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宝光塔英语作文Title: The Majestic Baoguang Pagoda: A Symbol of Timeless EleganceNestled amidst the tranquility of ancient landscapes and surrounded by the whispers of history, the Baoguang Pagoda stands tall as a testament to the rich cultural heritage of its region. This majestic structure, adorned with centuries of stories and legends, invites visitors from all corners of the world to embark on a journey through time, marveling at its architectural brilliance and spiritual significance.Origins and HistoryThe Baoguang Pagoda, with its roots deeply embedded in the past, was built during a period when Buddhism flourished in China. Its construction, believed to have commenced in the Song Dynasty (960-1279 AD), reflects the profound influence of Buddhist beliefs and practices on the local community. Over the years, the pagoda has undergone several renovations and restorations, yet it remains largely intact, preserving its original charm and grandeur.Architectural WondersArchitecturally, the Baoguang Pagoda is a masterpiece that showcases the ingenuity and craftsmanship of ancient Chinese artisans. It rises gracefully from its base, its tiers tapering upwards towards the sky, each layer adorned with intricate carvings and delicate decorations. The use of bricks, stones, and wooden beams in its construction not only ensures structural stability but also adds to its aesthetic appeal. The pagoda's exterior is painted in vibrant hues, symbolizing the various aspects of Buddhist teachings and the cycle of life.Cultural SignificanceBeyond its physical beauty, the Baoguang Pagoda holds immense cultural significance. It serves as a place of pilgrimage for Buddhists, who come to seek enlightenment, pray for blessings, and meditate amidst the serene surroundings. The pagoda also attracts scholars, historians, and tourists alike, who are fascinated by its historical value and the stories it whispers. During festivals and special occasions, the pagoda comes alive with the sounds of chanting, prayers, and the vibrant colors of traditional costumes, creating a harmonious blend of the past and present. Preservation and FutureRecognizing the importance of preserving this cultural treasure, local authorities and communities have taken measures to ensure the Baoguang Pagoda's longevity. Regular maintenance and restoration work are carried out to safeguard its structural integrity, while efforts are made to promote its cultural significance and attract visitors. With the help of modern technology and traditional knowledge, the pagoda continues to thrive, serving as a bridge between the past and the future.In conclusion, the Baoguang Pagoda stands as a shining example of China's rich cultural and architectural heritage. Its timeless elegance, profound historical significance, and spiritual aura make it a must-visit destination for anyone interested in exploring the depths of Chinese culture and history. As the years pass, the Baoguang Pagoda will continue to inspire awe and admiration, reminding us of the enduring power of tradition and the beauty of human creativity.。
兵马俑简介－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－敬的女士们、先生们：今天，我们将参观秦兵马俑博物馆。
秦兵马俑博物馆位于酉安市东35公里处，从喜来登酒店坐车到那里大约需要50分钟时间。
自秦兵马俑博物馆于1979年10月1日开馆至今，已有为效众多国家的党政首脑都参观过这个博物馆，更有数以百万计的中外游客不远干里来参观这个人类奇迹。
法国总统希拉克曾留言说过：“世界上原有七大奇迹，秦兵马俑的发现，可以说是第八大奇迹了。
不看金字塔．不算真正到过埃及…不看秦俑，不算真正到过中国。
”美国前副总统蒙代尔也说；“这是真正的奇迹。
全世界人民都应该到这里看一看。
”从这些高度凝练的话语中，我们不难看出秦俑的历史价值及艺术价值。
下面，我想把秦始皇这位成就一代伟业的历史巨人介绍给诸位，以便使大家能对秦始皇兵马俑和与之相关的历史有一个较详尽的了解。
谈秦兵马俑，必须先介绍一下面赢政。
公元前259年，一代伟人赢政诞生了。
13岁时，其父死后．他继承了王位。
按照泰国惯例．国王在22岁举行过“加冕礼”以后，才可以亲自执政。
当时．秦国的大权仍攀握在太后、相国吕不韦、宦官谬瘪(160 6i)手中。
特别是螺贞．手握大权，炙手可热，他见宛改逐渐长大，便阴谋叛乱．夺取政权。
公元前238年，22岁的赢政去故都雍城纳新年宫举行加冕礼。
谬酉在从霸年宫回咸阳的路上埋下伏兵，准备将赢政杀死。
当赢政察觉此事后，凭优势兵力订败锣盔，将其生擒并车裂致死。
赢政亲致以后．以吕不韦放纵嫁亮为名，放逐日不韦。
吕不韦后来饮毒洒自毙。
这样，赢政顺利地消灭了政权内部的两大敌对势力，巩固了自身的地位。
为了进一步巩固君主权力，赢政选用了一批精明强干酌文臣武将，如掌握军事的尉缭，攀管国欧的李斯。
接着，赢政制定了远交近攻、离间外敌、各个击破的战赂方针，开始统一中国，结束了春秋战国以来长达500多年各诸侯国割据称雄的混乱局面。
从公元前230年到公元前221年，在不到io年的时间里，泰国灭亡了6国，终于完成了统一大业，建立了第一个中央集权的封建国家，为封建社会经济、政治、思想文化的迅速发展奠定了基础。
听美文故事记六级单词第42期:罗德岛的太阳神巨像The Colossus of Rhodes罗德岛的太阳神巨像To you, O Sun, the people of Dorian Rhodes set up this bronze statue reaching to Olympus when they had pacified the waves of war and crowned their city with the spoils taken from the enemy.罗德岛的多利安人民谨以此高耸入云的铜像献给您，太阳神，以纪念他们平息了战争的狂涛并用从敌人手中夺来的战利品荣耀了他们的城市。
Not only over the seas but also on land did they kindle the lovely torch of freedom.Dedicatory inscription of the Colossus.他们不仅在海上而且在陆地上，点亮了自由的火炬---太阳神巨像上的提辞。
From its building to its destruction lies a time span of merely 56 years. Yet the colossus earned a place in the famous list of Wonders.太阳神巨像的建立到毁坏只有短促的56年时间。
然而这座巨像却在著名的世界七大奇观的名单里赢得了一席之位。
"But even lying on the ground, it is a marvel", said Pliny the Elder. The Colossus of Rhodes was not only a gigantic statue.老莆林尼说过:“但是即便在它倒塌在地时，仍不失是为一个奇观。
