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/324315 | 7 | Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e\_i)\_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\mathcal H\to\mathcal H$. Then one can apply the Gram-Schmidt process to the (linearly independent) vectors $Ae\_i$. That is... | https://mathoverflow.net/users/57840 | Is Gram-Schmidt on a separable Hilbert space operator norm continuous? | The answer to the main question is no. Working on $l^2$, let $A$ be the operator $A: e\_n \mapsto \frac{1}{n}e\_n$ and for each $i$ let $A\_i$ be $A$ followed by the unitary $U\_i$ that switches $e\_i$ and $e\_{i+1}$ and fixes the other standard basis vectors. Then $A\_i \to A$ in norm but $(U\_i)$ does not converge in... | 8 | https://mathoverflow.net/users/23141 | 324318 | 139,770 |
https://mathoverflow.net/questions/324319 | 0 | Suppose $T$ is a self-adjoint operator in $B(H)$ with $\sigma(T)$ a spectrum of $T$. $\mu$ is a spectral measure. For the operators having a generally continuous spectrum how to calculate the multiplicity function? Where multiplicity is usually called spectral multiplicity. Up to compact operators we know how to decomp... | https://mathoverflow.net/users/136400 | Computing multiplicity function for self adjoint operator with nonatomic spectral measure | A *measurable Hilbert bundle* is something of the form $\bigcup X\_n \times H\_n$ where $(X\_n)$ is a measurable partition of a $\sigma$-finite measure space $X$ and $H\_n$ is a Hilbert space of dimension $n$, for $n = 0, 1, 2, \ldots, \infty$. (I assume we're working with separable Hilbert spaces.) The associated Hilb... | 1 | https://mathoverflow.net/users/23141 | 324321 | 139,771 |
https://mathoverflow.net/questions/324325 | 5 | Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
1. Are there any lower bounds known for $\sum\_{p\leq x}|A(1,p)|^2$ or $\sum\_{n\leq x}|A(1,n)|^2$ ? (we know the lower bound $\sum\_{m^2n\leq x}|A(m,n)|^2\gg\_{\delta} x^{1-\delta}$
2. Is something known about the Dirichlet seri... | https://mathoverflow.net/users/136403 | Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms | For $(1)$, I believe the size would still be $\gg x^{1-\epsilon}$. However, if you allow $$\sum\_{p\sim x} |A(p,1)|^2+A|(p^2,1)|^2$$ a lower bound of similar sort is obtained by [Blomer-Maga's paper](https://academic.oup.com/imrn/article/2015/14/5311/779623) Corollary $4.3$. (In any case, one elementary way to know the... | 3 | https://mathoverflow.net/users/36735 | 324333 | 139,776 |
https://mathoverflow.net/questions/324337 | 3 | It is well-known that the BGG category $\mathcal{O}$ was introduced in the early 1970s by Joseph Bernstein, Israel Gelfand and Sergei Gelfand. I google for a while but I cannot find out the original paper for defining Category $\mathcal{O}$. Does anyone know where to find or what is the name of the paper? Please help m... | https://mathoverflow.net/users/110229 | Looking for access to original paper for Category O | Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand, [*Category of $\mathfrak{g}$-modules*,](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=2144&option_lang=eng) Functional Analysis and its Applications **10** (1976), 87–92. [MR0407097](https://mathscinet.ams.org/mathscinet-getitem?mr=... | 10 | https://mathoverflow.net/users/11260 | 324340 | 139,779 |
https://mathoverflow.net/questions/324341 | 0 | I have a following sum:
$S\_g=\sum\_{k=0}^g k\binom{4g+2}{2k}$
I can transform it into a different sum
$S\_g=(2g+1)\sum\_{k=1}^g\binom{4g+1}{2k-1}$
What is the closed form or what is the method to deal with any of above sums?
| https://mathoverflow.net/users/116167 | Specific partial sum of even/odd binomial coefficients | According to Maple,
$$ S\_g = \left( g + \frac12\right) \left(16^g - {4 g \choose 2g}\right) $$
| 1 | https://mathoverflow.net/users/13650 | 324343 | 139,781 |
https://mathoverflow.net/questions/324286 | 4 | I'm writing a software package to decompose group representations, and am struggling to find good examples of quaternionic-type representations of dimension > 4.
Reading [MathOverflow](https://mathoverflow.net/questions/54800/why-are-there-so-few-quaternionic-representations-of-simple-groups), I found that the [McLau... | https://mathoverflow.net/users/136343 | Which dimensions exist for irreducible quaternionic-type real representations of finite groups? | Let $\mathbf{H}$ be the skew field of real quaternions. Let $Q\subset \mathbf{H}^\*$ be the quaternion subgroup of order 8, namely $Q=\{\pm 1,\pm i,\pm j,\pm k\}$.
Let the wreath product $G\_n=\mathfrak{S}\_n\ltimes Q^n$ (of order $n!8^n$) act on $\mathbf{H}^n$ (viewed as right $\mathbf{H}$-module) as monomial matric... | 4 | https://mathoverflow.net/users/14094 | 324349 | 139,784 |
https://mathoverflow.net/questions/324351 | 2 | Suppose that $X$ is a general $(4,1)$ hypersurface in $\mathbb P^3 \times \mathbb P^1$, which we think of via $\pi : X \to \mathbb P^1$, the projection onto the second factor, as a family of K3 surfaces.
For a genenal $t \in \mathbb P^1$, $X\_t$ doesn't contain any $(-2)$ curves, but for special values of $t$ it does... | https://mathoverflow.net/users/128878 | Normal bundle of "extra" curve in a fiber | If you want to check the condition in practice, you can consider the following exact sequence
$$
0 \to N\_{L/X} \to N\_{L/\mathbb{P}^3 \times \mathbb{P}^1} \to N\_{X/\mathbb{P}^3 \times \mathbb{P}^1} \vert\_L \to 0.
$$
The second term is $\mathcal{O}(1) \oplus \mathcal{O}(1) \oplus \mathcal{O}$, the third term is $\mat... | 3 | https://mathoverflow.net/users/4428 | 324361 | 139,788 |
https://mathoverflow.net/questions/324371 | 2 | Consider the two sequences
$$a(n)=\sum\_{k=1}^n\binom{n}k\sum\_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$
and
$$b(n)=\sum\_{k=0}^n\binom{n}k^2k!$$
**QUESTION.** Is this true?
$$\frac{a(n)}{b(n)}\longrightarrow 0 \qquad \text{as} \qquad n\rightarrow\infty.$$
| https://mathoverflow.net/users/66131 | Decaying of a certain ratio of binomial sums | It is very much true. We may simplify the first sum by changing the order of summation:
$$\sum\_k \binom{n}k\binom{k}{2j}=\sum\_k\binom{n}{2j}\binom{n-2j}{k-2j}=2^{n-2j}\binom{n}{2j}.$$
Now the summand for $a(n)$ is $$2^{n-2j}\binom{n}{2j}\frac{(2j)!}{j!}=2^{n-2j}\frac{n!}{(n-2j)!j!}.$$
The denominator is not less th... | 4 | https://mathoverflow.net/users/4312 | 324375 | 139,792 |
https://mathoverflow.net/questions/263504 | 6 | Let $E/F$ be a quadratic field extension of p-adic fields. Let $V$ be a (skew-)Hermitian space and $U(V)$ be the unitary group. Let $GU(V)$ be the similitude unitary group. Given an irreducible smooth representation $\pi$ of $GU(V)$, do we know that the restriction $\pi|\_{U(V)}$ has multiplicity one?
For the pair $(... | https://mathoverflow.net/users/13466 | multiplicity one for restriction of representations from $GU$ to unitary group | In case you didn't already see it, this question has now been answered in the affirmative by Adler and Prasad in Theorem 12a here: $[$1$]$
$[$1$]$ Jeffrey D. Adler, Dipendra Prasad. Multiplicity upon restriction to the derived subgroup, 2018. ([arXiv link](https://arxiv.org/abs/1806.03635))
| 2 | https://mathoverflow.net/users/38495 | 324378 | 139,794 |
https://mathoverflow.net/questions/324366 | 8 | $\def\CC{\mathbb{C}}$Let $K = \CC(x\_1, \ldots, x\_n)$ and let $G$ be a countable group of automorphisms of $K$; in the cases I care about, $G \cong \mathbb{Z}$. Then the field of $G$-invariants, $K^G$, is an extension of $\CC$ of some transendence degree $d$. I would like to know under what circumstances I can say tha... | https://mathoverflow.net/users/297 | Dimension of orbit versus invariant functions | Yes for $G=\mathbb Z$, see Theorem 4.1 of this paper by [Amerik-Campana](https://arxiv.org/pdf/math/0510299.pdf).
| 7 | https://mathoverflow.net/users/605 | 324381 | 139,796 |
https://mathoverflow.net/questions/324011 | 7 | Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine permutations of $V$. That is for every $g\in G$ there exists a square invertible binary matrix $M$ and vector $b$ such that for eve... | https://mathoverflow.net/users/85489 | Cycle types of permutations from affine group |
>
> Are there any theoretical results, which provide information about all possible cycle types of affine permutations?
>
>
>
Yes: basic linear algebra.
There is no reason to restrict to a field on 2 elements, so let me assume that $K$ is an arbitrary finite field $K$.
(a) First, assume that $b=0$, i.e. let'... | 4 | https://mathoverflow.net/users/14094 | 324401 | 139,804 |
https://mathoverflow.net/questions/324407 | 4 | I am reading through the Dyer and Lashof's paper "Homology of Iterated Loop Spaces". They are quoting the following theorem:
If $A$ is a connected, free, associative, commutative, primitively generated mod $p$ Hopf algebra of finite type, then $A$ is isomorphic as a Hopf algebra to a tensor product of free monogenic ... | https://mathoverflow.net/users/123432 | "Free" Hopf algebra | 1) Yes, free means free as an algebra. (I don't think it could possibly mean anything else. "Free Hopf algebras" and "free coalgebras" as left adjoints to the forgetful functor don't exist.)
