数学术语表

算盘是一种计算工具，基于在金属丝上滑动的珠子，或凹槽中的球或石子。

An abacus is a calculating device based on either beads sliding on wires or balls or stones in grooves.

阿贝尔函数或超椭圆函数是椭圆函数的推广。它是具有四个周期的二元函数。
与椭圆函数类似，它也可以视为某些积分的反函数（称为阿贝尔或超椭圆积分），这些积分具有形式$\large{\int \frac{dz}{\sqrt{R(z)}}}$，其中$R$是次数为大于 4的多项式。

An abelian or hyperelliptic function is a generalisation of an elliptic function. It is a function of two variables with four periods.
In a similar way to an elliptic function it can also be regarded as the inverse function to certain integrals (called abelian or hyperelliptic integrals) of the form $\large{\int \frac{dz}{\sqrt{R(z)}}}$ where $R$ is a polynomial of degree greater than 4.

阿贝尔群是群运算为交换的群，即满足$a.b = b.a$。

An abelian group is a group in which the group operation is commutative. i.e. it satisfies $a.b = b.a$.

高中毕业考试（Abitur）是德国在文理中学（Gymnasium）或中学最后一年参加的考试集合。

The abitur is the set of examinations taken in the final year of the gymnasium or secondary school in Germany.

盈数是其真因数之和大于该数本身的整数。

例如，12的真因数是1、2、3、4、6，其和为16。
另见：完全数、亏数。

An abundant number is an integer for which the sum of its proper divisors is greater than the number itself.

For example, the proper divisors of 12 are 1, 2, 3, 4, 6 which sum to 16.
See also: perfect number, deficient number.

教师资格会考（Agrégation）是法国的一种竞争性考试，通过者可在公立中学或大学任教。

The agrégation is a competitive examination in France which allows the holder to teach in lycées or colleges.

亚历山大大帝或亚历山大三世（生于：公元前356年，马其顿的佩拉，卒于：公元前323年，巴比伦）是征服了波斯帝国的马其顿国王。

Alexander the Great or Alexander III (born: 356 BC in Pella in Macedonia, died: 323 BC in Babylon) was a King of Macedonia who conquered the Persian Empire.

代数是一个集合，它同时是一个环和一个向量空间，当然满足各种特殊性质或公理。

An algebra is a set which is both a ring and a vector space satisfying, of course various special properties or axioms.

代数几何研究由多项式方程组定义的曲线、曲面及其高维等价物，并将它们的性质与它们所确定的多项式环的代数性质联系起来。

Algebraic geometry studies curves, surfaces and their higher dimensional equivalents defined by systems of polynomial equations and relates their properties to the algebraic properties of the polynomial rings that they determine.

代数数是整系数（或有理的系数）多项式方程的根，且为实数。

例如，√2是$x^{2} = 2$的根，因此是代数数。

An algebraic number is a real number which is the root of a polynomial equation with integer (or rational) coefficients.

For example, √2 is a root of $x^{2} = 2$ and so is algebraic.

代数数域是$\mathbb{R}$或$\mathbb{C}$的子域，它包含有理数$\mathbb{Q}$以及某个有理系数多项式的根。
例如，记作$\mathbb{Q}$(√2)的域是实数集{$a + b√2 | a, b$为有理数}。它包含方程$x^{2} - 2 = 0$的根。

An algebraic number field is a subfield of $\mathbb{R}$ or $\mathbb{C}$ which contains the rational numbers $\mathbb{Q}$ as well as the roots of some polynomial with rational coefficients.
For example, the field written $\mathbb{Q}$ (√2) is the set of real numbers {$a + b√2 | a, b$ are rationals}. This contains the roots of the equation $x^{2} - 2 = 0$.

代数数论是不使用来自分析的无穷级数、收敛等方法研究的数论。
它与解析数论相对。

Algebraic number theory is number theory studied without using methods like infinite series, convergence, etc. taken from analysis.
It contrasts with analytic number theory.

代数拓扑学是拓扑学的一个分支，其中计算拓扑空间的不变量，这些不变量是代数结构，通常是群。
例如，一个空间的基本群就是这样的不变量，同伦和同调也是。

Algebraic topology is a branch of topology in which one calculated invariants of a topological space which are algebraic structures, often groups.
For example, the fundamental group of a space is such an invariant as are homotopy groups and homology groups.

一个交错群是有限集合的所有偶排列的群。
一个排列是偶，如果它可以写成偶数个对换的乘积，其中对换交换一对元素并保持其余元素不变。
大小为$n$的集合的交错群记作$A_{n}$。
交错群$A_{n}$包含对称群$S_{n}$的一半元素，其阶为$n! /2$。

An alternating group is the group of all even permutations of a finite set.
A permutation is even if it can be written as a product of an even number of transpositions where a transposition swaps a pair of elements and leaves the rest fixed.
The alternating group of a set of size $n$ is denoted $A_{n}$.
The alternating group $A_{n}$ contains half the elements of the symmetric group $S_{n}$ and has order $n! /2$.

亲和数是一对数a, b，其中a的真因子之和等于b，且b的真因子之和等于a。

例如，220和284是亲和数。

Amicable numbers are a pair of numbers a, b for which the sum of the proper divisors of a equals b and the sum of the proper divisors of b is equal to a.

For example, 220 and 284 are amicable numbers.

阿摩尼乌斯（生于：约公元550年）是一位希腊哲学家，主要研究逻辑学和科学。他写了许多关于亚里士多德的评论性著作。他曾是普罗克洛的学生，后来被任命为亚历山大里亚学派的负责人。

Ammonius (born: about 550 AD) was a Greek philosopher who worked mainly on logic and science. He wrote many critical works on Aristotle. He was a student with Proclus then later he was appointed the head of the Alexandrian school.

解析数论是使用来自分析、特别是复分析中的无穷级数、收敛等方法研究的数论。
它与代数数论相对。

Analytic number theory is number theory studied using methods like infinite series, convergence, etc. taken from analysis and especially from Complex analysis.
It contrasts with algebraic number theory.

阿那克西曼德（生于：公元前610年，米利都[今土耳其]，卒于：公元前546年）是一位希腊哲学家，是泰勒斯的学生。他被认为是天文学的创始人。他是第一个发展宇宙论的人。

Anaximander (born: 610 BC in Miletus [now in Turkey], died: 546 BC) was a Greek philosopher who was a pupil of Thales. He is considered as the founder of astronomy. He was the first to develop a cosmology.

希腊数学的经典问题之一是找到一种尺规作图来三等分任意角。
其他经典问题是化圆为方和倍立方。

One of the classic problems of Greek mathematics was to find a ruler and compass construction to divide any angle into three equal pieces.
The other classic problems were squaring the circle and duplicating the cube.

日环食是一种日食，其中太阳环仍然可见，因为月球离地球太近而无法完全遮挡太阳。

An annular eclipse is a solar eclipse in which a ring of the sun is still visible because the Moon is too close to the Earth to shield the Sun completely.

远地点是天体离其轨道中心最远的点。
最近的点称为近地点。

The apogee is the point where a heavenly body is furthest away from the centre of its orbit.
The nearest point is called the perigee.

阿波罗多罗斯（生于：约公元前130年）是一位希腊学者，以其希腊历史编年史而闻名。他是阿里斯塔克斯的学生。公元前146年，他离开亚历山大港前往帕加马，后来定居雅典。

Apollodorus (born: about 130 BC) was a Greek scholar who is best known for his Chronicle of Greek history. He was a pupil of Aristarchus. In 146 BC he left Alexandria for Pergamum and later settled at Athens.

一个余理数是欧几里得的无理数范畴之一。
形如$√(√A - √B)$的数称为余理数。
带+号的对应数称为二项式。

An apotome is one of Euclid's categories of irrational numbers.
A number of the form $√(√A - √B)$ is called a apotome.
The corresponding number with a + sign is called a binomial.

阿普列乌斯（生于：公元124年，努米底亚的马道罗斯，卒于：约公元175年）是一位柏拉图主义哲学家，曾在迦太基和雅典接受教育，然后在罗马教授修辞学，之后返回非洲。他撰写了重要的哲学论著，其中包括三部关于柏拉图的著作。

Apuleius (born: 124 AD in Madauros in Numidia, died: about 175 AD) was a Platonic philosopher, who was educated at Carthage and Athens, then taught rhetoric in Rome before returning to Africa. He wrote important philosophical treatises including three on Plato.

阿尔克西拉乌斯（生于：公元前316年，生于奥托里库斯，位于伊奥利亚[今土耳其]，卒于：约公元前241年）是一位哲学家，后来成为希腊学园的领袖。他是一位怀疑论者，认为智者只需知道自己的行为是合理的。

Arcesilaus (born: 316 BC in Pitane in Aeolis [now in Turkey], died: about 241 BC) was a philosopher who became head of the Greek Academy. He was a Sceptic who believed that a wise man need know only that his actions are reasonable.

阿基米德原理指出，任何完全或部分浸没在流体中的物体都会受到一个向上的力，该力等于物体所排开的流体的重量。

Archimedes' principle states that any body completely or partially submerged in a fluid is acted upon by an upward force which is equal to the weight of the fluid displaced by the body.

阿基米德螺旋由一根螺旋形的管子组成，其下端浸入水中。当装置旋转时，水沿管子上升。

Archimedes' screw consists of a pipe in the shape of a helix with its lower end dipped in the water. As the device is rotated the water rises up the pipe.

等差数列是一个数列，其中每个数都比前一个数增加相同的量（公差）。
例如：5, 8, 11, 14, 17, ...

An arithmetic progression is a sequence in which each number increases by the same amount (the common difference) over the previous one.
As for example: 5, 8, 11, 14, 17, ...

集合上的二元运算 . 如果满足恒等式 $(a.b).c = a.(b.c)$，则称为结合的。
函数的复合与 矩阵 的乘法是重要的结合运算。

An binary operation . on a set is associative if it satisfies the identity $(a.b).c = a.(b.c)$
Composition of functions and multiplication of matrices are important associative operations.

星盘是一种早期用于测量地平线与恒星或行星之间角度的仪器。它被八分仪和六分仪所取代。

An astrolabe is an early instrument for measuring the angle between the horizon and a star or planet. It was superceded by the octant and sextant

曲线的渐近线是一条直线（或更一般地，另一条曲线），它任意接近该曲线。
双曲线 有一对直线作为渐近线。

An asymptote of a curve is a straight line (or more generally, another curve) which approaches the curve arbitrarily closely.
A hyperbola has a pair of straight lines as asymptotes.

