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https://mathoverflow.net/questions/384047 | 9 |
>
> **Problem**: Let $x\_1 = 1$ and $x\_{n+1} = \frac{x\_n}{n^2} + \frac{n^2}{x\_n} + 2, \ n\ge 1$.
> Find the third term in the asymptotic expansion of $x\_n$.
>
>
>
I have posted it in MSE six months ago without solution for the third term
<https://math.stackexchange.com/questions/3801405/the-limit-and-asympto... | https://mathoverflow.net/users/141801 | Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$ | Consider the substitutions
\begin{equation\*}
x\_n=n+1/2+y\_n/n,\quad y\_n=u\_n+5/8.
\end{equation\*}
Then $u\_1=-9/8$ and
\begin{equation\*}
u\_{n+1}=f\_n(u\_n)
\end{equation\*}
for $n\ge1$, where
\begin{equation\*}
f\_n(u):=\frac{-64 n^4 u-8 n^3 (4 u-13)+n^2 (56 u+115)+n (96 u+76)+4 (8 u+5)}{8 n^2 \left(8 n^2+4 n... | 8 | https://mathoverflow.net/users/36721 | 384086 | 159,743 |
https://mathoverflow.net/questions/383898 | 5 | $\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-Macaulay local domain.
By Eisenbud's famous work "Homological algebra on a complete intersection, with an application... | https://mathoverflow.net/users/172430 | Euler characteristic and rational Poincaré series | $\DeclareMathOperator\Tor{Tor}$I studied this function in my thesis (gee, typing this answer makes me feel old!). In fact, this function can be defined even when the Tor modules have finite length *eventually*. One simply takes the difference $\theta^R(M,N):=\ell(\Tor\_{2e}(M,N)) - \ell(\Tor\_{2e-1}(M,N))$ for some $e\... | 2 | https://mathoverflow.net/users/2083 | 384087 | 159,744 |
https://mathoverflow.net/questions/384067 | 1 | Let $A \in \mathcal{M}\_{n\times n}(\mathbf{R})$ and $b \in \mathbf{R}^n$ with $n\geqslant2$ be given.
Now should $(A,b)$ satisfy the Kalman rank condition
\begin{equation}
\text{rank}[b \,\, Ab \,\, \ldots \,\, A^{n-1}b] = n,
\end{equation}
or equivalently
\begin{equation}
\text{span}\{b, Ab, \ldots, A^{n-1}b\} = \m... | https://mathoverflow.net/users/115381 | Similarity to companion matrix, uniqueness | Yes. Suppose there was another such matrix. Since $P$ is invertible, it could be written as $QP$ for some $n \times n$ matrix $Q$. Thus $A Q P = Q P {\bf A} = Q A P$,
so $Q$ commutes with $A$, and
$Q P e\_n = b$. But then $A^k b = A^k Q P e\_n = Q A^k P e\_n = Q A^k b$. Since the vectors $A^k b$ span $\mathbb R^n$, we ... | 2 | https://mathoverflow.net/users/13650 | 384089 | 159,745 |
https://mathoverflow.net/questions/384105 | 10 | The claim in the title is proved on pp.19-20 of [Topological rigidity for non-aspherical manifolds](https://arxiv.org/pdf/math/0509238.pdf)
by M. Kreck and W. Lueck. Is there an earlier (classical) reference?
| https://mathoverflow.net/users/1573 | Where was it first shown that every homotopy self-equivalence of $S^1\times S^2$ is homotopic to a homeomorphism? | My guess is that the oldest reference might be Pontryagin's 1941 paper on the homotopy classification of maps from a 3-dimensional complex to the 2-sphere, the English version of which is in Recueil Mathématique 51 pp. 331-359. The application to homotopy equivalences of $S^1\times S^2$ is in an example on page 356 at ... | 13 | https://mathoverflow.net/users/23571 | 384111 | 159,753 |
https://mathoverflow.net/questions/384017 | 0 | To formulate my question precisely: let $s\_k$ be the number of isomorphism types of automorphism groups of convex 3-dimensional polytopes with $k$ faces. Are there any references discussing the asymptotic growth of $(s\_k)$ as $k\rightarrow\infty$?
| https://mathoverflow.net/users/173919 | Growth of number of isomorphism types of automorphism groups of convex 3-dimensional polytopes | The possible symmetry groups of 3-dimensional polytopes have been complete classified (see [here](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions)).
In particular, there are only a few families of symmetry groups with arbitrarily large size (according to [this section](https://en.wikipedia.org/wiki/Poi... | 0 | https://mathoverflow.net/users/108884 | 384120 | 159,757 |
https://mathoverflow.net/questions/384034 | 3 | Let $U \sim \mathcal{N}(0, I\_K)$ be a Gaussian vector of dimension $K$ and $V \sim \mathcal{N}(0,1)$, independent of $U$. Let $\Delta$ be a diagonal matrix with non-negative diagonal elements, $c\in\mathbb{R}^K$ and $\sigma^2\geq 0$. Consider the ratio
$$R= \frac{c^T U + (\sigma^2 + U^T\Delta U)^{1/2} V}{(c^Tc + \sigm... | https://mathoverflow.net/users/174236 | Bound on the distribution of a ratio involving Gaussian distributions | Royen's proof of the Gaussian correlation conjecture (see [here](https://arxiv.org/pdf/1512.08776.pdf) yields the following general statement:
Let $(W\_1(t),W\_2(t))\in \mathbb R^{n\_1+n\_2}$ be a Gaussian vector for every fixed $t\in[0,1]$ with the correlation matrix $C(t)=\begin{bmatrix}C\_{11}&tC\_{12}\\ tC\_{21}&... | 4 | https://mathoverflow.net/users/1131 | 384128 | 159,759 |
https://mathoverflow.net/questions/384098 | 2 | Let $T$ be a fully symmetric tensor of rank $3$ and size $N$.
Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that:
\begin{equation}
\sum\_{jk}^NT\_{ijk}x\_kx\_j=\lambda x\_i
\label{eq1}
\end{equation}
with the constraint that $\sum\_i x\_i^2=1$.
In the literat... | https://mathoverflow.net/users/142153 | Can the eigenvalues of a real symmetric tensor be complex? | Here is a counter example of a complex eigenvector: $N=3$, the nonzero elements of $T$ are $T\_{111}=2$, $T\_{122}=T\_{212}=T\_{221}=1$, $T\_{133}=T\_{313}=T\_{331}=1$. Eigenvectors with eigenvalue $2$ are $x=(1,iz,z)$, for any $z\in\mathbb{C}$.
(The flaw in the argument of the OP is indicated by [@lambda](https://ma... | 3 | https://mathoverflow.net/users/11260 | 384129 | 159,760 |
https://mathoverflow.net/questions/384077 | 6 | Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}\_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N\_G(D)$. Let $c$ be the Brauer correspondent of $b$.
M. Broué conjectured in the 90's that $b$ and $c$ are derived equivalent under these assumption... | https://mathoverflow.net/users/12826 | What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified? | I don't know the group of smallest order for which the conjecture has not been verified.
But certainly it is known to be true for all groups of order less than 200. There are general results that deal with most small groups. For example,
* For $p$-soluble groups (and in particular soluble groups) it is known to be ... | 5 | https://mathoverflow.net/users/22989 | 384133 | 159,761 |
https://mathoverflow.net/questions/384136 | 4 | Let $X$ be a smooth plane projective quintic curve (over $\mathbb C$). Then we know that it has gonality $4$. Assume that it has genus $g(X)=6$. Then my question is the following:
>
> Is it necessarily true that for every line bundle $A$ on $X$ with
> $h^0(A) \geq 2$ one has $\text{deg}(A)\geq h^0(A)+2$?
>
>
>
... | https://mathoverflow.net/users/133832 | On degree and section of a line bundle on a smooth plane quintic | This is true, and can be shown by an induction argument on $h^0(A)$.
If $h^0(A)=2$, then $\deg(A)\geq 4$ since the gonality of $X$ is $4$.
If $h^0(A)>2$, let $p\in X$ be a point in the support of an effective divisor representing $A$. Since $\mathrm{H}^0(X,A(-p))\subset \mathrm{H}^0(X,A)$ is a subspace of codimension $... | 2 | https://mathoverflow.net/users/110362 | 384148 | 159,765 |
https://mathoverflow.net/questions/384124 | 4 | It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have a sequence of measures admitting moment-generating functions and I wish to prove something like convergence of the mome... | https://mathoverflow.net/users/18936 | Bochner-Minlos for moment-generating functions? | While waiting for more context around the question, one can already mention the main definitions and tools for this topic.
I will assume the space on which these probability measures live is the space $\mathscr{S}'(\mathbb{R}^d,\mathbb{R})$ of temperate real-valued Schwartz distributions on $\mathbb{R}^d$.
It is the ... | 2 | https://mathoverflow.net/users/7410 | 384168 | 159,772 |
https://mathoverflow.net/questions/384035 | 1 | In Lang's Algebraic Number Theory book he uses a certain bound on the discriminant of the Galois closure of a number field $K$ without proof stating that it is an easy exercise. Let $\tilde{K}$ be the Galois closure of $K$. Set $[K:\mathbb{Q}] = N$ and $[\tilde{K}:\mathbb{Q}] = \tilde{N}$. Let us also denote the absolu... | https://mathoverflow.net/users/167999 | Upper bound for discriminant of Galois closure | One approach is to prove the corresponding fact on the Artin conductor side. Let $G$ be the Galois group ok $\tilde{K}/\mathbb Q$ and $H$ the Galois group of $\tilde{K}/K$, so that $|G|= \tilde{N}$ and $|H|= \tilde{N}/N$.
>
> At a prime $p$, the Artin conductor of the regular representation of $G$ is at most $|G|/2... | 4 | https://mathoverflow.net/users/18060 | 384173 | 159,773 |
https://mathoverflow.net/questions/384172 | 1 | Let me be more specific.
Let $T>0$ be a finite real number.
We know that if $f\in C([0,T],L^p(\mathbb{R}^N))$ is complex valued then the statement $\|1\_{\{|f|>M\}}|f|\|\_{L^\infty([0,T], L^p(\mathbb{R}^N))}\rightarrow 0$ as $M\rightarrow \infty$ is true by dominated convergence theorem with the dominant function $... | https://mathoverflow.net/users/127918 | $\|1_{\{f>M\}}f\|_{L^\infty_t L^p}\rightarrow 0$ as $M\rightarrow \infty$ for $f\in L^\infty([0,T],L^p)$? | Fix $\chi$ an $L^p$ function.
Let $\chi\_s(x) = s^{N/p} \chi(sx)$ for $s > 0$.
