Short programs for functions on curves
Short Programs for functions on Curves
Victor https://www.wendangku.net/doc/157980212.html,ler
Exploratory Computer Science
IBM,Thomas J.Watson Research Center
Yorktown Heights,NY10598
May6,1986
Abstract
The problem of deducing a function on an algebraic curve having a given divisor is important in the?eld of inde?nite integration.Indeed,
it is the main computational step in determining whether an algebraic
function posseses an inde?nite integral.It has also become important
recently in the study of discrete elliptic logarithms in cryptography,and
in the construction of the new class of error-correcting codes which exceed
the Varshamov-Gilbert bound.It can also be used to give a partial answer
to a question raised by Schoof in his paper on computing the exact number
of points on an elliptic curve over a?nite?eld.
Heretofore,the best known algorithm for calculating such functions was exponential in the size of the input.In this paper I give an algorithm
which is linear in the size of the input(if arithmetic in the?eld under
consideration takes constant time).For this algorithm it is necessary to
represent the function as a straight-line program,for representation as a
rational function may be exponential in the size of the input.
1Introduction
I recall some standard terminology from the theory of algebraic curves.There are many good references for this theory,for example[Ful69].It contains all results quoted here.For simplicity in presentation I only work with plane curves, even though all of the results work for more general curves.
De?nition1Projective n-space over a?eld K:denoted by P n(K)is the set of equivalence classes{(a0,...,a n)=0|a i K}where two vectors are equivalent if they are non-zero scalar multiples of one another.The equivalence class con-tataining(a0,···,a n)is denoted by(a0:···:a n).The projective line is P1, and the projective plane is P2.
1
De?nition2A plane(projective)algebraic curve is the set of solutions to a homogeneous polynomial f(X,Y,Z)considered as points in boldP1.The curve is non-singular if the three partial derivatives of f never vanish simultaneously.
De?nition3The function?eld of the curve,C,is the set of functions from the curve to the projective line,which are given by rational functions of the coordinates and is denoted by k(C).
De?nition4A(geometric)point,P on C is a solution to the set of de?ning equations.
De?nition5At each point,P,of the curve one can de?ne the:hp2order, v P(f),of the function f at P:it is n>0if the function has a zero of order n at P,and is?n<0if the function has a pole of order n at P(1/f has a zero of order n).
De?nition6A divisor on a curve,C as being a formal sum of points on the curve with integral coe?cients: P a P(P),where only?nitely many of the a P are=0.Divisors will normally be denoted by script letters such as A or B.
De?nition7The degree of a divisor is the sum of its coe?cients:deg A=
a P.
P
A function f has only a?nite number of zeroes and poles.Therefore we may de?ne
De?nition8The divisor of a function f=0,denoted by(f)is P v P(f)(P). It is also true that deg(f)=0.The set of divisors of degree0is denoted by D0.
A divisor of a function is also called principal.
De?nition9Linear equivalence.If A and B are divisors we write A～B if there is a function f such that A=B+(f).Principal divisors are also called linearly equivalent to0.The set of all such is denoted by D .The group D0/D is known as the Picard group of C and is denoted by Pic(C).
De?nition10Genus.The genus g of a curve C is an integer≥0,which is a measure of the curve’s complexity.
It,for example,is the number of“donut-holes”in the Riemann-surface of the curve if we are working over the complex numbers.See[Ful69]for an exact de?nition.For our purposes,we only need to know that g≤(n?1)(n?2)/2 where n is the degree of the polynomial de?ning the curve.A curve whose genus is1is usually called an elliptic curve.
De?nition11Linear system.The linear system of a divisor A is L(A)= {0}∪{f k(C)|(f)+A≥0}.It is easily seen that L(A)is a vector space(in fact?nite-dimensional).We denote (A)=dim L(A).
De?nition12Support.The support supp A of a divisor A is the set{P|A P= 0}.
2
2The main algorithm
One of the fundamental problems in the theory of curves is to determine when a divisor of degree0is principal,and to construct a function whose divisor is the given divisor.In[Dav79].Davenport gives an algorithm for this problem which does not always work for curves of genus>1,and which is claimed to run in O(lg A where A is the largest multiplicity of a zero or a pole at any point. This claim seems to be unjusti?ed.The construction below gives a straight-line program instead of a ratio of two polynomials.This is in keeping with the spirit of[Kal85].
Algorithm1Input:
1.An algebraic curve C of genus g
2.a?xed point Q on C
3.a divisor A= P a P(P)of degree0
Output:
1.A straight-line program for a function f on the curve
2.A divisor of the form C?g(Q)where deg C=g and C≥0.
