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https://mathoverflow.net/questions/440648 | 6 | I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to something called "neck cutting." I'm not sure what this means specifically, but I'm aware that Seiberg-Witten theory, fo... | https://mathoverflow.net/users/314845 | "Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres | We cannot go as far as to say all of SW theory is insufficient, but the invariants that exist now are insufficient, via neck 'stretching' and 'pinching'. If you're aware of the proof (such as [No homotopy 4-sphere invariants using ECH=SWF](https://msp.org/agt/2021/21-5/p11.xhtml)), then I believe similar gauge theories... | 3 | https://mathoverflow.net/users/12310 | 440679 | 177,933 |
https://mathoverflow.net/questions/336494 | 26 | For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|\_r\leq \|f\|\_p\|g\|\_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. Equivalently
\begin{align\*}
\|f\|\_p=\|g\|\_q=\|h\|\_{r'}=1\Rightarrow \int\_{\mathbf{R}^d}\int\_{\mathbf{R}^d}f(x)g(y)... | https://mathoverflow.net/users/27767 | Proofs of Young's inequality for convolution | I found the proof given by Hormander in the Analysis of Linear Partial Differential Operators to be quite clear and instructive. It can be found in the first volume, pages 116-117, and for convenience I will reproduce it here.
>
> $|u\_1 \* u\_2 \* \dots \* u\_k(0)| \leq \|u\_1\|\_{p\_1}\dots\|u\_k\|\_{p\_k}$ if $\... | 3 | https://mathoverflow.net/users/nan | 440680 | 177,934 |
https://mathoverflow.net/questions/440685 | 2 | Let $A/\mathbb{Q}$ be an abelian variety with good reduction at a prime $p$. Assume $\mathcal{A}/\mathbb{Z}\_{(p)}$ is an integral model at $p$(hence proper smooth).
For any number field $K$ and any prime ideal $\mathfrak{p}$ of $K$ over $p$ with residue field $\kappa$, by valuation criterion on $\mathcal{O}\_{K,\mat... | https://mathoverflow.net/users/177957 | The reduction map on the $\ell$-primary torsion of abelian variety | This is true. One reference for this is Theorem C.1.4 in Hindry-Silverman's "Diophantine Geometry: An Introduction" which says that for an abelian variety $A$, number field $K$ and any prime $\mathfrak p$ above $p$, the reduction map $A(K)\to A(O\_K/\mathfrak p)$ is injective on prime-to-$p$-torsion. The result you ask... | 3 | https://mathoverflow.net/users/30186 | 440687 | 177,938 |
https://mathoverflow.net/questions/376603 | 13 | The following result is Proposition 2.4.3 in [1]:
>
> **Theorem.** Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if
> there is $r>0$ such that $K$ is the unioin of balls of radius $r$.
>
>
>
**Question.** Do you know who is the author o... | https://mathoverflow.net/users/121665 | Regularity of convex sets in $\mathbb{R}^n$ | Theorem 1.3 in:
<https://arxiv.org/pdf/1304.4179.pdf>
and the historical discussion below it is relevant. The result (in a form that applies without the convexity assumption) seems to first appear in 1957.
| 4 | https://mathoverflow.net/users/112954 | 440697 | 177,939 |
https://mathoverflow.net/questions/440613 | 8 | Let $B\_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B\_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B\_t\in C^s(x)$.
Moreover I read a claim that said the same about any translation by a continuous function: given $s>1/2$ and $f$ continu... | https://mathoverflow.net/users/39180 | Regularity of translations for Brownian motion | The result holds for any bounded function $f$, in the following sense: for any real $s>1/2$,
\begin{equation}
P^\*(A)=0,
\end{equation}
where
\begin{equation}
A:=\Big\{\exists t\_0\in[0,1]\ \limsup\_{t\to t\_0}\frac{|W\_f(t)-W\_f(t\_0)|}{|t-t\_0|^s}<\infty\Big\},
\end{equation}
$P^\*$ is the outer probability, $\limsu... | 3 | https://mathoverflow.net/users/36721 | 440698 | 177,940 |
https://mathoverflow.net/questions/440695 | 0 | This is a follow-up question from [my previous question](https://mathoverflow.net/questions/439653/maximum-number-of-vectors-with-bounds-on-inner-products/440643?noredirect=1#comment1136719_440643).
Suppose there are (2n+1) vectors $\{m\_1,m\_2,...,m\_n\}$, $\{p\_1,p\_2,...,p\_n\}$ and $p^\*$ in $R^{k+1}$. $m\_i$ are... | https://mathoverflow.net/users/498587 | Maximum number of vectors with bounds on inner products (follow up question) | This is not an answer to the question, but here are some upper/lower bounds. Firstly, if we let $A\_i\subseteq\{0,1,\dots,k+1\}$ be the non zero coordinates of $m\_i$, then we can't have $A\_i\subseteq A\_j$ for $i\neq j$, because then we would have $m\_ip\_j=0$ (you already mentioned this in the other question I think... | 1 | https://mathoverflow.net/users/172802 | 440700 | 177,941 |
https://mathoverflow.net/questions/440705 | 11 | A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose interior admits a hyperbolic metric. The map
$$\sigma\_M:\mathrm{Teich}(\partial M) \to \mathrm{Teich}(\overline{\partial M})$$ takes a c... | https://mathoverflow.net/users/499323 | Definition of Thurston's skinning map | Let’s simplify to the case where $M$ has exactly one boundary component, say $\partial M = S$. So the hyperbolic structures on $M$ are parametrised by the conformal structures on $S$. Fix one such conformal structure $\rho$ and lift the resulting hyperbolic structure on $M$ to the cover corresponding to $\pi\_1(S)$. As... | 13 | https://mathoverflow.net/users/1650 | 440707 | 177,942 |
https://mathoverflow.net/questions/439697 | 6 | I'm posting this question in hopes that someone more familiar with the literature will be able to point me in the right direction (or give an obvious answer).
Let $M^{d-2} \hookrightarrow \mathbb{R}^d$ be a smooth embedding and let $g$ be the metric induced on $M$ from the flat metric on $\mathbb{R}^d$. Under what co... | https://mathoverflow.net/users/104933 | Existence of local isometric embedding of smooth $(M^{d-2},g)$ in $\mathbb{R}^{d-1}$ | It's hard to answer the OP's question satisfactorily without considering what would actually constitute an answer. The most obvious answer is tautological: $M^{d-2}\hookrightarrow\mathbb{R}^d$ lies in a hyperplane if and only if there is a nonzero affine function $\ell:\mathbb{R}^d\to\mathbb{R}$ such that $\ell(M^{d-2}... | 5 | https://mathoverflow.net/users/13972 | 440714 | 177,944 |
https://mathoverflow.net/questions/440615 | 1 | I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm.
I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). These constraints designate the domain for variable $x$. Imagine I have one new constraint $cx\le d$, which may or may... | https://mathoverflow.net/users/499253 | Adding linear constraint to the domain | By LP duality, the new constraint $cx \le d$ is redundant iff there exists $u \ge 0$ such that
\begin{align}
u A &= c \tag1\label1 \\
u (b - y) &\le d \tag2\label2
\end{align}
To see the easy direction of the iff, note that \eqref{1} and \eqref{2} imply
$$cx = u A x \le u (b - y) \le d$$
You want to minimize $\sum\_i y... | 1 | https://mathoverflow.net/users/141766 | 440728 | 177,948 |
https://mathoverflow.net/questions/440692 | 2 | **Question.** Does there exist an entire function $h$ satisfying three following assertions:
* $h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
* $zh - 1$ belongs to $H^2(\mathbb{C}\_+)$, where $\mathbb{C}\_+ = \{\text{Im}(z) > 0\}$;
* $h$ has infinitely many zeroes in some horizontal strip ... | https://mathoverflow.net/users/498423 | Existence of the special entire Hardy space function with infinitely many zeros in the strip | Let
$$f(z)=\frac{e^{iz}-1}{iz}.$$
This function is in the Hardy class for any upper half-plane,
and has these properties: $f(0)=1,$ $f(2\pi n)=0$,
$$|f(z)|\leq C\frac{e^y+1}{|z|+1},$$
(this evidently holds for large and small $|z|$, therefore there is a constant $C$ so that this holds everywhere).
Since the $L^2$ nor... | 1 | https://mathoverflow.net/users/25510 | 440729 | 177,949 |
https://mathoverflow.net/questions/438836 | 1 | Let $(X\_n)\_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of
$$
\sum\_{k=1}^n c^k X\_k.
$$
I can prove that this converges a.s. for $n\to\infty$ iff $\mathbb{E}(\max(0,\log(|X\_1|)<\infty$.
To be more specific:
* Are there known cond... | https://mathoverflow.net/users/473107 | Asymptotic properties of weighted random walks / infinite convolutions of random variables | Let $(X\_n)\_{n\in\mathbb{N}}$ be i.i.d. real random variables and let $0<c<1$.
Then the following are equivalent:
(a) There exists $r>0$ such that $P(|X\_k|>e^{rk} \; \, \text{infinitely often} )=0$.
(b) $\mathbb{E}(\max(0,\log(|X\_1|)<\infty$.
(c) For all $r>0$, we have $P(|X\_k|>e^{rk} \; \, \text{infinitel... | 1 | https://mathoverflow.net/users/7691 | 440730 | 177,950 |
https://mathoverflow.net/questions/440744 | 6 | The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}\_{2,n}$ is given by the number of tuples $(\lambda\_1,\lambda\_2)$ satisfying
$$
n - 2 \geq \lambda\_1 \geq \lambda\_2 \geq 0.
$$
Explicitly this is given by
$$
\binom{n}{2}.
$$
This also happens to be the dimension of $V\_{\p... | https://mathoverflow.net/users/491434 | The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$ | This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over $\mathbb{C}$ there is no room for the differential in the cellular complex). In the cellular decomposition of the Grassmannian, the (Schubert) cells are indexed by... | 10 | https://mathoverflow.net/users/1306 | 440753 | 177,954 |
https://mathoverflow.net/questions/440770 | 1 | Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any physically important? (describing any non-smooth dynamics of the real world). Thank you!
| https://mathoverflow.net/users/60556 | Any example of a multi-valued monotone maximal operator without subdifferential? | My prime example of such an operator comes from saddle point problems of the form
$$
\min\_x\max\_y F(x) + \langle Kx,y\rangle - G(y)
$$
with $F,G$ being two proper, convex, lower-semicontinuous functions defined on Hilbert spaces $X$ and $Y$, respectively, and $K:X\to Y$ linear and bounded. The Fenchel-Rockafellar opt... | 1 | https://mathoverflow.net/users/9652 | 440777 | 177,959 |
https://mathoverflow.net/questions/440771 | 2 | Consider the matrix $$D=\begin{pmatrix}1&0\\0&e^{i\theta}\end{pmatrix}.$$ For the commonly used norms $\|\cdot\|$ on $\mathbb{C}^2$ or for $\theta=0$ the associated subordinate norm is $1$. Is it always true ? can a subordinate norm be strictly bigger than one ?
| https://mathoverflow.net/users/126690 | Existence of weird complex norms | Consider the norm $\Vert\cdot\Vert$ on $\mathbb{C}^2$ defined by $\Vert z \Vert^2 := |z\_1|^2+|z\_1+z\_2|^2$. Then
$$\left\Vert \left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = 1,$$
$$\left\Vert D\left(\begin{array}{c}1 \\ -1 \end{array}\right) \right\Vert^2 = \left\Vert \left(\begin{array}{c}1 \\ -e^{i... | 6 | https://mathoverflow.net/users/169474 | 440782 | 177,960 |
https://mathoverflow.net/questions/440758 | 1 | Given a probability space $(\Omega, \mathcal {A}, P)$, what are the minimum and maximum of the quantity
$$
P(A\_1 \cap \cdots \cap A\_n) - P(A\_1) \cdots P(A\_n)
$$
over $A\_1, \ldots, A\_n \in \mathcal {A}$, $n \geq 1$?
