Newest Questions
159,032 questions
2
votes
0
answers
59
views
graded reps of Lie algebras literature
I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2
grading), such as ...
- 21
4
votes
1
answer
303
views
On Giuga numbers
I am studying Giuga numbers and I am reading a paragraph that says all Giuga numbers $n$ satisfy the property $\left(\sum_{p|n}1/p\right) - 1/n \in \mathbb{N}$ and beside it says that all known Giuga ...
2
votes
2
answers
235
views
Theoretical/Practical Implications of DFT Eigenvectors
Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
- 4,058
6
votes
2
answers
2k
views
Cover letters for tenure-track jobs on mathjobs [closed]
The following past question attracted answers on what to put on a cover letter for postdoc positions advertised on mathjobs.org:
On mentioning recommenders' names in cover letter for postdoctoral ...
- 111
2
votes
1
answer
171
views
Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by
$$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$
over the integers as $3,43,163,907$. It says this list may not be complete and ...
- 458
5
votes
0
answers
264
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
- 405
16
votes
0
answers
426
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
11
votes
1
answer
225
views
Properties of the ring of all holomorphic modular forms
Let $R$ be the ring of modular forms on congruence subgroups, say
of integral or half integral weight. In other words
$$R=\bigcup_{N\ge1}\bigoplus_{k\in(1/2)\Bbb Z}M_k(\Gamma(N))\;.$$
The important ...
- 13.1k
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
0
answers
50
views
Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 23
0
votes
1
answer
171
views
For which $p>p^*$ does the inequality $\cos^2(−π/4+π/p)>1/2+π/p^2$ hold?
I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking ...
12
votes
2
answers
429
views
Trading Choice for Comprehension (or Replacement)
This question is basically a request for clarification about a remark made by Sam Sanders in a comment to another question: IIUC what he's saying, there are statements that can be proved either with a ...
- 32.5k
-5
votes
1
answer
284
views
Does the set of Goldbach numbers have positive density?
We say that an integer $n > 1$ is a Goldbach number if $n$ is the sum of two primes. The famous Goldbach conjecture says that every even integer greater than $2$ is Goldbach.
Consider the following ...
- 48.9k
2
votes
1
answer
161
views
Does allowing Ur-elements in $\sf ZF^*GC$ break bi-interpretability with $\sf ZF^*GC$?
Is $(\sf ZF^*GC - Extensionality)$ bi-interpretable with $\sf ZF^*GC$?
Where $\sf ZF^*GC$ is $\sf ZF - Replacement + Separation + Collection + Global \ Choice$
- 11.3k
5
votes
0
answers
179
views
Zigzag vs direct map in rational homotopy theory
I was reading these notions from "Rational Homotopy Theory" by Felix, Halperin, and Thomas.
The notion of weak homotopy type is as follows: two spaces $X$ and $Y$ are said to be weak ...
- 349
6
votes
1
answer
253
views
Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
- 243
2
votes
1
answer
147
views
On the square mean of Fourier coefficients of cusp forms
I have a question which may look naive for many experts here:
For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that:
$$\sum_{X<n\le 2X}...
- 563
0
votes
0
answers
245
views
What does the set of all fundamental coweights look like?
Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
- 2,751
9
votes
3
answers
2k
views
A dice probability question
Suppose you have a die with $n$ sides labeled $1,2,\ldots,n$. Each turn, you roll the die and add the number you get to the running total (which starts at $0$). You do this for an infinite number of ...
- 129
3
votes
2
answers
492
views
SDE driven by fractional Brownian motion
Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$
I am looking for references that ...
- 1,334
2
votes
1
answer
219
views
Looking for an example of a point $P$ on an abelian variety $X$ such that no curve on $X$ contains all multiples of $P$
Is there an example of an abelian variety $X$ defined over a number field $K$, with $\dim X > 1$, and a $K$-rational point $P$ on $X$, such that no curve $C$ on $X$ (say defined over a number field)...
- 658
2
votes
0
answers
133
views
Efficient decoding of the E8/Leech lattice
Background:
Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....
- 121
2
votes
1
answer
187
views
Law of iterated logarithm for quadratic variation of Brownian motion
Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...
- 41
4
votes
2
answers
319
views
Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
- 658
5
votes
0
answers
83
views
When does the ΔY-family of a simple graph contain multigraphs?
Given a graph $G$, its ΔY-family is the smallest family of graphs that contains $G$ and is closed under ΔY- and YΔ-transformations.
Since YΔ-transformations can introduce multi-edges, the ΔY-family of ...
- 13.6k
3
votes
3
answers
340
views
Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure.
Suppose $f: \mathbb R^...
