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There is some gap, in fact I can improve half of the argument of Discrete harmonic function , the pdf version is Discrete harmonic function in Z^n, but I still have some gap to deal with the residue half… 1. The statement of result First of all, we give the definition of discrete harmonic function. Definition 1 (Discrete harmonic function) We say a function $ {f: {\mathbb Z}^n \rightarrow {\mathbb R}}$ is a discrete harmonic function on $ {{\mathbb Z}^n}$ if and only if for any $ {(x_1,&#8…

  This is a note concentrate on the log average Sarnak conjecture, after the work of Matomaki and Raziwill on the estimate of multiplication function of short interval. Given a overview of the presented tools and method dealing with this conjectue.   1. Introduction Sarnak conjecture [1] assert that for any obersevable $latex {\{f(T^n(x_0))\}_{n=1}^{\infty}}&fg=000000$ come from a determination systems $latex {(T,X),T:X\rightarrow X}&fg=000000$, where $latex {h(T)=0}&fg=000…

The story of the infinite dimensional space of $\Delta$ is following, we eliminate ourself with compact smooth non-boundary manifold $M$ with metric $g$, then we have Betrami-Laplace operator $\Delta_g$. We could instead $\Delta_g$ by hodge laplace $dd^*+d^*d$, but let we consider $\Delta_g$ the eigenvalue problem: $$\Delta_g u=\lambda u$$ A classical way to investigate the eigenvalue problem is according to consider variational principle and max-min principle. We equip the path integral on the …

The pdf version is Uncertainty principle. The nice note of terrence tao seems given a nice answer for the problem below. 1. Introduction Is there a Brunn-Minkowski inequality approach to the phenomenon charged by uncertainty principle? More precisely, is it possible to say some thing about the Gaussian distribution $ G(x)=e^{-|x|^2} \ \ \ \ \ (1) $   to be the best choice that $ {\|\hat G-G\|_2} $ arrive minimum? Remark 1 Or some other suitable distance space on reasonable function (ma…

In this short note, I posed a conjecture on Brunn-Minkwoski inequality and explain why we could be interested in this inequality, what is it meaning for further developing of some fully nonlinear elliptic equation come from geometry. The main part of the note devoted to discuss several different proof of classical Brunn-Minkowski inequality. Brunn-Minkowski inequality 1. Introduction I believe, every type of Brunn-Minkowski inequality, type of Brunn-Minkowski inequality is in some special sense …

I already understand this material 3days ago but it is a little difficult for me to type the latex…   1. Introduction There is two space to understand a function’s behaviour, the physics space and the frequency space (Why thing going like this? Why there is such a duality?). Namely, we have: $latex \displaystyle \hat f(\xi)=\int_{{\mathbb R}^d}e^{2\pi i\xi x}f(x)dx \ \ \ \ \ (1)&fg=000000$   The key point is, waves is a parameter group of scaling of definition of a cons…

1. Introduction This is the first note of a series of notes concert on semiclassical analysis. Given the basic material on symplectic geometry. Including the following material, The case at a point, or we can look it as the case in $latex {{\mathbb R}^{2n}}&fg=000000$. The standard material in symplectic geometry, i.e. Hamiltonian mechanics, two approach, global one concentrating on lie derivative, and a locally one concentrating on the power of Darboux theorem, i.e. the existence of a canon…

There are some interesting problem, I post them at there in case I forget them. Excuse me if they are trivial, I have not took enough time to consider them about I think they are valuable to be consider. Problem 1: This problem is stated by graph coloring. there are two prat of it, in fact the first part I heard from someone else and I try to generate it to high dimension. there are finite lines $latex \{l_i\}_{i\in I}, l_i\subset \mathbb R^2$, crossing each other and the is a set $latex J$ of c…

1. Introduction We have talked about a very basic result in singular integral, i.e. if we have an additional condition, i.e. $latex {q-q}&fg=000000$ bounded condition, then by interpolation theorem we only need to establish the weak $latex {1-1}&fg=000000$ bound then we establish the $latex {p-p}&fg=000000$ bound of $latex {T}&fg=000000$, $latex {\forall 1< p< q }&fg=000000$. The category of of singular integral is very general, in fact the singular integral we interest…

I thinks there is some problem related to the solution of a 2 order differential equation given by Sturm-Liouville system which is nontrivial. It is well-know that the power of Sturm-Liouville theory see  wiki, is due to it is some kind of “spectral decomposition” in the solution space. Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURM-LIOUVILLE. PROBLEMS BY APPROXIMAT…