2) is elementary, I think. Let $H$ be a connected free monogenic Hopf algebra. Let $x$ be a generator. Thanks to the connectivi... | 10 | https://mathoverflow.net/users/36146 | 324411 | 139,807 |
https://mathoverflow.net/questions/324398 | 2 | I have a question about compact operators on Banach spaces.
Let $B$ be a real Banach space and $L$ a closed linear operator on $B$.
We assume that $L$ generates a contraction semigroup $\{T\_t\}\_{t>0}$ on $B$ .
**If $B$ is a Hilbert space** and $L$ is self-adjoint, the following assertions are equivalent:
(1) Th... | https://mathoverflow.net/users/68463 | Compact operators on Banach spaces and their spectra | The essential spectrum (and even the spectrum) of the generator of a contractive $C\_0$-semigroup on an $L^1$-space can be empty even if the generator does not have compact resolvent.
**Example.** Endow $[0,1]^2$ with the Lebesgue measure and define a $C\_0$-semigroup $(T\_t)\_{t \ge 0}$ on $L^1([0,1]^2)$ by
\begin{a... | 5 | https://mathoverflow.net/users/102946 | 324416 | 139,808 |
https://mathoverflow.net/questions/324415 | 1 |
>
> Given a finite connected graph, let $A$ be a set of edges such that each edge in $A$ is not part of a cycle. Suppose that no path contains all edges in $A$. Must it be true that for some three edges in $A$, no path contains all the three edges?
>
>
>
This is equivalent to showing that if every subset of thre... | https://mathoverflow.net/users/136454 | Three edges in a path | Extend $A$ to a spanning tree $T$ of $G$, which is possible greedily for any acyclic subgraph of a connected graph. Since $A$ cannot be covered by a single path, there is a vertex $v$ such that at least $3$ of the branches of $T$ at $v$ lead to components $C\_1, C\_2, C\_3$ containing edges $a\_1, a\_2, a\_3$ of $A$. W... | 2 | https://mathoverflow.net/users/25485 | 324429 | 139,812 |
https://mathoverflow.net/questions/324335 | 0 | Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection.
For example, if $X$ and $Y$ are embedded submanifolds of a manifold (or spaceform) $Z$, then the intersection of $X$ and $Y$ is a derived manifold of dimension $\dim X+\dim Y-\dim Z$, which can be n... | https://mathoverflow.net/users/111304 | Metrics on derived smooth manifolds | As far as I am aware, there is nothing in the literature that
treats Riemannian or pseudo-Riemannian metrics on derived smooth manifolds.
However, there is an extensive treatment of symplectic structures
on derived stacks by Pantev, Toën, Vaquié, and Vezzosi:
[Shifted symplectic structures](https://arxiv.org/abs/1111... | 6 | https://mathoverflow.net/users/402 | 324439 | 139,814 |
https://mathoverflow.net/questions/324237 | 4 | Consider the structure of the positive real numbers $(0, \infty) $ with its unit $1$, its addition $+$, its multiplication $\times $, and its strict ordering $> $.
Is this structure
$$( (0, \infty), 1, +, \times, >) $$
o-minimal?
| https://mathoverflow.net/users/136356 | Is the order arithmetic of the positive reals o-minimal? | Another way to see o-minimality is to note that $log$ induces an isomorphism
$$log : ((0,\infty), 1, +, \times, >) \cong (\mathbb R, 0, \oplus, +, >),$$
where $\oplus$ is the binary operation defined by $\oplus(x,y) = \log(e^x + e^y)$. But $\oplus$ is definable in $\mathbb R\_{exp}$. Hence the latter structure is a red... | 6 | https://mathoverflow.net/users/57712 | 324449 | 139,817 |
https://mathoverflow.net/questions/324373 | 11 | Is there a reason we consider [$\infty$-categories](https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category) to be the $\omega^{th}$ step in the 2-internalization inside **Cat** (or enrichment over **Cat** if you prefer)\* process made invertible above some finite ordinal, and don't continue on to higher steps in the... | https://mathoverflow.net/users/92164 | Higher $\infty$-categories | Using Street's "one type" definition of strict $\infty$-category one can see that the concept of "strict $P$-category" makes sense not justs for any ordinals $P$ but in fact for any posets $P$ (though I expect the definition below is not right when $P$ is not totally ordered, see the remark at the end)
**Definition**... | 14 | https://mathoverflow.net/users/22131 | 324468 | 139,822 |
https://mathoverflow.net/questions/324467 | 11 | I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
>
> **Do you know any suitable reference for the basics of complex cobordism?**
>
>
>
If possible, I would lik... | https://mathoverflow.net/users/12204 | Reference on complex cobordism | This is worked out in part 2 of
>
> *Adams, J. F.*, Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). [ZBL0309.55016](https://zbmath.org/?q=an:0309.55016).
>
>
>
(note that to understand part 2 you need to have... | 11 | https://mathoverflow.net/users/43054 | 324469 | 139,823 |
https://mathoverflow.net/questions/324465 | 12 | A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual finiteness of $G$ implies the same property for $G'$?
| https://mathoverflow.net/users/51663 | Is residual finiteness a quasi isometry invariant for f.g. groups? | No: let $Q$ be a non-abelian group of order 8. Then the standard lamplighter groups $(\mathbf{Z}/2\mathbf{Z})\wr\mathbf{Z}$ (which is RF) and the wreath product $Q\wr\mathbf{Z}$ (which is not RF: exercise; initially due to Gruenberg 1957) are QI.
Indeed, $(\mathbf{Z}/2\mathbf{Z})\wr\mathbf{Z}$ has a unique normal sub... | 17 | https://mathoverflow.net/users/14094 | 324470 | 139,824 |
https://mathoverflow.net/questions/324436 | 30 | By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a generalization of the semidirect product. As requested in the comments, we note that $10!=6!\cdot 7!$, so that the necessary cardi... | https://mathoverflow.net/users/128140 | Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$? | Here are a few comments and a slightly different approach, though we take advantage of some of the earlier comments. We first note the well-known (at least to people who work with factorizations) fact that if the finite group $G$ has a factorization of the form $G = AB$ with $A \cap B = 1 $ and $A,B$ subgroups, then we... | 20 | https://mathoverflow.net/users/14450 | 324474 | 139,826 |
https://mathoverflow.net/questions/324459 | 4 | We say that a hypergraph $(\mathbb{N}, E)$ where $E\subseteq {\cal P}(\mathbb N)$ is *perfectly dense* if
1. $\mathbb{N}\notin E$,
2. all $e\in E$ are infinite,
3. $e\_1, e\_2 \in E$ implies $|e\_1\cap e\_2| = 1$, and
4. for all $m\neq n\in \mathbb{N}$ there is $e\in E$ such that $\{m,n\}\subseteq e$.
If $(\mathbb... | https://mathoverflow.net/users/8628 | Are two "perfectly dense" hypergraphs on $\mathbb{N}$ necessarily isomorphic? | There are continuum-many pairwise non-isomorphic perfectly dense hypergraphs. Below is a sketch of a proof.
Given a countably infinite field $\mathbb{K}$, the projective plane $\mathbb{KP}\_2$ over $\mathbb{K}$ can be seen as a perfectly dense hypergraph, where vertices are points and edges are lines. I will show tha... | 3 | https://mathoverflow.net/users/120363 | 324475 | 139,827 |
https://mathoverflow.net/questions/324466 | 6 | Let $\mathcal{U}$ be (selective) ultrafilter and $\omega = \coprod\_{i<\omega}S\_i$ be partition of $\omega$ with small sets $S\_i\notin\mathcal{U}$. All $S\_i$ are infinite. Does there exist a system of bijections $\varphi\_i:\omega\to S\_i$ such that for any big set $B\in\mathcal{U}$ and any system of big sets $\{B\_... | https://mathoverflow.net/users/118366 | On infinite combinatorics of ultrafilters | Like your previous question, [Selective ultrafilter and bijective mapping](https://mathoverflow.net/questions/324254) , this fails for all nonprincipal ultrafilters $\mathcal U$ on $\omega$, and for essentially the same reason. If there were such bijections $\phi\_i$, then the function $f:\omega\to\omega$ that is const... | 7 | https://mathoverflow.net/users/6794 | 324478 | 139,829 |
https://mathoverflow.net/questions/324462 | 2 | In a directed graph, a vertex $a$ can reach every vertex, and every vertex can reach another vertex $b$. Can we always sort all the edges as $e\_1,e\_2,\ldots,e\_n$ so that every prefix $e\_1,e\_2,\dots,e\_i$ (when viewed as undirected edges) forms a connected subgraph, and similarly for every suffix $e\_i,e\_{i+1},\do... | https://mathoverflow.net/users/83212 | Vertex reachability in directed graph | Add to $G$ a new edge $e=\vec{ba}$, the new graph $G\cup e$ is strongly connected. Start with a directed cycle containing $e$, let it consist of edges $e\_1,e\_2,e\_3,\dots,e\_m,e$, where $e\_1$ is incident to $a$ and $e\_m$ is incident to $b$. Start with a sequence $e\_1,e\_2,\dots,e\_m$. Now I use an ear decompositio... | 3 | https://mathoverflow.net/users/4312 | 324487 | 139,833 |
https://mathoverflow.net/questions/324489 | 8 | Denote $\pmb{X}\_n=(x\_1,x\_2,\dots,x\_n)$. Consider the [symmetric polynomial](https://en.wikipedia.org/wiki/Symmetric_polynomial)
$$f\_n(\pmb X\_n)=\prod\_{1\leq i<j\leq n}(x\_i+x\_j).$$
Expand these in terms of [elementary symmetric polynomials](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial), say
$$f... | https://mathoverflow.net/users/66131 | The vanishing of sum of coefficients: symmetric polynomials | Choose $n$ numbers $x\_1,\dots,x\_n$ for which all elementary symmetric polynomials are equal to 1 and substitute them to our $f\_n$. We should get zero value for odd $n$. Well, what are these numbers? The roots of $x^{n}-x^{n-1}+x^{n-2}-\ldots-1=(x^{n+1}-1)/(x+1)$. This polynomial indeed has two roots with sum equal t... | 10 | https://mathoverflow.net/users/4312 | 324490 | 139,834 |
https://mathoverflow.net/questions/324437 | 17 | In the standard reference books *Locally presentable and accessible categories* (Adamek-Rosicky, Theorem 2.11) and *Accessible categories* (Makkai-Pare, $\S$2.3), it is shown that for regular cardinals $\lambda\le\mu$, the following are equivalent:
* Every $\lambda$-accessible category is $\mu$-accessible.
