Unified QCD Evolution Equations and the Dominant Behaviour of Structure Functions at Low X

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美籍华人崔琦获诺贝尔物理学奖

美籍华人崔琦获诺贝尔物理学奖1998年华裔美籍科学家崔琦获诺贝尔物理学奖，他是获此殊荣的第六位华人1998年10月13日，华裔美籍科学家崔琦因发现逊电子在强磁场、超低温条件下互相作用，能形成某种特异性质的量子流体，获得1998年诺贝尔物理学奖，他是继1997年朱棣文之后登上诺贝尔殿堂的第六位华人。

崔琦，1939年生于河南，已入美国籍。

1967年获美国芝加哥大学物理学博士，1982年起任普林斯顿大学教授。

他于1984年获得了美国物理学会颁发的奥列佛·伯克利奖，1998年还获得了世界著名的本杰明·富兰克林物理奖。

1998年12月，他对采访他的记者谈了治学为人之道：崔琦认为，“获得成功，要有一定的运气和时机，但勤奋是基础。

”他说，他每年都要带二三十个研究生，他们都很刻苦，往往把别人花在舞会上的时间花在了实验室，周末不休息，一天工作10至12小时是常有的事。

一般人看来，学物理非常枯燥，不容易出成果，与学法律与商业的人相比，也挣不到大钱。

但崔教授有自己的见解。

他认为，搞物理研究只要投入，就会趣味盎然，每当有新的发现，哪怕是很微小的发现，也会享受到无穷的乐趣。

崔琦还说，要想成功，千万不要受周围环境的影响，“要相信自己，相信自己的能力”，“我常常鼓励学生们往前看，相信自己从事的是对人类有用的事业。

”他说，“如果只为一日三餐，并不需要去做研究，从事简单的体力劳动，便可达到目的。

做学问可不是为了钱，而是为了能对别人有用。

”崔琦指出，在相信自己的同时，还要相信别人。

只有向别人敞开你的胸怀，才会赢得别人的信任和帮助。

做到这一点对于远离故土到异国他乡求学的中国人来说尤为重要，因为他们要承受比别人更大的压力，如果不能与周围的人接近，很容易陷入孤立。

崔琦谈到他的那些来自中国名牌大学的一流学生时说，“他们的考试成绩非常好，但我告诉他们，做学问可不是做作业，那只是重复前人做过的事。

”他打了一个比喻，就像在旷野或森林中寻找回家的路一样，需要有开创性的探索精神。

《2024年非线性耦合方程组的高阶无振荡有限体积方法》范文

《非线性耦合方程组的高阶无振荡有限体积方法》篇一一、引言在科学与工程领域，非线性耦合方程组的求解是一项关键技术。

其精确性与稳定性对多种问题，如流体动力学、电路模拟和材料科学等具有重要意义。

为了更好地处理这些问题，研究者们提出了一种高阶无振荡有限体积方法。

本文将探讨此方法在非线性耦合方程组中的应用。

二、非线性耦合方程组的基本概念非线性耦合方程组由多个非线性偏微分方程组成，它们在空间和时间上相互依赖和影响。

这种复杂性使得求解过程变得复杂且计算量大。

为了解决这一问题，我们需要寻找一种有效的数值求解方法。

三、高阶无振荡有限体积方法高阶无振荡有限体积方法是一种求解偏微分方程的有效数值方法。

它利用有限体积的概念将求解空间离散化，通过求解离散化后的方程来逼近原方程的解。

此方法具有高精度、无振荡的特性，特别适合于求解非线性耦合方程组。

四、高阶无振荡有限体积方法的实施步骤高阶无振荡有限体积方法的实施步骤主要包括以下几步：1. 空间离散化：将求解空间划分为一系列的有限体积单元，每个单元代表一个离散点或一组离散点。

2. 建立离散化方程：基于高阶导数的空间分布特性，在每个有限体积单元上建立离散化后的偏微分方程。

3. 时间推进：采用合适的时间推进策略（如Runge-Kutta方法）求解离散化后的方程。

4. 迭代与收敛：通过迭代过程逐步逼近原方程的解，同时需要确保解的稳定性和收敛性。

五、高阶无振荡有限体积方法在非线性耦合方程组中的应用将高阶无振荡有限体积方法应用于非线性耦合方程组时，需要考虑以下几点：1. 适当的离散化策略：根据问题的特性选择合适的空间离散化策略，以确保解的准确性和稳定性。

2. 耦合项的处理：对于非线性耦合方程组中的耦合项，需要采用适当的方法进行处理，以保持解的准确性。

3. 时间推进策略的选择：根据问题的特性和需求选择合适的时间推进策略，如显式或隐式时间积分方法等。

六、实验与结果分析我们采用了几种典型的非线性耦合方程组进行实验，并比较了高阶无振荡有限体积方法与其他方法的性能。

定量电子显微学方法与氧化钛纳米结构研究获国家自然科学二等奖

据介绍，包括美国、Ｈ本、欧盟、俄罗斯在内的５０多拓宽，释许多传统理论无法描述的物质体系，且将理论解并
个国家和地区都有各自明确的纳米科技发展战略，投入巨成果成功应用到纳米结构的研究领域。在科学家眼中，食盐不是食盐，而是一种典型的离子资、抢占战略制高点。美国国家纳米讨‘ （划ＮＮＩ）２１０１晶体。么一种最最普通的东西，电子显微镜去观察它的这用年财政预算ｌ美元，历年投入稳定增长。日本尤其重视８亿结构时，也没有好的理论去描述电子在这种离子晶体表面纳米技术在信息、金属、环境、能源、生命科学以及基础技的衍射现象。彭练矛团队给出了描述诸如离子晶体表面衍
仿生智能纳米界面材料、碳纳米管宏观薄膜及纤维结构的贝尔物理学奖就颁发给了成功地分离出单层石墨烯材料的
制备与性能研究、纳米结构金属材料达到高性能极限等诸两名英国物理学家。
多方面取得突破。
像科技研发和产业化的整体协调。同时，入并持续开展纳米重，多数情况下我们在电子显微镜中所看到的 ‘ ’不能直深接反映物体的真正结构。练矛说，用量子力学的原理， ”彭利科技领域战略研究，提高决策能力；整合资源，形成在纳米

陈运泰院士荣获美国地球物理联合会国际奖

讶，也感到十分愉快。在接到通知书几乎１个月之后，我才得知我是由多位国际著名地球物理学家联合提名而获得这一声誉极高的
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∈ 。
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陈运泰院士荣获美国地球物理联合会国际奖
颁奖词
鉴于陈运泰数十年来杰出的科学研究成就，鉴于他为中国地震学与大地构造学研究
１８９０年代。后来陈运泰返回中国并成为中国地震局的教授，现在他仍是名誉教授。
陈运泰于１９９１年被选为中国科学院院
并无私奉献地组织或者共同组织了许多次国
际学术研讨会，并在多处举办培训班，极大
（ＡＧＵ）会。大
士，于１９９９年被选为发展中国家科学院
（ＡＳ院士。从２０Ｔｗ）０１至２００８年，他在北
京大学地球与空间科学学院任院长。陈运泰在中国现代地震学的发展以及推
第５期
陈运泰院士荣获美国地球物理联合会国际奖
州大学（ＬＵＣＡ）李昂・诺波夫（ｅｎ与Ｌｏ
Ｋｎｐｆ一同研究地震破裂力学。尽管我ｏｏｆ）
害、地震危险性和强地面运动委员会委员，并且已为这些委员会服务多年。他坚持不懈

【省级联考】陕西省2023届高三第一次模拟联考理科综合试卷(全真演练物理部分)

【省级联考】陕西省2023届高三第一次模拟联考理科综合试卷（全真演练物理部分）一、单项选择题：本题共8小题，每小题3分，共24分，在每小题给出的答案中，只有一个符合题目要求。

(共8题)第(1)题随着“第十四届全国冬季运动会”的开展，各类冰雪运动绽放出冬日激情，下列说法正确的是（）A．评委给花样滑冰选手评分时可以将运动员看作质点B．滑雪比赛中运动员做空中技巧时，处于失重状态C．冰壶比赛中刷冰不会影响压力大小，则滑动摩擦力不变D．短道速滑转弯时是运动员重力的分力充当向心力第(2)题如图所示，A、B、C是位于匀强电场中某直角三角形的三个顶点，，。

现将电荷量的电荷P从A移动到B，电场力做功；将P从C移动到A，电场力做功，已知B点的电势，则（）A．将电荷P从B移动到C，电场力做的功为B．C点的电势为C．电场强度大小为，方向由C指向AD．电荷P在A点的电势能为第(3)题2022年诺贝尔物理学奖授予法国学者阿兰·阿斯佩（AlainAspect），美国学者约翰·克劳泽（JohnClauser）和奥地利学者安东·蔡林格（AntonZeilinger），既是因为他们的先驱研究为量子信息学奠定了基础，也是对量子力学和量子纠缠理论的承认，下列关于量子力学发展史说法正确的是（）A．普朗克通过对黑体辐射的研究，第一次提出了光子的概念，提出“光由光子构成”B．丹麦物理学家玻尔提出了自己的原子结构假说，该理论的成功之处是它保留了经典粒子的概念C．爱因斯坦的光电效应理论揭示了光的粒子性D．卢瑟福的原子核式结构模型说明核外电子的轨道是量子化的第(4)题下图为自动控制货品运动的智能传送带，其奥秘在于面板上蜂窝状的小正六边形部件，每个部件上有三个导向轮A、B、C，在单个方向轮子的作用下，货品可获得与导向轮同向的速度v，若此时仅控制A、C两个方向的轮子同时按图示箭头方向等速转动，则货品获得的速度大小为（）A．v B．C．D．2v第(5)题如图甲所示，质量分别为、的A、B两物体用轻弹簧连接构成一个系统，外力F作用在A上，系统静止在光滑水平面上（B靠墙面），此时弹簧形变量为x，撤去外力并开始计时，A、B两物体运动的图像如图乙所示，表示0到时间内图线与坐标轴所围面积大小，、分别表示到时间内A、B的图线与坐标轴所围面积大小，A在时刻的速度。

11月25日国家科技部下发《关于批准北京凝聚态物理等5个

国际会议做邀请报告（境外 11次），是该领域有国际影响的学者，曾14次在国际会议
邀请报告（境外11次），担任三个国际刊物的编委和
一系列国际会议的程序委员会委员，获中科院自然科
学奖一等奖、华人物理学会亚洲成就奖和求是青年学
者奖等学术奖励。主要学术成绩如下：
1.首次利用低温STM获得能够分辨碳-碳单键和双键
本着“公开、平等、竞争、择优”的原则，谨向国内外公开招聘合肥微尺度物质科学国家实验室主任1 人，研究部主任若干人。
1．国家实验室主任（招聘人数1人）
应聘资格：应聘人员应是微尺度物质科学研究或
相关领域国际知名学者，能够在学术方向、国内外合作与交流等方面为国家实验室的发展做出重要贡献。
提交材料：（1）基本信息表（网上下载）；（2）论文等学术成果清单；（3）代表性论文抽印本 5篇；（4）研究成果简介（限2000字）；（5）工作计划或建议（限2000字）。
化、化学反应动力学等领域的重要基础问题。利用
STM和STS相结合，发展了确定单分子在固体表面特
定位置吸附取向的方法，并首次确定了C60分子在Si表面的吸附取向与局域电子态，被PRL审稿人认为是一
项“精致的实验”（第一作者），美国物理学会新闻网
也专门加以介绍。并被评为99年中国基础科学研究十
大新闻。
国家科技部、教育部、安徽省政府和中国科学院关注“合肥微尺度物质科学国家实验室”的筹建
♦ 2003年5月28日，全国人大常委会副委员长、中国科学院院长路甬祥视察了合肥微尺度物质科学国家实验室（筹），充分肯定了实验室和部分学科带头人的工作成绩，并就有关工作给予了明确的指示。安徽省委副书记、常务副省长张平和省人大常委会副主任黄岳忠、副省长田维谦等有关领导陪同路甬祥院长视察。 ♦ 2003年9月3日上午，科技部副部长程津培、科技部基础司司长张先恩、基础建设处处长叶玉江等一行考察了合肥微尺度物质科学国家实验室（筹）。程津培副部长对我校在筹备国家实验室期间所取得的成果表示赞赏，认为科大有实力、有能力做得更好，他希望实验室继续保持科研团队的活力，争取有一些学科率先在国际上达到并保持一流。 ♦ 2003年9月19日，中国科学院综合计划局李志刚局长一行考察了我校合肥微尺度物质科学国家实验室（筹）建设运行情况。李志刚局长表示中科院将积极支持该实验室的建设和试运行，希望该实验室能够加快建设进程，早日通过论证。 ♦ 2003年10月28 日，教育部科技司雷朝滋副司长对合肥微尺度物质科学国家实验室（筹）的建设情况进行了考察和指导，听取了关于国家实验室建设及试运行情况的汇报。雷副司长表示教育部非常重视科大的发展，教育部已经考虑在高校试点建设国家实验室，对于科大正在建设的国家实验室，教育部将会非常坚决地表示支持。

