All Questions
152,882
questions
1
vote
0
answers
187
views
Group scheme representation from action of a group scheme on a variety?
Let f be some homogenous polynomial of d.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$
where $k$ is algebraically closed field of characteristic $p>0$.
Now $G$ is a group scheme acting on $X$.
...
16
votes
1
answer
451
views
Escaping from a centralizer
Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...
3
votes
0
answers
212
views
Renyi's theorem on mixing
I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations:
A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...
4
votes
1
answer
253
views
Invariant lifts of a closed curve on a surface of genus > 1
I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
0
votes
1
answer
247
views
Quasiconvexity property of quasinorms
Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.
If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
12
votes
1
answer
294
views
Factoring polynomials over the abelian closure of the rationals
What algorithms are known to perform the following task?
Input: a univariate polynomial over the rationals $f \in \mathbb{Q}[t]$.
Output: the factorization of $f$ into irreducible factors over the ...
5
votes
1
answer
277
views
Interesting properties in $...\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to ...$
Let $K(G,n)$ be the Eilenberg Maclane space.
Consider the map from
$$
K(\mathbb{Z}_2,1) \to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to \dots,
$$
It ...
8
votes
1
answer
591
views
Spectral and derived deformations of schemes
I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is.
Let $S = (X, ...
106
votes
2
answers
32k
views
What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
6
votes
0
answers
876
views
Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle
I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature.
I'm hoping there's a nice ...
5
votes
2
answers
378
views
Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces
I apologize in advance if this question has an obvious answer.
Let $(M,g)$ be a Riemannian manifold.
Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
4
votes
1
answer
126
views
On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
2
votes
0
answers
252
views
Cohomology and base change
Let $f$ be some homogenous polynomial of $d$.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$.
Now let $R$ be a $k$-algebra.
What ...
2
votes
0
answers
530
views
eigenvalues of a square block matrix
How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix}
A&B\\-B& 0
\end{bmatrix}
where A, B are real matrices, $A =\...
5
votes
1
answer
338
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
8
votes
0
answers
288
views
Relationships among constructions of fundamental group for schemes
There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
6
votes
0
answers
470
views
The $E_2$-page of the May spectral sequence
I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
4
votes
1
answer
285
views
Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators
I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}...
1
vote
0
answers
50
views
What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?
Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...
8
votes
1
answer
339
views
Characterization of KL divergence for continuous variables?
This is an analog of an older question:
What characterizations of relative information are known?
With the modification that I’m interested in the case when the distribution is over something that’s ...
1
vote
0
answers
41
views
Numerator-cancellable Modules
I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following :
Let $R$ be a ...
0
votes
1
answer
248
views
Weak convergences in Bochner spaces
I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $\Omega$:
Let $\{x_n\}$ be a bounded sequence in $L^2((0,T)\...
12
votes
2
answers
388
views
Has the arcsine law been generalised to higher order divisor functions?
The arcsine law for the distribution of the logarithms of the divisors of an integer $n$ states that
$$
\frac{1}{x}\sum_{n\leq x}\frac{1}{d(n)}\sum_{\substack{q|n\\q\leq n^{A}}}1\sim \frac{2}{\pi}\...
3
votes
1
answer
117
views
How many $2d$-elements subsets with specific property at most can we choose from $\{1,2,\cdots,n\}$
Are there any known results about the following problem:
Given any integer $n\geqslant4$, how many $4$-elements subsets at most can we choose from $\{1,2,\cdots,n\}$ such that the intersection of any ...
2
votes
1
answer
140
views
Are separability and ccc equivalent for closed subspaces of $\beta N$?
Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
1
vote
0
answers
73
views
relative compact on nonlinear term
On the paper: Decay of Solutions to Nonlinear Schrodinger
Equations.
Let $u$ be a solution of the equation
$$Hu+|u|^2u=0,$$
where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
2
votes
1
answer
158
views
Graphs with exactly $n$ perfect matchings
Is there for every $n\in\mathbb{N}$ a connected, simple, undirected graph $G_n=(V_n,E_n)$ such that $G_n$ has exactly $n$ perfect matchings?
3
votes
0
answers
243
views
Periodic orbit for certain Hamiltonian on the tangent bundle
In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
6
votes
0
answers
324
views
Explicit bounds for the Mertens function
It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
15
votes
2
answers
3k
views
Measures and differential forms on manifolds
Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$.
What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...
2
votes
0
answers
62
views
Negative association in a "k out of n" process
Suppose we have $n$ distinct balls labeled $1,2,\dots,n$ in a black box. Now we want to fetch $k$ balls from the box, one by one. Let event $E_i$ be that the label of the $i$-th ball we fetch is no ...
24
votes
2
answers
684
views
What's the maximum probability of associativity for triples in a nonassociative loop?
In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...
6
votes
0
answers
168
views
What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
2
votes
0
answers
95
views
Polynomials passing through points with tangential conditions
In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
1
vote
1
answer
240
views
Sage: Evaluation precision for elliptic curves over p-adic fields
Consider the elliptic curve $E: y^2 = x^3 + 23x+11$ over p-adic fields. In Sage I use:
k = GF(257)
E = EllipticCurve(k,[23,11])
kp = Qp(257,5) # 257-adic Field with capped relative ...
1
vote
1
answer
412
views
Sufficient conditions for a topological space to be regular $T_3$
There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be completely regular $T_{3^1/_2}$.
Please, let me know any known condition(s) that ...
4
votes
0
answers
131
views
Lower bound on $\epsilon$-covers of arbitrary manifolds
Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
4
votes
1
answer
273
views
Interpolation of families of local fields
Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
1
vote
1
answer
275
views
exact sequence of deformations
Suppose that $C\cong P^1$ and $Def(f)$ denote the first order deformation of pointed stable map $(C,{p_i},f:C\longrightarrow X)$. I read that we have short exact sequence:
$0\longrightarrow H^0(C,T_C)...
4
votes
0
answers
58
views
Separation principle in generic models?
Separation principle
$\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$
claims that any two disjoint pointsets of boldface class $\mathbf\Sigma^1_n$ are separated by a $\mathbf\Delta^1_n$ set.
...
11
votes
1
answer
568
views
“Algebraization" of $p$-adic fields
Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
0
votes
1
answer
91
views
Connected infinite graphs in which all matchings are "small"
Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
21
votes
1
answer
677
views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
12
votes
1
answer
549
views
Fast convolution of sparse functions
Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
8
votes
5
answers
463
views
Ideals on $\mathbb N$ and large sets that have small intersection
Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,...
3
votes
0
answers
105
views
Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
1
vote
1
answer
539
views
Reformulation of Grothendieck vanishing theorem
Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$?
...
3
votes
0
answers
101
views
Differential of p-divisible groups
Setting: Let $p$ be a prime number and let $S$ be a scheme such that $p$ is locally nilpotent on $\mathcal O_S$ ($p^N=0$). Let $X$ be a $p$-divisible group over $S$. Let $X[p^n] $ be the kernel of ...
5
votes
1
answer
491
views
A name for a group with finite abelization?
Let us recall that a group $G$ is called perfect if it coincides with its commutator subgroup $G'$, or equivalently, if its abelianization $G/G'$ is trivial.
Question. Is there any name for a group ...
1
vote
1
answer
265
views
About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$
It follows from Exercise 1.13 in Humphreys' Category $\mathcal{O}$ book that $M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$ has a direct sum decomposition $M=\oplus M_i$ such that all weights of each $...