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https://mathoverflow.net/questions/320182 | 5 | $O(n)$ is an extension of $\mathbb{Z}\_2$ by $SO(n)$,
$$1\to SO(n) \to O(n)\to \mathbb{Z}\_2 \to 1.$$
Below we denote the Stiefel-Whitney class of real vector bundle $V\_G$ of the group $G$ as:
$$
w\_j(V\_{G}) : =w\_j({G}).
$$
>
> My question is that how do the "generalized" Stiefel-Whitney class of $O(n)$ and $... | https://mathoverflow.net/users/27004 | Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) and SO(n) | First I will write up what your question is asking in terms of Arun Debray's comment. I strongly suggest that when discussing questions like this, you use precise notation as in the following; I found your question impossible to understand until that comment.
First, the Stiefel-Whitney classes are a set of generator... | 13 | https://mathoverflow.net/users/40804 | 320245 | 138,306 |
https://mathoverflow.net/questions/320190 | 20 | Reading through Bhatt-Morrow-Scholze's ["Topological Hochschild Homology and Integral p-adic Hodge Theory"](https://arxiv.org/abs/1802.03261) I encountered the following statement.
>
> We use only “formal” properties of THH throughout the paper, with the
> one exception of Bökstedt’s computation of $THH(\mathbb{F}... | https://mathoverflow.net/users/134242 | revisiting $THH(\mathbb{F}_p)$ | Alright, here is the promised answer. First, the Hopkins-Mahowald theorem states that $\mathbb{F}\_p$ is the free $\mathbb{E}\_2$-ring with $p=0$, i.e. it is the homotopy pushout in $\mathbb{E}\_2$-rings of $S^0 \leftarrow \mathrm{Free}\_{\mathbb{E}\_2}(x) \rightarrow S^0$ where we have $x \mapsto 0$ for the first arro... | 24 | https://mathoverflow.net/users/6936 | 320246 | 138,307 |
https://mathoverflow.net/questions/320220 | 9 | I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL\_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand this automorphism group, $Aut(GL\_n(\mathbb{Z}))$, explicitly?
| https://mathoverflow.net/users/19926 | Automorphisms of $GL_n(\mathbb{Z})$ | Hua and Reiner described this in their paper "Automorphisms of the unimodular group" (for them "unimodular" means determinant of absolute value 1, hence it's ${\rm GL}\_n(\mathbf Z)$ rather than just ${\rm SL}\_n(\mathbf Z)$) that appeared in 1951 [here](https://www.ams.org/journals/tran/1951-071-03/S0002-9947-1951-004... | 15 | https://mathoverflow.net/users/3272 | 320249 | 138,308 |
https://mathoverflow.net/questions/320252 | 0 | Let $\mathbb S\_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z\_1,\ldots,z\_n$ be a i.i.d sample from $\mathcal N(0, 1)$.
Question
========
Given $\epsilon > 0$ (may be assumed to be very small), what is a reasonable **upper bound** for the tail probability $P(\sup\_{a \in \mathbb S\_{n-1}}\sum\_{i=1}^na\_i^2z\_... | https://mathoverflow.net/users/78539 | Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$ | As pointed out by a user (Nate Eldgredge) in the comments under the question,
$$
P\left(\sup\_{a \in \mathbb S\_{n-1}}\sum\_{i=1}^n a\_i^2Z\_i^2 \ge \epsilon\right) = P\left(\max\_{1 \le i \le n}Z\_i^2 \ge \epsilon\right) = 1-P(Z\_1^2 \le \epsilon)^n = 1-P(\chi^2 \le \epsilon)^n.
$$
One can then invoke standard tail... | 1 | https://mathoverflow.net/users/78539 | 320253 | 138,309 |
https://mathoverflow.net/questions/320226 | 1 | Consider an open domain $U$ split in two non-overlapping subdomains: $U = U\_1 \cup U\_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
>
> Consider the following elliptic problem:
>
>
> \begin{align\*} -&\nabla ( A\_1(x)\nabla u ) =f\_1 \ &\text{ in } U\_1\\
> -&\nabla ( A\_2(x)\nab... | https://mathoverflow.net/users/nan | Elliptic interface problem without conditions on the interface | There are several aspects of your question.
In scientific computing, this amounts to "domain decomposition methods". Also "finite element methods" use such an idea, which is based on Ritz's method/Galerkin's method for a variational or weak formulation. Poincare's "balayage" also uses such decompositions.
Sometimes o... | 2 | https://mathoverflow.net/users/134264 | 320261 | 138,313 |
https://mathoverflow.net/questions/320263 | 1 | Let $\{0,1\}^{<\omega}$ denote the subgroup of $\{0,1\}^{\omega}$ where $f:\omega\to\{0,1\}$ is a member of $\{0,1\}^{<\omega}$ if there is $N\in\omega$ such that $f(n)= 0$ for all $n\in\omega$ with $n\geq N$.
If $(G,+)$ is an abelian group with $|G|=\aleph\_0$ and $g+g = 0$ for all $g\in G$, does this imply $G\cong ... | https://mathoverflow.net/users/8628 | Countable abelian group of exponent $2$ | Yes. An abelian group $(G,+)$ of exponent $2$ is the same as a module over $\mathbb{Z}/2\mathbb{Z}$, that is, a vector space over the finite field $\mathbb{F}\_2$. If $|G|$ is infinite, then the dimension of $G$ over $\mathbb{F}\_2$ is $|G|$ by basic cardinal arithmetic, whence $(G,+)$ is isomorphic to a direct sum of ... | 7 | https://mathoverflow.net/users/11919 | 320267 | 138,315 |
https://mathoverflow.net/questions/320274 | 0 | Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach algebra, that is, if $xA=(0)$ for some $x\in A$, then $x=0$. (right multiplier is defined in a similar way). However, som... | https://mathoverflow.net/users/85784 | The definiton of a multiplier on a Banach algebra | I would be interested to know *exactly* what sources you are using, as any standard textbook does prove the following.
I actually don't know what happens for just a left multiplier (though also do not know any counter-examples off the top of my head), but suppose we have a *double multiplier* $(L,R)$, that is, $L$ is... | 4 | https://mathoverflow.net/users/406 | 320284 | 138,320 |
https://mathoverflow.net/questions/320270 | 4 | Given the following polyhedron: All the $n\times n$ matrices $\boldsymbol{X}$ with elements $x\_{ij}\in(0,1)$ such that
$$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{X}=\boldsymbol{c}^T$$
For some given vectors $\boldsymbol{r}$ and $\boldsymbol{c}$.
Can I calculate the volume of t... | https://mathoverflow.net/users/103133 | Volume of polyhedron | When $\boldsymbol{r}=\boldsymbol{c}=(1,1,\dots,1)$ the polytope $X$ is the Birkhoff polytope of $n\times n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in <http://arxiv.org/abs/math/0701866>, but it is quite a... | 11 | https://mathoverflow.net/users/2807 | 320293 | 138,323 |
https://mathoverflow.net/questions/320295 | 5 | $G$ is an p-adic group, and $\pi$ is an irreducible representation of $G$, then do we naturally have
$Hom\_G(C\_c^{\infty}(G),\pi)\cong Hom\_{\mathbb{C}}(\pi^{\vee},\mathbb{C})$? I think it is true, but I do not have found the detailed proof.
| https://mathoverflow.net/users/121163 | $Hom_G(C_c^{\infty}(G),\pi)\cong Hom_{\mathbb{C}}(\pi^{\vee},\mathbb{C}) ?$ | For two smooth representations $\pi\_i$, $i=1,2$, of $G$, one has $\mathrm{Hom}\_G (\pi\_1 ,\pi\_2 ) \simeq \mathrm{Hom}\_G (\pi\_2^\vee ,\pi\_1^\vee )$. On the other hand the contragredient of $C\_c^\infty (G)$ is $C^\infty (G)$ (the space of smooth functions with arbitrary support), the pairing being given by $\langl... | 5 | https://mathoverflow.net/users/4767 | 320324 | 138,331 |
https://mathoverflow.net/questions/320321 | 11 | In an action-packed three pages of Lurie's [DAG-XIII: Rational and p-adic Homotopy Theory](http://www.math.harvard.edu/~lurie/papers/DAG-XIII.pdf), section 2.2: Power Operations on $\mathbb{E}\_{\infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate c... | https://mathoverflow.net/users/134242 | Power operations from a Tate construction | What you are looking for is probably [Lecture 24](https://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture24.pdf) of Lurie's [lecture notes on the Sullivan Conjecture](https://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-t... | 11 | https://mathoverflow.net/users/360 | 320327 | 138,332 |
https://mathoverflow.net/questions/320062 | 13 | In the following I present a conjecture on Nakayama algebras that I have for nearly 2 years now. Since I was not able to solve it and it can be stated purely combinatorically, I thought it might be worth a try to post it on mathoverflow. A proof of conjecture 1 would have very nice consequences and applications that I ... | https://mathoverflow.net/users/61949 | Combinatorial inequality for dominant dimension | Here I will prove a bound that is much stronger in general. If $b(A)$ denotes the number of bounded projective-injective modules (a special kind of projective-injective module defined below) then we have
$$\operatorname{sdomdim}(A)\le \operatorname{def}(A)+2b(A).$$
To see why this is much stronger notice that $b(A)$ is... | 4 | https://mathoverflow.net/users/2384 | 320329 | 138,333 |
https://mathoverflow.net/questions/320294 | 2 | Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy
$$
h(A)=\sum\_{\lambda} \max\{0, \log|\lambda| \}
$$
where $\lambda$ goes through all eigenvalue of $A$ with multiplicity.
Consider the case when $h(A)>0$. I would like to ask what are the lower bounds
$$
\inf\_{A\in SL(... | https://mathoverflow.net/users/45092 | Lower bound of positive entropies of automorphisms on tori | For fixed $d$ the best known lower bound is due to Dobrowolski, who showed that if $h(A)>0$ then $$h(A) > \log\Big[1+\frac{1}{1200}\Bigl(\frac{\log\log d}{\log d}\Bigr)^3\Bigr].$$
For this and much more, see Chris Smyth's article *The Mahler measure of algebraic numbers: a survey*, arXiv:math/0701397v2 [math.NT].
| 1 | https://mathoverflow.net/users/8112 | 320333 | 138,336 |
https://mathoverflow.net/questions/320176 | 6 | I am looking for references for theorems of the form:
1) Any proof of theorem $X$ requires $n$ applications of induction axioms
and especially
2) Any proof of theorem $X$ requires $n$ nested applications of induction axioms.