凸多边形的闵可夫斯基和分解及其最优估计

凸多边形的闵可夫斯基和分解及其最优估计
张松海;吴奕
【期刊名称】《中国科技论文》
【年(卷),期】2006(001)003
【摘要】本文讨论了闵可夫斯基和的逆问题(称为闵可夫斯基和分解),即将一个凸多边形分解为两个更为简单的凸多边形的问题,首先通过三角分解的方法证明了凸多边形阂可夫斯基和分解的存在性.在此基础上研究了在边个数和面积之和的意义下的最优分解.
【总页数】6页(P191-196)
【作者】张松海;吴奕
【作者单位】清华大学计算机科学与技术系,北京,100084;清华大学计算机科学与技术系,北京,100084
【正文语种】中文
【中图分类】TP391
【相关文献】
1.闵可夫斯基时空的命运 [J], 冯晓华;张耀;高策
2.闵科夫斯基内积空间上的矩阵分解 [J], 李洵;刘国境
3.凸多边形的闵可夫斯基和分解及其最优估计 [J], 张松海;吴奕;
4.活跃在不等式试题中的闵可夫斯基不等式 [J], 邹峰
5.直观与同情——闵可夫斯基现象学精神病学的方法论反思 [J], 黄旺
因版权原因，仅展示原文概要，查看原文内容请购买。