Define $f$ by considering its values on the dyadic interval $t\in [2^{k}, 2^{k+1})$. For
$$ f|\_{t\in [2^k, 2^{k+1})} = \chi\_{2^{-k}}(x) $$
Then you have $$\|f\|\_{L^\infty([0,T] ,L^p(\mathbb{R}^N))} = \|\chi\|\_{L^p} = \|1\_{\{|f|>M\}}... | 2 | https://mathoverflow.net/users/3948 | 384178 | 159,774 |
https://mathoverflow.net/questions/384170 | 9 | I was reading a paper from 1994 which claimed that the following statement was a conjecture of Hirzerbruch:
If a complex surface X is homeomorphic to either $S^2 \times S^2$ or $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ then it is biholomorphic to one of the Hirzerbruch surfaces.
>
> I would like to know if this... | https://mathoverflow.net/users/92483 | Status of a conjecture of Hirzebruch | Suppose $X$ is diffeomorphic to $S^2\times S^2$ or $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. Then $X$ is biholomorphic to a Hirzebruch surface.
Note that $b\_1(X) = 0$, so $X$ admits a Kähler metric. It follows from Seiberg-Witten theory that any compact Kähler surface with non-negative Kodaira dimension does not ad... | 10 | https://mathoverflow.net/users/21564 | 384180 | 159,775 |
https://mathoverflow.net/questions/384109 | 20 | ### Motivation:
While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number Theorem occurred to me. I thought I would share it here as I wonder whether other mathematicians have pursued thi... | https://mathoverflow.net/users/56328 | information-theoretic derivation of the prime number theorem | You may be interested in this arxiv paper [1], "Some information-theoretic computations related to the distribution of prime numbers", Ioannis Kontoyiannis, 2007.
It discusses Chebyshev's 1852 result,
$$ \sum\_{p \leq n} \frac{\log p}{p} \sim \log n , $$
which is related to the prime number theorem and used in pr... | 20 | https://mathoverflow.net/users/29697 | 384183 | 159,777 |
https://mathoverflow.net/questions/384185 | 0 | **Setting**
Although this detail is not relevant to my question, let me set the problem that my question arise.
We are considering an initial value problem
\begin{align\*}
\begin{cases}
u\in L^\infty(I,H^{1}\_0)\cap W^{1,\infty}(I,H^{-1})\\
iu\_t+\Delta u+\lambda |u|^\alpha u =0 \\
u(0)=\varphi
\end{cases}
\end{ali... | https://mathoverflow.net/users/127918 | An inequality for uniqueness proof of NLS | I assume that $u\_1,u\_2$ are complex and $\alpha\geqslant 0$ is fixed. Denote $u\_1=a$, $u\_2=a+b$, $f(z)=|z|^\alpha z$, then $$f(a+b)-f(a)=\int\_0^1 \frac{d}{dt}f(a+bt)dt\\=b\int\_0^1 |a+bt|^\alpha+\alpha (a+bt)|a+bt|^{\alpha-1}\frac{d}{dt}|ab^{-1}+t|dt.$$
Since $a+bt=:c$ is a point on the segment $[a,b]$ we get $|c|... | 1 | https://mathoverflow.net/users/4312 | 384186 | 159,778 |
https://mathoverflow.net/questions/384184 | 2 | Drutu, Sapir, Osin showed that
a finitely generated group $G$ is strongly hyperbolic relative to a finite collection $\mathcal{H}$ of subgroups if and only if any asymptotic cone is tree-graded with respect to the collection of pieces given by ultralimits of elements in $\mathcal{L}\mathcal{H}$, where $\mathcal{L}\math... | https://mathoverflow.net/users/173504 | Weakly relatively hyperbolicity and asymptotic cone | A finitely generated group is always weakly hyperbolic relative to itself. But here the hyperbolic space you get is just a point... A more interesting example: a free abelian group $\mathbb{Z}^n$ is weakly hyperbolic relative to $\mathbb{Z}^{n-1}$. But here the hyperbolic space you get is a line, so it still may be con... | 3 | https://mathoverflow.net/users/122026 | 384190 | 159,779 |
https://mathoverflow.net/questions/384193 | 3 | What is the expression of the (non $u \equiv 0$) solutions to
\begin{align\*}
(-\Delta)^s u &= 0 && x \in B\_r(0) \\
u&=0 && x \in \mathbb R^N \setminus B\_r(0),
\end{align\*}
where $$
(-\Delta)^s u(x) = \int\_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy,
$$ ($0<s<1$)
is the fractional Laplacian?
| https://mathoverflow.net/users/173196 | Solution of the fractional Laplace equation on a ball | *Martin kernel* and *Martin representation* is what you are after. Positive solutions are:
$$ u(x) = \int\_{\partial B\_r} \frac{(1 - |x|^2)^s}{|x - y|^N} \, \mu(dy) $$
for any positive measure $\mu$. Signed solution can also be of that form with signed $\mu$, but there are other (more singular) solutions, too.
In ot... | 6 | https://mathoverflow.net/users/108637 | 384195 | 159,781 |
https://mathoverflow.net/questions/384176 | 1 | Consider the following integral:
\begin{equation}
\int \mathrm{d}\rho \frac{1}{\rho} e^{N f(\rho)}
\end{equation}
Where:
\begin{equation}
f(\rho)=\ln \rho-\frac{1}{2} \rho^{2}+\frac{1}{2 p w^{2}} \rho^{2 p}\implies f^{\prime}(\rho)=\frac{1-\rho^{2}+\frac{1}{w^{2}} \rho^{2 p}}{\rho}
\end{equation}
To compute this in... | https://mathoverflow.net/users/142153 | Critical point of saddle point equation | You seek a solution $\rho$ of the equation $f'(\rho)=0$, hence
$$\rho^2=1+w^{-2}\rho^{2p}.$$
The solution should remain $>0$ when $w\rightarrow\infty$.
The OP says the solution should vanish as $1/w$, but that is mistaken, I think.
To gain some insight, take $p=2$, then the solution is
$$\rho=\frac{\sqrt{w^2-w \sq... | 1 | https://mathoverflow.net/users/11260 | 384206 | 159,783 |
https://mathoverflow.net/questions/362609 | 3 | While working on a variational problem I have reached to the following question:
Let $f:(0,\infty) \to [0,\infty)$ be a $C^1$ function satisfying $f(1)=0$. Suppose that $f(x)$ is strictly increasing on $[1,\infty)$ and strictly decreasing on $(0,1]$. Define $F:(0,1) \to [0,\infty)$ by
$$
F(s)=\min\_{xy=s,x,y\in(0,\in... | https://mathoverflow.net/users/46290 | When is the optimum of an optimization problem affine in the constraint parameter? | Say that $F$ is affine on an interval $I$, with $F'\equiv\lambda$, a constant. Then for $x=x(s)$ and $y=y(s)$, one has $f'(x)=\lambda y$ and $f'(y)=\lambda x$, thus
$$xf'(x)\quad (=\lambda xy)\quad=yf'(y).$$
This implies a functional equation
$$xf'(x)=\frac1\lambda f'(x)f'(\frac1\lambda f'(x)).$$
Simplifying, one finds... | 2 | https://mathoverflow.net/users/8799 | 384209 | 159,785 |
https://mathoverflow.net/questions/383356 | 1 | I am currently very confused about the real side of the Ward correspondence. Recall that the Ward correspondence gives a one-to-one correspondence between:
$M$-trivial holomorphic bundles $E$ on $Z$, and,
holomorphic bundles $\hat{E}$ on $M$ with holomorphic connection,
where one has an analytic family $Z\overset... | https://mathoverflow.net/users/109193 | Real part of the Ward correspondence | The isomorphism does not spell trouble, my argument is just flawed. I forgot that one only obtains a holomorphic connection on $M\setminus A$ where $A$ is some hypersurface. This is because an instanton bundle on $\mathbb{CP}^3$ may fail to $\text{Gr}\_2\left(\mathbb{C}^4\right)$-trivial on this hypersurface $A$. Then ... | 0 | https://mathoverflow.net/users/109193 | 384214 | 159,786 |
https://mathoverflow.net/questions/384218 | 23 | This is a question about a naming convention. The Barr-Beck theorem (or simply Barr-Beck) is used a lot in descent theory over the past 30 years, almost invariably without a reference, like folklore.
To make precise which theorem I am talking about: According to one source *the Barr-Beck monadicity theorem gives nece... | https://mathoverflow.net/users/89948 | What is Barr-Beck? | It is well-attested in the category theory literature (e.g. in Mac Lane's *Categories for the Working Mathematician*, Chapter VI) that the well-known theorem giving necessary and sufficient conditions for monadicity of a functor is due to Jon Beck. Indeed, most category theorists I know call this "Beck's monadicity the... | 25 | https://mathoverflow.net/users/57405 | 384225 | 159,788 |
https://mathoverflow.net/questions/384224 | 4 | Let $A = k[x\_1 , \dots , x\_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x\_1 , \dots , x\_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset of variables $(x\_{i\_1} , \dots , x\_{i\_\ell})$ also has a linear minimal free resolution?
The answer seems to be... | https://mathoverflow.net/users/73780 | In a commutative Koszul algebra, does every ideal generated by a subset of variables have linear resolution? | The ring $R= K[a,b,c,d]/(ac,ad,ab-bd,a^2+bc,b^2)$ is Koszul but the ideal $I=(b)$ is not Koszul as $bc^2=0$, and indeed $c^2$ appears in the presentation matrix of $I$.
| 7 | https://mathoverflow.net/users/2083 | 384227 | 159,789 |
https://mathoverflow.net/questions/384156 | 4 | Let $G\_0(x)=G(x,0)$ be the Green's function of the simple symmetric random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G\_0$ must always vary locally, i.e., whether
$$
\sum\_{\substack{y\in\mathbb Z^d:\\ |y-x|=1}} |G\_0(x)-G\_0(y)| >0
$$
holds for all $x\in\mathbb Z^d$.
The claim seems intuitive i... | https://mathoverflow.net/users/174306 | Does the Green's function of the simple random walk on $\mathbb Z^d$ always vary locally? | We will use the elementary fact that for $m \ge k \ge m/2$, the binomial coefficients satisfy
$${m\choose k+1} <{m \choose k}. \quad (\#)$$
The case $x=0$ is obvious so we may assume $x$ has some nonzero coordinate.
By symmetry, we may assume that $x\_1>0$. Then it suffices to show that for every $y \in \mathbb Z^d$ ... | 3 | https://mathoverflow.net/users/7691 | 384232 | 159,791 |
https://mathoverflow.net/questions/384137 | 5 | In Ireland and Rosen's book on number theory they give a proof of the finiteness of the class group of a number field which they attribute to Hurwitz, but which is essentially due to Kronecker (as I learnt from a comment by Franz Lemmermeyer to [this answer](https://mathoverflow.net/a/19035/2381) by KConrad). The proof... | https://mathoverflow.net/users/2381 | The Kronecker--Hurwitz property for rings of integers in global function fields | Let us take $f$ to be the norm (i.e. view $R$ as a finite rank module over $\mathbb F\_q[t]$. Each element of $R$ acts by multiplication on this module, so its determinant lies in $\mathbb F\_q[t]$. Take $f$ to be $q$ to the degree.
