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https://mathoverflow.net/questions/348731 | 0 | $P$ means polynomial complexity.
$S\_p$ is class of all $P$\_random sets, and $S\_{pc}$ is class of all $P$ incomputable sets, is $S\_{pc} \setminus S\_p$ empty? If not empty, any example?
what is the result, if we replace $P$ complexity with $NP$?
Moreover, $S$ is class of all random sets, and $S\_c$ is class of a... | https://mathoverflow.net/users/14024 | Are all $P$-noncomputable sets $P$-random? | No, the differences are not empty. As an example, take any noncomputable sequence such that every bit is repeated. That is, the $(2n)$th bit is the same as the $(2n+1)$th bit, for all $n$. This will fail to be $P$-random.
| 3 | https://mathoverflow.net/users/32178 | 348750 | 147,618 |
https://mathoverflow.net/questions/348745 | 1 | In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve. He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$ is no greater than its multiplicity as a pole of $x$. Then he says there is some natural number $k$ and some complex $a\neq 0$ such that ... | https://mathoverflow.net/users/38783 | Some simple algebra of rational functions by André Weil | Let us write the irreducible equation relating $x$ and $y$ as
$$P\_k(x)y^k+P\_{k-1}(x)y^{k-1}+\ldots+P\_0(x)=0.$$
Consider the Newton polygon (the graph of the smallest concave function $\phi$ with
$\phi(j)\geq \deg P\_j,\; 0\leq j\leq k$.
Condition on the poles of $x$ and $y$ tells us that
$P\_k=\mathrm{const}$, and... | 3 | https://mathoverflow.net/users/25510 | 348752 | 147,619 |
https://mathoverflow.net/questions/348682 | 14 | Let $m$ be Lebesgue measure on $\mathbb R$, and let $m\_i$ and $m\_o$ be the inner and outer measures respectively.
>
> Is it the case that for all $A \subset \mathbb R$ and all $x \in [m\_i(A), m\_o(A)]$ there exists a countably additive extension $m^+$ of $m$ to the powerset of $\mathbb R$ such that $m^+(A)=x$?
... | https://mathoverflow.net/users/96899 | On the existence of a family of countably additive extensions of Lebesgue measure | Suppose $\kappa$ is the least real-valued measurable cardinal and $\nu:\mathcal{P}(\kappa) \to [0, 1]$ is a witnessing $\kappa$-additive probability measure. Gitik and Shelah showed that the Maharam type of the measure algebra $\mathbb{M}$ of $\nu$ is $\geq \kappa^+$ (See Theorem 2.6 in Gitik-Shelah, Forcing with ideal... | 10 | https://mathoverflow.net/users/2689 | 348759 | 147,621 |
https://mathoverflow.net/questions/348755 | 3 | We say that a finite, simple, undirected graph $G=(V,E)$ is $k$-critical for $k\in\mathbb{N}$ if $\chi(G)=k$ and $\chi(G\setminus \{v\}) = k-1$ for all $v\in V(G)$. Let $\delta(G)$ denote the minimum degree of $G$.
It is easy to see that $\delta(G)\geq k-1$ for any $k$-critical graph $G$.
Is there a global constant... | https://mathoverflow.net/users/8628 | Minimal degree in a critical graph | The only $2$-critical graph is $K\_2$ and the only $3$-critical graphs are odd cycles, therefore the answer is yes for $k\in \{2,3\}$. For $k\geq 4$ the answer is no.
Let's start with the easy case, $k\geq 6$. The Dirac construction of critical graphs starts with two graphs $G\_1$ which is $k\_1$-critical, and $G\_2$... | 5 | https://mathoverflow.net/users/2384 | 348761 | 147,622 |
https://mathoverflow.net/questions/348585 | 7 | I've come upon [this](https://mathoverflow.net/questions/61141/generalization-of-borsuk-ulam) MO post, and I was wondering if there is a way to generalize what is said in the answer at the very bottom. Specifically, we have that given continuous maps $f: \mathbb{S}^n \to \mathbb{S}^n$, and $g: \mathbb{S}^n \to \mathbb{... | https://mathoverflow.net/users/143629 | About a generalization of the Borsuk-Ulam theorem | A first observation is that a map $f:S^n\to S^n$ with $f^k=\operatorname{Id}\_{S^n}$ generates a topological action of the cyclic group $\mathbb{Z}/k$ on $S^n$. So we are talking about generalising from $\mathbb{Z}/2$ actions to $\mathbb{Z}/k$ actions.
There are many generalisations of the Borsuk-Ulam theorem of this... | 3 | https://mathoverflow.net/users/8103 | 348769 | 147,623 |
https://mathoverflow.net/questions/348772 | 7 | Let $G$ be a finite group and $D(G)$ its quantum double. As in [my previous question](https://mathoverflow.net/questions/347944/classification-of-operatornamerep-dg), a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\theta$ is a conjugacy class of $G$ and ... | https://mathoverflow.net/users/124549 | Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$ | 1)2) is standard for an arbitrary f.d. Hopf algebra $H$, as you say it's not hard to idenfity $D(H)$-modules with Yetter-Drinfeld modules. Then, given two of those, say $V,W$ you can define a braiding by
$$V \otimes W \rightarrow H \otimes V \otimes W \rightarrow H \otimes W \otimes V \rightarrow W \otimes V$$
where t... | 4 | https://mathoverflow.net/users/13552 | 348777 | 147,624 |
https://mathoverflow.net/questions/348775 | 2 | I recently asked this question [Unbounded sectional curvature implies infinite diameter?](https://mathoverflow.net/questions/348751/unbounded-sectional-curvature-implies-infinite-diameter).
I would like now to ask something similar, but in another context.
Suppose you have a complete metric space $(M,d)$. Assume tha... | https://mathoverflow.net/users/94097 | Unbounded curvature implies infinite diameter on complete metric spaces | Here is counter example. Consider the surface of the unit cube in $\mathbb{R}^3$. This is compact and has infinite curvature at the corners. One can also construct a sequence of smooth compact manifolds that converge to this surface, some care needs to be taken of what kind of convergence you want.
**Edit:** Thanks t... | 2 | https://mathoverflow.net/users/58103 | 348779 | 147,625 |
https://mathoverflow.net/questions/348767 | 10 | I deleted by previous questions, seems they are too vague. Let me try to ask a more precise question.
Let $f:G\rightarrow K$ a morphism of simplicial groups such that $f$ is a weak homotopy equivalence of underlying simplicial sets. We will make to assumptions:
1) for each natural number $i$, $G\_{i}$ is a free group... | https://mathoverflow.net/users/141114 | abelianization and homotopy | The map is not necessarily a weak equivalence, even if $K\_i$ is actually the whole product of free groups rather than just a subgroup.
Let $K$ be the constant simplicial set which is $\Bbb Z^2$ in each degree. The map $K \to K\_{ab}$ is an isomorphism, and $K = K\_{ab}$ has no higher homotopy groups.
Let $G$ be a ... | 7 | https://mathoverflow.net/users/360 | 348786 | 147,628 |
https://mathoverflow.net/questions/348527 | 5 | A fusion ring $\mathcal{F}$ (of rank $r$) is given by a finite set $B = \{b\_1,b\_2, \dots, b\_r \}$ such that $b\_i b\_j = \sum\_k n\_{i,j}^k b\_k$ with $n\_{i,j}^k \in \mathbb{Z}\_{\ge 0}$, satisfying axioms slightly augmenting the group axioms (see the details [here](https://mathoverflow.net/q/344079/34538)). The fu... | https://mathoverflow.net/users/34538 | What is the smallest rank for a noncommutative fusion ring? | I think that a noncommutative fusion ring of rank 5 does not exist. Namely, let $a$ and $b$ be the formal codegrees (see <https://arxiv.org/pdf/0810.3242.pdf>) of such ring. Then $a$ and $b$ are positive (EDIT: and rational, see the explanation by Noah) integers satisfying $\frac1a+\frac2b=1$ (see Proposition 2.10 in <... | 6 | https://mathoverflow.net/users/4158 | 348789 | 147,630 |
https://mathoverflow.net/questions/348757 | 7 | Let $\mathcal{C}\_n$ be the monoid of self-maps $\alpha$ of $\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$ and decreasing ($\forall x$, $\alpha(x)\le x$).
Let N($\mathcal{C\_n})$ be the set of nilpotent elements of $\mathcal{C}\_n$, i.e., those $\alpha\in\mathc... | https://mathoverflow.net/users/132399 | Counting nilpotent self-maps of $\{1,\dots,n\}$ with image of a given cardinal | The answer is ${n-2\choose r-1}{n-1\choose r-1}-{n-2\choose r-2}{n-1\choose r}=\frac1{n-r}{n-2\choose r-1}{n-1\choose r}$.
Let $\{1=a\_0<a\_1<a\_2<\ldots<a\_{r-1}\}$ be the image of $\alpha\in X\_{n,r}$ and denote $p\_i:=\max \alpha^{-1}(a\_{i-1})$ for $i=1,\ldots,r-1$. Then $p\_1<p\_2<\ldots p\_{r-1}<n$ and $p\_i\ge... | 5 | https://mathoverflow.net/users/4312 | 348790 | 147,631 |
https://mathoverflow.net/questions/348810 | 16 | Suppose that $G, H$ are **finitely generated** groups such that $H$ is isomorphic to a finite index subgroup of $G$ and vice versa. Does it follow that $G$ is isomorphic to $H$?
I am sure that the answer is negative but cannot find an example. I am mostly interested in the case of finitely presented groups. The assu... | https://mathoverflow.net/users/39654 | Groups containing each other as finite index subgroups | Simple counterexample:
$G$ is the square of an infinite dihedral group,
consisting of symmetries of the ${\bf Z}^2$ lattice
of the form $(x,y) \mapsto (\pm x + a, \pm y + b)$
with $a,b \in \bf Z$;
and $H$ is the index-$2$ subgroup where $a \equiv b \bmod 2$.
