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https://mathoverflow.net/questions/308318 | 8 | Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi\_1(M)$ must be infinite?
Consider the case where $d=n-1$. Within Section 4 of [this paper by Grove and Ziller](https://www.researchgate.net/publication/22718... | https://mathoverflow.net/users/106283 | Torus action implying infinite fundamental group | Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of [this paper](https://msp.org/pjm/1974/53-2/p12.xhtml). However, I haven't checked that the action is smoothable. The authors are only considering locally smooth actions in the paper (see [p. 170 of Bredon](http://www.indiana.edu/~jf... | 6 | https://mathoverflow.net/users/1345 | 308333 | 134,295 |
https://mathoverflow.net/questions/308306 | 2 | Let $\mathcal{D}$ be a stack. An [atlas](https://mathoverflow.net/questions/303826/understanding-the-definition-of-atlas-of-a-stack) for stack $\mathcal{D}$ is given by
* a smooth manifold $X$ and
* a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$
such that, for any
* manifold $M$ and
* a map of stacks ... | https://mathoverflow.net/users/118688 | To check if a stack is coming from a manifold | Take $M = X$ and $f=p$, so that $\underline{P} = \underline{X} \times\_\mathcal{D} \underline{X}$. The Lie groupoid $P\rightrightarrows X$ you get should be proper, in the sense the source-target map $(s,t)\colon P\to X\times X$ is a proper map, and $(s,t)$ should be injective, so that there are no nontrivial automorph... | 2 | https://mathoverflow.net/users/4177 | 308337 | 134,296 |
https://mathoverflow.net/questions/308163 | 6 | Consider a collection of $m$ matrices $A\_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system
$$\left\{ x^T A\_i y = b\_i : i = 1,\dots,m \right\}$$
in variables $x,y$. Is there an *efficient* way of doing this?
This is both a theoretical and a practical question: the matrice... | https://mathoverflow.net/users/10481 | Solving system of bilinear equations | We have a system of $m$ bilinear equations in $\mathrm x, \mathrm y \in \mathbb R^n$
$$\begin{aligned} \mathrm x^\top \mathrm A\_1 \,\mathrm y &= b\_1\\ \mathrm x^\top \mathrm A\_2 \,\mathrm y &= b\_2\\ &\vdots\\ \mathrm x^\top \mathrm A\_m \,\mathrm y &= b\_m\end{aligned}$$
If matrices $\mathrm A\_1, \mathrm A\_2,... | 5 | https://mathoverflow.net/users/91764 | 308341 | 134,297 |
https://mathoverflow.net/questions/308342 | 3 | This question is inspired by question in reference.
**Question :** If $M$ is a simply connected closed Riemannian manifold of nonnegative sectional curvature, then there is a totally geodesic submanifold $S$ of codimension 1
**Def :** ${\rm conv}\ X$ is a smallest closed convex set containing $X$.
And a subset $... | https://mathoverflow.net/users/36572 | Totally geodesic submanifold of codimension 1 | There is no three-dimensional totally geodesic submanifold in $\mathbb{CP}^2$.
| 11 | https://mathoverflow.net/users/21123 | 308346 | 134,299 |
https://mathoverflow.net/questions/308336 | 1 | Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection?
Rem. It seems that when $A=B(H)$, the range of the projection $p-p\wedge(1-e)$
is just $\overline{peH}$ which is clea... | https://mathoverflow.net/users/84390 | The range projection of product of projections | By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is finite.
| 1 | https://mathoverflow.net/users/84700 | 308347 | 134,300 |
https://mathoverflow.net/questions/308038 | 7 | I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) SDEs and its applications in finance and I would like some help finding some recent papers related to or useful when doin... | https://mathoverflow.net/users/51480 | Good papers on stochastic differential equations with applications in finance | As indicated in the comments, the field is very wide, but I understand from the comment of the OP to zab's answer that there is a specific interest in the more narrow subtopic of applications of fractional Brownian motion to quantitative finance. Here are some overviews:
* [Fractional Brownian Motion in Finance](http... | 5 | https://mathoverflow.net/users/11260 | 308358 | 134,302 |
https://mathoverflow.net/questions/307788 | 10 | Let $S$ be the dyadic [solenoid](https://en.wikipedia.org/wiki/Solenoid_(mathematics)).
Let $x\in S$, and let $X$ be the union of all *arcs* (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a *composant* of $S$.
It is well-known that $X$ is a dense first category one-to-one continuous image... | https://mathoverflow.net/users/91061 | homeomorphisms induced by composant rotations in the solenoid | In [this paper of J.Kwapisz](http://www.math.montana.edu/jarek/documents/papers/solenoid.pdf) I have found the following
**Theorem 1.** Any homeomorphism $h$ of the dyadic solenoid $S$ is isotopic to the "affine" homeomorphism of the form $g:x\mapsto \pm(2^n x+b)$ for some $n\in\mathbb Z$ and some $b\in S$.
If $h$ ... | 5 | https://mathoverflow.net/users/61536 | 308364 | 134,305 |
https://mathoverflow.net/questions/308370 | 2 | I have questions about the definition of representation variety. In François Labourie's book "Lectures on representations of surface groups", [Section 3.5](https://math.unice.fr/~labourie/preprints/pdf/surfaces.pdf), the author gives four models of the representation variety. I am confused about the model using the lan... | https://mathoverflow.net/users/105481 | Flat R-bundles on surfaces | Any flat $\mathbb{R}^{+}$-bundle is precisely given, as in Tsemo Aristide's answer, by a representation of the fundamental group of the surface into $\mathbb{R}^+$. As $\mathbb{R}^+$ is abelian, this descends to an element of $H\_1(S,\mathbb{R}^+)$, whose logarithm is a uniquely determined element of $H\_1(S,\mathbb{R}... | 1 | https://mathoverflow.net/users/13268 | 308380 | 134,311 |
https://mathoverflow.net/questions/308385 | 3 | An Egyptian fraction expansion is a sum of reciprocals of integers, for example:
$$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$
Every positive rational number $p/q$ has such an expansion, although it is not unique:
$$\frac{4}{17} = \frac{1}{5} + \frac{1}{30} + \frac{1}{510}$$
... | https://mathoverflow.net/users/126543 | What is the shortest length of an Egyptian fraction expansion for a given $p/q$? | If there is an expansion with $k$ terms, one of the denominators is at most $kq/p$. So to check whether there is an expansion with at most $k$ terms: for each $m$ from $\lceil q/p \rceil$ to $\lfloor kq/p \rfloor$, check recursively whether $p/q - 1/m$ has an expansion with at most $k-1$ terms.
Whether there is a pol... | 3 | https://mathoverflow.net/users/13650 | 308386 | 134,313 |
https://mathoverflow.net/questions/308340 | 10 | I am trying to characterize when a semi-direct product of the form $(Z/pZ)^n \rtimes (Z/qZ)$ is isomorphic to a group generated by two elements. Here $p$ and $q$ are distinct odd primes.
I would be happy for a reference or even some examples of this happening for $n > 1.$
(I found a similar question [here](https://... | https://mathoverflow.net/users/22512 | When is the semidirect product of an elementary abelian group and a cyclic group generated by two elements? | This answer corroborates YCor's claim according to which the conditions $n\_i \le 1$ for $i > 1$ and $n\_1 \le 2$ on the irreducible modular representations'multiplicities $n\_i$, are necessary and sufficient for $G$ to be two-generated. We actually show a slightly more general result expressed in terms of the geometri... | 7 | https://mathoverflow.net/users/84349 | 308388 | 134,314 |
https://mathoverflow.net/questions/308324 | 0 | Let $u\_i \in C^1(\Omega)$ with $|\nabla u\_i|>0$ in a simply connected region $\Omega$ with connected boundary, and $u\_1=u\_2$ on $\partial \Omega$. Assume
$$ \nabla u\_i(x) \cdot V\_i (x)=|\nabla u\_i(x)||V\_i(x)|, \ \ \forall x \in \Omega$$
for two vector fields $V\_i\in L^{\infty}(\Omega)$ with $|V\_i|>0$, $i=1,2... | https://mathoverflow.net/users/42326 | A basic stability question | Let $\Omega$ be the unit ball in $\mathbb{R}^2$.
Let $u\_k(x,y) = \tan^{-1}(k^3 x)$.
Let $v\_k(x,y)$ be a function that agrees with $u\_k$ on $\partial\Omega$, and is constant on the level sets of $\{ (x- k)^2 + y^2\}$.
So $\nabla u\_k / |\nabla u\_k| = \partial\_x$, and
$ \nabla v\_k / |\nabla v\_k| = \parti... | 2 | https://mathoverflow.net/users/3948 | 308389 | 134,315 |
https://mathoverflow.net/questions/308351 | 5 | Let $S=S\_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x\_1,\ldots , x\_n\}$ be a set of $n$ distinguished points in the interior of $S$.
Define ${\rm Mod}(S,\{Q\})$ to be the mapping class group of orientation-preserving self-homeomorphisms... | https://mathoverflow.net/users/8103 | Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures | The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk and $Q$ is empty or a single point. If one chooses a basepoint in $\partial S$ then these Dehn twists induce the identity... | 2 | https://mathoverflow.net/users/23571 | 308391 | 134,317 |
https://mathoverflow.net/questions/308382 | 2 | When reading various literature on spectral sequences one always comes across two setups:
* A chain complex with an increasing filtration
* A cochain complex with a decreasing filtration
My question is why the other two options are never mentioned:
* A chain complex with an decreasing filtration
* A cochain compl... | https://mathoverflow.net/users/64302 | Why only consider decreasing filtrations on cochain complexes? | There is not a conceptual reason why increasing filtrations cannot happen on cochain complexes, or vice versa. A prominent example of this type of spectral sequence is the Eilenberg-Moore spectral sequence
$$
Tor^{H^\*(Z)}\_{\*\*}(H^\* Y, H^\* X) \Rightarrow H^\*(Y \times\_Z X)
$$
for the cohomology groups of a (homoto... | 4 | https://mathoverflow.net/users/360 | 308398 | 134,319 |
https://mathoverflow.net/questions/308393 | -2 | I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the category of groupoids of this type, ie, generated by such an adjunction? For instance, are there any interesting adjunctions b... | https://mathoverflow.net/users/10007 | Adjunctions between Groupoids and Hilbert spaces | One suggested variant was to ask whether there are any interesting adjunctions between the category $FinGpd$ of finite groupoids and the category $FinHilb$ of finite-dimensional Hilbert spaces, which is still undefined. The answer is still no.
