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https://mathoverflow.net/questions/343701 | 6 | I am learning TQFT from compact Lie groups by Freed, Hopkins, Lurie,
and Teleman: <https://arxiv.org/abs/0905.0731> , and got stuck very hard
even in the first section ($n = 1$), which was "trivial but included
for completeness".
In particular, I have several unfamiliar terms while it discusses
"1-dimensional pure ga... | https://mathoverflow.net/users/124549 | 1-dimensional pure gauge theory | A gauge theory in mathematical terms (as I understand it) is a field theory whose fields include a gauge field: a $G$-bundle with connection. In this case, $G$ is a finite group, so there are no non-trivial connections. Thus a gauge field is simply a $G$-bundle.
1) Here, by pure gauge theory, I think the authors mean... | 5 | https://mathoverflow.net/users/7762 | 343713 | 145,852 |
https://mathoverflow.net/questions/343707 | 12 | ### Questions
1. For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| \_{Y=0}.$$ The answer should directly relates to a **counting problem about Feynman diagrams**.
2. Is there a tutorial for how Feynman diagrams work in this context? I look forward to an answer a lot, since the question has been redu... | https://mathoverflow.net/users/124549 | A toy model in 0-d QFT | Draw a point from which $r$ lines emanate (a "vertex"). The ends of the lines are associated with the derivatives. Now let the derivatives act. Two things can happen:
a.) The derivative acts on the exponential. Represent this by attaching a dot to the end of the line (now the derivative is gone, but there's a factor ... | 12 | https://mathoverflow.net/users/134299 | 343714 | 145,853 |
https://mathoverflow.net/questions/343712 | 4 | I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois representation of a modular form or the Tate module of an abelian variety, but I have not considered the families, though I often... | https://mathoverflow.net/users/nan | Motivations of families of modular forms, elliptic curves and Galois representations? | Three applications which come to mind:
1. Greenberg and Stevens "p-adic L-functions and p-adic periods of modular forms" use the 2-variable p-adic L-function (associated to a Hida family) to prove a formula for $L\_p'(E,1)$ for an elliptic curve with split multiplicative reduction in terms of the L-invariant of the ... | 3 | https://mathoverflow.net/users/nan | 343717 | 145,855 |
https://mathoverflow.net/questions/343718 | 6 | It is well known that two $n\times n$ symmetric positive semidefinite matrices $A$, $B$ such that $AB=0$ are simultaneously diagonalizable.
My question is related to the existence of a specific simultaneous diagonalization in the following sense: Let $\{A\_k\}$, $\{B\_k\}$ be two sequences of symmetric matrices conv... | https://mathoverflow.net/users/1172 | On approximate simultaneous diagonalization | The answer is *no* in general.
For a $2\times 2$-counterexample, let $A = 0$, let $B$ be the diagonal matrix with diagonal entries $1$ and $0$ (i.e. $B$ is the projection onto the first component), choose $B\_k = B$ for each $k \in \mathbb{N}$ and
$$
A\_k =
\frac{1}{k}
\begin{pmatrix}
1 & 1 \\
1 & 1
\end{pmatrix}
$... | 7 | https://mathoverflow.net/users/102946 | 343723 | 145,858 |
https://mathoverflow.net/questions/343728 | 8 | The Grothendieck-Verdier duality:
$$
Rf\_\*\big(R\mathcal{H}\textit{om}\_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet\_Y(Rf\_\*\mathcal{E}^\bullet,\mathcal{G}^\bullet)
$$
is known to hold for $f:X\to Y$ being a proper map of noetherian schemes.
Is there a way to ge... | https://mathoverflow.net/users/109370 | Grothendieck-Verdier duality without the noetherian condition |
>
> Does one have the duality in this setting?
>
>
>
Yes, we do have duality in a very general setting. Your question is equivalent to asking for the existence of a right adjoint to the derived pushforward functor $\mathbf{R}f\_\*\colon \mathbf{D}\_{qc}(X) \to \mathbf{D}\_{qc}(Y)$. It turns out that it exists fo... | 9 | https://mathoverflow.net/users/115211 | 343739 | 145,860 |
https://mathoverflow.net/questions/343741 | 2 | Let the unit arc be,
$$\{x \in \mathbb{R}^2| x\_1^2 + x\_2^2 =1, x\_1 \geq 0, x\_2 \geq 0\}$$
There is something I found curious about the unit arc which is that,
* It has an empty interior viewed as a subset of $\mathbb{R}^2$
* It has an empty relative interior viewed as a subset of its affine
hull, which is aga... | https://mathoverflow.net/users/68973 | How to define "interior" for the unit arc? | This seems well described by the notion of interior $\mathrm{Int\ }M$ of a manifold with boundary $M$.
See the Wikipedia subentry [Manifold: Boundary and interior](https://en.wikipedia.org/wiki/Manifold#Boundary_and_interior).
| 0 | https://mathoverflow.net/users/4600 | 343745 | 145,862 |
https://mathoverflow.net/questions/343659 | 3 | What is the tightest lower bound currently known for the Carmichael function?
I imagine it must grow much more slowly than the Euler's totient function which according to [here](https://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below) is bounded as
$$ \phi(n) \ge \frac{n}{e^{\gamma}... | https://mathoverflow.net/users/46536 | Lower bound on Carmichael Function | An answer is given in Theorem 1 of the paper *[Carmichael's lambda function](https://math.dartmouth.edu/~carlp/PDF/lambda.pdf)*, by Erdos, Pomerance, and Schmutz.
**Theorem $1$**: For every $c < 1/\log{2}$, we have $\lambda(n) > (\log{n})^{c \log\log\log{n}}$ for all large enough $n$. On the other hand, for some cons... | 7 | https://mathoverflow.net/users/16510 | 343776 | 145,870 |
https://mathoverflow.net/questions/343749 | 0 | New foundations "NF" (formulated in the language of $\small \sf FOL(\in)$), can define a kind of ordered pair relation $``\rho"$ such that we can have a set $E$ of those pairs where NF proves the existence of an ordered pair $(V,E)$ and at the same time NF proves each sentence $\alpha'$ that replaces each formula $x \i... | https://mathoverflow.net/users/95347 | Can Godel's incompleteness theorems be in some sense circumvented this way? | The role of NF here seems to me a red herring. ZFC (say) already exhibits a similar phenomenon: there is a single formula $\varphi$ such that in every model $M$ of ZFC, $\varphi^M$ is a model of ZFC. PA exhibits a similar phenomenon. I'll talk about both of these below, and also end by addressing an issue in your post.... | 8 | https://mathoverflow.net/users/8133 | 343777 | 145,871 |
https://mathoverflow.net/questions/343763 | 1 | I am conducting an experiment on a mechanical device. The theoy is that there is a function that maps an input force ($F\_I$) to the otput ($F\_O$), i.e. $F\_O = f(F\_I)$ and vice versa (function $f$ is linear). My goal is to test this theory in real-life conditions with an influence of numerous factors. Due to some li... | https://mathoverflow.net/users/147163 | How to compare two experimental samples | $\newcommand{\si}{\sigma}$
Let $X\_1,X\_2,\dots$ be iid random variables (r.v.'s) representing the measurements of the input force. Let $Y\_1=a+bX\_1,Y\_2=a+bX\_2,\dots$ be iid r.v.'s representing the measurements of the output, for some real $a$ and $b$. We assume that $X\_i\sim N(\mu,\si^2)$.
Only the observed val... | 0 | https://mathoverflow.net/users/36721 | 343780 | 145,872 |
https://mathoverflow.net/questions/343779 | 2 | So I came across a formula that looks like:
$x\_n = \alpha x\_{n-1} + \beta$
Since I don't have a strong mathematical background I didn't recognize it was an AGP and as I tried to express $x\_n$ with only the first term $x\_0$ I concluded that this was probably the formula:
$x\_n = \alpha^nx\_0 + \frac{1-\alph... | https://mathoverflow.net/users/146473 | Explanation about arithmetico-geometric progression (AGP) | Let $a:=\alpha$ and $b:=\beta$. If $a=1$, then the
equation
$$x\_n=ax\_{n-1}+b \tag{1}$$
implies $x\_n=x\_0+bn$.
If $a\ne1$, let $y\_n:=x\_n-c$, so that $x\_n=y\_n+c$. Substituting $y\_n+c$ for $x\_n$ in (1), we have $y\_n+c=ay\_{n-1}+ac+b$, which simplifies to $y\_n=ay\_{n-1}$ if $c=ac+b$, that is, if we let $c:=... | 1 | https://mathoverflow.net/users/36721 | 343784 | 145,873 |
https://mathoverflow.net/questions/343791 | 6 | Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers?
That is, the height function would have only Bott-type extrema and saddle singularities. A Bott-type singularity is a non-degenerate singular circle: a circle where the derivative is zer... | https://mathoverflow.net/users/61824 | Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers | I wanted to prove that this is impossible but instead proved that this is possible...
Unfortunately, it is a bit hard to draw the picture but I'll try to explain how this should look like.
**Construction.** In this construction the Klein bottle will be included between the planes $z=0$ and $z=1$. The curves $\{z=... | 7 | https://mathoverflow.net/users/943 | 343792 | 145,875 |
https://mathoverflow.net/questions/343448 | 9 | I would like to know if the following conjecture is correct and if so what's a good citation for its proof.
Let $\mathsf{E}$ be the category of euclidean vector spaces, i.e. objects are finite-dimensional real vector spaces endowed with a scalar product and morphisms are isometries. The tensor powers $(-)^{\otimes a}... | https://mathoverflow.net/users/3041 | Characterising natural transformations between tensor functors | The second question has a well known, affirmative answer from invariant theory.
Note that $O(V)$ acts on every tensor power $V^{\otimes k}$ and by conjugation also on $Hom(V^{\otimes a},V^{\otimes b})$ and the homomorphisms commuting with $O(V)$ are exactly the fixed points of this conjugation action. We have a natur... | 7 | https://mathoverflow.net/users/3041 | 343797 | 145,876 |
https://mathoverflow.net/questions/343796 | 4 | I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an orthogonal $d\times d$ matrix $Q$ such that $QX = Y$. Then $\sim$ is an equivalence relation on $\mathbb{R}^{d\times m}$. I... | https://mathoverflow.net/users/123506 | Parametrizing quotient of matrices by the orthogonal group | It is not a manifold, but there are still ways to talk about the dimension of the quotient space in terms of principal orbits. Chapter 4 of G.E. Bredon's *Introduction to compact transformation groups*, Academic Press, 1972 is a useful reference on this topic, specifically Theorem 3.8 there (re-phrased below).
>
> ... | 2 | https://mathoverflow.net/users/118731 | 343800 | 145,878 |
https://mathoverflow.net/questions/343790 | 3 | As noted in the [recent answer by Yuval Peres](https://mathoverflow.net/questions/343769/is-the-sum-of-a-bunch-of-uniform-distributions-say-u-c-c-each-of-which-has-a), the sum of independent uniformly distributed random variables (r.v.'s) cannot have a normal distribution.
