FROM MAX-PLUS ALGEBRA TO NON-LINEAR PERRON-FROBENIUS THEORY
A-level数学词汇(P1-P4)
Pure mathematics 1 1 Algebraic expression 代数表达式Index laws 指数定律Indices (index的复数形式) 指数Notation 注释Simplify 化简Power 指数Base 底Exponent 指数Expression 表达式Term 项Numerator 分子Expand 展开Possible 可能Fraction 分数Bracket 括号Product 乘积Multiply 乘Collecting like terms 合并同类项Linear 一次的Diagram 图形Rectangle 长方形Square 正方形Length 长度Width 宽Side length 边长Area 面积Shade 阴影Cuboid 长方体Dimension 维Show that 证明V olume 体积Given that 已知Constant 常数Value 值Factorize 因式分解Factor 因子Opposite 相反的Completely 完全地Common factor 公因式Quadratic 二次的Form 形式Real number 实数Positive 正的Negative 负的Include 包含Surd 无理数Add 加Sum 和Take out 提取Difference 差Difference of two squares 平方差Cancel 取消，相互抵消Similarly 同样的Rational 有理的Rational number 有理数Integer 整数Square root 平方根Evaluate 求…的值Substitute 代替Calculator 计算器Square number 平方数Irrational number 无理数Decimal 小数的Expansion 展开式Never-ending 无限的Never repeat 不循环的Exact 准确的Answer 答案Manipulate 操作Denominator 分母Rationalizing denominator 分母有理化Rearrange 调整Prime 质数Work out 计算Hence 然后Fully 完全地State 陈述Solve 解决Equation 方程2 Quadratics 二次方程式Quadratic equation 二次方程Solution 解Real solution 实根Set 设置Root 根Distinct 不同的Repeated root 重根Case 情况Straightforward 简单直接的Symbol 符号Plus 加，正Minus 减，负Factorization 因式分解Shape 形状Section 部分Formula 公式Reading off 读取Coefficient 系数Necessary 必要的Significant figures 有效数字Choose 选择Suitable 适当的Method 方法Trapezium 梯形Height 高Discard 丢弃Completing the square 完全平方（配方）Frequently 经常的Useful 有用的Process 过程Original 最初的Determine 决定Otherwise 另外Function 函数Mathematical 数学上的Relationship 关系Map 映射Set 集合Input 输入Output 输出Single 单一的Notation 符号Represent 代表Domain 定义域Range 值域，范围Member 成员Define 定义Minimum 最小的Occur 发生Explain 解释Consider 考虑Graph 图像Curve 曲线Parabola 抛物线Sketch 画图Identify 确定Key 关键的Feature 特征Overall 整体的Cross 交叉，横过Axis 轴Coordinate 坐标Turning point 转折点（顶点）Maximum 最大的Since 因为Symmetrical 对称的Symmetry 对称性Line of symmetry 对称轴Half-way 位于中途的Explore 探测Technology 技术Plot 绘制Scale 刻度However 但是，不管怎样Smooth 平滑的Relevant 相关的Intercept 截距Label 标记Axes (axis的复数)坐标轴Discriminant 判别式Sign 符号Check 核实Inequality 不等式Calculate 计算Match 匹配Prove 证明Algebra 代数学Diver 跳水运动员Launch 发射Springboard 跳板Meter 米Pool 水池Second 秒Model 模型High 高的Hit 撞击Reach 达到Non-zero 非零3 Equations and inequalities 方程和不等式Simultaneous 联立的Linear simultaneous equations 一次方程组Elimination 消元法Substitution 置换Quadratic simultaneous equations 二次方程组Up to 直到，多达Make sure 确保Correctly 正确地Simplest 最简的Graphically 以图表形式As 因为Satisfy 满足Intersection 相交Simultaneously 同时地Intersect 相交Once 一次Twice 两次Result 结果，导致Produce 产生Graph paper 坐标纸Accurately 准确地Verify 验证Linear inequalities 一次不等式Set notation 集合符号Number line 数轴Overlap 重叠Separately 单独地Illustrate 图解，阐明Quadratic inequalities 二次不等式Corresponding 相应的Critical 临界的Require 要求Describe 描述Interpret 解释Region 区域，范围Coordinate grid 坐标网Dotted line 虚线Solid line 实线Vertex 顶点Vertices (vertex的复数)顶点Within 在内部，之内4 Graphs and transformations 图像和转换Cubic 三次的Cubic function 三次函数Several 几个Depend on 取决于Touch 接触Coordinate axes 坐标轴Indicate 表明，显示Reciprocal 倒数的Reciprocal function 反比例函数Such as 例如Asymptote 渐近线Approach 接近Reach 到达Quadrant 象限Point of intersection 交点Steeper 更陡峭的Eventually 最后，终于Reason 理由，原因Appropriate 恰当的Number 数量Translate 平移Transform 改变Alter 改动Subtract 减Outside 在外面Vertically 竖直地Translation 平移Vector 矢量Horizontally 水平地Direction 方向In terms of 用…来表示Slide 滑动Stretch 伸缩Scale factor 比例系数Double 两倍Halve 减半，对分Inside 在里面Triple 三倍的Reflection 反射(镜面对称) Alternatively 二选一Parallel 平行Lie on 坐落在Pass through 穿过Apply 应用Unfamiliar 陌生的，不熟悉的Specific 特殊的Origin 原点Position 位置Image 像Suggest 提议Mark 标记5 Straight line graphs 直线图像Gradient 斜率Straight line 直线Join 连接Distance 距离Formula 公式Collinear 共线的Intercept 截距Define 定义Either 两者中的任一个Condition 条件Triangle 三角形General equation 一般式Parallel 平行Perpendicular 垂直Whether 是否Quadrilateral 四边形Trapezium 梯形Right angle 直角Congruent 全等的Neither 两者都不Hypotenuse 直角三角形斜边Line segment 线段Scalene 不等边的Respectively 分别地Go through 通过6 Trigonometric ratios 三角比Cosine rule 余弦定理Miss 缺失Version 版本Exchange 交换Standard 标准Prove 证明Opposite 对边Adjacent 邻边Pythagoras’ theorem 勾股定理Letter 字母Round 四舍五入Final 最终的Coastguard 海岸警卫队Station 驻地Bearing 方位Away from 远离Appropriate 适当的Mark 标记Airport 机场Due north 正北Due east 正东Due west 正西Due south 正南Sail 航行Helicopter 直升飞机Tee 球座Flag 旗Particular 特定的Hole 孔，洞Golf course 高尔夫球场Yard 码（1码=3英尺）Tee shot 发球台Land 着陆Largest 最大的Farmer 农场Field 场地Fence 栅栏Cargo 货物Plane 平面Kilometer 千米Sine rule 正弦定理Refer to 涉及Data 数据Remain 剩余Located on 坐落于Zookeeper 动物管理员Enclosure 围场Llama 骆驼Diagonal 对角线Surveyor 检验员Measure 测量Elevation 高程，仰角Apart 相距Assumption 假设Mathematical 数学的Model 模型Obtuse 钝角Acute 锐角Isosceles 等腰的Circle 圆Radius 半径Centre 圆心Least 最小的Instead 代替Crane 吊车Anchored 固定Wreck 破坏Suspend 悬挂Cable 缆绳Rotate 旋转Level 对准Proof 证明Triangular plot 三角图Involve 涉及Trigonometry 三角函数Encounter 遇到Decide 决定Mast 桅杆In order that 为了Interfere 干扰Efficient 有效的Hiker 徒步旅行者Radar 雷达Perimeter 周长Tangent 正切Periodic 周期性的Repeat 重复的Certain 确定的Interval 间距Period 周期Undefined 无意义的Knowledge 知识Periodicity 周期性Verify 证明Variation 变化Rock pool 潮汐潭Midday 中午During 在…期间Non-exact 非精准的Significant figure 有效数字Windmill 风车Sail 帆Tower 塔Deduce 推导Dune 沙丘Realistic 现实的7 Radians 弧度Radian 弧度So far 到目前为止Probably 大概，可能Degree 度Revolution 循环Around 围绕Circle 圆Subtend 朝着Arc 圆弧Circumference 周长Convert 转换Without 没有Multiple 倍数Arc length 弧长Sector 扇形Radius 半径Contain 包含Perimeter 周长Border 边界Pond 池塘Consist 由…组成Edge 边缘Minor arc 劣弧Major arc 优弧Chord 弦Diameter 直径Template 模板Brooch 胸针Ferris wheel 摩天轮Pod 蚕茧，豆荚Estimate 估计Speed 速率Patio 露台Lawn 草坪Design 设计Earring 耳环Nearest 最近点（精确到）Segment 弓形Radii (radius的复数形式) A plot of …的一块Erect 建造Along 沿着Subtract 减Tangent 切线Ratio 比例Bound 关，围入Decimal place 小数Midpoint 中点Semicircular 半圆Drawer 抽屉Handle 把手Difference 差Badge 徽章Equilateral 等边的Railway 铁路Track 轨迹Prism 三棱镜Attempt 尝试Mistake 错误8 Differentiation 微分Gradient 斜率Constantly 不断地Although 然而Comment on 对…评论Copy 抄写，复制Complete 完成Table 表格Hypothesis 假设Derivative 导数Principle 原理Detail 细节Account 解释Originate 起源Formalize 确定，形成Approach 方式，方法Limit 极限Tend to 趋向Gradient function 斜率函数Evaluate 求…的值Fixed value 定值Limiting value 定值Definition 定义One-at-a-time 一次一个Turning point 转折点（顶点）Slope 斜率Disappear 消失Polynomial 多项式Normal 切线First order derivative 一阶导数Second order derivative 二阶导数Rate of change 变化率Respect to 关于Displacement 位移Acceleration 加速度Local 局部的9 Integration 积分Reverse 相反的Differ 不同Integrate 求积分Integral 积分Indefinite 不确定的Indefinite integral 不定积分Elongated 拉长的，伸长的Arrow 箭Fire 射击Castle 城堡Drop off 下降Cliff 悬崖Cyclist 骑行者Pure mathematics 2 1Algebraic methods 代数方法Division 除法Dividing polynomial 多项式除法Finite 有限的Whole number 整数Long division 长除法Quotient 商Remainder 余数Factor theorem 因式定理Remainder theorem 余数定理Logical 逻辑的Structured 有组织的Argument 论据Statement 命题Conjecture 猜想Previously 预先Establish 建立Deduction 推导Desired 想要的Conclusion 结论Odd number 奇数Demonstration 示范，演示Even number 偶数Identical 完全相等的Identity 恒等式Parallelogram 平行四边形Rhombus 菱形Congruent 全等的Exhaustion 穷举法Consecutive 连续的Square number 平方数Break into 拆分Is suited to 适合于Disprove 反驳Counter-example 反例Sufficient 充分的Prime number 质数Divisible 可整除的Either … or…二者择一的Cube number 立方数Hold 有效Claim 宣称Opposite edge 对边Hexagon 六边形Regular hexagon 正六边形Side length 边长Reason 原因2Coordinate geometry in the (x,y) plane 解析几何Bisector 二等分线Perpendicular bisector 中垂线Averaging 求平均值Endpoint 端点Circumcentre 外心Equidistant 等距的Fixed point 定点Vector 向量Property 性质Unique 独一无二的Circumcircle 外接圆3Exponentials and logarithms 指数和对数Exponential 指数的Decrease 减小Increase 增加Smooth 光滑的，平滑的Increasing function 增函数Decreasing function 减函数Justify 证明Logarithms 对数Specific 特定的Button 按钮Typically 典型的Natural logarithms 自然对数Instance 实例Multiplication law 乘法定律Division law 除法定律Power law 指数定律Recognize 识别Attention 注意Condition 条件Complicated 复杂的Whenever 无论何时Convenient 方便的Suppose 假设Notice 注意Particular 特别的4The binomial expansion 二项式展开Binomial 二项式Pascal’s triangle 杨辉三角（帕斯卡三角形）Immediately 直接地Pattern 图案Adjacent 相邻的Investment 投资Interest rate 利率Annum 年，岁Approximation 近似值Ignore 忽略Factorial notation 阶乘Combination 组合Superscript 上标Subscript 下标Probability 可能性Toss 投Likelihood 可能性Ascending powers 升幂Individual 个别的Estimation 估值Engineering 工程学Science 科学Percentage error 百分误差Microchip 微型集成电路片Faulty 有缺点的Chip 芯片Restrict 限制Achieve 达到School fair 学校园游会Prize 奖赏Digit 数字Display 显示5Sequences and series 数列和级数Arithmetic sequence 等差数列Arithmetic progression 等差数列Common difference 公差Arithmetic series 等差级数（等差数列前n 项求和）Exceed 超过Inclusive 包含的Stick 棒子Pentagon 五角形Geometric sequences 等比数列Geometric progression 等比数列Common ratio 公比Converge 收敛Alternating sequence 交错数列Million 百万Geometric series 等比级数（等比数列前n项求和）Sum to infinity 无限项求和Divergent 发散的Convergent 收敛的Recurring 循环的Sigma notation 求和符号Capital 首都，大写字母Signify 表示Recurrence relations 递推关系Previous term 前一项First term 初项Generate 生成，产生Periodic sequence 周期数列Period 周期Salary 薪水Profit 利润Predict 预言Annual 年度的Business 商业Financial 金融的Advisor 顾问Fold 折叠Thickness 厚度Unrealistic 不切实际的Investor 投资人Account 账户Thereafter 以后Deposit 存款，定金Wage 工资Rise 上升Gear 齿轮Successive 连续的Intermediate 中间的Valuable 有价值的Commission 佣金Insurance 保险Policy 政策Prospector 勘探者Drill 钻孔Subsequent 随后的Available 可获得的Payment 报酬Virus 病毒Infect 传染Diagnose 诊断Overfish 过度捕捞Chess 象棋Chessboard 棋盘Sponsored 赞助的Polygon 多边形Appointment 约会，任命6Trigonometric identities and equations 三角恒等式和方程Unit circle 单位圆Anticlockwise 逆时针Quadrant 象限Equivalent 相等的Equilateral triangle 等边三角形Isosceles right-angled triangle 等腰直角三角形Identity 恒等式Reflex 优角(大于180度，在第三、四象限)Principal value 主值Inverse trigonometric function 反三角函数Justification 理由7Differentiation 微分Strictly 严格地Interval 区间Stationary point 驻点Local maximum 局部最大Greatest value 最大值Local minimum 局部最小Least value 最小值Point of inflection 拐点，反曲点Immediate 最接近的Vicinity 邻近，附近Second derivative 二次求导Rate of change 改变的快慢Convex 凸Concave 凹Establish 建立，证实Liter 升Instant 瞬间Tank 水槽Cuboid 长方体的Sheet 薄片Metal 金属Sphere 球体Displacement 位移Cylinder 圆柱体Perimeter 周长Semicircular 半圆的Semicircle 半圆Frame 框架Split 分离，分开Motion 运动Damped 阻尼Spring 弹簧Bent 弯的Biscuit 饼干Tin 罐头Close-fitting 紧贴的Lid 盖子Thin 薄的，瘦的Wastage 损耗Obtain 获得Percentage 百分比Store 储存Capacity 容量Container 容器Calculus 微积分学8Integration 积分Definite integral 定积分Indefinite integral 不定积分Whereas 反之，然而Upper limit 上限Lower limit 下限Square bracket 中括号Magnitude 大小Negligible 可忽略的Straddle 跨坐Unless 除非Complicated 复杂的Trapezium 梯形Trapezium rule 梯形法则Beneath 在…下面Strip 条，带Boundary 边界Adjacent 相邻的Improve 改善Accuracy 精确度Approximation 近似值Underestimate 低估Overestimate 高估Compare 比较Pure mathematics 3 Common multiple 公倍数Improper fraction 假分数Partial fractions 部分分数Degree 次数Modulus function 模函数Absolute value 绝对值Argument 辐角Set notation 集合符号Piecewise-defined function 分段函数Composite function 复合函数Inverse function 反函数Secant 正割Cosecant 余割Cotangent 余切Interval 区间Symmetry 对称性Symmetrical 对称的Chord 弦Inverse trigonometric function 反三角函数Addition formulae 加法公式Compound-angle formulae 复合角公式Double-angle formulae 二倍角公式Round 四舍五入Exponential function 指数函数Natural logarithms 自然对数Trend 趋势Outlier 极值Chain rule 链式法则Product rule 乘法法则Quotient rule 除法法则Continuous 连续的Fixed point iteration 定点迭代Successive 连续的Converge 收敛Staircase diagram 梯形图Cobweb diagram 网状图Diverge 发散Pure mathematics 4 Contradiction 反驳Assert 主张Falsehood 虚假Negation 反论Prime number 质数Split 分解Separate 独立的Parametric equation 参数方程Variable 变量Parameter 参数Revolution 循环Plot 绘图Valid 有效的As long as 只要Condition 条件Accurate 精确的Ascending 上升的Approximation 近似值Implicit differentiation 隐函数微分Explicitly 明确的Implicit 隐含的Rate of change 变化率Hemisphere 半球Cylindrical 圆柱形的Conical 圆锥形的Concave 凹Convex 凸Integrand 被积函数Integration by substitution 换元积分法Integration by part 分部积分法Polynomial 多项式Separating the variables 分离变量General solution 通解Boundary condition 边界条件Directed line segment 有向线段Parallelogram 平行四边形Unit vector 单位向量Column vector 列向量Position vector 位置矢量Scalene 不等边的21Clockwise 顺时针Anticlockwise 逆时针Coplanar 共面的Parallelepiped 平行六面体Trisect 三等分Hexagon 六边形Regular hexagon 正六边形Direction vector 方向向量Anchor 固定Dot product 点乘22。
