parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/383272 | 5 | Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define
\begin{equation}
F(n) = (-1)^n \prod\_{1<i<n, p\nmid i} i.
\end{equation}
Since $\mathbb{Z}$ is dense in $\mathbb{Z}\_p$, we can extend $F$ uniquely to a continuous function on $\mathbb{Z}\_p$, which is the $p$-adic gamma function $\Gamma\_p$.
Is there any ... | https://mathoverflow.net/users/171396 | Computing the $p$-adic gamma function $\Gamma_p$ | The $p$-adic gamma function is implemented in Pari/GP (hence available in Sage).
The algorithm used is due to Fernando Rodriguez-Villegas (but probably predates him) and is explained in detail both in his book on experimental mathematics published at Oxford, and in my Springer GTM 240.
| 5 | https://mathoverflow.net/users/81776 | 383284 | 159,451 |
https://mathoverflow.net/questions/383270 | 1 | Let $X$ be a smooth projective curve of genus $g\geq2$, we can construct rank $2$ vector bundles on $X$ with determinant $\omega\_X$ by extension $$0\to \mathcal{O}\to E\to\omega\_X\to 0,$$ does such $E$ necessarily embeds in $\omega\_X\oplus\omega\_X$? (for general $E$?)
| https://mathoverflow.net/users/nan | Embedding extension of sheaves in direct sum | Ok, let me merge my comments into an answer. I claim that for a general extension $0\rightarrow \mathscr{O}\_X\rightarrow E\rightarrow \omega \_X\rightarrow 0$, we have $h^0(E)=1$; therefore $h^0(\mathscr{H}om(E,\omega \_X))=1$ by Riemann-Roch and Serre duality, so there is only one way (up to a scalar) to map $E$ to $... | 2 | https://mathoverflow.net/users/40297 | 383285 | 159,452 |
https://mathoverflow.net/questions/383261 | 1 | I was recently made aware (thanks to the answers on [Why does Riesz's Representation Theorem apply in quantum mechanics?](https://mathoverflow.net/questions/382809/why-does-rieszs-representation-theorem-apply-in-quantum-mechanics)) that the $C^\*$ algebra approach and the Hilbert space approach to quantum mechanics don... | https://mathoverflow.net/users/98901 | Is there a Hilbert space approach to commutative probability theory on locally compact spaces? | I am under the impression that part of the problem here is the relevant distinction between *states* and *normal states*. So let me briefly recall this distinction, before getting to the actual question.
**States vs. normal states**
Note that there are actually two *$C^\*$-algebra like* settings to describe a dual ... | 8 | https://mathoverflow.net/users/102946 | 383294 | 159,457 |
https://mathoverflow.net/questions/383296 | 2 | I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school.
A disk $D$ of radius $1$ contains disks $D\_i$ ($i \ge 1$) of radius $r\_i<1$ with pairwise disjoint interiors. Assuming the $D\_i$ "use up" the area of $D$ in the sense that $\s... | https://mathoverflow.net/users/123679 | On covering a disk by non-overlapping subdisks | Proven by [O. Wesler, “An infinite packing theorem for spheres,” PAMS Vol. 11, pp. 324-326, (1960).](https://sci-hub.do/10.2307/2032977)
| 1 | https://mathoverflow.net/users/20516 | 383298 | 159,459 |
https://mathoverflow.net/questions/383036 | 1 | Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x\_1, \ldots, x\_m\}$ and $j\_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism class of $(\Sigma, x, j\_0)$ in the moduli space $\mathcal{M}\_{g,m}$ of genus $g$ Riemann surfaces with $m$ marked poi... | https://mathoverflow.net/users/41200 | Deform a complex structure fixing marked points | This is true if $U$ satisfies an extra boundedness condition: There exists $\delta>0$ such that for all $(\Sigma',x', j\_0') \in U$, the distance between any two marked points $x\_i',x\_j'$ is at least $\delta$.
This boundedness condition follows from the mild boundedness condition that the closure of $U$ is compact.... | 1 | https://mathoverflow.net/users/18060 | 383300 | 159,460 |
https://mathoverflow.net/questions/383295 | 6 | In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R\_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving right adjoints is closed under limits in $\operatorname{Cat}\_\infty$. I would like to understand some detail of the p... | https://mathoverflow.net/users/144100 | Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$? | By lemma 5.4.5.5., the forgetful functor $Pr^R\_\omega\to Cat\_\infty$ preserves pullbacks : the projection functors in the pullback preserve filtered colimits.
It's easy to prove that the same thing holds for arbitrary products ( a colimit in $\prod\_\alpha C\_\alpha$ is computed pointwise, which means precisely tha... | 9 | https://mathoverflow.net/users/102343 | 383309 | 159,466 |
https://mathoverflow.net/questions/383303 | 5 | **Question:** let $x\_{i}>0$ $(i=1,2,\cdots,n)$, such that $x\_{i}\neq x\_{j},\forall i\neq j$, find all real numbers $p$ that satisfy the following inequality
$$\sum\_{i=1}^{n}\dfrac{x^p\_{i}}{\prod\_{j\neq i}(x\_{i}-x\_{j})}\ge 0$$
when $n=3$, I have solved it as $p\le 0$ or $p\ge 1$, because we only use this
$$x^p... | https://mathoverflow.net/users/38620 | How to show this symmetric function inequality | The conjecture for $n=4$ is correct. For general $n$ and any $x\_1,\ldots,x\_n$,
$$
f(p) := \sum\_{i=1}^{n}\dfrac{x^p\_i}{\prod\_{j\neq i}(x\_i-x\_j)}
$$
is positive for $p > n-2$ and switches sign at $p=0,1,2,\ldots,n-2$
and nowhere else. For example:
If $n=5$ then $f(p) \geq 0$ iff
$p \geq 3$ or $1 \leq p \leq 2$ ... | 10 | https://mathoverflow.net/users/14830 | 383311 | 159,467 |
https://mathoverflow.net/questions/383314 | 13 | Let $C\_n=\frac1{n+1}\binom{2n}n$ be the familiar Catalan numbers.
>
> **QUESTION.** Is there a combinatorial or conceptual justification for this identity?
> $$\sum\_{k=1}^n\left[\frac{k}n\binom{2n}{n-k}\right]^2=C\_{2n-1}.$$
>
>
>
| https://mathoverflow.net/users/66131 | Looking for a combinatorial proof for a Catalan identity | By the ballot theorem, $\frac{k}{n} \binom{2n}{n+k}$ is the number of Dyck paths, i.e. $(1,1), (1,-1)$-walks in the quadrant, from the origin to $(2n-1, 2k-1)$. You need to concatenate a pair of those to get a Dyck path to $(4n-2,0)$, and $k$ takes values between 1 and $n$.
| 38 | https://mathoverflow.net/users/47484 | 383319 | 159,471 |
https://mathoverflow.net/questions/383328 | 4 | Given an elliptic curve $E/\mathbb{Q}$, is it possible to determine whether or not $E$ has torsion points just by looking at it's Hasse-Weil L function $L(E,s)$? In general, what effects does an elliptic curve having torsion points have on its L function?
The effect of having free points is clearly seen in the Birch ... | https://mathoverflow.net/users/159298 | What effect do Torsion points have on an Elliptic Curve's L function? | This has been alluded to in one of the comments, but if $E(\mathbb Q)$ has an $\ell$-torsion point, then at every prime $p$ of good reduction we have
$$ p+1-a\_p = \#E(\mathbb F\_p) \equiv 0 \pmod \ell, $$
so the local factor of the $L$-function at $p$ satisfies
$$ L\_p(T) = 1 - a\_p T + p T^2 \equiv 1 - (p+1) T + p T^... | 6 | https://mathoverflow.net/users/11926 | 383332 | 159,472 |
https://mathoverflow.net/questions/383291 | 2 | I have a grid of 16 tiles face down. Half are good outcomes and half are bad outcomes. How would I calculate the probability of picking x number of Good outcomes before y number of bad outcomes are picked.
Once a tile is selected and it’s state is revealed (good or bad), it is left face up. That is to say there are n... | https://mathoverflow.net/users/173691 | Calculate the discrete probability of x number of good outcomes occurring before y number of bad outcomes | You get $x$ *good* before $y$ *bad* if you get $x$ or more *good* out of $x+y-1$ attempts. Let's call the probability of this $P(x,y)$ and using hypergeometric probabilities we have $$P(x,y)= \sum\_{n=x}^8 \frac{{8 \choose n}{8 \choose y-1}}{{16 \choose n+y-1}}$$
There are some fairly obvious symmetric and anti-symme... | 1 | https://mathoverflow.net/users/12565 | 383335 | 159,474 |
https://mathoverflow.net/questions/383150 | 8 | $\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$. Let $\Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$. Let $\O(n)\subseteq \... | https://mathoverflow.net/users/97652 | Action of symmetric matrices under $\mathrm{O}(n)$ | The adjoint action of $SO(n,\mathbb C)$ on $\mathrm{Sym}\_0(n,\mathbb C)$ (we can replace $O(n,\mathbb C)$ by $SO(n,\mathbb C)$, and $\mathrm{Sym}(n,\mathbb C)$
by the subspace of traceless matrices, since the
action splits off the scalar multiples of the identity) is the isotropy representation of the complex symmetri... | 7 | https://mathoverflow.net/users/15155 | 383338 | 159,476 |
https://mathoverflow.net/questions/382870 | 2 | If you remove any $2$ vertices from a complete graph, the chromatic number gets decreased by two. (The famous [double-critical graph conjecture](https://arxiv.org/abs/1610.00636) is about the existence of a non-complete graph such that any $2$ connected vertices can be removed such that the chromatic number of the grap... | https://mathoverflow.net/users/8628 | Decreasing the chromatic number by $2$ by removing $2$ well-chosen vertices | Let $G\_0=(V\_0,E\_0)$ be a complete graph of order $n\ge4$. Choose two distinct points $a,b\in V\_0$ and two distinct points $x,y\notin V\_0$. The graph $G=(V,E)$ with vertex set $V=V\_0\cup\{x,y\}$ and edge set $E=E\_0\cup\{\{a,x\},\{b,y\}\}$ satisfies your requirements with $\chi(G)=n$.
| 1 | https://mathoverflow.net/users/43266 | 383346 | 159,479 |
https://mathoverflow.net/questions/332695 | 16 | Call a poset locally countable if the set of predecessors of every member of the poset is countable. Is the following consistent?
There is no locally countable poset $P$ of size continuum such that every locally countable poset of size continuum embeds into $P$?
