数学外文翻译
数学专业外文文献翻译
第3章 最小均方算法3．1 引言最小均方(LMS ,least-mean-square)算法是一种搜索算法，它通过对目标函数进行适当的调整[1]—[2],简化了对梯度向量的计算。
由于其计算简单性，LMS 算法和其他与之相关的算法已经广泛应用于白适应滤波的各种应用中[3]-[7]。
为了确定保证稳定性的收敛因子范围，本章考察了LMS 算法的收敛特征。
研究表明，LMS 算法的收敛速度依赖于输入信号相关矩阵的特征值扩展[2]—[6]。
在本章中，讨论了LMS 算法的几个特性，包括在乎稳和非平稳环境下的失调[2]—[9]和跟踪性能[10]-[12]。
本章通过大量仿真举例对分析结果进行了证实。
在附录B 的B ．1节中，通过对LMS 算法中的有限字长效应进行分析，对本章内容做了补充。
LMS 算法是自适应滤波理论中应用最广泛的算法，这有多方面的原因。
LMS 算法的主要特征包括低计算复杂度、在乎稳环境中的收敛性、其均值无俯地收敛到维纳解以及利用有限精度算法实现时的稳定特性等。
3．2 LMS 算法在第2章中，我们利用线性组合器实现自适应滤波器，并导出了其参数的最优解，这对应于多个输入信号的情形。
该解导致在估计参考信号以d()k 时的最小均方误差。
最优(维纳)解由下式给出：10w R p-= (3.1)其中，R=E[()x ()]Tx k k 且p=E[d()x()] k k ，假设d()k 和x()k 联合广义平稳过程。
如果可以得到矩阵R 和向量p 的较好估计，分别记为()R k ∧和()p k ∧，则可以利用如下最陡下降算法搜索式(3．1)的维纳解：w(+1)=w()-g ()w k k k μ∧w()(()()w())k p k R k k μ∧∧=-＋２ (3.2) 其中，k ＝0，1，2，…,g ()w k ∧表示目标函数相对于滤波器系数的梯度向量估计值。
一种可能的解是通过利用R 和p 的瞬时估计值来估计梯度向量，即 ()x()x ()TR k k k ∧=()()x()p k d k k ∧= (3.3) 得到的梯度估计值为()2()x()2x()x ()()T w g k d k k k k w k ∧=-+2x()(()x ()())Tk d k k w k =-+ 2()x()e k k =- (3.4)注意，如果目标函数用瞬时平方误差2()e k 而不是MSE 代替，则上面的梯度估计值代表了真实梯度向量，因为2010()()()()2()2()2()()()()Te k e k e k e k e k e k e k w w k w k w k ⎡⎤∂∂∂∂=⎢⎥∂∂∂∂⎣⎦2()x()e k k =-()w g k ∧= (3.5)由于得到的梯度算法使平方误差的均值最小化．因此它被称为LMS 算法，其更新方程为 (1)()2()x()w k w k e k k μ+=+ (3.6) 其中，收敛因子μ应该在一个范围内取值，以保证收敛性。
高等数学术语英语翻译一览
高等数学术语英语翻译一览微积分英文词汇,高数名词中英文对照,高等数学术语英语翻译一览V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标，一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cau ch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
数学专业外文翻译---幂级数的展开及其应用
数学专业外文翻译---幂级数的展开及其应用In the us n。
we XXX its convergence n。
a power series always converges to a n。
We can use simple power series。
as well as XXX quadrature methods。
to find this n。
However。
this n will address another issue: can an arbitrary n f(x) be expanded into a power series?XXX n will address this XXX power series can be seen as an n of reality。
so we can start to solve the problem of expanding a n f(x) into a power series by considering f(x) and polynomials。
To do this。
we will introduce the following formula without proof:Taylor'XXX that if a n f(x) has derivatives of order n+1 in a neighborhood of x=x0.then we can use the following XXX:f(x)=f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)^2+。
+f^(n)(x0)(x-x0)^n+r_n(x)Here。
r_n(x) represents the remainder term.XXX (x) is given by (x-x)n+1.This formula is of the (9-5-1) type for the Taylor series。
数学专业外文翻译--多元函数的极值
外文文献EXTREME VALUES OF FUNCTIONS OF SEVERALREAL VARIABLES1. Stationary PointsDefinition 1.1 Let n R D ⊆ and R D f →:. The point a D a ∈ is said to be:(1) a local maximum if )()(a f x f ≤for all points x sufficiently close to a ;(2) a local minimum if )()(a f x f ≥for all points x sufficiently close to a ;(3) a global (or absolute) maximum if )()(a f x f ≤for all points D x ∈;(4) a global (or absolute) minimum if )()(a f x f ≥for all points D x ∈;;(5) a local or global extremum if it is a local or global maximum or minimum. Definition 1.2 Let n R D ⊆ and R D f →:. The point a D a ∈ is said to be critical or stationary point if 0)(=∇a f and a singular point if f ∇ does not exist at a .Fact 1.3 Let n R D ⊆ and R D f →:.If f has a local or global extremum at the point D a ∈, then a must be either:(1) a critical point of f , or(2) a singular point of f , or(3) a boundary point of D .Fact 1.4 If f is a continuous function on a closed bounded set then f is bounded and attains its bounds.Definition 1.5 A critical point a which is neither a local maximum nor minimum is called a saddle point.Fact 1.6 A critical point a is a saddle point if and only if there are arbitrarily small values of h for which )()(a f h a f -+ takes both positive and negative values.Definition 1.7 If R R f →2: is a function of two variables such that all second order partial derivatives exist at the point ),(b a , then the Hessian matrix of f at ),(b a is the matrix⎪⎪⎭⎫ ⎝⎛=yy yxxy xx f f f f H where the derivatives are evaluated at ),(b a . If R R f →3: is a function of three variables such that all second order partial derivatives exist at the point ),,(c b a , then the Hessian of f at ),,(c b a is the matrix⎪⎪⎪⎭⎫ ⎝⎛=zz zy zx yz yy yx xz xy xx f f f f f f f f f H where the derivatives are evaluated at ),,(c b a .Definition 1.8 Let A be an n n ⨯ matrix and, for each n r ≤≤1,let r A be the r r ⨯ matrix formed from the first r rows and r columns of A .The determinants det(r A ),n r ≤≤1,are called the leading minors of ATheorem 1.9(The Leading Minor Test). Suppose that R R f →2:is a sufficiently smooth function of two variables with a critical point at ),(b a and H the Hessian of f at ),(b a .If 0)det(≠H , then ),(b a is:(1) a local maximum if 0>det(H 1) = f xx and 0<det(H )=2xy yy xx f f f -;(2) a local minimum if 0<det(H 1) = f xx and 0<det(H )=2xy yy xx f f f -;(3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at ),(b a .Suppose that R R f →3: is a sufficiently smooth function of three variables with a critical point at ),,(c b a and Hessian H at ),,(c b a .If 0)det(≠H , then ),,(c b a is:(1) a local maximum if 0>det(H 1), 0<det(H 2) and 0>det(H 3);(2) a local minimum if 0<det(H 1), 0<det(H 2) and 0>det(H 3);(3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at ),,(c b a .In each case, if det(H )= 0, then ),(b a can be either a local extremum or a saddleExample. Find and classify the stationary points of the following functions:(1) ;1),,(2224+++++=xz z y y x x z y x f(2) ;)1()1(),(422++++=x y x y y x fSolution. (1) 1),,(2224+++++=xz z y y x x z y x f ,so)24),(3z xy x y x f ++=∇（i )2(2y x ++j )2(x z ++kCritical points occur when 0=∇f ,i.e. when(1) z xy x ++=2403(2) y x 202+=(3) x z +=20Using equations (2) and (3) to eliminate y and z from (1), we see that 021433=--x x x or 0)16(2=-x x ,giving 0=x ,66=x and 66-=x .Hence we have three stationary points: ）（0,0,0,）（126,121,66-- and ）（126,121,66--. Since y x f xx 2122+=,x f xy 2=,1=xz f ,2=yy f ,0=yz f and 2=zz f ,the Hessian matrix is⎪⎪⎪⎭⎫ ⎝⎛+=201022122122x x y x H At ）（126,121,66--, ⎪⎪⎪⎪⎭⎫ ⎝⎛=201023/613/66/11H which has leading minors 611>0, 039631123/63/66/11det >=-=⎪⎪⎭⎫ ⎝⎛ And det 042912322>=--=H .By the Leading Minor Test, then, ）（126,121,66--is a local minimum. At ）（126,121,66--, ⎪⎪⎪⎪⎭⎫ ⎝⎛--=201023/613/66/11H which has leading minors 611>0,039631123/63/66/11det >=-=⎪⎪⎭⎫ ⎝⎛ And det 042912322>=--=H .By the Leading Minor Test, then, ）（126,121,66--is also a local minimum. At ）（0,0,0, the Hessian is⎪⎪⎪⎭⎫ ⎝⎛=201020100HSince det 2)(-=H , we can apply the leading minor test which tells us that this is a saddle point since the first leading minor is 0. An alternative method is as follows. In this case we consider the value of the expressionhl l k k h h l k h f f D ++++=+++-=22240,0,00,0,0）（）（，for arbitrarily small values of h, k and l. But for very small h, k and l , cubic terms and above are negligible in comparison to quadratic and linear terms, sothat hl l k D ++≈22.If h, k and l are all positive, 0>D . However, if 0=k and 0<h and h l <<0,then 0<D .Hence close to ）（0,0,0,f both increases and decreases, so ）（0,0,0 is a saddle point.(2) 422)1()1(),(++++=x y x y y x f so))1(4)1(2(),(3+++=∇x y x y x f i ))1(2(2+++x y j .Stationary points occur when 0=∇f ,i.e. at )0,1(-.Let us classify this stationary point without considering the Leading Minor Test (in this case the Hessian has determinant 0 at )0,1(- so the test is not applicable). Let.0,10,1422h k h k k h f f D ++=++---=）（）（Completing the square we see that .43)2(222h h k D ++=So for any arbitrarily small values of h and k , that are not both 0, 0>D and we see that f has a local maximum at )0,1(-.2. Constrained Extrema and Lagrange MultipliersDefinition 2.1 Let f and g be functions of n variables. An extreme value of f (x )subject to the condition g (x) = 0, is called a constrained extreme value and g (x ) = 0 is called the constraint.Definition 2.2 If R R f n →: is a function of n variables, the Lagrangian function of f subject to the constraint 0),,,(21=n x x x g is the function of n+1 variables),,,,(),,,(),,,,(212121n n n x x x g x x x f x x x L λλ+=where is known as the Lagrange multiplier.The Lagrangian function of f subject to the k constraints0),,,(21=n i x x x g ,k i ≤≤1, is the function with k Lagrange multipliers,i λk i ≤≤1,∑=+=ki n n n x x x g x x x f x x x L 1212121),,,(),,,(),,,,( λλTheorem 2.3 Let R R f →2: and ),(00y x P = be a point on the curve C, withequation g(x,y) = 0, at which f restricted to C has a local extremum.Suppose that both f and g have continuous partial derivatives near to P and that P is not an end point of C and that 0),(00≠∇y x g . Then there is some λ such that ),,(000z y x is a critical point of the Lagrangian Function),(),(),,(y x g y x f y x L λλ+=.Proof. Sketch only. Since P is not an end point and 0≠∇g ,C has a tangent at P with normal g ∇.If f ∇ is not parallel to g ∇at P , then it has non-zero projection along this tangent at P .But then f increases and decreases away from P along C ,so P is not an extremum. Hence f ∇and g ∇are parallel and there is some¸such that g f ∇-=∇λ and the result follows.Example. Find the rectangular box with the largest volume that fits inside the ellipsoid 1222222=++cz b y a x ,given that it sides are parallel to the axes. Solution. Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one corner of the box be corner (x, y, z) in the positive octant, then the box has corners (±x,±y,±z) and its volume is V= 8xyz .We want to maximize V given that 01222222=-++cz b y a x . (Note that since the constraint surface is bounded a max/min does exist). The Lagrangian is⎪⎪⎭⎫ ⎝⎛-+++=18),,,(222222c z b y a x xyz z y x L λλ and this has critical points when 0=∇L , i.e. when,.280,28022b y zx y L a x yz x L λλ+=∂∂=+=∂∂=⎪⎪⎭⎫ ⎝⎛-++=∂∂=+=∂∂=10,2802222222c z b y a x z L c z xy z L λ (Note that λL will always be the constraint equation.) As we want to maximize V we can assume that 0≠xyz so that 0,,≠z y x .)Hence, eliminating λ, we get,444222zxy c y zx b x yz a -=-=-=λ so that 2222b x a y = and .2222c y b z =But then 222222c z b y a x ==so 2222222231ax c z b y a x =++= or 3a x =,which implies that 3b y = and 3c z = (they are all positive by assumption). So L has only one stationary point ),3,3,3(λc b a (for some value of λ, which we could work out if we wanted to). Since it is the only stationary point it must the required max and the max volume is3383338abc c b a =.中文译文 多元函数的极值1. 稳定点定义1.1 使n R D ⊆并且R D f →:. 对于任意一点D a ∈有以下定义:(1)如果)()(a f x f ≤对于所有x 充分地接近a 时，则)(a f 是一个局部极大值;(2)如果)()(a f x f ≥对于所有x 充分地接近a 时，则)(a f 是一个局部极小值;(3)如果)()(a f x f ≤对于所有点D x ∈成立，则)(a f 是一个全局极大值（或绝对极大值）;(4) 如果)()(a f x f ≥对于所有点D x ∈成立，则)(a f 是一个全局极小值（或绝对极小值）; (5) 局部极大（小）值统称为局部极值；全局极大（小）值统称为全局极值.定义 1.2 使n R D ⊆并且R D f →:.对于任意一点D a ∈，如果0)(=∇a f ，并且对于任意奇异点a 都不存在f ∇，则称a 是一个关键点或稳定点.结论 1.3 使n R D ⊆并且R D f →:.如果f 有局部极值或全局极值对于一点D a ∈, 则a 一定是:(1)函数f 的一个关键点, 或者(2)函数f 的一个奇异点, 或者(3)定义域D 的一个边界点.结论 1.4 如果函数f 是一个在闭区间上的连续函数，则f 在区间上有边界并且可以取到边界值.定义 1.5 对于任一个关键点a ，当a 既不是局部极大值也不是局部极小值时，a 叫做函数的鞍点.结论 1.6 对于一个关键点a 是鞍点当且仅当h 任意小时，对于函数)()(a f h a f -+取正值和负值.定义 1.7 如果R R f →2: 是二元函数，并且在点),(b a 处所有二阶偏导数都存在，则则根据函数f 在点),(b a 处导数，有f 在点),(b a 处的Hessian 矩阵为：⎪⎪⎭⎫ ⎝⎛=yy yx xy xxf ff f H . 推广：如果R R f →3: 是三元函数，并且在点),,(c b a 处所有二阶偏导数都存在，则根据函数f 在点),,(c b a 处导数，有f 在点),,(c b a 处的Hessian 矩阵为：⎪⎪⎪⎭⎫ ⎝⎛=zz zyzxyz yy yxxz xy xxf f f f f f f f f H . 定义 1.8 矩阵A 是n n ⨯ 阶矩阵，并且对于每一个都有n r ≤≤1,从矩阵A 中选取左上端的r 行和r 列，令其为r r ⨯阶的矩阵r A .则行列式det(r A ),n r ≤≤1,叫做矩阵A 的顺序主子式.定理 1.9 假如R R f →2:是一个充分光滑的二元函数，且在点),(b a 处稳定，其Hessian 矩阵为H .如果0)det(≠H ，则根据偏导数判定),(b a 点是：(1) 一个局部极大值点， 如果0>det(H 1) = f xx 并且0<det(H )=2xy yy xx f f f -; (2) 一个局部极小值点， 如果0<det(H 1) = f xx 并且0<det(H )=2xy yy xx f f f -;(3) 一个鞍点，如果点),(b a 既不是局部极大值点也不是局部极小值点. 假如R R f →3:是一个充分光滑的三元函数，且在点),,(c b a 处稳定，其Hessian 矩阵为H .如果0)det(≠H ，则根据偏导数判定),,(c b a 点是： (1) 一个局部极大值点， 如果当0>det(H 1), 0<det(H 2) 并且 0>det(H 3)时; (2) 一个局部极小值点， 如果当0<det(H 1), 0<det(H 2) 并且 0>det(H 3)时; (3) 一个鞍点，如果点),,(c b a 既不是局部极大值点也不是局部极小值点. 在不同的情况下 ,当det(H )= 0时, 点),(b a 是一个局部极值点，或者是一个鞍点.例. 确定下列函数的稳定点并说明是哪一类点: (1) ;1),,(2224+++++=xz z y y x x z y x f (2) ;)1()1(),(422++++=x y x y y x f 解. (1) 1),,(2224+++++=xz z y y x x z y x f ,so)24),(3z xy x y x f ++=∇（i )2(2y x ++j )2(x z ++k当0=∇f 时有稳定点,也就是说, 当(1) z xy x ++=2403 (2) y x 202+= (3) x z +=20时,将方程(2)和方程(3)带入到方程(1)可以消去变量y 和z, 由此可以得到021433=--x x x 即0)16(2=-x x ,得0=x ,66=x 和66-=x .因此我们可以得到函数的三个稳定点:）（0,0,0,）（126,121,66--和）（126,121,66--. 又因为y x f xx 2122+=,x f xy 2=,1=xz f ,2=yy f ,0=yz f 和2=zz f ,则Hessian 矩阵为⎪⎪⎪⎭⎫⎝⎛+=201022122122x x y x H在点）（126,121,66--处, ⎪⎪⎪⎪⎭⎫⎝⎛=201023/613/66/11H则顺序主子式611>0, 039631123/63/66/11>=-=并且行列式042912322>=--=H .根据主子式判定方法，则点）（126,121,66--是一个局部极小值点.在点）（126,121,66--处, ⎪⎪⎪⎪⎭⎫ ⎝⎛--=201023/613/66/11H则顺序主子式 611>0,039631123/63/66/11>=-=-- 并且行列式042912322>=--=H .根据主子式判定方法，则点）（126,121,66--也是一个极小值点.在点）（0,0,0处，Hessian 矩阵为⎪⎪⎪⎭⎫⎝⎛=201020100H因此det 2)(-=H ,根据主子式判定方法，第一主子式为0，由此我们可以知道该点是一个鞍点. 下面是另一种计算方法，在这种情况下，我们考虑现在下面函数表达式hl l k k h h l k h f f D ++++=+++-=22240,0,00,0,0）（）（，的值，对于任意h, k 和l 无限小时. 担当h, k 和l 非常小时, 三次及三次以上方程相对线性二次方程时可忽略不计,则原方程可为hl l k D ++≈22.当h, k 和l 都为正时,0>D .然而， 当0=k 、0<h 和h l <<0,则0<D .因此当接近）（0,0,0时,f 同时增加或者同时减少, 所以 ）（0,0,0是一个鞍点. (2) 422)1()1(),(++++=x y x y y x f so))1(4)1(2(),(3+++=∇x y x y x f i ))1(2(2+++x y j .当0=∇f 时有稳定点,也就是说, 当在)0,1(-时.现在我们在不考虑主子式判定方法的情况下为该稳定点进行分类（因为在)0,1(-时Hessian 矩阵的行列式为0，所以该判定方法在此刻无法应用）.令.0,10,1422h k h k k h f f D ++=++---=）（）（配成完全平方的形式为.43)2(222h h k D ++=所以对h 和k 为任意小时（h 和k 都不为0），有0>D ，因此我们可以确定函数f 在点)0,1(-处有局部极大值.2. 条件极值和Lagrange 乘数法定义 2.1 函数f 和函数g 都是n 元函数.对于限制在条件g (x) = 0下的函数f (x )的极值叫做函数的条件极值，函数g (x ) = 0叫做限制条件.定义 2.2 如果函数R R f n →: 是一个n 元函数, 则对应于函数f 的Lagrange 函数在限制条件0),,,(21=n x x x g 下的函数是一个n +1元函数),,,,(),,,(),,,,(212121n n n x x x g x x x f x x x L λλ+=这就是著名的Lagrange 乘数法.对应于函数f 的Lagrange 函数在k 个限制条件0),,,(21=n i x x x g ,k i ≤≤1时, 带有k 个i λk i ≤≤1,的Lagrange 函数为：∑+=kn n n x x x g x x x f x x x L 212121),,,(),,,(),,,,( λλ定理 2.3 使R R f →2:并且),(00y x P =是曲线C 上的一个点, 有方程 g(x,y) = 0成立，则在限制条件C 上函数f 有局部极值.假设函数f 和函数g 在点P 都有连续的偏导数，点P 不是曲线C 的端点，且0),(00≠∇y x g . 因此存在λ的值使得点),,(000z y x 是Lagrange 函数的关键点),(),(),,(y x g y x f y x L λλ+=.证明.仅仅描述. 因为点P 不是曲线C 的端点，且0≠∇g ,则曲线C 在点P 处的切线与g ∇有关.如果f ∇在点P 处与g ∇平行，则函数在点P 处的切线有非零值.但另一方面函数 f 的值随着P 在C 的运动增加减小,所以点P 不是极值点. 因为f ∇和g ∇平行，所以存在λ使得g f ∇-=∇λ成立.例. 求内接于椭球1222222=++cz b y a x 的体积最大的长方体的体积，长方体的各个面平行于坐标面解：明显地，当长方体的体积最大时，长方体的各个顶点一定在椭球上. 设长方体的一个顶点坐标为(x, y, z) (x>0, y>0, z>0), 则长方体的其他顶点坐标分别为(±x,±y,±z)，并且长方体的体积为V= 8xyz.我们要求V 在条件01222222=-++cz b y a x 下的最大值. (注意：因为约束条件是有边界的，故其一定存在极大或者极小值). 其Lagrange 函数为⎪⎪⎭⎫⎝⎛-+++=18),,,(222222c z b y a x xyz z y x L λλ并且存在稳定点当0=∇L 时，也就是说，当,.280,280,280222cz xy z L b yzx y L a x yz x L λλλ+=∂∂=+=∂∂=+=∂∂=⎪⎪⎭⎫ ⎝⎛-++=∂∂=10222222c z b y a x z L 时.(注意：λL 是约束方程.要想求得体积V 的最大值，假设0≠xyz ，则可得0,,≠z y x .)因此, 用其他式子表示λ, 我们可以得到,444222zxyc y zx b x yz a -=-=-=λ 消去λ，有2222b x a y =和.2222c y b z =进而得出 222222cz b y a x ==，因此有2222222231ax c z b y a x =++=或者得出3a x =,同理可得出3by =和3c z = (根据假设可得x, y, z 都是正值).所以函数 L 有且仅有一个稳定点),3,3,3(λc b a (λ为某一计算可得到的常数). 又因为该点是函数L 的唯一稳定点，则该稳定点一定是所要求的最大值点，故其体积的最大值为3383338abcc b a =.。
（完整版）各种数学名词的英语翻译
（完整版）各种数学名词的英语翻译各种数学名词的英语翻译数学 mathematics, maths(BrE), math(AmE)公理 axiom定理 theorem计算 calculation运算 operation证明 prove假设 hypothesis,hypotheses(pl.)命题 proposition算术 arithmetic加 plus(prep.), add(v.), addition(n.)被加数 augend, summand加数 addend和 sum减 minus(prep.),subtract(v.),subtraction(n.)被减数 minuend减数 subtrahend差 remainder乘 times(prep.),multiply(v.),multiplication(n.)被乘数 multiplicand, faciend 乘数 multiplicator 积 product除 divided by(prep.),divide(v.), division(n.)被除数 dividend除数 divisor商 quotient等于 equals, is equal to, is equivalent to 大于 is greater than小于 is lesser than大于等于 is equal or greater than小于等于 is equal or lesser than运算符 operator 平均数mean算术平均数arithmatic mean几何平均数geometric mean n个数之积的n次方根倒数（reciprocal） x的倒数为1/x有理数 rational number无理数 irrational number实数 real number虚数 imaginary number数字 digit数 number自然数 natural number整数 integer小数 decimal小数点 decimal point分数 fraction分子 numerator分母 denominator比 ratio正 positive负 negative零 null, zero, nought, nil十进制 decimal system二进制 binary system十六进制 hexadecimal system 权 weight, significance进位 carry截尾 truncation四舍五入 round下舍入 round down上舍入 round up有效数字 significant digit无效数字 insignificant digit 代数 algebra公式 formula, formulae(pl.) 单项式 monomial多项式 polynomial, multinomial系数 coefficient未知数 unknown, x-factor,y-factor, z-factor等式，方程式 equation一次方程 simple equation二次方程 quadratic equation 三次方程 cubic equation四次方程 quartic equation不等式 inequation阶乘 factorial对数 logarithm指数，幂 exponent乘方 power二次方，平方 square三次方，立方 cube四次方 the power of four, the fourth powern次方 the power of n, the nth power开方 evolution, extraction二次方根，平方根 square root 三次方根，立方根 cube root 四次方根 the root of four,the fourth rootn次方根 the root of n, the nth rootsqrt(2)=1.414sqrt(3)=1.732sqrt(5)=2.236常量 constant变量 variable坐标系 coordinates坐标轴 x-axis, y-axis,z-axis横坐标 x-coordinate纵坐标 y-coordinate原点 origin象限quadrant截距(有正负之分)intercede （方程的）解solution几何geometry点 point线 line面 plane体 solid线段 segment射线 radial平行 parallel相交 intersect角 angle角度 degree弧度 radian锐角 acute angle直角 right angle钝角 obtuse angle平角 straight angle周角 perigon底 base边 side高 height三角形 triangle锐角三角形 acute triangle直角三角形 right triangle直角边 leg斜边 hypotenuse勾股定理 Pythagorean theorem 钝角三角形 obtuse triangle 不等边三角形 scalenetriangle等腰三角形 isoscelestriangle等边三角形 equilateral triangle四边形 quadrilateral平行四边形 parallelogram矩形 rectangle长 length宽 width周长 perimeter面积 area相似 similar全等 congruent三角 trigonometry正弦 sine余弦 cosine正切 tangent余切 cotangent正割 secant余割 cosecant反正弦 arc sine反余弦 arc cosine反正切 arc tangent反余切 arc cotangent 反正割 arc secant 反余割 arc cosecant补充：集合aggregate元素 element空集 void子集 subset交集 intersection并集 union补集 complement映射 mapping函数 function定义域 domain, field ofdefinition值域 range单调性 monotonicity奇偶性 parity周期性 periodicity图象 image数列，级数 series微积分 calculus微分 differential导数 derivative极限 limit无穷大 infinite(a.)infinity(n.)无穷小 infinitesimal积分 integral定积分 definite integral不定积分 indefinite integral 复数 complex number矩阵 matrix行列式 determinant圆 circle圆心 centre(BrE),center(AmE)半径 radius直径 diameter圆周率 pi弧 arc半圆 semicircle扇形 sector环 ring椭圆 ellipse圆周 circumference轨迹 locus, loca(pl.)