On the notion of geometric realization
艺术与数学的关联英语作文
艺术与数学的关联英语作文Art and mathematics, seemingly distinct subjects, are intricately linked. While one is often associated with imagination and creativity, the other is aligned with logic and rational thinking. However, upon closer examination, it becomes evident that both disciplines complement and inform each other. In this essay, we will explore the relationship between art and mathematics, highlighting various ways in which they intersect and influence one another.One of the most striking connections between art and mathematics lies in geometry. Geometry, a branch of mathematics concerned with the properties of shapes and space, forms the foundation for many artistic compositions. Artists throughout history have employed geometric principles intheir works to create balance, harmony, and proportion. For instance, the use of the Golden Ratio, a mathematical ratioknown for its aesthetic appeal, can be observed in numerous art forms, including architecture, painting, and sculpture. This connection between geometry and aesthetics demonstrates how mathematics enhances the visual impact of artistic creations.Moreover, mathematics plays a role in the creation and analysis of patterns in art. Patterns are prevalent across various art forms, from textile designs to mosaics. The employment of mathematical concepts, such as symmetry and tessellation, enables artists to construct intricate and visually engaging patterns. Tessellation, in particular, relies on repeated geometric shapes to cover a surface without any gaps or overlaps. Artists like M.C. Escher are renowned for their use of tessellation, which creates captivating and mesmerizing designs. Therefore, mathematics provides artists with valuable tools for creating patterns that captivate and delight the audience.In addition to the creation process, mathematics alsoaids in the interpretation and appreciation of art. When examining a piece of artwork, observers often engage incritical thinking to decipher its meaning and symbolism. This process involves analysis, deduction, and logical reasoning—skills closely associated with mathematical thinking. By employing mathematical problem-solving techniques, viewerscan unravel the intricate messages conveyed by artists, enhancing their understanding and appreciation of the artwork.Furthermore, the relationship between art and mathematics can be seen in the concept of fractals. Fractals are complex mathematical shapes characterized by self-similarity at different scales. These intricate patterns can be found in nature, architecture, and art. Artists such as JacksonPollock have utilized fractal-inspired techniques to create visually stunning and thought-provoking paintings. Fractals also contribute to computer-generated art, where mathematical algorithms are programmed to generate intricate and detailedimages. This intersection between art and mathematics in the realm of fractals exemplifies the interconnectedness of the two disciplines.Lastly, art and mathematics converge in the realm of aesthetics. Aesthetics, the branch of philosophy concerned with beauty and taste, draws upon both subjective and objective elements. While personal preferences may vary, studies have shown that certain aesthetic experiences can be attributed to mathematical principles. For instance, research indicates that humans tend to find symmetrical and proportional objects more visually appealing. This preference for order and regularity is closely tied to mathematical concepts and is often reflected in artistic choices. Therefore, mathematics provides a basis for understanding and appreciating the aesthetics of art.In conclusion, the relationship between art and mathematics is multidimensional and profound. From geometryand patterns to interpretation and aesthetics, mathematics informs and enriches the world of art. This connection dispels the notion of a stark divide between the two disciplines and demonstrates how they intertwine to create visually captivating, intellectually stimulating, and emotionally moving works of art. Ultimately, embracing this symbiotic relationship allows for a deeper and more holistic appreciation of both art and mathematics.。
关于欧几里得的英语作文
关于欧几里得的英语作文Title: Euclid: The Father of Geometry。
Euclid, often referred to as the "Father of Geometry," left an indelible mark on the world of mathematics through his seminal work, "Elements." His contributions not only revolutionized the study of geometry but also laid the groundwork for many branches of mathematics and influenced scientific thought for centuries to come.Born around 300 BCE in Alexandria, Egypt, Euclid's life remains shrouded in mystery, with few details known about his personal life. However, his mathematical legacy endures through his magnum opus, "Elements," a comprehensive compilation of geometrical propositions and proofs."Elements" consists of thirteen books, covering various aspects of geometry, number theory, and mathematical rigor. Euclid's deductive approach, characterized by logical reasoning and axiomatic principles, remains a hallmark ofmathematical inquiry.One of Euclid's most significant contributions is his formulation of Euclidean geometry, which deals with the properties of points, lines, angles, and planes. In "Elements," Euclid establishes fundamental concepts such as the Pythagorean theorem, parallel postulate, and the concept of similarity, providing a systematic framework for geometric reasoning.The Pythagorean theorem, attributed to Pythagoras but rigorously proved by Euclid, states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem has applications not only in geometry but also in various fields such as physics, engineering, and computer graphics.Euclid's treatment of parallel lines, known as the parallel postulate, sparked centuries of debate and exploration in geometry. This postulate states that if a line intersects two other lines forming two interior angleson the same side that sum to less than two right angles, then the two lines, if extended indefinitely, will eventually intersect on that side. Euclid's parallel postulate, while intuitive, led to the development of non-Euclidean geometries, challenging the notion of a single, absolute geometry and broadening the horizons of mathematical inquiry.Furthermore, Euclid's emphasis on logical reasoning and rigorous proof profoundly influenced the development of mathematics and scientific thought. His axiomatic approach, where propositions are derived from a set of self-evident axioms, established a standard for mathematical rigor that continues to shape mathematical discourse today.Beyond geometry, Euclid's work in number theory also holds significance. His study of prime numbers and the fundamental theorem of arithmetic, which states that every integer greater than 1 either is prime itself or is the product of prime numbers, laid the groundwork for modern number theory and cryptography.Euclid's impact extends far beyond the realm of mathematics. His emphasis on clarity, logical reasoning,and the pursuit of truth serves as a model for scholarly inquiry in all disciplines. Moreover, his work has inspired generations of mathematicians, scientists, and philosophers, shaping our understanding of the universe and our place within it.In conclusion, Euclid's legacy as the "Father of Geometry" is indisputable. His monumental work, "Elements," remains a testament to his intellectual prowess and mathematical insight. Through his systematic approach to geometry, logical reasoning, and axiomatic method, Euclid not only revolutionized the study of mathematics but also laid the foundation for scientific inquiry for centuries to come. As we continue to explore the mysteries of the universe, we are indebted to Euclid for providing us with the tools to navigate the intricate beauty of geometric truths.。
映入眼帘的是一些形状各异的瓶子英语表达的高级表达
映入眼帘的是一些形状各异的瓶子英语表达的高级表达全文共6篇示例，供读者参考篇1When I Glanced Over, an Array of Intricate Vessels Arrested My GazeAs I ambled into the kitchen this morning, my eyes were instinctively drawn to the cluttered array of containers adorning the counter. At first glance, it appeared to be a seemingly random assortment of variedly shaped receptacles. However, upon closer inspection, I detected an inherent elegance and purposefulness underlying their outward designs. Each vessel possessed its own distinct personality and flair, almost as if it were a sculptural masterpiece deliberately crafted by a skilled artisan.The first item that captured my attention was a tall, willowy bottle composed of sleek, cylindrical glass. Its elongated neck tapered gracefully into a slender stem, evoking the gracile form of a serpentine dancer mid-pirouette. The body flared outwards towards the base, resembling the gentle swell of a bloomingflower bud. This streamlined silhouette exuded an air of refined minimalism and modern sophistication.In stark contrast, a squat, rotund jar demanded my focus next. Its spherical contours reminded me of a plump ripe fruit, heavy with juice and bursting at the seams. The glossy,amber-tinted glass glinted invitingly, as if beckoning me to uncap it and imbibe its sweet, viscous contents. I could almost envision thick rivulets of golden syrup languidly drizzling down its convex flanks.Nestled amongst the other containers was an intricately etched decanter, its multifaceted surface refracting the morning sunlight into a dazzling prism of colors. Each meticulously carved facet caught the light at a subtly different angle, casting an ever-shifting kaleidoscope of geometric patterns across the neighboring bottles. To me, this crystal chalice evoked the lavish opulence and ornate grandeur of a baroque cathedral, every inch embellished with painstaking artistry.My gaze then alighted upon a ruggedly handsome flask composed of mottled, sea-green glass. Its surface was beautifully marred by swirling eddies and organic imperfections that imbued it with an unmistakable rustic charm. I imagined this robust vessel had endured countless journeys and adventures,accumulating a rich patina of character with each passing year. It seemed to beckon me to raise it to my lips and imbibe a bracing draft of some roguish, full-bodied vintage.In the midst of this eclectic congregation, a solitary vase of delicate, hand-blown glass stood apart. Its diaphanous walls were so translucent and gossamer-thin, they appeared almost ethereal and insubstantial – like a fragile soap bubble moments from dissipating into the ether. Gazing into its hollow, crystalline depths, I felt as if I could somehow perceive the very fabric of space and time itself, stretching endlessly into an infinite cosmic continuum. This transcendent objet d'art filled me with an indescribable sense of reverent wonder.Finally, my eyes were drawn to a series of tinted laboratory flasks and beakers clust篇2The Magnificent Menagerie of Multifarious BottlesWhen I gaze upon the assemblage of bottles arrayed before me, a cavalcade of curiosities dances before my eyes. Each vessel is a singular marvel, a sculptural embodiment of form and function that tantalizes the senses and ignites the imagination.Let us commence our exploration with the amphora, that stately bottle whose silhouette harkens back to ancient civilizations. Its curvaceous contours, suggestive of a slender hourglass, exude an air of timeless elegance. The elongated neck tapers gracefully, as if craning to catch a glimpse of the world beyond its confines.Nestled amongst its brethren is the portly demijohn, a rotund and convivial companion. Its plump, globular body beckons with the promise of generous portions, while the slender neck extends upwards, a verdant bottleneck acting as a verdant sentinel. One can almost envision the liquid treasures sloshing merrily within its corpulent depths.Ah, but let us not overlook the sprightly carafe, a lithe and effervescent vessel that seems to dance upon the table. Its slender frame betrays a litheness that belies its capacity to contain, its elongated neck reaching skyward like a graceful swan. With each tilt and pour, it transforms into a fountain of libations, its contents cascading forth in a scintillating stream.Behold the amphora's distant cousin, the alabastron, whose name whispers of ancient Egyptian grandeur. This petite and precipitous bottle, adorned with intricate carvings or pigments,once cradled precious unguents and perfumed oils. Even today, its diminutive stature exudes an air of regal mystique.The squat, robust bottle known as the celandine cuts a striking figure, its stout proportions and broad shoulders lending it an air of stalwart reliability. Its wide mouth gapes invitingly, ready to receive any bounty that might be bestowed upon it. This is a vessel that does not mince words, but rather embraces its earthy, utilitarian charm.And how could we neglect the lithesome flute, that slender, elongated bottle whose graceful lines evoke the mellifluous tones of its musical namesake? Its sleek, cylindrical body tapering to a delicate neck is a study in minimalist elegance, inviting one to caress its smooth surfaces and revel in its understated beauty.But let us not forget the whimsical curvature of the onion bottle, whose bulbous belly swells outward in a delightful, organic embrace. This bottle is a reminder that nature's forms can be as captivating as any human artifice, its gentle undulations evoking the verdant landscapes from whence it might have sprung.As my gaze wanders further, I am transfixed by the angular allure of the cubical bottle, its bold, geometric planes defying theconventions of curvaceousness. This is a vessel that revels in its modernity, its sharp edges and clean lines a testament to the enduring appeal of minimalist design.And what of the serpentine marvel that is the snake bottle, its sinuous curves coiling and undulating in a mesmerizing dance? This bottle is a true sculptural feat, its twisting form inviting one to trace its contours with fingertips, imagining the secrets it might contain within its serpentine depths.Yet, amidst this panoply of shapes and sizes, there is one bottle that stands apart, a true maverick in the realm of vessels. The whimsical, eccentric form of the novelty bottle defies categorization, its unconventional silhouette a playful rebuke to convention. Perhaps it is a whimsical animal shape or a fanciful, abstract design – whatever its guise, it reminds us that the world of bottles is a canvas upon which creativity can run rampant.As I survey this magnificent menagerie, I am struck by the sheer diversity of forms that human ingenuity has conceived. Each bottle is a testament to the boundless creativity of the artisans who crafted them, their shapes and silhouettes a reflection of the cultures, aesthetics, and functional demands that birthed them.But beyond their physical forms, these bottles are vessels of stories untold, repositories of histories and traditions that span centuries and continents. They are time capsules, repositories of memories and moments, waiting to be uncorked and unveiled.So, let us raise a metaphorical glass to the wondrous world of bottles, these unsung guardians of liquid treasures. May their manifold shapes and forms continue to delight and inspire, reminding us that even the humblest of vessels can be a canvas for art, a receptacle for dreams, and a wellspring of endless fascination.篇3A World of Wondrous BottlesAs I gaze upon the array of bottles before me, I'm struck by the sheer diversity of their forms – a veritable kaleidoscope of shapes and silhouettes that captivates the eye and ignites the imagination. It's as if each vessel holds not only its liquid contents, but a universe of artistry and ingenuity waiting to be explored.Let us commence our odyssey with the sleek and streamlined contours of the cylindrical bottle. At first glance, its shape may seem deceptively simple, but upon closer inspection,one can discern the subtleties that elevate it beyond mere functionality. The parallel lines that run the length of its body exude a sense of order and precision, while the gently curved shoulders lend an air of elegance and refinement.Yet, this is but the prelude to the symphony of forms that awaits us. Behold the amphora, a shape that has graced the tables of civilizations past, its svelte neck tapering gracefully into a rotund body – a silhouette that seems to whisper tales of ancient banquets and revelries long since faded into the annals of history.And what of the curvaceous undulations of the flacon, its sensuous lines evoking the fluidity of a dancer's movements? Each gentle swell and dip is a brushstroke in a masterpiece of design, inviting the eye to trace its captivating contours with rapturous delight.But let us not overlook the whimsical charms of the decanter, a shape that defies convention with its playful twists and turns. Its serpentine neck coils and unfurls, beckoning us to peer into its depths and unravel the mysteries therein. Truly, it is a vessel that revels in its own eccentricity, a delightful departure from the ordinary.And yet, amidst this riot of forms, there stands a bastion of solidity and strength – the mighty demijohn. Its broad shoulders and sturdy base lend an air of indomitable resilience, while its bulbous body whispers of the bounty it can contain. Ah, but let us not be deceived by its robust appearance, for even the mightiest of vessels can conceal a delicate soul within.As our gaze wanders further, we encounter the enigmatic silhouette of the carafe, its graceful curves seeming to defy the very laws of physics. With its slender neck and voluptuous body, it cuts a figure that is at once refined and sensual, inviting us to ponder the depths of its allure.And what of the whimsical charms of the cruet, its diminutive stature belying the boundless potential of its contents? Its petite frame and delicate spout are a testament to the notion that great things can indeed come in small packages, reminding us to never underestimate the power of the unassuming.But perhaps the most captivating of all is the enigmatic silhouette of the flute – a shape that seems to defy gravity itself. Its slender stem rises like a delicate tendril, blossoming into a graceful bowl that beckons us to partake of its nectar. Truly, it is a vessel that embodies the very essence of elegance and sophistication.As our odyssey through this world of wondrous bottles draws to a close, I am left with a profound sense of awe and appreciation for the artistry that has been poured into these humble vessels. Each shape, each curve, each line is a masterclass in design, a testament to the boundless creativity of the human spirit.For in these bottles, we find not mere containers, but canvases upon which the dreams and visions of their creators have been etched – visions that transcend the practical and soar into the realms of pure, unadulterated beauty. And as we raise these vessels to our lips, we partake not only of their liquid treasures, but of the very essence of human ingenuity and imagination.So let us raise a toast to the humble bottle, that unsung hero of our daily lives, for in its myriad forms we find a celebration of the boundless potential of design, a reminder that even the most utilitarian of objects can be elevated to the heights of artistry when viewed through the discerning eye of a true visionary.