[数学]数学09级1班闻晶晶外文文献翻译
土木工程外文文献翻译
土木工程外文文献翻译数学模型预测水运输混凝土结构中的渗透性eluozo，s.n全文粗骨料细砂的混凝土构件，大孔隙混凝土率确定孔隙率和混凝土结构孔隙比，渗透系数的影响确定率水运混凝土。
数学模型来预测渗透率对水率交通是数学发展，该模型是监测水运输的混凝土率结构。
渗透性建立大孔上构成的影响下一种关系，即由混凝土制成，应用混凝土浇筑的决定渗透性的沉积速率混凝土结构，渗透性建立是大孔的混合物之间的影响下通过水泥净浆，考虑到系统中的变量，数学模型的建立是为了监测水通过通过具体的速度，也确定渗透系数的率对混凝土结构。
关键字:混凝土结构、渗透性和数学模型一、简介混凝土结构的耐久性依赖于通过频密迁移率的熔化的成分。
这种搬迁就是通过磁导率的影响。
在排序该条件混凝土混合物就是通过中存有的基质中的微孔隙的已连续网络混凝土协调比。
其他影响就是通过存有于的界面的孔隙率骨料的级分体式结构。
本研究中，其特征测量的快速和精确度在煮混凝土的渗透性，这包含创建理论模型的叙述渗透性对混凝土结构的影响。
实验中采用的就是瞬时顺利完成渗透性设备监控措施细骨料细沙和水是这种材料例如混凝土称作孔隙率和孔隙率中的组件之间的微孔混凝土结构中，渗透系数的影响确认水的速率运输在混凝土加水物水分搬迁混凝土，设备容许快速和精确测量在混凝土加水水中搬迁。
混凝土就是一种类型的多孔材料做成，并且可以由于在物理上和化学受损其曝露在各种环境中从混凝土浇筑至其使用寿命。
在特别就是，一些外部有毒元素，例如硫酸根，氯离子，和二氧化碳，扩散在混凝土少于长期周期做为溶液或气体状态，并引致物理侵害，由于化学反应。
这些反应可以影响应用领域中钢筋破损具体内容的，这减少了耐热寿命，例如钢筋和力量。
因此，它是非常重要的是插入腐蚀抑制剂为在超过临界恶化元件的情况下钢棒腐蚀的钢筋的位置量[1]。
然而，这是非常困难的保证在使用该应用传统技术钢筋位置的耐腐蚀性腐蚀抑制剂仅在混凝土[2-3]的表面上。
我们班的数学天才英语作文600字初中
我们班的数学天才英语作文600字初中全文共3篇示例，供读者参考篇1The Math Genius in Our ClassYou know that kid in every class who just gets math? Like, really gets it down to the core in a way the rest of us can't even fathom? Well, in my class that genius is Adam Smith. Math seems to flow through his veins more naturally than blood.From the first day of 6th grade when Mr. Patterson worked through some simple algebraic equations on the board, Adam's hand shot up before anyone could even processed what was going on. He explained the steps for solving the equations so clearly and effortlessly that even I, a kid who has always struggled with math, had a Light bulb moment."Oh, that's how you do it!" I thought as Adam walked us through algebraically isolating the variable. It all seemed so obvious when he laid it out systematically.That was just the start of Adam's math wizardry. As we moved into more complex concepts like graphing systems of equations, factoring polynomials, and applying the principles ofgeometry, he became increasingly indispensable. If anyone raised their hand with a question or got stuck on a problem set, we'd all turn our heads eagerly toward Adam waiting for him to share his insight.Despite being shy around school hallways, Adam transforms in math class. He leans forward in his seat, makes deliberate eye contact with the teacher, and articulates his thoughts with impressive coherence and precision. You can almost see the synapses firing in his brain as he works through the logicstep-by-step.Even Mr. Patterson regularly calls on Adam to explain certain problems or theorems to the rest of us when we look puzzled. I've lost count of how many times Adam has dropped a "Big Ah" moment on the class that allowed a breakthrough or clarified something we were all stuck on.What makes Adam even more remarkable is his humility and patience. Despite having a staggering mathematical ability, he never lorded it over anyone or acted arrogant. If someone didn't understand his explanation at first, Adam would re-word it orre-work the problem with a different approach until it clicked for that person.He spends plenty of free periods, study halls, and free time after school walking different classmates through the topics they struggled with. Almost everyone has taken advantage of Adam's tutoring at one point or another. His guidance has allowed so many of us to maintain solid grades in math rather than allowing the subject to become our downfall.While the rest of us dream about getting outdoors as soon as the weather turns nice each year, Adam looks most at peace and in his element when working through a challenge problem set or attempting to decipher a new mathematical proof. Books on advanced mathematics concepts are is his Preference for light reading over young adult fiction.Simply put, Adam is a savant when it comes to numbers, equations, geometry, and the wonderful world of mathematics. We're all just lucky he ended up in our class to shed is brilliant light on the subject and help raise the mathematical excellence of every student. Having a genius in our midst drove the rest of us to live up to our fullest potential as well.篇2My Class's Math GeniusThere's this kid in my class who is, simply put, a total math genius. His name is David and he has an incredible knack for numbers and problem-solving that is really mind-blowing. I'm not exaggerating when I say he could probably end up being one of the great mathematicians of our time if he keeps developing his skills.It's kind of crazy just how advanced David's math abilities are compared to the rest of us. I still have to use a calculator for most complicated calculations, but he can do enormously complex equations entirely in his head in just seconds. Once, when we were learning about algebraic expressions in class, the teacher put a really long and convoluted problem on the board to demonstrate. While the rest of us just stared at it blankly, David's hand shot up and he solved the whole thing, step-by-step, from memory. The teacher was stunned and so were we.David's parents told me he started showing an affinity for math basically from the moment he could talk. Apparently, as a toddler, he would constantly recite numbers and look for patterns and relationships between them. By the age of 4, he could already do basic addition, subtraction, multiplication and division. That's just not normal child development - the kid was clearly some kind of math prodigy from a very young age.In school, he breezes through math assignments and tests like they're nothing. He gets every question right, every time. Our math teacher once told me she plans to start giving David extra advanced coursework because the normal curriculum isn't enough of a challenge for his capabilities.Sometimes he even catches mistakes in the textbooks that the authors and editors all missed.You might think having such a talented student would be great, but it does create some difficulties too. Often the concepts we're supposed to be learning go completely over my head because David has already understood it in about 2 seconds. Then he gets bored waiting for the rest of us to catch up and starts distracting the class by joking around. The teacher has had to warn him about it multiple times.Still, David is a pretty humble, down-to-earth guy despite his incredible talents. He never brags or acts arrogant about his superior intellect. I've learned not to feel bad about not being as skilled as him because it's clear his brain is just wired differently for math aptitude. It would be like him feeling bad for not being as good at basketball as one of the players on the school team.For his future plans, David says he's torn between wanting to become a mathematics professor and researcher, or going into amore applied field like engineering or computer science. No matter what path he chooses, I'm confident he'll be hugely successful and make important contributions. David is truly one of the most gifted students in mathematics our school has ever seen. I'm just lucky I get to be in his class and witness his incredible abilities firsthand. Who knows, maybe I'm learning from the next Einstein or Ramanujan.篇3Our Class's Math GeniusYou know that kid in every class who just seems to get math effortlessly? The one who finishes tests in half the time while the rest of us are still struggling? Well, that person in our class is Tony. Tony is, without a doubt, a certified math genius.I've known Tony since we were little kids playing together at the neighborhood park. Even back then, you could tell he just thought differently about numbers and patterns. While the rest of us would count out objects one by one, Tony would immediately see the groupings and patterns. "That's two groups of four!" he would exclaim about a simple pile of rocks. His mind worked that way from the very start.In elementary school, you could already see the math genius emerging. He would race through worksheets of basic operations, making teachers do double-takes. By third grade, he was getting pulled out for an hour each day to work on higher level math meant for older kids. Can you imagine being 8 years old and doing algebra? That was Tony.When we got to middle school, that's when Tony's talents really went into overdrive. I'm in the highest level of math available, and even I can't keep up with that kid half the time. He breezes through complicated equations like a hot knife through butter. Just last week, our math teacher put an extremely tricky problem on the board involving exponential functions and logarithms. The rest of us sat there dumbfounded, but within 30 seconds, Tony's hand shot up with the correct solution fully worked out.Sometimes I think he gets bored in class because the material isn't challenging enough for his incredible mind. He'll start thinking about unsolved math problems or theorems that no one has cracked yet. Once Mrs. Robinson caught him scribbling some sort of crazy number theory conjecture during her lesson. When she asked him about it, he launched into thisintense explanation about prime numbers that went completely over our heads. Even she looked dazed by the end!While Tony's skills are clearly off the charts, he's also a really humble, down-to-earth guy which makes him even more likeable. You'd never know he was a genius unless you saw him do math. In a world where kid geniuses can sometimes be arrogant know-it-alls, Tony is the complete opposite. He never lords his abilities over others or makes us feel dumb in comparison. If someone is struggling, he's the first to try patiently explaining it in a new way. We all wish we had Tony's brain, but are just glad to be his friend.I have no idea what the future holds for our class's math whiz kid. Ivy League universities are definitely in his future, and perhaps he'll go on to win a Fields Medal or formulate the next big theory in mathematics. Wherever life takes him, I'm just happy to be able to witness Tony's incredible talents and be inspired by them. Having a bonafide genius in your midst makes you realize there's so much potential in this world still to be unlocked. Math may seem like a boring subject to some, but seeing it through Tony's eyes shows how exciting and full of possibilities it truly is.。
数学家--中英文对照3篇
数学家--中英文对照数学家数学家是指从事研究数学领域的专业人士。
数学家们通过发表论文、教课、参与研究项目等方式，不断推动数学知识的发展和应用。
数学家们使用符号、公式、模型等工具来描述和解决各种数学问题，如代数、几何、拓扑、概率等。
他们的工作不仅是基础研究，也可以为应用数学提供支持，如在物理、工程、计算机科学等领域中的应用。
数学家们需要具备良好的逻辑思维能力、创造性和耐心，他们需要不断探索数学领域中新的思想和方法。
他们的工作需要深入掌握大量的数学知识，因此需要进行长期的学习和实践。
著名数学家包括欧拉、高斯、牛顿、莫比乌斯、庞加莱等。
他们的工作对数学学科的发展和现代科技的进步起到了重要的作用。
作为未来的数学家，我们需要勤奋学习，探索数学领域中的新思想和方法，并在未来的研究工作中为数学学科的发展和社会的进步做出贡献。
MathematiciansMathematicians are professionals who engage inthe study of mathematics. Mathematicians continue to promote the development and application of mathematical knowledge through publishing papers, teaching, and participating in research projects.Mathematicians use tools such as symbols, formulas, and models to describe and solve a variety of mathematical problems, such as algebra, geometry, topology, and probability theory. Their work is not only basic research, but also provides support for applied mathematics, such as in physics, engineering, and computer science.Mathematicians need to possess good logical thinking ability, creativity, and patience. They need to constantly explore new concepts and methods in the field of mathematics. Their work requires a deep understanding of a large amount of mathematical knowledge, so they need to undergo long-term learning and practice.Famous mathematicians include Euler, Gauss, Newton, Mobius, Poincare, etc. Their work has played an important role in the development of mathematical disciplines and the progress of modern technology.As future mathematicians, we need to study diligently, explore new ideas and methods in the field of mathematics, and contribute to the development of mathematical disciplines and social progress in our future research work.数学家的职业生涯数学家是从事以数学领域为主要研究对象的专业人士，数学家的职业生涯从学生时代开始，到最后可以成为权威，分享了解数学领域中新的思想和技术。
有关数学故事
有关数学故事（中英文实用版）Story: The Detective and the Math Problem故事：侦探与数学问题Once upon a time, there was a famous detective who was known for solving the most difficult mysteries.One day, he received a strange letter from a wealthy collector who claimed that he had been robbed of a priceless artifact.The collector suspected that it was an inside job and that the thief was someone he trusted.从前，有一位著名的侦探，以解决最困难的谜团而闻名。
有一天，他收到了一封来自一位富有收藏家的奇怪信件，信中说他丢失了一件无价的文物。
收藏家怀疑这是一起内部作案，而且窃贼是他所信任的某人。
The collector provided the detective with a list of suspects and a description of the artifact.He also mentioned that the thief must have used some mathematical calculations to determine the best time to strike.Intrigued by the mathematical aspect of the case, the detective decided to use his own math skills to solve it.收藏家向侦探提供了一份嫌疑人名单和文物的描述。
班班幼儿园第二章英文
班班幼儿园第二章英文Chapter 2: English at Banban KindergartenIntroduction:English language learning is an important aspect of education at Banban Kindergarten. In Chapter 2, we will explore the various methods and strategies used to teach English to our young learners. The curriculum is designed to provide a comprehensive and engaging English learning experience, focusing on four key areas: listening, speaking, reading, and writing. Through a combination of interactive activities, games, and songs, children develop their English language skills in a fun and stimulating environment.Listening:At Banban Kindergarten, listening skills are developed through various activities that encourage active listening and comprehension. Teachers use audio materials, such as English stories or songs, to expose children to different accents, intonations, and vocabulary. This helps them to develop their listening skills and comprehension abilities gradually.Additionally, teachers engage students in classroom discussions, group activities, and storytelling sessions, which further enhance their listening skills. By actively participating in these activities, children become more confident in their ability to understand spoken English.Speaking:To foster speaking skills, teachers at Banban Kindergarten use a communicative approach. From the moment children step into the classroom, English is the primary language of instruction. Teachers use simple, clear language and encourage children to express themselves in English. Activities such as role-plays, pair work, and games are incorporated into the curriculum to encourage children to speak confidently and fluently. Through frequent practice and positive reinforcement, children gradually gain confidence in using English to communicate with their teachers and peers.Reading:Reading is an essential part of language development, and at Banban Kindergarten, we aim to cultivate a love for reading in our students. The curriculum includes a wide range of storybooks, picture books, and leveled readers that are suitablefor young learners. Teachers create a welcoming reading corner within the classroom, stocked with books that cater to different reading abilities and interests. Through shared reading sessions, guided reading activities, and independent reading time, children develop reading skills such as phonics, word recognition, and comprehension. The goal is to instill a lifelong love for reading and provide a solid foundation in literacy.Writing:Writing skills are introduced gradually at Banban Kindergarten, beginning with the basics of letter formation and phonics. Teachers provide guided writing activities and encourage children to write simple sentences and short paragraphs. Through interactive writing exercises, children learn to express their ideas and thoughts in English, gradually improving their writing abilities. Teachers provide constructive feedback and assist students in improving their grammar, vocabulary, and sentence structure. The focus is on creating a positive and supportive environment that encourages children to practice and develop their writing skills.Assessment:At Banban Kindergarten, assessment is an integral part of thelearning process. Teachers use a variety of assessment methods to track each child's progress and identify areas for improvement. These methods include observation, informal assessments, and periodic tests. Teachers provide feedback to both students and parents regularly, highlighting areas of strength and areas that need improvement. This feedback helps parents to stay informed about their child's progress and actively engage in their English language learning journey.Conclusion:English language learning at Banban Kindergarten is designedto be an immersive and enjoyable experience for young learners. By focusing on listening, speaking, reading, and writing skills, children develop a solid foundation in English language proficiency. Through a combination of interactive activities, engaging materials, and skilled teachers, students gain confidence in using English to communicate effectively. The goal is to equip children with the necessary language skills to succeed in their future academic endeavors and beyond.。