* For ever... | https://mathoverflow.net/users/49 | Raising the index of accessibility | Under GCH, if $\lambda < \mu$ are regular cardinals, then $\lambda \lhd \mu$ implies $\lambda \ll\mu$. The proof uses the following standard fact:
**Lemma.** *Suppose $\lambda \leq \gamma$ are infinite cardinals. Then $\gamma^{<\lambda} = 2^{<\lambda} \cdot \text{cf}(P\_\lambda(\gamma))$.*
*Proof.* Recall that si... | 9 | https://mathoverflow.net/users/102684 | 324492 | 139,836 |
https://mathoverflow.net/questions/324453 | 2 | Suppose $\Omega$ is a bounded smooth domain in $ R^N$ and $ a,b$ are bounded smooth vector fields in $ \Omega$. Suppose we have
$$ -\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in } \Omega$$
$$-\Delta v(x) + b(x) \cdot \nabla u(x) = g(x) \ge 0 \quad \mbox{ in } \Omega$$ with $u=v=0$ on $ \partial \Om... | https://mathoverflow.net/users/66623 | Maximum principle for an elliptic system | Without any further assumptions, there is no such maximum principle. For example, in dimension $N = 1$, the functions $$u(x) = -1 + \cos x , \qquad v(x) = -\sin x$$ satisfy the system of elliptic equations given in the question with $f(x) = g(x) = 0$, $a(x) = 1$ and $b(x) = -1$, and both are equal to zero at the endpoi... | 3 | https://mathoverflow.net/users/108637 | 324494 | 139,837 |
https://mathoverflow.net/questions/324391 | 14 | Suppose $G$ is a finitely generated group, and suppose $Rep\_k(G)$ is its category of representations over some field (or maybe even a ring) $k$, endowed with whatever extra structure is needed --- monoidal structure, fiber functor etc. Is it possible to reconstruct $G$ from this category?
Please note that my questio... | https://mathoverflow.net/users/43309 | Is it possible to reconstruct a finitely generated group from its category of representations? | Let $F: Rep\_k(G) \to k\mathsf{-mod}$ be the fibre functor. $G$ can be reconstructed from this functor and the monoidal structure as the group of automorphisms, because $t: G\to Aut(F,\otimes,k), g\mapsto (t\_g^V)\_{V\in Rep\_k(G)}$ is an isomorphism, where $t\_g^V$ is just the $k$-linear map $F(V)\to F(V), v\mapsto gv... | 13 | https://mathoverflow.net/users/3041 | 324500 | 139,840 |
https://mathoverflow.net/questions/285184 | 17 | Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open cover $\mathcal U$, we define the complex ${\check{\mathcal C}}^\bullet(\mathcal U,\mathbb Z)$ of Cech cochains relative the... | https://mathoverflow.net/users/110236 | Is there a complex which computes Cech cohomology? | This is the kind of thing sieves are good for. For an open cover, let $S$ denote the sieve it generates, so $S$ is poset of open subsets $V$ such that $V$ is contained in some element of the cover. A quasi-isomorphic model for the Cech complex is given by the "homotopy limit over the sieve", so the $0$th term in the co... | 21 | https://mathoverflow.net/users/3931 | 324509 | 139,843 |
https://mathoverflow.net/questions/324501 | 6 | The Cartan matrix $C$ of a finite quiver algebra $A$ with points $e\_i$ is defined as the matrix having entries $c\_{i,j}=\dim(e\_i A e\_j)$. The Cartan determinant is defined as the determinant of the Cartan matrix.
>
> **Question.** Who proved first that the Cartan determinant is an invariant of the derived categ... | https://mathoverflow.net/users/61949 | Derived invariance of the Cartan determinant | I suspect that this is one of those things that was essentially well-known to many people before anybody wrote it down, so it will be hard to pin down the first person to prove it.
But the main idea goes back to
*Happel, Dieter*, [**On the derived category of a finite-dimensional algebra**](http://dx.doi.org/10.10... | 7 | https://mathoverflow.net/users/22989 | 324515 | 139,844 |
https://mathoverflow.net/questions/324520 | 1 | Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|\_{X}$ by
$$
\| A \|\_{X} = \sup\_{x \ne 0} \frac{\|A x\|\_{X}}{\|x\|\_{X}}
$$
where the matrix $A$ is interpreted as an operator $X \to X$ in the obvious way.
I'm looking to define a minimal norm, namely
$$
f(A)
=
\inf\_{X}
\| A ... | https://mathoverflow.net/users/112001 | Minimal value of matrix norm induced by a norm | For finite matrices, your 'norm' is the spectral radius of $A$. Indeed, one can construct for each matrix $A$ a matrix norm induced by a vector norm such that $\|A\| \leq \rho(A) + \varepsilon$ for each $\varepsilon>0$. (And, on the other hand, $\|A\|\geq \rho(A)$ for each norm induced by a vector norm).
1. for each ... | 5 | https://mathoverflow.net/users/1898 | 324523 | 139,847 |
https://mathoverflow.net/questions/324510 | 5 | An equivalence class of permutations has come up in my research, and I'm wondering if anybody knows if it's named or has been studied before. If so, I'd appreciate being pointed towards more information.
Specifically, two permutations are considered equivalent if they have the same cycle decomposition, up to inverses... | https://mathoverflow.net/users/136502 | Equivalence class of permutations based on cycle decomposition and their inverses | If $f(n)$ is the number of equivalence classes, then
$$ \sum\_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{\exp\left(\frac x2+
\frac{x^2}{4}\right)}{\sqrt{1-x}}. $$
This goes back to Frobenius. It is also the number of distinct monomials in the expansion of the determinant of an $n\times n$ generic symmetric matrix. See *En... | 7 | https://mathoverflow.net/users/2807 | 324536 | 139,850 |
https://mathoverflow.net/questions/324512 | 5 | This is a [cross-post](https://math.stackexchange.com/questions/3103848/non-conformal-metrics-on-vector-bundles-where-nabla-g-omega-cdot-g).
Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\nabla$, s... | https://mathoverflow.net/users/46290 | Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$? | The answer is 'no'. For example, just take $M$ to be $\mathbb{R}^n$ (for $n>1$), and $E = M\times \mathbb{R}^r$ for some $r>1$. Let $\omega$ be *any* $1$-form on $M$, and define a connection $\nabla$ on $E$ by setting
$$
\nabla e\_i = -\tfrac{1}{2} \omega\otimes e\_i
$$
where $e\_i$ for $1\le i\le r$ is some basis for ... | 7 | https://mathoverflow.net/users/13972 | 324545 | 139,851 |
https://mathoverflow.net/questions/324497 | 2 | Let $p\_1, ... ,p\_k$ denote the probabilities of drawing bin $1, .. ,k$, where $\sum\_{i = 1}^{k} p\_i= 1$. My question is if we draw $n$ times, how can I show that the probability that all bins are drawn less than $C$ times is maximized by $p\_1 = p\_2 = ... = p\_k = 1/k$ ?
I've tried using conditional probability... | https://mathoverflow.net/users/136209 | Birthday problem extension to unequal probabilities and multiple collisions | Here is a proof using generating functions. Let $c\geq 2$ and $k\geq 2$ be fixed.
Let $X=(X\_1(n),\ldots,X\_k(n))$ be the $k$-tuple of occupancy numbers at time $n$, i.e. $X\_i(n)$ = number of bins of type $i$ drawn at time $n$.
Clearly $X$ has the multinomial distribution with parameters $n$ and $p=(p\_1,\ldots,p... | 3 | https://mathoverflow.net/users/48831 | 324548 | 139,852 |
https://mathoverflow.net/questions/324547 | 6 | Consider the first-order language $L\_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\mathrm{comp}$ is a ternary relation symbol. This is intended to be thought of as the language of a single category: $\mathrm... | https://mathoverflow.net/users/nan | Which branches of mathematics can be done just in terms of morphisms and composition? | I don't really know what you're after but here is an analogue of ETCS for topological spaces
>
> Dana I. Schlomiuk, *An elementary theory of the category of topological spaces*, Trans. Amer. Math. Soc. 149 (1970), 259-278, doi:[10.1090/S0002-9947-1970-0258914-7](https://doi.org/10.1090/S0002-9947-1970-0258914-7)
> ... | 10 | https://mathoverflow.net/users/4177 | 324550 | 139,853 |
https://mathoverflow.net/questions/312565 | 13 | I am searching for the article by Gerhard Frey, which has indicated a connection between Fermat's Last Theorem and the Taniyama-Shimura Conjecture. The reference is give as
>
> Gerhard Frey, Links between stable elliptic curves and certain diophantine equations, *Annales Universitatis Saraviensis* **1**, 1-40 (1986... | https://mathoverflow.net/users/63938 | Gerhard Frey, "Links between stable elliptic curves and certain diophantine equations" | Please find here the scanned version of the manuscript [Links between stable elliptic curves and certain diophantine equations](https://github.com/FrancescaRossi/frey) by Gerhard Frey.
| 10 | https://mathoverflow.net/users/136527 | 324554 | 139,854 |
https://mathoverflow.net/questions/324553 | 4 | Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
branching to Levi, or parabolic, subgroups.
| https://mathoverflow.net/users/125941 | GAP versus SageMath for branching to Lie subgroups | I don't know GAP but Sage has a nice tutorial for branching and is quite usable. It is however, slower than LiE which is on the other hand quite "basic" i.e. it requires you to write the branching code (example is in its documentation).
| 5 | https://mathoverflow.net/users/6818 | 324555 | 139,855 |
https://mathoverflow.net/questions/324564 | 1 | This question is motivated by Richard Stanley's answer to [this MO question](https://mathoverflow.net/questions/324510/equivalence-class-of-permutations-based-on-cycle-decomposition-and-their-inverse/324536#324536).