*Remark:* under ZFC+CH, there is such a universal loc... | https://mathoverflow.net/users/2689 | Universal locally countable partial order | The answer to this question, at least in ZFC, is "no." It is provable in ZFC that there is a universal locally countable partial order of size continuum. In other words, it is provable in ZFC that there is a locally countable partial order of size continuum into which every other such partial order embeds. This is true... | 7 | https://mathoverflow.net/users/147530 | 383351 | 159,482 |
https://mathoverflow.net/questions/383360 | 3 | I am interested in the following class of knots $K$:
$\{$$K$ has a braid presentation such that for any fixed position $k$, either only positive or negative powers of $\sigma\_k$ appear in the braid word$\}$ (For example, $\sigma\_1 \sigma\_2^{-1}\sigma\_1 \sigma\_2^{-1}$ would be an instance of such a braid.)
My f... | https://mathoverflow.net/users/45553 | Knots with a braid presentation with only positive or negative crossings on each fixed position | The braids that you're talking about are called *homogeneous* braids. I don't know that closures of homogeneous braids have a name, though. (There is something called "homogeneous knot", but the class of homogeneous knots strictly contains the set of closures of homogeneous braids.)
Stallings [proved](https://www.mat... | 5 | https://mathoverflow.net/users/13119 | 383365 | 159,483 |
https://mathoverflow.net/questions/383180 | 4 | Let $X$ be a smooth projective curve, let $E',E''$ be stable vector bundles on $X$, with $\mathrm{slope} (E'')>\mathrm{slope} (E')$.
Let $0\neq[E]\in \mathrm{Ext}^1(E'',E')$ be an extension,
$$0\to E'\to E\to E''\to 0$$
do we know if $E$ is necessarily semi-stable? If not, how about a general $E$ (a general point i... | https://mathoverflow.net/users/nan | Extension of stable vector bundles on curves | For the first question, the answer is negative even for a general extension. On $\mathbb P^1$, you can take $E' = \mathcal O(-1)$ and $E'' = \mathcal O$ so that a general extension splits.
In the special case $E' = \mathcal O\_X, E'' = \omega\_X$, $X$ a curve of genus $>2$, the general extension is stable. It suffice... | 2 | https://mathoverflow.net/users/18060 | 383383 | 159,489 |
https://mathoverflow.net/questions/383378 | 4 | Could you give an example of a unital simple $C^\*$-algebra that $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$?
| https://mathoverflow.net/users/137242 | $\tau (p) = \tau (q)$ for all normalized traces does not imply $p \sim q$ | If a simple $C^\*$-algebra admits an infinite projection $p$ (ie a projection that is equivalent to a proper subprojection $q$), then it does not carry any tracial trace and in particular it provides an example of the kind you are looking for. Indeed a trace $\tau$ would vanish on the nonzero projection $p-q$, and ther... | 6 | https://mathoverflow.net/users/10265 | 383387 | 159,491 |
https://mathoverflow.net/questions/383271 | 8 | For the purposes of this question, pretend that there is a platonic, "standard" model $V$ of ZFC - so that every set theoretic statement has a meaningful truth value.
Fix a bijection between $\mathbb{N}$ and the set of formulas $\varphi(x)$ with one free variable $x$ in the language of set theory, and let $\varphi\_i... | https://mathoverflow.net/users/2363 | How many reals can we construct by iteratively writing down truth tables for ZFC? | I think it is easier to think about the question generalized to an arbitrary transitive model of ZFC, resisting the natural urge to grasp towards the Absolute. So fix such a model $M$, and let $\mathcal T^M(\alpha)$, $F^M$, and $S^M(\alpha)$ be as you defined them, replacing $V$ with $M$. Let's omit the superscripts, t... | 5 | https://mathoverflow.net/users/102684 | 383409 | 159,499 |
https://mathoverflow.net/questions/383412 | 4 | $\newcommand\R{\mathbb R}$Suppose $f:\R^2 \to \R^2$ is a Whitney map with singularities (well, I'm not sure if this is the name for it, Whitney calls them *excellent maps* in his [1955 paper](https://www.jstor.org/stable/1970070)), i.e. it is an infinitely differentiable map such that the singularity set in the domain ... | https://mathoverflow.net/users/3949 | restricting the "Whitney" map | No this is false, here is a counter example. Let $f:(\frac{1}{2},1)\times(0,4\pi)\rightarrow \mathbb{R}^2$ be given by $f(r,\theta)=(r \cos(\theta),r\sin(\theta))$. Of course the domain is diffeomorphic to the plane and this map does not have any singularity. It is not injective.
| 5 | https://mathoverflow.net/users/12156 | 383417 | 159,501 |
https://mathoverflow.net/questions/383399 | 15 | Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the [shape](https://ncatlab.org/nlab/show/shape+of+an+%28infinity%2C1%29-topos) of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to\_\leftarrow \mathcal E: \Gamma$ is the the constant / global... | https://mathoverflow.net/users/2362 | Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos? | In subsection 13.8.10. of version 7 of the "Exodromy" paper by Barwick, Glasman and Haine (arXiv:[1807.03281v7](https://arxiv.org/abs/1807.03281v7)), the authors define the "pyknotic étale homotopy type". They say that it should recover the pro-étale fundamental group of Bhatt–Scholze and will explore this in a future ... | 9 | https://mathoverflow.net/users/98835 | 383424 | 159,504 |
https://mathoverflow.net/questions/383435 | -2 | I suppose the solid torus in $\mathbb{R}^3$ is not a geometric manifold. Since I am not an expert in this area, I would like to ask whether there is some easy way to see this.
| https://mathoverflow.net/users/41224 | Existence of a geometric structure on a solid torus | It all depends on your definition of a "geometric manifold."
1. One definition would require the existence of a complete finite volume locally homogeneous Riemannian metric (from Thurston's list of eight 3D geometries). This definition is used, for instance by Walter Neumann in
his [Notes on Geometry and 3-Manifolds]... | 8 | https://mathoverflow.net/users/39654 | 383440 | 159,508 |
https://mathoverflow.net/questions/383366 | 1 | I am trying to read [this](http://publications.ias.edu/sites/default/files/Letter%20-%20golden%20gates%20march_0.pdf) letter by Sarnak, where, among other things, he discusses the problem of intrinsic diophantine approximation on $S^3 = \left\{\mathbf{x} \in \mathbb{R}^4 : x\_1^2+x\_2^2+x\_3^2+x\_4^2 = 1\right\}$. In p... | https://mathoverflow.net/users/27791 | Diophantine approximation on spheres | Here is a proof that $S(\mathbb{Z}[\frac{1}{p}])$ lies dense in $S^3$ for all primes $p \equiv 1 \pmod{4}$.
Since this is a wholly algebraic/arithmetical question, it is easier to switch to algebro-geometric terminology. We consider the map
$$
\phi \, \colon S^3 \rightarrow \mathbb{A}^1
$$
where $\mathbb{A}^1$ is the... | 3 | https://mathoverflow.net/users/17907 | 383455 | 159,515 |
https://mathoverflow.net/questions/383418 | 8 |
>
> What are some good introductory references to constructive mathematics for non-specialist mathematicians?
>
>
>
I would like to learn more about constructive mathematics, just to improve my general mathematical knowledge, but I have some difficulties in finding good references for non-specialists. Are there ... | https://mathoverflow.net/users/122026 | Initiation to constructive mathematics | I have a soft spot for *Constructivism in Mathematics: An Introduction* (2 volumes) by Troelstra and van Dalen as an overview. And for what one loses compared to classical mathematics (as well as its historical importance), it's probably worth going straight to Bishop's *Foundations of constructive analysis*.
However... | 14 | https://mathoverflow.net/users/49 | 383458 | 159,517 |
https://mathoverflow.net/questions/383452 | 0 | Consider the definition of group scheme in *Stack Project* [[022R]](https://stacks.math.columbia.edu/tag/022R). In the paragraph following definition 39.4.1, it is said that
>
> We have morphisms of schemes over $S$: (identity) $e:S\rightarrow G$ and (inverse) $i:G\rightarrow G$ such that for every $T$ the quadrupl... | https://mathoverflow.net/users/133871 | Definition of group scheme | You need to use the condition that $m$ is a morphism of scheme. By that we can deduce $G(\cdot)$ is a functor.
| 0 | https://mathoverflow.net/users/133871 | 383463 | 159,519 |
https://mathoverflow.net/questions/383478 | 1 | To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L\_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and eigenfunctions of $L$ well.
Consider $L\_{per}=L+ikx$, where $k$ is an integer. What can we say about its spectrum ? Is there a ... | https://mathoverflow.net/users/173845 | Sum of positive self-adjoint operator and an imaginary "potential": literature request | Let me consider the case of the Schrödinger equation, $L=-\nabla^2+V(x)$. Then the operator $L+ikx$ has [PT-symmetry,](https://en.wikipedia.org/wiki/Non-Hermitian_quantum_mechanics) meaning that it is invariant under the combined action of inversion $x\mapsto-x$ (parity P) and complex conjugation (time-reversal T). Suc... | 2 | https://mathoverflow.net/users/11260 | 383483 | 159,524 |
https://mathoverflow.net/questions/383486 | 1 | I have the following question and I'd greatly appreciate any help!
Basically, I have an arbitrary probability distribution with pdf $f(x)$, we can assume it's continuous with support on $[0,\infty]$
Denote $g(x)$ as pdf of an exponential random variable with fixed, known rate $\lambda$, and $h(x) = f(x) \* g(x)$ wh... | https://mathoverflow.net/users/173849 | Uniqueness of deconvolution after convolution? | The answer is yes. Let $\hat p$ denote the characteristic function of a pdf $p$, so that
$$\hat p(t)=\int\_{\mathbb R}e^{itx}p(x)\,dx$$
for real $t$. Then $\hat h=\hat f\,\hat g$ and
$$\hat g(t)=\frac1{1-it/\lambda}$$
for real $t$, so that
$$\hat f(t)=\hat h(t)/\hat g(t)=\hat h(t)(1-it/\lambda)$$
for real $t$. [Inverti... | 2 | https://mathoverflow.net/users/36721 | 383493 | 159,528 |
https://mathoverflow.net/questions/383487 | 7 | Given a smooth, projective (complex) varieties $X$, is it true that the grothendieck group $K\_0(X)$ of equivalence classes of coherent sheaves on $X$, is generated by clases of invertible sheaves i.e., any class of a coherent sheaf on $X$ can be written as a linear combination of classes of invertible sheaves on $X$? ... | https://mathoverflow.net/users/45397 | Grothendieck group generated by classes of invertible sheaves | Here is a different, perhaps more elementary example.
