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One-way Functions In Reversible Computations

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预处理子空间迭代法的一些基本概念

预处理子空间迭代法的一些基本概念

CG算法的预处理技术：、为什么要对A进行预处理：其收敛速度依赖于对称正定阵A的特征值分布特征值如何影响收敛性：特征值分布在较小的范围内，从而加速CG的收敛性特征值和特征向量的定义是什么？（见笔记本以及收藏的网页）求解特征值和特征向量的方法：Davidson方法：Davidson 方法是用矩阵( D - θI)- 1( A - θI) 产生子空间，这里D 是A 的对角元所组成的对角矩阵。

θ是由Rayleigh-Ritz 过程所得到的A的近似特征值。

什么是子空间法：Krylov子空间叠代法是用来求解形如Ax=b 的方程，A是一个n*n 的矩阵，当n充分大时，直接计算变得非常困难，而Krylov方法则巧妙地将其变为Kxi+1=Kxi+b-Axi 的迭代形式来求解。

这里的K(来源于作者俄国人Nikolai Krylov姓氏的首字母)是一个构造出来的接近于A的矩阵，而迭代形式的算法的妙处在于，它将复杂问题化简为阶段性的易于计算的子步骤。

如何取正定矩阵Mk为：Span是什么？:设x_(1,)...,x_m∈V ,称它们的线性组合∑_(i=1)^m?〖k_i x_i \|k_i∈K,i=1,2...m〗为向量x_(1,)...,x_m的生成子空间，也称为由x_(1,)...,x_m张成的子空间。

记为L（x_(1,)...,x_m），也可以记为Span（x_(1,)...,x_m）什么是Jacobi迭代法：什么是G_S迭代法：请见PPT《迭代法求解线性方程组》什么是SOR迭代法：什么是收敛速度：什么是可约矩阵与不可约矩阵？：不可约矩阵（irreducible matrix）和可约矩阵（reducible matrix）两个相对的概念。

定义1：对于n 阶方阵A 而言，如果存在一个排列阵P 使得P'AP 为一个分块上三角阵，我们就称矩阵A 是可约的；否则称矩阵A 是不可约的。

定义2：对于n 阶方阵A=(aij) 而言，如果指标集{1，2，...，n} 能够被划分成两个不相交的非空指标集J 和K，使得对任意的j∈J 和任意的k∈K 都有ajk=0, 则称矩阵 A 是可约的；否则称矩阵A 是不可约的。

热力学第二定律

热力学第二定律

第三章热力学第二定律前面，所学的热力学第一律，是以“能量守恒原理”为基础，建立了U和H两个热力学函数，通过对过程ΔU和ΔH的计算，解决了过程的热效应问题。

然而，在一定条件下，一过程能否自动进行，进行到什么程度，亦即，过程的方向和限度问题，第一定律无能为力，这恰恰是第二定律所要解决的问题。

人类经验表明：一切自然界的过程都是有方向性的。

大家都知道：自然界中存在朝一定方向自发进行的过程，例如：热自动从高温物体传向低温物体，直至两物体温度相等；气体自动地从高压区流向低压区，直至各处压力相同，相互接触的不同气体，总是自动的相互混合均匀；电流总是从高电流处流向低电流处直至各处电势相等：浓度不均匀的溶液，自动地变成浓度均匀一致。

