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https://mathoverflow.net/questions/418549 | 3 | Consider the $D\_{\mathbb{A}^1}$-module $M:=D\_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^\*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which is equal to
$$\mathbb{C}[z]\otimes\_{\mathbb{C}[z^k]}(D/(x)),$$
with connection
$$\partial(z^q\otimes \partial^w)=qz... | https://mathoverflow.net/users/64302 | Explicit computation of D-modules pullback | If you write the degree of $z^q \otimes \delta^w$ as $ kw -q$ then there is a one-dimensional vector space of elements of each degree $\geq (1-k)$ (since if $q \geq k$ we can reduce by bring $z^q$ over to the other side) and $\partial$ takes elements of a given degree to elements of the same degree plus one.
So to ch... | 3 | https://mathoverflow.net/users/18060 | 418559 | 170,420 |
https://mathoverflow.net/questions/418556 | 2 | My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the full subcategory of the coherent objects. Then $\mathcal{X}^{coh}$ is a local $\infty$-pretopos.
In the proof, to show... | https://mathoverflow.net/users/156537 | Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos | If $X\_0\to X$ is an effective epimorphism and $X\_0$ is locally $n$-coherent, then $X$ is also locally $n$-coherent: every $Y$ over $X$ is covered by $Y\times\_XX\_0$, which is in turn covered by a coproduct of $n$-coherent objects. So in the proof of A.6.1.6 we know beforehand that $X$ is locally $n$-coherent for all... | 2 | https://mathoverflow.net/users/20233 | 418574 | 170,422 |
https://mathoverflow.net/questions/418534 | 9 | In MacLane's *Categories for the working mathematician*, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}\_{\mathrm{BMC}}(B,M)\simeq M\_0$ where $B$ is the braid category, $M$ is a braided monoidal category and $M\_0$ is the underlying (ordinary) category to $M$. As a result of... | https://mathoverflow.net/users/479089 | Coherence theorem in braided monoidal categories | Theorem 1 says precisely that for every braided monoidal category $M$ and every object $V \in M$, any isotopy class of braid on $n$ strands induces an isomorphism
$$
V^{\otimes n}\longrightarrow V^{\otimes n}.
$$
In particular this theorem also says this isomorphism is well defined, i.e. it does not depend one the part... | 2 | https://mathoverflow.net/users/13552 | 418583 | 170,424 |
https://mathoverflow.net/questions/418596 | -1 |
>
> Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm?
>
>
>
with $g^2=g \circ g $
If we can find $g$ then $F$ a closed of $A$, $id \in F$ with $g\circ F\subset F$, we have $F=A$?
| https://mathoverflow.net/users/110301 | The grail of functional analysis? | The answer is no. If there was, then for some $N$ we would have $\|g^N\|\leq 1/2$, in order for $g^N$ to be within distance $1/2$ of the constant $0$ function. This means that the range of $g^N$ is contained in $[0,1/2]$. But then the same holds for $g^n$ for all $n\geq N$, since $g^n=g^N\circ g^{n-N}$. In particular $... | 13 | https://mathoverflow.net/users/30186 | 418599 | 170,429 |
https://mathoverflow.net/questions/418509 | 2 | I am trying to understand this paper by [Chapelle and Li "An Empirical Evaluation of Thompson Sampling" (2011)](https://papers.nips.cc/paper/2011/file/e53a0a2978c28872a4505bdb51db06dc-Paper.pdf). In particular, I am failing to derive the equations in algorithm 3 (page 6). The first equation looks like an NLL of $p(x|w)... | https://mathoverflow.net/users/177845 | Derive equation for regularized logistic regression with batch updates | I found the solution (with the help of a friend: cudos!). The posterior is
$$\begin{align\*}
-\log p(\boldsymbol{w}|\boldsymbol{x}) &= -\log p(\boldsymbol{x}|\boldsymbol{w}) - \log p(\boldsymbol{w}) + \text{const.} \\
&= \sum\limits\_j \log \left( 1 + \exp(-y\_j \boldsymbol{w}^\top \boldsymbol{x}\_j) \right) + \sum\lim... | 3 | https://mathoverflow.net/users/177845 | 418600 | 170,430 |
https://mathoverflow.net/questions/418361 | 5 | Let $X$ be a connected compact complex manifold, $U$ an open subset of $X$ such that the complement of $U$ in $X$ is an analytic subset of codimension at least 2 in $X$. Let $O\_X$ (resp. $O\_U$) be the sheaf of holomorphic functions on $X$ (resp. on $U$). If $n$ is a nonnegative integer then there is a natural homomor... | https://mathoverflow.net/users/9658 | A cohomological variant of the second Riemann's extension theorem | For the first cohomology statement, you need the codimension to be at least three and for the first and second cohomology, codimension four. This theorem was proved by G Scheja in [1]. You can also find a proof in the book by Banica and Stanasila ([2] Chapter II, §II.3 pages 66-67).
**References**
[1] Constantin Ba... | 3 | https://mathoverflow.net/users/4696 | 418609 | 170,435 |
https://mathoverflow.net/questions/418607 | 4 | I am looking for a smooth curve $C$ of genus $g=2k+1 \geq 5$ over the complex numbers, endowed with a *free* $\mathbb{Z}/2$-action such that the following condition is satisfied: denoting by $$H^0(C, \, \omega\_C) = V^+ \oplus V^{-},$$ the decomposition of $H^0(C, \, \omega\_C)$ into invariant and anti-invariant subspa... | https://mathoverflow.net/users/7460 | Looking for a curve with a special, free $\mathbb{Z}/2$-action | Your examples are the *only* curves with fixed-point-free involution that don't have such a basis.
Let $C\_0$ be a curve of genus $k+1$ and let $L$ be a nontrivial line bundle on $C\_0$ with $L^2\cong \mathcal O\_{C\_0}$. Then $\mathcal O\_{C\_0}+L$ has a natural algebra structure defined using that isomorphism. The ... | 7 | https://mathoverflow.net/users/18060 | 418612 | 170,436 |
https://mathoverflow.net/questions/418603 | 2 | $\newcommand{\pt}{\mathit{pt}}$For $d>1$ is it possible to understand $\text{Aut}(\text{Sym}^n(\mathbb{P}^d\_{\mathbb{C}}))$? Automorphisms mean biregular morphisms from the variety to itself. Not let $\pt$ be a point on the projective space. Consider the point on $\text{Sym}^n(\mathbb{P}^d\_{\mathbb{C}})$ correspondin... | https://mathoverflow.net/users/127776 | Automorphisms of symmetric powers of projective space | $\DeclareMathOperator\PGL{PGL}$Yes, it's possible to understand this automorphism group, and yes, such a point is necessarily mapped to another point of the same form.
I claim the automorphism group is given by $\PGL\_{d+1}$, acting in the way induced by its action on $\mathbb P^d$. The statement about orbits follows... | 7 | https://mathoverflow.net/users/18060 | 418613 | 170,437 |
https://mathoverflow.net/questions/418592 | 9 | I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are some other somewhat surprising places(somewhat because I am aware of the subjectivity of the term) where games were/are use... | https://mathoverflow.net/users/467143 | Surprising applications of the theory of games? | I think evolutionary biology is a major application, if you're accepting answers outside of math. The central notion of an [evolutionarily stable strategy (ESS)](https://www.siue.edu/%7Eevailat/ev-gt-simple.htm) is a Nash equilibrium.
| 9 | https://mathoverflow.net/users/23141 | 418616 | 170,438 |
https://mathoverflow.net/questions/417015 | 2 | Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.
Denote $Y := \int\_0^1 f(t) \, dW\_t$.
For each $\varepsilon > 0$, consider the conditioned random variable $Y\_\varepsilon := \varepsilon Y | \{W\_1 \geq \frac{1}{\epsilon}\}.$
**Question:** Is it true that $Y\_\... | https://mathoverflow.net/users/173490 | A large noise limit | Let $\varphi$ be the standard normal density. Since
$P[W\_1 \ge x] =(1+o(1))\varphi(x)/x$ as $ x \to \infty$ by [1], we obtain for fixed $\delta>0$ that as $\epsilon \to 0$,
$$P[W\_1 \ge \epsilon^{-1}+\delta \,| \,W\_1 \ge \epsilon^{-1}] \le 2 \varphi(\epsilon^{-1}+\delta)/ \varphi(\epsilon^{-1}) \to 0 \,, $$
so in par... | 4 | https://mathoverflow.net/users/7691 | 418622 | 170,441 |
https://mathoverflow.net/questions/418403 | 9 | One characteristic of the surreal numbers is that they are a monster model of the first-order theory of real numbers, according to Joel David Hamkins in [this post](https://mathoverflow.net/questions/126158/a-mother-of-all-groups-what-kind-of-structures-have-mother-of-alls). Thus they are real-closed, and every other r... | https://mathoverflow.net/users/24611 | A "surnatural numbers" as a largest model of the natural numbers | I asked (and also answered) [a more general version of this question](https://mathoverflow.net/questions/304290/for-which-theories-does-zfc-without-global-choice-prove-the-existence-of-a-prope) a while ago. To summarize the answer, some [results of Kanovei and Shelah](https://arxiv.org/abs/math/0311165) have the follow... | 5 | https://mathoverflow.net/users/83901 | 418630 | 170,443 |
https://mathoverflow.net/questions/418629 | 1 | We consider the sum
$$ \sum\_{m \in \mathbb Z^2} \frac{1}{(3 m\_1^2+3m\_2^2+3(m\_1+m\_1m\_2+m\_2)+1)^2}. $$
Numerically, it is not particularly hard to see that the value of this series is well below $4$, indeed one gets numerically an upper bound of roughly $3.43$
I wonder if there is analytically a quick argume... | https://mathoverflow.net/users/457901 | Upper bound on double series | As noted in the comment by Beni Bogosel, the sum in question is
\begin{equation}
s:=\sum\_{x=-\infty}^\infty\sum\_{y=-\infty}^\infty\frac1{f(x,y)^2},
\end{equation}
where
\begin{equation}
f(x,y):=\frac32\, ((x + 1)^2 + (y + 1)^2 + (x + y)^2) - 2.
\end{equation}
Note that
\begin{equation}
f(x,y)\ge x^2+y^2+2\ge2\sq... | 3 | https://mathoverflow.net/users/36721 | 418636 | 170,445 |
https://mathoverflow.net/questions/418634 | 1 | A real random variable $X$ is said to be *subgaussian* if there exists an $a > 0$ such that $\mathbb{E}[e^{\lambda X}] < e^{a^2 \lambda^2}$ for all $\lambda \in \mathbb{R}$. The space of such random variables admits a Banach space structure, with an Orlicz norm given by $$\| X \|\_{\psi\_2} = \inf\left\{ t > 0 : \mathb... | https://mathoverflow.net/users/479167 | The central limit theorem in the subgaussian Orlicz norm | $\newcommand\ep\varepsilon$No. If this were so, then (by Lemma 1 below) $S\_n$ would converge to a normally distributed random variable $Y$ in probability, which is [false](https://math.stackexchange.com/questions/2145140/clt-cannot-be-enhanced-to-convergence-in-probability) for any iid $X\_i$'s.