平行六面体 parallelepiped立方体 cube七面体 heptahedron八面体 octahedron九面体 enneahedron十面体 decahedron十一面体 hendecahedron十二面体 dodecahedron二十面体 icosahedron多面体 polyhedron旋转 rotation轴 axis球 sphere半球 hemisphere底面 undersurface表面积 surface area体积 volume空间 space双曲线 hyperbola抛物线 parabola四面体 tetrahedron五面体 pentahedron六面体 hexahedron菱形 rhomb, rhombus, rhombi(pl.), diamond正方形 square梯形 trapezoid直角梯形 right trapezoid等腰梯形 isosceles trapezoid五边形 pentagon六边形 hexagon七边形 heptagon八边形 octagon九边形 enneagon十边形 decagon十一边形 hendecagon十二边形 dodecagon多边形 polygon正多边形 equilateral polygon相位 phase周期 period振幅 amplitude内心 incentre(BrE),incenter(AmE)外心 excentre(BrE),excenter(AmE)旁心 escentre(BrE),escenter(AmE)垂心 orthocentre(BrE), orthocenter(AmE)重心 barycentre(BrE), barycenter(AmE)内切圆 inscribed circle外切圆 circumcircle统计 statistics平均数 average加权平均数 weighted average 方差 variance 标准差 root-mean-square deviation, standard deviation比例 propotion百分比 percent百分点 percentage百分位数 percentile排列 permutation组合 combination概率，或然率 probability分布 distribution正态分布normal distribution 非正态分布abnormal distribution图表 graph条形统计图 bar graph柱形统计图 histogram折线统计图 broken line graph 曲线统计图 curve diagram扇形统计图 pie diagram--------------数学中常用单词术语abscissa 横坐标 absolute value 绝对值acute angle 锐角adjacent angle 邻角addition 加algebra 代数altitude 高angle bisector 角平分线arc 弧area 面积arithmetic mean 算术平均值（总和除以总数）arithmetic progression等差数列（等差级数）arm 直角三角形的股at 总计（乘法）average 平均值base 底be contained in 位于...上bisect 平分center 圆心chord 弦circle 圆形circumference 圆周长circumscribe 外切，外接clockwise 顺时针方向closest approximation 最相近似的combination 组合common divisor 公约数，公因子common factor 公因子complementary angles 余角（二角和为90度）composite number 合数（可被除1及本身以外其它的数整除）concentric circle 同心圆cone 圆锥（体积＝1/3*pi*r*r*h）congruent 全等的consecutive integer 连续的整数coordinate 坐标的cost 成本counterclockwise 逆时针方向cube 1.立方数2.立方体（体积＝a*a*a 表面积＝6*a*a）cylinder 圆柱体decagon 十边形decimal 小数decimal point 小数点decreased 减少decrease to 减少到decrease by 减少了degree 角度define 1.定义2.化简denominator 分母denote 代表，表示depreciation 折旧distance 距离distinct 不同的dividend 1. 被除数 2.红利divided evenly 被除数divisible 可整除的division 1.除 2.部分divisor 除数down payment 预付款，定金equation 方程equilateral triangle 等边三角形even number 偶数expression 表达exterior angle 外角 face （立体图形的）某一面factor 因子fraction 1.分数2.比例geometric mean 几何平均值（N个数的乘积再开N次方）geometric progression 等比数列（等比级数）have left 剩余height 高hexagon 六边形 hypotenuse 斜边improper fraction 假分数increase 增加increase by 增加了 increase to 增加到 inscribe 内切，内接intercept 截距integer 整数interest rate 利率 in terms of... 用...表达interior angle 内角 intersect 相交irrational 无理数 isosceles triangle 等腰三角形least common multiple 最小公倍数least possible value 最小可能的值leg 直角三角形的股length 长list price 标价margin 利润mark up 涨价 mark down 降价maximum 最大值median, medium 中数（把数字按大小排列，若为奇数项，则中间那项就为中数，若为偶数项，则中间两项的算术平均值为中数。
数学与应用数学论文中英文资料外文翻译文献
数学与应用数学论文中英文资料外文翻译文献UNITS OF M EASUR EM ENT AND FUNC TIONAL FOR M ( V o t i n g O u t c o m e s a n d C a m p a i g n E x p e n d i t u r e s )In the voting outcome equation in (2.28), R = 0.505. Thus, the share of campaign expenditures explains just over 50 percent of the variation in the election outcomes for this sample. This is a fairly sizable portionTwo important issues in applied economics are (1) understanding how changing theunits of measurement of the dependent and/or independent variables affects OLS estimates and (2) knowing how to incorporate popular functional forms used in e conomi c s i nt o regres s i o n analysis. The mathemati c s ne e ded for a ful l un de rs t anding of functional form issues is reviewed in Appendix A.The Effects of Changing Units of Measurement on OLSStatisticsIn Example 2.3, we chose to measure annual salary in thousands of dollars, and t he return on e quit y was mea s ured as a perc e n t (ra t her than a s a dec i ma l). I t is c ruci a l to know how salary and roe are measured in this example in order to make sense of the estimates in equation (2.39). We must also know that OLS estimates change in entirely expected ways when the units of measurement of the dependent and independent variables change. In Example2.3, suppose that, rather than measuring s a l ary in thousands of do l la rs, we m ea s u re it i n doll a rs. Let sal a rdol be sal a ry i n dollars (salardol =845,761 would be interpreted as $845,761.). Of course, salardol has a simple relationship to the salary measured in thousands of dollars: salardol ? 1,000? salary. We do not need to actually run the regression of salardol on roe to know that the estimated equation is: salaˆrdol = 963,191 +18,501 roe.We obtain the intercept and slope in (2.40) simply by multiplying the intercept and theslope in (2.39) by 1,000. This gives equations (2.39) and (2.40) the same interpretation.Looking at (2.40), if roe = 0, then salaˆrdol = 963,191, so the predicted salary is $963,191 [the same value we obtained from equation (2.39)]. Furthermore, if roe increases by one, then the predicted salary increases by $18,501; again, this isw hat w econcluded from our earlier analysis of equation (2.39).Generally, it is easy to figure out what happens to the intercept and slope estimates when the dependent variable changes units of measurement. If the dependent variable is multiplied by the constant c—which means each value in the s a m ple is multi pl i ed b yc—t h en t he OLS in t ercept a nd s lope esti m at es are als o multiplied by c. (This assumes nothing has changed about the independent variable.) In the CEO salary example, c ?1,000 in moving from salary to salardol.Chapter 2T he Sim pl e Re g re s sion ModelWe can also use the CEO salary example to see what happens when we change the units of measurement of the independent variable. Define roedec =roe/100 to be t he d e cimal equiva l ent of ro e; t hus, roedec =0.23 means a return o n equi ty of23 percent. To focus on changing the unitsof measurement of the independent variable, we return to our original dependent variable, salary, which is measured in thousands of dollars. When we regress salary onroedec, we obtain salˆary =963.191 + 1850.1 roedec.T he coef fi ci e nt on roedec is 100 times t he coe ffi cient on roe i n (2.39). This i s as it should be. Changing roe by one percentage point is equivalent to Δroedec = 0.01. From (2.41), if Δ roedec = 0.01, then Δ salˆary = 1850.1(0.01) = 18.501, which is what is obtained by using (2.39). Note that, in moving from (2.39) to (2.41), the independentv ariable was divided b y 100, and so t h e OLS slope estim a te was multiplied by 100, preserving the interpretation of the equation. Generally, if the independent variable is divided or multiplied by some nonzero constant, c, then the OLS slope coefficient is also multiplied or divided by c respectively.The intercept has not changed in (2.41) because roedec =0 still corresponds to a z ero retur n on equity. In ge n eral, changin g t he uni t s of m easurem e nt of only the independent variable does not affect the intercept.In the previous section, we defined R-squared as a goodness-of-fit measure for OLS regression. We can also ask what happens to R2 when the unit of measurement of either the independent or the dependent variable changes. Without doing any algebra, we should know the result: the goodness-of-fit of the model should not depend on the units of measurement of our variables. For example, the amount of variation in salary, explained by the return on equity, should not depend on whether salary is measured in dollars or in thousands of dollars or on whether return on equity is a percent or a decimal. This intuition can be verified mathematically: using the definition of R2, it can be shown that R2 is, in fact, invariant to changes in the units of y or x.Incor por a ting Nonlinear ities in Simple R egressionSo far we have focused on linear relationships between the dependent and independent variables. As we mentioned in Chapter 1, linear relationships are notn early gen er a l enou g h for a ll e co nomi c a pplications. F ortuna t ely, it is rathe r e a s y to incorporate many nonlinearities into simple regression analysis by appropriately defining the dependent and independent variables. Here we will cover two possibilities that often appear in applied work.In reading applied work in the social sciences, you will often encounter re gr es sion equati o ns w he re the de pende nt varia bl e a pp ears in l og arit hm i c f orm. W h y is this done? Recall the wage-education example, where we regressed hourly wage on years of education. We obtained a slope estimate of 0.54 [ see equation (2.27)], which means that each additional year of education is predicted to increase hourly wage by54 cents.B ecaus e of t he l i near n at ur e of (2.27), 54 c ents is the i ncrea s e f or e i ther the fi rst year of education or the twentieth year; this may not be reasonable.Suppose, instead, that the percentage increase in wage is the same given one m ore yea r of e ducation. Model (2.27) does no t im ply a c onst a nt per c entag e i nc re ase: the percentage increases depends on the initial wage. A model that gives (approximately) a constant percentage effect is log(wage) =β 0 +β 1educ + u,(2.42) where log(.) denotes the natural logarithm. (See Appendix A for a review of logarithms.) In particular, if Δu =0, then % Δwage = (100* β 1) Δ educ.