First let me suppose that the morphisms of $FinHilb$ are all linear maps. ... | 5 | https://mathoverflow.net/users/2362 | 308403 | 134,321 |
https://mathoverflow.net/questions/308366 | 12 | Let $$C\_n=\frac{1}{2n+1}\binom{2n+1}{n}$$ be a Catalan number. It is well-known that $$(\sum\_{n\ge{0}}C\_n x^n)^k=\sum\_{n\ge{0}}C(n,k)x^n$$ with $$C(n,k)=\frac{k}{2n+k}\binom{2n+k}{n}.$$
It is also known that the Hankel matrix $\left( {{C(i+j,2)}} \right)\_{i,j = 0}^{n - 1}$ can be factored in the form $$\left( {{C(... | https://mathoverflow.net/users/5585 | A matrix identity related to Catalan numbers | After unpacking the equation $$\left( {{C(i+j,k+2)}} \right)\_{i,j = 0}^{n - 1}=A\_{n}G\_{n,k} A\_{n}^T$$
we see that we want to prove the identity
$$C(i+j,k+2)=\frac{k+2}{(2i+2j+k+2)}\binom{2i+2j+k+2}{i+j}$$
$$=\sum\_{0\le r\le i,0\le s\le j} \left[\binom{2i+1}{i-r}-\binom{2i+1}{i-r-1}\right]\cdot\left[\sum\_{m=|r-s|-... | 12 | https://mathoverflow.net/users/2384 | 308405 | 134,322 |
https://mathoverflow.net/questions/308286 | 3 | In [this paper](http://www.math.lsa.umich.edu/~ablass/bbh.pdf) Bergelson, Blass, and Hindman prove the following
>
> **Theorem 1.2** Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. Then $\bar s$ has a variable extraction $\bar t$ such that the set... | https://mathoverflow.net/users/73874 | Partition theorems for located words | If I understand the definitions correctly, you can take $\bar s$ to be a constant sequence equal to $v$. Take a coloring that colors words containing only the letter $a$ in red and words containing only the letter $b$ in blue. Then any $\bar t$ obtained by concatenating members of $\bar s$ will be composed of words in ... | 2 | https://mathoverflow.net/users/19534 | 308408 | 134,323 |
https://mathoverflow.net/questions/308383 | 6 | Assume that we have a set system $\mathfrak T = \{\mathcal T\_1, \mathcal T\_2, \dots, \mathcal T\_N \}$ where each $\mathcal T\_k$ is a collection of subsets of $[n] := \{1,\dots,n\}$ of the form
$$ \mathcal T\_k = [m\_k, M\_k] := \{T \subseteq [n]:\; m\_k \subseteq T \subseteq M\_k \}. $$
Moreover, we know that $\ma... | https://mathoverflow.net/users/36687 | Algorithm to decide if the union of a set system covers the power set | Given $T\_k = [m\_k,M\_k]$ and $X\subseteq [n]$, it is easy to calculate the number of sets in $T\_k$ that contain $X$: $$Q\_{X,k}=|\{T\in T\_k \mid X\subseteq T\rbrace|.$$
So start with $\sum\_k Q\_{\emptyset,k}$. If it equals $2^n$, every subset of $[n]$ is covered.
Otherwise, there is $x\_1\in [n]$ such that $\s... | 4 | https://mathoverflow.net/users/9025 | 308414 | 134,326 |
https://mathoverflow.net/questions/308265 | 3 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We say that $C\subseteq E$ is a *cover* if $\bigcup C = V$. Let $H$ be a hypergraph with the following properties:
1. $\bigcup E = V$,
2. all members of $E$ are finite, and
3. $d,e\in E$ with $d\subseteq e$ implies $d=e$.
**Question.** Does... | https://mathoverflow.net/users/8628 | Minimal covers in hypergraphs with finite edges | Let $V:=\omega\times\omega$ and $E=\{E\_{n,m}:n,m\in\omega\}$ where $$E\_{n,m}:=(\{0,\dots,n\}\times\{m\})\cup\{(2n,m+1)\}.$$ It seems that the hypergraph $(V,E)$ has no minimal cover.
A simplification of this example was suggested by Gerhard Paseman in his comment:
Just take $V=\mathbb Z$ and $E=\{E\_n,F\_n:n\in\mat... | 5 | https://mathoverflow.net/users/61536 | 308417 | 134,328 |
https://mathoverflow.net/questions/308402 | 1 |
>
> Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
> $$\frac{1}{x+y^2+i0}?$$
>
>
>
Remark. **Equivalent question:** consider the Schroedinger equation one the line
$$i\frac{\partial}{\partial t}\Psi(x,t)=\frac{\partial^2}{\partial x^2}\Psi(x,t).$$
>
> Find ... | https://mathoverflow.net/users/16183 | Fourier transform of a generalized function on the plane | The Fourier transform vanishes for $u>0$, for $u<0$ instead
$$
I(u,v)=\frac{1}{2\pi}\int\_{-\infty}^\infty dx\int\_{-\infty}^\infty dy\,\frac{e^{\mathrm{i}(ux+vy)}}{x+y^2+\mathrm{i}0^+}$$
$$\qquad\qquad=-ie^{i\pi/4}\sqrt{\pi }(- u)^{-1/2}\exp\left({\frac{i v^2}{4 u}}\right)\;\;\text{for}\;\; u<0.
$$
For $u=0$ there is... | 2 | https://mathoverflow.net/users/11260 | 308420 | 134,329 |
https://mathoverflow.net/questions/308432 | 11 | Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results,
$$(S^{n-1} \wedge X)\_{hO(n)} \simeq X\_{hO(n-1)}$$
and
$$\Omega^\infty (X\_{hG)}) \simeq (\Omega^\infty X)\_{hG} $$
where $G$ is a compact Lie group.
I've spent the last few ... | https://mathoverflow.net/users/117088 | Homotopy orbits, spectra and infinite loop spaces | Both of these are false.
The first is close to true: if $S(n-1)$ is the unit sphere in $\Bbb R^{n}$ with its standard $O(n)$-action, then we can identify $S(n-1)$ with $O(n) / O(n-1)$ and so get the identification
$$
(S(n-1)\_+ \wedge X)\_{hO(n)} = EO(n)\_+ \wedge\_{O(n)} O(n)/O(n-1)\_+ \wedge X = EO(n)\_+ \wedge\_{O... | 13 | https://mathoverflow.net/users/360 | 308439 | 134,336 |
https://mathoverflow.net/questions/308443 | 3 | Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded.
Is it possible to remove the assumption "finite topology"? Have there been any progress in that direction?
| https://mathoverflow.net/users/86341 | On the Calabi-Yau conjecture for minimal surfaces | Properness is expected to hold for finite genus embedded minimal surfaces while it seems likely that there are infinite genus counterexamples. Both of these claims are completely open (and are extremely difficult)
Currently the best results are in a [preprint](http://wpd.ugr.es/~jperez/wordpress/wp-content/uploads/Ca... | 4 | https://mathoverflow.net/users/127803 | 308452 | 134,341 |
https://mathoverflow.net/questions/244611 | 9 | In a symmetric space of rank $k$ (and I'll take $k > 1$) *every* geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
**Question.** Are there non-symmetric homogeneous spaces that share this property?
In this [paper](http://link.springer.com/article/10... | https://mathoverflow.net/users/21123 | k-flats in homogeneous spaces | If the k-flats are compact, then the space must be symmetric
(Molina-Olmos, J. Differential geometry
45 (1997) 575-592; see also Proc. Amer. Math. Soc. 129 (2001), 3701-3709).
Homogeneous spaces (non-symmetric and irreducible) with the property that every geodesic is contained in a k-flat ($k\geq 2$) can be construct... | 7 | https://mathoverflow.net/users/127807 | 308454 | 134,342 |
https://mathoverflow.net/questions/308451 | 4 | Let $(\gamma\_n)\_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum\_{n \in \mathbb{N}}\gamma\_n = + \infty \text{ and }\sum\_{n \in \mathbb{N}}\gamma\_n^2 < + \infty$$
I would like to know if the sequence $(u\_n)\_{n \geq 1}$ defined by
$$\forall n \in \mathbb{N},~u\_n = \sum\_{i=1}^n \gamma\_i \p... | https://mathoverflow.net/users/125712 | Convergence of a real sequence (stochastic approximation) | Yes, the sequence is bounded. As you already suggested yourself, it's useful to estimate the product by $\exp \left( -2\sum\_{k=j+1}^n \gamma\_k \right)$.
The statement is clearest in its continuous version, for integrals: then it's immediately obvious that
$$
\int\_1^N f(x) e^{-2\int\_x^N f(t)\, dt}\, dx = \frac{1}{... | 5 | https://mathoverflow.net/users/48839 | 308468 | 134,345 |
https://mathoverflow.net/questions/308239 | 10 | The question was motivated by [this question](https://mathoverflow.net/questions/307534/running-most-of-the-time-in-a-connected-set) of Anton Petrunin.
By a *metric continuum* we understand a connected compact metric space.
Let $p$ be a positive real number. A metric continuum $X$ is called *$\ell\_p$-almost path-c... | https://mathoverflow.net/users/61536 | Is every metric continuum almost path-connected? | Yes there are such examples.
Assume the sequence $\varepsilon\_n$ is very fast converging to $0$.
Consider a sequence of short $\varepsilon\_n$-crooked maps between intervals $\mathbb{J}\_n\to \mathbb{J}\_{n-1}$.
Its inverse limit is a pseudoarc $\mathbb{J}\_\infty$; denote by $\phi\_n\colon\mathbb{J}\_\infty\to \ma... | 3 | https://mathoverflow.net/users/1441 | 308471 | 134,348 |
https://mathoverflow.net/questions/308497 | 4 | The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious from the right point of view.