---
>
> **Lemm... | 0 | https://mathoverflow.net/users/36721 | 418637 | 170,446 |
https://mathoverflow.net/questions/418579 | 5 | Let $X$ be a finite type scheme over $\mathbb{Z}\_p$ for some prime $p$. Assume that $X\_{\mathbb{Q}\_p}$ is smooth of dimension $n$, but not necessarily irreducible. Then is
$$X(\mathbb{Z}/p^k\mathbb{Z}) = O(p^{kn})$$
as $k \to \infty?$
| https://mathoverflow.net/users/5101 | Number of points on schemes modulo $p^k$ | Yes, this is true, for elementary reasons (i.e. not to do with the Igusa zeta function or something).
By passing to an open cover, we may assume $X$ is affine, say $X = \operatorname{Spec} \mathbb Z\_p[x\_1,\dots, x\_N]/ (f\_1,\dots, f\_m)$. The Jacobian of this system of equations is an $N \times m$ matrix.
Consid... | 5 | https://mathoverflow.net/users/18060 | 418638 | 170,447 |
https://mathoverflow.net/questions/417993 | 2 | Recall that $M\subseteq\omega$ is **maximal** if it is c.e., and can be only trivially extended by other c.e. sets, i.e. if $M\subseteq N$ and $N$ is c.e., then either $\overline{N}$ or $N\setminus M$ is finite. Similarly say a set $M$ is **$A$-maximal** if it is $A$-c.e. and only trivially extended by other $A$-c.e. s... | https://mathoverflow.net/users/172527 | Sets $A$ such that $A$-maximal sets are $\Delta^0_2$ | Turns out (as I suspected!) all such sets are $\Delta^0\_2$, so that this property exactly characterizes lowness.
Fix an $A$-maximal $M$. For any $B\leq\_T A$, one can build $C = \overline{M}\oplus\_B\emptyset = \{n \mid p\_B(n)\in\overline{M}\}$. Then any $W\_e^A$ you might wish to intersect with $C$ can be transfor... | 0 | https://mathoverflow.net/users/172527 | 418641 | 170,448 |
https://mathoverflow.net/questions/416977 | 7 | Let $B$ be a paracompact space with the property that any (topological) vector bundle $E \to B$ is trivial. What are some non-trivial examples of such spaces, and are there any interesting properties that characterize them?
For simple known examples we of course have contractible spaces, as well as the 3-sphere $S^3$... | https://mathoverflow.net/users/143629 | Examples and properties of spaces with only trivial vector bundles | Let $B$ be a closed manifold with such that every vector bundle is trivial. Then $H^1(B; \mathbb{Z}\_2) = 0$, otherwise there would be a non-trivial line bundle. Therefore every bundle over $B$ is orientable and $B$ itself is orientable. Orientable rank two bundles over $B$ are classified by $H^2(B; \mathbb{Z})$, so we... | 7 | https://mathoverflow.net/users/21564 | 418642 | 170,449 |
https://mathoverflow.net/questions/418656 | 0 | Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate
$$
\max\{\|u\|\_0,\|v\|\_0\}\leq d-s,
$$
where, as usual, for any vector $x \in \mathbb{R}^d$ we define $\|x\|\_0:=\sum\_{i=1}^d\,I\_{x\_i\neq 0}$.
Let $A$ be an $d\times d$-matrix solvi... | https://mathoverflow.net/users/36886 | How sparse can a matrix mapping between sparse vectors be? | $||v||\_0$, so $\leq d-s$: The matrix $A$ needs to have at least one entry for every entry of $v$ (otherwise it can't obtain that entry). It is also sufficient to have so many entries, as if we consider one $j$ with $u\_j \not= 0$ (which exists otherwise the system has a solution (the zero matrix) only if $v\_i = 0$ fo... | 1 | https://mathoverflow.net/users/101157 | 418657 | 170,454 |
https://mathoverflow.net/questions/418660 | 3 | I was reading [this answer](https://mathoverflow.net/a/32914/146831), which says that:
>
> In his Master's Thesis, Merlin Carl has computed a polynomial that is solvable in the integers iff ZFC is inconsistent. A joint paper with his advisor Boris Moroz on this subject can be found at <http://www.math.uni-bonn.de/p... | https://mathoverflow.net/users/146831 | What does it really mean for a polynomial to be solvable in $\mathbb{Z}$ iff $\mathsf{ZFC}$ is inconsistent? | Edit: After I wrote this, the paper was posted in the commentary. What I wrote below is what is meant, however they use $\mathsf{GBC}$ instead of $\mathsf{ZFC}$. Since these two theories are equiconsistent and have the same first order consequences, this technical difference is immaterial.
However, when people say th... | 4 | https://mathoverflow.net/users/114946 | 418663 | 170,455 |
https://mathoverflow.net/questions/418628 | 1 | Given a symmetric Hessenberg matrix $A = \left[\begin{matrix}\ddots& \vdots & \vdots\\\dotsb & a & b\\\dotsb& b & c\end{matrix}\right]$, the Wilkinson shift $\mu$ employed in some eigenvalue solvers is given by
$$\mu = c - \frac{b^{2} \operatorname{sign}{\left(\delta \right)}}{\sqrt{b^{2} + \delta^{2}} + \left|{\delta}... | https://mathoverflow.net/users/75761 | Does Wilkinson's shift need to be discontinuous? | **It must be discontinuous to ensure that all fixed points are attractive**.
We make the following assumptions about iterative eigenvalue algorithms over PSD matrices:
1. They consist of iterating some function $f$,
2. $f(M)$ is orthogonally similar to $M$
3. The sequence $(f^n(M))\_{n\in\mathbb N}$ converges for e... | 2 | https://mathoverflow.net/users/75761 | 418664 | 170,456 |
https://mathoverflow.net/questions/418632 | 4 | I think the question as expressed in the title should be clear. I do not know whether there is a known "characterization" of the weakly compact convex sets in $c\_0(\mathbb N\_0)$ but testing examples has lead me to conjecture that sequences in $\ell^1(\mathbb N\_0)$ converging to zero uniformly on these sets also conv... | https://mathoverflow.net/users/12643 | Are sequences in $\ell^1(\mathbb N_0)$ converging uniformly on convex weakly compact subsets of $c_0(\mathbb N_0)$ norm convergent? | The answer is yes.
*Proof.* Assume to the contrary that a sequence $(x\_n)$ in $\ell^1$ converges, say to $0$, uniformly on convex weakly compact subsets of $c\_0$, but is not norm convergent and hence not norm convergent to $0$.
Note that the sequence even converges uniformly on all weakly compact subsets of $c\_0... | 3 | https://mathoverflow.net/users/102946 | 418669 | 170,459 |
https://mathoverflow.net/questions/418554 | 51 | $\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The group theoretic explanation for this that I know (forgive me if it is an oversimplification) is that the state space of the... | https://mathoverflow.net/users/478715 | Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled? | In this answer, I'm going to crib from [this](https://math.ucr.edu/home//baez/hydrogen/4d/hydrogen_4d.pdf) presentation by @JohnBaez and the paper [On the Regularization of the Kepler Problem](https://arxiv.org/abs/1007.3695). Milnor's [paper](https://www.maa.org/programs/maa-awards/writing-awards/on-the-geometry-of-th... | 10 | https://mathoverflow.net/users/947 | 418677 | 170,462 |
https://mathoverflow.net/questions/418672 | 7 | In many areas of mathematics it is informative to conduct numerical experiments.
But, it not uncommon that the searches do not lead to the examples or data one was hoping for. Since the numerical searches can be quite time consuming, it seems useful to share these negative results, so that others avoid spending time ... | https://mathoverflow.net/users/469 | Reporting inconclusive experimental searches | An easy and reliable way to share code is via [Zenodo](https://en.wikipedia.org/wiki/Zenodo) --- works much like arXiv, you get a DOI, can update your files, and it's free. We use it regularly to document computer simulations in physics, I imagine computational mathematics is not that different.
Note that Zenodo expl... | 6 | https://mathoverflow.net/users/11260 | 418678 | 170,463 |
https://mathoverflow.net/questions/418578 | 2 | The current post comes from my previous post at [stackexchange](https://math.stackexchange.com/questions/4408614/sqrtf-has-bounded-tangential-derivatives-if-f-in-c1-1). However, I have not get any comment yet.
In a celebrated paper written by [Guan, Trudinger, and Wang](https://projecteuclid.org/journals/acta-mathema... | https://mathoverflow.net/users/105893 | Estimates on the second-order derivatives for degenerate Monge-Ampere equations | This inequality comes from scaling. Assume that $f$ satisfies $|D^2f| \leq 1$ on $\mathbb{R}^n$ and $f \geq 0$. It suffices to prove that
$$|\nabla f(0)|^2 \leq 2f(0).$$
Equality holds if $f(0) = 0$, so assume that $f(0) > 0$. We may assume that $f(0) = 1$ after taking the rescaling $\tilde{f}(x) = \lambda^{-2}f(\lambd... | 3 | https://mathoverflow.net/users/16659 | 418684 | 170,464 |
https://mathoverflow.net/questions/418676 | 7 | Suppose $\mathcal{A}$ is a Grothendieck abelian category with enough projectives, then $\mathcal{A}$ is tensored and cotensored over $\mathrm{Ab}$ with $\mathbb{Z}^{\oplus S}\otimes X\cong \bigoplus\_S X$ for any set $S$, and for any abelian group $A$, with presentation $\mathbb{Z}^{\oplus R}\xrightarrow{f} \mathbb{Z}^... | https://mathoverflow.net/users/479208 | Derived functor of functor tensor product | The answer is yes if you assume enough things. In particular, the notion of a left flat object of $\mathcal A$ comes up :
**Definition:** An object $L\in\mathcal A$ is left flat if $-\otimes L$ is exact.
My assumption will be that $\mathcal A$ has enough left flat objects. This allows you to even define $\mathbb L(... | 5 | https://mathoverflow.net/users/102343 | 418688 | 170,465 |
https://mathoverflow.net/questions/418438 | 2 | Let $K$ be a symmetric convex body in $\mathbb{R}^n$ (that is the unit ball of a norm). Let $h\_K$ be its support function, that is $h\_K(u) = \sup\_{x \in K}\langle x,u \rangle$. The quantity $w(K) = \int\_{S^{n-1}} h\_K(\theta)\,d\theta$ is called the mean width of $K$ and is well studied.