Hilfer_分数阶脉冲随机发展方程的平均原理

第 63 卷第 1 期2024 年 1 月Vol.63 No.1Jan.2024中山大学学报（自然科学版）（中英文）ACTA SCIENTIARUM NATURALIUM UNIVERSITATIS SUNYATSENIHilfer分数阶脉冲随机发展方程的平均原理*吕婷1，杨敏1，王其如21. 太原理工大学数学学院，山西太原 0300242. 中山大学数学学院，广东广州 510275摘要：利用分数阶微积分理论、半群性质、不等式技巧和随机分析理论，建立了分数布朗运动驱动的Hilfer 分数阶脉冲随机发展方程的平均原理，证明了原方程的适度解均方收敛于无脉冲平均方程的适度解，并通过实例说明了所得理论结果的适用性.关键词：平均原理；Hilfer分数阶导数；脉冲随机发展方程；分数布朗运动中图分类号：O211.63 文献标志码：A 文章编号：2097 - 0137（2024）01 - 0145 - 09Averaging principle for Hilfer fractional impulsivestochastic evolution equationsLÜ Ting1， YANG Min1， WANG Qiru21. School of Mathematics， Taiyuan University of Technology， Taiyuan 030024， China2. School of Mathematics， Sun Yat-sen University， Guangzhou 510275， ChinaAbstract：By using fractional calculus，semigroup theories，inequality techniques and stochastic analysis theories， an averaging principle for Hilfer fractional impulsive stochastic evolution equations driven by fractional Brownian motion is established. The mild solution of the original equations converges to the mild solution of the reduced averaged equations without impulses in the mean square sense is proved. And an example is presented to illustrate the applicability of our obtained theoretical results.Key words：averaging principle； Hilfer fractional derivative； impulsive stochastic evolution equations；fractional Brownian motion在实际生活中，系统常受外力影响或内部产生的“噪声”干扰，所以，随机微分方程可以更加准确的刻画系统的变化特征，因而研究随机微分方程是很有必要的且存在实际的应用价值. 另外，现实生活中的许多现象都有长期后效作用，Mandelbrot et al.（1968）研究表明分数布朗运动可以较好的描述长期后效现象，这推动了更多学者们对分数布朗运动驱动的随机微分方程的广泛关注. 分数布朗运动（fBm）最早是由Kolmogorov（1940）提出的一个依赖于Hurst参数H∈(0，1)的高斯随机过程，当H=1/2时，分数布朗运动简化为标准布朗运动；当H≠1/2时，分数布朗运动既不是半鞅也不是Markov过程；当H >1/2 时，分数布朗运动具有自相似性、长时记忆性等特征，这些性质使分数布朗运动可以引入到数理金融（Bollerslev et al.，1996）、网络通信（Leland et al.，1994）、生物医学工程（de la Fuente et al.，2006；Boudrahem et al.，2009）等随机模型中作为随机噪声项，得以更好的描述系统特征和保证模型性能. 除此之外，具有脉冲干扰的微分方程能准确的呈现出系统的瞬时变化规律，因此，脉冲随机微分方程吸引了很多学者的关注，详见文DOI：10.13471/ki.acta.snus.2023A006*收稿日期：2023 − 01 − 16 录用日期：2023 − 03 − 22 网络首发日期：2023 − 11 − 15基金项目：国家自然科学基金（12001393，12071491）；山西省自然科学基金（201901D211103）作者简介：吕婷（1999年生），女；研究方向：分数阶随机微分方程；E-mail：********************通信作者：杨敏（1986年生），男；研究方向：泛函微分方程理论及其应用；E-mail：******************第 63 卷中山大学学报（自然科学版）（中英文）献（Sakthivel et al.，2013；Ren et al.，2014；Liu et al.，2020）.另一方面，平均原理作为一种高效、准确的近似分析方法，在非线性动力系统的研究中发挥着重要作用. 它的主要思想是对原始动力系统进行简化得到一个平均系统，并且这个简化后的平均系统可以反映原系统的动力学行为. 目前为止，随机微分系统的平均原理理论已经获得了极大的发展. 例如，Cerrai et al.（2009）研究了一类随机反应扩散模型的平均原理；Ma et al.（2019）研究了Lévy 噪声驱动的脉冲随机微分方程的周期平均原理；Cui et al.（2020）在非Lipschitz 系数条件下，考虑了脉冲中立型随机微分方程的平均原理；Ahmed et al.（2021）探索出含泊松跳和时滞的Hilfer 分数阶随机微分方程的平均原理；Liu et al.（2022a ）在非Lipschitz 系数条件和无周期条件下，考虑了由分数布朗运动驱动的脉冲随机微分方程的平均原理.但现有研究存在两方面不足：一是大多数平均原理建立在有限维空间上，很少考虑空间是无穷维的情形（Xu et al.，2020；Liu et al.，2022b ），二是Caputo 分数阶脉冲随机微分方程已有相应的平均原理研究（Wang et al.，2020；Xu et al.，2011；刘健康等，2023），但Hilfer 分数阶脉冲随机发展方程的平均原理尚未见到研究结果. 基于上述讨论，本文在Hilbert 空间上考虑如下Hilfer 分数阶脉冲随机发展方程的平均原理ìíîïïïïïïD γ，β0+x (t )=Ax (t )+f (t ，x t )+h (t ，x t )d B H Q (t )d t ， t ≠t k ， t ∈J =(0，b ]，Δx (t k )=I k (x (t k ))=x (t +k )-x (t -k )， t =t k ，k =1，2，⋯，m ，x (t )=φ(t )， -λ≤t <0，I (1-β)(1-γ)0+x (0)=φ0，（1）其中D γ，β是Hilfer 分数阶导数，γ∈[]0，1，β∈()12，1，x (⋅)取值于实可分Hilbert 空间X . 闭线性算子A ：D (A )⊂X →X 是强连续算子半群{S (t )}t ≥0的无穷小生成元. B H Q (t )是定义在实可分Hilbert 空间Y 上的分数布朗运动，其中Hurst 参数H ∈()12，1. P C ()[]-λ，0；X 指从[]-λ，0到X 上所有具有càdlàg 路径的连续函数φ构成的空间，其范数 φP C =sup-λ≤t ≤0φ(t )<+∞，x t =x (t +τ)(τ∈[-λ，0])是P C -值的随机过程. x (t -k )和x (t +k )分别表示x (t )在t =t k 时的左极限和右极限，I k 表示x (t )在t =t k 时刻的脉冲扰动，脉冲时间序列{t k }满足0<t 1<⋯<t m <t m +1=b . 系数函数 f ：J ×P C →X ，h ：J ×P C →L 02(Y ，X ). 1 预备知识假设(Ω，F ，{F t }t ≥0，P )是一个带流的完备概率空间，其中{F t }t ≥0满足通常条件，即{F t }t ≥0是右连续的且F 0包含所有零测集. {B H (t )}t ∈R 是带有Hurst 参数H ∈()12，1的一维分数布朗运动，即B H (t )是一个中心高斯过程且具有以下协方差函数R H (t ，s )=E (B H (t )B H (s ))=12()t 2H +s 2H -|t -s |2H， t ，s ∈R =(-∞，+∞).记X 和Y 是两个实可分Hilbert 空间，L (Y ，X )是从Y 映射到X 上所有有界线性算子构成的空间. Q ∈L (Y )是一个非负自伴算子，满足Qe n =λn e n ，有限迹tr Q =∑n =1∞λn <+∞，其中{λn }≥0，(n =1，2，⋯)是一个非负有界实数序列，{e n }(n =1，2，⋯)是空间Y 上一组标准正交基. {B H n (t )}n ∈N +是独立于完备概率空间(Ω，F ，P )的一维标准分数布朗运动序列，现在我们在空间Y 上定义无穷维分数布朗运动如下：B HQ(t )=∑n =1∞B H n(t )Q 12e n =∑n =1∞B H n (t )λn e n ， t ≥0，则B H Q (t )∈L 2(Ω，Y )且在空间Y 中收敛，其中L 2(Ω，Y )表示所有强可测，平方可积的Y -值随机过程组成的146第 1 期吕婷，等：Hilfer 分数阶脉冲随机发展方程的平均原理空间.若ψ∈L (Y ，X )并且使得ψQ 12是Hilbert-Schmidt 算子，满足范数 ψ2L 02=∑n =1∞λn ψe n 2<+∞，则ψ被称为从Y 映射到X 的Q -Hilbert-Schmidt 算子. 记L 02≔L 02(Y ，X )是所有Q -Hilbert-Schmidt 算子ψ∈L (Y ，X )构成的空间，定义空间L 02的内积为ψ1，ψ2L 02=∑n =1∞ψ1e n ，ψ2e n ，则L 02(Y ，X )是一个可分Hilbert 空间. 引理1（Abouagwa et al.，2021）对任意ϕ：J →L 02(Y ，X )，∫0bϕ(s )2L 02d s <+∞成立，当t ∈J ，∑n =1∞ϕ(t )Q 12e n 一致收敛，则对任意t 1，t 2∈J 且t 2>t 1，有E∫t 1t 2ϕ(s )d B H Q(s )2≤2H (t 2-t 1)2H -1∫t 1t 2ϕ(s )2L 02d s .定义1（Yang et al.，2017a ）函数f ：[a ，+∞)→R 是一个Lebesgue 可积函数，对任意β∈(0，1)，函数f 的β阶Riemann-Liouville 积分定义为I βa +f (t )=1Γ(β)∫a t (t -s )β-1f (s )d s ， t >a ，β>0，其中Γ(⋅)是Gamma 函数.定义2（Yang et al.，2017a ）函数f ：[a ，+∞)→R 的β阶Riemann-Liouville 分数阶导数定义为LD βa +f (t )=1Γ(n -β)d nd t n∫at (t -s )n -1-βf (s )d s ， t >a ， n -1<β<n ，其中n ∈N +. 定义3（Yang et al.，2017a ）函数f ：[a ，+∞)→R 且f ∈C n [a ，+∞)，f 的β阶Caputo 分数阶导数定义为CD βa +f (t )=1Γ(n -β)∫at (t -s )n -1-βf (n )(s )d s ， t >a ， n -1<β<n ，其中C n [a ，+∞)表示在区间[a ，+∞)上n 次连续可微的函数构成的空间，n ∈N +.定义4（Sheng et al.，2022）函数f ：[a ，+∞)→R 的Hilfer 分数阶导数定义为D γ，βa+f (t )=I γ(1-β)a +d d t I (1-γ)(1-β)a+f (t )， 0≤γ≤1，0<β<1.注1（Sheng et al.，2022）当γ=0，0<β<1，a =0，则Hilfer 分数阶导数对应经典的Riemann-Liou ‐ville 分数阶导数D 0，β+f (t )=d d tI 1-β0+f (t )=L D β0+f (t ).当γ=1，0<β<1，a =0，则Hilfer 分数阶导数对应经典的Caputo 分数阶导数D 1，β0+f (t )=I 1-β0+dd tf (t )=C D β0+f (t ).引理2 方程（1）等价于如下的积分方程x (t ) = φ0t (γ-1)(1-β)Γ(γ(1-β)+β)+1Γ(β)∫0t (t -s )β-1(Ax (s )+f (s ，x s ))d s+1Γ(β)∫0t (t -s )β-1h (s ，x s )d B H Q(s )+t (γ-1)(1-β)Γ(γ(1-β)+β)∑0<t k <tIk(x t k). (2)证明可参考文献（Yang et al.，2017a ；Ahmed et al.，2018）. 为了给出方程（1）的适度解，引入以下Wright-type 函数M β(θ)=∑n =1∞(-θ)n -1(n -1)Γ(1-βn )， 0<β<1，θ∈C .147第 63 卷中山大学学报（自然科学版）（中英文）引理3（Yang et al.，2017a ）若积分等式（2）成立，其等价于如下的等式：x (t )=S γ，β(t )φ0+∫0t Tβ(t -s )f (s ，x s )d s +∫0t Tβ(t -s )h (s ，x s )d B H Q (s )+∑0<t k <tSγ，β(t -t k )I k (x t k)=S γ，β(t )φ0+∫0t (t -s )β-1P β(t -s )f (s ，x s )d s +∫0t (t -s )β-1P β(t -s )h (s ，x s )d B H Q (s )+∑0<t k <tSγ，β(t -t k )I k (x t k)，(3)其中P β(t )=∫∞βθM β(θ)S (t βθ)d θ，T β(t )=t β-1P β(t )，S γ，β(t )=I γ(1-β)0+T β(t ).定义5 若一个P C -值的随机过程x ：[-λ，b ]→X 满足以下条件，则称x (t )是方程（1）的适度解.（i ） x (t )是F t -适应的且∫0bE x (s )2d s <+∞几乎必然成立；（ii ） x (t )=φ(t )，-λ≤t ≤0；（iii ）当t ∈J 时，x (t )具有càdlàg 路径且对任意t ∈J 有x (t )=S γ，β(t )φ0+∫0t (t -s )β-1P β(t -s )f (s ，x s )d s+ ∫t(t -s )β-1P β(t -s )h (s ，x s )d B H Q (s )+∑0<t k <tSγ，β(t -t k )I k (x t k). (4)本文中，我们假设如下条件成立：（H0）当t ≥0时，S (t )是一致算子拓扑连续的，且S (t )是一致有界的，即存在M >1，使得supt ∈[0，+∞)S (t )<M .引理4（Yang et al.，2017b ）在条件（H0）下，对任意t >0，{P β(t )}t >0和{S γ，β(t )}t >0是线性算子，且对任意x ∈X 有P β(t )x ≤M Γ(β) x ， S γ，β(t )x ≤Mt (γ-1)(1-β)Γ(γ(1-β)+β) x .定义6（Liu ，2007）设X n (n ≥1)，X 是同一概率空间(Ω，F ，P )上的随机变量，若E (|X n |2)<+∞，且lim n →∞E (|X n -X |2)=0成立，则称X n 均方收敛于X .2 平均原理接下来，我们建立Hilfer 分数阶脉冲随机发展方程的平均原理.首先，定义方程（1）的扰动形式为ìíîïïïïïïD γ，β0+x ε(t )=Ax ε(t )+εf (t ，x ε，t )+εHh (t ，x ε，t )d B H Q (t )d t ， t ≠t k ，t ∈J =(0，b ]，Δx ε(t k)=x ε(t +k )-x ε(t -k )=εI k (x ε(t k ))， t =t k ，k =1，2，⋯，m ，x ε(t )=φ(t )， -λ≤t <0，I (1-β)(1-γ)0+x ε(0)=φ0.（5）然后根据方程（1）适度解的定义，可以得到方程（5）的适度解为：x ε(t )=S γ，β(t )φ0+ε∫0t (t -s )β-1P β(t -s )f (s ，x ε，s )d s+ εH∫0t (t -s )β-1P β(t -s )h (s ，x ε，s )d B HQ (s )+ε∑0<t k <tSγ，β(t -t k )I k (x ε，t k)， (6)其中ε∈(0，ε0]是一个很小的正参数，ε0是一个固定的常数.为了得出本文的主要结果，假设系数函数f ，h 具有周期T ，则存在正整数m ∈N +，使得0<t 1<…<t m <T ，那么对整数k >m ，有t k =t k -m +T ，I k =I k -m . 现引入可测的系数函数f ˉ：P C →X ，h ˉ：148第 1 期吕婷，等：Hilfer 分数阶脉冲随机发展方程的平均原理P C →L 02(Y ，X )，-I k ：P C →X ，其中f ˉ(x )=1T∫T f (s ，x )d s ，hˉ(x )=1T∫Th (s ，x )d s ，-I (x )=1T ∑k =1m I k (x ). 另外，我们做如下假设：（H1）对任意x ，y ∈P C ，t ∈J ，存在正常数M 1使得f (t ，x )-f (t ，y )2∨ h (t ，x )-h (t ，y )2L 02≤M 21 x -y 2.（H2）对任意的x ，y ∈P C ，存在正常数c k 和d k ，使脉冲函数I k 满足I k (x )2≤c k ， I k (x )-I k (y )2≤d k x -y 2.（H3）对所有T ∈J ，x ∈P C ，存在有界函数ρi (T )>0(i =1，2)使得1T ∫0T f (s ，x )-f ˉ(x )2d s ≤ρ1(T )()1+ x 2，1T∫0T h (s ，x )-h ˉ(x )2d s ≤ρ2(T )()1+ x 2，其中lim T →∞ρi (T )=0(i =1，2).则方程（5）对应如下无脉冲项平均系统：ìíîïïïïD γ，β0+z ε(t )=Az ε(t )+εf ˉ(z ε，t )+ε-I (z ε，t )+εH h ˉ(z ε，t)d B H Q (t )d t ， t ∈J =(0，b ]，I (1-β)(1-γ)0+z ε(0)=φ0，z ε(t )=φ(t )，-λ≤t <0.（7）参考文献（Gu et al.，2015）中引理2.12的证明，可以得到方程（7）的适度解z ε(t )为z ε(t )=S γ，β(t )φ0+ε∫0t (t -s )β-1P β(t -s )f ˉ(z ε，s )d s+ εH∫t (t -s )β-1P β(t -s )h ˉ(z ε，s )d B H Q (s )+ε∫t (t -s )β-1P β(t -s )-I (z ε，s )d s . (8)定理1 假设条件（H0）~（H3）成立，则当ε趋于零时，方程（5）的适度解x ε(t )均方收敛于平均方程（7）的适度解z ε(t ). 