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/393398 | 4 | Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:
a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.
b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U... | https://mathoverflow.net/users/175352 | O. Frink's characterization of completely regular spaces | You can copy any standard proof of Urysohn's Lemma and substitute "member of $\mathcal{B}$" for "open set" and "complement of member of $\mathcal{B}$" for closed set.
Let $\mathcal{C}$ denote $\{X\setminus B:B\in\mathcal{B}\}$. Then point a says: if $x\in X$ and $C\in\mathcal{C}$ are such that $x\notin C$ then there ... | 5 | https://mathoverflow.net/users/5903 | 393449 | 162,695 |
https://mathoverflow.net/questions/393215 | 2 | Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$.
1. Suppose $g = f$ at atmost finitely many points. Does there exist a polynomial $p$ such that $g \leq p \leq f$? As pointed out in comments below, if $f^{(n)}(x) = g^{(n)}(x)=0$ for some $x... | https://mathoverflow.net/users/151406 | Polynomial approximation (Weierstrass theorem) with bounds | As I have commented, if $f,g:[0,1]\rightarrow\mathbb{R}$ are $C^{\infty}$ functions, and $c\in(0,1)$ is a real number with $f^{(n)}(c)=g^{(n)}(c)$ for each $n$, and $f(x)\leq g(x)$ for each $x\in[0,1]$, then there can be at most one polynomial $p$ with $f(x)\leq p(x)\leq g(x)$ for each $x\in[0,1]$, and one can easily s... | 5 | https://mathoverflow.net/users/22277 | 393465 | 162,700 |
https://mathoverflow.net/questions/393464 | 5 | $B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B\_x:= \{y \in \mathbb{R}: (x,y) \in B \}$.
If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often include that fact that the function $\lambda(B\_x)$ is measurable.
**Question:** If $H^s$ denotes the $s$th Hausdorf... | https://mathoverflow.net/users/105628 | Fubini's theorem for Hausdorff measures | If $s>1$, then clearly $H^s(B\_x)=0$ so there is nothing to do. If $s=1$, $H^1$ is just the Lebesgue measure so measurability follows. If $0<s<1$ the situation is a way more complicated, but the answer is "yes" if $H^{1+s}(B)<\infty$ and it follows from the following result due to Federer [F, Theorem 2.10.25], commonly... | 6 | https://mathoverflow.net/users/121665 | 393472 | 162,703 |
https://mathoverflow.net/questions/393458 | 5 | According to the [Pinsker inequality](https://en.wikipedia.org/wiki/Pinsker%27s_inequality), we have the following inequality:
\begin{equation}
\delta\_{TV} (p, q)^2 \leq \frac{1}{2} D\_{KL}(p,q),
\end{equation}
where $\delta\_{TV} (\cdot, \cdot)$ and $D\_{KL}(\cdot, \cdot)$ are [total variation distance](https://en.wi... | https://mathoverflow.net/users/235487 | Is there an inequality relation between KL-divergence and $L_2$ norm? | Such inequality is impossible: consider $p(x)=1$, $q(x)=1/(2\sqrt{x})$, as probability densities on $(0,1)$. Then $D\_{KL}(p\parallel q)$ is finite, while $\|p-q\|\_2=\infty$, as $q\not\in L^2$.
The reverse direction is also impossible: take $p(x)=a e^{-ax}$, $q(x)=a^2e^{-a^2x}$ on $(0,\infty)$. Then $\|p-q\|\_2\to0$... | 6 | https://mathoverflow.net/users/219013 | 393477 | 162,706 |
https://mathoverflow.net/questions/392916 | 13 | Suppose we have a function $\phi\colon \mathfrak H \longrightarrow \mathbb C$ such that
1. $\phi^{24}$ is a modular function of level $5$.
2. $\phi(\tau)=\sum\_{n=-1}^{\infty}a\_{n}q^{n/5}$, $a\_{-1}\neq 0,q=e^{2\pi i\tau}$.
Does it follow that $\phi$ is a modular function of level $5$?
In particular, I am intere... | https://mathoverflow.net/users/122104 | Why is this function a modular function of level $5$? | Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.
**Claim**: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is a modular function for $\Gamma\_{0}(25) \cap \Gamma\_{1}(5)$. (This is straightforward to prove using properties ... | 5 | https://mathoverflow.net/users/48142 | 393485 | 162,710 |
https://mathoverflow.net/questions/393413 | 2 | Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)\_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the limit of $e^{-t(-\Delta)^\alpha} f$ when $\alpha \to 1^-$ with respect to an appropriate norm, e.g., $L^2(0,T;L^2(D))$... | https://mathoverflow.net/users/149793 | Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$ | *Edit: I was thinking about $(-\Delta)^{\alpha/2}$ rather than $(-\Delta)^\alpha$, so the $\alpha$ in the following answer is equal to $2\alpha$ with the notation of the statement of the question.*
As suggested by Nate Eldredge, if $u\_t(x) = e^{-t(-\Delta)^{\alpha/2}} f(x)$ and $v\_t(x) = e^{t \Delta} f(x)$, then
$$... | 1 | https://mathoverflow.net/users/108637 | 393493 | 162,712 |
https://mathoverflow.net/questions/77133 | 11 | How can I determine whether $A\_1,A\_2\in GL(n,\mathbb Z)$ conjugate in $GL(n,\mathbb Z)$ and if they are, how can I find a $P\in GL(n,\mathbb Z)$ for which $A\_2 = P^{-1}.A\_1.P$ ?
In $GL(n,\mathbb Q)$ one could achieve this by checking if the Frobenius normal forms (FNF) are equal and if they are
$\quad\quad FNF\... | https://mathoverflow.net/users/17551 | Conjugacy in $GL(n,\mathbb Z)$ | [The Conjugacy Problem in $\operatorname{GL}(n, \mathbb{Z})$](https://arxiv.org/abs/1811.06190) by Eick, Hofmann and O'Brien gives an algorithm for solving this problem, which has been implemented in Magma.
| 11 | https://mathoverflow.net/users/13025 | 393499 | 162,714 |
https://mathoverflow.net/questions/393509 | 4 | Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.
Is it known that $M\otimes\_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ and $N$ have finite projective dimension?
[Since $M\otimes\_R^{\mathbf L} N$ is represented by the chain complex $M... | https://mathoverflow.net/users/174552 | derived tensor product and finite projective dimension | Let me preface this by saying that I don't know a reference - so if that's what you're really looking for, someone else will have to answer.
Let $k:=R/m$ denote the residue field (I'm assuming "commutative" was implicit in your question).
>
> **Lemma 1** : Suppose $X$ is a bounded below chain complex of finitely ... | 7 | https://mathoverflow.net/users/102343 | 393518 | 162,719 |
https://mathoverflow.net/questions/393434 | 4 | I am currently working on the following paper by Lee Rudolph: <https://arxiv.org/abs/math/9307233>
Using Kronheimer-Mrowka's theorem, he proves in page 6 that the slice Euler characteristic of a given transverse $\mathbb C$-link $K\_f = V\_f \cap S^3$ with no singularities in $D^4$ is precisely the Euler characterist... | https://mathoverflow.net/users/234416 | Link at infinity of a complex algebraic curve transverse to S^3 and non-singular in D^4 | I think one should be a bit cautious when speaking of *the* link at infinity in this context, and rather talk about the link at infinity *with respect to some line in $\mathbb{CP}^2$*.
Having said this, to get $T(d,d)$ as the link at infinity, I claim that it suffices to take a generic line in $\mathbb{CP}^2$, i.e. a... | 3 | https://mathoverflow.net/users/13119 | 393529 | 162,723 |
https://mathoverflow.net/questions/393511 | 4 | Let $(\mathcal{E},\mathcal{S})$ be a realized limit sketch, i.e. a locally small category $\mathcal{E}$ with a class $\mathcal{S}$ of limit cones in it. It is not assumed that $\mathcal{E}$ is small, and $\mathcal{S}$ is allowed to be a proper class. We have the category $\mathrm{Mod}(\mathcal{S})$ of $\mathcal{S}$-mod... | https://mathoverflow.net/users/2841 | Example of a non-cocomplete model category of a realized limit sketch | Here is a nice trick to construct an example. But maybe there are more naturally occuring examples. I feel like there should be a better way to explain the construction, but I don't know how for now.
The core of the idea is the following observation:
* The category of suplattices (poset with arbitrary suprema) is m... | 7 | https://mathoverflow.net/users/22131 | 393536 | 162,725 |
https://mathoverflow.net/questions/393374 | 11 | In the process of computing inclusion constants for the complex matrix cube (which is a free spectrahedron), the following identity was proven: for all $n \geq 1$,
$$\mathbb E \Big| \sum\_{i=1}^{2n} x\_i^2 - \sum\_{j=1}^{2n} y\_j^2 \Big| = \mathbb E \Big| \sum\_{i=1}^{2n} x\_i^2 - \sum\_{j=1}^{2n-2} y\_j^2 \Big| = 4^{1... | https://mathoverflow.net/users/30138 | A remarkable identity involving $\chi^2$ random variables | I think I found an elementary proof of Question 2/3 for arbitrary probability distributions. In fact, it is not required that the components in the sums are squares, but general i.i.d. non-negative random variables work. Further, the requirement that both sums have an even number of terms ($2k$ and $2(n-k)$ in the ques... | 10 | https://mathoverflow.net/users/106046 | 393542 | 162,726 |
https://mathoverflow.net/questions/393554 | 3 | Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, from the series condensation theorem, that for $0< p\in\mathbb{R}, k\in\mathbb{N}$ and large enough $M\in\mathbb{N}$ that
$$
\sum\... | https://mathoverflow.net/users/237946 | Is there an asymptotic bound between converging and diverging series? | 1. Suppose $a\_n>0$ for all $n$ and $\sum\_n a\_n=\infty$. Let $s\_n:=\sum\_{j\le n}a\_j$. Let $n\_1<n\_2<\cdots$ be natural numbers such that $s\_{n\_{k+1}}-s\_{n\_k}>k$ for all $k$. Let $b\_n:=a\_n/k$ if $n\_k<n\le n\_{k+1}$, so that $t\_{n\_{k+1}}-t\_{n\_k}>1$ for all $k$, where $t\_n:=\sum\_{j\le n}b\_j$. Then $\su... | 4 | https://mathoverflow.net/users/36721 | 393557 | 162,730 |
https://mathoverflow.net/questions/393531 | 7 | For a Galois extension $K/\mathbb{Q}$, the Chebotarev Density Theorem predicts the density of primes with a certain splitting type.
I'm wondering if there is a similar result for non-Galois extension?
Any references are welcome!
| https://mathoverflow.net/users/177957 | Is there a Chebotarev‘s theorem for non-Galois extension over Q？ | Actually the usual Chebotarev density theorem in the Galois case can also be applied to the non-Galois case.