When $n = 2$, it is easily seen, from the Cauchy-Schwarz inequality (since
$$
P(A\_1 \cap A\_2... | https://mathoverflow.net/users/498800 | Gap to independence | The suggestions in the [comment](https://mathoverflow.net/questions/440758/gap-to-independence#comment1137007_440758) by usul are correct.
Indeed, let
\begin{equation}
p:=P(B),\quad B:=\bigcap\_1^n A\_j,\quad p\_j:=P(A\_j).
\end{equation}
We want to find the extreme values of
\begin{equation}
d:=p-\prod\_1^n p\_j.... | 2 | https://mathoverflow.net/users/36721 | 440790 | 177,962 |
https://mathoverflow.net/questions/440797 | 3 | Let $A\in M\_n(\mathbb{R})$ be a matrix and $\|\cdot\|$ be a norm on $\mathbb{C}^n$. When we look at the operator norm of $A$ with respect to $\|\cdot\|$ we can either consider the inclusion of $M\_n(\mathbb{R})$ in $M\_n(\mathbb{C})$ or the restriction of $\|\cdot\|$ to $\mathbb{R}^n$. Are these points of view equival... | https://mathoverflow.net/users/126690 | Real and complex operator norms | $\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$No. E.g., let
$$A=\begin{bmatrix}1&0\\0&0 \end{bmatrix}$$
and
$$\left\|\begin{bmatrix}z\_1\\z\_2 \end{bmatrix}\right\|=|z\_1|+|z\_1+iz\_2|.$$
Then the real norm of $A$ is $1$ and the complex norm of $A$ is $2$.
---
Indeed, if $x=\begin{bmatrix}x\_1\\x\_2\end{bmatr... | 2 | https://mathoverflow.net/users/36721 | 440802 | 177,965 |
https://mathoverflow.net/questions/440791 | 0 | I am trying to figure out if the infinite product $$\omega=\frac{5\sqrt{3}}{12}\prod\limits\_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits\_{\substack{p\equiv 2\pmod3 \\
p\ge 13}}\left(\frac{p}{p-1}\right)$$ is asymptotically equal to the infinite product $$c=\frac{5775}{2592\pi}\prod\... | https://mathoverflow.net/users/483436 | Asymptotic equivalence of two infinite products of prime numbers in residue classes | I don't know why you are restricting the products to $p \geq 13$ or where the factor $5\sqrt{3}/12$ is coming from. I am going to ignore that and discuss the following product over all primes $p$:
$$
C = \prod\_{p}\left(1- \frac{\chi(p)}{p-1}\right)
$$
where the terms in the product are in order of increasing $p$ and $... | 3 | https://mathoverflow.net/users/3272 | 440806 | 177,967 |
https://mathoverflow.net/questions/440804 | 3 | Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\setminus SL\_2(\mathcal{O}(X))$, where $B$ is the Borel subgroup of upper triangular matrices.
My question is if this func... | https://mathoverflow.net/users/174655 | Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf? | No, this isn't true. Take $X = \mathbb{P}^1$ with $U\_1$ and $U\_2$ being the standard affine cover of $\mathbb{P}^1$. Let the coordinate rings of $U\_1$ and $U\_2$ be $k[t]$ and $k[t^{-1}]$, so the coordinate ring of $U\_1 \cap U\_2$ is $k[t, t^{-1}]$.
Map $U\_1$ to $B \backslash \text{SL}\_2$ by $\begin{bmatrix} 1&... | 6 | https://mathoverflow.net/users/297 | 440807 | 177,968 |
https://mathoverflow.net/questions/440781 | 9 | $\newcommand{\A}{\mathcal{A}}\newcommand{\T}{\mathcal{T}}$The notions of *continuous map* of topological spaces and *measurable function* of measurable spaces are very similar:
* A map of topological spaces from $(X,\T\_X)$ to $(Y,\T\_Y)$ is **continuous** if for each $V\in\T\_Y$, we have $f^{-1}(V)\in\T\_X$.
* A map... | https://mathoverflow.net/users/130058 | Analogue of open/closed maps for measurable spaces | There are at least three different answers that can be given to this question, and in all three interpretations the answer essentially states that all maps are “open”, for the appropriate analogue of “open”.
In the first interpretation, we may ask: under what conditions is the image of a measurable set measurable?
Of... | 5 | https://mathoverflow.net/users/402 | 440814 | 177,971 |
https://mathoverflow.net/questions/440743 | 12 | Decades ago, Voevodsky constructed the six-functor formalism in motivic homotopy theory [[Ayoub's thesis](https://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF)]. This construction seems very technical, long and "hard".
Very recently [[Mann's thesis](https://arxiv.org/abs/2206.02022)], the six-functor formalism has been... | https://mathoverflow.net/users/173315 | Voevodsky's six functor formalism VS Lucas Mann's | There may be some confusion in this question about what exactly Voevodsky/Ayoub and Mann do, as they do very different things.
* Mann's thesis constructs a formalism of six operations in the setting of rigid-analytic geometry, using some $\infty$-categorical construction techniques developed for this purpose by Liu a... | 21 | https://mathoverflow.net/users/20233 | 440819 | 177,975 |
https://mathoverflow.net/questions/440754 | 1 | It is known that the closed unit ball of $L\_{\infty}(\mu)$ is weakly compact in $L\_{1}(\mu)$. A natural question arises in the case of spaces of Bochner integral functions:
Question. Let $X$ be a Banach space. In what cases the closed unit ball of $L\_{\infty}(\mu,X)$ is weakly compact in $L\_{1}(\mu,X)$ ?
I am n... | https://mathoverflow.net/users/41619 | Weak compactness of the closed unit ball of $L_{\infty}(\mu,X)$ in $L_{1}(\mu,X)$ | As Jochen commented, you need $X$ to be reflexive, and this is sufficient. It is enough to show that the unit ball of $L\_\infty(X)$ is closed in the reflexive space $L\_2(X)$. But the injection from $L\_\infty(X)$ into $L\_2(X)$ is the adjoint of the injection from $L\_2(X)$ into $L\_1(X)$, so this is automatic. (Reca... | 4 | https://mathoverflow.net/users/2554 | 440822 | 177,977 |
https://mathoverflow.net/questions/440824 | 5 | A classical result from Young in 1936 says that if $f\in C^\alpha$ and $g\in C^\beta$ with $\alpha+\beta>1$ then $\int f \, dg$ exists as a Riemann-Stieltjes integral.
However, I am interested in the converse. Clearly if the support of $f$ is disjoint with the support of $g$ then they can have as bad of analytic prop... | https://mathoverflow.net/users/479223 | Converse to Young's classical result on Riemann-Stieltjes integration | $\newcommand\al\alpha\newcommand\be\beta$Yes, of course. Take any $\al>0$ and $\be>0$ such that $\al+\be<1$. For $x\in[0,1]$, let
$$f(x):=\sum\_{j=1}^\infty 2^{-j}(x-r\_j)\_+^\al$$
and
$$g(x):=\sum\_{j=1}^\infty 2^{-j}(x-r\_j)\_+^\be,$$
where $(r\_1,r\_2,\dots)$ is an enumeration of the rational numbers in $[0,1)$ and ... | 6 | https://mathoverflow.net/users/36721 | 440829 | 177,979 |
https://mathoverflow.net/questions/440772 | 3 | Let $p \in [1, \infty)$. Let $\mathcal P\_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D\_p$ be the collection of all Borel measurable functions $f:\mathbb R^d \to \mathbb R\_{\ge 0}$ such that $\int\_{\mathbb R^d} f (x) \, \mathrm d x = 1$ and $\int\_... | https://mathoverflow.net/users/477203 | Is this map (from the space of probability densities to the Wasserstein space) Lipschitz? | $\newcommand{\vpi}{\varphi}\newcommand\R{\mathbb R}$
>
> **Claim 1:** The map $F$ is not Lipschitz if $p>1$.
>
>
>
>
> **Claim 2:** The map $F$ is $1$-Lipschitz if $p=1$: For all $f,g$ in $D\_p$,
> \begin{equation\*}
> W\_1(F(f),F(g))\le[f-g]\_1. \tag{1}\label{1}
> \end{equation\*}
>
>
>
*Proof of Clai... | 3 | https://mathoverflow.net/users/36721 | 440836 | 177,983 |
https://mathoverflow.net/questions/440206 | 13 | Recently I've been learning more about differential geometry, and I came upon the notion of a [diffeological space](https://en.wikipedia.org/wiki/Diffeology), which encompasses a number of already known extensions of smooth manifolds or related notions, like Banach and Frechét manifolds, complex and analytic manifolds,... | https://mathoverflow.net/users/130058 | Applications of diffeological spaces to ordinary differential geometry | As far as I am aware, you can find some applications of diffeology "merely" in differential geometry of manifolds in the following list (I am not sure this list is exhaustive):
* **The (internal) tangent space of the diffeomorphism group of a compact manifold is the space of its vector fields**, See [Hector G. Géomét... | 10 | https://mathoverflow.net/users/131015 | 440841 | 177,984 |
https://mathoverflow.net/questions/440783 | 2 | I am reading the research article *"The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets"* by Shishikura. The following two definitions are given without any examples in this paper. Therefore to understand the following definitions, I need examples of these two definitions:
Let $f$ be rational map... | https://mathoverflow.net/users/499397 | Examples of hyperbolic set and J-stable sets | Hyperbolic functions - for example, quadratic polynomials with an attractive periodic point - are examples of maps that are J-stable. The notion of J-stability arises from the famous article of Mañe, Sad and Sullivan. It is discussed in McMullen's book on renormalisation.
