Newest Questions
159,034 questions
5
votes
1
answer
192
views
Restriction of cuspidal representations of GL(2) to the Borel subgroup
Let $\mathbb{F}_q$ be a finite field and let $B$ be the standard Borel subgroup of $GL(2, \mathbb{F}_q)$ consisting of upper triangular matrices. Is there an explicit description of the restriction ...
- 53
4
votes
1
answer
294
views
Why should we expect this odd behavior of negative binomial distributions?
In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
0
votes
0
answers
146
views
Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
- 307
7
votes
1
answer
300
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The origin of a planar graph theorem of Steinitz and Rademacher
The subsequent statements are extracted from the article titled 'Generating r-regular graphs' (https://doi.org/10.1016/S0166-218X(02)00593-0).
A well-known classical theorem of Steinitz and ...
- 1,851
7
votes
0
answers
273
views
Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
- 6,025
1
vote
1
answer
259
views
Question about the proof of Propp-Wilson algorithm in Olle Häggström's book
Update: Oops! This is a stupid question and should be closed. The definition of the probability space that contains events $A_i$ requires using a single random stream.
I have difficulties ...
- 565
3
votes
1
answer
227
views
If $f$ is a derivative and $f=g$ a.e. for some Riemman integrable function $g$, then can we obtain the Riemann integrability of $f$?
Let $a,b\in\mathbb R$ with $a<b$ and $f:[a,b]\to\mathbb R$. Assume that there exists a Riemann integrable function $g:[a,b]\to\mathbb R$ such that $f=g$ almost everywhere.
Then we can NOT conclude ...
- 299
2
votes
2
answers
422
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A proof of $\dim(R)+1 \leq \dim(R[T]) \leq 2 \dim(R)+1$ with the Coquand-Lombardi characterization of Krull dimension
Question. Can we use the Coquand-Lombardi characterization$^1$ of Krull dimension to prove the well-known inequalities$^2$
$$\dim(R)+1 \leq \dim(R[T]) \leq 2 \cdot \dim(R)+1,$$
where $R$ is any ...
- 63.1k
4
votes
0
answers
112
views
Elliptic integral as quantity associated with Riemann surface?
There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 5,230
4
votes
0
answers
93
views
What is the Goldie dimension of the ring of stable stems?
Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
- 64k
10
votes
0
answers
598
views
Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 3,577
1
vote
0
answers
102
views
Collision among products of binomial coefficients
Let $k, \ell \geq 2$ be positive integers. Do there exist infinitely many positive integers $N$ such that the equation
$$\displaystyle \binom{m}{k} \binom{n}{\ell} = N$$
have more than four solutions? ...
- 26.9k
4
votes
1
answer
332
views
Question regarding the definition of linearization of line bundles
I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...
- 737
0
votes
1
answer
112
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Does any warped product metric admit a function with hessian proportional to the metric?
It is known that the existence of a function with hessian proportional to the metric implies that the metric is a warped product metric. Is the reciprocal true as well? I.e, if $(B \times N, g = g_B + ...
- 659
8
votes
0
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383
views
Is this set, defined in terms of an irreducible representation, closed under inverses?
$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
- 4,081
0
votes
1
answer
133
views
Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...
- 337
1
vote
0
answers
120
views
Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$
This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
- 1,126
0
votes
1
answer
89
views
$c_0(2^{\kappa})$ does not embed in $\ell_\infty(\kappa)$?
How to prove that $(c_0(2^\kappa),\|\cdot\|_\infty)$ does not embed into $(\ell_\infty(\kappa),\|\cdot\|_\infty)$? Recall that $(c_0(2^\kappa),\|\cdot\|_\infty)$ is the Banach space of all families $(...
15
votes
1
answer
1k
views
Is this a typo in Macdonald's paper "The Poincaré Series of a Coxeter Group"?
I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The ...
- 565
4
votes
1
answer
254
views
Frobenius and regular scheme
Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...
- 3,472
3
votes
1
answer
137
views
Thinning of (mixed) binomial point process
Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
- 133
1
vote
1
answer
181
views
Is $\sup_{f\in \mathcal{F}}\left|\int _Xfg \, d\mu\right|<\infty$ true for all $g\in L^\infty _\mathbb{C}(\mu )$?
Suppose that $(X,\mathcal{A},\mu )$ is a finite measure space. Let $\mathcal{F}\subseteq L^1_\mathbb{C}(\mu )$. If $\sup_{f\in \mathcal{F}}\left|\int _Xf\varphi \, d\mu\right|<\infty$ for all ...
