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https://mathoverflow.net/questions/386944 | 3 |
>
> Suppose that $X\_1,\ldots,X\_n$ are iid $N(0,1)$ random variables. Consider the random variable given by
> $$
> \xi\_n
> =\Bigl|\frac1{\sqrt{n}}\sum\_{t=1}^nX\_t\Bigr|^2-\frac1n\sum\_{t=1}^nX\_t^2
> =\frac1n\sum\_{s\ne t}X\_sX\_t.
> $$
> What is the distribution of $\xi\_n$?
>
>
>
My observations are as foll... | https://mathoverflow.net/users/46211 | Distribution of a certain functional of iid $N(0,1)$ random variables | For $X\_1,\dots, X\_n$ Gaussian random variables with variance-covariance matrix $M$, and $L$ a $m \times n$ matrix, the variables $Y\_1,\dots, Y\_m$ given by $Y\_i = \sum\_{j=1}^n L\_{ij} X\_j$ have variance-covariance matrix $L M L^T$.
This follows immediately from writing the covariance as an expectation of produc... | 1 | https://mathoverflow.net/users/18060 | 386994 | 160,748 |
https://mathoverflow.net/questions/386996 | 0 | I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
>
> Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field. Let $\mathcal{E}$ be a coherent sheaf over $X\times\_{Spec(K)} S=X\times S$ flat over $S$. Then, $Supp(\mathcal... | https://mathoverflow.net/users/129919 | Support of a coherent sheaf over a fiber product scheme | Take $X = S = \mathbb{A}^1$, $Z = \{xy = 1\} \subset X \times S$ (where $x$ is the coordinate on $X$ and $y$ on $S$), and let $\mathcal{E}$ be the structure sheaf of $Z$.
| 4 | https://mathoverflow.net/users/4428 | 386997 | 160,749 |
https://mathoverflow.net/questions/386971 | 7 | Let $(x\_n)$ be a monotonically decreasing sequence of positive real numbers that is also summable.
Let $(y\_n)$ be a sequence of positive real numbers such that
$\sum\_n x\_n y\_n$ converges.
Let $(z\_n)$ be a monotonically increasing sequence of positive real numbers such that $\sum\_n x\_n z\_n =\infty.$
Assum... | https://mathoverflow.net/users/150564 | Comparing divergent and convergent sums | No. For example $x\_n=2^{-n}$, $y\_n=n$, and we now keep $z\_n$ constant on long intervals. More precisely, consider first $\epsilon=1$, and set $z\_1= \ldots = z\_{N\_1}=c\_1$, with $N\_1$ taken so large that $N\_1 2^{-1\cdot 1}>\sum 2^{-1\cdot n}$.
We then continue in the same way: let $z\_{N\_1+1}=\ldots = z\_{N\_... | 4 | https://mathoverflow.net/users/48839 | 387003 | 160,750 |
https://mathoverflow.net/questions/386947 | 4 | The first few lines of this post is based on [this lecture notes](https://people.phys.ethz.ch/%7Ebabis/Teaching/QFT1/qft1.pdf), but similar expositions can be found in other physics books such as [Peskin & Schroeder's book](https://www.amazon.com.br/Introduction-Quantum-Field-Theory/dp/0201503972).
On chapter 8 of th... | https://mathoverflow.net/users/150264 | Representations of the Lorentz group | Since your question is now asking for references, here are a few standard ones.
For those who wish to study Lie groups and Lie algebras for the purposes of representation theory (one already mentioned by gmvh):
>
> * *Fulton, William; Harris, Joe*, Representation theory. A first course, Graduate Texts in Mathemat... | 3 | https://mathoverflow.net/users/2622 | 387011 | 160,755 |
https://mathoverflow.net/questions/386861 | 1 | Optimal vertex-disjoint cycle covers of weighted symmetric graphs with $n$ vertices can be calculated efficiently with the method of Tutte.
It is also possible to efficiently calculate optimal matchings with $2k<n$ edges.
**Question:**
Is it also possible to reduce the calculation of optimal vertex-disjoint cyc... | https://mathoverflow.net/users/31310 | Finding optimal cycle covers with fixed number of vertices | Just realized that the method for finding optimal matchings of fixed size can easily be generalized to arbitrary $f$-factors.
In order to calculate an optimal $1$-factor whose edges are adjacent to $k<n$ vertices, one adds $n-k$ vertices to the graph, connects each added vertex to all $n$ vertices of the original gra... | 0 | https://mathoverflow.net/users/31310 | 387013 | 160,756 |
https://mathoverflow.net/questions/387014 | 3 | Let $X$ be a random variable taking values on the real line.
Let $R(X) = max\{0, X\}$. Is it true that the covariance $Cov[X, R(X)] \ge 0$ irrespective of the distribution of $X$? Many experiments, as well as the intuition seem to suggest that their covariance must be non-negative. Is it true? If so, how can I prove it... | https://mathoverflow.net/users/176364 | On estimating Covariance between a random variable and its non-linear transform | $${\rm Cov}\,(X,R(X))= \int\_0^\infty x^2P(x)\,dx - \left(\int\_{-\infty}^\infty xP(x)\,dx\right)\left(\int\_{0}^\infty xP(x)\,dx\right)$$
$$=\int\_0^\infty x^2P(x)\,dx - \left(\int\_{0}^\infty xP(x)\,dx\right)^2- \left(\int\_{-\infty}^0 xP(x)\,dx\right)\left(\int\_{0}^\infty xP(x)\,dx\right)$$
$$={\rm Var}\,R(X)- \lef... | 2 | https://mathoverflow.net/users/11260 | 387015 | 160,757 |
https://mathoverflow.net/questions/387000 | 6 | Define a poset on the set of all finite binary strings, defined by $a \le b$ whenever $b = uav$ for (possibly empty) binary strings $u, v$.
What is the [Möbius function](https://en.wikipedia.org/wiki/Incidence_algebra#Special_elements) of this poset?
| https://mathoverflow.net/users/176357 | What is the Möbius function of substrings? | This problem was solved by [Anders Björner, The Möbius function of factor order, *Theoretical Computer Science* **117** (1993), 91-98](https://doi.org/10.1016/0304-3975(93)90305-D). In particular, the Möbius function assumes only the values $-1,0,1$. It is also Exercise 3.134(b) in *Enumerative Combinatorics*, vol. 1, ... | 9 | https://mathoverflow.net/users/2807 | 387017 | 160,758 |
https://mathoverflow.net/questions/386990 | 6 | Let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a finite $\Gamma$-module,
that is, a finite abelian group on which $\Gamma$ acts.
It is a hopeless problem to classify finite $\Gamma$-modules.
We consider $A^\Gamma=\{a\in A\mid{}^\gamma a=a\}$ and
$$H^2(\Gamma,A)=A^\Gamma/\{a'+{}^\gamma a'\mid a'\in A\}.$$... | https://mathoverflow.net/users/4149 | Finite $\Gamma$-modules with trivial $H^2$, where $\Gamma$ is a group of order 2 | The answer to question 2 is yes. Take $A=\mathbb Z/8\mathbb Z$ where $\gamma$ acts by multiplication by $5$. Then $A^\Gamma= \ker( \cdot 4)= 2\mathbb Z/8\mathbb Z$ but $(\mathrm{id} +\gamma)(3)=6\cdot 3=2$, thus $H^2(\Gamma,A)$=0 (and obviously $A$ doesn't contain any induced module).
For the question 1 I do not real... | 6 | https://mathoverflow.net/users/42606 | 387032 | 160,766 |
https://mathoverflow.net/questions/387021 | 45 | Let
$$ f(x) = \sum\_{j=1}^n c\_j e^{2\pi i\alpha\_j x}, g(x) = \sum\_{k=1}^m d\_k e^{2\pi i\beta\_k x}$$
be two (quasi-periodic) trigonometric polynomials, where the coefficients $c\_j, d\_k$ are complex and the frequencies $\alpha\_j,\beta\_k$ are real (but are not necessarily commensurable). Suppose that $f,g$ have i... | https://mathoverflow.net/users/766 | Is there a nullstellensatz for trigonometric polynomials? | I'm converting the [comment](https://mathoverflow.net/questions/387021/is-there-a-nullstellensatz-for-trigonometric-polynomials#comment986917_387021) above to an answer:
Shapiro made the following conjecture in the paper "[The expansion of mean-periodic functions in series of exponentials](https://doi.org/10.1002/cpa... | 31 | https://mathoverflow.net/users/2384 | 387037 | 160,767 |
https://mathoverflow.net/questions/387047 | 14 | For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r\_1+r\_2-1$.
It seems that standard algebraic number theory textbooks usually prove this theorem by using Minkowski's theory or using the Blichfeldt-Minkowski Lemm... | https://mathoverflow.net/users/95241 | How Dirichlet proved Dirichlet's unit theorem for general number fields? | Dirichlet's proof is described in [Number Theory: Algebraic Numbers and Functions](https://books.google.nl/books?id=qEwpwWyVPIAC) (starting on page 48).
Dirichlet did not use Minkowski’s theorem; he proved the unit theorem in 1846 while Minkowski’s theorem appeared in 1889. Dirichlet’s substitute for the convex-body ... | 15 | https://mathoverflow.net/users/11260 | 387053 | 160,772 |
https://mathoverflow.net/questions/387023 | 5 | Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
---
**Definition:** A pair $(A, \Delta)$ is called $C^\*$-algebraic compact quantum group if $A$ is a unital $C^\*$-a... | https://mathoverflow.net/users/nan | Relating different constructions of the universal compact quantum group | Yes, $\widetilde{A}\_r$ and $A\_r$ are canonically isomorphic. This follows from the following result in which you could take as $(B,\Delta\_B)$ the universal C$^\*$-algebraic compact quantum group $(A\_u,\Delta\_u)$ associated with $(A\_0,\Delta\_0)$.
**Proposition.** Let $(B,\Delta\_B)$ and $(A,\Delta\_A)$ be C$^\*... | 4 | https://mathoverflow.net/users/159170 | 387054 | 160,773 |
https://mathoverflow.net/questions/387025 | 2 | The $\mathbb{Z}$-extensions of $\mathbb{Z}/n\mathbb{Z}$
$$
0\longrightarrow \mathbb{Z}\longrightarrow E\longrightarrow \mathbb{Z}/n\mathbb{Z}\longrightarrow0
$$
are classifiied by $\operatorname{Ext}\_{\mathbb{Z}}^1(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z})$. But one can compute $\operatorname{Ext}\_{\mathbb{Z}}^1(\mathbb{Z}... | https://mathoverflow.net/users/32746 | $\mathbb{Z}$-extensions of $\mathbb{Z}/n\mathbb{Z}$ | Yes. May be a general remark: the group $\mathrm{Ext}^1$ classifies the short exact sequences, which is more structure than remembering $E$ or even $E$ with the $2$-term filtration. In any case, $0\rightarrow \mathbb Z \xrightarrow{\cdot n} \mathbb Z \rightarrow \mathbb Z/n\mathbb Z \rightarrow 0$ is a generator of $\m... | 5 | https://mathoverflow.net/users/42606 | 387063 | 160,775 |
https://mathoverflow.net/questions/387069 | 1 | Consider a set of i.i.d. (positive) random variables $\{X\_i\}\_{i=1}^N$. Each variable $X\_i$ has a distribution with finite mean but infinite variance. In particular, if $P\_{X\_i}(x)$ is the P.D.F. of the random variable $X\_i$ $P\_{X\_i}(x) \sim \frac{1}{x^{\alpha +1}}$ (with $1< \alpha <2$) for $x>\tilde{x}>0$ and... | https://mathoverflow.net/users/174176 | Rate of variance's decrease for the mean's distribution of infinite variance i.i.d. random variables | Assuming that $\langle X\rangle$ denotes the expectation of each of the iid $X\_i$'s, the variance of $W\_N$ is $\infty$. Indeed,
$$Var\,W\_N=\frac1{N^2}Var\sum\_{i=1}^N X\_i=\frac1{N^2}\,N\,Var\,X\_1=\infty,$$
since $Var\,X\_1=\infty$.