原子论是物质和空间不是无限可分的理论。

Atomic theory is the theory that matter and space are not infinitely divisible.

群 $G$ 的自同构群是由 $G$ 的自同构（从 $G$ 到自身的保持乘法的双射）形成的群。
类似地，可以通过考察保持其数学结构的双射来考虑其他结构如 环、图等的自同构群。

An automorphism group of a group $G$ is the group formed by the automorphisms of $G$ (bijections from $G$ to itself which preserve the multiplication).
Similarly, one can consider the automorphism groups of other structures like rings, graphs, etc. by looking at bijections which preserve their mathematical structure.

选择公理指出，给定任意一族两两不相交的非空集合$A$，可以通过从$A$中的每个集合中选取一个元素来构造一个新集合。它等价于关于无限集合的几个其他重要陈述（良序原理、马克斯·奥古斯特·佐恩引理……），并且已被证明独立于其他集合论公理。

The Axiom of Choice states that given any collection $A$ of mutually exclusive non-empty sets, one can create a new set by choosing an element out of each set in $A$. It is equivalent to several other important statements about infinite sets (the Well-Ordering Principle, Zorn's Lemma, ...) and has been shown to be independent of the other set-theory axioms.

方位角是子午线（北方方向）与通过天体的垂直圈之间的地平弧。

The azimuth is the arc of the horizon between the meridian (the direction North) and a vertical circle passing through a heavenly body.

巴拿赫空间是完备的赋范向量空间。

也就是说，带有度量的向量空间，该度量是完备的，即向量的 奥古斯丁·路易·柯西 序列总是收敛到空间中的一个确定极限。

A Banach space is a complete normed vector space.

That is, vector space with a metric that is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit in the space.

伯努利数 由 雅各布·伯努利 定义，与计算形如 ∑ $i^{k}$ 的和有关。
序列 $B_{0} , B_{1} , B_{2} , ...$ 可以用公式
$x/(e^{x} - 1) = \sum (B_{n}x^{n})/n!$
生成，尽管对它们使用了各种不同的记号。
最初几个是：$B_{0} = 1 , B_{1} = -{{1}\over{2}} , B_{2} = {{1}\over{6}} , B_{4} = -{{1}\over{30}} , B_{6} = {{1}\over{42}} , ...$
它们出现在数学的许多不同领域中，包括 $\tan(x)$、费马大定理、... 的级数展开。

Bernoulli numbers were defined by Jacob Bernoulli in connection with evaluating sums of the form ∑ $i^{k}$.
The sequence $B_{0} , B_{1} , B_{2} , ...$ can be generated using the formula
$x/(e^{x} - 1) = \sum (B_{n}x^{n})/n!$
though various different notations are used for them.
The first few are: $B_{0} = 1 , B_{1} = -{{1}\over{2}} , B_{2} = {{1}\over{6}} , B_{4} = -{{1}\over{30}} , B_{6} = {{1}\over{42}} , ...$
They occur in many diverse areas of mathematics including the series expansions of $\tan(x)$, Fermat's Last theorem, ...

贝塞尔函数是圆形鼓面振动模态的径向部分。它们满足弗里德里希·威廉·贝塞尔的partial differential equation，这是具有柱对称性的Laplace's equation的一种形式。

Bessel functions are the radial part of the modes of vibration of a circular drum. They satisfy Bessel's partial differential equation which is a form of Laplace's equation with cylindrical symmetry.

贝塔函数 B $(m, n)$ 定义为积分：
插图：beta_funct.gif ↗。
它可以通过 伽马函数 定义为：
插图：beta_identity.gif ↗。
许多积分可以化为 Beta 函数的求值。

The beta function B $(m, n)$ is defined to be the integral:
插图：beta_funct.gif ↗.
It can be defined in terms of the gamma function by:
插图：beta_identity.gif ↗.
Many integrals can be reduced to the evaluation of Beta functions.

比伯巴赫猜想 指出，如果 $f$ 是由级数
$f(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ...$
给出的复函数，且该级数以一一对应的方式将单位圆盘共形映射，则对每个 $n$ 有 $|a_{n}| ≤ n |a_{1}|$。
它可以表示为：
单叶函数的第 $n$ 个系数不能超过 $n$。

The Bieberbach conjecture states that if $f$ is a complex function given by the series
$f(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ...$
which maps the unit disc conformally in a one-one way then $|a_{n}| ≤ n |a_{1}|$ for each $n$.
It can be expressed as:
The $n$th coefficient of a univalent function can be no more than $n$.

双星是两颗恒星互相绕转的恒星系统

Binary Stars are star systems where two stars are orbiting each other

二项式 是 欧几里得 的 无理数 范畴之一。

形如 $√(√A + √B)$ 的数称为 二项式。
带负号的相应数称为 余理数。

A binomial is one of Euclid's categories of irrational numbers.

A number of the form $√(√A + √B)$ is called a binomial.
The corresponding number with a - sign is called a apotome.

二项式系数 是 $(1 + x)^{n}$ 的展开式中 $x$ 的幂的系数。
我们有 插图：binom1.gif ↗
其中二项式系数 插图：binom2.gif ↗ 是从大小为 $n$ 的集合中选取大小为 $k$ 的（无序）子集的方法数。
二项式系数是 帕斯卡三角形 中的元素。

The binomial coefficients are the coefficients of powers of $x$ in the expansion of $(1 + x)^{n}$.
We have 插图：binom1.gif ↗
where the binomial coefficient 插图：binom2.gif ↗ is the number of ways of choosing an (unordered) subset of size $k$ from a set of size $n$.
The binomial coefficients are the entries in the Pascal triangle.

二项式定理 是允许展开 二项式 的结果：
$(x + y)^{n} = x^{n} + a_{n-1}x^{n-1}y + a_{n-2}x^{n-2}y^{2} + ... + y^{n}$
其中系数 a i 称为 binomial coefficients。

The Binomial Theorem is the result which allows one to expand a binomial:
$(x + y)^{n} = x^{n} + a_{n-1}x^{n-1}y + a_{n-2}x^{n-2}y^{2} + ... + y^{n}$
where the coefficients ai are called binomial coefficients.

一个数$m$是一个双二次剩余模$n$，如果对某个$r$有$m = r^{4}$模$n$。

A number $m$ is a biquadratic residue modulo $n$ if $m = r^{4}$ modulo $n$ for some $r$.

布尔代数 是一种 代数，其中二元运算的选择是为了模拟集合论中的并与交运算。
对于任何集合 $A$，$A$ 的子集在并、交和补运算下构成一个布尔代数。

A Boolean algebra is an algebra in which the binary operations are chosen to model the union and intersection operations in Set Theory.
For any set $A$, the subsets of $A$ form a Boolean algebra under the operations of union, intersection and complement.

一个边值问题是连同边界条件一起给出的一个微分方程或偏微分方程，以确保解的唯一性。
这些边界条件通常涉及指定函数或其导数在方程定义区域边界上的值。

A boundary value problem is an ordinary or partial differential equation given together with boundary conditions to ensure a unique solution.
These boundary conditions usually involve specifying the value of the function or its derivative on the edges of the region on which the equation is defined.

最速降线是这样一条曲线：一个质点在重力作用下沿该曲线下降时，将从一点到另一点所需的时间最短。
摆线就是这样一条曲线。

The brachistochrone is a curve such that a particle descending along the curve under gravity will travel from one point to another in minimum time.
The cycloid is such a curve.

布朗运动是悬浮在液体或气体中的粒子由于与液体或气体中快速运动的原子或分子碰撞而产生的随机运动。
它以植物学家Robert Brown命名，他首次观察到了这种现象。

Brownian motion is the random motion of particles suspended in a liquid or a gas resulting from their collision with the fast-moving atoms or molecules in the gas or liquid.

It named after the botanist Robert Brown who first observed it.

伯恩赛德问题问的是：一个有限生成群，如果它的所有元素都具有有限阶，是否可能为无限群。这个版本通常称为General Burnside problem，1964年找到了这种群的一个例子。
Burnside group $B(d, n)$是最大的$d$生成群，其中每个元素都满足$x^{n} = 1$。

Restricted Burnside problem问的是：对于固定的$d$和$n$，是否存在最大的有限$d$生成群，其中每个元素都满足$x^{n} = 1$。受限威廉·伯恩赛德问题的肯定解决将表明，$B(d, n)$的有限商群只有有限多个。

The Burnside problem asks whether it is possible for a finitely generated group to be infinite if all its elements have finite order. This version is usually called the General Burnside problem and an example of such a group was found in 1964.
The Burnside group $B(d, n)$ is the largest $d$ generator group in which every element satisfies $x^{n} = 1$.

The Restricted Burnside problem asks whether, for fixed $d$ and $n$, there is a largest finite $d$ generator group in which every element satisfies $x^{n} = 1$. A positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of $B(d, n)$.

卡尔基狄乌斯——（或Chalcidius）（4世纪）是一位基督教作家，他制作了柏拉图著作的重要拉丁文译本和注释。

Calcidius - (or Chalcidius) (4th century) was a Christian writer who produced an important Latin translations and commentaries of Plato's works.

变分法是微积分的一种推广。它试图找到使给定函数具有平稳值（通常是最大值或最小值）的路径、曲线、曲面等。

The calculus of variations is a generalisation of the calculus. It attempts to find the path, curve, surface, etc., for which a given function has a stationary value (usually a maximum or minimum).

插图：catenary.gif ↗一个悬链线是一条均匀重链悬挂时形成的曲线。
它的方程是：
$y = \cosh(x) = \frac{1} {2} (e^{x} + e^{-x}) = 1 + \frac{1}{2} x^{2} + \frac{1} {24} x^{4} + ...$

插图：catenary.gif ↗A catenary is the curve in which a heavy uniform chain hangs.
Its equation is:
$y = \cosh(x) = \frac{1} {2} (e^{x} + e^{-x}) = 1 + \frac{1}{2} x^{2} + \frac{1} {24} x^{4} + ...$

反射光学是光学中处理反射的部分。

Catoptrics is the part of optics dealing with reflection.

卡图卢斯（公元前84年生于西阿尔卑斯高卢的维罗纳[今意大利]，卒于公元前54年罗马）是一位罗马诗人，他创作了关于爱与恨的诗歌，被认为是古罗马最优秀的抒情诗。他的一些诗表达了对西塞罗、庞培大帝和尤利乌斯·凯撒的蔑视或仇恨。

Catullus (born: 84 BC in Verona in Cisalpine Gaul [now Italy], died: 54 BC in Rome) was a Roman poet who wrote poems about love and hate which are thought to be the best lyric poetry written in ancient Rome. Some of his poems express of contempt or hatred for Cicero, Pompey and Julius Caesar.

