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https://mathoverflow.net/questions/330352 | 3 | I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}\_p)^k \to 1$, where $G$ is a compact lie group and $\Bbb{Z}\_p$ is the cyclic group of order $p$?
The only possible homomorphism $(\Bbb{Z}\_p)^k \to Aut(\mathbb{S... | https://mathoverflow.net/users/127053 | How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$? | Let's only consider central extensions (so we only miss a few cases when $p=2$). I denote by $C\_p=\mathbf{Z}/p\mathbf{Z}$, to avoid confusion with $p$-adics.
So extensions are classified by $H^2(C\_p^k,S^1)$. The commutator map yields a canonical homomorphism $\phi$ from $H^2(C\_p^k,S^1)$ onto $\mathrm{Hom}(\Lambda^... | 9 | https://mathoverflow.net/users/14094 | 330409 | 141,541 |
https://mathoverflow.net/questions/330406 | 2 | Let $\xi=(\xi\_1,\ldots,\xi\_n)$ be a sequence of independent random variables.
Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi\_\nu$ by a constant $c$. The rest of the $\xi\_i$ remain unchanged.
**Question:** Is it true that the altered sequence remains independent, even if $\nu$ is random... | https://mathoverflow.net/users/101180 | Is independence preserved if a random entry in an independent sequence is replaced by a constant? | The answer is no. E.g., assume that $n=2$, $\xi\_i=X\_i$, and $X\_1,X\_2,\nu$ are any random variables (r.v's) each with values in the set $\{1,2\}$ and with $P(\nu=1)=p\in(0,1)$.
With these conditions in place, the dependence between $X\_1,X\_2,\nu$ may be arbitrary. For instance, we may suppose that $\nu$ is indepen... | 3 | https://mathoverflow.net/users/36721 | 330410 | 141,542 |
https://mathoverflow.net/questions/330391 | 2 | Suppose we have the manifold $\mathbb{R}^3$ equipped with a Riemanian metric $g$ (not necessarily the Euclidean metric. And the induced metric on the $B\_1$ (the ball with radius $1$) is $\gamma$.
Suppose we are given $Ric\_g$, $\gamma$, and $tr\_{\gamma} K$, where $K$ is the second fundamental form. Can we find $K$... | https://mathoverflow.net/users/138705 | Given $Ric_g$ of 3-dim Riemannian manifold, induced metric $\gamma$ and mean curvature $tr_{\gamma}K$ on a hypersurface, do we have $K$? | The answer is 'no' even if $g$ is the standard flat metric. Indeed, it is well known that there are isometric *minimal* surfaces (the catenoid and the helicoid) such that the isometry between them does not align their second fundamental forms. Thus, knowing $Ric\_g=0$ and $\mathrm{tr}\_\gamma K = 0$ and knowing $\gamma... | 6 | https://mathoverflow.net/users/13972 | 330417 | 141,547 |
https://mathoverflow.net/questions/330291 | 3 | Did someone develop ZFC by means of ZF plus axioms for a binary well-ordering constant, say $\blacktriangleleft$? Are there results that suggested accounts are conservative extensions of ZFC?
| https://mathoverflow.net/users/37385 | Choice with well-ordering constant? | This theory is indeed a conservative extension of ZFC.
This can be seen by a class forcing argument. My understanding of the history is that several mathematicians independently noticed this, among them Cohen, Felgner, and Solovay. But only Felgner published the argument.\*
Let me sketch the argument. Consider the... | 7 | https://mathoverflow.net/users/64676 | 330419 | 141,548 |
https://mathoverflow.net/questions/330411 | 2 | Let $S\_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S\_{g,n}^b$ deformation retracts to a bouquet of $m:=2g+n+b-1$ circles and $\pi\_1(S\_{g,n}^b)$ is free on $m$ generators.
Let $m\geq 2$. If $S\_{g,n}^b$ is endowe... | https://mathoverflow.net/users/135446 | $PSL_2(\mathbb{R})$ representations of free groups | It suffices to count the punctures and boundary components. For each puncture or boundary component, the loop around it gives a conjugacy class in the fundamental group, well-defined up to inversion. It suffices to characterize which conjugacy classes arise from this construction applied to both punctures and boundary ... | 3 | https://mathoverflow.net/users/18060 | 330421 | 141,549 |
https://mathoverflow.net/questions/330396 | 3 | Let $F\subset \mathbb{R}^n$ be a finite set and $\sigma$ be uniformly distributed over $\{-1,1\}^n$. The usual Rademacher average of $F$ (modulo normalizing factors) is
$$ R\_n(F)=\mathbb{E}\_\sigma \max\_{f\in F}\sum\_{i=1}^n \sigma\_if\_i.
$$
Now let us define two operations on $F$: $\mathrm{conv}(F)$ and $[F]\_\vee$... | https://mathoverflow.net/users/12518 | Rademacher, maxima, convex hulls | It looks that no. Take $n=4$ and $F$ containing four vectors: $f=(2,-2,1,-5); g=(-2,2,1,-5); h=(0,0,-5,1)$. We have $f\vee g=(2,2,1,-5)$, $f\vee h=(2,0,1,1)$, $g\vee h=(0,2,1,1)$. Thus $\max\_{w\in [F]\_\vee} (w\_3+w\_4-w\_1-w\_2)=0$. On the other hand $\frac{f+g}2\vee h=(0,0,1,1)$, therefore $\max\_{w\in [\mathrm{conv... | 3 | https://mathoverflow.net/users/4312 | 330439 | 141,554 |
https://mathoverflow.net/questions/330436 | 4 | Let $S^{n-1} = \{x \in \mathbb{R}^n : x\_1^2 + \cdots + x\_n^2 = 1\}$. Consider a regular spherical simplex, obtained e.g. by taking a hyperspherical cap, picking $n$ equally-spaced points $P = \{p\_1, \dots, p\_n\}$ on the boundary (which is isomorphic to $S^{n-2}$), then taking the boundaries of the simplex to be the... | https://mathoverflow.net/users/48204 | Tiling the surface of a hypersphere with regular simplices | The regular simplex construction is related to 3-3-...-3 Coxeter groups, and the orthant construction is related to 3-3-...-4 Coxeter groups. Along the lines, there are constructions related to the 3-5 and 3-3-5 Coxeter groups:
For the 3-5 Coxeter group, project a regular icosahedron to $S^2$ with the same centre.
... | 4 | https://mathoverflow.net/users/125498 | 330440 | 141,555 |
https://mathoverflow.net/questions/330418 | 1 | Let $P \in \mathbb R^{n \times n}$ be a symmetric positive definite matrix with eigenvalues denoted by $\lambda\_i$ and corresponding eigenvectors denoted by $v\_i$. For each $j \in \{1, 2, 3, 4\}$, let $\alpha\_j$ be a non-zero real number. Let $x: [0, \infty) \rightarrow \mathbb R^{2n}$ be a continuous, differentiabl... | https://mathoverflow.net/users/108277 | Solve a linear matrix ODE involving symmetric blocks | Diagonalize $P=O\Lambda O^T$ with an $n\times n$ orthogonal matrix $O$ (containing the eigenvectors $v\_i$) and a diagonal matrix $\Lambda={\rm diag}\,(\lambda\_1,\lambda\_2,\ldots\lambda\_n)$. Define $X={{O\; 0}\choose{0\; O}}x$ and $Z={{O\; 0}\choose{0\; O}}z$. Then the differential equation becomes
$$ \frac{d}{dt}X(... | 1 | https://mathoverflow.net/users/11260 | 330441 | 141,556 |
https://mathoverflow.net/questions/330425 | 3 | Kobayashi formulated the analog of the main conjecture in Iwasawa's theory for elliptic curves which are supersingular at a prime p. This makes use of the $\pm$ Selmer groups which are shown to be cotorsion with the use of the $\pm$ Coleman maps, and the definition of $\pm$ L-functions (with bounded coefficients) due t... | https://mathoverflow.net/users/nan | State of the art on the main conjecture for supersingular elliptic curves/modular forms |
>
> **Theorem** (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a\_p(E) = 0$, then Kobayashi's $\pm$ Iwasawa main conjectures are true for $E$ (for both choices of sign).
