All Questions

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...

For a stack $\mathcal{X}$ in $(Sch/S)_{\textrm{ét}}$, there is a site $(Sch/\mathcal{X})_{\textrm{ét}}$ whose objects are $(T,t)$, where $T$ is an étale $X$-scheme and $t\in \mathcal{X}(T)$. A cover ...

So, I got this weird question stuck in my head...? Is there a general formula for the area of a regular n - sided polygon? I so far am not at research level ...

In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...

I'm trying to come up with a reasonable way to determine the weights in a mixture distribution. Let us consider the following example: There are two districts ($i=1,2$) in a city, both of which using ...

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form ...

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.

Dirichlet problem for Laplace equation as follows \begin{eqnarray} \Delta{u}&=&0\text{ in }B_r(0)\\ u&=&g\text{ on }\partial B_{r}(0), \end{eqnarray} where $ g $ is continuous. It is ...

Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$. In ...

Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...

I was having some difficulties understanding the Levy-Ito decomposition, so I summarised some results into one result. Could anyone please tell me if the following makes sense, i.e. is mathematically ...

Let $V$ be a $(2n+1)$-dimensional complex vector space endowed with a non-degenerate symmetric bilinear form $q:V\times V\to \mathbb C$. Fix the notation: $$ OG(n-1,n,V):=\{W_{n-1}\subset W_n\subset ...

It is well-known that the Hochschild cochain complex $\mathrm{CC}^*(A)$ of an associative algebra $A$ carries a lot of structure. In particular: a differential, a cup product, and a bracket, which ...

Consider the following process on $\mathbb{C}$: Start at the point 1. At each step, move by adding $e^{i\theta}$, where $\theta$ is uniformly drawn from $\mathbb{S}^1$. Stop at the first positive ...

On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1) offers a characterization of the category of (real and ...

Let $X$ be a complex projective $3$-fold whose singular locus consists of a unique $A_1$ point $p\in X$. Consider a rational curve $C\subset X$ generating the Mori cone of $X$ and passing through $p$, ...

A group object $G$ in smooth proper rigid spaces has formal semistable model (possibly after a finite extension). Hyodo-Kato give the cohomology of $G$ a $(\phi, N)$-model structure which by Fontaine-...

It is well-known that on the Minkowski spacetime $\mathbb{R}^4$, there exist a free quantum field of arbitrary spin. In the book "QFT : A Tourist Guide For Mathematicians" by Folland, a ...

Let $M$ be a compact homogeneous space given by $K/N$, where $ K=G \times \mathbb{T}^n / H $, $G$ is a simply connected compact Lie group, $\mathbb{T}^n$ the $n$-torus and $H$ is a central finite ...

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...

We have the following sum $$S=\sum_{\substack{0<a'\leq k'\\(a',k')=1\\a'\equiv b \bmod q}}e(hl'a'/k').$$ Here, $e(x):=\exp(2\pi i x)$, $h,k',q,a'$ are all natural numbers. We do know that $\gcd(h,l'...

Let $i:\mathbb F\hookrightarrow\mathbb P^4$ be a cubic scroll i.e. $\mathbb F\simeq \mathbb P(\mathcal O_{\mathbb P^1}(1)\oplus\mathcal O_{\mathbb P^1}(2))\overset{n}{\rightarrow} \mathbb P^1$ with $\...

Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?

I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...

Let $X$ be a proper Deligne-Mumford stack of finite type over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus. Question: Is the following statement true? ...

Let $\omega^\omega$ denote the collection of functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say that $g$ eventually dominates $f$, or $f\leq^* g$, if there is $N\in\omega$ such that for ...

Is there a smooth projective variety over a complete discretely valued field such that a regular model is known but no semistable model is known? Models after a finite possibly ramified extension also ...

My father, who held 4 post graduate degrees and was a lifetime student, passed away recently. He has an entire bookcase full of older mathematics books, including some on related topics such as ...

I know it might be a bit elementary but I‘ve just read about the defintion of nerve functor $N: Cat \rightarrow sSet$ in nLab : https://ncatlab.org/nlab/show/nerve And Proposition 3.12 states that ...

Let $k$ be a perfect field of characteristic $p$, and $L$ be a finite extension of $k$. For a smooth projective variety $X$ defined over $k$, we denote the base change $X \times_k L$ by $X_L$. In this ...

In some MO questions such as this and this, Hamkins gave some examples that is independent with ZF+V=L, however, all of them increase the consistency strength. Are there some propositions P, which is ...

Fix a finitely generated Coxeter system $(W, S)$, and let $W_J$ denote the standard parabolic proper subgroup generated by a subset $J \subset S$. It is well known that the poset of cosets $\{xW_J\}$ ...

As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram $$\require{AMScd} \begin{CD} A @>>> X;\\ @VV{i}V @VVV \\ B @>&...

I have known that derivations on the first Weyl algebra are all inner. However，for the higher order algebras，does all derivatioms are inner ？

As for minimization problem, What is the advantages of global lower bound over tight lower bound ( which is larger than the global one) with respect to the quality of solution in case of approximate ...

I'm sure this is either a standard result or false, but I don't have enough experience with the derived category to decide either way. I have tried looking in Kashiwara-Schapira's Sheaves on Manifolds ...

Suppose that $\Lambda \subseteq \mathbb{R}^n$ is a lattice and $H \subseteq \mathbb{R}^n$ is a hyperplane such that $H \cap \Lambda$ has rank $n - 1$. I would like to know an upper bound on the ...

Let $\sigma,b\in\mathcal{C}^1\big(]0,1[,\mathbb{R}\big)$, with $\sigma$ non-vanishing, and consider the SDE \begin{align} & X_0=x\in\,]0,1[ \\ & \mathrm{d}X_t=\sigma(X_t)\,\mathrm{d}B_t+b(X_t)\...

Let $k \subseteq \bar{k}$ be an extension of fields (Orlov in the reference below seems to indicate the same thing will hold for faithfully flat maps but the case of fields is enough for me). On page ...

I am interested in the poset of all ideals of the local ring $$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$ $n=1$ is trivial. $n=2$ takes little work and it is shown below....

I am conducting my research in optimization in manifold. I come up with the following question. Let $\mathcal{F}_{p,q}$ is the collection of all matrices $\mathbf{F} \in \mathbb{C}^{p \times q}$ such ...

$\DeclareMathOperator\lcm{lcm}$Let $p_k$ be the $k$th prime number. Set $$L(n) = \lcm(p_1-1, p_2-1, \dotsc, p_n-1). $$ What can we say about the growth of $L(n)$? Trivially, one has that $L(n) < ...

Giuga's conjecture (1950), which is still open and has strong numerical support, reads : Let $n$ be a positive integer. If $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime. What would ...

Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time? Geometrically this means we have to sort through a positive proportion of the ...

Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that $$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$ where $\mu$ is close (or going) to zero. We also assume that the moment ...

Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book): Let $\mu = \mathcal ...

For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every ...

Recall that Matérn covariance function $C_\nu(d)$ is defined as $$ C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...

This is a repost of this question: https://math.stackexchange.com/questions/4244128/comparing-velocities-of-two-distinct-parametrizations-of-the-same-curve, because I think this can be a research ...

Consider the statement A proper birational map of varieties with a normal target has connected fibers. Is there a proof of it that does not involve inverse limits at all?