(2.43) N otice ho w we mult i ply β 1 b y 100 t o g et the perc e ntage change in w a ge give n one additional year of education. Since the percentage change in wage is the same for each additional year of education, the change in wage for an extra year of education increases aseducation increases; in other words, (2.42) implies an increasing return to education.B y e x ponenttiat i ng (2.42), we c an w ri t e wage =ex p(β 0+β 1educ + u). T his equationis graphed in Figure 2.6, with u = 0.Estimating a model such as (2.42) is straightforward when using simple regression. Just define the dependent variable, y, to be y = log(wage). The i ndependent v ar i able is represented b y x = e duc. The mechanics of O L S are the sa m e as before: the intercept and slope estimates are given by the formulas (2.17) and (2.19). In other words, we obtain β ˆ0 andβ ˆ1 from the OLS regression of log(wage) on educ.E X A M P L E 2 . 1 0( A L o g W a g e E q u a t i o n )Using the same data as in Example 2.4, but using log(wage) as the dependent variable, we obtain the following relationship: log(ˆwage) =0.584 +0.083 educ(2.44) n = 526, R =0.186.The coefficient on educ has a percentage interpretation when it is multiplied by 100: wage increases by 8.3 percent for every additional year of education. This is what economists mean when they refer to the “return to another year of education.”It is important to remember that the main reason for using the log of wage in (2.42) is to impose a constant percentage effect of education on wage. Once equation (2.42) is obtained, the natural log of wage is rarely mentioned. In particular, it is not correct to say that another year of education increases log(wage) by 8.3%.The intercept in (2.42) is not very meaningful, as it gives the predicted log(wage), when educ =0. The R-squared shows that educ explains about 18.6 percent of the variation in log(wage) (not wage). Finally, equation (2.44) might not capture all of the non-linearity in the relationship between wage and schooling.If there are“diplomae ffects,”t hen t he twelft h ye a r of e ducat i on—gradu a ti on from hi gh s c hool—c ould be worth much more than the eleventh year. We will learn how to allow for this kind of nonlinearity in Chapter 7. Another important use of the natural log is in obtaining a constant elasticity model.E X A M P L E 2 . 1 1( C E O S a l a r y a n d F i r m S a l e s )We can estimate a constant elasticity model relating CEO salary to firm sales. The data set is the same one used in Example 2.3, except we now relate salary to sales. Let sales be annual firm sales, measured in millions of dollars. A constant elasticity model is log(salary =β 0 +β 1log(sales) +u, (2.45) where β 1 is the elasticity of s a l ary w ith respe c t to sal es. T h is model fa ll s under t he simple regressio n model by defining the dependent variable to be y = log(salary) and the independent variable to be x = log(sales). Estimating this equation by OLS givesPart 1Regression Analysis with Cross-Sectional Data l og(salˆary)= 4.822 ?+0.257 log(sa le s)(2.46)n =209, R = 0.211.The coefficient of log(sales) is the estimated elasticity of salary with respect to sales. It implies that a 1 percent increase in firm sales increases CEO salary by about 0.257 pe r cent—t he usual int e rp re tation of an e la s ti c ity.The two functional forms covered in this section will often arise in the remainder of this text. We have covered models containing natural logarithms here because they a ppear so freque nt ly in a ppl ied wo r k. The i nterpr e tat i on of such m odel s w i l l n ot be much different in the multiple regression case.It is also useful to note what happens to the intercept and slope estimates if we change the units of measurement of the dependent variable when it appears in logarithmic form.B ecaus e t he ch a nge t o log ar i thm i c form approx i mates a proportionate change, i t makes sense that nothing happens to the slope. We can see this by writing the rescaled variable as c1yi for each observation i. The original equation is log(yi) =β 0 +β 1xi +ui. If we add log(c1) to both sides, we get log(c1) + log(yi) + [log(c1) β 0] +β 1xi + ui, orlog(c1yi) ? [log(c1) +β 0] +β 1xi +ui.(Remember that the sum of the logs is equal to the log of their product as shown in Appendix A.) Therefore, the slope is still ? 1, but the intercept is now log(c1) ? ? 0. Similarly, if the independent variable is log(x), and we change the units of measurement of x before taking the log, the slope remains the same but the intercept does not change. You will be asked to verify these claims in Problem 2.9.We end this subsection by summarizing four combinations of functional forms available from using either the original variable or its natural log. In Table 2.3, x and y stand for the variables in their original form. The model with y as the dependent variable and x as the independent variable is called the level-level model, because each variable appears in its level form. The model with log(y) as the dependentv ariable a nd x as t he independent va r i able i s called t he l og-level m od el. We w i ll no t explicitly discuss the level-log model here, because it arises less often in practice. In any case, we will see examples of this model in later chapters.Chapter 2Th e Simpl e R e gr e s s i on M od e lTable 2.3The last column in Table 2.3 gives the interpretation of β 1. In the log-level model, 100* β 1 i s so m e t imes called the s emi-elasti c ity of y wit h re s pe ct to x. As we mentioned in Example 2.11, in the log-log model, β1 is the elasticity of y with respect to x. Table 2.3 warrants careful study, as we will refer to it often in the r em aind er of the text.The Meaning of“Linear”RegressionThe simple regression model that we have studied in this chapter is also called the simple linear regression model. Yet, as we have just seen, the general model also allows for certain nonlinear relationships. So what does “linear”mean here? You can se e b y looking a t equ a ti o n (2.1) tha t y =β 0 +β 1x + u. The key i s t hat t his equati on i s linear in the parameters, β 0 and β 1. There are no restrictions on how y and x relate to the original explained and explanatory variables of interest. As we saw in Examples 2.7 and 2.8, y and x can be natural logs of variables, and this is quite common in applications. But we need not stop there. For example, nothing prevents us from using simple regression to estimate a model such as cons =β 0 +β 1√inc+u, where cons is annual consumption and inc is annual income.While the mechanics of simple regression do not depend on how y and x are defined, the interpretation of the coefficients does depend on their definitions. For successful empirical work, it is much more important to become proficient at interpreting coefficients than to become efficient at computing formulas such as (2.19). We will get much more practice with interpreting the estimates in OLS regression lines when we study multiple regression.There are plenty of models that cannot be cast as a linear regression model because they are not linear in their parameters; an example is cons = 1/(β 0 +β 1inc) + u.E s t im a ti on of such mode l s ta ke s us into t he real m of t he nonli ne ar regressi on model, which is beyond the scope of this text. For most applications, choosing a model that can be put into the linear regression framework is sufficient.EXPECTED VAL UES AND VAR IANCES OF THE OLSESTIM ATOR SI n Sec t ion 2.1, we defined the popula t ion m ode l y =β 0 +β 1x +u, a nd w e claimed that the key assumption for simple regression analysis to be useful is that the expected value of u given any value of x is zero. In Sections 2.2, 2.3, and 2.4, we discussed the algebraic properties of OLS estimation. We now return to the population model and study the statistical properties of OLS. In other words, we now view β ˆ0 a nd β ˆ1 as e s timat ors for th e pa rameters ? 0 and ? 1 t ha t appear in t he popula t ion model. This means that we will study properties of the distributions of ? ˆ0 and ? ˆ1 over different random samples from the population. (Appendix C contains definitions of estimators and reviews some of their important properties.)Unbiasedness of OLSW e be g i n by establishing the unbi a s e dne s s of OLS unde r a simple set of assumptions.For future reference, it is useful to number these assumptions using the prefix“S LR”for simple linear regression. The first assumption defines the population model.测量单位和函数形式在投票结果方程（2.28）中，R²=0.505。
数学专业英语翻译24
This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction.
