12
votes
Reference request: Software for producing sounds of drums of specified shapes
The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on ...
- 183k
5
votes
Accepted
Number of planes generated by integer vectors
For $k=d-1$ this is a result of Bárány-Harcos-Pach-Tardos (2001). See Theorem 3 in the preprint version or the published version.
- 102k
5
votes
Fermat last theorem : proof of a criterion by Cauchy
Cauchy's criterion is a special case$^\ast$ of a more general criterion proven by Kummer in 1857 [1], and by Fueter in 1922 [2]. A description of Kummer's derivation and how it implies the Cauchy ...
- 183k
4
votes
Hilbert's approach to Riemann hypothesis using Fredholm's work
Since this question was bumped to the front page, I might address
Q1: Can someone provide historical references for it?
This goes back to André Weil, who writes in [1] that Ernst Hellinger, a student ...
- 183k
4
votes
Is there a version of Weyl's law for graph Laplacians?
There will certainly exist a Weyl law for random planar graphs. However, the nature of the law will depend very sensitively on exactly which model of random graphs one takes.
One can start to see this ...
- 141k
4
votes
Accepted
Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
If you take $R=k[[x,y]]$ and $M=k[[x]][x^{-1}]$, this should give you an example. $M$ is not injective because there is a non-split injective map $M \to R[x^{-1},y^{-1}]/yR[x^{-1}]$. But if you ...
- 13.9k
4
votes
Accepted
What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
Multi-dimensional generalizations (one player against $d$ other players) are explored by P. Lorek in Generalized Gambler's Ruin Problem: explicit formulas via Siegmund duality.
For the analogue on a ...
- 183k
4
votes
Accepted
Strong Liouville property of virtually abelian groups
I see that you added a symmetry assumption, but I'm still answering your original question, for sake of completeness.
So first of all, the following remark is very important. The probability measure $\...
- 2,050
4
votes
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
Erin Pearse's Introduction to dimension theory and fractal geometry may well be suited for this purpose. It introduces the various ways to define and measure a fractional dimension (box counting, ...
- 183k
3
votes
On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
Clearly $R_P=(R/P)_P$, and so we may replace $R$ by $R/P$. Since its dimension is $s$, one can find a system of parameters $y_1,y_2,\ldots,y_s$ in the maximal ideal of $R/P$. Then the inclusion $k[[...
- 6,242
3
votes
Is there a comprehensive survey of the discrete series representation of a real reductive group?
In Harish-Chandra’s classification of the discrete series representations for $G(\mathbf R)$, there are some unusual aspects of the formulas for their characters. Langlands confronted this in the work ...
- 238
2
votes
Possible new series for $\pi$
A related, and perhaps easier, question is whether there are other known series for 𝜋 that involve a complex parameter 𝜆 in the summand, but where the sum of the series is independent of the value ...
- 2,510
2
votes
Accepted
Extensions of bounded uniformly continuous functions
$\DeclareMathOperator{\R}{\mathbb R}
\DeclareMathOperator{\eps}{\varepsilon}$
If you prefer to define uniformities in terms of a family $D$ of pseudometrics you can reduce the theorem to pseudometric ...
2
votes
Action of $O(3,\mathbb{R})$ on the conic $\{x^2+y^2+z^2=0\}$
It is not doubly transitive. It is actually very easy to see if one works with an explicit parametrisation $\rho:\mathbb{P}^2\to \mathbb{P}^3$ of the conic: $(u:v)\mapsto (2ixy:i(u^2-v^2):u^2+v^2)$. ...
- 13.9k
2
votes
Accepted
What is the fastest algorithm for classical period finding?
This works only if you know the factorization of $N$ thus can compute $\phi(N)$ efficiently:
First find out the largest power of $2$ dividing $\phi(N):=M.$ This can be done in $\log N$ steps. Let $M=2^...
- 10.2k
1
vote
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
The book Fractal Geometry - Mathematical Foundations and Applications by Kenneth Falconer may be what you are looking for. As far as I recall, everything is done in $\mathbb{R}^n$ and he tries to keep ...
- 56
1
vote
Accepted
Transcendence degree and Krull dimension for homomorphic images of power series rings
Let $k$ be countable and let $I=(0)$. Then $R_P$ is the field $k((x_1,\dots,x_n))$, which is uncountable provided $n>0$, and therefore of uncountable transcendence degree over $k$, since an ...
- 13.9k
1
vote
Determinantal inequality for difference of substochastic matrices
I guess this type of inequality can be fast proven as follows:
Notice that the expression $\det(A-B)$ is an affine function in the entries of $A$ and $B$ considered as variables.
Each row of $A=[a_{i,...
- 660
1
vote
Is there a version of Weyl's law for graph Laplacians?
For $N_\lambda$ denoting the number of eigenvalues less than $\lambda$, Weyl's law gives the asymptotics of $N_\lambda$ as $\lambda$ tends to infinity. The usual approach to establish this asymptotics ...
- 183k
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