研究生英语读说写1课文翻译
研究生英语读说写1课文翻译一、A Working Community5、None of us, mind you, was born into these communities. Nor did we move into them, U-Hauling our possessions along with us. None has papers to prove we are card-carrying members of one such group or another. Y et it seems that more and more of us are identified by work these days, rather than by street.值得一提的是，我们没有谁一出生就属于这些社区，也不是后来我们搬了进来。
这些身份是我们随身携带的，没有人可以拿出文件证明我们是这个或那个群体的会员卡持有者。
然而，不知不觉中人们的身份更倾向于各自所从事的工作，而不是像以往一样由家庭住址来界定。
6、In the past, most Americans live in neighborhoods. W e were members of precincts or parishes or school districts. My dictionary still defines communtiy, first of all in geographic terms, as ―a body of people who live in one place.‖过去大多数彼邻而居的美国人彼此是同一个街区、教区、校区的成员。
今天的词典依然首先从地理的角度来定义社区，称之为“一个由居住在同一地方的人组成的群体”。
7、But today fewer of us do our living in that one place; more of us just use it for sleeping. Now we call our towns ―bedroom suburbs,‖ and many of us, without small children as icebr eakers, would have trouble naming all the people on our street.然而，如今的情况是居住和工作都在同一个地方的人极少，对更多的人来说家成了一个仅仅用来睡觉的地方。
数学与应用数学英文文献及翻译
（外文翻译从原文第一段开始翻译，翻译了约2000字）勾股定理是已知最早的古代文明定理之一。
这个著名的定理被命名为希腊的数学家和哲学家毕达哥拉斯。
毕达哥拉斯在意大利南部的科托纳创立了毕达哥拉斯学派。
他在数学上有许多贡献，虽然其中一些可能实际上一直是他学生的工作。
毕达哥拉斯定理是毕达哥拉斯最著名的数学贡献。
据传说，毕达哥拉斯在得出此定理很高兴，曾宰杀了牛来祭神，以酬谢神灵的启示。
后来又发现2的平方根是不合理的，因为它不能表示为两个整数比，极大地困扰毕达哥拉斯和他的追随者。
他们在自己的认知中，二是一些单位长度整数倍的长度。
因此2的平方根被认为是不合理的，他们就尝试了知识压制。
它甚至说，谁泄露了这个秘密在海上被淹死。
毕达哥拉斯定理是关于包含一个直角三角形的发言。
毕达哥拉斯定理指出，对一个直角三角形斜边为边长的正方形面积，等于剩余两直角为边长正方形面积的总和图1根据勾股定理，在两个红色正方形的面积之和A和B，等于蓝色的正方形面积，正方形三区因此，毕达哥拉斯定理指出的代数式是：对于一个直角三角形的边长a，b和c，其中c是斜边长度。
虽然记入史册的是著名的毕达哥拉斯定理，但是巴比伦人知道某些特定三角形的结果比毕达哥拉斯早一千年。
现在还不知道希腊人最初如何体现了勾股定理的证明。
如果用欧几里德的算法使用，很可能这是一个证明解剖类型类似于以下内容：六^维-论~文.网“一个大广场边a+ b是分成两个较小的正方形的边a和b分别与两个矩形A和B，这两个矩形各可分为两个相等的直角三角形，有相同的矩形对角线c。
四个三角形可安排在另一侧广场a+b中的数字显示。
在广场的地方就可以表现在两个不同的方式：1。
由于两个长方形和正方形面积的总和：2。
作为一个正方形的面积之和四个三角形：现在，建立上面2个方程，求解得因此，对c的平方等于a和b的平方和（伯顿1991）有许多的勾股定理其他证明方法。
一位来自当代中国人在中国现存最古老的含正式数学理论能找到对Gnoman和天坛圆路径算法的经典文本。
取遍数字 数学英文
取遍数字数学英文The Language of Mathematics" with a word count exceeding 1000 words, as requested:Numbers are the fundamental building blocks of our understanding of the world around us. They are the universal language that transcends cultures and barriers, allowing us to communicate, quantify, and make sense of the complexities of our existence. From the ancient civilizations of Mesopotamia and Egypt to the modern technological marvels of our time, the power of numbers has been a constant companion in our journey of discovery and progress.At the heart of this numerical odyssey lies the discipline of mathematics, a field that has evolved alongside human civilization, constantly expanding our horizons and challenging our perceptions. Mathematics is not merely a collection of formulas and equations; it is a way of thinking, a language that allows us to explore the intricate patterns and relationships that govern the universe.One of the most captivating aspects of mathematics is the sheer diversity of the numbers themselves. From the simplicity of the natural numbers to the complexity of irrational and imaginarynumbers, each type of number carries its own unique characteristics and applications. The natural numbers, for instance, form the foundation of our numerical system, allowing us to count, quantify, and compare. These familiar digits, from 1 to 9, and the subsequent powers of 10, have become the backbone of our everyday interactions with numbers, from tallying inventory to calculating the cost of our groceries.As we venture deeper into the realm of mathematics, we encounter numbers that defy the traditional boundaries of our understanding. Fractions, for example, represent the division of a whole into equal parts, enabling us to express and manipulate quantities with greater precision. The introduction of negative numbers, once considered a paradox, has revolutionized our ability to represent and solve complex problems, from tracking financial transactions to mapping the movements of celestial bodies.But the true wonders of mathematics lie in the realm of irrational and imaginary numbers. Irrational numbers, such as the famous pi (π), are numbers that cannot be expressed as a simple fraction, their decimal representations continuing on infinitely without repeating. These enigmatic figures have captivated mathematicians and scientists alike, as they reveal the inherent complexity and beauty of the natural world, from the perfect circles of planets to the intricate patterns of snowflakes.Imaginary numbers, on the other hand, represent a whole new dimension of mathematical exploration. These numbers, denoted by the symbol "i," are defined as the square root of -1, a concept that initially defied logical explanation. Yet, these seemingly abstract constructs have become indispensable in fields ranging from quantum mechanics to electrical engineering, allowing us to model and understand phenomena that defy traditional numerical representations.The power of numbers, however, extends far beyond their mathematical applications. In the realm of language and communication, numbers have become an integral part of our daily lives, serving as a universal medium for expressing ideas and conveying information. From the ubiquitous use of numerical codes in our digital world to the symbolic significance of numbers in various cultural and religious traditions, these seemingly simple entities have become woven into the fabric of human expression.In the realm of art and design, numbers have also played a pivotal role, inspiring and shaping the creative process. The golden ratio, a mathematical proportion found in nature and often employed in art and architecture, has long been celebrated for its aesthetic appeal and its ability to evoke a sense of harmony and balance. Similarly, the use of numerical patterns and symmetries in music and visualarts has been a source of fascination, as artists explore the interplay between the rational and the emotive.As we delve deeper into the world of numbers, we realize that they are not merely abstract constructs, but rather a reflection of the underlying order and structure of the universe. From the microscopic realm of subatomic particles to the vast expanses of the cosmos, numbers serve as the language through which we can understand and quantify the fundamental principles that govern our existence.In the field of science, numbers have been instrumental in unlocking the mysteries of the natural world. The precise measurement and analysis of physical phenomena, from the speed of light to the half-life of radioactive isotopes, have been made possible through the rigorous application of mathematical principles. These numerical insights have not only expanded our knowledge but have also enabled us to harness the power of nature for the betterment of humanity, from the development of life-saving medical treatments to the harnessing of renewable energy sources.As we continue to push the boundaries of our understanding, the role of numbers in shaping our future becomes ever more apparent. In the realm of artificial intelligence and machine learning, numbers are the foundation upon which complex algorithms and models are built, allowing us to make sense of vast troves of data and uncoverhidden patterns that were once beyond our reach.In the end, the true power of numbers lies not just in their ability to quantify and analyze, but in their capacity to inspire wonder, foster creativity, and unlock the secrets of our universe. As we take a tour of the numerical landscape, we are reminded of the enduring legacy of mathematics, a discipline that has been a constant companion in our quest to understand and shape the world around us. It is a journey of discovery that continues to captivate and challenge us, inviting us to explore the infinite possibilities that lie within the realm of numbers.。
中小学数学代数几何常用概念中英文数学对照
中英文数学对照代数Algebra正数positive负数negative零zero数字digit/number整数integer分数fractions假分数proper fraction带分数mixture fractions/improper fraction 分子numerator分母denominator小数decimal百分数percentage/percent数字1one2two3three4four5five6six7seven8eight9nine10ten11eleven12twelve13thirteen14fourteen15fifteen16sixteen17seventeen18eighteen19nineteen20twenty100hundred1000thousand10000million1000000000billion奇数odd偶数even质数prime合数composite最大公约数maximum common factor 最小公倍数least common multiples加法addition减法subtraction乘法multiple除法division被除数dividend除数divisor商quotient和sum乘积product因数factor结合律association交换律communication分配律distribution因式分解factoring因子factors简化simplify等式/方程equation不等式inequation倒数receiption符号symbol约等于/近似approximately估算estimation实数real numbers有理数rational numbers无理数irrational numbers一元二次方程linear equations二元一次方程quadratic equations绝对值方程absolute equations方程的根root方程组system of equations变量variable常量constant多项式polynomial单项式monomial反比例函数inverse proportional function 正比例函数proportional function指数函数exponential function对数函数logarithmic function三角函数trigonometric function消元法elimination代入法substitute集合set并集union set交集intersection set空集empty set坐标轴axis横轴x-axis纵轴y-axis截距x,y-intercepts象限quadrant抛物线parabola顶点vertex准线directrix对称轴symmetric axis主轴Major axis副轴Minor axis水平对称轴horizontal symmetric axis垂直对称轴vertical symmetric axis数列sequence/series等差数列arithmetic sequence等比数列geometric sequence几何geometric点point线line面plane曲线curve多边形polygon平行四边形parallelogram菱形rhombus长方形rectangular正方形square梯形trapezoid三角形triangle斜三角形skew triangle正三角形right triangle等腰三角形isosceles triangle锐角三角形acute triangle直角三角形right triangle钝角三角形obtuse triangle凹多边形concave polygon凸多边形convex polygon对边opposite site邻边adjacent side斜边hypotenuse side对角线diagonal髙height底面base中线midline垂直平分线perpendicular bisector 垂直perpendicular平分bisector重心gravity垂心orthocenter角angle锐角acute angle直角right angle钝角obtuse angle圆circle半径radius直径diameter弦chord弧arc优弧major arc劣弧minor arc切线tangent line割线secant line长方形rectangle正方形square边side椭圆ellipse抛物线parabola双曲线hyperbola相交intersection相切tangent正交orthogonal立体图形solid立方体cube三棱柱triangular prism棱柱prism棱锥pyramid圆锥cone圆柱cylinder球sphere规则多边体不规则多边体勾股定理Pythagorean theorem 边长side length面积area周长perimeter/circumference 体积volume表面积surface area侧面积lateral area底面积base area斜边slant立方体的高altitude位似变化transformation位移translation水平平移horizontal shift垂直平移vertical shift对称reflection放大/缩小dilation strectch/compress 旋转rotation公式formula定理theorem矩阵matrix行列式determinant行row列column排列permutation组合combination概率probability极限limit导数derivative微分differential积分integral平均数average/mean方差variance标准差standard variance中位数median众数mode。
数学专业外文翻译---幂级数的展开及其应用
数学专业外文翻译---幂级数的展开及其应用In the us n。
we XXX its convergence n。
a power series always converges to a n。
We can use simple power series。
as well as XXX quadrature methods。
to find this n。
However。
this n will address another issue: can an arbitrary n f(x) be expanded into a power series?XXX n will address this XXX power series can be seen as an n of reality。
so we can start to solve the problem of expanding a n f(x) into a power series by considering f(x) and polynomials。
To do this。
we will introduce the following formula without proof:Taylor'XXX that if a n f(x) has derivatives of order n+1 in a neighborhood of x=x0.then we can use the following XXX:f(x)=f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)^2+。
+f^(n)(x0)(x-x0)^n+r_n(x)Here。
r_n(x) represents the remainder term.XXX (x) is given by (x-x)n+1.This formula is of the (9-5-1) type for the Taylor series。
微积分中英文对照外文翻译文献
中英文对照外文翻译文献错误！未找到引用源。
中英文对照外文翻译牛顿与莱布尼兹创立微积分之解析摘要:文章主要探讨了牛顿和莱布尼兹所处的时代背景以及他们的哲学思想对其创立广泛地应用于自然科学的各个领域的基本数学工具———微积分的影响。
关键词:牛顿;莱布尼兹;微积分;哲学思想Abstract:This paper mainly discusses the background of the times of Newton and Leibniz, and their philosophy of its founder is widely used in various fields of natural science basic mathematical tools --- calculus.Key words: Newton; Leibniz; calculus; philosophical thought今天,微积分已成为基本的数学工具而被广泛地应用于自然科学的各个领域。
恩格斯说过:“在一切理论成就中,未有象十七世纪下半叶微积分的发明那样被看作人类精神的最高胜利了,如果在某个地方我们看到人类精神的纯粹的和唯一的功绩,那就正是在这里。
”[1] (p. 244) 本文试从牛顿、莱布尼兹创立“被看作人类精神的最高胜利”的微积分的时代背景及哲学思想对其展开剖析。
一、牛顿所处的时代背景及其哲学思想“牛顿( Isaac Newton ,1642 - 1727) 1642 年生于英格兰。
⋯⋯,1661 年,入英国剑桥大学,1665 年,伦敦流行鼠疫,牛顿回到乡间,终日思考各种问题,运用他的智慧和数年来获得的知识,发明了流数术(微积分) 、万有引力和光的分析。
”[2] (p. 155)1665 年5 月20 日,牛顿的手稿中开始有“流数术”的记载。
《流数的介绍》和《用运动解决问题》等论文中介绍了流数(微分) 和积分,以及解流数方程的方法与积分表。
过程装备与控制工程专业U形管换热器毕业论文外文文献翻译及原文