I've seen similar statements for applications of axioms other than induction axioms (an... | https://mathoverflow.net/users/129428 | Cases where multiple induction steps are provably required | Here is a reference for one way of making precise sense of your question and answering it:
>
> Stefan Hetzl and Tin Lok Wong (2017): "Some observations on the logical foundations of inductive theorem proving", Logical Methods in Computer Science, Volume 13, Issue 4, doi:[10.23638/LMCS-13(4:10)2017](https://doi.org/... | 11 | https://mathoverflow.net/users/35868 | 320336 | 138,339 |
https://mathoverflow.net/questions/320308 | 0 | Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of *geodesic connectedness on Semi-Riemannian manifolds*?
| https://mathoverflow.net/users/nan | Analytic approach to geodesic connectedness in Semi-Riemannian manifolds | I'd recommend taking a look at the following monograph and surveys (and at the references cited therein):
1. *Masiello, Antonio*, Variational methods in Lorentzian geometry, Pitman Research Notes in Mathematics Series. 309. Harlow, Essex: Longman Scientific & Technical. New York, NY: Wiley. xix, 175 p. (1994). [ZBL0... | 1 | https://mathoverflow.net/users/100351 | 320338 | 138,340 |
https://mathoverflow.net/questions/320301 | 3 | Let $f$ a function from $\{0, 1 \}^{2n}$ to $\{0, 1 \}^{n}$ uniformly picked at random. I would like to have an estimation of the expected size of the smallest premiage set of $f$, more formally $\mathbb{E}\_{f \leftarrow \left(2^n\right)^{2^{2n}}} \left(\min \left(|f^{<-1>}\left( i\right) |\right)\_{i \in \{ 0,1\}^{n}... | https://mathoverflow.net/users/90197 | Expected size of the smallest preimage set | This is not a full answer but easy to obtain. See the reference by @esg in the comments for the full answer.
I will give a very loose upperbound to this probability, which still tends to zero quite fast with increasing $N.$ You have $N^2$ balls thrown into $N$ bins where $N=2^n.$ So the probability that the lightest ... | 1 | https://mathoverflow.net/users/17773 | 320340 | 138,341 |
https://mathoverflow.net/questions/320339 | 5 | Does there exist an endomorphism of $(\mathbb Z\_p,+)$ of finite order different of $x\mapsto\xi x$ where $\xi$ is a root of unity in $\mathbb Z\_p$?
Thanks in advance
| https://mathoverflow.net/users/33128 | Endomorphisms of the p-adic group $(\mathbb Z_p,+)$ | I claim that any endomorphism of $\mathbb{Z}\_p$ is just multiplication by a constant.
Let $f$ be an endomorphism of $\mathbb{Z}\_p$. First observe that $p^n f(x) = f(p^n x)$, so $f$ takes $p^n \mathbb{Z}\_p$ to $p^n \mathbb{Z}\_p$. This shows that $f$ is continuous. For any $x \in \mathbb{Z}\_p$, we can find a seque... | 15 | https://mathoverflow.net/users/36579 | 320344 | 138,342 |
https://mathoverflow.net/questions/320323 | 3 | For simplicity this is about polynomials in just two variables.
Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials
$X^iY^j$ and therefore as a sum of polynomials $p\_{ij}\in\mathbb Q[X^iY^j]$ over one variable:
$$\displaystyle f(X,Y)=p\_{00}+\sum\_{\gcd(i,j)=1}p\_{ij}(X^iY^j).$$
>
>... | https://mathoverflow.net/users/57255 | Integer valued polynomials over several variables | Every polynomial function $f : \mathbb{Z}^2 \to \mathbb{Z}$ is a $\mathbb{Z}$-linear combination of the polynomials $\pmatrix{x \\ i} \pmatrix{y \\ j}$ with $i,j \geq 0$.
Indeed, let $f : \mathbb{Z}^2 \to \mathbb{Z}$ be a polynomial function. By interpolation, we know that $f$ comes from a polynomial in $\mathbb{Q}[x... | 8 | https://mathoverflow.net/users/6506 | 320363 | 138,349 |
https://mathoverflow.net/questions/139469 | 7 | Let $G \leq {\rm S}\_n$ be a finite permutation group, and let
$S = \{g\_1, \dots, g\_k\}$ be a generating set for $G$ which is closed
under inversion and which does not contain the identity.
The *growth function* of $G$ with respect to $S$ is the sequence
$(a\_0, a\_1, a\_2, \dots)$, where $a\_r$ is the number of elem... | https://mathoverflow.net/users/28104 | How hard is it to compute the diameter and the growth function of a finite permutation group of small degree? | I just came across this question and, even though I'm a bit late, I thought you might be interested in this reference:
*Even, S.; Goldreich, O.*, [**The minimum-length generator sequence problem is NP-hard**](http://dx.doi.org/10.1016/0196-6774(81)90029-8), J. Algorithms 2, 311-313 (1981). [ZBL0467.68046](https://zb... | 5 | https://mathoverflow.net/users/801 | 320365 | 138,350 |
https://mathoverflow.net/questions/320364 | 8 | I have a question about the argumentation at the beginning of section 15 in this paper. The goal is to estimate the sum
$$V(\beta) = 2 \sum\_{(z\_1,z\_2)=1} \beta\_{z\_1} \overline{\beta\_{z\_2}} \sum\_{\substack{de > X \\ r\_1s\_2 \equiv r\_2s\_1 \text{ } (4d)}} \frac{\varphi(de)}{de} f \Big( \frac{|\Delta|}{de} \Big... | https://mathoverflow.net/users/125074 | Question about Friedlander, Iwaniec: "The polynomial $X^2+Y^4$ captures its primes" | Note first the $\beta\_z$ are bounded by $\tau(|z|^2)$.
What you then have as a bound with removing the gcd in 15.2 is a sum over $g=\gcd(z\_1,z\_2)$ as
$$\sum\_{|g|\ge P}\tau(g)^2\sum\_{|z\_i|\le N/|g|\atop \text{$z\_i$ in box}}\tau(|z\_1|^2)\tau(|z\_2|^2)\sum\_{e|\gcd({\rm Re}(z\_1),{\rm Re}(z\_2))}\sum\_{d\le N\at... | 4 | https://mathoverflow.net/users/129840 | 320369 | 138,353 |
https://mathoverflow.net/questions/320337 | 10 | Let $\mathbb{F}\_2$ be the free group on two generators. Does $\mathbb{F}\_2 \times \mathbb{F}\_2$ embed as a subgroup of $\mathrm{SL}\_3(\mathbb{Z})$?
| https://mathoverflow.net/users/30721 | Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$ | Here is a brief elementary argument only relying on Burnside's theorem that proper unital subalgebras of matrix algebras are non-irreducible (over an algebraically closed field).
**Proposition:** Let $K$ be an algebraically closed field. Let $A,B$ be noncommuting matrices in $\mathrm{M}\_3(K)$. Then the centralizer o... | 13 | https://mathoverflow.net/users/14094 | 320375 | 138,357 |
https://mathoverflow.net/questions/320382 | 5 | Let ${\mathbb N}$ denote the set of positive integers, let $A,B\subseteq \mathbb{N}$. For $n\in\mathbb{N}$ we set $n+A:=\{n+a: a\in A\}$. We say that $A$ *is infinitely often in* $B$ if the set $$\big\{n\in\mathbb{N}:(n+A)\subseteq B\big\}$$ is infinite.
Moreover, for any $S\subseteq {\mathbb N}$ we set $\mu(S) = \lim ... | https://mathoverflow.net/users/8628 | On the "infinitely often in" relation between subsets of $\mathbb{N}$ | Unless I'm missing something, the answer is **yes** (even if you replace "inf" with "sup").
---
I'm going to switch from *sets* to *binary sequences* for simplicity. We define $\mu(B)$ for an infinite binary sequence $B$ as $\mu(\{n: B(n)=1\})$. Fix $n\in\mathbb{N}$ and $B$ an infinite binary string with $\mu(B)=... | 8 | https://mathoverflow.net/users/8133 | 320383 | 138,362 |
https://mathoverflow.net/questions/320378 | 1 | Given a number of normal distributions $N(\mu\_1, \sigma^2), N(\mu\_2, \sigma^2), ..., N(\mu\_n, \sigma^2)$ with fixed variance $\sigma^2$, but not necessary equal means. My question is how to approximate the variance given a number of samples of the normal distributions. Hence given samples
$$
X^1\_1, X^1\_2, ..., X^1... | https://mathoverflow.net/users/134354 | Approximate the variance of multiple normal distributions with the same standard deviation | $\newcommand{\si}{\sigma}$
Let us assume that the $n$ samples from the respective distributions $N(\mu\_1, \sigma^2), \dots, N(\mu\_n, \sigma^2)$ are independent. Let $X\_{ij}:=X^i\_j$. Everywhere here $i=1,\dots,n$ and $j=1,\dots,m\_i$. So, all the $X\_{ij}$'s are independent and $X\_{ij}\sim N(\mu\_i, \si^2)$. So, t... | 1 | https://mathoverflow.net/users/36721 | 320385 | 138,363 |
https://mathoverflow.net/questions/320386 | 0 | It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably transformed into the [canonical form of the Heun equation](https://en.wikipedia.org/wiki/Heun_function). Surely, giving this tran... | https://mathoverflow.net/users/46321 | Bringing a Heun equation into canonical form | The reference is
A. Ronveaux, Heun's differential equation, Oxford UP NY, 1995.
You describe the process correctly: first by a fractional linear change of the independent
variable move the singular points in the position $(0,1,a,\infty)$. (One can move
any 4 distinct points to such a position).
Second, by the tr... | 2 | https://mathoverflow.net/users/25510 | 320391 | 138,367 |
https://mathoverflow.net/questions/320184 | 6 | 1. Introduction.
----------------
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
>
> **Question.** What are the known regularity results for ***anisotropic*** minimal surfaces?
