# L^p (1<p<infty) 的一致凸性

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

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

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

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

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

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

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

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

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

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

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

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

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

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

# 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

## 代数初步

### 交换环上的代数

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

# 数论问题的价值

(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。

(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)。

(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。

【注记】

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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 at$x$. 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.