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https://mathoverflow.net/questions/390038 | 6 | Let $X$ be a compact Kähler manifold with $L$ denoting the Lefschetz operator $L(\bullet) = \bullet \wedge \omega$. The primitive cohomology groups are defined, for $k \in \mathbb{N}$, by $$P^k(X, \mathbb{C}) = \ker(L^{n-k+1}: H^{k-1}(X, \mathbb{C}) \longrightarrow H^{2n-k+1}).$$
How do I remember that the $k$th prim... | https://mathoverflow.net/users/174369 | How do I remember which power of the Lefschetz operator $L$ corresponds to the $k$th Primitive cohomology group? | The formula you give for the primitive cohomology group is not correct. I think you meant to write
$$P^k(X, \mathbb{C}) = \ker(L^{n-k+1}: H^{k}(X, \mathbb{C}) \longrightarrow H^{2n-k+2}(X, \mathbb C)).$$
What's going on here is that the hard Lefschetz theorem says that the map $L^{n-k}: H^k \to H^{2n-k}$ is an isom... | 8 | https://mathoverflow.net/users/18060 | 390044 | 161,478 |
https://mathoverflow.net/questions/390046 | 8 | Lately I have to use a lot of functional calculus. A question that keeps popping up and that I don't manage to resolve is the following:
>
> Let $A,B$ be self-adjoint (not necessarily bounded) operators such that $\pm A\leq B$. Is it true that $B^{-1} A$ is a bounded operator?
>
>
>
In case this is false, woul... | https://mathoverflow.net/users/91098 | Does $\pm A \leq B$ imply that $B^{-1} A$ is bounded? | On $R^2$, consider the matrices $B\_N=\pmatrix{N&0\cr 0&1}$, $A\_N=\pmatrix{0&\sqrt{N}\cr \sqrt{N}& 0}$. It is easily checked that $B\_N\pm A\_N$ is positive definite, but $B\_N^{-1}A\_N$ is of order $\sqrt{N}$. You can build infinite dimensional operators using $B\_N$ and $A\_N$ as diagonal blocks.
| 12 | https://mathoverflow.net/users/12120 | 390056 | 161,484 |
https://mathoverflow.net/questions/389578 | 4 | Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{M}(\mathbb{R}^d)$ be the space of finite signed measures on $\mathbb{R}^d$ endowed with the narrow topology (i.e. the initial topology w.r.t. $C\_b(\mathbb{R}^d)$, the set of real valued, continuous and bounded functions on $\mathbb{R}^d$)... | https://mathoverflow.net/users/142961 | Measurable selection involving measure valued random variable | I think this is straightforward by what I call the "lexicographic" method: fix a countable dense sequence $(f\_n)$ in $\_{0,1}$. Then as you point out, $\omega\mapsto|\mu\_\omega|=\sup\int f\_n\,d\mu\_\omega$. Let $N(\omega)=\min\{n:\int f\_n\,d\mu\_\omega\ge\frac12|\mu\_\omega|\}$. This is measurable because $N^{-1}\{... | 4 | https://mathoverflow.net/users/11054 | 390061 | 161,485 |
https://mathoverflow.net/questions/390069 | 0 | Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$.
E.g. with the equation $f(f(x))=x+x^2$ one could prove that the solution is between $l(x)=|x+0.5|+0.5$ and $h(x)=x+x^2$. However I'm unsure how to proc... | https://mathoverflow.net/users/179074 | Is it possible to numericaly solve functional equation | Select the one you like ($f^{[1/2]}(x)$ means half-iteration, iterative square root):
$$f^{[1/2]}(x)=\sum\_{m=0}^{\infty} \binom {1/2}m \sum\_{k=0}^m\binom mk(-1)^{m-k}f^{[k]}(x)$$
$$f^{[1/2]}(x)=\lim\_{n\to\infty}\binom {1/2}n\sum\_{k=0}^n\frac{1/2-n}{1/2-k}\binom nk(-1)^{n-k}f^{[k]}(x)$$
$$f^{[1/2]}(x)=\lim\_{n... | 0 | https://mathoverflow.net/users/10059 | 390075 | 161,490 |
https://mathoverflow.net/questions/390080 | 2 | I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int\_0^1 \ldots \int\_0^1 \prod\_{i=1}^d u\_i^{\frac{k\_i-1}{2}} \prod\_{m=1}^d (1-u\_m)^{\vert\frac{n}{2}-d\vert-\frac{1}{2}} \prod\_{j<l}^d \vert u\_l-u\_j \vert du\_1 \ldots du\_d,$$
where $k\_1, \ldots... | https://mathoverflow.net/users/149975 | Generalized Selberg integral | Your integrand is not symmetric in the variables. We must maneuver around that.
Begin by defining the integrand
$$f(u)=\prod\_i u\_i^{-1/2}(1-u\_i)^a |V(u)|$$
where $V(u)$ is the Vandermonde and $a$ is the constant you want it to be.
Your integral is $\int du f(u)(u\_1^{q\_1}u\_2^{q\_2}\cdots)$, with $q\_i=k\_i/2$.... | 4 | https://mathoverflow.net/users/78061 | 390088 | 161,494 |
https://mathoverflow.net/questions/390087 | 4 | Let $k$ be an **infinite perfect field in positive characteristic $p$**, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:
1. Let $L$ be a finite Galois extension of $k$. Is $\mathrm{Gal}(L/k)$ always cyclic?
2. Let $L$ and $L'$ be two... | https://mathoverflow.net/users/127497 | perfect fields in positive characteristic | To produce an obvious counterexample to (1), fix a finite field $F$ of characteristic $p$, $n\ge 3$ and $L=\bigcup\_m F[x\_1^{p^{-m}},\dots,x\_n^{p^{-m}}]$. This is a perfect field. The symmetric group $S\_n$ acts by permuting the variables. Let $K$ be the fixed point field. Then $L$ is a finite Galois extension of $K$... | 11 | https://mathoverflow.net/users/14094 | 390089 | 161,495 |
https://mathoverflow.net/questions/390070 | 8 | (This question was previously posted on [MSE](https://math.stackexchange.com/questions/4092639/malgrange-preparation-theorem-with-less-regularity)
and I decided to post it here too.)
I am studying the proof of the Malgrange preparation theorem given in the book "Stable mappings and their singularities" written by Gol... | https://mathoverflow.net/users/179094 | Malgrange preparation theorem with less regularity | *Edit*:
Here's a better reference, which actually states and proves more clearly the version that you are looking for. This one is due to Peter Michor (who is also on MO and may drop by?)
* P. MICHOR, The division theorem on Banach spaces, Österrich. Akad. Wiss. Math.- Natur. Kl. Sitzungsber II,189 (1980), pp. 1–18... | 11 | https://mathoverflow.net/users/3948 | 390096 | 161,500 |
https://mathoverflow.net/questions/390106 | 3 | Let $ k $ be a field and let $ X $ be a smooth projective variety over $ k $ of dimension $ d $.
We denote by $ \overline{X} = X \times\_k \overline{k} \ $ the base change of $ X $ to the algebraic closure $ \overline{k} $.
Then he Galois group $ G = \mathrm{Gal} ( \overline{k} / k ) $ acts on $ \overline{X} $ via the ... | https://mathoverflow.net/users/169088 | Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \, \mathbb{Q}_{ \ell } (r) )^G $? | This is false for a general field $k$. It is true for some special fields, like finite fields.
Counterexample: Take $k = \mathbb C((t))$, $E$ an elliptic curve over $\mathbb C$ base-changed to $\mathbb C((t))$, $r=1$. Because we're over $\mathbb C$, the twists don't matter and can be ignored.
$H^2(\overline{E}, \ma... | 11 | https://mathoverflow.net/users/18060 | 390110 | 161,507 |
https://mathoverflow.net/questions/390053 | 6 | This is related to the conjecture that all odd integers greater than $17$ can be written as the sum of 3 distinct primes.
Schinzel showed that the Goldbach conjecture implied this in 1959 and as the Goldbach conjecture has been verified up to $4\times10^{18}$ by Oliveria e Silva, Herzog and Pardi, this conjecture hol... | https://mathoverflow.net/users/178889 | Weak Goldbach conjecture with distinct primes for odd integers between $4\times 10^{18}$ and $10^{27}$ | The slight issue with Will Sawin's answer above for the second part where he suggests Bertrand's postulate is the case where the prime in between $n/2$ and $n$ equals $n-6$, $n-4$ or $n-2$. Therefore it seems better to use the fact that the largest prime gap less than $15\times10^{18}$ is less than $1526$ so for odd $3... | 1 | https://mathoverflow.net/users/178889 | 390111 | 161,508 |
https://mathoverflow.net/questions/390104 | 3 | I'm interested in the representation theory of symmetric groups.
I'm now trying to search for the formula for the characters of $\Omega^{k}$, the set of $k$-tuple of elements of $\Omega$ a set of $n$ elements where $S\_{n}$ acts in the standard way.
More precisely, I want to know the formula for the *expansion* of ... | https://mathoverflow.net/users/123226 | Is there any simple formula for the character of $S_{n}$ represented by the set of $k$-tuples of $\{1,2,...,n\}$? | There are several possible interpretations of $\Omega^k$ (admittedly some don't quite align with what you ask): ordered/unordered subsets of $k$ distinct/not necessarily distinct elements of $[n]$. There are two questions about them, "what is the character?" and "what are the multiplicities of irreducibles?".
Case 1:... | 6 | https://mathoverflow.net/users/159272 | 390117 | 161,511 |
https://mathoverflow.net/questions/390105 | 3 | Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}\_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i(E, F)$ is a non-zero class.
**Question:** Do there always exist integers $n, d$ such that the map $H^n(X, E\otimes\ma... | https://mathoverflow.net/users/39304 | Extension between vector bundles inducing non-zero map on cohomology | This fails even on curves, where one has more line bundles to play with.
Let $X$ be a curve of genus $>1$. Let $i=1$. Take $E$ a stable vector bundle of rank $2$ and degree $0$ and $F = E \otimes K\_X $.
We have a $Ext^1(E, F) = H^1( X, K\_X \otimes E \otimes E^\vee)$ which admits $H^1(X, K\_X) \neq 0$ as a summand... | 5 | https://mathoverflow.net/users/18060 | 390119 | 161,513 |
https://mathoverflow.net/questions/390121 | 7 | Fix $L \in (0,\infty)$ and consider $\mathcal{C}\_L$ defined as follows:
\begin{align\*}
\mathcal{C}\_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}.