毕业设计（论文）外文文献翻译文献、资料中文题目：U形管换热器文献、资料英文题目：文献、资料来源：文献、资料发表（出版）日期：院（部）：专业：过程装备与控制工程专业班级：姓名：学号：指导教师：翻译日期： 2017.02.14毕业设计（论文）外文翻译毕业设计（论文）题目： U形管式换热器设计外文题目： U-tube heat exchangers译文题目：指导教师评阅意见U-tube heat exchangersM. Spiga and G. Spiga, Bologna1 Summary:Some analytical solutions are provided to predict the steady temperature distributions of both fluids in U-tube heat exchangers. The energy equations are solved assuming that the fluids remain unmixed and single-phased. The analytical predictions are compared with the design data and the numerical results concerning the heat exchanger of a spent nuclear fuel pool plant, assuming distinctly full mixing and no mixing conditions for the secondary fluid (shell side). The investigation is carried out by studying the influence of all the usual dimensionless parameters (flow capacitance ratio, heat transfer resistance ratio and number of transfer units), to get an immediate and significant insight into the thermal behaviour of the heat Exchanger.More detailed and accurate studies about the knowledge of the fluid temperature distribution inside heat exchangers are greatly required nowadays. This is needed to provide correct evaluation of thermal and structural performances, mainly in the industrial fields (such as nuclear engineering) where larger, more efficient and reliable units are sought, and where a good thermal design can not leave integrity and safety requirements out of consideration [1--3]. In this view, the huge amount of scientific and technical informations available in several texts [4, 5], mainly concerning charts and maps useful for exit temperatures and effectiveness considerations, are not quite satisfactory for a more rigorous and local analysis. In fact the investigation of the thermomechanieal behaviour (thermal stresses, plasticity, creep, fracture mechanics) of tubes, plates, fins and structural components in the heat exchanger insists on the temperature distribution. So it should be very useful to equip the stress analysis codes for heat exchangers withsimple analytical expressions for the temperature map (without resorting to time consuming numerical solutions for the thermal problem), allowing a sensible saving in computer costs. Analytical predictions provide the thermal map of a heat exchanger, aiding in the designoptimization.Moreover they greatly reduce the need of scale model testing (generally prohibitively expensive in nuclear engineering), and furnish an accurate benchmark for the validation of more refined numerical solutions obtained by computer codes. The purpose of this paper is to present the local bulk-wall and fluid temperature distributions forU-tube heat exchangers, solving analytically the energy balance equations.122 General assumptionsLet m, c, h, and A denote mass flow rate (kg/s), specific heat (J/kg -1 K-l), heat transfer coefficient(Wm -2 K-l), and heat transfer surface (m2) for each leg, respectively. The theoretical analysis is based on classical assumptions [6] :-- steady state working conditions,-- equal flow distribution (same mass flow rate for every tube of the bundle),-- single phase fluid flow,-- constant physical properties of exchanger core and fluids,-- adiabatic exchanger shell or shroud,-- no heat conduction in the axial direction,-- constant thermal conductances hA comprehending wall resistance and fouling.According to this last assumption, the wall temperature is the same for the primary and secondary flow. However the heat transfer balance between the fluids is quite respected, since the fluid-wall conductances are appropriately reduced to account for the wall thermal resistance and thefouling factor [6]. The dimensionless parameters typical of the heat transfer phenomena between the fluids arethe flow capacitance and the heat transfer resistance ratiosand the number of transfer units, commonly labaled NTU in the literature,where (mc)min stands for the smaller of the two values (mc)sand (mc)p.In (1) the subscripts s and p refer to secondary and primary fluid, respectively. Only three of the previous five numbers are independent, in fact :The boundary conditions are the inlet temperatures of both fluids3 Parallel and counter flow solutionsThe well known monodimensional solutions for single-pass parallel and counterflow heat exchanger,which will be useful later for the analysis of U-tube heat exchangers, are presented below. If t, T,νare wall, primary fluid, and secondary fluid bulk temperatures (K), and ξ and L represent the longitudinal space coordinate and the heat exchanger length (m), the energy balance equations in dimensionless coordinate x = ξ/L, for parallel and counterflow respectivelyread asM. Spiga and G. Spiga: Temperature profiles in U-tube heat exchangersAfter some algebra, a second order differential equation is deduced for the temperature of the primary (or secondary) fluid, leading to the solutionwhere the integration constants follow from the boundary conditions T(0)=T i , ν(0)≒νifor parallel T(1) = Ti ,ν(0) = νifor counter flow. They are given-- for parallel flow by - for counterflow byWishing to give prominence to the number of transfer units, it can be noticed thatFor counterflow heat exchangers, when E = 1, the solutions (5), (6) degenerate and the fluidtemperatures are given byIt can be realized that (5) -(9) actually depend only on the two parametersE, NTU. However a formalism involving the numbers E, Ns. R has been chosen here in order to avoid the double formalism (E ≤1 and E > 1) connected to NTU.4 U-tube heat exchangerIn the primary side of the U-tube heat exchanger, whose schematic drawing is shown in Fig. 1, the hot fluid enters the inlet plenum flowing inside the tubes, and exits from the outlet plenum. In the secondary side the fluid flows in the tube bundle (shell side). This arrangement suggests that the heat exchanger can be considered as formed by the coupling of a parallel and a counter-flow heat exchanger, each with a heigth equal to the half length of the mean U-tube. However it is necessary to take into account the interactions in the secondary fluid between the hot and the cold leg, considering that the two flows are not physically separated. Two extreme opposite conditions can be investigated: no mixing and full mixing in the two streams of the secondary fluid. The actual heat transfer phenomena are certainly characterized by only a partial mixing ofthe shell side fluid between the legs, hence the analysis of these two extreme theoretical conditions will provide an upper and a lower limit for the actual temperature distribution.4.1 No mixing conditionsIn this hypothesis the U-tube heat exchanger can be modelled by two independent heat exchangers, a cocurrent heat exchanger for the hot leg and a eountercurrent heat exchanger for the cold leg. The only coupling condition is that, for the primary fluid, the inlet temperature in the cold side must be the exit temperature of the hot side. The numbers R, E, N, NTU can have different values for the two legs, because of thedifferent values of the heat transfer coefficients and physical properties. The energy balance equations are the same given in (2)--(4), where now the numbers E and Ns must be changed in E/2 and 2Ns in both legs, if we want to use in their definition the total secondary mass flow rate, since it is reduced in every leg to half the inlet mass flow rate ms. Of course it is understood that the area A to be used here is half of the total exchange area of the unit, as it occurs for the length L too. Recalling (5)--(9) and resorting to the subscripts c and h to label the cold and hot leg, respectively, the temperature profile is given bywhere the integration constants are:M. Spiga and G. Spiga: Temperature profiles in U-tube heat exchangersIf E, = 2 the solutions (13), (14) for the cold leg degenerate into4.2 Full mixing conditionsA different approach can be proposed to predict the temperature distributions in the core wall and fluids of the U-tube heat exchanger. The assumption of full mixing implies that the temperaturesof the secondary fluid in the two legs, at the same longitudinal section, are exactly coinciding. In this situation the steady state energy balance equations constitute the following differential set :The bulk wall temperature in both sides is thenand (18)--(22) are simplified to a set of three equations, whose summation gives a differential equation for the secondary fluid temperature, withgeneral solutionwhere # is an integration constant to be specified. Consequently a second order differential equation is deduced for the primary fluid temperature in the hot leg :where the numbers B, C and D are defined asThe solution to (24) allows to determine the temperaturesand the number G is defined asThe boundary conditions for the fluids i.e. provide the integration constantsAgain the fluid temperatures depend only on the numbers E and NTU.5 ResultsThe analytical solutions allow to deduce useful informations about temperature profiles and effectiveness. Concerning the U-tube heat exchanger, the solutions (10)--(15) and (25)--(27) have been used as a benchmark for the numerical predictions of a computer code [7], already validated, obtaining a very satisfactory agreement.M. Spiga and G. Spiga: Temperature profiles in U-tube heat exchangers 163 Moreover a testing has been performed considering a Shutte & Koerting Co. U-tube heat exchanger, designed for the cooling system of a spent nuclear fuel storage pool. The demineralized water of the fuel pit flows inside the tubes, the raw water in the shell side. The correct determination of the thermal resistances is very important to get a reliable prediction ; for every leg the heat transfer coefficients have been evaluated by the Bittus-Boelter correlation in the tube side [8], by the Weisman correlation in the shell side [9] ; the wall material isstainless steel AISI 304.and the circles indicate the experimental data supplied by the manufacturer. The numbers E, NTU, R for the hot and the cold leg are respectively 1.010, 0.389, 0.502 and 1.011, 0.38~, 0.520. The difference between the experimental datum and the analytical prediction of the exit temperature is 0.7％ for the primary fluid, 0.9% for the secondary fluid. The average exit temperature of the secondary fluid in the no mixing model differs from the full mixing result only by 0.6%. It is worth pointing out the relatively small differences between the profiles obtained through the two different hypotheses (full and no mixing conditions), mainly for the primary fluid; the actual temperature distribution is certainly bounded between these upper and lower limits,hence it is very well specified. Figures 3-5 report the longitudinal temperaturedistribution in the core wall, τw = (t -- νi)/(Ti -- νi), emphasizing theeffects of the parameters E, NTU, R.As above discussed this profile can be very useful for detailed stress analysis, for instance as anM. Spiga and G. Spiga: Temperature profiles in U-tube heat exchangersinput for related computer codes. In particular the thermal conditions at the U-bend transitions are responsible of a relative movement between the hot and the cold leg, producing hoop stresses with possible occurrence of tube cracking . It is evident that the cold leg is more constrained than the hot leg; the axial thermal gradient is higher in the inlet region and increases with increasing values of E, NTU, R. The heat exchanger effectiveness e, defined as the ratio of the actual heat transfer rate(mc)p (Ti-- Tout), Tout=Tc(O), to the maximum hypothetical rateunder the same conditions (mc)min (Ti- νi), is shown in Figs. 6, 7respectively versus the number of transfer units and the flow capacitance ratio. As known, the balanced heat exchangers E = 1) present the worst behaviour ; the effectiveness does not depend on R and is the same for reciprocal values of the flow capacitance ratio.U形管换热器m . Spiga和g . Spiga,博洛尼亚摘要:分析解决方案提供一些两相流体在u形管换热器中的分布情况。
数学 外文翻译 外文文献 英文文献 具体数学
Concrete MathematicsR. L. Graham, D. E. Knuth, O. Patashnik《Concrete Mathematics》，1.3 THE JOSEPHUS PROBLEM R. L. Graham, D. E. Knuth, O. Patashnik Sixth printing, Printed in the United States of America 1989 by Addison-Wesley Publishing Company，Reference 1-4pages具体数学R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克《具体数学》，1.3，约瑟夫环问题R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克第一版第六次印刷于美国，韦斯利出版公司，1989年，引用8-16页1.递归问题本章研究三个样本问题。
这三个样本问题给出了递归问题的感性知识。
它们有两个共同的特点：它们都是数学家们一直反复地研究的问题；它们的解都用了递归的概念，按递归概念，每个问题的解都依赖于相同问题的若干较小场合的解。
2.约瑟夫环问题我们最后一个例子是一个以Flavius Josephus命名的古老的问题的变形，他是第一世纪一个著名的历史学家。
据传说，如果没有Josephus的数学天赋，他就不可能活下来而成为著名的学者。
在犹太|罗马战争中，他是被罗马人困在一个山洞中的41个犹太叛军之一，这些叛军宁死不屈，决定在罗马人俘虏他们之前自杀，他们站成一个圈，从一开始，依次杀掉编号是三的倍数的人，直到一个人也不剩。
但是在这些叛军中的Josephus和他没有被告发的同伴觉得这么做毫无意义，所以他快速的计算出他和他的朋友应该站在这个恶毒的圆圈的哪个位置。
在我们的变形了的问题中，我们以n个人开始，从1到n编号围成一个圈，我们每次消灭第二个人直到只剩下一个人。
例如，这里我们以设n= 10做开始。
财务风险 外文翻译 外文文献 英文文献 财务风险重要性分析
外文原文How Important is Financial Risk?作者：Sohnke M. Bartram, Gregory W. Brown, and Murat Atamer起止页码：1-7出版日期（期刊号）：September 2009,V ol. 2, No. 4(Serial No. 11)出版单位：Theory and Decision, DOI 10.1007/s11238-005-4590-0Abstract：This paper examines the determinants of equity price risk for a large sample of non-financial corporations in the United States from 1964 to 2008. We estimate both structural and reduced form models to examine the endogenous nature of corporate financial characteristics such as total debt, debt maturity, cash holdings, and dividend policy. We find that the observed levels of equity price risk are explained primarily by operating and asset characteristics such as firm age, size, asset tangibility, as well as operating cash flow levels and volatility. In contrast, implied measures of financial risk are generally low and more stable than debt-to-equity ratios. Our measures of financial risk have declined over the last 30 years even as measures of equity volatility (e.g. idiosyncratic risk) have tended to increase. Consequently, documented trends in equity price risk are more than fully accounted for by trends in the riskiness of firms’ assets. Taken together, the results suggest that the typical U.S. firm substantially reduces financial risk by carefully managing financial policies. As a result, residual financial risk now appears negligible relative to underlying economic risk for a typical non-financial firm.Keywords：Capital structure；financial risk；risk management；corporate finance 1IntroductionThe financial crisis of 2008 has brought significant attention to the effects of financial leverage. There is no doubt that the high levels of debt financing by financial institutions and households significantly contributed to the crisis. Indeed, evidence indicates that excessive leverage orchestrated by major global banks (e.g., through the mortgage lending and collateralized debt obligations) and the so-called “shadow banking system” may be the underlying cause of the recent economic and financial dislocation. Less obvious is the role of financial leverage among nonfinancial firms. To date, problems in the U.S. non-financial sector have been minor compared to thedistress in the financial sector despite the seizing of capital markets during the crisis. For example, non-financial bankruptcies have been limited given that the economic decline is the largest since the great depression of the 1930s. In fact, bankruptcy filings of non-financial firms have occurred mostly in U.S. industries (e.g., automotive manufacturing, newspapers, and real estate) that faced fundamental economic pressures prior to the financial crisis. This surprising fact begs the question, “How important is financial risk for non-financial firms?” At the heart of this issue is the uncertainty about the determinants of total firm risk as well as components of firm risk.Recent academic research in both asset pricing and corporate finance has rekindled an interest in analyzing equity price risk. A current strand of the asset pricing literature examines the finding of Campbell et al. (2001) that firm-specific (idiosyncratic) risk has tended to increase over the last 40 years. Other work suggests that idiosyncratic risk may be a priced risk factor (see Goyal and Santa-Clara, 2003, among others). Also related to these studies is work by Pástor and Veronesi (2003) showing how investor uncertainty about firm profitability is an important determinant of idiosyncratic risk and firm value. Other research has examined the role of equity volatility in bond pricing (e.g., Dichev, 1998, Campbell, Hilscher, and Szilagyi, 2008).However, much of the empirical work examining equity price risk takes the risk of assets as given or tries to explain the trend in idiosyncratic risk. In contrast, this paper takes a different tack in the investigation of equity price risk. First, we seek to understand the determinants of equity price risk at the firm level by considering total risk as the product of risks inherent in the firms operations (i.e., economic or business risks) and risks associated with financing the firms operations (i.e., financial risks). Second, we attempt to assess the relative importance of economic and financial risks and the implications for financial policy.Early research by Modigliani and Miller (1958) suggests that financial policy may be largely irrelevant for firm value because investors can replicate many financial decisions by the firm at a low cost (i.e., via homemade leverage) and well-functioning capital markets should be able to distinguish between financial and economic distress. Nonetheless, financial policies, such as adding debt to the capital structure, can magnify the risk of equity. In contrast, recent research on corporate risk management suggests that firms may also be able to reduce risks and increase valuewith financial policies such as hedging with financial derivatives. However, this research is often motivated by substantial deadweight costs associated with financial distress or other market imperfections associated with financial