All Questions

0
votes
0answers
2 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 ...
1
vote
0answers
33 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? ...
1
vote
1answer
36 views

Infinite norm of two randomly picked points

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]...
1
vote
0answers
29 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
0answers
20 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 ...
2
votes
0answers
56 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 $K$-...
1
vote
0answers
96 views

A 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})$...
-1
votes
0answers
18 views

How to choose the suitable confidence interval for my data?

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
0answers
29 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
0answers
11 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(\...
1
vote
0answers
50 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
0answers
65 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
0answers
28 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
0answers
14 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
0answers
60 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 $X$.
0
votes
0answers
53 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
0answers
24 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>...
2
votes
1answer
165 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]^{<\...
0
votes
1answer
50 views

meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...
0
votes
0answers
41 views

R²/Plane Subset Equation With Plane Transormation [on hold]

Let's consider $H_k∶\ \left\{\begin{matrix}\mathbb{R}^2\rightarrow\mathbb{R}^2\\(x,y)\longmapsto(kx,ky)\\\end{matrix}\right.\ $ It is an homothetic transformation of $\mathbb{R}^2$ of center $(0,0)$...
6
votes
1answer
233 views

Closed form expression for a recursion relation with binomial coefficients

I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k \end{pmatrix} T_{k}, \ \ \ \ T_0 = C \in \mathbb{N} $$ I would like to express it as a function of n, ...
9
votes
3answers
1k views

Does anyone recognize this inequality?

In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
-1
votes
0answers
53 views

how to compute the number of possible trees in a tree graph? [on hold]

Let's suppose that I have a tree with n nodes. The root of my tree does not change in time. It is the same. However, the rest of nodes change their positions (...
1
vote
1answer
56 views

Minimization Proof of Conditioning on Gaussian is Gaussian

It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...
2
votes
0answers
78 views

Reference request: invariants/tableaux functions for 4 lines in $P^3$

Does anybody have a reference for invariants of configurations of linear subspaces in the projective space? In particular I would be curious to see an explicit expression of the invariant functions ...
5
votes
0answers
51 views

Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
5
votes
2answers
792 views

William Thurston's quote?

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. Is this from Thurston? If yes, where and when it has been said. I've checked "ON PROOF AND ...
4
votes
1answer
73 views

Closures of orbits in the space of representations of a quiver

Let $Q$ be a quiver, and let $d=(d_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \...
-6
votes
0answers
44 views

Could someone please explain how these questions can be solved without a calculator [on hold]

I am currently stuck on these practice questions and was wondering If I could get some advice to help solve them. Questions here
-4
votes
0answers
57 views

It is possible to draw 15 line segment such a way that each line segment intersect exactly 5 others ? Justify your answer [on hold]

i am stuck in above question. 5*5 line i draw then only it is possible to cut 5 points and all 5 need to be parallel horizontally and 5 vertically. how can we prove above.
7
votes
1answer
186 views

Does the cubical nerve preserve weak equivalences of simplicial sets?

The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N_\Box: sSet \to Set^{\Box^{op}}$. $N_\Box$ is a right Quillen equivalence, ...
0
votes
0answers
24 views

Fokker-Planck equations where drift and/or diffusion terms are not differentiable at some points

Fokker-Planck equations are given by Is this equation correct if drift ,$\mu(x,t)$, or diffusion term ,$D(x,t)$, are not differentiable with respect to $x$ at some points? If not, then how to drive ...
4
votes
1answer
67 views

A Riccati type integral inequality

Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality: $$ x'(t) \le \int_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds, $$ for any $t \ge 1$, where $k(t),t\in [1,\...
4
votes
1answer
152 views

Are complete minimal submanifolds closed?

Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?
5
votes
1answer
316 views

Explaining the consistency of PRA and ZF from predicative foundations

Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to ...
6
votes
0answers
122 views

Chromatic polynomial and the circle

In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
1
vote
0answers
30 views

Smallest eigenvalue for Gram matrix of unit norm matrices

Given $n$ symmetric matrices $A_1, \dots, A_n \in \mathbb{R}^{k\times k}$, such that $\|A_i\| \leq 1$ for all $i$, we consider the matrix $M \in \mathbb{R}^{n\times n}$, where $M_{ij} = \langle A_i, ...
0
votes
0answers
172 views

Are there any schools of mathematics that hold that one should know the proof to every theorem that they know? [on hold]

Disclosure: I am not a research mathematician, but an undergraduate in that field. I have noticed that the pedagogical model for my entire math education has depended on memorization of theorems ...
4
votes
1answer
182 views

Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$. Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...
5
votes
0answers
65 views

Differential equations satisfied by quasi modular forms?

It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question. Differential Equations Satisfied by Modular Forms ...
0
votes
0answers
20 views

Infinite spectral norm of linear mapping [on hold]

Suppose we have a linear mapping $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$. We define its $k$-spectral norm as: $\sigma_k(A)=\sup_{x} \frac{||Ax||_k}{||x||_k}$. We know that when $k=2$, $\sigma_2(A)$ ...
3
votes
1answer
125 views

Finding a curve through a zero-cycle and a dense subset

Let $k$ be a field. Is it true that for any smooth irreducible projective $k$-variety $X$ and a dense open set $U\subset X$, for any zero-cycle on $X$ one can find an irreducible curve containing its ...
2
votes
0answers
89 views

Hardy-Littlewood in Sobolev Spaces

For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
1
vote
0answers
59 views

On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p + \cdots + p^i)$, where $i > 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ of the ...
-4
votes
0answers
59 views

Euclidean Geometry [on hold]

In Euclidean Geometry we can create a square with side length of $\sqrt2$. as we know, $\sqrt2$ is a number which has no end. so it is physically impossible to have a square with this length in the ...
0
votes
0answers
31 views

Find and prove optimal solution for a linear programming problem

Given $C \in R^+$, $\epsilon \in R^+$, $vec \in R^n$ and $eps = [\epsilon \dots \epsilon]$ vector of n $\epsilon$ I have to $ \textrm{minimize } [vec, eps]^T [x_1, x_2]$ with $x_1, x_2 \in R^n$ ...
1
vote
0answers
50 views

Component Groups of Reductive Groups

Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can ...
0
votes
0answers
33 views

Estimate on Covariant Derivatives of Coordinate Derivatives

I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that $\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\...
-3
votes
0answers
25 views

Fundamentals of Networks and Graph theory: listing labelled graphs with given number of edges [on hold]

Under the assumption of simple graph, how it is possible to determine the list of all the labelled graph with order 4, and so vertex set {1,2,3,4} and three edges?
11
votes
1answer
189 views

Quasimorphisms and Bounded Cohomology: Quantitative Version?

Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...

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