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https://mathoverflow.net/questions/344392 | 1 | Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E$ a subset of the boundary of $V$. If $\omega(x,W)(E)=0$ ($\omega(x,W)$ is the harmonic measure of $W$ at $x\in W$), can we conclude that $\omega(x,V)(E)=0?$
| https://mathoverflow.net/users/100746 | A question about harmonic measure 2 | Not really. Take one of the usual counter-examples in potential theory: $$W = (0,1) \times (0,1) \setminus \biggl(\bigcup\_{n=2}^\infty\{\tfrac{1}{n}\} \times (\tfrac{1}{n}, 1-\tfrac{1}{n})\biggr).$$ Then $V = (0,1) \times (0,1)$. If $E = \{0\} \times (0, 1)$, then $\omega(x, W)(E) = 0$, but of course $\omega(x, V)(E) ... | 1 | https://mathoverflow.net/users/108637 | 344395 | 146,064 |
https://mathoverflow.net/questions/344377 | 7 | I've just read a [review paper](https://arxiv.org/pdf/1908.01639.pdf) about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds.
Briefly, if $\phi\_\lambda$, $\lambda$ are an eigenpair for the Laplace-Beltrami operator on a manifold $M$, i.e.
$$-\Delta\phi\_\lambda = \lambda\phi... | https://mathoverflow.net/users/49417 | Yau's conjecture on nodal sets for manifolds with boundary | Yau's conjecture was proved for the real analytic manifolds by Donelly and Fefferman, first for manifolds without boundary, then for manifolds with [boundary](https://www.sciencedirect.com/science/article/pii/B9780125742498500171) in the paper *Nodal Sets of Eigenfunctions: Riemannian Manifolds With Boundary*, Analysis... | 4 | https://mathoverflow.net/users/118731 | 344398 | 146,066 |
https://mathoverflow.net/questions/344401 | 5 | Consider the finite sums
$$F\_n(q)=\sum\_{k=1}^nq^{\binom{k}2}$$
with exponents the *triangular numbers* $\binom{k}2$. When $n$ is odd, it appears that $F\_n(q)$ does not factorize over $\mathbb{Z}[q]$. On the other hand, when $n=2m$ is even
>
> **QUESTION.** is it true that $F\_{2m}(q)$ is divisible by the product... | https://mathoverflow.net/users/66131 | Divisibility of certain polynomials | **Yes.**
Let $n = a\*2^b$ with $a$ odd. Then your question is whether $\prod\_{j = 1}^{b} \left((1 + q^{\frac{n}{2^j}})\right) \, | \, F\_n(q)$. Multiplying both by $q^a - 1$, the question becomes whether $F\_n(q)(q^a - 1)$ is divisible by $q^{n} - 1$.
Consider the multiset $\{{i(i - 1)} (\text{mod} \; 2n)\}\_{i =... | 9 | https://mathoverflow.net/users/44191 | 344407 | 146,070 |
https://mathoverflow.net/questions/344317 | 1 | Let $L$ be a link in $S^3$ and $\rho : \pi\_1(S^3 \setminus L) \to \operatorname{SL}\_2(\mathbb C)$ be a representation of its knot group. If the twisted homology $H^\rho(S^3 \setminus L)$ is acyclic, we can obtain the *twisted Reidemeister torsion* $\tau(S^3 \setminus L, \rho)$, which is closely related to the twisted... | https://mathoverflow.net/users/113402 | Twisted torsions of reducible representations of knot groups | This should be true, here is a (hoperfully correct) sketch of proof: the Reidemeister torsion is a continuous function on the set of acyclic representations (as it can be computed from determinants of matrices whose coefficients vary continuously with the representation). On the other hand one can simultaneously conjug... | 2 | https://mathoverflow.net/users/32210 | 344428 | 146,078 |
https://mathoverflow.net/questions/344421 | 12 | Cross posted from [here](https://math.stackexchange.com/questions/3391037/d3-in-the-atiyah-hirzebruch-spectral-sequence-for-twisted-ko) after no responses and a bounty being placed on the question.
Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-Hirzeb... | https://mathoverflow.net/users/121307 | $d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$ | Here are the first, more straightforward, differentials in the AHSS
$$H^p(X;KO^q(\ast)) \Rightarrow KO^{p+q}(X)$$
for real K-theory. Note $KO^q(\ast)$ is
$$
\begin{cases}
\Bbb Z &\text{if }q = 8k,\\
\Bbb Z/2 &\text{if }q = 8k-1,\\
\Bbb Z/2 &\text{if }q = 8k-2,\\
\Bbb Z &\text{if }q = 8k-4,\\
0 &\text{otherwise.}
\end{c... | 14 | https://mathoverflow.net/users/360 | 344431 | 146,079 |
https://mathoverflow.net/questions/344429 | 4 | The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta\_{\mathrm{d}} = \mathrm{d}^\*\mathrm{d} + \mathrm{d}\mathrm{d}^\*$ acting on functions. The eigenvalues of this operator must be very well known. Where can one see these values written out explicitly?
| https://mathoverflow.net/users/126606 | Laplace spectrum of the $2$-Sphere | There is a very simple way to compute the Laplace-Beltrami $\Delta\_S$ operator of a function on the sphere $f:\mathbb{S}^{n-1}\to\mathbb{C}$. You simply extend the function to $\mathbb{R}^n\setminus\{0\}$ by $F(x)=f(x/|x|)$ and define
$$
\Delta\_Sf(x)=(\Delta F)|\_{\mathbb{S}^{n-1}},\quad
\text{ where } \quad \Delta F... | 9 | https://mathoverflow.net/users/121665 | 344434 | 146,080 |
https://mathoverflow.net/questions/344388 | 3 | Let $(W,S)$ be a Coxeter system. Let $q=(q\_s)\_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q\_s=q\_t$ whenever $s$ and $t$ are conjugate to each other. Follwing [Davis', Proposition 19.1.1](https://people.math.osu.edu/davis.12/davisbook.pdf) we can build a (unique) $\ast$-algebra $\m... | https://mathoverflow.net/users/64444 | Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic | Section 68 of Curtis, Reiner "Methods of representation theory, volume II" explains this carefully. The active ingredient is the "Tits deformation theorem".
| 2 | https://mathoverflow.net/users/48296 | 344439 | 146,081 |
https://mathoverflow.net/questions/344053 | 30 | This is an incredibly naive question so this may be closed. Nevertheless, I have been reading about the problem asking *if every obtuse triangle admits a periodic billiard path*, which has been open for a very long time. As someone who has not worked on this problem, I am wondering why what (on the surface) appears to ... | https://mathoverflow.net/users/129192 | Why is the billiard problem for obtuse triangles so hard? | I asked Rich Schwartz, who is one of the experts in this area (as noted by the OP). Here, with Rich's permission, is his response:
>
> I am not sure why it is so hard. All I can really say is that, after
> a lot of experimentation, I can't really see any pattern to it. It
> might be hard in the same way that buil... | 19 | https://mathoverflow.net/users/11926 | 344448 | 146,083 |
https://mathoverflow.net/questions/342372 | 2 | I have an algebraic group $G$ acting on an affine variety $X$, the orbit $O(m)$ of an element $m \in X$, and an affine curve $C$ contained in the Zariski closure $\overline{O(m)}$ of $O(m)$, such that $m \in C$.
If we define $C^\prime = C \cap O(m) $ then it's not hard to see that $C^\prime$ is open in $C$. And ther... | https://mathoverflow.net/users/146352 | When does an affine subset of an orbit have affine preimage under the orbit map? | I was able to figure out an elementary answer with some help:
It seems like a good hint is to look at abstract varieties or even easier use that every quasi-affine variety is an abstract variety.
The preimage is presumably not affine, but it is a locally closed subset of $G$ since $C$ is locally closed. Therefore $... | 1 | https://mathoverflow.net/users/146352 | 344454 | 146,084 |
https://mathoverflow.net/questions/344433 | 7 | Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition?
$\sim\_{M,\mathscr{F}}\,=\bigl\{(x,y)\in M\times M\bigm|\forall A\in\mathscr{F}\,(x\in A\leftrightarrow y\in A)\bigr\}$.
| https://mathoverflow.net/users/101817 | Is there a name for this equivalence relation? | $\mathscr F$-indistinguishability.
In analogy with [Topological indistinguishability](https://en.m.wikipedia.org/wiki/Topological_indistinguishability).
| 7 | https://mathoverflow.net/users/4600 | 344456 | 146,086 |
https://mathoverflow.net/questions/344445 | 3 | I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int\_2^x\frac{dt}{\log t}$, that will be denoted in this post as $$\operatorname{ali}(x)$$ for real numbers $x\geq 2$.
>
> **Question.** A) Do you know... | https://mathoverflow.net/users/142929 | The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral | The answer to (B) is yes, at least conditionally. The sign of $p\_n-\mathop{\rm ali}(n)$ is determined by the sign of $\pi(p\_n) - \mathop{\rm li}(p\_n)$ (other than the negligible possibility that one equals $0$ while the other is tiny and nonzero). The logarithmic density of those primes $p$ for which $\pi(p) > \math... | 7 | https://mathoverflow.net/users/5091 | 344457 | 146,087 |
https://mathoverflow.net/questions/344460 | 7 | I have a subset $K\subset X^\ast$ of the dual of a Banach space $X$. (In fact $X$ is $C^1(M)$ for some smooth compact manifold $M$.) I hope that there exists $x\in X$ such that every $k\in K$ satisfies $k(x)>0$. I know that the convex hull of $K$ does not contain the origin, and I know that $K$ is compact. Is that enou... | https://mathoverflow.net/users/6666 | A hyperplane separation question | Here is a counterexample:
Let $M$ be the one-dimensional unit circle, so we can identify $C^1(M)$ with
\begin{align\*}
X = \{f \in C^1([0,1]): \, f(0) = f(1), \; f'(0) = f'(1)\}.
\end{align\*}
For each $z \in [0,1)$ let $d\_z \in X^\*$ be given by $\langle d\_z,f\rangle = f'(z)$ for each $f \in X$. Then the set
\b... | 5 | https://mathoverflow.net/users/102946 | 344473 | 146,089 |
https://mathoverflow.net/questions/344477 | 6 | Let $(S,\eta,s)=\mathrm{Spec}(R)$ be the spectrum of a DVR.
Let $f\colon P\to S$ be an abelian scheme. Taking restriction gives an injection from the group of $P$-torsors to the group of $P\_\eta$-torsors: $$\rho\colon\mathrm{H}^1(S,P)\to\mathrm{H}^1(\eta,P\_\eta).$$
How can we discribe the cokernel of $\rho$?
