Newest Questions
159,040 questions
3
votes
1
answer
362
views
Does a quasi-split reductive group scheme admit a maximal torus?
Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
- 605
0
votes
0
answers
96
views
The relation between two characteristic subgroups in finite p-group
Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...
- 171
2
votes
0
answers
115
views
Orthogonal bundles with values in a line bundle vs. reductions of structure group to $O(n)$
I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well.
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:...
- 737
1
vote
0
answers
143
views
When flat base change is reduced
I am sorry, if this a very standard fact.
Let $f\colon X\to S$ be a morphism between varieties over field of characteristic zero. Let $\psi \colon S'\to S$ be a flat morphism. Is it true that $X\...
- 219
3
votes
1
answer
108
views
How to represent morphisms in a fibration in the internal type theory
Given a fibration $p:\mathcal{E \to B}$, we can work with a minimal type theory with semantics in $p:\mathcal{E \to B}$, its internal type theory.
The type theory for $p$ is dependent, with contexts ...
- 1,083
4
votes
1
answer
291
views
Effective version for Silverman’s specialization theorem
In his paper, Silverman proves Theorem C (page 208, the indexes of the pages don't match the file), which says that the set $$\{t\in C^0(\bar{K})\mid \sigma_t \text{ is not injective}\}$$ is a set of ...
- 463
1
vote
0
answers
57
views
Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
- 6,853
9
votes
1
answer
444
views
Hochschild cohomology of a group algebra
Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
- 985
3
votes
1
answer
228
views
Are "ultra-regular" bipartite graphs complete?
Let $X, Y$ be non-empty, disjoint sets and let $R\subseteq X\times Y$ be a binary relation. For $x\in X$, we set $R(x) = \{y\in Y: (x,y) \in R\}$ and for $y\in Y$, let $R^{-1}(y) = \{x\in X:(x,y)\in R\...
- 48.9k
4
votes
2
answers
342
views
On a generalized homotopy transfer theorem
In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
- 215
1
vote
0
answers
142
views
Perfect complexes on a formal scheme
Let $X$ be a quasi separated, quasi compact scheme, $Z$ be a closed subscheme on $X$. We denote by $\hat{X}$ the formal scheme of $X$ along $Z$.
My questions are the following.
(1) How to define the ...
- 127
2
votes
1
answer
628
views
Does any finite group of order $2m$ with odd $m$ have a subgroup of index 2? [closed]
Let $G$ be a finite group of order $2m$ where $m>1$ is an odd natural number.
Question. Is it true that any such $G$ has a subgroup $H$ of index 2?
If yes, I would be grateful for a reference or ...
- 14.2k
6
votes
0
answers
232
views
Reference for Understanding Shelah's Proof of Vaught's Conjecture for $\omega$-stable Theories
I'm looking for a source to help me better understand Shelah's proof of Vaught's Conjecture for $\omega$-stable Theories (https://shelah.logic.at/files/95409/158.pdf). An obvious candidate is Makkai's ...
- 173
8
votes
0
answers
352
views
A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$
Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...
- 15.6k
0
votes
1
answer
99
views
Is it possible obtain an identity of the type $\dfrac{|x-y|}{t} + \dfrac{|y-z|}{s} \approx a(t,s) | y - b(t,x,s,z)| + c(t,s)|x-z|$?
using the Euclidean inner product, and by completing squares it is possible to prove that
$$\dfrac{|x-y|^2}{t} + \dfrac{|y-z|^2}{s} = \dfrac{t+s}{ts} \left| y - \dfrac{sx+tz}{s+t} \right|^2 + \dfrac{|...
- 677
2
votes
1
answer
394
views
Implementing the $\pi$ BBP algorithm
The formula
$$\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)$$
is a basis of the BBP algorithm for calculating arbitrary ...
- 317
7
votes
0
answers
222
views
Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
0
votes
0
answers
161
views
Number of solutions to $x^3 - y^3 + z^3 = k$?
For $k$ is a fixed number such that $-n^3 \leq k \leq 2n^3$.