Let $g(n)$ be the number of distinct monomials in the expansion of the determinant of an $n\times n$ g... | https://mathoverflow.net/users/66131 | Counting monomials in skew-symmetric+diagonal matrices | The correct generating function is
$$ \exp\left( x +\frac{x^2}{2}+\frac 12\sum\_{n\geq 2}\frac{x^{2n}}{2n}\right)
=\frac{\exp\left(x+\frac{x^2}{4}\right)}{(1-x^2)^{1/4}}. $$
This appears in <http://oeis.org/A243107>, but without a combinatorial or algebraic interpretation. For skew symmetric matrices just multiply... | 5 | https://mathoverflow.net/users/2807 | 324571 | 139,862 |
https://mathoverflow.net/questions/324576 | 7 | Let $K$ be a local field of characteristic $0$ with valuation $v$. I think
$$\lim\_{\substack{q\in K\\q\to1}}\sum\_{n\ge0}\prod\_{j=1}^n\frac{q^j-1}{q-1}$$ converges to $\sum\_{n\ge0}n!\in K$ but I did not manage to prove it. Is my guess correct and if yes, can I have a hint or a proof of this fact?
Thanks in advanc... | https://mathoverflow.net/users/33128 | Convergence of a $p$-adic series | Is this question being asked just out of curiosity? As far as I know, the series $\sum\_{n \geq 0} n!$ is not important in $p$-adic analysis.
For $j \in \mathbf N$ and $q \not= 1$, let $(j)\_q = (q^j-1)/(q-1) = 1 + q + \cdots + q^{j-1}$. As $q \rightarrow 1$ we have $(j)\_q \rightarrow j$ so set $(j)\_1 = j$ for $j \... | 10 | https://mathoverflow.net/users/3272 | 324584 | 139,866 |
https://mathoverflow.net/questions/324569 | 3 | Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E\_0,...,E\_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$. Define inductively $E\_{n+i} = R\_{<E\_{i+1},...,E\_{n+i-1}>} E\_i$ and $E\_{-i} = L\_{<E\_{-i+1},...,E\_{-i+n-1}>} ... | https://mathoverflow.net/users/54337 | How to check that exceptional sequence of vector bundles on Fano variety is helix foundation | It is not enough to check only $E\_0$, but checking $E\_0,\dots,E\_n$ is enough.
There are many ways of proving fullness, most of them require first to construct some more objects from the considered exceptional collection, and after that relate the variety with another one, where an exceptional collection is alread... | 3 | https://mathoverflow.net/users/4428 | 324585 | 139,867 |
https://mathoverflow.net/questions/324513 | 7 | Call a function $f: [0, \infty) \to \mathbb R$ **eventually almost periodic** with period $p > 0$ if for all $x \in [0, p)$, the sequence ${f(x + np)}\_{n \in \mathbb N}$ converges.
Suppose $f: [0, \infty) \to \mathbb R$ is continuous and eventually almost periodic of periods $1$ and $a$, where $a$ is irrational and ... | https://mathoverflow.net/users/132446 | Eventually almost periodic functions | $F$ must be constant. Consider an $\epsilon>0.$
The sets $$C\_N=\{x\in[0,a)\mid |f(x+an)-f(x+am)|\leq \epsilon/3\text{ for all }n,m\geq N\}$$ are closed and cover $[0,a),$ so by the Baire category theorem there is an interval $[c,d]\subset C\_N$ for some $0<c<d<a$ and some $N.$ Shrinking the interval $[c,d]$ if necessa... | 8 | https://mathoverflow.net/users/112284 | 324586 | 139,868 |
https://mathoverflow.net/questions/324589 | 0 | Is there any way to solve the integration below? or make it simple to eliminate the Dirac-delta function?
$$\int\_{-\infty}^\infty m(x)\delta(G(x)-g\_c)f\_X(x)dx $$
where $f\_X(x)$ is a probability density function (PDF) of random variable x.
It will be very helpful for any reference or clue to solve it.
Thank ... | https://mathoverflow.net/users/136540 | Integration for Dirac-delta function | The usual way to solve problems such as this is to Fourier transform: Call your function $f(g\_c)$, then its Fourier transform
$$F(\xi)=\int\_{-\infty}^\infty f(g\_c)e^{i\xi g\_c}dg\_c=\int\_{-\infty}^\infty m(x)f\_X(x) e^{i\xi G(x)}\,dx$$
no longer contains the Dirac delta function. You can then recover $f(g\_c)$ by ... | 2 | https://mathoverflow.net/users/11260 | 324591 | 139,872 |
https://mathoverflow.net/questions/323871 | 6 | Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring.
Is there a good notion of a "left brace module" over a brace algebra?
I do not think the definition of a module over an algebra over an operad is the definition I'm looking for.
It should take any brace tree with white vertices lab... | https://mathoverflow.net/users/124286 | "Left Brace Module" | $\newcommand\P{\mathtt{P}}\newcommand{\M}{\mathtt{M}}$In general, the structure of a "module over an algebra over an operad" (a mouthful) is encoded by a *moperad* (module + operad).
If $\P = \{\P(n)\}\_{n \ge 0}$ is an operad, then a $\P$-moperad is a monoid in the category of right $\P$-modules. Concretely, this is... | 4 | https://mathoverflow.net/users/36146 | 324592 | 139,873 |
https://mathoverflow.net/questions/324504 | 5 | Let $G=(V,E)$ be a graph with minimum degree $\delta(G)=n\lt\aleph\_0$. Does $G$ necessarily have a spanning subgraph $G'=(V,E')$ which also has minimum degree $\delta(G')=n$ and is minimal with that property?
This question is easily answered in the affirmative if $G$ is locally finite or if $n\le1$. It already seems... | https://mathoverflow.net/users/43266 | Graphs with minimum degree $\delta(G)\lt\aleph_0$ | [Fedor Petrov](https://mathoverflow.net/users/4312/fedor-petrov) pointed out in a comment that the hypergraph question for $n=1$ was settled nicely by [Taras Banakh](https://mathoverflow.net/users/61536/taras-banakh) in [his answer](https://mathoverflow.net/questions/308265/minimal-covers-in-hypergraphs-with-finite-edg... | 2 | https://mathoverflow.net/users/43266 | 324597 | 139,876 |
https://mathoverflow.net/questions/324606 | 1 | If one define the universal abelian covering $M\_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, then what can one say about $H\_1(M\_0)$ ? Note that Hurewicz Theorem gives us a group isomorphism between $H\_1(M\_0)$ ... | https://mathoverflow.net/users/124800 | Homology of universal abelian cover of a manifold | This has not much to do with $H\_1(M\_0)$. If $\pi :M\_0\rightarrow M$ is your abelian covering, we have $\int\_{\tau }\overline{\omega} =\int\_{\pi \_\*\tau }\omega $. But the exact sequence $0\rightarrow \pi \_1(M\_{0})\rightarrow \pi \_1(M)\rightarrow H\_1(M)\rightarrow 0$ shows that $\pi \_\*\tau $ is zero in $H\_1... | 4 | https://mathoverflow.net/users/40297 | 324611 | 139,880 |
https://mathoverflow.net/questions/324570 | 0 | There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
| https://mathoverflow.net/users/132446 | Reference request: Baire class 2 functions | You may find something useful in Kuratowski, Topology, Volume 1 $\S 31$
| 3 | https://mathoverflow.net/users/112448 | 324617 | 139,882 |
https://mathoverflow.net/questions/324572 | 2 | Given a sequence of reals $(a\_n)\_{n > 0}$, let $f: [0, 1] \to R$ be the **generalised raindrop function** defined:
$f(x) = a\_q$ if $x$ is rational, with denominator $q$ in lowest form; $0$ otherwise.
**Questions:**
* What are necessary and sufficient conditions on $a\_n$for $f$ to be differentiable a.e.?
* If ... | https://mathoverflow.net/users/132446 | Generalised raindrop function | For question 2:
If $a\_n \to 0$ as $n \to \infty$, then $f$ is continuous at all irrationals, and thus a.e., as $\lim\_{t \to x} f(t) = 0$ for every $x$.
If $\limsup\_{n \to \infty} a\_n > \varepsilon > 0$, then $\{x: f(x) > \varepsilon\}$ is dense, and $f$ is discontinuous everywhere.
For question $2$: if $a\_q... | 3 | https://mathoverflow.net/users/13650 | 324620 | 139,884 |
https://mathoverflow.net/questions/324557 | 22 | Isbell gave, in [*Two set-theoretic theorems in categories* (1964)](https://eudml.org/doc/213746), a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in [*Concreteness* (1973)](https://www.sciencedirect.com/science/article/pii/0022404973900315), showed that ... | https://mathoverflow.net/users/2273 | Has the Isbell–Freyd criterion ever been used to check that a category is concretisable? | An inverse category can be defined as a category where every $f$ admits a unique regular inverse, i.e. a map $g$ such that $fgf=f$ and $gfg=g$. In [1], Kastl proves that any locally small inverse category admits a faithful functor into $PInj$, the category of sets and partial injections. The proof first verifies Isbell... | 11 | https://mathoverflow.net/users/136562 | 324623 | 139,886 |
https://mathoverflow.net/questions/324615 | 3 | Let
$$I(\sigma,T)=\int\_0^T |\zeta(\sigma+ i t)|^2 dt.$$
Unconditional bounds and asymptotics for $I(\sigma,T)$, $1/2\leq \sigma <1$, have been known since Hardy and Littlewood (see Chapter 7 of Titchmarsh). What about $0<\sigma<1/2$?
One can of course use the functional equation to try to obtain something useful. I ... | https://mathoverflow.net/users/398 | Bounding the second moment of $|\zeta(\sigma+i t)|^2$ for $0<\sigma<1$ | Ingham has a nice discussion of this in the introduction of his 1926 [paper.](https://mathscinet.ams.org/mathscinet-getitem?mr=1575391)
| 2 | https://mathoverflow.net/users/2627 | 324628 | 139,888 |
https://mathoverflow.net/questions/324625 | 5 | Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a multi-fusion category cannot be written as a non-trivial direct sum. Are all such irreducible multi-fusion categories fus... | https://mathoverflow.net/users/115363 | Are there irreducible multi-fusion categories that are not fusion categories? | Matrix categories, $\mathrm{End}(\mathrm{Vec}^{\oplus n})$. (The identity on each copy of $\mathrm{Vec}$ are summands of the identity.)