(Indeed, by the strong law of large numbers, $W\_N\to0$ almost surely and hence i... | 1 | https://mathoverflow.net/users/36721 | 387072 | 160,776 |
https://mathoverflow.net/questions/387059 | 2 | I am looking for a large family (infinite pairs) of cospectral graphs with these condtions:
* The graphs are non-regular,
* Minimum degree is greater than $1$,
* The degree sequences of these cospectral graphs are the same.
I need cospectrality by adjacency matrix and the graphs are simple.
The motivation for ask... | https://mathoverflow.net/users/19885 | Non-regular cospectral graphs with same degree sequences | Let $D$ be a Steiner triple system on $v$ points. (So $v\equiv1,3$ mod 6). The incidence
graph is the bipartite graph with the $v$ points as one colour class and the $v(v-1)/6$
blocks as the second; a point is incident with the $(v-1)/2$ blocks that contain it.
Let $N$ be the point-block incidence matrix of the syste... | 1 | https://mathoverflow.net/users/1266 | 387076 | 160,778 |
https://mathoverflow.net/questions/387073 | 1 | I would like to ask a follow-up question on a [previous question](https://mathoverflow.net/questions/386802/is-the-harmonic-series-worse-than-any-summable-series) of mine here whose proof does not seem to carry over to this case in an obvious way:
We define the function $$F\_{\varepsilon}(x) = \sum\_{i=1}^{\infty} 2^... | https://mathoverflow.net/users/119875 | Comparing growth of sequences in weighted spaces | $\newcommand\ep\varepsilon\newcommand\de\delta$
Let us show more: for all $x\in A$,
\begin{equation\*}
\frac{F\_{\ep}((n))}{F\_{\ep}(x)}\to0\tag{$\*$}
\end{equation\*}
(as $\ep\downarrow0$).
Indeed, take any $x\in A$ and let
\begin{equation\*}
y\_n:=x\_n/n^2,
\end{equation\*}
so that $\sum\_n y\_n<\infty$ and
\begi... | 1 | https://mathoverflow.net/users/36721 | 387080 | 160,781 |
https://mathoverflow.net/questions/387058 | 7 | **Under which 'minimal' conditions on a symmetric monoidal category does an abelization functor from its monoids to its commutative monoids exist?**
More precisely: let $(\mathcal{C},\otimes,1)$ be a symmetric monoidal category, and let $U : \mathrm{CMon}(\mathcal{C}) \rightarrow \mathrm{Mon}(\mathcal{C})$ be the for... | https://mathoverflow.net/users/176383 | Abelianization of monoids in arbitrary (symmetric) monoidal categories | Here is an approach that has the advantage of avoiding to go in the details of a concrete construction. More explicit constructions of the abelianization might give even more general results, in terms of corollary 2 below they would corresponds to more explicit description of the object $X\_{ab}$, but I think corollary... | 3 | https://mathoverflow.net/users/22131 | 387082 | 160,782 |
https://mathoverflow.net/questions/386875 | 1 | Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ).
Can you tell me how to show that,
$$ \mathrm{CH}\_k (X) \otimes\_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega\_k (X) \otimes\_{ \mathbb{Z} } \mathbb{Q} $$ where,
$ \mathrm{CH}\_k ( X ) $ is the Chow group that is the f... | https://mathoverflow.net/users/169088 | How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $? | I haven't checked the book by Levine and Morel, but there is a short arXiv article by Levine at <https://arxiv.org/pdf/math/0304206.pdf>. There he writes $\mathbb{L}$ for the Lazard ring, which is the same as $MU\_\*$, and is polynomial over $\mathbb{Z}$ on countably many generators. There is a natural map $\Omega\_\*(... | 6 | https://mathoverflow.net/users/10366 | 387087 | 160,784 |
https://mathoverflow.net/questions/387089 | 5 | Let $(X,d\_X)$ be a compact metric space and let $\{K\_n\}\_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x\_n \in K\_n$ we have
$$
d\_X(x\_n,K)\leq \frac1{n},
$$
does this imply that $K$ is the Kuratowski lower limit ($\mathop{\mathrm{Li}}\_{n \to \inf... | https://mathoverflow.net/users/176409 | Criterion for Kuratowski Limit Inferior | The answer is no, or what am I missing. Let $X = K = \{0,1\}$ and $K\_n = \{0\}$, $n \in \mathbb{N}$ with $d\_X(0,1) = 1$. Then $\text{LI}\_{n \to \infty} K\_n = \{0\} \not= K$.
| 6 | https://mathoverflow.net/users/100904 | 387092 | 160,785 |
https://mathoverflow.net/questions/387081 | 4 | Working on a problem in the symmetric group I have stumbled upon the following equation:
$$\sum\_{\substack{\pi=(1^{c\_1},2^{c\_2},\ldots,n^{c\_n})\\\textrm{partition of }n}}(-1)^{n-\sum\_{i=1}^nc\_i}\frac{n!}{\prod\_{i=1}^ni^{c\_i}c\_i!}\left(\sum\_{\substack{\eta=(1^{b\_1},2^{b\_2},\ldots,k^{b\_k})\\\textrm{partiti... | https://mathoverflow.net/users/45242 | Identity involving binomial coefficients and partitions | If $A\subset B$, $b$ is a permutation (self-bijection) of $B$, and $a$ a permutation of $A$, we say that $a$ is a subpermutation of $b$ if any cycle of $a$ is a cycle of $b$. Your sum is the sum of ${\rm sign}(b)$ taken over all permutations $b$ of $B=\{1,2,\ldots,n\}$ and all subpermutations $a\_1,\ldots,a\_\ell$ of $... | 3 | https://mathoverflow.net/users/4312 | 387095 | 160,786 |
https://mathoverflow.net/questions/387033 | 6 | In Proposition 5.13 (ii) in Scholze's [Perfectoid Spaces](https://www.math.uni-bonn.de/people/scholze/PerfectoidSpaces.pdf), we have $R \to S$ a morphism of $\Bbb F\_p$-algebras and the **assumption** that the relative Frobenius $\Phi\_{S/R}$ induces an isomorphism $R\_{(\Phi)} \otimes\_R^{\Bbb L} S \to S\_{(\Phi)}$ in... | https://mathoverflow.net/users/91375 | Proposition 5.13 (ii) in Scholze's Perfectoid Spaces | Regarding the first and third question, what you say is correct. For the second question, you are looking for the base change compatibility of the cotangent complex: If $R\to R'$ is any map of rings and $S$ is an $R$-algebra such that $S'=S\otimes^L\_R R'$ sits in degree $0$, then
$$R'\otimes^L\_R \mathbb L\_{S/R}\cong... | 9 | https://mathoverflow.net/users/6074 | 387102 | 160,788 |
https://mathoverflow.net/questions/387093 | 10 | Is there a sequence of rational numbers $a\_0, a\_1, \dotsc$ such that $\sum\limits\_{i\geq 0}a\_i x^i$ converges absolutely to $2^x$ for every $x\in \mathbb{Z}$?
| https://mathoverflow.net/users/176392 | Taylor series with coefficients in $\mathbb{Q}$ | Yes.
You can construct such a series of the form
$$ 1 + \sum\_{i=0}^\infty (a\_i + b\_i x) \left( \frac{x^2}{(i+1)^2}\right)^{e\_i} \prod\_{j=-i}^i (x-j) $$
for some sequence $a\_i, b\_i$ of rational numbers and some sequence $e\_i$ of natural numbers (converging to $\infty$.)
This certainly defines a power ser... | 10 | https://mathoverflow.net/users/18060 | 387103 | 160,789 |
https://mathoverflow.net/questions/386828 | 5 | Let $(X\_t^x)\_{t\in [0,\infty),\,x\in \mathbb{R}^n}$ be a Markov process taking values in $\mathbb{R}^m$ and defined on some stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t\in [0,\infty}), \mathbb{P})$; where $X\_0^x=x$ $\mathbb{P}$-a.s. Suppose also that $\sup\_{t\geq 0} \mathbb{E}[\|X\_t^x\|]<\infty$ for ... | https://mathoverflow.net/users/172598 | Regularity of law of conditional law of a Markov process equivalent to regularity of its paths | No, this has no reason to be true. Take for example $X\_t^x = x+t$ for $x \ge 0$ and $x-t$ for $x < 0$ ($n=1$). Paths are smooth, but $f$ is discontinuous at $x=0$.
| 1 | https://mathoverflow.net/users/38566 | 387128 | 160,797 |
https://mathoverflow.net/questions/387108 | 3 | I found in the PhD thesis [Moments method for random matrices with applications to wireless communication](https://tel.archives-ouvertes.fr/tel-00805578) the following combinatorial formula to compute the free moments of the product of two random variables.
Given $a$ and $b$ random variables in free relation in a non... | https://mathoverflow.net/users/175894 | Combinatorial formula to compute the moments of the product of two free random variables | The formula as stated in your reference seems to be wrong. The Kreweras complement of the partitions should also show up. You can find the correct version in my original paper <https://link.springer.com/content/pdf/10.1023/A:1008643104945.pdf>
with Andu Nica on this, or also in my book with Andu, which you can find at ... | 6 | https://mathoverflow.net/users/112626 | 387129 | 160,798 |
https://mathoverflow.net/questions/387107 | 16 | The Brown representability theorem can be convenient way to construct a spectrum. But to get a ring spectrum of even a very unstructured form seems to be harder. There's even currently a statement [on the nlab](https://ncatlab.org/nlab/show/multiplicative+cohomology+theory#BrownRepresentabilityByRingSpectra) to the eff... | https://mathoverflow.net/users/2362 | Multiplicative Brown representability? | I'll write $h\_E$ for the functor from finite spectra to abelian groups given by $h\_E(X)=\pi\_0(E\wedge X)=[DX,E]$. This is an object of the category $\mathcal{A}$ of all additive functors from finite spectra to abelian groups. Given $A,B,C\in\mathcal{A}$, a *pairing* from $A$ and $B$ to $C$ means a natural map $A(X)\... | 15 | https://mathoverflow.net/users/10366 | 387132 | 160,799 |
https://mathoverflow.net/questions/387121 | 2 | Suppose $M\_{n}$ is an $n \times n$ matrix with independent ±1 entries. Recent [breakthrough](https://annals.math.princeton.edu/2020/191-2/p06) shows that the probability $\mathbb{P}(M\_{n} \text{ is singular})$ is
$$(1) \quad\quad\qquad \mathbb{P}(M\_{n} \text{ is singular})= \left(1/2 +o\_{n}(1)\right)^{n}.$$
bas... | https://mathoverflow.net/users/145559 | Lower bound of the probability of singular random matrix over $\{\pm1\}$ in ``Singularity of random Bernoulli matrices" | The first two rows are identical with probability $2^{-n}$, so $\mathbb{P}(\det M\_n = 0) \geq 2^{-n}$.