Consider the Grassmannian $\mathrm{Gr}(2,4)$ of lines in $\mathbf{P}^3$, and let $\mathscr{Q}$ be the universal quotient bundle. The line bundle $\mathrm{det}(\mathscr{Q})$ generates the Picard group of $\mathrm{Gr}(2,4)$. If $[\mathscr{Q}]$ were in the subgroup o... | 12 | https://mathoverflow.net/users/104669 | 383498 | 159,529 |
https://mathoverflow.net/questions/382179 | 9 | What combinatorial and number-theoretic propositions can $I\Delta\_0$ prove? Obviously there are an infinitude of them, but what are some well known theorems that can be proved in $I\Delta\_0$, if any?
| https://mathoverflow.net/users/163672 | What can $I\Delta_0$ prove? | Since no one else is biting, I'll answer, and thanks to comments I now this is accurate:
$I\Delta\_0$ can prove several basic theorems:
* Every square equals 0 or 1 mod 4
* No prime has a rational square root
* The only solutions to $x^3+y^3=z^3$ or $x^4+y^4=z^4$ are trivial
* Every $x$ is divisible by a prime $p$ ... | 9 | https://mathoverflow.net/users/nan | 383500 | 159,531 |
https://mathoverflow.net/questions/383490 | 2 | I know that in general exact-by-exact extensions of $C^\*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^\*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ is exact and $B$ is nuclear, then $A$ is exact?
If it is true, could you give a reference of the result?
If it is... | https://mathoverflow.net/users/173852 | About nuclear-by-exact extensions | Yes, if the quotient $B$ is nuclear, then the extension is locally semisplit by the Choi-Effros lifting theorem. Hence if (in addition) $I$ is exact, then $A$ is exact (see for instance Exercise 3.9.8 in the book of Brown and Ozawa).
---
NOTE: I wrote the following when I misread the question, and thought the que... | 5 | https://mathoverflow.net/users/126109 | 383513 | 159,533 |
https://mathoverflow.net/questions/383480 | 18 | Title. For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$?
If $G$ is a subgroup of either $S^0,S^1,S^3$ or $S^7$ this induces a free action on $S^{\infty}$ and thus a $G-$principal bundle $S^{\infty} \rightarrow BG$. Does the reverse direction hold? That is, ... | https://mathoverflow.net/users/127178 | For what topological groups $G$ can we take $EG \rightarrow BG$ to be of the form $S^{\infty} \rightarrow BG$? | I like to think of $EG$ and $BG$ in terms of configuration spaces.
The space $BG$ can be identified with the following configuration space. It consists of configurations of finitely many points in the open interval $(0,1)$ with the points having labels in the topological group $G$. It is topologized so that
* when ... | 20 | https://mathoverflow.net/users/184 | 383537 | 159,543 |
https://mathoverflow.net/questions/383534 | 3 | Let $R$ be a noetherian local ring; I say it has *isolated singularity* if its spectrum is regular outside the closed point. Such rings certainly don't need to be irreducible, for example the localisation of $k[x,y]/(x,y)$ at $(x,y)$. However, in dimension 2 I have to work a little harder to build an example; the first... | https://mathoverflow.net/users/4710 | Lci local rings with isolated singularity are irreducible? | Claim: A noetherian local normal ring $R$ is a domain.
Proof: recall that normal is equivalent to $(S\_2)$ and $(R\_1)$. Since $R$ is $(S\_2)$, $Spec(R)$ it is connected in codimension one (removing any subset of codimension at least $2$ leaves it connected). There is a pleasant characterization using the connectedne... | 2 | https://mathoverflow.net/users/2083 | 383543 | 159,544 |
https://mathoverflow.net/questions/383539 | 29 | It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a strong temptation which has been around for perhaps a hundred years.
>
> **Question.** *Suppose $\mathbb E[X^n] = 0$ for... | https://mathoverflow.net/users/78539 | Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments | This doesn't really require modern functional analytic tools, but we can prove a statement (due originally to Edelheit, according to Jochen Wengenroth in the comments) like
>
> Let $V$ be a Frechet space, complete with respect to a family of norms $||x||\_i$, $i=0,1,\dots$. Let $L\_1,L\_2,\dots$ be linear forms on ... | 13 | https://mathoverflow.net/users/18060 | 383559 | 159,550 |
https://mathoverflow.net/questions/383330 | 5 | I've come across two definitions of an equivalence of étale Lie groupoids, and I'd like to know whether they are equivalent.
Let $\mathcal{G}$ be an étale Lie groupoid with space of objects $\mathcal{G}\_0$ and space of arrows $\mathcal{G}\_1$.
Write $\alpha$ and $\omega\colon \mathcal{G}\_1 \to \mathcal{G}\_0$ for t... | https://mathoverflow.net/users/135175 | Equivalence of definitions of equivalence of étale Lie groupoids | Notice that because $\omega$ is étale and $\omega\pi\_1$ is a submersion, $\pi\_1\colon \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0 \to \mathcal{H}\_1$ is a submersion, as is $\alpha\pi\_1$. Since $\alpha$ is étale, $\pi\_2\colon \mathcal{H}\_1\times\_{\mathcal{H}\_0}\mathcal{G}\_0 \to \mathcal{G}\_0$ is étale... | 2 | https://mathoverflow.net/users/135175 | 383570 | 159,557 |
https://mathoverflow.net/questions/383592 | 3 | Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability measures $\mathcal{P}(X\times Y)$ on $X\times Y$. We equip this space with the uniform metric given on any two $f,g\in \mathcal... | https://mathoverflow.net/users/36886 | Is disintegration continuous? | No, at least if $X$ is uncountable and $Y$ has at least two points. Let $K$ be a compact subset of $Y$ with at least two points (any finite such subset will do). Under this assumption, there exists an atomless Borel probability measure $\nu$ on $X$. Let $\mathcal{P}\_{\nu,K}(X\times Y)$ be the closed set of Borel proba... | 1 | https://mathoverflow.net/users/35357 | 383598 | 159,564 |
https://mathoverflow.net/questions/383590 | 2 | Every $2$-form $\omega\in \Omega^2(\mathbb{R}^{2n+1})$ induces a skew-symmetric map
$$
\omega(-,-)\colon\Gamma(T\mathbb{R}^{2n+1})\otimes \Gamma(T\mathbb{R}^{2n+1}) \to C^\infty(\mathbb{R}^{2n+1})
$$
where $\Gamma(T\mathbb{R}^{2n+1})$ denotes the space of vector-fields on $\mathbb{R}^{2n+1}$. It is clear that at every ... | https://mathoverflow.net/users/89741 | $2$-Form inducing a non-degenerate form on $\Gamma(T\mathbb{R}^{2n+1})$ | The example in $\mathbb{R}^3$ given by
$$
\omega = x\,\mathrm{d}y\wedge\mathrm{d}z + y\,\mathrm{d}z\wedge\mathrm{d}x + z\,\mathrm{d}x\wedge\mathrm{d}y,
$$
shows that the kernel of $\omega$ need not be a line bundle. Away from the origin, the kernel is spanned by the radial vector field
$$
R = x\,\frac{\partial\ }{\part... | 6 | https://mathoverflow.net/users/13972 | 383599 | 159,565 |
https://mathoverflow.net/questions/383593 | 3 | In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and **assuming** the zeta values to be transcendental and algebraically independent, this means that the loop amplitudes live in extensions of $\mathbb{Q}$ of increa... | https://mathoverflow.net/users/45250 | Can one define a degree of a period? | Jianming Wan's preprint on "Degrees of Periods" should answer your question, using the notion of periods in the sense of Kontsevich-Zagier: <https://arxiv.org/abs/1102.2273>
| 0 | https://mathoverflow.net/users/13625 | 383603 | 159,568 |
https://mathoverflow.net/questions/294591 | 11 | Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is *uninteresting* if
1. $J(R)$ coincides with the nilradical, or
2. $J(R)$ is the intersection of a finite number of maximal ideals.
It seems as if most rings used in algebraic geometry have uninteresting Jacbson radical:
* Every fi... | https://mathoverflow.net/users/2362 | Do commutative rings with "interesting" Jacobson radicals turn up "in nature"? | Rings similar to the ring $R$ as you consider above, "show up" in real algebraic geometry. If $k:=\mathbb{R}, A:=k[x]$ is the ring of polynomials with real coefficients, and $S\subseteq A$ is the set of polynomials with no real roots ($f(x):=x^2+1$ is such a polynomial), then $B:=S^{-1}A$ may be seen as the global sect... | 3 | https://mathoverflow.net/users/nan | 383605 | 159,569 |
https://mathoverflow.net/questions/383597 | 2 | Let $H=(V,E)$ be a [hypergraph](http://en.wikipedia.org/wiki/Hypergraph), and $\kappa$ be a cardinal. We say that a map $c:V \to \kappa$ is a *coloring* if the restriction $c\restriction\_e$ is non-constant whenever $e\in E$ and $|e|\geq 2$. The smallest cardinal $\kappa$ such that there is a coloring $c:V\to\kappa$ is... | https://mathoverflow.net/users/8628 | Chromatic number of rainbow hypergraphs | In fact, every rainbow hypergraph has chromatic number $2$.
Let $H=(V,E)$ be a rainbow hypergraph, $E=\{e\_2,e\_3,\dots\}$, $|e\_n|=n$. Consider a random coloring $c:V\to\{0,1\}$, let $A$ be the event that $c$ is not a proper coloring of $H$, and let $A\_n$ be the event that $c$ is constant on $e\_n$. Then
$$P(A)=P\l... | 2 | https://mathoverflow.net/users/43266 | 383612 | 159,571 |
https://mathoverflow.net/questions/383596 | 1 | Do we know the Taylor expansion at $0$ of the *radial Mathieu functions* $(\mathsf{Mc}\_n^{(j)}(\,\cdot\,, \sqrt{q}))\_{n \ge 0}$ and $(\mathsf{Ms}\_n^{(j)}(\,\cdot\,, \sqrt{q}))\_{n \ge 1}$, for $q \in \mathbb{R}$ and $j \in \{1, 2, 3, 4\}$, with the definition and conventions of [[Sec. 28.20(iv)](https://dlmf.nist.go... | https://mathoverflow.net/users/153800 | Taylor expansion of Modified Mathieu functions | I have not found the series expansion worked out explicitly, but it can be obtained from the representation of the Mathieu functions as series of Bessel functions. I found this [collection of formulas](https://homepages.mty.itesm.mx/jgutierr/Mathieu/Mathieu.pdf) convenient.