(You can generalize this example by putting fusion categories along the diagonal of a matrix, and Morita equivalences between them into the off-diagonal entries. Over an algebraicall... | 9 | https://mathoverflow.net/users/22 | 324630 | 139,889 |
https://mathoverflow.net/questions/324618 | 1 | Let
$$
D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}
$$
that is the critical strip without critical line.
I have to find if the following equation, with Gamma Euler function, has any root in $D$
$$
\Gamma(z)
=
\dfrac
{\pi^z}
{\cos
\left(
\dfrac{\pi}{2} \cdot z
\right)
\cdot
2^{1-z}
}... | https://mathoverflow.net/users/108867 | An equation with Gamma Euler function in critical strip | Imprecise answer: if you denote by $f(z)$ the log of the ratio of the two sides, we have $f(1-z)=-f(z)$ (I assume you constructed your function in that way). One now uses an
old theorem of Hermite, unfortunately I don't remember the exact statement and reference (he states it for polynomials, but it is easily generaliz... | 3 | https://mathoverflow.net/users/81776 | 324644 | 139,892 |
https://mathoverflow.net/questions/324657 | 2 | Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x\_i \in \mathbb C^2:$
$$ x\_{i} = \left(\begin{matrix} z &&-1 \\ 1 && 0 \end{matrix} \right)^ix\_0.$$
I would like to understand whether one can obtain a sharp bound on $\Vert x\_i... | https://mathoverflow.net/users/nan | Discrete dynamical system and bound on norm | This is an extension and correction of my comment, made more explicit.
---
Denote the normalised eigenvectors of $A$ by $u$ and $v$, and the corresponding eigenvalues by $\lambda$ and $\mu$. Since all norms on $\mathbb{R}^2$ are equivalent, we have
$$ C\_1(z) \max\{|\alpha|, |\beta|\} \leqslant \|x\| \leqslant C\... | 0 | https://mathoverflow.net/users/108637 | 324687 | 139,903 |
https://mathoverflow.net/questions/324710 | 2 | Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T\_X),H^1(X,T\_X),H^2(X,T\_X)$? Here $T\_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T\_X)$ gives the dimension of the automorphism group of $X$.
| https://mathoverflow.net/users/nan | Cohomology of tangent sheaf of a hypersurface | Use the normal sequence
$$
0 \to T\_X \to T\_{\mathbb{P}^n}\vert\_X \to N\_{X/\mathbb{P}^n} \to 0,
$$
exact sequences
$$
0 \to \mathcal{O}\_{\mathbb{P}^n} \to \mathcal{O}\_{\mathbb{P}^n}(d) \to i\_\*N\_{X/\mathbb{P}^n} \to 0
$$
(we identify here $N\_{X/\mathbb{P}^n}$ with $\mathcal{O}\_X(d)$ and denote by $i$ the embed... | 4 | https://mathoverflow.net/users/4428 | 324713 | 139,909 |
https://mathoverflow.net/questions/324715 | 4 | Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $\mathrm{Spec}\,k$ vanish?
If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^{sep}\rightarrow\mathrm{Spec}\,k$ which, I believe, is the initial object of the category of etale hypercovers. If we ... | https://mathoverflow.net/users/136557 | Do higher etale homotopy groups of spectrum of a field always vanish? | The étale topos of a field $k$ is just the topos of sets with a continuous $\mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $\mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $\mathrm{Gal}(k)... | 14 | https://mathoverflow.net/users/43054 | 324718 | 139,910 |
https://mathoverflow.net/questions/323098 | 2 | Let $R$ be a ring. Recall that a module $M$ is called reflexive in case the natural evaluation map $M \rightarrow M^{\*\*}$ (with $M^{\*}=Hom\_R(M,R)$) is an isomorphism. A module is reflexive if and only if its indecomposable summands are reflexive (at least that should be true for noetherian semiperfect rings), and t... | https://mathoverflow.net/users/61949 | Rings with only finitely many indecomposable reflexive modules | For any ring $R$, the functor $\mbox{Hom}\_R(-,R)$ induces a duality between the categories of left and right reflexive $R$-modules (see Corollary 19.40 in Lam's *Lectures on Modules and Rings* for a more general statement). Since the category of reflexive right (or left) $R$-modules is closed under finite direct sums ... | 4 | https://mathoverflow.net/users/11791 | 324720 | 139,911 |
https://mathoverflow.net/questions/324717 | 4 | Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the classic books of Goursat and Titchmarsh.
| https://mathoverflow.net/users/126532 | Reference request: Oldest complex analysis books with (unsolved) exercises? | It seems the first textbook on complex analysis was J.C. Bouquet and C.A. Briot, 1859, [Théorie des fonctions doublement periodiques et, en particulier, des fonctions elliptiques](https://gallica.bnf.fr/ark:/12148/bpt6k99569b.image)
source: [The Real and t... | 5 | https://mathoverflow.net/users/11260 | 324729 | 139,914 |
https://mathoverflow.net/questions/324732 | 1 | MacMahon's enumeration of all [plane partions (PP)](https://en.wikipedia.org/wiki/Plane_partition) inside an $n$-cube generalizes to
$${\tt PP\_n}(q)=\prod\_{i,j,k=1}^n\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.$$
A $q$-analogue of symmetric plane partitions in an $n$-cube
$${\tt SPP\_n}(q)=\prod\_{i,j=1}^n\frac{1-q^{i+j+n-1}... | https://mathoverflow.net/users/66131 | $q$-plane partitions & specialization & interlinks | Stembridge's "$q=-1$ phenomenon" (the precursor to the cyclic sieving phenomenon) was developed precisely to explain these kind of evaluations of generating functions for plane partitions at $-1$. See Stembridge's ["Some hidden relations involving the ten symmetry classes of plane partitions"](https://www.sciencedirect... | 2 | https://mathoverflow.net/users/25028 | 324733 | 139,917 |
https://mathoverflow.net/questions/324734 | 2 | Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.
>
> **QUESTION.** Let $r=\sqrt{x^2+y^2}$. Is this true?
> $$\int\_Cr\,ds\geq 2\cdot Area(R).$$
> Equality iff $C$ is a circle centered at the... | https://mathoverflow.net/users/66131 | An isoperimetric inequality for curve in the plane? | Expanding on the comment of RBega2:
Let $(x(t),y(t))$, $t\in [0,1]$ be a parametrization of $C$. From [Green's Theorem](http://mathworld.wolfram.com/GreensTheorem.html),
$$\int\_C-y\,dx+x\,dy=\iint\_R2\,dxdy=2\cdot Area(R).$$
From [Cauchy-Schwartz](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality) ine... | 7 | https://mathoverflow.net/users/121665 | 324736 | 139,918 |
https://mathoverflow.net/questions/324595 | 1 | *Note: originally posted on [math.SE](https://math.stackexchange.com/questions/3132992/valid-metric-on-a-hyperbolic-space).*
I'm looking at the distance that's defined in [this paper on Poincaré Embeddings](https://papers.nips.cc/paper/7213-poincare-embeddings-for-learning-hierarchical-representations.pdf):
$d(\mat... | https://mathoverflow.net/users/136544 | Valid metric on a hyperbolic space | I'm not sure about directly proving the triangle inequality, but it *does* follow from the fact that the distance metric is induced by a Riemannian metric.
**In general a Riemannian metric induces a distance metric (necessarily satisfying the triangle inequality)**
See for example: <https://en.wikipedia.org/wiki/Ri... | 2 | https://mathoverflow.net/users/78645 | 324740 | 139,920 |
https://mathoverflow.net/questions/324745 | 0 | Problem
=======
This is the first time I have posted a question on this site and it may not be suitable for this venue, which is primarily used for research questions in maths. If someone finds it unsuitable, it could be migrated to other sites like Maths StackExchange.
In a [paper](https://arxiv.org/pdf/1712.06541... | https://mathoverflow.net/users/136423 | Existence of rank-1 weight matrix in some type of deep neural network | I just skimmed the paper, but it seems that this is later formalized into Theorem 3, a proof of which is in the appendix.
| 1 | https://mathoverflow.net/users/1898 | 324753 | 139,923 |
https://mathoverflow.net/questions/324723 | 1 | Let $\textrm{cd}(G)=\lbrace \chi(1)\,|\, \chi\in\textrm{Irr}(G)\rbrace$ denote the set of character degrees of a finite group $G$. Similarly, denote by $\textrm{mcd}(G)$ the set of monomial character degrees. I have proved the following statement:
There exists no finite group $G$ of odd order such that:
* $\textrm{... | https://mathoverflow.net/users/128914 | Even Counterexample to Statement About the Non Existence of Certain Groups with Two Irreducible Monomial Character Degrees | $\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\mcd}{mcd}
\newcommand{\C}{\mathbb{C}}$Such groups do not exist. Indeed, suppose that $G$ has even order and satisfies the three conditions. The third condition forces $p$ to be $2$. Let $\chi$ be an irreducible character of $G$ of degr... | 4 | https://mathoverflow.net/users/35416 | 324763 | 139,926 |
https://mathoverflow.net/questions/324756 | 4 | Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi\_1(S^3-K)\to SL(2,{\mathcal O}\_3)\subset SL(2,C)$. This representation is not conjugate into $SU(2)$, because traces are not r... | https://mathoverflow.net/users/39082 | Hyperbolic Dehn surgeries and SU(2)-representations | All Dehn surgeries on the figure eight knot $K$ admit irreducible $SU(2)$ representations. This can be proved using Corollary 4.8 of my paper with John Baldwin, "Stein fillings and $SU(2)$ representations", [arXiv:1611.05629](https://arxiv.org/abs/1611.05629). The case of 0-surgery follows from Kronheimer and Mrowka's ... | 8 | https://mathoverflow.net/users/428 | 324765 | 139,927 |
https://mathoverflow.net/questions/324747 | 4 | I'm largely following the definitions of [this](https://home.sandiego.edu/~shulman/papers/pstonecech.pdf) paper, but I will replicate the relevant ones here.