等等，这些过程都是可以自动进行的，叫“自发过程”。

显然，一切自然界的过程都是有方向性及一定的进行限度。

从未发现哪一自发过程可自动恢复原状。

为什么自发过程的逆过程不能自动进行？这就是第二定律所要解决的中心问题—判断过程的方向和限度问题。

究竟什么因素决定自发过程的方向和限度？从表面上看，似乎不同的过程，有着不同的决定因素。

如，决定热传导方向和限度的是温度T；决定气体流动的是压力p；决定电流的是电势V；等等。

决定化学反应的是什么？这就要找出：决定一切自发过程方向和限度的共同因素，以此作为判断的共同根据。

寻找一切自发过程方向和限度的判据，这就要研究自发过程的共同特征，根据经验总结热功转化规律，找出反映自发过程本质特征的状态函数—S,以ΔS判断过程的方向和限度。

进而又S据判据在特殊条件下，推演出了A、G状态函数，从而，得到更方便更实用的判据ΔA、ΔG。

§3.1自发变化的共同特征—不可逆性前已述及，一切自发过程都是有方向性的，亦即，自发过程进行之后，系统不能自动恢复原状。

若要让其恢复原状，环境中有什么变化？若让环境也复原，需要什么条件？现举例说明。

1. 理想气体向真空膨胀过程。

这是一个自发过程，当气体向真空膨胀时，Q = 0，W = 0，ΔU=0，ΔT=0。

数值分析英文课件

数值分析英文课件
ˆ ∆y = y − y = 1.4 − 1.41421L ≈ 0.0142
or relative forward error of about 1 percent. Since 1.96 = 1.4 , the absolute backward error is
ˆ ∆x = x − x = 1.96 − 2 = 0.04
Computational error = Truncation error + rounding error
• Propagated （传播） vs. computational error 传播）
– x = exact value, – f = exact function,
ˆ x = approx. value ˆ f = its approximation
Backward vs. forward errors
Suppose we want to compute y = f ( x ) , where f : ℜ → ℜ ˆ but obtain approximate value y
Forward Error:
ˆ ˆ ∆y = y − y = f ( x ) − f ( x )
Example of Ill-Posed Problem
x 1 x 1 x 11 1 + 2 + 3 = 2 3 6 1 1 1 13 x1 + x2 + x3 = 3 4 12 2 1 x1 + 1 x2 + 1 x3 = 47 3 4 5 60
2 significant digits rounding
• Problems that are not well-posed are ill-posed.

可逆操作皇后(FAROQ)密码的对称密码快速算法(IJCNIS-V9-N4-4)

可逆操作皇后(FAROQ)密码的对称密码快速算法(IJCNIS-V9-N4-4)

Prof. Abdul Monem S. Rahma
Computer Science Department, University of Technology-Baghdad, Iraq E-mail: Monem.rahma@
Assist. Prof Abdul Mohssen J. Abdul Hossen
I. J. Computer Network and Information Security, 2017, 4, 29-36
PublishΒιβλιοθήκη d Online April 2017 in MECS (/) DOI: 10.5815/ijcnis.2017.04.04
Computer Science Department, University of Technology-Baghdad, Iraq E-mail: Abdulmoohsen53@ Abstract—In this paper, we have proposed a new iterated symmetric cipher, which is designed with Substitution and Permutation Network (SPN) structure and depends on strong mathematical built. It uses a compact algorithm for encryption and decryption processes, which consists of four main stages that roughly similar in its work to the Advance Encryption Standard (AES) stages. Starting by the SubByte operation, ReversibleShiftrows operation, ReversibleMixcolumns operation, and Round key addition. The proposed operations in this cipher have implemented in a straightforward manner relatively in both Encryption/Decryption by an elegant way. These four stages designed to reduce the hardware requirements and to produces high efficiency, which keeps the encryption and decryption process at the same speed in the hardware devices and eliminates the difference of execution times as well as creates a balance in forward and backward operations. The proposed cipher interested with modern design by adopted new algebraic operations and clear mathematical notations to ensure a high level of security. The proposed cipher did not build suddenly or arbitrarily but it acts as a sequence of developments and represents as a long process of design for long time, since several proposed ciphers have been published recently by us that paved the way to its new design, so the designed cipher inherited a good properties from a proven famous algorithms’ features to produce high resistance against al l known attacks and to submit a high performance on many platforms and in a wide range of hardware and software applications. Index Terms—Block Cipher, Symmetric Cipher, Advance Encryption Standard (AES), Substitution and Permutation Network (SPN), Feistel Structure (FS). aspects and encourage on flexible style and intractable manner with robust structure by selection effective factors of cryptographic primitives for the algorithm built [1]. The National Institute of Standards and Technology (NIST) invited the cryptographers, cryptanalysts academic researchers and data security specialists all over the world to apply their own algorithms in order to select one to be a good alternative for the Data Encryption Standard (DES) and triple-DES, because the DES cipher suffered from many problems, represented by short key that not withstand the advance technology through nonresistance against brute force attacks, and the Triple-DES suffered from slowness with three ciphering key [2]. Many ciphers were originally submitted by designers from several countries. Fifteen algorithms were selected as initial selection, then after deep study and hard analysis processes reduced to the best five algorithms as a finalist candidate in the second selection [3]. The five algorithms were MARS, RC6, RIJNDAEL, SERPENT and TWOFISH which are considered non-superior at all. On October 2nd 2000, NIST announced that the Rijndael algorithm is the best one that has got the higher voting scores. The Rijndael cipher appears to be a very good in both hardware and software implementation through a wide range of platforms [4]. Our proposed model called FAROQ cipher similar in its work to the Rijndael cipher from several aspects and can be pictured as a byteoriented structure that iterated for a number of stable rounds. So the first stage is a Subbyte operation that designed by applying another irreducible equation and a new affine transformation that was chosen from the best selection of what we have gotten. They have been given acceptable results in randomness and non-linearity. The second stage is a ReversibleShiftRow that is based on the same rotate operations in forward and backward operations. The third stage is a ReversibleMixcolumn operation is also based on adoption new linear equation with Maximum Distance Separable (MDS) notation that has self-inverse, which uses a single 4*4 MDS matrix over GF(28)[5]. The ReversibleMixcolumns considers a

