Notes on probability measures on metric spaces

1 Borel sets

2 Borel probability measure

3 Weak convergence of measures

4 The Prohkorov metic

Let (X,d) be a metric space. Denoted by P(X) all the Borel probability measure on X.

We have defined the notation  of weak convergence in P(X), Define for \mu,\nu\in P(X)

\displaystyle d_P(\mu,\nu):=\inf\{\alpha>0:\mu(A)\leq \nu(A_\alpha)+\alpha\ \text{and}\ \nu(A)\leq \mu(A_\alpha)+\alpha\}

where

A_\alpha=\{x:d(x,A)<\alpha\}.

The function d_P is called the Prokhorov metric on P(X) (induced by  d).

Conclusion If (X,d) is a separable metric space, then so is P(X) with the induced Prokhorov metric. Moreover,  a sequence in P(X) converges in metric if and only if it converges weakly and to the same limit.

Inequality for 1/zeta(s), zeta'(s)/zeta(s), and log zeta(s)

Theorem We have

\displaystyle \zeta(s)=O(\log t)

uniformly in the region

\displaystyle 1-\frac{A}{\log t}\leq \sigma\leq 2,\ t>t_0

where A is any positive constant. In particular

\displaystyle \zeta(1+it)=O(\log t).

It is also easy to see that

\displaystyle \zeta'(s)=O(\log^2 t)

in the above region.

 

There is an alternative method, due to Landau obtaining results of this kind.

Lemma If f(s) is analytic, and

\displaystyle |f(s)|\leq |f(s_0)|e^M\quad (M>1)

in the circle |s-s_0|\leq r, then

\displaystyle |\frac{f'(s)}{f(s)}-\sum_{\rho}\frac{1}{s-\rho}|\leq \frac{AM}{r}\quad (|s-s_0|\leq \frac{1}{4}r),

where \rho runs through the zeros of f(s) such that |\rho-s_0|\leq \frac{1}{2}r.

Lemma 1  If f(s) satisfies the conditions of the previous lemma, and has no zeros in the right hand half of the circle |s-s_0|\leq r, then

\displaystyle -\mathrm{Re}(\frac{f'(s_0)}{f(s_0)})\leq \frac{AM}{r};

while if f(s) has a zero \rho_0 between s_0-\frac{1}{r} and s_0, then

\displaystyle -\mathrm{Re}(\frac{f'(s_0)}{f(s_0)})\leq \frac{AM}{r}-\frac{1}{s_0-\rho_0}.

Lemma Let f(s) satisfy the conditions of Lemma 1, and let

\displaystyle |\frac{f'(s_0}{f(s_0)}|\leq \frac{M}{r}.

Suppose also that f(s)\neq 0 in the part \sigma \geq\sigma_0-2r' of the circle |s-s_0|\leq r, where 0<r'<\frac{1}{4}r. Then

\displaystyle |\frac{f'(s)}{f(s)}|\leq A\frac{M}{r}\quad (|s-s_0|\leq r').

Jensen’s formula

Jensen’s formula relates the average magnitude of an analytic function on a circle with number of its zeroes inside the circle.

Theorem (Jensen’s formula) Suppose that f be an analytic function in a region in the complex plane which contains the closed disk D of radius r about the origin, a_1,a_2,\dots,a_n are zeros of f in the interior of D according to multiplicity, and f(0)\neq 0. Then

\displaystyle \log |f(0)|=\sum_{k=1}^n \log \frac{|a_k|}{r}+\frac{1}{2\pi}\int_0^{2\pi} \log f(re^{i\theta})\,d\theta.

Primitive Dirichlet Characters

Let \chi be a Dirichlet characters of modulus q. Thus \chi is a periodic function of period q. It is possible, that for values of a restricted by condition \gcd(a,q)=1, the function \chi may have d period less than q.

Definition (Primitive characthers) We say that a Dirichlet character \chi  of modulus q is imprimitive if \chi(a) restricted by \gcd(a,q)=1 has a period which is less than q, Otherwise we say that \chi is primitive.

For maximal clarity, let us write out exactly what we mean \chi(a) restricted (a,q)=1 has a period n\in\mathbf{Z}^+. This means: For any integers a,b with \gcd(a,q)=\gcd(b,q)=1 and a=b\ \text{mod}\ n we have \chi(a)=\chi(b).

Definition (Conductor) Given a Dirichlet character \chi of modulus q, the conductor of \chi is defined to be the smallest positive integer q' such that \chi(a) restricted by \gcd(a,q)=1 has a period q'.

Lemma If \chi is a Dirichlet character of modulus q and if q_1\in\mathbf{Z}^+ is a period of \chi(a) restricted by \gcd(a,q)=1, then c(\chi)\mid q_1. Hence in particular ,we have c(\chi)\mid q.

Lemma For each Dirichlet character \chi of modulus q, there is a unique Dirichlet character of \chi' of modulus c(\chi) such that

\displaystyle \chi(n)=\begin{cases}\chi'(n) & \text{if}\ \gcd(a,q)=1\\  0 & \text{if} \gcd(a,q)>1.  \end{cases}

This Dirichlet character \chi' is primitive

 

做数学一定要是天才吗?(陶哲轩)[转载自刘小川WordPress]

做数学一定要是天才吗?(陶哲轩)

(原文:Does
one have to be a genius to do maths?

http://liuxiaochuan.wordpress.com/2008/03/30/%E5%81%9A%E6%95%B0%E5%AD%A6%E4%B8%80%E5%AE%9A%E8%A6%81%E6%98%AF%E5%A4%A9%E6%89%8D%E5%90%97%EF%BC%9F-%EF%BC%88%E8%AF%91%E8%87%AA-%E9%99%B6%E5%93%B2%E8%BD%A9-%E5%8D%9A%E5%AE%A2%EF%BC%89/

[转载自刘小川WordPress]

做数学一定要是天才吗?