>
>
>
The proof is in [this paper](http://www.mcm.ac.cn/faculty/wx/20... | 3 | https://mathoverflow.net/users/2481 | 330442 | 141,557 |
https://mathoverflow.net/questions/330446 | 7 | Let $L\_1, \ldots, L\_m \in \mathbb{Z}[x\_1, \ldots, x\_n]$ be polynomials of the form $L\_i = l\_{i1} \cdot l\_{i2} \ldots \cdot l\_{ik}$, where every $l\_{ij}$ is an integer linear form.
Assume that magnitudes of all coefficients of all $l\_{ij}$ are bounded by some integer $H$. (So, every $l\_{ij}$ has form $A\_1... | https://mathoverflow.net/users/31356 | Coefficients of linear dependency | Yes. The coefficients of the $L\_j$ are $O(k! H^k)$ (the $k!$ is a bound for how many different products can contribute to the same term, a multinomial coefficient with $k$ on top). The $B\_i$ are given as the kernel of a matrix whose entries are the coefficients of $L\_j$. That kernel can be computed by Cramer's rule,... | 9 | https://mathoverflow.net/users/297 | 330452 | 141,560 |
https://mathoverflow.net/questions/330132 | 4 | Let $X$ be a topological space. A collection of closed subsets of $X$ is called a *family of supports* (in the sense of Cartan) if: (1) the union of any two elements of $\Phi$ is an element of $\Phi$, and (2) any closed subset of $X$ contained in an element of $\Phi$ is itself an element of $\Phi$.
Given a family of ... | https://mathoverflow.net/users/16046 | Difference between local cohomology and cohomology with support in a family | I think here's a simple example that might illustrate the difference. Let's take $X$ to be the real line and $Z$ to be the open interval $(0,1)$. Let $\Phi$ be the closed subsets of $X$. Then $\Phi\_Z$ consists of closed subsets of $\mathbb{R}$ contained in $(0,1)$. Since these are bounded, $\Phi\_Z$ winds up being the... | 2 | https://mathoverflow.net/users/6646 | 330479 | 141,567 |
https://mathoverflow.net/questions/330480 | 4 | For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities:
\begin{align\*}
[\partial,L] = 0, && [\overline{\partial},L] = 0, & & [\partial^\*,\Lambda] = 0, && [\overline{\partial}^\*, \Lambda] = 0, \\
[L,\parti... | https://mathoverflow.net/users/126606 | Lie super algebra presentation of the Kähler identities | Indeed there is. Apologies for tooting my own horn, but you can find it in [this paper](https://arxiv.org/abs/hep-th/9705161), cowritten with Chris Koehl and Bill Spence. Instead of repeating the explanation, I refer you to this [MathOverflow answer](https://mathoverflow.net/a/16215/394) from a decade ago.
**Added af... | 4 | https://mathoverflow.net/users/394 | 330487 | 141,572 |
https://mathoverflow.net/questions/330492 | 2 | Suppose a random positive definite matrix $A\in\mathbb{R}^{n\times n}$ has density function (with respect to the lebesgue measure on $\mathbb{R}^{n(n+1)/2}$) $f(A)=g(\lambda\_1(A),...,\lambda\_n(A))$ where $g$ is a function invariant under permutation of coordinates and $\lambda\_1(A),...,\lambda\_n(A)>0$ are eigenvalu... | https://mathoverflow.net/users/123075 | Density of random matrix only depends on its spectrum | $\newcommand{\tr}{\operatorname{\mathrm tr}}
\newcommand{\R}{\mathbb{R}}
$
The answer is (of course) no (because the Lebesgue measure on the set $\mathbb{R}^{n(n+1)/2}$ of all possible vectors corresponding to on-and-above-diagonal part of the symmetric matrix $A$ has little to do with with the spectrum of the matrix $... | 1 | https://mathoverflow.net/users/36721 | 330498 | 141,576 |
https://mathoverflow.net/questions/330456 | 1 | When are Carnot groups complete and negatively curved (in the sense of [$CAT(\kappa)$](https://en.wikipedia.org/wiki/CAT(k)_space) spaces)?
| https://mathoverflow.net/users/36886 | When are Carnot groups negatively curved and homeomorphic to Euclidean space | All Carnot groups are complete metric spaces, since they have all closed balls compact ("proper" metric space). In general, any metric space with a transitive isometry group, and having a compact subset with nonempty interior, is complete (easy exercise).
The result that every Carnot group of dimension $\ge 2$ is not... | 3 | https://mathoverflow.net/users/14094 | 330506 | 141,580 |
https://mathoverflow.net/questions/330513 | 3 | I am looking for a proof (or reference) of the fact that Namba Forcing preserves stationary subsets of $\omega\_1$. This fact is stated and used throughout the literature(whenever you google it you'll find examples), though no one is refering to a proof or giving one.
I will state a few relevant definitions regardin... | https://mathoverflow.net/users/133672 | Proof that Namba Forcing preserves stationary subsets of $\omega_1$ | See the appendix of the following paper:
[Topological spaces after forcing](https://www1.essex.ac.uk/maths/people/fremlin/n05622.pdf)
The ``**A2 Theorem**'' on page 86 is what you are looking for.
| 3 | https://mathoverflow.net/users/11115 | 330516 | 141,583 |
https://mathoverflow.net/questions/330512 | 2 | For an undirected graph, we know that nodes are adjacent to each other if there is a link that connects them. What about adjacency for directed graphs? Is it based on:
* outgoing links: node $n$ is adjacent to node $i$ if there is a link coming out from node $n$ to node $i$
* ingoing links: node $n$ is adjacent to n... | https://mathoverflow.net/users/140139 | Adjacency definition for a directed graph | It depends on the author.
Some authors use the outgoing link definition, e.g. [this one](https://www.cs.cmu.edu/~adamchik/21-127/lectures/graphs_1_print.pdf):
`In a directed graph vertex v is adjacent to u, if there is an edge leaving v and coming to u.`
Other authors use the ingoing link definition, e.g. [this on... | 2 | https://mathoverflow.net/users/125498 | 330520 | 141,584 |
https://mathoverflow.net/questions/328232 | 6 | Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits\_{i=1}^N X\_i$, whose irreducible components $X\_i$ are all smooth and of the same dimension $d$, and whose singular cohomology $H^\*(X)$ is generated by algebraic cycles (hence supported in even degrees). Note that va... | https://mathoverflow.net/users/114985 | Fixed points under a finite group action on projective variety | There is actually a simple answer. Pick X to be two $\mathbb{C}P^1$'s intersecting transversally, and a $\mathbb{Z}/2$ action that swaps two spheres. Then $\text{rk } H^\*(X)=3$ whereas $\text{rk } H^\*(X^G) = 1.$
| 1 | https://mathoverflow.net/users/114985 | 330525 | 141,585 |
https://mathoverflow.net/questions/330497 | 4 | Suppose $M$ is an object in the abelian category of mixed Tate motives over $\mathbb{Q}$, and it is an extension of $\mathbb{Q}(0)$ by $\mathbb{Q}(1)$
\begin{equation}
0 \rightarrow \mathbb{Q}(1) \rightarrow M \rightarrow \mathbb{Q}(0) \rightarrow 0.