天赤道是在北天极和南天极这两个固定点正中间的天球上的赤道。

The celestial equator is the equator on the celestial sphere midway between the fixed points of the North and South poles.

天球是恒星在天空中看起来运动所依附的球面。

The celestial sphere is the sphere on which the stars appear to move in the sky.

造父变星是脉动的非常明亮的恒星。它们强度的变化与它们的光度直接相关。

Cepheid Variables are very luminous stars that pulsate. Their variance in intensity is directly related to their luminosity.

一个群的特征标理论涉及研究该群的表示论的迹（对角元素之和）。
所获得的信息列在特征标表中，这些信息的性质可用于深入了解该群的性质。

The character theory of a group involves studying the traces (sums of the diagonal elements) of the matrix representations of the group.
The information gained is listed in character tables and the properties of these can be used to give insights into the properties of the group.

西塞罗（公元前106年生于拉丁姆的阿尔皮努姆[今意大利阿尔皮诺]，卒于公元前43年拉丁姆的福尔米亚[今福尔米亚]）在罗马和希腊接受教育。他后来成为罗马政治家、律师、学者和作家。他试图在摧毁罗马共和国的内战中维护共和原则，但未能成功。

Cicero (born:106 BC in Arpinum in Latium [now Arpino in Italy], died: 43 BC in Formiae in Latium [now Formia]) was educated in Rome and in Greece. He then became a Roman statesman, lawyer, scholar, and writer. He tried unsuccessfully to maintain republican principles in the civil wars that destroyed the Roman Republic.

插图：circumcircle.gif ↗如果多边形的顶点都在一个圆上，则称该圆外接的于该三角形或其他多边形。
此时称该多边形内接于该圆中。

三角形外接的（唯一）圆称为外接圆，其圆心称为外心。

插图：circumcircle.gif ↗A circle is said to be circumscribed to a triangle or other polygon if the vertices of the polygon lie on the circle.
The polygon is than said to be inscribed in the circle.

The (unique) circle circumscribed to a triangle is called the circumcircle and its centre is the circumcentre.

宇宙微波背景辐射是大爆炸遗留下来的辐射，目前测量其温度约为2.7K

Cosmic microwave background radiation is radiation left over from the big bang that we currently measure to be at around 2.7K

上同调计算拓扑学的代数不变量，这些不变量与同调形式对偶。
所得到的不变量一般比同调给出的更强，并且通常具有更多的代数结构。
对于拓扑学和纯代数结构，都已发展出Generalised cohomology theories，它们具有上同调的某些形式性质，但不具有相同的几何背景。

Cohomology calculates algebraic invariants of topological spaces which are formally dual to homology.
The invariants obtained are in general more powerful than those given by homology and usually have more algebraic structure.
Generalised cohomology theories, both for topological spaces and for purely algebraic structures, have been developed which have some of the formal properies of cohomology but which do not have the same geometric background.

给定集合的一个组合是忽略元素顺序的子集（与排列不同）。

例如，从集合{1, 2, 3, 4}中有6个大小为2的组合：
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}。

从大小为$n$的集合中取大小为$k$的组合数，是二项式系数插图：binom2.gif ↗，也写作$_{n}C_{k}$。

A combination is a subset of a given set where the order of elements is ignored (as distinct from a permutation).

For example there are 6 combinations of size 2 from the set {1, 2, 3, 4}:
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.

The number of combinations of size $k$ from a set of size $n$ is the binomial coefficient 插图：binom2.gif ↗ also written $_{n}C_{k}$.

如果两条直线或两个距离的长度之比是有理数，则称它们是可公度的的。
如果该比是无理数，则称它们是不可通约的。

Two lines or distances are commensurable if the ratio of their lengths is a rational number.
If the ratio is an irrational number, they are called incommensurable.

群中两个元素$x, y$的换位子是元素$x^{-1}y^{-1}x y$，记作$[x, y]$。
两个子群$H$和$K$的换位子群$[H, K]$是由所有换位子$[x, y]$生成的子群，其中$x \isin H$和$y \isin K$。

The commutator of two elements $x, y$ of a group is the element $x^{-1}y^{-1}x y$ and is written $[x, y]$.
The commutator subgroup $[H, K]$ of two subgroups $H$ and $K$ is the subgroup generated by all the commutators $[x, y]$ with $x \isin H$ and $y \isin K$.

如果空间的每个开集覆盖都有有限子覆盖，则称该拓扑学是紧的的。
在度量空间的情形下，紧空间具有有界序列有收敛子序列的性质。

A topological space is compact if every covering of the space by open sets has a finite sub-cover.
In the case of metric spaces compact spaces have the property that bounded sequences have cconvergent sub-sequences.

罗盘磁差或磁偏角是罗盘读数与真北之间的差值。

Compass variation or magnetic declination is the amount by which compass readings differ from the true North.

合数是不为素数的整数。
每个合数都可以写成素数的乘积。

A composite number is an integer which is not prime numbers.
Every composite number can be written as a product of prime numbers.

同心球是同心但半径不同的球面。

早期的宇宙论理论认为行星在同心球面上运动。

Concentric spheres are spheres with the same centre but different radii.

Early theories of cosmology had concentric spheres for the planets to move on.

共形映射是从平面（可视为$\mathbb{R}$2或$\mathbb{C}$）到自身的映射，它保角。也就是说，任意两条曲线之间的夹角与它们的像之间的夹角相同。
从$\mathbb{C}$到$\mathbb{C}$的任何解析映射在导数非零的点处是共形的。

A conformal mapping is a map from the plane (which can be regarded as $\mathbb{R}$2 or $\mathbb{C}$) to itself which preserves angles. That is, the angle between any two curves is the same as the angle between their images.
Any analytic map from $\mathbb{C}$ to $\mathbb{C}$ is conformal at a point where the derivative is non-zero.

一个圆锥曲线或圆锥曲线是以下三种曲线之一：抛物线、双曲线或椭圆，可以通过用平面与（双面）锥面相交得到。

A conic or conic section is one of the three curves: a parabola, hyperbola or ellipse which one can obtain by intersecting a plane with a (double sided) cone.

当两个天体在天空中移动到同一位置时，就发生合。
（这个词的字面意思是连接在一起）

A conjunction takes place when two heavenly bodies move into the same place in the sky.
(Literally, the word means joined together)

插图：continuedfrac.gif ↗数$r$的连分数展开是如下形式的表达式：

如果$r$是有理的数，则该展开式终止。

插图：continuedfrac.gif ↗The continued fraction expansion of a number $r$ is an expression of the form:

If $r$ is a rational number this expansion terminates.

连续统假设断言不存在基数介于自然数基数与实数基数之间的集合。
Generalised Continuum hypothesis断言如果$A$是任意集合，则不存在基数介于$A$的基数与$A$的所有子集构成的集合的基数之间的集合。
连续统假设已被证明独立于其他集合论公理。

The Continuum hypothesis states that there is no no set whose cardinality lies between that of the Natural numbers and that of the Reals.
The Generalised Continuum hypothesis states that if $A$ is any set, there is no set whose cardinality lies between the cardinality of $A$ and the cardinality of the set of all subsets of $A$.
The Continuum hypothesis has been shown to be independent of the other set-theory axioms.

控制理论是研究如何调整控制系统的方程中的参数以使其性能最大化的学科。

Control theory is the study of how to adjust the parameters in the equations controlling a system to maximise its performance.

两个整数如果除1以外没有公因数，则称它们是互素的。

例如，3和4互素，但25和15不互素，因为两者都有因数5。

Two integers are coprime if they have no factors in common.

For example, 3 and 4 are coprime, but 25 and 15 are not since both have a factor of 5.

科萨一词意为“事物”，曾用来表示方程中的未知量。

代数家曾被称为cossist学派，代数曾被称为cossic术，这种情况持续了许多年。

cosa与cube是一种特殊形式：带有$a, b > 0$的三次方程的$x^{3} + ax = b$。

The word cosa means "a thing" and was used for the unknown in an equation.

Algebraists were called cossists, and algebra the cossic art, for many years.

The cosa and cube was a particular form: $x^{3} + ax = b$ of a cubic equation with $a, b > 0$.

宇宙起源论是一种关于宇宙起源，尤其是关于恒星、星云等起源的神话理论。

Cosmogony is a theory of myth of the origin of the universe, especially of the stars, nebulae, etc.

宇宙论是把宇宙作为一个整体来研究的科学。

Cosmology is the science of the universe as a whole.

三次方程是最高次项为三次的方程：
一般地，$ax^{3} + bx^{2} + cx + d = 0$。

A cubic equation is an equation whose highest term is of degree three:
$ax^{3} + bx^{2} + cx + d = 0$
in general.

插图：cuboctahedron.gif ↗A 截角八面体 是一个可以通过切去立方体或 八面体 的角而得到的立体。它有 8 个等边三角形面和 6 个正方形面。

插图：cuboctahedron.gif ↗A cuboctahedron is a solid which can be obtained by cutting the corners off a cube or off an octahedron. It has 8 faces which are equilateral triangles and 6 faces which are squares.

柯蒂斯-沙普利辩论是1920年赫伯·柯蒂斯与哈罗·沙普利之间的一场辩论，争论我们的宇宙是否只包含银河系（沙普利主张），还是存在其他星系（当时称为“岛宇宙”，柯蒂斯主张）。辩论时没有明确的胜者，但回顾此事的天文学家一致认为柯蒂斯更接近正确。

The Curtis Shapley Debate was a debate in 1920 between Herber Curtis and Harlow Shapley about whether our universe consisted of only the milky way (advocated by Shapley) or if there were other galaxies (then called 'island universes', and advocated by Curtis). At the time of the debate there was no clear winner, but astronomers looking back agree that Curtis was closer to being correct.

达马斯基奥斯（生于约480年，卒于约550年）于约520年被任命为雅典学园的负责人。529年基督教皇帝查士丁尼关闭学园时，他仍是负责人。他是一位新柏拉图主义者哲学家，也是普罗克洛的追随者。

Damascius (born: about 480, died: about 550) was appointed head of the Academy in Athens about 520. He was still head when the Christian emperor Justinian closed the Academy in 529. He was a Neoplatonist philosopher and follower of Proclus.

赤纬是从天赤道起算的角距离。
另一个坐标是赤经。

The declination is the angular distance from the celestial equator.
The other coordinate is the right ascension.

