L^p (1<p<infty) 的一致凸性

定理. 设(X,\mathcal{X},\mu)是测度空间. 当1<p<\infty时, L^p(X)是一致凸的.

证明. 首先注意到2<p<\infty的情形根据Hanner不等式立刻得到.下面考虑1<p<2的情形, 根据Hanner 不等式, 我们有

\displaystyle \|2f\|_{L^p}^p+\|2g\|_{L^p}^p\geq (\|f+g\|_{L^p}+\|f-g\|_{L^p})^p+|\|f+g\|_{L^p}-\|f-g\|_{L^p}|^p.

在上式中令\|f\|_{L^p}=\|g\|_{L^p}=1, 那么有

\displaystyle  (\|f+g\|_{L^p}+\|f-g\|_{L^p})^p+|\|f+g\|_{L^p}-\|f-g\|_{L^p}|^p\leq 2\cdot 2^p\ \ \ \ \ (1)

注意到对于a\geq b\geq 0, 我们有下面的估计:

\displaystyle (a+b)^p+(a-b)^p\geq 2a^p+p(p-1)a^{p-2}b^2\ \ \ \ \ (2).

分为有两种情形:

(1) \|f-g\|_{L^p}\leq \|f+g\|_{L^p}, 根据(1)式和(2)式可得

\displaystyle 2\|f+g\|_{L^p}^p+p(p-1)\|f+g\|_{L^p}^{p-2}\|f-g\|_{L^p}^2\leq 2\cdot 2^p.

因此

\displaystyle \left\|\frac{f+g}{2}\right\|_{L^p}^p\leq 1-\frac{p(p-1)}{2}\left\|\frac{f+g}{2}\right\|_{L^p}^{p-2}\left\|\frac{f-g}{2}\right\|_{L^p}^2\leq 1-\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^2.

(2)  \|f+g\|_{L^p}\leq \|f-g\|_{L^p},则

\displaystyle 2\|f-g\|_{L^p}^p+p(p-1)\|f-g\|_{L^p}^{p-2}\|f+g\|_{L^p}^2\leq 2\cdot 2^p.

因此

\displaystyle \left\| \frac{f-g}{2} \right\|_{L^p}^p+\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^{p-2}\left\|\frac{f+g}{2}\right\|_{L^p}^2\leq 1\ \ \ \ \ (3).

\|\frac{f+g}{2}\|_{L^p}\leq \frac{1}{2}时, 我们平凡地有

\displaystyle \left\|\frac{f+g}{2}\right\|_{L^p}<1-\delta.

因此可以假设\| \frac{f+g}{2} \|_{L^p}\geq \frac{1}{2}, 那么由(3)式可得

\displaystyle \left\| \frac{f-g}{2} \right\|_{L^p}^p+\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^{p-2}\cdot \frac{1}{4}\leq 1.

注意到\|\frac{f-g}{2}\|_{L^p}\leq 1, 因此\|\frac{f-g}{2}\|_{L^p}^{p-2}\geq 1因为1<p<2, 因此

\displaystyle \left\|\frac{f-g}{2}\right\|_{L^p}^p+\frac{1}{8}(p-1)p\leq 1.

因此

\displaystyle \left\|\frac{f-g}{2}\right\|_{L^p}<1-\delta.

因此根据假设 \|f+g\|_{L^p}\leq \|f-g\|_{L^p}\leq 1-\delta

 

Topological properties of manifolds

We now proceed to the constructions that are directly related to what we shall later use

Theorem A smooth manifold has a compact exhaustion and is paracompact

A compact exhaustion is an increasing countable collection of compact sets K_1\subset K_2\subset\dots such that M=\bigcup K_i and K_i\subset\mathrm{int}K_{i+1} for all i.

The fundamental lemma we need is a smooth version of Urysohn’s lemma

Theorem (Smooth Urysohn Lemma) If M is a smooth manifold and A,B\subset M are disjoint closed sets, then there exist a smooth function f:M\to [0,1] such that A=f^{-1}(0 and B=f^{-1}(1).

