Newest Questions
159,035 questions
1
vote
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214
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Bound on Von Mangoldt for automorphic L-functions
Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote
$$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$
In terms of the local roots of the Euler product:
$$ \Lambda_f(...
- 13
2
votes
1
answer
291
views
When does uniqueness of a stable equilibrium imply it is globally stable?
Given a gradient dynamical system
$$\dot x=-\nabla f(x),$$
my question is:
(1) If there exists only one equilibrium $x^*$ which is stable (if necessary, this can be changed to stable asymptotically ...
- 405
3
votes
0
answers
311
views
What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
- 571
4
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0
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132
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Does a critical graph have to be product-irreducible?
A finite, simple, undirected graph $G=(V,E)$ is said to be (vertex-)critical if for all $v\in V$ we have $\chi(G\setminus\{v\}) < \chi(G)$.
Let me add that in this context, $\chi(\cdot)$ denotes ...
- 48.9k
3
votes
0
answers
250
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Grothendieck's vs Gruson and Raynaud's dévissages
In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage concept except that the first one works in "...
- 6,038
8
votes
1
answer
314
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Is there an elliptic curve over a number field with a point of order 64 and Mordell-Weil rank zero?
It seems to me that there ought to be elliptic curves over number fields with arbitrarily large torsion subgroups but Mordell-Weil rank zero. But I'll settle for a point of order 64. Does anyone ...
- 315
13
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2
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767
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Smooth Urysohn's lemma on Fréchet spaces
Let $V$ be a Fréchet topological vector space.
Let $K_0$ and $K_1$ be two closed subsets which are disjoint.
I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$
whose restriction ...
- 43.2k
1
vote
1
answer
118
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A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 3,477
3
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0
answers
104
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A depth version of a conjecture of Yamagata
Let $A$ be a finite-dimensional $K$-algebra.
Recall that the grade of an $A$-module $M$ is defined as the smallest $i$ such that $\operatorname{Ext}_A^i(M,A) \neq 0$ and the depth of $A$ is defined ...
- 26.5k
5
votes
5
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572
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Is every uniform hyperbolic linear space infinite?
I start with definitions.
Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms:
(L1) for any distinct ...
- 42k
4
votes
1
answer
253
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Cocartesian fibration classifying $\mathrm{Fun}(F,G)$
Throughout this question we consider $\infty$-categories.
Fix a cartesian fibration $p : \mathcal{F} \to \mathcal{C}$ and a cocartesian fibration $q : \mathcal{G} \to \mathcal{C}$ which straighten to $...
- 1,337
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0
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78
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Closed subgroups in Ratner's orbit closure theorem on unipotent flows
Let $G$ be a semisimple (real or $p$-adic) Lie group and $\Gamma$ a discrete and cocompact subgroup of $G$, as in the setting of Ratner's theorems on unipotent flows (see for example here \url{https://...
- 91
4
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73
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Small angles between independent centred random walks in $ \mathbb{Z}^d$
Let $W_n$ and $W'_n$ denote two independent random walks in $ \mathbb{Z}^d$ defined using a finitely supported centred (mean zero) probability measure on $\mathbb{Z}^d$.
For $N \ge 1$, let
$\theta_n$ ...
- 6,224
1
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1
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139
views
Does there exist a continuous choice of maximizing balls for the Hardy Littlewood maximal function?
Note: Here $\overline B_r (x)$ denotes the closed ball of radius $r$ around $x$.
Let $f \in L^1 (\mathbb R^d)$. We define the averages $A(x, r)$ for $x \in \mathbb R^d$ and $r > 0$ by
$$A(x, r) := \...
- 6,321
4
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0
answers
164
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When $X$ is homeomorphic to $\mathscr{F}[X]$?
While I was talking to some colleagues, one of them said that there exists a topological space $X$ such that $X$ is uncountable, non-discrete and homeomorphic to $\mathscr{F}[X]$ (the Pixley-Roy ...
