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英文文献翻译及原文

原文

Title: Improved Integral Inequalities for Producets of Convex Functions

A largely applied inequality for convex functions, due to its geometrical significance, is Hadamard’s inequality (see [3] or [2]) which has generated a wide range of directions for

extension and a rich mathematical literature. Below, we recall this inequality, together with its framework.

A function [],f a b R ?：, with [],a b R ì, is said to be convex if whenever

[],x a b " [],y a b ?,[]0,1t ? the following inequality holds

(1.1) ()()()()()11f tx t y tf x t f y +-?-

This definition has its origins in Jensen’s results from [4] and has opened up the most

extended, useful and multi-disciplinary domain of mathematics, namely, convex analysis. Convex curves and convex bodies have appeared in mathematical literature since antiquity and there are many important results related to them. They were known before the analytical foundation of convexity theory, due to the deep geometrical significance and many

geometrical applications related to the convex shapes (see, for example, [1], [5], [7]). One of these results, known as Hadamard’s inequality, which was first published in [3], states that a convex function f satisfies

(1.2) Recent inequalities derived from Hadamard’s inequality can be found in Pachpatte’s

paper [6] and we recall two of them in the following theorem, because we intend to improve them. Let us suppose that the interval [],a b has the property that 1b a - . Then the

following result holds.

Theorem 1.1. Let f and g be real-valued, nonnegative and convex functions on [],a b . Then

(1.3) ()()()12031(1)(1)2b b a a f tx t y g tx t y dtdydx b a ?-+--蝌

()()()()()

2,,118b a M a b N a b f x g x dx b a b a 轾+犏?犏--犏臌ò and

(1.4) ()()1

1122b a a b a b f tx t g tx t dtdx b a 骣骣骣骣++鼢琪琪珑+-+-鼢鼢珑珑鼢鼢珑珑桫桫-桫桫

蝌 ()()()122b a f a f b a b f f x dx b a 骣++÷?＃÷?÷?桫-ò

()()()()111,,4b a b a f x g x dx M a b N a b b a b a +-轾??臌--ò

where

(1.5) ()()()()(),M a b f a g a f b g b =+

and

(1.6) ()()()()(),N a b f a g b f b g a =+

Remark 1.2. The inequalities (1.3) and (1.4) are valid when the length of the interval [],a b does not exceed 1. Unfortunately, this condition is accidentally omitted in [6], but it is implicitly used in the proof of Theorem 1.1.

Of course, there are cases when at least one of the two inequalities from the previous theorem is satisfied for 1b a ->, but it is easy to find counterexamples in this case, as follows.

Example 1.1. Let us take [][],0,2a b =. The functions []:0,2f R ? and []:0,2g R ? are defined by ()f x x = and ()g x x =. Then it is obvious that (),4M a b =, (),0N a b =. Then, the direct calculus of both sides of (1.3) leads to

and, obviously, inequality (1.3) is false.

Remark 1.3. Inequality (1.3) is sharp for linear functions defined on []0,1, while inequality (1.4) does not have the same property.

In this paper we improve the previous theorem, such that the condition

1b a -

is eliminated and the derived inequalities are sharp for the whole class of linear functions.

()()()1203111(1)(1)26b b a a f tx t y g tx t y dtdydx b a ?-+-=-蝌 ()()()()()2,,1135824b a M a b N a b f x g x dx b a b a 轾+犏+=犏--犏臌

译文

题目：凸函数在积分不等式中的应用

由于凸函数的几何意义,Hadamard 不等式在很大程度上影响了不等式凸函数的应用，Hadamard 不等式在丰富的数学文化和广泛的深入研究等条件下形成的，以下,我们回忆一下这些不等式，

定义1 函数[],f a b R ?：,并且[],a b R ì为凸函数,若[],x a b " [],y a b ?,

[]0,1t ?,有不等式

()()()()()11f tx t y tf x t f y +-?- (1.1)

成立，

这个由Jensen 提出的定义起源于[4],并且由此发展出对数学和多学科专业领域有用的凸分析，凸曲线和凸面体在很古老的时候就已经出现在了数学领域,有很多重要的成果都和他们有关系，由于许多几何意义和几何应用与凹凸形状有关,认识他们的分析基础是凸性理论，（例如 [1],[5],[7]）其中一个成果是Hadamard 不等式,第一次在[3]出版,阐明了凸函数f 满足

()()()122b a f a f b a b f f x dx b a 骣++÷?＃÷?÷?桫-ò (1.2)

