
Metric entropy 1
Some basic thing, include the definition of metric entropy is introduced in my early blog. Among the other thing, there is something we need to focus on: 1.Definition of metric entropy, and more general, topological entropy. 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 $la…

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.

Hilbert 16th problem
Introduction the statement of Hilbert’s 16th problem: $ H(n)<\infty?$ definition of $ H(n)=max$ Limit cycle: Try beginning with BendixonPoincaré 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}…

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…

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…

khessian equation and kcurvature equation
here is the problem, how to understand khessian equation and kcurvature equation. khessian equation khessian equation is: $latex H_k(u)=\sigma_k(D^2(u))=f$ (*) where u is admissible, i.e. $latex \forall 1\leq i\leq k$, $latex \sigma_i(D^2(u))\geq 0$. this is just the condition to make (*) be a elliptic equation. The most important result is the following three: 1.sovable (*) with direchlet boundary condition. This is mainly the contribution of Caffaralli in 90’s. According flexible fun…

Note on Vinogradov main theorem
1.Introduction Question: Vinogradov mean value Let $latex k,s\in \mathbb N$,$latex x\in R^k $. $latex J_{s,k}(N)=\{(n_1,…,n_s,n_{s+1},…,n_{2s})n_1^j+…+n_s^j=n_{s+1}^j+…+n_{2s}^j) \forall 1\leq j\leq k,1\leq n_i\leq N(1\leq i\leq s) \}$ How to estimate $latex J_{s,k}(N)$? We assume $latex f_{k}(x,N)=\sum_{1\leq n\leq N}e(nx_1+n^2x_2+…+n^kx_k)$, then by following clear calculate: $latex \int_{[0,1]^k}f_k(x,N)^{2s}dx_1…dx_k =\int_{[0,1]^k}\sum_{1\leq…

Linear to Multilinear
The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful. such like: 1.The renormalization technique in complex dynamic system, and the generalization this is mainly the Ostrowoski representation,and something else. 2.Fouriour analysis this can be view when it is difficult to investigate a quality about a function $latex f$, it is always easier to take charge with some some part of $latex f$, in this case is given by $latex \hat f(\xi),…

How to compute the GromovHausdorff distance between spheres $latex S_n$ and $latex S_m$?
There is the question, because when we consider the GromovHausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $latex \mathbb{S}_n \to \mathbb{R}^{n+1}$. Is it possible for us to compute the GromovHausdorff distance $latex d_{GH}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $latex \mathbb{S}_n$ and $latex \mathbb{S}_m$, $latex m\neq n$? For example if we want to calculate $latex d_{GH}(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathb…

Regularity of kcurvature equation
this is a note after reading the article”” of Cafferalli. in his article,a large type of fully nonlinear elliptic equation has been established.in particular,including the kcurvature equation.and use the continue method,we just need to establish a ingredient estimate,$latex C^2$ estimate in the interior and $latex C^2$ estimate near the boundary.we establish these estimate step by step,base on construct special flexible function and use the maximum principle to establish the first a…