多元宇宙恶魔殿作文英文回答：The multiverse is a vast and mysterious place, filled with an infinite number of possibilities. It is a realm where anything is possible, and where the laws of physics and reality are constantly being tested. In the multiverse, there are countless worlds, each with its own unique set of rules and inhabitants. Some of these worlds are peaceful and idyllic, while others are filled with danger and chaos. There are worlds where magic is commonplace, and worlds where technology has reached dizzying heights. There are even worlds where the laws of nature themselves are different, and where time and space can be manipulated at will.The multiverse is also home to a wide variety of creatures, from the smallest insects to the largest cosmic entities. There are creatures that are kind and benevolent, and creatures that are cruel and malevolent. There arecreatures that are wise and intelligent, and creatures that are simple and dim-witted. There are even creatures that defy all known laws of nature, and whose powers are beyond human comprehension.The multiverse is a place of endless wonder and adventure, where anything is possible. It is a realm where the boundaries of reality are constantly being pushed, and where the limits of human imagination are tested.中文回答：多元宇宙是一片广袤而神秘的空间，充满了无限的可能性。
高二外星人课本上的连环画作文英语Imagine a world beyond our own, where the laws of physics as we know them do not apply. A world where beings with extraordinary capabilities roam the cosmos, their very existence a mystery to the inhabitants of Earth. This is the premise of the captivating comic strip that graces the pages of my high school textbook, a window into a realm where the extraordinary and the mundane collide in a captivating dance.The comic strip begins with the introduction of our protagonists, a group of high school students not unlike ourselves, going about their daily lives. They attend classes, socialize with their peers, and navigate the familiar challenges that come with adolescence. Yet, from the very first panel, there is a palpable sense of unease, a subtle undercurrent of something more profound lurking beneath the surface.As the story unfolds, the students begin to notice strange occurrences around their school. Unexplained lights in the sky, peculiar symbols appearing on the walls, and a sense of beingwatched that sends shivers down their spines. Gradually, they come to the realization that their school, their very world, has been infiltrated by extraterrestrial beings.The comic strip's artwork is a masterful blend of realism and whimsy, capturing the raw emotions and reactions of the students as they grapple with this extraordinary revelation. The panels shift seamlessly between the mundane routines of high school life and the awe-inspiring, otherworldly encounters that challenge the very fabric of their understanding.One particularly poignant scene depicts a group of students huddled together in a dimly lit classroom, their faces illuminated by the eerie glow of a hovering, translucent craft outside the window. Their expressions range from terror to fascination, a testament to the artist's ability to convey the complex psychological landscape of their experience.As the story progresses, the students embark on a journey of discovery, delving deeper into the mysteries of the alien presence. They uncover clues, piece together fragments of information, and even forge unlikely alliances with the very beings they initially feared. The comic strip's narrative arc is a captivating blend of suspense, wonder, and personal growth, as the characters navigate the uncharted waters of their newfound reality.One of the most compelling aspects of this comic strip is its ability to juxtapose the extraordinary with the mundane. The students' struggle to maintain a semblance of normalcy in the face of the alien invasion is both relatable and poignant. They grapple with the weight of their newfound knowledge, balancing the desire to share their discoveries with the fear of being labeled as outcasts or dismissed as delusional.The comic strip's exploration of the human condition in the face of the unknown is particularly striking. As the students delve deeper into the mysteries of the alien presence, they are forced to confront their own biases, preconceptions, and deeply held beliefs. The comic strip masterfully weaves these philosophical and existential themes into the narrative, inviting the reader to ponder the implications of extraterrestrial contact and the resilience of the human spirit.Ultimately, the comic strip in my high school textbook is a testament to the power of storytelling and the boundless imagination of its creators. It transports the reader to a world where the extraordinary and the familiar coexist, challenging our perceptions and inviting us to consider the vast possibilities that lie beyond the boundaries of our own understanding. As I turn the pages of my textbook, I am captivated by the rich tapestry of this captivating narrative, eager to see where the journey will lead next.。