>
>
>
1.2 Preliminaries
-----------------
For instance,... | https://mathoverflow.net/users/100976 | Anisotropic perimeter and regularity of anisotropic minimal surfaces | Very little is known in terms of regularity theory compared with the level it is known for area-minimizers. However there are recent breakthroughs in other senses i.e. existence and rectifiability.
See
De Philippis, De Rosa, Ghiraldin. "Rectifiability of varifolds with locally bounded first variation with respect t... | 5 | https://mathoverflow.net/users/122587 | 320404 | 138,370 |
https://mathoverflow.net/questions/320407 | 4 | In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\rangle$ has a unique accumulation point $a$ in the boundary $\partial X$. He then goes on to call $f$ proper if for any s... | https://mathoverflow.net/users/69797 | non-proper parabolic isometries of hyperbolic spaces | Yes. Let me call this "metrically proper" to avoid ambiguity.
Indeed, Edelstein (On non-expansive mappings of Banach spaces, Proc.
Cambridge Philos. Soc. 60 (1964), 439-447) observed that there infinite-dimensional Hilbert spaces admit unbounded isometries that are not metrically proper. This is not Gromov-hyperbolic... | 7 | https://mathoverflow.net/users/14094 | 320408 | 138,371 |
https://mathoverflow.net/questions/320403 | 4 | I'm curious whether lacunary functions (functions in the complex plane that are holomorphic in some open ball about the origin, but cannot be analytically extended past that ball) are typical or the exception.
**Are lacunary functions dense in the space of functions that are holomorphic on the open unit ball?**
| https://mathoverflow.net/users/134361 | Density of Lacunary Functions | I'm going to prove density in compact-open topology, i.e. that for any $f$ holomorphic on the disk there is a sequence of lacunary functions convergent to $f$ uniformly on compact sets.
Take any function $g$ lacunary in the unit disk satisfying $g(0)=0$. Then it's easy to see that for $g\_n(x)=g(x^n)$, $g\_n(x)$ is l... | 4 | https://mathoverflow.net/users/30186 | 320413 | 138,372 |
https://mathoverflow.net/questions/320419 | 6 | Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes\_{A}B, f\_{\ast})$ which is an equivalence. In this situation, $(-\otimes\_{A}B, f\_{\ast})$ is an equivalence precisely if the unit map
$$\eta\_{M} : M \rightarrow f\_{\ast}(M \otimes\_{A}B)$$
is a we... | https://mathoverflow.net/users/113828 | Quillen equivalent module categories | The counit map is cocontinuous in M, so using the fact
that any cofibrant object is a retract of a transfinite composition
of cobase changes of generating cofibrations of A-modules,
combined with the left properness of the model category of A-modules
and the fact that the left adjoint sends generating cofibrations to m... | 4 | https://mathoverflow.net/users/402 | 320425 | 138,374 |
https://mathoverflow.net/questions/320426 | 7 | I believe the following is true and well known.
**Theorem (?).** Let $X$ and $Y$ be smooth, irreducible, projective varieties over $\mathbb{C}$. Let
$$
f\colon X\rightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,\wedge^k\Omega\_X)\cong H^0(Y,\wedge^k\Omega... | https://mathoverflow.net/users/134370 | Outer Hodge groups of rationally connected fibrations | By spectral sequence, we just need to prove that $R^if\_\*\mathcal{O}\_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
| 9 | https://mathoverflow.net/users/42636 | 320427 | 138,375 |
https://mathoverflow.net/questions/317494 | 14 | Consider the product of $2n$ two-spheres $X\_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) central symmetry for each $S^2$. Is it possible to say for which $n$ the quotient manifold $X\_n/\mathbb Z\_2$ admits an almo... | https://mathoverflow.net/users/13441 | An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$ | Let $Y\_n = X\_n/\mathbb{Z}\_2$.
If $G$ is a finite group acting freely on a manifold $M$, and $\pi : M \to M/G$ denotes the quotient map, then $\pi^\* : H^\*(M/G; \mathbb{Q}) \to H^\*(M; \mathbb{Q})$ is injective; moreover, the image is $H^\*(M; \mathbb{Q})^G$.
Involution acts on $H^2(S^2; \mathbb{Q})$ by $-1$, so... | 20 | https://mathoverflow.net/users/21564 | 320435 | 138,378 |
https://mathoverflow.net/questions/320440 | 8 | Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local Langlands correspondence should give (1) a partition of the classes of irreducible admissible representations of $G(k)$ into fin... | https://mathoverflow.net/users/38145 | L-packets in the local Langlands correspondence: why finite sets? | The fiber $\mathcal{L}^{-1}(\rho)$ of an L-parameter $\rho:W\_k'\rightarrow ^LG$ is expected to be in bijection with the set of irreducible representations of a certain finite group attached to $\rho$. In greater detail, let $Z(\rho)$ denote the centralizer in $\hat{G}$ of the image of $\rho$ in $ ^LG$ and $Z$ the cent... | 7 | https://mathoverflow.net/users/nan | 320445 | 138,379 |
https://mathoverflow.net/questions/320452 | 5 | For any positive integer $n\in\mathbb{N}$ let $S\_n$ denote the set of all bijective maps $\pi:\{1,\ldots,n\}\to\{1,\ldots,n\}$. For $n>1$ and $\pi\in S\_n$ define the *neighboring number* $N\_n(\pi)$ as the minimum distance of $\pi$-neighbors, or more formally: $$N\_n(\pi) = \min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\... | https://mathoverflow.net/users/8628 | Neighboring number of a permutation | $\newcommand{\e}{\varepsilon}$
For any fixed $\e>0$, permutations with $N\_n(\pi)>\e n$ asymptotically have density zero. Indeed, consider values $\pi(1),\dots,\pi(k)$. There are $n$ choices for $\pi(1)$. For $\pi(2)$, we have at most $(1-\e)n$ choices, since it has to be at distance at least $\e n$ from $\pi(1)$. Simi... | 5 | https://mathoverflow.net/users/30186 | 320460 | 138,383 |
https://mathoverflow.net/questions/320423 | 6 | The **[thin shell conjecture](https://core.ac.uk/download/pdf/82335758.pdf)** states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies
$$\mathbb{P}\Big\{\big|\|X\|\_2-\sqrt{n}\big|\geq t\Big\}\leq Ce^{-ct}\qquad \foral... | https://mathoverflow.net/users/29873 | Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions? | Ahh, just take $X$ to be $0$ with probability $1/2$ and otherwise draw uniformly from the sphere of radius $\sqrt{2n}$.
If the support of the distribution were convex, then it wouldn't be able to "skip" the thin shell. I guess this is why logconcavity is important.
| 2 | https://mathoverflow.net/users/29873 | 320461 | 138,384 |
https://mathoverflow.net/questions/320475 | 6 | It is well known that the [Mertens function](https://en.wikipedia.org/wiki/Mertens_function)
$$M(x)=\sum \_{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a [short proof](https://math.stackexchange.com/a/2345591/109021).
**Are there known results about *how often* the Mertens function is 0?** (i... | https://mathoverflow.net/users/85239 | How often does the Mertens function vanish? | Quoting an answer to the question [https://mathoverflow.net/questions/273845/oscillation-of-the-summatory-möbius-function](https://mathoverflow.net/questions/273845/oscillation-of-the-summatory-m%C3%B6bius-function)
>
> Let $c=14.1347251…$. Then there are at least $(c/\pi-o(1))\log y$ sign changes in $M(x)$ in the ... | 6 | https://mathoverflow.net/users/17773 | 320479 | 138,389 |
https://mathoverflow.net/questions/320457 | 6 | One of the approaches to ["Special" meanders](https://mathoverflow.net/questions/146802/special-meanders) led (in particular) to the following question:
>
> What is the number $a\_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the following four kinds of steps: $(i,j)\mapsto$ $(i+1,j)$, $(i+1,j-1)$... | https://mathoverflow.net/users/41291 | Enumeration of lattice paths of a specific type | It seems the first solution to this problem appeared in Theorem 4 of
*Raschel, Kilian*, [**Counting walks in a quadrant: a unified approach via boundary value problems**](http://dx.doi.org/10.4171/JEMS/317), J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012). [ZBL1238.05014](https://zbmath.org/?q=an:1238.05014).
N... | 9 | https://mathoverflow.net/users/47484 | 320481 | 138,391 |
https://mathoverflow.net/questions/219588 | 9 | Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its *fibrewise suspension*, i.e. its suspension as an object of the slice $\newcommand{\SSet}{\mathbf{SSet}}\SSet/X$. Most obviously, one can take a homotopy pushout of $X \leftarrow Y \rightarrow X$, t... | https://mathoverflow.net/users/2273 | Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting) | There is an answer to this question in the comments that works for simplicial sets, and more generally for projective model structures on simplicial presheaves. Since 3 years later it hasn't been recorded as an "answer", I thought I would do the service, and provide the references.
For simplicial sets, it suffices to... | 1 | https://mathoverflow.net/users/49 | 320483 | 138,392 |
https://mathoverflow.net/questions/320490 | 3 | For $i= 1, \ldots, n$, let $A\_i \in \mathbb{R}^{d \times d}$ be random i.i.d. matrices with $E [A\_i] =0$.
Can we relate (upper bound) $E[\|\sum\_{i=1}^n A\_i \|\_F^4]$ to $E[\|A\_i\|^4\_F]$ ?
| https://mathoverflow.net/users/16976 | Powers of Frobenius norm of sum of random matrices | The $A\_i$'s are independent zero-mean random vectors in $\mathbb{R}^{d \times d}$, which is a Hilbert space with respect to the Frobenius norm $\|\cdot\|:=\|\cdot\|\_F$. So, by a vector version of Rosenthal's inequality (see e.g. [Theorem 5.2](https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477)), for some ... | 3 | https://mathoverflow.net/users/36721 | 320494 | 138,397 |
https://mathoverflow.net/questions/320474 | 7 | Let $(M,\omega)$ be a symplectic manifold and $N$ an isotropic submanifold. For a point $p\in N$, can we always find coordinates $(x\_1,\ldots,x\_n,y\_1,\ldots,y\_n)$ in a neighbourhood $U$ of $p$ such that both $\omega=\sum dx\_i\wedge dy\_i$ (Darboux coordinates) and
$$N\cap U=\{(x\_1,\ldots,x\_n,y\_1,\ldots,y\_n):x\... | https://mathoverflow.net/users/123207 | Normal coordinates for isotropic submanifolds | The answer is yes, and in fact, you can let $x\_1,\ldots,x\_k$ be any chosen coordinate patch on $N$.
