All Questions
23,893 questions
11
votes
1
answer
588
views
Malmsten-like integrals for $\zeta(n) $
Context
Let us define a Malmsten integral when it is of the form $$M_{P,Q} := \int_{0}^{1} \frac{P(x)}{Q(x)} \ln \ln \left( \frac{1}{x} \right) dx. \tag{1} $$ In a paper by Blagouchine [1], the author ...
- 4,829
11
votes
1
answer
1k
views
Classification of (not necessarily connected) compact Lie groups
I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
- 12.9k
1
vote
0
answers
133
views
Hall-Littlewood polynomials with sage
I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...
- 161
10
votes
1
answer
400
views
Rigorous treatment of Ostrogradsky's instability theorem?
The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
- 188k
2
votes
1
answer
153
views
Interpolation theorem for general rough paths
In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...
- 1,904
1
vote
0
answers
240
views
Papers showing two open problems are connected [closed]
I'm currently writing an article in which I connect two open problems from different fields of mathematics (group theory and combinatorics). While writing the introduction, I was trying to think of ...
- 127
4
votes
0
answers
284
views
Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 5,031
12
votes
11
answers
2k
views
References for literature from mathematicians who provided critiques and proposals concerning ethical aspects of mathematics research
I would like to know sources, articles, books or other, that provide information on ethical aspects in the research of mathematics, I wondered what is the literature that this community knows about ...
- 39.9k
1
vote
0
answers
30
views
Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
1
vote
1
answer
92
views
On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 6,262
3
votes
2
answers
141
views
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
- 81
16
votes
5
answers
3k
views
Simple random walk on a locally finite graph: when is it recurrent?
I'm giving a talk tomorrow about a result in computer science which I recently proved. It's a recurrence-transience result on a random process which is related in spirit to a simple random walk. My ...
- 6,711
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 60.5k
12
votes
1
answer
598
views
Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
- 188k
5
votes
1
answer
221
views
How big is the class of all closed range bounded linear operator?
Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
- 63.9k
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 128k
3
votes
0
answers
80
views
Local Class field theory and Artin map for the Weil group
I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...
- 367
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 60.5k
2
votes
1
answer
202
views
Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
- 2,090
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
- 3,054
4
votes
0
answers
127
views
A "resampling identity" for the Bessel(3) process
I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
- 10.3k
3
votes
1
answer
192
views
Extensions of bounded uniformly continuous functions
Let $X$ be a uniform space, $S\subseteq X$ and $f:S\to \mathbb{R}$ bounded uniformly continuous, then there exists a uniformly continuous extension of $f$ to $X$. (Katětov, 1951)
I am looking for ...
- 11.4k
1
vote
3
answers
359
views
For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?
Let $F$ be a tempered distribution on $\mathbb{R}^2$ and $n \in \mathbb{N}$ be a fixed natural number. I wonder what exactly it means by
$\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$ where $x,y \...
- 21
2
votes
1
answer
129
views
What is the fastest algorithm for classical period finding?
Let $N$ be a positive integer, and choose an integer $a$ such that $\gcd(a,N)=1$. Then $a^r \equiv 1 \,\text{mod}\, N$ for some $r$. What is the current fastest classical algorithm for finding the ...
- 10.4k
1
vote
0
answers
43
views
Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
- 420
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
10
votes
1
answer
297
views
Bibliography request: Entropy for continued fractions
Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and
$$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$
...
CommunityBot
- 1
13
votes
2
answers
751
views
Is there a version of Weyl's law for graph Laplacians?
Is there a version of Weyl's law or a local Weyl's law for eigenvectors of the graph Laplacian?
For some context, a colleague in statistics has encountered eigenvectors of the Laplacian for certain ...
- 148k
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
CommunityBot
- 1
2
votes
1
answer
177
views
Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
The action of the orthogonal group $O(3,\mathbb{R})$ on the conic
$C= \{ x^2+y^2+z^2=0 \}$ in $\mathbb{P}^2$ must be well-understood, but I could not find any reference.
Is it doubly transitive?
- 14k
8
votes
1
answer
584
views
Reference request: Software for producing sounds of drums of specified shapes
Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
- 188k
0
votes
0
answers
143
views
Introductory resources on rewriting logic
Hi I would like to grasp the theory behind Maude [1], [2]
Are there any recommended video lecture notes, talks or introductory notes?
I have been exposed to Functional Analysis, Topology and some Term ...
0
votes
1
answer
100
views
Spatially localised solution to the Schrödinger equation with potential is a combination of eigenfunctions
In this Terry Tao's blog post, he claims that if one has a solution to the Schrödinger equation
$$i\,\partial_t u +\Delta u=Vu $$
with a "reasonably smooth and localised $V$", $u$ has ...
3
votes
0
answers
117
views
2-cells in the double category of 2-functors
Mike Shulman has in the answer to my previous question argued that for 2-categories $C$ and $K$ there is a double category whose objects are 2-functors between them and morphisms are lax and colax ...
- 1,347
9
votes
2
answers
738
views
Hilbert's approach to Riemann hypothesis using Fredholm's work
I read somewhere that Hilbert wanted to prove Riemann hypothesis using Fredholm's work on integral equations but I can't find anything online.
Can someone provide historical references for it?
What ...
- 12.9k
0
votes
1
answer
123
views
Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
2
votes
1
answer
526
views
What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
- 188k
0
votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
2
votes
0
answers
84
views
Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
- 3,425
12
votes
1
answer
238
views
Number of planes generated by integer vectors
For fixed dimension $d$ and large $R$ consider all non-zero integer vectors in the ball $B(0,R)\subset \mathbb{R} ^d$ of radius $R$ centered at the origin. The number of such vectors grows as $c_d\...
- 105k
1
vote
0
answers
76
views
Cartan decomposition over a not-necessarily-discretely-valued field
Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
- 63
41
votes
8
answers
17k
views
What are some good group theory references?
I'm curious about what books people use for a group theory reference. I don't currently own a dedicated group theory book, and I think I'd find such a book very helpful in my research. What is your ...
- 850
0
votes
0
answers
149
views
Reference book for a probability course
In the next months I am planning to deliver a (more-or-less) advanced course in probability theory. My students will have had already a first encounter with discrete probability theory (discrete ...
- 869
1
vote
1
answer
88
views
Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
- 16.2k
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 6,367
11
votes
1
answer
383
views
Is there a comprehensive survey of the discrete series representation of a real reductive group?
Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group?
Motivation:
I am a master's student trying ...
- 869
2
votes
1
answer
112
views
Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 16.2k
0
votes
0
answers
79
views
Is the Bures metric equivalent to the Euclidean one?
Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
- 1,334
4
votes
1
answer
252
views
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 $...
- 2,292