A note

英语美文翻译(A note to my mom)

A love note to my momWhen i was a little girl,i wound often accompany you as you modeled for fashion photographers .It was years later that I finally understood what role modeling play in your life.Little did i know you were saving every penny you earned to go to law school.I cannot thank you enough for what you told me one autumn afternoon when i was nine.After finishing my homework, i wandered into the dining where you were buried under piles of law books.I was puzzled. Why were you doing what i do ---memorizing textbooks and studying for tests?When you said you were in law school, i was more puzzled.I did not know mom can be lawyers ,too.You smiled and said …In life ,you can do anything you want to do.”As young as i was,that statement kept in my ears. I watched as you faced the challenges of completing your studies,starting company with dad,while still being a model and a mom of five kids.I was exhausted just watching you i action.With your words of wisdom in my young mind,I suddenly felt unlimited freedom to dream.My whole world opened up.I set out to live my life filled with hope,i seeing endless possibilities for personal and professional achievements.Your words became my motto.I constantly found myself in the unique position of being either the first or one of the few women in my field.I gained strength every time i said “Yes,i will try.”Encouraged by your words,i have forged ahead with my life‟s journey,less afraid to mistakes,and eagerly meeting each challenge.You did it,now i am doing it.Sorry,got to run.So much to do,so many dreams to live.翻译：给妈妈的一封感谢信那是，我还是一个小女孩，时常陪着作为时尚模特的你。

A Note of Thanks的常用句型

A Note of ThanksUseful Expressions表达感谢I am writing to extend my thanks for your hospitality.I am writing to extend my sincere gratitude for your timely assistance.I am writing to extend my appreciation of your outstanding serviceI am writing to express my appreciation of your diligence in doing such a superb job. Would you like to extend my appreciation of your outstanding service ? You certainly deserve the credit.Would you like to convey my gratitude to your loyalty and devotion ?Thank you for your hospitality . I enjoyed every moment.Thanks for your excellent proposal on….Thank you for the kindness you have shown….How can I thank you enough for remembering my graduation ?I would like to convey in this letter my heartfelt thanks to you for your hospitality. Please convey our sincere thanks to Mr. Xxx, your insurance instructor who gives us a hand when our apartment flooded in May.My sincere thanks go to all of you for making such a great effort to….I am most grateful for your selfless donation.I really appreciate your taking the time to do so much for me.I especially appreciated your organizing the trip according to the agenda. Your efforts were an extraordinary accomplishment.I feel deeply indebted to you and I rea lly don’t know how to thank you enough for your help.How very sweet of you to send Clark and me such a beautiful gift.结束句Thanks again for helping us at this critical time.Thanks again for continuing your excellent service.Lots of thanks and love f or ….I am sure there are hundreds of people who would like to join me in saying "thank you" for the terrific job.We are grateful to the superb job you have done.I feel obliged to thank you once more.My true gratitude is beyond the word’s description.Please accept my gratitude, now and always.We are happy with superb job you have done.We are fortunate to have you with us.We are completely satisfied with your timely concern..We are delighted with the manner in which you played you part.8Expressing Thanks11.1Write a note of about 50-60 words based on the following situation:You have been hospitalized for two weeks. During the time your good friend, Jean, visited you everyday. Now write a note expressing your thanks to her for what she has done.Feb.15 Dear Jean,Many thanks for your everyday visit which has made me forget the sharp pains that the operation brought to me. What I am going to say that your daily visit, which to me, serves as pain relief. The thing that bothers me is that you are usually very busy so don’t see me everyday. I am feeling much better now. Thanks again.Sincerely yours,Mary11.2Write a note of about 50-60 words based on the following situation:Your friend, Tim, recommended you as a computer programmer. By his recommendation, you got the position. Now write a note expressing your thanks to him.April 10 Dear Tim,I am sincerely grateful to you for recommending me as a computer programmer at mypresent company. It was hard for me to get the position without your active recommendation. So I will do everything possible to work dutifully and try to be worthy of your recommendation. Hope to repay you for your kindness some day.Yours very truly,Sam11.3Write a note of about 50-60 words based on the following situation:You received the “red package” money gifts from your aunty during the Spring Festival. Now write a note thanking her for the gifts and saying what you intend to do with it.Feb.2 Dear aunty,Thanks for giving me the “red package” money gifts. These days I am considering how to do with all the “red package” money gifts from you and some other relatives. Now I decide to use the money to make a donation to a nationwide “Project Hope”to help the dropouts to go back to school. I think you will support me.Sincerely yours,Bob11.4Write a note of about 50-60 words based on the following situation:You are planning to return a magazine which you borrowed from your friend, Xiao-fang. Now write a note thanking her for lending it to you and telling her how much you enjoyed it.Oct.2 Dear Mary,Thank you for lending me the magazine, “Studio Classroom”(May 2004). I have been particularly touched by a passage, Giving Hope to China’s “Children of the Stars”, which says that one woman’s efforts to help her own son expanded into China’ first school for autistic children. Reading Studio Classroom has benefited me a lot. So I plan to buy it once a month.Yours truly,Linda11.5Write a note of about 50-60 words based on the following situation:You have just returned to Shanghai after an enjoyable visit to your cousin in Beijing. Write a note thanking her for her hospitality and telling her what you enjoyed most while you were in Beijing during the seven-day week long holidays.Oct. 7 Dear Lily,I am extremely grateful to you for your hospitality when I was in Beijing. Every day you drove me to some scenic spots. Every day you cook some wonderful meals for me. What I enjoyed most during the week were the Great Wall and the noodles with the sweet soybean paste you cooked. Thanks again.Yours,Dora11.6Write a note of about 50-60 words based on the following situation:Your friend, Mary, lent you a book which was helpful for you to win the first prize at the English Writing Contest. Now write a note expressing thanks.May 9 Dear Mary,Words are beyond me to express my thanks to you for lending me the book ENGLISH WRITING TECHNIQUES which was so much helpful for me to win the first prize at the English Writing Contest. You must be very annoyed with me for not writing earlier, but you are the first one to learn the good news because you are my best friend. support.Yours truly,。