Here's one way to go about it. The first point is that the symplectic normal bundle $TN^{\omega}/TN \rightarrow N$ is locally trivial (as a symplectic vector bundle). Feeding a particular trivialization into the stan... | 3 | https://mathoverflow.net/users/66405 | 320508 | 138,404 |
https://mathoverflow.net/questions/320456 | 2 | Some theories can lie about their own consistency (for example, if $PA$ is consistent then the theory $PA + \lnot CON(PA)$ is consistent, although it proves its own inconsistency).
Now working with ZFC, one can avoid problems like ZFC lying about its own consistency, by assuming ZFC is $\omega$ - consistent.
that ... | https://mathoverflow.net/users/123559 | ZFC ability to express truth and $\omega$ - consistency | The metatheory involved in proving "if ZFC is $\omega$-consistent then it doesn't prove its own inconsistency" is some tiny fragment of PA. I'd expect primitive recursive arithmetic (PRA) to be more than enough, since the proof is essentially just inspection of definitions. So we needn't worry that the metatheory argum... | 6 | https://mathoverflow.net/users/6794 | 320514 | 138,407 |
https://mathoverflow.net/questions/320516 | 5 | Fix constants $1\leq \alpha<\beta$. What is the density of the set of positive integers $n$ with at least two factors between $\alpha\sqrt{n}$ and $\beta\sqrt{n}$?
(I am specifically interested when $\alpha=\sqrt{2}$ and $\beta=\sqrt{3}$, and I am hoping the density is zero. I am not an expert in this field, so apol... | https://mathoverflow.net/users/16356 | Density of numbers with multiple factors near square root | Even one such factor gives you zero density. Indeed, if $d \mid n$, $\alpha \sqrt{n} \le d \le \beta \sqrt{n}$ then $\frac{1}{\beta}\sqrt{n} \le \frac{n}{d} \le \frac{1}{\alpha}\sqrt{n}$ therefore $n$ is a product of two numbers not greater than $\max(\frac{1}{\alpha}, \beta)\sqrt{n}$. Therefore amount of desired numbe... | 9 | https://mathoverflow.net/users/104330 | 320519 | 138,408 |
https://mathoverflow.net/questions/320526 | 4 | Artin in his book, Geometric Algebra, says the connection between the left
degree and right degree of a skew-field extension is unknown.
Since I'm not an expert, I was wondering if someone knew the answer to this question. The book is rather old and there must have been some developments since that time.
| https://mathoverflow.net/users/68719 | Left vs right degree of skew-field extensions | Anything that might happen does happen.
In
*Schofield, A. H.*, [**Artin’s problem for skew field extensions**](http://dx.doi.org/10.1017/S0305004100062538), Math. Proc. Camb. Philos. Soc. 97, 1-6 (1985). [ZBL0574.16008](https://zbmath.org/?q=an:0574.16008).
it is shown that for any integers $m,n>1$ there is a ske... | 4 | https://mathoverflow.net/users/22989 | 320535 | 138,414 |
https://mathoverflow.net/questions/320086 | 8 | In a [Krull--Schmidt category](https://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt_category), if
$$
X\_{1}\oplus X\_{2}\oplus \cdots \oplus X\_{r}\cong Y\_{1}\oplus Y\_{2}\oplus \cdots \oplus Y\_{s},
$$
where the $X\_{i}$ and $Y\_j$ are all indecomposable, then $r = s$, and there exists a permutation $\pi$ such that $... | https://mathoverflow.net/users/126606 | Infinite Krull-Schmidt categories? | The statement about simple Lie algebras is not true.
A (finitely generated right) module $P$ for a ring $R$ is **stably free** if $P\oplus R^m\cong R^n$ for some integers $m,n$.
Suppose $R$ has a non-free stably free module $P$, as above. If also $R$ is a (right) Noetherian domain, then the regular module $R$ is i... | 5 | https://mathoverflow.net/users/22989 | 320537 | 138,416 |
https://mathoverflow.net/questions/320459 | 4 | Consider the usual [bounded knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem#Definition), with the extra twist: you know that $k$ of the chosen items will get spoiled after the sack is packed. And this happens adversarially, i.e. $k$ most valuable items will be gone.
Is there still a pseudo-polynomial... | https://mathoverflow.net/users/11100 | complexity of bounded knapsack with spoilage | This was my question to start with, and I think I have figured out how to solve it in pseudopolynomial time. We can order the items by value, from largest to smallest, and guess what is the last (least valuable) item in this order that will get spoiled. Suppose this is the $i$-th item. Then pick $k-1$ lowest-weight ite... | 2 | https://mathoverflow.net/users/134399 | 320548 | 138,418 |
https://mathoverflow.net/questions/320545 | 7 | Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it?
Also, given this upper bound on $D\left(P\parallel Q\right)$, can I deduce some upper bound on $\left|P\left(A\right)-Q\left(A\right)\right|$ for some arbitrary event $A$?
... | https://mathoverflow.net/users/103133 | Relation between the two possible KL divergences of two distributions | The answer to your first question is no: $D(Q||P)$ may be however large while $D(P||Q)$ is however small. E.g., let $P$ have masses $s$ and $1-s$ at points $0$ and $1$, respectively, and let $Q$ have masses $t$ and $1-t$ at points $0$ and $1$, respectively, where $0<s,t<1$. Then
\begin{equation}
D(P||Q)=s\ln\frac st+... | 10 | https://mathoverflow.net/users/36721 | 320552 | 138,420 |
https://mathoverflow.net/questions/320487 | 7 | Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.
Brauer second main theorem states
>
> If $\chi\in Irr(\mathbb{C}G)$ is an ordinary character in the $p$-block $B\in Bl(\mathbb{F}G)$, $u\in G\_p$ a $p$-element, and ... | https://mathoverflow.net/users/3041 | Why is Nagao's theorem the "Module theoretic version of Brauer's second main theorem"? | To see the connection, it is easiest to work over a local ring $R$ of characteristic zero with residue field $R/J(R) \cong \mathbb{F}$ (there are some technicalities I am omitting here for the sake of brevity).
The key point is that if $B$ is a $p$-block with defect group $P$ (I avoid $D$ for the name of a defect gro... | 8 | https://mathoverflow.net/users/14450 | 320556 | 138,424 |
https://mathoverflow.net/questions/320583 | -1 | Let $G=(V,E)$ be a finite, simple, undirected graph such that $\bigcup E = V$ (that is, every vertex belongs to at least one edge).
For $v\in V$ we set $N(v) = \{w\in V:\{v,w\}\in E\}$, and for $S\subseteq V$ we let $N(S) = \bigcup \{N(v):v\in S\}$. A *bijective neighborhood map* is a bijection $\varphi:V\to V$ such ... | https://mathoverflow.net/users/8628 | Version of Hall's marriage theorem in arbitrary finite graphs | This is actually a special case of Hall's theorem itself, rather than an extension of it. To each such graph $G$ you can associate a bipartite graph $G'$ with vertex set two copies of $V$, which we can denote $V\_1,V\_2$, and edges between two vertices in different copies whenever their copies in $V$ were connected by ... | 7 | https://mathoverflow.net/users/2384 | 320586 | 138,428 |
https://mathoverflow.net/questions/320590 | 14 | *This question was originally asked [on Math StackExchange](https://math.stackexchange.com/questions/3064470/projective-invariant-differential-operator).*
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align\*}
&T(g) = 0 \Longleftrightarrow g... | https://mathoverflow.net/users/74578 | Projective-invariant differential operator | There's a straightforward abstract answer that you may not like, but, because it clarifies your question and explains a uniform way to answer similar questions, I'll sketch it here.
First, consider a simpler problem of this kind: Suppose that one wants to describe the group of isometries of a Riemannian metric $\rho$... | 16 | https://mathoverflow.net/users/13972 | 320597 | 138,431 |
https://mathoverflow.net/questions/320589 | 13 | The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemm... | https://mathoverflow.net/users/70190 | Famous theorems that are special cases of linear programming (or convex) duality | To elaborate on M. Winter's comment: Von Neumann's [minimax theorem](https://en.wikipedia.org/wiki/Minimax_theorem) for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem [did not make this connection explicit](https://www.jstor.org/stable/41134130).
| 5 | https://mathoverflow.net/users/3106 | 320601 | 138,433 |
https://mathoverflow.net/questions/317347 | 5 | Let $\ell$ and $d$ be two integers such that $\ell \le d$.
I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x\_1, \ldots, x\_n) := \sum\_{\substack{S \subseteq [n] \\ |S| = \ell}} \sum\_{\substack{z \in [d]^S \\ \sum\_i z\_i = d}} \prod\_{i \in S} x\_i \... | https://mathoverflow.net/users/8271 | Optimizing a multivariate symmetric (permutation-invariant) function | One can show that this function is *Schur-concave* using the *Schur-Ostrowski criterion*, which then implies the maximum is attained at the diagonal.
See also
<https://math.stackexchange.com/questions/439649/are-elementary-symmetric-polynomials-concave-on-probability-distributions>
I will omit the calculations here... | 2 | https://mathoverflow.net/users/8271 | 320611 | 138,438 |
https://mathoverflow.net/questions/320609 | 1 | LLL guarantees that we can find a basis $v\_1,\dots,v\_n$ of a lattice in $\mathbb R^n$ with
$$\|v\_i\|\leq \gamma\_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma\_i$ is a function only of $i$ and $n$.
1. Are there lattices where this cannot be improved to $\|v\_i\|\leq \gamma\_{i,n} \det(\Lambda)^{1/(n-i+2)}$?
2.... | https://mathoverflow.net/users/10035 | Lowering $i$th shortest vector of a lattice | There are lattices where your requirement cannot be met. In fact if you fix any lattice $\Lambda\_0$ and you dilate it by a factor of $R$, then its determinant gets scaled by $R^n$, while its shortest lattice vector gets scaled by $R$. However, your inequality $\|v\_1\|\ll\_n\det(\Lambda)^{1/(n+1)}$ would yield that th... | 4 | https://mathoverflow.net/users/11919 | 320612 | 138,439 |
https://mathoverflow.net/questions/320600 | 4 | Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ what is the largest $m$ we have such that in the interval $(p^\alpha,p-p^\alpha)$ there are $m$ integers $a\_1<\dots<a\_m$ such that
1. $a\_i-a\_j\neq a\_{i'}-a\_{j'}$ holds if $i\neq i'$ or $j\neq j'$ or both.
