# Fractals – A Very Short Introduction

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.

# Chapter 7 A little history

Geometry, with its highly visual and practical nature, is one of the oldest branches of mathematics. Its development through the ages has paralleled its increasingly sophisticated applications. Construction, crafts, and astronomy practised by ancient civilizations led to the need to record and analyse the shapes, sizes, and positions of objects. Notions of angles, areas, and volumes developed with the need for surveying and building. Two shapes were especially important: the straight line and the circle, which occurred naturally in many settings but also underlay the design of many artefacts. As well as fulfilling practical needs, philosophers were motivated by aesthetic aspects of geometry and sought simplicity in geometric structures and their applications. This reached its peak with the Greek School, notably with Plato (c 428–348 BC) and Euclid (c 325–265 BC), for whom constructions using a straight edge and compass, corresponding to line and circle, were the essence of geometric perfection.

As time progressed, ways were found to express and solve geometrical problems using algebra. A major advance was the introduction by René Descartes (1596–1650) of the Cartesian coordinate system which enabled shapes to be expressed concisely in terms of equations. This was a necessary precursor to the calculus, developed independently by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1714) in the late 17th century. The calculus provided a mathematical procedure for finding tangent lines that touched smooth curves as well as a method for computing areas and volumes of an enormous variety of geometrical objects. Alongside this, more sophisticated geometric figures were being observed in nature and explained mathematically. For example, using Tycho Brahe’s observations, Johannes Kepler proposed that planets moved around ellipses, and this was substantiated as a mathematical consequence of Newton’s laws of motion and gravitation.

The tools and methods were now available for tremendous advances in mathematics and the sciences. All manner of geometrical shapes could be analysed. Using the laws of motion together with the calculus, one could calculate the trajectories of projectiles, the motion of celestial bodies, and, using differential equations which developed from the calculus, more complex motions such as fluid flows. Although the calculus underlay Graph of a Brownian process8I to think of all these applications, its foundations remained intuitive rather than rigorous until the 19th century when a number of leading mathematicians including Augustin Cauchy (1789–1857), Bernhard Riemann (1826–66), and Karl Weierstrass (1815–97) formalized the notions of continuity and limits. In particular, they developed a precise definition for a curve to be ‘differentiable’, that is for there to be a tangent line touching the curve at a point. Many mathematicians worked on the assumption that all curves worthy of attention were nice and smooth so had tangents at all their points, enabling application of the calculus and its many consequences. It was a surprise when, in 1872, Karl Weierstrass constructed a ‘curve’ that was so irregular that at no point at all was it possible to draw a tangent line. The Weierstrass graph might be regarded as the first formally defined fractal, and indeed it has been shown to have fractal dimension greater than 1.

In 1883, the German Georg Cantor (1845–1918) wrote a paper introducing the middle-third Cantor set, obtained by repeatedly removing the middle thirds of intervals (see Figure 44). The Cantor set is perhaps the most basic self-similar fractal, made up of 2 scale copies of itself, although of more immediate interest to Cantor were its topological and set theoretic properties, such as it being totally disconnected, rather than its geometry. (Several other mathematicians studied sets of a similar form around the same time, including the Oxford mathematician Henry Smith (1826–83) in an article in 1874.) In 1904, Helge von Koch introduced his curve, as a simpler construction than Weierstrass’s example of a curve without any tangents. Then, in 1915, the Polish mathematician Wacław Sierpiński (1882–1969) introduced his triangle and, in 1916, the Sierpiński carpet. His main interest in the carpet was that it was a ‘universal’ set, in that it contains continuously deformed copies of all sets of ‘topological dimension’ 1. Although such objects have in recent years become the best-known fractals, at the time properties such as self-similarity were almost irrelevant, their main use being to provide specific examples or counter-examples in topology and calculus.