Then $H$ has index-$2$ subgroup consisting of the symmetries... | 35 | https://mathoverflow.net/users/14830 | 348812 | 147,637 |
https://mathoverflow.net/questions/348813 | 13 | Let $S$ be a compact connected orientable bordered surface of genus $g$ with $n$ holes (a hole is a component of the border homeomorphic to a circle). Consider a cell decomposition (the closure of each cell is a closed disk of the same dimension as the cell) with $f$ faces, $e\_i$ interior edges, $e\_b$ boundary edges,... | https://mathoverflow.net/users/25510 | Elementary topology of surfaces | The answer is negative. First, take two hexagons $A\_1 B\_1 C\_1 D\_1 E\_1 F\_1$ and $A\_2 B\_2 C\_2 D\_2 E\_2 F\_2$ and identify vertices $(A\_1,A\_2), (B\_1,B\_2)$, as well as glue the pairs of edges $(F\_1A\_1,F\_2A\_2), (B\_1C\_1, B\_2C\_2)$. Now add edges $D\_1D\_2, E\_1E\_2$ and a $2$-cell $D\_1D\_2E\_2E\_1$. You... | 13 | https://mathoverflow.net/users/2384 | 348820 | 147,638 |
https://mathoverflow.net/questions/348823 | 2 | I have a graph clustering problem I'm working on and it basically involves finding a factorization of the adjacency matrix $A$ such that the following equations are (approximately) satisfied:
$$
A \approx L \tilde{A} R,\quad
RL\approx \mathbb{I}\_k \\
A \in \mathbb{R}^{n \times n},\;
L \in \mathbb{R}^{n \times k},\;
R ... | https://mathoverflow.net/users/150152 | Matrix factorization for dimensional reduction similar to spectral decomposition/SVD | 1. I would suggest you to attach no particular meaning to $\tilde{A}$ being diagonal, because you have enough freedom to introduce changes of basis there: for any invertible $M$, you can replace $L,\tilde{A},R$ with $LM$, $M^{-1}\tilde{A}M$ and $M^{-1}R$. So $\tilde{A}$ can always be made diagonal, or at least in Jorda... | 3 | https://mathoverflow.net/users/1898 | 348831 | 147,642 |
https://mathoverflow.net/questions/348829 | 1 | I have studied some papers related to solving integer programs via Gröbner bases. I wonder if the other way is possible or not — i.e., given any ideal, can we find the Gröbner basis by translating this into an integer program and then solving it by the usual branch & bound method?
The professor teaching the course ha... | https://mathoverflow.net/users/150157 | Gröbner basis via integer programming | ILP is $NP$-complete, while computing a Grobner Basis is $EXPSPACE$-complete. As one has the containments $NP\subseteq PSPACE\subsetneq EXPSPACE$, one should expect that you can reduce ILP to computing a Grobner basis, but not the other direction.
| 3 | https://mathoverflow.net/users/101207 | 348834 | 147,643 |
https://mathoverflow.net/questions/348809 | 3 | Given a connected graph $G$, the *Cheeger constant* $h(G)$ (a.k.a. *Cheeger number* or *isoperimetric number*) roughly measures the "bottleneckedness" of $G$. See [Wikipedia](https://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)) for the precise definition.
I want to have an approximate value of the maximal C... | https://mathoverflow.net/users/17294 | Reference request: maximal Cheeger constant for 3-regular graphs | This is expander territory and someone will doubtless give a reference soon.
Meanwhile, here's a simple proof that $\liminf h\_n \le 1$.
Consider a connected induced subgraph $H$ with $n\_1,n\_2,n\_3$ vertices of degree 1,2,3, respectively. Since $H$ is connected, we have $n\_1+2n\_2+3n\_3\ge 2(n\_1+n\_2+n\_3)-2$ (... | 2 | https://mathoverflow.net/users/9025 | 348837 | 147,645 |
https://mathoverflow.net/questions/347212 | 2 | For $\kappa >1$ and $t,X\geq 1$ $$\sum \_{n\leq X}a\_n=\frac {1}{2\pi i}\int \_{\kappa \pm iT}\frac {\mathcal F(s)X^sds}{s}+\mathcal O\left (x^\kappa \sum \_{=1}^\infty \frac {1}{n^\kappa (1+T|\log (X/n)|)}\right )$$ where $a\_n\ll 1$ and $$\mathcal F(s)=\sum \_{n=1}^\infty \frac {a\_n}{n^s}.$$ This is a quantitative v... | https://mathoverflow.net/users/110603 | Quantitative Perron formula with weights | If $|a\_n|\ll 1$ and $c>1$, then
$\displaystyle\sum\_{n\leq x}(x-n)a\_n = \frac{1}{2\pi i}\int\_{c-iT}^{c+iT}\mathcal{F}(s)\frac{x^{s+1}}{s(s+1)}ds+O\Big(\frac{x^{c+1}(\log x)^2}{T^2}\Big)$.
A detailed proof can be found in Murty's "Problems in Analytic Number Theory", solution to Problem 4.1.8.
| 3 | https://mathoverflow.net/users/111215 | 348848 | 147,648 |
https://mathoverflow.net/questions/348832 | 1 | Let $(X,\mathscr X,\mathbb P)$ be a probability space, $(Y,\mathscr Y)$ a measurable space, and $h:X\times Y\to\mathbb R$ a real-valued function measurable with respect to the product $\sigma$-algebra $\mathscr X\otimes\mathscr Y$ (where $\mathbb R$ is endowed with the Borel $\sigma$-algebra).
Moreover, let $\mathscr... | https://mathoverflow.net/users/55976 | Almost identical $\sigma$-algebras and measurability | I believe the answer is yes. First of all, either $\mathscr F=\mathscr G$ or both $\sigma$-algebras consist of sets of measure $0$ or $1$. Indeed, suppose that a set $E$ is in $\mathscr F$ but not in $\mathscr G$ and $\mathbb P(E)=1$ (otherwise, take $E^c$). For every set $A\in \mathscr F$, either $A\cap E$ or $A^c\cap... | 1 | https://mathoverflow.net/users/143037 | 348850 | 147,649 |
https://mathoverflow.net/questions/348845 | 4 | Brocard's conjecture states that: If $p\_{k}$ and $p\_{k+1}$ are consecutive prime numbers greater than $2$, then between $p\_{k}²$ and $p\_{k+1}²$ there are at least four prime numbers.
I know that is statement is not yet proved. But I am asking on a **weaker** version:
Show that there is infinitely many indices $k... | https://mathoverflow.net/users/74668 | A weaker version of the Brocard's Conjecture | Theorem: For any constant $c$ there are infinitely many primes $p\_k$ such that there are at least $c$ primes between $p\_k^2$ and $p\_{k+1}^2$.
Proof: Fix a $c$. Assume that for sufficiently large $k$ there are never more than $c$ primes between $p\_k^2$ and $p\_{k+1}^2$. Then for sufficiently large $n$ there are n... | 10 | https://mathoverflow.net/users/127690 | 348855 | 147,650 |
https://mathoverflow.net/questions/348802 | 2 | Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )\_{n\in\mathbb{N}}$ becomes arbitrarily close to squares?
More precisely, what can be said about the set of $\alpha$ such that for any $\varepsilon >0$ there are infinitely many $n,k\in \mathbb{N}$ such that
$$... | https://mathoverflow.net/users/42864 | Approximation of a square with an irrational arithmetic progression | This holds for all $\alpha \in \bf R$. If $\alpha \in \bf Q$ it's easy,
so we may assume $\alpha$ irrational. Divide by $\alpha$ to get
$$
|\alpha^{-1} k^2 - n| < \alpha^{-1} \epsilon.
$$
So, we want to show that $\alpha^{-1} k^2$ comes arbitrarily close to integers.
This is a special case of Weyl's
[equidistribution
... | 5 | https://mathoverflow.net/users/14830 | 348859 | 147,651 |
https://mathoverflow.net/questions/348847 | 2 | I was considering the following problem. Let $\{(X\_i,Y\_i)\}\_{i=1}^n$ be i.i.d. zero-mean random vectors with covariance matrix
\begin{equation}
\mathrm{Cov}\{(X\_1,Y\_1)\}=\begin{pmatrix}
1 & \sigma\\
\sigma & 1
\end{pmatrix}.
\end{equation}
We assume the covariance matrix can be nearly degenerate, that is, $X$ ... | https://mathoverflow.net/users/78326 | Multivariate Berry-Esseen Theorem for possibly co-linear random vector | By Theorem 1.3 in the [article of Götze you linked to](https://projecteuclid.org/download/pdf_1/euclid.aop/1176990448), the answer is yes, because the result holds for all convex sets and by a linear transformation you can make the covariance an identity. In fact a result of Sazonov (1968) that Götze refers to already ... | 1 | https://mathoverflow.net/users/143037 | 348866 | 147,655 |
https://mathoverflow.net/questions/348634 | 4 | For concreteness, let's say that $(X,d)$ is a metric space homeomorphic to $\mathbb{R}^2$ whose Hausdorff 2-measure $\mathcal{H}\_d^2$ is locally finite.
We can pass from $(X,d)$ to the length metric, denoted by $\overline{d}$, defined by the infimum of the length of rectifiable curves joining two points in $X$. The ... | https://mathoverflow.net/users/126691 | Can passing to a length metric increase Hausdorff measure? | It's often the case that a question is easier to answer on my own after I've posted it to mathoverflow.
The answer seems to be "yes", that Hausdorff measure and even dimension can be increased. It might be some work to write down the details carefully (which I intend to do), but here is the idea: Pick a suitable Cant... | 0 | https://mathoverflow.net/users/126691 | 348877 | 147,658 |
https://mathoverflow.net/questions/348870 | 0 | The following question remain open and requests further research given function hypothesis.
Is it possible to obtain a closed expression to the inverse of a function integral.
$\int\_{0}^t \frac{1}{f(\tau)} d \tau = g(t) - g(0)$
Above the corresponding function is g(t). I thank in advance.
| https://mathoverflow.net/users/148215 | Integral of inverse of a function | Take $f(\tau):=\dfrac{1}{e^{-\tau^2}}$, the integral you get can not be expressed by means of usual functions. You can just get an expression with the error function $\mathrm{erf}$.
| 2 | https://mathoverflow.net/users/124904 | 348879 | 147,659 |
https://mathoverflow.net/questions/348852 | 1 | let $$\Pi\binom{n}{k}:=\mathrm{card}\left( \left\lbrace \lbrace \Pi\_1^n\,\cdots\,\Pi\_k^n\rbrace\,|\,0\leq \pi\_{r,c}\in\sum\_{i=1}^k\Pi\_i^n\ni\pi\_{r,c}\leq 1\right\rbrace\right)$$ be the number of sets with $k\leq n$ distinct $n\times n$ permutation matrices *without* common non-zero entry.
>
> **Questions:** ... | https://mathoverflow.net/users/31310 | Calculating the values of a generalization of binomials to permutations | It is a $k\times n$ latin rectangle: write the permutations one per row.
[This paper](https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/on-the-number-of-latin-rectangles/E093CA0EC0A723261F4D635AED40A567) has a nice summary of theoretical and practical methods.
The sum o... | 3 | https://mathoverflow.net/users/9025 | 348885 | 147,661 |
https://mathoverflow.net/questions/348602 | 0 | Let $A\_n$, $B\_n$ for $n \in \mathbb N$ be finte subsets of compact set $X$ in $\mathbb C$ such that
$A\_n \subset B\_n$.
Let $\delta\_{A\_n}:= \frac{1}{|A\_n|} \displaystyle\sum\_{x\in A\_n} \delta\_x$ and $\delta\_{B\_n}:=\frac{1}{|B\_n|} \displaystyle\sum\_{x\in B\_n} \delta\_x$ be normalized dirac probability me... | https://mathoverflow.net/users/130742 | Absolute continuity of limiting measures | Let $\nu:=\sigma$. This answer, based mainly on comments by Anthony Quas, provides a necessary and sufficient condition for $\mu\ll\nu$ (the absolute continuity of $\mu$ with respect to $\nu$) in terms of $|A\_n|$ and $|B\_n|$, assuming that $X$ contains at least two distinct non-isolated points.