亏数是其真因数之和小于该数本身的整数。

例如，8的真因数是1、2、4，其和为7。
另见：完全数、盈数。

A deficient number is an integer for which the sum of its proper divisors is less than than the number itself.

For example, the proper divisors of 8 are 1, 2, 4, which sum to 7.
See also: perfect number, abundant number.

提洛同盟是由雅典领导的希腊城邦联盟，总部设在提洛岛，成立于公元前478年的希波战争期间。所有希腊城邦都被邀请加入以保护自己免受波斯侵略。公元前404年斯巴达击败雅典后，他们解散了该联盟。

The Delian League was a confederation of Greek states led by Athens, with headquarters at Delos, founded in 478 BC during the Greco-Persian wars. All Greek states were invited to join to protect themselves from Persia. After Sparta defeated Athens in 404 BC they disbanded the league.

最著名的古代神谕是位于帕纳索斯山上的德尔斐阿波罗神谕。到公元前7世纪和6世纪，它的名声已传遍希腊世界。立法者、崇拜的创立者、想知道战争结果的人或寻求了解其行动结果的政客都会前往那里寻求建议。建议以编码形式给出，可以有不同解释。

The most famous ancient oracle was that of Apollo at Delphi, situated on Mount Parnassus. Its fame stretched across the Hellenic world by the 7th and 6th centuries BC. Lawmakers, founders of cults, those wanting to know the outcome of wars or politicians seeking to know the outcome of their actions went there for advice. It was given in a coded form that could be interpreted in different ways.

矩阵的行列式是矩阵元素的组合，其性质是行列式为0当且仅当矩阵不可逆。

The determinant of a matrix is a combination of the entries of the matrix with the property that the determinant is 0 if and only if the matrix is not invertible.

辩证法是对话、问答的艺术；根据柏拉图，辩证技能是就事物的本质提出和回答问题的能力。

Dialectic is the art of conversation, of question and answer; and according to Plato, dialectical skill is the ability to pose and answer questions about the essences of things.

差分方程是涉及整变量函数的相继值之差的方程。
差分方程可视为微分方程的离散版本。
例如，差分方程$f(n+1) - f(n) = g(n)$是微分方程$f'(x) = g(x)$的离散版本。

A difference equation is an equation involving the differences between successive values of a function of an integer variable.
A difference equation can be regarded as the discrete version of a differential equation.
For example the difference equation $f(n+1) - f(n) = g(n)$ is the discrete version of the differential equation $f'(x) = g(x)$

一个微分方程是涉及待解函数的一阶或高阶导数的方程。
如果方程只涉及一阶导数，则称为一阶方程，依此类推。
如果只涉及导数的$n$次幂，则称该方程具有次数$n$。次数为一的方程称为线性。
只含一个变量的方程称为常微分方程，以区别于偏微分方程。

A differential equation is an equation involving the first or higher derivatives of the function to be solved for.
If the equation only involves first derivatives it is called an equation of order one, and so on.
If only $n$-th powers of the derivatives are involved, the equation is said to have degree $n$. Equations of degree one are called linear.
Equations in only one variable are called ordinary differential equations to distinguish them from partial differential equations.

微分几何研究流形的性质，这些性质依赖于能够进行微分。
所研究的性质示例包括曲线的曲率和挠率以及曲面的各种曲率。

Differential geometry studies properties of manifolds which depend on being able to do differentiation.
Examples of the properties studied are curvature and torsion for curves and various curvatures for surfaces.

微分拓扑是拓扑学的一个分支，研究在连续变换下保持不变的微分几何的性质。

Differential topology is a branch of topology in which one investigates properties of differential geometry which are preserved by continuous transformation.

第欧根尼·拉尔修（公元前3世纪）是一位希腊作家，撰写了重要的希腊哲学史。他的主要著作Lives, Teachings, and Sayings of Famous Philosophers引用了数百位权威，内容从轶事到有价值的传记资料不等。

Diogenes Laertius (3rd century BC) was a Greek author who wrote an important history of Greek philosophy. His major work Lives, Teachings, and Sayings of Famous Philosophers quotes hundreds of authorities and ranges from gossip to valuable biographical data.

狄奥尼修斯一世（生于约公元前430年，卒于公元前367年）是叙拉古的僭主，他征服了西西里和意大利南部，使叙拉古成为希腊本土以西最强大的希腊城邦。他阻止了迦太基征服西西里，但其残暴统治损害了希腊的声誉。

Dionysius I (born: about 430 BC, died: 367 BC) was a tyrant from Syracuse who conquered Sicily and southern Italy making Syracuse the strongest Greek city west of the Greek mainland. He prevented Sicily from being conquered by Carthage but his cruel regime damaged the reputation of Greece.

狄奥尼修斯二世（约生于公元前400年，约卒于公元前330年）是狄奥尼修斯一世之子，公元前367年继承其父之位，并与迦太基缔结和约。公元前357年被狄翁驱逐后，他一直流亡在外，直到公元前346年重新控制了叙拉古。公元前344年，他投降并退居科林斯。

Dionysius II (born: about 400 BC, died: about 330 BC) was a son of Dionysius I he succeeded his father in 367 BC and made peace with Carthage. Driven out bt Dion in 357 BC he was in exile until 346 BC when he regained control of Syracuse. He surrendered and retired to Corinth in 344 BC.

丢番图逼近是用有理的（整数之比）逼近实数。

Diophantine approximation is approximating real numbers by rationals (ratios of integers).

丢番图方程是只需求整数解的方程。

A Diophantine equation is one which is to be solved for integer solutions only.

多项式方程的判别式是系数的函数，它允许人们判断该方程何时有一对重根。
二次方程：$ax^{2} + bx + c = 0$的判别式是$b^{2} - 4ac$。
三次方程：$x^{3} + qx + r = 0$的判别式是$4q^{3} + 27r^{2}$。

一个代数数域的判别式是一个重要的数值不变量，它（粗略地）衡量该域整数环的大小。

The discriminant of a polynomial equation is a function of the coefficients which allows one to say when it has a pair of equal roots.
The discriminant of the quadratic equation: $ax^{2} + bx + c = 0$ is $b^{2} - 4ac$.
The discriminant of the cubic equation: $x^{3} + qx + r = 0$ is $4q^{3} + 27r^{2}$.

The discriminant of an algebraic number field is an important numerical invariant that (roughly) measures the size of the ring of integers of the field.

讲师（Docent）或讲师或编外讲师是德国大学中一位被认可但不在领薪编制内的教师。这一职位通常由一位为获得教授资格论文（Habilitation）而工作的年轻数学家担任。

A docent or dozent or privatdozent was a recognised teacher at a German university who was not on the salaried staff. It was often a post held by a young mathematician working for his habilitation.

插图：dodecahedron.gif ↗A 正十二面体 是一个有 12 个面的 正多面体，每个面都是正五边形。

插图：dodecahedron.gif ↗A dodecahedron is a regular polyhedron with 12 faces each of which is a regular pentagon.

希腊数学的经典问题之一是找到2的立方根的尺规构造。这被称为倍立方。
这有时被称为Delian problem，源于德尔斐神谕要求进行此构造以阻止瘟疫的故事。
其他经典问题是化圆为方和三等分角。

One of the classic problems of Greek mathematics was to find a ruler and compass construction for the cube root of 2. This was called duplicating the cube
This is sometimes called the Delian problem from the story that the oracle at Delphi demanded that this construction be performed to stop a plague.
The other classic problems were squaring the circle and trisecting an angle.

一个动力系统提供了一种描述系统的一个状态如何演变为另一个状态的方法。
其研究可追溯到儒勒·昂利·庞加莱关于三体问题的工作，甚至更早到惠更斯关于复摆的工作。

A dynamical system gives a means of describing how one state of a system evolves into another.
Its study goes back to the work of Poincaré's work on the three body problem or even earlier to Huygen's work on the compound pendulum.

偏心理论是行星在中心不与地球重合的圆上运动的学说

Eccentric theory is the theory that planets move round in circles whose centres do not coincide with the Earth

黄道是地球轨道切割天球的平面。
因此太阳在恒星之间的视周年运动。

The ecliptic is the plane in which the orbit of the Earth cuts the celestial sphere.
Hence the apparent annual motion of the Sun among the fixed stars.

黄道坐标是用赤经和赤纬描述天体位置的方式。

这些是相对于黄道平面给出的。

Ecliptic coordinates are the description of the position of a heavenly body in terms of its right ascension and declination.

These are given relative to the ecliptic plane.

如果$T$是从一个向量空间到自身的线性变换，那么满足$T\mathbb{v} = \lambda \mathbb{v}$的非零向量$\mathbb{v}$称为本征向量，λ称为特征值/特征向量/特征函数。
如果空间是函数空间，该特征向量可以称为本征函数。

If $T$ is a linear transformation from a vector space to itself, then a non-zero vector $\mathbb{v}$ satisfying $T\mathbb{v} = \lambda \mathbb{v}$ is called an eigenvector and λ is called the eigenvalue.
If the space is a space of functions, the eigenvector may be called an eigenfunction.

埃利亚学派是来自芝诺的一个哲学学派，芝诺是意大利南部的一个希腊殖民地。其主要教义出自巴门尼德，他认为实在是单一实体，我们通过感官察觉到的众多对象只是表象或名称。

The Eleatic School was a school of philosophy from Elea which was a Greek colony in southern Italy. The main doctrines were due to Parmenides who argued that reality was a single entity, the apparent many objects we detect with our senses are only appearances or names.

插图：ellipse.gif ↗A 椭圆是圆锥曲线之一。
它可以用焦点准线性质定义为到定点的距离是到定直线距离的固定倍数$e < 1$的点的轨迹。
或者通过笛卡尔坐标定义为平面上满足方程的点的集合：
$ax^{2} + by^{2} = 1$

插图：ellipse.gif ↗A ellipse is one of the conic sections.
It may be defined using the focus directrix property as the locus of points whose distance from a fixed point is a fixed multiple $e < 1$ from a fixed line.
or via Cartesian coordinates as the set of points in a plane satisfying the equation :
$ax^{2} + by^{2} = 1$

椭圆函数是从$\mathbb{C}$到$\mathbb{C}$的双周期解析函数。也就是说，对于复数$w$的两个独立值，函数$f(z)$和$f(w + z)$相同。
它也可以视为某些积分（称为椭圆积分）的反函数，这些积分的形式为插图：ellip_int.gif ↗，其中$R$是次数为3或4的多项式。

An elliptic function is an analytic function from $\mathbb{C}$ to $\mathbb{C}$ which is doubly periodic. That is, for two independent values of the complex number $w$, the functions $f(z)$ and $f(w + z)$ are the same.
It can also be regarded as the inverse function to certain integrals (called elliptic integrals) of the form 插图：ellip_int.gif ↗ where $R$ is a polynomial of degree 3 or 4.

