Newest Questions
159,019 questions
4
votes
1
answer
293
views
Double q-analog of Pochhammer
Has the function
$$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$
been studied in the math literature? For example, does it obey any difference ...
- 533
3
votes
0
answers
66
views
The supermoduli space of supertori with odd spin structure and metaplectic group actions
I'm trying to understand the description of the supermoduli space of supertori with odd spin structure as a quotient of the super complex upper half plane $\mathbb{H}^{1|1}$. Such a description ...
- 6,777
10
votes
2
answers
750
views
Universal group such that every finite group is a quotient
We say that a permutation $\varphi:\mathbb{N}\to\mathbb{N}$ is finitary if there is $k\in\mathbb{N}$ such that $\varphi(i) = i$ for all $i\in\mathbb{N}$ with $i\geq k$. Let $I_\mathbb{N}$ denote the ...
- 48.9k
2
votes
1
answer
171
views
A priori estimates to $u_t - \Delta u = u^2$ [closed]
My research is now considering the a priori estimates on the equation
$$
\begin{cases}u_t - \Delta u = u \min(u,c) \\
u(0,y) = u(1,y)\\
u_x(0,y) = u_x(1,y)\\
\partial_n u(x,0) = \partial_n u(x,1) = 0
\...
- 159
1
vote
1
answer
166
views
Suggestions about the set of all irreducible complex character degrees of a finite group
Let $G$ be a finite group, $\operatorname{cd}(G)$ be the set of all irreducible complex character degrees of $G$, and $\rho(G)$ be the set of all prime divisors of integers in $\operatorname{cd}(G)$. ...
- 577
8
votes
2
answers
563
views
Is there a purely topological definition of $\text{Spin}(p,q)$?
I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty).
A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...
- 233
3
votes
0
answers
77
views
Distribution of waiting time conditioned on a fixed time length
FYI, this question is a duplicate from math stack exchange
I ask here again because I got no response.
Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
1
vote
2
answers
115
views
Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
- 297
2
votes
0
answers
82
views
Is there any work on the intersection loci of the universal theta divisor with torsion sections?
Let $Y$ be a Siegel modular variety of some non-stacky level and genus $g$, carrying over it a universal principally polarized family of dimension-$g$ abelian varieties $A\to Y$. Inside $A$, with fine ...
- 2,044
0
votes
0
answers
201
views
Kato's explicit reciprocity law paper
Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
- 2,044
2
votes
0
answers
203
views
Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 167
5
votes
0
answers
213
views
Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 26.5k
6
votes
0
answers
310
views
Geometry of syntomic cohomology
Deligne cohomology has a geometric interpretation. For example, $H^{2}_{\mathcal{D}}(X,\mathbb{Z}(1))$ is identified with the group $H^{1}(X,\mathcal{O}_{X}^{\ast})$ of isomorphism classes of line ...
- 1,404
1
vote
1
answer
98
views
Geometrically connected affinoid cover of geometrically connected smooth rigid space
I am interested in the following question:
Given a geometrically connected smooth rigid analytic space $X$ over a non-archimedean field $k$, is it always possible to find an affinoid open covering, ...
- 208
5
votes
1
answer
602
views
Intersection cohomology and Poincaré duality
When trying to learn about perverse sheaves I hand-wavingly thought that intersection cohomology is the ‘minimal’ way of fixing the failure of Poincaré duality. But I am very aware that it is risky to ...
- 85
4
votes
0
answers
184
views
How to think about Beilinson's gluing data?
Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves":
Given a perverse ...
- 5,711
2
votes
0
answers
119
views
Limit of a distribution using Hörmander’s theorem
Let $\alpha \in \mathbb{C}$. I want to prove that
$$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$
in $D’(\mathbb{R}^n\setminus \left\{0\right\})...
- 319
2
votes
0
answers
142
views
Unipotent closure in classical groups
Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
- 81
0
votes
1
answer
123
views
Length of truncated Farey sequence
Farey sequence $F_n$ of order $n$ is defined as a sequence of completely reduced fractions $a/b$ such that $0 \le a \le b \le n$.
$$ F_1 = \frac{0}{1}, \ \frac{1}{1}$$
$$ F_2 = \frac{0}{1}, \ \frac{...
- 345
4
votes
1
answer
308
views
Casson's knot invariant
$\DeclareMathOperator\SU{SU}$Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology $3$-sphere $M$ into the ...
- 954
0
votes
0
answers
74
views
Bramble with order 5 for the Wagner graph
For treewidth $3$, there are four forbidden minors: K5, the graph of the octahedron, the pentagonal prism graph, and the Wagner graph.
This implies that the Wagner graph should have tree-width at ...
- 101
2
votes
0
answers
111
views
Constructing Hamiltonian circuits in acyclic digraphs
Any directed graph $G$ lacking cycles can acquire a Hamiltonian circuit through the addition of a sufficient number of edges.
Q. Is there a method to minimize the addition of edges to achieve a ...
- 4,058
1
vote
1
answer
345
views
Gateaux differentiability of the norm in Banach spaces
I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
- 297
3
votes
0
answers
102
views
Stein manifolds admitting uniform strictly plurisubharmonic exhaustion functions
Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\...
- 31
1
vote
0
answers
260
views
Is every $\sigma$-algebra generated by some measurable function? [closed]
I think this statement should be intuitively true, but I can't prove it myself or find the proof elsewhere. Could you help me, please?
Consider a general measurable space $(X,\mathcal{B})$ and any ...
- 31
0
votes
0
answers
145
views
$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?
