parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/379724 | 2 | Since more than 4 months ago, I have posted a question on Mathstack and I haven't recieved any concrete answers. The link to the original post with the problem and my attempts are [here](https://math.stackexchange.com/questions/3776416/minimum-solution-over-closed-ball-of-h-01-omega).
To summarize, we need to prove t... | https://mathoverflow.net/users/75254 | Minimum solution over closed ball of $H_0^1(\Omega)$ | I think there's a mistake in the definition of your norm $\|\cdot\|\_\kappa$ : it does not seem to be equivalent to the $H^1(\Omega)$ norm (since there's no gradients involved).
I would more simply consider the following norm on $H^1\_0(\Omega)$ :
\begin{align\*}
N(v) := \left(\int\_\Omega \kappa |\nabla v|^2\right)^... | 4 | https://mathoverflow.net/users/27767 | 379726 | 158,128 |
https://mathoverflow.net/questions/379723 | 1 | Is there a database that has all the known particular values of the $j$-invariant?
| https://mathoverflow.net/users/87910 | Is there a database about the particular values of $j$-invariant? | What do you mean by "known"? For any $\tau\in\mathbb C$ with $\text{Im}(\tau)>0$, one can compute $j(\tau)$ to as much precision as one's computer allows, but presumably that's not what you mean. In general, if $\tau$ is algebraic and $[\mathbb Q(\tau):\mathbb Q]\ge3$, then $j(\tau)$ is transcendental over $\mathbb Q$,... | 8 | https://mathoverflow.net/users/11926 | 379727 | 158,129 |
https://mathoverflow.net/questions/365213 | 18 | Any homology sphere is [stably parallelizable](https://mathoverflow.net/questions/242412/are-homology-spheres-stably-trivial), hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}({\Omega^{\text{SO}}\_{5}}) \cong \mat... | https://mathoverflow.net/users/14233 | Oriented cobordism classes represented by rational homology spheres | The necessary condition pointed out by Jens Reinhold is also sufficient: any torsion class $x = [M] \in \Omega^{SO}\_d$ admits a representative where $M$ is a rational homology sphere.
**EDIT**: This is Theorem 8.3 in [$\Lambda$-spheres](http://sites.mathdoc.fr/PMO/PDF/B_BARGE_61_7405.pdf) by Barge, Lannes, Latour, a... | 11 | https://mathoverflow.net/users/171227 | 379729 | 158,130 |
https://mathoverflow.net/questions/379733 | 41 | For example, consider the following problem
$$\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2},\hspace{0.5cm} u(x,0)=f(x),\hspace{0.5cm} u(0,t)=0,\hspace{0.5cm} u(L,t)=0$$
Textbooks (e.g., [Paul's Online Notes](https://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx)) usually apply separ... | https://mathoverflow.net/users/124262 | Do we lose any solutions when applying separation of variables to partial differential equations? | Consider your purported solution $u(x,t)$ at fixed $t$, i.e., think of it as a function only of $x$. Such a function can be expanded in a complete set of functions $f\_n (x)$,
$$
u(x,t)=\sum\_{n} u\_n f\_n (x)
$$
What happens when you now choose a different fixed $t$? As long as the boundary conditions in the $x$ direc... | 39 | https://mathoverflow.net/users/134299 | 379736 | 158,132 |
https://mathoverflow.net/questions/379757 | 2 | Fix $\alpha, \epsilon \in(0,1)$. Take $(S\_n)\_n$ to be any sequence of sets with each $S\_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(S\_n^\ast)\_n$ by, for each $S^\ast\_n$, drawing $\lceil (n!)^{1-\alpha+\epsilon} \rceil$ permutations uniformly at ran... | https://mathoverflow.net/users/10668 | Do enough permutations of an initial set probably cover most permutations? | If $M\_n$ is all permutations of $n$ elements then $\{\pi \circ S\_n: \pi \in M\_n\}$ is precisely a $|S\_n|$-fold cover of $M\_n$. Choosing $S\_n^\ast$ and then forming the composition of $S\_n^\ast$ with $S\_n$ is like sampling this cover. The log likelihood that $\pi\in M\_n$ is not covered by the composition is $\l... | 0 | https://mathoverflow.net/users/10668 | 379765 | 158,142 |
https://mathoverflow.net/questions/379731 | 4 | It is well-known that algebraic $K$-theory is $\mathbb{A}^1$-invariant for regular Noetherian schemes. The way this is proved is usually to prove that $K$-theory of coherent sheaves i.e. $G$-theory matches with the $K$-theory of vector bundles. Then $\mathbb{A}^1$-invariance is proved for $G$-theory using the localizat... | https://mathoverflow.net/users/127776 | Homotopy invariance of $K_0$ | I will use capital letters to denote $R[t]$ modules and capital letters subscripted with $0$ to denote $R$ modules. All rings are assumed noetherian and all modules are assumed finitely generated. The key lemma is due to Swan:
**Lemma**: Let $M\subset N\_0[t]$. Then there exists a short exact sequence of the form
$$0... | 7 | https://mathoverflow.net/users/10503 | 379768 | 158,145 |
https://mathoverflow.net/questions/379732 | 1 | Define $x\_0=0$ and $x\_{i+1} = F(x\_i)$ for all integers $i \ge 0$, where $F(x)$ is a polynomial with integral coefficients such that $x\_{i}\rightarrow\infty$ as $i$ grows large.
Suppose $l(p)$ is the least positive integer such that $p|x\_{l(p)}$ for some prime $p$.
Then if we let
$$P\_{(0<\alpha<1)} = \{ p\in \ma... | https://mathoverflow.net/users/160943 | do we have any result proving a strong upper bound on the cardinality of set $P_{\alpha}(x)$ for some large parameter $x$? | As pointed out already in the comments, you need extra assumptions to ensure that you get the behavior you want. I shall comment under such assumptions.
Consider the simplest case of $F$ a linear polynomial, say for instance $F\_e(x):=ex+(e-1)$ for a fixed $e \in \mathbb N$. Then you are asking for the multiplicative... | 1 | https://mathoverflow.net/users/nan | 379773 | 158,146 |
https://mathoverflow.net/questions/379775 | 2 | ***Edit:*** *I prefer to formulate first the problem as Fedor Petrov suggests in the comments*:
We are given a multiset $F$, initially containing only the single integer $h$. Sequentially, at each time step, one of the currently maximal elements $t$ of $F$ is replaced by $0, 1, 2, \ldots, t-1$. The process ends when ... | https://mathoverflow.net/users/115803 | Combinatorial process on multisets of integers | I claim that $n(k,h)=1$ when $k\in \{h,h-1\}$ and $n(k,h)=2^{h-k-1}$ otherwise (if $k<h-1$).
Proof by induction. For $h=1$ this is clear. Induction step. Assume that for smaller values of $h$ this holds. For $k\in \{h-1,h\}$ this is clear. So assume that $k<h-1$. We have $n(k,h)=n(k,h-1)+n(k,h-2)+\ldots$. Thus the clai... | 3 | https://mathoverflow.net/users/4312 | 379781 | 158,148 |
https://mathoverflow.net/questions/379782 | 4 | [Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments]
Let $W,X$ be cancellative invertible-free [1] monoids. A map $e\colon W\rightarrow X$ is a *homography* [2] if it is non-decreasing in the p... | https://mathoverflow.net/users/47107 | Is every invertible-free cancellative monoid action represented by "shifting" certain maps? | This is an answer for the special case of a free monoid where this is somehow a different language for standard facts about transducers and automata although I can’t write an exact reference in this language. I strongly suspect that this case is indeed special and will not generalize but this is too long for a comment ... | 2 | https://mathoverflow.net/users/15934 | 379795 | 158,152 |
https://mathoverflow.net/questions/379737 | 6 | For a gaussian vector variable $w\sim N(0,I\_{n\times n})$, the moments of square norm are $\mathbb{E} \|w\|^{2 r} = \prod\_{t=0}^{r-1} (n + 2 t)$.
Based on [Isserlis' theorem](https://www.wikiwand.com/en/Isserlis%27_theorem), $\mathbb{E} \|w\|^{2 r}$ can also be evaluated as
$$\sum\_{\pi\in \mathcal{P}([r]), |\pi|\l... | https://mathoverflow.net/users/123836 | A combinatorics problem and the probability interpretation | Fix $n$. Let
$$ G(x) = \sum\_{i=0}^n \frac{n!}{(n-i)!}\frac{x^i}{i!} = (1+x)^n. $$
Let
$$ F(x) = \sum\_{j\geq 1}\frac 12 (2j-1)!!\frac{x^j}{j!} =
\frac{1}{2\sqrt{1-2x}}-\frac 12. $$
By the Compositional Formula (Theorem 5.1.4 of *Enumerative
Combinatorics*, vol. 2), the number you want is $r!$ times the
coefficient of... | 7 | https://mathoverflow.net/users/2807 | 379799 | 158,155 |
https://mathoverflow.net/questions/379803 | 7 | What is the geometric meaning of potential automorphy, and conceptually why is it so hard to go from potential automorphy to automorphy on the nose?
Is there an obstruction to descent from potential automorphy to automorphy and does it lie in some Galois cohomology group? Is there an example where a variety is potent... | https://mathoverflow.net/users/168668 | Potential automorphy vs. automorphy | $\DeclareMathOperator\GL{GL}$There is no obstruction to descent from potential automorphy to automorphy lying in some cohomology group. Conjecturally, potential automorphy should be equivalent to automorphy.
To see why, let us take as our primary object that can be automorphic or not, Galois representation instead of... | 10 | https://mathoverflow.net/users/9317 | 379805 | 158,157 |
https://mathoverflow.net/questions/379730 | 5 | There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$.
I'm particularly interested in the information geometric one (using the Fisher-Rao-Riemann metric) and the one induced by... | https://mathoverflow.net/users/170917 | Comparison of Information and Wasserstein Topologies | It is not the case that the Fisher-Rao distance dominates the Wasserstein distance. For instance, it fails for univariate normal distributions $\mathcal{N}(\mu,\sigma)$. In particular, the Wasserstein distance is the Euclidean distance on the half-plane $\mathbb{H}= \{(\mu,\sigma)~|~ \sigma >0\}$. On the other hand, th... | 10 | https://mathoverflow.net/users/125275 | 379811 | 158,158 |
https://mathoverflow.net/questions/379810 | 1 | Let $\phi: G \rightarrow \mathbb{C}$ be a continuous function. We say that $\phi$ is positive type if $\sum\_{i,j=1}^{n} c\_i\bar{c\_j}\phi(g\_{j}^{-1}g\_i)\geq 0$ for all $n \in N, c\_i \in \mathbb{C}, g\_i \in G$.
Let G be a topological group and H be an open subgroup of G and $\phi$ be the characteristic function ... | https://mathoverflow.net/users/137242 | Positive type function on open subgroup | Note that
$$\phi(g\_{j}^{-1}g\_i)=1 \Leftrightarrow g\_{j}^{-1}g\_i \in H$$
On $G$ define $g \equiv h \Leftrightarrow g^{-1}h \in H$.
Now, let $n \in N, c\_i \in \mathbb{C}, g\_i \in G$.