For example, consider a non-Galois cubic extension $K=\mathbb{Q}[x]/(f)$. I claim that the following splitting types of unramified primes $p$ occur with the following densities $\delta$:
$$\delta(1,1,1) = 1/6,... | 10 | https://mathoverflow.net/users/5101 | 393581 | 162,734 |
https://mathoverflow.net/questions/393545 | 1 | I am writing my thesis on the ACD model of Engle and Russel (1998). Because of the great similarities with GARCH, a common feature is non-degeneracy or non-degeneration. Apparently it is absolute basic knowledge, because I can figure out what non-degeneracy is. The ACD or GARCH process are basically stochastic processe... | https://mathoverflow.net/users/237823 | Non-degenerate stochastic process | A nondegenerate stochastic process has a nonzero variance, or equivalently, a nonzero diffusion coefficient.
| 1 | https://mathoverflow.net/users/11260 | 393586 | 162,736 |
https://mathoverflow.net/questions/392999 | 5 | Let $A$ be a $\mathbb{C}[[h]]$ algebra (not necessarily commutative). The Hochschild homology is then defined via a bar construction and that $HH\_0(A)=A/[A,A]$. Note that each $HH\_i(A)$ is a $\mathbb{C}[[h]]$-module. We can define $\overline{HH\_0}(A):=A/\overline{[A,A]}$, where the overline means taking the $h$-adic... | https://mathoverflow.net/users/111070 | Completed Hochschild (co)homology | The book:
Hübl, Reinhold:
*Traces of differential forms and Hochschild homology*.
Lecture Notes in Mathematics, **1368**. Springer-Verlag, Berlin, 1989.
treats the case of Hochschild co/homology for topological algebras. In the first chapter it gives an introduction to the basic constructions in the topological cas... | 0 | https://mathoverflow.net/users/6348 | 393589 | 162,738 |
https://mathoverflow.net/questions/393595 | 6 | I noticed [this](https://mathoverflow.net/questions/306564/current-vs-varifold) question posted on MO, hence I estimated that this may be an acceptable question even in MO (and not for MSE). I studied the notion of current and in a nutshell I understood "varifolds are weaker objects than currents."
My question is wha... | https://mathoverflow.net/users/151925 | Background for Varifold theory | The general prerequisites are almost the same as for currents, mainly a strong understanding of measure theory and a bit of geometrical intuition.
There is an aspect of multilinear algebra and some functional analysis involved as well, but a lot of that can be studied at the same time. The need for Riemannian geometr... | 9 | https://mathoverflow.net/users/51695 | 393601 | 162,741 |
https://mathoverflow.net/questions/393012 | 3 | I have asked this question on StackExchange but didn't get an answer, therefore I am asking again here.
If $M$ is smooth, and $T^\*M\to$ Spec$H^0(T^\*M,\mathcal{O}(T^\*M))$ is a projective birational map, then the conjecture predicts that $M=G/P$ for some semisimple group G and a parabolic $P$. If in addition that Sp... | https://mathoverflow.net/users/111070 | Demailly Campana Peternell Conjecture for isolated singularities | This is related to Mori's theorem through
Grauert's ampleness criterion in
Hartshorne's "Ample vector bundles" (Proposition 3.5).
Let's assume that $M$ is projective and $\dim M \ge 2$.
Let $\alpha : T^\*M \to Y$ denote the affinization of $T^\*M$.
To show that $TM$ is ample, according to the criterion
it suffices to s... | 3 | https://mathoverflow.net/users/14037 | 393603 | 162,742 |
https://mathoverflow.net/questions/393597 | 1 | $X$ is smooth Poisson. Kontsevich formality theorem says that there is a $L\_\infty$ quasi-isomorphism $$T\_{\text{poly}}\xrightarrow{L\_\infty}D\_{\text{poly}},$$ where $T\_{\text{poly}}:=(\bigwedge^\bullet\_{\mathcal{O}\_X}T^1(X))[1]$ is the dgla of (shifted) polyvector fields on $X$ and $D\_{\text{poly}}:=C^\bullet(... | https://mathoverflow.net/users/111070 | Is the map in Kontsevich Formality Theorem $\mathcal{O}$-linear? | $\DeclareMathOperator{\Br}{Br}$Willwacher proves (Theorem 4) in [The Homotopy Braces Formality Morphism](https://arxiv.org/abs/1109.3520) that there is a $\Br\_\infty$-structure on $T\_{poly}\mathbb R^n$ extending the $L\_\infty$-algebra structure defined by the Schouten bracket for which the first $L\_\infty$-morphism... | 3 | https://mathoverflow.net/users/35687 | 393616 | 162,749 |
https://mathoverflow.net/questions/393506 | 8 | $\newcommand{\F}{\mathbb{F}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}$
I would like to know if the following is true:
>
> **Proposition A** : Let $A\_1, A\_2$ be two abelian varieties over a finite field $k$. If $\End\_{\overline k}(A\_1) \otimes\_{\Z} \Q$ and $\End\... | https://mathoverflow.net/users/84923 | Isomorphic endomorphism algebras implies isogenous (for abelian varieties over finite fields)? | Here is a counterexample to Proposition A provided by 8-dimensional abelian varieties $A\_1$ and $A\_2$ over a finite field $F\_{p^2}$ where $p$ is any prime that is congruent to $1$ modulo $17$ (e.g., $p=103$). The corresponding endomorphism algebra is the $17$th cyclotomic field $E=Q(\zeta\_{17})$. The congruence con... | 9 | https://mathoverflow.net/users/9658 | 393619 | 162,751 |
https://mathoverflow.net/questions/393613 | 5 | *Note: For convenience, all sequences will be indexed by the positive integers $\mathbb Z\_+$.*
**Definitions and some motivation:**
The Riemann rearrangement theorem says that if we have a sequence that is conditionally but not absolutely convergent, we can rearrange it to converge to any desired value. Looking at... | https://mathoverflow.net/users/173490 | Riemann rearrangement theorem with restricted choices | No. For example, we can define $S$ by insisting that $\epsilon\_{2n+1}=\epsilon\_{2n}$. If we have no restrictions, then a good strategy to make $\sum \epsilon\_n a\_n$ convergent would be to choose the $\epsilon\_n$ recursively in such a way that the partial sums stay as close to $0$ as possible. A slightly modified v... | 7 | https://mathoverflow.net/users/48839 | 393622 | 162,752 |
https://mathoverflow.net/questions/393587 | 2 | Let $(E\_n,i\_n)\_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E\_{\infty}$ by their inductive limit. What is the family of semi-norms defining the LCS topology on $E\_{\infty}$?
Similarly, if $(E\_n,\pi\_n)$ is a projectiv... | https://mathoverflow.net/users/36886 | Semi-norms on LCS inductive limit of Banach Spaces | There is a simple **abstract** description of the semi-norms of an inductive limit of Banach or locally convex space $E\_n$ with
linking maps $i\_n^m:E\_n\to E\_m$ for $n\le m$ and $i\_n^\infty:E\_n\to E\_\infty$: Just take all semi-norms $p:E\_\infty\to[0,\infty)$ such that $p\circ i\_n^\infty$ is continuous on $E\_n$... | 2 | https://mathoverflow.net/users/21051 | 393623 | 162,753 |
https://mathoverflow.net/questions/393604 | 5 | For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{bmatrix}
$$
For example here is a quick list of different Sobel operators:
$$
\begin{matrix}
& a & b & c & d \\
\text{Vertical Sobel}\hfill & 1 & 2 ... | https://mathoverflow.net/users/239352 | Is there a name for this type of matrix? | These are called "skew-centrosymmetric" matrices. The term "centrosymmetric matrix" seems to be popular enough to have its own Wikipedia page: <https://en.wikipedia.org/wiki/Centrosymmetric_matrix>
References on the skew version of these matrices can be found by Googling.
| 9 | https://mathoverflow.net/users/11236 | 393626 | 162,754 |
https://mathoverflow.net/questions/393470 | 1 | Why
$$
\Pi\_+ \left(\frac{\overline{z}}{1-\overline{qz}}f\right)= \frac{f(z)-f(\bar{q})}{z-\overline{q}}, \quad f\in H^2(\mathbb D),$$
where
* $q\in \mathbb C$,
* $\Pi\_{+}$ is the Szegö projector:
$$\Pi\_{+}\left(\sum\_{k \in \mathbb{Z}} \widehat{f}(k) z^{k}\right)=\sum\_{k \in \mathbb{Z}\_{+}} \hat{f}(k) z^{k}
$$ a... | https://mathoverflow.net/users/157604 | Computation on the Hardy space | **Use nontangential boundary values.** Also, you want $|q| < 1$, since otherwise $f(\bar{q})$ need not be defined.
$f \in H^2(\mathbb{D})$ extends almost everywhere to $f \in L^2(\mathbb{T})$, using boundary values, where $\mathbb{T} = \{ z : |z| = 1 \}$.
On $\mathbb{T}$, we have $|z|=1$ and so $\bar{z} = 1/z$. So,... | 2 | https://mathoverflow.net/users/239844 | 393628 | 162,755 |
https://mathoverflow.net/questions/393630 | 2 | In [these notes of Kedlaya](https://www.math.arizona.edu/%7Eswc/aws/2007/KedlayaNotes11Mar.pdf), he calculates the de Rham cohomology of an affine part $X$ of an elliptic curve $E$ over a field $K$, given by $y^2 = P(x) = x^3 + ax + b$.
He uses these relations:
* $0 = y^2 - P(x)$
* $0 = d(y^2 - Pdx) = 2ydy - P'dx$
... | https://mathoverflow.net/users/94086 | Where does this clever choice of differential come from? (calculating $\mathrm{H}^1_{\mathrm{dR}}(E/k))$ | Yes, $\omega$ is an invariant differential, also characterized by being nowhere vanishing (including the point at infinity).
At every point, $dx$ and $dy$ together span the cotangent space (which is $1$-dimensional). From the second equation, you see that $dx$ vanishes precisely when $y$ vanishes, and $dy$ vanishes p... | 8 | https://mathoverflow.net/users/39747 | 393632 | 162,758 |
https://mathoverflow.net/questions/393602 | 1 | Let $N\_+$ denote the set of positive integers. Let $E$ be the collection of all finite $A\subseteq \mathbb{N}\_+$ such that $|A|\geq 3$ and $\min(A)$ is the greatest common divisor of $A\setminus\{\min(A)\}$.
**Question.** Is there a positive integer $n$ such that there is a map $f: \mathbb{N}\_+ \to \{1,\ldots,n\}$... | https://mathoverflow.net/users/8628 | "Coloring" the greatest common divisor relation | There is no such $n$.
Denote $f(1)=2$, $f(n)=2f(n-1)+2$, that is, $f(n)=2^{n+1}-2$.
