Neutral Network

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 …

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…

Sarnak conjecture is a conjecture lies in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of an entropy zero dynamic system by looking at the correlation of an observable and the Mobius function . We state it in a rigorous way: let $ (X,T)$ be an entropy zero topological dynamic system. Let the Mobius function be defined as $ \mu(n)=(-1)^t$, where $ $ is the number of different primes that occur in the decomposition of $ n$. Then for any continuo…

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…

1. Some Examples and Observations Let $latex (M^2,g)$, $latex g(t) = e^{2u(t)}g_0$, and $latex \frac{\partial u}{\partial t} = e^{-2u}\Delta u + \frac{r}{2} – e^{-2u}K_0$ Let $latex (M^n,g_{ij}(t))$ and $latex \frac{\partial g_{ij}(t)}{\partial t} = -2Ric(g_{ij})$ The given “smooth” initial: $latex \exists T>0$, the solution exists on $latex [0,T]$ Deturk Trick Threshold Type Theorem Ricci Flow: Mean Curvature Flow: HMF: Calabi Flow: Proof of Observation 4:If the threshold c…

Rough Outline of Heat Kernel Proof of Atiyah-Singer Index Theorem 1. Proof Strategy Theorem 1 (Mckean-Singer Formula):$$\text{ind}(D^+) = \text{Str}(e^{-tD^2}) = \int_{x \in M} \text{Str}(K(x,y)).$$ From this, we know Fredholm operator deformation invariance. At the same time, we need Chern-Weil theory. Main Challenge: Proof:Our proof road: Mckean-Singer formula $\rightarrow$ local-index thm $\rightarrow$ A-S index thm $\rightarrow$ Riemann-Roch-Hirzebunch theorem. $\Box$ 1.2. Preliminary Work S…