Split $g\_1,.., g\_n$ into left cosets. To make this clear, denote by $H\_1,.., H\_k$ the **non-repeated** left cosets $g\_1H,.... | 3 | https://mathoverflow.net/users/11552 | 379813 | 158,160 |
https://mathoverflow.net/questions/379814 | 4 | We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v\_0 \in V$ such that $\textrm{Stab}(v\_0)= H$. This gives a $G$-equivariant embedding \begin{equation} G/H \rightarrow V \\ gH \mapsto gv\_0. \end{equation}
A possible construc... | https://mathoverflow.net/users/171283 | Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation | You can just use the embedding $f\colon G/H=U(n)/(U(k)\times U(n-k))\to M\_n(\mathbb{C})$ given by $f(gH)=gpg^{-1}$, where $p=1\_k\oplus 0\_{n-k}$. This gives a homeomorphism from $G/H$ to the space
$$ P = \{q\in M\_n(\mathbb{C}) : q^2=q^\dagger=q,\;\text{trace}(q)=k\}. $$
The map $f$ is equivariant if we let $G$ act o... | 10 | https://mathoverflow.net/users/10366 | 379819 | 158,161 |
https://mathoverflow.net/questions/378629 | 0 | Given a compact set $E$ with non-empty interior in $R^d$ and some small positive number $r$, what kind of conditions should we put on the set $E$ so that for all $x\in E$, the volume of the intersection of $B(x,r)$ with $E$ is uniformly bounded away from $0$? Here $B(x,r)$ is the ball of radius $r$ centered at $x$.
| https://mathoverflow.net/users/124768 | conditions on the boundary of a compact set to ensure the volume of the intersection of a small ball with the set doesn't vanish | For any fixed $r$, $f(x):=vol(B(x,r)\cap E)$ is Lipschitz, hence continuous, with respect to $x$. Indeed, $\vert f(x)-f(y)\vert$ is at most the volume of the symmetrical difference of the balls. Hence, the minimum of $f$ is reached at some $x\in E$. So, it is enough that $f>0$. This will be the case if you suppose e.g.... | 0 | https://mathoverflow.net/users/105095 | 379843 | 158,167 |
https://mathoverflow.net/questions/379856 | 1 | Let $K:=\mathbb{C}$, and let $R:=K[x\_1,\dots , x\_n]$.
Then, a system of polynomial equations $p\_1=0, p\_2=0, \dots , p\_r = 0$, where the $p\_i$ are polynomials in the $x\_j$, has finitely many solutions $\Leftrightarrow$ the Krull dimension of $R/I$ is equal to $0$, where $I:=\langle p\_1, p\_2, \dots , p\_r \ran... | https://mathoverflow.net/users/91107 | Krull dimension and elimination theory over the integers | This should be a comment, but my reputation is low, so I have to post this.
I assume that you want the solutions of $p\_1=\dots=p\_r=0$ to be in $\mathbb{Z}^n$. Then the answer is no. Take, for example, $r=1$, $n\geq 2$ and $p\_1$ to be any polynomial in $\mathbb{Z}[X\_1,\dots,X\_n]$ that has no solutions in $\mathbb... | 2 | https://mathoverflow.net/users/171303 | 379861 | 158,172 |
https://mathoverflow.net/questions/379847 | 0 | I'm trying to prove that a sequence of functions $(k\_N)\_{N\in\mathbb{N}}$
$$k\_N(\vec{y}):=e^{-N^2r(\vec{y},Q\vec{y})}\sqrt{\frac{r}{\pi}}^kN^{k}$$
where $r>0$, $k>2$ and
**Edit**: I have forgot to say, that $k\in\mathbb{N}$ is the size of the matrix $Q$ i.e. $Q$ is $k\times k$. This means that $\det Q=0$ and $... | https://mathoverflow.net/users/171313 | Error function of multivariate Gaussian | $\newcommand\R{\mathbb R}$Let us write $x$ for $\vec x$ and $t$ for $\eta$. Then
$$\int\_{x\in\R^k\colon|x|>t}k\_n(x)\,dx=\infty$$
for all real $t>0$.
Indeed, we can write
$$Q=I\_k-\frac1k\,1\_k1\_k^T,$$
where the matrix $I\_k$ is the $k\times k$ identity matrix and $1\_k$ is the $1\times k$ column matrix with unit e... | 0 | https://mathoverflow.net/users/36721 | 379864 | 158,174 |
https://mathoverflow.net/questions/355308 | 6 | The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u\_{xxxx} + 8x\left(1+x^{2}\right)u\_{xxx} + 4\left(1+3x^{2}\right)u\_{xx}+ t\left(2xuu\_{xx} + \left(1+x^{2}\right)\left(uu\_{xx}\right)\_{x} - 4\left(1+3x^{2}\right)u\_{xxt} - 4x\left(1+x^{2}\right)u\_... | https://mathoverflow.net/users/99716 | Lax pair of an integrable non-linear PDE | You could try using the Wahlquist-Estabrook prolongation structure technique, per H.D. Wahlquist and F.B. Estabrook, J. Math. Phys **16** (1975) 1-7 (covering the Korteweg-deVries equation), & F.B. Estabrook and H.D. Wahlquist J. Math. Phys. **17** (1976) 1293-1297 (covering the nonlinear Schrödinger equation).
A non... | 2 | https://mathoverflow.net/users/106467 | 379878 | 158,178 |
https://mathoverflow.net/questions/379854 | 3 | There is a forgetful functor from condensed abelian groups to condensed sets. According to Scholze's notes, this has an adjoint $T \mapsto \mathbb{Z}[T]$ (which is the sheafification of the functor sending any extremally disconnected set $S$ to the free abelian group $\mathbb{Z}[T(S)]$).
Now in [Scholze's notes](http... | https://mathoverflow.net/users/170467 | Question about adjoint of forgetful functor from condensed abelian groups to condensed sets | Noting that $\underline{S} = \operatorname{Cont}(\cdot, S) = \operatorname{Hom}\_{\operatorname{ProFin}}(\cdot, S)$, this reduces
to the Yoneda lemma:
$$\operatorname{Nat}(h\_S, M) \xrightarrow{\sim} M(S), \eta \mapsto \eta\_{S}(\operatorname{id}\_S)$$
where $h\_S = \operatorname{Hom}(\cdot, S)$ denotes the contrav... | 4 | https://mathoverflow.net/users/116837 | 379885 | 158,183 |
https://mathoverflow.net/questions/379153 | 5 | Consider a set $V$ of $n$ vertices, and three degree sequences $a\_i$, $b\_i$ and $c\_i$ such that $c\_i = a\_i+b\_i$, $i=1..n$.
Assume these degree sequences are graphical: there exist *simple* graphs (no loop, no multiple edge) with degree sequence $a\_i$, $b\_i$, and $c\_i$.
**Does this imply that there exists a... | https://mathoverflow.net/users/158328 | Simple graphs with prescribed degrees as disjoint union of simple subgraphs with prescribed degrees | The answer to this question is ***No***.
Let us assume $V = \{1,2,3,4,5,6\}$ and consider degree sequences $a = [3,2,2,1,0,0]$, $b = [1,0,0,3,2,2]$ and $c = a+b = [4,2,2,4,2,2]$.
The only simple graph with degree sequence $a$ is given by $1-2$, $1-3$, $1-4$, and $2-3$. Similarly, the only one with degree sequence $... | 5 | https://mathoverflow.net/users/158328 | 379897 | 158,185 |
https://mathoverflow.net/questions/379890 | 3 | From *endomorphisms of rank 1* of the full transformation semigroup $[n]^{[n]}$ or *idempotents* in $[n]^{[n]}$, we have
$$c\_n:=\sum\_{m=1}^n\binom{n}mm^{n-m}.$$
Denote the $2$-adic valuation of $x$ by $\nu\_2(x)$.
>
> **QUESTION.** Is it true that $\nu\_2(c\_n)=\nu\_2(n-1)$, for $n\geq2$?
>
>
>
| https://mathoverflow.net/users/66131 | Divisibility question while enumerating endomorphisms: PART 2 | Yes, this is true. The idea is to get $n-1$ out of the sum as a multiple.
We have $$c\_n=1+\sum\_{m=1}^{n-1}m{n\choose m}m^{n-1-m}=1+\sum\_{m=1}^{n-1}n{n-1\choose m-1}m^{n-1-m}$$
We see that if $n$ is even, then $c\_n$ is odd.
Now let $n>1$ be odd. Then proceed this way:$$c\_n=1+n\sum\_{m=1}^{n-1}{n-1\choose m-1}+n... | 4 | https://mathoverflow.net/users/4312 | 379919 | 158,190 |
https://mathoverflow.net/questions/379921 | 28 | Context: A submission to a very good generalist journal X received one positive referee report recommending publication and two shorter opinions which both deemed the paper a solid and valuable contribution and thus worthy of publication but perhaps not a priority given the backlog this journal has, thus only weakly re... | https://mathoverflow.net/users/130882 | Can one reuse positive referee reports if paper ends up being rejected? | For many journals the referee is asked to tick a box when they submit their report to indicate whether or not they (1) allow the report to be used for another journal and (2) whether their identity may be disclosed to the editors of that other journal.
Given this practice, the answer to question 1 would be a "yes". W... | 23 | https://mathoverflow.net/users/11260 | 379922 | 158,191 |
https://mathoverflow.net/questions/379853 | 6 | One of the initial motivating factors for learning category theory, besides needing it for my work, was the idea that almost all mathematical notions I would encounter could be understood using categories one way or another.
That’s largely been borne out at the $1$-categorical level, and (almost?) completely vindicat... | https://mathoverflow.net/users/92164 | Multicategories vs Categories | Multicategories and bicategories, to me, are first of all completely
orthogonal generalisations of *monoidal* categories, with virtual
double categories as a common generalisation of multicategories and
(strict) $2$-categories (they are to multicategories as categories are
to monoids, or $2$-categories to monoidal cate... | 6 | https://mathoverflow.net/users/165619 | 379930 | 158,195 |
https://mathoverflow.net/questions/379912 | -2 | Let $E$ be a $\mathbb R$-Banach space and $\mathcal M\_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M\_+(E)$, let $$\left.\lambda\right|\_\delta(B):=\lambda\left((B\cap\left\{x\in E:\left\|x\right\|\_E\le\delta\right\}\right)$$ and $$\left.\lambda\right|\_\delta^c(B... | https://mathoverflow.net/users/91890 | If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent | I suppose here that $E$ is separable.
Let $\epsilon > 0$ and fix a sequence $\epsilon\_i > 0$ such that $\sum\_{i=1}^\infty \epsilon\_i < \epsilon$.
Let $B\_r$ denote the closed ball of radius $r$. Set $A\_1 = B\_1^c = \{x : \|x\| > 1\}$ and $A\_i = \{x : \frac{1}{i} < \|x\| \le \frac{1}{i-1}\}$ for $i \ge 2$, so t... | 1 | https://mathoverflow.net/users/4832 | 379933 | 158,196 |
https://mathoverflow.net/questions/379089 | 4 | In Chapter II of his book *Non-Euclidean Geometry* (1961; first published in Polish, 1956), Stefan Kulczycki defines a mapping of the hyperbolic plane onto the interior of a disk. Its construction begins by specifying a center *O* and an acute angle *α*. Given those, the image *P’* of an arbitrary point *P* is found as... | https://mathoverflow.net/users/165237 | Mapping the hyperbolic plane onto the interior of a disk | According to the ["Hjelmslev transformation" page](https://en.wikipedia.org/wiki/Hjelmslev_transformation)
on the English language Wikipedia, this mapping is due
to Johannes Hjelmslev himself who studied its properties
and proved it preserves straightness.