The claim follows from the following
**Proposition.** Assume that $N\geqslant f(n)$ and all subsets of an $N$-set $\Omega$ are colored with $n$ colors. Then there exist three distinct sets $A,B,C$ of the same color such that $A=B\cap ... | 4 | https://mathoverflow.net/users/4312 | 393637 | 162,760 |
https://mathoverflow.net/questions/393059 | 2 | Let $E$ be a Banach space, let $e\_{n}\in E$ and $g\_{n}\in E^{\*}$ be biorthogonal basic sequences (i.e. $\left<e\_n,g\_m\right>=\delta\_{mn}$ ). Moreover, both of these sequences are weakly null. (note that existence of these sequences is equivalent to negation of Dunford-Pettis property)
>
> Can we always find $... | https://mathoverflow.net/users/53155 | Biorthogonal weakly null basic sequences | I think I've found a counterexample in the literature. I would really appreciate if somebody verified that I didn't get confused about the terminology of Orlicz spaces.
1. Recall that a Banach space $E$ has *Dunford-Pettis* property if for every Banach space $F$ any weakly compact operator $T:E\to F$ operator is *com... | 1 | https://mathoverflow.net/users/53155 | 393643 | 162,761 |
https://mathoverflow.net/questions/393642 | 9 | I need an explicit lower bound for $\mathrm{Li}(x)$ in terms of $x$ and $\log x$. Say, Wikipedia gives
$$
\mathrm{li}(x) >\frac x{\log x}+\frac x{(\log x)^2}
$$
for $x>e^{11}$, see the [logarithmic integral entry](https://en.wikipedia.org/wiki/Logarithmic_integral_function#Asymptotic_expansion), and so
$$
\mathrm{Li}... | https://mathoverflow.net/users/138069 | $\mathrm{Li}(x)$ vs $x/\log x$ | Using ${\rm Li}(x) := \int\_2^x dt/\log t$, as usual, here is an elementary argument that ${\rm Li}(x) > x/\log x$ for $x \geq 7$, so no need to appeal to a lower bound valid only starting at $e^5 \approx 148.4$ as in the Rosser-Schoenfeld paper from the comments.
For $x > 1$, let $f(x) = {\rm Li}(x) - x/\log x$. The... | 11 | https://mathoverflow.net/users/3272 | 393652 | 162,764 |
https://mathoverflow.net/questions/393607 | 13 | We consider a matrix
$$M\_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily checks that $\operatorname{det}(M\_{\mu})=1$.
I however noticed something peculiar:
Consider a sequence of real numbers $\mu\_i$ then the four eigenvalues $\lambda\... | https://mathoverflow.net/users/239367 | Eigenvalue pattern | The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i & 0 & -i
\end{pmatrix}$$
we have
$$B^{-1}M\_{\mu}B =
\begin{pmatrix}
1+i\mu & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 1-i\mu & 1 \\
0 & 0 & -1 & 0
\end{pmatr... | 20 | https://mathoverflow.net/users/160416 | 393656 | 162,767 |
https://mathoverflow.net/questions/393625 | 2 | The motivation for this question is a statement about the Bellman-Ford algorithm, that doesn't agree with the definition of what a [path](https://mathworld.wolfram.com/PathGraph.html) in a graph is.
On wikipedia's description of the [Bellman-Ford Algorithm](https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm... | https://mathoverflow.net/users/31310 | What do shortest-path algorithms actually calculate? | Computer Science often (usually, in my experience) defines a *path* as a sequence of vertices with edges between them, i.e. what others call a *walk*. E.g. on my shelf, this definition appears in *Algorithm Design* by Kleinberg and Tardos, *Introduction to Algorithms* by CLRS, and *AI: A Modern Approach* by Russell and... | 3 | https://mathoverflow.net/users/29697 | 393657 | 162,768 |
https://mathoverflow.net/questions/393359 | 5 | I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a proof which cannot be readily found in published expositions).
Let $R$ be a Noetherian commutative ring (of finite Kr... | https://mathoverflow.net/users/2106 | Relative version of Hilbert syzygy theorem | The answers to the first two questions readily follow from Proposition 7.5.2 in [J. C. McConnell and J. C. Robson, "Noncommutative Noetherian rings", AMS, 1987]. Actually, they prove a much more general result which holds for skew polynomial rings and skew Laurent polynomial rings. The base ring $R$ can be arbitrary, n... | 4 | https://mathoverflow.net/users/17774 | 393671 | 162,772 |
https://mathoverflow.net/questions/393664 | 4 | In some calculations, I saw the following formula
$$\int\_{\mathrm{SU}(2)}\,\mathrm{d}g\,D^{j\_{1}}\_{m\_{1}n\_{1}}(g)D^{j\_{2}}\_{m\_{2}n\_{2}}(g)D^{j\_{3}}\_{m\_{3}n\_{3}}(g)=(-1)^{j\_{1}+j\_{2}+j\_{3}}\begin{pmatrix}j\_{1} & j\_{2} & j\_{3}\\m\_{1} & m\_{2} & m\_{3}\end{pmatrix}\begin{pmatrix}j\_{1} & j\_{2} & j\_... | https://mathoverflow.net/users/199422 | Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2) | You can find a fully worked-out derivation in these [lecture notes.](http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf) The formula you are looking for is equation (404), written in terms of the Wigner (small)-$d$ matrix. The relationship to the (large)-$D$ matrix goes via the... | 2 | https://mathoverflow.net/users/11260 | 393675 | 162,775 |
https://mathoverflow.net/questions/393596 | 1 | Let $n=a\_3z^3+a\_2z^2+a\_1z+1$ where $a\_1<z, \ a\_2<z, \ 1 \le a\_3<z, z>1$ are non negative integers. To obtain proper divisors of $n$ of the form $xz+1$, one may perform trial divisions $xz+1 \ | \ n$, for all $xz+1 \le \sqrt n$. Trial division however is inefficient as $z$ becomes large. The method below is much m... | https://mathoverflow.net/users/166404 | Finding all proper divisors of $a_3z^3 +a_2z^2 +a_1z+1$ of the form $xz+1$ | Such an extension is highly unlikely to exist. Already in the simple case of $z=2$, it's equivalent to just factoring a given odd integer $n$, which is a [famous hard problem](https://en.wikipedia.org/wiki/Integer_factorization).
| 2 | https://mathoverflow.net/users/7076 | 393691 | 162,781 |
https://mathoverflow.net/questions/393698 | 3 | Let $\Sigma$ be a Riemann surface of genus $g$. To it, we can associated $M\_{Dol}$ be the Higgs moduli space of rank $n$ and degree $d$. Fo simplicity let us take $(n,d)=1$. This quasiprojective variety admits a morphism $$h:M\_{Dol}\rightarrow \bigoplus\_{i=0}^nH^0(\Sigma,\Omega^i\_{\Sigma})=A\_n $$ where $\Omega^1\_... | https://mathoverflow.net/users/146464 | Integral locus of Hitchin morphism | To get the codimension, if the curve corresponding to $x = (x\_i \in H^0 (\Sigma, \Omega\_\Sigma^i))\_{i=1}^n$ fails to be integral, then the equation $T^n - x\_1 T^{n-1} + x\_2 T^{n-2} + \dots $ defining the curve splits as a product of two such equations, say $T^k - y\_1 T^{k-1} + \dots + (-1)^k y\_k $ and $T^{n-k} -... | 7 | https://mathoverflow.net/users/18060 | 393701 | 162,785 |
https://mathoverflow.net/questions/393685 | 8 | **Definitions:**
We say a sequence of continuous functions $f\_n: [0, 1] \to \mathbb R$ is *equicontinuous on average* if for every $x \in [0, 1]$ and $\varepsilon > 0$ there exists some $\delta > 0$ such that $\limsup\_{N \to \infty} \frac{1}{N} \sum\_{n = 0} ^{N-1} |f\_n (x) - f\_n (y)| < \varepsilon$ whenever $|x ... | https://mathoverflow.net/users/173490 | Almost Arzela Ascoli | $\newcommand{\ep}{\epsilon}\newcommand{\de}{\delta}$The answer is no. To show this, let us use a slightly modified suggestion in a comment by Mateusz Kwaśnicki, as follows. Let
\begin{equation}
f\_n:=g\_{a\_n,1/n},
\end{equation}
where
\begin{equation}
g\_{a,h}(x):=1(a<x\le a+h)\frac{x-a}h+1(x>a+h)
\end{equation}
and... | 6 | https://mathoverflow.net/users/36721 | 393707 | 162,786 |
https://mathoverflow.net/questions/393703 | 4 | I have recently read the problem named "Square of the distance function on a Riemannian manifold"([enter link description here](https://mathoverflow.net/questions/215573/square-of-the-distance-function-on-a-riemannian-manifold)) and I am interested in the formula
$ d^2(exp\_{x\_0}(tv),exp\_{x\_0}(tw))=|v-w|^2t^2-\fr... | https://mathoverflow.net/users/241460 | Taylor expansion of the square of the distance function on a Riemannian manifold | The "standard" proof using Jacobi fields can be found in section 1.3 here [https://www2.math.upenn.edu/~wziller/math660/TopogonovTheorem-Myer.pdf](https://www2.math.upenn.edu/%7Ewziller/math660/TopogonovTheorem-Myer.pdf)
| 5 | https://mathoverflow.net/users/1540 | 393708 | 162,787 |
https://mathoverflow.net/questions/393405 | 2 | The category of semisimplicial sets has the structure of a monoidal category by the geometric product $\otimes$, see for example Rourke and Sanderson's paper '$\Delta$-sets I: Homotopy Theory'. This geometric product has the property that for the left adjoint $L$ of the forgetful functor $U$ from simplicial sets to sem... | https://mathoverflow.net/users/137822 | Does the monoidal structure on semisimplicial sets preserve fibrant objects? | It is not the case: the terminal semi-simplicial set $1$ is obviously fibrant but as I will show below the geometric product $1 \otimes 1$ is not fibrant.
**1) What does $1 \otimes 1$ look like ?**
So, $1 \otimes 1$ identifies with the subset of non-degenerate cells of $L(1 \otimes 1) = L 1 \times L 1$.
In genera... | 2 | https://mathoverflow.net/users/22131 | 393709 | 162,788 |
https://mathoverflow.net/questions/393680 | 2 | Consider the matrix, for some $\lambda \in \mathbb R$ .
$$A=\begin{pmatrix} i \lambda & -1 & i & 0 \\ 1 & 0 & 0& 0 \\ i & 0 & - i \lambda & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$
I would like to know if there exists a matrix $B$, independent of $\lambda$ such that
$$ B A B^{-1} = \begin{pmatrix} K\_1(\lambda) & 0\\0... | https://mathoverflow.net/users/150564 | Similarity of two matrices | I will show that it is not possible for $\phi=\pi/2$, so it is certainly not for general $\phi$. (actually, I don't think that it is possible for any single $\phi$ except $0$ and $\pi$, by an analogous argument).
