I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \mathbb{R}$, $$ F(x - a) e^{x a} + (1 - F(x+a)) e^{-x a} = e^{a^2/2}. $$ I had never seen anything ...

Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the platonic realm of sets. The platonic realm of sets is supposed to be the world of truly existing sets, ...

Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication. Does there always exist a differential equation for which $ G $ is the differential Galois group? For example, $ \...

In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain properties, is canonically a stratified space with a smooth structure turning the projection $\pi:M\right...

Let $L$ be a countable language and $M$ be a model of $L^N$ (the realization of $L$ in the natural numbers $N$) in which every recursive unary relation is expressible. Show that $M$ is not recursive.

Let $\mathbb{F_q}$ be some finite field and let $f,g: \mathbb{F_q} \to \mathbb{C}$. By $\hat{f}, \hat{g}$ let's denote the Fourier coefficients of $f,g$ with respect to the additive characters of the field. How good an upper bound can we exhibit for the following sum (if it's possible to find a n...

Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes of string diagrams (without crossings) contained in a planar strip up to the relation that any c...

Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere $$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$ Let us denote $$(f,g) = \int_{\mathbb{R}^d} fg\,dx$$ and $\hat{f}$ is the Fourier transform on $\mathb...

Let $P$ and $Q$ be complete boolean algebras. Suppose that $\dot H$ is a $P$-name such that $1_P\Vdash\dot H$ is $Q$-generic. For each $p\in P$, let $A_p$ be the set of $q\in Q$ such that $p\Vdash q\in\dot H$. Then the map $\sigma:P\rightarrow Q$, given by $\sigma(p)=\prod A_p$, is a projection, ...

Given integers $k$ and $l$ and matrix $A$ of rank $kl$, can we always find matrix $B$ of rank $k$ and matrix $C$ of rank $l$, such that $A$ is the Hadamard product of $B$ and $C$, namely $A=B \odot C$? For example, when $k=2$ and $l=2$ and $A$ is the 4 by 4 identity matrix, \begin{equation*} A = ...