篇4As I stepped into the dimly lit room, a kaleidoscope of shapes and colors dazzled my eyes. The first objects thatensnared my attention were an assortment of vessels, each one as unique and captivating as the next. They were not mere bottles, but rather sculptural masterpieces that seemed to defy the conventions of form and function.At the forefront stood a towering cylindrical bottle, its elongated silhouette exuding an air of elegance and sophistication. Its surface was adorned with intricate etchings that resembled the delicate lace patterns found on my grandmother's tablecloths. As the soft light danced across its facets, the bottle appeared to shimmer with an ethereal radiance, beckoning me to draw nearer and lose myself in its mesmerizing intricacies.Nestled beside this statuesque figure was a rotund, globular bottle that evoked images of a plump, jolly fellow from a storybook. Its curved sides were smooth and inviting, almost as if they were begging to be cradled in one's embrace. The rounded shape seemed to exude a sense of warmth and comfort, like a cherished childhood teddy bear that had weathered countless nights of tender hugs and whispered secrets.Just beyond this spherical marvel, a cluster of triangular bottles caught my eye, their sharp angles and pointed tops creating a striking contrast against the softer contours of theirneighbors. Each one was a study in precision and boldness, their faceted surfaces refracting the light in a kaleidoscope of brilliant hues. They reminded me of the majestic peaks of a mountain range, their jagged silhouettes standing as sentinels against the vast expanse of the sky.In a shadowy corner, a group of amorphous bottles huddled together, their shapes seeming to shift and morph with every passing moment. Some appeared to be melting, their forms drooping and sagging like candle wax left too long in the sun's embrace. Others resembled twisted vines, their contours winding and undulating in serpentine patterns that seemed to defy the laws of physics. These enigmatic vessels were a testament to the boundless creativity of their creators, challenging my perception of what a bottle could be.As my gaze wandered further, I encountered a series of bottles that appeared to be fashioned from the very fabric of nature itself. One evoked the image of a towering oak tree, its sturdy base tapering upwards into a graceful, twisting trunk that culminated in a crown of delicate branches. Another seemed to mimic the undulating curves of a rolling hillside, its gentle slopes beckoning me to lie down and bask in the warmth of the imaginary sun.In a corner, a cluster of bottles resembled the jagged peaks of a mountain range, their sharp edges and angular forms a stark contrast to the soft, organic shapes that surrounded them. These bottles were a study in geometry and precision, their facets catching the light and refracting it into a dazzling array of colors that danced across the room like a kaleidoscopic ballet.Amidst this choreography of shapes and hues, a solitary bottle stood apart, its form a seemingly impossible fusion of opposing elements. Its base was a solid, grounded cylinder, but as it rose, it blossomed into a series of undulating curves that appeared to defy the laws of physics. This bottle was a living embodiment of the concept of dichotomy, a seamless melding of opposites that challenged my perception of what was possible in the realm of design.As I slowly made my way through this captivating gallery of vessels, I found myself in awe of the boundless creativity and ingenuity of their creators. Each bottle was a unique expression of artistry, a physical manifestation of the human spirit's ability to transcend the boundaries of form and function.篇5From the Lens of a Child: Bottles of Whimsy and WonderAs I ambled into the room, a kaleidoscope of shapes and hues dazzled my curious eyes. An array of bottles, each a unique masterpiece, beckoned me closer with their captivating designs. It was as if a talented artisan had breathed life into ordinary glass, sculpting it into wondrous forms that defied convention.The first bottle that caught my fancy was a rotund fellow, its ample girth reminiscent of a jovial fellow sporting a round belly. Its plump silhouette exuded a sense of joviality, as though it yearned to be cradled in one's arms like a cherished friend. The glass shimmered with a warm amber hue, evoking images of golden sunsets and the rich aromas of autumn leaves.Nestled alongside its portly companion was a willowy beauty, its slender neck stretching skyward with graceful poise. This elegant vessel seemed to dance on delicate feet, its svelte figure swaying gently as if caught in an ethereal breeze. The glass shimmered with a soft, pearlescent luster, evoking visions of moonlit waves caressing sandy shores.Yet another bottle captured my imagination with its undulating curves, reminiscent of rolling hills or the gentle swell of ocean waves. Its sides dipped and swelled in a mesmerizing rhythm, almost as if the glass itself had been sculpted by the ebb and flow of nature's rhythms. A vibrant azure hue adorned thissinuous form, conjuring images of crystalline waters and endless summer skies.Nearby, a cluster of slender, tapered bottles stood erect, their forms reminiscent of regal sentinels standing guard. Each one boasted a narrow neck that flared outward, resembling the trumpets of triumphant heralds. Their gleaming surfaces reflected the world around them, mirroring the room's contents in a kaleidoscope of distorted shapes and colors.One bottle, in particular, caught my eye with its intricate, almost baroque design. Swirling patterns adorned its surface, as if a master glassblower had etched intricate lace into the very fabric of the glass. This ornate vessel seemed fit for a royal court, its elaborate embellishments hinting at tales of grandeur and opulence.Yet, amidst this array of fanciful forms, a humble, unassuming bottle stood out in its simplicity. Its cylindrical shape was unadorned, yet it exuded a quiet elegance, like a zen master radiating serenity. The translucent glass allowed glimpses of the bottle's contents, hinting at mysteries waiting to be unveiled.As I reveled in this captivating display, I couldn't help but marvel at the ingenuity and artistry that had breathed life into these inanimate objects. Each bottle was a masterpiece in its ownright, a canvas upon which the glassblower had painted their unique vision.In that moment, I realized that the world around us is a tapestry woven with beauty and wonder, if only we take the time to appreciate the intricate details. These bottles, once mere vessels for containment, had transcended their humble origins to become objects of fascination and delight.With childlike wonder, I found myself drawn into this realm of shapes and hues, eager to explore the stories each bottle yearned to tell. Perhaps the portly fellow cradled a warm, comforting elixir, while the willowy beauty housed a fragrant bouquet of exotic blooms. The undulating curves might have embraced a vintage wine, its ruby depths holding the secrets of sun-drenched vineyards and the passage of time.As my imagination took flight, I realized that these bottles were not mere receptacles; they were portals into worlds of whimsy and enchantment, inviting us to shed the shackles of adulthood and embrace the boundless curiosity of a child's mind.In that moment, I vowed to carry this sense of wonder with me, to approach each day with the wide-eyed awe of a child discovering the world anew. For in that room of bottles, I hadglimpsed a profound truth – that beauty lies not only in grand vistas and majestic landscapes but also in the humble,oft-overlooked details that surround us.Those bottles, once dismissed as mere containers, had become talismans of joy and wonderment, reminding me to savor the simple pleasures and revel in the magical moments that life presents. With a heart brimming with gratitude, I bid farewell to that captivating array, carrying with me the knowledge that the world is a canvas awaiting our appreciation, and that the true artistry lies in the lens through which we choose to view it.篇6An Enchanting Sight to BeholdAs I skipped merrily down the cobblestone path, my curious gaze was instantly enraptured by a mesmerizing spectacle that adorned the window of the quaint little shop. An eclectic assortment of peculiarly fashioned vessels, each one a unique masterpiece, beckoned to me like a siren's call. Their very existence seemed to defy the laws of physics, as if they had been plucked from the whimsical realms of a fairytale.My eyes danced from one captivating form to the next, each bottle more entrancing than the last. Some were adorned with intricate etchings that resembled the delicate lace of a spider's web, while others boasted vibrant hues that put even the most brilliant of rainbows to shame. There were tall, slender ones that seemed to stretch towards the heavens, and squat, rotund ones that appeared to be the embodiment of jolly contentment.One particular vessel caught my undivided attention, a true masterwork of artistry. Its surface was a kaleidoscope of swirling colors, each hue blending seamlessly into the next, creating a mesmerizing vortex that threatened to draw me in. The curves and angles of its form defied conventional logic, twisting and turning in ways that made my young mind spin with delight.As I pressed my eager nose against the glass, I fancied that I could perceive the faintest hint of exotic fragrances wafting from within those enchanted containers. Perhaps they housed the essences of far-off lands, or the distilled dreams of ancient alchemists. Or, maybe, just maybe, they contained the very essence of magic itself, waiting to be uncorked and unleashed upon the world.My childlike imagination ran wild with possibilities, each more fanciful than the last. In my mind's eye, I could envisionmyself as a brave explorer, embarking on grand adventures with a trusty bottle by my side. Would it grant me the power of flight, allowing me to soar among the clouds like a majestic eagle? Or perhaps it would bestow upon me the ability to commune with the creatures of the forest, unlocking the secrets of their ancient tongues.Alas, my reverie was abruptly shattered by the gentle tug of my mother's hand, urging me onward. Reluctantly, I tore my gaze away from that spellbinding display, but not before silently vowing to return one day and unravel the mysteries that those enigmatic vessels held.For you see, in the wondrous realm of childhood, even the most mundane of objects can take on an air of enchantment. A simple collection of bottles, to the world-weary adult, might be but a fleeting curiosity, but to a child's fertile imagination, it becomes a gateway to realms of pure, unadulterated magic.In those whimsical vessels, I saw not mere containers, but portals to fantastical worlds where anything was possible. They were talismans imbued with the power to unlock the boundless potential of a young mind, inviting me to dream, to explore, and to embrace the infinite wonder of the universe.Even now, years later, whenever I catch a glimpse of an oddly shaped bottle, my heart flutters with the same sense of childlike wonder that captivated me on that fateful day. For in those peculiar forms, I am reminded of the profound truth that magic is not merely the stuff of fairy tales, but a ever-present force that resides within the hearts and minds of those who dare to believe.So, my dear friends, I implore you: never lose that sense of wide-eyed amazement that allows you to perceive the extraordinary in the ordinary. Embrace the magic that surrounds you, for it is that very enchantment that makes life worth living, one captivating bottle at a time.。
comparison theorem in riemannian geometry
Comparison Theorems in Riemannian GeometryJ.-H.Eschenburg0.IntroductionThe subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when(mainly lower)bounds for the sectional or Ricci curvature are given?Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidis-tant hypersurfaces,we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov.An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are ing convexity arguments,we prove the”soul theorem”of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative cur-vature.If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll.The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature(Milnor).We also discuss briefly Bochner’s method.The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work:The comparison theory for the Riccati ODE, which probably goes back to L.Green[15]and which was used more systematically by Gromov[20],the triangle inequality for the Riemannian distance,the method of support function by Greene and Wu[16],[17],[34],the maximum principle of E.Hopf,generalized by E.Calabi[23],[4],and the idea of critical points of the distance function which was first used by Grove and Shiohama[21].We have tried to present the ideas completely without being too technical.These notes are based on a course which I gave at the University of Trento in March 1994.It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures.We also thank Evi Samiou and Robert Bock for many valuable corrections.Augsburg,September1994J.-H.Eschenburg11.Covariant derivative and curvature.Notation:By M we always denote a smooth manifold of dimension n.For p∈M, the tangent space at p is denoted by T p M,and T M denotes the tangent bundle.If M is another manifold and f:M→M a smooth(i.e.C∞)map,its differential at some point p∈M is always denoted by d f p:T p M→T f(p)M .For v∈T p M we writed f p(v)=d f p.v=∂v f.If c:I→M is a(smooth)curve,we denote its tangent vector by c (t)=dc(t)/dt=dc t.1∈T c(t)M(where1∈T t I=I R).If f:M→I R,then d f p∈(T p M)∗.If M is a Riemannian manifold,i.e.there exists a scalar product<,> on any tangent space of M,this gives an isomorphism between T p M and(T p M)∗;the vector∇f(p)corresponding to d f p is called the gradient of f.Let M be a Riemannian manifold.We denote by<,>the scalar product on M and we define the norm of a vector byv =√1.D(fX+gX )Y=fD X Y+gD X Y;2.D X(fY+gY )=(∂X f)Y+fD X Y+(∂X g)Y +gD X Y ;3.D X Y−D Y X=[X,Y]=”Lie bracket”;4.∂Z<X,Y>=<D Z X,Y>+<X,D Z Y>.Definition1.2The Riemannian curvature tensor(X,Y,Z)→R(X,Y)Z is defined as follows:R(X,Y)Z=D X D Y Z−D Y D X Z−D[X,Y]ZIt satisfies certain algebraic identities(”curvature identities”),namely<R(X,Y)Z,W>=−<R(Y,X)Z,W>=−<R(X,Y)W,Z>=<R(Z,W)X,Y>and the Bianchi identityR(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0(cf.[29]).In particular,R V:=R(.,V)Vis a self adjoint endomorphism of TM for any vectorfield V on M.Several notions of curvature are derived from this tensor:1.Sectional curvature K(,):For every linearly independent pair of vectors X,Y∈T p M,<R(X,Y)Y,X>K(X,Y)=where{E i}n i=1is a local orthonormal basis.There is a close relationship between R V=R(.,V)V and the sectional curvature: Let V =1.For X orthogonal to V we have<R V X,X>=<R(X,V)V,X>=K(V,X) X 2Hence the highest(”λ+”)and lowest(”λ−”)eigenvalues of R V give a bound to K(V,X), sinceλ−(R V)≤<R V X,X>dt:=Dγ (t)Yis defined(just extendγ and Y arbitrarly outsideγ).Similar,ifγ:I1×...×I k→M depends on k variables,we have k partial derivatives∂γ∂t i (i=1,...,k)alongγ.(Formally,a vectorfield alongγis a section of thepull-back bundleγ∗T M,and D induces a covariant derivative on this bundle.)Definition1.3A vectorfield Y along a curveγ:I→M is called parallel if Y =0.A curveγis a called a geodesic in M ifγ is parallel,i.e.if(γ ) =Dγ γ =0.(1.1)(1.1)is a2nd order ODE.In fact,if x=(x1,...,x n):M→I R n is a coordinate chart with E i=∂(summation convention!),thenγ =(γi) E i whereγi:=x i◦γ,andD2.Jacobi and Riccati equations;equidistant hypersurfaces.Equation(1.1)is a nonlinear ODE which in general cannot be solved explicitly. Therefore,we consider its linearization.This is the ODE satisfied by a variation of solutions of(1.1),i.e.of geodesics.So letγ(s,t)=γs(t)be a smooth one-parameterfamily of geodesicsγs.Put V=∂γ∂s .Then J is the variation vectorfield and V the tangentfield of the geodesicsγs,hence D V V=O.Fig.1.Then we haveJ =D∂tJ=D∂t∂γ∂s D∂t+R(V,J)V,J +R(J,V)V=0.(2.1) Equation(2.1)is called Jacobi equation.Definition2.1A vectorfield J along a geodesicγis called a Jacobifield if it satisfies the Jacobi equation.Remark2.2J is a Jacobifield alongγif and only ifJ(t)=dfor some one-parameter family of geodesicsγs withγ0=γ.Implication”⇐”was shown above.To prove the opposite implication,we have to construct the familyγs.Letα(s)=expγ(0)sJ(0).Let X be a vectorfield alongαsuch that X(0)=γ (0)and X (0)=J (0)and putγs(t)=expα(s)tX(s)(2.3).If we put˜J=∂∂t ∂∂s∂∂sX(s)|0=X (0)=J (0),we get J=˜J by uniqueness of the solution.Next,we want to split this2nd degree equation in a system of1st degree equations. To do this,we embed the1-parameter family of geodesics describing the Jacobifield into an(n−1)-parameter family.I.e.we choose a suitable smooth mapγ:S×I→Mwhere S is an(n−1)-dimensional manifold,such thatγs(t)=γ(s,t)is a geodesic for any s∈S.Ifγis a regular map,then V=dγ(∂Hence we have(D V A)X=D V D X V−A(D X V+[V,X])=D X D V V+R(V,X)V+D[V,X]V−A2X−A[V,X]=D X D V V+R V(X)−A2X.ThereforeD V A+A2+R V=D(D V V).(2.5) If we suppose D V V=0(i.e.the integral curvesγs are geodesic),then we get an ODE for A,the so called Riccati equationA +A2+R V=0.Thus we have split the Jacobi equation J =−R V J in two equations as follows:J =AJ(2.6)A +A2+R V=0.(2.7) We note that the second equation can be solved independently of thefirst.Let us consider now the important special case where(DV)∗=DV,that is<D X V,Y>=<X,D Y V>for all vectorfields X,Y.Then V is locally a gradient,i.e.locally V=∇f for some function f:M→I R.Consequently,<V,V>is constant,since∂X<V,V>=2<D X V,V>=2<D V V,X>=0.Thus we may assume that<V,V>=1.Now let us consider the level hypersurfacesS t={x∈M:f(x)=t}.Since V=∇f=0,the S t are regular hypersurfaces and V|St is the unit normal vectorfield on S t.Thus in this case,our(n−1)-parameter family of geodesicsγ:S×I→M is given byγ(s,t)=exp(t−t0)V(s)8where S=S tfor some t0∈I,and f(γ(s,t))=t,or in other words,S t=φt(S)where φt(s):=γ(s,t).Such a family of hypersurfaces S t is called equidistant,and the function f−t0is called the signed distance function of the hypersurface S=S t.In fact we have|f(x)−t0|=|x,S|:=inf s∈S|x,s|(2.8) for x in a small neighborhood of ly,if c:[a,b]→γ(S×I)⊂M is a curve withc(a)∈S t0and c(b)∈S t1,then we have c(u)=γ(s(u),t(u))with t(a)=t0,t(b)=t1,andc (u) 2= dγ.s (u) 2+t (u)2≥t (u)2,hence its length isL(c)≥ b a|t (u)|du≥|t(b)−t(a)|≥|t0−t1|.Fig.2.In this case,all the quantities discussed above have geometric meanings.The Jacobifields J(t)=dγ(s,t)(x,0)=dφt x for x∈T s S measure the change of the metric of S t=φt(S)when t is changed;in fact, J(t) / J(t0) is the length distortion between S t and S.Moreover,A=DV,restricted to the hypersurface S t,is the shape operator of S t since V|S t is a unit normal vectorfield on S t.Its eigenvalues are called principal curvatures,their average the mean curvature of S t.Since Equation(2.7)is nonlinear, A(s,t)can develop singularities which are called focal points of S.Let us see some examples.Example2.3Let S t=∂B t(p),where B t(p)={x∈M:|x,p|<t}is the Riemannian ball.9Then V is radial and1A(t)∼√k cot(,a(t)=0.