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https://mathoverflow.net/questions/299235 | 0 | I don't know if this question is considered research-related. If not, I will move it to [Math SE](https://math.stackexchange.com).
---
I am searching for matrices with the property
$$|A|\_F^2 = \deg( \chi\_A(t) ) = 2 \deg( m\_A(t)), tr(A) = 1$$
where $\chi\_A(t)$ is the characteristic polynomial and $m\_A(t)... | https://mathoverflow.net/users/nan | Searching for matrices with some property | Take $J \oplus O$, where $J$ is the Jordan block with eigenvalue $0$ and size $n$, and $O$ is the $n\times n$ zero matrix. Replace $n+1$ zeros with ones anywhere in the upper triangular part of $J$ (which does not change the fact that $J^n=0$, $J^{n-1}\neq 0$).
| 0 | https://mathoverflow.net/users/1898 | 299251 | 131,326 |
https://mathoverflow.net/questions/299267 | 0 | Let $A\_k$ be the $n\times n$ matrix defined by
$$
A\_k=\left[
\begin{array}{}
1 & x\_1 & x\_1^2 & \cdots & x\_1^{n-2} & x\_1^k \\
1 & x\_2 & x\_2^2 & \cdots & x\_2^{n-2} & x\_2^k \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
1 & x\_n & x\_n^2 & \cdots & x\_n^{n-2} & x\_n^k
\end{array}\right].
$$
If $k=n... | https://mathoverflow.net/users/124008 | Prove a special form of Schur polynomial identity | This is exactly Exercise 6.16 in [Darij Grinberg, *Notes on the combinatorial fundamentals of algebra*, 10 January 2019](https://github.com/darijgr/detnotes/releases/tag/2019-01-10), except that the order of columns in my exercise is the other way round (which is why I end up with $\prod\limits\_{1\leq i<j\leq n} \left... | 5 | https://mathoverflow.net/users/2530 | 299282 | 131,337 |
https://mathoverflow.net/questions/299244 | 4 | A well-known result of Solovay states that $ZFC$ + "the continuum is real-valued measurable" is equiconsistent with $ZFC$ + "there is a measurable cardinal", over $ZFC$. That the continuum is mentioned here is no coincidence. A theorem of Ulam back in 1930 gave evidence that the continuum is some sort of a dividing lin... | https://mathoverflow.net/users/29231 | Consistency of the size of the least real-valued measurable cardinal, vis-a-vis the continuum | All of them are equiconsistent with the existence of a measurable cardinal. I find that a nice reference for this stuff, in addition to Solovay's article, is Jech's *Set Theory: Third Millennium Edition*.
I will discuss 3 first. Any measurable cardinal is real-valued measurable (it just happens not to use any of the... | 3 | https://mathoverflow.net/users/61785 | 299284 | 131,338 |
https://mathoverflow.net/questions/299266 | 4 |
>
> Is any finitely generated module $M$ over a complete commutative ring $A$ complete?
>
>
>
Here complete is with respect to the $I$-adic topology, where $I$ is a given ideal of $A$.
Edit: as Luc Guyot mentioned, the answer is positive in the noetherian case.
| https://mathoverflow.net/users/111816 | Finitely generated modules over a complete ring | First off, a general comment.
**Remark.**
It is not always the case that for any ring $A$, and any ideal $I$ of $A$, the $I$-adic completion of an $A$-module (no matter if finitely generated) exists. (By "exists" I mean: it is not always the case that for an $A$-module $M$, $\varprojlim\_{k\ge 1} M/I^kM$ is $I$-adica... | 7 | https://mathoverflow.net/users/nan | 299285 | 131,339 |
https://mathoverflow.net/questions/299286 | 13 | In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified simplicial sets spanned by the strict complicial sets is equivalent to the category of globular strict $\omega$-categories.
A ... | https://mathoverflow.net/users/1353 | Weak complicial sets: Are the morphisms too strict? | Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of [this paper of Steve Lack](http://tac.mta.ca/tac/volumes/18/1/18-01abs.html)). The solution to your troubling corollary is that the "correct" models for weak $\omega$-categories are not the weak complicial sets... | 11 | https://mathoverflow.net/users/57405 | 299294 | 131,343 |
https://mathoverflow.net/questions/299287 | 2 | I have the following minimization problem in $z \in \mathbb R^n$, which contains $x\_1, \dots, x\_t, y \in \mathbb R$.
$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & xQx'= y\\ & 0 \leq x\_i \leq1\\ & Az \leq b\end{array}$$
where $Q$ is diagonal and has positive diagonal integer values, $A \in \mathbb... | https://mathoverflow.net/users/10035 | Linear optimization with one positive definite quadratic equality condition in P? | The following conditions
$$
\begin{array}{l}
y=\sum x\_i^2\\
0\leq x\_i\leq 1\\
y=\sum x\_i
\end{array}
$$
are equivalent to $x\_i\in \{0,1\}$, which means your construction allows you to introduce binary variables. Then you can use the minimization objective to encode something NP-hard, like the smallest dominatin... | 3 | https://mathoverflow.net/users/120143 | 299296 | 131,344 |
https://mathoverflow.net/questions/299302 | 5 | Sylvester's law of inertia (here I quote wikipedia)
>
> If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number of negative entries in the diagonal of D is always the same, for all such S; and the same goes for the number of pos... | https://mathoverflow.net/users/119834 | higher order analogues of sylvester's law of inertia? | A generalization of Sylvester's classification of canonical quadratic forms (which is the "law of inertia") to cubic forms has been presented in [Canonical forms for symmetric tensors](https://www.sciencedirect.com/science/article/pii/0024379584901927) (1984). The matrices $A,D$ are now $2\times 2\times 2$ tensors, and... | 8 | https://mathoverflow.net/users/11260 | 299305 | 131,346 |
https://mathoverflow.net/questions/299308 | 1 | According to R.A.Kazmi's dissertation ["Isogenies and Cryptography"](https://eprint.iacr.org/2010/550.pdf) (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as
$$G\_l(x,y)=\sum\_{r=0}^{l+1}\sum\_{k=0}^{v}a\_{r,k}x^ry^k\in\mathbb Z[x,y]$$
where $v=\frac{s(l-1)}{12}$, $s=\frac{12}{\g... | https://mathoverflow.net/users/103866 | How to compute the Müller modular polynomials? | The definition of the coefficients $a\_{r,k}$ is given by theorem III.17 on page 53 of ["Elliptic curves in cryptography"](https://www.cambridge.org/core/books/elliptic-curves-in-cryptography/16A2B60636EFA7EBCC3D5A5D01F28546) by I.F. Blake, G. Seroussi, N.P. Smart (Cambridge University Press, 1999) (also on [Google Boo... | 5 | https://mathoverflow.net/users/54780 | 299310 | 131,347 |
https://mathoverflow.net/questions/299283 | 0 | We have a set of $n^2$ bins, partitioned into two subsets: $A\neq\emptyset$ and $B\neq\emptyset$. Assume $n$ goes to infinity.
We continue to throw balls independently and uniformly at random into the $n^2$ bins, until $\log(n)$ balls are placed in bins belonging to subset $A$, then we stop. Let $m$ be the total numb... | https://mathoverflow.net/users/115803 | Minimum number of balls to obtain accurate information about a bipartition of the bin set | This is a form of law of large numbers, and Chernoff's bound seems to work.
Denote $n\_A/n^2=p$, $\log n=k$, then we deal with Bernoulli i.i.d. $\xi\_1,\xi\_2,\dots$, $\xi\_i\in \{0,1\}$, $\mathbb{E} \xi\_i=p$, and $m$ is the minimal number for which $\xi\_1+\dots+\xi\_m\geqslant k$.
Then the event $\{m\geqslant c... | 2 | https://mathoverflow.net/users/4312 | 299316 | 131,348 |
https://mathoverflow.net/questions/291822 | 10 | Suppose $A$ and $A'$ are symmetrizable (generalized) Cartan matrices, in the sense of Kac's book *Infinite-dimensional Lie algebras*. Say $A$ *dominates* $A'$ if every entry of $A$ has weakly greater absolute value than the corresponding entry of $A'$. Write $\Phi(A)$ for the Kac-Moody root system associated to $A$. I ... | https://mathoverflow.net/users/5519 | Dominance relation among Cartan matrices implies containment of root systems: Is this known? | I suggest to have a look at Lemma 3.5 in <https://arxiv.org/abs/1509.01976>
| 2 | https://mathoverflow.net/users/106751 | 299336 | 131,353 |
https://mathoverflow.net/questions/299317 | 3 | Let $X,Y \geq 1$. I am interested in the number of solutions of the following diophantine equations:
$$S\_1\colon \, \, x\_1y\_1^3 = x\_2 y\_2^3 $$
Let $N\_1(X,Y) $ denote the number of solutions to $S\_1$ with $1\leq x\_i \leq X$ and $1\leq y\_i \leq Y$. There are the obvious solutions $x\_1=x\_2$ and $y\_1=y\_2$ whic... | https://mathoverflow.net/users/41685 | Diophantine equations and 'quasi-paucity' | For $S\_1$ you can take $A=0$.
Setting $g=\gcd(x\_1,x\_2)$ and $v\_i = x\_i/g$, and $h=\gcd(y\_1,y\_2)$ and $w\_i = y\_i/h$, one obtains $v\_1w\_1^3 = v\_2w\_2^3$ with $\gcd(v\_1,v\_2)=\gcd(w\_1,w\_2)=1$; this implies that $v\_1=w\_2^3$ and $v\_2=w\_1^3$. Therefore all solutions to $S\_1$ are of the form $(x\_1,x\_2,... | 7 | https://mathoverflow.net/users/5091 | 299343 | 131,356 |
https://mathoverflow.net/questions/299337 | 4 | In his article [Group quasi-representations and index theory](https://pdfs.semanticscholar.org/9521/ebb0f8e1bec139e74345d454b60e6be36e82.pdf) (p. 11), Dadarlat claims that a \*-representation $\phi\_1\colon C(Y)\to L(H)$ of a commutative C\*-algebra "is a compact perturbation of a \*-representation of $C(Y)$ which is a... | https://mathoverflow.net/users/57840 | Representation is a compact perturbation of a direct sum of one-dimensional representations | In practice, there are several non-obvious corollaries of Voiculescu's theorem that are collectively called "by Voiculescu's Theorem." In this case it is a consequence of Voiculescu's theorem (the one you mention) that every representation of a separable C\*-algebra is approximately unitarily equivalent modulo the comp... | 4 | https://mathoverflow.net/users/34640 | 299347 | 131,358 |
https://mathoverflow.net/questions/299338 | 9 | Let $(X;U,V)$ be an excisive triad and consider the corresponding part of the Mayer-Vietoris sequence $H^{\bullet-1}(U\cap V)\stackrel{\delta^\*}{\to} H^\bullet(X)\to H^\bullet(U)\oplus H^\bullet(V)$. Now let $\alpha,\beta\in H^{\bullet-1}(U\cap V)$. Can we say anything about $\delta^\*(\alpha\smile\beta)$?
| https://mathoverflow.net/users/124042 | Cup products in the Mayer-Vietoris sequence | Here are some comments.
