MA 1505 Tutorial 1: Derivative

Definition of Derivative:

f^{'}(x)=\lim_{\Delta x\rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

Rule: Assume f(x) and g(x) are two differentiable functions, the basic rules of derivative are

(f\pm g)^{'}(x)=f^{'}(x)\pm g^{'}(x)

(f\cdot g)^{'}(x)= f^{'}(x) g(x) + f(x)g^{'}(x)

(f/g)^{'}(x)=(f^{'}(x)g(x)-f(x)g^{'}(x))/(g(x))^{2}

(f\circ g)^{'}(x)=f^{'}(g(x))g^{'}(x)

Definition of Critical Point: x_{0} is called a critical point of f(x), if f^{'}(x_{0})=0.

If f^{'}(x)>0 on some interval I, then f(x) is increasing on the interval I. Similarly, if f^{'}(x)<0 on some interval I, then f(x) is decreasing on the interval I.

Tangent Line: Assume f(x) is a differentiable function on the interval I, then the tangent line of f(x) at the point x_{0}\in I is y-f(x_{0})=f^{'}(x_{0})(x-x_{0}), where f^{'}(x_{0}) is the slope of the tangent line.

Derivative of Parameter Functions: Assume y=y(t) and x=x(t), the derivative y^{'}(x) is y^{'}(t)/x^{'}(t), because the Chain Rule of derivatives.

Question 1. Calculate the tangent line of the curve x^{\frac{1}{4}} + y^{\frac{1}{4}}=4 at the point (16,16).

Method (i). Take the derivative of the equation x^{\frac{1}{4}}+y^{\frac{1}{4}}=4 at the both sides, we get

\frac{1}{4}x^{-\frac{3}{4}} + \frac{1}{4}y^{-\frac{3}{4}} y^{'}=0.

Assume x=y=16, we have the derivative y^{'}(16)=-1. That means the tangent line of the curve at the point (16,16) is y-16=-(x-16). i.e. y=-x+32.

Method (ii). From the equation, we know y(x)=(4-x^{\frac{1}{4}})^{4} , then calculating the derivative directly. i.e.

y^{'}(x)=4(4-x^{\frac{1}{4}})^{3}\cdot (-1)\cdot \frac{1}{4}x^{-\frac{3}{4}}

Therefore, y^{'}(16)=-1.

Method (iii). Making the substitution x=4^{4}\cos^{8}\theta, y=4^{4}\sin^{8}\theta, then (16,16) corresponds to \theta=\pi/4. From the derivative of the parameter functions, we know

\frac{dy}{dx}= \frac{dy/d\theta}{dx/d\theta}=\frac{4^{4}\cdot 8\sin^{7}\theta\cdot \cos\theta}{4^{4}\cdot 8\cos^{7}\theta\cdot (-\sin\theta)}

If we assume \theta=\pi/4, then y^{'}(16)=-1.

Method (iv). Geometric Intuition. Since the equation x^{\frac{1}{4}}+y^{\frac{1}{4}}=4 is a symmetric graph with the line y=x, and (16,16) is also on the symmetric line. Therefore, the slope of the curve at the point (16,16) is -1. Hence, the tangent line is y=-x+32.

Question 2. Let y=(1+x^{2})^{-2} and x=\cot \theta. Find dy/dx and express your answer in terms of \theta.

Method (i). y=\frac{1}{1+x^{2}}= \sin^{2}\theta

\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta} = \frac{2\sin\theta \cos \theta}{-\sin^{-2}\theta}= - \sin^{2}\theta\sin2\theta.

Method (ii). \frac{dy}{dx}=-\frac{2x}{(1+x^{2})^{2}} = -\frac{2\cot \theta}{(1+\cot^{2}\theta)^{2}}=-\sin^{2}\theta\sin 2\theta.