More specifically, ... | 1 | https://mathoverflow.net/users/36721 | 348893 | 147,663 |
https://mathoverflow.net/questions/348886 | 1 | I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between exponentiation and taking the derivative.
(From [Wikipedia](https://en.wikipedia.org/wiki/Derivation_(differential_algebra... | https://mathoverflow.net/users/150174 | Link btw. exponential and derivatives from an algebraic perspective | Take a look at the Heaviside operational calculus and its relation to the Laplace transform solns. of differential eqns. by transforming them into algebraic equations.
As far as the derivation formula goes, just look at action on $1 = 1 \cdot 1$ and $x = x \cdot 1$ then $x^2 = xx$, etc. to develop a formula for deriv... | 0 | https://mathoverflow.net/users/12178 | 348895 | 147,664 |
https://mathoverflow.net/questions/348484 | 1 | Let $m\geq 3$ be fixed and $n\to\infty$. Consider $v=(v\_j)\_{j\leq m}$ with $v\_1,\ldots,v\_m\in \{-1,+1\}^n$. Let:
* $N\_I(v)$ be the number of sequences $u\_1,\ldots,u\_m\in \{-1,+1\}^n$ isometric to $v$ in $\mathbb{R}^n$, i.e. $u\_j=Qv\_j$ for some orthogonal matrix $Q$.
* $N\_S(v)$ be the number of sequences $u\... | https://mathoverflow.net/users/143037 | How typical are integer isometries on a hypercube? Littlewood-Offord problem for Bernoulli Gram matrices | By the Chernoff bound, we see that for each $1 \leq i < j \leq m$, one has $u\_i \cdot u\_j = O(\sqrt{n})$ with probability at least $1-\frac{1}{10m^2}$ (say), where implied constants are allowed to depend on the fixed constant $m$. Thus, with probability at least $1-\frac{1}{10}$, the random variable $U$ takes values ... | 3 | https://mathoverflow.net/users/766 | 348899 | 147,665 |
https://mathoverflow.net/questions/348906 | 4 | Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories
$$
D^b(coh(X))\overset{\sim}{\to} D^b\_{coh}(Qcoh(X))
$$
where $D^b(coh(X))$ is the derived category of bounded complexes of coherent sheaves, and $D^b\_{coh}(Qcoh(X))$ is the derived catego... | https://mathoverflow.net/users/24965 | A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$ | Indeed, the proof in the book shows that if $G$ is a bounded complex of quasi-coherent sheafs with coherent cohomologies, then there is a subcomplex $G\_1\subseteq G$ with same cohomologies of $G$, where $G\_1$ is bounded complex of coherent sheafs. Then you can prove directly that two bounded categories have same hom ... | 3 | https://mathoverflow.net/users/106580 | 348910 | 147,669 |
https://mathoverflow.net/questions/348903 | 8 | In a footnote to the 2018 Zerbes-Loeffler [lecture notes](http://swc.math.arizona.edu/aws/2018/2018LoefflerZerbesNotes.pdf) from the Arizona Winter School, it's stated that Euler systems constructed from algebraic cycles "cannot give a full Euler system, only an anticyclotomic one." Why is this the case? It is not obvi... | https://mathoverflow.net/users/120548 | Why can Euler systems constructed from algebraic cycles only be anticyclotomic? | Let me explain a bit more what that footnote was supposed to mean.
As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1(L, V)$, as $L$ varies over a suitable class of abelian extensions of $K$. If we're willing to temporarily forget about... | 13 | https://mathoverflow.net/users/2481 | 348922 | 147,674 |
https://mathoverflow.net/questions/348862 | 7 | I would like a reference for the result [here](https://en.wikipedia.org/wiki/Weyl%27s_inequality#Weyl's_inequality_in_number_theory). Having that $t$ there makes me happy. I would prefer not to have to, in my paper, run through and (not trivially but not too greatly) alter the proof of the standard Weyl inequality to g... | https://mathoverflow.net/users/129185 | Quick reference for general Weyl's inequality in number theory | This form of Weyl's inequality is due to [Ivan Matveevich Vinogradov](https://en.wikipedia.org/wiki/Ivan_Vinogradov) and the relevant reference is the 1927 paper [3]. Precisely, **Lemma III** at pages 568-569 states the following equivalent form: if
$$
S=\sum\_{x=N+1}^{N+P} e^{2\pi i f(x)},\quad f(x)=\lambda x^n+\ldots... | 12 | https://mathoverflow.net/users/113756 | 348923 | 147,675 |
https://mathoverflow.net/questions/348917 | 4 | Let $C\_d$ be a smooth curve of degree $d$ in $\mathbb{CP}^2$.
If we pick some homogeneous coordinates $[z\_0:z\_1:z\_2]$ on $\mathbb{CP}^2$, then
$C\_d$ is the zero set of a generic polynomial of degree $d$.
Further, let $Y \subset \mathbb{CP}^2$ be the real projective plane
on which the coordinates $z\_i$ are real... | https://mathoverflow.net/users/150186 | Real points on a projective curve | It looks like the lower bound is $0$ if $d$ is even and $1$ if $d$ is odd.
**Construction.** Suppose $d=2p$. Take the curve $F=(z\_1^2+z\_2^2+z\_3^2)^p=0$. It doesn't have real points at all. Taking a small perturbation $F'$ of the polynomial $F$ we get a smooth curve $F'=0$ also disjoint from the real plane.
Supp... | 8 | https://mathoverflow.net/users/943 | 348924 | 147,676 |
https://mathoverflow.net/questions/348908 | 0 | Consider $s = \Theta(n^{\delta})$ for a $\delta\in (0,1)$ and let $p\in (0,1)$ with $m = \lfloor pn\rfloor$. Consider the random variable $Y$ which chooses $m$ elements from $\{1,\ldots,n\}$ such that any set of $m$ elements is equally likely. Then define $X$ to be $|Y \cap \{1,\ldots,s\}|$ where $|A|$ denotes the size... | https://mathoverflow.net/users/105971 | Coupling between two distributions | According to [formula (2)](https://arxiv.org/pdf/1905.03009), the total variation distance in question is bounded from above by $\dfrac{m-1}{n-1}$ assuming that $p=m/n$. Obviously, this bound does not depend on $s$. According to [this paper](ftp://ftp.stat.math.ethz.ch/U/hkuensch/hypergeom.pdf), this bound is optimal, ... | 3 | https://mathoverflow.net/users/36721 | 348938 | 147,682 |
https://mathoverflow.net/questions/55735 | 8 | Let $R$ be a ring and $X,Y$ two $R$-schemes, which you may assume to be noetherian or anything reasonable you like. Is it possible to "construct" $\text{Qcoh}(X \times\_R Y)$ out of $\text{Qcoh}(X)$ and $\text{Qcoh}(Y)$ in the $2$-category of all cocomplete $R$-linear tensor categories?
Perhaps it is the $2$-coproduc... | https://mathoverflow.net/users/2841 | Description of quasi-coherent modules on a product | More generally, I have proven that for quasi-compact and quasi-separated schemes $\mathrm{Qcoh}(X \times\_S Y)$ is the bicategorical pushout of $\mathrm{Qcoh}(X)$ and $\mathrm{Qcoh}(Y)$ over $\mathrm{Qcoh}(S)$ in the bicategory of cocomplete linear tensor categories. The technique of the proof has many other applicatio... | 2 | https://mathoverflow.net/users/2841 | 348940 | 147,683 |
https://mathoverflow.net/questions/348943 | 10 | In Higher topos theory and Higher algebra Lurie defines (see section 2.3.1 of HTT) a correspondence between two $\infty$-categories $C$ and $D$ as being an $\infty$-category $\mathcal{M}$ over $\Delta[1]$ whose fiber over $0$ and $1$ are respectively $C$ and $D$. Of course this is supposed to be equivalent to presheave... | https://mathoverflow.net/users/22131 | Correspondences of $\infty$-categories | Your question is answered by the following result, for which I will give a few references.
---
**Theorem.** For each pair of simplicial sets $A$ and $B$, there is a functor $$a\_{A,B}^\* \colon \mathbf{Cyl}(A,B) \longrightarrow \mathbf{sSet}/(A^\mathrm{op} \times B)$$ which is both a left and a right Quillen equi... | 12 | https://mathoverflow.net/users/57405 | 348954 | 147,688 |
https://mathoverflow.net/questions/348883 | 4 | In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), though of course it's of considerable interest to consider them at the PDE level as well.
The two standard examples of wh... | https://mathoverflow.net/users/121692 | Examples of particle systems with higher-order collisions | I understand that the OP's original focus is classical statistical mechanics. However, i think that the question is of interest from a more general viewpoint including the dynamical systems/integrability and/or the quantum statistical mechanics point of view.
In this sense, i am not sure if this is the kind of answe... | 5 | https://mathoverflow.net/users/85967 | 348959 | 147,690 |
https://mathoverflow.net/questions/348957 | 2 | For any prime $p\equiv\pm1\pmod5$, we can write $p$ uniquely in the form $x\_p^2+3x\_py\_p+y\_p^2$ with $x\_p,y\_p\in\mathbb Z$ and $x\_p>y\_p>0$.
I have the following conjecture.
**Conjecture.** We have
$$\lim\_{N\to+\infty}\frac{\sum\_{p\le N\atop p\equiv\pm1\pmod5}(3x\_p^2+2x\_py\_p)}
{\sum\_{p\le N\atop p\equi... | https://mathoverflow.net/users/124654 | A conjecture for primes $p\equiv\pm1\pmod5$ | Let us consider $\mathbb{Q}(\sqrt{5})$ as a subfield of $\mathbb{R}$. Let us also consider the positive fundamental unit $\epsilon:=(1+\sqrt{5})/2$, whose square generates the group of totally positive units. The conditions on $x\_p$, $y\_p$ can be rewritten as
$$p=x\_p^2+3x\_py\_p+y\_p^2\qquad\Longleftrightarrow\qquad... | 10 | https://mathoverflow.net/users/11919 | 348965 | 147,693 |
https://mathoverflow.net/questions/348050 | 3 | Let $\sigma\_d:\mathbb{P}^2\to\mathbb{P}^n$ be the d-th Veronese map and let $X=\sigma\_d(\mathbb{P}^2)$. Let $W\subset\mathbb{P}^n$ be a 2-plane such that $W\cap X=\emptyset$. For a line $L\subset \mathbb{P}^2$ let $X\_L=\sigma\_d(L)$.