即任意给定一个很小的数δ>0，存在M 0>0，α∈(0，1)以及ε1∈(0，ε0]，使得当ε∈(0，ε1]时有E()sup t ∈[-λ，M 0ε-α]x ε(t )-z ε(t )2≤δ.证明由式（6）和式（8），有x ε(t )-z ε(t )=ε∫0t(t -s )β-1P β(t -s )[f (s ，x ε，s)-f ˉ(z ε，s)]d s + εH∫0t(t -s )β-1P β(t -s )[]h (s ，x ε，s )-hˉ(z ε，s)d B HQ(s )+ ε()∑0<t k <tSγ，β(t -t k )I k (x ε，t k)-∫0t (t -s )β-1P β(t -s )-I (z ε，s )d s ，(9)从而对任意ν∈(0，b ]，利用基本不等式得到E ()sup 0<t ≤νx ε(t )-z ε(t )2≤3ε2E ()sup 0<t ≤ν∫0t (t -s )β-1P β(t -s )[]f (s ，x ε，s )-f ˉ(z ε，s )d s 2+ 3ε2HE ()sup 0<t ≤ν∫0t (t -s )β-1P β(t -s )[h (s ，x ε，s )-h ˉ(z ε，s )]d B H Q(s )2+ 3ε2E ()sup 0<t ≤ν ∑0<t k<tS γ，β(t -t k )I k (x ε，t k)-∫0t(t -s )β-1P β(t -s )-I (z ε，s )d s 2≤N 1+N 2+N 3. (10)对于第1项，由引理4可得149第 63 卷中山大学学报（自然科学版）（中英文）N 1≤6M 2Γ2(β)ε2E ()sup 0<t ≤ν∫0t(t -s )β-1[]f (s ，x ε，s )-f (s ，z ε，s )d s 2 +6M 2Γ2(β)ε2E ()sup 0<t ≤ν∫0t(t -s )β-1[]f (s ，z ε，s )-f ˉ(z ε，s )d s 2≔N 11+N 12 . ()11利用假设条件（H1）和Cauchy-Schwarz 不等式得到N 11≤6M 2M 21ε2ν2β-1(2β-1)Γ2(β)∫νE()sup 0<s 1≤sx ε，s 1-z ε，s12d s =Λ11ε2ν2β-1∫νE()sup0<s 1≤sx ε，s 1-z ε，s12d s ，（12）其中Λ11=6M 2M 21(2β-1)Γ2(β).由假设条件（H3）得到N 12≤6M 2ε2ν2β-1(2β-1)Γ2(β)E ()sup 0<t ≤νt ⋅1t ∫0t f (s ，z ε，s )-f ˉ(z ε，s )2d s ≤Λ12ε2ν2β，（13）其中Λ12=6M 2(2β-1)Γ2(β)sup 0<t ≤νρ1(t )()1+E ()sup 0<t ≤νz ε，t2. 对于第2项，由引理4可以推出N 2≤6M 2Γ2(β)ε2HE ()sup 0<t ≤ν∫0t (t -s )β-1[]h (s ，x ε，s )-h (s ，z ε，s )d B HQ (s )2+ 6M 2Γ2(β)ε2HE ()sup 0<t ≤ν∫0t (t -s )β-1[]h (s ，z ε，s )-h ˉ(z ε，s )d B H Q(s )2≔N 21+N 22 . (14)由引理1、假设条件（H1）和Cauchy-Schwarz 不等式得到N 21≤12M 2H Γ2(β)ε2H ν2H -1E ()sup0<t ≤ν∫t (t -s )2(β-1) h (s ，x ε，s )-h (s ，z ε，s )2d s≤Λ21ε2H ν2(H +β-1)∫0νE()sup0<s 1≤sxε，s 1-z ε，s12d s ，（15）其中Λ21=12M 2M 21H(2β-1)Γ2(β).由引理1、假设条件（H1）和假设条件（H3）得到N 22≤12M 2H (2β-1)Γ2(β)ε2H ν2(H +β-1)E ()sup 0<t ≤νt ⋅1t ∫0th (s ，z ε，s)-h ˉ(z ε，s )2d s ≤Λ22ε2H ν2H +2β-1，（16）其中Λ22=12M 2H(2β-1)Γ2(β)sup 0<t ≤νρ2(t )()1+E ()sup 0<t ≤νz ε，t 2. 对于第3项，由基本不等式得到N 3≤6ε2E ()sup 0<t ≤ν∑0<t k<t S γ，β(t -t k )I k (x ε，t k)2+ 6ε2E ()sup 0<t ≤ν∫0t (t -s )β-1P β(t -s )-I (z ε，s )d s 2≔N 31+N 32， (17)由引理4、假设条件（H2）和Cauchy-Schwarz 不等式得到N 31≤6ε2M 2ν2(γ-1)(1-β)Γ2(γ(1-β)+β)E ()sup 0<t ≤ν∑0<t k<t I k (x ε，tk)2≤6ε2M 2mν2(γ-1)(1-β)Γ2(γ(1-β)+β)E ()sup 0<t ≤ν∑k =1m I k (x ε，t k)2≤6ε2M 2m 2c k ν2(γ-1)(1-β)Γ2(γ(1-β)+β)=Λ31ε2ν2(γ-1)(1-β)， (18)其中Λ31=6m 2M 2c kΓ2(γ(1-β)+β).150第 1 期吕婷，等：Hilfer 分数阶脉冲随机发展方程的平均原理N 32≤6M 2ε2Γ2(β)E ()sup 0<t ≤ν∫0t(t -s )β-1I ˉ(z ε，s )d s 2≤6M 2ε2ν2β-1(2β-1)Γ2(β)E ()sup 0<t ≤ν∫0tI ˉ(z ε，s)2d s≤6M 2mε2ν2β-1(2β-1)T 2Γ2(β)E ()sup 0<t ≤ν∑k =1m∫0tI k(zε，s)2d s ≤6M 2m 2c k ε2ν2β-2(2β-1)Γ2(β)=Λ32ε2ν2(β-1)， (19)其中Λ32=6M 2m 2c k(2β-1)Γ2(β).将估计式（11）~（19）代入式（10），则对任意ν∈(0，b ]，得到不等式E ()sup 0<t ≤νx ε(t )-z ε(t )2≤Λ12ε2ν2β+Λ22ε2H ν2H +2β-1+Λ31ε2ν2(γ-1)(1-β)+Λ32ε2ν2(β-1)+ (Λ11ε2ν2β-1+Λ21ε2Hν2(H +β-1))∫0νE()sup0<s 1≤sxε，s 1-z ε，s12d s . (20)令Ξ(ν)=E ()sup 0<t ≤νx ε(t )-z ε(t )2，由于E()sup -λ≤t <0x ε(t )-z ε(t )2=0，则Ξ(s +τ)=E()sup0<s 1≤sx ε，s 1-z ε，s12=E()sup 0<s 1≤sx ε(s 1+τ)-z ε(s 1+τ)2， τ∈[-λ，0)，因此，Ξ(ν)≤Λ12ε2ν2β+Λ22ε2H ν2H +2β-1+Λ31ε2ν2(γ-1)(1-β)+Λ32ε2ν2(β-1) + (Λ11ε2ν2β-1+Λ21ε2Hν2(H +β-1))∫0νΞ(s +τ)d s . (21)对任意ν∈(0，b ]，令Θ(ν)=sup -λ≤t ≤νΞ(t )，则Ξ(t )≤Θ(t )，Ξ(t +τ)≤Θ(t )，τ∈[-λ，0). 从而得到Θ(ν)=sup -λ≤t ≤νΞ(t )≤max{}sup -λ≤t ≤0Ξ(t )+sup 0<t ≤νΞ(t )≤Λ12ε2ν2β+Λ22ε2H ν2H +2β-1+ Λ31ε2ν2(γ-1)(1-β)+Λ32ε2ν2(β-1)+()Λ11ε2ν2β-1+Λ21ε2Hν2(H +β-1)∫0νΘ(s )d s . (22)由Gronwall 不等式，可推出Θ(ν)≤()Λ12ε2ν2β+Λ22ε2H ν2H +2β-1+Λ31ε2ν2(γ-1)(1-β)+Λ32ε2ν2(β-1)exp ()Λ11ε2ν2β+Λ21ε2H ν2H +2β-1，（23）即有E()sup -λ<t ≤νx ε(t )-z ε(t )2≤()Λ12ε2ν2β+Λ22ε2H ν2H +2β-1+Λ31ε2ν2(γ-1)(1-β)+Λ32ε2ν2(β-1)× exp ()Λ11ε2ν2β+Λ21ε2H ν2H +2β-1. (24)即存在M 0>0和α∈(0，1)，使得对所有t ∈(0，M 0ε-α]⊂(0，b ]满足E()sup 0<t ≤M 0ε-αx ε(t )-z ε(t )2≤με1-α，其中常数μ=(Λ12M 2β0ε1+α-2αβ+Λ22M 2H +2β-10ε2α(1-H -β)+2H -1+Λ31M 2(γ-1)(1-β)0ε2α(1-γ)(1-β)+α+1)+Λ32M 2(β-1)0ε3α-2αβ+1exp ()Λ11M 2β0ε2-2αβ+Λ21M 2H +2β-10ε2H -α(2H +2β-1) . (25)所以对任意给定的数δ>0，存在ε1∈(0，ε0]，使得对任意ε∈(0，ε1]和t ∈[-λ，M 0ε-α]⊂ [-λ，b ]，有E()supt ∈[-λ，M 0ε-α]x ε(t )-z ε(t )2≤δ.定理1证毕.注2 现有文献考虑的是有限维空间上含泊松跳以及Wiener 过程的无脉冲扰动的Hilfer 分数阶随机微分方程的平均原理（Ahmed et al.，2021；Luo et al.，2021），与之相比，本文考虑了分数布朗运动驱动的含脉冲项的Hilfer 分数阶随机微分方程. 更为重要的是，我们在Hilbert 空间上建立了具有算子的Hilfer 分数阶151第 63 卷中山大学学报（自然科学版）（中英文）脉冲随机发展方程的平均原理，一定程度上丰富了Hilfer 分数阶随机微分方程的平均原理的相关理论.3 实例为了说明所得结果的适用性，我们考虑以下含脉冲的Hilfer 分数阶随机发展方程ìíîïïïïïïïïïïD γ，23x ε(t ，z )=∂2∂z2x ε(t ，z )+εsin 2(t )x ε，t (z )+2εH cos 2(t )x ε，t(z )d B H Q (t )d t ， I 13(1-γ)x ε(0，z )=φ0，Δx ε(t k ，z )=εx ε(t k ，z )(4+k )(5+k )， z ∈[0，π]，k ∈1，2，⋯，m ，x ε(θ，z )=φ(θ，z )， θ∈[-λ，0]，z ∈[0，π]，x ε(t ，0)=x ε(t ，π)=0， t ∈(0，m π].（26）令空间X =Y =L 2[0，π]，系数函数f ()t ，x ε，t =sin 2(t )x ε，t (z )，h ()t ，x ε，t =2cos 2(t )x ε，t (z )，脉冲函数I k =x ε(t k ，z )(4+k )(5+k ).定义算子A ：D (A )→X ，Ax ε(t ，z )=∂2∂z2x ε(t ，z )，其中定义域D (A )={}x ∈X ， x ，x '全连续，x ″∈X ， x (0)=x (π)=0，则A 是强连续算子半群{S (t )}t ≥0的无穷小生成元，且对任意t ≥0，S (t )是紧的、解析且自伴的，由一致有界定理可知存在一个常数M >0，使得 S (t )≤M ，且A 有离散谱，其特征值是-n 2，n ∈N +，对应的标准正交特征向量为ωn (z )=(nz )，n =1，2，⋯，则当x ∈D (A )时，Ax =-∑n =1∞n 2x ，ωn ωn . 为了定义算子Q ：Y →Y ，选择一组非负有界实数序列{λn }n ≥1，并在Y 中选取标准正交基{e n }n ≥1，使得Qe n =λn e n 成立，并且假设tr (Q )=∑n =1∞λn <+∞，从而可以定义随机过程B H Q(t )=∑n =1∞λn B H n (t )e n ，其中H ∈()1/2，1，{B Hn(t )}n ∈N +，是一个独立于完备概率空间(Ω，F ，P )的一维标准分数布朗运动序列.取T =π，则f ˉ(x )=1π∫πf (s ，x )d s =12x ， h ˉ(x )=1π∫0πh (s ，x )d s =x ，I ˉ(x )=1π∑k =1m x (4+k )(5+k )=mx 5(5+m )π，于是方程（26）的平均系统为ìíîïïïïïïïïïïïïD γ，23y ε(t ，z )=Ay ε(t ，z )+ε()m 5(5+m )π+12y ε，t (z )+y ε，t εH (z )d B H Q (t )d t ，I 13(1-γ)y ε(0，z )=φ0，y ε(θ，z )=φ(θ，z )， θ∈[-λ，0]，z ∈[0，π]，y ε(t ，0)=y ε(t ，π)=0.（27）显然，平均系统（27）比原系统（26）简单. 假设条件（H0）~（H3）满足，根据定理1，当ε趋于零时，系统（26）的适度解均方收敛于平均系统（27）的适度解.参考文献：刘健康，王进斌，徐伟，2023. Caputo 分数阶中立型微分方程的随机平均原理［J ］. 山西大学学报（自然科学版），46（2）：304-308.ABOUAGWA M ，BANTAN R A R ，ALMUTIRY W ，et al ，2021. Mixed Caputo fractional neutral stochastic differential equationswith impulses and variable delay ［J ］. Fractal Fract ，5（4）：239.152153第 1 期吕婷，等：Hilfer分数阶脉冲随机发展方程的平均原理AHMED H M，EL-BORAI M M，EL-OWAIDY H M，et al，2018. Impulsive Hilfer fractional differential equations［J］. Adv Differ Equ，2018（1）：1-20.AHMED H M，ZHU Q，2021. The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps［J］. Appl Math Lett， 112： 106755.BOLLERSLEV T， OLE MIKKELSEN H， 1996. Modeling and pricing long memory in stock market volatility［J］. J Econom， 73（1）： 151-184.BOUDRAHEM S，ROUGIER P R，2009. Relation between postural control assessment with eyes open and centre of pressure visual feedback effects in healthy individuals［J］. Exp Brain Res， 195（1）： 145-152.CERRAI S， FREIDLIN M， 2009. Averaging principle for a class of stochastic reaction-diffusion equations［J］. Probab Theory Relat Fields， 144（1）： 137-177.CUI J，BI N，2020. Averaging principle for neutral stochastic functional differential equations with impulses and non-Lipschitz coefficients［J］. Stat Probab Lett， 163： 108775.de la FUENTE I M， PEREZ-SAMARTIN A L， MARTÍNEZ L， et al， 2006. Long-range correlations in rabbit brain neural activity ［J］. Ann Biomed Eng， 34（2）： 295-299.GU H， TRUJILLO J J，2015. Existence of mild solution for evolution equation with Hilfer fractional derivative［J］. Appl Math Comput， 257： 344-354.KOLMOGOROV A N，1940.The Wiener spiral and some other interesting curves in Hilbert space［J］. Dokl Akad Nauk SSSR， 26（2）：115-118.LELAND W E， TAQQU M S， WILLINGER W， et al， 1994. On the self-similar nature of Ethernet traffic （extended version）［J］.IEEE/ACM Trans Netw， 2（1）： 1-15.LIU B D，2007. Uncertainty theory［M］. 2nd ed. Berlin：Springer-Verlag.LIU J，WEI W，XU W，2022a. An averaging principle for stochastic fractional differential equations driven by fBm involving impulses ［J］. Fractal Fract，6（5）：256.LIU J，XU W，GUO Q，2020. Global attractiveness and exponential stability for impulsive fractional neutral stochastic evolution equations driven by fBm［J］. Adv Differ Equ：1-17.LIU J，XU W，GUO Q，2022b. Averaging of neutral stochastic partial functional differential equations involving delayed impulses ［J］. Appl Anal，101（18）：6435-6450.LUO D，ZHU Q，LUO Z，2021. A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients［J］. Appl Math Lett， 122： 107549.MA S， KANG Y， 2019. Periodic averaging method for impulsive stochastic differential equations with Lévy noise［J］. Appl Math Lett， 93： 91-97.MANDELBROT B B， van NESS J W， 1968. Fractional Brownian motions， fractional noises and applications［J］. SIAM Rev， 10（4）： 422-437.REN Y，CHENG X，SAKTHIVEL R，2014. Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm［J］. 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普林斯顿方程