Let $H=L^2(-\infty,\infty)$ and let $S:H \supset {\rm dom}(S) \to H$ be the densely-defin... | https://mathoverflow.net/users/763 | Can this self-adjoint operator have an infinite-dimensional compression with compact inverse? | Sure, for instance let $P$ be the orthogonal projection onto the closed span of the characteristic functions $\chi\_{[n,n+1)}$ for $n \in \mathbb{N}$. You get property 1 because each of these functions is in the domain of $S$, and you get property 2 because, identifying $V$ with $l^2$ in the obvious way, the operator $... | 7 | https://mathoverflow.net/users/23141 | 308502 | 134,363 |
https://mathoverflow.net/questions/308458 | 3 | In this [reference](https://www.researchgate.net/profile/Roshdi_Khalil/publication/268629246_Tensor_Product_Semigroups/links/5616772308ae0f2140071ba8/Tensor-Product-Semigroups.pdf) the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are ... | https://mathoverflow.net/users/nan | Closure of tensor product /tensor product semigroup | So... It seems to me that the 1st claim is in Lemma 6 of the main paper. This actually references the following:
Ichinose, Takashi
Operators on tensor products of Banach spaces.
Trans. Amer. Math. Soc. 170 (1972), 197–219. [MR0322553](https://mathscinet.ams.org/mathscinet-getitem?mr=322553). [Available on the TAMS ar... | 2 | https://mathoverflow.net/users/406 | 308515 | 134,366 |
https://mathoverflow.net/questions/308514 | 2 | I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d\_2$ differentials of the spectral sequence (associated with a double complex). I know how to construct it using the big bad machinery of filtered complexes, but in t... | https://mathoverflow.net/users/1353 | Construction of differentials in the spectral sequence for double complexes | Write $d = d^v + d^h$ for the vertical and the horizontal differential in the double cochain complex $C^{\bullet,\bullet}$. Take $x \in E\_2^{pq}$ and lift it to an element $x' \in E\_1^{pq}$. Then $d\_1(x')=0$, which means precisely that if we lift $x'$ further to an element $x'' \in C^{pq}$ then $d^h(x'')$ is in the ... | 4 | https://mathoverflow.net/users/1310 | 308518 | 134,367 |
https://mathoverflow.net/questions/308507 | 1 | Let $p$ be a prime and let $\{A\_n\}\_{n > 0}$ be an inverse limit of abelian groups such that $A\_n$ is $p^n$-torsion with $A\_n/p^{n - 1} \cong A\_{n - 1}$ (these isomorphisms are part of the data). Let $A = \varprojlim A\_n$ be the inverse limit, which surjects onto each $A\_n$. Is it necessarily true that $A/p^n \c... | https://mathoverflow.net/users/63877 | Inverse limit of $p^n$-torsion abelian groups | See [Tag 09B8](https://stacks.math.columbia.edu/tag/09B8). Here are some more characters.
| 1 | https://mathoverflow.net/users/127835 | 308519 | 134,368 |
https://mathoverflow.net/questions/308511 | 6 | The question is to prove:
$$
\int\_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to prove this analytically but have all failed.
I also wonder if there is a systematic approach to solve this kind of probl... | https://mathoverflow.net/users/122571 | Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$ | Here is a proof of the inequality for $x\geq 2$. For the remaining range, see the Added section below.
Let $\lambda:=1-1/\sqrt{2}$, then by convexity we have
$$\frac{1}{\sqrt{1+s^2}}\leq 1-\lambda s^2,\qquad 0\leq s\leq 1.$$
Using this bound we can estimate
\begin{align\*}\int\_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}... | 7 | https://mathoverflow.net/users/11919 | 308534 | 134,376 |
https://mathoverflow.net/questions/308374 | 18 | Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^\*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some two-sided ideal of $A$ of finite codimension. Then $A^∗$ has a natural Hopf algebra structure.
Question: Is the Hopf ... | https://mathoverflow.net/users/125941 | Hopf dual of the Hopf dual | I am going to give three counterexamples to your first question. (The third counterexample is courtesy of @Adrien, who did most of the job.) While none of them leads to a full answer of your second question, at least they strongly restrict the possibilities.
1. The first counterexample: binate groups
================... | 12 | https://mathoverflow.net/users/2530 | 308546 | 134,379 |
https://mathoverflow.net/questions/308075 | 5 | Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The group $\Gamma$ of elements of order $2$ in $J$
acts on $PH^0(J^1, L\_\Theta^2)$ in a natural way; let $A$ be the associ... | https://mathoverflow.net/users/48420 | Confusion in known result about moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2 | EDIT: After I posted my answer, I realized that you were very close to answering your own question---the vector bundle you want is the one whose fiber over $\alpha \in {\rm Jac}(C)$ is $H^{0}(\mathcal{O}(2\Theta) \otimes \alpha)^{\vee}.$ However, this is equivalent via Strange Duality to what I have written below.
In... | 3 | https://mathoverflow.net/users/5496 | 308559 | 134,388 |
https://mathoverflow.net/questions/308558 | 24 | Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was looking for injective immersions or at least for immersions that do not factor through a covering onto a compact surfa... | https://mathoverflow.net/users/21123 | Immersions of the hyperbolic plane | Yes, it immerses isometrically into certain [solvmanifolds.](https://en.wikipedia.org/wiki/Solvmanifold) Take an Anosov map of $T^2$, such as $\left[\begin{array}{cc}2 & 1 \\1 & 1\end{array}\right]$. The mapping torus admits a locally homogeneous metric modeled on the [3-dimensional unimodular solvable Lie group](https... | 32 | https://mathoverflow.net/users/1345 | 308560 | 134,389 |
https://mathoverflow.net/questions/308565 | 3 | Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not canonical? If so, how bad can they be?
| https://mathoverflow.net/users/73650 | Singularities of a central fibre of a flat family of smooth surfaces | The cone over a plane curve of degree $d$ deforms to a smooth surface in $\mathbb P^3$ of degree $d$. Take $d\ge 5$ to see that things can be arbitrarily bad.
| 6 | https://mathoverflow.net/users/8726 | 308568 | 134,390 |
https://mathoverflow.net/questions/308562 | 2 | Question 1
==========
Given probability measures $\mu$ and $\nu$ on the same metric space $X=(X,d)$, and $\alpha \in [0, 1]$, is it always possible to find another probability measure $\lambda\_\alpha$ on $X$ such that $W\_1(\mu,\lambda\_\alpha) \le \alpha W\_1(\mu,\nu)$ and $W\_1(\nu,\lambda\_\alpha) \le (1 - \alpha... | https://mathoverflow.net/users/78539 | Wasserstein interpolation between two probability measures on a metric space | The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).
Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). Let $\gamma$ be the optimal transport plan between $\mu$ and $\nu$ under the $p$-Wasserstein distance. Denote by $\pi^i$ be ... | 6 | https://mathoverflow.net/users/42644 | 308570 | 134,392 |
https://mathoverflow.net/questions/308579 | 1 | Let $X$ be a compact metric space.
Let $\{X\_\alpha:\alpha\lt \mathfrak c\}$ be a partition of $X$ into $\mathfrak c=|\mathbb R|$ dense first category $F\_\sigma$-subsets of $X$.
Let $A$ be a non-empty closed subset of $X$ such that $A\cap X\_\alpha$ is first category in $A$ for each $\alpha<\mathfrak c$.
It is ... | https://mathoverflow.net/users/95718 | Quantity of partition sets intersecting a compact set | **Counterexample.**
Let $\{\alpha:\alpha\lt\mathfrak c\}=I\cup J$ where $I\cap J=\emptyset,\ |I|=|J|=\mathfrak c.$
Let $A=\{t\_\alpha:\alpha\in J\}$ be the Cantor ternary set; $t\_\alpha\ne t\_\beta$ for $\alpha\ne\beta$.
Let $S$ be a dense $G\_\delta$-subset of $[0,1]$ which has Lebesgue measure zero and is disj... | 1 | https://mathoverflow.net/users/43266 | 308583 | 134,395 |
https://mathoverflow.net/questions/308536 | 13 | I am interested the following element of the group algebra $\mathbb{Q}S\_n$:
\begin{align}
\phi\_n=2e+(1\ 2)+(1\ 2\ 3)+\dotsb+(1\ldots n)
\end{align}
where $e$ is the identity permutation. My question is whether $\phi\_n$ is a unit.
For small $n$ I can see numerically that $\phi\_n$ is a unit, but I have no idea how ... | https://mathoverflow.net/users/74448 | Is this sum of cycles invertible in $\mathbb QS_n$? | $\newcommand{\cyc}{\operatorname{cyc}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\BB}{\mathbf{B}}
\newcommand{\AA}{\mathbf{A}}
\newcommand{\kk}{\mathbf{k}}
\newcommand{\ww}{\mathbf{w}}
$
PART 1 OF 3
===========
[This is part of a long answer, which I had to split into 3 posts.
[Go to part 1](https://mathove... | 16 | https://mathoverflow.net/users/2530 | 308600 | 134,403 |
https://mathoverflow.net/questions/307303 | 0 | A subspace $Y$ of $X$ is said to be semi M-ideal if $\exists$ a projection $P$ (not necessarily linear) from $X^\*$ to $Y^\perp$ such that $\|x^\*\|=\|Px^\*\|+\|x^\*-Px^\*\|$. And also, $P(\lambda x^\*+Py^\*)=\lambda Px^\*+Py^\*$ , $\forall x^\*, y^\*\in X^\*$.
It is known that $ker(\mathbb{1})$ as a subspace of $\e... | https://mathoverflow.net/users/76412 | Why ker(1) is a semi M-ideal in $\ell_1$? | For $a=(a\_n) \in \ell\_\infty$ let $m(a) = (\sup a\_n + \inf a\_n)/2$. Then $a\mapsto P(a)=m(a) 1$ is the projection you're looking for.
