# All Questions

100,225 questions

**3**

votes

**1**answer

118 views

### Bicommutant theorem for commutative operator algebras

Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says:
Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...

**4**

votes

**0**answers

154 views

### Confusion about topological Hochschild homology and $\mathbb{Z}_p$-topological Hochschild homology

Let $R$ be the ring of integers in a perfectoid field of mixed characteristic $p$. Is $\pi_*THH(R)$ (as defined in Bhatt--Morrow--Scholze) $p$-complete (as an abelian group)?

**0**

votes

**0**answers

39 views

### Relating the components of the Riemann curvature tensor to the second partials of the components of the metric

I am currently reading through a proof of Proposition 6 in
Chernoff's theorem and discrete time approximations of Brownian motion on manifolds OG Smolyanov, H Weizsäcker, O Wittich - Potential ...

**7**

votes

**0**answers

65 views

### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity.
If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...

**-3**

votes

**0**answers

29 views

### I want to obtain 'r' from the equation second time derivative of r is equal to a constant divided by square of r plus a constant? [on hold]

The solution expression should be like 'r=....', where the right hand side of the solution expression should not contain r. what to do? please help.

**3**

votes

**2**answers

63 views

### Why we use Caputo fractional derivative in application?

I'm working on some papers which use Caputo fractional evolution equation as application for thier main result:
For example:
$$\left\{\begin{matrix}
^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{...

**1**

vote

**0**answers

101 views

### Non-linear Lie algebra

Several versions of non-linear Lie algebras exist - at least in the physics litt. One version is just an ordinary Lie algebra but where the underlying vector space is a polynomial algebra.
Another is ...

**3**

votes

**0**answers

53 views

### Quartic link in a 5-sphere

In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...

**13**

votes

**1**answer

305 views

### Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$
Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO.
It is a well-known fact that a smooth projective ...

**2**

votes

**0**answers

72 views

### Is every orientable $I$-bundle over an orientable surface trivial?

Is every orientable $I$-bundle over an orientable surface $F$ trivial, $I \times F$? Is this also the case for vector all bundles?
Similarly, is every orientable $I$-bundle over an nonorientable ...

**1**

vote

**0**answers

31 views

### Continuous self-maps of the plane are semiconjugate or conjugate?

Let $f : X → X$ and $g : Y → Y$ be continuous functions. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $α : X → Y$ such that $$f∘α=α∘g$$.
A related idea is the ...

**2**

votes

**0**answers

31 views

### Enrichment of lax monoidal functors between closed monoidal categories

Let $\mathscr C,\mathscr D$ be (right) closed monoidal categories. Then both of them can be considered as enriched over themselves via their internal homs, which I will denote by $\textbf{Maps}$. Now ...

**3**

votes

**0**answers

81 views

### Is there an affine embedding X for every normal singularity, so that Pic(X)=0?

More formal: Let a normal algebraic singularity be given by a local ring $R$ of finite type. Is there always an affine variety $X$ with a point $x$, so that
$\widehat{\mathcal{O}}_{X,x} \cong \...

**3**

votes

**0**answers

59 views

### Transport Distance between Level Sets of a Convex Function

Suppose I have a well-behaved, strictly convex function $f : \mathbf{R}^d \to [0, \infty)$, and assume that $f$ has its unique minimiser at $x = 0$, with $f(0) = 0$.
For $y > 0$, I define the ...

**2**

votes

**2**answers

208 views

### Applications of flat submanifolds to other fields of mathematics

Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).
I am a curious about potential or actual applications to other ...

**-4**

votes

**0**answers

41 views

### External squares on sides of a triangle [on hold]

For the sides of a triangle ABC we construct external squares with center in P for AB side , Q for AC side and R for BC side.Proof that PQ=AR.

**0**

votes

**1**answer

86 views

### Regarding extreme point in a Banach space

Let $X$ be a Banach space. And let $X^* $ be the dual space of $X$. Let $E_X$ and $E_{X^*}$ denote the extreme points of the unit ball of $X$ and $X^*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E_{...