It was in 1918 that Felix Hausdorff proposed a natural way of ‘measuring’ the middle-third Cantor set and related sets, utilizing a general approach due to Constantin Carathéodory (1873–1950). Hausdorff showed that the middle-third Cantor set had dimension of log2/log3 = 0.631, and also found the dimensions of other self-similar sets. This was the first occurrence of an explicit notion of fractional dimension. Now termed ‘Hausdorff dimension’, his definition of dimension is the one most commonly used by mathematicians today. (Hausdorff, who did foundational work in several other areas of mathematics and philosophy, was a German Jew who tragically committed suicide in 1942 to avoid being sent to a concentration camp.) Box-dimension, which in many ways is rather simpler than Hausdorff dimension, appeared in a 1928 paper by Georges Bouligand (1889–1979), though the idea underlying an equivalent definition had been mentioned rather earlier by Hermann Minkowski (1864–1909), a Polish mathematician known especially for his work on relativity.

For many years, few mathematicians were very interested in fractional dimensions, with highly irregular sets continuing to be regarded as pathological curiosities. One notable exception was Abram Besicovitch (1891–1970), a Russian mathematician who held a professorship in Cambridge for many years. He, along with a few pupils, investigated the dimension of a range of fractals as well as investigating some of their geometric properties.

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.

# Shape of Inner Space

shing-tung_yau_nadis_s._the_shape_of_inner_space

String Theory and the Geometry of the Universe’s Hidden Dimensions

Shing-Tung YAU and Steve NADIS

Chapter 3: P.39

My personal involvement in this area began in 1969, during my first semester of graduate studies at Berkeley. I needed a book to read during Chrismas break. Rather than selecting Portnoy’s Complaint, The Godfather, The Love Machine, or The Andromeda Strain-four top-selling books of that year-I opted for a less popular title, Morse Theory, by the American mathematician John Milnor. I was especially intrigued by Milnor’s section on topology and curvature, which explored the notion that local curvature has a great influence on geometry and topology. This is a theme I’ve pursued ever since, because the local curvature of a surface is determined by taking the derivatives of that surface, which is another way of saying it is based on analysis. Studying how that curvature influences geometry, therefore, goes to the heart of geometric analysis.

Having no office, I practically lived in Berkeley’s math library in those days. Rumor has it that the first thing I did upon arriving in the United States was visiting that library, rather than, say, explore San Francisco as other might have done. While I can’t remember exactly what I did, forty years hence, I have no reason to doubt the veracity of that rumor. I wandered around the library, as was my habit, reading every journal I could get my hands on. In the course of rummaging through the reference section during winter break, I came across a 1968 article by Milnor, whose book I was still reading. That article, in turn, little else to do at the time (with most people away for the holiday), I tried to see if I could prove something related to Preissman’s theorem.

Chapter 4: P.80

From this sprang the work I’ve become most famous for. One might say it was my calling. No matter what our station, we’d all like to find our true calling in life-that special thing we were put on this earth to do. For an actor, it might be playing Stanley Kowalski in A Streetcar Named Desire. Or the lead role in Hamlet. For a firefighter, it could mean putting out a ten-alarm blaze. For a crime-fighter, it could mean capturing Public Enemy Number One. And in mathematics, it might come down to finding that one problem you’re destined to work on. Or maybe destiny has nothing to do with it. Maybe it’s just a question of finding a problem you can get lucky with.

To be perfectly honest, I never think about “destiny” when choosing a problem to work on, as I tend to be a bit more pragmatic. I try to seek out a new direction that could bring to light new mathematical problems, some of which might prove interesting in themselves. Or I might pick an existing problem that offers the hope that in the course of trying to understand it better, we will be led to a new horizon.

The Calabi conjecture, having been around a couple of decades, fell into the latter category. I latched on to this problem during my first year of graduate school, though sometimes it seemed as if the problem latched on to me. It caught my interest in a way that no other problem had before or has since, as I sensed that it could open a door to a new branch of mathematics. While the conjecture was vaguely related to Poincare’s classic problem, it struck me as more general because if Calabi’s hunch were true, it would lead to a large class of mathematical surfaces and spaces that we didn’t know anything about-and perhaps a new understanding of space-time. For me the conjecture was almost inescapable: Just about every road I pursued in my early investigations of curvature led to it.

