Fixed-point theorem for asymptotic contractions of Meir Keeler type in complete metric spaces

Nonlinear Analysis64(2006)971–978

https://www.wendangku.net/doc/2518361380.html,/locate/na Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces

Tomonari Suzuki?,1

Department of Mathematics,Kyushu Institute of Technology,1-1,Sensuicho,Tobata-ku,

Kitakyushu8048550,Japan

Received21January2005;accepted25April2005

Abstract

In this paper,we introduce the notion of asymptotic contraction of Meir–Keeler type,and prove a?xed-point theorem for such contractions,which is a generalization of?xed-point theorems of Meir–Keeler and Kirk.In our discussion,we use the characterization of Meir–Keeler contraction proved by Lim[On characterizations of Meir–Keeler contractive maps,Nonlinear Anal.46(2001) 113–120].We also give a simple proof of this characterization.

?2005Elsevier Ltd.All rights reserved.

MSC:Primary54H25;Secondary54E50

Keywords:Meir–Keeler contraction;L-function;Asymptotic contraction;Fixed point;Complete metric space

1.Introduction

Throughout this paper we denote by N the set of all positive integers.

In1969,Meir and Keeler[7]proved the following very interesting?xed-point theorem, which is a generalization of the Banach contraction principle[2].See also[8,9].

?Tel.:+81938843417;fax:+81938843417.

E-mail address:suzuki-t@mns.kyutech.ac.jp.

1The author is supported in part by Grants-in-Aid for Scienti?c Research from the Japanese Ministry of Education,Culture,Sports,Science and Technology.

0362-546X/$-see front matter?2005Elsevier Ltd.All rights reserved.

doi:10.1016/j.na.2005.04.054

972T.Suzuki/Nonlinear Analysis64(2006)971–978

Theorem1(Meir and Keeler[7]).Let(X,d)be a complete metric space and let T be a mapping on X.Assume that for every >0,there exists >0such that

d(x,y)< + implies d(T x,T y)<

for x,y∈X.Then T has a unique?xed point.

On the other hand,in2003,Kirk[5]introduced the notion of asymptotic contraction on a metric space,and proved a?xed-point theorem for such contractions.Asymptotic contraction is an asymptotic version of Boyd–Wong contraction[3].See also[1].

De?nition1(Kirk[5]).Let(X,d)be a metric space and let T be a mapping on X.Then T is called an asymptotic contraction on X if there exists a continuous function from[0,∞) into itself and a sequence{ n}of functions from[0,∞)into itself such that

(i) (0)=0,

(ii) (r)

(iii){ n}converges to uniformly on the range of d,and

(iv)for x,y∈X and n∈N,

d(T n x,T n y) n(d(x,y)).

Theorem2(Kirk[5]).Let(X,d)be a complete metric space and let T be a continuous, asymptotic contraction on X with{ n}and in De?nition1.Assume that there exists x∈X such that the orbit{T n x:n∈N}of x is bounded,and that n is continuous for n∈N. Then there exists a unique?xed point z∈X.Moreover,lim n T n x=z for all x∈X. Jachymski and Jó′z wik showed that the continuity of T is needed in Theorem2;see Example1in[4].They also proved a result similar to Theorem2.The assumption in[4]is that is upper semicontinuous with lim t→∞(t? (t))=∞and T is uniformly continuous. In this paper,we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a?xed-point theorem for such contractions,which is a generalization of both Theorems1and2.In our discussion,we use the characterization of Meir–Keeler contraction proved by Lim[6].We also give a simple proof of this characterization.

2.Meir–Keeler contraction

In this section,we discuss Meir–Keeler contraction.

De?nition2.Let(X,d)be a metric space.Then a mapping T on X is said to be a Meir–Keeler contraction(MKC,for short)if for any >0,there exists >0such that

d(x,y)< + implies d(T x,T y)<

for all x,y∈X.

In[6],Lim introduced the notion of an L-function and characterized MKC.

T.Suzuki/Nonlinear Analysis64(2006)971–978973 De?nition3(Lim[6]).A function from[0,∞)into itself is called an L-function if (0)=0, (s)>0for s∈(0,∞),and for every s∈(0,∞)there exists >0such that (t) s for all t∈[s,s+ ].

We give a simple proof of Lim’s characterization.

Proposition1(Lim[6]).Let(X,d)be a metric space and let T be a mapping on X.Then T is an MKC if and only if there exists an(nondecreasing,right continuous)L-function such that

d(T x,T y)< (d(x,y))(1) for all x,y∈X with x=y.

Proof.We only show the necessity because we can prove the suf?ciency easily;see[6]. Assume that T is an MKC.From the de?nition of MKC,we can de?ne a function from (0,∞)into itself such that

d(x,y)< +2 ( )implies d(T x,T y)<

for ∈(0,∞).Using such ,we de?ne a nondecreasing function from(0,∞)into[0,∞) by

(t)=inf{ >0:t + ( )}

for t∈(0,∞).Since t t+ (t),we note that (t) t for t∈(0,∞).De?ne a function 1from[0,∞)into itself by

1(t)= 0if t=0,

(t)if t>0and min{ >0:t + ( )}exists, ( (t)+t)/2otherwise.