**1**

vote

**0**answers

30 views

### Equivalence of Sobolev spaces for different metrics

Consider $M$ a manifold and $g_1, g_2$ two different Riemannian metrics. I want to know how the condition $|\nabla^{g_1,k}(g_1-g_2)|_{g_1}\leq C$ implies that the norms of $|\nabla^{g_1,i}u|_{T^{\...

**2**

votes

**0**answers

46 views

### Holonomic sections $C^\infty(M)$-generate jet bundle

Given a vector bundle $E \to M$ with a corresponding $k$-th jet bundle $J^kE \to M$, denote by $j^k : \Gamma(E) \to \Gamma(J^kE)$ the $k$-th jet prolongation $(k \in \mathbb{N} \cup \{0\})$ and recall ...

**6**

votes

**1**answer

373 views

### The Hilbert symbols of quaternion algebras over a totally real field

Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as
$$B = \left(\frac{a,b}{k}\right), $$
for some constants $a,b \in k^\times$. My question is, can I always ...

**2**

votes

**0**answers

55 views

### Associated bundle construction and classifying space

Let $\theta:G\rightarrow H$ be a morphism of Lie groups.
Given $G$ we have classifying space $BG$ and given $H$ we have classifying space $BH$. This $\theta:G\rightarrow H$ gives a map $B\theta:BG\...

**-1**

votes

**1**answer

49 views

### CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation*}
f_U(u)=\exp\...

**1**

vote

**0**answers

53 views

### Sobolev embedding in complete manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry.
Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we ...

**5**

votes

**1**answer

115 views

### How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?

I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...

**1**

vote

**0**answers

43 views

### coboundary in the slow mixing systems

given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...

**4**

votes

**0**answers

196 views

### Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$

Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...

**4**

votes

**0**answers

63 views

### Invariant polynomials in curvature tensor vs. characteristic classes

Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weyl theory. For any sequence of numbers $k_1, \dots, k_l$ such ...

**3**

votes

**1**answer

132 views

### Reference request: Dynamical systems

I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read ...

**1**

vote

**0**answers

72 views

### Locally compact metric spaces whose group of isometries generated by isometries of small displacement is transitive enough

I wasn't sure how to make the title any more precise than that.
Let $(X,d)$ be a locally compact metric space. For any $\varepsilon>0$ let $\mathrm{Aut}_\varepsilon (X)$ be the subgroup of the ...

**0**

votes

**0**answers

30 views

### Are these conditions sufficient for a self-distributive algebra to occur in the algebras of elementary embeddings?

Suppose that $\mathcal{E}_{\lambda}$ is the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ and $*$ is the operation on $\mathcal{E}_{\lambda}$ defined by $j*k=\bigcup_{\alpha&...

**0**

votes

**0**answers

55 views

### Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$
where $i$ is the imaginary unit number. We see that the integral of a Fourier transform
$$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...

**2**

votes

**0**answers

24 views

### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

**4**

votes

**0**answers

54 views

### Is there a notion of tunnel number for 2-knots?

Given an embedded circle $K$ in $S^3$, the tunnel number of $K$ is the minimum number of embedded arcs one needs to add to $K$ so that the complement of $K$ and the arcs is a handlebody.
For an ...

**6**

votes

**0**answers

101 views

### A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?
...

**0**

votes

**1**answer

100 views

### Infinite norm of two randomly picked points [on hold]

Let X and Y be points in the 4000-dimensional unit cube, picked at random with uniform distribution, which means from I what I understand that all locations in the cube are equally likely. $X \in [0,1]...

**3**

votes

**0**answers

70 views

### Arbitrary distortion of cyclic subgroups of solvable groups

Suppose $G$ is a finitely generated solvable group, and let $H$ be an infinite cyclic subgroup of $G$. What possible functions may arise as the subgroup distortion of $H$ in $G$?
For polycyclic ...