Pachpatte 不等式理论是由Hadamard 不等式理论推断而来，我们称他们两个为定理因为我们还要优化他们，设区间 [],a b 满足1b a - ,有如下结果

定理1.1 设函数f 和g 是定义在非负实数区间[],a b 的凸函数，

那么

()()()12031(1)(1)2b b a a f tx t y g tx t y dtdydx b a ?-+--蝌 (1.3) ()()()()()2,,118b a M a b N a b f x g x dx b a b a 轾+犏?犏--犏臌

ò 和

()()1

1122b a a b a b f tx t g tx t dtdx b a 骣骣骣骣++鼢琪琪珑+-+-鼢鼢珑珑鼢鼢珑珑桫桫-桫桫

蝌 (1.4)

()()()()111,,4b a b a f x g x dx M a b N a b b a b a +-轾??臌--ò

其中

()()()()(),M a b f a g a f b g b =+ (1.5)

()()()()(),N a b f a g b f b g a =+ (1.6)

注意1.2 当[],a b 中1b a - 的时候,不等式(1，3)和 (1，4)成立,虽然这个条件被忽略,但在证明定理过程中已经被隐含的使用，

当然,至少有一两个不等式在以上定理中当1b a ->时成立的,但它很容易找到反例说明在这种情况下,如下

令[][],0,2a b =,函数[]:0,2f R ?和[]:0,2g R ?被定义为()f x x =和()g x x =,显然有(),4M a b =,(),0N a b =，带入（1，3）有

()()()1203111(1)(1)26

b b a a f tx t y g tx t y dtdydx b a ?-+-=-蝌 ()()()()()

2,,1135824b a M a b N a b f x g x dx b a b a 轾+犏+=犏--犏臌ò 很显然,不等式(1.3)是错误的，

定理1.3 不等式(1.3)是定义在[]0,1上的线性函数,但是不等式(1.4)是不存在这种性质的，

本论文将优化早先的定理，比如除去1b a - 条件,并且衍生的不等式满足所有的线性函数，

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英文文献及中文翻译

毕业设计说明书英文文献及中文翻译学院：专 2011年6月电子与计算机科学技术软件工程

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英文论文及中文翻译

International Journal of Minerals, Metallurgy and Materials Volume 17, Number 4, August 2010, Page 500 DOI: 10.1007/s12613-010-0348-y Corresponding author: Zhuan Li E-mail: li_zhuan@https://www.sodocs.net/doc/c310988683.html, ? University of Science and Technology Beijing and Springer-Verlag Berlin Heidelberg 2010 Preparation and properties of C/C-SiC brake composites fabricated by warm compacted-in situ reaction Zhuan Li, Peng Xiao, and Xiang Xiong State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China (Received: 12 August 2009; revised: 28 August 2009; accepted: 2 September 2009) Abstract: Carbon fibre reinforced carbon and silicon carbide dual matrix composites (C/C-SiC) were fabricated by the warm compacted-in situ reaction. The microstructure, mechanical properties, tribological properties, and wear mechanism of C/C-SiC composites at different brake speeds were investigated. The results indicate that the composites are composed of 58wt% C, 37wt% SiC, and 5wt% Si. The density and open porosity are 2.0 g·cm–3 and 10%, respectively. The C/C-SiC brake composites exhibit good mechanical properties. The flexural strength can reach up to 160 MPa, and the impact strength can reach 2.5 kJ·m–2. The C/C-SiC brake composites show excellent tribological performances. The friction coefficient is between 0.57 and 0.67 at the brake speeds from 8 to 24 m·s?1. The brake is stable, and the wear rate is less than 2.02×10?6 cm3·J?1. These results show that the C/C-SiC brake composites are the promising candidates for advanced brake and clutch systems. Keywords: C/C-SiC; ceramic matrix composites; tribological properties; microstructure [This work was financially supported by the National High-Tech Research and Development Program of China (No.2006AA03Z560) and the Graduate Degree Thesis Innovation Foundation of Central South University (No.2008yb019).] 温压-原位反应法制备C / C-SiC刹车复合材料的工艺和性能李专，肖鹏，熊翔粉末冶金国家重点实验室，中南大学，湖南长沙410083，中国（收稿日期：2009年8月12日修订：2009年8月28日;接受日期：2009年9月2日）摘要：采用温压?原位反应法制备炭纤维增强炭和碳化硅双基体(C/C-SiC)复合材

