parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/437645 | 9 | I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear component of the free Lie algebra (Theorem 8.1 in Reutenauer's book on the subject), but my general confusion is more tha... | https://mathoverflow.net/users/497171 | Using Schur-Weyl duality | This answer is a response the the prompt
>
> Any help or general information about the relationship between Schur-Weyl duality and symmetric functions you could provide would be greatly appreciated.
>
>
>
If you have more questions (e.g. about the free Lie algebra), feel free to ask.
For convenience I will l... | 6 | https://mathoverflow.net/users/159272 | 437878 | 176,896 |
https://mathoverflow.net/questions/427315 | 0 | Let $N = p^k m^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$.
Descartes (1638), Frenicle (1657), and subsequently [[Sorli (2003) - Conjecture 3, Chapter 5 on page 89]](https://opus.lib.uts.edu.au/bitstream/10453/20034/2021/02Whole.pdf) conjectured that $... | https://mathoverflow.net/users/10365 | If $p^k m^2$ is an odd perfect number with special prime $p$, then under what other conditions on $\sigma(p^k)/2$ does $k=1$ follow? | This is a partial response, as it does not directly answer the original question that was asked. Additionally, what follows are actually some remarks that would be too long to fit in the *Comments* section.
---
This answer builds on the results in [this MSE question from January 03, 2023](https://math.stackexchan... | 0 | https://mathoverflow.net/users/10365 | 437890 | 176,901 |
https://mathoverflow.net/questions/437911 | 1 | The eigenvalues of a circulant matrix $C$ can be extracted as $$
\Lambda=F^{-1} C F
$$
where the $F$ matrix is a discrete Fourier transform matrix and $\Lambda$ is a diagonal matrix of eigenvalues.
Since $F^{-1}F$=$FF^{-1}=I$, is it possible to write $$
\Lambda=FC F^{-1}
?$$
For example, if our circulant matrix ... | https://mathoverflow.net/users/142414 | Extracting eigenvalues of a circulant matrix using discrete Fourier matrix | The discrete Fourier transform matrix $F$ is special because it squares to a permutation matrix,$^\ast$ $F^2=P$ with $P^2=I$. I insert $I=F^2 P=P F^{-2}$,
$$F^{-1}\Lambda F= F^{-1}(F^2 P) \Lambda (P F^{-2})F=F(P\Lambda P)F^{-1}.$$
So you see that you can either write $\Lambda=FCF^{-1}$ or $\Lambda'=F^{-1}CF$, the diffe... | 2 | https://mathoverflow.net/users/11260 | 437919 | 176,907 |
https://mathoverflow.net/questions/437913 | 0 | A *nice* density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
* (1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
* (2) $\int\_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
* (3) $\phi$ is even, i.e $\phi(-x) = \phi(x)$ for all $x \in \mathbb R$, and
* (4) $\phi$ is differentiable and $\|\phi'\|\_\... | https://mathoverflow.net/users/78539 | Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$ | Of course, not.
Indeed, suppose that $t\phi(\epsilon/t) \ge c \|\phi'\|\_\infty$ for all positive $t$.
Then, letting $s:=\epsilon/t$, we get $\phi(s)\ge Cs$ for $C:=c \|\phi'\|\_\infty/\epsilon>0$ and all real $s>0$. So, $\int\_{\mathbb R}\phi=\infty$ and hence $\phi$ cannot be a pdf.
| 4 | https://mathoverflow.net/users/36721 | 437920 | 176,908 |
https://mathoverflow.net/questions/437916 | 5 | We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.
I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the hyperbolic metric on $\mathbb{H}^n$ is a stationary solution (constant in time) of the flow ?
My second question,
L... | https://mathoverflow.net/users/497380 | Ricci flow negative curvature | The answer to your first question is Yes. The equation
$$
\tag{\*} \partial\_t g = -2\textrm{Ric} - 2(n-1)g
$$
is a Ricci flow type equation that admits hyperbolic space as a static solution. In fact, if you have solution to (\*) you can rescale time/space to obtain a Ricci flow and vice versa. This is proven in Lemma ... | 8 | https://mathoverflow.net/users/1540 | 437922 | 176,909 |
https://mathoverflow.net/questions/437874 | 6 | Call a set $X$ *hesive* if for every infinite computable set $C$, both $C \cap X$ and $C \setminus X$ are infinite.
It's not hard to see that every hyperimmune degree computes a hesive set, but this isn't a characterization, since also any random set is hesive (in fact, Church stochasticity suffices).
Does every no... | https://mathoverflow.net/users/32178 | Sets meeting and avoiding computable sets | $X$ is hesive iff $X$ is bi-immune.
Jockusch showed that a Sacks generic has bi-immune-free degree.
*Jockusch, C. G. Jr.*, [**The degrees of bi-immune sets**](http://dx.doi.org/10.1002/malq.19690150707), Z. Math. Logik Grundlagen Math. 15, 135-140 (1969). [ZBL0184.02002](https://zbmath.org/?q=an:0184.02002).
So n... | 2 | https://mathoverflow.net/users/4600 | 437941 | 176,914 |
https://mathoverflow.net/questions/437914 | 0 | I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at the good and legal sources.
If anyone knows a textbook that contemplates the Bowen formula, that'd be great. I found Io... | https://mathoverflow.net/users/88795 | Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism | There's Dimension Theory in Dynamical Systems by Yakov Pesin.
| 1 | https://mathoverflow.net/users/152429 | 437943 | 176,916 |
https://mathoverflow.net/questions/437945 | 5 | On 8:38 of [Session 9: Masterclass in Condensed Mathematics](https://www.youtube.com/watch?v=gZ4ES3vjAw4&t=3107s) an $\infty$-category is defined as a simplicial set $\mathcal{S}$ (i.e a functor $\Delta^{op}\rightarrow Sets$) such that for every horn $\wedge\_{i}^{n}\rightarrow \mathcal{S}$ with $1<i<n$ there is at lea... | https://mathoverflow.net/users/476832 | On the definition of infinity-category | *Partial monoids* (see Definition 2.2) play a useful role in
```
Segal, Graeme
Configuration-spaces and iterated loop-spaces.
Invent. Math. 21 (1973), 213–221.
```
| 2 | https://mathoverflow.net/users/9684 | 437946 | 176,918 |
https://mathoverflow.net/questions/436117 | 4 | I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}\_{m|m}(\mathbb{C})$.
The real forms of the *simple* complex Lie superalgebras were classified by Serganova in the paper *Classification of real simple Lie superalgebras and symmetric spaces*. For $\mathfrak{gl}\... | https://mathoverflow.net/users/29738 | Real forms of the general linear Lie superalgebra | It turns out that one *does* get all real forms of $\mathfrak{gl}\_{m|m}(\mathbb{C})$ in the way described in the question. To see this, one can show that a real form of $\mathfrak{gl}\_{m|m}(\mathbb{C})$ is uniquely determined by the induced real form of $\mathfrak{psl}\_{m|m}(\mathbb{C})$. This same method also works... | 0 | https://mathoverflow.net/users/29738 | 437953 | 176,919 |
https://mathoverflow.net/questions/437948 | 2 | $\DeclareMathOperator\Cov{Cov}$**Backround of my Question**
Let $Y$ be the response variable, $\mathbb{X}$ be the explanatory variables. The ultimate goal of prediction is finding a function $f^{\*}$ that minimize $\mathbb{E}[(Y - f^{\*}(\mathbb{X})^2)]$, we know that the solution is $f^{\*}(\mathbb{X}) = \mathbb{E}[... | https://mathoverflow.net/users/497398 | Justification of the use of residual plot | $\newcommand\ep\epsilon\newcommand{\de}{\delta}$Getting rid of the instances of $\approx$, one can state the question as follows:
>
> Let $f(X):=E(Y|X)$, where $X$ and $Y$ are random variables (r.v.'s) ($X$ possibly a multivariate one) such that $EY^2<\infty$. Note that $E\ep=0$ and $Cov(f(X),\ep)=0$, where $\ep:=Y... | 2 | https://mathoverflow.net/users/36721 | 437954 | 176,920 |
https://mathoverflow.net/questions/436932 | 6 | Which contractible spaces appear as retracts of the Hilbert cube or of $\Bbb R^\omega$ ?
One wants to think that a sufficiently “nice” contractible space is necessarily
a retract of the Hilbert cube or of $\Bbb R^\omega$ (if non-compact).
Is this view supported by theorems ?
Such a space would have to be both contrac... | https://mathoverflow.net/users/494312 | When is a contractible space a retract of the Hilbert cube or $\Bbb R^\omega$? | Recall that a space $X$ is called an *absolute (neighbourhood) extensor for a property(=class) $\mathcal{P}$ of spaces*, abbreviated $AE(\mathcal{P})$ (respectively, $ANE(\mathcal{P})$), if for every space $Y$ with property $\mathcal{P}$ and every closed subspace $A\subset Y$, every continuous function $f:A\to X$ can b... | 3 | https://mathoverflow.net/users/54788 | 437956 | 176,921 |
https://mathoverflow.net/questions/437957 | 5 | Within ZFC, the [von Neumann hierarchy](https://en.wikipedia.org/wiki/Von_Neumann_universe) consists of sets $V\_\alpha$ indexed by ordinals, subject to the following conditions:
* $V\_0=\varnothing$.
* $V\_{\alpha+1}=\mathcal P(V\_\alpha)$.
* $V\_\lambda=\bigcup\_{\beta<\lambda}V\_\beta$ for limit $\lambda$.
My qu... | https://mathoverflow.net/users/147705 | Replacement axiom and the von Neumann hierarchy | This is really just a long comment, but the phrasing on the wikipedia page for replacement is verbose and perhaps obscuring how to use it here. Consider this version of replacement:
>
> For any set $X$ and binary predicate $\phi(-,-)$ such that for each element $x\in X$ there exists a unique set $y\_x$ such that $\... | 6 | https://mathoverflow.net/users/92164 | 437962 | 176,923 |
https://mathoverflow.net/questions/436938 | 2 | Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some $p \le H(n)$ such that $r$ is a $p$-th power mod $n$. (As usual, $p$ denotes a prime number.)
Is $H\_0$ such a function, wh... | https://mathoverflow.net/users/6043 | Mod n, are all higher powers also lower powers? | So your question is as follows: let $n$ be a positive integer, $q$ the smallest prime such that $q\sharp > n$, $e>q$ be an integer and $r$ be an $e$-th power mod $n$. Then is $r$ a $p$-th power for some $p\leq q$?
