数学专业英语

学号：10901040201 姓名：曹菁 lg LinearA ebra

Non-trivial linear combination. This is a contradiction and proves 1).Part 2) follows from 1) because dim(V)=n.

Exercise Let A=0110?? ?-??

()2R ∈. Find an invertible 2C C ∈ such that 1

c AC - is diagonal.

Show that C cannot be selected in 2R .Find the characteristic polynomial of A .

Exercise Suppose V is a 3-dimensional vector space and :f V V → is an endomorphism with ()()3

f CP x x λ=-.Show that ()f I λ- has characteristic polynomial and is thus a

nilpotent endomorphism. Show there is a basis for V so that the matrix representing

is 001001λλλ?? ? ? ??? ,001000λλλ?? ? ? ???or.000000λλλ?? ? ? ???

. We could continue and finally give an ad hoc proof of the Jordan canonical form,but in this chapter we prefer to press on to inner product spaces.The Jordan form will be developed in Chapter 6 as of the general theory of finitely generated modules over Euclidean domains.The next section is included only as a convenient reference.

This section should be just skimmed or omitted entirely.It is unnecessary for the rest of this chapter,and is not properly part of the flow of the chapter.The basic facts of Jordan form are summarized here simply for reference.

The statement that a square matrix B over a field F is a Jordanblock means that F λ?∈

Such that B is a lower triangular matrix of the form 0101B λλ

λ??

? ?

?= ? ?

???

.B gives a homomorphism :m m g F F →with ()m m g e e λ= and ()1i i i e e e λ+=+ for 1i m ≤<.Note that ()()m

B CP x x m =-and so λ is the only eigenvalue of B ,and B satisfies its characteristic polynomial,i.e.,()0B CP B =.

Definition A matrix n D F ∈ is in Jordan if ?Jordan blocks i ni B F ∈ such that

120000000000

000000000

00t B B D B ??

?= ?

? ??

.Suppose D is of this form and i ni B F ∈has eigenvalue i λ.Then

1..t n n n += and ()()()1

1..n nt

D t CP x x x λλ=--.Note that a diagonal matrix is a special case

of Jordan form. D is a diagonal matrix iff each i n ,iff each Jordan block is 11? a matrix. Fundamentals of Statistics

The preceding chapter was mainly concerned with the theory of probability,including distribution theory. In practice researchers have to find methods to choose among distributions and to estimate distribution parameters from real data. The subject of sampling brings us now to the theory of statistics. Whereas probability assumes the distributions are known, statistics attempts to make, inferences from actual data

Here we sample from the distribution of a population, say the change in the Exchange rate,to make inferences about the population.The questions are ,what Is the best distribution for this random variable and what are the best parameters for This distribution ? Risk measurement, however, typically deals with large numbers of random variable. So, we also want to characterize the relationships between the risk factors to which the portfolio is exposed. For example, do we observe that movements in the yen/dollar rate are correlated with the dollar/euro rate? Another type of problem a to develop decision rule to rest some hypotheses, for instance Wherher the volatility remains stable over time.

These examples illustrate two important problems in statistical inference, i.e, estimation and tests of hypotheses. With estimation, we wish to estimate the value of an unknown parameter from ample data. With tests of hypotheses, we wish to Verify a conjecture about the data.

This examples reviews the fundamental tools of statistics theory for risk man- agers. Section 3.1 discusses s the sampling of real data and the construction

of returns. The problem of parameter estimation is presented in Section 3.2. Section 3.3 then turns to regression analysis's,summarizing important results as Well as common pitfalls in their interpretation. REAL DATA

To start with an example, let us say that we observe movement in the daily Yen/dogar exchange rate and wish to characterize the distribution of tomorrow's Exchange rate

The risk manager ’s job is ro assess the range of potential gains and lasses on a

trader's position . He or she observes a sequence a of past spot prices 0,1,...t S S S from which we have to infer the distribution of tomorrow's price, 1t S +. Measuring Returns

The truly random component in tomorrow ’s prices is not its level, but rather its Change relative to today's prices. We measure the relative rate of change in the spot price:

()11/t t t t r S S S --=-

Alternatively, we could construct the logarithm of the price ratio:

[]1ln t t t R S S -=-

which is equivalent to using continuous instead of discrete compounding. This is also

()[]11ln 1/ln 1t t t t t R S S S r --??=+-=+??

