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https://mathoverflow.net/questions/449188 | 1 | I come across an interesting question.
>
> Let $ B\_r=\{x\in\mathbb{R}^3:|x|\leq r\} $ be the ball in $ \mathbb{R}^3 $ with radius $ r $. Assume that $ u \in C(\mathbb{R}^3\setminus B\_1) $ satisfies
> $$
> \Delta u\leq -u^3,\,\,u\geq 0,\,\,\forall |x|\geq 1.
> $$
>
> where $ \Delta u $ is defined in the sense o... | https://mathoverflow.net/users/241460 | How to show that $ u $ is vanishing in $ \mathbb{R}^3\setminus B_1 $? | A proof that $u=0$ is as follows. Assume that such $u \geq 0 $ is not identically zero. Since $\Delta u \leq 0$, $u$ can never vanish, otherwise it would have an interior minimum. Taking averages on the unit sphere of $\mathbb R^3$ we may assume also that $u$ is radial (the inequality in the differential equation is pr... | 2 | https://mathoverflow.net/users/150653 | 449212 | 180,808 |
https://mathoverflow.net/questions/449206 | 2 | Let $\{V\_n\}\_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V\_n \subset V\_{n+1}$ and $\bigcup\_{n=1}^\infty V\_n$ is dense in $L^2[0,1]$. Suppose further that each $V\_n$ consists of smooth functions.
Consider a collection of invertible linear mappings $\{T\_n\}\_{n=... | https://mathoverflow.net/users/56524 | LF or LB space that happens to be finite dimensional | The expression
\begin{equation}
\tilde W\_m:=\bigcup\_{n=1}^\infty T\_n(V\_m)
\end{equation}
is undefined in general for $m\ge2$, because $T\_n$ is defined (and is invertible) only on $V\_n$ and hence $T\_n(V\_m)$ may be undefined for $n<m$.
So, instead of $\tilde W\_m$, one may want to consider
\begin{equation}
W\... | 4 | https://mathoverflow.net/users/36721 | 449214 | 180,809 |
https://mathoverflow.net/questions/449213 | 1 | I am writing up some notes and the following occurred to me and I would like to see if there are **a variety of ways to prove it**. Just for reference, the identity pops out of equality between constant term evaluations of [Laurent polynomials from my earlier quest](https://mathoverflow.net/questions/449007/have-you-se... | https://mathoverflow.net/users/66131 | Product/quotient of factorials beget dyadic powers | This is easy. Start with $$n!.2^n=(2n)(2n-2)\dots 2.$$ This implies that $$((2n-1)!(2n-3)!\dots 1!).n!.2^n=(2n)!(2n-2)!\dots 2!.$$ Then $$((2n)!(2n-1)!\dots 1!).n!.2^n=((2n)!(2n-2)!\dots 2!)^2.$$ Thus $$\left(\prod\_{j=0}^n (n+j)!j!\right).2^n=\prod\_{j=0}^n (2j)!^2. $$ Finally, divide both sides by the product on the ... | 3 | https://mathoverflow.net/users/460592 | 449215 | 180,810 |
https://mathoverflow.net/questions/449217 | 2 | Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say anything about the scheme-theoretic image of $g$?
For example, let $Z$ be the scheme theoretic image of $g$, with $i :... | https://mathoverflow.net/users/nan | Images of smooth schemes under lci morphisms | It's very hard to say anything. In fact, if $S = \operatorname{Spec} \mathbf C$, then *any* (integral) projective $k$-variety $Z$ arises in this way. Indeed, since $Z$ is projective, there exists a closed immersion $i \colon Z \hookrightarrow \mathbf P^n$ for some $n$, so we may take $Y = \mathbf P^n$. Then we may choo... | 6 | https://mathoverflow.net/users/82179 | 449223 | 180,814 |
https://mathoverflow.net/questions/448915 | 4 | *Disclaimer*: This question [was initially](https://math.stackexchange.com/q/4718589) asked yesterday in Mathematics Stack Exchange but left unanswered there.
---
I am interested in learning about differential graph theory or differential operators on graphs, something related to what [E. Bautista](https://mathov... | https://mathoverflow.net/users/106458 | Reference request for differential graph theory | Well, here's a recent monograph on different flavours of graph Laplacians by Kostenko and Nicolussi: [https://www.mat.univie.ac.at/~kostenko/list/GraphLaplInf.pdf](https://www.mat.univie.ac.at/%7Ekostenko/list/GraphLaplInf.pdf)
But I'd say that the topic is too vast to be sufficiently covered in any single coherent b... | 1 | https://mathoverflow.net/users/81055 | 449232 | 180,815 |
https://mathoverflow.net/questions/449234 | 5 | Let $f: \mathbb R \to \mathbb R$ be a locally integrable measurable function.
We say $f$ satisfies the *intermediate value property* if given any $a, b\in \mathbb R$ with $a < b$, whenever $u \in \mathbb R$ is such that $\min(f(a), f(b)) \leq u \leq \max(f(a), f(b))$, there exists some $x \in [a, b]$ such that $f(x) ... | https://mathoverflow.net/users/173490 | If every point is a Lebesgue point of $f$, does $f$ satisfy the intermediate value property? | If $F(x)=\int\_a^x f(x)dx$ (for some fixed $a$), then $x$ being a Lebesgue point of $f$ yields $F'(x)=f(x)$; and the derivatives enjoy the intermediate value property by Darboux theorem.
| 6 | https://mathoverflow.net/users/4312 | 449239 | 180,817 |
https://mathoverflow.net/questions/449238 | 3 | Let $P(n)$ denote the largest prime factor of $n$. I wanted to know if an analogue of Baker-Harman estimate for smooth shifted primes in arithmetic progressions with fixed moduli is there in literature.
Baker-Harman proved that there exist infinitely many primes $p$ such that $P(p-1)<p^{0.2961}$ and the exponent was ... | https://mathoverflow.net/users/160943 | Infinitude of smooth shifted primes in arithmetic progression with fixed moduli | The only relevant result I found is [a paper](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-52/issue-3/Smooth-values-of-shifted-primes-in-arithmetic-progressions/10.1307/mmj/1100623415.full) by Banks, Harcharras, and Shparlinski. They did not establish existence of such smooth numbers but inst... | 1 | https://mathoverflow.net/users/449628 | 449262 | 180,822 |
https://mathoverflow.net/questions/449277 | 2 | Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
>
> Is the fiber product scheme $X\_T:=X\times\_YT$ Cohen-Macaulay?
>
>
>
What conditions on $f$ would ensure that $X\_T$ is Cohen-Macaulay?
... | https://mathoverflow.net/users/501361 | Cohen-Macaulay fiber products | Ok, $T \hookrightarrow Y$ is locally a complete intersection, right? (Closed regular subscheme of a regular scheme). So you'd want that regular sequence (locally) to become a regular sequence on $X$. That's not always true, but if $f : X \to Y$ is *flat* it is ok (since a weakly regular sequence will stay weakly regula... | 3 | https://mathoverflow.net/users/3521 | 449279 | 180,827 |
https://mathoverflow.net/questions/449265 | 5 | Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}\_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and consider the map $f:\mathbb{R}^n\times X \rightarrow \mathcal{P}\_2(\mathbb{R}^n)$ given by
$$
f(a,B) = N(a,B)
$$
where $N... | https://mathoverflow.net/users/496781 | Local Lipschitzness of parameterization of Gaussians in Wasserstein space | $\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$The answer is yes.
Indeed, it is easy to see (cf. e.g. [Proposition 7](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-31/issue-2/A-class-of-Wasserstein-metrics-for-probability-distributions/10.1307/mmj/1029003026.full) or the begi... | 5 | https://mathoverflow.net/users/36721 | 449287 | 180,828 |
https://mathoverflow.net/questions/449285 | 3 | I am wondering if there is an analog of the following theorem by Morgan and White:
Suppose $g\_E$ is the Euclidean metric on $\mathbf R^3$. If $\gamma$ is a closed $C^{k,\alpha}$ curve in $(\mathbf R^3,g\_E)$ that bounds a unique area minimizing surface, then every closed curve sufficiently closed to $\gamma$ in $C^{k,... | https://mathoverflow.net/users/175594 | Minimal surface on $R^3$ with with non Euclidean metric | Below I sketch the proof of the following theorem:
>
> Theorem: Suppose that $\Sigma^n\subset (M^{n+1},g\_0)$ is smooth and uniquely area-minimizing relative to it's boundary $\Gamma : = \partial\Sigma$ and is strictly stable (no nontrivial Jacobi fields with Dirichlet boundary conditions). Then if $g$ is a suffici... | 4 | https://mathoverflow.net/users/1540 | 449291 | 180,829 |
https://mathoverflow.net/questions/449302 | 2 | I'm looking for an example of a function $u \in H\_2$ such that $u \notin H\_\infty$, where $H\_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a complete definition.
Let $\mathbb{C}^+ := \{s\in \mathbb{C} \mid \text{Re}(s) > 0 \}$ and $\bar{\mathbb{C}}^+ := \{s\in ... | https://mathoverflow.net/users/7667 | Hardy space inclusion in the right-half plane | Here is a suggestion (but I have not worked through the details). What follows is given for the upper-half plane (UHP for short) but of course a trivial rotation will convert the example to one on the right-half plane as you requested.
There is a weighted composition operator giving an isometry of Hilbert spaces from... | 3 | https://mathoverflow.net/users/763 | 449303 | 180,834 |
https://mathoverflow.net/questions/449308 | 3 | Let $p$ be a prime and let $w$ be a primitive $p$-th root of unity in $\mathbb{C}$. There is an element $e$ of order $p$ in $G=\operatorname{PGL}\_n(\mathbb{C})$ where $n=pk$ and
$$e=\left[\left(\begin{matrix}I\_k&&&&\\&{wI}\_k&&&\\&&{w^2I}\_k&&\\&&&\ddots&\\&&&&{w^{p-1}I}\_k\\\end{matrix}\right)\right].$$
My focus is ... | https://mathoverflow.net/users/488802 | normalizer quotient is $\operatorname{GL}_2(p)$ | Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL\_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order... | 3 | https://mathoverflow.net/users/460592 | 449310 | 180,836 |
https://mathoverflow.net/questions/393787 | 7 | From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact convex domain) in the $3$-space form of constant curvature $\kappa$.
For $\kappa=0$, the surface is unique up to isometry... | https://mathoverflow.net/users/111820 | Rigidity for convex surfaces in elliptic/hyperbolic space | Yes, closed convex surfaces in spaces of constant curvature are rigid. See Chapter V of Pogorelov's book [Extrinsic Geometry of Convex Surfaces](https://www.ams.org/books/mmono/035/mmono035-endmatter.pdf).
As described at the beginning of the chapter on p. 270-271, the main idea is to reduce the problem to the Euclid... | 3 | https://mathoverflow.net/users/68969 | 449321 | 180,839 |
https://mathoverflow.net/questions/449329 | 8 | Let me preface this by saying that I have next to no background in representation theory. I come from geometry but the following representations showed up naturally in my work.