[He... | https://mathoverflow.net/users/nan | When does a torsor of the generic fiber extend? | Here's a suggestion (not a full answer): take a geometric point $\bar{s} \rightarrow s$ above $s$. Then the stalk $\left(R^1 j\_\* P\_{\eta}\right)\_{\bar{s}}$ is computed as the Galois cohomology $H^1(K^{sh},P\_{\eta})$ where $K^{sh}$ is the fraction field of the strict henselization of $R$. This follows from the disc... | 0 | https://mathoverflow.net/users/110362 | 344480 | 146,093 |
https://mathoverflow.net/questions/344487 | 0 | Let $X\subset\mathbb{P}^n$ be a smooth nondegenerate (i.e. not contained in any hyperplane) curve over $\mathbb{C}$. Is it possible that every collection of $n-3$ points on $X$ lies on a $n-1$-secant (i.e. a linear space spanned by $n-1$ points on $X$) that has dimension at most $n-3$? If yes, can $X$ even be chosen to... | https://mathoverflow.net/users/36563 | Special secants to curves | I will give an example for $n = 4$. Let
$$
S = \mathbb{P}\_{\mathbb{P}^1}(\mathcal{O}(-1) \oplus \mathcal{O}(-2)) \subset \mathbb{P}^4
$$
be a cubic scroll and let
$$
X \subset S
$$
be a curve which is a 3-section of the morphism $S \to \mathbb{P}^1$. Then every point of $X$ lies on a trisecant line (the ruling of t... | 3 | https://mathoverflow.net/users/4428 | 344488 | 146,094 |
https://mathoverflow.net/questions/344491 | 6 | I moved this question from Math StackExchange.
I am trying to compute homology of $Conf(n, \mathbb{R}^2)$ - ordered configurations of $n$ points on the plane - using Serre spectral sequence. I know that this computations has been done for cohomology in Arnold's *The Cohomology Ring of Colored Braid Group*.
There is... | https://mathoverflow.net/users/123432 | Zero differential in Serre spectral sequence for configuration spaces | I'll write $C\_n$ for the configuration space, and $X\_n$ for $\mathbb{R}^2$ with $n$ points removed. You are presumably thinking about the spectral sequence
$$ E\_2^{pq} = H^p(C\_{n-1};H^q(X\_{n-1})) \Longrightarrow H^{p+q}(C\_n), $$
with differentials $d\_r\colon E\_r^{pq}\to E\_r^{p+r,q+1-r}$. It is a key point that... | 13 | https://mathoverflow.net/users/10366 | 344495 | 146,097 |
https://mathoverflow.net/questions/344493 | 1 | The [Kimberling sequence](http://oeis.org/A007063) is a recursively defined "shuffling sequence" (pictorial description [here](https://codegolf.stackexchange.com/questions/67542/the-kimberling-sequence)). Let $k:\mathbb{N}\to \mathbb{N}$ be the Kimberling sequence. Does $k$ map members of $\mathbb{N}$ arbitrarily far a... | https://mathoverflow.net/users/8628 | Does the Kimberling sequence map numbers "arbitrarily far away"? | The answer is yes. Indeed, as noted at [A007063](http://oeis.org/A007063),
$$k(\theta\_j)=3\theta\_j-(j+1),
$$
where
$$\theta\_j:=\sum\_{i=0}^{j-1}2^{\lfloor i/3\rfloor}\ge2^{\lfloor(j-1)/3\rfloor}.
$$
So,
$$k(\theta\_j)-\theta\_j=2\theta\_j-(j+1)\underset{j\to\infty}\longrightarrow\infty,
$$
as desired.
| 3 | https://mathoverflow.net/users/36721 | 344502 | 146,098 |
https://mathoverflow.net/questions/344483 | 1 | Let $A$ be a Nakayama algebra, that is an Artin algebra such that any indecomposable module has a unique composition series. The easiest examples of such algebras are $K[x]/(x^n)$.
>
> Question 1: In case $Ext^1(X,Y) \neq 0$ for two indecomposable $A$-modules $X,Y$, can one write down an explicit nice non-split sho... | https://mathoverflow.net/users/61949 | Ext in Nakayama algebras | I'll answer Question 2 first. The answer is yes.
If $Z$ has more than two indecomposable summands, then $\text{soc}(Z)$ has at least three summands, but $X$ has simple socle since it's uniserial, so $Y$, the kernel of $Z\to X$, has at least two summands. But $Y$ is also uniserial, and so has simple socle.
For Quest... | 2 | https://mathoverflow.net/users/22989 | 344516 | 146,102 |
https://mathoverflow.net/questions/344484 | 1 | Johnson in cohomology of Banach algebra proved the following proposition.
I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this part of the proof?
Proposition 8.2. Let $A$ be a commutative amenable Banach algebra and $X$ a commutative Banach $A$-mo... | https://mathoverflow.net/users/27066 | A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras | I will attempt to give an expanded version of Johnson's proof, based on some private unpublished notes I once made (but I claim no originality, I'm sure other people have worked through this themselves).
---
The first thing to note is that for any Banach algebra $A$, any quotient map of Banach $A$-bimodules $q: Y... | 4 | https://mathoverflow.net/users/763 | 344517 | 146,103 |
https://mathoverflow.net/questions/341511 | 4 | This is the lemma 4.25 of [Vistoli's note](https://arxiv.org/pdf/math/0412512.pdf#theorem.4.25)
>
> Let $S$ be a scheme, $\mathscr{F} \to \mathscr{S}ch/S$ a fibred category.
> Then $\mathscr{F}$ is a stack over the fpqc site on $S$ iff
>
> (1) $\mathscr{F}$ is a stack over the Zariski site on $S$, and
>
> ... | https://mathoverflow.net/users/128235 | Reducing the stack condition (descent condition) over an fpqc site to the case of single coverings | I have understood.
First by the construction, the diagram commutes on $V' \times\_{U'} V'$ for every affine open $U'$ of $U$ and its inverse image $V'$ in $V$.
Now $V \times\_U V = \cup V' \times\_{U'} V'$, where $U'$ runs through over affine opens of $U$.
So by Zariski descent, the diagram commutes on whole of $V \t... | 0 | https://mathoverflow.net/users/128235 | 344530 | 146,107 |
https://mathoverflow.net/questions/344157 | 5 | Let $X$ be a scheme over $K = k((t))$, where $k$ is a field. We define the loop scheme $LX$ to be the functor from the category of $k$-algebras to sets by $R \mapsto LX(R) := X(Spec (R((t))))$.
Do we need sheafification in order to make $LX$ a sheaf? A particular interest is when $X$ is a connected linear reductive g... | https://mathoverflow.net/users/130879 | Sheafification of loop scheme/group | Despite Will Sawin's answer in comment, it was not so obvious for me at first glance, so I give a detailed (at least for $X$ affine) proof here.
>
> **Lemma.** Let $A$ be a $K$-algebra and let $A\to B$ be fpqc. Then the following sequence is exact $$
> A((t))\to B((t))\rightrightarrows (B\otimes\_{A}B)((t)).
> $$
>... | 5 | https://mathoverflow.net/users/38052 | 344549 | 146,109 |
https://mathoverflow.net/questions/344562 | 11 | Let $\Gamma = \operatorname{SL}\_2(\mathbb{Z})$ be the usual modular group. It is well-known that $\Gamma$ contains infinitely many distinct (non-conjugate even) subgroups which are isomorphic to the infinite cyclic group $(\mathbb{Z}, +)$. Indeed, for any square-free integer $d > 1$, the unit group of the quadratic fi... | https://mathoverflow.net/users/10898 | Infinite cyclic subgroups of $\text{SL}_2(\mathbb{Z})$ | The maximal infinite cyclic subgroups are of the form you mentioned (to get all infinite cyclic subgroups you also need to include their finite index subgroups). This follows from Theorem 1.4 in [Sarnak, Peter, Class numbers of indefinite binary quadratic forms. J. Number Theory 15 (1982), no. 2, 229--247](https://www.... | 18 | https://mathoverflow.net/users/422 | 344564 | 146,111 |
https://mathoverflow.net/questions/344450 | 5 | For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".
<https://en.wikipedia.org/wiki/Hilbert_series_and_Hilbert_polynomial>
Concepts like modules and graded algebras are rather unfamiliar to me. I have not encountered these before.
I... | https://mathoverflow.net/users/nan | Reference book for understanding Hilbert Series/functions | In addition to Eisenbud's book which has already been pointed out, I would like to suggest also the following references.
* Chapter 11 of **Atiyah-MacDonald's "Introduction to Commutative Algebra"** develops the theory of dimension for graded and local rings and its relation with the Hilbert function. The prerequisit... | 1 | https://mathoverflow.net/users/106706 | 344567 | 146,112 |
https://mathoverflow.net/questions/341843 | 2 | Let $F$ be the spherically complete extension of $\mathbb C\_p$ and $(a\_n)\_{n\in\mathbb N}$ be a sequence of $\mathbb C\_p$ such that for all $r\in\mathbb R$, one has $$\lim\_{n\to+\infty}|a\_n|\_pr^n=0.$$ Put
$f(z)=\sum\_{n\ge0}a\_nz^n$. For a non-negative real number $r$ define the *modulus of growth in $F$* by $$... | https://mathoverflow.net/users/33128 | Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$ | It is known that for an entire function $f(x)=\sum\_{n\geq 0} a\_nx^n\in{\mathbb C}\_p[[x]]$, the function
\begin{equation\*}
r\longrightarrow M(f,r)=\inf\_{n\geq 0} v(a\_n)+n r=\inf\_{v(x)x\geq r} v(f(x))
\end{equation\*}
is continuous and piecewise affine, cf. A. Robert,\emph{A course in $p$-adic analysis}, Springer ... | 2 | https://mathoverflow.net/users/45381 | 344568 | 146,113 |
https://mathoverflow.net/questions/344548 | 4 | Let $u$ be a real-valued harmonic function on $\mathbb{D}$, which extends continuously to the boundary. I wonder how to prove the inequality
$$\int\_\mathbb{D} e^{2u} dxdy\leq\dfrac{1}{4\pi}\left(\int\_{\partial\mathbb{D}}e^uds\right)^2.$$
There are a lot of properties with harmonic function $u$ (like maximum princip... | https://mathoverflow.net/users/137193 | an Integral Inequality about harmonic function | This is the isoperimetric inequality in disguise. Define the Riemannian metric
with length element $e^{u(z)}|dz|$. The curvature of this metric is
$$-e^{-2u}\Delta u=0,$$
so the metric is flat. Then your inequality says that the area of the disk is
at most $1/4\pi$ times the square of the length of the boundary, the us... | 6 | https://mathoverflow.net/users/25510 | 344569 | 146,114 |
https://mathoverflow.net/questions/344570 | 0 | I am looking for a proof of the following result: Let $\mathfrak{g}$ be a Lie algebra and $I$ an injective $\mathfrak{g}$-module. Then $\mathrm{H}^q(\mathfrak{g},I)=0$ $\forall q>0$. More precisely, I am looking for a proof in a textbook so that I can cite it.
The result shows up in Weibel's book (Exercise 2.5.1), but ... | https://mathoverflow.net/users/131014 | Lie algebra cohomology with values in injective module | If you define Lie algebra cohomology via injective resolutions, then your claim is trivially true, because you can take the injective resolution which is I in degree 0 and 0 in higher degrees.