I'm taking a pseudotopological space to be a set $X$ together with a relation $\rightarrow$ on the set $\mathscr{UF}X\times X$ (called "converges to" or "conver... | https://mathoverflow.net/users/83901 | Nonexistence of a 'product universal' compact Hausdorff pseudotopological space? | Given such a space $Y$ we can find a space $X$ with distinct $a,b\in X$ such that every continuous $f:X\to Y$ satisfies $f(a)=f(b).$ This rules out any such embedding.
Let $\kappa$ be a regular cardinal larger than $\max(\aleph\_0,|Y|)$ and take $X$ to be the pseudotopological space on $\kappa\cup \{a,b\}$ defined by... | 5 | https://mathoverflow.net/users/112284 | 324773 | 139,928 |
https://mathoverflow.net/questions/324746 | 6 | This question quotes from this [article](https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/beilin.pdf), but I've noticed this pattern in the literature I've read.
>
> "The values or
> better the leading coefficients at integral arguments of the L-functions of algebraic varieties over number fields seem to be clos... | https://mathoverflow.net/users/123309 | Is there research on the special values of the zeta function outside the integers? | One research paper that might qualify for what you are searching: [On the values of the Riemann zeta-function at rational arguments](https://hal.archives-ouvertes.fr/hal-01109799/document), S. Kanemitsu, Y. Tanigawa and M. Yoshimoto (2001).
>
> We give a closed form evaluation of Ramanujan’s type of the values of
>... | 4 | https://mathoverflow.net/users/11260 | 324775 | 139,930 |
https://mathoverflow.net/questions/324797 | 17 | A recently asked question (linked [here](https://mathoverflow.net/q/324705/9924)) deals with the remarkable identity
$$ \sum\_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$
where $\mathrm{sinc}(x)=\sin(x)/x$.
It is easy to construct functions $f$ other than $\mathrm{sinc}(x)$ such that $\sum\_{n\i... | https://mathoverflow.net/users/47453 | When is $\sum_{n\in\mathbb Z} f(x+n)$ constant? | If the Fourier transform $F(k)$ of $f(x)$ vanishes outside of the interval $(-1,1)$ then, by virtue of [Poisson summation,](https://en.wikipedia.org/wiki/Poisson_summation_formula)
$$\sum\_{n=-\infty}^\infty f(x+n)=\sum\_{n=-\infty}^\infty F(n)e^{2\pi inx}=F(0)$$ independent of $x$.
An example is $F(k)=k^2-a^2$ for $... | 34 | https://mathoverflow.net/users/11260 | 324799 | 139,935 |
https://mathoverflow.net/questions/324792 | 5 | Why does each integer $x$ between two consecutive primes have at least one non-trivial divisor that *unique on set* of all integers between these two consequtive primes except $x$?
We call a divisor $d$ of a integer $x$ *unique on a set* of integers $Q$, if there is no number from $Q$ divisible by it.
Perhaps this ... | https://mathoverflow.net/users/37289 | Why does each integer between two consecutive primes have at least one "unique" non-trivial divisor? | If we consider divisors to mean prime divisor, then this holds for most short intervals (less than length 6) and fails for some long intervals. 120 and 125 and 126 are smooth numbers between the same two consecutive primes; 24 and 27 are two others. In general, if you have an interval of 2k+1 consecutive composite numb... | 4 | https://mathoverflow.net/users/3402 | 324803 | 139,936 |
https://mathoverflow.net/questions/324793 | 9 | Let $A = (a\_0,a\_1,\ldots,a\_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \ast A$).
Is it true that for any such sequence $A$, there exists $n$ such that $A^{\ast n}$ is log-concave?
As an e... | https://mathoverflow.net/users/69564 | Log-concavity of repeated convolution | Odlyzko and Richmond proved your conjecture:
<http://www.dtc.umn.edu/~odlyzko/doc/arch/unimodal.convolut.pdf>
Odlyzko, A. M.(1-BELL); Richmond, L. B.(3-WTRL)
On the unimodality of high convolutions of discrete distributions.
Ann. Probab. 13 (1985), no. 1, 299–306.
| 5 | https://mathoverflow.net/users/69800 | 324811 | 139,939 |
https://mathoverflow.net/questions/324767 | 3 | Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff space $X$ such that $A \cong C(X)$, the $\*$-algebra of complex-valued continuous functions on $X$. The space $X$ is in fact... | https://mathoverflow.net/users/127523 | Ultraweak topology in abelian von Neumann algebras | მამუკა ჯიბლაძე hints that "hyperstonian" is an important definition here. I have struggled to find a good internet reference, and so am following Section 1 of Chapter III of Takesaki's book.
A *Stonian* space is a compact Hausdorff [Extremally disconnected space](https://en.wikipedia.org/wiki/Extremally_disconnected_... | 5 | https://mathoverflow.net/users/406 | 324814 | 139,941 |
https://mathoverflow.net/questions/324824 | 2 | I have a question related to [this one](https://mathoverflow.net/questions/141999/how-to-efficiently-sample-uniformly-from-the-set-of-p-partitions-of-an-n-set). For $n,p \in \mathbb{N}\_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all *equipartitions* of $n$ in $p$ sets; i.e., in sets of equal size ... | https://mathoverflow.net/users/37266 | How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set? | Randomly permute $n$ and then divide into blocks of size $n/p$.
| 5 | https://mathoverflow.net/users/1907 | 324825 | 139,944 |
https://mathoverflow.net/questions/324742 | 9 | Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
>
> $(\ast) \quad$ $\mathrm H^0 \big( X, \Omega\_X^p \otimes \mathscr L^{-1} \big) = 0$ for all $p < \dim X$ and ample line bundles $\mathscr L$... | https://mathoverflow.net/users/44860 | Nakano vanishing in positive characteristic | There exists singular Fano varieties of dimension $n\ge 3$ in characteristic $p$ ($p$ small when compared to $n$) with a non-zero section of $\Omega^{n-1}\_X\otimes \mathcal L^\*$ for an ample $\mathcal L$.
The existence of these examples is established in Kollár’s paper [Nonrational hypersurfaces](https://www.ams.org/... | 4 | https://mathoverflow.net/users/605 | 324827 | 139,945 |
https://mathoverflow.net/questions/324819 | 2 | I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:
For a compact set $E\subseteq \mathbb R^n$, we define its distance set $\Delta(E)\subseteq [0,\infty)$ to be:
$$
\Delta(E)=\{|x-y|:x,y\in E\... | https://mathoverflow.net/users/111012 | Failure of Falconer distance problem in one dimension | The answer to the second question (and thus the first) is yes, and the reference is Section 4.12 of P. Mattila, *Geometry of Sets and Measures in Euclidean Spaces*, CUP, 1995. The set in question is a Cantor type set, constructed by taking intersections of an increasing number of spaced intervals of decreasing lengths.... | 3 | https://mathoverflow.net/users/118731 | 324829 | 139,946 |
https://mathoverflow.net/questions/324796 | 3 | I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."
Let $X$ be a proper geodesic hyperbolic space. $X$ is called cobounded if there exists $R>0$ such that for all $x,y \in X$ there e... | https://mathoverflow.net/users/134603 | Criterion for visuality of hyperbolic spaces | Let $x,y \in X$ be two points. If $X$ is bounded, there is nothing to prove. Otherwise, $X$ must have a boundary of cardinality at least two. So we can fix two distinct points $\zeta, \xi \in \partial X$, and a bi-infinite geodesic $\gamma$ between them. As $X$ is cobounded, up to translating $\gamma$ by an isometry, w... | 3 | https://mathoverflow.net/users/122026 | 324832 | 139,947 |
https://mathoverflow.net/questions/324807 | 3 | Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A [theorem](https://projecteuclid.org/euclid.kjm/1250524785) by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth then all the automorphisms of $X$ are induced by automorphisms of $\ma... | https://mathoverflow.net/users/nan | Automorphisms of singular hypersurfaces | According to Theorem 1.1 of
M. J. Bradley, H. J. D’Souza: *On the orders of automorphism groups of complex projective hypersurfaces*, Lanteri, A. (ed.) et al., Geometry of complex projective varieties. Proceedings of the conference, Cetraro, Italy, May 28-June 2, 1990. Rende: Mediterranean Press, Semin. Conf. 9, 75-8... | 3 | https://mathoverflow.net/users/7460 | 324837 | 139,948 |
https://mathoverflow.net/questions/324808 | 8 | Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $F:M\leftrightarrow N:i$ where $i$ is the embedding functor and $(i \circ F)^{2}=i \circ F=L$. If $N$ is a combinatorial ... | https://mathoverflow.net/users/136683 | Localization, model categories, right transfer | The statement is unfortunately not true in full generality, I will give a counter example at the end.
What I know is that:
* One in general gets a "Right semi-model structure" on $M$ with the properties you want (see below)
* If one assumes the localization is left exact then one gets a Quillen model structure on ... | 7 | https://mathoverflow.net/users/22131 | 324843 | 139,949 |
https://mathoverflow.net/questions/324800 | 9 | This question was asked on MSE some time ago, [here](https://math.stackexchange.com/questions/3128810/characters-of-orthogonal-groups-as-symmetric-functions), but got no attention.
The Schur functions are characters of irreps of the unitary group, $s\_\lambda(U)=Tr(R\_\lambda(U))$. They are symmetric functions of the... | https://mathoverflow.net/users/78061 | Characters of orthogonal groups as symmetric functions | I might be totally wrong, but the analogues of Schur functions in the orthogonal case, the so called [orthogonal characters](https://www.math.upenn.edu/~peal/polynomials/schurMisc.htm#schurOrthogonal)
are not polynomials in just the $x\_i$, but polynomials in $x\_i^{\pm 1}$. You can perhaps treat the negative alphabet ... | 4 | https://mathoverflow.net/users/1056 | 324844 | 139,950 |
https://mathoverflow.net/questions/324845 | 4 | I would like some examples of groups $G$ satisfying all of the following criteria:
1. $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive.
2. $G$ has a regular suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts regularly.