We let $ V = C^\infty \left( \mathbb{R}^2 \setminus \left\{ 0 \right\} \right) $ be equipped with its standard Frechet space topology. Then, ... | https://mathoverflow.net/users/507338 | Is the GL(2,R)-representation of smooth, odd and 0-homogeneous functions on the punctured plane irreducible? | Here's the general set up, as I understand it. Recall that a character of $\mathbb{R}^{\times}$ is a continuous homomorphism from $\mathbb{R}^{\times}$ to $\mathbb{C}^{\times}$. Every character of $\mathbb{R}^{\times}$ is of the form $\operatorname{sgn}(x)^{\kappa} |x|^s$ for some $\kappa \in \{0,1\}$ and $s \in \mathb... | 6 | https://mathoverflow.net/users/3803 | 449340 | 180,842 |
https://mathoverflow.net/questions/449337 | 6 | **Setting:** Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, sample $\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}$ and then orthogonalize and normalize $\mathbf{v}$).
Let $\lceil p/2\rceil\l... | https://mathoverflow.net/users/100796 | Expectation of the inner product of a subset of two random orthonormal vectors | Denote $\alpha=\mathbb{E} u\_1^2v\_1^2$, $\beta=\mathbb{E} u\_1v\_1u\_2v\_2$. Then by the symmetry and linearity of expectation we have $$f(m):=\mathbb{E} (u\_1v\_1+\ldots+u\_mv\_m)^2=m\alpha+m(m-1)\beta.$$
We have $f(p)=0$, thus $\beta=-\alpha/(p-1)$, and $f(m)=\alpha m(p-m)/(p-1)$.
It remains to bound $\alpha$. Cho... | 3 | https://mathoverflow.net/users/4312 | 449362 | 180,849 |
https://mathoverflow.net/questions/449324 | 2 | For reference, the linked paper is Composite parameterization and Haar measure for all unitary and special unitary groups by Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr (J. Math. Phys. 53, 013501 (2012)).
I am implementing an algorithm that "scrambles" a Hamiltonian by a random (w.r.t. to the Haar measur... | https://mathoverflow.net/users/505917 | Does there exist a Python package that samples random special unitary matrices such that the matrices are parameterized | I would just use Euler angles to parameterise the unitary, and then vary the angles according to the Haar measure. Below I copy the relevant equations from [Zyczkowski and Kus](http://yaroslavvb.com/papers/zyczkwoski-random.pdf). I do not have a Python code (when I used this method we were still using Fortran...), but ... | 2 | https://mathoverflow.net/users/11260 | 449365 | 180,851 |
https://mathoverflow.net/questions/449358 | 3 | Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by
\begin{equation}
Hf(x)=x^2f(x)-f''(x)
\end{equation}
for any smooth function $f$ compactly supported on the interval $(-a,a)$. Then, we may extend $H$ as an unbounded operator on $L^2(-a,a)$.
Extending from the Wikipedia... | https://mathoverflow.net/users/56524 | First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$? | Your formula for the first Dirichlet eigenvalue cannot be correct: for $a=1/\sqrt{2}$, the first eigenvalue is $5$.
Indeed, the eigenfunction
$$y(x)=e^{-x^2/2}(2x^2-1)$$
is positive on $(-1/\sqrt{2},1/\sqrt{2})$, zero at the ends,
and satisfies
$$x^2y-y''=5y,$$
therefore $5$ is the smallest eigenvalue.
Your formula giv... | 5 | https://mathoverflow.net/users/25510 | 449368 | 180,852 |
https://mathoverflow.net/questions/448818 | 4 | This question is "take 2" of [this older one](https://mathoverflow.net/questions/420048/comparing-bornologies-for-domination-escaping), following a suggestion of Francois Dorais. Consider the following [bornologies](https://en.wikipedia.org/wiki/Bornology) $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functio... | https://mathoverflow.net/users/8133 | Comparing bornologies for cardinal characteristics via Borel maps | A bornomorphic map $i\colon\mathcal{N}\to\mathcal{N}$ for these bornologies cannot be Borel.
First, I shall use $f<^\infty g$ to mean that $(\forall m\in\mathbb{N})(\exists n>m)f(n)<g(n)$, and $f<^\*g$ to mean that $(\exists m\in\mathbb{N})(\forall n>m)f(n)<g(n)$.
For $f\in\mathcal{N}$, let $\mathbb{D}\_f=\{g\in\ma... | 6 | https://mathoverflow.net/users/478588 | 449381 | 180,854 |
https://mathoverflow.net/questions/449375 | 7 | The [replica trick](https://en.wikipedia.org/wiki/Replica_trick) attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim\_{n\to 0}\frac{Z^n -1}{n}, \quad\text{or}\quad \log(Z) = \lim\_{n\to 0} \frac{\partia... | https://mathoverflow.net/users/122659 | Proving the Replica Trick works | **Q:** *Am I overlooking something important?*
I think you are ignoring the role played by the thermodynamic limit.
There are two interplaying limits here, the replica limit $n\rightarrow 0$ and the thermodynamic limit $N\rightarrow \infty$, where $N$ quantifies the system size. The usual practice in the replica me... | 10 | https://mathoverflow.net/users/11260 | 449387 | 180,858 |
https://mathoverflow.net/questions/449355 | 0 | The $\|\cdot\|\_{\infty}$-norm on $\mathbb{R}^n$ for $n\in \mathbb{Z}^+$ is not a smooth function. However, I came across [this post](https://stats.stackexchange.com/questions/298849/soft-version-of-the-maximum-function) which essentially says that a pointwise approximation to the maximum function, and therefore $\|\cd... | https://mathoverflow.net/users/496781 | Is this a smooth approximation to the $\ell$-infinity distance actually a quasi-metric? | $\newcommand\R{\mathbb R}$By rescaling, without loss of generality $\lambda=1$.
So, the question becomes the following: is there a real $C$ not depending on $u=(u\_1,\dots,u\_n)\in\R\_+^n$, $v=(v\_1,\dots,v\_n)\in\R\_+^n$, and $w=(w\_1,\dots,w\_n)\in\R\_+^n$ such that for all such $u,v,w$
$$w\le u+v\implies m(w)\le C... | 1 | https://mathoverflow.net/users/36721 | 449395 | 180,860 |
https://mathoverflow.net/questions/449389 | 3 | For a polynomial $f \in \mathbb{R}[x\_1, \cdots, x\_n]$, we say that $f$ is *coercive* (see my earlier question: [Real polynomials that go to infinity in all directions: how fast do they grow?](https://mathoverflow.net/questions/444925/real-polynomials-that-go-to-infinity-in-all-directions-how-fast-do-they-grow)) if
... | https://mathoverflow.net/users/10898 | Lower bound for coercive polynomials | A counterexample is given by
$$
f(x,y)=(x^4-y^3)^2+y .
$$
Clearly, $f(R, R^{4/3})=R^{4/3}\ll R^2 = \left( \min \{ R, R^{4/3} \} \right)^2$, so $f$ doesn't satisfy your condition.
However, $f$ is coercive: If $y\le 0$, then $f(x,y)\ge x^8+y^6 +y$ and now either $|y|$ is not large and there are no problems or if $|y|\g... | 3 | https://mathoverflow.net/users/48839 | 449411 | 180,864 |
https://mathoverflow.net/questions/449415 | 1 | Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L\_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is allowed.
>
> Are there second (or even higher) order infinitary languages? I mean something like $\mathcal L^2\_{\... | https://mathoverflow.net/users/95347 | Are there second (or higher) order infinitary logic languages? References? | Sure there are. They even come up in practice from time to time; e.g. to show that assuming Vopenka's Principle the modal analogue of *second-order* logic (a la [Hamkins/Woloszyn](https://arxiv.org/abs/2009.09394)) has definable-in-$V$ semantics, the only argument I'm aware of goes through $\mathcal{L}\_{\theta,\theta}... | 2 | https://mathoverflow.net/users/8133 | 449416 | 180,867 |
https://mathoverflow.net/questions/448607 | 5 | Let $\mathfrak S\_w(x\_1,\ldots,x\_n)$ be a Schubert polynomial. It's known that if we pick an index $i$, there are nonnegative integer coefficients $c\_{w'}^w(i,j)$ such that
$$\mathfrak S\_w(x\_1,\ldots,x\_n)=\sum\_{w'\in S\_\infty,j}c\_{w'}^w(i,j)x\_i^j\mathfrak S\_{w'}(x\_1,\ldots,x\_{i-1},x\_{i+1},\ldots,x\_n)$$
W... | https://mathoverflow.net/users/62135 | Pulling out a variable from a Schubert polynomial | This is in a Section five of a paper of mine with Bergeron in the Transactions of the AMS, which identifies it.
| 2 | https://mathoverflow.net/users/507423 | 449425 | 180,870 |
https://mathoverflow.net/questions/448489 | 3 | Let $X$ be a K3 over $\overline{\mathbb{F}\_p}$. The (crystalline version's) Tate conjecture predicts:
$c\_1: Pic(X)\otimes\mathbb{Q}\_p\rightarrow H^2\_{crys}(X/W)^{\Phi=p}\otimes\mathbb{Q}\_p$
is an isomorphism.
If $\lambda=1-\frac{1}{h}$ is the smallest slope of $H^2\_{crys}(X/W)$, then $H^2\_{crys}(X/W)$ must... | https://mathoverflow.net/users/177957 | (crystalline cohomology version's) Tate's conjecture for K3 surfaces | This is related to a phenomenon called 'hypersymmetry' (or the lack of it). The term comes from Chai and Oort, who explored this for abelian varieties. The essential point is that the category of F-isocrystals over the algebraic closure of a finite field is *not* the filtered colimit of the corresponding categories ove... | 3 | https://mathoverflow.net/users/7868 | 449426 | 180,871 |
https://mathoverflow.net/questions/449282 | 0 | I'm trying to construct lists with elements of type $A$ as the initial algebra over a base endofunctor in $\mathsf{Set}/\mathcal{P}(A)$, such that the list is indexed by the set of its elements.
My idea is to model `Cons` using a bifunctor $C\colon\ \mathsf{Set}/\mathcal{P}(A)\times\mathsf{Set}/\mathcal{P}(A)\to \mat... | https://mathoverflow.net/users/502814 | Is this kind of functor $\mathsf{Set}/M×\mathsf{Set}/M\to \mathsf{Set}/M$, with $M$ a monoid, a known construction? | In general, for any monoidal category $(\mathcal M, \otimes, I)$ and a monoid $(m, \*, i) \in \mathcal M$, the [slice category](https://ncatlab.org/nlab/show/over+category#examples) $\mathcal M/m$ inherits the monoidal structure pointwise, i.e. the tensor product is given by taking morphisms $f : x \to m$ and $g : y \t... | 5 | https://mathoverflow.net/users/152679 | 449441 | 180,879 |
https://mathoverflow.net/questions/449451 | 8 | Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d\_k$ be a closed immersion into a projective space of high enough dimension.