I will multiply $A$ by $-i$ and conjugate it by
$\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -i & 0 & 0 \\
0 & 0... | 4 | https://mathoverflow.net/users/160416 | 393714 | 162,789 |
https://mathoverflow.net/questions/393732 | 3 | Suppose for every $x \in [0, 1]$, we have a subset $S\_x$ of the natural numbers with asymptotic density $1$ such that if
$n \in S\_x$, there is an open neighbourhood $U$ of $x$ (depending on $x$ and $n$) such that $n \in S\_y$ for all $y \in U$.
**Question:**
For any $\varepsilon > 0$ can I find a subset $M$ of $[... | https://mathoverflow.net/users/173490 | A density lemma for families of sequences indexed by the unit interval | Define $S\_x=\{n\colon \langle n!x\rangle>\frac 1n\}$, where $\langle \cdot\rangle$ is the fractional part of $\cdot$. Another way to think of this is you can (essentially uniquely) write a number as
$$
x=\sum\_{n=2}^\infty \frac{a\_n(x)}{n!},
$$
where $a\_n(x)\in\{0,\ldots,n-1\}$. This is the expansion of $x$ where th... | 1 | https://mathoverflow.net/users/11054 | 393737 | 162,797 |
https://mathoverflow.net/questions/393743 | 8 | The page of ncatlab on [group object](https://ncatlab.org/nlab/show/group+object) states that:
>
> A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf
> algebra.
>
>
>
**Question:** Is a (noncommutative) Hopf algebra a group object of some category?
[let assume finite dimensional, if nece... | https://mathoverflow.net/users/34538 | Is a Hopf algebra a group object of some category? | Not with their definition, where they assume the underlying category to be cartesian. You can define a notion of "Hopf object" in arbitrary symmetric monoidal categories, where you also need to specify the existence of a coproduct map. It's a matter of taste, but I think this general definition should actually be calle... | 12 | https://mathoverflow.net/users/13552 | 393744 | 162,799 |
https://mathoverflow.net/questions/393706 | 15 | Suppose we have $x\_1^2 + y\_1^2 + x\_2^2 + y\_2^2 + x\_3^2 + y\_3^2 + x\_4^2 + y\_4^2 = 1$ and we define $z\_j = x\_j + iy\_j$, where $j = 1,\,2,\,3,\,4$.
The problem is finding or approximating the following integral (The actual problem is more complex than this !)
$$I(k) = \int\limits\_{\mathbb{S}^7}\lvert(z\_1z\_... | https://mathoverflow.net/users/130850 | Integration of a function over 7-sphere | This is not a complete answer but shows how to simplify your integral:
As pointed out by mlk in the comments the Hopf fibration plays a crucial role: Consider $\mathbb S^7\subset\mathbb C^4.$ On $\mathbb C^4=\mathbb C^2\oplus\mathbb C^2$ the special unitary group $\mathrm{SU}(2)$ acts by the direct sum of the standard ... | 6 | https://mathoverflow.net/users/4572 | 393749 | 162,800 |
https://mathoverflow.net/questions/393758 | 11 | Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is for $\mu \in \mathbb R$ skew hermitian, i.e. all the eigenvalues are imaginary.
Let $(\mu\_i)\_i$ be a sequence of real numbers.
We consider the product
$$M=\p... | https://mathoverflow.net/users/150564 | Imaginary eigenvalues | Define the unitary and Hermitian matrices
$$U=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
\end{array}
\right),\;\;
V=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & i \\
i & 0 & 0 & 0 \\
0 & -i & 0 & 0 \\
\end{array}
\right),
\;\;U^2=I=V^2,$$
and not... | 15 | https://mathoverflow.net/users/11260 | 393760 | 162,802 |
https://mathoverflow.net/questions/393729 | 1 | Consider the following ODE eigenproblem of $y(x)$
\begin{equation}
y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0
\end{equation}
with eigenvalue $\varepsilon$, real constants $a,b$. The boundary condition is $y(\pm\infty)=0$. Numerically, this turns out to have well-behaved eigensolutions.
My question i... | https://mathoverflow.net/users/121010 | Asymptotic behavior of an ODE | The quick-and-dirty way to guess the asymptotic behavior is to substitute in a WKB ansatz $y = e^{S(x)}$ and keep only the leading order terms, which here would be the highest power of $x$ and the highest power of $S'(x)$, namely $(S')^2 - (x^2 b^2/2)^2 = \text{l.o.t}$. The solution is $S(x) \sim \pm x^3 b^2/6$ as $|x|... | 1 | https://mathoverflow.net/users/2622 | 393762 | 162,803 |
https://mathoverflow.net/questions/393681 | 7 | Recall that triangular numbers are those $T(n)=n(n+1)/2$ with $n\in\mathbb N=\{0,1,2,\ldots\}$. Fermat ever proved that the equation $x^4+y^4=z^2$ has no positive integer solution. So I think it's natural to investigate triangular numbers of the form $x^4+y^4$ with $x,y\in\mathbb N$. Clearly,
$$T(0)=0^4+0^4\ \ \mbox{an... | https://mathoverflow.net/users/124654 | Triangular numbers of the form $x^4+y^4$ | An exhaustive search up to $10^7$ finds that the only solutions of
$x^4+y^4=T(n)$ in integers with $0 \leq x \leq y \leq 10^7$ are:
$$
0^4 + 0^4 = T(0), \quad
0^4 + 1^4 = T(1), \quad
15^4 + 28^4 = T(1153),
$$
noted by OP **Zhi-Wei Sun**;
$$
3300^4 + 7712^4 = T(85508608),
$$
already found by **Tomita**; and the new exam... | 20 | https://mathoverflow.net/users/14830 | 393763 | 162,804 |
https://mathoverflow.net/questions/393764 | 4 | Let $G$ be a Lie group acting smoothly on a smooth manifold $M$. Suppose that the orbit space $M / G$ is a topological manifold, and is endowed with a smooth structure such that:
1. the quotient map $\pi : M \to M / G$ is smooth, and
2. the pullback map $\pi^\* : C^\infty(M / G) \to C^\infty(M)^G$ is an isomorphism, ... | https://mathoverflow.net/users/243217 | Quotient map is a submersion | **Edit**: This answer is wrong, I was misled by the complex case. The map $\pi$ is not surjective, its image is a semi-algebraic set. In fact it would seem that $M/G$ is never a manifold.
No. Take $M=\mathbb{R}^2$, $G=\mathbb{Z}/2$ acting by swapping the coordinates. The quotient is isomorphic to $\mathbb{R}^2$, with... | 0 | https://mathoverflow.net/users/40297 | 393765 | 162,805 |
https://mathoverflow.net/questions/393751 | 1 | Let $n\in\mathbb{N}$ and consider the Sobolev space $W^{1,\infty}(\mathbb{R}^n)=\lbrace u\in L^{\infty}(\mathbb{R}^n):\partial\_iu\in L^{\infty}(\mathbb{R}^n) \rbrace$. A function is in $W^{1,\infty}$ iff it is bounded and Lipschitz continuous. We know also from Rademacher theorem that any Lipschitz function is differe... | https://mathoverflow.net/users/125729 | Norms in Sobolev space $W^{1,\infty}$ | Write, using the Fundamental Theorem of Calculus for a $C^1$ function, $$\frac{u(x)-u(y)}{x-y} = \frac1{x-y} \int\_C Du d\sigma,$$ where $C$ is the line segment from $x$ to $y$. This gives $$\left|\frac{u(x)-u(y)}{x-y}\right|\leq \|Du\|\_{L^\infty}.$$ Then by density you obtain $$\| \cdot \|\_2 \leq \| \cdot \|\_1.$$
... | 3 | https://mathoverflow.net/users/40120 | 393767 | 162,806 |
https://mathoverflow.net/questions/393537 | 2 | Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu\_y$ by
$$\mu\_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu\_y$ is a translation of the measure $\mu\_0$ in the direction $y$.
Now, let $V: \mathbf{R}^d \to \mathbf{R}$ be another con... | https://mathoverflow.net/users/121692 | Reweighting probability measures by convex potentials, and contraction in transport distance | There is a simple sufficient condition: If $\nabla W$ is $L$-Lipschitz, then $y \mapsto \nu\_y$ is $(L/m)$-Lipschitz with respect to the quadratic Wasserstein distance $d=\mathcal{W}\_2$. You thus have a contraction if $L<m$, and this condition fits with your "bit of intuition." Though this may be a stronger assumption... | 1 | https://mathoverflow.net/users/44169 | 393776 | 162,808 |
https://mathoverflow.net/questions/362921 | 3 | $\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}$I'm trying to read the physics paper [Two Dimensional QCD as a String Theory](https://arxiv.org/abs/hep-th/9212149). I'm struggling with my ignorance about some computational aspects regarding Lie algebras.
Section 2.3 of the aforementio... | https://mathoverflow.net/users/157706 | On how to diagonalize a Casimir element | See <https://arxiv.org/abs/0807.3696> for derivation.
Let $E\_i^j$ be the ${\rm U}(N)$ generators and $V$ be the fundamental representation. Decompose $V^{\otimes k}$ by the Schur-Weyl duality and apply the second Casimir $C\_2 = E\_i^j E\_j^i$ to $V\_R^{{\rm U}(N)} \otimes V\_R^{S\_k}$.
After some computation, $C\... | 1 | https://mathoverflow.net/users/243537 | 393789 | 162,814 |
https://mathoverflow.net/questions/393750 | 5 | A Banach space $X$ is ***subprojective*** if every infinite dimensional closed subspace $Y$ of $X$ contains an infinite dimensional subspace $Z$ which is complemented in $X$.
I am interested in conditions on an Orlicz function $M$ implying that the Orlicz sequence space $\ell\_M$ is subprojective.
A concrete case: $M... | https://mathoverflow.net/users/39421 | Subprojective Orlicz sequence spaces | As Bill pointed out the answer is Yes. In this particular example, the situation is even simpler: every normalized block sequence $(u\_i)$ has a subsequence equivalent to either uvb of $\ell\_M$ or to of $\ell\_p$. This is easy to see since the set $C\_{M}=\{t^p\}$. Specifically, if $\|u\_i\|\_{\infty}>0$ then the sequ... | 5 | https://mathoverflow.net/users/3675 | 393802 | 162,820 |
https://mathoverflow.net/questions/393692 | 2 | This is a [cross-post](https://math.stackexchange.com/questions/4148059/if-any-two-triangles-of-equal-area-can-be-mapped-via-affine-maps-what-can-we-sa).
