All Questions

Consider a $n$-branched $\pi: S \rightarrow M$, where both $S$ and $M$ are algebraic curves. Take $L\rightarrow S$ a line bundle. I want to understand why $\pi_{*}L$ is a $n$ dimensional vector bundle....

A closed subspace $M$ of $L_2(0,1)$ is said to be strongly embedded if the norms $\|\cdot\|_2$ and $\|\cdot\|_1$ are equivalent on $M$. Let $(f_n)_{n\in \mathbb N}$ be a orthonormal basis of $L_2(...

I've read that a degenerate LP can make the simplex algorithm slower. In the following example however, it appears that it even fails to converge. I am sure I am making a basic mistake or a wrong ...

In this post I consider the abc conjecture, the formulation ABC Conjecture II from the Wikipedia abc conjecture, where the product of the distinct prime numbers dividing an integer $m>1$ is denoted ...

How much time does a referee take for a journal in pure mathematics? I have submitted a paper almost 10 and 1/2 months back and the paper consists of less than 15 pages. I have already sent two ...

I still don't get why the axiom of choice isn't trivial. I mean- if our domain $X$ contains only one non-empty set S, the proof of the AOC, in this case, is trivial- $\exists x\in S$ so our choice ...

Let $f \left (x_1, \ldots x_n; u, \frac{\partial u}{\partial x_1}, \ldots \frac{\partial u}{\partial x_n}; \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots \frac{\partial^2 u}{\partial x_1 \...

For a field $K\subset \mathbb{Q}(\zeta_p)$ $~$($\zeta_p$ a primitive pth root of unity, p a prime), it seems to be the case that the discriminant of $K$ is $p^{[K:\mathbb{Q}]-1}$ (according to ...

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...

Premise Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed determinants and I find their definitions ...

Let $E$ be a (Dedekind $\sigma$-complete) Riesz space and $H\subseteq E$ a subspace. A Daniell integral $I\colon H\to\mathbb R$ is defined to be a positive linear functional which is continuous with ...

Suppose $(M,g)$ is a smooth globally hyperbolic Lorentzian manifold of dimension $n$. Let $\beta:I \to M$ be a finite null geodesic in $M$, that is to say: $$ \nabla^g_{\dot{\beta}}\dot{\beta} = 0, \...

Let $X$ be a regular finite connected graph and $\Gamma$ its fundamental group. Fix a character $\chi:\Gamma\to\mathbb{C}^\times$. This defines a locally constant sheaf on $X$ and there is an ...

In other words, does the first order theory of $(\mathbb{R},+,\times,0,1)$ have a computable countable model? What do we know more generally about countable models of real arithmetic?

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...

Let $\mathbb{D}^n = \{ x \in \mathbb{R}^n: |x| \leq 1 \}$ be the unit ball in $\mathbb{R}^n$, and let $ \omega$ be the usual volume-form on $\mathbb{D}^n$ inherited from $\mathbb{R}^n$. What is the ...

Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...

Suppose M is a compact manifold and Met(M) is the Frechet manifold of all Riemannian metrics on M, does there exist a set of metrics with distinguished properties (e.g. metrics with constant scalar or ...

We say, following this definition, that a domain $\Omega\subset \mathbb{R}^{n}$ is weakly Lipschitz if it can locally be flattened by a Lipschitz homeomomorphism $\phi$ (i.e., a Lipschitz continuous ...

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...

Consider a vector $X=[x_1,x_2,\cdots,x_n]$. It is given that: a) $x_i > 0, \forall i$ b) $\mathcal{l}_2$-norm $(X) = 1$, i.e $\sum_{i=1}^{n} x_i^2 = 1$ I would like to know what is maximal (...

It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...

I have a question: Suppose that a knight makes a “random walk” on an infinite chessboard. Specifically, every turn the knight follows standard chess rules and moves to one of its eight accessible ...

Similar questions have already been asked here and here but not exactly in the direction I need. I have a (small) index category $\mathcal{I}$ which is not cofiltered, and I need to consider ...

Let $p: \mathcal{F} \rightarrow \mathcal {C}$ be the data of a fibred category. Then, for arrows $f: U \rightarrow V$ in $\mathcal{C}$, a morphism $\phi: \xi \rightarrow \eta$ in $\mathcal{F}$ is said ...