- 15
4
votes
2
answers
413
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
- 985
4
votes
2
answers
374
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
5
votes
2
answers
458
views
Logarithm of a bounded operator
Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...
- 16.2k
8
votes
1
answer
588
views
Con(PA) via non-well-foundedness?
Lumsdaine made the following interesting
comment:
if Con(PA) fails in a non-standard model, it means it contains a
“proof of non-standard length” of a contradiction from PA. With a
little work, one ...
- 16.6k
2
votes
0
answers
147
views
Extension of isotopies
In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
- 641
10
votes
1
answer
572
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
2
votes
0
answers
125
views
Inequality for 2-associated Stirling numbers of the second kind
Let $S_2(n,k)$ denote the 2-associated Stirling number of the second kind for $n$ objects and $k$ blocks, with $n$ being at least two. That is, we partition $n$ labeled objects into $k$ unlabeled ...
1
vote
1
answer
186
views
Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
- 33
5
votes
1
answer
452
views
Cohomology of Shimura varieties before and after completion at some prime
Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
- 769
4
votes
1
answer
172
views
The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$
$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
- 43
4
votes
4
answers
405
views
Groups acting non-properly cocompactly on hyperbolic spaces
A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...
- 349
1
vote
0
answers
104
views
Amenability of $\textrm{w}_0(L^1(G))$
Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...
- 2,605
2
votes
0
answers
106
views
Hereditarily just-infinite pro-$2$ groups
An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
- 667
4
votes
1
answer
418
views
Series convergence if $\sum a_n^2 < \infty$
There are quite a few simple results about convergent/divergent series derived from similar ones. Here is a question in the same spirit that I saw posted on another forum. Unfortunately, I don't have ...
- 689
2
votes
0
answers
68
views
Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
- 167
2
votes
0
answers
158
views
Infinitesimal criteria for unramified morphism on stacks
In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...
- 605
7
votes
0
answers
239
views
Chip firing on hypergraphs
A (finite) hypergraph is a pair $(V, \mathcal{E})$ where $V$ is a finite set of vertices and $\mathcal{E}\subseteq\mathcal{P}(V)$ with each $E\in\mathcal{E}$ having at least two elements; a ...
- 20.5k
1
vote
0
answers
90
views
What do $\gamma$-radonifying operators radonify?
In the second volume of their Analysis in Banach Spaces, Hytönen et al. introduce the notion of $\gamma$-radonifying operator more or less as follow.
Let $(\gamma_j)_{j\in\mathbf N}$ be a sequence of ...
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2
votes
2
answers
356
views
Is a Hausdorff separable topological space that is uniform and complete necessarily a Polish space?
Is an Hausdorff separable topological space that is uniform and complete necessarily a Polish space ?
- 2,655
1
vote
0
answers
62
views
Dispersive equations at low frequencies and time oscillations
It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
The Schrodinger dispersion relation is $\omega(...
- 111
30
votes
0
answers
814
views
Interpretation of "1089-number trick" in terms of symmetric group action on cohomology group?
I tried posting the following on math.stackexchange, but no answers. I can of course delete if inappropriate.
The "1089 number trick" (see e.g. here) says that if you take a three-digit ...
- 301
0
votes
0
answers
107
views
An integral and a Jensen-type formula
For $f$ be analytic on the disc $\overline{D}(0,R)$ centred at $0$ with radius $R>0$ and such that $f(0)\neq 0$, then the following formula is well-known
\begin{align}
\frac{1}{2\pi}\int_{-\pi}^{\...
- 59
0
votes
1
answer
161
views
Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
- 31
3
votes
1
answer
251
views
Do regularity structures involve infinite "Taylor" series?
I have been learning about the theory of regularity structures, for which the common motivation is Taylor series. However, I keep seeing direct sums in the definition of a regularity structure, which ...
- 447
4
votes
1
answer
297
views
Why is this continuum circle-like?
A continuum is a compact connected metrizable space.
A continuum $X$ is called arc-like if for every $\varepsilon>0$ there is an open cover $U_1,\ldots,U_n$ of $X$ such that the diameter of $U_i$ ...
- 3,011
2
votes
1
answer
211
views
How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?
I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:
Let $M$ be a smooth quasi-...
- 41
1
vote
0
answers
101
views
Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves
In this article in Compositio (2011), Munshi proves a mean value result for
$$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$
where here $f$ is a primitive holomorphic form of level $q$ with ...
- 1,354
2
votes
0
answers
152
views
Trying to decode a module functor
This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.
Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
- 1,347