How many solutions to the equation $x^3 - y^3 + z^3 = k$ with $1 \leq x, y, z \leq n$?
One of the trivial bounds is $n^{1 + \epsilon}$ ...
- 431
1
vote
1
answer
260
views
A short exact sequence regarding Kähler differentials and an invertible ideal on an algebraic curve
$\def\sO{\mathcal{O}}
\def\sK{\mathcal{K}}
\def\sC{\mathscr{C}}$I am trying to understand what the maps are on a certain s.e.s. of sheaves of modules on an algebraic curve. It is \eqref{ses} on Conrad'...
3
votes
0
answers
186
views
Combinatorial interpretation of Sylvester–Lipschitz formula?
If we denote the Bernoulli numbers by $B_n$, then
$$ {a^{k+1}(a^{2k} - 1) B_{2k} \over 2k}\in \mathbb{Z}$$
for any $a\in \mathbb{Z}$ and $k\in\mathbb{N}$. This fact is often called the Sylvester–...
- 82.7k
0
votes
2
answers
287
views
Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 612
1
vote
0
answers
87
views
Effective bound of Fourier coefficients of weakly modular forms
Assuming $$f=\sum_{n=n_0}^\infty c_f(n/m)e^{{2\pi inz}/{m}},\quad (n_0\in\mathbb Z, m\in\mathbb Z_{\geq1})$$ is a weakly modular form with weight $k$ and congruence subgroup $\Gamma=\Gamma_0(N),\...
- 111
12
votes
1
answer
994
views
General solution of the quartic $a^4+b^4=c^4+d^4$?
The background to the question:
$$a^4+b^4=c^4+d^4 \label{1}\tag 1 $$
Tito Piezas, Tomita & others have recently given some parametric solutions on Math stack exchange & Math overflow. In math ...
- 127
3
votes
1
answer
225
views
Does base change respect Galois correspondence between $\ell$-adic sheaves and representations of the fundamental étale group?
It is known that for $X$ a connected scheme there is an equivalence of categories
$$\left\lbrace \text{$\ell$-adic smooth sheaves over $X$} \right\rbrace \leftrightarrow \left\lbrace \text{$\ell$-adic ...
- 1,270
5
votes
1
answer
253
views
Collapsing every cardinal outside the Prikry sequence
All variants of Prikry forcing with collapses that i have been able to find preserve some points outside of the generic sequence (at least the successors). This is done for two reasons, (1) to obtain ...
- 1,799
5
votes
2
answers
764
views
How to effectively search Internet for graphs not for function graphs? [closed]
So, is there any way to distinguish graphs and plots in Internet?
I was looking for (Olivier-)Ricci curvature of graphs and found a lot about Ricci curvature of function graphs and not so much about ...
- 1,422
2
votes
1
answer
252
views
Sum of Bessel function with integer parameters and fixed argument
Question.
Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
- 577
14
votes
0
answers
920
views
Explicit and complete list of Lean's Axioms
I'm a big fan of the idea of fully formalizing mathematics. So the Lean proof checker appeals to me.
Relating to this, one of the biggest appeals of mathematics to me is that there is a (largely) ...
- 406
1
vote
0
answers
45
views
Existence of real solutions to nonlinear algebraic equation: conditions on coefficients
Good day. I am dealing with the following system of nonlinear algebraic equations:
$$
x_j = \prod_{k=1}^N (1 + x_k)^{A_{j,k}}\,,\quad j=1,\ldots,N\,,
$$
where $A_{j,k}\in\mathbb{Z}$.
I would like to ...
- 105
1
vote
1
answer
245
views
Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...
- 178
8
votes
0
answers
192
views
Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?
I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...
- 60.6k
4
votes
2
answers
410
views
The associated graded algebra of a finite dimensional algebra
$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).
Denote by $A_G$ the associated ...
- 26.5k
2
votes
1
answer
110
views
Why is the set of singular points of starlike boundary $\Gamma$ closed?
I'm reading Geometric Inequalities of Yu. D. Burago and V. A. Zalgaller. I don't understand why $E_1$ is closed in the proof of the following lemma.
