描写时间的英语作文及翻译汇集20篇

Good morning/afternoon/evening, my name isXXX . It is really a great honor to have this opportunity/chance to introduce myself. I would like to answer whatever you may raise, and I hope I can make a good performance today.

I am a third year master major in automation at Shanghai Jiao Tong University, P. R. China. Withtremendous interest in Industrial Engineering, I am writing to apply for acceptance into your Ph.D. graduate program.

In 1995, I entered the Nanjing University of Science & Technology (NUST) -- widely considered one of the China’s best engineering schools. During the following undergraduate study, my academic records kept distinguished among the whole department. I was granted First Class Prize every semester,In 1999, I got the privilege to enter the graduate program waived of the admission test.

At the period of my graduate study, my overall GPA(3.77/4.0) ranked top 5% in the department. In the second semester, I became teacher assistant that is given to talented and matured students only. This year, I won the Acer Scholarship as the one and only candidate in my department, which is the ultimate accolade for distinguished students endowed by my university. Presently, I am preparing my graduation thesis and trying for the honor of Excellent Graduation Thesis.

Research experience and academic activity

When a sophomore, I joined the Association of AI Enthusiast and began to narrow down my interest for my future research. With the tool of OpenGL and Matlab, I designed a simulationprogram for transportation scheduling system. It is now widely used by different research groups in NUST. I assumed and fulfilled a sewage analysis & dispose project for Nanjing sewagetreatment plant. This was my first practice to convert a laboratory idea to a commercial product.

In retrospect, I find myself standing on a solid basis in both theory and experience, which has prepared me for the Ph.D. program. My future research interests include: Network Scheduling Problem, Heuristic Algorithm research (especially in GA and Neural network), Supply chain network research, Hybrid system performance analysis with Petri nets and Data Mining.

译文：

早上好/下午好/晚上好,我的名字isXXX。真的很荣幸有这个机会进行自我介绍的机会。我想回答任何你可能会提高,我希望我能成为一个好今天的表现。

我是一个第三年硕士主修自动化在上海交通大学,中国公关。Withtremendous对工业工程的兴趣,我写这封信是为了申请接受到你的博士研究生项目。

1995年,我进入了南京科技大学(社交)——被广泛认为是中国最好的工程学校之一。在接下来的本科学习,我的学习成绩保持杰出的在整个部门。我每学期获得了头等奖,1999年,我有幸进入了研究生入学考试的放弃。

在我研究生学习期间,我的总平均成绩排名(3.77/4.0)前5%。在第二学期,我成为老师的助理,只给有才华的和成熟的学生。今年,我赢得了宏碁奖学金在我部门唯一的候选人,这是著名的终极荣誉学生赋予我的大学。目前,我准备我的毕业论文和优秀毕业论文的荣誉。

研究经验和学术活动

大二的时候,我加入了AI爱好者协会,开始为我的未来研究缩小我的兴趣。OpenGL的工具Matlab,我设计了一个simulationprogram运输调度系统。由不同的研究小组现在广泛使用的雄厚。我认为,实现污水分析&处理项目为南京sewagetreatment工厂。这是我第一次实践实验室的想法转换为一个商业产品。

回想起来,我发现自己站在一个坚实的基础的理论和经验,已准备好了我拿不到博士学位。我的未来的研究兴趣包括:网络调度问题,启发式算法研究(特别是遗传算法和神经网络),供应链网络研究,混合动力系统性能分析与佩特里网和数据挖掘。

[关于自我介绍英语作文带中文翻译

My name is Xia Shunyao. Im eight years old, grade two in primary school. I have a round face, big black eyes, big ears. My hair is black and bright, also please. In fact, I want to keep a long braid, but mother is not let me stay, I do not agree, but there is no way, had to leave a short hair.

My mother told me that my name was very meaningful. Because of the Xia Dynasty is the first dynasty Chinese, Emperor Yao and shun China is the earliest and most kind-hearted monarch, mom and dad gave me this name, is the hope that Ill do.

I am a good student in my class. I can finish my homework every day on time. I can check it after I finish it. In class, I can speak actively and help the students, so the teacher chose me as her little assistant.

At home I often help my mother do the housework, such as tow the floor, wipe the table, help my mother to buy food. When my parents are out of work, I can also go to the canteen to eat.