PHD身边的时间陷阱

对于一个读博士的学生来说,一天的科研工作可以从早晨八点开始,直至深夜,但是以什么样的状态,什么样的心情来进行这一天的科研就是一个关键的问题。通常来说,在办公室就有一个坑,一个专门坑PHD的时间陷阱,而挖这个陷阱的往往就是自己。

一个PHD可以在早晨8:00起床,洗漱完了之后就可以在8:30左右到达办公室。走到楼下,去楼下的小卖部买了一个小面包就上楼了。到了之后,先去打一壶开水,去卫生间把昨天的杯子清洗一下,然后泡一杯咖啡放在办公桌前面。

8:30这个时候通常都会不由自主的打开电脑,然后链接无线网络,心想:一天的工作开始了。但是打开电脑之后,第一个打开的往往不是LaTex或者Word文档,而是各种各样的浏览器。首先登录的就是自己的邮箱,看一看有没有学校发来的邮件,老板发来的邮件。OK,如果没有,那就看一下有没有各种打折促销的邮件。如果有,那就看一下各类网购的网站,看看有没有自己需要的东西。这样在网上晃一下,就已经一个小时过去了。

9:30。这个时候facebook上面一个消息弹了出来,原来是一个朋友留了一个言,登陆上去回复一下,顺便更新一下自己的最新状态,并且回复一些朋友的新鲜事。这个时候看到了一个不错的新闻,登录百度搜索了一下,去网易,新浪上面看一下网友们的评论,顺便灌了一点水。

坐了一个多小时,应该起来晃一下了,就哼着小曲去了卫生间,慢慢的洗了个手,就已经10:00了。这个时候,系里面的邮件来了,让大家后天交一份报告,于是慢慢的打开了Word文档。纠结了十分钟之后,不知道如何下手,突然想到Google是最好的老师,上去搜索了一大堆资料,储存在电脑硬盘里面,觉得先看一下再开始写自己的报告比较合适,于是已经过了10:30了。这个时候想到早晨自己起来得早,饭没有吃饱,就半个小时就可以吃今天的午饭了,不需要在早上科研了。这样理所当然的就觉得玩一会算了,就顺理成章的说服自己打开了Diablo 3,上去看了下自己过去的装备,今天没有完成的任务是什么,打造了一些新的装备,拆卸了一些没用的东西。

好了,11:15到了,Office突然想起了敲门声,另外办公室的一个PHD来敲门了,喊大家去吃午饭,因为12:00的时候食堂里面人山人海。下楼吃饭,吃完饭了,12:00刚刚好,看着下面排队的人群,觉得自己提前下来吃饭非常的明智。

回到办公室,觉得刚吃完饭,可以休息一下,不应该马上投入苦逼的科研工作,于是再次打开了电脑,点击了浏览器,输入了新浪的网址,看了一下最新的体育新闻,最新的NBA比分结果,最新的足球转会消息。看了新闻之后,觉得不满足,打开了YouTube,看一看自己昨天没有看完的电视剧。

好,此时已经是14:00了。应该开始写自己应该写的论文了,打开LaTex,发现自己昨天思考的方法错了,应该重新审视自己的论文步骤。哎,还是翻一下别人遇到类似问题怎么处理的吧。再次打开GOOGLE搜索,搜了一会,发觉Google的功能非常的强大,搜到了自己需要的论文,但是如果没有登录学校的数据库,就没法下载论文。那就打开了学校的图书馆网址,填写了自己的用户名和密码,ok,一切顺利,下载好了论文并且去打印室打印出来。

15:00,突然瞅了一眼自己的日程安排,突然想起来这个时候是系里面的Tea Break,有免费的食物和咖啡,于是拿着自己的杯子就下楼,找到一些PHD同学,系里面的老师聊了一下天,吃了一些事物,谈论了一下最近的新闻。一个小时后,Tea Break结束,拿着自己的杯子返回Office。

现在已经16:00了,突然觉得自己的桌子有点脏,于是拿着抹布去了卫生间,弄湿了来Office把自己的桌子擦拭干净。16:15,这个时候想起来,炉石传说每天新的任务就在这个时候更新。拿出了Ipad,打开了炉石传说的app,看了一下今天的任务,就觉得自己今天肯定能够完成,于是玩了几把炉石传说。

在17:00觉得貌似可以开始下楼吃晚饭了,于是就喊上office的几位好友,一同前往楼下的食堂。在电梯里的时候,有人提议去外面吃一顿吧,改善下自己的伙食,在下面的食堂已经吃腻了。于是就去了地铁站,直奔外面的餐馆。吃完饭,就可以收拾收拾自己在office没有完成的工作返回宿舍了。

其实在PHD的办公室中,一直都有这种时间陷阱,会让人有一种自己总处于很忙碌的错觉。看着自己电脑右上角的时间一点一点过去,但是自己应该做的事情却一点都没有干。这种就是拖延症,让自己无法从这种状况下逃离,只能在一个漩涡里面越陷越深。

Manjul Bhargava and his 290 theorem

Fight with Infinity

ICM 2014今天在韩国首尔召开。正如之前所预测的那样,Manjul Bhargava获得了2014年的Fields Medal. 一同获奖的还有Artur Avila, Martin HairerMaryam Mirzakhani.