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that **for any** two geodesic triangles $A,B$ **having ... | https://mathoverflow.net/users/46290 | If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry? | Using the structure equations, it is not difficult to show that, if $f:(M,g)\to(N,h)$ is a diffeomorphism of (not necessarily complete) connected surfaces that is affine in the OP's sense, i.e., $\nabla(\mathrm{d}f)=0$, then $f$ has constant singular values and $L\bigl(f(p)\bigr) = K(p)/|\det(f)|$ for any $p\in M$, whe... | 5 | https://mathoverflow.net/users/13972 | 393810 | 162,821 |
https://mathoverflow.net/questions/393795 | 2 | I was reading [these](http://people.math.harvard.edu/%7Egaitsgde/grad_2009/SeminarNotes/Sept17(Bun(G)).pdf) notes by D. Gaitsgory, and I don't understand a claim he makes about relative affine schemes. Namely, on page 3 he says that if $f: Y \rightarrow X$ is an affine scheme over $X$, then there exist two vector bundl... | https://mathoverflow.net/users/91572 | Relative affine schemes | This is closely related to the *resolution property* [Tag [0F85](https://stacks.math.columbia.edu/tag/0F85)], so it won't be true in complete generality. Indeed, if $\mathscr F$ is a coherent sheaf on $X = S$ that cannot be given as a quotient of a vector bundle $\mathscr E$ on $S$, then $Y = \mathbf{Spec}\_X \operator... | 2 | https://mathoverflow.net/users/82179 | 393812 | 162,822 |
https://mathoverflow.net/questions/393804 | 8 | In section X.7 of [Reed & Simon's book](https://www.amazon.com.br/II-Fourier-Analysis-Self-Adjointness-2/dp/0125850026) there is a nice rigorous construction of the free scalar field theory which applies to the Klein-Gordon field.
**Question:** Are there references which discuss, in analogous fashion, the constructio... | https://mathoverflow.net/users/150264 | Rigorous construction of fermionic field theory? | There is the construction of the C${}^\*\!$-algebra of canonical anticommutation relations (CAR's), which is actually somewhat easier than the construction of free bosonic fields: given any complex pre-Hilbert space $\mathfrak{h}$, which may be thought of as our "one-particle" space, define the unital \*-algebra $\text... | 16 | https://mathoverflow.net/users/11211 | 393819 | 162,824 |
https://mathoverflow.net/questions/393811 | 3 | Is it possible to find an estimate of the summation
$$s(n)=\sum\_{k=1}^n\frac1{\varphi(k\cdot p\_k)}$$
where $\varphi(n)$ is the totient function and $p\_k$ the k-th prime?
The corresponding series seems to converge to the value
$$\lim\_{n\rightarrow\infty}s(n)=1.86491\ldots$$
but I don't see a simple way to prove it... | https://mathoverflow.net/users/150698 | Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$? | $\newcommand\vpi\varphi$[It is known](https://en.wikipedia.org/wiki/Euler%27s_totient_function#Growth_rate) that
$$\vpi(n)\ge\frac n{c\ln\ln(n+10)}=:\psi(n)$$
for some real $c>0$ and all natural $n$. Also, $\psi$ is increasing on the interval $[N,\infty)$ for some natural $N$.
Therefore and because $p\_k\ge k$ for all ... | 4 | https://mathoverflow.net/users/36721 | 393821 | 162,825 |
https://mathoverflow.net/questions/392883 | 14 | The category of complete atomic boolean algebras $\mathbf{CABA}$ is equivalent to $\mathbf{Set}^{\mathrm{op}}$ via
$$\mathbf{Set}^{\mathrm{op}} \to \mathbf{CABA}, ~ X \mapsto (P(X),\bigcup,\bigcap).$$
The inverse takes the atoms. The functor $P : \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ is monadic, its left adjoint... | https://mathoverflow.net/users/2841 | What are internal complete atomic boolean algebras, intuitively? | **(A)** Here is how to get an internal CBA-structure from an internal CABA-structure. A reference is Formula 1.5.22 in E. Manes, *Algebraic theories*.
Notice that for $f : X \to Y$ the induced map $\tilde{f} : P(P(X)) \to P(P(Y))$ is $\tilde{f}(S) = \{A \in P(Y) : f^\*(A) \in S\}$.
Assume $B \in \mathcal{C}$ and we... | 2 | https://mathoverflow.net/users/2841 | 393823 | 162,826 |
https://mathoverflow.net/questions/393818 | 3 | Let $M\_g$ be the moduli space of genus $g$ curves, $A\_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli map is a map $\tau\_g: M\_g \to A\_g$ that takes a curve to it's Jacobian.
For $g=3$, $M\_g$ and $A\_g$ have the same ... | https://mathoverflow.net/users/58001 | What is the involution on the moduli space of genus 3 curves induced by the Torelli map | The Torelli morphism being a double cover is purely a stacky phenomenon. It is not visible on coarse moduli spaces.
The involution you ask about is supposed to act on the fibers of the Torelli morphism. Here is how this works out. Pick a geometric point of $A\_3$, corresponding to a ppav $(A,\Theta)$. The fiber over ... | 9 | https://mathoverflow.net/users/1310 | 393828 | 162,828 |
https://mathoverflow.net/questions/393786 | 2 | In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more properties of such a map). They also say that such a manifold $N$ has "non-positive" curvature.
My question is whether one can... | https://mathoverflow.net/users/89741 | Hyperbolization with word-hyperbolic fundamental group | Charney-Davis in [Strict hyperbolization](https://www.sciencedirect.com/science/article/pii/004093839400027I) showed how to make $N$ locally CAT($-1$), provided $M$ is PL.
Ontaneda in [Riemannian hyperbolization](https://link.springer.com/article/10.1007/s10240-020-00113-1) showed how to make $N$ a Riemannian manifol... | 3 | https://mathoverflow.net/users/1573 | 393829 | 162,829 |
https://mathoverflow.net/questions/393826 | 16 | One way to approach QFT in mathematical terms is provided by the so-called Gårding-Wightman axioms, which defines in rigorous mathematical terms what a quantum field theory is supposed to be. If I'm not mistaken, this is what is called *axiomatic QFT*. Although it is a precise mathematical theory, it is widely known th... | https://mathoverflow.net/users/150264 | QFT and mathematical rigor | As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at the Preface of the book of Streater and Wightman, *PCT, Spin and Statistics, and All That*. More precisely, it lists a minimal set ... | 19 | https://mathoverflow.net/users/11211 | 393830 | 162,830 |
https://mathoverflow.net/questions/393832 | 14 | Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$.
**Question: Is the top singular cohomology group $H^n(M,\mathbb Z)$ zero?**
This naïve question does not seem to be answered in the standard algebraic topology treatises, like those by Bredon, Dold, Hatcher, Massey, Spanier... | https://mathoverflow.net/users/450 | What is the top cohomology group of a non-compact, non-orientable manifold? | I believe you can deduce this from the corresponding statement in the orientable case. Let $\tilde M$ be the oriented double cover. Make an exact sequence of cochain complexes
$$
0 \to C^\bullet(M;\mathbb Z^t)\to C^\bullet(\tilde M;\mathbb Z)\xrightarrow{p\_!} C^\bullet(M;\mathbb Z)\to 0,
$$
where $\mathbb Z^t$ is the ... | 9 | https://mathoverflow.net/users/6666 | 393834 | 162,831 |
https://mathoverflow.net/questions/393833 | 1 | I'm currently reading [this paper](https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p10.pdf) and the authors define the set $C^{k,1}(\mathbb{R}^n)$ as consisting of all functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ having $k$ derivatives and for which:
$$
\|f\|:= \max\_{|\beta|<k}\left[ \sup\_{x \... | https://mathoverflow.net/users/36886 | Examples of $C^{k,1}$ functions which are not $C^{k+1}$? | For $n=1$ and $k=1$, let
$$f(x)=x^2 \sin(1/x)$$
for real $x\ne0$, with $f(0)=0$. Then $f\in C^{k,1}(\mathbb R^n)\setminus C^{k+1}(\mathbb R^n)$.
It should be easy to extend this example to the other $n$ and $k$.
| 3 | https://mathoverflow.net/users/36721 | 393835 | 162,832 |
https://mathoverflow.net/questions/393836 | 3 | Denote by $[0,\infty]\equiv [0,\infty)\cup \{\infty\}$ the one-point compactification of $[0,\infty)$, i.e. all the open sets related to $[0,\infty]$ are either the open sets of $[0,\infty)$ or the sets of the form $G\cup \{\infty\}$, where $G\subset [0,\infty)$ is an open subset s.t. $[0,\infty)\setminus G$ is compact... | https://mathoverflow.net/users/nan | Weak convergence of probability measures on the one-point compactification of $[0,\infty)$ | $\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand{\ep}{\epsilon}$The implications 1$\iff$3 follow because $[0,\infty]$ is homeomorphic to $[0,1]$, with the preservation of the order. Such an order-preserving homeomorphism $g\colon[0,\infty]\to[0,1]$ is given by $g(x):=x/(1+x)$ for $x\in[0,\infty)$, with $g... | 3 | https://mathoverflow.net/users/36721 | 393844 | 162,834 |
https://mathoverflow.net/questions/393839 | 6 | I have an elementary question concerning zeros of polynomials, which must be well-known.
Fix a base field $ k$ (can assume to be characteristic zero if it makes a difference). Consider the affine space $ P\_n \times \mathbb A^1\_k $, where $ P\_n $ denotes the space of polynomials of degree $ n $ over $ k$ (so $ P\_n... | https://mathoverflow.net/users/438 | Closure of the locus of polynomials vanishing to a given order at two points | For a commutative ring $R$, write $P\_{n,R}$ for the affine $(n+1)$-space of polynomials of degree $\leq n$ over $R$. In other words, its $S$-points for an $R$-algebra $S$ are given by $S[x]\_{\leq n}$. Note that $P\_{n,R} = P\_{n,\mathbf Z} \times \operatorname{Spec} R$.
**Lemma.** *Let $R$ be a domain, and $g \in R... | 2 | https://mathoverflow.net/users/82179 | 393851 | 162,836 |
https://mathoverflow.net/questions/393856 | 2 | **Motivation:**
If a continuous function on the real line is periodic with periods $p\_1, p\_2 > 0$ such that $\frac{p\_1}{p\_2}$ is irrational, then the function is constant. Is there a probabilistic analogue of this statement?
**Problem set up:**
Let $X$ be a continuous stochastic process with $X\_t$ in $L^1$ f... | https://mathoverflow.net/users/173490 | If a process is periodic on average with mutually incommensurable periods, is the process a martingale? | **Yes**, it is. For each $t$, let $S\_t$ be the set of those $s \geqslant t$ for which $X\_t = \mathbb E(X\_s | \mathcal F\_t)$. By assumption (and induction), for all $k, n \geqslant 0$ such that $k p\_1 - n p\_2 \geqslant 0$ we have
\[ X\_t = \mathbb E(X\_{t + k p\_1} | \mathcal F\_t) \qquad \text{and} \qquad X\_{t +... | 3 | https://mathoverflow.net/users/108637 | 393860 | 162,840 |
https://mathoverflow.net/questions/393858 | 2 | Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim\_k H^q(X,\Omega\_{X/k}^p)$ be the Hodge numbers.
If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i.e. $h^{p,q}=h^{q,p}$, for the following reason: we can regard $X$ as a base change of a smooth projective variety $... | https://mathoverflow.net/users/177957 | Do we have Hodge symmetry for char $p$? | Hodge symmetry fails in positive characteristic in general; see Serre's Mexico paper [Ser58, Prop. 16] (for a more modern/conceptual version of that argument, see e.g. [vDdB21, Prop. 1.4]).