The map $\delta^\ast$ is a composition of the form
$\require{AMScd}$
\begin{CD}
H^{\ast-1}(U\cap V) @> \sigma >\cong > H^{\ast}(\Sigma (U\cap V)) \to H^\ast X
\end{CD}
where the first map is given by the suspension isomorphism and the second is induced by the map $X\to \Sigma(U\cap V)$ wh... | 6 | https://mathoverflow.net/users/8032 | 299358 | 131,360 |
https://mathoverflow.net/questions/299351 | 3 | One can define the convergence of a sequence $(\Lambda\_k)\_k$ of full rank lattices as folow : $(\Lambda\_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists B\_k\in (\mathbb{R}^n)^n \text{ a $\mathbb{Z}$-basis of $\Lambda\_k$}, \exists B\in (\mathbb{R}^n)^n \text{ a $\math... | https://mathoverflow.net/users/123269 | Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$ | Fixing any right invariant metric $D$ on $G=\text{GL}\_n(\mathbb{R})$ you get a metric $d$ on $G/\Gamma$ ($\Gamma=\text{GL}\_n(\mathbb{Z})$) be letting $d(g\Gamma,h\Gamma)=D(g,h\Gamma)$. If $D$ generates the standard topology on $G$ then (as you can check) $d$ will generate the standard topology on $G/\Gamma$.
A way to... | 2 | https://mathoverflow.net/users/89334 | 299366 | 131,364 |
https://mathoverflow.net/questions/299361 | 9 | By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "equivalent"; for me (braided) fusion categories are just F-tensors (and R-tensors), in which case it's really "equal" (up ... | https://mathoverflow.net/users/115363 | Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal? | In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of [this paper](https://arxiv.org/pdf/0809.3031.pdf). Note that they say the result was already known to Kitaev and Müger. This is also an iff, though the ... | 12 | https://mathoverflow.net/users/22 | 299370 | 131,365 |
https://mathoverflow.net/questions/299090 | 4 | In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\phi:\mathbb{P}^1\rightarrow C$ we consider the rational normal scroll
$$S\_{(1,k)} = \bigcup\_{p\in \mathbb{P}^1}\left... | https://mathoverflow.net/users/nan | Linear sections of Segre varieties and rational normal scrolls | The scroll $S\_{(1,k)}$ can be locally parametrized by the map
$$
\begin{array}{cccc}
\phi: & \mathbb{A}^1\times\mathbb{P}^1 & \longrightarrow & \mathbb{P}^{k+2}\\
& (u,[\alpha\_0:\alpha\_1]) & \mapsto & [\alpha\_0 u:\alpha\_0:\alpha\_1 u^k:\alpha\_1 u^{k-1}:\dots:\alpha\_1 u:\alpha\_1].
\end{array}
$$
Now, conside... | 2 | https://mathoverflow.net/users/14514 | 299376 | 131,368 |
https://mathoverflow.net/questions/299378 | 6 | As a consequence of the loop theorem, if $F$ is a closed surface in the boundary of a 3-manifold and if the kernel $N = \ker(\pi\_1(F) \to \pi\_1(M))$ is nonempty then there is a nontrivial element of $N$ that can be represented by an embedded curve. I read in Hempel's book that there are lots of normal subgroups of $\... | https://mathoverflow.net/users/99414 | Examples of normal subgroups of a surface group with no embedded elements | One source of interesting examples is the lower central series. In my paper
J. Malestein, A. Putman,
On the self-intersections of curves deep in the lower central series of a surface group,
Geom. Dedicata 149 (2010), no. 1, 73–84.
my coauthor and I show that the minimal number of self-intersections among nontrivial... | 8 | https://mathoverflow.net/users/317 | 299379 | 131,369 |
https://mathoverflow.net/questions/299364 | 4 | From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique hyperbolic metric which can be considered as a canonical metric on the surface. I am wondering if similar results exist... | https://mathoverflow.net/users/64606 | Canonical Metrics on 3- and 4-Manifolds | There is no canonical Lorentzian metric in $[g]$, because that would be a diffeomorphism invariant Lorentzian metric. The diffeomorphism group of any manifold has infinite dimension, and infinite dimensional stabilizer, so does not preserve any affine connection (by Bochner's lemma), or any rigid geometric structure. R... | 4 | https://mathoverflow.net/users/13268 | 299385 | 131,371 |
https://mathoverflow.net/questions/299368 | 8 | I would like to prove that the integral cohomology of $BPU\_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.
We have a fiber sequence of the form $BSU\_{n}\rightarrow BPU\_{n} \rightarrow K(\mathbb{Z}/n,2)$. Then we can consider the Serre spectral sequence associated to... | https://mathoverflow.net/users/121001 | Torsion in the integral cohomology of $BPU_{n}$ | You may want to have a look at this paper:
* X. Gu. On the cohomology of classifying spaces of projective unitary groups. arXiv:1612.00506, [(link to arXiv)](https://arxiv.org/abs/1612.00506)
The spectral sequence involving $BSU\_n$ appears in the discussion in Section 5. The paper contains a lot of spectral sequen... | 10 | https://mathoverflow.net/users/50846 | 299393 | 131,375 |
https://mathoverflow.net/questions/299400 | 1 | 1. If
$$\min x'Qx + Rx$$
$$Ax\leq b$$
$$x\in\mathbb Z^n$$
is a quadratic program with $x'Qx$ is convex is there a polynomial time algorithm for this if $A$ is totally unimodular?
2. In particular if we ask that the smallest $x'Qx$ that touches on a vertex of $AX\leq b$ is that in $\mathcal P$?
We know that it is in $... | https://mathoverflow.net/users/10035 | Convex integer programming on totally unimodular polytope? | This sort of problems are NP-hard even if $A=0$. For instance integer least squares <https://web.stanford.edu/~boyd/papers/pdf/int_least_squares.pdf>
Your other question (the optimal vertex) is just as hard - consider binary least squares (ie. the restrict to the unit cube).
| 1 | https://mathoverflow.net/users/120143 | 299404 | 131,378 |
https://mathoverflow.net/questions/299369 | 1 | Let $W^{p\times p}$ be a normal data matrix with $W\_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is, for $\lambda \in \mathbb{R}$, $E[\exp (\lambda ||W||)]$, where $||W|| = \sqrt{\lambda\_1 (W^T W)}$ with $\lambda\_1$ ... | https://mathoverflow.net/users/66585 | Moment generating function of spectral norm of iid N(0,1) data matrix | The asymptotic behavior of the operator norm of Wishart matrices ($W^TW$ in your notation) has been studied by Johnstone, see Theorem 1.1 in [his paper](https://projecteuclid.org/euclid.aos/1009210544). For finite $N$ results, Vershynin's [review](https://arxiv.org/abs/1011.3027v7) is a gold mine, see e.g. Corollary 5.... | 2 | https://mathoverflow.net/users/30138 | 299406 | 131,379 |
https://mathoverflow.net/questions/299407 | 2 | Given $\tau\in H$ (up-half plane) and $q=e^{2\pi i \tau}$, Weber polynomail is defined as
$$f(\tau)=q^{-\frac{1}{48}}\prod\_{i=0}^{\infty}(1+q^{i-\frac{1}{2}}).$$
My question is: How can I compute a product of unlimited sequence? Anyway, I have to finish $f(\tau)$ in finite steps. Of course, the faster the better.... | https://mathoverflow.net/users/103866 | How to compute Weber polynomials efficiently? | An efficient construction of Weber polynomials, including a publicly available code, is given on page 340 and following of [On the Efficient Generation of Elliptic Curves over Prime Fields](https://www.ceid.upatras.gr/webpages/faculty/zaro/pub/conf/C29-ches2002-gen-ec.pdf) (2003).
| 1 | https://mathoverflow.net/users/11260 | 299412 | 131,381 |
https://mathoverflow.net/questions/299411 | 3 | For $p\in (0,1)$ and $\epsilon>0$ a small enough constant, consider the function $f:\mathbb{N}\to\mathbb{R}$ given by
$$f(y)=\int\_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt.$$
Is it true that $f(y)=O(1)$ for all $p\in(0,1)$? Numerical experiments seem to support this asserti... | https://mathoverflow.net/users/106097 | Is $\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt$ bounded by a constant for large $y$? | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\renewcommand{\th}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\P... | 2 | https://mathoverflow.net/users/36721 | 299427 | 131,386 |
https://mathoverflow.net/questions/299428 | 9 | Suppose $b\_1, b\_2, b\_3, \dots \in \Bbb{R}$ satisfy the Riccati-type recurrence
$$b\_{k+1}=\frac{1+kb\_k}{k-b\_k},\quad k\ge 1.$$
>
> Is it true that such a sequence reaches infinitely many positive as well as negative values? It appears to be so.
>
>
>
| https://mathoverflow.net/users/66131 | Riccati-type recurrence: infinitely many sign changes? | Yes, choose $a\_1$ such that $\frac{1}{b\_1}=\tan(a\_1)$, and let $a\_{k+1}=a\_k-\arctan\frac{1}{k}$, then we have $\frac{1}{b\_k}=\tan(a\_k)$, since $\sum\_k \arctan\frac{1}{k}=\infty$, the conjecture follows.
| 8 | https://mathoverflow.net/users/108486 | 299429 | 131,387 |
https://mathoverflow.net/questions/298728 | 5 | In the context of finite von Neumann algebras, I see many results involving the choice of a faithful normal tracial state $\tau$. For instance, the standard representation $L^2(R, \tau)$ of a finite von Neumann algebra $R$ is often used to prove results about $R$ which are independent of the choice of the state $\tau$.... | https://mathoverflow.net/users/123735 | Conditional expectation preserving center-valued trace | Answer to 2: No. Here is an example where existence fails:
Let $M$ be a finite factor with tracial state $\tau$. Let $R = M \oplus M$. Then the center-valued trace is given by $Tr(x,y) = (\tau(x),\tau(y))$.
