Geometrical Hyperbolic Systems for General Relativity and Gauge

Art and structural engineering-Art of structural engineeringMotro René 1SummaryBeyond the classical dispute between architects and engineers, it is a matter of fact that art and engineering are not only compatible, but also for some cases indissociable. This communication aims to contribute to the debate. In the first part the experience of artists is exemplified in terms of influence on structural engineering : Pevsner and Snelson opened the way to complex surfaces and innovative structural composition. Following David P. Billington proposal for a so called “Structural Art”, some few examples are described enhancing the symbolic dimension of realizations, whose morphology reflects artistic processes. Some explanations provided by cognitive sciences help to understand the mental process of designers and the required conditions for a creative process. Historic examples of artistic movements like “Constructivism”, “Bauhaus” and the experiment of “Black Mountain College” illustrate the benefice of crossing experiences of designers, whatever can be their field.KeywordsArt, structural engineering, structural compositionThemeStructural and architectural design1.IntroductionBeyond the classical dispute between the architect and the designer, there is another one debate which is of interest for structural engineers: can we speak of Art when Structural Engineering ends with some solutions which are recognized as pieces of art, like it is for the Eiffel Tower, and more recently the Millau Viaduct ? This communication intends to provide some arguments following previous publications reflecting our interrogations (Motro [1]), and willingness to associate in the same project at the training level architects, artists and engineering (Aanhaanen [2]). We firstly present some cases enhancing a close relationship between art and structural engineering, and secondly we describe some of the features and conditions of situations where structural engineering becomes an art.2.Art and structural engineering2.1.Pevsner and Xenakis : from constructivism to Pavillon Philips in BrusselsConstructivism is an artistic movement born in Russia at the beginning of the XX° century. Brothers Pevsner and Gabo wrote its manifesto. This movement proclaims a geometrical construction of the space, using especially elements such as the circle, the rectangle and the straight line. This way of thinking adapts itself also well to the sculpture as to the design even in the architecture. A somewhat recent exhibition held at the Guggenheim Museum, “The Russian and Soviet Avant-Garde, 1915-1932”, gives more details on the work of constructivists who organised their first exhibition manifestation, the Obmokhu (the Society of Young Artists) in May 1921. Rodchenko, one of the constructivists, claimed in January 1921 (Lodder [3]):“All new approaches to art arise from technology and engineering and move towards organization and construction”.Artists and engineers are indistinctly members of this group. Vladimir Grigorievitch Choukhov, a famous Russian engineer was member of the Constructivism. He developed and achieved several hyperbolic towers. In the field of metallic structures design (Figure 1 A). Choukhov is one of first to develop practical methods of calculation of the efforts and the elastic deformations of the beams, the shells and the membranes. His design is based on ruled surfaces.Following the same geometrical principle Antoine Pevsner realized many sculptures based on ruled surfaces and sometimes developable (Figure 1-B). Such artistic achievments opened the way to double curved surfaces generated by straight lines.A BFigure 1: Constructivism : A-Hyperbolic Tower Moscou 1922 (Choukhov) B-Developable surface (Pevsner)In 1958, for the Universal Exhibition in Brussels, Yannis Xenakis designs the so-called “Pavilion Philips”. Yannis Xenakis, mathematician, musician and architect worked at that times with Le Corbusier. It is a matter of fact that the two men disagreed during this event, but this dispute is beyond our text. What we want to underline is the formal analoghat we want to underline is the formal analog between this Pavilion and the sculptures presented by Pevsner. It is obvious that the geometrical design, based on ruled surfaces (Figure 2 A) is clearly in the line of Pevsner’s sculptures. The sketch of Figure 2 B, even if difficult to read has been drawned by Xenakis (his name is mentioned on it), the third illustration (Figure 2 C) represents the completed structure.A B CFigure 2 : Pavillon Philips : A Ruled surface - B Sketch by Xenakis - C RealizationIt is interesting to note that the double negative curvature surfaces result from the assembly of double negative curved paving stones poured on sand on the ground of a rough dimension of 1 m 50 aside. These paving stones are 5 cms in thickness ; they are supported by a double network of prestressed cables 8 mm in thickness. Following this realization, and keeping the same concept of surfaces with negative double curvature, Yannis Xenakis will design steel and membrane system for his famous “Polytope” raised behind the Pompidou Center in Paris.2.2.Kenneth Snelson, Forces made visibleKenneth Snelson is an artist who is at the key point between art and structural engineering. Many papers have been devoted to his pionneer work in the field of tensegrity structures. His controversy with Buckminster Fuller he met in Black Mountain College in the early fifties is beyond the scope of our paper. He himself explained his view in a public letter inserted as annex of the book that I devoted to tensegrity (Motro [4]). The most important thing to note is that he had a pure artistic behaviour and he reached a first concretisation of the tensegrity concept by working successively on three sculptures ending in the well known “Double X”. Every “X” component described in his patent is strut –like element. Using an assembly of three (Figure 3) gives access to the classical simplex (Motro et al [5]).Figure 3 : Simplex generation by assembly of three “X” componentsA BFigure 4 : Snelson sculptures A Easy Land, Boston – B Tensegrity TowerIn 2009 Snelson had an exhibition in Marlborough Gallery, Chelsea, New York. The title was “K enneth Snelson Forces Made Visible, and this is also the title of the book edited for this opportunity (Hartney [6]). The best title for the work of this artist who is able to make forces visible. Forces are a mechanical concept useful for engineers who want to size their structures and they are by nature invisible. On the other hand forms are visible and measurable, and they are the product of the artistic process. Why are forces made visible? Some keypoints may be put forward:x By differenciating clearly cables and struts, Snelson’s sculptures provide an information on whether tension or compression is present..This is not the same for classical reticulate systems x If the level of tension and/or compression can be qualitatively evaluated according to the external diameter size of components, it is insufficient since this level is depending upon the material, and the thickness of tubes, or the arrangement of cables.x The very specific structural composition surprises and fascinates everyone seeing them for the first time : struts seem to float in the air. And this is also a key point, since people, and engineers morethan the others, are surprised by this new kind of flow of forces. They are accoutumated to gravity effects, and in this case gravity seems to be absent. The artist provokes interrogations by submitting an unknown process for transmitting forces.x Last but not least : art is a source of emotional feeling. In case of tensegrity systems people feel that at every end of each strut, cables contribute to the equilibrium. And it cannot be possible without an amount of stress of tension in cables, and compression in struts: struts are “lifted” inside the structure by a continuum of cables. But or course it is not easy to understand how all these forces are distributed, we only know that the whole is in equilibrium, stressed and stableFinally it can be said that the artistic work by Snelson obliges the structural engineers to question their structural approach, and to enrich it.3..Art of structural engineering3.1 StructuralartSome engineers brought up their practice at the level of the art. Several were celebrated during the exhibition The Art of Engineer, Builder, Contractor, Inventor, held in Paris 1997 (Picon [7]).