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https://mathoverflow.net/questions/333225 | 3 | I have a question about the definition of the graph Fourier transform. Let me start with definition.
Let $A$ be the adjacency matrix of a graph $G$ with vertex set $V = \{1, 2, \dots, n\}$. The Laplacian matrix of $G$ is defined as $L = D - A$, where $D$ is a diagonal degree matrix with $d\_{ii} = deg(i)$. Let $\varp... | https://mathoverflow.net/users/141468 | Graph Fourier transform definition | I saw the definition I mentioned in the question in many places. (E.g. [here](http://www.norbertwiener.umd.edu/Research/lectures/2014/MBegue_Prelim.pdf).) In [Graph Structured Data Viewed Through a Fourier Lens](http://digitalassets.lib.berkeley.edu/techreports/ucb/text/EECS-2013-209.pdf) the definition is different:
\... | 1 | https://mathoverflow.net/users/141468 | 333236 | 142,535 |
https://mathoverflow.net/questions/333241 | 4 | I am trying to understand various ways in which one can modify the Langlands correspondence. Hopefully I will be able to learn something from you. First, one can categorify/decategorify.
It is my understanding the the geometric Langlands over complex numbers is by default categorical (yet there has been attempt to de... | https://mathoverflow.net/users/nan | Modifying the Langlands correspondence | You have asked a lot of questions all at once, but I will try to answer some of them.
1) A very minor point: I don't think it's quite right to say that Frenkel and Langlands attempt to decategorify it. I would say that Langlands is attempting to define a new, non-categorical correspondence and Frenkel is attempting t... | 6 | https://mathoverflow.net/users/18060 | 333248 | 142,539 |
https://mathoverflow.net/questions/333239 | 14 | Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times\_Z Y$ can be canonically identified with the fiber of the map $f - g \colon X \oplus Y \rightarrow Z$?
I am sure that the q... | https://mathoverflow.net/users/130024 | Describing fiber products in stable $\infty$-categories | In fact what you need is that your ∞-category is *additive* (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories are additive, so I'll just prove it in this case.
**Step 1**: The pullback $X\times\_Z Y$ is equivalent to the pullb... | 20 | https://mathoverflow.net/users/43054 | 333249 | 142,540 |
https://mathoverflow.net/questions/333215 | 1 | Given a $d$-regular graph with $n$ vertices is there a known (non-trivial) upper bound on the length of [chordless cycles](http://mathworld.wolfram.com/ChordlessCycle.html) in it (presumably as a function of $d$ and $n$)? I wasn't able to find anything after some online searches. Thank you.
| https://mathoverflow.net/users/141463 | Upper bound on the length of chordless cycles in d-regular graphs | Say a chordless cycle $C$ has length $k$.
The number of edges between $C$ and $V-C$ is
$k(d-2)$. Also, the number of edges from $V-C$ to $C$ is at most $(n-k)d$. The solution is
$$k\le \biggl\lfloor
\frac{nd}{2(d-1)}\biggr\rfloor.$$
This is often realised but I'm too lazy to figure out if it is always realised.
| 1 | https://mathoverflow.net/users/9025 | 333254 | 142,543 |
https://mathoverflow.net/questions/333258 | 7 | I am not sure, if this is a research problem. If not I will move this question to ME:
Let $\Omega(n) = \sum\_{p|n} v\_p(n)$, which we might view as a random variable.
Let $E\_n = \frac{1}{n} \sum\_{k=1}^n\Omega(k)$ be the expected value and $V\_n=\frac{1}{n} \sum\_{k=1}^n(E\_n-\Omega(k))^2$ be the variance.
Then
$$\pi... | https://mathoverflow.net/users/nan | A curious prime counting approximation or just data overfitting? | Your heuristic approximation is not correct. It was proved by Turán (1934) that
$E\_n$ and $V\_n$ are both asymptotically $\log\log n$. As a result, the RHS of your display is
$$n\frac{\gamma\left(1+o(1),\frac{1.4854177+o(1)}{\log\log n}\right)}{\Gamma\bigl(1+o(1)\bigr)}=n\frac{1.4854177+o(1)}{\log\log n}.$$
On the oth... | 22 | https://mathoverflow.net/users/11919 | 333260 | 142,545 |
https://mathoverflow.net/questions/333221 | 4 | For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of certain flow on $M$ is the main object of this paper ["Number theory and dynamical system of foliated manifolds](https://arxiv... | https://mathoverflow.net/users/36688 | Leafwise de Rham cohomology (A true definition of differential forms along leaves) | The thing you are confused about is that leafwise differential forms are not invariant under foliation charts. Basic forms are invariant under change of foliation charts and foliated maps (in your notation, forms in the y-variables). One way of defining leafwise differential forms is by first choosing a metric and then... | 2 | https://mathoverflow.net/users/74129 | 333263 | 142,546 |
https://mathoverflow.net/questions/333262 | 5 | I believe I've encountered the statement below, but I've lost my reference and am unable to find another one. So, I'm posting this question to see if someone can give a reference, or at least confirm the statement is true (or make a correction).
>
> **Proposition:** Let $\mathcal{C}$ be a class of schemes with the ... | https://mathoverflow.net/users/nan | Quasi-compact quasi-separated induction? | This is proposition 3.3.1 and remark 3.3.2 of *Generators and representability of functors in commutative and noncommutative geometry* by A. Bondal, M. van den Bergh.
I originally found this reference via the MO question [The biggest class of schemes which the reduction principle holds](https://mathoverflow.net/q/157... | 4 | https://mathoverflow.net/users/nan | 333266 | 142,547 |
https://mathoverflow.net/questions/333270 | 4 | Rank-into-rank cardinals have the rather intriguing property that they reflect upwards. I would be interested to know how far the upward reflection goes:
1) Does "There exists a rank-into-rank cardinal (of type I3, say)" imply that "There exists an unbounded class of rank-into-rank cardinals (in V) ?"
2) If 1) is f... | https://mathoverflow.net/users/94232 | Upward reflection of rank-into-rank cardinals | The answer is negative.
The existence of a rank-to-rank cardinal $j:V\_\lambda\to V\_\lambda$ is $\Sigma\_2$ expressible, since it is witnessed inside any sufficiently large $V\_\alpha$. Therefore, if one cuts off at any inaccessible or worldly cardinal above $\lambda$, one still has the rank-to-rank cardinal, but t... | 7 | https://mathoverflow.net/users/1946 | 333271 | 142,549 |
https://mathoverflow.net/questions/248488 | 16 | ***Introduction:***
Given two univariate and coprime integer polynomials $f(x), g(x)$, we can always write
\begin{equation}
u(x)f(x)+v(x)g(x)=c
\end{equation}
for a unique pair of integer polynomials $u(x), v(x)$, and a unique positive integer $c$, such that $\deg u<\deg g$ and $\deg v<\deg f$. (This is a consequenc... | https://mathoverflow.net/users/85967 | Reduced resultants and Bezout's identity | First off, it can be seen that the reduced resultant must divide $c$. Indeed, if $u\_1(x)f(x)+v\_1(x)g(x)=c$ and $u\_2(x)f(x)+v\_2(x)g(x)=d$, then from the Bezout identity for integers $c,d$, we can construct the identity $u(x)f(x)+v(x)g(x)=\gcd(c,d)$. Hence, if $d$ is the reduced resultant, then $d=\gcd(c,d)$, i.e. $d... | 7 | https://mathoverflow.net/users/7076 | 333279 | 142,552 |
https://mathoverflow.net/questions/332876 | 7 | I'm interested in the following strange question: for some $d > 1$, what is the minimum dimension of a commutative $\mathbb{C}$-algebra containing infinitely many elements that square to zero, but where the product of any $d$ of these elements is nonzero?
The best upper bound I know is $d\cdot \binom{\lceil 3d/2 \rce... | https://mathoverflow.net/users/141277 | Finite dimensional commutative algebras containing infinitely many nilpotents whose $d$-way products are nonzero | $\def\CC{\mathbb{C}}\def\fm{\mathfrak{m}}\def\PP{\mathbb{P}}$Here are some ideas, but not a solution. First, I will reduce the problem to the sort of examples the OP is already considering. I will then make some connections to secant varieties. I will prove the optimum value for $d=3$ to be $12$, given by the OP's cons... | 5 | https://mathoverflow.net/users/297 | 333284 | 142,553 |
https://mathoverflow.net/questions/333253 | 3 | For a locally convex space $E$ let $E\_\beta$ be the space $E$ endowed with the locally convex topology $\beta(E,E')$ whose neighborhood base at zero consists of barrels, i.e., closed absolutely convex absorbing sets.
Observe that a space $E$ is barrelled if and only if $E=E\_\beta$.
>
> **Question 1.** Is $(E\_... | https://mathoverflow.net/users/61536 | Is the strong topology of a locally convex space always barrelled? | The answer to question 1 is negative. There are Frechet spaces $X$ whose strong duals $(X',\beta(X',X))$ are not barrelled (equivalently, not bornological by a theorem of Grothendieck). The first examples of such *non-distinguished* Frechet spaces were constructed by Köthe and Grothendieck but there are also examples w... | 6 | https://mathoverflow.net/users/21051 | 333289 | 142,556 |
https://mathoverflow.net/questions/333326 | 3 | I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $AA^=$ and $||A||\_0 << nm$.
It is not hard to generate orthogonal matrices through an SVD, but the orthogonal components o... | https://mathoverflow.net/users/126743 | Parametrising a sparse orthogonal matrix | The smallest number of nonzero entries in an $n\times n$ fully indecomposable$^\*$ orthogonal matrix is $4n−4$. A method to construct such a matrix is described in [Sparse orthogonal matrices](https://core.ac.uk/download/pdf/82782666.pdf) (2003).
$^\*$ A fully indecomposable matrix does not have a $p\times q$ zero su... | 3 | https://mathoverflow.net/users/11260 | 333330 | 142,568 |
https://mathoverflow.net/questions/333332 | 8 | I’m currently reading Burago, Burago, Ivanov’s book *A Course in Metric Geometry*. I’m really enjoying it so far - what would be a good continuation to the book once I’m done?
| https://mathoverflow.net/users/132446 | Textbook recommendation: Metric Geometry | As you have been reading Loh's book, I recommend you take a look at *Metric spaces of non-positive curvature* by M. Bridson and A. Häfliger.
See also [An invitation to Alexandrov geometry: CAT(0) spaces](https://anton-petrunin.github.io/invitation/) by S. Alexander, V. Kapovitch and A. Petrunin, or its [older version... | 10 | https://mathoverflow.net/users/131919 | 333337 | 142,570 |
https://mathoverflow.net/questions/333336 | 6 | Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE\_{k}$ from $a$ to $b$.
