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A discret to continuous approach to the Dirichlet principle.
Direchlet principle: $ \Omega \subset R^n$ is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on $ \partial \Omega$. Perron lifting and barrier function We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary. The key point is if we define the variation energy $ E(u)=\int_{\Omega}|\nabla u|^2$, then it is easy to see for $ u_1,u_2$ is in perron set, $ E(sup (…
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Weyl law
In the 1911 year, when Weyl was a young mathematician specializing in integrable systems and PDE, He proved the important result about the asystomztion of eigenvalues of Dirichlet problem in $latex \Omega\subset R^n$ is a compact domain;i.e. $N(\lambda)=(2\pi)^d Vol(\Omega)\lambda^{\frac{d}{2}}(1+o(1))$ Which in fact is a conjecture of *** in *** in published in 1910. This is a very amazing achievement of mathematicians, The realist meaning we can actually charge with the spectrum assessment. In…
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Sarnak conjecture, understand with standard model
Sarnak conjecture is a conjecture lies in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of an entropy zero dynamic system by looking at the correlation of an observable and the Mobius function . We state it in a rigorous way: let $ (X,T)$ be an entropy zero topological dynamic system. Let the Mobius function be defined as $ \mu(n)=(-1)^t$, where $ $ is the number of different primes that occur in the decomposition of $ n$. Then for any continuo…
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Baragar-Bourgain-Gamburd-Sarnak conjecture
$latex M$ is the markov triple $(x,y,z)$: $x^2+y^2+z^2=xyz$ and $latex (x,y,x)\in \mathbb Z^3 \ \ \ \ (*)$. It is easy to see: $latex R_1: (x,y,z)\to (3yz-x,y,z)$. map markov triple to markov triple. This is also true for $latex R_2,R_3$. and the transform $latex R_1,R_2,R_3$ and permutation a classical result of Markov claim that all solutions of (*) could be generated from $latex (1,1,1)$. I got a similar result for a similar algebraic equation 1 half years ago when considering a …
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Metric entropy 2
I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”. Now we focus on the upper semi-continuity of the metric entropy map. The object we investigate is $(X,T,\mu)$, where $\mu$ is a $T$-invariant measure. The insight that makes us interested in this kind of problem is a part of a variational problem, something about the existence of a certain object that combines a certain moduli space to make some quantity attain a critical value (maximum or minimum). The simplest example may…
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Metric entropy 1
Some basic thing, including the definition of metric entropy, is introduced in my early blog. Among the other thing, there is something we need to focus on: 2. Spanning set and separating set describe of entropy. 3.amernov theorem: $latex h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$. Now we state the result of Margulis and Ruelle: Let $latex M$ be a compact Riemannian manifold, $latex f: M\to M$ is a diffeomorphism and $latex \mu$ is a $latex f$-invariant measure. Entropy is always bounded above by the s…
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Gromov’s idea applicate to parabolic equation
Gromov’s idea applicate to parabolic equation. Key point: 1.rescaling+renormalization. 2.analysis it on every scale.
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Hilbert 16th problem
Introduction the statement of Hilbert’s 16th problem: $ H(n)<\infty?$ definition of $ H(n)=max$ Limit cycle: Try beginning with Bendixon-Poincaré theorem, which is classical stuff and belongs to a lot of textbooks on vector fields. Affine invariance The number of limit cycle is invariant under affine map. Classification of singular point Bezout theorem Example 1.$ \frac{dx}{dt}=y,\frac{dy}{dt}=x$.The graph is just like: 2.$ \frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=x-y$.The graph is just like…
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A determinantal formula
I see a similar formula I wish to be true and merely have a proof in mind occur as a MO’s problem: In my research, I encounter the following formula which I believe is correct (checked for $ n\le3$). Is it classical ? I am given a real symmetric matrix $ S:=\int Y(t)Y(t)^Td\mu(t),$ where $ \mu$ is a probability and $ Y(t):\Omega\rightarrow{\mathbb R}^n$. Let $ \sigma_k(S)$ be the elementary symmetric polynomial in the eigenvalues of $ S$. For instance, $ \sigma_1(S)$ is the trace and $ \si…
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Isoperimetric inequality
Introduction the statement go isometry inequality is very simple: $latex \Omega\subset R^n$, iff $latex \Omega$ is a ball, $latex \frac{Vol(\Omega)}{Surf(\Omega)}$ arrive a minimum . This is a classical problem in variation theory. The difficult is divide into two parts. The first is to create a “flow” which descrement the energy and the “flow” is compatible with the feature of a ball, i.e. every set under the flow will tend to like a “ball”. The second one i…