leverage. Empirical research provides conflicting accounts of how costly financial distress can be for a typical publicly traded firm.We attempt to directly address the roles of economic and financial risk by examining determinants of total firm risk. In our analysis we utilize a large sample of non-financial firms in the United States. Our goal of identifying the most important determinants of equity price risk (volatility) relies on viewing financial policy as transforming asset volatility into equity volatility via financial leverage. Thus, throughout the paper, we consider financial leverage as the wedge between asset volatility and equity volatility. For example, in a static setting, debt provides financial leverage that magnifies operating cash flow volatility. Because financial policy is determined by owners (and managers), we are careful to examine the effects of firms’ asset and operating characteristics on financial policy. Specifically, we examine a variety of characteristics suggested by previous research and, as clearly as possible, distinguish between those associated with the operations of the company (i.e. factors determining economic risk) and those associated with financing the firm (i.e. factors determining financial risk). We then allow economic risk to be a determinant of financial policy in the structural framework of Leland and Toft (1996), or alternatively, in a reduced form model of financial leverage. An advantage of the structural model approach is that we are able to account for both the possibility of financial and operating implications of some factors (e.g., dividends), as well as the endogenous nature of the bankruptcy decision and financial policy in general.Our proxy for firm risk is the volatility of common stock returns derived from calculating the standard deviation of daily equity returns. Our proxies for economic risk are designed to capture the essential characteristics of the firms’ operations and assets that determine the cash flow generating process for the firm. For example, firm size and age provide measures of line of- business maturity; tangible assets (plant, property, and equipment) serve as a proxy for the ‘hardness’ of a firm’s assets; capital expenditures measure capital intensity as well as growth potential. Operating profitability and operating profit volatility serve as measures of the timeliness and riskiness of cash flows. To understand how financial factors affect firm risk, we examine total debt, debt maturity, dividend payouts, and holdings of cash andshort-term investments.The primary result of our analysis is surprising: factors determining economic risk for a typical company explain the vast majority of the variation in equity volatility. Correspondingly, measures of implied financial leverage are much lower than observed debt ratios. Specifically, in our sample covering 1964-2008 average actual net financial (market) leverage is about 1.50 compared to our estimates of between 1.03 and 1.11 (depending on model specification and estimation technique). This suggests that firms may undertake other financial policies to manage financial risk and thus lower effective leverage to nearly negligible levels. These policies might include dynamically adjusting financial variables such as debt levels, debt maturity, or cash holdings (see, for example, Acharya, Almeida, and Campello, 2007). In addition, many firms also utilize explicit financial risk management techniques such as the use of financial derivatives, contractual arrangements with investors (e.g. lines of credit, call provisions in debt contracts, or contingencies in supplier contracts), special purpose vehicles (SPVs), or other alternative risk transfer techniques.The effects of our economic risk factors on equity volatility are generally highly statistically significant, with predicted signs. In addition, the magnitudes of the effects are substantial. We find that volatility of equity decreases with the size and age of the firm. This is intuitive since large and mature firms typically have more stable lines of business, which should be reflected in the volatility of equity returns. Equity volatility tends to decrease with capital expenditures though the effect is weak. Consistent with the predictions of Pástor and Veronesi (2003), we find that firms with higher profitability and lower profit volatility have lower equity volatility. This suggests that companies with higher and more stable operating cash flows are less likely to go bankrupt, and therefore are potentially less risky. Among economic risk variables, the effects of firm size, profit volatility, and dividend policy on equity volatility stand out. Unlike some previous studies, our careful treatment of the endogeneity of financial policy confirms that leverage increases total firm risk. Otherwise, financial risk factors are not reliably related to total risk.Given the large literature on financial policy, it is no surprise that financial variables are,at least in part, determined by the economic risks firms take. However, some of the specific findings are unexpected. For example, in a simple model of capital structure, dividend payouts should increase financial leverage since they represent an outflow of cash from the firm (i.e., increase net debt). We find thatdividends are associated with lower risk. This suggests that paying dividends is not as much a product of financial policy as a characteristic of a firm’s operations (e.g., a mature company with limited growth opportunities). We also estimate how sensitivities to different risk factors have changed over time. Our results indicate that most relations are fairly stable. One exception is firm age which prior to 1983 tends to be positively related to risk and has since been consistently negatively related to risk. This is related to findings by Brown and Kapadia (2007) that recent trends in idiosyncratic risk are related to stock listings by younger and riskier firms.Perhaps the most interesting result from our analysis is that our measures of implied financial leverage have declined over the last 30 years at the same time that measures of equity price risk (such as idiosyncratic risk) appear to have been increasing. In fact, measures of implied financial leverage from our structural model settle near 1.0 (i.e., no leverage) by the end of our sample. There are several possible reasons for this. First, total debt ratios for non-financial firms have declined steadily over the last 30 years, so our measure of implied leverage should also decline. Second, firms have significantly increased cash holdings, so measures of net debt (debt minus cash and short-term investments) have also declined. Third, the composition of publicly traded firms has changed with more risky (especially technology-oriented) firms becoming publicly listed. These firms tend to have less debt in their capital structure. Fourth, as mentioned above, firms can undertake a variety of financial risk management activities. To the extent that these activities have increased over the last few decades, firms will have become less exposed to financial risk factors.We conduct some additional tests to provide a reality check of our results. First, we repeat our analysis with a reduced form model that imposes minimum structural rigidity on our estimation and find very similar results. This indicates that our results are unlikely to be driven by model misspecification. We also compare our results with trends in aggregate debt levels for all U.S. non-financial firms and find evidence consistent with our conclusions. Finally, we look at characteristics of publicly traded non-financial firms that file for bankruptcy around the last three recessions and find evidence suggesting that these firms are increasingly being affected by economic distress as opposed to financial distress.In short, our results suggest that, as a practical matter, residual financial risk is now relatively unimportant for the typical U.S. firm. This raises questions about the level of expected financial distress costs since the probability of financial distress islikely to be lower than commonly thought for most companies. For example, our results suggest that estimates of the level of systematic risk in bond pricing may be biased if they do not take into account the trend in implied financial leverage (e.g., Dichev, 1998). Our results also bring into question the appropriateness of financial models used to estimate default probabilities, since financial policies that may be difficult to observe appear to significantly reduce risk. Lastly, our results imply that the fundamental risks born by shareholders are primarily related to underlying economic risks which should lead to a relatively efficient allocation of capital.Before proceeding we address a potential comment about our analysis. Some readers may be tempted to interpret our results as indicating that financial risk does not matter. This is not the proper interpretation. Instead, our results suggest that firms are able to manage financial risk so that the resulting exposure to shareholders is low compared to economic risks. Of course, financial risk is important to firms that choose to take on such risks either through high debt levels or a lack of risk management. In contrast, our study suggests that the typical non-financial firm chooses not to take these risks. In short, gross financial risk may be important, but firms can manage it. This contrasts with fundamental economic and business risks that are more difficult (or undesirable) to hedge because they represent the mechanism by which the firm earns economic profits.The paper is organized at follows. Motivation, related literature, and hypotheses are reviewed in Section 2. Section 3 describes the models we employ followed by a description of the data in Section 4. Empirical results for the Leland-Toft model are presented in Section 5. Section 6 considers estimates from the reduced form model, aggregate debt data for the no financial sector in the U.S., and an analysis of bankruptcy filings over the last 25 years. Section 6 concludes.2 Motivation, Related Literature, and HypothesesStudying firm risk and its determinants is important for all areas of finance. In the corporate finance literature, firm risk has direct implications for a variety of fundamental issues ranging from optimal capital structure to the agency costs of asset substitution. Likewise, the characteristics of firm risk are fundamental factors in all asset pricing models.The corporate finance literature often relies on market imperfections associated with financial risk. In the Modigliani Miller (1958) framework, financial risk (or more generally financial policy) is irrelevant because investors can replicate the financialdecisions of the firm by themselves. Consequently, well-functioning capital markets should be able to distinguish between frictionless financial distress and economic bankruptcy. For example, Andrade and Kaplan (1998) carefully distinguish between costs of financial and economic distress by analyzing highly leveraged transactions, and find that financial distress costs are small for a subset of the firms that did not experience an “economic” shock. They conclude that financial distress costs should be small or insignificant for typical firms. Kaplan and Stein (1990) analyze highly levered transactions and find that equity beta increases are surprisingly modest after recapitalizations.The ongoing debate on financial policy, however, does not address the relevance of financial leverage as a driver of the overall riskiness of the firm. Our study joins the debate from this perspective. Correspondingly, decomposing firm risk into financial and economic risks is at the heart of our study.Research in corporate risk management examines the role of total financial risk explicitly by examining the motivations for firms to engage in hedging activities. In particular, theory suggests positive valuation effects of corporate hedging in the presence of capital market imperfections. These might include agency costs related to underinvestment or asset substitution (see Bessembinder, 1991, Jensen and Meckling, 1976, Myers, 1977, Froot, Scharfstein, and Stein,1993), bankruptcy costs and taxes (Smith and Stulz, 1985), and managerial risk aversion (Stulz,1990). However, the corporate risk management literature does not generally address the systematic pricing of corporate risk which has been the primary focus of the asset pricing literature.Lintner (1965) and Sharpe (1964) define a partial equilibrium pricing of risk in a mean variance framework. In this structure, total risk is decomposed into systematic risk and idiosyncratic risk, and only systematic risk should be priced in a frictionless market. However, Campbelletal (2001) find that firm-specific risk has increased substantially over the last four decades and various studies have found that idiosyncratic risk is a priced factor (Goyal and Santa Clara,2003, Ang, Hodrick, Xing, and Zhang, 2006, 2008, Spiegel and Wang, 2006). Research has determined various firm characteristics (i.e., industry growth rates, institutional ownership, average firm size, growth options, firm age, and profitability risk) are associated with firm-specific risk. Recent research has also examined the role of equity price risk in the context of expected financial distress costs (Campbell and Taksler, 2003, Vassalou and Xing, 2004, Almeida and Philippon, 2007, among others). Likewise, fundamental economicrisks have been shown to be to be related to equity risk factors (see, for example, Vassalou, 2003, and the citations therein). Choiand Richardson (2009) examine the volatility of the firm’s assets using issue-level data on debt and find that asset volatilities exhibit significant time-series variation and that financial leverage has a substantial effect on equity volatility.How Important is Financial Risk?财务风险的重要性作者：Sohnke M. Bartram, Gregory W. Brown, and Murat Atamer起始页码：1-7出版日期（期刊号）：September 2009,Vol. 2, No. 4(Serial No. 11)出版单位：Theory and Decision, DOI 10.1007/s11238-005-4590-0外文翻译译文：摘要：本文探讨了美国大型非金融企业从1964年至2008年股票价格风险的决定小性因素。
Eudora Welty
Soon after she returned to Jackson in 1931, her father died of leukemia. She took a job at a local radio station and wrote about Jackson society for the Memphis newspaper Commercial Appeal. In 1935, she began work for the Works Progress Administration. As a publicity agent, she collected stories, conducted interviews, and took photographs of daily life in Mississippi. It was here that she observed the Southern life and human relationships that she would later use in her short stories. During this time she also held meetings in her house with fellow writers and friends, a group she called the Night-Blooming Cereus Club. Three years later, she left her job to become a full-time writer.
Novels
Autobiographical essays
The Optimist's Daughter
The Optimist's Daughter is a Pulitzer Prize for Fiction
201205-外文书单勾选结果-数学