Assume that $\operatornam... | https://mathoverflow.net/users/125325 | Distribution of the support function of convex bodies: beyond mean width | Actually the statement is pretty strong, in particular it implies the Bourgain-Milman (reverse Blaschke Santalo inequality) theorem, which says that the Mahler conjecture is true up to constant, that is, there exists $c\_0>0$ such that for all convex body $K$ :
$$\operatorname{vol}(K)\operatorname{vol}(K^\circ) \geq ... | 1 | https://mathoverflow.net/users/125325 | 418689 | 170,466 |
https://mathoverflow.net/questions/418671 | 4 | In "M. B. Nathanson - Elementary Methods in Number Theory" is shown (Theorem 7.14) that if $A$ is a set of positive integers such that $\sum\_{a \in A} 1 / a$ converges then the set of multiples of $A$ has a natural density (a set of positive integers $S$ has a natural density if exists $\lim\_{x \to \infty} |S \cap [1... | https://mathoverflow.net/users/160943 | Relative density of primes in certain congruence classes | This is too long for a comment and possibly it still answers your question.
Fix a positive integer $b \in \mathbb{N}$ and set $A\subseteq \mathbb{N}$ such that $\sum\_{n \in A}1/n<\infty$ and $1\notin A$. Let $\{a\_1,a\_2,\ldots\}$ be the increasing enumeration of $A$. Let also $\mathbb{P}$ be the set of primes and d... | 3 | https://mathoverflow.net/users/32898 | 418691 | 170,468 |
https://mathoverflow.net/questions/418680 | 6 | Write $\psi(x) = \sum\_{n\le x} \Lambda(n)$. The classical omega theorem says that
$\psi(x) - x = \Omega\_{\pm}(x^{1/2})$.
Question: How often does this hold? For example, what do we know about the size of the set
$\{ n\le x: \psi(x) - x > c x^{1/2} \}$
for some $c$? Ditto for $\psi(x) - x < c x^{1/2}$. What ab... | https://mathoverflow.net/users/66397 | How often does the omega theorem hold? | For any $\varepsilon>0$, there exist $c(\varepsilon)>0$ and $X\_0(\varepsilon)>0$ such that for any $X>X\_0(\varepsilon)$ we have
$$\sup\_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad
\inf\_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(... | 7 | https://mathoverflow.net/users/11919 | 418713 | 170,475 |
https://mathoverflow.net/questions/418714 | 4 | Let $P$ be a "nice" distribution on $\mathbb R^m$ (e.g., multivariate Gaussian, etc.), with density $p$. Let $H := \{x \in \mathbb R^m \mid x^\top w = b\}$ be a hyperplane in $\mathbb R^m$ with unit-normal $w \in \mathbb R^m$. Let $R$ be the *Radon transform* of $p$ w.r.t $H$ by
$$
R := \int\_H p(x)\,ds(x),
$$
where $d... | https://mathoverflow.net/users/78539 | Consistent empirical estimation of Radon transform of a multivariate density function | $\newcommand\th\theta\newcommand\R{\mathbb R}$Suppose that we have a parametric setting, that is, the unknown distribution $P$ belongs to a known parametric family $(P\_\th)$ of distributions parameterized by a sufficiently low-dimensional parameter $\th$; this may be the case if $P$ is Gaussian.
Then you can get (say)... | 4 | https://mathoverflow.net/users/36721 | 418715 | 170,476 |
https://mathoverflow.net/questions/418402 | 16 | In Conway's "On Numbers And Games," page 44, he writes:
>
> **NON-STANDARD ANALYSIS**
>
>
> We can of course use the Field of all numbers, or rather various small
> subfields of it, as a vehicle for the techniques of non-standard
> analysis developed by Abraham Robinson. Thus for instance for any
> reasonable fun... | https://mathoverflow.net/users/24611 | Interpreting Conway's remark about using the surreals for non-standard analysis | Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA
puzzling and I am not sure what he had in mind. What I think Conway might have said is: "$\mathbf{No}$ is really irrelevant to nonstandard analysis", and, vice versa. After all, wherea... | 8 | https://mathoverflow.net/users/18939 | 418718 | 170,478 |
https://mathoverflow.net/questions/418727 | 8 | It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to solving Diophantine equations or factoring an integer.
The restrictions are as follows: $f(xy)=f(x+y)$ if $x,y\geq 3$ are p... | https://mathoverflow.net/users/140356 | Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations | Too long for a comment. My guess is that for $x\geq 7$ we that that $f(x)=c$ by strong induction. We need to check the cases $7\leq x\leq 20$ by hand .
Let us now suppose we have $y\geq 20$ and integer. If it can be factored as $a\cdot b$ with $a,b\geq 3$ coprime then obviously $6\leq a+b<ab=y$ and we are done. Thus ... | 9 | https://mathoverflow.net/users/41010 | 418730 | 170,480 |
https://mathoverflow.net/questions/418721 | 2 | I am a bit confused by the following question and I hope someone could help me out.
Let $u$ be the solution of the following initial value problem
$$
u''(t) = g(t) \; \text{ in } (0,\infty), \quad\quad u(0)=a, \quad u'(0)=0. \label{1}\tag{1}
$$
Let $U$ and $G$ be the extensions of $u$ and $g$ as zero to $(-\infty, \i... | https://mathoverflow.net/users/479159 | Restore initial condition for distributions | $\newcommand{\R}{\mathbb R}$In accordance with comments by Willie Wong and the OP, let us extend $u$ by $a$ to the left of $0$:
\begin{equation\*}
U(t) :=
\begin{cases}
u(t) & \text{ if }t\ge0, \\
a & \text{ if }t<0.
\end{cases}
\tag{3}\label{3}
\end{equation\*}
Then
\begin{equation\*}
U''=G, \tag{4}\label{4}
\end{e... | 2 | https://mathoverflow.net/users/36721 | 418755 | 170,490 |
https://mathoverflow.net/questions/418706 | 1 | It is [well known](https://i.stack.imgur.com/EJ9gG.png) that for the diffusions
\begin{align\*}
dX&=f(X)dt+&cdB\\
dY&=&cdB
\end{align\*}
the density of the law of $X$ with respect to the law of $Y$ is
\begin{align\*}
\frac{d\mu\_X}{d\mu\_Y}(c B)&=\exp\left(\int\_0^T\frac{f(cB(t))}{c}dB(t)-\frac12\int\_0^T\frac{f^... | https://mathoverflow.net/users/479223 | Where does the extra term in the density of a diffusion with respect to $c B(t)$ come from? | In a nutshell you are asking the following:
When $F(x)=\int\_0^xf(y)\,dy$ we get from Ito
$$
F(cB(t))=\int\_0^Tf(cB(t))\,d(cB(t))+\frac{1}{2}\int\_0^Tf'(cB(t))\,c^2\,dt\,
$$
so that the expressions
\begin{align}
&F(cB(t))-\frac{c^2}{2}\int\_0^Tf'(cB(t))\,dt\,,\tag{1}\\
&\int\_0^Tf(cB(t))\,d(cB(t))\tag{2}
\end{align}
... | 2 | https://mathoverflow.net/users/340165 | 418756 | 170,491 |
https://mathoverflow.net/questions/418746 | 5 | Let $A$ be a unital complex algebra with the unit $\bf1$. Let $\mathcal{N}$ be the family of all norms on $A$ making it a unital normed algebra with the same unit $\bf1$. Let us put
$B\_{\|\cdot\|}=\{x\in A : \|x-{\bf1}\|<1\}$ where $\|\cdot\|\in \mathcal{N}$.
Clearly, the intersection $\bigcap\_{\mathcal{N}}B\_{\|\c... | https://mathoverflow.net/users/84390 | Permanent invertible elements | Here is a non-trivial condition:
>
> If $x$ belongs to this intersection, then $x$ commutes with every nilpotent element.
>
>
>
*Proof*. For every invertible $a\in A$, the function
$$N(z):=\lVert a^{-1}za\rVert$$
is a norm of unital algebra. Let $n$ be a nilpotent element, of order $k$, and choose $a\_t=\mathb... | 8 | https://mathoverflow.net/users/8799 | 418762 | 170,494 |
https://mathoverflow.net/questions/418726 | 3 | An operator $T$ on a separable Hilbert space $H$ is called *unicellular* if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many examples of (compact) unicellular operators, for example, the Volterra operator.
I am looking at a weaker property.... | https://mathoverflow.net/users/69275 | Unicellular compact operators | $V\oplus V\in B(L^2(0,1)\oplus L^2(0,1))$, with $V$ being the Volterra operator, is an operator without eigenvalues that does not have your property. The invariant subspaces are $L^2(0,a)\oplus L^2(0,b)$, and for any such space $M=A\oplus B$, we can find another one $N=C\oplus D$, with $C\subsetneq A$, $D\supsetneq B$,... | 1 | https://mathoverflow.net/users/48839 | 418769 | 170,498 |
https://mathoverflow.net/questions/418770 | 5 | Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard model are "charged spinors", which are fermionic particles, which transform under a non-trivial representation. Now, the p... | https://mathoverflow.net/users/199422 | Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles" | I'll assume that the vector space "$V$" occuring in constructions (1) and (2) doesn't have to be the same. In that case I'll rename vector space in construction (2) to "$W$."
Then I claim that construction (1) is a special case of construction (2) with $W=V\otimes\Delta\_n$. That follows from the general fact that if... | 6 | https://mathoverflow.net/users/163893 | 418781 | 170,503 |
https://mathoverflow.net/questions/418778 | 5 | Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^\*$ by conjugation.
1. One can describe the orbits of $\mathfrak{g}^\*$ explicitly (e.g. using Jordan blocks for $\operatorname{GL}\_n$),
2. There is an especially interesting class, of *nilpotent* orbits. There are f... | https://mathoverflow.net/users/119012 | Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$ | Generally they can be classified when the action of $G$ on $V$ is *visible*, which by definition means there are finitely many orbits in the nullcone. Irreducible visible pairs $(G, V)$ were classified in [Kac - Some remarks on nilpotent orbits](https://doi.org/10.1016/0021-8693(80)90141-6) (with some corrections in [D... | 5 | https://mathoverflow.net/users/38434 | 418788 | 170,504 |
https://mathoverflow.net/questions/418780 | 6 | How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper.
>
> Let $U$ be a scheme of finite type over $\mathbb{F}\_q$. Let $\mathcal{L}\_1$, $\mathcal{L}\_2$ be two local systems of pure weight $i$. Suppose for any $u \in U$ and $k \in \mathbb... | https://mathoverflow.net/users/130879 | Chebotarev density theorem and pure weight local systems | As Piotr says, we must assume $U$ normal. The purity assumption is not needed.
