# All Questions

128,653
questions

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### A consequence of the Min-Max Principle for self-adjoint operators

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 $...

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20 views

### Definition of $\mathcal{O}_{\mathcal{X}}$-modules over a stack$\mathcal{X}$

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 ...

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17 views

### Area of N sided polygon?

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 ...

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32 views

### Minimal $b_2$ in Sarkisov's construction

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}...

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18 views

### Weights in a mixture distribution

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 ...

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**1**answer

53 views

### Determining polynomial approximations of piecewise constant functions

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
...

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**2**answers

59 views

### Different Bialgebra/Hopf algebra structures on coalgebras

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.

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**1**answer

63 views

### Is the Poisson formula valid when the boundary condition is $ L^2 $?

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 ...

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72 views

### Non-singular variety covered by pairwise disjoint singular subvarieties

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 ...

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37 views

### Least positive value of a random polynomial

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]...

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31 views

### Levy Ito decomposition

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 ...

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32 views

### Vector bundle associated to orthogonal flag

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 ...

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51 views

### Explicit $L_\infty$-operations on Hochschild cochains of $A_\infty$-algebra

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 ...

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**1**answer

56 views

### Distribution of stopping time for a 2D random walk

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 ...

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41 views

### Link between a categorical and a n algebraic characterization of (infinite-dimensional) hilbert space

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 ...

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37 views

### Inverse flips of $3$-folds

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 $p$, ...

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57 views

### Direct way to get the Hyodo-Kato Galois action

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-...

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23 views

### Is there an analogous notion of 'free quantum field of arbitrary spin' on a $4-$dimension finite lattice?

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 ...

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60 views

### Invariant metrics on homogeneous spaces

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 ...

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63 views

### Finding an injective envelope containing another injective envelope

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 ...

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103 views

### What does this exponential sum evaluate to?

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'...

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**1**answer

66 views

### Conics on a cubic scroll

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 $\...

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**1**answer

187 views

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

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

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45 views

### Eigenvector to zero eigenvalue of general Laplacian

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 ...

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92 views

### Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

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?
...

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63 views

### Restricting the quasi-order of eventual dominance on $\omega^\omega$ to "almost (strictly) increasing" functions

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 ...

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45 views

### Is there a variety such that a regular model is known but not a semistable model?

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 ...

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162 views

### What to do with antique/older mathematics books? Throw away or something else?

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 ...

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29 views

### How to prove the fully-faithfulness of nerve functor？ [migrated]

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 ...

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58 views

### Base change of Hodge-Witt cohomology

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 ...

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144 views

### Are there some interesting propositions independent with ZF+V=L that do not increase consistency strength?

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 ...

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**1**answer

61 views

### coset poset of reflection subgroup

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 $\{xW_J\}$ ...

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57 views

### A fibration is a map which has the right lifting property with respect to injections that are weak equivalences

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 @>&...

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23 views

### Derivations of n-th Weyl algebra

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

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13 views

### Difference between global lower bound and tight lower bound in minimization problem [closed]

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 ...

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55 views

### Product and coproduct in derived category

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 ...

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39 views

### Short lattice vectors in the complement of a hyperplane

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 ...

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23 views

### SDE on an interval with non regular boundary points: weak uniqueness?

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)\...

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59 views

### Perfect dg-modules under faithfully flat extension

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 ...

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**1**answer

215 views

### Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$

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....

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34 views

### Unit modulus manifolds

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 ...

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votes

**1**answer

140 views

### Asymptotics of $\operatorname{lcm} ((2-1), (3-1), (5-1), (7-1), (11-1), \dotsc, p_n-1 )$

$\DeclareMathOperator\lcm{lcm}$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) < ...

**8**

votes

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342 views

### What is a function field analog of Giuga's conjecture?

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 ...

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**1**answer

100 views

### Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)$ time

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 ...

**1**

vote

**1**answer

122 views

### Random walk always stays below a level $a$

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 ...

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29 views

### Generalization of a Gaussian measure continuity result from Hilbert to Banach space

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 ...

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**2**answers

181 views

### Is it improper to define matrices as being $n \times m$ rather than $m \times n$? [closed]

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 ...

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49 views

### Eigenvalues of Matérn covariance function

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), ...

**-1**

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23 views

### Comparing velocities of two distinct parametrizations of the same curve

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 ...

**2**

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131 views

### Zariski's main theorem without inverse limits

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?