在复分析中，整函数（有时称为整函数）是在整个复平面上全纯的复值函数。

这样的函数在C中处处有复导数。

In complex analysis, an entire function (sometimes called an integral function) is a complex-valued function that is holomorphic over the whole complex plane.

Such a function then has a complex derivative everywhere in C.

一条曲线如果切线族中的每一条曲线，那么它就是该曲线族的包络。

A curve is the envelope of a family of curves if it touches every curve in the family.

爱比克泰德（生于：约55年，位于Phrygia的Hierapolis[今土耳其Pamukkale]，卒于：约135年，位于Epirus[希腊]的Nicopolisin）是一名奴隶，曾参加斯多葛派Musonius Rufus的讲座。他获得了自由，但一生饱受疾病之苦。90年，他与其他斯多葛派哲学家一起被皇帝Domitian驱逐出罗马。他在Nicopolis度过了余生。他的教义因其宗教色彩而受到基督徒的欢迎。

Epictetus (born: about 55 in Hierapolis, in Phrygia [now Pamukkale in Turkey], died: about 135 in Nicopolisin in Epirus [Greece]) was a slave who attend lectures by the Stoic Musonius Rufus. He gained his freedom but spent his life suffering ill health. He was expelled from Rome in 90 with other Stoic philosophers by the emperor Domitian. He spent the rest of his life at Nicopolis. His teachings were popular with Christians because of their religious tone.

伊壁鸠鲁学派是伊壁鸠鲁的追随者，他们认为快乐是幸福生活的基础。尽管他们相信诸神太忙而不会关注人类，但人们仍必须向往像诸神那样，因为只有这样他们才能接近完美。

Epicureans were followers of Epicurus who believed pleasure was the basis of a happy life. Although they believed that the Gods were too busy to pay attention to humans, yet men had to aspire to like the Gods since only in this way could they approach perfection.

伊壁鸠鲁（生于：公元前341年，希腊萨摩斯岛；卒于：公元前270年，雅典）是Nausiphanes的学生，而Nausiphanes是德谟克利特的弟子。他一生中前35年大部分时间在亚洲度过，然后于公元前306年前往雅典并建立了“花园”。他的哲学基于简单的快乐和友谊。他还提出了原子理论。

Epicurus (born: 341 BC in Samos in Greece, died: 270 BC in Athens) was a pupil of Nausiphanes, a disciple of Democritus. He spent most of the first 35 years of his life in Asia, then went to Athens in 306 and established The Garden. His philosophy was based on simple pleasure and friendship. He also proposed an atomic theory.

本轮是一个圆，其圆心沿另一个圆的圆周运动。
本轮最初被引入是为了解释行星体在恒星之间的轨道。

An epicycle is a circle whose centre is carried around the circumference of another circle.
Epicycles were introduced originally to explain the orbits of planetary bodies among the fixed stars.

赤道坐标是相对于地球赤道平面给出的天体位置的描述。
它们与黄道坐标不同，因为地轴相对于黄道平面有倾角或黄赤交角。

Equatorial coordinates are the description of the position of a heavenly body given relative to the plane of the Earth's equator.
They differ from ecliptic coordinates because of the inclination or obliquity of the Earth's axis of rotation to the plane of the ecliptic.

二分点是一年中正午太阳直射赤道、使昼夜等长的时间。
春分和秋分大约发生在3月21日和9月23日。

The equinox is the time of the year when the noon sun is overhead at the equator making day and night equal in length.
Equinoxes occur about 21st March and 23rd September.

如果两个数m和n的真因数之和相同，则称它们为等价。

例如，159、559和703的真因数之和都等于57。

Two numbers m and n are called equivalent if sums of their proper divisors are the same.

For example, the proper divisors of 159, 559 and 703 all sum to 57.

遍历理论研究系统的统计行为和定性行为。
它起源于对统计力学的研究，但现在在许多其他领域有应用。

Ergodic theory investigates the statistical and qualatitive behaviour of a system.
It originated in the study of statistical mechanics but now has applications in many other fields.

婆罗摩笈多批评阿耶波多先前给出的、且好得多的π值。他给出3作为实际计算的近似值，√10作为精确值。

Brahmagupta criticises Aryabhata for his earlier, and much better, value of π. He gave 3 as the approximate value for practical calculation and √10 as the exact value.

伦理学是道德的科学：哲学中处理人的品格和行为的分支。

Ethics is the science of morals: the branch of philosophy dealing with human character and conduct.

麦加拉的欧几里得（生于公元前430年，卒于约公元前360年）是苏格拉底的学生。他的哲学是“善是一，尽管它有许多名称，有时称为智慧，有时称为神，有时称为理性”，并且“与善相反的东西没有实在性”。

Euclid of Megara (born: 430 BC, died: about 360 BC) was a pupil of Socrates. His philosophy was that "the good is one, though it is called by many names, sometimes wisdom, sometimes God, and sometimes reason" and that "the contrary to the good has no reality".

欧几里得算法是一种高效程序，用于计算一对$m, n$的最大公因数$d$，并将其写成$d = pm + qn$的形式，其中$p, q$为整数。

The Euclidean algorithm is an efficient procedure to calculate the highest common factor $d$ of a pair $m, n$ and write it in the form $d = pm + qn$ with $p, q$ integers.

欧拉-麦克劳林求和公式将积分与级数的和联系起来。它可用于将积分求值为和，或通过积分求无穷级数的和。它由莱昂哈德·欧拉和Maclaurin独立发现。

The Euler-Maclaurin Summation Formula relates integrals and sums of series. It can be used to evaluate integrals as sums or sums of infinite series by integrals. It was discovered independently by Euler and Maclaurin.

1769年，莱昂哈德·欧拉猜想不可能给出三个四次幂，其和为四次幂，四个五次幂，其和为五次幂，对于更高次幂也类似。直到20世纪才找到反例。

In 1769 Euler conjectured that it is impossible to exhibit three fourth powers whose sum is a fourth power, four fifth powers whose sum is a fifth power, and similarly for higher powers. Counterexamples were not found until the 20th Century

曲线的渐屈线是正规到该曲线的包络。这也可以看作曲率中心的轨迹。

The evolute of a curve is the envelope of the normals to the curve. This can also be thought of as the locus of the centres of curvature.

如果$N$是群$G$的正规子群，那么可以定义一个商群$G/N$，其中$N$的元素都“变为恒等元”。
然后可以尝试用$N$和$G/N$的结构来理解$G$的结构。
如果$G$是有限的，那么$G/N$的阶是$|G|/|N|$。

If $N$ is a normal subgroup of a group $G$ then one can define a factor group $G/N$ in which the elements of $N$ all "become the identity".
One can then try to understand the structure of $G$ in terms of the structures of $N$ and of $G/N$.
If $G$ is finite then the order of $G/N$ is $|G|/|N|$.

整数n的阶乘是乘积1.2.3. ... .n
它用$n!$表示
推广到非整数时，它就是伽马函数。

The factorial of an integer n is the product 1.2.3. ... .n
It is denoted by $n!$
Generalised to non-integers, it is the Gamma function.

费马大定理（如此称呼是因为它是皮埃尔·德·费马声称但尚未证明的最后一个结果）断言：如果$n > 2$，那么方程$x^{n} + y^{n} = z^{n}$没有正整数解。

Fermat's last theorem (so called because it was the last of the results which Fermat claimed and which had not been proved) states that if $n > 2$ then the equation $x^{n} + y^{n} = z^{n}$ has no positive integer solutions.

斐波那契数列是由规则$f_{1} = f_{2} = 1,  f_{n+1} = f_{n} + f_{n-1}$生成的数列1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...。
它由比萨的斐波那契在Liber abaci年引入，源于一个涉及兔子种群增长的问题。

The Fibonacci sequence is the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... generated by the rule $f_{1} = f_{2} = 1,  f_{n+1} = f_{n} + f_{n-1}$.
It was introduced by Fibonacci of Pisa in Liber abaci in a problem involving the growth of a population of rabbits.

域是其中非零元素都有乘法逆元的环。
最重要的域包括实数$\mathbb{R}$、复数$\mathbb{C}$以及数论和组合数学中使用的各种有限域。

A field is a ring in which non-zero elements have multiplicative inverses.
Among the most important fields are the Real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$ and various finite fields which are used in Number Theory and Combinatorics.

菲尔兹奖每四年颁发一次，授予40岁或以下最杰出的数学家。由于数学没有诺贝尔奖，它被视为数学家所能获得的最高专业荣誉。
它由加拿大数学家J C 约翰·查尔斯·菲尔兹资助，于1936年首次颁发。

Fields Medals are given every four years to the most distinguished mathematicians aged 40 or under. In the absence of a Nobel prize in mathematics, they are regarded as the highest professional honour a mathematician can attain.
They were funded by the Canadian mathematician J C Fields and were first awarded in 1936.

圆锥曲线的焦点准线定义是：一动点的轨迹，该点移动时到一定点（焦点）的距离是到一定直线（准线）的距离的固定倍数$e$。
如果$e = 1$，则曲线是抛物线；如果$e < 1$，则它是椭圆；如果$e > 1$，则它是双曲线。

The focus directrix definition of a conic section is as the locus of a point which moves so that its distance from a fixed point (the focus) is a fixed multiple $e$ from a fixed line (the directrix).
If $e = 1$ then the curve is a parabola, if $e < 1$ then it is an ellipse, if $e > 1$ then it is a hyperbola.

四色定理指出，平面或球面上任何区域地图都可以用至多四种颜色着色，使得共享边界的区域颜色不同。
尽管这一结果早在1852年就被猜想，但其证明直到1976年才完成。

The four colour theorem states that any map of regions on a plane or sphere can be coloured with at most four colours in such a way that regions sharing a boundary are coloured differently.
Although the result was conjectured as early as 1852 its proof was not completed until 1976.