同素异形体的英语Title: Understanding Allotropes: The Fascinating World of Chemical StructuresAllotropes, a term derived from the Greek words "allos," meaning other or different, and "trope," meaning form or type, refer to different structural forms in which the same chemical element can exist. Despite being composed of identical atoms, allotropes exhibit varying properties due to differences in their molecular structures. This phenomenon is observed among several elements, with carbon and its allotropes—diamond, graphite, fullerenes, and graphene—being the mostwell-known examples. This essay explores the concept of allotropes, delving into their formation, properties, and significance.The concept of allotropism challenges the traditional understanding of elements as singular substances with fixed properties. Instead, it reveals that an element's properties can vary dramatically depending on how its atoms are arranged. For instance, diamond and graphite—both pure carbon—display vastly different physical characteristics. Diamond, known for its exceptional hardness and brilliant sparkle, is highly valued as a gemstone. In contrast, graphite, with itsslippery texture and electrical conductivity, is widely used in pencil lead and battery electrodes. These differences arise from the distinct ways in which carbon atoms bond in each structure. In diamond, carbon atoms form a rigid tetrahedral lattice, while in graphite, they arrange in layers of hexagonal rings, allowing them to slide over one another.Beyond carbon, other elements also display allotropism. Phosphorus, for example, exists as white, red, and black phosphorus, each with unique properties. White phosphorus is highly reactive and is used in match heads and fireworks, whereas red and black phosphorus are less reactive and have applications in safety matches and semiconductors respectively. Similarly, oxygen exists in two common allotropic forms: diatomic oxygen (O₂) and ozone (O₃). While O₂constitutes the majority of Earth's atmosphere, ozone is concentrated in the stratosphere, where it plays a crucial role in absorbing harmful ultraviolet radiation.The transition between different allotropes often involves changes in temperature, pressure, or catalysis. For instance, converting graphite to diamond requires extremely high pressure and temperature conditions found deep within the Earth or mimicked in laboratory settings using hydraulicpresses. Such transformations underscore the dynamic nature of atomic arrangements and highlight the potential for manipulating matter at the molecular level.The study of allotropes has practical implications across various fields. Materials science benefits significantly from the exploration of allotropes, leading to the development of novel materials with tailored properties. Carbon nanotubes and graphene, discovered more recently, showcase extraordinary strength, flexibility, and electrical conductivity, promising advancements in electronics, energy storage, and composite materials. Additionally, understanding allotropism aids in environmental science by elucidating the behavior and transformation of elements in natural processes, such as the carbon cycle.However, the existence of multiple forms for a single element also poses challenges in terms of nomenclature, classification, and understanding. Traditionally, elements were defined by a fixed set of properties, but allotropism necessitates a more nuanced approach. Each allotrope must be recognized as a distinct entity with its own set of characteristics, yet they all share a common chemical identity based on their elementary composition.In conclusion, allotropes represent a fascinating dimension of chemical diversity, demonstrating that an element's properties are not fixed but can vary extensively with changes in atomic arrangement. From the familiar contrasts between diamond and graphite to the specialized uses of different phosphorus allotropes, this phenomenon underscores the complexity and versatility of the natural world. As research into allotropes progresses, it opens new avenues for innovation and deepens our comprehension of the fundamental nature of matter. Embracing the study of allotropes not only enriches our scientific knowledge but also paves the way for technological advancements and practical applications across numerous domains.。