Then it suffices to prove for $\gamma$ in the field of fractions of $R$ that there is... | 3 | https://mathoverflow.net/users/18060 | 384263 | 159,808 |
https://mathoverflow.net/questions/384261 | 6 | So as it says in the title, how can one explicitly calculate the comodule structures on $BP\_\*\mathbb{C}P^n$ and $BP\_\*\mathbb{C}P^{\infty}$ for a prime $p$?
For example, $\mathbb{C}P^2$ sits in a cofiber sequence of spectra
$$\Sigma^2\mathbb{S}\to \mathbb{C}P^2\to \Sigma^4\mathbb{S}
$$
giving rise to a SES of ... | https://mathoverflow.net/users/152458 | Explicit $BP_*BP$-comodule structure on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$ | A concise formula
$$
\mu(\beta) = \beta(c(t^F))
$$
for the $BP\_\* BP$-coaction $\mu$ on $BP\_\* CP^\infty$ is given in the "Note added in proof" on page 279 of
```
Ravenel, Douglas C.; Wilson, W. Stephen
The Hopf ring for complex cobordism.
J. Pure Appl. Algebra 9 (1976/77), no. 3, 241–280.
```
A more explicit fo... | 9 | https://mathoverflow.net/users/9684 | 384269 | 159,811 |
https://mathoverflow.net/questions/384270 | 4 | Suppose I have a trivial rank $k$ bundle $E$ over $\mathbb C^n$. Suppose that on $\mathbb C^n\setminus 0$ I have two algebraic sub-bundles $V\_1,V\_2\subset E$ of ranks $l$ and $k-l$ such that $V\_1\oplus V\_2=E$. Is it possible then to extend $V\_1$ and $V\_2$ as sub-bundles of $E$ to the whole $\mathbb C^n$? If not, ... | https://mathoverflow.net/users/13441 | Splitting a trivial bundle over punctured $\mathbb C^n$ | The decomposition $\mathscr E|\_U = V\_1 \oplus V\_2$ gives projectors $\pi\_i \colon \mathscr E|\_U \to \mathscr E|\_U$ such that $\operatorname{im}(\pi\_i) = V\_i$. By Hartog's lemma, the restriction $\Gamma(X,\mathscr End(\mathscr E)) \to \Gamma(U,\mathscr End(\mathscr E))$ is a ring isomorphism, so $\pi\_i$ extend ... | 5 | https://mathoverflow.net/users/82179 | 384281 | 159,817 |
https://mathoverflow.net/questions/380692 | 4 | This is a reference request. There is a large body of work, I'm familiar with, that describes the existence of bi-Hölder embeddings of finite metric spaces into Euclidean space (such as this snowflaking business in Assouad's theorem).
My question is: Given a finite metric space $(X,d\_X)$ are there known results outl... | https://mathoverflow.net/users/36886 | Bi-Hölder embeddings of finite metric spaces | The thing about Assouad's theorem is that the distortion of the embedding provided depends on the metric dimension of the space aka the doubling constant. If you take a Hamming cube which is a set $\{0,1\}^n$ considered with $L\_1$ metric and fix some $1/2 < a < 1$ then for any embedding $f:\{0,1\}^n \rightarrow L\_2$ ... | 2 | https://mathoverflow.net/users/32454 | 384295 | 159,820 |
https://mathoverflow.net/questions/384175 | 14 | Let $G$ be a finite group admitting an automorphism $\sigma$ of prime order $p$. Define the norm map $N:G\rightarrow G$ with respect to $\sigma$ by $N(g)= g\sigma(g)\sigma^2(g)\dotsb\sigma^{p-1}(g)$.
>
> **Question.** If we know all elements of $G$ have norm equal to $e$, what can be said about the structure of the... | https://mathoverflow.net/users/128502 | What is known about the structure of finite groups admitting an automorphism where all elements have "norm" one? | First a note about terminology. It seems in the literature if $\sigma$ is an automorphism of $G$ such that $\sigma^n = 1$ and $g\sigma(g)\sigma^2(g) \cdots \sigma^{n-1}(g) = 1$ for all $g \in G$, we call $\sigma$ a **splitting automorphism**.
After some searching (experts could probably give more detail), I found tha... | 8 | https://mathoverflow.net/users/38068 | 384299 | 159,821 |
https://mathoverflow.net/questions/384289 | 7 | Let $p$ be a prime number and $\mathbb{F}\_p$ be a finite field with $p$ elements. Let $\chi$ be a multiplicative character from $\mathbb{F}\_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}\_p^{\times}=\{x\in\mathbb{F}\_p: x\ne 0\}$.
Recently, I met the following sum:
$$\sum\_{x=0}^{p-1}\chi(a^x-1),$$
where $a>1$ is a... | https://mathoverflow.net/users/nan | Character sums concerning $a^x-1$ | The best result on the market is the one of [Yu](https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/97/3/82426/estimates-of-character-sums-with-exponential-function), who proves that your sum is less than $$p^{1/2} \left( \frac 2 \pi \log p + \frac 7 5 \right).$$ If $a$ is not a primitive... | 7 | https://mathoverflow.net/users/nan | 384310 | 159,823 |
https://mathoverflow.net/questions/384332 | -1 | I am working with the function $f(z) = \frac{z+1}{z-1}$, for a complex variable $z$. I understood that for $z$ in the unit disc, i.e $\lvert z\rvert \le 1$, $\mathrm{Re}(f(z)) \le 0$.
What if $z$ is in a disc, say $D = \left\{ z \in \mathbb C: \, \lvert z\rvert \le a \right\}$ for a given constant $a \in \mathbb R\_{... | https://mathoverflow.net/users/143783 | Image of a complex disc by this function? | This question might have been better suited to math.SE but since it's here I may as well post an outline of an answer. Your function $f$ is an example of a Moebius transformation, and these are known to take circles to circles provided that you allow straight lines as circles passing through "the point at infinity". So... | 3 | https://mathoverflow.net/users/763 | 384333 | 159,829 |
https://mathoverflow.net/questions/384217 | 3 | We fix a binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree $d = 2g+2$, and consider the hyperelliptic curve
$$C\_F: \displaystyle z^2 = F(x,y).$$
We say that a point $(x,y,z)$ is a *quadratic point* if there exists a quadratic extension $K/\mathbb{Q}$ over which $x,y,z$ are defined.
Then $... | https://mathoverflow.net/users/10898 | Density of quadratic points on a hyperelliptic curve | Yes, this is (provably!) dominant.
As Jackson Morrow pointed out, quadratic points on $C$ are the same as rational points on the symmetric square of $C$. The obvious points are the rational points on a $\mathbb P^1$. Contracting this $\mathbb P^1$, we get a surface in an abelian variety, hence by Falting's theorem (a... | 8 | https://mathoverflow.net/users/18060 | 384349 | 159,841 |
https://mathoverflow.net/questions/383783 | 7 | $\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller isomorphism by Fausk-Hu-May [FHM] in *isomorphisms between left and right adjoints* and by Balmer-Dell'Ambrogio-Sanders... | https://mathoverflow.net/users/144100 | Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al | $\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\tensor}{\otimes}$
Let me write down what I (think I) know so far.
**Answer to 1:** I think that this does not follow, i.e. that the additional assumptions that $f^\*$ has a left adjoint $f\_{(1)}$ (denoted $f... | 2 | https://mathoverflow.net/users/144100 | 384356 | 159,845 |
https://mathoverflow.net/questions/384346 | 24 | By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $\*$-homomorphisms between them (I don't want to restrict to separable predual as I think it would prevent the existence of finite products).
Of course, I don't really believe it is a t... | https://mathoverflow.net/users/22131 | Is the opposite category of commutative von Neumann algebras a topos? | The opposite category of commutative von Neumann algebras is not a topos
because categorical products with a fixed object do not always preserve small colimits.
See Theorem 6.4 in Andre Kornell's [Quantum Collections](https://arxiv.org/abs/1202.2994v2).
| 22 | https://mathoverflow.net/users/402 | 384357 | 159,846 |
https://mathoverflow.net/questions/384262 | 4 | Let $G$, $H$ be finite groups. Consider the group algebra $\mathbb{C}G$ acting on $L^2(G)$, making $\mathbb{C}G$ into a C\* algebra, and the resulting positive elements, say $P\_G\subset \mathbb{C}G$. Similarly we have $P\_H\subset \mathbb{C}H$. Now take any function $f:G\to H$, and this induces a linear map which we a... | https://mathoverflow.net/users/29625 | Positive maps on finite group algebras and group homomorphisms | I think the "modified conjecture" (that is, one gets a homomorphism or anti-homomorphism) can be proved by elementary calculations, in this setting. (Walter's theorem, which Yemon alludes to, uses only that we have an isometry, not that the map has the rather special form). However, my tolerance for such calculations i... | 2 | https://mathoverflow.net/users/406 | 384360 | 159,847 |
https://mathoverflow.net/questions/384369 | 2 | $\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the [special unitary group $\SU(5)$](https://en.wikipedia.org/wiki/Special_unitary_group) and the [unitary group $\U(16)$](https://en.wikipedia.org/wiki/Unitary_group).
Below I specify a *specfic* way to embed $\SU(5) \subse... | https://mathoverflow.net/users/27004 | The normalizer of SU(n) in U(m)? | $\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$It's $\U(5) \times \U(1) \times \U(1)$.
We have a natural map from the normalizer of $G$ to the outer automorphism group of $G$. The outer automorphism group of $G$ is $\mathbb Z/2$, generated by the inverse transpose / complex conjugation.
This map is trivial, ... | 4 | https://mathoverflow.net/users/18060 | 384375 | 159,852 |
https://mathoverflow.net/questions/384397 | 2 | Let $D\subseteq X$ be a dense subset of a separable metric space $X$. Let $P(D)$ and $P(X)$ respectively denote the probability measures on $D$ and on $X$ with their weak topologies. Then, if we view $P(D)$ as a subset of $P(X)$ via the "inclusion" $\iota:P(D)\rightarrow P(X)$ defined by:
$$
\iota(\mu)\mapsto \mu(\cdot... | https://mathoverflow.net/users/36886 | Probability measures on a dense subset | Yes. First the probability measures with finite support are dense in $P(X)$. Second, if $P = \sum\_{i=1}^n p\_i \delta\_{x\_i}$ with $x\_i \in X$ and $\sum\_{i=1}^n p\_i = 1$, let $(x\_{im})\_{m \in \mathbb{N}}$ be sequences in $D$ with $\lim\_{m \to \infty} x\_{im} = x\_i$. Then $\lim\_{m \to \infty} \sum\_{i=1}^n p\_... | 3 | https://mathoverflow.net/users/100904 | 384400 | 159,859 |
https://mathoverflow.net/questions/361698 | 2 | Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if
$$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$
My question is whether given a Lorentzian manifold $(M,g)$, it is always possible to find a metric $\hat{g}=cg$ such that the curvature of $(M,\hat{g})$ is non-positive.