The divisor C?g(Q)will=0if and only if A～0.We will further have A= (f)+C?g(Q).The length of this program is O(L)where L= P lg(|a P|+1) and the running time of the algorithm is O(L)arithmetic operations in the?eld of de?nition.The program gives the value unde?ned on a set of points of size ≤2gL.
We use the following fundamental theorem:
Theorem1The Riemann Theorem:Given a non-singular curve C,and a divisor A,then
(A)≥deg A+1?g
with equality holding if deg A>2g?2.
Using the above we have
Lemma1Any divisor A of degree0is equivalent to a divisor of the form C?g(Q),where deg C=g and C≥0.
Proof:Using the Riemann Theorem we?nd that (A+g(Q))≥1.Thus there is a function f=0,f L(A+g(Q)).Now set C=(f)+A+g(Q).It is readily seen that C has the desired properties.2 We use1to make calculations in the Picard Group,by representing each equiv-alence class by a divisor of the form C?g(Q).The fundamental point is that
3
to calculate a representative for the sum of two such divisors takes O(g3)?eld
operations(basically the solution of a system of linear equations).We then
write the divisor A= P a P(P)as P a P((P)?(Q)).We then can calculate a straight-line program of length≤2lg(|a P|+1)and a divisor C?g(Q)as
above by means of the binary method of addition(or any other addition chain).
The program will be unde?ned at most at all points of the supports of the di-
visors occuring in the intermediate steps.We then?nish o?by adding up all
of the individual pieces,while multiplying the functions(this concatenates the
straight-line programs with a multiplication in between).
In actual calculation one must proceed more explicitly:First calculate a
basis for the space L(3g(Q))in the form f1,...,f d where ord Q(f1)>ord Q(f2)>
···>ord Q(f d).When adding the two divisors A?g(Q)and B?g(Q)we?rst
?nd f L(3g(Q)?A?B).De?ne C=(f)+3g(Q)?A?B then it is e?ective
of degree g.Now?nd h L(2g(Q)?C),and set D=(h)+2g(Q)?C.This
divisor is also e?ective of degree g and A?g(Q)+B?g(Q)=D?g(Q)+(f/h).
We further exploit the fact that there is a one-to-one correspondence between
e?ective divisors and integral ideals of the ring of functions whose only poles
are at Q.We represent each ideal by means of it Grobner Basis.In this way we
avoid the necessity for root?nding and factorization.
In the case of genus1we do not even need to do this.All zeroes and poles of
intermediate divisors must be rational.If the curve is given in Weierstrass form
y2=x3+ax+b then1,x,y form a basis of L(3g(0)).Finding the function f
above amounts to?nding the coe?cients of the line y=cx+d passing through
the points A and B.
3Applications
In this section I discuss some applications of the above construction.The?rst
is the calculation of the“Weil”e N pairing.
Andre Weil[Wei46]introduced a pairing on points of?nite order on elliptic
curves in his?rst proof of the“Riemann Hypothesis”for curves over?nite?elds.
Following are its de?nition and properties:
De?nition13Given an elliptic curve C and a non-negative integer N there is a unique function e N de?ned on points of exponent N(P is of exponent N if NP=0,it is of order N if N is an exponent and no smaller integer is).It has the following properties:
1.e N(P,Q)is an N-th root of unity.
2.Skew-symmetry:e N(P,Q)=e N(Q,P)sup?1
3.Linearity:e N(P+R,Q)=e N(P,Q)e N(R,Q)
4.Non-degeneracy:Given a point P of order N there is a point Q of order
N such that e N(P,Q)=1.
4
This pairing may be de?ned as follows (see [Lan73,pages 243–245]):If A is a divisor of order N in the divisor class group D 0/D ,then there is a function f A such that (f A )=N A .This function is de?ned up to multiplication by a constant.If f is any function,and A = P a P (P )is a divisor of degree 0,then we de?ne f (A )= P f (P )a P .Note that this depends only on (f ).If P and Q are points of exponent N ,de?ne f P and f Q as above.Then
e N (P,Q )=
f A (B )/f B (A )
where A ～(P )?(0)and B ～(Q )?(0)and whose support is disjoint.This is independent of the choice of A and B .This yields:
Algorithm 2Calculation of e N pairing Input:
1.An Elliptic Curve E
2.A natural number N
3.Two points P,Q on C of exponent N
Output:
1.The value e N (P,Q )
Method:First ?x an addition chain 1=a 1,...,a t =N .