Let $S = \mathbb{C}\cdot1\_{M\oplus M}$. Any conditional expectation $\mathbb{E}: R \rightarrow S$ of $R$ on... | 1 | https://mathoverflow.net/users/124006 | 299439 | 131,389 |
https://mathoverflow.net/questions/299434 | 17 | Let $t\in\Bbb{N}$ and consider the sequences $p\_t(n)$ defined by
$$\sum\_{n\geq0}p\_t(n)x^n=\prod\_{i\geq1}\frac1{(1-x^i)^t}=(x;x)\_{\infty}^{-t}.$$
The numbers $p\_t(n)$ can be regarded as enumerating partitions of $n$ into parts that come with $t$ colors.
Furthermore, $p\_t(n)=\sum\_{\lambda\vdash n}\prod\_{j\geq1}... | https://mathoverflow.net/users/66131 | Congruences Ramanujan-style | More general versions of this have been established: see in particular Theorem 2 of [Kiming and Olsson](https://link.springer.com/content/pdf/10.1007/BF01197051.pdf), and for other work see (for example) [Locus and Wagner](http://www.mathcs.emory.edu/~mlocus/CongruencePPF.pdf).
To answer the question fully, as Ofir ... | 14 | https://mathoverflow.net/users/38624 | 299444 | 131,393 |
https://mathoverflow.net/questions/299435 | 5 | Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi\_1$-injective closed surfaces in $M$, and which do not intersect?
| https://mathoverflow.net/users/50629 | Intersection of $\pi_1$-injective surfaces | No, this is impossible. Assume that the surfaces $\Sigma\_1, \Sigma\_2$ are immersed and disjoint realized by immersions $f\_i:\Sigma\_i\to M$. Take a region $N$ in the complement of these
surfaces, and whose boundary intersects both surfaces (such a region
must exist if the manifold is connected, which is implicit in ... | 6 | https://mathoverflow.net/users/1345 | 299445 | 131,394 |
https://mathoverflow.net/questions/299438 | 6 | Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f\_n\colon X\to X$ converging point-wise to the identity mapping on $X$?
| https://mathoverflow.net/users/84390 | Point-wise limit of finite valued functions | To answer a question raised in comments: if $X$ is merely separable then this can fail. Let $X = \mathbb{R}^\mathbb{R}$ with the product topology, which is a Hausdorff topological vector space. Then $X$ is separable. ([Any product of continuum many separable spaces is separable.](https://dantopology.wordpress.com/2009/... | 7 | https://mathoverflow.net/users/4832 | 299449 | 131,397 |
https://mathoverflow.net/questions/299436 | 4 | Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. One way to construct a complex structure on $G/T$ is to choose a nilpotent subalgebra $\mathfrak{n}^+$ corresponding to some choice of positive roots of the complexified Lie algebra of $G$. We let $J\_0$ be the corresponding almost complex structure on $\ma... | https://mathoverflow.net/users/31556 | Equivalence of complex structures on flag manifold | Under the natural identification $G/T\to G^c/B$ they coincide. This is Borel-Hirzebruch ([1958](//ams.org/mathscinet-getitem?mr=102800), §14.3) $=$ Bourbaki ([2005](//ams.org/mathscinet-getitem?mr=2109105), §IX.4, Exercise 8 *g*)).
| 1 | https://mathoverflow.net/users/19276 | 299455 | 131,398 |
https://mathoverflow.net/questions/296492 | 15 | I work in algebraic geometry. Lately, the answer to most of my questions seems to be "you should read Lurie's *Higher Algebra*." I took this advice seriously, however it turned out not to be an easy task.
From what I gather, I should learn how to work with $E\_\infty$-rings. I understand the definition of a symmetric... | https://mathoverflow.net/users/3847 | $\infty$-operads and $E_\infty$-algebras | Here are two examples where $E\_k$-algebras show up in algebraic geometry, for $1<k<\infty$ (actually, just $k=2$):
* The work of Gaitsgory and Lurie on Weil's conjecture for function fields. See [here](http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf), [here](http://www.math.harvard.edu/~lurie/papers/tamagawa-... | 11 | https://mathoverflow.net/users/nan | 299464 | 131,403 |
https://mathoverflow.net/questions/299422 | 5 | I have a basic (probably naive) question about Tits buildings. Let $G$ be a (connected) linear algebraic group over a field $k$ (I am interested in the case where $k$ is algebraically closed but I appreciate information for general $k$ also).
When $G$ is a semisimple or reductive group, the (spherical Tits) building ... | https://mathoverflow.net/users/21491 | Tits building of a linear algebraic group | Yes, the definition you make works for a general linear algebraic group $G$. The reason you haven't seen it mentioned is that the solvable radical $S$ of $G$ is contained in any of its parabolic subgroups, thus the building associated with $G$ coincides with the building associated its natural semisimple factor $G/S$.
... | 5 | https://mathoverflow.net/users/89334 | 299468 | 131,405 |
https://mathoverflow.net/questions/299470 | 2 | I begin my question with a multilinear question then I will consider two local smooth analogies:
Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\overbrace{\mathbb{R}^n\times\mathbb{R}^n\times\ldots\times \mathbb{R}^n}^{k\; \text{times}} \to \mathbb{R}$ which is unchanged ... | https://mathoverflow.net/users/36688 | A kind of "Curvature tensor" for higher dimensional tensors | Any $k$-linear symmetric map $\alpha:(\mathbb R^n)^k \to \mathbb R$ is the polarization of a homogeneous polynomial $\bar \alpha: x\mapsto \alpha(x,x,x\dots,x)$ of degree $k$ on $\mathbb R^n$.
First question: You ask whether any homogeneous polynomial of degree $k$ is of the form $\lambda\_1 x\_1^k + \lambda\_2 x\_2^... | 3 | https://mathoverflow.net/users/26935 | 299476 | 131,407 |
https://mathoverflow.net/questions/299482 | 7 | Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a general element of the linear system $|\mathcal{L}|$ is a smooth $K$-variety? If not true in general, is there any condition ... | https://mathoverflow.net/users/43198 | Bertini's theorem over non-algebraically closed field | This is true both over finite and infinite fields.
* For infinite fields, see [Jou, **Cor. I.6.11(2)**]. It works for a general section of any very ample line bundle $\mathscr L$, using that over an infinite field a nonempty open subset of $|\mathscr L| \cong \mathbb P^N$ contains a rational point.
* For finite field... | 12 | https://mathoverflow.net/users/82179 | 299484 | 131,410 |
https://mathoverflow.net/questions/299488 | 4 | In the Grothendieck ring of varieties, there are ways of distinguishing classes of varieties, for example $\ell$-adic cohomology. The Grothendieck ring of stacks is a localization of the Grothendieck ring of varieties, more precisely it is obtained by inverting $\mathbb{L}$ and the cyclotomic polynomials in $\mathbb{L}... | https://mathoverflow.net/users/123358 | Inequality in the Grothendieck ring of stacks | I'm not sure why you think there is a problem with applying $\ell$-adic cohomology here. If we want to show that ${X} \neq {Y}$ in $K\_0(Stacks\_k)$, and we know their associated classes in $K\_0$ of Galois representations are different, it suffices to show that cyclotomic polynomials in the Lefschetz motive/cyclotomic... | 3 | https://mathoverflow.net/users/18060 | 299497 | 131,415 |
https://mathoverflow.net/questions/299483 | 9 | Let $d\mu\_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha\_n=\frac{\int\_0^{\pi/2}x^4d\mu\_n(x)}{\int\_0^{\pi/2}d\mu\_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum\_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^2(\alpha\_{n-1}-\alpha\_n)
=\frac{21}8\zeta(4)-\frac32\zeta(2);$$
where $\zeta... | https://mathoverflow.net/users/66131 | Summing moments and Riemann zeta values | After my first failed attempt I now follow the route suggested by Nemo --- which works smoothly.
Starting from Nemo's identities
$$F(b)\equiv\int\_0^{\pi/2}\cos^{2n}x\cos bx\,dx=\frac{\pi (2n)!}{2^{2n+1}\Gamma(n+1+b/2)\Gamma(n+1-b/2)}$$
$$\int\_0^{\pi/2}x^4\cos^{2n}x\,dx=\lim\_{b\rightarrow 0}\frac{d^4}{db^4}F(b),$$
I ... | 11 | https://mathoverflow.net/users/11260 | 299501 | 131,417 |
https://mathoverflow.net/questions/268850 | 8 | Suppose $Y$ is a projective variety over a field $k$. Fix an embedding $\iota: Y \hookrightarrow \mathbb{P}^n\_k$ for some $n$, and consider the local cohomology *sheaves* $\mathcal{H}^j\_Y(\omega\_{\mathbb{P}^n})$ of the canonical sheaf on $\mathbb{P}^n$ supported on $Y$. I believe that the sheaf cohomology groups
... | https://mathoverflow.net/users/31771 | Independence of embedding for higher sheaf cohomology of local cohomology on projective space | It is false! As mentioned in the edit, a positive answer to this question would imply the Lyubeznik numbers $\lambda\_{i,j}$ of a projective scheme are independent of the defining projective embedding for $i \geq 2$. The recent preprint
<https://arxiv.org/abs/1803.07448>
by T. Reichelt, M. Saito, and U. Walther, gi... | 2 | https://mathoverflow.net/users/31771 | 299502 | 131,418 |
https://mathoverflow.net/questions/261907 | 9 | The restriction of a continuous map $D^2\to S^1$ to $\partial D^2\to S^1$ must have degree zero. Is that statement true or false if the map is only $W^{1,2}(D^2;S^1)$ and continuous on $\partial D^2$?
In two dimensions, the Sobolev space $W^{1,2}$ is at the borderline regularity and does not embed into the space of c... | https://mathoverflow.net/users/76361 | Can a $W^{1,2}$ map from the disk to the circle restrict to a degree one map on the boundary? | **The restriction of $f$ to the boundary has degree zero.** It is true also in higher dimensions. The proof presented below is based on the proof of density of $C^\infty(M,N)$ in $W^{1,p}(M,N)$, $p\geq \operatorname{dim}M$, due to Schoen and Uhlenbeck [3] (see also Theorem 2.1 in [1]).