\end{align\*}
I am interested in the "blow-up" of $\gamma$, denoted $\gamma\_{+r}$, defined as follows: For any ... | https://mathoverflow.net/users/118316 | Isoperimetric type inequality in $\mathbb{R}^2$ | The problem of determining the volume of the tubular neighborhood of a submanifold of Euclidean spaces was studied by Hotelling and Weyl (and then others). See <https://www.jstor.org/stable/2371513?seq=1>
The amazing result of Weyl is that, as long as the tubular neighborhood has no "overlaps" (in the sense of non-in... | 5 | https://mathoverflow.net/users/3948 | 390144 | 161,521 |
https://mathoverflow.net/questions/390132 | 1 | Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m\_RS=m\_S$, namely, the ideal in $S$ generated by $m\_R$ (= the maximal ideal of $R$) is $m\_S$ (= the maximal ideal of $S$).
Further assume that $R$ and $S$ are $\mathbb{C}$-algebras, $R \subseteq S$ is fla... | https://mathoverflow.net/users/72288 | Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$ | In general if $R$ is a local ring, then its henselization $R^h$ is flat and "algebraic" over $R$, but rarely integral. The intuition is that the henselization is built out of localizations of etale extensions of $R$. Both localizations and etale extensions are flat and "algebraic" in your sense, but localizations are r... | 1 | https://mathoverflow.net/users/15242 | 390145 | 161,522 |
https://mathoverflow.net/questions/386880 | 7 | Is it known that $\Pi^1\_1$-induction is independent of ATR$\_0$? Simpson's book shows this for $\Pi^1\_1$ transfinite induction ($\Pi^1\_1$-TI), but I'm only interested in inducting on $\omega$.
I can show that ATR$\_0$ + $\Pi^1\_1$-induction implies $\Sigma^1\_1$-TI, but unlike simpler inductions, it's not clear th... | https://mathoverflow.net/users/32178 | Independence of $\Pi^1_1$-induction from ATR$_0$ | Let me sketch the proof that $\mathsf{ATR}\_0+\Pi^1\_1\textsf{-Ind}\vdash\mathsf{Con}(\mathsf{ATR}\_0)$ (by Gödel's 2-nd incompleteness this implies that $\mathsf{ATR}\_0$ doesn't prove $\Pi^1\_1\textsf{-Ind}$).
We reason in $\mathsf{ATR}\_0+\Pi^1\_1\textsf{-Ind}$. Assume for a contradiction that $\mathsf{ATR}\_0$ is... | 4 | https://mathoverflow.net/users/36385 | 390164 | 161,528 |
https://mathoverflow.net/questions/390154 | 5 | It is mentioned on [Wikipedia](https://en.wikipedia.org/wiki/Approximately_finite-dimensional_C*-algebra#Finite-dimensional_C*-algebras) that every unital \*-homomorphism $\Phi:M\_i\to M\_j$ is necessarily of the form $\Phi(a)=U^\*(a\otimes I\_r)U$ for some unitary $U$ and some $r$. (Here $M\_i$ are the $i\times i$ com... | https://mathoverflow.net/users/101775 | Unital *-homomorphisms between matrices | **Theorem:** For Hilbert spaces $H,K$, every normal unital \*-homomorphism $\Phi:\mathcal B(H)\to\mathcal B(K)$ is of the form $\Phi(a)=U(a\otimes 1\_{K\_0})U^∗$ for some Hilbert space $K\_0$ and some unitary $U$.
Here is a super down-to-earth proof, from a functional analysis / operator-algebras perspective. I'll st... | 5 | https://mathoverflow.net/users/406 | 390180 | 161,535 |
https://mathoverflow.net/questions/138870 | 15 | (For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.)
Vopěnka's Principle ($VP$) states that, given any proper class $\mathcal{C}$ of structures in the same (set-sized, relational) signature $\Sigma$, there are some distin... | https://mathoverflow.net/users/8133 | Vopěnka's Principle for non-first-order logics | This is a really late answer, but the answer to your question is "No." As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky [proved](https://www.jstor.org/stable/2273786?seq=1) that Vopěnka's Principle is equivalent to the statement "All logics have a com... | 2 | https://mathoverflow.net/users/178292 | 390182 | 161,536 |
https://mathoverflow.net/questions/390157 | 4 | Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let $f:\mathbb R \to \mathbb R$ be a continuous function which is twice-continuously differentiable at $0$ (you may assume ... | https://mathoverflow.net/users/78539 | Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$ | $\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}$The covariance of two random variables (r.v.'s) does not change if one of them is shifted by a constant. So, without loss of generality $f(0)=0$. Let $n:=d$.
To compute the asymptotics, we need to assume that
\begin{equation\*}
f(x)=Ax+Bx^2+Cx^3+Dx^4+e\_1x^5+O(x... | 3 | https://mathoverflow.net/users/36721 | 390195 | 161,541 |
https://mathoverflow.net/questions/389109 | 2 | In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:
* **Homotopy dimension** (henceforth h.dim.), which is $\leq n$ if $n$-connective objects admit global sections.
* **Local Homotopy dimension** $\leq n$ if there exist objects $\{ U\_\alpha \}$... | https://mathoverflow.net/users/156537 | Counterexamples concerning $\infty$-topoi with infinite homotopy dimension | To close this thread off, I will try to expand Lurie's helpful comment into an answer:
First of all concerning examples of $\infty$-topoi that are locally, but not globally, of finite homotopical dimension, an easy counterexample is the slice $\infty$-topos $\mathcal{S}\_{/X}$ with $X$ a space that is not a retract o... | 1 | https://mathoverflow.net/users/156537 | 390196 | 161,542 |
https://mathoverflow.net/questions/390179 | 5 | If $R\mathrel{:=}\mathbb{R}[x\_1,\dotsc,x\_{n+1}]/(x\_1^2+\dotsb+x\_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^6$. An [almost complex structure](https://en.wikipedia... | https://mathoverflow.net/users/nan | An almost complex structure on the real $n$-sphere $S^n$ | Let $M=SU\_3$, the compact semisimple Lie group.
By request of the OP, for those unfamiliar with [Maurer-Cartan](https://en.wikipedia.org/wiki/Maurer%E2%80%93Cartan_form) form, let me define it. Write each point of $SU\_3$ as a matrix $g$. Left translation by $g^{-1}$ takes $g$ to $I$, so takes $T\_g SU\_3$ to $T\_I ... | 11 | https://mathoverflow.net/users/13268 | 390198 | 161,544 |
https://mathoverflow.net/questions/390191 | 1 | Let $B = \{x\_1,\dots,x\_{d-2},y\_1,\dots,y\_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x\_1,\dots,x\_{d-2}$ to a fixed point $p\_0$ and $y\_1,\dots,y\_k$ to general points $q\_1,\dots,q\_k\in\mathbb{P}^2$.
Assume that there exists a morp... | https://mathoverflow.net/users/14514 | Tangent space to spaces of maps | I think this is not true, at least if $k\geq 6$. The Euler exact sequence pulled back to $\mathbb{P}^1$ is
$$0\rightarrow \mathscr{O}\_{\mathbb{P}^1}\rightarrow \mathscr{O}\_{\mathbb{P}^1}(d)^3\rightarrow f^\*T\_{\mathbb{P}^1}\rightarrow 0\,.$$Thus $H^1(f^\*T\_{\mathbb{P}^1}\otimes \mathscr{I}\_B)=H^1(f^\*T\_{\mathbb{P... | 3 | https://mathoverflow.net/users/40297 | 390209 | 161,549 |
https://mathoverflow.net/questions/390218 | 6 | Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B\_{p,q}^s(\mathbb{R}^d)$ defined by the norm
$$\|f\|\_{B\_{p,q}^s} := (\sum\_{j=0}^\infty \|P\_{j} f\|\_{L^p}^q)^{1/q},$$
where $\{P\_j\}\_{j=0}^\infty$ is an inhomogeneous Littlewood-Paley partition of unity with the convention that $... | https://mathoverflow.net/users/54316 | Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra? | You may want to take a look at
* Herbert Koch and Winfried Sickel, "Pointwise multipliers of Besov spaces of
smoothness zero and spaces of
continuous functions", Rev. Mat. Iberoamericana 18 (2002), 587–626.
They established that the set of all distributions $f$ such that $g \mapsto fg$ is a bounded linear map from ... | 5 | https://mathoverflow.net/users/3948 | 390225 | 161,555 |
https://mathoverflow.net/questions/381893 | 3 | Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K\_0=W(k)[\frac{1}{p}]$ and, $K/K\_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.
Consider the Kummer/Breuil extension $K\_{\infty}:=\bigcup\_{n \geq 0} K\_n$, where $K\_n:=K(\pi\_n)$ and define $G\_{K\_{\i... | https://mathoverflow.net/users/122445 | To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$ | I think Kisin's article "Crystalline representations and $F$-crystals" may be helpful. If you haven't read this, the following is a quick mention of some results.
There are several ways to express $T\_pG$ in the form of a $\mathbf{Z}\_p[G\_{K\_{\infty}}]$-module.
For example, using Breuil-Kisin modules (cf. Corollary... | 2 | https://mathoverflow.net/users/112230 | 390249 | 161,561 |
https://mathoverflow.net/questions/390255 | 4 | Given a commutative ring, the rank of a free module is unique. This is the well known statement that commutative rings have invariant basis numbers. Does an analogue of this property hold for free modules over $\mathbb{E}\_\infty$-ring spectra?
More precisely, I want to know the following: let $R$ be an $\mathbb{E}\_... | https://mathoverflow.net/users/160648 | Is the rank of free module spectra unique? | $\pi\_0: Sp\to Ab$ is a direct sum preserving functor, and it sends $E\_1$-ring spectra to rings, and modules over them to modules over them.
In particular you get a functor $\pi\_0: Mod\_R\to Mod\_{\pi\_0(R)}$. If $R^n\simeq R^m$ as $R$-modules, then $\pi\_0(R)^n\cong \pi\_0(R)^m$ as $\pi\_0(R)$-modules. So if $\pi\... | 6 | https://mathoverflow.net/users/102343 | 390258 | 161,564 |
https://mathoverflow.net/questions/390236 | 1 | The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w\_1,w\_2,...,w\_i$, positive values $v\_1,v\_2,...,v\_i$, and a bag with capacity $B$, we want to insert items into the bag without exceeding the capacity $B$ while maximising the total values (i.e., maximising $\sum\_{h=1}^... | https://mathoverflow.net/users/168850 | Knapsack problem with value range constraint | It is easy to enforce this constraint for a generic knapsack problem.
Indeed, it is enough just to multiply all $w\_h$ and $B$ by $c:=\max\_h \frac{v\_h}{w\_h}$. Then for any $h$,
$$v\_h = \frac{v\_h}{w\_h} w\_h\leq cw\_h,$$
i.e. $v\_h\in[0,cw\_h]$, while the constraint $
\sum\_h p\_hw\_h\leq B$ is equivalent to $\sum\... | 2 | https://mathoverflow.net/users/7076 | 390259 | 161,565 |
https://mathoverflow.net/questions/390078 | 7 | In his book *Introduction to arithmetic groups*, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/\mathbb{Q}$ be a cubic Galois extension and $\sigma$ a generator of its Galois group.If $p \in \mathbb{Z}^+$ and $p \... | https://mathoverflow.net/users/168129 | Explicit construction of division algebras of degree 3 over $\mathbb{Q}$ | Looking more carefully in Pierce - *Associative algebras*, I found the answer I was looking for, which I'm going to describe here for future reference.