集合$A$上的自由群（生成元）是由$A$中元素的所有有限词及其逆元组成的群。
因此，例如，如果$A = {x, y, z}$，那么诸如$x y^{2}z^{-1}x$或$x^{2}z y^{-1}$这样的词就是自由群的元素。
人们通过连接这些词（将它们并排写在一起）然后消去任何与其逆元并排的元素来相乘这样的词。
空字是这个群的单位元。
大小为$n$的集合上的自由群记作$F_{n}$。

A free group on a set $A$ of generators is the group consisting of all finite words in elements of $A$ and their inverses.
So if, for example, $A = {x, y, z}$ then words such as $x y^{2}z^{-1}x$ or $x^{2}z y^{-1}$ are elements of the free group.
One multiplies such words by concatenating them (writing them alongside one another) and then cancelling out any elements which are alongside thir inverses.
The empty word is the identity of this group.
The free group on a set of size $n$ is denoted $F_{n}$.

傅里叶分析是将（例如）区间 [ 0, 2π ] 上的周期函数写成函数 $\sin(nx)$ 与 $\cos(nx)$ 的倍数之和的过程。
$f(x) = a_{0} + a_{1}\cos(x) + b_{1}\sin(x) + a_{2}\cos(2x) + b_{2}\sin(2x) + ...$
这称为 $f$ 的 Fourier series，而数 $a_{i}$ 与 $b_{i}$ 称为 Fourier coefficients。

Fourier analysis is the process by which one may write a periodic function on (say) the interval [ 0, 2π ] as the sum of multiples of the functions $\sin(nx)$ and $\cos(nx)$.
$f(x) = a_{0} + a_{1}\cos(x) + b_{1}\sin(x) + a_{2}\cos(2x) + b_{2}\sin(2x) + ...$
This is called the Fourier series of $f$ and the numbers $a_{i}$ and $b_{i}$ are called the Fourier coefficients.

泛函分析研究无穷维 向量空间 以及它们之间的映射。
这些空间的元素本身常常就是函数，例如区间上的连续函数空间。

Functional analysis studies infinite dimensional vector spaces and mappings between them.
The elements of these spaces are very often themselves functions, as for example, the space of continuous functions on an interval.

代数基本定理是任何实系数或复系数多项式在复平面上都有根这一结果。

The fundamental theorem of algebra is the result that any polynomial with real or complex coefficients has a root in the complex plane.

拓扑空间 $X$ 的 基本群 是一个群，其构造方式是考察 $X$ 中的闭路径如何组合以得到新路径。
在一种适当的路径等同方式（“同伦”）下，可以在该集合上得到群结构，从而给出空间 $X$ 的一个代数不变量。

The fundamental group of a topological space $X$ is a group constructed by looking at how closed paths in $X$ can be combined to get new paths.
Under a suitable way of identifying paths ("homotopy") one can get a group structure on the set which gives an algebraic invariant of the space $X$.

Gaius 西门·马里乌斯（生于：约公元前157年，Cereatae，靠近Latium[今意大利]的Arpinum[Arpino]；卒于：公元前86年，罗马）是一位好士兵和技巧娴熟的将军，凭借其士兵的选票七次当选执政官。他是一位糟糕的政治家。

Gaius Marius (born: about 157 BC in Cereatae, near Arpinum [Arpino] in Latium [now in Italy], died: 86 BC in Rome) was a good soldier and a skilful general who was elected consul seven times on the strength of the votes from his soldiers. He was a poor politician.

伽罗瓦理论 是研究与多项式方程可以相关联的某些 群 的学科。
一个方程的解能否用有理函数以及平方根、立方根等写出，取决于这个 Galois group 的某些群论性质。

Galois theory is the study of certain groups which can be associated with polynomial equations.
Whether or not the solutions to an equation can be written down using rational functions and square roots, cube roots, etc. depends on certain group-theoretic properties of this Galois group.

博弈论处理对“博弈”（即涉及利益冲突的情形）的分析。除了可以被完全分析的简单博弈外，该理论还应用于像扑克、国际象棋等“真实博弈”，以及政治、经济等领域。

Game theory deals with the analysis of "games" (i.e., situations which involve conflicts of interests). As well as simple games, which can be analysed completely, the theory has applications in "real games" like poker, chess, etc as well as in areas like politics, economics etc.

伽马函数 是 阶乘 到实数线和复平面的推广。
它由下式定义：
插图：gamma_funct.gif ↗
如果 $n$ 是整数，那么 $\Gamma (n+1) = n!$

The gamma function is a generalisation of the factorial function to the Real line and to the Complex plane.
It is defined by:
插图：gamma_funct.gif ↗
If $n$ is an integer then $\Gamma (n+1) = n!$

1929年，亚历山大·格尔丰德 猜想：如果 $a_{n}$ 和 $b_{n}$ 对于 $1 ≤ m ≤ n$ 是 代数数，使得 {$\log_{e} a_{n}, 1 ≤ m ≤ n$} 在 Q 上线性无关，那么
$b_{1} \log_{e} a_{1} + b_{2} \log_{e} a_{2} + ... + b_{n} \log_{e} a_{n} ≠ 0$。
其一个特例是：如果 $a$ 是代数数（$a ≠ 0, 1$）且 $b$ 是 无理数 代数数，则 $a^{b}$ 是 超越的。

In 1929 Gelfond conjectured that:-

If $a_{n}$ and $b_{n}$ for $1 ≤ m ≤ n$ are algebraic numbers such that {$\log_{e} a_{n}, 1 ≤ m ≤ n$} are linearly independent over Q then
$b_{1} \log_{e} a_{1} + b_{2} \log_{e} a_{2} + ... + b_{n} \log_{e} a_{n} ≠ 0$.

一般拓扑学或点集拓扑学是拓扑学的一个分支，研究如何在集合上赋予一个结构，使得从实数$\mathbb{R}$到自身的映射的连续性概念得到推广。
集合$X$上的一个拓扑学是集合$X$的某些所谓“开子集”的集合，这些开子集满足各种公理。集合$X$连同这个拓扑称为一个拓扑空间。

General topology is a branch of topology in which one investigates how to put a structure on a set in such a way as to generalise the idea of continuity for maps from the reals $\mathbb{R}$ to itself.
A topology on a set $X$ is a certain set of so-called "open subsets" of the set $X$ which satisfy various axioms. The set $X$ together with this topology is called a topological space.

等比数列是每一项都是前一项的相同倍数（公比）的数列。
例如：3, 6, 12, 24, 48, ...

A geometric progression is a sequence in which each number is the same multiple (the common ratio) of the previous one.
As for example: 3, 6, 12, 24, 48, ...

1742年，在给莱昂哈德·欧拉的一封信中，克里斯蒂安·哥德巴赫猜想任何大于2的偶数都可以写成两个素数之和。尽管取得了许多进展，这个猜想仍未得到证明。

In 1742, in a letter to Euler, Goldbach conjectured that any even integer > 2 can be written as the sum of two primes. Despite much progress, the conjecture is still unproved.

黄金比例或黄金数或黄金比是实数(√5 + 1)/2 = 1.6180...。
它出现在数学的许多领域中，包括正五边形的几何、连分数、欧几里得算法、……

The golden ratio or golden number or golden mean is the real number (√5 + 1)/2 = 1.6180... .
It occurs in many areas in mathematics, including the geometry of the regular pentagon, continued fractions, the Euclidean algorithm, ...

引力波

Gravitational Waves

大圆是球面上最短距离的线：球面与过原点的平面的交线。

A great circle is a line of shortest distance on a sphere: the intersection of the sphere with a plane through the origin.

群是一个集合连同一种将元素组合以得到新元素的方法（加法或乘法或……），该方法满足某些性质，使其适用于广泛的应用。
排列的群、对称群、矩阵、……广泛用于数学和物理的许多领域。

A group is a set together with a method of combining elements to get new ones (addition or multiplication or ...) which satisfies certain properties making it suitable for a wide variety of applications.
Groups of permutations, symmetries, matrices, ... are widely used in many areas of mathematics and physics.

群的表现是将一个群表示为形式$< A | R >$的一种方式，其中$A$是该群的一组生成元，$R$是生成元的字所满足的一组关系。
因此，例如$< a, b | a b = b a >$是具有两个生成元的自由阿贝尔群。

A group presentation is a way of representing a group in the form $< A | R >$where $A$ is a set of generators for the group and $R$ is a set of relations which are satisfied by words in the generators.
So, for example$< a, b | a b = b a >$is a free abelian group with two generators.

文理中学（Gymnasium）是德国以及欧洲大陆某些其他国家的高中。

A Gymnasium is a senior secondary school in Germany and certain other countries of mainland Europe.

教授资格论文（Habilitation）是在德国大学授课所需的额外博士后资格。作为其中一部分必须提交的论文是Habilitationsschrift。

The Habilitation is the extra post-doctoral qualification needed to lecture at a German university. The thesis that has to be presented as part of this is the Habilitationsschrift.

日心说是行星围绕太阳而非地球运动的学说

Heliocentric theory is the theory that planets move around the Sun rather than the Earth

赫拉克利特（约公元前540年生于安纳托利亚以弗所，约公元前480年卒）是一位希腊哲学家，他试图基于火构成有序宇宙的基本物质本原这一信念来解释世界。他强调人们必须在社会和谐中共同生活。他以将生命类比为河流而闻名。

Heracleitus (born: about 540 BC in Ephesus in Anatolia, died: about 480 BC) was a Greek philosopher who tried to explain the world based on the belief that fire forms the basic material principle of an orderly universe. He stressed that men must live together in social harmony. He is famous for his analogy of life to a river.

希罗多德（公元前484年生于小亚细亚的哈利卡纳苏斯[今土耳其博德鲁姆]，约公元前425年卒）广泛游历波斯帝国，但曾在雅典和意大利南部的雅典殖民地图里逗留。他作为历史学家而闻名，其最著名的著作是关于公元前499年至公元前479年希腊与波斯之间战争的历史。

Herodotus (born: 484 BC in Halicarnassus in Asia Minor [now Bodrum in Turkey], died: about 425 BC) travelled widely through the Persian Empire but spent time in Athens and Thurii, a colony of Athens in southern Italy. He is famed as an historian whose most famous work is his history of the wars between Greece and Persia from 499 BC to 479 BC.

希尔伯特空间是$\mathbb{R}$或$\mathbb{C}$上的配备内积的向量空间。
最重要的实例如平方可和序列空间连同通常的点积。

A Hilbert space is a vector space over $\mathbb{R}$ or $\mathbb{C}$ equipped with an inner-product.
The most important real example is the space of square summable sequences together with the usual dot-product.

马鞍形曲线是Eudoxus在其行星理论中使用的一种8字形曲线。
（该词意为马绊）。

A hippopede is a figure of eight curve Eudoxus used in his planetary theory.
(The word means a horse-fetter).