**0**

votes

**0**answers

29 views

### Different solution sets of Partial Differential equations

Consider Laplace equation $∇^2u=0$. We can find a set of solutions for that by assuming $u=f(x)g(y)$. Also we can find another set of solutions by assuming $u=f(x)+g(y)$ that is not the same as the ...

**3**

votes

**1**answer

138 views

### Graded Grothendieck Group and Hilbert Polynomial

I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group.
Let $A$ be a noetherian graded $K$-...

**2**

votes

**0**answers

153 views

### Triple link in a 5-sphere — Proposal

In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...

**-3**

votes

**0**answers

24 views

### How to choose the suitable confidence interval for my data? [on hold]

Let's suppose that I have a set of dataset. My data representsFor every dataset, I compute its regression line: y=slope*x +intercept. I extract different slopes.
I ...

**0**

votes

**0**answers

40 views

### Smallest eigenvalues of block Kronecker product

Let $D \in \mathbb{R}^{n \times n}$ defined as
\begin{equation}
D := \begin{pmatrix}
1 & 0 & \cdots & \cdots & 0 \\
-1 & 1 & \ddots & \ddots & 0 \\
\vdots & \ddots &...

**0**

votes

**1**answer

24 views

### Error metric for joint estimation of mean and variance

Background:
Let $\mu:\mathbb{R}^n\to\mathbb{R}$ and $\sigma:\mathbb{R}^n\to\mathbb{R}_+$ be two unknown functions, and consider a stochastic model of the form
$$
\mathbb{E}[Y|\mathbf{x}] = \mu(\...

**4**

votes

**0**answers

83 views

### Characterisation of special Cohen-Macaulay modules

Let $A$ be an $R$-order of Krull dimension $d$ that is an isolated singularity, see for example section 3 of https://arxiv.org/pdf/0803.2841.pdf for the relevant definitions.
With $D_d(-):=Hom_R(-,R)$...

**0**

votes

**0**answers

106 views

### Request for an English proof of Robin's result on $\sigma(n)$

Define $\sigma(n)=\sum_{d\mid n} d$. It is a result of Robin that if the Riemann Hypothesis is false, then there exists constants $0<\beta<1/2$ and $C>0$ such that
$$\sigma(n)\geq e^{\gamma}...

**-4**

votes

**0**answers

32 views

### I am looking for a working function in python to numerically calculate an N dimentional integral [on hold]

https://drive.google.com/file/d/1QJU6-MX5X0CJWesQTCcL_Ws5p_UQgtFW/view?usp=sharing
This is the article I am reading and trying to implement,
on page 22 this formula
The issue I have is that this ...

**1**

vote

**0**answers

16 views

### Optimal control problem with spike source and “split” state

For $p \in \mathbb{R}$, consider the following problem:
\begin{equation} \label{1}
\begin{cases}
\operatorname{div}(a \nabla u ) = p\delta_{x_0} \quad \text{in } \Omega \\
u=0 \quad \text{on } \...

**0**

votes

**0**answers

85 views

### Can someone give a closed immersion of schemes $f: Z\to X$ such that $f(Z)\subset U$ with $U$ a proper open subset of $X$? [on hold]

Please give a closed immersion of schemes $f: Z\to X$ such that $f(Z)\subset U$ with $U$ a proper open subset of $X$.

**1**

vote

**0**answers

76 views

### Geometric meaning of residue constraints

$\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having ...

**0**

votes

**1**answer

49 views

### Inequality involving product-of-minus vs minus-of-product for positive integers

I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...

**3**

votes

**1**answer

244 views

### Is it consistent that $|[\kappa]^{<\kappa}| > \kappa$?

Let $\kappa>0$ be a cardinal, and let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality strictly less than $\kappa$. Is it consistent that $$|[\kappa]^{<\...