Dirk
| 3 | https://mathoverflow.net/users/127871 | 308611 | 134,409 |
https://mathoverflow.net/questions/308609 | 1 | Let $X$ and $Y$ be compact metrizable spaces with $f:Y\rightarrow X$ an open surjection. Suppose that $G\subseteq Y$ is a closed set. How topologically complicated can the set $\{x\in X : f^{-1}(x)\cap G\text{ is clopen in }f^{-1}(x)\}$ be? Note that it's the same set if we replace clopen with open.
I can find exampl... | https://mathoverflow.net/users/83901 | Complexity of set of fibers on which a set is relatively clopen | If a map $f:Y\to X$ is an open surjection, then the inverse map $f^{-1}:X\to \mathcal K(Y)$ to the hyperspace $\mathcal K(Y)$ is continuous. The hyperpsace $\mathcal K(Y)$ is the space of nonpempty compact subsets endowed with the Vietoris topology.
So, your problem reduces to evaluating the Borel complexity of the ... | 1 | https://mathoverflow.net/users/61536 | 308628 | 134,418 |
https://mathoverflow.net/questions/308605 | 7 | During my research, I came across the following question.
Let $(f\_n)\_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that:
$\forall n\in\mathbb N, f\_n''<h$, where $h$ is locally integrable on $]0,1[$.
Is it true that $\lim \int\_0^1 f\_n=\int\_0^1 g$ ?
| https://mathoverflow.net/users/110301 | Dominated convergence 2.0? | **Counterexample.** Let $f: \mathbb{R} \to \mathbb{R}$ denote your favourite test function with support in $(0,1)$ and with integral $1$. We define $f\_n(x) := n f(nx)$ for all $n \in \mathbb{N}$ and all $x \in [0,1]$.
Then $f\_n(x) \to 0$ as $n \to \infty$ for all $x \in [0,1]$ and $\int\_0^1 f\_n = 1$ for all $n$. ... | 8 | https://mathoverflow.net/users/102946 | 308630 | 134,419 |
https://mathoverflow.net/questions/308578 | 4 | I have $n$ objects $O\_i$, each of them having $3$ values, $O\_i = (A\_i, B\_i, C\_i)$. I am trying to group them into $k$ groups $P\_u$ such as $P\_u =(A\_u, B\_u, C\_u)$ such that
$$\text{minimize} \quad \sum\_u^k M\_k \left( a A\_u + b B\_u + c C\_u \right)$$
for all $O\_i \in P\_u$, $A\_i \leq A\_u$, $B\_i \leq... | https://mathoverflow.net/users/127853 | A structural optimization problem | I think what is wanted is a k-means algorithm or something similar for optimizing the cost. Let's look at it in terms of packing books.
I am able to order k boxes for packing my (long) shelf of n books. Many aspects of my order have low cost, but one aspect which I want to optimize is space-height. So when I pack boo... | 1 | https://mathoverflow.net/users/3402 | 308634 | 134,420 |
https://mathoverflow.net/questions/206508 | 6 | I am looking for an electronic copy of this volume:
Advanced studies in Pure Mathematics, Volume 17
Algebraic Number Theory - in honor of K. Iwasawa
Edited by J. Coates, R. Greenberg, B. Mazur and I. Satake
August, 1989
Does anyone know where I can find it?
Thank you!
| https://mathoverflow.net/users/49492 | Looking for a copy of Algebraic Number Theory in honor of Iwasawa | You can download PDF files at project euclid <https://projecteuclid.org/euclid.aspm/>
for volumes 1 to 25.
| 7 | https://mathoverflow.net/users/127883 | 308638 | 134,422 |
https://mathoverflow.net/questions/308585 | 2 | In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This category is absolutely key because it massively influences the behaviour of the resulting homology theory.
In Definition 3.4 ... | https://mathoverflow.net/users/119240 | Coefficient (or target) category for factorization homology | One rich source of examples is that every combinatorial symmetric
monoidal model category gives rise to such a category V.
In particular, this covers all the examples in the main post,
including cdgas, which do form such a model category with
tensor product as the monoidal structure.
| 2 | https://mathoverflow.net/users/402 | 308644 | 134,426 |
https://mathoverflow.net/questions/308631 | 6 | Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$
where
$$\operatorname{Ri}(x) = \sum\_{n = 1}^\infty \frac{\mu(n)}{n} \operatorname{li}(x^{1/n}) = 1 + \sum\_{k = 1}^\infty \frac{(\log x)^k}{k \cdot k!\ \zeta(k+1)}$$
for all $x > 0$? If so, how ca... | https://mathoverflow.net/users/17218 | asymptotic for li(x)-Ri(x) | Yes, the stated asymptotics (and much more) is true. The idea is to truncate $\operatorname{Ri}(x)$ appropriately.
Let us use the series representation (see [here](https://en.wikipedia.org/wiki/Logarithmic_integral_function#Series_representation))
$$\operatorname{li}(t)=\gamma+\log\log t+\sum\_{k=1}^\infty\frac{(\lo... | 6 | https://mathoverflow.net/users/11919 | 308646 | 134,427 |
https://mathoverflow.net/questions/308627 | 6 | I'm having trouble parsing a definition in Lurie's "Rotation Invariance in Algebraic $K$-Theory". The definition os for the notion of center of an associative algebra object, and occurs in Remark 2.1.3.
The setting is as follows. We have a symmetric monoidal $\infty$-category $\mathcal{C}$. We write $\mathrm{Alg}(\ma... | https://mathoverflow.net/users/94624 | Parsing the definition of center of an algebra in a higher-categorical setting | Let us try to figure out what's happening on *discrete* rings, where $E\_2=E\_\infty$. The category $\mathrm{LMod}^{(2)}$ is, as you surmised, the category of pairs $(A,B)$ where $A$ is a commutative algebra and $B$ is an associative $A$-algebra (i.e. an algebra object in the monoidal category of $A$-modules). That is ... | 4 | https://mathoverflow.net/users/43054 | 308651 | 134,430 |
https://mathoverflow.net/questions/308655 | 10 | It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric description of the projection to the sphere?
This manifold is actually a complex algebraic variety (namely a plane with one poin... | https://mathoverflow.net/users/9833 | Geometric description of a certain sphere bundle | Yes. If $p\in\mathbb{CP}^2$ is a point, you can consider the blowup $X\_p$ of $\mathbb{CP}^2$ at $p$ as the space of pairs $(L,q)$ such that $L\subset\mathbb{CP}^2$ is a line passing through $p$ and $q\in L$ is any point. Now let $M\subset\mathbb{CP}^2$ be any line *not* passing through $p$. Then one can define a map $... | 14 | https://mathoverflow.net/users/13972 | 308658 | 134,432 |
https://mathoverflow.net/questions/308652 | 5 | Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$
A family $\mathcal N$ of subsets of $\omega$ is called a *network* for $\mathcal F$ if for any $F\in\mathcal F$ and $E\in\mathcal F^+$ there exists a set $N\in\mathcal N$ such that $N\sub... | https://mathoverflow.net/users/61536 | On filters possessing a countable network | Maybe I posed this question too quickly: for the 6 hours that passed since the time of asking this question I have found a (relatively simple) counterexample to my Problem 2.
**Example.** There exists a non-diagonalizable free filter with countable network on a countable set.
*Proof.* Consider the space $X=2^{<\ome... | 0 | https://mathoverflow.net/users/61536 | 308680 | 134,443 |
https://mathoverflow.net/questions/308683 | 4 | I have a question about a property of Lipschitz domain.
>
> Let $D \subset \mathbb{R}^d$ be a bounded domain (connected open subset ). $D$ is called **a
> bounded Lipschitz domain** if there exist positive constants $\delta$,
> $M$ such that for each $x\_0 \in \partial \Omega$ there exist a
> neighborhood $U\_{x... | https://mathoverflow.net/users/68463 | A property of Lipschitz domains | This is probably somewhat over-the-top, but anyway: The nice paper [1] by Hajlasz, Koskela and Tuominen says that your desired inequality is true for Sobolev extension domains, so for domains $D$ for which there exists a continuous linear operator $E \colon W^{1,p}(D) \to W^{1,p}(\mathbb{R}^n)$ such that $(Eu)\_{\restr... | 3 | https://mathoverflow.net/users/85906 | 308688 | 134,445 |
https://mathoverflow.net/questions/308686 | 5 | After this question : [Dominated convergence 2.0?](https://mathoverflow.net/questions/308605/dominated-convergence-2-0)
I want to know, what about the case when $h\in L^1([0,1])$.
The completed question :
Let $(f\_n)\_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$ and $\forall x \in [... | https://mathoverflow.net/users/110301 | Dominated convergence 2.1? | I claim that under these assumptions, the functions $f\_n$ are uniformly bounded. Then the conclusion follows from the dominated convergence theorem.
First set $H(x) = \int\_0^x \int\_0^t h(s)\,ds$, which is $C^1$. Letting $u\_n = f\_n-H$, we have that $u\_n$ is concave (i.e. $-u\_n$ is [convex](https://en.wikipedia.... | 3 | https://mathoverflow.net/users/4832 | 308700 | 134,450 |
https://mathoverflow.net/questions/308350 | -1 | Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp\_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$.
Is there any reference for such kind of results?
| https://mathoverflow.net/users/19114 | $A[x]$ points of an algebraic group | The comments show that the answer to the question is negative if $G$ contains a copy of $\mathbb G\_a$. The examples $O(n)$ and $Sp(n)$ in the question suggest, though, that the emphasis is on anisotropic $G$. It is interesting that in this case the answer is affirmative to some extent.
Let, e.g., $K=\mathbb R$, let... | 3 | https://mathoverflow.net/users/89948 | 308704 | 134,452 |
https://mathoverflow.net/questions/307766 | 5 | Let $k$ be a field of characteristic $0$.