To see that your preferred definition agrees with that via injective resolutions, you have to check two things:
1. any two ... | 1 | https://mathoverflow.net/users/39082 | 344575 | 146,116 |
https://mathoverflow.net/questions/344520 | 1 | Let $S \subset \mathbb{R}^3$ be a compact and locally $C^1$ simply-connected surface with a $C^1$ boundary with no self intersection. Is there a $C^1$ bijection $F: \overline{B(0,1)} \rightarrow \overline{S}$ such that $ a|\xi|^2 \leq (D F)\_{i,j} \xi\_i \xi\_j \leq A |\xi|^2$ (a,A>0), $F$ maps $\partial B(0,1)$ to $\p... | https://mathoverflow.net/users/115905 | A basic question about compact $C^1$ surfaces with boundary | By the classification of surfaces, $S$ is $C^1$ diffeomorphic to a closed disk (Morris Hirsch, Differential Topology, p. 205 theorem 3.7, for the $C^{\infty}$ case, combined with Whitney's smoothing theorem [p. 51 theorem 2.9] to get from $C^{\infty}$ to $C^1$). But any $C^1$ diffeomorphism $F \colon B \to S$ from the ... | 1 | https://mathoverflow.net/users/13268 | 344583 | 146,121 |
https://mathoverflow.net/questions/344464 | 9 | Is there a common notation for the set of all injections from $A$ into $B$?
Some set-theorists use $B^{(A)}$, e.g., A. Levy in his book *Basic Set Theory*.
But some combinatorists use $B^{\underline{A}}$ or $(B)\_A$, e.g. JMoravitz's answer in [this question](https://math.stackexchange.com/questions/1896815).
Som... | https://mathoverflow.net/users/101817 | Notation for the set of all injections from $A$ into $B$ | The notation suggested by cardinal equalities such as
\begin{array}{l|l|l}
\text{concept} & \text{notation} & \text{cardinality} \\
\hline
\text{disjoint union of $A$ and $B$} & A + B & |A + B| = |A| + |B| \\
\text{Cartesian product of $A$ and $B$} & A \times B & |A \times B| = |A| \times |B| \\
\text{set of function... | 14 | https://mathoverflow.net/users/74578 | 344590 | 146,124 |
https://mathoverflow.net/questions/344325 | 8 | **Edit:** According to answer and comments by Prof. Valette we edite the question.
The Kadison Kaplansky conjecture says:
**Kadison-Kaplansky conjecture:** If $G$ is a torsion-free discrete group then $C^\*\_{\mathrm{red}}(G)$ has no nontrivial projection.
It is a particular case of a more general conjecture, [Th... | https://mathoverflow.net/users/36688 | A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u ) | Let $G$ be a discrete abelian group, denote by $\epsilon$ the trivial character. Let $u\in C^\*\_r(G)$ be a self-adjoint unitary element such that $\epsilon(u)=1$. If $g\in G$ is such that $\|u-g\|<2$, then $g$ has finite order. Indeed, by Pontryagin duality $C^\*\_r(G)\simeq C(\hat{G})$, with $\hat{G}$ the Pontryagin ... | 6 | https://mathoverflow.net/users/14497 | 344609 | 146,132 |
https://mathoverflow.net/questions/344612 | 5 | Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y^\*:X\to\mathbb R$ such that $y^\*(y)=1=\|y^\*\|$. This unique functional $y^\*$ will be called the *supporting function... | https://mathoverflow.net/users/61536 | An extremal property of points on the unit sphere of a 2-dimensional Banach space | The answer is *no*, in general.
For a counterexample, consider the $\ell^p$-norm on $\mathbb{R}^2$ with $p=4$, and let $x = e\_1 = (1,0)$.
We first note that the vectors $e\_2 = (0,1)$ and $-e\_2$ do not maximize the function
\begin{align\*}
f: S \ni s \mapsto \|s-x\|+\|s+x\| \in [0,\infty).
\end{align\*}
Indeed... | 8 | https://mathoverflow.net/users/102946 | 344629 | 146,140 |
https://mathoverflow.net/questions/344634 | 6 | For the $D\_n$-series simple Lie algebra $\frak{so}\_{2n}$
a curious phenomenon occurs for the fundamental representations corresponding to the spinor nodes of the Dynkin diagram, which is to say the spinor representations $V\_{\pi\_{n-1}}$, and $V\_{\pi\_{n}}$: In the case where $n$ is even both $V\_{\pi\_{n-1}}$ and ... | https://mathoverflow.net/users/147728 | Duals of the spinor representations of $\frak{so}_{2n}$ | Here is one explanation, although I am not sure if this is the conceptual explanation you are looking for.
Let $E = \mathbb{R}^n$ with orthonormal basis $\varepsilon\_1, \ldots, \varepsilon\_l$. You can realize a root system of type $D\_n$ as $\Phi = \{ \pm (\varepsilon\_i \pm \varepsilon\_j) : i \neq j \}$.
The ... | 4 | https://mathoverflow.net/users/38068 | 344638 | 146,143 |
https://mathoverflow.net/questions/344560 | 0 | Quine-Rosser ordered pair is type-level; i.e., it is definable by a stratified formula that assigns to it the same type it assigns to its projections.
It is known that if $\langle A,B \rangle$ is type-level, then we can define a triplet $ \langle A,B,C \rangle$ using this type-level merit as $\langle \langle A,B \ran... | https://mathoverflow.net/users/95347 | Are there known type-level infinite tuple implementations in ZFC? | I'll post this as an answer because it can be thought of being closely related to known work on pairs, although I'm not sure really if it has been worked out before.
I realized that Quine-Rosser pairs can be adapted to suite extending them to implement tuples of any ordinal length! The tuples I'll describe here would... | 2 | https://mathoverflow.net/users/95347 | 344647 | 146,146 |
https://mathoverflow.net/questions/344651 | 3 |
>
> Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
>
>
>
How would I attack this problem? The goal is to eliminate points by knowing whether they are encompassed by any 3 other points.
| https://mathoverflow.net/users/61792 | On 4 random points in a rectangle | As explained in [Square Triangle Picking](http://mathworld.wolfram.com/SquareTrianglePicking.html), the mean area of a triangle picked inside a rectangle of unit area is 11/144. So the probability that the fourth point lands inside this triangle is $11/144=0.0764$.
See [Polygon Triangle Picking](http://mathworld.wolf... | 10 | https://mathoverflow.net/users/11260 | 344652 | 146,149 |
https://mathoverflow.net/questions/344639 | 1 | Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f\_{\epsilon}$ which is locally defined on an open set $U$ containing $p$ such that (i) $f(p)=f\_{\epsilon}(p)$, (ii) $f \ge f\_{\epsilo... | https://mathoverflow.net/users/105900 | Smooth approximation of a subharmonic function in the barrier sense | Withtout loss of generality, we can assume $p=0$ and $f(0)=0$.
Also, the problem is purely local : we can assume that $f$ and all the functions $f\_\epsilon$ are compactly supported in the unit ball $\text{B}$, which contains also all the neighborhoods $U\_\epsilon$. You can also assume that for all $\epsilon>0$ the... | 3 | https://mathoverflow.net/users/27767 | 344655 | 146,151 |
https://mathoverflow.net/questions/344309 | 0 | $\def\cube{\mathbf Q}$Let $\cube$ be the Hilbert cube. Give an example of a closed subset $B$ of $\cube$ such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set.
If anyone has any idea to solve this problem, it would help me a lot. I write down some information:
**Definition**. Let $\cube= \prod\_{n=1}... | https://mathoverflow.net/users/147514 | A closed subset $B$ of the Hilbert cube such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set | $A=\{0\} \times [-1,1]^{\Bbb N}$ works as an example, e.g.
For more info on $Z$-sets in $Q$, see *Infinite-dimensional Topology* by Jan van Mill.
| 0 | https://mathoverflow.net/users/2060 | 344661 | 146,153 |
https://mathoverflow.net/questions/344641 | 3 | Let $(X\_n)\_{n\in\mathbb N}$ be a discrete time stochastic process taking values in a Banach space $E.$ Suppose there exist constants $C,\alpha,\beta>0$ such that $\mathbb E\|X\_n-X\_m\|^\alpha\leq C|m-n|^{1+\beta}$ for all $m,n\geq 1.$ Is it true that there exist a almost surely equal version of $(X\_n)$ say $(Y\_n)$... | https://mathoverflow.net/users/136860 | Hölder continuity for discrete time process | The answer is no. E.g., let $X\_n=n$ for all natural $n$. Let $C=1$ and $\alpha=1$, and take any $\beta\in(0,1)$ and any natural $m<n$. Then
$$E|X\_n-X\_m|^\alpha=E|X\_n-X\_m|=n-m\le C|m-n|^{1+\beta}.
$$
However,
$$|X\_n(\omega)-X\_m(\omega)|=n-m>C(\omega)|m-n|^\gamma
$$
for all $\omega\in\Omega$, all real $C(\omega)... | 1 | https://mathoverflow.net/users/36721 | 344666 | 146,154 |
https://mathoverflow.net/questions/344604 | 9 | Binomial coefficients have a well known asymptotics (<https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas>) given by $$\binom nk\sim\binom{n}{\frac{n}{2}} e^{-d^2/(2n)} \sim \frac{2^n}{\sqrt{\frac{1}{2}n \pi }} e^{-d^2/(2n)}.$$
Is there corresponding asymptotics for multinomials (I am mo... | https://mathoverflow.net/users/136553 | Asymptotics of multinomial coefficients | Suppose that $k$ is a fixed natural number, $n\to\infty$, and
\begin{equation\*}
a\_i=\frac nk+o(n^{2/3})
\end{equation\*}
for each $i$; here in what follows, $i\in\{1,\dots,k\}$.
Let
\begin{equation\*}
h\_i:=\frac kn\,a\_i-1=o(n^{-1/3}), \tag{1}
\end{equation\*}
so that $h\_i\to0$ and
\begin{equation}
a\_i=\frac nk... | 9 | https://mathoverflow.net/users/36721 | 344669 | 146,156 |
https://mathoverflow.net/questions/344675 | 14 | This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual understanding and related references if they exist.
Let $R$ be a commutative ring. For two vectors $v=(a,b,c,d), w=(A,B,C,D)\i... | https://mathoverflow.net/users/2083 | Why does this matrix have zero determinant? | Three vectors $v\_1,v\_2,v\_3$ lie in a hyperplane $H:\alpha x+\beta y+\gamma z+\delta t=0$, in this plane we have $Q(v,v):=(\alpha x+\beta y)^2-(\gamma z+\delta t)^2=0,\forall v\in H$. Thus by polarization $Q(v,w)=\frac 14 (Q(v+w,v+w)-Q(v-w,v-w))=0$ for all $v,w\in H$ that yields a relation between columns of your mat... | 21 | https://mathoverflow.net/users/4312 | 344676 | 146,159 |
https://mathoverflow.net/questions/344667 | 8 | We consider the solution of $x^2=x+1$ and denote them as $\phi=\frac{1}{2}(1-\sqrt{5}),\bar\phi=\frac{1}{2}(1+\sqrt{5})$.
