Relationship of relationship

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. Key point: 1.rescaling+renormalization. 2.analysis it on every scale.

  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}…

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…

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…

here is the problem, how to understand k-hessian equation and k-curvature equation. k-hessian equation k-hessian 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…

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…

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),…

There is the question, because when we consider the Gromov-Hausdorff 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 Gromov-Hausdorff distance $latex d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ 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_{G-H}(\mathbb{S}_2,\mathbb{S}_3)=\inf_{M,f,g}d_{M}(\mathb…

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 k-curvature 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…