max-plus algebra 通俗理解
Max-Plus代数是一种数学运算方法，它扩展了矩阵运算的性质，将矩阵中的每个元素视为对应路径的最大权重。
这种方法可以应用于各种领域，如系统控制、生产计划、交通运输等。
在Max-Plus代数中，矩阵中的每个元素表示的是路径的最大权重，因此可以通过对矩阵进行运算来获得不同路径的最大权重。
这种运算方法在处理复杂问题时非常有用，因为它可以将问题分解为较小的子问题，并将这些子问题的解组合起来得到最终的解。
Max-Plus代数还具有线性周期性的性质，即对于一个序列，如果其中任何一个元素被另一个元素替换，那么这个新序列仍然具有与原序列相同的线性周期。
这个性质使得Max-Plus代数在处理一些周期性问题时更加有效。
总之，Max-Plus代数是一种非常有用的数学运算方法，它可以扩展矩阵运算的性质，并应用于各种领域。
通过理解Max-Plus代数的概念和性质，可以更好地理解和应用这种方法来解决复杂问题。
LINEAR AND MULTILINEAR ALGEBRA
2
Eigenvalues and Their Multiplicities
In this section we examine the relationship between the multiplicities of the eigenvalues of the quasi{blocks of a matrix, and the matrix itself. We prove our results for matrices consisting of several quasi{blocks with cut{points. We then give several corollaries in the two{block case. 3
y Work
supported by an NSERC Research Grant
1
1
Introduction
One important direction of research in graph theory is the investigation of graphs with cut{points and their associated matrices. A cut{point is a vertex of a connected graph, deletion of which leads to a disconnected graph. Any matrix associated with such a graph is referred to as a a matrix with a cut{point. Matrices of this kind appear in many applications from combinatorial matrix theory to chemistry. For example, these matrices arise in the investigations of extremal matrices with certain speci ed properties and given sparsity pattern (see e.g. 4], 6]). They are useful in calculations of determinants of matrices based on their digraphs (see e.g. 10]). Cut points appear in construction di erence schemes for the numerical solution of minimax problems using Nelder-Mid-like methods. In addition, it is clear that each tridiagonal matrix can be viewed as a matrix with cut-point(s). Since there are a variety of spectrum preserving methods reducing matrices of different types to tridiagonal form, dealing with submatrices of such matrices will reduce necessary computer memory and CPU operating time. Another important area of investigation with these types of matrices is the description of spectra of graphs (see , 5], 7]) and characteristic vertices of weighted trees (see 8]). Let R 2 Mn(I ) be a complex matrix of the form C 1 0 A b 0 R = B c d c C; A @ 0 b A
Introduction to Linear Algebra
»a = 5 a= 5
A vector is a mathematical quantity that is completely described by its magnitude and direction. An example of a three dimensional column vector might be 4 b= 3 5 uld easily assign bT to another variable c, as follows:
»c = b' c= 4 3 5
A matrix is a rectangular array of scalars, or in some instances, algebraic expressions which evaluate to scalars. Matrices are said to be m by n, where m is the number of rows in the matrix and n is the number of columns. A 3 by 4 matrix is shown here 2 A= 7 5 5 3 2 3 2 0 6 1 3 (3)
»a = 5;
Here we have used the semicolon operator to suppress the echo of the result. Without this semicolon MATLAB would display the result of the assignment:
»A(2,4) ans = 1
The transpose operator “flips” a matrix along its diagonal elements, creating a new matrix with the ith row being equal to the jth column of the original matrix, e.g. T A = 2 5 3 6 7 3 2 1 5 2 0 3
中科院机器学习题库-new
机器学习题库一、 极大似然1、 ML estimation of exponential model (10)A Gaussian distribution is often used to model data on the real line, but is sometimesinappropriate when the data are often close to zero but constrained to be nonnegative. In such cases one can fit an exponential distribution, whose probability density function is given by()1xb p x e b-=Given N observations x i drawn from such a distribution:(a) Write down the likelihood as a function of the scale parameter b.(b) Write down the derivative of the log likelihood.(c) Give a simple expression for the ML estimate for b.2、换成Poisson 分布：()|,0,1,2,...!x e p x y x θθθ-==()()()()()1111log |log log !log log !N Ni i i i N N i i i i l p x x x x N x θθθθθθ======--⎡⎤=--⎢⎥⎣⎦∑∑∑∑3、二、 贝叶斯假设在考试的多项选择中，考生知道正确答案的概率为p ，猜测答案的概率为1-p ，并且假设考生知道正确答案答对题的概率为1，猜中正确答案的概率为1，其中m 为多选项的数目。
数学中英语专业名词
数学中英语专业名词Aabelian group：阿贝尔群；absolute geometry：绝对几何；absolute value：绝对值；abstract algebra：抽象代数；addition：加法；algebra：代数；algebraic closure：代数闭包；algebraic geometry：代数几何；algebraic geometry and analytic geometry：代数几何和解析几何；algebraic numbers：代数数；algorithm：算法；almost all：绝大多数；analytic function：解析函数；analytic geometry：解析几何；and：且；angle：角度；anticommutative：反交换律；antisymmetric relation：反对称关系；antisymmetry：反对称性；approximately equal：约等于；Archimedean field：阿基米德域；Archimedean group：阿基米德群；area：面积；arithmetic：算术；associative algebra：结合代数；associativity：结合律；axiom：公理；axiom of constructibility：可构造公理；axiom of empty set：空集公理；axiom of extensionality：外延公理；axiom of foundation：正则公理；axiom of pairing：对集公理；axiom of regularity：正则公理；axiom of replacement：代换公理；axiom of union：并集公理；axiom schema of separation：分离公理；axiom schema of specification：分离公理；axiomatic set theory：公理集合论；axiomatic system：公理系统；BBaire space：贝利空间；basis：基；Bézout's identity：贝祖恒等式；Bernoulli's inequality：伯努利不等式；Big O notation：大O符号；bilinear operator：双线性算子；binary operation：二元运算；binary predicate：二元谓词；binary relation：二元关系；Boolean algebra：布尔代数；Boolean logic：布尔逻辑；Boolean ring：布尔环；boundary：边界；boundary point：边界点；bounded lattice：有界格；Ccalculus：微积分学；Cantor's diagonal argument：康托尔对角线方法；cardinal number：基数；cardinality：势；cardinality of the continuum：连续统的势；Cartesian coordinate system：直角坐标系；Cartesian product：笛卡尔积；category：范畴；Cauchy sequence：柯西序列；Cauchy-Schwarz inequality：柯西不等式；Ceva's Theorem：塞瓦定理；characteristic：特征；characteristic polynomial：特征多项式；circle：圆；class：类；closed：闭集；closure：封闭性或闭包；closure algebra：闭包代数；combinatorial identities：组合恒等式；commutative group：交换群；commutative ring：交换环；commutativity:：交换律；compact：紧致的；compact set：紧致集合；compact space：紧致空间；complement：补集或补运算；complete lattice：完备格；complete metric space：完备的度量空间；complete space：完备空间；complex manifold：复流形；complex plane：复平面；congruence：同余；congruent：全等；connected space：连通空间；constructible universe：可构造全集；constructions of the real numbers：实数的构造；continued fraction：连分数；continuous：连续；continuum hypothesis：连续统假设；contractible space：可缩空间；convergence space：收敛空间；cosine：余弦；countable：可数；countable set：可数集；cross product：叉积；cycle space：圈空间；cyclic group：循环群；Dde Morgan's laws：德·摩根律；Dedekind completion：戴德金完备性；Dedekind cut：戴德金分割；del：微分算子；dense：稠密；densely ordered：稠密排列；derivative：导数；determinant：行列式；diffeomorphism：可微同构；difference：差；differentiable manifold：可微流形；differential calculus：微分学；dimension：维数；directed graph：有向图；discrete space：离散空间；discriminant：判别式；distance：距离；distributivity：分配律；dividend：被除数；dividing：除；divisibility：整除；division：除法；divisor：除数；dot product：点积；Eeigenvalue：特征值；eigenvector：特征向量；element：元素；elementary algebra：初等代数；empty function：空函数；empty set：空集；empty product：空积；equal：等于；equality：等式或等于；equation：方程；equivalence relation：等价关系；Euclidean geometry：欧几里德几何；Euclidean metric：欧几里德度量；Euclidean space：欧几里德空间；Euler's identity：欧拉恒等式；even number：偶数；event：事件；existential quantifier：存在量词；exponential function：指数函数；exponential identities：指数恒等式；expression：表达式；extended real number line：扩展的实数轴；Ffalse：假；field：域；finite：有限；finite field：有限域；finite set：有限集合；first-countable space：第一可数空间；first order logic：一阶逻辑；foundations of mathematics：数学基础；function：函数；functional analysis：泛函分析；functional predicate：函数谓词；fundamental theorem of algebra：代数基本定理；fraction：分数；Ggauge space：规格空间；general linear group：一般线性群；geometry：几何学；gradient：梯度；graph：图；graph of a relation：关系图；graph theory：图论；greatest element：最大元；group：群；group homomorphism：群同态；HHausdorff space：豪斯多夫空间；hereditarily finite set：遗传有限集合；Heron's formula：海伦公式；Hilbert space：希尔伯特空间；Hilbert's axioms：希尔伯特公理系统；Hodge decomposition：霍奇分解；Hodge Laplacian：霍奇拉普拉斯算子；homeomorphism：同胚；horizontal：水平；hyperbolic function identities：双曲线函数恒等式；hypergeometric function identities：超几何函数恒等式；hyperreal number：超实数；Iidentical：同一的；identity：恒等式；identity element：单位元；identity matrix：单位矩阵；idempotent：幂等；if：若；if and only if：当且仅当；iff：当且仅当；imaginary number：虚数；inclusion：包含；index set：索引集合；indiscrete space：非离散空间；inequality：不等式或不等；inequality of arithmetic and geometric means：平均数不等式；infimum：下确界；infinite series：无穷级数；infinite：无穷大；infinitesimal：无穷小；infinity：无穷大；initial object：初始对象；inner angle：内角；inner product：内积；inner product space：内积空间；integer：整数；integer sequence：整数列；integral：积分；integral domain：整数环；interior：内部；interior algebra：内部代数；interior point：内点；intersection：交集；inverse element：逆元；invertible matrix：可逆矩阵；interval：区间；involution：回旋；irrational number：无理数；isolated point：孤点；isomorphism：同构；JJacobi identity：雅可比恒等式；join：并运算；K格式：Kuratowski closure axioms：Kuratowski 闭包公理；Lleast element：最小元；Lebesgue measure：勒贝格测度；Leibniz's law：莱布尼茨律；Lie algebra：李代数；Lie group：李群；limit：极限；limit point：极限点；line：线；line segment：线段；linear：线性；linear algebra：线性代数；linear operator：线性算子；linear space：线性空间；linear transformation：线性变换；linearity：线性性；list of inequalities：不等式列表；list of linear algebra topics：线性代数相关条目；locally compact space：局部紧致空间；logarithmic identities：对数恒等式；logic：逻辑学；logical positivism：逻辑实证主义；law of cosines：余弦定理；L??wenheim-Skolem theorem：L??wenheim-Skolem 定理；lower limit topology：下限拓扑；Mmagnitude：量；manifold：流形；map：映射；mathematical symbols：数学符号；mathematical analysis：数学分析；mathematical proof：数学证明；mathematics：数学；matrix：矩阵；matrix multiplication：矩阵乘法；meaning：语义； measure：测度；meet：交运算；member：元素；metamathematics：元数学；metric：度量；metric space：度量空间；model：模型；model theory：模型论；modular arithmetic：模运算；module：模；monotonic function：单调函数；multilinear algebra：多重线性代数；multiplication：乘法；multiset：多样集；Nnaive set theory：朴素集合论；natural logarithm：自然对数；natural number：自然数；natural science：自然科学；negative number：负数；neighbourhood：邻域；New Foundations：新基础理论；nine point circle：九点圆；non-Euclidean geometry：非欧几里德几何；nonlinearity：非线性；non-singular matrix：非奇异矩阵；nonstandard model：非标准模型；nonstandard analysis：非标准分析；norm：范数；normed vector space：赋范向量空间；n-tuple：n 元组或多元组；nullary：空；nullary intersection：空交集；number：数；number line：数轴；Oobject：对象；octonion：八元数；one-to-one correspondence：一一对应；open：开集；open ball：开球；operation：运算；operator：算子；or：或；order topology：序拓扑；ordered field：有序域；ordered pair：有序对；ordered set：偏序集；ordinal number：序数；ordinary mathematics：一般数学；origin：原点；orthogonal matrix：正交矩阵；Pp-adic number：p进数；paracompact space：仿紧致空间；parallel postulate：平行公理；parallelepiped：平行六面体；parallelogram：平行四边形；partial order：偏序关系；partition：分割；Peano arithmetic：皮亚诺公理；Pedoe's inequality：佩多不等式；perpendicular：垂直；philosopher：哲学家；philosophy：哲学；philosophy journals：哲学类杂志；plane：平面；plural quantification：复数量化；point：点；Point-Line-Plane postulate：点线面假设；polar coordinates：极坐标系；polynomial：多项式；polynomial sequence：多项式列；positive-definite matrix：正定矩阵；positive-semidefinite matrix：半正定矩阵；power set：幂集；predicate：谓词；predicate logic：谓词逻辑；preorder：预序关系；prime number：素数；product：积；proof：证明；proper class：纯类；proper subset：真子集；property：性质；proposition：命题；pseudovector：伪向量；Pythagorean theorem：勾股定理；QQ.E.D.：Q.E.D.；quaternion：四元数；quaternions and spatial rotation：四元数与空间旋转；question：疑问句；quotient field：商域；quotient set：商集；Rradius：半径；ratio：比；rational number：有理数；real analysis：实分析；real closed field：实闭域；real line：实数轴；real number：实数；real number line：实数线；reflexive relation：自反关系；reflexivity：自反性；reification：具体化；relation：关系；relative complement：相对补集；relatively complemented lattice：相对补格；right angle：直角；right-handed rule：右手定则；ring：环；Sscalar：标量；second-countable space：第二可数空间；self-adjoint operator：自伴随算子；sentence：判断；separable space：可分空间；sequence：数列或序列；sequence space：序列空间；series：级数；sesquilinear function：半双线性函数；set：集合；set-theoretic definition of natural numbers：自然数的集合论定义；set theory：集合论；several complex variables：一些复变量；shape：几何形状；sign function：符号函数；singleton：单元素集合；social science：社会科学；solid geometry：立体几何；space：空间；spherical coordinates：球坐标系；square matrix：方块矩阵；square root：平方根；strict：严格；structural recursion：结构递归；subset：子集；subsequence：子序列；subspace：子空间；subspace topology：子空间拓扑；subtraction：减法；sum：和；summation：求和；supremum：上确界；surreal number：超实数；symmetric difference：对称差；symmetric relation：对称关系；system of linear equations：线性方程组；Ttensor：张量；terminal object：终结对象；the algebra of sets：集合代数；theorem：定理；top element：最大元；topological field：拓扑域；topological manifold：拓扑流形；topological space：拓扑空间；topology：拓扑或拓扑学；total order：全序关系；totally disconnected：完全不连贯；totally ordered set：全序集；transcendental number：超越数；transfinite recursion：超限归纳法；transitivity：传递性；transitive relation：传递关系；transpose：转置；triangle inequality：三角不等式；trigonometric identities：三角恒等式；triple product：三重积；trivial topology：密着拓扑；true：真；truth value：真值；Uunary operation：一元运算；uncountable：不可数； uniform space：一致空间；union：并集；unique：唯一；unit interval：单位区间；unit step function：单位阶跃函数；unit vector：单位向量；universal quantification：全称量词；universal set：全集；upper bound：上界；Vvacuously true：??；Vandermonde's identity：Vandermonde 恒等式；variable：变量；vector：向量；vector calculus：向量分析；vector space：向量空间；Venn diagram：文氏图；volume：体积；von Neumann ordinal：冯·诺伊曼序数；von Neumann universe：冯·诺伊曼全集；vulgar fraction：分数；ZZermelo set theory：策梅罗集合论；Zermelo-Fraenkel set theory：策梅罗-弗兰克尔集合论；ZF set theory：ZF 系统；zero：零；zero object：零对象；。