>
> Is there a smooth projective variety $X'$ that is a global complete intersection in some projective space, together with an ... | https://mathoverflow.net/users/501361 | Alterations and smooth complete intersections | As you guessed, there are cohomological obstructions. Indeed, if $f \colon Y \twoheadrightarrow X$ is a surjective morphism of smooth projective varieties and $H$ is a Weil cohomology theory (with coefficients containing $\mathbf Q$), then the pullback $f \colon H^i(X) \to H^i(Y)$ is *injective*; see for instance [Klei... | 11 | https://mathoverflow.net/users/82179 | 449452 | 180,881 |
https://mathoverflow.net/questions/449445 | 3 | Let $X$ be a smooth and proper scheme over $\mathbb{Q}$ and choose integers $n,i$ such that $n>\frac{i}{2}+1$. Then we have
$$ ord\_{s=i+1-n}L(H^i(X),s)=\dim H^{i+1}\_{\mathcal{D}}(X\_\mathbb{R},\mathbb{R}(n))=\text{Ext}^1\_{\mathbb{R}-MHS}(\mathbb{R},H^i(X(\mathbb{C}),\mathbb{R}(n))$$
(see for instance 3.1.4 in [this ... | https://mathoverflow.net/users/152554 | Order of vanishing of $L$-function and mixed Hodge-structures | There is a good reason why *this particular form* of the Beilinson conjecture cannot possibly be valid for the particular $i$ and $n$ you mention.
The real $\mathbb{R}$-MHS $H^1(X(\mathbb{C}), \mathbb{R})$ is the same for all elliptic curves $X$: it's always a 2-dimensional pure Hodge structure of weight 1, with $(0,... | 4 | https://mathoverflow.net/users/2481 | 449453 | 180,882 |
https://mathoverflow.net/questions/449455 | 2 | I am looking for the english translation of the paper in russian [Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001)](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1116&option_lang=eng) by Gusarov.
There is a .ps file on [Dror Bar-Natan webs... | https://mathoverflow.net/users/504366 | English version of a paper by Gusarov | Try this [link](https://drive.google.com/file/d/14uqJjQ8a8Cckza5ctofDJF6c4k_Zxs0O/view?usp=sharing). I put the paper into pdf format; it looks ok to me. I didn't have any trouble reading the original postscript file with ghostscript on Ubuntu. If you can't read the pdf or the postscript file, maybe your computer is mis... | 4 | https://mathoverflow.net/users/13268 | 449456 | 180,883 |
https://mathoverflow.net/questions/449463 | 1 | Suppose $n\in\mathbb{N}$ and set $A\subseteq\mathbb{R}^{n}$.
If we define a sequence of sets $\left(F\_r\right)\_{r\in\mathbb{N}}$ with a [set theoretic limit](https://en.wikipedia.org/wiki/Set-theoretic_limit) of $A$; how do we define the rate *at* which $\left(F\_r\right)\_{r\in\mathbb{N}}$ converges to $A$?
| https://mathoverflow.net/users/87856 | Convergence rate of a sequence of sets to a set-theoretic limit? | $\newcommand\R{\mathbb R}\newcommand\si\sigma\newcommand\Si\Sigma$If $\mu$ is a finite measure on a $\si$-algebra $\Si$ over $\R^n$ and $(F\_r)$ is sequence in $\Si$ such that $\lim\_r F\_r=A$, then ($A\in\Si$ and) $\lim\_r \mu(F\_r)=\mu(A)$.
So then, one may define the rate of convergence of $(F\_r)$ to $A$ as the rat... | 2 | https://mathoverflow.net/users/36721 | 449464 | 180,885 |
https://mathoverflow.net/questions/449361 | 25 | Let $a(n)$ be [A301897](https://oeis.org/A301897), i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I\_n(b) + EX\_n(b) \leqslant D\_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{n}+\sum\limits\_{k=1}^{n-2}\sum\limits\_{j=1}^{n-k-1}\binom{n}{k-1}\binom{n-1}{k+j}\bino... | https://mathoverflow.net/users/231922 | Elegant recursion for A301897 | Here is an expanded version of the generating function argument I sketched in a comment.
For $i=1,2,3$, define the generating functions $F\_i(x,y) := \sum\_{n=0}^\infty \sum\_{q=0}^\infty R(n,3q+i) x^n y^q$, which are well defined for $x,y$ small. If one starts with the recursive identities
\begin{align\*} R(n,3q) &=... | 36 | https://mathoverflow.net/users/766 | 449471 | 180,886 |
https://mathoverflow.net/questions/449477 | 2 | Do there exist some non-zero rational numbers $x, y$ such that $x \neq \pm y$ and
$$4y^p = x^2 + 3 \tag{1}$$
for some odd prime $p$?
If one lets $y=a/b$ and $x=c/d$ where $a, b, c, d$ are non-zero integers and $\gcd(a, b)=1=\gcd(c, d)$, you obtain $\frac{a^p}{b^p} = \frac{c^2 + 3d^2}{4d^2}$ thus both $c^2 + 3d^2$... | https://mathoverflow.net/users/492235 | On the equation $4y^p= x^2 + 3$ | Note that $(y,p,x)=(7,3,37)$ is a solution since $4(7^3) = 37^2 + 3 = 1372$.
You got that
$$\frac{a^p}{b^p} = \frac{c^2 + 3d^2}{4d^2}$$
Since $\gcd(c,d) = 1$, if $d$ is even, then $\gcd(4d^2, c^2 + 3d^2) = 1$, so the rest of your analysis would then be correct.
However, if $d$ is odd (e.g., $d = 1$ in my exampl... | 4 | https://mathoverflow.net/users/129887 | 449479 | 180,889 |
https://mathoverflow.net/questions/449421 | 6 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}$The braid groups $ B\_1=1 $ and $ B\_2\cong \mathbb{Z} $ have no perfect quotients.
$ B\_3 $ has quotients $ \SL(2,\mathbb{Z}) $ and $ \PSL(2,\mathbb{Z})\cong B\_3/Z(B\_3) $. As a result, $ B\_3 $ has inf... | https://mathoverflow.net/users/387190 | Perfect quotients of braid groups | A’Campo showed that the finite symplectic group $Sp(2m,p)$, $p>2$ prime, is a quotient of the braid group $B\_n$ for some $m$ depending on $n$. Hence the finite groups $PSp(2m,p)$ are quotients of $B\_n$. [These groups are simple](https://groupprops.subwiki.org/wiki/Projective_symplectic_group_is_simple) non-abelian he... | 4 | https://mathoverflow.net/users/1345 | 449481 | 180,890 |
https://mathoverflow.net/questions/449293 | 3 | Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it true that the specialization of a general member of this family is also geometrically integral. More precisely,
Let $... | https://mathoverflow.net/users/45397 | Degeneration of curves in smooth families | I am just writing my comments as one answer. Without further hypotheses, there are counterexamples. Even without a specific example of $\mathcal{X}$, there are plenty of examples of a $K$-scheme $B\_K$ and a family of smooth, projective, geometrically connected relative curves $\mathcal{C}\_K\to B\_K$ such that for eve... | 2 | https://mathoverflow.net/users/13265 | 449486 | 180,891 |
https://mathoverflow.net/questions/449462 | 3 | Recall that two elements $h\_1,h\_2$ of a finite group $G$ are called *conjugate* when $h\_2 = gh\_1 g^{-1}$ for some $g \in G$, and *algebraic-conjugate* when $h\_2 = gh\_1^a g^{-1}$ for some $a \in (\mathbb{Z}/\mathrm{order}(g))^\times$; equivalently when the cyclic subgroups $\langle h\_1\rangle$ and $\langle h\_2 \... | https://mathoverflow.net/users/78 | How often does algebraic-conjugacy imply conjugacy? | It is reasonably standard to call a finite group $G$ a rational group if all its complex irreducible characters are rational-valued ( equivalently, if $g \in G,$ then $g$ is conjugate within $G$ to all generators of $\langle g \rangle )$. Thanks to work of Feit-Seitz, and of J.G. Thompson,( see Feit-Seitz "On rational ... | 11 | https://mathoverflow.net/users/14450 | 449488 | 180,892 |
https://mathoverflow.net/questions/449501 | 0 | Let $(a\_n)\_n$ be an increasing real sequence with $a\_n=O(\sqrt n)$.
Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits\_{k=1}^{\phi(n)}\cos(a\_k)\right|+\left|\sum\limits\_{k=1}^{\phi(n)}\sin(a\_k)\right|=\infty?$$
<https://artofproblemsolving.com/... | https://mathoverflow.net/users/110301 | Series involving sine and cosine | Yes, it suffices that $a\_n=o(n)$.
Denote $h(n)=|\sum\_{j=1}^n \cos a\_j|+|\sum\_{j=1}^n \sin a\_j|$.
For any fixed integer $M>0$ there exists arbitrarily large $n$ for which $a\_{n+M}-a\_n<1/M^2$. Thus by triangle inequality and 1-Lipschitz property of functions $\cos$ and $\sin$ we have
$$h(n+M)+h(n)\geqslant |\c... | 2 | https://mathoverflow.net/users/4312 | 449505 | 180,896 |
https://mathoverflow.net/questions/449137 | 0 | Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:
1. The facet complex of any [simplicial polytope](https://en.wikipedia.org/wiki/Simplicial_polytope) is a simplicial complex.
2. The facet complex completely det... | https://mathoverflow.net/users/142698 | Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes? | A simplicial polytope is determined by the 1-skeleton of its dual *as
a polytope*, not as a simplicial complex. For instance, take a
sufficiently large simplicial polytope of dimension at least three,
and identify two far away vertices (so that it remains a simplicial
complex). The dual polyhedral complexes of these tw... | 2 | https://mathoverflow.net/users/2807 | 449509 | 180,897 |
https://mathoverflow.net/questions/449517 | -1 | Let V and W be any two subspaces of $(\mathbb{C}^d)^{\otimes n}$ such that there exists two irreducible and non-isomorphic representations $\rho\_V: G \to GL(V)$ and $\rho\_W: G \to GL(W)$. Does this imply: V and W are orthogonal subspaces?
I am new to representation theory, so apologies if this is basic knowledge. I... | https://mathoverflow.net/users/173056 | Orthogonality of irreducible and non-isomorphic representations | I wasn't sure what the question meant; to make sense of it, $V$ and $W$ would have had to be subspaces of a common space with some sort of form with respect to which we could measure orthogonality, and two abstract representations need not be presented in this way.