Suppose $\phi \not\in \mathbb{F}\_p$. In other words, $\sqrt{5} \not \in \mathbb{F}\_p\Leftrightarrow p = \pm 2 \bmod 5$. Arbitrary element of $\mathbb{F}\_p(\phi)$, $a$ and $b$ satisfies $ab=0\Left... | https://mathoverflow.net/users/142913 | How to prove $(\phi-1)(\phi-2)...(\phi-p) = \sqrt{5} + p\left(\frac{1}{2}+A\sqrt{5}\right) \bmod p^2$? | Start with $$(\phi-i)(\phi-(p+1-i))=\phi^2-\phi(p+1)+i(p+1-i)=p(i-\phi)+(1+i-i^2).$$
Using this for $i=1,2,\ldots,(p-1)/2$ we get
$$
\prod\_{j=1}^p(\phi-j)=\left(\phi-\frac{p+1}2\right)\prod\_{i=1}^{(p-1)/2}\left(p(i-\phi)+(1+i-i^2)\right).
$$
Expand the brackets and take it modulo $p^2\mathbb{Z}[\phi]$. We get
$$
M:=... | 6 | https://mathoverflow.net/users/4312 | 344680 | 146,161 |
https://mathoverflow.net/questions/344678 | 7 | Let $S\_{g,n}$ be a Riemann surface of genus $g$, with $n$ points removed. The mapping class group of $S\_{g,n}$ is denoted by $\Gamma\_{g,n}$.
Is there a reference where the abelianization of $\Gamma\_{g,n}$ calculated (or at least for $g$ sufficiently large, are they trivial)?
| https://mathoverflow.net/users/nan | Abelianization of mapping class groups $\Gamma_{g,n}$ | The following statement can be found in Section 5 of [Low-dimensional homology groups of mapping class groups: a survey](https://arxiv.org/abs/math/0307111):
>
> **Theorem:** Let $g \geq 1$. Then $$H\_1(\Gamma\_{g,r}^n,\mathbb{Z}) \simeq \left\{ \begin{array}{cl} \mathbb{Z}\_{12} & \text{if $(g,r)=(1,0)$} \\ \mathb... | 8 | https://mathoverflow.net/users/122026 | 344681 | 146,162 |
https://mathoverflow.net/questions/344626 | 3 | If $X$ is a uniformly rotund space , then for any closed subspace $M$ of $X$, $X/M$ is uniformly rotund. Does this hold for a locally uniformly rotund space? That is if $X$ is locally uniformly rotund, is it true that $X/M$ is locally uniformly rotund? I couldn't prove it nor could I found a counter example to disprove... | https://mathoverflow.net/users/41137 | Quotient space of a locally uniformly rotund space | Consider any set $\Gamma$ and the Banach space $X = \ell\_1(\Gamma)$. Then the norm $$\|x\|^2 = \|x\|\_{\ell\_1(\Gamma)}^2 + \|x\|\_{\ell\_2(\Gamma)}^2$$ is LUR (and equivalent to the original norm on $\ell\_1(\Gamma)$). Now pick a space $Y$ without a LUR renorming (for example, $\ell\_\infty$). If $\Gamma$ has cardina... | 6 | https://mathoverflow.net/users/15129 | 344682 | 146,163 |
https://mathoverflow.net/questions/344673 | 3 | Let $X$ be a scheme, let $\mathcal{A}$ be a sheaf of locally free algebras on $X$. We say $\mathcal{A}$ is an azumaya algebra, if the natural map $$\mathcal{A}\otimes\_{\mathcal{O}\_X}\mathcal{A}^{opp}\to\mathcal{E}nd\_{\mathcal{O}\_X}(\mathcal{A}), $$ $$a\otimes b\mapsto (x\mapsto axb)$$ is an isomorphism.
Two azuma... | https://mathoverflow.net/users/nan | Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$? | The answer is yes and one can take $$\mathcal{H}:=\mathcal{Hom}\_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A,
\mathcal G).$$ Here, $\mathcal F\otimes \mathcal A$ is viewed as a left $\mathcal{End}(\mathcal G)$-module via the isomorphism $\mathcal{End}(\mathcal G)\cong \mathcal{End}(\mathcal F)\otimes \math... | 5 | https://mathoverflow.net/users/86006 | 344684 | 146,164 |
https://mathoverflow.net/questions/344687 | 3 | Let $B$ be a cellular (simplicial, semi-simplicial etc) complex having $\mathbb{Z}^n$-symmetry (that is, there is a free action of $\mathbb{Z}^n$ on $B$, commuting with the boundary operators) and let $R$ be a commutative ring. In this case it is possible to establish an isomorphism (as $R$-modules) of the homology mod... | https://mathoverflow.net/users/147764 | Homology modules and symmetry | Suppose a group $G$ acts `properly discontinuously' on a friendly topological space $B$, so that $G$ is acting freely on $B$, and, if we let $B^{\prime} = B/G$, then $B \rightarrow B^{\prime}$ is a covering map. It is a nice exercise in the standard lifting theorem of covering space theory to show that cellular structu... | 2 | https://mathoverflow.net/users/102519 | 344692 | 146,167 |
https://mathoverflow.net/questions/344657 | 1 | Let $A=KQ/I$ be a finite dimensional quiver algebra with an admissible ideal $I$.
Is it true that in case $A$ is representation-finite, $Q$ has to be planar?
In case it is true a possible approach would be to use Kuratowski's criterion which says that a graph is planar iff it does not contain a subgraph homeomorphi... | https://mathoverflow.net/users/61949 | Representation-finite implies planar for quiver algebras? | Consider path algebras $KQ$ modulo "radical square zero" relations (i.e., paths of length two are zero). It is well known that these have finite representation type if and only if the separated quiver of $Q$ is a disjoint union of Dynkin quivers.
If $Q$ is the complete graph $K\_5$ on five vertices (nonplanar), orien... | 3 | https://mathoverflow.net/users/22989 | 344700 | 146,168 |
https://mathoverflow.net/questions/344706 | 1 | Consider a B&D process with infinitely many states, State 0 absorbing. Probability of ultimate absorption when starting in State $n$ (denoted $a\_n$) is computed by solving $$a\_n=\frac{\lambda\_na\_{n+1}+\mu\_na\_{n-1}}{\lambda\_n+\mu\_n}$$ for $n\ge1$, where $\lambda\_n$ and $\mu\_n$ are the up and down rates (respec... | https://mathoverflow.net/users/141969 | Computing probability of ultimate absorption in B&D processes | $\newcommand{\intr}[2]{\overline{#1,#2}}$
The desired result follows immediately from
>
> **Theorem**
>
>
> (I) If $a\_1=1$, then $a\_j=1$ for all $j\in\intr1\infty$.
>
>
> (II) If $a\_1<1$, then $S\_\infty<\infty$ and $a\_j=(S\_\infty-S\_j)(1-a\_1)$ for all $j\in\intr1\infty$, where
> \begin{equation\*}
>... | 2 | https://mathoverflow.net/users/36721 | 344708 | 146,172 |
https://mathoverflow.net/questions/344713 | 18 | We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is:
$$
S\_n = \sum\_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j
$$
We have observed numerically that $S\_n \approx 2 n e^{-n}$. We would like to establish whether this conjecture is true. More precisely, we would lik... | https://mathoverflow.net/users/147778 | A finite alternating sum | I have obtained a formula for the generating function of your sequence.
Let $S\_n$ be defined as in the quesion. We extend the definition to $n = 0$ by demanding $0^0 = 0$, hence $S\_0 = 0$.
Consider $S(t) = \sum\_{n\geq 0} S\_n t^n$. I will work out a formula for $S(t)$.
\begin{eqnarray\*}
S(t) &=& \sum\_{n=0}^\... | 23 | https://mathoverflow.net/users/76332 | 344716 | 146,173 |
https://mathoverflow.net/questions/344709 | 19 | Suppose $p$ is a prime number such that $p\equiv 7 \pmod{12}$.
Since $p \not \equiv 1 \mod{4}$, the ring of integers of $\mathbb{Q}(\sqrt{p})$ is $\mathbb{Z}\oplus\mathbb{Z}(\sqrt{p})$ with fundamental unit of the form $a+b\sqrt{p}$, where $a, b > 0$
It is of the norm $+1$, because $a^2 - pb^2 \equiv -1 \pmod{4}$ d... | https://mathoverflow.net/users/65714 | First coefficient of totally positive fundamental unit modulo 3 | By Theorem 1.1 of Zhang-Yue: Fundamental units of real quadratic fields of odd class number, J. Number Theory 137 (2014), 122-129, we have that $a\equiv 0\pmod{2}$. From here it is a simple matter to prove that $a\equiv 2\pmod{3}$, hence in fact $a\equiv 2\pmod{6}$. To see this, let us write
the unit equation as
$$(a-1... | 14 | https://mathoverflow.net/users/11919 | 344722 | 146,175 |
https://mathoverflow.net/questions/344725 | 2 | Do we know anything about sums over primes in arithmetic progressions like the following:
$$\sum\_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$
where $q$ is a prime and $\alpha > 0$? If we consider the average over this sum:
$$ \frac{1}{\pi(x,l,a)} \sum\_{\substack{q \equiv a (\text{mod } l) \\ q \le ... | https://mathoverflow.net/users/73880 | Sums over primes in arithmetic progressions | The standard analytic proof of the prime number theorem for arithmetic progressions will work here as well, just replacing $L(s,\chi)$ with $L(s-\alpha,\chi)$; the asymptotic size of your first sum will be
$$
\frac{x^{1+\alpha}}{(1+\alpha)\phi(l)\log x}.
$$
| 4 | https://mathoverflow.net/users/5091 | 344726 | 146,177 |
https://mathoverflow.net/questions/344501 | 0 | I was wondering if someone would be willing to suggest an alternative reference to Davenport's book Analytic Methods for Diophantine Equations and Diophantine Equations. I like the book but I would like to read up from a different source some theorems about the geometry of numbers which are important to analytic number... | https://mathoverflow.net/users/nan | Alternative reference to Davenport's Analytic Methods for geometry of numbers? | Some classical books on the subject, I use them all:
* Cassels: An introduction to the geometry of numbers
* Gruber-Lekkerkerker: Geometry of numbers
* Siegel-Chandrasekharan: Lectures on the geometry of numbers
| 1 | https://mathoverflow.net/users/11919 | 344728 | 146,179 |
https://mathoverflow.net/questions/344731 | 0 | Let $p \in \mathbb{Z}$ be a prime and consider the number field $k = \mathbb{Q}[x]/(x^{(p^2 -1)/2} - p)$. We shall denote by $O\_k$ the ring of integers of $k$. Let $\beta \in O\_k$ be such that $\beta^{(p^2-1)/2} = p$. Then it is clear that $(\beta)$ is a prime ideal of $O\_k$. Let $\widehat{O}\_k$ be the completion o... | https://mathoverflow.net/users/11392 | Roots of unity, in the completion of the ring of integers of a number field, at a prime ideal | Let $K = \mathbf Q\_p(\sqrt[n]{p})$, where $n \geq 1$. The polynomial $x^n-p$ is Eisenstein at $p$, so $K/\mathbf Q\_p$ is totally ramified at $p$, so its residue field has size $p$, as you indicate. In a local field with residue field of characteristic $p$ and size $q$, its roots of unity with order relatively prime t... | 4 | https://mathoverflow.net/users/3272 | 344733 | 146,181 |
https://mathoverflow.net/questions/344746 | 28 | This is a very basic question, but I can't find a clean answer anywhere.