The return defined so far is the capital appreciation return, which ignores the income payment on the asset. Define the dividend or coupon as Dt.In the case of An exchange rate position, this is the interest payment in the foreign currency over the holding period. The rural mum av the asset is

()11/TOT t t t t t r S D S S --=+-

When the horizon is very short,the income return is typically very small compared to the capital appreciation return.

the next question is whether the sequence of variables R can be viewed as independent observations. if so ,one could hypothesize,for instance,that the random variables are drawn from a normal distribution N(u,).we could then proceed to estimate u and --from the data and use this information to create a distribution for tomorrow;s spot price change.

Independent observations have the very nice property that their joint distribution is the product of their marginal distribution ,which considerably simplifies the analysis.the obvious question is whether this assumption is a workable approximation. in fact,thee are good economic reasons to believe that rates of change on financial prices are close to independent .

the hypothesis of efficient markets postulates that current prices convey all relevant information about the asset. if so,any change in the asset price must be due to news,or events which are by definition impossible to forecast(otherwise,it would not be news).this implies that changes in prices are unpredictable and,hence,satisfy our definition of independent random variables.

conditional distribution of returns only on current prices, and not on the previous history of prices .if so,technical analysis must be a fruitless exercise. technical analysts try to forecast price movements from past price patterns.

If in addition the distribution of return is constant over time, the variables are said to be independently and identically distributed(i.i.d.).so,we could consider that observations RT are

independent draws from the same distribution ()

,N u σ.

Later, we will consider deviations from this basic model,distributions of financial returns typically display fat tails. also, variances are not constant and display some persistence; expected returns can also slightly vary over time. Time Aggregation

It is often necessary to translate parameters over a given horizon to anther horizon . For

example, we may have raw data for daily returns , from which we compute a daily volatility that we want to extend to a monthly volatility .

Returns can be easily related across time when we use the log of the price ratio, because the log of a product is the sum of the logs of the individual terms.The two-day return, for example, can be decomposed as

[]()[][]02211012210112ln //ln /ln /R S S S S S S S S R R =?=+=+

This decomposition is only approximate if we use discrete returns, however. The expected return and variance are then ()()()020112E R E R E R =+ and

()()()()020********,V R V R V R Cov R R =++.Assuming returns are uncorrelated and have

identical distributions across days, we have ()()02012E R E R =and ()()02012V R V R =. Generalizing over T days, we can relate the moments of the T-day returns T R to those of the 1-day returns:1R

()()1T E R E R T = ()()1T V R V R T =

Expressed in terms of volatility, this yields the square root of time rule:

()()1T SD R SD R T =

KEY CONCEPT

When successive returns are uncorrelated, the volatility increase as the horizon extends following the square root of time .

More generally, the variance can be added up from different values across different periods . For instance, the variance over the next year can be computed as the average monthly variance over the first three months, multiplied by 3, plus the average variance over the last nine months, multiplied by 9 .This type of analysis is routinely used to construct a term structure of implied volatilities , which are derived from option data for different maturities .

It should be emphasized that this holds only if returns have constant parameters across time and are uncorrelated . When there is non-zero correlation across days, the two-day variance is

()()()()()()21111221V R V R V R V R V R ρρ=++=+

Because we are considering correlations in the time series of the same variable ,P is called the autocorrelation coefficient ,or the serial autocorrelation coefficient .A positive value for P implies that a movement in one direction in one day is likely to be followed by another movement in the same direction the next day. A positive autocorrelation signals the existence of a trend. In this case, Equation(3.8)shows that the two-day variance is greater than the one obtained by the square root of time rule.

A negative value for P implies that a movement in one direction in one day is likely to be followed by a movement in the other direction in one day is likely to be followed by a movement in the other direction the next day .So,prices tend to revert back to a mean value .A negative autocorrelation signals

译文：

非线性组合，这是一个矛盾和证明1)第2部分)遵循从1)因为V 的秩=n.