Note-Writing英语专业四级便条写作

Marks will be awarded for content organization, language and appropriateness
2012
April 22，2012 Dear Lucy, I’m writing to express my appreciation for all that you did for me yesterday. It was so kind of you to meet me at the railway station when I came back from hometown. What’s more, your help in cleaning my dorm really saved me from getting exhausted. I am so lucky to have such a warmhearted friend. I will be willing to reciprocate your kindness if you need any help in the future. Yours, Juliet
评分构成
分值
要求右上角， April 17th, 2011 17 April, 2011 Apr(.) 17, 2011 17/4/2011 4/17/2011 April 17th Apr(.)17
扣分标准
日期
格式 3
称呼
在日期下一行左侧， Dear Jimmy, Dear Jimmy: Jimmy, Jimmy: 右下方， Yours sincerely, Sincerely yours, Yours,
Write on ANSWER SHEET THREE a note of about 50-60 words based on the following situation: Marks will be awarded for content, organization, language and appropriateness.

A note on negligible functions

快速治口臭的最好方法口臭是一种常见的口腔健康问题，给人们的生活和社交带来了很多困扰。

许多人都希望能够找到一种快速有效的方法来治疗口臭，以恢复自信和舒适感。

下面将介绍一些治疗口臭的最好方法，希望对您有所帮助。

首先，保持良好的口腔卫生习惯是治疗口臭的基础。

定期刷牙、使用牙线和漱口水是非常重要的。

刷牙可以有效清除口腔中的食物残渣和细菌，使用牙线可以清洁牙齿间的难以触及的部位，漱口水可以杀灭口腔中的细菌并保持口气清新。

此外，定期洁牙也是保持口腔清洁的重要手段，可以有效预防口臭的产生。

其次，注意饮食习惯对口臭的影响也是非常重要的。

食物中的大蒜、洋葱等会在消化后释放出刺激性气味物质，而过多的咖啡因和酒精也会对口气产生不良影响。

因此，适当控制这些食物的摄入量，多吃一些清新口气的食物，如苹果、胡萝卜等，对改善口臭有很大帮助。

另外，定期去口腔科检查也是治疗口臭的重要手段。

口腔科医生可以帮助您检查口腔健康状况，及时发现口腔问题并进行治疗。

有些口腔疾病，如牙龈炎、牙周炎等，如果不及时治疗，会导致口腔异味，甚至影响全身健康。

此外，定期饮用温水也是改善口臭的好方法。

饮用温水可以有效促进口腔血液循环，清洗口腔细菌，并且可以帮助口腔黏膜保持湿润，减少口腔异味的产生。

最后，保持良好的生活习惯也是治疗口臭的关键。

戒烟限酒、保持充足的睡眠、减少压力等都对口臭的改善有积极作用。

因为吸烟和酗酒会导致口腔内细菌增加，口腔黏膜受损，从而产生口臭。

而长期的睡眠不足和过度的压力也会影响口腔健康，导致口臭问题加重。

总之，治疗口臭需要综合考虑口腔卫生、饮食习惯、口腔健康和生活习惯等多个方面的因素。

希望以上方法对您治疗口臭有所帮助，祝您拥有清新自信的口气！。

note作文英语模板

note作文英语模板英文回答：What is a Note?A note is a brief written record of information, often used to help memory or communicate ideas. Notes can be written for a variety of purposes, such as:Recording information: Notes can be used to record information from lectures, meetings, books, or other sources.Generating ideas: Notes can be used to generate new ideas, explore different perspectives, and develop strategies.Planning and organizing: Notes can be used to plan and organize projects, events, or tasks.Problem-solving: Notes can be used to identify and solve problems.Communication: Notes can be used to communicate ideas, information, or instructions to others.How to Take Effective Notes。