严格地说，这种关于正整数的描述是不完整的，因 为我们没有详细解释“等等”或者“1的重复累加” 的含义。
Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.
New Words & Expressions:
irrational number 无理数 rational 有理的
the order axiom 序公理 rational number 有理数
ordered 有序的
reasoning 推理
product 积
应用数学专业外文翻译
本科毕业论文外文翻译外文译文题目（中文）：具体数学：汉诺塔问题学院:专业:学号:学生姓名:指导教师:日期: 二○一二年六月1 Recurrent ProblemsTHIS CHAPTER EXPLORES three sample problems that give a feel for what’s to c ome. They have two traits in common: They’ve all been investigated repeatedly by mathe maticians; and their solutions all use the idea of recurrence, in which the solution to eac h problem depends on the solutions to smaller instances of the same problem.1.1 THE TOWER OF HANOILet’s look first at a neat little puzzle called the Tower of Hanoi,invented by the Fr ench mathematician Edouard Lucas in 1883. We are given a tower of eight disks, initiall y stacked in decreasing size on one of three pegs:The objective is to transfer the entire tower to one of the other pegs, movingonly one disk at a time and never moving a larger one onto a smaller.Lucas furnished his toy with a romantic legend about a much larger Tower of Brah ma, which supposedly has 64 disks of pure gold resting on three diamond needles. At th e beginning of time, he said, God placed these golden disks on the first needle and orda ined that a group of priests should transfer them to the third, according to the rules abov e. The priests reportedly work day and night at their task. When they finish, the Tower will crumble and the world will end.It's not immediately obvious that the puzzle has a solution, but a little thought (or h aving seen the problem before) convinces us that it does. Now the question arises：What's the best we can do？That is，how many moves are necessary and suff i cient to perfor m the task?The best way to tackle a question like this is to generalize it a bit. The Tower of Brahma has 64 disks and the Tower of Hanoi has 8；let's consider what happens if ther e are TL disks.One advantage of this generalization is that we can scale the problem down even m ore. In fact, we'll see repeatedly in this book that it's advantageous to LOOK AT SMAL L CASES first. It's easy to see how to transfer a tower that contains only one or two di sks. And a small amount of experimentation shows how to transfer a tower of three.The next step in solving the problem is to introduce appropriate notation：NAME ANO CONQUER. Let's say that T n is the minimum number of moves that will t ransfer n disks from one peg to another under Lucas's rules. Then T1is obviously 1 , an d T2= 3.We can also get another piece of data for free, by considering the smallest case of all：Clearly T0= 0，because no moves at all are needed to transfer a tower of n = 0 disks! Smart mathematicians are not ashamed to think small,because general patterns are easier to perceive when the extreme cases are well understood(even when they are trivial).But now let's change our perspective and try to think big；how can we transfer a la rge tower? Experiments with three disks show that the winning idea is to transfer the top two disks to the middle peg, then move the third, then bring the other two onto it. Thi s gives us a clue for transferring n disks in general：We first transfer the n−1 smallest t o a different peg (requiring T n-1moves), then move the largest (requiring one move), and finally transfer the n−1 smallest back onto the largest (req uiring another T n-1moves). Th us we can transfer n disks (for n > 0)in at most 2T n-1+1 moves：T n≤2T n—1+1，for n > 0.This formula uses '≤' instead of '=' because our construction proves only that 2T n—1+1 mo ves suffice; we haven't shown that 2T n—1+1 moves are necessary. A clever person might be able to think of a shortcut.But is there a better way? Actually no. At some point we must move the largest d isk. When we do, the n−1 smallest must be on a single peg, and it has taken at least Tmoves to put them there. We might move the largest disk more than once, if we're n n−1ot too alert. But after moving the largest disk for the last time, we must trans fr the n−1 smallest disks (which must again be on a single peg)back onto the largest；this too re quires T n−1moves. HenceT n≥ 2T n—1+1，for n > 0.These two inequalities, together with the trivial solution for n = 0, yieldT0=0；T n=2T n—1+1 , for n > 0. (1.1)(Notice that these formulas are consistent with the known values T1= 1 and T2= 3. Our experience with small cases has not only helped us to discover a general formula, it has also provided a convenient way to check that we haven't made a foolish error. Such che cks will be especially valuable when we get into more complicated maneuvers in later ch apters.)A set of equalities like (1.1) is called a recurrence (a. k. a. recurrence relation or r ecursion relation). It gives a boundary value and an equation for the general value in ter ms of earlier ones. Sometimes we refer to the general equation alone as a recurrence, alt hough technically it needs a boundary value to be complete.The recurrence allows us to compute T n for any n we like. But nobody really like to co m pute fro m a recurrence，when n is large；it takes too long. The recurrence only gives indirect, "local" information. A solution to the recurrence would make us much h appier. That is, we'd like a nice, neat, "closed form" for Tn that lets us compute it quic kly，even for large n. With a closed form, we can understand what T n really is.So how do we solve a recurrence? One way is to guess the correct solution,then to prove that our guess is correct. And our best hope for guessing the solution is t o look (again) at small cases. So we compute, successively,T3= 2×3+1= 7; T4= 2×7+1= 15; T5= 2×15+1= 31; T6= 2×31+1= 63.Aha! It certainly looks as ifTn = 2n−1，for n≥0. (1.2)At least this works for n≤6.Mathematical induction is a general way to prove that some statement aboutthe integer n is true for all n≥n0. First we prove the statement when n has its smallest v alue,no; this is called the basis. Then we prove the statement for n > n0,assuming that it has already been proved for all values between n0and n−1, inclusive; this is called th e induction. Such a proof gives infinitely many results with only a finite amount of wo rk.Recurrences are ideally set up for mathematical induction. In our case, for exampl e，(1.2) follows easily from (1.1)：The basis is trivial，since T0 = 20−1= 0.And the indu ction follows for n > 0 if we assume that (1.2) holds when n is replaced by n−1：T n= 2T n+1= 2(2n−1−1)+1=2n−1.Hence (1.2) holds for n as well. Good! Our quest for T n has ended successfully.Of course the priests' task hasn't ended；they're still dutifully moving disks,and wil l be for a while, because for n = 64 there are 264−1 moves (about 18 quintillion). Even at the impossible rate of one move per microsecond, they will need more than 5000 cent uries to transfer the Tower of Brahma. Lucas's original puzzle is a bit more practical, It requires 28−1 = 255 moves, which takes about four minutes for the quick of hand.The Tower of Hanoi recurrence is typical of many that arise in applications of all kinds. In finding a closed-form expression for some quantity of interest like T n we go t hrough three stages：1 Look at small cases. This gives us insight into the problem and helps us in stages2 and 3.2 Find and prove a mathematical expression for the quantity of interest.For the Tower of Hanoi, this is the recurrence (1.1) that allows us, given the inc lination，to compute T n for any n.3 Find and prove a closed form for our mathematical expression.For the Tower of Hanoi, this is the recurrence solution (1.2).The third stage is the one we will concentrate on throughout this book. In fact, we'll fre quently skip stages I and 2 entirely, because a mathematical expression will be given to us as a starting point. But even then, we'll be getting into subproblems whose solutions will take us through all three stages.Our analysis of the Tower of Hanoi led to the correct answer, but it r equired an“i nductive leap”；we relied on a lucky guess about the answer. One of the main objectives of this book is to explain how a person can solve recurrences without being clairvoyant. For example, we'll see that recurrence (1.1) can be simplified by adding 1 to both sides of the equations：T0+ 1= 1;T n + 1= 2T n-1+ 2, for n >0.Now if we let U n= T n+1，we haveU0 =1；U n= 2U n-1，for n > 0. (1.3)It doesn't take genius to discover that the solution to this recurrence is just U n= 2n；he nce T n= 2n −1. Even a computer could discover this.Concrete MathematicsR. L. Graham, D. E. Knuth, O. Patashnik《Concrete Mathematics》，1.1 ，The Tower Of HanoiR. L. Graham, D. E. Knuth, O. PatashnikSixth printing, Printed in the United States of America1989 by Addison-Wesley Publishing Company，Reference 1-4 pages具体数学R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克《具体数学》，1.1，汉诺塔R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克第一版第六次印刷于美国，韦斯利出版公司，1989年，引用1-4页1 递归问题本章将通过对三个样本问题的分析来探讨递归的思想。