Since $\dim(L)=1$ the space $U\_L$ spanned by $X\_L$ has dimension $d$ which is bas... | https://mathoverflow.net/users/149726 | Projecting onto the span of a generic Veronese variety | This is not true as stated in the comments. Consider $W=x\_3^{d-2}\mathbb{C}[x\_1,x\_2]\_2$ and $l$ a generic linear form. Then $W(x\_1,x\_2,l)=l^{d-2}\mathbb{C}[x\_1,x\_2]\_2$ and then setting $x\_3=0$ gives $\mathbb{C}[x\_1,x\_2]\supseteq W'=l'^{d-2}\mathbb{C}[x\_1,x\_2]\_2$. But $W'$ contains $l'^d$. If $l$ does not... | 0 | https://mathoverflow.net/users/149726 | 348975 | 147,695 |
https://mathoverflow.net/questions/348842 | 0 | Let $G$ be a hyperbolic group. Let $M$ be a vN algebra in standard form. Can there exist a faithful action of $G$ on $M$ such that
\begin{align\*}
\sigma\_{g\_n} \rightarrow I
\end{align\*}
for some sequence $(g\_n)$ of hyperbolic elements.?
| https://mathoverflow.net/users/145907 | Action of hyperbolic group on von Neumann algebra | Such actions abound. For instance, one can embed $F\_2$ in $\mathrm{SO}(3)$ (à la [Banach–Tarski](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#A_sketch_of_the_proof)) and let the latter act on the hyperfinite $\mathrm{II}\_1$-factor $R$ by realising the latter in terms of [canonical anticommutation relat... | 3 | https://mathoverflow.net/users/1275 | 348977 | 147,696 |
https://mathoverflow.net/questions/283889 | 8 | In [page 67](https://books.google.com/books?id=nKbwBwAAQBAJ&pg=PA67) of *Topology and Analysis* by *Booss and Bleecker*, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically trivial, but it appears to be no reason to believe that this... | https://mathoverflow.net/users/12233 | Smooth trivialization of smooth Hilbert bundles | I believe, the answer is (essentially) contained in the main theorem of the paper [*Equivalences of smooth and continuous principal bundles with infinite-dimensional structure group* by Christoph Müller and Christoph Wockel](https://edoc.hu-berlin.de/bitstream/handle/18452/11495/214.pdf?sequence=1):
>
> Let $K$ be ... | 2 | https://mathoverflow.net/users/1275 | 348984 | 147,698 |
https://mathoverflow.net/questions/348876 | 5 | Let $\mathcal{F}$ be a set algebra (or a Boolean algebra). Following Kalton, let me call a function $f\colon \mathcal{F}\to \mathbb R$ $\delta$-*additive* ($\delta \geqslant 0$), whenever $f(\varnothing) = 0$ and
$$| f(A) + f(B) - f(A\cup B) | \leqslant \delta$$
as long as $A\cap B=\varnothing$ for $A,B\in \mathcal... | https://mathoverflow.net/users/15129 | Do 1-additive maps admit tensor products? | As you guessed in the comments, it does not exist. The idea is the following: you are searching for some values that have small distance from some given values (the one on rectangles). If for example a new value $v$ must be close to some given values $v\_1,v\_2$, then necessarily $v\_1, v\_2$ must be close.
Conditio... | 3 | https://mathoverflow.net/users/140013 | 348985 | 147,699 |
https://mathoverflow.net/questions/348962 | 7 | Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties? Does this matrix have a name? In particular is there a formula for its inverse?
| https://mathoverflow.net/users/113992 | What are known properties of matrices where off-diagonal elements are 1? | Such a matriix has the form $J +D,$ where $D$ is a diagonal matrix, and $J$ is a square matrix with all entries $1$. One small remark is that if $D$ has two of its diagonal entries equal to $\lambda$, then $\lambda$ is also an eigenvalue of $J+D$. This is because the $\lambda$-eigenspace of $D$ is at least two-dimensio... | 6 | https://mathoverflow.net/users/14450 | 348987 | 147,700 |
https://mathoverflow.net/questions/348994 | -1 | Does there exist a smooth compactly supported function $$f \in C^{\infty}\_c((0,1))$$
such that
$$ \|D^k f\|\_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
for some $\alpha \in (0,1)$?
| https://mathoverflow.net/users/50438 | Existence of a function with slow growth on derivatives | For all natural $k$, we have $\|D^kf\|\_1\le\|D^kf\|\_2\le k!$, where $\|\cdot\|\_p:=\|\cdot\|\_{L^p(0,1)}$. So, for all $x\in(0,1)$ we have
$$|(D^kf)(x)|\le\int\_0^x |(D^{k+1}f)(u)|\,du\le\|D^{k+1}f\|\_1\le(k+1)!.
$$
So, for the Lagrange remainder
$$R\_n(a,x)=\int\_a^x (D^{n+1}f)(u)\frac{(x-u)^n}{n!}\,du
$$
for the... | 1 | https://mathoverflow.net/users/36721 | 349005 | 147,704 |
https://mathoverflow.net/questions/349010 | 7 |
>
> Do there exist such three non-constant pairwise independent random variables $X, Y, Z$ such that $X + Y + Z = 0$?
>
>
>
I managed only to prove the following two facts:
>
> If such $X, Y, Z$ exist, they are not independent.
>
>
>
*Proof:*
If they are, then $X$ and $-X = Y + Z$ are also independent... | https://mathoverflow.net/users/110691 | Do there exist three pairwise independent random variables, such that their sum is zero? | Replace $Z$ by $-Z$, so that $Z=X+Y$. Let $f\_X$ and $f\_Y$ the characteristic functions of $X$ and $Y$, so that $f\_X(s)=Ee^{isX}$ for real $s$. Suppose the pairwise independence.
Then for all real $s$ and $u$
$$f\_X(u)f\_Y(u)f\_X(s)=f\_Z(u)f\_X(s)=Ee^{iuZ+isX} \\
=Ee^{i(u+s)X+iuY}=f\_X(u+s)f\_Y(u). \tag{1}
$$
Theref... | 10 | https://mathoverflow.net/users/36721 | 349014 | 147,708 |
https://mathoverflow.net/questions/56887 | 76 | Call a category $C$ *rigid* if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets, commutative monoids, groups, abelian groups, commutative rings, but also the category of topological spaces. The cat... | https://mathoverflow.net/users/2841 | Rigidity of the category of schemes | As requested by the OP in the comments of the (correct and complete) accepted answer of user131755: it's possible to say more.
>
> **Theorem** [Mochizuki 2004, vDdB 2019]. *Let $S$ and $S'$ be schemes. Then the natural functor
> $$\operatorname{Isom}(S,S') \to \mathbf{Isom}(\mathbf{Sch}\_{S'},\mathbf{Sch}\_S)$$
> i... | 15 | https://mathoverflow.net/users/82179 | 349015 | 147,709 |
https://mathoverflow.net/questions/348798 | 6 | Suppose $G$ is a finite group and that $\rho: G\rightarrow O(d)$ is a faithful orthogonal representation, with action on $\mathbb{R}^d$ denoted $\cdot$. Let's say that $\rho$ is "strongly" angle preserving if for each $g\in G$ one has
\begin{equation}\langle g\cdot v, v\rangle = \langle g\cdot w, w\rangle \end{equati... | https://mathoverflow.net/users/78458 | Free linear group actions on spheres with "strong" angle preservation | I believe, the examples you gave are essentially the only ones. Indeed, let $\rho\colon G\to O(d)$ be a faithful irreducible “strongly angle-preserving” representation.
**Claim.** The image of $\mathbb{R}G$ under (the natural linear extension of) $\rho$ is a division algebra.
*Proof.* By linearity of the scalar pro... | 3 | https://mathoverflow.net/users/1275 | 349021 | 147,712 |
https://mathoverflow.net/questions/349023 | 9 | Torelli's theorem states:
>
> Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces $R$ and $R'$ are isomorphic if and only if $(J(R), \Theta)$ and $(J(R'), \Theta')$ are isomorphic as principally p... | https://mathoverflow.net/users/29836 | Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem | Consider the case of curves of genus $2$. If $\mathrm{A}$ is an abelian surface and $\mathrm{C}$ a smooth curve in $\mathrm{A}$ of genus $2$, then $\mathrm{A}\simeq\mathrm{J}(\mathrm{C})$ and $\mathrm{C}$ is the theta divisor of $\mathrm{J}(\mathrm{C})$. The special case $\mathrm{A}=\mathrm{E}\times\mathrm{E}$ (where $... | 6 | https://mathoverflow.net/users/104669 | 349026 | 147,714 |
https://mathoverflow.net/questions/348803 | 2 | Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x\_0$, where $A$ generates an **analytic** $C\_0$-semigroup on a Banach space $X$. How we can prove an inequality of this type:
$$\int\_0^T \|x'(t)\|\_X dt \le C\int\_0^T \|x(t)\|\_X dt +C\|x(T)\|\_X,$$
for some constant $C$, which is... | https://mathoverflow.net/users/146543 | An inequality for abstract Cauchy problem | The inequality in question does not hold in general. To show this, we shall consider a case when the initial condition $x\_0$ is given by a highly oscillating function $f\_0\colon\mathbb R\to\mathbb R$. The oscillations of $x\_0=f\_0$ will affect $\|x'(t)\|$ much more than they will affect $\|x(t)\|$, which will lead t... | 3 | https://mathoverflow.net/users/36721 | 349030 | 147,715 |
https://mathoverflow.net/questions/348981 | 4 | Let $G$ be an infinite profinite group, so $$G=\lim\_{\longleftarrow}G/N$$ where $N$ runs through the open normal subgroups. I have two questions:
1. Is $G$ of Haar measure zero in the compact group $\prod\_NG/N$?
2. What is the relation between the Haar measure of a subset $E$ of $G$ and the numbers $\frac{|EN/N|}{... | https://mathoverflow.net/users/84700 | Measure of subsets of profinite groups | 1) The measure of a closed subgroup $H$ of a profinite group $G$ is $\frac{1}{\vert G:H \vert}$. So $G$ has measure zero in $\prod G/N$ if and only if it has infinite index. This way you should be able to show that $G$ always has measure zero in $\prod G/N$.
2) As Yves mentioned, you always have the inequality $\mu(S... | 3 | https://mathoverflow.net/users/68337 | 349047 | 147,718 |
https://mathoverflow.net/questions/349040 | 1 | Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}\_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.
>
> Does the obvious... | https://mathoverflow.net/users/16183 | Relative version of the cohomology product | This answer is a translation of the much more general [Stacks, [Tag 0B68](https://stacks.math.columbia.edu/tag/0B68)] to this setup:
Write $\mathcal O\_X = \mathbf F\_X$ and $\mathcal O\_Y = \mathbf F\_Y$. Then $f \colon (X,\mathcal O\_X) \to (Y,\mathcal O\_Y)$ is a morphism of ringed spaces, with $f^{-1}\mathcal O\_... | 2 | https://mathoverflow.net/users/82179 | 349055 | 147,721 |
https://mathoverflow.net/questions/344761 | 17 | As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by Picado and Pultr. I am curious for a comparison between the emphases of these three books. As far as prerequisites go, I ha... | https://mathoverflow.net/users/nan | Best introductory texts on pointless topology | Topology via Logic - theoretical computer scientist
Stone Spaces - pure mathematician
Both are really good. Topology via logic as it gives a good account of domain theory, including power domains.