半线性空间的基与基数
半线性空间的基与基数舒乾宇;王学平【摘要】半环上的线性代数技术在选择理论,分离事件网络模型,以及半环是除半环时在图论上都有重要的应用.然而不同于经典线性代数的是,半环上(∮)-半线性空间vn中不同基可能有不同的数量的元素.主要讨论了交换的零和自由半环上(∮)-半线性空间vn中的基数问题.首先给出每组基有相同基数的充要条件,回答了A.Di Nola 等在其论文(Fuzzy Sets and Systems,2007,158:1-22)中提出的开问题.其次证明文中给出的充要条件和已有充要条件之间的关系.最后证明在一些交换零和自由半环上不同基也有相同的基数.【期刊名称】《四川师范大学学报（自然科学版）》【年(卷),期】2014(037)002【总页数】5页(P143-147)【关键词】零和自由;基;基数【作者】舒乾宇;王学平【作者单位】四川师范大学数学与软件科学学院,四川成都610066;四川师范大学数学与软件科学学院,四川成都610066【正文语种】中文【中图分类】O153研究半环上的半线性结构已经有很长的历史.1979年,R.A.Cuninghame-Green等[1]在 minplus代数中构建了类似于线性代数的一系列理论:线性方程系统、特征值问题、向量组的线性相关性与线性无关性、秩与维数等.1985年,P.Butkoviˇc等[2]引用线性相关与线性无关以及向量组的秩等经典线性代数中的概念来讨论强正则矩阵的相关性质.随后研究者们将这些理论应用到相应的领域,比如选址问题[3]、控制系统问题[4-5]、分离事件系统[6]以及一些代数基本问题[7-14].而随后P.Butk oviˇc[15-16]和K.Cechl rov 等[17]则将线性相关与线性无关、特征值、线性方程的求解等相关的定义和结论类似的引入到max-plus代数中.而在2004年,R.A.Cuninghame-Green等[18]在max-plus代数中证明当空间是有限生成时,该空间有基且每组基的基数相等,最后给出在有限生成的空间中求基的方法.在2007年,A.Di Nola等[19]在MV-代数上建立半线性空间,引入向量组的线性相关、线性无关及基的概念,并解决了相应线性方程组有解的充要条件等问题.同时也提出一些公开问题:在半线性空间中,不同基的基数是否相等的问题,线性无关的向量组能否扩张成基的问题等.2010年,S.Zhao等[20]在join-半环中给出不同基的基数相等的充要条件,而Q.Y.Shu等[21]则在交换的零和自由半环上给出不同基有相同基数的一些充要条件.本文将主要讨论一些零和自由半环上,半线性空间基的一些性质,首先证明半线性空间中不同基有相同基数的充要条件,然后在一类特殊的零和自由半环上证明在其对应的半线性空间中,不同基有相同基数.1 预备知识以下假定读者对半环及半环中一些基本概念和符号已经熟悉[22],仅给出文中常用的一些基本概念.定义1.1 设L=〈L,+,·,0,1〉为半环.若对∀r,r′∈ L,都有r·r′=r′·r,则称 L 为交换半环.若a+b=0蕴含a=b=0,∀a,b∈L,则称半环L是零和自由的.定义1.2 设L=〈L,+,·,0,1〉为半环,a=(A,+A,oA)为一个加法交换幺半群.若外积∗:L×A→A满足对∀r,r′∈L和a,a′∈A都有(i)(r·r′)∗a=r·(r′∗a);(ii)r∗(a+Aa′)=r∗a+Ar∗a′;(iii)(r+r′)∗a=r∗a+Ar′∗a;(iv)1∗a=a;(v)o∗a=r∗oA=oA,则称〈L,+,·,0,1;∗;A,+A,oA〉为左L- 半模.类似地,还可以定义右L-半模,其中外积的定义为A×L→A.后面的定义是对文献[1]中定义的半线性空间的一种推广.定义1.3 设L=〈L,+,…,0,1〉是半环,称L上的半模为L-半线性空间.注意,在定义1.3中,半模即是指左L-半模.方便起见,以下令表示集合{1,2,·s,n},其中n是任意正整数.例 1 设L=〈L,+,·,0,1〉是半环,对n≥1,令其中(x1,x2,…,xn)T表示(x1,x2,…,xn)的转置.对∀x = (x1,x2,…,xn)T,y =(y1,y2,…,yn)T ∈Vn(L)和r∈L,定义运算为则 Vn= 〈L,+,·,0,1;∗;Vn(L),+,on×1〉为 L-半线性空间,其中on×1=(0,0,…,0)T.也称 Vn为半环L上的n维向量空间.为方便起见,下面在不会引起混淆的情况下,在L-半线性空间〈L,+,·,0,1;∗;A,+A,oA〉中,将用ra来代替r∗a,其中∀r∈L,a∈A.定义 1.4 设〈L,+,·,0,1;∗;A,+A,oA〉是L-半线性空间,称表达式为A中向量组a1,…,an的线性组合,其中λ1,λ2,…,λn∈L为标量(也称系数).若向量x能表示成向量组a1,a2,…,an的线性组合,则称向量x能被向量组a1,a2,…,an线性表出或线性表示.定义1.5 在L-半线性空间中,单个向量a是线性无关的.若向量组a1,a2,…,an(n≥2)中的任一向量都不能被其余向量线性表出,则称该向量组是线性无关的,否则,称向量组a1,a2,…,an是线性相关的.若无限集合的任意有限子集都是线性无关的,则称此无限集合是线性无关的.注意到,半线性空间或半模中相应的线性相关和线性无关的概念曾被许多学者研究过[2,19,22,24-25].设S是L-半线性空间的一个非空子集,若L-半线性空间中的任意向量都能表由集合S中的向量线性表出,则称S是L-半线性空间的一个生成集[19].令S表示L-半线性空间a的生成集,则可记作a= 〈S〉.特别地,若S={a1,…,ap},则记作a=〈a1,…,ap〉.定义1.6[22] 称L-半线性空间a中线性无关的生成集为a的基.定义1.7[21] 在L-半线性空间a中,若每一组基都有相同的基数,则称每组基的基数为a的维数,记作dim(a).设矩阵A∈Mn(L),令P表示集合{1,2,…,n}上的所有置换.定义矩阵 A的行列式,记作Det(A).由以上定义易知Det(A)=Det(AT).定义1.8 设向量x=(x1,x2,…,xn)T,y=(y1,y2,…,yn)T∈Vn,则x和y的内积记作(x,y),等于它们对应分量乘积的和(x,y)=x1·y1+x2·y2+ … +xn·yn.定义1.9 在半环L=〈L,+,·,0,1〉中,设a∈L,若存在b∈L使得ab=ba=1,则称元素a是可逆的,b为a的逆元,记作a-1.用U(L)表示半环L中所有可逆元构成的集合. 定义1.10 矩阵A∈Mn(L)称为左可逆(或右可逆)的,如果存在矩阵B∈Mn(L)使得AB=In(或BA=In).若矩阵A既是左可逆的又是右可逆的,则称它是可逆的.自现在起,都假设L=〈L,+,·,0,1〉是交换的零和自由半环.引理 1.1[26] 设A,B∈Mn(L),若存在k∈使得对∀j∈都有ajk=0,那么Det(A)=0. 引理1.2[27] 设A∈Mn(L),则下列条件等价.1)A是左可逆的;2)A是右可逆的;3)A是可逆的;4)AAT是可逆的对角阵;5)ATA是可逆的对角阵.引理1.3[27] 设矩阵A,B∈Mn(L),若A是可逆的,则有Det(AB)=Det(A)·Det(B)和Det(BA)=Det(B)·Det(A)都成立.2 L-半线性空间Vn的基的基数显然向量组e1,e2,…,en是L-半线性空间Vn的一组基,其中称e1,e2,…,en为Vn的标准基[20].一般的零和自由半环上的半线性空间Vn中,不同基可能有不同的基数.例如在图1所示的半环上(只需将半环中的加法和乘法运算分别取成格中的并和交运算),易知U(L)={1}.在L-半线性空间V2中,向量组都是V2的基.因此将首先讨论Vn中不同基的基数的充要条件.引理2.1 在L-半线性空间Vn中,不同的基有相同的基数的充要条件是:任何一个基中的向量都可以由其所在的基唯一线性表出.证明充分性设x1,x2,…,xs是Vn的任意一组基,由已知有∀xi,i∈,都可由x1,x2,…,xs唯一的线性表出.只需证n=s.若n≠s,则必有n<s或s<n.若n<s,则同定理2.2的证明,不妨设因此又由已知,任何一个基中的向量都可以由其所在的基唯一线性表出,可知AB=Is.由于s>n,则将矩阵A补上s-n列O,而将矩阵B补上s-n行O,使之都变成方阵,则有一方面,由引理1.2知方阵(AO)和(B)都是可逆矩阵,另一方面,两边取行列式,由引理1.1和1.3知矛盾.同理,也可由s<n推出矛盾,因此必然有n=s.必要性若L-半线性空间Vn中,不同基有相同的基数,则不妨设{x1,x2,…,xn}为Vn 的任意一组基.对∀xi∈ Vn,i∈,设其中ri∈L,i∈.由充分性的证明,不妨设其中C,D∈Mn(L),则有因此DC=In,也就是说,D是可逆矩阵.从而因此D(r1,…,ri,…,rn)T=D(0,…,1,…,0)T.而D是可逆的,即(r1,…,ri,…,rn)T=(0,…,1,…,0)T,也就是说,任一向量 xi都能被其所在基{x1,x2,…,xn}唯一的线性表出.引理2.2[21] 在L-半线性空间Vn中,每组基有相同的基数的充要条件是:任一向量都可以由基唯一的线性表出.由引理2.2易得推论2.1.推论2.1[23] 在L-半线性空间Vn中,下列条件等价:(1)每组基有相同的基数;(2)任一向量都可以由基唯一的线性表出;(3)任何一个基中的向量都可以由其所在的基唯一线性表出.显然定理2.1是对引理2.1的改进.从定理2.1前的例子可以看出,并不是所有的零和自由半环上的半线性空间Vn中不同基都有相同的基数,下面将给出一种特殊的零和自由半环,使得其对应的半线性空间Vn中不同基有相同的基数.定理2.2 若U(L)=L\{0},则dim(Vn)=n.证明只需证Vn中任意一组基都含有n个向量.设x1,x2,…,xs是Vn的任意一组基,若n≠s,则要么n>s,要么n<s.若n>s,由于{x1,x2,…,xs}是Vn的一组基,因此∀ei,i∈都能表示成向量组x1,x2,…,xs的线性组合,即由定义1.4知,存在元素aij∈L使得,从而有同理,由{e1,e2,…,en}是Vn的一组基可知,存在元素bjk∈L,使得对.从而从而BA=In.由于n>s,将矩阵A补上n-s行O,而将矩阵B补上n-s列O,使之都变成方阵,则有一方面,由引理1.2知方阵(BO)和都是可逆矩阵,另一方面,两边取行列式,由引理1.1和1.3知矛盾.若n<s.由x1,x2,…,xs线性无关可知矩阵A的每一行都有非零元,又从而由L是零和自由的,有aikajt(xk,xt)=0,其中i,j∈,i≠j,k,t∈.特别地,aikajk(xk,xk)=0,其中i,j∈,i≠j,k∈.另一方面,由向量组x1,x2,…,xs是线性无关的,显然有(xk,xk)≠0,其中k∈又U(L)=L\{0},从而aikajk=0,其中i,j∈,i≠j,k∈，这就意味着矩阵A的每一行恰好有一个非零元.因此不妨设因为半环L是零和自由的,所以由(5)式可知aikkbkt=0,t≠ik,k∈从而由(4)式有bkt=0,t≠ik,k∈,即矩阵B的每一列至多有一个非零元.而由x1,x2,…,xs线性无关可知矩阵B的每一列都有一个非零元.又由n<s可知存在j∈使得bhj,blj≠0,h≠l,h,l∈且xh=bhjej且xl=bljej,也就是说向量组xh,xl线性相关,矛盾.从而必有n=s.由定理2.2的证明可得出推论2.2.推论2.2 若U(L)=L\{0},则向量集{a1,a2,…,an}是L-半线性空间Vn的一组基当且仅当其中,对∀i∈,都有aii≠ 0.参考文献[1]Cuninghame-Green R A,Minimax A.Minimax Algebra(Lecture Notes in Economics and Mathematical Systems)[M].Berlin:Springer-Verlag,1979. [2]Butkoviˇc P,Hevery F.A condition for the strong regularity of matrices in the minimax algebra[J].Discrete Appl Math,1985,15:133-155.[3]Francis R L,McGinnis L F,White J A.Locational analysis[J].European J Oper Res,1983,12:220-252.[4]Carré B A.Graphs and Networks[M].Oxford:Oxford University Press,1979.[5]Karp R M.A characterization of the minimum cycle mean in adigraph[J].Discrete Math,1978,23:309-311.[6]Cuninghame-Green R A,Huisman F.Convergence problems in minimax algebra[J].J Math Anal Appl,1982,88:196-203.[7]Butkoviˇc P,Cuninghame-Green R A.On the regularity of matrices in min-algebra[J].Linear Algebra and Its Applications,1991,145:127-139.[8]Perfilieva I.Fuzzy function as an approximate solution to a system of fuzzy relation equations[J].Fuzzy Sets and Systems,2004,147:363-383. [9]Perfilieva I.Semi-linear spaces[C]//Noguchi H,Ishii H,et al.Proc of Seventh Czech-Japanese Seminar on Data Analysis and Decision Making under Uncertainty.Hyogo:Japan,2004:127-130.[10]Perfilieva I,Novàk V.System of fuzzy relation equations as a continuous model of if-then rules[J].Information Sciences,2007,177:3218-3227. [11]Perfilieva I,Noskov L.System of fuzzy relation equations with inf- → composition:complete set of solutions[J].Fuzzy Sets andSystems,2008,159:2256-2271.[12]Gondran M,Minoux M.Dioid and semirings:links to fuzzy sets and other applications[J].Fuzzy Sets and Systems,2007,158:1273-1294.[13]Gondran M,Minoux M.Graphs,Dioids and Semirings[M].NewYork:Springer-Verlag,2008.[14]熊清泉,舒乾宇.完备格上区间值t-半模及其R-蕴含算子[J].四川师范大学:自然科学版,2013,36(2):165-171.[15]Butkoviˇc P.Simple image set of(max,+)linear mappings[J].Discrete Appl Math,2000,105:73-86.[16]Butkoviˇc P.Max-algebra:the linear algebra of combinatorics[J].Linear Algebra and Its Applications,2003,367:313-335.[17]Cechl rov K,Cuninghame-Green R A.Soluble approximation of linear systems in max-plus algebra[J].Kybernetika(Prague),2003,39:137-141. [18]Cuninghame-Green R A,Butkoviˇc P.Bases in max-algebra[J].Linear Algebra and Its Applications,2004,389:107-120.[19]Di Nola A,Lettieri A,Perfilieva I,et al.Algebraic analysis of fuzzy systems[J].Fuzzy Sets and Systems,2007,158:1-22.[20]Zhao S,Wang X P.Invertible matrices and semilinear spaces over commutative semirings[J].Information Sciences,2010,180:5115-5124. [21]Shu Q Y,Wang X P.Bases in semilinear spaces over zerosumfree semirings[J].Linear Algebra and Its Applications,2011,435:2681-2692. [22]Golan J S.Semirings and Their Applications[M].Dordrecht:Kluwer Academic Publishers,1999.[23]Zimmermannn U.Linear and Combinatorial Optimization in Ordered Algebraic Structures[M].Amsterdamn:North-Holland Press,1981. 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introduction to linear algebra 每章开头方框-概述说明以及解释
introduction to linear algebra 每章开头方框-概述说明以及解释1.引言1.1 概述线性代数是数学中的一个重要分支，主要研究向量空间和线性变换的性质及其应用。
它作为一门基础学科，在多个领域如物理学、计算机科学以及工程学等都有广泛的应用。
线性代数的研究对象包括向量、向量空间、矩阵、线性方程组等，通过对其性质和运算法则的研究，可以解决诸如解线性方程组、求特征值与特征向量等问题。
线性代数的基本概念包括向量、向量空间和线性变换。
向量是指在空间中具有大小和方向的量，可以表示为一组有序的实数或复数。
向量空间是一组满足一定条件的向量的集合，对于向量空间中的任意向量，我们可以进行加法和数乘运算，得到的结果仍然属于该向量空间。
线性变换是指将一个向量空间映射到另一个向量空间的运算。
线性方程组与矩阵是线性代数中的重要内容。
在实际问题中，常常需要解决多个线性方程组，而矩阵的运算和性质可以帮助我们有效地解决这些问题。
通过将线性方程组转化为矩阵形式，可以利用矩阵的特殊性质进行求解。
线性方程组的解可以具有唯一解、无解或者有无穷多解等情况，而矩阵的行列式和秩等性质能够帮助我们判断线性方程组的解的情况。
向量空间与线性变换是线性代数的核心内容。
向量空间的性质研究可以帮助我们理解向量的运算和性质，以及解释向量空间的几何意义。
线性变换是一种将一个向量空间映射到另一个向量空间的运算，通过线性变换可以将复杂的向量运算问题转化为简单的矩阵运算问题。
在线性变换中，我们需要关注其核、像以及变换的特征等性质，这些性质可以帮助我们理解线性变换的本质和作用。
综上所述，本章节将逐步介绍线性代数的基本概念、线性方程组与矩阵、向量空间与线性变换的相关内容。
通过深入学习和理解这些内容，我们能够掌握线性代数的基本原理和应用，为进一步研究更高级的线性代数问题打下坚实的基础。
1.2文章结构在文章结构部分，我们将介绍本文的组织结构和各章节的内容概述。
MATHEMATICS（PUREANDAPPLIED）：（纯粹数学和应用）