The algebra $D$ in the question is a realisation of a special type of crossed product algebra, which gives precisely the central division algebras of degree 3 over $\... | 5 | https://mathoverflow.net/users/168129 | 390265 | 161,568 |
https://mathoverflow.net/questions/212172 | 4 | If $X \sim Normal(0,1)$, then we have the tail bound:
$$ (\*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$
Now for general variables $X$, a nice condition is that $X$ be *subgaussian*, meaning that $\mathbb{E}[e^{tX}] \leq e^{t^2/2}$. In this case we traditionally get the tail bound
$$ \Pr... | https://mathoverflow.net/users/29697 | Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound? | $\newcommand{\si}{\sigma}\newcommand{\R}{\mathbb{R}}\newcommand{\ch}{\operatorname{ch}}$Let $X$ be a $\sigma$-sub-Gaussian random variable (r.v.) for some real $\sigma>0$, that is,
\begin{equation\*}
M\_X(t):=Ee^{tX}\le e^{\sigma^2t^2/2}\quad\text{for all real $t$}. \tag{1}
\end{equation\*}
Then the standard upper bou... | 1 | https://mathoverflow.net/users/36721 | 390268 | 161,570 |
https://mathoverflow.net/questions/385582 | 3 | It has been [proved](https://www.semanticscholar.org/paper/Sur-l%E2%80%99existence-locale-de-certaines-metriques-Gasqui/28e72863cb2fdaebe98288cce8427bc368229bfd) that,
If $\lambda\_1,\,\lambda\_2,\cdots,\lambda\_n$ are real analytic functions from $\mathbb{R}^n$ to $\mathbb{R}$, such that $\lambda\_i(0)\neq \lambda\_... | https://mathoverflow.net/users/171439 | Local existence of flat metrics with degenerate singular values | Classifying the solutions is a non-trivial problem when two of the $\lambda\_i$ are equal, even when they are constants. The reason is that this is essentially an overdetermined problem when two of the $\lambda\_i$ are equal.
The point is this: When the $\lambda\_i$ are distinct, a choice of an orthonormal coframing ... | 5 | https://mathoverflow.net/users/13972 | 390270 | 161,571 |
https://mathoverflow.net/questions/390279 | 0 | I'm looking at the fundamental group $\pi\_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi\_{1}(M)\cong S\_{n}$ (symmetric group) for $d\geq3$ and $\pi\_{1}(M)\cong B\_{n}$ (braid group) for $d=2$.
Let $\Pi(a,b)$ denote the homotopy classes of pa... | https://mathoverflow.net/users/135817 | Fundamental group to groupoid : bijection between homotopy classes? | Just fix a basepoint $a$ and choose a path $u\_x$ from $a$ to $x$ for each $x$. Then define $h\_{xy}\colon \Pi(a,a)\to\Pi(x,y)$ by $h\_{xy}(p)=u\_y\circ p\circ u\_x^{-1}$. These maps are bijections and satisfy $h\_{yz}(q)\circ h\_{xy}(p)=h\_{xz}(q\circ p)$. This is an instance of the general fact that any connected gro... | 3 | https://mathoverflow.net/users/10366 | 390280 | 161,573 |
https://mathoverflow.net/questions/390274 | 4 | Let $X$ be a Banach space and let $P$ be a bounded, linear projection on $X$. Is $P[B\_X]$ closed in $X$? Here $B\_X$ is the closed unit ball of $X$.
This is trivial if $X$ is reflexive, but otherwise is it true?
| https://mathoverflow.net/users/181739 | Closedness of the image of the unit ball | Define $P: c\_0 \to c\_0$ by $P(a\_0, a\_1, a\_2, \ldots) = (\sum \frac{a\_n}{2^n}, 0, 0, \ldots)$. Then $P(a\_0, 0, 0, \ldots) = (a\_0, 0, 0, \ldots)$, so this is a projection onto the first coordinate. But the image of the closed unit ball of $c\_0$ under this map is the open interval $(-2, 2)$.
| 8 | https://mathoverflow.net/users/23141 | 390286 | 161,577 |
https://mathoverflow.net/questions/390291 | 3 | When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x\_{1}, x\_{2}, \dots, x\_{n})^{\intercal} $, we can solve for each component of this vector by means of Cramer's Rule:
\begin{equation}
\tag{1}\label{1}
x\_{i} = \frac{\det(A\_{i})}{\det(A)}.
\end{equation}
Here, ... | https://mathoverflow.net/users/93724 | Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations? | Denote the Fredholm determinant $\Delta=\det(I+K)$, then the solution of the Fredholm equation $g=(I+K)f$ is given formally by
$$f=(I+K)^{-1}g=\frac{1}{\Delta}\left(\frac{\partial\Delta}{\partial K}\right)^\dagger g.$$
This can be seen as the infinite-dimensional, or "continuous", version of Cramer's rule.
For a der... | 5 | https://mathoverflow.net/users/11260 | 390294 | 161,578 |
https://mathoverflow.net/questions/390296 | 1 | Let $$A\_n=\int\_{n^{-\frac{1}{2}}}^{1}\frac{\log(nx)}{nx(\log\log(nx)-\log\log(1+x))}dx.$$
I want to discribe the order of $A\_n$, by geting a progressive formula or a good lower bound for it. The order is at most $\frac{\log^2n}{n\log\log n}$ if we loose the denominator to $nx(\log\log(nx))$ but maybe the answer is... | https://mathoverflow.net/users/160959 | Decide the order of of an integration involving the $\log$ function | $\newcommand{\de}{\delta}$Write
\begin{equation}
A\_n=\frac{\ln^2n}n\, J\_n,
\end{equation}
where
\begin{equation}
J\_n:=\int\_{1/2}^1\frac{t\,dt}{\ln t +\ln\ln n-\ln\ln(1+n^{t-1})}.
\end{equation}
Since $\ln(1+x)\asymp x$ for $x\in(0,1]$, we have
\begin{equation}
J\_n=\int\_{1/2}^1\frac{t\,dt}{O(1) +\ln\ln n-\ln(n... | 1 | https://mathoverflow.net/users/36721 | 390303 | 161,581 |
https://mathoverflow.net/questions/390276 | 1 | Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace $.
I'm looking for an a simple the proof of the claim which says that $G/T$ has finitely many $G^\theta (\mathbb{C})$-orbits or just a s... | https://mathoverflow.net/users/172459 | $G/T$ has finitely many $G^\theta$ orbits | First of all, one has to worry about how $H:=G^\theta(\mathbb C)$ is supposed to act on $G/T$. The only way which comes to my mind is to use the well-known fact that $G/T\cong G(\mathbb C)/B=:X$ where $B$ is a Borel subgroup. This holds because $G$ acts transitively on $G(\mathbb C)/B$ with isotropy group $T$.
Assumi... | 8 | https://mathoverflow.net/users/89948 | 390304 | 161,582 |
https://mathoverflow.net/questions/390076 | 13 | I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the subject, and in particular the following two issues keep surfacing in my work, for which I'd appreciate to be able to cite som... | https://mathoverflow.net/users/134438 | Modern proofs for simplicial localizations | For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's *Higher Topos theory* that $N\_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor
$$(\*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N\_\D... | 10 | https://mathoverflow.net/users/1017 | 390308 | 161,584 |
https://mathoverflow.net/questions/389910 | 1 | Not research but advertising [this](https://math.stackexchange.com/questions/4098133/measures-coming-from-pontryagin-duality-v-s-fourier-inversion) question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first equat... | https://mathoverflow.net/users/105628 | Haar measure coming from Pontryagin duality v/s Fourier inversion | If I understand author correctly - For $f \in L^1(G)$ with $\hat{f}\in L^1(\hat{G})$, you get
$$
f(x) = c^{-1}\hat{\hat{f}}(\Phi(x^{-1}))
$$
where $\phi\_\*(d\_G) = cd\_{\hat{\hat{G}}}$.
On the other hand, if $h \in \mathcal{B}^1(G)$, from 'inversion I' we know $\hat{h}\in L^1(\hat{G})$ and that,
$$
h(x) = \int \xi(x... | 0 | https://mathoverflow.net/users/105628 | 390313 | 161,587 |
https://mathoverflow.net/questions/390299 | 2 | Let X be an infinite dimensional normed linear space. A sequence $(e\_n)$ in $X$ is called a **basis** if for every $x \in X$ there is a unique sequence of scalars $(a\_n)$ such that $x=\sum\_n a\_n e\_n $ (equality in norm).
We say that $(e\_n)$ is a **Weak basis** if for every $x \in X$ there is a unique sequence of ... | https://mathoverflow.net/users/155342 | Weak basis of normed linear space | We will show that if $F$ is a Banach space, and $\{f\_n\}\_{n\in\mathbb{N}}$ is a weak basis, it is a basis. For $n\in\mathbb{N}$ define $P\_{n}$ on $F$ by $P\_{n}f=\sum\limits\_{k=1}^{n}a\_{k}e\_{k}$, where $f=\sum\limits\_{n\in\mathbb{N}}a\_{n}e\_{n}$ (weak limit).
Define a norm $|||\cdot|||$ on $F$ by $|||f|||=\su... | 2 | https://mathoverflow.net/users/53155 | 390315 | 161,589 |
https://mathoverflow.net/questions/363790 | 4 | Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his [1980 paper](https://link.springer.com/article/10.1007/BF01420529 "Modular forms of half-integral weight on Gamma_0(4), Math. Annal.") and then generalized the work in a later [1982 paper](https://eudml.... | https://mathoverflow.net/users/127239 | Necessity of conditions $N$ odd, square-free and $\chi$ quadratic in Kohnen's plus space - modular forms of half-integral weight | This is not an answer and I cannot comment due to lack of reputation, but it seems to first pop up in his Lemma 4 on p. 50. This occurs after a long, very technical discussion about double coset operators. The goal is to show that two specific representations of the Hecke algebra are equivalent (one whose image lies in... | 4 | https://mathoverflow.net/users/181689 | 390316 | 161,590 |
https://mathoverflow.net/questions/390320 | 11 | I do not know where this question is on the trivial to intractable spectrum.
Consider the set of isomorphism classes of groups finitely generated by elements of finite order. What is the cardinality of this set?
| https://mathoverflow.net/users/35482 | How many finitely-generated-by-elements-of-finite-order-groups are there? | Grigorchuk showed that there are uncountably many growth degrees of 2-generated infinite p-groups for any prime p. Hence there are uncountably many isomorphisms classes and even quasi-isometry classes of finitely generated torsion groups. See page 116 of [Grigorchuk - On the Gap Conjecture concerning group growth](http... | 18 | https://mathoverflow.net/users/15934 | 390323 | 161,593 |
https://mathoverflow.net/questions/390317 | 2 | By a result of Klein-Nagata rings of the form $A\_Q=K[x\_1,...,x\_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form.