There are a few key principles to follow when taking effective notes:Be selective: Don't try to write down everything you hear or read. Focus on the most important information.Be organized: Use headings, subheadings, and bullet points to organize your notes. This will make them easier to read and understand later.Use abbreviations and symbols: Use abbreviations and symbols to save time and space. However, make sure you understand what they mean later.Review your notes: Review your notes regularly to reinforce the information and improve your understanding.Different Types of Notes。

英文note的汉语是什么意思

英文note的汉语是什么意思英文单词note的汉语意思英 [nəʊt] 美 [noʊt]名词笔记; 音符; 便笺; 钞票及物动词注意; 记录; 指出; 对…加注释相关词组1. compare notes : 交换意见;2. of note : 著名的;3. take note of : 注意到;4. strike the right note : 说得恰当;英文单词note的词典解释1. 强调指出;特别提到If you note something, you mention it in order to draw people's attention to it.e.g. The report notes that export and import volumes picked up in leading economies...报告特别指出经济大国的.进出口量攀升。

e.g. The yearbook also noted a sharp drop in reported cases of sexually transmitted disease.年鉴同时指出，性传播疾病报告病例数显著下降。

2. 记录;记下When you note something, you write it down as a record of what has happened.e.g. 'He has had his tonsils out and has been ill, too,' she noted in her diary...“他切除了扁桃体，一直病着，”她在日记里写道。

e.g. One policeman was clearly visible noting the number plates of passing cars...能清楚地看到一名警察记下了经过车辆的车牌号。

3. see also: noted;promissory note;sleeve note4. 交换意见;交流情况If you compare notes with someone on a particular subject, you talk to them and find out whether their opinion, information, or experience is the same as yours.e.g. The women were busily comparing notes on the queen's outfit...女人们忙着交换对女王衣着的看法。

【英语作文】一个便条 A Note

【英语作文】一个便条 A Note
亲爱的妈妈，
我知道你每天都很忙，所以我写了这个便条，希望你看到后能够知道我的心意。

我想对你说声抱歉。

我知道这段时间你工作很辛苦，家里的事情也很多，可是我经常会让你操心。

我会尽量做到让你少操心一点，让你放心一些。

我要感谢你。

感谢你每天为我做的一切，感谢你无私的付出。

我知道你为了我付出了很多，而我却很少表达对你的感激之情。

所以我想在这个便条里告诉你，我真的很感激你。

我希望你能够放松一下。

工作虽然很重要，但是我的妈妈更重要。

我希望你能够在工作之余多花一些时间和我在一起，多和我交流，多陪陪我。

妈妈，我爱你。

希望你能够收到这个便条，读到我的心意。

每天辛苦了，也记得照顾好自己。

爱你的女儿收到者：妈妈。

note的用法

note的用法1. Note作为名词的用法：- A written or printed piece of information or documentation, usually brief and informal. (一条书面或印刷的信息或文档，通常简短而非正式)例句：I wrote a note to remind myself to buy milk. (我写了一个便条提醒自己要买牛奶。

)2. Note作为动词的用法：- To notice or pay attention to something. (注意到或关注某物)例句：Please note that the meeting has been rescheduled to next Monday. (请注意会议已经改期到下周一。

)- To mention or make a comment about something. (提及或评论某事)例句：I'd like to note that our sales have significantly increased this quarter. (我想提到我们本季度的销售额显著增加。

)3. Note作为形容词的用法：- Used to describe the type or quality of a sound or musical tone. (用于描述声音或音乐音调的类型或质量)例句：He played the piano with a beautiful, melodic note. (他用美妙的旋律演奏了钢琴。

)4. Note作为动词的用法：- To make a short written record or summary of something. (做一份关于某事情的简短书面记录或摘要)例句：Can you please note down the main points of the presentation? (你能否记下演示的要点？)。

便条用英语怎么说

便条用英语怎么说便条指日常生活中，如果我们有什么事情要告诉另一方，或委托他人办什么事，在不面谈的情况下书写的一种条据。

内容简单大多是临时性的询问、留言、通知、要求、请示等，往往只用一两句话。

那么你知道便条用英语怎么说吗?下面来学习一下吧。

便条的英语说法1：note便条的英语说法2：memo便条的相关短语：便条本 note pad ; Writing pads一张便条 A Note通知便条 Notice留言便条 Message Writing便条纸盒 NotepaperHolder道歉便条 Note of Apology便条纸 Note Paper ;便条的英语例句：1. She scribbled a note to tell Mum she'dgone out.她匆匆写了个便条告诉妈妈她已外出。