The question is, what happens without the i... | https://mathoverflow.net/users/36721 | Normality of the sum of uniformly distributed random variables | Yes. Let the random variables $V\_i$ be independent, with $V\_i$ uniform on $[-\frac 1i,\frac 1i]$. Let $N$ be an independent standard normal. Inductively define $Z\_i\in \{\pm 1\}$ by
$$
Z\_i=
\begin{cases}
1&\text{if $\text{sgn} (N-\sum\_{j<i}Z\_jV\_j)=\text{sgn}(V\_i)$}\\
-1&\text{otherwise.}
\end{cases}
$$
Notice ... | 6 | https://mathoverflow.net/users/11054 | 343810 | 145,882 |
https://mathoverflow.net/questions/343782 | 3 | Inspired by [this MSE question](https://math.stackexchange.com/questions/3383009/on-equation-exp-x-y-exp-x-exp-y-on-a-lie-group-and-its-lie-algebra) we ask the following question:
Is there a noncommutative $C^\*$-algebra $A$ for which the following identity holds for all $x,y \in A$?
$$e^{(xy-yx)}= e^xe^y e^{-x}e^{... | https://mathoverflow.net/users/36688 | On equation $e^{xy-yx}=e^xe^ye^{-x}e^{-y}$ in $C^*$ algebras | Yes:
>
> A $C^\*$-algebra satisfies the identity $e^{[xy-yx]}=e^xe^ye^{-x}e^{-y}$ iff it is commutative.
>
>
>
This follows from two independent facts (I write $[x,y]=xy-yx$)
>
> 1) A (real/complex) unital Banach algebra satisfies the identity $e^{[xy-yx]}=e^xe^ye^{-x}e^{-y}$ $\Leftrightarrow$ it satisfie... | 14 | https://mathoverflow.net/users/14094 | 343814 | 145,884 |
https://mathoverflow.net/questions/343766 | 2 | For a complex Lie group $G$, with $B$ a choice of Borel subgroup. The line bundles over the flag manifold $G/B$ are indexed by elements of the weight lattice of $\frak{g}$. Which of these line bundles is ample?
| https://mathoverflow.net/users/143172 | Classifying ample line bundles over the flag manifold $G/B$ | As Donu said, line bundles on $G/B$ correspond to elements of the weight lattice. The ample bundles correspond to the dominant regular weights. This means that the pairing of the weight $\lambda$ with any positive root is strictly positive (or strictly negative depending on your conventions for indexing the line bundle... | 5 | https://mathoverflow.net/users/7762 | 343827 | 145,886 |
https://mathoverflow.net/questions/343824 | 3 | I'm interested in understanding how one may associate modular symbols to the L-functions and $p$-adic L-functions associated to the Rankin Selberg convolution of two modular forms/ elliptic curves and the symmetric square of a modular form/elliptic curves. Looking through the literature I see that the $\operatorname{GL... | https://mathoverflow.net/users/nan | Modular symbols associated to Rankin Selberg convolutions and the symmetric square | I do not think there is a reference for this theory, because as far as I know no such theory exists. I have spent a substantial portion of my career studying the arithmetic of the special values of the $GL\_2 \times GL\_2$ Rankin--Selberg L-function, and I am not aware of a theory of modular symbols in this setting.
| 4 | https://mathoverflow.net/users/2481 | 343839 | 145,888 |
https://mathoverflow.net/questions/343409 | 10 | Let $\mathrm{X}$ be a cubic $d$-fold, and $\mathrm{F}(\mathrm{X})$ its Fano variety of lines. *Is the integral cohomology of $\mathrm{F}(\mathrm{X})$ torsion-free?* For $d=3$ A. Collino (`The fundamental group of the Fano surface, I') proves that there exists an exact sequence $$[\pi\_1,\pi\_1]\rightarrow\pi\_1\rightar... | https://mathoverflow.net/users/104669 | Torsion in the cohomology of Fano varieties of lines | Here is an expanded version of my comments. Let's work over the complex numbers which I suppose is assumed in the question. Let $K\_0(Var)$ be the Grothendieck ring of varieties (see e.g. <https://arxiv.org/abs/1405.5154>). Furthemore, let $K\_0(HS)$ be the Grothendieck group of integral polarizable Hodge structures wi... | 7 | https://mathoverflow.net/users/111491 | 343842 | 145,889 |
https://mathoverflow.net/questions/335637 | 3 | Let $G$ be an infinite countable group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is [ACC](https://en.wikipedia.org/wiki/Ascending_chain_condition) of infinite length, and for every $K \in (H,G]$, $G$ is generated by a single $K$-coset (i.e. there is $g \in G... | https://mathoverflow.net/users/34538 | Infinite group generated by a single coset | The answer is "no". Let $G$ be a Tarski torsion-free monster, $H$ be the trivial subgroup. Then $(H,G]$ consists of $G$ and all cyclic subgroups of $G$, it has ACC and infinite length. If $K$ is a cyclic subgroup and $a$ is not in the cyclic centralizer $C(K)$, then the coset $aK$ contains two non-commuting elements, s... | 3 | https://mathoverflow.net/users/nan | 343850 | 145,892 |
https://mathoverflow.net/questions/343861 | 6 | Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ such that $X(\mathbb{C})$ is homeomorphic to $Y(\mathbb{C})$. Is it true that $\dim\_{\mathbb{C}} H^k(X,\mathcal{O}\_X)=\dim\_{\mathbb{C}} H^k(Y,\mathcal{O}\_Y)$ for $k\ge 2$? (Note that for $k=1$ it follows by Hodge theory).
| https://mathoverflow.net/users/116075 | Is $h^{0,k}$ a topological invariant? | The answer is *no*.
There are counterexamples already for surfaces, due to X. Gang and F. Campana (unpublished). The link to Campana's article is [here](http://www.iecl.univ-lorraine.fr/~Pierre-Yves.Gaillard/DIVERS/hodgenumbers.pdf), and the relevant result is
>
> **Proposition 0.1** Il existe des surfaces proj... | 13 | https://mathoverflow.net/users/7460 | 343862 | 145,893 |
https://mathoverflow.net/questions/343378 | 2 | I am wondering about a natural generalization of theorem 1.4 in the article [*Dvoretzky's theorem — Thirty years later*](https://link.springer.com/article/10.1007/BF01896663) by Milman. My first thought was to look at Milman's paper that he cites for the result (*On a property of functions defined on infinite-dimension... | https://mathoverflow.net/users/83901 | Concentration of measure on finite powers of $S^\infty$ | I was able to track down the paper and after Theorem 3, which is the result in question in the $k=2$ case he says:
>
> Аналогичный теореме 3 факт имеет место и для функции к переменных. Приведем формулировку для трех переменных, поскольку дальнейшие обобщения очевидны.
>
>
>
Which a certain website translates ... | 0 | https://mathoverflow.net/users/83901 | 343886 | 145,898 |
https://mathoverflow.net/questions/343854 | 3 | Look at Bernoulli percolation on $\mathbb{Z}^2$ with $p> p\_c$ ($p$ can be arbitrarily close to 1).
I am interested in the probability that there exists an infinite cluster starting at $(0,0)$ and it stays above the graph $y=x^n$ where $n$ is a large even number.
Is there a relation between $p$ and $n$ so that thi... | https://mathoverflow.net/users/49551 | Bernoulli percolation, infinite path from (0,0) in a "cone" | This is true for every positive $n$. I assume that for $n$ odd or fractional, the bounding curve is $y=|x|^n$. If $(0,0)$ is not connected to infinity, then there must be a blocking contour in the dual lattice. The exponential decay iof connectivity in subcritical percolation and the Borel Cantelli Lemma preclude that.... | 6 | https://mathoverflow.net/users/7691 | 343888 | 145,899 |
https://mathoverflow.net/questions/343625 | 2 | For any embedding of *smooth* varieties $X\subset Y$ given by an ideal sheaf $I\subset \mathcal {O}\_Y$, it is well known that the *normal cone* $C \_{X/Y}= \textbf{Spec}(\oplus\_{\geq 0} I^i/I^{i+1})$ is isomorphic to the normal bundle of $X$ to $Y$. Therefore, its fiber over each point $x \in X$ is the affine space a... | https://mathoverflow.net/users/145172 | Fibers of the normal cone | Yes, $X$ must be smooth. The question is local at $x \in X$. Write $R$ for the local ring $\mathcal{O}\_{Y,x}$, and $\mathfrak{m}$ for the maximal ideal of $R$. Write $I$ for the ideal defining $X$. Then $(C\_{X/Y})\_x = \mathrm{Spec}(\oplus I^n/mI^n)$. Its Krull dimension is called the analytic spread of $I$; it is at... | 0 | https://mathoverflow.net/users/14895 | 343895 | 145,903 |
https://mathoverflow.net/questions/343607 | 4 | If $G(n,p)$ is a random graph of the [Erdös-Rényi](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model) model,
what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$
Please feel free to relate answers to other models of random graphs.
| https://mathoverflow.net/users/31310 | Probability of a vertex being a "degree-celebrity" in a random graph | Calculating analogously to the friendship paradox:
In the large $N$ limit with fixed average degree $d=np$ the degree of each node is distributed as Poisson with mean $d$ while each neighbor is Poisson with mean $d+1$. So we condition on $k$ and compute the probability that all the neighbors have degree less than $k$... | 4 | https://mathoverflow.net/users/39187 | 343901 | 145,906 |
https://mathoverflow.net/questions/343577 | 1 | I asked similar question before, after some modification, I have a new question. Suppose $M^{n\geq 4}$ is a connected compact smooth manifold with connected nonempty boundary. Suppose $i\_\*: H\_1(\partial M)\rightarrow H\_1(M)$ is not injective, for simplicity, assume that $\dim (\ker(i\_\*))=1$, that means, there is ... | https://mathoverflow.net/users/120509 | Injectivity of homomorphism between homology groups of manifold and its boundary | I am not sure whether this is the kind of thing you want to know, but here is a procedure to make $\partial M \rightarrow M$ an isomorphism on $\pi\_1$ if $\text{dim}(M) \geq 5$.
As $M$ is compact, $\pi\_1 M$ is finitely presented. Given one of the finitely many generators of $\pi\_1 M$, represent it by an embedded c... | 1 | https://mathoverflow.net/users/134967 | 343905 | 145,908 |
https://mathoverflow.net/questions/343876 | 7 | Related to Hilbert's Tenth problem.
Is there polynomial with integer coefficients $P(a,x\_1,...,x\_n)$
such that $P(A,X\_i)=0$ has rational solutions $X\_i$ iff $A$ is
not the square of integer (or as another question not the square
of rational)?