“The Tower and the Bridge” ( Billington [8]) has as subtitle “The New Art of Structural Engineering”. In this book many famous engineers are presented in their artistic way of designing: Thomas Telford and Gustave Eiffel, Robert Maillart, Felix Candela and Heinz Isler among others. Let Billington speak about this structural art:“The conservative, plodding, hip-booted technicians might be, as the architect Le Corbusier said, “healthy and virile, active and useful, balance and happy in their work, but only the architect, by his arrangement of forms realizes an order which is a pure creation of his spirit…it is then that we experience the sense of beauty”. The belief that the happy engineer, like the noble savage, gives us useful things but only the architect can make them into art is one that ignores the centrality of aesthetics to the structural artist”Following this affirmation he describes the three “dimensions of structure” : scientific, social and symbolic. The symbolic dimension is closely related to aesthetics. These classical virtues Firmitas,Utilitas,Venustas enhanced by Vitruve could appear as an old history without any interest. Nevertheless if “Venustas” is recognized as one of the characteristics of artistic manifestation, some engineers practiced a real art.Figure 5 : Garabit Viaduct – Gustave Eiffel3.2 Morphology in questionThere is always a close relationship between the resulting morphology of a design process and the personality of the designer. The following quotation is generally attributed to Victor Hugo :“La forme c’est le fond qui remonte à la surface”A direct translation would be without meaning. The idea is that every form is somewhat the visible result of an invisible content.The external appearance is insufficient to conclude, and it is a matter of fact that in the case of the Millau Viaduct (Figure 6), known as Foster’s project, the resulting morphology is strongly related to Michel Virlogeux’s work. We can perhaps claim that Virlogeux has been the soul of this project. The soul assesses the expression of the designer through the resulting morphology among other parameters. At the beginning of the process there is at least one symbolic choice that is not related to scientific or social dimensions. Michel Virlogeux said that in this case the main idea was to design a viaduct not to span the river Tarn, but to span the entire valley.Figure 6 : Millau Viaduct – Foster and VirlogeuxFor other cases the morphologic appearance allows one to identify the designer. Isler’s (Figure 7) morphologies are characteristics of their strong expression as designers, but the morphology is also the result of a strong coupling between form and forces: they are somewhat “funicular shapes” as Gaudi’s Colonia Güell.Figure 7 : Concrete Shell by Isler and experimental funicular model3.3 Which way to Art of Structural Engineering?3.3.1 New design situationsNowadays, the scientific aspects of design are helped by the increasing power of available numerical tools, but if designers can use them at the different stages of the process, since the initial idea to the realization of the project, they cannot reduce their work to manipulation of tools. They are now freer for expressing their own personality by a continuous process, and as artists can experiment, they may simulate by prototypes and numerical modeling the continuous materialization of their project. Besides, let say, the classical artists they have to take into specificdimensions as claimed by Billington: scientific (in terms of mechanics of material and structures), and social (adequacy with the ongoing progresses of technology, the cost necessities, and the apparition of new constraints like environmental ones). Nevertheless some of them are sufficiently imaginative and also creative to submit new solutions evolving from their experience and meeting the actual constraints.If the design process in structural engineering is governed by the scientific dimension, true designers do not provide the same solution to a given problem. Their own experience and way of thinking are conditioning the quality of their proposal. There are many differences with classical manifestations of art like size of construction and the necessity of permanence in terms of security, but the mental process is of the same kind. Similarly training conditions are also very important, and common training with artists and architects may contribute to increase the level of their art of engineering.3.3.2 Contribution of the cognitive sciencesAccording to the preceding remarks, it appeared interesting to try to identify some characteristics of the mental behavior of a designer regardless of his training as architect, engineer or artist. Taking advantage of cognitive sciences results we could investigate more precisely some aspects of the design process.One major issue is related to the mental representation for the designer of an existing or a projected object. Perception, and conception are two faces of this issue.The perception of real 3D objects is firstly the source of a mental representation in the so called “working memory” of the designer. These mental representations are progressively stored in the long term memory of the designer. The link between mental and physical worlds, during design process and morphogenesis, takes advantage of “a knowledge tank” (so called “Long term memory”). This tank is filled step by step during designer’s life according to his perception of the physical world and his own skills. During the conceptual design phase mental representations of previously perceived objects arise in the working memory of the designer,coming from his long term memory. This long term memory is like “a memory tank” which is filled by perception operations and training, and which is also dependent of heredity, culture, training, travels, discussions…An iterative work is then operated and requires a progressive materialization of the projected solutions, generally by means of 2D representations (computer’s screen or sheet of paper) and/or by means of physical models. This iterative process is a sequence of problem solvings in working memory. They give access to solutions (hypotheses) which are analyzed and compared with known solutions stored in the long term memory . This long term memory is continuously enriched by perceptions and training. A major point is that the designer is not always filling his long term memory with elements, he is also building links between the information of increasing complexity, he is building procedures and he memorizes these procedures.On the basis of this cognitive approach undertook by Silvestri [9], and associated with an experimental study (concerning more than thirty people), we have some information useful for a better understanding of the mental process of designers.3.3.3 Creative movementsWe evoked at the beginning of this paper the role of Constructivism. There was also a similar attempt with “De Stijl” with Pietr Mondrian and Theo Van Doesburg (Figure 8). A true and free dialog between architects, engineers and artists was clearly fruitful in other places where they worked together like the Bauhaus or Black Mountain College inviting designers to exchange their thoughts for a better mutual understanding.In Black Mountain College (Figure 9), where Snelson was sculptor student in 1948, the experimentation was the main adopted principle as it can be understood by reading the essay by Diaz [10] who writes :These three models of experiment- the methodological testing of the appearance and construction of form in the interest of designing new visual experiences (Albers), the organization of aleatory processes and the anarchical acceptance of accident (Cage), and “comprehensive, anticipatory design science’ that propels, teleologically current limited understanding towards a finite totality of universal experience (Fuller)- represent important incipient yet disparate directions of post-war art practice, elements of which would be sampled, if not wholly adopted, by Black Mountain students and subsequent practitioners.Again, like for the famous Bauhaus, cross pollination between creative people creates the best condition for improving the experience of every one, and enriching his knowledge gained by very separate practices that have in common creativity and artistic attitude in common. Generally engineers who shared this kind of training have more chance to reach the Art of Structural Engineering.A BFigure 8 : A Piet Mondrian painting – B Project by Theo Van Doesburg.Figure 9 : Buckminster Fuller teaching at Black Moutain College.ConclusionsAt the era of computers and numeric models, the engineers have an impressive set of tools that they can use during their design process. But they remain, as human, the key element able to make their practice an art. Some of them reached this level for the benefit of mankind. Memory, experimentation, own culture, imagination, creativity are the prerequisite for this art of structural engineering that needs to provide appropriate answers to the three dimensions quoted by D. P. Billington : scientific, social and symbolic.References[1] Motro R., Oliva Salinas G. J., Who is the designer ?, Journal of the IASS,Vol. 51, No. 3 September n. 165 , p. 207-216, 2010[2] Aanhaanen J., Demuyter J.Y., Bagneris M., Silvestri C., Motro R., Introducing pascalian forms to large scale physical models. IASS-SLTE, Acapulco Mexico, Full text in CD of proceedings, Book of abstracts p. 33-34, 2008[3] Lodder C., The Transition to Constructivism. The Great Utopia. The Russian and Soviet Avant-Garde, 1915-1932,Exhibition. Guggenheim Museum,1992.[4] Motro R., Tensegrity : Structural Systems for the future, Edition Hermés Penton Sciences.ISBN 1903996376, 2003[5]Motro R., Smaili A., Foucher O., Form controlled method for tensegrity formfinding: Snelson and Emmerich examples. International IASS Symposium on Lightweight Structures in Civil Engineering, Contemporary problems, Warsaw, Poland, Edited by Jan B. Obrebski, Micropublisher, ISBN 83-908867-6-6, p.243-248, 2002[6]Hartney E., Kenneth Snelson - Forces made visible, Hard press editions, 2009[7] Exhibition « L’art de l’Ingénieur, constructeur, entrepreneur, inventeur » Catalog supervised by Antoine Picon . Centre Georges Pompidou, Paris, Editions du Moniteur, 1997[8] Billington D.P.,The Tower and the Bridge The New Art of Structural Engineering, Princeton University Press, 1983[9] Silvestri C., Perception et conception en architecture non-standard (une approche expérimentale pour l’étude des processus de conception spatiale des formes complexes). PhD Thesis, Université Montpellier 2, 2009.[10] Diaz E., Experiment, Expression and the paradox of Black Mountain College, ISBN 0-907738-78-8, Kettle’s Yard Arnolfi, 2008。