3. $G$ has a non-faithful suborbit, i... | https://mathoverflow.net/users/801 | Example of primitive permutation group with a regular suborbit and a non-faithful suborbit | It is easy to find examples that satisfy conditions 1 - 3. I searched through the Atlas of Finite Simple Groups, looking for simple groups with maximal subgroups that might provide examples, where we take $G$ to be the image of the action of the simple group $S$ on the cosets of the chosen maximal subgroup $H$, and I f... | 3 | https://mathoverflow.net/users/35840 | 324852 | 139,951 |
https://mathoverflow.net/questions/324724 | 1 | Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $b$ copies of the $n$-pushoff of $K$ with the opposite orientation as $K$. In particular, $K(1,0) = K$ and $K(0,1)$ is the... | https://mathoverflow.net/users/99414 | Signature/nullity function for a link obtained by parallel pushoffs of a knot? | As Marco mentions, it appears you are studying cables. These are particular examples of satellite operations. As you already noticed, Litherland proves that if $S$ is a winding number $n$ satellite with pattern $P$ and companion $C$, then
$$ \sigma\_S(\omega)=\sigma\_P(\omega)+\sigma\_C(\omega^n). $$
For the nullity, ... | 5 | https://mathoverflow.net/users/36098 | 324859 | 139,954 |
https://mathoverflow.net/questions/324840 | 0 | I have read in the paper by Einsiedler, Lindenstrauss, Michel and Venkatesh on Duke's Theorem the following bound that I don't understand:
Let $d$ be a positive non-square interger and set let $K = \mathbb{Q}(\sqrt{d})$ and denote by $\mathcal{O}\_K$ the ring of integers of $K$. We furthermore assume that $d$ is a d... | https://mathoverflow.net/users/122635 | Bound on number of proper ideals of norm equal to n | Here is a solution for counting ideals of a given norm in $\mathcal O\_K$ with $K$ being a general number field. By relating the count of *proper* ideals with a given norm in a non-maximal order to counting ideals with a given norm in the maximal order containing it, I suspect the case of non-maximal orders could be de... | 3 | https://mathoverflow.net/users/3272 | 324861 | 139,955 |
https://mathoverflow.net/questions/324830 | 2 | The center of $K(H)$ is 0 and $K(H)$ has no nonzero finite dimensional representation. Can we conclude that if the center of a $C^\*$-algebra $A$ is zero, then $A$ has no nonzero finite dimensional representation?
If the above conclusion is not true, does there exist a counterexample?
| https://mathoverflow.net/users/63864 | center of $C^*$-algebra and finite dimensional representation | Let $P$ be a rank $1$ projection in $K(H)$ and let $\mathcal{A}$ be the set of continuous functions $f: [0,1]\to K(H)$ with $f(0) = \alpha P$ for some $\alpha \in \mathbb{C}$. Operations are pointwise. The map $f \mapsto \alpha$ is then a complex homomorphism, i.e., a one dimensional representation, but the center of $... | 3 | https://mathoverflow.net/users/23141 | 324863 | 139,956 |
https://mathoverflow.net/questions/324868 | 5 | I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here.
Let $X\_1,X\_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X\_i]=0$, $0<\operatorname{Var}(X\_i)<\infty$ ($m$-dependent means that each $X\_i$ is independen... | https://mathoverflow.net/users/112489 | Variance of sum of $m$ dependent random variables | First, the random variable (r.v.) $Y$ plays no role here, since $Y/\sqrt n\to0$.
Second, $\sigma^2$ may be zero. However, in the abstract of [Janson](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&ved=2ahUKEwjzvpeOs_DgAhUh4IMKHcsUDvUQFjAHegQIABAC&url=https%3A%2F%2Farxiv.org%2Fpdf%2F1312.1563&usg=AOv... | 6 | https://mathoverflow.net/users/36721 | 324872 | 139,961 |
https://mathoverflow.net/questions/324867 | 40 | Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known:
* $X$ canonically embeds into $\beta X$ (by taking principal ultrafilters);
* If $X$ is finite, then there are no non-principal ultrafilters, so $\beta X = X$.
* If $X$ is infinite, then (assuming choice) we have $|\beta ... | https://mathoverflow.net/users/39521 | Ultrafilters as a double dual | This is a quite standard idea in functional analysis. Let $X$ be any set and let $c\_0(X)$ be the space of all functions from $X$ to $\mathbb{C}$ which go to zero at infinity. Then the algebra homomorphisms from $c\_0(X)$ to $\mathbb{C}$ are precisely the point evaluations at elements of $X$, i.e., the *spectrum* of $c... | 32 | https://mathoverflow.net/users/23141 | 324879 | 139,964 |
https://mathoverflow.net/questions/324285 | -4 | Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"):
>
> Definition 8. A cardinal $\kappa$ is measurable iff it is the critical point of a non-trivial elementary embedding $j$: $V$ $\rightarrow$ $M$, for some transi... | https://mathoverflow.net/users/20597 | Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism? | Paring the philosophy away (which I think just obscures things) this seems to just be a question of **definability** - the key point being that Hamkins' recursively-defined translation **works for any formula in the language of set theory**. The elementary embeddings definition of measurability *isn't* such a formula s... | 3 | https://mathoverflow.net/users/8133 | 324880 | 139,965 |
https://mathoverflow.net/questions/324887 | 8 | I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about ***algebraic varieties*** (page 479). Since I still don't have the permission to add images I quote the relevant excerpt:
>
> ...The notion of properness has been introduced in 9.5/4. It means that the
> stru... | https://mathoverflow.net/users/108274 | Why is, for a group scheme of finite type, "smooth" (resp. irreducible) equivalent to "geometrically reduced" (resp. geometrically irreducible)? | Let $G/K$ be a group scheme of finite type.
1. $G/K$ is smooth if and only if $\bar G / \bar K$ is smooth. Suppose $\bar G$ is reduced, then it has a smooth $\bar K$-point $x$ (because we are over an algebraically closed field). But $\bar G(\bar K)$ acts transitively on itself, so now every closed point of $\bar G$ i... | 8 | https://mathoverflow.net/users/3847 | 324889 | 139,969 |
https://mathoverflow.net/questions/324874 | 5 | While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have
$${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D} {\mathbf n}$$
for all ${\mathbf n}$, where ${\mathbf R}$ is a $K$ times $K$ positive definite matrix and the diagonal... | https://mathoverflow.net/users/38730 | Inverse of a matrix and the inverse of its diagonals | Let me change the notations and reformulate the problem. You have a positive definite $n\times n$ ($n$ is your $K$) matrix $R$ with diagonal $D$ (your $D$ is $n$ times less than mine), and you have to prove that $nR^{-1}-D^{-1}$ is non-negative definite. Denote $R=D^{1/2}QD^{1/2}$, then $Q=D^{-1/2}RD^{-1/2}$ is a posit... | 5 | https://mathoverflow.net/users/4312 | 324892 | 139,970 |
https://mathoverflow.net/questions/324894 | 6 | Let $p\_n$ be a probability distribution on the positive
integers $n$. Let
$$ \frac{1}{1-\sum\_{n\geq 1} p\_nx^n}=\sum\_{k\geq 0}a\_kx^k. $$
Suppose there does not exist an integer $d>1$ such that
$d|n$ whenever $p\_n\neq 0$. I remember once seeing a proof
of the result
$$ \lim\_{k\to\infty} a\_k = \frac{1}{\sum\_{n\... | https://mathoverflow.net/users/2807 | A limit obtained from a probability distribution on the positive integers | This is proved in Spitzer's "Principles of Random Walk", claim P3 of Section 9 (p.100 of the second edition).
| 7 | https://mathoverflow.net/users/8588 | 324896 | 139,973 |
https://mathoverflow.net/questions/324885 | 3 |
>
> Is there a division algebra $D$ with center $K$ that satisfies the
> following 3 conditions?
>
>
>
1) $D$ is of infinite dimension over $K$;
2) every element of $D$ is algebraic over $K$;
3) $D$ is finitely generated (as division $K$-algebra).
| https://mathoverflow.net/users/10482 | Infinite dimensional finitely generated algebraic division algebra | This is a fairly well known old and open (as far as I know) problem: Kurosh’s Problem for division rings. See, for example, Question 3 in Agata Smoktunovicz’s [2006 ICM talk](http://icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_12.pdf).
| 8 | https://mathoverflow.net/users/22989 | 324899 | 139,976 |
https://mathoverflow.net/questions/324855 | 7 | Given the Dedekind eta function $\eta(\tau)$, define,
$$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$
$$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$
$$\quad\gamma(\tau) =\frac{\alpha(\tau)}{\beta(\tau)} =\frac{\sqrt2\,\eta(4\tau)}{\eta(\tau)}$$
Note that $\alph... | https://mathoverflow.net/users/12905 | Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$? | For your question $2$ all that is needed is a quick search of 'eta07.gp' for three terms identities with even rank and with all even exponents. I turned
up $t\_{12,12,40}$ and the other $5$ members of the $6$ cluster. They are: $$t\_{12,12,40},\; t\_{12,12,56},\; t\_{12,24,72},\; t\_{12,24,90},\; t\_{12,24,120},\; t\_{... | 5 | https://mathoverflow.net/users/113409 | 324904 | 139,979 |
https://mathoverflow.net/questions/324907 | 2 | There are various kinds of ([convergence of random variables](https://en.wikipedia.org/wiki/Convergence_of_random_variables)) but I have never read about convergence of density functions.
Let $X\_1, X\_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $f\_1, f\_2, \dots, f$ be those density functions $\mat... | https://mathoverflow.net/users/136731 | Convergence of probability density function | Such theorems, called local (central) limit theorems, are well known; see e.g. [The Encyclopedia of Mathematics](https://www.encyclopediaofmath.org/index.php/Local_limit_theorems).
| 1 | https://mathoverflow.net/users/36721 | 324909 | 139,981 |
https://mathoverflow.net/questions/324905 | 4 | The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S\_n=X\_1+\cdots+X\_n$ of independent random variables $X\_i$ with bounded third moments. Consider the following example: let $\varepsilon\_{i}$ stand for independent Rademac... | https://mathoverflow.net/users/24494 | Rate of decay in the multivariate Central Limit Theorem | It follows from the main result in the paper
>
> MR1309710 (95k:60015) Nagaev, S. V.; Chebotarëv, V. I. On the
> Edgeworth expansion in a Hilbert space. (Russian) Limit theorems for
> random processes and their applications, 170--203, 304, Trudy Inst.