We think if $P$ is homogeneous and ask about integer solutions,
scali... | https://mathoverflow.net/users/12481 | Rational Diophantine set for the non-squares | The set $A$ of non-squares (of rationals) is Diophantine in $\mathbb{Q}$ by [1]. The set $B:=\mathbb{Q}\smallsetminus\mathbb{Z}$ is also Diophantine by [2]. The set of non-squares of integers is equal to $A\cup B$, hence Diophantine.
For a generalization of [1], see also [3].
**[EDIT]** The paper [1] treats arbit... | 10 | https://mathoverflow.net/users/7666 | 343911 | 145,910 |
https://mathoverflow.net/questions/343890 | 7 | A friend and I are writing a paper that uses the BGG resolution of $L(\lambda)$ (where $\mathfrak g$ is a semisimple complex Lie algebra, $\lambda \in P^+$ is a dominant integral weight, and $L(\lambda)$ the simple finite module with highest weight $\lambda$).
A point is unclear for us : one needs to choose appropri... | https://mathoverflow.net/users/104742 | Unicity of the BGG complex | Theorem 33 in the preprint [1] gives the uniqueness of BGG resolutions (= direct sums of Verma modules resolving a simple module) in category $\mathcal{O}$, both in regular and singular blocks. (However, this does not apply directly to parabolic versions of category $\mathcal{O}$).
---
[1]: Mazorchuk-Mrđen: BGG c... | 5 | https://mathoverflow.net/users/15292 | 343929 | 145,916 |
https://mathoverflow.net/questions/343937 | 0 | **Definitions :**
>
> $(E,d)$ a metric space is finished-compact if any covering of $E$ by open, we can extract a finite subcover
>
>
> $(E,d)$ is $\aleph\_0$-compact if for any infinite covering of $E$ by open, we can extract a countable subcover
>
>
>
**Remark :**
>
> we can imagine what's $\aleph\_i$-... | https://mathoverflow.net/users/110301 | About the finished, $\aleph_0$...-compactness | I think you are asking about *cardinal functions* on metric spaces. First, the property you call $\aleph\_0$-compact is more usually known as *Lindelöf*. As you say, it is well-known that for a metric space $E$, it is Lindelöf iff it is separable (iff it is second-countable iff it is c.c.c.). Of course, for topological... | 3 | https://mathoverflow.net/users/61785 | 343945 | 145,920 |
https://mathoverflow.net/questions/343942 | 3 | Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times\_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic of $E$ in terms of the inducing representation $V$?
I am also interested in the question of how the $G$-module of ho... | https://mathoverflow.net/users/126606 | Euler characteristic of a holomorphic homogeneous vector bundle | In the case $G$ is a complex semisimple Lie group and $P$ its parabolic subgroup, the answer is given by Kostant's version of Borel-Bott-Weil theorem [K]. Any homogeneous vector bundle is given by a associated bundle construction from some $P$-representation $\mathbb{F}$. In general, one can consider also $P$-represent... | 4 | https://mathoverflow.net/users/6818 | 343947 | 145,921 |
https://mathoverflow.net/questions/343914 | 5 | Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}\_p$ with coefficients in $\mathbb{Q}\_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are elements in $\overline{\mathbb{Q}}\_p$.
**My question is**: are the generalized Hodge-Tate weight always elements of $... | https://mathoverflow.net/users/143589 | Generalized Hodge-Tate weights of an arbitrary p-adic Galois representation | If $K$ is a finite extension of $Q\_p$, let $G\_K = Gal(Q\_p^{alg}/K)$. Let $\chi: G\_K \to Z\_p^\times$ be the cyclotomic character. If $n \geq 1$ and $K$ is large enough, then $\chi(G\_K) \subset 1+p^{n+1} Z\_p$ so that $\chi(g)^{1/p^n}$ (using the binomial series) converges and makes sense if $g \in G\_K$. This give... | 7 | https://mathoverflow.net/users/5743 | 343949 | 145,922 |
https://mathoverflow.net/questions/343910 | 2 | Helmholtz (-Hodge) decomposition commonly used in physics includes decomposition of a (sufficiently smooth) vector field $F = -\mathrm{grad}(U) + \mathrm{curl}(W)$ on bounded simply connected domain $\Omega \subseteq \mathbb{R}^3$ (with smooth boundary), with scalar $U$ and vector field $W$ which are explicitly given b... | https://mathoverflow.net/users/114094 | Proof of Helmholtz-Hodge decomposition, poor man's version | For avoiding $\delta$, you should be able to just do as follows:
Let $G(x)$ denote the Newton potential $\frac{1}{4\pi |x|}$, and $G\_y(x) = G(x-y)$
Let $r'(x)$, for $x \in \Omega$, denote $\frac12 d(x, \Omega^c)$.
Given $y\in \Omega$, consider the integral for $\lambda\in (0,1)$
$$ \tilde{F}(\lambda,y) = - \... | 2 | https://mathoverflow.net/users/3948 | 343963 | 145,926 |
https://mathoverflow.net/questions/343959 | 3 | I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K\_\mathfrak{A}$ as the group
$\{ g \in G \mid g\mathfrak{A}g^{-1} = \mathfrak{A} \} $, and it can be proved that $K\_\mathfrak{A... | https://mathoverflow.net/users/141462 | A detail in Bushnell and Henniart's book, “The local Langlands conjecture for GL(2)” | I do not give all details, but the main steps of the calculation. Put $E=F[\alpha ]$ and $m=[n/2]+1$.
First $K\_{\mathfrak A}/J\_\alpha =U\_{\mathfrak A}/(U\_{\mathfrak A}\cap E^\times )U\_{\mathfrak A}^m$.
Next observe that $U\_{\mathfrak A}\cap E^\times = U\_E$, where $U\_E$ is the group of units of the ring of ... | 2 | https://mathoverflow.net/users/4767 | 343967 | 145,928 |
https://mathoverflow.net/questions/343966 | 3 | **EDIT** Let $\mathcal{O}$ be the ring of integers in a non-Archimedean local field. Let $GL\_n(\mathcal{O})$ be the (compact) group of $n\times n$ matrices with entries in $\mathcal{O}$ such that its inverse also has entries in $\mathcal{O}$.
>
> Consider a continuous finite dimensional representation of $GL\_n(\m... | https://mathoverflow.net/users/16183 | Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$ | Call the representation $\pi$. Let $U$ be a neighbourhood of the identity in $\operatorname{GL}(V)$ that contains no non-trivial subgroup. Then the pre-image of $U$ under $\pi$ is a neighbourhood of the identity there, hence contains some open subgroup. That is, $\ker(\pi)$ contains an open subgroup, hence is open, hen... | 5 | https://mathoverflow.net/users/2383 | 343969 | 145,929 |
https://mathoverflow.net/questions/343746 | 0 | Define: $n$-skew pair of $x,y$, denoted by $\langle x,y \rangle^n$, as: $(singleton^n(x), y)$
Define: $(-n)$-skew pair of $x,y$, denoted by $\langle x,y \rangle^{-n}$, as: $(x, singleton^n(y))$
Where $(-,-)$ is the Kuratwoski ordered pair implementation, and $n$ is a natural
where: $singleton^0(x) = x$
$single... | https://mathoverflow.net/users/95347 | Is Cantor-Bernstein-Schroeder theorem for skew cardinality, consistent with NF? | No. A counterexample to skew CBS follows from the existence of any cardinals
T(kappa) < lambda < kappa [cardinals in NF(U) are Frege cardinals; T(kappa) is
the cardinality of elementwise images of elements of kappa under the singleton map.]
This is a counterexample because an element of lambda will have an ordinary i... | 6 | https://mathoverflow.net/users/130007 | 343985 | 145,932 |
https://mathoverflow.net/questions/343977 | 0 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact metric space. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable functions such that for each $\omega$, $F(\omega,\,\cdot\,) \colon X \to X$ is a homeomorphism with inverse $\bar{F}(\omega,\,... | https://mathoverflow.net/users/15570 | If a probability measure is stationary in both forward time and reverse time, does this imply that the measure is incompressible? | Not at all. Let me reformulate your question in terms of the group $G$ of homeomorphisms of $X$ (and replacing your $\mathbb P$ with $\mu$ - which is more conventional in this context). Essentially you are saying that there is a measure $\mu$ on $G$ and a measure $\rho$ on $X$ which is stationary with respect to both $... | 1 | https://mathoverflow.net/users/8588 | 343988 | 145,934 |
https://mathoverflow.net/questions/343646 | 0 | The [HoTT Book](https://homotopytypetheory.org/book/) states in the first chapter that universes are cumulative and that every universe is in some other universe.
Obviously, there needs to be an infinite number of universes then, but universes indexed by natural numbers a-priori seem to suffice.
It also states that... | https://mathoverflow.net/users/124996 | A sequence in the hierarchy of universes | I think you'll find that your definition of $T$ doesn't require as many universes as you think. It's mainly unclear because you've elided subscripts. For instance, what if you wrote:
$$
T : ℕ → \mathcal{U}\_1 \\
T(0) = ℕ \\
T(n+1) = T(n) → \mathcal{U}\_0
$$
That's a valid way of assigning subscripts, and it only re... | 1 | https://mathoverflow.net/users/5880 | 343991 | 145,935 |
https://mathoverflow.net/questions/342457 | 7 | Suppose you have a zero-dimensional ideal $I=(f\_1,...,f\_n)$ in a polynomial ring $R=k[x\_1,...,x\_n]$ over a field $k$, so that $\dim\_k(R/I)<\infty$. Take indeterminates $\alpha\_1,...,\alpha\_n$ and consider the ideal $J=(f\_1-\alpha\_1,...,f\_n-\alpha\_n)$ of the ring $S=K[x\_1,...,x\_n]$ over the field $K=k(\alph... | https://mathoverflow.net/users/146401 | Given a zero-dimensional ideal $(f_1,...,f_n)$, is the ideal $(f_1-\alpha_1,...,f_n-\alpha_n)$ also zero-dimensional? | If $f$ is not generically finite-to-one then there are easy counterexamples (e.g. take $f\_1 = x\_1$, $f\_2 = x\_1 - 1 \in k[x\_1, x\_2]$). So assume $f$ is generically finite-to-one.
**Claim 1:** If $f$ is generically finite-to-one and $\dim\_k(R/I) < \infty$, then $\dim\_k(R/I) \leq \dim\_K(S/J) < \infty$.
*... | 4 | https://mathoverflow.net/users/1508 | 343994 | 145,937 |
https://mathoverflow.net/questions/343990 | 1 | Let $H$ be a finite dimensional semisimple Hopf algebra over $\mathbb{C}$. Can one choose a basis $\{h\_1, \dots, h\_n \}$ of $H$, where $h\_1 = 1$, such that if we write
$$
\Delta(h\_i) = \sum\_{1 \leq j,k \leq n} \alpha\_{ijk} h\_j \otimes h\_k
$$
for some $\alpha\_{ijk} \in \mathbb{C}$, then $\alpha\_{ij1} = 0$ for... | https://mathoverflow.net/users/147317 | "Nice" bases for finite dimensional semisimple Hopf algebras | Given a basis for $H$ of the sort you are looking for, let $V$ be the linear span of $h\_2, \dots, h\_n$. Then $V \cap \mathbb C1 = 0$ and $\Delta(V) \subset V \otimes V$. Conversely, given such a $V$, any basis for $V$, together with 1, will do what you ask.