Incidentally, there are $2 \times \binom{n}{2}$ events like this to consider, though not quite independent, so it is reasonable to expect $\mathbb{P}(\det M\_n = 0) \approx n(n-1) 2^{-n}$, and this has been conject... | 2 | https://mathoverflow.net/users/20598 | 387136 | 160,800 |
https://mathoverflow.net/questions/386633 | 1 |
>
> Let $\mathcal{D}$ be a $k$-linear, Hom-finite triangulated category with a Serre functor $\mathbb{S}$. An important class of objects in $\mathcal{D}$ are the cluster-tilting objects, which have many nice properties.
>
>
> **[Definition 2.10.](https://arxiv.org/abs/1504.00093)**
> (1) An object $T$ in $\mathcal{... | https://mathoverflow.net/users/118028 | Rigid, maximal rigid and cluster-tilting objects | For rigid and maximal rigid, you can think instead in the category of quiver representations.
The term "rigid" comes from the fact that the vanishing of Ext^1 can be understood as the absence of any true deformation of the module. It is a general fact in the study of quiver representations that the tangent space of t... | 2 | https://mathoverflow.net/users/10881 | 387139 | 160,801 |
https://mathoverflow.net/questions/386689 | 10 | Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum\_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\rho$ is a real representation.
A theorem by Brauer states that every irreducible representation over $\mathbb{C}$ can ... | https://mathoverflow.net/users/136343 | Realizability of a real representation using real cyclotomic coefficients | Edit: yes
---------
After doing a bit of work, I can now say, yes, it's always possible
to realise $\rho$ over $\mathbb{Q}(\zeta\_n)\cap\mathbb{R}$, see
<https://arxiv.org/abs/2107.03452>
The main ingredient in the latter is Serre's induction theorem,
from Jean-Pierre Serre. Conducteurs d’Artin des caractères réels... | 3 | https://mathoverflow.net/users/11100 | 387143 | 160,802 |
https://mathoverflow.net/questions/387079 | 14 | I am a PhD student in algebraic topology, and I would like to learn something about **group cohomology**.
The final goal would be to present one or two seminars on this topic, in order to give my mates a gently introduction to this subject and at the same time showing them some striking result/application of this the... | https://mathoverflow.net/users/169319 | A road map through group cohomology | Brown: *Lectures on the cohomology of groups*.
Adem: *Lectures on the cohomology of finite groups.*
Carlson: *The cohomology of groups* (from Handbook of Algebra, Vol.1, 1996).
Rotman: *Homology of groups* (chapter 9 from one of his algebra books that I forget the name).
| 3 | https://mathoverflow.net/users/12310 | 387144 | 160,803 |
https://mathoverflow.net/questions/387146 | 7 | We know that every finite simple group can be generated by $2$ elements.
This (correct me if I'm wrong) was proved, as far as I know, by Steinberg (Steinberg, R. (1962). Generators for Simple Groups. Canadian Journal of Mathematics, 14, 277-283) for Chevalley groups and by Aschbacher and Guralnick (M. Aschbacher, R. ... | https://mathoverflow.net/users/5710 | CFSG-free bound for the number of generators of a finite simple group | Still elementary, but marginally stronger than the $\log\_{2}(n)$ bound is: every non-Abelian finite simple group $G$ of order $n$ can be generated by fewer than $\log\_{p}(n)$ elements, where $p$ is the largest prime divisor of $\lvert G\rvert$. In particular (using Burnside's $p^{a}q^{b}$-theorem), such a group $G$ c... | 7 | https://mathoverflow.net/users/14450 | 387148 | 160,806 |
https://mathoverflow.net/questions/387012 | 2 | The Ramsey function $R(k,x)$ is defined as the minimal integer $n$ such that any graph on $n$ vertices contains either a clique of size $k$ or an independent set of size $x$. [Miklós Ajtai, János Komlós and Endre Szemerédi(1980)](https://linkinghub.elsevier.com/retrieve/pii/0097316580900308) shows that $R(3,x)≤cx^2/\ln... | https://mathoverflow.net/users/160959 | Best known upper bound for the Ramsey function $R(k,x)$ | I don't have the reputation to comment, but I don't think there has been any improvement on the result you mentioned. See pg 5 of this survey by Conlon, Fox and Sudakov from 2015 <https://arxiv.org/pdf/1501.02474.pdf>
| 3 | https://mathoverflow.net/users/165057 | 387162 | 160,811 |
https://mathoverflow.net/questions/387157 | 2 | Let $S\subseteq \{0,\ldots,n\}^d$ be a set of $d$-dimensional vectors of with bounded, natural, coordinates.
We are given that
$$v'+v\_1+\ldots+v\_t=u'+u\_1+\ldots+u\_s$$
where $v\_1,\ldots,v\_t,u\_1,\ldots,u\_s,v',u'\in S$ (and the vectors are not necessarily distinct).
That is, two sets of vectors whose sums are ... | https://mathoverflow.net/users/64542 | Equal subset-sums of bounded vectors | As I understand, your $n$ and $d$ are fixed, and you want to prove the existence of corresponding non-empty subsets $I$, $J$ provided that both $t$, $s$ are large enough (greater than some constant $C(n,d)$).
This is true.
We find a positive integer $M$ satisfying the following condition: whenever $U,V$ are finite ... | 5 | https://mathoverflow.net/users/4312 | 387172 | 160,815 |
https://mathoverflow.net/questions/387169 | 4 | Repost from math.SE since no answer after two months, but feel free to close if not appropriate:
---
Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its category of modules.
Suppose now that we *fix* a rigid structure on $B\text{-mod}$ such that the dual of a module... | https://mathoverflow.net/users/41706 | Bialgebras with rigid representation theory | Yes, it is true that if a quasi-Hopf algebra has a trivial coassociator, then it's equivalent to an actual Hopf algebra (with $\alpha=\beta=1$). In other words, if you know the category is rigid (i.e. if you know duals exists) then there is a particular choice for this duality (canonically isomorphic to any other one) ... | 3 | https://mathoverflow.net/users/13552 | 387176 | 160,817 |
https://mathoverflow.net/questions/387140 | 2 | In the paper "The Kadison-Singer Problem" by Marcin Bownik (<https://arxiv.org/pdf/1702.04578.pdf>), the following Lemma (3.8) is proven:
**Lemma:**
Let $p, q\in \mathbb{R}[x]$ be stable monic polynomials of the same degree. Suppose that
every convex combination $(1 − t)p + tq, 0 \leq t \leq 1$, is also stable. Then ... | https://mathoverflow.net/users/104165 | Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem) | Indeed, the proof in the linked paper does not seem valid.
Here is a "real" proof of the lemma in question, without using complex analysis:
Let $m:=$maxroot and $q\_t:=(1-t)p+tq$, so that $q\_0=p$ and $q\_1=q$.
As in your post, without loss of generality (wlog) $m(p)\le m(q)$. If $m(p)=m(q)$, then $m(p)$ is a commo... | 1 | https://mathoverflow.net/users/36721 | 387177 | 160,818 |
https://mathoverflow.net/questions/386603 | 6 | Suppose that $X$ is a complex, irreducible, projective variety with at most terminal singularities. Furthermore, assume that $\mathbb{C}^\*$ acts on $X$ with exactly $k$ fixed points, where $k>0$.
>
> **Question.** Is it true that $k > \dim\_{\mathbb{C}}(X)$?
>
>
>
| https://mathoverflow.net/users/99732 | About the number of fixed points of a torus action | This is true in a much more general setting. Let $X$ be any normal projective $T$-variety defines over an algebraically closed field. Then I claim that $\# X^T>\dim X$. We show this in two steps.
(1) According to a theorem of Sumihiro, there is an equivariant embedding of $X$ into a projective space ${\bf P}^n$. Norm... | 11 | https://mathoverflow.net/users/89948 | 387182 | 160,822 |
https://mathoverflow.net/questions/387175 | 8 | Question:
>
> Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}\_2$, $G=D\_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite dihedral group?
>
>
>
I'm also interested in other fields $K$ (in which case one can focus on describing the quot... | https://mathoverflow.net/users/14094 | Units of group algebra of dihedral group | A description of the group of units of the group algebra $\mathbb{F}\_2 D\_\infty$ can be found in Theorem 4.1 of the following paper:
M. Mirowicz: Units in group rings of the infinite dihedral group, Can. Math. Bull. 34 (1991), 83-89
[DOI link](https://doi.org/10.4153/CMB-1991-013-4) (the paper is accessible with ... | 13 | https://mathoverflow.net/users/14653 | 387186 | 160,824 |
https://mathoverflow.net/questions/386305 | 5 | Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$
\begin{align}
L u=\operatorname{div}[\nabla u-v u]
\end{align}
with a reflecting boundary condition
$$
\boldsymbol{n} \cdot(\nabla u-... | https://mathoverflow.net/users/175818 | Spectrum of an elliptic operator in divergence form with a reflecting boundary condition | To get the analytic semigroup estimate, we note first of all that the operator $M$: $Mu=Lu+(v\cdot\nabla)u$, with the same boundary condition as $L$, is self-adjoint and hence generates an analytic semigroup. We can then use results on perturbation of analytic semigroups and elliptic regularity to show that $L$ also ge... | 1 | https://mathoverflow.net/users/12120 | 387190 | 160,826 |
https://mathoverflow.net/questions/386454 | 9 | Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. The first-order Sobolev space $W^{1,2}(D)$ on $D$ is defined by
\begin{align\*}
W^{1,2}(D)=\{f \in L^2(D,m) \mid \partial f/\partial x\_i \in L^2(D,m),\, 1\le i \le d\}.
\end{align\*}
Here, $m$ is the Lebesgue measure and ... | https://mathoverflow.net/users/68463 | Core for a Sobolev space | (Too long for a comment.)