\end{equation}
Suppose the Hodge realisation of $M$ is the one associ... | https://mathoverflow.net/users/87910 | $p$-adic realisation of Kummer motive and Frobenius matrix | The matrix entry $\*$ is $\log(u^{1-p})/p$ (where $\log$ is the $p$-adic logarithm). This can be obtained from the description of the inverse of Frobenius given in sections 2.9-2.10 of [1]. You can find more information about the crystalline realization of mixed Tate motives in section 4 of [2] (the Kummer motive shows... | 2 | https://mathoverflow.net/users/5263 | 330530 | 141,586 |
https://mathoverflow.net/questions/330502 | 8 | Let $A=k[x\_1,\cdots,x\_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$.
**Question 1.** Given an element $e\in A^n$, is there a method to tell whether $e$ can be admitted as an element of some $A$-basis for $A^n$?
Obviously, the components of $e$ need to generate $A$, but I ... | https://mathoverflow.net/users/127776 | Basis for free modules over an affine domain | 1) In the positive direction: If any entry in $e$ is an $(n-1)!$ power, then $e$ is an element of a basis. (More generally, it's enough to have $e=(z\_1^{m\_1},\ldots z\_n^{m\_n})$ with $(n-1)!$ dividing the product of the $m\_i$.) The case $n=3$ appears in a paper of Swan and Towber and is proved by explicitly writing... | 11 | https://mathoverflow.net/users/10503 | 330538 | 141,588 |
https://mathoverflow.net/questions/330408 | 4 | In the article *[Centralizers in R. Thompson group $V\_n$](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-7207&vol=7&iss=4&rank=2)*, the following question is asked:
>
> **Question:** Is the centraliser of an element of $V\_n$ always finitely presented?
>
>
>
I am wondering: is this question still... | https://mathoverflow.net/users/122026 | Centralisers in Thompson's group $V_n$ | I finally found the answer to my question. In *[Centralizers in R. Thompson group $V\_n$](https://www.ems-ph.org/journals/show_abstract.php?issn=1661-7207&vol=7&iss=4&rank=2)*, it is proved that the centraliser of an element of $V\_n$ decomposes as
$$ \left( \prod\limits\_{i=1}^s K\_{m\_i} \rtimes G\_{n,r\_i} \right) ... | 2 | https://mathoverflow.net/users/122026 | 330550 | 141,592 |
https://mathoverflow.net/questions/330549 | 4 | Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda\_1,...,\lambda\_n$ be given real numbers. What is the distribution of
$$X=\sum\_{i=1}^n\lambda\_iv\_i^2\;?$$
Does it happen to belong to any known family of distributions? I think this is a very flexible way to model the distribution with... | https://mathoverflow.net/users/123075 | Linear combination of coordinates of random unit vector | The distribution of $X$ is the distribution of the ratio
$$\frac{\sum\_{i=1}^n\lambda\_iZ\_i^2}{\sum\_{i=1}^nZ\_i^2}
$$
of two quadratic forms in iid standard normal random variables $Z\_1,\dots,Z\_n$ (because the distribution of $(v\_1,\dots,v\_n)$ is the same as that of $(Z\_1,\dots,Z\_n)\big/\sqrt{\sum\_{i=1}^nZ\_i... | 3 | https://mathoverflow.net/users/36721 | 330560 | 141,595 |
https://mathoverflow.net/questions/330526 | 6 | Let $\tau>0$, and let $T\in \mathcal{D}'(\mathbb{R})$ be a $\tau$-periodic distribution (that is,
$
\langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle
$
for all $\varphi \in \mathcal{D}(\mathbb{R})$). Then
$$
T=\sum\_{n\in \mathbb{Z}} c\_n e^{i 2\pi t/\tau},
$$
for some $c\_n\in \mathbb{C}$, and where ... | https://mathoverflow.net/users/140146 | Fourier coefficients of a periodic distribution? | Just a quick complement to what Paul said, in order to explain more concretely what "descends" means. Take $\tau=1$ for simplicity. Let $\rho$ be a function in $\mathscr{D}(\mathbb{R})$ ($\mathscr{S}(\mathbb{R}$) would work too) which gives a partition of unity
of the form
$$
\sum\_{n\in\mathbb{Z}}\rho(t+n)=1
$$
for al... | 3 | https://mathoverflow.net/users/7410 | 330562 | 141,596 |
https://mathoverflow.net/questions/330557 | 3 | Let $F$ be a finite field, and $T$ be a torus over $F$. Assume that $T\_1,T\_2$ are two $F$-subtori of $T$, such that $T\_1 \times T\_2 \to T,(t\_1,t\_2) \mapsto t\_1 t\_2$ is surjective with finite kernel $K$. I wonder whether $T\_1(F)$ and $T\_2(F)$ would generate $T(F)$.
Of course the case that $T$ is $F$-split is t... | https://mathoverflow.net/users/75756 | A rationality problem about $F$-points in tori | I think the F-points of the two subtori do not, in general, generate the group of F-points of the bigger torus. Take $F=\mathbb{F}\_3$ and $L=\mathbb{F}\_9=\mathbb{F}\_3[\sqrt{-1}]$. Then let $T$ denote the Weil restriction from $L$ to $K$ of the split rank $1$ torus. We can view $T$ as the group of invertible $2\times... | 5 | https://mathoverflow.net/users/19432 | 330564 | 141,597 |
https://mathoverflow.net/questions/330552 | 4 | Let $d$ be an integer. It is a well-known theorem, attributed to Hermite and Minkowski, which asserts that the number of number fields $K$, allowed to have *any* degree over $\mathbb{Q}$, having discriminant $\Delta\_K = d$ is finite.
Let $S(d)$ be the number of isomorphism classes of number fields of discriminant $... | https://mathoverflow.net/users/10898 | Uniform boundedness of the number of number fields having fixed discriminant | The answers to both questions is no. For instance for $n=3$, if $p\_1$,... $p\_k$
are primes congruent to $1$ modulo $3$ there exist $(3^k-1)/2$ cyclic cubic fields
of discriminant equal to $(p\_1...p\_k)^2$ by elementary class field theory.
| 8 | https://mathoverflow.net/users/81776 | 330569 | 141,600 |
https://mathoverflow.net/questions/330573 | 14 | How to show that $$\left\lfloor\zeta\left(1+\frac{1}{n}\right)\right\rfloor=n$$ for every positive integer $n$?
| https://mathoverflow.net/users/117091 | Floor of Riemann zeta function | It is known that (see Corollary 1.14 in Montgomery-Vaughan: Multiplicative number theory I)
$$\frac{1}{\sigma-1}<\zeta(\sigma)<\frac{\sigma}{\sigma-1},\qquad \sigma\in(0,1)\cup(1,\infty).$$
In particular, taking $\sigma=1+\frac{1}{n}$, we get
$$n<\zeta\left(1+\frac{1}{n}\right)<n+1.$$
This is slightly stronger than you... | 23 | https://mathoverflow.net/users/11919 | 330574 | 141,601 |
https://mathoverflow.net/questions/330593 | 2 | Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $\mathbb{Z}[G]$ be a Prüfer domain?