It is clear that 1(0)=0and0< 1(s) s for s∈(0,∞).Fix s∈(0,∞).In the case of 1(t) s for all t∈(s,s+ (s)],we can put = (s).In the other case,there exists ∈(s,s+ (s)]with 1( )>s.Since s+ (s),we have ( ) s.If ( )=s,then 1( )= ( )=s< 1( ).This is a contradiction.Thus,

( )

We can choose u∈( ( ),s)with u+ (u),and put =s?u>0.Fix t∈[s,s+ ]. Since

t s+ =2s?u<2( ( )+ )/2? ( )= u+ (u),

we have (t) u.Hence

1(t) ( (t)+t)/2 (u+s+ )/2=s.

Therefore 1is an L-function.Fix x,y∈X with x=y.From the de?nition of 1, for every t∈(0,∞),there exists ∈(0, 1(t)]such that t + ( ).So,there exists

974T.Suzuki/Nonlinear Analysis64(2006)971–978

∈(0, 1(d(x,y))]such that d(x,y) + ( ).Therefore

d(T x,T y)< 1(d(x,y))

holds.That is, 1satis?es(1).De?ne functions 2and 3by

2(t)=sup{ 1(s):s t}

and

3(t)=inf{ 2(s):s>t}

for t∈[0,∞).Then we have

0< 1(t) 2(t) 3(t) t

for all t∈(0,∞).Hence, 2and 3also satisfy(1).It is not dif?cult to verify that 2is a nondecreasing L-function and 3is a nondecreasing,right continuous L-function.This completes the proof.

3.ACMK

In this section,we discuss the following notion,which is a generalization of both asymp-totic contraction and MKC.

De?nition4.Let(X,d)be a metric space.Then a mapping T on X is said to be an asymp-totic contraction of Meir–Keeler type(ACMK,for short)if there exists a sequence{ n}of functions from[0,∞)into itself satisfying the following:

(A1)lim sup n n( ) for all 0.

(A2)For each >0,there exist >0and ∈N such that (t) for all t∈[ , + ]. (A3)d(T n x,T n y)< n(d(x,y))for all n∈N and x,y∈X with x=y.

We obtain the following.

Proposition2.Let(X,d)be a metric space.Let T be an MKC on X.Then T is also an ACMK on X.

Proof.By Proposition1,there exists an L-function from[0,∞)into itself satisfying(1). De?ne a sequence{ n}of functions by n= for all n∈N.It is obvious that{ n}satis?es (A1)and(A2).Fix x,y∈X with x=y.We note

d(T n+1x,T n+1y) d(T n x,T n y) ··· d(T x,T y) d(x,y)

for all n∈N.So we obtain

d(T n x,T n y) d(T x,T y)< (d(x,y))= n(d(x,y)).

This implies(A3).This completes the proof.

T.Suzuki /Nonlinear Analysis 64(2006)971–978975

Proposition 3.Let (X,d)be a metric space and let T be an asymptotic contraction on X .Then T is also an ACMK .

Proof.We put E ={d(x,y):x,y ∈X },that is,E is the range of d .Let and { n }be as in De?nition 1.De?ne a sequence { n }of functions from [0,∞)into itself by n (t)= n (t)+t/n if t ∈E,0if t /∈E.

We shall show that { n }satis?es (A1)–(A3).It is obvious that { n }satis?es (A1).Fix n ∈N and x,y ∈X with x =y .Since d(x,y)>0,we have

d(T n x,T n y) n (d(x,y))< n (d(x,y))+d(x,y)/n = n (d(x,y)).

Therefore (A3)holds.Let us prove (A2).Fix >0.Since ( )< and is continuous,we can choose such that 0< <( ? ( ))/2and

(t)? ( )<( ? ( ))/2

for t ∈[ , + ].For such ,we also choose ∈N such that

( + )/ < /2and sup {| (t)? (t)|:t ∈E }< /2.

We ?x t ∈[ , + ].In the case of t /∈E ,we have (t)=0 .In the case of t ∈E ,we have

(t)= (t)+t/

(t)+ /2+( + )/

( )+( ? ( ))/2+2 /2

( )+2( ? ( ))/2= .

We have shown (A2).This completes the proof.

Remark.We only use the right upper semicontinuity of .So this proposition is the af?rmative answer of the problem raised by Jachymski and Jó′z wik;see Remark 4in [4].

4.Fxed-point theorems

In this section,we prove a ?xed-point theorem which is a generalization of both Theorems 1and 2.

Theorem 3.Let (X,d)be a complete metric space .Let T be an ACMK on X .Assume that T is continuous for some ∈N .Then there exists a unique ?xed point z ∈X .Moreover ,lim n T n x =z for all x ∈X .

Proof.Let { n }be as in De?nition 4.We note

d(T n x,T n y) n (d(x,y))

976T.Suzuki /Nonlinear Analysis 64(2006)971–978

for all x,y ∈X and n ∈N .We ?rst show

lim n →∞d(T n x,T n y)=0for all x,y ∈X .(2)Fix x,y ∈X .Let T 0be the identity mapping on X .In the case of T x =T y for some ∈N ∪{0},(2)clearly holds.In the other case of T x =T y for all ∈N ∪{0},we assume :=lim sup n d(T n x,T n y)>0.From (A2),we can choose 1∈N satisfying 1(d(x,y)) d(x,y).We have

d(T 1x,T 1y)< 1(d(x,y)) d(x,y).

By (A1),we have

=lim sup n →∞d(T n ?T 1x,T n ?T 1y)

lim sup n →∞ n (d(T 1x,T 1y))

d(T 1x,T 1y)

By a similar argument,we obtain 0and 2∈N such that 2(t) for all t ∈[ , + 2].We choose 3∈N with d(T 3x,T 3y)< + 2.Then we have

d(T 2+ 3x,T 2+ 3y)=d(T 2?T 3x,T 2?T 3y)< 2(d(T 3x,T 3y)) .