Your reference to [Stembridge - Orthogonal sets of Y... | 4 | https://mathoverflow.net/users/2383 | 449519 | 180,899 |
https://mathoverflow.net/questions/449305 | 2 | Let $\omega\_1,\cdots,\omega\_n$ be $n$ elements of $\overline{\mathbb F\_q(T)}$ that are $\mathbb F\_q(T)$ linearly independant. Denote by $\Lambda$ the lattice $\Lambda=\mathbb F\_q[T]\omega\_1+\cdots+\mathbb F\_q[T]\omega\_n$. Can one estimate (with at least two significant terms) the number of elements of $\Lambda$... | https://mathoverflow.net/users/33128 | Numbers of points in lattice | There are two possibilities. If $\omega\_1,\dots, \omega\_n$ are not $\mathbb F\_q((\frac{1}{T}))$-linearly independent, then the count is infinite for all $r$ sufficiently large. Indeed, if $\sum\_{i=1}^n a\_i \omega\_i$ is a relation, then for $f$ in $\mathbb F\_q[T]$ any polynomial, if we take the power series $fa\_... | 1 | https://mathoverflow.net/users/18060 | 449523 | 180,901 |
https://mathoverflow.net/questions/449504 | 5 | *Crossposted from <https://math.stackexchange.com/questions/4717613>*
---
An $\omega$-cover $\mathscr U$ of a space $X$ is a collection of open sets so that $X \not\in\mathscr U$ and every finite subset of $X$ is contained in a member of $\mathscr U$. Similarly, a $k$-cover $\mathscr U$ of a space $X$ is a collec... | https://mathoverflow.net/users/57800 | Is there an $\varepsilon$-space which is not $k$-Lindelöf? | (MA) There exists a countable, hereditarily (strongly) paracompact space
X, with ω<kL(X) (Example 6.4 in "Tightness, character and related properties
of hyperspace topologies" Topology and its Applications 142 (2004) 245–292).
| 0 | https://mathoverflow.net/users/112417 | 449529 | 180,903 |
https://mathoverflow.net/questions/449490 | 6 | Let $H\_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H\_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A\_{i,j} : 1 \leq i,j \leq n]$ in $n^2$ variables, and for $m \geq 0$, define the polynomial
\begin{equation\*}
\phi\_{m;i,j} := \su... | https://mathoverflow.net/users/380543 | The combinatorics of the Nullstellensatz for the variety of nilpotent matrices | Thanks everyone for your replies! As Darij suggested in his answer, it appears the clever trick used in [Gert Almkist's generalisation of a mistake of Bourbaki](https://sites.math.rutgers.edu/%7Ezeilberg/mamarim/mamarimhtml/gert.html) can be generalised to tackle the problem. Thanks Darij for the suggestion and for the... | 5 | https://mathoverflow.net/users/380543 | 449531 | 180,905 |
https://mathoverflow.net/questions/449525 | 5 | Let $D$ be a possibly unbounded domain in $\mathbb{R}^d$, $d \ge 2.$
We say that $D$ has the property P if there exists $C>0$ such that such that any pair of points $x,y \in D$ can be joined by a curve $\gamma$ in $D$ satisfying $\text{length}(\gamma) \le C|x-y|.$ Here, $\text{length}(\gamma)$ denotes the length of $... | https://mathoverflow.net/users/68463 | On the property P in the Whitney extension theorem | No. The complement $M$ in $\mathbb R^2$ of the union of the closed unit ball and the *half strip* $\{(x,y)\in\mathbb R^2: x\ge 0, |y|\le 1\}$ has $C^1$-boundary but curves in $M$ joining $(n,2)$ and $(n,-2)$ (which have distance $4$) have length $>2n$.
| 6 | https://mathoverflow.net/users/21051 | 449535 | 180,906 |
https://mathoverflow.net/questions/449493 | 4 | Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U\_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup \{(x+\sqrt{2}+y, -y) : 0 < y < 1/n\}.$$ The space $X$ is called Mysior plane, and it's an example of a space which is a unio... | https://mathoverflow.net/users/150060 | Mysior plane is not realcompact | The proof that I can think of applies a Baire-category argument to the normal topology of the real line.
As in your argument for realcompactness you need to look at zero-sets that are subsets of the $x$-axis. Let $Z\_f$ be such a zero-set, and look at $Z=\{x:f(x,0)=0\}$. There are two cases: (1) every nonempty open int... | 1 | https://mathoverflow.net/users/5903 | 449540 | 180,907 |
https://mathoverflow.net/questions/449377 | 9 | Crossposted from [MSE.](https://math.stackexchange.com/questions/4721350/changing-variables-in-discrete-calculus)
In discrete calculus one defines the $h$-difference operator $$\Delta\_h[f(x)] = f(x+h) - f(x)$$
and we often define $\Delta = \Delta\_1.$ We can similarly define the [indefinite sums](https://en.wikipedi... | https://mathoverflow.net/users/506963 | Change of variable formulas in discrete calculus? | In terms of the differential operator $\partial\_x\equiv d/dx$ one has $f(x+h)=e^{h\partial\_x}f(x)$, hence
$$\Delta\_h=e^{h\partial\_x}-1.$$
Upon Fourier transformation, $\hat{f}(k)=\int\_{-\infty}^\infty e^{ikx}f(x)\,dx$, one has $\partial\_x\mapsto -ik$, hence
$$\hat{\Delta}\_h=e^{-ihk}-1\Leftrightarrow \hat{\Delta}... | 5 | https://mathoverflow.net/users/11260 | 449547 | 180,908 |
https://mathoverflow.net/questions/449560 | 2 | This question was originally posted in ME: <https://math.stackexchange.com/questions/4725157/what-is-an-explicit-subset-of-mathbbz3-that-makes-bigl-sinn-cdot-x>
but more and more I think about it, this problem looks nontrivial. So, I ask for help here.
Basically I would like to find an explicit orthonormal basis of... | https://mathoverflow.net/users/56524 | What is a subset of $\mathbb{Z}^3$ making $\Bigl( \sin(n \cdot x),\cos(n \cdot x) \Bigr)_{n \in \mathbb{Z}^3}$ linearly independent? | One can use, for example, $n \in \mathbb{Z}^3$ with the following restrictions (edited upon comment by Alexei Kulikov to be careful about cases with $n\_i =0 $): $n\_1 >0 \ \ \lor \ (n\_1 =0 \land n\_2 > 0) \ \lor \ (n\_1 = n\_2 =0 \land n\_3 \geq 0) $.
To abbreviate things, introduce the notation
$$
sss = \sin 2\pi ... | 3 | https://mathoverflow.net/users/134299 | 449564 | 180,916 |
https://mathoverflow.net/questions/449572 | 11 | Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of $Z$ is an element in $H^{2k}(X,\mathbb{C})$. Denote by $\phi\_Z \in H^{2n-2k}(X,\mathbb{C})^{\vee}$ the element corresp... | https://mathoverflow.net/users/45397 | Does Poincaré duality preserve algebraic cycles? | A positive answer to your question\* would imply one of Grothendieck's standard conjectures (conjecture A) for complex varieties. See his note: [Standard conjectures on algebraic cycles](https://web.archive.org/web/20210822125536/http://www.math.tifr.res.in/~publ/studies/SM_04-Algebraic-Geometry.pdf). It's safe to say ... | 11 | https://mathoverflow.net/users/4144 | 449574 | 180,920 |
https://mathoverflow.net/questions/449380 | 3 | Is it consistent with $\sf ZFC + \text{ countable models of } ZFC \text { exist}$, that every countable model of $\sf ZFC$ is a subset of some parameter free definable pointwise-definable model of $\sf ZFC$?
| https://mathoverflow.net/users/95347 | Is every countable model of ZFC a subset of some parameter free definable pointwise-definable model of ZFC? |
>
> The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) is consistent.
>
>
>
Wojowu's answer already points out that the proof of the following fact (S) can be found ... | 6 | https://mathoverflow.net/users/9269 | 449576 | 180,921 |
https://mathoverflow.net/questions/449211 | 7 | Consider the finite Boolean lattice $B\_n$ of subsets of $[n]:=\lbrace 1,\dots,n\rbrace$ ordered by inclusion, let $1\leq j,k\leq n$ and consider the poset:
$$A\_{j,k}=\lbrace\emptyset\neq U\in B\_n\mid (1,\dots,j)\nsubseteq U, (n-k,\dots,n)\nsubseteq U \rbrace$$
I want to compute the geometric realization for every $j... | https://mathoverflow.net/users/482329 | Geometric realization of a poset | Put $P\_0=\{1,\dotsc,n-k-1\}$ and $P\_1=\{n-k,\dotsc,j-1\}$ and $P\_2=\{j,\dotsc,n\}$, so $[n]=P\_0\amalg P\_1\amalg P\_2$ with each $P\_i$ nonempty. Any subset $U$ can be decomposed as $\coprod\_{i=0}^2U\_i$ with $U\_i\subseteq P\_i$. You are looking at the space
$$ A = \{(U\_0,U\_1,U\_2): U\_0\cup U\_1\cup U\_2\neq\e... | 2 | https://mathoverflow.net/users/10366 | 449596 | 180,924 |
https://mathoverflow.net/questions/449251 | 1 | I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not decomposable? Which homotopy groups does one need to look at to construct such an example?
(This problem has been edited u... | https://mathoverflow.net/users/15197 | A torus bundle whose vertical tangent bundle is indecomposable | Put $T=\{(z\_0,z\_1,z\_2)\in(S^1)^3:z\_0z\_1z\_2=1\}$, so $T$ is homeomorphic to $S^1\times S^1$ and has an obvious action of the symmetric group $\Sigma\_3$. The action of $\Sigma\_3$ on $H\_1(T;\mathbb{R})$ is indecomposable, and this homology group can also be identified with the tangent space to $T$ at the identity... | 2 | https://mathoverflow.net/users/10366 | 449601 | 180,925 |
https://mathoverflow.net/questions/449578 | 1 | **Imprecise Question**: Suppose I have a function defined on non-codimension-zero strata of a smooth manifold with a stratification, and I know the function is smooth when restricted to each of these strata. Is there any setting in which the function extends smoothly to the whole smooth manifold?
Here is the baby cas... | https://mathoverflow.net/users/507571 | Smooth extension of piecewise smooth function on a corner | You have a function defined on the boundary of $\mathbb R^n\_{\ge 0}$. Its restriction to each face is smooth. You can extend it to all of $\mathbb R^n\_{\ge 0}$ by writing
$$
f(x)=\sum\_P (-1)^{|P|-1}f\_P(x\_P),
$$
where $x\mapsto x\_P$ is the projection $\mathbb R^n\_{\ge 0}\to C\_P$.
| 2 | https://mathoverflow.net/users/6666 | 449607 | 180,927 |
https://mathoverflow.net/questions/449580 | 2 | The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k\_i\}\_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. For each $k\in \{1,2,3,4\}$, consider the function $f\_k:]-r\_k,r\_k[\rightarrow \mathbb{R}$ given by $x\mapsto\sum\_{i=... | https://mathoverflow.net/users/32135 | Local equality of functions implies global equality? | Consider $$\eqalign{f\_1(x) &= (x+1)^2-1\cr f\_2(x) &= \sqrt{x+1}-1\cr f\_3(x) &= f\_4(x) = x
}$$
$f\_1, f\_3$ and $f\_4$ being polynomials, their radius of convergence is $\infty$, while $f\_2(x)$ has a Maclaurin series with radius of convergence $1$. $f\_2(f\_1(x)) = f\_4(f\_3(x)) = x$ for $x \in (-1,1)$. But sinc... | 4 | https://mathoverflow.net/users/13650 | 449608 | 180,928 |
https://mathoverflow.net/questions/449598 | 0 | Assume I have the following function
$f\left(n\right)=\frac{-2\sqrt{n}}{A}e^{-A\left(\frac{k}{\sqrt{n}}\right)}\left(e^{-\frac{A}{\sqrt{n}}}-1\right)$
Where $n-k\gg1$ and $k\gg\sqrt{n}$ and $A=2$ is a constant.