In my case $G=GL\_n(A)$ for $A=k[x\_1,\cdots, x\_n]/I$ for some field $k$ and pri... | https://mathoverflow.net/users/127776 | Flatness of submodules of free modules | The augmentation ideal of $\mathbb Z[G]$ is flat if and only if $H\_i(G, \mathbb Z) = 0$ for all $i > 1$. I think this should rule out most cases of $GL\_n(A)$, since these groups generally have interesting and nontrivial homology.
| 6 | https://mathoverflow.net/users/52918 | 330595 | 141,610 |
https://mathoverflow.net/questions/330587 | 6 | I want to show that
$9\*\left[\frac{xy}{x+y}-q(1-q)\right]-12\*[xy-q(1-q)]+(1-q-x)^{3}+(x+y)^{3}+(q-y)^{3}-1\geq0$ where
$0<q<1$
$0<x<1-q$
$0<y<q$
$(x+y)\left[1+max\{\frac{1-q}{y},\frac{q}{x}\}\right]\leq3$
I play with it numerically. It is right. But don't know how to prove it analytically. Anybody can hel... | https://mathoverflow.net/users/140169 | Want to prove an inequality | [Iosif's answer](https://mathoverflow.net/a/330591) is very interesting and if anyone knows how to do the same thing in Maple I'd like to know.
However I disagree with Iosif about the difficulty. Mathematica will use a systematic procedure that is guaranteed to work in a wide variety of cases, and that may take many ... | 10 | https://mathoverflow.net/users/9025 | 330600 | 141,611 |
https://mathoverflow.net/questions/330607 | 3 | 1. First, a combinatorial question fit for an undergrad course. Say I have a collection $\mathcal{C}$ of non-empty subsets of $S=\{1,\dotsc,n\}$ such that every element of $\mathcal{C}$ has at most $k$ elements and every element of $S$ is contained in no fewer than $1$ and no more than $k$ elements of $\mathcal{C}$. Th... | https://mathoverflow.net/users/398 | Maximal disjoint collections and matrix rank | In 2) you may get $n/k$ as follows:
consider the random linear ordering $\Pi$ on the set of columns. In each row $\alpha$, mark the non-zero element which is the $\Pi$-maximal, if it belongs to the column $s$, say that $s$ is the leader of the row $\alpha$: $s=L(\alpha)$. For $s=1,\ldots,n$ denote $\xi(s)=\mathbb{1}... | 4 | https://mathoverflow.net/users/4312 | 330612 | 141,615 |
https://mathoverflow.net/questions/330606 | 3 | Let $X$ be a compact Riemann surface, and let $P \rightarrow X$ be a holomorphic $\mathbb P^1$-bundle over $X$. Then we know that $P$ is of form $\mathbb P(E)$ for some vector bundle $E \rightarrow X$ of rank $2$. At [the end of section 5](http://www.numdam.org/article/SB_1958-1960__5__193_0.pdf), Grothendieck proved t... | https://mathoverflow.net/users/40042 | Automorphism of ruled surfaces associated to stable vector bundles | Not necessarily, but the counter-examples are quite particular. If $E\cong E\otimes T$, taking determinants give $T^{\otimes r}\cong \mathcal{O}\_X$, with $r=\operatorname{rk}(E) $. Let $\pi :\tilde{X}\rightarrow X $ be the degree $r$ cyclic étale covering associated to $T$. The isomorphism $E\rightarrow E\otimes T$ de... | 5 | https://mathoverflow.net/users/40297 | 330613 | 141,616 |
https://mathoverflow.net/questions/330567 | 2 | Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\left[\exp\left(\int\_0^tV(X\_s)~ds\right)f(X\_t)\bigg|X\_0=x\right]
\tag{1}$$
for $(t,x)\in[0,\infty)\times\mathbb Z^d$... | https://mathoverflow.net/users/50406 | Feynman-Kac formula for lattice heat equation with non-diagonal potential | A Feynman-Kac formula for (3) is given by (1) with $V$ replaced with $$\left[
\begin{array}{ccccc}
&\ddots&&\\
&&V(-2)+U(-1)+V(-1)&&\\
&&&V(-1)+U(0)+V(0)&&\\
&&&&V(0)+U(1) + V(1)&&\\
&&&&&\ddots&&
\end{array}\right].$$ and the stochastic process $X\_t$ being the one generated by the following infinitesimal generator $$... | 1 | https://mathoverflow.net/users/64449 | 330623 | 141,618 |
https://mathoverflow.net/questions/329879 | 8 | In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know of any reference for this result? Does it follow easily from a known result? Here is the result:
Let $N$ be a finite... | https://mathoverflow.net/users/47820 | Global to local principle for f.g. $\mathbb{Z}[x]$ modules | Your lemma easily follows from the [Smith Normal Form Theorem](https://en.wikipedia.org/wiki/Smith_normal_form), a result you already referred to. The **short heuristic argument** that you gave can indeed be turned into **a short proof**.
Nothing in the sequel should be new to you. But it is shorter and it also makes... | 7 | https://mathoverflow.net/users/84349 | 330627 | 141,619 |
https://mathoverflow.net/questions/330354 | 1 | This is a crosspost of [this MSE question](https://math.stackexchange.com/q/2994508/223002).
---
Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration.
* The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$
furnishes for each path $b\to b^\prime$ in the base a continuous map
$\alpha ^{-1}(b)\... | https://mathoverflow.net/users/69037 | Relation between transport functor of a fibration and a Hurewicz connection on it | Let $p: E\to B$ be a map. Define $\Lambda(p) = E \times\_B B^I$; this is the space of pairs $(x,\gamma)$ consisting of a point $x\in E$ and a path $\gamma: [0,1] \to B$ such that $\gamma(0) = p(x)$. There is an evident restriction map
$$
\rho: E^I \to \Lambda(p)
$$
where $E^I$ is the free path space of $E$.
The map... | 2 | https://mathoverflow.net/users/8032 | 330628 | 141,620 |
https://mathoverflow.net/questions/330277 | 9 | Finitely generated nilpotent groups are always finitely presented. This is true for abelian groups, and can be shown by induction for nilpotent ones, using the classical lift of a presentation of $N$ and a presentation of $G/N$ to a presentation of $G$.
As a consequence, the free $c$-nilpotent group of rank $n$, $F\_n... | https://mathoverflow.net/users/130856 | What is the simplest known finite presentation of a free nilpotent group? | Taking "simplest" to mean "having fewest generators", the question is how many generators you need to normally generate $\Gamma\_n$ in a free group. As mentioned already in the comments, $\Gamma\_n / \Gamma\_{n+1}$ is free abelian with rank given by Witt's formula, so certainly you need at least this many generators. I... | 2 | https://mathoverflow.net/users/20598 | 330631 | 141,622 |
https://mathoverflow.net/questions/330640 | 1 | Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v\_{i+1}\le v\_i$. Let $P(\cdot)$ be a discrete probability distribution over $V$.