This is a contradiction.Therefore we have shown (2).Let u ∈X and de?ne a sequence {u n }in X by u n =T n u for n ∈N .From (2),we have lim n d(u n ,u n +1)=0.We shall show that

lim n →∞sup m>n d(u n ,u m )=0.(3)Let >0be ?xed.Then there exist 4∈(0, )and 4∈N such that 4(t) for all t ∈[ , + 4].For such 4,there exists 5∈N such that d(u n ,u n +1)< 4/ 4for every n 5.Arguing by contradiction,we assume that there exist ,m ∈N with m > 5and d(u ,u m )>2 .Then we put

k =min {j ∈N :

It is obvious that k m .Since

2 4< + 4 d(u ,u k ) k ?1

j = d(u j ,u j +1) k ?1 j = 4/ 4=(k ? ) 4/ 4,

we have 2 4

d(u ,u k ? 4) d(u ,u k )?d(u k ? 4,u k )

d(u ,u k )? 4?1

j =0d(u k ?j ?1,u k ?j )

+ 4? 4 4/ 4= .

T.Suzuki/Nonlinear Analysis64(2006)971–978977

Since d(u ,u k?

4

)< + 4,we have

d(u +

4,u k)=d(T 4u ,T 4u k?

4

)<

4

(d(u ,u k?

4

)) .

Hence

d(u ,u k)

4

j=1d(u +j?1,u +j)+d(u +

4

,u k)< 4 4/ 4+ = 4+ .

This contradicts the de?nition of k.Therefore m>n 5implies d(u n,u m) 2 and hence (3)holds.So{u n}is a Cauchy sequence.Since X is complete,there exists z∈X such that {u n}converges to z.Then from the continuity of T ,we have

z=lim

n→∞T +n u=lim

n→∞

T ?T n u=T

lim

n→∞

T n u

=T z.

That is,z is a?xed point of T .Since

lim n→∞d(T n +1u,T z)=lim

n→∞

d(T n +1u,T n +1z)=lim

n→∞

d(T n u,T n z)=0

by(2),we have

T z=lim

n→∞T n +1u=lim

n→∞

T n u=z.

That is,z is a?xed point of T.If T x=x,then

d(z,x)=lim

n→∞

d(T n z,T n x)=0

by(2),and hence x=z.Therefore a?xed point of T is unique.Since u is arbitrary, lim n T n x=z holds for every x∈X.This completes the proof.

We can state Theorem2as follows:

Theorem4(Kirk[5]).Let(X,d)be a complete metric space and let T be a continuous, asymptotic contraction on X.Then there exists a unique?xed point z∈X.Moreover, lim n T n x=z for all x∈X.

Acknowledgements

The author wishes to express his gratitude to Professor W.A.Kirk for giving the historical comment.

References

[1]I.D.Arandelovi′c,On a?xed point theorem of Kirk,J.Math.Anal.Appl.301(2005)384–385.

[2]S.Banach,Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales,Fund.

Math.3(1922)133–181.

978T.Suzuki/Nonlinear Analysis64(2006)971–978

[3]D.W.Boyd,J.S.W.Wong,On nonlinear contractions,Proc.Amer.Math.Soc.20(1969)458–464.

[4]J.R.Jachymski,I.Jó′z wik,On Kirk’s asymptotic contractions,J.Math.Anal.Appl.300(2004)147–159.

[5]W.A.Kirk,Fixed points of asymptotic contractions,J.Math.Anal.Appl.277(2003)645–650.

[6]T.C.Lim,On characterizations of Meir–Keeler contractive maps,Nonlinear Anal.46(2001)113–120.

[7]A.Meir,E.Keeler,A theorem on contraction mappings,J.Math.Anal.Appl.28(1969)326–329.

[8]T.Suzuki,Several?xed point theorems in complete metric spaces,Yokohama Math.J.44(1997)61–72.

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初中英语介词用法总结

初中英语介词用法总结 介词(preposition)：也叫前置词。在英语里，它的搭配能力最强。但不能单独做句子成分需要和名词或代词（或相当于名词的其他词类、短语及从句）构成介词短语，才能在句中充当成分。 介词是一种虚词，不能独立充当句子成分，需与动词、形容词和名词搭配，才能在句子中充当成分。介词是用于名词或代词之前，表示词与词之间关系的词类，介词常与动词、形容词和名词搭配表示不同意义。介词短语中介词后接名词、代词或可以替代名词的词（如：动名词v-ing）.介词后的代词永远为宾格形式。介词的种类： （1）简单介词：about, across, after, against, among, around, at, before, behind, below, beside, but, by, down, during, for, from, in, of, on, over, near, round, since, to, under, up, with等等。 （2）合成介词：inside, into, outside, throughout, upon, without, within （3）短语介词：according to, along with, apart from, because of, in front of, in spite of, instead of, owing to, up to, with reguard to （4）分词介词：considering, reguarding, including, concerning 介词短语：构成 介词+名词We go to school from Monday to Saturday. 介词+代词Could you look for it instead of me? 介词+动名词He insisted on staying home. 介词+连接代/副词I was thinking of how we could get there. 介词+不定式/从句He gives us some advice on how to finish it. 介词的用法： 一、介词to的常见用法 1.动词+to a)动词+ to adjust to适应, attend to处理；照料, agree to赞同,