普林斯顿方程引言普林斯顿方程，又称为普林斯顿方程组，是描述等离子体动态行为的一组非线性偏微分方程。

它由数学家M.G. 普林斯顿（M. G. Prandtl）于20世纪初提出，是等离子物理学中的重要理论工具。

本文将对普林斯顿方程进行全面、详细、完整且深入的探讨。

普林斯顿方程的概述普林斯顿方程组是描述等离子体中电离、扩散、湍流运输等现象的一组非线性偏微分方程。

它包括了等离子体的连续性方程、动量守恒方程、能量守恒方程和泊松方程。

连续性方程连续性方程描述了等离子体的质量守恒关系，用于描述等离子体中粒子的扩散和输运过程。

它可以写成以下形式：∂n∂t+∇⋅(nv)=S n其中，n是等离子体的粒子数密度，v是等离子体的速度场，S n是粒子源项。

动量守恒方程动量守恒方程描述了等离子体中动量的输运和转换过程，用于揭示等离子体中的湍流行为和推动力的产生机制。

它可以写成以下形式：∂v ∂t +(v⋅∇)v=−∇pm+qm(nE+v×B)+ν∇2v+F其中，v是等离子体的速度场，p是等离子体的压力，m是等离子体的质量，q是等离子体的电荷，E和B分别是电场和磁场，ν是等离子体的动力粘性系数，F是外力项。

能量守恒方程能量守恒方程描述了等离子体中能量的输运和转换过程，用于研究等离子体的加热、辐射和能量损失机制。

它可以写成以下形式：∂T ∂t +(v⋅∇)T=23n(∂q∂t+∇⋅q)+23n∇⋅(κ∇T)+Q其中，T是等离子体的温度，q是等离子体的热流密度，κ是等离子体的热导率，Q 是能量源项。

泊松方程泊松方程描述了等离子体中电势场的分布和电场的生成机制，用于研究等离子体中的电磁行为。

它可以写成以下形式：∇2ϕ=−ρϵ0其中，ϕ是电势场，ρ是等离子体的电荷密度，ϵ0是真空介电常数。

普林斯顿方程的应用普林斯顿方程在等离子体物理学的研究中具有广泛的应用。

以下是一些普林斯顿方程的典型应用领域：1.等离子体控制–利用普林斯顿方程可以研究等离子体在磁约束聚变装置中的控制方法，从而实现稳定的等离子体状态，为聚变实验提供可靠的等离子体环境。

William_jennings_bryan全文

The second time(1900) :approved anti-imperialism(反帝国主义) McKinley won the electoral college with a count of 292 votes compared to Bryan's 155.
The third time (1913):He lost the electoral college 321 to 162, his worst defeat yet.
Fundamentalism :a religious movement of conservative Protestants in the U.S.A. in the early 1920s;
Its purpose : to maintain the traditional Christian view of the Bible and to assert the literal interpretation of the Biblical narrative
Three times of Presidential election
In1896,at the age of 36, Bryan became (and still remains) the youngest presidential nominee of a major party in American history.
politician—democrat， the 41st United States Secretary of State
one of the best known orators
a Presbyterian(长老教会员)t(禁酒主义者)

麦克斯韦方程组英文

麦克斯韦方程组英文English: The Maxwell's equations are a set of fundamental equations in classical electromagnetism that describe how electric and magnetic fields interact with each other and with electric charges and currents. They were formulated by the physicist James Clerk Maxwell in the 19th century and played a crucial role in the development of electromagnetic theory. The equations consist of four equations: Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law of electromagnetic induction, and Ampère's law with Maxwell's addition. These equations are mainly concerned with the spatial and temporal changes of electric and magnetic fields, and they are usually written in differential form or integral form. In the differential form, the equations express how the fields change at each point in space, while in the integral form, they describe the flux of the fields through closed surfaces or the circulation of the fields along closed paths. The Maxwell's equations have important implications in many areas of physics and engineering, as they govern the behavior of electromagnetic waves, the propagation of signals through transmission lines, the behavior of antennas, the operation of electric motors and generators, andvarious other electromagnetic phenomena. In addition, the Maxwell's equations also played a crucial role in the development of the theory of relativity, as they led to the realization that electric and magnetic fields are two different manifestations of a single electromagnetic field, and they can transform into each other under certain conditions. Overall, the Maxwell's equations are of fundamental importance in understanding the behavior of electric and magnetic fields and their interactions with matter, and they have paved the way for numerous technological advancements and scientific discoveries.中文翻译: 麦克斯韦方程组是经典电磁学中描述电场、磁场与电荷电流相互作用的一组基本方程。

超分子化学读物

超分子化学读物以下是一些关于超分子化学的读物推荐：1.《分子自组装与超分子化学导论》（Introduction to Molecular Self-Assembly and Supramolecular Chemistry） by Akira Harada and Hiroyuki Furuta这本书介绍了超分子化学的基本概念和原理，以及分子自组装的理论与应用。

2.《超分子化学》（Supramolecular Chemistry） by Jonathan W. Steed and Jerry L. Atwood这是一本系统介绍超分子化学领域的教材，涵盖了自组装、配位化学、分子识别和分子动态等核心内容。

3. 《拓扑结构导向的超分子化学》（Topological Structure-Directed Supramolecular Chemistry） by Wei Zhang and Guo-Xin Jin这本书侧重于介绍超分子化学中的拓扑结构导向策略，即通过特定的分子拓扑结构来调控超分子组装和性能。

4. 《超分子化学的原理与设计》（Principles and Design of Supramolecular Chemistry） by George W. Gokel这本书详细介绍了超分子化学中的各类分子识别机制和分子组装策略，并且探讨了与超分子化学相关的应用领域。

5. 《超分子合成自组装与应用》（Supramolecular Synthesis, Self-Assembly and Applications） by Edite M. G. P. da Silva andVitor S. Amaral该书涵盖了超分子化学中合成和自组装的基本方法和应用，以及分子开关、化学传感器和药物输运等前沿领域的发展。

这些读物都可以帮助你深入了解超分子化学的基本原理和最新研究进展，选择其中一本根据自己的兴趣和需求来阅读。

自由基化学

自由基化学刘有成(中国科学技术大学化学系　合肥　230026) 刘中立(兰州大学应用有机化学国家重点实验室　兰州　730000)刘有成　男(1920111—),安徽舒城人,教授,博士,中国科学院院士,研究方向:有机化学。

刘中立　男(194116—),湖北汉阳人,教授,研究方向:有机化学。

1999210210收稿摘　要　近20年来,中国的自由基化学研究在若干领域取得了重要的进展,包括单电子转移反应,自旋离域取代基参数的建立,自由基的热力学稳定性研究等。

哌啶氮氧自由基与一些生物小分子,如半胱氨酸、谷胱甘肽、抗坏血酸等,在溶液及胶束中的反应经过动力学研究证明反应的单电子转移机理。

哌啶氧铵盐氧化芳香胺和含氮、硫芳杂环为相应的自由基正离子,首次报道了噻蒽及甲基吩噻嗪自由基正离子与其中性母体之间电子转移的同位素效应。

通过对对位取代的Α,Β,Β2三氟苯乙烯热环加成的动力学研究,建立了一套新的自旋离域取代基参数ΡJJ ・,与取代基的极性效应分开,并用一个双参数方程对一些自由基反应进行相关分析,取得了成功。

研究了吸电子取代基和给电子取代基对自由基热力学稳定性的影响,证明了C lass O 自由基的存在。

关键词　自由基　电子转移　取代基效应Abstract D u ring the past tw o decades i m po rtan t advances have been m ade in som e areas of freeradiacal research in the Peop le’s R epub lic of Ch ina ,i .e .single electron tran sfer reacti on s ,theestab lishm en t of sp in 2delocalizati on sub stituen t con stan ts ,and the study of thermodynam ic stab ilities ofradicals.It has been show n th rough k inetic studies that the reacti on s betw een p i peridine n itrox ides w ith som e b i o logical s m all mo lecu les ,e .g .cysteine ,glu tath i one ,asco rb ic acid ,in so lu ti on o r m icelles p roceed by electron tran sfer m echan is m .P i peridine oxoammon ium salts ox idize arom atic am ines and N ,S 2con tain ing heterocycles to the co rresponding radical cati on s ,and the iso tope effects in electron tran sfer reacti on s betw een th ian th rene and N 2m ethylpheno th iazine radical cati on s and their respective neu tral paren t mo lecu les have been repo rted fo r the first ti m e .T h rough k inetic studies on the therm alcycloadditi on reacti on s of Y 2sub stitu ted Α,Β,Β2trifluo ro styrenes ,a new scale ΡJJ ・of sp in 2delocalizati on sub stituen t con stan ts is estab lished ,w h ich is separated from the sub stituen t po lar effect ,and a dual 2param eter equati on is u sed fo r co rrelati on analysis of a num ber of free radical reacti on s w ith success.By study of the effects of E W G and ED G sub stituen ts on the thermodynam ic stab ilities of free radicals ,the ex istence of C lass O radicals has been p roved .Key words free radical ,electron tran sfer ,sub stituen t effect .新中国成立以来,尤其是改革开放以来,我国的有机自由基化学研究取得了重要的进展,引起了国际学术界的关注。

信息资源检索答案(完整)

1111111111111111111111试题1 单项选择题 (5.0分得分:5.0 难度:基本题)不同的数据库系统的字段代码是有所不同的，其中题名的常见表述有正确答案学生答案 TIAU KW AB试题2 判断题 (5.0分得分:5.0 难度:水平题)期刊是有固定名称，有一定的出版规律，标有刊期序号的一种论章成册的出版物。

正确答案学生答案真假试题3 单项选择题 (5.0分得分:5.0 难度:基本题)在进行信息检索的过程中，公式“（检索出的相关信息量/检索出的信息总量）×100%”计算的是：正确答案学生答案查准率查全率漏检率误检率试题4 单项选择题 (5.0分得分:5.0 难度:基本题)信息侵权主要包括侵犯______________和侵犯个人隐私权等。

正确答案学生答案社会隐私权著作权知识产权人权试题5 多项选择题 (5.0分得分:5.0 难度:难度题) 下列哪些语句能被检索式“information N/2 retrieval”命中（）正确答案学生答案―information retrieval‖―retrieval information‖―information computer aided retrieval‖ ―retrieval of education information‖试题6 单项选择题 (5.0分得分:5.0 难度:基本题)按照信息的载体类型，可将信息资源划分为：______、缩微型、声像型、电子型和多媒体。

正确答案学生答案印刷型书本型视频型音频型试题7 多项选择题 (5.0分得分:0.0 难度:难度题)以下类型的文献属于三次文献的是（）正确答案学生答案目录词典百科全书科技报告试题8 判断题 (5.0分得分:5.0 难度:水平题)信息可以记载在印刷纸本中，也可以存储在声、光、电、磁等介质中，如磁带、光盘、硬盘等。