I have just learned that my colleagues, Bartłomiej Dyda and Michał Kijaczko, wrote a paper [1] on that particular problem for *fractional* Sobolev spaces. In their work, they cite Theorem 3.25 in McLean's book [2], which reads as follows (with the original notation):
**Theorem:** *For any ... | 2 | https://mathoverflow.net/users/108637 | 387191 | 160,827 |
https://mathoverflow.net/questions/387106 | 5 | In *On the Axiom of Extensionality, Part II*, The Journal of Symbolic Logic, Vol. 24, No. 4 (Dec., 1959), <https://doi.org/10.2307/2963897>, pp. 287-300, R. O. Gandy shows that a class theory X containing NBG *minus extensionality* is not weaker than NBG; X includes a use of class-abstraction denoted with $\lambda$, so... | https://mathoverflow.net/users/37385 | Is this set theory used by Gandy first-order with signature $(\in, \lambda)$? | A signature of first order [logic](https://en.wikipedia.org/wiki/Signature_(logic)) is usually taken to be a list of extra-logical symbols that range over specific elements (for constants) or over specific subsets (for n+1 ary functions, predicates) of the universe of discourse. The class-abstraction symbol $\lambda$ h... | 3 | https://mathoverflow.net/users/95347 | 387196 | 160,828 |
https://mathoverflow.net/questions/387163 | 12 | What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants [here](https://mathoverflow.net/questions/386903/does-there-exist-a-type-of-discriminant-not-only-for-irreducible-polynomials-but). We have an ine... | https://mathoverflow.net/users/nan | Testing whether $e^x+ax^2+bx+c$ has a zero | First let's consider a simpler problem on testing whether $g(x) := e^x + ux + v$ has a zero in the given interval $(L,U]$. It does when one of the following cases takes place:
* $g(L)<0$ **and** $g(U)\geq 0$;
* $g(L)>0$ **and** $g(U)\leq 0$;
* $g(L)\geq 0$ **and** $g(U)\geq 0$ **and** $u<0$ **and** $L<\log(-u)\leq U$... | 5 | https://mathoverflow.net/users/7076 | 387197 | 160,829 |
https://mathoverflow.net/questions/387099 | 8 | Let $\omega\_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w\_1 : \cdots : w\_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's paper [On real bisectional curvature for Hermitian manifolds](https://arxiv.org/abs/1610.07165) they claim the followin... | https://mathoverflow.net/users/174369 | An integration identity on $\mathbb{P}^{n-1}$ | If I remember correctly, you can find this in the book "Complex Differential Geometry" by Fangyang Zheng in Chapter 7. The analogous result "with real coefficients" (i.e. for real vectors and with $S^{n-1}$ instead of projective space) is originally due to Marcel Berger in Lemma 7.4 [here.](https://mathscinet.ams.org/m... | 6 | https://mathoverflow.net/users/13168 | 387208 | 160,834 |
https://mathoverflow.net/questions/387212 | 3 | Is there examples of non-isomorphic cospectral graphs having
1. Non-isomorphic automorphism groups?
2. Isomorphic automorphism groups?
| https://mathoverflow.net/users/10035 | On cospectral graphs | Yes, there are lots of such examples of strongly regular graphs. E.g. [Chang graphs](https://en.m.wikipedia.org/wiki/Chang_graphs) give an example of for 1).
For 2), many cospectral graphs have trivial automorphism groups (thus isomorphic as groups).
Or, for a concrete example 2), take point graphs of [generalised ... | 3 | https://mathoverflow.net/users/11100 | 387214 | 160,835 |
https://mathoverflow.net/questions/387245 | 3 | **Definition**:
The Hausdorff distance associated with a distance $d$ on a space $E$ between two sets $A\subset E$ and $B \subset E$ is $d\_H(A, B) = \max(\sup\_{x\in B}\{d(x, A)\}, \sup\_{y\in A}\{d(y, B)\})$, i.e. this is the maximum distance any point of A can be from B and vice versa.
**Question**:
For a fixed di... | https://mathoverflow.net/users/175383 | Does convergence of convex sets in Hausdorff distance implies convergence of the complementary sets? | Yes. For each unit vector $v$ in $\mathbb{R}^d$ and for any set $B \subseteq \mathbb{R}^d$, let $$r(B,v) = \inf\_{b \in B} \langle b,v\rangle,$$ where $\langle x, y \rangle$ is the inner product of $x$ and $y$. It is not too hard to show that for any $B,C \subseteq \mathbb{R}^d$, $$|r(B,v) - r(C,v)| \leq d\_H(B,C),$$ w... | 3 | https://mathoverflow.net/users/83901 | 387262 | 160,848 |
https://mathoverflow.net/questions/387255 | 3 | I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at [MSE](https://math.stackexchange.com/questions/4064375/is-the-kernel-of-an-action-of-a-hopf-algebra-on-an-algebra-a-biideal?noredirect=1#comment8397383_4064375), but without success.
[S.Dascalescu, C.Nastas... | https://mathoverflow.net/users/18943 | Is the kernel of an action of a Hopf algebra on an algebra a biideal? | No, it is not true (in general).
For a specific counterexample, let me first recall that what you call an
algebra $A$ equipped with an action of $H$ is the same as what is classically
called a (left) $H$-module algebra. When $H$ is the group algebra $k\left[
G\right] $ of a finite group (where $k$ is the base field),... | 5 | https://mathoverflow.net/users/2530 | 387269 | 160,852 |
https://mathoverflow.net/questions/387272 | 37 | In *Algebraic Number Theory*, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and
the reciprocity law become obvious once the machinery of algebraic
geometry is available."
Inspired by this, I wonder if there is some (pref... | https://mathoverflow.net/users/131975 | Using algebraic geometry to understand class field theory | The usual reference here is Serre's *Algebraic Groups and Class Fields*. But it still felt to me like there was a lot of magic left over after I read this.
We had an [earlier question along similar lines](https://mathoverflow.net/questions/73054) with many good answers. I remember really liking [David Ben Zvi's lectu... | 17 | https://mathoverflow.net/users/297 | 387275 | 160,857 |
https://mathoverflow.net/questions/387280 | 5 | Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ induce morphisms between $\mathrm{HH}\_\bullet(A)$ and $\mathrm{HH}\_\bullet(B)$ and $\mathrm{HC}\_\bullet(A)$ and $\mathrm... | https://mathoverflow.net/users/91245 | Morphisms of Hochschild (or cyclic) homology induced by homotopic maps | $\newcommand{\dd}{\mathrm d}\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\id}{id}$If $f,g:A\to B$ are dga morphisms which are homotopic as chain maps, the maps induced by $f$ and $g$ on Hochschild homology and its variants can be different. I give an example above the line; the easiest way to construct it, at le... | 5 | https://mathoverflow.net/users/35687 | 387307 | 160,866 |
https://mathoverflow.net/questions/387304 | 2 | I was reading this wonderful sequence of posts:
[nlab: *manifold with boundary*](https://ncatlab.org/nlab/show/manifold+with+boundary)
and [nlab: *collar neighbourhood theorem*](https://ncatlab.org/nlab/show/collar+neighbourhood+theorem)
and I couldn't help but wonder. Is there an extension of the [Collar neighborhood ... | https://mathoverflow.net/users/36886 | Collar neighborhood theorem for manifold with corners | This is an extended comment.
The way your question is stated, the answer is
"yes," of course. Each (topological) manifold with corners is also a manifold with boundary. Your subset $X\_0$ is just the boundary. Now, the claim follows from the existence of a collar neighborhood of the boundary.
There is a more refine... | 3 | https://mathoverflow.net/users/39654 | 387309 | 160,867 |
https://mathoverflow.net/questions/387279 | 4 | This question is already posted in [math.stackexchange,](https://math.stackexchange.com/questions/4066212/hopf-algebra-structure-of-a-groupoid-convolution-algebra) but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer.
To make things a... | https://mathoverflow.net/users/54507 | Hopf "algebroid" structure of a groupoid convolution algebra? | The group ring $\mathbb{C}[G]$ is a Hopf algebra that is cocommutative but not commutative. The dual is the ring $R=\text{Map}(G,\mathbb{C})$. This is a Hopf algebra with $(uv)(g)=u(g)v(g)$ and $\Delta(u)(g,h)=u(gh)$ (where we identify $R\otimes R$ with $\text{Map}(G\times G,\mathbb{C})$). This is commutative but not c... | 2 | https://mathoverflow.net/users/10366 | 387313 | 160,869 |
https://mathoverflow.net/questions/387322 | 5 | $\DeclareMathOperator\colim{colim}$This is inspired by [Clausen's answer](https://mathoverflow.net/a/387158/).
**Question:** Recall that $\mathbb Z\_p$ is endowed with the $p$-adic topology. Consider the countable sum $M:=\bigoplus\_{n=0}^\infty\mathbb Z\_p$ as a topological abelian group. One way to describe the top... | https://mathoverflow.net/users/176381 | Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group | The sum $M=\bigoplus\_{\mathbb N} \mathbb Z\_p$ is not first-countable, but it is Cauchy complete. More precisely, $M$ maps isomorphically to $\varprojlim\_{U\subset M} M/U$ where $U$ runs over open subgroups (so $M/U$ is discrete). This was observed in Example 7.1.7 [here](https://www.math.uni-bonn.de/people/scholze/p... | 9 | https://mathoverflow.net/users/6074 | 387324 | 160,872 |
https://mathoverflow.net/questions/387331 | 3 | The question from Remark (2) below Lemma 7 in [Enno Lenzmann: uniqueness of ground states for the pseudorelativistic Hartree equations](https://msp.org/apde/2009/2-1/p01.xhtml). Let $l \ge 1$, and consider the following operator
\begin{equation}
H=-\partial\_{rr} - \frac{2}{r} \partial\_r + 1 + \frac{l(l+1)}{r^2} - \fr... | https://mathoverflow.net/users/137915 | How to prove that there are infinitely many eigenvalues below $1$ of the following differential operator? | You can find a mathematically precise treatment in [On the Energy Levels of the Hydrogen Atom](https://cds.cern.ch/record/251291/files/9305102.pdf). For $l=0$ the eigenvalues $1-\kappa^2$ of $H$ are given by the square integrable solutions of
$$\left(-\frac{d^2}{dr^2}-\frac{\alpha}{r}\right)v(r)=-\kappa^2v(r).$$
The so... | 7 | https://mathoverflow.net/users/11260 | 387334 | 160,875 |
https://mathoverflow.net/questions/387298 | 9 | Sometimes, results on automorphic representations are available only under local assumptions. Typically, one could require the representation to be a $\xi$-*cohomological* cuspidal representation, and I would like to understand the meaning of it.
Let $G$ be a connected reductive group over $\mathbb{Q}$. Let $\xi$ be ... | https://mathoverflow.net/users/43737 | Relation between $\xi$-cohomological and discrete series | This condition comes up because of $(\mathfrak{g}, K)$-cohomology, which is an extremely important invariant of automorphic representations.
