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15 views

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....
1
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0answers
6 views

Subsequences of an orthonormal basis generating a strongly embedded subspace in $L_2(0,1)$

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(...
-1
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0answers
5 views

Consequences of degenerate LP

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 ...
0
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0answers
12 views

On inequalities that can arise from a combination of the abc conjecture and the Riemann hypothesis

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 ...
-4
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0answers
85 views

Average time for the referee report of a journal in pure mathematics [on hold]

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 ...
-1
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0answers
84 views

Why the axiom of choice isn't trivial? [on hold]

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 ...
3
votes
0answers
23 views

Integral of Differential Operator Continuous?

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 \...
1
vote
1answer
45 views

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 ...
0
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0answers
36 views

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 ...
3
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0answers
17 views

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 ...
0
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0answers
18 views

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 ...
0
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0answers
46 views

Is it possible for null geodesics to suddenly “stop”?

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, \...
0
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0answers
11 views

Twisted Laplace eigenvalues on graph

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 ...
0
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0answers
29 views

Is there an analog of Tennenbaum's theorem for real arithmetic?

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?
0
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0answers
41 views

Frobenius twist of a field

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 $k$-...
1
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0answers
34 views

Volume-preserving diffeomorphism group of ball $\mathbb{D}^n$

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 ...
1
vote
1answer
40 views

Is there an integral fusion ring which is not of Frobenius type?

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 = \...
1
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0answers
28 views

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 ...
2
votes
1answer
24 views

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 ...
2
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0answers
40 views

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 ...
-3
votes
0answers
16 views

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 (...
0
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0answers
25 views

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 ...
0
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0answers
62 views

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 ...
0
votes
1answer
59 views

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 ...
0
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0answers
58 views

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 ...
0
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0answers
54 views

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 $B$-...
1
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0answers
39 views

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 ...
-1
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0answers
9 views

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?
-6
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0answers
29 views

Proof verification using COQ [on hold]

Can someone verify this https://arxiv.org/abs/1910.02954 using COQ or any other similar software ?
3
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0answers
95 views

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 ...
1
vote
0answers
57 views

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 ...
14
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0answers
183 views

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 ...
9
votes
0answers
128 views

'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 ...
0
votes
0answers
99 views

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 $M$ ...
0
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0answers
35 views

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 ...
-1
votes
0answers
94 views

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 ...
1
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0answers
55 views

Relative dualizing sheaf and automorphisms

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_{...
-4
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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 ...
1
vote
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(\...
3
votes
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 $n$-...
5
votes
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 $j$; ...
1
vote
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 ...
0
votes
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. ...
-1
votes
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 \begin{equation} |p_{i}-q_{i}|\le c\left|\ln\frac{...
2
votes
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 ...
-1
votes
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 $\mathfrak O_\tau$ ...
0
votes
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 ...
2
votes
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 $\...
0
votes
0answers
15 views

Obtain a sparse solution for a bad conditioned linear system with either or constraints

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 $\vec{c}_n$ ...
0
votes
0answers
68 views

Are torsion and torsor related in differential geometry

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 ...

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