For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \gamma$ has two components. Say that $A$ is the left-component of $\mathbb{D} \setminus \gamma$ when we traverse $\gamma$... | https://mathoverflow.net/users/41873 | The distribution of the area of a region cut out by chordal SLE? | The expected area of $A$ is easy to compute, in principle explicitly:
$$
\mathbb{E}(\text{Area}(A))=\mathbb{E}\left(\int\_\mathbb{D}\mathbb{1}\_{z\in A}\right)=\int\_\mathbb{D}\mathbb{P}(z\in A).
$$
The probability inside the integral is given by so-called Schramm's formula and can be computed by standard SLE technique... | 3 | https://mathoverflow.net/users/56624 | 333341 | 142,571 |
https://mathoverflow.net/questions/333235 | 0 | Good codes (those with positive rate $r=k/n$ and positive relative distance $\delta=d/n$) will achieve capacity on $BSC$ (binary symmetric channel) if the codes have lower rates than capacity where positive relative distance is seen. However this requires very long codes to drive the error to reasonably low value.
To... | https://mathoverflow.net/users/10035 | Shortest possible good codes? | Upper and lower bounds as well as approximations are given in "Channel Coding Rate in the Finite Blocklength Regime" by Yury Polyanskiy, H. Vincent Poor, and Sergio Verdu.
| 3 | https://mathoverflow.net/users/82838 | 333342 | 142,572 |
https://mathoverflow.net/questions/333210 | 2 | In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (<https://arxiv.org/abs/1207.3648>), there is the following assertion whose proof I would like to understand.
Let $e\_1, ..., e\_n$ be natural numb... | https://mathoverflow.net/users/141457 | Regularity of certain schemes | Here is another argument to prove the desired regularity, along the lines of Piotr Achinger's comment.
As explained in the linked seminar notes, the essential case to consider is $V(S, \pi, e\_1, \dots, e\_n)$ (for which we write $V$), where for some $i$, the exponent $e\_i$ is invertible in $\eta$.
From the hypoth... | 2 | https://mathoverflow.net/users/141530 | 333345 | 142,573 |
https://mathoverflow.net/questions/333312 | 2 | This is a question taken (inferred) from Ex. 19, chap 3 in Rudin's Real and Complex Analysis book.
Let $\mu$ be Legesgue's measure on $X=[0,1]$. Given a measurable $L^{\infty}$ function $f:X\to C$, we denote by $R\_f$ the essential image of $f$ (that is, the set of complex numbers $z\in C$ such that $\mu(f^{-1}(D(z,... | https://mathoverflow.net/users/65954 | A question about measure-weighted barycenters | Claim: For every $f \in L^1[0,1]$, the set $A\_f$ is convex.
Proof: Let $\mu\_1=\mu$ be Lebesgue measure on $[0,1]$ and consider the signed measures $\mu\_2,\mu\_3$ on $[0,1]$ defined using the real and imaginary parts of $f$ by $\mu\_2(E)= \int\_E {\mathrm Re} (f) \,d\mu$ and $\mu\_3(E)= \int\_E {\mathrm Im} (f) \,d... | 2 | https://mathoverflow.net/users/7691 | 333360 | 142,578 |
https://mathoverflow.net/questions/333267 | 3 | Let $K$ be a finite extension of $\mathbb{Q}$. Let $T$ be a finite dimensional $G\_{K}$ module over $\mathbb{Z}\_p$. Does the Bloch-Kato Selmer group $H^{1}\_{f}$ satisfy $H^1\_f(\mathbb{Q},Ind^{G\_{\mathbb{Q}}}\_{G\_K}T) = H^{1}\_{f}(K,T) $?
If not, then can you tell me what kind of condition on $K/\mathbb{Q}$ as we... | https://mathoverflow.net/users/141485 | Shapiro lemma for Bloch Kato Selmer group | This is a nice exercise. You start by making some reduction steps. Firstly, the BK Selmer group for $T$ is by definition saturated (i.e. it is the preimage in $H^1(K, T)$ of a $\mathbf{Q}\_p$-subspace of $H^1\_{\mathrm{f}}(K, T \otimes \mathbf{Q}\_p)$) so you can invert $p$ everywhere and ask the question for finite-di... | 3 | https://mathoverflow.net/users/2481 | 333372 | 142,583 |
https://mathoverflow.net/questions/333245 | 7 | I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p\_1, \dots, p\_n)$ (i.e. independent on the $q$s) by a canonical transformation.
Many persons told me that something simil... | https://mathoverflow.net/users/138060 | Practical example of Hamiltonian reduction | The reduction you require *is* a (very) special case of Marsden-Weinstein ([1974](http://www.cds.caltech.edu/~marsden/bib/1974/01-MaWe1974/)). Your one constant of the motion, say $\psi$, is the moment map of an action of the additive group $G=\mathbf R$ on the symplectic manifold with coordinates say $(x\_1,y\_1,\dots... | 2 | https://mathoverflow.net/users/19276 | 333374 | 142,584 |
https://mathoverflow.net/questions/333355 | 4 | During my research, I checked using Maple that we have numerically for $ 2 \leq n \leq 20,$ $ \forall t $ integer satisfying $ 0 \leq t \leq n-1$
$$ \displaystyle \frac{1}{2} \displaystyle \sum\_{p=1}^{n-t} C\_{n}^{p+t} C\_{n+p+t}^{n} \displaystyle \frac{(-1)^{p}}{p}= \displaystyle C\_{n+t}^n\displaystyle \sum\_{p=1}... | https://mathoverflow.net/users/136807 | About binomial identity | We have $$C\_{n}^{p+t}C\_{n+p+t}^n=[y^{n-t}x^tz^t](x+y+z)^{n+t}\cdot \left(\frac{(x+y+z)y}{xz}\right)^p,$$
where $[M]f$ is a coefficient of monomial $M$ in the polynomial or series $f$. Multiplying by $(-1)^p/p$ and summing by all $p=1,2,\ldots$ we get
$$\sum\_{p=1}^\infty (-1)^p p^{-1}C\_{n}^{p+t}C\_{n+p+t}^n=[y^{n-t}... | 11 | https://mathoverflow.net/users/4312 | 333377 | 142,585 |
https://mathoverflow.net/questions/333379 | 5 |
>
> **Definition.** Let us define a Banach space $X$ to be *co-Sobczyk* if every linear bounded operator $T:Z\to c\_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c\_0$.
>
>
>
By the classical Sobczyk Theorem, each separable Banach space is co-Sobczyk.
But the class of c... | https://mathoverflow.net/users/61536 | What is a name for co-Sobczyk Banach spaces? | Nigel Kalton studied [a similar but stronger notion](https://books.google.cz/books?id=8pwDl-jWe4YC&pg=PA160&lpg=PA160&dq=c0%20EXTENSION%20PROPERTY&source=bl&ots=WUHwUcGc6p&sig=ACfU3U0YLIKq0yi7CpFgXHW4EiG7H_E7cQ&hl=pl&sa=X&ved=2ahUKEwismPy-ytTiAhXh0qYKHa7oCcwQ6AEwCnoECAUQAQ#v=onepage&q=c0%20EXTENSION%20PROPERTY&f=false)... | 5 | https://mathoverflow.net/users/15129 | 333386 | 142,587 |
https://mathoverflow.net/questions/333223 | 2 | I am studying PDEs whose symbols satisfy
\begin{equation}
|\partial^\alpha\_\xi\partial^\beta\_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|}
\end{equation}
for all multi-indices $\alpha,\beta$ and $(x,\xi) \in \mathbb{R}^{2n}$.
It is said that we must look at the metric
\begin{equatio... | https://mathoverflow.net/users/102092 | Metric on the phase space | You can in fact reformulate the conditions on the symbol $p$ by saying that
$$
\vert p^{(k)}(X) T^k\vert\le C\_k M(X)g\_X(T)^{k/2},
$$
with $X=(x,\xi)$. Here $p^{(k)}(X) $ stands for the $k$-th derivative (a $k$-multilinear form) and $T$ is a vector in $\mathbb R^{2n}$. Using Lars Hörmander's notations in Chapter 18 of... | 2 | https://mathoverflow.net/users/21907 | 333387 | 142,588 |
https://mathoverflow.net/questions/333384 | 4 | **Setting**
In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections
$$\mathcal{M}\_k = \{A \in L^{2}\_{k}(X, \Lambda^1 \otimes\mathrm{ad}(P))\ | \ F\_A^+ = 0\}/\mathcal{G}\_{k+1}$$
(the subscript is a Sobolev parameter)
is equivalent to the trivial... | https://mathoverflow.net/users/99042 | Orientability of moduli space and determinant bundle of ASD operator | You want a line bundle that's well-defined over the entire space of irreducible connections $\mathcal B^\*$; this is how you see, for instance, that a choice of orientation at one point in the moduli space induces an orientation at every point (even if the moduli space is not connected). Similarly, one proves orientabi... | 2 | https://mathoverflow.net/users/40804 | 333399 | 142,591 |
https://mathoverflow.net/questions/333404 | 3 | I encountered the following value in my research:
>
> Let $n,m$ be some integer. Suppose $\alpha\_1,\dots,\alpha\_m$ are unit vectors in $\mathbb{R}^n$.
> Denote
> $$
> L = \mathop{\mathrm{E}}\_x[ \prod\_{1\leq j\leq m} \langle \alpha\_j,x \rangle^2],
> $$
> where $x \in \mathbb{R}^n$ is a random vector whose $\... | https://mathoverflow.net/users/22954 | Lower bound of the expectation of the product of inner products of random vectors | The exact lower bound on $L$ is $0$ for any $n\ge2$ and $m\ge2$. Indeed, for $j=1,\dots, n$, let $a\_j:=\alpha\_j$ be any unit vectors in $\mathbb R^n$ such that $a\_1=\frac1{\sqrt{2}}(1,1,0,\dots,0)$ and $a\_2=\frac1{\sqrt{2}}(1,-1,0,\dots,0)$. Then
$\prod\_{1\leq j\leq m} \langle \alpha\_j,x \rangle^2=0$ and hence $... | 3 | https://mathoverflow.net/users/36721 | 333417 | 142,595 |
https://mathoverflow.net/questions/250978 | 6 | This question is an extension of something I asked earlier here:
[Ordering of large cardinals by cardinality](https://mathoverflow.net/questions/219132/ordering-of-large-cardinals-by-cardinality?noredirect=1&lq=1)
I have seen large cardinals ordered by consistency strength in several places but no ordering by cardina... | https://mathoverflow.net/users/94232 | Large cardinals ordered by cardinality of least instance | I added some information elsewhere, but it was to long to put in one post: <http://metaordinals.azurewebsites.net/?p=261>
**Part 1:** Your ordering is correct, but the size of the gaps are much more complex than just $LC\_1<LC\_2$ can express, and that list might be misleading. There are two main types of limits. Sta... | 8 | https://mathoverflow.net/users/141402 | 333418 | 142,596 |
https://mathoverflow.net/questions/332780 | 7 | I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:
<https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf>
She writes in this remark the following:
**Remark 0.5** *A compact Kähler manifold $X$ which is rationally connected satisfies $H^2(X, {\cal O}\_X) = 0$, hen... | https://mathoverflow.net/users/13441 | Rationally connected Kähler manifolds are projective | This result follows from Corollaire (p.212) of [this paper](http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00209696X) by F. Campana, *Coréduction algébrique d'un espace analytique faiblement kählérien compact,* Invent. Math. 63 (1981), no. 2, 187–223.