>
> Question 1: When is $A\_Q$ a principal ideal domain or even an euclidean ring for $n \geq 2$ for a general quadratic polynomial $Q$?
>
>
>
>
... | https://mathoverflow.net/users/61949 | When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic? | PID's have Krull dimension $1$ (or $0$, if you call a field a PID); $A\_Q$ will have Krull dimension $n-1$. So the only option is $n=2$ (the case $n=1$ doesn't apply since $k[x]/x^2$ is not a domain).
However, the $n=2$ case will also not give a PID. The ideal $\langle x\_1, x\_2, \ldots, x\_n \rangle$ is not even lo... | 8 | https://mathoverflow.net/users/297 | 390324 | 161,594 |
https://mathoverflow.net/questions/390327 | 6 | A [cogenerator](https://ncatlab.org/nlab/show/cogenerator) in a category $\mathcal{C}$ is an object $\Omega$ such that for any pair of parallel arrows $f,g:X\rightrightarrows Y$ in $\mathcal{C}$ we have
$$
\forall h:Y\to\Omega\big(h\circ f=h\circ g\big)\implies f=g,
$$
so morphisms into $\Omega$ are sufficient to d... | https://mathoverflow.net/users/92164 | Are the Surreals a cogenerator in the category of ordered fields? | On page 21 of
Conway's Field of Surreal Numbers
Norman L. Alling
Transactions of the American Mathematical Society
Jan., 1985, Vol. 287, No. 1, pp. 365-386.
one finds the statement that, on page 43 of
J. H. Conway, On numbers and games, Academic Press, London, 1976,
Conway states that ``As an abstract Field, $... | 8 | https://mathoverflow.net/users/75735 | 390332 | 161,595 |
https://mathoverflow.net/questions/389687 | 4 | In enriched category theory over a base monoidal category $(\mathcal{V},\otimes\_{\mathcal{V}},\mathbf{1}\_{\mathcal{V}})$, one can consider $\mathcal{V}$ itself as a $\mathcal{V}$-enriched category when it has a closed monoidal structure $[-,-]\_{\mathcal{V}}$.
Is there a similar procedure in internal category theor... | https://mathoverflow.net/users/130058 | Internalising the base in internal category theory | Internal categories are too limited for self-internalisation.
Think of the most fundamental category, $\textbf{Set}$: an internal category is a small category, and $\textbf{Set}$ is (usually) not even essentially small.
(In NF and related set theories with a universal set, $\textbf{Set}$ has a set of objects but fails ... | 8 | https://mathoverflow.net/users/11640 | 390348 | 161,603 |
https://mathoverflow.net/questions/390113 | 2 | Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that every $C(x)$ principal bundle in the etale topology is also a Zariski principal bundle. Everything should be considered her... | https://mathoverflow.net/users/146464 | Properties of stabilizers of adjoint action general linear group | This is true and follows from:
**Claim:** Let $x$ be a $n\times n$ matrix with $\mathbb{C}$-coefficients. Then the centralizer $C(x)$ of $x$ in $GL\_n(\mathbb{C})$ fits into a short exact sequence $1\rightarrow U\rightarrow C(x) \rightarrow \prod\_{n\_i} GL\_{n\_i}\rightarrow 1$, where $U$ is a unipotent group and $n... | 5 | https://mathoverflow.net/users/110362 | 390351 | 161,604 |
https://mathoverflow.net/questions/390363 | 5 | Probably an easy question, but here goes:
In a concrete von Neumann algebra $M \subseteq B(H)$, every element $m \in M$ has a polar decomposition $m= p|m|$ where $p$ is a partial isometry and $|m|= \sqrt{m^\*m}$. Imposing extra conditions on $p$ ensures that $p$ is unique. For example, one can ask that $\ker p = \ker... | https://mathoverflow.net/users/nan | Polar decomposition in abstract von Neumann algebra | I would say that the polar decomposition of $m \in M$ is the unique pair $(v,a)$ of elements in $M$ satisfying the following (algebraic) properties.
1. $m = va$.
2. $v$ is a partial isometry and $a$ is positive.
3. $a^2 = m^\* m$.
4. Whenever $p \in M$ is a projection satisfying $mp = 0$, we have $vp=0$.
| 9 | https://mathoverflow.net/users/159170 | 390367 | 161,610 |
https://mathoverflow.net/questions/390355 | 4 | Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it to be "terminal" in some way, or to play a role in factorizing certain (all?) (co)fibrations.
Note: I'm aware that $\m... | https://mathoverflow.net/users/183020 | Universal property of the codomain fibration | First a note about terminology: older literature sometimes uses the term "cofibration" for a functor $p:E\to B$ such that $p^{\rm op} : E^{\rm op} \to B^{\rm op}$ is a fibration, but it's more common nowadays to call this an "opfibration", with "cofibration" used instead for a functor whose corresponding morphism $B\to... | 8 | https://mathoverflow.net/users/49 | 390368 | 161,611 |
https://mathoverflow.net/questions/390370 | 3 | Let $A$ be an $R$-algebra.
Suppose $A$ has a $R$-coalgebra structure compatible with the algebra structure.
(I.e. there is a comultiplication map $\Delta$ and counit map $\epsilon$ compatible with the multiplication map and unit map of $A$.)
Then I am wondering whether $A$ can have another $R$-coalgebra structure o... | https://mathoverflow.net/users/29422 | Is a comultiplication structure unique? | Yes, $A$ can have multiple compatible $R$-coalgebra structures. Let $G$ be a finite set and let $A=R(G)$ be the commutative algebra of functions on $G$ with pointwise multiplication. Then any group structure on $G$ gives an Hopf structure on $A$ and non-isomorphic group structures on $G$ leads to non-isomorphic Hopf st... | 14 | https://mathoverflow.net/users/13552 | 390371 | 161,612 |
https://mathoverflow.net/questions/390389 | 1 | I am looking for results on the probabilities of common roots of $f(x)$ and $f'(x)$ for $f(x) = \sum\_{s \in S\_f} u\_s x^s$ and $u\_s$ i.i.d $\mathcal{N}(0,1)$-distributed, for $x \in [0,1]$. Or, put differently, how likely is it for a root of $f(x)$ to have multiplicity larger than 1?
Thank you for any ideas or rea... | https://mathoverflow.net/users/180542 | Multiple roots of Random Polynomials | You did not let us know what $S\_f$ is. Since the title of your post is "Multiple roots of Random Polynomials", it seems reasonable to assume that $S\_f$ is the set of integers $n\_0,\dots,n\_k$ such that $0\le n\_0<\dots<n\_k$.
One of the following cases must occur:
*Case 1: $n\_0\ge2$.* Then $0$ is a multiple roo... | 1 | https://mathoverflow.net/users/36721 | 390398 | 161,619 |
https://mathoverflow.net/questions/390397 | 9 | Connes showed in [Cohomologie cyclique et foncteurs $Ext^n$ (1983)](https://alainconnes.org/wp-content/uploads/n83.pdf) that the classifying space of his [cycle category](https://ncatlab.org/nlab/show/cycle+category) $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$.
Connes' proof is not quite as co... | https://mathoverflow.net/users/2362 | Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$ | This statement appears as Corollary B.4 in Nikolaus-Scholze's [On topological cyclic homology](https://arxiv.org/abs/1707.01799); essentially, $\Lambda$ is the quotient of the *paracyclic* category $\Lambda\_\infty$ by a free $B\mathbb Z$-action, which receives a final map from the simplicial category, hence is contrac... | 10 | https://mathoverflow.net/users/35687 | 390399 | 161,620 |
https://mathoverflow.net/questions/390412 | 3 | Let $f\in\mathcal{S}(\mathbb{R}^n)$, Schwartz class. Consider the function $g$ defined on $[0,\infty)$ by $$g(r)=\int\_{S^{n-1}}f(rw)d\mu(w),$$
where $d\mu$ is the normalised surface measure of $S^{n-1}.$
1)Is $\sup\_r|r^kg(r)|<\infty,$ for any $k\in\mathbb{N}$?
2. Is $g\in\mathcal{S}([0,\infty))?$
Answer of only... | https://mathoverflow.net/users/184109 | Is radial part of a Schwartz class function also in Schwartz class? | Wikipedia's definition of the Schwartz class is a bit awkward and I think that is causing some of the difficulty. They define $f \in \mathcal{S}(\mathbb{R}^n)$ if $$\sup\_{x \in \mathbb{R}^n} |x^\beta D^\alpha f| < \infty \label{1}\tag{\*}$$ for all multi-indices $\alpha, \beta$. A more convenient equivalent definition... | 4 | https://mathoverflow.net/users/4832 | 390414 | 161,624 |
https://mathoverflow.net/questions/390391 | 1 | Define a *pre-ordinal* as a transitive set of transitive sets.
Is it consistent with [Zermelo](https://en.wikipedia.org/wiki/Zermelo_set_theory#The_axioms_of_Zermelo_set_theory) set theory (without choice) to have a nonempty set $S$ such that: for every element $s \in S$ there exists a pre-ordinal $\beta$ and a set $... | https://mathoverflow.net/users/95347 | Is existence of this set consistent with Zermelo set theory minus choice? | First note that if $s\in S$, then there is a unique $r$ such that your condition holds. Simply because $\mathcal P(X)=\mathcal P(X')$ if and only if $X=X'$. So we can write $s^-$ for that $r$ and call it a predecessor.
Next, consider Hartogs' theorem, but without Replacement. Namely, for every set $X$ there is a smal... | 5 | https://mathoverflow.net/users/7206 | 390421 | 161,626 |
https://mathoverflow.net/questions/390377 | 18 | In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ over $k[[h]]$ with an $\overline{R}$ which satisfies the unitary condition. I describe the process here:
$\overline{R} = R\*(R^{21... | https://mathoverflow.net/users/183353 | Why does Drinfeld Unitarization work? | The short answer to your question is that if $x,y$ are elements in an algebra in topologically free $k[[\hbar]]$-modules whose constant term is 1, then they have a unique square root whose constant term is also one, and if $x,y$ commute then say the square root of $x$ also commutes with $y$. Indeed if $a$ is the square... | 7 | https://mathoverflow.net/users/13552 | 390422 | 161,627 |
https://mathoverflow.net/questions/390427 | 11 | 1. Are there mathematical theories that do not use intuitive integers? [That is, do not use integers to write statements.]
2. Can you propose a theory that describes natural integers, without using intuitive integers to state axioms, definitions, and propositions?
| https://mathoverflow.net/users/110301 | Can you do math without knowing how to count? | A "philosophy of math" tag would have been a good idea.