数学 外文翻译 外文文献 英文文献 具体数学
Concrete MathematicsR. L. Graham, D. E. Knuth, O. Patashnik《Concrete Mathematics》，1.3 THE JOSEPHUS PROBLEM R. L. Graham, D. E. Knuth, O. Patashnik Sixth printing, Printed in the United States of America 1989 by Addison-Wesley Publishing Company，Reference 1-4pages具体数学R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克《具体数学》，1.3，约瑟夫环问题R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克第一版第六次印刷于美国，韦斯利出版公司，1989年，引用8-16页1.递归问题本章研究三个样本问题。
这三个样本问题给出了递归问题的感性知识。
它们有两个共同的特点：它们都是数学家们一直反复地研究的问题；它们的解都用了递归的概念，按递归概念，每个问题的解都依赖于相同问题的若干较小场合的解。
2.约瑟夫环问题我们最后一个例子是一个以Flavius Josephus命名的古老的问题的变形，他是第一世纪一个著名的历史学家。
据传说，如果没有Josephus的数学天赋，他就不可能活下来而成为著名的学者。
在犹太|罗马战争中，他是被罗马人困在一个山洞中的41个犹太叛军之一，这些叛军宁死不屈，决定在罗马人俘虏他们之前自杀，他们站成一个圈，从一开始，依次杀掉编号是三的倍数的人，直到一个人也不剩。
但是在这些叛军中的Josephus和他没有被告发的同伴觉得这么做毫无意义，所以他快速的计算出他和他的朋友应该站在这个恶毒的圆圈的哪个位置。
在我们的变形了的问题中，我们以n个人开始，从1到n编号围成一个圈，我们每次消灭第二个人直到只剩下一个人。
例如，这里我们以设n= 10做开始。
数学术语翻译.doc
一，代数部分1.有关数学运算add，plus加 subtract减 difference差 multiply,times乘 product积 divide除divisible可被整除的 dividedevenly被整除 dividend被除数，红利 divisor因子,除数quotient商 remainder余数 factorial阶乘 power乘方 radicalsign,rootsign根号roundto四舍五入 tothenearest四舍五入2.有关集合union并集 pro per subset真子集 solution set解集3. 有关代数式、方程和不等式algebraic term代数项 like terms,similar terms同类项 numerical coefficient数字系数literal coefficient字母系数 inequality不等式triangle inequality三角不等式range值域 original equation原方程 equivalent equation同解方程，等价方程linear equation线性方程(e.g.5 x +6=22)4. 有关分数和小数proper fraction真分数 improper fraction假分数 mixed number带分数 vulgar fraction，common fraction普通分数 simple fraction简分数 complex fraction繁分数numerator分子 denominator分母 (least)common denominator(最小)公分母quarter四分之一 decimal fracti o n纯小数 infinite decimal无穷小数recurring decimal循环小数 tenthsunit十分位5.基本数学概念arithmetic mean算术平均值 weighted average加权平均值 geometric mean几何平均数exponent指数，幂 base乘幂的底数,底边 cube立方数，立方体 square root平方根 cuberoot立方根 common logarithm常用对数 digit数字 constant常数 variable 变量inversefunction反函数 complementary function余函数 linear一次的，线性的factorization因式分解 absolute value绝对值，e.g.｜-32｜=32 round off四舍五入6. 有关数论natural number自然数 positive number正数 negative number负数 odd integer,odd number奇数 even integer,even number偶数 integer,whole number整数 positive whole number正整数negative whole number负整数 consecutive number连续整数 real number, rational number实数,有理数 irrational(number)无理数 inverse倒数 composite number合数e.g.4,6,8,9,10,12,14,15…… prime number质数e.g.2,3,5,7,11,13,15……注意：所有的质数(2除外)都是奇数，但奇数不一定是质数reciprocal倒数 common divisor公约数multiple倍数 (least)common multiple(最小)公倍数 (prime)factor(质)因子common factor公因子ordinaryscale,decimalscale十进制 nonnegative非负的 tens十位 units个位 mode众数（可以不止一个） median中数（如果有偶数个，取中间两个数的平均数） common ratio公比 standard diviation标准方差，方差是一组数据中的每一个数与这组数据的平均数的差的平方的和再除以数据的个数，标准方差是方差开方7. 数列arithmetic progression(sequence)等差数列 geometric progression(sequence)等比数列8. 其它approximate近似 (anti)clockwise(逆)顺时针方向 cardinal基数 ordinal序数directproportion正比 distinct不同的 estimation估计，近似 parenthe ses括号proportion比例 permutation排列 combination组合 table表格 trigonometricfunction三角函数 unit单位,位二，几何部分1.所有的角alternate angle内错角 corresponding angle同位角 vertical angle对顶角central angle 圆心角 interior angle内角 exterior angle外角 supplementary angles补角 complementary angle余角 adjacent angle邻角 acute angle锐角obtuse angle钝角 right angle直角round angle周角 straight angle平角included angle夹角2. 所有的三角形equilateral triangle等边三角形 scalene triangle不等边三角形 isosceles triangle等腰三角形right triangle直角三角形 oblique斜三角形 inscribed triang le内接三角形3. 有关收敛的平面图形，除三角形外semicircle半圆concentric circles同心圆 quadrilateral四边形 pentagon五边形 hexagon六边形 heptagon 七边形 octagon八边形 nonagon九边形 decagon十边形 polygon多边形parallelogram平行四边形 equilateral等边形 plane平面 square正方形，平方rectangle长方形 regular polygon正多边形 rho mbus菱形 trapezoid梯形4. 其它平面图形arc弧 line,straight line直线 line segment线段 parallel lines平行线 segment of a circle弧形5. 有关立体图形cube立方体，立方数 rectangular solid长方体 regular solid/regular polyhedron正多面体circular cylinder圆柱体 cone圆锥 sphere球体 solid立体的6. 有关图形上的附属物altitude高 depth深度 side边长 circumference,perimeter周长 radian弧度 surface area 表面积volume体积 arm直角三角形的股 cross section横截面 center of acircle圆心chord弦radius半径 angle bisector角平分线 diagonal对角线 diameter直径 edge棱face of a solid立体的面 hypotenuse斜边 included side夹边 leg三角形的直角边medianofatriangle三角形的中线 base底边，底数(e.g.2的5次方，2就是底数)opposite直角三角形中的对边 midpoint中点 endpoint端点 vertex(复数形式vertices)顶点tangent切线的 transversal截线 intercept截距7. 有关坐标coordinate system坐标系 rectangular coordinate直角坐标系 origin原点 abscissa横坐标ordinate纵坐标 numberli n e数轴 quadrant象限 slope斜率 complex plane复平面8. 其它plane geometry平面几何 trigonometry三角学 bisect平分 circumscribe外切 inscribe内切intersect相交 perpendicular垂直 pythagorean theorem勾股定理congruent全等的multilateral多边的三，其它1. 单位类cent美分 penny一美分硬币 nickel5美分硬币 dime一角硬币 dozen打(12个) score廿(20个)Centigrade摄氏 Fahrenheit华氏 quart夸脱 gallon加仑(1gallon=4quart) yard码meter米micron微米 inch英寸 foot英尺 minute分(角度的度量单位，60分=1度) squaremeasure平方单位制cubicmeter立方米 pint品脱(干量或液量的单位)2. 有关文字叙述题，主要是有关商业intercalary year(leapyear)闰年(366天) common y ear平年(365天) depreciation折旧 down payment直接付款 discount打折 margin利润 profit利润 interest利息 simple interest单利compounded interest复利 dividend红利 decrease to减少到 decrease by减少了 increase to增加到 increase by增加了 denote表示 list price标价 markup涨价 per capita每人ratio比率 retai l price零售价 tie打平units个位十位tens 十分位tenths to the nearest hundredths近似到百分位digit number数位，页码，比如：47的digit是2位product乘积consecutive positive integer连续正整数the least common multiple最小公倍数the greatest common divisor最大公约数凸多边形内角和公式：（n-2）×180各314，157，628里面有个派同一段孤或弦对应的圆心角是圆周角的两倍圆锥体积=1/3×底面积×高圆锥侧面积=l×派×r/2 球体积=3/4派r的3次方球面积=4派r的平方抛物线开口朝向排列组合：Pnk Cnk 概率：目标事件除以总事件，比如：打字、拿球，投骰子，走格子，照相图表题：要仔细读题，特别是星号部分是解题线索数学总共错2道题，还是800分。
常用数学词汇英语翻译
常用数学词汇abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直 与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线 ；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线 ；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西- 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切 commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲 ; 圆锥曲conic section 二次曲 ; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲 conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer's rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre's theorem 棣美弗定理De Morgan's law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法) determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns (variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler's formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of…以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat's last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron's formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder's Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke's law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线 ；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分)invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分)kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L'Hospital's rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami's law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直 ；最大斜率line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度measurement 量度mechanical energy 机械能median (1)中位数；(2)中线meet 相交；相遇mensuration 计量；求积法method 方法method of completing square 配方法method of interpolation 插值法; 内插法method of least squares 最小二乘法; 最小平方法method of substitution 代换法；换元法method of successive substitution 逐次代换法; 逐次调替法method of superposition 迭合法metric unit 十进制单位mid-point 中点mid-point formula 中点公式mid-point theorem 中点定理million 百万minimize 极小minimum point 极小点minimum value 极小值Minkowski Inequality 闵可夫斯基不等式minor (1)子行列式；(2)劣；较小的minor arc 劣弧；小弧minor axis 短轴minor of a determinant 子行列式minor sector 劣扇形；小扇形minor segment 劣弓形；小弓形minus 减minute 分mixed number(fraction) 带分数modal class 众数组mode 众数model 模型modulo (1)模; 模数; (2)同余modulo arithmetic 同余算术modulus 模; 模数modulus of a complex number 复数的模modulus of elasticity 弹性模(数)moment arm (1)矩臂; (2)力臂moment of a force 力矩moment of inertia 贯性矩momentum 动量monomial 单项式monotone 单调monotonic convergence 单调收敛性monotonic decreasing 单调递减monotonic decreasing function 单调递减函数monotonic function 单调函数monotonic increasing 单调递增monotonic increasing function 单调递增函数motion 运动movable pulley 动滑轮multinomial 多项式multiple 倍数multiple angle 倍角multiple-angle formula 倍角公式multiple root 多重根multiplicand 被乘数multiplication 乘法multiplication law (of probability) (概率)乘法定律multiplicative inverse 乘法逆元multiplicative property 可乘性multiplicity 重数multiplier 乘数；乘式multiply 乘multi-value 多值的mulually disjoint 互不相交mutually exclusive events 互斥事件mutually independent 独立; 互相独立mutually perpendicular lines 互相垂直n factorial n阶乘n th derivative n阶导数n th root n次根；n次方根n the root of unity 单位的n次根Napierian logarithm 纳皮尔对数; 自然对数natural logarithm 自然对数natural number 自然数natural surjection 自然满射necessary and sufficient condition 充要条件necessary condition 必要条件negation 否定式negative 负negative angle 负角negative binomial distribution 负二项式分布negative index 负指数negative integer 负整数negative number 负数negative vector 负向量; 负矢量neighborhood 邻域net 净(值)net force 净力Newton-Cote's rule 牛顿- 高斯法则Newton-Raphson's method 牛顿- 纳逊方法Newton's formula 牛顿公式Newton's law of motion 牛顿运动定律Newton's method 牛顿方法n-gon n边形nonagon 九边形non-collinear 不共线non-commutative 非交换的。