推荐（必需、推荐1、推荐2、推isbn书名中文译名作者出版日期出版社Kirkwood, Jam数学物理学与偏2012Academic PresMathematical余用江 史仁坤-推荐19780123869111实分析课程Mcdonald, Joh2012Academic PresA Course In R史仁坤-推荐297801238777412012Oxford UniverLopes, Helena不可压缩流动的余用江 史仁坤9780198528548MathematicalCarlo Berzuin因果性－统计观2012-05WileyCausality - S吴耀琨9780470665565George Dassio2012Cambridge Uni椭球谐波:理论Ellipsoidal H史仁坤-推荐29780521113090Edited by Luc代数几何最新趋2012-04Cambridge UniCurrent Devel张月辉-推荐9780521768252Mark Levi2012-04Princeton Uni为什么猫安全着Why Cats Land麻志浩97806911485402012-03Princeton Uni数值方法Anne Greenbau汤洪阳9780691151229Numerical Met2012-02Princeton UniAmy N. Langvi谁是第一？评定麻志浩9780691154220Who''s #1? :2012Birkhäuser BoRinaldo B. Sc史仁坤-推荐19780817682880从微积分到分析From CalculusLacra Pavel2012Birkhäuser Bo光网络控制的博Game Theory f吴耀琨9780817683214Carlos S. Kub2012Birkhäuser BoHilbert空间算余用江-推荐1Spectral Theo史仁坤9780817683276Joaquin Perez几何分析：偏微2012-06American MathGeometric Ana余用江-推荐1史仁坤97808218499272012-05American Math热势理论导论（Neil A. WatsoIntroduction史仁坤-推荐19780821849989余用江-推荐12012-04American MathDominic JoyceA Theory of G广义唐纳森 -张月辉-推荐97808218527982012-01American MathEdited by P.New Trends In非交换代数新趋张月辉-推荐9780821852972Victor Goryun实奇点与复奇点2012-06American MathReal and Comp余用江-推荐1史仁坤-推荐197808218535972012-05American MathAntonio CampiZeta Function代数和几何中的张月辉-推荐9780821869000调和分析：从傅2012-06American MathMaria CristinHarmonic Anal余用江 史仁坤9780821875667A. B. Sossins几何学（丛书）2012-06American MathGeometries (史仁坤-推荐29780821875711Pascal Cherri2012-08American Math希尔伯特空间中Linear and Qu余用江 史仁坤9780821875766Graham J. LeuCohen-Macaula2012-06American MathCohen-Macaula张月辉9780821875810Loo-Keng Hua2012Cambridge Uni高等数学导论（Introduction史仁坤-推荐29781107020016Geometric Ana几何分析Peter Li2012Cambridge Uni 史仁坤-推荐29781107020641余用江-推荐1概率学者的偏微2012Cambridge UniDaniel W. StrPartial Diffe史仁坤-推荐19781107400528孤子（丛书）T. Miwa2012-04Cambridge Uni 史仁坤-推荐19781107404199Solitons : Di2012Cambridge Uni真实分析讲座Finnur LárussLectures on R余用江 史仁坤-推荐29781107608528边界条件下得偏J. L. Synge2012Cambridge UniThe Hypercirc史仁坤-推荐29781107666559J. J. Thomson2012Cambridge UniJames Clerk M詹姆斯·克拉克史仁坤-推荐29781107670952微积分，国际研Howard Anton2012WileyCalculus, 10T史仁坤-推荐29781118092484常微分方程Greenberg2012John Wiley汤洪阳9781118230022Ordinary Diff2012WH Freeman适量微积分Jerrold E. MaVector Calcul史仁坤-推荐29781429224048Sasho Kalajdz2012Taylor & Fran一本图文并茂的An Illustrate吴耀琨9781439848159David Jerison现代数学的发展2012InternationalCurrent Devel史仁坤-推荐19781571462282Zoltán ádám M线性规划－理论2012-02Nova ScienceLinear Progra吴耀琨9781612095790Edited by Lyn2012Curran Associ2011 第29届系29th Internat史仁坤-推荐29781618392299Inequality ThCho, Yeol Je2012-02Nova Science不等式理论与应9781621008491麻志浩 余用江-推荐1史仁坤-推荐2012-02Alpha Science统计算法Rajan ChattamStatistical A吴耀琨9781842657270Rafael Ayala2012-02Alpha Science代数拓扑学：导Algebraic Top吴耀琨9781842657362Erwan Faou (E2012-01European Math几何数值积分和Geometric Num史仁坤-推荐29783037191002B?rm, Steffen特征值问题的数2012-05de GruyterNumerical Met汤洪阳9783110250336Edited by Sat交换代数进展2012-03de GruyterProgress in C张月辉9783110250343线性与半线性偏Precup, Radu2012-05de GruyterLinear and Se史仁坤-推荐29783110269048余用江-推荐12012-05de Gruyter分布：索伯列夫BhattacharyyaDistributions史仁坤-推荐19783110269277余用江-推荐12012-05de GruyterGodet-Thobie,Young Measure测量空间中的新史仁坤-推荐19783110276404拓扑分析：从基V?th, Martin2012-05de GruyterTopological A余用江-推荐19783110277227Edited by Sat2012-03de Gruyter交换代数进展Progress in C张月辉9783110278590Edited by Giu2012-02Edizioni dellTopics in Mod现代正则性理论余用江 史仁坤-推荐197888764242672012Atlantis PresRamachandran,随机微分方程的史仁坤-推荐29789491216466Stochastic DiHibi Takayuki2012World ScientiGRÖBNER基与现Harmony Of Gr吴耀琨9789814383455Xue Yifeng2012World Scienti算子的稳定扰动史仁坤9789814383592Stable Pertur语言模式的数学Blasi Damian 2012World ScientiMathematical吴耀琨97898143836912012-05World ScientiSinai Yakov E沃尔夫数学奖余用江-推荐19789814390279Wolf Prize InSinai Yakov E2012-05World ScientiWolf Prize In沃尔夫数学奖余用江-推荐19789814390293Chua Leon O2012World ScientiWOLFRAM新学科余用江-推荐19789814390514Nonlinear DynMiller J J H 2012World Scienti奇异摄动问题的Fitted Numeri余用江-推荐19789814390736classno版本卷次币制价格副标题编者页数装帧O1USD89.95432Hardback O1USD134.95688Hardback O1Hardback O1USD90384ClothO1USD110Theory and Applications390Hardback O1USD90428Hardback O1USD18.95240Paperback O1USD95464ClothO1USD29.95266ClothO1EUR54.95250Hardback O1EUR84.95265Hardback O1EUR49.95200Hardback O1USD74191Softcover O1USD86266Hardcover O1USD86199Softcover O1USD99297Paperback O1USD74202Softcover O1USD108347Softcover O1USD58411Softcover O1USD48301Softcover O1USD75378Hardcover O1USD96367Hardcover O18USD200900Hardback O1USD75415Hardback O18USD29.99232Paperback O1USD35118Paperback O1USD39.99130Paperback O1USD76.99A Method for the Approx436Paperback N9USD35A Commemoration Volume 156Paperback O1USD86.95Paperback O1USD114.95544Hardback O1GBP60.99676Hardback O1GBP57.99500Hardback O1USD48182Hardback O1USD89389Hardcover N94USD2954312Softcover O1USD89323Paperback O1GBP29.95304Hardcover O1GBP49.95385Hardcover O1EUR32146Paperback O1EUR29.95190Paperback O1EUR129.95300Hardcover O1EUR49.95290Paperback O1EUR69.95575Paperback O1EUR119.95350Hardcover O1EUR119.95450Hardcover O1EUR129.95300Hardcover O1EUR25200Softcover O1EUR79.95248Hardback O1USD180The Second Crest-Sbm In500Hardback O1USD98300Hardback O1USD94Organization And Evolut250Hardback O1USD188600Hardcover O1USD182580Hardcover N80USD158350HardbackO1USD68Error Estimates In The 192Hardback主题词丛书名描述附件精简装对照语种Mathematical Physics with Partial Differential Equations is for advanced undeThe second edition of A Course in Real Analysis provides a solid foundation oFocussing on the mathematical theory of viscous compressible fluids, this text for gra Time Series Collating a broad collection on co EnglishThe sphere is what might be called a perfect shape. Unfortunately nature is i EncyclopediaMathematics-Topology an Algebraic geometry is one of the m EnglishMathematics Ever wonder why cats land on their EnglishMathematics Numerical Methods provides a clear EnglishMathematics A website''s ranking on Google can EnglishThis comprehensive textbook is intended for a two-semester sequence in analysis. The f Static & DynaOptical networks epitomize complex communication systems, and they comprise tThis work is a concise introduction to spectral theory of Hilbert space operators. Its Math This volume contains research and EnglishMath This book is the first to be devot EnglishMath This book studies generalized Dona EnglishMATHEMATICS / General EnglishMath This volume is a collection of pap EnglishMath The volume contains the proceeding EnglishMath In the last 200 years, harmonic an EnglishMath The book is an innovative modern e EnglishMath This book considers evolution equa EnglishMath This book is a comprehensive treat EnglishThe self-taught mathematician Hua Loo-Keng (1910–1985) spent most of his wor The CambridgeThe aim of this graduate-level text is to equip the reader with the basic too Cambridge StuThis book deals with equations that have played a central role in the interpl Cambridge StuMathematics-Topology an This book was first published in 2EnglishThis is a rigorous introduction to real analysis for undergraduate students, Australian MaOriginally published in 1957, this book was written to provide physicists and engineerOriginally published in 1931, this volume was created to mark the centenary of James CThe new edition of Calculus continues to bring together the best of both new and tradiAfter a brief review of first-order differential equations, this book focuses on seconThis textbook by respected authors helps students foster computational skills and intuThis text explores the beauty of topology and homotopy theory in a direct, engaging, aPapers based on selected lectures given at the Current Development Mathematics Confere Mathematics At the crossroads of mathematics, EnglishElectrical & Computer E Sponsor: System Dynamics SocietyEnglishMathematics This book introduces and exchanges EnglishMathematics STATISTICAL ALGORITHMS integrates EnglishMathematics ALGEBRAIC TOPOLOGY: An Introductio EnglishMath The goal of geometric numerical in EnglishMathematics Eigenvalues and eigenvectors of ma EnglishMathematics This is the first of two volumes o EnglishMathematics The text is intended for students EnglishMathematics This textbook is addressed to the EnglishMathematics In recent years, technological pro EnglishMathematics This monograph aims to give a self EnglishMathematics This is the second of two volumes EnglishPartial Differential Eq This book contains lecture notes o EnglishThe subject theory is important in finance, economics, investment strategies, Atlantis StudThis volume consists of research papers and expository survey articles presented by thThis book provides a broad introduction to the generalized inverses, Moore–Penrose inThe Mathematical Structure of Language Patterns provides a comprehensive comp World ScientiMathematics-Geometry & This invaluable book features bibl EnglishMathematics-Geometry & This invaluable book features bibl EnglishWhen not immersed in science, he relaxes by searching for Wagner's leitmotifs World ScientiSince the first edition of this book, the literature on fitted mesh methods fundergraduate and beginning graduate students taking a course on mathematical physics taught out of mion of real analysis concepts and principles, presenting a broad range of topics in a clear and concisuids, this text for graduates and researchers in applied mathematics develops the most recent ideas and concepts in the ly nature is imperfect and many bodies are better represented by an ellipsoid. The theory of ellipsoidal harmonics, orience in analysis. The first four chapters present a practical introduction to analysis by using the tools and concepts hey comprise the Internet’s infrastructural backbone. The first of its kind, this book develops the mathematical frame rt space operators. Its emphasis is on recent aspects of theory and detailed proofs, with the primary goal of offering ost of his working life in China and suffered at first hand the turbulence of twentieth-century Chinese politics. His i the basic tools and techniques needed for research in various areas of geometric analysis. Throughout, the main theme in the interplay between partial differential equations and probability theory. Most of this material has been treated ate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any precon physicists and engineers with a means of solving partial differential equations subject to boundary conditions. The tex he centenary of James Clerk Maxwell's birth. Comprised of ten essays dealing with various aspects of Maxwell's life and t of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. The s book focuses on second-order equations with constant coefficients that derive their general solution using only resul ational skills and intuitive understanding with a careful balance of theory, applications, historical development and o n a direct, engaging, and accessible manner while illustrating the power of the theory through many, often surprising, ent Mathematics Conference, held in November 2010 at Harvard University.</p>nt strategies, health sciences, environment, industrial engineering, etc.rticles presented by the invited speakers of the conference on “Harmony of Gröbner Bases and the Modern Industrial Soc rses, Moore–Penrose inverses, Drazin inverses and T–S outer generalized inverses and their perturbation analyses in t rehensive compilation, description, and discussion of recent advancements on quantitative linguistics carried out from r's leitmotifs, musing over Kandinsky's chaos, and contemplating Wittgenstein's inner thoughts.This penultimate volumemesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes haof math departments. The text presents some of the most important topics and methods of mathematicaloncise manner. The book is excellent at balancing theory and applications with a wealth of examples aas and concepts in the theory of compressible fluids, in particular, the question of existence for optimal values of th psoidal harmonics, originated in the nineteenth century, could only be seriously applied with the kind of computationalhe tools and concepts of calculus. The last five chapters present a first course in analysis. The presentation is clear the mathematical framework needed from a control perspective to tackle various game-theoretical problems in optical net mary goal of offering a modern introductory textbook for a first graduate course in the subject. The coverage of topics hinese politics. His influence has been credited with inspiring generations of mathematicians, while his papers on numb ghout, the main theme is to present the interaction of partial differential equations and differential geometry. More s rial has been treated elsewhere, but it is rarely presented in a manner that makes it easy to understand for people who book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete order ry conditions. The text gives a systematic and unified approach to a wide class of problems, based on the fact that the of Maxwell's life and achievements, the text includes contributions from the following figures: J. J. Thomson; Max Pla eaching calculus. The author team's extensive experience teaching from both traditional and innovative books and their ution using only results described previously. Higher-order equations are provided since the patterns are more readily ical development and optional materials.ny, often surprising, applications. It offers a comprehensive presentation that uses a combination of rigorous argument Modern Industrial Society”. Topics include computational commutative algebra, algebraic statistics, algorithms of D-m urbation analyses in the spaces of infinite-dimensional. This subject has many applications in operator theory, operatotics carried out from the perspective of complex systems. It provides an overview of the emergence and construction of is penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Pears, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential etical physics. The premise is to study in detail the three most important partial differential equatioles and exercises. The authors take a progressive approach of skill building to help students learn toor optimal values of the adiabatic exponent, which up to now has been unsolved.e kind of computational power available in recent years. This, therefore, is the first book devoted to ellipsoidal harm e presentation is clear and concise, allowing students to master the calculus tools that are crucial in understanding a problems in optical networks. In doing so, it aims to help design control algorithms that optimally allocate the resour The coverage of topics is thorough, as the book explores various delicate points and hidden features often left untreahile his papers on number theory are regarded as 'virtually an index to the major activities in that subject during the ential geometry. More specifically, emphasis is placed on how the behavior of the solutions of a PDE is affected by the derstand for people whose background is probability theory. Many results are given proofs designed for readers with lim nts of a complete ordered field. All of the standard topics are included, as well as a proper treatment of the trigonom ed on the fact that the solution may be viewed as a point in function-space, this point being the intersection of two l J. J. Thomson; Max Planck; Albert Einstein; Joseph Larmor; James Jeans; William Garnett; Ambrose Fleming; Oliver Lodge vative books and their expertise in developing innovative problems put them in an unique position to make this new curr tterns are more readily grasped by students. Stability and fourth order equations are also discussed since these topicon of rigorous arguments and extensive illustrations to facilitate an understanding of the material. The author covers tics, algorithms of D-modules and combinatorics. This volume also provides current trends on Gröbner bases and will sti perator theory, operator algebras, global analysis and approximation theory and so on.S table Perturbations of Operatorsce and construction of linguistic structures, addressing those concepts that have become the key ideas of complexity: o al cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules,partial differential equations. In the revised version of this book, the reader will find an introduction to the basicuations in the field - the heat equation, the wave equation, and Laplace’s equation. The