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比较级的语法

斯托尔兹-切萨罗定理

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∙/∞ 情况的叙述

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以及 为两个实数数列。假设 是个严格单调且发散的数列(亦即严格递增并接近无穷大,或者严格递减并接近负无穷大),以及下述极限存在:

那么,可以推得极限

0/0 情况的叙述

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以及 为两个实数数列。假设 以及 ,并且 严格递减。如果

[1]

证明

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的情况

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第一种: 已知 为严格递增并发散至 ,而且 ,因此我们有 使得

也就是说

因为 是严格递增,所以 。故以下式子成立:

接着可以注意到

因此,将刚刚的不等式套入 的等式中,我们可以得到

现在因为当 时,,因此存在 使得 。接着在 的不等式两边同除以 可以得出

接着我们定义数列

观察这两个数列,可以发现因为 以及分子部分不是变数,所以皆收敛至 。因此我们有 使得

附注:因为现在只专注在 本身,并没有考虑 与不等式的关系,故 不必大于 。又因为可能找到一个 使得 ,故 一定要大于
最后,将以上不等式结合起来可以得到 使得

亦即,
换成是严格递减并发散至 的话,其证明的方法和上面的过程类似。

第二种: 已知 为严格递增并发散至 ,而且 ,因此我们有 使得

一样地,将刚刚的不等式套入 的等式中,我们可以得到

以及

接着我们定义的数列

会收敛至 ,故我们有 使得

最后,将以上不等式结合起来可以得到 使得

亦即,
若考虑 为严格递增或严格递减,以及发散至 的话,其证明的方法皆和上面的过程相同。

的情况

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第一种: 已知 为严格递减并收敛至 ,而且 ,因此我们有 使得

也就是说

因为 为严格递减,所以

接着可以注意到对于每一个

因此,将刚刚的不等式套入 的等式中,我们可以得到

现在因为 为严格递减并收敛至 ,所以可以保证 故以下式子成立:

接着我们定义数列

观察这两个数列,可以发现若只考虑 在变动并将其取的很大,则 会收敛至 。因此我们有 使得

最后,将以上不等式结合起来可以得到 使得

亦即,

第二种: 已知 为严格递减并收敛至 ,而且 ,因此我们有 使得

一样地,将刚刚的不等式套入 的等式中,我们可以得到

以及因为 ,所以

接着我们定义的数列

会收敛至 ,故我们有 使得

最后,将以上不等式结合起来可以得到 使得

亦即,

英文

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比较级用来描述“两个人事物”之间的不同差异,具有修饰其他词性的词性会有比较级的用法,例如形容词副词

形容词

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形容词的比较级由其原级变化而来,变化方法如下:

  • 多数一个音节及部分两个音节的形容词
    • 字尾直接加 - er ,如:young - younger、fast - faster、tall - taller、high - higher、clean - cleaner、low - lower、small - smaller...等。
    • 字尾是“e”时,直接加 - r ,如:fine - finer、cute - cuter、large - larger、nice - nicer、white - whiter、close - closer...等。
    • 字尾是子音+“y”时,去“y”加 - ier ,如:dry - drier、easy - easier、happy - happier、funny - funnier、busy - busier、dirty - dirtier、healthy - healthier...等。
    • 字尾结构是‘子音+短母音+子音’时,重复字尾再加 - er ,如:thin - thinner、hot - hotter、big - bigger、sad - sadder、fat - fatter、wet - wetter、red - redder...等。
  • 大多数两个音节以上( ),或字尾已含有字根ful, less, ing, ed, ous, ive, ible, able, ent, ant, ish, ic, ary, al等的形容词,直接在形容词字前加 more (更),如:wonderful - more wonderful、useless - more useless、exciting - more exciting、interested - more interested、difficult - more difficult、famous - more famous、expensive - more expensive...等。
  • 不规则变化,如:good/well - better、bad/ill/sick - worse、many/much - more、little - less、far - farther(表距离)/further(表程度)、late - later(表时间)/latter(表顺序)、old - older(表年纪、新旧)/elder(表长幼)...等。

其中,有些形容词的意思已表达“绝对”的含义,故这些形容词没有比较级的功能,例如:alone, full, favorite, basic, superior, correct, total, immortal, wrong, perfect...等。