The first few chapters of Stone Spaces really have no equal. Comprehensive, starting at the basics, excellent narrat... | 5 | https://mathoverflow.net/users/45669 | 349060 | 147,723 |
https://mathoverflow.net/questions/349025 | 4 | Here is an apparent gap in a discreteness result of Lang that is a preliminary step in his proof of Dirichlet’s $S$-unit theorem.
I have been working on a Minkowski-free approach to algebraic number theory, the goal being to rewrite Neukirch’s chapter on the Rieman-Roch theorem in his book Algebraic Number Theory in t... | https://mathoverflow.net/users/149974 | Is there a gap or flaw in Lang's proof of Dirichlet's $S$-units theorem? | I think what is meant here is indeed not a monic polynomial over $\mathbb Z$, but rather one whose coefficients are (jointly) coprime. This, of course, is uniquely determined by the minimal polynomial of an algebraic number $x$.
Now, if $x$ satisfies the equation
$$
a\_nx^n + \dots + a\_1 x + a\_0 = 0
$$
with $a\_i\i... | 9 | https://mathoverflow.net/users/1275 | 349068 | 147,727 |
https://mathoverflow.net/questions/349039 | 11 | For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}\_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}\_q$ such that $f$ has three distinct roots in $\mathbb{F}\_q$, one of which is a quadratic residue and the other two are non-residues. I have not ... | https://mathoverflow.net/users/nan | Cubic polynomials over finite fields whose roots are quadratic residues or non-residues | EDIT: Following a clever observation of user44191 in the comments:
If $f(x)$ is a monic polynomial, and $c$ a number, then the polynomial $xf(x)^2+c$ has a similar property to your example (the case $f(x)=x+1/2$). Indeed, we have $x = \frac{-c}{f(x)^2}$ so
* If $-c$ is a nonzero square then all rational roots are ... | 13 | https://mathoverflow.net/users/18060 | 349070 | 147,728 |
https://mathoverflow.net/questions/322378 | 10 | Suppose that $X$ is an inclusion-minimal finite set of non-zero complex numbers
such that $\sum\limits\_{x\in X}x^n=0\ $ for infinitely many integers $n$.
>
> 1. Can the cardinality of $X$ be a composite number?
>
>
> ~~2. Can $X$ be something different from $\root^p\of c$
> (for some $c\in\mathbb C$ and prime $p... | https://mathoverflow.net/users/24165 | Perfectly balanced sets of complex numbers | First of all, thank you Gerry Myerson for bringing this problem to my attention at West Coast Number Theory 2019.
The answer to this question 1 is **No**.
**Edit on 12/26** : With a few more detailed analysis of the paper by Lam and Leung, it is possible to complete the proof, a crucial point is indicated by ***bo... | 5 | https://mathoverflow.net/users/21090 | 349085 | 147,734 |
https://mathoverflow.net/questions/349076 | 1 | Let $\mathit{Profinite}\_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}\_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor
$$\mathit{Profinite}\_{\mathrm{Ab}} \to \mathit{Profinite}\_{Sets}$$
admit a left adjoint?
I am a beginner to this kind of... | https://mathoverflow.net/users/150275 | Is there a free profinite abelian group on a profinite set? | Yes, the free functor (i.e. the left adjoint to the "forgetful" functor) exists.
Let $Ab^{fin}$ be the category of finite abelian groups and $Set^{fin}$ the category of finite sets. Because each of these categories are essentially small and have finite limits, the categories of [pro-objects](https://ncatlab.org/nlab/... | 1 | https://mathoverflow.net/users/2362 | 349086 | 147,735 |
https://mathoverflow.net/questions/349042 | 3 | Let $f\colon E\to B$ locally trivial bundle of 'nice' topological spaces (say finite CW-complexes) with fiber $F$. Assume also that the base $B$ is simply connected.
Assume that either the cohomology spectral sequence (with coefficients in a field) degenerates in the second term $H^\*(B)\otimes H^\*(F)$ or that the p... | https://mathoverflow.net/users/16183 | Cohomology algebra of a fibration whose spectral sequence degenerates in the second term | The statement is false, here is a counterexample. First note that for a Lie Group $G$ and its closed subgroup $H$, we have a fibration $G/H\to BH\to BG$. $BG$ and $BH$ are not finite, but they are almost just as good, if you really want finite CW complex, you can restrict the fibration over a finite skelton of $BH$. No... | 5 | https://mathoverflow.net/users/43326 | 349101 | 147,742 |
https://mathoverflow.net/questions/349046 | 1 | **Disclaimer.** This is follow up to the question <https://math.stackexchange.com/q/3486130/168758>.
Let $X=(X,d)$ be a **Polish** metric space equiped with the Borel $\sigma$-algebra and let $\mu$ be a nonnegative measure. For a nonempty measurable subset $A$ of $X$ and $\varepsilon > 0$, define the $\varepsilon$-en... | https://mathoverflow.net/users/78539 | Comparing $(((A^\varepsilon)')^\varepsilon)'$ and $int(A)$, where $A' := X\setminus A$ and $A^\varepsilon := \{x \in X \mid d(x,A) \le \varepsilon\}$ | Here is an example for question 2. Given $\varepsilon>0$, consider $X=[-1,0] \cup (\varepsilon,1+\varepsilon]$ with the usual metric.
Then $A=[-1,0]$ is closed and open in $X$, and $ A^\varepsilon =A$. Note that $0 \in (A')^\varepsilon$, so $(A^\varepsilon)^{-\varepsilon}=[-1,0)$ does not contain $\overset{\circ}{A} =A... | 3 | https://mathoverflow.net/users/7691 | 349121 | 147,749 |
https://mathoverflow.net/questions/349122 | 11 | Let $f \in S\_k(\Gamma\_0(N))$ be a normalized newform with Fourier expansion
$$f(z) = \sum\limits\_{n=1}^{\infty} a\_n e^{2\pi i z n}$$
and $a\_1 = 1$. Then $f$ is an eigenfunction of all Hecke operators $T\_p$, not just those with $(p,N) = 1$, and for the normalized L-function
$$L^{S\_{\infty}}(f,s) = \sum\limi... | https://mathoverflow.net/users/38145 | Are the L-functions of a normalized newform and the corresponding cuspidal representation equal? | $L(\pi,s)$ agrees with $L(f,s)$ if $f\in\pi$ is a newform, and this is even true for $\mathrm{GL}\_n$. Of course, things are complicated by the fact that there are many ways to define $L(\pi,s)$ and $L(f,s)$. I found the following papers very useful to check various consistencies: [Schmidt](http://www.math.unt.edu/%7Es... | 10 | https://mathoverflow.net/users/11919 | 349124 | 147,750 |
https://mathoverflow.net/questions/345107 | 1 | Let $(C,\otimes,1)$ be a [symmetric monoidal category](https://ncatlab.org/nlab/show/symmetric+monoidal+category). Let $(M,\mu,\eta)$ be an [internal commutative monoid object](https://ncatlab.org/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category).
The functor $X\mapsto M\otimes X$ has a canonical [monad](... | https://mathoverflow.net/users/30366 | Internal commutative monoid gives commutative monad | Directly refering to the definition at the nlab, the morphism $\alpha : T(A) \otimes T(B) \to T(A \otimes B)$ is given by the composition
$$\small (A \otimes M) \otimes (B \otimes M) \cong (A \otimes (B \otimes M)) \otimes M \cong ((A \otimes B) \otimes M) \otimes M \cong (A\otimes B) \otimes (M \otimes M) \to (A\otime... | 2 | https://mathoverflow.net/users/2841 | 349131 | 147,753 |
https://mathoverflow.net/questions/349126 | 4 | The [Hilbert curve](https://en.wikipedia.org/wiki/Hilbert_curve) is a continuous space-filling curve that maps $I$ to $I^n$ where $I$ denotes the unit interval [1]. Like all other space-filling curves, it is not one-to-one. I am wondering if the Hilbert curve, or a common variant thereof, becomes a continuous bijective... | https://mathoverflow.net/users/24274 | Is the Hilbert space-filling curve bijective over computable numbers? | The Hilbert curve, due to its fractal nature, is mapping certain subintervals of the unit interval to certain squares in the unit square. On any given resolution, we have a bijection between the subintervals we consider on that scale, and the squares we consider at that scale. The non-injectivity of the actual map then... | 6 | https://mathoverflow.net/users/15002 | 349133 | 147,755 |
https://mathoverflow.net/questions/341650 | 10 | I always tried to understand how the [*finite reflection groups*](https://en.wikipedia.org/wiki/Reflection_group) of $\Bbb R^d$ (of some fixed dimension $d$) relate to the [*point groups*](https://en.wikipedia.org/wiki/Point_group) of the same space $\smash{\Bbb R^d}$ (finite subgroup of the orthogonal group $\smash{\m... | https://mathoverflow.net/users/108884 | How are reflection groups related to general point groups? | Here is a more worked-out and concrete version of the proposed counterexample mentioned in the "Update" of the post.
1. We start from a symmetric arrangement of 12 great circles $F\_1,\ldots,F\_{12}$ on the 3-sphere $\mathbb{S}^3$:
[12 circles from the Hopf fibration, shown in stereographic projection to $\mathbb{R}^... | 3 | https://mathoverflow.net/users/30800 | 349139 | 147,757 |
https://mathoverflow.net/questions/349141 | 1 | Let $M\_k(\mathrm{SL}\_2\mathbb{Z})$ be the space of modular forms of (integer) weight for the full modular group. Let $\mathbf{H}$ denote the Algebra generated by the Hecke operators $T\_n$. Is $\mathbf{H}$ an integral domain? Specifically, let $T\_mT\_n=0,$ can we conclude either $T\_m=0$ or $T\_n=0$ on $M\_k(\mathrm... | https://mathoverflow.net/users/127239 | Algebra of Hecke operators on $M_k(\mathrm{SL}_2\mathbb{Z})$ is an integral domain? | You mean the (commutative, normal for the Petersson inner product thus diagonalizable) complex algebra $\Bbb{T(C)}$ of endomorphisms of the complex vector space $M\_k(SL\_2(Z))$ ($k$ even) generated by the identity and the $T\_n$.
If $\dim(M\_k(SL\_2(Z))=1$ then it is $= \Bbb{C}$, otherwise it is not an integral doma... | 4 | https://mathoverflow.net/users/84768 | 349143 | 147,759 |
https://mathoverflow.net/questions/349147 | 8 | What is known regarding which hyperbolic groups are cubulated?
I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.
My impression is that not all of them are, but I didn't manage to find references with a counterexample.