数学外文+中文翻译
SIAM J. DISCRETE MATH.V ol. 26, No. 1, pp. 193–205ROMAN DOMINATION ON 2-CONNECTED GRAPHS∗CHUN-HUNG LIU†AND GERARD J. CHANG‡Abstract. A Roman dominating function of a graph G is a function f: V (G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f is w(f) = . The Roman domination number of G is the minimum weight of a Roman dominating function of G Chambers,Kinnersley, Prince, and West [SIAM J. Discrete Math.,23 (2009), pp. 1575–1586] conjectured that ≤[2n/3] for any 2-connected graph G of n vertices.This paper gives counterexamples to the conjecture and proves that≤max{[2n/3], 23n/34}for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which = 23n/34 when 23n/34 > [2n/3].Key words. domination, Roman domination, 2-connected graphAMS. subject classifications. 05C69, 05C35D O I. 10.1137/0807330851. Introduction. Articles by ReVelle [14, 15] in the Johns Hopkins Magazine suggested a new variation of domination called Roman domination; see also [16] for an integer programming formulation of the problem. Since then, there have been several articles on Roman domination and its variations [1, 2, 3, 4, 5, 7, 8, 9, 10,11, 13, 17, 18, 19]. Emperor Constantine imposed the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armies, stationary and traveling. Each vertex (city) that has no army must have a neighboring vertex with a traveling army. Stationary armies then dominate their own vertices; a vertex with two armies is dominated by its stationary army, and its open neighborhood is dominated by the traveling army.In this paper, we consider (simple) graphs and loopless multigraphs G with vert ex set V (G) and edge set E(G). The degree of a vertex v∈V (G) is the number of edges incident to v. Note that the number of neighbors of v may be less than degGv in a loopless multigraph. A Roman dominating function of a graph G is a function f:V(G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f, denoted by w(f), is defined as.For any subgraph H of G, let w(f,H) =. The Roman dominationnumber of G is the minimum weight of a Roman dominating function.Among the papers mentioned above, we are most interested in the one by Chambers et al. [2] in which extremal problems of Roman domination are discussed.In particular, they gave sharp bounds for graphs with minimum degree 1 or 2 and boundsof + and . After settling some special cases, they gave the following conjecture in an earlier version of the paper [2].Conjecture (Chambers et al. [2]). For any 2-connected graph G of n vertices, ≤[2n/3]。
数学与应用数学英文文献及翻译
数学与应用数学英文文献及翻译-勾股定理(外文翻译从原文第一段开始翻译,翻译了约2000字)勾股定理是已知最早的古代文明定理之一。
这个著名的定理被命名为希腊的数学家和哲学家毕达哥拉斯。
毕达哥拉斯在意大利南部的科托纳创立了毕达哥拉斯学派。
他在数学上有许多贡献,虽然其中一些可能实际上一直是他学生的工作。
毕达哥拉斯定理是毕达哥拉斯最著名的数学贡献。
据传说,毕达哥拉斯在得出此定理很高兴,曾宰杀了牛来祭神,以酬谢神灵的启示。
后来又发现2的平方根是不合理的,因为它不能表示为两个整数比,极大地困扰毕达哥拉斯和他的追随者。
他们在自己的认知中,二是一些单位长度整数倍的长度。
因此2的平方根被认为是不合理的,他们就尝试了知识压制。
它甚至说,谁泄露了这个秘密在海上被淹死。
毕达哥拉斯定理是关于包含一个直角三角形的发言。
毕达哥拉斯定理指出,对一个直角三角形斜边为边长的正方形面积,等于剩余两直角为边长正方形面积的总和图1根据勾股定理,在两个红色正方形的面积之和A和B,等于蓝色的正方形面积,正方形三区因此,毕达哥拉斯定理指出的代数式是:对于一个直角三角形的边长a,b和c,其中c是斜边长度。
虽然记入史册的是著名的毕达哥拉斯定理,但是巴比伦人知道某些特定三角形的结果比毕达哥拉斯早一千年。
现在还不知道希腊人最初如何体现了勾股定理的证明。
如果用欧几里德的算法使用,很可能这是一个证明解剖类型类似于以下内容:六^维-论~文.网“一个大广场边a+ b是分成两个较小的正方形的边a和b分别与两个矩形A 和B,这两个矩形各可分为两个相等的直角三角形,有相同的矩形对角线c。
四个三角形可安排在另一侧广场a+b中的数字显示。
在广场的地方就可以表现在两个不同的方式:1。
由于两个长方形和正方形面积的总和:2。
作为一个正方形的面积之和四个三角形:现在,建立上面2个方程,求解得因此,对c的平方等于a和b的平方和(伯顿1991)有许多的勾股定理其他证明方法。
一位来自当代中国人在中国现存最古老的含正式数学理论能找到对Gnoman和天坛圆路径算法的经典文本。
数学专业英语第二版的课文翻译
1－A What is mathematics Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics. 数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。
反过来，数学服务于实践，并在各个领域中起着非常重要的作用。
没有应用数学，任何一个现在的科技的分支都不能正常发展。
From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying . To deal with some more complex practical problems, man established and then solved equation with unknown numbers ,thus algebra occurred. Before 17th century, man confined himself to the elementary mathematics, i.e. , geometry, trigonometry and algebra, in which only the constants are considered. 很早的时候，人类的需要产生了数和形式的概念，接着，测量土地的需要形成了几何，出于测量的需要产生了三角几何，为了处理更复杂的实际问题，人类建立和解决了带未知参数的方程，从而产生了代数学，17世纪前，人类局限于只考虑常数的初等数学，即几何，三角几何和代数。
数学名词英文翻译(Mathematical terms, English translation)
数学名词英文翻译（Mathematical terms, English translation）代数 algebra1. 数论Natural number positive number negative number 负数自然数正数 odd odd integer.Number 奇数 even integer, even number 偶数整数 positive whole number integer.Whole number 正整数 negative whole number 负整数 consecutive number 连续整数realNumber and rational number 实数, 有理数 irrational (number) 无理数 inverse 倒数compositeNumber 合数 e.g. 4,6,8,9,10,12,14,15... Prime number 质数e.g. 2,3,5,7,11,13,15...Reciprocal 倒数common 公约数 divisor multiple 倍数 (minimum) common multiple (最小) 公倍数(prime) factor (质) 因子 common factor 公因子ordinary decimal scale,Scale 十进制 nonnegative 非负的 have 十位 units 个位mode 众数 mean平均数 median中值 commonThe ratio 公比2. 基本数学概念算术平均值 arithmetic mean geometric mean 几何平均数exponent weighted average 加权平均值指数, 幂 base 乘幂的底数, 底边 cube 立方数, 立方体 square root 平方根cube root 立方根 common logarithm常用对数 digit 数字 constant 常数 variable 变量inverse function 反函数 complementaryFunction 余函数 linear 一次的, 线性的 factorization 因式分解 absolute value绝对值, e.g.｜ - 32｜ = 32 round off 四舍五入数学3. 基本运算Add, plus 加 subtract 减 difference 差 multiply, times 乘product 积 divides除divisible 可被整除的 divided evenly 被整除 dividend 被除数, 红利 divider 因子, 除数, 公约数商 quotient remainder 余数 factorial 阶乘 power 乘方radical sign, root sign 根号Round to 四舍五入 to the nearest 四舍五入4. 代数式, 方程, 不等式Algebraic term 代数项 like terms, similar terms 同类项numerical coefficient数字系数literal coefficient 字母系数 inequality 不等式triangle inequality 三角不等式 range 值域The original equation 原方程 equivalent equation 同解方程, 等价方程 linear equation 线性方程 (e.g.5x + 6 = 22)5. 分数, 小数Proper fraction 真分数 improper fraction 假分数 mixed number 带分数 vulgarFraction, common fraction simple fraction 简分数 complex fraction 普通分数繁分数numerator 分子 denominator 分母 (least) common denominator (最小) 公分母 quarter四分之一 decimal fraction 纯小数 infinite decimal recurring decimal 循环小数无穷小数Tenths unit 十分位6. 集合Union 并集 proper subset 真子集 solution set 解集Arithmetic progression (sequence) 等差数列 geometric progression (sequence) 等比数列8. 其它Approximate 近似 (anti clockwise) (逆) 顺时针方向 cardinal ordinal 基数序数 directProportion 正比 distinct 不同的 estimation 估计, 近似parentheses 括号 proportion 比例Permutation 排列 combination 组合 table 表格 trigonometric function 三角函数 unit 单位, 位几何 geometry1. 角Alternate angle 内错角 corresponding angle 同位角 vertical angle 对顶角 central angle.圆心角 interior angle 内角 exterior angle 外角 supplementary angles 补角complementaryAngle 余角 adjacent angle 邻角 acute angle obtuse angle 钝角right angle 直角锐角周角 round angle straight angle 平角 included angle 夹角Scalene triangle equilateral triangle isosceles triangle 等腰三角形不等边三角形等边三角形直角三角形直角三角形斜斜三角形内接三角形内接三角形三.收敛的平面图形，除三角形外半圆半圆同心圆同心圆四边形四边形五角大厦五边形六边形六边形七边形七边形八角八边形九边形十边形多边形多边形十边形平行四边形等边平行四边形等边形平面平面平方正方形，平方矩形长方形正多边形正多边形菱形菱形梯形梯形4。
高等数学教材翻译成英语
高等数学教材翻译成英语Translation of Advanced Mathematics Textbooks into EnglishIntroduction:Mathematics is a fundamental discipline that plays a crucial role in various fields, including science, engineering, and finance. In order to promote international communication and facilitate global academic exchanges, the translation of advanced mathematics textbooks into English is of paramount importance. This article will discuss the significance, challenges, and strategies involved in translating advanced mathematics textbooks accurately and effectively.1. Significance of Translating Advanced Mathematics Textbooks into English:1.1 Global Communication: English has become the lingua franca of academia, and translating advanced mathematics textbooks into English allows for effective communication and knowledge sharing among scholars around the world.1.2 Accessible Education: English translations enable students and researchers who are not fluent in the source language to access comprehensive and in-depth knowledge of advanced mathematics.1.3 Research Collaboration: Translation facilitates global research collaborations, as scholars can understand and discuss mathematical concepts, theories, and findings from various sources written in English.2. Challenges in Translating Advanced Mathematics Textbooks into English:2.1 Technical Terminology: Advanced mathematics contains a vast array of specialized terms and notation, which can be difficult to translate accurately while preserving their precise meaning.2.2 Cultural Context: Mathematical concepts can be influenced by cultural norms and assumptions, which may not directly translate into English. Translators must ensure that the translated text conveys the same mathematical principles without distortion.2.3 Style and Structure: Translating textbooks requires striking a balance between maintaining the original style and