If $\xi$ is an algebraic rep, then $\xi$ defines a locally-constant sheaf on the locally-symmetric space $Y\_G(U)$ for any open compact $U \subset G(\mathbf{A}\_f)$. The Betti c... | 6 | https://mathoverflow.net/users/2481 | 387351 | 160,881 |
https://mathoverflow.net/questions/387363 | 8 | Let $(A, \Delta)$ be a $C^\*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^\*$-algebra. By elementary $C^\*$-algebra theory, we known that
$$A\cong M\_{n\_1}(\mathbb{C}) \oplus \dots \oplus M\_{n\_k}(\mathbb{C})$$ as $C^\*$-algebras.
If $X$ is a f... | https://mathoverflow.net/users/nan | Finite compact quantum groups | Another example apart from the example of Kac & Paljutkin (reference below) are the quantum groups of Sekine:
>
> Y. Sekine, *[An example of finite-dimensional Kac algebras of
> Kac-Paljutkin type](https://www.ams.org/journals/proc/1996-124-04/S0002-9939-96-03199-1/)*, Proc. Amer. Math. Soc. **124** (1996), no. 4,
... | 8 | https://mathoverflow.net/users/35482 | 387367 | 160,887 |
https://mathoverflow.net/questions/387199 | 4 | In characteristic not $2$, the Theorem of Cartan-Dieudonné states:
1. [Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every element in the orthogonal group $O(q)$ can be written as a product of at most $n$ reflections.
**In a nutshell:** I a... | https://mathoverflow.net/users/127497 | Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2 | I suggest reading the result 6.2.17 in the book written by A.J.Hahn and O.T.O'Meara called "The Classical Groups and K-Theory". The result is a general version of the Cartan-Dieudonné-Scherk theorem (which implies the Cartan-Diuedonné theorem). The result is stated in the language of "Quadratic spaces over form rings".... | 2 | https://mathoverflow.net/users/56010 | 387373 | 160,889 |
https://mathoverflow.net/questions/387384 | 2 | Consider $X$ to be a compact set in $[0,5]^2\subset \mathbb{R}^2$. Assume that $X$ is a union of translations of a unit ball. Then prove that $\partial X$ has a length bounded by $C$ where $C$ is independent of $X$.
| https://mathoverflow.net/users/36572 | Length of a boundary of union of unit balls in $\mathbb{R}^2$ | 1. Fix small $\epsilon>0$. We shall show that the union of balls with center in a given ball of radius $\epsilon$ must have bounded perimeter. Note each ball $D\_i$ has boundary expressible as a polar graph $$r=f\_i(\theta)$$ where $f\_i$ are uniformly bounded and uniformly Lipschitz. Then the boundary of $\cup\_iD\_i$... | 4 | https://mathoverflow.net/users/142740 | 387388 | 160,893 |
https://mathoverflow.net/questions/387407 | 0 | Working out circulant graphs that have eigenvalue 0 of multiplicity 1 (the so-called "nut" graphs), I came to the polynomial
$$
x^6 - x^5 - 4x^4 + 3x^2 + 3x - 1,
$$
for which I would like to show that it does not have roots in the form $2\cos(a\pi/b)$, where $a/b$ is an arbitrary rational number. Any suggestion how to ... | https://mathoverflow.net/users/168200 | Polynomial roots in the form 2cos of a rational multiple of pi | If $2\cos \frac{a\pi}b$ is a root of your polynomial $f(x)$, then $e^{i\pi a/b}$ is a root of $g(x):=x^6f(x+1/x)$. But the minimal polynomial of $e^{2i\pi k/n}$, where $k$ and $n$ are coprime integers (and $n>0$), is $\Phi\_n(x)$, the cyclotomic polynomial. So we should check whether $g(x)$ has cyclotomic divisors. Sin... | 3 | https://mathoverflow.net/users/4312 | 387408 | 160,894 |
https://mathoverflow.net/questions/387412 | 1 | Let $\mathbf{c}\_i,\mathbf{s}\_i$ be given entry-wise positive $n\times 1$ vectors for $i\in[1,\dots,d]$. Let $\tau, \alpha\_1,\dots, \alpha\_d$ be given positive constants.
Consider the linear programming problem to find vectors $\mathbf{x}\_1,\dots,\mathbf{x}\_d$ for the joint optimization problem
\begin{align}
\max\... | https://mathoverflow.net/users/27249 | Optimality gap between a joint linear program and decoupled sub programs | This idea is the essence of *Dantzig-Wolfe decomposition*, which is an exact algorithm for solving linear and mixed integer linear programming problems with such *block-angular* structure. The $\le 0$ constraints in your original problem are called *complicating* or *linking constraints*. If these constraints are remov... | 3 | https://mathoverflow.net/users/141766 | 387425 | 160,898 |
https://mathoverflow.net/questions/387393 | 3 | Consider $P\_i$, which is a regular $i$-gon in $\mathbb{R}^2$ having diameter $1$.
Assume that a compact set $X$ in $[0,10]^2\subset \mathbb{R}^2$ is a union of convex hulls of $P\_i$. I want to prove that there is $0<r\_0$ s.t. $0<r<r\_0$ implies that area of $r$-tubular neighborhood $U\_r(X)$ of $X$ is less than ${... | https://mathoverflow.net/users/36572 | Area of tubular neighborhood of union of convex sets having diameter $1$ with vertices | $\DeclareMathOperator{\area}{area}$ The region $X$ is a non-convex polygon and for simplicity I will asume it is simply connected. Its vertices are of two types: convex vertices and non-convex vertices. Denote by $V\_+$ the set of convex vertices and by $V\_-$ the set of non-convex vertices. For a vertex $v$ denote by ... | 2 | https://mathoverflow.net/users/20302 | 387434 | 160,900 |
https://mathoverflow.net/questions/387431 | 1 | There is [this sequence](https://oeis.org/A002895) listed on OEIS - named *Domb numbers*. I'm curious about
>
> **QUESTION.** Is there a direct combinatorial proof for the identity
> $$\sum\_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}
> =\sum\_{a+b+c+d=n}\binom{n}{a,b,c,d}^2,$$
> where the RHS is sum over all ... | https://mathoverflow.net/users/66131 | In search of a combinatorial proof for a multinomial sum | The [Richmond and Shallit paper](https://arxiv.org/pdf/0807.5028.pdf) linked from OEIS shows that both sides count the number of Abelian squares of length $2n$ on an alphabet of $4$ letters.
| 4 | https://mathoverflow.net/users/141766 | 387437 | 160,901 |
https://mathoverflow.net/questions/387432 | 3 | If we did not know it before, then wikipedia teaches us the [generalized geometric series](https://en.wikipedia.org/wiki/Binomial_series)
$$\sum\_{n \ge 0} \binom{n+k}{n} (1-\mu)^{n} \mu^k = \frac{1}{\mu}.$$
We can then study for $0 <\varepsilon < \mu,\nu <1-\varepsilon$ the Lipschitz bound
$$\sum\_{n \ge 0} \bin... | https://mathoverflow.net/users/108483 | Optimal scaling of Lipschitz estimates in generalized geometric series | $\newcommand\ep\varepsilon\newcommand\de\delta$
The best possible value for the Lipschitz constant $C\_\ep(k)$ is
\begin{equation\*}
C^\*\_\ep(k):=\sup\_{\ep<\mu<1-\ep}c\_\mu(k),\tag{-1}
\end{equation\*}
where
\begin{align\*}
c\_\mu(k)&:=\sum\_{n\ge0} \binom{n+k}n \Big|\frac d{d\mu}{(1-\mu)^n} \mu^k\Big| \\
&=\s... | 3 | https://mathoverflow.net/users/36721 | 387458 | 160,908 |
https://mathoverflow.net/questions/387419 | 3 | I need to evaluate the following finite sum:
$$
\sum\_{j=0}^{h}(-1)^j\binom{h}{j}(jx)\_{k},\qquad k\geq h,\, x\in\mathbb{R}^{+}
$$
and
$$
(jx)\_{k}=jx(jx-1)\cdots(jx-k+1)
$$
is the falling factorial. Any hints would be appreciated!
| https://mathoverflow.net/users/78781 | Finite sum with falling factorial | This sum is $k!$ times the coefficient of $z^k$ in $[1-(1+z)^x]^h$. So it can be simplified for a few values of $x$ (specifically $x=0$, $1$, $-1$, $2$, or $1/2$) but not in general.
| 8 | https://mathoverflow.net/users/10744 | 387464 | 160,909 |
https://mathoverflow.net/questions/387460 | 0 | I am curious about the irreducible representations $\rho: D\_{2p} \rightarrow GL\_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two questions, especially the first question:
Given a prime number $p$, what are the faithful irreducible representations... | https://mathoverflow.net/users/166540 | Faithful irreducible representations of the dihedral group $D_{2p}$, of dimension at most $p-1$ | Over $\mathbb{Q}$, $\mathbb{Z}/p$ has two irreducible representations: the trivial one, and one of dimension $p-1$. Indeed, the group algebra is $\mathbb{Q}[t]/(t^p-1) = \mathbb{Q} \times \mathbb{Q}[t]/(1+t+\ldots+t^{p-1}) = \mathbb{Q} \times \mathbb{Q}(\zeta\_p)$ by the irreducibility of the cyclotomic polynomial.
A... | 8 | https://mathoverflow.net/users/144469 | 387465 | 160,910 |
https://mathoverflow.net/questions/387345 | 1 | In their arXiv preprint, "Infinite Time Turing Machines" ([arXiv:math/9808093v1](https://arxiv.org/abs/math/9808093v1) [math.LO] 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM's as follows:
>
> Lost Melody Theorem 4.9 [pg. 28 in the preprint above—my comment]. There is a real, $c$, such that ... | https://mathoverflow.net/users/20597 | Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why not? | Note that reals have an odd "double role" in the ITTM setting; besides being sets of natural numbers, they are also individual inputs to type-$2$ functionals. In the latter role they are analogous to *natural numbers* in ordinary recursion theory; more accurately, though, IT recursion theory simply has a distinct "flav... | 4 | https://mathoverflow.net/users/8133 | 387466 | 160,911 |
https://mathoverflow.net/questions/387463 | 0 | I want to understand [this entry](https://mathoverflow.net/questions/145678/expected-value-of-logarithm-of-a-binomial-random-variable), but do not understand how the $\mathcal{O}\left(\frac{1}{n^2}\right)$ in the accepted answer comes into play.
I reproduce the question here: We have $x \sim \mathrm{Bin}\_{p,n}$ and ... | https://mathoverflow.net/users/166974 | Expected value of Taylor series with central moments of binomial variate | There are many ways to bound the Binomial central moments. A good (very general) estimate is provided by the Marcinkiweicz-Zygmund inequality, in the sharp form due to Burkholder [1], Writing $X-np$ as a sum of $n$ i.i.d. mean zero variables $Y\_i$ taking values
$1-p$ and $-p$, observe that the square function S(X) def... | 1 | https://mathoverflow.net/users/7691 | 387469 | 160,912 |
https://mathoverflow.net/questions/387424 | 9 | This question was asked on math.stackexchange and didn't receive an answer. But I think it's interesting, and I at least would love to know the answer.