We are given the expected number $\mu$ of positive (i.e. non-null) vector components of a vector selected fr... | https://mathoverflow.net/users/115803 | Random optimization problem | Your inequality holds with $\alpha=1/2$. Indeed, let $U=(U\_1,\dots,U\_n)$ be a random vector in $V$ with $N$ non-null coordinates, so that $U\_1\ge\dots\ge U\_N>0=U\_{N+1}=\dots=U\_n$; here $N$ is also random. Let $\nu:=\lceil \mu \rceil$. Let
$$M:=\max\_{{\bf u}\in V} \sum\_{i=1}^\nu u\_i.
$$
Then
$$M\ge \sum\_{i=... | 2 | https://mathoverflow.net/users/36721 | 330644 | 141,624 |
https://mathoverflow.net/questions/330638 | 8 | This previous question traces the notion of group homomorphism to Jordan (1870) and the term "homomorphic" to Fricke and Klein (1897) and to earlier lectures of Klein:
[Whence “homomorphism” and “homomorphic”?](https://mathoverflow.net/questions/280261/whence-homomorphism-and-homomorphic)
What about the concept of ... | https://mathoverflow.net/users/33757 | History of the kernel of a homomorphism? | The *word* at least, seems to originate with **Pontryagin** ([1931](//zbmath.org/?q=an:57.0717.01), [p. 186](//hsm.stackexchange.com/a/5867)):
>
> 28) Wenn eine Gruppe $A$ auf eine Gruppe $B$ homomorph abgebildet ist, so heißt die Untergruppe von $A$, die aus allen Elementen besteht, welche auf das Einheits- (Null-... | 12 | https://mathoverflow.net/users/19276 | 330646 | 141,625 |
https://mathoverflow.net/questions/330632 | 15 | This is a [cross post](https://math.stackexchange.com/questions/3192540/is-a-hnn-extension-of-a-virtually-torsion-free-group-virtually-torsion-free) from Math.StackExchange after 2 weeks without an answer and a bounty being placed on the question.
Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually tor... | https://mathoverflow.net/users/121307 | Is an HNN extension of a virtually torsion-free group virtually torsion-free? | Yes, here's an example with an HNN over finite index subgroups as requested. It's based on constructing an amalgam of two f.g. virtually free groups, that has no proper finite index subgroups, using Burger-Mozes groups.
Fact (proved below): *for every $n\ge 3$ there exists a non-torsion-free, virtually free group $G$... | 11 | https://mathoverflow.net/users/14094 | 330658 | 141,628 |
https://mathoverflow.net/questions/330650 | 9 | Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}\_2(G/H,\mathbb{Z}) \cong \frac{\overline{H} \cap [\overline{G},\overline{G}]}{[\overline{H},\overline{G}]}$$
Here $\overline{G}$ is a [Schur covering grou... | https://mathoverflow.net/users/140206 | Reference for Schur multiplier identity | Let $G = F/R$ with $F$ free and $H = S/R$.
Since everything is happening modulo $[F,R]$, I am just going to work modulo $[F,R]$.
Then, by the Hopf formula, $M(G)$ (the Schur Multiplier) is isomorphic to $[F,F] \cap R$, which has free abelian complements $C$ in $R$ with $R = ([F,F] \cap R) \times C$, and $\bar{G} = ... | 6 | https://mathoverflow.net/users/35840 | 330661 | 141,629 |
https://mathoverflow.net/questions/330668 | 4 | Let $G\_r(n)$ be the real Grassmannian which is the collection of all $r$ dimensional subspace in $\mathbb{R}^n$ equipped with the usual invariant metric $g$.
Let $U\_A\in\mathbb{R}^{n\times r}$ and $U\_B\in\mathbb{R}^{n\times r}$ be the orthonormal basis of $A$ and $B$. Let $1\geq\sigma\_1\geq...\geq\sigma\_r\geq0$... | https://mathoverflow.net/users/123075 | Comparing two Riemannian metrics on Grassmannian | If we multiply $U\_A$ and $U\_B$ on the left by the same element of $U(n)$, this will preserve $U\_A^T U\_B$, thus preserve $\sigma\_1,\dots, \sigma\_r$ and preserve the metric distance between $A$ and $B$. Because the distance is $U(n)$-invariant, if it it arrises from a Riemannian metric, the Riemannian metric must b... | 4 | https://mathoverflow.net/users/18060 | 330670 | 141,632 |
https://mathoverflow.net/questions/330671 | 1 | Let us consider the group of continuous functions $C(S^1, S^1)$ from the circle to itself with the compact open topology. Does it have a chance to contain a dense cyclic subgroup?
Unfortunately it is not compact so we cannot apply the standard argument about compact, connected groups.
| https://mathoverflow.net/users/140215 | When are function groups monothetic? | No. Actually, for every, say metrizable space $X$ that is not totally disconnected, $C(X,S^1)$ is not even monothetic for the pointwise convergence topology (that induced by inclusion in $(S^1)^X$ with the Tychonov topology). Indeed, let $C$ be a connected component of $X$ not reduced to a point.
If $f$ is constant o... | 3 | https://mathoverflow.net/users/14094 | 330674 | 141,634 |
https://mathoverflow.net/questions/330435 | 6 | Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)$, we have:
* $\text{Z}(SU(2)) =\mathbb Z\_2$,
* $\text{Inn}(SU(2)) = SO(3)$,
* $\text{Out}(SU(2)) = 0$,
And so $\... | https://mathoverflow.net/users/106497 | Automorphism group of the special unitary group $SU(N)$ | Let $G$ be a compact simple simply connected Lie group. Then any automorphism of $G$ determines an automorphism of its Lie algebra $\mathfrak{g}$ and visa versa. So $\mathrm{Aut}(G)$ is naturally isomorphic to the linear group $\mathrm{Aut}(\mathfrak{g})$.
The sequence $1\to \mathrm{Inn}(\mathfrak{g})\to \mathrm{Aut}... | 13 | https://mathoverflow.net/users/12218 | 330676 | 141,636 |
https://mathoverflow.net/questions/330680 | 3 | Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V\_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V\_{\lambda}^w=U(\mathfrak{b}).v\_{w \lambda}$. There are many nice descriptions of these modules (sections of line bundles res... | https://mathoverflow.net/users/119460 | Convex Hulls of Demazure Modules | If I understand the question correctly, there is a nice description of the faces. They show up as so called reduced Kogan faces.
However, taking the convex hull is not the right operation,
as Demazure characters (or key polynomials) are given as the integer point transform of a *union* of reduced Kogan faces. In parti... | 3 | https://mathoverflow.net/users/1056 | 330687 | 141,637 |
https://mathoverflow.net/questions/330672 | 9 | For $g \in \operatorname{SL}\_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by
$$f[g](z) = f(g.z) j(g,z)^{-k}$$
where $j(g,z) = (cz+d)^{-1}$, $g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}$.
In... | https://mathoverflow.net/users/38145 | The correct determinant exponent of the weight $k$-operator for defining Hecke operators/adelizing modular forms | This is a question which has no "right" answer.
A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'\_v \Pi\_v$ of $GL\_2(\mathbb{A}\_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\de... | 15 | https://mathoverflow.net/users/2481 | 330691 | 141,639 |
https://mathoverflow.net/questions/330543 | 5 | Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the **polynomial Hopf algebra** Pol$(\mathcal{G})$.
For example, consider the famous $C(SU\_q(2))$, the $q$-deformation of $SU(2)$, with generators $\alpha$ an... | https://mathoverflow.net/users/128876 | Zero divisors in compact quantum groups | Concerning the first question, consider the quantum permutation group $S\_N^+$. Its polynomial algebra is generated by an $N\times N$ matrix $u=(u\_{ij})$ of self-adjoint projections which sum up to $1$ on each row and column. As a consequence, it has lots of zero divisors.