However, it is true for abelian varieties; see for example [Mum08, §13, Cor. 2]. As you note, this boils down to the computation... | 5 | https://mathoverflow.net/users/82179 | 393861 | 162,841 |
https://mathoverflow.net/questions/393872 | 0 | It is well known that the variance of the sum of independent random variables (not necessarily i.i.d.) is the sum of the variance of each random variable (i.e. $Var[X\_1 + X\_2 ... X\_n] = \sum\_{i=1}^{n} Var[X\_i]$). What about the higher absolute moment? For instance, does the following equation hold?
\begin{equation... | https://mathoverflow.net/users/245020 | High absolute moment of sum of the independent random variables | The generalization to higher powers of the additivity statement of the variance goes via the cumulants: If two variables $X$ and $Y$ are independent, then the cumulants $\kappa\_n$ are additive, $\kappa\_n(X+Y)=\kappa\_n(X)+\kappa\_n(Y)$.
It follows from this additivity that third central moments are additive, $$E\le... | 5 | https://mathoverflow.net/users/11260 | 393875 | 162,846 |
https://mathoverflow.net/questions/393878 | 1 | Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ with $\vert\vert f \vert\vert\_{1,\infty}<1$ is given and $I:\mathbb{R}^n\rightarrow\mathbb{R}^n$ denotes the identity ... | https://mathoverflow.net/users/125729 | Classical fixed-point argument and invertible function | $\newcommand{\R}{\mathbb R}$Let $F:=W^{1,\infty}(\R^n)$, with $\|f\|\_{1,\infty}:=\|f\|\_\infty+L(f)$ for $f\in F$, where $L(f)$ is the Lipschitz constant of $f$.
Take any $f\in F$ with $q:=\|f\|\_{1,\infty}<1$.
Let us show that then $I-f$ is invertible. Consider the Banach space
\begin{equation}
F\_0:=\{h\in\R^\R\c... | 1 | https://mathoverflow.net/users/36721 | 393890 | 162,850 |
https://mathoverflow.net/questions/393884 | 4 | I write the question for algebraic cobordism but I have the analogue question for classic cobordism.
The spectrum representing algebraic cobordism
$$
\mathbf{MGL}=(\*, \mathrm{Th}(1) , \ldots , \mathrm{Th}(n), \ldots)
$$
is given by the Thom spaces $\mathrm{Th}(n)$ of the universal bundles of the Grassmanians $\mathr... | https://mathoverflow.net/users/12204 | (Algebraic) cobordism and the rank function | Let me first write what happens for classical cobordism. You are basically asking whether the map $\operatorname{MU}\to H\mathbb{Z}$ factors through the projection $\operatorname{ku}\to H\mathbb{Z}$. But this is clear, since all spectra in sight are connective and we have an equivalence
$$\operatorname{Map}(E,H\mathbb{... | 3 | https://mathoverflow.net/users/43054 | 393897 | 162,852 |
https://mathoverflow.net/questions/393894 | 7 | As far as I understand, it is easy to see (and find in the literature) that the affine variety
$$z\_1^2+z\_2^2+z\_3^2=1$$
with the restriction of the standard $\omega\_{std}$ of $\mathbb{C}^3$ is symplectomorphic to $T^\*S^2$ with the standard complex structure. Now, as mentioned here ([https://www.math.stonybrook.edu/... | https://mathoverflow.net/users/117946 | Cotangent bundles of surfaces as varieties | This is not possible. There is something called the growth rate of symplectic cohomology which is subexponential for affine varieties and exponential for cotangent bundles of higher genus surfaces (amongst other things). This was proved by McLean:
<https://arxiv.org/abs/1011.2542>
| 8 | https://mathoverflow.net/users/10839 | 393901 | 162,853 |
https://mathoverflow.net/questions/393837 | 4 | $\DeclareMathOperator{\comp}{comp}\DeclareMathOperator{\refl}{refl}\DeclareMathOperator{\transp}{transp}$I've been reading about Cubical Type Theory and playing around with the Agda implementation of it\*. I'm wondering about whether one can add an additional equality judgement to composition operations that states tha... | https://mathoverflow.net/users/64294 | In cubical type theory, can we insist that "constant" compositions are the identity? | That is regularity. It is consistent, via a non constructive proof, for example taking an Orton-Pitts model (from Orton, Pitts, [*Axioms for Modelling Cubical Type Theory in a Topos*](https://arxiv.org/abs/1712.04864)) where cofibrations are all locally decidable monomorphisms. Using the law of excluded middle, every m... | 8 | https://mathoverflow.net/users/30790 | 393906 | 162,854 |
https://mathoverflow.net/questions/393888 | 2 | This is probably a simple question but I can't seem to find any references for it. Call a Kan complex $X$ $k$-truncated if $\pi\_n(X, x) = 0$ for all $x \in X\_0$ and $n > k$.
Claim: $X$ is $k$-truncated iff $\text{Fun}(S,X)$ is $k$-truncated for every simplicial set $S$, where $\text{Fun}(S,X) = X^S$ is the mapping ... | https://mathoverflow.net/users/124010 | A space $X$ is $k$-truncated iff $\text{Fun}(S,X)$ is $k$-truncated | Let $S^n := \Delta^n/\partial \Delta^n$; and let me assume for simplicity that $X$ is connected.
We have a (homotopy) fiber sequence $\Omega^n X \to X^{S^n} \to X$.
In particular, for $n>k$, $\Omega^n X$ is contractible (it is a Kan complex and its homotopy groups are trivial by assumption), so that for $n>k$, $X^{... | 5 | https://mathoverflow.net/users/102343 | 393907 | 162,855 |
https://mathoverflow.net/questions/393892 | 1 | In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested in the relation between the solutions of the equations $v\_t=Lv$ and $v\_t=Av$. For example, if the first has an unbounde... | https://mathoverflow.net/users/51290 | Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$ | **General references.**
The references that you're probably looking for are books on the theory of $C\_0$-semigroups. Some classics are:
[1] Amnon Pazy: *Semigroups of Linear Operators and Applications to Partial Differential Equations*, 1983
[2] Klaus-Jochen Engel and Rainer Nagel: *One-Parameter Semigroups of L... | 2 | https://mathoverflow.net/users/102946 | 393911 | 162,857 |
https://mathoverflow.net/questions/393908 | 1 | Let $m\geq1$ and $P$ be an arbitrary poset with vertex set $V=\{v\_1,\dots,v\_n\}$, edge set $E,$ and set $O$ of orbits under $\text{Aut}(P).$ Can we efficiently generate all inequivalent nonnegative weight assignments $\alpha:V\to\mathbb{N}$ such that $\sum\_{i=1}^n\alpha(v\_i)=m$ from this information alone or do we ... | https://mathoverflow.net/users/5090 | inequivalent vertex weights on finite poset | Let's see if I understand the question. You have a vertex set $V$ with an automorphism group $G$. You want to assign a nonnegative integer weight to each vertex, with sum fixed to $m$, and you want to list all possible weight assignments when two assignments related by an element $g\in G$ are considered equivalent. Als... | 1 | https://mathoverflow.net/users/171662 | 393915 | 162,858 |
https://mathoverflow.net/questions/393919 | 3 | Let $\mu(n)$ be the Moebius function. Let $a$ be a positive integer. For odd $n$ (this is possibly true for all $n$, but I only care for odd ones) I get that
$$\sum\_{d|n} \mu\left(\frac{n}{d}\right) a^d \equiv 0 \pmod n.$$
Is there a nice reason for this that can be seen purely from the formula? I have a proof for som... | https://mathoverflow.net/users/1378 | Why does this formula always give me $0$ modulo $n$? | Your expression, divided by $n$, is the number of aperiodic necklaces of length $n$ with $a$ colors. Since this number is an integer, the sum must be divisible by $n$.
See *Aperiodic necklaces* [here](https://en.wikipedia.org/wiki/Necklace_(combinatorics)?wprov=sfti1).
| 7 | https://mathoverflow.net/users/25 | 393921 | 162,861 |
https://mathoverflow.net/questions/393797 | 43 | Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:
>
> Are there any good reasons to use Tate's theory of rigid-analytic spaces, given that Huber's theory of adic spaces seems to be superior in all regards?
>
>
>
The only poss... | https://mathoverflow.net/users/30186 | Are rigid-analytic spaces obsolete, since adic spaces exist? | There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it's very much a matter of taste. Adic spaces are genuinely topological spaces, whereas Tate's G-topological spaces aren'... | 30 | https://mathoverflow.net/users/2481 | 393922 | 162,862 |
https://mathoverflow.net/questions/351531 | 9 | According to the nlab, [horizontal categorification](https://ncatlab.org/nlab/show/horizontal+categorification) is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized to the same type of categories with an arbitrary number of... | https://mathoverflow.net/users/2841 | Horizontal categorification: Two questions | 1. Often when a horizontal categorification is introduced, the author has intended examples, which the author gives as the motivation. It thus makes it a little tricky to determine when a structure has been motivated explicitly by horizontal categorification. However, here are a few examples where (1) the concept had n... | 2 | https://mathoverflow.net/users/152679 | 393933 | 162,864 |
https://mathoverflow.net/questions/393887 | 3 | A Mersenne number is a number of the form $2^k-1$ for some $k \in \mathbb{N}$. Consider the set of $2^n-1$ products of Mersenne numbers
$$M\_n=\left\{ \prod\_{k\in S} (2^k-1) : S \subseteq [n], S\neq \emptyset\right\}.$$
**Question:** What is the minimum $r \in \mathbb{N}$ for which there exists $\alpha\_1,\dots, \al... | https://mathoverflow.net/users/150898 | Products of Mersenne numbers as sums of real numbers | We can indeed get a superlinear lower bound. I prove a lower bound of $\tilde\Omega(n^2+n)$ (ignoring log factors). I thank Gerry Myerson for pointing out the following helpful reference in the comments:
Moulton, David. (2001). Representing Powers of Numbers as Subset Sums of Small Sets. Journal of Number Theory - J ... | 3 | https://mathoverflow.net/users/150898 | 393945 | 162,868 |
https://mathoverflow.net/questions/393929 | 4 |
>
> Proof or counterexample: if $\mathcal F\subseteq \mathbb{N}^{(r)}$ is intersecting then $\exists A\subseteq \mathbb{N}: A$ is finite and $\{F\cap A:F\in\mathcal F\}$ is intersecting.
>
>
>
I know this result is false for $\mathcal F\subseteq \mathcal P( \mathbb{N})$ because you can find a bijection $f:\mathb... | https://mathoverflow.net/users/157270 | Does an intersecting $r$-family in $\mathbb{N}$ have a finite underlying base set? | It is easy to prove by induction the following stronger statement.