Chapter 5: P.104

A mathematical proof is a bit like climbing a mountain. The first stage, of course, is discovering a mountain worth climbing. Imagine a remote wilderness area yet to be explored. It takes some wit just to find such an area, let alone to know whether something worthwhile might be found there. The mountaineer then devises a strategy for getting to the top-a plan that appears flawless, at least on paper. After acquiring the necessary tools and equipment, as well as mastering the necessary skills, the adventurer mounts an ascent, only to be stopped by unexpected difficulties. But others follow in their predecessor’s footsteps, using the successful strategies, while also pursuing different avenues-thereby reaching new heights in the process. Finally someone comes along who not only has a good plan of attack that avoids the pitfalls of the past but also has the fortitude and determination to reach the summit, perhaps planting a flag there to mark his or her presence. The risks to life and limb are not so great in math, and the adventure may not be so apparent to the outsider. And at the end of a long proof, the scholar does not plant a flag. He or she types in a period. Or a footnote. Or a technical appendix. Nevertheless, in our field there are thrill as well as perils to be had in the pursuit, and success still rewards those of us who’ve gained new views into nature’s hidden recesses.

# 2014 International Congress of Mathematics: Awards

Fields Medalist:

Artur Avila

CNRS, France & IMPA, Brazil

[Artur Avila is awarded a Fields Medal] for his profound contributions to dynamical systems theory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

Avila leads and shapes the field of dynamical systems. With his collaborators, he has made essential progress in many areas, including real and complex one-dimensional dynamics, spectral theory of the one-frequency Schródinger operator, flat billiards and partially hyperbolic dynamics.

Avila’s work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory. His work in complex dynamics led to a thorough understanding of the fractal geometry of Feigenbaum Julia sets.

In the spectral theory of one-frequency difference Schródinger operators, Avila came up with a global de- scription of the phase transitions between discrete and absolutely continuous spectra, establishing surprising stratified analyticity of the Lyapunov exponent.

In the theory of flat billiards, Avila proved several long-standing conjectures on the ergodic behavior of interval-exchange maps. He made deep advances in our understanding of the stable ergodicity of typical partially hyperbolic systems.

Avila’s collaborative approach is an inspiration for a new generation of mathematicians.

Manjul Bhargava

Princeton University, USA

[Manjul Bhargava is awarded a Fields Medal] for developing powerful new methods in the geometry of numbers and applied them to count rings of small rank and to bound the average rank of elliptic curves.

Bhargava’s thesis provided a reformulation of Gauss’s law for the composition of two binary quadratic forms. He showed that the orbits of the group SL(2, Z)3 on the tensor product of three copies of the standard integral representation correspond to quadratic rings (rings of rank 2 over Z) together with three ideal classes whose product is trivial. This recovers Gauss’s composition law in an original and computationally effective manner. He then studied orbits in more complicated integral representations, which correspond to cubic, quartic, and quintic rings, and counted the number of such rings with bounded discriminant.

Bhargava next turned to the study of representations with a polynomial ring of invariants. The simplest such representation is given by the action of PGL(2, Z) on the space of binary quartic forms. This has two independent invariants, which are related to the moduli of elliptic curves. Together with his student Arul Shankar, Bhargava used delicate estimates on the number of integral orbits of bounded height to bound the average rank of elliptic curves. Generalizing these methods to curves of higher genus, he recently showed that most hyperelliptic curves of genus at least two have no rational points.

Bhargava’s work is based both on a deep understanding of the representations of arithmetic groups and a unique blend of algebraic and analytic expertise.

