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4 votes
1 answer
284 views

Do the two orientations on an orientable manifold $M$ uniquely witness lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?

For an orientable $n$-manifold $M$ and its (orthonormal) frame bundle classifying map $\tau_M : M \to BO(n)$, we have a lift diagram of the following sort: There are two orientations on $M$. Is it ...
9 votes
3 answers
726 views

Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
3 votes
1 answer
228 views

Compute generators for group of totally positive units of a number field?

Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$. Update: I've tried some code (details below), which I've received some help on in ...
9 votes
1 answer
346 views

Software for working with fusion categories

One way to describe fusion categories is via a fusion system: several lists of numbers that define the fusion ring, associator, braiding (if it exists), etc. Often, these sets of numbers are quite big,...
0 votes
0 answers
51 views

Translation Invariants of Polynomials

The function $f(k)$ is a (numerical) polynomial in $\mathbb{Q}[k]$, and the set $ S_f = \{ f_d : d \in \mathbb{N} \} $ is a set associated with $ f $, where $f_d(k)=f(k+d)$. I am interested in finding ...
16 votes
5 answers
3k views

Embedding Klein bottles in 4-space

A question about topology from an ignorant logician, so please be kind if this is obvious! We all know that the Klein bottle, unlike the torus, cannot be embedded in 3-space. And we all know (because ...
34 votes
1 answer
3k views

Closed formula for the factorial over naturals

How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on natural numbers, powers of natural numbers, and fixed natural numbers? The same question over the ...
-2 votes
0 answers
89 views

Can this prime pyramid reveal deeper insights into prime distribution? Has someone seen this pattern before? [closed]

Definition The Burz Prime-Number Pyramid is a triangular arrangement of consecutive integers, structured such that the length of each row corresponds to a prime number, except the first row which ...
1 vote
0 answers
60 views

Computing with the Picard group of non-integral curves

Let $C$ be a curve over $\mathbb{Q}_p$ and let $\mathcal{C}$ be a regular model of $C$ over $\mathbb{Z}_p$, with $\mathcal{J}$ the Neron model of the Jacobian of $C$. Raynaud's theorem asserts that $\...
3 votes
0 answers
51 views

Semisimple elements and fixed points

The following statement seems to be well-known: Let $X$ be a variety on which an affine algebraic group $H$ acts with finitely many orbits and let $s \in H$ be semisimple. Then $H_s = \{h \in H \mid ...
1 vote
1 answer
419 views

Densest safe disk packing

Inspired by current regulations regarding the minimal distance to be kept among people to prevent spreading of the COVID-19 virus and the maximal number of people in a group that is not subjected to ...
3 votes
0 answers
98 views
+100

Can I get a spherical coordinate from a real cocycle?

The Setting I am currently working on a project in Topological Data Analysis (TDA), where I aim to construct a density-robust spherical coordinate associated with a dataset $X$, sampled from a ...
-2 votes
0 answers
30 views

Convergence of $ \vec{x_{n+1}} = A^{-1} f(\vec{x_{n}},\vec{y_{n}}), \:\:\: \vec{y_{n+1}} = A^{-1} g(\vec{x_{n}},\vec{y_{n}}) $ [closed]

I have two systems $$ A \vec{x} = f(\vec{x}, \vec{y}), \:\:\: A \vec{y} = g(\vec{x}, \vec{y}) $$ Both have the same constant, square, invertible matrix $A$. I implemented an iterative algorithm with ...
11 votes
2 answers
201 views

For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?

More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
5 votes
0 answers
125 views
+50

Dimension of the intersection of the commuting variety with a particular subspace

Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as: $$ \mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}. $$ It is well known that $\...
5 votes
1 answer
84 views

Measure dependance of groupoid von Neumann algebra

Let $(G,\mu)$ be a measured groupoid and denote by $\nu,\nu^{-1}$ the measures on all of $G$ induced by $\mu$ and Haar system $\{\lambda^x\}$. I have a question regarding the dependance of the ...
3 votes
0 answers
78 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
16 votes
3 answers
1k views

Is there a natural topology for sets of topological spaces?

The Gromov–Hausdorff metric makes a set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
-1 votes
0 answers
19 views

What is the expected value of the set when N elements are chosen from the same probability distribution?

Suppose we have a parameter that follows some probability distribution $f(x)$. When simulating an $N$-body with that parameter as an attribute, how should values of the parameter be chosen? Let each ...
-1 votes
0 answers
47 views

How to prove the following theorem by distribution function and series

Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e., $$ \mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0. $$ Let $\eta>...
0 votes
0 answers
74 views

Co-boundary crossed homomorphism & "sign" preserving. Why 2-valued components is special?

Suppose $h_{g}: \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a coboundary crossed homomorphism with action $g$ as a cyclic permutation of coordinates on $\mathbb{R}^n$ vectors. So, the acting group is a ...
4 votes
2 answers
205 views

Lower bounding a sumset quantity

Given $A,B \subset[0,...,d]^n$ such that $A \cap B = \phi$. Can we show $$ |(2A \cup 2B) \triangle (A + B)| \geq \Omega_d({\rm poly}(|A|,|B|))$$ where $2A = A+A, 2B = B+B$ and we are taking the ...
6 votes
1 answer
168 views

Equivariant KR-theory of representation sphere

I would like to say my question first. Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...
-1 votes
0 answers
48 views

Homomorphism from field of hyperreals to field of reals?