- 151
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1
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154
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Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time
Consider the following stochastic integral:
$$
X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s.
$$
Is $X_t$ almost-surely non-negative?
Using this answer, it seems that
$$
X_t = \max( W_t, 0) - \...
- 109
6
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1
answer
360
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The Dirichlet series of the harmonic numbers
I'm curious about the Dirichlet series $$F(s) = \sum_{n = 1}^\infty \frac{H_n}{n^s}$$
of the sequence $H_n = \sum_{k = 1}^n \frac{1}{k}$ of harmonic numbers. Its abscissa of convergence is $1$. ...
- 4,985
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 75
2
votes
2
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208
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On the number of values with exactly $k$ prime factors of a given polynomial
This is surely be a well studied problem. Let $f(x) \in \mathbb{Z}[x]$. Is there some $k \in \mathbb{N}$ such that there are infinitely many $n \in \mathbb{Z}$ where $f(n)$ has exactly $k$ prime ...
- 1,763
4
votes
1
answer
193
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Binary codes with upper and lower bound on pairwise distance
The Gilbert-Varshamov bound provides a lower bound for codes of length $n$ with minimum pairwise distance (say $\frac{n}8$). If we wish for the codes to also have pairwise distances bounded above (say ...
- 141
1
vote
1
answer
125
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Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
- 31
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1
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351
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"Compactness length" of Baire space
Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...
- 20.5k
3
votes
2
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409
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If $p_1$ and $p_2$ are prime numbers, then either $p_1$ divides $\sum_{i=1}^{p_1-1} i^{p_1p_2-1}$ or $p_2$ divides $\sum_{i=1}^{p_2-1} i^{p_1p_2-1}$?
I feel like it's true as for small cases I couldn't find counterexample.
In general, whether it's true that if we have prime number, $p_{1}, p_{2},\dotsc, p_{k}$ and $n=p_{1}p_{2}p_{3}\dotsb p_{k}$ ...
1
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0
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68
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Galois connection for homeomorphisms
let $M = \mathbb{R^2}$ and $X = \{0\}$ and $G = Aut_X(M)$ the group of homeorphisms fixing $X$ (pointwise). Then we have, in analogy to classical Galois theory for field extensions, a Galois ...
- 11
3
votes
1
answer
220
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Conditional expectation as square-loss minimizer over continuous functions
It is well-known that the conditional expectation of a square-integrable random variable $Y$ given another (real) random variable $X$ can be obtained by minimizing the mean square loss between $Y$ and ...
- 463
1
vote
0
answers
128
views
Centralizer of Frobenius on filtered $\phi$ module
Suppose $K$ is an unramified extension of $\mathbb Q_p$ of degree $m$, and $\sigma$ is the $p$ power frobenius on $K$. Suppose $V$ is a $2$ dimensional admissible filtered $\phi$ module over $K$.
I ...
- 785
1
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1
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253
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About a result by P. Erdős and H. Sachs on graph with large girth
I recently came across the following result:
For any integers $d,r\geq 2$ and $n\geq 4d^{r(d+1)}$, there is a $d$-regular graph on $n$ vertices with girth at least $(d+1)r+1$.
referring to
"P. ...
- 281
1
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2
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350
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Finding lectures PDF "Four lectures on simple groups and singularities"
I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
3
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0
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139
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proper smooth dg-categories and colimit
Let $I$ be a filtered category and $\{k_i\}_{i\in I}$ be a system of commutative rings over $I$. Toen proved that there is an equivalence of categories
$$
\text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\...
- 349
11
votes
1
answer
341
views
Density of linear subspaces in $C(K)$
Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...
- 467
2
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0
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120
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On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 69
3
votes
2
answers
256
views
Flat norm of currents and minimal surfaces
Let $A$ be a $k \leq n$ integral current with compact support over $\mathbb{R}^n$ (for conciseness).
Its flat norm $F(A)$ can be defined via
$ F(A) = \inf \{ M(T) + M(S) \, | A = T + \partial S \}$
...