全纯函数是一个或多个复变量的复值函数，在其定义域的每一点的邻域内复可微。

A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

荷马（公元前7世纪以前）是创作了著名《伊利亚特》和《奥德赛》的诗人。他是否真实存在远未确定，但如果他确实存在，他必须被视为有史以来最伟大的诗人之一。这些作品是早期希腊发展中的统一因素，荷马必定在其成功中发挥了很大作用。

Homer (before 7th century BC) was the poet who wrote the famous Iliad and Odyssey. It is far from certain that he actually existed, but if he did he must be regarded as one of the greatest poets of all time. These works were unifying factors in the development of early Greece and Homer must have played a large part in its success.

同心球系由若干旋转的球面组成，每个球面绕一条穿过地心的轴旋转。
每个球面的旋转轴并非固定于空间之中，但对大多数球面而言，这条轴本身也在旋转，因为它由固定在另一个旋转球面上的点所确定。

The homocentric sphere system consisted of a number of rotating spheres, each sphere rotating about an axis through the centre of the Earth.
The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere.

同调是将阿贝尔群（或更复杂的代数对象）附着于一个拓扑学以得到代数不变量的方法。在某种意义上，它探测空间中各种维数的“洞”的存在。
为处理这一问题而发展起来的方法导致了如今所谓的同调代数，并且可以对许多纯代数结构计算同调不变量。

Homology is a way of attaching abelian groups (or more elaborate algebraic objects) to a topological space so as to obtain algebraic invariants. In some sense it detects the presence of "holes" of various dimensions in the space.
The methods developed to handle this led to what is now called homological algebra and homological invariants can be calculated for many purely algebraic structures.

同伦是从一个拓扑学中的一条路径到另一条路径的连续变换，或者更一般地，是从一个函数到另一个函数的连续变换。
由同伦相连接的路径称为同伦，并说它们处于同一个同伦类中。
在这种同伦下保持不变的性质称为同伦不变量。
路径的同伦类可以复合以构成基本群或第一同伦群。其他映射可用于构成高阶同伦群。

A homotopy is a continuous transformation from one path in a topological space to another or more generally of one function to another.
Paths which are connected by a homotopy are called homotopic and are said to be in the same homotopy class.
Properties which are left unchanged by such homotopies are called homotopy invariants.
Homotoy classes of paths can be composed to form the fundamental group or first homotopy group. Other maps can be used to form higher homotopy groups.

插图：hyperbola.gif ↗A 双曲线是圆锥曲线之一。
它可以用焦点准线性质定义为到定点的距离是到定直线距离的固定倍数$e > 1$的点的轨迹。
或者通过笛卡尔坐标定义为平面上满足方程的点的集合：
$ax^{2} - by^{2} = 1$

插图：hyperbola.gif ↗A hyperbola is one of the conic sections.
It may be defined using the focus directrix property as the locus of points whose distance from a fixed point is a fixed multiple $e > 1$ from a fixed line.
or via Cartesian coordinates as the set of points in a plane satisfying the equation :
$ax^{2} - by^{2} = 1$

由卡尔·弗里德里希·高斯引入的超几何函数是超几何级数的和：
插图：hypergeometric_ser.gif ↗
它是超几何方程的解：
插图：hypergeometric_eqn.gif ↗
许多常见函数可以写成超几何函数。

The hypergeometric function introduced by Gauss, is the sum of the hypergeometric series:
插图：hypergeometric_ser.gif ↗
It is the solution of the hypergeometric equation:
插图：hypergeometric_eqn.gif ↗
Many common functions can be written as hypergeometric functions.

直角三角形的斜边是直角所对的边。

The hypotenuse of a right-angled triangle is the side opposite the right angle.

扬布利科斯（生于约公元250年，出生地为科埃勒叙利亚的哈尔基斯[今属黎巴嫩]，卒于约330年）是一位叙利亚哲学家，在新柏拉图主义的发展中起了重要作用。他留存下来的著作包括On the Pythagorean Life、The Exhortation to Philosophy、On the General Science of Mathematics、On the Arithmetic of Nicomachus和Theological Principles of Arithmetic。他对普罗克洛产生了重大影响。

Iamblichus (born: about 250 AD in Chalcis in Coele Syria [now in Lebanon], died: about 330) was a Syrian philosopher who played a large part in the development of Neoplatonism. His writings which have survived include On the Pythagorean Life; The Exhortation to Philosophy; On the General Science of Mathematics; On the Arithmetic of Nicomachus; and Theological Principles of Arithmetic. He was a major influence on Proclus.

插图：icosahedron.gif ↗正二十面体是一个正多面体，它有20个面，每个面都是等边三角形。

插图：icosahedron.gif ↗An icosahedron is a regular polyhedron with 20 faces each of which is an equilateral triangle.

环或半群的元素$r$若满足$r^{2} = r$，则称为幂等。

An element $r$ of a ring or semigroup is idempotent if it satisfies $r^{2} = r$.

无穷小是一种任意小的量，早期数学家在缺乏适当的极限理论时认为有必要将其纳入自己的理论中。
Infinitesimal calculus是微分与积分学。

An infinitesimal is an arbitrarily small quantity which early mathematicians found it necessary to incorporate into their theories in the absence of a proper theory of limits.
Infinitesimal calculus is the Differential and Integral Calculus.

如果一个圆与一个三角形或其他多边形的边相切，则称该圆内切于该三角形或多边形。
此时称该多边形外切于该圆。
内切于一个三角形的（唯一）圆称为内切圆，其圆心称为内心。

A circle is said to be inscribed to a triangle or other polygon if the edges are tangents to the circle.
The polygon is than said to be circumscribed to the circle.
The (unique) circle inscribed to a triangle is called the incircle and its centre is the incentre.

积分方程是包含待求解函数的积分的方程。

例如，给定$g, w$和$a$，人们可能希望就$f$求解方程插图：int_eqn.gif ↗。

An integral equation is an equation which involves an integral of the function which is to be solved for.

For example, being given $g, w$ and $a$, one might wish to solve the equation 插图：int_eqn.gif ↗ for $f$.

积分变换是涉及积分的过程，可应用于一个函数以得到另一个函数。
它通常具有形式插图：int_trans.gif ↗，其中$w$是某个“权”函数，积分在区间$I$上取。
积分变换的例子有拉普拉斯变换变换或约瑟夫·傅里叶变换。

An integral transform is a process involving an integral which can be applied to a function to get another function.
It is usually of the form 插图：int_trans.gif ↗ for some "weight" function $w$ where the integral is taken over an interval $I$.
Examples of integral transforms are Laplace transforms or Fourier transforms.

积分因子是乘以微分方程以使其可解的东西。

An integrating factor is something which one multiplies an ordinary differential equation by to make it possible to solve it.

直觉主义是逻辑学的一个分支，它强调数学优先于逻辑，数学对象由数学家在头脑中构造和操作，并且不可能仅仅通过建立若干公理来定义数学对象的性质。
特别地，直觉主义者拒绝排中律，而排中律允许使用反证法。

Intuitionism is a branch of logic which stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.
In particular, intuitionists reject the Law of the Excluded Middle which allows proof by contradiction.

不变量是在某类函数作用下保持不变的东西。特别地，不变量理论研究的是与多项式方程相关联且在变量变换下保持不变的量。
例如，判别式：$b^{2} - 4ac$是二次型：$ax^{2} + bxy + cy^{2}$的一个不变量。

An invariant is something that is left unchanged by some class of functions. In particular, invariant theory studied quantities which were associated with polynomial equations and which were left invariant under transformations of the variables.
For example, the discriminant: $b^{2} - 4ac$ is an invariant of the quadratic form: $ax^{2} + bxy + cy^{2}$.

如果$C$是一条曲线，$C'$是它的渐屈线，则$C$称为$C'$的渐伸线。
任何与$C$平行的曲线也是$C'$的渐伸线。
因此一条曲线有唯一的渐屈线，但有无穷多条渐伸线。
或者，渐伸线可以看作是与给定曲线的所有切线都正交的任意曲线。

If $C$ is a curve and $C'$ is its evolute, then $C$ is called an involute of $C'$.
Any parallel curve to $C$ is also an involute of $C'$.
Hence a curve has a unique evolute but infinitely many involutes.
Alternatively an involute can be thought of as any curve orthogonal to all the tangents to a given curve.

无理数是不能写成整数之商的无理数。

例如，√2是无理数。

An irrational number is a real number which is not rational and so cannot be written as a quotient of integers.

For example, √2 is irrational.

伊西多鲁斯（约公元537年）是来自米利都的建筑师，他主持了537年圣索菲亚大教堂的建造。

Isidorus (about 537 AD) was an architect from Miletus who directed the building of the Hagia Sophia in 537.

等周问题是比较具有相同周长的图形的面积或具有相同表面积的立体的体积。

Isoperimetry is the comparison of the areas of figures with the same perimeter of of the volumes of solids with the same surface areas.

约翰内斯·开普勒推测，球体在三维空间中最有效的排列就是用来堆放橙子、炮弹等的最显而易见的排列方式。这一点最终在1998年被证明是正确的。

Kepler speculated in that the most efficient arrangement of spheres in three-dimensions is the most obvious arrangement used to stack oranges, cannon-balls, ... . This was eventually proved to be true in 1998.

纽结理论是研究闭曲线如何嵌入三维空间而不自交的学科。
直观上，可以通过在一根普通绳子上打一个结，然后将绳子的自由端融合在一起来“打一个结”。
与任何纽结相关联的是它的纽结群，即从它所嵌入的$\mathbb{R}$3中移除该纽结后所得空间的基本群。

Knot theory is the study of the way a closed curve can be embedded in three dimensional space without intersecting itself.
Intuitively, one may "make a knot" by tying a knot in an ordinary piece of string and then fusing together the free ends of the string.
Associated with any knot is its knot group which is the fundamental group of the space obtained by removing the knot from the $\mathbb{R}$3 in which it is embedded.

函数f的拉普拉斯变换由积分定义：插图：laplace_trans.gif ↗

The Laplace transform of a function f is defined by the integral: 插图：laplace_trans.gif ↗

拉普拉斯方程是物理学中重要的偏微分方程之一。
在二维中它是：插图：laplace_eqn2.gif ↗，而在三维中它是插图：laplace_eqn3.gif ↗。

Laplace's equation is one of the important partial differential equations of physics.
In two dimensions it is: 插图：laplace_eqn2.gif ↗ while in three dimensions it is 插图：laplace_eqn3.gif ↗.

