 弧arc component 弧分量arc cosecant 反余割arc cosine 逆余弦arc cotangent 逆余切arc hyperbolic function 反双曲函数[ˌhaɪpəˈbɒlɪk] arc length 弧长arc of a circle 圆弧arc secant 反正割arc set 弧集arc sine 逆正弦arc sine distribution 反正弦分布arc sine law 反正弦定律arc sine transformation 反正弦变换arc tangent 反正切arch 拱形archaeometry 考古测量学archimedean 阿基米德性的archimedean group 阿基米德群archimedean semigroup 阿基米德半群archimedean space 阿基米德空间archimedean total order 阿基米德全序archimedean valuation 阿基米德赋值archimedes axiom 阿基米德公理[ˈæksiəm] archimedes spiral 阿基米德螺线archimedically ordered field 阿基米德有序域archimedically ordered number field 阿基米德有序数域arcwise connected set 弧连通集arcwise connected space 弧连通空间arcwise connectedness 弧连通性are 公亩area 面积area function 面积函数area of a circle 圆面积area preserving mapping 保面积映射areal coordinates 重心坐标areal derivative 面积导数areal element 面积元素areal integral 面积分areal velocity 面积速度[vəˈlɒsəti]argand plane 复数平面argument 自变数;辐角argument function 辐角函数argument of a function 函数的自变数argument principle 辐角原理argumentation 论证aristotelian logic 亚里斯多德逻辑学arithmetic 算术[əˈrɪθmətɪk]arithmetic al function 数论函数arithmetic difference 算术差arithmetic division 算术除法arithmetic element 运算元素arithmetic expression 算术表达式arithmetic function 数论函数arithmetic genus 算术狂arithmetic geometric mean 算术几何平均arithmetic geometric series 算术几何级数arithmetic logic of pocket calculator 袖珍计算机的算术逻辑arithmetic mean 算术平均arithmetic number 正实数arithmetic of algebraic number fields 代数数域的数论arithmetic of algebras 代数的数论arithmetic of local fields 局部域的数沦arithmetic operation 算术操作算术运算arithmetic progression 算术级数arithmetic subgroup 算术子群arithmetic unit 运算元素arithmetical hierarchy 算术谱系[ˌærɪθˈmetɪkl] arithmetical predicate 算术谓词arithmetical triangle 帕斯卡三角形arithmetics 算术arithmetization 算术化arithmometer 四则计算机arrangement 排列array 排列arrow 射artificial variable 人工变量artificial variable method 人工变量法artin conductor 阿廷前导子artin conjecture 阿廷猜想[kənˈdʒektʃə(r)]artin reciprocity law 阿廷互反禄[ˌresɪˈprɒsəti] n. 互惠;互助;互换artinian module 阿丁模artinian ring 阿丁环ascending 上升的ascending chain condition 升链条件ascending difference 后向差分ascending power series 升幂级数ascending powers 升幂ascending sequence 递升序列aspherical space 非球面空间asphericity 非球面牲assemblage 集[əˈsemblɪdʒ]assembler 汇编assertion sign 断定号assignable cause 可指定的原因assignment problem 配置问题associate equation 相伴方程associated equation 相伴方程associated fiber bundle 相伴的纤维丛associated form 连带形式associated function 连带函数associated graded module 相伴分次模associated graded ring 形式环associated homogeneous equation 相伴齐次方程associated homogeneous system 相伴齐次组associated laguerre polynomial 连带的拉盖尔多项式associated legendre function 相伴勒让德函数associated legendre polynomial 连带的勒让德多项式associated minimal surface 相伴极小曲面associated power series 相伴幂级数associated prime ideal 相伴素理想associated radius of convergence 相伴收敛半径associated space 相伴空间associated spherical harmonic 相伴球面低associated surface 连带曲面associated system 相伴系associated undirected graph 相伴无向图association 结合associative algebra 结合代数associative law 结合律associative law for series 级数的结合律associativity 结合性assume 假定assumption 假定assumption formula 假定公式asterisk 星号asteroid 星形线asymmetric 非对称的asymmetric relation 非对称关系asymmetric variety 非对称簇asymmetrical 非对称的[ˌæsɪˈmɛtrɪkəl] asymmetrical graph 恒等图asymmetry 非对称性asymptote 渐近线asymptote of curve 曲线的渐近线asymptotic 浙近的asymptotic behavior 渐近状态asymptotic circle 渐近圆asymptotic cone 渐近锥面asymptotic convergence 渐近收敛asymptotic curvature 渐近曲率asymptotic curve 渐近曲线asymptotic density 渐近密度asymptotic direction 渐近方向asymptotic efficiency 渐近效率asymptotic expansion 渐近展开asymptotic formula 渐近公式asymptotic line 渐近线asymptotic mean value 渐近平均值asymptotic minimal basis 渐近极小基asymptotic order 渐近阶asymptotic path 渐近路线asymptotic plane 渐近平面asymptotic point 渐近点asymptotic rate of convergence 渐近收敛速度asymptotic series 渐近级数asymptotic solution 渐近解asymptotic stability 渐近稳定性asymptotic surface 渐近曲面asymptotic unbiased estimator 渐近无偏估计量asymptotic value 渐近值asymptotically efficient estimator 渐近有效估计量asymptotically equal 渐近相等asymptotically equal sequence 渐近相等序列asymptotically equivalent function 渐近等价函数asymptotically normal distribution 渐近正态分布asymptotically normally distributed 渐近正规分布的asymptotically stable 渐近稳定的asymptotically stable solution 渐近稳定解asynchronous computer 异步计算机atiyah singer index theorem 阿蒂亚辛格指数定理atlas 坐标邻域系[ˈætləs] n. 地图册;地图集atom 原子atomic element 原子元素atomic formula 原子公式atomic lattice 原子格[ˈlætɪs]atomic proposition 原子命题atomicity 原子性attaching map 接着映射attenuation 衰减[əˌtenjʊˈeɪʃn]attenuation constant 衰减常数attenuation factor 衰减因数attenuator 衰减器attraction 引力attractive force 引力attractor 吸引区attribute 属性augend 被加数？augmentation 扩张[ˌɔːgmɛnˈteɪʃ(ə)n] augmentation preserving map 增广保存映射augmented complex 扩张复形augmented matrix 增广矩阵austausch 交换autocorrelation 自相关autocorrelation coefficient 自相关系数autocorrelation function 自相关函数autocorrelogram 自相关图autocovariance 自协方差autocovariance function 自协方差函数autodistributivity 自分配性automata 自动机[ɔːˈtɒmətə] automatic check 自动检验automatic coding 自动编码automatic computation 自动计算automatic computer 自动计算机automatic control 自动控制automatic control system 自动控制系统automatic control theory 自动控制理论automatic programming 自动程序设计automatic testing 自动检验automation 自动化automaton 自动机automaton graph 自动机图automorphic form 自守形式automorphic function 自守函数automorphism 自同构automorphism group 自同构群autonomous system 自治系统autoparallel curve 自平行曲线autoregression 自回归autoregression equation 自回归方程autoregressive process 自回归过程autoregressive transformation 自回归变换auxiliary 辅助的[ɔːɡˈzɪliəri]auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 相伴齐次方程auxiliary function 辅助函数auxiliary line 辅助线auxiliary variable 辅助变数average 平均值average deviation 平均偏差average error 平均误差average life 平均寿命average outgoing quality 平均出厂质量average outgoing quality limit 平均出厂质量极限average quality protection 平均品质保护average sample number 平均样本数average speed 平均速度average term 普通项average time 平均时间average value 平均值averaging 取平均数averaging method 平均法averaging operator 平均算子axes of coordinates 座标轴axial 轴的axial symmetry 轴对称[ˈsɪmətri]axial vector 轴向量axially symmetric flow 轴对称流axialsymmetric vector field 轴对称向量场axiom 公理axiom of accessibility 可达性公理axiom of addition 加法公理axiom of choice 选择公理axiom of completeness 完备性公理axiom of comprehension 概括公理axiom of constructibility 可构成性公理axiom of constructivity 可构成性公理axiom of continuity 连续公理axiom of extensionality 外延性公理axiom of infinity 无穷性公理axiom of power set 幂集公理axiom of reducibility 可化归性公理axiom of regularity 正则性公理axiom of subsets 子集公理axiom of substitution 替换公理axiom of sum set 并集公理axiom of the empty set 空集公理axiom of union 并集公理axiom scheme 公理格式[skiːm]axiomatic 公理的axiomatic method 公理法axiomatic set theory 公理论的集论;公理集合论axiomatic system 公理系统axiomatics 公理学axiomatization 公理化axiomatize 公理化axioms of congruence 叠合公理[ˈkɒŋɡruəns] axioms of continuity 连续公理axioms of denumerability 可数公理axioms of incidence 关联公理axioms of order 次序公理axis 轴axis of a cone 锥轴axis of abscissas 横坐标轴axis of absolute convergence 绝对收敛轴axis of affinity 仿射轴axis of convergence 收敛轴axis of coordinate 坐标轴axis of curvature 曲率轴axis of ordinates 纵坐标轴axis of projection 射影轴axis of reals 实轴axis of revolution 回转轴axis