In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably. A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to... | https://mathoverflow.net/users/147807 | Smooth vs regular vs non-singular | In the general context, "regular" is a property of a scheme (or a ring, or local ring), and "smooth" is a property of a morphism of schemes.
"Regular" means exactly that at every point, the dimension of the (Zariski) tangent space is equal to the (Krull) dimension (of the local ring at that point).
A map $f: X \to ... | 33 | https://mathoverflow.net/users/18060 | 344749 | 146,183 |
https://mathoverflow.net/questions/344720 | 3 | Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The [nLab entry](https://ncatlab.org/nlab/show/site) suggests the term **Morita equivalence**, however I have not yet found any other sour... | https://mathoverflow.net/users/35353 | What to call a morphism of sites inducing an equivalence on categories of sheaves? | Johnstone's *Sketches of an Elephant* (2 volumes) is a standard reference which uses "Morita equivalence" in this way. In fact, Jonstone systematically uses "Morita equivalence" in a similar way across many types of categorical logic. Relevant here is the case of *geometric logic*.
Although some of the terminology in... | 2 | https://mathoverflow.net/users/2362 | 344750 | 146,184 |
https://mathoverflow.net/questions/344739 | 1 | In [Calculus of functors and model categories II](https://arxiv.org/abs/1305.2834) Biedermann and Rondigs claim in Corollary 6.18 that the $n$-homogeneous model structure on $\mathrm{Fun}(\mathcal{C}, \mathcal{D})$ is stable if $\mathcal{D}$ admits a set of generating cofibrations with cofibrant domain.
Can anyone s... | https://mathoverflow.net/users/117088 | Calculus of Functors and Model categories | The assumptions for Corollary 6.18 are not just the ones stated in the question. They are really a whole list of sometimes rather technical assumptions on the source and target category. Still, they are satisfied in many cases of interest.
Now about stability. On top of the page 2909 of our paper (the same page of Co... | 2 | https://mathoverflow.net/users/102493 | 344756 | 146,186 |
https://mathoverflow.net/questions/344686 | 2 | I am studying the compositions $(n\_1,...,n\_r)$ of an integer $m$ such that $i\vert n\_i$ for all $i=1,...,r$. (Recall that a composition $(n\_1,...,n\_r)\vDash m$ of $m$ is just a sequence $(n\_1,...,n\_r)\in \mathbb{P}^r$ of positive integers for which $n\_1+n\_2+\cdots+n\_r=m$.) Here the length $r$ of the compositi... | https://mathoverflow.net/users/128914 | Compositions $(n_1,...,n_r)$ of an integer $m$ such that $i$ divides $n_i$ | The number $f(m)$ of such sequences is the number $q(m)$ of partitions of $m$ into distinct parts (<https://oeis.org/A000009>). For a bijection, take your sequence $(n\_1,\dots,n\_r)$ and let $\lambda=(\lambda\_1,\dots,\lambda\_r)$ be the conjugate partition (using the standard definition of partition) of the partition... | 6 | https://mathoverflow.net/users/2807 | 344765 | 146,187 |
https://mathoverflow.net/questions/344711 | 1 | Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of section of $\wedge^{\bullet}E$ for a vector bundle $E$, we get a natural algebra structure on $H^{\bullet}(X, F)$.
For e... | https://mathoverflow.net/users/104742 | Multiplicative structure for sheaf cohomology of flag varieties | I think to obtain the product you need to use Massey products. Let me illustrate this in the case of $G = \mathrm{SL}\_3$. Consider the element
$$
x\_1 \otimes x\_2 \otimes x\_1 \in
\mathrm{Ext}^1(\mathcal{O},\mathcal{L}\_{-\alpha\_1}) \otimes
\mathrm{Ext}^1(\mathcal{L}\_{-\alpha\_1},\mathcal{L}\_{-\alpha\_1-\alpha\_2... | 2 | https://mathoverflow.net/users/4428 | 344771 | 146,188 |
https://mathoverflow.net/questions/344768 | 1 | Given the hyperbolic Vlasov equation
$$ \frac{\partial f }{\partial t} +v\nabla\_x f + F(t,x)\nabla\_vf =0$$
where $f=f(t,x,v)$ and $(t,x,v)\in \mathbb{ R}\times\mathbb{R}^{n}\times \mathbb{R}^{n} $. I wonder how can be proved that $$ \Vert f(t,x,v)\Vert\_{L^p(\mathbb{R}^{2n})} = \Vert f(0,x,v)\Vert\_{L^p(\mathbb{R}^... | https://mathoverflow.net/users/137336 | Conservated quantity and hyperbolic equation | The "one phrase answer" is "divergence theorem".
Slightly wordier but a bit formally (for ease of typing I write $dz = dx~dv$ for the volume on phase space)
$$ \partial\_t \int f^p dz = \int \partial\_t f^p dz $$
Next,
$$ 0 = \int \nabla\_x \cdot (vf^p) dz $$
assuming $f$ decays suitably fast at infinity, a... | 4 | https://mathoverflow.net/users/3948 | 344773 | 146,189 |
https://mathoverflow.net/questions/344775 | 8 | There is no such thing as a free field, because there are no morphisms between fields of different characteristics. However, ordered fields seem to be much better behaved: There is an initial object ($\mathbb{Q}$, the rational numbers) and a terminal object (**No**, the surreal numbers). Does this mean a free ordered f... | https://mathoverflow.net/users/51063 | Free ordered field? | I interpret a "free ordered field" to mean the existence of a left adjoint functor to the$^1$ forgetful functor from ordered fields either to sets or to totally ordered sets. In both cases, the answer is "no". You don't even need two points x,y and compare them, a singleton $X=\lbrace\ast\rbrace$ is enough.
I claim t... | 15 | https://mathoverflow.net/users/3041 | 344778 | 146,190 |
https://mathoverflow.net/questions/344780 | 0 | Is this statement true. A bounded half-superharmonic function in $\mathbb R$ is a constant. That is $(-\Delta)^{1/2} u\geq 0$ implies $u\equiv 0.$
| https://mathoverflow.net/users/139853 | Fractional super-harmonic functions | This claim true, but the details depend on your favourite definition of $(-\Delta)^{1/2} u$ and regularity assumptions on $u$.
One argument could be as follows: Suppose that $u \geqslant 0$ and $(-\Delta)^{1/2} u \geqslant 0$. Denote by $P\_r(x,z)$ the Poisson kernel for $(-\Delta)^{1/2}$ in $(-r,r)$ and by $G\_r(x,y... | 1 | https://mathoverflow.net/users/108637 | 344785 | 146,193 |
https://mathoverflow.net/questions/344784 | 4 | In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, the following characterization of countable groups: $G$ is countable if and only if it has a left-invariant metric with finite balls. See Proposition 1.A.1. and Lemma 2.B.5 of <https://arxiv.org/pdf/1403.3796.pdf> ,
... | https://mathoverflow.net/users/147032 | Characterization of countable groups as groups with a left-invariant distance with finite balls | I have not found a proof in the book. But the statement is not difficult. Assign to each element of a countable group $G$ a natural number $0,1,2,...$ (for every $n$ only finite number of elements $g$ are assigned the same numbers $n(g)=n$, $n(1)=0$ and 1 is the only element with number 0, $n(g^{-1})=n(g)$). Then defin... | 7 | https://mathoverflow.net/users/nan | 344788 | 146,195 |
https://mathoverflow.net/questions/344804 | 12 | Let $C(\mathbb{R})$ be the space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ with the compact-open topology, and the associated Borel $\sigma$-algebra. Consider the function $p$ from $C(\mathbb{R})$ to $\mathbb{R}\_{\geq 0} \cup \{\infty\}$ that maps a continuous function to its period, with the conventio... | https://mathoverflow.net/users/56183 | Is the map sending a continuous function to its period measurable? | Isn't the set $p^{-1}([0, T\_0])$ closed for every finite $T\_0$? Suppose that $f\_n$ has period $T\_n \leqslant T\_0$ and it converges locally uniformly to $f$. By passing to a subsequence, we may assume that $T\_n$ has a limit $T$. Uniform convergence of $f\_n$ on $[x, x + T\_0]$ implies that $$f(x + T) = \lim f\_n(x... | 21 | https://mathoverflow.net/users/108637 | 344805 | 146,197 |
https://mathoverflow.net/questions/344810 | 1 | I am interested in a reference and proof for some version of the following (folklore?) statement:
``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ be an (irreducible) arithmetic lattice. Then $\mathrm{Comm}\_G(\Gamma)$ is a simple group."
I have yet been unable to ... | https://mathoverflow.net/users/121307 | Reference request: The commensurator of an arithmetic lattice is a simple group | Hmm, that's not exactly true. For example, $Comm\_{SL\_2(\mathbb{R})}(SL\_{2}(\mathbb{Z}))$ contains the normal subgroup $\pm I$.
For a special case describing the commensurator (when the complexified algebraic group has trivial center), see Ex. 4, §5.2. <https://arxiv.org/abs/math/0106063> For the general case, map... | 2 | https://mathoverflow.net/users/1345 | 344814 | 146,201 |
https://mathoverflow.net/questions/162375 | 19 | In "Automorphic representations of GSp(4)" (2004) (see <http://www.math.toronto.edu/arthur/>), James Arthur announces a classification of discrete automorphic representations of GSp(4). There are no proofs however, these are to appear in a monograph "Automorphic representations of classical groups" (in preparation), wh... | https://mathoverflow.net/users/1310 | What's the status of Arthur's announced classification for GSp(4)? | This question is answered pretty definitively by the following recent paper:
>
> *Gee, Toby; Taïbi, Olivier*,
> [**Arthur’s multiplicity formula for $\mathrm{GSp}\_4$ and restriction
> to $\mathrm{Sp}\_4$**](http://dx.doi.org/10.5802/jep.99), Journal de
> l'École polytechnique — Mathématiques, Volume 6 (2019), p... | 11 | https://mathoverflow.net/users/2481 | 344818 | 146,202 |
https://mathoverflow.net/questions/344815 | 6 | Let $(X,d)$ be an $\mathcal{H}^n$-rectifiable metric space, i.e. there exits a collection of Lipschitz maps from measurable subsets of $\mathbb{R}^n$ to $X$ such that $ \mathcal{H}^n(X \backslash \cup\_i f\_i(A\_i)) = 0 $.
Is it true that for any subset $A \subset X$,
$$ \mathcal{H^n}(A) = \mathcal{H}^n\_\infty (A) ... | https://mathoverflow.net/users/91442 | Is Hausdorff Measure equal to Hausdorff Content on rectifiable (metric) spaces? | In general, no. For example, $X$ may be a countably infinite collection of lines through the origin in $\mathbb{R}^2$. Then $X$ is $1$-rectifiable.