(t−t0)10Fig.4.These solutions correspond to the three umbilical parallel hypersurface families in eu-clidean space:concentric spheres with increasing(t>t0)or decreasing(t<t0)radii and parallel hyperplanes.Finally,if k<0,the space M is hyperbolic.The solutions are given bya(t)= |k|(t−t0),a(t)= |k|(t−t0),a(t)=±parison theory.We want to derive a comparison theorem for solutions of the Riccati equation A +A2+R V=0(cf.2.7).Fixing an integral curveγof V(which is a geodesic)and identifying all tangent spaces Tγ(t)M by parallel displacement(i.e.via an orthonormal basis(E i(t))of vectorfields alongγwhich are parallel,i.e.E i=0),we consider A(t) as a self adjoint endomorphism on a single vector space E=Tγ(0)M.More generally, let E be afinite-dimensional real vector space with euclidean inner product , .The space S(E)of self adjoint endomorphisms inherits the inner productA,B =trace(A·B)(3.1) for A,B∈S(E).We get a partial ordering≤on S(E)by putting A≤B if Ax,x ≤ Bx,x for every x∈E.Theorem3.1(cf.[14],[9])Let R1,R2:I R→S(E)be smooth with R1≥R2.For i∈{1,2}let A i:[t0,t i)→S(E)be a solution ofA i+A2i+R i=0(3.2) with maximal t i∈(t0,∞].Assume that A1(t0)≤A2(t0).Then t1≤t2and A1(t)≤A2(t)on(t0,t1).Proof.Let U=A2−A1;then U(t0)≥0andU =A 2−A 1=A21−A22+R1−R2.(3.3) We define S=R1−R2≥0and X=−112Remark3.2Theorem3.1still holds if A1,A2are singular at t0,but U=A2−A1has a continuous extension to0with U(0)≥0.See[14]for the proof.A similar argument also shows that t1<t2if A1(t0)<A2(t0);for a different proof of this fact see[11], Lemma3.1.The geometric interpretation of Theorem3.1is:principal curvatures(i.e.eigenvalues of the shape operator)of equidistant hypersurfaces decrease faster on the space of larger curvature.In particular,this is true for Riemannian spheres,as follows by Remark3.2).Now we want tofind a comparison theorem for equation(2.6).For A∈S(E), denote byλ−(A)the lowest eigenvalue and byλ+(A)the highest eigenvalue of A.Theorem3.3Let A1,A2:(t0,t )→S(E)such thatλ+(A1(t))≤λ−(A2(t))everywhere.(3.7) Let J1,J2:(t0,t )→E be nonzero solutions of J i=A i·J i.Then J1 / J2 is monotoneously decreasing.Moreover,iflimt t0J1J i =<J i,J i><J i,J i>∈[λ−(A i),λ+(A i)]so thatlog( J1 ) = J1 J2 =log( J2 ) ,hence log J113We consider the most important special cases due to Rauch and Berger (called Rauch I and Rauch II in [5]):Rauch ISuppose that J i for i =1,2are solutions of J i +R i J i =0with λ−(R 1)≥λ+(R 2)andJ i (0)=0, J 1(0) = J 2(0) .Then J 1 ≤ J 2 up to the first zero of J 1.Rauch IISuppose that J i for i =1,2are solutions of J i +R i J i =0with λ−(R 1)≥λ+(R 2)andJ i (0)=0, J 1(0) = J 2(0) .Then J 1 ≤ J 2 up to the first zero of J 1.In fact we apply the theorems 3.1and 3.3where in the first case,A i (t )∼t −1I as t →0and in the second case,A i (0)=0.Corollary 3.4Let M be a complete manifold with K ≥0,p 0,p 1∈M and γ:[0,1]→M a shortest geodesic segment connecting p 0and p 1.Let X ⊥γ be a parallel vector field along γ.Put p s (t )=exp tX (s )for all s ∈[0,1].Then|p 0(t ),p 1(t )|≤|p 0,p 1|with equality for some t >0only if p 0,p 1,p 1(t ),p 0(t )bound a flat totally geodesic rectangle.Proof.We have|p 0(t ),p 1(t )|≤10 ∂∂s p s (t )is a Jacobi field along the geodesic γs (t )=p s (t )with J s (0)=0.Thus comparing with the euclidean case we get from Rauch II that J s (t ) ≤ J s (0) which shows the inequality.If we have equality at t 1>0,the equality discussion of Theorem 3.3shows that J s is parallel along γs |[0,t 1].Moreover,the curves s →p s (t )are shortest geodesics of constant length for 0≤t ≤t 1.Thus the surface p :(s,t )→p s (t )is a flat rectangle in M with D ∂s =D ∂s =D∂t=0,so it is also totally geodesic ,i.e.covariant derivatives of vector fields tangent to p remain tangent to p .4.Average comparison theorems.Now we consider the trace of the Riccati equation A +A2+R V=0for self adjoint A.Since trace and derivative commute,we gettrace(A) +trace(A2)+Ric(V)=0.(4.1)This is unfortunately not a differential equation for trace(A),because of the term trace(A2).However,puttrace(A)a=n−1 A0 2+Ric(V) ≥1a(t0)wherea +with t2≤+∞maximal.Leta=trace(A)a(t)for t∈[t0,t1).Proof.Apply theorem3.1with(R1,A1,R2,A2)replaced with(r,a,k,t−t0I anda=s /s,where s is the solution ofs +ks=0,s(t0)=0,s (t0)=1.Next,let J1,...,J n−1be a basis of solutions of J =A·J,and putj=det(J1,...,J n−1).Since(J1∧...∧J n−1) =n−1k=1J1∧...∧A·J k∧...∧J n−1,we getj =(n−1)a·j.(4.6) Geometrically,equation(4.6)says how the volume element of S t,namely det(dφt)(see page9of chapter2),changes with t.Theorem4.3Let A:[t0,t1)→S(V)be given witha≤n−1trace(A),and let j be as above.Choose¯j such that¯j =(n−1)a,¯j).165.Bishop-Gromov inequalityLet M be a complete connected Riemannian manifold.By the theorem of Hopf and Rinow(cf.[29]),any two points p,q∈M can be connected by a shortest geodesic γ,i.e.L(γ)=|p,q|.Let S p M={v∈T p M: v =1}be the unit sphere in T p M.For any v∈S p M,we definecut(v)=max{t:γv|[0,t]is shortest}.This defines a function cut:S p M→(0,∞],the cut locus distance,which is continuous (cf[5],p.94).LetC p={tv:v∈S p M,t≤cut(v)}.(5.1) This is a closed subset of T p M,and its boundary∂C p(sometimes also exp p(∂C p)⊂M) is called the cut locus of the point p.It follows from this definition thatB r(p)=exp p(B r(0))=exp p(B r(0)∩C p)∀r>0.(5.2)In fact,if we choose q∈B r(p),there exists a shortest geodesicγv joining p and q;the length ofγv should be≤cut(v),hence v∈C p(theorem of Hopf-Rinow).Example5.1On the unit sphere we have cut(v)=πfor every v.In fact,in every direction,the geodesic is a meridian,hence it is shortest up to the opposite(”antipodal”) point.Example5.2On the cylinder S1×I R,we have cut(v)=π/cosαwhereαis the angle between v and the S1-direction.Fig.6.17There are two ways how a geodesicγ=γv:[0,∞)→M(where v∈S p M)can cease to be shortest beyond the parameter t0=cut(v)(cf.[5],p.93):Either there exists a nonzero Jacobifield J alongγwhich vanishes at0and t0-in this case,γ(t0)is called a conjugate point of p(cf.Example5.1),or there exists a second geodesicσ=γof the same length which also connects p andγ(t0)(cf.Example5.2).Hence q=γ(t0) is in the cut locus of p=γ(0)iffp is in the cut locus of q.Moreover,there are no conjugate points onγ|[0,cut(v)).The conjugate points in turn are the singular values of the exponential map exp p;more precisely,we have:Lemma5.3Let J(t)be the Jacobifield alongγv defined by J(0)=0,J (0)=w.Then we haved(exp p)tv.tw=J(t).In particular,d(exp p)tv is singular if and only if exp p(tv)is a conjugate point of p. Proof.Let w∈T v T p M≡T p M.Then we haved(exp p)v.w=dds s=0γv+sw(1).(5.3)If we letJ(t)=∂∂t 0∂∂s 0∂∂s 0(v+sw)=w .Therefore we getd(exp p)v·w=J(1),(5.5) and generallyd(exp p)tv·tw=J(t).(5.6)Remark5.4Consequently,on the interior of C p,the exponential map exp p is injective and regular,hence a diffeomorphism.Note that Int(C p)is star-shape,thus it is con-tractive;hence also its image is contractive.But by Hopf-Rinow,the whole manifold M is the image of exp p:C p→M,so the topology of M is given by the image of the boundary∂C p.After these preparations,we come to the main theorem of this section. Theorem5.5Let us consider a manifold M n with Ricci curvature satisfyingRicVol¯B r r(5.7) i.e.this quotient is monotonely decresing with r.Moreover,for r→0,the quotient goes to one.Corollary5.6For any two positive real numbers R>r we haveVol B R(p).(5.8)Vol¯B rRemark5.7Corollary5.6gives an upper bound for the growth of the metric balls in M.Moreover,if equality holds for some r<R,then B R(p)is isometric to¯B r(this can be seen from the proof).Proof of the theorem.By(5.3)we haveVol B r(p)= B r(0)∩C p det d(exp p)u du.(5.9) Passing to polar coordinates and denoting r(v)=min{r,cut(v)},we getVol B r(p)= S r(v)0det d(exp p)tv t n−1dt dv(5.10)19where S:=S1(0)⊂T p M.If we consider a basis e1,...,e n−1of v⊥⊂T p M,then byLemma5.3,d(exp p)tv e i=1tJ i(t),where J i is the Jacobifield alongγv with J i(0)=0and J i(0)=e i.Hencedet d(exp p)tv =1Vol(S) S(j v/¯j)dvis still monotone.Moreover,Vol¯B r= S r0¯j(t)dtdv=Vol(S) r0¯j(t)dt.(5.14)Therefore we have thatVol B r(p)r0¯j(t)dt(5.15)is a monotone decreasing function in r,because the mean of a monotone function on growing intervals is still monotone.If r→0,both volumes approximate the euclidean ball volume,hence the quotient goes to one.206.Toponogov’s Triangle Comparison TheoremLet usfix o∈M and letρ=|o,·|.We already know that near o,precisely in exp o(Int(C o)\{0}),ρis a C∞function andρ(exp o(v))= v .(6.1) Let us consider the unit radialfield V=∇ρ.Then S r=∂B r(o)is a family of equidistant hypersurfaces,as in chapter2.Suppose that the sectional curvature K of M is≥k.If˜M is the standard space of sectional curvature k,then,by the comparison theorem3.1,we getA≤˜A=ssI,(6.3) whileD∇ρ|I R V=0,(6.4) becauseρgrows linearly along the integral curves of V.Analogous relations hold for˜ρ:D∇˜ρ|V⊥= swhere C is somefixed constant.Analogously,for˜σ=f◦˜ρ,we get from(6.5)and(6.6)D∇˜σ=−k˜σI+C.(6.9) Theorem6.1(Toponogov’s triangle comparison theorem)[18],[5],[24]Let M be a complete Riemannian manifold with sectional curvature K≥k.Let˜M be the standard space of constant curvature k.Let p0,p1,o∈M,and choose correspond-ing points˜p0,˜p1,˜o∈˜M.Letγbe a geodesic from p0to p1,andβi a shortest geodesic from p i to o,i=0,1,all parametrized by arc length,and let˜γ,˜βi be the corresponding curves in˜M,with L(γ)=L(˜γ)=L and L(βi)=L(˜βi).Let us suppose that all the√lengths are smaller thanπ/(β 0,γ (0)),α1=Letβt be the shortest geodesic joining o toγ(t);the corresponding one˜βt is a shortest geodesic in˜M(for t close to0),henceL(βt)≥|o,γ(t)|,L(˜βt)=|˜o,˜γ(t)|.We haveL(˜βt)=|o,p0|+t ddt 0L(βt)+O(t2)(6.13) and,by thefirst variation formula for curves(cf.[5],p.5),we getddt 0L(˜βt)=−<˜γ (0),˜β 0(0)>.Since we supposedα0<˜α0,for small t we get L(γt)<L(˜γt),which implies|o,γ(t)|≤L(γt)<L(˜γt)=|˜o,˜γ(t)|.Thus,by Toponogov’s theorem,we get a contradiction.If p0happens to be a cut locus point of o,we choose oε=β0(ε)onβ0close to o.Then certainly p0is not in the cut locus of oε.Now we putβt the broken geodesic β|[0,ε]∪βε,t whereβε,t denotes the shortest geodesic from oεtoγ(t),and the same argument holds.Proof of theorem6.1Let us defineρ=|o,·|,˜ρ=|˜o,·|,andσ=f◦ρ,˜σ=f◦˜ρ.Consider the functionδ=σ◦γ−˜σ◦˜γ.(6.14) Hence we have to prove thatδ≥0on[0,L].(6.15)23Fig.8.We prove(6.15)by contradiction.Suppose that there is t∈[0,L]such thatδ(t)<0, and let m=min[0,L]δ(t)<0.We choose k >k sufficiently close to k andτ>0such thatL<πk−τ.(6.16)It is easy tofind a solution a0of the equation a 0+k a0=0,with a0(−τ)=0and a0|[0,L]≤m.Then there existsλ>0such that a=λa0satisfies the following proper-ties:1.a≤δ2.a(t0)=δ(t0)for some t0∈(0,L).Case1:γ(t0)is not a cut locus point of o.Thusδis of class C∞in a neighborhood of t0and(σ◦γ) =<Dγ ∇σ,γ >≤−k(σ◦γ)+C,(6.17) where the inequality follows from(6.8).By eqution(6.9)we get(˜σ◦˜γ) =<D˜γ ∇˜σ,˜γ >=−k(˜σ◦˜γ)+C.(6.18) Henceδ ≤−kδ.(6.19) On the other hand a =−k a.Moreover,in t0we haveδ(t0)=a(t0)<0,which implies(δ−a) (t0)≤δ(t0)(k −k)<0.(6.20) This is a contradiction becauseδ−a takes a minimum at t0.24Case2:γ(t0)is a cut locus point of o.Letβbe a shortest geodesic from o toγ(t0).We choose oεonβclose to o,say|oε,o|=ε.Then we replaceρbyρε(x):=|x,oε|+|oε,o|. By triangle inequality,ρε(x)≥ρ(x),(6.21) and equality holds at x=γ(t0).In other words,ρεis an upper support function ofρat γ(t0).Sinceβis shortest from o toγ(t0),oεis not a cut point ofγ(t0),and therefore,γ(t0)is not a cut point of oε(cf.Ch.5).Puttingσε=f◦ρε,we get the same estimates as in Case1forσεin place ofσ,up to a small error which goes to zero asε→0:(σε◦γ) ≤−k(σε◦γ)+C+error.(6.22) Nowσεis an upper support function ofσatγ(t0)as f is monotoneously increasing. Henceδε−a is an upper support function ofδ−a at t0whereδε=σε−˜σ.Thus it also takes a minimum at t0.But this is a contradiction,because(δε−a) (t0)<0by(6.20).7.Number of generators and growth of the fundamental groupLet M be a complete Riemannian manifold andˆM its universal covering.The fundamental groupπ1(M)will be viewed as group of deck transformations acting onˆM. In other words,M is the orbit space of a discrete groupΓ∼=π1(M)of isometries ofˆM acting freely onˆM,i.e.if g∈Γwith g(p)=p for some p∈M,then g=1.Remark7.1The fundamental group of any compact Riemannian manifold M isfinitely generated.Proof.There exists a compact fundamental domain F(see definition below)for the action ofΓonˆM;e.g.one may take the so called Dirichlet fundamental domainF={x∈ˆM;|x,o|≤|x,go|∀g∈Γ}.We say that g∈Γis small if gF∩F=∅,i.e.if the fundamental domains F and gF are neighbors.If d(F)denotes the diameter of F,i.e.the largest possible distance within F,then gF⊂B2d(F)(o)for all small g,for somefixed o∈F.Since the subsets g(Int(F))are all disjoint with equal volume,there can be onlyfinitely many of them in this ball,hence there exist onlyfinitely many small g.We claim that they form a set of generators.In fact,let g∈Γarbitrary.Choose a geodesic segmentγfrom o to go.Thenγis covered byfinitely many fundamental domains g0F,...,g N F where g0=1 and g N=g,and g i−1F,g i F are neighbors.Thus g−1i g i−1is small,and hence g is a composition of small group elements.Proof.We prove only part a);the second part is similar,but more technical(see Remark at the end of the proof).We define a”norm”inΓas follows:|g|=|p,g(p)|for somefixed p∈ˆM.There exists g1∈Γ\{1}with|g1|minimal(not necessarily unique).By induction,we can construct a sequence(g j):given g1,...,g k,we defineΓk= g1,...,g k ⊂Γand choose g k+1∈Γ\Γk such that|g k+1|has minimum norm inΓ\Γk.Tofinish the proof,we only have to show√Claim:Γk=Γfor some k≤c0(n):=2Fig.10. There are at most2√2are disjoint and their inner half balls are contained inB√5/2(0)≥k2hencek≤2Vol B√22=2√Definition7.6LetΓbe afinitely generated group and G afinite set of generators of Γwith G=G−1and1∈G.We define the growth function N(k)(depending onΓand G)as follows:N(k)= {g∈Γ|∃g1,...,g k∈G such that g=g1·...·g k}.(7.2) So N(k)is the number of group elements which can be written as a product of k elements of G.The dependence of N(k)on G is easy to estimate:If G is another such generating set,then there are numbers p,q such that any element of G can be expressed by p elements of G and each element of G by q elements of G.Thus we haveN (k)≥N(qk),N(k)≥N (pk).Theorem7.7(Milnor’68,[30])Let M be a complete manifold with Ric≥0and letΓ⊂π1(M)anyfinitely generated subgroup of the fundamental group.Then the growth ofΓcan be estimated byN(k)<c·k n.(7.3)where the constant c depends onˆM and the chosen set of generators ofΓ.Proof.Let G be a set of generators as above;it has N(1)elements.Fix a point o∈ˆM. For all g∈Γ,let|g|=|o,go|.Put R =max{|g|;g∈G}.Choose some r>0small enough,so thatB r(go)∩B r(o)=∅∀g∈Γ\{1}(7.4) Put R=R +r.Then the family of balls{B r(go);g∈G}is disjoint and its union is contained in B R(o)so thatVol B R(o) ≥N(1)·Vol B r(o) .(7.5) We can iterate this argument as follows:At the second step,we considerG2:={g1g2;g1,g2∈G}.with (G2)=N(2).Then all balls B r(go)with g∈G2are disjoint and contained in B2R(o)so thatVol B2R(o) ≥N(2)·Vol B r(o) .(7.6) In general,we obtain thatVol B kR(o) ≥N(k)·Vol B r(o) .(7.7)29Recall that we have the Bishop-Gromov inequality(cf.Corollary5.6),Vol B kR(o) ≤ωn k n R n,whereωn denotes the volume of the euclidean unit ball,henceN(k)≤ ωn R n308.Gromov’s estimate of the Betti numbersHomology is a main tool to measure the complexity of topology.Fix afield F and let H q(M)denote the q-th singular homology of M with coefficients in F.Further,let H∗(M)=⊕q≥0H q(M)be the total homology of M.The total Betti number of M is given byb(M)=dim F H∗(M).(8.1)Theorem8.1Gromov,1980(cf.[15],[1],[28])There is a constant C(n)such that:(a.)any complete n-dimensional manifold M with nonnegative curvature K satisfiesb(M)≤C(n);(8.2)(b.)any compact n-dimensional manifold M with curvature K≥−k2,and boundeddiameter,diam(M)≤D,satisfiesb(M)≤C(n)1+kD.(8.3)We will give the proof of part(a.),following ideas of Abresch[1]and W.Meyer [28].(Part(b.)is similar,cf.Remark7.2.)The proof uses the estimates of Bishop-Gromov and Toponogov.It can be viewed as an application of some sort of Morse theory for the distance functionρ(x)=|o,x|where o∈M isfixed.In ordinary Morse theory,one considers a smooth function f:M→I R with isolated critical points with nondegenerate Hessian(p critical means that∇f(p)=0),and one observes how the topology of M c={x∈M;f(x)<c}is changed as c grows.There are two main facts in Morse theory(cf[29]):(1.)If M b\M a contains no critical points,then M b and M a are diffeomorphic.(2.)If M b\M a contains exactly one critical point p,then M b is homotopic to M a witha k-cell attached,where k is the index of the Hessian of f at p.The distance functionρ=|o,|:M→I R is no longer smooth,but we still have the notions of critical and regular points:Definition8.2A point x∈M is called a regular point ofρif there exists v∈T x M such thatv,γ (0) <0(8.4) for any shortest geodesicγfrom x to o.Any such vextor v is called gradientlike.31A point x∈M is a critical point forρif it is non-regular,i.e.if for any v∈T x M there is a shortest geodesicγfrom x to o such thatv,γ (0) ≥0.Remark8.3These notions make sense also if the point o is replaced by a closed subset Σ⊂M.This will be needed in Ch.10.Fact(1.)is still valid:Since the set of initial vectors of shortest geodesics to o is closed,the gradientlike vectors form an open subset of T M and moreover a convexcone at any regular point.Thus we may cover the closure of M b\M a=B b(o)\B a(o) byfinitely many open sets with gradientlike vectorfields and past them together usinga partition of unity,thus getting a gradientlike vectorfield in a neighborhood of theclosure of B b(o)\B a(o).This has the property thatρis strictly increasing along its integral curves.Hence,pushing along the integral curves,we may deform the bigger ball B b(o)into the smaller one B a(o).(See Lemma10.9for details.)We will use this in Lemma8.10below.However,Fact(2.)has no meaning and has to be replaced by another idea:Large balls can be covered by a bounded number of small balls(Bishop-Gromov inequality), and the jump of the Betti number when passing from a small ball to a large ball can be controlled using Toponogov’s theorem.First of all,critical points ofρare not necessarily isolated,but still in some sense, we have to take onlyfinitely many into account:Lemma8.4Let M be a complete manifold with nonnegative curvature.For any L>1 there exists afinite number c(n,L)such that there are at most c(n,L)critical points {q i}forρsatisfying|o,q i+1|≥L|o,q i|.(8.5)√E.g.for L=2we have c(n,2)=2(c (0),v)≤90◦.Applying Toponogov’s theorem(Corollary6.3)with the standard space˜M=I R n,we get˜β≤90◦. Considerfirst the limit case˜β=90◦.Let˜αbe the angle in˜o.It follows thatcos(˜α)=|q i,o|L.32。
A geometric proof of con uence by decreasing diagrams
2
Figure 1: The Escher diagram Actually, we consider an enrichment of mere reduction diagrams, namely diagrams with a `tree covering'. A tree covering of a diagram determines an ancestor-descendant relation between the edges appearing in a reduction diagram. By means of a tree covering an edge can be traced back to its ancestor edge on one of the original divergent reduction sequences. The theorem proved in this paper states the impossibility of certain in nite diagrams with a tree-covering. Since an in nite reduction diagram composed of (trace-)decreasing diagrams would give rise in a natural way to a tree covering|of the impossible kind|we have as an immediate corollary then the theorem of con uence by decreasing diagrams. The method of proof of our theorem is purely geometric. It employs topological notions such as condensation points of point sets in the real plane.