of rotation 回转轴axis of symmetry 对称轴axisymmetric 轴对称的axonometric perspective 轴测投影法axonometric projection 轴测投影法axonometry 轴测法azimuth 方位角azimuthal 方位角的Bb measurability b可测性b measurable function 波莱尔可测函数babylonian numerals 巴比伦数字back substitution 逆计算backward difference 后向差分backward difference operator 后向差分算子backward difference quotient 后向差商backward solution 后向解法baire function 贝利函数baire measure 贝利测度baire set 贝利集baire space 贝利空间baire theorem 贝利定理balance 平衡balanced category 平衡范畴balanced functor 平衡函子balanced hypergraph 平衡超图balanced neighborhood 平衡邻域balanced sample 平衡样本balanced set 平衡集balancing method 平衡法balayage 扫除ball 球ballistic curve 弹道banach algebra 巴拿赫代数banach lie group 巴拿赫李群banach space 巴拿赫空间band 带band chart 带状图band matrix 带状矩阵bar construction 棒构成bar diagram 条线图bar graph 条线图barrel 桶集barrel shape 桶型barrelled space 桶型空间barrier 闸barycenter 重心barycenter of a simplex 单形的重心barycentric 重心的barycentric complex 重心复形barycentric coordinates 重心坐标barycentric mapping 重心映射barycentric subdivision 重心重分base 底base angle 底角base line 底线base number 底数base of logarithms 对数的底base point 基点base register 基址寄存器变址寄存器base space 底空间base vector 基向量basic 基础的basic block 基本块basic field 基域basic form 基本形式basic point 基础点basic representation 基本表示basic ring 基环basic solution 基本解basic symbol 基本符号basic variable 基本变量basis 基basis for cohomology 上同爹basis for homology 同爹basis of linear space 线性空间的基basis of vector space 向量空间的基basis replacement procedure 基替换过程basis theorem of hilbert 希耳伯特基定理。

GTM研究生参考书

GTM(Graduate Texts of Mathematics)丛书目录1 Introduction to Axiomatic Set Theory, Gaisi Takeuti, W. M. Zaring2 Measure and Category, John C. Oxtoby (1997, ISBN 978-0-387-90508-2)3 Topological Vector Spaces, . Schaefer, . Wolff (1999, ISBN 978-0-387-98726-2)4 A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, ISBN 978-0-387-94823-2)5 Categories for the Working Mathematician, Saunders Mac Lane (1998, ISBN 978-0-387-98403-2)6 Projective Planes, Hughes, Piper7 A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7)8 Axiomatic Set Theory, Gaisi Takeuti, Zaring9 Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7)10 A Course in Simple-Homotopy Theory, M. M. Cohen11 Functions of One Complex Variable I, John B. Conway (1995, ISBN 978-0-387-90328-6)12 Advanced Mathematical Analysis, R. Beals (1973, ISBN 978-0-387-90065-0)13 Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, ISBN978-0-387-97845-1)14 Stable Mappings and Their Singularities, Golubitsky, Guillemin15 Lectures in Functional Analysis and Operator Theory, S. K. Berberian16 The Structure of Fields, D. Winter17 Random Processes, M. Rosenblatt18 Measure Theory, Paul R. Halmos (1974, ISBN 978-0-387-90088-9)19 A Hilbert Space Problem Book, Paul R. Halmos (1982, ISBN 978-0-387-90685-0)20 Fibre Bundles, Dale Husemoller (1994, ISBN 978-0-387-94087-8)21 Linear Algebraic Groups, James E. Humphreys (1998, ISBN 978-0-387-90108-4)22 An Algebraic Introduction to Mathematical Logic, Barnes, Mack23 Linear Algebra, Werner H. Greub (1981, ISBN 978-0-387-90110-7)24 Geometric Functional Analysis and Its Applications, Holmes25 Real and Abstract Analysis, Edwin Hewitt, Karl Stromberg (1975, ISBN 978-0-387-90138-1)26 Algebraic Theories, Manes27 General Topology, John L. Kelley (1975, ISBN 978-0-387-90125-1)28 Commutative Algebra I, Oscar Zariski, Pierre Samuel, Cohen (1975, ISBN 978-0-387-90089-6)29 Commutative Algebra II, Oscar Zariski, Pierre Samuel30 Lectures in Abstract Algebra I, Nathan Jacobson31 Lectures in Abstract Algebra II, Nathan Jacobson32 Lectures in Abstract Algebra III, Nathan Jacobson33 Differential Topology, Morris W. Hirsch34 Principles of Random Walk, Frank Spitzer35 Several Complex Variables and Banach Algebras, Herbert Alexander, John Wermer36 Linear Topological Spaces, John L. Kelley, Isaac Namioka37 Mathematical Logic, J. Donald Monk38 Several Complex Variables, Grauert, Fritzsche39 An Invitation to C * -Algebras, William Arveson40 Denumerable Markov Chains, John George Kemeny, Snell, Knapp et al.41 Modular Functions and Dirichlet Series in Number Theory, Tom M. Apostol42 Linear Representations of Finite Groups, Jean-Pierre Serre, Scott43 Rings of Continuous Functions, Gillman, Jerison44 Elementary Algebraic Geometry, K. Kendig45 Probability Theory I, M. Loève46 Probability Theory II, M. Loève47 Geometric Topology in Dimensions 2 and 3, Edwin E. Moise48 General Relativity for Mathematicians, . Sachs, H. Wu49 Linear Geometry, . Gruenberg, . Weir50 Fermat's Last Theorem, Harold M. Edwards51 A Course in Differential Geometry, Klingenberg52 Algebraic Geometry, Robin Hartshorne53 A Course in Mathematical Logic, Yu. I. Manin54 Combinatorics with Emphasis on the Theory of Graphs, Graver, Watkins55 Introduction to Operator Theory I, Brown, Pearcy56 Algebraic Topology: An Introduction, William S. Massey57 Introduction to Knot Theory, Richard H. Crowell, Ralph H. Fox (1977, ISBN 978-0-387-90272-2)58 P-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz59 Cyclotomic Fields, Serge Lang60 Mathematical Methods of Classical Mechanics, V. I. Arnold61 Elements of Homotopy Theory, George W. Whitehead62 Fundamentals of the Theory of Groups, Kargapolov, Merzljakov, Burns63 Graph Theory, Béla Bollobás64 Fourier Series I, Edwards65 Differential Analysis on Complex Manifolds, . Wells Jr.66 Introduction to Affine Group Schemes, . Waterhouse67 Local Fields, Jean-Pierre Serre, Greenberg68 Linear Operators on Hilbert Spaces, Weidmann, Szuecs69 Cyclotomic Fields II, Serge Lang70 Singular Homology Theory, William S. Massey71 Riemann Surfaces, Herschel Farkas, Irwin Kra72 Classical Topology and Combinatorial Group Theory, John Stillwell73 Algebra, Thomas W. Hungerford74 Multiplicative Number Theory, Harold Davenport, Hugh Montgomery75 Basic Theory of Algebraic Groups and Lie Algebras, G. P. Hochschild76 Algebraic Geometry, Iitaka77 Lectures on the Theory of Algebraic Numbers, Hecke, Brauer, Goldman et al.78 A Course in Universal Algebra, Burris, Sankappanavar (Online [1])79 An Introduction to Ergodic Theory, Peter Walters80 A Course in the Theory of Groups, Derek . Robinson81 Lectures on Riemann Surfaces, Forster, Gilligan82 Differential Forms in Algebraic Topology, Raoul Bott, Loring Tu83 Introduction to Cyclotomic Fields, Lawrence C. Washington84 A Classical Introduction to Modern Number Theory, Ireland, Rosen (1995, ISBN 0-387-97329-X)85 Fourier Series A Modern Introduction, R. E. Edwards86 Introduction to Coding Theory, . van Lint (3rd ed 1998, ISBN 3-540-64133-5)87 Cohomology of Groups, Kenneth S. Brown88 Associative Algebras, . Pierce89 Introduction to Algebraic and Abelian Functions, Serge Lang90 An Introduction to Convex Polytopes, Arne Brondsted91 The Geometry of Discrete Groups, Alan F. Beardon92 Sequences and Series in Banach Spaces, J. Diestel93 Modern Geometry - Methods and Applications I, Dubrovin, Fomenko, Novikov et al.94 Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner95 