There are two steps to this proof
(1) Suppose for any $u \in U$ and $k \in \mathbb{N}$, we have
$Tr(\sigma\_u^k,\mathcal{L}\_1)=Tr(\sigma\_u^k,\mathcal{L}\_2)$, where $\sigma\_u$ is the Frobenius conjugacy class of $u$ in $\pi\_1(U)$. Th... | 11 | https://mathoverflow.net/users/18060 | 418793 | 170,506 |
https://mathoverflow.net/questions/418801 | 4 | Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
| https://mathoverflow.net/users/170286 | Can local $0^\#$ exists in L? | Take a countable elementary submodel of $L\_\kappa[0^\#]$, code that into a real, note that "There is a real coding a well-founded model of $V=L[0^\#]$" is a $\Sigma^1\_2$ statement, remember what Shoenfield said about such statements: they are also true in $L$.
Now take $M$ to be a transitive collapse of some real c... | 8 | https://mathoverflow.net/users/7206 | 418803 | 170,508 |
https://mathoverflow.net/questions/417589 | 1 | Consider a normal complex analytic space $X$ and a projective birational resolution $f:Y\rightarrow X$. Let $T$ be a closed positive current of bi-dimension $(p,p)$ on $X$. Is there always a closed positive current $S$ on $Y$ such that $f\_\*S=T$?
| https://mathoverflow.net/users/58308 | Do all closed positive currents lift to a resolution? | I believe the answer is no.
If I am reading the article below by Méo correctly, he shows that if
$\pi : Y \to X$ is the blowup of the polydisc $X=D^m$ along $Z := \{ z\_1 = \dots = z\_k = 0 \}$, and if $k \leq p \leq n-2$, then there exists a closed positive $(p,p)$-current $T$ on $X$ such that $T' := \pi^\* (T|\_{X\... | 2 | https://mathoverflow.net/users/49151 | 418811 | 170,510 |
https://mathoverflow.net/questions/418789 | 3 | Let $p $ be an odd prime. Assume that we have the following perfect pattern:
all the primes below $p$ are successively quadratic residues and quadratic non-residues. What can we say about $p$? Is it possible that only finitely many such $p$ exist?
Edit: Mathematically I mean: denote by $p\_n$ the $n$-th smallest prim... | https://mathoverflow.net/users/9232 | Perfect equidistribution for the Legendre symbol | It is very likely that there are only finitely many primes with this property. Heuristically, the probability of this happening for the $N$th prime is $2^{2-N}$, and $\sum\_{N \geq 2} 2^{2-N} =2$, so we expect only a couple primes where this happens.
I can't see how one would ever hope to prove this. However, we can ... | 6 | https://mathoverflow.net/users/18060 | 418819 | 170,514 |
https://mathoverflow.net/questions/418749 | 3 | Suppose that $r>1$ and $s>1$ are irrational numbers, and let $a\_n=\lfloor nr \rfloor$ and $b\_n=\lfloor ns \rfloor$. Assume that $r$ and $s$ are numbers for which $\{a\_n\}\cap\{b\_n\}$ is infinite, and let $(c\_n)$ be the increasing sequence of numbers in $\{a\_n\}\cap\{b\_n\}$. It appears that the number
$$t = \li... | https://mathoverflow.net/users/172419 | Limit associated with two Beatty sequences that are not a Beatty pair | I will give a complete answer when $1,1/r$ and $1/s$ are linearly independent over $\mathbb Q$ and a recipe to compute your $t$ otherwise.
First of all, notice that $n$ lies in a Beatty sequence $\lfloor m\alpha\rfloor$, where $\alpha>1$ if and only if $\{n/\alpha\}>1-\frac{1}{\alpha}$. Indeed, if $n=\lfloor m\alpha \r... | 4 | https://mathoverflow.net/users/101078 | 418824 | 170,518 |
https://mathoverflow.net/questions/418753 | 3 | Let $\mu$ be a centred Radon Gaussian measure on a locally convex space $X$ and $q : X \to \mathbb{R}$ a seminorm that is $\mathcal{B}(X)\_\mu$-measurable, where $\mathcal{B}(X)\_\mu$ is the Lebesgue completion of the Borel $\sigma$-algebra on $X$.
Consider a linear functional $\phi : X \to \mathbb{R}$ with $\vert \phi... | https://mathoverflow.net/users/18936 | Is a functional bounded by a measurable seminorm also measurable? | I have found an answer exploiting the equivalence of Lusin and Borel measurability as stated in [Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures](https://books.google.de/books/about/Radon_measures_on_arbitrary_topological.html?id=19zuAAAAMAAJ) on page 6, theorem 5:
>
> Let $H : X \to Y$. If ... | 1 | https://mathoverflow.net/users/18936 | 418826 | 170,520 |
https://mathoverflow.net/questions/418479 | 5 | Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus.
Let us assume that it admits a maximal torus after a finite surjective (resp. finite flat) cover, is it possible to replace it by a finite étale cover?
| https://mathoverflow.net/users/27398 | Group scheme with an isotrivial maximal torus | **Edit.** I realized after the original post that there are even easier examples. These examples also show that the Quillen–Suslin Theorem fails already for smooth affine quadric hypersurfaces (in some sense, the "simplest" smooth affine varieties after affine spaces, where Quillen–Suslin does hold).
Let $\overline{B... | 6 | https://mathoverflow.net/users/13265 | 418827 | 170,521 |
https://mathoverflow.net/questions/418809 | 8 | For $n\in \mathbb{N}$ with prime decomposition $n=p\_1^{r\_1}\cdots p\_k^{r\_k},p\_i\neq p\_j$, let $A=\{p\_1,\cdots,p\_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all prime factors of } q\,\in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q)
\end{equation}
where $d(n,Q):=|\{m\in \mathbb{N... | https://mathoverflow.net/users/172051 | Estimates about prime numbers: a lemma in Bourgain's article | Let $q<Q$ be such that all its prime divisors are in $A$. We can write $q=q\_0\times r\_1^{n\_1}\cdots r\_s^{n\_s}$, where all prime factors of $q\_0$ are $<\log Q$ and $\log Q\leq r\_1<\cdots<r\_s\in A$. Clearly, $q$ is uniquely determined once we have fixed a) $q\_0$, b) the set $\{r\_1,\ldots,r\_s\}$, and c) the int... | 10 | https://mathoverflow.net/users/385 | 418839 | 170,524 |
https://mathoverflow.net/questions/418837 | 1 | Let $S$ be a K3 surface and $h:=c\_1(i^\*\mathcal{O}\_{\mathbb{P^3}}(1))$, then we can compute that $c\_1(S)=0,c\_2(S)=6h^2$. Hence
\begin{align}
\sqrt{\text{td}(S)}=1+\frac{c\_2(S)}{24}=1+\frac{1}{4}h^2
\end{align}
According to materials e.g. *Lectures on K3 Surfaces*, the Mukai vector of a vector bundle $E$ on $S$ is... | https://mathoverflow.net/users/nan | Understand the Mukai vector | $c\_2(S)$ should not depend on the choice of polarization. See Corollary 3.3 of Huybrecht's Lectures on K3 surfaces.
$c\_2(S) = 24 c\_s$ where $c\_s$ is the generator of $H^2(S, \mathbb{Z})$. Then note that $\sqrt{td(S)} = (1+2c\_S)^{\frac{1}{2}} = 1 + c\_S$.
| 2 | https://mathoverflow.net/users/164620 | 418840 | 170,525 |
https://mathoverflow.net/questions/418705 | 4 | Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]\_x\hookrightarrow\mathbb{C}\{X\}\_x\hookrightarrow\widehat{\mathbb{C}[X]\_x}$; that is, the ring of holomorphic germs is sandwiched between the ring of polynomial germ... | https://mathoverflow.net/users/158123 | Comparing the first-order theories of different kinds of local rings of a complex variety | It turns out that all of these except the ring of polynomial germs are elementary equivalent. The following answer is from Elliot Kaplan (posted with permission):
>
> I think it's hard to say in general what an elementary extension of commutative rings looks like, but a necessary condition is certainly that the sma... | 1 | https://mathoverflow.net/users/158123 | 418841 | 170,526 |
https://mathoverflow.net/questions/418806 | 5 | I recently learned about the [MIP^\*=RE result](https://arxiv.org/abs/2001.04383). I have to admit that I don't understand big parts of this paper and I am barely familiar with quantum physics. I hope my questions below make sense.
I copy the basic definitions:
IP= the class of languages with an randomized interact... | https://mathoverflow.net/users/13694 | MIP^*=RE and quantum computation | **Q1:** *"Do you have a reference for `quantum computable = computable by a classical Turing machine' ?*
**A1:** This follows immediately from the fact that any quantum computation can be simulated by a classical computer with exponential overhead.
**Q2:** Is there any result (or conjecture) that quantum machines ... | 5 | https://mathoverflow.net/users/11260 | 418854 | 170,531 |
https://mathoverflow.net/questions/418857 | 3 | The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the only condition is to preserve the metric structure of the graph). Any numerical algorithms like MDS are not suitable. U... | https://mathoverflow.net/users/479332 | Explicit formula for embedding Cayley graph of free group into hyperbolic space | The subgroup $\Gamma < \mathrm{SL}(2, \mathbb{R})$ generated by the matrices
$
a =
\begin{pmatrix}
1 & 2 \\
0 & 1
\end{pmatrix}
$
and
$
b =
\begin{pmatrix}
1 & 0 \\
2 & 1
\end{pmatrix}
$ is free of rank two. It acts on the upper half plane model of $\mathbb{H}^2$ via Mobius transformations. The orbit of $i$ gives the... | 7 | https://mathoverflow.net/users/1650 | 418860 | 170,532 |
https://mathoverflow.net/questions/418863 | 7 | It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified operator:
$$ Lu=\Delta u+ fu,$$
where $f$ is a smooth function.
**More specifically:** If $f$ equals to minus an eigenvalue o... | https://mathoverflow.net/users/81414 | Kernel of the Laplacian + a function | **Q:** Can we conclude that $Lu=\Delta u+ fu=0$ has only zero (or constant solutions) if we assume $f$ non-constant?
**A:** No, a counter example in one dimension is the [Mathieu equation](https://en.wikipedia.org/wiki/Mathieu_function), which has non-constant $\pi$-periodic or $2\pi$-periodic solutions $u(x)$ when $... | 11 | https://mathoverflow.net/users/11260 | 418867 | 170,533 |
https://mathoverflow.net/questions/418829 | 8 | There is so much literature on the relation between the multiplicative structure of a finite field and elements having zero trace, that I am hoping that the following is known.