In general, *no*. Take a large prime $q$ and $n$ be the greatest power of $2$ less than $q\sharp$: then ... | 1 | https://mathoverflow.net/users/165657 | 437966 | 176,925 |
https://mathoverflow.net/questions/437324 | 1 | Let $R$ be a commutative ring with unit. Let $M\_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I want to know if the following statement is true:
1. $M\_A = M\_B$ if and only if there exists an invertible matrix $... | https://mathoverflow.net/users/151406 | Condition for equality of modules generated by columns of matrices | Interpreting the various matrices as maps between free modules in the usual way, the question becomes:
If $M$ is a submodule of $R^m$, and $\alpha,\beta: R^k\to M$ are epimorphisms, then must $\alpha$ and $\beta$ be equivalent, in the sense that there is an automorphism $f:R^k\to R^k$ with $\alpha = \beta f$?
For $... | 2 | https://mathoverflow.net/users/22989 | 437972 | 176,927 |
https://mathoverflow.net/questions/437992 | 2 | The Hecke group of level two, $\Gamma\_{0}(2)$, is an index-$2$ subgroup of the Fricke group of level two, $\Gamma\_{0}^{+}(2)$, i.e. $\left[\Gamma\_{0}^{+}(2):\Gamma\_{0}(2)\right] = 2$. The index of $\Gamma\_{0}(2)$ in the modular group, $\text{SL}(2,\mathbb{Z})$ is $\left[\text{SL}(2,\mathbb{Z}):\Gamma\_{0}(2)\right... | https://mathoverflow.net/users/99716 | Fractional group index? | $\Gamma\_0^+(2)$ is not a subgroup of ${\rm SL}\_2(\mathbb{Z})$: the elements of $\Gamma\_0^+(2)$ not in $\Gamma\_0(2)$ are fractional linear transformations such as $z \mapsto -1/(2z)$ that are represented by integer matrices (such as $\left(0\;-1\atop2\;\phantom-0\right)$) of determinant $2$. [Likewise for $\Gamma\_0... | 3 | https://mathoverflow.net/users/14830 | 437996 | 176,937 |
https://mathoverflow.net/questions/437987 | 2 | I'm looking for a reference for the following:
Suppose that $f\_1,f\_2\colon S^n\rightarrow S^n$ are smooth maps. Let $i\colon S^n\rightarrow \mathbb{R}^{n+1}$ be the inclusion, and suppose that $F\colon S^n\times I \rightarrow \mathbb{R}^{n+1}$ is a smooth homotopy of $i\circ f\_1$ and $i\circ f\_2$. Moreover, suppo... | https://mathoverflow.net/users/497433 | Calculating degree via homotopy | I believe this follows from the fact that 'bordant' maps induce the same map on homology. So, for example: suppose $W$ is an oriented $(n+1)$-manifold with boundary and $W \to \mathbb{R}^{n+1}-\{0\}$ is a map. Then the composite $\partial W \to W \to \mathbb{R}^{n+1}-\{0\}$ is trivial on $H\_n$.
In your case you'd li... | 6 | https://mathoverflow.net/users/6936 | 437999 | 176,938 |
https://mathoverflow.net/questions/437993 | 1 | Suppose $I(X;Y)$ denotes mutual information and on the other hand there is a relationship as follows.
\begin{align}
|p(y)-p(y|x)|<\delta p(y),\qquad\forall x,y.
\end{align}
Then we can say about mutual information:
\begin{align}
I(X;Y)=\sum\_{x,y}p(x,y)\log\frac{p(y|x)}{p(y)}\leq\sum\_{x,y}p(x,y)\log\frac{(1+\delta)p(y... | https://mathoverflow.net/users/68835 | Does bounding mutual information restrict the defined meter? | $\newcommand\ep\varepsilon$Of course not.
E.g., suppose that $P(X=0,Y=0)=t\ep$, $P(X=0,Y=1)=1/2-t\ep$, $P(X=1,Y=0)=(1-t)\ep$, and $P(X=1,Y=1)=1/2-(1-t)\ep$, where $t$ and $\ep$ are in the interval $(0,1)$. Then
$$I(X;Y)\sim c\_t\ep\to0$$
as $\ep\downarrow0$, where $c\_t:=\ln2+t\ln t+(1-t)\ln(1-t)$.
However, here $P... | 1 | https://mathoverflow.net/users/36721 | 438002 | 176,939 |
https://mathoverflow.net/questions/429008 | 11 | Given a well-ordering of $\mathbb{R}$, there is a natural way to define an associated partition of pairs of real numbers into two pieces: one assigns the value $0$ to a pair $r<s$ if the well-ordering agrees with the standard ordering on the pair, and gives it the value $1$ if not.
This construction is due originally... | https://mathoverflow.net/users/18128 | Higher-dimensional Sierpiński partitions | We may assume $r\ge3$. Let $\prec$ be a well-ordering of $\mathbb R$. For $n\in\mathbb N$ let $S\_n$ denote the set of all permutations of the set $[n]=\{1,2,\dots,n\}$.
Consider a set $X=\{x\_1,\dots,x\_r\}\in\binom{\mathbb R}r$ with $x\_1\lt x\_2\lt\cdots\lt x\_r$ and let $d\_i=|x\_i-x\_{i+1}|$ for $1\le i\le r-1$.... | 3 | https://mathoverflow.net/users/43266 | 438013 | 176,940 |
https://mathoverflow.net/questions/438001 | 14 | Let $C\_p$ b the cyclic group of order $p$, with $p$ a prime.
Is is possible for $C\_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points?
Standard Smith Theory implies that for this to be possible, $H\_\*(X;\mathbb Z/p)$ would have to be infinite dimensional. (So, e... | https://mathoverflow.net/users/102519 | Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and have no fixed points? | Yes, I think you can make an example like this (for $p=2$, but it generalizes).
Let $R$ be the group ring $\mathbb Z[C\_2]=\mathbb Z[x]/(x^2-1)$. Make a chain complex of free $R$-modules
$$
M\_0 \leftarrow M\_1 \leftarrow M\_2
$$
as follows.
$M\_0=R$.
$M\_1$ has a basis $(a\_n)$ indexed by $n\ge 0$.
$M\_2$ has ... | 17 | https://mathoverflow.net/users/6666 | 438014 | 176,941 |
https://mathoverflow.net/questions/437607 | 1 | Let $X$ be completely regular space, $\beta X$ be Stone-Čech
compactification of $X$, and $\upsilon X$ be Hewitt realcompactification of $X$.
Then $X\subset \upsilon X\subset \beta X$.
If the remainder $\beta X\setminus X$ is countably compact space, then can $%
\beta X\setminus \upsilon X$ be countably compact?
Un... | https://mathoverflow.net/users/86099 | A question about realcompact spaces | There exists a Tychonoff space $X$ such that $\beta X\setminus X$ is countably compact but $\beta X\setminus \upsilon X$ is locally compact, $\sigma$-compact and not countably compact. Such a space can be constructed as follows.
In the Stone-$\check{\mathrm C}$ech remainder $\omega^\*=\beta\omega\setminus \omega$ cho... | 4 | https://mathoverflow.net/users/61536 | 438023 | 176,943 |
https://mathoverflow.net/questions/438011 | 11 | Nirenberg's paper [*On elliptic PDEs*](http://www.numdam.org/item/?id=ASNSP_1959_3_13_2_115_0) claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality
$$
\lVert∇ f \rVert\_{L^{2p}} ≤ C \left(\lVert f\rVert\_{L^\infty} \lVert ∇^2 f... | https://mathoverflow.net/users/123448 | Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis? | Here is a Littlewood-Paley + interpolation style proof. If $P\_k$ denotes a Littlewood-Paley projection to frequencies $|\xi| \sim 2^k$, then the standard Littlewood-Paley characterisations of Sobolev spaces give
$$ \| \nabla f \|\_{L^{2p}} \sim \| (\sum\_k 2^{2k} |P\_k f|^2)^{1/2} \|\_{L^{2p}}$$
and
$$ \| \nabla^2 f \... | 12 | https://mathoverflow.net/users/766 | 438037 | 176,948 |
https://mathoverflow.net/questions/438038 | 1 | Let $\mu$ be *a finite complex valued measure* on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int\_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
**Question:** Does $\hat{\mu}$ uniquely determine $\mu$? I am fairly sure that it does. However, I was not able to locate my standard r... | https://mathoverflow.net/users/69661 | Uniqueness of Fourier–Stieltjes transform for finite complex valued measures | I assume that $\mu$ is a regular complex Borel measure. Assume that $\widehat{\mu}=0$. Let $f \in \mathcal{S}(\mathbb{R})$ be a Schwartz class function. Then, writing $f=\widehat{g}$ for $g\in \mathcal{S}(\mathbb{R})$, we have from Fubini's theorem that
$$
\int\_{\mathbb{R}}f(t)d\mu(t) = \int\_{\mathbb{R}}\int\_{\mathb... | 4 | https://mathoverflow.net/users/78745 | 438039 | 176,949 |
https://mathoverflow.net/questions/438010 | 1 | suppose we have a $p$-stabilized newform $f$ of classical level $\Gamma\_0(p)$; then there is a unique ordinary deformation into a Hida family $\mathbf{f}$ defined over a finite flat extension $R/\Lambda$, and a corresponding representation $\rho: G\_{\mathbb{Q}}\to \text{GL}\_2(\text{Frac}(R))$ on a big vector space $... | https://mathoverflow.net/users/120548 | Reference for auto-duality of nearly ordinary deformations associated to Hida families | [Section 1.6 in Fukaya-Kato](https://www.math.ucla.edu/%7Esharifi/sharificonj.pdf) spells it out very nicely; the same material appeared earlier in section 4.1 of Ohta's *On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms*.
| 1 | https://mathoverflow.net/users/120548 | 438043 | 176,950 |
https://mathoverflow.net/questions/437909 | 5 | $\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading [this article by Boyd and Chua [1]](https://web.stanford.edu/%7Eboyd/papers/pdf/fading_volterra.pdf), in which they prove the approximability of arbitrary time-invariant (TI) operators (or filters) with polynomials of some basis filters ... | https://mathoverflow.net/users/123104 | Boyd & Chua 1985: Is the proof of Lemma 2 correct? | Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not mean that the functional $G$ constructed by the kernel $g\_0$ separates all the points $u, v\in K\_-$ such that $u\neq v$: *it means only that, for any such two functions, you can construct a kernel such that the associated ... | 6 | https://mathoverflow.net/users/113756 | 438052 | 176,954 |
https://mathoverflow.net/questions/435828 | 0 | I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone provide some useful references?
| https://mathoverflow.net/users/65795 | Tensor nuclear norm for a binary 3rd-order tensor | I apologize that I'm not more knowledgeable about tensor calculus, but I did find this on arXiv: [Friedland and Lim - Nuclear Norm of Higher-Order Tensors](http://arxiv.org/abs/1410.6072). I hope you find an answer to your question.
| 0 | https://mathoverflow.net/users/497473 | 438056 | 176,955 |
https://mathoverflow.net/questions/438053 | 0 | Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian submanifold $(M\_{\kappa},g)$ of the $k$-fold product of the hyperbolic plane $\prod\_{i=1}^k\, \mathbb{H}^2$ with sectional curvature... | https://mathoverflow.net/users/491352 | Hadamard submanifolds of $k$-fold product of hyperbolic plane | For the $i$th factor ${\mathbb H}^2$ in the product of hyperbolic planes, pick a complete geodesic $c\_i$, $i=1,...,k$. The product
$$
F=c\_1\times ... \times c\_k\subset X=\prod\_{i=1}^k {\mathbb H}^2
$$
is a $k$-flat, i.e. a totally-geodesic (although we do not need this) isometrically embedded Euclidean subspace of... | 4 | https://mathoverflow.net/users/39654 | 438058 | 176,956 |
https://mathoverflow.net/questions/438067 | 3 | A topological space $X$ is called a **$\sigma$-space** if every $F\_{\sigma}$-subset of $X$ is $G\_{\delta}$.