You see, am I great? Would you like to be a friend with me?

我叫夏舜尧，今年八岁，上小学二年级。我有一张圆圆的脸，乌黑的大眼睛，大大的招风耳。我的头发又黑又亮，还有点卷。其实，我很想留长辫子，可是妈妈就是不让我留，我不同意，但也没办法，只好留了个短发。

妈妈告诉我，我的名字很有意义。因为夏朝是中国第一个朝代，尧帝和舜帝是中国最早的最善良的君主，爸爸妈妈给我取这个名字，是希望我将来大有作为。

我在班上是个好学生，每天的作业都能按时完成，做完后也能自己检查。上课时能积极发言，爱帮助同学，所以老师才选我当她的小助手。

在家里我经常帮妈妈干家务，比如拖地板、抹桌子，帮妈妈买菜。爸妈没下班的时候，我还能自己到食堂打饭吃呢。

你看，我是不是很棒?你愿意和我做朋友吗?

I love summer very much. In summer, its hot, and sometimes its rainy. People wear T-shirts, shorts and sunglasses. Girls love to wear dresses and skirts.

I love smimming in summer. So I always go swimming with my parents in the sea. Sometimes, I like to play basketball on the playground with my friends if its not very hot.

My favourite food in summer is ice-cream. Its very cold and sweet. Its delicious for me in summer.

Wow, summer is really my favourite season!

我非常喜欢夏天。在夏天,很热,有时下雨。人们穿着t恤、短裤和太阳镜。女孩喜欢穿裙子,裙子。

我喜欢在夏天游泳。所以我总是和我的父母一起去游泳在海里。有时候,我喜欢打篮球和我的朋友在操场上如果不是很热。

在夏天我最喜欢的食物是冰淇淋。它很冷又甜。夏天对我来说是美味的。

哇,夏天是我最喜欢的季节!

I went to the Peoples Park with my family today. We had picnic in the

park. My mother and grandparents were preparing food, father was playing

badminton with his friend while I was reading under a tree.

今天我跟家人去公园。我们在公园野餐。我妈妈和爷爷奶奶准备吃的。爸爸跟朋友在打羽毛球，我在树下看书。

when it comes to english learning, people may start pouring out their woes about how it difficult to learn. well, as an english major student, i think there are several ways we can do to improve our english skills and make it easier to learn. some english learners believe that extensive reading is an important way to learn english. i really agree with that. for every english learner, extensive reading is a indispensable part to enhance their english. there are some of the advantages of reading:

当提到学习英语时,人们就开始大倒苦水,抱怨英语是有多么的难学。好吧,作为一名英语专业的学生,我认为我们有几种方法可以使我们的英语水平提高,让其更容易学。一些英语学习者认为泛读是学习英语的重要方法。我同意这种看法。对于每个英语学习者来说,泛读是提高英语水平不可缺的一部分。关于阅读,有这些好处:

first of all, reading on different kinds of topics can broaden our horizon. we can know about the things we don’t know before and learn about the cultural differences. to a large extend, we acquire our knowledge through reading. second, reading english books or novel can avoid we speak “chinglish”. we can know the way how the native english speakers express themselves. third, reading is a good way to enlarge our vocabulary. if we encounter a new word in the book, we can refer to the dictionary, and then write it down on the notebook. what’s more, the more we read, the better our reading comprehension will be. it’s a very useful skill for english learner. last but not least, reading can provide us with a lot of pleasure. we will not feel lonely if we read an interesting book.

首先,阅读不同题材的文章可以扩宽我们的眼界。我们可以知道那些我们不曾了解的事物,并了解文化差异。从很大的程度上来说,我们通过阅读来获取知识。其二,阅读可以避免我们说“中式英语”。我们可以了解到地道的英语应该怎么样说。第三,阅读可以帮助我们扩展自己的词汇量。如果我们遇到一个新的单词,可以查阅字典,然后把新的单词抄在笔记本中。这对于英语学习者来说非常有用。最后,阅读可以提供给我们很多快乐。当我们阅读一本有趣的书时,就不会觉得孤独了。

in a word, reading is a good way to improve our english skills. try to read extensively, you’ll have a lot of benefit from it.