most commonrn to absorb the abstract. Real world applications, probability theory, harmonic analysis, and dynamied to ellipsoidal harmonics. Topics are drawn from geometry, physics, biosciences and inverse problems. It contains cla ial in understanding analysis. From Calculus to Analysis prepares readers for their first analysis course—important be ly allocate the resources of these networks. </p>With its fresh problem-solving approach, 'Game Theory for Control of O ures often left untreated. </p>Spectral Theory of Operators on Hilbert Space</i> is addressed to an interdisciplinary ahat subject during the first half of the twentieth century.' Introduction to Higher Mathematics is based on the lecture PDE is affected by the geometry of the underlying manifold and vice versa. For efficiency the author mainly restricts h d for readers with limited expertise in analysis. The author covers the theory of linear, second order partial differen atment of the trigonometric functions, which many authors take for granted. The final chapters of the book provide a ge intersection of two linear subspaces orthogonal to one another. Using this method the solution is located on a hyperci Fleming; Oliver Lodge; Richard Glazebrook; Horace Lamb. This book will be of value to anyone with an interest in Maxwe to make this new curriculum meaningful for those going into mathematics and those going into the sciences and engineer ssed since these topics typically appear in further study for engineering and science majors. In addition to applicatial. The author covers basic topology, ranging from the axioms of topology to proofs of important theorems. He also discK ey Features: • Compre ner bases and will stimulate further development of many research areas surrounding Gröbner bases.urbations of Operators and Related Topics is self-contained and unified in presentation. It may be used as an advanced ideas of complexity: order and entropy, information, correlation, self-organization, network patterns, evolution, and st only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from aroduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations.ommon techniques of solving such equations are developed in this book, including Green’s functions, tynamical systems theory are included, offering considerable flexibility in the choice of material to c oblems. It contains classical results as well as new material, including ellipsoidal bi-harmonic functions, the theory is course—important because many undergraduate programs traditionally require such a course. Undergraduates and some a Theory for Control of Optical Networks' is a unique resource for researchers, practitioners, and graduate students in a an interdisciplinary audience of graduate students in mathematics, statistics, economics, engineering, and physics. Itis based on the lectures given by Hua at the University of Science and Technology of China from 1958. The course reflec thor mainly restricts himself to the linear theory and only a rudimentary background in Riemannian geometry and partial order partial differential equations of parabolic and elliptic type. Many of the techniques have antecedents in proba f the book provide a gentle, example-based introduction to metric spaces with an application to differential equations is located on a hypercircle in function-space, and the approximation is improved by reducing the radius of the hypercir th an interest in Maxwell and his key role in the development of physics and mathematics.e sciences and engineering. This new text exhibits the same strengths from earlier editions including an emphasis on n addition to applications to engineering systems, applications from the biological and life sciences are emphasized. theorems. He also discusses the classification of compact, connected manifolds, ambient isotopy, and knots, which leadK ey Features:• A guid • Comprehensive treatment of Gröbner bases • Articles by leading figures in the mathematics worlds.be used as an advanced textbook by graduate students. It is also suitable for researchers as a reference. The proofs ofT he focus of quantitative linguistics is to detect, characterize, and explain the patter terns, evolution, and strategy.174 local rules from an arbitrary initial bit-string configuration. This is contrary to the well-known folklore that 2differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perns, the Fourier transform, and the Laplace transform, which all have applications in mathematics andto cover in the classroom. The accessible exposition not only helps students master real analysis, b functions, the theory of images in ellipsoidal geometry and vector surface ellipsoidal harmonics, which exhibit an inte ergraduates and some advanced high-school seniors will find this text a useful and pleasant experience in the classroom graduate students in applied mathematics and systems/control engineering, as well as those in electrical and computer e ering, and physics. It will also be useful to working mathematicians using spectral theory of Hilbert space operators,958. The course reflects Hua's instinctive technique, using the simplest tools to tackle even the most difficult proble n geometry and partial differential equations is assumed. Originating from the author's own lectures, this book is an i e antecedents in probability theory, although the book also covers a few purely analytic techniques. In particular, a c ifferential equations on the real line. The author's exposition is concise and to the point, helping students focus on radius of the hypercircle. The complexities of calculation are illuminated throughout by simple, intuitive geometricalcluding an emphasis on modeling and a flexible approach to technology.nces are emphasized. Ecology and population dynamics are featured since they involve both linear and nonlinear equatio and knots, which leads to coverage of homotopy theory.• A guidebook for graduate students • The reader can see a panoramic view of Gröbner baseshematics worldference. The proofs of statements and explanations in the book are detailed enough that interested readers can study it and explain the patterns of language by means of mathematical tools. The last two decades have witnessed a sharp increa-known folklore that 256 local rules are necessary, leading to the new concept of quasi-global equivalence.Another surpnts for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwiseand physics far beyond solving the above equations. The book’s focus is on both the equations and tis, but also makes the book useful as a reference., which exhibit an interesting analytical structure. Extended appendices provide everything one needs to solve formally rience in the classroom or as a self-study guide. The only prerequisite is a standard calculus course.ectrical and computer engineering.</p>lbert space operators, as well as for scientists wishing to apply spectral theory to their field. </p>e most difficult problems, and contains both pure and applied mathematics, emphasising the interdependent relationships ures, this book is an ideal introduction for graduate students, as well as a useful reference for experts in the field. ues. In particular, a chapter is devoted to the DeGiorgi-Moser-Nash estimates and the concluding chapter gives an intro ping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to th intuitive geometrical pictures. This book will be of value to anyone with an interest in solutions to boundary value pr and nonlinear equations, and these topics form one application thread that weaves through the chapters. Diffusion ofK ey Features: • The stable perturbation theory of generalized inverses, Drazin inv ed readers can study it by themselves.itnessed a sharp increase in the interest of physicists and applied mathematicians on language, and it is not surprisinquivalence.Another surprising result is the introduction of a simple, yet explicit, infinite bit string called the supesured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but thend their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Fuolve formally boundary value problems. End-of-chapter problems complement the theory and test the rea ependent relationships between different branches of the discipline.experts in the field.chapter gives an introduction to the theory of pseudodifferential operators and their application to hypoellipticity, i y of them adding to the theory in the text. The book is perfect for second-year undergraduates and for more advanced st ns to boundary value problems in mathematical physics.hapters. Diffusion of material, heat, and mechanical and electrical oscillators are also important in biological and e d inverses, Drazin inverses and index of a Fredholm element relative to a homomorphismare are of unique interest in thind it is not surprising that this explosion was parallel to the expansion of the interest in complex systems. This book string called the super string S, which contains all random bit strings of finite length as sub-strings. As an illustra。
数学教研室英文
The Department of Mathematics at [University Name] stands as a beacon of academic excellence and innovation in the field of mathematics. With a rich history of nurturing talent and fostering research, the department has earned a reputation for its rigorous curriculum, distinguished faculty, and vibrant academic community. This article aims to provide an overview of the department's achievements, its curriculum, faculty, research initiatives, and the impact it has on both students and the broader academic community.I. History and BackgroundEstablished in [Year], the Department of Mathematics has grown from a small group of dedicated faculty and students to a vibrant and diverse academic community. Over the years, the department has played a pivotal role in shaping the mathematical landscape of [University Name]. It has produced numerous notable alumni who have gone on to achieve great success in various fields, including academia, industry, and public service.The department's commitment to excellence is evident in its long-standing partnership with leading research institutions and industry partners. This collaboration has not only enriched the department's research portfolio but has also provided students with invaluable opportunities for internships, co-op programs, and real-world problem-solving experiences.II. Curriculum and Academic ProgramsThe Department of Mathematics offers a comprehensive curriculum that spans undergraduate and graduate programs, catering to a diverse range of interests and career aspirations. The curriculum is designed to provide students with a strong foundation in mathematical theory and practical skills, enabling them to excel in their chosen fields.A. Undergraduate ProgramsThe undergraduate program in mathematics provides students with a solid understanding of the fundamental concepts and techniques of mathematics. The curriculum includes a variety of courses, such as calculus, linearalgebra, real analysis, and abstract algebra. Students can also choose from a range of elective courses to specialize in areas such as applied mathematics, statistics, and computational mathematics.The department also offers a dual-degree program in mathematics and another field, such as engineering or computer science, to provide students with a broader skill set and enhance their employability.B. Graduate ProgramsThe graduate program in mathematics is designed to prepare students for careers in research, academia, and industry. The program offers both a Master's and a Ph.D. degree, with concentrations in pure mathematics, applied mathematics, and interdisciplinary fields.The curriculum includes advanced courses in algebra, analysis, geometry, and topology, as well as research seminars and workshops. Graduate students are encouraged to engage in original research under the guidance of experienced faculty members.III. Faculty and Research InitiativesThe Department of Mathematics boasts a distinguished faculty of scholars and educators who are committed to excellence in teaching and research. The faculty members are actively involved in various research initiatives, contributing to the advancement of mathematical knowledge and its applications in various fields.A. Research AreasThe department's research initiatives span a wide range of topics, including:1. Algebra and Number Theory2. Analysis and Differential Equations3. Geometry and Topology4. Probability and Statistics5. Computational Mathematics6. Mathematical PhysicsB. Collaborations and PartnershipsThe faculty members collaborate with researchers from other departments and institutions, both within and outside the country. These partnerships have led to numerous joint publications, research grants, and international conferences.IV. Impact on Students and the Academic CommunityThe Department of Mathematics has a profound impact on its students, preparing them for successful careers and fostering their intellectual growth. The department's commitment to excellence is reflected in the following aspects:A. Student SuccessThe department has a strong track record of student success, with a high percentage of graduates finding employment in their chosen fields. Many alumni have gone on to pursue advanced degrees and achieve notable accomplishments in their respective careers.B. Academic Community EngagementThe department actively engages with the academic community through various events, such as seminars, workshops, and conferences. These events provide a platform for faculty, students, and researchers to share their knowledge, exchange ideas, and foster collaboration.C. Public EngagementThe department is committed to promoting mathematics and itsapplications to the general public. The department organizes outreach programs, workshops, and lectures aimed at inspiring and educating the public about the beauty and importance of mathematics.V. ConclusionThe Department of Mathematics at [University Name] is a premier institution that continues to excel in teaching, research, and publicengagement. With a dedicated faculty, rigorous curriculum, and vibrant academic community, the department has established itself as a leading center for mathematical education and research. As the field of mathematics continues to evolve, the department is poised to play an even more significant role in shaping the future of mathematics and its applications.。
英语作文我们班上的一位模范学生简单