A topological space $X$ is called a **$Q$-space** if any subset of $X$ is $F\_{\sigma}$.
**Definition.** A topological space $X$ is called a **hereditary $\sigma$-space** if every subset of $X$ is $\sigma$-... | https://mathoverflow.net/users/112417 | Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space? | Every $S\_1(B\_\Gamma,B\_\Gamma)$ space is a $\sigma$-space, and the property $S\_1(B\_\Gamma,B\_\Gamma)$ is hereditary for subsets (B. Tsaban and M. Scheepers, [The combinatorics of Borel covers](https://doi.org/10.1016/S0166-8641(01)00078-5), Topology and its Applications 121 (2002), 357-382.)
For example, a Sierpi... | 4 | https://mathoverflow.net/users/2415 | 438073 | 176,960 |
https://mathoverflow.net/questions/438070 | -2 | For given $N$ and $K$ I need to compute:
$\displaystyle\sum\_{}\prod\_{K\_1}^{K\_N}\binom{N}{k\_i}$ such that $\displaystyle\sum\_{1}^Nk\_i=K$
The outer $\displaystyle\sum\_{}$ is to indicate that I need sum of all such products,
for example if $N=3$ and $K=5$
the following are sets of $k\_i$ = {0,0,5}, {0,5,0}, ... | https://mathoverflow.net/users/497485 | How can I efficiently compute the following series | It is a coefficient of $x^K$ in $(1+x)^N\cdot (1+x)^N\cdots (1+x)^N=(1+x)^{N^2}$, thus ${N^2\choose K} $
| 4 | https://mathoverflow.net/users/4312 | 438074 | 176,961 |
https://mathoverflow.net/questions/438059 | -1 | Let $N = q^k n^2$ be an odd perfect number with *special prime* $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the *classical sum of divisors* of the positive integer $x$ by $\sigma(x)=\sigma\_1(x)$.
By the definition of a *perfect number* $N$, we have $\sigma(N)=2N$. Since $\gcd(q,n)=1$ and b... | https://mathoverflow.net/users/10365 | Given that $H = \frac{n^2}{\sigma(q^k)/2} = G \times J^2$, where $q^k n^2$ is an odd perfect number, then what is the value of $\gcd(G, J)$? | **This is a partial answer, which uses the ideas in my earlier comments.**
---
We compute
$$\gcd(G,J)={p\_1}^{\min\left(\min(b\_1,2a\_1 - b\_1),2a\_1 - b\_1 - \min(a\_1,2a\_1 - b\_1)\right)} \cdots {p\_m}^{\min\left(\min(b\_m,2a\_m - b\_m),2a\_m - b\_m - \min(a\_m,2a\_m - b\_m)\right)}$$
$$={p\_1}^{\min\left(2a\_... | 0 | https://mathoverflow.net/users/10365 | 438080 | 176,965 |
https://mathoverflow.net/questions/437979 | 3 | Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int\_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{2}\pi^{3/2}}{2\Gamma{(1/4)^{2}}}.$$
The proof I have is based on the following formula concerning the elliptic i... | https://mathoverflow.net/users/nan | Some Log integrals related to Gamma value | I also played around with this integral. My solution is a bit shorter than the OPs: First use the trick by @Claude and define
$$\tag{1}
I(a)=\int\_0^1 \mathrm dx \sqrt\frac{x}{1-x^2}\log(a+\sqrt{1+x}),
$$
such that
$$\tag{2}
I(1) = I(0) + \int\_0^1 \mathrm da \, I'(a).
$$
Partial fraction decomposition of $I'(a)$ gives... | 4 | https://mathoverflow.net/users/90413 | 438084 | 176,966 |
https://mathoverflow.net/questions/438068 | 1 | Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray\*}
N\_1(A,B)
&=&\sup\left\{\sqrt{\|Ax\|^2+\|Bx\|^2},\;x\in E,\;\|x\|=1\;\right\},
\end{eqnarray\*}
and
$$N\_2(A,B)=\sqrt{\|A\|^2+\|B\|^2}.$$
Clearly,... | https://mathoverflow.net/users/113054 | Sufficient condition for two norms to be equal | $\newcommand\la\lambda\newcommand\ep\varepsilon\newcommand\ip[2]{\langle #1,#2\rangle}\newcommand\Span{\operatorname{span}}$As noted in a comment by [Yemon Choi](https://mathoverflow.net/questions/438068/sufficient-condition-for-two-norms-to-be-equal/438089#comment1129532_438068), the question is not well posed.
Appa... | 3 | https://mathoverflow.net/users/36721 | 438089 | 176,968 |
https://mathoverflow.net/questions/438020 | 4 | Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free?
I know such an action must induce an injection $\rho:\pi\_1(S^1)\to\pi\_1(T^n)$.
Another related question is: Is the image of $\rho$ saturated?
Thanks in advance!
| https://mathoverflow.net/users/110093 | Faithful locally free circle actions on a torus must be free? | This follows from Theorem 9.3 (page 216) of the book "Compact tranformation groups" by Bredon (note that it is freely available online).
More is true: Any effective group action of a torus on a torus is free. The proof starts along the lines you mentioned via fundamental group considerations.
The result is original... | 10 | https://mathoverflow.net/users/99732 | 438101 | 176,976 |
https://mathoverflow.net/questions/438064 | 1 | I encountered in my research on dynamical systems a problem, which considers for some $L>0$ on the $C\_n=[0,L]^n$ the set $\mathcal{C}\_n=\{(x\_1,\ldots,x\_n)\mid\exists j,k:\,x\_{i\_j}=x\_{i\_k}\}$. I am looking for an interpretation of $\mathcal{C}\_n$ on the $n$-torus, when imposing periodic boundary conditions on $... | https://mathoverflow.net/users/333230 | Curves on $n$-torus analogous to curve implied by diagonal in square for torus | I suppose I should add a caveat before turning my comment into an answer. In your definition, you have the $\exists j,k$ qualifier, which I believe means that $\mathcal C\_n$ is the union of $n \choose 2$ tori, i.e. one for every choice of $j \neq k$. If you allow $j=k$ then I suppose you would have $\mathcal C\_n = C\... | 1 | https://mathoverflow.net/users/1465 | 438103 | 176,978 |
https://mathoverflow.net/questions/378539 | 5 | Let $(M,g)$ be a Riemannian manifold. Let $S\_g$ be the corresponding Sasaki metric on $TM$. For every $p\in M$, $V\_p\in T\_pM$, is it true and obvious that $0\_p$ is the closest point of the zero section to $V\_p$?
With some abuse of terminology a rephrase of the question would be: Is the height of a right triangle... | https://mathoverflow.net/users/36688 | The distance to the zero section of $TM$ | *Let me expand my comment to remove the question from unanswered.*
Any point in $w\in\mathrm{T}M$ is a pair $w=(V,p)$ where $p\in M$ and $V\in\mathrm{T}\_pM$.
Let $t\mapsto w(t)=(V(t),\gamma(t))$ be a curve in $\mathrm{T}M$.
Let $V=V(0)$ and $p=\gamma(0)$.
Note that
$$w'(0)=\nabla\_{\gamma'(0)}V\oplus \gamma'(0)\in... | 2 | https://mathoverflow.net/users/1441 | 438108 | 176,980 |
https://mathoverflow.net/questions/438111 | 5 | Suppose one is given an odd prime $p$, a generator $g$ of $(\mathbb Z/p \mathbb Z)^\*$ and two integers $a$ and $b$. Is there an efficient method to determine whether $\log\_g a < \log\_g b$? (Here we are defining $\log\_g x$ as between one and $p-1$ inclusive.)
Bonus question added later: What if we restrict the pro... | https://mathoverflow.net/users/7089 | Discrete log problem modified | If there were a way of doing this in time polynomial in $\log(p)$, you could solve discrete logarithm in time polynomial in $\log(p)$ by doing a binary search. That is, to find $\log\_g(a)$, first see if $\log\_g a < \log\_g (g^{(p-1)/2})$. If yes, next compare $\log\_g(a)$ to $\log\_g (g^{\lfloor (p-1)/4\rfloor})$, if... | 7 | https://mathoverflow.net/users/13650 | 438116 | 176,981 |
https://mathoverflow.net/questions/422392 | 2 | I found myself with the following integral
$$ \int\_{b\_1}^{b\_2} \sqrt{\frac{(b-b\_1)(b\_2-b)(b\_3-b)}{(b\_4-b)}} \ db $$
with $ b\_1 < b\_2 < b\_3 < b\_4 $. I know that
$$ \int\_{b\_1}^{b\_2} \frac{db}{\sqrt{(b-b\_1)(b\_2-b)(b\_3-b)(b\_4-b)}} $$
is equal to
$$ \frac{2}{(b\_4-b\_2)(b\_3-b\_1)} K(k) $$
wher... | https://mathoverflow.net/users/482168 | Definite integral of the square root of a polynomial ratio | The expression obtained by @Robert can be simplified with the Imaginary-Argument Transformation from [DLMF 19.7.7](https://dlmf.nist.gov/19.7). Then, the limit $t\to 1^-$ can be performed. The result can be further simplified and written in many ways due to the large number of elliptic functions relations. The simplest... | 3 | https://mathoverflow.net/users/90413 | 438127 | 176,986 |
https://mathoverflow.net/questions/438105 | 4 | Working in $\sf ZF$
Define: $W\_0 = \emptyset \\ W\_{\alpha+1} = H\_{\leq |W\_\alpha|} =\{x \mid \forall y: y \in \operatorname {trcl} (\{x\}) \ |y| \leq |W\_\alpha| \} \\ W\_\lambda= \bigcup W\_{\alpha < \lambda}, \text { for limit ordinal } \lambda$
Where cardinality "| |" is defined after Scott.