一句话总结,阅读是提高我们英语学习能力的好方法。试着去广泛的阅读吧,你会从中收获很多的。

[大学生英语日记带翻译：英语学习之我见

The rabbits family

hello, my friends! my name’s sandy. today, i will tell you an interesting story about rabbits. look at me , now , i’m not sandy but a mother rabbit. i have a happy family because of my three daughters. they are very clever and lovely. in the morning , they have good habits . after they get up , they have good habits. after they get up in the morning, they wash the face , brush the hair , clean the ears and blow the noses. they like singing and dancing , too . now , let’s have a share , children , are you ready ? (yes, let’s go.) action , please! ( sing and dance)

兔子的家

你好,我的朋友!我的名字叫桑迪。今天,我将告诉你一个有趣的故事关于兔子。看着我,现在,我不是沙,但母亲的兔子。我有一个幸福的家庭,因为我的三个女儿。他们非常聪明和可爱。早上,他们有良好的习惯。起床之后,他们有良好的习惯。早上起床之后,他们洗的脸,刷头发,清洁耳朵吹鼻子。他们喜欢唱歌和跳舞。现在,让我们来分享,孩子们,你准备好了吗?(是的,我们走吧。)行动,谢谢!(唱歌跳舞)

I caught a cold today

In this morning，I felt very uncomfortable when i wake up,after i got up，i

sneezed continuously,so i realized that i had caught a cold.Then i went to the

clinic nearby my home to see the doctor, the doctor prescribed some cold

medicine for me and told me to drink lots of hot water. After taking the pills i

had a good sleep,now i feel better,I hope i will be better soon.

参考翻译

我今天感冒了

今天早上醒来的时候我感到很不舒服，我起床之后一直打喷嚏，于是我意识到我感冒了。于是我就去我家附近的诊所看医生，医生给我开了一些感冒药，并且嘱咐我多喝热水。我吃了药之后就睡了个好觉。现在感觉好多了，我希望我快点好起来。

I have a good friend. Her English name is Ann. Her Chinese name is Wang Xiaolan. She is in Nanzhuang Primary School. She is twelve. She has black eyes and black hair. Her father is a doctor and her mother is a nurse. Her brother is five years old. She likes dancing, singing and watching TV. She is friendly and she likes to help others. What a good girl!我有一个好朋友。她的英文名字是安。她的中午名字是王小兰。她在南庄小学。她十二岁。她有黑眼睛和黑头发。她的父亲是一个医生和她的母亲是一个护士。她的弟弟五岁 。她喜欢跳舞、唱歌和看电视。她很友好而且她喜欢帮助别人。多么好的一个女孩！

[关于我的好朋友的英语作文带翻译

A kind of little car may someday take the place of todays big ones. If people drive such cars in the future, there will be less pollution in the air. There will be more space for parking in cities.

The streets will be less crowded. And driving will be safer as these little cars can go only 65 km per hour.

The future cars will be fine for getting round a city, but they will not be useful for long trips. Little cars will go 200km, before needing to stop for more gas. If big carsare still used along with the small ones, two sets of roads will be needed in the future.

Some roads will be used for the big fast cars and other roads will be needed for the slow and small ones. Its said that three such little cars fit in the space now needed for o Summer is the second season in the year.

In the Northern Hemisphere, summer goes from June to August.

Summer is the hottest season in the year. Sometimes, the temperature reaches40°C.

Many people can not stand such hot weather. So they always stay indoors. At night, people would go out for a walk. Swimming is the most popular sport in summer. It brings cool to people.ne car the usual size.

On Christmas morning， the children wake up very early. Some even turn on the light at two o‘clock， and most of them are awake by six o’clock although it is not light in England for another hour or two at this time of the year.

Children look for their presents ， and the young ones play while the dinner is prepared. At about one o‘clock in the afternoon， the Christmas dinner is brought in . The turkey or chicken is quickly eaten . Children search in their Christmas pudding for new coins which are hidden in it . The rest of the day is full of games and eating until the happiest of all Christmas hollidays comes to an end.