Martin Hairer

University of Warwick, UK

[Martin Hairer is awarded a Fields Medal] for his outstanding contributions to the theory of stochastic partial differential equations, and in particular created a theory of regularity structures for such equations.

A mathematical  problem that  is important  throughout science is to understand the influence of noise on differential equations, and on the long time behavior of the solutions. This problem was solved for ordinary differential equations by Itó in the 1940s. For partial differential equations, a comprehensive theory has proved to be more elusive, and only particular cases (linear equations, tame nonlinearities, etc.)  had been treated satisfactorily.

Hairer’s work addresses two central aspects of the theory.  Together with Mattingly  he employed the Malliavin calculus along with new methods to establish the ergodicity of the two-dimensional stochastic Navier-Stokes equation.

Building  on the rough-path approach of Lyons for stochastic ordinary differential equations, Hairer then created an abstract theory of regularity structures for stochastic partial differential equations (SPDEs). This allows Taylor-like expansions around any point in space and time. The new theory allowed him to construct systematically solutions to singular non-linear SPDEs  as fixed points of a renormalization procedure.

Hairer was thus able to give, for the first time, a rigorous intrinsic meaning to many SPDEs arising in physics.

Maryam Mirzakhani

Stanford University, USA

[Maryam Mirzakhani is awarded the Fields Medal] for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.

Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from diverse fields, such as algebraic geometry, topology and probability theory.

In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.

In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and mixing.

Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that – while the closure of a real geodesic in moduli space can be a fractal cobweb, defying classification – the closure of a complex geodesic is always an algebraic subvariety.

Her work has revealed that the rigidity theory of homogeneous spaces (developed by Margulis, Ratner and others) has a definite resonance in the highly inhomogeneous, but equally fundamental realm of moduli spaces, where many developments are still unfolding

Nevanlinna Prize Winner:

Subhash Khot

New York University, USA

[Subhash Khot is awarded the Nevanlinna Prize] for his prescient  definition of the “Unique Games” problem, and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.

Subhash Khot defined the “Unique Games” in 2002 , and subsequently led the effort to understand its complexity and its pivotal role in the study of optimization problems. Khot and his collaborators demonstrated that the hardness of Unique Games implies a precise characterization of the best approximation factors achievable for a variety of NP-hard optimization problems. This discovery turned the Unique Games problem into a major open problem of the theory of computation.

The ongoing quest to study its complexity has had unexpected benefits. First, the reductions used in the above results identified new problems in analysis and geometry, invigorating analysis of Boolean functions, a field at the interface of mathematics and computer science. This led to new central limit theorems, invariance principles, isoperimetric inequalities, and inverse theorems, impacting research in computational complexity, pseudorandomness, learning and combinatorics. Second, Khot and his collaborators used intuitions stemming from their study of Unique Games to yield new lower bounds on the distortion incurred when embedding one metric space into another, as well as constructions of hard families of instances for common linear and semi- definite programming algorithms. This has inspired new work in algorithm design extending these methods, greatly enriching the theory of algorithms and its applications.

Gauss Prize Winner:

Stanley Osher

University of Califonia, USA

[Stanley Osher is awarded the Gauss Prize] for his influential contributions to several fields in applied mathematics, and his far-ranging inventions have changed our conception of physical, perceptual, and mathematical concepts, giving us new tools to apprehend the world.

1. Stanley Osher has made influential contributions in a broad variety of fields in applied mathematics. These include high resolution shock capturing methods for hyperbolic equations, level set methods, PDE based methods in computer vision and image processing, and optimization. His numerical analysis contributions, including the Engquist-Osher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for Hamilton-Jacobi type equations have revolutionized the field. His level set contribu- tions include new level set calculus, novel numerical techniques, fluids and materials modeling, variational approaches, high codimension motion analysis, geometric optics, and the computation of discontinuous so- lutions to Hamilton-Jacobi equations; level set methods have been extremely influential in computer vision, image processing, and computer graphics. In addition, such new methods have motivated some of the most fundamental studies in the theory of PDEs in recent years, completing the picture of applied mathematics inspiring pure mathematics.