- 93
4
votes
1
answer
189
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Sign of error in the central limit theorem
Let $X_n$ and $Y_n$ be independent copies of two random variables $X$ and $Y$ with domain $\{-1,0,1\}$ for $n\in \mathbb{N}$. For a given $k\in \mathbb{N}$, I would like to find conditions on $X$ and $...
- 53
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2
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304
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How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?
The following question was posted to MathStackExchange (original here). As there were no comments/answers on the original, I have ported it unedited.
I am interested in determining the cardinality of $...
- 2,206
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0
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65
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Can the regularity argument for the solution of a parabolic PDE in Pinsky's paper be generalized?
In this paper Pinsky shows existence, uniqueness and regularity for the problem
$$
u_t=\Delta u-a(x) u^p |\nabla u|^q
$$
where $a\in C^2( \mathbb{R}^d)$ satisfies the condition $ a(x)|\leq (1+|x|^2)^N$...
- 677
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0
answers
185
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What is the fastest algorithm for multiplying one given number with many others?
When multiplying two numbers with each other, which are $n$-bit numbers, there are several algorithms like the one of Karatsuba ($O(n^{\log_2 3})$) and a new one doing it even better (Harvey - Van der ...
- 749
1
vote
1
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345
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Topological degree of differentiable map using line integrals?
Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
3
votes
1
answer
519
views
Group action of $\text{SL}(2, \mathbb{C})$
Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$
Then a group action $\rho :...
user515818
1
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1
answer
65
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Boundedness of maximisers of parametric strictly concave functions
Let $L:[0,1]\times \mathbb R^m\times \mathbb R^n\to \mathbb R$ be defined by
$$L(\lambda, x,y):=\sum_{1\le i\le m}\alpha_i x_i + \sum_{1\le j\le n}\beta_j y_j -\sum_{1\le i\le m, 1\le j\le n} p_{i,j}\...
- 1,399
1
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0
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109
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$K_0$ of finite graphs
We have two operations on finite graphs, first the disjoint union and the categorical product. I want to use these operations to associate a r(i)ng $R$ to finite graphs. An element of that ring is an ...
- 11.1k
12
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0
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384
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What are some examples of 3-dualizable $(\infty,2)$ categories?
From the cobordism hypothesis, we know that (the space of) symmetric monoidal functors from the $(\infty,3)$ category of framed cobordisms into a symmetric monoidal $(\infty,3)$ category is the same ...
- 2,356
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0
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165
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Upper bound of special Cheeger constant on $(S^2,g)$
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric.The Cheeger constant of $(S^2,g)$ is
$$
h(S^2,g)=\inf_{\gamma} \frac{|\gamma|_g}{\min\{|A_1|_g, |A_2|_g\}}
$$
take the infimum over all closed ...
- 165
2
votes
0
answers
253
views
Xi function representation
Would it be correct to write down the following, or is it completely wrong?
$$ \Xi(z) = \frac{1}{2} \int_{-\infty}^{\infty} e^{-\pi x^2} \theta''(x) \sin(zx) \, dx, $$
with
$$ \theta(x) = \sum_{n=-\...
- 27
0
votes
1
answer
113
views
Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
The inequality (2.3) in this ...
- 677
6
votes
2
answers
241
views
Does "perpendicular phase incoherence" satisfy the triangle inequality?
I asked this question at https://math.stackexchange.com/q/4783968/222867, but even after a 200-point bounty, no solution was provided, only some thoughts regarding possible directions. So I'm now ...
- 708
5
votes
1
answer
338
views
To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?
If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable.
How much can be said about the ...
- 1,481
0
votes
1
answer
204
views
Equivalence of dihedral and symmetric group actions on a specialized real algebra
Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...
- 151
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
2
votes
0
answers
174
views
Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
7
votes
1
answer
416
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Classifying abelian (but non-central) group extensions using homotopy theory
Let $G$ be a group and let $A$ be an abelian group equipped with an action of $G$. Group extensions
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
inducing the ...
- 44.8k