>
> **Theorem.** If $f\in W^{... | 4 | https://mathoverflow.net/users/121665 | 299512 | 131,421 |
https://mathoverflow.net/questions/299507 | 7 | Fix $h \in (0,\pi/2)$. I am trying to explicitly exhibit a holomorphic function $f\colon \Sigma \to \mathbb{C}$, where $\Sigma$ is the punctured sector
$$\Sigma := \{z \in \mathbb{C} \:|\: z\neq 0, 0\leq\arg z\leq h\},$$
and where we require $f$ to satisfy the following conditions on $\partial\Sigma$:
* $f(x) \in... | https://mathoverflow.net/users/20391 | Looking for holomorphic function on a sector with specified boundary behavior | Such a function can be written with the help of the Poisson integral.
First, your function must extend by reflection to $\{ z:|\arg z|\leq h\}$.
Let $g(z)=f((-iz)^{2h/\pi})$, then $g$ must be analytic in the upper half-plane,
and
$$g(t)=\sqrt{1+|t|^{4h/\pi}}.\quad\quad\quad\quad\quad (1)$$
To construct $g$,
first de... | 7 | https://mathoverflow.net/users/25510 | 299517 | 131,425 |
https://mathoverflow.net/questions/299448 | 3 | I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be answered without reading what follows.
Let $j(\omega)$ be the modular invariant, $\Gamma$ be the modular group and $... | https://mathoverflow.net/users/122104 | The degree of the cube root of the $j$-invariant | Let $N=\frac {9-D}{4}$ and let $\bar\omega$ be the complex conjugate of $\omega$. In particular $\omega+\bar\omega = -3$ and $\omega\bar\omega=N$.
For every $A=\bigl(\begin{smallmatrix}a & b\\c&d\end{smallmatrix}\bigr)\in\Gamma$ we have that $ad=bc+1$, so $$A\omega = \frac{(a\omega +b)(c\bar\omega+d)}{(c\omega+d)(c\b... | 3 | https://mathoverflow.net/users/58242 | 299535 | 131,427 |
https://mathoverflow.net/questions/299532 | 4 | Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.
Under these conditions, must $f$ have a fixed point in $B^-$?
| https://mathoverflow.net/users/101577 | Brouwer fixed-point for open ball and bijective uniformly continuous function? | No, for instance a piecewise-linear (say with two pieces) bijection $f:(0,1)\to(0,1)$ whose graph lies on one side of the diagonal $y=x$.
| 6 | https://mathoverflow.net/users/120143 | 299538 | 131,428 |
https://mathoverflow.net/questions/299524 | 3 | **KG** (the Kleene Getaway) is the name I improvised (in my answer to a [question on MO on constructive Perron-Frobenius](https://mathoverflow.net/questions/299318/constructive-proof-of-a-rational-version-of-perron-frobenius/299340#299340)) for a constructive principle which enables the direct constructive use of a cla... | https://mathoverflow.net/users/101577 | Unorthodox constructive reasoning: The Kleene Getaway | See Michael Beeson, “Some Relations between Classical and Constructive Mathematics”, Journal of Symbolic Logic 1978, on [JStor](http://www.jstor.org/stable/2272821).
Unfortunately, he doesn’t give these principles a good name, and in my experience, people distrust this metamathematics unnecessarily.
| 4 | https://mathoverflow.net/users/nan | 299543 | 131,432 |
https://mathoverflow.net/questions/299542 | 2 | I believe that anyone who can answer this question knows the terminology, but $\beta\omega$ is the Čech-Stone compactification of the integers and a point $p$ is a weak P-point in a space $X$ iff it is not in the closure of any countable subset of $X-\{p\}$. I admit to lacking intuition with $\beta\omega$ and maybe thi... | https://mathoverflow.net/users/29491 | Does any subset of $\beta\omega$ of cardinality $\mathfrak{c}$ have a weak P-point in its closure? | The answer is no, because every infinite closed subset of $\beta \omega$ has cardinality $2^\mathfrak{c}$. So for example, if $\{x\_1,x\_2,\dots\}$ is any countably infinite set of non-weak-$P$-points, then its closure $X$ contains no weak $P$-points either. Any $Y \subseteq X$ of cardinality $\mathfrak{c}$ provides a ... | 3 | https://mathoverflow.net/users/70618 | 299545 | 131,433 |
https://mathoverflow.net/questions/299548 | 3 | Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}\_{g,n}$ represents the functor $$\overline{\mathcal{M}}\_{g,n}(S)$$
assigning to a scheme $S$ the groupoid of proper stable nodal curves $C\to S$ over $S$ with genus $g$ and $n$ mar... | https://mathoverflow.net/users/7108 | Moduli problem of stable nodal curves over the integers | Already Deligne and Mumford's original paper constructs $\overline{\mathcal M}\_g$ over $\operatorname{Spec} \mathbb Z$ [DM, §5]. They do not do the pointed version, so they restrict themselves to $g \geq 2$ (otherwise it will only exist as an Artin stack). Note also that they write $\mathcal M\_g$ for what is now comm... | 4 | https://mathoverflow.net/users/82179 | 299555 | 131,437 |
https://mathoverflow.net/questions/299546 | 5 | Let $(X,T)$ be a Hausdorff topological space. Let $C\_b(X)$ be its algebra of continuous bounded functions. Let $T'$ be the initial topology on $X$ given by $C\_b(X)$. It is known that $T=T'$ if and only if $(X,T)$ is completely regular. If $\mathcal B(X,T)$ and $\mathcal B(X,T')$ are the Borel $\sigma$-algebras genera... | https://mathoverflow.net/users/54780 | Do a Hausdorff space and its associated completely regular space have the same Borel subsets? | There are Hausdorff spaces all of whose real-valued functions are constant. A classical example (of a countable space) was given by Uryshon ([*Über die Mächtigkeit der zusammenhängenden Mengen*, Math.Annalen, 1925](https://eudml.org/doc/159111)). This implies that the initial topology on $X$ induced by $C(X)$ is trivia... | 8 | https://mathoverflow.net/users/6101 | 299556 | 131,438 |
https://mathoverflow.net/questions/299536 | 1 | Let $\mathbb{G}\_m$ be the multiplicative group and $T$ a maximal torus of a semisimple group. Let $X^\*(T)=\{ \phi: T \to \mathbb{G}\_m \}$ be the set of characters and $X\_\*(T)=\{ \phi^{\vee}: \mathbb{G}\_m \to T \}$ the set of co-characters. Let
\begin{align}
L(T) = \{ \phi: \mathbb{G}\_m \to T \},
\end{align}
and ... | https://mathoverflow.net/users/11877 | How to decompose an map $\phi: \mathbb{G}_m \to T$ as the product of a cocharacter $\phi'$ and a map $\phi'':\mathbb{G}_m \to T$? | Although the notation and explanation of the paper is terrible, it's not supposed to be a very deep fact.
Indeed, let's assume for simplicity that $T = \mathbb G\_m^n$ is a *split* torus. Then arbitrary morphisms of schemes $\mathbb G\_m \to T$ correspond to $n$-tuples $(f\_1,\ldots,f\_n) \in (k[x^{\pm 1}]^\times)^n$... | 1 | https://mathoverflow.net/users/82179 | 299567 | 131,441 |
https://mathoverflow.net/questions/299553 | 4 | I am reading the paper ["$p$-adic cohomology: from theory to practice"](http://swc.math.arizona.edu/aws/2007/KedlayaNotes11Mar.pdf) by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack proper backgrounds needed for this paper, and I am very gra... | https://mathoverflow.net/users/87910 | Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya | For question 1, you are correct that the map
$$\bar f^\* \colon H^i\_{\text{dR}}(X',Z') \to H^i\_{\text{dR}}(X,Z)$$
is the one obtained from
$$\bar f^\* \colon H^i\_{\text{crys}}(\bar X',\bar Z') \to H^i\_{\text{crys}}(\bar X,\bar Z)$$
using the comparison isomorphism. This is exactly the point of Remark 2.1.3: it is r... | 2 | https://mathoverflow.net/users/82179 | 299575 | 131,444 |
https://mathoverflow.net/questions/299581 | 14 | The [q-Vandermonde identity](https://en.wikipedia.org/wiki/Q-Vandermonde_identity) reads:
$$ \binom{m + n}{k}\_{\!\!q} =\sum\_{j} \binom{m}{k - j}\_{\!\!q} \binom{n}{j}\_{\!\!q} q^{j(m-k+j)} $$
The [q-binomial coefficients](https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient):
$$ \binom{ a }{ b}\_{\!\!q} ... | https://mathoverflow.net/users/10446 | Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$? | Assume $V$ is a vector space of dimension $m+n$, $M \subset V$ is a subspace of dimension $m$, and $N = V/M$. Let $p:V \to N$ be the projection. Consider the Grassmannian $X = Gr(k,V)$ and its stratification by the dimension of intersection with $M$, i.e., set
$$
X\_j = \{ U \in Gr(k,V) | \dim(U \cap M) = k - j \}.
$$
... | 20 | https://mathoverflow.net/users/4428 | 299582 | 131,448 |
https://mathoverflow.net/questions/299587 | 6 | An object $P$ in a topos $\mathcal{E}$ is said to be projective if $Hom\_{\mathcal{E}}(p,-)$ preserves epis, internally projective if $(-)^P$ preserves epis.
Can anyone give an example of a topos where every object is internally projective but not every object is projective? Can such a topos be a presheaf topos? Gen... | https://mathoverflow.net/users/124163 | A topos where every object is internally projective but not every object is projective | See Johnstone's *Sketches of an Elephant*, section D.4.5 for the following and more examples.
If $G$ is a group, then the topos of $G$-sets satisfies the internal axiom of choice -- every object is internally projective. But the only projective objects are the free $G$-sets. So if $G$ is nontrivial, then this topos d... | 10 | https://mathoverflow.net/users/2362 | 299591 | 131,450 |
https://mathoverflow.net/questions/299570 | 7 | What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$?
**EDIT**: Also assume that the residue field of the DVR is algebraically closed.
| https://mathoverflow.net/users/43198 | index of smooth varieties | Let $d\geq 3$ be any integer that is relatively prime to the residue characteristic of $R$. Let $s$ be any generator of the maximal ideal $\mathfrak{m}$ of $R$. Let $\widetilde{R}$ be the finite, flat extension $R[\sigma]/\langle \sigma^d -s \rangle$ with its natural action of the group scheme $\mu\_d=\text{Spec} \ \ma... | 4 | https://mathoverflow.net/users/13265 | 299594 | 131,451 |
https://mathoverflow.net/questions/299599 | 5 | I was reading John Baez’s paper on [operads and phylogenetics trees](https://arxiv.org/abs/1512.03337) where he formalizes a Jukes–Cantor model of phylogenetics. Because biological questions receive different answers depending on the model used, I was wondering if the operad that represents that model could be compared... | https://mathoverflow.net/users/123650 | Can operads (or category theoretic structures more generally) be compared? | Yes, operads can be compared.