美国正在开发用于生物分离的活性纳米表面
佚名
【期刊名称】《纳米科技》
【年(卷),期】2006(3)6
【摘要】美国国家科学基金会（NSF）资助了一个研究团队100万美元，用于研究生物分离的改进方法。

在Rensselaer Polytechnic研究所的化学和生物工程学副教授Ravi Kane的领导下，该团队计划开发在DNA存在的情况下重组的活性纳米表面，并最终开发出能更有效地分离基因组和蛋白质组的工具。

【总页数】2页(P73-74)
【关键词】美国国家科学基金会;生物分离;开发;表面;纳米;活性;生物工程学;蛋白质组
【正文语种】中文
【中图分类】Q503
【相关文献】
1.新型有序纳米介孔生物活性玻璃在模拟体液中的表面生物活性研究 [J], 陈建伟;张权;严晓霞;黄煌渊
2.超滤与泡沫分离内耦合应用于表面活性物质浓缩分离的实验研究 [J], 王斐;南碎飞;窦梅;胡岸松
3.新型生物表面活性剂甘露糖赤藓糖醇脂（MEL）的分离与表面性质研究 [J], 华兆哲;方云
4.海洋极端微生物的分离及其开发研究——Ⅰ 嗜碱、嗜冷微生物的分离及其产生
的活性物质 [J], 方金瑞;黄维真
5.帝斯曼和美国生物基表面活性剂开发商建立合作关系 [J], 庞晓华
因版权原因，仅展示原文概要，查看原文内容请购买。

专利名称：用于疾病和病症分析的无细胞DNA甲基化模式专利类型：发明专利
发明人：向红·婕思敏·周,康舒里,李文渊,史蒂文·杜比尼特,李青娇
申请号：CN201780047763.3
申请日：20170607
公开号：CN110168099A
公开日：
20190823
专利内容由知识产权出版社提供
摘要：本文公开了利用测序读取来检测并定量由血液样品制备的无细胞DNA中组织类型或癌症类型的存在的方法和系统。

申请人：加利福尼亚大学董事会,南加利福尼亚大学
地址：美国加利福尼亚州
国籍：US
代理机构：北京柏杉松知识产权代理事务所(普通合伙)
更多信息请下载全文后查看。

专利名称：体内细胞表面工程化
专利类型：发明专利
发明人：H·什尔万,K·G·埃尔佩克,E·S·尤尔库申请号：CN200680052511.1
申请日：20061207
公开号：CN101426532A
公开日：
20090506
专利内容由知识产权出版社提供
摘要：本发明提供了使用一种或多种免疫共刺激多肽在体内工程化细胞表面如肿瘤细胞表面的方法和组合物。

该方法、组合物和工程化的细胞例如可用于刺激对抗细胞的免疫应答。

当工程化的细胞表面是肿瘤细胞表面时，该方法、组合物和工程化的细胞可用于提高患者对抗癌症的免疫应答以及用于降低肿瘤大小和抑制肿瘤生长。

申请人：路易斯维尔大学研究基金会有限公司
地址：美国肯塔基州
国籍：US
代理机构：中国专利代理(香港)有限公司
更多信息请下载全文后查看。

羟基磷灰石纳米粒子对人肝癌细胞增殖及细胞周期的作用研究
(英文)
曹献英;李世普;张然;闫玉华
【期刊名称】《肿瘤防治杂志》
【年(卷),期】2003(10)3
【摘要】目的 :研究羟基磷灰石 (HAP)纳米粒子体外抗肝癌作用的机理。

方法 :采用形态学观察和MTT法检测HAP纳米粒子作用于人肝癌细胞系Bel 74 0 2的量效和时效关系。

流式细胞仪分析细胞周期的时相变化。

结果 :HAP纳米粒子在体外对人肝癌细胞系Bel 74 0 2具有明显的抑制作用 ,并呈现良好的量效和时效关系。

肝癌细胞的生长阻滞于G1期。

结论 :HAP纳米粒子具有体外抗肝癌作用 ,细胞增殖阻滞于G1期 ,阻断细胞周期的进展 ,导致癌细胞胀亡。

【总页数】3页(P256-258)
【关键词】羟基磷灰石类;纳米粒子;肝肿瘤;胀亡
【作者】曹献英;李世普;张然;闫玉华
【作者单位】武汉理工大学生物中心
【正文语种】中文
【中图分类】R735.7
【相关文献】
1.羟基磷灰石纳米粒子对人肝癌和结肠癌细胞生长的抑制作用 [J], 刘志苏;唐胜利;艾中立;胡军
2.羟基磷灰石纳米粒子诱导人肝癌细胞凋亡的研究 [J], 唐胜利;刘志苏;艾中立;郑从义;姚相杰;朱忠超
3.羟基磷灰石纳米粒子对A431细胞增殖抑制作用的研究 [J], 张东波;王建力;陈莉;吴志鹏
4.羟基磷灰石纳米粒子对人胃癌细胞增殖及侵袭的影响 [J], 陈晓娟;李小平;唐忠志;刘非凡
5.羟基磷灰石纳米粒子对HL-60细胞增殖抑制作用的研究 [J], 李戈;符仁义
因版权原因，仅展示原文概要，查看原文内容请购买。