There are four classes of radial Mathieu fu... | 1 | https://mathoverflow.net/users/11260 | 383614 | 159,572 |
https://mathoverflow.net/questions/383316 | 1 | [Lectures on Condensed Mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf), Theorem 3.3 says that for any compact Hausdorff space $S$, the cohomology $H\_{\mathrm{cond}}^i(S,\mathbb R)=0$ for $i>0$ and $H\_{\mathrm{cond}}^0(S,\mathbb R)=C(S,\mathbb R)$ the *space* of continuous functions on $S$. Fur... | https://mathoverflow.net/users/nan | When is a complex of Banach spaces exact as condensed abelian groups? | Thanks for the nice question. As you say in the update, the issue is basically the following: suppose given a map of Banach spaces $f:V\rightarrow W$. When is the induced map of condensed $\mathbb{R}$-modules $\underline{V}\rightarrow\underline{W}$ an epimorphism? This is not obvious to answer from first principles.
... | 4 | https://mathoverflow.net/users/3931 | 383617 | 159,573 |
https://mathoverflow.net/questions/383582 | 4 | Given some absolute constant $C$ (In my case, $C=4$ would suffice) and an elliptic curve $E/\mathbb{Q}$, are there upper bounds on $|L(E,s)|$ that are uniform for $|s|<C$? Using the functional equation we see that $|L(E,s)|\gg N\_{E/\mathbb{Q}}$ for at least some points $s$, and so would it be possible to prove some so... | https://mathoverflow.net/users/159298 | Are there upper bounds on the L function $|L(E,s)|$ for $|s|<C$? | Assuming the modularity theorem, apply the maximum modulus principle to $$\Lambda(E,s)=N^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$$ which is entire and $\Lambda(E,s)=\pm \Lambda(E,2-s)$ (where $N$ is the conductor).
The Hasse bound gives a bound for $\log L(E,s)$ on $\Re(s)=C+2$, this gives a bound for $\Lambda(E,s)$ on $\Re(... | 3 | https://mathoverflow.net/users/84768 | 383618 | 159,574 |
https://mathoverflow.net/questions/383610 | 9 | The question is: does the set of prime numbers $q$ such that $\sum\limits\_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers, contain infinitely many elements? You can find the first elements here (<http://oeis.org/A329403>). Any insight on this would be welcomed.
Thanks in advance for your time and effort!
| https://mathoverflow.net/users/172800 | Set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers | Nice question! The answer is affirmative. If $\sigma(x)$ denotes the sum of primes up to $\sqrt{x}$, then it suffices to show that $\pi(x)-\sigma(x)$ changes sign infinitely often, because
$$\pi(x)-\sigma(x)<0<\pi(x+1)-\sigma(x+1)$$
never holds. I thank Juan Moreno and Will Sawin for this simple but crucial observation... | 15 | https://mathoverflow.net/users/11919 | 383620 | 159,576 |
https://mathoverflow.net/questions/383621 | 12 | Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.
| https://mathoverflow.net/users/36688 | Holomorphic Urysohn Lemma | Yes, since the morphism of sheaves $\mathcal{O}\_{\mathbb{C}^n} \to \mathcal{O}\_{M \cup N}$ is surjective, it is surjective on global sections by Cartan's theorem B. Thus, any holomorphic function on $M \cup N$ has a holomorphic extension to $\mathbb{C}^n$, in particular, this holds for the function $f$ which is $\equ... | 12 | https://mathoverflow.net/users/49151 | 383638 | 159,579 |
https://mathoverflow.net/questions/383628 | 1 | Let $N \in \mathbb N$ and $c\_n \in \mathbb C$, $t\_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t\_n$ and coefficients $c\_n$, i.e.
$$
f=\sum\_{n=1}^Nc\_n\delta\_{t\_n}.
$$
The $k$-th Fourier coefficient of $f$ is defined by
$$
\hat f (k) = \sum\_... | https://mathoverflow.net/users/106425 | How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely? | Assuming you only care about $t\_n, s\_n$ mod $1$, then $2N$ values of $k$ suffice. (A dimension argument shows this is optimal). In fact, we can take $K= \{0,\dots, 2N-1\}$.
Form a $2N \times N$ matrix whose entry in the $j$th column and the $k+1$st rown is $e^{- 2 \pi i k t\_j}$. Add at most $N$ additional column t... | 5 | https://mathoverflow.net/users/18060 | 383647 | 159,583 |
https://mathoverflow.net/questions/383626 | 0 | Given $E\to B$ a non-isotrivial (compact) Riemann surface-bundle (of genus $g>1$) between two complex manifolds and $E'\to E$ is a finite branched cover. Then is the composition map $E'\to E\to B$ still non-isotrivial?
| https://mathoverflow.net/users/88180 | Is the composition of a finite branched cover and a non-isotrivial Riemann surface bundle still non-isotrivial | If $E'$ is isotrivial, then the fibers $E'\to B$ are all isomorphic to a single fiber, so the Jacobians of every fiber are isomorphic to a single Jacobian abelian variety $A$. Then the Jacobian of every fiber of $E \to B$ is isomorphic to a quotient of $A$. But $A$ has only countably many distinct quotients, up to isom... | 2 | https://mathoverflow.net/users/18060 | 383653 | 159,585 |
https://mathoverflow.net/questions/383650 | 0 | Suppose that $X$ and $Y$ are Cauchy-distributed with $\gamma=1$, i.e., with PDF $\frac 1 \pi \frac 1 {1+x^2}$. I tried to find the distribution of $R = \sqrt{X^2+Y^2}$. The PDF of $R$ should be given by integrating over an annulus $A: r^2 < x^2+y^2 < (r+dr)^2$ in the $x-y$ plane.
$P(r)dr = \int\_A \frac 1 {\pi^2} \fr... | https://mathoverflow.net/users/114143 | Distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Cauchy | The integral over $\theta$ should be
$$ \frac{1}{\pi^2} \int\_0^{2\pi} \frac{r\; d\theta}{1+r^2 + r^4 \cos^2(\theta) \sin^2(\theta)} = \frac{4 r}{\pi (r^2+2)\sqrt{r^2+1}}$$
The integral of this from $r=0$ to $\infty$ is indeed $1$.
| 0 | https://mathoverflow.net/users/13650 | 383654 | 159,586 |
https://mathoverflow.net/questions/383532 | 2 | Let $X\_1, X\_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S\_n = (X\_1 + \cdots + X\_n)/\sqrt{n}$ to be their normalized sum. Define $D\_n$ to the total variation (TV) distance between $S\_n$ and a standard Gaussian.
**Question:** Is $D\_n$ always mon... | https://mathoverflow.net/users/83122 | Is total variation distance of normalized sum of random variables to Gaussian monotonic decreasing? | Unless I made a mistake, this is not true.
Take the random variable $X$ to be $M$ with probability $\epsilon$, $-M$ with probability $-\epsilon$, and distributed as $N(0, \frac{ 1- 2 \epsilon M^2}{ 1-2\epsilon} )$ with probability $1-2\epsilon$. Here we take $M$ very large and $\epsilon M^2$ somewhat small, for insta... | 2 | https://mathoverflow.net/users/18060 | 383658 | 159,588 |
https://mathoverflow.net/questions/383547 | 6 | Consider $\mathbb CP^n$ and let $H\subset \mathbb CP^n$ be a hyperplane. Suppose $\varphi: \mathbb CP^n\to H$ is a rational map that fixes $H$ pointwise. I believe that $\varphi$ must be a projection from a point $p\in \mathbb CP^n\setminus H$ to $H$. How to prove this?
**Added.** F\_L gave a nice counterexample to t... | https://mathoverflow.net/users/13441 | Rational maps from $\mathbb CP^n$ to $\mathbb CP^{n-1}$, fixing $\mathbb CP^{n-1}$ | With the additional restriction, $\varphi$ is indeed a linear projection.
Applying a linear change of coordinates, $H$ is given by $x\_0=0$, so $\varphi$ is
$[x\_0:\cdots:x\_n]\mapsto [0:P\_1(x\_0,\ldots,x\_n):\cdots: P\_n(x\_0,\ldots,x\_n)]$
for some homogeneous polynomials $P\_1,\ldots,P\_n\in \mathbb{C}[x\_0,\... | 6 | https://mathoverflow.net/users/23758 | 383660 | 159,589 |
https://mathoverflow.net/questions/383548 | 5 | This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum\_{n=0}^K \frac1{n!} x^n
$
converges uniformly to $g$ on $[0,1]$ and that for $K$ large enough
$$
\max\_{x \in [0,1]}\, \|\sum\_{n=0}^K \frac1{n!} ... | https://mathoverflow.net/users/170917 | Approximation of analytic function by a fixed number of monomials | The general description of all continuous functions which are uniform limits of fewnomials seems to be given by the following
**Theorem.** Fix $0<a<b$. The following two properties of a continuous function $f\in C[a,b]$ are equivalent:
(i) $f$ is log-quasipolynomial, that is, $f(x)=P(x,\log x)$ for a certain quasi-... | 4 | https://mathoverflow.net/users/4312 | 383665 | 159,592 |
https://mathoverflow.net/questions/383546 | -1 | How do sub-convex bounds turn back into number theory? As of around 2010, we get estimates of L-functions like this:
$$ L(\tfrac{1}{2}, \pi \times \chi ) \ll\_{F, \epsilon, \pi} C(\chi)^{\frac{1}{2}} $$
$\pi$ is a "cuspidal automorphic representation", and $\chi$ is a character. The twisted L-function would look like t... | https://mathoverflow.net/users/1358 | How does the sub-convex bound $ L(\tfrac{1}{2}, \pi \times \chi ) \ll_{F, \epsilon, \pi} C(\chi)^{\frac{1}{2}} $ relate to "elementary" number theory? | The most straightforward application of subconvexity is to bounds to sums of the form
$$ \sum\_{ n < X} \chi(n) a\_n(f) $$ where $f(q) = \sum\_n a\_n(f) q^n$ is a modular form, or to the analogous sequences of Fourier coefficients arising from other modular forms. It's slightly easier to work with sums of the form $$... | 3 | https://mathoverflow.net/users/18060 | 383670 | 159,595 |
https://mathoverflow.net/questions/383677 | 3 | Let $M\subset B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$. There is a norm $\|.\|\_\tau$ on $M$ given by $\sqrt{\tau(xx^\*)}$. How to show the $\|.\|\_\tau$-topology coincides with the strong operator topology on $M\_1$, i.e. the operator-norm unit ball?
| https://mathoverflow.net/users/169800 | Coincidence of two topology on a bounded subset of a finite von Neumann algebra | Some errors identified in the comments by Mateusz and Matt — will try to fix these later, although I believe that the overall *strategy* outlined here can be made to work.
---
It's late here so I haven't checked through the details, but I think the following is an outline of one possible approach. I'm leaving it ... | 3 | https://mathoverflow.net/users/763 | 383680 | 159,598 |
https://mathoverflow.net/questions/383624 | 9 | I'm dealing with finite groups $G$ in which the maximal order of an element is $6$.