例如：矩阵A=0110?? ?-??

的秩是2，找到一个可逆，2C C ∈,比如1

C AC -是对角线。

表明C 的秩不是2，得到特征多项式A .

例如假设V 是一个三维向量空间，:f V V →是一个自同态与()()3

f CP x x λ=-。因此有特征多项式()f I λ-,因此是一个幂零自同态。显示有一个基础,以便矩阵表示是

001001λλλ?? ? ? ??? ,001000λλλ?? ? ? ???or.000000λλλ?? ? ? ???

我们可以继续,最后给出一个特设的 Jordan 证明规范形式,但在这一章中,我们更喜欢到内积空间。约旦的形式将被运用在第6章的一般理论的有限生成模块在欧几里得域。接下来的部分是只包括一个方便的参考。

这个部分应该只是脱脂或全部删除了。这是不必要的,剩下的这一章,是不正确的部分流动的一章。约旦的基本事实,总结了在这里只是形式为参考。

该声明,一个方阵超过一个字段是一个意味着那F λ?∈这样是一个下三角矩阵的形式

0101B λλλ??

? ?

?= ? ?

???

给一个同态:m m g F F →和()m m g e e λ= 且()1i i i e e e λ+=+ for 对1i m ≤<. 既然()()m

B CP x x m =-，因此λ 是矩阵B 是唯一的特征值 ,且 B 满足其特征多项式，即()0B CP B =

定义一个矩阵n D F ∈是在 Jordan 如果存在i n i B F ∈在Jordan 区，比如

120000000000

000000000

00t B B D B ??

?= ?

? ??

，假设D 是这种形式和i ni B F ∈有特征值i λ。然后1..t n n n

和()()()11..n nt

D t CP x x x λλ=--。注意,一个对角矩阵是一个特例的约旦形式。D 是一个对角矩阵敌我识别每个i n ,识别每个 Jordan block 是一个11?矩阵。

前面的章节，主要是关注与概率论，包括分布理论。在实践中，研究人员必须找到方法来选择之间的分布和分布参数估计真实数据。采样的主题，我们现在统计的理论。而假定的分布是已知的概率，统计试图从实际的数据，推论

在这里，我们从人口分布的采样，说的变化汇率，使推断群体的问题是什么，这个随机变量，什么是最好的分布是最好的参数这种分配？风险测量，然而，通常涉及大量随机变量。所以，我们也希望表征之间的关系投资组合暴露的危险因素。例如，我们观察到，在日元/美元汇率的走势与美元/欧元的汇率？另一问题类型发展的决策规则放置一些假设，例如随着时间的推移，Wherher 的波动保持稳定。

这些例子说明了两个重要的问题，在统计推断，即估计和假设检验。通过估算，我们希望估计值从大量数据的未知参数。随着测试的假设，我们希望验证猜想的数据。这个例子的点评统计理论的基本工具风险的人不停。第3.1节讨论s 的真实数据的采样和建造的回报。在3.2节中的参数估计问题。第3.3节，然后回归分析，总结了重要成果，常见的陷阱，在他们的解释。

开始一个例子，让我们说，我们观察到在日常的运动的日元/ dogar 的汇率，并希望明天的分布特征汇率

风险经理的工作是ro 的潜在收益和情人们一个评估范围

交易者的位置。他或她观察到一个过去的现货价格S0 S1序列....... ST ，我们推断明天的价格，ST+1的分布。

在明天的价格是不是真正的随机成分的水平，而是其更改相对于今天的价格。我们在现场测量的相对变化率价格：

另外，我们可以构建价格比的对数：

Undo edits

这是相当于使用连续的，而不是离散复利。这是还

定义至今的回报是资本增值的回报，而忽略了

对资产的收入支付。定义红利或优惠券为Dt.In 的情况下，汇率位置，这是外币支付利息持有时间。农村妈妈A V 的资产

在地平线上时，是很短，收益回报通常是非常小的资本增值回报。接下来的问题是，的顺序

是否变量

在地平线上时，是很短，收益回报通常是非常小的资本增值回报。

接下来的问题是，是否可以被看作是独立的观测序列变量r 。如果是这样，你可以假设，例如，来自正态分布N （U ）的随机变量。然后，我们可以继续进行，估计u 和 - 的数据和使