... | https://mathoverflow.net/users/50438 | conformal changes to Lorentzian curvature | No, this is not possible if the Lorentzian manifold has a null geodesic admitting a pair of conjugate points.
The notion of conjugate points for null geodesics does not depend on the conformal factor.
By the Lorentzian version of the Cartan-Hadamard theorem (see e.g. the book by Beem et al. 1996 Global Lorentzian Geome... | 3 | https://mathoverflow.net/users/40549 | 384401 | 159,860 |
https://mathoverflow.net/questions/384358 | 11 | We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
>
> A. J. MacLeod, *A Simple Method for High-Rank Families of
> Elliptic Curves with Specified Torsion*, arXiv, Number Theory [math.NT] (2014), arXiv:[14... | https://mathoverflow.net/users/95511 | Z/8Z elliptic curve with a missing generator | By pts := PointsQI(fourcovers[1], 3\*10^12 : OnlyOne := true);
I found the fifth point with the first coordinate
```
184125172284095573254772251800166095132866268069994053071269775232967258156536458883390620097481326304558948736/5569478916890701609111892835084489778974169869180762045545087780442851900762058308974543... | 7 | https://mathoverflow.net/users/21337 | 384402 | 159,861 |
https://mathoverflow.net/questions/384355 | 7 | This question arose from [Amdeberhan's question](https://mathoverflow.net/q/384145/11260), the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$\sum \_{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\right)^2 \Gamma \left(n+\frac{s}{2}\right)}{\Gamma (n+1)^2 \Gamma (n+s)}=\frac{... | https://mathoverflow.net/users/11260 | Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?) | Let's rewrite the given problem
$$\sum \_{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\right)^2 \Gamma \left(n+\frac{s}{2}\right)}{\Gamma (n+1)^2 \Gamma (n+s)}=\frac{\pi ^2 2^{1-s} \Gamma \left(\frac{s}{2}\right)}{\left[\Gamma \left(\frac{3}{4}\right) \Gamma \left(\frac{s}{2} +\frac{1}{4}\right)\right]^2}.\tag1$$
Ch... | 10 | https://mathoverflow.net/users/66131 | 384409 | 159,866 |
https://mathoverflow.net/questions/384391 | 16 | One can study the standard semantics of classical propositional logic within classical logic set theory, so we can say that the semantics of classical logic is meta-theoretically "self-hosting". This property is probably a big part of why classical logic is so easy to accept as the default/implicit background/foundatio... | https://mathoverflow.net/users/117787 | Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"? | I would argue that intuitionistic logic is perfectly self-hosting: working in an intuitionistic set theory, one can define a sound semantics of intuitionistic logic relative to models built out of plain sets in the (intuitionistic) metatheory, without any need for Kripke-ness or anything complicated. Just as the interp... | 12 | https://mathoverflow.net/users/49 | 384422 | 159,872 |
https://mathoverflow.net/questions/384088 | 2 | It is known that the ratio of the probability of infinitesimal tubes around paths of Itō diffusion processes converges to the Onsager--Machlup (OM) functional. I wonder whether the ratio of the joint density of the diffusion along two paths, evaluated at a partition of the time interval, also converges to the to the On... | https://mathoverflow.net/users/24041 | Onsager--Machlup functional as the density across a mesh of discrete points | In preparing the material for asking the question, I understood it better
and ended out figuring out an answer it.
As [answering your own question](https://mathoverflow.net/help/self-answer)
is encouraged in MathOverflow, I'm posting it here. The proof relies on the fact
that, using Levy's modulus of continuity, when r... | 1 | https://mathoverflow.net/users/24041 | 384423 | 159,873 |
https://mathoverflow.net/questions/384432 | 3 | If $(Q,\leq)$ is any preordered set (that is, $\leq$ is a reflexive and transitive, but not necessarily anti-symmetric relation), then we say that $S\subseteq Q$ is
1. *unbounded* if for all $q\in Q$ there is $s\in S$ such that $s \not \leq q$, and
2. *dominating* if for all $q\in Q$ there is $d \in S$ such that $q\l... | https://mathoverflow.net/users/8628 | ${\frak b}$ and ${\frak d}$ in the Rudin-Keisler preordering | Since every ultrafilter has at most $\mathfrak{c}$ predecessors in the Rudin-Keisler ordering, it follows the the relevant dominating number is $2^{\mathfrak{c}}$, the cardinality of the entire collection.
On the other hand, we have:
a) every maximal chain in the RK order has cardinality exactly $\mathfrak{c}^+$ (s... | 7 | https://mathoverflow.net/users/18128 | 384435 | 159,876 |
https://mathoverflow.net/questions/384108 | 6 | In [this paper](https://arxiv.org/abs/math/9906097) by Katz and Tao, the following bounds were established.
Let $A,B$ be finite subsets of an abelian group, with $|A|,|B|\le N$. We fix some $G \subset A\times B$. We define $C = \{a+b:(a,b) \in G\},D=\{a+2b:(a,b)\in G\}, X = \{a-b:(a,b)\in G\}$.
1. If $|C|\le N$, th... | https://mathoverflow.net/users/130484 | Bounding size of partial difference sets given size of partial sumsets | Some improvements to the lower bounds appear in
*Lemm, Marius*, [**New counterexamples for sums-differences**](http://dx.doi.org/10.1090/S0002-9939-2015-12603-2), Proc. Am. Math. Soc. 143, No. 9, 3863-3868 (2015). [ZBL1358.42006](https://zbmath.org/?q=an:1358.42006).
In particular for 1, there are now examples with... | 5 | https://mathoverflow.net/users/766 | 384436 | 159,877 |
https://mathoverflow.net/questions/384292 | 49 | A semigroup $S$ is defined to be *squared* if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
>
> **Problem:** Is each squared finite group trivial?
>
>
>
**Remarks (corrected in an Edit).**
0. I learned this problem from my former Ph.D. student ... | https://mathoverflow.net/users/61536 | Is each squared finite group trivial? | I think, as implicitly suggested by Yemon Choi, it is possible to explain the proof of the answer of user49822 by making more use of idempotents. Suppose that the finite group $G$ is squared via the subset $A$. The element
$ e = \frac{1}{|G|}\sum\_{g \in G} g$ is a primitive idempotent of $\mathbb{C}G.$
Let $ f= \fra... | 27 | https://mathoverflow.net/users/14450 | 384437 | 159,878 |
https://mathoverflow.net/questions/384307 | 3 | Sect. 6.7 of the HoTT Book establishes in the context of CW complexes that
>
> "we can obtain an n-dimensional path as a continuous family of 1-dimensional paths parametrized by an (n − 1)-dimensional object"
>
>
>
I hope I'm correct in taking this to be also the basic principle that allows to construct $S^n$ ... | https://mathoverflow.net/users/167839 | Higher-dimensional paths as parametrizations of 1-dimensional paths | I'm not quite sure what you're asking for with the first question, but have you looked at some basic algebraic topology books that work with CW-complexes, e.g. [Hatcher](https://pi.math.cornell.edu/%7Ehatcher/AT/AT.pdf)? For instance, I believe Hatcher discusses the topological fact that the product of two CW-complexes... | 4 | https://mathoverflow.net/users/49 | 384440 | 159,879 |
https://mathoverflow.net/questions/384438 | 3 | Could someone please indicate me some reference that contains the proof of the following theorem?
Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$.
**Theorem:** Let $M\subset \mathbb{R}^N$ be a $k$-dimensional manifold of class $C^1$, $1\leq k\leq N$.
1. Let $\varphi$ be... | https://mathoverflow.net/users/143671 | Hausdorff dimension and surface measure | Section 3.3.4D in Evans and Gariepy "Measure Theory and Fine Properties of Functions". Chapter 3 of Federer's "Geometric Measure Theory" is also a canonical reference.
| 4 | https://mathoverflow.net/users/156492 | 384448 | 159,882 |
https://mathoverflow.net/questions/384455 | 0 | I came across the following problem:
>
> What are conditions such that the polynomial $x^2+1$ divides $p(y)+q(z)+ax+b=F(x,\,y,\,z)$,?
>
>
>
Here $p$ and $q$ are also polynomials and $a$, $b$ are real numbers. The main difficulty is that $F(x,\, y, \,z)$ has three variables, and the idea of using roots cannot a... | https://mathoverflow.net/users/74668 | What are conditions such that the polynomial $x^2+1$ divides $p(y)+q(z)+ax+b=F(x,\, y, \,z)$? | Suppose $(x^2+1)$ divides $F(x,y,z)$, that is, $F(x,y,z) = (x^2+1)G(x,y,z)$ for some polynomial $G(x,y,z)$.
Then, setting $x = \pm \sqrt{-1}$, we see that
$$
F( \pm i, y,z) = 0.
$$
In your particular case, we must have that
$$
p(y)+q(z)\pm ai + b =0,
$$
Taking the difference of these two equations, we see that $2ai=0$,... | 2 | https://mathoverflow.net/users/1056 | 384458 | 159,884 |
https://mathoverflow.net/questions/384464 | -2 | The closed-loop dynamics of a linear optimal controller are simple but have interesting properties. From a starting state $\mathbf{v}(0)$ the dynamic can be iterated to reach a final state $\mathbf{v}(N)$ as in
$$
\mathbf{v}(N) = (A - BC)^N\mathbf{v}(0).
$$
I would like to optimize the transition parameter matrix $... | https://mathoverflow.net/users/151410 | Gradient Descent for Markov Dynamics | Note that $A$ only appears in the combination $M=A-BC$, so the derivative with respect to $A$ equals the derivative with respect to $M$; The function $f(A)$ is given by
$$f(A)=||(A - BC)^Nv - w||\_2^2=(M^Nv)^T M^Nv+w^Tw-2w^TM^Nv.$$
For a simple case, let me first consider a scalar perturbation, $f(A+\epsilon I)=f(A)+\e... | 0 | https://mathoverflow.net/users/11260 | 384468 | 159,886 |
https://mathoverflow.net/questions/384469 | 3 | Let $k>1$ be a positive integer.
>
> **Question 1.** Does there exist infinitely many positive integers $n$ such that $$n \, | \, (1^n + 2^n + \cdots + k^n)?$$
>
>
>
And, more generally:
>
> **Question 2.** Does there exist infinitely many positive integers $n$ such that $$n \,| \, (\epsilon\_11^n + \epsil... | https://mathoverflow.net/users/70464 | Are there infinitey many $n$ dividing $\epsilon_11^n + \epsilon_22^n + \cdots + \epsilon_kk^n$? | Yes (for the first question). Assume at first that $k$ is even, and let $p$ be a prime divisor of $k+1$. Choose $n=p^s$. I claim that $a^n+(k+1-a)^n$ is divisible by $n$ for all $a=1,2,\ldots,k/2$. This is clear if $p$ divides $a$. If $a$ and $p$ are coprime, we get $$a^n+(k+1-a)^n=a^n-(a-k-1)^n$$
is divisible by $p^{s... | 7 | https://mathoverflow.net/users/4312 | 384473 | 159,888 |
https://mathoverflow.net/questions/384478 | 1 | An integer is called **utterly odd** if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is utterly odd iff $k$ is odd. The first few utterly odd positive numbers $1,5,7,9,13,17,21,23,25,29,31,\dots$ are liste... | https://mathoverflow.net/users/66131 | A binomial convolution of Catalan numbers vs "utterly odd numbers" | $C\_k$ is odd iff $k=2^s-1$, $s=0,1,\ldots$. $\binom{n}{2^s-1}$ is odd iff $s$ terminal binary digits of $n$ are 1's (by [Kummer's theorem](https://en.wikipedia.org/wiki/Kummer%27s_theorem)). Thus the result.
| 4 | https://mathoverflow.net/users/4312 | 384482 | 159,890 |
https://mathoverflow.net/questions/384447 | 2 | Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(\mathbb{R})$, and let $\Omega$ be a fundamental domain for the left-action of $G(\mathbb{Z})$ on $G(\mathbb{R})/K$.