Now choose T,U points on C at random until T and T +P are distinct from a i U and a i (U +Q )and U and U +Q are distinct from a i T and a i (T +P )for all i .This choice guarantees that the straight-line programs constructed below are de?ned at input values used.Set A =(T +P )?(T )and B =(U +Q )?(Q ).Use 1to construct straight-line programs for the functions f A and f B .Now the number of pairs (T,U )which do not satisfy the two conditions above is ≤8tN p where N p is the total number of points on the curve in GF (p ).Because there always is an addition chain with t ≤2lg N we see that when N ≥256that the probability of success is gt 1/2.We then set
e N (P,Q )=
f A (U +Q )f B (T )
f B (T +P )f A (U )
If one is willing to work a little harder,a deterministic algorithm for the Weil pairing with the same running time as above may be obtained.The main idea is to modify the straight line program to give the value of the function at all points.The program will no longer be straight-line,but will be allowed to have “case”statements.One handles zeroes and poles by representing the value there by a pair (α,n )with the property that α=0and that the Laurent series of the function at the point P starts with αu n where u is a uniformizer at P .Such functions may be multiplied without any problems:one multiplies their α’s and adds their n ’s.If one uses this representation it is possible to dispense with the usual restriction that the divisors A and B above have disjoint supports.
5
One application of the e N pairing is to give a partial answer to a question raised by Schoof[Sch85].In that paper,Schoof gives a determinstic algorithm, polynomial in lg p for calculating the exact order of the group of points on an elliptic curve mod p.He points out that he does not determine the structure of the underlying group.It is known that this group is always either cyclic,or the product of two cyclic groups of orders d and e where d|e.I give a random algorithm which runs in expected time polynomial in lg p if the factorization of gcd(p?1,N)is known,where N is the order computed by Schoof’s algorithm. Algorithm3 1.Calculate N p=N0N1where gcd(N0,N1)=1,and the set of prime divisors of N0is the same as the set of prime divisors of gcd(p?1,N p).
https://www.wendangku.net/doc/157980212.html,ing the Euclidean Algorithm?ndαandβsuch thatαN0+βN1=1.
3.Pick two points P ,Q on the curve C,and set P=βN1P ,Q=βN1Q .
4.Find the exact order of P and Q(this is where we need the factorization).
Say they are m and n.
5.Set r=lcm(m,n).
6.Set a=e r(P,Q)
7.Find the exact order of a.Say it’s s.
8.If rs=N then stop,the orders of the two groups are r and s.
9.If not go to step3.
The analysis of the algorithm depends on the non-degeneracy of e N,and its skew-symmetry.The probability of failure in the last step is O(lg lg?1p). References
[Dav79]James H.Davenport.On the Integration of Algebraic Functions,volume 102of Lecture Notes in Computer Science.Springer-Verlag,Berlin,
1979.
[Ful69]William Fulton.Algebraic Curves.W.A.Benjamin,Inc.,New York, 1969.
[Kal85]Erich https://www.wendangku.net/doc/157980212.html,puting with polynomials given by straight-line pro-grams i;greatest common divisors.In Proc.17th ACM Symp.Theory
Comp.,pages131–142.ACM Press,1985.
[Lan73]Serge Lang.Elliptic Functions.Addison-Wesley,Reading,1973.