Carefully dualizing, we are asking if the dual Hopf algebr... | 1 | https://mathoverflow.net/users/102519 | 343998 | 145,938 |
https://mathoverflow.net/questions/343972 | 2 | Let $S\subseteq \mathbb D:=\{z\in \mathbb C:|z|<1\}$. Suppose that $\overline S$, the Euclidean closure of $S$
meets $\mathbb T=\{z\in\mathbb C:|z|=1\}$. A point $\xi=e^{i\theta}\in \overline S\cap \mathbb T$ is called a non-tangential boundary
point, if there is a cone
$$S\_\alpha(\theta):=\{z\in \mathbb D: |\arg(... | https://mathoverflow.net/users/61993 | Measurability of the set of non-tangential boundary points | For given $\alpha$ the corresponding set of nontangential boundary points is a $G\_{\delta}$ set, since the cone must contain points of $S$ in annuli arbitrarily close to the unit circle. Here $G\_n$ consists of boundary points $\xi$ where the cone $S\_\alpha(\theta)$ intersects $\{z: 1-1/n<|z|<1\} \cap S$. Thus
the s... | 5 | https://mathoverflow.net/users/7691 | 344000 | 145,939 |
https://mathoverflow.net/questions/343997 | 2 | For a labeled tree $T$ on $\{1,2,...,n\}$ an *inversion* of $T$ is a pair $1 < i < j \leq n$ such that $j$ belongs to the unique path from $1$ to $i$ (we think of $T$ as being rooted at $1$). Let $\mathrm{inv}(T)$ denote the number of inversions of $T$.
Define the generating function $f(q) := \sum\_{T} q^{\mathrm{inv... | https://mathoverflow.net/users/25028 | Sign-reversing involution proof of tree inversion generating function at $-1$ equals number of alternating permutations? | J.-J. Pansiot, [Nombres d'Euler et Inversions dans les Arbres](https://www.sciencedirect.com/science/article/pii/S0195669882800371), Europ. J. Combin. 3 (1982), 259–262, uses a sign-reversing involution to show that $f(-1)$ is the number of increasing trees on $[n]$ in which every vertex other than the root has an even... | 5 | https://mathoverflow.net/users/10744 | 344001 | 145,940 |
https://mathoverflow.net/questions/343979 | 6 | I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $\*$-algebras". There he have discussed the Arens regularity of common Banach algebras like $L^1(G)$, $C^\*$-algebras,$M(G)$, $K(H)$ etc. Some other primary Banach algebras that com... | https://mathoverflow.net/users/145729 | Arens regularity of Banach algebras | For the question about trace-class operators, you could look at my [New York Journal article](http://nyjm.albany.edu/j/2007/13_215.html) There is a survey of sorts; see Theorem 5.39 for your question. For the trace-class operators in particular, I cite Dales's book, Theorem 2.6.23, where the result is attributed to Ulg... | 5 | https://mathoverflow.net/users/406 | 344006 | 145,943 |
https://mathoverflow.net/questions/157519 | 6 | Let $DF$ denote the category whose objects are categories and whose morphisms $F\colon R\to S$ are the discrete fibrations. This category has applications to the real-world problem of structuring data. You can think of any discrete fibration $R\to S$ as providing a schematic structure $S$ for more raw data $R$. This qu... | https://mathoverflow.net/users/2811 | Does there exist a terminal surjective discrete fibration out of $C$? | In general, the answer is "No": the category $DFS\_{C/}$ of surjective discrete fibrations under $C$ need not have a terminal object. This is due to the following:
**Lemma.** Let $C$ be the [codiscrete category](https://ncatlab.org/nlab/show/indiscrete+category) with $n$ objects. For any group $G$ with $n$ elements, ... | 3 | https://mathoverflow.net/users/5768 | 344007 | 145,944 |
https://mathoverflow.net/questions/343889 | 2 | Suppose that $(X,d)$ is a Polish metric space and $A$ is a set of continuous bounded functions $f:X\to \mathbb{R}$.
Suppose that $\mu:X\to[0,1]$ is a Borel probability measure.
Define
$$\sup A:X\to (-\infty,+\infty],\quad \sup A(x):=\sup\{f(x)\colon f\in A\},
$$
On the other hand, we have the essential suppremum, t... | https://mathoverflow.net/users/70540 | Supremum of continuous functions and essential supremum of continuous functions | The claim now is correct.
For the following proof we only need that $X$ is separable metric and $A$ a family of lower semicontinuous (l.s.c.) functions on $X$. (Of course these assumptions may be further weakened.) To simplify notation we may assume that $0 \leq f \leq 1$ for each $f \in A$. Let $g$ be an arbitrary r... | 1 | https://mathoverflow.net/users/100904 | 344018 | 145,948 |
https://mathoverflow.net/questions/343719 | 3 | In the paper "Orbispaces, orthogonal spaces, and the universal compact Lie group" by Stefan Schwede, he studies (spaces with an action of) the topological monoid $\mathbf{L}(\mathbb R^\infty,\mathbb R^\infty)$ of linear isometric self-embeddings of ${\mathbb R}^\infty$ equipped with the subset topology of $\operatornam... | https://mathoverflow.net/users/147135 | Homotopy type of linear isometric self-isomorphisms of ${\mathbb R}^\infty$ | There are two related questions for which the answer is known. Put $H\_0=\mathbb{R}^\infty$, and let $H$ be the Hilbert space completion of $H\_0$. Let $G\_0$ be the colimit of the orthogonal groups $O(\mathbb{R}^n)$. Let $G\_1$ be the group of pairs $(g,h)\in\mathbf{L}(H\_0,H\_0)^2$ with $fg=gf=1$, topologised as a su... | 6 | https://mathoverflow.net/users/10366 | 344027 | 145,949 |
https://mathoverflow.net/questions/343488 | 4 | (First asked in [MSE](https://math.stackexchange.com/questions/3385521/equations-in-the-brauer-algebra))
The Brauer algebra $B\_n(x)$ is an algebra of matchings whose product is described [here](https://en.wikipedia.org/wiki/Brauer_algebra).
Given $A$ and $B$ two elements of $B\_n(x)$, and given an integer $m$, the... | https://mathoverflow.net/users/83671 | Solving equations in the Brauer algebra | This answer does not give a full answer to the question. I started working out what I could; so, I'll post it in case it is of use to anyone.
Let $[n] = \{1,2,\dots, n\}$ and $[n'] = \{1',2', \dots, n'\}$. We will use the convention that $i'' = i$. A diagram $D$ is a perfect matching of the the complete graph on the ... | 1 | https://mathoverflow.net/users/51668 | 344032 | 145,950 |
https://mathoverflow.net/questions/344034 | 6 | Suppose I have a compact set $K \subset B\_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B\_{r\_j}(x\_j)\}$ such that
1. $x\_j \in K$ and $B\_{r\_j}(x\_j) \subset B\_1(0)$ for each $j$;
2. $K \subset \bigcup\_j B\_{r\_j}(x\_j)$; and
3. The collection $\{B\_{2r\_j}(x\_j)\}$ has bounded overla... | https://mathoverflow.net/users/122587 | Can I cover a compact set by balls {B} such that {2B} has bounded overlap? | Let $r$ be less than half the distance from $K$ to the complement of $B\_1(0)$. Start with a ball of radius $r$ centered at each point of $K$. By <https://en.wikipedia.org/wiki/Besicovitch_covering_theorem> there is a subcover of $K$ by balls that is a union of $C\_n$ collections $A\_i$, where each $A\_i$ consists of p... | 6 | https://mathoverflow.net/users/7691 | 344036 | 145,952 |
https://mathoverflow.net/questions/343851 | 2 | Let $n$ be an odd positive integer, Let $o=\operatorname{ord}\_n 2$ be the order of 2 modulo $n$ and $m$ the period of $1/n, k$ is number of distinct odd residues contained in set $\{2^1,2^2,...,2^{n−1}\}$ modulo $n$.
If $o,m$ and $k$ are even power of 2 and $k$ divide $n-1$, then $n$ is item in the sequence $17, 257... | https://mathoverflow.net/users/33646 | Question on odd positive integers and Fermat factors | *Hmm, unfortunately you deleted instructive portions of your OP, but let me answer based on that earlier version which included explicitela the iterations scheme.*
You give a game of iteration on numbers, which I'd like to rewrite for some given positive odd number $z$ and an initial $a=1$ :
$$ b = {z+a\over 2^A ... | 4 | https://mathoverflow.net/users/7710 | 344040 | 145,953 |
https://mathoverflow.net/questions/344042 | 1 | Let $\Omega\_x$ and $\Omega\_y$ be sets of finite Lebesgue measure.
We can then look at the space $X\_1:=L^2(\Omega\_x \times \Omega\_y).$
This space is contained in the larger space
$$X\_0:=L^2(\Omega\_x; L^1(\Omega\_y)).$$
On the other hand, the space $$X\_2:=L^2(\Omega\_y;L^{\infty}(\Omega\_x))$$ is smaller... | https://mathoverflow.net/users/nan | Interpolation of $L^p$ spaces | No. Notice that since you assumed finiteness of the $\Omega\_x$ measure, we can restrict to looking at functions that are constant in $x$.
Let $\Omega\_y = [0,1]$ for example, and take $\chi\_n(y) = \sqrt{n} \mathbb{1}\_{[0,1/n]}$.
The sequence of functions $\chi\_n \overset{L^1}{\longrightarrow} 0$, and is bound... | 3 | https://mathoverflow.net/users/3948 | 344043 | 145,955 |
https://mathoverflow.net/questions/344011 | 5 | Is there an injective map $j:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ satisfying: For every $z\in \mathbb{Z}$ we have $$\lim\_{N\to \infty}\sum\_{k=-N}^Nj(k,z) = 0 = \lim\_{N\to \infty}\sum\_{k=-N}^Nj(z,k)\text{ ?}$$
That is, for every $z\in \mathbb{Z}$ there is $N\_0=N\_0(z)\in \mathbb{N}$, such that for every in... | https://mathoverflow.net/users/8628 | Magic $\mathbb{Z}\times\mathbb{Z}$-square | I'll try to show that we can make a square such that all rows and columns give zero sum. (Where "sum" is meant in the sense described in the question.) I.e., this is the answer to the stronger variant.1
The description is a bit informal, but I hope it could be clear how the construction goes.