基本的形容词比较级句型为“主词1 + Be动词 + 形容词比较级 + than + 主词2 (+ Be动词)”,如:“Health is more important than money.”(健康比金钱重要)。须要注意表示此句型时:
(1)两个主词的性质必须相同,才可加以比较,
例如:“Your house is larger than mine.”(你的房子比我的还要大)。
(2)than当连接词使用时,后应接“主词2 + Be动词”才算完整,但在口语上会把Be动词省略掉,此时主词2如果为人称代名词,可用受词代替主格,例如:“Tom is more handsome than he (is).”=“Tom is more handsome than him.”(汤姆比他还要英俊)。
(3)在不影响句意的情况下,比较的对象(than之后的内容)可以完全省略,例如:“It was colder yesterday.”(昨天比较冷)。

形容词比较级前面可以用程度副词来修饰,例如:much, a lot和far(比...得多)表示差异很大、even和still(甚至更)表示来强调比较,与a little(...一点点)表示差异很小,除此之外也可以用数字修饰,以表示具体的不同。
例1:“The radio is much more expensive than the fan.”(收音机比电扇还要昂贵很多)
例2:“Ted is even thinner than Kelly.”(泰迪甚至比凯莉还要瘦)
例3:“Linda is a little taller than her sister.”(琳达比她的姐姐高一点)
例4:“Kevin is one year older than his brother.”(凯文比他的弟弟大一岁)

其它常见的比较级句型及用法:

  • “主词1 + 一般动词 + 形容词比较级 + 名词 + than + 主词2 (+ 助动词)”,如:“Nick eats more cookies than his cousin.”(尼克饼干吃得比他表弟还多)。
  • 表示“两者之间来做比较”时,可用“the + 形容词比较级 + of the two”,此处加the是因为后面的“of the two”的关系表示限定,如:“He is the happier of the two.”(他是两人之中比较快乐的)。
  • 表示“越...(就)越...”时,可用“the + 形容词比较级, the + 形容词比较级”或“the + 形容词比较级 + 主词 + 动词, the + 形容词比较级 + 主词 + 动词”,此处的第一个和第二个形容词比较级是两个不同的形容词,如:“The more, the better.”(越多越好);“The harder you study, the better grades you will get.”(你越努力用功,你就会得到越好的成绩)。
  • 表示“越来越...”时,可用“形容词比较级 + and + 形容词比较级”,此处的第一个和第二个形容词比较级是相同的,如:
    (1)“The weather is getting warmer and warmer.”(天气变得越来越暖和)
「原級 + - er」類型的比較級,用「比較級 + 比較級」表示「越來越...」

(2)“This kind of cellphone is becoming more and more popular.”(这款手机变得越来越受欢迎)

「more + 原級」類型的比較級,用「more and more + 原級」表示「越來越...」

如果是表示“越来越不...”时,则可用“less and less + 形容词原级”如:
(1)“It is getting less and less cold.”(天气变得越来越不冷了)。
(2)“This man is less and less handsome.”(这位男人越来越不英俊)。

  • 表示“比任何其他的都...”时,可用“形容词比较级 + than + any other + 单数名词”;表示“比所有其他的都...”时,则用“形容词比较级 + than + all the other + 复数名词”,此两种句型都是用形容词比较级来表达形容词“最高级”的概念。

例1:“Jason is taller than any other student in his class.”(杰森比他班上任何其他男生都来得高)

此句意同Jason is the tallest student in his class. 

例2:“To David, math is more difficult than all the other subjects.”(对大卫来说,数学比所有其他的科目还要来得困难)

此句意同To David, math is the most difficult subject of all. 

如果是在同范围内做比较时,比较的对象用“any other + 单数名词”或“all the other + 复数名词”表达,是为了将自己排除在外,避免造成自己和自己比较的逻辑错误,但若是在不同范围之间做比较时,则名词前不加other,如下所示。
例如:“Emily is shorter than any boy in her class.”=“Emily is shorter than all boys in her class.”(艾蜜丽比她班上其他男生都来得矮)

因為艾蜜麗是女生,不是和班上其他男生一樣,屬於不同範圍之間的比較,因此boy/boys前不加other。

其中上述的“any other + 单数名词 / all the other + 复数名词”还可以替换成“No other + 单数名词/复数名词”,
例如:“No other student(s) in his class is(are) more handsome than Gavin.”(在他班上没有其他同学比盖文还要俊俏)