The trivial examples of hyperbolic sets are r... | 1 | https://mathoverflow.net/users/3651 | 440849 | 177,987 |
https://mathoverflow.net/questions/440726 | 7 | Let $\Sigma\_n$ be a genus $n$ surface, let $\mathcal{H}\_n$ be a genus $n$ handle body, and let $F\_n$ be a free group of rank $n$. Fix an identification of $\pi\_1(\mathcal{H}\_n)$ with $F\_n$. I know several proofs of the following result:
**Theorem**: Let $\phi\colon \pi\_1(\Sigma\_n) \rightarrow F\_n$ be a surje... | https://mathoverflow.net/users/499341 | Surjections from genus $n$ surface group to free group of rank $n$ | I think the first formal proof is due to Zieschang, Stallings probably knew it:
<https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=187195&sort=Newest&vfpref=html&r=101&mx-pid=161901>
There is a discussion at the end of the paper that refers to a correspondence with Lyndon. There it is mentioned ... | 4 | https://mathoverflow.net/users/69797 | 440851 | 177,988 |
https://mathoverflow.net/questions/440852 | 12 | Let $X$ be a smooth projective variety over a number field $K$. Then there are two cohomology groups we can attach to $X$: the algebraic de Rham cohomology group
$H^k\_{\text{dR}}(X/K), $
which is a finite dimensional $K$-vector space, and the singular cohomology group
$H^k\_{\text{sing}}(X(\mathbf{C}), \mathbf{Q}), $
... | https://mathoverflow.net/users/394740 | Comparing singular cohomology with algebraic de Rham cohomology | This is the subject of *periods*: recall that the de Rham isomorphism between $H^k\_{\text{dR}}(X/K) \otimes\_K \mathbf C = H^k\_{\text{dR}}(X\_{\mathbf C}/\mathbf C)$ and $H^k\_{\text{sing}}(X(\mathbf C),\mathbf Q) \otimes\_{\mathbf Q} \mathbf C = H^k\_{\text{sing}}(X(\mathbf C),\mathbf C)$ is defined by integrating $... | 16 | https://mathoverflow.net/users/82179 | 440854 | 177,989 |
https://mathoverflow.net/questions/440405 | 7 | On affine space, a sufficiently smooth continuous-time [Hamiltonian dynamic system](https://en.wikipedia.org/wiki/Hamiltonian_system) $\dot p = \nabla\_q H, \dot q = -\nabla\_p H$ conserves $H$, preserves the volume form (e.g. if we are looking for ergodicity), and lends itself to a limited physical intuition of positi... | https://mathoverflow.net/users/140723 | Hamiltonian-ization of a dynamic system | This construction is somewhat similar to the Martin-Siggia-Rose formalism for writing expectation values over solutions of a stochastic differential equation
$$
\dot{p}(t)=f(p,t)+\xi(p,t)
$$
with $\xi$ a Gaussian noise with correlation function
$$
\mathbb{E}[\xi(p,t)\xi(p',t')]=G(p,t,p',t')
$$
as a path integral
$$
\ma... | 0 | https://mathoverflow.net/users/45250 | 440855 | 177,990 |
https://mathoverflow.net/questions/440869 | 2 | Forgive me for asking what is undoubtedly an elementary question.
The Weil representation (defined below) of the metaplectic group $\operatorname{Mp}\_2(\mathbb{Z})$ can be defined in terms of the generators traditionally denoted $(T,1)$ and $(S,\sqrt{\tau})$. On a superficial level the images of these generators see... | https://mathoverflow.net/users/167073 | Does the Weil representation depend only on the discriminant group? | The Weil representation depends only on the discriminant form, as you already observed.
The thesis of Alfes-Neumann and the paper that you cite use various theta lifts, which do *not* depend only on the discriminant form. The lattice $L$ needs to be given, not because the Weil representation depends specifically on $... | 3 | https://mathoverflow.net/users/499465 | 440870 | 177,993 |
https://mathoverflow.net/questions/440848 | 7 | Simon Plouffe found experimentally a series for $\pi$ that can be written as
$$\frac{\pi}{24} = \sum\_{n=1}^\infty \frac{1}{n} \left( \frac{3}{e^{\pi n}-1} -\frac{4}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-1}\right) $$
A related series for $\log(2)$, easily found with lindep in PARI, is
$$\frac{\log(2)}{8} = \sum\_{n=1}... | https://mathoverflow.net/users/85758 | Proving a series for $\pi$ by Plouffe | It is immediate to show that
$S\_1(x)=-x/24-\log(\eta(ix/(2\pi)))$
and $T\_1(x)=S\_1(x)-2S\_1(2x)$, so all these formulas are immediate
consequences of the properties of the Dedekind $\eta$ function.
| 3 | https://mathoverflow.net/users/81776 | 440881 | 177,998 |
https://mathoverflow.net/questions/439672 | 3 | There are well-known results about nilpotent and solvable (=virtually nilpotent) Kähler groups coming from the work of (to name a few) Campana, Carlson-Toledo, Arapura-Nori, Delzant...
1. Are there any interesting known restrictions on *amenable* Kähler groups?
2. What about interesting known examples?
More general... | https://mathoverflow.net/users/105615 | Kahler groups with no non-abelian free groups? | For 1: beyond the case of nilpotent/solvable groups, I know of no restrictions on amenable kähler groups.
For 3: this is an interesting question but again I know of no restriction. This is probably due to our lack of ideas or to the lack of any suitable technology.
For 2: besides the 2-step nilpotent Kähler groups ... | 2 | https://mathoverflow.net/users/61960 | 440884 | 178,000 |
https://mathoverflow.net/questions/440756 | 1 | I'm interested in the following Cauchy problem for a linear diffusion equation
$$
\begin{cases}
{^C}\!D^{a}\_tu(t,x) = \sigma\Delta u(t,x),\\
u(0)=u\_0\in X.
\end{cases}
$$
where ${^C}\!D^{\sigma}\_t$
denotes the Caputo fractional derivative of order $a\in (0,1)$, and $X$ is a Banach space.
I wonder if there are a... | https://mathoverflow.net/users/98050 | Fractional reaction-diffusion with Caputo derivative | In $L^2(\Omega)$, see
>
> K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl.,382 (2011), 426–447.
>
>
>
For a general theory in Banach spaces, see
>
> E. G. Bajlekova, Fractional evolutio... | 1 | https://mathoverflow.net/users/124904 | 440888 | 178,002 |
https://mathoverflow.net/questions/440411 | 1 | I do not understand the following proof in the paper [Abelian varieties](http://van-der-geer.nl/%7Egerard/AV.pdf) by Edixhoven, van der Geert, and Moonen:
**(1.12) Rigidity Lemma.** Let $X$, $Y$ and $Z$ be algebraic varieties over a field $k$. Suppose that $X$ is complete. If $f: X \times Y \to Z$ is a morphism with ... | https://mathoverflow.net/users/108274 | Reduction step to $k=\bar{k}$ in the proof of rigidity lemma | One way to argue is as follows: given a morphism $f \colon X \times Y \to Z$, consider its graph $\Gamma\_f \subseteq X \times Y \times Z$. Let $W$ be the scheme-theoretic image of $\Gamma\_f$ under the projection $\pi\_{Y \times Z} \colon X \times Y \times Z \to Y \times Z$, and let $X \times W \subseteq X \times Y \t... | 1 | https://mathoverflow.net/users/82179 | 440890 | 178,003 |
https://mathoverflow.net/questions/440892 | 3 | I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the boundary such that there is a basis of neighborhoods of $x$ in $B$ whose intersections with $B'$ are connected.
This seems... | https://mathoverflow.net/users/68565 | Boundaries of subsets of simply-connected domains | It seems if you take $B=\mathbb{R}^2$ and $B'$ the complement of the closure of $\Big\{\big(x,\sin\big(\frac{1}{x}\big)\big);x\in(0,\infty)\Big\}$ this is a counterexample. (Added bonus: $B'$ is also simply connected)
| 9 | https://mathoverflow.net/users/172802 | 440893 | 178,004 |
https://mathoverflow.net/questions/440897 | 5 | Let $M$ be a compact complex manifold of dimension three.
Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that
$$ h^0(M, \mathcal O\_M(nD)) > C n^3 $$
for sufficiently large $n \in \mathbb N$.
Assume that $M$ has a big divisor. My question is
>
> Is $M$ projective?
>
>
>
If th... | https://mathoverflow.net/users/69559 | Big divisors and projectivity | The existence of a big divisor implies that $M$ is Moichizon so the "only obstruction" to projectivity is that $M$ may not be Kahler. In fact, this can happen even for nonprojective varieties.
Hironaka's construction gives for any projective $3$-fold $X$ equipped with curves $C, C' \subset X$ intersecting transversal... | 5 | https://mathoverflow.net/users/154157 | 440900 | 178,007 |
https://mathoverflow.net/questions/440877 | 2 | $\begin{align}x \text { is predicatively} &\text{ definable } \iff \exists x\_1,..,x\_n \exists \varphi:\\ & \rho(x\_1) < \rho(x) ,.., \rho(x\_n) < \rho(x) \ \land \\&\forall y \, (y \in x \iff V\_{\rho(x)} \models \varphi(y,x\_1,..,x\_n))\end{align} $
Where $\rho(x)$, the rank of $x$, is defined as the ordinal index... | https://mathoverflow.net/users/95347 | Which fragment of ZF does the class of all hereditarily predicatively definable sets capture? | HPD satisfy extensionality and regularity trivially.
It satisfy Pairing, as if $x,y\in HPD$ we can explicitly write down the definition of $(x,y)$ using only $x,y$ as parameters (and they have strictly smaller rank).
It satisfy infinity (it has all of the Ordinals and it computes Ordinals correctly).
It has power... | 2 | https://mathoverflow.net/users/113405 | 440906 | 178,009 |
https://mathoverflow.net/questions/440905 | 2 | Let us assume $<$ is some class relation without minimal elements, meaning $\forall a, \exists b, b< a$. This means that for every $n\in\omega$, one can build a decreasing function $f$ with domain $n+1$, meaning that $f(k+1)<f(k)$ for every $k\in n$. This is a very simple inductive argument.
My question is, how can w... | https://mathoverflow.net/users/147705 | Infinite decreasing sequence for class relation without minimal elements | Why do you need a global choice function, though? Yes, it's neater, but unnecessary.
Define, for each $x\in V\_\alpha$ an ordinal $\alpha\_x$ such that it is the minimal for which $V\_{\alpha\_x}$ contains witnesses that $x$ is not minimal in $<$. Now let $\alpha\_0$ be defined for witnesses that $\varnothing$ is not... | 4 | https://mathoverflow.net/users/7206 | 440907 | 178,010 |
https://mathoverflow.net/questions/440898 | 5 | In this [paper](https://statweb.stanford.edu/%7Ecgates/PERSI/papers/zabell82.pdf) by Diaconis and Zabell from 1982, Theorem 2.1 and the remark after essentially stated that
>
> Given two probability measures $P$ and $Q$ on the same probability space $\Omega$. If $Q\ll P$ and their
> RN-derivative $\frac{dQ}{dP} \in... | https://mathoverflow.net/users/49551 | Radon-Nikodym derivative and conditional probability | Theorem 2.1 in the quoted paper of Diaconis and Zabell actually states that boundedness of the ratios
$Q(\omega)/P(\omega)$ is a **necessary and sufficient condition** for obtaining $Q$ from $P$ by conditioning in the discrete case. This is true in the general case as well.
Indeed, **conditioning** here means that th... | 2 | https://mathoverflow.net/users/8588 | 440920 | 178,013 |
https://mathoverflow.net/questions/440914 | 2 | For $A\subseteq \omega$ we let the *lower and upper density* be defined as $$\mu^-(A):= \lim\inf\_{n\to\infty}\frac{|A\cap n|}{n+1} \text{ and } \mu^+(A):= \lim\sup\_{n\to\infty}\frac{|A\cap n|}{n+1}$$ respectively.