I had the same problem of finiding a citable reference... | 4 | https://mathoverflow.net/users/13168 | 333419 | 142,597 |
https://mathoverflow.net/questions/333401 | 1 | In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the usual definition is given bellow. More information could be found, for example, in [Dafermos's book](https://www.springer.com/la/book/9783662494493).
Let's consider a probl... | https://mathoverflow.net/users/117762 | Banach space-valued test functions in the definition of a weak solution of a PDE problem | Writing your equation (3) as $a(u,\psi)=0$, it is indeed common to call $u\in L^1\_\mathrm{loc}$ a `weak solution' to your problem if and only it satisfies
$$
a(u,\psi) = 0 \mbox{ for all } \psi \in C\_c^\infty(\mathbb{R}\times [0,T))
$$
However, this is a *definition* which implicitly encodes $C^\infty\_c(\mathbb{... | 2 | https://mathoverflow.net/users/61771 | 333423 | 142,598 |
https://mathoverflow.net/questions/333175 | 3 | I'd like to ask the following question:
>
> Are the **Brauer character values** of $kG$-modules (where $k$ and $G$ are finite) in MAGMA computed with respect to the standard $p$-modular system described in the book of Lux and Pahlings\* (chapter 4 in Representations of Groups: A computational Approach)?
>
>
>
... | https://mathoverflow.net/users/12826 | Does MAGMA use a standard p-modular system? | Paragraph 3.2.1 of [On Certain Subgroups of $E\_8(2)$ and their Brauer Character Tables](https://www.research.manchester.ac.uk/portal/files/84018931/FULL_TEXT.PDF) explains precisely how the Brauer characters are defined in MAGMA. The ambiguity in the definition refers to the choice of a definite root of unity. MAGMA u... | 5 | https://mathoverflow.net/users/11260 | 333431 | 142,601 |
https://mathoverflow.net/questions/333378 | 0 | Let $k$ be an infinite field. Let $f:X\rightarrow \mathrm{Spec}\:k$ be a morphism of finite type. Assume that $X$ is not the empty scheme and that $f$ is not of relative dimension $\leq 0$ ([definition](https://stacks.math.columbia.edu/tag/02NJ)).
Do the following sets have equal cardinalities:
* the set of elemen... | https://mathoverflow.net/users/nan | Counting points on a scheme of finite type over an infinite field | *Remark: I am not sure if this is non-trivial enough to be on MO, feel free to downvote/vote to delete/transfer to MSE if you think this is trivial. I tried to write an obnoxiously detailed answer, primarily for my own understanding. Please point out if the argument is non-optimal in some places.*
Throughout the answ... | 2 | https://mathoverflow.net/users/nan | 333442 | 142,605 |
https://mathoverflow.net/questions/333444 | 5 | **Motivation (informal).** When trying to generate a random bit-stream, we expect that "half of the" bits are $0$, and the "other half" are $1$. So, how about $010101\ldots$? Well, we would also expect that if we look at every second member of the sequence, then "half of" those bits are $0$ and the other half are $1$. ... | https://mathoverflow.net/users/8628 | Arithmetically random bitstreams | The [Thue–Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) is such an example, as was (first, I believe) [proved by Dumont](https://mathscinet.ams.org/mathscinet-getitem?mr=725391).
If you take a uniformly random real number in $[0,1]$, its binary expansion will have this property with proba... | 7 | https://mathoverflow.net/users/5091 | 333445 | 142,606 |
https://mathoverflow.net/questions/333436 | 6 | I'm trying to understand something about the Monge problem. The Monge problem is:
Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu\_1,\mu\_2) = \{ T: \mathbb{R}^d \rightarrow \mathbb{R}^d | \text{ Borel maps with condition} \, \, T\#\mu\_1 = \mu\_2 \}$$
where $\mu\_1$ and ... | https://mathoverflow.net/users/69441 | Why is it difficult to solve the Monge problem directly? | 1. It is not true that $T\_n\to T$ in any sense. However, since $L^\infty=(L^1)^\*$, the unit ball in $L^\infty$ is weak\* compact by Banach-Alaogu theorem. Hence, one can extract from $T\_n$ a weak\* convergent subsequence.
2. Yes, this is a standard notion of lower semi-continuity from calculus of variations.
3. For ... | 3 | https://mathoverflow.net/users/56624 | 333457 | 142,608 |
https://mathoverflow.net/questions/333430 | 12 | Let $V$ be a set of positive integers whose natural density is 1. Is it necessarily true that $V$ contains an infinite arithmetic progression?—i.e., that there are non-negative integers $a,b,\nu$ with $0\leq b\leq a-1$ so that: $$\left\{ an+b:n\geq\nu\right\} \subseteq V$$
In addition to an answer, any references on... | https://mathoverflow.net/users/120369 | Is there a set of positive integers of density 1 which contains no infinite arithmetic progression? | Another construction is to let $n \notin V$ if and only if $n$ begins with at least $\sqrt{\log{n}}$ consecutive '9's when written in decimal. This satisfies the stronger property that there is no non-constant polynomial $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is contained entirely in $V$.
| 13 | https://mathoverflow.net/users/39521 | 333470 | 142,612 |
https://mathoverflow.net/questions/333449 | 3 | If $A$ is a square matrix with positive integer coefficients with a biggest real eigenvalues $\lambda>1$, then $\lambda$ is a *Perron-number*, which means that it is a real algebraic integer bigger than $1$, whose Galois conjugates $\mu$ satisfy $\lvert \mu\rvert<\lambda$. This follows from the Perron-Frobenius theorem... | https://mathoverflow.net/users/23758 | Analogue of Perron numbers for non-negative matrices | You are looking for "weak Perron numbers", which are numbers such that some positive integer power is a Perron number. In your example $\lambda=\sqrt{2}$, its square $\lambda^2=2$ is a Perron number, being an eigenvalue of a square matrix whose entries are all $2$.
For information, see Doug Lind's paper "The entropie... | 4 | https://mathoverflow.net/users/20787 | 333471 | 142,613 |
https://mathoverflow.net/questions/333476 | 2 | If we shift the set $\mathbb P=\{p\_1,...,p\_n,...\}$ of all prime numbers by some natural number $2a$ to obtain a set $\mathbb P+2a=\{p\_1+2a,...,p\_n+2a,...\}$ then I expect that $\mathbb P +2a$ contains an infinite number of prime numbers and that it contains an infinite number of composite numbers.
I would like t... | https://mathoverflow.net/users/nan | Density of primes in the set $\{p_1+2a,...,p_n+2a,...\}$ for every natural $a$, any conjectures? | Fix $a\geq 1$. It is elementary to see that $\mathbb P+2a$ contains infinitely many composite numbers -- indeed, if this were not the case, then for sufficiently large primes $p$, $p+2a$ would also be a prime, hence so would be $p+4a,p+6a,\dots,p+2pa=p(1+2a)$ which is clearly absurd.
Indeed, it is true that $f(a)=0$ ... | 6 | https://mathoverflow.net/users/30186 | 333479 | 142,616 |
https://mathoverflow.net/questions/333473 | 4 | Let $\kappa < \beth\_2$ and $\lambda<\beth\_1$ be cardinals. What can we say about $\kappa^{\lambda}$ without assuming CH? Is it true that $\kappa^{\lambda} < \beth\_2$ or $\kappa^{<\beth\_1} < \beth\_2$?
| https://mathoverflow.net/users/141323 | Cardinal exponentiations inequality | Even if CH holds, this can break.
Suppose $\beth\_1=\aleph\_1$ and $\beth\_2=\aleph\_{\omega+1}$. Let $\lambda=\aleph\_0$ and $\kappa=\aleph\_\omega$. By [Konig's theorem](https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_(set_theory)) we know that $\kappa^\lambda=\kappa^{cf(\kappa)}>\kappa$; since $\kappa^+=\beth... | 6 | https://mathoverflow.net/users/8133 | 333481 | 142,617 |
https://mathoverflow.net/questions/333487 | 1 | I try to understand Iarrobinos example of a nonsmoothable 0-dimensional scheme with the help of Artins notes on it:
<http://www.math.tifr.res.in/~publ/ln/tifr54.pdf> (pages 4-6)
But I have some difficulties with this topic. So here are my questions:
1. Very supid question: Are "smooth" and "nonsingular" schemes t... | https://mathoverflow.net/users/141610 | Example of a nonsmoothable scheme | 1. In the situation at hand, non-singular and smooth are interchangeable.
2. As @Asura Path mentioned, the total space need not be smooth.
3. Parameter space can be any scheme (over $k$), since one is looking at deformations of a finite dimensional $k$-algebra. If such an algebra can be smoothened, then you may assume ... | 1 | https://mathoverflow.net/users/9502 | 333489 | 142,619 |
https://mathoverflow.net/questions/333458 | 3 | I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus".
---
Denote by $\mathcal{F}^X\_{t\_0}$ the natural filtration corresponding to a process $X:[0,\infty)\times \Omega \to \mathbb{R}^k$. Suppose that every sample path of $X$ is RCLL (**R**ight **C**ontinuous on $[0,\in... | https://mathoverflow.net/users/141594 | Filtration exercise | Continuity at zero is included in the RCLL property.