2. $\min\_{1\leq i<j\leq m}|a\_i-a\_j|>b$
h... | https://mathoverflow.net/users/10035 | An extremal combinatorics problem | Taking [Fedor Petrov](https://mathoverflow.net/users/4312/fedor-petrov)'s observation a little further, I believe the right question to ask is as follows:
$$ \text{What is the largest size of a $b$-separated Sidon set in the interval $[0,L]$?} $$
Here *"$b$-separated"* means that the difference between any two conse... | 4 | https://mathoverflow.net/users/9924 | 320620 | 138,442 |
https://mathoverflow.net/questions/320570 | 0 | Suppose $A$ is a $C^\*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for the strict topology on $M(M(A))$?
| https://mathoverflow.net/users/63864 | strict topology on multiplier algebras | So, if I understand the question, you are asking this: Let $S$ be a subset of $M(A)$ which is compact for the strict topology. (The strict topology is such that a net $(x\_i)$ converges to $x$ exactly when $x\_ia\rightarrow xa, ax\_i\rightarrow ax$ in norm, for each $a\in A$). Now set $B=M(A)$, a unital algebra, and co... | 3 | https://mathoverflow.net/users/406 | 320623 | 138,443 |
https://mathoverflow.net/questions/307450 | 8 | I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the literature. Unfortunately, I could not find them. Here is the construction.
Let $\mathbb P^4$ over $\mathbb C$ with hom... | https://mathoverflow.net/users/29730 | Explicit descriptions of a flop | You have a threefold hypersurface $Y\subset \mathbb{P}^4$ of the form $V(ax\_1 - bx\_2)$, for some quartic polynomials $a,b\in\mathbb{C}[\mathbb{P}^4]$.
You blow up the plane $Z=V(x\_1,x\_2)$ in the ambient space, to obtain $X\subset \mathbb{P}^1\_{\lambda:\mu}\times\mathbb{P}^4$. As you say, this makes a small resol... | 3 | https://mathoverflow.net/users/104695 | 320626 | 138,444 |
https://mathoverflow.net/questions/286218 | 2 | It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $\lceil \log\_2 n\rceil$. (See Fact 2.5 [here](http://theory.stanford.edu/~megiddo/pdf/uzx.pdf).)
What about when the tournament is chosen probabilistically? What are some known results about the size of the minimum ... | https://mathoverflow.net/users/83212 | Minimum dominating sets in tournaments | The probability that a given set of size $k$ is dominating equals $(1-2^{-k})^{n-k}<e^{-(n-k)2^{-k}}$. So, the expectation of the number of dominant sets of size $k$ does not exceed $$\binom{n}ke^{-(n-k)2^{-k}}\leqslant \left(\frac{en}k\right)^ke^{-(n-k)2^{-k}}\leqslant e^{-(n-k)2^{-k}+k\log(en/k)}.$$
If $2^k<\frac{n}{... | 0 | https://mathoverflow.net/users/4312 | 320629 | 138,445 |
https://mathoverflow.net/questions/320625 | 8 | A topological space $X$ is called *analytic* if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.
We say that a topological space $X$ is a *Borel image of a Polish space* if $f(P)=X$ for some Borel surjective function $f:P\to X$ defined on ... | https://mathoverflow.net/users/61536 | Is a Borel image of a Polish space analytic? | I have found a simple counterexample to Problems 1 and 2 (maybe it will be helpful to other researchers):
>
> **Fact.** *The Sorgenfrey line $\mathbb S$ does not have countable network (and hence is not analytic), but it is the image of the real line under the identity map $\mathbb R\to\mathbb S$, which is Borel (... | 9 | https://mathoverflow.net/users/61536 | 320637 | 138,447 |
https://mathoverflow.net/questions/320627 | 7 | Le $A$ be an abelian category and $B$ a Serre subcategory of $A$. Under which conditions do we have $$D^\*(A/B) \simeq D^\*(A)/D^\*(B),$$ where $D^\*$ stands for the derived category with $\* = +,-,b$ or nothing ?
| https://mathoverflow.net/users/86286 | Derived category of a quotient | What it is true is that
$$
D^\*(A/B) \simeq D^\*(A)/D\_B^\*(A),
$$
where the category $D\_B^\*(A)$ is the subcategory of $D^\*(A)$ whose homologies lie in $B$. In general it may not agree with $D^\*(B)$.
Notice that $D\_B^\*(A)$ is identified with the full subcategory of $D^\*(A)$ formed by objects whose image under ... | 12 | https://mathoverflow.net/users/6348 | 320643 | 138,449 |
https://mathoverflow.net/questions/320641 | 0 | Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}\_m = (\mathbb{Z}/m\mathbb{Z}, E\_m)$ where $$E\_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ and } \exists k\in\{1,\ldots n\}: (x+k = y \text{ or } y+k = x)\big\}.$$
It is easy to see that $\mathbb{Z}\_m$ is ... | https://mathoverflow.net/users/8628 | $2n$-regular graphs with maximal chromatic number | **Example 1.** Let $n=1$, $m=7$. Then $c=3$, and $G=C\_3+C\_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z\_7$.
**Example 2.** Let $n=2$, $m=15$. Then $c=5$, as $G=3K\_5$ is a $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z\_{15}$ has chro... | 7 | https://mathoverflow.net/users/43266 | 320645 | 138,450 |
https://mathoverflow.net/questions/320630 | 10 | I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), the expected distance after $n$ steps from the starting point is approximated by $\sqrt{n\pi}/2$
(source: <https://pd... | https://mathoverflow.net/users/93775 | Random Walks on high dimensional spaces | Let $X\_1,X\_2,\dots$ be iid random vectors each uniformly distributed on $S^{d-1}$. Let $S\_n:=\sum\_1^n X\_i$. By the symmetry, $EX\_1=0$. Also, $1=|X\_1|^2=\sum\_{j=1}^d X\_{1j}^2$, where $X\_1=(X\_{11},\dots,X\_{1d})$. Since the $X\_{1j}$'s are exchangeable and $1=E|X\_1|^2=\sum\_{j=1}^d EX\_{1j}^2$, we conclude th... | 13 | https://mathoverflow.net/users/36721 | 320649 | 138,452 |
https://mathoverflow.net/questions/320646 | 18 | We all know that
>
> A monoidal category is a bicategory with one object.
>
>
>
How do we fill in the blank in the following sentence?
>
> A multicategory is a ... with one object.
>
>
>
The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straig... | https://mathoverflow.net/users/70015 | A multicategory is a ... with one object? | This has been called a "fc-multicategory" by Tom Leinster, for example [here](https://arxiv.org/abs/math/9903004).
I think this as also been called a "Hypervirtual double category" [here](https://arxiv.org/pdf/1511.04070.pdf), but I don't remember if this is exactly the same notion or if there are some additional as... | 14 | https://mathoverflow.net/users/22131 | 320650 | 138,453 |
https://mathoverflow.net/questions/320640 | 0 | Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow that the spectrum of $D$ contains only eigenvalues?
EDIT: Diagonalisable means the existence of a countable ONB $e\_n... | https://mathoverflow.net/users/121660 | Spectrum equals eigenvalues for unbounded operator | I agree with Andreas that the obvious straightforward interpretation of "the eigenvalues grow to infinity" is that the sequence of eigenvalues $(\lambda\_n)$ increases to infinity. (And, counter to Sascha's interpretation, that "diagonalisable" means that the eigenvectors form a basis for $H$.) Under this interpretatio... | 4 | https://mathoverflow.net/users/23141 | 320657 | 138,456 |
https://mathoverflow.net/questions/320653 | 1 | Let $X \subset \mathbb{R}^{n}$ be compact and convex. Moreover, let $f:X \rightarrow \mathbb{R}$ be a differentiable map with $\sup\_{x \in X} \|\nabla f(x)\| = K < \infty$, where $\|\cdot\|$ denotes the Euclidean norm and $\nabla f$ the gradient.
Both [in this question](https://mathoverflow.net/questions/115458/sma... | https://mathoverflow.net/users/134473 | Smallest Lipschitz Constant of a Differentiable Function | This is essentially Corollary 1.42 in my book *Lipschitz Algebras* (second edition).
| 3 | https://mathoverflow.net/users/23141 | 320664 | 138,460 |
https://mathoverflow.net/questions/320666 | 9 | I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read through Schulten’s *Introduction to 3 manifolds*. Are there any good texts about these that focus on the geometric aspects ... | https://mathoverflow.net/users/132446 | Reference request for wild 3-manifolds | The only two books that I know of that focus on wild/pathological aspects of 3-manifolds are Bing's "Geometric topology of 3-manifolds" and Moise's "Geometric topology in dimension 2 and 3". But if you want to learn "Bing style topology", the best sources expand their focus to high-dimensional phenomena as well. Here t... | 14 | https://mathoverflow.net/users/317 | 320669 | 138,462 |
https://mathoverflow.net/questions/320683 | 1 | Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|\_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If I have the three bounds
$$\Bigg{|}\iiint\_{\mathbb{R}^3} K(x,y,z)f(y,z)g(x,z)h(x,y)dxdydz\Bigg{|}\le
\begin{cases} & C\... | https://mathoverflow.net/users/41581 | Interpolation of a trilinear functional | If you take the bilinear operator $T:(f,g) \mapsto \int K(x,y,z) f(y,z) g(x,z) ~\mathrm{d}z$, your three boundedness statements are equivalent to
* $T: L^2 \times L^4 \to L^{4/3}$ with norm $C\_1$
* $T: L^4 \times L^2 \to L^{4/3}$ with norm $C\_2$
* $T: L^4 \times L^4 \to L^2$ with norm $C\_3$
By multilinear inter... | 3 | https://mathoverflow.net/users/3948 | 320685 | 138,464 |
https://mathoverflow.net/questions/320687 | 1 | Lets say that I have a function $F(x,y)$ that is defined on nonnegative integers (or at least those are the values I care about) and is symmetric, so that $F(x,y)=F(y,x)$. Moreover, I know that for any fixed value of $y$ I have that $F(x,y)$ is a polynomial in $x$ of degree $y$. What can I say about the form of the fun... | https://mathoverflow.net/users/51694 | Function on two variables that restricts to a polynomial | With the polynomiality assumption a characterization is possible. I doubt that much can be said without it.