Let $s:\omega\to\{0,1\}$ be an infinite binary string, $n\in\omega\setminus\{0\}$ a positive integer a... | https://mathoverflow.net/users/8628 | Strongly uniform infinite binary strings | Yes, the Champernowne binary string is [normal](https://en.wikipedia.org/wiki/Champernowne_constant#Properties) (or strongly uniform, in your terms); see also [here](https://en.wikipedia.org/wiki/Normal_number#Properties_and_examples).
[Almost all infinite binary string are normal](https://en.wikipedia.org/wiki/Norma... | 6 | https://mathoverflow.net/users/36721 | 440922 | 178,014 |
https://mathoverflow.net/questions/440903 | 8 | Let $M\_1,M\_2 \subset \mathbf{S}^n$ be two smoothly embedded, connected hypersurfaces of the round sphere, which are realized as the zero sets of two homogeneous polynomials $P\_1,P\_2$ in $\mathbf{R}^{n+1}$:
\begin{equation}
M\_i = \{ P\_i = 0 \} \cap \mathbf{S}^n.
\end{equation}
>
> Is there an upper bound for t... | https://mathoverflow.net/users/103792 | Intersection of two hypersurfaces via... Bezout's theorem? | In Proposition 4.13 of Coste’s introduction to semi-algebraic geometry, a bound of $d(2d-1)^{s+k-1}$ is given for the number of connected components of a system of $s$ real polynomial equations and inequations of degree at most $d\ge 2$ in $k$ variables. In your case, $k=n+1$ and $s=3$.
| 15 | https://mathoverflow.net/users/3404 | 440924 | 178,015 |
https://mathoverflow.net/questions/440954 | 3 | Let $\sigma=(\sigma\_1,...,\sigma\_m)$ be i.i.d. uniform binary 0-1 valued variables.
I'm trying to figure out what is the order of $E[||\sigma||\_p]$ with respect to $m$.
Jensen's inequality gives an upper bound of $(m/2)^{1/p}$, but how do I get a lower bound? (I'm hoping for a lower bound of the same order.)
| https://mathoverflow.net/users/61472 | Expected p-norm of binary vector | $\newcommand\si\sigma$
For any real $p>0$,
$$m^{-1/p}\,E\|\si\|\_p=E\Big(\frac1m\sum\_{j=1}^m\si\_j\Big)^{1/p}.$$
By the law of large numbers, $\frac1m\sum\_{j=1}^m\si\_j\to2^{-1}$ in probability (as $m\to\infty$). So, by Fatou's lemma,
$$\liminf\_{m\to\infty}m^{-1/p}\,E\|\si\|\_p\ge2^{-1/p}$$
and hence
$$E\|\si\|\_p\g... | 3 | https://mathoverflow.net/users/36721 | 440956 | 178,025 |
https://mathoverflow.net/questions/440945 | 2 | Let $X$ be a finite set and let $\emptyset\neq \mathcal{H}\subseteq \{ 0,1 \}^{\mathcal{X}}$. Let $\{(X\_n,L\_n)\}\_{n=1}^N$ be i.i.d. random variables on $X\times \{0,1\}$ with law $\mathbb{P}$. Without knowing more of $\mathcal{H}$, what is the best risk-bounds available via VC-dimension or Rademacher-complexity theo... | https://mathoverflow.net/users/491352 | VC-based risk bounds for classifiers on finite set | $\newcommand\HH{\mathcal H}\newcommand\ep\varepsilon$Let
\begin{equation}
D\_N:=\sup\_{h\in\HH}\Big(E\,I(h(X)=L) - \frac1{N}\sum\_{n=1}^N\, I(h(X\_n)=L\_n)\Big).
\end{equation}
It was shown in (the proof of) [Long's main result, 1999](https://link.springer.com/article/10.1023/A:1007666507971) (which he ascribed to T... | 3 | https://mathoverflow.net/users/36721 | 440959 | 178,027 |
https://mathoverflow.net/questions/440939 | 2 | I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this topic.
Let me briefly sketch the setting: $X^n \subset \mathbb{P}^N$ is a projective variety of dimension $n$ (we can ass... | https://mathoverflow.net/users/146431 | Linear system giving the projective embedding of the tangential variety | Consider the Euler exact sequence
$$0\rightarrow \mathcal{O}\_{\mathbb{P}}\rightarrow \mathcal{O}\_{\mathbb{P}}(1)^{N+1}\rightarrow \mathcal{T}\_{\mathbb{P}}\rightarrow 0\,.$$Restrict to $X$, and pull-back by the inclusion $ \mathcal{T}\_X\subset \mathcal{T}\_{\mathbb{P}|X}$. This gives an exact sequence
$$0\rightarrow... | 1 | https://mathoverflow.net/users/40297 | 440966 | 178,028 |
https://mathoverflow.net/questions/440909 | 1 | I asked a simillar question with the weaker restriction:
[On the equation $a^4+b^4+c^4=2d^4$ in positive integers $a\lt b\lt c$ such that $a+b\ne c$](https://mathoverflow.net/questions/438196/on-the-equation-a4b4c4-2d4-in-positive-integers-a-lt-b-lt-c-such-that)
.
---
I couldn't find any solution to this equa... | https://mathoverflow.net/users/nan | On the equation $a^4+b^4+c^4=2d^4$ in natural numbers with $a<b<c<d$ | If we take a point $3Q(m)$ then we get a solution as follows.
For more details, please see [$a^4+b^4+c^4=2d^4$](https://mathoverflow.net/questions/438196/on-the-equation-a4b4c4-2d4-in-positive-integers-a-lt-b-lt-c-such-that).
```
a = 100954906225546184690686373445232988785377384105455647295649619
... | 4 | https://mathoverflow.net/users/150249 | 440969 | 178,030 |
https://mathoverflow.net/questions/440913 | 3 | **Problem:** Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e\_{(0,\infty)}(m)$ be the spectral projection of $m$ on $(0,\infty)$. How can we show that $e\_{(0,\infty)}(m)m^{-1/2}$ is $\tau$-measurable?
I got stuck with this pr... | https://mathoverflow.net/users/477204 | $\tau$-measurable operator | I think that this is simply not true. Take $M = \ell^\infty(\mathbb{N})$ with the semifinite trace $\tau(F) = \sum\_n F(n)$. When $p \in M$ is a nonzero projection, we have $\tau(p) \geq 1$. So the only $\tau$-measurable operators are the elements of $M$ itself. Taking $m \in M$ given by $m(k) = 1/k$, the element $m^{-... | 3 | https://mathoverflow.net/users/159170 | 440990 | 178,038 |
https://mathoverflow.net/questions/440992 | 1 | I am working with Ash's Probability and Measure Theory, Second Edition, specifically on theorem 6.2.1 (some convergence criteria for a random variable sequence).
We are given a sequence $(X\_i)\_{i \ge 1}$of random variables, and:
1. $\sum\_{i=1}^\infty \text{Var} X\_i$ converges.
2. Wlog, $\forall i.\:E(X\_i)=0$, ... | https://mathoverflow.net/users/499564 | Equivalence of unions in probability theory | Let $\epsilon>0$ and $n \ge 1$. Then
$$\bigcap\_{k=1}^\infty\{|S\_{n+k}-S\_n|<\epsilon = \bigcap\_{j \geq n}\{|S\_j-S\_n|<\epsilon\} \subset \bigcap\_{j,k\geq n}\{|S\_j-S\_k|<2\epsilon\} .$$
Hence, taking complements,
$$\bigcup\_{j,k\geq n}\{|S\_j-S\_k|\geq 2\epsilon\} \subset \bigcup\_{k=1}^\infty\{|S\_{n+k}-S\_n|\geq... | 3 | https://mathoverflow.net/users/169474 | 440993 | 178,040 |
https://mathoverflow.net/questions/440988 | 1 | Let $z>0$ be fixed. Consider the function $p\_a: \mathbb R^2\_+\to\mathbb R\_+$ given as
$$
p\_a(t,x):=\frac{1}{\sqrt{2\pi N\_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N\_a(t)}\right)-\exp\left(-\frac{(x+z)^2}{2N\_a(t)}\right)\right],
$$
where $N\_a:\mathbb R\_+\to\mathbb R\_+$ is defined by
$$N\_a(t):=\int\_0^t\frac{ds... | https://mathoverflow.net/users/493556 | Numerical solution to some functional equation | $\newcommand\erf{\operatorname{erf}}\newcommand\R{\mathbb R}$The functional equation in question is
\begin{equation\*}
a=F(a) \tag{1}\label{1}
\end{equation\*}
on $(0,\infty)$, where $a$ is in the closed convex set, say $A$, of all nonincreasing functions from $(0,\infty)$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|\_\... | 1 | https://mathoverflow.net/users/36721 | 441004 | 178,042 |
https://mathoverflow.net/questions/441005 | 3 | Let $V(\pi\_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}\_n$, for $n > 2$. A basic fact is the tensor product $V(\pi\_1) \otimes V(\pi\_1)$ decomposes as
$$
V(\pi\_1) \otimes V(\pi\_1) \simeq V\_{2\pi\_1} \oplus V\_{\pi\_2}.
$$
For higher tensor powers
$$
V(\pi\_1)^{\otimes k}
$$
does the... | https://mathoverflow.net/users/499575 | Decomposition of tensor powers of the vector representation of $\frak{sl}_n$ | The multiplicity of the Young diagram of shape $\lambda$ is the number of standard Young tableaux of shape $\lambda$, which can be computed with the [hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula). This can be deduced from [Schur-Weyl duality](https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_du... | 7 | https://mathoverflow.net/users/297 | 441006 | 178,043 |
https://mathoverflow.net/questions/440853 | 2 | It could be a naive question. Probably, it is not true.
However, this question makes sense in the setting of function spaces.
For example, for $L\_\infty (0,1)$, we have $L\_p(0,1)\supset L\_\infty (0,1)$
and $L\_p(0,1)$ is reflexive when $p>1$.
On the other hand, $L\_1(0,1)$ itself is weakly sequentially complete and ... | https://mathoverflow.net/users/91769 | For a Banach space $X$, can we find a reflexive (or weakly sequentially complete) space $Y$ such that $X\subset Y$? | Let $K$ be a compact scattered space and $X=C(K)$ the space of continuous functions on $K$. We want to show that there is no injective bounded linear $T:X\to Y$ into a weakly sequentially complete (w.s.c.) Banach space $Y$ if $K$ has *large*(see below) cardinality.
Let $T:X\to Y$ be as above. $C(K)$ has Pelczynski pr... | 5 | https://mathoverflow.net/users/164350 | 441010 | 178,044 |
https://mathoverflow.net/questions/441012 | 2 | I am reading this paper <https://arxiv.org/abs/1608.04797>
Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G\_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh\_P$ be the stack of coherent sheaves parameterized by $X$ and with Hilbert polynomial $P$.