To answer your question, take a look at Hatry Field's "Science without Numbers". Also, Edward Nelson has developed a theory of "proto-integers", which is discussed in "Diffusion, Quantum Theory, and Radically Elementary Mathematics" (MN-47).
In my opinion, mos... | 19 | https://mathoverflow.net/users/33505 | 390433 | 161,631 |
https://mathoverflow.net/questions/390425 | 1 | **Introduction**
So far, I have [found](https://arxiv.org/abs/1408.3902) (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind:
\begin{equation} \tag{1} \label{1} \sum\_{l=1}^{n} |S\_{1}(n,l)|z^{l} = (z)\_{n} = \prod\_{k=0}^{n-1} (z+k) =: g(z) , \end{equation} and \begin{equat... | https://mathoverflow.net/users/93724 | What is the inverse Laplace transform of $\frac{(1/s)_{n}}{s} $? | Your question is weird because $\frac{(1/s)\_{n}}{s}$ is a rational function vanishing at $\infty$, zero problem to apply the residue theorem to its inverse Laplace transform integral: $$\mathcal{L}^{-1}[\frac{(1/s)\_{n}}{s}](t)=\operatorname{Res}(\frac{(1/s)\_{n}}{s}e^{st},s=0) 1\_{t >0}=\sum\_{l=1}^{n} |S\_{1}(n,l)| ... | 1 | https://mathoverflow.net/users/84768 | 390435 | 161,633 |
https://mathoverflow.net/questions/390441 | 4 | It seems that there is no digital copy of Leon Karp's Ph.D. thesis
*L. Karp*, **Vector fields on manifolds**, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief and without any detailed proof. (I wonder that peers read the thesis or they trust to the advisory committee).... | https://mathoverflow.net/users/90655 | Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis) | The full text of Karp's thesis (a scanned PDF file) is available here:
<https://search.proquest.com/pqdtglobal/docview/302809402/>
| 5 | https://mathoverflow.net/users/402 | 390448 | 161,637 |
https://mathoverflow.net/questions/390437 | 1 | Let $A\subset \mathbb{R}$ be measurable such that there are $a,b\in \mathbb{R}$, $a<b$ fulfilling $[b,\infty)\subset A\subset [a,\infty)$. The right rearrangement of $A^{\*}$ of $A$ is defined as $A^{\*}=[c,\infty)$ where $c:=b-|A\cap [a,b]|$. Now we can define the increasing rearrangement as follows: For a function $u... | https://mathoverflow.net/users/182809 | Convergence of increasing rearrangment | How about $u\_n = 1\_{[-n-1, -n] \cup [1,\infty)}$ (the characteristic function of $[-n-1, -n] \cup [1,\infty)$)? Then $u\_n^\* = 1\_{[0,\infty)}$ for all $n$ but $u\_n$ converges pointwise to $u = 1\_{[1,\infty)}$, which has $u^\* = 1\_{[1,\infty)}$.
| 3 | https://mathoverflow.net/users/23141 | 390454 | 161,639 |
https://mathoverflow.net/questions/390462 | 8 | Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n\_p$ in the group $E(\mathbb F\_p)$. Has there been any work on the asympotitcs of the average of $n\_p$ for $p < X$ as $X \to \infty$?
M... | https://mathoverflow.net/users/58001 | What's the average order of the reduction of a section of an elliptic curve | Let $E/\mathbb{Q}$ be an elliptic curve. There exist positive integers $d\_p$ and $e\_p$, with $d\_p|e\_p$, such that group $E(\mathbb{F}\_p)$ is isomorphic to $\mathbb{Z}/d\_p\mathbb{Z} \times \mathbb{Z}/e\_p\mathbb{Z}$. Kowalski conjectured that there exists a constant $c\_E>0$ such that $\sum\_{p\leq x}d\_p\sim c\_E... | 4 | https://mathoverflow.net/users/111215 | 390466 | 161,641 |
https://mathoverflow.net/questions/390419 | 2 | Consider the integers $\{1,\dots, N\}$ for some positive integer $N$. Let us suppose that for each $\{1, \dots, N\}$ there is an associated probability $p\_1, \dots, p\_N$. We also define an integer threshold $1 \leq n < N$.
We sample independently and repeatedly **without** replacement from $\{1,\dots, N\}$. For eac... | https://mathoverflow.net/users/45564 | A fast algorithm for a probabilistic counting problem without replacement? | I'm not sure why you ask for "distinct integers" when sampling without replacement guarantees distinctness.
Let $q\_i=1-p\_i$. The ordinary generating function
$$F(u,y) = \prod\_{i=1}^n (q\_i+p\_i uy) \prod\_{i=n+1}^N (q\_i+p\_iy)$$
counts subsets with $u$ monitoring the choices in $[1..n]$ and
$y$ monitoring all cho... | 2 | https://mathoverflow.net/users/9025 | 390470 | 161,643 |
https://mathoverflow.net/questions/376791 | 3 | I borrow notation from [this answer](https://math.stackexchange.com/a/1672996/748775). Let $X,Y\subseteq\mathbb{P}^{n}$ be two irreducible varieties over an algebraically closed field $k$. Consider
$$
S^{0}\_{X,Y}:=\{(x,y,z)\in X\times Y\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\},
$$
and
$$
S\_{X,Y}:=\overl... | https://mathoverflow.net/users/150898 | Is the pre-closure of the join of two projective varieties quasi-projective? | No.
===
The pre-closure of the join of two irreducible projective varieties is **NOT** necessarily quasi-projective.
Let $X$ be a smooth plane conic and let $Y$ be a single point of $X$. The pre-closure of the join of $X$ and $Y$ is the union of all the lines through points $x \in X$, $y \in Y$, $x \neq y$. Of cour... | 3 | https://mathoverflow.net/users/88133 | 390471 | 161,644 |
https://mathoverflow.net/questions/388693 | 6 | The classical [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number)
$$ C\_n = \frac{1}{n+1} \binom{2n}{n}, $$
well-known for their numerous combinatorial interpretations (the second volume of Stanley's *Enumerative Combinatorics* famously lists a total of 66), essentially arise as the even moments $E[X^{2n}] ... | https://mathoverflow.net/users/21655 | Are the “generalized Catalan numbers” of Dumitrescu–Mulase the "moments" of some "multivariate Wigner semicircle distribution"? | I'm new here and not sure if this constitutes an answer, but I saw this was open for a while and figure I can shed some light (please remove, edit, etc. if needed).
You say that the $k$-th Catalan number is equal to $\int x^{2k}d\mu$ where $\mu$ is the Wigner SC distribution. Let me rephrase this in terms of the dens... | 4 | https://mathoverflow.net/users/186416 | 390477 | 161,645 |
https://mathoverflow.net/questions/390483 | 8 | "[A refinement of the Peter–Weyl theorem](https://link.springer.com/chapter/10.1007/978-0-8176-4717-9_29)" is the title of Chapter 29 in Lusztig's "[Introduction to quantum groups](https://doi.org/10.1007/978-0-8176-4717-9)" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is inside Part IV ("Canonical basi... | https://mathoverflow.net/users/41291 | Trying to understand "a refinement of the Peter–Weyl theorem" by Lusztig | For a complex reductive algebraic group $ G $, the Peter-Weyl theorem gives an isomorphism of $ G \times G $ representations
$$
\mathbb C[G] \cong \oplus\_{\lambda} V(\lambda)^\* \otimes V(\lambda)
$$
Here $ V(\lambda) $ is the irreducible representation with highest weight $\lambda $, $ \mathbb C[G] $ is the ring of r... | 5 | https://mathoverflow.net/users/438 | 390489 | 161,648 |
https://mathoverflow.net/questions/390495 | 7 | Apologies in advance if this is well-known; a google search did not produce anything useful.
Let $(M,p)$ be a pointed real analytic manifold. Are the (free or pointed) loop spaces of continuous, smooth and analytic loops in $M$ all homotopy equivalent?
| https://mathoverflow.net/users/64302 | Homotopy type of continuous/smooth/analytic loop spaces? | Suppose $S$ and $M$ are smooth manifolds. For simplicity let us also suppose that $S$ is compact. Then the inclusion $C^\infty(S, M)\hookrightarrow C^0(S, M)$ is a weak equivalence. This is a "standard" fact that follows from the Whitney approximation theorem. A proof can be found the book of Hirsch. There is a discuss... | 15 | https://mathoverflow.net/users/6668 | 390500 | 161,652 |
https://mathoverflow.net/questions/390497 | 2 | Let $f:X\to Y$ be an affine morphism of locally Noetherian schemes. By [this](https://stacks.math.columbia.edu/tag/0G9R), we know that $Rf\_\*\mathcal{F}=f\_\*\mathcal{F}$ for any quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}\_X$-modules (the equality holds in the derived category of $\mathcal{O}\_Y$-modules).
L... | https://mathoverflow.net/users/66686 | Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms? | I never liked the definition of formal completion (as defined in EGA or Hartshorne), so I'll use the following definition from Brian's formal GAGA [paper](https://math.stanford.edu/%7Econrad/papers/formalgaga.pdf).
Definition: Let $X$ be a locally Noetherian scheme, and $I \subseteq \mathcal{O}\_X$ a coherent ideal. ... | 5 | https://mathoverflow.net/users/21278 | 390505 | 161,653 |
https://mathoverflow.net/questions/390512 | 2 | This is by no means a research question. But asking here I hope for the most expert opinion.
A friend of mine, who is a working economist, asked me for advice about a book which uncovers wealth and mightiness of Markov chains in a possibly better way.
Since I am not an expert in the area, I have to ask the same to ... | https://mathoverflow.net/users/73577 | The reference on Markov chains uncovering the power of the subject in a better way for a working macro-economist | This may not be a viable route, but if your friend is familiar with Python, a hands-on course might be an effective way to explore Markov chains. The [course](https://python.quantecon.org/intro.html) designed by Sargent and Stachurski guides you step-by-step through the basics and a variety of applications from economi... | 3 | https://mathoverflow.net/users/11260 | 390517 | 161,657 |
https://mathoverflow.net/questions/378370 | 10 | I read the following passage in [Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves](https://arxiv.org/abs/1506.00742) by Kedlaya-Temkin:
"One can construct many algebraic extensions of $\mathbb{Q}\_p$ whose completions $K$ tilt to the completed perfect closure of a power series field ov... | https://mathoverflow.net/users/119928 | How many untilts? | This specific question is probably not addressed in the literature; let's try to figure it out!
Let $K$ be an algebraic extension of $\mathbb Q\_p$ such that the tilt of $\widehat{K}$ is isomorphic to $\mathbb F\_p((t^{1/p^\infty}))$. We can observe the following:
1. Tilting preserves residue fields, so necessarily... | 15 | https://mathoverflow.net/users/6074 | 390522 | 161,660 |
https://mathoverflow.net/questions/390531 | 1 | We are provided a single Diophantine equation
$$f(x\_1,\dots,x\_n)=0$$
having degree $\geq2$ and having the promise it has $\leq1$ solutions in the set $\{0,\dots,m-1\}^n$ and $t$ is the number of terms in the polynomial.