There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$, and a functor $P$ from the category of Hopf algebras over $k$ to the category of $k$-Lie algebras sendin... | https://mathoverflow.net/users/30211 | The Ungraded Milnor-Moore Theorem | The "ungraded" version of the theorem -which is actually the version for the Hopf algebras- can be found in most of the classical references on the subject, although its statement and proof appears scattered among paragraphs or several different sections. For example see:
* Sweedler's book: Hopf algebras, Theorem 8.... | 5 | https://mathoverflow.net/users/85967 | 308711 | 134,454 |
https://mathoverflow.net/questions/300302 | 4 | Let $T$ be a conservative measure preserving (non-invertible!) transformation of a measure space $(X, \mathscr{F}, m)$ with *infinite* measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup\_{k=0}^\infty T^{-k} A \pmod{m}$ and $0<m(A)<\infty$. Then the first hitting time $\tau(x):= \inf\{k \ge 1: T^k x \in A\}$ is... | https://mathoverflow.net/users/116098 | Lifting back the induced invariant measure / general version of Kac's formula for occupation times | I was not able to find a reference and eventually proved the statement by myself, see Lemma A.3 (in the Appendix) of my paper <https://arxiv.org/abs/1808.05010>
The idea of the proof reminds that of Aaronson's.
| 0 | https://mathoverflow.net/users/116098 | 308720 | 134,458 |
https://mathoverflow.net/questions/308726 | 0 | We write $A\subseteq^\* B$ if $A\setminus B$ is finite.
Let $(A\_n)\_{n\in\omega}$ be a sequence of subsets of $\omega$ such that for all $n\in\omega$ we have $A\_n \subseteq^\* A\_{n+1}$ and $A\_{n+1}\not\subseteq^\* A\_n$.
Let $D\_n:= A\_{n+1}\setminus A\_n$ for all $n\in\omega$. Is it possible that $$\bigcup\_{n... | https://mathoverflow.net/users/8628 | Can the union of difference sets in towers equal $\omega$? | Take any increasing tower, but then modify it by adding all the numbers below $n$ to $A\_n$, when $n$ is even, and removing them when $n$ is odd. This is a finite change to each set in the tower, and so it doesn't affect any $\subseteq^\*$ relation, but now every number will eventually appear in the difference sets, ju... | 2 | https://mathoverflow.net/users/1946 | 308727 | 134,459 |
https://mathoverflow.net/questions/306273 | 8 | Is the following proposition correct?
$X\_1, X\_2, X\_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement.
$f : \mathcal X^2 \rightarrow \mathbb R\_{\ge0}$ is symmetric:
$
f(x, y) = f(y, x)
$, then:
$$
\mathbb E\_{X\_1, X\_2, X\_3} f(X\_1, X\_2) f(X\_1, X\_3) f(X\_2, X\_3)
\le ( \... | https://mathoverflow.net/users/126729 | Expectation inequality for sampling without replacement | Let $\lambda\_1,\dots,\lambda\_n$ be eigenvalues of the symmetric matrix $(f\_{ij})$, where $f\_{ii}=0$ by definition. They are real, $\sum \lambda\_i=0$ and the inequality rewrites as
$$
\left(\frac{\sum \lambda\_i^3}{n(n-1)(n-2)}\right)^2\leqslant
\left(\frac{\sum \lambda\_i^2}{n(n-1)}\right)^3,
$$
or $(\sum \lamb... | 2 | https://mathoverflow.net/users/4312 | 308729 | 134,460 |
https://mathoverflow.net/questions/308717 | 18 | Let $X$ be a smooth compact 4-manifold. Then every element of $H\_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H\_2(X; \mathbb{Z}) \to \mathbb{Z}\_{\geq 0}$ which assigns to a homology class the smallest genus of such a smooth surface needed to... | https://mathoverflow.net/users/99414 | Behavior of genus function on a 4-manifold for sums | In the case that $x\cdot x \neq 0$, topological methods based on the G-signature show that the genus goes to infinity more or less quadratically in $n$. (I'll be more specific below.) This goes back to Rochlin (Two-dimensional submanifolds of four-dimensional manifolds) and Hsiang-Szczarba (On embedding surfaces in 4-m... | 11 | https://mathoverflow.net/users/3460 | 308731 | 134,461 |
https://mathoverflow.net/questions/308151 | 2 | Let $X$ be a subset of $\{0,1\}^\*$ with the following property: for every pair of distinct strings $x\_1$, $x\_2$ from $X$
$x\_1$ is not a substring of $x\_2$ and $x\_2$ is not a substring of $x\_1$.
How much can be the log-density of $X$,
i.e. $\lim \frac{ \log X\_n}{n}$, where $X\_n$ is the cardinality of all... | https://mathoverflow.net/users/31356 | Set of strings $S$ such that no string from $S$ is a substring of another one | I'm going to use $\log\_2$ instead of $\log$ throughout; this only results in constant factor change (and the largest possible answer becomes $1$, which is convenient).
Fedor Petrov's answer implies that $\lim \frac{\log\_2 X\_n}{n} < 1$ for any $X$. We can, however, construct a family $X$ with $\lim \frac{\log\_2 X\... | 3 | https://mathoverflow.net/users/106512 | 308769 | 134,473 |
https://mathoverflow.net/questions/308300 | 34 |
>
> Here is a revised version: [On a revised quantum Riemann
> hypothesis](https://mathoverflow.net/q/364311/34538).
>
>
>
---
[Robin's theorem](https://en.wikipedia.org/wiki/Robin%27s_theorem) (1984) states that
$$ \sigma(n) < e^\gamma n \log \log n$$
for all $n > 5040$ if and only if the [Riemann hypothe... | https://mathoverflow.net/users/34538 | On a quantum Riemann Hypothesis | Can you clarify whether there exists a notion of direct product in this setting with the desired properties?
If so, the asymptotics you are predicting only seem consistent with the hypothesis that there are no such objects besides groups. (I guess they do actually exist or you wouldn't ask this question.) The first ... | 6 | https://mathoverflow.net/users/127955 | 308781 | 134,480 |
https://mathoverflow.net/questions/308450 | 7 | A paradox:
* Goodwillie calculus considers only finitary functors.
* $TC$ isn't finitary.
* Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(Here, a finitary functor is one preserving filtered colimits[1]. $\partial$ denotes the first Goodwillie derivative... | https://mathoverflow.net/users/2362 | What does it mean to say the first Goodwillie derivative of $TC$ is $THH$? | This answer addresses Question 1.
Let "ring" mean associative unital ring spectrum, say in the $A^\infty$ sense. For a functor $F$ from rings to spectra (such as $TC$), differentiating $F$ at the ring $R$ means finding the best excisive approximation to the functor from rings-having-$R$-as-a-retract to spectra,
$$
... | 12 | https://mathoverflow.net/users/6666 | 308783 | 134,481 |
https://mathoverflow.net/questions/308785 | 0 | Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K\_{i}\right\} \_{i=1}^{M}, U\_{i}D\_{i}U\_{i}^{T}=K\_{i},\quad i=1,\dots,M,$$ is there any $O(N^{2})$ method for computing the eigenvalues of a given convex combination$$\mathcal{D}^{T}=\mathca... | https://mathoverflow.net/users/97437 | Computing spectrum of convex combination of SPD matrices given individual spectral decompositions | No, there isn't such a method. It's difficult to find a reference that "proves a negative", but I can tell you that many people in my field (numerical linear algebra) would be very happy to know about one. :)
A partial argument to convince you could be: if this were possible, then you could write for any $2n\times 2n... | 4 | https://mathoverflow.net/users/1898 | 308790 | 134,486 |
https://mathoverflow.net/questions/308779 | 5 | Let $X\to \Delta$ be a projective family, smooth over $\Delta^\*$, such that all fibers over $t\in \Delta^\*$ are isomorphic. Does the monodromy representation factor through the algebraic automorphism group of the smooth fiber, $Aut(X\_t)$?
This is certainly false for non-isotrivial families, since Dehn twists on cu... | https://mathoverflow.net/users/30554 | Isotrivial Monodromy | I think the answer is yes, depending on how you are defining the monodromy\*. I take "isotrivial" to mean that you have a smooth fibre bundle over the punctured plane where the fibres have complex structures and any two fibres are biholomorphic. Take the covering space of $X/X\_0$ corresponding to the subgroup of $\pi\... | 3 | https://mathoverflow.net/users/10839 | 308794 | 134,488 |
https://mathoverflow.net/questions/308793 | 0 | The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2\_{min}I)^{-1}A^Hb$$
where $\sigma^2\_{min}$ is the minimum of the singular values of $A$.
Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma V^H$$
then the solution is,
$$x=(V\Sigma^2 V^H+\sigma^2\_{min} I)^{-1}V\Sigma U^Hb$$
But on the textbook, it says ... | https://mathoverflow.net/users/127575 | How to derive the solution of Tikhonov Regularization via SVD | No need for the Woodbury identity. Just replace $I$ with $VV^H$, and factor out the $V$s.
| 3 | https://mathoverflow.net/users/1898 | 308803 | 134,495 |
https://mathoverflow.net/questions/308761 | 12 | A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called
$\bullet$ *minimal* if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$;
$\bullet$ *minimizable* if $\mathcal C$ contains a minimal subcover of $X$.
For example, any cover of the plane by parallel lines is minimal... | https://mathoverflow.net/users/61536 | Is each cover of the plane by lines minimizable? | Unfortunately, you have two questions in one post. The one about $\mathbb R^2$ is too hard for me. The question about $\mathbb Q^2$ seems to have an easy affirmative answer, unless I'm making some dumb mistake.
Let $P=\{p\_0,p\_1,p\_2,\dots\}$ be the set of points, and let $L$ be the set of lines. (In general, the c... | 9 | https://mathoverflow.net/users/43266 | 308807 | 134,497 |
https://mathoverflow.net/questions/308805 | 5 | The following question is extracted from [this question on MSE](https://math.stackexchange.com/questions/2886186/are-these-two-definitions-stably-isomorphic-modules-k-0a-equivalent/), which got no answer so far, probably because it was a bit hidden by another question which a posteriori was totally obvious.
Let $A$ b... | https://mathoverflow.net/users/36683 | Are these two constructions of $K_0(A)$ isomorphic? | The two constructions give the same result, namely the universal group with a monoid homomorphism from $Proj(A)$ aka the Grothendieck group. The first definition gives this because a short exact sequence with $M\_3$ projective always splits so that the relations are simply "$[M\_2] = [M\_1]+[M\_3]$ in the group wheneve... | 7 | https://mathoverflow.net/users/3041 | 308820 | 134,506 |
https://mathoverflow.net/questions/308821 | 5 | Let $(M, g)$ be a (complete) Kähler manifold with Ricci curvature $\geq c$.