信息学奥赛试题精选33题(附带题解)

信息学奥赛试题精选33题(附带题解)

第1~10题为基础题，第11~20题为提高题，第21~33为综合题注：因为在本文档中需要用到一些特殊的数学符号（如：求和号、分数等），所以当您在百度文库中浏览时，一些数学符号可能会显示不出来，不过当您把本文档下载下来在本地浏览时，所有的符号即可全部都显示出来。

^_^基础题：【1 Prime Frequency】【问题描述】给出一个仅包含字母和数字（0-9, A-Z 以及a-z）的字符串，请您计算频率（字符出现的次数），并仅报告哪些字符的频率是素数。

输入：输入的第一行给出一个整数T( 0<T<201)，表示测试用例个数。

后面的T行每行给出一个测试用例：一个字母-数字组成的字符串。

字符串的长度是小于2001的一个正整数。

输出：对输入的每个测试用例输出一行，给出一个输出序列号，然后给出在输入的字符串中频率是素数的字符。

这些字符按字母升序排列。

所谓“字母升序”意谓按ASCII 值升序排列。

如果没有字符的频率是素数，输出“empty”（没有引号）。

注：试题来源：Bangladesh National Computer Programming Contest在线测试：UV A 10789提示先离线计算出[2‥2200]的素数筛u[]。

然后每输入一个测试串，以ASCLL码为下标统计各字符的频率p[]，并按照ASCLL码递增的顺序（0≤i≤299）输出频率为素数的字符（即u [p[i]]=1且ASCLL码值为i的字符）。

若没有频率为素数的字符，则输出失败信息。

【2 Twin Primes】【问题描述】双素数（Twin Primes）是形式为(p, p+2)，术语“双素数”由Paul Stäckel (1892-1919)给出，前几个双素数是(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)。

在本题中请你给出第S对双素数，其中S是输入中给出的整数。

review_material11_12

review_material11_12

Illustration of histogram equalization
No. of pixels
2 4 3 2
3 2 2 4
3 4 3 2
2 3 5 4
6 5 4 3
2
1
4x4 image Gray scale = [0,9]
Gray level
0 1 2 3 4 5 6 7 8 9
histogram
4-component
4-component
Different ways of measuring distance
1. The Euclidean distance between p and q is defined 1 as: 2 2 2
De ( p, q) [( x s) ( y t ) ]
1 F (u, v) MN
M 1 x 0
M 1 N 1 x 0 y 0
f ( x, y) e
f ( x, y)e
j 2vy/N
j 2 ( ux/M vy/N )
can be expressed in the separable form
1 F (u, v) M
1 0
255=>11111111 235=>11101011 188=>10111100 155=>10011011
0
1
0 1
124=>01111100 90 =>01011010
……
Moving window example: find the minimum
Padded image Original image
Storage requirement calculation:

科技英语综合练习复习资料

科技英语综合练习复习资料阅读理解Unit 11.The term “games” in game theory all shares the characteristic of______?A.integrityB.interferenceC.interactionD.interdependence参考答案：D2.According to the passage, recent research of game theory lays its emphasis onA.zero-sum gamesB.pure conflictC.pure cooperationpetition and cooperation参考答案：D3.In simultaneous-move games, the player shouldA.watch how the other players respond and then respond in turnB.be conscious of the other players5 current actionsC.be thinking in a linear chain of reasoning and calculate the possible outcomeD.try to select his best response to what the others do参考答案：D4.In the sentence ''Other games were considered in a cooperative form. That is, the participants were supposed to choose and implement their actions jointly”，the underlined word “implement”means .____A.selectB.fulfill 履行C.diversifyD.quantify参考答案：B5.We can infer from “The Prisoner's Dilemma” of game theory that_____:A.it belongs to the so-called zero-sum games because the interests af the prisoners conflict totallyB.an equilibrium can be reached in repeated plays of the game in order to avoid mutual lossC.the prisoners are ignorant about whether their partner cooperates with them or confesses to the authoritiesD.a precondition has been set that each prisoner^ personal best option will turn to a jointly optimal outcome6.According to the authors, which of the following statements il true?A.Nash’s concept of equilibrium addresses the issues of both sequential-move and simultaneous-move games.B.The pursuit of an equilibrium should end up looking for dominant strategies or eliminating dominated players.C.Nash’s concept of equilibrium doesn’t specify the dynamic process which can result in an equilibrium.D.An equilibrium will ultimately be arrived at in simultaneous-move games.参考答案：C7.To keep the rival guessing or losing, the use of mixing moves can be found in ______A.lumberjack chopping woodB.Cortes conquering Mexico in 16th centuryC.Polaroid securing a monopoly marketD.a tennis player hitting a passing shot一个网球运动员参考答案：D8.In the sentence "When we say that an outcome is an equilibrium, there is no presumption that each person's privately best choice will lead to a collectively optimal result”, the underlined word “presumption” means _____.A.hypothesis 假设B.conclusionC.meditationD.anticipation参考答案：A9.Which of the following ways can enhance the credibility of threats and promise to affect the opponent in a strategic interaction?A.Keeping the rival guessing.ing the strategy of brinkmanship.C.Bargaining to have a better outcome.D.Concealing and revealing information.参考答案：B10.In the sentence “A player can use threats and promises to alter other players’expectations of his future actions, and thereby induce them to take actions favorable to him or deter them from making moves that harm him”，the underlined word “deter”means___A.alterB.remindC.stimulateD.prevent 阻止Unit 441.Cable operators can offer their subscribers services like reserving bandwidth for specific application by using _______specification.A.DOCSISB.PCMMC.TDMD.VoIP参考答案：B42.The PCMM standard creates a framework that allows the_________ to control and change the allocated bandwidth for various services.A.cable modemB.record keeping serverC.application managerD.end-user devices参考答案：D43.In the sentence “The seamless Wi-Fi roaming technology would alert the mobile carrier when a user migrates into a Wi-Fi zone”, the underlined word “migrates” meansA.switchesB.residesC.settlesD.dwells参考答案：A44.Which of the following-is a compelling reason for promoting cellular roaming?A.Saving operating expenditure.B.Expanding cell, phone's markets.C.Increasing entertaining functions.D.Improving overall communications. 通信参考答案：D45.Meshed networks are moving toward the concept "________"A.hot spotB.access pointC.metro spot 地铁站点D. network node参考答案：C46.Which of the following is the characteristic of meshed wireless network?A.Self-searching.B.Self-configuring.C.Self-checking.,D.Self-upgrading.47.In the sentence “A hybrid TDM/IP approach can incorporate IP features in conjunction with an existing TDM-based PBX”,the underlined word "hybrid"A.integratedbinedC.relatedD.divided参考答案：B48.What does the IP Centrex service use to provide all the IP calling features to the enterprise customer?A A context switch and feature server architecture.B A softswitch and management server architecture.C A context switch and management server architecture.D. A softswitch and feature server architecture. 