For any ball $B$ centered at the origin, $B\cap X$ has finite Hausdorff $1$-content but infinite Hausdorff $1$-measure.
If you have some kind of Ahlfors regularity of t... | 7 | https://mathoverflow.net/users/147874 | 344820 | 146,204 |
https://mathoverflow.net/questions/344802 | 1 | I've heard that if you assume the existence of (Dedekind) infinitely many objects, you can derive -- in second-order logic, given suitable definitions -- the (second-order) Peano axioms for arithmetic. I was wondering how much second-order logic is actually needed for that result, and how exactly the natural numbers ar... | https://mathoverflow.net/users/147858 | Interpreting PA2 in second-order logic + existence of infinitely many objects | First note that monadic second-order logic (i.e. the variant of second-order logic with second-order quantifiers only over unary predicates) isn't sufficient. This is implied by the fact that the monadic theory $\mathsf{MSO}(\mathbb{N},0,S)$ is decidable. Thus further I consider second-order logic with quantifiers $\fo... | 1 | https://mathoverflow.net/users/36385 | 344822 | 146,205 |
https://mathoverflow.net/questions/344797 | 0 | I have a question on asymptotic behavior of distributions of Brownian hitting times.
Let $B\_t$ and $W\_t$ be independent one-dimensional Brownian motions starting at the origin. The joint law is denoted by $P$.
For $x,y>0$, we set
\begin{align\*}
\sigma\_x&=\inf\{t>0 \mid B\_t=x\} \\
\tau\_y&=\inf\{t>0 \mid W\_t \n... | https://mathoverflow.net/users/68463 | Asymptotics of distributions of hitting times | OK, here is the extended version of my comment.
---
Let $u(x,y)$ be the (bounded) harmonic function in $D = (-1,1) \times (0, \infty)$, with boundary value $1$ along $\{-1,1\} \times (0, \infty)$ and $0$ along $(-1,1) \times \{0\}$. Consider a 2-D Brownian motion $(X\_t, Y\_t)$, started at $(X\_0,Y\_0) = (x,y)$, ... | 1 | https://mathoverflow.net/users/108637 | 344825 | 146,207 |
https://mathoverflow.net/questions/344167 | 19 | Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My questions are:
>
> Question 1: has this function been studied before? Any reference? It looks related to [Sidon sets](https... | https://mathoverflow.net/users/2083 | Size of sets with complete double | Let $S\subset [1,N]$ with $S+S=2N$. I will show that $S$ must have at least $(2.033 +o(1))\sqrt{N}$ elements for large $N$. The argument can certainly be improved, but I don't know what the right answer should be.
Assume that $N$ is large and that $|S| =O(\sqrt{N})$ (else we are done of course). Let $r(n)$ denote th... | 11 | https://mathoverflow.net/users/38624 | 344827 | 146,208 |
https://mathoverflow.net/questions/344811 | 0 | Let $p\_l$ the $l$-th prime number. I've considered the formula
$$\frac{N\_{n+1}}{N\_n}+\frac{N\_{n+2}}{p\_{n+1}N\_n}\pm1$$
where $N\_k=\prod\_{l=1}^k p\_l$ is the primorial of order $k$. Previous formula can be simplified thus as $$p\_{n+1}+p\_{n+2}\pm 1.\tag{1}$$
I wondered, for $n\geq 1$ running over positive i... | https://mathoverflow.net/users/142929 | On the quantity of twin prime pairs of a given form | Indeed, standard conjectures imply that there should be infinitely many such twin prime pairs even if we insist that $p\_{n+1},p\_{n+2}$ are themselves twin primes! In other words, there should be infinitely many integers $p$ such that $p$, $p+2$, $p+(p+2)-1=2p+1$, and $p+(p+2)+1=2p+3$ are all prime—this is a special c... | 5 | https://mathoverflow.net/users/5091 | 344835 | 146,210 |
https://mathoverflow.net/questions/344848 | 3 | The Dilworth theorem for finite posets implies that a finite poset contains either a "large" chain or a "large" antichain. I am sure I saw an infinite version of this :
An infinite poset has either an infinite chain or an infinite antichain.
But I can't find a reference to that statement. What is the reference or a... | https://mathoverflow.net/users/nan | An infinite version of the Dilworth theorem | This is studied in Reverse Mathematics as the [Chain Antichain Principle](https://pdfs.semanticscholar.org/e849/71951a12a859da7d7f46acc78b5374aef557.pdf) (CAC)
and it is observed that it follows from Ramsey's theorem.
| 5 | https://mathoverflow.net/users/4600 | 344851 | 146,212 |
https://mathoverflow.net/questions/344852 | 1 | Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization
$\mathbf p:\mathbb R\to \partial B$ assigning to each real number $t$ the unique point of the intersection of $\partial B$ with the ray $\{re^{it}:r>0\}$. The function $\... | https://mathoverflow.net/users/61536 | Convex-like properties of the polar parametrization of the boundary a convex body on the plane | If $f(x)=\min\{s>0:x/s\in B\}$ is Minkowski functional of $B$, then $f$ is a convex function on the plane and ${\bf p}(t)=\frac{e^{it}}{f(e^{it})}$. I think your claims now follow from the properties of convex functions. For example, you may locally choose a smooth function $h(t)\in (0,\infty)$ such that $h(t)e^{it}$ r... | 1 | https://mathoverflow.net/users/4312 | 344856 | 146,214 |
https://mathoverflow.net/questions/344857 | 7 | *Inspired by [this question](https://mathoverflow.net/questions/344775/free-ordered-field) I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)?*
For example for the function field $\mathbb{Q}(X)$ there are the following examples of total or... | https://mathoverflow.net/users/3969 | What is the topology on the set of field orders | The topology you are looking for is called the *Harrison topology*. If we denote the set of ordering of a field $F$ with $\mathrm{Sper}\,F$ (more on that in a moment), this is the subspace topology given by the embedding
$$\mathrm{sign}:\mathrm{Sper}\,F\to \prod\_{x\in F^\times}\{+1,-1\}$$
sending an order to the colle... | 12 | https://mathoverflow.net/users/43054 | 344859 | 146,215 |
https://mathoverflow.net/questions/344632 | 2 | The question is about the existence of a number $x$ for which we know the existence of $c>0$ such that for all $u>0,n\in\mathbb{N}^\*$ that $$
\frac {1}{nu}\sum\_{j=1}^{n}1\_{d(jx,\mathbb Z)<u}<c
$$
This holds for each $u>0$ for $x$ irrational (i.e. with $c$ depending on $u$, see references on "well distributed numbers... | https://mathoverflow.net/users/16934 | Is there a number for which we know precisely the approximation by rationals? | Yes, quadratic irrationalities satisfy such inequality. Denote $\|t\|=d(t,\mathbb{Z})$. We know that if $x$ is a fixed quadratic irrationality, say $x=\sqrt{2}$, then $\|mx\|\geqslant C/m$ for fixed $C$ depending only on $x$ and any positive integer $m$. Thus if $\|j\_1x\|<u$ and $\|j\_2x\|<u$ for two non-negative inte... | 2 | https://mathoverflow.net/users/4312 | 344861 | 146,216 |
https://mathoverflow.net/questions/344862 | 3 | Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum\_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$ but I was hoping for something better for example $\sum\_{n\le x} \frac{n}{\tau(n)}=c\cdot x^2+\mathcal O(x^{smal... | https://mathoverflow.net/users/38851 | How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$? | It was proven by Ivic that $$
\sum\_{n \leq x} \frac{1}{d\_{k}(n)}=b\_{k, 1} x \log ^{1 / k-1} x+\cdots+b\_{k, N} x \log ^{1 / k-N} x+O\left(x \log ^{1 / k-N-1} x\right)$$ where $k \geq 2$ and $N$ is arbitrary, fixed, natural number; the constants $b\_{k, 1}, \ldots, b\_{k, N}$ depend only on $ k$ (see *A. Ivic. On the... | 5 | https://mathoverflow.net/users/5712 | 344869 | 146,218 |
https://mathoverflow.net/questions/344846 | 3 | Let $f: X \to [ -\infty, \infty]$ be some function,
Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$,
$$\partial f(x)$$ is "unbounded"? (trivial examples included the improper functions)
A rough definition of an unbounded subgradient is that there exists some sequence $v\... | https://mathoverflow.net/users/145832 | What is a non-trivial example of an unbounded subdifferential? | A simple example is the convex function
$$
f(x) = \begin{cases} \infty, & \text{if}\ x<0\\
x & \text{if}\ x\geq 0.
\end{cases}
$$
It holds that $\partial f(0) = ]-\infty,1]$.
There are no examples without the value $\infty$: If $f$ is convex and bounded in a neighborhood at some point, then $f$ is locally Lipschitz a... | 3 | https://mathoverflow.net/users/9652 | 344877 | 146,221 |
https://mathoverflow.net/questions/344879 | 3 | Definition: a set $X$ is to be labeled as "*non-exhaustively overlapping*" if and only if each element of $X$ is not a subset of any other element of $X$; formally:$$ X \text { is non-exhaustively overlapping } \equiv\_{df}\\ \forall x,y \in X (x \not \subsetneq y) $$
>
> Question 1: Is it consistent with ZF to hav... | https://mathoverflow.net/users/95347 | Can a power set be equal in cardinality to a subset of it that is non-exhaustively overlapping? | The answer to question 1 is yes. Take the binary binary branching tree $2^{<\omega}$ and label every node with a different natural number. Now, for each branch $b$ through the tree, let $X\_b$ be the set of labels on those nodes. Distinct branches $b$ will give continuum many set $X\_b$, and none of these is a subset o... | 7 | https://mathoverflow.net/users/1946 | 344881 | 146,223 |
https://mathoverflow.net/questions/344868 | 0 | Let $C\_0([0,1];\mathbb{R}^d)$ be the classical Wiener space (of continuous paths with initial value $0$) and let $\nu$ be the Wiener measure on this space. Does there exist a countable family $\left\{U\_n\right\}\_{n \in \mathbb{N}}$ of open subsets of $C\_0([0,1];\mathbb{R}^d)$ such that
* $\nu(U\_n)= \frac1{2^n}$
... | https://mathoverflow.net/users/36886 | "Geometric" Decomposition of Wiener Space | For natural $n$, let
$$U\_n:=\{x=(x\_1,\dots,x\_d)\in C\_0([0,1];\mathbb{R}^d)\colon \Phi(x\_1(1))\in\delta\_n\},
$$
where $\Phi$ is the standard normal pdf and $\delta\_n:=(1-1/2^{n-1},1-1/2^n)$. Then the family $(U\_n)$ has all the desired properties.