**Theorem:** Assuming that $F(x,y)$ is a polynomial in $x$ of degree $y$ for any $y\in \mathbb Z\_{\geq 0}$ we have that
$$F(x,y)=\sum\_{k\geq 0}\alpha\_k (x+y)^k\binom{x+y-2k}{y-k}$$
for some arbitrary sequenc... | 5 | https://mathoverflow.net/users/2384 | 320692 | 138,468 |
https://mathoverflow.net/questions/320633 | 6 | I am having trouble understanding this part in *Lurie's Higher Topos Theory*. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
>
> **Lemma 2.4.4.1.** Let $p : \mathcal{C} \rightarrow \mathcal{D}$ be an inner fibration of $\infty$-categories and
> let $X, Y \in \mathcal{C}$. The induced map
> $$\ph... | https://mathoverflow.net/users/131022 | Need help understanding comment in Higher Topos Theory | The version that you've given has a typo. In the latest version of HTT from Lurie's website, it instead defines $F$ as the fibre of $\mathcal{C}\_{/e}\to \mathcal{D}\_{/\bar{e}}\times\_{\mathcal{D}} \mathcal{C}$ over the point $(\sigma, x)$.
The comparison map that you want now is to show that actually $\phi^{-1}(\ba... | 4 | https://mathoverflow.net/users/1353 | 320693 | 138,469 |
https://mathoverflow.net/questions/320504 | 13 | Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W\_n(k))\_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of smooth projective $k$-varieties.
We know there is a morphism of (small) crystalline topoi
$$f\_{\rm cris} = (f^{-1}\_... | https://mathoverflow.net/users/nan | Functoriality of crystalline cohomology | Let's first figure out why the definition given in Berthelot-Ogus coincides with the one from the Stacks project.
Unraveling the definition 5.8.3 we see that for a sheaf $G$ on $(Y/W)\_{cris}$ the inverse image $f^{-1}G$ is the sheafification of the presheaf $$\varphi(G):(X\supset U\to T)\mapsto \mathrm{colim}\_{T\t... | 3 | https://mathoverflow.net/users/39304 | 320696 | 138,471 |
https://mathoverflow.net/questions/320647 | 5 | I am starting to study Lie algebras and when I reached the notion of **solvable Lie algebra**, I tryed to find concrete applications ( in physics for exemple) and I couldn't find one.
For exemple, **Solvable group** are very important for the unsolvability of quintic equation ( and by the way, it's the only application... | https://mathoverflow.net/users/134472 | Solvable Lie algebra application | There are actually lots of applications of solvable Lie algebras, especially in the field of integrable systems where the solvabiltiy of Hamilton's equations of motion is frequently related to the solvability of the Lie algebra of the integrals of motion of the system.
For example:
>
> for any Hamiltonian syste... | 1 | https://mathoverflow.net/users/85967 | 320701 | 138,473 |
https://mathoverflow.net/questions/264574 | 7 | Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated:
1. $\mathcal{C}$ is right proper.
2. There is an explicitly-describable set of generating acyclic cofibrations for $\mathcal{C}$.
(Of course, "explicitly-describable" is vague, but ... | https://mathoverflow.net/users/2362 | Explicit generating acyclic cofibrations and right properness of a model category | If $\mathcal C$ is the category of simplicial presheaves on a small category $A$ equipped with the injective model structure, it is proper, but it is very unlikely that we will get "explicit generating (trivial) cofibrations" without further assumptions (for instance on the indexing category $A$, such as being elegant ... | 6 | https://mathoverflow.net/users/1017 | 320702 | 138,474 |
https://mathoverflow.net/questions/320704 | 1 | Given an irrational $a$, the sequence $b\_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified.
Given a point $p$ in $\mathbb S^1$, can we obtain a quantitative upper bound (that can depend on $a, p, e$) on the smallest $n$ such that $na$ is in ... | https://mathoverflow.net/users/132446 | Quantitative bound on irrational rotation recurrence time | This is probably suboptimal, but one can use the convergents of the continued fraction expansion of $a$ to efficiently find $n \in \mathbb{N}$ such that $||na||\_\mathbb{Z} \leq e$, where $||\cdot||\_\mathbb{Z}$ denotes the distance to the nearest integer. Then one can find $m \in \mathbb{N}$ with $m \leq ||na||\_\math... | 5 | https://mathoverflow.net/users/30721 | 320707 | 138,475 |
https://mathoverflow.net/questions/320706 | 16 | Let $G\subset \mathrm{SU}(2)$ be a finite group. (These are famously classified through the McKay correspondence.) The Lie group framing of $\mathrm{SU}(2) = S^3$ descends to the quotient manifold $S^3 / G$, at least after getting some "left"s and "right"s in the correct places.
Every framed $k$-manifold determines a... | https://mathoverflow.net/users/78 | What is the value of $[S^3/G] \in \pi_3(Sphere)$ for a finite subgroup $G \subset SU(2)$? | The answer can be found in Theorem 2.1 from a paper of José Seade and Brian Steer (*Complex singularities and the framed cobordism class of compact quotients of $3$-dimensional Lie groups by discrete subgroups*, Comment. Math. Helv. **65** (1990), no. 3, 349–374, available [here](http://www.digizeitschriften.de/dms/res... | 18 | https://mathoverflow.net/users/17846 | 320708 | 138,476 |
https://mathoverflow.net/questions/320698 | 7 | In the discussion on the [nLab article](https://ncatlab.org/nlab/show/monadic+adjunction) for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called the semantics-structure adjunction.
It is later asked what exactl... | https://mathoverflow.net/users/92164 | Semantics-structure adjunction | I am not sure about what prof. Shulman refers to, but I am pretty confident that the answer to your question is on page 74 of *[Kan Extensions in Enriched Category Theory](https://doi.org/10.1007/BFb0060485)*, Lecture Notes in Mathematics **145**, by Eduardo Dubuc:
>
> There is an adjunction $\text{Str} \dashv \tex... | 7 | https://mathoverflow.net/users/104432 | 320722 | 138,481 |
https://mathoverflow.net/questions/320718 | 4 | If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to make the new function sum to $1$. (Rule out the degenerate case where $p(x\_0) = 1$ for a unique $x\_0$.)
It seems that... | https://mathoverflow.net/users/74610 | Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$ | Denote $f(p)=p(1-p)$, $H(p)=-p\log p$, let $p\_1,\dots,p\_n$ denote all positive probabilities of our distribution, then $\sum p\_i=1$, finally denote $s=\sum\_i f(p\_i)$. Then we need to prove the inequality
$$
\sum\_i H(f(p\_i)/s)\geqslant \sum\_i H(p\_i).
$$
Since $H$ is concave, it suffices to prove that
the multis... | 8 | https://mathoverflow.net/users/4312 | 320726 | 138,482 |
https://mathoverflow.net/questions/320725 | 6 | Here's a question I should know the answer to but don't:
>
> Suppose $1\to F \to G \to G/F \to 1$ is a short exact sequence of abelian groups with $F$ finite and $G/F$ torsion-free. Must the sequence split?
>
>
>
This is not true if you merely assume that $F$ is torsion. A counterexample is given by YCor here:... | https://mathoverflow.net/users/20598 | Finite-by-torsion-free abelian groups (or compact abelian groups with finitely many components) | Here’s a quick homological proof.
Suppose $F$ is finite and $H$ torsion free. Then $F\cong\text{Hom}(F,\mathbb{Q}/\mathbb{Z})$, so
$$\text{Ext}^1(H,F)\cong\text{Ext}^1\left(H,\text{Hom}(F,\mathbb{Q}/\mathbb{Z})\right)
\cong\text{Hom}\left(\text{Tor}\_1(H,F),\mathbb{Q}/\mathbb{Z}\right),
$$
which is zero since torsion... | 13 | https://mathoverflow.net/users/22989 | 320727 | 138,483 |
https://mathoverflow.net/questions/320716 | 2 | Does there exist a $C^\*$-algebra $A$ such that the center of $A$ is $0$ and $A$ also has a tracial state?
I know the fact that the center of $\mathcal{K}(H)$ is $0$, but $\mathcal{K}(H)$ has no tracial states.
| https://mathoverflow.net/users/63864 | center of a $C^*$-algebra | There are lots of AF simple nonunital C$^\*$-algebras with (finite) traces (these have trivial centre). For example, begin with the usual $2^{\infty}$ UHF algebra, call it $B$, and take an infinite strictly increasing sequence of projections, $(p\_n)$, and form $\cup\_n p\_n B p\_n$; let $A$ be its closure. Then $A$ is... | 7 | https://mathoverflow.net/users/42278 | 320738 | 138,487 |
https://mathoverflow.net/questions/152565 | 7 | The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a\_j\geq 0$, for a collection of balls $\{B\_j\}\_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum\_j a\_j\chi\_{\lambda B\_j}\bigg\|\_p\leq C(n,p,\lambda)\bigg\|\sum\_j a\_j\chi\_{B\_j}\bigg\|\_p.$$
I saw in some papers that people call this Bojarski'... | https://mathoverflow.net/users/26608 | Does anyone know what is the right reference for the following simple lemma from harmonic analysis? | The lemma is due to:
**J. O. Strömberg, and A. Torchinsky**.
Weights, sharp maximal functions and Hardy spaces. *Bull. Amer. Math. Soc. (N.S.)* 3 (1980), no. 3, 1053–1056.
The lemma is stated there without proof, but the proof is in the paper by Boman:
**J. Boman,** $L^p$-estimates for very strongly elliptic sy... | 6 | https://mathoverflow.net/users/121665 | 320747 | 138,491 |
https://mathoverflow.net/questions/320759 | 2 | Consider the following quasilinear elliptic equation
$$\nabla\_x (A(x,u(x),\nabla\_x u(x))) + f(u,x) = 0 $$
on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|\_{\partial \Omega} = 0.$$
* Where can I find references on this kinds of problems (in particular about a variational approa... | https://mathoverflow.net/users/nan | Quasilinear elliptic problem: Ellipticity-type conditions | Here is a basic result. For a detailed proof, see Theorem 5.4 in [2].