超全的英语介词用法归纳总结

超全的英语介词用法归纳总结常用介词基本用法辨析 表示方位的介词：in, to, on 1. in 表示在某地范围之内。 Shanghai is/lies in the east of China. 上海在中国的东部。 2. to 表示在某地范围之外。 Japan is/lies to the east of China. 日本位于中国的东面。 3. on 表示与某地相邻或接壤。 Mongolia is/lies on the north of China. 蒙古国位于中国北边。 表示计量的介词：at, for, by 1. at 表示“以……速度”“以……价格”。 It flies at about 900 kilometers an hour. 它以每小时900公里的速度飞行。 I sold my car at a high price. 我以高价出售了我的汽车。 2. for 表示“用……交换，以……为代价”。 He sold his car for 500 dollars. 他以五百元把车卖了。 注意：at表示单价(price) ，for表示总钱数。

3. by 表示“以……计”，后跟度量单位。 They paid him by the month. 他们按月给他计酬。 Here eggs are sold by weight. 在这里鸡蛋是按重量卖的。 表示材料的介词：of, from, in 1. of 成品仍可看出原料。 This box is made of paper. 这个盒子是纸做的。 2. from 成品已看不出原料。 Wine is made from grapes. 葡萄酒是葡萄酿成的。 3. in 表示用某种材料或语言。 Please fill in the form in pencil first. 请先用铅笔填写这个表格。They talk in English. 他们用英语交谈。 表示工具或手段的介词：by, with, on 1. by 用某种方式，多用于交通。 I went there by bus. 我坐公共汽车去那儿。 2. with表示“用某种工具”。 He broke the window with a stone. 他用石头把玻璃砸坏了。注意：with表示用某种工具时，必须用冠词或物主代词。

(完整版)介词for用法归纳

介词for用法归纳 用法1：(表目的)为了。如： They went out for a walk. 他们出去散步了。 What did you do that for? 你干吗这样做？ That’s what we’re here for. 这正是我们来的目的。 What’s she gone for this time? 她这次去干什么去了？ He was waiting for the bus. 他在等公共汽车。 【用法说明】在通常情况下，英语不用for doing sth 来表示目的。如： 他去那儿看他叔叔。 误：He went there for seeing his uncle. 正：He went there to see his uncle. 但是，若一个动名词已名词化，则可与for 连用表目的。如： He went there for swimming. 他去那儿游泳。(swimming 已名词化) 注意：若不是表目的，而是表原因、用途等，则其后可接动名词。(见下面的有关用法) 用法2：(表利益)为，为了。如： What can I do for you? 你想要我什么？ We study hard for our motherland. 我们为祖国努力学习。 Would you please carry this for me? 请你替我提这个东西好吗？ Do more exercise for the good of your health. 为了健康你要多运动。 【用法说明】(1) 有些后接双宾语的动词(如buy, choose, cook, fetch, find, get, order, prepare, sing, spare 等)，当双宾语易位时，通常用for 来引出间接宾语，表示间接宾语为受益者。如： She made her daughter a dress. / She made a dress for her daughter. 她为她女儿做了件连衣裙。 He cooked us some potatoes. / He cooked some potatoes for us. 他为我们煮了些土豆。 注意，类似下面这样的句子必须用for： He bought a new chair for the office. 他为办公室买了张新办公椅。 (2) 注意不要按汉语字面意思，在一些及物动词后误加介词for： 他们决定在电视上为他们的新产品打广告。 误：They decided to advertise for their new product on TV. 正：They decided to advertise their new product on TV. 注：advertise 可用作及物或不及物动词，但含义不同：advertise sth＝为卖出某物而打广告；advertise for sth＝为寻找某物而打广告。如：advertise for a job=登广告求职。由于受汉语“为”的影响，而此处误加了介词for。类似地，汉语中的“为人民服务”，说成英语是serve the people，而不是serve for the people，“为某人的死报仇”，说成英语是avenge sb’s death，而不是avenge for sb’s death，等等。用法3：(表用途)用于，用来。如： Knives are used for cutting things. 小刀是用来切东西的。 This knife is for cutting bread. 这把小刀是用于切面包的。 It’s a machine for slicing bread. 这是切面包的机器。 The doctor gave her some medicine for her cold. 医生给了她一些感冒药。 用法4：为得到，为拿到，为取得。如： He went home for his book. 他回家拿书。 He went to his friend for advice. 他去向朋友请教。 She often asked her parents for money. 她经常向父母要钱。

计量经济学术语(国际经济与贸易)