这份名单相当有趣:史上第一位女性获奖者(或许是为了赶在美国第一位女总统之前?);4人广义上的“祖国”(印度、巴西、奥地利以及伊朗)此前均无Fields奖得主(何其政治正确!当然这也反映了现今欧美基础学科研究人员的“去欧美化”趋势);3人的研究工作和遍历理论紧密相关;2人有参加IMO的经历(有此经历的Fields奖得主越来越多,当然,其中并没有华人的身影——丘成桐或许会愿意就数学研究和奥林匹克数学的关系作进一步的评论);等等。

本文将介绍获奖者Manjul Bhargava的一项“初等”工作:简化了Conway-Schneeberger 15定理的证明,并进一步证明了Conway的290猜想。

1.
我们感兴趣的是在整格$latex Bbb Z^n$上取整值的$latex n$元多项式$latex f$。若$latex f$是齐次的,这相当于要求$latex f$的系数为整数。对可表示集$latex R_f:=f(Bbb N^n) subset Bbb Z$(约定$latex 0 in Bbb N$)的研究贯穿了整个数论史:
(1.1)Fermat集中研究了用2元2次整系数多项式表示素数$latex p$的问题,并发现
若$latex f(x,y)=x^2+y^2$,则$latex p in R_f$当且仅当$latex p$形如$latex 4a+1$;
若$latex f(x,y)=x^2+2y^2$,则$latex p in R_f$当且仅当$latex p$形如$latex 8a+1$或$latex 8a+3$;
若$latex f(x,y)=x^2-2y^2$,则$latex p in R_f$当且仅当$latex p$形如$latex 8a+1$或$latex 8a+7$;
若$latex f(x,y)=x^2+3y^2$,则$latex p in R_f$当且仅当$latex p$形如$latex 3a+1$;
若$latex f(x,y)=x^2+5y^2$,则$latex p in R_f$当且仅当$latex p$形如$latex 20a+1$或$latex 20a+9$;
此类现象是代数数论乃至类域论的渊薮。
(1.2)由Fermat二平方和定理开始,Euler等数学家获得了一系列经典结果。
(Fermat二平方和定理, 由Euler证明) 若$latex f(x,y)=x^2+y^2$,则自然数$latex k in R_f$当且仅当$latex k$的奇素因子(若有)均形如$latex 4a+1$。
(Lagrange四平方和定理) 若$latex f(x,y,z,w)=x^2+y^2+z^2+w^2$,则$latex R_f=Bbb N$。
(Legendre三平方和定理) 若$latex f(x,y,z)=x^2+y^2+z^2$,则自然数$latex k in R_f$当且仅当$latex k$不能写成$latex 2^{2a}(8b+7)$的形式。
(1.3) 平方数有一类推广,即所谓的多边形数:填满正多边形内部的点的个数。
(Gauss三角数定理,“Eureka定理”)令$latex f(x,y,z)=frac{x(x+1)}{2}+frac{y(y+1)}{2}+frac{z(z+1)}{2}$,则$latex R_f=Bbb N$。
推广Gauss三角数定理和Lagrange四平方和定理,我们有如下结果:
(Fermat多边形数定理,由Cauchy证明) 任意自然数均可表示为不超过$latex n$个$latex n$边形数之和。
(1.4)从Lagrange四平方和定理出发,我们也可以研究高次幂多项式的表示问题:
(Waring问题,由Hilbert解决) 给定$latex k geq 2$,$latex f=sum_{1 leq i leq g} x_i^k$。对于充分大的$latex g$,$latex R_f=Bbb N$。
关于$latex g$的下确界$latex…

View original post 234 more words

Khot, Osher, Griffiths

What's new

In addition to the Fields medallists mentioned in the previous post, the IMU also awarded the Nevanlinna prize to Subhash Khot, the Gauss prize to Stan Osher (my colleague here at UCLA!), and the Chern medal to Phillip Griffiths. Like I did in 2010, I’ll try to briefly discuss one result of each of the prize winners, though the fields of mathematics here are even further from my expertise than those discussed in the previous post (and all the caveats from that post apply here also).