英语作文我们班上的一位模范学生简单全文共3篇示例，供读者参考篇1One of the model students in our class is Lily. She is a diligent and responsible student who always impresses us with her excellent performance both academically and socially.First of all, Lily is an outstanding student in terms of academic achievement. She consistently ranks at the top of our class in all subjects. She is always well-prepared for class, participates actively in discussions, and completes her assignments with great care and attention to detail. Whenever we have group projects, Lily is the one everyone wants to work with because we know she will do her part and deliverhigh-quality work.In addition to her academic excellence, Lily is also a kind and compassionate person. She is always willing to lend a helping hand to her classmates, whether it's helping them understand a difficult concept in class or supporting them through a tough time. She is well-liked by everyone in our class because of her friendly and approachable demeanor.Furthermore, Lily is a natural leader and is always willing to take on responsibilities. She is an active member of the student council and is involved in organizing various events and activities in our school. She demonstrates excellent leadership skills by coordinating with her peers, teachers, and other staff members to ensure that everything runs smoothly.Overall, Lily is a role model for all of us in our class. Her dedication to her studies, her compassion towards others, and her leadership skills make her an exemplary student. We all look up to her and aspire to be more like her. We are truly lucky to have Lily as a classmate and a friend.篇2One Model Student in Our ClassIn our class, there is a student who stands out as a model of excellence and dedication. His name is Jack, and he embodies all the qualities of a top student.First and foremost, Jack is incredibly hardworking. He is always the first one to arrive in class and the last one to leave. He spends countless hours studying and preparing for exams, and his dedication to his schoolwork is truly inspiring. Whenever we have a group project, Jack is the one who takes the lead andensures that everything is done on time and to the highest standard.Not only is Jack hardworking, but he is also extremely intelligent. He consistently receives top marks in all of his subjects and never hesitates to ask questions or seek help when he doesn't understand something. His thirst for knowledge is unmatched, and he is always eager to learn new things and expand his horizons.In addition to his academic achievements, Jack is also a role model in terms of his behavior and attitude. He is always respectful to his teachers and classmates, and he never hesitates to lend a helping hand to those in need. His positive attitude and kind demeanor make him a pleasure to be around, and he is truly a shining example of good character.Overall, Jack is a true model student in every sense of the word. His hard work, intelligence, and positive attitude make him an invaluable member of our class, and we are lucky to have him as a peer and a friend. I have no doubt that Jack will continue to achieve great things in the future, and I am excited to see where his path leads him.篇3In our class, there is a student who stands out as a role model for others. His name is Peter, and he is an exceptional student who excels in both academics and extracurricular activities.Peter is known for his outstanding academic performance. He consistently scores the highest marks in all subjects and is always willing to help his peers understand difficult concepts. He has a thirst for knowledge and is constantly seeking ways to learn and improve himself. Peter is diligent and hardworking, never hesitating to put in extra hours of study to achieve his goals.Aside from academics, Peter is also actively involved in various extracurricular activities. He is a talented musician who plays the violin in the school orchestra. He also participates in debates and public speaking competitions, showcasing his excellent communication skills and leadership abilities. Peter is a well-rounded student who excels not only in the classroom but also in other areas of school life.Moreover, Peter is a kind and considerate individual who is always willing to lend a helping hand to those in need. He is well-liked by his classmates and teachers for his friendly and approachable nature. Peter is a true role model who inspiresothers to strive for excellence and be the best version of themselves.In conclusion, Peter is truly a model student who embodies the qualities of hard work, dedication, and kindness. He sets a high standard for his peers to follow and serves as an inspiration to all who know him. We are lucky to have him in our class, and I have no doubt that he will continue to achieve great things in the future.。
班级物品作文英语翻译
In the realm of education,the classroom is not merely a space for learning but also a hub of shared resources and communal responsibility. One aspect of this shared environment is the management and use of classroom items.These items,ranging from textbooks and stationery to technological devices and furniture,play a crucial role in enhancing the learning experience for students.Here,we delve into the significance of these items and the practices adopted in managing them within a classroom setting.Classroom items are essential tools that facilitate the educational process. Textbooks,for instance,are the cornerstone of any classroom,providing a structured and comprehensive source of information on a given subject. They are carefully selected by educators to align with the curriculum and cater to the learning needs of the students.The presence of these books in the classroom ensures that students have access to a wealth of knowledge at their fingertips.Stationery,another category of classroom items,includes pens,pencils, erasers,and notebooks.These are the basic tools that students use to engage with the material and express their understanding.The availability of such items in the classroom is vital,as it allows students to actively participate in the learning process without being hindered by the lack of necessary supplies.Technological devices,such as computers and tablets,have become increasingly prevalent in classrooms.These items not only provide access to a vast array of digital resources but also enable interactive learningexperiences.They can be used for research,presentations,and even collaborative projects,fostering a more dynamic and engaging learning environment.Furniture,including desks,chairs,and storage units,is another critical component of the fortable and functional furniture contributes to a conducive learning atmosphere,allowing students to focus on their studies without distractions.Moreover,wellorganized storage solutions ensure that classroom items are easily accessible and wellmaintained.The management of these classroom items is a shared responsibility among teachers,students,and sometimes even parents.Teachers are often tasked with the procurement and distribution of these items, ensuring that they are available when needed.They may also establish rules and guidelines for the use and care of these items to prevent misuse or damage.Students,on the other hand,are expected to treat classroom items with respect and care.This includes handling them gently,using them responsibly,and returning them to their designated places after use.By doing so,they contribute to the sustainability of these resources and help maintain a clean and organized learning environment.In some cases,parents may also be involved in the management of classroom items,particularly when it comes to fundraising for new resources or volunteering to help with the upkeep of the classroom.Theirsupport can be invaluable in ensuring that the classroom is wellequipped and that students have access to the tools they need for learning.In conclusion,classroom items are an integral part of the educational experience.They provide the necessary tools for learning and contribute to a conducive learning environment.The management of these items is a collective effort that requires the cooperation of teachers,students,and parents.By working together,they can ensure that these resources are effectively utilized and maintained,ultimately enhancing the learning experience for all.。
数学专业英语课文翻译(吴炯圻)
第一章：数学方程与比例1-A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。
与数学反过来，为实践服务和所有字段中的伟大作用。
没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。
然后，开发出的几何土地和三角测量的问题来自测量的问题。
若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。
17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。
17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。
数学家研究理念和主张。
所有命题公理、假设、定义和定理都。
符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。
公式、数字和图表是阿拉伯数字1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。
数学中的结论得到主要由逻辑推理和计算。
长期的数学史上，以中心地点的数学方法被占领逻辑扣除。
现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。
在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。
1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。
因此(a-5）= 一5a 和x 3 = 5 是方程。
方程的两种——身份和方程的条件。
方程的算术或代数的身份。
这种方程中两名成员是相似的或成为相似的指示操作的性能。
因此12-2=2+8,(m+n)(m-n) = m n 是身份。
1—c 比与测量今天的思想沟通往往根据编号和数量的比较。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
河南理工大学本科毕业设计（论文）外文文献资料翻译院（系部）数学信息科学学院专业名称数学与应用数学年级班级 2009级01班学生姓名闻晶晶学生学号 3109110101082013年6月3 日一类负相伴随机阵列部分和的精致大偏差汪世界 王伟 王文胜（安徽大学数学科学院，合肥，230039） （华东师范大学金融统计学院，上海，200241）摘要本文在一些适当的条件下得到了多风险模型中负相伴随机阵列的精致大偏差，推广了一些已知的结果，同时表明在多风险模型中负相伴结构对精致大偏差同样不具有敏感性.关键词：负相伴随机阵列，大偏差，一致变化尾 学科分类号：O212.3.§1. 引言近年来，很多学者都总结出重尾分布和的精致大偏差，因为用大偏差概率的损失过程来描述破产概率的估计，是一个非常重要的目标风险管理.为此，我们参阅了一些最新文献，如Ng et al.(2004)，Tang （2006），Wang et al.(2006)，Liu,(2007)，Chen and Zhang （2007），,Yang et al.（2009），Liu(2009)等.然而，他们只研究单一类型的风险，即他们总是假定保险公司只提供一种保险合同.在实际生活中，这种假设是不存在的，所以，研究多风险模型的大偏差问题是很有价值的.为此，Wang and Wang （2007）首次把精致大偏差的相关结论扩展到独立索赔多风险模型中.显然，Wang and Wang （2007）的独立性假设是极其不符合现实的.Alam and Saxena （1981）及Joag-Dev and Proschan （1983）中介绍到这种较弱的结构是负相关的.定义1.1 d 是正整数，{}dn n X N∈;是有限的实值随机变量.我们称一维随机变量是负相伴的，如果对dN 任意两个不相交的非空子集T S ,都成立()()()0j ;,;≤∈∈T X g S i X f Cov j i其中()S i X f i ∈;和()T j X g j ∈;是任意两个使得协方差存在且对任意变量都增加的函数.在本文中，我们称{}ki j X 1ij 1,=≥是NA 序列，其中{}1,≥j X ij ，k i ,,1 =，表示关于i 的同分布损失函数()x F i ，满足∞<=i ij EX μ，()()01i >-=x F x F i ，()∞∞-∈，x .我们同样可以假定，对任意k i ,,1 =，C F ∈i ，如果满足()()1limliminf1y x F xy F x →∞= 或 ()()1lim limsup1y x F xy F x →∞=，我们说分布函数F 属于重尾子集C ，其中分布函数F 具有一致变化尾.Cline et al.（1994）也曾研究过重尾子集C ，他称其为‘中间正规变量’.另一个著名的重尾子集被称为控制变量集（D 族）. 一个分布函数F 支撑在(),-∞∞上且属于D ，当且仅当对任意0y<1<（或某些()0,1y ∈），()()l i m s u p→∞<∞x F xy F x 成立.对于像R ，S ，L 等其他重尾子集的更多细节，参考文献Ng et al.（2004）或者Wang and Wang （2007）.集合()log =inf ,1log **⎧⎫->⎨⎬⎩⎭：FF y J y y ，其中，()()()liminf x F y F xy F x *→∞=.在Tang （2006）的专业用语中，F J *被称为F 的上 Matuszewska指数.{},1,,=i n i k 是k 正整数序列.为方便起见，令∑==ii n j ij n X S 1，k i ,,1 =，()111;,,===∑∑in kk ij i i S k n n X .(){},1,,=i N t i k 是一列关于索赔次数的独立非负整数计数过程，我们假定{}1,1kij i X j =≥和(){},1,,=i N t i k 是相互独立的，且当()1,,→∞=t i k 时，()()i i EN t t λ=→∞.令()()11;,0i N t kij i i S k t X t ===≥∑∑，Tang(2006)研究了带有一致变化尾的负相伴随机变量和的精致大偏差，Chen et al.（2007）和Liu （2007）把Tang （2006）的研究结果扩展到负相伴随机变量的随机和，它们各自具有一致变化尾.在本文中，我们研究多风险模型中的负相伴随机阵列部分和的精致大偏差.我们对一些已知的结论进行推广，发现在多风险模型中精致大偏差的渐近同样呈现负相伴结构.后面的章节安排如下：在第二节中，我们介绍一些预备知识，主要的结果和证明将在第三章节给出，第四章将会给出一个应用程序的主要结果.§2预备知识在这一章节，我们按照惯例用符号n 1nii S X==∑，()()1t N t N ii S X==∑以及G F ≈表示0liminflimsup <≤<∞F FG G. 显然，如果F G ∈，那么，对任意0>c ，()()≈F cx F x .这在T ang and Yan （2002）中同样也可以看到.下面我们给出一些证明定理的引理，引理2.1是对Joag-Dev 和Proschan(1983)的轻微调整.引理2.1 设{},1k X k n ≤≤为一NA 随机变量序列，1,,m A A 为{}1,,n 的任意一列两两不交子集.如果{},1,,=i f i m 为对每个分量不降（或不增）函数，()()11,,,,∈∈j m j m f X j A f X j A 仍为NA 序列，且对任意 ,2,1=n 以及12,,,n x x x ，有()11==⎛⎫>≤> ⎪⎝⎭∏n ni i k k k i P X x P X x 以及()11n ni i k k k i P X x P X x ==⎛⎫≤≤≤ ⎪⎝⎭∏引理2.2 设{},1,2,=k X k 是一列同分布的NA 随机变量，共同发布()F x D ∈，期望为 μ，且()()(),.F x F x x ο-=→∞如果存在某1>r ，使得()1-<∞rE X ，{}11max 0,X X -=-.则对任意给定的常数0γ>，当n →∞时，对x n γ≥一致地有()()(),n P S n x nF x μο-≤=对x n γ≥一致成立，即()()limsup 0n n x nP S n x nF x γμ→∞≥-≤-=.证明：由于{},1,2,=k X k 为NA 序列，根据定义，{},1,2,-=k X k 同样是NA 序列.由Tang（2006）的引理2.3得，对任意0γ>，*>F p J ，必存在某正常数0υ与C ，使得对任意γ>x n ，1,2,=n 有()()()10μμμυ--≤-=-+≥≤-+>+p n n P S n x P S n x nP X Cx()0pnF Cxυμ-≤-++. (2.1)显而易见，对任意给定的*>F p J ，则当x →∞时，有()()p x F x ο-=；对于较大的x ，()0F v x()F x .在（2.1）中，利用条件()()()F x F x ο-=，我们得到()()()()0p n P S n x n x Cx nF x nF x μυμ--≤--++≤()()()0pn F Cx nF x ου-+=()()()1ο⎛⎫≈+ ⎪⎝⎭F x C n F x . 从而引理2.2证毕.注1（1）在引理2.2的证明中，对任意0ε>，用x ε替换x ，当n →∞时，()()()n P S n x nF x μεο-≤-= （2.2） 对x n γ≥一致成立.（2）设{}1,1kij i X j =≥是负相伴序列，且()()1,,=i F x i k 满足定理2.2 的条件.我们可以用数学归纳法证明，对任意0ε>，当i n →∞时，()111i k kk n i i i i i i i P S n x n F x μεο===⎛⎫⎛⎫-≤-= ⎪ ⎪⎝⎭⎝⎭∑∑∑ （2.3）对所有{}max ,1,,γ≥=i x n i k 一致成立.事实上，对2k =和任意()0,12δ∈，由引理2.1,引理2.2和负相伴性质，有2211i n i i i i P S n x με==⎛⎫-≤- ⎪⎝⎭∑∑()()()()12112211n n P S n x P S n x μδεμδε≤-≤--+-≤--()()121122n n P S n x P S n x μδεμδε+-≤--≤- ()()()()()()()()11221122n F x n F x n F x n F x οοοο=++()()()1122n F x n F x ο=+ （2.4） 因此，（2.3）可以直接由（2.4）用归纳假设证出.§3 主要结论及其证明定理 3.1 设{}1,1kij i X j =≥为NA 随机阵列，对任意{}1,,,,1=≥ij i k X j 具有相同的分布()i F x C ∈，有限期望为i μ，且满足()(),i i xF x F x x -=→∞.{},1,2,,=i n i k 为任意给定的k 个正整数，如果对任意的1,,=i k ，存在某1>r 使得<∞rij E X .则对任意给定的0>r ，对所有的1,,=i k ，当n →∞时，有()()111;,,μ==⎛⎫-> ⎪⎝⎭∑∑kkk i i i ii i P S k n n n x n F x , （3.1）对所有{}()max ,1,,k γ≥==∆：i x n i k 一致成立.注2 假定所有()()1,,=i F x i k 是同分布函数，那么(3.1)可以推出Tang(2006)的定理1.1.特别的，如果我们已知{}1,1kij i X j =≥是非负随机变量序列，很容易可以验证定理3.1的条件一定成立.因此，（3.1）验证Liu(2007)的定理2.1.如果{},1,1,,≥=ij X j i k 是独立随机阵列，由(3.1)推出Wang and Wang(2007)的引理3.1.证明 我们用数学归纳法证明(3.1).当2k =时，首先，显然有()()()()122121,212212;,lim inf 1μ=→∞≥∆⎛⎫-> ⎪⎝⎭≥+∑i i i n n x P S n n n x n F x n F x . （3.2）注意，对任意01ε<<，任意0>x ，()()1211222;,μμ-->P S n n n n x()()1211221,μεμε≥->+->-n n P S n x S n x()()2122111,μεμε+->+->-n n P S n x S n x()()()1211221,1μεμε-->+->+n n P Sn x S n x123:I I I =+-. （3.3）先估计1I ,注意到，()()12112221,μεμε=->+->-n n I P S n x S n x()()()1211221μεμε≥->+--≤-n n P S n x P S n x . （3.4） 由Tang （2006）定理2.1得，对任意01δ<<，当1n →∞时， ()()()1111111sup11γμεδε≥->+-<+n x n P S n xn F x. （3.5）又()()()22xF x F x ο-=，则更有()()()22F x F x ο-=成立，由引理2.2，对2γ>x n 一致的有，()()()22222n P S n x n F x μεο-≤-=.综合以上各式，对充分大的1n ，2n ，()()()()()1112211I n F x n F x δεο≥-++ （3.6） 对()2x ≥∆一致成立.同理亦有对充分大的12,n n ，()()()()222111I n F x n F x εο≥++.对()2x ≥∆一致成立.最后我们估计3I ，由于{}12111212,,,,,n n X X X X 为NA ，则由Wang andWang(2007)得，()()()01lim lim sup10,1,2i ii n x ni F x i F x εγε↓→∞≥+-== （3.7）注意到{}12111212,,,,,n n X X X X 是NA ，1n S ，2n S 也是NA.因此，由T ang(2006)的引理2.1和（3.11）得，()()()()123112211μεμε≥->+->+n n I P S n x P S n x()()()()()21122111n F x n F x δεε≤+++()()()211221n F x n F x δ≤+ （3.8） 联合(3.3)-(3.8)得，当12,n n →∞时，对()2x ≥∆一致地有， ()()1211222;,μμ-->P S n n n n x()()()()()()()2112211221n F x n F x n F x n F x δο≥-+++此外，令0δ↓，我们得到(3.2).下面，我们再证()()()()()12121122,211222;,limsup sup1μμ→∞≥∆-->≤+n n x P S n n n n x n F x n F x . （3.9）任意给定()210，∈ε以及0>x ，由NA 性质、引理2.1和T ang （2006）的定理2.1,有， ()()1211222;,μμ-->P S n n n n x()()()()12112211μεμε≤->-+->-n n P S n x P S n x()()121122μεμε+->->n n P S n x P S n x()()()()()()11221122ο≈++++n F x n F x n F x n F x （3.10） 从而（3.9）成立.这样（3.1）对2k =时成立.假定（3.1）对1k -时成立，下面往证结果对k 时也成立.我们采用类似（3.3）的分解法，可得到()11;,...,μ=⎛⎫-> ⎪⎝⎭∑kk i P S k n n n x()1111,μεμε--==⎛⎫≥->+->- ⎪⎝⎭∑∑i k k i k n i i n k k i i P S n x S n x()1111,μεμε--==⎛⎫+->+->- ⎪⎝⎭∑∑k i k i k n k k n i i i i P S n x S n x()()⎪⎭⎫ ⎝⎛+>-+>--∑∑-=-=x n S x n S P i i n k i i i i k i n i i εμεμ1,1111由NA 性质，注1和归纳假设得，()()()111,...,1;,,liminf inf1μ=→∞≥∆=⎛⎫-> ⎪⎝⎭≥∑∑k kk i i i kn n x k i ii P S k n n n x n F x .（3.12）另一方面，利用归纳假设表明，()()()111,...,1;,,limsup sup1μ=→∞≥∆=⎛⎫-> ⎪⎝⎭≤∑∑k kk i i i kn n x k i ii P S k n n n x n F x （3.13）结合（3.12）（3.13），定理证明成立.定理3.2 设{}1,1kij i X j =≥为一负相伴随机阵列，对1,,=i k ，具有相同的分布()i F x C ∈，期望为0μ>i ，且满足()(),i i xF x F x x -=→∞，如果对任意的1,,=i k ，存在某1>r 使得<∞rij E X .再令(){}1ki i N t =为一列相互独立的非负正整数值计数过程()()λ=→∞i EN t t ，()n i ,,2,1 =，且{}1,1kij i X j =≥与(){}1ki i N t =相互独立.如果(){}1ki i N t =满足：对任意0δ>，均存在+>Fp J ，当t →∞时，使得 ()()()()()()()1+t δλολ>=iipii N t EN t I t . （3.14）则对任意固定的{}max ,1,2,,γμ>=i i k ，当t →∞时，有()()()()11;μλλ==⎛⎫-> ⎪⎝⎭∑∑kki i iii i P S k t t x t F t （3.15）对(){}()max,1,,γλ≥==Γ：i x t i k k 一致成立.注3 如果假定所有的()()1,,=i F x i k 为同一分布，则由（3.15）可推出Chen 和Zhang 定理1.2.特别地，如果我们假定{},1,1,,≥=ij X j i k 是非负随机变量序列，可以很轻易的看出满足定理3.2的条件.所以，（3.15）验证了Liu （2007）定理2.2.如果假定{},1,1,,≥=ij X j i k 是一列相互独立的序列，可由（3.15）证出Wang 和Wang （2007）的定理4.1.证明 我们仍然采用数学归纳法证明本定理的结论，其证明思路与定理3.1完全相同，为简洁起见，这里我们只证明2k =情形.为此，我们首先证 ()()()()()()()()()1122211222;liminf inf1λμλμλλ→∞≥Γ-->≥+t x P S t t t x t F x t F x . （3.16）同理，对任意01ε<<以及0>x ，()()()()11222;λμλμ-->P S t t t x()()()()()()1211221,λμελμε≥->+->-t t N N P S t x S t x()()()()()()2122111,λμελμε+->+->-t t N N P S t x S t x()()()()()()()1211221,1λμελμε-->+->+t t N N P S t x S t x123J J J =+-： （3.17）先估计1J ，由于()()()()()()()12111221λμελμε≥->+--≤-t tN N J P S t x P S t x . （3.18）由Chen 和Zhang （2007）的定理1.2易知()()()()()()()1111111lim sup101γλλμελε→∞≥->+-=+t N t x t P S t xt F x. （3.19）现在对任意()0,βε∈，令0μ<，()()()222λμε-≤-tN P S t x()()()()()22221n n P S n x n t P N t n μελμ∞==-≤-+-=∑()()()()2222ελμβελμβ-+-≤--+->-=+∑∑x n txx n tx12:K K =+ （3.20） 首先，运用引理2.2，我们得到()()()()()()22222n x n t xK P S t x P N t n ελμβλμβ-+-≤-≤-≤-=∑()()()()()()2222x n t xnF x P N t n ελμβο-+-≤-==∑()()()()()()()22221n F x n PN t n t F x οολ∞=≤==∑ （3.21） 现在我们估计2K ,为简单起见，我们声明在下文中()2εβμ-属于C .