I have two infinite-dimensional (Frechet) linear spaces $X$ and $Y$ over $\mathbb R$, and a nondegenerate pairing $\langle\cdot,\cdot\rangle:X\times Y\to \mathbb R$. Neither $X$ no …

It's fine and nice and wonderful when a part of learning mathematics is chaotic, ad hoc, spontaneous, social, ... However it would be perhaps of fundamental value to know a very c …

I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve. I want to predict the nature of use …

Let $M$ be a $n$-dimensional smooth connected not compact manifold s.t. groups (singular cohomology) $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a suff …

I have two relations R and R' satisfying following property - R(a,b) & R'(a,a') & R(a',b') => R(a,b') Pictorially, it looks like this - a --(R)-> b | \ (R') \ (* n …

Hi All, here is my question. I'm given a directed graph $(V,E)$ with $|V| = n$ vertices and in-degrees $d_1$, $d_2$ ... $d_n$ (so that $\sum_i d_i = |E|$). Can we upper bound the …

My question is related to the Riemann Mapping Theorem. A function $f$ is biharmonic if $\Delta^2f=0$. Let $D$ be a simply connected domain in $\mathbb{R}^2$ and denote by $B$ the …

Let $$\hat{f}(\lambda):= \int_0^{+\infty}K(x,\lambda)\ f(x)\ dx, \text{where } K(x,\lambda)=\sqrt{\frac{2}{\pi}}\frac{\lambda \cos(\lambda x)+h\sin(\lambda x)}{\lambda^2+h^2}$$ be …

Maybe one of the strange things about quantum mechanics comes from the mental dificulty of reconciling the fact that a system in a superposition of several states, when measured or …

Consider the class of rooted trees and suppose I have at my disposal a lowest common ancestor (LCA) operator given by $$ \textrm{lca}(u,v) = \text{the lowest common ancestor of $u$ …

In learning about the K-theory of $C^*$-algebras, I have encountered the following 3 proofs of Bott periodicity: $\bullet$ An argument based on Moyal quantization found in "Elemen …

In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independe …

In the book of Evans the transport equation, $$\frac{d}{dt} u + b\cdot \nabla u = 0, \quad u(t=0)=u_0,$$ is solved by the method of charateristics for $b$ and $u_0$ smooth enogh (i …

Any closed form for series like $$F(x)=\Sigma_{i=2}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$? More generally,we can obtain a power series from decim …

I am attempting to work through "Shahverdiev, Sivaprakasam, and Shore (2002) Lag synchronization in time-delayed systems", but I'm missing something basic up front. The problem is …