As for a general criterion, note that for a... | 5 | https://mathoverflow.net/users/131654 | 330692 | 141,640 |
https://mathoverflow.net/questions/328666 | 5 | The title is a reference to [this article by Martin Escardo](http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/), referring to work by originally by Ulrich Berger. It occurred that the programs described in this article can interpreted in the Turing machine model for higher order functions, whe... | https://mathoverflow.net/users/41947 | Do "seemingly impossible functional programs" work with arrow types interpreted as Turing machines? | $\newcommand{\E}[1]{\mathtt{E}\_{#1}}$
You are indeed using the model of computability that corresponds to the effective topos. It is also called *hereditarily effective operators* (HEO), *Type I* computability, and *Russian constructivism*. These things got invented several times.
Let us first recall a couple of d... | 9 | https://mathoverflow.net/users/1176 | 330699 | 141,643 |
https://mathoverflow.net/questions/330681 | 2 | Let $S\_1, \dots, S\_k \subset \mathbb{R}^n$ be a set of (non-regular) simplices. Let $m\_i$ indicate the number of vertices of simplex $S\_i$ (we do not assume it is equal to $n-1$).
Is there a simple upper bound on the maximum number of vertices of the intersection $\bigcap\_i S\_i$, stated in terms of the set $\{ ... | https://mathoverflow.net/users/76565 | Upper bound on number of vertices in intersection (and union) of simplices | In the literature, the dimension is usually $d$ (rather than your $n$),
and the number objects is $n$ (rather than your $k$).
The intersection of $n$ halfspaces in dimension $d$ can have
$n^{\lfloor d/2 \rfloor}$ vertices.
This is achieved by the dual of cyclic polytopes.
See the MO question
[How many vertices can ... | 1 | https://mathoverflow.net/users/6094 | 330705 | 141,644 |
https://mathoverflow.net/questions/330229 | 5 | Let $(A,H,D)$ be a spectral triple and let $B$ be an algebra which is Morita equivalent to $A$. Then there exists a finitely generated, projective $A$ module $E$ such that $B=End\_A(E)$. Endow $E$ with $A$-valued inner product $(\cdot,\cdot)$ and form a Hilbert space $H':=E \otimes\_A H$ with the inner product given by... | https://mathoverflow.net/users/24078 | Dirac operator on a Morita equivalent algebra | Your question is entirely covered by Section 2 of [Brain–Mesland–Van Suijlekom](https://arxiv.org/abs/1306.1951), but the fgp case is simple enough to ultimately boil down to folklore proved by [Chakraborty–Mathai](https://arxiv.org/abs/0804.3232). Let me summarise what happens, while incorporating some technical simpl... | 5 | https://mathoverflow.net/users/6999 | 330717 | 141,646 |
https://mathoverflow.net/questions/330641 | 4 | Let $P\_1$,...,$P\_m$ be polynomials in $n$ variables with coefficients in $\mathbb{Z}$ and consider the set
$$X(\mathbb{Z})=\{(x\_1,...,x\_n)\in \mathbb{Z}^n \ |\ P\_i(x\_1,...,x\_n)=0 ~ ,\ \forall i \in\{1,...,m\}\}.$$
In algebraic geometry, we are often interested in the number of elements in
$$X(\mathbb{F}\_{p})... | https://mathoverflow.net/users/66686 | $X(\mathbb{Z}/p\mathbb{Z})$ versus $\{X(\mathbb{Z})\pmod{p}\}$ | I found the MSRI survey *[Strong approximation for algebraic groups](http://library.msri.org/books/Book61/files/70rapi.pdf)* by
Andrei Rapinchuk very well-written and informative (and it has a lengthy bibliography for further exploration).
To answer your question: the cubic hypersurface $X\subset \mathbb{A}^3$ define... | 3 | https://mathoverflow.net/users/12218 | 330719 | 141,647 |
https://mathoverflow.net/questions/330720 | 4 | Let $\tilde{Gr}(k,n)$ be the affine cone of the Grassmannian $Gr(k,n)$. I think that the following set $S$ is an additive basis of $\mathbb{C}[\tilde{Gr}(k,n)]$:
\begin{align}
S = \{e\_T: T \text{ is a rectangular semi-standard Young tableau with $k$ rows}\},
\end{align}
where $e\_T = P\_{T\_1} \cdots P\_{T\_n}$, wher... | https://mathoverflow.net/users/11877 | Reference request: additive basis of coordinate ring of Grassmannians | The result you mention is very classical, but it also fits within the more general and conceptual framework of Standard Monomial Theory: <https://en.wikipedia.org/wiki/Standard_monomial_theory>.
| 5 | https://mathoverflow.net/users/25028 | 330721 | 141,648 |
https://mathoverflow.net/questions/330467 | 5 | [Morrey and Campanato space](https://en.wikipedia.org/wiki/Morrey%E2%80%93Campanato_space) is some subspace of $L^p$. We know that for a bounded domain $\Omega$, $L^p$ space characterize how the function blow up at some point. I want to know what Morrey and Campanato space characterize?
---
For bounded domain $\O... | https://mathoverflow.net/users/46341 | What Morrey and Campanato space characterize | The [lecture notes](https://www.math.uzh.ch/index.php?id=ve_mfs_sem_vor0&key1=0&key2=371&key3=1053&L=1) by Melanie Rupflin answer the question *"What is a Morrey Space? What is a Campanato Space?"*
The **Morrey space** $L^{p,\lambda}$ is a subset of $L^p$ containing functions $f$ on a domain $\Omega\in\mathbb{R}^m$ s... | 4 | https://mathoverflow.net/users/11260 | 330732 | 141,650 |
https://mathoverflow.net/questions/328576 | 4 | Has any work been done about *numerical methods* for the continuity equation
$$
\partial\_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $b \in L^1\_tW^{1,p}\_x$?
---
A related equation has been asked on [Mathematica StackExchange](https://mathemat... | https://mathoverflow.net/users/122620 | Numerics for continuity equation with Sobolev vector field | While the standard numerical methods (at least some of them) work even in the Sobolev regularity setting, the analysis of convergence is far from being trivial (due to non-smoothness of $b$). It is addressed for instance in the papers
1. A. Schlichting, C. Seis: *[Convergence rates for upwind schemes with rough coef... | 0 | https://mathoverflow.net/users/44463 | 330739 | 141,652 |
https://mathoverflow.net/questions/330108 | 2 | Is there a singular holomorphic foliation $F$ of $\mathbb{C}P^2$ which does not admit a global transverse holomorphic curve? More precisely there is no an immersed holomorphic submanifold of $\mathbb{C}P^2$ which intersect all regular leaves, transversely? If there exist such an example $F$, does this foliation admit a... | https://mathoverflow.net/users/36688 | A Singular Foliation of $\mathbb{C}P^2$ which does not admit a global transverse submanifold | The answer to this question is negative. If we require the transversal surface to be an immersed holomorphic curve then the only foliation for which such a surface exists is the pencil of lines. This is proven the Claim below. I will consider more generally the second part of your question where we require the transver... | 2 | https://mathoverflow.net/users/943 | 330751 | 141,656 |
https://mathoverflow.net/questions/330750 | 2 | A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak equivalences are homotopy equivalences and cofibrations are defined in terms of a lifting property for Hurewicz fibratio... | https://mathoverflow.net/users/7108 | Is the Hurewicz model category left proper? | The answer is yes. I say this because I recall that the way you form homotopy pushout of a `prepushout' diagram $C\gets A\to B$ in the Hurewicz structure is
1. map a diagram $\bar C \gets \bar A \to \bar B$ in which both arrows are (Hurewicz) cofibrations into the given one by a pointwise homotopy equivalence
2. for... | 3 | https://mathoverflow.net/users/3634 | 330753 | 141,657 |
https://mathoverflow.net/questions/330686 | 5 | We consider the standard symplectic structure $\omega=\sum dx\_i\wedge dy\_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle $E$ over $M$ as follows:
$$E=\{(x,v)\in M\times \mathbb{R}^{2n}\mid \omega(v,N\_x)=0,\;\;N\_x\perp T\_x M\}$$
>
> Does th... | https://mathoverflow.net/users/36688 | A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold | We have to choose a compatible Riemannian metric $g$ to define the normal bundle $N$. Then the almost complex structure $J$ defined by $\omega$ and $g$ induces an isomorphism between $E$ and $TM$.