唯美的中文翻译成英文

唯美的中文翻译成英文 Abandon 放弃 Disguise 伪装 Abiding 持久的，不变的~friendship Indifferent 无所谓 Forever 最爱 I know what you want 我知道你想要什么 See you forget the breathe 看见你忘了呼吸 Destiny takes a hand.命中注定 anyway 不管怎样 sunflower high-profile向日葵，高姿态。 look like love 看起来像爱 Holding my hand, eyes closed you would not get lost 牵着我的手，闭着眼睛走你也不会迷路 If one day the world betrayed you, at least I betray the world for you! 假如有一天世界背叛了你，至少还有我为你背叛这个世界! This was spoiled child, do not know the heart hurts, naive cruel. 这样被宠惯了的小孩子，不知道人心是会伤的，天真的残忍。

How I want to see you, have a look you changed recently, no longer said once, just greetings, said one to you, just say the word, long time no see. 我多么想和你见一面，看看你最近的改变，不再去说从前，只是寒暄，对你说一句，只说这一句，好久不见。 In fact, not wine, but when the thought of drinking the unbearable past. 其实酒不醉人，只是在喝的时候想起了那不堪的过去。 The wind does not know clouds drift, day not know rain down, eyes do not understand the tears of weakness, so you don't know me 风不懂云的漂泊，天不懂雨的落魄，眼不懂泪的懦弱，所以你不懂我 Some people a lifetime to deceive people, but some people a lifetime to cheat a person 有些人一辈子都在骗人，而有些人用一辈子去骗一个人 Alone and lonely, is always better than sad together 独自寂寞，总好过一起悲伤 You are my one city, one day, you go, my city, also fell 你是我的一座城，有一天，你离开了，我的城，也就倒了。

经典中文英文翻译

经典中文的英译但愿人长久，千里共婵娟。 We wish each other a long life so as to share the beauty of this graceful moonlight, even though miles apart. 独在异乡为异客，每逢佳节倍思亲。 A lonely stranger in a strange land I am cast, I miss my family all the more on every festive day. 大江东去，浪淘尽，千古风流人物。 The endless river eastward flows; with its huge waves are gone all those gallant heroes of bygone years. 二人同心，其利断金。 If two people are of the same mind, their sharpness can cut through metal. 富贵不能淫，贫贱不能移，威武不能曲，此之谓大丈夫。 It is a true great man whom no money and rank can confuse, no poverty and hardship can shake, and no power and force can suffocate. 海内存知己，天涯若比邻。 A bosom friend afar brings distance near.

合抱之木，生于毫末，九层之台，起于累土；千里之行始于足下。 A huge tree that fills one’s arms grows f rom a tiny seedling; a nine-storied tower rises from a heap of earth; a thousand li journey starts with the first step. 祸兮，福之所依；福兮，祸之所伏。 Misfortune, that is where happiness depends; happiness, that is where misfortune underlies. 见贤思齐焉，见不贤而内自省也。 On seeing a man of virtue, try to become his equal; on seeing a man without virtue, examine yourself not to have the same defects. 江山如此多娇，引无数英雄尽折腰。 This land so rich in beauty has made countless heroes bow in homage. 举头望明月，低头思故乡。 Raising my head, I see the moon so bright; withdrawing my eyes, my nostalgia comes around. 俱往矣，数风流人物，还看今朝。 All are past and gone; we look to this age for truly great men.

中英文翻译与文献

Monolithic integrated circuit history The monolithic integrated circuit was born in the late-1970s, has experienced SCM, MCU, the SOC three big stages. SCM namely monolithic microcomputer (Single Chip Microcomputer) the stage, mainly seeks the best monolithic shape embedded system's best architecture. “the innovation pattern” obtains successfully, has established SCM and the general-purpose calculator completely different development path. In founds on the embedded system independent development path, Intel Corporation has lasting achievements. MCU namely micro controller (Micro Controller Unit) the stage, the main technological development direction is: Expands unceasingly when satisfies the embedded application, the object system request's each kind of peripheral circuit and the interface circuit, underline its object intellectualization control. It involves the domain is related with the object system, therefore, develops the MCU heavy responsibility to fall inevitably on electrical, the electronic technology factory. Looking from this angle, Intel fades out the MCU development also to have its objective factor gradually. Is developing the MCU aspect, the most famous factory family belongings count Philips Corporation. Philips Corporation by it in embedded application aspect huge superiority, MCS-51 from monolithic microcomputer rapidly expand to micro controller. Therefore, when we review the embedded system development path, do not forget Intel and the Philips historical merit. Monolithic integrated circuit is the embedded system's road of independent development, to the MCU stage development's important attribute, seeks application system's on chip maximized solution;