The corresponding ["Transformación de Hjelmslev" page](https:... | 2 | https://mathoverflow.net/users/38711 | 379943 | 158,201 |
https://mathoverflow.net/questions/379842 | 10 | Suppose $(G,+)$ is a countable abelian group and $p$ is a prime number such that:
1. The subgroup $pG$ has finite index in $G$, and
2. For every $n \in \mathbb{N}$, $G$ contains an element of order $p^n$.
Must $G$ contain a subgroup isomorphic to the Prüfer $p$-group?
I have tried carrying out a pigeonhole-type a... | https://mathoverflow.net/users/171304 | Do these properties of a countable abelian group guarantee a Prüfer subgroup? | Yes, it must. And $G$ doesn't need to be countable.
Let $H$ be the $p$-primary component of the torsion subgroup of $G$. Then the natural map $H/pH\to G/pG$ is injective, so $H$ also satisfies (1), and clearly satisfies (2). So, replacing $G$ by the subgroup $H$, we shall assume that $G$ is a $p$-group.
Let $X$ be ... | 11 | https://mathoverflow.net/users/22989 | 379945 | 158,202 |
https://mathoverflow.net/questions/379946 | 17 | I am looking for an *explicit* (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation
$$\frac{\partial L}{\partial q^i}=\frac{d}{dt}\frac{\partial L}{\partial \dot q^i}, \, \, i=1,2,3,$$
with the same solutions.
I prefer to ... | https://mathoverflow.net/users/16183 | Example of ODE not equivalent to Euler-Lagrange equation | **Note:** *I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first reading of his paper). This also addresses the question of time-dependent Lagrangians originally raised by the OP.*
... | 22 | https://mathoverflow.net/users/13972 | 379952 | 158,204 |
https://mathoverflow.net/questions/379947 | 2 | Let $S=\sum\_{n=1}^\infty a\_n$, be a semi-convergent series with $T=\sum\_{n=1}^\infty a\_n^2 < \infty$ and $\sum\_{n=1}^\infty |a\_n|=\infty$. Under which conditions are the following formulas valid? They are of course valid for finite sums, but not necessarily for infinite sums.
$$S^2 = T + 2 \sum\_{n=1}^\infty a\... | https://mathoverflow.net/users/140356 | Squaring a semi-convergent series | For
$$S^2 = T + 2 \sum\_{n=1}^\infty a\_n w\_n \mbox{ with } w\_n=\sum\_{m=n+1}^\infty a\_m\tag{1}$$
to be valid, it is enough that $\sum\_n a\_n$ be any alternating series with $\sum\_{n=1}^\infty a\_n^2<\infty$ and $|a\_n|$ converging to $0$ (eventually) monotonically.
Indeed, since (1) holds for finite sums, this ... | 3 | https://mathoverflow.net/users/36721 | 379958 | 158,206 |
https://mathoverflow.net/questions/379791 | 8 | Large set axioms are notions corresponding to large cardinals on constructive set theories like $\mathsf{IZF}$ or $\mathsf{CZF}$. The notion of *inaccessible sets*, *Mahlo sets*, and *2-strong sets* correspond to inaccessible, Mahlo, and weakly compact cardinals on $\mathsf{ZFC}$.
(See Rathjen's [*The Higher Infinite i... | https://mathoverflow.net/users/48041 | Reference request: proof-theoretic strength of $\mathsf{KP}$ with recursively large ordinals and $\mathsf{CZF}$ with large set axioms | Rathjen stated rough sketch of proof that intuitionistic theory (e.g. $\mathsf{CZF}+(\forall x)(\exists I)[x\in I\land I \text{ is inaccessible}]$) has at least the strength of the classical one (e.g. $\mathsf{KP}\omega+(\forall\alpha)(\exists\kappa)[\alpha<\kappa\land\kappa\text{ is recursively inaccessible}]$) in [Th... | 2 | https://mathoverflow.net/users/149565 | 379969 | 158,214 |
https://mathoverflow.net/questions/379944 | 2 | Let $g(x)$ be a polynomial with integral coefficients.We define $\gamma(g(x))$ to be the degree of the non constant polynomial $r(x)$ which divides $g(x)$ for all $x$ and also has minimal degree.
For $r\geq 1$, We define the sequence $a\_{g}$ for some polynomial $g(x)$ as follows:
$\clubsuit)a\_{g}(1)=g(x)$
$\clu... | https://mathoverflow.net/users/160943 | If $a_{g}(1)=g(x)$ and $a_{g}(r)=g(a_{g}(r-1))$ for $r>1$ then is it true that $\limsup\limits_{r\to\infty}\gamma(a_{r})=\infty?$ | Yes, it is true, and I'm guessing it's well-known. (For example, it might be a theorem in Chapter 3 of Silverman's "The Arithmetic of Dynamical Systems", but I don't own that book yet.)
It is useful to use the terminology of arithmetic dynamics. We call the polynomials $a\_{f}(r)$ the iterates of $f$, and for a compl... | 3 | https://mathoverflow.net/users/48142 | 379980 | 158,216 |
https://mathoverflow.net/questions/379978 | 0 | Suppose $G$ is a finite group and $V$ is a finite-dimensional representation of $G$ over a field $k$. I'd like to write $V$ as a tensor product $V\_1 \otimes V\_2 \otimes \dots V\_n$ satisfying
(1) Each $V\_i$ is a representation of $G$.
(2) No $V\_i$ can be written as a tensor product of two or more $G$-representa... | https://mathoverflow.net/users/171437 | Prime factorization of a group representation | These factorizations are not unique, and there is already a counterexample for the smallest nontrivial group $G = C\_2$. The trivial representation has character $(1, 1)$ (where the first entry is the value of the character on the identity and the second entry is the value of the character on $-1$), and the sign repres... | 6 | https://mathoverflow.net/users/290 | 379987 | 158,220 |
https://mathoverflow.net/questions/379985 | 2 | I would like to read article(s) that provide the “state of the art” on the following open problem:
“Enumerate all convex uniform 5-polytopes.”
This problem is posted on the “Open Problem Garden” (<http://garden.irmacs.sfu.ca/op/convex_uniform_5_polytopes>), but no citation is given. (The same open problem is also l... | https://mathoverflow.net/users/112337 | State-of-the-art article on "uniform 5-polytopes?" | There is some information in M.Winter's MSE question, [How many uniform polytopes are there in higher dimensions?](https://math.stackexchange.com/q/3063822/237). There I quote Egon Shulte's "Semiregular and Uniform Convex Polytopes,"
emphasizing [Wythoff's construction](https://en.wikipedia.org/wiki/Wythoff_constructio... | 1 | https://mathoverflow.net/users/6094 | 379992 | 158,222 |
https://mathoverflow.net/questions/379970 | 4 | Let $\mathcal{D}\_{i}$ be a family of triangulated categories, labelled by a countable poset $I$ with a lowest element. Suppose further that for $i\leq j$, we have exact functors $F\_{i,j}: \mathcal{D}\_{i} \to \mathcal{D}\_{j} $ (and $F\_{i,i}=Id$).
Is the 2-(co)limit $\operatorname{colim}\_{I}\mathcal{D}\_{i}$ a tr... | https://mathoverflow.net/users/171433 | 2-limits of triangulated categories | This is the kind of problem where working with stable $\infty$-categories is much easier than triangulated categories. Lurie considers the $\infty$-category of $\infty$-categories $Cat\_\infty$, and the subcategory $Cat\_\infty^{Ex}$ of stable $\infty$-categories and exact functors, and shows that $Cat\_\infty^{Ex}$ is... | 5 | https://mathoverflow.net/users/1310 | 379993 | 158,223 |
https://mathoverflow.net/questions/380000 | 11 | Consider a variant of set theory with these axioms:
* Extensionality,
* Regularity (foundation),
* Separation,
* Powerset,
* Axiom of Choice, and
* ~~[Transitive closure of a set-like relation is set-like](https://proofwiki.org/wiki/Transitive_Closure_of_Set-Like_Relation_is_Set-Like).~~ ***Update:*** *This did not e... | https://mathoverflow.net/users/9550 | Is this set theory equivalent to ZFC? | Accepting the convention that it is a logical axiom that the universe is nonempty, the answer is yes. We will formalize the transitive closure axiom schema (TC) as follows: for any definable (with parameters) binary relation $R,$ if for all $x,$ $\{y: y R x\}$ is a set, then for all $x,$ there is a set $T$ such that $x... | 12 | https://mathoverflow.net/users/109573 | 380004 | 158,229 |
https://mathoverflow.net/questions/380025 | 0 | Let $\mathbb{S}^n$ be the $n$-sphere. I would like to know if anyone knows of the following result in the literature (or whether anyone knows a proof/counterexample).
>
> Let $f\colon\mathbb{S}^1\times[0,1]\to\mathbb{S}^2$ be a $C^1$-function such that $f(x,0)\ne p$ for all $x\in\mathbb{S}^1$. Then there exists a c... | https://mathoverflow.net/users/166628 | Existence of certain continuous curves | Here's a counter-example in the smooth case: take $p$ as the North pole, $f(\cdot,0)$ the Equator, and as $t$ grows, $f(\cdot,t)$ is a parallel of higher and higher latitude, until $f(\cdot,1)=p$.
If $f$ starts from the South pole instead and goes northwards to the North pole, no matter what $g(0)$ is chosen to be, i... | 5 | https://mathoverflow.net/users/165657 | 380027 | 158,230 |
https://mathoverflow.net/questions/380011 | 9 | I want to construct the Hilbert class field of $K=\Bbb Q(\zeta\_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.
| https://mathoverflow.net/users/165415 | how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$? | The cyclotomic field $K=\mathbb{Q}(\zeta\_{23})$ contains the quadratic field $F=\mathbb{Q}(\sqrt{-23})$, and $F$ has class number $3$ (it is the first quadratic field, when those are ordered by absolute value of the discriminant, that has class number divisible by $3$). Thus, $F$ has an unramified cubic extensions $H$... | 19 | https://mathoverflow.net/users/35416 | 380031 | 158,234 |
https://mathoverflow.net/questions/380015 | 0 | Note: this is [cross-posted](https://math.stackexchange.com/questions/3964161/coding-a-function-g-kappa-to-v-zeta1-by-an-element-of-v-zeta1) from MSE.