计量经济学术语 A 校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的方差时对添加的解释变量用?一个自由度来调整。 对立假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。 AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。 渐近置信区间（Asymptotic Confidence Interval）：大样本容量下近似成立的置信区间。 渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。 渐近性质（Asymptotic Properties）：当样本容量无限增长时适用的估计量和检验统计量性质。 渐近标准误（Asymptotic Standard Error）：大样本下生效的标准误。 渐近t 统计量（Asymptotic t Statistic）：大样本下近似服从标准正态分布的t统计量。 渐近方差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须用以除估计量的平方值。 渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的?一致性估计量，有最小渐近方差的估计量。 渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。 衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因而有衰减偏误的估计量的期望值小于参数的绝对值。 自回归条件异方差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异方差性模型，即给定过去信息，误差项的方差线性依赖于过去的误差的平方。 ?一阶自回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：?一个时间序列模型，其当前值线性依赖于最近的值加上?一个无法预测的扰动。 辅助回归（Auxiliary Regression）：用于计算检验统计量——例如异方差性和序列相关的检验统计量——或其他任何不估计主要感兴趣的模型的回归。 平均值（Average）：n个数之和除以n。 B 基组、基准组（Base Group）：在包含虚拟解释变量的多元回归模型中，由截距代表的组。 基期（Base Period）：对于指数数字，例如价格或生产指数，其他所有时期均用来作为衡量标准的时期。 基期值（Base Value）：指定的基期的值，用以构造指数数字；通常基本值为1或100。 最优线性无偏估计量（Best Linear Unbiased Estimator, BLUE）：在所有线性、无偏估计量中，有最小方差的估计量。在高斯—马尔科夫假定下，OLS是以解释变量样本值为条件的贝塔系数（Beta Coef?cients）：见标准化系数。 偏误（Bias）：估计量的期望参数值与总体参数值之差。 偏误估计量（Biased Estimator）：期望或抽样平均与假设要估计的总体值有差异的估计量。 向零的偏误（Biased Towards Zero）：描述的是估计量的期望绝对值小于总体参数的绝对值。 二值响应模型（Binary Response Model）：二值因变量的模型。 二值变量（Binary Variable）：见虚拟变量。 两变量回归模型（Bivariate Regression Model）：见简单线性回归模型。 BLUE（BLUE）：见最优线性无偏估计量。 Breusch-Godfrey 检验（Breusch-Godfrey Test）：渐近正确的AR（p）序列相关检验，以AR（1）最为流行；该检验考虑到滞后因变量和其他不是严格外生的回归元。 Breusch-Pagan 检验（Breusch-Pagan Test）：将OLS残差的平方对模型中的解释变量做回归的异方差性检验。 C 因果效应（Causal Effect）：?一个变量在其余条件不变情况下的变化对另?一个变量产生的影响。 其余条件不变（Ceteris Paribus）：其他所有相关因素均保持固定不变。 经典含误差变量（Classical Errors-in-Variables, CEV）：观测的量度等于实际变量加上?一个独立的或至少不相关的测量误差的测量误差模型。 经典线性模型（Classical Linear Model）：全套经典线性模型假定下的复线性回归模型。 经典线性模型（CLM）假定（Classical Linear Model (CLM) Assumptions）：对多元回归分析的理想假定集，对横截面分析为假定MLR.1至MLR.6，对时间序列分析为假定 对参数为线性、无完全共线性、零条件均值、同方差、无序列相关和误差正态性。 科克伦—奥克特（CO）估计（Cochrane-Orcutt (CO) Estimation）：估计含AR（1）误差和严格外生解释变量的多元线性回归模型的?一种方法；与普莱斯—温斯登估计不同，科克伦—奥克特估不使用第?一期的方程。 置信区间（CI）（Con?dence Interval, CI）：用于构造随机区间的规则，以使所有数据集中的某?一百分比（由置信水平决定）给出包含总体值的区间。 置信水平（Con?dence Level）：我们想要可能的样本置信区间包含总体值的百分比，95%是最常见的置信水平，90%和99%也用。 不变弹性模型（Constant Elasticity Model）：因变量关于解释变量的弹性为常数的模型；在多元回归中，两者均以对数形式出现。 同期外生回归元（Contemporaneously Exogenous）：在时间序列或综列数据应用中，与同期误差项不相关但对其他时期则不?一定的回归元。 控制组（Control Group）：在项目评估中，不参与该项目的组。 控制变量（Control Variable）：见解释变量。 协方差平稳（Covariance Stationary）：时间序列过程，其均值、方差为常数，且序列中任意两个随机变量之间的协方差仅与它们的间隔有关。 协变量（Covariate）：见解释变量。 临界值（Critical Value）：在假设检验中，用于与检验统计量比较来决定是否拒绝虚拟假设的值。 横截面数据集（Cross-Sectional Data Set）：在给定时点上从总体中收集的数据集 D 数据频率（Data Frequency）：收集时间序列数据的区间。年度、季度和月度是最常见的数据频率。 戴维森—麦金农检验（Davidson-MacKinnon Test）：用于检验相对于非嵌套对立假设的模型的检验：它可用相争持模型中得出的拟合值的t检验来实现。 自由度（df）（Degrees of Freedom, df）：在多元回归模型分析中，观测值的个数减去待估参数的个数。 分母自由度（Denominator Degrees of Freedom）：F检验中无约束模型的自由度。 因变量（Dependent Variable）：在多元回归模型（和其他各种模型）中被解释的变量。

【备战高考】英语介词用法总结(完整)

【备战高考】英语介词用法总结(完整) 一、单项选择介词 1． passion, people won't have the motivation or the joy necessary for creative thinking. A．For . B．Without C．Beneath D．By 【答案】B 【解析】 【详解】 考查介词辨析。句意：没有激情，人们就不会有创新思维所必须的动机和快乐。A. For 对于；B. Without没有； C. Beneath在……下面 ; D. By通过。没有激情，人们就不会有创新思维所必须的动机和快乐。所以空处填介词without。故填without。 2．Modern zoos should shoulder more social responsibility _______ social progress and awareness of the public. A．in light of B．in favor of C．in honor of D．in praise of 【答案】A 【解析】 【分析】 【详解】 考查介词短语。句意：现代的动物园应该根据社会的进步和公众的意识来承担更多的社会责任。A. in light of根据，鉴于；B. in favor of有利于，支持；C. in honor of 为了纪念；D. in praise of歌颂，为赞扬。此处表示根据，故选A。 3．If we surround ourselves with people _____our major purpose, we can get their support and encouragement. A．in sympathy with B．in terms of C．in honour of D．in contrast with 【答案】A 【解析】 【详解】 考查介词短语辨析。句意：如果我们周围都是认同我们主要前进目标的人，我们就能得到他们的支持和鼓励。A. in sympathy with赞成；B. in terms of 依据；C. in honour of为纪念； D. in contrast with与…形成对比。由“we can get their support and encouragement”可知，in sym pathy with“赞成”符合句意。故选A项。 4．Elizabeth has already achieved success_____her wildest dreams. A．at B．beyond C．within D．upon