And $n\rightarrow\infty$.
My thoughts is that $f\left(n\right)\rightarrow 0$ (I have ran some example... | https://mathoverflow.net/users/491400 | asymptotic behavior of function $f\left(n\right)=\frac{-2\sqrt{n}}{A}e^{-A\left(\frac{k}{\sqrt{n}}\right)}\left(e^{-\frac{A}{\sqrt{n}}}-1\right)$ | You have
$$f(n)=\sqrt{n} (1-e^{-2/\sqrt{n}})
e^{-2k/\sqrt{n}}\sim2e^{-2k/\sqrt{n}}\to0,
$$
since $k>>\sqrt{n}$. So, $f(n)\to0$.
| 1 | https://mathoverflow.net/users/36721 | 449612 | 180,930 |
https://mathoverflow.net/questions/449628 | 1 | $\sf V=HOD$ is stated as:
$\forall X \, \exists \theta \, \exists \alpha < \theta \, \exists \varphi : X=\{y \in V\_\theta\mid V\_\theta\models\varphi(y,\alpha)\}$
This use two ordinal parameters (other than the code for $\varphi$) $``\theta; \alpha"$.
Can we do with just ONE parameter (other than the code for $\... | https://mathoverflow.net/users/95347 | Can we state $\sf V=HOD$ using a single ordinal parameter(other than the formula code)? | Yes, because there is a definable ordinal pairing function.
Specifically, if you want to get the set $\{y\in V\_\theta\mid V\_\theta\models\varphi(y,\alpha)\}$, then let $\beta=\langle\theta,\alpha\rangle$ be the ordinal coding the pair, and then look at $V\_{\beta+1}$. Inside this structure, we have $\beta$ as the l... | 6 | https://mathoverflow.net/users/1946 | 449629 | 180,933 |
https://mathoverflow.net/questions/449638 | 3 | I'm writing a Master's thesis on knot invariants and I'm trying to chase down the original source that introduced the unknotting number and perhaps proved that it is a knot invariant. The texts I've consulted that reference the unknotting number already discuss it in terms of being an established invariant. But I can't... | https://mathoverflow.net/users/137916 | Straightforward reference on the unknotting number being a knot invariant | I agree with Tom Goodwillie that it is immediate from the definitions that the unknotting number is well-defined. I think that what might be tripping you up is being taught that “knot invariants” are really diagram invariants that are invariant under the Reidemeister moves. That makes invariants like this one that cann... | 6 | https://mathoverflow.net/users/317 | 449643 | 180,936 |
https://mathoverflow.net/questions/449553 | 8 | I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for large cardinal axioms that only remain consistent because they "got off on a technicality".
In particular, I am looking... | https://mathoverflow.net/users/22277 | Large cardinal near inconsistencies | I would argue that a "restricting versions" of large cardinals are such.
Starting from the top down, we have the inconsistent Berkeley cardinals:
* $κ$ is Berkeley if for every transitive $M\ni\kappa$ and $ν<κ$ we have an elementary embedding on $M$ with critical point between $\nu$ and $κ$.
A way to make this ca... | 7 | https://mathoverflow.net/users/113405 | 449648 | 180,940 |
https://mathoverflow.net/questions/449610 | 5 | Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:
1. If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a monomorphism in $\mathcal{C}$ at every component, is it a monomorphism in $Fun(K,\mathcal{C})$?
2. If every 1-simplex in th... | https://mathoverflow.net/users/11546 | Monomorphisms of diagrams in an $\infty$-category | For completeness, and because I cannot figure out the general case (cf. my comment below Daniel's answer), let me prove the following: if $f:\Delta\to C$ is a functor which, when restricted to $\Delta\_{inj}$ takes values in monomorphisms, then $\lim\_\Delta f\to f([0])$ is a monomorphism.
This is simply the followin... | 4 | https://mathoverflow.net/users/102343 | 449649 | 180,941 |
https://mathoverflow.net/questions/449656 | 6 | $\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Mat{Mat}$This question was originally asked on [MSE](https://math.stackexchange.com/questions/4723838/are-algebroids-just-matrices) but may be better here.
[Algebroids](https://ncatlab.org/nlab/show/algebroid) are particularly interesting structures: they are basic... | https://mathoverflow.net/users/24611 | Are algebroids "just matrices"? | If we take the large but locally small category $\mathcal{C}$ described by Isbell in Example 2.4 of
* *Two set-theoretical theorems in categories*, Fundamenta Mathematicae **53** Issue 1 (1964) pp 43-49, ([EuDML](https://eudml.org/doc/213746)),
namely, with class of objects $(\{1,2\}\times\mathrm{ORD}) \sqcup \{X,Y... | 8 | https://mathoverflow.net/users/4177 | 449660 | 180,944 |
https://mathoverflow.net/questions/449657 | 5 | The statement in the title seems to be generally accepted as true, but I have not seen proof. They are?
The strict formulation I have in mind is the following. By an algebraic category we mean the category of algebras of some monad on $\mathrm{Set}^S$. In particular, this includes all the usual finitary algebraic cat... | https://mathoverflow.net/users/148161 | Can we prove that any number of algebraic data cannot be a complete invariant of a homotopy type? | I'll answer the corresponding question for the homotopy category $\mathcal{S}$ of spectra. I doubt that this makes much difference, but I have not checked the details. We can choose a list $X\_0,X\_1,\dotsc$ containing one representative of every homotopy equivalence class of finite spectra, and then put $X=\bigvee\_iX... | 10 | https://mathoverflow.net/users/10366 | 449669 | 180,947 |
https://mathoverflow.net/questions/449670 | 2 | I'm reading about the Weierstrass zeta function. In this context,
$\phi(z)=\zeta(z)-\pi\bar{z}$
is periodic over the lattice
$$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
If we take $w\in\mathcal{L}\setminus\{\mathbf{0}\}$, then
$$\phi(z+w)-\phi(z)=w\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}-\pi\bar{w}.$$ Since $\phi... | https://mathoverflow.net/users/482837 | The sum of $q^{-2}$ over nonzero Gaussian integers | The series
$$\sum\_{q\in\mathcal{L}\setminus\{0\}}q^{-2}$$
is not absolutely convergent. If we arrange the terms in groups of four of shape $\{\pm q,\pm iq\}$, then the series converges to zero since
$$q^{-2}+(iq)^{-2}+(-q)^{-2}+(-iq)^{-2}=0,\qquad q\in\mathcal{L}\setminus\{0\}.$$
For more general sums of similar shape... | 6 | https://mathoverflow.net/users/11919 | 449673 | 180,949 |
https://mathoverflow.net/questions/449676 | 1 | Let $X,Y,Z$ be Banach spaces and let $m\,:\,X\times Y\to Z$ be a bilinear map such that $\|m(x,y)\|\leq C \|x\|\|y\|$ for some fixed constant $C$. Moreover, let $\mu$ be a Borell vector measure on $\mathbb{R}^d$ valued in $Y$ with finite total variation and let $f\,:\,\mathbb{R}^d\to X$ be bounded and measurable (in th... | https://mathoverflow.net/users/47256 | Integration of vector function against vector measure | $\newcommand\R{\mathbb R}$Suppose that $X$ is separable. Let
$$\int\_{\R^d}m(x\,1\_B,d\mu):=m(x,\mu(B))$$
for all $x\in X$ and Borel $B\subseteq\R^d$. Then, for simple functions
$$s\_n:=\sum\_{j=1}^n x\_j\,1\_{B\_j},$$
with $x\_j\in X$ and Borel $B\_j\subseteq\R^d$ for all $j$,
let
$$\int\_{\R^d}m\big(s\_n,d\mu\big):=\... | 1 | https://mathoverflow.net/users/36721 | 449681 | 180,950 |
https://mathoverflow.net/questions/449679 | 3 | Is the following a theorem of $\sf ZF+[V=HOD]$?
If $Q$ is a property definable in a parameter free manner, then: $$\operatorname {Con}(\sf ZF + [V=HQD]) \implies V=HQD$$
where $\sf V=HQD$ means:
$$\forall X \exists v\_0 \exists v\_1: Q(v\_0) \land Q(v\_1) \land \rho(v\_0) > \rho(v\_1) \land \exists \varphi:\\ X=\... | https://mathoverflow.net/users/95347 | Does V=HOD prove all kinds of consistent universal hereditary definability? | The answer is no. Indeed, one can rarely move from consistency to truth in this way.
For a counterexample, let $Q(v)$ be the property "CH holds and $v$ is an ordinal."
If CH holds, then $Q$ expresses the property of being an ordinal, but if CH fails, then $Q$ never holds. Consider a model of ZF+V=HOD in which CH fa... | 6 | https://mathoverflow.net/users/1946 | 449682 | 180,951 |
https://mathoverflow.net/questions/449613 | 3 | I already asked this on [Math.SE](https://math.stackexchange.com/questions/4722773/how-to-determine-the-type-of-a-divisor-d-on-a-product-of-elliptic-curves), but didn't receive an answer yet.