Professor & Head of DepartmentNT Bishop, MA(Cambridge), PhD(Southampton), FRASSenior LecturersJ Larena, MSc(Paris), PhD(Paris)D Pollney, PhD(Southampton)CC Remsing, MSc(Timisoara), PhD(Rhodes)Vacant LecturersEOD Andriantiana, PhD(Stellenbosch)V Naicker, MSc(KwaZulu-Natal)AL Pinchuck, MSc(Rhodes), PhD(Wits)Lecturer, Academic Development M Lubczonok, Masters(Jagiellonian)Mathematics (MA T) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as major subjects for the degrees of BSc, BA, BJourn, BCom, BBusSci, BEcon and BSocSc, and for the diploma HDE(SEC).To major in Mathematics, a candidate is required to obtain credit in the following courses: MAT1C; MAM2; MAT3. See Rule S.23.To major in Applied Mathematics, a candidate is required to obtain credit in the following courses: MAT1C, MAM2; MAP3. See Rule S.23.The attention of students who hope to pursue careers in the field of Bioinformatics is drawn to the recommended curriculum that leads to postgraduate study in this area, in which Mathematics is a recommended co-major with Biochemistry, and for which two years of Computer Science and either Mathematics or Mathematical Statistics are prerequisites. Details of this curriculum can be foundin the entry for the Department of Biochemistry, Microbiology and Biotechnology.See the Departmental Web Page http://www.ru.ac.za/departments/mathematics/ for further details, particularly on the content of courses.First-year level courses in MathematicsMathematics 1 (MAT1C) is given as a year-long semesterized two-credit course. Credit in MAT1C must be obtained by students who wish to major in certain subjects (such as Applied Mathematics, MATHEMATICS (PURE AND APPLIED)Physics and Mathematical Statistics) and by students registered for the BBusSci degree.Introductory Mathematics (MAT1S) is recommended for Pharmacy students and for Science students who do not need MAT1C or MAT1C1.Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.Mathematics 1L (MA T1L) is a full year course for students who do not qualify for entry into any of the first courses mentioned above. This is particularly suitable for students in the Social Sciences and Biological Sciences who need to become numerate or achieve a level of mathematical literacy. A successful pass in this course will give admission to MA T1C.First yearMAT1CThere are two first-year courses in Mathematics for candidates planning to major in Mathematics or Applied Mathematics. MAT1C1 is held in thefirst semester and MAT1C2 in the second semester. Credit may be obtained in each course separately and, in addition, an aggregate mark of at least 50%will be deemed to be equivalent to a two-creditcourse MAT1C, provided that a candidate obtains the required sub-minimum (40%) in each component. Supplementary examinations may be recommended in either course, provided that a candidate achieves a minimum standard specified by the department. Candidates obtaining less than 40% for MAT1C1 are not permitted to continue with MAT1C2.MAT1C1 (First semester course): Basic concepts (number systems, functions), calculus (limits,continuity, differentiation, optimisation, curvesketching, introduction to integration), propositional calculus, mathematical induction, permutations, combinations, binomial theorem, vectors, lines andplanes, matrices and systems of linear equations.MAT1C2 (Second semester course): Calculus (integration, applications of integration, improper integrals), complex numbers, differential equations, partial differentiation, sequences and series.MAT1S (Semester course: Introductory Mathematics) (about 65 lectures)Estimation, ratios, scales (log scales), change of units, measurements; Vectors, systems of equations, matrices, in 2-dimensions; Functions: Review of coordinate geometry, absolute values (including graphs); Inequalities; Power functions, trig functions, exponential functions, the number e (including graphs); Inverse functions: roots, logs, ln (including graphs); Graphs and working with graphs; Interpretation of graphs, modeling; Descriptive statistics (mean, standard deviation, variance) with examples including normally distributed data; Introduction to differentiation and basic derivatives; Differentiation techniques (product, quotient and chain rules); Introduction to integration and basic integrals; Modeling, translation of real-world problems into mathematics.MAT 1L: Mathematics Literacy This course helps students develop appropriatemathematical tools necessary to represent and interpret information quantitatively. It also develops skills and meaningful ways of thinking, reasoning and arguing with quantitative ideas in order to solve problems in any given context.Arithmetic: Units of scientific measurement, scales, dimensions; Error and uncertainty in measure values.Fractions and percentages - usages in basic science and commerce; use of calculators and spreadsheets. Algebra: Polynomial, exponential, logarithmic and trigonometric functions and their graphs; modelling with functions; fitting curves to data; setting up and solving equations. Sequences and series, presentation of statistical data.Differential Calculus: Limits and continuity; Rules of differentiation; Applications of Calculus in curvesketching and optimisation.Second Year Mathematics 2 comprises two semesterized courses,MAM201 and MAM202, each comprising of 65 lectures. Credit may be obtained in each course seperately. An aggregate mark of 50% will grant the two-credit course MAM2, provided a sub-minimumof 40% is achieved in both semesters. Each semester consists of a primary and secondary stream which are run concurrently at 3 and 2 lectures per week, respectively. Additionally, a problem-based course in Mathematical Programming contributes to the class record and runs throughout the academic year.MAM201 (First semester):Advanced Calculus (39 lectures): Partial differentiation: directional derivatives and the gradient vector; maxima and minima of surfaces; Lagrange multipliers. Multiple integrals: surface and volume integrals in general coordinate systems. Vector calculus: vector fields, line integrals, fundamental theorem of line integrals, Green’s theorem, curl and divergence, parametric curves and surfaces.Ordinary Differential Equations (20 lectures): First order ordinary differential equations, linear differential equations of second order, Laplace transforms, systems of equations, series solutions, Green’s functions.Mathematical Programming 1 (6 lectures): Introduction to the MATLAB language, basic syntax, tools, programming principles. Applicationstaken from MAM2 modules. Course runs over twosemesters.MAM202 (Second semester):Linear Algebra (39 lectures): Linear spaces, inner products, norms. Vector spaces, spans, linear independence, basis and dimension. Linear transformations, change of basis, eigenvalues, diagonalization and its applications.Groups and Geometry (20 lectures): Number theory and counting. Groups, permutation groups, homomorphisms, symmetry groups in 2 and 3 dimensions. The Euclidean plane, transformations and isometries. Complex numbers, roots of unity and introduction to the geometry of the complex plane.Mathematical Programming 2 (6 lectures): Problem-based continuation of Semester 1.Third-year level courses inMathematics and Applied Mathematics Mathematics and Applied Mathematics are offered at the third year level. Each consists of four modules as listed below. Code TopicSemester Subject AM3.1 Numerical analysis 1 Applied MathematicsAM3.2 Dynamical systems 2 Applied Mathematics AM3.4 Partial differentialequations 1 Applied Mathematics AM3.5 Advanced differentialequations 2 Applied MathematicsM3.1 Algebra 2 MathematicsM3.2 Complex analysis 1 MathematicsM3.3 Real analysis 1 MathematicsM3.4 Differential geometry 2 Mathematics Students who obtain at least 40% in all of the above modules will be granted credit for both MAT3 and MAP3, provided that the average of the Applied Mathematics modules is at least 50% AND the average of the Mathematics modules is at least 50%. Students who obtain at least 40% for any FOUR of the above modules and with an average mark over the four modules of at least 50%, will be granted credit in either MAT3 or MAP3. If three or four of the modules are from Applied Mathematics then the credit will be in MAP3, otherwise it will be in MAT3.Module credits may be carried forward from year to year.Changes to the modules offered may be made from time-to-time depending on the interests of the academic staff.Credit for MAM 2 is required before admission to the third year courses.M3.1 (about 39 lectures) AlgebraAlgebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.Syllabus: Sets, equivalence relations, groups, rings, fields, integral domains, homorphisms, isomorphisms, and their elementary properties.M3.2 (about 39 lectures) Complex Analysis Building on the first year introduction to complex numbers, this course provides a rigorous introduction to the theory of functions of a complex variable. It introduces and examines complex-valued functions of a complex variable, such as notions of elementary functions, their limits, derivatives and integrals. Syllabus: Revision of complex numbers, Cauchy- Riemann equations, analytic and harmonic functions, elementary functions and their properties, branches of logarithmic functions, complex differentiation, integration in the complex plane, Cauchy’s Theorem and integral formula, Taylor and Laurent series, Residue theory and applications. Fourier Integrals.M3.3 (about 39 lectures) Real AnalysisReal Analysis is the field of mathematics that studies properties of real numbers and functions on them. The course places great emphasis on careful reasoning and proof. This course is an essential basis for any further study in mathematics.Syllabus: Topology of the real line, continuity and uniform continuity, Heine-Borel, Bolzano-Weierstrass, uniform convergence, introduction to metric spaces.M3.4 (about 39 lectures) Differential Geometry Roughly speaking, differential geometry is concerned with understanding shapes and their properties in terms of calculus. This elementary course on differential geometry provides a perfect transition to higher mathematics and its applications. It is a subject which allows students to see mathematics for what it is - a unified whole mixing together geometry, calculus, linear algebra, differential equations, complex variables, calculus of variations and topology.Syllabus: Curves (in the plane and in the space), curvature, global properties of curves, surfaces, the first fundamental form, isometries, the second fundamental form, the normal and principal curvatures, the Gaussian and mean curvatures, the Gauss map, geodesics.AM3.1 (about 39 lectures) Numerical Analysis Many mathematical problems cannot be solved exactly and require numerical techniques. These techniques usually consist of an algorithm which performs a numerical calculation iteratively until certain tolerances are met. These algorithms can be expressed as a program which is executed by a computer. The collection of such techniques is called “numerical analysis”.Syllabus: Systems of non-linear equations, polynomial interpolation, cubic splines, numerical linear algebra, numerical computation of eigenvalues, numerical differentiation and integration, numerical solution of ordinary and partial differential equations, finite differences,, approximation theory, discrete Fourier transform.AM3.2 (about 39 lectures) Dynamical Systems This module is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields of applied mathematics (like control theory and the Lagrangian and Hamiltonian formalisms of classical mechanics). The emphasis is on the mathematical aspects of various constructions and structures rather than on the specific physical/mechanical models. Syllabus Linear systems; Linear control systems; Nonlinear systems (local theory); Nonlinear control systems; Nonlinear systems (global theory); Applications : elements of optimal control and/or geometric mechanics.AM3.4 (about 39 lectures) Partial Differential EquationsThis course deals with the basic theory of partial differential equations (elliptic, parabolic and hyperbolic) and dynamical systems. It presents both the qualitative properties of solutions of partial differential equations and methods of solution. Syllabus: First-order partial equations, classification of second-order equations, derivation of the classical equations of mathematical physics (wave equation, Laplace equation, and heat equation), method of characteristics, construction and behaviour of solutions, maximum principles, energy integrals. Fourier and