在圣诞节的早晨，孩子们很早就起床。有些人甚至在两点钟的时候开灯，但大多数人还是在六点钟醒的，虽然在英国这个时间就算再过一两个小时也还没有天亮。

在准备晚宴的时候，孩子寻找自己的礼物，年轻的玩耍。在下午约一点钟的时候，圣诞晚餐开始。火鸡或鸡肉被迅速吃掉。孩子们在他们的圣诞布丁寻找哪个隐藏了新币。在一天剩下的时间都充满了游戏和吃的，直到最幸福的各个圣诞假期结束。

January 16 20xx

20xx年1月16号

Today, I still went to my mothers office. My mother was very busy, so was my father. They always live in Beijing. They must get up early. Because they will manage the factory. So I know they are very laborious. So I should save my money. Also I should help them. Although I cant do something useful, but I think I should share the work with them. I am one of my family member. In the future, I will take a job and work. It time for me to begin to learn how to work.

今天,我还是去了我母亲的办公室。我母亲很忙碌,所以是我父亲。他们一直生活在北京。他们必须早起。因为他们将管理工厂。所以我知道他们很辛苦。因此,我要节省我的钱。此外,我应该帮助他们。虽然我不能做一些有用的,但我认为我应该与他们共同工作。我对我的家庭成员之一。今后,我将工作和工作。它的时候我开始学习如何工作。

Everybody wishes (hopes) to get money so that he can maintain his

livelihood。 In other words， money is so useful that it is impossible for mankind

to live without it。 For this reason， we have an interesting proverb。 It says

“Money can make the world go around。” In a word， money is more powerful than

anything else。

However， money should not be wasted。 It must be used for (some) proper

purposes。 We young men should form the good habit of not spending money in the

wrong way。

每个人为了能够维持生活，所以都期望得到钱。换句话说，钱是十分有用，所以没有它人类即无法生活。基于此种理由，我们有一句搞笑的谚语。它说有“钱能使鬼推磨”。一言以蔽之，钱比任何另外的东西更有力量。

然而，钱不就应被浪费掉。它务必用于正当的用途上。我们青年人就应养成不乱花钱的好习惯。

to introduce myself（介绍我自己）

hello,every one!（大家好）

my name is **** . （我叫****）

im a 15 years old boy. （我是一个15岁的男孩）（具体情况自己改）

i live in the beautiful city of rizhao.（我住在美丽的rizhao城）（你可以把rizhao改成自己家乡的城市的名称的拼音）

im an active ,lovely and clever boy.(我是一个活跃的可爱的聪明的男孩） in the school , my favourite subject is maths . （在学校，我最喜欢数学） perhaps someone thinks its difficult to study well .（也许有些人认为这很难学） but i like it.（但我喜欢他）

i belive that if you try your best, everything can be done well.（我相信每件事付出努力就会有害结果）

i also like sports very much.（我也很喜欢运动）

such as,running,volleyball and so on. （像跑步、排球等等）

im kind-hearted.（我很热心）

i hope we can be good friends!（我希望我们能成为好朋友）

ok.this is me .a sunny boy.（好了，这就是我，一个阳光男孩）

Dear Ms. Fu,

亲爱的付老师，

I would like you to know that I appreciate the help and support you gave to

me when I was in junior high school. I enjoyed being taught by you immensely.

You are an excellent teacher and inspired me to continue learning with an open

and positive mind.

我想让您知道我非常感激在我上初中的时候你给予的帮助和支持。我非常喜欢你做我的老师。你是一个优秀的教师，鼓励我以开放和积极的态度继续学习。

I would like to convey my heartfelt gratitude to you for teaching me what

is responsibility. You helped me how to find the best place for myself and look

for the personal values. I remember you gave me a book as a present of the

Spring Festival. You are the first teacher who gave me a present which I was

moved for a long time.

我衷心的感谢你教会了我什么是责任。你帮我找到最适合自己的位置并和个人价值。我记得你曾给我一本书作为春节的礼物。你是第一个给我礼物的老师，这让我感动了很久。

I would further say that you gave me so good memories that I wish I could

go back to that time again. It was you who encouraged me to overcome obstacles.

It was you who cared about my emotions. I appreciate all you have done for me,

which meant so much to me. I hope that this letter brings my best wishes to you

and I once again thank you for giving me those precious memories of my life,

which I shall cherish forever.