Let ${\sf ZFCU}$ be the axioms of ${\sf ZFC}$ modified to allow for urelements in the usual way. We do not assume that the urelements form a set.
The question is wh... | https://mathoverflow.net/users/17968 | Does the axiom schema of collection imply schematic dependent choice in ZFCU? | (Remark: Sam Roberts pointed out to me an error in the first version of the proof I posted earlier -- there was no reason that the desired automorphisms exist, as there was no specification of them on the atoms outside of the sets of atoms under consideration. The following is a significantly modified version, and I th... | 5 | https://mathoverflow.net/users/160347 | 387471 | 160,914 |
https://mathoverflow.net/questions/387462 | -1 | Let $G=(V,E)$ be a finite, simple, undirected graph. For each $v\in V$ we define the *path sequence* $\text{ps}^G\_v: \omega\to \omega$ where $\text{ps}\_v(n)$ is the number of vertices that can be reached from $v$ via a [path](https://en.wikipedia.org/wiki/Path_(graph_theory)) of length at most $n$. (So $\text{ps}^G\_... | https://mathoverflow.net/users/8628 | Path sequences and isomorphism of graphs | Take two nonisomorphic $k$-regular graphs of order $n$ and diameter $2$. For any vertex $v$ in such a graph, $|\{w:d(v,w)\le r\}|$ is $1$ if $r=0$, $k+1$ if $r=1$, and $n$ if $r\gt1$.
With $k=3$ and $n=6$ you can take $K\_{3,3}$ and the [triangular prism graph](https://en.wikipedia.org/wiki/Prism_graph). They are non... | 3 | https://mathoverflow.net/users/43266 | 387474 | 160,916 |
https://mathoverflow.net/questions/386845 | 0 | I have a slightly open-ended question about the Barban-Halberstam-Davenport theorem and hope that it is not off-topic. The theorem itself states that for any $A>0$ and $Q$ lying the range $x\log^{-A}x\leq Q\leq x$, we have $$\sum\_{q\leq Q}\sum\_{\substack{a=1,\\\gcd(a,q)=1}}^q\left(\psi(x;q,a)-\frac{x}{\phi(q)}\right)... | https://mathoverflow.net/users/nan | Why is the Barban-Halberstam-Davenport theorem important? | Barban-Davenport-Halberstam provides evidence for Montgomery's conjecture on the distribution of primes in progressions. This is important for many reasons.
For example, Montgomery's conjecture is sufficient to imply the Elliott-Halberstam conjecture.
The left-hand side is an average over $\sim \frac{3}{\pi^2}Q^2$ te... | 2 | https://mathoverflow.net/users/111215 | 387482 | 160,920 |
https://mathoverflow.net/questions/385786 | 4 | Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\mathrm{GL}(\Bbb R^d)}$ that fixes $P$ set-wise).
This symmetry induces an automorphism $\phi:\mathcal F(P)\to\mathcal F(P)... | https://mathoverflow.net/users/108884 | A combinatorial characterization of the central inversion of a polytope? | The answer is *Yes* and it also applies not only to convex polytopes, but also to any other bounded full-dimensional polytopal complex with a well-defined interior (not necessarily spherical or convex). For simplicity, say that $P$ contains the origin in its interior.
Suppose that $T\in\mathrm{GL}(\Bbb R^d)$ induces ... | 2 | https://mathoverflow.net/users/108884 | 387489 | 160,921 |
https://mathoverflow.net/questions/387491 | 6 | Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a *half-dimensional* subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have infinite dimension.
Fix one half-dimensional subspace $W\_0$. The Grassmannian of $H$ is
$$\mathrm{Gr}(H, W\_0) = \{W... | https://mathoverflow.net/users/16702 | Nonvanishing section of infinite-dimensional tautological bundle | Let $X$ be any paracompact space. Then Hilbert vector bundles over $X$ are classified by homotopy classes of maps $[X, BU(\mathcal H)]$. But when $\mathcal H$ is infinite-dimensional, the group $U(\mathcal H)$ is contractible (this is Kuiper's theorem), and hence every infinite-dimensional Hilbert bundle over a paracom... | 8 | https://mathoverflow.net/users/40804 | 387493 | 160,922 |
https://mathoverflow.net/questions/387492 | 9 | Recall that the [$q$-binomial coefficient](https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient) $\big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]$ is the Laurent polynomial in $q$ given by
$$
\big[\begin{smallmatrix}a\\b\end{smallmatrix}\big]=\frac{[a]!}{[b]![a-b]!}
$$
where $[n]!=[1][2][3]...[n]$, and $[i]=\... | https://mathoverflow.net/users/5690 | sl(2)-reps categorifying q-binomials | Yes there is. In my article with Chipalkatti ["On the Wronskian combinants of binary forms"](https://www.sciencedirect.com/science/article/pii/S0022404906001812) in J. Pure Appl. Algebra, we gave an explicit construction in Section 2.5. We called it the Wronskian isomorphism because when followed (on the sym-sym side) ... | 4 | https://mathoverflow.net/users/7410 | 387501 | 160,925 |
https://mathoverflow.net/questions/387508 | 1 | Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c\_1=1, c\_2=6, c\_3=0$.
>
> **Question.** Is it true that $E(-1)=E^\*$?
>
>
>
What I am able to show is that there is equality at the level of Chern characters, namely $\operatorname{ch}(E(-1))= \operatornam... | https://mathoverflow.net/users/153842 | Dual of stable vector bundle on a Fano threefold | For any vector bundle $E$ of rank 2 there is an isomorphism
$$
E^\* \cong E \otimes \det(E^\*).
$$
If $\det(E) = \mathcal{O}(1)$, this boils down to $E^\* \cong E(-1)$.
| 6 | https://mathoverflow.net/users/4428 | 387511 | 160,928 |
https://mathoverflow.net/questions/387521 | 9 | I am looking for an example (if such exist) of a smooth projective variety $X$ whose $\mathbb{Q}$-homology $H\_\*(X,\mathbb{Q})$ is generated by algebraic cycles, and yet does not have a second homotopy group, $\pi\_2(X)=0.$ Thus, algebraic cycles that span $H\_2(X,\mathbb{Q})$ are coming from some non-rational curves.... | https://mathoverflow.net/users/114985 | Smooth projective variety with no second homotopy group | [Fake projective planes](https://en.wikipedia.org/wiki/Fake_projective_plane) have $H\_2(X,\mathbb{Z}) \cong \mathbb{Z}$. They have metrics of pinched negative curvature, so they have $\pi\_2(X) \cong \{0\}$. Thus $H\_2(X,\mathbb{Q}) \cong \mathbb{Q}$ is generated by a hyperplane section, and this is not a rational cur... | 18 | https://mathoverflow.net/users/142269 | 387523 | 160,933 |
https://mathoverflow.net/questions/387500 | 5 | *This is a follow-up question on a [previous question](https://mathoverflow.net/questions/387491/nonvanishing-section-of-infinite-dimensional-tautological-bundle/387495#387495) of mine, which ended up to be trivial, because I overlooked the obvious problem with Hilbert space bundles, which I fix here.*
---
Let us... | https://mathoverflow.net/users/16702 | Nonvanishing section of infinite-dimensional tautological bundle II | $\DeclareMathOperator{\Gr}{Gr}\newcommand{\C}{\mathbb{C}}\DeclareMathOperator\*{\colim}{colim}$The total space of your bundle is the colimit of the spaces $\tau\_n\times\C^{[N,\infty)}$, where $\tau\_n\to \Gr\_n(E\_{[-n,n]})$ is the tautological bundle over $\Gr\_n(E\_{[-n,n]})$ and the transition functions are the pro... | 2 | https://mathoverflow.net/users/35687 | 387527 | 160,936 |
https://mathoverflow.net/questions/387533 | 33 | The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.
This is in the vein of many other questions on mathoverflow:
* [What is the Levi-Civita connection trying to describe?](https://mathoverflow.net/questions/376486/what-is-the-levi-civita... | https://mathoverflow.net/users/98901 | How should you explain parallel transport to undergraduates? | This may not reallly be an answer that you like, but I think that, maybe you misunderstood what Ben McKay was trying to describe. Here is a more explicit, extrinsic description that may help:
Suppose that $M^m\subset\mathbb{E}^n$ is an isometrically embedded submanifold of Euclidean $n$-space. Let $\gamma:(a,b)\to M^... | 27 | https://mathoverflow.net/users/13972 | 387538 | 160,937 |
https://mathoverflow.net/questions/387455 | 5 | Let $H$ be a cocommutative hopf algebra.
Let $M$ be the category of $H$-bimodules.
Does the category $M$ form a braided monoidal category with tensor product $\otimes\_{H}$ ?
| https://mathoverflow.net/users/141953 | Braided monoidal category, example | The answer is no in general.
Here is a counter example. Let us work over a ground field $k$, and let $ H = \oplus\_n k$ be the direct sum of $n$ copies of $k$, with $n \geq 2$. This is a commutative, cocommutative Hopf algebra. Of course, as others have commented, the monoidal category $({}\_H Mod\_H, \otimes\_H)$ on... | 2 | https://mathoverflow.net/users/184 | 387542 | 160,940 |
https://mathoverflow.net/questions/387447 | 1 | Working in $$\sf ZF + GC + j:V \xrightarrow {auto} V + \exists \alpha: V\_{j(\alpha)} \subsetneq V\_\alpha \land \alpha \text{ is limit }$$
Of course $j$ is *external* in the sense that it is not used in instances of separation nor replacement.
Now suppose that $S \subsetneq V\_\alpha$ and such that the ordinal ran... | https://mathoverflow.net/users/95347 | Would automorphisms cause nested subset-hood? | Regarding the first question, no:
Let $\beta=j(\alpha)$ and $\beta+\gamma=\alpha$. Define a version of the cumulative hierarchy with the empty set replaced with $\beta$. (That is, let $V\_0’=\beta$, $V\_1’=\{\beta\}$, and then $V\_{\xi+1}’=\mathcal{P}(V\_\xi’)\cup\{\beta\}\backslash\{\emptyset\}$ for $\xi>0$, and tak... | 5 | https://mathoverflow.net/users/160347 | 387549 | 160,943 |
https://mathoverflow.net/questions/387537 | 14 | Assume we look at $n\in\mathbb N$ people that can have anywhere between $1$ to $k\in\mathbb N$ hairs on their head.
Formally, I look at $n$ independent (in fact, this is not really true in real life because of inheritance etc., but bear with me) uniformly (might also not be true in real life) distributed random varia... | https://mathoverflow.net/users/129831 | How many people have the same exact number of hairs? | This question has been studied extensively in the computer science literature under the name "balls in bins"; see [1] which gives quite tight bounds in Theorem 1, page 161 and also describes prior work before 1999.