Laplace transforms, introduction to dynamical systems.AM3.5 (about 39 lectures)Advanced differential equationsThis course is an introduction to the study of nonlinearity and chaos. Many natural phenomena can be modeled as nonlinear ordinary differential equations, the majority of which are impossible to solve analytically. Examples of nonlinear behaviour are drawn from across the sciences including physics, biology and engineering.Syllabus:Integrability theory and qualitative techniques for deducing underlying behaviour such as phase plane analysis, linearisations and pertubations. The study of flows, bifurcations, the Poincare-Bendixson theorem, and the Lorenz equations.Mathematics and Applied Mathematics Honours Each of the two courses consists of either eight topics and one project or six topics and two projects.A Mathematics Honours course usually requires the candidate to have majored in Mathematics, whilst Applied Mathematics Honours usually requires the candidate to have majored in Applied Mathematics. The topics are selected from the following general areas covering a wide spectrum of contemporary Mathematics and Applied Mathematics: Algebra; Combinatorics; Complex Analysis; Cosmology; Functional Analysis; General Relativity; Geometric Control Theory; Geometry; Logic and Set Theory; Measure Theory; Number Theory; Numerical Modelling; Topology.Two or three topics from those offered at the third-year level in either Mathematics or Applied Mathematics may also be taken in the case of a student who has not done such topics before. With the approval of the Heads of Department concerned, the course may also contain topics from Education, and from those offered by other departments in the Science Faculty such as Physics, Computer Science, and Statistics. On the other hand, the topics above may also be considered by such Departments as possible components of their postgraduate courses.Master’s and Doctoral degrees in Mathematics or Applied MathematicsSuitably qualified students are encouraged to proceed to these degrees under the direction of the staff of the Department. Requirements for these degrees are given in the General Rules.A Master’s degree in either Mathematics or Applied Mathematics may be taken by thesis only, or by a combination of course work and a thesis. Normally four examination papers and/or essays are required apart from the thesis. The whole course of study must be approved by the Head of Department.。
代数英语
(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。
0+||zero-dagger; 读作零正。
1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。
AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。
BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。
B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。
C0 类函数||function of class C0; 又称“连续函数类”。
CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。
Cp统计量||Cp-statisticC。
algebra2 知识点总结
algebra2 知识点总结Linear Equations and FunctionsOne of the fundamental concepts in Algebra 2 is linear equations and functions. A linear equation is an equation of the form y = mx + b, where m is the slope and b is the y-intercept. Students learn how to graph linear equations, find the slope and y-intercept, and solve systems of linear equations using various methods such as substitution, elimination, and graphing.Functions are a core concept in Algebra 2, and students study various types of functions such as linear, quadratic, exponential, and logarithmic functions. They learn how to analyze the behavior of functions, find their domain and range, and determine whether a function is even, odd, or periodic. Students also explore transformations of functions, such as shifts, stretches, and reflections, and how they affect the graph of a function.Inequalities and Absolute Value EquationsIn Algebra 2, students also study inequalities and absolute value equations. They learn how to solve and graph linear inequalities, quadratic inequalities, rational inequalities, and absolute value inequalities. They also explore the concept of compound inequalities and how to solve systems of inequalities.Absolute value equations are another important topic in Algebra 2. Students learn how to solve and graph absolute value equations, as well as inequalities involving absolute value expressions. They also study the properties of absolute value functions and their applications in real-life scenarios.Polynomials and Polynomial FunctionsPolynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. In Algebra 2, students learn how to add, subtract, multiply, and divide polynomials, as well as factor and solve polynomial equations. They also study the properties of polynomial functions, such as end behavior, zeros, and turning points.Students delve into advanced topics such as polynomial long division, synthetic division, the remainder theorem, and the factor theorem. They also explore the relationship between polynomial functions and their graphs, and how to use this information to solve real-world problems.Rational Expressions and Rational FunctionsRational expressions are fractions that contain polynomials in the numerator and denominator. In Algebra 2, students learn how to simplify, multiply, divide, add, and subtract rational expressions, as well as solve rational equations. They also study the properties of rational functions, such as asymptotes, intercepts, and end behavior.Students explore the relationship between rational functions and their graphs, and how to use this information to analyze and solve real-world problems. They also study advanced topics such as partial fraction decomposition, complex fractions, and applications of rational functions in areas such as economics, physics, and engineering.Exponential and Logarithmic FunctionsExponential and logarithmic functions are essential in Algebra 2, and students learn how to solve exponential and logarithmic equations, as well as graph exponential and logarithmic functions. They study the properties of exponential and logarithmic functions, such as growth and decay, domain and range, and asymptotic behavior.Students also explore the relationship between exponential and logarithmic functions, and how to use this information to solve real-world problems. They study applications of exponential and logarithmic functions in areas such as finance, population growth, radioactive decay, and pH levels.Sequences and SeriesSequences and series are important topics in Algebra 2, and students learn how to find the nth term of a sequence, as well as the sum of a finite or infinite series. They study arithmetic sequences and series, geometric sequences and series, and other types of sequences such as Fibonacci and recursive sequences.Students explore the properties of sequences and series, such as common difference, common ratio, and convergence. They also study applications of sequences and series in areas such as finance, physics, and computer science.Complex NumbersComplex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). In Algebra 2, students learn how to perform operations with complex numbers, such as addition, subtraction, multiplication, division, and simplification. They also study the properties of complex numbers, such as the conjugate, modulus, and argument.Students explore the relationship between complex numbers and their graphs on the complex plane, and how to use this information to solve equations involving complex numbers. They also study applications of complex numbers in areas such as electrical engineering, signal processing, and quantum mechanics.Matrices and DeterminantsMatrices are arrays of numbers arranged in rows and columns, and they are used to represent and solve systems of linear equations. In Algebra 2, students learn how to add, subtract, multiply, and invert matrices, as well as find the determinant of a matrix. They also study the properties of matrices, such as the identity matrix, transpose, and rank.Students explore the relationship between matrices, determinants, and systems of linear equations, and how to use this information to solve real-world problems. They also study applications of matrices in areas such as computer graphics, cryptography, and economics.Conic SectionsConic sections are curves obtained by intersecting a cone with a plane, and they include the circle, ellipse, parabola, and hyperbola. In Algebra 2, students learn how to graph and analyze conic sections, as well as find their equations given certain properties.Students study the properties of conic sections, such as the focus, directrix, eccentricity, and asymptotes. They also explore the relationship between conic sections and their equations, and how to use this information to solve real-world problems. They study applications of conic sections in areas such as astronomy, engineering, and architecture.ConclusionAlgebra 2 is a challenging but rewarding branch of mathematics that builds upon the concepts learned in Algebra 1. In this article, we have provided a comprehensive summary of the key topics in Algebra 2, including linear equations and functions, inequalities and absolute value equations, polynomials and polynomial functions, rational expressions and rational functions, exponential and logarithmic functions, sequences and series, complex numbers, matrices and determinants, and conic sections.By mastering these topics, students will develop a deeper understanding of algebraic concepts and techniques, as well as their applications in various fields such as science, engineering, economics, and finance. Algebra 2 is an essential foundation for further study in mathematics and related disciplines, and it provides students with the analytical and problem-solving skills necessary for success in the modern world.。
Theorem 1.1. (Fundamental Theorem of Algebra) A non-constant polynomial