Let $q$ be a prime power, let $n$ be an odd prime number, let $\mathbb{F}\_{q^{2n}}$ be the field of cardinality $q^{2n}$ and let $\mathrm{Tr... | https://mathoverflow.net/users/45242 | Zero trace elements in finite fields | I can show that exceptions occur at most for $n=3$. (Primality of $n$ is never used.)
Since $n$ is odd, $\mathbb F\_{q^{2n}} = \mathbb F\_{q^2} \otimes\_{\mathbb F\_q} \mathbb F\_{q^n}$. The trace map $\operatorname{Tr}:\mathbb F\_{q^{2n}} \to \mathbb F\_{q^2}$ is obtained by tensoring the identity map $\mathbb F\_{q... | 5 | https://mathoverflow.net/users/18060 | 418868 | 170,534 |
https://mathoverflow.net/questions/418875 | 4 | I am reading a [paper](https://link.springer.com/content/pdf/10.1007/s11222-015-9598-x.pdf) about constructing a non-reversible Metropolis Hastings Markov chain from a reversible one as described at a high level in paragraph $3$ of page $1$.
>
> But I don't understand how, given a reversible Markov chain $P$ with s... | https://mathoverflow.net/users/479350 | About non-reversible Metropolis Hastings Markov chain | As defined after lemma 2.1, any skew-symmetric matrix $\Gamma$ (i.e. $\Gamma=-\Gamma^T$) that satisfies $\Gamma {\mathbb 1}=0$ is called a vorticity matrix. It need not be of the form $\pi(x)P(xy)-\pi(y)P(y,x)$.
| 4 | https://mathoverflow.net/users/7691 | 418879 | 170,540 |
https://mathoverflow.net/questions/418880 | 4 | Helmut Wielandt [discussed](https://books.google.com/books/about/Finite_Permutation_Groups.html?id=npviBQAAQBAJ) an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):
Let $g\neq 1$ be a permutation in some finite primitive permutation group $G$ of degree $n$. The minimal degree $m$ is defi... | https://mathoverflow.net/users/18286 | Minimal degree of primitive permutation group | You seem to be aware of the answer to your own question, since you give the reference to the paper of Guralnick and Magaard, which classifies groups of minimal degree $\leq n/2$. Therefore $n \leq 2m$ with explicit exceptions list in the G--M paper. See also the previous paper of Liebeck and Saxl, which is easier and d... | 5 | https://mathoverflow.net/users/20598 | 418891 | 170,543 |
https://mathoverflow.net/questions/418892 | 1 | I want to generate all strongly connected tournament of size $n \in \{4, 11\}$.
As a strongly connected tournament has an hamiltonian path I may assume that $v\_i v\_{i+1}$ is always an arc, and $v\_n v\_1$ is an arc. Thus, I have to decide the outcome of $m:= n\frac{(n-1)}{2} -n$ games, ie I have $2^m$ tournaments... | https://mathoverflow.net/users/342793 | Generate all strongly connected tournament | Brendan McKays program “gentourng” can generate all tournaments up to isomorphism - I recently used it for all 11-vertex tournaments.
It is part of the nauty package:
[https://users.cecs.anu.edu.au/~bdm/nauty/](https://users.cecs.anu.edu.au/%7Ebdm/nauty/)
| 1 | https://mathoverflow.net/users/1492 | 418896 | 170,545 |
https://mathoverflow.net/questions/418898 | 3 | In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $f: X \to B$ with smooth base $B$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because Lagrangian fibrations have equidimensional fibers. Then they write
>
> *Remark 1.18.* Note that the conclusion that $f$ is... | https://mathoverflow.net/users/111897 | Does miracle flatness always fail for a non-regular base? | The answer lies in *Theorem 23.7* from Matsumura's *Commutative Ring Theory*:
>
> *Theorem 23.7.* Let $(A, \mathfrak m, k)$ and $(B, \mathfrak n, k')$ be local Noetherian local rings, and $A \to B$ a local homomorphism [...]. We assume that $B$ is flat over $A$. (i) If $B$ is regular then so is $A$.
>
>
>
As $... | 2 | https://mathoverflow.net/users/111897 | 418904 | 170,547 |
https://mathoverflow.net/questions/418866 | 3 | Quantum groups are useful for making knot/link invariants: for example, $U\_q(\mathfrak{sl}\_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U\_q(\mathfrak{sl}\_2)$ is a braided monoidal category, which is **not** symmetric, hence gives us interesting knot invariants.... | https://mathoverflow.net/users/5323 | Motivating quantum groups from knot invariants | Let $\mathcal{C}$ be the category of finite-dimensional representations of a semisimple Lie algebra $\mathfrak{g}$ and $\mathcal{C}[\![\hbar]\!]$ the ribbon category you mention (which depends on the choice of a Drinfeld associator).
Given an irreducible $\mathfrak{g}$-representation $V$, the corresponding knot invar... | 7 | https://mathoverflow.net/users/18512 | 418906 | 170,548 |
https://mathoverflow.net/questions/418894 | 6 | I have been interested in the following paper ("On systems of linear indeterminate equations and congruences" by
Henry J. Stephen Smith,
Philosophical Transactions of the Royal Society of London, Vol. 151 (1861), pp. 293-326) [JSTOR open link](https://www.jstor.org/stable/108738?seq=1).
On page 300, the author states... | https://mathoverflow.net/users/84272 | What is Euler's method in linear algebra? | Euler's method is described, for example, by James Fogo in [Linear indeterminate problems](https://www.jstor.org/stable/27957030?seq=1). This applies to systems of equations where there are more unknowns than there are equations, and a solution can be found by restricting the solutions to integers. The method was publi... | 9 | https://mathoverflow.net/users/11260 | 418908 | 170,549 |
https://mathoverflow.net/questions/418903 | 2 | Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := \int\_H f(x)ds(x) = \int\_{\mathbb R^n}f(x)\delta(b-x^\top w)\,dx,
$$
where $\delta$ is the Dirac distribution and $d... | https://mathoverflow.net/users/78539 | Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$ | $\newcommand{\si}{\sigma}\newcommand{\Si}{\Sigma}\newcommand{\ep}{\varepsilon}\newcommand{\vpi}{\varphi}\newcommand{\R}{\mathbb R}$In this "Gaussian" setting especially, it is convenient to approximate the delta function by the normal distribution $N(0,\ep^2)$ with $\ep\downarrow0$, so that
\begin{equation\*}
R[f](... | 2 | https://mathoverflow.net/users/36721 | 418914 | 170,551 |
https://mathoverflow.net/questions/418905 | 2 | When comparing two sub-$\sigma$-algebras on a probability space $(\Omega,\Sigma,\pi)$, say $\mathcal{X}$ and $\mathcal{Y}$, say that $\mathcal{X}$ is strictly coarser than $\mathcal{Y}$ if the completion of $\mathcal{X}$ does not contain $\mathcal{Y}$. Here completion always refers to the restriction of $\pi$. Do there... | https://mathoverflow.net/users/479356 | Spaces with atomless independent $\sigma$-sub-algebras | The answer to the first question is yes. There is a class of probability spaces known under various names such as superatomless, saturated, nowhere countably-generated, $\aleph\_1$-atomless, and a couple of other names that have exactly this property. Note that the restriction to sub-$\sigma$-algebras admitting sets of... | 3 | https://mathoverflow.net/users/35357 | 418916 | 170,553 |
https://mathoverflow.net/questions/255246 | 5 | In [this](https://mathoverflow.net/questions/250926/characterization-of-exact-groups-via-the-existence-of-amenable-actions-on-unital) recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C\*-algebras. The answer, provided by Caleb Eckhardt in a comment,... | https://mathoverflow.net/users/97532 | Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2 | The answer to the question is positive: see Remark 6.6 in the paper
<https://arxiv.org/pdf/1904.06771.pdf>
The approximation property implies amenability in the sense of Claire Anantharaman Delaroche, this has been proved in
<https://arxiv.org/pdf/1907.03803.pdf>
for discrete groups, see also
<https://arxiv.o... | 3 | https://mathoverflow.net/users/75215 | 418918 | 170,554 |
https://mathoverflow.net/questions/384312 | 5 | $\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\newcommand{\Sp}{Sp}$
$\newcommand{\iHom}{\underline{\rm Hom}}$
(This is the follow-up of [this question](https://mathoverflow.net... | https://mathoverflow.net/users/144100 | Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$ | Consider the projective line $\mathbb{P}^1\_k$ over a field $k$. The structure morphism $f:\mathbb{P}^1\_k\rightarrow \mathrm{Spec}(k)$ induces a functor $f^\*:D(k) \rightarrow D\_{qc}(\mathbb{P}^1\_k)$ which is fully faithful (equivalently, $f\_\*(1) \simeq 1$) and which satisfies Grothendieck--Neeman duality. Since $... | 4 | https://mathoverflow.net/users/1148 | 418934 | 170,559 |
https://mathoverflow.net/questions/418658 | 9 | Consider the sequence defined by
\begin{align}
c\_0 &{}= 1 \\
c\_n &{}= 2\,n\,c\_{n-1}-\frac{1}{2}\sum\_{m=1}^{n-1}c\_m\,c\_{n-m}.
\end{align}
How can you prove that it has the following asymptotics (strongly suggested by numerics)
$$
c\_n\sim \frac{2}{\pi}\Gamma(n)\,2^n\ \ \ ?
$$
To make the question more motivated: t... | https://mathoverflow.net/users/174308 | Asymptotics of a quadratic recursion | **TL;DR:** I have a proof of your conjectured asymptotic formula, modulo the correctness of a certain alternative description of your $c\_n$ sequence.
---
I tried to complete Iosif Pinelis's elegant analysis by finding a way to derive the value of the constant $2/\pi$ (denoted $a$ in Iosif's answer) from the quad... | 7 | https://mathoverflow.net/users/78525 | 418935 | 170,560 |
https://mathoverflow.net/questions/418835 | 3 | Let $(X\_n)\_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X\_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin.
>
> Does there exist a (non-random) Borel-measurable function $h \colon \{0,1\}^{\mathbb{Z}\_{\geq 0}} \to \mathbb{Z}\_{\geq 0}$ such... | https://mathoverflow.net/users/15570 | Does a sequence of coin-tosses a.s. have a subsequence on which the remainder of the sequence can be identified with the position in the sequence? | Such a function $h$ does not exist.