软交换和特征服务参考答案：D49.In the sentence “PON has been heralded as the chosen FTTP path for RBOCs”, the underlined word "heralded" means .A.proclaimed 宣告B.advancedC.guidedD.signaled参考答案：A410.The actual number of active devices in an active network is less than ________ more thain those in a passive network.A.1%B.2%C.10%D.20%参考答案：AUnit 661.What is the main point of the article?A.The laws of thermodynamics will eventually cause computer system performance to plateau.B.Reversible computing is inevitable in the near future.C.Reversible computing will unlock the power o f tomorrow’s computing platforms.D.The principles of reversible computing.参考答案：C62.Reversible computers, or adiabatic systems .A.cause existing technology to falterB.are an alternative computing technique 代替计算技术C.are quantum computerse at a steep price参考答案：B63.The core elements to create a fully functional reversible computer system involveA.reversible circuits, reversible power supply and reversible switchesB.reversible power supply and reversible switchesC.reversible circuits, reversible switches and reversible chipsD.reversible circuits, reversible power supply and reversible chips可逆电路、可逆电源、可逆芯片参考答案：D64.Which of the following states why improvements in computing come at a steep price?A.Conventional computing architectures need to be changed.B.Deleting information inevitably produces heat.C.Hardware designs need to be re-engineered.D.Software and development tools need to be re-designed.参考答案：A65.Researchers have been investigating alternative computing techniques such as _______to stave off the problems of conventional computing performance.A.switchesB.routersC.quantum computing 量子计算D.algorithmic-intensive operations参考答案：C66.In the sentence “To stave off these problems, researchers have been investigating alternative com puting techniques”, the underlined phrase-&stave off’ meansA.give way toB.keep offC.reduceD.solve 解决参考答案：D67.Which of the following lists all the fundamental computing performance bottlenecks?A.Hardware and software designs and leakage.B.Tremendous heat dissipation and hardware and software designs.C.Tremendous heat dissipation and thermal noise.1D.Tremendous heat dissipation, thermal noise and leakage.参考答案：D68.Which of the following statements would the author most probably agree withT^dA.Clock speed needs to be reduced to improve the energy efficiency of computing systems.B.A future where systems based on reversible chips become necessary is out of reach.C.Frank’s power supply sparks circuit nodes, to new levels,D.Knight persisted in adiabatic systems, knowing the technology would be, important soon.参考答案：A69.Which of the following statements supports the idea that “the energy consumption of laptops remains the same”?A.Do not confuse adiabatic methods with today’s power-saying features on laptops.B.The computations of laptops may consume less power, but take twice as long to complete.完成C.Overall energy of laptops is conserved since much of the time they are not running.ptops run the idle loop fast, thus laptops might conserve power.参考答案：B610.It can be implied from the passage that______.A.adiabatic systems are more necessary for general-purpose applicationsB.a future where reversible systems become necessary is inevitableC.reversible chips will not be more efficient for practical applications even if optimizedD.Knight is not in favor of adiabatic research参考答案：BUnti 771.In the sentence “As the enterprise considers VoIP and more significantly VoIP over Wi-Fi, the decision to implement location-aware technology will become amatter of legal compliance”, the underlined word “compliance” means _____.A.flexibilitymand 命令C.issueD.assent参考答案：B72.When a caller dials 911, a selective router switch within the phone network sends the call to a local ______A.base stationB.PSAP 公共安全应答站C.access pointD.PSTN参考答案：B73.What is the percentage of people in the U.S. who will get E911 service?A.About 93 percent.B.About 95 percent.C.About 88 percent. 