## Tagged Questions

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### Sequences satisfying gcd(S(x), S(y)) = S(gcd(x,y))

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 …
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### Are $\infty$-topoi determined by their localic points ?

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 …
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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( … 1answer 198 views ### Homotopy left-exactness of a left derived functor Let $$F: \mathcal{C} \leftrightarrows \mathcal{D} :G$$ be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors $$\mathb … 1answer 131 views ### Probability density function of the node positions in a random walk after N time slots Hello, my question basically is how do I find the probability density function of the position of the nodes in a given area after N discrete time slots when the nodes move followin … 1answer 184 views ### Probability Density Optimization I am working on an optimization problem which I am stuck on towards the end. Essentially, I have two probability density functions in \mathbb{R}^2, call them q(x,y) and p(x, … 0answers 143 views +100 ### New differintegral formula: how is it related to other differintegral formulas? Lets define new differintegral formula as$$\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 … 1answer 233 views ### derivative in the Wasserstein space 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 … 2answers 247 views ### In What Sense is Set Theory a ‘Foundation’ for Mathematics? 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 … 0answers 31 views ### n balls, k colors, expected color change problem 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 … 0answers 46 views ### How is the expected fraction of zeros correctly calculated when throwing bits? 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 … 0answers 18 views ### Smoothness and curvature of geodesics in a length space 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 … 1answer 33 views ### translating a given boolean function to universal boolean function 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 … 3answers 2k views ### What would be some major consequences of the inconsistency of ZFC? 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 … 2answers 194 views ###$\Delta f \le - \lambda f$then${\lambda _1}\left( M \right) \ge \lambda$? 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\$?

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