There are lots of kinds of operad (enriched in various categories, symmetric or plain or defined with respect to a monad, one-colored or many-colored, and don't even get me started on models of $\infty$-operads...) and I haven't looked to see which flavor Baez is using.
But one thing t... | 8 | https://mathoverflow.net/users/2362 | 299600 | 131,453 |
https://mathoverflow.net/questions/299304 | 37 | I had posted the following problem on [stack exchange](https://math.stackexchange.com/questions/2722893/the-modulus-of-a-polynomial-are-the-same-is-1) before.
>
> Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the polynomial $$
> f\left ( x ... | https://mathoverflow.net/users/38620 | A family of polynomials whose zeros all lie on the unit circle | This is a special case of the Lee-Yang theorem. Let $G$ be a finite graph with vertex set $V$ and edge set $E$. Let $\beta \in (0,1)$ be a real number. Let $\sigma$ denote a ``spin function" $\sigma: V \to \{ -1, 1\}$. Put $m(\sigma)$ to be the number of vertices with positive spin, and $d(\sigma)$ to be the number of ... | 55 | https://mathoverflow.net/users/38624 | 299609 | 131,455 |
https://mathoverflow.net/questions/299612 | 4 | The merit of a prime gap equals $(p\_{n+1}-p\_n)/\ln p\_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very definition got rigged to make the mean of merit equal 1 in the limit. What about higher moments? Do we have theorems, standard... | https://mathoverflow.net/users/10909 | Moments of merit | The standard conjecture is that the statistics of $(p\_{n+1}-p\_n)/\ln n$ are, in the limit, precisely an exponential distribution with parameter $1$; one can recover conjectures about individual moments from this, although evaluating the moments is the natural way to approach the conjecture. This is the natural heuris... | 11 | https://mathoverflow.net/users/5091 | 299615 | 131,458 |
https://mathoverflow.net/questions/299613 | 3 | For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x, y\in X\text{ and } x\neq y\big\}$.
Let $G=(V,E)$ be a simple, undirected graph, and suppose ${\cal V}$ is a collection of subsets of $V$ such that for all $W, W' \in {\cal V}$ we have either $W\subseteq W'$ or $W' \subseteq W$.
If for all $W\in {\cal V}$ the induce... | https://mathoverflow.net/users/8628 | Ascending chain of vertex-transitive graphs | Let $Q$ be a chain (totally ordered set). Let $Y\_Q$ be the graph whose set of vertices is the set of pairs $(x,y)$ with $x,y\in Q$ and $x<y$, and an edge between $(x,y)$ and $(x',y')$ iff the intervals $]x,y[$ and $]x',y'[$ are disjoint.
If $Q$ is a 2-transitive chain then $Y\_Q$ is a vertex-transitive graph.
Let... | 5 | https://mathoverflow.net/users/14094 | 299626 | 131,461 |
https://mathoverflow.net/questions/299618 | -2 | Let $R$ be a ring and $R((x))$ the ring of [formal Laurent series](https://en.wikipedia.org/wiki/Formal_power_series#Formal_Laurent_series). The elements in the ring $R((x))$ are series of the form
$$
f = \sum\_{n\in\mathbb{Z}} a\_n x^n,
$$
where ${\displaystyle a\_{n}=0}$ for all but finitely many negative indices $n$... | https://mathoverflow.net/users/11877 | What is the spectrum of the ring $R((x))$ of formal Laurent series over a ring $R$? | If $R$ is a field, then the same page that you link to states that [$R((x))$ is a field](https://en.wikipedia.org/wiki/Formal_power_series#Formal_Laurent_series), too - in fact, it is the field of fractions of the ring $R[[x]]$ of formal series. Since it is a field, its only ideals are $0$ and $R((x))$ - of which, by c... | 4 | https://mathoverflow.net/users/54780 | 299633 | 131,463 |
https://mathoverflow.net/questions/299619 | 2 | Let $T = (V, E)$ be a tree. Let $\Sigma$ be a finite set of labels. Given a label function $\ell : V \to \Sigma$, the cost of $\ell$ is given by
$$\mu(\ell) = \left| \{(u,v) \in E \mid \ell(u) \neq \ell(v) \} \right|$$
i.e., $\mu(\ell)$ is the number of edges whose vertices have different labels. Given the labels o... | https://mathoverflow.net/users/75053 | How to label a tree with minimum cost? | **Notation** Given a tree $T$, let $V(T)$ be the set of its vertices, let $L(T)\subseteq V(T)$ be the set of its leaves. Denote by $T'=T\setminus L(T)$, i.e. let it be the subtree obtained from $T$ removing its leaves. For every $v\in T'$ let $L\_v(T)\subseteq L(T)$ be the set of leaves (of $T$) adjacent to $v$.
**Th... | 3 | https://mathoverflow.net/users/58242 | 299634 | 131,464 |
https://mathoverflow.net/questions/299572 | 30 | $\def\Cl{\mathcal C\ell}
\def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$
I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question.
A well known definition of Clifford algebras is the following:
>
> Fix a quadratic form $q$ on a vector space $V$[¹] and ... | https://mathoverflow.net/users/7952 | Clifford algebras as deformations of exterior algebras | As Igor mentions in the comments, this is really a question about deformations of the multiplication map of the exterior algebra in the space of associative multiplications. Since this is a pretty general story, let me try to sketch how it works in this case.
Let's fix some field $k$ and a $k$-algebra $(A,\mu)$. We w... | 28 | https://mathoverflow.net/users/35687 | 299645 | 131,467 |
https://mathoverflow.net/questions/299617 | 4 | Suppose $X$ is a metric space with a free group action by a topological group $G$, which is also a metric space, such that $\pi\colon X \to X/G$ is a fiber bundle.
Does the quotient inherit the paracompactness of $X$? If this is not true in general I'm interested in the case, where $X$ is the space of embeddings of ... | https://mathoverflow.net/users/89741 | Paracompactness of Quotient by Group Action | For $k=\infty$ this is well-known: See 44.1 of [here](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf), or 13.14 of [here](http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf), or [here](http://www.mat.univie.ac.at/~michor/mf-smo-3.pdf) since the base of the principal fiber bundle is a smoo... | 3 | https://mathoverflow.net/users/26935 | 299651 | 131,468 |
https://mathoverflow.net/questions/299636 | 3 | This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the case $R=k[[h]]$ for $k$ a field of characteristic 0). Recall that an $R$-module is complete if the canonical map
$$ M\r... | https://mathoverflow.net/users/13552 | Is a filtered colimit of complete module complete? | The answer is no. Here is another example. Let the ring $R=k[[h]]$ and the modules be $M\_n=k[h]/h^n$ with the bounding maps $M\_n\to M\_{n+1}$ sending 1 to $h$.
Then all $M\_n$ are complete (they are $h$-torsion), but $M=\mathrm{colim}\_n M\_n=k[h^{\pm1}]/k[h]$ (including $M\_n$ as the submodule spanned by $h^{-n}$)... | 1 | https://mathoverflow.net/users/43054 | 299660 | 131,471 |
https://mathoverflow.net/questions/299656 | 1 | So far as I know:
1. The power iteration method can only get the eigenvector corresponding to the largest eigenvalue;
2. The inverse power iteration method requires that the matrix is invertible;
3. The QR algorithm requires too much storage space.
I'm not sure if I'm understanding the above points right.
Since t... | https://mathoverflow.net/users/124186 | Compute the eigenvectors corresponding to the $k$ smallest eigenvalues w.r.t high dimensional symmetric sparse matrix? | This would get better answers at [scicomp.se], anyway:
* if your matrix happens to be positive semidefinite, then after a suitable shift the smallest eigenvalues become the largest ones in modulus and can be computed with the power method. Similar tricks apply in cases when the sought eigenvalues are "at the border o... | 4 | https://mathoverflow.net/users/1898 | 299667 | 131,474 |
https://mathoverflow.net/questions/299662 | 5 | Do you know any reference where you have a **formal justification** for the following statement that appears in nLab?
<https://ncatlab.org/nlab/show/suspensions+are+H-cogroup+objects>
"Let $\mathcal{C}$ be an $(∞,1)$-category with finite $(∞,1)$-colimits and with a zero object. Write $\Sigma \colon X \mapsto 0 \und... | https://mathoverflow.net/users/124190 | Suspensions are H-cogroup objects | Ok, let me try to give you a proof of something that is a lot stronger than what you asked for, but which hopefully is a bit more natural. I am basically going to smother the problem under the abstract nonsense, so I recommend you look at special cases of this proof, e.g. when $X=S^n$ in pointed spaces, in order to get... | 8 | https://mathoverflow.net/users/43054 | 299675 | 131,477 |
https://mathoverflow.net/questions/299657 | 1 | Let $\Gamma$ be a regular boundary of a $C^{k,1}$ domain $\Omega$ and $H^s(\Gamma)$, $s\in(0,1)$, denote the fractional Sobolev space on $\Gamma$. Suppose I define a multiplication operator $M\_\phi:H^s(\Gamma)\to H^s(\Gamma)$ where $M\_\phi v=\phi v$. What should be the minimal regularity of $\phi$ for the map to be c... | https://mathoverflow.net/users/124188 | Continuity of a multiplication operator in fractional Sobolev space |
>
> **Theorem.** Multiplication by a Lipschitz function defines a bounded operator in $H^{1/2}(\partial\Omega)$.
>
>
>
**First proof.**
More generally, if $\Omega\subset\mathbb{R}^n$ is a bounded Lipschitz domain, then the fractional Sobolev space $W^{1-1/p,p}(\partial\Omega)$ (the trace space for $W^{1,p}(\Omeg... | 1 | https://mathoverflow.net/users/121665 | 299678 | 131,479 |
https://mathoverflow.net/questions/252313 | 11 | The [inverse Galois problem](https://en.wikipedia.org/wiki/Inverse_Galois_problem) asks whether every finite group appears as the Galois group of some finite extension of $\mathbb Q$. I was wondering to what extent the analogous problem for ideal class groups has been investigated. More precisely, consider the followin... | https://mathoverflow.net/users/30186 | Is every group an ideal class group of a number field? | The recent paper by [Homlin, Jones, Kurlberg, McLeman, and Petersen](https://arxiv.org/pdf/1510.04387.pdf) (Experimental Math., to appear) is devoted to these questions especially in the context of imaginary quadratic fields. One should expect that every natural number arises as the class number of an imaginary quadrat... | 9 | https://mathoverflow.net/users/38624 | 299694 | 131,485 |
https://mathoverflow.net/questions/299625 | 3 | I ask a question about $\prod k$ in Mathematics about several days.<https://math.stackexchange.com/q/2766054/453628>.