> Mat., 20, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., No... | 4 | https://mathoverflow.net/users/36721 | 324911 | 139,982 |
https://mathoverflow.net/questions/324406 | 30 | In a recent lecture at the *Vladimir Voevodsky's Memorial Conference*, Deligne wondered whether "the axioms used are strong enough for the intended purpose." (See his remark from roughly minute 40 in this video: [What do we mean by "equal" ? - by Pierre Deligne](https://www.youtube.com/watch?v=WfDcrN5_1wA).)
Can any... | https://mathoverflow.net/users/12884 | Deligne's doubt about Voevodsky's Univalent Foundations | It is a bit difficult to understand what he is asking. The [already-linked nForum discussion](https://nforum.ncatlab.org/discussion/9059/what-is-deligne-asking/) includes some clarification about his example, which at the meeting took us a while to figure out.
More broadly, I think that what he is asking is whether e... | 24 | https://mathoverflow.net/users/49 | 324914 | 139,984 |
https://mathoverflow.net/questions/324692 | 4 | In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
<http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/branching_rules.html#sage.combinat.root_system.branching_rules.BranchingRule.Stype>
Explicitly, one is branchin... | https://mathoverflow.net/users/125941 | Branching to Levi subgroups in SAGE and the circle action | Branching to Levi subalgebras should really take into account the central part of the Levi subalgebra but it is not the case. The problem is that the `WeylCharacterRing` is defined only for semisimple Lie algebras and not for reductive ones. The mathematical side of the problem is easy -- decompose orthogonally your Ca... | 3 | https://mathoverflow.net/users/6818 | 324926 | 139,989 |
https://mathoverflow.net/questions/324921 | 1 | In the context of percolation, e.g., bond/site percolation, random graph connectivity in 2-3 dimensions, etc., once the percolation threshold is reached, that is the system is spanned by an infinite cluster (or in the finite case a cluster connecting two boundaries of the system), the behaviour of backbone of this clus... | https://mathoverflow.net/users/115841 | Figuring out a consistent definition for the percolation backbone | 
[source](https://arxiv.org/abs/cond-mat/9910236)
Both striped sites and black sites belong to the percolation cluster, but only the black sites are part of the backbone. The backbone can be defined as the set of current carrying paths from A to B, or equivalently... | 2 | https://mathoverflow.net/users/11260 | 324928 | 139,990 |
https://mathoverflow.net/questions/324769 | 7 | Let $N$ be a closed, connected, oriented hypersurface of $\mathbb{R}^n$. Such a manifold inherits a volume form from the usual volume from on $\mathbb{R}^n$ and has an associated volume given by integration of the inherited volume form over $N$.
I would like to know if there is a proof of the following intuivitely ob... | https://mathoverflow.net/users/86065 | Volume of manifolds embedded in $\mathbb{R}^n$ | I presume by the volume form on $N$ inherited from $\mathbb{R}^n$, you mean the induced hypersurface area measure. I'll write $N\_{\epsilon} = \psi\_{\epsilon}^X (N)$
The answer to your question is
* **Yes** if $N$ is mean convex,
* **Maybe** otherwise depending on $\int\_{\partial N} \langle X, \nu \rangle H d\sig... | 5 | https://mathoverflow.net/users/78645 | 324929 | 139,991 |
https://mathoverflow.net/questions/324925 | 1 | This question is motivated by a real life task (which is briefly described after the question.)
Let $G=(V,E)$ be an infinite graph. For $v\in V$ let $N(v) = \{x\in V: \{v,x\}\in E\}$. If $S\subseteq V$ with $\emptyset\neq S\neq V$ we say that $v\in V$ is *happy with respect to $S$* if $$N(v)\cup \{v\}\subseteq S \tex... | https://mathoverflow.net/users/8628 | Maximizing "happy" vertices in splitting an infinite graph | I think, no. Imagine that we have infinitely many boys $b\_1,b\_2,\dots$ and girls $g\_1,g\_2,\dots$ such that all boys are mutual friends, all girls are mutual friends and $b\_i,g\_j$ are friends if and only if $i\geqslant j$. If $b\_1$ is happy, all boys and $g\_1$ must be in the same part (without loss of generality... | 2 | https://mathoverflow.net/users/4312 | 324930 | 139,992 |
https://mathoverflow.net/questions/324927 | 4 | Given any pre-ordering $\preceq$ of an arbitrary set $X$ is the following lemma:
$$\text{There exists an inclusion minimal set }S\text{ satisfying }\{a\preceq b:b\in S\}=X\\\iff \text{ Every chain in }(X,\preceq)\text{ has a lower bound}$$
Equivalent to the axiom of choice? Also assuming the axiom of choice if ever... | https://mathoverflow.net/users/38626 | Is this lemma equivalent to the axiom of choice? | You seem to flip some directions of the order, take $\Bbb N$, every chain has a lower bound, but there is no set of maximal elements. You also want to quantify over *all* $X$, in the sense that choice is equivalent to that for any preordered $X$ etc.
Now, the answer is simple, yes. This is equivalent to choice. Simpl... | 10 | https://mathoverflow.net/users/7206 | 324932 | 139,994 |
https://mathoverflow.net/questions/324915 | 3 | If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating cofibrations is a cofibration, that the Leibniz product of a generating cofibration with a generating acyclic cofibration is ... | https://mathoverflow.net/users/2362 | Monoidalness of a model category can be checked on generators | This is Corollary 4.2.5 of Hovey's book. The proof is not that involved.
| 5 | https://mathoverflow.net/users/11540 | 324936 | 139,996 |
https://mathoverflow.net/questions/324826 | 9 | If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{\*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W\_{+})$
and
2) $K[T^\vee]^W$ is a polynomial ring,
where $K$ is a field of characteristic zero. I found this in page 2 [here](http://www.am... | https://mathoverflow.net/users/16356 | Borel's presentation for the cohomology of a Flag Variety | The answer to your question is contained in Demazure's paper: <https://eudml.org/doc/142233>
In particular, as you noted, the group is special iff the Borel presentation holds in any charactertistic.
If you want to know the structure in characteristic $p$, there is another useful reference by Victor Kac: <https://lin... | 9 | https://mathoverflow.net/users/5107 | 324937 | 139,997 |
https://mathoverflow.net/questions/324938 | 2 | I guess this is something pretty standard in calculus, but I was unable to google the answer.
Assume I have unit hypercube $C\_n = [0,1]^n$. I also have a function $f : \mathbb{R}^n \to \mathbb{R}^{n+1}$. The components $f\_i : \mathbb{R}^n \to \mathbb{R}$ are polynomials $\forall i = 1,...,n+1$, so it's all quite we... | https://mathoverflow.net/users/81305 | How to compute volume of parametric regions? | For $t=(t\_1,\dots,t\_n)$ in the cube $C\_n$, let $A(t)=(a\_{ij}(t))\_{i,j=1}^{n+1}$ be the matrix whose entries $a\_{ij}(t)$ are defined as follows:
$a\_{ij}(t):=\partial\_i f\_j(t)$ for $i=1,\dots,n$ and $j=1,\dots,n+1$, where $\partial\_k f\_j(t)$ denotes the partial derivative of the function $f\_j$ with respect... | 1 | https://mathoverflow.net/users/36721 | 324941 | 139,998 |
https://mathoverflow.net/questions/324835 | 4 | From the answer of Andreas Blass and comments of Ali Enayat on my question [Selective ultrafilter and bijective mapping](https://mathoverflow.net/questions/324254/selective-ultrafilter-and-bijective-mapping) it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\nsim\mathcal{U}\otimes\mathcal{... | https://mathoverflow.net/users/118366 | The example of the idempotent filter or subsets family with finite intersections property | Following the link given by Martin Sleziak in comment I have got the answer:
There exists filter $\mathcal{F}$ with the following properties:
1. $\mathcal{N}\subset\mathcal{F}$
2. $\mathcal{F}\sim\mathcal{F}\otimes\mathcal{F}$
Existence proof is easy. But construction of $\mathcal{F}$ is complicated.
| 2 | https://mathoverflow.net/users/118366 | 324947 | 140,001 |
https://mathoverflow.net/questions/228966 | 6 | Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, etc.). This category is also still a monoidal category (the opposite of a monoidal category is canonically a monoidal catego... | https://mathoverflow.net/users/11546 | What structure of a monoidal simplicial model category is preserved by taking the opposite category? | The general statement is that if $V$ is a monoidal model category (I will assume the unit object of $V$ is cofibrant, so that there are no funny extra axioms related to the unit) and $M$ is a $V$-model category, then $M^{\mathrm{op}}$ is also a $V$-model category. The enrichment comes from that of $M$, and the roles of... | 5 | https://mathoverflow.net/users/126667 | 324950 | 140,003 |
https://mathoverflow.net/questions/324916 | 3 | If $T=V|T|\text { and } S=W|S|$ is the polar decomposition of $T$. Is it true that the polar decomposition of $T\otimes S$ is $T\otimes S=(V\otimes W)(|T| \otimes |S|)$. If $T$ and $S$ are self-adjoint what is $f(T\otimes S)$ where f is continous function appropriately defined on the domain?
| https://mathoverflow.net/users/136400 | Polar decomposition of tensor product of operators in von Neumann algebra | For the 1st part, the answer is "yes".
Let $T,S$ be bounded operators on $H$ and $K$ respectively. As $(T\otimes S)^\*(T\otimes S) = |T|^2\otimes|S|^2$ it follows that $|T\otimes S| = |T|\otimes |S|$. With polar decompositions $T=V|T|, S=W|S|$ we have that $(V\otimes W)|T\otimes S| = T\otimes S$. Clearly $V\otimes W$... | 5 | https://mathoverflow.net/users/406 | 324955 | 140,004 |
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