With GAP I found out that for all groups with order $<1000$ the number of elements of order 6 $k := |\{x\in G : ord(x)=6\}|$ is $0$ or $2$ $\mod 6$.
I'm trying to understand, why $k \not\equiv 4 \mod 6$.
I'm collecting some facts... | https://mathoverflow.net/users/170420 | Groups with maximal element order 6 | Let $P$ be a Sylow $3$-subgroup of $G$ (which we may assume to be nontrivial) and consider the conjugation action of $P$ on the set $X$ of all elements of order six in $G$. Remove from $X$ all of its $P$-fixed points. The resulting set $Y$ is a union of non-trivial $P$-orbits and therefore $|Y|$ divisible by three. As ... | 12 | https://mathoverflow.net/users/36466 | 383683 | 159,599 |
https://mathoverflow.net/questions/383687 | 7 | **PRELIMINARY DEFINITIONS:**
Let $E^\*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special element $t\in\tilde{E^2}(S^2)$ which corresponds to $1\_E\in E^0(pt)$ under the above isomorphism.
$E^\*$ is sai... | https://mathoverflow.net/users/169319 | Motivation for the definition of complex orientable cohomology theory | As you wrote, complex orientability can be characterized by the cohomology of $\mathbb{C}P^\infty$: $E$ is complex orientable if $E^\*(\mathbb{C}P^\infty)$ splits according to the cell structure of $\mathbb{C}P^\infty$, i.e. $E$ doesn't "see" the attaching maps of $\mathbb{C}P^\infty$.
For example, this happens wheneve... | 12 | https://mathoverflow.net/users/39747 | 383694 | 159,602 |
https://mathoverflow.net/questions/383263 | 6 | One can use Thue's 1909 result to show that the Diophantine equation $Ax^3 + By^3 = C$ ($A,B$ not perfect cubes, $C\neq 0$) has finitely many integer solutions $(x,y)$.
But does there exist a simple way to prove this equation (because I think this is a special equation?
| https://mathoverflow.net/users/38620 | Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions | I think both of the answers given avoid the modern treatment of Thue equations, which despite being newer is probably simpler in many ways. As my doctoral advisor is a pioneer in this area I feel obligated to explain these ideas.
Let us first emphasize that Thue's theorem, extending to the theorems of Siegel, Dyson, ... | 3 | https://mathoverflow.net/users/10898 | 383714 | 159,606 |
https://mathoverflow.net/questions/383636 | 2 | I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is:
Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=0$ and
$$
\int\_0^1 \left(\log \frac{1}{u}\right)^{\!1/2} \, dp(u) < + \infty.
$$
Show that
$$
\lim\_{h\to 0} \frac{\displayst... | https://mathoverflow.net/users/173966 | Asymptotic of an improper integral | I think you reproduced the "elementary" claim correctly (for the case $n=1$). However, as stated in my earlier comment, the claim is false in general, even if you assume that $p>0$ on $(0,1]$.
Indeed, the claim was that the limit
\begin{equation\*}
\lim\_{h\downarrow0}r(h) \tag{1}
\end{equation\*}
exists and is fini... | 3 | https://mathoverflow.net/users/36721 | 383717 | 159,607 |
https://mathoverflow.net/questions/383695 | 2 | Let $\theta \colon R \to S$ be a morphism of commutative rings (with unit). We assume that:
* $R$ and $S$ are [coherent](https://en.wikipedia.org/wiki/Coherent_ring).
* $S$ is [finitely presented](https://en.wikipedia.org/wiki/Finitely_generated_module#Finitely_presented_module) as an $R$-module.
Now, let $M$ be an... | https://mathoverflow.net/users/20883 | Preservation of finite presentation along restriction of scalars | There is an exact sequence of $S$-modules,
$$ 0 \to K \to S^n \to M \to 0 $$
where $K$ is a finite $S$-module. Restricting to the category of $R$-modules, $S^n$ is finitely presented and $K$ is finite (because $S$ is a finitely presented $R$-module). Therefore, by [Tag 0519](https://stacks.math.columbia.edu/tag/0519), ... | 4 | https://mathoverflow.net/users/154157 | 383723 | 159,610 |
https://mathoverflow.net/questions/383718 | 6 | For a well-based space $X$ denote by $C(\mathbb{R};X)$ the unordered configuration space of points on the real line with labels in $X$, and a point can vanish if its label reaches the basepoint. (Alternatively, you can think about the free $E\_1$-algebra over the based space $X$.) Note that $C(\mathbb{R};X)$ is filtere... | https://mathoverflow.net/users/124042 | Is there a filtered splitting of product labelling spaces? | The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras
$$ H\_\*(C(\mathbb R; X \vee Y \vee (X\wedge Y)); \mathbb F) \simeq H\_\*(C(\mathbb R; X \times Y); \mathbb F)$$
for any field coefficients $\mathbb F$. Also, one ... | 8 | https://mathoverflow.net/users/102519 | 383724 | 159,611 |
https://mathoverflow.net/questions/383721 | 5 | What happens if we define a functor $F:C \to D$ to be **injective** when it is injective on isomorphism classes, or equivalently when it gives an injection from the objects of the skeleton of $C$ to the skeleton of $D$?
Edit: To be more specific, how does this definition relate to a that of a fully faithful functor?
... | https://mathoverflow.net/users/160055 | Injectivity of functors in terms of skeletons? | Call $F : \mathbf C \to \mathbf D$ *[essentially injective](https://ncatlab.org/nlab/show/essentially+injective+functor)* (mirroring the definition of [essentially surjective functor](https://ncatlab.org/nlab/show/essentially+surjective+functor)) if $F(A) \cong F(B)$ implies that $A \cong B$. This matches your definiti... | 8 | https://mathoverflow.net/users/152679 | 383726 | 159,612 |
https://mathoverflow.net/questions/383666 | 2 | Let $X$ be a three-dimensional variety over $\mathbb{C}$ with a nodal singularity at a point, say $P$. Is the exceptional divisor of the blow-up of $X$ at $P$ isomorphic to a smooth quadric in $\mathbb{P}^3$? I read this statement in an article, but am not able to find a proof.
| https://mathoverflow.net/users/32151 | Blow-up of a three-dimensional variety at a node | There is no need to take analytic coordinates.
There is a general principle for blowing up the solution set of a hypersurface $f(x\_1,\dots,x\_n)=0$ at a point $x\_1,....,x\_n=0$.
Recall that blowing up involves introducing projective coordinates $(y\_1: \dots :y\_n)$ in addition to our original ones, satisfying th... | 4 | https://mathoverflow.net/users/18060 | 383734 | 159,614 |
https://mathoverflow.net/questions/383686 | 2 | Reading the proof of the Hales-Jewett theorem the author defines $W\_L$ as the set
of finite words over some alphabet $L$, $W\_{L\_v}$ as the set of variable-words
over $L$, i.e. finite words over $L \cup \{v\}$ where $v \notin L$ and $v$ occurs
at least once, $S = W\_L \cup W\_{L\_v}$ and the substitution map from $S$... | https://mathoverflow.net/users/153785 | The extension of the substitution map of the semigroup of variable words to its Stone–Čech compactification is a homomorphism | As I mentioned in comments, I will prove a more general fact: that for any for any semigroup homomorphism $f\colon S→T$, its extension $βf\colon βS→βT$ is a continuous homomorphism of topological semigroups. Here, I use the definitions
$$f(\mathcal{U}) = \{ A \subseteq
T: f^{-1}(A) \in \mathcal{U} \}$$
and, using ... | 3 | https://mathoverflow.net/users/136473 | 383735 | 159,615 |
https://mathoverflow.net/questions/383742 | 0 | A terminal time $\tau$ is a stopping time satisfying $$\omega \in \{\tau(\omega) > t\} \text{ implies that } \tau(\omega) = t + \tau(\theta\_t\omega), $$ for all $t\ge 0$. Here $\theta\_t$ is the shift operator. I am wondering why the first hitting time is a terminal time but the second hitting time is not? Thanks.
| https://mathoverflow.net/users/34483 | A question concerning terminal time | $\newcommand\om\omega\newcommand\th\theta$Let $\tau$ be the first hitting time of a set $B\subseteq S$, so that for each "path" $\om\colon[0,\infty)\to S$ we have
$$\tau(\om)=\inf\{u\colon u\ge0,\om(u)\in B\}.$$
For each real $t\ge0$, the $t$-shifted path $\th\_t\om$ is defined by the formula
$$(\th\_t\om)(s):=\om(t+s)... | 3 | https://mathoverflow.net/users/36721 | 383752 | 159,617 |
https://mathoverflow.net/questions/383746 | 7 | [Wikipedia says](https://en.wikipedia.org/wiki/Closed_monoidal_category):
>
> The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system.
>
>
>
["A Fibrational Framework for Substructural and Modal Logics" says](https://dlicata.wescreates.wesleya... | https://mathoverflow.net/users/112755 | Ordered logic is the internal language of which class of categories? | Yes, ordered logic is the internal language of non-symmetric monoidal categories. As with linear and nonlinear logic, if the ordered logic contains function-types then they correspond to internal-homs making the monoidal category closed, although one has to be a bit careful since in the non-symmetric case there are two... | 6 | https://mathoverflow.net/users/49 | 383753 | 159,618 |
https://mathoverflow.net/questions/383708 | 11 | A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which are perhaps a bit mysterious to an audience of general mathematicians.
One common way to get QFT into a precise language ... | https://mathoverflow.net/users/119114 | Alternative approaches to topological QFTs | It sounds like what you would like is a rigorous version of Witten's original Feynman integral & Wilson loop approach. This is not a totally unreasonable thing to ask for, since QFTs have been rigorously constructed along these lines, by demonstrating the existence of moments of the Euclidean-signature Feynman integral... | 10 | https://mathoverflow.net/users/35508 | 383760 | 159,620 |
https://mathoverflow.net/questions/382571 | 3 | What is an example of a gradient vector field $X$ on a Riemannian manifold $(M,g)$ which cannot be converted to a divergence free vector field via the following processes:
1. First we remove the singularities $S$ from $M$ then we set $M:=M\setminus S$
2. We are allowed to reparameterize $X$ to $X:=fX$ for some positi... | https://mathoverflow.net/users/36688 | Is every gradient vector field a divergence free vector field? | I think the answer is **no** as soon as your gradient vector field admits a saddle point where the divergence is non-zero.
Let $\omega$ denote the volume form associated to the Riemann metric. We have
$$\mathrm{div}(X) \omega = X\cdot \omega$$
where $X\cdot \omega$ denotes the Lie derivative. The goal is to find posi... | 2 | https://mathoverflow.net/users/173096 | 383765 | 159,622 |
https://mathoverflow.net/questions/383766 | 8 | Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-submanifold $\Sigma^p \subset N$ which is nontrivial in $H\_p(N; \mathbb{Z})$, the preimage $\pi^{-1}(\Sigma)$ is nontrivial ... | https://mathoverflow.net/users/85934 | Conditions under which the preimage of a submanifold in nontrivial in homology | A sufficient condition with $\mathbf{Q}$ coefficients should be that the fiber $F$ has non-vanising Euler characteristic.