This question is about the following remark (modified to be self-contained), found in Donald Martin's [book](https://www.math.ucla.edu/%7Edam/booketc/D.A._Martin,_Det... | https://mathoverflow.net/users/171458 | Coding a function $g:\kappa\to V_{\zeta+1}$ by an element of $V_{\zeta+1}$ | Fix a flat pairing function $p : V\times V\to V$. For any infinite ordinal $\alpha$, if $S$ is a binary relation on $V\_\alpha$,
let $A\_S = p[S]$, which is an element of $V\_{\alpha+1}$ coding $S$. The coding in the second sentence can be implemented by setting $g(0)$ equal $A\_R$ and $g(1+\alpha) = \tilde{g}(\alpha)$... | 3 | https://mathoverflow.net/users/102684 | 380045 | 158,237 |
https://mathoverflow.net/questions/380006 | 2 | Given a hyperplane $\alpha^T x = \beta$ in $\mathbb R^n$, with $\beta > 0, \alpha\_i > 0$ **for all** $i \in [n]$. Then for any $\{v^i\} \subseteq \{x \in \mathbb Z^n\_+ \mid \alpha^T x = \beta\}$, it's obvious to see that there must have: $\{v\_i\}$ forms an antichain with respect to the component-wise order. My quest... | https://mathoverflow.net/users/129960 | Existence of a hyperplane with strictly positive coefficients to contain an antichain in $\mathbb{Z}^n_+$ | Here is a counterexample. Consider vectors $v^1 = (4,4,1,1)$, $v^2 = (1,1,4,4)$, and $v^3 =(3,3,3,3)$. They form an antichain, as required. Further, $d = 3 < n =4$. However, there is no hyperplane $\{x:\alpha x = \beta\}$ with all $\alpha\_i > 0$ that contains all vectors $v^i$.
**Discussion.** It's not hard to see t... | 2 | https://mathoverflow.net/users/26349 | 380046 | 158,238 |
https://mathoverflow.net/questions/380041 | 12 | In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H\_1, H\_2 \subseteq G$ are subgroups that are **not conjugate** as subgroups of $G$ but **are conjugate** as subgroups of $G^{\mathbb{... | https://mathoverflow.net/users/4730 | Non-conjugate subgroups that are conjugate in complexification | Define $M\_s=\begin{pmatrix}0 & 0 & 1 & 0\\ 0 & 0 & 0 & s\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$, and $G\_s=\exp(\mathbf{R}M\_s)\subset\mathrm{SL}\_4(\mathbf{R})$. Then $G\_1$ and $G\_{-1}$ are not conjugate in $\mathrm{SL}\_4(\mathbf{R})$ while they are conjugate in $\mathrm{SL}\_4(\mathbf{C})$.
---
$M\_1... | 14 | https://mathoverflow.net/users/14094 | 380048 | 158,239 |
https://mathoverflow.net/questions/380049 | 2 | I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors write down the optimality conditions for the problem
$$\min\_{x\in Q} f(x) \text{ subject to } G(x)\in K,$$
where $Q$ and $K$ ... | https://mathoverflow.net/users/110674 | Normal cones and KKT conditions | The answer to my question became clear when I stumbled upon Example 2.62 and Equation 2.110 from the book, which are located in pages 50/51. Let me state that as a lemma for completeness:
**Lemma:** Let $X$ be a banach space, let $K\subseteq X$ be a closed convex cone, and let $x\in K$, then
$$N\_K(x)= K^\circ \cap... | 2 | https://mathoverflow.net/users/110674 | 380050 | 158,240 |
https://mathoverflow.net/questions/380047 | 4 | Is there an exotic $\mathbb{R}^4$ admitting an integrable almost complex structure?
| https://mathoverflow.net/users/8003 | Exotic $\mathbb{R}^4$ with a complex structure? | It is a result of Gromov that an open manifold of dimension six or less admits a complex structure if and only if it admits an almost complex structure; see the corollary on page 103 of his book *Partial Differential Relations*. As $\mathbb{R}^4$ is contractible, every exotic $\mathbb{R}^4$ is parallelisable. Therefore... | 15 | https://mathoverflow.net/users/21564 | 380051 | 158,241 |
https://mathoverflow.net/questions/379942 | 10 | What is the minimal $\delta$ such that the hyperbolic plane is $\delta$-hyperbolic, in the sense of the four point definition of Gromov?
**Four point definition of Gromov:** A metric space $(X, d)$ is $\delta$-hyperbolic if, for all $w, x, y, z \in X$,
$$ d(w, x) + d(y, z) \leq \text{max}\{d(x, y) + d(w, z), d(x, z) ... | https://mathoverflow.net/users/16654 | Optimal $\delta$ for Gromov's $\delta$-hyperbolicity of the hyperbolic plane | Indeed, the hyperbolic plane is $\log(2)$-hyperbolic (with the 4-point definition of hyperbolicity) and this is the optimal constant. The result is nontrivial and first appeared as Corollary 5.4 in
*Nica, Bogdan; Špakula, Ján*, [**Strong hyperbolicity**](http://dx.doi.org/10.4171/GGD/372), Groups Geom. Dyn. 10, No. 3... | 7 | https://mathoverflow.net/users/39654 | 380055 | 158,243 |
https://mathoverflow.net/questions/380056 | 4 | Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere.
Maybe it is just an easy consequence of properties of characters that I'm not aware of, anyway thank you in advance for any help/answers/suggestions... | https://mathoverflow.net/users/146431 | Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$ | Summary: I consider the limit $\lim\_{n\to\infty}\prod\_{i=1}^n i^{\chi(i)}$ (let me drop the $\mod s\mathbb Z$ for brevity). If we restrict to $n$ divisible by $s$, then the limit will always be equal to zero. If we consider the limit over all $n$, the limit will never exist. [The question was edited to only ask about... | 8 | https://mathoverflow.net/users/30186 | 380060 | 158,245 |
https://mathoverflow.net/questions/380005 | 4 | Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup\_{g\in G} \ g\cdot P)$$ be the convex hull.
**Question**: does there exist a compact rational polytope $Q \s... | https://mathoverflow.net/users/29730 | Existence of a fundamental domain for the convex hull of group action on a rational polytope | I will assume that by an "arithmetic group" you mean a subgroup $\Gamma$ commensurable to the intersection $GL(n, {\mathbb Z})\cap G$, where $G< GL(n, {\mathbb R})$ is an algebraic subgroup defined over a number field $F$; actually, $F={\mathbb Q}$ will suffice for my purposes. (I prefer to use Greek letters to denote ... | 1 | https://mathoverflow.net/users/39654 | 380062 | 158,246 |
https://mathoverflow.net/questions/380061 | 4 | This is Example 6.47 in Saveliev's book *Invariants for homology $3$-spheres*:
>
> Let us consider a two-component link $\mathcal L = L\_1 \cup L\_2$ in
> $S^3$ such that $\mathrm{lk}(L\_1,L\_2) = \pm 1$ and the component $L\_1$ is an
> unknot. Let $\Sigma\_p (\mathcal L)$ be the integral homology sphere
> obtained... | https://mathoverflow.net/users/nan | Mazur homology spheres | Saveliev writes:
>
> they generalize the original Mazur's *example*
>
>
>
and not "exampleS". The original example he's referring to is a single contractible 4-manifold with a boundary that's not the 3-sphere. Then there is a wealth of Mazur (4-)*manifolds* which are obtained, as you say, by attaching a 2-hand... | 5 | https://mathoverflow.net/users/13119 | 380063 | 158,247 |
https://mathoverflow.net/questions/380036 | 4 | Let $V,W$ be vector spaces and $X\subset V\times W.$ If $X$ is the zero set of a collection of bilinear maps then it satisfies the following properties:
1. $(0,w),(v,0)\in X$ for all $v,w.$
2. If $(v,w)\in X$ then $(\alpha v,w),(v, \alpha w)\in X$ for any scalar $\alpha.$
3. If $(v\_1,w),(v\_2,w)\in X$ then $(v\_1+v\... | https://mathoverflow.net/users/170979 | Characterizing zero sets of bilinear maps | This will not always work. Take $V=W=\mathbb R^2$, and then something like $v\_1=(1,0)$, $v\_2=(0,1)$, $v\_3=(1,1)$, $v\_4=(1,2)$, $v\_5=(2,1)$, and let $X$ be the smallest set that contains $(v\_j, v\_j)$ ($j=1,2,3$), $(v\_4,v\_5)$ and satisfies your conditions. In other words, the vectors here can be multiplied by ar... | 2 | https://mathoverflow.net/users/48839 | 380066 | 158,248 |
https://mathoverflow.net/questions/380064 | 7 | I noticed on the OEIS that there are various sequences (e.g. A050515-A050520) that describe arithmetic progressions whose totients are all equal. For example, we have
$$\varphi(\{1,2\}) = 1$$
$$\varphi(\{8,10,12\}) = 4$$
$$\varphi(\{72,78,84,90\}) = 24$$
$$\varphi(353640 + [0,4]\cdot 210) = 80640$$
$$\varphi(583200 + [... | https://mathoverflow.net/users/168142 | Arithmetic progressions in inverse image of totient function | We have a conditional result.
See Theorem 4 of S. W. Graham, J. J. Holt, C. Pomerance, On the Solutions to $\phi(n) = \phi(n + k)$,
Number Theory in Progress, eds. K. Gyory, H. Iwaniec, and J. Urbanowicz, eds.,
Vol. 2 (de Gruyter, Berlin, 1999), pp. 867-882.
Theorem 4 in above:
Suppose that $j$, $j+k$, $\ldots$, ... | 5 | https://mathoverflow.net/users/21090 | 380073 | 158,250 |
https://mathoverflow.net/questions/379967 | 2 | Let $p(x)$ be a polynomial of degree $n>2$, with roots $x\_1,x\_2,\dots,x\_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V\_m(p)=\sum\_{1\leq i<j\leq n}(x\_i-x\_j)^m.$$
>
> **QUESTION.** For $\deg p(x)=n>2$ and $p'(x)$ its derivative, can you express
> $$\frac{V\_... | https://mathoverflow.net/users/66131 | Ratios of polynomials and derivatives under a certain functional | Here a [SageMath code](https://sagecell.sagemath.org/?z=eJx9jrFuwkAQRPv7ipFSsBsMxJRILnGHlEskNxGJHDis03n3osMu_Pc54qB02W6kN2_WWkWFOrWnwUetvevP9Bz7SaP4tn_x2pG1BRa6YDY31Fpdd06JzSGn10nEDcmf6lF_DNeMK68lzoKM7f_BXO_E6dCmKYPGnN0FDQnv8HcPaD6EvtjcQsqu6ygEWpX8HvCIT_87JEXgnA9vsgpFOOISEwK8IrXaOSoLyGazxRIl4y5bViDNprl3vA-G6mkG3DAmxZ5S... | 3 | https://mathoverflow.net/users/7076 | 380079 | 158,253 |
https://mathoverflow.net/questions/380088 | -2 | We say a set $A\subseteq\mathbb{Z}$ is *arithmetical* if there are integers $a>0,b\geq 0$ such that $A=\{ax+b:x\in\mathbb{Z}\}$.
Is there $S\subseteq\mathbb{Z}$ such that $$S\cap A\neq\varnothing \neq (\mathbb{Z}\setminus S)\cap A$$
for every arithmetical set $A\subseteq\mathbb{Z}$?
| https://mathoverflow.net/users/8628 | Set "crossing" all arithmetical integer sets | Yes. Enumerate all arithmetic sets $A\_1, A\_2, \ldots$. Choose $n\_k\in A\_k$ with $|n\_k|>2^k$. The set $\{n\_1, n\_2, \ldots\}$ intersects any arithmetic set by construction, but it has zero density, thus does not contain a whole arithmetic set.