2. Stanley Osher has unique mentoring qualities: he has influenced the education of generations of outstanding applied mathematicians, and thanks to his entrepreneurship he has successfully brought his mathematics to industry.

Trained as an applied mathematician and an applied mathematician all his life, Osher continues to surprise the mathematical and numerical community with the invention of simple and clever schemes and formulas. His far-ranging inventions have changed our conception of physical, perceptual, and mathematical concepts, and have given us new tools to apprehend the world.

Chern Medalist:

Phillip Griffiths

Institute for Advanced Study, USA

[Phillip Griths is awarded the 2014 Chern Medal] for his groundbreaking and transformative development of transcendental methods in complex geometry, particularly his seminal work in Hodge theory and periods of algebraic varieties.

Phillip Griffiths’s ongoing work in algebraic geometry, differential geometry, and differential equations has stimulated a wide range of advances in mathematics over the past 50 years and continues to influence and inspire an enormous body of research activity today.

He has brought to bear both classical techniques and strikingly original ideas on a variety of problems in real and complex geometry and laid out a program of applications to period mappings and domains, algebraic cycles, Nevanlinna theory, Brill-Noether theory, and topology of K¨ahler manifolds.

A characteristic of Griffithss work is that, while it often has a specific problem in view, it has served in multiple instances to open up an entire area to research.

Early on, he made connections between deformation theory and Hodge theory through infinitesimal methods, which led to his discovery of what are now known as the Griffiths infinitesimal period relations. These methods provided the motivation for the Griffiths intermediate Jacobian, which solved the problem of showing algebraic equivalence and homological equivalence of algebraic cycles are distinct. His work with C.H. Clemens on the non-rationality of the cubic threefold became a model for many further applications of transcendental methods to the study of algebraic varieties.

His wide-ranging investigations brought many new techniques to bear on these problems and led to insights and progress in many other areas of geometry that, at first glance, seem far removed from complex geometry. His related investigations into overdetermined systems of differential equations led a revitalization of this subject in the 1980s in the form of exterior differential systems, and he applied this to deep problems in modern differential geometry: Rigidity of isometric embeddings in the overdetermined case and local existence of smooth solutions in the determined case in dimension 3, drawing on deep results in hyperbolic PDEs(in collaborations with Berger, Bryant and Yang), as well as geometric formulations of integrability in the calculus of variations and in the geometry of Lax pairs and treatises on the geometry of conservation laws and variational problems in elliptic, hyperbolic and parabolic PDEs and exterior differential systems.

All of these areas, and many others in algebraic geometry, including web geometry, integrable systems, and Riemann surfaces, are currently seeing important developments that were stimulated by his work.

His teaching career and research leadership has inspired an astounding number of mathematicians who have gone on to stellar careers, both in mathematics and other disciplines. He has been generous with his time, writing many classic expository papers and books, such as “Principles of Algebraic Geometry”, with Joseph Harris, that have inspired students of the subject since the 1960s.

Griffiths has also extensively supported mathematics at the level of research and education through service on and chairmanship of numerous national and international committees and boards committees and boards. In addition to his research career, he served 8 years as Duke’s Provost and 12 years as the Director of the Institute for Advanced Study, and he currently chairs the Science Initiative Group, which assists the development of mathematical training centers in the developing world.

His legacy of research and service to both the mathematics community and the wider scientific world continues to be an inspiration to mathematicians world-wide, enriching our subject and advancing the discipline in manifold ways.

Leelavati Prize Winner:

Adrián Paenza

University of Buenos Aires, Argentina

[Adrian Paenza is awarded the Leelavati Prize] for his contributions have definitively changed the mind of a whole country about the way it perceives mathematics in daily life. He accomplished this through his books, his TV programs, and his unique gift of enthusiasm and passion in communicating the beauty and joy of mathematics.