Probability, Shiryaev, Boas96 A Course in Functional Analysis, John B. Conway97 Introduction to Elliptic Curves and Modular Forms, Neal Koblitz98 Representations of Compact Lie Groups, Broecker, Dieck99 Finite Reflection Groups, Grove, Benson100 Harmonic Analysis on Semigroups, Berg, Christensen, Ressel101 Galois Theory, Harold M. Edwards102 Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan103 Complex Analysis, Serge Lang104 Modern Geometry - Methods and Applications II, Dubrovin, Fomenko, Novikov et al.105 SL2®, Serge Lang106 The Arithmetic of Elliptic Curves, Joseph H. Silverman107 Applications of Lie Groups to Differential Equations, Peter J. Olver108 Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range109 Univalent Functions and Teichmüller Spaces, O. Lehto110 Algebraic Number Theory, Serge Lang111 Elliptic Curves, Dale Husemöller112 Elliptic Functions, Serge Lang113 Brownian Motion and Stochastic Calculus, Ioannis Karatzas, Steven Shreve114 A Course in Number Theory and Cryptography, Neal Koblitz115 Differential Geometry, Berger, Gostiaux, Levy116 Measure and Integral, John L. Kelley, Srinivasan117 Algebraic Groups and Class Fields, Jean-Pierre Serre118 Analysis Now, Gert K. Pedersen119 An Introduction to Algebraic Topology, Joseph J. Rotman120 Weakly Differentiable Functions, William P. Ziemer121 Cyclotomic Fields I-II, Serge Lang, Karl Rubin122 Theory of Complex Functions, Remmert, Burckel123 Numbers, Lamotke, Ewing, Ebbinghaus et al.124 Modern Geometry - Methods and Applications III, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Petrovich Novikov et al. (1990, ISBN 978-0-387-97271-8)125 Complex Variables, Berenstein, Gay126 Linear Algebraic Groups, Armand Borel127 A Basic Course in Algebraic Topology, William S. Massey128 Partial Differential Equations, Jeffrey Rauch129 Representation Theory, William Fulton, Joe Harris130 Tensor Geometry, Dodson, T. Poston131 A First Course in Noncommutative Rings, . Lam132 Iteration of Rational Functions, Alan F. Beardon133 Algebraic Geometry, Joe Harris134 Coding and Information Theory, Steven Roman135 Advanced Linear Algebra, Steven Roman136 Algebra, Adkins, Weintraub137 Harmonic Function Theory, Axler, Bourdon, Ramey138 A Course in Computational Algebraic Number Theory, Henri Cohen (1996, ISBN 0-387-55640-0) 139 Topology and Geometry, Glen E. Bredon140 Optima and Equilibria, Jean-Pierre Aubin141 Gröbn er Bases, Becker, Weispfenning, Kredel142 Real and Functional Analysis, Serge Lang, (1993, ISBN 14)143 Measure Theory, . Doob144 Noncommutative Algebra, Farb, Dennis145 Homology Theory, James W. Vick146 Computability, Douglas S. Bridges147 Algebraic K-Theory and Its Applications, Jonathan Rosenberg148 An Introduction to the Theory of Groups, Joseph J. Rotman149 Foundations of Hyperbolic Manifolds, John G. Ratcliffe150 Commutative Algebra with a View Toward Algebraic Geometry, David Eisenbud 151 Advanced Topics in the Arithmetic of Elliptic Curves, Joseph H. Silverman 152 Lectures on Polytopes, Günter M. Ziegler153 Algebraic Topology, William Fulton154 An Introduction to Analysis, Brown, Pearcy155 Quantum Groups, Christian Kassel156 Classical Descriptive Set Theory, Alexander S. Kechris157 Integration and Probability, Malliavin, Airault, Kay et al.158 Field Theory, Steven Roman159 Functions of One Complex Variable II, John B. Conway160 Differential and Riemannian Manifolds, Serge Lang161 Polynomials and Polynomial Inequalities, Borwein, Erdelyi162 Groups and Representations, Alperin, Bell163 Permutation Groups, Dixon, Mortimer164 Additive Number Theory I, Melvyn B. Nathanson165 Additive Number Theory II, Melvyn B. Nathanson166 Differential Geometry, . Sharpe, Shiing-Shen Chern167 Field and Galois Theory, Patrick Morandi168 Combinatorial Convexity and Algebraic Geometry, Guenter Ewald169 Matrix Analysis, Rajendra Bhatia170 Sheaf Theory, Glen E. Bredon171 Riemannian Geometry, Peter Petersen172 Classical Topics in Complex Function Theory, Remmert, Kay173 Graph Theory, Reinhard Diestel174 Foundations of Real and Abstract Analysis, Douglas S. Bridges175 An Introduction to Knot Theory, W. B. Raymond Lickorish176 Riemannian Manifolds, John M. Lee177 Analytic Number Theory , Donald J. Newman178 Nonsmooth Analysis and Control Theory, Clarke, Ledyaev, Stern et. al179 Banach Algebra Techniques in Operator Theory, Ronald G. Douglas180 A Course on Borel Sets, . Srivastava181 Numerical Analysis, Rainer Kress182 Ordinary Differential Equations, Walter, Thompson183 An Introduction to Banach Space Theory, Robert E. Megginson184 Modern Graph Theory, Béla Bollobás185 Using Algebraic Geometry, Cox, Little, O Shea186 Fourier Analysis on Number Fields, Ramakrishnan, Valenza187 Moduli of Curves, Harris, Morrison188 Lectures on the Hyperreals, Robert Goldblatt189 Lectures on Modules and Rings, Tsit-Yuen Lam190 Problems in Algebraic Number Theory, Esmonde, Murty191 Fundamentals of Differential Geometry, Serge Lang192 Elements of Functional Analysis Hirsch, Lacombe, Levy193 Advanced Topics in Computational Number Theory, Henri Cohen (2000, ISBN 0-387-98727-4) 194 One-Parameter Semigroups for Linear Evolution Equations, Engel, Nagel195 Elementary Methods in Number Theory, Melvyn B. Nathanson196 Basic Homological Algebra, M. Scott Osborne197 The Geometry of Schemes, Eisenbud, Harris198 A Course in p-adic Analysis, Alain Robert199 Theory of Bergman Spaces, Hedenmalm, Korenblum, Zhu200 An Introduction to Riemann-Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen 201 Diophantine Geometry, Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4)202 Introduction to Topological Manifolds, John M. Lee203 The Symmetric Group, Bruce E. Sagan204 Galois Theory, Jean-Pierre Escofier205 Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-0)206 Problems in Analytic Number Theory, M. Ram Murty (2001, ISBN 978-0-387-95143-0)207 Algebraic Graph Theory, Godsil, Royle (2001, ISBN 978-0-387-95241-3)208 Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6)209 A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7)210 Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9)211 Algebra, Serge Lang212 Lectures on Discrete Geometry, Jiri Matousek213 From Holomorphic Functions to Complex Manifolds, Fritzsche, Grauert214 Partial Differential Equations, Juergen Jost215 Algebraic Functions and Projective Curves, David Goldschmidt216 Matrices, Denis Serre217 Model Theory: An Introduction, David Marker218 Introduction to Smooth Manifolds, John M. Lee (2003, ISBN 978-0-387-95448-6)219 The Arithmetic of Hyperbolic 3-Manifolds, Maclachlan, Reid220 Smooth Manifolds and Observables, Jet Nestruev221 Convex Polytopes, Branko Grünbaum (2003, ISBN 0-387-00424-6)222 Lie Groups, Lie Algebras, and Representations, Brian C. Hall223 Fourier Analysis and its Applications, Anders Vretblad224 Metric Structures in Differential Geometry, Walschap, G., (2004, ISBN 978-0-387-20430-7) 225 Lie Groups, Daniel