- 6,321
15
votes
1
answer
582
views
Are some numbers more equidistributed than others?
Weyl's theorem says that $n\alpha$ is equidistributed mod 1 for any irrational $\alpha$. One corollary is that, if I consider the fractional part $\{n\alpha\}$ for $n \leq N$, and look at the indices ...
- 1,069
3
votes
0
answers
171
views
Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants
$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
- 1,541
0
votes
0
answers
75
views
Conditions for logarithmic upper bound on size of antichains in a partial order
Suppose I have a partial order $(P,\le)$ (in practice I have a very specific one, but let's start with a generic question).
Are there known conditions that would imply that the size of any antichain ...
- 121
2
votes
0
answers
140
views
Asymptotics of a "non-constant order" quadratic recurrence relation in two variables
Consider the following recurrence relation defined for two integer variables $H,n \geq 0$:
\begin{equation}
\gamma(H,n) = \sum_{K=0}^{\lfloor H/2 \rfloor} \gamma(K,n-1) \gamma(H-K,n-1)
\end{equation}
...
- 143
0
votes
2
answers
262
views
Is the Boltzmann entropy lower semi-continuous in the weak topology induced by $C_b (\mathbb R^d)$?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 825
4
votes
2
answers
255
views
Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 825
2
votes
2
answers
832
views
Is there a definition of $\log(x)$ for quaternion/octonion $x$?
I'm trying to implement $\log({\bf q})$ in python, where ${\bf q} = (q_0,\ldots,q_7) \in \mathbb{O}$ is an octonion. There is a well known definition of $\log({\bf q})$ for quaternions ${\bf q} = (s,v)...
- 2,101
2
votes
1
answer
175
views
Are there nonzero morphisms between families of stable sheaves, with slopes decreasing?
Consider over $\mathbb{C}$. Let $C$ be a smooth projective curve. Fix degrees $d_1,d_2$ and ranks $r_1,r_2$ such that $\mu_1=\frac{d_1}{r_1}>\mu_2=\frac{d_2}{r_2}$.
It is well known that if $E,F$ ...
- 930
1
vote
1
answer
88
views
Bounded $C_0$-semigroups on barrelled spaces are equicontinuous
I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, ...
- 129
1
vote
0
answers
123
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
0
votes
0
answers
90
views
Invariants of primary groups
In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
- 31
5
votes
1
answer
256
views
Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure
Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
- 91
12
votes
3
answers
995
views
Area of a smooth complex projective curve
Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\...
- 2,858
4
votes
0
answers
286
views
Dévissage for a stratification in Grothendieck's Esquisse d’un programme: What is it?
I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have ...
- 6,038
7
votes
1
answer
117
views
Picture of the isotopy class of a degree $d$ smooth complex curve
All smooth complex curves of degree $d$ in $\mathbb{C}P^2$ are isotopic. Let $C$ be such a curve. I often picture $\mathbb{C}P^2$ as a 2-dimensional disk bundle over $S^2$ (of Euler class 1) which ...
- 1,075
12
votes
2
answers
787
views
Examples of non-polynomial comonads on Set?
Question: What are examples of comonads on $\mathbf{Set}$ that are not polynomial?
Background: polynomial functors and comonads on Set
A functor $F\colon\mathbf{Set}\to\mathbf{Set}$ is called ...
- 8,669
1
vote
0
answers
33
views
Nonholonomic version of bijection between $r$th order connection on $TM$ and principal connection on $r$th order frame bundle $P^r M$?
Given a smooth manifold $M$, Kolář - On the torsion of linear higher order connections showed that there is an equivalence between a linear, $r$-th order connection the tangent bundle $TM$ of $M$, and ...
- 250
2
votes
0
answers
151
views
When do Fourier coefficients vanish in Hida families?
Suppose you have a Hida family with $q$-expansion $F = \sum_{n=1}^{\infty} a_n(T) q^n$, where the coefficients $a_n(T)$ are power series in $\mathbb{Z}_p [[T]]$. Assume that $F$ is a cuspidal ...
- 1,949
2
votes
2
answers
211
views
Limit of a integral whose integrand diverges under the limit
I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
- 43
6
votes
0
answers
209
views
"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
- 890
1
vote
1
answer
214
views
Bound on Von Mangoldt for automorphic L-functions
Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote
$$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$
In terms of the local roots of the Euler product:
$$ \Lambda_f(...
- 13
2
votes
1
answer
291
views
When does uniqueness of a stable equilibrium imply it is globally stable?
Given a gradient dynamical system
$$\dot x=-\nabla f(x),$$
my question is:
(1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...
- 405
3
votes
0
answers
311
views
What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
- 571