Adrián Paenza has been the host of the long-running weekly TV program “Cient´ıficos Industria Argentina” (“Scientists Made in Argentina”), currently in its twelfth consecutive season in an open TV channel. Within a beautiful and attractive interface, each program consists of interviews with mathematicians and scientists of very different disciplines, and ends with a mathematical problem, the solution of which is given in the next program.

He has also been the host of the TV program “Alterados por Pi” (“Altered by Pi”), a weekly half-hour show exclusively dedicated to the popularization of mathematics; this show is recorded in front of a live audience in several public schools around the country.

Since 2005, he has written a weekly column about general science, but mainly about mathematics, on the back page of P´agina 12, one of Argentinas three national newspapers. His articles include historical notes, teasers and even proofs of theorems.

He has written eight books dedicated to the popularization of mathematics: five under the name “Matem´atica

. . . ¿est´as ah´ı?” (“Math . . . are you there?”), published by Siglo XXI Editores, which have sold over a million copies. The first of the series, published in September 2005, headed the lists of best sellers for a record of 73 consecutive weeks, and is now in its 22nd edition. The enormous impact and influence of these books has extended throughout Latin America and Spain; they have also been published in Portugal, Italy, the Czech Republic, and Germany; an upcoming edition has been recently translated also into Chinese.

# 失之毫厘，差之千里

1961年，作为天气预报员的Lorenz在利用计算机来做气象预测时，为了省事，就在第二次计算的时候，直接从第一次程序的中间开始运算。但是两次的预测结果产生了巨大的差异。Lorenz看到这个结果之后大为震惊，然后经过不断地测试，发觉在自己的模型当中，只要初始的数据不一样，就会产生不同的结果，而且结果大相径庭。在1979年的科学会议上，Lorenz简单的描述了“蝴蝶效应”:

$x^{'}(t) =\sigma(y-x)$

$y^{'}(t)=x(\rho-z)-y$

$z^{'}(t)=xy-\beta z$

# 分形几何学：复杂简单化

$z \mapsto z^{2}+c$

# 科普文：从人人网看网络科学（Network Science）的X个经典问题

http://blog.renren.com/share/270937572/16694767796?from=0101010202&ref=hotnewsfeed&sfet=102&fin=3&fid=24297752024&ff_id=270937572&platform=0&expose_time=1386225841

### 科普文：从人人网看网络科学（Network Science）的X个经典问题作者： 邓岳

长文，写了N个小时写完的。你肯定能看懂，所以希望你能看完，没看完就分享/点赞没有意义。有图有超链接，不建议用手机看。相关内容我想应该可以弄成一个小项目加到某门课中。

1、链路预测（Link Prediction）

（1）最简单的指标（Common Neighbors，CN）

CN为两人共同好友的个数，直观感觉CN越大，此二人是好友的可能性越大。

CN(x,y)=|N(x)∩N(y)|        N(x)为节点x的所有邻居（计算原因也可以加上x自己）

（2）改进指标（Jaccard index）

Jaccard(x,y)=|N(x)∩N(y)| / |N(x)∪N(y)|

我刚注册人人，一个好友都没有，何谈共同好友？

这里还有问题大家可以考虑：推荐的都是目前和你没加好友的人，但整个人人里和你不是好友的有几千万人，总不能给这些人全都和你算一个共同好友数目，然后排序推荐。如何能圈定一个大致的范围？

==================================

2、社团发现（Community Detection）

==================================

3、中心性（Centrality）

==================================

4、复杂网络的一些拓扑特性

（1）小世界（Small World）

（2）无标度（Scale-free）

（3）Giant Component

Facebook的数据显示约99.7%的用户处在一个超大的Giant Component中。

==================================

5、高影响力节点（high influential nodes）/传播（spread）

1、首先删除网络中所有度为1的节点。删完以后检查，原来某些度大于1的节点会变成度为1的，就继续删，删完再检查，再删……直到没有度为1的节点为止。最终，认为刚才所有删掉的的节点属于第一层，即ks=1的节点（上图外围蓝色圈）；