---
Say $E\_1, \dotsc, E\_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E\_1 \times \dotsc \times E... | https://mathoverflow.net/users/111897 | How to determine the type of a divisor on a product of elliptic curves? | I managed to calculate my examples. In the first one, $D\_0$ has indeed polarization type $(2,2)$. To see this, let $E = \mathbb C / (\mathbb Z + \tau \mathbb Z)$ be an elliptic curve, and consider the isogeny
$$\varphi: E \times E \to E \times E, (z\_1, z\_2) \mapsto (z\_1 + z\_2, z\_1 - z\_2).$$
Then $\varphi^2(z\_1,... | 1 | https://mathoverflow.net/users/111897 | 449683 | 180,952 |
https://mathoverflow.net/questions/449634 | 7 | I’m wondering what is known about the cohomology of $\operatorname{GL}\_3(\mathbb{Z}\_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}\_2$. There are results in Chapter I of Knudson’s “[Homology of Linear Groups](https://doi.org/10.1007/978-3-0348-8338-2)” as well as Chapter VII of... | https://mathoverflow.net/users/503849 | Cohomology of $\operatorname{GL}_3(\mathbb{F}_2)$ | As you mention in your update, you have a general answer, but if you want a concrete answer for the low-dimensional integral cohomology of $G = \operatorname{GL}(3,2)$ (or any other finite group!), you can use [GAP](https://www.gap-system.org/index.html), specifically the `hap` package. The following code computes $H^i... | 11 | https://mathoverflow.net/users/120914 | 449684 | 180,953 |
https://mathoverflow.net/questions/439203 | 0 | The Crust algorithm by Amenta, Bern, and Eppstein computes a polygonal reconstruction of a smooth curve $C$ without boundary from a discrete set of sample points $S$. It is known that if $S$ is an a $\epsilon$-sample of $C$ with $\epsilon < \frac{1}{5}$, then the Crust algorithm computes the correct polygonal reconstru... | https://mathoverflow.net/users/48162 | What is the most dense sample for which the Crust algorithm returns an incorrect polygonal reconstruction? | In 2022, Håvard Bakke Bjerkevik gave a 0.72-sample of a particular curve for which reconstruction is not possible. Moreover, Bjerkevik showed that curve reconstruction is always possible from a 0.66-sample. This work appears in "Tighter Bounds for Reconstruction from ϵ-samples," <https://doi.org/10.4230/LIPIcs.SoCG.202... | 0 | https://mathoverflow.net/users/48162 | 449691 | 180,955 |
https://mathoverflow.net/questions/449687 | 4 | This question is somewhat similar to [Minimizing the L1 norm of odd-term trigonometric polynomial](https://mathoverflow.net/questions/260077/minimizing-the-l1-norm-of-odd-term-trigonometric-polynomial). The context of the question is based on the paper [Hardy's Inequality and the $L^1$ norm of Exponential Sums](https:/... | https://mathoverflow.net/users/482554 | A lower bound for the $L^1$ norm of real trigonometric polynomials | This is not true.
Consider the [Fejer Kernel](https://en.wikipedia.org/wiki/Fej%C3%A9r_kernel) $F\_n$. From the explicit formula for $F\_n$ we see that $$\|F\_n\|\_{L^1} \leq C $$ independent of $n$.
On the other hand since $ F\_n = 2 \sum\_{0\leq k \leq n-1} \left(1 - \frac{|k|}{n} \right) \cos(k x)$ we have that ... | 4 | https://mathoverflow.net/users/630 | 449699 | 180,956 |
https://mathoverflow.net/questions/449697 | 3 | $\sf ZF + Def$ is the theory that extends $\mathcal L(=,\in)\_{\omega\_1,\omega}$ with axioms of $\sf ZF$ (written in $\mathcal L(=,\in)\_{\omega, \omega}$) and the axiom of definability:-
$\textbf{Define: } Dx \iff \bigvee x= \{ y \mid \Phi \}$
where $\Phi$ range over all formulas in $\mathcal L(=,\in)\_{\omega, \... | https://mathoverflow.net/users/95347 | Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples? | Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model.
If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the Skolem function that gives the least witness using that well ordering, the class $M\_0$ of definable (without paramet... | 6 | https://mathoverflow.net/users/113405 | 449700 | 180,957 |
https://mathoverflow.net/questions/449611 | 6 | A finite group $G$ is called *rational* if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ *real* if every $g$ is conjugate to its inverse. The ratio of these notions is something I'll call *rational-relative-to-rea... | https://mathoverflow.net/users/78 | Which finite simple groups are rational-relative-real? | Rational-relative-to-real groups are frequently called inverse semi-rational groups (or cut groups). Inverse semi-rational groups are a subclass of semi-rational groups. A finite group $G$ is *semi-rational* if for every $g \in G$ there exists a positive integer $m\_0$ such that every primitive power $g^a$ ($a \in (\ma... | 6 | https://mathoverflow.net/users/153043 | 449702 | 180,958 |
https://mathoverflow.net/questions/449677 | 7 | I am looking for the analytic continuation of
\begin{align\*}
& f\_m(v,w) := \sum\limits\_{k,l=0}^\infty v^k w^l {k+l+m \choose k} {k+l+m \choose l} \ ,
\end{align\*}
where $m \in \{1,2,...\}$ is fixed. The sums converge for small enough $|v|$ and $|w|$ and I have already made a few observations:
We can write
\begin{... | https://mathoverflow.net/users/409412 | Help finding an analytic continuation | Your $f\_m(v,w)$ is a special case of [Appell's $F\_4$ hypergeometric function](https://mathworld.wolfram.com/AppellHypergeometricFunction.html),
$$f\_m(v,w) = F\_4(m+1;m+1;m+1,m+1;v,w).$$
Some information about analytic continuation of $F\_4$ can be found in <https://arxiv.org/abs/2005.07170> and the references cited ... | 12 | https://mathoverflow.net/users/10744 | 449703 | 180,959 |
https://mathoverflow.net/questions/449696 | 3 | Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int\_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following inequality holds:
$$
\int\_0^1 \log(1+f(x)) \,{\mathrm d}x \; +\; \int\_0^1 (1-f(x)) \log(1-f(x)) \, {\mathrm d} x \;\stackrel... | https://mathoverflow.net/users/83189 | Bound on an integral representing a difference of two relative entropies | Yes, the inequality is true. Indeed, the inequality in question can be rewritten (or, if you prefer, generalized) as
\begin{equation}
Eg(Y)\ge\ln(1-a^3), \tag{10}\label{10}
\end{equation}
where
$$g(t):=\ln(1+t)+(1-t)\ln(1-t)$$
and $Y$ is a random variable (r.v.) such that
\begin{equation}
P(|Y|\le a)=1\quad\text{and... | 3 | https://mathoverflow.net/users/36721 | 449706 | 180,960 |
https://mathoverflow.net/questions/449704 | 5 | My question stems from the following result about holomorphic functions on the unit disc:
>
> "A function, continuous on the closed unit disc, holomorphic inside, and vanishing on an open subset of the boundary, vanishes identically. A simple proof of this well-known proposition is obtained by considering its Cauch... | https://mathoverflow.net/users/157422 | Boundary zeros of a holomorphic function $f: \Omega \to \Bbb C$ | It is not clear in your question what "Lebesgue measure on $\partial\Omega$" really means.
Let us begin with the unit disk. In the unit disk, every bounded holomorphic function which is zero on a set $E$ of positive measure, in the sense that
$$\limsup\_{r\to 1}|f(re^{i\theta})|=0,\quad e^{i\theta}\in E,$$ vanishes. ... | 5 | https://mathoverflow.net/users/25510 | 449718 | 180,964 |
https://mathoverflow.net/questions/449729 | 3 | Let $A,B$ be $C^\*$-algebras and $E$ be a right $A$-Hilbert $C^\*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^\*$, there is a unique bounded map
$$\iota \otimes \omega: E \otimes B \to E$$
extending $\iota \odot \omega$.
Now, let $F$ be a closed submod... | https://mathoverflow.net/users/216007 | Property that follows from conditions involving slice maps on Hilbert module | Theorem 8 from the paper "A Pathology in the Ideal Space of $L(H)\otimes L(H)$" by Simon Wassermann states that there exists an element $x \in B(H)\otimes B(H)$ such that all slices belong to the algebra of compact operators, but $x \notin K(H) \otimes B(H)$, so in general it does not hold.
The property you are after... | 5 | https://mathoverflow.net/users/24953 | 449740 | 180,967 |
https://mathoverflow.net/questions/449723 | 2 | Consider a projective morphism of Noetherian schemes $p:X\to \mathrm{Spec}(A)$. Let $\mathcal{E},\mathcal{F}$ be coherent $\mathcal{O}\_X$-modules flat over $A$. For every (Noetherian) ring map $A\to B$, denote $X\_B:=X\times\_{\mathrm{Spec}(A)}\mathrm{Spec}(B)$, $\mathcal{E}\_B:=(X\_B\to X)^\*\mathcal{E}$, and $p\_B:X... | https://mathoverflow.net/users/105537 | If $\mathrm{Ext}^i(E,F)$ commutes with base change, then is $\mathrm{Ext}^{i+1}(E,F)$ representable? | **Edit.** I started this answer earlier, but then I had to do something else.
**Positive answer.**
Without any further hypothesis, the answer is positive if $i$ equals $0$. This is one of the theorems the follows from the "Exchange Property" in Section 7.7 of EGA III\_2. If $i$ equals $1$, at least there is a well-de... | 3 | https://mathoverflow.net/users/13265 | 449741 | 180,968 |
https://mathoverflow.net/questions/449341 | 5 | Suppose $X\_1, X\_2, X\_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X\_2\mid X\_2 \geq X\_1 - a, X\_1 \leq X\_3 + b] < 1,$$
where the inequality is strict. I've run some simulations and this seems to be true for various choices of $a$ and $b$... | https://mathoverflow.net/users/128729 | Bounding the variance of a truncated Gaussian random variable | The inequality is a special case of the following claim.
Claim:
If $X = (X\_1, \dotsc, X\_d) \sim N(\mu, \Sigma)$ is an $\mathbb{R}^d$-valued normal random variable with invertible covariance matrix $\Sigma$, $L : \mathbb{R}^d \to \mathbb{R}$ is linear, and $K \subseteq \mathbb{R}^d$ is convex and has a nonempty inte... | 4 | https://mathoverflow.net/users/42355 | 449743 | 180,969 |
https://mathoverflow.net/questions/449739 | 2 | Let $X\subseteq \mathbb{P}^n$ be a closed irreducible subvariety, with vanishing ideal $I(X)\subseteq k[x\_0,\ldots,x\_n]$, where $k$ is the ground field, assumed to be algebraically closed. Let $F\in k[x\_0,\ldots,x\_n]$ be a homogeneous polynomial. If $F$ belongs to the ideal $I(X)^2$, then $F$ is singular along $X$ ... | https://mathoverflow.net/users/23758 | What is the ideal of hypersurfaces singular at a given irreducible variety? | If $X=\mathbb{V}(I)$ is given by the ideal $I$, then the $m$th symbolic power $I^{[m]}$ consists of all those functions vanishing to multiplicity $m$ at the generic point of $X$. Thus a hypersurface $\mathbb{V}(F)$ will be singular along $X$ if and only if $F\in I^{[2]}$. (Thanks to Zach Teitler for pointing this out i... | 9 | https://mathoverflow.net/users/104695 | 449744 | 180,970 |
https://mathoverflow.net/questions/449290 | 0 | Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi\_{2s}(n)$ to count the number of such pairs $p,p+2s$ below or equal to $n$.
Does this conjecture hold ?
$$ \pi\_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi\_{4a}(n)$$
f... | https://mathoverflow.net/users/80790 | Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples? | It is false, try $a=1, n=250\,003\,639$.
| 2 | https://mathoverflow.net/users/6043 | 449747 | 180,971 |
https://mathoverflow.net/questions/371232 | 3 | If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^\*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to A)$ the $S$-algebra $(A,a\circ\lambda:SA\to A)$.