TOPICS IN p-ADIC FUNCTION THEORYWILLIAM CHERRY1.Picard TheoremsI would like to begin by recalling the Fundamental Theorem of Algebra. Theorem1.1.(Fundamental Theorem of Algebra)A non-constant polyno-mial of one complex variable takes on every complex value.Moreover,if the poly-nomial is of degree d,then every complex value is taken on d times,countingmultiplicity.Because entire functions have power series expansions,they are sort of like poly-nomials of infinite degree.Picard’s well-known theorem is a complex analytic analogof the Fundamental Theorem of Algebra.Theorem1.2.(Picard’s(Little)Theorem)A non-constant entire functiontakes on all but at most one complex value.Moreover,a transcendental entirefunction must take on all but at most one complex value infinitely often.The function e z shows that a complex entire function can indeed omit one value.Lately,it has become fashionable to prove p-adic versions of value distributiontheorems,of which Picard’s Theorem is an example,though not a recent one.Morerecent examples can be found in the works listed in the references section.Recallthat the p-adic absolute value||p on the rational numberfield Q is defined as follows.If x∈Q is written p k a/b,where p is a prime,k is an integer,and a andb are integers relatively prime to p,then|x|p=p−pleting Q with respect to this absolute value results in thefield of p-adic numbers,denoted Q p.Takingthe algebraic closure of Q p,extending||p to it,and then completing once moreresults in a complete algebraically closedfield,denoted C p,and often referred toas the p-adic complex numbers.Recall that the absolute value||p satisfies a very strong form of the trian-gle inequality,namely|x+y|p≤max{|x|p,|y|p}.This is referred to as a non-Archimedean triangle inequality,and this non-Archimedean triangle inequality is what accounts for most of the differences between function theory on C p and on C.Recall that an infinite series a n converges under a non-Archimedean norm if and only if lima n=0.By an entire function on C p,one means a formaln→∞∞ n=0a n z n,where a n are elements of C p,and lim n→∞|a n|p r n=0,for power seriesevery r>0,so that plugging in any element of C p for z results in an absolutely convergent series.Most of what I will discuss here is true over an arbitrary algebraically closedfield complete with respect to a non-Archimedean absolute value,but for simplicity’s sake,I will stick with the concrete case C p here.12WILLIAM CHERRYIf one tries to prove Picard’s Theorem for p-adic entire functions,what one gets is the following theorem.Theorem1.3.(p-Adic Case)A non-constant p-adic entire function must take on every value in C p.Moreover,a transcendental p-adic entire function must take on every value in C p infinitely often.Proof.Let f(z)= a n z n be a p-adic entire function,so lim n→∞|a n|p r n=0,for all r>0.Denote by|f|r=sup|a n|p r n.The graph of{log|a n|p+n log r}log r→log|f|r=supn≥0is piecewise linear and closely related to what’s known as the Newton polygon.In particular,ther zeros of f occur at the“corners”of the graph of log r→log|f|r (c.f.,[Am]and[BGR]).For r close to zero,|f|r=|a0|p,provided a0=0.Moreover,it is clear that if f is not constant,then for all r sufficiently large,|f|r=|a0|p.Hence,the graph of log r→log|f|r has a corner,and hence f has a zero.If f is transcendental,then f has infinitely many non-zero Taylor coefficients, and thus for every n,there exists r n such that for all r≥r n,we have|f|r>|a n|p r n. Hence,log r→log|f|r must have infinitely many corners,and so f has infinitely many zeros.2Note that Theorem1.3is an even closer analogy to the Fundamental Theorem of Algebra than Picard’s Theorem was,since p-adic entire functions,like polynomials, cannot omit any values.Thus,in this respect,the function theory of p-adic entire functions is more closely related to the function theory of polynomials than it is to the function theory of complex holomorphic functions.That will be the theme of this survey.2.Algebraic CurvesMy second illustration that p-adic function theory is more like that of polyno-mials comes from considering Riemann surfaces.Let X be a projective algebraic curve of genus g.Then,the three analogous theorems we have are:Theorem2.1.(Polynomial Case)If f:C→X is a non-constant polynomial mapping,then g=0.Theorem2.2.(Complex Case)If f:C→X is a non-constant holomorphic mapping,then g≤1.Theorem2.3.(p-Adic Case)If f:C p→X is a non-constant p-adic analytic mapping,then g=0.The polynomial case follows from the Riemann-Hurwitz formula,which says that the genus of the image curve cannot be greater than the genus of the domain.The complex case was again proved by Picard.Riemann surfaces of genus≥2 have holomorphic universal covering maps from the unit disc,and thus any holo-morphic map form C to a Riemann surface of genus≥2lifts to a holomorphic map to the unit disc,which must then be constant by Liouville’s Theorem.The p-Adic analog of this theorem was proven only recently,by V.Berkovich[Ber].One of the major difficulties in p-adic function theory is the fact that the nat-ural p-adic topology is totally disconnected,and therefore analytic continuationTOPICS IN p -ADIC FUNCTION THEORY 3in these circumstances is a delicate task.Moreover,geometric techniques that are commonplace in complex analysis cannot be applied in the p -adic case.In order to prove his p -adic analog of Picard’s Theorem,Berkovich developed a theory of p -adic analytic spaces that enlarges the natural p -adic spaces so that they become nice topological spaces,and geometric techniques,such as universal covering spaces,can be used to prove theorems.3.Berkovich TheoryBerkovich’s theory is somewhat deep,and I do not have ther required space to go into it in much detail here.However,the reader may find the following brief description of his theory helpful.The interested reader is encouraged to look at:[Ber],[Ber 2],and [BGR].The last reference covers the more traditional theory of rigid analytic spaces.Although one can associate a Berkovich space to any p -adic analytic variety,we will concentrate here on the special case of the unit ball in C p ,which is the local model for smooth p -adic analytic spaces,at least in dimension one.Consider the closed unit ball B ={z ∈C p :|z |p ≤1}.The p -adic analytic func-tions on B are of the form a n z n ,with lim n →∞|a n |p =0.These functions form aBanach algebra A under the norm |f |0,1=sup n |a n |p .The Berkovich space associated to B consists of all bounded multiplicative semi-norms on A .This space is provided with the weakest topology such that all maps of the form ||→|f |,f ∈A are continuous maps to the real numbers with their usual topology.Here ||denotes one of the bounded multiplicative semi-norms in the Berkovich space.Berkovich spaces have many nice topological properties,such as local compact-ness and local arc-connectedness.They also have universal covering spaces,which are again Berkovich spaces.For f ∈A ,z 0∈B ,and 0≤r ≤1,define |f |z 0,r by |f |z 0,r =sup n |c n |p r n ,where f =c n (z −z 0)n ,or in other words,the c n are the coefficients of the Taylor expansion of f about z 0.Note that if r =0,then |f |z 0,0=|f (z 0)|p ,and note that by the non-Archimedean triangle inequality,if |z 0−w 0|p ≤r,then ||z 0,r =||w 0,r .There are in fact more bounded multiplicative semi-norms on B than these,but these are the main ones to thinkabout.||0,1|z 0,0Figure 1.4WILLIAM CHERRYFigure 1gives a sort of intuitive “tree-like”representation for the Berkovich space associated to B .The dots at the top correspond to the totally disconnected points in B .Of course there are infinitely many of these,and there are points arbitrarily close together,much like a Cantor set.The lines represent the connected continuum of additional multiplicative semi-norms connecting the Berkovich space.There are of course infinitely many places where lines join together,and the junctures are by no means discrete.Finally,the point at the bottom corresponds to the one semi-norm ||z 0,1which is the same for all points z 0in B .We say that two points z 0and w 0in B are in the same residue class if |z 0−w 0|p <1.This leads to a concept called “reduction,”whereby the space is “reduced”to the space of residue classes.The reduction of B can be naturally identified with A 1F alg p ,the affine line over the algebraic closure of the field of p elements.This process of reduction extends to the Berkovich space associated to B ,and there is a reduction mapping πfrom the Berkovich space B to A 1F alg p .The reduction mapping πhas what I would call an anti-continuity property,in thatπ−1of a Zariski open sets in A 1F algpwill be closed in the Berkovich topology and π−1of a Zariski closed set will be open in the Berkovich topology.In Figure 1,two points in the Berkovich tree are in different residue classes if their branches do not join except at the one point ||0,1,which is kind of like a “generic”point in algebraic geometry,and is in fact the inverse image of the genericpoint in A 1F alg p under the reduction map.Thus,three residue classes are shown in Figure 1. 4.Abelian VarietiesIn my Ph.D.thesis [Ch 1],I extended Berkovich’s Theorem to Abelian varieties.See also:[Ch 2]and [Ch 3].Theorem 4.1.(Cherry)If f :C p →A is a p -adic analytic map to an Abelian variety,then f must be constant.Proof sketch.Tis a product of multiplicative groups (i.e.a multiplicative torus).Gis the universal cover of A in the sense of Berkovich,and a semi-Abelian variety.Bis an Abelian variety with good reduction,meaning it has a reduction mapping πB to an Abelian variety e B over F alg p .T1G B 1A C pf BπB f !Figure 2.Step 1.First,we use Berkovich theory to lift f to a map f !:C p →G to the universal covering of A.TOPICS IN p-ADIC FUNCTION THEORY5 Step2.Next we use p-adic uniformization([BL1],[BL2],[DM])to identify Gas a semi-Abelian variety,as in Figure2.Step3.Then,we use reduction techniques.We get a mapC p→G→B→ B.This map must be constant because if it were not we would induce a non-constant rational map from the projective line over F alg p to the Abelian variety B.Thus,the image in B lies above a single smooth point in B.The inverse image of a smooth point in B is isomorphic to an open ball in C n p,where n is the dimension of B. Thus,the map to B is also constant,by the p-adic version of Liouville’s Theorem,for example.Step4.Thus,we only need consider mappings from C p to T.But,T∼=C p\{0}×···×C p\{0}.The projection onto each factor is constant by the p-Adic version of Picard’s LittleTheorem.2Because p-adic analytic maps to Abelian varieties must be constant,the follow-ing conjecture seems plausible.Conjecture4.2.Let X be a smooth projective variety.If there exists a non-constant p-adic analytic map from C p to X,then there exists a non-constantrational mapping from P1to X.5.Value SharingOne of the more striking consequences of Nevanlinna theory is Nevanlinna’stheorem that if two non-constant meromorphic functions f and g sharefive values,then f must equal g,[Ne].The polynomial version of this was taken up by Adamsand Straus in[AS].Theorem5.1.(Adams and Straus)If f and g are two non-constant polynomi-als over an algebraically closedfield of characteristic zero such that f−1(0)=g−1(0) and f−1(1)=g−1(1),then f≡g.Proof.Assume deg f≥deg g and consider[f (f−g)]/[f(f−1)].This is a polynomial because if f(z)=0or1,then f(z)=g(z)by assumption,and hencethe zeros in the denominator are canceled by the zeros in the numerator,and thef in the numerator takes care of multiple zeros.On the other hand,the degree of the numerator is strictly less than the degree of the denominator,so the numeratormust be identically zero.In other words f is constant,or f is identically equal tog.2Theorem5.2.(Adams and Straus)If f and g are non-constant p-adic(char-acteristic zero)analytic functions such that f−1(0)=g−1(0),and f−1(1)=g−1(1), then f≡g.Proof.We may assume without loss of generality that there exist r j→∞such that|f|rj≥|g|r j.Let h=[f (f−g)]/[f(f−1)].Then,h is entire since,as in the polynomial case,zeros in the denominator are always matched by zeros in thenumerator.On the other hand,by the non-Archimedean triangle inequality,wehave for r j sufficiently large that|h|r j= f f r j·|f−g|r j|f−1|r j≤ f f r j·|f|r j|f|r j= f f r j.6WILLIAM CHERRYNow,I claim |f /f |r ≤r −1,and therefore |h |r j →0as r j →∞.Hence,h ≡0,and again,either f is constant of f ≡g.2The claim that |f /f |r ≤1/r is the p-adic form of the Logarithmic Derivative Lemma,and note this is much stronger than what is true in the complex case.Theorem 5.3.(p -Adic Logarithmic Derivative Lemma)If f is a p -Adic analytic function,then |f /f |r ≤1/r.Proof.Write f = a n z n .Then,since |n |p ≤1,we have|f |r =sup n ≥1{|na n |p r n −1}=1r sup n ≥1{|na n |p r n }≤1r sup n ≥0{|a n |p r n }=1r |f |r 2Notice the similarity in both the proof and the statement of both of Adams and Straus’s theorems.An active topic of current research has to do with so called “unique range sets.”Rather than considering functions which share distinct values,one considers finite sets and functions f and g such that f −1(S )=g −1(S ).Here,Boutabaa,Escassut,and Haddad [BEH]gave a nice characterization for unique range sets of polynomials,in the counting multiplicity case.Theorem 5.4.