Several definition.
Suppose $ \Omega $ is a ...
- 195
2
votes
0
answers
178
views
Product subvariety of a simple abelian variety
Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
- 615
1
vote
0
answers
90
views
Cohomological dimension of the kernel of a homomorphism induced by a singular fibration
I have a very concrete question. Let $N={\Bbb C}-\{\pm 1\}$, and ${\Bbb Z}_2$ is acting on $N$ by rotation around $0$. Consider $M=\{(x,y)\in N^2\ |\ {\Bbb Z}_2x\neq {\Bbb Z}_2y\}$. Let $p:M\to N$ be ...
- 585
3
votes
1
answer
329
views
Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction
I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
- 785
3
votes
1
answer
346
views
Second order theory of a real-closed field
It is well known that the first-order theory of any real-closed field is complete, and consequently not capable of interpreting the majority of modern mathematics.
Is this still true for the second-...
- 10.1k
1
vote
1
answer
100
views
Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 13
1
vote
1
answer
250
views
Drinfeld modules and shtukas
Let $\rho$ be a Drinfeld module on $\mathbb F_q(T)$ defined by
$$\rho_T=\sum_{î=0}^da_i(T)F^i\qquad(a_i\in\mathbb F_q(T))$$
where $F$ is the $q$-Frobenius.
Let $f$ be the shtuka function associated to ...
- 3,998
3
votes
1
answer
231
views
Average size of the Fourier--Stieltjes transform of the fractal measures
For $0<\theta<1/2$ define $\mu_\theta$ to be the uniform (self-similar) measure on the Cantor set obtained from the dissection pattern $(1-2\theta,\theta)$. For example, when $\theta=1/3$ the $\...
- 1,692
0
votes
0
answers
55
views
Modeling player interactions in multi-dimensional rating systems
In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the ...
- 1
3
votes
1
answer
273
views
Inflection point calculation for cubic Bézier curve encounters division by zero
I've been working on finding the inflection points of a cubic Bezier curve using the method described in a paper Hain, Venkat, Racherla, and Langan - Fast, Precise Flattening of Cubic Bézier Segment ...
- 133
6
votes
1
answer
676
views
Ricci flow + Nash embedding
I have a basic question about how geometric flows such as the Ricci flow interact with the Nash embedding theorem.
Say you have a 1-parameter family of Riemann metrics on a compact manifold $N$. If ...
- 44.4k
1
vote
1
answer
113
views
Minimal dominant permutation in weak order
Consider $S_\infty$ as a Coxeter group with Coxeter generators the adjacent transpositions $s_i$, $i\geq 1$. We view elements of $S_\infty$ as functions $u:\mathbb{N}\to\mathbb{N}$. Recall the Lehmer ...
- 2,168
3
votes
1
answer
128
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
2
votes
2
answers
210
views
An identity for the ratio of two partial Bell polynomials
Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...
- 1,101
0
votes
1
answer
179
views
Question in a paper by Erdős on divisibility properties of central binomial coefficient
In Erdős, Graham, Ruzsa, and Straus - On the prime factors of $\binom{2n}n$, at the beginning of the proof of theorem 1, they consider the case where $\log p$ and $\log q$ are commensurable numbers (...
- 111
6
votes
1
answer
203
views
Odd integral Stiefel–Whitney classes in terms of even ones
As computed in various places (e.g. in Brown - The Cohomology of $B\mathrm{SO}_n$ and $B\mathrm O_n$ with Integer Coefficients), the integral cohomology ring of $B\mathrm{O}(n)$ (and similarly $B\...
- 9,853
2
votes
1
answer
151
views
Is the slice of a subcanonical site also subcanonical?
A subcanonical site is one for which every representable functor is a sheaf.
For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...
- 237
3
votes
0
answers
267
views
Does the orbit in geometric invariant theory have natural scheme structure
Let $X$ be a scheme locally of finite type over a sufficiently "nice" base scheme $S$ (nice in sense of reasonable "finiteness conditions", for sake of simplicity let's start as ...
- 6,048