adapting it to suit the target language's preferences and academic conventions.2.4 Mathematical Illustrations: Graphs, diagrams, and equations must be accurately reproduced in the translated version to ensure comprehension and consistency.3. Strategies for Accurate and Effective Translation:3.1 In-depth Subject Knowledge: Translators must possess a strong understanding of advanced mathematics concepts to accurately convey the meaning of technical terms and equations.3.2 Extensive Research: Translators should consult multiple sources, including existing translated mathematics texts, to ensure consistency and accuracy in their translations.3.3 Collaboration: Collaboration between mathematicians and linguists can bridge the gap between the mathematical and linguistic aspects of translation, ensuring both accuracy and fluency.3.4 Proofreading and Editing: A thorough review of the translated text by experts in both mathematics and language will help identify and rectify any errors or ambiguities.3.5 Contextual Adaptation: While preserving the original mathematical content, translators should adapt cultural and contextual aspects to ensure comprehension and relevance to the target audience.3.6 Localization: Adapting the translated textbooks to fit the particular educational system and syllabus of the target country will enhance their practicality and acceptance.Conclusion:The translation of advanced mathematics textbooks into English has significant benefits for global academic communication, accessible education, and research collaboration. Overcoming challenges such as technical terminology, cultural context, style, and mathematical illustrations requires strategies such as subject knowledge, extensive research, collaboration, proofreading, contextual adaptation, and localization. By following these strategies, translators can produce accurate and effective translations that contribute to the advancement of mathematics education worldwide.。
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Power Series Expansion and Its ApplicationsIn the previous section, we discuss the convergence of power series, in its convergence region, the power series always converges to a function. For the simple power series, but also with itemized derivative, or quadrature methods, find this and function. This section will discuss another issue, for an arbitrary function ()f x , can be expanded in a power series, and launched into.Whether the power series ()f x as and function? The following discussion will address this issue. 1 Maclaurin (Maclaurin) formulaPolynomial power series can be seen as an extension of reality, so consider the function ()f x can expand into power series, you can from the function ()f x and polynomials start to solve this problem. To this end, to give here without proof the following formula.Taylor (Taylor) formula, if the function ()f x at 0x x = in a neighborhood that until the derivative of order 1n +, then in the neighborhood of the following formula :20000()()()()()()nn f x f x x x x x x x r x =+-+-++-+… (9-5-1)Among|10()()n n r x x x +=-That ()n r x for the Lagrangian remainder. That (9-5-1)-type formula for the Taylor.If so 00x =, get2()(0)()nn f x f x x x r x =+++++…, (9-5-2)At this point,(1)(1)111()()()(1)!(1)!n n n n n f f x r x x x n n ξθ+++++==++ (01θ<<).That (9-5-2) type formula for the Maclaurin.Formula shows that any function ()f x as long as until the 1n +derivative, n can be equal to a polynomial and a remainder.!We call the following power series()2(0)(0)()(0)(0)2!!n nf f f x f f x x x n '''=+++++…… (9-5-3) For the Maclaurin series.So, is it to ()f x for the Sum functions? If the order Maclaurin series (9-5-3) the first 1n + items and for 1()n S x +, which()21(0)(0)()(0)(0)2!!n nn f f S x f f x x x n +'''=++++…Then, the series (9-5-3) converges to the function ()f x the conditions1lim ()()n n s x f x +→∞=.Noting Maclaurin formula (9-5-2) and the Maclaurin series (9-5-3) the relationship between theknown;1()()()n n f x S x r x +=+Thus, when()0n r x =There,1()()n f x S x +=Vice versa. That if1lim ()()n n s x f x +→∞=,Units must*()0n r x =.This indicates that the Maclaurin series (9-5-3) to ()f x and function as the Maclaurin formula (9-5-2) of the remainder term ()0n r x → (when n →∞).In this way, we get a function ()f x the power series expansion:()()0(0)(0)()(0)(0)!!n n n nn f f f x x f f x x n n ∞='==++++∑……. (9-5-4) It is the function ()f x the power series expression, if, the function of the power series expansion is unique. In fact, assuming the function f (x ) can be expressed as power series20120()n n n n n f x a x a a x a x a x ∞===+++++∑……, (9-5-5)Well, according to the convergence of power series can be itemized within the nature of derivation, and then make 0x = (power series apparently converges in the 0x = point), it is easy to get()2012(0)(0)(0),(0),,,,,2!!n nn f f a f a f x a x a x n '''====…….(Substituting them into (9-5-5) type, income and ()f x the Maclaurin expansion of (9-5-4) identical.In summary, if the function f (x ) contains zero in a range of arbitrary order derivative, and in thisrange of Maclaurin formula in the remainder to zero as the limit (when n → ∞,), then , the function f (x ) can start forming as (9-5-4) type of power series.Power Series()20000000()()()()()()()()1!2!!n n f x f x f x f x f x x x x x x x n '''=+-+-++-……,Known as the Taylor series.Second, primary function of power series expansionMaclaurin formula using the function ()f x expanded in power series method, called the direct expansion method.Example 1 `Test the function ()xf x e =expanded in power series of x .Solution because()()n x f x e =,(1,2,3,)n =…Therefore()(0)(0)(0)(0)1n f f f f '''====…,So we get the power series21112!!n x x x n +++++……, (9-5-6) Obviously, (9-5-6)type convergence interval (,)-∞+∞, As (9-5-6)whether type ()x f x e=is Sumfunction, that is, whether it converges to ()xf x e = , but also examine remainder ()n r x .<Because1e ()(1)!xn n r x x n θ+=+ (01θ<<)，且x x x θθ≤≤,Therefore11e e ()(1)!(1)!xx n n n r x x x n n θ++=<++,Noting the value of any set x ,xe is a fixed constant, while the series (9-5-6) is absolutely convergent, sothe general when the item when n →∞, 10(1)!n xn +→+ , so when n → ∞,there10(1)!n xx e n +→+, From this{lim ()0n n r x →∞=This indicates that the series (9-5-6) does converge to ()xf x e =, therefore21112!!x n e x x x n =+++++…… (x -∞<<+∞). Such use of Maclaurin formula are expanded in power series method, although the procedure is clear,but operators are often too Cumbersome, so it is generally more convenient to use the following power series expansion method.Prior to this, we have been a functionx-11, xe and sin x power series expansion, the use of these known expansion by power series of operations, we can achieve many functions of power series expansion. This demand function of power series expansion method is called indirect expansion .Example 2Find the function ()cos f x x =,0x =,Department in the power series expansion. Solution because！(sin )cos x x '=,And3521111sin (1)3!5!(21)!n n x x x x x n +=-+-+-++……，（x -∞<<+∞） Therefore, the power series can be itemized according to the rules of derivation can be342111cos 1(1)2!4!(2)!n n x x x x n =-+-+-+……,（x -∞<<+∞） Third, the function power series expansion of the application exampleThe application of power series expansion is extensive, for example, can use it to set some numericalor other approximate calculation of integral value.Example 3 Using the expansion to estimate arctan x the value of π. <Solution because πarctan14= Because of357arctan 357x x x x x =-+-+…, (11x -≤≤),So there1114arctan14(1)357π==-+-+…Available right end of the first n items of the series and as an approximation of π. However, the convergence is very slow progression to get enough items to get more accurate estimates of πvalue.！此外文文献选自于：Walter.Rudin.数学分析原理(英文版)[M].北京：机械工业出版社.幂级数的展开及其应用】在上一节中，我们讨论了幂级数的收敛性，在其收敛域内，幂级数总是收敛于一个和函数．对于一些简单的幂级数，还可以借助逐项求导或求积分的方法，求出这个和函数．本节将要讨论另外一个问题，对于任意一个函数()f x ，能否将其展开成一个幂级数，以及展开成的幂级数是否以()f x 为和函数?下面的讨论将解决这一问题．一、 马克劳林(Maclaurin)公式幂级数实际上可以视为多项式的延伸，因此在考虑函数()f x 能否展开成幂级数时，可以从函数()f x 与多项式的关系入手来解决这个问题．为此，这里不加证明地给出如下的公式．泰勒(Taylor)公式 如果函数()f x 在0x x =的某一邻域内，有直到1n +阶的导数，则在这个邻域内有如下公式：()20000000()()()()()()()()()2!!n n n f x f x f x f x f x x x x x x x r x n '''=+-+-++-+…,(9-5-1)其中(1)10()()()(1)!n n n f r x x x n ξ++=-+．称()n r x 为拉格朗日型余项．称(9-5-1)式为泰勒公式．*如果令00x =，就得到2()(0)()n n f x f x x x r x =+++++…， (9-5-2)此时，(1)(1)111()()()(1)!(1)!n n n n n f f x r x x x n n ξθ+++++==++, (01θ<<)．称(9-5-2)式为马克劳林公式．公式说明，任一函数()f x 只要有直到1n +阶导数，就可等于某个n 次多项式与一个余项的和． 我们称下列幂级数()2(0)(0)()(0)(0)2!!n nf f f x f f x x x n '''=+++++…… (9-5-3)(为马克劳林级数．那么，它是否以()f x 为和函数呢?若令马克劳林级数(9-5-3)的前1n +项和为1()n S x +，即()21(0)(0)()(0)(0)2!!n nn f f S x f f x x x n +'''=++++…，那么，级数(9-5-3)收敛于函数()f x 的条件为1lim ()()n n s x f x +→∞=．注意到马克劳林公式(9-5-2)与马克劳林级数(9-5-3)的关系，可知1()()()n n f x S x r x +=+．于是，当()0n r x =。