用这些信息来创建一个分发明天;的现货价格的变化。

独立的观察有很不错的属性，它们的联合分布的边缘分布，大大简化了分析。明显的问题是产品的是这个假设是否是一个可行的近似。其实，你是好经济有理由相信，金融价格的变化率接近独立。

有效市场假说假设目前的价格转达该资产的所有相关信息。如果是这样，资产价格的任何变动，必须是由于新闻或事件是由定义的不可能预测（否则，它不会成为新闻）的，这意味着价格的变化是不可预测的，因此，满足我们的定义独立的随机变量。

这一假说，也被称为随机漫步理论，隐含的条件分布只返回当前的价格，而不是以前的历史价格。如果是这样，技术分析必须是无果而终的运动。技术分析师预测价格走势，从过去的价格模式。

如果在此外回报的分布是随时间变化的变量是独立同分布（IID ）。因此，我们可以认为是独

立的观察RT 吸引了来自相同分布()

2,N u σ

后，我们会考虑偏离这个基本模型，财务回报典型的displayfet 尾巴的分布。也，差异不是恒定的，并显示一些持久性的预期回报也slghtly 随着时间的推移而变化

时间聚合

通常是必要的翻译参数超过一个给定的地平线花药地平线。例如，我们可能每天的回报，我们计算出每天的波动，我们要扩展到每月波动的原始数据。

返回时，可以很容易地跨越时间有关，我们的价格比使用日志，因为该日志的产品是个人terms.The 为期两天的回报的总和的日志，例如，可以分解为这种分解是近似的，如果我们使用离散的回报，但是。

预期收益和方差，然后[]()[][]02211012210112ln //ln /ln /R S S S S S S S S R R =?=+=+ 假设回报是不相关的和具有相同的分布跨天，我们有()()()020112E R E R E R =+和

()()()()020********,V R V R V R Cov R R =++。

在T 日的推广，我们可以涉及的日退货()()02012E R E R =()()02012E R E R =天的回报率R1的时刻：

()()02012E R E R =()()1T E R E R T =

以波动，这会产生时间的平方根规则：()()1T SD R SD R T =

KET 概念

在连续的回报是不相关的，在地平线上的波动性增加，延长时间的平方根。

更一般地，可以添加方差从不同的值在不同的期间。例如，在未来的一年方差计算的前三个月，每月的平均方差乘以3，再加上平均方差在过去的9个月，再乘以9。这种类型的分析经常被用来建设隐含波动率期限结构，这是来自不同到期日的期权数据。

应当强调的是，这认为仅当回报跨越时间和具有恒定的参数是不相关的。当有非零相关跨天，2天的方差是

因为我们考虑在时间序列中的相同变量的相关性，P 被称为自相关系数，或序列的自相关系数，用于P 甲正值意味着在一天之内，在一个方向上的运动是可能要遵循由另一个运动在相同的方向，第二天。一个积极的自相关信号存在的一个趋势。在这种情况下，公式（3.8）表示，2天的方差是大于1，通过以下方式获得的时间的平方根规则。

()()()()()()21111221V R V R V R V R V R ρρ=++=+

为P 甲负值意味着在一天之内，在一个方向上的运动是可能应遵循的在另一个方向上的运动在一天之内是可能应遵循的在另一个方向上的运动，第二天，所以价格往往恢复到负自相关信号的平均值。

数学专业英语

数学专业英语课后答案

2.1数学、方程与比例词组翻译 1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation 2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction 3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart 4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation 5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction 6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。 Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches. （2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。 No modern scientific and technological branches could be regularly developed without the application of mathematics. （3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。 Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. （4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。Before 17th century, man confined himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. （5）方程与算数的等式不同在于它含有可以参加运算的未知量。 Equation is different from arithmetic identity in that it contains unknown quantity which can join operations. （6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. （7）方程很有用，可以用它来解决许多实际应用问题。