李代数是一种代数，其乘法满足类似于由$[A, B] = AB - BA$给出的矩阵上所谓的括号运算的性质，其中右侧的运算是矩阵的普通乘法和减法。
该运算不是结合的。

A Lie algebra is an algebra in which the muliplication satisfies properties similar to the so-called bracket operation on matrices given by $[A, B] = AB - BA$ where the operation on the right-hand side are ordinary multiplication and subtraction of matrices.
The operation is not associative.

李群是同时为流形的群。
实矩阵群自然地给出李群的例子。
李群的单位元处的切空间自然地构成一个李代数。

A Lie group is a group which is also a manifold.
Groups of real matrices give naturally occurring examples of Lie groups.
The tangent space at the identity element of a Lie group forms a Lie algebra in a natural way.

李环是一种环，其乘法满足类似于由$[A, B] = AB - BA$给出的矩阵上所谓的括号运算的性质，其中右侧的运算是矩阵的普通乘法和减法。
该运算不是结合的。

A Lie ring is a ring in which the muliplication satisfies properties similar to the so-called bracket operation on matrices given by $[A, B] = AB - BA$ where the operation on the right-hand side are ordinary multiplication and subtraction of matrices.
The operation is not associative.

一个线性群是矩阵在矩阵乘法下的一个群。

A linear group is a group of matrices under matrix multiplication.

一个链环是将若干 纽结理论嵌入$\mathbb{R}$3 的结果。

A link is the result of embedding several knots in $\mathbb{R}$3 .

李维（生于：公元前62年，意大利威尼托的帕塔维乌姆；卒于：公元17年，帕塔维乌姆）一生大部分时间在罗马度过。他是伟大的罗马历史学家之一，写了一部142卷的罗马史，在他有生之年即成为经典。这部著作对近2000年的历史写作风格和哲学产生了重大影响。

Livy (born: 62 BC in Patavium in Venetia in Italy, died: 17 AD in Patavium) spent most of his life in Rome. He was one of the great Roman historians and he wrote a history of Rome in 142 books that became a classic in his lifetime. The work had a major influence on the style and philosophy of historical writing for nearly 2000 years.

轨迹是满足某种条件的点或线的所有位置构成的线或面。
例如，到定点等距的点的轨迹是一个圆。

A locus is the line or surface made up of all the positions of a point or line satisfying some condition.
For example, the locus of points equidistant from a fixed point is a circle.

斜驶线是球面上与每条经线相交成相同角度的线。它是沿固定罗盘方位行进时所走的路径，在墨卡托投影上是一条直线。

A loxodrome is a line on a sphere cutting each meridian at the same angle. It is the path followed when following a fixed compass bearing and is a straight line on the Mercator's projection.

卢奇安（生于约公元120年，生于叙利亚科马基尼的萨莫萨塔[今土耳其萨姆萨特]，卒于公元180年后，卒于[希腊]雅典）在小亚细亚西部接受教育，在那里接受了希腊文学教育。他游历了希腊、意大利，然后是高卢。在雅典，他成为著名的修辞学家、小册子作者和讽刺作家。

Lucian (born: about 120 AD in Samosata in Commagene in Syria [now Samsat, in Turkey], died: after 180 in Athens [Greece]) was educated in western Asia Minor where he received a Greek literary education. He travelled in Greece, Italy and then Gaul. In Athens he became a famous rhetorician, pamphleteer and satirist.

插图：lune_alone.gif ↗一个月形是一个圆从较小圆内部截出的区域。

插图：lune_alone.gif ↗A lune is the area cut off by one circle from the interior of a smaller one.

幻方是一组整数（通常是$1, 2, ... , n^{2}$）排成正方形，使得每一行、每一列（通常还有两条对角线）的和都相同。
例如：插图：magic_sq.gif ↗是一个3×3幻方。

A magic square is a set of integers (often $1, 2, ... , n^{2}$) arranged in a square in such away that each row, each column (and often the two diagonals as well) sum to the same number.
For example:  插图：magic_sq.gif ↗   is a 3 × 3 magic square.

磁偏角或罗盘磁差是罗盘读数与真北之间的差值。

Magnetic declination or compass variation is the amount by which compass readings differ from the true North.

绝对星等是恒星在10秒差距处的亮度。
视星等是恒星在地球上看起来的亮度。

The absolute magnitude is how bright the star is from 10 parsecs away.
The apparent magnitude is how bright the star appears on earth.

一个$n$维流形是一个局部欧几里得的拓扑空间。也就是说，每一点都位于一个看起来像空间$\mathbb{R}$$^{n}$的区域中。
1维流形是曲线，2维流形是曲面，等等。
具有允许进行微分运算的结构的流形称为可微流形，如果微分可以任意多次进行，则称为光滑流形。

An $n$-dimensional manifold is a topological space which is locally Euclidean. That is, every point lies in a region which looks like the space $\mathbb{R}$$^{n}$.
A 1-manifold is a curve, a 2-manifold is a surface, etc.
Manifolds which have a structure which allow differentiation to be performed are called differentiable manifolds and if differentiation can take place arbitrarily often they are called smooth manifolds.

马塞卢斯（生于：约公元前268年，卒于：公元前208年，地点在阿普利亚的维努西亚附近）是一位罗马将军，曾五次担任执政官。公元前216年罗马人在坎尼战败后，他指挥军队，使一些地区免于汉尼拔的侵袭。从公元前214年到公元前211年，他在西西里任职，在那里经过两年围城，他在一次围城中攻占了叙拉古，阿基米德在这次围城中死亡。后来他与汉尼拔进行了无决定性结果的战斗。

Marcellus (born: about 268 BC, died: 208 BC near Venusia in Apulia) was a Roman general who was consul five times. After the Romans were defeated at Cannae in 216 BC, he commanded the army and saved regions from Hannibal. From 214 to 211 he served in Sicily, where, after a two-year siege, he took Syracuse in a siege in which Archimedes died. He later fought inconclusive battles with Hannibal.

在概率论中，鞅是一个公平博弈的模型，其中对过去事件的了解永远无助于预测未来收益的均值。

最初它是一种赌博系统，不断加倍赌注，希望最终获胜从而必定产生净利润。

In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings.

Originally it was a gambling system of continually doubling the stakes in the hope of an eventual win that must yield a net profit.

矩阵是数字或更复杂事物的二维数组或“盒子”。
有（简单的）规则用于加减形状相同的矩阵，有时有（更复杂的）规则用于相乘甚至相除它们。
方阵的集合常常形成有趣的代数对象，如群或更一般的结构。
一些例子：插图：matrix.gif ↗

A matrix is two dimensional array or "box" of numbers or more complicated things.
There are (easy) rules for adding and subtracting matrices of the same shape and sometimes (more complicated) rules for multiplying or even dividing them.
Sets of square matrices often form interesting algebraic objects like groups or more general structures.
Some examples:插图：matrix.gif ↗

测度论研究积分得以进行的条件。
它考虑估计一个集合的“大小”的各种方式。

Measure theory investigates the conditions under which integration can take place.
It considers various ways in which the "size" of a set can be estimated.

一个中间的是欧几里得的无理数范畴之一。
如果一个矩形的边仅平方可通约（例如3与√2），那么与该矩形面积相等的正方形的边（在此情形下为3√2）是一个称为中间的的无理数。

A medial is one of Euclid's categories of irrational numbers.
If the sides of a rectangle are commensurable in square only (e.g., 3 and √2), then the side of the square equal to the rectangle (3√2, in this case) is an irrational called a medial.

皮拉的梅内德谟是柏拉图在世时学园的成员。

Menedemus of Pyrrha was a member of the Academy in Plato's lifetime.

亚纯函数是没有本质奇点的复变函数。也就是说，它最坏的奇点是极点。

A meromorphic function is a function of a complex variable which has no essential singularities. That is, its worst singularities are poles

形而上学是研究自然与思想第一原理的哲学分支。

Metaphysics is the branch of philosophy that investigates the first principles of nature and thought.

穷竭法是通过用一列多边形的面积逼近来计算面积。

例如，通过内接边数越来越多的多边形来填满圆的内部。

The method of exhaustion is calculating an area by approximating it by the areas of a sequence of polygons.

For example, filling up the interior of a circle by inscribing polygons with more and more sides.

银河是构成宇宙的数十亿个星系之一。它是一个巨大的盘状恒星系统，太阳位于其中，它表现为一条环绕天空的不规则发光带。这条发光带是由大量各自太暗而无法单独看到的恒星造成的，但大量的恒星在天空该区域的星系平面内形成了背景辉光。

The Milky Way is one of billions of galaxies that make up the universe. It is a large disk-shaped system of stars in which the sun is situated and appears as an irregular luminous band that circles the sky. The luminous band is caused by large numbers of stars each too faint to be seen in individually, but the large number gives a background glow in that part sky in the plane of the galaxy.

模型论是研究满足特定一组公理的数学结构，尤其是在逻辑学领域。

Model theory is the study of those mathematical structures which satisfy a particular set of axioms -- particularly in the area of Logic.

如果$p$整除$m - n$，则称数$m$与$n$相等模$p$。

We say that numbers $m$ and $n$ are equal modulo $p$, if $p$ divides exactly into $m - n$.

新柏拉图主义者是古希腊最后一个伟大的哲学学派。该学派由普罗提诺于3世纪创立。普罗提诺及其追随者自认为是柏拉图主义者，但他们的哲学既包含柏拉图的思想，也包含亚里士多德学派和斯多葛学派的思想。

Neoplatonists were the last great ancient Greek school of philosophy. The school was founded by Plotinus in the 3rd century. Plotinus and his followers considered themselves Platonists but their philosophy contained ideas from Aristotelian and the Stoics as well as from Plato.

环或半群的元素若其某个幂为0，则称为幂零。

An element of a ring or semigroup is nilpotent if some power of it is 0.

一个幂零群是一个群，它有一条子群${1} = \gamma _{n+1} \subset \gamma _{n} \subset ... \subset \gamma _{1} = G$链，其中每个子群$\gamma _{i}$都是换位子$[ \gamma _{i-1}, G]$。
使$\gamma _{n+1} = {1}$成立的最小$n$是幂零指数。
每个幂零群都是对应于不同素数$p$的$p$-groups的直积。

A nilpotent group is a group which has a chain of subgroups ${1} = \gamma _{n+1} \subset \gamma _{n} \subset ... \subset \gamma _{1} = G$ in which each subgroup $\gamma _{i}$ is the commutator subgroup $[ \gamma _{i-1}, G]$.
The smallest $n$ for which $\gamma _{n+1} = {1}$ is the index of nilpotence.
Every nilpotent group is a direct product of $p$-groups for different primes $p$.