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[Sch85]R.Schoof.Elliptic curves over?nite?elds and the computation of square roots mod https://www.wendangku.net/doc/157980212.html,p.,pages483–494,1985.
[Wei46]Andre Weil.Sur les fonctions algebriques a corps de constantes?ni.C.
R.Academie des Sciences,1946.
7
词汇学
名词解释 Extension of meaning, also known as generalization, is the name given to the widening of meaning which some words undergo. It is a process by which a word which originally had a specialized meaning has now become generalized.词义的扩大也被称为普遍化，是指词的意义的扩展。这是一个词的词义从表达某一专门的意义转化到表示概括意义的过程。 Narrowing of meaning, also called specialization, is the opposite of widening 16 meaning. It is a process by which a word of wide meaning acquires a narrower or specialized sense. 词义的缩小也称为词义的具体，与词义的扩大相反。这是意义范围较大的词向意义较小的的方向演变的过程 Elevation or amelioration refers to the process by which words rise from humble beginnings to positions of importance.词义的的升格是指原来表示低微意义的词转而成为表示重要意义的词的词义演变过程。 Degradation or pejoration of meaning is the opposite of semantic elevation. It is a process whereby words of good origin fall into ill reputation or non-affective words come to be used in derogatory sense. 词义的降格与语义的升格相反，这是词从表示褒扬或中性的意义向表示贬义的意义的演变的过程 Blending is the formation of new words by combining parts of two words or a word plus a part of another word. Words formed in this way are called blends or pormanteau words. 拼缀法是通过合并两个词的某些部分或一个词与另外一个词的另一部分相加起来构成新词。以此种方式构成的词被称作为拼缀词或拼缀。 Another common way of making a word is to shorten a longer word by cutting a part off the original and using what remains instead. This is called clipping.-另一个常见的构词法时将原词的一部分截去，只用剩下来的一部分构成新词，这种方法叫截短法。 Acronymy is the process of forming new words by joining the initial letters of names of social and political organizations or special noun phrases and technical terms. Words formed in this way are called initialisms or acronyms, depending on the pronunciation of the words.首字母拼音法是将社会组织、特殊的名词词组或技术术语的首字母连起来构成新词的方法。以这种方法构成的词根根据其读音不同又分别叫做首字母缩略词和首字母拼音词。 Back-formation is considered to be the opposite process of suffixation. Words created through back-formation are mostly verbs. 逆生法被认为是后缀法的队里过程。由逆生法生成的词多半是动词。 Morphemes are abstract units, which are realized in speech by discrete units known as morphs. They are actual spoken, minimal carriers of meaning. The morpheme is to the morph what a phoneme is to a phone. -词素（形位）是抽象的单位，是由分立的形素在话语中具体实现的。“形素”是实际说出来的最小的意义携带体形位与形素的关系同音位与因素的关系一样。 Some morphemes, however, are realized by more than one morph according to their position in a word. Such alternative morphs are known as allomorphs.有些词素根据它们在词中的位置不同可有一个以上的不同形素实现，这些不同的形素叫词素变体。 Derivational affixes. As the term indicates, derivational affixes are affixes added to other morphemes to create new words. Derivational affixes can be further divided into prefixes and suffixes. Prefixes come before the ward and the suffixes after the word.派生词缀。是加在其他词
背词根记单词
利用词根线索记忆单词 A ac do 做，动 ag to do active 行动作用 actual 实际上的 exact 确切的 react 反应 interact 相互作用 agent 代理人，经办人 agency 代办处 agenda 议程 agile 敏捷的 audi hear 听 audible 听得见的 audio 音频的 andio and video 音频和视频 video 视频的 B bi bio life 生命 biology 生物学 biography 传记 biotic 生命的 antibiotic 抗生素 biology n.生物 biological 生物学的联想: ecological 生态学的 metabolic 新陈代谢的 metabolism 新陈代谢 brev short 短 abbreviate 缩写 breviary 短暂 brief 简短的 brief introduction 简要介绍C cap head 头 cap 帽子 captain 上尉 capital 首都；资本 caption 标题，加标题 capture 捕获，俘虏 cas 降fall 落 casual 偶然的 casualty 伤亡 occasion 场合，时机，机会 occasional 偶然 ced ceed