数学专业英语论文(含中文版)

Differential Calculus Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus. In this article, we give su ?cient conditions for controllability of some partial neutral functional di ?erential equations with in?nite delay. We suppose that the linear part is not necessarily densely de?ned but satis?es the resolvent estimates of the Hille -Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result. Key words Controllability; integrated semigroup; integral solution; in?nity delay 1 Introduction In this article, we establish a result about controllability to the following class of partial neutral functional di ?erential equations with in?nite delay: 0,) ,()(0≥?? ???∈=++=?? t x xt t F t Cu ADxt Dxt t βφ (1) where the state variable (.)x takes values in a Banach space ).,(E and the control (.)u is given in []0),,,0(2>T U T L ,the Banach space of admissible control functions with U a Banach space. C is a bounded linear operator from U into E, A : D(A) ? E → E is a linear operator on E, B is the phase space of functions mapping (?∞, 0] into E, which will be speci?ed later, D is a bounded linear operator from B into E de?ned by B D D ∈-=????,)0(0 0D is a bounded linear operator from B into E and for each x : (?∞, T ] → E, T > 0, and t ∈ [0, T ], xt represents, as usual, the mapping from (?∞, 0] into E de?ned by ]0,(),()(-∞∈+=θθθt x xt F is an E-valued nonlinear continuous mapping on B ??+. The problem of controllability of linear and nonlinear systems repr esented by ODE in ?nit dimensional space was extensively studied. Many authors extended the controllability concept to in?nite dimensional systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, [4, 7, 10, 21]. There are many systems that can be written as abstract neutral evolution equations with in?nite delay to study [23]. In recent years, the theory of neutral functional di ?erential equations with in?nite delay in in?nite dimension was deve loped and it is still a ?eld of research (see, for instance, [2, 9, 14, 15] and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, [5, 8]. The objective of this article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely de?ned but satis?es the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the su ?cient conditions for controllability of some partial neutral functional di ?erential equations with in?nite delay. The results are obtained using the integrated semigroups theory and Banach ?xed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory. Treating equations with in?nite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that ).,(B B is a (semi)normed abstract linear space of functions mapping (?∞, 0] into E, and satis?es the following fundamental axioms that were ?rst introduced in [13] and widely discussed

关于数学专业英语课程的研究与探讨

第34卷第10期2017年10月吉林化工学院学报 JOURNAL OF JILIN INSTITUTE OF CHEMICAL TECHNOLOGY V〇1.34N〇.10 Oct.2017 文章编号：1007-2853(2017) 10-0069-03 关于数学专业英语课程的研究与探讨许洁 (吉林化工学院理学院，吉林吉林132022) 摘要：通过介绍数学专业英语课程开设目的，结合专业本身的特点对数学专业英语课程进行研究，分析当前数学专业英语课程在教与学过程中存在的问题，并对相应问题的解决提出思考。希望通过对授课方法，评价体系等方面的改革不断提高数学专业英语的实用性，培养出适应社会发展需要的专业化人才。关键词：数学专业英语;教学方法;评价体系中图分类号：H319 文献标志码：A D0l：10.16039/https://www.360docs.net/doc/61367542.html,22-1249.2017.10.017 随着计算机科学技术的迅速发展，人们进入了高速发展的信息时代。信息时代拉近了人与人之间的距离，增进了国际间的交流合作。社会生活的信息化、经济的全球化，使英语的重要性日益突出。英语成为许多领域重要的通用语言。绝大多数学科前沿的学术论文都是用英文撰写。许多领域的学术、科技交流会议也以英语作为官方语言的首选。培养具有国际交流能力的人才势在必行，掌握具有国际交流能力的专业人才又成为高校培养人才的重中之重。一、专业英语课程开设的目的伴随着人类社会进入21世纪，我国的教育也面临着如何进一步与国际接轨的问题。教育部提出了高等学校各专业逐步使用英文教材，培养学生阅读英文版专业文献的能力[1]。为适应人才培养的需要，高等院校根据各专业的实际情况开设适应各专业的专业英语、科技外语阅读等课程。通过类似课程的学习使学生增加本专业的专业词汇的英文表达方式。数学，作为古老的学科为适应新形式下教学改革的需要同样面临着如何与国际接轨的问题。探讨数学专业英语的特点，如何很好的开设这门课程成为很多从事该课程的一线教师关注的热点[2-6]。数学专业英语具有科技英语的共性、科学内容的客观麵性、表达形式的完整性和简练性要求[7]。数学专业英语作为高等院校的一门重要课程，是以大学英语为基础，是数学专业的基础课程之一。通过本课程的学习，使学生能够适应国际、国内数学教育的发展，了解本专业的最新发展动态，开拓学生的视野。通过教师讲解，结合学生课后查阅英文资料，培养学生听、说、写的综合能力，掌握本专业的当前动态和前沿发展，为进一步的学习、工作打下坚实的基础。二、数学专业英语的特点数学专业英语与许多其他专业的专业英语类似，不能简单的定义为一门专业基础课程或者是英语课程。数学的专业知识和大学英语课程的基础都是学好数学专业英语的关键。本课程是对于数学专业学生专业英语能力训练和培养的一门重要课程，是对大学高年级学生继公共英语课程之后的一个重要补充和提高。数学专业英语与大学英语既有区别又有联系。数学专业英语课程中，数学的专业性十分典型。数学专业英语以叙述的方式介绍数学的方法、推导过程及主要结论。其学科本身的特点决定了其内容通常与特定的时间无关。数学课程或是数学文献中涉及到的结论有时是很久以前给出的，但在叙述的过程中一細现时絲表示。收稿日期：017-04-05 基金项目：吉林化工学院2016年一般教研项目作者简介:许洁（1980-)，女，吉林省吉林市人，吉林化工学院副教授，博士，主要从事矩阵代数方面的研究。