2. She had written him a note a couple of weeks earlier.她几周前给他写过一个便条。

3. He gave the girl at the desk the message.他把便条递给了桌旁的女孩。

4. Stevens wrote him a note asking him to come to his apartment.史蒂文斯给他写了张便条，请他去自己的公寓。

5. Before she ran away, she left a note explaining her actions.逃跑前，她留下一张便条解释原委。

6. There was a note. Well, not really a note.那儿有张便条，噢，其实不算是便条。

7. He has written a note asking me to pass on his thanks. 他写了张便条让我转达他的谢意。

8. I sellotaped the note to his door.我用透明胶带把便条贴在他的门上。

ANOTE说明书简介

现代教育与心理测量通用分析系统简介《现代教育与心理测量通用分析系统》，英文名为ANALYSIS 0F TEST，简记为ANOTE。

该系统有18项测验编制和分析功能，包括经典测验理论的所有常规分析、经典测验理论的等组设计和非等组设计下的各种线性和非线性等值、项目反应理论单、双、叁参数l0gistic摸型、Samejima单、双参数等级反应模型项目参数和能力参数估计、各种模型下的项目参数等值、模型-数据资料拟合检验、项目和测验信息函数分析计算。

具体各项功能简叙如下：1、经典测验理论项目质量分析在经典测验理论框架下分析试题的难度、区分度，作出试题质量评价，绘制题分、总分回归图与难度、区分度散点图。

2、经典测验理论被试质量分析在经典测验理论框架下分析被试的分数分布、整体水平、离散程度、个体相对位置、构建常模表。

3、经典测验理论测验质量分析：在经典测验理论框架下分析计算测验试卷的多种信度与效度。

4、经典测验理论等组设计线性等值在等被试组设计下采用线性等值法完成两不同测验上被试得分的等值转换。

5、经典测验理论等组设计等百分位等值在等被试组设计下采用等百分位等值完成两不同测验上被试得分的等值。

6、经典测验理论Tucker线性等值在铆测验非等组设计下采用Tucker线性等值计算方法完成不同测验分数间的等值。

7、经典测验理论频数估计等百分位等值在铆测验非等组设计下采用先估计联合频数分布，再用等百分位方法完成测验分数间的等值计算。

8、单参数模型参数的估计与模型资料拟合检验在项目反应理论单参数L0gistic模型下，应用EM算法完成1，0记分题难度参数和被试能力参数的估计，并作模型资料拟合检验。

9、双参数模型参数估计与模型资料拟合检验在项目反应理论双参数L0gistic模型下，应用EM算法完成1，0记分题难度与区分度参数和被试能力参数的估计，并作模型资料拟合检验。

10、叁参数模型参数估计与模型资料拟合检验在项目反应理论叁参数L0gistic模型下，应用EM算法完成1，0记分题难度、区分度、猜测度和被试能力参数的估计，并作模型资料拟合检验。

英语note的中文意思是什么

英语note的中文意思是什么英语note的中文意思是什么由单词note构成的单词有很多，因此我们必须要先知道note的中文意思。

下文是店铺为大家准备的英语单词note的中文意思的内容，希望能对大家有所帮助!英语单词note的中文意思名词笔记; 音符; 便笺; 钞票及物动词注意; 记录; 指出; 对…加注释1. 音符In music, a note is the sound of a particular pitch, or a written symbol representing this sound.e.g. She has a deep voice and doesn't even try for the high notes...她嗓音低沉，从没试过高音。

e.g. If the note of D is sounded on a harp, all the corresponding D strings of other octaves will likewise resonate.如果用竖琴演奏D音符，八度音阶所有相应的D弦都将发出共鸣。

2. (说话的)声调，语气You can use note to refer to a particular quality in someone's voice that shows how they are feeling.e.g. There is an unmistakable note of nostalgia in his voice when he looks back on the early years of the family business...他回忆早些年的家族生意时，无疑是带着一种怀旧的语气的。

e.g. It was not difficult for him to catch the note of bitterness in my voice.他不难听出我话语中的愤愤不平。