cess go, walk 行走 acces 接近 precede 先行 antecedence 居先的 recession 萧条衰退 retrocede 后退 intercede 调停 ceive ，cept take 拿，取 accept 接受 intercept 截取 incept 摄入 deceive 欺骗,行骗 deception 欺骗 except 除….外,都 receipt 收据 deceive 欺骗 except 除了….之外 receive 得到 reception 接待处 cid cis kill,cut, fall 切割，掉落 accident 事故 incident 事 incise 切割，雕刻 suicide 自杀 concise 简明的 precise 精确的 circ ring 环，圈 circus 马戏团 circle 圆 semicircle 半圆 circulate 循环 circumstances 环境，条件（复数）civ city urb 城市 civil 全民的，市民的 civil war 内战 civilize 使文明，使开化 civilization 文明 uncivilized 未开化的 claim clam cry, shout 哭喊，叫喊 acclaim 喝彩,欢呼 claim 提主张，要求，声称 declaim 演讲 exclaim 呼喊 proclaim 宣布声明 喝彩，称赞
初中英语介词用法归纳总结
初中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”。
英语介词用法详解
英语常用介词用法与辨析 ■表示方位的介词：in, to, on 1. in 表示在某地范围之内。如： 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。如： 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。如： 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”。如： It flies at about 900 kilometers a 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”。如： 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。如： 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。如： This box is made of paper. 这个盒子是纸做的。 2. from成品已看不出原料。如： 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。如： 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(from 。 注意：in指用材料，不用冠词；而with指用工具，要用冠词。请比较：draw in penc il/draw with a pencil。 ■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。如by bus乘公共汽车，by e-mail. 通过电子邮件。
英语词汇构词知识词根篇二十cur,curs,cour,ccurs
英语词汇构词知识词根篇二十cur，curs，cour，ccurs 20、cur，curs，cour，ccurs cur，curs，cour，ccurs=run跑 occur [oc-表示to ，或towards，cur跑；“跑来”→来临] 出现，发生 occurrence [见上，-ence名词后缀] 出现，发生，发生的事 occurrent [见上，-ent形容词后缀，…的] 偶然发生的，正在发生的 current [cur跑→行，-ent形容词后缀，…的] 流行的，通行的，流通的，现行的，当前的，现在的；[-ent 名词后缀] 流，水流，气流 undercurrent [under-底下，current流] 暗流，潜流currency [cur跑→行，-ency名词后缀] 流行，流通，流通货币，通货 excurse [ex-外，出，curs跑→行走；“跑出去”，“出行”] 远足，旅游，旅行 excursion [见上，-ion名词后缀] 远足，旅行，旅游 excursionist [见上，-ist表示人] 远足者，旅游者excursive [ex-外，出，curs跑→行走；“走出”→走
离→走离正题；-ive形容词后缀，…的] 离题的，扯开的excursus [见上，-us名词后缀；“离题”的话] 离题语，附注，附记 course [cours跑→行进] 行程，进程，路程，道路，课程 intercourse [inter-在…之间，cours跑→行走→来往；“彼此之间的来往”] 交往，交际，交流 concourse [con-共同，一起，cours跑；“跑到一起来”] 汇合，集合，合流 courser [cours跑，-er者；“善跑的人或动物”] 跑者，追猎者，猎犬，骏马 courier [cour跑，-ier 表示人，“跑路的人”] 送急件的人，信使 succour [suc-后，随后，cour跑；“随后赶到”] 救助，救援，援助 cursory [surs跑→急行，-ory形容词后缀，…的；“急行奔走的”] 耸促的，草率的，粗略的 cursorial [见上，-ial形容词后缀，…的] （动物）疾走的，善于奔跑的 cursive [curs跑→速行，-ive形容词后缀，…的，“疾行的”，“速走的”] （字体）草写的，行书，草书，草写体incursion [in-内，入内，curs跑→行走，-ion名词
介词by用法归纳-九年级
页脚.
. . 教学过程 一、课堂导入 本堂知识是初中最常见的介词by的一个整理与总结，让学生对这个词的用法有一个系统的认识。页脚.
. . 二、复习预习 复习上一单元的知识点之后，以达到复习的效果。然后给学生一些相关的单选或其他类型题目，再老师没有讲解的情况下，让学生独立思考，给出答案与解释，促进学生发现问题，同时老师也能发现学生的盲点，并能有针对性地进行后面的讲课。 页脚.
. . 三、知识讲解 知识点1： by + v.-ing结构是一个重点，该结构意思是“通过……，以……的方式”，后面常接v.-ing形式，表示“通过某种方式得到某种结果”，即表示行为的方式或手段。 I practice speaking English by joining an English-language club. 我通过加入一个英语语言俱乐部来练习讲英语。 Mr Li makes a living by driving taxis.先生靠开出租车为生。 页脚.
. . 页脚. 介词by + v.-ing 结构常用来回答How do you...?或How can I...?之类的问题。 —How do you learn English? 你怎样学习英语呢？ —I learn English by reading aloud. 我通过大声朗读来学英语。 —How can I turn on the computer? 我怎样才能打开电脑呢？ —By pressing this button. 按这个按钮。 知识点2：by 是个常用介词，其他用法还有： 1【考查点】表示位置，意思是“在……旁边”，“靠近……”，有时可与beside互换。 The girls are playing by (beside) the lake. 女孩们正在湖边玩。 此时要注意它与介词near有所不同，即by 表示的距离更“近”。比较： He lives by the sea. 他住在海滨。 He lives near the sea. 他住在离海不远处。