I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?) Assuming that ...
2 votes
1 answer
163 views

Existence of Kähler metric of bounded geometry on the Hermitian vector bundle on projective spaces

$\DeclareMathOperator\Tot{Tot}$A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity ...
4 votes
1 answer
186 views

Projective automorphisms of a plane cubic curves

Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$. What is the group of the projective transformations preserving $E$ ? In characteristic $0$ the answer is known ...
1 vote
0 answers
58 views

$\mathbb{P}_1$-finite element as convolution of $\mathbb{P}_0$-finite element

For a vector $\vec{u}\in\mathbb{R}^N$ let's denote $\pi_N\left(\vec{u}\right)$ the unique piecwise linear and $1$-periodic function matching the components of $\vec{u}$ on the discretization $x_k = \...
16 votes
5 answers
2k views

"Classical" consequences of Bezout's theorem in dimensions $>2$

By Classical I mean something that could have been found before 1900 (say). A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
7 votes
1 answer
211 views

Is there a more natural way to define the Young symmetrizer and the Specht module?

It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups. For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
2 votes
0 answers
98 views

Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?

Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
3 votes
0 answers
50 views

Martingale problem for the Wiener process

Consider $\Omega \triangleq \mathbf{C}([0,T];\mathbb{R})$, $\mathbf{F} \triangleq \mathbf{B}(\mathbf{C}[0,T];\mathbb{R})$ (Borelian $\sigma$-algebra) and $\mathbf{F}_t \triangleq \sigma \left \{ W_s, \...
9 votes
1 answer
2k views

The definition of ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ in Inter-universal Teichmüller theory

$\newcommand{\Vect}{\mathit{Vect}}$I am reading Mochizuki's INTER-UNIVERSAL TEICHMÜLLER THEORY I to III and I hardly understand this theory, but there is a thing particularly I bother. Given an ...
1 vote
0 answers
141 views

Measurability of a map involving probability measures

Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
2 votes
1 answer
155 views

Converse of Scherk–Segre theorem on the number of vertices of a convex space curve

It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
12 votes
0 answers
120 views

When could a diligent calculus student compute all Picard iterates algebraically?

As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
7 votes
1 answer
205 views

Inductive type constructors with the defined type appearing in non-strictly positive position

In the Homotopy Type Theory book, §5.6 “The general syntax of inductive definitions”, there is a proof that the existence of an inductive type $T$ with a constructor $t : ((T \to \mathsf{Prop}) \to \...
14 votes
4 answers
2k views

The ten most fundamental topics in geometric group theory

What are the ten most fundamental topics in geometric group theory? This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
60 votes
72 answers
9k views

When is 2 qualitatively different from 3?

I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to ...
0 votes
0 answers
46 views

Fractal dimension using wavelets [closed]

I'm trying to estimate the fractal dimension of a function. I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation $$ \alpha = -...
1 vote
1 answer
137 views

About dimensions of quotients of quasi projective varieties

This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
9 votes
1 answer
334 views

An elementary proof of the equivalence of the Bol and Moufang identities

By a well-known result of Bol (1937) and Bruck (1946), for any loop the following two identities are equivalent: B: $x(y(xz))=((xy)x)z$ M: $(xy)(zx)=(x(yz))x$. A proof of the equivalence (B)$\...
7 votes
2 answers
429 views

closed form for an alternating cosecant sum

Is there any closed form for the following finite sum $$\sum_{j=1}^{n-1}\frac{(-1)^j}{\sin (\frac{j\pi}{n})}$$ where $n$ is an even number? Any comment or reference is welcome.
5 votes
0 answers
65 views

Underlying noncommutative topologies of noncommutative complex varieties

Let $X$ be a (separated) complex algebraic variety. Then we can view its analytification $\newcommand\topo{\text{top}}X^{\topo}$ as a locally compact Hausdorff space. I wonder whether the same ...
3 votes
1 answer
210 views

Comparison of solutions of Hamilton–Jacobi equations with different initial conditions

Consider a Hamilton–Jacobi equation: $$u_{t} + f(u_{x}) = 0 \quad (x,t) \in \mathbb{R}\times [0,+\infty)$$ with two possible initial conditions $u(x,0) = g_{i}(x)$ for $x \in \mathbb{R}$ and $i=1,2$. ...
0 votes
2 answers
147 views

Is $1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$?

For $p>2,m>2$, is $$1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$$ ? Motivation: I am trying to ...
0 votes
0 answers
100 views

Algebraic relations for $\Gamma$ function

Let $N$ and $n$ be positive integers with $\mathrm{GCD}(n,N)\ne1$. I want to prove the following claim: $\Gamma\left(\frac nN\right)$, $\pi$ and the $\Gamma\left(\frac uN\right)$ ($u\in[1,N-1]$, $\...
3 votes
1 answer
266 views

$\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$ ("eventually&...
0 votes
0 answers
77 views

Can a generalized root formula exist for polynomials with finite degrees? [closed]

Let $p\in\mathbb{Z}[x]$, set $d = \textbf{Deg}(p)$, and write $$p(x) = \sum_{i=0}^{d}{a_ix^i}$$ for some sequence $\{a_0,a_1,a_2,...., a_{d-1}\}$. Is there a mapping $\mathcal{F}$ so that $$\mathcal{F}...
9 votes
1 answer
430 views

Renewal process: domination by product measure

Consider a stationary process $(X(i), i\in\mathbb{Z})\in \{0,1 \}^\mathbb{Z}$ with the following structure; runs of 0s alternate with runs of 1s, with the length of all runs independent, and with each ...
10 votes
1 answer
460 views

Extending models of topological set theory

$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see the SEP article for more detail (or this MO post, which ...

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