Actually you may construct such $S$ for each infinite sequence $A\_1,... | 5 | https://mathoverflow.net/users/4312 | 380089 | 158,257 |
https://mathoverflow.net/questions/379977 | 1 | I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, real, harmonic function. So the Beltrami coefficient is just a phase.
Among the almost infinite literature about th... | https://mathoverflow.net/users/171439 | Beltrami equation with harmonic coefficient | Note that, if you take $\phi=0$, then the equation reduces to $w\_y =0$, i.e., if $D\subset C$ is the domain of $w$ and $x:D\to\mathbb{R}$ is the projection on the $x$-axis and has connected fibers, then $w= h(x)$ for some $C^1$ function $h:x(D)\to\mathbb{C}$, and this is the general solution on such $D$.
Something s... | 2 | https://mathoverflow.net/users/13972 | 380094 | 158,260 |
https://mathoverflow.net/questions/309826 | 8 | By a polytope I mean the convex hull of finitely many points. The graph of a polytope is the graph isomorphic to its 1-skeleton. By equivalence of polytopes I mean combinatorial equivalence, i.e. their face lattices are isomorphic.
I know that two polytopes can have isomorphic graphs while being non-equivalent, e.g. ... | https://mathoverflow.net/users/108884 | Can two non-equivalent polytopes of same dimension have the same graph? | There are many non-equivalent neighborly polytopes, already in dimension $d=4$. See for example
* [Arnau Padrol. *"Many Neighborly Polytopes and Oriented Matroids"*](https://link.springer.com/article/10.1007/s00454-013-9544-7)
| 4 | https://mathoverflow.net/users/108884 | 380096 | 158,262 |
https://mathoverflow.net/questions/380081 | 2 | Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$.
>
> I would like to know whether we can show that if $(\mu\_n)\_{n\in\mathbb N}$ is convergent with respect to the topology of [weak converg... | https://mathoverflow.net/users/91890 | If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well? | $\newcommand\R{\mathbb R}$The answer to the second question (and hence to the first one) is no.
Indeed, let $E:=\R$.
For all odd natural $n$, let $\mu\_n:=\mu$, where $\mu$ is the uniform distribution on the interval $[2,3]$, so that $\mu\_n(dx)=\mu(dx)=1(2\le x\le3)\,dx$, $\exp^\*(\mu\_n)=\exp^\*(\mu)$, $\|\exp^\*... | 2 | https://mathoverflow.net/users/36721 | 380107 | 158,264 |
https://mathoverflow.net/questions/380113 | 1 | Let $E$ be a metric space, $\mathcal M(E)$ denote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$, $(\mu\_t)\_{t\in I}$ be a net in $\mathcal M(E)$ and $\mu\in\mathcal M(E)$ with $(\mu\_t)\_{t\in I}\to\mu$ with respect to the topology of [weak con... | https://mathoverflow.net/users/91890 | If $\mu_t\to\mu$ weakly, then $\limsup_t|\mu_t|(A)\le|\mu|(A)$ for all closed $A$ | $\newcommand\R{\mathbb R}$What you want to show is false in general. We reason here similarly to [this previous answer](https://mathoverflow.net/a/380107/36721).
Indeed, let $E:=\R$.
For all even natural $n$, let $\mu\_n(dx):=\mu(dx)(1+c\cos nx)$, where $\mu$ is the uniform distribution on the interval $[0,1]$ and $c... | 2 | https://mathoverflow.net/users/36721 | 380116 | 158,269 |
https://mathoverflow.net/questions/379630 | 9 | The goal of this question is to fill in the gap in [this old answer of mine](https://mathoverflow.net/a/348082/8133).
For a transitive set $M$, thought of as an $\{\in\}$-structure, we define the following ordinals (this is not the notation used in the linked answer above, but on reflection I like it more, and to the... | https://mathoverflow.net/users/8133 | Can two versions of $\omega_1^{CK}(\mathsf{Ord})$ ever coincide? | Every ordinal $α$ that is the least ordinal satisfying a given $Σ^1\_1$ property (about $α$, equivalently, about $L\_α$) is a Gandy ordinal, so $\mathsf{Def}(L\_α)=\mathsf{Ad}(L\_α)$. This includes the least $L\_α$ satisfying ZFC, and even includes the least $L\_α$ with $α$ an inaccessible cardinal in $L\_{α^{+\text{CK... | 3 | https://mathoverflow.net/users/113213 | 380117 | 158,270 |
https://mathoverflow.net/questions/380072 | 11 | The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What is the precise statement of the conjecture for higher genus curves? What are the modular/automorphic forms we expect to ... | https://mathoverflow.net/users/168668 | Modularity of higher genus curves | Following the suggestion of Faris I looked at [Abelian Surfaces over Totally Real Fields are Potentially Modular *by Boxer, Calegari, Gee & Pilloni*](https://arxiv.org/abs/1812.09269), whose section 1.4.1 discusses the modularity conjecure for higher genus curves and points to [On the Langlands Correspondence for Sympl... | 3 | https://mathoverflow.net/users/168668 | 380119 | 158,271 |
https://mathoverflow.net/questions/379906 | 1 | On page 271 of Trefethen and Bau's *Numerical Linear Algebra*, it is constructed a matrix
$$A=2I\_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$
for $m=200$, where rand(m) is an array with $m\times m$ measures of a normal random variable with mean zero and variance one.
The eigenvalues of this matrix are c... | https://mathoverflow.net/users/169665 | Norm of a matrix with clustered eigenvalues | The book does not actually state a norm equality, rather they state that
$$ \| (I - \tfrac{1}{2} A)^n \| \sim (1/4)^n$$
Here, $n$ is the degree of the Krylov polynomial used, for the purpose of analyzing $n$ steps of the GMRES iteration. Thus the approximation above describes the exponential decay behavior as $n$ incre... | 1 | https://mathoverflow.net/users/7378 | 380125 | 158,275 |
https://mathoverflow.net/questions/380083 | 1 | I have previously asked the question [A truncated divisor function sum](https://mathoverflow.net/questions/92967/a-truncated-divisor-function-sum)
where the sum
$$
S\_f(x)=\sum\_{n\leq x} \min\{f(x),d(n)\}\quad (1)
$$
was of interest, and it was answered satisfactorily.
Here, I am interested in estimating the followi... | https://mathoverflow.net/users/17773 | Moments of number of interval restricted divisors | We assume $m\leq x$. Your $S\_1(x,m)$ is in fact, $x\log m + O(m)$.
This answer finds an estimate of $S\_2(x,m)$.
$$
\begin{align}
S\_2(x,m)&=\sum\_{n\leq x} \left(\sum\_{d|n, d\leq m} 1 \right)^2=\sum\_{d\_1\leq m, d\_2\leq m} \sum\_{n\leq x, [d\_1,d\_2]|n}1\\
&=\sum\_{d\_1\leq m, d\_2\leq m} \frac x{[d\_1,d\_2]}+... | 1 | https://mathoverflow.net/users/21090 | 380130 | 158,277 |
https://mathoverflow.net/questions/380152 | 0 | How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with
$$
\big(xJ\_a'(\ell x)
\big)'+\left(\ell^2-\frac{a^2}{x}\right)J\_a(\ell x)=0
$$
and
$$
\big(xJ\_a'(\ell' x)
\big)'+\left(\ell'^2-\frac{a^2}{x}\right)J\_a(\el... | https://mathoverflow.net/users/nan | Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$ | For, $bl$ and $bl'$ being distinct zeroes of Bessel function , we have
$(xJ\_a(lx)')'+(l^2x-q^2/X)J\_a(lx)=0$ and
$(xJ\_a(l'x)')'+(l'^2x-q^2/X)J\_a(l'x)=0$ .
Multiplying first with $J\_a(l'x)$ and second with $J\_a(l'x)$ and then subsratcting second from first ,
$-[J\_a(l'x)(xJ\_a(l'x))'-J\_a(lx)(xJ'\_a(l'x))'... | 0 | https://mathoverflow.net/users/156029 | 380159 | 158,286 |
https://mathoverflow.net/questions/380161 | 1 | Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set, and let $\phi:\Omega\to\mathbb{R}^N$ be a $C^1$ function with the property that $\phi^{-1}(0)\neq\emptyset$, and $\nabla\phi(x)\neq 0,\ \forall\ x\in \phi^{-1}(0)$.
How can we prove or disprove that:
$$\mathcal{H}^{N-1}\left (\overline{\phi^{-1}(0)}\setm... | https://mathoverflow.net/users/61629 | Why is the Hausdorff measure of this set zero? | The next result answers the question in the negative.
>
> **Theorem.** There is $\phi:\mathbb{R}^n\supset\Omega\to\mathbb{R}^n$ of class $C^\infty$ such that
> $\phi$ is a local diffeomorphism in a neighborhood of $\phi^{-1}(0)$, but
> the Lebesgue measure of the following set is positive:
> $$
> (\*)\quad \mathcal... | 3 | https://mathoverflow.net/users/121665 | 380166 | 158,288 |
https://mathoverflow.net/questions/380170 | -3 | Suppose a tree with nodes located at levels $1,2,3...$. At each level the nodes branch into several nodes or do not branch.
Does the cardinality of the set of all infinite paths in this tree depend on the growth rate of the nodes by levels? Well, yes.
I mean, if the nodes do not branch at all, then the path is 1.
... | https://mathoverflow.net/users/10059 | Continuum hypothesis and cardinality of infinite tree paths | The set of all branches is a closed set of reals. Cantor proved that closed sets are either countable or of size continuum.
| 7 | https://mathoverflow.net/users/11145 | 380172 | 158,290 |
https://mathoverflow.net/questions/378430 | 1 | I do have a problem in continuum mechanics for which I kindly ask for your help. How can I obtain the isotropic scalar-valued functional $\psi(\boldsymbol{U})$ that satisfies the derivative
$$\frac{\partial \psi}{\partial \boldsymbol{U}}=\ln (\det(\boldsymbol{U})) ~\boldsymbol{1}$$
in which $\boldsymbol{1}$ represe... | https://mathoverflow.net/users/170150 | How to obtain an isotropic functional of a symmetric tensor which satisfies a given first derivative? | If I understand the question, we can try the case where $U$ is a two by two
matrix $$U = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$
and we want a function $\psi(U)$ that satisfies
$$ \psi\_a = \ln(ad-bc), \qquad \psi\_b = 0, \qquad \psi\_c = 0, \qquad \psi\_d = \ln(ad-bc).$$ There does not seem to be any solution u... | 1 | https://mathoverflow.net/users/6998 | 380176 | 158,293 |
https://mathoverflow.net/questions/380173 | 3 | Let $A$ be the set of numbers whose sum of digits is prime (<http://oeis.org/A028834>).
I would like to know if $A$ has zero natural density, that is, if $$\lim\_{n \to +\infty} \frac{A(n)}{n} = 0,$$ where $A(n)$ is the number of elements of $A$ which are less than or equal to $n$.