(**Edit** -- here i... | https://mathoverflow.net/users/76440 | Stabilizer in $G(\mathbb{Z})$ of point in fundamental domain $G(\mathbb{Z}) \backslash G(\mathbb{R}) / K$ | Yes, the stabilizer is the intersection of the center of $G$ with $G(\mathbb Z)$.
The center in this setting is compact, hence lies in every maximal compact subgroup and stabilizes every point. So it suffices to show that only elements of the center stabilize points in the interior of $\Omega$. Let $g$ be a non-centr... | 4 | https://mathoverflow.net/users/18060 | 384483 | 159,891 |
https://mathoverflow.net/questions/384491 | 6 | Is there a simply-connected smooth closed 4-manifold with a characteristic class $x \in H\_2(X; \mathbb{Z})$ such that $x$ can not be represented by a disjoint union of tori in $X$?
I would not know how to prove this without the characteristic hypothesis either so any thoughts on that would also be appreciated.
| https://mathoverflow.net/users/99414 | Characteristic class that cannot be represented by disjoint tori | In $H\_2(CP^2)$, every class $nH$ where $H$ is a generator and n is odd is characteristic. However, if $n >3$, then such a class is not represented by a torus. It is not represented by a disjoint union of tori, either. For non-zero classes in $H\_2(CP^2)$ have non-zero intersection numbers. So if you had a disjoint uni... | 15 | https://mathoverflow.net/users/3460 | 384494 | 159,893 |
https://mathoverflow.net/questions/384502 | 2 | Consider $f\_n(t)=\sum\_{i=1}^{n}e^{ik\_{i}t}$ with all $k\_i$ some distinct integers for $t\in [-\pi,\pi)$. For $p>2$ I am interested in the maximum possible value of $$||f\_n||\_p,$$
where $f\_n$ runs through all possible such sums with $n$ terms. Of particular interest is the case when $p$ is an even integer and $n\... | https://mathoverflow.net/users/24494 | $L_p$ norms of $0-1$ exponential sums | For even integer exponents, say $p=2k$ and $p \geq2$, the quantity is just the $k$-order additive energy of the set $S \subset \mathbb{Z}$ of non-zero Fourier coefficients. It is easy to see that this is maximized by any arithmetic progression of the desired length (which coincides with the $n$-order Dirichlet kernel).... | 4 | https://mathoverflow.net/users/630 | 384504 | 159,895 |
https://mathoverflow.net/questions/384489 | 1 | Let $X$ be a Banach space with a centered Gaussian measure $\mu\_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|\_X$ and $\|\cdot \|\_E$. It is well known (see prop 3.30 here <https://arxiv.org/pdf/0907.4178.pdf>) that
$$\|u\|\_X\leq C\|u\|\_E$$
for all $u\in X$. There is no ... | https://mathoverflow.net/users/168590 | Local inverse bound of Cameron Martin and Banach norms | If $X$ is infinite dimensional, then the inclusion $E\subset X$ is strict. So, the inequality
$$\|u\|\_X\le C\|u\|\_E \tag1$$
makes sense (and holds) only for $u\in E$, where $C$ is a positive real constant, depending on the measure $\mu\_0$.
Anyhow, the answer to your question (corrected in view of the above remark)... | 2 | https://mathoverflow.net/users/36721 | 384509 | 159,898 |
https://mathoverflow.net/questions/384323 | 2 | This is an export of <https://math.stackexchange.com/questions/4016545/non-central-tensor-product-of-central-algebras> which despite a bounty has sadly attracted no answer.
I repeat the question here: for (unital associative) algebras over a field $K$, it is easy to show that $Z(A\otimes\_K B)=Z(A)\otimes\_K Z(B)$. I... | https://mathoverflow.net/users/68479 | Non-central tensor product of central algebras | I'm not sure what "central $R$-algebra" means, I have been taking it to mean that the natural map $R \to Z(A)$ is an isomorphism. If it just has to be surjective, the OP has already given a solution. I think the following is an answer to my interpretation where we have to have $R = Z(A)$.
---
Let $R$ be the commu... | 2 | https://mathoverflow.net/users/297 | 384513 | 159,900 |
https://mathoverflow.net/questions/384533 | 4 | I am trying to understand the following theorem from the book Geodesic flows :
>
> Given a metric $g$ on a simply connected manifold $X$, there exists a constant $C\_1>0$ such that given any pair of points $x,y\in X$ and any positive integer $i$, any element in $H\_i(\Omega(X,x,y))$ can be represented by a cycle wh... | https://mathoverflow.net/users/nan | Question about the proof of Gromov's theorem in geodesic flows | 1. Since the manifold is Riemannian, it is in particular smooth and has a $C^1$ triangulation: see Whitehead's "On $C^1$ Complexes" for example (the original triangulation theorem in this case seems to be due to Cairns). If a manifold can't be triangulated it has no smooth structure.
2. Recall the cell structure on a p... | 1 | https://mathoverflow.net/users/75344 | 384538 | 159,908 |
https://mathoverflow.net/questions/384535 | 4 | I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today (as an introduction and overview of their place in mathematics). So I've been entertaining myself with reading the earl... | https://mathoverflow.net/users/174567 | Finding Motzkin's original paper on copositive quadratic forms | [Hall and Newman (1963)](https://doi.org/10.1017/S0305004100036951) cite this work as
>
> Motzkin, T., Copositive quadratic forms. National Bureau of Standards Report 1818 (1952), pp. 11–12.
>
>
>
This cited part of the NBS report is available online at [this link on pages 259-260](https://babel.hathitrust.org... | 3 | https://mathoverflow.net/users/7076 | 384545 | 159,910 |
https://mathoverflow.net/questions/384531 | 3 | Let $\Delta\_R:D(\Delta\_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta\_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,
where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)... | https://mathoverflow.net/users/61629 | Neumann/Robin Laplacian semigroup well-known estimate | The estimate the OP is looking for is called an *ultracontractivity estimate*. A characterisation of semigroups that satisfy such an estimate can be found in the Theorem on page 65, Subsection 7.3.2, of the following survey article:
Wolfgang Arendt: *Semigroups and Evolution Equations: Functional Calculus, Regularity... | 5 | https://mathoverflow.net/users/102946 | 384547 | 159,911 |
https://mathoverflow.net/questions/384534 | 5 | [Moved here from MSE]
Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$.
>
> **Question.** Is it true that there exists a stratification $\tau$ of $X$ which is $G$-equivariant and such that $F$ is $\tau$-const... | https://mathoverflow.net/users/92131 | Are equivariant perverse sheaves constructible with respect to the orbit stratification? | Every complex of sheaves has a unique maximal open subset on which it is locally constant, because if it is locally constant on two open sets, it is locally constant on their union.
Let $U$ be then maximal open subset of $X$ where $F$ is locally constant. Then $U$ is $G$-invariant, because $gU$ is also the maximally ... | 9 | https://mathoverflow.net/users/18060 | 384551 | 159,913 |
https://mathoverflow.net/questions/384543 | 1 | Let $x\_1,\dots,x\_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w\_1,\dots,w\_n)\in\Delta$ to the finitely supported measure $\sum\_{k=1}^n w\_k\delta\_{x\_k}$. This map is clearly continuous with respect to the Wasserstein distance, but i... | https://mathoverflow.net/users/172598 | Modulus of continuity of parameterizing Wasserstein | $\newcommand\De\Delta\newcommand\de\delta$Yes, this map is Lipschitz. Indeed, the map is
\begin{equation\*}
\De\ni w=(w\_1,\dots,w\_n)\mapsto\mu\_w:=\sum\_{k=1}^n w\_k\de\_{x\_k}. \tag{1}
\end{equation\*}
Let $d$ denote the metric on $X$, and then let
\begin{equation\*}
D:=\max\_{i,j\in[n]}d(x\_i,x\_j),
\end{equation... | 1 | https://mathoverflow.net/users/36721 | 384554 | 159,914 |
https://mathoverflow.net/questions/384540 | 4 | It is known (cf. [Wikipedia, Noncentral\_chi\_distribution](https://en.wikipedia.org/wiki/Noncentral_chi_distribution)) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).
There are many proofs but all are basically the same (see ... | https://mathoverflow.net/users/147603 | Intuition behind the noncentral chi square as Poisson mixing | [Patnaik](https://academic.oup.com/biomet/article-abstract/36/1-2/202/200823) writes on p. 203:
>
> The [non-central $\chi^2$ distribution] has been [sic] obtained by [Fisher, R. A. (1928). Proc. Roy. Soc. A, **121**, 654]
> as a particular case of the distribution of the multiple correlation
> coefficient. A purel... | 1 | https://mathoverflow.net/users/36721 | 384555 | 159,915 |
https://mathoverflow.net/questions/384525 | 8 | Are there any good approximations (especially upper bounds) for the quantity $E(\|X\_1-X\_2\|$), where each $X\_i$ is uniformly distributed in a rectangle $[a\_i,b\_i]\times[c\_i,d\_i]$? It does not appear that I can do this analytically, but I am in a situation where I need to compute hundreds of thousands of these. O... | https://mathoverflow.net/users/70190 | Expected distance between two uniform points in distinct rectangles | I tried to implement my proposal in a C-code. That is a mixture of analytic and numeric integration. It does $10^6$ rectangles with half-percent relative precision in about 16 seconds, which is a bit better than the corresponding Iosif's 30 minutes. You can play with parameters to trade speed for precision and vice ver... | 4 | https://mathoverflow.net/users/1131 | 384568 | 159,919 |
https://mathoverflow.net/questions/384552 | 5 | In Chapter 2 of Lurie's Higher Topos Theory, the first main theorem establishes a connection between categories cofibered in groupoids and left fibrations and asserts the importance of studying left fibrations in $\infty$-category theory. However, I lack understanding of the importance of categories cofibered in groupo... | https://mathoverflow.net/users/164702 | Examples of categories cofibered in groupoids | Before getting into high-falutin' stacky considerations, I think there's something much more basic to say.