我还想说你给了我这么美好的回忆，我希望我能再次回到那个时候。是你鼓励我克服困难。是你关心我的情绪。我很感激你为我所做的一切，这对我来说是很重要的。我希望这封信能给你带去我最好的祝福，我再次感谢你给我的生活带来那些珍贵的回忆，我会永远珍惜它的。

Sincerely yours,

Yang

杨

I have rested for 10 days. In these days, I felt very bored. I didn’t know

to do what. Although I had a lot of things to do, I felt uncomfortable. I was

ill because of the hot weather. I was tired, sleepy and had no strength. My

parents are worried about my health. in fact, it didn’t matter. I was always in

the room with air-conditioner and opened it in a low temperature. So when I went

out, the high temperature disagreed to me.At last, I was ill.

我已经休息了10天。在这些日子里，我觉得很无聊。我不知道做什么。虽然我有很多事情要做，我觉得不舒服。我是因为炎热的天气而生病。我累了，困了，没有实力。我的父母担心我的健康。事实上，这并不重要。我总是在房间里有空调，并在低温下打开了它。所以，当我走了出去，高温不给me.At最后，我病了。

Winter is like your room, if you can clean it very well, you will like it, but if you throw things in a mess, you dont want to know, you must hate it. "Winter" is the same, you use a good attitude to look at it, you will definitely like me, step by step slowly falling in love with the "different winter".

I like winter, because only winter can make trees appear different styles. From spring and summer to autumn, leaves are always changing and changing. But winter can make trees become bare. I like to accompany my brother to practice football with my family in winter, not only to make my body warm, but also to give my brother an opportunity to practice more. There are also, in winter, I dont know if you have noticed that mosquitoes are missing a lot. At last, they can make me calm down and do well in homework. This is also a great benefit of winter. The last reason is my favorite, the family out to buy Hot pot material, in supermarkets to all of a sudden to the streams of people busily coming and going, in winter, many fewer people, do not have to queue for a check out, is really good, in the winter, the whole family sitting around the round table, each other help each other to resist the cold wind up, this feeling is only a winter feel, together with my family, so hot fireplace, warm atmosphere, in addition to "winter", any season cant taste good.

Anyway, there are also its shortcomings in winter. Of course, there are cold and cold currents that make the nose and water direct. They will make us want to hide in the quilt, but occasionally, this is not a bad thing. In a word, I like it in spring, summer, autumn and winter, and each season has its own characteristics, of which, winter is a very special one. But every trip to winter makes me feel different.

“冬季”就像你的房间，如果能整理的很干净，自己就会喜欢它；但如果你把东西丢的乱七八糟，那不用想也知道，你一定很讨厌它。“冬季”也是一样，你用好的心态去看待它，你绝对会如同我，一步一步的慢慢爱上那“不一样的冬季”。

我喜欢冬季，因为只有冬季，能使树木出现不一样的风格，从春天、夏天在到秋天，叶子都一直变多、变色，但“冬季”却能让树木成为光秃秃的一片。我喜欢在冬季与家人一同去陪弟弟练足球，不仅让我的身体变暖和，也给弟弟多一次练习的机会呢！还有还有，在冬季时，不知道你有没有发现，蚊子顿时少了很多，总算能让我静下心来好好写功课了，这也是冬季的一大好处唷！最后一个理由也是我最喜爱的原因了，全家一起出动去买火锅料，在员来熙熙攘攘的大卖场，一下子到了冬季，人少了许多，结帐都不必排队排个大老远，真是不错，在冬季，全家人一起围着圆桌，互相帮对方把冷飕飕的风给抵挡起来了，这种感觉也只有冬季感受的到 了，和家人一起围炉，如此这般热融融的气氛，是除了“冬季”之外，任何一季都品尝不到的好滋味。

反正不管怎么说，当然冬季也有它的缺点，像是有冷到令人鼻水直流的寒流等，都会使我们想躲在被窝里不要出来，但偶尔这样，也不是件坏事吧！总而言之，春夏秋冬我都喜欢，每一季都有它的特色，其中，冬季是很特别的一个喔！但是每经过一趟“冬之旅”，都让我有不同的感受。

My name is Qiou Ye. I am twelve.My favorite sport is ping-pong and swimming. I like basketbll and I like volletball. I can play the piano and draw, I can play them very well. This is Me.

Can you tell me about yourself ?

我的名字是丘叶。我十二岁，我最喜欢的运动是乒乓球和游泳。我喜欢篮球，我喜欢volletball。我可以弹钢琴和画画，我可以打得很好。这是我。

你能告诉我你自己吗?