This cumulativ... | https://mathoverflow.net/users/95347 | Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one? | The answer is yes, in a very general way.
What I claim, first, is that the [Lévy-Montague reflection theorem](https://en.wikipedia.org/wiki/Reflection_principle) holds in ZF for any definable continuous cumulative hierarchical representation of the set-theoretic universe $V$. That is, if you have defined sets $U\_\al... | 7 | https://mathoverflow.net/users/1946 | 438131 | 176,988 |
https://mathoverflow.net/questions/438133 | 2 | Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
>
> $(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$ such that for all $x\in [0,1]^n$ there is $y\in S$ such that $\| x-y \|< \frac{p}{q}$.
>
>
>
**Question.**... | https://mathoverflow.net/users/8628 | Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls | The question whether $(p,q,n,s)\in R$ in any instance can be expressed as a sentence in the language of the structure $\langle\mathbb{R},+,\cdot,0,1,<\rangle$, a real-closed field, and by Tarski's theorem on real closed fields, there is a computable uniform decision procedure to decide the truth of all such sentences.
... | 5 | https://mathoverflow.net/users/1946 | 438135 | 176,989 |
https://mathoverflow.net/questions/438129 | 6 | Can choice be proved with ZF+[Tarski axiom](https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory)?
| https://mathoverflow.net/users/51212 | Can the axiom of choice be proved with ZF+Tarski axiom? | Following the link found in the Wikipedia article about the [Tarski–Grothendieck set theory](https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory), the required proof (by Tarski himself!) can be found beginning on p.181 of his article ["On the well-ordered subsets of any set"](http://matwbn.icm.edu.pl/k... | 12 | https://mathoverflow.net/users/54780 | 438137 | 176,990 |
https://mathoverflow.net/questions/438124 | 5 | Let $G\_{\mathbb{Q}\_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}\_{p}$. Also, their structure as abstract groups is completely known.
It is well known that this group embeds into the absolute Galois group of the rationals, $G\_{\mathbb{Q}}$.
My question is: are there any known relations b... | https://mathoverflow.net/users/174655 | Relation between $G_{\mathbb{Q}_p}$ for different primes | As noted in the remarks above, there are different (although conjugate) embeddings $G\_{\mathbb{Q}\_p}\rightarrow G\_\mathbb{Q}$. Let us fix one of them and denote its image by $G\_p$.
If we fix prime numbers $p\_1,\dots,p\_n$ (not necessarily distinct), then
the set of $\sigma=(\sigma\_1,\dots,\sigma\_n)\in G\_\math... | 3 | https://mathoverflow.net/users/50351 | 438138 | 176,991 |
https://mathoverflow.net/questions/437668 | 1 | Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X\_0$ be a reduced (Cartier) divisor with simple normal crossings. Here a reduced divisor is a Weil divisor $D=\sum\_{i}n\_i D\_i$ with all $n\_i=1$. T... | https://mathoverflow.net/users/141609 | Determine the coefficient of the exceptional divisor | Ok, here's an expanded version of what I said in the comment.
SNC case
--------
First, I'm going to create a special log resolution of the pair $(X, X\_0)$. We assume $d = \dim X$. We are assuming that $X$ is nonsingular and $X\_0$ is simple normal crossings.
We begin by letting $S\_{n}$ denote the stratum of thi... | 1 | https://mathoverflow.net/users/3521 | 438145 | 176,994 |
https://mathoverflow.net/questions/437885 | 4 | Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}\_{n}(\mathbb C)$ be a finite dimensional complex representation.
Is there an elementary way (undergrad level) to see that $\rho(T)$ is diagonalizable?
The standard way uses algebra of functions on $k[T]$, tensor products and symmetric a... | https://mathoverflow.net/users/27398 | Action of complex torus on a vector space | You need to put some niceness hypothesis on the function $\rho$, such as "algebraic" or "analytic". Otherwise, the map $\mathbb{C}^{\ast} \to \text{GL}\_2(\mathbb{C})$ by $z \mapsto \begin{bmatrix} 1 & \log |z| \\ 0 & 1 \end{bmatrix}$ is a representation and not diagonalizable. Since you didn't tell me which hypothesis... | 6 | https://mathoverflow.net/users/297 | 438146 | 176,995 |
https://mathoverflow.net/questions/437039 | 1 | Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable maps $T:X \to Y$ such that $T\_\sharp \mu = \nu$. Let
$$
\mathbb M (T) := \int\_X c(x, T(x)) \mathrm d \mu(x) \quad \for... | https://mathoverflow.net/users/99469 | Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map | First a comment: you write (before stating your Theorem 3.14) that "The existence and uniqueness of the solution of the Monge Problem is guaranteed if $c$ is strictly convex and the supports of $\mu,\nu$ are compact", but this is absolutely not true: you really need some conditions on the starting point, i-e that $\mu$... | 2 | https://mathoverflow.net/users/33741 | 438148 | 176,996 |
https://mathoverflow.net/questions/427565 | 8 | In W.S.Massey's singular homology (Graduate Texts in Mathematics 70 Springer (1980)) there is a formula for the boundary of a double slant product on page 176
$$\partial(\phi\backslash\backslash a\otimes b\otimes g) = (\delta\phi)\backslash\backslash a\otimes b\otimes g + (-1)^{|\phi|}\phi\backslash\backslash\partial(a... | https://mathoverflow.net/users/124943 | Calculation of boundary of slant product in W. S. Massey's Singular Homology textbook | You and Tyrone are correct that this is a sign mismatch.
Just to be clear: Massey writes this sign for the double slant product because it is the desirable one. It would imply that we have a map of chain complexes
$$C^\*(Y, G\_1) \otimes C\_\*(X) \otimes C\_\*(Y) \otimes G\_2 \to C\_\*(X) \otimes G\_1 \otimes G\_2$$
... | 6 | https://mathoverflow.net/users/360 | 438150 | 176,997 |
https://mathoverflow.net/questions/430546 | 3 | I am reading this paper
>
> "A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows"
> (<https://arxiv.org/abs/1602.04457>)
>
>
>
and in the proof of Proposition 2.2, basically, if the measure $\rho$ is smooth such that $\rho = \rho(x)dx$, i.e., we think about the measure as its density then one can ... | https://mathoverflow.net/users/124759 | Equivalent definition of the Kantorovich-Fisher-Rao distance | Well, when we wrote the paper we were not really concerned with full rigor at this stage, all we wanted to emphasize was that the "KFR" distance (by now rather the WFR or HK distance, as in Wasserstein-Fisher-Rao or Hellinger-Kantorovich) constructed in 3 independent papers was really the same, at least formally. To th... | 1 | https://mathoverflow.net/users/33741 | 438151 | 176,998 |
https://mathoverflow.net/questions/438157 | 2 | I am looking at the convergence of the series
$$ \cos(t\theta) = \frac{\sin(\pi t)}{\pi} \cdot \Bigg[\frac{1}{t} + 2t \sum\_{k=1}^\infty (-1)^k \frac{\cos(k\theta)}{t^2 - k^2}\Bigg].$$
Here $t\in\mathbb{R}$. The above equality is rather trivial, but the convergence of the right side towards the left side is not strai... | https://mathoverflow.net/users/140356 | Convergence of series related to partial fraction expansion of cotangent function | Here's a complex analysis proof. For $|\theta|\leq\pi,$ we have that
$$
F(t)=\frac{\cos(\theta t)\pi}{\sin (\pi t)}
$$
is an odd meromorphic function for $t\in\mathbb{C}$ with simple poles at $k\in\mathbb{Z}$, which moreover is bounded as $|\Im t|\to\infty$.
We claim that the series in brackets
$$
G(t)=\frac{1}{t} + ... | 3 | https://mathoverflow.net/users/56624 | 438165 | 177,002 |
https://mathoverflow.net/questions/438081 | 17 | Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. Note that $M$ is minimal iff $∀M' \, M'⊈M$ and minimum iff $∀M' \, M'⊇M$.
It may be tempting to consider ZFC + $0^\#$ (a... | https://mathoverflow.net/users/113213 | Minimum transitive models and V=L | Yes, I claim you can in fact get one whose minimum model is a set forcing extension of a segment of $L$. Let $L\_\alpha$ be least modelling ZFC. Let $\mathbb{P}=\mathbb{P}^{L\_\alpha}$ be Jensen's forcing for adding a $\Pi^1\_2$-singleton, as defined over $L\_\alpha$; ZFC proves that forcing with $\mathbb{P}^L$ over $L... | 9 | https://mathoverflow.net/users/160347 | 438178 | 177,005 |
https://mathoverflow.net/questions/437901 | 3 | If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory?
The wholeness axiom $\sf WA$, [introduced](http://pcorazza.lisco.com/papers/MVS/WholenessAxiom.pdf) by Paul Corazza, found to be [consistent with $\sf V=HOD$](https:... | https://mathoverflow.net/users/95347 | If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength? | The statement beginning "The rationale beyond" is not quite correct. The critical sequence can be constucted by exploiting the facts that $\bigcup j(x)$ will be assigned the same number as $x$ in a stratification assignment, $\bigcup x=x$ for any limit ordinal, and $\omega$ is well ordered by proper subset relation $\s... | 5 | https://mathoverflow.net/users/133981 | 438182 | 177,006 |
https://mathoverflow.net/questions/438164 | 2 | Let $\Omega\_1$ and $\Omega\_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega\_2 \right)=0$, consider the system of PDEs
\begin{align}
\left(\frac{\partial X}{\partial x} \right)^2+\left(\frac{\p... | https://mathoverflow.net/users/171439 | Hyperbolic system of PDEs with elliptic-like boundary contions | Here is an example for which there is no solution: Let $\Omega\_1$ be defined by $x^2+y^2\le 1$ and $\Omega\_1$ be defined by $X^2+Y^2\le R^2$, where $R>0$ is large. Take $Z(X,Y) = 0$. Then one is asking for a map $f:\Omega\_1\to\Omega\_2$ that preserves lengths of $x$-lines and $y$-lines and carries the circle $x^2+y^... | 3 | https://mathoverflow.net/users/13972 | 438183 | 177,007 |
https://mathoverflow.net/questions/438184 | 10 | Given two sets $A$ and $B$, the function set $B^A$ is characterized by the universal property that the functor $(-)^A:\mathrm{Set} \to \mathrm{Set}$ is the right adjoint of the functor $(-)\times A:\mathrm{Set} \to \mathrm{Set}$. What is the universal property of the set of injections between two sets $A$ and $B$ in th... | https://mathoverflow.net/users/483446 | Universal property of the set of injections in the category of sets | $\newcommand{\Inj}{\operatorname{Inj}}\newcommand{\Set}{\mathrm{Set}}$**Maps $X \to \Inj(A,B)$ correspond to monomorphisms $A \times X \to B \times X$ in the slice $\Set/X$, which can be thought of as “$X$-indexed monomorphisms $A \to B$”.**
This isn’t hard to check: the main tool is that for any objects over $X$, sa... | 16 | https://mathoverflow.net/users/2273 | 438186 | 177,008 |
https://mathoverflow.net/questions/438189 | 1 | **Disclaimer.** Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE.
Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) := \sqrt{(t-1)^2 + a^2} + b|t|$. It is clear that $g$ is convex.
**Question.** What is an analytic formula for th... | https://mathoverflow.net/users/78539 | Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$ | First, a few simplifications. Note that $g(-t)\ge g(t)$ and $|-t-1|\ge|t-1|$ if $t\ge0$. So, without loss of generality (wlog) $t\ge0$ and
$$g(t)=\sqrt{(t-1)^2 + a^2} + bt. \tag{1}\label{1}$$
Since $g(t)$ is increasing in $t\ge1$, wlog $t\le1$.
Next, change the variables and the constants according to the formulas
$$... | 4 | https://mathoverflow.net/users/36721 | 438213 | 177,015 |
https://mathoverflow.net/questions/436898 | 8 | Let $X$ be an algebraic stack of finite type over a field.
Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$?
By intrinsic here I mean using constructions such as the inertia stack, the stabilizer group construction, etc.
A natural conjecture is that this minimum should be... | https://mathoverflow.net/users/496640 | Smallest atlas for algebraic stack | Professor Jarod Alper provided me an answer to this question in separate correspondence. When the stabilizer at a point of the stack is smooth, my conjecture above is true by the theory of miniversal deformations. This is Theorem 3.6.1 in Professor Alper's notes here: [https://sites.math.washington.edu/~jarod/moduli.pd... | 1 | https://mathoverflow.net/users/496640 | 438223 | 177,018 |
https://mathoverflow.net/questions/438218 | 2 | The *Euclidean distortion* of a metric space $X$, denoted $c\_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that
$$d\_X(x,y) \leq \|f(x)-f(y)\|\_{\ell^2} \leq c\cdot d\_X(x,y).$$
In the introduction of their paper "[Nonembeddability theorems via Fourier analysis](https://arxiv.... | https://mathoverflow.net/users/29873 | When is Euclidean distortion finitely determined? | The problem is that you misdefined "Euclidean distortion". The infimum is over all maps into arbitrary Hilbert spaces, not just $\ell\_2$. Of course, you can use only $\ell\_2$ if the metric space is separable, but not when it is non separable. Now if every finite subset $F $of an infinite metric space $X$ embeds into ... | 5 | https://mathoverflow.net/users/2554 | 438226 | 177,019 |
https://mathoverflow.net/questions/438199 | 3 | I am looking at trying to show that a complex symmetric matrix always has a complex symmetric square root. Showing a square root exists is fairly easy if the matrix is also invertible by using the Jordan Canonical Form.
I have seen on here that showing that the square root of a matrix A is a (Hermite) polynomial in A... | https://mathoverflow.net/users/497608 | Why is the square root of a complex symmetric matrix also complex symmetric | If your complex symmetric matrix $A$ is not invertible, it might not have a square root at all, e.g. $$ \pmatrix{i & 1\cr 1 & -i\cr}$$
If $A$ is invertible, let $\lambda\_j$ be the eigenvalues of $A$ and $m\_j$ their multiplicities. Let $P(z)$ be a polynomial such that $P(z)$ and the first $m\_j-1$ of its derivatives a... | 6 | https://mathoverflow.net/users/13650 | 438232 | 177,020 |
https://mathoverflow.net/questions/438222 | 8 | It's well known that sentences about the real closed field can be decided by algorithm and the complexity of this is about $d^{2^{O(n)}}$ where $d$ is the product of the degrees of polynomials in the sentence. See [here](https://en.wikipedia.org/wiki/Real_closed_field#Complexity_of_deciding_%F0%9D%98%9Brcf)
Now addin... | https://mathoverflow.net/users/46536 | What theories are larger than the real closed field but still decidable? | Thanks to a 1958 paper by Abraham Robinson (whose impetus was a question of Alfred Tarski), an example of such a theory that properly extends RCF is the theory of the structure $(\mathbb{R},~+,~\cdot,~A)$, where $A$ is the collection of algebraic real numbers.
More generally, Robinson's proof shows that the theory of... | 12 | https://mathoverflow.net/users/9269 | 438235 | 177,021 |
https://mathoverflow.net/questions/438035 | 9 | Given the [*Ramanujan theta function*](https://en.wikipedia.org/wiki/Ramanujan_theta_function),
$$f(a,b) = \sum\_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$.
---
**I. Degree 5**
\begin{align}
a &= q^{11/60}\;\frac{f(-q,-q^4)}{f(-q)} \,=\, q^{11/6... | https://mathoverflow.net/users/12905 | On the Klein quartic and the similar $a^2b+b^2c+c^2a$? | The questions are:
>
> 1. Is there anything else special about $a^2b+b^2c+c^2a = 0$?
> 2. In what other contexts does it appear?
>
>
>
For question 1, Given the $q$-series of degree $9$,
$$ \frac ac=a^3-b^3,\quad\frac ba=b^3-c^3,\quad\frac cb=c^3-a^3 $$
which leads to the telescoping sum
$$ \frac ac+\frac ba+\... | 3 | https://mathoverflow.net/users/113409 | 438238 | 177,023 |
https://mathoverflow.net/questions/438225 | 1 | So on nLab the definition of a trivial (Grothendieck) topology is the following: "The Grothendieck topology on any category for which only the identity morphisms are covering is the trivial topology. Its sheaves are all the presheaves."
I am having trouble understanding what is meant exactly by only the "identity mor... | https://mathoverflow.net/users/497629 | Trivial Grothendieck topology and identity morphisms | Technically, a Grothendieck topology is specified by its covering sieves, not its covers, so it would have been more accurate to say the covering sieves of the trivial topology are those generated by identity morphisms. The cover of $X$ by the sieve $hom(-,X)$ is indeed the sieve generated by the identity morphism of $... | 2 | https://mathoverflow.net/users/32 | 438239 | 177,024 |
https://mathoverflow.net/questions/438234 | 3 | By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\small f(x)=\int\_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-(0.1)^2)+2 \pi (0.1) s^x )}{(2 (0.1) s^x \cos(\pi x)+s^{2 x}+(0.1)^2)^2} ds.$$
Note that the integrated function c... | https://mathoverflow.net/users/149793 | Integral of a function changing sign | In fact, this conjecture fails to hold for $x$ in a right neighborhood of $0$.
Indeed, let $g(x,s)$ denote the integrand. Note that $g(x,s)$ is continuous in $x\in[0,1)$ for each real $s>0$, and
$$g(0+,s)=g(0,s)=\frac{100 \pi e^{-2 s}}{121 s}$$
for all real $s>0$. So, by the Fatou lemma,
$$f(0+)\ge\int\_0^\infty g(0+... | 5 | https://mathoverflow.net/users/36721 | 438245 | 177,027 |
https://mathoverflow.net/questions/352806 | 5 | The answer to another question ([Upper bound of the fraction of Gamma functions](https://mathoverflow.net/q/313103/125166)) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\!\leq \,C\,\frac{a+b}a, \forall a,b\geq\frac12$$
What is the b... | https://mathoverflow.net/users/122182 | The exact constant in a bound on ratios of Gamma functions | The optimal $C$ is $\mathrm{e}$.
**Proof**:
We have
$$\ln C \ge \ln a - \ln(a + b)
+ \frac{\ln \Gamma(a + b)
-\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$
Let
$$F(a, b) := \ln a - \ln(a + b)
+ \frac{\ln \Gamma(a + b)
-\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$
We have
$$\frac{\partial F}{\partial b}
= - \frac{1}{... | 2 | https://mathoverflow.net/users/141801 | 438254 | 177,031 |
https://mathoverflow.net/questions/438258 | 35 | (This *must* have been asked before and exist somewhere in Community Wiki, but I can't find it...)
Where can you post open (math) problems? And what are the advantages and disadvantages?
* Example: This place (and Math StackExchange), duh.
* Example: The journal AMM had a corner "Unsolved Problems", but no longer. ... | https://mathoverflow.net/users/11504 | Places where one can post open problems | If you can motivate the problem and make some partial progress on it, you can try and publish it as a paper in a specialized journal, or at the very least upload it to the arXiv.
If you only have empirical evidence, there are journals that are receptive to this kind of this ("Mathematics of Computation" and "Experime... | 22 | https://mathoverflow.net/users/31469 | 438262 | 177,034 |
https://mathoverflow.net/questions/430766 | 3 | Is there any result like the [Bramble-Hilbert lemma](https://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert_lemma) for Bochner spaces?
More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)$ for $Y$ a normed space. If $L$ vanishes on $k-1$ poly... | https://mathoverflow.net/users/155442 | Approximation in Bochner spaces | Writing $$v(t)=p(t)+\frac{1}{(k-1)!}\int\_0^t (t-s)^{k-1}v^{(k)}(s)\, ds$$ with $p$ the Taylor polynomial at $0$ of degree $k-1$, then $\|v-p\|\_{H^k} \leq C\|v^{(k)}\|\_{L^2}$. It follows that
$$\|Lv\|=\|L(v-p)\| \leq \|L\|\|v-p\|\_{H^k} \leq C\|L\|\|v^{(k)}\|\_{L^2}.
$$
| 2 | https://mathoverflow.net/users/150653 | 438275 | 177,037 |
https://mathoverflow.net/questions/438269 | 6 | Let $f: \mathbb R^n \to \mathbb R$ be a measurable function.
We say $f$ is *precise* if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B\_\delta (x)|>0$ for every $\delta>0$, we have :
* $\lim\_{\delta \to 0^+} \text{essinf}\_{K \cap B\_\delta (x)} f = \lim\_{\del... | https://mathoverflow.net/users/173490 | A characterisation of continuous real functions | Edit: the proof can be made a little simpler.
Yes, this condition is equivalent to $f$ being continuous. The reverse direction is easy because if $f$ is continuous at $x$ then all of the limits in question equal $f(x)$. For the forward direction, suppose $f$ is not continuous at some point $x$. Wlog $f(x) > \lim\_{\d... | 6 | https://mathoverflow.net/users/23141 | 438276 | 177,038 |
https://mathoverflow.net/questions/438280 | 14 | A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint union of more than one and at most $\kappa$ many closed subsets. In particular a space is connected iff it is $2$-connecte... | https://mathoverflow.net/users/49381 | How “disconnected” can a continuum be? | The answer to all three of your questions is yes.