If $X(\cdot,\omega)$ is discontinuous at some $t \in (0,t\_0)$, then the left and right limits at $t$ (which exist) must differ. Thus there must exist an integer $n>0$ such that
$$
\|\lim\_{s\to t^{-}} X(s,\omega)-\lim\_{s\to t^{+}} X(s,\omega)\|>1/n \,.
$$
Since t... | 3 | https://mathoverflow.net/users/7691 | 333490 | 142,620 |
https://mathoverflow.net/questions/333493 | 5 | Let $f\_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int\_{\mathbb{R}^{n}}f\_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\epsilon>0$, does there exist a sequence $x\_n\in\mathbb{R}^n$ such that
$$\lim\_{n\to+\infty}a^n\frac{\int\_{\|x-x\_n\|^2\leq ... | https://mathoverflow.net/users/123075 | Ratio of integrals with increasing dimension over Euclidean balls | Such a sequence $(x\_n)$ always exists. Indeed, the displayed ratio expression under the limit sign equals $R\_n(x\_n)$, where
\begin{equation\*}
R\_n(y):=\frac{g\_{n,r\_n}(y)}{g\_{n,s\_n}(y)},\quad
r\_n:=n^{(1-\epsilon)/2}, \quad
s\_n:=n^{1/2},
\end{equation\*}
\begin{equation\*}
g\_{n,r}(y):=\int f\_n(x)1\_{|... | 2 | https://mathoverflow.net/users/36721 | 333497 | 142,623 |
https://mathoverflow.net/questions/333519 | 7 | There are various different notions of the proof-theoretic ordinal of a theory; most of these are "notation-dependent" in that they're only nontrivial once we restrict attention to a class of "natural" notations. There are, however, ways to attack this in a notation-independent way (e.g. the supremum of the ordinals wh... | https://mathoverflow.net/users/8133 | Proof-theoretic ordinals: inevitable consistency? | The ineventable consistency ordinals do not exist for all the computably axiomatizable extensions of $\mathsf{RCA}\_0$. Indeed, for any given notation system $\alpha$ I'll construct a notation system $\alpha^\*$ such that $\alpha$ and $\alpha^\*$ are notations for the same ordinal and $\mathsf{RCA\_0}\vdash \lnot \math... | 7 | https://mathoverflow.net/users/36385 | 333522 | 142,628 |
https://mathoverflow.net/questions/333511 | 0 | $\textbf{Definitions:}$
Let $M$ be an $A$-module. The collection $\mathcal{S}(M)$ of all submodules of $M$ is partially ordered by the inclusion relation. The collection $\mathcal{S}(M)$ is said to be distributive if and only if the following equivalent identities hold for all choices of $N,P,Q\in\mathcal{S}(M)$:
(... | https://mathoverflow.net/users/80991 | Equivalent condition for distributive property of $\mathcal{S}(M)$ | Let $H$ be an $A$-submodule of $M$. Note that the condition cannot hold if there exist two distinct $A$-submodules $P$ and $Q$ of $M$ containing $H$ such that $P/H$ and $Q/H$ are simple and isomorphic. Therefore the desired conclusion follows from Theorem 1 of the following paper by V. Camillo: <https://www.sciencedire... | 3 | https://mathoverflow.net/users/14653 | 333527 | 142,629 |
https://mathoverflow.net/questions/333510 | 3 | this might be a dumb question, but I was working on a problem and ran into the following (sub)problem.
Suppose we have a nonnegative vector $\pi \in \mathbf{R}^n$ that satisfies $\sum\_{i=1}^n \pi\_i = 1$, i.e., it is a discrete probability density. We want to choose a unit vector $v \in \mathbf{R}^n$, $\|v\|=1$, whe... | https://mathoverflow.net/users/126373 | A unit vector that maximizes variance in a discrete probability distribution | Define the $n\times n$ symmetric matrix $M$ with elements
$$M\_{ij}=\pi\_i \delta\_{ij}-\pi\_i\pi\_j.$$
We seek to maximise the quadratic form
$$f(v)=\sum\_{ij} v\_i M\_{ij} v\_j$$
where $v$ is a vector on the unit $n$-sphere. An extremum is reached for an eigenvector $v\_0$ of $M$ and $f(v\_0)=\mu\_0$ is the correspon... | 1 | https://mathoverflow.net/users/11260 | 333532 | 142,631 |
https://mathoverflow.net/questions/301279 | 1 | Suppose we have abstract polytope $F$ of dimension $d$ (that is the greatest rank facet has rank $n$). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A\_i(F)$, and in some cases such realisation may have properly defined d-dimensional volume $V\_i$ (with various values, depend... | https://mathoverflow.net/users/3811 | Is volume of abstract polytope realisation bounded by length of edges? | When $M\_{d-1}^s$ is the facet volume of the $s$-th facet, then it would contribute the largest part to the unknown polytope volume $M\_d$ if it would be assumed to be a $(d-1)$-dimensional hyperball volume and thus contribute to $M\_d$ as the according cone. That is, the volume of this cone, $V\_d^{\operatorname{cone}... | 1 | https://mathoverflow.net/users/118679 | 333539 | 142,633 |
https://mathoverflow.net/questions/333537 | 2 | Let $\mathcal{D}'\_+:=\{T\in \mathcal{D}'(\mathbb{R}): \textrm{supp}(T)\subset [0,\infty)\}$. Here $\mathcal{D}'(\mathbb{R})$ is the usual space of distributions on $\mathbb{R}$, equipped with the weak$\ast$-topology induced by $\mathcal{D}(\mathbb{R})$, and $\mathcal{D}\_+'$ is given the subspace topology induced from... | https://mathoverflow.net/users/140146 | Continuity of convolution on $\mathcal{D}'_+$ | **Yes**, it is.
The convolution on $\mathcal{D}'\_+$ can be defined as follows. Fix a $C^\infty$ function $\psi$ such that $\psi(x) = 1$ for $x \geqslant 0$ and $\psi(x) = 0$ for, say, $x \leqslant -1$. If $\phi \in \mathcal{D}$, define
$$\tilde\phi(x, y) = \phi(x + y) \psi(x) \psi(y) . $$
Note that $\tilde{\phi} \in... | 2 | https://mathoverflow.net/users/108637 | 333559 | 142,644 |
https://mathoverflow.net/questions/260249 | 4 | *This is a revised version of a [post](https://math.stackexchange.com/q/2098832/212120) on Math.SE. It is a rather basic question (which I'd be glad to delete if the community regards as off-topic).*
---
Is there a way to prove that (if consistent) $\mathsf{ZFC}$ can't prove that there exists a weakly inaccessibl... | https://mathoverflow.net/users/66044 | Consistency strength of weakly inaccessibles without $\mathsf{GCH}$ | Suppose $\kappa$ is weakly inaccessible. Then it is immediate $L\_\kappa$ satisfies pairing, separation, extensionality, regularity, infinity, union, and choice. To see that $L\_\kappa$ satisfies Replacement, let $h(x)=min\{\alpha|x\subseteq L\_\alpha\}$. Let $X\in L\_\kappa$, and $F(X)=\{f(x)|x\in X\}$. Let $Z=\{h(x)|... | 2 | https://mathoverflow.net/users/141402 | 333561 | 142,645 |
https://mathoverflow.net/questions/325854 | 6 | [Ackermann](https://en.wikipedia.org/wiki/Ackermann_set_theory) in 1956 proposed an axiomatic set theory.
Reinhardt proved that [Ackermann's set theory equals ZF](https://www.sciencedirect.com/science/article/pii/0003484370900112)
It's clear that Zermelo set theory can be interpreted in Ackermann's set theory minus... | https://mathoverflow.net/users/95347 | Is Ackermann's set theory minus class comprehension equal to ZF? | Let $V$ be the $V$ from $A$. It is clear $V$ satisfies exstensionality, seperation, and regularity. Pairing follows from reflection ($x=a\lor x=b$). Inductively, we can prove each $n\in V$, and so we have $\omega\in V$. If $X\in V$, then $\cup X\subseteq V$. Then define $\phi(x)\leftrightarrow \exists y(y\in X\land x\i... | 1 | https://mathoverflow.net/users/141402 | 333572 | 142,648 |
https://mathoverflow.net/questions/219725 | 23 | $\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Homeo{Homeo}$Let $M$ be a $1$-connected, closed, smooth manifold with $\dim(M)>4$ and let us set $\MCG(M)=\pi\_0(\Diff(M))$. Dennis Sullivan proved that $\MCG(M)$ is commensurable to an arithmetic group.
*Edit:* regarding A. Kupers' rem... | https://mathoverflow.net/users/27816 | Mapping class groups in high dimension | Let me assume that M is at least 5-dimensional.
Sullivan's proof only uses surgery theory and properties of O(n) that also hold for Top(n), so the answer to your first question is yes.
Regarding your other questions: the homotopy fiber Homeo(M)/Diff(M) of the map BDiff(M) -> BTop(M) is subject of smoothing theory. ... | 11 | https://mathoverflow.net/users/32022 | 333583 | 142,653 |
https://mathoverflow.net/questions/333405 | 6 | Let $(X,d)$ be a metric space having Hausdorff dimension $\alpha>0$ and let $0<\beta<\alpha$. Is there a metric subspace of $X$ having Hausdorff dimension $\beta$?
| https://mathoverflow.net/users/130919 | Subspaces of metric spaces having prescribed dimension | By Corollary 7 in [How95], every analytic subset of a complete separable metric
space which has positive (or infinite) Hausdorff measure of dimension s contains a compact
set which has finite and positive Hausdorff measure of dimension s.
[How95] J.D. Howroyd. On dimension and on the existence of sets of finite posit... | 2 | https://mathoverflow.net/users/7691 | 333589 | 142,656 |
https://mathoverflow.net/questions/333534 | 3 | I have been lately thinking about the feasibility of creating a "mediocre algebraic geometer" AI. I thought that to train it, one could feed it some large chunks of algebraic geometry presented in an accessible form.
I do not think that human-written text counts as an accessible form, even one using very limited num... | https://mathoverflow.net/users/nan | A complete formalization of EGA in Lean | Schemes have been formalized in Lean, with the aim of verifying formally some parts of the Stacks project: see [here](https://github.com/kbuzzard/lean-stacks-project) and [here](https://github.com/ramonfmir/lean-scheme). They have schemes but I'm not sure they have morphisms of schemes yet. This should give you a feeli... | 3 | https://mathoverflow.net/users/6506 | 333600 | 142,658 |
https://mathoverflow.net/questions/333599 | 0 | On p.24 of the book **"Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$"**.
**I quote:**
"For example, the fixed point $-\rho$ under the dot action lies in a linkage
class by itself (and no other weight does). The usual notion of regular
weight for $\lambda \in\mathfrak{h}^\*$, requiring t... | https://mathoverflow.net/users/110229 | About weights in $\mathfrak{h}^*$ | 1. Any element of $\mathfrak{h}^\*$ is called weight. I don't know why in this case the author used definite article. As a non-native speaker I would use indefinite one, i.e. "a weight is called regular if ..."
2. That sounds wrong. One can speak about weights which are integral (or dominant) with respect to some (posi... | 4 | https://mathoverflow.net/users/6818 | 333602 | 142,659 |
https://mathoverflow.net/questions/333548 | 13 | Start with the product
$$(1+x+x^2) (1+x^2)(1+x^3)(1+x^4)\cdots$$
(The first polynomial is a trinomial..The others are binomials..)
Is it possible by changing some of the signs to get a series all of whose coefficients are $ -1,0,$or $1$?
A simple computer search should suffice to answer the question if the answer... | https://mathoverflow.net/users/40145 | Can you make an identity from this product? | Proof of Doriano Brogloli's answer:
Call $a(n)$ the $n$th coefficient of $A(x)=(1-x)(1-x^2)\cdots$. By Euler's pentagonal theorem
we have $a(n)=0$ unless $n=m(3m\pm1)/2$ for some $m$, in which case $a(m)=(-1)^m$.