英语介词for的用法归纳总结.doc

英语介词for的用法归纳总结用法1：(介词for表目的)为了 They went out for a walk. 他们出去散步了。 What did you do that for? 你干吗这样做? That s what we re here for. 这正是我们来的目的。 What s she gone for this time? 她这次去干什么去了? He was waiting for the bus. 他在等公共汽车。 【用法说明】在通常情况下，英语不用for doing sth 来表示目的 他去那儿看他叔叔。 误：He went there for seeing his uncle. 正：He went there to see his uncle. 但是，若一个动名词已名词化，则可与for 连用表目的 He went there for swimming. 他去那儿游泳。(swimming 已名词化) 注意：若不是表目的，而是表原因、用途等，则其后可接动名词。(见下面的有关用法) 用法2：(介词for表利益)为，为了 What can I do for you? 你想要我什么? We study hard for our motherland. 我们为祖国努力学习。 Would you please carry this for me? 请你替我提这个东西好吗?

Do more exercise for the good of your health. 为了健康你要多运动。 【用法说明】(1) 有些后接双宾语的动词(如buy, choose, cook, fetch, find, get, order, prepare, sing, spare 等)，当双宾语易位时，通常用for 来引出间接宾语，表示间接宾语为受益者 She made her daughter a dress. / She made a dress for her daughter. 她为她女儿做了件连衣裙。 He cooked us some potatoes. / He cooked some potatoes for us. 他为我们煮了些土豆。 注意，类似下面这样的句子必须用for： He bought a new chair for the office. 他为办公室买了张新办公椅。 (2) 注意不要按汉语字面意思，在一些及物动词后误加介词for： 他们决定在电视上为他们的新产品打广告。 误：They decided to advertise for their new product on TV. 正：They decided to advertise their new product on TV. 注：advertise 可用作及物或不及物动词，但含义不同：advertise sth=为卖出某物而打广告;advertise for sth=为寻找某物而打广告advertise for a job=登广告求职。由于受汉语为的影响，而此处误加了介词for。类似地，汉语中的为人民服务，说成英语是serve the people，而不是serve for the people，为某人的死报仇，说成英语是avenge sb s death，而不是avenge for sb s death，等等。 用法3：(介词for表用途)用于，用来 Knives are used for cutting things. 小刀是用来切东西的。

介词for用法完全归纳

用法1：(表目的)为了。如： They went out for a walk. 他们出去散步了。 What did you do that for? 你干吗这样做? That’s what we’re here for. 这正是我们来的目的。 What’s she gone for this time? 她这次去干什么去了? He was waiting for the bus. 他在等公共汽车。 【用法说明】在通常情况下，英语不用for doing sth 来表示目的。如：他去那儿看他叔叔。 误：He went there for seeing his uncle. 正：He went there to see his uncle. 但是，若一个动名词已名词化，则可与for 连用表目的。如： He went there for swimming. 他去那儿游泳。(swimming 已名词化) 注意：若不是表目的，而是表原因、用途等，则其后可接动名词。(见下面的有关用法) 用法2：(表利益)为，为了。如： What can I do for you? 你想要我什么? We study hard for our motherland. 我们为祖国努力学习。 Would you please carry this for me? 请你替我提这个东西好吗? Do more exercise for the good of your health. 为了健康你要多运动。 【用法说明】(1) 有些后接双宾语的动词(如buy, choose, cook, fetch, find, get, order, prepare, sing, spare 等)，当双宾语易位时，通常用for 来引出间接宾语，表示间接宾语为受益者。如：