Let $C$ be a 1-category. There is an equivalence between [discrete fibrations](https://ncatlab.org/nlab/show/discrete+fibration) over $C$ and functors $C^{op} \to Set$, i.e. *presheaves*. Here, "discrete fibrati... | 8 | https://mathoverflow.net/users/2362 | 384577 | 159,922 |
https://mathoverflow.net/questions/384569 | 4 | I asked this question on [Stack Exchange](https://math.stackexchange.com/questions/4032940/projective-modules-restricted-to-smooth-curves), but no one answered this.
I want to prove a coherent sheaf $M$ on $X$ is locally free if and only if this is true for $M|\_{X'}$ ,
for all smooth curves $X'$ mapping to $X$. I th... | https://mathoverflow.net/users/111070 | Projective modules restricted to smooth curves | Assume that $X$ is integral and smooth (as in Gaitsgory's notes). For any two points $x,y$ in $X$, there is a smooth connected curve $C$ passing through $x$ and $y$. Since $M\_{|C}$ is locally free, this implies $\dim M\_x/\mathfrak{m}\_xM\_x=\dim M\_y/\mathfrak{m}\_yM\_y$, where $\mathfrak{m}\_x$ is the maximal ideal ... | 8 | https://mathoverflow.net/users/40297 | 384580 | 159,924 |
https://mathoverflow.net/questions/384597 | 3 | I am not sure about the term "strictly" subharmonic.
What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits\_{|x|\rightarrow\infty}\psi(x)=0$.
I tried several times but still failed at the origin. I took $\psi=\exp\left(-\frac{1}{|x|}\right)-1$ with
$$\Delta\psi=\frac{1}{|... | https://mathoverflow.net/users/174600 | Find strictly subharmonic function that vanishes at infinity | For an explicit example on $\mathbb{R}^n$ with $n > 2$:
Let $\phi(x) = -(\sqrt{1 + |x|^2})^{2-n}$, then
$$ \nabla \phi(x) = (n-2) x(\sqrt{1 + |x|^2})^{-n} $$
and
$$ \triangle \phi = (n-2)n (\sqrt{1+|x|^2})^{-n} - (n-2)n |x|^2 (\sqrt{1+|x|^2})^{-n-2} = (2-n)n (\sqrt{1 + |x|^2})^{-n -2} > 0$$
---
Here's a s... | 5 | https://mathoverflow.net/users/3948 | 384601 | 159,925 |
https://mathoverflow.net/questions/384607 | 2 | The following question was motivated by [this MO-post](https://mathoverflow.net/q/384488/61536).
I hope that the answer should be known to experts (because of very simple formulation)...
>
> **Problem.** Let $n\ge 2$. Is the set of complex numbers $\{e^{i\pi k/2^n}:0\le k<2^n\}$ linearly independent over the fiel... | https://mathoverflow.net/users/61536 | Are half of the $2^n$-th roots of the unit rationally independent? | Denote by $\omega$ the order $2^{n+1}$-th primitve root of unity $\omega=e^{i\pi/2^n}$. The linear dependence of the above set would imply that there is a polynomial of degree at most $2^n-1$ with $\mathbb{Q}$ coefficients which vanishes on $\omega$. But its minimal polynomial is the cyclotomic polynomial $\Phi\_{2^{n+... | 9 | https://mathoverflow.net/users/41010 | 384609 | 159,926 |
https://mathoverflow.net/questions/384600 | 3 | My question is about Corollary 6.1(ii) in [Lectures on Condensed Mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) by Scholze (page 41). Here is the claim:
>
> The derived category $D(\mathrm{Solid})$ is compactly generated, and the full subcategory $D(\mathrm{Solid})^{\omega}$ of compact obje... | https://mathoverflow.net/users/54637 | Duality between $D^b(\mathbb{Z})$ and $D(\mathrm{Solid})^\omega$ | The notation is actually amibiguous which I guess causes the confusion. In fact this $\underline{RHom}(C,\mathbb{Z})$ stands for the internal hom from $C$ to $\mathbb{Z}$ in derived $\textit{condensed}$ abelian groups. Here $C$ and $\mathbb{Z}$ start in $D(\mathbb{Z})$ but we can view that as the ``discrete" full subca... | 3 | https://mathoverflow.net/users/3931 | 384613 | 159,928 |
https://mathoverflow.net/questions/384521 | 24 | I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a smooth map from $\mathrm{P}^2\_\mathbb{C}$ into $S^5$, which I thought was a topological embedding. Removing a point from $S... | https://mathoverflow.net/users/81645 | On a curious map from the complex projective plane into $S^5$ | I think I can prove the following
**Claim** There is no topological embedding of $\mathbb CP^2$ into $\mathbb R^6$.
The proof uses the van Kampen obstruction. Let me review the idea. Suppose there is a topological embedding $f\colon \mathbb CP^2\hookrightarrow\mathbb R^6$. Then $f$ induces a $\Sigma\_2$-equivariant... | 29 | https://mathoverflow.net/users/6668 | 384621 | 159,931 |
https://mathoverflow.net/questions/384586 | 2 | In arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I show that there is a 1:1 correspondence between $\mathbb{N} = \{0,1,2,3,4,\ldots\}$ and $\mathcal{D}\_{r\_{\text{min}}} = \{\epsilon,(),(()),()(()),((())), \ldots\}$. So what exactly is $\mathcal{D}\_{r\_{\text{min}}}$? It's the set of *Dy... | https://mathoverflow.net/users/174548 | What is the cardinality of the set of Dyck natural numbers of semilength $k$? | Let $N\_n$ be the number of words of semilength $n$ in
$$\{d \in \mathcal{D} \mid (\;{'})()){'} \text{ is not a substring of } d\;)\;\wedge\;(\;{'})(){'} \text{ is not a suffix of } d\;) \}.$$
Then $N\_0 = N\_1 = 1$, and similarly to Catalan numbers for $n\geq 2$, we have a recurrence:
\begin{split}
N\_n &= \sum\_{i=... | 3 | https://mathoverflow.net/users/7076 | 384634 | 159,938 |
https://mathoverflow.net/questions/384565 | 8 | In
* Erik Dofs, *Solutions of $x^3 + y^3 + z^3 = nxyz$*, Acta Arithmetica **73** (1995) pp. 201–213, doi:[10.4064/aa-73-3-201-213](https://doi.org/10.4064/aa-73-3-201-213), [EuDML](https://eudml.org/doc/206818)
the author has studied the Diophantine equation
\begin{equation}
x^3+y^3+z^3=nxyz\tag{1}
\end{equation}
w... | https://mathoverflow.net/users/122633 | Status of $x^3+y^3+z^3=6xyz$ | (Collecting comments into a community wiki answer.)
There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book [Rational Points on Elliptic Curves](https://www.math.brown.edu/johsilve/RPECHome.html). It is also implemented i... | 15 | https://mathoverflow.net/users/3106 | 384644 | 159,943 |
https://mathoverflow.net/questions/384561 | 2 | $\DeclareMathOperator\maj{maj}\DeclareMathOperator\inv{inv}$Major index, $\maj$, of a permutation on $1,2,\dotsc,n$ is defined as
$$
\maj(\pi) \mathrel{:=} \sum\_{i=1}^{n-1} i \cdot \chi(\pi(i)\gt \pi(i+1))
$$
where $\chi$ is 1 if the statement inside is true, 0 otherwise.
Let $t\_{a,b}$ be the numbers
$$
t\_{a,b} \... | https://mathoverflow.net/users/1056 | Number of permutations in $S_{a+b}$ with $\operatorname{maj}(\pi)=a$ and $\operatorname{maj}(\pi^{-1})=b$ | Here is a derivation of $(\ast)$ from the displayed equation
$$\sum \frac{z^n}{(1-q)^n [n]\_q! (1-t)^n [n]\_t!} \sum\_{\pi \in S\_n} q^{\mathrm{maj}(\pi)} t^{\mathrm{maj}(\pi^{-1})} = \prod\_{i,j \geq 0} \frac{1}{1-zt^i q^j}.$$
Taking the coefficient of $z^n$ on both sides, we have
$$\frac{\sum\_{\pi \in S\_n} q^{\ma... | 3 | https://mathoverflow.net/users/297 | 384657 | 159,948 |
https://mathoverflow.net/questions/384650 | 6 | I'm currently reading *Traces of Hecke Operators* by Knightly and Li, while simultaneously revisiting the adelic/representation-theoretic point of view on automorphic forms.
In Knightly and Li, they give a familiar definition of a representation. That is, for a locally compact group $G$ and a normed vector space $V$,... | https://mathoverflow.net/users/14508 | With a linear representation, how does the continuity of $G \to \mathrm{GL}(V)$ relate to that of $G \times V \to V$? | For locally compact groups continuity of $\pi$, joint continuity of the action map $G\times V\to V$ and separate continuity in both variables are equivalent. See Theorem 2.3 of Karl H. Hofmann, Sidney A. Morris
The Structure of Compact Groups (edition 3).
| 9 | https://mathoverflow.net/users/15934 | 384660 | 159,951 |
https://mathoverflow.net/questions/384653 | 1 | Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering map $\pi:X\rightarrow Y$.
Suppose that $G$ is a Lie group and that $\rho:\Gamma \rightarrow G$ is a group homomorphis... | https://mathoverflow.net/users/143492 | The notion of a "relatively" flat connection | Your bundle is of the form $E\_\rho$ if and only if it admits a flat connection whose holonomy $\rho:\pi\_1(Y) \to G$ is trivial in restriction to $\pi\_1(X)$.
Even without this additional assumption, I don't know a general answer to when a principal bundle admits a flat connection. There are necessary conditions (na... | 3 | https://mathoverflow.net/users/173096 | 384665 | 159,954 |
https://mathoverflow.net/questions/384404 | 2 | Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u\_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(\Lambda)$ be an integrable representation of $g$ with h... | https://mathoverflow.net/users/174472 | Action of the Casimir on highest weight modules for Kac-Moody algebra | You should be a bit careful, as this isn't precisely the action of the Casimir on $v \otimes v$, but instead follows from it.
For each positive root $\alpha$, let $e\_\alpha^{(1)}, \dots, e\_\alpha^{(n\_\alpha)}$ be a basis of the root space $\mathfrak{g}\_\alpha$, and let $\{f\_\alpha^{(i)}\}$ be the corresponding d... | 1 | https://mathoverflow.net/users/138296 | 384673 | 159,958 |
https://mathoverflow.net/questions/384140 | 6 | Let $C$ be a compact convex set in $\mathbb R^n$ that intersects the strictly positive orthant $\mathbb R\_+^n$. Does $C\cap \mathbb R\_+^n$ have to contain a point $x$ such that some vector $v\in\mathbb R\_+^n$ is normal to $C$ at $x$?
Here, a vector $v$ is normal to a convex set $C$ at $x$ iff for all $y\in C$, $\l... | https://mathoverflow.net/users/26809 | If a compact convex set meets the positive orthant does it meet it at a point with a normal in the positive orthant? | While fedja gave an answer in the comments, here is a different approach that yields a more general answer (the non-negative orthant is self-dual).
