For $n\in\mathbb{N}$, denote by $\mathcal{Q}_n$ the set of all probability measures on $\mathbb{R}$ that are supported on at most $n$ points. Question: Is it known which element in $\mathcal{Q}_n$ is closest to the standard normal distribution with respect to the $p$-Wasserstein distance (for som...

I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these questions are too trivial or silly for this site. I've been working on understanding the proof of...

The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear combinations of Hodge cycles coincide with the $ \ell $-adic Tate cycles. Question. Would that mean tha...

If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma_n$, which we can call the universal multiset for $X$. Now let $X$ instead be an affine scheme $\text{spec}(A)$ over a base scheme $S=\text{spec}(k)$, and suppose t...

I am frequently interested to find less technical proofs of results which already appear in the literature, at least in some special cases of these results. Sometimes a published proof shows that an object with some properties exists, but actually the proof does not (at least not without addition...

Let $\alpha,\beta, \gamma \in \mathbb{R}^+$ be and the function $$ F(m,n)= \begin{cases} 1, & \text{if $m n=0$ }; \\ \alpha F(m ,n-1)+ \beta F(m-1,n )+ \gamma F(m-1,n-1), & \text{ if $m n>0$. }% \end{cases} $$ Please, a proof for $$\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n)} =\lim_{n\righ...

Let $\pi: V\to S$ be a standard conic bundle of a threefold $V$ to a surface $S$, i.e., $\pi$ is relative minimal. Assume that everything is nonsingular and is over $\mathbb{C}$. We may assume that $V$ is embedded in a $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{E})$ over $S$, where $\mathcal{E}$ ...

Consider the following ODE with parameters $\alpha,\beta,\gamma \in \mathbb R$ $$f'(t)= \begin{pmatrix} \alpha-\beta t & \gamma t \\ \gamma t & -(\alpha-\beta t) \end{pmatrix} f(t).$$ This ODE is non-autonomous and the matrix also does not commute with its derivatives, so diagonalization is not g...