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### Understanding the direct image of a line bundle

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....
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### discriminant of subfield of $\mathbb{Q}(\zeta_p)$

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 ...
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### Analytic formula for leading eigenvector of $uu^T + vv^T$?

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 ...
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### Are mixed determinants and hyper-determinants the same thing?

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 ...
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### Daniell integral of “generalized (of some sort)” functions?

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 ...
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### Which set of Riemannian metrics is compact in the manifold of Riemannian metrics?

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 ...
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### Is the projection of a weakly Lipschitz domain still a Lipschitz domain?

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 ...
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### Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

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 ...
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### maximal and minimal value of entries of : X*l1-norm(X)

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 (...
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### Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces

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 ...
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### The 'knight on an infinite chessboard' puzzle [on hold]

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 ...
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### Need of filtered indexed categories

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 ...
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### Regarding the definition of $f$-morphisms/cartesian arrows in a fibred category $\mathcal{F} \rightarrow \mathcal{C}$

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 ...
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### Modules over quasiisomorphic DG algebras

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 $B$-...
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### Is there an internal notion of “flat continuous presheaf”?

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

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?
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### Proof verification using COQ [on hold]

Can someone verify this https://arxiv.org/abs/1910.02954 using COQ or any other similar software ?
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### Extension of $FP_{n}$ group

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 ...
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### Real-Complex warped product

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 ...
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### Why is the billiard problem for obtuse triangles so hard?

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 ...
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### 'Continuity' of the étale topos

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 ...
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### Degree of the direct image of a line bundle

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 $M$ ...
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### Parametric Seifert surfaces for parametric families of knots in $\mathbb{R}^3$

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 ...
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### Cohomology of (complex) varieties

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 ...
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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 $L_{... 0answers 33 views ### Question about proportionality urgent! [on hold] I always come here when I have a programming question so I figured only the professionals answer here :) So the question is: 60 workers will complete a task for 28 days. If after 10 days 12 more ... 1answer 81 views ### Interpolation of$L^p$spaces Let$\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(\... 0answers 81 views ### Extending a map from$S^n\to M^n$to a nice map from$B^{n+1}\to M^n$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$n$-... 1answer 172 views ### Can I cover a compact set by balls {B} such that {2B} has bounded overlap? 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$j$; ... 0answers 63 views ### Algebraic definition of the “pseudo complement” of algebraic curve 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 ... 0answers 28 views ### Partial sums involving Gregory coefficients that cannot be an integer 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. ... 1answer 48 views ### Lower bound for log-Ratios 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 |p_{i}-q_{i}|\le c\left|\ln\frac{... 0answers 66 views ### Self intersection number for special fibers 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 ... 0answers 49 views ### Modular functions and integral closure of a valuation ring 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$\mathfrak O_\tau$... 0answers 64 views ### Zeta functions, zeros, and extrema and cycle index polynomials 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 ... 0answers 51 views ### Is there any geometric interpretation for the trace of Fisher information matrix? 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 ...