Is it true that the volume ratio of geodesic balls in $M$ with respect to balls in the corresponding (simply connected) complex space form with Ricci $\equiv c$ is a decreasing function?
I suspect the answer is no. Examples would be nice, ... | https://mathoverflow.net/users/127247 | Bishop-Gromov for Kähler metrics | It looks like the answer is indeed no with the quadric $\mathbb CP^1\times \mathbb CP^1$ a counterexample. Here the corresponding complex model space is $\mathbb CP^2$.
Recall that to get an Einstein metric with coefficient $\lambda=1$ we should choose it is as curvature of the anti-canonical bundle $-K$. Now, for $... | 3 | https://mathoverflow.net/users/943 | 308833 | 134,512 |
https://mathoverflow.net/questions/308857 | 0 | Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\prime }\times K^{\prime }$
?.\ In the case that the answer is "no": Are there any reasonable constraints to $H$ and $K$ s... | https://mathoverflow.net/users/123061 | Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? | No. Let $C\_p$ (the cyclic group of order $p$) act on $C\_p^p$ by permuting the basis vectors $e\_1$, $e\_2$, \dots, $e\_p$. Writing $\sigma$ for a generator of the $C\_p$ that acts, we have $(\sigma, e\_1)^p = (0,e\_1+e\_2+\cdots+e\_p) \neq (0,0)$ and $(\sigma,e\_1)^{p^2} = (0,0)$, so the subgroup generated by $(\sigm... | 4 | https://mathoverflow.net/users/297 | 308859 | 134,523 |
https://mathoverflow.net/questions/308845 | 1 | I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?
| https://mathoverflow.net/users/36886 | Intuition for coercive functions | Coercive function, where I have met such things, is one that grows sufficiently fast as the absolute value of its argument grows.
For example: A function $f$ from a normed space $X$ to real numbers might be called coercive iff $\lim\_{|x| \to \infty } f(x) = \infty$.
This means that the function eventually grows to i... | 3 | https://mathoverflow.net/users/1445 | 308876 | 134,528 |
https://mathoverflow.net/questions/308856 | 20 | A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m\_{\*}(E)$ and outer measure $m^{\*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure.
A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesgue int... | https://mathoverflow.net/users/nan | Why is Lebesgue measure theory asymmetric? | I have a (possibly idiosyncratic) view that the natural form of measure theory is for finite measure spaces and bounded functions. Other cases are obviously very important, but we have to work harder to get them. You can see this is many of the proofs, where the finite case is easier, and we have to work a bit more to ... | 21 | https://mathoverflow.net/users/3711 | 308888 | 134,532 |
https://mathoverflow.net/questions/308872 | 3 | This is a follow-up of [this question](https://math.stackexchange.com/questions/2889556/continuity-of-the-kernel-of-bounded-operators-under-perturbation).
>
> In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
>
>
>
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed ve... | https://mathoverflow.net/users/46290 | Is the kernel of a Fredholm operator stable under perturbation? | The kernel of a Fredholm operator is not continuous with respect to small norm perturbations: For $t\geq 0$, consider the operator $S\_t:X\times Y \to X\times Y$ defined by $S(x,y)=(tx,y)$ where $X,Y$ are Banach spaces and $X$ is finite dimensional.
However, when the operator $T:X\to Y$ has closed range, you have se... | 1 | https://mathoverflow.net/users/39421 | 308890 | 134,534 |
https://mathoverflow.net/questions/192257 | 2 | I'm interested in a possible generalization of [Tiling relation on the set of partitions](https://mathoverflow.net/questions/192203/tiling-relation-on-the-set-of-partitions) (the question has only been partially answered).
Let $x$ be an infinite set and let $\text{Part}(x)$ be the collection of
all partitions of $x$.... | https://mathoverflow.net/users/8628 | Optimal tiling for a collection of partitions | 1) This question (as posed) has a simple negative answer. Just take any countable set $x$ and put $M$ be the family of finite subsets of $x$. Then for the family $\mathcal A$ of all possible partitions of $x$ into finite subsets there is no partition $Z$ of $x$ into finite sets such that $Z\triangleleft A$ for every $A... | 1 | https://mathoverflow.net/users/61536 | 308891 | 134,535 |
https://mathoverflow.net/questions/308900 | 1 | Consider the following matrix
$$
A=\left[
\begin {array}{cccc}
1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0
\end {array}
\right].
$$
Assume that $B=A^k$ for some positive integer $k$.
*My Question:*
How to prove there is no $k$ such that
all entries of $B$ are odd numbers.
In terminology of graph theory, we shoul... | https://mathoverflow.net/users/124008 | Walks of odd Lengths in a Matrix | This matrix is invertible modulo 2, thus so is each its power, but all-ones matrix is singular.
| 8 | https://mathoverflow.net/users/4312 | 308903 | 134,539 |
https://mathoverflow.net/questions/308880 | 12 | Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will be great if it will be mentioned the key ideas and people of the past, the problems which are to be solved nowadays and ... | https://mathoverflow.net/users/61629 | History of ODE and PDE reference request | Another useful reference on the history of PDE theory is ["The Prehistory of The Theory of Distributions"](https://www.springer.com/us/book/9781461394747) by Jesper Lützen.
| 1 | https://mathoverflow.net/users/7410 | 308909 | 134,542 |
https://mathoverflow.net/questions/308782 | 3 | I asked a very similar question [here](https://mathoverflow.net/q/308340/22512) and got a wonderful answer. But now I need to change the question slightly (this is the last question like this, I promise).
I would like to characterize when $(\mathbb{Z}/p\mathbb{Z})^n \rtimes (\mathbb{Z}/q\mathbb{Z})^2$ is a centerless... | https://mathoverflow.net/users/22512 | When is the semidirect product of $(Z/pZ)^n$ and $(Z/qZ)^2$ generated by two elements? | Here is a proof of YCor's claim that the group is $2$-generated if and only if $V:=(Z/pZ)^n$ is isomorphic as a $(Z/qZ)^2$-module to a direct sum of distinct nontrivial irreducible modules.
For the "only if" part it is sufficient to consder the case when $V = W \oplus W$ is a direct sum of two isomorphic modules, and... | 6 | https://mathoverflow.net/users/35840 | 308916 | 134,545 |
https://mathoverflow.net/questions/308819 | 8 | How can we use elementary methods to prove that
$$\sum\_{i = 2}^{n}{{n \choose i} i! n^{n - i}} = \sum\_{i = 1}^{n - 1}{{n \choose i}i^i (n - i)^{n - i}}$$
for any integer $n \geq 0$?
The values of each side for fixed $n$ are 0, 0, 2, 24, 312, 4720, ... ([A001864 - OEIS](https://oeis.org/A001864)).
| https://mathoverflow.net/users/98438 | Proof of a combinatorial equation | Everything is already contained in OEIS comments for [A001864](http://oeis.org/A001864) and [A000435](http://oeis.org/A000435) (a remarkable comment is that A000435 *is the sequence that started it all: the first sequence in the database!*)
We take $n$ labelled vertices, consider all trees on them, and sum up the dis... | 12 | https://mathoverflow.net/users/4312 | 308918 | 134,547 |
https://mathoverflow.net/questions/308912 | 23 | Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
>
> Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
>
>
>
I am interested in knowing the answer for closed manifolds.
One example which I can think of is an exotic $\mathbb R... | https://mathoverflow.net/users/33064 | What can we say about the Cartesian product of a manifold with its exotic copy? | Your question seems to be about simply connected exotic 4-manifolds, in which the answer is yes. That's because $M$ and $M^E$ are h-cobordant (by Wall), say via an h-cobordism W. Then $M \times W$ is an h-cobordism between $M \times M$ and $M\times M^E$, which is trivial by the high-dimensional h-cobordism theorem.
... | 21 | https://mathoverflow.net/users/3460 | 308929 | 134,550 |
https://mathoverflow.net/questions/308925 | 3 | Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad.)
Then cover $M$ by open sets $\cup\_iU\_i=M$.
In a local coordinate chart, $(U\_i,\phi\_i)$, where $\phi\_i:U\_i\ri... | https://mathoverflow.net/users/128036 | non-existence of global coordinates | One particular condition, to give some examples, arises from the [Cartan-Hadamard theorem](https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem): if a simply connected manifold admits a complete metric of nonpositive sectional curvature, then it is diffeomorphic to a ball, so admits global coordinates.
Anoth... | 7 | https://mathoverflow.net/users/13268 | 308936 | 134,552 |
https://mathoverflow.net/questions/308938 | 1 | Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp\_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle\_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp\_{H^{-1}}}:=\left\{ \xi \in H^{-1}; \langle \widehat{\xi}, \widehat{x} \rangle\_{H^{-1},H^1} =0 \ \forall x \in X \right... | https://mathoverflow.net/users/128044 | Orthogonal complement vector space | Take $d=1$. In this case all functions in $H^1(\mathbb{R})$ are (absolutely) continuous, so evaluation at a point is well defined . Let $X = \{f \in H^1 : f(0) = 0\}$ which is a well-defined closed subspace of $H^1$ with codimension 1, but is dense in $L^2$. So $X^{\perp\_{L^2}} = 0$ but $X^{{\perp}\_{H^{-1}}}$ is one-... | 2 | https://mathoverflow.net/users/4832 | 308939 | 134,554 |
https://mathoverflow.net/questions/308835 | 4 | I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute value less than 1 for certain choices of $N$.
Note that the explicit formula for entry $(i,j)$ is $-i(N+1-j)$ when $... | https://mathoverflow.net/users/67016 | How to find the analytical representation of eigenvalues of the matrix $G$? | It's straightforward to show that this is the inverse of $1/(N+1)$ times the tridiagonal matrix $T\_N$ with $-2$ on its main diagonal and $1$ on its super- and sub-diagonals.
