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

  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}…

1. some example and observations $latex {(M^2,g)}&fg=000000$,$latex {g(t)=e^{2u(t)}g_0}&fg=000000$, $latex \displaystyle \frac{\partial u}{\partial t}=e^{-2u}\tilde\Delta u+\frac{r}{2}-e^{-2u}K_0 &fg=000000$ $latex {(M^n,g_{ij}(t))}&fg=000000$ $latex \displaystyle \frac{\partial g_{ij}(t)}{\partial t}=-2Ric(g_{ij})&fg=000000$ The given “smooth” initial : $latex {\exists }&fg=000000$ T small ,$latex {T>0}&fg=000000$,the solution exists on $latex {[0,T]}&…

1. rough outline of heat kernel proof of Atiyah-singer index theorem 1.1. proof strategy Theorem 1 (Mckean-Singer formula.) $latex \displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)). &fg=000000$ from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory. Main challenge: 1. in the expansion on heat kernel ,we need to proof when $latex { t \rightarrow 0}&fg=000000$, the limit exist and find a way to calculate it. 2. indentify …