2、现在网络中肯定没有度为1的节点了（都删掉了），那就开始删除度为2的节点（和上一步方法一致）。这次删掉的就是ks=2的节点（上图绿圈）；

3、依此类推，接着删度为3的节点，然后是度为4的节点……最终网络删干净了，网络中所有的点都被分配了一个ks值。

==================================

6、网络的比对（alignment）、去匿名化（de-anonymization）

==================================

7、动态网络（Dynamic Networks）/演化（Evolution）

（1）Growth：同一个社团，在相邻时刻出现了增长（如节点变多了）。和Growth相对的就是Contraction。

（2）Merging：t时刻的两个社团，在t+1时刻合并成了一个社团。和Merging相对的就是Splitting。

（3）Birth：t+1时刻新出现了一个社团（该社团在t时刻不存在）。和Birth相对的就是Death。

【未完待续】

==================================

1、《网络科学导论》《链路预测》《网络科学》 、《Network Science》 入门且全面，正统的Network Science

2、《推荐系统实践》《推荐系统》 主流的Web应用里都有推荐系统，算是网络科学的主要应用方向

3、《网络、群体与市场》 也是入门书，结合经济学、社会学、计算与信息科学以及应用数学的有关概念与方法，考察网络行为原理及其效应。

4、《链接》 巴拉巴西早期经典著作

1、Social Network Analysis 2013.3开课时我全程跟下来并拿到了成绩。2013.10再次开课。强烈建议英文差不多的10级不考研同学10月跟一下这个。

2、网络、群体与市场 中文公开课，在Coursera也有。对软件方向的同学，建议重点看一下课程的第2、3、9、11、13~17章。

3、Social and Economic Networks: Models and Analysis，斯坦福，2014.1开课（来自文后留言）

# 转载：世界十个著名悖论的最终解答

（一）电车难题（The Trolley Problem）

Das曰：

Das这样驳斥这种观点：

Das曰：

Das认为：

Das来讲一个现实生活中的真实的故事：

A显然对这种威胁不屑一顾：“我真的不知道你问什么。”

A这一次没有回答。

Das曰：

Das曰：

das曰：

Das曰：

“中文房间”最早由美国哲学家John Searle于20世纪80年代初提出。这个实验要求你想象一位只说英语的人身处一个房间之中，这间房间除了门上有一个小窗口以外，全部都是封闭的。他随身带着一本写有中文翻译程序的书。房间里还有足够的稿纸、铅笔和橱柜。写着中文的纸片通过小窗口被送入房间中。根据Searle，房间中的人可以使用他的书来翻译这些文字并用中文回复。虽然他完全不会中文，Searle认为通过这个过程，房间里的人可以让任何房间外的人以为他会说流利的中文。

Searle创造了“中文房间”思想实验来反驳电脑和其他人工智能能够真正思考的观点。房间里的人不会说中文；他不能够用中文思考。但因为他拥有某些特定的工具，他甚至可以让以中文为母语的人以为他能流利的说中文。根据Searle，电脑就是这样工作的。它们无法真正的理解接收到的信息，但它们可以运行一个程序，处理信息，然后给出一个智能的印象。

“中文房间”问题足够著名，这是塞尔为了反击图灵设计的一个思想实验。

“我不知道。”

Das曰：除非你脑袋里头首先有必要的相关知识、概念，并且能够使用这些知识、概念对感觉到的事实、现象、真理进行分类整理、分析判断，得出相应的结论，否则你不可能“知道”任何东西。