Bump, (2004, ISBN 978-0-387-21154-1)226 Spaces of Holomorphic Functions in the Unit Ball, Zhu, K., (2005, ISBN 978-0-387-22036-9) 227 Combinatorial Commutative Algebra, Ezra Miller, Bernd Sturmfels, (2005, ISBN978-0-387-22356-8)228 A First Course in Modular Forms, Fred Diamond, J. Shurman, (2006, ISBN 978-0-387-23229-4) 229 The Geometry of Syzygies, David Eisenbud (2005, ISBN 978-0-387-22215-8)230 An Introduction to Markov Processes, Stroock, ., (2005, ISBN 978-3-540-23499-9)231 Combinatorics of Coxeter Groups, Anders Björner, Francisco Brenti, (2005, ISBN978-3-540-44238-7)232 An Introduction to Number Theory, Everest, G., Ward, T., (2005, ISBN 978-)233 Topics in Banach Space Theory, Albiac, F., Kalton, ., (2006, ISBN 978-0-387-28141-4)234 Analysis and Probability · Wavelets, Signals, Fractals, Jorgensen, (2006, ISBN978-0-387-29519-0)235 Compact Lie Groups, M. R. Sepanski, (2007, ISBN 978-0-387-30263-8)236 Bounded Analytic Functions, Garnett, J., (2007, ISBN 978-0-387-33621-3)237 An Introduction to Operators on the Hardy-Hilbert Space, Martinez-Avendano, ., Rosenthal, P., (2007, ISBN 978-0-387-35418-7)238 A Course in Enumeration, Aigner, M., (2007, ISBN 978-3-540-39032-9)239 Number Theory - Volume I: Elementary and Algebraic Methods for Diophantine Equations, Cohen, H., (2007, ISBN 978-0-387-49922-2)240 Number Theory - Volume II: Analytic and Modern Tools, Cohen, H., (2007, ISBN978-0-387-49893-5)241 The Arithmetic of Dynamical Systems, Joseph H. Silverman, (2007, ISBN 978-0-387-69903-5) 242 Abstract Algebra, Grillet, Pierre Antoine, (2007, ISBN 978-0-387-71567-4)243 Topological Methods in Group Theory, Geoghegan, Ross, (2007, ISBN 978-0-387-74611-1)244 Graph Theory, Bondy, ., Murty, (2007, ISBN 978-)245 Complex Analysis: Introduced in the Spirit of Lipman Bers,Gilman, Jane P., Kra, Irwin, Rodriguez, Rubi E. (2007, ISBN 978-0-387-74714-9)246 A Course in Commutative Banach Algebras, Kaniuth, Eberhard, (2020, ISBN 978-0-387-72475-1) 247 Braid Groups, Kassel, Christian, Turaev, Vladimir, (2008, ISBN 978-0-387-33841-5)248 Buildings Theory and Applications, Abramenko, Peter, Brown, Ken (2008, ISBN978-0-387-78834-0)249 Classical Fourier Analysis, Grafakos, Loukas, (2008, ISBN 978-0-387-09431-1)250 Modern Fourier Analysis, Grafakos, Loukas, (2008, ISBN 978-0-387-09433-5)251 The Finite Simple Groups, Wilson, Robert, Parker, Christopher W., (2009, ISBN 978-)252 Distributions and Operators, Grubb, Gerd, (2009, ISBN 978-0-387-84894-5)。

abel范畴的可裂短正和列 -回复

abel范畴的可裂短正和列-回复Abelian Category and Split Short Exact SequencesIn mathematics, an abelian category is a category that satisfies a set of axioms inspired by the study of abelian groups. These axioms capture the essence of linearity and additivity, which are fundamental concepts in algebraic structures.One important concept in abelian categories is the notion of a short exact sequence. A short exact sequence is a sequence of morphisms between objects in the category, such that the images of consecutive morphisms form an exact sequence. This means that the composition of any two consecutive morphisms gives the zero morphism, and the kernel of each morphism is equal to the image of the previous morphism.To understand split short exact sequences in the context of abelian categories, it is important to first understand what it means for a sequence to be split. A split sequence is a sequence in which there exist morphisms that can "undo" the effects of the original morphisms. In other words, the original sequence can be decomposed into sub-sequences that are isomorphic to eachother.In the context of abelian categories, a short exact sequence is said to be split if there exists a morphism that can "undo" the effect of the original morphisms, resulting in a sequence that is isomorphic to the original sequence. This split morphism is called a splitting morphism, and it is usually denoted by a dashed arrow, indicating that it is a retraction of the original morphisms.Let's take a concrete example to illustrate this concept. Consider the category of abelian groups, where the objects are abelian groups and the morphisms are group homomorphisms. In this category, a short exact sequence is a sequence of group homomorphisms of the form:0 →A →B →C →0where 0 denotes the trivial group. This sequence is exact if the image of the first morphism A →B is equal to the kernel of the second morphism B →C, and the image of the second morphism B →C is equal to the kernel of the third morphism C →0.Now, let's suppose that the short exact sequence above is split. This means that there exists a morphism C →B that "undoes" the effect of the morphism B →C, resulting in a sequence that is isomorphic to the original sequence:0 →A →B →C →0^v0 →A →B →C →0In other words, we have a splitting morphism C →B that satisfies the following conditions:1. The composition of the morphisms B →C and C →B is equal to the identity morphism on C.2. The composition of the morphisms C →B and B →C is equal to the identity morphism on B.This splitting morphism allows us to decompose the short exactsequence into two sub-sequences:0 →A →B →C →0and0 →C →B →A →0where the first sub-sequence is isomorphic to the original sequence, and the second sub-sequence is isomorphic to the reversed sequence.Split short exact sequences are important in the study of abelian categories because they provide a way to decompose complex structures into simpler ones, making them easier to analyze and manipulate. They also have important applications in algebraic topology, algebraic geometry, and representation theory.In conclusion, split short exact sequences play a fundamental role in the study of abelian categories. They allow us to decompose complex structures into simpler ones, providing insights and toolsfor further analysis. By understanding the concept of split short exact sequences, mathematicians are able to explore and uncover the deep connections between different areas of mathematics.。

faltings

The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles
Gerd Faltings
T
he proof of the conjecture mentioned in the title was finally completed in September of 1994. A. Wiles announced this result in the summer of 1993; however, there was a gap in his work. The paper of Taylor and Wiles does not close this gap but circumvents it. This article is an adaptation of several talks that I have given on this topic and is by no means about my own work. I have tried to present the basic ideas to a wider mathematical audience, and in the process I have skipped over certain details, which are in my opinion not so much of interest to the nonspecialist. The specialists can then alleviate their boredom by finding those mistakes and correcting them.
Gerd Faltings is affiliated with the Max-Planck-Institute für Mathematik in Bonn, Germany. Translated from Testausdruck DMV Mitteilungen 27. März 1995 für 2/95 by Uwe F. Mayer, University of Utah.