东方国家对美的定义英语作文
东方国家对美的定义英语作文The concept of beauty has been a subject of fascination and debate across cultures and civilizations throughout human history. While the Western world has long been enamored with ideals of physical perfection, the Eastern perspective on beauty often takes a more holistic and spiritual approach. In the Eastern tradition, beauty is not merely a matter of aesthetics, but a reflection of inner harmony, balance, and connection with the natural world.In many Eastern countries, the definition of beauty is deeply rooted in the philosophical and religious traditions that have shaped their cultural identity. In China, for instance, the principles of Taoism and Confucianism have profoundly influenced the way beauty is perceived and appreciated. The Taoist emphasis on the harmony between yin and yang, the constant flow of energy, and the appreciation of natural forms and cycles are all reflected in the Chinese aesthetic sensibility.One of the most striking examples of this can be seen in the art of Chinese landscape painting. Rather than focusing on the realisticdepiction of a specific location, Chinese landscape painters sought to capture the essence of the natural world, the interplay of light and shadow, and the rhythm of the elements. The resulting paintings are not merely representations of the physical landscape, but rather meditations on the deeper connections between humanity and the natural environment.Similarly, in Japanese culture, the concept of wabi-sabi – the appreciation of the imperfect, the impermanent, and the incomplete – is central to the understanding of beauty. This aesthetic principle celebrates the inherent beauty in the natural process of aging and decay, and encourages a mindful engagement with the world around us. In the tea ceremony, for example, the use of simple, unadorned utensils and the careful attention to the preparation and presentation of the tea are all manifestations of this wabi-sabi sensibility.In India, the notion of beauty is deeply intertwined with the spiritual and religious traditions of Hinduism and Buddhism. The intricate and ornate temples, sculptures, and paintings that dot the Indian landscape are not merely aesthetic displays, but rather expressions of the divine and the sacred. The human form, too, is celebrated not just for its physical attributes, but for its capacity to embody the divine essence.The concept of the "third eye" in Hinduism, for instance, represents the spiritual and intuitive aspect of perception, which is considered essential to the appreciation of true beauty. Similarly, the Buddhist concept of "mindfulness" emphasizes the importance of being fully present and engaged with the world around us, rather than being caught up in the superficial or the transient.In the Islamic world, the appreciation of beauty is often closely tied to the divine and the sacred. The ornate calligraphy, intricate geometric patterns, and harmonious use of color that characterize Islamic art and architecture are all expressions of the belief that beauty is a reflection of the divine order and the unity of all creation.Moreover, the emphasis on modesty and restraint in Islamic culture has also shaped the way beauty is perceived and expressed. Rather than focusing on the overt display of physical attributes, the Islamic tradition encourages a more subtle and understated approach to beauty, one that values inner harmony and spiritual connection over external appearances.Across the diverse cultures of the East, the definition of beauty is often characterized by a deep respect for the natural world, a reverence for the sacred and the divine, and a holistic understanding of the human experience. Rather than focusing solely on physical attributes, the Eastern perspective on beauty emphasizes theimportance of balance, harmony, and the cultivation of the inner self.In a world that is increasingly dominated by the ideals of Western beauty, it is important to recognize and appreciate the rich and varied perspectives that the Eastern tradition has to offer. By embracing a more holistic and spiritually-grounded understanding of beauty, we can not only deepen our own appreciation of the world around us, but also foster a greater sense of connection and understanding between different cultures and civilizations.。
欧几里得几何推理 英文
欧几里得几何推理英文Euclidean Geometry ReasoningGeometry has been a fundamental branch of mathematics since ancient times, and Euclidean geometry, named after the renowned Greek mathematician Euclid, has played a pivotal role in its development. Euclid's work, compiled in his seminal text "Elements," laid the foundation for our understanding of the properties and relationships of geometric shapes and their behavior in a two-dimensional plane.At the heart of Euclidean geometry lies the concept of axioms, which are self-evident truths that serve as the starting point for logical deductions. These axioms, combined with a set of postulates, form the basis for the construction of a coherent and consistent system of geometric principles. From these fundamental building blocks, Euclid and his successors were able to derive a vast array of theorems and proofs that have stood the test of time and remain integral to our understanding of the physical world.One of the most fundamental principles in Euclidean geometry is the concept of congruence. Two geometric figures are said to becongruent if they have the same size and shape, and can be superimposed on one another without any distortion. This idea of congruence is essential in many areas of geometry, from the construction of triangles and quadrilaterals to the study of transformations and symmetry.Another key concept in Euclidean geometry is the notion of parallel lines. Parallel lines are those that never intersect, no matter how far they are extended. Euclid's fifth postulate, often referred to as the parallel postulate, states that given a line and a point not on that line, there exists a unique line passing through the point that is parallel to the original line. This seemingly simple yet profound statement has been the subject of much debate and exploration throughout the history of mathematics.The study of triangles is a fundamental aspect of Euclidean geometry, and many of the theorems and proofs within this field revolve around the properties and relationships of these three-sided figures. From the Pythagorean theorem, which relates the lengths of the sides of a right triangle, to the various congruence and similarity criteria, the study of triangles has provided a rich tapestry of geometric understanding.Quadrilaterals, too, play a crucial role in Euclidean geometry. The classification and properties of these four-sided figures, such asrectangles, squares, rhombi, and trapezoids, have led to a deeper understanding of the underlying symmetries and relationships within the geometric plane.Beyond the study of individual shapes, Euclidean geometry also explores the concept of transformations, which are the various ways in which a figure can be moved, rotated, or scaled without altering its essential properties. These transformations, such as translations, reflections, and rotations, have important applications in fields ranging from art and design to computer graphics and engineering.The elegance and logical rigor of Euclidean geometry have made it a cornerstone of mathematical education and a powerful tool for problem-solving and reasoning. Its principles have been applied in a wide range of disciplines, from architecture and engineering to physics and astronomy. Moreover, the study of Euclidean geometry has paved the way for the development of more advanced geometric systems, such as non-Euclidean geometries, which have expanded our understanding of the nature of space and the universe.In conclusion, Euclidean geometry, with its foundational axioms, theorems, and proofs, remains a vital and dynamic field of study. Its ability to provide a coherent and logical framework for understanding the properties and relationships of geometric shapes has made it an enduring and essential component of mathematicaleducation and research. As we continue to explore the depths of this ancient discipline, we uncover new insights and applications that deepen our appreciation for the beauty and power of geometric reasoning.。
《the story of math》中出现的所有数学问题(二)
《the story of math》中出现的所有数学问题(二)The Story of MathIntroductionIn the documentary “The Story of Math”, various mathematical problems are discussed. These problems shedlight on the development and application of mathematics throughout history. Below, I have listed and explained some of the key mathematical questions covered in the documentary.1. What is Prime Factorization?•Prime factorization refers to the process of breaking down a number into its prime factors.•The documentary explains how prime factorization is crucial for various mathematical concepts, such asfinding the greatest common divisor or simplifyingfractions.•It also explores the notion of unique factorization theorem and its significance in number theory.2. How does Geometry Influence Architecture?•This question delves into the connection between geometry and architecture.•The documentary showcases examples of ancientarchitectural wonders, such as the Pyramids of Egypt,that were built using mathematical principles.•It highlights the importance of geometric shapes, ratios, and symmetry in constructing magnificent structures.3. What is the Golden Ratio?•The concept of the golden ratio, denoted by the symbol φ (phi), is explored in this section.•The documentary explains how the golden ratio appears in art, nature, and even in the proportions of the humanbody.•Its significance in aesthetic design and its historical relevance in various cultures are also discussed.4. Can We Solve Fermat’s Last Theorem?•Fermat’s Last Theorem, proposed by Pierre de Fermat, states that there are no three positive integers a, b,and c that satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.•The documentary delves into the long and arduous journey mathematicians undertook to finally prove the theorem in 1994.•It discusses the complexity of the problem and the ingenious techniques used to solve it.5. How Does Calculus Transform Mathematics?•This section explores the advent of calculus and its transformative impact on mathematics.•The documentary explains how calculus revolutionized fields such as physics and engineering by providing the tools to study change and motion.•It highlights the contributions of mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz to thedevelopment of calculus.6. What is the significance of Zero?•The concept and significance of zero in mathematics are explored in this segment.•The documentary discusses the origins of zero in different ancient civilizations and its role inpositional numeral systems.•It showcases how zero revolutionized mathematics by introducing the concept of nothingness and the power ofplace value notation.7. How is Probability Calculated?•Probability theory and its applications are discussed in this section.•The documentary delves into the concept of probability, its calculation, and its uses in various fields likegambling, insurance, and decision-making.•It explores the work of mathematicians like Blaise Pascal and Pierre-Simon Laplace in developing the theory of probability.ConclusionThe documentary “The Story of Math” covers a wide range of mathematical questions, showcasing their historical significance and impact on various fields. From prime factorization to the calculus revolution, these questions demonstrate the beauty and versatility of mathematics. Byunderstanding the answers to these questions, we can appreciate the extensive and profound contribution of mathematics to our world.。
大小形状颜色英语口诀
大小形状颜色英语口诀An English Mnemonic":Shapes, Sizes, and Colors: An English MnemonicThe English language is rich with descriptive words that allow us to paint vivid pictures with our words. Among the most fundamental of these are the terms we use to describe the physical attributes of the world around us - the shapes, sizes, and colors that define the objects and environments we encounter every day. Mastering these basic linguistic building blocks can be an invaluable tool for improving one's command of the language, whether for the purposes of creative expression, effective communication, or simply expanding one's vocabulary.Let us begin with the concept of shape. The most elementary geometric forms are the circle, the square, the triangle, and the rectangle. A circle is a continuous curved line that encloses a flat, round area. Circles are often seen in everyday objects like wheels, plates, and clocks. A square is a four-sided shape with four right angles and four equal sides. Squares are common in architectural design, from windows to floor tiles. A triangle is a three-sided shapewith three angles. Triangles can be found in roofs, road signs, and even the structure of molecules. Finally, a rectangle is a four-sided shape with four right angles and two pairs of equal, parallel sides. Rectangles abound in the built environment, from doors and windows to tabletops and computer screens.Beyond these basic shapes, there is a wealth of other geometric forms that add richness and complexity to our visual world. The oval is an elongated circular shape, often seen in decorative elements and the shapes of certain fruits and leaves. The hexagon is a six-sided shape, commonly found in natural structures like honeycombs and mineral formations. The octagon, with its eight sides and angles, is a distinctive shape utilized in stop signs and gazebo designs. The pentagon, with its five sides and angles, is a less common but no less important geometric form, manifesting in the shapes of some flowers and the arrangement of seeds within certain fruits.The concepts of size and scale are equally fundamental to our understanding and description of the physical world. Words like "small," "large," "tiny," "gigantic," "miniature," and "colossal" allow us to convey the relative dimensions of the objects and living things we encounter. A mouse is small, while an elephant is large. A dust mite is tiny, whereas a blue whale is gigantic. A model train is miniature, but a skyscraper is colossal. These size-related terms enable us to make meaningful comparisons and paint precise mental pictures.Closely related to size is the notion of proportion - the relative scale and dimensions of an object's various parts. We might describe something as "tall and thin," "short and wide," or "long and slender." These proportional descriptors give us a more nuanced understanding of an item's physical attributes. A beanstalk is tall and thin, a beach ball is short and wide, and a giraffe's neck is long and slender.Finally, we come to the realm of color - perhaps the most evocative and expressive of all physical attributes. The spectrum of hues available to us in the English language is truly vast, ranging from the primary colors of red, blue, and yellow to the infinite gradients and blended tones in between. Colors can be warm or cool, bright or muted, deep or pale. They can inspire emotions, evoke memories, and convey symbolic meaning.A ripe apple is a vibrant red, while the sky on an overcast day is a somber gray. The grass is a rich, verdant green, but a chameleon's skin can shift through a rainbow of color-changing hues. The petals of a sunflower are a cheerful yellow, contrasting with the inky black of its central seed head. In the world of fashion and design, color choices hold immense power to influence mood, style, and cultural associations.Mastering the vocabulary of shapes, sizes, and colors is not merely an academic exercise - it is a gateway to richer, more nuanced self-expression and a deeper appreciation of the visual world that surrounds us. Whether we are describing the shape of a skyscraper, the size of a mountain, or the color of a sunset, these fundamental linguistic building blocks allow us to paint vivid mental pictures and communicate with greater precision and clarity.By committing these shape, size, and color terms to memory through repetition and practice, we equip ourselves with a versatile toolbox of descriptive language that can be applied across a wide range of contexts - from creative writing to technical documentation, from casual conversation to formal presentations. The English mnemonic outlined here provides a solid foundation upon which to build one's fluency and confidence in this essential realm of the language.So the next time you find yourself in need of the right word to capture the essence of a physical object or phenomenon, let this shape, size, and color vocabulary be your guide. With these linguistic tools at your disposal, you will be empowered to describe the world around you with precision, clarity, and artistry.。
魔方的英文而带出来的笑话
魔方的英文而带出来的笑话Certainly, I will provide the answer in both English and Chinese, with the English response first, followed by the Chinese response. The total word count will not be less than 1200 words, and I will not reveal your prompt within the text.English Response:Rubik's Cube-Inspired Jokes: A Twist of HumorRubik's Cube, the iconic three-dimensional puzzle that has captivated the minds of millions, has also become a source of endless amusement and laughter. The intricate twists and turns of this colorful cube have inspired a plethora of jokes and puns, each one more clever and entertaining than the last.One of the most common Rubik's Cube-inspired jokes revolves around the frustration of solving the puzzle. "Why was the Rubik's Cube so popular? Because it was a bigtwist!" This simple yet effective joke pokes fun at the seemingly endless struggle many people face when trying tosolve the cube, often feeling like they're caught in a never-ending loop of twisting and turning.Another popular joke plays on the cube's ability to confuse and perplex. "Why did the Rubik's Cube go to the doctor? Because it had a twisted personality!" This quip highlights the perplexing nature of the puzzle, with its constantly shifting colors and patterns that can leave even the most experienced puzzlers scratching their heads.Puns are another favorite when it comes to Rubik's Cube-inspired humor. "Why did the Rubik's Cube cross the road? To get to the other side... of the cube!" This clever play on words combines the classic "why did the chicken cross the road?" joke with the cube's inherent ability to be turned and twisted in different directions.One particularly amusing joke takes a jab at the cube's complexity: "Why did the Rubik's Cube go to the gym? To work on its abs!" This humorous remark plays on the notion that solving the Rubik's Cube requires a significant amount of mental and physical dexterity, akin to the core-strengthening exercises performed at the gym.Extending the physical metaphor, another joke suggests, "Why did the Rubik's Cube join the circus? Because it was a master of twists and turns!" This quip humorously imagines the Rubik's Cube as a talented acrobat, showcasing its impressive range of motion and flexibility.The Rubik's Cube's association with intelligence and problem-solving skills has also spawned some witty jokes. "Why did the Rubik's Cube go to the math class? Because it needed to work on its square roots!" This play on words combines the cube's geometric nature with the mathematical concepts often required to solve it.In a similar vein, another joke pokes fun at the cube's perceived difficulty: "Why did the Rubik's Cube go to the library? Because it needed to check out a book on problem-solving!" This humorous remark suggests that even the most intelligent puzzlers may need a little extra help when it comes to conquering the Rubik's Cube.Ultimately, the Rubik's Cube's enduring popularity and the ongoing challenge it presents have made it a rich source of comedic inspiration. From puns and wordplay to physical metaphors and references to intelligence, thehumble Rubik's Cube has become a cultural icon that continues to twist and turn its way into the hearts and minds of humor enthusiasts worldwide.中文回应:魔方启发的笑话:幽默的扭转魔方,这个引发了数百万人迷恋的三维拼图,也成为了无穷无尽乐趣和笑声的源泉。