英文单词记忆法。词根法。
请大家想一想，英语是谁发明的？英国人呗！英国人认不认识汉语？不认识！那么英国人在学英语单词的时候需不需要记住单词的汉语意思？不需要，英国人的英语课本里根本就没有汉字，何谈记住单词的汉语意思？那么既然英国人学英语不需要记住(甚至根本就见不到)单词的汉语意思，那么中国人学英语为什么要去记住单词的汉语意思呢？这种做法大家不觉得奇怪吗？ 然而由于中国人学英语时都在背单词的汉语意思，因此大家反而觉不出“背汉字”有什么奇怪的了。其实仔细想一想，这个行为真的很奇怪，奇怪的根源不在于行为本身，而在于中国人普遍不会直接识别英语单词的意思，因而只好靠汉语符号来机械地帮助记忆英语单词的意思，这样去学英语不仅多此一举，而且必然会陷入苦海无边的符号记忆灾难中。 其实英语单词和汉字一样，存在着很多的“偏旁部首”，知道了偏旁部首你就可以根据它们直接来猜测单词的意思，虽不说百分之百猜准，但起码可以猜测个大概，至少在别人告诉过你单词的意思后你可以恍然大悟地领会它，这样就可以大大增强你对英语单词“见字识意”的能力，做到真正认识一个单词，而把它的汉语意思仅做为一般参考。 举几个例子来说吧： 比如单词representative，请别急着告诉我你认识这个单词，其实你不见得“认识”这个单词，你仅是凭着你的记忆力记住了这串英语字母和两个汉字符号“代表”之间的对应关系，这样去学英语你会多费劲？下面我来告诉你这个单词为什么是“代表”的意思。re在英语里是一个偏旁部首，它是“回来”的意思；pre也是一个偏旁部首，是“向前”的意思；sent也是一个偏旁部首，是“发出去、派出去”的意思；a仅是偏旁部首之间的一个“连接件”，没了它两个辅音字母t就要连在一起了，发音会分不开，会费劲，因此用一个元音字母a隔开一下；tive也是一个偏旁部首，是“人”的意思。那么这几个偏旁部首连在一起是什么意思呢？re-pre-sent-a-tive，就是“回来-向前-派出去-的人”，即“回来征求大家的意见后又被派出去替大家讲话的人”，这不就是“代表”的意思吗！这么去认识一个单词才是真正“认识”了这个单词，把它认识到了骨子里。 再举一个例子吧：psychology。 psy=sci，是一个偏旁部首，是“知道”的意思；cho是一个偏旁部首，是“心”的意思；lo是一个偏旁部首，是“说”的意思；gy是一个偏旁部首，是“学”的意思，logy合起来是“学说”的意思。因此psy-cho-logy连起来就是“知道心的学说”，因此就是“心理学”的意思。 依此类推，不多举例了，我要表达的观点已经清楚了，那就是，不要去死记硬背单词的汉语意思，而要用识别“偏旁部首”的方法去真正认识一个单词，真正认识了单词后，你会发现单词表里的汉语翻译原来其实很勉强，有时甚至根本翻译不出来，因为汉语和英语是两种不同的文字体系，两者在文字上本来就不是一一对应的，只背英语单词的汉字意思是不能真正认识这个单词的，会造成很多的后续学习困难，会造成你一辈子看英语单词如雾里看花，永远有退不掉的陌生感。 那么接下来的问题是，英语里有多少个“偏旁部首”，怎样知道和学会它们？ 回答这个问题时我才发现中国人对英语偏旁部首陌生的两个主要原因，一是这些重要内容不在学校的英语教材当中，大家在课堂上学不到(这是目前学校英语教材急需弥补的缺陷)；二是少数书店里销售的有关这方面内容的书过分复杂化，动辙几百上千页，内容苦涩庞大，影响了这些常识的普及，使得本来是常识的东西不常识。其实英语里偏旁部首的学名叫“字根”，常用的也就二百多个，它们就像26个字母一样普通而重要，就像汉语里的偏旁部首那样普通而重要，它们是学英语第一课里就应该学习的重要内容，学英语者应及早地掌握这些重要的常识，及早地摆脱死记硬背的蛮干状态，及早地进入科学、高效的识字状态。 【英语字根】 1,ag=do,act 做，动 2,agri=field 田地，农田(agri也做agro,agr) 3,ann=year年 4,audi=hear听 5,bell=war战争 6,brev=short短 7,ced,ceed,cess=go行走
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。
【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
词汇学词根讲解
最常用英语词根、词缀 ------ 词根----- ag,act 做，驱动 agent代理人active活动的，积极的 agr 农田 agriculture农业agrarian田地的 am 爱，亲爱 amicable友善的amiable和蔼可亲的 anim 心灵，精神，生命 animal动物animate有生命的 ann，enn 年 annual一年的centennial一世纪的 astro 星 astronomy天文学astronaut宇宙航行员 audi 听 audience听众audible听得见的 bell 战争 rebellion反叛，反抗bellicose好战的 bio 生命，生物 biology生物学antibiotic抗生的,抗生素 brev 短brevity 简洁
1．ag,act 2．agr 3．am 4．anim 5．ann，enn 6．astro 7．audi 8．bell 9．bio 10．brev 1．做，驱动 2．农田 3．爱，亲爱 4．心灵，精神，生命5．年 6．星 7．听 8．战争 9．生命，生物10．短
agent代理人，agrarian田地的，amicable 友善的amiable和蔼可亲的animate有生命的centennial一世纪的astronomy天文学astronaut宇宙航行员audience听众audible听得见的rebellion反叛，反抗bellicose好战的antibiotic抗生的,抗生素brevity简洁 agent，agrarian，amicable，amiable，animate，centennial，astronomy，astronaut，audience，audible，rebellion，bellicose，antibiotic，brevity cede 走 precedent先行的，在前的precede先行 cent 百 centimeter厘米centigrade百分度的 center,centr 中心 concentrate集中centrifugal离心力的 cide,cis 杀，切 suicide自杀bactericide杀菌剂
英语单词常见前后缀词根总结
英语单词常见前后缀词根总结 中国人在学习英语的过程中，感到最困难的就是记忆单词，这在很大程度上是因为我们对英语的构词规律缺乏熟悉和分析。假如我们对英语单词的结构像对汉语的偏旁、部首一样熟悉，记忆英语单词就不会那么枯燥、乏味了。在汉语中，当我们见到"财"、"贩"、"购"、"资"、"货"等字时，从部首"贝"就知道它们都与钱有关。英语单词同样是由词缀和词根构成的，词缀分为前缀和后缀 常见前缀 1. a，an: ①无，不，非astable不稳定的acentric无中心的 ②含有in,on,at,by,with,to等意义asleep在熟睡中ahead向前 2.ab: 脱离abnormal不正常的abaxial离开轴心的 3.anti: 反抗antiwar反战的anti?imperialist反帝的 4.auto:自动，自己autobiography自传autostable自动稳定 5.be: 使…加强belittle使缩小befriend友好相待 6.bi:二，双biweekly双周刊biligual两种语言的 7.co,col,com,con,cor 共同cooperation协作collaboration协作，勾结combine联合correlation相互关系 8.counter,contra: 反，对应counteraction反作用contrast对比,对照 9.di,dif,dis: 否定，相反diffident不自信的dislike不喜欢 10.en,em: 使… enlarge扩大enable使…能empower使…有权力 11.e,ex: 外，出external外部的erupt喷出 12.extra: 以外，超过extraordinary格外的extrasolar太阳系以外的 13.hyper: 在上，超hyperfrequency超高频hypersonic超声的 14.il,im,in,ir: 否定illogical不合逻辑的impossible不可能的invisible 不可见irrational不合理的 15.inter:互相interchange交换interlock连锁 16.intra,intro: 在内，内部intrapersonal个人内心的introspect内省 17.mal: 恶，不良maltreat虐待malfunction机能失常