克隆巴赫系数的英文

克隆巴赫系数的英文English:The Bernstein–Sato polynomial of a germ of a complex analytic function at a point is an object of central importance in singularity theory and complex analysis. It encodes crucial information about the singular behavior of the function near the given point. The Bernstein–Sato polynomial is intimately related to the Bernstein–Sato ideal, which is a fundamental concept in the study of D-modules and p-adic differential equations. The study of Bernstein–Sato polynomials and ideals has deep connections to various areas of mathematics, including algebraic geometry, representation theory, and harmonic analysis. These polynomials have been extensively studied in the context of local zeta functions, where they play a crucial role in understanding the structure of singularities and in the computation of zeta functions associated with singularities. In recent years, there has been significant progress in understanding the properties and applications of Bernstein–Sato polynomials, leading to new insights into the nature of singularities and their interactions with other mathematical objects.中文翻译:克隆巴赫多项式是复解析函数在某一点处的一个基本概念，它在奇点理论和复分析中具有重要的地位。

信息与计算科学专业英语训练

专业英语训练学校：学院：理学院专业：信息与计算科学姓名：学号：日期：2012-6-19一、专业英语词汇abelian algebra 阿贝耳代数abelian differential 阿贝耳微分abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分abelian summation 阿贝耳求和法abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号abscissa 横坐标abscissa of absolute convergence 绝对收敛坐标abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute 绝对形absolute address 绝对地址absolute class field 绝对类域absolute coding 绝对编码absolute cohomology 绝对上同调absolute conic 绝对二次曲线absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率absolute geometry 绝对几何absolute homology group 绝对同岛absolute homotopy group 绝对同伦群absolute inequality 绝对不等式absolute instability 绝对不稳定性absolute maximum 绝对极大值absolute minimum 绝对极小值absolute moment 绝对矩absolute neighborhood 绝对邻域absolute neighborhood retract 绝对邻域收缩核absolute norm 绝对范数absolute number 不名数absolute parallelism 绝对平行性absolute quadric 绝对二次曲面absolute ramification index 绝对分歧指数absolute rotation 绝对旋转absolute singular homology group 绝对奇异同岛absolute space 绝对空间absolute space time 绝对时空absolute stability 绝对稳定性absolute term 常数项absolute unit 绝对单位absolute value 绝对值absolute value sign 绝对值符号absolute velocity 绝对速度absolutely additive measure 完全加性测度absolutely compact set 绝对紧集absolutely complete system 绝对完备系absolutely continuous 绝对连续的absolutely continuous distri but ion 绝对连续分布absolutely continuous function 绝对连续函数absolutely continuous measure 绝对连续测度absolutely continuous part 绝对连续部分absolutely continuous transformation 绝对连续变换absolutely convergent 绝对收敛的absolutely convergent integral 绝对收敛积分absolutely convergent series 绝对收敛级数absolutely convex hull 绝对凸包absolutely discontinuous function 绝对不连续函数absolutely integrable 绝对可积的absolutely irreducible character 绝对不可约特征absolutely irreducible representation 绝对不可约表示absolutely irreducible variety 绝对不可约簇absolutely normal number 绝对范数absolutely prime ideal 绝对素理想absolutely semisimple algebra 绝对半单代数absolutely simple group 绝对单群absolutely summable sequence 绝对可和序列absolutely unbiased estimator 绝对无偏估计量absolutely unramified extension 绝对非分歧扩张absorbing barrier 吸收障碍absorbing medium 吸收媒体absorbing set 吸收集absorbing state 吸收态absorption 吸收absorption coefficient 吸收系数absorption curve 吸收曲线absorption factor 吸收因数absorption index 吸收指数absorption law 吸收律absorption probability 吸收概率abstract 抽象的abstract algebra 抽象代数abstract algebraic geometry 抽象代数几何abstract automaton 抽象自动机abstract category 抽象范畴abstract code 理想码abstract complex 抽象复形abstract group 抽象群abstract interval function 抽象区间函数abstract mathematics 抽象数学abstract number 不名数abstract ordered simplicial complex 抽象有序单纯复形abstract simplex 抽象单形abstract simplicial subcomplex 抽象单子复形abstract space 抽象空间abstraction 抽象abstraction operator 抽象算子absurd 谬论的absurdity 谬论abundance ratio 丰度比abundant number 过剩数accelerated motion 加速运动acceleration 加速度acceleration of convergence 收敛性的加速acceleration of gravity 重力加速度acceptable quality level 合格质量水平acceptance 肯定acceptance inspection 接受检查acceptance limit 接受界限acceptance line 接受线acceptance number 接受数acceptance probability 接受概率acceptance region 接受区域acceptance zone 接受带access 存取access speed 存取速度access time 存取时间accessibility 可达性accessible boundary point 可达边界点accessible ordinal number 可达序数accessible point 可达点accessible set 可达集accessible vertex 可达顶点accessory extremal 配连极值accidental 偶然的accidental coincidence 偶然符合accidental error 随机误差accommodation 第accumulated error 累积误差accumulating point 聚点accumulation 累积accumulation point 聚点accumulator 累加器存储器accuracy 准确性accuracy grade 准确度accuracy of measurement 测量精确度accuracy rating 准确度acnod 孤点acount 计算action integral 酌积分action variable 酌变量active restriction 有效限制actual 真实的actual infinity 实无穷acute 尖锐的acute angle 锐角acute angled triangle 锐角三角形acute triangle 锐角三角形acuteness 锐度acyclic 非循环的acyclic complex 非循环复形acyclic graph 环道自由图acyclic model theorem 非循环模型定理ad infinitum 无穷地adams circle 阿达姆斯圆adams extrapolation method 阿达姆斯外插法adaptability 适应性adaptation 适应adapted basis 适应基add 加added circuit 加法电路addend 加数adder 加法器addition 加法addition formulas 加法公式addition sign 加号addition system 加法系addition table 加法表addition theorem 加法定理addition theorem of probability 概率的加法定理additional 加法的additional code 附加代码additional condition 附加条件additional error 附加误差additive 加法的additive category 加性范畴additive class 加性类additive functional 加性泛函数additive functional transformation 加性泛函变换additive group 加法群additive interval function 加性区间函数additive inverse element 加性逆元素additive number theory 堆垒数论additive operator 加性算子additive process 加性过程additive relation 加性关系additive separable 加法可分的additive theory of numbers 堆垒数论additive valuation 加法赋值additively commutative ordinal numbers 加性交换序数additivity 加法性address 地址address part 地址部分address register 地址寄存器addressing 指定箱位adele 阿代尔adele group 阿代尔群adequate 适合的adherent point 触点adhesion 附着adjacency 邻接adjacency matrix 邻接矩阵adjacent angles 邻角adjacent edge 邻棱adjacent side 邻边adjacent supplementary angles 邻角adjacent vertex 邻顶adjoint boundary value problem 伴随边值问题adjoint determinant 伴随行列式adjoint difference equation 伴随差分方程adjoint differential equation 伴随微分方程adjoint differential expression 伴随微分式adjoint form 伴随形式adjoint function 伴随函数adjoint functor 伴随函子adjoint graph 导出图adjoint group 伴随群adjoint hilbert problem 伴随希耳伯特问题adjoint integral equation 伴随积分方程adjoint kernel 伴随核adjoint linear map 伴随线性映射adjoint matrix 伴随阵adjoint operator 伴随算子adjoint process 伴随过程adjoint representation 伴随表示adjoint space 伴随空间adjoint surface 伴随曲面adjoint system 伴随系adjoint system of differential equations 微分方程的伴随系adjoint transformation 伴随算子adjoint vector 伴随向量adjunct 代数余子式adjunction 附加adjunction of an identity element 单位元的附加adjustable point 可胆点adjustment 蝶admissibility limit 容许界限admissible 容许的admissible category 容许范畴admissible chart 容许图admissible control 可行控制admissible decision function 容许判决函数admissible decision rule 容许判决函数admissible deformation 容许形变admissible domain 容许区域admissible function 容许函数admissible homomorphism 容许同态admissible hypothesis 容许假设admissible lifting 容许提升admissible map 容许映射admissible sequence 容许序列admissible space 容许空间admissible strategy 容许策略admissible subgroup 容许子群admissible test 容许检定admissible value 容许值affine algebraic set 仿射代数集affine collineation 仿射直射变换affine connection 仿射联络affine coordinates 平行坐标affine curvature 仿射曲率affine differential geometry 仿射微分几何学affine distance 仿射距离affine figure 仿射图形affine function 仿射函数affine geometry 仿射几何学affine group 仿射群affine group scheme 仿射群概型affine isothermal net 仿射等温网affine length 仿射长度affine line 仿射直线affine normal 仿射法线affine parameter 仿射参数affine principal curvature 仿射助率affine rational transformation 仿射有理变换affine space 仿射空间affine sphere 仿射球面affine surface 仿射曲面affine transformation 仿射变换affine