Let $t\_N$ be the characteristic polynomial of $T\_N$. We have $t\_0(x)=1$, $t\_1(x)=x+2$, and by cofactor expansion $t\_N(x)=(x+2)t\_{N-1}(x)-... | 4 | https://mathoverflow.net/users/112641 | 308940 | 134,555 |
https://mathoverflow.net/questions/308746 | 1 | I am trying to solve for K in the following problem:
$ 3I = A\_1 + A\_2 + A\_3$
$ A\_1 K A\_1 = K\_1 $
$ A\_2 K A\_2 = K\_2 $
$ A\_3 K A\_3 = K\_3 $
Where $I$ is the identity, $K, K\_1, K\_2, K\_3, A\_1, A\_2, A\_3$ are known to be symmetric and positive definite. $K, A\_1, A\_2, A\_3$ are unknown. $K\_1, K\_... | https://mathoverflow.net/users/127940 | How to solve this system of Matrix equations? (Coupled riccati equations?) | Not sure if I misunderstood something, but the following seems to work. Let us change the unknown positive-definite variables from the tuple $(A\_{1},A\_{2},A\_{3},K) \mapsto (X\_{1},X\_{2},X\_{3},K)$ where $X\_{i} := A\_{i}K^{1/2}$ for $i=1,2,3$ (all positive definite matrices have unique pos. def. square root).
The... | 1 | https://mathoverflow.net/users/18526 | 308942 | 134,557 |
https://mathoverflow.net/questions/308951 | 3 | Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious enriched full embedding of the subcategory of fibrant objects $\mathbf{S}^\circ \hookrightarrow \mathbf{S}$.
Since $\... | https://mathoverflow.net/users/1353 | Excellent monoidal model categories admit enriched fibrant replacement functors? | I think the answer is yes (to both questions). In Emily Riehl's book "Categorical homotopy theory", chapter 13 is all about the enriched small object argument. Theorem 13.2.1 on page 177 (I hope I'm looking at a version close to the final one) explains when the enriched small object argument works. A condition Riehl ca... | 4 | https://mathoverflow.net/users/11540 | 308973 | 134,562 |
https://mathoverflow.net/questions/308260 | 8 | I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given a representative cocycle of $H^2(Q,N)$ corresponding to such an extension. I will use the example of $\mathbb Z\_4$, wh... | https://mathoverflow.net/users/125997 | Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence | The action of the quotient on the cohomology groups of the normal subgroup is the trivial action, because the normal subgroup is central. (Think of group cohomology as a functor of the group: the conjugation action of $Q$ on $N$ induces the action of $Q$ on $H^\*(N;\mathbb{Z})$.)
What happens for the spectral sequen... | 5 | https://mathoverflow.net/users/124004 | 308983 | 134,565 |
https://mathoverflow.net/questions/308989 | 19 | I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the matrix M whose entry $M\_{i,j}$ (with $0\leq i,j\leq deg(p)-1$) is the coefficient of $X^iY^j$ in the polynomial $$\frac{P... | https://mathoverflow.net/users/61910 | Counting real zeros of a polynomial | By Remark 9.21 page 340 of the [book by Basu, Pollack and Roy on real algebraic geometry](https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.pdf) the matrix $H$ is the expansion of the [Bezoutiant](https://arxiv.org/abs/math/0406410) of $P$ and $P'$ in the Horner basis of $P$ instead of the basis of usua... | 13 | https://mathoverflow.net/users/7410 | 308995 | 134,569 |
https://mathoverflow.net/questions/306572 | 1 | Let $M$ be an $n\times m$ matrix, say with entries in $\left\{0,1\right\}$ ; and let $\mathcal C(M)$ be the $n\times m$ matrix such that there exists $P$, $m\times m$ permutation matrix such that $M.P=\mathcal C(M)$ and such that the columns of $\mathcal C(M)$ are *lexicographically increasing* (1) (for a formal defini... | https://mathoverflow.net/users/112382 | order of a permutation and lexicographic order | There are counterexamples for $Q=J$.
Here is a binary $6\times7$ binary matrix $M$ that belongs to an orbit of $\mathcal{L}\_J$ with period $3$:
$\begin{matrix}
1&1&1&0&0&0&0\\
1&1&0&1&1&0&0\\
0&0&1&1&0&1&0\\
0&0&1&0&1&0&1\\
1&0&0&1&0&1&1\\
0&1&0&0&1&1&1\\
\end{matrix}$
Here is another with period $4$:
$\begin... | 1 | https://mathoverflow.net/users/127616 | 308996 | 134,570 |
https://mathoverflow.net/questions/308267 | 3 | Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?
Edit: My goal is to understand a bit better what the intuition behind compactness / non-compactness of the isometry of a closed Semi-Riemannian manifold is. For example are there topo... | https://mathoverflow.net/users/99468 | Closed Semi-Riemannian manifolds with non-compact isometry group | Of course there is D’Ambra's 1988 paper "Isometry groups of Lorentz manifolds", from which the theorem you state is taken.
A later paper, taking a more general perspective, is [this one](http://www.ihes.fr/~/gromov/PDF/1[74].pdf) by D’Ambra and Gromov from 1991.
Zimmer's school had its impact then, with Kowalski's th... | 4 | https://mathoverflow.net/users/89334 | 309024 | 134,578 |
https://mathoverflow.net/questions/309019 | 53 | In my research I came across the following question :
>
> Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x\_n)\_n$, taking infinitely many values, converging to some real number $c$, such that the sequence $(f(x\_n))\_n$ converges to $f(c)$ ?
>
>
>
| https://mathoverflow.net/users/110301 | Does every real function have this weak continuity property? | Suppose $f$ were a counterexample. Then, for any $c$, we could find a little interval $(a,b)$ containing $c$ and we could find some $\varepsilon>0$ such that all points $x\in(a,b)$ except $c$ have $|f(x)-f(c)|>\varepsilon$. (Otherwise, by taking smaller and smaller intervals and $\varepsilon$'s, we could produce a sequ... | 64 | https://mathoverflow.net/users/6794 | 309025 | 134,579 |
https://mathoverflow.net/questions/309018 | 2 | Let $X$, $Y$ be sets and let G, H be groups which act on $X$, $Y$ (respectively). Denote the set of functions from $X$ to $Y$ by $X^Y$. I will use $f$ for functions in $X^Y$ and $g, h$ for elements of $G, H$, respectively.
As I understand it, the Polya Enumeration Theorem allows us to count the number of orbits of fu... | https://mathoverflow.net/users/128120 | Is there a generalisation of the Polya Enumeration Theorem to actions of wreath products? | A good starting point with some useful references is
[Enumeration under two representations of the wreath product](https://link.springer.com/content/pdf/10.1007/BF02392038.pdf) by Palmer and Robinson.
| 2 | https://mathoverflow.net/users/7076 | 309028 | 134,581 |
https://mathoverflow.net/questions/308920 | 0 | Edit: the question was answered to the negative because $ZF$ proves the existence of Hartog numbers. So this calls for a modification of the question to be in just $Z-\text{Regularity}$
Is it consistent with $Z-\text{Regularity}$ [instead of $ZF-\text{Regularity}$ in the original question] to have a set that is stric... | https://mathoverflow.net/users/95347 | Is it consistent with Z - Regularity to have a set that is bigger than any set in the cumulative hierarchy of Z? | I think I have the answer to this question, the general idea I got from Noah Schweber's prior answer.
It is consistent with ZF-Regularity to have a class $Q$ of Quine atoms of any size, even we can have a proper class of Quine atoms, so let $$|Q|=|V\_{\omega+\omega}|$$, we simply construct the set $V^Q\_{\omega+\ome... | 0 | https://mathoverflow.net/users/95347 | 309042 | 134,585 |
https://mathoverflow.net/questions/308992 | 7 | An answer to this question would also answer [Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms](https://mathoverflow.net/questions/301236/isotopy-of-periodic-homeomorphisms-of-a-surface-along-periodic-homeomorphisms)
Let $M$ be a topological manifold and let $f,g$ be two orientation prese... | https://mathoverflow.net/users/43097 | Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$? | No: Let $M$ be the 2-dim. surface consisting of tori welded to each other so that they form a string (sorry, I do not know how to draw here). Let $f$ be the the time 1 flow of a vector field supported in a small part of the $i$-torus, and let $g$ be the time 1 flow of the same vector field, but now supported in the $i+... | 4 | https://mathoverflow.net/users/26935 | 309048 | 134,589 |
https://mathoverflow.net/questions/309036 | 13 | 2d topological field theories $Z : \mathrm{Cob}(2) \to \mathrm{Vect}$ are classified by commutative Frobenius algebras.
What can be said about $(\infty,1)$ 2d TFTs $Z: \mathrm{Cob}(2) \to \mathcal{S}$ with values in a symmetric monoidal $(\infty,1)$-category $\mathcal{S}$? I am interested in different targets $\mathc... | https://mathoverflow.net/users/119240 | $(\infty,1)$ 2d TFTs | As you point out, if you just look at the operad of bordisms with exactly one output disc, you get the *framed* $E\_2$-operad (framed here means you can rotate the discs) and so the value of the circle is a framed $E\_2$ algebra. (Aside: if we were working in categories an $E\_2$-algebra is a braided monoidal category,... | 8 | https://mathoverflow.net/users/22 | 309053 | 134,590 |
https://mathoverflow.net/questions/308665 | 7 | Proposition A.3.3.9. in Higher Topos Theory is as follows:
>
> Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every combinatorial $S$-enriched model category $A$, the pullback $f^\*:A^{C'}\rightarrow A^C$ preserves projective cofibr... | https://mathoverflow.net/users/51424 | Proposition in HTT on cofibrations of categories | You can argue as follows. Suppose that $g: D \to D'$ is a retract of $f: C \to C'$ (in the category of $S$-enriched categories) via maps $D \stackrel{i}{\to} C \stackrel{r}{\to} D$ and $D' \stackrel{i'}{\to} C' \stackrel{r'}{\to} D'$. Assume that $g^\*: A^{C'} \to A^C$ preserves projective cofibrations and that the uni... | 5 | https://mathoverflow.net/users/51164 | 309054 | 134,591 |
https://mathoverflow.net/questions/309050 | 10 | Does $SL(2,\mathbb{Z})$ have a finite-dimensional faithful unitary representation? No such representation exists for $SL(2,\mathbb{R})$, but I don't see a reason why one shouldn't exist for $SL(2,\mathbb{Z})$.
| https://mathoverflow.net/users/98045 | Finite-dimensional faithful unitary representations of SL(2,Z) | Here a non-explicit proof of the existence of a faithful representation of $\mathrm{SL}\_2(\mathbf{Z})$ in $\mathrm{SU}(2)$, using basic algebraic geometry and topology, and relying on the amalgam decomposition of $\mathrm{SL}\_2(\mathbf{Z})$.