时光一去不复返的英文
时光一去不复返的英文Title: The Irreversible Flow of TimeIn the grand tapestry of existence, time weaves its intricate patterns with an unwavering hand, oblivious to human desires and dreams. It is a force majestically indifferent, moving forward with relentless precision, leaving behind only memories and echoes of what once was. This essay explores the profound concept that time, once lost, is irretrievably gone, urging us to reflect on its implications for our lives.Time, in its essence, is a linear progression, a one-way street stretching from birth to death, devoid of detours or return lanes. Unlike the ebb and flow of tides or the cyclical nature of seasons, time does not offer second chances or opportunities for revision. Once a moment has passed, it becomes part of the past, forever beyond our reach. This immutable truth underscores the significance of each instant, making every decision, action, and inaction a critical component of our life's narrative.The realization that time cannot be reclaimed imbues our lives with a sense of urgency and purpose. Knowing that tomorrow is promised to no one compels us to live fully in the present, to cherish the moments we have with loved ones, andto pursue our passions with vigor. It teaches us the value of seizing opportunities as they arise, for hesitation and procrastination can lead to missed chances and lingering regrets. In this light, time acts as both a guide and a teacher, leading us towards growth, wisdom, and fulfillment.Moreover, the awareness of time's irreversibility fosters a deep appreciation for the fleeting nature of beauty and happiness. Just as a sunset fades into night, so do moments of joy and contentment give way to new challenges and experiences. This ephemeral quality of life encourages us to find joy in the simple things, to cultivate gratitude for our blessings, and to embrace change as a natural part of our journey. By doing so, we learn to navigate the complexities of existence with grace and resilience.However, the notion that time is irreversible also carries with it a sense of melancholy and loss. As we age, we inevitably look back on our lives with a mixture of pride and sorrow, reflecting on paths not taken and dreams unfulfilled. The passage of time reveals the impermanence of all things, reminding us that even the most cherished memories will eventually fade. Yet, within this bittersweet reality lies the potential for redemption and renewal. Each day offers a freshstart, a chance to learn from past mistakes and to create new legacies.In conclusion, the irreversible flow of time is a fundamental aspect of human existence, shaping our perceptions, actions, and aspirations. It serves as a reminder of the preciousness of life and the importance of living intentionally. While we may long to turn back the clock or relive happy moments, such desires are futile. Instead, by accepting the transient nature of time and embracing the present, we can lead richer, more meaningful lives. Let us therefore tread carefully upon the sands of time, mindful of its swift passage and committed to making the most of every fleeting moment.。
学习上的困惑请求老师帮助英语作文
全文分为作者个人简介和正文两个部分：作者个人简介：Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文：学习上的困惑请求老师帮助英语作文全文共3篇示例，供读者参考篇1Dear Teacher,I am writing to you today with a heavy heart and a mind full of worries about my academic performance. As a student, I have always strived for excellence, driven by an insatiable thirst forknowledge and a burning desire to succeed. However, the path I have trodden thus far has been fraught with challenges that seem to grow more daunting with each passing day.From the very onset of my academic journey, I have encountered obstacles that have threatened to impede my progress. The subjects that once seemed so alluring and captivating have gradually transformed into labyrinthine mazes, leaving me disoriented and uncertain. Mathematics, a discipline that I once embraced with fervor, now appears to be a cryptic language, its symbols and equations evading my comprehension. Science, which once ignited my curiosity and wonder, now feels like a vast ocean of knowledge, its depths too vast for me to fathom.It is not for a lack of effort or dedication that I find myself in this predicament. I have diligently attended every class, meticulously taken notes, and spent countless hours poring over textbooks and study materials. Yet, despite my best efforts, the concepts seem to elude me, slipping through the cracks of my understanding like grains of sand through clenched fingers.The weight of these academic struggles has taken a toll on my mental well-being. I find myself plagued by self-doubt, questioning my capabilities and worthiness. The once-vibrantflame of motivation that burned within me has been reduced to a flickering ember, struggling to stay alight amidst the relentless winds of discouragement.Compounding these challenges is the pressure I feel to excel, not only for my own sake but also for the pride and joy of my family. Their unwavering support and belief in my potential have been the bedrock upon which I have built my dreams, yet I fear that my struggles might disappoint them, shattering their hopes and expectations.It is with a heavy heart that I confess my shortcomings, but it is also with a glimmer of hope that I turn to you, my teacher, for guidance and support. Your wisdom and experience are beacons in the darkness, illuminating the path forward and offering solace to those who stumble.I implore you to lend me your insight and expertise. Guide me through the labyrinth of knowledge, help me decipher the cryptic codes of mathematics, and unveil the mysteries of science. Provide me with the tools and strategies that will enable me to overcome these obstacles and unlock the full potential that lies dormant within me.Perhaps I require a different approach to learning, one that resonates more deeply with my unique strengths and learningstyle. Or perhaps I simply need a patient mentor who can break down complex concepts into digestible morsels, filling the gaps in my understanding and illuminating the connections that have eluded me thus far.I am not seeking shortcuts or easy paths to success – for I understand that true learning requires dedication, perseverance, and a willingness to embrace challenges. Instead, I seek your wisdom and guidance, your ability to inspire and motivate, and your belief in my capacity to overcome these hurdles.Together, let us forge a path through this academic wilderness, navigating the treacherous terrain of knowledge and emerging victorious on the other side. With your support and my unwavering determination, I am confident that I can conquer these struggles and reclaim the joy and passion that once fueled my academic pursuits.I eagerly await your counsel, for in your guidance lies the key to unlocking the doors of understanding and propelling me towards the academic heights I have long dreamed of attaining.Respectfully yours,[Your Name]篇2Dear Teacher,I am writing to you today because I have been struggling in your class and I need your help to get back on track. As someone who has always taken great pride in being a good student, it's been really hard for me to admit that I'm having such a difficult time. But I know that if I don't reach out for support, I'm only going to fall further and further behind.The truth is, I'm feeling completely overwhelmed and confused by the material we've been covering lately. I've never had such a hard time understanding the concepts and theories we're learning about. It feels like no matter how many times Ire-read the chapters or go over my notes, nothing is really sinking in or making sense. The information just isn't sticking in my brain the way it usually does.I think part of the issue is that this subject matter is very different from anything I've studied before. The ideas and terminology are brand new to me, and a lot of it seems really abstract and hard to visualize or apply to real-world scenarios. I'm a very hands-on, practical learner, so I tend to grasp things better when I can see concrete examples of how the knowledge can be used. The textbook explanations and lecture content hasbeen great, but I could really use some more tangible ways to understand and engage with the material.Another factor that has made it tough for me is how rigorous and fast-paced this course is. We cover so much ground in such a short period of time, which has made it hard for me to fully digest and master one concept before we're already moving on to the next. I feel like I'm constantly playing catch-up and scrambling to keep my head above water. The workload and number of assignments has been really overwhelming too. I'm used to being one of the strongest students in the class, but lately I've been struggling to even finish the bare minimum work, let alone go above and beyond like I normally do.I've never been the type of student who just coasts by doing the bare minimum though. I really want to succeed in this class and have a true understanding of the concepts, not just surface-level knowledge. This subject is something I'm genuinely interested in, but my frustration and confusion has started to make me dread coming to class. I'm worried that if I don't get some extra support soon, I'm going to completely check out mentally. I don't want to develop a negative attitude or give up on a subject I care about just because I'm struggling.That's why I'm reaching out to you for help - because I know you want me to succeed just as much as I want to succeed myself.I was wondering if you could suggest some additional resources, study strategies, or tutoring options that could help reinforce the material for me in a different way. Maybe you could recommend some supplemental readings, online tutorials, or practice exercises that could give me another angle on the conceptswe're covering. I'd also be really grateful if you could take some time to go over my notes with me one-on-one to clarify anything I'm still fuzzy on. Getting some personalized guidance and explanations tailored to how my brain learns best could make a huge difference.I'm also open to any other advice or guidance you can provide. Is there anything I should be doing differently to absorb or engage with the material more effectively? Maybe I need to adjust my time management, study habits, or general approach to this class. I'm seriously at a loss here, so any wisdom or strategies you can share would be invaluable.I know I have the capability and work ethic to turn things around in this class. I just need to get some tools and support to help overcome this overwhelming barrier I've hit. With a little extra assistance from you, I'm confident I can get back on theroad to understanding the content, regaining my footing, and achieving my full potential. I appreciate you taking the time to read this letter and consider how you can help. Thank you in advance for your guidance and support.Sincerely,[Your Name]篇3Dear Teacher,I find myself in a position where I need to reach out for your guidance and expertise. As a diligent student, I have always strived to excel academically, but recently, I have encountered a challenging obstacle that has left me feeling perplexed and uncertain.The subject matter that has been causing me considerable distress is the intricate realm of advanced calculus. While I have a strong foundation in mathematics and have excelled in previous courses, the concepts and applications we are currently exploring seem to elude my comprehension. It is as if I am standing before an impenetrable fog, unable to discern the path that lies ahead.Permit me to elaborate on the specific areas where I am experiencing difficulties. The notion of multivariable functions and their derivatives has proven to be a formidable hurdle. I struggle to visualize the geometric interpretations and grasp the intricate calculations involved. Each time I attempt to solve a problem, I find myself entangled in a labyrinth of symbols and equations, unsure of the correct approach or the underlying reasoning.Moreover, the concept of multiple integrals has left me utterly bewildered. The process of integrating over regions in higher-dimensional spaces is a concept that seems to evade my grasp, no matter how diligently I study the material or practice the examples provided. I find myself questioning the very foundations of these principles, unable to comprehend their significance or practical applications fully.Compounding my frustration is the fact that I have dedicated countless hours to studying and reviewing the course material. I have meticulously read and re-read the textbook chapters, watched supplementary video lectures, and sought guidance from my peers. Yet, despite my unwavering efforts, the veil of confusion remains stubbornly in place.This predicament has not only challenged my academic abilities but has also taken a toll on my confidence and motivation. I find myself questioning my aptitude for mathematics, a subject that has always been a source of pride and fulfillment for me. The once-familiar feeling of satisfaction upon solving a complex problem has been replaced by a sense of despondency and self-doubt.It is with a heavy heart that I acknowledge my struggles, but it is also with a glimmer of hope that I turn to you, my esteemed mentor. Your vast knowledge, patient guidance, and ability to shed light on even the most obscure concepts have been a constant source of inspiration throughout my academic journey.I humbly request your assistance in navigating these turbulent waters of advanced calculus. Perhaps through your insightful explanations and personalized guidance, the fog will dissipate, and the path forward will become clear. I am eager to learn from your expertise and regain the confidence that has been shaken by this challenging experience.Please, if you have the time and inclination, could we arrange a meeting or additional tutoring sessions? I am willing to dedicate as much time and effort as necessary to overcome thisobstacle, for I firmly believe that with your invaluable guidance, I can conquer these complexities and emerge victorious.Thank you for your unwavering support and for being a beacon of knowledge in my academic journey. I eagerly await your response and the opportunity to work together to overcome this formidable challenge.Sincerely,[Your Name]。
几何美学的英语作文
几何美学的英语作文Title: The Aesthetic Appeal of Geometry。
Geometry, often referred to as the language of the universe, holds a profound aesthetic appeal that transcends cultures and epochs. Its beauty lies not only in its precise measurements and logical deductions but also in its capacity to evoke wonder and admiration. In this essay, we delve into the enchanting realm of geometric aesthetics, exploring its manifestations in art, architecture, nature, and beyond.First and foremost, geometry embodies an intrinsic harmony that resonates with human sensibilities. From the elegant symmetry of a snowflake to the perfect proportions of the Parthenon, geometric forms captivate our visual perception and elicit a sense of order and balance. The symmetry inherent in geometric shapes, such as circles, squares, and triangles, instills a feeling of completeness and stability, appealing to our innate desire forequilibrium and coherence.Moreover, geometry serves as a powerful tool for creative expression in various artistic endeavors. Artists throughout history have drawn inspiration from geometric principles to create masterpieces that captivate the imagination. The works of renowned artists like Leonardo da Vinci, with his meticulous renderings of geometric solids, and Piet Mondrian, with his abstract compositions of lines and rectangles, demonstrate the aesthetic potency of geometric motifs. By harnessing geometric forms, artists can imbue their creations with a sense of clarity, order, and rhythm, transcending the confines of representation to evoke deeper emotional responses.In architecture, geometry plays a central role in shaping the built environment and defining spatial experiences. Ancient civilizations, such as the Egyptians and Greeks, employed geometric principles to design enduring architectural marvels that stand as testaments to human ingenuity. The geometric precision of the Great Pyramid of Giza and the mathematical harmony of theParthenon exemplify the timeless appeal of geometric design in architecture. Furthermore, contemporary architects continue to explore the aesthetic possibilities of geometry, employing advanced computational tools to push the boundaries of form and structure.Beyond the realms of art and architecture, geometry permeates the natural world, giving rise to mesmerizing patterns and phenomena. From the spirals of seashells tothe hexagonal symmetry of beehives, nature abounds with geometric motifs that inspire awe and fascination. The Fibonacci sequence, a mathematical pattern found in sunflowers, pinecones, and nautilus shells, exemplifies the underlying geometric order inherent in natural processes.By recognizing and appreciating these geometric patterns in nature, we gain a deeper understanding of the interconnectedness and beauty of the universe.In addition to its aesthetic allure, geometry plays a crucial role in scientific inquiry and technological innovation. The laws of geometry govern the motion of celestial bodies, the structure of molecules, and thedesign of intricate machinery. From the discovery of the Pythagorean theorem to the development of fractal geometry, mathematical principles have revolutionized our understanding of the world and fueled advancements in various fields.In conclusion, the aesthetic appeal of geometry lies in its ability to evoke a sense of harmony, order, and beauty across diverse domains. Whether expressed through art, architecture, nature, or science, geometric forms captivate the human imagination and illuminate the inherent elegance of the universe. By embracing the aesthetic richness of geometry, we embark on a journey of discovery and appreciation that transcends cultural boundaries and enriches our understanding of the world.。
Lectures on the Geometric Group Theory
Corollary 3. Suppose that G is a group quasi-isometric to a nilpotent group. Then
G itself is virtually nilpotent, i.e. contains a nilpotent subgroup of nite index.