As for your second question, assume that $M=f^{-1}(0)$ for some function $f$ that has $0$ as a regular value. Then $E\cap... | 1 | https://mathoverflow.net/users/70808 | 330780 | 141,665 |
https://mathoverflow.net/questions/330738 | 15 | It is well known that topology and algebraic geometry assign different meanings to the word "proper".
Let us recall the relevant definitions from topology (and we work in the context of topological spaces):
* A map $f:X\to Y$ is called **separated** iff the diagonal $X\hookrightarrow X\times\_YX$ is a closed inclusio... | https://mathoverflow.net/users/35353 | Which definition of "proper" is better? | I think the separated condition should be included, at least for purposes of sheaf cohomology. First let's consider the case where Y is a point. if X is a compact Hausdorff space, then sheaf cohomology over X commutes with filtered colimits. This is a nice property which fails if X is non-Hausdorff. For example, take X... | 20 | https://mathoverflow.net/users/3931 | 330788 | 141,667 |
https://mathoverflow.net/questions/330787 | 1 | Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman ([Invariant functions on Lie groups and Hamiltonian flows of surface group representations](http://terpconnect.umd.edu/~wmg/InvariantFunctions.pdf)) proved that, for a fairly general class of groups $G$, $M=Hom(\pi\_1(S),G)/G$ admits a symplectic str... | https://mathoverflow.net/users/37286 | Lie bracket on the complex valued functions of the space of representations of a Riemann surface | In the complex case, Goldman's symplectic form is a $(2,0)$-form, and the Poisson bracket is in terms of that form (the bracket is determined by a bivector and the bivector corresponds to the form).
The Goldman bracket formula works the same way as it does in the real case.
You can find a proof about this latter f... | 3 | https://mathoverflow.net/users/12218 | 330791 | 141,668 |
https://mathoverflow.net/questions/330767 | 0 | I have been trapped in solving the following ODE for a long time. I wonder if it has unique analytical solution
\begin{equation}
[b+c\_B(\bar{\beta}^H-\bar{\beta}^L)]\frac{dF(x)}{dx}+c\_BF(x)-c\_BF(x+\bar{\beta}^H-\bar{\beta}^L-b/c\_B)-c\_B=0.
\end{equation}
I could try to assume $F(x)$ is linear. But I was wondering... | https://mathoverflow.net/users/140317 | Solving a specific differential equation | To continue Liviu Nicolaescu's simplification: put $F(x):=f(x/B)$ so the equation writes
$$f'(x)+f(x)-f(x+1)+1=0.$$
A particular solution of it is simply $f(x):=x^2$, so we are left with the homogeneous equation
$$u'(x)+u(x)-u(x+1)=0.$$
If we put $u(x)=v(x)e^{-x}$ this becomes the (well-known)
$$v'(x)=\lambda v(x+1)$$... | 6 | https://mathoverflow.net/users/6101 | 330805 | 141,669 |
https://mathoverflow.net/questions/330669 | 4 | I believe this should be some standard result in random matrices theory, but my initial search failed to find a definitive answer.
The question is given a random sparse matrix $M\in\mathbb{R}^{n\times m}$ (in general $m\geq n$ having as any of its entries either 0 with probability $1/2$ and a positive number with pr... | https://mathoverflow.net/users/49821 | Rank of a random sparse matrix with nonnegative reals | For $m$ large and $n < m$, such a matrix will have full rank with probability approximately $1 - n/2^m$. If $n = m$, then it will have full rank with probability approximately $1 - 2m/2^m$.
It turns out that your question is equivalent to the following:
>
> What is the expected size of the maximum matching in a r... | 1 | https://mathoverflow.net/users/113535 | 330820 | 141,674 |
https://mathoverflow.net/questions/327520 | 3 | It's well-known that in a totally transcendental ($\omega$-stable) theory, $p(x)\subseteq q(x)$ is a non-forking extension if and only if $\text{MR}(p) = \text{MR}(q)$. In my answer to [this Math Stackexchange question](https://math.stackexchange.com/questions/3179518/non-forking-extension-is-equivalent-to-sameness-of-... | https://mathoverflow.net/users/2126 | Morley rank and forking in arbitrary theories | I think the answer is yes: if $p\subseteq q$ is non-forking, then $MR(p)=MR(q)$. To see this, let $p$ be over $A$ and assume that $\phi(x;b)\in q$ has ordinal Morley rank. It follows that $\phi(x;y)\wedge tp\_y(b/A)$ is stable and then by compactness, there is $\psi(y)\in tp(b/A)$ such that $\phi(x;y)\wedge \psi(y)$ is... | 3 | https://mathoverflow.net/users/19534 | 330821 | 141,675 |
https://mathoverflow.net/questions/330809 | 2 | Given $a>b>0$, is there any upper bound of the following ratio of [hypergeometric function](https://en.wikipedia.org/wiki/Hypergeometric_distribution)?
$$\frac{\_2F\_1(a,1-b;a+1;x)}{\_2F\_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some powers of $x/y$？
| https://mathoverflow.net/users/123075 | Ratio of hypergeometric function | If $b$ is not an integer, this ratio is bounded by a constant.
Indeed, Remark 1.2.4 pp 34-35 of "From Gauss to Painlevé" by Iwasaki-Kimura-Shimomura-Yoshida can be applied to your case, in which $\gamma-\alpha-\beta=b$. It yields that the function $f={}\_2F\_1(\alpha,\beta;\gamma,\cdot)$ you consider is continuous on... | 3 | https://mathoverflow.net/users/36752 | 330823 | 141,676 |
https://mathoverflow.net/questions/327062 | 2 | Let $L$ be a boolean lattice, $A$ and $B$ sublattices of $L$ that are
themselves boolean lattices, and suppose that $I = A \cap B$ is
nonempty.
Is $I$ a boolean sublattice of $L$? Is it a homomorphic image or
retract of $L$? If so, is there an explicit characterization of its
upper and lower bounds?