The cardinal $\mathrm{disc}([0,1])$ is discussed in [this](https://mathoverflow.net/questions/285780/are-the-sierpi%C5%84ski-cardinal-acute-mathfrak-n-and-its-measure-modification?rq=1) MO question of Taras Banakh. He calls this cardinal the *Sierpiński cardinal* and ... | 15 | https://mathoverflow.net/users/70618 | 438283 | 177,043 |
https://mathoverflow.net/questions/438287 | 1 | * Suppose with have a topological manifold $X$ and a group $G$, is there a way to compute the fundamental group of $X/G$ in function of $\pi(X)$ and $\pi(G)$?
* are there any settings on X that can simplify the computing of $\pi(X/G)$ in function of $\pi(X)$ and $\pi(G)$?
could you please suggest to me a reference wh... | https://mathoverflow.net/users/497616 | fundamental group of $X/\mathbb{R}^n$ | See Proposition 8.10 in Chapter I of [Transformation Groups and Algebraic K-Theory](https://link.springer.com/book/10.1007/BFb0083681) by Lück. When a Lie group $G$ acts properly on a connected manifold $X$, the proposition provides an exact sequence
$$
\pi\_1(X,x\_0)\rightarrow\pi\_1(X/G,x\_0G)\rightarrow\pi\_0(G)/M\r... | 3 | https://mathoverflow.net/users/128556 | 438289 | 177,045 |
https://mathoverflow.net/questions/438290 | 8 | By the downward Lowenheim-Skölem theorem we can find two countable ordinals $\alpha < \beta$ such that $L\_\alpha \prec L\_{\omega\_1}$ and $L\_\beta \prec L\_{\omega\_1}$. That is, $L\_\alpha$ and $L\_\beta$ are elementary submodels of $L\_{\omega\_1}$. Consequently, $L\_\alpha \prec L\_\beta$.
Hence my question, th... | https://mathoverflow.net/users/138089 | Elementary countable submodels in Gödel's universe | Very clearly not. Take some countable elementary submodel $M\_0$ of $L\_{\omega\_2}$, and take $M\_1$ to be another one, but with $M\_1$ a end extension of $M\_0$. We can find such models by first finding two uncountable $\gamma<\delta$ such that $L\_\gamma\prec L\_\delta\prec L\_{\omega\_2}$, and then taking $M\_1$ be... | 7 | https://mathoverflow.net/users/7206 | 438292 | 177,046 |
https://mathoverflow.net/questions/438306 | 3 | Call a family of sets $\mathcal{F} \subseteq [\omega]^\omega$ *maximal* if there does not exist some $X \in [\omega]^\omega \setminus \mathcal{F}$ such that $X$ is almost disjoint with all elements of $\mathcal{F}$. Let $\mathcal{A} \subseteq \mathcal{F}$ be an almost disjoint family. Does there always exist some $\mat... | https://mathoverflow.net/users/146831 | Extending almost disjoint family in a maximal set | Every family $\mathcal{F}$ containing $\omega$ itself as a member is trivially maximal, since no infinite set is almost disjoint from $\omega$. But the family could otherwise consist of an almost disjoint non-maximal family $\mathcal{A}$. That is, $\mathcal{F}=\mathcal{A}\cup\{\omega\}$ is a counterexample.
| 3 | https://mathoverflow.net/users/1946 | 438318 | 177,055 |
https://mathoverflow.net/questions/438312 | 2 | Let $W\_t$ be a Wiener process with $W\_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).
Assume that $W\_t$ stayed above $L$ up to time $T$. What is the PDF of $W\_T$ under this assumption? Does it have a closed form?
| https://mathoverflow.net/users/64302 | Density of $W_t$ assuming it stayed above a line $L$ | The problem can be restated as follows:
>
> For a real $a>0$ and a real $b$, let
> $$X\_t:=a+bt+W\_t$$
> for real $t\ge0$, where $W$ is a standard Wiener process. Let
> $$\tau:=\inf\{t>0\colon X\_t=0\}.$$
> For a real $t>0$, find the joint distribution of $X\_t$ and $\tau$.
>
>
>
The answer is well known:
$$P(... | 3 | https://mathoverflow.net/users/36721 | 438322 | 177,059 |
https://mathoverflow.net/questions/438227 | 2 | One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C\_{\rm erg}$. During one of my studies, I found the expression below for such a metric considering a Rayleigh channel and AWGN.
\begin{align}\label{eq:CAPACIDADEERGODICA}
C\_{\rm erg} & =
\fra... | https://mathoverflow.net/users/103291 | Asymptotic analysis of an expression involving a Fox's H function | Your expression can be simplified as follows: As $m=q=4$, the denominator parameter $(0,1)$ cancels with the numerator parameter $(0,1)$. Furthermore, as all linear coefficients are one, the Fox H function reduces to the simpler Meijer G function. I'll denote
$$\tag{1}
\xi = \frac 1 {\bar\gamma h\_1^2 A\_0^2},\quad \ze... | 5 | https://mathoverflow.net/users/488821 | 438324 | 177,060 |
https://mathoverflow.net/questions/438263 | 5 | Is there a concrete example of a $4$ tensor $R\_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather\*}
R\_{ijkl} = - R\_{ijlk},\quad R\_{ijkl} = R\_{jikl},\quad R\_{ijkl} = R\_{klij}, \\
R\_{ijkl} + R\_{iklj} + R\_{iljk} = 0.
\end{gather\*}
for which there is no metric for which it i... | https://mathoverflow.net/users/497575 | Example of a curvature with no associated metric | A simple example (which just uses Deane Yang/Robert Bryant's idea) is to consider any space of dimension at least three and consider the tensor field
$$ R\_{ijkl} = f(x)(\delta\_{ik}\delta\_{jl}-\delta\_{il}\delta\_{jk})$$
where $f(x)$ is your favorite function which changes sign and whose derivative is non-vanishing w... | 6 | https://mathoverflow.net/users/125275 | 438332 | 177,062 |
https://mathoverflow.net/questions/438273 | 1 | A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}\_{X} $-linear map $ \phi: F\_{\ast}(\mathcal{O}\_{X}) \to \mathcal{O}\_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}\_{\mathcal{O}\_{X}} $. Such a map $ \phi $ is called a splitting. A closed sub-scheme $ Y $ of $ X $ is compatibly split if t... | https://mathoverflow.net/users/470753 | Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split? | Actually, on an affine variety, if it is $F$-split, then it is compatibly $F$-split with every point (possibly changing the splitting). This is not true in the projective case (an ordinary projective elliptic curve is $F$-split, but not compatibly split with any point). It is also not true for non-closed points (for in... | 2 | https://mathoverflow.net/users/3521 | 438337 | 177,064 |
https://mathoverflow.net/questions/438196 | 4 | **Background:** The equation
$$a^4+b^4+c^4=2d^4$$
has infinitely many positive integral solutions if we take $c=a+b$ and $a^2+ab+b^2=d^2$ further assuming that $GCD(a,b,c)=1$.
**Main problem:** Find some positive integral solutions to the equation
$$a^4+b^4+c^4=2d^4$$
with $a\lt b\lt c\ne a+b$ and $GCD(a,b,c)... | https://mathoverflow.net/users/nan | On the equation $a^4+b^4+c^4=2d^4$ in coprime positive integers $a\lt b\lt c$ such that $a+b\ne c$ | Let $A=\frac{a}{d}, B=\frac{b}{d}, C=\frac{c}{d},$ then we get
$$A^4+B^4+C^4=2\tag{1}$$
Let $A=x+y, B=x-y, z=C^2$ then
$$2x^4+12y^2x^2+2y^4+z^2 = 2\tag{2}$$
Hence
$$y^2 = -3x^2 \pm \frac{\sqrt{32x^4+4-2z^2}}{2}\tag{3}$$
So we find the rational solutions of $(4)$.
$$v^2=32x^4+4-2z^2\tag{4}$$
$(4)$ can be... | 13 | https://mathoverflow.net/users/150249 | 438347 | 177,068 |
https://mathoverflow.net/questions/438342 | 7 | In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is called "lax" and the other "colax" (or "oplax"). We have to make at least one arbitrary choice for which of the two to call... | https://mathoverflow.net/users/49 | Which direction does a lax dinatural transformation go? | This is a *very* loose answer since it's based on something that, as far as I know, has not been fully formalized. The key idea is that of what I might call a "dependent arrow" between two objects.
The closest analogy is to that of a dependent path in HoTT, where if we have $a, a' : A$, $p : a = a'$, $b : B(a)$ and $... | 3 | https://mathoverflow.net/users/178774 | 438361 | 177,073 |
https://mathoverflow.net/questions/438352 | 5 | (This is a fairly basic question, not really research level, except that I am a research mathematician working on other things who is trying to understand more topology for use in my own work.)
Let $\Sigma$ be a smooth, connected, compact 2d manifold, henceforth simple a "surface". We don't assume anything more (not ... | https://mathoverflow.net/users/497719 | Meaning of the first Chern class of the unit tangent bundle of a surface | The Poincaré-Hopf holds for non-orientable manifolds also. For a closed non-orientable surface $\Sigma$ the structure group of the sphere tangent bundle does not reduce to $SO(2)=S^1$, so it's classified by a map $\Sigma\to BO(2)$ rather than a map $\Sigma\to BS^1$. The obstruction to finding a section of the sphere ta... | 6 | https://mathoverflow.net/users/8103 | 438370 | 177,075 |
https://mathoverflow.net/questions/438369 | 2 | Not counting equivalent braids, are there finite or infinite numbers of 3-braids whose closures are trivial knot or links? If the answer is infinite, are there some patterns in those infinite numbers of braids, e.g. there exists some repeated parts?
| https://mathoverflow.net/users/492606 | Are there infinite number of 3-braids with trivial closure? | Suppose that $\beta$ is a three-braid whose closure is trivial. Suppose that $\gamma$ is any three-braid. Then $\gamma \beta \gamma^{-1}$ is again a three-braid with trivial closure. However, if $\beta$ is non-trivial, then for "generic" $\gamma$, the braids $\beta$ and $\gamma \beta \gamma^{-1}$ will not be equivalent... | 2 | https://mathoverflow.net/users/1650 | 438372 | 177,076 |
https://mathoverflow.net/questions/438368 | 6 | A quick search for "diagonalisable matrix" on Wikipedia immediately gives the result that the set of real-diagonalisable matrices is not dense in the set of real matrices.