>
> **Theorem.** If $\Omega\subset\mathbb{R}^n$ is a bounded Lipschitz domain and $$
> F=F(x,u,\xi):\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R} $$
> satisfies
>
>
> * $F$ and $\nabla\_\xi F$ are continuous,
> * $\xi\mapsto F(x,u,\x... | 2 | https://mathoverflow.net/users/121665 | 320772 | 138,496 |
https://mathoverflow.net/questions/320779 | 4 | In the paper ["Normal Subgroups in the Cremona Group"](https://link.springer.com/content/pdf/10.1007/s11511-013-0090-1.pdf), under remark 5.1 they stated that for any generic set $\Sigma \subset \mathbb{P}^2\_\mathbb{C}$ of $k$ points, and $h$ is an automorphism of $\mathbb{P}^2\_\mathbb{C}$, then $h$ is the identity a... | https://mathoverflow.net/users/134269 | $h$ is identity as soon as $h(\Sigma)\cap \Sigma$ contains at least 5 points | In other words, if $\Sigma$ is generic, there does not exist a non-identity automorphism $h$ for which $|\Sigma\cap h(\Sigma)|\geqslant 5$. Denote $\Sigma=\{p\_1,\dots,p\_k\}$ and assume that $h(p\_i)=p\_{f(i)}$ for $i=1,2,3,4,5$ and a certain (we may supposed that fixed) injective function $f:\{1,2,3,4,5\}\rightarrow ... | 2 | https://mathoverflow.net/users/4312 | 320780 | 138,498 |
https://mathoverflow.net/questions/320781 | 0 | Suppose $X,Y$ are topological spaces with $Y$ [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space) and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$
Does this imply that there is a homeomorphism $\varphi:Y\to Y$ such tha... | https://mathoverflow.net/users/8628 | On a pair of continuous functions "connected" by continuous functions | **Counterexample.** Take $X=\{0,1,2,3\}$ to be the discrete space with four points and $Y=S=\{e^{it}; 0\le t<2\pi\}$ to be a circle. Then you can simply take different orderings or the images. (Or, to be more precise, sufficiently different orderings. This will not work, for example, if you simply shift the images cycl... | 2 | https://mathoverflow.net/users/8250 | 320783 | 138,499 |
https://mathoverflow.net/questions/320785 | 7 | Let $f\colon M \to N$ and $g\colon A \to N$ be two smooth maps between manifolds $A,M,N$.
Can one perturb $f$ to be transverse to $g$ (without touching $g$)?
Transverse meaning: For every $y\in f(M)\cap g(A)$ and every $x\in f^{-1}(y)$ and $a\in g^{-1}(y)$, we have
$$ Df(T\_x M) + Dg(T\_a A) = T\_yN. $$
I always ... | https://mathoverflow.net/users/67031 | perturbing one map to be transverse to a second map | I'm not sure what exactly you mean by perturb, but you can always make $f$ transverse to $g$ by a homotopy and this is often enough. This is proved in Section IV.2 of Kosinski's "Differential Manifolds" book, in particular see Corollary IV.2.5.
| 6 | https://mathoverflow.net/users/8103 | 320791 | 138,501 |
https://mathoverflow.net/questions/320422 | 4 | I would like to understand the proof of Lemma 1, page 339 in [this](http://www.alainconnes.org/docs/book94bigpdf.pdf) book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator chosen and we would like to construct a Fredholm module based on this dat... | https://mathoverflow.net/users/24078 | Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma | Here is the answer to the questions. (Recall $V$ is a closed oriented conformal manifold of dimension $2n$). According to the Hodge-deRham decomposition, any $\omega\in L^2(V,\Lambda^nT^\*V) $can be written
$$\omega=h+d\alpha+d^\*\beta$$ where $h\in L^2(V,\Lambda^nT^\*V)$ is harmonic, $\alpha\in W^{1,2}(V,\Lambda^{n-... | 2 | https://mathoverflow.net/users/48525 | 320792 | 138,502 |
https://mathoverflow.net/questions/320709 | 11 | I'll be working in the stable world. It's an easy observation that any 2-cell complex (over the sphere) with bottom cell in dimension zero is a Thom spectrum: any such complex is the cofiber of some element $\alpha\in \pi\_n S$, so it is the Thom spectrum of the map $S^{n+1} \to B\mathrm{GL}\_1 S$ classifying $\alpha$.... | https://mathoverflow.net/users/102390 | Finite complexes which are not Thom spectra | The proposed argument for why $Q = S \cup\_2 e^1 \cup\_\eta e^3$ is not a Thom spectrum seems to use that the Thom isomorphism commutes with the Steenrod operations, which is often false. The deviation is measured by the Stiefel-Whitney classes.
If $Q$ were the Thom spectrum $B^\gamma$ of a stable spherical fibration... | 7 | https://mathoverflow.net/users/9684 | 320805 | 138,511 |
https://mathoverflow.net/questions/261966 | 8 | Let $(X,\*),(Y,\*),(Z,\*)$ be finite algebras. The binary operations $\*$ are not required to satisfy any identities though I am interested in the special case where $\*$ is associative. Suppose that $f:X\rightarrow Z,g:Y\rightarrow Z$ are homomorphisms. Then is there already an optimized computer program that searches... | https://mathoverflow.net/users/22277 | Is there a good computer program for searching for endomorphisms between finite algebras which make diagrams commute? Is this problem NP-complete? | I mentioned this problem to one of my students, Kevin Berg, who proved that this homomorphism factorization problem is NP-complete when considered for the class of all finite algebras in a language containing at least 2 unary operations or containing at least one operation of arity at least 2. He also proved that the p... | 6 | https://mathoverflow.net/users/75735 | 320808 | 138,512 |
https://mathoverflow.net/questions/320786 | 2 | Let $X,Y$ be sets, $f, g:X\to Y$ be functions. We say $u:Y\to Y$ is a *transfer function* for $g$ to $f$ if $$f = u \circ g.$$ In that case we write $f \leq\_t g$. Let $\mathrm{Fct}(X,Y)$ denote the collection of all functions from $X$ to $Y$. So $\leq\_t$ is reflexive and transitive. The relation $\simeq\_t\subseteq \... | https://mathoverflow.net/users/8628 | Infima and suprema in the "transfer" function ordering | Here is an answer to the question as literally asked.
For a function $f: X \to Y$, let $\ker(f) = \{(x, x') \in X \times X: f(x) = f(x')\}$. This is of course an equivalence relation.
**Proposition:** For $f, g: X \to Y$, we have $f \leq\_t g$ iff $\ker(g) \subseteq \ker(f)$.
**Proof:** For "only if", notice t... | 2 | https://mathoverflow.net/users/2926 | 320810 | 138,513 |
https://mathoverflow.net/questions/320825 | 4 | A cardinal $\kappa$ is *real-valued measurable* if there is a $\kappa$-additive probability measure on $2^\kappa$ which vanishes on singletons. The existence of measurable $\kappa$ is independent of ZFC.
Question: if $\kappa$ is assumed to be real-valued measurable, does it necessarily follow that $2^\kappa$ is real-... | https://mathoverflow.net/users/12518 | Does measurability of cardinal $\kappa$ imply measurability of $2^\kappa$? | No, this is not necessarily true. In fact, this is almost never true - it can hold only if $2^\kappa$ is equal to the continuum.
Indeed, assume $2^\kappa$ is strictly greater than the continuum. By standard results (see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Measurable_cardinal#Real-valued_measurable)), a car... | 9 | https://mathoverflow.net/users/30186 | 320827 | 138,516 |
https://mathoverflow.net/questions/320815 | 2 | I am interested in an upper bound for
$$\sum\_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^3}$$
for some positive constant C. However I would like to do better. I think that the upper bound should be around
$... | https://mathoverflow.net/users/134544 | A truncated divisor sum | I will prove below that your bound $\frac{\exp\left(C\frac{\log N}{\log \log N}\right)}{A^3}$ (which follows from $\sum\_{d\mid N, d > A} \frac{1}{d^3} \le \frac{d(N)}{A^3}$) is optimal at least in the regime $A = N^c$, where $0 < c < 1$ is fixed. (note that we can't have $A$ very small since $\sum\_{d\in \mathbb{N}} \... | 4 | https://mathoverflow.net/users/104330 | 320831 | 138,518 |
https://mathoverflow.net/questions/320400 | 5 | A knot $K$ is said to have Property P if every nontrivial Dehn surgery on $K$ yields a 3-manifold that is not simply connected. It is known that every knot except the unknot has Property P. I am wondering what can be said about a link that admits a nontrivial Dehn surgery that yields $S^3$.
An $n$-component link $L$ ... | https://mathoverflow.net/users/99414 | Is there a generalized Property P - what can we say about framed link descriptions of $S^3$? | I know of no analogue of property P.
In fact, there are many links in $S^3$ which admit infinitely many fillings which are also $S^3$. The simplest is probably the Whitehead link: each component is unknotted, and one can "twist" along an unknotting disk bounding one component to obtain infinitely many non isotopic l... | 2 | https://mathoverflow.net/users/1345 | 320839 | 138,520 |
https://mathoverflow.net/questions/320841 | 8 | Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int\_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
Unfortunately one has no control over the error term $o(\log t)$ (other than what is implied by the above asymptotic behavio... | https://mathoverflow.net/users/70686 | Differentiating an integral that grows like log asymptotically | The answer is no, even in the smooth case. Take for example:
$$
f(x) = \frac{2}{x} + \frac{\cos(\log(x))}{x}
$$
Alter it on a small neighborhood of $0$ in such a way that there is no singularity there, preserving smoothness (this will be irrelevant for the asymptotics). This function is decreasing and, for $t$ suff... | 11 | https://mathoverflow.net/users/13915 | 320843 | 138,522 |
https://mathoverflow.net/questions/320172 | 7 | The following paper is well known, but hard to find:
**J. Boman**, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper the so called **Boman chain condition** has been introduced. However, most of the people who quote this paper hav... | https://mathoverflow.net/users/121665 | Famous but unavailable paper of Jan Boman | I went and scanned it in our library. Here's a Dropbox link.