**Proposition:** Let $C\subseteq \mathbb R^n$ be a compact convex set that meets some closed convex cone $K$. Then $C\cap K$ contains a point with a non-zero normal in th... | 3 | https://mathoverflow.net/users/26809 | 384676 | 159,960 |
https://mathoverflow.net/questions/384636 | 2 | Let $W$ be a Coxeter group, and let $V$ be its geometric representation (as defined for instance in Section 5.3 of Humphreys' book *Reflection groups and Coxeter groups*). Let $v\in V\backslash\{0\}$ (in the case in which I am interested one may assume that $v$ is a $\mathbb{Q}$-linear combination of positive roots).
... | https://mathoverflow.net/users/26751 | Algorithm to determine if a vector in the geometric representation of a Coxeter group is proportional to a root | To avoid repetition, I'll just say $v$ "is a root" instead of $v$ "is proportional to a root", because the algorithm won't care about scaling much.
Here is how I would try to program this. (I have a "bad" habit of putting simple roots and fundamental weights in spaces dual to each other, contrary to the usual Lie the... | 2 | https://mathoverflow.net/users/5519 | 384684 | 159,964 |
https://mathoverflow.net/questions/384687 | 1 | I want to get some deep understanding on *closed* *orientable* Riemannian manifolds admitting $k$-forms ($k\geq 2$) $\omega$ that satisfices the following conditions:
$$\nabla \omega\neq 0,\quad \Delta\omega=0.$$
where $\Delta=(\delta +d)^2$ is the Laplace-Beltrami operator (Hodge Laplacian). What other useful properti... | https://mathoverflow.net/users/90655 | Reference for non-parallel harmonic $k$-forms | There are too many of these for them to have any particularly interesting structure. For example, consider any metric $g$ on the $3$-torus $\mathbb{T}^3$. By the Hodge theorem, the space of $g$-harmonic $2$-forms has dimension $3$. However, if, say, all of them were $g$-parallel, then their duals would be a basis of $g... | 5 | https://mathoverflow.net/users/13972 | 384698 | 159,967 |
https://mathoverflow.net/questions/384695 | 2 | What is an example of a Hopf algebra $(H,\Delta,\epsilon)$ containing an invertible element $h$ which is not grouplike: An element $h \in H$ such that
$$
\Delta(h) \neq h \otimes h\qquad\text{(not grouplike)}
$$
and such that there exists a $h^{-1} \in H$ with $hh^{-1} = h^{-1}h = 1$ (invertible).
| https://mathoverflow.net/users/153228 | Hopf algebra with a non-grouplike invertible element | Let $L$ be a finite-dimensional $p$-nilpotent restricted Lie algebra over a field of characteristic $p>0$ and consider its restricted enveloping algebra $u(L)$. Then the only group-like element of the Hopf algebra $u(L)$ is 1. On the other hand, every element of $u(L)$ that is not in the kernel of the counit is inverti... | 11 | https://mathoverflow.net/users/14653 | 384699 | 159,968 |
https://mathoverflow.net/questions/384615 | 7 | We know that in general, there is no smooth manifold structure on $Hom(X,\, Y)$ where $X$ and $Y$ are smooth manifolds, but under *certain nice conditions* (see <https://ncatlab.org/nlab/show/manifold+structure+of+mapping+spaces>) we can give a smooth structure on $Hom(X, \, Y)$.
Let $\mathcal{G}$ and $\mathcal{H}$ b... | https://mathoverflow.net/users/86313 | Is there any Lie groupoid structure on $Hom(\mathcal{G}, \mathcal{H})$ where $\mathcal{G}$ and $\mathcal{H}$ are Lie groupoids? | As Dmitri points out, given a cartesian closed category $S$, the groupoid of functors and natural transformations between fixed internal groupoids $X$ and $Y$ is again an internal groupoid: this result goes back to Charles Ehresmann, but it is not difficult to write down this construction directly.
However, since you... | 3 | https://mathoverflow.net/users/4177 | 384704 | 159,970 |
https://mathoverflow.net/questions/384619 | 3 | Watching (the begining of) a lecture on *free probability theory* by Dimitri Shlyakhtenko <https://www.youtube.com/watch?v=F8Urtr39jM0>, I'm led to consider the following question
>
> **Question.** How can one build classical probability theory (measurable functions on measure space, expectations, etc.) from the fr... | https://mathoverflow.net/users/78539 | Quick derivation of classical probability theory from von Neumann algebraic framework | At the beginning of his talk [What actually is free probability theory?](https://www.youtube.com/watch?v=IewAq-RNJFI) Tobias Mai explains how classical probability theory fits into the context of non-commutative probability theory.
| 4 | https://mathoverflow.net/users/112626 | 384711 | 159,973 |
https://mathoverflow.net/questions/384719 | 8 | In Sept. 2013, I investigated the determinant
$$D\_n=\det[\gcd(i-j,n)]\_{1\le i,j\le n}$$
and computed the values $D\_1,\ldots,D\_{100}$ (cf. <http://oeis.org/A228884>). To my surprise, they are all positive!
**Question.** Does $D\_n>0$ hold for all $n=1,2,3,\ldots$?
I believe that $D\_n$ is always positive. How to... | https://mathoverflow.net/users/124654 | On the determinant $\det[\gcd(i-j,n)]_{1\le i,j\le n}$ | Denote $f(k)={\rm gcd}(k,n)$. Clearly $f$ is an $n$-periodic function. $D\_n$ is the circulant $D\_n=\det(f(i-j):0\leqslant i,j\leqslant n-1)$ which equals $\prod\_{k=0}^{n-1}h(\omega^k)$ where $\omega=e^{2\pi i/n}$ and $h(t)=f(0)+f(1)t+\ldots+f(n-1)t^{n-1}$. We have
$$
h(t)=\sum\_{d|n} \varphi(d)(1+t^d+t^{2d}+\ldots+t... | 13 | https://mathoverflow.net/users/4312 | 384723 | 159,976 |
https://mathoverflow.net/questions/384728 | 2 | Notations :
$R$ is a commutative ring with unity. $P(R)$ is the category of finitely generated projective $R-$ modules, $Ch^{b}(P(R))$ is the the category of bounded chain complexes on $P(R)$ and $C^q(P(R))$ is the category of bounded exact chain-complexes on $P(R)$.
Each of the above mentioned categories are exact c... | https://mathoverflow.net/users/nan | Does the inclusion functor induce an injection in this case? | Yes, it's injective. Both $K\_0$ groups are free with infinite countable basis. The basis of the first one is formed by the chain complexes
$$\cdots\rightarrow0\rightarrow\mathbb{F}\rightarrow\mathbb{F}\rightarrow0\rightarrow\cdots$$
concentrated in two consecutive degrees (the middle arrow is the identity). The basis ... | 1 | https://mathoverflow.net/users/12166 | 384731 | 159,981 |
https://mathoverflow.net/questions/384702 | 7 | This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "freshman's intuition" for exponential works.
If $Q$ is some bounded linear operator on some Banach space(and maps to the same space),... | https://mathoverflow.net/users/174600 | Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$ | Convergence to $0$ is simple:
**Proposition 1.**
The following are equivalent:
(i)
The operator $e^{tQ}$ converges to $0$ with respect to the operator norm as $t \to \infty$.
(ii)
The spectrum of $Q$ is contained in the open left halfplane $\{\lambda \in \mathbb{C}: \, \operatorname{Re} \lambda < 0\}$.
*Sketch ... | 7 | https://mathoverflow.net/users/102946 | 384733 | 159,982 |
https://mathoverflow.net/questions/225835 | 5 | In a Jordan algebra elements $a$ and $b$ are said to **operator-commute**, whenever $a \circ (b \circ x) = b \circ (a \circ x)$ for every other element $x$. (That is: $T\_aT\_b = T\_bT\_a$, writing $T\_x(y) = x \circ y$.) In a JB-algebra elements $a$ and $b$ operator-commute if and only if they generate an associative ... | https://mathoverflow.net/users/66745 | Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative? | Yes, this is true. I couldn't find any proof of the statement you quote in the article, and even after emailing the authors I didn't get any wiser, so I decided to work out the details myself, see my paper [Commutativity in Jordan Operator Algebras](https://doi.org/10.1016/j.jpaa.2020.106407).
My main result is that if... | 2 | https://mathoverflow.net/users/174691 | 384735 | 159,983 |
https://mathoverflow.net/questions/384706 | 6 | Working in ZfC + [Wholeness](https://en.wikipedia.org/wiki/Wholeness_axiom):
Can we have a countable sequence of non-trivial elementary embeddings of the universe to itself, such that the range of each embedding is a subclass of the range of its successor, and such that the union of all the ranges of those embeddings... | https://mathoverflow.net/users/95347 | Can there exist such a sequence of elementary embeddings of the universe to itself? | It was pointed out by Monroe that if $\lambda$ is a limit and $j:V\_\lambda\to V\_\lambda$ is elementary and $j\_n$ is the $n$th iterate, i.e. $j\_0=j$ and $j\_{n+1}=j\_n(j\_n)$, and $\kappa\_n=\mathrm{crit}(j\_n)$, then since $\lim\_{n<\omega}\kappa\_n=\lambda$, we get every $x\in V\_\lambda$ is in $\mathrm{rng}(j\_n)... | 7 | https://mathoverflow.net/users/160347 | 384738 | 159,985 |
https://mathoverflow.net/questions/384749 | 1 | Clarification: Here $\mu$ being absolutely continuous means being absolutely continuous with respect to the Lebesgue measure $dx$: $\mu(A)=\int\_A fdx$ for some $f$ for all Lebesgue measurable $A$. Having bounded density means the density functions of these probability measures are uniformly bounded by a constant. Also... | https://mathoverflow.net/users/169471 | Is the set of probability measures on $\mathbb{R}$ absolutely continuous with bounded density a closed subset? | The answer is yes. Indeed, a probability measure $\mu$ over $\mathbb R$ has a density bounded by a real $K>0$ iff the cdf of $\mu$ is $K$-Lipschitz, that is, Lipschitz with the Lipschitz constant $K$.
So, you have a sequence $(\mu\_n)$ of probability measures over $\mathbb R$ with $K$-Lipschitz cdf's $F\_n$ convergin... | 4 | https://mathoverflow.net/users/36721 | 384751 | 159,987 |
https://mathoverflow.net/questions/384752 | 2 | Let $S\_\omega$ be the collection of bijections $f:\omega\to \omega$. Endow $\omega$ with the discrete topology and let $S\_\omega$ be endowed with the subspace topology of $\omega^\omega$, where $\omega^\omega$ carries the product topology.