有关宇宙的文案英文短句Title: Exploring the Vast Mysteries of the UniverseIntroduction:The universe, a realm of infinite wonders and mysteries, has captivated humanity's imagination since time immemorial. Spanning countless light-years, this cosmic expanse stirs curiosity and fascination in both scientists and dreamers alike. In this article, we embark on a journey to delve into the captivating realm of the universe, a spectacle that continues to astonish us with its endless possibilities.1. Overview:1.1 The Vastness of the CosmosThe universe is a breathtaking canvas of celestial bodies,from shimmering galaxies and sparkling stars to enigmaticblack holes and dazzling nebulae. With its sheer vastness, it encompasses everything we know and yet holds countlesssecrets waiting to be unveiled.1.2 The Birth of Stars and PlanetsDeep within the cosmic abyss, stars are born. As clouds ofgas and dust collapse under gravity, they ignite to form blazing beacons scattered throughout the universe. These celestial bodies pave the way for planetary systems to emerge, potentially harboring life in the vastness of space.1.3 The Expanding UniverseThe universe is ceaselessly expanding. The famous Big Bang theory suggests that the universe originated from a singular point, rapidly expanding and giving rise to what we know today. Evidence of this expansion lies within the redshift oflight from distant galaxies, painting a vivid picture of the ever-expanding cosmos.1.4 Dark Matter and Dark EnergyDespite the tremendous advancements in astrophysics, a large portion of the universe's composition remains unknown. Dark matter and dark energy, elusive entities that cannot bedirectly observed, are believed to account for the majorityof the universe's mass and energy. Their nature continues to perplex scientists and drive research forward.2. The Quest for Extraterrestrial Life2.1 The Possibility of Habitable ExoplanetsExoplanets, planets orbiting stars outside our solar system, offer tantalizing prospects for the existence ofextraterrestrial life. Through space telescopes and other observational tools, scientists have identified numerous exoplanets within habitable zones, where liquid water – akey ingredient for life – may potentially exist.2.2 The Search for Intelligent LifeThe age-old question of whether intelligent life existsbeyond Earth remains unanswered. Initiatives such as the Search for Extraterrestrial Intelligence (SETI) actively scan the skies, sifting through the cosmic noise in hopes of detecting signals or signs of intelligent alien civilizations. While no conclusive evidence has emerged, the searchcontinues unabated.3. Cosmic Mysteries and Unanswered Questions3.1 Black Holes and Event HorizonsBlack holes, gravitational powerhouses with such immense density that nothing, not even light, can escape their grasp, captivate both scientists and the public. Exploring their nature, behavior, and the existence of event horizons is anongoing pursuit, shedding light on the boundaries of our understanding.3.2 The Fate of the UniverseWhat lies ahead in the future of our universe? Scientists ponder over the possibilities, considering scenarios such as the Big Crunch, where the universe collapses upon itself, or the eventual heat death, where all energy is evenly distributed and extinguished. The ultimate fate of our universe is yet unknown.Conclusion:The universe, a captivating enigma, continues to beckon us to explore its vast mysteries. From the birth of stars and potential for extraterrestrial life to the enigmas of dark matter and the fate of the cosmos, the universe remains an intriguing realm awaiting further discovery. As we strive to unravel its secrets, we gain not only scientific knowledge but also a deeper appreciation for the magnificence and grandeur of the cosmos that surrounds us.。
宇宙共同体
2019-03-04
作曲 : Ballrockin
作词 : RoccWay
宇宙共同体
在星空中穿梭让火箭的尾焰划破
低温包裹的星系几百万光年的谜题
陨石经过飞船解体
人类的寂寞在宇宙留下垃圾
重构解体时间定做
稀稀落落留下痕迹宇宙共同体
勤勤恳恳敬畏生灵
走走停停锁定外星生命躲躲藏藏
未知现象踉踉跄跄匆忙落在星球上
正如灵魂不停被降解
亮点焦点逐渐被削减
远在苍穹的星斑放任天狗垂涎
放眼在放电远在天际
但星火落在身边
为升迁畏升天畏身边位高权重怕胶片
准备翻篇整备相片投稿会被吞没的邮件卫星精准查找的维度
疑似落在银河的九天
共同体共生的画面
触电快出现
第三类接触电影的画面和ET的照面
笑靥如花的经历要用爵士的配乐
兑现和平的诺言暴力的宣泄
一切的一切消散在世界
透过神的嘴巴扩散宗教的一面
串联神话故事惊人相似的情节
存在世上的神迹人类阴谋的游戏
深谙黑暗丛林法则游弋逡巡奴役
被洗脑的族群扭曲真理禁锢人性
穿越黑洞时空旅行
道法自然
最终觉醒
阴阳相生
打破禁忌
宇宙共同体。
生或死五大奶子英文版Five Breasts":To Live or to Die: Five BreastsFrom the moment we are born, we are faced with the constant struggle of life and death. This eternal battle is the foundation of our existence, shaping the very essence of who we are and how we navigate the world around us. In the grand scheme of this cosmic dance, the concept of "five breasts" emerges as a poignant metaphor for the delicate balance between survival and oblivion.At the heart of this essay lies the question of what it truly means to be alive. Is it merely the physical act of drawing breath, or is there a deeper significance that transcends the corporeal realm? The notion of "five breasts" challenges us to contemplate the very nature of life and the fragility of our mortal existence.In the realm of the living, the breast holds a multifaceted significance. It is a symbol of nourishment, a source of sustenance