>
> If $\cal F$ and $\cal G$ are two families of sets of size at most $r$ and $s$, respectively, that *cross-intersect*, i.e., for all pair of sets $F\in \mathcal F, G\in\cal G$ we have $F\cap G\ne\emptyset$, then $\exists A\subseteq \mathbb{N}: A$ ... | 6 | https://mathoverflow.net/users/955 | 393951 | 162,869 |
https://mathoverflow.net/questions/393939 | 1 | Let $X\_t$ be a continuous real valued stochastic process on $\mathbb R\_+$. Then it is not necessarily true that $E[X\_t]$ is continuous in $t$.
**Question:**
What is known about the discontinuity set of $E[X\_t]$? Namely:
* Can it have removable discontinuities?
* Can it have essential discontinuities?
* Can it... | https://mathoverflow.net/users/173490 | Discontinuity set of the expected value of a continuous process | Without further assumptions on $X\_t$, this is a question "what type of functions one gets if one averages continuous functions". Let $\phi\_n(x) = x$ when $0 \leqslant x \leqslant n$, $\phi\_n(x) = n$ if $x > n$, and $\phi\_n(x) = 0$ when $x < 0$. By the dominated convergence theorem,
\[ f\_n^+(t) := \mathbb E \phi\_n... | 1 | https://mathoverflow.net/users/108637 | 393958 | 162,871 |
https://mathoverflow.net/questions/393969 | 0 | Consider a positive Hermitian $N \times N$ matrix $A$ with complex valued coefficients. We list its eigenvalues in increasing order and with their multiplicities, $\mu\_{1} \leq \mu\_{2} \leq \cdots \leq \mu\_{N}$ and consider the one parameter family of matrices $A+\lambda$. How can I verify that for any $\lambda>-\mu... | https://mathoverflow.net/users/157604 | Computation to differentiate a determinant | The eigenvalues of $A+\lambda$ are $\{\mu\_j+\lambda\}$ which are positive by assumption. So
$$\frac{d}{d\lambda} \log\det(A+\lambda)
= \frac{d}{d\lambda} \sum\_j \log (\lambda+\mu\_j)
= \sum\_j (\lambda+\mu\_j)^{-1},
$$
which is the sum of the eigenvalues (i.e. the trace) of
$(A+\lambda)^{-1}$.
| 6 | https://mathoverflow.net/users/9025 | 393973 | 162,877 |
https://mathoverflow.net/questions/393977 | 4 | I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance.
Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. Consider the ordinary elliptic $k$-curve $E\!: y^2 = x^3 + 1$ (of $j$-invariant $0$) and its cubic power $E^3$. There are qui... | https://mathoverflow.net/users/69852 | Mirror partners of some Calabi-Yau threefolds | Your two examples are actually of very different characters. The first has Hodge numbers $h^{2,1} = 3$ and $h^{1,1} = 51$; the second is rigid. This means that in the first case you're looking for a Calabi-Yau threefold with "mirror" Hodge numbers $h^{2,1} = 51$ and $h^{1,1} = 3$, while the mirror of the second may not... | 4 | https://mathoverflow.net/users/60708 | 393981 | 162,879 |
https://mathoverflow.net/questions/393980 | 9 | To be a bit more precise and fix notations, let $X$ be a Banach space (over $\mathbb{R}$ or $\mathbb{C}$), $X^{\ast\ast}$ its second dual (as a Banach space). Here and in the following we identify $X$ as a (norm) closed subspace of $X^{\ast\ast}$ via the canonical embedding $J : X \hookrightarrow X^{\ast\ast}$. Now let... | https://mathoverflow.net/users/128540 | Is the unit sphere of a Banach space dense in the unit sphere of its second dual with respect to the weak-$\ast$ topology | I think a simple rescaling argument works. I do Question 1, Q3 being similar. Given $F\in S\_{X^{\*\*}}$, by Goldstine there is a net $(x\_i)$ in $B\_X$ converging weak$^\*$ to $F$. Given $\epsilon>0$ there is $f\in S\_{X^\*}$ with $|F(f)|>1-\epsilon$ and so
$$ 1-\epsilon < |F(f)| = \lim\_i |f(x\_i)| \leq \|f\|\liminf\... | 11 | https://mathoverflow.net/users/406 | 393983 | 162,880 |
https://mathoverflow.net/questions/393290 | 2 | I am looking for a general formula for hitting times in a standard birth-and-death chain.
I'm absolutely convinced that I've seen a paper with such a formula in it in the past, but I cannot for the life of me find it now.
The formula looks something like this:
$$
E\_{i-1}(\tau\_i) =
\prod\_{j < i} \frac{q\_{j,j+1} ..... | https://mathoverflow.net/users/59264 | Reference Request: Hitting Times in Birth-and-Death Chains | There is a discussion of birth an death chains in [1]; see page 27 for the hitting time formulas. See also [2], [3], [4].
[1] Markov Chains and Mixing Times: Second Edition by Levin and Peres, with contributions by Wilmer, <https://bookstore.ams.org/mbk-107>
[https://darkwing.uoregon.edu/~dlevin/MARKOV/mcmt2e.pdf](ht... | 3 | https://mathoverflow.net/users/7691 | 393992 | 162,883 |
https://mathoverflow.net/questions/393999 | 22 | Does there exist an archive somewhere of posts to the USENET newsgroup `sci.math.research`?
The best approximation I'm aware of is [Google Groups](https://groups.google.com/g/sci.math.research). However, despite the Google brand name, the search capability of Google Groups is abysmal, and it is hard to tell how much ... | https://mathoverflow.net/users/3106 | sci.math.research archive? | There's a bit of content [on Archive.org](https://archive.org/details/usenet-sci.math): the link says `sci.math`, but among the files in the collection there is a `sci.math.research.20140626.mbox.gz` which contains ~11k messages apparently posted between 2003 and 2014. Sadly, this does not seem to contain the specific ... | 11 | https://mathoverflow.net/users/17064 | 394001 | 162,885 |
https://mathoverflow.net/questions/393923 | 2 | In his celebrated paper, "On Computable Numbers, With An Application To the Entscheidungsproblem", Turing defines a "computable number" as follows:
>
> The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means....According to my definition, a... | https://mathoverflow.net/users/20597 | Are ITTM's necessary to compute Turing's "computable numbers" and what does that mean for ordinary recursion theory? | No, classical computability theory as you point is quite capable of dealing with infinitary computable enumerations and computability-in-the-limit from its earliest stages. I believe that Turing is to be credited with the fundamental distinction between a computably decidable decision problem and one that is merely sem... | 5 | https://mathoverflow.net/users/1946 | 394011 | 162,892 |
https://mathoverflow.net/questions/393416 | 2 | Let $J\_\nu$ be a Bessel function of the first kind and let $\{\lambda\_{n, \nu}\}\_{n\ge 1}$ be a sequence of its zeroes. I claim that
$$
\inf\_{n\ge 1}\bigg|\sqrt{\lambda\_{n,\nu}} J\_{\nu+1}(\lambda\_{n,\nu})\bigg|>0.
$$
The reason I believe this is true is that:
(1) we have $|J\_{\nu+1}(x)|\lesssim x^{-1/2},\qqua... | https://mathoverflow.net/users/157356 | Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type | You have, using the asymptotics for $\lambda\_{n,\nu}$,
\begin{align\*}
& \cos\left(\lambda\_{n,\nu} - \frac{2\nu+3}{4}\pi \right) \\
&= \cos\left(\lambda\_{n,\nu} - \frac{2\nu-1}{4}\pi+\pi \right) \\
&= \cos\left(\lambda\_{n,\nu} - \frac{2\nu-1}{4}\pi-n\pi \right) (-1)^{n-1}\\
&= \cos\left(O(n^{-1}) \right) (-1)^{n-1}... | 1 | https://mathoverflow.net/users/40120 | 394015 | 162,894 |
https://mathoverflow.net/questions/393718 | 4 | A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this [question](https://mathoverflow.net/questions/86627/what-is-a-pointed-hopf-algebra) for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional comodule, namely the trivial comodule for a f... | https://mathoverflow.net/users/153228 | Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule | **Q:** *Is there a name for a Hopf algebra that admits no one-dimensional comodule other than the trivial comodule?*
**A:** Not in the literature, but if you would like to coin a specific name for such a Hopf algebra, at the opposite extreme of pointed, you could call it ***blunt***.
A class of *blunt Hopf algebras... | 2 | https://mathoverflow.net/users/11260 | 394019 | 162,896 |
https://mathoverflow.net/questions/394017 | 3 | Suppose $M= \mathbb R \times M\_0$ with a Lorentzian metric $g(t,x)=-dt^2+ g\_0(t,x)$ where
$M\_0$ is a compact manifold with a smooth boundary and $g\_0$ is a family of smooth Riemannian metrics on $M\_0$ depending smoothly on $t$. Assume furthermore that $M$ is nontrapping, that is to say, all inextendible null geode... | https://mathoverflow.net/users/50438 | A question on light cones in Lorentzian manifolds with timelike boundary | **Edit:** Right now I will *not* give a complete argument in what follows (the previous version of the answer missed the part of the question asking for $S$ to be the boundary of a spacelike hypersurface in $M$), I will add the missing ingredients and/or additional hypotheses (if needed) as time allows in a future edit... | 1 | https://mathoverflow.net/users/11211 | 394026 | 162,898 |
https://mathoverflow.net/questions/393990 | 2 | I wonder whether it is always possible to bring a 2D Riemannian metric to a diagonal form with determinant one by changing the coordinates, i.e. for the line element
$$
ds^2 = A(x,y)\, dx^2 + B(x,y)\, dy^2
$$
to obtain $A(x,y)\, B(x,y) = 1$ everywhere.
It is known that one can bring the metric to a diagonal and conf... | https://mathoverflow.net/users/161595 | 2D-metric to diagonal form with determinant 1 | Locally, this is always possible. Constructing such a coordinate system is equivalent to solving a first-order hyperbolic PDE system for two unknowns of two variables, so it always has local smooth solutions.
Here is a sketch of how this can be proved: You want to find a $g$-orthonormal coframing $g = {\omega\_1}^2+{... | 3 | https://mathoverflow.net/users/13972 | 394030 | 162,901 |
https://mathoverflow.net/questions/393934 | 28 | Let $G$ be a finite group and let $\phi:G\to Z\_2$ be a homomorphism to the group with two elements. Is it always the case that there are more conjugacy classes in the kernel of $\phi$ than conjugacy classes not in the kernel of $\phi$?
