Newest Questions
159,037 questions
5
votes
2
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If every ascending chain of ideals leading up to an ideal stabilises, is the ideal finitely generated?
I'm a fourth-year undergraduate currently studying a master's degree. In the last couple of weeks, I have been wondering about the interaction of the Noetherian condition with the prime ideals of a ...
- 53
4
votes
0
answers
155
views
Two other variants of Arhangel'skii's Problem
This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is ...
- 4,413
4
votes
1
answer
209
views
Chain components and posets
Let $(X,f)$ be a topological dynamical system ($f$ continuous, $X$ compact, metric with distance $d$).
Let $C\subseteq X^2$ indicate the chain recurrence relation:
$$xCy\iff \forall \epsilon>0\ \...
- 4,459
7
votes
1
answer
451
views
When is a module a filtered colimit of finitely presented submodules?
For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both:
a filtered colimit of finitely generated $R$-submodules (by considering all finite subsets of $M$ and ...
- 2,040
2
votes
1
answer
112
views
Local rings whose the endomorphism rings of E(R/J) is division ring
Let $R$ be a local ring with maximal ideal $J$. Assume that ${\rm End}_{R}({\rm E}(R/J))$ is a division ring (${\rm E}(R/J)$ means the injective envelope of $R/J$). Does $R/J$ is injective?
- 321
5
votes
1
answer
357
views
Does the category of integral domains admit a symmetric monoidal structure?
Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...
- 63.1k
1
vote
1
answer
188
views
T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via:
$$
T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...
- 13
2
votes
2
answers
596
views
What is the relationship between Hölder spaces and differentiability?
I'm porting this question over from MSE as it did not get any responses other than one comment on there.
Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...
- 721
0
votes
1
answer
28
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Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
- 13.2k
5
votes
1
answer
453
views
von Neumann subalgebra having separable predual
Let $M$ be a von Neumann algebra.
Let $x,y$ be two self-adjoint operators in $M$.
Are there any von Neumann subalgebra $A$ of $M$ containing $x,y$ such that the predual of $A$ is separable?
- 617
2
votes
0
answers
41
views
Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
- 4,115
2
votes
0
answers
26
views
Functor from Leibniz algebra category to Lie-Yamaguti algebra category
Is there any functor from $\operatorname{Leib}_{\mathbb{K}}$ (Leibniz algebra category) to $\operatorname{LYA}_{\mathbb{K}}$ (Lie-Yamaguti algebra category)?
From Kinyon and Weinstein's paper I saw ...
- 229
1
vote
0
answers
136
views
Exotic automorphisms of an extension of Thompson's group $V$
Recall that R. Thompson's group $V$ acts transitively on the set $\mathbb{Q}_2$ of dyadic rationals contained in the unit interval $[0,1)$.
Main question. Does there exist a non-trivial $V$-...
5
votes
0
answers
171
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
7
votes
1
answer
269
views
Efficiently computing $\sum_k x^{k^2}$ modulo $p$
Let $p$ be prime. There is a whole host of "large" degree polynomials that can be computed efficiently modulo $p$. I was wondering if:
$$q(x) = \sum_{k=0}^{p-1} x^{k^2}$$
is a polynomial ...
- 591
3
votes
0
answers
84
views
Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) mapping such that
\begin{equation}
\lVert F(f) \rVert \leq \lVert f \rVert
\end{equation}
for all $f \in L^2(S^1)$. For the space of smooth periodic ...
- 3,477
3
votes
0
answers
65
views
Is there a fixed point theorem that applies to $f: \sum_k x_k 10^k \mapsto \sum_k x_k!$?
Let $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $f:x=\sum_k x_k 10^k \mapsto \sum_k x_k!$ where $x_k$ is the $k$-th digit of $x$ in base ten. This function came up in a Project Euler problem. The question ...
- 571
2
votes
2
answers
283
views
Expressions for binomial residue sum $\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$
I'm interested in the sum:
$$\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$$
where $q$ is a prime number. This is just the binomial expansion with an extra weight on quadratic residues ...
- 591
6
votes
1
answer
1k
views
Full expansion of $\det(I+\varepsilon A)$
It is well known that given a $n \times n$ matrix $A$, it holds that
$$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$
I would need a full representation of $ \...
- 739
4
votes
1
answer
282
views
Borel cohomology for circle actions on odd spheres
Suppose we have a $S^1$-action on the odd sphere $S^3$ as follows:
$$ \lambda \cdot (z_1, z_2) = (\lambda \cdot z_1, \lambda^2 . z_2)$$
I would like to understand the (Borel) equivariant cohomology of ...
- 169
2
votes
0
answers
84
views
Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
3
votes
1
answer
145
views
Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?
We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
- 825
4
votes
0
answers
197
views
Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
1
vote
1
answer
464
views
Variance of Itô integral on general time interval
I have a question concerning the variance of the Itô integral on general time intervals, i.e. I want to calculate
\begin{align*}\operatorname{Var}\left(\int_s^T f(t)dW_t\right),\end{align*}
where $f: [...