The paper says : groupoid of maps... | https://mathoverflow.net/users/111666 | Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves | Maps $\mathrm{B} G \to \mathcal{X}$ correspond to an object of $\mathcal{X}(k)$ along with a $G$-action. Indeed, the map $\* \to \mathrm{B} G \to \mathcal{X}$ selects an object $x$ and for each test scheme $T$ we get a natural map,
$$ \{ T \text{-torsors} \} \to \mathcal{X}(T) $$
so that the trivial torsor maps to ... | 2 | https://mathoverflow.net/users/154157 | 441016 | 178,046 |
https://mathoverflow.net/questions/441002 | 0 | Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. Let $\textbf{n}$ denote the normal to the surface $S$. Does the surface integral over $S$ ... | https://mathoverflow.net/users/353746 | Does surface integral preserve the curl operation? | $\renewcommand\r{\mathbf r}\newcommand\n{\mathbf n}\newcommand\F{\mathbf F}\newcommand\0{\mathbf 0}\newcommand\curl{\operatorname{\mathbf{curl}}}$No. E.g., let $S$ be the unit sphere and let $\F:=(1,0,0)$, so that $\curl\F=\0$ and hence the right-hand side of the identity in question is $\0$.
On the other hand, the l... | 1 | https://mathoverflow.net/users/36721 | 441018 | 178,047 |
https://mathoverflow.net/questions/441008 | -2 | Consider $$\left\|2\sum\_{i<j}L\_{ij}+4\sum\_i \operatorname{diag}e\_i \right\|,$$ where
(1) $L\_{ij}=\operatorname{diag}e\_i+\operatorname{diag}e\_j-e\_ie\_j^T-e\_je\_i^T$
(2) $e\_i$ denotes $n$-by-$1$ vector with only $i$-th element equals to $1$ and others are $0$
(3) $\operatorname{diag}e\_i$ is a $n$-by-$n$ ... | https://mathoverflow.net/users/494410 | How to compute the spectral norm of this matrix | Making through the terrible notations, we see that
$$M\_n:=2\sum\_{i<j}L\_{ij}+4\sum\_{i}\text{diag}\,e\_i=(2n+4)I\_n-2\,1\_n1\_n^\top,$$
where $I\_n$ is the $n\times n$ identity matrix and $1\_n$ is the $n\times1$ column matrix of $1$'s.
The eigenvectors of the symmetric matrix $M\_n$ are the nonzero multiples of $1... | 2 | https://mathoverflow.net/users/36721 | 441021 | 178,048 |
https://mathoverflow.net/questions/440997 | 3 | Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for braid index 3 is 4, the smallest crossing number for braid index 4 is 6, the smallest crossing number for braid index 5... | https://mathoverflow.net/users/6043 | Bounds for the crossing number in terms of the braid index? | As in the comment of dvitek, as for the relation of the braid index and the crossing number, Ohyama proved $c(L) \geq 2b(L)-2$ in [On the Minimal Crossing Number and the Braid Index of Links](https://doi.org/10.4153/CJM-1993-007-x).
Here I add three additional information.
**(a)** A simpler proof of $c(L) \geq 2b(L... | 3 | https://mathoverflow.net/users/193957 | 441026 | 178,049 |
https://mathoverflow.net/questions/441040 | 3 | Following the terminology of
*Drozd, Yuriy A.*, [**Derived tame and derived wild algebras**](https://doi.org/10.48550/arXiv.math/0310171), Algebra Discrete Math. 2004, No. 1, 57-74 (2004). [ZBL1067.16028](https://zbmath.org/1067.16028).
let $A$ and $R$ be algebras over a field $k$. A strict family of $A$-complexes ... | https://mathoverflow.net/users/157483 | Explicit proof that algebra is derived wild | A few such examples are constructed in
*Bekkert, Viktor; Drozd, Yuriy; Futorny, Vyacheslav*, [**Derived tame local and two-point algebras**](https://doi.org/10.1016/j.jalgebra.2009.05.023), J. Algebra 322, No. 7, 2433-2448 (2009). [ZBL1191.16017](https://zbmath.org/1191.16017).
| 0 | https://mathoverflow.net/users/157483 | 441044 | 178,052 |
https://mathoverflow.net/questions/441043 | 6 | Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A **homogeneous vector bundle** over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times\_{\rho} V \to G/K, ~~~ (g,v) \mapsto [g],
$$
where $(V,\rho)$ is a $K$-module. Can there exist line bundles over $G/K... | https://mathoverflow.net/users/499575 | Non-homogeneous line bundles over a homogeneous space | Yes. This happens whenever $G$ admits nontrivial vector bundles $E$ which can be equipped with an equivariant structure for the $K$-action. Then $E$ descends in the same way to $G \times\_{\rho} E \to G / K$. The homogenous ones are exactly the case that $E$ is a trivial $G$-bundle.
A trivial example is $G = S^1$ and... | 9 | https://mathoverflow.net/users/154157 | 441045 | 178,053 |
https://mathoverflow.net/questions/441052 | 1 | Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and correlation are known, $\mathbf{C}\_{\mathbf{X}}$, $\mathbf{C}\_{\mathbf{y}}$, and $\mathbf{C}\_{\mathbf{Xy}}$. The question ... | https://mathoverflow.net/users/496172 | Bound for expectation of random matrix | No, of course not.
Indeed, consider a simplest case when $m=n=1$, and $X:=\mathbf X$ and $y:=\mathbf y$ are iid standard normal random variables. Then
$$E(\mathbf{X}^{\mathrm{H}} \mathbf{X})^{-1} \mathbf{X}^{\mathrm{H}} \mathbf{y}=E\frac yX$$
does not even exist (because here $\frac yX$ has [the standard Cauchy distr... | 2 | https://mathoverflow.net/users/36721 | 441053 | 178,055 |
https://mathoverflow.net/questions/441062 | 1 | Let $z>0$ be fixed and $A$ be the set of non-increasing functions from $\mathbb R\_+$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|\_\infty$. Define by $F$ the operator on $A$ by
\begin{equation\*}
F(a)(t):=\text{Erf}\left(\frac{z}{\sqrt{2N\_a(t)}} \right),\quad \forall t\ge 0,
\end{equation\*}
where $\text{Erf}$ is... | https://mathoverflow.net/users/493556 | Implementable numerical scheme for the equation $a=\text{Erf}\big(z/\sqrt{2N_{a}}\big)$ | $\newcommand\erf{\operatorname{erf}}\newcommand\R{\mathbb R}\newcommand{\de}{\delta}\newcommand\ep\epsilon
$In the [previous answer](https://mathoverflow.net/questions/440988/numerical-solution-to-some-functional-equation), it was shown that the operator $F$ on $A$ is $r$-Lipschitz for a certain universal constant $r\i... | 1 | https://mathoverflow.net/users/36721 | 441069 | 178,058 |
https://mathoverflow.net/questions/441064 | 2 | Let $n=2m$. What is the order of the following permutation $\sigma$?
$$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$
| https://mathoverflow.net/users/84390 | If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$ | By adding a fixed point at $0$ (which preserves the order), the permutation $\sigma$ considered is just the multiplication by $2$ modulo $2m+1$. Thus, for $k \ge 0$, $\sigma^k$ is the identity map if and only if it fixes $1$, namely if and only if $2m+1$ divides $2^k-1$.
Hence, the order of $\sigma$ is the order of $... | 8 | https://mathoverflow.net/users/169474 | 441073 | 178,060 |
https://mathoverflow.net/questions/218855 | 9 | All sheaf topoi have [W-types](http://ncatlab.org/nlab/show/W-type) and in fact there's an explicit construction given by [Benno van den Berg & Ieke Moerdijk](http://www.phil.cmu.edu/projects/ast/Papers/vdBM_Wtypes.pdf), but the construction is quite involved.
I would like to know whether the inverse image part of a ... | https://mathoverflow.net/users/10875 | W-types and inverse image functor | We have a canonical map in one direction, namely $f^\*(W(p)) \to W(f^\*(p))$, but this map can fail to be an isomorphism. Here is an explicit counterexample.
Let $X$ be the set of countably-brancing trees, so $X = W(p)$ where $p : \mathbb{N} \to \mathbf{2}$ is a constant map. A tree is either a leaf or a node with co... | 2 | https://mathoverflow.net/users/31233 | 441081 | 178,062 |
https://mathoverflow.net/questions/441080 | 3 | Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. Let $\textbf{n}$ denote the normal to the surface $S$. It is well known using Helmhotz dec... | https://mathoverflow.net/users/353746 | Are all Helmholtz decompositions related? | **Q:** *How are two Helmholtz decompositions related?*
**A:** The scalar fields differ by a *harmonic function*.
Starting from a first decomposition $\sigma\_1,\Gamma\_1$, you can construct a second one by adding to $\sigma\_1$ a harmonic function $h$,
$$\sigma\_2=\sigma\_1+h,\;\;\text{with}\;\;\nabla^2 h=0.$$
Then... | 7 | https://mathoverflow.net/users/11260 | 441083 | 178,063 |
https://mathoverflow.net/questions/441084 | 1 | Let $f: X \to Y$ be a map of CW-complexes with continuous maps as morphisms.
How would one show that homotopy pullback $\mathcal D/Y → \mathcal D/X$ is right adjoint?
Here $\mathcal D$ is the derived category (edit, I mean the category of CW-complexes with continuous maps).
| https://mathoverflow.net/users/30211 | Homotopy pullback is right adjoint in the derived category | There is no such functor $\mathcal D/Y\to \mathcal D/X$. It's clear what is meant to be on objects, but it is not well-defined on morphisms.
Let $f$ be the inclusion of a point $p$ into a circle $C$. Let $g$ and $h$ be the inclusions of two points, say $q$ and $r$, into $C$. Regard $g$ and $h$ as objects of $\mathcal... | 6 | https://mathoverflow.net/users/6666 | 441088 | 178,064 |
https://mathoverflow.net/questions/441082 | 1 | Before stating the question, I would like to first use an example for the type of formulation that I'm interested in.
Suppose we consider the continuity equation $\partial\_t \rho + \mathrm{div}( \rho v ) = 0$ with boundary conditions $\rho(0) = \rho\_0$ and $\rho(1) = \rho\_1$.
Now, if we were to test this equation ... | https://mathoverflow.net/users/170491 | Are there PDEs in which Hessian appears in the weak formulation | Integration by parts of $\nabla^{2} f$ against a symmetric field $\sigma=(\sigma\_{ij})$ yields eventually the formula,
$$
\int\_{\Omega}(\nabla^{2}f,\sigma)dx=\int\_{\Omega}f\,\nabla^{\*}\nabla^{\*}\sigma\,dx+\int\_{\partial\Omega}\partial\_nf\,(\sigma,n\otimes n)\,dS+\int\_{\partial\Omega}f\,T\sigma\,dS
$$
where $(... | 1 | https://mathoverflow.net/users/144247 | 441097 | 178,068 |
https://mathoverflow.net/questions/441099 | 3 | Let $X$ be a locally compact Hausdorff space. Denote $C\_c(X)$ and $C\_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation theorem, the dual space $C\_0(X)^\*$ is isomorphic to the space of finite Radon measures $M(X)$. Since $C\_c(X)$ is dense in... | https://mathoverflow.net/users/49284 | Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures? | The dual unit ball of a normed space $E$ is weakly compact à la Alaoglu and hence there is no strictly coarser Hausdorff topology. Applying this to $E=C\_0(X)$ you get that the topologies $\sigma(M(X),C\_0(X))$ and $\sigma(M(X),C\_c(X))$ coincide on all bounded sets of $M(X)$.
| 4 | https://mathoverflow.net/users/21051 | 441100 | 178,069 |
https://mathoverflow.net/questions/440994 | 12 | Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V\_\kappa$ is a partial order and $p \in P$. We define the ordering $(Q,q) \leq (P,p)$ to hold when $P$ is a regular suborder of $Q$, and $q \leq\_Q p$.