We will proceed by ind... | 5 | https://mathoverflow.net/users/8250 | 344049 | 145,956 |
https://mathoverflow.net/questions/344067 | 1 | Similar questions have already been asked [here](https://mathoverflow.net/q/268743/18238) and [here](https://mathoverflow.net/q/53548/18238) but not exactly in the direction I need.
I have a (small) index category $\mathcal{I}$ which is **not** cofiltered, and I need to consider categories of projective systems, inde... | https://mathoverflow.net/users/18238 | Need of filtered indexed categories | The fact that categories of projective (or inductive) systems over arbitrary categories in abelian categories are again abelian is explained in Grothendieck's Tohoku paper (Sections 1.6 and 1.7).
| 2 | https://mathoverflow.net/users/11025 | 344070 | 145,959 |
https://mathoverflow.net/questions/344029 | 1 | Can we find a universal constant $c>0$ such that for all $p,q\in\Delta:=\lbrace x\in (0,1)^{n}\ \colon\ x\_{1}+\dots+x\_{n}=1\rbrace$ it is true that
\begin{equation}
|p\_{i}-q\_{i}|\le c\left|\ln\frac{p\_{i}}{g(p)}-\ln\frac{q\_{i}}{g(q)}\right|,\quad i=1,\dots,n
\end{equation}
where $g(p):=\left(\prod\_{i=1}^{n}p\_{i}... | https://mathoverflow.net/users/85194 | Lower bound for log-Ratios | Already in dimension $n=3$ the answer is negative.
Let $p=(1/3,1/3,1/3)$ and $q=(1/7,2/7,4/7)$. Then $g(p)=1/3$ and $g(q)=2/7$ so $p\_2/g(p)=q\_2/g(q)$.
| 1 | https://mathoverflow.net/users/7691 | 344077 | 145,962 |
https://mathoverflow.net/questions/343958 | 16 | Let $M$ be a smooth *compact* manifold of dimension $n$, and let $U$ be a smooth *compact* manifold with boundary, of the same dimension $n$, embedded in $M$.
The embedding induces maps on $\pi\_1$.
If $\pi\_1(\partial U) \to \pi\_1(M)$ is injective, does this imply that $\pi\_1(U) \to \pi\_1(M)$ is injective?
I... | https://mathoverflow.net/users/14105 | Does injectivity of $\pi_1(\partial U) \to \pi_1(M)$ imply injectivity of $\pi_1(U) \to \pi_1(M)$? | The answer is 'yes' by *Britton's lemma* (see [wikipedia](https://en.wikipedia.org/wiki/HNN_extension#Britton's_Lemma) and, more generally, Serre's book *Trees* and Scott and Wall's article 'Topological methods in group theory').
Since $M$ and $U$ are smooth and compact, $\partial U$ has a product neighbourhood $N(\... | 26 | https://mathoverflow.net/users/1463 | 344084 | 145,965 |
https://mathoverflow.net/questions/344095 | 6 | For a field $K\subset \mathbb{Q}(\zeta\_p)$ $~$($\zeta\_p$ a primitive pth root of unity, p a prime), it seems to be the case that the discriminant of $K$ is $p^{[K:\mathbb{Q}]-1}$ (according to Sage). How can I prove that/ is the proof implied by something written down somewhere?
I have a feeling this is somewhat d... | https://mathoverflow.net/users/147392 | discriminant of subfield of $\mathbb{Q}(\zeta_p)$ | The Führerdiskriminantenproduktformel tells you that is it the product of conductors of characters, but all but the trivial character must have conductor $p$.
| 10 | https://mathoverflow.net/users/5015 | 344099 | 145,970 |
https://mathoverflow.net/questions/344056 | 5 | I am reading a paper (*[Finitely presented residually free groups](https://arxiv.org/abs/0809.3704)* by Bridson, Howie, Miller III, and Short, Theorem 5.2) where they write the following:
Since $S\_{0}$ is a group of type $FP\_{n}(\mathbb{Q})$ and there is a series $$S\_{0}\triangleleft S\_{1}\triangleleft \cdots \tr... | https://mathoverflow.net/users/142651 | Extension of $FP_{n}$ group | If $0\to A \to B \to C\to 0$ is a short exact sequence of groups and $A$, $C$ are of type $FP\_{n}$, then so is $B$. Reference: Proposition 2.2 in *Homological finiteness properties of fibre products* by Kochloukova and Ferreira Lima ([arXiv link](https://arxiv.org/abs/1611.03759)).
| 7 | https://mathoverflow.net/users/142651 | 344105 | 145,971 |
https://mathoverflow.net/questions/324125 | 5 | Consider the natural action of $W\_1=k\left\langle x,\frac{d}{dx}\right\rangle$ on $X=\mathbb C[x]$. Then $\frac{d}{dx}, x\frac{d}{dx},x^2\frac{d}{dx}$ is essentially a $\mathfrak{sl}\_2$-tuple ($\left[x\frac{d}{dx},\frac{d}{dx}\right] =-\frac{d}{dx}$, $\left[x\frac{d}{dx},x^2\frac{d}{dx}\right]= x^2\frac{d}{dx}$, $\le... | https://mathoverflow.net/users/102104 | Construction of non-split extension of simple modules of Lie algebras using linear differential operators | Perhaps you are already aware of this, but your observation is essentially an example of the [Beilinson–Bernstein theory](https://en.wikipedia.org/wiki/Beilinson%E2%80%93Bernstein_localization) connecting $\mathfrak g$-modules to D-modules on the flag variety.
In your example, the 3 first-order operators you have wri... | 4 | https://mathoverflow.net/users/7762 | 344118 | 145,975 |
https://mathoverflow.net/questions/344121 | 5 | Let $k$ be an algebraically closed field, $A$ a finitely generated $k$-algebra. Let $x,y$ be two distinct closed points of $\mathrm{Spec}(A)$. Is there an affine $k$-scheme of finite type obtained from $\mathrm{Spec}(A)$ by "gluing $x$ and $y$"? More precisely: let $\mathfrak{m}\_x$ and $\mathfrak{m}\_y$ be the maximal... | https://mathoverflow.net/users/116075 | Gluing two points in an affine algebraic variety | $B$ is finitely generated. First notice that $B\subset A$ is an integral extension, since if $f\in A$, then $(f-f(x))(f-f(y))\in I$, giving you an integral equation. Then, it is easy to find a finitely generated algebra $C\subset B$ such that $C\subset A$ is integral. Then, $A$ is a finite $C$-module and thus so is $B$... | 5 | https://mathoverflow.net/users/9502 | 344122 | 145,977 |
https://mathoverflow.net/questions/344102 | 5 | A closed subspace $M$ of $L\_2(0,1)$ is said to be *strongly embedded* if the norms $\|\cdot\|\_2$ and $\|\cdot\|\_1$ are equivalent on $M$.
>
> Let $(f\_n)\_{n\in \mathbb N}$ be a orthonormal basis of $L\_2(0,1)$. Suppose that $\limsup\_{n\to\infty}\|f\_n\|\_1>0$. Is it possible to find a subsequence $(f\_{n\_k})... | https://mathoverflow.net/users/39421 | Subsequences of an orthonormal basis generating a strongly embedded subspace in $L_2(0,1)$ | The answer is ``yes". Slightly more generally, if $(f\_n)$ is an orthonormal sequence in $L\_2:= L\_2(0,1)$ whose $L\_1$ norms are bounded away from zero, then there is a subsequence that spans a strongly embedded subspace. It is equivalent (by extrapolation) to get a subsequence that in the $L\_p$ norm with $p:= 3/2$ ... | 6 | https://mathoverflow.net/users/2554 | 344127 | 145,979 |
https://mathoverflow.net/questions/344129 | 14 | I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this spirit, from the Wikipedia on [functional equations for L-functions](https://en.wikipedia.org/wiki/Functional_equation_(L-fun... | https://mathoverflow.net/users/123015 | How do functional equations for zeta functions arise from the structure of a homology group? | There are examples that don't come from number theory, although it's not much simpler. Specifically, the [Lefschetz zeta function](https://en.wikipedia.org/wiki/Lefschetz_zeta_function).
Let $X$ be a compact manifold of dimension $d$ and let $f: X\to X$ be a map. Let $L(f^n)$ be the number of fixed points of $f^n$, c... | 19 | https://mathoverflow.net/users/18060 | 344132 | 145,980 |
https://mathoverflow.net/questions/342877 | 16 | Could anyone cite some applications or developments in mathematical physics or string theory that use schemes?
I find curious the fact that while things like derived algebraic geometry and stacks have certain applications to mathematical physics I cannot find such applications for the case of (underived) schemes.
J... | https://mathoverflow.net/users/131104 | Applications of schemes to mathematical physics | In string theory, gauge symmetries on D-branes can be studied using classical scheme theory.
This was introduced around the late 1990s and early 2000s, cf. [this 1998 paper](https://arxiv.org/abs/hep-th/9811197), [this 2000 note](https://arxiv.org/abs/hep-th/0008150), [these 2003 notes](https://arxiv.org/abs/hep-th/0... | 3 | https://mathoverflow.net/users/130493 | 344140 | 145,982 |
https://mathoverflow.net/questions/344116 | 1 | Is there any way to estimate the following function, which is a result of sum of ratios of Gamma functions?
$$
{\_3F\_2}\begingroup
\renewcommand\*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
\begin{array}{c}
-q, \frac{M}{2}, \frac{1}{2}+\frac{M}{2}\\
\frac{1}{2}, -q-\frac{n-M}{2}+1
\end{a... | https://mathoverflow.net/users/122182 | Estimation of Hypergeometric function ${_3F_2}$ | Here are explicit forms of
$$f(q,n,m)={\_3F\_2}\begingroup
\renewcommand\*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
\begin{array}{c}
-q, \frac{m}{2}, \frac{1}{2}+\frac{m}{2}\\
\frac{1}{2}, -q-\frac{n-m}{2}+1
\end{array} ;& 1
\end{array}\right]\endgroup,$$
for small values of $q$:
$$f(1... | 0 | https://mathoverflow.net/users/11260 | 344151 | 145,985 |
https://mathoverflow.net/questions/344146 | 1 | The answer of the following question may be well-known in the field of Geometric Topology, so I ask for help in here.
>
> Does the total space of circle bundle over a closed hyperbolic surface admit a Riemannian metric with non-positive sectional curvature?
>
>
>
In particular, the circle bundle which comes f... | https://mathoverflow.net/users/90512 | Non-positive sectional curvature in 3-dimensional manifold | Edited.
From Theorem 5.3 in <https://homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf> a circle bundle over a hyperbolic surface has a geometry locally modeled on either $H^2\times R$ or $\widetilde{SL(2,R)}$.