b
ab
a -2
a -1
1
a
a2
b -1
Figure 1: Free abelian group.
Example 5. Let ? be the free group on two generators s1; s2 . Take S = fsi; i = 1; 2g.
The Cayley graph C = C (?; S ) is the 4-valent tree (there are four edges incident to each vertex). See Figures 1, 2.
Gromov's theorem and its corollary will be proven in chapter 5. Proving these theorems are the main objectives of this course. Along the way we will introduce several tools of the geometric group theory: coarse topology, ultralimits, quasiconformal mappings.
1.3 Quasi-isometries
Let X be a metric space. We will use the notation NR (A) to denote R-neighborhood of a subset A X , i.e. NR (A) = fx 2 X : d(x; A) < Rg. Recall that Hausdor distance between subsets A; B X is de ned as
数学专业英语翻译2-4
New Words & Expressions:
irrational number 无理数 the order axiom 序公理 ordered 有序的 product 积 quotient 商 rational 有理的 rational number 有理数 reasoning 推理 scale 尺度，刻度 尺度， sum 和
This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction.
现在我们来定义正整数，就是属于每一个归纳集的实数。 现在我们来定义正整数，就是属于每一个归纳集的实数。
Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive set, we refer to P as the smallest inductive set. 表示所有正整数的集合。 用 P表示所有正整数的集合。那么 本身是一个归纳 表示所有正整数的集合 那么P本身是一个归纳 集 ， 因为其中含1， 满足(a)； 只要包含x就包含 因为其中含 ， 满足 ； 只要包含 就包含x+1, 就包含 满足(b)。由于P中的元素属于每一个归纳集 因此P 中的元素属于每ห้องสมุดไป่ตู้个归纳集， 满足 。由于 中的元素属于每一个归纳集，因此 是最小的归纳集。 是最小的归纳集。
differential manifold 英文书籍
differential manifold 英文书籍A Brief Introduction to Differential ManifoldsIntroductionDifferential manifold theory is an essential branch of mathematics and plays a crucial role in various disciplines such as physics and engineering. In this article, we will explore the concept of a differential manifold, its properties, and its applications. This article aims to provide a concise yet comprehensive understanding of differential manifolds for readers interested in the subject.What is a Differential Manifold?A differential manifold is a mathematical space that locally resembles Euclidean space but may have a complicated global structure. It can be defined in terms of charts, which are coordinate systems that map points in the manifold to points in Euclidean space. These charts allow us to define smooth functions on the manifold, and the compatibility conditions between different charts ensure a smooth transition between coordinate systems.Properties of Differential ManifoldsOne of the fundamental properties of differential manifolds is their dimension. A manifold can be n-dimensional if it can be locally parametrized by n real numbers. The local linearity property of manifolds ensures that tangent vectors can be defined at each point, allowing for the study of derivatives and vector fields.Another key property of differential manifolds is their topology. A manifold can be either open or closed, connected or disconnected, compact or non-compact, depending on its underlying topology. The notion of continuity and differentiability can be extended to functions defined on manifolds, leading to the study of smooth maps and flows.Applications of Differential ManifoldsDifferential manifold theory finds applications in numerous fields, including physics, engineering, and computer science. In physics, differential manifolds provide a geometric framework for studying space-time in general relativity. The curvature and topology of manifolds play a crucial role in understanding the gravitational field.In engineering, differential manifolds are used to model and analyze various systems. Electrical circuits, control systems, and signal processing can all be represented as manifolds, allowing for the application of differential equations and control theory.In computer science, differential manifolds are employed in computer vision, robotics, and machine learning. Manifold learning algorithms aim to uncover the underlying structure and dimensionality of high-dimensional data by mapping it onto a lower-dimensional manifold.ConclusionIn conclusion, differential manifolds are fascinating mathematical objects that provide a flexible framework for studying smooth functions, derivatives, and geometric properties. They have broad applications in various scientific and engineering fields, making them an indispensable toolfor understanding and solving complex problems. By grasping the fundamental concepts and properties of differential manifolds, readers can delve into more advanced topics and further explore the beauty and elegance of this mathematical discipline.。
有关对西方现代艺术的英语作文
有关对西方现代艺术的英语作文全文共3篇示例，供读者参考篇1The Enigmatic World of Western Modern ArtAs a student of art history, I have always been fascinated by the vast and diverse realm of Western modern art. This revolutionary movement, spanning from the late 19th century to the present day, has challenged conventional notions of beauty, form, and expression, leaving an indelible mark on the cultural landscape.At the heart of modern art lies a profound rejection of the traditional artistic values that had dominated the art world for centuries. The pioneers of this movement, such as Pablo Picasso, Henri Matisse, and Wassily Kandinsky, sought to break free from the shackles of realism and academic traditions, embracing a new era of experimentation and abstraction.One of the most striking features of Western modern art is its embrace of subjectivity and personal expression. Artists no longer aimed to depict reality as it appeared to the naked eye; instead, they sought to capture the essence of their inner worlds,emotions, and unique perspectives. This shift towards introspection and self-exploration gave rise to a multitude of artistic styles, each with its own distinct visual language and conceptual underpinnings.Take, for instance, the revolutionary movement of Cubism, pioneered by Picasso and Georges Braque. By fragmenting and reassembling objects onto a two-dimensional plane, these artists challenged the conventional notions of perspective and spatial representation. Their bold and geometric compositions, which simultaneously depicted multiple viewpoints, invited viewers to engage with the artwork in a profoundly different way, forcing them to question their own perceptions of reality.Similarly, the Expressionist movement, spearheaded by artists like Edvard Munch and Egon Schiele, sought to convey intense emotional states through distorted forms, bold brushstrokes, and vivid colors. Their works, often imbued with a sense of angst and psychological turmoil, reflected the turbulent social and political climate of the early 20th century, resonating with audiences on a deeply personal level.As the 20th century progressed, modern art continued to evolve and diversify, giving birth to a myriad of movements and styles, each pushing the boundaries of artistic expression further.The Surrealists, led by Salvador Dalí and René Magritte, delved into the realms of the subconscious and the dreamlike, creating enigmatic and thought-provoking works that challenged our perceptions of reality.The Abstract Expressionists, such as Jackson Pollock and Willem de Kooning, embraced pure abstraction, emphasizing the physical act of painting and the spontaneous expression of emotion through their bold and gestural brushwork. Their canvases became vessels for raw, primal energy, inviting viewers to engage with the work on a visceral level.While some may dismiss Western modern art as mere chaos or a departure from traditional aesthetics, I believe that this movement has profoundly enriched our understanding of art and its role in society. Modern artists have challenged us to expand our perspectives, to question our assumptions, and to embrace the limitless possibilities of artistic expression.Through their bold experimentation and unwavering pursuit of innovation, these artists have opened our eyes to new ways of seeing and interpreting the world around us. Their works invite us to engage in a dialogue, to contemplate the deeper meanings and emotions embedded within each brushstroke, each line, and each color.Moreover, Western modern art has played a crucial role in reflecting and shaping the cultural and social landscapes of its time. From the tumultuous upheavals of the world wars to the civil rights movements and the ongoing pursuit of individual freedom, modern art has served as a powerful means of expression, giving voice to the experiences, struggles, and aspirations of various communities and individuals.As a student of art history, I am constantly in awe of the richness and diversity of Western modern art. Each time I delve into the works of these pioneering artists, I am reminded of the boundless potential of human creativity and the power of art to challenge, provoke, and inspire.While some may find the enigmatic nature of modern art perplexing or even off-putting, I believe that embracing this complexity is a crucial step towards expanding our understanding of the world and our place within it. Modern art invites us to question our preconceived notions, to engage in critical thinking, and to embark on a journey of self-discovery and personal growth.As I continue my studies and immerse myself in the captivating world of Western modern art, I am certain that this movement will continue to challenge, inspire, and reshape ourperceptions for generations to come. Its enduring legacy lies not only in the masterpieces it has produced but also in the enduring questions it poses about the nature of art, beauty, and human expression.篇2The Puzzling World of Western Modern ArtArt has been a part of human civilization for tens of thousands of years, with ancient cave paintings and sculptures dating back to prehistoric times. Throughout history, art has served as a means of storytelling, cultural expression, and creative outlet for societies around the world. However, the early 20th century marked a radical shift in the Western art world that left many people baffled and challenged long-held notions about the purpose and definition of art.This era, known as the modern art movement, introduced a plethora of new artistic styles, techniques, and philosophies that diverged dramatically from the traditional, representational approach to art that had dominated for centuries. As a student trying to wrap my head around this perplexing period in art history, I often find myself asking: what exactly is modern art, and why did it spark such controversy and polarization?At its core, modern art rejected the idea that art should simply imitate reality or conform to conventional standards of beauty and technique. Instead, modern artists sought to express their innermost emotions, experiences, and interpretations of the world through abstract, non-representational forms. They believed that art should challenge the viewer, provoke thought, and push the boundaries of what was considered "art."One of the pioneering movements of modern art was Cubism, spearheaded by Pablo Picasso and Georges Braque in the early 1900s. Cubist paintings deconstructed and fragmented objects, depicting them from multiple perspectives simultaneously. This radical departure from traditional perspective and representation was initially met with bewilderment and criticism, but it paved the way for even more abstract and experimental styles to emerge.The Dada movement, which arose during World War I, took a more anarchic and anti-bourgeois stance, rejecting reason and aesthetics in favor of nonsense, chance, and provocation. Dada artists like Marcel Duchamp challenged the very notion of what constituted art by presenting ordinary objects, such as a urinal or a bottle rack, as art pieces, a concept known as "readymades."Surrealism, pioneered by artists like Salvador Dalí and René Magritte, delved into the realms of the subconscious and the irrational, creating dreamlike, bizarre, and often unsettling imagery. Surrealist works were meant to unleash the power of the imagination and explore the boundaries between reality and fantasy.As a student grappling with these unconventional artistic movements, I often find myself torn between appreciation and confusion. On one hand, I admire the boldness and creativity of modern artists who dared to challenge traditional norms and push the boundaries of artistic expression. Their works force us to question our perceptions, confront our preconceptions, and engage with art on a deeper, more intellectual level.On the other hand, the abstract and often enigmatic nature of modern art can be frustrating and alienating, especially for those accustomed to more representational forms of art. It's tempting to dismiss some modern works as mere splashes of color or random assemblages of objects, devoid of any deeper meaning or artistic merit.Yet, as I delve deeper into the study of modern art, I've come to appreciate the complex philosophies, socio-political contexts, and personal narratives that underlie these seemingly bizarrecreations. Modern artists were not simply creating nonsensical or meaningless works; rather, they were responding to the rapidly changing world around them, grappling with the traumas of war, the rise of industrialization, and the shifting cultural and societal norms of the early 20th century.For instance, the Cubist deconstruction of form can be seen as a reflection of the fragmentation and multiple perspectives that characterized modern life in rapidly urbanizing cities. The Dada movement's embrace of nonsense and anti-art sentiments was a reaction against the societal structures and bourgeois values that led to the devastation of World War I.Surrealism, with its exploration of the subconscious and the irrational, sought to tap into the depths of human psychology and challenge the dominance of reason and rationality in an increasingly industrialized and mechanized world.As I continue to study and engage with modern art, I've come to appreciate its ability to provoke thought, challenge conventions, and offer unique perspectives on the human experience. While some works may initially appear impenetrable or lacking in substance, a deeper understanding of the artists' intentions, historical contexts, and underlying themes can reveal profound layers of meaning and significance.Moreover, the impact of modern art extends far beyond the confines of galleries and museums. Its influence can be seen in various aspects of contemporary culture, from design and fashion to advertising and popular media. The bold experimentation and boundary-pushing spirit of modern art have paved the way for new forms of creative expression and have challenged us to expand our perceptions of what constitutes art.In conclusion, the world of Western modern art remains a puzzling and often polarizing subject, eliciting both admiration and bewilderment from students and art enthusiasts alike. While the abstract and unconventional nature of modern art can be challenging to comprehend, a deeper exploration of its historical contexts, philosophies, and artistic intentions can reveal profound layers of meaning and significance. Modern art has forever changed the way we perceive and engage with artistic expression, challenging us to question our preconceptions, embrace the power of the imagination, and embrace theever-evolving nature of art itself.