Are there known ways to create non-cubula... | https://mathoverflow.net/users/149916 | Examples of non-cubulated hyperbolic groups | If a group $G$ satisfies Kazhdan's property (T), then any action of $G$ on a CAT(0) cube complex has a global fixed point. See Niblo and Roller's article *Groups acting on cubes and Kazhdan's Property (T)*. Examples of hyperbolic groups which satisfy this property include:
* Uniform lattices in quaternionic hyperboli... | 11 | https://mathoverflow.net/users/122026 | 349150 | 147,760 |
https://mathoverflow.net/questions/349154 | 2 | Howard Masur's research asserts that if $S\_g$ is a closed surface of genus $g\geq2$, then the Teichmuller space $T(S\_g)$ does not have nonpositive curvature. His proof relies on the existence of similar Strebel rays. However, similar Strebel rays does not exist in $T(S\_g)$ if $g=1$. On the other hand, $T(S\_1)=\math... | https://mathoverflow.net/users/143284 | What is the Teichmuller metric on the Teichmuller space of a closed surface of genus 1? | The answer is yes, up to renormalization.
Precisely, the bijection $\mathbb{H}^2\to Teich(\mathbb{T})$ you refer to induces an isometry from $(\mathbb{H}^2,d\_{\mathbb{H}^2})\to (Teich(\mathbb{T}),2d\_{Teich})$.
This is exactly Theorem 11.20 in Farb and Margalit's book A primer on mapping class groups, available [here]... | 6 | https://mathoverflow.net/users/111917 | 349158 | 147,761 |
https://mathoverflow.net/questions/349123 | 0 | A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A *coloring* is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal, such that for every $e\in E$ with $|e|\geq 2$ the restriction $c|\_e$ is non-constant.
**Question.** Is every hypergraph $H=(V,E)$ wi... | https://mathoverflow.net/users/8628 | Are hypergraphs $H=(V,E)$ with $|E|=|V|$ $2$-colorable? | This is essentially done by the Bernstein set construction: if one has $\kappa$ many sets each of size $\kappa$, then order them into ordinal $\kappa$ and recursively choose 2 points from each, so that all these points are distinct. That is, we have $x\_\alpha,y\_\alpha\in A\_\alpha$ with all $x\_\alpha,y\_\alpha$ dist... | 7 | https://mathoverflow.net/users/6647 | 349160 | 147,762 |
https://mathoverflow.net/questions/348881 | 12 | Since this subject is full of misunderstandings (see [here](https://math.stackexchange.com/q/94422/3217), [here](https://math.stackexchange.com/q/800913/3217), [here](https://math.stackexchange.com/questions/2713073/locally-free-module), and [here](https://math.stackexchange.com/q/1905409/3217)) let us fix a precise te... | https://mathoverflow.net/users/450 | Is every locally free module of rank $1$ over a commutative ring concretely invertible? | The answer is yes. Recall that given an invertible $A$-module $P$ and $n \in \mathbf{Z}$ there is an invertible $A$-module $P^{\otimes n}$ such that $P^{\otimes 0} = A$, $P^{\otimes 1} = P$, and $P^{\otimes n} \otimes\_A P^{\otimes m} = P^{\otimes n + m}$. Set $B = \bigoplus\_{n \in \mathbf{Z}} P^{\otimes n}$; this is ... | 10 | https://mathoverflow.net/users/150339 | 349177 | 147,768 |
https://mathoverflow.net/questions/349175 | 2 | I am quite sure I have seen somewhere the connection between the characteristic polynomial of a (finite undirected) graph and its dual. I am not able to find it currently. Could you please refer me to the result?
(eigenvalues would be enough)
By dual, I mean the graph where edges become vertices and adjacent if th... | https://mathoverflow.net/users/142777 | Characteristic polynomial of the line graph (originally dual graph) | The term for the graph construction you are talking about is "line graph" (see [the Wikipedia article](http://en.wikipedia.org/wiki/Line_graph)). The eigenvalues of the adjacency matrix of the line graph $L(\Gamma)$ are closely related to the signless Laplacian eigenvalues of the original graph $\Gamma$, as explained f... | 2 | https://mathoverflow.net/users/25028 | 349185 | 147,772 |
https://mathoverflow.net/questions/349196 | 0 | Let $V$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $W$ be an open neighborhood of the boundary $\partial V$ of $V$. If $u$ is superharmonic on $W$, is there a way to extend $u$ to a function $\tilde{u}$ that is superharmonic on $\complement V\cup W$ and is equal to $u$ at least on $\overline{V}\cap W$?... | https://mathoverflow.net/users/100746 | Extension of superharmonic functions | In general, this is not possible. Consider the case $n=2$ take the unit disk for $V$,
and some ring, for example $1/2<|z|<2$ for $W$. Function $u(z)=\log|z|$ is harmonic in
$W$ but cannot be extended from any neighborhood of the unit circle to the closure
of the unit disk as a superharmonic function. The obstacle is cl... | 2 | https://mathoverflow.net/users/25510 | 349201 | 147,777 |
https://mathoverflow.net/questions/349259 | 5 | From this page:
<https://en.wikipedia.org/wiki/Chebyshev_function#Asymptotics_and_bounds>
A theorem due to Erhard Schmidt states that, for some explicit positive constant $K$, there are infinitely many natural numbers $x$ such that $$ψ(x)>x+K√x$$
Since the original paper is in German. I read this paper: <https://p... | https://mathoverflow.net/users/74668 | Is the result of Schmidt conditional to RH | **1.** It is known unconditionally that, as $x$ tends to infinity,
$$\psi(x)-x=\Omega\_{\pm}(x^{1/2}).\tag{$1$}$$
This is Corollary 15.4 in Montgomery-Vaughan: Multiplicative number theory I.
**2.** In fact Hardy and Littlewood proved the stronger result
$$\psi(x)-x=\Omega\_{\pm}(x^{1/2}\log\log\log x).\tag{$2$}$$
Th... | 13 | https://mathoverflow.net/users/11919 | 349260 | 147,791 |
https://mathoverflow.net/questions/335873 | 7 | Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}\_p$. As in II.4.3 in [Colmez - Fonctions d'une variable $p$-adique](https://webusers.imj-prg.fr/~pierre.colmez/fonctionsdunevariable.pdf), I can restrict $\mu$ to $1+p\mathbf Z\_p$, and there is a formula which tells me how to compute the Amice tran... | https://mathoverflow.net/users/108588 | Change of variables for $p$-adic integral | If I understand correctly, the answer is probably not, since it would imply that the Iwasawa power series of a $p$-adic $L$-function has a "simple" relation with the original measure on $1+p\mathbb{Z}\_p$. But the Iwasawa power series is usually hard to compute (explicitly). But, there are relations between your two me... | 1 | https://mathoverflow.net/users/109085 | 349271 | 147,795 |
https://mathoverflow.net/questions/283376 | 3 | Let $X$ be a set and let $\Phi(X)$ denote the collection of [filters](https://en.wikipedia.org/wiki/Filter_(mathematics)) on $X$. For $x\in X$ we denote by $P\_x$ the filter $P\_x=\{A\subseteq X:x\in A\}$. A *convergence space* is a pair $(X,\to)$, where $X$ is a set, and $\to$ is a subset of $\Phi(X)\times X$ with the... | https://mathoverflow.net/users/8628 | Adjoints for the functor ${\bf Top}\to {\bf Conv}$ | At @DominicvanderZypen's [request](https://mathoverflow.net/questions/283376/adjoints-for-the-functor-bf-top-to-bf-conv#comment699664_283376), here is
@მამუკა-ჯიბლაძე's answer from the comments [1](https://mathoverflow.net/questions/283376/adjoints-for-the-functor-bf-top-to-bf-conv#comment699473_283376) [2](https://mat... | 2 | https://mathoverflow.net/users/2383 | 349293 | 147,807 |
https://mathoverflow.net/questions/349245 | 5 | I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be complicated but could there still be some reasonable version of the spectral theorem? Why or why not? Some elaboration ... | https://mathoverflow.net/users/136074 | Is there a reasonable notion of spectral theorem on a pre-Hilbert space? | Here is a simple example that shows that the idea of spectral theory on pre-Hilbert spaces in the sense you are asking is hopeless. Consider the pre-Hilbert space consisting of the restrictions of all complex polynomials to $[0,1]$, as a dense subspace of $L^2[0,1]$. Then let $A$ be the operator of multiplication by $x... | 8 | https://mathoverflow.net/users/23141 | 349296 | 147,809 |
https://mathoverflow.net/questions/349274 | 6 | Let $E\_0$ be a matrix with non-negative entries.
Given $E\_n$, we apply the following two operations in sequence to produce $E\_{n+1}$.
A. Divide every entry by the sum of all entries in its column (to make the matrix column-stochastic).
B. Divide every entry by the sum of all entries in its row (to make the mat... | https://mathoverflow.net/users/120987 | Limit of alternated row and column normalizations | When $E\_0$ is square (i.e., $r = c$) this procedure is called Sinkhorn iteration or the Sinkhorn-Knopp algorithm (see [this Wikipedia page](https://en.wikipedia.org/wiki/Sinkhorn%27s_theorem)). You can find a wealth of results by Googling those terms, the most well-known of which is that if $E\_0$ has strictly positiv... | 5 | https://mathoverflow.net/users/11236 | 349299 | 147,812 |
https://mathoverflow.net/questions/349327 | 6 | I am interested in what is known about the topology (diffeomorphism type) or geometry (is it complex? non-complex? symplectic?) of either the compact space of orthonormal complex structures on $\mathbb{R}^{2n}$, $O(2n)/U(n)$, or the full space of complex structures, $Gl(2n,\mathbb{R})/Gl(n,\mathbb{C})$ (which retracts ... | https://mathoverflow.net/users/131760 | Topology/geometry of $O(2n)/U(n)$ | You may be remembering papers by Vogan ([1987](//ams.org/mathscinet-getitem?mr=89h:22034), p. 262; [2008](//ams.org/mathscinet-getitem?mr=2009f:22013), prop. [6.9](http://math.mit.edu/~dav/paper.html)). There he describes:
(a) $\mathrm{GL}(2n,\mathbf R)/\mathrm{GL}(n,\mathbf C)\cong\{\!$complex structures on $\mathbf... | 9 | https://mathoverflow.net/users/19276 | 349332 | 147,819 |
https://mathoverflow.net/questions/349328 | 1 | I've been reading the first section Furstenberg's [Noncommuting Random Products](https://www.ams.org/journals/tran/1963-108-03/S0002-9947-1963-0163345-0/S0002-9947-1963-0163345-0.pdf) and I am confused with how he is defining conditional distribution.