介词的用法总结归纳
介词的用法总结归纳 一、 In 介词 prep. 1.(表示位置)在…里面; 在, 于; 在…部位上 I could feel the tension in the room.我可以感觉到房间里的紧张气氛。 They live in France.他们住在法国。 2.(表示时间)在…时期, 在…之后, 在过程中 In her twenties and thirties she had had no difficulty getting jobs. 她在二三十岁时找工作一点也不困难。 I cannot see you now, e back in half an hour.我现在不能见你, 半小时后回来。 3.(表示方向)往…内, 朝…方向 I saw him go in the shop.我看到他走进了商店。 4.(表示状态)处于…之中, 在…情况下 Martin was in his pyjamas.马丁穿着睡衣。 They were living in terrible poverty.他们生活在极度贫困之中。 5.(表示方式)用, 以, 按, 乘, 以…形式 They were speaking in Italian.他们在讲意大利语。 They went up in the lift.他们乘电梯上楼了。
6.(表示原因)由于, 为了 He went in fear of his life.他为自己的性命担忧, 所以走了。 7.(表示领域, 范围)在…以内 It is not in my power to do that.做那事非我力所能及。 8.(表示结果)当做, 作为 What did you give him in return?你给他什么作为报答呢? 9.(表示目的)为了 They set off in search of the lost child.他们出发去寻找走失的孩子。 10.[表示职业、活动等]从事于，参加 11.[表示数量、程度、比例]按，以;从…中 12.[表示品质、能力等]在…之中;在…身上 I don't think he had it in him.我认为他没这个本事。 二、Into 介词 prep. 1.(表示时间)持续到, 进行到 The meeting carried on into the afternoon.会议一直延续到下午。 2.(表示方向)进入…中, 到…里 Anney dived into the water.安尼潜入水中。 He came into the room.他到房子里面。 3.(表示状态)进入…状态, 欠…债
《英语词汇学》重要术语中英文对照
《英语词汇学》重要术语 One: 1. Native words 本族词 Words of Anglo-Saxon origin or of Old English are native words. 2. Loan words 借词 Words borrowed from other languages are loan words or borrowed words. 3. Slang words 俚语 Slang words are those words of a vigorous, colourful, facetious, or taboo nature, invented for specific occasions, or uses, or derived from the unconventional use of the standard vocabulary. 4. Function words 功能词 Function words are often short words such as determiners, conjunctions, prepositions, auxiliaries that serve grammatically more than anything else. 5. Content words 实义词 Content words are used to name objects, qualities, actions, processes or states, and have independent lexical meaning. 6. Free forms 自由形式 Forms which occur as sentences are free forms. Two: 1. Morphemes 语素 Morphemes are the smallest meaningful linguistic units of English language, not divisible or analyzable into smaller forms. 2. Allomorphs 语素变体 Allomorphs are any of the variant forms of a morpheme as conditioned by position or adjoining sounds. 3. Free morpheme 自由语素 Free morpheme is one that can be uttered alone with meaning. 4. Bound morpheme 粘着语素 Bound morpheme cannot stand by itself as a complete utterance and must appear with at least one other morpheme, free or bound. 5. Root 词根 Root is the basic unchangeable part of a word and it conveys the main lexical meaning of the word. 6. Affix 词缀 Affix is a collective term for the type of formative that can be used only when added to another morpheme. 7. Inflectional affix 屈折词缀 Inflectional affix serves to express such meanings as plurality, tense, and the comparative or superlative degree. 8. Derivational affix 派生词缀 Derivational affix is the kind of affixes that has specific lexical meaning hand can derive a word when it is added to another morpheme. 9. Prefixes 前缀 Prefixes are affixes added before words.