Call $b(n)$ the $n$th coefficient of $B(x)=(1-x^2)(1-x^3)...$. Since $A(x)=(1-x)B(x)$
we have $b(n)-b(n-1... | 12 | https://mathoverflow.net/users/81776 | 333604 | 142,661 |
https://mathoverflow.net/questions/333605 | 7 | Is every flat morphism of schemes $X\rightarrow S$ such that the fiber over any point is smooth necessarily formally smooth? [There are](https://mathoverflow.net/a/200/141498) formally smooth morphisms that are not flat so the converse fails. If we assume that the morphism is locally of finite presentation, then it is ... | https://mathoverflow.net/users/nan | Flat with smooth fibers implies formally smooth? | Welcome new contributor. This is too long for a comment. Let $R$ be a DVR with a uniformizing element $t$, e.g., $k[[t]]$ where $k$ is a field. Let $C$ be the $R$-algebra $$C=R[x\_n:n\in \mathbb{Z}\_{\geq 0}].$$ Let $J$ be the ideal in $C$, $$J=\langle x\_n-tx\_{n+1} :n\in \mathbb{Z}\_{\geq 0} \rangle.$$ Let $A$ be the... | 6 | https://mathoverflow.net/users/13265 | 333607 | 142,662 |
https://mathoverflow.net/questions/333620 | 12 | This is an extension of a question I asked [here on Math.SE](https://math.stackexchange.com/q/3251019/274352)
---
Assume that I have a finitely generated residually finite centerless group $G$. Is it true that the profinite completion $\hat{G}$ also has trivial center?
In the linked question, user YCor was abl... | https://mathoverflow.net/users/95243 | Does a (nice) centerless group always have a centerless profinite completion? | The answer is No in general. Let $n\geq 3$ be odd (it is not necessary that $n$ be odd) and suppose $G=\mathrm{SL}\_n({\mathbb Z})$. There exists a subgroup $\Gamma \subset \mathrm{SL}\_n({\mathbb Z})$ of finite index which is torsion-free and centreless (the centre can only be $\pm 1$ and because $n$ is odd the centre... | 20 | https://mathoverflow.net/users/23291 | 333622 | 142,669 |
https://mathoverflow.net/questions/331083 | 6 | There is a problem about Cartan's development, arising from the paper ['Kinetic Brownian motion on Riemannian manifolds', Subsection 2.4.1](https://projecteuclid.org/euclid.ejp/1465067216). To be precise, let $(M,g)$ be a $d$-dimensional complete Riemannian manifold, $\pi:OM\to M$ be its orthonormal frame bundle with s... | https://mathoverflow.net/users/90082 | How to prove two curves in the frame bundle to project to the same curve on base manifold? | **Method 1:** Let us think about it reversely.
>
> We want to find a curve $\tilde z = \{z\_t\}$ on $OM$ such that $\pi(\tilde z\_t) = x\_t$ and the horizontal component of the tangent vector field of $\tilde z$ is corresponding to the constant vector $\epsilon\_1\in\mathbb R^d$, that is,
> \begin{equation}\tag{... | 1 | https://mathoverflow.net/users/90082 | 333624 | 142,670 |
https://mathoverflow.net/questions/325598 | 0 | From what I read of time-scale calculus literature, most results of continuous-time and discrete-time systems can be generalized to arbitrary time-scales by considering the generalized derivative operator instead of the forward difference operator or the standard derivative. In particular, the time-scale concepts can b... | https://mathoverflow.net/users/22389 | Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems? | The answer is no. As a former student of Davis who works in control theory, I can tell you that (1) comes from examining the "energy" of the system: i.e. by examining the (in this case delta) derivative of the scalar quantity ||x(t)||^2 . Take a look at Jackson's and Dacuhna's dissertations from Baylor and you will see... | 2 | https://mathoverflow.net/users/141665 | 333625 | 142,671 |
https://mathoverflow.net/questions/333565 | 2 | The number of ways to color the vertices of an $n$-dimensional cube can be obtained from the Redfield-Pólya theorem by obtaining the cycle index of the relevant permutations, of which there are $n!2^n$. A method for doing this using conjugacy classes of the $n!$ permutations of the axes leads to quick results. Both the... | https://mathoverflow.net/users/14207 | Quick enumeration for the coloring of the vertices of an n-dimensional cube | The group of symmetries of a hypercube is known as the [hyperoctahedral group](https://en.wikipedia.org/wiki/Hyperoctahedral_group). For $n$-dimensional hypercube, it is formed by the wreath product of symmetric groups $S\_2$ and $S\_n$. The classic reference for enumeration under actions of wreath products is [Enumera... | 2 | https://mathoverflow.net/users/7076 | 333635 | 142,675 |
https://mathoverflow.net/questions/333632 | 1 | I have a technical question on a continuity of green function.
**Setting**
Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$.
Let $X=(\{X\_t\}\_{t \ge 0},\{P\_x\}\_{x \in E})$ be a **transient** diffusion process. We assume that there exists a continuous function $p\_{t}(... | https://mathoverflow.net/users/68463 | Continuity of green functions | *(A comment too long for a comment).*
---
I am not an expert, but I guess an answer will depend on how specific you want it to be. If this is about reflecting Brownian motions, then we know a lot, including heat kernel bounds for nice enough domains; see [Bass–Hsu, *Some potential theory for reflecting Brownian m... | 1 | https://mathoverflow.net/users/108637 | 333640 | 142,677 |
https://mathoverflow.net/questions/333642 | 6 | A result of Bhargava-Skinner-Zhang says that a majority of elliptic curves over $\mathbb{Q}$ satisfy the BSD rank conjecture.
The are infinitely many isomorphism classes of $\mathbb{Q}$ so one naturally has to be more precise about what "a majority" means; in their case, the elliptic curves are done by height.
The... | https://mathoverflow.net/users/nan | Why do Bhargava-Skinner-Zhang consider the ordering by height? | Roughly speaking, the height of an arithmetic object (number, variety, ...) is a natural measure of its complexity, say in the sense of "how many bits does it take to describe the object." (This is not meant to be rigorous, but you seem to want to know why people use "heights".) One can then ask for heights (complexity... | 18 | https://mathoverflow.net/users/11926 | 333646 | 142,679 |
https://mathoverflow.net/questions/314214 | 8 | I want to collect different proofs of Erdös-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$.
I know the original proof of Erdös-Rényi, the proof that uses Galton-Watson processes and the short proof of Sudakov and Krivelevich.
Are there other (essentially different) pr... | https://mathoverflow.net/users/46573 | Collecting proofs of the birth of the giant component | Nachmias, Asaf, and Yuval Peres. "The critical random graph, with martingales." Israel Journal of Mathematics 176, no. 1 (2010): 29-41.
<https://arxiv.org/abs/math/0512201>
| 3 | https://mathoverflow.net/users/7691 | 333653 | 142,683 |
https://mathoverflow.net/questions/333675 | 4 | I'm reading this [paper](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-55.1.59) and at page 67, he states that for any line bundle $L$ over a Rieman surface there is a connection $A$ whose curvature is
$$
F(A)=(\deg L)\omega,
$$
where $\omega$ is a positive form. Does anyone know how can I prove i... | https://mathoverflow.net/users/128860 | Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$ | Let me expand a bit Henri's (hi Henri!) answer, even if this is completely standard. In general, given a compact Kähler manifold $X$ of any dimension, given a holomorphic line bundle $L\to X$, and given any $(1,1)$-form $\eta$ representing $c\_1(L)$, there exists a smooth hermitian metric $h\_\eta$ on $L$ such that its... | 5 | https://mathoverflow.net/users/9871 | 333693 | 142,694 |
https://mathoverflow.net/questions/333708 | 3 | Remember that a Bernstein set is a set
$B\subseteq \mathbb{R}$ with the property that for any uncountable closed set, $S$, in the real line both
$B\cap S$ and $(\mathbb{R}\setminus B)\cap S$ are non-empty. Some known results are the following: Bernstein sets are Baire spaces, also the Banach-Mazur game played in a Bern... | https://mathoverflow.net/users/138770 | Product of Bernstein sets | There exist some general results which show when a product of Baire spaces is Baire. Specifically, Theorem 2 [here](http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49113.pdf) states
>
> If $X,Y$ are two Baire spaces and at least one has a *locally countable pseudo-base*, then $X\times Y$ is a Baire space.
>
>
>
A *... | 5 | https://mathoverflow.net/users/30186 | 333710 | 142,699 |
https://mathoverflow.net/questions/333700 | 0 | Suppose $Y$ is a subspace of a normed linear space $X$ and let $y\in S\_Y, y^\*\in S\_{Y^\*}$ such that $y^\*(y)=1$, where $S\_Y$ denotes the closed unit sphere in $Y$. My question is the following:
Is it possible to find an extension $x^\*$ of $y^\*$ to $X$ such that $$\{x\in X:x^\*(x)=1\}\subset Y.$$ Any suggestion... | https://mathoverflow.net/users/41137 | Can a hyperplane be contained in a subspace? | The answer is "No, unless $X=Y$".
In fact, if there is $u\in X\backslash Y$, then the vector $v=u-x^\*(u)y$ is not in $Y$ and satisfies $x^\*(v)=0$. Therefore all vectors $y+tv, t\ne 0$ satisfy $x^\*(y+tv)=1$, but $y+tv$ are not in $Y$ for $t\ne 0$.
| 3 | https://mathoverflow.net/users/85406 | 333722 | 142,702 |
https://mathoverflow.net/questions/289025 | 2 | Given a directed graph $G=(V,E)$, with no self-loops, with a vertex that has a maximal out-degree $\le d\in O(\log |V|)$, and with a diameter $\text{diam}(G)\in O(\text{poly }d)$, consider converting the adjacency matrix of $G$ into a lazyish, doubly-stochastic transition matrix, with the following transition probabili... | https://mathoverflow.net/users/8927 | If the diameter of a bounded degree, directed graph is polynomial in the degree of the graph, is the mixing time also polynomial? | If your graph is too close to bipartite, that could greatly increase the mixing time.
If you make the walk more lazy, a general upper bound on the order of the mixing time is the diameter times the number of edges. See Chapter 10 in [1]. But the answer to your question is negative.
Let $G$ be a random connected graph o... | 2 | https://mathoverflow.net/users/7691 | 333729 | 142,705 |
https://mathoverflow.net/questions/333659 | 6 | Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(F)\in K\_1(C^\*(|\mathbb{R}|)),$$
where $C^\*(|\mathbb{R}|)$ is the Roe algebra of the metric space (as opposed to ... | https://mathoverflow.net/users/78729 | Coarse index of Dirac operator on $\mathbb{R}$ | There are a number of ways to do this calculation, but at risk of shamelessly plugging my own work there is a nice way to see it using a Mayer-Vietoris principle.