Numerical experiments seems to in... | https://mathoverflow.net/users/171588 | Density of the set of numbers whose sum of digits is prime | Yes, $A(n)$ has zero natural density. It suffices to prove this for $n$ which is a power of $10$.
and it is possible to make this more precise. To see this, first let $n=10^k$ and note that for $X$ chosen uniformly among integers in $[0,n-1]$, the sum $S(X)$ of base 10 digits is the sum of $k$ i.i.d. random variables u... | 16 | https://mathoverflow.net/users/7691 | 380177 | 158,294 |
https://mathoverflow.net/questions/380180 | 29 | I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I am wondering, then, is there any record of his attempts to prove RH?
| https://mathoverflow.net/users/8435 | Riemann's attempts to prove RH | The short answer is no. If anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes. However, [Siegel wrote](https://arxiv.org/abs/1810.05198):
>
> Approaches to a proof of the so-called “Riemann hypothesis” or even to a proof of the exi... | 28 | https://mathoverflow.net/users/3106 | 380182 | 158,295 |
https://mathoverflow.net/questions/380185 | 5 | Let $(M,+,e)$ be a commutative monoid with unit $e$. An element $a\in M$ is called cancellative element if
for any $b,c \in M$ such that $a+b=a+c$ implies that $b=c$.
Let $(\mathbf{N},+,0)$ the commutative monoid of natural numbers.
suppose that
1. we have two morphisms of monoids $f:(\mathbf{N},+,0)\rightarrow... | https://mathoverflow.net/users/128371 | Cancellation property for commutative monoid | The answer is no. Let $U=\{0,1\}$ under multiplication. Let $P$ be the semigroup of positive integers under $+$. Consider $S=P\times U$, the direct product and let $M=S\cup \{I\}$ where $I$ is an adjoined identity. Then $M$ is torsion-free, there is a homomorphism $f\colon \mathbf N\to M$ given by $f(0)=I$ and $f(n)=(n... | 7 | https://mathoverflow.net/users/15934 | 380188 | 158,298 |
https://mathoverflow.net/questions/380193 | 1 | For any $a, b\in\mathbb{N}$ with $a+2b\not\equiv 0\pmod 3$, we define $\delta(a, b)$ as follows:
\begin{align\*}
\delta(a, b)={\left\{\begin{array}{rl}
1,\ \ \ \ &{\rm if} \ a+2b\equiv 1\pmod 3,\\
0,\ \ \ \ &{\rm if} \ a+2b\equiv 2\pmod 3.
\end{array}\right.}
\end{align\*}
Furthermore, for any $m, n\in\mathbb{N}$, we l... | https://mathoverflow.net/users/126115 | A special congruence | We can rewrite the sum as
$$\sum\_{a=0}^{m} (-1)^a \binom{m}{a}\sum\_{\substack{b=0 \\ 3 \nmid a+2b}} (-1)^b 2^{\delta(a,b)} \binom{n}{b}$$
1. Now, when, $a=3k+1$, then we have $b=3k$ or $3k-1$.
Then for, $b=3k$, $\delta(a,b)=1$ and for $b=3k-1$ and $\delta(a,b)=0$.
2. Similarly, when $a=3k-1$, $b=3k, \delta(a,... | 2 | https://mathoverflow.net/users/156029 | 380195 | 158,300 |
https://mathoverflow.net/questions/380206 | 37 | At the age of 16, Leonhard Euler defended his Master's Thesis, where he discussed and compared Descartes' and Newton's approaches to planet motion. I don't know anything else about it. In particular, I don’t know what position the young Euler supported.
>
> Is there any modern account of this dissertation? In Engli... | https://mathoverflow.net/users/8799 | Euler's Master's Thesis | Martin Mattmüller, in his article *[Leonhard Euler, seine Heimatstadt und ihre
Universität](https://www.researchgate.net/publication/283452904_Leonhard_Euler_seine_Heimatstadt_und_ihre_Universitat)* on Euler's hometown Basel, writes that this public talk (not a dissertation or written thesis), which Euler gave in 1724,... | 39 | https://mathoverflow.net/users/3503 | 380209 | 158,305 |
https://mathoverflow.net/questions/380205 | 1 | Let $F$ be a finite extension of $\mathbb{Q}\_2$, and let $(-,-)\_F$ be the quadratic Hilbert symbol over $F$. Is the following true?
$(-1,-1)\_F=1$ if and only if $\sqrt{-1}\in F$
| https://mathoverflow.net/users/32746 | Hilbert symbol over 2-adic field | No. For example, $(-1,-1)\_F = 1$ for *all* quadratic extensions $F/\mathbf Q\_2$. This was discussed a few days ago on math.stackexchange [here](https://math.stackexchange.com/questions/3965273/over-which-extensions-of-mathbb-q-2-is-x2y2z2-isotropic).
| 1 | https://mathoverflow.net/users/3272 | 380213 | 158,307 |
https://mathoverflow.net/questions/380181 | 3 | Given a short exact sequence of vector bundles on a projective variety, after tensoring with an $\mathcal{O}(n)$ with high $n$ that makes all terms globally generated (so that taking global sections becomes an exact functor), we can lift short exact sequence to a short split exact sequence of trivial bundles. This mean... | https://mathoverflow.net/users/127776 | Resolution of short exact sequences by the split ones | Every short exact sequence $0\to X\stackrel{\alpha}{\to}Y\stackrel{\beta}{\to}Z\to0$ is the quotient of a split short exact sequence by a split short exact subsequence. This answer is copied from [my answer](https://math.stackexchange.com/a/1794602/88262) on math.stackexchange.
$$\require{AMScd}\begin{CD}
@.0@.0@.0\\... | 7 | https://mathoverflow.net/users/22989 | 380216 | 158,308 |
https://mathoverflow.net/questions/380212 | 1 | Gottfried Leibniz completed his habilitation dissertation (as part of his book *De Arte Combinatoria*) in philosophy somewhere in the mid-1660s. Prior to that, he had acquired his master’s degree; what I would like to know is the following:
>
> Did Leibniz’s work for his habilitation come from ideas in his previous... | https://mathoverflow.net/users/166628 | Leibniz habilitation dissertation in philosophy | Leibniz's [De Arte Combinatoria](https://en.wikipedia.org/wiki/De_Arte_Combinatoria) from 1666 was influenced by the Ars Magna of the 13th century philospher [Ramón Llull](https://en.wikipedia.org/wiki/Ramon_Llull). The overarching principle that complex ideas can be reformulated in terms of a small number of elementar... | 4 | https://mathoverflow.net/users/11260 | 380219 | 158,310 |
https://mathoverflow.net/questions/378480 | 1 | Let $\mathfrak{g}$ be a simple finite-dimensional complex Lie algebra, and let $\theta$ be a complex linear involution on $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subspace, and choose a $\theta$-stable Cartan subalgebra $\mathfrak{h}$ containing $\mathfrak{a}$. Finally, make a choice of positive restricted roots ... | https://mathoverflow.net/users/97652 | Existence of commuting Chevalley involution | I've realized that the answer is no. I should have checked some examples! The case of $(\mathfrak{gl}(4),\mathfrak{gl}(3)\times\mathfrak{gl}(1))$ gives a counterexample. Indeed, this is induced by the involution $\theta$ given by conjugation by
$$\begin{bmatrix}0 & 0 & 0 & i\\0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\-i &0 &0&0\e... | 1 | https://mathoverflow.net/users/97652 | 380222 | 158,311 |
https://mathoverflow.net/questions/380220 | 3 | Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial part of $u$, defined by
$$
\tilde u(r):=\frac{1}{S\_{N-1}}\int\_{\partial B(0,1)} u(ry)d\sigma\_y
$$
is radially symmetric ... | https://mathoverflow.net/users/121404 | Estimate of the norm of the radial part of a function | Yes, it is true. Take a function $u \in L^2(R^N)$ and expand for $x=r\omega$
$$
u(r\omega)=\sum\_{k=0}^\infty u\_k(r)P\_k(\omega)$$
where $(P\_k)$ an orthonormal basis of spherical harmonics in $L^2(S^{N-1})$ and $u\_0$ is your $\tilde u$. Then for $\xi=s\eta$
$$\hat u(s\eta)
=\sum\_{k=0}^\infty U\_k(s)P\_k(\eta),$$ se... | 1 | https://mathoverflow.net/users/150653 | 380226 | 158,314 |
https://mathoverflow.net/questions/380237 | 19 | This question is inspired by [this physics stack exchange post](https://physics.stackexchange.com/questions/604644/how-to-compute-gauge-potential-a-from-the-field-strength-f), which is recent and has not received an answer yet, nontheless I feel that there is a better way to ask this question here with a larger scope t... | https://mathoverflow.net/users/85500 | A non-Abelian de Rham complex? | What is being described in the main post is simply the de Rham (crossed) complex valued in a Lie group (not necessarily commutative).
See, for example, Section 6.2 in Anders Kock's [Synthetic Geometry of Manifolds](https://users-math.au.dk/%7Ekock/SGM-final.pdf).
One easy way to see what is going on is as follows.
... | 16 | https://mathoverflow.net/users/402 | 380243 | 158,318 |
https://mathoverflow.net/questions/380235 | 22 | Multiple times in talks about condensed mathematics (e.g. [the Masterclass talks](https://www.math.ku.dk/english/calendar/events/condensed-mathematics/), Clausen's [RAMpAGe](http://math.bu.edu/people/jsweinst/rampage/) talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes" the non... | https://mathoverflow.net/users/134334 | Condensed criterion for sheafiness of adic spaces | Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of [www.math.uni-bonn.de/people/scholze/Analytic.pdf](http://www.math.uni-bonn.de/people/scholze/Analytic.pdf) (see especially Proposition 13.16, Proposi... | 26 | https://mathoverflow.net/users/6074 | 380249 | 158,320 |
https://mathoverflow.net/questions/380241 | 3 | The following question makes sense in a more general setting but for sake of simplicity let me stick to a particular case.
Consider the degree three Veronese embedding $V\subset\mathbb{P}^9$ of $\mathbb{P}^2$ via the complete linear system of plane cubics.
Does there exist a point $p\in\mathbb{P}^9$ not lying on an... | https://mathoverflow.net/users/nan | Linear spaces secant to Veronese varieties | Here is an answer in terms of power sum decompositions of polynomials. A point $p \in \mathbb{P}^9$ corresponds to a homogeneous polynomial $P$ of degree $3$ in $3$ variable, defining a plane cubic. Points of the Veronese $q \in V$ correspond to pure powers of linear forms, $q = \ell^3$. A point $p$ lies in the span of... | 8 | https://mathoverflow.net/users/88133 | 380266 | 158,328 |
https://mathoverflow.net/questions/380272 | 2 | Let $f\colon\overline{\mathbb{D}}\to\mathbb{C}$ be a continuous function but that $f\colon\mathbb{D}\to\mathbb{C}$ is holomorphic. My question is
>
> Can the restriction of $f$ to $\mathbb{S}$ assume its values in the unit interval $[0,1]$, that is $f(\mathbb{S})\subseteq[0,1]$?