for的用法完全归纳

for的用法完全归纳 用法1：(表目的)为了。如： They went out for a walk. 他们出去散步了。 What did you do that for? 你干吗这样做？ That’s what we’re here for. 这正是我们来的目的。 What’s she gone for this time? 她这次去干什么去了？ He was waiting for the bus. 他在等公共汽车。 在通常情况下，英语不用for doing sth 来表示目的。如：他去那儿看他叔叔。 误：He went there for seeing his uncle.正：He went there to see his uncle. 但是，若一个动名词已名词化，则可与for 连用表目的。如： He went there for swimming. 他去那儿游泳。(swimming 已名词化) 注意：若不是表目的，而是表原因、用途等，则其后可接动名词。 用法2：(表利益)为，为了。如： What can I do for you? 你想要我什么？ We study hard for our motherland. 我们为祖国努力学习。 Would you please carry this for me? 请你替我提这个东西好吗？ Do more exercise for the good of your health. 为了健康你要多运动。 (1)有些后接双宾语的动词(如buy, choose, cook, fetch, find, get, order, prepare, sing, spare 等)，当双宾语易位时，通 常用for 来引出间接宾语，表示间接宾语为受益者。如： She made her daughter a dress. / She made a dress for her daughter. 她为她女儿做了件连衣裙。 He cooked us some potatoes. / He cooked some potatoes for us. 他为我们煮了些土豆。 注意，类似下面这样的句子必须用for： He bought a new chair for the office. 他为办公室买了张新办公椅。 (2) 注意不要按汉语字面意思，在一些及物动词后误加介词for： 他们决定在电视上为他们的新产品打广告。 误：They decided to advertise for their new product on TV. 正：They decided to advertise their new product on TV. 注：advertise 可用作及物或不及物动词，但含义不同：advertise sth＝为卖出某物而打广告；advertise for sth＝为寻找某物而打广告。如：advertise for a job=登广告求职。由于受汉语“为”的影响，而此处误加了介词for。类似地，汉语中的“为人民服务”，说成英语是serve the people，而不是serve for the people，“为某人的死报仇”，说成英语是avenge sb’s death，而不是avenge for sb’s death，等等。 用法3：(表用途)用于，用来。如： Knives are used for cutting things. 小刀是用来切东西的。 This knife is for cutting bread. 这把小刀是用于切面包的。 It’s a machine for slicing bread. 这是切面包的机器。 The doctor gave her some medicine for her cold. 医生给了她一些感冒药。 用法4：为得到，为拿到，为取得。如： He went home for his book. 他回家拿书。 He went to his friend for advice. 他去向朋友请教。 She often asked her parents for money. 她经常向父母要钱。 We all hope for success. 我们都盼望成功。 Are you coming in for some tea? 你要不要进来喝点茶？ 用法5：给(某人)，供(某人)用。如： That’s for you. 这是给你的。 Here is a letter for you. 这是你的信。 Have you room for me there? 你那边能给我腾出点地方吗？ 用法6：(表原因、理由)因为，由于。如：

介词for 的常见用法归纳

介词for 的常见用法归纳 贵州省黔东南州黎平县黎平一中英语组廖钟雁介词for 用法灵活并且搭配能力很强，是一个使用频率非常高的词，也是 高考必考的重要词汇，现将其常见用法归纳如下，供参考。 1.表时间、距离或数量等。 ①意为“在特定时间，定于，安排在约定时间”。如： The meeting is arranged for 9 o’clock. 会议安排在九点进行。 ②意为“持续达”，常于last、stay 、wait等持续性动词连用，表动作持续的时间，有时可以省略。如： He stayed for a long time. 他逗留了很久。 The meeting lasted （for）three hours. 会议持续了三小时。 ③意为“（距离或数量）计、达”。例如： He walked for two miles. 他走了两英里。 The shop sent me a bill for $100.商店给我送来了100美元的账单。 2. 表方向。意为“向、朝、开往、前往”。常与head、leave 、set off、start 等动词连用。如： Tomorrow Tom will leave for Beijing. 明天汤姆要去北京。 He put on his coat and headed for the door他穿上大衣向门口走去。 介词to也可表示方向，但往往与come、drive 、fly、get、go、lead、march、move、return、ride、travel、walk等动词连用。 3.表示理由或原因,意为“因为、由于”。常与thank、famous、reason 、sake 等词连用。如: Thank you for helping me with my English. 谢谢你帮我学习英语。 For several reasons, I’d rather not meet him. 由于种种原因，我宁可不见他。 The West Lake is famous for its beautiful scenery.西湖因美景而闻名。 4.表示目的,意为“为了、取、买”等。如: Let’s go for a walk. 我们出去散步吧。 I came here for my schoolbag.我来这儿取书包。 He plays the piano for pleasure. 他弹钢琴是为了消遣。 There is no need for anyone to know. 没必要让任何人知道。 5.表示动作的对象或接受者,意为“给、为、对于”。如: Let me pick it up for you. 让我为你捡起来。 Watching TV too much is bad for your health. 看电视太多有害于你的健康。 Here is a letter for you. 这儿有你的一封信。

英语介词的用法总结

介词的用法 1.表示地点位置的介词 1)at ,in, on, to，for at (1)表示在小地方; (2)表示“在……附近，旁边” in (1)表示在大地方; (2)表示“在…范围之内”。 on 表示毗邻，接壤，“在……上面”。 to 表示在……范围外，不强调是否接壤；或“到……” 2)above, over, on 在……上 above 指在……上方,不强调是否垂直，与below相对； over指垂直的上方,与under相对,但over与物体有一定的空间，不直接接触。 on表示某物体上面并与之接触。 The bird is flying above my head. There is a bridge over the river. He put his watch on the desk. 3)below, under 在……下面 under表示在…正下方 below表示在……下，不一定在正下方 There is a cat under the table. Please write your name below the line. 4)in front [frant]of, in the front of在……前面 in front of…意思是“在……前面”，指甲物在乙物之前，两者互不包括；其反义词是behind（在……的后面）。There are some flowers in front of the house.(房子前面有些花卉。) in the front of 意思是“在…..的前部”，即甲物在乙物的内部.反义词是at the back of…（在……范围内的后部）。 There is a blackboard in the front of our classroom. 我们的教室前边有一块黑板。 Our teacher stands in the front of the classroom. 我们的老师站在教室前.(老师在教室里) 5）beside，behind beside 表示在……旁边 behind 表示在……后面 2.表示时间的介词 1)in , on，at 在……时 in表示较长时间，如世纪、朝代、时代、年、季节、月及一般（非特指）的早、中、晚等。 如in the 20th century, in the 1950s, in 1989, in summer, in January, in the morning, in one’s life , in one’s thirties等。 on表示具体某一天及其早、中、晚。 如on May 1st, on Monday, on New Year’s Day, on a cold night in January, on a fine morning, on Sunday afternoon等。 at表示某一时刻或较短暂的时间，或泛指圣诞节，复活节等。 如at 3:20, at this time of year, at the beginning of, at the end of …, at the age of …, at Christmas，at night, at noon, at this moment等。 注意：在last, next, this, that, some, every 等词之前一律不用介词。如：We meet every day. 2)in, after 在……之后 “in +段时间”表示将来的一段时间以后； “after+段时间”表示过去的一段时间以后； “after+将来的时间点”表示将来的某一时刻以后。 3)from, since 自从…… from仅说明什么时候开始，不说明某动作或情况持续多久；