I need, however, to know whether the set of complex-diagonalisable real matrices, i.e. matrices with purely real entries that become diagonalisabl... | https://mathoverflow.net/users/113020 | Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices? | The answer is yes. Recall that the discriminant of a polynomial $x^n + a\_{n-1} x^{n-1} + \cdots + a\_1 x + a\_0$ is a polynomial $\Delta(a\_0, a\_1, \ldots, a\_{n-1})$ which vanishes if and only if the complex roots of $x^n + a\_{n-1} x^{n-1} + \cdots + a\_1 x + a\_0$ are not mutually distinct. Recall also that, if th... | 9 | https://mathoverflow.net/users/297 | 438380 | 177,080 |
https://mathoverflow.net/questions/438343 | 5 | I have been thinking about the validity of the following inequality:
if $P(z)=\sum\_{k=0}^na\_kz^k, a\_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$
\begin{align\*}
&\int\_{0}^{2\pi}\left|n\left(1-\frac{|a\_0|-|a\_n|}{n(|a\_0|+|a\_n|)}\right)P(e^{i\theta})+(\eta-e^{i\theta}... | https://mathoverflow.net/users/128472 | An inequality for polynomials | $\newcommand\ep\varepsilon$This inequality does not hold in general.
Indeed, by continuity/denseness, the condition $a\_n\ne0$ can be dropped (in response to a comment by the OP, details on this are now given at the end of this answer).
Now just take $p=2$, $\eta=-1$, $n=2$, $P(z)=1-z+0z^2$ (so that $a\_0=1$ and $a... | 2 | https://mathoverflow.net/users/36721 | 438382 | 177,081 |
https://mathoverflow.net/questions/438174 | 6 | A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. The starting position of the player is at the center (i.e. at the $n$th row and the $n$th column) of the map. For each s... | https://mathoverflow.net/users/114036 | What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map? | So the following is a bit naive, but perhaps can be a starting point and an actual percolation theorist can do more.
The event you're considering is $E(n,p)$ = there exists a connected component $C$ containing the origin and some site on the boundary of $[-n,n]^2$ whose sum is positive. (This seems equivalent to your... | 1 | https://mathoverflow.net/users/116357 | 438383 | 177,082 |
https://mathoverflow.net/questions/438243 | 2 | Let $G$ be a linearly reductive algebraic group, and let $X$ be an irreducible affine variety, over an algebraically closed field $\mathbb{K}$, with a regular action of $G$. By Rosenlicht's result, we know that there's a $G$-invariant open set $U\subseteq X$ such that the geometric quotient $\Phi:U\rightarrow U/\!/G$ e... | https://mathoverflow.net/users/338456 | Orbits in the open set given by Rosenlicht's result | Let me add more details to my [comment](https://mathoverflow.net/questions/438243/orbits-in-the-open-set-given-by-rosenlichts-result#comment1130011_438243) above. Let $S$ be a scheme. Let $\overline{X}$ be a proper $S$-scheme, and let $X\subset \overline{X}$ be a dense Zariski open subscheme.
A closed subset $R\subse... | 2 | https://mathoverflow.net/users/13265 | 438396 | 177,086 |
https://mathoverflow.net/questions/438305 | 19 | Let's denote the Fibonacci numbers by $F\_0=0,F\_1=1,F\_{n+2}=F\_{n+1}+F\_n \; \forall n \ge 0$. According to [Zeckendorf's theorem](https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem), every positive integer can be represented uniquely as the sum of some (at least $1$) distinct Fibonacci numbers where no two consecu... | https://mathoverflow.net/users/166298 | Grothendieck group of the Fibonacci monoid | This operation is not mysterious at all! The monoid $(\mathbf N,\circ)$ is isomorphic to a multiplicative submonoid $T$ of the commutative ring $\mathbf Z[\varphi] = \mathbf Z[t]/(t^2-t-1)$, where $\varphi = \tfrac{1+\sqrt{5}}{2}$ is the golden ratio. In particular, it is isomorphic to a submonoid of $\mathbf Z \oplus ... | 9 | https://mathoverflow.net/users/82179 | 438398 | 177,088 |
https://mathoverflow.net/questions/438410 | 0 | Let
* $X := \mathbb R^n$,
* $C\_b(X)$ the space of all real-valued bounded continuous,
* $C\_c(X)$ the space of all real-valued continuous functions with compact supports, and
* $C\_c^\infty(X)$ the space of all real-valued smooth functions with compact supports.
Let $\mu, \mu\_n$ be Borel probability measures on $... | https://mathoverflow.net/users/477203 | Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence? | Any $f\in C\_c(X)$ can be uniformly approximated by functions $f\_n\in C\_c^\infty(X)$, say by convolving $f$ with appropriate mollifiers $\psi\_n\in C\_c^\infty(X)$.
So, your desired conclusion indeed follows.
| 4 | https://mathoverflow.net/users/36721 | 438412 | 177,092 |
https://mathoverflow.net/questions/438407 | 8 | When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, and contemplate a salvage by forsaking choice altogether like in Reinhardt's cardinals setting.
>
> Is there a known w... | https://mathoverflow.net/users/95347 | Is there a form of choice that can elude Kunen's inconsistency theorem? | [Work of Usuba](https://arxiv.org/abs/2004.01515) combined with [work of Woodin](https://www.worldscientific.com/doi/10.1142/S021906131000095X) shows that if there is a Reinhardt cardinal $\kappa$ that is a limit of Lowenheim-Skolem cardinals, then there is a forcing extension in which $\kappa$ remains a Reinhardt card... | 22 | https://mathoverflow.net/users/102684 | 438413 | 177,093 |
https://mathoverflow.net/questions/438405 | 5 | For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \mathcal{P}D$ and $\mathcal{E} \hookrightarrow \mathcal{P}C$ such that the inverse image functor $f^\* : \mathcal{E} \to... | https://mathoverflow.net/users/49 | Fibrations of sites for $\infty$-topoi | Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see internal site barely hidden in plain sight, but the point is you don't need to see them to follow the proof)
As a firs... | 5 | https://mathoverflow.net/users/22131 | 438417 | 177,095 |
https://mathoverflow.net/questions/438418 | -5 | This is more of a curiosity than a research question, but I could not find it answered anywhere. What is the largest $N$ for which the statement in the title is true? I have recently read that the largest known prime is $2^{82589933} − 1$, but I imagine it is not known if, for example, $2^{82589933} − 3$ is prime.
A cl... | https://mathoverflow.net/users/66323 | It is known if $n$ is prime for all $n\leq N$ | As explained in the comments, there is no well-defined answer to this question, because there is no organized effort to test primality of all numbers up to a certain bound.
However, there is a (more or less) well-defined answer (at any given point in time) to a related question: What is the largest $N$ for which the ... | 1 | https://mathoverflow.net/users/3106 | 438429 | 177,098 |
https://mathoverflow.net/questions/438404 | 3 | Let $p$ be a prime. The *minimal ramification problem* is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $r(G)$ ramified primes where $r(G)$ is the minimal number of generators of $G$. This problem has an affirmative solut... | https://mathoverflow.net/users/492970 | On the refined minimal ramification problem for $p$-groups | The answer is affirmative at least for $p\ge 11$ by the following construction (for the smaller primes, the extension constructed is not of the demanded degree $>p^9$, but surely there will be some alternative construction):
Let $q\_1\equiv 1$ mod $p$, $q\_1\ne 1$ mod $p^2$, and let $K/\mathbb{Q}$ be the $C\_p$-subex... | 5 | https://mathoverflow.net/users/127660 | 438437 | 177,101 |
https://mathoverflow.net/questions/438451 | 0 | Axiality has been studied under a definition given here: <https://en.wikipedia.org/wiki/Axiality_(geometry)>
Consider an *alternative definition* of axiality as follows: For a convex region C, consider a chord L and the set of chords of C that are perpendicular to L. For each perpendicular chord, consider the ratio: ... | https://mathoverflow.net/users/142600 | On 'axiality' of planar convex regions | Too many bodies have vanishing axiality ratio.
Indeed, if the ratio for $L$ is positive, then the whole body projects to $L$.
So you may start with convex smooth body, locate these exceptional chords and modify the body in a small neighborhood of its ends making the ratio = zero.
In particular, you get a body with ze... | 0 | https://mathoverflow.net/users/1441 | 438455 | 177,105 |
https://mathoverflow.net/questions/438181 | 12 | Roughly speaking, given a set-sized logic $\mathcal{L}$ let $\mathcal{L}'$ be gotten by adding to $\mathcal{L}$ the ability to quantify over $\mathcal{L}$-definable relations. (The details are a bit tedious, so I've put the rigorous definition at the end of this question.) Note that $\mathcal{L}'$ is always strictly st... | https://mathoverflow.net/users/8133 | What is the "iterated definability" limit of first-order logic? | If I understand the question properly (I'm not sure whether I do), then it looks like your conjecture for question 2 is correct, i.e. $L\_{\beta\_0}\cap\mathcal{P}(\omega)$. Here is a hastily written sketch.
For in fact, letting $N\_\alpha$ be the set of $\mathcal{L}^{(\alpha)}$ relations on $\omega$, then $N\_\alpha... | 6 | https://mathoverflow.net/users/160347 | 438464 | 177,107 |
https://mathoverflow.net/questions/438442 | 3 | The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism
$$A \to R^{\mathrm{Spec}\_R(A)}$$
is an isomorphism.
Now, we don't assume that the Kock-Lawvere axiom holds for t... | https://mathoverflow.net/users/483446 | Analogue of Kock-Lawvere axiom for power series rings? | Yes, it is consistent, it even follows from the Kock-Lawvere axiom, as follows.
We defined $\mathrm{Spf}(R[[\epsilon]]) := \mathrm{colim}\_n \mathrm{Spec}(R[\epsilon]/(\epsilon^n))$, so we have
$$R^{\mathrm{Spf}(R[[\epsilon]])} = R^{\mathrm{colim}\_n \mathrm{Spec}(R[\epsilon]/(\epsilon^n))} = \mathrm{lim}\_n R^{\math... | 4 | https://mathoverflow.net/users/166281 | 438467 | 177,109 |
https://mathoverflow.net/questions/438462 | 6 | Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G\_\varepsilon:=\{x\in G\mid B\_\varepsilon(x)\subseteq G\}$$
be the set of all points in $G$ with distance at least $\varepsilon$ from the boundary of $G$.
I am looking for results of the form
$$\m... | https://mathoverflow.net/users/81206 | Concentration of volume towards the boundary | $\newcommand\ep\varepsilon\newcommand\R{\mathbb R}$Suppose that $G$ is a measurable subset of $\R^N$ with volume $|G|>0$ such that
$$|G|\le C^N|B|,$$
where $C>0$ is a real constant and $B$ is the unit ball in $\R^N$.