(Boutabaa,Escassut,and Haddad)If f and g are polynomi-als over an algebraically closed field F of characteristic zero,and if S is a finite subset of F such that f −1(S )=g −1(S ),counting multiplicity,then either f ≡g or there exist constants A and B,A =0,such that g =Af +B and S =AS +B.Proof.Let S ={s 1,...,s n }and let P (X )=(X −s 1)···(X −s n ).Then,P (f )and P (g )are polynomials with the same zeros,counting multiplic-ity by the assumption f −1(S )=g −1(S ).Thus,P (f )/P (g )is some non-zero con-stant C,and if we set F (X,Y )=P (X )−CP (Y ),we have F (f,g )=0.Thus,z →(f (z ),g (z ))is a rational component of the possibly reducible algebraic curve F (X,Y )=0.Because F (X,Y )=0has n distinct smooth points at infinity in P 2(characteristic zero!)and because (f (z ),g (z ))has only one point at infinity,(f (z ),g (z ))must in fact be a linear component of F (X,Y )=0.2Boutabaa,Escassut,and Haddad also made a preliminary analysis of the p -adic entire analog of their theorem,and solved the case when the cardinality of S equals three completely.C.-C.Yang and I,[CYa],combined Berkovich’s Picard theorem with their argument to complete the p -adic entire case.Theorem 5.5.(Cherry and Yang)If f and g are p -adic entire functions and S is a finite subset of C p such that f −1(S )=g −1(S ),counting multiplicity,thenthere exist constants A and B,with A =0,such that g =Af +B,and S =AS +B.Proof.Again,setP (X )=(X −s 1)···(X −s n ).Again,P (f )/P (g )is a constant C =0.Again,set F (X,Y )=F (X )−CF (Y ).By Berkovich’s p-Adic Picard Theorem,(f (z ),g (z ))is contained in a rational compo-nent of F (X,Y )=0.Thus,there exist rational functions u and v,and a p -adic entire function h,such that f =u (h )and g =v (h ).It is then easy to see that u and v must in fact be polynomials,and we are then back to the polynomial case,thinking of h as a variable.2TOPICS IN p-ADIC FUNCTION THEORY76.Concluding RemarksIn many respect,it appears that algebraic geometry,rather than complex Nevan-linna theory,is the appropriate model for p-adic value distribution theory.At least, that is what I hope this survey has conveyed to the reader.This leads me to a gen-eral principle.Principle6.1.Appropriately stated theorems about the value distribution of poly-nomials should also be true for p-adic entire functions.Similarly,theorems for rational functions should also be true for p-adic meromorphic functions.Conjecture4.2is a special case of this principle.With some luck,solving a p-adic problem based on the above principle might help us better understand complex Nevanlinna theory.For example,it would be reasonable to make the following conjecture.Conjecture6.2.If f:C p→X is a p-adic analytic map to a K3surface X,the the image of f must be contained in a rational curve.This conjecture can be thought of as a special case of a p-adic version of the Green-Griffiths conjecture[GG]that says a holomorphic curve in a smooth pro-jective variety of general type must be algebraically degenerate.One might hope to attack Conjecture6.2since much is known about K3surfaces and they have a close connection to Abelian varieties.It might also be thatfinding a proof for Conjecture6.2would shed some light on an attack of the general Green-Griffiths conjecture over the complex numbers.References[AS]W.Adams and E.Straus,Non-Archimedian analytic functions taking the same values at the same points,Illinois J.Math.15(1971),418–424.[Am]Y.Amice,Les nombres p-adiques,Presses Universitaires de France,1975.[Ber]V.Berkovich,Spectral Theory and Analytic Geometry over Non-Archimedean Fields, AMS Surveys and Mographs33,1990.[Ber2]V.Berkovich,Etale cohomology for non-Archimedean analytic spaces,Inst.Hautes Etudes Sci.Publ.Math.78(1993),5–161.[BGR]S.Bosch,U.G¨u ntzer and R.Remmert,Non-Archimedean Analysis,Springer-Verlag, 1984.[BL1]S.Bosch and W.L¨u tkebohmert,Stable Reduction and Uniformization of Abelian Vari-eties I,Math.Ann.270(1985),349–379.[BL2]S.Bosch and W.L¨u tkebohmert,Stable Reduction and Uniformization of Abelian Vari-eties II,Invent.Math.78(1984),257–297.[Bo1] A.Boutabaa,Theorie de Nevanlinna p-Adique,Manuscripta Math.67(1990),251–269. [Bo2] A.Boutabaa,Sur la th´e orie de Nevanlinna p-adique,Th´e se de Doctorat,Universit´e Paris 7,1991.[Bo3] A.Boutabaa,Applications de la theorie de Nevanlinna p-adique,Collect.Math.42 (1991),75–93.[Bo4] A.Boutabaa,Sur les courbes holomorphes p-adiques,Annales de la Facult´e des Sciences de Toulouse V(1996),29–52.[BEH] A.Boutabaa,A.Escassut,and L.Haddad,On uniqueness of p-adic entire functions, Indag.Math.(N.S.)8(1997),145–155.[Ch1]W.Cherry,Hyperbolic p-Adic Analytic Spaces,Ph.D.Thesis,Yale University,1993. [Ch2]W.Cherry,Non-Archimedean analytic curves in Abelian varieties,Math.Ann.300 (1994),393–404.[Ch3]W.Cherry,A non-Archimedean analogue of the Kobayashi semi-distance and its non-degeneracy on abelian varieties,Illinois J.Math.40(1996),123–140.8WILLIAM CHERRY[CYa]W.Cherry and C.-C.Yang,Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicity,Proc.Amer.Math.Soc.,to appear.[CYe]W.Cherry and Z.Ye,Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem,Trans.Amer.Math.Soc.349(1997), 5043–5071.[Co1] C.Corrales-Rodrig´a˜n ez,Nevanlinna Theory in the p-Adic Plane,Ph.D.Thesis,Univer-sity of Michigan,1986.[Co2] C.Corrales-Rodrig´a˜n ez,Nevanlinna Theory on the p-Adic Plane,Annales Polonici Math-ematici L VII(1992),135–147.[DM]P.Deligne and D.Mumford,The irreducibility of the space of curves of given genus,Inst.Hautes Etudes Sci.Publ.Math.No.36(1969),75–109.[GG]M.Green and P.Griffiths,Two applications of algebraic geometry to entire holomorphic mappings,The Chern Symposium1979(Proc.Internat.Sympos.,Berkeley,Calif.,1979), Springer-Verlag1980,41–74.[H`a1]H`a Huy Kho´a i,On p-Adic Meromorphic Functions,Duke Math.J.50(1983),695–711. [H`a2]H`a Huy Kho´a i,La hauteur des fonctions holomorphes p-adiques de plusieurs variables,C.R.Acad.Sci.Paris S´e r.I Math.312(1991),751–754.[HMa]H`a Huy Kho´a i and Mai Van Tu,p-Adic Nevanlinna-Cartan Theorem,Internat.J.Math.6(1995),719–731.[HMy]H`a Huy Kho´a i and My Vinh Quang,On p-adic Nevanlinna Theory,in Lecture Notes in Mathematics1351,Springer-Verlag1988,146–158.[Ne]R.Nevanlinna,Le th´e or`e me de Picard-Borel et la th´e orie des fonctions m´e romorphes, Paris,1929.Department of Mathematics,P.O.Box305118,University of North Texas,Denton, TX76203-5118,USAE-mail address:wcherry@。
读数学专业英文作文
读数学专业英文作文Mathematics is a field that has captivated the minds of scholars and thinkers throughout history. From the ancient Greek philosophers to the modern-day mathematicians, the pursuit of mathematical knowledge has been a driving force in the advancement of human understanding. For those who choose to embark on the journey of studying mathematics, the experience can be both challenging and rewarding.Pursuing a mathematics degree is a path that requires dedication, perseverance, and a deep appreciation for the intricacies of the subject. The curriculum is often rigorous, encompassing a wide range of topics, from calculus and linear algebra to abstract algebra and real analysis. Students are expected to develop a strong foundation in these core areas, as well as explore specialized fields such as number theory, topology, or computational mathematics.One of the key aspects of studying mathematics is the development of critical thinking and problem-solving skills. Mathematics is not merely about memorizing formulas and theorems; it is aboutunderstanding the underlying principles and applying them to solve complex problems. This process requires students to think logically, analyze information, and devise innovative solutions. These skills are highly valued in a wide range of industries, making a mathematics degree a versatile and valuable asset.Another important aspect of studying mathematics is the opportunity to engage in research and collaboration. Many mathematics programs offer undergraduate research opportunities, where students can work alongside faculty members on cutting-edge projects. This experience not only deepens their understanding of the subject but also provides valuable exposure to the research process, including the formulation of hypotheses, the design of experiments, and the communication of findings.In addition to the academic challenges, studying mathematics also requires a strong commitment to self-discipline and time management. The coursework can be demanding, with a heavy emphasis on problem-solving and mathematical proofs. Students must be able to manage their time effectively, prioritize their tasks, and maintain a consistent study routine to succeed. This discipline and time management skills are not only essential for academic success but also translate well to the professional world.One of the unique aspects of studying mathematics is theopportunity to collaborate with peers and engage in lively discussions. Mathematics classrooms often foster a sense of community, where students can share ideas, challenge one another's thinking, and learn from each other. This collaborative environment not only enhances the learning experience but also prepares students for the teamwork and communication skills required in many professional settings.Beyond the academic realm, studying mathematics can also open up a wide range of career opportunities. Mathematics graduates are highly sought after in fields such as finance, data analysis, computer science, engineering, and even government and policy-making. The problem-solving and analytical skills developed through a mathematics degree are highly valued in these industries, making graduates attractive candidates for a variety of roles.Furthermore, studying mathematics can be a gateway to advanced degrees and research opportunities. Many students who complete a mathematics degree go on to pursue graduate studies in fields such as mathematics, statistics, or operations research. These advanced programs offer the opportunity to delve deeper into the theoretical and applied aspects of mathematics, potentially leading to careers in academia, research, or specialized industries.In conclusion, reading a mathematics degree is a challenging butrewarding experience. It requires dedication, critical thinking, and a deep appreciation for the subject matter. However, the skills and knowledge gained through a mathematics education can be invaluable in a wide range of professional and academic pursuits. For those who are willing to embrace the rigor and intellectual challenge, a mathematics degree can be a transformative and empowering experience.。
LINEAR ALGEBRA 线性代数 课文 翻译
4 LINEAR ALGEBRA 线性代数“Linear algebra” is the study of linear sets of equations and their transformation properties. 线性代数是研究方程的线性几何以及他们的变换性质的Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.线性代数也研究空间旋转的分析，最小二乘拟合，耦合微分方程的解，确立通过三个已知点的一个圆以及在数学、物理和机械工程上的其他问题The matrix and determinant are extremely useful tools of linear algebra.矩阵和行列式是线性代数极为有用的工具One central problem of linear algebra is the solution of the matrix equation Ax = b for x. 线性代数的一个中心问题是矩阵方程Ax=b关于x的解While this can, in theory, be solved using a matrix inverse x = A−1b，other techniques such as Gaussian elimination are numerically more robust.理论上，他们可以用矩阵的逆x=A-1b求解，其他的做法例如高斯消去法在数值上更鲁棒。
