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80 views

On some loci of rings

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set $$ P(R) =\{\mathfrak p \in Spec(R)\,\,\, |\,\,\, R_{\mathfrak p}\, \, \mbox{is } P\},$$ $$ nP(R) =\{\mathfrak p \in ...
2
votes
0answers
97 views

When is the following a formula for local cohomology?

Suppose $R$ is a Noetherian local ring, and $\kappa$ its residue field. For $R$ module $M$, we can consider the module $$N:=\kappa \otimes_S RHom(\kappa,M)$$ where $S$ is the derived ring of ...
4
votes
1answer
327 views

How do working constructivists get by with out the zero product property?

It is stated by Douglas Bridges in Constructive mathematics: a foundation for computable analysis that the following property, which I will call the zero product property: If $x,y \in \mathbb{R}$ and $...
2
votes
2answers
141 views

Name of an inductively defined sequence of graphs

Let $G_k$ be the graph obtained by applying the following procedure k-times: Start with a graph with single vertex $v$ (Call this graph $H$) Add a vertex $u$ such that $u$ is not adjacent to any ...
2
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0answers
46 views

Does the following corollary of Mackey's tensor product theorem hold for smooth representations?

Let $G$ be a locally profinite group, and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth representation of $G$, and let $\tau$ be a smooth representation of $H$ (henceforth, every ...
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0answers
97 views

Why do we assume that a stopping time is a random variable?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a probability space and $\tau: \Omega \rightarrow [0,\infty]$ be a stopping time. By definition this should be random variable so ...
2
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0answers
67 views

Tate module whose maximal semisimple subrepresentation is a line

An abelian variety over $\mathbb{Q}_p$ is cool if the maximal semisimple subrepresentation of its Tate module is a line. Are there cool abelian varieties of arbitrarily high dimension? What about the ...
2
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0answers
131 views

Words with critical exponent $< \frac 73$

In a comment made by Gjergji Zaimi to this older question, it is conjectured that $\frac 73$ is the threshold separating countability and uncountability of the sets of infinite binary words having a ...
4
votes
1answer
209 views
+100

$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$

Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that $$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
9
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1answer
230 views

Revisiting Gordon-Luecke theorem

$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also ...
2
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0answers
74 views

All Galois characters showing up in cohomology of one family of varieties

Fix a prime $p$. Can we find a smooth proper map $X\to Y$ of $\mathbb{Q}_p$-varieties such that any given representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to \mathrm{GL}_1(\mathbb{F}...
4
votes
2answers
152 views

When does a finite metric induce a matrix norm?

If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit ...
4
votes
1answer
115 views

Computation of cusp shape from vertex invariants

Following Takahashi ("On the concrete construction of hyperbolic structure of 3-manifolds"), I was able to construct the Euclidean cusp cross-section for the 5_2 knot complement (please see ...
4
votes
1answer
327 views

Total sum of characters of the symmetric group $\frak{S}_n$

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...
0
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0answers
81 views

Scott-replacement and transitive closure

In D. Scott: More on the axiom of extensionality, in Essays on the foundations of mathematics, dedicated to A. A. Fraenkel on his seventieth anniversary, edited by Y. Bar-Hillel, E. I. J. Poznanski, M....
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0answers
69 views

A special case of Frankl's conjecture. A question about known results

Let's recall a Frankl's conjecture. Consider a finite family of finite sets $\mathcal{F}$, such for every pair of sets $A\in \mathcal{F}$ and $B\in\mathcal{F}$, we have $A\cup B\in\mathcal{F}$(the ...
1
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1answer
40 views

Conditions such that split coequalizers are a symmetric notion

Consider the notion of a split coequalizer (see the nLab for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes ...
2
votes
1answer
103 views

Distance formula for continued fractions

In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated: $$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
3
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0answers
132 views

When is the space of maps between varieties a finite CW complex?

$\DeclareMathOperator\Cont{Cont}$Given two algebraic varieties over $\mathbb{C}$ denoted by $X$ and $Y$ where $Y$ is projective and $X$ is either projective or affine/Stein. The space of continuous ...
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0answers
35 views

Reference request: Pascal type octagon theorem

I am looking for a reference to a generalisation of the celebrated hexagon theorem of Pascal which states that if $A$, $B$, $C$, $A_1$, $B_1$ and $C_1$ are the (distinct) vertices of a hexagon in ...
2
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1answer
103 views

Integer sequences with unique $k$-subsets sum

let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the ...
5
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0answers
118 views

Center of Grothendieck differential operators in positive characteristic

Let $k$ be a field of characteristic $p$. Consider the algebra $A:=\mathcal{D}(k[x])^{S_2}$ consisting of Grothendieck differential operators invariant under the $S_2$ action $x\mapsto -x$. The ...
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0answers
16 views

Closed-form expression for the integral of a Gamma CDF times a Rice PDF

Given the following CDF and PDF \begin{equation} {F_{\gamma_d}}(\gamma_d)=Q\left(\kappa ,0,\sqrt{\frac{{\lambda_d}}{\theta ^2}}\right)~~\text{for $\gamma_d>0$} , \end{equation} \begin{equation} ...
-1
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0answers
53 views

Changing base field in MAGMA [closed]

How do I change base field in MAGMA? I am working with field extensions $\Bbb Q \subseteq K \subseteq L$. I want to change the basefield of $L$ from $\Bbb Q$ to $K$.
26
votes
3answers
2k views

How can I seek help in preparing a very long research article for publication?