Das在很多帖子里多次谈到薛定谔的猫，这个悖论的重要性不言而喻。薛定谔的猫和麦克斯韦的妖并列为科学史上的两大奇观。不同的是麦克斯韦的妖是一个已经解决的问题，薛定谔的猫至今仍悬而未决。有人说薛定谔猫态在介观尺度早已实现了，有人说哥本哈根解释早已崩溃了，公说公有理，婆说婆有理。很多人不愿意介入这场争论——尽管这是现阶段人类面临的最为重要的问题——不是他们不感兴趣，而是他们根本不愿意花费数年的生命去搞清楚量子力学的基本原理。
Das曾经立志要让毫不懂得量子力学的人在二十分钟之内了解薛定谔的猫，可是我失败了。失败了不要紧，我们从头再来。这一次das不再用现实世界中的例子来比喻，而是用一个如假包换的量子力学的真实事例来说明：

“反对称”是什么意思？

“纠缠态”、“叠加态”真的存在吗？或者仅仅是数学对我们不了解的原因给与了近似的描述？

10．缸中的大脑（Brain in a Vat）

# Five (math) things to do before you die

An interesting question was posed to me recently. If you were told you were going to die tomorrow, which 5 math topics/questions would you be most sad you never got to learn about/have answered? First of all, I must admit I freeze any time people ask me to rank my top five anything. It feels so final, and I really want to think about it carefully before I answer. Also, honestly, if I were told I had 24 hours to live I would be sad and upset but probably not about the math I was going to miss. But that is not the point of the question, I guess. In this post I will attempt to answer this question, with full awareness that I may change my mind in a few days. But I will also pose a few other questions and then leave it to you, my readers, to ponder them.

1. My immediate response to the question was the Riemann hypothesis. Not that given 100 more years to live I would have any hope of solving this problem, but I would like to see it proved in my lifetime. Especially because we are all pretty certain that it’s true.

Of course, then one can go through the list of Millenium Problems and I would add two more things:

2. the Birch and Swinnerton-Dyer Conjecture, and

3. the P vs NP problem.

Again, I am not saying I have any chance of solving them, just that I would like to see the solutions to these problems. But this is where it gets tricky. I basically have a list of three things that probably anyone could have made (these are some of the most famous problems in math!). So how do I add two more things to it? Nothing will seem as important (nothing else I can think of would make you a millionaire!). OK, there are three other millenium problems, but I’m just not as interested in them. So then I started thinking about the math topics I would be sad not to have learned if I were to die tomorrow.

4. I have gotten interested in mirror symmetry and its relation to physics and number theory, so I guess I would be sad if I died tomorrow without learning more about it.

5. Arithmetic dynamics, since I am very interested but kind of new to it.

But doesn’t the list become weak after I add these two things? Anyway, please share your Top 5 in the comments below.

The original question got me thinking about other fun questions on might ask:

– Which 5 math books would you take to a desert island? The funny thing is that I can’t think of a top 5 but I can always think of at least one or two things. For example, I would bring Serre’s A Course in Arithmetic. But of course, if you asked me to bring just one I would be stuck.

– Who are your Top 5 mathematicians of all time? Gauss? Ramanujan?

-Slight variation: which 5 mathematicians would you take to a desert island? See, here I would probably pick some fun/handy mathematicians. I don’t know if Gauss would be very good at building a hut.

– What are the best 5 math formulas? Euler’s formula is widely regarded as one of the most beautiful formulas in mathematics. Do you agree? Can you think of others?

– What are your 5 favorite functions? I know one: hypergeometric functions!

As a final comment, I wanted to say the first question was suggested by my friend Casey Douglas, who is an Assistant Professor at St. Mary’s College of Maryland. He thought of this question as he was preparing a talk for the SMCM math department’s annual “MATH WEEK OF AWESOME”, which sounds, indeed, awesome.

So now I open it to you. Do you have answers to these questions? Do you also find it slightly frustrating when these questions are posed (if so, I apologize)? Can you think of other questions like this?

http://blogs.ams.org/phdplus/2012/03/23/five-math-things-to-do-before-you-die/