Suppose there is a quasiisomorphism $q: A \to B$ between DG algebras. Is there some reasonable description of induced functor $q^*: B-mod \to A-mod$? Can we say something better if it was a $B$-...

I have asked a version of this question here, but have been unable to receive an answer or devise one of my own, and am venturing that it may be appropriate for Math Overflow. The "internal form" of ...

Let g be a pseudo-Riemannian metric on a Lie algebra A. How to find the adjoint of the endomorphism ad (x) compared with the metric g?

Can someone verify this https://arxiv.org/abs/1910.02954 using COQ or any other similar software ?

I am reading a paper ('Finitely presented residually free groups', http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.210.3212&rep=rep1&type=pdf, Theorem 5.2) where they write the ...

I have a warped product $M=N_1\times_f N_2$ where $N_1$ and $N_2$ are Riemannian manifolds. The dimension of $N_2$ is $2n$ (for n integer) and $N_2$ is an almost Hermitian manifold, i.e., is ...

This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking if every obtuse triangle admits a periodic billiard path, which has been open ...

In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative ...

Consider a $n$-branched cover $\pi:S\rightarrow M$, where $S$ and $M$ are both algebraic curves. If $\pi_{0}: L\rightarrow S$ is a line bundle over $S$, we can define the bundle $\pi_{*}L$ on $M$ ...

Let $K_t$ be certain $1-$ parametric family of knots in $\mathbb{R}^3$. I am wandering what are the precise obstructions for a parametric Seifert surface to exist; i.e. a $1-$parametric family of ...

I am trying to understand a version of Lotthar Göttsche's computation of the betti numbers of the punctual Hilbert Scheme of a smooth projective surface. However, before I can even begin, I am unsure ...

Let $\pi:\mathcal{C}\rightarrow S$ we a flat family of stable curves parametrized by $S$. Consider the relative dualizing sheaf $\omega_{\pi}$ of $\pi:\mathcal{C}\rightarrow S$ and the line bundle $L_{...

I always come here when I have a programming question so I figured only the professionals answer here :) So the question is: 60 workers will complete a task for 28 days. If after 10 days 12 more ...

Let $\Omega_x$ and $\Omega_y$ be sets of finite Lebesgue measure. We can then look at the space $X_1:=L^2(\Omega_x \times \Omega_y).$ This space is contained in the larger space $$X_0:=L^2(\...

Let $S^n$ and $B^{n+1}$ be the unit sphere and unit ball in $\mathbb{R}^{n+1}$, and let $M^n$ be a contractible space of dimension $n$. If necessary, assume that $M^n$ is a contractible simplicial $n$-...

Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that $x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$; ...

Not sure if this makes sense. Let $K$ be field and $C : f(x,y)=0$ algebraic curve curve over $K$. Define the "pseudo complement" $\hat{C}$ to be the rational surface $z f(x,y) - 1=0$ with ...

For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia Gregory coefficients. I think that I was inspired in [1] to ask the question. ...

Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x_{1}+\dots+x_{n}=1\rbrace$ it is true that \begin{equation} |p_{i}-q_{i}|\le c\left|\ln\frac{...

Let $\pi\colon X\to Y$ be a proper morphism of smooth complex algebraic varieties with $\dim X = 2n$ and general fibers of dimension $<n$. Assume that $F := \pi^{-1}(p)$ is a an irreducible and ...

Let $j$ be the modular invariant and let $\tau$ be a point in the upper half-plane. Let $\mathfrak o_\tau$ consist of all $f\in \mathbf Q(j)$ which are defined at $\tau$. Let $\mathfrak O_\tau$ ...

Zeta functions and the cycle index polynomials (CIPs, $p_n(\sigma)$) of the symmetric groups are intimately related as described in the MO-Q "Cycling through the zeta garden". In that question a ...

Consider a parametric family $p_\theta(x)$ of distributions, with parameter $\theta \in \Theta \subseteq \mathbb R^p$. If the mapping $\theta \mapsto p_\theta(x)$ is continuously differentiable at $\...

What is the best way to obtain a sparse solution for a linear system $\mathbf{A}\vec{x}=\vec{b}$ with $x_n \in \mathbb{R}$? The linear system is special, because I know that: some columns $\vec{c}_n$ ...

Given a manifold $M$ and a connection $\nabla$ on the tangent bundle $TM\rightarrow M$, we can talk about torison of the connection $\nabla$. The question What is torsion in differential geometry ...