>
>
>
Specific/explicit example... | https://mathoverflow.net/users/166628 | A boundary behaviour of holomorphic functions | As Dan Petersen anticipated, any function satisfying the requirements is constant. Indeed, the function defined as $g(z):=f(z)$ for $|z|\leq 1$ and $g(z):=\overline{f(1/\overline{z})}$ for $|z|\geq 1$ is entire and bounded, hence constant by Liouville's theorem.
| 8 | https://mathoverflow.net/users/11919 | 380275 | 158,331 |
https://mathoverflow.net/questions/380260 | 5 | Let $\phi\in C\_c^\infty(\mathbb{R})$ be an even function such that $\chi\_{(-1/2,1/2)}\le\phi\le \chi\_{(-1,1)}$, where $\chi\_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $\lambda>0$ consider the oscillatory integral
$$
I(\lambda)=\int\_\mathbb{R} \phi(x)\, \exp \left(i\lambda(x+\epsilon|x|... | https://mathoverflow.net/users/157356 | A simple oscillatory integral with a non-smooth phase | $\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand{\ep}{\epsilon}\newcommand{\tI}{\tilde I}$
Take any $a\in(1,2)$ and then any nonzero $\epsilon\in(-1/a,1/a)$. Then
\begin{align\*}
I(t)&:=\int\_\mathbb{R} \phi(x)\, \exp(it(x+\epsilon|x|^a))\, dx \\
&\sim\frac{2\epsilon\,\Gamma(a+1)}{it^a}\,\sin\frac{\pi a... | 8 | https://mathoverflow.net/users/36721 | 380276 | 158,332 |
https://mathoverflow.net/questions/379839 | 9 | I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First of all, to define the etale site of an adic space, we only consider adic spaces that are locally of the form $Spa(A,A^+)$ where ... | https://mathoverflow.net/users/171303 | On the definition of the etale site of an adic space | Great question!
The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat obscure for some time, so he probably didn't want to further scare people by allowing the most general non-noetherian... | 13 | https://mathoverflow.net/users/6074 | 380289 | 158,336 |
https://mathoverflow.net/questions/380268 | 3 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$.
A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A *coloring* is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal, such that for every $e\in E$ with $|e|\geq 2$ the restri... | https://mathoverflow.net/users/8628 | A sequence of cardinal characteristics constructed with hypergraph coloring | The cardinals $\bf k\_n$ ($2\le n\lt\omega$) are all equal.
**Lemma.** Let $\kappa$ be an infinite cardinal. Given a set $A\subseteq[\omega]^\omega$ with $|A|=\kappa$ and $\chi(\omega,A)\gt n$, we can construct a set $B\subseteq[\omega]^\omega$ with $|B|=\kappa$ and $\chi(\omega,B)\gt n^2$.
**Proof.** For each $a\i... | 6 | https://mathoverflow.net/users/43266 | 380293 | 158,337 |
https://mathoverflow.net/questions/380270 | 0 | The multivariate generalised central limit theorem with their domains of attraction was given by [Rvačeva](https://books.google.com.hk/books?id=5d__hmE9q34C&pg=PA183&lpg=PA183&dq=%22On%20domains%20of%20attraction%20of%20multi-dimensional%20distributions%22&source=bl&ots=ilMkRwGxz8&sig=ACfU3U0pqpq33nNTpI8Ya0Jkx7Lny0uKYg... | https://mathoverflow.net/users/115114 | Stable law and the domains of attraction | 1. As stated in the [linked post](https://mathoverflow.net/a/368974/36721), the domain of attraction to multidimensional stable distributions with for $\alpha<2$ was characterized by Rvačeva's [Theorem 4.2 (p. 196)](https://books.google.com/books?id=5d__hmE9q34C&pg=PA183&lpg=PA183&dq=%22On%20domains%20of%20attraction%2... | 1 | https://mathoverflow.net/users/36721 | 380300 | 158,342 |
https://mathoverflow.net/questions/379602 | 7 | Suppose given $n\ge 1$ and a subspace $U$ in $\mathbb{Q}^n$. It is given as $\mathbb{Q}$-span of certain known vectors.
For $x \in U$, we let the Hamming weight of $x$ be the number of its nonzero entries.
Is it possible to find an element of minimal Hamming weight in $U\smallsetminus\{0\}$? Is there an algorithm?
... | https://mathoverflow.net/users/9300 | Vectors with minimal Hamming weight in a rational vector space? | Here is an Integer Linear Programming approach to this problem.
First, let's multiply given vectors by a suitable integer to make them all having integer components. Second, we notice that if certain rational coefficients deliver the minimum Hamming weight, then by scaling them, we can obtain integer coefficients del... | 3 | https://mathoverflow.net/users/7076 | 380304 | 158,343 |
https://mathoverflow.net/questions/380169 | 4 | Suppose an anti-holomorphic involution $\sigma$ is defined in a neighbourhood of $0\in \mathbb C^2$. Suppose that $\sigma$ fixes a real two-dimensional surface $\Sigma$ containing $0$. Is it true that locally close to $0$ the involution $\sigma$ is holomorphically conjugate to the map $(z\_1,z\_2)\to (\bar z\_1,\bar z\... | https://mathoverflow.net/users/13441 | A normal form of local anti-holomorphic involutions of $\mathbb C^2$? | I claim that the answer is positive in the sense that $\sigma$ is locally holomorphically conjugate to the standard antiholomorphic involution, assuming that when you say "a real surface" you mean "a totally real surface."
I will need:
Lemma. Let $S\subset {\mathbb C}^n$ be a totally-real real-analytic $k$-dimensio... | 3 | https://mathoverflow.net/users/39654 | 380305 | 158,344 |
https://mathoverflow.net/questions/380302 | 18 | Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some interesting examples?
A couple of remarks: when $I : 1 \to \mathscr V$ has a left adjoint, then $\mathscr V$ is semicartes... | https://mathoverflow.net/users/152679 | Monoidal categories whose tensor has a left adjoint | Just to clean up the $\epsilon$ of room left after Qiaochu's answer -- we can get rid of the extra hypotheses. I'll write $I$ for the monoidal unit and $1$ for the terminal object.
Assume that $(\ell,r) \dashv \otimes$. Then the natural isomorphisms $A \cong I \otimes A \cong A \otimes I$ give rise, by adjunction, to... | 20 | https://mathoverflow.net/users/2362 | 380315 | 158,349 |
https://mathoverflow.net/questions/372533 | 8 | I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X\_n]$ where the spaces $X\_n$ are defined as follows:
Fix the background inner product space $\mathbb{R}^{\infty}$ with the standard i... | https://mathoverflow.net/users/40180 | What is the inverse in K-theory represented by Clifford module extensions? | For what it's worth, I did eventually get the answer. The idea is to notice that if $\overline{W}$ denotes the background Clifford module $W$ (above $\mathbb{R}^n$) endowed with the opposite module structure, i.e., where the operators $e\_i$ act instead by $-e\_i$, then for any map $f:M \to X\_n(W)$, the map $f \oplus ... | 2 | https://mathoverflow.net/users/40180 | 380324 | 158,352 |
https://mathoverflow.net/questions/380327 | 0 | There is a well known infinite product both for $\phi(x)=\sin x$ and $\phi(x)=\cos x$. These are particular cases of the [Weierstrass factorization theorem](https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem). What about
$\phi(x)=a\_1\cos b\_1 x + a\_2\cos b\_2 x + a\_3\cos b\_3 x$, where all coefficients ... | https://mathoverflow.net/users/140356 | Infinite products for linear combinations of sines or cosines | The product you wrote for a finite linear combination of cosine is divergent,
unless you group opposite zeros. Since your function is even, the correct product is
this:
$$c\prod\left(1-\frac{z^2}{\rho^2}\right).$$
This follows from Hadamard's theorem. Since your function is even, you can write it
as $f=g(z^2)$ where $g... | 3 | https://mathoverflow.net/users/25510 | 380330 | 158,355 |
https://mathoverflow.net/questions/380323 | 1 | This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact sequence of projective $A$-modules $l\_1:\quad0\rightarrow P\_1 \rightarrow P\_2 \rightarrow P\_3 \rightarrow 0$. Let's assume... | https://mathoverflow.net/users/127776 | When splitting of short exact sequence preserves the kernels | The answer is "no" unless $A=k$.
Let $a\in A\setminus k$, and let $l\_2$ and $l\_1$ be the first and second rows of the commutative diagram
$$\require{AMScd}
\begin{CD}
0@>>>A@>\begin{pmatrix}1\\0\end{pmatrix}>>A^2
@>\begin{pmatrix}0&1\end{pmatrix}>>A@>>>0\\
@.@VV1V@VV\begin{pmatrix}1&a\end{pmatrix}V@VVV\\
0@>>>A@>1>... | 3 | https://mathoverflow.net/users/22989 | 380340 | 158,358 |
https://mathoverflow.net/questions/380346 | 9 | I know this is not a research question, but I searched somewhat thoroughly and could not find the exact answer I want. But I've always wondered the following: suppose that $(X,\mathcal{M})$ is a measurable space and $Y$ is a real topological vector space equipped with the Borel $\sigma$-algebra $\mathcal{B}$. Let
$$L^0... | https://mathoverflow.net/users/105925 | When is the set of measurable functions a vector space? | The usual thing to do, even when $Y$ is a Banach space, is to define "measurable" in such a way that it works. (A while back I posted a counterexample to the general case. See below.)
**Bochner measurable**, meaning there exist simple functions $f\_n$ that converge a.e. to $f$. "Simple" functions have finite range. I... | 9 | https://mathoverflow.net/users/454 | 380351 | 158,361 |
https://mathoverflow.net/questions/380347 | 2 | I am trying to find a copy of H. Shulman's 1972 Berkeley thesis 'On Characteristic Classes'. I've seen it referenced in Bott's ['On the de Rham theory of Certain Classifying Spaces'](https://core.ac.uk/download/pdf/82496263.pdf) but I can't seem to find it anywhere online. Does anyone know where I can find it?
| https://mathoverflow.net/users/122319 | Shulman's Thesis on Characteristic Classes | the full title of Shulman's thesis is "On characteristic classes and foliations"; a few libraries have it, see [WorldCat](https://www.worldcat.org/title/characteristic-classes-and-foliations/oclc/21923876?referer=br&ht=edition), it seems you can order a copy from [the British Library.](http://explore.bl.uk/primo_librar... | 4 | https://mathoverflow.net/users/11260 | 380352 | 158,362 |
https://mathoverflow.net/questions/380345 | 3 | Consider the set of finite sequences (of bounded length $\leq k$, if necessary) whose elements are taken from some finite alphabet $\Sigma$. We define a partial order on this set so that
$X = (X\_1,...,X\_{m}) \prec Y = (Y\_1,...,Y\_{n})$ whenever $X$ is a subsequence of $Y$.
Formally, this means that there's a stric... | https://mathoverflow.net/users/119381 | Is there an explicit linear extension for the subsequence partial order? | Note that two words of the same lengths are comparable if and only if they are equal. So you can order the words in the following way:
$$X\prec^\* Y\iff |X|<|Y|\text{ or } (X<\_{\rm Lex}Y \text{ and } |X|=|Y|).$$
Here $|X|$ is the length of the word $X$. By $<\_{\rm Lex}$ we mean that we fix an enumeration of the a... | 6 | https://mathoverflow.net/users/7206 | 380354 | 158,364 |
https://mathoverflow.net/questions/380335 | 5 | I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid is constructed such that $x^2=x$ and so the ring would be boolean and thus commutative). My question is:
>
> Do we ... | https://mathoverflow.net/users/91098 | Extending monoids to a ring | This was a classical question in semigroup theory. I suggest looking at this paper <http://www.numdam.org/article/SD_1969-1970__23_2_A12_0.pdf> which is interested in finite semigroups. It gives a reference to a Russian paper that in some sense says it impossible to axiomatize such rings but I don't have access to the ... | 7 | https://mathoverflow.net/users/15934 | 380356 | 158,365 |
https://mathoverflow.net/questions/380363 | 2 | Let $k$ be a field. Let $X$ be a smooth projective equidimensional variety over $k$. How to show the following proposition?
