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An Fourier coefficient decay estimate.
f is a matrix value analytic function on $\mathbb T$, we know $latex h(f)>\alpha$, this is just mean $\forall k\in \mathbb Z$ , assume $latex | \hat f(k)|\leq e^{-|k|\alpha}$ , $g=log(f)$ for which we assume $latex g$ is a lifting of f use the inverse of ramification map , $M_{n\times n} \to M_{n\times n} , A \to e^A$. Then exists $\beta=c(\alpha)>0$ such that , $\forall k \in \mathbb Z$, $ |\hat g(k)| \leq e^{-k\beta}$.
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Geometric intuition of mean value property of nonlinear elliptic equation
I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition. Linear elliptic equation case First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e. MVP of laplace equation $\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$ I need to point out first, this property is not difficult to proof by standard integral by part meth…
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Diophantine approximation of algebraic number
An important theorem in Diophantine approximation is the theorem of Liuoville: **Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $latex p,q\in N^*$ where $latex q>0$. This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated la…
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Rotation number
Consider a compact one-dimensional dynamic system. We focus on $S_1$, it does not mean $S_1$ is the only compact one-dimensional system, but it is a typical example. Let $T: S_1 \rightarrow S_1$. If $T$ is a homomorphism, then $T$ stays the order of $S_1$ (by continuity and the zero point theorem). This just means: (may be do a reflexion $latex e^{2\pi i\theta}\to e^{-2\pi i\theta}$). In the homomorphism case, we try to define the rotation number to describe the expanding rate of the dynamic sys…
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Eloquent theory
Consider matrix ODE: $$\dot{\phi}(t)=A(t)\phi(t)$$ Where $A(t)$ is a given periodic matrix with period $T$, i.e. $A(x)=A(x+T)$, for all $x\in \mathbb{R}$. Then the solution $\phi(t)$ satisfies the identity: $$\phi(t+T)=\phi(t)\phi^{-1}(0)\phi(T)$$ This could be explained as $\phi^{-1}\phi(T)=\int_{0}^T\phi(t)$. Now we consider solving the equation: $e^{TB}=\phi^{-1}(0)\phi(T)$. At least formally, it could be solved: $$B=\frac{1}{T}\log\left(\frac{\phi(T)}{\phi(0)}\right)$$ (Unfortunately, $\log$…
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Van der curpurt trick
There is the statement of Van der carport theorem: Given a sequences $ \{x_n\}_{n=1}^{\infty}$ in $ S_1$, if $ \forall k\in N^*$, $ \{x_{n+k}-x_n\}$ is uniformly distributed, then $ \{x_n\}_{n=1}^{\infty}$ is uniformly distributed. I do not know how to establish this theorem with no extra condition, but this result is true at least for polynomial flow. $ |\sum_{n=1}^Ne^{2\pi imQ(n)}|= \sqrt{(\sum_{n=1}^Ne^{2\pi imQ(n)})(\overline{\sum_{n=1}^Ne^{2\pi imQ(n)}})}$ $ = \sqrt{\sum_{h_1=1}^N\sum_{n=1}…
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Multiplication function on short interval
The most important background of analytic number theory is the new understanding of the multiplication function on the shared interval. This result is established by Kaisa Matomäki & Maksym Radziwill, two very young and intelligent superstars. The main theorem in their article is: Theorem (Matomaki, Radziwill):As soon as $H \to \infty$ when $x \to \infty$, one has:$$\sum_{x\leq n\leq x+H}\lambda(n)= o(H)$$for almost all $x\sim X$. In my understanding of the result, the main strategy is: Step…
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transverse intersections
https://en.wikipedia.org/wiki/Transversality_(mathematics) Question 1: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+1}$, and take a point $p \notin M$. Prove that there is always a line $l_p$ passing through $p$ such that $l_p \cap M \neq \emptyset$, and $l_p$ intersects transversally with $M$. Question 2: Let $M$ be a compact $n$-dimensional smooth manifold in $\mathbb{R}^{n+m}$, and take a point $p \notin M$. Prove that for all $1\leq k\leq m$, there is always a hyper…
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An approach to Vinogradov estimate
Vinogradov estimate is: $ |\sum_{n=1}^{N}e^{2\pi i\alpha P(n)}|\leq c_A\frac{N}{log^A N}$ For fix $ \alpha$ is irrational and $ \forall A>0 … (*)$. Assume $ deg(P)=n$, this could view as a effective uniformly distribute result of dynamic system: $ ([0,1]^n,T)$, where $ T: x\to (A+B)x$, $ b$ is a nilpotent matrix, matrix $ A$ is identity but with a irrational number $ \alpha$ in the $ (n, n)$ elements. First approach we could easily to get a “uniform distribute on fiber”…
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From periodic to quasi periodic