Indeed, given a Borel-measurable function $h \colon \{0,1\}^{\mathbb{Z}\_{\geq 0}} \to \mathbb{Z}\_{\geq 0},$ define for each $k \ge 0$ the events
$$A\_k=\Bigl\{h \Bigl((X\_{n})\_{n \geq 0}\Bigr)=k \Bigr\}$$
and
$$B\_k=\Bigl\{h \Bigl((X\_{k+n})\_{n \geq 0}\Bigr)=k \Bigr\} \,.$$
Cle... | 2 | https://mathoverflow.net/users/7691 | 418938 | 170,561 |
https://mathoverflow.net/questions/418932 | 12 | My question is about Simpson's motivicity conjecture, that is the conjecture that for any (cohomogically) rigid irreducible connection $(M,\nabla)$ on a smooth complex scheme $X$ is of geometric origin in the sense that there exists $Y\overset{f}{\to}X$ such that $(M,\nabla)$ is a subquotient of $R^nf\_\*\mathcal{O}\_Y... | https://mathoverflow.net/users/152554 | Simpson's motivicity conjecture | At the time Simpson formulated his conjecture, he had proved that rigid local systems correspond to rational (in a suitable sense) variations of Hodge structure. And, as you point out, we now know that they are integral (Esnault-Groechenig). So rigid local systems have many of the earmarks of geometric local systems.
... | 11 | https://mathoverflow.net/users/4144 | 418940 | 170,562 |
https://mathoverflow.net/questions/418941 | 1 | The paper spatial interaction and the statistical analysis of lattice systems by Besag (1974) presents an alternative proof for the Hammersley-Clifford theorem.
In order to prove the HM theorem, Besag says that that under the following assumptions
1. The number of possible values at each site/vertex is finite.
2. $... | https://mathoverflow.net/users/137267 | Hammersley-Clifford theorem | $\newcommand{\x}{\mathbf x}\newcommand{\tx}{\tilde{\mathbf x}}\newcommand{\0}{\mathbf0}\newcommand{\R}{\mathbb R}$You are citing Besag's paper ([formulas (3.1) and (3.3) there](http://links.jstor.org/sici?sici=0035-9246%281974%2936%3A2%3C192%3ASIATSA%3E2.0.CO%3B2-3)) incorrectly. What the paper actually has is this:
\b... | 2 | https://mathoverflow.net/users/36721 | 418945 | 170,563 |
https://mathoverflow.net/questions/418844 | 7 | Define "$\alpha$ starts a gap of order $n+1$ and length $\beta$" iff $\mathcal P^n(\omega)\cap (L\_{\alpha+\beta}\setminus L\_\alpha)=\emptyset\land\forall\gamma\in\alpha: L\_\alpha\setminus L\_\gamma\neq\emptyset$ where $\mathcal P^n$ is the powerset operation iterated $n$ times.
Define "$\alpha$ is in a gap of orde... | https://mathoverflow.net/users/120848 | Can countable ordinals start gaps of every order in the constructible universe? | I presume that by "starts a gap (of order $n$)" you mean "starts a gap of positive length (of order $n$)", since length $0$ is trivial.
Note that the answers given by Monroe Eskew and Fedor Pakhomov were written when there was a different definition of "starts a gap" than what is now there.
1. Yes (by $L\_\alpha\mo... | 7 | https://mathoverflow.net/users/160347 | 418951 | 170,564 |
https://mathoverflow.net/questions/418944 | 2 | Let $K$ be compact Hausdorff, let $U\subset K$ be open and dense, and let $x\in K\backslash U$. Can we find a disjoint collection $\{U\_i,~ i\in I\}$ of open subsets of $U$ and a collection $\{K\_i,~ i\in I\}$ of compact sets such that $K\_i\subset U\_i$, for every $i$, and $x\in \overline{\bigcup K\_i}$?
| https://mathoverflow.net/users/53155 | Can a point of a compact set be approximated by a disjoint union? | Yes. This is possible. Recall that a subset of a topological space $X$ of the form $f^{-1}[\{0\}]^{c}$ for some continuous function $f:X\rightarrow \mathbb{R}$ is known as a cozero set.
Let $\mathcal{V}$ be a maximal collection of disjoint cozero subsets of $K$ subject to the condition that $\bigcup\mathcal{V}\subset... | 3 | https://mathoverflow.net/users/22277 | 418952 | 170,565 |
https://mathoverflow.net/questions/318890 | 2 | The method of steepest descent provides an asymptotic approximation for integrals of the form:
$$I = \int\_C \exp(M f(z))\mathrm dz$$
for large positive $M$, where $f(z)$ is analytic in the region of interest, $C$ a contour and $f(z)$ goes to zero at the endpoints of the contour. The asymptotic approximation is:
... | https://mathoverflow.net/users/16615 | Steepest descent integration in several dimensions | You can look for **Multivariable Morse Lemma** to get an extension of Steepest Descent into multiple complex variables $z=[z\_1, z\_2, ...,z\_n]$. Higher dimensional asymptotics as $M\rightarrow\infty$ for this multiple integral with $f:\mathbb{C}^n\rightarrow\mathbb{C}$ and $C=C\_1\times C\_2\times ...\times C\_n$ a m... | 2 | https://mathoverflow.net/users/141375 | 418958 | 170,567 |
https://mathoverflow.net/questions/418947 | 2 | I asked this question on MSE some time ago but didn't get a response.
Everything can be assumed in $ \mathbb{C} $, or atleast in characteristic $ 0 $.
Consider degree $ d $ hypersurfaces in projective space $ \mathbb{P}^n $. These correspond, upto nonzero scalars, to homogeneous polynomials $ h $ of degree $ d $ in... | https://mathoverflow.net/users/152391 | Singular locus of the discriminant variety | (Details of what follows can be found in any exposition of dual varieties such as [Lamotke's paper](https://www.sciencedirect.com/science/article/pii/0040938381900136).)
Given a smooth projective variety $X\subset\mathbb{P}^M$ we can look at the subvariety $D(X)\subset\mathbb{P}^M\times\mathbb{P}^{M\*}$ which is the ... | 6 | https://mathoverflow.net/users/124862 | 418961 | 170,568 |
https://mathoverflow.net/questions/418899 | 3 | Let $G$ be a connected simple graph. For any spanning tree $T$ of $G$, let $l(T)$ be the number of leaves of the graph $T$. Consider $\ell=\min\_Tl(T)$, can I find a spanning tree $T$ with $l(T)=\ell$, such that the set of leaves $A$ of $T$ is very close to an independent set.
For example, I guess that there exists a... | https://mathoverflow.net/users/160959 | Property of the spanning tree with minimal leaves | If I am not mistaken, and if I understand you correctly, it seems to me that you are right.
The following statement is true.
>
> Let $G$ be a connected graph and $T$ be the spanning tree with the
> smallest number of leaves and $\ell=l(T)$. Let $A$ be the set of all
> leaves of tree $T$. If $|A|=\ell>2$, then $A$... | 3 | https://mathoverflow.net/users/173068 | 418968 | 170,569 |
https://mathoverflow.net/questions/418799 | 1 | The New York Times, [reporting on Dennis Sullivan's Abel prize](https://www.nytimes.com/2022/03/23/science/abel-prize-mathematics.html), recounts the incident that lured Sullivan from chemical engineering to mathematics:
>
> One day during an advanced calculus lecture, the professor drew two shapes on the blackboa... | https://mathoverflow.net/users/10503 | Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure |
>
> In the first quoted paragraph, what is the meaning of "fit on"?
>
>
>
There is a [homeomorphism](https://en.wikipedia.org/wiki/Homeomorphism) between any pair of (open, Riemannian) disks.
>
> In the second quoted paragraph, what is the meaning of the professor's statement?
>
>
>
There is an (essenti... | 3 | https://mathoverflow.net/users/1650 | 418973 | 170,571 |
https://mathoverflow.net/questions/418977 | 5 | I've copied over this question from [what I asked on StackExchange](https://math.stackexchange.com/questions/4412970/if-k-rtimes-mathbbz-is-a-finitely-generated-group-but-k-isnt-must-the-f), in the hope that an expert here can readily answer the question.
Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfy... | https://mathoverflow.net/users/105730 | If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group? | No. Fix $p\ge 2$. Take the group
$$G=\{M(x,y,z;n):(x,y,z)\in\mathbf{Z}[1/p],n\in\mathbf{Z}\}$$where $$M(x,y,z;n)=\begin{pmatrix}1 & x & z \\ 0 & p^n & y\\ 0 & 0 & 1\end{pmatrix}$$
and $K$ the set of such $M(x,y,z;n)$ for $n=0$, and identify $\mathbf{Z}$ to powers of $M(0,0,0,1)$.
Then $G$ is finitely generated (nam... | 11 | https://mathoverflow.net/users/14094 | 418978 | 170,572 |
https://mathoverflow.net/questions/418975 | 1 | A similar post on [MSE](https://math.stackexchange.com/questions/4407255/the-nearness-of-transversal-pre-images-of-slices-of-a-trivial-tubular-neighborho) without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold o... | https://mathoverflow.net/users/363264 | Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood | The condition $\mathcal P(\epsilon)$ is needlessly complicated. It's always true without passing to components.
Let $U$ be a tubular neighborhood of $S$ diffeo to $[-1,1]\times S$. Let $\pi: U\to \mathbb R$ be the projection onto the first factor. Let $V=f^{-1}(U)$. Then $V$ is compact and the transversality assumpti... | 3 | https://mathoverflow.net/users/18050 | 418992 | 170,578 |
https://mathoverflow.net/questions/418962 | 3 | Pierre-Gilles Lemarie-Rieusset, *The Navier-Stokes Problem in the 21st Century* treats the heat equation on $\mathbb{R}^3$ for time $t\geq 0$, and proves uniqueness of suitably smooth solutions by a kind of energy argument. For a solution $u(t, x)$ with $u(0,x)=0$ he looks at the integral over all $x\in \mathbb{R}^3$ ... | https://mathoverflow.net/users/38783 | Unique solutions to the heat equation on $\mathbb{R}^3$ | Sorry, maybe my previous comment was not clear enough. You have $$\frac{d}{dt} \int |u|^2 e^{-|x|}\, dx=-2\int |\nabla u|^2 e^{-|x|}\, dx +2 \int u \nabla u \cdot \frac{x}{|x|}e^{-|x|}\, dx.$$
Now use $2|u \nabla u| \leq \frac 12 |u|^2+2|\nabla u|^2$
to estimate the last integral.
| 8 | https://mathoverflow.net/users/150653 | 419003 | 170,581 |
https://mathoverflow.net/questions/418996 | 4 | There are $n$ men, standing one at each vertex of a convex $n$-gon. If they are allowed to move together along sides or diagonals of the polygon to reach another vertex, how many different ways are there to do so without meeting another one?
See OEIS [A350599](http://oeis.org/search?q=A350599) for the first few numer... | https://mathoverflow.net/users/479476 | Combinatorics related plane geometry | Assume that the paths may not cross and each man must move.