that ensures the continuation of our species. The breast is the first point of contact between a newborn and the world, a tangible expression ofthe bond between mother and child. Yet, this same source of life can also be the harbinger of death, as diseases such as breast cancer threaten to extinguish the very spark that ignites our being.The concept of "five breasts" introduces a level of complexity that forces us to grapple with the duality of existence. In a world where the norm is the presence of two breasts, the emergence of five breasts can be seen as a deviation from the expected, a disruption of the natural order. This anomaly, whether real or imagined, challenges us to confront our preconceptions about the boundaries of life and the parameters of acceptable physical form.As we delve deeper into this exploration, we are confronted with the unsettling possibility that the "five breasts" may not be a symbol of abundance or fertility, but rather a harbinger of something far more sinister. What if the presence of this extra appendage is not a manifestation of life, but a harbinger of impending death? This unsettling thought forces us to confront the stark reality that even the most fundamental aspects of our biology are not immune to the capricious nature of existence.The duality of life and death is further amplified by the cultural and societal implications of the "five breasts" concept. In some societies, the presence of an unusual physical trait may be seen as a sign of divinity, a mark of special favor from the divine. Yet, in others, it maybe viewed as a curse, a deformity that ostracizes the individual from the accepted norms of the community.This dichotomy speaks to the complex and often contradictory ways in which we, as human beings, grapple with the unknown and the unexplained. Our response to the "five breasts" phenomenon is a reflection of our own biases, fears, and deeply held beliefs about the nature of life and the boundaries of acceptability.As we ponder the significance of this curious anatomical anomaly, we are forced to confront the fragility of our own existence. The "five breasts" serve as a stark reminder that even the most fundamental aspects of our being are subject to the whims of chance and the capricious nature of the universe.In the end, the "five breasts" metaphor forces us to confront the essential truth of our mortality. It challenges us to contemplate the delicate balance between life and death, and to grapple with the uncomfortable reality that even the most profound and deeply cherished aspects of our existence are ultimately beyond our control.As we navigate this complex and often unsettling terrain, it is incumbent upon us to embrace the inherent duality of our existence. We must acknowledge the fragility of life while also celebrating its resilience, and strive to find meaning and purpose in the face of theever-present specter of mortality.For in the end, the "five breasts" are not just a physical anomaly, but a profound metaphor for the human condition – a testament to the enduring power of the spirit and the relentless march of time, forever intertwined in an eternal dance of life and death.。
第40卷第5期2021年5月中国野生植物资源Chinese Wild Plant ResourcesVol.40No.5May.2021doi：10.3969/j.issn.1006-9690.2021.05.009侧柏叶、何首乌复合提取物防脱发功效研究马世宏，单承莺，聂轉，束成杰(南京野生植物综合利用研究院，江苏南京211110)摘要制备侧柏叶、何首乌复合提取物，采用体外1,1-二苯基-2-三霸基苯M(DPPH)自由基清除率实验、5a -还原酶抑制实验和小鼠动物模型对其进行防脱发功效研究。
结果表明，该复合提取物在100mg/mL的有效浓度下可有效清除DPPH,体外可显著抑制5a-还原酶活性，动物试验结果表明能明显改善小鼠雄激素源性脱发。
该复合提取物具有防脱发功效，可用于吩脱发个人护理用品的开发。
关键词侧柏叶;何首乌;5a-还原酶;雄激素源性脱发中图分类号:R282文献标识码:A文章编号：1006-9690(2021)05-0043-05Study on the Anti-hair Loss Effects of the CompoundExtract of Platycladus orientalis and Polygonum multiflorumMa Shihong,Shan Chengying,Nie Wei,Shu Chengjie(Nanjing Institute of Comprehensive Utilization of Wild Plants,Nanjing211110,China) Abstract The compound extract of Platycladus orientalis and Polygonum multiflorum was prepared,and the anti-hair loss effects were studied by1,1-Diphenyl-2-picrylhydrazylradical2,2-Diphenyl-1一(2,4,6—trinitrophenyl)hydrazyl(DPPH)clearance test in vitro,5a-reductase inhibition test in vitro and mice experimental model.The results showed that the composite extract could effectively eliminate DPPH,and significandy inhibit5a-reductase activity at an effective concentration of100mg/mL in vitro.The animal experiment result showed that it could significandy improve androgen-induced alopecia in mice.The compound extract has anti一hair loss effect and can be used in the development of anti一hair loss personal care products.Key words Platycladus orientalis；Polygonum multiflorum；5a-reductase；Androgen-induced Alopecia脱发是一种常见现象，分为生理性脱发和病理性脱发两种。
