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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…
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k-hessian equation and k-curvature equation
Understanding the k-Hessian Equation and k-Curvature Equation k-Hessian Equation The k-Hessian equation is given by: $H_k(u)=\sigma_k(D^2(u))=f$ (*) where u is admissible, i.e. $\forall 1\leq i\leq k$, $\sigma_i(D^2(u))\geq 0$. This condition ensures that (*) is an elliptic equation. The most important results are as follows: This result could be used to establish some results on singularity points that can be removable. k-Curvature Equation My Ideas: Looking at it as the “average” o…
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Note on Vinogradov main theorem
1. Introduction Question:Vinogradov mean valueLet $k, s \in \mathbb{N}$, $x \in \mathbb{R}^k$. $J_{s,k}(N) = |{(n_1, …, n_s, n_{s+1}, …, n_{2s}) | n_1^j + … + n_s^j = n_{s+1}^j + … + n_{2s}^j, \text{for all } 1 \leq j \leq k, 1 \leq n_i \leq N (1 \leq i \leq s)}|$ How to estimate $J_{s,k}(N)$? We assume $f_{k}(x,N) = \sum_{1 \leq n \leq N} e(nx_1 + n^2x_2 + … + n^kx_k)$, then by following clear calculate: $\int_{[0,1]^k} |f_k(x,N)|^{2s} dx_1…dx_k = \int_{[0,1]^k} |\sum_{1 \leq n \leq N} e(nx_1 +…
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Linear to Multi-linear
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),…
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How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?
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 $\mathbb{S}n o \mathbb{R}^{n+1}$. Is it possible for us to compute the Gromov-Hausdorff distance $d_{G-H}(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $meq n$? For example, if we want to calculate $d_{G-H}(\mathbb{S}2,\mathbb{S}3)=\inf{M,f,g}d{M}(\mathbb{S}_2,\mathbb{S}_3)$, where $M$ ranges over…
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Regularity of k-curvature 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 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…