And I have the following question:
1.What is the global dimension about $\prod k$?
2.is there an example of a ring which is absolutely flat and the global dimension is infinite?of course,if this exi... | https://mathoverflow.net/users/106580 | question about infinite global dimension | As pointed out in comments, the answer to question 1 has come up here before. But just for completeness, in
*Osofsky, B.L.*, [**Homological dimension and cardinality**](http://dx.doi.org/10.2307/1995518), Trans. Am. Math. Soc. 151, 641-649 (1970). [ZBL0209.07101](https://zbmath.org/?q=an:0209.07101),
Osofsky prove... | 6 | https://mathoverflow.net/users/22989 | 299700 | 131,488 |
https://mathoverflow.net/questions/299702 | 4 | If we consider $x\_t$ an Ornstein-Uhlenbeck process (with $W\_t$ the Wiener process), does anyone know what would be the variance of the convolution of $x\_t$ with a given filter $A$ i.e. $V(x\_t \star A)$ ?
I've been trying to derive the solution starting from the analytical solution such as
\begin{align}
x\_t = {... | https://mathoverflow.net/users/124219 | Variance of convolution between filter $A$ and Ornstein-Uhlenbeck process $x_t$ | Welcome to MO! This is not quite a research level question, but here is the answer anyway.
Represent the Orenstein-Uhlenbeck process as white noise passing through a low-pass (this is really the representation in the equation for $x\_t$ you wrote). Call the transfer function of the low-pass $H$. Then your process is ... | 2 | https://mathoverflow.net/users/35520 | 299703 | 131,489 |
https://mathoverflow.net/questions/299709 | 1 | Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it:
$$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$
while from Sokhotski–Plemelj theorem it follows (seemingly) that
$$ H(x+a)\to~PV\frac{-i}{w} +\pi\delta(w)+ |a|e^{iax/2}\operatorname {sinc} \left({\frac {w a }{2\p... | https://mathoverflow.net/users/10059 | Which Fourier transform is the correct one? | application of the Sokhotski–Plemelj formula does in fact give the first of the two expressions:
$$\int\_{-\infty}^\infty H(x+a)e^{-ix w}dx=\int\_{-a}^\infty e^{-i xw}dx=e^{iwa}\int\_0^\infty e^{-ix w}dx$$
$$=e^{iwa}\left(\text{PV}\frac{-i}{w}+\pi\delta(w)\right)=-\text{PV}\frac{ie^{i aw}}{w}+\pi\delta(w)\qquad(\ast)... | 2 | https://mathoverflow.net/users/11260 | 299710 | 131,491 |
https://mathoverflow.net/questions/299714 | -2 | I'm looking for a reference that treats basic module theory over non-associative rings, the isomorphism theorems and so on. I imagine the theory is known, but have not been able to find a reference.
Edit: Let $R$ be a non-associative ring. By a left $R$-module (similarly for a right $R$-module), I mean an abelian gro... | https://mathoverflow.net/users/124224 | Non-associative module theory | A Google search of this term brings a number of references, did you try it?
In any case, a very obvious relevant reference is the old paper of Osborn called [Modules over nonassociative rings](https://www.tandfonline.com/doi/abs/10.1080/00927877808822293?journalCode=lagb20).
More recent works define modules over ... | 4 | https://mathoverflow.net/users/1306 | 299716 | 131,493 |
https://mathoverflow.net/questions/299718 | 4 | Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X\_K$ be an étale $G$-torsor.
>
> For what field extensions $K/k$ does any such $Y$ descend to $k$? I.e.: for what field extensions $K/k$ does there exist $Y\_0\to X$ a... | https://mathoverflow.net/users/nan | Descent of étale torsors | Let $L$ be a finite separable extension of $k$. Let $X$ be the Weil restriction from $L$ to $k$ of $\mathbb G\_m$. Then for any field extension $K$ of $k$ (including $k$ itself),if $L \otimes\_k K$ is a product of $n$ distinct fields, then $H^1(X\_K, \mathbb Z/2) / H^1(K, \mathbb Z/2) = (\mathbb Z/2)^n$. This can be ca... | 6 | https://mathoverflow.net/users/18060 | 299720 | 131,495 |
https://mathoverflow.net/questions/299699 | 16 | Brunnian links are well known, where deleting any component allows you to isotope the rest to an unlink. It's common to construct them by taking an $n-1$ component unlink and defining the $n$th component as a chain of commutators in the Wirtinger presentation such as $[x\_1,[x\_2,[x\_3,x\_4]]]]$
Can you construct $n ... | https://mathoverflow.net/users/50693 | Link such that deleting any two components leaves an unlink | Yes, this is done in
*Penney, D.E.*, [**Generalized Brunnian links**](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077378123), Duke Math. J. 36, 31-32 (1969). [ZBL0176.22201](https://zbmath.org/?q=an:0176.22201).
Call a link *$(n,k)$-Brunnian* if it has $n$ components, and every sublink with $m$ components... | 21 | https://mathoverflow.net/users/8103 | 299721 | 131,496 |
https://mathoverflow.net/questions/299713 | 4 | It is well known that the equation $$(\*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra.
I search for some reasonable generalization of this equation in higher variable for investigation of solvability of such generalized equations. The above equation can be read as $$(\*\*)\;\;\;\;\;\sum\_{\sigma \in S\_2} s(\s... | https://mathoverflow.net/users/36688 | A generalization of unsolvable equation $ab-ba=1$ in a Banach algebra | In the algebra of real $2\times 2$ matrices, take
$$ a=\left[ \begin{array}{cc} 0 & 1\\ 0 & 0\end{array} \right],\
b=\left[ \begin{array}{cc} 0 & 0\\ 1 & 0\end{array}\right],\
c=\left[ \begin{array}{rr} 1/3 & 0\\ 0 & -1/3\end{array}\right]. $$
In the other direction, is it true that if for all $n$ an equation has ... | 8 | https://mathoverflow.net/users/2807 | 299726 | 131,497 |
https://mathoverflow.net/questions/299732 | 5 | Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in
a) Kassel C: Quantum Groups. Springer. (For oriented tangles.)
b) Freyd P.J, Yetter DN. Braided compact closed categories with... | https://mathoverflow.net/users/99088 | Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations | As luck would have it and at the risk of shameless self-promotion, I wrote a paper answering this very question for framed virtual tangles: <https://arxiv.org/abs/1602.03080> .
Let $T$ be the category of framed oriented tangles and $V$ the category of framed oriented virtual tangles. Then $V$ is in fact **symmetric**... | 5 | https://mathoverflow.net/users/13552 | 299736 | 131,500 |
https://mathoverflow.net/questions/299534 | 2 | The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int\_{-T}^T\varphi\_n(x) \text{sinc}(t-x) dx = \lambda\_n \varphi\_n(t)$$
where $\text{sinc}(t) = \sin(\pi t)/ \pi t$. Alternatively (as discovered by Slepian et al.) they are also the eigenfunctions of the following diffe... | https://mathoverflow.net/users/18560 | Determinant of a matrix involving the Prolate Spheroidal Wave Functions | I have worked for some time on the problem and made some progress. The argument is basically inductive. First all I would like to tell that this problem arises in Gabor Theory if one wants to prove that a Gabor frame can be generated by multiple prolate spheroidal wave functions i.e. if one wants to prove that the sequ... | 1 | https://mathoverflow.net/users/18560 | 299741 | 131,502 |
https://mathoverflow.net/questions/299737 | 7 | For a commutative ring $R$, a matrix $A \in M\_n(R)$ is invertible iff $\det (A)$ is a unit in $R$. Is there a similar criterion to determine invertibility (having two-sided inverse) of a matrix over a noncommutative ring ? I am particularly interested in determining invertibility of matrices in $M\_n(F \langle x, y \r... | https://mathoverflow.net/users/123735 | invertibility of matrix over free associative algebra | You are asking whether the following holds.
>
>
> >
> > **Claim.** Let $F$ be a field. Then the [free associative algebra](https://en.wikipedia.org/wiki/Free_algebra) $F\langle x,y\rangle$ is a $GE\_2$-ring in the sense of P. M. Cohn [2].
> >
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The answer is **yes**. Indeed, free associative a... | 6 | https://mathoverflow.net/users/84349 | 299745 | 131,503 |
https://mathoverflow.net/questions/299728 | 2 | Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $H\subseteq G$ be the group subscheme stabilizing $Y$ (i.e. $gY(R)=Y(R)$ for every $g\in H(R)$ and every $k$-algebra $R$).... | https://mathoverflow.net/users/124233 | Subschemes in group action | I am writing up my comment as an answer. Let $p$ be a prime integer. Let $k$ be field of characteristic $p$. Denote by $\mathbb{A}^2\_k$ the $k$-scheme $\text{Spec}\ k[s,t]$ with its usual structure of group $k$-scheme, $$m:\mathbb{A}^2\_k \times\_{\text{Spec}\ k}\mathbb{A}^2\_k \to \mathbb{A}^2\_k, \ \ m^\*(x) = x\oti... | 3 | https://mathoverflow.net/users/13265 | 299746 | 131,504 |
https://mathoverflow.net/questions/299727 | 2 | Let $p\neq5$ be a prime number such that $q=(p-1)/2$ is prime. Does there exist an Alternating group of degree $p$ in which every minimal subgroup of order $p$ is contained properly in exactly one maximal subgroup of $A\_{p}$?
Note that if such Alternating groups exist, then the mentioned maximal subgroup is isomorph... | https://mathoverflow.net/users/119365 | Maximal subgroups of Alternating groups of degree $p$, for some prime $p$ | Yes, this is the case for most primes of this form. The smallest example is with $p=47$.
Burnside proved that a transitive group of prime degree $p$ either has a normal subgroup of order $p$ or is doubly transitive. As a consequence of the classification of finite simple groups, the doubly transitive groups are now k... | 7 | https://mathoverflow.net/users/35840 | 299747 | 131,505 |
https://mathoverflow.net/questions/299743 | 3 | I am trying to understand the notation $\rho^{\vee}(-1)$. Let $T$ be a maximal torus of a semi-simple algebraic group $G$ and $\mathbb{G}\_m$ the multiplicative group. I think that $\rho^{\vee}$ is a map $\mathbb{G}\_m \to T$ which satisfies $\alpha\_i(\rho^{\vee}(s))=s$ for all $s \in \mathbb{G}\_m$, where $\alpha\_i:... | https://mathoverflow.net/users/11877 | What is $\rho^{\vee}(-1)$? | In general $\rho^\vee$ is an element of the co-weight lattice. The notation $\rho^\vee(-1)$ only makes sense if $\rho^\vee$ is in fact in the co-character lattice, i.e. the lattice $\text{Hom}(\mathbb G\_m,T)$.