The product metric on $H^2\times R$ is nonpositively curved, while (by Igor‘s comment below)... | 1 | https://mathoverflow.net/users/39082 | 344154 | 145,986 |
https://mathoverflow.net/questions/344164 | 3 | We know that a line bundle $L$ on the complex flag variety $G/P$ is trivial iff $c\_1(E) = 0$. But if we have a homogeneous vector bundle $E$ of higher rank, then is it true that $c\_i(E) = 0$ $ \forall i$ implies that $E$ is trivial? If not, is there anything analogous we can say about such vector bundles? I apologize... | https://mathoverflow.net/users/nan | Homogeneous vector bundles with zero chern classes | Take for instance $G/P = Gr(k,n)$, and let
$$
0 \to U \to V \otimes \mathcal{O} \to Q \to 0
$$
be the tautological exact sequence. Then
$$
c\_\bullet(U \oplus Q) = c\_\bullet(U) \cdot c\_\bullet(Q) = c\_\bullet(V \otimes \mathcal{O}) = 1,
$$
but $U \oplus Q$ is a nontrivial vector bundle.
A similar example can be co... | 5 | https://mathoverflow.net/users/4428 | 344165 | 145,987 |
https://mathoverflow.net/questions/344104 | 5 | For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally flat imbeddings of topological manifolds", Morton Brown, 1962. Let $M$ be a *non-metrizable* manifold-with-boundary. Doe... | https://mathoverflow.net/users/32487 | Collared boundary of a non-metrizable manifold | A nice recent reference for questions about non-metrisable manifolds is David Gauld's book aptly named "non-metrisable manifolds". For instance it is shown that any metrisable component of the boundary of a manifold (metrisable or not) is collared (Corollary 3.11 on page 44, it is an almost immediate consequence of R. ... | 8 | https://mathoverflow.net/users/29491 | 344169 | 145,988 |
https://mathoverflow.net/questions/343971 | 2 | I have searched, but only managed to turn up the presentation in interaction nets. I'd be equally interested in a categorical model of DiLL.
| https://mathoverflow.net/users/147310 | What is the sequent calculus for differential linear logic? | Although this is well-known to experts, it is surprisingly difficult to find a paper explicitly presenting the sequent calculus of (classical) differential linear logic (DiLL). The following one is a rare example:
Michele Pagani, [The Cut-Elimination Theorem for Differential Nets with Boxes](https://www.irif.fr/~mich... | 1 | https://mathoverflow.net/users/45027 | 344170 | 145,989 |
https://mathoverflow.net/questions/344155 | 20 | Any element of $H^1(M,\mathbb{Z}/2)$ is the $w\_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are the $w\_2(E)$ of a real vector bundle $E$ over $M$.
Considering tensor products and tensoring by line bundles, it i... | https://mathoverflow.net/users/5420 | Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$? | This is an obstruction theory problem; you regard $v\in H^2(M;\mathbb{Z}/2)$ as a homotopy class of maps $v: M\to K(\mathbb{Z}/2,2)$, then ask if $v$ lifts through the universal Stiefel-Whitney class $w\_2:BSO\to K(\mathbb{Z}/2,2)$. The first obstruction is $\beta(v^2)\in H^5(M;\mathbb{Z})$, as mentioned by Theo Johnso... | 19 | https://mathoverflow.net/users/8103 | 344172 | 145,990 |
https://mathoverflow.net/questions/344166 | 8 | I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the canonical sheaf represents the relevant functor through some very non-explicit homological algebra gymnastics. The name of the... | https://mathoverflow.net/users/113369 | Can one determine the trace map for a nonsingular projective variety explicitly? | This is a good question. It's one that my colleague Joe Lipman spent a lot of time thinking about. You can look at some of his papers for a more explicit answer for computing the trace. Probably you should start with his book "Dualizing sheaves, differentials and residues on algebraic varieties." As for name, the Groth... | 7 | https://mathoverflow.net/users/4144 | 344173 | 145,991 |
https://mathoverflow.net/questions/343286 | 4 | I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, *Approximating Euler's Constant*, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the Meissel-Mertens constant (as reference I add the Wikipedia link [Meissel–Mertens constant](https://en.wikipedia.org/wiki/Me... | https://mathoverflow.net/users/142929 | Sharp estimates for Meissel-Mertens constant | Already Mertens has proved in 1874 (*Ein Beitrag zur analytyischen Zahlentheorie*, J. Reine Angew. Math. 78) a refined variant of the result, which can be found in the book of Apostol. He showed that
$$\tag{1}\sum\_{p \le x} \frac{1}{p} = \log \log x +M + \mathcal{O}\Big(\frac{1}{\log{x}}\Big).$$
A proof can be found i... | 2 | https://mathoverflow.net/users/122316 | 344175 | 145,992 |
https://mathoverflow.net/questions/344160 | 5 | Consider the generating function of Legendre polynomials:
$$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}\_{n=0} P\_n(x)t^n$$
Is it true that for $0<x<1, t=1$ series of Legendre polynomials converges to the function on the left-hand side, i.e.
$$\frac{1}{\sqrt{2-2x}} \stackrel{?}{=} \sum\limits^{\infty}\_{n=0... | https://mathoverflow.net/users/147443 | Convergence of the series of Legendre polynomials | The answer is yes. By a theorem of Fatou
Theorem [Fatou]
If $a\_n\to0$ and the function $f(z)=\sum\_{n=0}^\infty a\_nz^n$ is analytic at
the point $z=1$, then the series $\sum\_{n=0}^\infty a\_n$ converges with value
$f(1)$.
we only have to show that $\lim\_{n\to\infty}P\_n(x)=0$ for $0<x<1$. To see this we note... | 6 | https://mathoverflow.net/users/7402 | 344179 | 145,994 |
https://mathoverflow.net/questions/344107 | 4 | I was inspired in Lemma 3.1 of [1] and in the Theorem 4.12 of [2] to ask about a similar statement that shows Lagarias in his paper as Lemma 3.1.
The Lemma from Lagarias's paper is that if $H(n)=\sum\_{i \leq n} \frac{1}{i}$, then for $n\geq 3$ we have $$e^{H(n)}\log H(n) \geq e^{\gamma}n \log \log n,$$ where $\gamma... | https://mathoverflow.net/users/142929 | A similar lemma to a lemma due to Lagarias, for the partial sums of reciprocal of primes | As pointed out in the comments $h\_n\sim \log\log n +M,$ so for $n$ large enough we can write
$$e^{h\_n}\log h\_n \sim e^M \log n \log(\log \log n+M)\quad(1).$$
For $n$ large the right hand side can be approximated by
$$
e^M \log n \left(\log \log\log n+\frac{M}{\log \log n}\right)
$$
where $M$ is the Mertens-Meiss... | 4 | https://mathoverflow.net/users/17773 | 344180 | 145,995 |
https://mathoverflow.net/questions/344182 | 2 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $S$ be a countable set, and we define a family of sub-$\sigma$-algebras $(F\_A)\_{A \subseteq S}$ such that $A \subseteq B \Rightarrow \mathcal{F}\_A \subseteq \mathcal{F}\_B$ for all $A, B \subseteq S$. We define a random subset $\zeta$ of $S$ to be a m... | https://mathoverflow.net/users/146831 | Properties of Random and Stopping Sets | $\newcommand{\si}{\sigma}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\F}{\mathcal{F}}$
The answer to your first question is negative. E.g., let $\Omega:=2^S=\mathcal P(S)$, with $\mathcal F:=2^\Omega$, and let $\zeta$ be the identity map of $\Omega$, so that $\zeta(A)=A$ for all $A\subseteq S$. For e... | 1 | https://mathoverflow.net/users/36721 | 344185 | 145,997 |
https://mathoverflow.net/questions/344189 | 2 | Let $X$ be a compact metric space and $f:X\to X$ is a homeomorphism. A point $x$ is aid to be nonwandering if for any open set $U$ containing $x$, there is an $N>0$ such that $f^N(U)\cap U\neq \emptyset$. Donote by $NW(f)$ the set of all nonwandering points of $f$. A point in $X$ is called a $\omega$-limit point for $x... | https://mathoverflow.net/users/36604 | Non wandering sets and limit sets | Consider the shift space $X \subset \{0,1\}^{\mathbb{Z}}$ obtained by forbidding the words $1 0^m 1^n 0$ for all $m, n \geq 1$, and denote the shift map on $X$ by $\sigma$.
Since a point of $X$ can contain at most three transitions from $0$ to $1$ or back, the only $\omega$-limit points of $X$ are the two uniform point... | 6 | https://mathoverflow.net/users/66104 | 344193 | 145,999 |
https://mathoverflow.net/questions/344131 | 8 | Let $n$ be a positive integer, $a\_1,\ldots,a\_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$,
$$
\tag{$\star$}
\sum\_{i=1}^{n-1} i a\_{(ui\!\!\!\mod n)} = 0.
$$
Then the product of values of the Gamma function
$$
\tag{$\star\star$}
\prod\_{i=1}^{n-1}\Gamma\left(\frac{i}{n}\right)^{a\... | https://mathoverflow.net/users/5263 | Converse of a result of Koblitz and Ogus on algebraic products of gamma values | A convenient way to formulate this kind of questions is to use the language of *distributions* introduced by Kubert and Lang. I gave a short account in [this answer](https://mathoverflow.net/a/318995/6506) to a previous MO question.
Your question is equivalent to asking whether the $\Gamma$ distribution is universal.... | 6 | https://mathoverflow.net/users/6506 | 344203 | 146,002 |
https://mathoverflow.net/questions/344196 | 2 | I was recently thinking about links where each component plays the same role: for every permutation of components, there is an isotopy permuting these components in the prescribed way. In the vein of knot/link invariants, we might ask how to tell when this is not the case. The obvious way to do this is by removing comp... | https://mathoverflow.net/users/147463 | Link invariants distinguishing components | The multivariable Alexander polynomial has the potential to do this. It has one variable $t\_i$ for each link component, and if there is an isotopy interchanging the $i^{th}$ and $j^{th}$ component then the polynomial is symmetric with respect to $t\_i$ and $t\_j$. (Maybe you have to take account of orientations and so... | 2 | https://mathoverflow.net/users/3460 | 344208 | 146,005 |
https://mathoverflow.net/questions/344219 | 30 | Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at the moment we don't have a more satisfactory version."
Is this still a fair description of the situation?
| https://mathoverflow.net/users/468 | Replacing triangulated categories with something better | My opinion, and that of many other people although not of everyone, is that the "correct" notion is that of [stable ∞-category](https://en.wikipedia.org/wiki/Stable_%E2%88%9E-category).
Now, this is not a category in the strictest sense, rather a generalization of the notion of category known as an [(∞,1)-category](h... | 36 | https://mathoverflow.net/users/43054 | 344221 | 146,009 |
https://mathoverflow.net/questions/344206 | 6 | Wikipedia and Borceux (Handbook of Categorical Algebra, Part I) give the following definitions of subobjects and well-powered categories:
>
> A *subobject* of an object $X$ of a category $\mathsf{C}$ is an equivalence class of the equivalence relation $\equiv$ on the class of all monomorphisms with codomain $X$ whe... | https://mathoverflow.net/users/83143 | Adjusting the definition of a well-powered category to category theory with universes: size issues | Borceux's Definition 4.1.1
>
> An equivalence class of monomorphisms with codomain A is called a subobject of A.