variety 仿射簇affinely connected manifold 仿射连通廖affinely connected space 仿射连通空间affinity 仿射变换affirmation 肯定affirmative proposition 肯定命题affix 附标after effect 后效酌aggregate 集aggregation 聚合agreement 一致air coordinates 空间坐标airy function 亚里函数airy integral 亚里积分aitken interpolation 艾特肯插值aitken interpolation formula 艾特肯插值公式albanese variety 阿尔巴内斯簇aleph 阿列夫aleph zero 阿列夫零alexander cohomology module 亚历山大上同担alexander cohomology theory 亚历山大上同帝alexander matrix 亚历山大阵alexander polynomial 亚历山大多项式algebra 代数学algebra of events 事件场algebra of logic 逻辑代数algebra of tensors 张量代数algebra over k 环k上的代数algebraic 代数的algebraic adjunction 代数的附加algebraic affine variety 仿射代数集algebraic algebra 代数的代数algebraic branch point 代数分歧点algebraic calculus 代数计算algebraic closure 代数闭包algebraic closure operator 代数闭包算子algebraic complement 代数余子式algebraic cone 代数锥algebraic correspondence 代数对应algebraic curve 代数曲线algebraic equation 代数方程algebraic expression 代数式algebraic extension 代数扩张algebraic form 代数形式algebraic fraction 代数分式algebraic function 代数函数algebraic function field 代数函数域algebraic geometry 代数几何学algebraic group 代数群algebraic hull 代数包algebraic hypersurface of the seconed order 二阶代数超曲面algebraic integer 代数整数algebraic irrational number 代数无理数algebraic lie algebra 代数的李代数algebraic logic of pocket calculator 袖珍计算机的代数逻辑algebraic multiplicity 代数重度algebraic number 代数数algebraic number field 代数数域algebraic number theory 代数数论algebraic operation 代数运算algebraic polynomial 代数多项式algebraic singularity 代数奇点algebraic space 代数空间algebraic spiral 代数螺线algebraic structure 代数结构algebraic sum 代数和algebraic surface 代数曲面algebraic system 代数系algebraic variety 代数簇algebraically closed field 代数闭域algebraically dependent elements 代数相关元algebraically equivalent 代数等价的algebraically independent elements 代数无关元algebraization 代数化algebro geometric 代数几何的algebroid function 代数体函数algebroidal function 代数体函数algorithm 算法algorithm of division 辗转相除法algorithm of euclid 欧几里得算法algorithm theory 算法论algorithmic language 算法语言algorithmization 算法化aligned systematic sampling 列系统抽样alignment chart 列线图aliquot part 整除部分alligation 混合法allocation problem 配置问题allowable 容许的allowable defects 容许靠allowable error 容许误差allowance 允许almost all 几乎处处almost bounded function 殆有界函数almost certain convergence 几乎必然收敛almost complex manifold 殆复廖almost convergent sequence 殆收敛序列almost equivalent 殆等价almost everywhere 几乎处处almost impossible event 殆不可能事件almost invariant set 殆不变集almost periodic function 殆周期函数almost periodicity 殆周期性almost significant 殆显著的alpha capacity 容量alpha limit set 极限集alphabetical 字母的alphanumeric 字母数字式alphanumeric representation of information 信息的字母数字表示alternate angles 错角alternating chain 交错链alternating differential form 外微分形式alternating differential of differential form 微分形式的交错微分alternating direction method 交替方向法alternating form 交错形式alternating function 反对称函数alternating group 交错群alternating harmonic series 莱布尼兹级数alternating knot 交错纽结alternating matrix 交错矩阵alternating method 交错法alternating product 外积alternating sequence 交错序列alternating series 交错级数alternating series test 交错级数检验alternating sum 交错和alternating tensor 交错张量analytic proof 解析证明analytic proposition 解析命题analytic set 解析集analytic space 解析空间analytic transformation 解析变换analytical differential 解析微分analytical geometry 分析几何学analytical hierarchy 解析分层analytical mapping 全纯映射analytical transformation 全纯映射analytically continuable 可解析开拓的analytically independent 解析无关的analytically irreducible variety 解析不可约簇analytically representable function 解析可表示的函数anastigmatic 去象散的anchor ring 环面ancillary statistic 辅助统计量and circuit 与电路andre permutation 安得列置换andre polynomial 安得列多项式anger function 安格尔函数angle 角angle at center 圆心角angle between chord and tangent 弦和切线的角angle function 角函数angle of 落后角angle of advance 超前角angle of attack 迎角angle of contact 接触角angle of contingence 切线角angle of declination 俯角angle of diffraction 衍射角angle of incidence 入射角angle of inclination 斜角angle of intersection 相交角angle of lead 超前角angle of reflection 反射角angle of refraction 折射角angle of rotation 旋转角angle of torsion 挠率角angle preserving 保角的angle preserving map 保角映象angular 角的angular acceleration 角加速度angular coefficient 角系数angular coordinates 角坐标angular correlation 角相关angular derivative 角微商angular dispersion 角色散angular displacement 角位移angular distance 角距angular distri but ion 角分布angular domain 角域angular excess 角盈angular frequency 角频率angular magnification 角放大率angular measure 角测度angular metric 角度量angular momentum 角动量angular momentum conservation law 角动量守恒律angular motion 角运动angular neighborhood 角邻域angular transformation 角变换angular unit 角的单位angular velocity 角速度anharmonic oscillation 非低振动anharmonic ratio 交比anisotropic 蛤异性的anisotropic body 蛤异性体anisotropy 蛤异性annihilator 零化子annular 环annulator 零化子annulus 圆环anomalous magnetic moment 反常磁矩anomalous propagation 反常传播anomalous scattering 反常散射anomaly 近点角antecedent 前项anti automorphism 反自同构anti hermitian form 反埃尔米特形式anti isomorphic lattice 反同构格anti isomorphism 反同构anti position 反位置anti reflexiveness 反自反性anti semiinvariant 反半不变量antianalytic function 反解析函数antichain 反链anticlockwise 逆时针的anticlockwise revolution 逆时针回转anticlockwise rotation 逆时针回转anticoincidence 反重合anticoincidence method 反重法anticommutation 反交换anticommutative 反交换的anticommutativity 反交换性anticommutator 反换位子antiderivative 不定积分的antiholomorphic 反全纯的antihomomorphism 反同态antiisomorphy 反同构antilinear 反线性的antilinear mapping 反线性映射antilinear transformation 反线性变换antilogarithm 反对数antimode 反方式antimodule 反模antinode 反结点antinomy 二律背反antiorder homomorphism operator 反序同态算子antiordered set 反有序集合antiparallel 逆平行的antiplane 反平面antipodal map 对映映射antipodal point 对映点antipodal set 对映集antipode 对映点antipoints 反点antiradical 反根式antiresonance 反共振antisymmetric 反对称的antisymmetric function 反对称函数antisymmetric matrix 反对称矩阵antisymmetric relation 反对称关系antisymmetric tensor 反对称张量antisymmetrical state 反对称态antithesis 反题antitone mapping 反序映射antitone sequence 反序列antitonic function 反序函数antitonicity 反序性antitony 反序性antitrigonometric function 反三角函数antiunitary 反酉的aperiodic damping 非周期衰减aperiodic motion 非周期运动aperiodicity 非周期性aperture ratio 口径比apex 顶点apex angle 顶角apical angle 顶角apolar 非极性的apolarity 从配极性apothem 边心距apparent error 貌似误差apparent force 表观力apparent motion 表观运动apparent orbit 表观轨迹apparent singularity 貌似奇性application 应用applied mathematics 应用数学applied mechanics 应用力学discontinuity on the left 左方不连续性discontinuity on the right 右方不连续性discontinuous function 不连续函数discontinuous group 不连续群discontinuous random variable 不连续变量discontinuous set 不连续集discontinuous term 不连续项discontinuous variate 不连续变量discontinuum 密断统discount 折扣discount factor 折扣因子discrete 分立的discrete category 离散范畴discrete continuous system 离散连续系统discrete distri but ion 离散分布discrete distri but ion function 离散分布函数discrete flow 离散流discrete fourier transform 离散傅里叶变换discrete group 离散群discrete mathematics 离散数学discrete optimization 离散最佳化discrete optimization problem 离散最优化问题discrete problem 离散问题discrete process 离散随机过程discrete programming 离散规划discrete random variable 离散随机变量discrete series 离散序列discrete set 离散集discrete spectrum 离散谱discrete state 离散状态discrete system 离散系统discrete time 离散时间discrete topological space 离散拓扑空间discrete topology 离散拓扑discrete uniform distri but ion 离散均匀分布discrete valuation 离散赋值discreteness 离散性discretization 离散化discretization error 离散化误差discrimator 判别式函数discriminant 判别式discriminant analysis 判别分析discriminant function 判别式函数discriminant of a polynomial 多项式的判别式discriminatory analysis 判别分析disjoint elements 不相交元素disjoint relations 不相交关系disjoint sets 不相交集disjoint sum 不相交并集disjoint union 不相交并集disjointed set 不相交集disjunction 析取disjunction sign 析取记号disjunction symbol 析取记号disjunctive normal form 析取范式disjunctive proposition 选言命题disk 圆盘disorder 无秩序disorder order transformation 无序有序变化dispersion 方差dispersion matrix 方差矩阵dispersion relations 分散关系dispersive 扩散的displacement 