The Amoeba's body grows lar... | https://mathoverflow.net/users/11145 | Amoeba collapse | $\kappa$ is preserved, and moreover all reals are added by the small generics.
Let $(P\_0,p\_0)$ be a condition and let $\sigma$ be a name for a real. First, enumerate the elements of $P\_0$ below $p\_0$ as $\langle p\_i : 0<i< \lambda \rangle$. Let $(Q\_1,q\_1) \leq (P\_0,p\_1)$ decide $\sigma(0)$. Then let $(Q\_2,q... | 6 | https://mathoverflow.net/users/11145 | 441103 | 178,071 |
https://mathoverflow.net/questions/441102 | 2 | I have a really stupid question that I don't seem to know the answer to and have been too embarassed to ask. In some number theory papers, I encounter sums of the form $$\sum\_{\substack{{x \asymp B}\\P(x) =0}}1$$ to count solutions "of size B" to some quadratic form $P$ (often $x \in \mathbb{Z}^n$ but let us ignore th... | https://mathoverflow.net/users/nan | What does it mean to have a number of size $B$? | This is a good question, and the answer is that writing $x\asymp B$ under a sum **is sloppy notation without further clarification**. It can mean that $c\_1 B<x<c\_2 B$ for any fixed constants $c\_1$ and $c\_2$ (and then the sum will depend on those constants), or it can mean the same for some constants $c\_1$ and $c\_... | 6 | https://mathoverflow.net/users/11919 | 441104 | 178,072 |
https://mathoverflow.net/questions/441122 | 4 | In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich ([book](https://link.springer.com/book/10.1007/978-3-662-03644-0), page 104) it is asserted that to explicit a transcendence measure of a complex number $w$, it is sufficient to prove that there is an infinite sequence of polynomia... | https://mathoverflow.net/users/33128 | Transcendence measure: of $\ln(a/b)$ | Warning: this is for irrationaity measure, not transcendence measure.
Let $a/b$ be an approximation of $w$ such that $|w-a/b|=b^{-\kappa}$. Then $$P\_m(a/b)=P\_m(w)+(w-a/b)P\_m'(\theta)$$
for certain $\theta$ between $a/b$ and $w$. Note that $P\_m(a/b)$ is either 0 or at least $b^{-d}$ in absolute value, where $\deg ... | 1 | https://mathoverflow.net/users/4312 | 441123 | 178,077 |
https://mathoverflow.net/questions/441114 | 6 | According to some authors, it is built in *A.A.Beilinson "Higher regulator of modular curves"* a class $\mathbf{Eis}\_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a Schwartz function over the finite adeles. Since the modular curve is only quasi-projective, I assume it is mixed-motivic cohomolo... | https://mathoverflow.net/users/169282 | References for the construction of Beilinson's motivic Eisenstein classes | The Eisenstein classes $\mathrm{Eis}^k\_\phi$ live in the motivic cohomology $H^{k+1}\_{\mathcal{M}}(E^k, \mathbf{Q}(k+1))$, where $E \to Y(N)$ is the universal elliptic curve over the open modular curve $Y(N)$. For example the classes $\mathrm{Eis}^0\_\phi$ are the Siegel modular units. At the time, Beilinson defined ... | 5 | https://mathoverflow.net/users/6506 | 441126 | 178,080 |
https://mathoverflow.net/questions/441128 | 3 | This posting is related to a recent question asked in [MSE](https://math.stackexchange.com/q/4640948/121671): Suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\nu(X)=\mu(X)$, and $\nu\ll\mu$, is there a measurable map $T:(X,\mathscr{B})\rightarrow(X,\maths... | https://mathoverflow.net/users/78591 | A type of coupling problem I | $\newcommand\de\delta$The answer to your first question is no.
E.g., let $\mu:=2\de\_{1/3}+2\de\_{2/3}$ and $\nu:=\de\_{1/3}+3\de\_{2/3}$, where $\de\_a$ is the Dirac measure with support $\{a\}$. Then all the conditions imposed on $\mu$ and $\nu$ hold. However, $\nu(\{1/3\})=1$ is an odd integer, whereas all the val... | 2 | https://mathoverflow.net/users/36721 | 441132 | 178,081 |
https://mathoverflow.net/questions/441096 | -4 | **Nested Selection:** For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and each element of $x$ has a proper superset of it in $y$" or "$x$ is a set of proper supersets of elements of $y$ and each el... | https://mathoverflow.net/users/95347 | Is Nested Selection equivalent to AC? | For a counterexample to Nested Selection, construct sets $A\_\alpha\subseteq\omega$ ($\alpha\lt\omega\_1$) so that, for $\alpha\lt\beta\lt\omega\_1$, we have $|A\_\alpha\setminus A\_\beta|\lt\aleph\_0=|A\_\beta\setminus A\_\alpha|$. Let $\mathcal S\_\alpha=\{X\subseteq\omega:|X\triangle A\_\alpha|\lt\aleph\_0\}$ and le... | 3 | https://mathoverflow.net/users/43266 | 441138 | 178,084 |
https://mathoverflow.net/questions/440984 | 6 | In [1] Fedorcuk, using diamond, proved that there is a hereditarily separable compact space of cardinality $2^{2^\omega}$.To my best knowledge, Kunen created a humanly digestible proof, but he has not published it (and he passed away). Can I find Kunen's proof somewhere? Or do you know any other proof of the theorem of... | https://mathoverflow.net/users/71011 | How to construct large, hereditarily separable compact spaces? | I was informed that the following joint paper of Dzamoja and Kunen contains the proof I was looking for.
[Dz̆amonja, Mirna; Kunen, Kenneth Measures on compact HS spaces.
Fund. Math. 143 (1993), no. 1, 41–54](http://matwbn.icm.edu.pl/ksiazki/fm/fm143/fm14314.pdf).
| 3 | https://mathoverflow.net/users/71011 | 441158 | 178,088 |
https://mathoverflow.net/questions/441161 | 3 | Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$.
Assume now that $K'$ is another compact Lie group that admits a homomorphic embedding $i:K\hookrightarrow K'$ verifying the condition t... | https://mathoverflow.net/users/409881 | Sub-coroot systems | This can fail. Consider the natural embedding $\operatorname{Sp}\_4(\mathbb C) \to \operatorname{SL}\_4(\mathbb C)$, where $\operatorname{Sp}\_4$ is taken with respect to the symplectic form $(x, y) \mapsto x\_1 y\_4 + x\_2 y\_3 - x\_3 x\_2 - x\_4 y\_1$. (This is a map of complex Lie groups, but it carries maximal comp... | 6 | https://mathoverflow.net/users/2383 | 441172 | 178,091 |
https://mathoverflow.net/questions/441168 | 6 | Given a simplicial set $X\_\bullet$, define its **powerset simplicial set** $\mathcal{P}\_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X\_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\mathsf{Set},$$
where $\mathcal{P}$ is the covariant powerset functor.
**How homotopically well-behaved is the power... | https://mathoverflow.net/users/130058 | Homotopical properties of powersets of simplicial sets | The first question has a negative answer, given by the simplicial set $\def\Exi{{\sf Ex}^{\sf\infty}}X=\Exi Y$, where $Y$ is a simplicial set generated by vertices $a,b,b',c,c',d,d'$, 1-simplices $ab,ab',ac,ac',ad,ad'$, 2-simplices $abc,acd,abd,ab'c',ac'd',ab'd',abc'$, and 3-simplices $abcd,ab'c'd'$.
We specify a 3-h... | 2 | https://mathoverflow.net/users/402 | 441182 | 178,095 |
https://mathoverflow.net/questions/441180 | 0 | [This is a sequel to the previous question [sub-coroot systems](https://mathoverflow.net/questions/441161/sub-coroot-systems), that has been answered! :-) ]
Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Lambda \subset {\mathfrak t}$ be the coroot lattice for $(K,T)$, where $\mathfrak t$ is the Lie ... | https://mathoverflow.net/users/409881 | Sub-coroot lattices | As for your [other question](https://mathoverflow.net/questions/441161/sub-coroot-systems), I will work with complex Lie groups, but you can pass to maximal compact subgroups if you prefer.
Consider the natural embedding $\operatorname{SO}\_4(\mathbb C) \to \operatorname{SL}\_4(\mathbb C)$, where $\operatorname{SO}\_... | 1 | https://mathoverflow.net/users/2383 | 441185 | 178,096 |
https://mathoverflow.net/questions/441194 | 2 | The following is inspired from the most recent [riddle of the week](https://www.spiegel.de/karriere/fuenf-baelle-und-fuenf-eimer-raetsel-der-woche-a-6b30bd21-3487-4ecf-aa34-bcd7efe66509) of the German news magazine [Der Spiegel](https://www.spiegel.de).
For any positive integer $n\in\mathbb{N}$, let $[n]$ denote the ... | https://mathoverflow.net/users/8628 | Approximate size of the image of functions $f:[n]\to[n]$ | For each $i \in [n]$, the probability that $i$ is in the image of a random function $f: [n] \to [n]$ is $1 - \frac{(n-1)^n}{n^n}$. By linearity of expectation, the expected size of the image of $f$ is
$$n-\frac{(n-1)^n}{n^{n-1}}.$$
Thus, $$\lim\_{n \to \infty} \frac{E\_n}{n}= \lim\_{n \to \infty} 1 - \frac{(n-1)^n}{n... | 15 | https://mathoverflow.net/users/2233 | 441196 | 178,099 |
https://mathoverflow.net/questions/439835 | 2 | I've tried to find counterexamples or results in this direction, but I haven't found what I'm after (except for the $\mathbb{R}^2$ case).
Allard's regularity theorem guarantees that $(\Lambda,r\_0)$-perimeter minimisers are $C^{1,\frac{1}{2}-\epsilon}$ for all $0<\epsilon<\frac{1}{2}$. But as I understand it, Allard'... | https://mathoverflow.net/users/106263 | Are $(\Lambda,r_0)$-perimeter minimising sets $C^{1,1}$? | No.
---
Let $f(x,y):=(x^2-y^2)\log(x^2+y^2)$. Then $f$ has bounded mean curvature on bounded sets, and $f\in C^{1,1-\epsilon}$ for all $\epsilon>0$, but $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are not Lipschitz.
| 2 | https://mathoverflow.net/users/106263 | 441205 | 178,102 |
https://mathoverflow.net/questions/441190 | 4 | The papers *Periods of integrals on algebraic manifolds* by Griffiths is often quoted as the first instance where the Hodge ring of a smooth projective hypersurface (say defined by the homogenous polynomial $f$) is related to the jacobian ring of $f$.