[1] Raab, Martin, and Angelika Steger. "“Balls into bins”—A simple and tight analysis." In Internationa... | 15 | https://mathoverflow.net/users/7691 | 387552 | 160,945 |
https://mathoverflow.net/questions/387530 | 6 | Is there a continuous function $f:[0,+\infty) \to \operatorname{SO}(n)$ whose image is dense in $\operatorname{SO}(n)$ and that is well behaved in certain ways?
* For each $\varepsilon>0$ it doesn't take longer than necessary, or not much, to come within distance $\varepsilon$ of every point.
* It is not too hard to ... | https://mathoverflow.net/users/6316 | Grand tour of the special orthogonal group | [**EDIT:** Dan Asimov notified me that this construction is similar to a construction in his 1985 paper entitled "The Grand Tour: a Tool for Viewing Multidimensional Data". The construction in the 1985 paper is somewhat more elegant than this one, avoiding the use of the exponential map and the sine function.]
We'll ... | 6 | https://mathoverflow.net/users/39521 | 387560 | 160,948 |
https://mathoverflow.net/questions/387557 | 4 | Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"?
For purpose to prove this we might have to prove the green function of basin of attraction to infinity is Holder continuous.
| https://mathoverflow.net/users/176741 | Hausdorff dimension of Julia set | This argument is from [Eremenko-Lyubich survey](https://www.math.purdue.edu/%7Eeremenko/dvi/AAsurveyp2.pdf): Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By [Ledrappier-Young entropy formula](http://www.scholarpedia.org/article/Pesin_entropy_formula#The_Ledrappier-Young_E... | 5 | https://mathoverflow.net/users/128556 | 388558 | 160,949 |
https://mathoverflow.net/questions/388563 | 2 | I am interested in learning about Mathematical Instanton bundles on Projective varieties and their moduli. The earliest paper that I can locate online is [This paper](http://www.numdam.org/article/ASENS_1982_4_15_2_365_0.pdf). Indeed there are several other papers after that. Is there any book or survey paper on Mathem... | https://mathoverflow.net/users/133832 | References on Mathematical Instanton bundles | The book [Okonek, Christian; Schneider, Michael; Spindler, Heinz. Vector bundles on complex projective spaces] is a very nice source, but it is a bit old, of course.
| 5 | https://mathoverflow.net/users/4428 | 388565 | 160,950 |
https://mathoverflow.net/questions/387551 | 5 | Let $(A, \Delta)$ be a $C^\*$-algebraic compact quantum group (in the sense of Woronowicz).
***Definition:*** A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a\_{i,j}) \in M\_n(A)$ such that
$$\Delta(a\_{i,j}) = \sum\_k a\_{ik}\otimes a\_{kj}$$
for all $i,j$.
The matrix $a$ is called non-degenerate (unit... | https://mathoverflow.net/users/nan | Matrix coefficients of a compact quantum group | In general, the converse inclusion $A\_1 \subseteq A\_0$ does not hold.
As the counterexample below shows, it is not a good idea to define a corepresentation matrix as in the question. To really be considered as a corepresentation matrix, one should require $a$ to be invertible as an element in $M\_n(A)$. Below is an... | 4 | https://mathoverflow.net/users/159170 | 388577 | 160,952 |
https://mathoverflow.net/questions/388571 | 1 | Define for $u\in C\_c^\infty (\mathbb R^n), 0<s<1$ the integral
$$
I\_s(u) = \int\_{(x,y)\in \mathbb R^{n+n}} \frac{(u(x+y)-u(x))^2}{|y|^{d+2s}} dxdy.
$$
I wish to prove that for some $C=C(s)>1,$
$$
C^{-1}\|u\|\_{\dot H^s} \leq I\_s(u) \leq C\|u\|\_{\dot H^s} .
$$
See [this Wikipedia page](https://en.wikipedia.org/wiki... | https://mathoverflow.net/users/121404 | Double space integral formulation of homogeneous Sobolev norm | You have done half of the job. We have the absolutely convergent integral which is such that
$$
f(\xi)=\int\_{\mathbb R^d} \frac{\vert e^{2iπ y\cdot \xi}-1\vert^2}{\vert y\vert^{d+2s}} dy
=c\_{s,d}\vert \xi\vert^{2s},
$$
since for $A\in O(d)$, $f(A\xi)=f(\xi)$ (change of variables $y=Ay'$) and moreover for $\lambda ... | 2 | https://mathoverflow.net/users/21907 | 388579 | 160,953 |
https://mathoverflow.net/questions/388572 | 3 | Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?
| https://mathoverflow.net/users/177755 | Absolute Galois group with unique closed non-open subgroup | Eventually this is purely a fact of profinite groups.
**Proposition.** *Let $G$ be an infinite profinite group in which every nontrivial closed subgroup is open. Then $G$ is isomorphic to $\mathbf{Z}\_p$ for some prime $p$.*
Proof. First, the case when $G$ is abelian. Then its Pontryagin dual is a discrete torsion ... | 2 | https://mathoverflow.net/users/14094 | 388580 | 160,954 |
https://mathoverflow.net/questions/388585 | 18 | The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are finitely presented.
Are there simpler examples of intermediate growth if we drop the requirement that there exists an inverse... | https://mathoverflow.net/users/174368 | Are there any "simple" monoids with intermediate growth? | Yes. Jan Okninski showed that $$\begin{bmatrix} 1 & 1 \\ 0 &1\end{bmatrix}\ \text{and}\ \begin{bmatrix} 1 & 0\\ 1 & 0\end{bmatrix}$$ generate a semigroup of intermediate growth. Details can be found in [Nathanson](http://www.theoryofnumbers.com/melnathanson/pdfs/nath1999-94.pdf). The growth was estimated there to be li... | 22 | https://mathoverflow.net/users/15934 | 388586 | 160,956 |
https://mathoverflow.net/questions/388587 | 4 | Let $X$ be a normal, projective (complex) variety with at worst rational singularities. Let $\pi:Y \to X$ be the resolution of singularities obtained by blowing-up the singular points. Is $R^1 \pi\_\*\mathbb{Z}=0$? I am mainly interested in the case when $X$ is of dimension $3$.
| https://mathoverflow.net/users/45397 | Higher direct image for rational singularities | This should be true. Use the exponential sequence to obtain
$$\pi\_\*\mathcal{O}\_Y\to \pi\_\*\mathcal{O}\_Y^\*\to R^1\pi\_\*\mathbb{Z}\to R^1\pi\_\*\mathcal{O}\_Y$$
The last term is zero since you have rational singularities. You can check that the first map is surjective as follows: By normality
$$\pi\_\*\mathcal{O}\... | 6 | https://mathoverflow.net/users/4144 | 388589 | 160,957 |
https://mathoverflow.net/questions/388590 | 0 | Suppose that $M\subseteq \mathbb R^D$ is $d$-dimensional compact submanifold with $0\in M$, having reach $\tau>0$. Thus, for every $p\in M$, $\exp\_p:B\_{T\_pM}(p,\tau)\rightarrow B\_M(p,\tau)$ is a diffeomorphism. Can we say there is $\phi:B\_{T\_0M}(0,\tau)\rightarrow \mathbb R^{D-d}$ such that $\exp\_0(u)=(u,\phi(u)... | https://mathoverflow.net/users/nan | implicit function theorem on manifold | No, consider for example $M=\{x\in \mathbb{R}^2| |(x\_1,x\_2)-(0,1)|=1\}$. Then $exp\_0(u)=(sin(u),1-cos(u))$.
| 1 | https://mathoverflow.net/users/35593 | 388595 | 160,959 |
https://mathoverflow.net/questions/388605 | 6 | Let $(A, \Delta\_A)$ and $(B, \Delta\_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta\_A) \to (B, \Delta\_B)$ to be a unital $\*$-morphism
$$\pi: B \to A$$
such that $$(\pi \otimes \pi)\circ \Delta\_B = \Delta\_A \circ \pi.$$
However, in the literatu... | https://mathoverflow.net/users/nan | Morphisms between compact quantum groups | When $\pi : B \to A$ is a unital $\*$-homomorphism respecting the comultiplication, then automatically $\pi(B\_0) \subseteq A\_0$, because $\pi$ maps a corepresentation of $(B,\Delta\_B)$ to a corepresentation of $(A,\Delta\_A)$.
The converse need not hold: if $\pi : B\_0 \to A\_0$ is a unital $\*$-homomorphism respe... | 7 | https://mathoverflow.net/users/159170 | 388608 | 160,962 |
https://mathoverflow.net/questions/388598 | 6 | I'm studying the article "An alternative proof of Lickorish–Wallace theorem" ([doi link](https://doi.org/10.1016/j.topol.2013.06.009))
and I got stuck in a problem.
Let $H\_g$ be a 3 dimensional handlebody of genus $g$, a primate curve in $H\_g$ is a knot in $\partial H\_g$ that intersects an essential disk of $H\_g$... | https://mathoverflow.net/users/153263 | Dehn surgery along primitive knot in 3-dimensional handlebody | Since $c\subset H\_g$ intersects an essential disc $D$ in a single point, the boundary of a regular neighbourhood of $D\cup c$ is another disc $D'$, which splits $H\_g$ into a solid torus containing $D\cup c$ and the rest. You can forget about the rest (this is a $\partial$-connected sum) and consider the solid torus a... | 9 | https://mathoverflow.net/users/6205 | 388615 | 160,964 |
https://mathoverflow.net/questions/387513 | 2 | The permutation group $S\_{2n}$ has $H\_{2,n}=S\_2\wr S\_n$ as a subgroup. The plethysm $h\_n(h\_2)=\sum\_{\lambda\vdash n}s\_{2\lambda}$ is well known.
The zonal spherical functions $\omega\_\lambda(g)=\frac{1}{H\_{2,n}}\sum\_{h\in H\_{2,n}}\chi\_{2\lambda}(gh)$ are responsible for the transition between power sum s... | https://mathoverflow.net/users/83671 | Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials | The corresponding orthogonal polynomials should be closely related to the *Eberlein polynomials*, i.e., their coefficients should be closely related to the zonal spherical functions of the Gelfand pair $(S\_{2n},S\_{n} \times S\_n)$, see VII.1 Ex. 13 of Macdonald's text, for example.