使用“形容词原级”做比较的句型

  • 如果比较的两个对象其性质是一致的,也就是阐述“主词1和主词2一样...”时,可以用as...as... 表示,其中第一个as含义为“像”;第二个as含义为“一样”,基本句型为“主词1 + Be动词 + as + 形容词原级 + as + 主词2 (+ Be动词)”,如:“I am as young as her.”(我和她一样年轻)。
  • 如要表示“主词1比主词2还不...”时,可以用“主词1 + Be动词 + less + 形容词原级 + than + 主词2 (+ Be动词) ”,

例如:“Mandy is less smart than Ethan.”=“Mandy isn't as smart as Ethan.”=“Mandy isn't smarter than Ethan.”=“Mandy is stupider than Ethan.”(蔓蒂没比伊森聪明)

副词

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副词的比较级由其原级变化而来,变化方法如下:

  • - ly 结尾的副词,在该副词之前加 more,如:quickly - more quickly、expensively - more expensively、sadly - more sadly、easily - more easily、seriously - more seriously、merrily - more merrily...等。
  • 与形容词同形的副词,其变化和形容词一致,如:early - earlier、late - later、fast - faster、hard - harder、high - higher、low - lower、first - firster...等。
  • 不规则变化,如:well - better、badly - worse、much - more、little - less、far - farther(表距离)/further(表程度)...等。

副词比较的用法和形容词大致相同,基本的副词比较级句型为“主词1 + 一般动词 + 副词比较级 + than + 主词2 (+ 助动词)”,如:“Tom can speak Japanese better than me.”(汤姆讲日文比我还要好)。

副词比较级前面也可以用程度副词来修饰,如:“We eat out much less expensively than you.”(我问了他一个问题)。

Comparative coordination vs. comparative subordination

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At times the syntax of comparatives matches the syntax of coordination, and at other times, it must be characterized in terms of subordination.

Comparative coordination

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The syntax of comparatives can closely mirror the syntax of coordination. The similarity in structure across the following a- and b-sentences illustrates this point. The conjuncts of the coordinate structures are enclosed in square brackets:[2]

a. [The boys] and [the girls] sent flowers to him today.
b. More [boys] than [girls] sent flowers to him today.
a. [The boys sent] and [the girls dropped off] flowers for him today.
b. [More boys sent] than [girls dropped off] flowers for him today.
a. The boys sent [flowers to him] and [chocolates to her] today.
b. More boys sent [flowers to him] than [chocolates to her] today.
a. The boys sent [flowers to him today] and [chocolates to her yesterday].
b. More boys sent [flowers to him today] than [chocolates to her yesterday].

The structure of the b-sentences involving comparatives is closely similar to the structure of the a-sentences involving coordination. Based on this similarity, many have argued that the syntax of comparatives overlaps with the syntax of coordination at least some of the time.[3] In this regard, the than in the b-sentences should be viewed as a coordinator (coordinate conjunction), not as a subordinator (subordinate conjunction).

Comparative subordination

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Examples of the comparative that do not allow an analysis in terms of coordination (because the necessary parallel structures are not present) are instances of comparative subordination.[4] In such cases, than has the status of a preposition or a subordinator (subordinate conjunction), e.g.

a. We invited more people than wanted to come.
b. A better striker was playing for them than we have.
c. More passengers than the airline had issued tickets tried to board the plane.[5]
d. More guests than we had chairs showed up.
e. Who did he eat more hotdogs than?

Since the parallel structures associated with coordinate structures, i.e., the conjuncts, cannot be acknowledged in these sentences, the only analysis available is one in terms of subordination, whereby than has the status of a subordinator (as in sentences a-d) or of a preposition (as in sentence e). What this means is that the syntax of comparatives is complex because at times an analysis in terms of coordination is warranted, whereas at other times, the analysis must assume subordination.

  1. ^ Choudary, A. D. R.; Niculescu, Constantin. Real Analysis on Intervals. Springer India. 2014: 59–60. ISBN 978-81-322-2147-0 (英语). 
  2. ^ The examples are taken from Osborne (2009:428).
  3. ^ For examples of accounts that argue that the syntax of comparatives overlaps with the syntax of coordination at least some of the time, see Pinkham (1982), Napoli (1983), McCawley 1988, Lechner (2004), Corver (2006), and Osborne (2009).
  4. ^ The distinction between comparative coordination and comparative subordination is discussed at length by Osborne (2009).
  5. ^ The example is taken from Pinkham (1982:50).