Consider the sequence S(x) = 2^x - 1. This sequence has two interesting properties: a) If the GCD of S(x) and S(y) is S(gcd(x,y)), and b) For any prime p, S(p-1) is divisible by …

Hello ! If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This as …

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu( …

Let $$ F: \mathcal{C} \leftrightarrows \mathcal{D} :G $$ be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors $$ \mathb …

Hello, my question basically is how do I find the probability density function of the position of the nodes in a given area after N discrete time slots when the nodes move followin …

I am working on an optimization problem which I am stuck on towards the end. Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x, …

Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x …

Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space : $$\nabla_W F(\rho) = -\nabla.(\rho \n …

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "th …

I was asked this question during my interview recently and despite the amount of thinking i put into this, I am yet to figure it out: Given $n$ balls which are painted by $k$ co …

Here is a random sequence of 25 bits: 0101001100100011011010111 A sequence of any desired length can be obtained here http://www.random.org/integers/?num=25&min=0&max=1&amp …

Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to R^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p(x) dx$, and the distance …

A Boolean function U($z_1$, $z_2$ ..... , $z_m$) is universal for given n > 1 if it realizes all Boolean functions f($x_l$ ..... $x_n$) by substituting for each $z_i$ with a variab …

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computa …

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?