Does a similar phenomenon happen for pseudomonads and their pseudoal... | https://mathoverflow.net/users/30366 | Morphism of pseudomonads induces pullback functors between pseudoalgebras | Yes, Theorem 3.4 of Gambino–Lobbia's [On the formal theory of pseudomonads and pseudodistributive laws](https://arxiv.org/abs/0907.1359) establishes that pseudomonad morphisms are in correspondence with liftings to pseudoalgebras. They work more generally in the setting of pseudomonads in a Gray-category, and with pseu... | 1 | https://mathoverflow.net/users/152679 | 449755 | 180,972 |
https://mathoverflow.net/questions/449735 | 0 | Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int\_\Omega(|\nabla u|^2+h^{-1}|u-u\_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ u\_0\in H\_0^1(\Omega) $. For the minimizing problem
$$
\min\_{u\in H\_0^1(\Omega)}\mathcal{F}(u),
$$
it is easy to get ... | https://mathoverflow.net/users/241460 | Limit of minimizers of a class of functionals | By simple calculations, we can obtain that $ u\_{h} $ and $ u\_{h\_0} $ satisfy the following Euler-Lagrange equation
\begin{align}
\frac{u\_h-u\_0}{h}-\Delta u\_{h}=0,\\
\frac{u\_{h\_0}-u\_0}{h}-\Delta u\_{h\_0}=0.
\end{align}
Then it can be got that
$$
\begin{aligned}
0&=\frac{u\_h-u\_0}{h}-\frac{u\_{h\_0}-u\_0}{h\_0... | 1 | https://mathoverflow.net/users/241460 | 449766 | 180,975 |
https://mathoverflow.net/questions/449719 | 1 | Are there any examples where a usual algebraic 1-stack $X$ and the corresponding derived stack enhancement $\mathbb{R}X$ coincide?
Let me take an example from notes of Bertrand Toen, page 41 of <https://arxiv.org/pdf/math/0604504.pdf>.
Let $C$ be a smooth projective curve and consider the functor $i: \mathrm{St}(k)... | https://mathoverflow.net/users/465579 | Examples when algebraic 1-stack = derived enhancement? | If $S$ is a smooth projective variety of dimension $d$, then the derived stack $X=Bun\_G(S)$ has cotangent complex perfect of amplitude $[-(d-1), 1]$. If $S=C$ is a curve then it is in $[0,1]$, in particular $H^\*(\mathbb{L}\_X)=0$ for $\*<0$. For stacks, this condition (+ finite presentation) is exactly smoothness.
... | 3 | https://mathoverflow.net/users/85136 | 449769 | 180,976 |
https://mathoverflow.net/questions/448614 | 2 | Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}\_{\text{H}}(A)$ is the Hausdorff dimension of set $A$.
**Motivation:** If the set $A\subseteq[0,1]\times[0,1]$, I would like to measure set $A$'s deviatio... | https://mathoverflow.net/users/87856 | Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$ | Here is another possible approach, perhaps closer to what the OP had in mind.
Let $S:=[0,1]^2$ be the unit square. "Partition" $S$ naturally into four congruent squares $S\_{1,j}$ (with side length $1/2$ each), where $j=1,\dots,4$; the quotation marks are used here because the $S\_{1,j}$'s will have some common bound... | 2 | https://mathoverflow.net/users/36721 | 449772 | 180,977 |
https://mathoverflow.net/questions/449776 | 13 | Does the following infinite series have a closed form?
$$
\sum\_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$
| https://mathoverflow.net/users/105725 | Closed form of an infinite series | **Q:** Does the following infinite series have a closed form?
It does, according to Mathematica:
$$\sum\_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac{2\pi n}{3}\right)}=-2^{-10/3} \Gamma \left(\tfrac{1}{3}\right)\Big[2 \sqrt{3}+9\pi^{-1} \, \_2F\_1\left(\tfrac{2... | 11 | https://mathoverflow.net/users/11260 | 449778 | 180,979 |
https://mathoverflow.net/questions/449313 | 0 | If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the **joint** characteristic function of this process and its quadratic variation process?
If not possible in general, are there specific examples of compound Poisson processes for which this function is known in ... | https://mathoverflow.net/users/109513 | Characteristic function of quadratic variation of compound Poisson process | Suppose that $N=\{N(t), t\in\mathbb{R}\_+\}$ is a Poisson process with rate $\lambda $, and $\{D\_j,j\ge 1\} $ are i.i.d. random variables, with distrbution function $F$, which are also independent of $N$. Let
\begin{equation\*}
X(t) = \sum\_{j=1}^{N(t)}D\_j, \quad t\in\mathbb{R}\_+, \tag{1}
\end{equation\*}
then the... | 1 | https://mathoverflow.net/users/103256 | 449787 | 180,983 |
https://mathoverflow.net/questions/449764 | 1 | I am reading the following two papers:
* Pappas, *On the arithmetic moduli schemes of PEL Shimura varieties*, 1999 (it seems to be difficult to find online nowadays - only a .ps file remains available),
* Krämer, *Local models for ramified unitary groups*, 2003.
Let $n\geq 1$ and consider integers $r,s\geq 0$ such ... | https://mathoverflow.net/users/125617 | Is Krämer's local model for ramified unitary groups isomorphic to the blow-up of Pappas' flat model at the singular point? | In Yousheng Shi's [paper](https://arxiv.org/abs/2004.07158):
**Proposition 2.2.** $N^{\text{Kra}}$ is the blow-up of $N^{\text{Pap}}$ along its singular locus Sing.
Also see the words before the Proposition: "The following fact should be well-known to experts. However due to the lack of
a precise reference, we prov... | 1 | https://mathoverflow.net/users/486528 | 449792 | 180,985 |
https://mathoverflow.net/questions/449705 | 1 | Let $X$ be a $\mathbb R^d$ valued continuous stochastic process. I am interested in bounding $$P(\|X\|\_\gamma>R).$$
The standard technique to do so, is to apply Markov inequality and then Garsia-Rodemich-Rumsey inequality. Recall that GRR inequality says that
>
> For any $\alpha>1$, $\delta>1/\alpha$ there is so... | https://mathoverflow.net/users/479223 | Garsia-Rodemich-Rumsey without Markov | Apologies for the earlier error. Upon examining what went wrong, I think I have discovered that no such bound can hold - there is in general no relationship between $\Phi$ and the Holder norm of $X$.
The idea is to define $X$ to be a small bump placed uniformly at random. Changing the parameters of this bump gives us... | 2 | https://mathoverflow.net/users/173490 | 449797 | 180,986 |
https://mathoverflow.net/questions/449798 | 0 | Suppose we have a parameter $\theta \in R^{n}$ that defines some noisy observation $z=\mu(\theta)+\eta, z\in R^{m}$ where the noise follows a Gaussian distribution whose covariance is a function of the parameter, i.e. $\eta\sim N(0,\Sigma(\theta))$. Then the observation random variable is $z \sim N(\mu(\theta),\Sigma(\... | https://mathoverflow.net/users/506618 | Derivative of log-likelihood function for Gaussian distribution with parameterized variance | $\newcommand\th\theta\newcommand\si\sigma\newcommand\p\partial\newcommand\ol\overline$There is no reason to get confused here. Indeed, that "the first term of the derivative does not depend on $(z\_j-\mu\_j(\theta))$" does not at all prevent the derivative from taking the zero value.
If e.g. $\mu(\th)=\th$ and $\si\_... | 2 | https://mathoverflow.net/users/36721 | 449803 | 180,989 |
https://mathoverflow.net/questions/449785 | 5 | Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t\_i \to t$ in the *strong operator topology* if $t\_i \xi \to t \xi$ for every $\xi \in H$.
**Question:** Let $B := (B(H)\_1, SOT)$ be the unit ball of $B(H)$ equipped with the strong operator to... | https://mathoverflow.net/users/147609 | Is the unit ball of $B(H)$ a Baire space (with the SOT)? | I would say it is. Below is a sketch to more-or-less reduce to the separable case.
Given a finite set $V$ of unit vectors and some $\epsilon>0$, let
$$
U\_{V,\epsilon}=\{t\in B(H)\_1\mid \|tv\|\leq\epsilon \text{ for every }v\in V\}.
$$
The sets $\bar t + U\_{V,\epsilon}$ form a basis neighborhood for the strong oper... | 3 | https://mathoverflow.net/users/54309 | 449809 | 180,990 |
https://mathoverflow.net/questions/449811 | -1 | I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that
$$ \int\_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \, \mathrm{d}z \ll \frac{x^{-\sigma}}{2(n+1)\pi-|t|/x} \tag{1}$$
where $\sigma=\Re(s), t=\Im(s), x$ is a half-positive integer and $n$ is an arbitrary positive integer.
... | https://mathoverflow.net/users/507786 | On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$ | Your inequality does not make sense, since the RHS has $n$ in it, while the LHS does not. What Titchmarsh claims is the bound
$$\int\_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \mathrm{d}z \ll \frac{x^{-\sigma}}{2\pi-|t|/x}.$$
And he provides a detailed proof stretching 5 lines: "In the second integral we put $z=x+ir$ etc."... | 0 | https://mathoverflow.net/users/11919 | 449820 | 180,994 |
https://mathoverflow.net/questions/449770 | 2 | In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:
1. The sphere spectrum is the unit object
2. The tensor product bifunctor Sp x Sp -> Sp preserves colimits in each variable
Let R be an $E\_\inft... | https://mathoverflow.net/users/131360 | Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique? | If by "$R$ is the unit object" you mean : 1- the object $R$ in $Mod\_R$ is the unit **and** 2- The induced commutative algebra structure on $R = map(1,1)$ is the given commutative algebra structure on $R$, then the answer is yes.
Without the "and", namely if you only have 1-, then there are as many such symmetric mon... | 2 | https://mathoverflow.net/users/102343 | 449829 | 180,997 |
https://mathoverflow.net/questions/449808 | 7 | I've seen it stated in the $\infty$-categorical literature (without proof or reference) that every object in the $\infty$-category $\operatorname{Fun}(\Delta^1, \mathcal{C})$ of morphisms in a stable $\infty$-category $\mathcal{C}$ is a geometric realization of arrows of the form $A \rightarrow A \oplus B$. I may be mi... | https://mathoverflow.net/users/507783 | Are morphisms in a stable $\infty$-category generated by split injections? | Any map $f:A\to B$ fits in a cofiber sequence of arrows $(0\to A)\to (A\to A\oplus B)\to (A\to B)$
In other words, any map is a cofiber of split inclusions. But now cofibers (as any colimit, by the Bousfield-Kan formula) can be rewritten as geometric realizations of coproducts of the involved terms, and thus we can c... | 7 | https://mathoverflow.net/users/102343 | 449831 | 180,999 |
https://mathoverflow.net/questions/449722 | 0 | * Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
* Let
$$
F(x)=\sum\limits\_{m\geqslant 0}f(m)x^m
$$
* Define the operator $\operatorname{SR}$, which is associated with the [series reversion](https://mathworld.wolfram.com/SeriesReversion.html).