Decompose $\mathbb{R}$ as the union of the rays $\mathbb{R}^+ = [0, \infty)$ and $\mathbb{R}^- = (-\infty, 0]$, intersecting at a point. This decomposition... | 6 | https://mathoverflow.net/users/4362 | 333731 | 142,706 |
https://mathoverflow.net/questions/333298 | 5 | Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$. I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$
Let $ v \in V $. Assume that the orbit $ Gv $ is closed in $ V $. Assume also that the stabilizer of $ v $ in $ G $ is finite.
*... | https://mathoverflow.net/users/438 | Fibre of GIT morphism | I am just recording what was said in the comments so this question does not appear *completely* unanswered.
Let $\pi\_X:X\to X//G$ be the GIT quotient of an affine variety over $\mathbb{C}$ by a reductive group $G$. WLOG assume the action is effective.
First, a point is properly stable if its orbit is closed and it... | 2 | https://mathoverflow.net/users/12218 | 333744 | 142,711 |
https://mathoverflow.net/questions/333756 | 5 | We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}\_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M\_h$ the coarse moduli space of $\mathcal{M}\_h$. For any $X\in \mathscr{M}\_h({\rm Spec}(\mathbb{C}))$, we know that its Kuranishi fami... | https://mathoverflow.net/users/62735 | Coarse moduli space versus Kuranishi family | The answer is *yes*: the germ of complex space $(M\_h, \, [X])$ is analytically isomorphic to the quotient $S/\mathrm{Aut}(X)$.
This is true not only for moduli spaces of hyperbolic curves, but also in the (much more difficult) context of Gieseker moduli space of (canonical models of) surfaces of general type, see R... | 4 | https://mathoverflow.net/users/7460 | 333759 | 142,714 |
https://mathoverflow.net/questions/333747 | 1 | As an outgrowth of [this](https://mathoverflow.net/questions/319496/on-a-theorem-of-chetwynd-and-hilton-in-graphs) question, I have another question, that is, why not the definition of conformability includes a $\Delta$ vertex coloring also, instead of only $\Delta+1$ coloring of vertices. This is because, the square o... | https://mathoverflow.net/users/100231 | Clarifications regarding conformability in graph colorings | There is nothing to be missed. In fact, a $\Delta$-conformable(or any $k$-conformable coloring, where $k\le\Delta+1$) induces a $\Delta+1$-conformable coloring in case of even order graphs, as one(or more) color class(es) can be taken to have $0$ vertices, which has same parity as the total number of vertices. However,... | 0 | https://mathoverflow.net/users/100231 | 333763 | 142,715 |
https://mathoverflow.net/questions/333748 | 4 | If I collapse a (say, closed) set in a length space, I obtain a length space: is there some literature on this?
We consider length spaces as defined by Gromov and others. [However the case of a Riemannian distance already leads to interesting examples of what I am writing]. If $X$ is one such space, it has a distance... | https://mathoverflow.net/users/130919 | Quotient metrics in length spaces | If you take $X=\mathbb R$ and $E=[1,2]\cup[3,4]$ you see that $D\_E(0,5) = 3$ while the geodesic distance between $0$ and $5$ in $X\setminus E \cup\{E\}$ is $2$ because you can join $0$ to $E$ with a curve of length $1$ and $E$ to $5$ with a curve of length $1$. I think you need to require $E$ to be geodesic-convex if ... | 1 | https://mathoverflow.net/users/36826 | 333765 | 142,716 |
https://mathoverflow.net/questions/333757 | 10 | The paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl called *Linear spaces with flag transitive automorphism groups* (Geom. Dedicata) from 1990 annonces a very powerful classification result for the objects mentioned in the title. However, it does not contain any proofs.
In a recent paper (Feng 20... | https://mathoverflow.net/users/37021 | About the paper by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl | The proof for this appeared over a series of papers. The final one was
Jan Saxl, `[On Finite Linear Spaces with Almost Simple Flag-Transitive Automorphism Groups](http://dx.doi.org/10.1006/jcta.2002.3305)' Journal of Combinatorial Theory, Series A 100, 322–348 (2002).
which includes references for all the papers.
... | 13 | https://mathoverflow.net/users/3214 | 333766 | 142,717 |
https://mathoverflow.net/questions/333695 | 9 | It is well-known that the order type of any countable non-standard model of arithmetic $\mathfrak{A}$ is $\omega+(\omega^\*+\omega)\eta$. My question is what could be said about the order types of ordinal notation systems within non-standard models of arithmetic?
The question is motivated by my answer to the questio... | https://mathoverflow.net/users/36385 | Ordinal notations within non-standard models of arithmetic | I'll provide a description of $(\alpha)^{\mathfrak{A}}$ in the case of countable $\mathfrak{A}$. I base my answer on observations by Emil Jeřábek (see discussion below the initial question), but of course any mistakes here are due to me.
From Cantor's normal form theorem it is easy to conclude that each ordinal $\alp... | 6 | https://mathoverflow.net/users/36385 | 333769 | 142,718 |
https://mathoverflow.net/questions/333424 | 7 | Let $\mathcal{L}$ be an *infinite* relation language (this question is trivial in a finite relational language). Suppose that $\mathcal{K}$ is the class of finite models of some $\mathcal{L}$-theory (since we're only concerned with finite models this is equivalent to saying that for each $n$ the class of structures in ... | https://mathoverflow.net/users/83901 | Quantifier elimination in uncountable elementary "Fraïssé classes" | I'm thinking that a certain choice of "generalized metric space" should accomplish this, but it depends on if the "generic theory" ends up right.
First I'll describe the general setup. Let $S$ be a fixed subset of $\mathbb{Q}^{\geq 0}$. Assume $0\in S$. Let $L$ be a language with a binary relation symbol $d\_s(x,y)$ ... | 5 | https://mathoverflow.net/users/38253 | 333783 | 142,723 |
https://mathoverflow.net/questions/333796 | 3 | It is known that there is a natural $\pi\_1(X,x)$ action on $\pi\_n(X,x)$ which also induces a bijection $\pi\_n(X,x)/\pi\_1(X,x) \cong [S^n, X]$.
Now, let $(X,A)$ be a pair of path-connected spaces and $x\in A$. We also have a $\pi\_1(A,x)$ action on the relative homotopy group $\pi\_n(X,A,x)$.
>
> Is there any... | https://mathoverflow.net/users/69190 | $\pi_1$ action on relative homotopy groups $\pi_n(X,A)$ | I'd say the only tricky part is showing the action is well-defined, but if you trust it is here is a proof following Hatcher's proof for the basepointed version:
Let's consider the general case of maps $(X,A,x\_0) \rightarrow (Y,B,y\_0)$ with $Y$ and $B$ path-connected. To be explicit the action is the following: giv... | 4 | https://mathoverflow.net/users/134512 | 333800 | 142,731 |
https://mathoverflow.net/questions/333794 | 5 | Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma\_{V}\left(z\right)\overset{\textrm{def}}{=}\sum\_{v\in V}z^{v}$$ Let $T\_{V}$ denote the set of all $t\in\left[0,1\right)$ for which the limit:
$$c\_{V}\left(t\right)\overset{\textrm{def}}{=}\lim\_{x\uparrow1}\left(1-x\right)\varsi... | https://mathoverflow.net/users/120369 | Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients | The set can be infinite (but only countable). For an example choose any $t\_j$ linearly independent over $\mathbb Q$ and let $V$ be the set of all $v$ such that $vt\_j\mod 1 \notin (\frac 12-a\_j,\frac 12+a\_j)$ for all $1\le j\le J(v)$ where $a\_j$ decrease so fast that $\sum\_j a\_j<+\infty$ and the function $J$ incr... | 8 | https://mathoverflow.net/users/1131 | 333801 | 142,732 |
https://mathoverflow.net/questions/333753 | 6 | Let $G=GL\_2(\mathbb{Q}\_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$.
**Question 1:** Is $\mathrm{ind}\_K^G \rho$ cuspidal?
Here cuspidal is meant in the sense that matrix coefficients are compactly supported.
More precisely, I know already from... | https://mathoverflow.net/users/105652 | When is compact induction cuspidal? | Suppose
(1) $K$ is open and contains a finite index subgroup of the center
(2) For every parabolic subgroup $P$ of $G$ with unipotent radical $N$, $\rho^{ N \cap K}=0$.
Then $\rho$ is cuspidal (i.e. a finite direct sum of super cuspidal representations).
I think the argument is (or finishes on) page 28 of the ... | 4 | https://mathoverflow.net/users/18060 | 333803 | 142,733 |
https://mathoverflow.net/questions/333810 | 2 | Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$. Let $$E = \{\{a,b\}: a,b\in [\omega]^\omega\text{ and } |a\cap b| = 1\}.$$
It is clear that $G = ([\omega]^\omega, E)$ has no uncountable cliques, but do we also have $\chi(G) = \aleph\_0$?
| https://mathoverflow.net/users/8628 | Chromatic number of the linear graph on $[\omega]^\omega$ | Yes.
Take the two smallest elements of a vertex $V\in[\omega]^\omega$ as its color. The number of colors is $\aleph\_0$, and any two vertices with the same color shares at least two elements, so they are not connected.
| 4 | https://mathoverflow.net/users/125498 | 333813 | 142,736 |
https://mathoverflow.net/questions/333779 | 5 | A few weeks ago, Bhatt and Scholze uploaded a preprint of their paper 'Prisms and Prismatic Cohomology' to [arxiv](https://arxiv.org/abs/1905.08229).
In Theorem 6.3 they state their Hodge-Tate comparison. Recently, I started reading on Hodge decomposition and Hodge-Tate composition. However, all Hodge-Tate compariso... | https://mathoverflow.net/users/62127 | Prisms and Hodge-Tate comparisons | **tl;dr** The Hodge-Tate comparison isomorphism relates the reduction mod $I$ of prismatic cohomology to something similar to the "Hodge-Tate cohomology" $\bigoplus\_{i+j = k} H^i(X, \Omega^j\_{X/K})$. Together with the étale comparison theorem relating prismatic cohomology away from $V(I)$ to étale cohomology, this gi... | 11 | https://mathoverflow.net/users/56878 | 333818 | 142,738 |
https://mathoverflow.net/questions/333816 | 0 | Suppose that $p\_1,...,p\_k$ are distinct prime numbers.
Let $f(n,l)$ be equal to the number of elements from set $\{n+1,n+2,...,n+l\}$ that are divisible by some $p\_1,...,p\_k$. Is it true that
$$\sup\_{n,m,l \in \mathbb{N}} |f(n,l)-f(m,l)| = O(k).$$
| https://mathoverflow.net/users/141787 | The maximum difference between the number of elements in the two sets of equal length of consecutive numbers that divisible by some prime numbers | No. There are various results which give counterexamples.
For example, Rankin's construction of large prime gaps boils down to the fact that if $p\_1, \ldots, p\_k$ denote all prime numbers below $x$, then there exists some $n$ with $f\left(n, \frac{x\log x\log\log\log x}{(\log\log x)^2}\right)=0$.