This holds, for example, if $G$ is adjoint.
If this condition holds, $\rho^\vee(s)$ is an element of $T$ ... | 6 | https://mathoverflow.net/users/6030 | 299758 | 131,507 |
https://mathoverflow.net/questions/299742 | 5 | For $f\in\mathcal{S}$ a Schwartz function on $\mathbb{R}^n$ and $m$ a bounded function, define $Tf$ by $\widehat{T f}=m\cdot \widehat{f}$.
Fix $1<p<\infty$, $p\not=2$. Suppose we have proved that there exists a constant $C>0$ such that
$$ \|Tf\|\_p \le C \|f\|\_p$$
holds for all $f\in\mathcal{S}$ satisfying $\widehat... | https://mathoverflow.net/users/124244 | $L^p$ estimates and functions with positive Fourier transform | I apologize for answering my own question (and for asking a trivial question).
The answer is yes. If $\widehat {f}\ge 0$, then we always have by the triangle inequality,
$$\|Tf\|\_4=\|\widehat{Tf}\*\widehat{Tf}\|\_2^{1/2}\le C \|\widehat{f}\*\widehat{f}\|\_2^{1/2}=C\|f\|\_4 $$
so any multiplier that is not bounded on $... | 2 | https://mathoverflow.net/users/124244 | 299763 | 131,508 |
https://mathoverflow.net/questions/299738 | 5 | The join $\ast$ and the product $\times$ are both important monoidal structures on simplicial sets, but the way they interact is not so simple. For instance, neither distributes over the other. However, I believe there is a comparison map $(A\times B) \ast (C \times D) \to (A \ast C) \times (B \ast D)$, suggesting that... | https://mathoverflow.net/users/2362 | Do simplicial join and product form a duoidal category structure? | I probably should have read more closely. If $C$ is any monoidal category which is also cartesian monoidal, then the combination of the two monoidal structures is duoidal. This is on the nlab page.
| 4 | https://mathoverflow.net/users/2362 | 299765 | 131,509 |
https://mathoverflow.net/questions/299766 | 26 |
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> 1. Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{F}\_p$) in terms of generators and relations.
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For concreteness, let's say the input is given by a ge... | https://mathoverflow.net/users/38434 | Is the cohomology ring of a finite group computable? | As I understand it this follows from Benson's Regularity Conjecture, proved by Symonds fairly recently. It says that $b\_p = 2(|G|-1)$ will do.
| 19 | https://mathoverflow.net/users/318 | 299777 | 131,512 |
https://mathoverflow.net/questions/214570 | 4 | The following general definition of subharmonic function comes from the classical text book [*elliptic partial differential equations of second order*] by Gilbarg and Trudinger.
We call a function $u$ subharmonic in $\Omega$ if $u \in C^0(\Omega)$ and for every ball $B \subset \subset \Omega$ and every function $h$ h... | https://mathoverflow.net/users/51546 | Second order differentiability of subharmonic function almost everywhere? | Almost everywhere is too strong, but a Lusin-like theorem holds true :
Let $u$ be a subharmonic function in a domain $D\subset\mathbb{R}^n$,
$K\subset D$ a compact set, and $\epsilon>0$. There exist an open set $G\_\epsilon$ whose Lebesgue measure is less than $\epsilon$ and a twice continuously differentiable funct... | 7 | https://mathoverflow.net/users/89429 | 299792 | 131,516 |
https://mathoverflow.net/questions/299789 | 5 | Is it true that the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated?
| https://mathoverflow.net/users/16183 | Fundamental group of Alexandrov space. | Under these hypotheses the systole of $X$ is clearly bounded from below, and the usual comparison arguments would give an upper bound on the number of points in an $\epsilon$-net in $X$. Then every loop can be discretized and the finite-generation follows.
| 4 | https://mathoverflow.net/users/28128 | 299795 | 131,517 |
https://mathoverflow.net/questions/299794 | 2 | Let $X$ be a scheme over a separably closed field, equipped with an action of a constant group scheme $G$.
Let $H$ be a finite group whose size is invertibile on $X$, and $Y\to X$ an $H$-torsor with respect to the étale topology.
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> Is $Y\to X$ equipped with a $G$-equivariant structure?
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| https://mathoverflow.net/users/nan | Étale torsors and equivariant structures | If $G$ fixes a basepoint $x\in X(k)$, then your question roughly asks whether $G$ acts trivially on (the prime-to-$p$ completion of) $\pi\_1(X, x)$. This suggests a negative answer.
For example, let $X=\mathbf{G}\_m^2$, $G=\mathbf{Z}/2$ acting on $X$ by swapping the coordinates, let $H=\mu\_\ell(k)$, and let $Y\to X$... | 1 | https://mathoverflow.net/users/3847 | 299796 | 131,518 |
https://mathoverflow.net/questions/299735 | 3 | First, let us fix some Notation:
Let $n\in\mathbb{N}$ and $x\_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let
\begin{align}L\_i(x)={\displaystyle\prod\_{\substack{0\leq j\leq n\\i\neq j}}}\frac{x-x\_j}{x\_i-x\_j}, \end{align}
$i=0,\dots,n$, be the Lagrange polynomials. Now let $f\in C^... | https://mathoverflow.net/users/124237 | Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis | A bound of the kind you're looking for is not possible, and it doesn't even matter what interpolation points you use (much less which particular polynomial basis you choose to work with, since that has no effect on the $c\_{n,j}$ anyway).
For a given value of $n$, let $f$ be (for instance) the Chebyshev polynomial of... | 1 | https://mathoverflow.net/users/20507 | 299797 | 131,519 |
https://mathoverflow.net/questions/299719 | 1 | Suppose that $Z,Z\_1,Z\_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z\_t=Z\_tI\_{\{|Z\_t|\le n^{1/s} \}}$ with $1\le t\le n$ be the triangular array of the truncated random variables, where $I\_A$ is the indicat... | https://mathoverflow.net/users/46211 | Maximum of the periodogram of a truncated sequence | Let $A\_N$ be the event defined by
$$
A\_N:=\bigcup\_{n=2^{N-1}+1}^{2^N}\left\{\max\_{1\le j\le q}I\_{n,Z}(\omega\_j)\neq \max\_{1\le j\le q}I\_{n,\tilde Z^{(n)}}(\omega\_j)\right\}.
$$
Then the following inclusion holds
$$
A\_N\subset \bigcup\_{n=2^{N-1}+1}^{2^N}\bigcup\_{t=1}^n\left\{ Z\_t\neq \widetilde{Z\_t}^{(n)}... | 3 | https://mathoverflow.net/users/17118 | 299798 | 131,520 |
https://mathoverflow.net/questions/299778 | 3 | I'm trying to use Magma to do a double coset calculation on the group M10, but the answer does not make sense to me. Your help and comments are most appreciated. First, here's the calculation:
(1) M10 has one conjugacy class of order 72 subgroups; pick one and call it T. There is also one class of order 2 subgroups; ... | https://mathoverflow.net/users/66397 | Double coset representatives and Magma | The problem with inconsistent results in the final part of your calculation is not just due to different choices of random elements, as I said in my comment, but probably due to a confusion about the meaning of $S^t$, which is defined to be $t^{-1}St$ in Magma.
Think of the group $S$ as acting by right multiplication... | 4 | https://mathoverflow.net/users/35840 | 299807 | 131,524 |
https://mathoverflow.net/questions/299423 | 3 | Cockayne and Lorimer ("The Ramsey number for stripes" 1975) prove that in every $r$-colored complete graph on $n=\sum\_{i=1}^rm\_i+m\_1-(r-1)$ vertices, where $m\_1\geq \dots\geq m\_r\geq 1$, has a monochromatic matching in color $i$ of size $m\_i$ (and this is tight). When $m\_1=\dots=m\_r$, we get $n=(r+1)m-(r-1)$. T... | https://mathoverflow.net/users/17798 | A direct proof that every $r$-colored complete graph on $n=(r+1)m-(r-1)$ vertices has a monochromatic matching of size $m$? | Here goes a direct proof of a general fact. It is not inductive, so you may substitute $m\_1=m\_2=\dots=m\_r=m$ into it, but it does not become any shorter. It uses Tutte/Berges formula of the maximal matching, as you ask for.
We use two easy lemmas.
**Lemma 1.** A graph on $N$ vertices with at least $N-k$ compon... | 3 | https://mathoverflow.net/users/4312 | 299813 | 131,525 |
https://mathoverflow.net/questions/299772 | 1 | I am studying the [Law of large numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers) for [independent and identically distributed (i.i.d)](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) random variables.
Assume there are i.i.d variables $(\xi\_k)\_{k\ge 1}$ taking valu... | https://mathoverflow.net/users/124254 | LLN large number law of Probability | If the $\xi$ take values arbitrarily close to 1 (e.g. the $\xi\_i$ are Unif[0,1]), this quantity is infinite: for any $\epsilon>0$, there exist arbitrarily long strings where all of the $\xi\_j$’s are at least $1-\epsilon$, which means that the sup is larger than$1/\epsilon$.
| 2 | https://mathoverflow.net/users/11054 | 299818 | 131,528 |
https://mathoverflow.net/questions/299563 | 8 | Let $n$ be a positive integer and $S\_n$ be the symmetric group on $\{1,2,\ldots,n\}$. Let $\mathcal{A}\_n$ be the anti-symmetrization operator on $\mathbb{Z}[x\_1,x\_2,\ldots,x\_n]$ such that for any $f(x\_1,x\_2,\ldots,x\_n)\in \mathbb{Z}[x\_1,x\_2,\ldots,x\_n]$,
$$\mathcal{A}\_n(f)=\sum\_{w\in S\_n}\varepsilon(w)f(x... | https://mathoverflow.net/users/58096 | The anti-symmetrization of a kind of polynomials in $\mathbb{Z}[x_1,x_2,\ldots,x_n]$ | Hope that now it works, but please check.