介词的归纳

介词的归纳 一、单项选择介词 1．（重庆）Last year was the warmest year on record, with global temperature 0.68 ℃ ________ the average. A．below B．on C．at D．above 【答案】D 【解析】 【详解】 考查介词。句意：去年是有纪录以来最热的一年，全球平均气温上升0.68度。A. below低于；B. on在……之上；C. at在；D. above超过，多于。根据前一句Last year was the warmest year on record推知，温度应该是上升了，故用介词above。 【点睛】 with的复合结构中，复合宾语中第一部分宾语由名词和代词充当，第二部分补足语由形容词，副词，介词短语，动词不定式或分词充当。而本题考查with +名词/代词+介词短语,而介词的使用则根据当时语境的提示来做出相应的变化即句中的the warmest year on record 起重要作用，可知高出平均气温。 2．According to Baidu, the high-quality content of Cloud Music will reach massive users _______ Baidu’s app and video platform. A．in honor of B．in view of C．by virtue of D．by way of 【答案】C 【解析】 【详解】 考查介词短语。句意：根据百度的说法，云音乐的高质量内容将借助于百度应用和视频平台到达广大用户。A. in honor of向……致敬；B. in view of考虑到；C. by virtue of借助于；D. by way of通过。根据句意可知，此处要表达“借助于”。故选C项。 3．We charge parcels ________ weight, rather than individual units. A．in honor of B．in contact with C．in terms of D．in connection with 【答案】C 【解析】 【详解】 考查介词短语。句意：我们根据包裹的重量，而不是包裹的件数收费。A. in honor of为了对……表示敬意；B. in contact with与……有联系，接触；C. in terms of根据，在……方面；D. in connection with与……有关，有联系。表示根据什么计费。故选C。 【点睛】

介词用法归纳

介词（preposition） 又称前置词，是一种虚词。介词不能单独做句子成分。介词后须接宾语，介词与其宾语构成介词短语。 一、介词从其构成来看可以分为： 1、简单介词（Simple prepositions）如：at ,by, for, in, from, since, through等； 2、复合介词（Compound prepositions）如：onto, out of, without, towards等； 3、短语介词（phrasal prepositions）如；because of, instead of, on account of, in spite of, in front of等； 4、二重介词（double prepositions）如：from behind, from under, till after等； 5、分词介词（participial prepositions），又可称动词介词（verbal prepositions）如：during, concerning, excepting, considering, past等。 二、常见介词的基本用法 1、 about 关于 Do you know something about Tom？ What about this coat？（……怎么样） 2、 after 在……之后 I’m going to see you after supper. Tom looked after his sick mother yesterday.（照看） 3、 across 横过 Can you swim across the river. 4、 against 反对 Are you for or against me? Nothing could make me turn against my country.（背叛） 5、 along 沿着 We walked along the river bank. 6、 before 在……之前 I hope to get there before seven o’clock. It looks as though it will snow before long.（不久） 7、behind 在……后面 The sun is hidden behind the clouds. 8、by 到……时 We had learned ten English songs by the end of last term. 9、during 在……期间 Where are you going during the holiday. 10、except 除了 Everyone except you answered the question correctly. 11、for 为了 The students are studying hard for the people. 12、from 从 I come from Shanghai. 13、in 在……里 on 在……上面 under在……下面 There are two balls in/on/under the desk. 14、near 在……附近 We live near the park. 15、of ……的 Do you know the name of the winner. 16、over 在……正上方 There is a bridge over the river. Tom goes over his English every day.（复习） 17、round/around 围绕 The students stand around the teacher. 18、to 朝……方向 Can you tell me the way to the cinema. 19、towards朝着 The car is traveling towards Beijing.

for的用法完全归纳

f o r的用法完全归纳 TTA standardization office【TTA 5AB- TTAK 08- TTA 2C】

f o r的用法完全归纳用法1：(表目的)为了。如： They went out for a walk. 他们出去散步了。 What did you do that for 你干吗这样做 That’s what we’re here for. 这正是我们来的目的。 What’s she gone for this time 她这次去干什么去了 He was waiting for the bus. 他在等公共汽车。 在通常情况下，英语不用 for doing sth 来表示目的。如：他去那儿看他叔叔。 误：He went there for seeing his uncle.正：He went there to see his uncle. 但是，若一个动名词已名词化，则可与 for 连用表目的。如： He went there for swimming. 他去那儿游泳。(swimming 已名词化) 注意：若不是表目的，而是表原因、用途等，则其后可接动名词。 用法2：(表利益)为，为了。如： What can I do for you 你想要我什么 We study hard for our motherland. 我们为祖国努力学习。 Would you please carry this for me 请你替我提这个东西好吗 Do more exercise for the good of your health. 为了健康你要多运动。 (1)有些后接双宾语的动词(如 buy, choose, cook, fetch, find, get, order, prepare, sing, spare 等)，当双宾语易位时，通常用 for 来引出间接宾语，表示间接宾语为受益者。如：