All Questions

Filter by
Sorted by
Tagged with
0
votes
0answers
3 views

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 $...
1
vote
0answers
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 ...
-3
votes
0answers
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 ...
3
votes
0answers
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}...
-1
votes
0answers
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 ...
1
vote
1answer
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 ...
4
votes
2answers
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.
0
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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]...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
4
votes
0answers
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 ...
1
vote
1answer
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 $p$, ...
2
votes
0answers
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-...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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'...
1
vote
1answer
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 $\...
4
votes
1answer
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?
0
votes
0answers
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 ...
4
votes
0answers
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? ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
4
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
5
votes
0answers
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 ...
0
votes
1answer
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 $\{xW_J\}$ ...
0
votes
0answers
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 @>&...
0
votes
0answers
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 ?
0
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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)\...
3
votes
0answers
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 ...
6
votes
1answer
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....
0
votes
0answers
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 ...
3
votes
1answer
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
2answers
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 ...
1
vote
1answer
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
1answer
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 ...
0
votes
0answers
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 ...
0
votes
2answers
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 ...
2
votes
0answers
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
votes
0answers
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
votes
0answers
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?

15 30 50 per page
1
2 3 4 5
2574