生物大分子的原位可视化和分子成像技术随着科技的不断发展，生物学领域也在进行着革命性的变化，其中，原位可视化和分子成像技术的突飞猛进，引领着生物学研究的新方向。

生物大分子是构成生物体的基本单位，其功能涉及生命的各个方面。

然而，由于其微小的尺度和复杂的结构，使得人们很难准确捕捉其活动状态。

因此，科学家们致力于开发新的技术，以便能够更好地了解生物大分子的行为和特征。

原位可视化技术能够在生物样本内部，实时、高分辨地观察生物大分子复杂的结构和互动方式。

相较于以往的显微镜观察方法，原位可视化技术更能满足现代科学家对于非侵入性、高灵敏度、高精度、快速成像的要求。

其中比较流行的技术包括荧光显微成像技术、电子显微成像技术、近红外成像技术等。

荧光显微成像技术能够将标记染色的生物大分子视觉化，并且具有较高的时间分辨率和灵敏度。

电子显微成像技术可以在 nm 级别的分辨率下观察样本内部的细节。

近红外成像技术则可以在体内实现对生物分子的追踪和观察。

分子成像技术则是通过标记分子的方式，来获得关于该分子分布、形态、组成及动力学行为等信息的技术。

例如，蛋白质分子成像技术、核磁共振成像技术等，能够在细胞和组织水平上表征生物分子的本地化和相互作用。

通过这些技术，科学家们能够深度挖掘生物体内部的奥秘以及成像技术能够承载，不断探索更多的新领域。

自 2000 年来，大量生物大分子的原位成像研究涉及到神经科学、药物学、生态学等多个领域。

比如，在神经科学领域中，科学家们不断探究脑部中神经元的功能以及神经信号传输的机制。

近年来，基于原位成像技术的脑图谱研究取得了长足进展，对于人类脑部复杂结构的建立和我们对疾病的认识奠定了重要基础。

在药物学领域，分子成像技术还可以帮助人们更好地评估药物治疗效果及其生物代谢产物的分布及治疗效果。

总之，生物大分子的原位可视化和分子成像技术为我们提供了一个更准确、更细化、更可靠的观察生物的方式。

相信未来，在这方面的研究将会有更为广阔的空间，引领出更多有价值的成果，这也为生物学研究的不断推进创造了新的机遇。

Dynamical systemThe Lorenz attractor is an example of a non-lineardynamical system. Studying this system helped giverise to Chaos theory.A dynamical system is a concept in mathematics where a fixedrule describes the time dependence of a point in a geometricalspace. Examples include the mathematical models that describethe swinging of a clock pendulum, the flow of water in a pipe, andthe number of fish each springtime in a lake.At any given time a dynamical system has a state given by a set ofreal numbers (a vector) that can be represented by a point in anappropriate state space (a geometrical manifold). Small changes inthe state of the system create small changes in the numbers. Theevolution rule of the dynamical system is a fixed rule thatdescribes what future states follow from the current state. The ruleis deterministic; in other words, for a given time interval only onefuture state follows from the current state.OverviewThe concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times —each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system .Once the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a trajectory or orbit .Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:•The systems studied may only be known approximately —the parameters of the system may not be knownprecisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent.The operation for comparing orbits to establish their equivalence changes with the different notions of stability.•The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic,whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.•The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as inthe transition to turbulence of a fluid.•The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.It was in the work of Poincaré that these dynamical systems themes developed.Basic definitionsA dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions Φ t that for any element of , the time, map a point of the phase space back into the phase space.The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.ExamplesThe evolution function Φ t is often the solution of a differential equation of motionThe equation gives the time derivative, represented by the dot, of a trajectory x(t) on the phase space starting at some point x. The vector field v(x) is a smooth function that at every point of the phase space M provides the velocity 0vector of the dynamical system at that point. (These vectors are not vectors in the phase space M, but in the tangent M of the point x.) Given a smooth Φ t, an autonomous vector field can be derived from it.space TxThere is no need for higher order derivatives in the equation, nor for time dependence in v(x) because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule:is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.The differential equations determining the evolution function Φ t are often ordinary differential equations: in this case the phase space M is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.Further examples•Logistic map•Dyadic transformation•Tent map•Double pendulum•Arnold's cat map•Horseshoe map•Baker's map is an example of a chaotic piecewise linear map•Billiards and outer billiards•Hénon map•Lorenz system•Circle map•Rössler map•List of chaotic maps•Swinging Atwood's machine•Quadratic map simulation system•Bouncing ball dynamicsLinear dynamical systemsLinear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).FlowsFor a flow, the vector field Φ(x) is a linear function of the position in the phase space, that is,with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x= 0, then the orbitremains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x,When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.Linear vector fields and a few trajectories.MapsA discrete-time, affine dynamical system has the formwith A a matrix and b a vector. As in the continuous case, the change of coordinates x → x + (1 − A) –1b removes the term b from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of . The solutions for the map are no longer curves, but points that hop in the phase space. The the linear system A n xorbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. For is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the example, if u1points along α u, with α ∈ R, is an invariant curve of the map. Points in this straight line run into the fixed point.1There are also many other discrete dynamical systems.Local dynamicsThe qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.RectificationA flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where v(x) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches. Near periodic orbitsIn general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed anin the orbit γ and approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point x). These points are a Poincaréconsider the points in phase space in that neighborhood that are perpendicular to v(x), of the orbit. The flow now defines a map, the Poincaré map F : S → S, for points starting in S and section S(γ, xreturning to S. Not all these points will take the same amount of time to come back, but the times will be close to the .time it takes xThe intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0. The Taylor series of the map is F(x) = J · x + O(x2), so a change of coordinates h can only be expected to simplify F to its linear partThis is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in theprocess discovered the non-resonant condition. If λ1, ..., λνare the eigenvalues of J they will be resonant if oneeigenvalue is an integer linear combination of two or more of the others. As terms of the form λi– ∑ (multiples ofother eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.Conjugation resultsThe results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically becomplex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point xof F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.In the hyperbolic case the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x. The hyperbolic case is also structurally stable. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.Bifurcation theoryWhen the evolution map Φt (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phasespace until a special value μis reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter μ. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.The bifurcations of a hyperbolic fixed point x0 of a system family Fμcan be characterized by the eigenvalues of thefirst derivative of the system DFμ(x) computed at the bifurcation point. For a map, the bifurcation will occur whenthere are eigenvalues of DFμon the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory.Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.Ergodic systemsIn many dynamical systems it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset A into the points Φ t(A) and invariance of the phase space means thatIn the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.In a Hamiltonian system not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms. One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region A is vol(A)/vol(Ω).The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ t. This introduces an operator U t, the transfer operator,By studying the spectral properties of the linear operator U it becomes possible to classify the ergodic properties of Φ t. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ t gets mapped into an infinite-dimensional linear problem involving U.The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−βH). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.Nonlinear dynamical systems and chaosSimple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random. (Remember that we are speaking of completely deterministic systems!). This seemingly unpredictable behavior has been called chaos. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"Note that the chaotic behavior of complicated systems is not the issue. Meteorology has been known for years to involve complicated—even chaotic—behavior. Chaos theory has been so surprising because chaos can be foundwithin almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.Geometrical definitionA dynamical system is the tuple , with a manifold (locally a Banach space or Euclidean space),the domain for time (non-negative reals, the integers, ...) and an evolution rule t → f t (with ) such thatf t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain .Measure theoretical definitionSee main article Measure-preserving dynamical system.A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, thequadruplet . Here, X is a set, and Σ is a sigma-algebra on X, so that the pair is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet is a probability space. A map is said to be Σ-measurable if and only if, for every , one has . A map τ is said topreserve the measure if and only if, for every , one has . Combining the above, a mapτ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple , for such a τ, is then defined to be a dynamical system.The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for integer n are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.Examples of dynamical systemsInternal links•Arnold's cat map•Baker's map is an example of a chaotic piecewise linear map•Circle map•Double pendulum•Billiards and Outer Billiards•Hénon map•Horseshoe map•Irrational rotation•List of chaotic maps•Logistic map•Lorenz system•Rossler mapExternal links•Interactive applet for the Standard and Henon Maps [1] by A. LuhnMultidimensional generalizationDynamical systems are defined over a single independent variable, usually thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.References[1]http://complexity.xozzox.de/nonlinmappings.htmlFurther readingWorks providing a broad coverage:•Ralph Abraham and Jerrold E. Marsden (1978). Foundations of mechanics. Benjamin–Cummings.ISBN 0-8053-0102-X. (available as a reprint: ISBN 0-201-40840-6)•Encyclopaedia of Mathematical Sciences (ISSN 0938-0396) has a sub-series on dynamical systems with reviews of current research.•Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge.ISBN 0-521-57557-5.•Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana (2005). Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer. ISBN 3-540-22066-6.•Stephen Smale (1967). "Differential dynamical systems". Bulletin of the American Mathematical Society73: 747–817. doi:10.1090/S0002-9904-1967-11798-1.Introductory texts with a unique perspective:•V. I. Arnold (1982). Mathematical methods of classical mechanics. Springer-Verlag. ISBN 0-387-96890-3.•Jacob Palis and Wellington de Melo (1982). Geometric theory of dynamical systems: an introduction.Springer-Verlag. ISBN 0-387-90668-1.•David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press.ISBN 0-12-601710-7.•Tim Bedford, Michael Keane and Caroline Series, eds. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. ISBN 0-19-853390-X.•Ralph H. Abraham and Christopher D. Shaw (1992). Dynamics—the geometry of behavior, 2nd edition.Addison-Wesley. ISBN 0-201-56716-4.Textbooks•Steven H. Strogatz (1994). Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering. Addison Wesley. ISBN 0-201-54344-3.•Kathleen T. Alligood, Tim D. Sauer and James A. Yorke (2000). Chaos. An introduction to dynamical systems.Springer Verlag. ISBN 0-387-94677-2.•Morris W. Hirsch, Stephen Smale and Robert Devaney (2003). Differential Equations, dynamical systems, and an introduction to chaos. Academic Press. ISBN 0-12-349703-5.•Julien Clinton Sprott (2003). Chaos and time-series analysis. Oxford University Press. ISBN 0-19-850839-5.•Oded Galor (2011). Discrete Dynamical Systems. Springer. ISBN 978-3-642-07185-0.Popularizations:•Florin Diacu and Philip Holmes (1996). Celestial Encounters. Princeton. ISBN 0-691-02743-9.•James Gleick (1988). Chaos: Making a New Science. Penguin. ISBN 0-14-009250-1.•Ivar Ekeland (1990). Mathematics and the Unexpected (Paperback). University Of Chicago Press.ISBN 0-226-19990-8.•Ian Stewart (1997). Does God Play Dice? The New Mathematics of Chaos. Penguin. ISBN 0140256024.External links• A collection of dynamic and non-linear system models and demo applets (.au/ simulations/non-linear/) (in Monash University's Virtual Lab)•Arxiv preprint server (/list/math.DS/recent) has daily submissions of (non-refereed) manuscripts in dynamical systems.•DSWeb (/) provides up-to-date information on dynamical systems and its applications.•Encyclopedia of dynamical systems (/article/Encyclopedia_of_Dynamical_Systems) A part of Scholarpedia — peer reviewed and written by invited experts.•Nonlinear Dynamics (http://www.egwald.ca/nonlineardynamics/index.php). Models of bifurcation and chaos by Elmer G. Wiens•Oliver Knill () has a series of examples of dynamical systems with explanations and interactive controls.•Sci.Nonlinear FAQ 2.0 (Sept 2003) (/faculty/jdm/faq-Contents.html) provides definitions, explanations and resources related to nonlinear scienceOnline books or lecture notes:•Geometrical theory of dynamical systems (/pdf/math.HO/0111177). Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level.•Dynamical systems (/online_bks/coll9/). George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.•Chaos: classical and quantum (). An introduction to dynamical systems from the periodic orbit point of view.•Modeling Dynamic Systems (/2000/0008/0008feat2.htm). An introduction to the development of mathematical models of dynamic systems.•Learning Dynamical Systems (/research/ai/dynamics/tutorial/home.html).Tutorial on learning dynamical systems.•Ordinary Differential Equations and Dynamical Systems (http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ ). Lecture notes by Gerald TeschlResearch groups:•Dynamical Systems Group Groningen (http://www.math.rug.nl/~broer/), IWI, University of Groningen.•Chaos @ UMD (/). Concentrates on the applications of dynamical systems.•Dynamical Systems (/dynamics/), SUNY Stony Brook. Lists of conferences, researchers, and some open problems.•Center for Dynamics and Geometry (/dynsys/), Penn State.•Control and Dynamical Systems (/), Caltech.•Laboratory of Nonlinear Systems (http://lanoswww.epfl.ch/), Ecole Polytechnique Fédérale de Lausanne (EPFL).•Center for Dynamical Systems (http://www.math.uni-bremen.de/ids.html/), University of Bremen•Systems Analysis, Modelling and Prediction Group (/samp/), University of Oxford •Non-Linear Dynamics Group (http://sd.ist.utl.pt/), Instituto Superior Técnico, Technical University of Lisbon •Dynamical Systems (http://www.impa.br/), IMPA, Instituto Nacional de Matemática Pura e Applicada.。