正确答案学生答案真假试题9 多项选择题 (5.0分得分:5.0 难度:难度题)Chun，Lu，and n，“Dynamic Analysis of Clamped Laminated Curved Panels.”Composite Structures 30.4(1995)：375-388. 这是一篇期刊论文描述(著录)的书写格式,其中包含的信息包括了（）。

Nonlinear dynamics in one dimension On a criterion for coarsening and its temporal law

a r X i v :c o nd -m a t /0512055v 2 [c o n d -m a t .s t a t -me c h ] 22 F e b 2006Nonlinear dynamics in one dimension:On a criterion for coarsening and its temporal lawPaolo Politi 1,∗and Chaouqi Misbah 2,†1Istituto dei Sistemi Complessi,Consiglio Nazionale delle Ricerche,Via Madonna del Piano 10,50019Sesto Fiorentino,Italy 2Laboratoire de Spectrom´e trie Physique,CNRS,Univ.J.Fourier,Grenoble 1,BP87,F-38402Saint Martin d’H`e res,France(Dated:February 2,2008)We develop a general criterion about coarsening for a class of nonlinear evolution equations de-scribing one dimensional pattern-forming systems.This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is ﬁxed in the course of time.An intermediate scenario may occur,namely ‘interrupted coarsening’.The power of the criterion on which a brief account has been given [P.Politi and C.Misbah,Phys.Rev.Lett.92,090601(2004)],and which we extend here to more general equations,lies in the fact that the statement about the occurrence of coarsening,or selection of a length scale,can be made by only inspecting the behavior of the branch of steady state periodic solutions.The criterion states that coarsening occurs if λ′(A )>0while a length scale selection prevails if λ′(A )<0,where λis the wavelength of the pattern and A is the amplitude of the proﬁle (prime refers to diﬀerentiation).This criterion is established thanks to the analysis of the phase diﬀusion equation of the pattern.We connect the phase diﬀusion coeﬃcient D (λ)(which carries a kinetic information)to λ′(A ),which refers to a pure steady state property.The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically for several classes of equations.An-other important and new result which emerges here is that the exploitation of the phase diﬀusion coeﬃcient enables us to determine in a rather straightforward manner the dynamical coarsening exponent.Our calculation,based on the idea that |D (λ)|∼λ2/t ,is exempliﬁed on several nonlinear equations,showing that the exact exponent is captured.We are not aware of another method that so systematically provides the coarsening exponent.Contrary to many situations where the one dimensional character has proven essential for the derivation of the coarsening exponent,this idea can be used,in principle,at any dimension.Some speculations about the extension of the present results are outlined.PACS numbers:05.70.Ln,05.45.-a,82.40.Ck,02.30.JrI.INTRODUCTIONPattern formation is ubiquitous in nature,and espe-cially for systems which are brought away from equi-librium.Examples are encountered in hydrodynam-ics,reaction-diﬀusion systems,interfacial problems,and so on.There is now an abundant literature on this topic [1,2].Generically,the ﬁrst stage of pattern for-mation is the loss of stability of the homogeneous solu-tion against a spatially periodic modulation.This gen-erally occurs at a critical value of a control parameter,µ=µc (where µstands for the control parameter)and at a critical wavenumber q =q c .The dispersion rela-tion about the homogeneous solution (where perturba-tions are sought as e iqx +ωt ),in the vicinity of the critical point assumes,in most of pattern-forming systems,the following parabolic form (Fig.1,inset)ω=δ−(q −q c )2(1)δcorresponding to unstable modes (Fig.1),so that inﬁnitesimal perturbations grow exponentially with time until nonlinear eﬀects can no longer be ignored.In the vicinity of the bifurcation point (δ=0)only the princi-pal harmonic with q =q c is unstable,while all other har-monics are stable.For example,Rayleigh-B´e nard convec-tion,Turing systems,and so on,fall within this category,and their nonlinear evolution equation is universal in the vicinity of the bifurcation point.If the ﬁeld of interest (say a chemical concentration)is written as A (x,t )e iq c x ,where A is a complex slowly varying amplitude,then A obeys the canonical equation∂t A =A +∂xx A −|A |2A(2)where it is supposed that the coeﬃcient of the cubic term is negative to ensure a nonlinear saturation.Because the band of active modes is narrow and centered around the principal harmonic,no coarsening can occur,and the pat-tern will select a given length,which is often close to that of the linearly fastest growing mode.However,the ampli-tude equation above exhibits a phase instability,known under the Eckhaus instability [1],stating that among theband ofallowed states,|∆q|=√δ/3are stable withrespect to a wavelength modulation.There are many other situations where the bifurca-tion wavenumber q c→0and therefore a separation ofa slow amplitude and a fast oscillation is illegitimate.Contray to the case(1),where theﬁeld can be writtenas A(x,t)e iq c x with A being supposed to vary slowly inspace and time,if q c→0the supposed fast oscillation,e iq c x,becomes slow as well and a separation of A doesnot make a sense anymore.In this case,a generic formof the dispersion relation is(Fig.2,main)ω=δq2−q4.(3)A third situation is the one where the dispersion relationtakes the form(Fig.2,inset)ω=δ−q2.(4)In both cases,Eqs.(3,4),the instability occurs forδ>0,and the band of unstable modes extends from q=0toq=√1In case(4)an equation similar to(2)may arise,but it describesthe fullﬁeld and not just the envelope.is c(dotted line).Forδ>0,the unstable band extends fromq=q1to q=q2(full line).With increasingδthe unstableregion widens(dashed line).The parabolic shape ofω(q)isan approximation,valid close to its maximum.This applies,e.g.,to the dispersion curve(38)of the Swift-Hohenberg eq.(see the mainﬁgure):ω=δ−(1−q2)2[q c=1].Whenδ=1(dashed line)the unstable band extends down to q=0andω(q)resembles the dispersion curve of the Cahn-Hilliard eq.(see(3)and Fig.2).(δ<0).Full line:just above threshold.Dashed line:wellabove threshold.The vanishing ofω(0)for anyδis a con-sequence of the traslational invariance of the CH eq.in the‘growth’direction.Inset:The Ginzburg-Landau eq.is onecase where such invariance is absent and the dispersion curvehas the form(4):ω=δ−q2.which leads to spatio-themporal chaos.Note that by set-ting u=∂x h we obtain an equivalent form of this equa-tion,namely∂t h=−∂xx h−∂xxxx h+(∂x h)2/2.Thisequation arises in several contexts:liquidﬁlmsﬂowingdown an inclined plane[5],ﬂame fronts[7],stepﬂowgrowth of a crystal surface[3].Complex dynamics such as chaos,coarsening,etc...,are naturally expected if modes of arbitrarily large wave-length are unstable.However,these dynamics may occurfor systems characterized by the dispersion relation(1)3as well,if the system is further driven away from the critical point(i.e.,if q2≫q1,see Fig.1)because higher and higher harmonics become active.We may expect, for example,coarsening to become possible up to a total wavelength of the order of2π/q1.For systems which are at global equilibrium the nonlin-earity u∂x u is not allowed,and a prototypical equation having the dispersion relation(3)is the Cahn-Hilliard equation∂t u=−∂xx[u+∂xx u−u3].(6) The linear terms are identical to the KS one,and the dif-ference arises from the nonlinear term.Note that if dy-namics is not subject to a conservation constraint,(−∂xx) on the right hand side is absent,and the dispersion rela-tion is given by Eq.(4).The resulting equation is given by(2)for a real A and it is called real Ginzburg-Landau (GL)equation or Allen-Cahn equation.The KS equation,or its conserved form(obtained by applying∂xx on the right hand side),was suspected for a long time to arise as the generic nonlinear evolution equation for nonequilibrium systems(the quadratic term is non variational in that it can not be written as a func-tional derivative)whenever a dispersion relation is of type (3).Several recent studies,especially in Molecular Beam Epitaxy(MBE),have revealed an increasing evidence for the occurrence of completely new types of equations,with a variety of dynamics:besides chaos,there are ordered multisoliton[8,9]solutions,coarsening[10],freezing of the wavelength accompanied by a perpetual increase of the amplitude[11].Moreover,equations bearing strong resemblance with each other[12]exhibit a completely diﬀerent dynamics.Thus it is highly desirable to extract some general criteria that allow one to discriminate be-tween various dynamics.A central question that has remained open so far, and which has been the subject of a recent brief expo-sition[13],was the understanding of the general condi-tions under which dynamics should lead to coarsening, or rather to a selection of a length scale.In this paper we shall generalize our proof presented in[13]to a larger number of classes of nonlinear equations,for which the same general criterion applies:the sign of the phase dif-fusion coeﬃcient D is linked to a property of the steady state branch.More precisely,the sign of D is shown to be the opposite of the sign ofλ′(A),the derivative of the wavelengthλof the steady state with respect its ampli-tude A.Therefore,coarsening occurs if(and only if)the wavelength increases with the amplitude.Another important new feature that constitutes a sub-ject of this paper,is the fact that the exploitation of the phase diﬀusion coeﬃcient D(λ)will allow us to derive an-alytically the coarsening exponent,i.e.the law according to which the wavelength of the pattern increases in time. For all known nonlinear equations whose dispersion rela-tion has the form(3)or(4)and display coarsening,we have obtained the exact value of the coarsening expo-nent,and we predict exponents for other non exploited yet equations.An important point is that this is expectedto work at any dimension.Indeed,the derivation of the phase equation can be done in higher dimension as well.If our criterion,based on the idea that|D(λ)|∼λ2/t, remains valid at higher dimensions,it should become aprecious tool for a straightforward derivation of the coars-ening exponent at any dimension.II.THE PHASE EQUATION METHODA.GeneralityCoarsening of an ordered pattern occurs if steady state periodic solutions are unstable with respect to wave-lengthﬂuctuations.The phase equation method[14]al-lows to study in a perturbative way the modulations ofthe phaseφof the pattern.For a periodic structure of periodλ,φ=qx,where q=2π/λis a constant.If weperturb this structure,q acquires a space and time de-pendence and the phaseφis seen to satisfy a diﬀusion equation,∂tφ=D∂xxφ.The quantity D,called phasediﬀusion coeﬃcient,is a function of the steady state so-lutions and its sign determines the stable(D>0)orunstable(D<0)character of a wavelength perturba-tion.A negative value of D induces a coarsening process,2whose typical time and length scales are related by |D(λ)|∼λ2/t,as simply derived from the solution of the phase diﬀusion equation:this relation allows toﬁndthe coarsening lawλ(t).Therefore,the phase equation method not only allows to determine if certain classes of partial diﬀerential equations(PDE)display coarsen-ing or not;it also allows toﬁnd the coarsening laws, when D<0.In the rest of this section,we are going to oﬀer a short exposition of the phase equation method without referring to any speciﬁc PDE.Explicit evolution equations will be treated in the next sections,with some calculations relegated to the appendix.Let us consider a general PDE of the form3∂t u(x,t)=˜N[u](7)where˜N is an unspeciﬁed nonlinear operator,which is assumed not to depend explicitly on space and time. u0(x)is a periodic steady state solution:˜N[u0]=0and u0(x+λ)=u0(x).When studying the perturbation of a steady state,it is useful to separate a fast spatial variable from slow time and space dependencies.The stationary solution u0does4 not depend on time and it has a fast spatial dependence,which is conveniently expressed through the phaseφ=qx.Once we perturb the stationary solution,u=u0+ǫu1+...,(8)the wavevector q=∂xφgets a slow space and time de-pendence:q=q(X,T),where X=ǫx and T=ǫαt.Because of the diﬀusive character of the phase variable,the exponentαis equal to two.Space and time deriva-tives now read∂x=q∂φ+ǫ∂X(9a)∂t=ǫ(∂Tψ)∂φ(9b)where the second order term in the latter equation(ǫ2∂T)has been neglected.Finally,along with the phaseφit isuseful to introduce the slow phaseψ(X,T)=ǫφ(x,t),sothat q=∂Xψ.Replacing the u−expansion(8)and the derivates(9)with respect to the new variables in Eq.(7),weﬁnd anǫ−expansion which must be vanished term by term.Thezero order equation is trivial,˜N0[u0]=0:this equation isjust the rephrasing of the time-independent equation interms of the phase variableφ(the subscript in˜N0meansthat Eqs.(9)have been applied at zero order inǫ,i.e.∂x=q∂φ).Theﬁrst order equation is more complicated,becauseboth the operator˜N and the solution u areǫ−expanded.On very general grounds,we can rewrite∂t u(x,t)=˜N[u]asǫ(∂Tψ)∂φu0=(˜N0+ǫ˜N1)[u0+ǫu1](10)where˜N1comes fromﬁrst order contributions to thederivatives(9).If we use the Fr´e chet derivative[15],˜L0,deﬁned through the relation˜N0[u0+ǫu1]=˜N0[u0]+ǫ˜L0[u1]+O(ǫ2)(11)we get˜L0[u1]=(∂Tψ)∂φu0−˜N1[u0]≡g(u0,q,ψ).(12)Atﬁrst order,therefore,we get an heterogeneous linearequation(the Fr´e chet derivative of a nonlinear operatoris linear).The translational invariance of the operator˜N guarantees that∂φu0is solution of the homogeneousequation:according to the Fredholm alternative theo-rem[16],a solution for the heterogeneous equation mayexist only if g is orthogonal to the null space of the ad-joint operator˜L†.In simple words,if˜L†[v]=0,v and gmust be orthogonal.This condition,see Eq.(12),readsv,∂φu0 ∂Tψ= v,˜N1[u0] ,(13)where4 f,g =(2π)−1 2π0dφf∗g.5 If we deﬁne w=vG(u0),the equation˜L†0[v]=0is iden-tical to˜L0[w]=0,so that we can choose w=∂φu0andv=∂φu0/G(u0).The orthogonality condition between v and g reads(∂Tψ) v,∂φu0 −(∂X Xψ) v,G(u0)(2q∂q+1)∂φu0 =0(22)and replacing the explicit expression for v,we get thephase diﬀusion equation∂Tψ=D∂X Xψ(23)withD=∂q q(∂φu0)2G(u0)≡D12π λ0dx(u′0)2=J2π∂J4π2 ∂λ4π2B(A)∂A−1(26)where A is the amplitude of the oscillation,i.e.the(pos-itive)maximal value for u0(x).If G(u)≡1,a compact formula for D isD=−λ2B(A)2,called the most unstable wavevector.The linear regime corresponds to an exponential unsta-ble growth of such mode,with a rateω(q u),followed by a logarithmic coarsening.5The above equation can be made of wider application by considering the following generalized Cahn-Hilliard (gCH)equation∂t u=−C(u)∂xx[B(u)+G(u)u xx].