---
Indeed,
(i) for each natural $n$, the set $U\_n$ is ... | 1 | https://mathoverflow.net/users/36721 | 344884 | 146,224 |
https://mathoverflow.net/questions/344886 | 0 | Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily exist a countable open cover $\{U\_i\}\_{i \in \mathbb{N}}$ of $X$ and a sequence of natural numbers $\{N\_i\}\_{i \in \mathbb... | https://mathoverflow.net/users/36886 | Topologically transitive dynamical system mapping space into ball | **No.**
Let $\phi$ be the left shift on the set $X = \{0,1\}^\mathbb{Z}$ of bi-infinite binary sequences with the prodiscrete topology, and let $V = \{ x \in X : x\_0 = 0 \}$ be the set of sequences that have $0$ at the central coordinate.
Then $X$ is Hausdorff (even metrizable and compact), $\phi$ is transitive (eve... | 1 | https://mathoverflow.net/users/66104 | 344888 | 146,226 |
https://mathoverflow.net/questions/344839 | 2 | In the paper
>
> *Vickers, J.; Welch, P. D.*, [**On elementary embeddings from an inner model to the universe**](http://dx.doi.org/10.2307/2695094), J. Symb. Log. 66, No. 3, 1090-1116 (2001). [ZBL1025.03049](https://zbmath.org/?q=an:1025.03049).
>
>
>
it is stated to that if $Ord$ is Ramsey (I.e. there is a pr... | https://mathoverflow.net/users/141402 | Possible inconsistency related to embeddings $j: M\prec V$ | There is no contradiction here.
Look at Theorem $2.3$:
>
> Suppose $I\subseteq On$ witnesses $On$ is Ramsey. Then, definably
> over $\langle V,\in, I\rangle$, there is a transitive class $M$, and an elementary embedding $j :\langle M,\in\rangle\rightarrow\langle V,\in\rangle$ with $j \not= id$.
>
>
>
Note t... | 4 | https://mathoverflow.net/users/8133 | 344892 | 146,228 |
https://mathoverflow.net/questions/344898 | 2 | Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space.
A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., a metric $d'$ such that if $p,q \in X$, there exists a point $r\in X$ such that $d'(p,r) = d'(q,r) = d'(p,q)/2$.
I w... | https://mathoverflow.net/users/56667 | Explicit construction of a convex metric | As far as I know the most accessible proof of Bing's construction is in Section 2 of the paper:
**J. C. Mayer, L. G. Oversteegen, E, D. Tymchatyn,**
The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets.
*Dissertationes Math. (Rozprawy Mat.)* 252 (1986), 45 pp.
... | 1 | https://mathoverflow.net/users/121665 | 344902 | 146,231 |
https://mathoverflow.net/questions/344867 | 3 | Let $G$ be a reductive algebraic group over $k$, and $V\_i$ be (finite-dimensional) representations of $G$. Are the irreducible components of $\bigotimes\_i V\_i$ always generated by indecomposable vectors? That is, vectors of the form $v\_1 \otimes \cdots \otimes v\_n$ where $v\_i \in V\_i$.
| https://mathoverflow.net/users/126543 | Are irreducible subrepresentations of a tensor product always generated by indecomposable vectors? | It seems unlikely, given that already the assertion is not literally true for the tensor product of two copies of the standard representation $V$ of $G=SL\_2(\mathbb R)$. Namely, with $v$ a highest-weight vector in $V$, and $w$ a lowest-weight vector in $V$, with $Lv=w$ with lowering operator $L$, in $V\otimes V$, by L... | 3 | https://mathoverflow.net/users/15629 | 344919 | 146,237 |
https://mathoverflow.net/questions/344891 | 3 | Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel probability measure on $X$.
Then, for every $\delta \in (0,1)$, does there exist:
* $\{V\_i\}\_{i \in \mathbb{N}}$ are ... | https://mathoverflow.net/users/36886 | Reversal of open cover with topologically transitive dynamical system | **No**, even if we assume $\nu$ to be invariant under $\phi$.
Let $X = \{0,1\}^\mathbb{Z}$ be the set of two-way infinite binary sequences with the prodiscrete topology, and let $\phi$ be the left shift on $X$.
Let $\nu = (\mu\_1 + \mu\_2)/2$ where $\mu\_1$ is the uniform Bernoulli measure on $X$ and $\mu\_2$ is an a... | 3 | https://mathoverflow.net/users/66104 | 344939 | 146,245 |
https://mathoverflow.net/questions/343748 | 9 | Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and that $\text{rank}(df) \ge n-1 $ on $\partial \mathbb{D}^n$.
>
> **Question:** Do there exist $f\_n \in C^{\infty}(\mat... | https://mathoverflow.net/users/46290 | Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank? | I believe the answer is positive.
First step: reduce to the case that $\mathrm{rank}(df\_x)\geq n-1$ except on a finite union of submanifolds of dimension at most $n-4.$
*Proof:* For $r=0,\dots,n-2,$ let $M\_r\subset \mathbb R^{n\times n}$ be the set of matrices of rank exactly $r.$ Each of these is a smooth manifo... | 1 | https://mathoverflow.net/users/112284 | 344952 | 146,247 |
https://mathoverflow.net/questions/344949 | 8 | During these first months in my PhD, I realized how my computational problems can be drastically reduced to *one* single problem:
>
> Find an efficient way to sample from a Gibbs measure.
>
>
>
Let me elaborate: if $H$ is a Hilbert space, $\mu$ a gaussian measure on it, then I need to numerically approximate a... | https://mathoverflow.net/users/147956 | Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)? | Any probability measure $\mu\_1$ absolutely continuous with respect to $\mu\_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below, $G$ will be bounded. So you're basically asking about how to sample from
a probability measure. This is a big field o... | 13 | https://mathoverflow.net/users/13650 | 344953 | 146,248 |
https://mathoverflow.net/questions/344960 | 4 | Let $F$ be a bounded subset of ${\bf M}\_n({\mathbb C})$. G.-C. Rota & G. Strang defined the joint spectral radius of $F$ as follows. For $k\ge1$, denote $F\_k$ the set of all products of $k$ elements of $F$. Set $\|F\_k\|$ the supremum of some matrix norm over $F\_k$. The sequence $\|F\_k\|^{\frac1k}$ converges to its... | https://mathoverflow.net/users/8799 | Joint spectral radius of $\{M,M^T\}$ | Your suspicion is correct. The lower bound is obtained by simply alternating $M$ and its transpose. The upper bound follows from sub-multiplicativity of the norm and the fact that
the operator norm $\|M\|\_2$ equals the top singular value of $M$, namely $\rho(M^\*M)^{\frac12}$.
| 4 | https://mathoverflow.net/users/7691 | 344964 | 146,252 |
https://mathoverflow.net/questions/344968 | 21 | Let $G$ be a finite group and $\Lambda = (\lambda\_{i,j})$ its character table with $\lambda\_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for all triple $(j,k,\ell)$ $$\sum\_i \frac{\lambda\_{i,j}\lambda\_{i,k}\lambda\_{i,\ell}}{\lambda\_{i,1}} \ge 0.$$
It is a... | https://mathoverflow.net/users/34538 | A new combinatorial property for the character table of a finite group? | By standard manipulations with the group algebra, your sum has a combinatorial/probabilistic interpretation that makes its nonnegativity clear.
The element $ \frac{1}{|G|} \sum\_{ g\in G} [g hg^{-1} ]$ in the group algebra is conjugacy invariant, and so acts by scalars on each irreducible representation. Because its ... | 25 | https://mathoverflow.net/users/18060 | 344969 | 146,253 |
https://mathoverflow.net/questions/337964 | 7 | Let $M\_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A\_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties.
Can someone provide a reference for the fact that the Torelli map extends to a map $M\_g^{ct} \to A\_g$ which is proper?
Here are some comments: Th... | https://mathoverflow.net/users/75970 | Does the compactified Torelli map extend to a proper map of stacks? | I wrote an answer to the question above on my website. It is currently available at <http://web.stanford.edu/~aaronlan/assets/properness-of-compact-type.pdf>.
The main points were explained in the comments above, but let me say them again. One can first extend the usual Torelli map by the general fact that over a nor... | 2 | https://mathoverflow.net/users/75970 | 344971 | 146,255 |
https://mathoverflow.net/questions/344989 | 4 | Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable.
We define the distribution function of $X$ by
$$F(x) = P(X \leq x)$$
and the quantile function of $X$ by
$$Q(\alpha)= \inf \{x \in \mathbb R : F(x) \geq \alpha \}$$
... | https://mathoverflow.net/users/147725 | Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$? | Let $F^{-1}:=Q$. It is [well known](https://en.wikipedia.org/wiki/Inverse_transform_sampling#Definition) that $Y:=F^{-1}(U)=Q(U)$ equals $X$ in distribution, where $U$ is any random variable uniformly distributed on $(0,1)$. Therefore, for any real $p>0$,
$$\int\_0^1|Q(u)|^p\,du=E|Y|^p=E|X|^p.$$
So, $Q\in L^p((0,1))$... | 6 | https://mathoverflow.net/users/36721 | 344990 | 146,258 |
https://mathoverflow.net/questions/344895 | 7 | It is known and quite easy to prove that $S\_{\mathbb N}(x) = \sum\_{n\in\mathbb N, n\leq x} \frac 1 n$ grows as $\ln x$. Even more, $\lim\_{x\rightarrow\infty} S\_{\mathbb N}(x)-\ln x = \gamma$, the Euler-Mascheroni constant.
It was also proved by Mertens that $S\_{\mathbb P}(x)$, the sum of reciprocals of the prime... | https://mathoverflow.net/users/114143 | A very slowly diverging series | A nice example was found by Erdos; "[On a problem of G. Golomb](https://www.renyi.hu/~p_erdos/1961-10.pdf)". Let $p\_1 = 3$, and for $i > 1$, let $p\_i$ be the least odd prime exceeding $p\_{i-1}$ which is not congruent to $1$ mod $p\_{j}$ for any $j < i$. That this sequence of primes has the property you seek follows ... | 4 | https://mathoverflow.net/users/16510 | 344994 | 146,260 |
https://mathoverflow.net/questions/344966 | 0 | **Edit:** According to comment conversations we revise the question.
Let $G$ be a group. We consider the following subset of $G$:
$$\{g\in G \mid e^{\lambda\_g} \in \mathbb{C}\lambda (G)\},$$
where $\lambda\_g\in C^\*\_{\text{red}}(G)$ is the left regular representation of $g$.
Under which conditions is this subset... | https://mathoverflow.net/users/36688 | A subset (or subgroup) associated to a group | Let $(\delta\_g)\_{g\in G}$ be the canonical basis of $\mathbf{C}G$. Define $\exp(g)=\sum\frac1{n!}\delta\_g^n$ as an element of the reduced $C^\*$-algebra.
>
> I claim that $\exp(g)\in\mathbf{C}G$ iff $g$ has finite order.