Consider the Gysin map (aka as fiber integraion) $$\pi^{!} = \int\_{\pi}\colon H^{\ast}(M;\mathbf{Q}) \to H^{\ast-k}(N;\mathbf{Q});$$ then $\langle \alpha , \pi^{-1}[S]\rangle = \langle \pi^{!} \a... | 8 | https://mathoverflow.net/users/14233 | 383784 | 159,625 |
https://mathoverflow.net/questions/383803 | 4 | The avoidance principle for mean curvature flow says that, given two disjoint compact, embedded hypersurfaces, when we run the mean curvature flow, they will remain disjoint.
In fact it is known that we can relax the assumptions to “one is compact and embedded, the other one is just embedded”.
Can we relax the assu... | https://mathoverflow.net/users/174092 | Avoidance principle of mean curvature flow for non-compact hypersurfaces? | You can't make such a relaxation, at least in the world of weak set flows. See Example 7.3 (specifically comment iv) of Ilmanen's paper [Generalized Flow of Sets by Mean Curvature on a Manifold](https://www.jstor.org/stable/24897049?seq=1#metadata_info_tab_contents)
I'd be surprised if restricting to classical flows ... | 2 | https://mathoverflow.net/users/127803 | 383824 | 159,645 |
https://mathoverflow.net/questions/383815 | 0 | I am looking for a dense sub-algebra $B$ in $C\_{b}((0,1))$ in uniform topology such that it satisfy following requirements:
1. $B\cap C^{\infty}\_{b}((0,1))=\mathbb{R}$ (No polynomial, no bump function).
$(0,1)$ can be replaced with other connected open set.
The only thing I can come up with is nowhere differentia... | https://mathoverflow.net/users/172458 | Dense sub-algebra of $C_{b}((0,1))$ which is not smooth | Such a subalgebra can be defined. See also the end of this answer for a more general result.
Let $I=(0,1)$. In the following, an open interval means a nonempty open interval of $I$. Fix a function $f\in C\_b(I)$ with values in $[1,2]$ such that $f$ is not smooth over any open interval; for instance, $f:t\mapsto\alpha... | 2 | https://mathoverflow.net/users/129074 | 383828 | 159,648 |
https://mathoverflow.net/questions/383062 | 25 | As with many of you, I've been following Peter Scholze's recent [question about universes](https://mathoverflow.net/questions/382270/reflection-principle-vs-universes) with great interest. In ring theory, we don't often have to deal with proper classes, but they occasionally pop their heads. For instance, one cannot ta... | https://mathoverflow.net/users/3199 | Is there a metamathematical $V$? | (taken from a comment)
To me, the idea of ordinals being a completed infinity contradicts the idea of ordinals (I mean the informal idea of ordinals, that is, that after every "completed collection" of ordinals there should be another ordinal).
| 9 | https://mathoverflow.net/users/65995 | 383831 | 159,650 |
https://mathoverflow.net/questions/383835 | 2 | Are there a set of theorems dealing with "amount of divergence" series?
Let me explain by example. The Dirchlet $\eta$ series $\sum\_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say amount of divergences when $x>0$ is $0$. When $x=0$, the partial sum oscillates between $1$ and $0$, thus amount of divergence at... | https://mathoverflow.net/users/123507 | Are there theorems dealing with the "amount of oscillatory divergence" of series? | The study of divergent series up to the early twentieth century was masterfully summarized in Hardy's book [1], see also [2]. Significant extensions were developed by Boshernitzan in the 1980's, see [3], [4], [5], with some overlapping work by Rosenlicht [6].
[1] Hardy, Godfrey Harold. Divergent series. Vol. 334. Ame... | 3 | https://mathoverflow.net/users/7691 | 383839 | 159,653 |
https://mathoverflow.net/questions/383837 | 2 | Let $ F $, $ F\_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F\_i \subseteq E $ and $ E = F\_1 \cdot F\_2 $ is the composite field of $ F\_1 $ and $ F\_2 $. Let $ Q $ be a place in $ E $, $ P\_i = Q \cap F\_i $ and $ P = Q \cap F $. Assume that ... | https://mathoverflow.net/users/132492 | Wild ramification in composite fields | Your intuition is good - it is not true.
We can take $F = \mathbb F\_q(t)$ and define $F\_1$ and $F\_2$ to be adjoining distinct roots of a cubic $x^3 - t x =1$. Then if $E = F\_1 ( \sqrt{t})$ since $x, x+ \sqrt{t}, x- \sqrt{t}$ are the three roots.
The prime $t=0$ has ramification degree $3$ in $F\_1$ and $F\_2$, ... | 4 | https://mathoverflow.net/users/18060 | 383842 | 159,655 |
https://mathoverflow.net/questions/7537 | 21 | In a paper I need to make reference to two conjectures by Gabber, from
* Ofer Gabber, *On purity for the Brauer group*, in: *Arithmetic Algebraic Geometry*, MFO Report No. 37/2004, doi:[10.14760/OWR-2004-37](https://doi.org/10.14760/OWR-2004-37)
(see Conjectures 2 and 3, page 1975)
1. Let $R$ be a strictly hensel... | https://mathoverflow.net/users/2083 | Two conjectures by Gabber on Brauer and Picard groups | Happy news: there have been some progress over the last 10 years. The hypersurface case of Conjecture 2 was proved in
* H. Dao, *Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free*. Compositio Mathematica, **148**(1) (2012) pp. 145-152. doi:[10.1112/S0010437X11005513](https://d... | 12 | https://mathoverflow.net/users/2083 | 383853 | 159,661 |
https://mathoverflow.net/questions/383850 | 7 | In Choquet-Bruhat's solution to the Cauchy problem for Einstein's equation, one reduces the Einstein equations to a quasidiagonal quasilinear hyperbolic system on $ M := [0, T] \times \bar M$ where $T > 0$ and $\bar M$ is some initial spacelike 3-manifold for unknowns $g\_{\alpha\beta}$ on $M$. Hyperbolic PDE theory th... | https://mathoverflow.net/users/147016 | Preservation of metric signature in Cauchy problem for the Einstein equations | 1. The dynamic metric **is** the metric for the quasidiagonal system; and so for self consistency the solution can only be proven to exist (and be unique) when the metric (i.e. the solution itself) is hyperbolic/Lorentzian.
2. Noncompactness has zero impact whatsoever. Hyperbolic equations have finite speed of propagat... | 7 | https://mathoverflow.net/users/3948 | 383854 | 159,662 |
https://mathoverflow.net/questions/383857 | 4 | For a Noetherian local ring $R$, Koszul complex is a useful tool to construct finite free resolution of any quotient of $R$ by regular sequences $R/(x\_1,\dots, x\_r)$.
Let $R=\mathbb C[x,y,z,w]/(xy-zw)$, and $M=R/(x-z,y-w) \cong \mathbb C[x,y]$. $x-z, y-w$ don't form a regular sequence of $R$, which can be checked d... | https://mathoverflow.net/users/172430 | (Infinite) free resolution of $R/(x-z, y-w)$ for $R=\mathbb C[x,y,z,w]/(xy-zw)$ | By a well-known [paper](https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf) by Eisenbud, over hypersurfaces the resolution of any module becomes periodic of period at most $2$ once after $depth(R)-depth(M)$ steps.
In your case the first map is just embedding of $I=(x... | 5 | https://mathoverflow.net/users/2083 | 383859 | 159,663 |
https://mathoverflow.net/questions/383810 | 0 | For a graph $G=(V,E)$ ($V$ set of vertices and $E$ set of edges ), $\mathcal{I}$ is defined as all of the subsets $E´\subseteq E$ where the components of $(V,E´)$ that are connected are simple paths. I want to show that $(E,\mathcal{I})$ does not satisfy the exchange property of a matroid.
I have tried to find an exa... | https://mathoverflow.net/users/174089 | Prove that a definition of $\mathcal{I}$ does not satisfy the exchange property | Take two paths $1234,5678$. Assume that there are no edges between their endpoints (so that you can not enlarge this independent set of 6 edges) but there exists a Hamiltonian path, say 13657428. You get two inclusion-maximal subsets of different size, thus it is not a matroid.
| 0 | https://mathoverflow.net/users/4312 | 383865 | 159,664 |
https://mathoverflow.net/questions/383866 | 2 | In my research work, I need to show that the sequence $(nu\_n)$ tends to 0 where $ (u\_n)$ is defined by $$u\_{n+1}=u\_{n} \cos^{2}(n),\quad u\_{0}=1$$
$(u\_n)$ is a positive and decreasing sequence. My adempt \begin{align\*}
u\_{n+1}&= \prod\_{k=1}^n\cos^2(k) =(\prod\_{k=1}^n \frac{e^{k i } + e^{- k i }}{2} )^2 \\
&=... | https://mathoverflow.net/users/126827 | About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$ | It follows from the irrationality of $\pi$ and [Weyl's criterion](https://en.wikipedia.org/wiki/Equidistributed_sequence#Weyl%27s_criterion) that the positive integers are equidistributed modulo $\pi$. In particular, asymptotically one-third of the integers $k\in\{1,\dotsc,n-1\}$ satisfy that $k\bmod\pi$ lies in $[\pi/... | 10 | https://mathoverflow.net/users/11919 | 383867 | 159,665 |
https://mathoverflow.net/questions/383841 | 3 | I have recently read
* Watts, D., Strogatz, S., *Collective dynamics of ‘small-world’ networks*, Nature **393** (1998) pp. 440–442, doi:[10.1038/30918](https://doi.org/10.1038/30918),
on small-world networks, and is still not very clear to me how is the "small-world network regime" defined. Meaning that for a fixed... | https://mathoverflow.net/users/168628 | Small world network regime | An increase of the "rewiring probability" $p$ (the probability that an edge is disconnected from one of its nodes and then randomly connected to another node) reduces both the mean path length $L$ and the [clustering coeffcient](https://en.wikipedia.org/wiki/Clustering_coefficient) $C$. There is a range of $p$ where th... | 2 | https://mathoverflow.net/users/11260 | 383877 | 159,668 |
https://mathoverflow.net/questions/383873 | 6 | Let $\mathcal C$ be a finitely complete, finitely cocomplete category. Then the following are equivalent:
1. $\mathcal C$ is regularly well-powered (i.e. every $C \in \mathcal C$ has a small set of [regular subobjects](https://ncatlab.org/nlab/show/regular+monomorphism));
2. $\mathcal C$ is regularly co-well-powered ... | https://mathoverflow.net/users/2362 | Regularly well-powered iff regularly co-well-powered? | Your first question is handled in Freyd’s paper, which shows that the notion of “a binary relation controlled by a set” is richer than you would think. Freyd says a relation $\square$ from $L$ to $R$ between large sets is resolvable if there is a small subset $L’\subseteq L$ such that every $l\in L$ has the same $\squa... | 4 | https://mathoverflow.net/users/43000 | 383882 | 159,671 |
https://mathoverflow.net/questions/383779 | 17 | In most introductory courses to category theory, the precise definition of a set is more-or-less ignored. The idea being that all basic results in the subject hold for any reasonable definition of a set.