(29)In Sec.III D we will discuss thoroughly the coarsening of this class of models,because of its relevance for the crystal growth of vicinal surfaces.6In that case,the local height z(x,t)of the steps satisﬁes the equation∂t z=−∂x{B(m)+G(m)∂x[C(m)∂x m]}(30) where m=∂x z.If we pass to the new variable u(m)= m0dsC(s)and take the spatial derivative of the above equation,we get the gCH equation(29).It is worthnoting that steady states are given by the equa-tion B(u0)+G(u0)u′′0=j0,where j0is a constant de-termined by the condition u0 =m0that imposes the (conserved)average value of the order parameter.If steps are oriented along a high-symmetry orientation, m0=0=j0.In the following we are considering this case only,so the equation determining steady states, B(u0)+G(u0)u′′0=0,is the same as for the gGL equa-tion.If we proceed along the lines explained in Sec.II A and keep in mind notations used in Sec.II B1,theﬁrst order equation in the small parameterǫreads−q2∂φφ˜L0[u1]=(∂Tψ)∂φu0C(u0) (∂Tψ)(33) +q2 v,∂φφ[G(u0)(2q∂q+1)∂φu0] (∂X Xψ)=0.6The quantity multiplying(∂X Xψ)can be rewritten as v,∂φφ[G(u0)(2q∂q+1)∂φu0 =∂φu0v,∂φu0˜D2.(35) In App.B we prove that the denominator˜D2is always positive.If C and G are(positive)constants the proof isstraightforward,because− v∂φu0 = (∂φv)u0 = u20 . The diﬀusion coeﬃcient(35)for the gCH equation istherefore similar to the diﬀusion coeﬃcient(24)for the nonconserved gGL equation:their sign is determined by the increasing or decreasing character ofλ(A),the wave-length of the steady state,with respect to its amplitude. The q2term in the numerator of(35)is evidence of the conservation law,i.e.,of the second derivative∂xx in Eq.(29).The denominators D2and˜D2diﬀer:this is irrelevant for the sign of D,but it is relevant for the coarsening law.If C(u)≡G(u)≡1,formulas simplify:D1=−λ3B(A)/4π2(∂Aλ)and˜D2= u20 =I/λ,where I= u20(x)has the same role as J= (∂x u0)2in the non-conserved model.Putting everything together we obtain D=−λ2B(A)1−√1+√2).The most unstable wavevector is q u=1for anyδ.For smallδthe unstable band is narrow;in fact,forδ<0.36,q1>q2/2and period doubling is not allowed.In other words study-ing coarsening for the Swift-Hohenberg equation close to the thresholdδ=0is not very interesting:nonetheless we will write the phase diﬀusion equation for anyδand for a generalized form of the Swift-Hohenberg equation as well.The zero order equation is easy to write˜N0[u0]≡−q4∂4φu0−2q2∂2φu0−(1−δ)u0−u30=0(39) and theﬁrst order equation has the expected form ˜L0[u1]=g,where˜L0≡−q4∂4φ−2q2∂2φ−(1−δ)−3u20(40) is the Fr´e chet derivative of˜N0andg≡(∂Tψ)∂φu0(41)+2(∂X Xψ)[(2q3∂q+3q2)∂3φu0+(2q∂q+1)∂φu o]. The operator˜L0is self-adjoint,so the solution of the homogeneous equation˜L†0[v]=0is immediately found, because of the translational invariance of˜N0along x: v=∂φu0.We therefore have(∂Tψ) (∂φu0)2 =(42)−(∂X Xψ)[ ∂φu0(4q3∂q+6q2)∂3φu0+2 ∂φu0(2q∂q+1)∂φu0 ].It is easy to check that both terms appearing in square brackets on the right hand side can be written as∂q(...):∂φu0(4q3∂q+6q2)∂3φu0 =−∂q 2q3(∂2φu0)2 (43a)∂φu0(2q∂q+1)∂φu0 ]=∂q q(∂φu0)2 (43b) so that the phase diﬀusion coeﬃcients readsD=∂q[2q3 (∂2φu0)2 −2q (∂φu0)2 ]2 (∂φu0)2n(−1)k+1nc n∂q q n−1(∂n/2φu0)2 (46)The standard Swift-Hohenberg equation corresponds to c2=−2and c4=1.The quantity(−1)k+1nc n q n−1/2 therefore gives2q3for n=4ans−2q for n=2,as shown by Eq.(44).7 III.THE COARSENING EXPONENTWe now want to use the results obtained in the previ-ous section for the phase diﬀusion coeﬃcient D in orderto get the coarsening lawλ(t).In one dimensional sys-tems,noise may be relevant and change the coarseninglaw.In the following we will restrict our analysis to thedeterministic equations.A negative D implies an unstable behavior of the phasediﬀusion equation,∂tψ=−|D|∂xxψ,which displays anexponential growth(we have reversed to the old coordi-nates for the sake of clarity):ψ=exp(t/τ)exp(2πix/λ),with(2π)2|D|=λ2/τ(in the following the time scaleτwill just be written as t).The relation|D(λ)|≈λ2/t willtherefore be used to obtain the coarsening lawλ(t):itwill be done for several models displaying the scenario ofperpetual coarsening(i.e.,λ′(A)>0for divergingλ).A.The standard Ginzburg-Landau andCahn-Hilliard modelsIt is well known[17]that in the absence of noise,boththe nonconserved GL equation(15)and the conservedCH equation(28)display logarithmic coarsening,λ(t)≈ln t.Let us remind that steady states correspond to thetrajectories of a classical particle moving in the potentialV(u)=u2/2−u4/4.The wavelength of the steady state,i.e.the oscillation period,diverges as the amplitude Agoes to one.This limit corresponds to the‘late stage’regime in the dynamical problem,and the proﬁle of theorder parameter is a sequence of kinks and antikinks.Thekink(antikink)is the stationary solution u+(x)(u−(x))which connects u=−1(u=1)at x=−∞to u=1(u=−1)at x=∞,u±(x)=±tanh(x/√√2)solution,7Q(x)≃Q0exp(7This may also be directly seen by expanding the diﬀerential equa-tion∂xx u+u−u3=0about u=1.Expansion to leading orderin Q=1−u yields∂xx Q−2Q=0,from which the solution√ QβQ=Q0e81.The nonconserved caseThe relationλ2J (∂A λ)(50)givest ∼Q −β/2β−2(51)so that λ(t )∼t n with n =(β−2)/(3β−2).2.The conserved caseIf the order parameter is conserved,wesimplyneed to replace J ∼1with I ∼λin Eq.(50),so as to obtaint ∼λλ3β−2β−2.(52)The coarsening exponent is therefore equal to n =(β−2)/(4β−4).We observe that in the limit β→2we recover logarithmic coarsening (n =0)both in the non-conserved and conserved case,as it should be.We also remark that in the opposite limit β→∞we get n =1/3in the nonconserved case and n =1/4in the conserved case,which make a bridge towards the models discussed in the next section.C.Models without uniform stable solutionsThe models considered in the previous subsection have uniform stable solutions,u =±1:the linear instabil-ity of the trivial solution u =0leads to the formation of domains where the order parameter is alternatively equal to ±1,separated by domain walls,called kinks and antikinks.This property is related to the fact that B (u )=u −u 3vanishes for ﬁnite u (up to a sign,B (u )is the force in the mechanical analogy for the steady states).In the following we are considering a modiﬁed class of models,where B (u )vanishes in the limit u →∞only,so that the potential V (u )= duB (u )does not have maxima at ﬁnite u .Therefore,it is not possible to deﬁne ‘domains’wherein the order parameter takes a constant value.These models [18],which may be relevant for the epitaxial growth of a high-symmetry crystal surface [10],are deﬁned as follows (α>1):∂t u =B (u )+u xx(53)∂t u =−∂xx [B (u )+u xx ](54)B (u )=u2(α−1)1λ2∼1α−1,which is slighty more complicated,be-cause the asymptotic contribution vanishes for α>2:in this case the ﬁnite,constant contribution coming from the motion within the ‘close region’dominates.There-fore,J ∼λ2α+1.Finally,B (A )∼λ1α−2α−1λ1−1α−2α∼λ22.For αlarger than two,λ23α−2.We can sum upour results,λ(t )∼t n ,withn =13α−2α>2(59b)The coarsening exponent varies with continuity from n =1/2for α<2to n =1/3for α→∞.These results conﬁrm what had already been found by one of us with a diﬀerent approach [18].2.The conserved caseFrom Eq.(36)we haveD ∼λ2λ1λ2α∼14.(61)The constant coarsening exponent n=1/4clashes with numerical results found in Ref.[18],n=1/4for α<2and n=α/(5α−2)forα>2.The opinion of the authors of Ref.[18]is that forα>2a crossover should exist from n=α/(5α−2)to n=1/4,the correct asymptotic exponent.Details and supporting arguments will be given elsewhere[19].D.Conserved models for crystal growthIt is interesting to consider a model of physical interest which belongs to the class of the full generalized Cahn-Hilliard equations,meaning that all the functions B(u), C(u)and G(u)appearing in(29)are not trivial.The starting point is Eq.(30),∂t z=−∂x{B(m)+G(m)∂x[C(m)∂x m]}which describes the meandering of a step,or—more generally—the meandering of a train of steps moving in phase.z is the local displacement of the step with re-spect to its straight position and m=∂x z is the local slope of the step.We do not give details here about the origin of the previous equation,which is presented in[12],but just write the explicit form of the functions B,G and C: B(m)=m1+m2(1+c)(1+m2)3/2(62b)and deﬁne the meaning of the two adimensional,positive parameters appearing there:βis the relative strength between the two relaxing mechanisms,line diﬀusion and terrace diﬀusion;c is a measure of the elastic coupling between steps.If we pass to the new variable u= m0ds C(s),we get Eq.(29),∂t u=−C(u)∂xx[B(u)+G(u)u xx](29) whose steady states,for high-symmetry steps,are given by the equation u xx=−B(u)/G(u).In App.C we study the potential V(u)= du[B(u)/G(u)]and the dynami-cal scenarios emerging from∂Aλ.We give here the results only.If c=0(see Fig.3),∂Aλ<0,while there is asymptotic coarsening if c>0(see Figs.4,5).Asymptotic coarsening means that∂Aλ>0for large enough A:according to the values of c andβ,λ(A)may be always increasing or it may have a minimum followed by∂Aλ>0:the distinction between the two cases is not relevant for the dynamics and it will not be further considered.Let us now determine the asymptotic behavior of all the relevant quantities,when c>0.In the limit of large m,we have C(m)∼m and B(m)∼1/m.As for G,G(m)∼1/m2and G(m)∼1/m,for V(u)=−√u so that C(u)∼√u and G(u)∼1/u(β=0)or G(u)∼1/√A forβ=0 (Fig.5).Similar and straightforward relations can be determined and the following general expression for the phase diﬀusion coeﬃcient is established,D∼√λ2∼λ26β=0(64a)λ(t)∼t1B(A)(65a)∂t u=−∂xx[B(u)+u xx]⇒t∼I(∂Aλ)10√|u |3/2for large u and λ′(A )>0(main,full circles).For large amplitude,λ≈A 1/4(main,full line).The dashed line in the inset is the harmonic potential,u 2/2.If c >0there is m 2A (main,full line).The dashed line in theinset is the harmonic potential,u 2/2.For β>0,λ(t )≈t 1/4.Passing from the standard GL/CH models (Sec.III A),to models where V (u )have (non quadratic)maxima at ﬁnite u (Sec.III B)and to models where V (u )has no maxima at all at ﬁnite u (Sec.III C),the coarsening ex-ponents change with continuity from n =0(logarithmic coarsening)to n =1/2for the nonconserved models and from n =0to n =1/4for the conserved models.The conservation law,as expected,always slows down the coarsening process.Formally,this corresponds to re-place the action J with the quantity I in the denominator of D .In most cases,J is a constant while I increases as λκ,with κ≥1:a smaller D implies a lower coarsening.We remark that only in a very special case (models with-out uniform stable solutions and α<2),I/J ∼λ2:when this happens,the double derivative ∂xx —which charac-terizes the conserved models—is equivalent (as for the coarsening law)to the factor 1/λ2.We stress again that this is an exception,it is not the rule.III D has been devoted to a class of conserved mod-are relevant for the physical problem of a grow-In that case the full expression (35)for D be considered (the result is reported in Eqs.(64a-It is remarkable that for all the models we have we found n ≤1/4and n ≤1/2for conserved models,respectively.It would be inter-to understand how general these inequalities are.9SWIFT-HOHENBERG EQUATION ANDCOARSENINGus start from the standard Swift-Hohenberg equa-tion (37),∂t u =−∂4x u −2∂2x u −(1−δ)u −u 3(66)linear dispersion curve is ω(q )=δ−(q 2−1)2.Thediﬀusion coeﬃcient (see Eq.(44))isD =2q∂q ( u 2xx − u 2x )2ω′′+(ω′)22ωdqωd ωd 9The condition n ≤1/2for the nonconserved models is equivalent to say that |D (λ)|does not diverge with increasing λ,which seems to be a fairly reasonable condition.11The amplitude A is deﬁned as A=[ dx u2(x)/λ]1/2. vanishes at q=q2and at q=q1,asωdoes,so that A un-dergoes a fold singularity at the center of the band.This is imposed by symmetry,since in the vicinity of threshold the band of active modes is symmetric.In this case we are in the situation with the dispersion relation(1).The phase diﬀusion coeﬃcient must also be symmetric with respect to the center,and therefore it can change sign at the fold,due to this symmetry.Thus D does not have the sign of A′in this case.Nonetheless,in the vicinity of q2the sign of D is still given by A′,as shown here below. Our speculation is that the existence of a fold is likely to destroy the simple link between D and A′.A meaningful expression for D can be found also for ﬁniteδ,close to q2= δ(let us remind that ω(q1,2)=0and q2>q1).In this limit we getω′(q)≃ω′(q2)=4q2−4q32andD=2(q32−q2)1dq=D E(72)so that D is equal to a positive quantity times dω/dq. Sinceω(q)=A2(q)/2,the sign of D is equal to the sign of dA/dq.This result follows from a perturbative scheme where use has been made of the fact that u∼cos(qx). This is legitimate as long as one considers small devia-tions from the threshold.Ifδis not small,or ifδ=1we fall in the dispersion relation(3).As one deviates from q2towards the center of the band,higher and higher har-monics become active,and one should in generalﬁnd nu-merically the steady state solutions in order to ascertain whether D is positive or negative.In the general case, we have not been able to establish a link between D and the slope of the steady state branch as done in the previ-ous sections.Our belief,on which some evidences will be reported on in the future,is that depending on the class of equations,it is not always the slope of the steady state solution that provides direct information on the nonlinear dynamics,but somewhat a bit more abstract quantities, as we have found,for example,by investigating the KS equation,another question on which we hope to report in the near future[20].Numerical solutions of the SH equation in the limitδ=1reveal a fold singularity inbranchλ(A),as shown in Fig.6.V.EQUATIONS WITH A POTENTIAL Some of the equations discussed in Sec.II B are deriv-from a potential:it is therefore possible to deﬁne function F which is minimized by the dynamics.This always the case for the generalized Ginzburg-Landau(16),which can be written as∂t u=B(u)+G(u)u xx=−G(u)δF2(u x)2−V(u) (74)where V(u)is the potential entering in the study of the stationary solutions,i.e.V′(u)=B(u)/G(u).If we eval-uate the time derivative of F weﬁndd Fδuu t=− dxG(u) δFδu].(76) If C(u)=G(u),weﬁndd Fδu2≤0.(77)We now want to evaluate F for the steady states.The pseudo free-energy F is nothing but the integral of the Lagrangian function L for the mechanical analogy deﬁn-ing the stationary solutions.If E=(u x)2/2+V(u)is the ‘energy’in the mechanical analogy and J is the action,F[uλ(x)]=ℓJdλ=ℓdλ−E=−ℓJ。