88D.About 79 percent.参考答案：C74.Wireless technologies and VoIP present unique problems for E911 because__________.A.VoIP phone users are usually able to roam in a large areaB.VoIP phones only use base stations to connect to the InternetC.VoIP phones can only work across Wi-Fi networkD.VoIP phones’ area codes are tied to its’ geographic location参考答案：A75.Which of the following is included in the deployment of 911 across cellular networks?A.Phase 0 ensures that E911 calls will get through to a PSAP.B.Phase 1 specifies that VoIP 911 calls will be routed to the local PSAP.C.Phase 2 will bring full E911 services to VoIP across the PSTN.D.Phase 3 will deliver E911 over IP to devices without telephone numbers.参考答案：A76.In the sentence “It could then offset those costs by passing them on to the application service provider, which could in turn charge the end user”, the underlined word “offset” means_______.A.counteractB.softenC.decreaseD.slash参考答案：A77.Which of the following is the advantage of A-GPS over GPS?A.The search time is reduced by 10 to 12 seconds.B.It takes several minutes to get the location information.C.It saves precious battery life for mobile phone users.D.It works equally well both inside and outside buildings.参考答案：C78.Wi-Fi uses_________.A.licensed bandsB.unlicensed bandsC.free bandsD.limited bands参考答案：B79.Which of the following is the function of triangulation?A.To measure the returning signal’s attenuation.B.To emit test signals into the environment.C.To compare multiple signal strengths.D.To account for multiple propagation.参考答案：C710.Which of the following examples shows that location-aware presence improves Wi-Fi security?A.It can locate interlopers penetrating the defenses of the Wi-Fi network.B.Within the Wi-Fi environment, consumers can be directed to restaurants.C.Shipping companies can now track the movement of their vans.D.In hospitals doctors and medical equipment can be quickly located.参考答案：A完形填空Unit 41. A.profits B. handicaps C. benefits促进 D. hindrances2. A. while B. when C. however D. though3. A. everywhere B. anywhere C. however D. though4. A. on B. with C. by D. over5. A. accede B. accelerate促进 C. access D. accept6. A. exceeds超越 B. precedes C. antecedes D. cedes7. A. decline B. arise C. grow种植 D. descend8. A. converting B. converging合并 C. convecting D. convolving9. A. interact B. interaction C. interacting D. interactive互动的10. A. coupling B. couple C. coupled连结的 D. being coupled11. A. potential潜能B. latency C. efficiency D. specialty12. A. but B. however C. therefore D. hence13. A. That B. It C. Which D. What14. A. constitutes构成B. consists C. comprises D. contains15. A. minimum最低限度的B. maximum C. medium D. average16. A. upon B. for C. against反对 D. beneath17. A. nevertheless B. though C. even平的 D. nonetheless18. A. approach B. access接入C. avail D. admission19. A. uncertainty不确定的B. probability C. certainty D. likelihood20. A. survive B. evolve使发展 C. revolve D. extendUnit61. A. on B. to C. for D. at2. A. award B. reward C. awards D. rewards报酬3. A. though B. so C. if D. worse4. A. punishment惩罚B. punish C. information D. judge5. A. detects B. detected C. is detected D. be detected6. A. punished B. unpunished C. punishing D.unpunishing7. A. say B. note记事本 C. understand D. calculate8. A. by accident偶然地B.by instinct C. by intention D. by all accounts9. A. proof B. evidence C. examine D. inspections检查10.A. surrenders B. reasons C. victims受害者D. causes11.A. having to stay B. having stayed C. staying D. of having to stay12.A. cleared B. revealed显示 C. victims D. causes13.A. lying B. untruthful C. dishonest不诚实的D. unhonest14.A. Unlike B. Like C. With D. For15.A. take B. commit犯罪 C. submit D. perform16.A. be charged B. charged C. charge D. charging17.A. depend B. command C. recommend D. demands18.A. exist B. result C. obtain D. gain19.A. hesitate犹豫 B. refuse C. stop D. lost20.A. confident B. private C. confidential机密的D. valuable句子翻译Unit 11.The essence of a game is the interdependence of player strategies.There are two distinct types of strategic interdependence:sequential and simultaneous.博弈的实质是博弈者采取策略之间的相互依赖性。

