Newest Questions

Consider the polynomial $f(x)= x^2+1$. Can you prove that there are there infinitely many integers $x$ such that $f(x)$ has no prime divisor congruent to $1 \bmod 3$? Obviously the prime divisors are ...

Let $\Omega\subset\mathbb{R}^n$ be a bounded Euclidean domain with Lipschitz (possibly smoother) boundary and consider an Elliptic Dirichlet problem of the form \begin{align} \mathcal{L} u + \alpha(x,...

Let $\Gamma$ be a torsion free Zariski-dense discrete subgroup of $\operatorname{SL}_3(\mathbb{R})$. Then one can show that the cohomological dimension of $\Gamma$ is less or equal to 5. The equality ...

I am still now stumped on deriving the series equivalence $$\zeta(3)=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$ Like even I did not get the series for $\frac1{n^2}$ mentioned ...

$\DeclareMathOperator{\A}{\mathcal{A}}\DeclareMathOperator{\C}{\mathcal{C}}\DeclareMathOperator{\coker}{coker}$Let $\A$ be a an abelian category and let $\C$ be an additive category such that $\C$ is ...

Suppose $\Omega \subsetneqq S^{n}$ is a connected open domain and carries the complete metric $ \tilde{g} = u^{\frac{4}{n-2}}\, g_{\mathrm{st}}\!\mid_{\Omega}, $ where $u : \Omega \to \mathbb{R}^+$ is ...

Let $V$ be the standard $2g$-dimensional representation of $\mathrm{Sp}(V)$ (with $g \ge 1$), and for each $m \ge 1$ let $S^m V$ denote the $m$-th symmetric power. Consider the Schur functor $S^{(2,1)}...

I have an economics and statistics background. However, I often find the two subjects too narrow in the sense that they employ very specific sets of mathematics. I would like to explore something more ...

The following material is quoted from A Crèche Course in Model Theory by Domenico Zambella, Section 15.3. $\mathcal{U}$ is how we denote the Monster model. For every $a\in\mathcal{U}^{x}$ and $b\in\...

The Eilenberg Watts theorem tells us that every $k$-linear right-exact cocontinuous functor $F:\text{Rep}(A)\to \text{Rep}(B)$ comes from tensoring with a bimodule. In particular if $F$ commutes with ...

In Theorem 2.1 of this paper, Tam and Yu prove that: if a Kähler manifold $M$ has ${\rm bisec}\geq2k$, $p\in M$ and $r(x)=d(p,x)$, then for any unit vector $v\in T_xM$ perpendicular to $\nabla r$, we ...

For a smooth Fano variety $Z$, let $X$ be the total space of its canonical bundle. Let $\operatorname{Coh}_Z(X)$ be the category of coherent sheaves that support on $Z$ set-theorically. How to show ...

Let $M^n$ be a closed smooth $n$-manifold with $n \ge 3$. Suppose its universal cover is diffeomorphic to $\widetilde{M} \cong S^{n-1} \times \mathbb{R},$ where $S^{n-1}$ carries the standard smooth ...

I’m implementing a version of the Levinson recursion that should handle sub-singular Hermitian Toeplitz matrices. My code works perfectly when the Toeplitz entries are real, but it fails as soon as ...

Let $G$ be a finite perfect group with non-trivial center. I am trying to show that a Sylow $2$-subgroup of $G$ is non-abelian. I think one needs to use transfer homomorphisms here. However I can not ...