域扩张

我们考虑域的一般结构。域论探讨的基本课题是域嵌入,或者说域扩张。我们主要考察代数扩张,采取的角度是系统地利用代数闭包的存在性,对代数扩张尽量广泛的处理。

代数初步

代数学中出现的许多环结构同时是域上的向量空间:环的加法来自向量空间的加法,而乘法(x,y)\mapsto xy是向量空间上的双线性型。典型的例子是域k上的n\times n矩阵环M_n(k)

交换环上的代数

以下设R是非零的交换幺环。所谓的代数都是要幺的结合代数。

定义 环R上的代数式一个具有环与R-模结构的结合$latex $A,使得环乘法(x,y)\mapsto xy是平衡积,亦即

\displaystyle (rx)y=x(ry)=r(xy),\ \ x,y\in A,\ r\in R.

数论问题的价值

Fight with Infinity

我以为判断一个数学问题的价值大致有2种途径:一是看它是否有趣,一是看它是否重要。

所谓“有趣”,大多与一个命题出乎意料的程度有关。一般来说,一个表述简单却难以证明或证否的问题通常是有趣的。如果这个问题最终得到证明,事情就更加有趣了:从一片混沌中诞生出简单的图景,提示我们必然有某种值得深入研究的机理存在。此时这个问题开始变得“重要”了:围绕着它,数学家们构筑起理论,试图把他们在解决这个问题的过程中所获得的经验推广到更多的问题上。新的数学产生了。

一个问题的重要性取决于它在我们对现象的理解(即“理论”)中占据何种位置。挡在通衢大道上的石头是谁都想搬开的,躺在路边的石头则不会有多少人注意。当然,如果路边的石头固执地抗拒一切推开它的努力,它将以另一种方式变得“重要”:这种重要性由种种失败中所产生的新数学的多少来衡量。

数论是数学中最特别的分支:我们对整数惊人的无知,几乎所有数论问题都在某种程度上是“有趣”的——正是凭借这一点数论吸引了人类最优秀的头脑。另一方面,除了少数几个经典问题外,似乎很难先验地知道哪些问题在整体图景中是“重要”的:此时我们只能转而求助第2种重要性的定义,希望从中产生尽可能多的数学。

下面是一个粗糙的分类。我们仅给出每一类中最具代表性(往往也最有价值)的例子,并不代表所有同类问题都具有同等价值。横向上看,通常(1)(3)中的问题都是值得珍视的(在数论中并不多见),(2)(4)的价值次之,充斥整个数论的(5)则不那么重要:

(1)第一类问题包括Riemann猜想Langlands纲领。它们在2种意义上都是重要的:既处在现代数论的核心,又催生了大量“好的数学”,是整个现代数论前进的定向标。

未解决的Hilbert问题中,第9问题第12问题(Kronecker’s Jugendtraum)都可以归入此类。除了本身的理论价值外,它们还是类域论复乘理论和Langlands纲领的渊薮。

(2)另一类问题有理论上的重要性,却因为太难或者太偏而没有产生太多主流数学,或者必须借助(1)中的问题才能得到迂回的理解:例子包括Gauss的类数猜想Artin的原根猜想,等等。如果有人能以“正确的方式”理解它们,则此类问题可能提升为(1)中的问题。

(3)Fermat大定理本身并不重要,但它在第二种意义上极端重要:例如,它催生了Kummer的理想理论,从而建立了代数数论和代数几何的基础。Wiles的证明则增进了对Langlands纲领的理解,由此产生的系列数学工具也极具威力(参见Richard Taylor的工作)。

目前看来,比Fermat大定理更强的abc猜想应该也属于此类。望月新一最近的工作能否像Wiles的工作那样推动整个领域的进步,我们拭目以待。

(4)同样,Goldbach猜想孪生素数猜想本身也没有太大的重要性(尽管它们是“有趣”的典型例子)。人们因此发展了加性数论(华罗庚的“堆垒数论”)。经典工具(例如筛法)的应用范围狭窄,和Fermat大定理衍生出的数学相比,眼下处在边缘位置。这解释了为什么某些数学家轻视这方面的工作。当然,(4)中的问题也有可能提升到(3):例如,加性数论最近接受了来自遍历理论的新想法,似有重新回归主流的趋势(参见陶哲轩的工作),而后者又依赖于从到van der Waerden定理Szemerédi定理的提升。