By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences.
A natural question then is: ...
- 141
2
votes
1
answer
242
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,071
1
vote
1
answer
98
views
Selection terms in the untyped lambda calculus
In the untyped $\lambda$-calculus, are there terms $S$ and $T$ such that for any $n$ and any terms $t_1, \dotsc, t_n$,
$$S(T(t_1)\dots(t_n)) \twoheadrightarrow_{\beta} t_1$$
Of course, if $n$ is fixed ...
- 358
1
vote
0
answers
292
views
Solutions of a Gauss–Codazzi-like system of nonlinear PDEs
Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
- 134
2
votes
0
answers
194
views
Calculate $D_{\mathrm{cris}}(V)$ for a crystalline representation
$\newcommand{\cris}{\mathrm{cris}}$In my setting, $K/\mathbb Q_p$ is finite and unramified, and $V$ is a $2$-dimensional crystalline representation of $G_K$. Then we have $D_{\cris}(V)$, which is $2$-...
- 785
3
votes
0
answers
77
views
Is it known whether a homeomorphism close to the identity of a compact manifold with nonzero Euler characteristic necessarily has a fixed point?
I recently saw in Kirby's list of open problems that it isn't known if two commuting homeomorphisms of a compact manifold close to the identity necessarily share a common fixed point, when the ...
- 31
1
vote
1
answer
87
views
Potentially elementary question on affine functions on Banach spaces
In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by
$ \varphi(x^*) = \left\{
\begin{array}{...
- 297
2
votes
1
answer
257
views
Grassmannian $\mathrm{Gr}(k, \pm \infty)$ in infinite dimension
$\DeclareMathOperator\Gr{Gr}$The Grassmnnian variety $\Gr(k,n)$ is the set of $k$-dimensional subspaces of $\mathbb{C}^n$. The coordinate ring $\mathbb{C}[\Gr(k,n)]$ is generated by Plucker ...
- 6,211
7
votes
4
answers
966
views
Random sample of spanning trees
In a complete graph with $n$ vertices there are $n^{n-2}$ spanning trees.
I want to get a random sample of size $k$ from the set of all spanning trees.
The most basic and naive idea is to generate all ...
- 49
0
votes
0
answers
94
views
Canonical bundle of a general surface section
Let $X$ be a smooth projective variety of dimension $n \geq 2$. $H$ be the very ample divisor on $X$ giving the embedding. Let $S$ be a general surface section under $H$. Then we know that $K_S:=k_X|...
- 1
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
1
vote
2
answers
404
views
A closed formula for a sum involving hypergeometric function
Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...
- 531
3
votes
0
answers
84
views
Clique number and spectrum of a graph
In the Wikipedia article on Grassmann graph it is stated that in this graph:
$$\omega=1-\frac{\lambda_{max}}{\lambda_{min}}$$
where $\omega$ is the clique number of the graph, and $\lambda_{max}$ and $...
- 175
1
vote
1
answer
116
views
A question on classification of quadratic polynomials in even characteristic
$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...
- 343
2
votes
1
answer
60
views
Left-shift cycle generating maps $f:\{0,1\}^{c_0}\to\{0,1\}$ for fixed length $c_0$
This is a strengthening of an older question.
Is there a positive integer $c_0$ with the following property?
For every integer $n\geq c_0$ there is a function $f:\{0,1\}^{c_0}\to\{0,1\}$ such that ...
- 48.9k
1
vote
1
answer
204
views
Curve length in the Sasaki metric
I am trying to read Appendix II.A.2 (Distances in the tangent bundle) in Canary, Epstein, Marden (eds.), Fundamentals of Hyperbolic Manifolds: Selected Expositions and am stumbling over a calculation ...
- 187
0
votes
1
answer
264
views
A question about the prime counting function
I was playing around with the prime counting function and came across something that seemed correct to me, maybe it's already been proven but I don't know so I decided to ask here.
maybe a stupid ...
- 61
4
votes
1
answer
453
views
Detecting a "bad map" in Fintushel-Stern knot surgery
Background
Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
- 159
1
vote
0
answers
122
views
Singularity of curves on very general surfaces
I want to ask if there is a known classification of possible singularities of curves on a general (or very general) surface in $\mathbb{P}^3$.
It was shown in Proposition 3 of "Subvarieties of ...
- 19
4
votes
1
answer
237
views
Homogeneous polynomials cutting out complex abelian varieties
This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one.
...
- 1,763
-3
votes
1
answer
374
views
On Haar measure and Spherical measure [closed]
Let $d$-dimensional complex sphere be
$$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$
We can define the Haar measure on this sphere by regarding the unitary group $U(d)$.
We can regard the $d$-...
- 1,503
17
votes
1
answer
1k
views
What variational problem does the parabolic suspension bridge solve?
(Posted to MSE here, no answers)
The catenary curve $y(x)$ minimizes the gravitational potential energy
$$\int \rho g y\,ds=\int \rho g y \sqrt{1+y'^2}dx,$$
subject to a fixed length, $L=\int \sqrt{1+...
- 1,549
6
votes
1
answer
331
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
2
votes
0
answers
109
views
Reference for numerically non-negative polynomials for nef vector bundles
Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
user513306
4
votes
1
answer
281
views
Example for simply connected variety with trivial holomorphic Euler characteristic
Are there simply connected smooth complex projective manifolds $X$ (of general type) such that the holomorphic Euler characteristic $\chi(\mathcal{O}_{X})=0$ ?
This is certainly not true for surfaces.
- 91