a r X i v :c o n d -m a t /9610135v 1 [c o n d -m a t .m e s -h a l l ] 17 O c t 1996Multifractality beyond the Parabolic Approximation:Deviations from the Log-normalDistribution at Criticality in Quantum Hall SystemsImre Varga,J´a nos PipekDepartment of Theoretical Physics,Institute of Physics,Technical University of Budapest,H–1521Budapest,HungaryMartin Janssen,Krystian PraczInstitut f¨u r Theoretische Physik,Universit¨a t zu K¨o ln,Z¨u lpicher Str.77,D-50937K¨o ln,Germany Based on differences of generalized R´e nyi entropies nontrivial constraints on the shape of thedistribution function of broadly distributed observables are derived introducing a new parameter in order to quantify the deviation from lognormality.As a test example the properties of the two–measure random Cantor set are calculated exactly and finally using the results of numerical simulations the distribution of the eigenvector components calculated in the critical region of the lowest Landau–band is analyzed.PACS numbers:71.23.AN,73.40.HmNumerical simulations of disordered critical systems always face the problem of dealing with system sizes smaller then the relevant length scale therefore highly complex fluctuations appear.Consequently expectation values of observables do not represent the full picture for the description of the dynamics and physics involved,in-stead the whole distribution function of these observables should be studied.This problem has become the central issue of theoretical research over the past decade [1–4].Multifractal analysis [5]has now become a standard tool to handle the aforementioned fluctuations aris-ing near criticality.Especially the disorder induced localization-delocalization (LD)transition is an impor-tant and widely studied example of such critical sys-tems [6].The general method of multifractal analysis starts with the calculation of the measure Q (ℓ)of the observable over boxes of linear size ℓin a system of length L divided into M pieces of these boxes.In the thermodynamic limit of ℓ/L →0the q th moment of this distribution scales with a nontrivial exponent D q that is,in general,strongly q –dependent.These moments are often called as generalized inverse participation numbers (IPN).The quantity (q −1)D q ,also known as the mass exponent τq ,is a monotonous,nonlinear function of q .Another way of describing these fluctuations is using the f (α)function,the Legendre–transform of τq .The f (α)is a smooth,single–humped,concave function.It is the distribution of the exponents αq =dτ/dq that charac-terize the µq (ℓ)∼(ℓ/L )αq scaling of the box–observables where µq (ℓ)is the total measure of the q th power of the observables Q over the boxes with linear length ℓ.The f (αq )gives the scaling of the number of boxes with mea-sure (ℓ/L )αq in the interval (α,α+dα)ρ(t,α)=g (t,α)e t f (α)(1)where we have introduced the variable t =−ln(ℓ/L ).The above distribution describes the multifractal in thesense that ln ρ(t,α)/t →f (α)if t →∞.The f (α)func-tion obeys certain general rules:f ′(αq )=q ,where prime denotes derivation with respect to α.For some specific values of qf ′(α0)=0,f (α0)=D 0(2a)f ′(α1)=1,f (α1)=α1=D 1(2b)where D 0is the Hausdorff–dimension of the support of the observable Q .Furthermore the support of the f (α)spectrum is a finite interval [α−∞,α+∞],and at the two limiting values the derivative of the f (α)is infinite.The importance of the study of the function f (α)is enhanced due to its connection to the properties of the local distri-bution function of the Q variables as [6]Π(t,ln Q )∼exp[t (D 0−f (α))],α=ln Q/t(3)The shape of the f (α)curve is close to a parabola [6,8]which means that the distribution Π(t,ln Q )is a Gaus-sian and the Π(t,Q )is lognormal.Therefore any devi-ation from the parabolic approximation (PA)shows up as a deviation of Π(t,Q )from lognormality.Numerical experiments show that there is a substantial deviation of the distribution function (3)from the simple Gaus-sian especially for large values of ln Q ,hence it is even asymmetrical [7](see Fig.1).The calculation of f (α)and D q functions in numerical experiments suffers from several difficulties.First this analysis is suitable only if ℓ≪L <ξis satisfied,where ξis the localization length and still the box–length should be larger that any microscopic length scale of the system.Second,higher moments of the distribution,|q |>3are usually calculated with low precision and the PA is usu-ally satisfactory.The PA is fixed by the values of α0and D 0.In spite of the simplicity of the PA it breaks down at values of q where the monotonicity of the τq is lost.FIG.1.The joint probability distribution function of the local amplitudes Q=|ψ|2of the eigenstates taken in the critical region in a Quantum Hall System.The dashed line represents the parabolic approximation,the solid one is the improvement given in the text.Apart from the results of numerical simulations,re-cently it has been shown analytically that the moments of the distribution function of the eigenstate components in a two