位移displacement operator 位移算符display statusconcomitant 相伴式disposition 配置disproportion 不相称disproportionate 不成比例的dissection 剖分dissimilar terms 不同类项dissipation 散逸dissipation of energy 消能dissipative function 散逸函数dissipative measurable transformation 散逸可测变换dissipative system 耗散系dissociation 解离dissociation constant 分离常数distance axioms 距离公理distance between two points 两点间距distance circle 距离圆distance function 距离函数distance matrix 距离矩阵distance meter 测距仪distance point 距离点distinction 差别distinguish 辨别distinguished polynomial 特异多项式distortion 畸变distortion angle 畸变角distortion theorem 畸变定理distortionless 无畸变的distri but ed constant 分布常数distri but ed parameter 分布参数distri but ion 分布distri but ion coefficient 分布系数distri but ion curve 分布曲线distri but ion family 分布族distri but ion function 分布函数distri but ion law 分布律distri but ion of prime numbers 素数分布distri but ion parameter 分布参数distri but ion ratio 分布系数distri but ion rule 分布规则distri but ion space 广义函数空间distri but ion with negative skewness 负偏斜分布distri but ion with positive skewness 正偏double cusp 双尖点double element 二重元素double exponential distri but ion 二重指数分布double folium 双叶线double fourier series 二重傅里叶级数double integral 二重积分double laplace transformation 二重拉普拉斯变换double layer 双层double layer potential 双层位势double limit 二重极限double line 二重线double loop 双环路double negation 双重否定double orthogonal system 二重正交系double periodicity 双周期性double plane 二重面double point 重点double point of curve 曲线的二重点double poisson distri but ion 二重泊松分布double product 二重积double ratio 交比double root 重根double sequence 二重数列double series 二重级数double subscript 双下标double sum 二重和double tangent 二重切线double valued function 双值函数double vector product 二重向量积doubly periodic function 双周期函数dozen 一打draw 拉drum 磁鼓dual abelian variety 对偶阿贝耳簇dual automorphism 逆自同构dual base 对偶基dual basis 对偶基dual category 对偶范畴dual cell 对偶胞腔dual complex 对偶复形dual cone 对偶锥dual curve 对偶曲线dual figure 对偶图dual form 对偶形式dual formula 对偶公式dual graph 对偶图dual group 特贞群dual ideal 对偶理想dual isomorphism 对偶同构dual lattice 对偶格dual mapping 对偶映射dual module 对偶模dual number 对偶数dual operation 对偶运算dual operator 对偶算子dual problem 对偶问题dual relation 对偶关系dual representation 对偶表示dual simplex method 对偶单形法dual spaces 对偶空间dual system 对偶系统dual theorem 对偶定理dual vector space 对偶向量空间duality 对偶性duality principle 对偶原理duality relation 对偶关系duality theorem 对偶定理duel 竞赛dummy index 哑指标duodecimal notation 十二进记数法duodecimal system 十二进制duodecimal system of numbers 十二进数系duplication formula 倍角公式duplication of the cube 倍立方duration 持久时间dyad 并向量dyadic expansion 二进展开dyadic product 并向量积dyadic rational 二进有理数dynamic optimization 动态最优化dynamic programming 动态规划dynamic store 动态存储器dynamic system 动力系统dynamical variables 动态变数dynamics 力学dynkin diagram 丹金图形eccentric angle 离心角eccentric angle of an ellipse 椭圆的离心角eccentric anomaly 离心角eccentricity 离心率eccentricity of a hyperbola 双曲线的离心率echelon matrix 梯阵econometrics 计量经济学eddy 涡流旋涡edge 边edge connectivity 边连通度edge homomorphism 边缘同态edge of a solid 立体棱edge of regression 回归边缘edge of the wedge theorem 楔的边定理editcyclic markov chain 循环马尔可夫链effective area 有效面积effective convergence 有效收敛effective cross section 有效截面effective differential cross section 有效微分截面effective divisor 非负除数effective interest rate 有效利率effective number of replications 有效重复数effective variance of error 有效误差方差effectively computable function 能行可计算函数efficiency 效率efficiency factor 效率因子efficient estimator 有效估计量efficient point 有效点egyptian numerals 埃及数字eigenelement 特摘素eigenfunction 特寨数eigenspace 本照间eigenvalue 矩阵的特盏eigenvalue problem 特盏问题eigenvector 特镇量eigenvector of linear operator 线性算子的特镇量einstein equation 爱因斯坦方程einstein metric 爱因斯坦度量elastic coefficient 弹性常数elastic constant 弹性常数elastic deformation 弹性变形elastic limit 弹性限度elastic modulus 弹性模数elastic scattering 弹性散射elasticity 弹性elastodynamics 弹性动力学electrodynamics 电动力学electromagnetism 电磁electronic computer 电子计算机electronic data processing machine 电子数据处理机electronics 电子学electrostatics 静电学element 元件element of area 面积元素element of best approximation 最佳逼近元素element of finite order 有限阶元素element of surface 面元素elementary 基本的elementary chain 初等链elementary circuit 基本回路elementary conjunction 基本合取式elementary divisor 初等因子elementary divisor theorem 初等因子定理elementary event 简单事件elementary formula 原始公式elementary function 初等函数elementary geometry 初等几何elementary matrix 初等阵elementary number theory 初等数论elementary operation 初等运算elementary path 有向通路elementary set 初等集elementary subdivision 初等重分elementary symmetric function 初等对称函数equipotent set 等势集equipotential 等势的equipotential line 等位线equipotential surface 等位面equivalence 等价equivalence class 等价类equivalence problem 等价问题equivalence relation 等价关系equivalent 等价的equivalent equation 等价方程equivalent fiber bundle in g g等价纤维丛equivalent form 等价形式equivalent functions 等价函数equivalent knot 等价纽结equivalent mapping 保面积映射equivalent matrix 等价阵equivalent metric 等价度量equivalent neighborhood system 等价邻域系equivalent norm 等价范数equivalent point 等价点equivalent proposition 等值命题equivalent states 等价状态equivalent stochastic process 等价随机过程equivalent terms 等价项equivalent transformation 初等运算equivariant map 等变化映射erasing 擦除ergodic chain 遍历马尔可夫链ergodic hypothesis 遍历假说ergodic markov chain 遍历马尔可夫链ergodic property 遍历性ergodic state 遍历态ergodic theorem 遍历定理ergodic theorem in the mean 平均遍历定理ergodic theory 遍历理论ergodic transformation 遍历变换ergodicity 遍历性error 误差error analysis 误差分析error band 误差范围error coefficient 误差系数error component 误差分量error curve 误差曲线error equation 误差方程error estimation 误差估计error function 误差函数error in the input data 输入数据误差error law 误差律error limit 误差界限error mean square 误差方差error model 误差模型error of estimation 估计误差error of first kind 第一类误差error of measurement 测量误差error of observation 观测误差error of reading 读数误差error of second kind 第二类误差error of the third kind 第三类误差error of truncation 舍位误差error originated from input 输入误差error probability 误差概率error sum of squares 误差平方和error variance 误差方差escribe 旁切escribed 旁切的propositional inference 命题推演propositional logic 命题逻辑propositional variable 命题变元protractor 量角器分度规provability 可证迷provable 可证媚;可证的provable formula 可证公式prove 证明论证proximity 附近proximity relation 邻近关系proximity space 邻近空间pseudo abelian category 伪阿贝耳范畴pseudo euclidean space 伪欧几里得空间pseudo random number 伪随机数pseudo tensor 伪张量pseudoanalytic function 伪解析函数pseudoarc 伪弧pseudobasis 伪基pseudocatenary 伪悬链线pseudocatenoid 伪悬链曲面pseudocode 伪码pseudocoherent sheaf 伪凝聚层pseudocompact 伪紧的pseudocomplement 伪余pseudocomplemented lattice 伪有余格pseudocomplex manifold 伪复廖pseudoconformal geometry 伪保角几何pseudoconformal mapping 伪保角映射pseudoconvex 伪凸的pseudoconvex domain 伪凸域pseudoconvex function 伪凸函数pseudoconvexity 伪凸性pseudocycle 伪循环pseudocycloid 伪旋轮线pseudodifferential operator 伪微分算子pseudodistance 伪距离pseudoelliptic function 伪椭圆函数pseudoelliptic integral 伪椭圆积分pseudofunctor 伪函子pseudogroup of transformations 伪变换群pseudoharmonic function 伪低函数pseudoinverse operator 伪逆算子pseudomanifold 伪廖pseudomeasure 伪测度pseudometric 伪度量pseudometric space 伪度量空间pseudometric uniformity 伪度量一致性pseudometrizable uniform space 伪可度量化的一致空间pseudonorm 伪范数二、英文文献原文Math problems about BaseballAbstract：Baseball is a popular bat-and-ball game involving both athletics and wisdom. There are strict restrictions on the material, size and manufacture of the bat. It is vital important to transfer the maximum energy to the ball in order to give it the fastest batted speed during the hitting process.Firstly, this paper locates the center-of-percussion (COP) and the viberational node based on the single pendulum theory and the analysis of bat vibration. With the help of the synthesizing optimization approach, a mathematical model is developed to execute the optimized positioning for the “sweet spot”, and the best hitting spot turns out not to be at the end of the bat. Secondly, based on the basic model hypothesis, taking the