We are to decide if $$|\{(x\_1,\dots,x\_n)\in\{0,1\}^n:f(x\_1,\dots,x\_n)=0\}|>0?$$
Is there a... | https://mathoverflow.net/users/10035 | Complexity of a Diophantine equation having $\leq1$ solutions | If you could solve this problem in polynomial time, then NP would be contained in BPP, which is viewed as being approximately as unlikely as P = NP. Too see this, pick your favorite encoding of SAT into diophantine equations on $\{0,1\}^n$ (for instance, you can take $f$ to be a sum of squares of expressions correspond... | 4 | https://mathoverflow.net/users/2363 | 390532 | 161,663 |
https://mathoverflow.net/questions/390530 | -2 | I have tried to get representations of some integers as sum of three cubic of the form $x^3+(k\*10^n)^3+z^3$ with $k$ is integer and $n$ is a postive integer, I took this example : $(48807585839879)^3-(4\times10^{14})^3+z^3=0$, I have got $z=399757627176339$ look [here](https://www.wolframalpha.com/input/?i=solve+in+in... | https://mathoverflow.net/users/51189 | Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $? | Not that I'm a great fan of Mathematica, but Wolfram Alpha yields the same thing as Will Jagy got with PARI:
* Input: 48807585839879^3 + 399757627176339^3
* Result: 63999999999999992173445324722427124758794658
I think that the issue is that the default precision in Mathematica is not set high enough to handle these... | 4 | https://mathoverflow.net/users/11926 | 390536 | 161,664 |
https://mathoverflow.net/questions/390537 | 2 | $\newcommand{\Cstar}{C^\*\_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let
$\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let
$$
\H = \{\gamma \in \G: r(\gamma ), s(\gamma )\in U\}
$$
be the reduction of $\G$ to $U$.
It is well known that any element of $\Cstar(\G)$ ... | https://mathoverflow.net/users/97532 | Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid? | I'm not an expert of groupoids and not sure what "Fourier coefficients" exactly means, but perhaps Wassermann's classical counterexample ([mr1127480](https://mathscinet.ams.org/mathscinet-getitem?mr=1127480)) serve (against) the purpose?
Take an infinite residually finite Kazhdan (T) group $\Gamma$ with finite quotient... | 2 | https://mathoverflow.net/users/7591 | 390542 | 161,665 |
https://mathoverflow.net/questions/388796 | 12 | Let $K$ be a [perfectoid field](http://www.math.uni-bonn.de/people/scholze/PerfectoidSpaces.pdf) of mixed characteristic $(0, p)$, i.e. $K$ has characteristic $0$ but its residue field has characteristic $p$. Further suppose that $K$ contains all the $p$th roots of unity.
A subgroup of $K^\times/K^{\times p}$ corresp... | https://mathoverflow.net/users/91375 | An explicit isomorphism $K^\times/K^{\times p} \cong K^\flat/\wp K^\flat$ where $K \supset \mu_p$ is perfectoid field of mixed characteristic $(0, p)$ | Here is an explicit description of the isomorphism: It takes $a\in K^\flat$ to the class of $1+(\zeta\_p-1)^p a^{1/p^n}\in K^\times$ for any large enough $n$ (the image modulo $p$-th powers is independent of the choice).
Here's an explanation. Let's first analyze the quotient $K^\times/(K^\times)^p$. As $K$ is perfec... | 9 | https://mathoverflow.net/users/6074 | 390555 | 161,667 |
https://mathoverflow.net/questions/390550 | 2 | **Definitions and some motivation:**
Let $\mathcal B$ be the set of bounded measurable functions from $[0, 1]$ to $\mathbb R$. Denote by $\mathcal N$ the set of measurable subsets of $[0, 1]$ with Lebesgue measure $0$.
Given a function $f \in \mathcal B$, define the function $\mathcal Of$ by
$\mathcal Of(x) := \i... | https://mathoverflow.net/users/173490 | Do we have full control of the oscillation of a function by modifying it on a small set? | Let $f,g: [0,1] \to \mathbf{R}$ be two given, bounded measurable functions and $\epsilon > 0$ be an arbitrary constant. By Lusin's theorem there exist continuous functions $F,G: [0,1] \to \mathbf{R}$ so that together
\begin{equation}
\lvert \{ F \neq f \} \rvert + \lvert \{ G \neq g \} \rvert < \epsilon.
\end{equation}... | 1 | https://mathoverflow.net/users/103792 | 390558 | 161,670 |
https://mathoverflow.net/questions/389873 | 1 | Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism
$$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \otimes A)$$
Suppose I want to show that ($1$ is the unit of $M(A))$:
$$(\iota \otimes \Delta)(x \otimes 1) = x \otimes 1... | https://mathoverflow.net/users/nan | About extensions between morphisms on the multiplier algebra | I think there is *something* to prove. But asking "... something to prove here?" is always going to be in the [eye of the beholder](https://www.collinsdictionary.com/dictionary/english/in-the-eye-of-the-beholder). One mathematician's checking of details is another's tedium; one mathematician's concise writing is anothe... | 2 | https://mathoverflow.net/users/406 | 390569 | 161,674 |
https://mathoverflow.net/questions/390581 | 2 | Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.
>
> **Question:** Does every function in the Sobolev space $W^{1,1} (\Omega)$
> admit a representative whose graph in $\Omega \times \mathbb R$ has a
> path connected component whose projection to $\Omega$ has full me... | https://mathoverflow.net/users/173490 | Is the graph of a Sobolev function path connected? | As pointed out by Mateusz Kwaśnicki, the answer is yes.
Applying the ACL characterisation of Sobolev functions (see, for example page 36 here: [https://math.aalto.fi/~jkkinnun/files/sobolev\_spaces.pdf](https://math.aalto.fi/%7Ejkkinnun/files/sobolev_spaces.pdf)) to each of the coordinates successively, we obtain a f... | 0 | https://mathoverflow.net/users/173490 | 390586 | 161,679 |
https://mathoverflow.net/questions/390565 | 3 | This question is in part related to a [question that I have already posed](https://mathoverflow.net/questions/390293/uniform-continuity-of-cholesky-decomposition).
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}\_A \mathbf{L}\_A^{\top}$ and $\m... | https://mathoverflow.net/users/181840 | Operator norm of difference of matrix decompositions | The answer is **negative**, and this happens as soon as $n=2$. The question is whether the composition $X\mapsto L:=L\_{X^2}$ is globally Lipschitz over ${\bf SPD}\_n$. Let $x\_j\in{\mathbb R}^n$ denote the $j$th column of $X$. Then we have the following formulae
$$\ell\_{11}=\|x\_1\|,\quad \ell\_{j1}=\frac{\langle x\_... | 3 | https://mathoverflow.net/users/8799 | 390589 | 161,681 |
https://mathoverflow.net/questions/390587 | 7 | Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$.
>
> **Question:** Can the homeomorphism $\phi :|\Delta|\to\mathbf S^d$ be chosen in a way, so that all combinatorial sym... | https://mathoverflow.net/users/108884 | Are there simplicial spheres with "non-geometric symmetries"? | The answer is negative. Already in dimension 4 there are fake real-projective spaces, which are smooth 4-manifolds homotopy-equivalent but not homeomorphic to $RP^4$. These correspond to smooth free involutions $\sigma: S^4\to S^4$ which are not topologically conjugate to orthogonal transformations. Similar examples ex... | 8 | https://mathoverflow.net/users/39654 | 390591 | 161,683 |
https://mathoverflow.net/questions/390579 | 2 | I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?
| https://mathoverflow.net/users/15934 | English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie | E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and the Theory of Representations], Math. Zeit. 30 (1929), 641–692; informal translation of section 23.
>
> Please feel free to edit as you see fit
>
>
>
[original](https://ilorentz.org/beenakker/MO/Noether_par_23.pdf)
**Dete... | 4 | https://mathoverflow.net/users/11260 | 390592 | 161,684 |
https://mathoverflow.net/questions/390597 | 12 | It is easy to see that every countable collection of sets $A\_n\subseteq\mathbb{N}$ has an upper bound in the Turing degrees, since we can just take a copy of their disjoint sum $\oplus\_n A\_n=\{\langle n,m\rangle\mid m\in A\_n\}$, using a computable pairing function $\langle\cdot,\cdot\rangle$. From this, we can easi... | https://mathoverflow.net/users/1946 | Does every countable set of Turing degrees have an upper bound, without AC? | No. It is quite possible to get a counterexample, even without having the reals as a countable union of countable sets. Indeed, we only need to add $\omega$ Cohen reals in order to find such a symmetric extension!
Let's first make a small observation:
>
> Suppose that there is a sequence of equivalence classes mo... | 12 | https://mathoverflow.net/users/7206 | 390609 | 161,686 |
https://mathoverflow.net/questions/390600 | 8 | I want to believe that this has an easy answer, but I’ve never considered it before and can’t seem to answer it now either.
>
> Does every infinite-dimensional Banach space admit a locally convex vector topology that is strictly coarser than the norm topology and strictly finer than the weak topology?
>
>
>
If... | https://mathoverflow.net/users/166628 | Strict topology between weak and norm topologies | In a normed space, every weak neighborhood of zero contains a finite codimensional subspace. Consequently, if you have a weak to norm continuous linear operator between normed spaces, then the operator has finite rank.
Suppose that $X$ is an infinite dimensional Banach space. Take a closed subspace $Y$ that has infin... | 14 | https://mathoverflow.net/users/2554 | 390610 | 161,687 |
https://mathoverflow.net/questions/390613 | 0 | I know that if $P\_n$ are continuous functions and $P\_n \rightrightarrows P$, $P$ is also continuous function. But I can't see in which direction I should dig to prove that $P$ is polynomial.
I will appreciate any hint and help.
| https://mathoverflow.net/users/188659 | If $P_n \rightrightarrows P$ in $\mathbb{R}$ and $P_n$ are polynomials proof that $P$ is polynomial | We have $(P\_n-P\_m)\rightrightarrows 0$ if $n,m$ tend to infinity. Since $P\_n-P\_m$ is a polynomial, this yields that degrees of $P\_n$ are uniformly bounded, say they do not exceed $d$. Now even the pointwise convergence in $d+1$ points yields the coefficientwise convergence (by Lagrange interpolation, for example),... | 1 | https://mathoverflow.net/users/4312 | 390615 | 161,688 |
https://mathoverflow.net/questions/390607 | 0 | Let $n$ be a fixed natural number.
Let $H$ be a complex Hilbert space and $H\_1,\dotsc,H\_n$ be closed subspaces of $H$.