Subhash Khot is best known for his Unique Games Conjecture, a problem in complexity theory that is perhaps second in importance only to the $latex {P neq NP}&fg=000000$ problem for the purposes of demarcating the mysterious line between “easy” and “hard” problems (if one follow standard practice and uses “polynomial time” as the definition of “easy”). The $latex {P neq…

View original post 1,244 more words

Lindenstrauss, Ngo, Smirnov, Villani

What's new

As is now widely reported, the Fields medals for 2010 have been awarded to Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani. Concurrently, the Nevanlinna prize (for outstanding contributions to mathematical aspects of information science) was awarded to Dan Spielman, the Gauss prize (for outstanding mathematical contributions that have found significant applications outside of mathematics) to Yves Meyer, and the Chern medal (for lifelong achievement in mathematics) to Louis Nirenberg. All of the recipients are of course exceptionally qualified and deserving for these awards; congratulations to all of them. (I should mention that I myself was only very tangentially involved in the awards selection process, and like everyone else, had to wait until the ceremony to find out the winners. I imagine that the work of the prize committees must have been extremely difficult.)

Today, I thought I would mention one…

View original post 3,867 more words

Avila, Bhargava, Hairer, Mirzakhani

What's new

The 2014 Fields medallists have just been announced as (in alphabetical order of surname) Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani (see also these nice video profiles for the winners, which is a new initiative of the IMU and the Simons foundation). This time last year, I wrote a blog post discussing one result from each of the 2010 medallists; I thought I would try to repeat the exercise here, although the work of the medallists this time around is a little bit further away from my own direct area of expertise than last time, and so my discussion will unfortunately be a bit superficial (and possibly not completely accurate) in places. As before, I am picking these results based on my own idiosyncratic tastes, and are not necessarily the “best” work of these medallists.

Artur Avila works in dynamical systems and in the…

View original post 1,412 more words

2014 International Congress of Mathematics: Awards

Fields Medalist:

Artur Avila

p_1

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

p_2

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

p_3

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

p_4

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

p_5

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

p_6

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

p_7

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

p_8

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.

NUS UTown成为登革热病危险区

http://www.channelnewsasia.com/news/singapore/nus-university-town/1298832.html

到目前为止,作为NUS的主要住宿区之一,University Town已经有十个登革热的病例。根据National Environment Agency(NEA)的报道,新加坡国立大学的University Town已经成为登革热的爆发区之一,目前已经有十个登革热的病例。总共有1800名学生住在University Town,这里是学生和教职人员的主要活动场所之一。在新加坡目前已经被划出21个登革热的危险地区,最大的区域在Choa Chua Kang,那里有527个病例。今年新加坡总共报告了12438个登革热的病例。根据NEA的网站统计,在上周日和这周周二,总共有200个病例爆发。

utown-dengue-cluster-data

2014年8月6日

 

何谓登革热?

登革热﹝俗称“断骨热”﹞是一种由登革热病毒引起的急性发热传染病,由蚊子传播给人类。病原体为登革热病毒( 可分为 1、2、3、4 型 )。全球每年约有五千万宗登革热个案,常见于热带和亚热带地域。近年登革热转趋活跃,影响全球各地,在东南亚部分国家,登革热已成为地方性流行病,国内有输入病例或局部暴发疫情出现。

登革热的病媒是什么?

登革热通过带有登革热病毒的雌性伊蚊叮咬而传染给人类。主要传播媒介为埃及伊蚊、白纹伊蚊。其中白纹伊蚊(俗称“花斑蚊”)在我省分布广泛,主要在清水容器中孳生,大多数在屋外或野外阴暗处流连,但亦会在户内活动。雌蚊嗜吸人血,吸血高峰在日落前两小时(约为下午五、六时),及早上八、九时。室外及室内皆可叮咬人。

登革热的传染途径是什么?

当人被带病毒蚊子叮咬后,病毒会从蚊子之唾液进入人体血液而感染。如果病者在刚发烧前至退烧期内(大约六至七日)被蚊叮,病毒就有可能传给蚊子继而传播开去。此病并不会经由人与人之间传播,与患者接触是不会被传染的。

典型登革热的病征是什么?

感染登革热病毒后,经过3至15天的潜伏期(通常为5至8日),患者多以突然发热为首发症状,持续发热3~5天,严重头痛,四肢酸痛、关节痛、肌肉痛、背痛、后眼窝痛。发病后3、4日出现红疹,恶心、呕吐,轻微的流牙血和流鼻血。病后有可能出现极度疲倦及抑郁症状,极少数病者会恶化至出血性登革热,并进一步出血、休克,严重时可引致死亡。

个人如何做好登革热防护?