事实上，对任意2*>Fp J ，20μ<，用Tchebychef 不等式，我们可得出()()()()()()222222ελμβλμε-+->-=-≤-=∑n x n t xK P S t x P N t n()()()()(){}()()222+2222λεβλμλ>⎛⎫-≤>+≤ ⎪ ⎪+⎝⎭pN t Cx t p EN t I P N t x t Cx t ()()()(){}()()()22212γλλ>+--≤=O p N t C t p p pEN t I C x t Cx()()22t F x ολ=. （3.22）由Tang （2006）引理2.1中的()()2p x F x ο-=，最后一个等式成立.联合(3.18)-(3,22)得，对任意0δ>都有()()()()()()()1112211J t F x t F x δλεολ≥-++. （3.23）同理可得()()()()()()()2221111J t F x t F x δλεολ≥-++.最后我们估计3J，类比（3.7）我们易得()()()()1limlim sup 10,1,2.ii t x t i F x i F x εγλε↓∞→∞≥±-== （3.24）注意到(){}21i i N t =相互独立以及{}21iji X =为NA 序列，由引理2.1，Chen 和Zhang （2007）以及（3.24）可得，()()()()()()()()123112211λμελμε≤->+->+N t N t J P S t x P S t x()()()()()()112211t F x t F x λελε++()()()()()1122t F x t F x ολλ=+. （3.25） 所以，用（3.23）-（3.25），令0δ↓，对任意充分大的t ，()2x ≥Γ，由()()()()()()()()()1122211222;liminf inf1λμλμλλ→∞≥Γ-->≥+t x P S t t t x t F x t F x证出（3.16）另一方面，我们再证明 ()()()()()()()()()1122211222;limsup sup1λμλμλλ→∞≥Γ-->≤+t x P S t t t x t F x t F x . （3.26）注意到对任意()0,12ε∈以及0>x 并利用NA 性质和（3.10）相同的方法，当(),2t x →∞≥Γ时，有()()()()11222;λμλμ-->P S t t t x()()()()()()()()12112211λμελμε≥->-+->-t t N N P S t x P S t x()()()()()()121122λμελμε+->->t t N N P St x P S t x()()()()()()()()()()1122112211t F x t F x t F x t F x λελελελε-+-+()()()()()()()()()11221122t F x t F x t F x t F x λλολλ++ （3.27）这样我们得到（3.26）.联合（3.16）（3.26）（3.15）定理对2k =时成立.定理3.2证明完毕.§4应用本节我们给出一个例子对本章主要结果加以应用.假定某保险公司经营着两个不同险种，而与第一个险种对应的索赔额记为{},1j X X j =≥，为一列独立同分布的非负随机变量，共同分布为F C ∈，有限期望为μ.若该索赔到来的时刻{},1j j σ≥为一更新过程(){}sup 1,σ=≥≤i N t n t ，0>t 其中对任意0t ≥，满足()()λ=<∞i i t EN t .再令{},1j I j ≥为一列Bernoulli 随机变量序列（即,(1)j I j ≥服从两点分布），且j I 的期望为q ，其中01<≤q ，q 表示第j 个索赔到来的概率.假定与公司第二个险种相应的索赔额为{},1j Y j ≥，为另外一列独立同分布的非负随机变量，共同分布为()G F C ≠∈，有限期望为ν.再令()()()2N t N t =Λ为一Cox 过程，其中()N t 为由一列独立同分布的非负随机变量序列{},1j Z j ≥生成的更新过程，且满足1j EZ =，令(),0t t Λ≥为另一个右连续的非降的随机过程，且满足()00Λ=.若()t Λ与()2N t 相互独立，对任意0t ≥()()1Λ<∞=P t .假定上述随机变量序列{}{}{},1,,1,,1j j j X j I j Y j ≥≥≥以及(){}(){}12,0,,0N t t N t t ≥≥相互独立，{},1j I j ≥为NA 序列，则可以看出到t 时刻时公司的累计索赔额为 ()()()1211,0.N t N t jj j j j S t XI Y t ===+≥∑∑ （4.1）这里我们假定公司同时经营着两种不同的险种，因此该模型是Denuit 等（2002）与Ng （2004）所研究的一维风险模型的推广.这里我们假设随机过程()t Λ满足，对任意0t ≥，()():λ*=Λ<∞t E t ，且当t →∞时，()t λ*→∞，对任意0θ>，存在+>Gp J 使得， ()()()()()()()11.θλλ**Λ>+Λ=O p t t E t t记{}1sup ,1,0n n N t I t σ*=≥=≥，则(){}1,0N t t *≥表示在[]0,t 时刻内发生索赔的真实次数.易见()()111N t j j N t I *==∑，以及()()1,0EN t q t t λ*=≥.因此（4.1）式可以被重新改写为 ()()()1211N t N t jjj j S t XY*===+∑∑.用和Wang 等（2007）中第五节相同的方法和定理3.2，我们得到，当t →∞时，有()()()()()()()()11λμνλλλ**-->+P S t q t t x q t F x t G x对任意0γ>，及()(){}1max ,x t t γλγλ*≥一致成立.参考文献[1] Alem, K. and Sexena, K.M.L., Positive dependence in multivariate distribution, Comm. Statist. A | Theory Methods, 10(1981), 1183-1196.[2] Bingham, N., Goldie, C. and Teugels, J., Regular Variation, Cambridge University Press, 1987. [3] Chen, Y . and Zhang, W.P., Large deviations for random sums of negatively dependent random variables with consistently varying tails, Stat. Prob. Lett., 77(2007), 530-538.[4] Cline, D.B.H. and Samorodnitsky, G ., Subexponentiality of the product of independent random variables, Stoch. Proc. Appl., 49(1994), 75-98.[5] Joag-Dev, K. and Proschan, F., Negative association of random variables with applications, Ann.Statist., 11(1983), 286-295.[6] Liu, L., Precise large deviations for dependent random variables with heavy tails, Stat. Prob. Lett.,79(2009), 1290-1298.[7] Liu, Y ., Precise large deviations for negatively associated random variables with consistently varying tails, Statist. Prob. Lett., 77(2007), 181-189.[8] Ng, K.W., Tang, Q.H., Yan, J.A. and Yang, H.L., Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Prob., 41(2004), 93-107.[9] Tang, Q.H., Insensitivity to negative dependence of the asymptotic behavior of precise large devia-tions, Electron. J. Pro., 11(2006), 107-120.[10] Tang, Q.H. and Yan, J.A., A sharp inequality for the tail probabilities of sums of i.i.d. r.v.'s with dominatedly varying tails, Sci. China Ser. A., 45(2002), 1006-1011.[11] Wang, S.J. and Wang, W.S., Precise large deviations for random variables with consistently varying tails in multi-risk models, J. Appl. Prob., 44(2007), 889-900.[12] Wang, Y .B., Wang, K.Y . and Cheng, D.Y ., Precise large deviations for sums of negatively associated random variables with common dominatedly varying tails, Acta Math. Sin. (English Ser.), 22(2006),1725-1734.[13] Yang, Y . and Wang, Y .B., Large deviations for random variables with two-sided distributions, Acta Math. Sin. (Chinese), 52(2009), 289-300.ψ-混合相依变量线性形式的强稳定性杨延召 刘妍岩（青岛科技大学数学系，青岛，266061）（武汉大学数学与统计学院，武汉，430072）线性形式的强稳定性在科学技术上存在着广泛应用.本文讨论了ψ-混合随机变量列线性形式的强稳定性.通过对ψ-混合随机变量列运用截尾术，借助于ψ-混合随机变量的性质以及 Borel- Cantelli 引理，得到了ψ-混合随机变量线性形式具有强稳健性的充分条件.同时也给出了一些其它形式的结果.关键词：强稳定性，ψ-混合，线性形式. 学科分类号：O211.4§1 序言概率密度估计，非参数非线性回归可能是研究最为广泛的非参数估计问题.许多研究方法已经在独立的观察下独立发展起来.近年来，一些论文因为广泛存在的独立随机变量产生的如强稳定性的线性形式等大量概率问题，就把这些方法扩展到不独立的情况.强稳定的线性形式在生态学、分子生物学、生物化学等领域都有应用.研究线性的强稳定性被大量的定律推动，在线性模型的兼容的最小平方估计中很有用.因此，研究线性强稳定性的重要性是毋庸置疑的.2004年，Gan(2004)研究了几乎收敛的-ρ混合随机变量.对于严平稳序列，-ψ很合序列首次在Blum 等（1963）中首次被提出.-ψ混合序列包括一些被广泛应用的例子，比如可数状态空间马尔可夫过程，在Blum 等（1963）中可以发现更多的-ψ混合序，列的例子.众所周知，极少的关于-ψ混合序列的研究可以被找出.在本文中，我们首先通过使用终止来研究变量，然后通过Broel-Cantelli 引理和ψ-混合序列的性质找到通常情况下ψ-混合序列的强稳定线性形式的充分条件，基于以上结果，我们给出在ψ-混合序列中其他线性形式的一些结果.接下来，我们证明ψ-混合序列强稳定线性形式的一些结果.本文的其他部分组织如下：在第二节中，我们陈述和证明主要的结论，然后在第三节中，我们证明ψ-混合序列中强稳定性的其他线性形式.§2.n X 的强稳定性线性形式在我们叙述主要结论之前，我们先复习几定义个下文即将用到的定义.定义2.1 设{}1,≥n X n 是定义在概率空间（Ω，F ，P)的一列稳定变量.分别用nm F F ,表示σ代数生成的{}m x X i ≤≤1,和{}n i X i ≥,.令()()()()()()1supsup,1-=≠∈∈≥+B P A P AB P r B P A P F B F A p pr p ψ， 如果当0→r 时，()0→r ψ，我们就说{}1,≥n X n 是ψ-混合随机序列.()r ψ是ψ-混合相关系数.定义2.2 一随机变量序列{},1≥n X n ，是强稳定的，如果存在两列常数{}{}∞↑<n n n b d b 0,，则01→--n n d X b ..s a （2.1) 定义2.3一随机变量序列{}1≥n X n ，被非负变量X 所控制，如果存在整数0>c ，则有 ()()1,0,≥∀>∀>≤>n t t X cP t X P n ， （2.2） 记为{}X X n <.除特别说明外，全文假定{}1≥n X n ，是ψ-混合随机变量序列，相应的混合系数满足()∞<∑∞=1r r ψ. （2.3）下面的定理是对ψ-混合序列{}n X 线性强稳定性的总结.定理2.1 设{}1,≥n X n 是一列零均值ψ-混合随机变量，{}n b 是一列正数∞↑n b ，若存在某个∞<≤≤∑∞=121n pn pn b X E p ，，则..,011s a Xbi i→∑∞=-为证明定理2.1，需先介绍以下引理.引理2.1（[3],引理1.2.11）设{}1,≥n X n 是ψ-混合随机变量，r k kF Y F X +∈∈,，∞<∞<Y E X E ,,则()Y E X E EXEY EXY XY E τψ≤-∞<,.引理2.2 设{}1,≥n X n 是一列零均值ψ-混合随机变量，∑==ni in XS 1，∞<2n EX ，1≥∀n .则0>∀ε，有()2121141max εψε∑∑=∞=≤≤⎥⎦⎤⎢⎣⎡+≤⎪⎭⎫ ⎝⎛>n i il j n j EX l S P . 证明：对0>∀ε，令(){}εωω>=Λ≤≤j nj S 1max :.对任意Λ∈ω，有()(){}εωω>≤≤=j S n j j v ,1:min ，(){}k v k ==Λωω:，当1=k 时，()0max 1=≤≤ωj kj S ，这样j i j i ≠Φ=Λ⋂Λ,且 nk k1=Λ=Λ.由()[]∑⎰∑⎰⎰=Λ=ΛΛ-+==nk k n k k n k nnkkdP S S S S dP S dP S 1221222,进行如[3]和引理2.1同样的讨论，得到()()∑∑∑=∞==≤-ni i l nk k nkEX l S SES 1211ψ，可得()()()Λ≥≥+≥+∑⎰∑∑⎰∑∑=Λ=∞=Λ=∞=P dP S EX l dP S EX l ES nk n ni il nni il k2121212121222εψψ （2.4） 由引理2.1，可得()()∑∑∑∑=∞==⎥⎦⎤⎢⎣⎡+≤-+≤ni i l j i j i ni inEX l X E X E i j EX ES 121122212ψψ ，由以上公式得()()212141εψ∑∑=∞=⎥⎦⎤⎢⎣⎡+≤Λn i i l EX l P . 引理2.3 设{}1,≥n X n 是一列ψ-混合随机变量，满足（2.3）的条件.若 （i ）∞<∑∞=1n nEX;（ii ）()∞<∑∞=1n nX Var ,则序列∑=nk kX1收敛.证明：对序列{}j j EX X -由引理2.2得知，对任意正整数m,21n n ≤有()()()∑∑∑=∞==≤≤⎥⎦⎤⎢⎣⎡+≤⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧≥-2112112411max n n i i l kn j j j n k n X Var l m m EX X P ψ， 由（ii ）知，对任意m 有，()01max lim 12121,=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧≥-∑=≤≤∞→∞→m EX X P kn j jj nk n n n . 所以{}∑-nn nEX X尾部收敛到0 a.s.，即{}∑-nn n EX X 收敛，由（i)知∑nn X 收敛.定理2.1的证明 设()x F n 是n X 的分布函数，令()n n n n b X I X Y ≤=，()⋅I 是示性函数，则()()∑∑⎰∑⎰∑≤≤=⎪⎪⎭⎫ ⎝⎛≤≤n p n pn n b x n n nb x n n n n b X E x dF b x x dF b x b Y E n n 222222，于是∞<≤⎪⎪⎭⎫⎝⎛≤⎪⎪⎭⎫ ⎝⎛∑∑∑n pnpn n n n n n n b X E b Y E b Y Var 22.因此，由定理2.3得，∑-nn nn b EY Y 收敛a.s. （2.5） 因为0=n EX ，故()()∑⎰∑⎰∑==≤nn b x n nn b x n nnn b x xdF b x xdF b EY nn()()x dF bxx dF b xn nb x pnpn nb x nnn∑⎰∑⎰≤≤∞<≤∑np npn bX E . （2.6）由（2.5）和（2.6）知∑nn nb Y一致收敛.且()()()∞<≤≤=≠∑∑⎰∑⎰∑np npn n nb x pnpnb x n nn nbX E x dF bxx dF Y XP nn由Borel-Cantelli 定理，∑nn nb X收敛a.s..应用Kronecker 定理，在每一个概率为1的集合上的任意一个样本点ω有()011→∑=-ωni inX b故有011→∑=-ni inXba.s.§3 其他线性形式的稳定性在这一节中，我们将给出ψ-混合随机变量的其他线性形式的稳定性.所有的证明建立在定理2.1的结果中.定理3.1 设{}{}n n b a 是两列正数，∞↑=n n n n b a b c ，，{}1≥n X n ，是一列ψ-混合随机变量，{}X X n <,令(){}R x x c n Card x N n ∈≤=,:.若下列条件满足(a)()∞<X EN (b)()()()21101≤≤∞<>⎰⎰∞+∞-p dydt y y N t X P ttp p ，则存在R d n ∈ ，，21=N 有011→-∑=-ni n ii nd Xa b a.s.证明 设()∑∑====≤=ni ii nni iin n n n n Y a T X a S c X I X Y 11,,，则()()()()∞<≤>≤>=≠∑∑∑∞=∞=∞=X cEN c X P c c X P Y XP n n n n n n n n111.由Broel-Cantelli 引理知:对任意实数列{}n d ，{}n n n d T b --1和{}n n n d S b --1在相同的集合上收敛到相同的极限.只须证明()011→-∑=-iini inEY Y a b，a.s.就有了定理的∑=-=ni iinnEY a bd11由于(){}1,≥-n EY Y a n n n 是一列零均值ψ-混合随机变量且()∞<∑∞=1l l ψ，得()∑∑∞=--∞=≤-11n pnpn n pn pn n n Y E c c b EY Y a E()∑⎰∞=-->≤`101n c n p p n ndt t X P t pc c(){}⎰∑∞>-->≤0:1tc n pnp n dt c t X P t cp()()⎰⎰∞+∞->≤tp p dydt y y N t X P t cp 1012最后不等式成立是由于以下事实{}{}()⎰∑∑-∞←<<-∞←>-==utp u u c t n p n u t c n p n y dN y c c n n lim lim::()()()⎥⎦⎤⎢⎣⎡+-=⎰≤≤+--∞→u y t p p pu dy y y N p t N t t N u1lim且()()∞→→≤⎰∞+-u dy y t N p u N u up p ,01由条件（2）和定理2.1即可证得.定理3.2 如果我们用如下条件替换定理3.1的条件（1）（2）： （3）()∞<X EN ； （4）()∞<⎰∞1ds s X EN ；（5）()n ccnj p jpjnj ο=∑∞=-≤≤1max ；此外假定0=n EX ，得011→∑=-ni ii nXa ba.s.证明：由定理3.1的n n n T S Y ,,.同理有()()∞<≤≠∑∞=X cEN Y XP n n n1.为了证明所要的结果只需须证明()..,011s a EY Y a bni iii→-∑=-.由条件（3）（4）易证011→∑=-ni ii Y a b .这样，我们只需证()011→-∑=-ni iiinEY Y a b.因为(){}n n n EY Y a -是一列ψ-混合随机变量且满足（2.3）得()()∑∑∞=-∞=≤≤-11n n n pn pn n ppn n n c X I X E c c b EY Y a E ()()()∑∞=-≤+>≤1n npnp npnc X I EX c X P c c c()()∑∑∞=-∞=≤+>=11n n p p nn nc X I EX cc c X P c.令0,max 01==≤≤d c d j nj n ，则()()()j j p n nj pn n n ppnn n pp nd X d I EX c d X I EX c c X I EX c≤≤=≤≤≤-∞==-∞=-∞=-∑∑∑∑11111()()()()()∞<⎪⎪⎭⎫⎝⎛>+≤>=<<≤<<≤<<=∑∑∑∑∑∑∑∞=∞=-∞=-∞=∞=--∞=-∞=-1111111111j j j j j j j j jn p np jj j jn p nj j j pc X P cd X P c d X djP ccdd X d P cd X d I EX由定理2.1得知，()∑=-→-ni iiinEY Y a b110证毕.在下文中，令()++→R R x :α为正的不增函数.令()n n nni i n n a b ca b n a ===∑=,,1α，假定（Ⅰ）()()∞<≤≤-∞→-∞→n n n n n n c c n c c n log sup lim log inf lim 011αα；（Ⅱ）对0>x ，()x x +log α是不增的. 在条件（Ⅰ）和（Ⅱ）下，我们有如下定理：定理3.3设{}n X 与()∞<+11log X X E α同分布，则存在n d ，使011→-∑=-n ni ii d Xa ba.s.证明 由n n n c b a ,,的定义及假定（Ⅱ）知，存在0,0,0>>∈βαN m ，对任意0m n ≥，有()n n n c c n βαα≤≤log .故()()1log -≥n n c n c αα，它能确保对任意0m m ≥都有()m c cm mj j222log αα≤∑∞=-.仿照定理3.1，令()n n n n c X I X Y ≤=，则当0m m ≥时，()()()()()()()()()()()()()()()()∞<≤≤+≤≤≤+≤≤≤+≤≤≤+≤⎥⎦⎤⎢⎣⎡≤≤+≤=≤=≤≤--∞=+--∞=--∞=--∞==--=--∞=-∞=-∞=-∞=∑∑∑∑∑∑∑∑∑∑i i m i i i m i ii i mi i m j jmi i i jjm i i i m m j jj mj j j j mj j jmj jj jjc X c I X X E c c X cI EX c c c X c I EX c i c c X c I EX cc c X c I EX c X I EX c c c X I X E c c c X I X E c c b EY Ya E 111121121211212121121211211121212122222log1log 1log 11αβαοαβαοααοο根据定理2.1，得到()011→-∑=-ni iiinEY Y a ba.s.另一方面，()()()()()()()()()()()∞<≥+-≤≥+-≤>+>=>=≠∑∑∑∑∑∑∞=+∞=+∞=-=∞=∞=0000log 1log log 1001111m i i i m i i i i i m i i i m i i ii i i i i ii X X P m c c X X P m c X P c XP c X P Y XP αααα由Borel-Cantelli 定理知，定理3.3的结论正确 ，ini inn EY a b d ∑=-=11.致谢 作者在此呜谢相关文献作者，他们做出的极有价值的研究.参考文献[1] Gan, S.X., Almost sure convergence for ½-mixing random variable sequences, Statist. Prob. L 67(4)(2004), 289-298.[2] Blum, J.R., Hanson, D.L. and Koopmans, L., On the strong law of large number for a cla stochastic processes, Z. Wahrsch., Verw. Gebiete, 2(1963), 1-11.[3] Lu, C.Y . and Lin, Z.Y ., The Limiting Theory of Mixing-Dependent Random Variables, China demic Press, 1997.[4] Chung, K.L., A Course in Probability Theory (2nd Ed.), Academic Press, New York, 1974.三参数威布尔、对数正态及伽马分布下的估计Russell F. KAPPENMAN西北和阿拉斯加渔业中心,国家海洋渔业局,国家海洋气象局，美国华盛顿州西雅图981121984年4月收到 1984年11月修订摘要：威布尔分布、对数正态分布、伽马分布的的位置、尺度、形状参数的新估计是发达的.估计是在封闭的形式，他们并不需要同步的非线性方程组求解.模拟研究结果和其他人已经提出的新的估计的性能进行比较.这些研究表明，新的估计更好，至少考虑到可能的偏差和均方误差.关键词：位置、尺度和形状参数，最大似然估计法、矩估计法，参数估计，仿真§1引言概率密度函数下三参数威布尔、对数正态和伽马分布的形式分别为：()()()[]()[]{}()()[]()[]{}()()()[]()[]()[]b a x b a xc b c b a x f c b a x a x c c b a x f b a x a x c c b a x f c c ---Γ=---=--=--exp 1,,;2ln exp 21,,;ln exp ,,;12221π 在每种情况下,a 是位置参数,b 是尺度参数,c 是形状参数.在这里所考虑的问题是给定一个随机从这些分布中观测的样本，来评估a,b,c.我们提出各个分布的参数估计，进行模拟仿真，然后和其他人提出的估计量的性能作对比.在一系列的论文中，Cohen and Whitten [2-4]报道他们运用最大似然法、改进的最大似然法和矩估计法估计三参数威布尔分布、对数正态分布、伽马分布的研究成果.在考虑多种不同的可能性，他们提出对每个分布的估计建议.对于对数正态分布的位置、尺度、形状参数估计，已经由Munro and Wixley [11]提出及LaRiccia 和Kindermann[10]进一步研究.检查估计在这些论文中设计使用迭代或搜索过程，来同时解决三个非线性方程.迭代或搜索过程有时无法找到方程的解.因此，它很有可能，在任何给定情况下，该程序将无法得到参数估计，即使样品来自假定分布.尤其是来自一个真实的中小程度大小的样本.本文提出的估计有三个重要优势，相比那些仅仅是引用的.首先，它们是相对比较简单的，也就是说，它们是在封闭形式而涉及非线性方程的数值解.第二，估计总是能被发现.最后，最重要的是，这些估计量似乎完成一些东西比那些到这个时候提出的好，至少考虑到偏差和均方误差.这里考虑的三种分布的参数估计都是来自一种相似的方式中.然而，不同的情况可以根据基本的开发方案稍作修改或调整.估计量的发展本质上源于 Wyckoff, Bain, and Engelhardt [12]，因他们的工作单单只是威布尔案例.他们的发展是稍做改进的威布尔分布产生较好的估计值，并产生了类似的对数正态分布和伽马分布的参数估计量.对于三个分布中任何一个，我们产生参数估计通过（1）以最初的、非参数估计量的位置为初始位置；（2）假设等于它的初始估计，并寻找形状参数的初始估计；（3）设置第一个次序统计量等于其预期值；（4）在形状参数被其初始估计替换后方程的解为位置参数，位置参数的一个函数替换尺度参数；（5）估计的尺度和形状参数通过假定位置参数等于（4）中得到的参数值.在以下三部分我们将详细介绍，我们将轮流检验威布尔分布、对数正态分布和伽马分布.在下文中，x1，x2，…，X ，N ，表示一个随机样本的N 个观测的分布的参数估计量的次序统计量，Y 代表样本均值，2s 表示样本的方差，F 表示经验分布函数.通常，100p 百分位数是样品q x ，其中np q =，若np 是整数，则[]np q +=1，否则，[]np 表示不超过np 的最大整数.100p 分布百分位上是Up ，()p Up F =.§2威布尔分布Wyckoff, Bain, and Engelhardt [12]提出了下列程序估计这三参数威布尔分布.第一阶段的统计作为位置参数的初始估计.形状参数的初始估计通过假设1x a =，利用估计的c 由 Dubey[6]得到两参数威布尔分布.这个估计为其中，k x 和h x 分别是第94和第17样本百分位数.则威布尔分布为的新的估计解用aˆ来表示，假定a a ˆ=，然后估计参数b 和c,根据Engelhardt and Bain [7]，利用b 和c 的估计推出两参数威布尔分布.注意，对于这个程序，b 本质上是由其重新估计的矩估计量替代的，似乎三个参数的估计都因此略有改善；反之，b 由a x m -替代，其中m x 是第63样本百分位数.a 的函数通过等同于第63百分位数样本和分布的百分比得到，后者是b a +.由此得到的a ，b ，c 分别为()[]()[]111ˆ1ˆˆc r c r x x am --= （2.2a ） ()[]()⎭⎬⎫⎩⎨⎧----=∑∑=+=s i in s i i n a x a x s n s nk c 11)ˆln(ˆln ˆ （2.2b ） ()()⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡-+=∑=n a x c b n i i 1ˆln ˆ5772.0exp ˆ （2.2c ）数.我们可以根据Engelhardt and Bain [7]中的一个表确定k 值.Cohen and Whitten [4] 研究了威布尔分布参数的最大似然估计法和矩估计法并进行了改进.他们建议在求解非线性方程组的同时找到估计.方程组的前两个方程通过使似然函数关于b 和c 的偏导数为零得到.第三个方程是第一次序统计量与其期望值相等，或()1x F 等于期望值()11+n . Cohen and Whitten [4]提出以比较估计的性能（2.2）和改进的最大似然估计（MMLE ）进行模拟统计研究.完成一个相当全面的统计量相对性能的研究所需的工作量，可以利用分布上的()c b b a aˆ,ˆ,ˆ-与a ，b 的取值无关而显著减少，其中c b a ˆ,ˆ,ˆ表示的估计量在（2.1）或最大似然估计中已经给出.这样，为了比较统计量的相对性，需要假定a=0和b=1.形状参数值c=0.5、1、1.5、2、2.5、3.5常被用于仿真研究中.对c 的每一个取值，分别由生a=0，b=1，形状参数相同的威布尔分布生成500个随机样本，每个样本容量20.对于每一个样本，可得到最大似然估计和（2.2）中的估计.在这一次研究中，我们第一次尝试通过1x 与其期望值相等求得最大似然估计.如果这些估计无法找到，我们将使用其他求解最大似然估计的方法.如果没有方法成功找到所求估计，样品将被舍弃.我们使用和Cohen and Whitten [4] 同样的标准，决定是否停止某一个方案.经过500次的实验，对于每个形状参数值，有如下结果：当c=3.5时，有77次未能找到估计值；当c=2.5时，27次未找到；当c=2，3，1.5时，有8次未找到估计值.表1给出了基于未丢弃的样本的最大似然估计和（2.2）估计的偏差和均方误差.a 和c 估计量（2.2）的均方误差总是远远小于那些用最大似然估计得到的a 和c 的估计值的均方误差.这同样适用与b 估计，除了c=0.5的情况.这些结论也基本上适用于偏差比较，除了c=3.5的情况.最大似然估计的偏差在这种情况下较小.表一偏差和均方误差的威布尔分布参数估计使用从a= 0，b= 1的情况下威布尔分布的500个随机样本，样本容量为20.然而，我们再次指出，在c=3.5的情况下，77个样本被丢弃，因此，这并不有助于表1.此外，当a ，b ，c 利用废弃的样本（2.2）的估计结果结合不采用废弃样本的估计结果，（2.2）估计的偏差大大降低了.同样适用于均方误差，除了c ＝3.5的情况，c 的估计的均方误差略为从1.6599增加到1.8398，这仍然是远远小于其MMLE 的情况.§3对数正态分布对一个对数正态分布的位置参数的初始估计，我们令∑==ni ii x a a 1ˆ， （3.1） 其中()[]n n a 1111-+=，()[]()[]nn i n i n i a 111----= n i ,...,2=.这种随机变量下界的非参数估计是由Cooke [5]推导出来的，它似乎能实现我们猜想最好的几个可能性.对于形状参数的初始估计c ，有()()[]{}()k k n k z E a x a x c 2ˆˆln ˆ1111--=+-， （3.2） 其中1ˆa由（3.1）给出，k x 和1+-k n x 分别是第25和第75个样本百分位数，k z 是一个正态分布下一个样本容量为n 的随机抽样的第二十五样本百分位数.标准正态分布的次序统计量已经被Harter [9]提出.统计量c 的估计是通过假设a 已知，则())ln(ln 1a x a x r k k n ---=+-是来自一个以b ln 为均值，c 为标准偏差的正态分布下四分位变化的随机样本.进一步()()k z cE r E 2-=.如果我们假设1ˆaa =，r 等同于它的期望值，求解c ，所得解即为（3.2）.重新估计a ，设()a x -1ln 等同于其自身期望值)(ln 1z cE b +.1z 是标准正态分布下一个样本量为n 的随机样本的第一次序统计量.在生成的方程中，我们用1ˆc替换c ，a 的函数替换b.对我们而言，最佳的函数是y-a ，其中，y 是样本中位数；即()1+=n x y ，如果n 是奇数；()[]2122+-=n n x x y ，如果n 是偶数.这个函数是通过等同()a y -ln 和它的期望值b ln 得到的，求解b.a 修正后的估计可由求解方程的解a 得到.如果这个解是2ˆa ，则b 和c 的估计可以有假定2ˆaa =和寻找这些参数的最大似然估计求得，分别表示参数估计11ˆ,ˆcb .事实证明，对于对数正态分布，如果我们更进一步的话，可以改善估计量a ，b ，c.另一种修订的估计量a 可以通过用1ˆb 和2ˆc 分别替换b 和c ，及()a x -1ln 等于其期望值的等式求得.如果使用最大似然估计和假定aa ˆ=可以重新估计a ，b ，c 表示. 由上可知，最终得到的a ，b ，c 分别为。