Some background first. I recently graduated (a couple of years ago) with a master's degree in applied mathematics. During graduate school I began working on a paper, which I continued to work on in my ...
14
votes
1answer
374 views

Reference request: Moore graphs

It is clear that the term Moore graph was coined by Hoffman and Singleton in their paper On Moore graphs with diameters $2$ and $3$, where they write E. F. Moore has posed the problem of describing ...
0
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0answers
96 views

Relation between Gamma function and base 2 exponential

In my work, I keep coming across the term $$ f(x) = \frac{1}{\sqrt{2\pi}} 2^{\frac{x-1}{2}} \Gamma(\frac{x+1}{2}), $$ in particular for $x \in [0,1]$. I have been working on bounding the density ...
1
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0answers
89 views

Dimension of the moduli space of conformal classes

Let $\Sigma$ be a closed orientable surface of genus $\gamma \geqslant 2$. By the Uniformization theorem the moduli space of conformal classes on $\Sigma$ can be identified with the space of ...
2
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0answers
59 views

Self study guide to Hamiltonian Monte Carlo

I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo. In this paper The Geometric Foundations of Hamiltonian Monte Carlo it is ...
2
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1answer
143 views

A simple question on the Navier-Stokes system

The Navier-Stokes system for incompressible fluids in $\mathbb R^3$ reads as \begin{align} &\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=...
5
votes
3answers
546 views

Is a spin structure on a knot complement the same thing as an orientation of the knot?

There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them: Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
3
votes
1answer
115 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
6
votes
0answers
62 views

How are symmetric functions related to Koszul duality?

Staying within the world of linear algebra, we have the following two "dualities" between exterior powers and symmetric powers. The first is that of Kozsul duality, so these two graded ...
4
votes
0answers
117 views

Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field. Let $A$ be a local integral domain with maximal ideal $M$, residue ...
2
votes
1answer
99 views

LLPO as constructivity/computability for dense subsets

LLPO is the statement $\forall x \in \mathbb R. x \leq 0 \vee x \geq 0.$ The statement should be understood as a fragment of the Law of Excluded Middle, rather than a statement about the ordering of ...
0
votes
1answer
73 views

A general Turan-like question

Thinking of an edge as of a $2$-clique, it's natural to consider a slightly more general question than Turan considered in his celebrated theorem: given $r \le k \le n$, what is the maximal possible ...
1
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0answers
150 views

Fields such that every finite Galois extension is solvable

What are the fields such that every finite Galois extension is solvable? We have algebraically closed fields, real closed fields, p-adic fields. Anything else? A more pointed question after comments: ...
1
vote
2answers
278 views

Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?

According to numerical simulation, the relationship $$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$ where $\Gamma$ is the Gamma function seems to be true. Do you ...
-4
votes
0answers
88 views

Makoto and hanmakoto numbers

Under Goldbach's conjecture, write $r_{0}(n):=\inf\{r>0\mid (n-r,n+r)\in\mathbb{P}\}$ and $r_{k+1}(n):=\inf\{r>r_{k}(n)\mid (n-r,n+r)\in\mathbb{P}\}$. Say a prime $p$ is a $k$-makoto if it can ...
1
vote
1answer
58 views

Hölder continuous dependence on parameters for solutions of ODE

This is a cross-post from Stackexchange Mathematics (https://math.stackexchange.com/questions/3893961/h%c3%b6lder-continuous-dependence-on-parameters-for-solutions-of-ode). We have the following ...
8
votes
0answers
102 views

Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that $$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
1
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0answers
59 views

Global sections appearing in Dolbeault complex with values in vector bundle

Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex ...
0
votes
0answers
43 views

Lower bound for expectation of minimum eigenvalue

Let $X$ be a random (symmetric) matrix drawn from an unknown distribution. I have an estimate of $\lambda_{\min}(\mathbf{E}[X])$. Specifically, I have $$\lambda_{\min}(\mathbf{E}[X]) \geq c$$ a ...
0
votes
0answers
63 views

Expectation of the inverse of random principal submatrices

The goal of this question is finding the concentration point of the inverse of random principal submatrices, which is posed as follows. Consider $\mathbf{S}\in\mathbb{S}^{n}_{++}$ to be a strictly ...
2
votes
1answer
109 views

Are vector bundles acyclic for $\Gamma_c$?

Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
2
votes
0answers
86 views

On odd perfect numbers and a GCD - Part III

Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
5
votes
1answer
149 views

Does every open set contain a dense $F_{\sigma}$ subset?

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set). [ In my specific situation $U$ is of the form $\operatorname{int} f^{-1}(0)$, where $f$ ...
1
vote
1answer
71 views

$K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$

Let $K$ be a number field and $v$ be it's one of $K$'s non-archimedian valuation. Then, I would like to prove $K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$. This is from Silverman's ...
8
votes
1answer
314 views

VC dimension of Borel sets [duplicate]

Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is a Borel set $U$ with $D=S\cap U$? I'm asking merely out of curiosity, but I'll mention that ...
5
votes
0answers
97 views

Bi/tricategorical coherence in terms of surface diagrams

Is there a typed-up version of the coherence theorem for bicategories in terms of surface diagrams? What about the GPS tricategorical coherence theorem in terms of 'volume diagrams'? I'm aware of ...

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