- 35
8
votes
1
answer
617
views
$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$
Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...
user515519
21
votes
1
answer
1k
views
Bubblesort with a twist: a tricky termination
Consider an $n$-tuple $\left(a_1, a_2, \ldots, a_n\right)$ of real numbers. We are allowed to perform the following two moves:
S-moves: We pick two adjacent entries $a_i$ and $a_{i+1}$ satisfying $...
- 33.8k
1
vote
1
answer
203
views
Moving of sphere embedding and its interior defined by Jordan-Brouwer separation theorem
Let $f_1:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be a continuous embedding, where $\mathbb S^{n-1}$ is the unit sphere of dimension $n-1$, and a point $x$ in the interior of $f_1(\mathbb S^{n-1})$ ...
- 435
2
votes
2
answers
617
views
In what sense does the Hermite expansion of a bounded smooth function converge?
Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function.
If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as
\...
- 3,477
6
votes
1
answer
396
views
Relationship between canonical topology on a topos and its site of definition
The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf.
According to First Order Categorical Logic Lemma 1....
- 237
2
votes
1
answer
196
views
Hausdorff dimension of the curve of a continuous nowhere differentiable function
It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
- 16.2k
0
votes
0
answers
50
views
pseudo inverse of a holomorphic multivariate injective map
Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
- 265
1
vote
0
answers
64
views
Cohomology of an intermediate extension (perverse) sheaf on the affine line
Let $\mathbb{A}^1$ be defined over a finite field or $\mathbb{C}$, $j: \mathbb{G}_m \rightarrow \mathbb{A}^1$ and $\mathcal{F}$ a local system on $\mathbb{G}_m$. I wonder what is known about the ...
- 195
22
votes
1
answer
4k
views
A challenging (for me) limit calculation
How to calculate the following limit
$$
\lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?}
$$
${}{}$
- 549
0
votes
0
answers
115
views
Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
- 471
2
votes
1
answer
288
views
About the filtration of crystalline cohomology
Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ ...
- 785
-2
votes
1
answer
213
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Is there any case of remormalization in which we have to solve it by ways in two different systems? [closed]
In renormalization of physics, $$\sum_{j=1}^{\infty}j=-\frac{1}{12}$$ We may obtain the result in two ways: first we may redifine the sum so we have used two system of math with different definition ...
- 3,104
1
vote
0
answers
46
views
Extension of a cylinder functor on C to the S_n C
I was looking at Waldhausen's definition of a cylinder functor and reading his proof that a cylinder functor on $C$ induces cylinder functors on $S_n C$ for all $n$. It seems to me that he is using ...
4
votes
1
answer
394
views
Does the rank of a subfunctor not exceed the rank of a functor?
It is known that Vopenka's principle is equivalent to the statement “a subfunctor of a accessible functor is accessible” (Adámek and Rosický, Cor 6.31 in Locally Presentable and Accessible Categories)....
- 6,962
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 4,115
5
votes
2
answers
536
views
Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
3
votes
1
answer
562
views
On the convergence in total variation
$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...
- 128k
0
votes
1
answer
282
views
Action of the Haar measure on the Heisenberg group
The Heisenberg group is $\mathbb{H}^N=\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the group operation
\begin{equation}
(...
- 677
4
votes
1
answer
260
views
Kummer's congruence at $p=3$
Let $B_{2k}$ be the Bernoulli numbers of even index and $\varphi(n)$ be Euler's totient function. We recall one instance of Kummer's congruences: for each integer $m\geq1$ and a prime number $p\geq5$, ...
- 43.2k
0
votes
1
answer
170
views
Help in understanding the singular system of linear forms and non escape of mass
I am having some trouble in understanding certain portions of the following paper by KKLM
https://link.springer.com/article/10.1007/s11854-017-0033-4
So in proposition 3.1, they proved the estimate ...
- 337
1
vote
0
answers
75
views
if a^x + b^y = c^z, 1/x + 1/y + 1/z < 1, how do we call this numbers?
I have equation
$a^x + b^y = c^z, 1/x + 1/y + 1/z < 1$, where $a$, $b$, $c$, $x$, $y$, $z$ are positive integers.
Are there any special name for solutions of this equation?
- 161
5
votes
0
answers
360
views
Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
1
vote
0
answers
203
views
Clarifications about the Iwasawa Main Conjecture
I would like to clarify a couple of things regarding the Iwasawa main conjecture.
In the paper where Mazur and Wiles prove the main conjecture, on page 182, it is written that $h_p(\omega^i, T)$ is ...
- 63
1
vote
1
answer
138
views
Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...
- 4,058
1
vote
0
answers
162
views
Bloch's higher Chow group as relative ordinary Chow group
If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of ...
- 219
1
vote
1
answer
100
views
Is there literature on the existence of solutions to elliptic systems on unbounded manifolds?
Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on ...
- 419