大学英语四级单词词根词缀
*******前缀********* a ，an ①无，不，非astable不稳定的acentric无中心的 ②含有in,on,at,by,with,to等意义asleep在熟睡中ahead向前 ab 脱离abnormal不正常的abaxial离开轴心的 anti 反抗antiwar反战的anti imperialist反帝的 auto 自动，自己autobiography自传autostable自动稳定 be 使…加强belittle使缩小befriend友好相待 bi 二，双biweekly双周刊biligual两种语言的 co,col,com,con,cor 共同cooperation协作collaboration协作，勾结combine联合correlation相互关系 counter ,contra 反，对应counteraction反作用contrast对比,对照 di ,dif ,dis 否定，相反diffident不自信的dislike不喜欢 en,em 使… enlarge扩大enable使…能empower使…有权力 e ,ex 外，出external外部的erupt喷出 extra 以外，超过extraordinary格外的extrasolar太阳系以外的 hyper 在上，超hyperfrequency超高频hypersonic超声的 il ,im ,in ,ir 否定illogical不合逻辑的impossible不可能的invisible 不可见irrational不合理的 inter 互相interchange交换interlock连锁 intra ,intro 在内，内部intrapersonal个人内心的introspect内省 mal 恶，不良maltreat虐待malfunction机能失常 micro 微microscrope显微镜microware微波 mid 中，中间mid air半空中midstream中流 mini 小minibus小公共汽车ministate小国 mis 错，坏mistake错误misspell拼错 multi 多multiparty多党的multilingual多种语言的 non 否定nonsmoker不抽烟的人nonexistent不存在的 out 超过，过度，外，出，除去 outgo走得比…远outgrow长得太大outdoor户外的outroot除根 over 上，过度overwork工作过度overbridge天桥 post 后postwar战后postnatal诞生后的 pre 前，领先prewar战前的prefix前缀 re 回，再return返回restart重新开始 semi 半semicircle半圆semiconductor半导体 sub ,s uc ,suf ,sup 次，亚，在下，低于 substandard低于标准规格的succeed继承suffix后缀supplement增补 trans 转换，变换，横过，越过transmit传送transatlantic横渡大西洋的 tri 三tricar三轮车triangle三角(形) un 否定unstable不稳定的unknown未知的
初中英语介词用法归纳整理
初中英语介词用法归纳整理 表示时间的介词 at：用于表示时刻，时间的某一点。 on:用于星期，某天，某一天的上午，下午，晚上指具体的某一天时，一律用on in:用于表示周，月，季节，年，泛指上午，下午，晚上 before:在...之前 after:在...之后 by:在....前时间截止到... untiltill：直到.....为止 for：达...之久表示过了多少时间 during：在....期间 through：一直..从开始到结束 from：从...起时间 since:自从...以来表示从以前某时一直到现在仍在继续 in：过...后未来时间 within：不超过...的范围 表示场所，方向的介词 at :在某地点表示比较狭窄的场所 in:在某地表示比较宽敞的场所 on:在...上面，有接触面 above：在...上方 over：在...正上方，是under的反义词 under：在..下面，在...之内 below :在...下方不一定是正下方
near：近的，不远的 by:在...的旁边，比near的距离要近 between：在两者之间 among：在三者或者更多的之中 around：环绕，在...的周围，在....的四周 in front of:在...的前面 behind：在...后边 in：在..之内，用于表示静止的位置 into：进入 out of :和into一样，也表示有一定的运动方向 along：沿着 across：横过平面物体 through：贯通，通过 to :达到..地点目的地或方向 for：表示目的，为了..... from：从...地点起 其他介词 with：和..在一起; 具有，带有; 用某种工具或方法 in：表示用什么材料例如：墨水，铅笔等或用什么语言。表示衣着.声调特点时，不用with而用in。 by:通过...方法，手段 of:属于...的，表示...的数量或种类 from：来自某地，某人，以...起始 without：没有，是with的反义词 like :像...一样