方程的英语知识点总结
方程的英语知识点总结Key Concepts of Equations:1. Definition of an Equation: An equation is a mathematical statement that asserts the equality of two expressions, typically denoted as LHS = RHS, where LHS (left-hand side) and RHS (right-hand side) are mathematical expressions containing variables and constants.2. Variables and Constants: In an equation, variables are symbols that represent unknown quantities, while constants are fixed numerical values. Equations allow us to solve for the value of the variable by manipulating the given information and applying various mathematical operations.3. Solutions of an Equation: The solution of an equation is the value or set of values for the variable that make the equation true. A solution to an equation satisfies the equality relationship between the LHS and RHS.4. Solving Equations: The process of finding the solutions to an equation involves using algebraic techniques to manipulate the given expressions and isolate the variable. Common methods for solving equations include combining like terms, applying inverse operations, and factoring.5. Equivalent Equations: Two equations are said to be equivalent if they have the same solution set. Algebraic manipulations such as adding or subtracting the same quantity from both sides, multiplying or dividing both sides by the same non-zero number, and applying the properties of exponents can be used to derive equivalent equations.6. Applications of Equations: Equations are used to model various real-world scenarios, such as calculating the trajectory of a projectile, determining the growth of populations, analyzing the behavior of electrical circuits, and predicting the spread of infectious diseases. Types of Equations:1. Linear Equations: A linear equation is an equation of the form ax + b = c, where x is the variable, a and b are constants, and c is a constant term. The graph of a linear equation is a straight line, and the solutions to a linear equation form a single point, a line, or no points (in the case of parallel lines).2. Quadratic Equations: A quadratic equation is an equation of the form ax^2 + bx + c = 0, where x is the variable, and a, b, and c are constants with a ≠ 0. Quadratic equations have solutions that can be found using the quadratic formula, factoring, or completing the square. The graph of a quadratic equation is a parabola.3. Exponential Equations: An exponential equation is an equation in which the unknown variable appears as an exponent. Exponential equations arise in situations involving exponential growth or decay, such as population growth, radioactive decay, and compound interest problems.4. Trigonometric Equations: Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. These equations often arise in problems related to periodic phenomena, wave functions, and harmonic motion.Properties of Equations:1. Reflexive Property: For any real number a, a = a.2. Symmetric Property: If a = b, then b = a.3. Transitive Property: If a = b and b = c, then a = c.4. Addition Property of Equality: If a = b, then a + c = b + c.5. Subtraction Property of Equality: If a = b, then a - c = b - c.6. Multiplication Property of Equality: If a = b, then ac = bc.7. Division Property of Equality: If a = b and c ≠ 0, then a/c = b/c.8. Multiplicative Property of Zero: For any real number a, a × 0 = 0.9. Multiplicative Property of One: For any real number a, a × 1 = a.10. Distributive Property: For any real numbers a, b, and c, a(b + c) = ab + ac.In conclusion, equations are a vital aspect of mathematics and are used to express and solve a wide range of problems in various fields. Understanding the key concepts, types, and properties of equations is essential for mastering algebra and applying mathematical principles to real-world situations. By studying equations and their properties, one can develop problem-solving skills and analytical thinking, which are invaluable in academic, professional, and everyday life.。
IntroductionToLinearAlgebra第三版Wellesley-CambridgeP
Introduction To Linear Algebra 第三版 Wellesley-Cambridge Press线性代数是高等数学中一个非常重要而且基础的分支，它是数学中处理线性方程组和线性变换的工具箱。
在现代计算机科学、物理学、经济学等应用学科中也占有重要地位。
本文主要介绍“Introduction To Linear Algebra”第三版，该书由Gilbert Strang编著，由Wellesley-Cambridge Press 出版。
作者简介Gilbert Strang是MIT的应用数学教授，他在教学和研究中一直致力于线性代数的教学。
此外，他还出版了许多线性代数相关的著作，其中包括“Introduction To Linear Algebra”第三版。
书籍简介本书主要介绍的是线性代数的基本概念及其应用，包括向量、矩阵、线性变换、行列式、特征值和特征向量等。
同时，本书也包括了一些实际应用的例子，如利用线性代数解决最小二乘法和压缩图片等问题。
值得注意的是，第三版相比于第二版，增加了许多新内容，如介绍了一些新的应用，如机器学习和数据分析等。
内容也更加深入和详细，对于初学者来说是一个非常有帮助的指导。
书籍结构本书共分为十二章，每章均包含许多例子和练习题。
下面是每章的简要介绍：1.Introduction To Vectors：介绍向量和向量的基本运算。
2.Solving Linear Equations：介绍如何应用线性代数求解线性方程组。
3.Vector Spaces And Subspaces：介绍向量空间和子空间的概念及其性质。
4.Orthogonality：介绍正交向量、正交矩阵、Gram-Schmidt正交化过程以及投影的概念。
5.Determinants：介绍行列式及其性质，以及如何计算行列式。
6.Eigenvalues And Eigenvectors：介绍特征值和特征向量以及它们的应用。
IntroductiontoLinearAlgebra第二版课程设计
Introduction to Linear Algebra第二版课程设计1. 课程简介本课程是关于线性代数的介绍。
线性代数在数学和计算机科学中都有着重要的应用。
本课程将涵盖线性方程组、矩阵、向量空间、线性变换、特征值和特征向量等概念和基本技能。
在课程中，我们将会探索从高斯消元法和向量计算到线性变换和矩阵分解的一系列主题。
2. 课程目标•理解线性代数的基本原理和概念，包括向量和矩阵的基本操作，行列式、矩阵的特征值和特征向量、线性变换等。
•掌握线性代数的基本技能，如求解线性方程组、计算矩阵的逆、计算行列式和特征值特征向量等。
•能够使用线性代数解决实际问题，如建立系统方程模型、进行数据分析和机器学习。
3. 课程大纲3.1 向量和矩阵•向量的概念和基本操作•矩阵的概念和基本运算•线性组合和生成子空间3.2 线性方程组•高斯消元法及其变形•LU分解法和矩阵求解线性方程组•矩阵的逆和行列式3.3 基本矩阵操作和应用•矩阵乘法和转置•矩阵的秩和行空间、列空间、零空间•最小二乘法和矩阵的伪逆3.4 特征值和特征向量•线性变换和特征值特征向量的概念•对角化和相似矩阵•应用：对于连续变量模型的解析解和感性理解4. 教学方法本课程采用讲授、课堂互动和实验等多种教学方法。
•讲授：讲解课程重点和难点，引导学生理解和掌握知识点。
•课堂互动：通过提问和讨论等方式加强师生互动，促进学生主动参与和思考。
•实验：设置实验环节，让学生通过实践巩固知识点，培养解决问题的能力。
5. 评估方式本课程采用多种评估方式，包括作业、考试和小组项目等。
•作业：布置课堂和课后作业，帮助学生深入理解和掌握知识点。
•考试：每学期末进行闭卷考试，测试学生对于知识点的掌握和应用能力。
•小组项目：小组合作完成一个基于线性代数的实际问题或者算法实现，评估学生对于知识的实际运用能力。
6. 参考书目•Gilbert Strang. Introduction to Linear Algebra. [第二版]。
Solving Linear Equations in Algebra I求解线性方程组的代数I (2)
Recognize, express and solve problems that can be modeled using single-variable linear equations; one- or two-variable inequalities; or two-variable systems of linear equations.
Goals from the New Jersey Algebra I Core Content Standards
Students should
◦ Understand the big ideas of equivalence and linearity
◦ Modeling real situations with variables ◦ Use appropriate tools such as algebra tiles and
Recognize, express and solve problems that can be modeled using linear functions. Interpret solutions in terms of the context of the problem.
Solve single-variable linear equations and inequalities with rational coefficients.
These problems appear pretty simple because we are giving all the directions in short steps and you are performing them in the order in which they are described.
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FROM MAX-PLUS ALGEBRA TO NON-LINEARPERRON-FROBENIUS THEORYSt´e phane GaubertINRIA,Domaine de Voluceau,78153Le Chesnay C´e dex.Stephane.Gaubert@inria.frThe max-plus(or tropical)algebra is obtained by replacing the addition by the maximisation(or the minimisation)and the multiplication by the addition.It arises in the dynamic programming approach to deterministic optimal control.In particu-lar,the evolution semigroup of afirst order Hamilton-Jacobi equation is linear in the max-plus sense if the Hamiltonian is convex in the adjoint variable.This obser-vation has been made by several authors,including Maslov.It has motivated the development of a max-plus analogue of the theory of linear operators or linear semi-groups,including spectral theory.The max-plus eigenvalue gives the optimal mean payoffper time unit and the eigenvectors can be used to parametrise the stationary optimal strategies.Stochastic control or zero-sum game problems yield more general dynamic program-ming operators,which are no longer max-plus linear,but preserve the order and have some homogeneity or nonexpansiveness properties.Such operators belong to non-linear Perron-Frobenius theory,which allows one to include in a common pers-pective the positive linear operators and the max-plus linear ones.The goal of this talk is to present this perspective,which has inspired some results of non-linear spectral theory,has suggested algorithms,and has been helpful in applications. The max-plus spectral theorem allows one to represent any eigenvector as a linear combination of extremal eigenvectors.In thefinite dimensional case,the extremal eigenvectors correspond to“recurrence classes”;in the case of a noncompact state space,extremal eigenvectors correspond either to recurrence classes or to the limits of geodesics leading offto infinity(Busemann points)[1],[2].This is reminiscent of the Martin representation of harmonic functions,and this is related to the representation of weak-KAM solutions of Fathi in terms of the Aubry set,although the setting is different.The former results have been extended to non-linear eigenproblems arising in the study of stochastic control problems,but only in thefinite dimensional case[3]:the eigenspace is sup-norm isometric to a convex set,the dimension of which is bounded by the number of recurrences classes.Since convex duality plays an essential role in this result,the extension to zero-sum games seems out of range,however,some uniqueness results can still be obtained using semidifferentiability techniques(work in progress with Akian and Nussbaum).Using the same tools,one also obtains some information on the asymptotic behaviour of the iterates(i.e.,on value iteration),see in particular[3],[1],[4].These results have algorithmic consequences.For instance,the notions of[3]have been used in[5]to design a policy iteration algorithm for stochastic games with mean payoff,adapted to degenerate cases.We will describe two applications.Thefirst one shows a somehow unexpected occur-rence of techniques from optimisation and game theory in computer science,more precisely,in static analysis,the goal of which is to determine the invariants of pro-grams.The abstract interpretation method of Cousot reduces this problem to com-puting the smallestfixed point of an order preserving map.The latter is essentially the dynamic programming operator of a zero-sum game,to which policy iteration algorithms can be applied[6].The second application is to nonhomogeneous Markov chains arising in web ranking[7].Bibliography[1]M.Akian,S.Gaubert,and C.Walsh.Discrete max-plus spectral theory.In G.L. Litvinov and V.P.Maslov,Eds,Idempotent Mathematics and Mathematical Physics, Contemp.Maths,19–51.AMS,2005,arxiv:math.SP/0405225.[2]M.Akian,S.Gaubert,and C.Walsh.How tofind horizon-independent optimal strategies leading offto infinity:a max-plus approach.In Proc.of the45th IEEE Conference on Decision and Control,San Diego,2006.arXiv:math.OC/0609243.[3]M.Akian and S.Gaubert.Spectral theorem for convex monotone homogeneous maps,and ergodic control,Nonlinear Anal.,52(2):637–679,2003.[4]M.Akian,S.Gaubert,B.Lemmens,and R.Nussbaum.Iteration of order pre-serving subhomogeneous maps on a cone,Math.Proc.Cambridge Philos.Soc., 140(1):157–176,2006.[5]J.Cochet-Terrasson and S.Gaubert.A policy iteration algorithm for zero-sum stochastic games with mean payoff,C.R.Math.Acad.Sci.Paris,343(5):377–382, 2006.[6]S.Gaubert,E.Goubault,A.Taly,and S.Zennou.Static analysis by policy iteration in relational domains.In Proc.of ESOP’07,Springer LNCS,to appear.[7]M.Akian,S.Gaubert,and L.Ninove.The T-PageRank:a model of self-validating effects of web surfing.In Positive systems,Springer LNCIS341,2006.。