篇3The Puzzling World of Modern Western ArtWhen you think of art, you probably picture classic paintings and sculptures - the Mona Lisa's enigmatic smile, Michelangelo's chiseled David, or Van Gogh's swirling Starry Night. But what about a canvas splattered with seemingly random colors? Or a sculpture made from scrap metal and garbage? That's the perplexing realm of modern Western art, where the traditional ideas of beauty and skill are challenged and redefined.As a student, I'll admit that I initially found much of modern art confusing and even disturbing. How could a solid black square hanging askew on a gallery wall be considered high art? What creative genius was behind an all-white painting or a urinal turned upside down? I felt like I was missing some vital context that would help me "get" these bizarre works.However, the more I studied modern art movements like Cubism, Surrealism, Abstract Expressionism, and Pop Art, the more I began to appreciate their bold rejection of conventional aesthetics. Rather than pursuing precise realism, these artists tapped into the subconscious mind, playing with color, form, and perception itself in new and radical ways.Consider Pablo Picasso's groundbreaking Les Demoiselles d'Avignon from 1907, a work that ushered in the Cubist movement. With its fragmented shapes and multiple perspectivepoints, it shattered traditional composition and representation. Looking at the geometric shards of color and distorted female forms is like seeing the world through a broken mirror - jarring yet fascinating.The Surrealists like Salvador Dalí took this concept even further by bringing dreams and the subconscious into vivid, bizarre tableaus. His iconic painting The Persistence of Memory depicts limp, melting clocks draped across a desolate landscape, provoking questions about the nature of time, perception, and reality itself. To the uninitiated, these Surrealist works seem like trippy nonsense. But looked at through the lens of exploring the human psyche and subverting norms, they become profound statements.While early modern artists were still tethered to some degree of representation, the Abstract Expressionists blew past that boundary into pure abstraction. Jackson Pollock's drip paintings like Number 1 (Lavender Mist) are basically just colored splatters on canvas, created by his innovative "action painting" technique of dripping and pouring paint. To a casual observer, it may look like a huge mess. But to the art critics, it represents the raw expression of the unconscious mind through bold colors and spontaneous composition.Pop artists like Andy Warhol took an entirely different approach by elevating mundane, mass-produced commercial objects into artistic statements on consumerism and celebrity culture. His famous screenprints of Campbell's Soup Cans and Marilyn Monroe portraits speak volumes about American obsessions with brands, products, and pop icons, while simultaneously poking fun at the high art world.At first glance, a lot of modern Western art can seem like a bizarre joke, created by either con artists or disturbed individuals. Believe me, I get that reaction! But the more I learn, the more I've come to appreciate the deeper ideas andemotional/psychological exploration behind these groundbreaking art movements.Modern artists weren't just creating pretty pictures to match the living room drapes. They were rebelling against the rigid constraints and traditions of the past. They were wrestling with big philosophical questions about reality, identity, meaning, and the role of the artist in society. Their art represents a passionate quest to depict new perspectives and the inner dimensions of the human experience.I still don't love or "get" every single piece of modern art out there. Some works are definitely more accessible than others. ButI've developed a deep respect for the avant-garde artists who had the courage and vision to completely upend conventional notions of beauty, meaning, and creativity in art. Ahead of their time, they faced backlash, confusion, and disdain, but persisted in their bold experimentation and innovations.So while I may not want to hang a urinal or a drip painting in my living room, I can't deny the profundity and significance of the modern art revolution in pushing artistic boundaries and exploring deeper realms. In reshaping how we experience and interpret art, these iconoclasts fundamentally changed how we view the world itself. And that makes their perplexing yet profound creations worth studying and appreciating.。
高等数学教材英文版
高等数学教材英文版Higher Mathematics Textbook (English Edition)Introduction:In the realm of mathematics, higher mathematics plays a pivotal role in shaping our understanding of complex mathematical concepts and their applications. As an essential component of advanced education, the Higher Mathematics Textbook in its English edition provides a comprehensive resource for students seeking to master this subject. This article highlights the importance and key features of the Higher Mathematics Textbook in its English edition.Chapter 1: Functions and GraphsThis chapter delves into the fundamental concept of functions and their graphical representation. It explores various types of functions, such as linear, quadratic, exponential, and logarithmic functions, along with their properties and behaviors. The chapter also introduces the notion of limits and continuity, laying the groundwork for future mathematical analysis.Chapter 2: Differentiation and IntegrationBuilding upon the foundation of functions, Chapter 2 delves into the concepts of differentiation and integration. It covers the principles of finding derivatives and calculating integrals, as well as their respective applications in various fields, including physics, economics, and engineering. Through carefully curated examples and exercises, students gain a solid understanding of these essential calculus techniques.Chapter 3: Sequences and SeriesThis chapter emphasizes the study of sequences and series, encompassing arithmetic, geometric, and power series. It examines convergence and divergence criteria and explores the concepts of convergence tests, including the ratio test, root test, and integral test. By mastering these concepts, students develop a deep comprehension of the behavior of sequences and series.Chapter 4: Differential EquationsDifferential equations play an integral role in modeling various natural phenomena and engineering systems. This chapter introduces students to ordinary differential equations and their applications. It covers topics such as first-order equations, linear differential equations, and higher-order differential equations. By comprehending the principles governing differential equations, students become equipped to tackle real-world problem-solving scenarios.Chapter 5: Multivariable CalculusMultivariable calculus extends the principles of differentiation and integration to functions of multiple variables. This chapter explores partial derivatives, multiple integrals, and vector calculus. By understanding the intricacies of multivariable calculus, students gain the ability to tackle complex mathematical problems involving multiple variables.Chapter 6: Linear AlgebraLinear algebra provides a powerful framework for solving systems of linear equations and studying vector spaces. This chapter introduces studentsto matrices, determinants, vector spaces, and linear transformations. It explores topics such as eigenvalues and eigenvectors, diagonalization, and the applications of linear algebra in diverse fields, including computer graphics and network analysis.Conclusion:The Higher Mathematics Textbook in its English edition serves as an indispensable resource for students pursuing a deeper understanding of higher mathematics. By covering a wide range of topics and providing clear explanations, examples, and exercises, this textbook equips students with the necessary skills to tackle complex mathematical problems. With its well-organized content and comprehensive approach, the Higher Mathematics Textbook facilitates effective learning and fosters a solid foundation in advanced mathematics.。
数学里的断点英语
数学里的断点英语Discontinuities in MathematicsMathematics is often viewed as a precise and unambiguous discipline where concepts and theorems build upon each other in a seamless progression. However, the history of mathematics is punctuated by moments of discontinuity where new ideas and approaches have disrupted the established order. These discontinuities have played a crucial role in the evolution of mathematical thought, challenging existing paradigms and paving the way for groundbreaking discoveries.One of the earliest and most profound discontinuities in mathematics occurred with the introduction of irrational numbers. The ancient Greek mathematicians had long believed that all quantities could be expressed as the ratio of two integers. This belief was shaken when they encountered the problem of squaring the circle. The square root of 2 emerged as a quantity that could not be represented as a ratio of two integers, and thus was deemed "irrational." This discovery was deeply unsettling, as it challenged thevery foundations of Greek mathematics and the notion of the completeness of the number system.The acceptance of irrational numbers was a slow and arduous process, as it required a fundamental shift in mathematical thinking. It was not until the 19th century that mathematicians such as Richard Dedekind and Georg Cantor developed a rigorous mathematical framework for understanding and working with irrational numbers. This paved the way for the development of more advanced mathematical concepts, such as the real number system and the theory of limits.Another significant discontinuity in mathematics arose with the development of non-Euclidean geometries. For centuries, the Euclidean geometry, with its familiar axioms and theorems, had been the dominant and seemingly unassailable framework for understanding space and spatial relationships. However, in the early 19th century, mathematicians such as Nikolai Lobachevsky and János Bolyai began to explore alternative geometric systems that violated one or more of Euclid's axioms.The emergence of non-Euclidean geometries was a revolutionary concept, as it challenged the long-held belief that Euclidean geometry was the only valid and "true" representation of space. These new geometries, which included hyperbolic and ellipticgeometries, demonstrated that there were multiple consistent and coherent ways of describing the structure of space. This realization had far-reaching implications, not only in mathematics but also in fields such as physics and philosophy.The discontinuity introduced by non-Euclidean geometries was further amplified by the development of relativity theory in the early 20th century. Albert Einstein's theories of special and general relativity incorporated non-Euclidean geometric concepts, demonstrating that the structure of space and time could not be adequately described by the Euclidean framework alone. This marked a profound shift in our understanding of the physical universe, as the previously held notions of absolute space and time were replaced by a more complex and dynamic view of the cosmos.Another significant discontinuity in mathematics is the emergence of set theory and the subsequent development of modern abstract algebra. Prior to the 19th century, mathematics was largely focused on the study of specific mathematical objects, such as numbers, geometric shapes, and functions. However, the work of mathematicians like Georg Cantor and Emmy Noether led to the recognition that mathematics could be studied at a more fundamental level by examining the underlying structures and relationships between these objects.The introduction of set theory provided a unifying framework for understanding the various branches of mathematics, from number theory to topology. This shift towards a more abstract and structural approach to mathematics opened up new avenues of exploration and allowed for the development of powerful new mathematical tools and techniques. The rise of abstract algebra, which focuses on the study of algebraic structures such as groups, rings, and fields, further solidified this discontinuity, as it moved away from the traditional focus on specific numerical operations and towards a more general and versatile understanding of mathematical structures.The discontinuities in mathematics have not been limited to the conceptual realm; they have also manifested in the development of new mathematical techniques and technologies. The invention of calculus by Newton and Leibniz in the 17th century was a profound discontinuity, as it introduced a powerful new set of tools for analyzing change and motion, which had previously been beyond the reach of classical mathematics.Similarly, the development of digital computers and the subsequent rise of computational mathematics have ushered in a new era of mathematical exploration. Computers have enabled the rapid processing of vast amounts of data, the simulation of complex mathematical models, and the exploration of previously intractable problems. This has led to the emergence of new fields, such ascomputational biology, computational finance, and data science, which have challenged traditional mathematical approaches and introduced new ways of thinking about and solving problems.The discontinuities in mathematics have not been without their challenges and controversies. The acceptance of irrational numbers, non-Euclidean geometries, and abstract algebraic structures was often met with resistance and skepticism, as they challenged the established norms and required a fundamental shift in mathematical thinking. Similarly, the rise of computational mathematics has raised concerns about the role of human intuition and the potential for computer-generated results to be misinterpreted or misused.However, despite these challenges, the discontinuities in mathematics have been essential for the discipline's continued growth and evolution. They have pushed the boundaries of mathematical knowledge, challenged existing paradigms, and opened up new avenues of exploration. By embracing these discontinuities and the opportunities they present, mathematicians have been able to uncover deep insights into the nature of the universe and the fundamental structures that govern it.In conclusion, the discontinuities in mathematics have been a crucial driver of the discipline's progress, challenging our assumptions and pushing us to rethink our understanding of the world around us.From the introduction of irrational numbers to the development of non-Euclidean geometries and abstract algebra, these moments of discontinuity have played a vital role in shaping the course of mathematical thought. As we continue to explore the frontiers of mathematics, we can expect to encounter more such discontinuities, each one offering the potential for new discoveries and a deeper understanding of the mathematical universe.。
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a rX iv:mat h /3464v3[mat h.CT]2J un23ON THE NOTION OF GEOMETRIC REALIZATION VLADIMIR DRINFELD To Borya Feigin on his 50th birthday Abstract.We explain why geometric realization commutes with Carte-sian products and why the geometric realization of a simplicial set (resp.cyclic set)is equipped with an action of the group of orientation pre-serving homeomorphisms of the segment [0,1](resp.the circle).2000Math.Subj.Class.18G30,55U10,19D55Key words:simplicial set,cyclic set,geometric realization,cyclic homology,fiber functor The next simplest curve is a circle.Even so simple a figure as this has given rise to so many and such profound investigations that they could constitute a course by themselves.D.Hilbert and S.Cohn-Vossen,Geometry and the imagination (Anschauliche Geometrie),ch.1,page 1.In this note there are no theorems,its only goal is to clarify the notion of geometric realization for simplicial sets [14,13,6,7]and cyclic sets [4,2,11].We reformulate the definitions so that the following facts become obvious:(i)geometric realization commutes with finite projective limits (e.g.,with Cartesian products);(ii)the geometric realization of a simplicial set (resp.cyclic set)is equipped with an action of the group of orientation preserving homeomor-phisms of the segment I :=[0,1](resp.the circle S 1).In the traditional approach [14,13,6,7,4,2,11]these statements aretheorems,and understanding their proofs requires some efforts.Example.To a small category C there corresponds a simplicial set NC (the nerve of C )and a cyclic set N cyc C (the cyclic nerve).It follows from our formula (1.1)that a point of the geometric realization |NC |is a piecewise constant functor I →C .Here it is assumed that the category structure on I comes from the standard order on I ,and the definition of piecewise constant functor is explained in (1.4).In §3we give a similar description of |N cyc C |in which I is replaced by S 1.2VLADIMIR DRINFELDConventions.The set of non-negative integers is denoted by Z+.Unless stated otherwise,an ordered set I is always equipped with the category structure such that the number of morphisms from i∈I to j∈I equals1 if i≤j and0otherwise.The author thanks D.Arinkin,A.Beilinson,D.Grayson,and P.May for useful remarks and references.The epigraph is due to R.Bezrukavnikov.Having written this article I learned that a very similar approach had been developed by A.Besser[1]and D.Grayson[8].It is used in[8]to treat not only products of simplicial sets but also Quillen’s edgewise subdivision.1.Simplicial setsPut I:=[0,1].This is an ordered set and therefore a category.Recall that a simplicial set is a functor X:∆◦→Sets,where∆is the category whose objects are the ordered sets[n]:={0,...,n},n∈Z+,and whose morphisms are nondecreasing maps.One can extend such X to a functor∆◦big→Sets,where∆big is the category of non-emptyfinite linearly ordered sets.Such an extension is unique up to unique isomorphism.We define the geometric realization of a simplicial set X to be the setX(π0(I\F)),(1.1)|X|:=lim−→Fwhere F runs through the set of allfinite subsets of I andπ0(I\F)is equipped with the natural order(soπ0(I\F)∈∆big is a quotient of I\F). At this point|X|is viewed merely as a set,the definition of the topology on |X|will be recalled later.One can also rewrite(1.1)as),(1.2)|X|:=X(I←is the pro-object of∆big which is the projective limit of the objects where I←π0(I\F)∈∆big over allfinite subsets F⊂I.It immediately follows from the definition that geometric realization com-mutes withfinite projective limits(e.g.,the map|X×Y|→|X|×|Y|is bijective)and that the group Aut I of orientation preserving homeomor-phisms of I acts on|X|.Example.The geometric realization|[n]|of[n](i.e.,of the functor∆◦→Sets represented by[n])is the set of piecewise constant nondecreasing func-tions f:I→[n]modulo the following equivalence relation:f1∼f2if f1 and f2are equal outside afinite set.We identify|[n]|with the standard simplex(1.3)Sim n:={(x1,...,x n)∈I n|x1≤...