这个问题的回答是一个大写的:!为了达到对数学有一个良好的,有意义的贡献的目的,人们必须要刻苦努力;学好自己的领域,掌握一些
其他领域的知识和工具;多问问题;多与其他数学工作者交流;要对数学有个宏观的把握。当然,一定水平的才智,耐心的要求,以及心智上的成熟性是必须的。但
是,数学工作者绝不需要什么神奇的“天才”的基因,什么天生的洞察能力;不需要什么超自然的能力使自己总有灵感去出人意料的解决难题。

大众对数学家的形象有一个错误的认识:这些人似乎都使孤单离群的(甚至有一点疯癫)天才。他
们不去关注其他同行的工作,不按常规的方式思考。他们总是能够获得无法解释的灵感(或者经过痛苦的挣扎之后突然获得),然后在所有的专家都一筹莫展的时
候,在某个重大的问题上取得了突破的进展。这样浪漫的形象真够吸引人的,可是至少在现代数学学科中,这样的人或事是基本没有的。在数学中,我们的确有很多
惊人的结论,深刻的定理,但是那都是经过几年,几十年,甚至几个世纪的积累,在很多优秀的或者伟大的数学家的努力之下一点一点得到的。每次从一个层次到另
一个层次的理解加深的确都很不平凡,有些甚至是非常的出人意料。但尽管如此,这些成就也无不例外的建立在前人工作的基础之上,并不是全新的。(例
如,Wiles 解决费马最后定理的工作,或者Perelman 解决庞加莱猜想的工作。)

今天的数学就是这样:一些直觉,大量文献,再加上一点点运气,在大量连续不断的刻苦的工作中慢慢的积累,缓缓的进展。事实上,我甚至觉得现实中的情
况比前述浪漫的假说更令我满足,尽管我当年做学生的时候,也曾经以为数学的发展主要是靠少数的天才和一些神秘的灵感。其实,这种“天才的神话”是有其缺陷
的,因为没有人能够定期的产生灵感,甚至都不能保证每次产生的这些个灵感的正确性(如果有人宣称能够做到这些,我建议要持怀疑态度)。相信灵感还会产生一
些问题:一些人会过度的把自己投入到大问题中;人们本应自己的工作和所用的工具有合理的怀疑,但是上述态度却使某些人对这种怀疑渐渐丧失;还有一些人在数
学上极端不自信,还有很多很多的问题。

当然了, 如果我们不使用“天才”这样极端的词汇,我们会发现在很多时候,一些数学家比其他人会反应更快一些,会更有经验,会更有效率,会更仔细
,甚至更有创造性。但是,并不是这些所谓的“最好”的数学家才应该做数学。这其实是一种关于绝对优势和相对优势的很普遍的错误观念。有意义的数学科研的领
域极其广大,决不是一些所谓的“最好”的数学家能够完成的任务,而且有的时候你所拥有的一些的想法和工具会弥补一些优秀的数学家的错误,而且这些个优秀的
数学家们也会在某些数学研究过程中暴露出弱点。只要你受过教育,拥有热情,再加上些许才智,一定会有某个数学的方面会等着你做出重要的,奠基性的工作。这
些也许不是数学里最光彩照人的地方,但是却是最健康的部分。往往一些现在看来枯燥无用的领域,在将来会比一些看上去很漂亮的方向更加有意义。而且,应该先
在一个领域中做一些不那么光彩照人的工作,直到有机会和能力之时,再去解决那些重大的难题。看看那些伟大的数学家们早期的论文,你就会明白我的意思了。

有的时候,大量的灵感和才智反而对长期的数学发展有害,试想如果在早期问题解决的太容易,一个人可能就不会刻苦努力,不会问一些“傻”的问题,不会
尝试去扩展自己的领域,这样迟早造成灵感的枯竭。而且,如果一个人习惯了不大费时费力的小聪明,他就不能拥有解决真正困难的大问题所需要耐心,和坚韧的性
格。聪明才智自然重要,但是如何发展和培养显然更加的重要。

要记着,专业做数学不是一项运动比赛。做数学的目的不是得多少的分数,获得多少个奖项。做数学其实是为了理解数学,为自己,也为学生和同事,最终要
为她的发展和应用做出贡献。为了这个任务,她真的需要所有人的共同拼搏!

Convergence of random variable

In this post, we will pursue a approach to the strong law of large numbers based on the convergence of random variables.

Definition (Tail \sigma-algebra) Let  (X_n)_{n\geq 1} be a sequence of random variables. Let \mathcal{F}_n=\sigma(X_n,X_{n+1},\dots) be the smallest \sigma-algebra with respect to which all X_n:m\geq n are measure. The tail \sigma-algebra \mathcal{T}:=\cap_n \mathcal{F}_n is the intersection of all \mathcal{F}_n.

Theorem (Kolmogorov’s 0-1 law) If X_1,X_2,\dots are independent and A\in\mathcal{T}, then P(A)=0 or 1.

Theorem (Kolmogorov’s maximal inequality) Suppose that X_1,\dots,X_n are independent with \mathbf{E}X_i=0 and \mathbf{Var}X_i<\infty. If S_n=X_1+\cdots+X_n then

\displaystyle \mathbf{P}\left(\max_{1\leq k\leq n}|S_k|\geq x\right)\leq \frac{\mathbf{Var}S_n}{x^2}.

Theorem Suppose that X_1,X_2,X_3,\dots are independent and have \mathbf{E}X_n=0. If

\displaystyle \sum_{n=1}^\infty \mathbf{Var} X_n<\infty,

then with probability one \sum_{n=1}^\infty X(\omega) converges.