数学专业英语课文翻译(吴炯圻)第二章 2.

数学专业英语课文翻译(吴炯圻)第二章 2. 数学专业英语3—A 符号指示集一组的概念如此广泛利用整个现代数学的认识是所需的所有大学生。集是通过集合中一种抽象方式的东西的数学家谈的一种手段。集，通常用大写字母：A、B、C、进程运行·、X、Y、Z ；小写字母指定元素：a、 b 的c、进程运行·，若x、y z.我们用特殊符号x∈S 意味着x 是S 的一个元素或属于美国的x如果x 不属于S，我们写xS.≠当方便时，我们应指定集的元素显示在括号内；例如，符号表示的积极甚至整数小于10 集{2,468} {2，，进程运行·} 作为显示的所有积极甚至整数集，而三个点等的发生。点的和等等的意思是清楚时，才使用。上市的大括号内的一组成员方法有时称为名册符号。涉及

到另一组的第一次基本概念是平等的集。DEFINITIONOFSETEQUALITY。两组A 和B，据说是平等的如果它们包含完全相同的元素，在这种情况下，我们写A = B。如果其中一套包含在另一个元素，我们说这些集是不平等，我们写 A = B。EXAMPLE1。根据对这一定义，于他们都是构成的这四个整数2，和8 两套{2,468} 和{2,864} 一律平等。因此，当我们用来描述一组的名册符号，元素的显示的顺序无关。动作。集{2,468} 和{2,2,4,4,6,8} 是平等的即使在第二组，每个元素 2 和 4 两次列出。这两组包含的四个要素2,468 和无他人；因此，定义要求我们称之为这些集平等。此示例显示了我们也不坚持名册符号中列出的对象是不同。类似的例子是一组在密西西比州，其值等于{M、我、s、p} 一组单词中的字母，组成四个不同字母M、我、s 和体育3 —B 子集S.从给定的集S，我们