≤x n}(a function f:I→[n]such that f(x)=i for x i<x<x i+1is identified with(x1,...,x n)∈∆n;here we assume that x0:=0,x n+1:=n).ON THE NOTION OF GEOMETRIC REALIZATION3 Recall that the nerve NC of a small category C is defined by NC(δ):= Funct(δ,C)forδ∈∆big,where Funct(δ,C)is the set of functorsδ→C.So |NC|is the set of piecewise constant functors I→C,more preciselyFunct(π0(I\F)),C).(1.4)|NC|:=lim−→FTo pass to the usual definition of|X|use the canonical representation of a functor X:∆◦→Sets as an inductive limit of representable ones,namelyΦ,(1.5)X=lim−→∆/Xwhere∆/X is the category of pairs consisting ofδ∈∆and a morphism ξ:δ→X(i.e.,an elementξ∈X(δ)),andΦis the functor from∆/X to the category of simplicial sets that sends(δ,ξ)toδ.As geometric realization commutes with inductive limits and the functor n→|[n]|identifies with the usual“standard simplex”functor Sim:∆→Sets we get the usual formula(1.6)|X|=lim|Φ|,|Φ|(δ,ξ):=standard simplex|δ|,−→∆/Xwhich can be rewritten as(1.7)|X|=X×∆Sim,where X×∆Sim denotes the coend of the bifunctor X×Sim:∆◦×∆→Sets (which is a kind of“tensor product of the right∆-module X and the left ∆-module Sim”,see§IX.6of[12]or Appendix B.7of[11]for more details on the×∆bifunctor).One uses the topology on the standard simplices and either(1.6)or(1.7) to define the standard topology on the geometric realization,then one checks that geometric realization considered as a functor from the category of sim-plicial sets to that of compactly generated spaces commutes withfinite pro-jective limits;see[6,7,13]for more details.One can describe the topology on|X|using the following metric.To define itfirst choose a measureµon I such that the measure of every point is zero and the measure of every non-empty open set is non-zero.For afinite F⊂I denote byµF the measure onπ0(I\F)induced byµ.If u,v∈X(π0(I\F)) define the distance dµ(u,v)to be the minimum ofµF(π0(I\F)\A)for all subsets A⊂π0(I\F)such that the images of u and v in X(A)are equal.If F′⊃F and u′,v′are the images of u,v in X(π0(I\F))then dµ(u′,v′)=dµ(u,v),so we get a well defined metric dµon|X|.It is easy to see that dµis continuous for the standard topology of|X|.It follows that this topology is Hausdorff.It also follows that if X isfinite the standard topology coincides with the one defined by the metric dµ.For any X the topological space|X|is the direct limit of the geometric realizations of thefinite subsets of X.4VLADIMIR DRINFELDRemarks.(i)Already[14]and[6]suggest that in the theory of simplicial sets one should use the realization(1.3)of the standard simplex rather than the“baricentric”realization y0+...y n=1,y i≥0.This is also natural in view of the theory of iterated integrals[10].(ii)Here is a way to think of the pro-object I←from(1.2).Consider thefollowing category∇.The objects of∇arefinite linearly ordered sets J oforder≥2(so the minimal element¯0=¯0J∈J does not equal the maximalelement¯1=¯1J∈J).The morphisms of∇are nondecreasing maps f:J→J′,J,J′∈Ob∇,such that f(¯0J)=¯0J′and f(¯1J)=¯1J′.It is essentially explained in§III.1.1of[6]that∇is antiequivalent to∆.More precisely,thefunctor∆◦big→∇that sends I∈∆big to the set I∗:=Hom∆big(I,{0,1}) equipped with the natural(“pointwise”)order is an equivalence.To see this first notice that it has a left adjoint(namely the functor∇→∆◦big that sends J∈∇to the set J∗:=Hom∇(J,{0,1})equipped with the natural order)and then show that for all I∈Ob∆big and J∈Ob∇the adjunction morphisms I→I∗∗and J→J∗∗are isomorphisms.The functor I→I∗induces an antiequivalence between the category ofpro-objects of∆big and that of ind-objects of∇.The latter category iden-tifies with the category∇∞of all linearly ordered sets J having a minimalelement¯0=¯0J∈J and a maximal element¯1=¯1J∈J such that¯0=¯1(the morphisms of this category are nondecreasing maps such that¯0→¯0and ¯1→¯1).So one can consider I=[0,1]as an ind-object of∇.It is easy to see that the pro-object I←from(1.2)corresponds to I=[0,1]viewed as an ind-object of∇.Therefore(1.2)implies Theorem III.1.3of[6],which says that the group of automorphisms of the geometric realization functor from the category of simplicial sets to that of sets equals the group of automor-phisms of the ordered set I(which is the same as the group of orientation preserving homeomorphisms of I).(iii)Formula(1.2)is natural in view of the general theory offiber functorson a topos developed in§6.8of[9](especially cf.§6.8.6-§6.8.7).We need only the case of a topos of the formˆC:=the category of presheaves of sets on a small category C(i.e.,ˆC is the category of functors C◦→Sets).According to[9],a functorˆC→Sets is said to be afiber functor if it commutes with finite projective limits and arbitrary inductive limits.Let Fib(ˆC)denote the full subcategory of the category of functorsˆC→Sets that consists offiber functors.It is closed under inductive limits,so the functor F:C◦→Fib(ˆC) defined by(F c)(X):=X(c),c∈C,X∈ˆC,canonically extends to a functor(Pro-C)◦=Ind-C◦→Fib(ˆC),where Pro-C(resp.Ind-C)denotes the category of pro-objects(resp.ind-objects).According to§6.8of[9], the functor(Pro-C)◦→Fib(ˆC)is an equivalence.As explained in[9],the quasi-inverse functor is constructed as follows.LetΦ:ˆC→Sets be afiber functor.Let I be the category of pairs(c,ξ),c∈C◦,ξ∈Φ(c).According to[9],I is afiltering category,and the functor I→C◦defines the desiredON THE NOTION OF GEOMETRIC REALIZATION5 pro-object of C.To show that I isfiltering the authors of[9]first notice that the categoryˆI of pairs(c,ξ),c∈ˆC◦,ξ∈Φ(c),isfiltering(because Φcommutes withfinite projective limits)and then deduce from this that I is alsofiltering using the fact thatΦcommutes with direct limits(this fact and the analog of formula(1.5)with∆replaced by C imply that every object ofˆI can be mapped to an object of I).2.Z+-categories and the categoryΛIn the theory of cyclic sets the role of I=[0,1]and[n]∈∆is played by certain Z+-categories.We define a Z+-category to be a category C equipped with a morphism of monoids Z+→End id C(the operation in Z+is addition).In other words, each object c∈C should be equipped with an endomorphism1c so thatf1c1=1c2f for every morphism f:c1→c2in C.If C,C′are Z+-categoriesthen a Z+-functor C→C′is a functorΦ:C→C′such thatΦ(1c)=1Φ(c) for all c∈C.A Z+-isomorphism is a Z+-functor which is an isomorphism.A full subcategory of a Z+-category is a Z+-category.Examples.In this work we will work only with the following examples of Z+-categories.All of them belong to the class described in Proposition2.1.1.Unless stated otherwise,we will always equip S1:=R/Z with the following structure of Z+-category:morphisms from c∈S1to c′∈S1are homotopy classes of“oriented”paths I→S1(i.e.,of those paths which can be represented as a composition I f→R→R/Z=S1with f nondecreasing), and the composition of morphisms is usual(so our category is a subcategory of the fundamental groupoid of S1,and1x is the homotopy class of a degree 1path from x∈S1to x).The category S1is generated by morphisms−→xy,x=y,where−→xy is the homotopy class of the shortest oriented path from x∈S1to y∈S1.If x=y then−→yx−→xy=1x,and if x,y,z∈S1are distinct points in the correct cyclic order then−→yz−→xy=−→xz.This is a complete set of relations between the generators−→xy and1x.2.For every n∈Z+embed[n]:={0,...,n}into S1by k→k/(n+1) mod Z.The set[n]equipped with the Z+-category structure induced from S1will be denoted by[n]cyc.3.For every non-emptyfinite F⊂S1there is a unique Z+-category C equipped with a Z+-functorπF:S1\F→C whose restriction to any system of representatives R of connected components of S1\F is an isomorphism (R is considered as a full subcategory of S1).We will denote C simply byπ0(S1\F).Of course,π0(S1\F)is(non-canonically)Z+-isomorphic to[n]cyc,n=Card F−1.If F⊂F′there is a unique Z+-functorΦ:π0(S1\F′)→π0(S1\F)such thatΦπF′=πF.4.Given a small category A we will define a Z+-category A cyc with the same set of objects(in fact,we will use this construction only for categories6VLADIMIR DRINFELDthat come from linearly ordered sets,and it is not clear if it is reasonable in general).By definition,A cyc is generated by A and new generators f∗corresponding to f∈Mor A.More precisely,if f is a morphism c1→c2 then f∗is a morphism c2→c1(intuitively,f∗=1f−1=f−11).Defining relations in A cyc:(gf)∗g=f∗,f(gf)∗=g∗.A cyc becomes a Z+-category if one defines1c:c→c by1c:=(id c)∗.To see this notice that for everyA-morphism f:c1→c2one has f1c1=ff∗f=1c2f and f∗1c2=f∗ff∗=1c1f∗(because f∗f=(f id c1)∗f=(id c1)∗=1c1and ff∗=f(id c2f)∗=(id c2)∗=1c2).Clearly A→A cyc is a functor from the category of smallcategories to that of small Z+-categories.If A is the ordered set[n],n∈Z+,considered as a category then A cycidentifies with the Z+-category[n]cyc from Example2.If F⊂I:=[0,1]is afinite set containg{0,1}and A is the ordered setπ0(I\F)considered as acategory then A cyc identifies with the Z+-categoryπ0(S1\¯F)from Example 3,where¯F⊂S1=R/Z is the image of F.If A is the ordered set[0,1)thenA cyc identifies with S1.Proposition2.1.The following properties of a small Z+-category C areequivalent:(i)C is Z+-isomorphic to A cyc for some linearly ordered set A;(ii)every non-emptyfinite full subcategory of C is Z+-isomorphic to[n]cycfor some n∈Z+.(iii)every full subcategory of C with n objects,n∈{1,2},is Z+-isomor-phic to[n−1]cyc.Proof.It suffices to show that(iii)⇒(i).For every x,y∈C there existsf:x→y such that every g:x→y is uniquely representable as g=f1n x,n∈Z+.This f(which is clearly unique)will be denoted by f xy.Nowfixc∈C.For every x,y∈C one has f xy f cx=f cy1m c for some m=m(x,y)∈Z+.Clearly m(x,y)+m(y,z)=m(x,z)and m(x,y)+m(y,x)=1.Let A⊂C be the subcategory with Ob A=Ob C whose morphisms are thef xy’s corresponding to those x,y for which m(x,y)=0.Then A is a linearlyordered set and A cyc=C. LetΛhuge be the category whose objects are small Z+-categories satisfying the conditions of Proposition2.1and whose morphisms are Z+-functors.Let Λ(resp.Λbig)be the full subcategory ofΛhuge formed by the Z+-categories [n]cyc,n∈Z+(resp.by Z+-categories which are Z+-isomorphic to[n]cyc for some n∈Z+).The following remarks are used only at the end of§3.Remarks.(i)A.Connes[3]showed thatΛ◦is equivalent toΛ(see alsoProposition6.1.11of[11]).A more understandable proof of this equiva-lence was given by Elmendorf[5].Here is a modification of it based onan idea of D.Arinkin.If C1,C2are Z+-categories and C1is small thenZ+-Funct(C1,C2)(i.e.,the full subcategory of Z+-functors in the category of functors C1→C2)has an obvious structure of Z+-category.So we getON THE NOTION OF GEOMETRIC REALIZATION7 a functor(Λhuge)◦→Λhuge defined by C→C∗:=Z+-Funct(C,[0]cyc),where[0]cyc is the Z+-category from Example2(it has a single object0,and End0=Z+).As([0]cyc)◦=[0]cyc one has Z+-Funct(C1,C∗2)=(Z+-Funct(C2,C1))◦,so one gets a canonical Z+-functor F C:C→C∗∗.Finally,if C=[n]cyc then C∗is Z+-isomorphic to[n]cyc and F C is anisomorphism.So the functor(Λhuge)◦→Λhuge induces an equivalence(Λbig)◦→Λbig.(ii)It is easy to see that in the situation of Example3the Z+-categoryF∗defined in Remark(i)canonically identifies withπ0(S1\F).Here is an abstract way to identify them.Let Z+-tors denote the Z+-category ofZ+-torsors,i.e.,Z+-sets isomorphic to Z+.For C∈Λhuge one has the Z+-bifunctor C◦×C→Z+-tors defined by(c,c′)→Mor(c,c′).Composing itwith the unique Z+-functor Z+-tors→[0]cyc one gets a Z+-functorΦ=ΦC:C→C∗.One also has the Z+-functorΦ′=(ΦC◦)◦:C→((C◦)∗)◦=C∗(which is usually different fromΦC).In the situation of Example3the−→(S1)∗→F∗compositions S1\F֒→S1Φ−→(S1)∗→F∗and S1\F֒→S1Φ′are equal.This Z+-functor can be taken asπF(see Example3).(iii)The category of ind-objects ofΛbig(or ofΛ)identifies withΛhuge.The category of pro-objects ofΛbig can be identified with(Λhuge)◦using the equivalence(Λbig)◦∼−→Λbig from Remark(i).It gives a class of equivalences −→Λeach two of which are related by a unique isomorphism.Λ◦∼3.Cyclic setsIn§2we defined the categoriesΛhuge⊃Λbig⊃Λ.According to A.Connes,a cyclic set is a functor X:Λ◦→Sets.One can extend such X to a func-torΛ◦big→Sets,and the extension is unique up to unique isomorphism. The functor C→C cyc from Example4of§2can be considered as a func-tor∆big→Λing this functor one can consider any cyclic set as a simplicial set.The geometric realization|X|of a cyclic set X is defined in[6,7,13]tobe the geometric realization of X considered as a simplicial set.By(3.1),this means that|X|is the direct limit of X((π0(I\F))cyc),where F runsthrough the set of allfinite subsets of I:=[0,1].We will get the sameanswer if F runs only through the set offinite subsets of I containing0and1.For such F one has(π0(I\F))cyc=π0(S1\¯F),where¯F⊂S1=R/Z is the image of F(see Example4of§2).SoX(π0(S1\F)),(3.1)|X|:=lim−→Fwhere F runs through the set of all non-emptyfinite subsets of S1andπ0(S1\F)is equipped with the Z+-category structure from Example3of§2.The reader may prefer to rewrite(3.1)as),(3.2)|X|:=X(S1←8VLADIMIR DRINFELDwhere S1is the pro-object ofΛbig which is the projective limit of the objects ←π0(S1\F)∈Λbig over allfinite subsets F⊂S1.Formula(3.1)or(3.2)makes it clear that the group Aut S1of orientation preserving homeomorphisms of S1acts on|X|.In particular,the group of rotations SO(2)acts.Let us recall that the cyclic nerve N cyc C of a small category C is defined by N cyc C(λ)=Funct(λ,C)forλ∈Λbig,where Funct(λ,C)is the set of functorsλ→C.So|N cyc C|is the set of piecewise constant functors S1→C, more preciselyFunct(π0(S1\F)),C).(3.3)|N cyc C|:=lim−→FExample.The geometric realization of[n cyc](i.e.,of the functorΛ◦→Sets represented by[n cyc])is the set of piecewise constant Z+-functors S1→[n cyc].It identifies with the set Sim n cyc of points(x0,...,x n)∈(S1)n+1 such that x0,...,x n are in the correct cyclic order(the points are not re-quired to be distinct):to x0,...,x n∈Sim n cyc we associate the Z+-functor Φ:S1\{x0,...,x n}→[n cyc]such that if x,y∈S1\{x0,...,x n}andxy)is x,x i,...,x j−1,y,x j,...,x i−1are in the correct cyclic order thenΦ(−→the product of the indecomposible morphisms i→i+1→...→j.Clearly Sim n cyc is homeomorphic to the product of S1and the n-dimensional sim-plex.If X is a cyclic set then|X|is the geometric realization of X considered as a simplicial set,so it is equipped with a topology.The action of Aut S1 on|X|is continuous if Aut S1is equipped with the compact-open topology. This can be proved using the metric dµon|X|(see§1)or the following ar-gument.Notice that similarly to(1.7)and(1.6)one has canonical bijections(3.4)|X|=X×ΛΨ,ΨX,(3.5)|X|=lim−→Λ/XwhereΨ:Λ→Sets,Ψ(λ):=the geometric realization of the functor repre-sented byλ,andΨX is the restriction ofΨtoΛ/X.The bijections(3.4) and(3.5)are ing any of them we see that it suffices to show that the action of Aut S1on the geometric realization|[n cyc]|of[n cyc] (i.e.,of the functorΛ◦→Sets represented by[n cyc])is continuous.It is easy−→Sim n cyc is a homeomorphism,and to see that the above bijection|[n cyc]|∼the action of Aut S1on Sim n cyc is continuous,QED.from(3.2) Remarks.(i)By Remark(iii)at the end of§2,the pro-object S1←can be considered as an object ofΛhuge.This object canonically identifies with S1(e.g.,one can use Remark(iii)from§2).ON THE NOTION OF GEOMETRIC REALIZATION9 (ii)The group of automorphisms of the geometric realization functor from the category of cyclic sets to that of sets equals Aut S1.This follows,e.g., from the previous remark.References[1]A.Besser,A simple approach to geometric realization of simplicial and cyclic sets.E-print available at http://www.math.bgu.ac.il/˜bessera[2]D.Burghelea,Z.Fiedorowicz,Cyclic homology and algebraic K-theory of spaces.II.Topology,vol.25(1986),no.3,303–317.[3]A.Connes,Cohomologie cyclique et foncteurs Ext n.C.R.Acad.Sci.Paris S´e r.IMath.,vol.296(1983),no.23,953–958.[4]W.G.Dwyer,M.J.Hopkins,D.M.Kan,The homotopy theory of cyclic sets.Trans.Amer.Math.Soc.,vol.291(1985),no.1,281–289.[5]A.D.Elmendorf,A simple formula for cyclic duality.Proc.Amer.Math.Soc.,vol.118(1993),no.3,709–711.[6]P.Gabriel,M.Zisman,Calculus of fractions and homotopy theory.Ergebnisse derMathematik und ihrer Grenzgebiete,Band35Springer-Verlag,New York,1967.Rus-sian translation:“Mir”,Moscow,1971.[7]P.Goerss,J.F.Jardine,Simplicial homotopy theory.Progress in Mathematics,174.Birkh¨a user Verlag,Basel,1999.[8]D.Grayson,Algebraic K-theory,Lecture notes available at/˜dan/Courses/2003/Spring/416/GraysonKtheory.ps [9]A.Grothendieck et J.L.Verdier,Topos.In:SGA4,tome1,Lecture Notes in Mathe-matics,vol.269,299–519.Springer-Verlag,1972.[10]R.M.Hain,Iterated integrals and homotopy periods.Mem.Amer.Math.Soc.,vol.47(1984),no.291.[11]J-L.Loday,Cyclic homology.Grundlehren der Mathematischen Wissenschaften,301.Springer-Verlag,Berlin,1998.[12]S.Mac Lane,Categories for the working mathematician.Graduate Texts in Mathe-matics,5.Springer-Verlag,1998.[13]J-P.May,Simplicial objects in algebraic topology.Van Nostrand,Princeton-Toronto-London,1967.University of Chicago Press,1992.[14]nor,The geometric realization of a semi-simplicial complex.Ann.of Math.(2),vol.65(1957),no.2,357–362.Dept.of Math.,Univ.of Chicago,5734University Ave.,Chicago,IL60637, USAE-mail address:drinfeld@。