Here he is considering a group $G$ acting on a space $M$. For a $M... | https://mathoverflow.net/users/71233 | Defining the conditional distribution of $Z$ as $E^{*}[Z| \mathcal{F}](f):=E[f(Z)| \mathcal{F}]$ | $\newcommand{\M}{\mathcal M}$
$\newcommand{\G}{\mathcal G}$
$\newcommand{\F}{\mathcal F}$
$\newcommand{\P}{\mathsf P}$
$\newcommand{\E}{\mathsf E}$
Suppose that $M$ is a Polish (i.e., complete separable metrizable) space with the Borel sigma-algebra $\M$ over it. Let $Z$ be an $M$-valued random variable (r.v.) defined ... | 2 | https://mathoverflow.net/users/36721 | 349335 | 147,822 |
https://mathoverflow.net/questions/348159 | 24 | The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The conjecture is:
>
> Let $R$ be an $N \times N$ rational orthogonal matrix. Define a sublattice $\Lambda \subseteq \mathbb{Z}^... | https://mathoverflow.net/users/149789 | Simple conjecture about rational orthogonal matrices and lattices | Proof
=====
Let $R$ be any matrix. We have the obvious exact sequence
$$ 0 \longrightarrow\mathbb{R}^N \xrightarrow[\left(\begin{matrix} I \\ R \end{matrix}\right)]{} \mathbb{R}^N \oplus \mathbb{R}^N \xrightarrow[\left(\begin{matrix} I & -R^{-1} \end{matrix}\right)]{} \mathbb{R}^N \longrightarrow 0 $$
This contai... | 13 | https://mathoverflow.net/users/149789 | 349340 | 147,825 |
https://mathoverflow.net/questions/348839 | 1 | Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.
A real matrix $A=(a\_{ij})\_{i,j\in I}$ is said to be a generalized Cartan matrix
if the following conditions are satisfied:
1. $A$ is symmetric;
2. $a\_{ii}=... | https://mathoverflow.net/users/33047 | Doubt in the Serre relation and the odd/even roots of a Lie superalgebra | Although i am not an expert in the topic, i did some studying on the references you provided together with Kac's monograph on [Infinite dimensional Lie algebras](https://www.cambridge.org/core/books/infinitedimensional-lie-algebras/053FE77E6E9B35C56B5AEF7336FE7306#fndtn-contents).
I do not have very clear answers to... | 3 | https://mathoverflow.net/users/85967 | 349343 | 147,826 |
https://mathoverflow.net/questions/349337 | 1 | If $F$ is a totally real number field with $[F:\mathbb{Q}] = d>1$, $X$ is the moduli space of Hilbert-Blumenthal Abelian varieties for $F$, and $\overline{X}$ is the projective toroidal compactification, then a paper I am reading refers to $D = \overline{X} \setminus X$ as the boundary divisor. But the cusps of $X$ sho... | https://mathoverflow.net/users/141571 | Boundary divisor of projective toroidal compactification | Toroidal compactifications of Shimura varieties are smooth, and $D$ will certainly be of codimension one (when it is not empty). There are in fact a number of different compactifications of Shimura varieties to consider each with different desirable properties. For example, there is a smaller minimal compactification w... | 3 | https://mathoverflow.net/users/150465 | 349351 | 147,830 |
https://mathoverflow.net/questions/349350 | 2 |
>
> Let $R=\oplus\_{I\geq 0}R\_i$ be a positive graded ring(maybe not commutative), where $R\_0$ is a commutative Noetherian ring. If $R$ is finite generated $R\_0$-algebra, is $R$ Noetherian?
>
>
>
In here, [Is every (left) graded-Noetherian graded ring (left) Noetherian?](https://mathoverflow.net/q/303005/1065... | https://mathoverflow.net/users/106580 | Is Hilbert basis theorem true for positive graded ring? | The answer is **no** by Exercice 26 in the 2012 edition of Bourbaki's *Algèbre* VIII.1. (This seems moreover to have nothing to do with graduations.)
(Translation of the exercise: Let $K$ be a commutative field, let $A$ be the polynomial ring $K[T]$, and let $\sigma$ be the endomorphism $P(T)\mapsto P(T^2)$ of $A$. T... | 3 | https://mathoverflow.net/users/11025 | 349359 | 147,833 |
https://mathoverflow.net/questions/349364 | 3 | Is there any way to represent every element of the mapping class group of a surface as an arc on that surface?
| https://mathoverflow.net/users/150179 | Arcs and elements of the mapping class group | The natural action (of mapping classes act on isotopy classes of arcs) has large stabilisers. So the "correct" answer to your question is "no".
Now, the mapping class group is countable. The set of isotopy classes of arcs is also countable. With a bit of work, you can construct a bijection between them. So in that s... | 4 | https://mathoverflow.net/users/1650 | 349365 | 147,834 |
https://mathoverflow.net/questions/349355 | 43 | The gnu (or Group NUmber) function describes how many groups there are of a given order. The number of groups of each order are known up to 2047, see <https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf>
Has any progress been made on the number of groups of order 2048? This case is particularly difficult due to 20... | https://mathoverflow.net/users/38744 | Has gnu(2048) been found? | No, it is unknown, and I don't think we will find it anytime soon. For the state of the art, see our 2017 paper "Constructing groups of ‘small’ order: Recent results and open problems" [DOI](http://dx.doi.org/10.1007/978-3-319-70566-8_8) (here is a [PDF](https://www.quendi.de/data/papers/EHH2018-small-groups.pdf)). I c... | 49 | https://mathoverflow.net/users/8338 | 349367 | 147,835 |
https://mathoverflow.net/questions/347853 | 9 | In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on the category of Banach spaces...". He gave no clue how to prove it, but it is known that the same fact is not true for ... | https://mathoverflow.net/users/140292 | Completed tensor product is exact | To make things precise, let me add the end of the quoted sentence: "with admissible linear operators as morphisms". Moreover, I believe that Berkovich refers here to tensor products over a fixed base field.
In this case, the exactness result you are looking for may be found in Gruson's paper "Théorie de Fredholm $p$... | 3 | https://mathoverflow.net/users/4069 | 349369 | 147,836 |
https://mathoverflow.net/questions/50343 | 113 | **EDIT** (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no longer relevant".
---
As some of you may already know, there are plans in the making for a [Museum of Mathematics]... | https://mathoverflow.net/users/3106 | What would you want to see at the Museum of Mathematics? | <https://www.scribd.com/document/479581247/Letter-to-MoMath-Board>
------------------------------------------------------------------
**Update: many of us got together to take a stand against unethical practices at the museum. See the above open letter to the Board of Trustees which recommends the replacement of the ... | 21 | https://mathoverflow.net/users/14835 | 349376 | 147,839 |
https://mathoverflow.net/questions/349354 | 19 | Is there a way to define perverse sheaves categorically/geometrically? Definitions like the following from lectures by Sophie Morel:
>
> The category of perverse sheaves on $X$ is $\mathrm{Perv}(X,F):=D^{\leq 0} \cap D^{\geq 0}$. We write $^p\mathrm{H}^k \colon D\_c^b(X,F) \to \mathrm{Perv}(X,F)$ for the cohomology... | https://mathoverflow.net/users/143390 | So what exactly are perverse sheaves anyway? | An $\infty$-categorical perspective is given here <https://arxiv.org/abs/1507.03913> and a triangulated expansion of those ideas is here <https://arxiv.org/abs/1806.00883>
More or less, perverse sheaves are the heart of a certain $t$-structure that you build "gluing along a perversity datum".
| 7 | https://mathoverflow.net/users/7952 | 349378 | 147,840 |
https://mathoverflow.net/questions/349371 | 3 | Let $H(p) = \sum\_i p\_i\log\frac{1}{p\_i}$ be the entropy of $p$
and $KL(p, q) = \sum\_i p\_i\log\frac{p\_i}{q\_i}$ be the KL divergence between $p$ and $q$. Does it hold that $H(p) \le H(q) + KL(p, q)$?
If this is not true, can we bound $H(p)$ using $H(q)$ and $KL(p, q)$ in certain form?
Edit 1: The motivation of... | https://mathoverflow.net/users/17589 | $H(p) \le H(q) + KL(p, q)$? | There are already nice negative answers by [Steve](https://mathoverflow.net/a/349379/23297) and [Rémi Peyre](https://mathoverflow.net/a/349383/23297). In the comments, user111 mentioned [this post](https://mathoverflow.net/a/133774/23297) by David Reeb who gives a bound on the difference of entropies in terms of the KL... | 5 | https://mathoverflow.net/users/23297 | 349390 | 147,844 |
https://mathoverflow.net/questions/342061 | 9 | **Update**:
I have found reference to this problem. It is known as "the Rédei-de Bruijn-Schönberg theorem", which is proved in the following papers:
* N. G. de Bruijn: *On the factorization of cyclic groups*, Indag. Math.15(1953), 370-377.
* L. Rédei: *Ein Beitrag zum Problem der Faktorisation von Abelschen Gruppen... | https://mathoverflow.net/users/76332 | Is an integral sum of periodic vectors always a sum of integral periodic vectors? | A beautiful question! Though I don't have the time or the space to fill in all the details, I think one can answer the question in the following way. The answer is yes.
We can reformulate the question as follows: let $I \subset \mathbb{Z}[x]$ be the ideal generated by the polynomials
$$
\frac{x^n-1}{x^d-1}, \quad d \... | 4 | https://mathoverflow.net/users/105625 | 349402 | 147,849 |
https://mathoverflow.net/questions/349413 | 1 | Does there exist an algebraic number $\alpha$ such that
$$\left|\frac{\alpha^n+\alpha^n\_1}{n!}\right|\sim\_{n\to+\infty}\frac1{(n!)^2}$$ where $\alpha\_1$ is a conjugate of $\alpha$?
Obviously $\alpha$ can not be a rational number.
Thanks in advance for any answer.
| https://mathoverflow.net/users/33128 | algebraic numbers with small norms | For any fixed nonzero complex numbers $z\_1,\dotsc,z\_m$, there are infinitely many $n$'s such that the arguments of $z\_1^n,\dotsc,z\_m^n$ all lie in $[-\pi/4,\pi/4]$. This follows from Dirichlet's theorem on simultaneous diophantine approximation. For such $n$'s,
$$|z\_1^n+\dotsb+z\_m^n|\geq\Re(z\_1^n+\dotsb+z\_m^n)\... | 4 | https://mathoverflow.net/users/11919 | 349414 | 147,853 |
https://mathoverflow.net/questions/349421 | 3 | Akbari, Ghodrati, Hosseinzadeh (2017), *On the structure of graphs having a unique k-factor*, Aust. J. Combin. ([pdf](https://ajc.maths.uq.edu.au/pdf/69/ajc_v69_p063.pdf)) show:
>
> ... we prove that there is no r-regular graph (r≥2) with a unique perfect matching.
>
>
>
It seems natural to explore the stronge... | https://mathoverflow.net/users/48278 | Does there exist an r-regular graph (r≥2) with a unique maximum matching? | Take a maximum matching $M$ and a vertex $v$ not in $M$. If $v$ has a neighbour $w$ not in $M$, then $M+vw$ is a larger matching. So $v$ has a neighbour $x$ which is in an edge $xy$ of $M$. Now $M-xy+vx$ is another maximum matching.
| 6 | https://mathoverflow.net/users/9025 | 349422 | 147,856 |
https://mathoverflow.net/questions/349423 | 10 | A projective plane $Π$ is a 3-tuple $(P,L,I)$ where $P$ and $L$ are sets, and $I$ is a relation between $P$ and $L$, such that:
* For every two elements $p\_1$, $p\_2\in P$, there exists a unique element $l \in L$ such that $p\_1 I l$ and $p\_2 I l$.