数学专业英语第二版-课文翻译-converted

2.4 整数、有理数与实数 4－A Integers and rational numbers There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers. 有一些R 的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。 To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers. 我们从数字 1 开始介绍正整数，公理 4 保证了 1 的存在性。1+1 用2 表示，2+1 用3 表示，以此类推，由 1 重复累加的方式得到的数字 1,2,3，…都是正的，它们被叫做正整数。 Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. 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. 虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。 DEFINITION OF AN INDUCTIVE SET. A set of real number s is cal led an i n ductiv e set if it has the following two properties: (a) The number 1 is in the set. (b) For every x in the set, the number x+1 is also in the set. For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set. 现在我们来定义正整数，就是属于每一个归纳集的实数。 Let P d enote t he s et o f a ll p ositive i ntegers. T hen P i s i tself a n i nductive set b ecause (a) i t contains 1, a nd (b) i t c ontains x+1 w henever i t c ontains x. Since the m embers o f P b elong t o e very inductive s et, w e r efer t o P a s t he s mallest i nductive set. 用 P 表示所有正整数的集合。那么 P 本身是一个归纳集，因为其中含 1，满足(a)；只要包含x 就包含x+1, 满足(b)。由于 P 中的元素属于每一个归纳集，因此 P 是最小的归纳集。 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.

数学专业英语论文

课文9-B Terminology and notation when we work with a differential equation such as(9.1),it is customary to write y in place of f（x) and y' in place of f'(x)，the higher derivatives being denoted by y"，y'''，etc.Of course ，other letters such as u，v，z，etc.are also used instead of y. By the order of an equation is meant the order of the highest derivatives which appears.For example ,(9.1)is first-order equation which may be written as y'=y.The differential equation ) sin(xy" y x y'3+ =is one of second order. In this chapter we shall begin our study with firs-order equations which can be solved for y' and written as follows: (9.2) y'=f(x,y), Where the expression f(x,y) on the right has various special forms. A defferentiable function y=Y(x) will be called a solution of (9. 2) on an interval I if the function Y and and its derivative Y' satisfy the relation Y'=f[x,Y(x)] For every x in I. The simplest case occurs when f(x,y)is independent of y.In this case , (9.2) becomes (9.3) y'=Q(x), Say, where Q is assumed to be a liven function defined on some interval I. To solve the differential equation(9. 3) means to find a primitive of Q.The Second fundamental theorem of calculus tells us how to do it when Q is continuous on an open interval I. We simply integrate Q and add any constant.Thus，every solution of (9.3) is included in the formula （9.4）y=∫Q(x)dx + C, where C is any constant ( usually called an arbitrary constant of integration). The differential equation(9.3) has infinitely many 课文9—B 术语和符号当我们在求解像(9.1)式的微分方程时，习惯用y代替f(x)，用y’代替f'(x)，用高阶导数y''和y'''等表示。当然，其他的字母如u,v,z等等，同样可以用来代替y。微分方程和阶数指的是现在其中的高阶导数的阶。例如，（9.1）式是一个一次方程可以写成y'=y。微分方程 ) s i n(x y" y x y'3+ =是一个二阶的。在这章我们将会学习到可以求解y'的一阶微分方程。一阶方程可以被写成这样：（9.2）y'=f(x,y), 其中，右边有各个特殊形式表示。如果对于区间I中的每一个x函数y和他的倒数满足 Y'=f[x,Y(x)] 那么可微函数就为（9.2）在区间I中的一个解，最简单的形式是f（x，y）与y无关。在这种情况下，（9.2）式变成了（9.3）y'=Q(x), 表明，其中Q是假定在区间中的一个给定函数，对于一个给定的函数定义在各个区间I.求解微分方程（9.3）就意味着找到原始的区间Q。第二基本积分定理告诉我们，当Q位于一个连续的开放的区间I 时该怎么做。我们直接对Q积分并加上任意常数。因此，y=∫Q(x)dx + C包含了（9.3）式的所有解（9.4）y=∫Q(x)dx + C, 其中C为任意常数（通常被称为积分下限的任意常数），微分方程（9.3）有无穷多个解，每个解对应一个C。