>
> For any coherent sheaf $F\_1$, $F\_2$ on $X$, $\operatorname{Ext}^i(F\_1,F\_2)=0$ if $i\notin [0,n]$.
>
>
>
This question is from Lemma 5.6 in this [paper](https://arxiv.org/pdf/1901.... | https://mathoverflow.net/users/118028 | For any coherent sheaf $F_1$, $F_2$ on $X$, $\operatorname{Ext}^i(F_1,F_2)=0$ if $i\notin [0,n]$ | First, $\mathrm{Ext}^i(F\_1,F\_2)$ for $i < 0$ vanish by definition. Second, for $i > n$ the vanishing follows from Serre duality
$$
\mathrm{Ext}^i(F\_1,F\_2) \cong \mathrm{Ext}^{n-i}(F\_2,F\_1 \otimes \omega\_X)^\vee
$$
and the above vanishing.
| 6 | https://mathoverflow.net/users/4428 | 380364 | 158,366 |
https://mathoverflow.net/questions/380333 | 1 | Consider the following subset of the unit cube in $\mathbb R^n$:
$$
\mathcal D = \{ p = (p\_1,p\_2,\dots,p\_n) \in [0,1]^n:\; p\_1 \le p\_2 \le \cdots \le p\_n\}.
$$
We would like to construct a probability measure supported on $\mathcal D$, continuous w.r.t. to
the $n$-dimensional Lebesgue measure and "somehow" diffus... | https://mathoverflow.net/users/36687 | A diffuse probability distribution in high dimensions with order constraints | Let $X$ be any nondegenerate random variable with values in the interval $[0,1]$. Let $p:=(p\_1,\dots,p\_n)$, where $p\_1=\dots=p\_n=X$. Then $p$ takes values in $\mathcal D$ and $\min\_i Var\,p\_i=Var\,X>0$ for all $n$.
This answers positively your second question. Letting $X$ be uniformly distributed on $[0,1]$, we... | 4 | https://mathoverflow.net/users/36721 | 380365 | 158,367 |
https://mathoverflow.net/questions/380348 | 13 | Let $D \colon \mathbf{J} \to \mathbf{Cat}$ be a filtered diagram of categories and functors. It has a colimit $\mathbf{C} = \mathrm{colim}\;D$. If you replace the diagram by a naturally isomorphic one $D' \colon \mathbf{J} \to \mathbf{Cat}$, then the colimit
$\mathbf{C'} = \mathrm{colim}\;D'$ is isomorphic to $\mathbf{... | https://mathoverflow.net/users/10368 | Are equivalences of categories stable under filtered colimits? | For filtered diagram (as asked in the question) the answer is **yes**. Of course this fails for general diagram as mentioned in Harry's answer.
Of course the "equivalence" has to be implemented by a pseudo-natural equivalence $f\_i:F\_i \to F'\_i$ otherwise it is not really an equivalence of diagram.
**First a very... | 16 | https://mathoverflow.net/users/22131 | 380367 | 158,368 |
https://mathoverflow.net/questions/380299 | 3 | Among many descriptions of the [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number) $C\_n$, let's use the recursive format $C\_0=1$ and
$$C\_{n+1}=\sum\_{i=0}^nC\_iC\_{n-i}.$$
Then, the [$2$-adic valuation](https://en.wikipedia.org/wiki/P-adic_order) of $C\_n$ is computed by $\nu\_2(C\_n)=s(n+1)-1$ where $s(... | https://mathoverflow.net/users/66131 | Tweaking the Catalan recurrence and $2$-adic valuations | This is not true. In fact, we can show that $\nu\_2(u\_{6,t}) = t+1$.
Indeed, computing first terms modulo $2^{t+2}$ for $t\geq 2$, we have
\begin{split}
u\_{0,t} &= 1,\\
u\_{1,t} &= 1,\\
u\_{2,t} &= 2,\\
u\_{3,t} &= 1 + 2^{t+1},\\
u\_{4,t} &= 2+2^{t+1}+O(2^{t+2}),\\
u\_{5,t} &= 2 + 2^{t+1} + O(2^{t+2}),\\
u\_{6,t} &... | 4 | https://mathoverflow.net/users/7076 | 380372 | 158,370 |
https://mathoverflow.net/questions/380378 | 0 | Let $(\delta\_k)\_{k\in\mathbb N}\subseteq(0,\infty)$ be nonincreasing with $\delta\_k\xrightarrow{k\to\infty}0$ and $(\varepsilon\_k)\_{k\in\mathbb N}\subseteq(0,\infty)$ with $\sum\_{k\in\mathbb N}\varepsilon\_k<1/2$ and $\sum\_{k\in\mathbb N}\varepsilon\_k/\delta\_k\le1/2$.
Let $E$ be a Banach space, $(\mu\_n)\_{n... | https://mathoverflow.net/users/91890 | How can we show this estimate for the convolution of two probability measures? | $\newcommand\ep\varepsilon\newcommand\de\delta$Let $\mu:=\mu\_n$, $\nu:=\nu\_n$, $B:=B\_n$,
$$C\_k:=\{x\in E:\mu(K\_k-x)>1-\de\_k\},$$
so that
$$B=\bigcap\_k C\_k.$$
We have
$$1-\ep\_k<(\mu\*\nu)(K\_k)=\int\mu(K\_k-x)\nu(dx) \\
=\int\_{C\_k}\mu(K\_k-x)\nu(dx)+\int\_{C\_k^c}\mu(K\_k-x)\nu(dx) \\
\le\nu(C\_k)+(1-\de\_k)\... | 1 | https://mathoverflow.net/users/36721 | 380384 | 158,373 |
https://mathoverflow.net/questions/380382 | 10 | The two powerhouse schemata of set theory are Replacement and Collection:
>
> **Replacement.** For every definable function $f$ and every set $x$, $f"x$ is a set.
>
>
> **Collection.** For every definable relation $R$ and every set $x$, there is a set $y$ such that for every $u\in x$ there is $v\in y$ such that $... | https://mathoverflow.net/users/7206 | Is Collection really stronger than Replacement? | The theories are equiconsistent and have the same strength as second order arithmetic $\text{Z}\_2$. Since we have an $L$-definable well-ordering of the constructible universe $L$, replacement implies collection and ZFC\P in $L$.
| 12 | https://mathoverflow.net/users/113213 | 380390 | 158,374 |
https://mathoverflow.net/questions/371997 | 1 | Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from the real line.
Let $X=\{z\in Z:\nu\_3(x)=0\}$.
Let $Y=\{z\in Z:\nu\_3(x)=-1\}$.
Now define an equivalence relation ... | https://mathoverflow.net/users/91341 | What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance? | One gets the trivial semi-distance. Let $d$ denote the semi-distance on $Z$ that defines $d\_\sim$ on $Z/\sim$, that is $d\_\sim([z],[z']):=d(z,z')$ for all $z$ and $z'$ in $Z$. Thus $d(z,z')\le |z-z'|$ for all $z$ and $z'$ in $Z$ and $d(x,f(x))=0$ for all $x\in X$.
It is convenient to extend the definition of $f$ to... | 0 | https://mathoverflow.net/users/6101 | 380398 | 158,376 |
https://mathoverflow.net/questions/380380 | 5 | Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in using the Leray spectral sequence to calculate the cohomology of the constant sheaf on $Y$ . My goal is to better understand spectra... | https://mathoverflow.net/users/160425 | Cohomology of doubly pinched torus via spectral sequences | I think the most direct way to figure out the mystery differential is using the edge map
$$ \mathbf Z^2 \cong H^1(X,\mathbf Z) \to H^0(Y,R^1 f\_\ast \mathbf Z) \cong \mathbf Z^2.$$
Let's first think about how this map is defined: a class in $H^q(X)$ restricts to a class in $H^q(F\_x)$ for each fiber $F\_x$, and $H^q(F\... | 3 | https://mathoverflow.net/users/1310 | 380400 | 158,378 |
https://mathoverflow.net/questions/380396 | 2 | Assume we are given a probability space $(\mathbb{X}, \mathcal{X}, \mathbb Q)$ and a measurable distance function defined on it $d:\mathbb{X}\times \mathbb{X}\to \mathbb{R}^+\cup\{0\}$ that conforms to the usual definition of distance on metric spaces. I am trying to understand (in lay terms) the "expected value of the... | https://mathoverflow.net/users/171715 | Expected measure of a ball in a probability space with a metric | Let $S:=\mathbb X$. Let $X$ and $Y$ be iid random elements of $S$ each with distribution $\mathbb Q$. It is more transparent to rephrase the question as follows:
>
> Can one give a good lower bound on $P(d(X,Y)\le r)$?
>
>
>
Let $(A\_i)$ be any countable measurable partition of $S$ with each $A\_i$ of diameter... | 2 | https://mathoverflow.net/users/36721 | 380403 | 158,379 |
https://mathoverflow.net/questions/379072 | 6 | Let
$$q = a\_1 x\_1^2 + \cdots + a\_n x\_n^2$$
be a quadratic form over some $p$-adic field $\mathbb{Q}\_p$. We thus have its Hasse invariant
$$\mathcal{h}(q) = \prod\_{1 \leq i < j \leq n} (a\_i,a\_j)\_p \in \{\pm 1\},$$
where $(a\_i,a\_j)\_p$ is the usual Hilbert symbol.
Let $\mathcal{C}(q)$ be the Clifford algebra... | https://mathoverflow.net/users/170182 | Hasse invariant and the Clifford algbera | You can find some information about this in Lam's book "Introduction to quadratic forms over fields," particularly in the 3rd chapter. I'll give the answer in a "field agnostic" way, not specifically for $p$-adic fields.
In general, the Hasse invariant is a slight modification of either the Brauer class of the Cliffo... | 2 | https://mathoverflow.net/users/17857 | 380405 | 158,381 |
https://mathoverflow.net/questions/380292 | 2 | The following problem arose when studying the same type of questions in Algebraic Geometry that led me to my previous question [MO379272](https://mathoverflow.net/questions/379272/combinatorial-problem-in-mathsfs-4).
Let us consider the group $G$ of order $32$ whose label in GAP4 database is $G(32, \, 6)$. It is a se... | https://mathoverflow.net/users/7460 | Combinatorial problem in $G(32, \, 6)$ | Here is a proof by contradiction. Assume an $8$-tuple $E$ from $G$ satisfies all of your relations.
Let $\Phi=\langle w^2,x,y \rangle \leq G$. It is straightforward to observe that
(A) $\Phi$ is abelian.
With a little more work, one can see that, for each $g \in G$,
(B) the centralizer $C\_G(g)$ is nonabelian i... | 2 | https://mathoverflow.net/users/36466 | 380410 | 158,384 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.