Set $H\_0\mathrel{:=}H\_1\cap H\_2\cap\dotsb\cap H\_n$ and let
$P\_i$ be the orthogonal projection onto $H\_i$, $i=0,1,2,\dotsc,n$.
I study the functions $f\_n:[0,1]\to\mathbb{R}$ defined by
$$
f\_n(... | https://mathoverflow.net/users/48157 | Can we choose a sequence of Hilbert spaces? | For every $m\ge1$ we have a proper class $A\_m$ of tuples $(H,H\_1,…,H\_n)$ satisfying the described condition.
Now invoke [Scott's trick](https://ncatlab.org/nlab/show/Scott%27s+trick)
and construct a subset $A'\_m⊂A\_m$ for each $m\ge1$,
consisting of tuples with the minimal rank in $A\_m$.
(This part of the constr... | 3 | https://mathoverflow.net/users/402 | 390625 | 161,692 |
https://mathoverflow.net/questions/390622 | 5 | The question concerns chess. I call a move *forced* if, in a given position, is the unique move consistent with the rules of the game. I wonder what is the largest integer $n$ such that there exists a legal position in which:
1. both black and white have forced moves for $n$ consecutive times;
2. the position is neve... | https://mathoverflow.net/users/167834 | How many consecutive forced moves are possible in chess? | This question has been answered at [chess.stackexchange.com](https://chess.stackexchange.com/q/4963/). It seems that if you allow promoted pieces, the current record is $n=9$.
| 10 | https://mathoverflow.net/users/3106 | 390632 | 161,694 |
https://mathoverflow.net/questions/390629 | 1 | Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W\_t$ usual Brownian motion (with $W\_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W\_t|^p$$
purely in terms of $\mathbb E|X|^p$ and $\mathbb E|X+W\_1|^p$ (which I can assume to exist).
I was able to show the following, which looks a bit simila... | https://mathoverflow.net/users/88505 | Bound moments wrt. known initial and final moments | With $Y:=W\_1$ and $t\in[0,1]$, the expectation in question is
$$
\begin{aligned}
E|X+\sqrt t\,Y|^p&=E|(1-\sqrt t)X+\sqrt t\,(X+Y)|^p \\
&\le(1-\sqrt t)^p E|X|^p+(\sqrt t)^p E|X+Y|^p \\
&\le E|X|^p+E|X+Y|^p \\
&=E|X|^p+E|X+W\_1|^p. \\
\end{aligned}
$$
| 2 | https://mathoverflow.net/users/36721 | 390634 | 161,696 |
https://mathoverflow.net/questions/390603 | 3 | In an online webinar, I heard (not directly) the statement that **(closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces**. Is this a valid theorem? Any reference that contains its proof? I am not sure that in the above statement whether "positive curvature" is a part of assumptio... | https://mathoverflow.net/users/90655 | Closed manifolds of nonnegative curvature operator are symmetric spaces | As Igor Belegradek commented, the correct statement is as follows:
>
> **Theorem (classification of closed simply connected manifold with nonnegative curvature operator):** A closed simply connected manifold with nonnegative curvature operator is isometric to a Riemannian product of
>
>
> 1. standard spheres with... | 3 | https://mathoverflow.net/users/90655 | 390636 | 161,697 |
https://mathoverflow.net/questions/390638 | 2 | Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}\_R(k, R)\cong k$, and so
$\text{Hom}\_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes\_R R^{\oplus n} , \forall n \ge 0.$ Let $\mod (R) $ be the category of finitely generated $R$-modules, and $\mathcal F(R)$ be the subcategory of a... | https://mathoverflow.net/users/127118 | On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$ | Yes.
We may as well take $\mathcal F(R)$ to be the category with objects $R^n$ for $n\in \mathbb N$ and take morphism $\operatorname{Hom}(R^m,R^n) =M\_{n,m}(R) $.
Both functors take $R^n$ to $k^n$ and take a homomorphism in $M\_{n,m}(R)$ to that matrix modulo the maximal ideal, so are naturally isomorphic.
Altern... | 3 | https://mathoverflow.net/users/18060 | 390639 | 161,698 |
https://mathoverflow.net/questions/390649 | 1 | Let $R \subseteq S$ be local rings with maximal ideals $m\_R$ and $m\_S$.
Assume that:
**(1)** $R$ and $S$ are (Noetherian) integral domains.
**(2)** $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull dimension.
**(3)** $R$ is regular ([hence](https://stacks.math.columbia.edu/tag/0AG0) a UFD).
**(4)** $S$ is... | https://mathoverflow.net/users/72288 | Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular | Condition (8) implies that the map is unramified, and since it is also flat by (6) (which by the way implies $S$ is CM by Miracle Flatness), it is etale, and so $S$ is regular. Note you don't need (5) here.
Without condition (8) you can take $R=\mathbf{C}[[x,y]]$ and $S=R[w]/(w^2 - xy)$.
| 5 | https://mathoverflow.net/users/3847 | 390651 | 161,703 |
https://mathoverflow.net/questions/390642 | 8 | By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups $\mathrm{Aut}(F\_n)$ and $\mathrm{Out}(F\_n)$ of free group of rank $n$ are *not* relatively hyperbolic if $n \geq 3$ (Theorem 9.2 of [BDM]). If $n = 2$, then $\mathrm{Out}(F\_2)$ is isomorphic to $\mathrm{GL}(2,Z)$ so that it is virtual... | https://mathoverflow.net/users/173504 | Is the automorphism group of free group of rank two relatively hyperbolic? | The group $\mathrm{Aut}(F\_2)$ is not relatively hyperbolic. This is contained in (the proof of) Theorem 8.1 of Behrstock-Drutu-Mosher.
We first pass to $\mathrm{Aut}^+(F\_2)$, the preimage of $\mathrm{SL}(2, \mathbb{Z})$. By Remark 7.2 of Behrstock-Drutu-Mosher it is enough to consider this index two subgroup.
Let... | 6 | https://mathoverflow.net/users/1650 | 390653 | 161,704 |
https://mathoverflow.net/questions/390544 | 3 | Suppose that $\alpha\_n$ is a sequence of positive numbers converging to $0$.
>
> **Question.** Is there a bounded measurable function $f$, say $1$-periodic, such that $f\_n(x)=f(x-\alpha\_n)$ does not converge to $f(x)$ for any $x$ in some set $A$ of positive Lebesgue measure?
>
>
>
This of course is not the ... | https://mathoverflow.net/users/78591 | Almost surely convergence of translations of a measurable function | These questions boil down to questions about maximal inequalities. If you define $(T\_nf)(x)=f(x-\alpha\_n)$, then maximal inequalities are concerned with the operator $Mf(x)=\sup\_n |T\_nf(x)|$ and the finite approximations $M\_nf(x)=\max\_{k\le n}|T\_kf(x)|$.
One then asks whether the (non-linear) operator $M$ is b... | 3 | https://mathoverflow.net/users/11054 | 390654 | 161,705 |
https://mathoverflow.net/questions/390664 | 5 | Let $k$ be an algebraically closed field and $\mathcal M\_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C\_g \to \mathcal M\_g$, there is a natural vector bundle $\pi\_\* \Omega\_{\mathcal C\_g / \mathcal M\_g}$ of rank $g$ called the Hodge bun... | https://mathoverflow.net/users/164782 | Tangent Space of the Hodge bundle on the moduli space of curves | The universal property of the total space $$\mathbf{V}(E^\vee) = \operatorname{Spec}\_M \operatorname{Sym} E^\vee $$ of a vector bundle (locally free sheaf) $E$ on some scheme $M$ is: giving a map $T\to \mathbf{V}(E^\vee)$ corresponds to giving a map $f\colon T\to M$ and a section $\omega$ of $f^\* E$.
We apply this ... | 4 | https://mathoverflow.net/users/3847 | 390667 | 161,707 |
https://mathoverflow.net/questions/385277 | 2 | I know from [this paper by Ward](https://arxiv.org/abs/1208.5543) that one can obtain the (signs of) the Gerstenhaber bracket using operadic suspension on any operad $\mathcal{O}$. More precisely, the insertion $\tilde{\circ}$ of the new operad has the appropriate sign when compared to the original insertion $\circ$ on... | https://mathoverflow.net/users/144957 | Bigraded operadic suspension | In the paper [Derived A-infinity algebras in an operadic context](https://arxiv.org/abs/1110.5167), Section 5.1, there is a star operation which is the convolution operation on $\mathrm{Hom}(C,P)$ where $C$ is a cooperad and $P$ is an operad. In the case of $C=dAs^¡$ and $P=\mathrm{End}\_A$, this complex is in bijectio... | 0 | https://mathoverflow.net/users/144957 | 390670 | 161,709 |
https://mathoverflow.net/questions/390482 | 3 | Let $V$ be a real vectors space, and $W$ be a linear subspace.
Say $W$ is *obviously closed* if, for every topology on $V$ that makes $V$ a Hausdorff locally convex topological vector space, the subspace $W$ is closed in $V$.
We know $V$ is obviously closed, and any finite-dimensional subspace of $V$ is obviously c... | https://mathoverflow.net/users/145424 | When is a linear subspace to be closed in all compatible topologies | Only the subspaces mentioned by the OP are obviously closed.
Let $V$ be a real vector space and $W\subset V$ a proper infinite-dimensional linear subspace. We shall endow $V$ with a norm so that $W$ will not be closed in $(V, \|\;.\;\|)$. For this we consider a Hamel basis for $W$ that we partition into a countable p... | 6 | https://mathoverflow.net/users/127871 | 390674 | 161,710 |
https://mathoverflow.net/questions/390665 | 0 | Let $A \in M\_n(\mathbb{R})$ be a **symmetric** matrix and assume we perform on $A$ an elementary row operation $A \xrightarrow{R\_i = R\_i + cR\_j} B$ where $i \neq j$ and $c \in \mathbb{R}$ to get $B$. Are all the eigenvalues of $B$ necessarily real?
---
This might look like a strange question to ask so here is... | https://mathoverflow.net/users/23902 | Are the eigenvalues of a symmetric matrix stay real after performing a row-addition operation? | No. Example
$$A=\left(\begin{array}{cc}-2&1\\ 1&-2\end{array}\right),\quad c=1.$$
| 3 | https://mathoverflow.net/users/25510 | 390675 | 161,711 |
https://mathoverflow.net/questions/390645 | 0 | Suppose that $M\subseteq\mathbb R^D$ is a compact smooth Riemannian submanifold of dimension $d$, having normal injectivity radius $\tau$. Let $x\_0\in M$ be a point, and $\delta\in (0,\tau)$ sufficiently small. I would like to know if $M\cap B\_{\mathbb R^D}(x\_0,\delta)$ has $d$-dimensional Hausdorff measure at least... | https://mathoverflow.net/users/nan | largest geodesic ball inside a small portion of Euclidean submanifold | For all $x,y \in M$, $d(x,y) \geq \lvert x - y \rvert$, where $d$ is the distance function on the manifold. Therefore for every point $x\_0 \in M$ and every radius $\rho > 0$,
\begin{equation}
B\_M(x\_0,\rho) = \{ x \in M \mid d(x,x\_0) < \rho \}
\subset \{ x \in M \mid \lvert x - x\_0 \rvert < \rho \}.