现时并没有一种有效疫苗来预防登革热。预防登革热的最佳方法就是清除积水,防止伊蚊孳生,以避免给蚊子叮咬,有关预防蚊咬的措施如下:

到登革热流行区旅游或生活,应穿着长袖衣服及长裤,并于外露的皮肤及衣服上涂上蚊虫驱避药物。

如房间没有空调设备,应装置蚊帐或防蚊网。

使用家用杀虫剂杀灭成蚊,并遵照包装指示使用适当的份量。切勿向运作中的电器用品或火焰直接喷射杀虫剂,以免发生爆炸。

避免在“花斑蚊”出没频繁时段在树荫、草丛、凉亭等户外阴暗处逗留。

防止积水,清除伊蚊孳生地:

尽量避免用清水养殖植物。

对于花瓶等容器,每星期至少清洗、换水一次,勿让花盆底盘留有积水。

把所有用过的罐子及瓶子放进有盖的垃圾桶内。

将贮水容器、水井及贮水池加盖。

所有渠道要保持畅通。

将地面凹陷的地方全部填平,以防积水。

怀疑自己感染登革热时最要紧是请教医生。

如何预防控制登革热暴发?

预防登革热暴发的有效措施就是杀灭伊蚊,其中除紧急杀灭带毒成蚊外,翻盆倒罐清除伊蚊孳生地,迅速降低蚊媒密度更为重要。

灭蚊要重在落实,检查评估考核效果:定期检查有否妥善弃置可积水的器皿杂物?(例如将空罐、发泡胶盒、杯、水樽、汽水罐等放进有盖的垃圾桶内)。有否盖好贮水容器?沟渠是否畅通?有否定期清洗积水容器?(例如花樽、花盆碟、冷气机盛水器、水缸、贮水池、废旧轮胎等)有否填平凹陷的地面以防积水?要注意检查花园、园艺角、花盆盛水碟、水栽植物,小卖部、垃圾站、沟渠、洗手间、水箱、贮水池等。

近期学校开学,教育部门要做好登革热防控工作。各学校要做好开学前的杀灭成蚊、清理伊蚊孳生地。加强健康教育、增加大家对蚊虫传播疾病的认识,提高对预防登革热的意识及评估其风险,鼓励全校积极参与预防蚊患。

旅游者如何防护登革热?

登革热常出现在热带与亚热带地区。夏秋季到东南亚旅游时,,提高自我保护意识。要做好防蚊、个人保护措施。旅游后半个月内如出现发热,应尽早就医治疗,并向医生说明旅行史。

[转载]美国是天堂吗?

南大校友王庆根,原为奥赛金牌得主,斯坦福大学化学博士,Paypal的首席工程师,可以说学业和事业都很成功,却因抑郁症,本月初自杀,留下一双儿女。这是多么惨痛的悲剧!王博士的经历和我惊人地相似,同年出生,都在南大上过学,后来到美国闯荡,他的孩子和我的孩子一样大。我自己还苟活着,但同病相怜,觉得我们这些在美国生活得时间比较久的人,有必要多说说自己实际的生活状况,让其余的人做选择的时候,起码多一些参考。我和同样在美国生活的涂子沛兄(《大数据大趋势》一书作者,现居匹兹堡)相约,就这个话题,展开一些讨论。

在美国的生活,起码对中国移民来说,是“儿童的天堂,中年人的战场,老年人的地狱”。这种概括虽笼统,却不离谱。为了给儿童一个天堂,我们闯进了战场。在美国生活的不易,很少有当事人自己说过。对外人,大家要面子,家丑不外扬,有问题不暴露。对家人,大家报喜不报忧。久了不说,问题就可能酿成悲剧。

我不知王博士的离世究竟是什么原因,但不妨借题发挥,顺着“压力”这个话题,说说在美国生存的压力。我只说自己认为比较重要的几点,抛砖引玉,希望其他海外朋友补充。

国内报道,多强调王走上绝路,是因工作压力太大。表面上看这似是最合理的解释,但未必有普遍意义。就我自己的体验,海外中国人的隐形压力不止工作。事实上,工作压力有时候还算次要。美国职场环境相对宽松,大部分美国上司处事随和。做同类工作,可能在国内的压力更大。当然我这里说的一切话都是笼统的说法,具体情况因人而异。