In particular, the zonal spherica... | 2 | https://mathoverflow.net/users/32968 | 388626 | 160,971 |
https://mathoverflow.net/questions/387543 | 5 | We are given a *multiset* $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
* two *distinct instances* $x\_r$ and $y\_r$ of $M$'s numbers are selected uniformly at random from $M$ (which implies that they cannot be the *same instance* of any number cont... | https://mathoverflow.net/users/115803 | Convergence speed of a random dyadic rational generator | This is a cute problem... Let's normalise your multiset $M$ to a probability measure $\mu$ by setting $\mu = {1\over |M|} \sum\_{x \in M} \delta\_x$ (repeated elements are repeated in the sum). Write also $h \colon x \mapsto x/2$. If we then set $t = \log N$ for $N$ the number of steps, a very good model for the evolut... | 6 | https://mathoverflow.net/users/38566 | 388632 | 160,974 |
https://mathoverflow.net/questions/387556 | 5 | Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a\_1, a\_2, b\_1, b\_2$ with $a\_1+b\_1=a\_2+b\_2=1$ consider the quantity
$$N(f)=\int\_{[0,1]} \int\_{[0,1]} f(a\_1x+b\_1y)f(a\_2x+b\_2y) \, dx \, dy.$$
If $a\_1=a\_2=1$ and $b\_1=b\_2=0$ then $N(f)^{1/2}$ is ... | https://mathoverflow.net/users/24494 | Which averages of products of a function give a norm? | It is a norm when $a\_1=a\_2$, $b\_1=b\_2$. Otherwise making the change of variables $u=a\_1x+b\_1y, v=a\_2x+b\_2y$ we get the integral of $f(u)f(v)$ over certain parallelogram $P$ with a diagonal joining $(0, 0) $ and $(1, 1) $. Assume that $f$ has a small support $\Delta$ so that $\Delta^2\subset P$. Then the integra... | 4 | https://mathoverflow.net/users/4312 | 388635 | 160,976 |
https://mathoverflow.net/questions/388641 | 8 | Let $\varphi\colon A\to B$ be a bounded, linear map between C\*-algebras. Is the bitranspose $\varphi^{\*\*}\colon A^{\*\*}\to B^{\*\*}$ continuous when the von Neumann algebras $A^{\*\*}$ and $B^{\*\*}$ are equipped with their $\sigma$-strong topologies?
Motivation/Background: Note that $\varphi^{\*\*}$ is clearly c... | https://mathoverflow.net/users/24916 | Is the bitranspose continuous for the $\sigma$-strong topology? | I think the transpose map on the compacts gives a counterexample.
Let $K(H)$ be the compacts on a separable infinite dimensional Hilbert space with orthonormal basis $\{ e\_n \}.$ Let $T:K(H)\rightarrow K(H)$ be the transpose map (i.e. $T(e\_{n,m})=e\_{m,n}$ on matrix units). Then $T$ is $\sigma$-weakly continuous so... | 6 | https://mathoverflow.net/users/34640 | 388647 | 160,980 |
https://mathoverflow.net/questions/388594 | 1 | Let $\alpha\_i\in[0,1]^k$, $x\_i\in\mathbb{R}^d$ for all $i\in[k]$, with $k \geq d$. Define $X: \operatorname{col}(X) = \{x\_i\}\_{i\in[k]}$, $\Lambda(\alpha) = \operatorname{diag}(\alpha)$, $y\in\mathbb{R}^d$. Is the following expression convex?
$$ f(\alpha) = y^\top\left(\sum\_{i=1}^k \alpha\_i x\_ix\_i^\top\right)... | https://mathoverflow.net/users/156139 | Convexity of inverse quadratic form | Claim 1: *If $A$ is positive semidefinite then $\langle Ax, x \rangle \langle Ay,y\rangle \geq \langle Ax,y\rangle^{2}$*.
This Cauchy--Schwartz inequality can be proved by considering quadratic function $\lambda \mapsto \langle x+\lambda y, A(x+\lambda y) \rangle \geq 0$ and writing down that its discriminant $D = 4 ... | 2 | https://mathoverflow.net/users/50901 | 388656 | 160,984 |
https://mathoverflow.net/questions/388652 | 32 | I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your mathematical field of research.
I am wondering however whether behind a computer algebra system like mathematica that does sy... | https://mathoverflow.net/users/119875 | How does Mathematica do symbolic integration? | An overview by one of the developers of Mathematica, focusing on definite integrals, is at [Symbolic definite integration: methods and open issues.](https://library.wolfram.com/infocenter/Conferences/5832/DefiniteIntegration.pdf)
Mathematica knows all the entries in Gradshteyn-Ryzhik, and more generally uses the Mari... | 40 | https://mathoverflow.net/users/11260 | 388662 | 160,986 |
https://mathoverflow.net/questions/388648 | 10 | First a disclaimer. This is an old question that I considered years ago and that I recently remembered. Since I am no longer in active research it may be considered as 'idle curiosity', although I feel I could probably learn a good deal from an answer (and maybe others could find it instructive as well).
Now the ques... | https://mathoverflow.net/users/17907 | Brauer-Manin obstruction on an open subset of an elliptic curve | Yes you can get:
$$
X(\mathbb{A}\_\mathbb{Q})^{\operatorname{Br}(X)} = \varnothing
$$
in fact we can prove
$$
X(\mathbb{A}\_\mathbb{Q})^{\operatorname{Br}(E)} = \varnothing
$$
and we can even make this (possibly larger) set to be empty.
The Brauer Manin obstruction for Abelian varieties is nicely explained in... | 7 | https://mathoverflow.net/users/23501 | 388670 | 160,989 |
https://mathoverflow.net/questions/388673 | 0 | For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in x\big\}$.
Let $G=(\omega,E)$ be a connected simple, undirected graph, where $E\subseteq [\omega]^2$.. For $m<n\in \omega$ we let $G/\_{\{m,n\}}$ be the graph that we get from removing $n$, and $m$ gets all the neighbours from $n$. Formally, let
$$G/\_{\{m,n\}} = ... | https://mathoverflow.net/users/8628 | Connected graphs on $\omega$ that "doesn't change" whenever 2 points are collapsed | There are many such graphs:
* The [(countable) **random graph**](https://en.wikipedia.org/wiki/Rado_graph).
* The "infinite depth-$1$ tree" given by $E=\{\{0,i\}: i>0\}$.
* The "majestic sunflower" consisting of infinitely many $n$-cycles, for each $n$, joined at a common point.
| 2 | https://mathoverflow.net/users/8133 | 388675 | 160,990 |
https://mathoverflow.net/questions/388672 | 3 | Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal functor functor.
Suppose that $g: C\rightarrow D$ a monoidal lax monoidal functor between small monoidal categories.
I ... | https://mathoverflow.net/users/17895 | Lax monoidal functor | Not necessarily; however, if $F$ is a $2$-functor (so it sends natural transformations to identities), then the answer is yes.
If you weaken it to $F$ sending naturally isomorphic functors to equal arrows, then the answer is no in general, but yes if $g$ is strong monoidal.
If you don't have any assumption of that ... | 5 | https://mathoverflow.net/users/102343 | 388677 | 160,992 |
https://mathoverflow.net/questions/388668 | 1 | Let $\phi$ be a nonnegative $C\_c^\infty(B(0,1))$ function, where $B(0,1)\in \mathbb R^n$ is the unit ball, and $\int \phi =1$. Let $\phi\_{\epsilon}(x) =\epsilon^{-n} \phi(x/\epsilon).$ For any $L^2$ function $u$, we can get a $C^\infty$ approximation by taking $u\ast \phi\_\epsilon.$ The problem I am focusing on here... | https://mathoverflow.net/users/121404 | Convolution mollification of $H^s$ functions uniformly in the unit ball of this sobolev space | This works for all $s>0$.
If you take Fourier transforms (and write $\widehat{\varphi}=\psi$), then you are asking if
$$
\lim\_{\epsilon\to 0}\sup\_{\|(1+|t|^s)\widehat{u}\|=1}\|\widehat{u}(\psi(\epsilon t)-1)\| =0 .
$$
That's the same as asking if $T\_{\epsilon}\to 0$ in operator norm on $L^2$, where
$$
(T\_{\epsilon}... | 0 | https://mathoverflow.net/users/48839 | 388679 | 160,994 |
https://mathoverflow.net/questions/388667 | 3 | Let $f:X\rightarrow Y$ be a morphism of scheme over $\mathbb{C}$. Assume that $Y$ and the the general fiber $F\_y = f^{-1}(y)$ of $f$ are irreducible.
Does there exists an irreducible component $X'$ of $X$ such that $f' := f\_{|X'}:X'\rightarrow Y$ satisfies $(f')^{-1}(y) = f^{-1}(y)$ for $y\in Y$ a general point?
| https://mathoverflow.net/users/nan | Irreducibility of the base and of the general fiber | If you assume $f$ is of finite type and $X$ has finitely many irreducible components, then the answer is yes. There's a nonempty open neighborhood $V$ of $\eta\_Y$ so that for $y\in V$, we have that $\dim X\_y=\dim X\_{\eta\_y}$ ([ref](https://stacks.math.columbia.edu/tag/05F7)), so there's a nonempty open subset $U\su... | 4 | https://mathoverflow.net/users/112114 | 388681 | 160,995 |
https://mathoverflow.net/questions/387138 | 4 | *Previously asked and bountied without response at [MSE](https://math.stackexchange.com/q/4056043/28111).*
---
This question is a companion to [this one](https://mathoverflow.net/questions/386478/how-many-steps-does-it-take-to-tarski-vaughtify-second-order-logic), about a tame(?) fragment of second-order logic wi... | https://mathoverflow.net/users/8133 | Compactness number for a fragment of second-order logic | I think that if you have a logic $\mathcal{L}$ which has downward Lowenheim-Skolem (for theories of arbitrary cardinality, i.e. if $T$ has cardinality $\lambda$ and $N$ is a model $A$ of size $\theta\geq\lambda$, and $\lambda\leq\gamma\leq\theta$, then we can find a sub-model of size $\gamma$ which is elementary in $A$... | 8 | https://mathoverflow.net/users/160347 | 388686 | 160,998 |
https://mathoverflow.net/questions/386010 | 2 | I'm currently reading Mar's and Springer's *Character Sheaves*. In Chapter 2 (Kummer local systems on tori), they provide a construction of Kummer local systems on a torus $T$ by way of the $m^{th}$ power isogeny, whose Galois covering $\_m T$ is the group of elements of $T$ with order dividing $m$. For a character $x$... | https://mathoverflow.net/users/175051 | Character constructed from Kummer local system lifts to representation of algebraic torus | For anyone interested: it turns out that the fundamental group in question is the *étale* fundamental group, which is defined as a projective limit along the automorphism groups of a pro-representing system for the finite étale covering spaces of $T$. As such, $\pi\_1(T,e)$ by definition comes equipped with a map to $\... | 1 | https://mathoverflow.net/users/175051 | 388687 | 160,999 |
https://mathoverflow.net/questions/388674 | 2 | Let $(X,D)$ be a log pair, with $X$ a projective manifold (or quasi-projective) and $D$ a divisor with simple normal crossings.
>
> I'd like to construct an example, or be pointed to a reference, for an
> example where $\Omega\_X^1(\log D)$ is big, but $\Omega\_X^1$ is not
> big.
>
>
>
There are metric charact... | https://mathoverflow.net/users/174369 | Examples of complex manifolds for which the logarithmic cotangent bundle is big, but the cotangent bundle is not big | Given any smooth projective variety $X$, it is always possible to find a normal crossing divisor $D$ such that $\Omega^1(\log D)$ is big (actually, such that it satisfies a stronger condition called *almost ampleness*).
This follows from the following result, see [**BD18**, Theorem A].
>
> **Theorem.** Let $X$ be... | 7 | https://mathoverflow.net/users/7460 | 388713 | 161,008 |
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