Label the vertices $1,2,\dots,n$ in clockwise order. Let the man at
vertex $i$ move to vertex $\pi(i)$, so $\pi$ is a permutation of
$1,2,\dots,n$. If we draw an arrow from vertex $i$ to $\pi(i)$, then
we get a disjoint union of noncrossing cycles of length $\... | 6 | https://mathoverflow.net/users/2807 | 419016 | 170,583 |
https://mathoverflow.net/questions/418972 | 7 | $\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}\_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices of size 2. Recall that a matrix $(a\_{ij})$ with octonionic entries is called Hermitian if $a\_{ji}=\bar a\_{ij}$.
O... | https://mathoverflow.net/users/16183 | Quadratic forms on $\mathbb{R}^{16}$ coming from octonions | I'm revising my answer because whether the image of the map $j$ that the OP defines is equal to the $10$-dimensional subspace $H$ of $\mathrm{Sym}^2(\mathbb{R}^{16})$ that is invariant under $\mathrm{Spin}(9,1)$ depends on which particular action of $\mathrm{Spin}(9,1)$ on $\mathbb{R}^{16}$ one chooses.
First, indepe... | 6 | https://mathoverflow.net/users/13972 | 419038 | 170,589 |
https://mathoverflow.net/questions/418662 | 0 | Consider roots $f = 0$ of a nicely-behaved real function $f(x, t)$ of two (real) variables.
Namely, points $(x, t)$ on which $f$ vanishes, $f(x, t) = 0$.
Suppose that $x$ can be written as function of time $t$, $x = x(t)$. By the multivariate chain rule,
$$
\partial\_x f \cdot x'(t) + \partial\_t f = 0 \;.
$$
This yiel... | https://mathoverflow.net/users/17083 | Calculating derivatives of arbitrary-order at an operator's root | Although this question sounds quite innocent, a systematic treatment of higher order derivatives of implicit functions is quite involved. On a second thought, this is no surprise if you think about how complicated higher derivatives of concatenations $f\circ g$ get (see [Faà di Bruno's formula](https://en.wikipedia.org... | 2 | https://mathoverflow.net/users/9652 | 419039 | 170,590 |
https://mathoverflow.net/questions/419031 | 7 | Suppose $n$ is a positive integer greater than 2, and $F$ is an arbitrary field with at least 4 elements.
Denote $\text{GL}\_n(F)$ the general linear group in the usual sense and $U\_n(F)$ the unipotent group in the sense that it consists of $n\times n$ upper-triangular matrices with 1's on the diagonal.
The inclus... | https://mathoverflow.net/users/471160 | How can I detect the homology image of a unipotent group in the general linear group? | Suppose first that $F$ is a finite field of characteristic $p$. Then $U\_n(F)$ is a Sylow $p$-subgroup of $GL\_n(F)$, and so using the transfer in group homology one sees that the image of $f\_k$ (for $k>0$) is precisely the $p$-torsion in $H\_k(GL\_n(F);\mathbb{Z})$. Unstably, this is not well understood, but it canno... | 10 | https://mathoverflow.net/users/318 | 419040 | 170,591 |
https://mathoverflow.net/questions/419030 | 4 | Let $κ$ be an infinite cardinal, $S$ a set of cardinality $κ$, and let
$I = [0, 1]$ be the closed unit interval. Define an equivalence
relation $E$ on $I × S$ by $(x,α) E (y,β)$ if either $x = 0 = y$
or $(x,α) = (y,β)$. Let $H(κ)$ be the set of all equivalence
classes of $E$; in other words, $H(κ)$ is the quotient set ... | https://mathoverflow.net/users/112417 | Hedgehog of spininess $κ$ is an absolute retract? | The answer is positive because the hedgehog $J(κ)$ is $AE$.
| 1 | https://mathoverflow.net/users/112417 | 419049 | 170,595 |
https://mathoverflow.net/questions/419052 | 12 | For which algebraic numbers $\alpha$ is there a valuation on the number field ${\mathbb {Q}}(\alpha)$ for which the infinite series $\sum\_{n=0}^\infty \alpha^n$ converges to $1/(1-\alpha)$?
| https://mathoverflow.net/users/3621 | Geometric series in algebraic number fields | This holds precisely for elements which aren't roots of unity. Indeed, by the product formula we have $\prod\_v|\alpha|\_v=1$, where the product runs over all (finite and infinite) primes. This shows that either all $|\alpha|\_v$ are equal to $1$, or at least one of them is smaller than $1$. If all $|\alpha|\_v$ are eq... | 18 | https://mathoverflow.net/users/30186 | 419055 | 170,597 |
https://mathoverflow.net/questions/419064 | 8 | **Edit:** In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only finitely dominated spaces which are homotopy equivalent to compact ENRs are those finitely dominated spaces whose Wall fi... | https://mathoverflow.net/users/8032 | Finite domination and compact ENRs | It was originally conjectured by Borsuk that every compact ANR should be homotopy equivalent to a finite CW complex. While it was known that every separable ANR has the homotopy type of a countable CW complex, the finer statement was an open problem for some time in the '60s and '70s.
It was J. West
>
> *[Mapping... | 11 | https://mathoverflow.net/users/54788 | 419065 | 170,598 |
https://mathoverflow.net/questions/419088 | 2 | Can such a set $A=$ {$a\_1,.. a\_k$} exist, such that:
1. $\sum\_i a\_i = 1$ and $a\_i $ are rational positive numbers
2. $k$ is and odd number, and is at least $3$.
3. We can partition $A$ in two parts of value $ \frac{1}{2}$ each
4. $\forall a\_j \in A$, let $B\_j := A - a\_j$. We can partition $B\_j$ into two grou... | https://mathoverflow.net/users/342793 | Odd partition with extra properties | Multiply by the least common multiple $M$ of the denominators to get the equivalent problem:
1. The $a\_i$ are positive integers.
2. $k$ is an odd number.
3. We can partition $A$ into two parts of equal sum.
4. If we remove any element of $A$, we can partition the remaining elements into two parts of equal sum.
In ... | 7 | https://mathoverflow.net/users/46140 | 419099 | 170,608 |
https://mathoverflow.net/questions/419097 | 5 | Let $k=\mathbb{F}\_q$ be a finite field with $q$ elements and let $X$ be a quasi-projective $k$-scheme. I saw somewhere claims the following results (without explanation):
1. Let $N$ be a positive integer and let $i: Z\hookrightarrow X$ be the closed immersion of the (finite) disjoint union of ${\rm Spec}(\kappa(x))$... | https://mathoverflow.net/users/42571 | About closed points in symmetric product schemes over a finite field | There are probably many ways to answer your question depending on what your preferred point of view on schemes and symmetric products are. Let me offer the following approach.
Forget that $k$ is a finite field, let it just be a perfect field, call $\newcommand{\alg}{\operatorname{alg}}k^{\alg}$ a fixed chosen algebra... | 5 | https://mathoverflow.net/users/17064 | 419104 | 170,610 |
https://mathoverflow.net/questions/410625 | 10 | Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$. Let $\ell\_{ij}$ denote the length of the edge of $P$ corresponding to $ij\in E(G)$.
>
> **Question.** Let $p\_1,\... | https://mathoverflow.net/users/108884 | Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger? | The answer to the question is unfortunately no, in the case of $n=m=3$. There is a simple example to illustrate this.
Let $P$ be the cube with vertices $(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$. Very obviously, every edge is of length $1$, and a quick calculation shows this is inscribed upon a sphere of rad... | 3 | https://mathoverflow.net/users/445044 | 419107 | 170,611 |
https://mathoverflow.net/questions/419059 | 4 | The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing):
>
> Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \varnothing$. An *$\eta$-submanifold* $(M,f)\subset V$ of a manifold $V$ is a pair $(M,f)$ where $M\subset V$ is a compact ... | https://mathoverflow.net/users/89166 | A cobordism theory from Hirsch's "Differential Topology" (reference request) | As noted in the comments this is in Tom Diecks book (section 21.2), and in Wall's differential topology (Section 8.1).
| 4 | https://mathoverflow.net/users/12156 | 419108 | 170,612 |
https://mathoverflow.net/questions/419106 | 1 | Let $G$ be a $d$-generated group. Then my first question is how to see free reduced group $FV(G)$ in the variety containing $G$. What I understood is: "Let $W \subset F\_d$ ($d$-generated free group) be the collection of words which are laws for $G$, i.e. for every $w \in W$ image of word map $w$ on $G$ is identity. Le... | https://mathoverflow.net/users/117282 | Reduced free group | I'll give a more detailed answer. My comment above about your first question was wrong because I read it too quickly and didn't catch the difference between $m$ and $d$. Your description of the free object in question 1 is off.
Take the cyclic group $C\_n$ of order $n\geq 3$. Then $C\_n$ is $1$-generated and the laws... | 5 | https://mathoverflow.net/users/15934 | 419110 | 170,613 |
https://mathoverflow.net/questions/418900 | 8 | I am curious about the reverse math status of the below statement. Note that we work in second-order RM, i.e. 'closed set' is interpreted as in Simpson's excellent SOSOA.
*For any closed $E\subset [0,1]$ and $\epsilon >0$, there are $x\_0, \dots, x\_k \in E$ such that $\cup\_{i\leq k} B(x\_i, \epsilon)$ covers $E$.*
... | https://mathoverflow.net/users/33505 | Reverse Mathematics strength of fixed radius covering theorem | The statement in provable in $\mathrm{WKL}\_0$.
Consider the following proof. Fix $k>1/\epsilon$, let $I=\{i<k:E\cap[i/k,(i+1)/k]\ne\varnothing\}$, and let $\{x\_i:i\in I\}$ be such that $x\_i\in E\cap[i/k,(i+1)/k]$. Then for any $x\in E$, we have $x\in[i/k,(i+1)/k]$ for some $i\in I$, hence $|x-x\_i|\le1/k<\epsilon$... | 11 | https://mathoverflow.net/users/12705 | 419111 | 170,614 |
https://mathoverflow.net/questions/418451 | 4 | Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in [Agore - Categorical Constructions for Hopf Algebras](https://arxiv.org/abs/0905.2613v3). Maybe the question should be: what does the spectral sequence that compute... | https://mathoverflow.net/users/143549 | Mayer–Vietoris sequence for coproduct of Hopf algebras | In the case of discrete groups, the MV sequence follows from the following argument: if you have $G=H\*K$ then you have a short exact sequence of the form:
$$0\to \mathbb{Z}G\stackrel{i}{\to} \mathbb{Z}G/H\oplus \mathbb{Z}G/K\stackrel{p}{\to} \mathbb{Z}\to 0,$$ where $i$ sends $g$ to $(gH,gK)$ and $p$ sends $gH$ and $g... | 2 | https://mathoverflow.net/users/41644 | 419122 | 170,619 |
https://mathoverflow.net/questions/419117 | 5 | I am interested in the asymptotics of the integral
$$I(a):=\int\_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$
for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\lesssim e^{-a^2}$ for large $a$. Numeric integration indicates that
$$ \lim\_{a\to\infty} I(a) e^{a^2} a^{5/2} =C \in(0,1)$$
... | https://mathoverflow.net/users/89934 | Asymptotics of error function integral with square root | $$\int\_0^\infty \sqrt{x}\,\mathrm{Erfc}(x+a)\,\mathrm{d}x\rightarrow \frac{e^{-a^2}}{(2a)^{5/2}},\;\;\text{for}\;\;a\rightarrow \infty,$$
so $C=2^{-5/2}.$ Corrections are smaller by a factor $1/a$.