Erdős是“趣味主义”的代言人,他提出的猜想大多属于(4)。概率数论(Erdős–Kac定理, etc.)和随机图(Erdős–Rényi模型, etc.)等工作是成功提升到(3)的例子,上面提到的陶哲轩的工作可能使Erdős猜想(若$latex sum 1/a_i$发散,则整数序列$latex {a_i}$中包含任意长的算术级数)获得提升。Ramanujan在模函数方面的工作中,Ramanujan猜想已通过Weil猜想成功提升。古老的同余数问题并无重要性,但它通过与BSD猜想联系获得了重要性。另一个相对近代的例子是经由Vojta的工作,Roth定理成功融入了算术几何的理论框架。

(5)证否和反例不一定是不重要的(尤其在第二种意义上):例如,Littlewood证否了Gauss猜想$latex pi(n)<mathrm{Li}(n)$,这增进了我们对$latex zeta$函数的理解,值得划入(4)。在寻找Euler猜想反例的过程中,Elkies和Frye等人发现了椭圆曲线理论的一个意外应用,这有一定的算法价值 (更不要说类似的构造椭圆曲线的方法提供了从谷山-志村猜想推出Fermat大定理的途径)。

Guy的Unsolved Problems in Number Theory中收录的问题也不一定是不重要的。事实上,它们中相当大的一部分都有某种程度的重要性。我们已在(2)(4)中提到部分例子,尝鼎一脔,其余可知。

很遗憾,在我看来Guy, F26不属于上述两类,而属于最不重要(也最常见)的一类数论问题:既在整个理论中没有位置,也不太可能产生有意思的数学。反例并不巨大(这意味着问题并不是那么难),同时,由于找到反例的方式是完全初等的,其潜在的算法价值也相当有限——这还是在不考虑Fuller,Iraids等人已得到好得多的结果的情况下。

【注记】
本文写成之后,豆瓣上的魔术师同学提醒我Guy, F26和素数的Kolmogorov复杂度有关。这样看来,一个Littlewood式的证否原本可能将它提升到(4)。我同意他的看法:“这个解法把他降低到了(5)。……一个昭示如何构造反例,或者证明仅有穷多反例,或者无穷多反例才可算是好的回答。”

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The Kakeya problem in finite fields

The kakeya problem, in its best known formulation, is the following. Let E\subset\mathbf{R}^n be set which contains a translate of every unit segment; equivalently, for every direction e\in S^{n-1}, E contains a unit line segment parallel to e. An n-dimensional ball of radius 1/2 is a simple example of  a set with this property, but there are many other such sets, some of which have n-dimensional measure 0.

Can E be even smaller than that and have Hausdorff dimension strictly smaller that n?

Ordered field

An ordered filed is a field together with a total ordering of its elements that is compatible with the field operations.

An ordered field necessarily has characteristic 0 since the elements 0<1<1+1<1+1+1<\dots necessarily are all distinct$. Thus, an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.

Every ordered field contains an ordered subfield that is isomorphic to rational numbers. Any Dedekind complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered filed.

Definition A filed (F,+,\times) together with a total order \leq on F is an ordered field if the order satisfies the following properties for all a,b and $c$ in F:

  • if a\leq b then a+c\leq b+c, and
  • if 0\leq a and 0\leq b then 0\leq a\times b.

Tangent space

One can attach to every point x of a differentiable manifold a tangent space, a real vector space that intuitively contains the possible “directions” at which one can tangentially pass through x. The elements of the tangent space are called tangent vectors atx. All tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold

Definition as velocities of curves

Suppose M is a C^k manifold (k\geq 1) and x is a point in M. Pick a chart \varphi:U\to\mathbf{R}^n where U is an open subset of M containing x. Suppose two  curves \gamma_1:(-1,1)\to M and \gamma_2:(-1,1)\to M with \gamma_1(0)=\gamma_2(0)=x are given such that \varphi\circ\gamma_1 and \varphi\circ\gamma_2 are both differentiable at 0. Then \gamma_1 and \gamma_2 are called equivalent at 0 if the ordinary derivatives of \varphi\circ\gamma_1 and \varphi\circ\gamma_2 coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vector of M at x. The equivalence class of the curve \gamma is written as \gamma'(0). The tangent space of M at x, denoted by T_xM, is defined as the set of all tangent vectors, it does not depened on the choice of chart \varphi.