THE STABLE MANIFOLD THEOREM FOR SEMILINEAR STOCHASTIC EVOLUTION EQUATIONS AND STOCHASTIC PA

research of this author is supported in part by NSF Grants DMS-9703852, DMS-9975462 and DMS-0203368. ∗∗ The research of this author is supported in part by EPSRC Grant GR/R91144. ∗∗∗ The research of this author is supported in part by EPSRC Grants GR/R69518 and GR/R93582. AMS 1991 subject classiﬁcations. Primary 60H10, 60H20; secondary 60H25. Key words and phrases. Stochastic semiﬂow, cocycle, stochastic evolution equation (see), stochastic partial diﬀerential equation (spde), multiplicative ergodic theorem, stationary solution, hyperbolicity, local stable (unstable) manifolds.
∗ The
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S.-E.A. MOHAMMED, T.S. ZHANG AND H.Z. ZHAO
(sde’s and sfde’s), this concept has been used extensively in previous work of one of the authors with M. Scheutzow ([M-S.1], [M-S.2-4]). Our main objective is to characterize the pathwise local structure of solutions of semi-linear see’s and spde’s near stationary solutions. We introduce the concept of hyperbolicity for a stationary solution of an see. Hyperbolicity is deﬁned by the non-vanishing of the Lyapunov spectrum of the linearized cocycle. The hyperbolic structure of the stochastic semiﬂow leads to local stable manifold theorems (Theorems 4.1-4.4) for semi-linear see’s and spde’s. For a hyperbolic stationary solution of the see, this gives smooth stable and unstable manifolds in a neighborhood of the stationary solution. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiﬂow. The proof of the stable manifold theorem uses inﬁnite-dimensional multiplicative ergodic theory techniques ([Ru.1], [Ru.2]) together with interpolation and perfection arguments ([Mo.1], [M-S.4]). In particular, we will assume that the reader is familiar with the results and the techniques in Ruelle’s articles [Ru.1] and [Ru.2]. Our results cover semilinear stochastic evolution equations, stochastic parabolic equations, stochastic reaction-diﬀusion equations, and Burgers equation with additive inﬁnite-dimensional noise. We recall below the deﬁnition of a cocycle in Hilbert space. Let (Ω, F , P ) be a complete probability space. Suppose θ : R × Ω → Ω is a group ¯ the P -completion of of P -preserving ergodic transformations on (Ω, F , P ). Denote by F F. Let H be a real separable Hilbert space with norm | · | and Borel σ -algebra B (H ). Take k to be any non-negative integer and ∈ (0, 1]. Recall that a C k, perfect

2024届湖北省高考物理核心考点考向预测卷(一)

2024届湖北省高考物理核心考点考向预测卷（一）学校：_______ 班级：__________姓名：_______ 考号：__________（满分：100分时间：75分钟）总分栏题号一二三四五六七总分得分评卷人得分一、单项选择题（本题包含8小题，每小题4分，共32分。

在每小题给出的四个选项中，只有一项是符合题目要求的）(共8题)第(1)题如图所示，理想变压器原、副线圈匝数之比，定值电阻，滑动变阻器R2的最大值为10Ω，阻值恒定的小灯泡L 的规格为“6V 6W”，电流表是理想交流电表，输入端接入的交流电压，下列说法正确的是（）A．通过电流表的电流方向每秒钟改变20次B．小灯泡正常工作时，滑动变阻器的阻值为6ΩC．滑动变阻器阻值为6.5Ω时，变压器输出功率最大且为12.5WD．滑片自上而下滑动时，电流表示数先增大再减小第(2)题2022年诺贝尔物理学奖授予法国学者阿兰·阿斯佩（AlainAspect），美国学者约翰·克劳泽（JohnClauser）和奥地利学者安东·蔡林格（AntonZeilinger），既是因为他们的先驱研究为量子信息学奠定了基础，也是对量子力学和量子纠缠理论的承认，下列关于量子力学发展史说法正确的是（）A．普朗克通过对黑体辐射的研究，第一次提出了光子的概念，提出“光由光子构成”B．丹麦物理学家玻尔提出了自己的原子结构假说，该理论的成功之处是它保留了经典粒子的概念C．爱因斯坦的光电效应理论揭示了光的粒子性D．卢瑟福的原子核式结构模型说明核外电子的轨道是量子化的第(3)题如图所示，甲、乙是两个完全相同的闭合导线线框，a、b是边界范围、磁感应强度大小和方向都相同的两个匀强磁场区域，只是a区域到地面的高度比b高一些。

甲、乙线框分别从磁场区域的正上方距地面相同高度处同时由静止释放，穿过磁场后落到地面。

下落过程中线框平面始终保持与磁场方向垂直。

以下说法正确的是（）A．甲乙两框同时落地B．乙框比甲框先落地C．落地时甲乙两框速度相同D．穿过磁场的过程中甲线框中通过的电荷量小于乙线框第(4)题如图所示是主动降噪耳机的降噪原理图。