>
>
>
in combination with Definition 4.1.2
>
> A category A is well-powered when the subobjects of every object constitute a set.
>
>
>
and the subsequent claim that the category of sets is... | 2 | https://mathoverflow.net/users/402 | 344222 | 146,010 |
https://mathoverflow.net/questions/344218 | 14 | [This answer](https://mathoverflow.net/a/243908/34538) of [Geoff Robinson](https://mathoverflow.net/users/14450/geoff-robinson) shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A\_5$ or $\mathrm{PSL}(2,7)$.
*Question:* Is there ... | https://mathoverflow.net/users/34538 | On the finite simple groups with an irreducible complex representation of a given dimension | I will write this an an answer, though the answer to the basic question is provided by one of the oldest results in finite group theory.
There is a theorem of C. Jordan, proved in the 19th century, that there is a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $n \in \mathbb{N}$ and $G$ is a finite subgr... | 21 | https://mathoverflow.net/users/14450 | 344223 | 146,011 |
https://mathoverflow.net/questions/344231 | 0 | In a paper published by Bruhn and Schaudt, as well as in a [presentation](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.081/Henning/unionclosedtalk.pdf) given by Bruhn, they point out how the union-closed sets conjecture is tight for power sets, implying it is still open. I am confused however, as I have n... | https://mathoverflow.net/users/129192 | Is the union-closed sets conjecture open for the power sets case? | It is trivially true for power sets, since these are union-closed families with $2^n$ sets such that every element appears in exactly $2^{n-1}$ sets.
This is what they mean by the conjecture being tight - power sets show that the $1/2$ of the conjecture cannot be increased to any greater constant.
| 8 | https://mathoverflow.net/users/385 | 344232 | 146,015 |
https://mathoverflow.net/questions/344115 | 5 | Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1.
>
> Is it possible to express $X$ as a filtering projective limit of finite graphs?
>
>
>
Here finite graphs means topological realizations of finite graphs, with (surjective?) continuous maps between them.
| https://mathoverflow.net/users/147412 | One-dimensional compacta as projective limits | References to Engelking's Dimension Theory (1978) ISBN 0-444-85176-3.
The answer is yes for compact metrizable spaces, see Section 1.13.
In general it is no in general, see Example 3.3.8 (Lokucievskii's example of a compact space $X$ with $\dim X=1$ and $\mathop{\mathrm{ind}}X=2$).
| 5 | https://mathoverflow.net/users/5903 | 344234 | 146,017 |
https://mathoverflow.net/questions/344176 | 5 | Let $k > 1$ be a positive, square-free integer. Consider the quadratic form $f\_k(x,y) = x^2 - ky^2$. When $k$ is composite, is it easy to determine for which proper divisors $k^\prime$ of $k$ does the equation
$$\displaystyle f\_k(x,y) = k^\prime$$
solvable in integers $x,y$? Here we do allow $k^\prime$ to be poss... | https://mathoverflow.net/users/10898 | Representation of integers by principal binary quadratic forms | There are precisely two such proper divisors of $k$, and their product equals $-k$. This result is essentially due to Gauss, see Theorem 1 in Pall: Discriminantal divisors of binary quadratic forms, J. Number Theory 1 (1969), 525-533. Determining this pair of divisors is the subject of the quoted article in very specia... | 8 | https://mathoverflow.net/users/11919 | 344242 | 146,019 |
https://mathoverflow.net/questions/344243 | 20 | As far as I understand, there are Lipschitz functions $f:\mathbb{R}\to\ell^\infty$ that are nowhere differentiable in the Frechet sense. Where can I find such an example?
| https://mathoverflow.net/users/121665 | Non-differentiable Lipschitz functions | Define $f\_n(t)=\sin(nt)/n$ and $\phi(t)=(f\_1(t),\ldots)$. This has Lipschitz constant 1, but has no Fréchet derivative as far as I can tell.
| 18 | https://mathoverflow.net/users/11054 | 344244 | 146,020 |
https://mathoverflow.net/questions/344205 | 4 | In their paper [Two Approximation Algorithms for 3-Cycle Covers](http://wwwhome.math.utwente.nl/~mantheyb/conferences/APPROX2002_BlaeserManthey_3CycleCovers.pdf) of Markus Bläser and Bodo Manthey it is stated that:
*"...deciding whether an unweighted directed graph has a 3-cycle cover is already NP-complete. This follo... | https://mathoverflow.net/users/31310 | How does the complexity of calculating the Permanent imply the NP completeness of directed 3-cycle cover? | When I look in Garey and Johnson I find that [GT 13] is **PARTITION INTO HAMILTONIAN SUBGRAPHS** which is the problem of deciding if a $3$-cycle cover exists in a digraph. So, this could be used as a reference for the complexity. Also, Garey and Johnson cited the Valiant paper for this NP-complete problem.
I took a l... | 4 | https://mathoverflow.net/users/51668 | 344246 | 146,021 |
https://mathoverflow.net/questions/344190 | 3 | I am reading a paper, [Yano and Ishihara, “Submanifolds with Parallel Mean Curvature Vector”](https://projecteuclid.org/euclid.jdg/1214430219) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=298598)), where the authors have constructed a linear operator, say $A$, on vector fields. They claim that this operator ... | https://mathoverflow.net/users/119117 | How does the constancy of an operator’s eigenvalues imply the integrability of its eigenvector distribution? | After much deliberation, I believe my problem is that I have misunderstood the definition of integrability. From <https://www.encyclopediaofmath.org/index.php/Distribution_of_tangent_subspaces>,
it looks like the bracket $[v,w]$ does not have to be itself an eigenvector, but only a linear combination of eigenvectors.
... | 1 | https://mathoverflow.net/users/119117 | 344249 | 146,022 |
https://mathoverflow.net/questions/344254 | 1 | I need some results on expected roots of polynomial with random coefficients. I searched for $\textit{roots of polynomial with random coefficients}$ online and found some papers, but they are dealing with more complicated problems than I need. I think my particular problem is more specific: suppose we have a, say, cubi... | https://mathoverflow.net/users/128503 | Expected roots of polynomials with randomness in coefficients |
>
> Is $\mathbb{E}[\tilde{x}] = x$?
>
>
>
No --- in the case of just $x^3-d$, the root is $\sqrt[3]{d}$, and $E(\sqrt[3]{d})\ne\sqrt[3]{E(d)}$ typically. For instance let $d$ be 1, 8, 27 with probability $1/3$ each, then
$$E(\sqrt[3]{d})=2\ne\sqrt[3]{12}=\sqrt[3]{E(d)}.$$
| 2 | https://mathoverflow.net/users/4600 | 344256 | 146,024 |
https://mathoverflow.net/questions/344248 | 6 | Let $k$ be a field, let $X$ be a smooth quasi-projective $k$-variety, let $Z\subset X$ be a closed subscheme of codimension at least $2$, it is [shown](https://projecteuclid.org/euclid.dmj/1558145274)
that the restriction map $\mathrm{H}^2(X,\mathbb{G}\_m)\to\mathrm{H}^2(X-Z,\mathbb{G}\_m)$ is an isomorphism.
Let $... | https://mathoverflow.net/users/nan | Purity of Brauer group for stacks | The answer seems to be positive and actually at least in the context of regular (locally) noetherian Deligne--Mumford stacks. (Actually, Artin stack should also be enough as we can compute the Brauer group also as the fppf-cohomology of $\mathbb{G}\_m$.)
Let $p\colon X \to \mathcal{X}$ be an \'etale cover. We obtain ... | 6 | https://mathoverflow.net/users/2039 | 344258 | 146,026 |
https://mathoverflow.net/questions/344245 | 2 | **Background**
The fact that there is no suborder of $\mathbb R$ which is of type $\omega\_1$ suggests (to me) that the continuum $c$ cannot be very far from $\omega\_1$: How could $c$ be far away from $\omega\_1$ if there is no room for an order embedding of $\omega\_1$ in $\mathbb R$? Of course, this fact is a cons... | https://mathoverflow.net/users/9825 | Consistency of embedding cardinals in linear orderings | Your axiom is inconsistent. (Or perhaps I have misunderstood it.)
Let $L\_n:= \aleph\_n$ with the reverse order, and let $L:= L\_1 + L\_2 + \cdots$ (horizontal sum); equivalently, let $L$ be the lexicographic order on $\bigcup\_k \{k\}\times L\_k$. Then $L$ is $\aleph\_\omega$-unbounded, yet there is no order-preserv... | 6 | https://mathoverflow.net/users/14915 | 344263 | 146,029 |
https://mathoverflow.net/questions/344141 | 0 | Let $M$ be a $m$ rows and $n$ columns matrix over GF(2). And Let $M^{'}$ be a $r$ rows and $n$ columns submatrix of $M$ ($r <= m$). Note that rows of $M^{'}$ is randomly selected from $M$. Is it possible that the $M^{'}$ is full row-rank?
If it is, why?
If it is not, how about the case where $r < m$?
Edit:
1) I... | https://mathoverflow.net/users/147423 | Full row-rank submatrix of full column-rank matrix over GF(2) | There's a formula to compute the probablity of an $m×n$ $(m≥n)$ matrix to be of full rank: if the matrix has i.i.d uniformly distributed elements drawn from $\{0,1\}$, the probability for it to be of full rank (i.e. rank n) is $∏^{n−1}\_{k=0}\frac{2^m−2^k}{2^m}$.
The formula can be proved by induction: Take the colu... | 2 | https://mathoverflow.net/users/125498 | 344275 | 146,033 |
https://mathoverflow.net/questions/344241 | -1 | Suppose we have three sets $A, B,C$ and a span $S := A \leftarrow C \rightarrow B$. There is a special case when the data of the span $S$ exactly specifies a function $f: A \rightarrow B$. In every other case, we have some data that may be used to define a function. It's a bit like a probabilistic model of a function o... | https://mathoverflow.net/users/10007 | Is it possible to regress an arbitrary function from a span? | There is no natural way to convert a span $A\leftarrow C\rightarrow B$ into a (partial) function $f:A\to B$ if, whenever an $a\in A$ is connected to some element(s) of $B$ by the span, $f(a)$ must be one of those elements.