J.reine angew.Math.549(2002),47—77Journal fu¨r die reine undangewandte Mathematik(Walter de GruyterBerlinÁNew York2002Discrete constant mean curvature surfaces andtheir indexBy Konrad Polthier at Berlin and Wayne Rossman at KobeAbstract.We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization.These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions,the simplicial structures, and(in the nonminimal case)the volume to one side of the surfaces.We thenfind explicit formulas for complete examples,such as discrete minimal catenoids and helicoids.We use these discrete surfaces to study the index of unstable minimal surfaces,by nu-merically evaluating the spectra of their Jacobi operators.Our numerical estimates confirm known results on the index of some smooth minimal surfaces,and provide additional in-formation regarding their area-reducing variations.The approach here deviates from other numerical investigations in that we add geometric interpretation to the discrete surfaces.1.IntroductionSmooth submanifolds,and surfaces in particular,with constant mean curvature(cmc) have a long history of study,and modern work in thisfield relies heavily on geometric and analytic machinery which has evolved over hundreds of years.However,nonsmooth sur-faces are also natural mathematical objects,even though there is less machinery available for studying them.For example,consider M.Gromov’s approach of doing geometry using only a set with a measure and a measurable distance function[9].Here we consider piecewise linear triangulated surfaces—we call them‘‘discrete surfaces’’—which have been brought more to the forefront of geometrical research by com-puter graphics.We define cmc for discrete surfaces in R3so that they are critical for volume-preserving variations,just as smooth cmc surfaces are.Discrete cmc surfaces have both in-teresting di¤erences from and similarities with smooth ones.For example,they are di¤erent in that smooth minimal graphs in R3over a bounded domain are stable,whereas discrete minimal graphs can be highly unstable.We will explore properties like this in section2.In section3we will see some ways in which these two types of surfaces are similar. We will see that:a discrete catenoid has an explicit description in terms of the hyperboliccosine function,just as the smooth catenoid has;and a discrete helicoid can be described with the hyperbolic sine function,just as a conformally parametrized smooth helicoid is;and there are discrete Delaunay surfaces which have translational periodicities,just as smooth Delaunay surfaces have.Pinkall and Polthier [17]used Dirichlet energy and a numerical minimization proce-dure to find discrete minimal surfaces.In this work,we rather have the goal to describe dis-crete minimal surfaces as explicitly as possible,and thus we are limited to the more funda-mental examples,for example the discrete minimal catenoid and helicoid.We note that these explicit descriptions will be useful test candidates when implementing a procedure that we describe in the next paragraphs.Discrete surfaces have finite dimensional spaces of admissible variations,therefore the study of linear di¤erential operators on the variation spaces reduces to the linear algebra of matrices.This advantage over smooth surfaces with their infinite dimensional variation spaces makes linear operators easier to handle in the discrete case.This suggests that a useful procedure for studying the spectra of the linear Jacobi operator in the second variation formula of smooth cmc surfaces is to consider the corre-sponding spectra of discrete cmc approximating surfaces.Although similar to the finite ele-ment method in numerical analysis,here the finite element approximations will have geo-metric and variational meaning in their own right.As an example,consider how one finds the index of a smooth minimal surface,that is the number of negative points in the spectrum.The standard approach is to replace the metric of the surface with the metric obtained by pulling back the spherical metric via the Gauss map.This approach can yield the index:for example,the indexes of a complete catenoid and a complete Enneper surface are 1([7]),the index of a complete Jorge-Meeks n -noid is 2n À3([12],[11])and the index of a complete genus k Costa-Ho¤man-Meeks surface is 2k þ3for every k e 37([14],[13]).However,this approach does not yield the eigenvalues and eigenfunctions on compact portions of the original minimal surfaces,as the metric has been changed.It would be interesting to know the eigenfunctions associated to negative eigenvalues since these represent the directions of variations that reduce area.The above procedure of approximating by discrete surfaces can provide this information.In sections 5and 6we establish some tools for studying the spectrum of discrete cmc surfaces.Then we test the above procedure on two standard cases—a (minimal)rectangle,and a portion of a smooth minimal catenoid bounded by two circles.In these two cases we know the spectra of the smooth surfaces (section 4),and we know the discrete minimal sur-faces as well (section 3),so we can check that the above procedure produces good approx-imations for the eigenvalues and smooth eigenfunctions (section 7),which indeed must be the case,by the theory of the finite element method [4],[8].With these successful tests,we go on to consider cases where we do not a priori know what the smooth eigenfunctions should be,such as the Jorge-Meeks 3-noid and the genus 1Costa surface (section 7).The above procedure can also be implemented using discrete approximating surfaces which are found only numerically and not explicitly,such as surfaces found by the method in [17].And in fact,we use the method in [17]to find approximating surfaces for the 3-noid and Enneper surface and Costa surface.Polthier and Rossman,Curvature surfaces48We note also that Ken Brakke’s surface evolver software [3]is an e‰cient tool for numerical index calculations using the same discrete ansatz.Our main emphasis here is to provide explicit formulations for the discrete Jacobi operator and other geometric proper-ties of discrete surfaces.Many of the discrete minimal and cmc surfaces introduced here are available as in-teractive models at EG-Models [19].2.Discrete minimal and cmc surfacesWe start with a variational characterization of discrete minimal and discrete cmc sur-faces.This characterization will allow us to construct explicit examples of unstable discrete cmc surfaces.Note that merely finding minima for area with respect to a volume constraint would not su‰ce for this as that would produce only stable examples.We will later use these discrete cmc surfaces for our numerical spectra computations.The following definitions for discrete surfaces and their variations work equally well in any ambient space R n but for simplicity we restrict to R 3.Definition 2.1.A discrete surface in R 3is a triangular mesh T which has the topology of an abstract 2-dimensional simplicial surface K combined with a geometric C 0realization in R 3that is piecewise linear on each simplex.The geometric realization j K j is determined by a set of vertices V ¼f p 1;...;p m g H R 3.T can be identified with the pair ðK ;V Þ.The simplicial complex K represents the connectivity of the mesh.The 0,1,and 2dimensional simplices of K represent the vertices,edges,and triangles of the discrete surface.Let T ¼ðp ;q ;r Þdenote an oriented triangle of T with vertices p ;q ;r A V .Let pq denote an edge of T with endpoints p ;q A V .For p A V ,let star ðp Þdenote the triangles of T that contain p as a vertex.For an edge pq ,let star ðpq Þdenote the (at most two)triangles of T that contain the edge pq .Definition 2.2.Let V ¼f p 1;...;p m g be the set of vertices of a discrete surface T .A variation T ðt Þof T is defined as a C 2variation of the vertices p iFigure 1.At each vertex p the gradient of discrete area is the sum of the p 2-rotated edge vectors J ðr Àq Þ,as in Equation(1).p i ðt Þ:½0;e Þ!R 3so that p i ð0Þ¼p i E i ¼1;...;m :The straightness of the edges and the flatness of the triangles are preserved as the vertices move.In the smooth situation,the variation at interior points is typically restricted to nor-mal variation,since the tangential part of the variation only performs a reparametrization of the surface.However,on discrete surfaces there is an ambiguity in the choice of normal vectors at the vertices,so we allow arbitrary variations.But we will later see (section 7)that our experimental results can accurately estimate normal variations of a smooth surface when the discrete surface is a close