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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…
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Kakeya conjecture (Tomas Wolff 1995)
There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the result established by Tomas Wolff in 1995 is almost the best result in $ R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate. For $ f\in L_{loc}^1(R^d)$,for $ 0<\delta<1$: $ f_{\delta}^*:P^{d-1}\longrightarrow R$.$ f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$. $ T$ is varise in all cylinders with length 1.radius $ \delta$.axis in…
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Julia set
There is a major open problem: Is there a polynomial $f(z)$ such that the Julia set $T(J)$ of map$T:z\longrightarrow f(z)$ satisfied the Hausdorff dimension of J equal to 2?
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Covering a non-closed interval by disjoint closed intervals
this note will talk about the Ostrowski representation and approximation by continue fraction. As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$. i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have: $latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$. but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$. we will give a approach…
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Heat flow and the zero of polynomial-a approach to Riemann Hypesis
There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the result established by Tomas Wolff in 1995 is almost the best result in $ R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate. For $ f\in L_{loc}^1(R^d)$,for $ 0<\delta<1$: $ f_{\delta}^*:P^{d-1}\longrightarrow R$.$ f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$. $ T$ is varise in all cylinders with length 1.radius $ \delta$.axis in…
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Kakeya conjecture in R^3
Kakeya conjecture in $latex R^3$ is very subtle.in fact wolff stay the best(but not very difficult to get,just use the structure so-called hairbrush)result $latex \frac{5}{2}$ until the result of Katz and Tao $latex \frac{5}{2}+\epsilon$.Where $latex \epsilon$ is a constant independent with kakeya set.and in the article of Tao,they proved $latex \epsilon>\frac{1}{10^{10}}$. Two-dimensional case first we overview the case of dimension 2,these is the only case that is proved.and the key point i…