If $n=x, \ell=x^... | 2 | https://mathoverflow.net/users/37555 | 333823 | 142,739 |
https://mathoverflow.net/questions/76108 | 10 | Fix integer $n\ge 1$, and let $E=\{e\_1,...,e\_n\}$ denote the standard basis of the vector space ${\mathbb F}\_2^n$. Thus, for a set $A\subset{\mathbb F}\_2^n$, the sumset $A+E:=\{a+e\colon a\in A,\ e\in E\}$ consists of all those elements of ${\mathbb F}\_2^n$ which are at Hamming distance $1$ from an element of $A$.... | https://mathoverflow.net/users/9924 | An isoperimetric problem on the hypercube | When A consists of even vectors only, the problem was probably solved by Korner and Wei [Odd and even Hamming spheres also have minimum boundary, Discrete Math. 51 (1984), 147–165]. See also Lemma 1.10 in [D. Galvin, On homomorphisms from the Hamming cube to Z, Israel J. of Math 138 (2003), 189-213].
| 6 | https://mathoverflow.net/users/141767 | 333832 | 142,743 |
https://mathoverflow.net/questions/333409 | 1 | This question is related to a [previous one](https://mathoverflow.net/questions/333245/practical-example-of-hamiltonian-reduction); now I better understand the problem and I can more clearly state what is the question.
**Background**
I refer to the following concepts:
Liouville integrability: a Hamiltonian with $... | https://mathoverflow.net/users/138060 | Global reduction of Hamiltonian with an integral of motion (Poincare' reduction) | It's probably easier to think about this geometrically: suppose we have a function $F:M\to\mathbb{R}$ (where $M$ is your phase space) such that $\lbrace H, F\rbrace = 0$, and also the level sets $F^{-1}(c)$ are all (embedded) submanifolds of $M$. Then we can take $P\_n=F$, and from that construct canonical coordinates ... | 1 | https://mathoverflow.net/users/17945 | 333834 | 142,745 |
https://mathoverflow.net/questions/333593 | 2 | I am interested in studying the category of [typical representations](https://link.springer.com/chapter/10.1007/BFb0063691) over basic classical simple Lie superalgebras. In particular, I want to know
1) is this category semisimple (character determines a typical representation)?
2)is this category closed under te... | https://mathoverflow.net/users/33047 | Category of typical representations for Lie superalgebras | Kac (in the linked article) talks about finite dimensional representations (of basic Lie superalgebras). So lets assume that.
1) Yes, the category is semisimple. This is contained in the article by Kac (Theorem 1).
2) No. Kac shows that the typical representations are projective. Therefore a tensor product of typic... | 2 | https://mathoverflow.net/users/129060 | 333852 | 142,747 |
https://mathoverflow.net/questions/333716 | 5 | Let $X=\lambda\_0u\_0v\_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda\_0\in\mathbb{R}$, $u\_0,v\_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\arg\min\_{\substack{Y:Y=aU\\U\in O(n)\\a\in\mathbb{R}}}\|Y-X\|\_F^2$$
where $U$ is orthogonal and $\|\|\_F$ is the F... | https://mathoverflow.net/users/123075 | Best orthogonal approximation of rank 1 matrix | As suggested by Federico Poloni in a comment, it suffices via the SVD to consider the case when $X$ has $x\_{1,1} > 0$ and $x\_{i,j} = 0$ when $(i,j) \neq (1,1)$. Then for any orthogonal matrix $U$ with columns $\mathbf{u}\_1,\ldots,\mathbf{u}\_n$ and scalar $a \in \mathbb{R}$ we have
\begin{align\*}
\|X - aU\|\_F^2 &... | 5 | https://mathoverflow.net/users/11236 | 333853 | 142,748 |
https://mathoverflow.net/questions/333318 | 13 | **I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces.** I need this to justify some computations I did assuming some formal properties which hold in the algebraic category. **Here is a list of properties I want** (some might be deducible from others, I haven't ch... | https://mathoverflow.net/users/110236 | Cotangent Complex in Analytic Category | I will attack the problem three ways, in increasing level of elaboration.
1. Here is a [recent paper](https://arxiv.org/abs/1704.01683) that proves all this. Compared to older sources, this paper uses more machinery. It uses stabilization / Goodwillie calculus, which is more directly about deformations and has to be... | 6 | https://mathoverflow.net/users/4639 | 333871 | 142,752 |
https://mathoverflow.net/questions/333868 | -2 | Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely generated submodule $N$ of $M$ is a direct factor of $N$) ?
| https://mathoverflow.net/users/99246 | Module such that every finitely generated submodule is semisimple | @Mohan has already given an example in the comments. If you ask that the ring $A$ injects into $End\_A(M)$, then here is an example. Let $M=\oplus {\mathbb Z}/p{\mathbb Z}$ be the $\mathbb Z$-module (= an abelian group). Here $p$ runs over all primes. Then $M$ is not free and every finitely generated submodule is of th... | 4 | https://mathoverflow.net/users/23291 | 333873 | 142,753 |
https://mathoverflow.net/questions/333877 | 3 | Consinder a smooth manifold $M$ and $\omega$ a smooth $1$-form on $M$. Assume that there is an open set $U\subset M$ such that $\omega$ never vanishes on $U$. One can define a smooth distribution $\cal F$ on $TU$ via the kernel of $\omega$. Frobenius theorem tells us that such distribution promotes a foliation on $M$ p... | https://mathoverflow.net/users/94097 | An extra condition on Frobenius theorem for $1$-forms | Locally, yes. By the Frobenius theorem, if $\omega\wedge d\omega=0$ then there is a local coordinate system, near each point, in which $\omega=f \, dx$ for some function $f$, and then we can take $\alpha=df/f$.
| 3 | https://mathoverflow.net/users/13268 | 333881 | 142,755 |
https://mathoverflow.net/questions/333838 | 4 | Let $(A,\Delta)=:F(G)$ be a **finite dimensional** $\mathrm{C}^\*$-Hopf algebra, and so the algebra of functions on a *quantum group* $G$.
Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$ the quotient map. Suppose that $J$ is such that
$$\Delta(J)\subset \ker(\pi\otimes\pi).$$
Such ideals are call... | https://mathoverflow.net/users/35482 | Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode | If you continue reading [Wang's paper (project Euclid link)](https://projecteuclid.org/download/pdf_1/euclid.cmp/1104272163), it becomes a little clearer I think. His Theorem 2.11 says that if $J$ is a Woronowicz $C^\*$-ideal is a Woronowicz $C^∗$-algebra $A$, then $A/J$ has canonically the structure of a Woronowicz $C... | 3 | https://mathoverflow.net/users/406 | 333889 | 142,759 |
https://mathoverflow.net/questions/333879 | 4 | Do you know a left-noetherian ring $R$ with a two-sided ideal $I$ such that:
* $I=I.I$;
* $I$ is not projective as a left $R$-module (and better, the tensor product over $R$ of $I$ with itself is not a projective left $R$-module)?
| https://mathoverflow.net/users/76506 | Noetherian ring with a "strange" idempotent ideal | Take any idempotent $e$ in a finite dimensional quiver algebra $KQ/L$ , then the ideal $I=AeA$ will be idempotent but only in rare cases it will be projective.
Here an explicit example:
Take the quiver $Q$ with two vertices 1 and 2 (with corresponding primitive idempotents $e\_1$ and $e\_2$) and an arrow a from 1 to 2 ... | 2 | https://mathoverflow.net/users/61949 | 333890 | 142,760 |
https://mathoverflow.net/questions/333908 | -1 | [EDIT] This posting had been edited to assert that we are speaking about regular mutual stages.
Let $H\_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than cardinality $\kappa$. Formally
$H\_\kappa = \{x: |TC(x)|<\kappa\}$
Where $TC(x)$ is the transitive closure of $x$ def... | https://mathoverflow.net/users/95347 | What is the strength of claiming that the class of all $V_\kappa$ stages that are $H_\kappa$ when $\kappa$ is regular, is inaccessible? | The axiom in your question adds no strength; it's provable in ZFC. The claim in your title, involving inaccessibility, has the same strength as the existence of an inaccessible cardinal.
| 6 | https://mathoverflow.net/users/6794 | 333909 | 142,765 |
https://mathoverflow.net/questions/333875 | 9 | The unpointed version is easy: the model $X = EG \times X \to (EG \times X)/G = X^{un}\_{hG}$ is a fibration with fiber $G$. But when we go pointed, $X = EG\_+ \wedge X \to (EG\_+ \wedge X) / G = X\_{hG}$ is no longer a fibration: its fiber
changes from $G$ over non-basepoints to $\ast$ over the basepoint.
Of course... | https://mathoverflow.net/users/2362 | What is the homotopy fiber of $X \to X_{hG}$, where this is a pointed homotopy orbit? | Here is a special case which gives a partial answer:
(i). Suppose $G$ acts in a *homotopically trivial* way on $X$. This means that there is a trivial $G$-space $Y$ and a pair of $G$-equivariant maps $X \overset\sim\leftarrow X' \overset\sim\to Y$ each which is a weak homotopy equivalence of underlying spaces.
(ii... | 7 | https://mathoverflow.net/users/8032 | 333911 | 142,766 |
https://mathoverflow.net/questions/333863 | 3 | I asked this question on math stack exchange, but it didn't receive any answers. Consider a countably infinite set of variables called $PROP$. We augment $PROP$ with a finite set of boolean connectives, that is, a finite set of n-ary connectives on {0,1}. We say that well-formed formulas of the language are generated a... | https://mathoverflow.net/users/43439 | Is the consequence relation of a finite set of boolean connectives finitely generated? | The answer is yes, but the proof involves tedious checking of various cases, and I will only sketch the main points below. Let $L$ be the given finite set of connectives, and $C$ the clone that it generates (i.e., the set of Boolean functions definable by $L$-formulas). I will name clones using the notation from [Wikip... | 5 | https://mathoverflow.net/users/12705 | 333919 | 142,771 |
https://mathoverflow.net/questions/333920 | 0 | Let $\mathcal{H}\_i$, for $i=1,2$, be Hilbert spaces, and $T:{\frak Dom}(T) \subseteq \mathcal{H}\_1 \to \mathcal{H}\_2$ a densely-defined closed operator. If the kernel of $T$, and the kernel of its adjoint $T^\*:{\frak Dom}(T^\*) \subseteq \mathcal{H\_2} \to \mathcal{H}\_1$, are both finite dimensional, then does it ... | https://mathoverflow.net/users/125790 | An adjoint characterization of (unbounded) Fredholm operators | No. Consider $P:\ell^2(\mathbb Z)\supset D(P)\rightarrow\ell^2(\mathbb Z)$ defined by $P(e\_n) = e^n e\_{n} $ with $$D(P)=\{ (\xi\_n)\in\ell^2(\mathbb Z) : \sum\_{n=-\infty}^\infty e^{2n} |\xi\_n|^2 < \infty \}.$$ Then $P$ is positive, self-adjoint (closed and densely defined) injective, and $P$ does not have closed ra... | 2 | https://mathoverflow.net/users/406 | 333923 | 142,772 |
https://mathoverflow.net/questions/333888 | 8 | Let $\Phi$ be a *Youngs's function*, i.e.
$$ \Phi(t) = \int\_0^t \varphi(s) \,\mathrm d s$$
for some $\varphi$ satifying
1. $\varphi:[0,\infty)\to[0,\infty]$ is increasing
2. $\varphi$ is lower semi continuous
3. $\varphi(0) = 0$
4. $\varphi$ is neither identically zero nor identically infinite
and define the *Luxe... | https://mathoverflow.net/users/141833 | Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$ | The conjectured inequality does not hold.