* For every two elements $l\_1$, $l\_2\in P$, there exists a unique... | https://mathoverflow.net/users/125498 | Does every $C_4$-free bipartite graph lies in some finite projective plane? | This is an open problem posed by Erdos in ["Some old and new problems in various branches of combinatorics"](https://users.renyi.hu/~p_erdos/1979-18.pdf) (see section 6). There hasn't been any substantial progress since then. After posing the question Erdos writes "I have no idea how to attack this problem", and that s... | 12 | https://mathoverflow.net/users/2384 | 349425 | 147,857 |
https://mathoverflow.net/questions/349387 | 3 | I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}\_{X}$-algebras over $X$. Let's suppose that for every geometric point $p$, the pullback of $\mathcal{F}$ to $X\_{p}$ is a finitely generated sheaf of $\mathcal{O... | https://mathoverflow.net/users/140062 | Finitely generated sheaf of algebras over geometric points | This is false for $X=S=\operatorname{Spec}\mathbb{Z}$. For example, take the sheaf of $\mathcal{O}\_X$-algebras corresponding to the $\mathbb{Z}$-algebra
$$
A=\mathbb{Z}+ \mathbb{Q}\varepsilon\subseteq \mathbb{Q}[\varepsilon\,|\,\varepsilon^2=0].
$$
It is easy to see that $A\otimes\_{\mathbb{Z}}K$ is finitely generated... | 6 | https://mathoverflow.net/users/86006 | 349430 | 147,858 |
https://mathoverflow.net/questions/298077 | 4 | Let $f:X \to S$ be a smooth morphism and $S$ the spectrum of a discrete valuation ring. If the generic fiber of $f$ is rationally (chain) connected then is the special fiber of $f$ also rationally (chain) connected? We know this to be true if $S$ is over a field of characteristic zero, due to Kollar, Miyaoka and Mori. ... | https://mathoverflow.net/users/45397 | Deformation invariance of rational connectedness in positive/mixed characteristic | In characteristic $ p>0 $, RCC and SRC (separable rational connectedness) do differ (see Janos Kollar's (Rational curves) book: V.5.19, I know that Janos Kollar refers to this even though I do not have the book.). In **Higher Dimensional Varieties and Rational Points** pg. 41 and pg. 43, Carolina Araujo and Janos Kolla... | 1 | https://mathoverflow.net/users/113893 | 349435 | 147,860 |
https://mathoverflow.net/questions/292538 | 9 | In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all points that are "near" it.
In particular, in a topological space $(X,T)$, the monad of $x \in X$ is defined as $$\mu (... | https://mathoverflow.net/users/65915 | A definition of topology using monads (a.k.a. halos) | Two researchers in the 1980s have independently discovered the necessary axioms for defining a topology out of halo axioms. The relevant papers are:
Vakil, N. Monadic binary relations and the monad systems at near-standard points. *The journal of Symbolic Logic*, 52(3):689-697, Sep 1987. ([link](https://www.jstor.org... | 3 | https://mathoverflow.net/users/150509 | 349439 | 147,861 |
https://mathoverflow.net/questions/349223 | 2 | For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A *Galois connection* is a pair of maps between posets $X$ and $Y$
$$ f\_{\bullet}: X \rightleftharpoons Y: f^{\bullet}$$
such that $f\_{\bullet} x \preceq y$ if and only if $x \preceq f^{\bu... | https://mathoverflow.net/users/128639 | A formula for a right adjoint in terms of a left | Just so that it is recorded here as an answer, here's the formula from the "naive" adjoint functor theorem that directly generalizes the one given in the post for posets:
$$f^\bullet(y) = \lim\_{(x,\alpha)\in (f\_\bullet \downarrow y)} x$$
where the limit is over the comma category $(f\_\bullet \downarrow y)$, whos... | 9 | https://mathoverflow.net/users/49 | 349446 | 147,864 |
https://mathoverflow.net/questions/349341 | 4 | It is true that in the category of spaces there exists a characterization of homotopy pullbacks in terms of homotopy fibers ([Proposition 4.1](https://ncatlab.org/nlab/show/homotopy+pullback)).
I want to know a category (or $\infty$-category) where I can find a square diagram where there is an equivalence in all the ... | https://mathoverflow.net/users/95695 | Homotopy pullbacks and fibers | For the record, the statement in spaces is this. A diagram
$$
\array{
A &\stackrel{f}{\longrightarrow}& B
\\
\downarrow && \downarrow^{p}
\\
C &\stackrel{g}{\longrightarrow}& D
}
$$
is a homotopy pullback if and only if, for every point $b: \ast \to B$, the map of homotopy fibers $hofib\_b(f) \to hofib\_{p(b)}(g)... | 6 | https://mathoverflow.net/users/360 | 349453 | 147,867 |
https://mathoverflow.net/questions/349437 | 2 | Let $T$ be an effectively generated (recursively enumerable) theory written in a first order language that has infinitely many extra-logical primitives.
Is it always the case that there is a theory $T^\*$ that is:
* effectively generated
* bi-interpretable with $T$
* written in a first order language that has fini... | https://mathoverflow.net/users/95347 | Can we re-write every effective first order theory using finitely many primitives? | The theory of the (full) random hypergraph is a counterexample. (Full meaning we are allowing any arity.)
The language consists of a relation symbol $E\_n$ for each $n \geq 1$ (sometimes people start at $2$ but it doesn't really matter). For each $n$, $E\_n$ is an $n$-ary relation symbol. The structure is a hypergrap... | 6 | https://mathoverflow.net/users/83901 | 349458 | 147,871 |
https://mathoverflow.net/questions/349420 | 5 | The unoriented bordism theory $MO$ has a map to $H\mathbb{F}\_2$ which is easily described for a space $X$ by pushing forward the fundamental class of a singular manifold to $H\_\*(X)$. Since $MO$ and $H\mathbb{F}\_2$ both factor through chain complexes, it is tempting to ask if this can be realized as a map of chain c... | https://mathoverflow.net/users/134512 | Relation between "triangulated bordism", MO, and $H\mathbb{F}_2$ | There is an implicit assumption in your question, namely that one can define a chain complex calculating the functor $X\mapsto MO\_\*(X)$ which is based on maps from unoriented manifolds into $X$. As far as I know, this is not known to be the case. Though, as you point out, since $MO$ is a wedge sum of shifts of copies... | 6 | https://mathoverflow.net/users/35353 | 349460 | 147,873 |
https://mathoverflow.net/questions/349455 | 2 | I'm trying to figure out the following problem:
Let $x\_1,\ldots,x\_k\in\mathbb{R}^n$ be some points for some $k<n$. Let $\mbox{conv} (x\_1,\ldots,x\_k)$ be their convex hull. I'm looking for a tight (possibly with an example) upper bound for the number of orthants that such convex hull can intersect with (depending ... | https://mathoverflow.net/users/150522 | Number of orthants intersected by a convex hull | Consider the $k-1$ dimensional simplex given by $\alpha\_1+\alpha\_2+\cdots \alpha\_k=1, \alpha\_i\geq 0$. The equations $e\_i\cdot (\sum\_{j=1}^k \alpha\_j x\_j)=0$ for $1\le i\le n$ describe $n$ hyperplanes that cut our simplex into several regions. Here $\cdot$ is the dot product and $e\_i\in \mathbb R^n$ is the $i$... | 6 | https://mathoverflow.net/users/2384 | 349469 | 147,880 |
https://mathoverflow.net/questions/349363 | 4 | Is it possible to tile an equilateral integer-sided triangle with smaller equilateral integer-sided triangles, with no congruent triangles touching? This has been answered in the negative for the case that no like sizes repeat, I would like to know if it can be done with repeats but like sizes not touching, even at a c... | https://mathoverflow.net/users/142399 | Triangling the triangle | It is not possible. Tutte shows in ["The dissection of equilateral triangles into equilateral triangles"](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/dissection-of-equilateral-triangles-into-equilateral-triangles/5D2608ED5B662B77D4984CC7D7CB0BD9) that ... | 7 | https://mathoverflow.net/users/2384 | 349483 | 147,888 |
https://mathoverflow.net/questions/349475 | 8 | As explained [here](https://ncatlab.org/nlab/show/relation+between+type+theory+and+category+theory), simply typed lambda calculus can be viewed as a syntactic language for category theory. My question is, can the following modification make it equally well a formal syntactic language for 2-category theory?
It goes so... | https://mathoverflow.net/users/30211 | Type theory - category theory correspondence | For the 2-categorical part there is Robert Seely's paper [Modelling computations: 2-categorical framework](http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf) from LICS 1987, and of course a bunch of papers that came afterwards [that cite the paper](https://scholar.google.si/scholar?cites=14241625227633821950&as_sdt=2005&sc... | 6 | https://mathoverflow.net/users/1176 | 349485 | 147,889 |
https://mathoverflow.net/questions/349486 | 12 | I am not sure if this is a well known problem, but I was not able to find anything online that answered my question.
Is it known how to tell whether two elements of the mapping class group of a surface are conjugate?
| https://mathoverflow.net/users/150179 | Conjugacy classes of the mapping class group | An exponential-time solution to the conjugacy problem in the mapping class group was given by Jing Tao, in:
Tao, Jing(1-OK)
Linearly bounded conjugator property for mapping class groups. (English summary)
Geom. Funct. Anal. 23 (2013), no. 1, 415–466.
Tao's main contribution is to prove that two conjugate periodic m... | 16 | https://mathoverflow.net/users/1463 | 349493 | 147,894 |
https://mathoverflow.net/questions/348919 | 2 | **The First question**
Let $A$ be a Banach or a $C^\*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \to A$ with $\phi(e)\psi(f)=\psi(f)\phi(e)=0$.
>
> 1) Is there an automorphism $\rho:A\to A$ with $\rho(e)=\phi(e... | https://mathoverflow.net/users/36688 | Automorphism of algebras with certain initial conditions on given idempotents | **Edit:** The answer to both of your questions is "no".
The counterexamples below all carry a similar flavor.
---
**Counterexample to the topological version of the first question.** (Taking the $C^\*$-algebra of continuous $\mathbb{C}$-valued functions on the counterexample gives a counterexample to the $C^\*... | 3 | https://mathoverflow.net/users/86006 | 349499 | 147,898 |
https://mathoverflow.net/questions/273963 | 4 | Assume that we have a $n-1$ dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a foliation $\mathcal{F}$. We fix an orientation for $D$.(For $n=2$ we assume that it is orientable, that is a vector field generates the distribution. For $n>2$ the distribution is automatically orien... | https://mathoverflow.net/users/36688 | A certain generalization of the Poincare Bendixson theorem | If I'm not mistaken, the [Reeb stability theorem](https://en.wikipedia.org/wiki/Reeb_stability_theorem) prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. *Feuilletages: études géométriques*, Théorème 3.1:
>
> **Theorem (Reeb global stability).** Let $... | 3 | https://mathoverflow.net/users/1275 | 349516 | 147,908 |
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