dimensional disordered system shows multifrac-tality[9],however,only the PA has been derived.In another work the multifractal properties of a relativistic fermion moving in a two dimensional random vector po-tentialfield has been derived[10]without the restrictions of the PA.Therefore it seems to be of growing importance of deriving as much information from numerical experi-ments as possible.In this Letter we show that the deviations from the simple lognormal distribution can be measured with the application of specific R´e nyi–entropies[11].These R´e nyi entropies are proportional to the variable t only for strictly self–similar observables in the thermodynamic limit,e.g.deterministic multifractals such as the two–measure Cantor set,however,they are connected to some specific values of the scaling dimensions of the general-ized IPNs and hence impose new conditions on the shape of the f(α)function.Hereby we use two differences of R´e nyi entropies[11]. Thefirst one describes the deviation of the generalized IPN ln P2of the distribution Eq.(1)from the Shannon entropy H(a special R´e nyi entropy)and the other one is the normalized value of ln P2S2=H−ln P2and m2=ln M−ln P2(4) Generalizations for higher moments are also possi-ble[17],however,numerical simulations for such cases are presently not too reliable.The above two parame-ters have already been successfully applied in a number of systems[12,14,15]for the shape–analysis of the com-plex distribution function of eigenvector components and energy spacing distributions.The calculation of the quantities S2and m2involve in-tegration over the measure of the distribution in question. In our case,we have to perform integration of Eq.(1)over the allowed range of theαvalues[α−∞,α+∞]P−12(ℓ)∼ αmaxαminℓ2αρ(ℓ,α)dα(5a)H(ℓ)∼−lnℓ αmaxαminαℓαρ(ℓ,α)dα(5b)Furthermore,being interested only in the smallℓ,i.e. large t limit,we can apply an asymptotic method due originally to Laplace[16]and obtain an expansion of t of the form[17]S2(t)=(D1−D2)t+∞k=0s2,k t−k(6a) m2(t)=(D−D2)t+∞k=0m2,k t−k(6b)From Eqs.(6)it is clear that their ratio tends to a con-stant as t→∞γ2=limt→∞S2(t)D0−D2.(7) This is the key result of our Letter.The D1and D2val-ues in Eqs.(6,7)are related to the f(α)function,since (2b)α1=f(α1)=D1and D2=2α2−f(α2).On the other hand one can calculate the quantities S2/m2di-rectly using the observable Q at any length scale.The latter value imposes a further condition on the f(α) through the dimensions D1and D2using relation Eq.(7): f(α2)=2α2−(α1−γ2D0)/(1−γ2).Note that for a log-normal distributionγ2=1/2independently of the details of the distribution as it is alsoγ2=1/2for a parabolic f(α),hence a deviation from the value1/2is a sign of non-lognormal distribution and non-parabolic f(α)[13]. Furthermore,the denominator of Eq.(7)is the quantity that has been recently connected to the anomalous dif-fusion exponent by Refs.[8]and[18].As an illustrationfirst let us consider the two-measure Cantor set representing a binary process[5].In this con-struction one divides the[0,1]interval into two equal parts receiving different,p and1−p measures.While it-erating this procedure for each subhalf of the interval we get a multifractal in D0=1.This is a strictly selfsimilar multifractal if we keep the0≤p≤1/2valuefixed,be-cause for the n th stage S(n)2=n S(1)2and m(n)2=n m(1)2 where S(1)2and m(1)2are the values for the initial config-uration.Hence all the s2,n and the m2,n in Eq.(6)are zero,and theγ2quantity is independent of n and is a function of p[17].Next we consider a generalization of this Cantor set by choosing at every stage a different value for p n then we obtain a non–selfsimiliar multifractal the random bi-nary Cantor set(RBCS).It is straightforward to derivethe f (α)function once the distribution function of the p n sequence is further specified.Taking a uniform distri-bution over the interval [1/2−δ,1/2+δ]we have found that γ2(δ)is a smooth function of 0≤δ≤1/ing this simple mathematical construction we will show that the PA can beeasily improved with the application of parameter γ2Eq.(7).As an example we took δ=1/3.The PA in Fig.2is denoted as a dotted line while the exact relation is represented by the solid symbols.We can see that adding a fourth order term (α−α0)4to the PA fulfilling both Eqs.(2)and (7)in f (α)gives the solid line that is clearly a much better approximation than the simple PA.FIG.2.The distribution of the random binary Cantor set (RBCS)with δ=1/3.The dashed line stands for the parabolic approximation and the solid one is the fourth order one using parameter γ2(see.Eq.