数学专业英语(Doc版).10

学专业英语－How to Write Mathematics? How to Write Mathematics? ------ Honesty is the Best Policy The purpose of using good mathematical language is, of course, to make the u nderstanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the se nse of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss. The emphasis in the preceding paragraph, while perhaps necessary, might see m to point in an undesirable direction, and I hasten to correct a possible misin terpretation. While avoiding pedantry and fuss, I do not want to avoid rigor an d precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way t o get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his s ympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.” Here is the sort of the thing I mean by less than complete honesty. At a certa in point, having proudly proved a proposition P, you feel moved to say: “Not e, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be per fectly pure, but the reader may feel cheated just the same. If he knew all abo ut the subject, he wouldn’t be reading you; for him the nonimplication is, qui te likely, unsupported. Is it obvious? (Say so.) Will a counterexample be suppl ied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean th at you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into y our confidence. There is nothing wrong with often derided “obvious”and “easy to see”, b ut there are certain minimal rules to their use. Surely when you wrote that so mething was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still thi nk that something was obvious? (A few months’ripening always improves ma nuscripts.) When you explained it to a friend, or to a seminar, was the someth ing at issue accepted as obvious? (Or did someone question it and subside, mu ttering, when you reassured him? Did your assurance demonstration or intimida

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, . , 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。The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics----analytic geometry and calculus, which belong to the higher mathematics. Now there are many branches in higher mathematics, among which are mathematical analysis, higher algebra, differential equations, function theory and so on. 17世纪工业的快速发展推动了经济技术的进步，从而遇到需要处理变量的问题，从常数带变量的跳跃产生了两个新的数学分支-----解析几何和微积分，他们都属于高等数学，现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful

数学专业英语

第二章精读课文-- 入门必修 2.1 数学方程与比例（Mathematics，Equation and Ratio）一、词汇及短语： 1. Cha nge the terms about变形 2. full of ：有许多的充满的例The StreetS are full of people as on a holiday像假日一样，街上行人川流不息） 3. in groups of ten?? 4. match SOmething against sb. “匹配” 例Long ago ,when people had to Count many things ,they matChed them against their fingers. 古时候，当人们必须数东西时，在那些东西和自己的手指之间配对。 5. grow out of 源于由…引起例Many close friendships grew out of common acquaintance 6. arrive at 得出（到达抵达达到达成）例We both arrived at the Same COnclusion我们俩个得出了相同的结论） 7. stand for “表示，代表” 8. in turn “反过来，依次” 9. bring about 发生导致造成 10. arise out of 引起起源于 11. express by “用…表示” 12. occur 发生，产生 13. come from 来源于，起源于 14. resulting method 推论法 15. be equal to 等于的相等的

数学专业英语2-11C

数学专业英语论文英文原文：2-12C Some basic principles of combinatorial analysis Many problems in probability theory and in other branches of mathematics can be reduced to problems on counting the number of elements in a finite set. Systematic methods for studying such problems form part of a mathematical discipline known as combinatorial analysis. In this section we digress briefly to discuss some basic ideas in combinatorial analysis that are useful in analyzing some of the more complicated problems of probability theory. If all the elements of a finite set are displayed before us, there is usually no difficulty in counting their total number. More often than not, however, a set is described in a way that makes it impossible or undesirable to display all its elements. For example, we might ask for the total number of distinct bridge hands that can be dealt. Each player is dealt 13 cards from a 52-card deck. The number of possible distinct hands is the same as the number of different subsets of 13 elements that can be formed from a set of 52 elements.Since this number exceeds 635 billion, a direct enumeration of all the possibilities is clearly not the best way to attack this problem; however, it can readily be solved by combinatorial analysis. This problem is a special case of the more general problem of counting the number of distinct subsets of k elements that may be formed from a set of n elements (When we say that a set has n elements,we mean that it has n distinct elements.Such a set is sometimes called an n-element set.),where k n ≥. Let us denote this number by ),(k n f .It has long been known that )1.12( ,),(??? ? ??=k n k n f where, as usual ??? ? ??k n denotes the binomial coefficient, )!(!!k n k n k n -=??? ? ?? In the problem of bridge hands we have 600,559,013,6351352)13,52(=??? ? ??=f different hands that a player can be dealt. There are many methods known for proving )1.12(. A straightforward approach is to form each subset of k elements by choosing the elements one at a time. There are n possibilities for the first choice, 1-n possibilities for the second choice, and )1(--k n possibilities for the kth choice. If we make all possible choices in this