I obtained this asymptotics by integrating the large-$a$ expansion of the error-function,
$$\text{Erfc}\,(x+a)\rightar... | 7 | https://mathoverflow.net/users/11260 | 419123 | 170,620 |
https://mathoverflow.net/questions/419127 | 2 | The question:
(1) Is every compact Lie group $G$ isomorphic (as a topological group) to some quotient $H/N$ where $H$ is a torsion-free compact metrizable group?
Or equivalently:
(2) Is every compact metrizable group $G$ isomorphic (as a topological group) to some quotient $H/N$ where $H$ is a torsion-free compac... | https://mathoverflow.net/users/479121 | Compact Lie groups as quotients of torsion-free compact metrizable groups | The answer to all questions is "no" (and even without the metrizability requirements).
a) A negative answer to (3) implies a negative answer to (1).
Indeed, if $f:G\to H$ is a surjective continuous homomorphism between compact groups, then $G^\circ\to H^\circ$ is surjective as well. So a non-abelian connected compa... | 3 | https://mathoverflow.net/users/14094 | 419130 | 170,621 |
https://mathoverflow.net/questions/419094 | 3 | This question just came to my mind and I have no idea as to how to approach it. Let $z\_1,z\_2,\dots,z\_n$ be $n$ be any complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the unit disc wiith real values
$$ f(z)=\sum\_{i=1}^n \frac{|z-z\_i|}{n}.
$$
My questions:
* Does there exist a $z \in ... | https://mathoverflow.net/users/158175 | A question about average deviation of given $n$ complex numbers | The answer is no. E.g., let $n=3$ and $z\_j=e^{i(j-1)2\pi/3}$ for $j=1,2,3$. Then $f(z)>|z|+15/100>|z|$ if $|z|\le1$.
---
This counterexample generalizes to any $n\ge3$. Indeed, take any $n\ge3$ and let $z\_j=e^{i(j-1)2\pi/n}$ for $j=1,\dots,n$. Then
$$f(z)=\frac1n\,\sum\_{j=1}^n f\_j(z),$$
where $f\_j(z):=|z-z\_... | 1 | https://mathoverflow.net/users/36721 | 419132 | 170,622 |
https://mathoverflow.net/questions/419075 | 5 | I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
**Edited**
The spaces I mentioned above are defined in section two of this paper:
<https://arxiv.org/pdf/0708.2011.pdf>.
| https://mathoverflow.net/users/471464 | Looking for references to study $U^p$ and $V^p$ spaces | You can take a look at Herbert Koch's contribution in
*Koch, Herbert; Tataru, Daniel; Vişan, Monica*, [**Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps**](http://dx.doi.org/10.1007/978-3-0348-0736-4), Oberwolfach Seminars 45. Basel: Birkhäuser... | 8 | https://mathoverflow.net/users/3948 | 419134 | 170,623 |
https://mathoverflow.net/questions/418295 | 3 | Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper <https://arxiv.org/abs/1501.02215> it is stated that, for every flat $O(X)^{\circ}$-module $Q$, the functor
$$A\mapsto \widehat{A\otimes\_{O(X)... | https://mathoverflow.net/users/476832 | On the exactness of some completed tensor products | This proposition has been removed from the paper in the published paper, available in the author's webpage.
| 1 | https://mathoverflow.net/users/476832 | 419135 | 170,624 |
https://mathoverflow.net/questions/419131 | 10 | A [**geometric theory**](https://ncatlab.org/nlab/show/geometric%20theory) is made up of sequents of restricted form: It may only be of the form $$\phi \vdash \psi$$
possibly with free variables (which are implicitly taken universal closure). $\phi, \psi$ are *geometric formulae* which can only involve $\top, \bot, =, ... | https://mathoverflow.net/users/136535 | Deductive system involving only geometric sequents | For geometric theories in a countable fragment, (axiomatized by countably many sequents where the disjunctions are also countable), there exists a sound and complete system that appears already in Makkai and Reyes "First-order categorical logic". In page 159, it is described in the section "Completeness of a one-sided ... | 12 | https://mathoverflow.net/users/12976 | 419137 | 170,626 |
https://mathoverflow.net/questions/419133 | 1 | Fix a half-space $H = \{x\_1 \geq 0: ~ (x\_1,\dots,x\_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) = \Pr\_{\mathbf{x} \sim p}\left[\mathbf{x} \in H\right]$ using random Voronoi partitions.
Take $m$ i.i.d. samples ... | https://mathoverflow.net/users/158769 | Approximating the probability of a half-space using random Voronoi diagrams | No. E.g., suppose that $n=1$, $m=2$, and the pdf $f$ of each of the $m=2$ iid sample points $X\_i:=\mathbf c^{(i)}$ ($i=1,\dots,m$) is given by the formula $f(x)=e^{-x-1}1(x>-1)$ for real $x$.
Then, by straightforward calculations,
$$p(H)=e^{-1}= 0.367\ldots\ne0.506\ldots=\frac{-2+6 e^2-8 e^3+6 e^4+e^6}{3 e^6}=E\hat p\... | 2 | https://mathoverflow.net/users/36721 | 419143 | 170,627 |
https://mathoverflow.net/questions/419074 | 4 | Let $n<d$ be positive integers, and let $0< \epsilon\leq 1-\frac{n}{d}$ be a real number. I would like a succinct description of the following quantity:
$$
f\_{\epsilon}(n,d):=\min\_{\substack{x\_1\geq \dots \geq x\_d\geq 0\\ x\_1+\dots+x\_n =1-\epsilon\\ x\_{n+1}+\dots+x\_d=\epsilon}} \left(\sum\_{1 \leq i\_1 < \dot... | https://mathoverflow.net/users/150898 | Bound on sum of products of positive real numbers with prescribed subset sums | Here is a succinct description of $f\_{\epsilon}(n,d)$ (I only have time to sketch the argument-- hopefully it is right).
Let
$$
h(x\_1,\dots, x\_d)=\left(\sum\_{1 \leq i\_1 < \dots < i\_{n+1} \leq d} x\_{i\_1}\cdots x\_{i\_{n+1}}\right),
$$
and let $x \in \mathbb{R}^d$ be a feasible element of the minimization (i.e.... | 3 | https://mathoverflow.net/users/150898 | 419153 | 170,631 |
https://mathoverflow.net/questions/418576 | 9 | For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is *homogeneous with respect to $f$* if the restriction $f|\_{[H]^\mu}: [H]^\mu \to \kap... | https://mathoverflow.net/users/8628 | Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$? | The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega\_2$$ does not imply the axiom of choice.
This was shown by Kleinberg and Seiferas in 1973, see
>
> [MR0340025](https://mathscinet.ams.org/mathscinet-getitem?mr=340025) (49 #4782)
> Kleinberg, E. M.; Seiferas, J. I.
> Infinite... | 11 | https://mathoverflow.net/users/6085 | 419160 | 170,634 |
https://mathoverflow.net/questions/419162 | 7 | Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then ϕ induces a homeomorphism between their boundaries.
The proof of the above statement is well-written in Bridson and Haefliger's book.
My question is that `can we drop the condit... | https://mathoverflow.net/users/173504 | Induced homeomorphism from a quasi-isometry between hyperbolic spaces | Properness is already needed to have a well-defined boundary at infinity, i.e., with a topology not depending on the chosen base point. This is Proposition III.3.7 in Bridson-Haefliger, which builds on some previous lemmata that are applications of the Arzela-Ascoli theorem. To apply the Arzela-Ascoli theorem one needs... | 8 | https://mathoverflow.net/users/39082 | 419166 | 170,635 |
https://mathoverflow.net/questions/419090 | 1 | Let $X,Y$ be metric space and suppose that $f:X\rightarrow Y$ is a *uniform embedding*; i.e.:
$$
\omega(d\_X(x,z))\leq d\_Y(f(x),f(z)) \leq \Omega(d\_X(x,z)),
$$
where $\omega\leq \Omega$ are both strictly increasing continuous functions mapping $[0,\infty)$ to itself and which fix $0$.
Is $f$ a quasisymmetry? I.e.: ... | https://mathoverflow.net/users/469470 | When are uniform embeddings quasisymetric | This is false. For example take $f: [0,\infty)\to [0,\infty)$ given by
$$
f(x)=\begin{cases}0& \text{ if } x=0\\
e^{-1/x}& \text{ if } x>0
\end{cases}$$
Then $\lim\_{x\to 0}\frac{f(2x)}{f(x)}=\infty$ so $f$ is not quasisymmetric. But $f$ is continuous and monotone hence gives a homeomorphism onto the image when restr... | 1 | https://mathoverflow.net/users/18050 | 419190 | 170,641 |
https://mathoverflow.net/questions/419158 | 4 | I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a\_1, ..., a\_n \} \subset D$ be the set of branch points of $f$ and the ramification degree of $a\_j$ is $m\_j \in [2, \infty) \cap \mathbb{... | https://mathoverflow.net/users/143311 | Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees | This (existence and uniqueness) is proved in the paper in much more general setting (in fact, the result is due to E. Picard):
M. Heins, ‘On a class of conformal metrics’, Nagoya Math. J. 21 (1962) 1–50.
For a more recent and simpler proof of the special case that you ask, see
MR1479037 (99d:30009)
Zakeri, Saeed
... | 7 | https://mathoverflow.net/users/25510 | 419194 | 170,642 |
https://mathoverflow.net/questions/419195 | 2 | Consider two dg-algebras $A,B$ and their respective derived categories $D(A),D(B)$. A natural way to give a covariant functor is to take an $(A,B)$-bimodule $X$ and to tensor with it, that is
$$D(A)\to D(B), \quad M\mapsto M\otimes^{\mathbb{L}}\_A X.$$
However it is "well known" (according to Toen) that there exist t... | https://mathoverflow.net/users/152554 | "Approximating" functors by Hom/Tensor product | I'm not sure how to phrase it in this language, so this answer is more of a comment (but too long for comments)
Note that it is related to your question via the slogan "dg-categories are the same as $\mathbb Z$-linear stable $\infty$-categories".
Taking this slogan for granted, I will answer your question in the se... | 3 | https://mathoverflow.net/users/102343 | 419198 | 170,644 |
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