自动机理论 语言和计算导论课后习题答案

证明：通过对|y|进行归纳，来证明δˆ (q , xy)=δˆ (δˆ (q , x) , y) ，具体过程如下：
Basis: If y = ε, then the statement is δ-hat(q,x) = δ-hat(δ-hat(q,x),ε). This statement follows from the basis in the definition of δ-hat. Note that in applying this definition, we must treat δ-hat(q,x) as if it were just a state, say p. Then, the statement to be proved is p = δ-hat(p,ε), which is easy to recognize as the basis in the definition of δ-hat.
01 ->s d q1 *q0 q0 q1 q1 q2 q3 q2 q4 q0 q3 q1 q2 q4 q3 q4
dd d
Exercise 2.2.9
Part (a) is an easy induction on the length of w, starting at length 1.
Basis: |w| = 1. Then δ-hat(q0,w) = δ-hat(qf,w), because w is a single symbol, and δ-hat agrees with δ on single symbols.
Definition of δ-hat
δˆ 的定义
δˆ (q,xy)
y=za
Exercise 2.2.4(a)

《科技英语阅读》课后名词解释和翻译

Unit1 mathematics名词解释绝对补集absolute complement / 代数algebra /代数式algebraic expression / 代数方程algebraic equation / 代数不等式algebraic inequality / 任意常数arbitrary constant / 数组array / 底数；基数base number / 连续函数continuous function / 函数function / 复合函数function of function / 函数记号functional notation / 集合aggregate / 子集subset /迭代函数iterative function/优先权之争priority battle/分形特征fractal properties/有意义make sense/以越来越小的规模重复同一模式patterns repeat themselves at smaller and smaller scales/混沌理论chaos theory/季刊a quarterly journal/数学界the mathematics community/波纹线crisp lines/会议公报proceedings of a conference翻译3. Translate the sentences into Chinese.1)他主要是因为用分形这个概念来描述（海岸线、雪花、山脉和树木）等不规则形状等现象而闻名于世，这些不规则形状在越来越小的规模上不断重复同一模式。

2)如果再仔细观察，就可以发现集的边界并没有呈波纹线，而是像火焰一样闪光。

3)但是，克朗兹在这场辩论中引入了一个新东西，他说曼德布洛特集不是曼德布洛特集发明的，而是早在“曼德布洛特集”这个术语出现几年以前就已经明确地在数学文献中出现了。

4)曼德布洛特同时也暗示即使布鲁克斯和马特尔斯基的论文先于他发表，但因为他们没有领会到其价值，仍然不能将他们看作是曼德布洛特集的发现者。

PhysRevA.87.042115

Instituto de F´ ısica, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Gragoat´ a, 24210-346, Niter´ oi, RJ, Brazil (Received 20 March 2013; published 22 April 2013) Geometric quantum discord is a well-defined measure of quantum correlation if Schatten one-norm (trace norm) is adopted as a distance measure. Here, we analytically investigate the dynamical behavior of the one-norm geometric quantum discord under the effect of decoherence. By starting from arbitrary Bell-diagonal mixed states under Markovian local noise, we provide the decays of the quantum correlation as a function of the decoherence parameters. In particular, we show that the one-norm geometric discord exhibits the possibility of double sudden changes and freezing behavior during its evolution. For nontrivial Bell-diagonal states under simple Markovian channels, these are new features that are in contrast with the Schatten two-norm (Hilbert-Schmidt) geometric discord. The necessary and sufficient conditions for double sudden changes as well as their exact locations in terms of decoherence probabilities are provided. Moreover, we illustrate our results by investigating decoherence in quantum spin chains in the thermodynamic limit. DOI: 10.1103/PhysRevA.87.042115 PACS number(s): 03.65.Ud, 03.67.Mn, 75.10.Jm
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