However, this paper is divided in three parts of equal length (abo... | https://mathoverflow.net/users/37214 | Precise reference in Griffiths' papers : computation of the Hodge theory of a smooth projective hypersurface | As mentioned in the post pointed out by Jason, the correct reference is *On the Periods of Certain Rational Integrals II* by
P. Griffiths,
Ann. Math. 90, no. 3 (1969), pp. 496-541. The best place to look at is §10, where the results are explained using sheaf cohomology.
| 2 | https://mathoverflow.net/users/40297 | 441210 | 178,104 |
https://mathoverflow.net/questions/441063 | 23 | Is the Salem prize discontinued? On the [relevant Wikipedia entry](https://en.wikipedia.org/wiki/Salem_Prize), I don't see anyone since 2018 on there. Why was it discontinued?
| https://mathoverflow.net/users/499631 | What happened to the Salem prize? | I inquired with the IAS. The Salem prize has not been discontinued, but the pandemic has interrupted operations. It should reopen for nominations later this year.
You can email me for the full message I received, which includes some personal details.
| 30 | https://mathoverflow.net/users/11260 | 441212 | 178,105 |
https://mathoverflow.net/questions/441130 | 5 | $\newcommand{\Aut}{\operatorname{Aut}}$Let $G$ be an abelian group.
It seems to be a well-known fact (for example [here](https://mathoverflow.net/questions/349880/classifying-space-for-fibrations-with-eilenberg-maclane-space-fibers-and-nontriv?noredirect=1&lq=1)) that $B\Aut(K(G,1))$, the classifying space of the top... | https://mathoverflow.net/users/141291 | Computing the homotopy type of $B\operatorname{Aut}(K(G,1))$ using a fibration sequence: why is $G \to \text{Aut(G)}$ given by conjugation? | This argument is rather elementary. Maybe we should later move this to MathStackExchange. Anyway:
As mentioned above in comments, $K(G,1)$ is not a topological monoid and $K(G,2)$ doesn't exist, so the initial setting is the fibration
$Aut\_\*(K(G,1))\hookrightarrow Aut(K(G,1))\twoheadrightarrow K(G,1)$
where $Au... | 3 | https://mathoverflow.net/users/12166 | 441226 | 178,107 |
https://mathoverflow.net/questions/441234 | 11 | For $A,B\in{\cal P}(\omega)$ we say $A\subseteq^\* B$ if $A\setminus B$ is finite (that is, $A$ is "almost contained" in $B$). We write $A\simeq\_{\text{fin}} B$ if $A\subseteq^\* B$ and $B\subseteq^\* A$ (that is, the sets $A, B$ are "almost the same set" except for finitely many elements). It is easy to see that $\si... | https://mathoverflow.net/users/8628 | Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$? | Yes! The fact that this is consistent is originally due to Laver. Later, Baumgartner, Frankiewicz, and Zbierski strengthened Laver's result to the following theorem:
**Theorem:** Is it consistent that $\mathsf{MA}\_{\sigma\text{-linked}}$ holds and that every Boolean algebra of size $\mathfrak{c}$ can be order-embedd... | 13 | https://mathoverflow.net/users/70618 | 441237 | 178,110 |
https://mathoverflow.net/questions/441167 | 2 | I have two smoothly embedded orientable surfaces $S\_1,S\_2\subset S^3 \times [0,1]$ with boundary such that
$(i)$ $S\_1\cap S\_2$ is a smoothly embedded surface without boundary and
$(ii)$ $\overline{S\_1}\cap \overline{S\_2}=\overline{S\_1\cap S\_2}$
Now I want to prove that $S\_1 \cup S\_2$ is a smoothly embed... | https://mathoverflow.net/users/171941 | Sufficient condition for the union of two submanifolds to be a submanifold | In the meantime, a very similar question of mine has been answered here <https://math.stackexchange.com/a/4642619/857154> , which answers these questions aswell. Moishe Kohan has provided a counterexample to my claim for 1-manifold which most likely carries over to surfaces. Therefore 1) cannot be shown, the answer to ... | 0 | https://mathoverflow.net/users/171941 | 441239 | 178,111 |
https://mathoverflow.net/questions/441106 | 5 | **Problem:** Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\tau$ on $\widetilde{M}\_+$ by $\tau(a):=\int\_0^\infty\lambda\tau(e\_\lambda)$ where $a=\int\_0^\infty\lambda\,de\_\... | https://mathoverflow.net/users/477204 | Continuity of the extension of a tracial state with respect to the strong operator topology | With the specific definition of strong convergence in the comment (namely, a sequence $x\_n \in M\_+$ is said to converge strongly to $x \in \widetilde{M}\_+$ if and only if $x\_n \xi \to x \xi$ for all $\xi \in D(x)$), both properties indeed hold.
Take such a sequence $x\_n \in M\_+$ converging strongly to $x \in \w... | 2 | https://mathoverflow.net/users/159170 | 441242 | 178,112 |
https://mathoverflow.net/questions/441224 | 2 | I noticed [this](https://mathoverflow.net/questions/380070/is-a-mixture-of-real-analytic-functions-again-analytic) post. But still I'd like to follow up with a specific case I have in mind. Say $p(x| \theta)$ is the density of a Gaussian distribution on $\mathbb{R}^n$ with mean $\theta$ and known covariance $\Sigma$. L... | https://mathoverflow.net/users/157159 | Mixture of gaussian density agree with another gaussian on positive measure | $\newcommand\R{\mathbb R}\renewcommand\th{\theta}\newcommand{\Th}{\Theta}\newcommand{\Si}{\Sigma}
\newcommand\La\Lambda\newcommand{\C}{\mathbb C}$The answer is: This will be so if (and only if) $\Lambda$ itself is a (possibly degenerate) Gaussian distribution.
Let us prove the "only if" part. Here it does not really ... | 1 | https://mathoverflow.net/users/36721 | 441256 | 178,118 |
https://mathoverflow.net/questions/441231 | 1 | I have been recently trying to look at substitution tilings with finite local complexity by examining their admissible patch\pattern atlas, which is sometimes called their language. I have also seen the term dictionary used.
To the best of my understanding, given a finite collection of proto-tiles $\mathcal{P}'$ and ... | https://mathoverflow.net/users/143153 | Computing admissible patches of a substitution | The higher dimensional situation isn't very different to the one-dimensional situation. Of course, there are probably quicker ways to do it than the following, but this at least works and is reasonably fast for most purposes. Also, keep in mind that this method is to check a single word/patch to see if it's legal. If y... | 3 | https://mathoverflow.net/users/21271 | 441276 | 178,124 |
https://mathoverflow.net/questions/441260 | 6 | I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:
Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E\_2^{p,q}(\mathcal{F})=\mathrm{H}^q(Y,R^q\pi\_\*\mathcal{F})\implies \mathrm{H}^{p+q}(X,\mathcal{F})$ and $(E')\_2^{p,q}... | https://mathoverflow.net/users/492820 | Leray spectral sequence and pullbacks | I apologize for the self promotion, but page 570 of my article *The Leray spectral sequence is motivic* has a very brief discussion of the functoriality of Leray.
**Added** In a bit more detail, here are the key points:
* To every object $(C, F)$ in the (bounded below, biregularly filtered) filtered derived categor... | 8 | https://mathoverflow.net/users/4144 | 441288 | 178,128 |
https://mathoverflow.net/questions/440785 | 6 | The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\boldsymbol{r}\_1$, $\boldsymbol{r}\_2$, ..., $\boldsymbol{r}\_N$ which can move on a plane pierced by a uniform and consta... | https://mathoverflow.net/users/101308 | Solution of an ODE upon singular perturbation | Let me change the notations to fit the mathematical literature. I will denote by $x(t) \in \mathbb{R}^{3N}$ the positions of the particles, by $y(t) \in \mathbb{R}^{3N}$ their velocities and by $0 < \varepsilon \ll 1$ their common small mass. Your *massive* problem can be written as a first order ODE as:
$$
\begin{case... | 2 | https://mathoverflow.net/users/50777 | 441296 | 178,130 |
https://mathoverflow.net/questions/441207 | 5 | Let $\mathbb{S}\_n$ denote the set of $n \times n$ symmetric positive semidefinite matrices. I am trying to figure out whether $k: \mathbb{S}\_n \times \mathbb{S}\_n \to \mathbb{R}\_+$ defined as:
$$k(A, B) = \text{Tr}[(A^{1/2} B A^{1/2})^{1/2}]$$
is a positive definite kernel. Can anyone find a counterexample show... | https://mathoverflow.net/users/59128 | Is $k(A, B) = \text{Tr}[(A^{1/2} B A^{1/2})^{1/2}]$ a positive definite kernel? | Counterexample for $n = 2$ :
Let $A\_k$ be the orthonormal projection on the span of $$(\cos(2 \pi (k-1) / 5), \sin(2 \pi (k-1) / 5))^\mathsf{T} , \quad k = 1...5.$$
Then $k(A\_k,A\_l) = \vert \cos(2 \pi (k-l) / 5) \vert$ .
The corresponding matrix has the eigenvalue $-0.11803398874989484820458683436563811772$ wi... | 3 | https://mathoverflow.net/users/17261 | 441297 | 178,131 |
https://mathoverflow.net/questions/441131 | 2 | **Theorem.** *Let $m$ be an integer and $P\_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P\_m$,
$$\|p\|\_{L^\infty(a,b)} \leq {C \over b-a} \|p\|\_{L^1(a,b)}$$*
*Proof.* Since $P\_m$ is finite-dimensional, there is a norm equivalence $... | https://mathoverflow.net/users/73890 | Most general reverse Hölder inequality for polynomials | Although Robert Israel's answer is completely accurate, I was able to find the generalization I was looking for. In case someone is looking for this exact thing in the future, I'm writing it up here.
**Theorem.** *Let $m$ be an integer and $P\_m$ the space of polynomials of degree $m$ in one real variable. Let $\kapp... | 1 | https://mathoverflow.net/users/73890 | 441312 | 178,133 |
https://mathoverflow.net/questions/441306 | 5 | A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is isomorphic to the $S^1$-equivariant cohomology of the fixed points after inverting the Euler class of the normal bundle $\n... | https://mathoverflow.net/users/148223 | Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself? | I don't know what you would mean by the normal bundle of $M$ in $LM$, but the answer to one question is "no": When the $\mathbb Q[u]$-module $H^\ast\_{S^1}(LM)$ is localized by inverting $u$, it does not become the same as the cohomology of $M$.