(7)).The symbols represent the exact relation.Finally we will apply our analysis to the widely studied LD transition that takes place in a two dimensional disor-dered system subject to a perpendicular strong magnetic field:the quantum Hall system [6,19].This LD tran-sition is responsible to the integer quantum Hall effect.It is described by the critical index νof the localization length and the f (α)spectrum,that is believed to be uni-versal [6],however,up to now the focus has been set on the PA only and the value of α0fixing the position of the maximum of it.The value of α0describes the scaling behavior of the typical local electron density.We show how the deviation from the lognormal distribution can be measured using Eq.(7)and propose an optimal mod-ification of both the f (α)and the distribution function Π(t,ln Q ).For that purpose we have performed calcu-lations of two dimensional systems of linear size of 200magnetic lengths.We have obtained 134eigenfunctions and calculated the overall joint probability distribution function of the local amplitudes Q =|ψ|2.In Fig.1we have plotted the logarithm of the histogram of ln Q as a function of ln Q/t .On the same figure the PA is also presented.The discrepancy is clear especially for largervalues of Q .On the same figure we have given a possible improvement of the analytical f (α)that shows a much better resemblance to the numerical data.FIG.3.Scaling dependence of parameters m 2(t )/t and S 2(t )/t as a function of 1/t .Solid symbols stand for the val-ues obtained in the Quantum Hall System and continuous curves show several [K/L ]order Pad´e approximants:dashed line [0/2],dotted line [0/3]and solid line [0/4].The improvement in Fig.1was achieved similarly as for the case of the RBCS.For that purpose we have cal-culated the m 2(t )/t and S 2(t )/t values and plotted as a function of 1/t .Unfortunatelly the low system size is a severe limitation on our problem,since it is desired to obtain the values of m 2(t )/t and S 2(t )/t for t →∞.However,it is not possible to go beyond t max =ln L and current computation possibilities impose a limit of t max ≈6.0.In order to obtain an acceptable value for the values m 2(t )/t and S 2(t )/t as 1/t →∞we have per-formed an [K/L ]order Pad´e approximation [20]for both quantities.Besides finite size scaling this type of approx-imation is ready to give an estimate of the coefficients in the expansions Eq.(6).Not having data obtained for dif-ferent system sizes at hand we can show that the Pad´e ap-proximation indeed gives acceptable results,as well.We assumed that the [K/L ]order approximation should have no singularity in the given interval and has to fall to zero for 1/t →∞since on the scale ℓ≈L the distribution is homogeneous,thus reducing to the possibilities to K <L and K =0.In Fig.3we have plotted the calculated quantities together with several Pad´e approximants.The limit 1/t →0gives the approximate value for the param-eter γ2.This way we obtained 0.445for [0/2]0.439for [0/3]and 0.436for [0/4].Therefore we set the value of γ2≈0.44.On the other hand,using standard multi-fractal analysis,our data gives α0=<ln Q >/t =2.24,D 1=1.78D 2=1.61that are all well in the generally accepted range [6,19].The latter values yield γ2≈0.44again.Having α0and γ2at hand we may extend the func-tional form of f (α)beyond the PA in order to get a bet-ter aggreement with the numerical distribution function (see Eq.(3)).For the improvement of the functional form of the f(α)entering in Eq.(3)there should be a num-ber of possibilities.We used the addition of the term (α−α0)3and(α−α0)5to the PAfixing their coefficients requiring the constraints Eqs.(2)and(7)within an er-ror of10−4.Therefore theΠ(x)function seems to be modified asΠ(x)=ΠPA(x)exp[−t(c3x3+c5x5)]where x=ln Q/t−α0,c3≈−0.03and c5≈0.12.In this Letter we have introduced the differences of R´e nyi entropies in order to describe the behavior of any multifractal as a function of t=−ln(ℓ/L).These pa-rameters enable to look beyond the parabolic approxi-mation and obtain further constraints on the shape of the f(α)function as well as the distribution function of the multifractal quantity.In order to get insight in the t dependence we have applied Laplace’s method and tried the results on exactly known deterministic and random multifractals.In the former case the self similarity prop-erty is also obtained while the latter case was a good test how the improvement from PA can be done.Finally we have analyzed the joint distribution func-tion of the local amplitudes of wave functions calculated in the critical regime of the Quantum Hall System.We have applied the Pad´e approximation in order to mimic the thermodynamic limit.The results show a possi-ble tail of the distribution function that is irregular for smaller amplitudes.Acknowledgements One of the authors(I.V.)is grate-ful for the warm hospitality at the Institut f¨u r The-oretische Physik,Universit¨a t zu K¨o ln where part of this work has been completed.Financial support from Orsz´a gos Tudom´a nyos Kutat´a si Alap(OTKA),Grant Nos.T7238/1993and T014413/1994is gratefully ac-knowledged.。