[The basic idea is that if all representations in $\mathrm{SU}(2)$ were no... | 18 | https://mathoverflow.net/users/14094 | 309064 | 134,595 |
https://mathoverflow.net/questions/309063 | 4 | Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and small $p$, what does the sum of the lengths of these edges look like (in a distributional sense)? Geometric intuition says tha... | https://mathoverflow.net/users/70190 | Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$ | Here is a heuristic that agrees with the power proposed by @Bullet51 in the comments above, showing that $C(p)$ should grow like $p^{3/2}$. The sum should look like $pn$ times the typical order of the $(pn)$th smallest distance.
To estimate that distance, consider a simpler problem, where the points are put in $Kn$ ... | 4 | https://mathoverflow.net/users/11054 | 309067 | 134,596 |
https://mathoverflow.net/questions/308990 | 4 | The starting point of this post is an [earlier question](https://mathoverflow.net/questions/162076/is-this-riemann-zeta-function-product-equal-to-the-fourier-transform-of-the-von), where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambd... | https://mathoverflow.net/users/25104 | Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture | Nice question! As I explained in the comments, Conjectures 1-2 follow easily. So let me answer your main question much more generally, and also the question that you implicitly asked at [OEIS A298825](https://oeis.org/A298825). I will denote by $\tau$ the divisor function.
**Theorem.** Let $r=\operatorname{rad}(h)$ b... | 5 | https://mathoverflow.net/users/11919 | 309073 | 134,599 |
https://mathoverflow.net/questions/293237 | 8 | For given block sizes $a<b<c<d$, consider the complete 4-partite graph $K\_{a,b,c,d }$.
>
> * Can such a graph be integral, i.e. have only integer eigenvalues?
>
>
>
It is easy to see that the four nonzero eigenvalues of $K\_{a,b,c,d }$ are the same as those of $$\begin{pmatrix} 0&b&c&d\\ a&0&c&d\\ a&b&0&d\\ ... | https://mathoverflow.net/users/29783 | Integral complete 4-partite graphs |
>
> * Can such a graph be integral, i.e. have only integer eigenvalues?
>
>
>
Yes. Two examples are $(a,b,c,d) = (441, 744, 1225, 5635)$,
with eigenvalues $-945$, $-525$, $-3038$, $4058$, and
$(a,b,c,d) = (1575, 1900, 4500, 33516)$,
with eigenvalues $-1710$, $-2940$, $-14250$, $18900$.
There are infinitely many ... | 7 | https://mathoverflow.net/users/14830 | 309079 | 134,603 |
https://mathoverflow.net/questions/309086 | 14 | Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}\_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $\text{ord}\_{s=1}L(E, s)\le 1$.
What is known in the case $\text{rank}... | https://mathoverflow.net/users/128164 | BSD conjecture for rank 1 elliptic curves | The following theorem is due to Chris Skinner, in [this 2014 paper](https://arxiv.org/abs/1405.7294).
>
> Let E/Q be an elliptic curve such that rank E(Q) = 1 **and** the
> Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
> assumptions hold. Then $ord\_{s = 1} L(E, s) = 1$, and in particular
... | 14 | https://mathoverflow.net/users/2481 | 309092 | 134,609 |
https://mathoverflow.net/questions/309037 | 2 | I would like to know the irreducible representations of the group $G\_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible representations of the general group $G\_m = \langle a,b \mid a^{2^{m}}, b^2, baba^{-d}\rangle$ where $d=2^{m-1}-1$.
I... | https://mathoverflow.net/users/94832 | Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups | There are good answers to this question both in formal answers and in the comments, but I'll make a couple of general remarks. If a finite group $G$ has a (necessarily normal) Abelian subgroup $A$ of index $2,$ then any complex irreducible character $\chi$ of $G$ has degree at most $2,$ for by Clifford's Theorem, ${\rm... | 5 | https://mathoverflow.net/users/14450 | 309095 | 134,611 |
https://mathoverflow.net/questions/309096 | 15 |
>
> Consider the additive abelian group $(\mathbb{R},+)$. Does there exists a binary operation $\circ:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that the following holds
>
>
> * $(\Bbb{R},\circ)$ is a group.
> * For all $S\subseteq \mathbb{R}$ the subgroup generated by $S$ in $(\mathbb{R},+)$ is **equal** (as ... | https://mathoverflow.net/users/nan | Does there exists a group structure on $\circ$ on $(\mathbb{R},\circ)$ such that $(\mathbb{R},\circ)$ is non-isomorphic to $(\mathbb{R},+)$? | The answer is no: every such group law is isomorphic to the standard law.
Let me prove something stronger: the poset structure of the lattice of subgroups $\mathrm{Sub}(\mathbf{R})$ characterizes $\mathbf{R}$ up to group isomorphism.
Let us first check it among abelian groups. Let $G$ be a group with $\mathrm{Sub}(... | 32 | https://mathoverflow.net/users/14094 | 309100 | 134,612 |
https://mathoverflow.net/questions/309098 | 5 | Given $n\in\Bbb{N}$, the number of (unrestricted) [integer partitions](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$ are given by
$$\sum\_{n\geq0}p(n)x^n=\prod\_{j\geq1}\frac1{1-x^j}.$$
Define the collapsed partitions of $n$ to be the partitions of $n$ with multiplicities removed. For example,
if $n=... | https://mathoverflow.net/users/66131 | Collapsed partitions and generating functions | It is easy to see that
$$cp\_k(n) = \sum\_{i=1}^n p(n-i)\cdot i^k,$$
where $p(n-i)$ stands for the number of (collapsed) partitions of $n$ that contain $i$ as a part.
Since $i^k$ is the coefficient of $x^{i-1}$ in $\frac{A\_k(x)}{(1-x)^{k+1}}$, we conclude that $cp\_k(n)$ equals the coefficient of $x^{n-1}$ in $\fra... | 4 | https://mathoverflow.net/users/7076 | 309110 | 134,617 |
https://mathoverflow.net/questions/309093 | 9 | Let $C$ be a symmetric monoidal $n$-category. An extended framed $C$-valued TQFT is a symmetric monoidal functor from the framed bordism category $\mathrm{Cob}^{fr}\_n(n)$ to $C$.
The cobordism hypothesis, first formally written down by Baez--Dolan, states that an extended framed $C$-valued TQFT is determined up to ... | https://mathoverflow.net/users/nan | Physical consequences of cobordism hypothesis? | Yes. The physical motivation is that topological field theories, as examples of quantum field theories, should be
fully local, meaning that one should be able to calculate any information about a (fully extended) TQFT $Z$ on a
manifold $M$ by cutting $M$ into pieces, formulating $Z$ on these pieces, and gluing. The tak... | 12 | https://mathoverflow.net/users/97265 | 309116 | 134,618 |
https://mathoverflow.net/questions/300754 | 7 | It's well known that a countable theory is uncountably categorical if and only if it is $\omega$-stable and has no Vaughtian pairs. One of the old definitions of unidimensional theory (all $\omega\_1$-saturated models of the same (sufficiently large) size are isomorphic) is a weak categoricity notion, so it's natural t... | https://mathoverflow.net/users/83901 | Is there a Baldwin-Lachlan style characterization of countable unidimensional theories? | I believe the answer to the second question is that all weakly minimal groups are unidimensional.
**Proof:** Let $T$ be the theory of a weakly minimal group. It suffices to show that if $p$ and $q$ are non-algebraic $1$-types over a sufficiently saturated model $G$, then $p$ and $q$ are not orthogonal. So fix such $... | 3 | https://mathoverflow.net/users/38253 | 309118 | 134,619 |
https://mathoverflow.net/questions/307800 | 3 | Consider the Cauchy problem:
$$
\frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D\_x) u = f \quad 0< t < T; \qquad u = u\_0 \quad \text{when}\; t = 0,
$$
where $A$ has real principal symbol $a(x,\xi)$. This problem is discussed
in a number of sources (Hormander v.iii, Taylor's $\Psi$DO, etc. ).
Let $S(t,s)$... | https://mathoverflow.net/users/110208 | Wavefront set and Duhamel's principle | Your equation reads $Pu=f$ where $P=D\_t+A(x,D\_x)$ is a real principal type operator which is scalar and of first order. (I assume that $D\_x=-i\partial\_x$ is meant.) Moreover, $u=0$ for $t<0$. I claim that the wavefront set of $u$ is contained in the union of the wavefront set of $f$ and the bicharacteristics issuin... | 1 | https://mathoverflow.net/users/nan | 309119 | 134,620 |
https://mathoverflow.net/questions/309122 | 6 | Is there any example of a complete atomless Boolean algebra with a **non-trivial** abelian automorphism group?
This is equivalent, by Stone duality, to asking for an extremally disconnected compact Hausdorff space with no isolated points, having abelian homeomorphism group. Note that no such example can be metrizable... | https://mathoverflow.net/users/16107 | Complete atomless Boolean algebras with abelian automorphism group | If A is a rigid complete BA, then the automorphism group of AxA is isomorphic to the direct sum of |A| copies of the two-element group.This is proved in the remark following Lemma 1.9 in a paper of McKenzie and Monk (Colloq. Math.
Soc. Janos Bolyai, 1973, 951-988); the result is attributed to de Groot; see the referenc... | 8 | https://mathoverflow.net/users/90095 | 309131 | 134,624 |
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