* Let $a(n)$ be an integer sequence with generating fun... | https://mathoverflow.net/users/231922 | Series reversion using something like continued fraction | We assume $F(0) \neq 0$, since otherwise we don't satisfy the assumptions for the series reversion. Let $G = G(0)$ be the fixpoint of the recurrence given:
$$G(x) = F\left(\frac{x}{G(x)}\right)$$
Multiply both sides by $\frac{x}{G(x)}$:
$$x = \frac{x}{G(x)} F\left(\frac{x}{G(x)}\right)$$
By inspection of the st... | 1 | https://mathoverflow.net/users/46140 | 449836 | 181,001 |
https://mathoverflow.net/questions/449841 | 5 | Let $A \subset [0, 1]$ be a measurable set, and $\mathbf 1\_A$ its indicator function, viewed as a function on $\mathbb R$. Define for each $\delta > 0$, the function $f\_{A, \varepsilon}: \mathbb R \to [0, 1]$ given by
$$f\_{A, \varepsilon} (x) = \sup\_{r \leq \varepsilon} \frac{1}{2r} \int\_{x-r}^{x+r} |\mathbf 1\_... | https://mathoverflow.net/users/173490 | Quantitative Lebesgue density theorem | Split $[0,1]$ into $n$-dyadic intervals and consider the set of alternating intervals:
$$A\_{n}\triangleq\Big[0,\frac{1}{2^{n}}\Big]\cup \Big[\frac{2}{2^{n}},\frac{3}{2^{n}}\Big]\cup\cdots\cup \Big[1-\frac{2}{2^{n}},1-\frac{1}{2^{n}}\Big]$$
and the function $f\_{n}(x)=1\_{A\_{n}}(x)$. Then we have that for $x\in A\... | 6 | https://mathoverflow.net/users/99863 | 449844 | 181,004 |
https://mathoverflow.net/questions/449838 | 10 | It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a valuation ring if it is integrally closed?
| https://mathoverflow.net/users/500859 | Is it a valuation ring? | This is (essentially) a conjecture of Krull; a counter-example was given by P. Ribenboim: [*Sur une note de Nagata relative à un problème de Krull*](https://link.springer.com/article/10.1007/BF01166564), Math. Zeit. 64, 159-168 (1956). Note that "primaire" means "local of dimension 1", and that "complètement intégralem... | 17 | https://mathoverflow.net/users/40297 | 449845 | 181,005 |
https://mathoverflow.net/questions/449789 | 3 | Given $k \in \mathbb N$, we define $f\_k: \mathbb N \longrightarrow \mathbb N$ by
$$ f\_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } x \text{ is odd} \end{cases} $$
For $k=0$, we have the function of the infamous Collatz conjecture.
*Does there exist f... | https://mathoverflow.net/users/4556 | A mutation of the Collatz disease | Maybe it is worth to write my comments into an answer:
1. The question is equivalent to asking whether all elements of the form $n/3^k$ end up in the collatz cycle.
2. It is still an open question whether there is an integer $n$ where the collatz iterates go to infinity
3. It is still an open question whether there a... | 7 | https://mathoverflow.net/users/3969 | 449849 | 181,007 |
https://mathoverflow.net/questions/449834 | 2 | Let $X$ be any topological space and denote by $\tau\_X$ the topology on $C\_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|\_\psi\mid\psi\in B\_0(X))$ with $\|f\|\_\psi:=\sup\_{x\in X}|f(x)\psi(x)|$ and
$$\tag{1}B\_0(X):=\{\psi:X\rightarrow\mathbb{R} \text{ bounded}\mid \forall\,\varepsilon>0... | https://mathoverflow.net/users/472548 | Does global boundedness ruin Stone-Weierstrass denseness? | If $c\notin\mathfrak{A}$ in general it is not true: e.g.: consider the case where $X$ is the real line , $\mathfrak{A}$ is the algebra of the polynomials, and $c$ is the function $-e^x$. Then $\mathfrak{A}\_c$ is empty.
If $\mathfrak{A}$ is also a lattice, it is true:
Note that for any $u,v\in C\_b(X;\mathbb{R})$ one... | 2 | https://mathoverflow.net/users/6101 | 449853 | 181,008 |
https://mathoverflow.net/questions/436711 | 5 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$A closed subgroup $ \Gamma $ of a Lie group $ G $ is called Lie primitive if it is not contained in any proper positive dimensional closed subgroup.
Let $ G $ be a compact group. Then a Lie primitive subgroup $ \Gamma $ of $ G $ must b... | https://mathoverflow.net/users/387190 | Compact Lie group has finitely many Lie primitive subgroups | Yes, it is true. Let us show that, in a compact Lie group $G$ with $G^0$ nonabelian, finite subgroups that are not contained in any proper subgroup of positive dimension have bounded cardinal.
By contradiction, let finite subgroups $G\_n$ have cardinal tending to infinity, and not contained in any proper subgroup of ... | 2 | https://mathoverflow.net/users/14094 | 449855 | 181,009 |
https://mathoverflow.net/questions/449602 | 5 | I apologize as I am certain this is not research-level, but several days have gone by without an answer on stackexchange (<https://math.stackexchange.com/questions/4724245/jacobian-criterion-for-zariski-cotangent-space-over-arbitrary-field>).
Section 7.2 of the notes on etale cohomology (<https://people.dm.unipi.it/t... | https://mathoverflow.net/users/91041 | Jacobian criterion for Zariski cotangent space over arbitrary field (X-post from SE) | Here's an attempt at solution.
Let $I=(f\_i)$ be the ideal defining the scheme $Spec(A)$ and $\mathtt{n} \subset k[\underline{x}\_i]$ be the maximal ideal s.t. $\mathtt{n}/I=\mathtt{m}$. Then we have the following commutative diagrams
$\require{AMScd}$
\begin{CD}
I/(I \cap \mathtt{n}^2) @>\alpha>> \mathtt{n}/\matht... | 2 | https://mathoverflow.net/users/495875 | 449877 | 181,013 |
https://mathoverflow.net/questions/449860 | 0 | I have scouted books on statistics, probability, and consulted with mathematicians whom I know, and I have not succeeded in getting a satisfactory answer to this basic question:
**Question:** Exactly what does it mean for a sequence of points to be sampled from a given probability measure?
I am looking for the prop... | https://mathoverflow.net/users/507776 | Definition of sequence sampled from a measure | What you are asking about is called **equidistribution** with respect to a given probability measure $\mu$ on a topological space $X$. Namely, a sequence $(x\_n)$ is $\mu$-equidistributed if the empirical distributions $(\delta\_{x\_1} + \dots + \delta\_{x\_n})$ weakly converge to $\mu$. Whether $X=\mathbb R^n$, or whe... | 0 | https://mathoverflow.net/users/8588 | 449891 | 181,019 |
https://mathoverflow.net/questions/449904 | 3 | Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-invariant characters of $T$, and the bijection is given by restriction. Is this correct? If so, how could it be proven?
| https://mathoverflow.net/users/507634 | Describing characters of a reductive group in terms of characters of a maximal torus | I thought at first that you meant "trace character of a finite-dimensional, irreducible representation of $G$" by "character of $G$". Then this is false; for example, if $G$ is $\operatorname{SL}\_2$, then $G$ certainly has non-trivial, finite-dimensional, irreducible representations (e.g., the defining representation)... | 5 | https://mathoverflow.net/users/2383 | 449906 | 181,024 |
https://mathoverflow.net/questions/435265 | 10 | I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a *measure* as opposed to a $p$-adic analytic function or as a power series.
**My question is:** What is the motivation behind viewing a $p$-adic $L$-function as a... | https://mathoverflow.net/users/394740 | Why $p$-adic measures? | This is extremely late, but hopefully it's still of some use/interest to you.
p-adic L-functions are usually described as 'p-adic interpolations of special values of L-functions'. For Kubota--Leopoldt's p-adic interpolation of the Riemann zeta function, the relevant special values are all the values $\zeta(1-k)$, whe... | 8 | https://mathoverflow.net/users/61424 | 449907 | 181,025 |
https://mathoverflow.net/questions/449889 | 3 | $\DeclareMathOperator\Fl{Fl}$It is known that $H^\*(\Fl(m)) \cong R^{\mathbb Z}(m)$, where $\Fl(m)$ denotes the variety of complete flags in $\mathbb C^m$, and $R^{\mathbb Z}(m)$ is the coinvariant algebra, which may be defined by $$\frac{\mathbb Z[x\_1,\dotsc,x\_m]}{e\_1(x\_1,\dotsc,x\_m),e\_2(x\_1,\dotsc,x\_m),\dots,... | https://mathoverflow.net/users/131046 | Alternative bases of symmetric polynomials in cohomology ring of flag varieties and coinvariant algebras | I assume we compute cohomology with $\mathbb{Q}$ coefficients. Let $x\_i$ map to $c\_1(U\_i/U\_{i-1})$, like you suggest. Then the polynomial $p\_k(n)$ gets sent to $k!$ times the $k$-th graded piece of the Chern character of the trivial bundle.
| 1 | https://mathoverflow.net/users/470175 | 449908 | 181,026 |
https://mathoverflow.net/questions/449884 | 3 | Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $M$ of permutation (that is a free $\mathbf{Z}$-module that admits a basis permuted by the action of $G$), the Tate-Shaf... | https://mathoverflow.net/users/170999 | The second Tate-Shafarevich group of a permutation module is trivial | We write $G\_w={\rm Gal}(L\_w/K\_v)$.
>
> **Definition.** For $n\ge 1$, we denote
> $$Ш\_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod\_C H^n(C,M)\Big)$$
> where $C$ runs over the cyclic subgroups of $G$.
>
>
>
>
> **Remark.** $Ш^2(L/K,M)\subseteq Ш\_\omega^2(G, M)$. Indeed,
> by the Chebotarev density theorem, fo... | 6 | https://mathoverflow.net/users/4149 | 449921 | 181,029 |
https://mathoverflow.net/questions/449927 | 0 | Let
* $H$ be a infinite dimensional, separable, complex Hilbert space,
* $\{v\_{1\_n}\}\_{n \in \mathbb{N}}$ be a sequence in $H$,
* $V\_1=\operatorname{span}\{v\_{1\_n}\}\_{n \in \mathbb{N}}$
* $U\_1=\overline{V\_1}$.
We know that $\{v\_{1\_n}\}\_{n \in \mathbb{N}}$ is a Schauder basis of $U\_1$
Let $\{v2\_n\}\_... | https://mathoverflow.net/users/108867 | Intersection of Hilbert spaces with Schauder basis | No, the intersection of the $U$'s might even be full. Note that it suffices to find $V$'s of countable dimension, since we can then use the Gram-Schmidt algorithm to find an orthonormal basis, which will be automatically Schauder.
Set $H=L^2([0,1])$, $V\_1$ the space generated by the $\mathbb1\_{[k/n,(k+1)/n)}$, and ... | 4 | https://mathoverflow.net/users/129074 | 449929 | 181,031 |
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