<https://www.dropbox.com/s/ks9gdgi0xwl5j65/Boman%20-%20Lp-estimates%20for%20very%20strongly%20elliptic%20systems.pdf?dl=0>
| 13 | https://mathoverflow.net/users/1310 | 320853 | 138,526 |
https://mathoverflow.net/questions/320867 | 3 | *Context: Deligne, Theorie de Hodge II, section 1.4.8.*
Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\bullet}$ a resolution of $\mathcal{F}$ by a complex of $f\_{\*}$-acyclic sheaves.
The canonical filtration of $f\_{\*}\... | https://mathoverflow.net/users/77909 | Leray spectral sequence from hypercohomology | This is really a comment, but too long to write as one. Deligne leaves some steps to the reader. The filtration $\tau$ is increasing. To write the standard spectral sequence, we need to switch signs (as you seem to surmise) to get a decreasing filtration. Let me write $T^p = \tau\_{\le -p}$. Then we have
$$E\_1^{p+q} =... | 5 | https://mathoverflow.net/users/4144 | 320870 | 138,534 |
https://mathoverflow.net/questions/320856 | 1 |
>
> Suppose that $X$ is an unbounded random variable such that $\operatorname EX=0$ and $\operatorname E|X|^q=1$ with some $q>2$. Only the distribution of $X$ matters, so the probability space can be chosen freely. Choose any $p\in(0,1)$. I want to prove that there exists an event $F$ such that
>
>
> 1. $P(F)=p$;
>... | https://mathoverflow.net/users/46211 | Existence of certain event | For real $a$, let $Q(a):=P(|X|>a)$. Then the function $Q$ is right-continuous. So,
\begin{equation\*}
b:=a\_p:=\inf\{a\in\mathbb R\colon Q(a)\le p\}=\min\{a\in\mathbb R\colon Q(a)\le p\}\in\mathbb R
\end{equation\*}
and hence
\begin{equation}
P(|X|>b)=Q(b)\le p\le Q(b-)=P(|X|\ge b)\le1/b^q, \tag{1}
\end{equation}
... | 1 | https://mathoverflow.net/users/36721 | 320876 | 138,538 |
https://mathoverflow.net/questions/320539 | 1 | I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and since the PDE has some boundary conditions, we need to compute its hitting time $\tau$.
Numerically speaking, I start ... | https://mathoverflow.net/users/nan | Time discretization in the Feynman-Kac formula with boundary conditions | [Computing Exit Times with the Euler Scheme](https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2003/2003-02.pdf) by F. Buchmann addresses this problem in detail.
For a random walk with position $x\_{k+1}=x\_k+\delta x\_k$, incremented by a normally distributed stochastic variable $\delta x\_k$ with varia... | 0 | https://mathoverflow.net/users/11260 | 320883 | 138,544 |
https://mathoverflow.net/questions/320863 | 2 | Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^\*$ which is obtained from the first one plus some injected noise $\eta$, where every entry $\eta\_{i,j}$ is i.i.d. and comes from a normal distribution with mean $0$ and ... | https://mathoverflow.net/users/93775 | Are Linear Maps resistant to Noise? | I'm assuming the vectorized $\eta$ is uniformly distributed on the sphere of radius $1/10$ in $\mathbb R^{n^2}$.
For fixed $x$,
$$ \mathbb E \| \eta x \|\_2^2 = \mathbb E
\sum\_i \sum\_j \sum\_k \eta\_{ij} \eta\_{ik} x\_j x\_k = \sum\_j \sum\_k \mathbb E (\eta^T \eta)\_{jk} x\_j x\_k $$
Now for $j = k$,
$$ \mathb... | 2 | https://mathoverflow.net/users/13650 | 320888 | 138,546 |
https://mathoverflow.net/questions/320774 | 6 | Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $K$ and $p$ be a prime number. For a positive integer $i>0$, consider the etale cohomology $V=H^i(X\_{K^{alg}},\mathbb Q\... | https://mathoverflow.net/users/102104 | How to compute Galois representations from etale cohomology groups of a generalized flag variety? | Over $\overline{K}$ we can decompose $G/P$ into Schubert cells. There is an action of $\operatorname{Gal}(K)$ on this set of Schubert cells. This is not completely obvious, as the Galois conjugate of a Schubert cell may not be a Schubert cell for the same Borel, but we can fix this by noting that it is a translate of o... | 2 | https://mathoverflow.net/users/18060 | 320892 | 138,547 |
https://mathoverflow.net/questions/291430 | 7 | A standard example of Nakajima quiver varieties are type A Springer resolutions $\widetilde{\mathcal{N}} \to \mathcal{N}$. In the theory of Springer resolutions it is often beneficial to consider the full Grothendieck-Springer resolution, i.e. to work with the commutative diagram
$$\require{AMScd}\begin{CD}
\widetild... | https://mathoverflow.net/users/459 | Quiver variety analogue of Grothendieck-Springer resolution | There are of course two moment maps to vary - the complex one and the real one. In most treatments of quiver varieties one fixes the complex level set to be zero and the real level set to a nonzero multiple of the identity, with the zero multiple giving the "quiver affine variety" $\mathfrak M\_0$ (best not referred to... | 2 | https://mathoverflow.net/users/391 | 320904 | 138,551 |
https://mathoverflow.net/questions/320906 | 18 | I am looking for information on the mathematician Anne Marie
Whitney. She wrote a number of significant papers related to total positivity with her thesis adviser Isaac Schoenberg. All I could find on the internet is
<https://www.genealogy.math.ndsu.nodak.edu/id.php?id=36202>.
| https://mathoverflow.net/users/2807 | Biographical information on Anne Marie Whitney | There is more under her married name, **Anne Calloway** (October 2, 1921, December 27, 2008). Here is a [photograph](https://www.langelands.com/notices/Anne-Calloway). Anne Whitney married her graduate school class mate [Jean Calloway,](https://books.google.nl/books?id=RfXGJBB1HvoC&pg=PA69) who himself became a profess... | 20 | https://mathoverflow.net/users/11260 | 320909 | 138,553 |
https://mathoverflow.net/questions/320858 | 11 | Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to determine whether this algebra has global dimension 3 or not.
Let $A=K\langle a,b\rangle/I$ with $I$ the ideal generated b... | https://mathoverflow.net/users/61949 | Does this algebra have finite global dimension ? (Human vs computer) | Let $M = P \oplus I \oplus \tau\_2 \oplus \tau\_2^2 \oplus \tau\_2^3$, where the notation is the obvious one. One way of computing the global dimension of $B=\operatorname{End}(M)$ is to find the projective resolution of all the simple $B$-modules. The simple $B$-modules are given by for each indecomposable direct summ... | 11 | https://mathoverflow.net/users/130741 | 320918 | 138,556 |
https://mathoverflow.net/questions/320913 | 3 | Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \subset \mathbb R^2$ and a "nice" function $\varphi \colon \partial \Omega \to \mathbb R$ I want to study the problem
$$
\... | https://mathoverflow.net/users/119793 | Uniqueness of minimizers in the Calculus of Variations | There's no uniqueness in general. If $\Omega$ is a unit square, and $$\varphi(x, y) = (1-x-y)^+,$$ then obviously any $u$ which is decreasing with respect to both $x$ and $y$, and which matches the boundary condition, has energy zero, and hence it is a minimiser. However, there are [many $u$ like that](https://www.wolf... | 5 | https://mathoverflow.net/users/108637 | 320919 | 138,557 |
https://mathoverflow.net/questions/320916 | 13 | This might be forced to migrate to math.SE but let me still risk it.
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a *natural* isomorphism between them?
To make the question more MOish - choosing a basis, elements of $\operatorname{S}^2(V)$ can be ident... | https://mathoverflow.net/users/41291 | Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$? | Let $E$ be a $2$-dimensional $k$-vector space. The Wronksian isomorphism is an isomorphism of $\mathrm{SL}(E)$-modules $\bigwedge^m \mathrm{S}^{m+r-1}(E)\cong \mathrm{S}^m \mathrm{S}^r(E) $. It is easiest to deduce it from the corresponding identity in symmetric functions (specialized to $1$ and $q$), but it can also b... | 21 | https://mathoverflow.net/users/7709 | 320927 | 138,560 |
https://mathoverflow.net/questions/320933 | 11 | While a great book about measure theory and real analysis in $\mathbb R^n$, the only downside is the lack of exercises. Can anyone provide a good book to supplement it with exercises? I plan to use it for self study.
| https://mathoverflow.net/users/132446 | Textbook recommendation request: Exercises to supplement Evans and Gariepy | If you look for a generic collection of problems in measure theory and functional analysis, I would highly recommend:
**A. Torchinsky,** *[Problems in real and functional analysis](https://books.google.com/books?id=wB-jCgAAQBAJ).* [Graduate Studies in Mathematics, 166](https://bookstore.ams.org/gsm-166/). American Ma... | 6 | https://mathoverflow.net/users/121665 | 320935 | 138,564 |
https://mathoverflow.net/questions/320929 | 3 | Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F\_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y \subset X \times \mbox{Spec}(A)$ the scheme-theoretic support of $F\_A$. Then, does there exist a non-empty open subse... | https://mathoverflow.net/users/38832 | Is the support of a flat module generically flat? | Let $A$ be a non-reduced, Artinian ring with maximal ideal $\mathfrak{m}$. The underlying topological space of $\text{Spec}\ A$ is a one-point space. Thus, the unique nonempty open is the entire space $\text{Spec}\ A$. Thus, the problem asks, for every $A$-flat coherent sheaf on a finite type $A$-scheme, whether the sc... | 4 | https://mathoverflow.net/users/13265 | 320936 | 138,565 |
https://mathoverflow.net/questions/320845 | 7 | When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, we know the associated L-function $L(s, \chi)$ can be completed into
$$\Lambda(s,\chi) = \left(\frac{\pi}{q}\right)^{-\f... | https://mathoverflow.net/users/128718 | Functional equation for general number fields | As Kimball mentioned, this is all in Godement-Jacquet's monograph "Zeta Functions of Simple Algebras". For the case $n = 1$, this is just Tate's thesis. When the field is $\mathbb{Q}$, a good reference is Goldfeld-Hundley "Automorphic Representations and $L$-Functions for the General Linear Group".
The story is rough... | 7 | https://mathoverflow.net/users/3803 | 320937 | 138,566 |
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