For a counterexample, consider $\Phi(t)=\max(t^2,t^3)$ and $\Omega=(0,1)$.
Let $f=a\chi\_{(0,b)}$ for $a,b\in (0,1)$.
It can be calculated that $\|f\|\_{L^\Phi}= a b^{1/3}$.
Then the inequality can be written as
\begin{equation\*}
a^2 b^{2/3} \leq C a^2 b
\end{equation\*}
wh... | 5 | https://mathoverflow.net/users/124831 | 333925 | 142,774 |
https://mathoverflow.net/questions/333902 | 14 | I asked [this question](https://math.stackexchange.com/q/3252981/660) on Mathematics Stackexchange, but got no answer.
Let $\mathcal A$ and $\mathcal B$ be nonempty categories whose categories $\mathcal A^{\mathcal A}$ and $\mathcal B^{\mathcal B}$ of endofunctors are equivalent.
>
> Are $\mathcal A$ and $\mathc... | https://mathoverflow.net/users/461 | $\mathcal A^{\mathcal A}\sim\mathcal B^{\mathcal B}\implies\mathcal A\sim\mathcal B\ ?$ (Does A^A ~ B^B imply A ~ B? --- A, B categories) | (Edited to reflect your edit to the question!)
The answer to your original statement (without the "nonempty" assumption) is no because we can let $A=\varnothing$ and $B=\ast$. Their endofunctor categories are each discrete with one object, but the categories themselves are not equivalent.
The question becomes a lot... | 21 | https://mathoverflow.net/users/132451 | 333930 | 142,776 |
https://mathoverflow.net/questions/333876 | 3 | Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|\_\infty,\|b\|\_\infty)<T$ and $\|c\|\_\infty<T^2$ and an $\epsilon>0$.
>
> Assume $a$ and $b$ are coordinatewise coprime (that is every $a\_i$ and $b\_i$ are coprime at every $i\in\{1,\dots,n\}$). Then do we always have such $A$ and $B$ of absol... | https://mathoverflow.net/users/10035 | Approximately satisfying simultaneous vector linear diophantine equations? | Even allowing $A$ and $B$ to be *real* numbers, the vectors $A a + B b$ will all lie in some fixed plane $P$. But then, if $n \ge 3$, for all but $\epsilon$ of the possible values of $c$ one will have
$$\| A a + B b - c \|\_{\infty} \gg T$$
where the constant depends only on $\epsilon$. So for $n \ge 3$ the probabi... | 2 | https://mathoverflow.net/users/141887 | 333940 | 142,777 |
https://mathoverflow.net/questions/333936 | 7 | For an ordinal $\alpha$, let $L\_\alpha$ be the $\alpha^{th}$ set of Gödel's constructible hierarchy and let $\omega\_1^{\mathrm{CK}}$ be the first non-recursive ordinal or the first admissible ordinal after $\omega$.
Question : does there exists a $\Sigma\_2$ formula able to capture that we reached, in Gödel's cons... | https://mathoverflow.net/users/138089 | Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy | This couldn't be achieved even by $\Sigma\_3$ sentences. First note that $L\_{\omega\_1^{CK}}$ (as any other model of $\mathsf{KP}\omega+L=V$) satisfies the scheme of $\Sigma\_3$-reflection:
$$\varphi(\vec{p})\to \exists a \;(\mathsf{Trans}(a)\land \vec{p}\in a\land (\varphi(\vec{p}))^a),\text{ where $\varphi$ is $\Sig... | 6 | https://mathoverflow.net/users/36385 | 333941 | 142,778 |
https://mathoverflow.net/questions/333939 | 7 | I am trying to get the hang of the available software for computing automorphism groups of plane curves over finite fields. I am using this Magma code to test it out on $y^2 = x^3 - x$ over $\mathbb{F}\_3$, which we know is of order 12 by (pg. 410, Prop 1.2(c)) Silverman's *Arithmetic of Elliptic Curves*.
The follow... | https://mathoverflow.net/users/56462 | Why does MAGMA claim that the automorphism group of an elliptic curve is order 24 when it is order 12? | Magma is computing the automorphism group of the associated **projective curve** $E$ defined over the base field (according to the link you gave). You are thinking of the automorphism group of $E$ **as an elliptic curve** i.e. with $\infty$ fixed.
The group magma is computing will also include the translations which ar... | 16 | https://mathoverflow.net/users/141887 | 333944 | 142,779 |
https://mathoverflow.net/questions/326495 | 7 | A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form
$$ f(x\_1,\dots,x\_n)=\sum\_{q=0}^{2n}\Phi\_q\left(\sum\_{p=1}^n\phi\_{p,q}(x\_p)\right),$$
where $\Phi\_p$ and $\phi\_{p,q}$ are unary continuous functions.
I'm curious about analogous resul... | https://mathoverflow.net/users/83901 | Kolmogorov superposition on the Hilbert Cube | After finally getting around to learning the proof of the Kolmogorov–Arnold representation theorem (thanks to [this recorded talk](http://www.math.toronto.edu/~drorbn/Talks/Fields-0911/) by Bar-Natan) I now know that the functions can be chosen so that the rearranged sequence in my edit converges uniformly (in particul... | 2 | https://mathoverflow.net/users/83901 | 333950 | 142,780 |
https://mathoverflow.net/questions/332972 | 4 | It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (say, over $\mathbb C$), the category of coherent sheaves of $\mathcal O\_X$-modules is abelian.
Now, a rigid analytic sp... | https://mathoverflow.net/users/69190 | Do coherent sheaves on rigid analytic spaces form an abelian category? | The answer is yes.
An abelian category is a category $\mathcal C$ with the following properties:
1. $\mathcal C$ is additive.
2. $\mathcal C$ has kernels and cokernels.
3. Images and coimages coincide. That is, for every morphism $f$ in $\mathcal C$, the canonical map $coker(ker(f)) \to ker(coker(f))$ is an isomorp... | 5 | https://mathoverflow.net/users/2362 | 333954 | 142,781 |
https://mathoverflow.net/questions/333965 | 7 | Notation: $[m] := \{1, 2, \dots, m \}$.
How many functions are there $f: [a] \to [b]$? The answer is easily seen to be $b^a$.
How many $1$-to-$1$ functions are there $f: [a] \to [b]$? Again the answer is well known, and it is sometimes called the falling factorial:
$$b(b-1) \dots (b-a+1).$$
How many functions are... | https://mathoverflow.net/users/4558 | Estimating the number of functions which are at most $c$-to-$1$ for some constant $c \ge 2$ | Let us just consider $g(n)$, and the general problem admits a similar treatment.
Note that $g(n)$ equals
$$
\sum\_{\substack{a\_1, \ldots, a\_n \\ a\_i \le 8 \\ a\_1+\ldots +a\_n = 5n}} \frac{(5n)!}{a\_1! a\_2! \cdots a\_n!}.
$$
We may see this by thinking of the inverse image of $1$ (a set of size $a\_1$) etc.
... | 7 | https://mathoverflow.net/users/38624 | 333970 | 142,784 |
https://mathoverflow.net/questions/333770 | 7 | Suppose $A$ is an integral domain and a finite type $\mathbb{C}$-algebra. Let $X := \text{Spec}(A)$ and $K := \text{Frac}(A)$ be the fraction field. Suppose $h \in K$ is a rational function that extends to a global complex analytic function on $X(\mathbb{C}).$ Can we conclude that $h \in A$?
If $A$ is integrally clo... | https://mathoverflow.net/users/141755 | Rational functions on reduced complex varieties that extend to global holomorphic functions | Let $A$ be a noetherian integral domain, $K$ its field of fractions, and $f \in K$. Assume that for each maximal ideal $\frak m$ of $A$ the element $f \in K \subseteq K\otimes\_{A}\hat{A}\_{\frak m}$ is in $\hat{A}\_{\frak m} \subseteq K\otimes\_{A}\hat{A}\_{\frak m}$ (here $\hat{A}\_{\frak m}$ denotes the completion o... | 2 | https://mathoverflow.net/users/4790 | 333973 | 142,786 |
https://mathoverflow.net/questions/333915 | 4 | Let me define the following groups
$$G(k,l,n):=\langle a,b\mid a^k=1=b^l, (ab)^n=(ba)^n\rangle$$
Fixed $k$ and $l$ (WLOG we can assume those are prime), I would like to know, whether the groups are mutually unequal. More precisely, prove that $(ab)^m\neq (ba)^m$ for $m<n$.
For $k=l=2$ those are the dihedral groups,... | https://mathoverflow.net/users/141867 | Proving some finitely presented groups being unequal | Ok, I think I have a proof of this. First of all, note that $D(k,l,n)$ is a quotient of $G(k,l,n)$ with respect to $(ab)^n=1$ (indeed, we have $(ba)^n=b(ab)^nb^{-1}=1=(ab)^n$ in $D(k,l,n)$). So, it is enough to prove in $D(k,l,n)$ that $(ab)^m\neq (ba)^m$. Now, we know that $D(k,l,n)$ [acts](https://en.wikipedia.org/wi... | 2 | https://mathoverflow.net/users/141867 | 333977 | 142,787 |
https://mathoverflow.net/questions/333980 | 6 | Ehresmann's theorem says that a proper smooth submersion is a fiber bundle. The proofs I know rely on the existence of connections locally on the base, and this is furnished by partitions of unity.
[This question](https://mathoverflow.net/q/56737/69037) gives a counterexample in the holomorphic category which is prob... | https://mathoverflow.net/users/69037 | In the real analytic category, are the fibers of a proper submersion isomorphic? | There exists an analytic Riemann metric on a real-analytic manifold, which follows from embeddability of real analytic manifolds (see The Analytic Embedding of Abstract Real-Analytic Manifolds Charles B. Morrey, Jr.) - maybe can also be proved easier.
So, you can probably just take orthogonal connection.
| 6 | https://mathoverflow.net/users/33286 | 333981 | 142,788 |
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