A Note on Quantum Hamming Bound

a r X i v :0711.4603v 1 [q u a n t -p h ] 29 N o v 2007A Note on Quantum Hamming BoundSalah A.AlyDepartment of Computer Science,Texas A&M University,College Station,TX 77843-3112,USAEmail:salah@Proving the quantum Hamming bound for degenerate nonbinary stabilizer codes has been an open problem for a decade.In this note,I prove this bound for double error-correcting degenerate stabilizer codes.Also,I compute the maximum length of single and double error-correcting MDS stabilizer codes over ﬁnite ﬁelds.1Bounds on Quantum CodesQuantum stabilizer codes are a known class of quantum codes that can protect quantum information against noise and decoherence.Stabilizer codes can be constructed from self-orthogonal or dual-containing classical codes,see for example [3,8,11]and references therein.It is desirable to study upper and lower bounds on the minimum distance of classical and quantum codes,so the computer search on the code parameters can be minimized.It is a well-known fact that Singleton and Hamming bounds hold for classical codes [10].Also,upper and lower bounds on the achievable minimum distance of quantum stabilizer codes are needed.Perhaps the simplest upper bound is the quantum Singleton bound,also known as the Knill-Laﬂamme bound.The binary version of the quantum Singleton bound was ﬁrst proved by Knill and Laﬂamme in [12],see also [1,2],and later generalized by Rains using weight enumerators in [16].Theorem 1(Quantum Singleton Bound).An ((n,K,d ))q stabilizer code with K >1satisﬁesK ≤q n −2d +2.(1)Codes which meet the quantum Singleton bound with equality are called quantum MDS codes.If we assume that K =q k ,then this bound can be stated as k ≤n −2d +2.In [11]It has been shown that these codes cannot be indeﬁnitely long and showed that the maximal length of a q -ary quantum MDS codes is upper bounded by 2q 2−2.This could probably be tightened to q 2+2.It would be interesting to ﬁnd quantum MDS codes of length greater than q 2+2since it would disprove the MDS conjecture for classical codes [10].A related open question is regarding the construction of codes with lengths between q and q 2−1.At the moment there are no analytical methods for constructing a quantum MDS code of arbitrary length in this range (see [9]for some numerical results).Another important bound for quantum codes is the quantum Hamming bound.The quantum Hamming bound states (see [6,7])that:1Theorem2(Quantum Hamming Bound).Any pure((n,K,d))q stabilizer code satisﬁes⌊(d−1)/2⌋i=0 n i (q2−1)i≤q n/K.(2)The previous quantum Hamming bound holds only for nondegenerate(pure)quantum codes. However,the degenerate(impure)quantum codes are particularly interesting class of quantum codes because they can pack more quantum information.In addition,the errors of small weights do not need active error correction strategies.So far no degenerate quantum code has been found that beats this bound.Gottesman showed that impure single and double error-correcting binary quantum codes cannot beat the quantum Hamming bound[8].It is proved in[11]that Hamming bound holds for quantum stabilizer codes with distance d=3.In general,does Hamming bound exist for any distance d in((n,K,d))q stabilizer codes?This has been an open question for a decade.In this note we prove Hamming bound for double error-correcting stabilizer codes with distance d=5and also give a sketch to prove it for general distance d.2Quantum Hamming Bound Holds for Distance d=5There have been several approaches to prove bounds on the quantum code parameters.In[1] Ashikhmin and Litsyn derived many bounds for quantum codes by extending a novel method originally introduced by Delsarte[5]for classical ing this method they proved the binary versions of Theorems1,2.We use this method to show that the Hamming bound holds for all double error-correcting quantum codes.See[11]for a similar result for single error-correcting codes.But ﬁrst we need Theorem3and the Krawtchouk polynomial of degree j in the variable x,K j(x)=js=0(−1)s(q2−1)j−s x s n−x j−s .