approximation to the smooth surface.In the following we derive the evolution equations for some basic entities under sur-face variations.The area of a discrete surface isarea ðT Þ:¼PT A T area T ;where area T denotes the Euclidean area of the triangle T as a subset of R 3.Let T ðt Þbe a variation of a discrete surface T .At each vertex p of T ,the gradient of area is‘p area T ¼12P T ¼ðp ;q ;r ÞA star pJ ðr Àq Þ;ð1Þwhere J is rotation of angle p 2in the plane of each oriented triangle T .The first derivative of the surface area is then given by the chain ruled dt area T ¼P p A Vh p 0;‘p area T i :ð2ÞThe volume of an oriented surface T is the oriented volume enclosed by the cone of the surface over the origin in R 3vol T :¼16P T ¼ðp ;q ;r ÞA T h p ;q Âr i ¼13P T ¼ðp ;q ;r ÞA Th ~N ;p i Áarea T ;where p is any of the three vertices of the triangle T and~N¼ðq Àp ÞÂðr Àp Þ=jðq Àp ÞÂðr Àp Þj is the oriented normal of T .It follows thatPolthier and Rossman,Curvature surfaces50‘p vol T¼PT¼ðp;q;rÞA star p qÂr=6ð3Þandd dt vol T¼Pp A Vh p0;‘p vol T i:ð4ÞRemark2.1.Note also that‘p vol T¼PT¼ðp;q;rÞA star p À2Áarea TÁ~NþpÂðrÀqÞÁ=6.Furthermore,if p is an interior vertex,then the boundary of star p is closed and PT A star ppÂðrÀqÞ¼0.Hence the qÂr in Equation(3)can be replaced with2Áarea TÁ~N whenever p is an interior vertex.In the smooth case,a minimal surface is critical with respect to area for any variation thatfixes the boundary,and a cmc surface is critical with respect to area for any variation that preserves volume andfixes the boundary.We wish to define discrete cmc surfaces so that they have the same variational properties for the same types of variations.So we will consider variations TðtÞof T thatfix the boundary q T and that additionally preserve volume in the nonminimal case,which we call permissible variations.The condition that makes a discrete surface area-critical for any permissible variation is expressed in the fol-lowing definition.Definition2.3.A discrete surface has constant mean curvature(cmc)if there exists a constant H so that‘p area¼H‘p vol for all interior vertices p.If H¼0then it is minimal.This definition for discrete minimality has been used in[17].In contrast,our definition of discrete cmc surfaces di¤ers from[15],where cmc surfaces are characterized algorithm-ically using discrete minimal surfaces in S3and a conjugation pare also [2]for a definition via discrete integrable systems which lacks variational properties.Remark2.2.If T is a discrete minimal surface that contains a simply-connected dis-crete subsurface T0that lies in a plane,then it follows easily from Equation(1)that the dis-crete minimality of T is independent of the choice of triangulation of the trace of T0.2.0.1.Notation from th e th eoryoffinite elements.Consider a vector-valued functionv pj A R3defined on the n interior vertices V int¼f p1;...;p n g of T.We may extend thisfunction to the boundary vertices of T as well,by assuming v p¼~0A R3for each boundaryvertex p.The vectors v pj are the variation vectorfield of any boundary-fixing variation ofthe formp jðtÞ¼p jþtÁv pj þOðt2Þ;ð5Þthat is,p0jð0Þ¼v pj.We define the vector~v A R3n by~v t¼ðv t p1;...;v tp nÞ:ð6ÞThe variation vectorfield~v can be naturally extended to a piece-wise linear continuous R3-valued function v on T,with v in the following vector space:Polthier and Rossman,Curvature surfaces51Definition2.4.On a discrete surface T we define the space of piecewise linear functionsS h:¼f v:T!R3j v A C0ðTÞ;v is linear on each T A T and v j q T¼0g: This space is named S h,as in the theory offinite elements.Note that any compo-nent function of any function v A S h has bounded Sobolev H1norm.For each triangle T¼ðp;q;rÞin T and each v A S h,v j T ¼v p c pþv q c qþv r c r;ð7Þwhere c p:T!R is the head function on T which is1at p and is0at all other vertices ofT and extends linearly to all of T in the unique way.The functions c pj form a basis(withscalars in R3)for the3n-dimensional space S h.2.0.2.Non-uniqueness of discrete minimal disks.Uniqueness of a bounded mini-mal surface with a given boundary ensures that it is stable.For smooth minimal surfaces, uniqueness can sometimes be decided using the maximum principle of elliptic equations, which ensures that the minimal surface is contained in the convex hull of its boundary, and,if the boundary has a1-1projection to a convex planar curve,then it is unique for that boundary and is a minimal graph.The maximum principle also shows that any mini-mal graph is unique even when the projection of its boundary is not convex.More gener-ally,stability still holds when the surface merely has a Gauss map image contained in a hemisphere,as shown in[1](although their proof employs tools other than the maximum principle).However,such statements do not hold for discrete minimal surfaces.Consider the surface shown in the left-hand side of Figure2,whose height function has a local maxi-mum at an interior vertex.This example does not lie in the convex hull of its boundary and thereby disproves the general existence of a discrete version of the maximum principle.Also, the three surfaces on the right-hand side in Figure3are all minimal graphs over an annular domain with the same boundary contours and the same simplicial structure,and yet they are not the same surfaces,hence graphs with given simplicial structure are not unique.And the left-hand surface in Figure3is a surface whose Gauss map is contained in a hemisphere but which is unstable(this surface is not a graph)—another example of this property is the first annular surface in Figure3,which is also unstable.(We define stability of discrete cmc surfaces in section5.)The influence of the discretization on nonuniqueness,like as in the annular examples of Figure3,can also be observed in a more trivial way for a discrete minimal graph over a simply connected convex domain.The two surfaces on the right-hand side of Figure2have the same trace,i.e.they are identical as geometric surfaces,but they are di¤erent as discrete surfaces.Interior vertices may be freely added and moved inside the middle planar square without a¤ecting minimality(see Remark2.2).In contrast to existence of these counterexamples we believe that some properties of smooth minimal surfaces remain true in the discrete setting.We say that a discrete surface is a disk if it is homeomorphic to a simply connected domain.Conjecture2.1.Let T H R3be a discrete minimal disk whose boundary projects in-jectively to a convex planar polygonal curve,then T is a graph over that plane.The authors were able to prove this conjecture with the extra assumption that all the triangles of the surface are acute,using the fact that the maximum principle(a height function cannot attain a strict interior maximum)actually does hold when all triangles are acute.One can ask if a discrete minimal surface T with given simplicial structure and boundary is unique if it has a1-1perpendicular or central projection to a convex polygonal domain in a plane.The placement of the vertices need not be unique,as we saw in Remark 2.2,however,one can consider if there is uniqueness in the sense that the trace of T in R3is unique:Conjecture2.2.Let G H R3be a polygonal curve that eitherðAÞ:projects injec-tively to a convex planar polygonal curve,orðBÞ:has a1-1central projection from a point p A R3to a convex planar polygonal curve.Let K be a given abstract simplicial disk,and let g:q K!G be a given piecewise linear map.If T is a discrete minimal surface that is a geometric realization of K so that the map q K!q T equals g,then the trace of T in R3is uniquely determined.Furthermore,T is a graph in the caseðAÞ,and T is contained in the cone of G over p in the caseðBÞ.We have the following weaker form of Conjecture2.2,which follows from Corollary5.1of section5in the case that there is only one interior vertex:Conjecture 2.3.If a discrete minimal surface is a graph over a convex polygonal do-main ,then it is stable .3.Explicit discrete surfacesHere we describe explicit discrete catenoids and helicoids,which seem to be the first explicitly known nontrivial complete discrete minimal surfaces (with minimality defined variationally).3.1.Discrete minimal catenoids.To derive an explicit formula for embedded com-plete discrete minimal catenoids,we choose the vertices to lie on congruent planar polygo-nal meridians,with the meridians placed so that the traces of the surfaces will have dihedral symmetry.We will find that the vertices of a discrete meridian lie equally spaced on a smooth hyperbolic cosine curve.Furthermore,these discrete catenoids will converge uniformly in compact regions to the smooth catenoid as the mesh is made finer.We begin with a lemma that prepares the construction of the vertical meridian of the discrete minimal catenoid,by successively adding one horizontal ring after another starting from an initial ring.Since our construction will lead to pairwise coplanar triangles,the star of each individual vertex can be made to consist of four triangles (see Remark 2.2).We now derive an explicit representation of the position of a vertex surrounded by four such triangles in terms of the other four vertex positions.The center vertex is assumed to be coplanar with each of the two pairs of two opposite vertices,with those two planes becoming the plane of the vertical meridian and the horizontal plane containing a dihedrally symmetric polygonal ring (consisting of edges of the surface).See Figure 4.Lemma 3.1.Suppose we have four vertices p ¼ðd ;0;e Þ,q 1¼ðd cos y ;Àd sin y ;e Þ,q 2¼ða ;0;b Þ,and q 3¼ðd cos y ;d sin y ;e Þ,for given real numbers a ,b ,d ,e ,and angle y so that b 3e .Then there exists a choice of real numbers x and y and a fifth vertex q 4¼ðx ;0;y Þso that the discrete surface formed by the four triangles ðp ;q 1;q 2Þ,ðp ;q 2;q 3Þ,ðp ;q 3;q 4Þ,and ðp ;q 4;q 1Þis minimal ,i.e.