**EDIT.** The following statement of mine from the original post is false:
... | https://mathoverflow.net/users/8628 | "Haar-like" measure on $S_\omega$ | Whenever $G$ is a non locally compact Polish group, there does not exist any nonzero $\sigma$-finite Borel measure $\mu$ on $G$ such that all left-translates of $\mu$ are absolutely continuous with respect to $\mu$. This is due to Weil; there is a nice proof in a 1946 paper of Oxtoby.
| 4 | https://mathoverflow.net/users/174731 | 384767 | 159,992 |
https://mathoverflow.net/questions/384771 | 0 | In [Iwaniec's paper](https://link.springer.com/content/pdf/10.1007/978-1-4612-0605-7_12.pdf) presenting the Gehring Lemma, the embedding used is $W^{1,p}\hookrightarrow L^2$ with $p=\frac{2d}{d+2}$.
>
> **Question.** What about dimension 2: can we actually go down to $p=1$?
>
>
>
| https://mathoverflow.net/users/40120 | Gehring Lemma in dimension 2 | The embedding is just the standard Sobolev embedding theorem. And yes, $p = 1$ is ok.
However, I am guessing you feel confused because many presentations of the embedding theorem use the Hardy-Littlewood-Sobolev lemma in the course of the proof, and the argument therefore fails for $p = 1$. Instead, you should follow... | 6 | https://mathoverflow.net/users/3948 | 384774 | 159,994 |
https://mathoverflow.net/questions/383978 | 9 | Let $ZC$ be [Zermelo set theory](https://en.wikipedia.org/wiki/Zermelo_set_theory) with choice, which differs from $ZFC$ in omitting the axiom scheme of replacement. **EDIT:** I think I want to include foundation in the axioms, which apparently isn't normally considered to be part of Zermelo set theory.
As is well-kn... | https://mathoverflow.net/users/2362 | Large cardinals without replacement | Overall, the large cardinal axiom hierarchy is very similar between ZC (ZFC minus replacement; we are including regularity) and ZFC. A large cardinal axiom (unprovable in ZFC) satisfied by $κ$ typically implies in ZC that $V\_κ$ satisfies ZFC + weaker large cardinal axioms. However, the axioms typically do not imply ad... | 5 | https://mathoverflow.net/users/113213 | 384779 | 159,996 |
https://mathoverflow.net/questions/384783 | 0 | Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y\_0$, let $h\_{\epsilon}(x) = \max\_{y^{'}: \| y^{'} - y\_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $h\_{\epsilon}(x)$ is continuous in $\epsilon$ on $(0 , \infty)$, that is, for any $\epsilon\_n \rightarrow \epsilon\_0 >... | https://mathoverflow.net/users/174738 | Is the pointwise supremum of a continuous function continuous? | Yes, the function is continuous. Let $R$ be a large parameter; then $f$ is uniformly continuous in $B(y\_0,R)$. Suppose $\epsilon<\epsilon'\leq R$. Then $h\_\epsilon(x)\leq h\_{\epsilon'(x)}$ (supremum over a larger set is no larger). Now let $y'$ be a point such that $h\_{\epsilon'}(x) = f(x,y')$. Then there is a poin... | 1 | https://mathoverflow.net/users/327 | 384788 | 159,998 |
https://mathoverflow.net/questions/384769 | 1 | So, I'm reading the classical paper
* Gidas, B., Ni, WM. & Nirenberg, L., *Symmetry and Related Properties via the Maximum Principle*, Commun. Math. Phys. **68** (1979) pp. 209–243, doi:[10.1007/BF01221125](https://doi.org/10.1007/BF01221125), [Project Euclid](https://projecteuclid.org/journals/communications-in-math... | https://mathoverflow.net/users/113406 | Some questions on the famous Gidas-Ni-Nirenberg 1979 paper | For your first question: you just need that given a solution $u$ there exists a function $c(x)$ such that $f(u(x)) - f(0) = c(x) u(x)$. This is possible because by the Mean Value Theorem applied to $f$ there exists a function $v(x)$ between $u(x)$ and $0$ such that $f(u(x)) - f(0) = f'(v(x)) u(x)$.
For your second qu... | 4 | https://mathoverflow.net/users/3948 | 384796 | 160,002 |
https://mathoverflow.net/questions/384807 | 10 | Let $X$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we set
$\textrm{sep}(A)=\inf\{\epsilon>0: A\subseteq K+\epsilon B\_{X}$ for some countable subset $K$ of $X\}$.
Clearly, $A$ is separable if and only if $\textrm{sep}(A)=0$... | https://mathoverflow.net/users/41619 | A quantity measuring the separability of Banach spaces | For the unit ball $B\_X$ of the Banach space there are only two possibilities:
sep$(B\_X)= 1$, if $B\_X$ is not separable, and sep$(B\_X)=0$ if $B\_X$ is separable. Indeed, if sep$(B\_X)<1$ there are $\varepsilon <1$ and a countable subset $K\subseteq B\_X$ with
$B\_X\subseteq K + \varepsilon B\_X$. But this can be i... | 20 | https://mathoverflow.net/users/21051 | 384810 | 160,008 |
https://mathoverflow.net/questions/384806 | 3 | Multiplication of natural numbers can be understood as iterated addition, and we can understand binary Cartesian products as set-indexed coproducts; for sets $X$ and $Y$,
$$X\times Y\cong\coprod\_XY\cong\coprod\_YX.$$
The internalization of indexed coproducts is dependent sums, so we need these to ask the question
... | https://mathoverflow.net/users/92164 | When aren't products iterated coproducts? | A discussion of the relation between dependent coproducts and cartesian products is found at
<https://ncatlab.org/nlab/show/dependent+sum>
| 3 | https://mathoverflow.net/users/39747 | 384811 | 160,009 |
https://mathoverflow.net/questions/384863 | 5 | Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A *fractional ideal* of $A$ is a finitely generated $A$-submodule of $K$. The set of all non-zero fractional ideals of $A$ is called $I... | https://mathoverflow.net/users/9317 | Computations of divisor class monoids | Here are a few remarks about $DC(A)$ (assuming $A$ is a complete Noetherian local domain of dimension $1$).
1. The equivalence relation in $D(A)$ is just isomorphism as $A$-modules. So you can view $DC(A)$ as the monoid of isomorphism classes of nonzero ideals $I$ in $A$ under multiplication.
2. For any $x\in DC(A)$,... | 5 | https://mathoverflow.net/users/2083 | 384874 | 160,030 |
https://mathoverflow.net/questions/384870 | 6 | $\DeclareMathOperator\Sym{Sym}$Recently, I have been thinking about the space $\Sym^2(S^n)$ of pairs of points in the sphere (including the repeated pairs $(p,p)$ for each $p\in S^n$) and possible ways to describe it. This is topologized as the quotient of $S^n\times S^n$ under the involution that switches components. ... | https://mathoverflow.net/users/147463 | The symmetric square of a sphere | That cofiber description in the old short paper of James and other famous folks tells you a lot.
The map you have turns out to be adjoint to the standard map $\mathbb RP^{n-1} \rightarrow \Omega^n S^n$.
Continuing the cofibration sequence one place to the right gives a cofibration sequence
$$ S^n \rightarrow SP^2(S... | 8 | https://mathoverflow.net/users/102519 | 384890 | 160,032 |
https://mathoverflow.net/questions/384099 | 10 | Does [condensed / pyknotic mathematics](https://ncatlab.org/nlab/show/condensed+set) afford an (yet!) another approach to orbifold theory?
Let me admit up-front that I don't know much about either condensed / pyknotic mathematics or about orbifold theory. But I think the following is a precise question which gets at ... | https://mathoverflow.net/users/2362 | Condensed / pyknotic approach to orbifolds? | Great question! I think the answer ought to be yes. But I must also make a disclaimer that I don't really know what orbifolds are, and the comments by David Roberts make be believe that what I thought they should be is not what they actually are.
To me, an orbifold should be to a manifold as a Deligne--Mumford stack ... | 14 | https://mathoverflow.net/users/6074 | 384898 | 160,035 |
https://mathoverflow.net/questions/384318 | 9 | Something that usefully emerged for me from this [discussion](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html) and follow-up [MO question](https://mathoverflow.net/q/356618/447) is that rather than see [cohesiveness](https://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos) and c... | https://mathoverflow.net/users/447 | Cohesion relative to a pyknotic/condensed base | Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of smooth manifolds say, based on which one defines differentials, and all sorts of de Rham cohomology etc.; I believe the "differential c... | 12 | https://mathoverflow.net/users/6074 | 384900 | 160,036 |
https://mathoverflow.net/questions/384829 | 3 | Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$, let $N(v) = \{w\in V:\{v,w\}\in E\}$ and let $\text{deg}(v) = |N(v)|$. Moreover, we set $L(v) = \{w\in N(v): \text{deg}(w) < \text{deg}(v)\}$, and we say that $v\in V$ is a *celebrity* if more than half of $v$'s neighbors have fewer neighbors than $v$ d... | https://mathoverflow.net/users/8628 | Celebrity vertices in graphs | The only bound is $r=1$. I will construct graphs below whose celebrity ratio is arbitrarily close to $1$.
Fix an arbitrarily large parameter $k$. The graph will have vertex set $V=V\_0\cup V\_1\cup\dots\cup V\_k$, where all vertices in $V\_i$ have degree $4k-2i+1$. Their neighbours will be distributed so that
* eac... | 4 | https://mathoverflow.net/users/12705 | 384902 | 160,037 |
https://mathoverflow.net/questions/384444 | 1 | This question is a simplified version of the one in the MO post [Superharmonic extension](https://mathoverflow.net/questions/383749/superharmonic-extension).
Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ if $m>2$. We know that this function is harmonic ... | https://mathoverflow.net/users/100746 | Superharmonic extension 2 | For $m = 2$ you can simply use Kelvin transform to exchange the roles of $y\_0$ and $\infty$, as pointed out by Alexandre Eremenko. Thus, we assume that $m \geqslant 3$.
Let $u$ be a superharmonic function in a neighbourhood of a compact set $K$ such that the complement of $K$ is connected. In your case $u$ is equal ... | 2 | https://mathoverflow.net/users/108637 | 384910 | 160,038 |
https://mathoverflow.net/questions/384896 | 0 | Let $\chi$ be a primitive quadratic Dirichlet character of d modulus $m$, and consider the product
$$\prod\_{\substack{p \text{ prime} \\ \chi(p) = 1}} (1-p^{-2})^{-1}.$$
What can we say about the value of this product? Do we have good upper or lower bounds?
**Some observations, ideas, and auxiliary questions**
*... | https://mathoverflow.net/users/145167 | Sum of inverse squares of numbers divisible only by primes in the kernel of a quadratic character | No need to use Chebotarev or Dirichlet: set $Z(k)=\prod\_{p\nmid m}(1-p^{-k})^{-1}$ and $L(k)=\prod\_p(1-\chi(p)p^{-k})^{-1}$. Then if $P(k)=\prod\_{\chi(p)=1}(1-p^{-k})^{-1}$ we have the recursion
$$P(k)=(Z(k)L(k)/Z(2k))^{1/2}P(2k)^{1/2}$$
from which you can deduce bounds that you want.
| 2 | https://mathoverflow.net/users/81776 | 384914 | 160,039 |
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