>
>
>
It's clear if $g$ has finite order. For the converse, I claim that more generally... | 3 | https://mathoverflow.net/users/14094 | 345003 | 146,263 |
https://mathoverflow.net/questions/344872 | -1 | We say that a sequence $(\mu\_{n})$ of measures in $M\_{b}(Q)$ converges **tightly** (or, equivalently, in the narrow topology of measures) to a measure $\mu$ in $M\_{b}(Q)$ if
$$\lim\_{n\to\infty}\int\_{Q}\varphi d\mu\_{n} \to \int\_{Q}\varphi d\mu\quad (1.1)$$
for every $\varphi\in C\_{b}(Q)$. If (1.1) holds only for... | https://mathoverflow.net/users/147907 | Convergence in the narrow topology of measures and strongly converge for signed measures | You claim that weak (i.e. tight) and strong convergence are equivalent for non-negative measures, but this is false: take $Q = (0,T) \times (-1,1)$, let $t \in (0,T)$ and let $\mu\_n$ be the Dirac measure concentrated at $(t, \frac 1 n)$ and $\mu$ be the Dirac measure concentrated at $(t,0)$. If $\varphi \in C\_b (Q)$ ... | 1 | https://mathoverflow.net/users/54780 | 345012 | 146,265 |
https://mathoverflow.net/questions/345010 | 8 | Let $SU$ denote the infinite unitary group. Does the quotient space $SU/SU(n)$ admit a delooping $X$? One could also ask that this space $X$ sit in a fiber sequence $BSU(n)\to BSU\to X$, but this is not strictly part of the question. Note that $SU/SU(n)$ is not a topological group, because $SU(n)$ is not normal in $SU$... | https://mathoverflow.net/users/102390 | Delooping the quotient space $SU/SU(n)$ | I'll work with mod $2$ cohomology. Note that $H^\*(BSU(2))$ is polynomial on $c\_2$ (in degree $4$) and $H^\*(BSU)$ is polynomial on $c\_k$ for $k\geq 2$. Here $c\_k$ has degree $2k$ and so $H^6(BSU)=\{0,c\_3\}$. If $X$ exists then it seems we should have $H^\*(X)$ polynomial on generators in degrees $6,8,10,\dotsc$. I... | 10 | https://mathoverflow.net/users/10366 | 345023 | 146,269 |
https://mathoverflow.net/questions/345007 | 2 | I posted this question on MSE (link: [Eventual Writability (general)](https://math.stackexchange.com/questions/3403361)) about 10 days ago. The current version of this question is a highly abridged version of the one posted there. Let's write "accidentally writable" and "eventually writable" as AW and EW respectively. ... | https://mathoverflow.net/users/112385 | Relation between $\eta$ and $\omega^L_1$ | Let me assume that you are concerned with ordinal time Turing
machines, using a tape of order type Ord.
My first observation is that the accidentally writable reals are
exactly the constructible reals.
**Theorem.** The OTM accidentally writable reals are exactly the
constructible reals, that is, the reals in $\math... | 4 | https://mathoverflow.net/users/1946 | 345037 | 146,271 |
https://mathoverflow.net/questions/345033 | -2 | Sometimes when one tries to capture the abstract aspect of some notion that is intuitively considered as being truly of abstract nature, in set theoretic terms, then this can extend the theory in a huge manner!
An obvious example of that is [Mathias principle about ordered pairs, p2](https://www.dpmms.cam.ac.uk/%7Etf... | https://mathoverflow.net/users/95347 | Is the principle of indifference of hierarchical construction consistent? What's its consistency strength? | This question can be more clearly phrased as:
>
> Is the principle "We can iterate powerset along any (definable) class-well-ordering" (which is really a scheme, appropriately) consistent with ZFC?
>
>
>
Unless I'm missing something, the example you give in fact demonstrates inconsistency: $\mathcal{H}(0)$ wou... | 2 | https://mathoverflow.net/users/8133 | 345046 | 146,273 |
https://mathoverflow.net/questions/345044 | 7 | Given a simplicial category $\mathcal{C}\_{\ast}$ (if necessary, you may assume it's fibrant), denote as $\mathcal{C}$ its underlying ordinary category, and as $\mathcal{W}$ the class of all equivalences in $\mathcal{C}\_{\ast}$ (in the simplicially enriched sense, i.e. $f: A \to B$ is an equivalence if there exists $g... | https://mathoverflow.net/users/134438 | Is the simplicial nerve a localization? | This is not true. Here is a counter example. We let $\mathcal{C}\_\*$ be the following simplicial category. It has two objects 0 and 1. Their only endomorphisms are the identity. There are no morphisms from 1 to 0. The simplicial set of morphisms from 0 to 1 is the simplicial circle $\Delta[1] / \partial \Delta[1]$. if... | 10 | https://mathoverflow.net/users/184 | 345047 | 146,274 |
https://mathoverflow.net/questions/345014 | 9 | Let $R$ be a one-dimensional Noetherian domain with fraction field $K$, let $\tilde{R}$ be the integral closure of $R$ in $K$, and assume that $\tilde{R}$ is finitely generated as an $R$-module. (In this situation one sometimes says that $R$ is an "order" in the Dedekind domain $\tilde{R}$.) For a fractional $R$-ideal ... | https://mathoverflow.net/users/1149 | Is an overring of an order reflexive as a module over the order? | The answer to Question 1 is yes by (the proof of) Proposition 2.14 in [this paper](https://arxiv.org/abs/math/0401306).
The answer to Question 2 is not. Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subsetneq \tilde R$ is **not** reflexive.
Let $m=(t^3,t^4... | 10 | https://mathoverflow.net/users/2083 | 345050 | 146,275 |
https://mathoverflow.net/questions/345051 | 13 | Kőnig's lemma states that any *finitely-branching* tree with infinitely many nodes contains an infinite path. Weak Kőnig's lemma states the same thing about *binary* trees.
It's known that these are not equivalent over the base system $RCA\_0$, but I'm struggling to see what goes wrong with the following construction... | https://mathoverflow.net/users/39521 | Why is weak Kőnig's lemma weaker than Kőnig's lemma? | The issue is that for a finitely branching subtree $T$ of $\omega^{<\omega}$, the function $f$ mapping $\sigma$ to the greatest $n$ such that the concatenation $\sigma ^\frown n$ is in $T$ may not be computably bounded.
So $f$ may not "exist" in your model. (Even though the model knows that for each $\sigma$ such an ... | 18 | https://mathoverflow.net/users/4600 | 345053 | 146,276 |
https://mathoverflow.net/questions/345049 | 19 | I recently refereed a paper that I returned to the author(s) for revision. The thrust of their argument relied on a claim whose justification I felt was lacking. I dutifully raised the issue in my report and, in addition, I corrected another portion of their proof.
The author(s) have yet to revise their work and, in ... | https://mathoverflow.net/users/104633 | Ethics questions concerning a referee assignment | The answer to your question really depends on how much work required justifying their claim. In my personal experience:
* It was many times that a referee provided me with an argument that lead to a simplification of my poof or even provided me with new results that I included in my paper.
* I did the same many times... | 22 | https://mathoverflow.net/users/121665 | 345054 | 146,277 |
https://mathoverflow.net/questions/339029 | 14 | Let $(\xi\_n)\_{n\ge 1}$, $(\eta\_n)\_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that
$$\sum\_n\mathbb P(\xi\_n\in A)\le\sum\_n\mathbb P(\eta\_n\in A)$$for any Borel set $A\subset X\setminus\{0\}$.
Let $1\le p<\infty$. Is there a constant $C$ (perhaps depending on $p$... | https://mathoverflow.net/users/105651 | On sums of independent random variables in Banach spaces | Certainly not always. The most trivial example seems to be $X=\ell^\infty$, $\eta\_n=\pm e\_n$ (with probability $1/2$ for each sign), and $\xi\_n$ being uniformly distributed on $\pm e\_1,\dots,\pm e\_N$ with large $N$ for $n=1,\dots,N$ (as usual, $e\_n$ is the vector with the $n$-th coordinate $1$). The rest of $\xi\... | 7 | https://mathoverflow.net/users/1131 | 345059 | 146,278 |
https://mathoverflow.net/questions/345066 | 6 | An *abstract Jordan decomposition* of an element of a Lie algebra L is a decomposition
of the form a = a$\_{s}$ + a$\_{n}$, where
(a) ad a$\_{s}$ is a diagonalizable (equivalently semisimple) endomorphism of L.
(b) ad a$\_{n}$ is a nilpotent endomorphism.
(c) [a$\_{s}$, a$\_{n}$] = 0 .
[This note](http://math.m... | https://mathoverflow.net/users/40640 | Abstract Jordan decomposition maybe not exist | In $\mathfrak g=\left\{\begin{pmatrix}x&x&y\\0&x&z\\0&0&0\end{pmatrix}:x,y,z\in \mathbf R\right\}$, $\mathrm{ad}$-semisimple and $\mathrm{ad}$-nilpotent elements all have $x=0$; so they don’t span.
| 11 | https://mathoverflow.net/users/19276 | 345070 | 146,282 |
https://mathoverflow.net/questions/345067 | 4 | Can one characterize the $a\in\mathbb F\_q\left(\left(\frac1T\right)\right)$ such that $a(T+1)=a(T)$? Although this seems elementary, I did not manage to find a answer.
Thanks in advance for any help.
| https://mathoverflow.net/users/33128 | Equality in $\mathbb F_q\left(\left(\frac1T\right)\right)$ | This is a field automorphism of order $p$, since $T+p=T$, so it fixes an index $p$ subfield. $\mathbb F\_q (( \frac{1}{T^p - T } ))$ is an index $p$ subfield and is fixed, hence is the fixed field. This is the simplest example of a wildly ramified extension of local fields in equal characteristic, and is in particular ... | 14 | https://mathoverflow.net/users/18060 | 345081 | 146,285 |
https://mathoverflow.net/questions/345072 | 2 | A positive integer $n$ is extremely abundant if either $n=10080$, or $n>10080$ and
$$σ(n)/(n×log(log (n)))>σ(m)/(m×log(log (m)))$$
for all $10080≤m<n$. Here $σ(n)$ is the sum-of-divisors function and $log$ is the natural logarithm.
My **question** is: About any known result relating prime numbers with extremely a... | https://mathoverflow.net/users/74668 | Results relating prime numbers with extremely abundant numbers | [This paper](https://cs.uwaterloo.ca/journals/JIS/VOL17/Nazar/nazar4.pdf) lists several statements relating extremely abundant numbers and prime numbers, for example:
* There is an infinite number of primes which cannot be the largest
prime factor of any extremely abundant number.
* The largest prime factor $p(n)$ of... | 3 | https://mathoverflow.net/users/11260 | 345085 | 146,288 |
https://mathoverflow.net/questions/345096 | -1 | Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ with $n>1$ and $E$ a subset of its boundary. We know that if $E$ has capacity zero I.e. it is a polar set , then the harmonic measure of $E$ with respect to $D$ is also zero: $\omega(x,D)(E)=0$ for all $x\in D$. Now suppose the capacity of $E$ is $<\epsilon$. Can we s... | https://mathoverflow.net/users/100746 | Capacity and harmonic measure | We can say that for the Martin capacity, see <https://projecteuclid.org/euclid.aop/1176988187>
For the classical capacity there is a correction factor of the minimum
of the potential kernel from $x$ to $D$ as explained in that reference.
All this is in dimension 3 and higher. There are some special effects in 2D.... | 2 | https://mathoverflow.net/users/7691 | 345105 | 146,297 |
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