(3)Theorem 3.Let Q be an((n,K,d))q stabilizer code of dimension K>1.Suppose that S is a nonempty subset of{0,...,d−1}and N={0,...,n}.Letf(x)=ni=0f i K i(x)(4)be a polynomial satisfying the conditionsi)f x>0for all x in S,and f x≥0otherwise; ii)f(x)≤0for all x in N\S.ThenK≤1f x.(5) 2Proof.See[11].We demonstrate usefulness of the previous Theorem by showing that quantum Hamming bound holds for impure codes when d=5.Lemma4(Quantum Hamming Bound).An((n,K,5))q stabilizer code with K>1satisﬁesK≤q n n(n−1)(q2−1)2/2+n(q2−1)+1 .(6) Proof.Let f(x)= n j=0f j K j(x),where f x=( e j=0K j(x))2,S={0,1,...,4}and N={0,1,...,n}. Calculating f(x)and f x gives usf0=(1+n(q2−1)+n(n−1)(q2−1)2/2)2f1=12(n−3)(n−2)(q2−1)2−(n−2)(q2−1))2f3=(1−2(n−3)(q2−1)+12(n−5)(n−4)(q2−1)2)2and,f(0)=q2n(1+n(q2−1)+1f x(7)So,there are four diﬀerent comparisons where f(0)/f0≥f(x)/f x,for x=1,2,3,4.Weﬁnd a lower bound for n that holds for all values of q.From Lemmas5,6,7,and8,shown below,for n≥7 it follows thatmax{f(0)/f0,f(1)/f1,f(2)/f2,f(3)/f3,f(4)/f4}=f(0)/f0(8)While the above method is a general method to prove Hamming bound for impure quantum codes,the number of terms increases with a large minimum distance.It becomes diﬃcult toﬁnd the true bound using this method.However,one can derive more consequences from Theorem3; see,for instance,[1,2,13,15].3Lemma5.The inequality f(0)/f0≥f(1)/f1holds for n≥6and q≥2.Proof.Let f(0)/f0≥f(1)/f1then1(n−1)2(n−2)2(q2−1)4(n−1)2(n−2)2(q2−1)4≥(1+n(q2−1)n(n−1)+(q2−1)2)(q2(q2−1)(n−1)+1)2divide both sides by(q2−1)2(q2−1)2and approximate1)(n−1)2by approximating both sides,theﬁnal result is(n−2)≥4orn≥6Lemma6.The inequality f(0)/f0≥f(2)/f2holds for n≥7and q≥2.Proof.Letq2n(−(n−2)(q2−1)+(n−3)(n−2)(q2−1)2/2)2 by simplifying both sides(−(n−2)(q2−1)+(n−3)(n−2)(q2−1)2/2)2≥(q4+2(n−2)(q2−1))(1+n(q2−1)+n(n−1)(q2−1)2/2)Simplifying L.H.S,(n−2)to(n−3)then(q2−1)4((n−3)2/2−(n−2))2≥(q4+2(n−2)(q2−1))(1+n(q2−1)+n(n−1)(q2−1)2/2)4by simplifying both sides((n−3)2/2−(n−2))2≥(q2q2−1)(1+n+n(n−1)/2)((n−3)2/2−(n−2))2≥2(2n+1)(n2+2n+2)(n−3)2((n−3)/2−1)2≥2(2n+1)((n+1)2+1)(n−5)2/4≥2(2n+1)n≥7Lemma7.The inequality f(0)/f0≥f(3)/f3holds for n≥7and q≥2.Proof.Letq2n(1−2(n−3)(q2−1)+(n−3)(n−4)(q2−1)2/2)2 by simpliﬁcation(1−2(n−3)(q2−1)+(n−3)(n−4)(q2−1)2/2)2≥6(q2−1)(1+n(q2−1)+n(n−1)(q2−1)2/2)((n−4)2(q2−1)2−4(n−4)(q2−1)+2)24((n−5)4≥3(q2−1)3(2+2n+n2)(n−5)4≥4(2+2n+n2)(n−5)2≥2n≥7Lemma8.The inequality f(0)/f0≥f(4)/f4holds for n≥7and q≥2.Proof.Letq2n(3−3(n−4)(q2−1)+(n−4)(n−5)(q2−1)2/2)2divide by q2n and simplifying(3−3(n−4)(q2−1)+(n−4)(n−5)(q2−1)2/2)2≥6(1+n(q2−1)+n(n−1)(q2−1)2/2)5then by approximating(n−4)to(n−5)in L.H.S and(n−1)to4n in R.H.S,we canﬁnd that (−3(n−4)(q2−1)+(n−5)2(q2−1)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2)((n−5)2(q2−1)+(n−5)2(q2−1)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2) (q2−1)4((n−5)2+(n−5)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2)dividing both sides by(q2−1)4and simplifying6(1+n(q2−1)+n2(q2−1)2/2)(9/4)(n−5)4≥(9)1+n(q2−1)If the Hamming bound is tighter than the Singleton bound for any((n,K,3))q quantum code,then it means that MDS codes cannot exist for that set of n,K.This occurs whenq nq n−2d+2=q n−4≥3.2Upper Bound on the Maximal Length of Double Error-correctingMDS CodesLemma10.The maximal length of double error-correcting quantum MDS codes is upper bounded by:(q2−3)+ 2(q2−1)(11)n≤Proof.It is known that the Hamming bound for d=5is given by:q nK≤1+n(q2−1)+n(n−1)(q2−3)2+8(q8−1)(q2−3)2+8(q8−1)(q2−3)2+8(q8−1)[3]A.R.Calderbank,E.M.Rains,P.W.Shor,and N.J.A.Sloane.Quantum error correction viacodes over GF(4).IEEE rm.Theory,44:1369–1387,1998.