那看不见的让人崩溃的压力究竟来自何方?我最近就遇到几个人,也抱怨说自己快得忧郁症了。原因和工作本身无关,倒是都牵涉到海外生活的孤立无助,或是紧张的家庭关系。

这种紧张来源有很多,比如孩子上学。美国学校通常三点下课。很多地方又规定,不到法定年龄,孩子不可无大人陪伴,单独在家。如果夫妻双方都上班,孩子的接送和安全就成了大问题。另外,美国的暑假长达三个多月,这中间孩子怎么办?有的送回国,有的请国内祖父母来带,有的花钱请人, 有的送往暑期的各种夏令营。每一种方法,都非常折腾。总的来说,我建议,能用钱解决的问题,争取用钱解决。人情债,以及由此产生的不和谐关系,能避免尽量避免,不要贪小便宜,最终后患无穷。

美国人自己也有这些问题,但他们毕竟是本地人,解决办法更多。有些是夫妇的一方把工作辞掉,或者换成兼职工作,时间上灵活起来,以便照顾孩子的起居和接送。目前来说,美国经济萧条,双职工家庭越来越多。即便这样,妇女在家不上班,也是常态,所有人都理解,她们自己也坦然。共和党的总统候选人罗姆尼的太太,被人指责“一辈子没工作过”,能谈什么经济?这个说法,反倒让罗姆尼得分。罗姆尼太太说她家中要负责五个孩子,这不叫工作什么叫工作?这个说法赢得了很多选民的认同。美国的纳税是根据全家收入来算的。除非真能挣到钱,否则,考虑到纳税、雇人看孩子成本,孩子成长中家长参与的欠缺等多方面原因,去工作反而得不偿失。如果孩子多,夫妻一方收入不高,那还不如别去上班。一定有那么一个公式,让我们计算到工作与不工作的成本-收益平衡点在哪里。

这也不仅是经济问题,个人自我认知和心态调整也很重要。来自我们大陆的家庭,心态一关就很难过。不少家庭里,夫妻在美国生活久了,可因地制宜,适应当地环境。但国内父母甚至其他亲戚的聒噪,则是新的一重压力。有些老年人一辈子下来,除了工作挣钱,找不到还有什么别的东西,可以去寄托人生的意义,也无法理解美国这边的情况,用国内环境下的心态,乱出主意。他们有的是为了面子,希望告诉他人自己的孩子在美国某某地方上班,不希望邻居同事亲戚朋友知道自己的孩子在美国“没工作”。他们不知道,这有时候是为了家庭的整体利益作出的一种主动选择。子女有时候出于孝顺,只好依从,好让国内父母显摆,小家庭的苦只有自己去尝。也有的父母观念错误,比如“不要在家吃闲饭”,“不要吃丈夫的饭”,硬是劝子女去上班。中国家庭,很多是一方出来读书,一方陪读,有了机会另外一方去读书,本来拿学位就有早晚,不是都能顺利找到理想工作。

有的家庭为了省几个钱,让国内老人过来带孩子。这会使得带孩子的问题表面上缓解,但是这会生出很多新的问题。最大的问题是医疗,在美国,保险通常只保“核心家庭”,亦即配偶和孩子。来访的父母不算dependent. 只能去另买保险。保险公司遇到这种既不是美国公民又不是年轻力壮的人投保,保险费通常很昂贵。很多家庭看情况还行,就去侥幸赌一把,不去买,但一旦父母在美国生病,又没有保险,最终医疗费惊人,甚至一下子就能把小家庭拖垮。这种风险,一些来访老人可能并不知道,有时候也不能理解。遇到这类问题,甚至在子女本来就已经压力重重的时候,因为自己不满而抱怨,让子女的家庭平添矛盾,使得人到中年、夹心饼干一样的他们痛苦不堪。

和其他任何地方的家庭一样,几代人之间的冲突总是难免,比如生活习惯,子女教育等多方面,大家都可能有差异。和国内不同的是,由于来一趟不易,很多父母一来,就把旅游签证所允许的半年用足。有了摩擦,无处可走,所以长时间困在一个地方,矛盾处理不当可能激化,影响小家庭夫妻关系。这样一来,美国生活不仅成为老人的地狱,也会成为三代人共同的地狱。