physical and material attributes of the bat as parameters, the moment of inertia and the highest batted ball speed (BBS) of the “sweet spot” are evaluated using different parameter values, which enables a quantified comparison to be made on the performance of different bats. Thus finally expl ained why Major League Baseball prohibits “corking” and metal bats.In problem I, taking the COP and the viberational node as two decisive factors of the “sweet zone”, models are developed respectively to study the hitting effect from the angle of energy conversion.Because the different “sweet spots” decided by COP and the viberational node reflect different form of energy conversion, the “space-distance” concept is introduced and the “Technique for Order Preferenceby Similarity to Ideal Solution (TOPSIS) is used to locate the “sweet zone” step by step.And thus, it is proved that the “sweet spot” is not at the end of the bat from the two angles of specific quantitative relationship of the hitting effects and the inference of energy conversion. In problem II, applying new physical parameters of a corked bat into the model developed in Problem I, the moment of inertia and the BBS of the corked bat and the original wood bat under the same conditions are calculated. The result shows that the corking bat reduces the BBS and the collision performance rather than enhancing the “sweet spot” effect. On the other hand, the corking bat reduces the moment of inertia of the bat, which makes the bat can be controlled easier. By comparing the two Team # 8038 Page 2 of 20 conflicting impacts comprehensively, the conclusion is drawn that the corked bat will be advantageous to the same player in the game, for which Major League Baseball prohibits “corking”.In problem III, adopting the similar method used in Problem II, that is, applying different physical parameters into the model developed in Problem I, calculate the moment of inertia and the BBS of the bats constructed by different material to analyze the impact of the bat material on the hitting effect. The data simulation of metal bats performance and wood bats performance shows that the performance of the metal bat is improved for the moment of inertia is reduced and the BBS is increased. Our model and method successfully explain why Major League Baseball, for the sake of fair competition, prohibits metal bats.In the end, an evaluation of the model developed in this paper is given, listing itsadvantages s and limitations, and providing suggestions on measuring the performance of a bat.Restatement of the ProblemExpl ain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit.Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect.Augment your model to confirm or deny this effect.Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?2.1 Analysis of Problem IFirst explain the “sweet spot” on a baseball bat, and then develop a model that helps explain why this spot isn’t at the end of the bat.[1]There are a multitude of definitions of the sweet spot:1) the location which produces least vibrational sensation (sting) in the batter's hands2) the location which produces maximum batted ball speed3) the location where maximum energy is transferred to the ball4) the location where coefficient of restitution is maximum5) the center of percussionFor most bats all of these "sweet spots" are at different locations on the bat, so one is often forced to define the sweet spot as a region.If explained based on torque, this “sweet spot” might be at the end of the bat, which is known to be empirically incorrect.This paper is going to explain this empirical paradox by exploring the location of the sweet spot from a reasonable angle.Based on necessary analysis, it can be known that the sweet zone, which is decided by the center-of-percussion (COP) and the vibrational node, produces the hitting effect abiding by the law of energy conversion.The two different sweet spots respectively decided by the COP and the viberational node reflect different energy conversions, which forms a two-factor influence.2.2 Analysis of Problem IIProblem II is to explain wheth er “corking” a bat enhances the “sweet spot” effect and why Major League Baseball prohibits “corking”.[4]In order to find out what changes will occur after corking the bat, the changes of the bat’s parameters should be analyzed first:1) The mass of the corked bat reduces slightly than before;2) Less mass (lower moment of inertia) means faster swing speed;3) The mass center of the bat moves towards the handle;4) The coefficient of restitution of the bat becomes smaller than before;。

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a r X i v :m a t h /9806077v 1 [m a t h .C V ] 13 J u n 1998Abelian functions satisfy an Algebraic Addition
Theorem
Mark B.Villarino Escuela de Matem´a tica,Universidad de Costa Rica,2060San Jos´e ,Costa Rica April 16,1998Abstract We prove that n independent abelian functions admit an algebraic addition theorem,with no appeal to theta functions.A famous and classical theorem of Weierstrass states:Theorem 1(Weierstrass,1869).Any n analytically independent Abelian functions in n complex variables that belong to the same 2n -dimensional period lattice satisfy an algebraic addition theorem.All published proofs of this theorem depend on the Weierstrass–Riemann “Theta-satz”which aﬃrms that any abelian function is globally representable as the quotient of two (generalized)theta functions [2].We show in this note that the theorem has nothing to do with theta functions per se ,but is a simple (indeed,almost trivial)consequence of another,even more famous,theorem of Weierstrass [4],Thimm [3]and Siegel [1]on algebraic dependence.Theorem 2(Weierstrass–Thimm–Siegel).Any n +1meromorphic functions on a compact complex space of dimension n are algebraically dependent.
Deﬁnition 1.A set of n meromorphic functions of n complex variables,ϕk (u 1,...,u n ),k =1,...,n ,satisfy an algebraic addition theorem if and only if there exist n polynomials G 1,...,G n in 2n +1complex variables with complex coeﬃcients,such that the equations
G k [ϕk (u +v );ϕ1(u ),...,ϕn (u );ϕ1(v ),...,ϕn (v )]=0
for k =1,...,n ,hold identically for all u =(u 1,...,u n )∈C n and v =(v 1,...,v n )∈C n where the ϕi are deﬁned.
Deﬁnition2.The meromorphic functionsϕ1,...,ϕk on the compact complex space X are called algebraically dependent iﬀthere exists a polynomial Q(z1,...,z k)≡0 with complex coeﬃcients such that
Q(ϕ1(u),...,ϕk(u))=0
holds identically for all u∈X where theϕk are deﬁned.
Proof of Theorem1.Let X be the period parallelotope for the latticeΩof periods of the abelian functionsϕ1(u),...,ϕn(u).Then the cartesian product X×X is a compact complex connected manifold of dimension2n.
The functionϕk(u+v)is a meromorphic function of2n variables(u1,...,u n, v1,...,v n)on X×X,as are the2n functionsϕk(u)andϕk(v).Therefore,the2n+1 meromorphic functionsϕk(u+v);ϕ1(u),...,ϕn(u);ϕ1(v),...,ϕn(v)on the2n-dimensional compact complex space X×X are necessarily algebraically dependent, by Theorem2.This conclusion holds for eachϕk(u+v),k=1,...,n.But,by Deﬁnition1,this means thatϕ1,...,ϕk satisfy an algebraic addition theorem. References
[1]C.L.Siegel,“Meromorphe Funktionen auf kompakten analytischen Mannig-
faltigkeiten”,Nachr.Akad.Wiss.G¨o ttingen4(1955),71–77.
[2]C.L.Siegel,Lectures on Complex Function Theory III,Addison-Wesley,Reading,
MA,1972.
[3]W.Thimm,“Meromorphe Abbildungen von Riemannschen Bereichen”,Math.Z.
60(1954),435–457.
[4]K.Weierstrass,“Untersuchungen¨u ber die2r-fach periodischen Funktionen von r
Ver¨a ndlichen”,J.reine angew.Math.89(1880),1–8.(Collected Works,vol.II, pp.125–133.)。