[4]P.Delsarte.Four fundamental parameters of a code and their combinatorial signiﬁcance.Information and Control,23(5):407-438,December1973.[5]P.Delsarte.Bounds for unrestricted codes by linear programming.Philips Res.Reports,27:272–289,1972.[6]K.Feng and Z.Ma.Aﬁnite Gilbert-Varshamov bound for pure stabilizer quantum codes.IEEE rm.Theory,50(12):3323–3325,2004.[7]D.Gottesman.A class of quantum error-correcting codes saturating the quantum Hammingbound.Phys.Rev.A,54:1862–1868,1996.[8]D.Gottesman.Stabilizer codes and quantum error correction.Caltech Ph.D.Thesis,eprint:quant-ph/9705052,1997.[9]M.Grassl,T.Beth,and M.R¨o tteler.On optimal quantum codes.Internat.J.QuantumInformation,2(1):757–775,2004.[10]W.C.Huﬀman and V.Pless.Fundamentals of Error-Correcting Codes.University Press,Cambridge,2003.[11]A.Ketkar,A.Klappenecker,S.Kumar,and P.K.Sarvepalli.Nonbinary stabilizer codes overﬁniteﬁelds.IEEE rm.Theory,52(11):4892–4914,2006.[12]E.Knill and ﬂamme.A theory of quantum error–correcting codes.Physical Review A,55(2):900–911,1997.[13]V.I.Levenshtein.Krawtchouk polynomials and universal bounds for codes and designs inHamming spaces.IEEE rm.Theory,41(5):1303–1321,1995.[14]J.H Van Lint.Introduction to coding theory.Third Edition,Springer-Verlag1999.[15]R.J.McEliece,E.R.Rodemich,jr.H.Rumsey,and L.R.Welch.New upper bounds on the rateof a code via the Delsarte-MacWilliams inequalities.IEEE rm.Theory,23(2):157, 1977.[16]E.M.Rains.Nonbinary quantum codes.IEEE rm.Theory,45:1827–1832,1999.84AppendixAn Approach (Sketch)to Prove Hamming Bound for Degen-erate Nonbinary Stabilizer Codes with Minimum Distance dOne way to prove the quantum Hamming bound for impure nonbinary stabilizer codes with d ≤(n −k +2)/2is to expand f (x )/f x in terms of Krawtchouk polynomials.Let f (x )= n j =0f j K j (x )and f x =( ei =0K i (x ))2.The Krawtchouk polynomial of degree e in the variables x and q is given byK e (q,x )=e j =0(−1)j (q 2−1)e −jx j n −xe −j(17)Theorem 11.Let Q be an ((n,K,d ))q stabilizer code of dimension K >1.Suppose that S is anonempty subset of {0,1,...,d −1}and N ={0,1,...,n }.Letf (x )=n i =0f i K i (x )be a polynomial satisfying the conditions:i)f x >0for all x ∈S ,and f x ≥0otherwise;ii)f (x )≤0for all x ∈N \S .ThenK ≤1f x.Notice that f (x )=ni =0f i K i (x )can be written as f i =q−2nnx =0f (x )K x (i ).Lemma 12(Sketch).Let Q be an ((n,K,d ))q stabilizer code of dimension k ≥1.Suppose that S is a non-empty subset of {0,1,2,....,2e },where e =⌊d −1f x equals toK ≤q nf(x)f x(19) And our goal is toﬁnd max{f(0)/f0,f(1)/f1,...,f(d−1)/f d−1}that may equal to f(0)/f0.Now,for x=0,weﬁnd thatf(0)f0=K0(q,0)+f1K1(q,0)f0(20)orf(0)e i=0K i(0) 2and for any other value of y∈{1,2,...,d−1},weﬁnd thatf(y)f y=f0K0(q,y)f y+......+f n K n(q,y) f y= n j=0 e i=0K i(j) 2K j(y)f0−f(y)f0−f(y)e i=0K i(0) 2− n j=0 e i=0K i(j) 2K j(y)e i=0K i(0) 2−K j(y)f0−f j K j(q,y)in the previous equation,f j>0and f y>0,so,if we prove thatf j K j(q,0)≥0,(24)f ythen the claim holds.As shown in[4],[14],we seek a constant value for the left side in24,so, multiplying both sides by K e(i)K e(i)K i(q,0)≥0(25)f yand take n i=0ni=0 K e(i)K i(q,0)f y ≥0n i=0K e(i)K i(q,0)f y≥0(26) from[14],given that n i=0K e(i)K i(q,j)=q nδej,by substitution,q nδe0≥0f yδe0≥0(27)f yNow,δe0=1,andδey=1or0;and obviously f y≥f0.So,if y=e=⇒δe0/f0≥0,and similarly,δey=1=⇒f y−f0≥0.11。