My dogs character is weird. Say it is gentle, it is really obedient. Let it lie there tease it to play, the United States and the United States looked at me, soon as not out. However, when it is eating, no matter who, that is, usually the best friend to its side will not work. It is so fierce, it is true ah, or how friends do not go to it around it? However, it came to see my grandfather will hide, for fear that grandfather see, because grandpa played it, it is so Gall bladder

If it is happy, can be cute than anyone else: turn around and turn around, the back of the tail shake to shake, as if you are affectionate. Or when you walk, rushed to your pants, let you stop walking.

It is a child can be pleasant to love! Just came to our house just full moon, the road is still bad, they learned to play their own. Walking around in the house, I do not know what to find, round little ass twisted, people laugh. Where is he beautiful, suddenly, hit the mans legs, back a few steps back, buttocks hit the wall again. Although it hit two, but it did not cry at all.

Then I do not know where to find a small ball, went to the yard to play. The small ball Diao Diao to go. It bite it all can not see the way.

I always like it and never say it. Look at it so lively and lively, not enough I like it, how could it be said?

August 12，2012 Friday Today， I did housework for grandmother. She is not very healthy and I decide to do housework for her more frequently. And she will feel happy and bee better. Suddenly， an idea hit me that it’s terrible to grow old becuase you have to worry about both your health and others’ opinions. Once again， I make a decision to be a good girl. 译文：这天，我帮忙了奶奶做家务.奶奶的身体不好.我以后要常帮她做事.这样她就会高兴身体也会好一些.突然觉得，人老了真不好.不但身体不会好，而且一不留意就会惹得大家厌恶.我又做了一个决定，以后必须要做个乖孩子。

Article two: visit the Great Wall composition 300 words

The Great Wall is my dream place, today I came to the the Badaling Great Wall in Beijing.

We began to climb the Great Wall. The Great Wall was covered with a piece of brick, very smooth. The Great Wall is very wide, five or six horses can be parallel, like a wide and long street. Every section of the road there is a Beacon Tower, Beacon Tower used to light fireworks convey important military, such as the enemy occurred, table is connected, transmitting military.

I look much against the wall, the Great Wall looks like a long dragon winding prone in the lofty mountains and high ranges between stretch as far as eye can see, under the blue sky, the queue is mighty, magnificent. To the distance, the mountain peaks on the rolling, tree verdant. Like a sea of green. I looked down, only a car such as beetles, a pedestrian as ants.

Standing on the Great Wall, tread the brick, leaning on the wall of stone, it is natural to think of the working people of ancient construction of the Great Wall. No car, train, crane handling tools of ancient, two thousand or three thousand pound brick rely on ancient craftsmen a express to this steep mountains, its not easy! Dont know how many people are dead in the the Great Wall, because of the ancient working people have today this magnificent dragon.

The Great Wall, you are so powerful, how magnificent, I love you.

Class, and Liu wrote on the blackboard "goldfish story" these four words, suddenly, the students are talking about each other, and some said: "What happened to the goldfish ah?" Some said: "How was the last goldfish?"

Liu said: "The story of the goldfish is a riddled question." Then, Liu wrote on the blackboard: "a goldfish tank in the original 10 goldfish, less 3, but also a few? Mr. Liu just finished. I am exclaimed: "This is not a math problem! Even kindergarten children will do, that is, 10 minus 3 is equal to 7 Well!" Liu also spoke: "This is not a simple math problem, its answer is A variety of! "Liu teacher voice faded, the class is like a blasted pot, and some say that 0 because the three goldfish died, the remaining seven goldfish to see also scared to death; Say there are seven, because 10 minus 3 equals 7 ...

Liu asked Luo Yukang answer, Luo Yu Kang immediately stood up and said: "There are 10, because the title said less 3, but did not say what three fish, so there are 10 goldfish." Song Wenxiu said : "I do not agree, if the three dead, the remaining seven goldfish to see their partner died, they do not want to live, so all suicide." "I do not agree," Zhang Ruifu said loudly : "If the fish tank has a big fish, it put the three fish to eat, the results are still six or a big fish to eat, so only one fish." Xiaohua said: "I do not agree with your idea, the fish tank should have 14, although the death of three, but if the three goldfish is a small goldfish, seven goldfish saw, and gave birth to seven small goldfish, so there are 14 "...

The riddles of the answer is really varied, the students say no matter how the upper and lower, mathematics is really wonderful!