For $n=4k$ or $n=4k+1$ we consider the following graph $H\_n$ on the set $V\_n$, $|V\_n|=n$ with $n(n-1)/4$ edges: $V\_n=\{a\_i,b\_i,c\_i,d\_i,1\leqslant i\leqslant k\}$ for $n=4k$ and $V\_n=V\_{n-1}\cup \{e\}$ for $n=4k+1$; the edges of $H\_n$ are the edges of $H\_{n-4}$ plu... | 5 | https://mathoverflow.net/users/4312 | 299825 | 131,530 |
https://mathoverflow.net/questions/299817 | 2 | I would like an example of maps between a smooth curve $C$ and a singular curve $B $, $f:C \rightarrow B$, where the genus $p\_a(C)=p\_a(B)$ and greater than or equal to 2.
| https://mathoverflow.net/users/94832 | Example of maps between a smooth curve and a singular curve | I am just writing my comment as an answer. Let $k$ be a field. Let $C$ be a smooth, projective, connected $k$-curve. Let $B$ be a projective, reduced, $k$-curve, and let $$f:C\to B$$ be a finite $k$-morphism. By the universal property of normalization, $f$ factors uniquely through the normalization, $$\nu:\widetilde{B}... | 3 | https://mathoverflow.net/users/13265 | 299829 | 131,532 |
https://mathoverflow.net/questions/299824 | 1 | I've posted this [to Math.SE](https://math.stackexchange.com/q/2733614/214353) about a month ago:
Seems like
$$
\Delta(a\_0+a\_1t^d+a\_2t^{2d}+...+a\_nt^{nd})=(-1)^{n\frac{d(d-1)}2}d^{nd}(a\_0a\_n)^{d-1}[\Delta(a\_0+a\_1t+a\_2t^2+...+a\_nt^n)]^d,
$$
where $\Delta$ is the discriminant.
Presumably this is not difficu... | https://mathoverflow.net/users/41291 | What would be a standard reference for the formula of the discriminant of $f(t^d)$? | You say "presumably not difficult to prove." Did you try? It seems like a pretty easy exercise, using
$$ \Delta(F(t)) = \prod\_{F(a)=0} F'(a). $$
Taking $F(t)=f(t^d)$, the roots of $F$ are the $d$'th roots of the roots of $f$, while $F'(t)=dt^{d-1}f'(t^d)$. So I doubt you'll find this formula in a reference, but if you... | 5 | https://mathoverflow.net/users/11926 | 299833 | 131,534 |
https://mathoverflow.net/questions/299834 | 5 | Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$.
Convention. For given Hilbert spaces $H$ and $K$, let us write $H\leq K$ if the Hilbertian dimension of $H$ is less than $K$.
... | https://mathoverflow.net/users/84390 | The largest topological copy of a Hilbert space contained in $\ell^1$ | No infinite dimensional reflexive space can be embedded into $\ell\_1$, because every infinite dimensional closed subspace of $\ell\_1$ has a non separable dual.
| 4 | https://mathoverflow.net/users/6101 | 299835 | 131,535 |
https://mathoverflow.net/questions/299392 | 4 | A Lindelöf space is a topological space in which every open cover has a countable subcover.
1. Does there exists a Lindelöf locally convex space which is not second countable?
2. I am also looking for a separable locally convex which is not second countable!
| https://mathoverflow.net/users/84390 | Separable Lindelöf locally convex spaces that are not second-countable | A space of the form $C\_p(X)$ (the space of continuous functions on a Tychonoff space $X$, in the pointwise-topology (i.e. the subspace topology induced from the product topology on $\mathbb{R}^X$) is often an example; note that such spaces are locally convex spaces:
$C\_p(X)$ is separable when $X$ is second countabl... | 5 | https://mathoverflow.net/users/2060 | 299846 | 131,536 |
https://mathoverflow.net/questions/299812 | 1 |
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> Is it possible to prove this matrix family only contains totally unimodular matrices?
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The matrix has dimensions $\frac{3n(n-1)}2$ rows and $n+\frac{n(n-1)}2$ columns.
To every pair $(i,i')$ with $1\leq i<i'\leq n$ we have an unique integer $f(i,j)$ from $\big\{n+1,\dots,n+\frac{n(n-1)}2\big\}$ associ... | https://mathoverflow.net/users/10035 | Is this totally unimodular family? | I don't see that your example matrix $M$ is TU. Taking the first three columns of rows 1, 4 and 7 gives the submatrix
$$\begin{pmatrix}1&1&0\\ 1&0&1\\ 0&1&1\end{pmatrix}$$
which has derterminant $-2$.
| 3 | https://mathoverflow.net/users/12674 | 299847 | 131,537 |
https://mathoverflow.net/questions/297433 | 3 | Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; particularly, the assertion that sheaves with respect to a coverage are the same as sheaves with respect to the generate... | https://mathoverflow.net/users/31420 | Why are sheaves of a coverage the same as those on its generated Grothendieck coverage? | Extend $g$ to any generating set for $\varphi^\*F$, say $\widehat{f} := \{ b\_k \overset{\widehat{f}\_k}{\longrightarrow} b \} \supseteq g := \{ b\_{k\_j} \overset{\widehat{f}\_{k\_j}}{\longrightarrow} b \}$, we will need to demonstrate that a sheaf with respect to the coverage $\mathcal{J}$ must satisfy descent for $\... | 0 | https://mathoverflow.net/users/31420 | 299853 | 131,538 |
https://mathoverflow.net/questions/299854 | 8 | Given a countable group $G$ and a generating subsemigroup $S\subset G$, let us consider the increasing sequence of alternating products of $S$ and $S^{-1}$ beginning with $S$
$$
A\_1=S, A\_2=S S^{-1}, A\_3=S S^{-1}S, A\_4=S S^{-1}S S^{-1}, A\_5=S S^{-1}S S^{-1}S, \dots \;,
$$
and put
$$
\kappa(S) = \min \{k: G=A\_k \} ... | https://mathoverflow.net/users/8588 | How many "steps" does one need to obtain a group from a semigroup | Marek Kuczma asked in 1980 whether for every positive integer $n$ there exists a subsemigroup $M$ of a group $G$ such that $G$ is equal to the $n$-fold product $MM^{-1}MM^{-1}\cdots M^{(-1)^{n-1}}$, but not to any proper initial subproduct of the product.
George Bergman proved that the answer is affirmative for all ... | 10 | https://mathoverflow.net/users/3959 | 299861 | 131,539 |
https://mathoverflow.net/questions/299866 | 3 | Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n\_R$ a projective embedding.
>
> Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n\_R$ of $p(X)$ in $\mathbf{P}^n\_R$ is quasi-compact?
>
>
>
| https://mathoverflow.net/users/nan | Projective embeddings and quasi-compactness | Take for $X$ a closed point of $S:=\mathrm{Spec}(R)$. Then $p$ extends to a section $\tilde{p}:S\to \mathbf{P}^n\_R$. If $U$ is quasicompact, so is $\tilde{p}^{-1}(U)=S\smallsetminus X$. So there are plenty of counterexamples, e.g. $R=k[t\_1,\dots,t\_n,\dots]$ and $X=$ the origin.
On the other hand, if $X$ is of fini... | 5 | https://mathoverflow.net/users/7666 | 299867 | 131,541 |
https://mathoverflow.net/questions/299870 | 1 | Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| = \lambda$?
| https://mathoverflow.net/users/8628 | Connected Hausdorff spaces with different cardinalities of open sets | Let $C$ be Bing's [countable connected Hausdorff space](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf). Specify some point $0 \in C$. Let $D(\alpha)$ be the discrete space of cardinality $\alpha$.
Let $F(\alpha)$ be the product $C \times D(\alpha)$ with the set $\{... | 3 | https://mathoverflow.net/users/58082 | 299873 | 131,542 |
https://mathoverflow.net/questions/299869 | 1 | Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$.
Suppose $X = \varprojlim X\_j$, for $X\_j$ qcqs schemes over $k$, with affine transition maps.
Do there exist, for $j$ large enough onwards, $F\_j$ ... | https://mathoverflow.net/users/nan | Approximation of constructible abelian étale sheaves | <https://stacks.math.columbia.edu/tag/09YU>
I found an answer to my question in part (3) of the Lemma in the SP.
| 1 | https://mathoverflow.net/users/nan | 299882 | 131,546 |
https://mathoverflow.net/questions/299839 | 2 | Let $Z\_2$ and $Z\_3$ be second and third order arithmetics respectively.
In $Z\_2$'s language, $\text{AD}$ (the axiom of determinacy) and $\text{PD}$ (projective determinacy) are stated the same way (since projective sets are precisely the ones defined by formulas in $Z\_2$'s language). But in $Z\_3$'s language we c... | https://mathoverflow.net/users/78441 | Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ | The only reference I know for precisely these matters is the handbook chapter
>
> [MR2768702](https://mathscinet.ams.org/mathscinet-getitem?mr=2768702). Koellner, Peter; Woodin, W. Hugh. *[Large cardinals from determinacy](https://link.springer.com/chapter/10.1007%2F978-1-4020-5764-9_24)*. In **Handbook of set the... | 7 | https://mathoverflow.net/users/6085 | 299889 | 131,548 |
https://mathoverflow.net/questions/299864 | 4 | What's the largest constant $c >1$ such that all triangle free graphs with chromatic number $n$ has atleast $\Theta(c^n) $ vertices?
| https://mathoverflow.net/users/115227 | Size of triangle free graph with chromatic number $n$ | It follows from the result in [A note on Ramsey numbers](https://www.sciencedirect.com/science/article/pii/0097316580900308) by M. Ajtai, J. Komlós, E. Szemerédi that a triangle free graph on $s$ vertices has chromatic number at most $\Theta(\sqrt\frac{s}{\log s})$.
Moreover, in [The Ramsey Number $R(3, t)$ has Order... | 6 | https://mathoverflow.net/users/2384 | 299891 | 131,549 |
https://mathoverflow.net/questions/299856 | 0 | In the proof of the full exact embedding theorem of the "Handbook of Categorical Algebra 2" of F. Borceux at the step 2, there is something I do not understand. It is at page 82 just after the colimits of the previous equalizer diagram, he says :
>
> Since $\downarrow \delta\_A$ is an initial part of $\mathcal{D}$
... | https://mathoverflow.net/users/123778 | Misunderstanding of the proof of the Embedding Theorem in Borceux | Sorry, it was not that complicated : it comes from the fact that $ \mathcal{D}$ is a $\wedge$-semilattice. Thus for $D \in \mathcal{D}$ we just have to compute $D'=\delta\_A \wedge D$ to have an element in $\downarrow \delta\_A$ such that $D' \leq D$.
| 1 | https://mathoverflow.net/users/123778 | 299894 | 131,552 |
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