\end{equation}
... | 0 | https://mathoverflow.net/users/103792 | 390678 | 161,713 |
https://mathoverflow.net/questions/390693 | 3 | I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have
\begin{align}\label{EqI}\tag{I}
\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)} \qquad\text{or equaly} \qquad \theta \leq \phi... | https://mathoverflow.net/users/112207 | Looking for non-polynomial functions: with the growth condition: $\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)}$ | With $f:=\phi$, $u:=s/t$, and $c:=\theta$, the desired inequality can be rewritten as
$$f(cu)f(t)\le f(tu) \tag{1}$$
for $u\in[0,1]$ and real $t\ge0$.
Let us show that (1) holds with $c=1$ if $f(x)\equiv\ln(1+x)$. That is, we have to show that
$$g(t):=\ln(1+u)\ln(1+t)-\ln(1+tu)\le0$$
for $u\in[0,1]$ and real $t\ge0$.... | 2 | https://mathoverflow.net/users/36721 | 390697 | 161,720 |
https://mathoverflow.net/questions/390662 | 12 | Let $X$ be a compact Riemann surface. I would like to find a somehow complete reference for the proof of the so called non-Abelian Hodge correspondence relating Dolbeaut, Betti and Higgs bundle moduli spaces.
I've tried to read the original articles by Hitchin (1987) or Simpson (1990) but it seems to me that I've not... | https://mathoverflow.net/users/146464 | Non-Abelian Hodge theory | I personally like the notes by Eugene Xia: [Abelian and Non-Abelian Cohomology](https://arxiv.org/pdf/1404.5025.pdf) to build intuition.
But for a definitive source, I would read Simpson: [Moduli of representations of the fundamental group of a smooth projective variety I](http://www.numdam.org/article/PMIHES_1994__7... | 14 | https://mathoverflow.net/users/12218 | 390701 | 161,722 |
https://mathoverflow.net/questions/390705 | 2 | This question is motivated by [my earlier (unanswered) MO post](https://mathoverflow.net/questions/390446/being-even-or-odd-in-the-product-expansion-prod1xkxk1).
The *number of partitions into distinct parts* is generated by $\sum\_{n\geq0}Q(n)x^n=\prod\_{k\geq1}(1+x^k)$. Focusing on parity of coefficients $Q(n) \,\,... | https://mathoverflow.net/users/66131 | Partity of partitions with distinct parts of parts $>1$ | We have
$$\prod\_{k\geq 2} (1+x^k) = \frac{\prod\_{k\geq 1} (1+x^k)}{1+x} \equiv \sum\_{j\geq 0} \frac{x^{j(3j+1)/2} + x^{(j+1)(3j+2)/2}}{1+x}$$
$$\equiv \sum\_{j\geq 0} x^{j(3j+1)/2}\frac{1 - x^{2j+1}}{1-x}\equiv \sum\_{j\geq 0} x^{j(3j+1)/2}(1+x+\dots+x^{2j}) \pmod{2}.$$
| 1 | https://mathoverflow.net/users/7076 | 390709 | 161,723 |
https://mathoverflow.net/questions/390668 | 2 | This question is inspired by this one:
[Can you do math without knowing how to count?](https://mathoverflow.net/questions/390427/can-you-do-math-without-knowing-how-to-count)
Let $M\_2$ be the set of words constructed by concatenation of the letters $a\_1$ and $a\_2$, with :
(\*) : for any $x$ word of $M\_2$ $xx ... | https://mathoverflow.net/users/110301 | Is the number of words finite, when you don't know how to count? | I'm upgrading my [comment](https://mathoverflow.net/questions/390668/is-the-number-of-words-finite-when-you-dont-know-how-to-count#comment995922_390668) to an answer, because I've found the source for the result asserted in the comment: For every $n$, the semigroup $M\_n$, presented by $n$ generators subject to the rel... | 10 | https://mathoverflow.net/users/6794 | 390722 | 161,729 |
https://mathoverflow.net/questions/390719 | 1 | I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all frequencies. This doesn't seem to be the case for many symmetrical distributions with fat tails (like the double Gamma di... | https://mathoverflow.net/users/100101 | Positivity of exponentially bounded characteristic functions | The rate of decrease of the tails of a probability distribution has to do with the degree of smoothness of the corresponding characteristic function (c.f.).
The rate of decrease of the tails has nothing to do with the positivity of the c.f. E.g., the uniform distribution over the interval $[-1,1]$ has zero tails, but... | 2 | https://mathoverflow.net/users/36721 | 390732 | 161,732 |
https://mathoverflow.net/questions/286932 | 19 | For convenience we work with commutative rings instead of commutative algebras.
---
Fix a commutative ring $R$. Consider the functor $\mathsf{Mod}\longrightarrow \mathsf{CRing}$ defined by taking an $R$-module $M$ to $R\ltimes M$ (with dual number multiplication). Following the [nlab](https://ncatlab.org/nlab/sho... | https://mathoverflow.net/users/69037 | "Formally unramified iff trivial Kähler differentials" using only universal properties? | Your question can be rephrased as follows: I know something about split square-zero extensions. How can I (categorically) conclude something about all square-zero extensions? The key observation is that square-zero extensions are the *torsors* for the split square-zero extensions. I first learned about this general ide... | 2 | https://mathoverflow.net/users/178527 | 390733 | 161,733 |
https://mathoverflow.net/questions/390739 | 8 | This question arose during my Differential Geometry course. Possibly there is an obvious answer, but I do not see it, and I could not find it in the literature. The same question was [asked](https://math.stackexchange.com/questions/4109257/non-orientable-closed-manifold-covered-by-exactly-two-charts) yesterday on MSE, ... | https://mathoverflow.net/users/7460 | Non orientable, closed manifold covered by two simply-connected charts | Take a non-orientable $S^n$ bundle over $S^1$ with $n \geq 2$ (\*), (sometimes called generalised Klein bottles) then covering $S^1$ by two intervals and taking preimages should work.
(\*) Let $\tau : S^n \rightarrow S^{n}$ be a non-orientable diffeomorphism of $S^n$ given by $(x\_{1},\ldots,x\_{n+1}) \mapsto (x\_1,\... | 22 | https://mathoverflow.net/users/99732 | 390740 | 161,734 |
https://mathoverflow.net/questions/390718 | 3 | I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is [the same question on MSE](https://math.stackexchange.com/questions/4108555/bounded-sequence-in-spatial-tensor-product-and-boundedness-of-simple-tensor-summ).
Let $E\subset A$ be a finite dimensi... | https://mathoverflow.net/users/164203 | Elements of the minimal tensor product of a finite dimensional operator system and a $C^*$-algebra | One way to see it is to use the dual basis of $\{x\_1,\dots,x\_n\}$. Let $\{\varphi\_1,\dots,\varphi\_n\} \subset E^{\ast}$ be functionals such that $\varphi\_i(x\_j)=\delta\_{ij}$. Because functionals are automatically completely bounded, we get bounded maps $\Phi\_i:= \varphi\_i \otimes Id : E\otimes B \to B$. On the... | 7 | https://mathoverflow.net/users/24953 | 390749 | 161,736 |
https://mathoverflow.net/questions/386787 | 7 | In his milestone paper on general relativity, Einstein not only introduces the Einstein summation convention, but also (formula (45) in [1]) abbreviates a minus at the Christoffel symbols away by introducing the Gamma notation for the connection coefficients of his variant of the covariant derivative, constructed on co... | https://mathoverflow.net/users/9161 | The Einstein minus convention, lost | Since the question has now narrowed down to "who lost the minus sign" in the Christoffel symbol, let me start a new thread: The OP asks for a reputable source, later than Einstein's 1916 paper, in which the minus sign is abandoned. I propose that it was Einstein himself who dropped it. Below I copy from his [1921 lectu... | 3 | https://mathoverflow.net/users/11260 | 390757 | 161,739 |
https://mathoverflow.net/questions/390764 | 3 | A *hypergraph* $H=(V,E)$ consists of a set $V$ and $E\subseteq {\mathcal P}(V)$, that is, $E$ consists of subsets of $V$ of arbitrary size. Obviously, a graph is a special kind of hypergraph.
Let $H=(V,E)$ be a hypergraph and $\kappa\neq \emptyset$ be a cardinal.
Then a map $c:V\to \kappa$ is said to be a *coloring* ... | https://mathoverflow.net/users/8628 | Coloring the uncountable Lebesgue-measurable sets of $\mathbb{R}$ | It is continuum. The coloring with continuum many colors is clear (all points may have different color). Assume that we have $\kappa<c$ colors. Consider the Cantor set $K$. All its subsets are Lebesgue measurable. If some color contains uncountably many points from $K$, it constitutes a monochromatic edge. So, each col... | 9 | https://mathoverflow.net/users/4312 | 390766 | 161,741 |
https://mathoverflow.net/questions/390768 | 1 | I have a probability vector $p$ s.t. $1^Tp=1$ and $p\geq 0$.
I want to compute the spectral radius of the matrix $M=\text{diag}(p)-pp^T$ where $\text{diag}(p)$ has its diagonal elements as $p$ and its off-diagonals as 0.
| https://mathoverflow.net/users/178766 | What is the spectral norm of the matrix $\text{diag}(p)-pp^T$? | A more general problem is addressed in section 5 of [Golub, G. H. "Some modified matrix eigenvalue problems". *SIAM Review* **15**, 318 (1973)](https://doi.org/10.1137%2F1015032).
For background and other references see the Wikipedia article on the [Bunch-Nielsen-Sorensen formula](https://en.wikipedia.org/wiki/Bunch%... | 3 | https://mathoverflow.net/users/1847 | 390769 | 161,742 |
https://mathoverflow.net/questions/390758 | 6 | We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)\_\*=\mathbb{F}\_p[v\_n^{\pm 1}]$ with $|v\_n|=2(p^n-1)$. I distinctly recall that we should have $\pi\_\*(L\_{K(n)}BP)\cong (v\_n^{-1}BP\_\*)^{\wedge}\_{I\_n}$, yet I am unable to find an explicit referen... | https://mathoverflow.net/users/131453 | Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum | See Lemma 2.3 of the following paper, and the surrounding discussion:
```
@incollection {MR1320994,
AUTHOR = {Hovey, Mark},
TITLE = {Bousfield localization functors and {H}opkins' chromatic
splitting conjecture},
BOOKTITLE = {The \v{C}ech centennial ({B}oston, {MA}, 1993)},
SERIES = {Contemp. Math.},
VOLU... | 10 | https://mathoverflow.net/users/10366 | 390770 | 161,743 |
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