At some point one feels that this must cease to be the case. So what is the simplest (a simple) example of a categ... | https://mathoverflow.net/users/160055 | When the definition of a set starts to matter in category theory | I didn't have time to write up a proper answer on initially seeing your question and much of what I have to say has been said in the comments and Tim's answer, but I'll still offer some specifics on things mentioned in the comments.
The definition of a set is, as Tim Campion pointed out, the axioms of whatever set th... | 17 | https://mathoverflow.net/users/92164 | 383883 | 159,672 |
https://mathoverflow.net/questions/383855 | 18 | I have found a conjecture in a research article (published in a good journal) on number theory, which is not well known but very reasonable. Let me be clear that, there is no counter-example that vote down the conjecture, rather, its trueness has been proved in some special situation so far. I need this conjecture to d... | https://mathoverflow.net/users/122445 | Is it a reasonable way to write a research article assuming truth of a conjecture? | A research article which assumes the truth of a conjecture also counts (indirectly) as research on the conjecture itself. If you show that certain results follow from a conjecture and those results are later shown to be false, then the original conjecture would have been shown to be false. Another possibility is that t... | 17 | https://mathoverflow.net/users/89084 | 383885 | 159,674 |
https://mathoverflow.net/questions/332823 | 3 | If I understand correctly, in the following thread
[Are There Primes of Every Hamming Weight?](https://mathoverflow.net/questions/22629/are-there-primes-of-every-hamming-weight)
two users of the site claim that it has been already proven that, for every sufficiently large $n \in \mathbb{N}$, there exist primes numb... | https://mathoverflow.net/users/99957 | Primes with given Hamming weight | As Wojowu and alpoge have said in the comments, this follows immediately from Theorem 1.1 of Drmota-Mauduit-Rivat paper, but here is more detail on exactly why, as requested.
Theorem 1.1 of the Drmota-Mauduit-Rivat paper with $q=2$ and $k=n$ and $\epsilon=1/4$ gives that the number of primes $p\leq x$ with Hamming we... | 3 | https://mathoverflow.net/users/385 | 383895 | 159,676 |
https://mathoverflow.net/questions/383911 | 4 | Let $Topos$ be the $(2,1)$-category of Grothendieck toposes and geometric morphisms. This is a $V$-sized, locally $V$-sized, locally locally small $(2,1)$-category with all small (2,1)-colimits (=pseudocolimits) and terminal object given by the topos $Set$.
**Question 1:** Is $Set$ a [tiny](https://ncatlab.org/nlab/s... | https://mathoverflow.net/users/2362 | Is $Set$ a tiny topos? | The answer to your question is unfortunately **[no](https://mathoverflow.net/questions/322396/is-set-a-finitely-presentable-object-in-topoi)**. The terminal topos is not even finitely presentable.
| 6 | https://mathoverflow.net/users/104432 | 383912 | 159,680 |
https://mathoverflow.net/questions/383892 | 0 | Is there an analogue of Quillen-Suslin theorem for power series? Let $A$ be a regular noetherian ring over a field. Consider the power series ring $A[[T]]$. Are projective modules on $A[[T]]$ extended from $A$?
| https://mathoverflow.net/users/127776 | Is there a version of Quillen-Suslin-Lindel for power series? | This is true for any $A$ (assuming projective means "projective of finite type", otherwise I don't know). Let $P$ be projective of finite type over $A[[T]]$. Put $Q:=(P/TP)\otimes\_{A} A[[T]]$. Then $P\cong Q$.
More generally, let $R$ be a ring and $I$ an ideal contained in the Jacobson radical of $R$. If $P$ and $Q$... | 5 | https://mathoverflow.net/users/7666 | 383943 | 159,692 |
https://mathoverflow.net/questions/383975 | 2 | Values appearing with density in an ergodic system
Let $(X,\mu)$ be a probability space with invertible, measure preserving, totally-ergodic map $T:X \to X$. ($(X,\mu,T)$ is a $\mathbb{Z}$ dynamical system). Suppose $f:X \to \mathbb{R}$ is a measurable function such that $\mu(x \colon f(x) = a) = 0$ for all $a \in \m... | https://mathoverflow.net/users/47510 | Values appearing with density in an ergodic system | Indeed $D(x)$ is empty for almost all $x$. It suffices to show that for all integers $b,\ell>0$, the set
$$
D\_b(x,\ell) = \left\{ a \in [b,b+1) \colon \liminf\_{n \to \infty} \frac{ \# \{0 \leq k < n \colon f(T^{k}x) = a \}}{n} > \frac{1}{\ell} \right\}
$$
is empty for almost all $x$, and then take a countable union o... | 1 | https://mathoverflow.net/users/7691 | 383986 | 159,706 |
https://mathoverflow.net/questions/383984 | 4 | I am wondering about an analog of the homotopy extension property in the setting of a pair $(\mathcal{C}, \mathcal{S})$ where $\iota : \mathcal{S} \to \mathcal{C}$ is a subcategory.
Explicitly, I say the pair $(\mathcal{C}, \mathcal{S})$ has the homotopy extension property if for any functors $F : \mathcal{C} \to \ma... | https://mathoverflow.net/users/154157 | Homotopy extension property of subcategory | In the natural isomorphism case, such an inclusion functor is also called a cofibration of categories. In fact they are the cofibrations in the [canonical model structure](https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat) on $Cat$, and they are characterized as the functors that are injective on objects (... | 9 | https://mathoverflow.net/users/49 | 383987 | 159,707 |
https://mathoverflow.net/questions/383964 | 2 | Let $x\_i, y\_i \in \mathbb{R}^n$ for $i=1, \dots, k < n$ satisfy
$$
\sum\_{i=1}^k x\_i^\top y\_i = 0.
$$
Let $E$ be a random subspace of dimension $m < n$ in $\mathbb{R}^n$ distributed uniformly on the Grassmannian $G\_m(\mathbb{R}^n)$ and let $P\_E$ be orthogonal projection onto $E$. What is the value of
$$
\mathbb{E... | https://mathoverflow.net/users/102255 | Variance of projection of vectors onto random subspace | It suffices to calculate $\mathbb E ( (P\_E)\_{ab} (P\_E)\_{cd})$ for all $1\leq a,b,c,d\leq n$, or, equivalently, calculate $\mathbb E( P\_E \otimes P\_E ) \in \mathbb R^n \otimes \mathbb R^n \otimes \mathbb R^n \otimes \mathbb R^n$.
This is an $O(n)$-invariant tensor. The space of $O(n)$-invariant tensors in $\math... | 3 | https://mathoverflow.net/users/18060 | 383990 | 159,708 |
https://mathoverflow.net/questions/383991 | 1 | Let us consider Minkowski inner product on $\mathbb R^{1+n}$, defined by
$$ \langle v,w \rangle = -v\_0w\_0+\sum\_{j=1}^n v\_j w\_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$
We say that a vector $v$ is null if $\langle v,v\rangle =0$. Moreover, given a symmetric matrix $A$, we say that $A$ is null positive definite i... | https://mathoverflow.net/users/50438 | A linear algebra question for semi-Euclidean norm | Consider the case $n = 1$.
Let $A = \begin{pmatrix} -1 \\ & -2t\end{pmatrix}$
If $v = (1,1)$, then $Av = (-1,-2t) \neq 0$.
If $w = (1,-1)$, then $Aw = (-1, 2t) \neq 0$.
But $\langle Av,v\rangle = \langle Aw,w\rangle = 1-2t$, and so $A$ is null positive definition for $t \in (0,\frac12)$, null semi-positive defini... | 3 | https://mathoverflow.net/users/3948 | 383994 | 159,710 |
https://mathoverflow.net/questions/384008 | 1 | Assume a filtered probability space $(\Omega,\{\mathcal F\_t\}\_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W\_t\}\_{t\in[0;T)}$ and the filtration $\{\mathcal F\_t\}\_{t\in[0;T)}$ being the filtration generated by the Brownian motion, augmented by the nullsets.
Assume an $\mathbb R^n$-va... | https://mathoverflow.net/users/130906 | Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval? | A counterexample should be just the deterministic $$Z\_t = \frac{1}{\sqrt{T-t}}$$ with $$\tau := \inf{\biggl\{s>0 : \int\_0^s Z\_u \, dW\_u =1\biggr\}}$$. You have $\tau < T$ a.s.and thus $$\mathbb{E}\biggl[ \int\_0^\tau Z\_u \, dW\_u\biggr] =1$$.
| 3 | https://mathoverflow.net/users/20026 | 384009 | 159,715 |
https://mathoverflow.net/questions/384010 | 2 | This is inspired by an older (as yet unanswered) [question](https://mathoverflow.net/questions/383891/graph-of-number-pairs-summing-to-a-square-number).
Let us call a set $S\subseteq\omega$ *thin in the 1st sense* if $$\lim\sup\_{n\to\infty}\frac{|S\cap n|}{n+1}=0$$ where $\omega$ is the first infinite cardinal, and ... | https://mathoverflow.net/users/8628 | A notion of thinness for subsets of $\omega$, using chromatic number | The two notions are incomparable.
To see that the first notion does not imply the second, let's construct a set $S$ with asymptotic density $0$, but with infinite chromatic number. We place infinitely many increasingly large intervals into $S$, but spaced very far apart, so that the density is zero. If $S$ has an int... | 9 | https://mathoverflow.net/users/1946 | 384012 | 159,716 |
https://mathoverflow.net/questions/384002 | 1 | $\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\U(2^{N-1})\supset \Spin(2N)
$$
when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^{N-1})$.