Without loss of generality (wlog), $|G\_\ep|>0$. Also, $G\_\ep+\ep B\subseteq G$ for any real $\ep>0$... | 5 | https://mathoverflow.net/users/36721 | 438481 | 177,110 |
https://mathoverflow.net/questions/438426 | 5 | Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}\_m$-gerbe over $K(C)$ splits, i.e., the etale cohomology $H^2\_{et}(K(C),\mathbb{G}\_m)=0.$
Now, let $T$ be a *not-necessarily split* to... | https://mathoverflow.net/users/497756 | Torus gerbes over curves | I am just posting my comment as one answer. Let $K$ be a field, and let $T$ be a $K$-group scheme such that there exists a field extension $K'/K$ that is finite and separable (i.e., étale), and there exists an isomorphism $i$ of $k'$-group schemes from $T\_{K'}:=\text{Spec}\ K'\times\_{\text{Spec}\ K}T$ to the split to... | 3 | https://mathoverflow.net/users/13265 | 438491 | 177,113 |
https://mathoverflow.net/questions/438492 | 3 | Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE
$$dX\_t = \sigma(X\_t) \, dW\_t \, , \, X\_0 = 0$$
with $\sigma: \mathbb R \to \mathbb R$ Lipschitz continuous.
For each $c > 0$, define the process $Y^c$ on $[0, 1]$ by
$$Y^c\_t := c^{-1/2} X\_{ct}$$
**Question:** I... | https://mathoverflow.net/users/173490 | Blow up limits for SDE | The answer to your question is yes, at least for the one- dimensional marginals. That said, I'm confident that the convergence you want actually holds on the level of stochastic processes (i.e. $c^{-1/2}X\_{ct}\xrightarrow{(d)}B\_t$ as functions on $C[0,T]$ for any $T>0$), but this requires some additional work. That s... | 3 | https://mathoverflow.net/users/80052 | 438499 | 177,116 |
https://mathoverflow.net/questions/438477 | 3 | Let $F$ be a non-archimedean field and let $\pi$ be a self-dual supercuspidal representation of $\mathrm{GL}\_n(F)$ (which, by a result of Adler exists only when $n=1$ or $n$ is even). Then, under LLC for $\mathrm{GL}\_n$ we obtain a $n$-dimensional irreducible self-dual representation $\varphi\_\pi$ of the Weil group ... | https://mathoverflow.net/users/123673 | Frobenius-Schur indicator of a self-dual L-parameter | Prasad and Ramakrishnan ([arXiv link](https://arxiv.org/abs/0807.0240)) study how the signs of (discrete series) representations behave along the local Langlands correspondence, not just for GL($n$), but for all inner forms. Assume the base field has characteristic 0.
If $n$ is odd, $\pi$ and its Langlands parameter ... | 1 | https://mathoverflow.net/users/6518 | 438505 | 177,117 |
https://mathoverflow.net/questions/317633 | 4 | Let $Y$ be a simplicial complex and let $\{Y\_i\}\_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup\_{i\in I}Y\_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each finite $J\subset I$, we have that $\bigcap\_{j\in J}Y\_j$ is either empty or contractible.
One version of the *nerv... | https://mathoverflow.net/users/76590 | Nerve theorem for locally infinite covers by subcomplexes | I asked myself exactly this question the other day (while looking back at Björner's handbook article), and I poked around in Björner's papers looking for an answer. My guess is that Björner was referring to the argument, attributed to Quillen, that is found on p. 92 of his article [Homotopy type of posets and lattice c... | 3 | https://mathoverflow.net/users/4042 | 438506 | 177,118 |
https://mathoverflow.net/questions/438488 | 7 | Let $0<\alpha<1$ and define
$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$
The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of
$Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that
$$ \|Tf\|\_{q}\... | https://mathoverflow.net/users/116555 | $L^p-L^q$ boundedness of this simple singular oscillatory integral operator | Let me consider instead the operator
$$Sf(x):=\int e^{-i x y} \frac{f(y)}{|x-y|^{\alpha}}dy$$
which has the same properties as $T$ since $Tf(x)=Sf(-x)$. Writing
$$e^{-i x y}=e^{i|x-y|^{2}/2}e^{-i|x|^{2}/2}e^{-i|y|^{2}/2}$$
the operator $S$ can be written
$$Sf(x)=e^{-i|x|^{2}/2}\int \frac{e^{i|x-y|^{2}/2}}{|x-y|^{a}}
e... | 8 | https://mathoverflow.net/users/7294 | 438507 | 177,119 |
https://mathoverflow.net/questions/438371 | 5 | Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ *absorbs* $a$? Can we say that $a$ is $b$-invariant? Or $b$ is $a$-invariant?
---
Now, for my question.
Let $C(\mathbb{G})$ be an algebra o... | https://mathoverflow.net/users/35482 | States "absorbed" by a Haar idempotent on a compact quantum group | Without making the assumption that $C(\mathbb{H})$ is the universal C$^\*$-algebra of the compact quantum group $\mathbb{H}$, the answer is negative. Whenever $\mathbb{G}$ is not co-amenable, we could take $C(\mathbb{H}) = C\_r(\mathbb{G})$ and $C(\mathbb{G}) = C\_u(\mathbb{G})$. We define $\pi : C\_u(\mathbb{G}) \to C... | 2 | https://mathoverflow.net/users/159170 | 438517 | 177,125 |
https://mathoverflow.net/questions/438523 | 2 | An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?
An abelian category is a very special type of $\mathbf{Ab}$-category. So can a category be given the structure of an abelian category in... | https://mathoverflow.net/users/491434 | Can a category be enriched over abelian groups in more than one way? | You can easily find examples among categories with one element: a category with one element is a (multiplicative) monoid, and $Ab$-enrichment over it is a choice of an addition which turns it into a ring. And there can be multiple such additive structures: you can for instance consider pullback along a permutation whic... | 15 | https://mathoverflow.net/users/30186 | 438525 | 177,126 |
https://mathoverflow.net/questions/438504 | 2 | $\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime and let $ \SL^1\_2(\mathbb{Z}\_p)$ denote the kernel of the natrual surjective morphism $\SL\_2(\mathbb{Z}\_p)\rightarrow \SL\_2(\mathbb{Z}\_p/p\mathbb{Z}\_p)$. Then the abelianization of $\SL^1\_2(\mathbb{Z}\_p)$ is $(\mathbb{Z}/p\mathbb{Z})^3$. Recall that there ar... | https://mathoverflow.net/users/149460 | Finite quotients of $p$-adic congruence subgroups of $\operatorname{SL}_2$ | $\DeclareMathOperator\SL{SL}\newcommand{\Z}{\mathbf{Z}}$Indeed, $G\_1$ is a quotient, but not $G\_2$.
First, one checks that the third term in the lower central series is the kernel of $\SL\_2^1(\Z\_p)\to\SL\_2^1(\Z/p^3\Z)$. So this converts this into a question about quotients of a group of order $p^6$, namely $\SL\... | 4 | https://mathoverflow.net/users/14094 | 438526 | 177,127 |
https://mathoverflow.net/questions/436414 | 1 | Suppose we have a sympletic toric manifold $(M,\omega)$ of dimension $4$ and let $\triangle$ be its corresponding Delzant polytope. Suppose that this polytope is "nice" enough so that we are able to defined a map, using action-angle coordinates $\Psi:\triangle \times \mathbb{T}^2\rightarrow \triangle \times \mathbb{T}^... | https://mathoverflow.net/users/155363 | Could this be a Hamiltonian symplectomorphism on a symplectic toric manifold | Sometimes it is, sometimes it isn't.
The spheres living over the edges generate the second homology, so you can read off the action on $H\_2$ from that. For $S^2\times S^2$ (square) the action on homology is trivial (because opposite edges are homologous) and the symplectomorphism is indeed Hamiltonian. For a hexagon... | 2 | https://mathoverflow.net/users/10839 | 438531 | 177,130 |
https://mathoverflow.net/questions/438203 | 6 | Suppose that $\mathbb{R}^n$ is the maximal group that can act properly on a manifold $N$ and $\mathbb{R}^m$ is the maximal group that can act properly on a manifold $M$ ( i.e, $\mathbb{R}^{m+1}$ can't act properly on $M$ ).
Question: is $\mathbb{R}^{n+m}$ the maximal group that can act properly on the manifold $M\tim... | https://mathoverflow.net/users/497616 | Proper action on product manifold | First, let's formulate the question properly:
Given a topological space $X$, define be
$$
d(X):=\sup \{ n: X~ \hbox{is homeomorphic to} ~Y\times {\mathbb R}^n\}.
$$
**Lemma.** The following quantities are equal when $X$ is a manifold:
(1) $d(X)$
(2) P(X):=$\max\{n: \exists ~ \hbox{a principal}~ {\mathbb R}^n\h... | 8 | https://mathoverflow.net/users/39654 | 438532 | 177,131 |
https://mathoverflow.net/questions/438494 | 0 | I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta\_1)+f(z+\eta\_2)+\ldots+f(z+\eta\_n)=0,\ \ \ \eta\_1 \cdots\eta\_n \in \mathbf{C}
\end{equation}
look like. It seems easy when $\eta\_i$ are all integers, but I don't know the general case.
If we cannot get the exac... | https://mathoverflow.net/users/497814 | Solutions of complex linear difference equations | Look for a solution of the form $f(z)=e^{\lambda z}$. Plugging this to your equation, you obtain that
$\lambda$ must be a zero of the entire function
$$F(\lambda)=\sum\_{j=1}^n e^{\lambda\eta\_j}.$$
When there are at least two distinct $\eta\_j$, this entire function $F$ has infinitely many zeros $\lambda\_k$.
Then any... | 2 | https://mathoverflow.net/users/25510 | 438533 | 177,132 |
https://mathoverflow.net/questions/438511 | 4 | Let $A$ be an $n\times n$ random matrix with i.i.d entries (say standard Gaussian) $A\_{ij}$. I know that there is a CLT type result known for the determinant of $A$. More precisely there is a CLT for $\log(|\det(A)|)$. For any fixed $(i, j)$, If we look at the minor $M\_{ij}$ of $A$, it follows that $\log(|M\_{ij}|)$ ... | https://mathoverflow.net/users/69849 | Joint distribution of minor of Wigner Hermitian matrices | There is certainly no asymptotic independence between $\det M\_{11}, \det M\_{22}$. From the base times height formula for parallelepipeds we see that
\begin{align\*} \frac{|\det M\_{12}|}{|\det M\_{22}|} &= \frac{|v\_1 \wedge v\_3 \wedge \dots \wedge v\_n|}{|v\_2 \wedge v\_3 \wedge \dots \wedge v\_n|}\\
&= \frac{\math... | 9 | https://mathoverflow.net/users/766 | 438535 | 177,133 |
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