‘p area ðstar p Þ¼0;if and only if2ad >ðe Àb Þ21þcos y:Figure 4.The construction in Lemma 3.1and a discrete minimal catenoid.Polthier and Rossman,Curvature surfaces54Furthermore,when x and y exist,they are unique and must be of the formx¼2ð1þcos yÞd3þðaþ2dÞðeÀbÞ2 2adð1þcos yÞÀðeÀbÞ2;y¼2eÀb:Proof.First we note that the assumption b3e is necessary.If b¼e,then one may choose y¼b,and then there is a free1-parameter family of choices of x,leading to a trivial planar surface.For simplicity we apply a vertical translation and a homothety about the origin of R3 to normalize d¼1,e¼0,and by doing a reflection if necesary,we may assume b<0.Let c¼cos y and s¼sin y.We derive conditions for the coordinate components of‘p area to vanish.The second component vanishes by symmetry of star ing the definitionsc1:¼ðaÀ1Þs2Àb2ð1ÀcÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b2ð1ÀcÞþðaÀ1Þ2s2q;c2:¼abþbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b2ð1ÀcÞþðaÀ1Þ2s2q;thefirst(resp.third)component of‘p area vanishes ifc1¼y2ð1ÀcÞÀðxÀ1Þs2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2y2ð1ÀcÞþðxÀ1Þ2s2q;resp:c2¼ÀðxÀ1ÞyÀ2yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2y2ð1ÀcÞþðxÀ1Þ2s2q:ð8ÞDividing one of these equations by the other we obtainxÀ1¼c2yð1ÀcÞþ2c1c2sÀc1yy;ð9Þso x is determined by y.It now remains to determine if one canfind y so that c2s2Àc1y30.If xÀ1is chosen as in equation(9),then thefirst minimality condition of equation(8)holds if and only if the second one holds as well.So we only need to insert this value for xÀ1into thefirst minimality condition and check for solutions y.When c130, wefind that the condition becomes1¼c2s2Àc1yj c2s2Àc1y jyj y jÀð1ÀcÞy2À2s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1ÀcÞc22s4þ4c21s2þÀ2ð1ÀcÞc21þs2ð1ÀcÞ2c22Áy2 q:SinceÀð1ÀcÞy2À2s2<0,note that this equation can hold only if c2s2Àc1y and y have opposite signs,so the equation becomes1¼ð1ÀcÞy2þ2s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1ÀcÞc22s4þ4c21s2þÀ2ð1ÀcÞc21þs2ð1ÀcÞ2c22Áy2q;Polthier and Rossman,Curvature surfaces55which simplifies to1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1ÀcÞy2þ2s2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1ÀcÞc22s2þ2c21 q:This implies y2is uniquely determined.Inserting the valuey¼G b;onefinds that the above equation holds.When y¼b<0,wefind that c2s2Àc1y<0, which is impossible.When y¼Àb>0,wefind that c2s2Àc1y<0if and only if 2að1þcÞ>b2.And when y¼Àb and2að1þcÞ>b2,we have the minimality condition whenx¼2þ2cþab2þ2b2 2aþ2acÀb:Inverting the transformation we did at the beginning of this proof brings us back to the general case where d and e are not necessarily1and0,and the equations for x and y be-come as stated in the lemma.When c1¼0,we haveðaÀ1Þð1þcÞ¼b2andðxÀ1Þð1þcÞ¼y2,so,in particular, we have a>1and therefore2að1þcÞ>b2.The right-hand side of equation(8)implies y¼Àb and x¼a.Again,inverting the transformation from the beginning of this proof, we have that x and y must be of the form in the lemma for the case c1¼0as well.rThe next lemma provides a necessary and su‰cient condition for when two points lie on a scaled cosh curve,a condition that is identical to that of the previous lemma.That these conditions are the same is crucial to the proof of the upcoming theorem.Lemma3.2.Given two pointsða;bÞandðd;eÞin R2with b3e,and an angle y with j y j<p,there exists an r so that these two points lie on some vertical translate of the modified cosh curvegðtÞ¼0@r cosh teÀbarccosh1þ1rðeÀbÞ21þcos y!"#;t1A;t A R;if and only if2ad>ðeÀbÞ2 1þcos y.Proof.Define^d¼eÀb1þcos y.Without loss of generality,we may assume0<a e dand e>0,and henceÀe e b<e.If the pointsða;bÞandðd;eÞboth lie on the curve gðtÞ, thenarccosh1þ^d2r2!¼arccoshdrÀsignðbÞÁarccoshar;Polthier and Rossman,Curvature surfaces 56where signðbÞ¼1if b f0and signðbÞ¼À1if b<0.Note that if b¼0,then a must equalr(and so arccosh a r¼0).This equation is solvable(for either value of signðbÞ)if and only ifd r þffiffiffiffiffiffiffiffiffiffiffiffiffiffid2r2À1r!arþffiffiffiffiffiffiffiffiffiffiffiffiffia2r2À1r!¼1þ^d2r2þ^drffiffiffiffiffiffiffiffiffiffiffiffiffi2þ^d2r2swhen b e0,ord r þffiffiffiffiffiffiffiffiffiffiffiffiffiffid2rÀ1 sa r þffiffiffiffiffiffiffiffiffiffiffiffiffia2rÀ1s¼1þ^d2rþ^drffiffiffiffiffiffiffiffiffiffiffiffiffi2þ^d2rswhen b f0,for some r Að0;a .The right-hand side of these two equations has the follow-ing properties:(1)It is a nonincreasing function of r Að0;a .(2)It attains somefinite positive value at r¼a.(3)It is greater than the function2^d2=r2.(4)It approaches2^d2=r2asymptotically as r!0.The left-hand sides of these two equations have the following properties:(1)They attain the samefinite positive value at r¼a.(2)Thefirst one is a nonincreasing function of r Að0;a .(3)The second one is a nondecreasing function of r Að0;a .(4)The second one attains the value d=a at r¼0.(5)Thefirst one is less than the function4ad=r2.(6)Thefirst one approaches4ad=r2asymptotically as r!0.It follows from these properties that one of the two equations above has a solution for some r if and only if2ad>^d2.This completes the proof.rWe now derive an explicit formula for discrete minimal catenoids,by specifying the vertices along a planar polygonal meridian.Then the traces of the surfaces will have dihe-dral symmetry of order k f3.The surfaces are tessellated by planar isosceles trapezoids like a Z2grid,and each trapezoid can be triangulated into two triangles by choosing a di-Polthier and Rossman,Curvature surfaces57agonal of the trapeziod as the interior edge.Either diagonal can be chosen,as this does not a¤ect the minimality of the catenoid,by Remark 2.2.The discrete catenoid has two surprising features.First,the vertices of a meridian lie on a scaled smooth cosh curve (just as the profile curve of smooth catenoids lies on the cosh curve),and there is no a priori reason to have expected this.Secondly,the vertical spacing of the vertices along the meridians is constant.Theorem 3.1.There exists a four-parameter family of embedded and complete discrete minimal catenoids C ¼C ðy ;d ;r ;z 0Þwith dihedral rotational symmetry and planar meridians .If we assume that the dihedral symmetry axis is the z-axis and that a meridian lies in the xz-plane ,then ,up to vertical translation ,the catenoid is completely described by the following properties :(1)The dihedral angle is y ¼2p k,k A N ,k f 3.(2)The vertices of the meridian in the xz-plane interpolate the smooth cosh curvex ðz Þ¼r cosh 1raz ;witha ¼r d arccosh 1þ1r 2d 21þcos y!;where the parameter r >0is the waist radius of the interpolated cosh curve ,and d >0is the constant vertical distance between adjacent vertices of the meridian .(3)For any given arbitrary initial value z 0A R ,the profile curve has vertices of the form ðx j ;0;z j Þwithz j ¼z 0þj d ;x j ¼x ðz j Þ;where x ðz Þis the meridian in item 2above .(4)The planar trapezoids of the catenoid may be triangulated independently of each other (by Remark 2.2).Proof.By Lemma 3.1,if we have three consecutive vertices ðx n À1;z n À1Þ,ðx n ;z n Þ,and ðx n þ1;z n þ1Þalong the meridian in the xz -plane,they satisfy the recursion formulax n þ1¼ðx n À1þ2x n Þ^d 2þ2x 3n 2x n x n À1À^d 2;z n þ1¼z n þd ;ð10Þwhere d ¼z n Àz n À1and ^d¼d =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þcos y p .As seen in Lemma 3.1,the vertical distance be-Polthier and Rossman,Curvature surfaces58。