I've tried a little bit of messing around algebraically and written down some exa... | https://mathoverflow.net/users/246663 | Are there always more conjugacy classes in the kernel of a morphism to $Z_2$ than not? | $\DeclareMathOperator\tr{tr}\DeclareMathOperator\Int{Int}\DeclareMathOperator\Cent{Cent}$The OP (@ClarkLyons) gave a lovely [generalisation](https://mathoverflow.net/a/394013) of an [answer](https://math.stackexchange.com/a/4153721) of @diracdeltafunk over on MSE. I believe that the technique can be pushed still furthe... | 11 | https://mathoverflow.net/users/2383 | 394036 | 162,903 |
https://mathoverflow.net/questions/394035 | 3 | Consider the two statements:
1. "Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra", as stated in [1506.03546](https://arxiv.org/abs/1506.03546) page 4. The above paper refers to (I think) Theorem 7.6 of this [paper](https://core.ac.uk/download/pdf/8275535... | https://mathoverflow.net/users/138208 | Realizing a fusion category as endomorphisms of an algebra | Every monoidal category is monoidally equivalent to a strict one. This is MacLane’s famous coherence theorem.
What the example $\mathrm{Vec}(G, \omega)$ shows is that you can’t always make it simultaneously strict and skeletal. But you can do either separately if you like.
| 5 | https://mathoverflow.net/users/22 | 394038 | 162,904 |
https://mathoverflow.net/questions/394054 | 26 | I begin by saying that while I understand what a triangulated / derived category is pretty well, I know nothing about Higher Algebra stuff and not even $\infty$-categories.
I've heard some people say that stable $\infty$-categories are a "more natural" point of view to derived categories. **Can anyone explain in simp... | https://mathoverflow.net/users/131975 | Why stable $\infty$-categories? | I already answered some version of this question in [this answer](https://mathoverflow.net/a/344221/43054), but let me try to expand a bit on the *concrete* advantages in mathematical practice. For understanding the following you need to take on faith that ∞-categories exist and have roughly the same properties as ordi... | 46 | https://mathoverflow.net/users/43054 | 394056 | 162,909 |
https://mathoverflow.net/questions/394059 | 6 | Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).
Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[0,3l]$, and we let $C\_1:=C\cap[0,l]$, $C\_2:=C\cap[l,2l]$, and $C... | https://mathoverflow.net/users/9924 | Trisecting $3$-fold sumsets: is the middle part always thick? | No. Take $A = \{0,1,\ldots,9,10,20,30,\ldots,90,100,200,300,\ldots,900,1000\}$. Then $|C\_1|=1001$, $|C\_2|=272$ and $|C\_3|=29$.
A smaller counterexample in the same spirit is $\{0,1,2,3,4,5,10,15,20,25,50,75,100\}$, with sizes $101, 53, 13$.
| 6 | https://mathoverflow.net/users/171662 | 394065 | 162,910 |
https://mathoverflow.net/questions/394093 | 1 | It is known (see Ch. 4 in Struwe's Variational Methods) that Radon measures on $\mathbb{R}^n$ satisfy the concentration-compactness principle. Does the same hold true for Radon measures on a general metric space?
| https://mathoverflow.net/users/100163 | Concentration-compactness for Radon measures on a metric space | $\newcommand{\bR}{\mathbb{R}}$ The concentration compactness principle as P.L. Lions first formulated (roughly) states that a sequence of probability measures on $\mathbb{R}^n$ absolutely continuous with respect to the Lebesgue measure, fails to be relatively compact only due to the actions of the noncompact group of h... | 1 | https://mathoverflow.net/users/20302 | 394104 | 162,919 |
https://mathoverflow.net/questions/394096 | 4 | Let $(M^n,g\_i)$ be a sequence of smooth complete Riemannian manifold with $|sec\_{g\_i}| \le 1$. Suppose $(M\_i^n,g\_i)$ converges to a limit space $(X^{n-1},d)$ in the Gromov-Hausdorff sense, where the Hausdoff dimension of $X$ is $n-1$.
Can we show that $X$ contains no boundary point? Here, a point is a boundary p... | https://mathoverflow.net/users/16323 | Does codimension-1 collapsing with bounded curvature have boundary? | Flat Klein bottles can collapse to a line segment.
| 4 | https://mathoverflow.net/users/1441 | 394130 | 162,927 |
https://mathoverflow.net/questions/394121 | 8 | For context, I am rather new to the whole business of abstract Weil cohomology theories and motives in general, so if I am not making sense somewhere, do let me know!
1. In many of the literature that I am consulting while trying to learn about Weil cohomology theories and motives, it is often said that these cohomol... | https://mathoverflow.net/users/143390 | Verifying the Lefschetz Conditions for crystalline cohomology | The Hard Lefschetz theorem can certainly not be deduced formally from the axioms of a Weil cohomology theory given in the Stacks Project. The reason it is called "hard" Lefschetz is that it is a really hard theorem, and deeper than the other properties.
The situation is rather that the axiomatization of what it means... | 11 | https://mathoverflow.net/users/1310 | 394133 | 162,930 |
https://mathoverflow.net/questions/394101 | 94 | I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:
*Has someone already tried this? If not, who in the mathematical commu... | https://mathoverflow.net/users/250603 | Peer review 2.0 | I'm the founder of <https://papers-gamma.link>, an Internet place
to discuss scientific articles, mentioned by Matthieu Latapy. I have
been supporting this site for 6 years now. I hope that one day it will
become popular (in a good sense of the word) and useful for the entire
scientific community. As you may imagine, I... | 27 | https://mathoverflow.net/users/31830 | 394137 | 162,932 |
https://mathoverflow.net/questions/393467 | 3 | I am looking for a proof that:
if $A\_{11}A\_{12}...A\_{1n}$; $A\_{21}A\_{22}...A\_{2n}$; $\cdots$; $A\_{i1}A\_{i2}...A\_{in}$; $\cdots$; $A\_{m1}A\_{m2}...A\_{mn}$ are $m$ oriented regular polygons ($n$-gons), where $n=2k$, then$\DeclareMathOperator\Area{Area}$
$$
\begin{align\*}
& \Area(A\_{11}A\_{21}...A\_{m1})+\A... | https://mathoverflow.net/users/122662 | Equal area of sum of pair opposite polygons conjecture | One fixes the counterclockwise orientation for polygons. Let $\omega\_1\omega\_2\cdots\omega\_m$ ($m\geq 3$) be a polygon, where $\omega\_i$'s are its vertices represented by complex numbers. It is standard to show that the area of the polygon is given by the imaginary part of the expression $$-\frac 1 2\sum\_{i=1}^m\o... | 2 | https://mathoverflow.net/users/217437 | 394143 | 162,934 |
https://mathoverflow.net/questions/394146 | 13 | Choose some $x > 1$. Then
$$
\lim\_{T\to\infty} \sum\_{\Im(\rho)<T}\cos(\ln(x)\Im(\rho))=-\infty
$$ where $\rho$ ranges over all zeros of the zeta function iff $x$ is prime or the power of some prime. This implies that the zeta function's zeros are in a sort of arithmetic progression, and are more likely to appear when... | https://mathoverflow.net/users/253504 | Are there any papers about this observation of the distribution of the zeros of the zeta function? | This is called Landau's formula. More precisely, if we extend the von Mangoldt function $\Lambda(n)$ to the function $\Lambda:\mathbb R\_+\to \mathbb R$ by $\Lambda(x)=0$ for non-integer $x$, then
$$
\sum\_{|\mathrm{Im}\,\rho|\leq T}x^{\rho}=-\frac{T}{2\pi}\Lambda(x) +O\_x(\ln T)
$$
For $x=2$ and $T=1000$, as the Riem... | 19 | https://mathoverflow.net/users/101078 | 394149 | 162,935 |
https://mathoverflow.net/questions/394125 | 7 | This is a refined version of the [question I asked yesterday](https://mathoverflow.net/q/394059/9924).
Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[0,3l]$, and I write $C\_1:=C\cap[0,l)$, $C\_2:=C\cap[l,2l]$, and $C\_3:=C\c... | https://mathoverflow.net/users/9924 | Trisecting $3$-fold sumsets, II: is the middle part ever thin? | The title and body are asking the question in opposite senses. For the title, the answer is "yes" (it can be thin), and for the body, the answer if "no" (it is not true that it is never thin).
An example is
$$
A = \{0,1,2,3,8,11,26,38,56,69,85,89,179,189,197,221,226,243,254,257,264,266,269,270\}
$$
where the sumset p... | 7 | https://mathoverflow.net/users/171662 | 394155 | 162,938 |
https://mathoverflow.net/questions/394161 | 11 | Let $X$ be a compact Kähler manifold, with $j\_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \hookrightarrow \widetilde{X}$. The restriction $\tau\_E: E\to Z$ is then a $\mathbb{CP}^{r-1}$-bundle.
Let $h=c\_1(\... | https://mathoverflow.net/users/192152 | What's the cohomology ring structure of a blow-up? | This is straightforward. Using $j^\*[E]=-h$ and the projection formula $j\_\*(x\cdot j^\*y)=j\_\*x\cdot y$, we get $[E]^p=j\_\*1\cdot [E]^{p-1}= (-1)^{p-1}j\_\*(h^{p-1})$. Then
$$[E]^{p}\cdot \tau ^\*\alpha = (-1)^{p-1}j\_\*(h^{p-1}\cdot j^\*\tau ^\*\alpha )= (-1)^{p-1}j\_\*(h^{p-1}\cdot p^\*\alpha\_{|Z} )\,,$$ where $... | 11 | https://mathoverflow.net/users/40297 | 394163 | 162,941 |
https://mathoverflow.net/questions/394144 | 0 | If I've got $f: x \to f(x)$, one may define the Arithmetic Dissidence $\delta\_A[f(x)]$ as the real value of the difference between the length following the curve of $f$ and the length of the $x$ axis (we could define it on intervals or on the whole axis). You may also want to define the Geometric Dissidence $\delta\_G... | https://mathoverflow.net/users/248399 | Getting a finite value for curve Dissidence | You seek the quantity
$$\delta\_G(x,\lambda)=\frac{1}{x}\int\_0^x \sqrt{1+f'(y)^2}\,dy,\;\;\text{with}\;\;f(x)=\lambda e^{-x^2/2},$$
where I have rescaled $x\mapsto x/\sigma$ to remove the parameter $\sigma$. There is no closed form answer for this integral, but for large $x$ it decays as
$$\delta\_G(x,\lambda)\approx ... | 2 | https://mathoverflow.net/users/11260 | 394171 | 162,944 |
https://mathoverflow.net/questions/394157 | 0 | Let $f(x)$ be any continuous function, then is it true that $$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)$$
where $\alpha>1$ is a real number and $\beta$ is any positive integer. $Z^+$ denotes the number of positive zeros.
Note: If $f(x)$ is a polynomial then it's easy to see that the ... | https://mathoverflow.net/users/126770 | Change in the number of positive zeros of a continuous function | No. Let $a:=\alpha$ and $b:=\beta$. If e.g. $f(x)=(x+b)^{-a}$, then $a f(x) + (x + b) f'(x)=0$ for all real $x\ge0$, so that the left-hand side (lhs) of your inequality is infinite, whereas the right-hand side (rhs) is $0$.
---
If you now insist that the lhs be a finite number, we can modify the above example as ... | 3 | https://mathoverflow.net/users/36721 | 394177 | 162,945 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.