In fact, in my paper in Topology (1985) "Cyclic homology, derivations, a... | 4 | https://mathoverflow.net/users/6666 | 441344 | 178,141 |
https://mathoverflow.net/questions/441334 | 4 | Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using [Helmholtz decomposition](https://en.wikipedia.org/wiki/Helmholtz_decomposition) that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = \nabla \sigma + \nabla \times \Gamma,$$ where $\nabla \sigma$ ... | https://mathoverflow.net/users/353746 | Are the irrotational and solenoidal parts of a smooth vector field linearly independent? | $\newcommand\R{\mathbb R}\newcommand\na{\nabla}\newcommand\om{\boldsymbol{\omega}}\newcommand\si{\sigma}\newcommand\Ga{\Gamma}\newcommand\F{\mathbf F}\newcommand\x{\mathbf x}\newcommand\0{\mathbf 0}$The answer is no, in general. E.g., take any nonzero $\om\in\R^3$ and let
$$\si(\x):=\om\cdot\x\quad\text{and}\quad\Ga(\x... | 9 | https://mathoverflow.net/users/36721 | 441347 | 178,142 |
https://mathoverflow.net/questions/441354 | 8 | It is known that every orientable 3-manifold has a spin structure, because its tangent bundle is trivial. Also it is known that if a manifold $X$ has a spin structure, then the number of distinct spin structures is equal to the order of $H^1(X;\Bbb Z\_2)$. In particular, the real projective space $\Bbb RP^3$ has exactl... | https://mathoverflow.net/users/164671 | Two different spin structures of the real projective space $\Bbb RP^3$ | I'm editing this to reflect the discussion.
You might think that there cannot be an automorphism of ${\mathbb R}P^3$ that acts non-trivially on the degree one cohomology. But the bijection between the spin structures and degree one cohomology is not canonical, it's a torsor. So it may be possible for an automorphism ... | 5 | https://mathoverflow.net/users/460592 | 441365 | 178,144 |
https://mathoverflow.net/questions/440378 | 1 | Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}\_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}\_1, \ldots \mathbf{x}\_n]^\top.
$$
By the singular decomposition, we can obtain
$$
\mathbf{X} = \mathbf{U} \mathbf{D} \mathbf{V}^\top = \mathbf{U}\_{:r}... | https://mathoverflow.net/users/159685 | Asymptotic property of the left singular vectors of i.i.d. data matrix | This is for the self-reference.
>
> **Claim.**
> There exists some $C>0$ such that
> $$\frac{1}{n} \mathbf{y}^\top \mathbf{U}\_{:r} \mathbf{U}\_{:r}^\top \mathbf{y} \xrightarrow{\mathbb{P}} C.$$
>
>
>
*Proof.*
The main idea of the proof is that linear approximation of the left singular vectors $\mathbf{U}\_{:r... | 0 | https://mathoverflow.net/users/159685 | 441368 | 178,146 |
https://mathoverflow.net/questions/441230 | 0 | I have a question about Theorem 3.7.25. of *Computational commutative algebra I* by M. Kreuzer and L. Robbiano.
Let $K$ be a perfect field, $I \subseteq K[x\_1, \ldots, x\_n]$, be a zero dimensional radical ideal in normal $x\_n$ position, let $g\_n \in K[x\_n]$ be the monic generator of the elimination ideal $I \cap... | https://mathoverflow.net/users/152308 | Are zero dimensional ideals radical? | From the comments, I got my answer and I will write it here for future reference.
In general, if $ I = (x\_1 - g\_1, \ldots, x\_{n-1}-g\_{n-1}, g\_n)$ with $g\_1, \ldots, g\_n \in K[x\_n]$, then $K[x\_1, \ldots, x\_n]/I \cong K[x\_n]/(g\_n)$. So $I$ is radical iff $K[x\_n]/(g\_n)$ is reduced iff $g\_n$ is separable.
... | 2 | https://mathoverflow.net/users/152308 | 441375 | 178,148 |
https://mathoverflow.net/questions/441362 | 2 | Consider the following exponential of matrices $\exp(X+\delta Y)$, where $\delta$ is a smaller number, and $X,Y$ are non-commuting matrices. I am interested in expanding it in such a way that
$$
\exp(X+\delta Y) = e^Xe^{\delta Y}e^{\delta A\_1}e^{\delta^2 A\_2}e^{\delta^3 A\_3}...,
$$
namely that organizing the terms o... | https://mathoverflow.net/users/476103 | Reorganizing the terms in the Baker–Campbell–Hausdorff formula (or Zassenhaus formula) for $\exp(X+\delta Y)$ for small $\delta$ | This expansion is derived by K. Kumar in [On Expanding the Exponential](https://doi.org/10.1063/1.1704742), see equation (9) (with $t=1$) and section 6.
| 2 | https://mathoverflow.net/users/11260 | 441380 | 178,149 |
https://mathoverflow.net/questions/441363 | 1 | $E(i+1)=(I-AT)E(i)+1/2(AT)^2$
How to find the maximum value of $E$ in this expression without using the iterative method? An approximate estimation is also acceptable. Only the $E$ vector is unknown, and the others are known matrix vectors.
It would be better if we could get a formula of $E$ about $T$.
| https://mathoverflow.net/users/499742 | How to find the maximum value of the following difference equation without using iterative method？ | Concerning your request "It would be better if we could get a formula of $E$":
By induction on $i$,
$$E(i)=B^i E(0)+\sum\_{j=0}^{i-1} B^jC \tag{1}\label{1}$$
for $i=0,1,\dots$, where $B:=I-AT$ and $C:=1/2(AT)^2$.
If $(I-B)^{-1}$ exists, then $\sum\_{j=0}^{i-1} B^j=(I-B^i)(I-B)^{-1}$ and hence \eqref{1} can be rewri... | 0 | https://mathoverflow.net/users/36721 | 441391 | 178,153 |
https://mathoverflow.net/questions/440856 | 5 | We recall two definitions. Let $A$ be a linearly topologized ring which is complete and Hausdorff.
>
> We say that $A$ is *pseudo-compact* if, for every open ideal $I\subset A$, the ring $A/I$ is artinian. We say that $A$ is *admissible* if $A$ has an ideal of definition (an open ideal $I\subset A$ such that every ... | https://mathoverflow.net/users/131975 | What's the relation between pseudo-compact and admissible rings? | Neither of these properties implies the other:
**There is an admissible algebra that is not pseudo-compact**
If $A$ is a ring with the discrete topology that is not artinian then it is admissible but not pseudocompact.
Less degenerately if $A$ is a noetherian ring and $I$ is an ideal such that $A/I$ is not artini... | 4 | https://mathoverflow.net/users/345 | 441398 | 178,158 |
https://mathoverflow.net/questions/441399 | 2 | I am trying to better understand the straightening-unstraightening equivalence of Lurie in the $\infty$-categorical setting. In the case that I am interested in, this equivalence states that
$$
\mathrm{LFib}(\mathcal{C}) \simeq \mathrm{Fun}(\mathcal{C}, \mathrm{Spaces}),
$$
where $\mathcal{C}$ is an $\infty$-category a... | https://mathoverflow.net/users/322094 | Morphisms in category of left fibrations | The answer is yes, but $\mathrm{LFib}(\mathcal C)$ is *also* the full subcategory of $(\mathrm{Cat}\_\infty)\_{/\mathcal C}$, it just so happens that you can prove that any morphism between such is a left fibration (this does not remain true in the case of cocartesian fibrations, though).
Here is a proof: Let $f:\mat... | 1 | https://mathoverflow.net/users/102343 | 441404 | 178,159 |
https://mathoverflow.net/questions/403450 | 6 | There's a theory of algebraic geometry over $\mathbb{Z}\_2$-graded commutative rings, often called "[algebraic supergeometry](https://arxiv.org/abs/2008.00700)" or the theory of [superschemes](https://ncatlab.org/nlab/show/super-scheme). From what I understand, there's also a variant theory of $\mathbb{Z}$-graded algeb... | https://mathoverflow.net/users/130058 | Applications of $\mathbb{Z}$-graded algebraic geometry to algebraic topology | Lars Hesselholt and Piotr Pstrągowski have since posted a paper to the arXiv doing *exactly this*!
>
> Hesselholt–Pstrągowski, *Dirac geometry I: Commutative algebra*. [[arXiv]](https://arxiv.org/abs/2207.09256)
>
>
>
In their paper, they develop a theory of $\mathbb{Z}$-graded-commutative algebraic geometry i... | 6 | https://mathoverflow.net/users/130058 | 441426 | 178,166 |
https://mathoverflow.net/questions/441420 | 1 | Define: $\operatorname {wo}^n(x) \iff \forall y (y \in^n x \to \operatorname {wo} (y))$
Where $\operatorname {wo}(y)$ refers to $y$ being well orderable.
Where $y \in^0 x \iff y=x \\ y \in^{n+1} x \iff \exists z (z \in^n x \land y \in z)$
**n-well ordered choice:** for $n=0,1,2,...$, for every set $x$ of nonempty... | https://mathoverflow.net/users/95347 | Does n-well ordered choice schema imply the axiom of choice? | $2$-well ordered choice is enough to imply AC.
Let $α$ be any ordinal, and look at $\mathcal P^2(α)\setminus\{\emptyset\}$.
We have $x\in^2 \mathcal P^2(α)\setminus\{\emptyset\}⇒∃z⊆\mathcal P(α)\;(x\in z)⇒x\subseteq α⇒x\text{ is well orderable}$.
A choice function on $\mathcal P^2(α)\setminus\{\emptyset\}$ induce... | 5 | https://mathoverflow.net/users/113405 | 441427 | 178,167 |
https://mathoverflow.net/questions/441441 | 0 | The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted with 'toy' versions, but I'm having some difficulty seeing how it generalizes.
I would greatly appreciate any help or ... | https://mathoverflow.net/users/105727 | Infinite limit of sums of gamma functions is constant? | For each $k>0$, $c\in\mathbb{C}$ it holds that
$$\lim\_{\alpha\rightarrow\infty}\frac{\Gamma(\alpha+c)}{\Gamma(\alpha)\alpha^c}=1\Rightarrow\lim\_{\alpha\rightarrow\infty} \left(\log\frac{\Gamma(k\alpha+c)}{\Gamma(k\alpha)}-c\log k\alpha\right)=0.$$
Apply this to
$$I= - \log \frac{\Gamma(k\alpha + L)}{\Gamma(k\alpha)} ... | 2 | https://mathoverflow.net/users/11260 | 441442 | 178,172 |
https://mathoverflow.net/questions/441000 | 7 | Let $Y=\Sigma(\alpha\_{1},\dots,\alpha\_{n})$ be a Seifert fibered homology 3-sphere, let $\pi:Y\to\Sigma$ denote the projection to the orbifold surface, and let $T\subset Y$ be the pre-image under $\pi$ of a small disk neighborhood of the singular point of order $\alpha\_{i}$, which we identify with a standard fibered... | https://mathoverflow.net/users/133991 | Calculating the Seifert framing for an exceptional fiber in a Seifert-fibered integer homology 3-sphere | Here's an answer to my own question (thanks to Matt Hedden for the approach):
It will be helpful to fix a presentation of $\pi\_{1}(Y)$. Let $T\_{1},\dots,T\_{n}$ be regular neighborhoods of the exceptional fibers. We can write $Y'=Y\setminus(\cup\_{j}T\_{j})$ as a circle bundle over $S^{2}\setminus(\cup\_{i}D\_{i})$... | 2 | https://mathoverflow.net/users/133991 | 441464 | 178,178 |
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