To define the vector space operations on T_xM, we use a chart \varphi:U\to\mathbf{R}^n and define the map d\varphi_x(\gamma'(0))=\frac{d}{dt}(\varphi\circ\gamma)(0). It turns out that this map is bijective and can thus be used to transfer the vector space operation from \mathbf{R}^n over T_xM, turning the latter into an n-dimensional real vector space. Again, one need to check that this construction does not depend on the particular char \varphi chosen, and in fact is does not.

Definition via derivations

Suppose that M is a C^\infty manifold. A real valued function f:M\to\mathbf{R} belongs to C^\infty(M) if f\circ\varphi^{-1} is infinitely differentaible for every chart \varphi:U\to \mathbf{R}. C^\infty(M) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

Pick a point x in M. A derivation at x is a liner map D:C^\infty(M)\to\mathbf{R} that has the property that for all f,g in C^\infty(M):

\displaystyle D(fg)=D(f)\times  g(x)+f(x)\times D(g)

modeled on the product rule of calculus. If we define addition and scalar multiplication for such derivations by

\displaystyle (D_1+D_2)(f)=D_1(f)+D_2(f)\ \text{and}\ (\lambda D)(f)=\lambda D(f)

we get a real vector space which we define as the tangent space T_xM.

The relation between the  tangent vectors defined earlier and derivatoin is as follows: if \gamma is a curve with tangent vector \gamma'(0), then the corresponding derivation D(f)=(f\circ\gamma)'(0).

\displaystyle \gamma'(0)\mapsto D_\gamma \  \text{where}\ D_\gamma(f)=\frac{d}{dt}(f\circ \gamma)\Big|_{t=0}.

Differentiable manifold

The notation of  a differentiable manifold refines that a manifold by requiring the functions that transform between charts to be differentiable. A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a liner space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

Atlas

One describes a manifold using an atlas.

Defintion An atlas for a topological spae M is a collection \{(U_\alpha,\varphi_\alpha)\} of charts on M such that \bigcup U_\alpha=M.

Fréchet derivative

The Fréchet derivative is a derivative defined on Banach spaces, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus variations.

Definition Let V and W be Banach spaces, and U\subset V be an open subset of V. A function f:U\to W is called Fréchet differentiable at x\in U if there exists a bounded linear operator A:V\to W such that

\displaystyle \lim_{h\to 0}\frac{\|f(x+h)-f(x)-Ah\|_W}{\|h\|_V}=0.

The limit here is meant in the usual sense of a limit of a function defined on  a metric space.

Relation to the Gâteaux derivative

Definition A function f:U\subset V\to W is called Gâteaux differentiable at x\in U if f has a directional derivative along all direction at x. This means that the limit 

\displaystyle \lim_{t\to 0}\frac{f(x+th)-f(x)}{t}

exists for any choosen vector h in V, where is t is from the scalar filed associated with V.

If f is Fréchet differentiable at x, is is also Gâteaux differentiable there, and the limit is just Df(x)(h).

Higher derivatives

If f:U\subset V\to W is a differentiable function at all points in an open subset U of V, it follows that its derivative

\displaystyle Df:U\to L(V,W)

is a function from U to the space L(V,W) of all bounded liner operators from V to W. This function may also have a derivative, the second order derivative of f, which, by the definition of derivative, will be a map

\displaystyle D^2f:U\to L(V,L(V,W)).

To make it easier to work with second order derivatives, the space on the right-hand side is identified with the Banach space L^2(V\times V,W) of all continuous bilinear map from V to W. An element \varphi in L(V,L(V,W)) is thus identified with \psi in L^2(V\times V,W) such that for all x and y in V

\displaystyle \varphi(x)(y)=\psi(x,y).

Primitive character, coductor and Jacobi sum

Dirichlet characters

Recall that is a character \chi of modulo q is said to be induced by a character \chi' of modulo d if \chi(n)=\chi'(n) for every n\in\mathbf{Z} with \gcd(n,q)=1, here d is a divisor of q.

Jacobi sum

Jacobi sum is a type of character sum formed with Dirichlet characters. The Jacobi sms for Dirichlet characters \chi,\chi' modulo a prime number p, defined by

\displaystyle J(\chi,\chi')=\sum_{n\in \mathbf{Z}/p\mathbf{Z}} \chi(n)\chi(1-n),

Jacobi sums are the analogues for finite fields of the beta function.