这种折磨,最终极为损害夫妻关系。再恩爱的夫妻,也架不住这种水滴石穿的长期冲突。很多海外中国夫妻关系紧张。美国人不像我们这样死要面子活受罪,遇到这些问题,会去找婚姻咨询等地方寻求帮助,商量解决夫妇双方解决不了的冲突,所有的工作都做完了还不行就索性散伙。中国人本来就含蓄,有问题自己相互都不说,更不要说寻求专业帮助,所以通常是带着问题过日子,如同两腿绑着沙袋去踢球。美国人在家庭关系中把夫妻关系摆在首要位置,夫妻关系和谐,子女会生活在幸福的环境之下,父母亲也可放心地安度晚年。中国舍本逐末,教孝不教慈,把孝道摆第一位,甚至孝道压倒人道,把很多其它的关系给扭曲了。久而久之,家庭的裂痕越来越大,小家庭又为了儿女或者父母的面子,强忍着在一起,形成过也过不好,离也离不了的亚婚姻。到了海外,在新的文化环境之下,这种冲突越发明显。我的下一本书,《生活意见》(暂定名,将由华师大出版),就谈到了很多这方面的话题。

另外一个压力源是工作许可的问题。美国的移民是一个复杂、漫长而又头痛的过程,人在美国扎下根来并不容易,有时候也没有必要,因为现在在国内,出国旅游、访学、商务也越来越容易。如果选择移民,除结婚和投资的渠道外,大部分中国留学生未来面临的是职业移民。这方面大部分人的过程相差无几:大家先读书,然后找工作,根据工作,一层层办工作许可,每一次都是一场小小的战斗。要是读博士,起码得四五年时间搭进去。然后利用一年到一年半(因专业而异)的“职业实习期”(OPT),此间可合法找工作。OPT是比较临时的工作身份,找到合适的工作后,得尽快转成工作签证。工作签证需要雇主帮你申请,雇主不肯,你只好再去找肯帮你办的雇主。工作签证三年一延,最多七年。这期间,大家努力去办理绿卡。绿卡办理分几个优先顺序,杰出人才办得很快,这要看你的学位(多为博士学位),学术成果等。余下类别多有根据国别的签证配额排期(相当于“入户指标”),排到了才可办理。这排期三五年是常事。排期中,不可轻易改变工作,这让很多人只好接受不满意的工作,这中间离开美国再回来,还要花钱申请Advance parole, “Parole”也是犯人假释的意思,真是“移民监”了。

几番折腾下来,到最终不再受“身份”限制,搞不好就八九年甚至十几年过去了。此时已人到中年,花的心,藏在蕊中,空把花季都错过了。好多人当年的梦想,早已灰飞烟灭。大家只好把兴趣放到孩子和房子身上。自我的丧失,对一个人来说是很凄凉的事。坦白地说,很多第一代移民的人生基本上就这样荒废了。能自我安慰的一点,是给孩子提供了一个好的环境,让他们不需要这样再来折腾。

不过如果心态调整好,学会享受生活,不与人攀比,不盲目追求出人头地,能看菜下饭,找到美国生活的美好之处,倒也海阔天空 ——事实上大部分美国人自己正是这么做的,能做到这样,才叫真正地“融入主流社会”了,很多人以为赚了大钱买了大房子,出人头地了,才觉得心里有底,似乎是在国外融入主流,对国内来说光宗耀祖了,这是一种极大的谬误。其实谁在乎?想想看你自己活得怎样?你开心吗?你对下一代尽到责任了吗?这些才是最重要的。

出国是有风险的事,各位需要慎重,大学刚毕业的人倒无所谓,到哪里不是重新开始?我倒不反对移民这件事本身,但是需要好好权衡。那些在国内事业有成的朋友,可能要认真考虑。千万不要有童话思维,认为到了美国,就可以“永远幸福生活”了。很多问题,不会因为你飞越了太平洋,就可以永久地留在身后。

目前喊移民的中产家庭很多。遇到一个问题,人们解决的办法,一为抗争,一为逃离(fight or flight)。来美国十年了,我发觉flight也不是长久之计。但愿我们都去努力,让中国的教育和各方面大环境能好起来,日后大家的志向,会从“美国梦”,转移到“中国梦”上面。我想我能做的,是尽量去介绍美国的教育,好让我们教育各界人士去取长补短,让日后的教育者、家长和儿童,对他人的模式,不再是只能望洋兴叹。

作者:一南大老师