Question
--------
* My question is what is the [normalizer](https://en.wikipedia.org/wiki/Centralizer_and_normal... | https://mathoverflow.net/users/27004 | The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$? | You can work out the answers to these questions using the material in Chapter 11 of the book *Spinors and Calibrations* by F. Reese Harvey. You will also need to recall that, for $N\not=4$, the group of automorphisms of $\mathrm{Spin}(2N)$ is $\mathrm{O}(2N)$ rather than $\mathrm{SO}(2N)$. The answers depend to some ex... | 10 | https://mathoverflow.net/users/13972 | 384025 | 159,718 |
https://mathoverflow.net/questions/384018 | 8 | Let $f(z)=\sum\_{n\ge 1}a\_nq^n$ be a cusp form, where $q=e^{2\pi i z}$. Let $
L(s) = \sum\_{n\ge 1} a\_nn^{-s}
$ be its corresponding L-function. The completed L-function of $L(s)$, $\Lambda(s)$, should satisfy a functional equation $\Lambda(s)=\bar \Lambda(a-s)$ for some integer $a$.
The analytic rank of $f(s)$ is ... | https://mathoverflow.net/users/125498 | Do odd-weight cusp forms have analytic rank 0? | It is conjectured that $L$-functions of motives do not have zeros or poles on the real line, except possibly at integers. In particular, if $f$ is a newform of odd weight $k$, then $L(f,s)$ should not vanish at $s=k/2$.
This conjecture appears in Fontaine, Perrin-Riou, *Autour des conjectures de Bloch et Kato: cohomo... | 9 | https://mathoverflow.net/users/6506 | 384026 | 159,719 |
https://mathoverflow.net/questions/384024 | 6 | Let $A$, $B\in\mathbb{R}^{n\times n}$ be full-rank random matrices and define the Kronecker products $P=A\otimes B$ and $Q=B\otimes A$. Through example-based examination, I have found that
$\text{rank}(P-Q)=n^2-n$,
but I am struggling to formulate a proof of this. Thus far, I have focused on the fact that $P$ and $... | https://mathoverflow.net/users/174229 | Rank of $A\otimes B - B\otimes A$ | For generic $A,B$ the matrix $A$ is invertible and $B=CA$, where $C$ is also generic. We have $(A\otimes B-B\otimes A)(u\otimes v)=Au\otimes CAv-CAu\otimes Av$. The vectors $Au$ run over the whole $\mathbb{R}^n$, so the image of the operator $A\otimes B-B\otimes A$ is the same as the image of $I\otimes C-C\otimes I$ (w... | 14 | https://mathoverflow.net/users/4312 | 384028 | 159,720 |
https://mathoverflow.net/questions/383996 | 3 | It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace Transform of a positive function of form $s^{-\alpha}$?
The answer is trivial when $\alpha>0$, because $\mathcal{L}\{t^{\al... | https://mathoverflow.net/users/172600 | Is inverse Laplace Transform of a power of $s$ a positive function? | The inverse Laplace transform of $s$ is the derivative $\delta'(x)$ of the Dirac delta function (a distribution). This is not a positive "function", as you can check by evaluating the integral with a test function $f(x)$,
$$\int\_{-\infty}^\infty f(x)\delta'(x)\,dx=-\int\_{-\infty}^\infty f'(x)\delta(x)\,dx=-\lim\_{x\r... | 3 | https://mathoverflow.net/users/11260 | 384029 | 159,721 |
https://mathoverflow.net/questions/384020 | 13 | The question is in the title. If $\Gamma\subset\mathrm{GL}(\Bbb R^d)$ is a *finite* matrix group, can it be generated by (at most) $d$ elements?
I suspect that this hope is too naive, but I have no counterexamples. I would also be interested in bounds on the number of generators.
| https://mathoverflow.net/users/108884 | Is every finite $d$-dimensional matrix group generated by $d$ elements? | The answer to the question is yes.
In Theorem 1.2 of [this paper](https://www.sciencedirect.com/science/article/pii/S0021869313002160). the authors (Colva Roney-Dougal and myself) prove that if $G \le {\rm GL}(n,F)$ with $F$ a field, where
(a) $G$ is finite;
(b) either $G$ is completely reducible or ${\rm char} F... | 19 | https://mathoverflow.net/users/35840 | 384042 | 159,725 |
https://mathoverflow.net/questions/384003 | 22 | I'm trying to show that $\sum\_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $0 \leq n \leq p-2$ spans the vector space $\mathbb{F}\_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}\_p$.
In other words, I would like to show that the determinant of the following matrix is a unit in $\mathbb{F}\_p$. I've shown this ... | https://mathoverflow.net/users/56462 | How to see that the determinant of this matrix is nonzero for all primes? | As discussed in the comments, I don't see how to extract the desired matrix from the original question (about spanning the vector space). However, the matrix being nonzero IS equivalent to the following:
The polynomials $\sum\_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ for $0\leq n\leq p-2$, along with the polynomial $t^{p-1}... | 11 | https://mathoverflow.net/users/51424 | 384046 | 159,726 |
https://mathoverflow.net/questions/384044 | 4 | I wonder if there is a simple instance of the following phenomena : an abstract subgroup S $\subset GL\_n(\mathbb{C})$ whose $\mathbb{Q}$-Zariski closure isn't a group ?
Is there some criteria to ensure this is not happening ? (Maybe under some reductivity hypothesis, e.g. Mumford-Tate groups ?)
(there is a non int... | https://mathoverflow.net/users/174239 | $\mathbb{Q}$-Zariski Closure not equal to smallest Q-subgroup | Sure. Take some elements $A\_1,\dots, A\_n$ that generate a dense subset $\Gamma$ of a non-abelian connected reductive group $G \subseteq GL\_n$.
For instance, I can take $$A\_1=\begin{pmatrix} 1 &1 \\ 0 & 1 \end{pmatrix}, A\_2 =
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ which generate $\Gamma = SL\_2(\mathbb... | 5 | https://mathoverflow.net/users/18060 | 384048 | 159,727 |
https://mathoverflow.net/questions/384045 | -1 | I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$.
I also know that if $h(x)$ is positive, then $g(x)$ is also positive.
I define the induction
$$h(x)=\int\_0^x f(t)g(t)h(t) dt$$
I imagine that is possible to say that $h(x)$ ... | https://mathoverflow.net/users/172600 | Seek help to formalize an argument to positiveness of function defined inductively by integral | Taking the derivative of your "induction" w.r.t. $x$, one gets
$$
h'(x) = f(x)g(x)h(x)
$$
and thus $h'(x)>0$ if $h(x)>0$, according to your conditions. Since $h(0)>0$, one concludes that $h(x)>0$ for $x\in[0;\infty)$.
| 1 | https://mathoverflow.net/users/45250 | 384052 | 159,729 |
https://mathoverflow.net/questions/384058 | 3 | Taking the doctrine of computational trinitarianism ( <https://ncatlab.org/nlab/show/computational+trinitarianism> ), if one understands the incompleteness theorems as the "logic" version, and the answer to the halting problem as the "language theory" version, is there a known expression of this "pair of abstract compu... | https://mathoverflow.net/users/146463 | Is there an equivalent of the incompleteness theorems/halting problem in category theory? | There is a category theoretic version of the incompleteness theorem originally due to André Joyal that has been unavailable for a long time, but has been written up not so long ago by [Joost van Dijk and Alexander Gietelink Oldenziel](https://arxiv.org/pdf/2004.10482.pdf).
It is a much more technical statement than w... | 10 | https://mathoverflow.net/users/22131 | 384062 | 159,733 |
https://mathoverflow.net/questions/383786 | 7 | For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined modulo $1/2$, i.e. as an element of $\mathbb R / \frac{1}{2}\mathbb Z$. (This is mentioned [here](https://arxiv.org/abs/... | https://mathoverflow.net/users/113402 | Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$? | Remark 7.2 in Christian's paper suggests that for the choice of a lift of the discrete-faithful $PSL\_2(\mathbb{C})$ representation to $SL\_2(\mathbb{C})$, there ought to be a lift of the Chern-Simons invariant to $\mathbb{R}/\mathbb{Z}$. As he points out, this is equivalent to the choice of a spin structure (which is ... | 4 | https://mathoverflow.net/users/1345 | 384063 | 159,734 |
https://mathoverflow.net/questions/383773 | 0 | I need to know if the following system is consistent, because I want to use it in presenting automorphisms over stratified versions of it.
The system I'd label as "*Acyclic ZF*", which is $\small \sf ZF-Reg.+ acyclic \ AFA + Rank$, is obtained by adding a two place predicate symbol $\mathcal R$ symbolizing *is the ra... | https://mathoverflow.net/users/95347 | Is acyclic ZF consistent? | Acyclic ZF is inconsistent. Let D be the directed graph whose vertices are the natural numbers where (a,b) is an edge if a=0 and a<b, or b>0 and a>b. By Acyclic construction, there is set {x0.x1,x2,...} such
that xi∈j iff (i,j) is in D. Each xi is a transitive set and thus each xi is an ordinal.
(1)Suppose R(xi,yi) f... | 3 | https://mathoverflow.net/users/133981 | 384070 | 159,736 |
https://mathoverflow.net/questions/384066 | 3 | What are the current best known upper bounds on
$$\sum\_{n<x}a\_n$$
where $a\_n$ are defined implicitly by $L(E,s)=\sum\_{n=1}^{\infty}\frac{a\_n}{n^s}$ where $L(E,s)$ is an Elliptic L Function and $E/\mathbb{Q}$ is an elliptic curve over $\mathbb{Q}$? The notion of "best" might be a bit fuzzy here since on one han... | https://mathoverflow.net/users/159298 | What upper bounds on $\sum_{n<x}a_n$ are known where $L(E,s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is an Elliptic L Function? | The Weil bound $a\_n \leq d\_2(n)\sqrt{n}$ where $n$ is the divisor function gives $ \sum\_{n < x} a\_n < \sum\_{n\_1n\_2<x} \sqrt{n\_1n\_2} = O ( x^{3/2} \log x )$. This is the trivial bound in this setting.
To do better than that, the main approach will be to use the modularity theorem, which recognizes $a\_n$ as F... | 6 | https://mathoverflow.net/users/18060 | 384071 | 159,737 |
https://mathoverflow.net/questions/384068 | 3 | Let $X$ be a path-connected, semilocally simply connected space.
Let $\tilde X$ be its universal convering and $\Gamma$ the group of deck transformations. Let $(\Gamma\_\alpha)\_{\alpha\in A}$ be a family of subgroups, closed under finite intersections, such that $\bigcap\_{\alpha\in A}\Gamma\_\alpha=\{1\}$. Let $X\_\a... | https://mathoverflow.net/users/nan | Projective limit of coverings | I'll start by saying that these types of maps $\Omega\to X$ are pretty well-understood even beyond spaces $X$ that admit simply connected covering spaces. Every inverse limit of covering maps (over any space) is a Hurewicz fibration with the unique path-lifting property. This class of fibrations is studied in Spanier's... | 4 | https://mathoverflow.net/users/5801 | 384085 | 159,742 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.