A note on random walks in a hypercube

arXiv:0711.2675v1 [math.PR] 16 Nov 2007
A note on random walks in a hypercube
Stanislav Volkov∗ Timothy Wong†
February 2, 2008
Abstract We study a simple random walk on an n-dimensional hypercube. For any starting position we ﬁnd the probability of hitting vertex a before hitting vertex b, whenever a and b share the same edge. This generalizes the model in [2] (see Exercise 1.3.7 there).
4
ቤተ መጻሕፍቲ ባይዱ
to n vertices (1 − x1 (x1 , 1 − x2 ... (x1 , x2 , . . . , 1 − xn−1 , 0) (x1 , x2 , . . . , xn−1 , 1) , x2 , . . . , xn−1 , 0) , . . . , xn−1 , 0)
˜ k , and among the ﬁrst n − 1 vertices k lie in Wk−1 where the last one lies in W and n − 1 − k in Wk+1 (since there are exactly k ones and n − 1 − k zeros in the set {x1 , x2 , . . . , xn−1 }). From Kirchhoﬀ’s and Ohm’s Laws, stating that the sum of all currents from a vertex is zero, and the current that ﬂows through an edge equals the diﬀerence in voltages divided by the resistance of that edge (which are all one in our case), we conclude that k× w k −1 − w k wk+1 − wk w ˜ k − wk + (n − 1 − k ) × + =0 1 1 1

strike a note of造句

strike a note of造句
"strike a note of" 是一个比较常见的英语表达，意思是"强调或突出一种情感或主题"。

以下是一些使用"strike a note of" 的句子：
1.The article struck a note of optimism in the face of adversity.
这篇文章在逆境中表现出乐观的态度。

2.The politician's speech struck a note of authenticity and sincerity.
这位政治家的讲话具有真实性和真诚性
3.The music struck a note of sadness and longing.
音乐发出悲伤和渴望的音符。

4.The teacher's tone struck a note of concern and care.
老师的语气带着关心和关怀。

5.Let me strike a note of hopefulness; this job will not be as hard as you think.
我对这工作抱着一种希望的态度，它不会像你想象的那么艰苦。

在这些句子中，"strike a note of" 后面接的是表示情感或主题的名词，意思是强调或突出这种情感或主题。

note的用法和短语例句文意思是什么优秀3篇

note的用法和短语例句文意思是什么优秀3篇note的用法篇一note的用法1：note的基本意思是“笔记，摘记”,指用来记录平时一些需要记住的事情的短小记录；也可指为了通知某人某事而写下的“短小信笺”。

“记笔记”可以说make a note (of),也可说take notes (of),后者更常见。

note的用法2：note也可指提醒人们对某事加以重视，即“注意”。

note 引申可指“照会”“评注，注释”“暗示，含义”“做注释”等。

note的用法3：note还可作“纸币”解， in notes指“以纸币的形式”,介词in不可用其他介词代替，且notes须用复数形式。

note后常可接介词短语、动词不定式或分词短语作定语。

note的用法4：take a note of是“做笔记”的意思，但如果变成take note of时则变成了“注意”。

note的用法5：note用作动词时指“将某事写在纸上”,以便于以后查找，即“记录”; 还可指被某人或某物所吸引而把精力投入在其中，即“注意”。

note的用法6：note多用作及物动词，后接名词、代词、带疑问词的动词不定式或that/wh-从句作宾语，还可接以形容词或不带to的动词不定式充当补足语的复合宾语。

可用于被动结构。

note的常用短语篇二用作名词 (n.)compare notesmake〔take〕 notes (of)of notestrike〔sound〕 a notetake note of用作动词 (v.)note down( v.+adv. )be noted for( v.+prep. )note的用法例句篇三1. Somehow he tells these stories without a note of horror.不知为什么他讲这些故事时一点都不害怕。

2. She scribbled a note to tell Mum she'dgone out.她匆匆写了个便条告诉妈妈她已外出。

A Note on Memristors

It is welI known that an immittance (impedance or admittance) matrix of a real resistance network can always be decomposed into the product ADAt, where A is a unimodular matrix and D is a diagonal matrix with real positive diagonal elements. If the network corresponds to an immittance matrix Q(= ADAt), then the diagonal D gives directly the values of the impedance or admittance for all branches of the network. Cederbaum [l] has presented a method of decomposition of an immittance matrix into the product ADAt and shown that a matrix can be decomposed if and only if it can be decomposed by the method. This particular method is applicable to finding the pair of matrices A and D from the given matrix Q. However, there is a problem left to determine unimodularity of A which is obtained by the method. We need to get the conditions for the unimodular decomposition of an lmmittance matrix directly. In this letter, we show two necessary conditions for existence of unimodular decomposition of an immittance matrix. First we show that the sum of ail elements of a principal submatrix of a matrix, which can be decomposed, is positive. Next we investigate a property of the minors of such a matrix. Theorem 1: Let Q = [qij] = ADA’be a nonsingular matrix of order n, where A = (aij] is an n X m totally unimodular matrix and D = [dtj] is an m X m diagonal matrix with real positive elements. Then the sum of ail elements of a principal submatrix of Q is positive. ProofiLetT~:~~::::~) be the sum of all elements of the sub-