Consider the case where $A=B=\{1,2,3\}$, where $C=\{(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\}$, an... | 3 | https://mathoverflow.net/users/6794 | 344289 | 146,036 |
https://mathoverflow.net/questions/344292 | 1 | I am reading the paper *Nodal sets of eigenfunctions of the Laplacian on Surfaces* by Donnelly and Fefferman available [here](https://www.ams.org/journals/jams/1990-03-02/S0894-0347-1990-1035413-2/S0894-0347-1990-1035413-2.pdf). I have a problem understanding Lemma 5.10. To my understanding, what follows is the content... | https://mathoverflow.net/users/118316 | Lemma from Donnelly-Fefferman's paper | $\newcommand{\al}{\alpha}
\newcommand{\G}{\mathcal{G}}
\newcommand{\PP}{\mathcal{P}}$
You are missing condition (ii) in formula (5.9) in that paper. That formula is
\begin{equation}
\begin{aligned}
&\text{(i)}\quad \sup\_{|\al|\le c\_4} \Big| \frac{\partial^{\al}g}{\partial z^{\al}}(0) \Big|\ge b\_6, \\
&\text{(ii)}\... | 1 | https://mathoverflow.net/users/36721 | 344300 | 146,039 |
https://mathoverflow.net/questions/344293 | 2 | Roth's theorem states that for an algbraic number $a$, $a$ is badly approximated by rationals: for every $\alpha>0$ there is $C>0$ such that for $l\in \mathbb Z$, $$d(la,\mathbb Z)>Cl^{-1-\alpha}.$$
I am wondering if there are some numbers which are even less well approximated: there is $\epsilon(l)$ going to $0$ su... | https://mathoverflow.net/users/16934 | Very badly approximable numbers | Yes, indeed one can even take $\epsilon(l)$ to be constant: $d(la,\Bbb Z)\gg\_a l^{-1}$ is equivalent to [the continued fraction of $a$ having bounded partial quotients](https://en.wikipedia.org/wiki/Diophantine_approximation#Badly_approximable_numbers). So, for example, every quadratic irrational numbers $a$ has this ... | 5 | https://mathoverflow.net/users/5091 | 344320 | 146,043 |
https://mathoverflow.net/questions/344299 | 0 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact connected $C^\infty$-smooth manifold. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable functions such that for each $\omega$, $F(\omega,\,\cdot\,) \colon X \to X$ is a $C^\infty$-diffeomorp... | https://mathoverflow.net/users/15570 | In smooth stochastic dynamics, if a Lebesgue-like measure is both forward-time and reverse-time stationary, is the measure necessarily incompressible? | Discontinuity is not crucial at all in the example given in the answer to [this question](https://mathoverflow.net/questions/343977/if-a-probability-measure-is-stationary-in-both-forward-time-and-reverse-time-do), and the same phenomenon is present in the smooth setup as well. Namely, the free group can be replaced wit... | 1 | https://mathoverflow.net/users/8588 | 344337 | 146,046 |
https://mathoverflow.net/questions/344220 | 0 | Is the trace of a finite hypercubic tensor defined?
Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal:
$$\operatorname {tr} (\mathbf {A}\_{2D} )=\sum \_{i=1}^{n}a\_{ii}=a\_{11}+a\_{22}+\dots +a\_{nn}$$
Is there any sound attempt at generalizing t... | https://mathoverflow.net/users/147390 | Trace of a finite hypercubic tensor | Although this is rather off-topic for mathoverflow, I'll provide an answer, given that no objections have been raised to the OP and since it's being explicitly requested:
First, one should carefully distinguish between tensors and matrices. These are distinct concepts, each with their own properties and with their ow... | 1 | https://mathoverflow.net/users/134299 | 344340 | 146,047 |
https://mathoverflow.net/questions/344339 | 2 | I am trying to understand where nonconstructive reasoning occurs in this passage from Skolem’s (1922) proof of the Löwenheim-Skolem theorem. As background, Skolem’s “solutions” are assignments of truth-values to the atomic components of a propositional formula U, and each step considers progressively more instances of ... | https://mathoverflow.net/users/116705 | Nonconstructive reasoning in Skolem's proof of the Löwenheim-Skolem Theorem | If I understand the proof correctly, it uses the following fact:
>
> **Statement:** *A monotone sequence $b : \mathbb{N} \to \{0,1\}$ stabilizes, i.e., there is $d \in \{0, 1\}$ and $n$ such that $b\_m = d$ for all $m \geq n$.*
>
>
>
This statement is not constructively provable because it implies [LPO](https:... | 3 | https://mathoverflow.net/users/1176 | 344342 | 146,049 |
https://mathoverflow.net/questions/344351 | 2 | We say that the sequence $a\_{k+1}(n)$ is a complete sequence of $a\_k(n)$ if:
(1) Every term of $a\_k(n)$ can be written as a sum of *distinct* terms of $a\_{k+1}(n)$.
(2) $\lim\_{n\to\infty} \frac{a\_k(n)}{a\_{k+1}(n)}=0$.
(Possibly there are more than one choices for $a\_{k+1}(n)$ if $a\_k(n)$ is given).
... | https://mathoverflow.net/users/38851 | Chain of sequences, such that $a_{k+1}(n)$ completes $a_k(n)$ | No, such $a\_4$ does not exist.
Surely, we may assume $a\_4$ to be increasing. Now consider a representation of some $a\_3(n)$. If it contains only elements of $a\_4$ which appear in the representations of $a\_3(k)$ with $k<n$, then $a\_3(n)\leq a\_3(1)+\dots+a\_3(n-1)$ which is wrong.
Therefore, the representati... | 3 | https://mathoverflow.net/users/17581 | 344353 | 146,051 |
https://mathoverflow.net/questions/344362 | 0 | Is there an algorithm to determine whether a given simple graph $G$ is a product graph, typically, say a cartesian product graph of two smaller simple graphs $G\_1, G\_2$, such that the two simple graphs have the cartesian product of their vertex sets having the same cardinality of the graph $G$, that is, $V(G\_1)\time... | https://mathoverflow.net/users/100231 | Recognition of a graph as a product of its quotients | The best algorithm is linear in the number of edges:see [Imrich, Wilfried; Peterin, Iztok (2007), "Recognizing Cartesian products in linear time", Discrete Mathematics 307 (3-5): 472–483](https://www.sciencedirect.com/science/article/pii/S0012365X06005280).
An easy-to-understand algorithm can be found [here](http://d... | 4 | https://mathoverflow.net/users/125498 | 344364 | 146,053 |
https://mathoverflow.net/questions/344358 | 7 | Does every infinite cardinal $\kappa$ have the following property?
>
> There is a simple, undirected graph $G\_0=(\kappa, E\_0)$ such that every simple, undirected graph $G=(\kappa, E)$ is isomorphic to an induced subgraph of $G\_0$.
>
>
>
| https://mathoverflow.net/users/8628 | Embeddability of all graphs of cardinality $\kappa$ into one graph of cardinality $\kappa$ | This turns my comments (and those of Will Brian) into an answer. The summary is that the answer to the question is consistently "yes", and consistently "no".
It is reasonable to treat the question one cardinal at a time:
>
> Let $P(\kappa)$ be the property that there is a graph of cardinality $\kappa$ that embeds... | 10 | https://mathoverflow.net/users/38253 | 344365 | 146,054 |
https://mathoverflow.net/questions/344359 | 4 | I am trying to understand Bestvina's "A Bers-like proof of the existence of train tracks for free group automorphisms". I'm going to ask a probably trivial question ... Here we go:
The automorphism $\Phi\in Out(F\_n)$ is called reducible if there is some $\phi:\Gamma\to\Gamma$ that represents $\Phi$ and leaves a homo... | https://mathoverflow.net/users/145318 | outer automorphism classification | If I've understood correctly, your concern is that for an optimal representative of a hyperbolic automorphism, the tension subgraph is a homotopically non-trivial subgraph left invariant by $\phi$, showing that the automorphism is reducible.
In fact, hyperbolic (meaning that the infimum of the displacement is positiv... | 7 | https://mathoverflow.net/users/95340 | 344372 | 146,056 |
https://mathoverflow.net/questions/163206 | 5 | More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially *not having* the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . The Bessel potential space $H={\rm H\,}s=H^{s,p}(\mathbb R^N,F)$ of tempered distributions on $\mathbb R^N$ with values... | https://mathoverflow.net/users/12643 | Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$? | Generally, the answer is **no**. I have known this already for some years but have not bothered to write the answer here since the question seems not having raised much interest. Now having some spare time, I put it here in case someone possibly be interested. The assertion follows from (1) in Theorem 5.6.12 on page 45... | 3 | https://mathoverflow.net/users/12643 | 344373 | 146,057 |
https://mathoverflow.net/questions/344148 | 1 | Let’s define the set of *outer-commutator group words* $OC \subset F\_\infty = F[x\_0, x\_1, …, x\_n, …]$ using the following recurrence:
$$\forall i \in \mathbb{N} \text{ } x\_i \in OC$$
$$\forall u, v \in OC \text{ } [u, v] \in OC$$
Let’s call a group variety *outer commutator variety* iff it can be defined by ... | https://mathoverflow.net/users/110691 | Bounds for Khukhro-Makarenko theorems | The Second, Third, and many others "Khukhro--Makarenko theorems" were proved
[here](http://dx.doi.org/10.1016/j.jalgebra.2014.10.029), see also
[arXiv](http://arxiv.org/abs/1309.0571) (so, they are rather
"Klyachko--Milentyeva theorems").
The bound is always the same:
$$
\log\_2|G:N|\leqslant f^{d(w)-1}(\log\_2(|G:H|... | 4 | https://mathoverflow.net/users/24165 | 344384 | 146,059 |
https://mathoverflow.net/questions/344313 | 4 | Expansion of a real function on 2-sphere in spherical harmonics, so-called Laplace series, converges uniformly for continuously differentiable functions (see e.g. <https://projecteuclid.org/euclid.bbms/1103408694>).
Is there any explicit example of a real function on 2-sphere, which is merely continuous, and for whic... | https://mathoverflow.net/users/114094 | Counterexample to uniform convergence of Laplace series (expansion in spherical harmonics) | One can just adapt an example from the circle to the sphere by lifting such a function from $\mathbb{S}^1$ to zonal spherical harmonics on $\mathbb{S}^2$.
It is then sufficient to give an example of a continuous function with Fourier series which does not converge pointwise, much less uniformly. On the circle, take ... | 3 | https://mathoverflow.net/users/118731 | 344386 | 146,061 |
https://mathoverflow.net/questions/344344 | 8 | Given a function $f$, let us define the translates $f\_t(x)=f(x-t)$. A (Bochner) almost-periodic function is a bounded continuous function on $\mathbb R^\nu$ such that the set of functions $\{f\_t\vert t\in\mathbb R^\nu\}$ form a precompact set with respect to the supremum norm (a precompact set is a set whose closure ... | https://mathoverflow.net/users/20838 | Can we characterize a periodic function by the compactness of the set of its translates? | I would also like to focus on the case $\nu =1$ (to avoid some slightly annoying but probably trivial bookkeeping issues).
Suppose that $\{f\_t\}$ is compact. This shows first of all that $f$ is uniformly continuous, or else we could shift problematic points to zero, say, to obtain a sequence with no convergent subse... | 10 | https://mathoverflow.net/users/48839 | 344389 | 146,062 |
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