$I$ is clearly ... | https://mathoverflow.net/users/137888 | Is the intersection of Boolean sublattices a Boolean sublattice? | Let $L$ be the lattice of all subsets of $\{1,2,...\}$. Let $X=\{2^k3^l:k,l\in\omega\}$ and $Y=\{2^k5^l:k,l\in\omega\}$. Let $A=\{a\subset X:$ $a$ is finite or $X\backslash a$ is finite$\}$, and $B=\{a\subset Y:$ $a$ is finite or $Y\backslash a$ is finite$\}$. Then $A$ and $B$ are Boolean sublattices of $L$, but $A\cap... | 2 | https://mathoverflow.net/users/90095 | 330825 | 141,678 |
https://mathoverflow.net/questions/330819 | 9 | Does a conservativity conjecture (e.g. Conjecture 2.1 of <http://user.math.uzh.ch/ayoub/PDF-Files/Article-for-Steven.pdf>) imply the standard conjectures? Specifically I am confused with Beilinson's article arXiV:1006.1116 "Remarks on Grothendieck’s standard conjectures", where it seems to show that the conservativity ... | https://mathoverflow.net/users/140298 | Does a conservativity conjecture imply the standard conjectures? | Here is a partial answer: In characteristic 0 it is known that conservativity + algebraicity *(edit: modulo rational equivalence)* of the Künneth projectors implies the remaining standard conjectures.
Indeed, if the Künneth projectors are algebraic, then one may use conservativity to show that the inverse of the Lefs... | 9 | https://mathoverflow.net/users/21815 | 330838 | 141,683 |
https://mathoverflow.net/questions/330785 | 4 | If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|\_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by
\begin{align}
\text{conv} (\mathcal{A}) = \{ M \in \text{Mat}\_{d \times d} (\mathbf{R}): M = M^T, \| M \|\_\* = 1\},
\end{align}
wher... | https://mathoverflow.net/users/121692 | Convex Hull of Outer Products of (Normalised) Nonnegative Vectors | Your characterization of $\text{conv} (\mathcal{A})$ needs one additional restriction---that $M$ is positive semidefinite (the equivalence of these two sets follows fairly quickly from the spectral decomposition).
For $\text{conv} (\mathcal{A}\_+)$, the convex hull is the exact same, but with the positive semidefinit... | 3 | https://mathoverflow.net/users/11236 | 330840 | 141,685 |
https://mathoverflow.net/questions/330810 | 7 | [This is a double of my question of math.stackexchange <https://math.stackexchange.com/questions/3214962/multiplication-in-deligne-cohomology-explicit-formula-for-p-q-1]>
In the very beginning of [1] the geometric meaning of Deligne cohomology $H^q(X, \mathbb{Z}(p))\_D$ and multiplicative structure on it is being dis... | https://mathoverflow.net/users/82309 | Multiplication in Deligne cohomology: explicit formula for $p=q=1$ | There is indeed a "universal" holomorphic bundle with connection on $\mathbb{C}^\times \times \mathbb{C}^\times$ which induces the bundles $r(f,g)$ defined in Esnault-Viehweg. This universal bundle has been constructed by D. Ramakrishnan using the Heisenberg group (Bulletin AMS vol. 5 n. 2, 1981, <https://doi.org/10.10... | 8 | https://mathoverflow.net/users/6506 | 330846 | 141,686 |
https://mathoverflow.net/questions/330848 | 8 | In any setting where we have a notion of space and a notion of cohomology theory, we in principle could ask "what are motives in this setting?". In some settings the question can be interesting (i.e. algebraic varieties and Weil cohomology theories), in some settings not so much (i.e. topological spaces and cohomology ... | https://mathoverflow.net/users/nan | Motives of complex-analytic spaces | See theorem 1.8 of (Joseph Ayoub. *Note sur les opérations de Grothendieck et la réalisation de Betti.*
J. Inst. Math. Jussieu.)
You can set up a theory of analytic motives, and the resulting category is equivalent to the derived category of vector spaces (over your field of coefficients, typically $\mathbb{Q}$).
I... | 8 | https://mathoverflow.net/users/21815 | 330852 | 141,687 |
https://mathoverflow.net/questions/330855 | 4 | This question is partially inspired by this question: *[Does every non-empty set admit a group structure (in ZF)?](https://mathoverflow.net/q/12973/12218)*
It was also inspired by my desire to explain the importance of quotient morphisms when discussing algebraic quotients. I wanted to point out to a student that the... | https://mathoverflow.net/users/12218 | Does every non-empty set admit an (affine) scheme structure (in ZFC)? | Take a [field of arbitrary infinite cardinality](https://math.stackexchange.com/q/1296889/29145); let $K$ be its algebraic closure (which has the same cardinality) and consider Spec(K[X]), which has the same cardinality plus one, i.e. the same cardinality.
| 12 | https://mathoverflow.net/users/10503 | 330856 | 141,689 |
https://mathoverflow.net/questions/195851 | 12 | There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane partitions of shape $n\times n$. See for instance the exposition given here: <http://www-users.math.umn.edu/~shopkins/do... | https://mathoverflow.net/users/25028 | Dynamics of RSK | The paper "Minuscule reverse plane partitions via quiver representations" by Garver, Patrias, and Thomas (<https://arxiv.org/abs/1812.08345>) does a good job explaining, from the perspective of quiver representations, how the piecewise-linear RSK map is related to another invertible piecewise-linear operation called "r... | 1 | https://mathoverflow.net/users/25028 | 330858 | 141,691 |
https://mathoverflow.net/questions/330865 | 3 | I am looking for a proof based on characteristic functions for the generalized central limit theorem when the second moment does not exist, in which case one ends up with a power law rather than a Gaussian state.
This Theorem is described on the wikipedia page of the CLT [click me](https://en.wikipedia.org/wiki/Cent... | https://mathoverflow.net/users/140314 | Heavy tail central limit theorem | Welcome to MO! This theorem is e.g. [Theorem 14 on page 91 of Petrov's book](https://books.google.com/books/about/Sums_of_Independent_Random_Variables.html?id=zSDqCAAAQBAJ&printsec=frontcover&source=kp_read_button#v=onepage&q&f=false), with further references to [24], [50].
| 2 | https://mathoverflow.net/users/36721 | 330867 | 141,693 |
https://mathoverflow.net/questions/330873 | 37 | I don't know anything about **type theory** and I would like to learn it.
I'm interested to know how we can found mathematics on it.
So, I would be interested by any text about type theory whose angle is similar to the one of Russel and Whitehead in the Principia, or similar to the one of Bourbaki (for instance).
... | https://mathoverflow.net/users/115872 | Good introductory book to type theory? | It seems that the HoTT book and Vladimir Voevodsky’s program for Univalent Foundations of Mathematics is made for you !
You will find everything from here:
<https://homotopytypetheory.org/>
| 8 | https://mathoverflow.net/users/110166 | 330875 | 141,695 |
https://mathoverflow.net/questions/330797 | 7 | Let $G$ be a Lie group and $\pi:E\_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E\_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{ad}(E\_G)\rightarrow \text{At}(E\_G)\rightarrow T M\rightarrow 0$$
where $\text{ad}(E\_G)$ is the adjoint vector... | https://mathoverflow.net/users/118688 | Atiyah Sequence and Connections on a Principal Bundle | First note that the adjoint bundle $ad(E\_G)$ can be canonically identified with the vertical tangent bundle $V E\_G / G$: send the pair $(p, \xi)$ consisting of a point $p \in E\_G$ and a Lie algebra element $\xi$ to the value $p \cdot \xi$ at $p$ of the fundamental vector field generated by $\xi$. Moreover, the Atiya... | 5 | https://mathoverflow.net/users/17047 | 330876 | 141,696 |
https://mathoverflow.net/questions/330859 | 3 | Let $A$ be an abelian torsion-free group. If $A$ is isomorphic with the group of rational numbers whose denominators are powers of, say, $2$, then its automorphism group is isomorphic with $\mathbb{Z}\_2\times\mathbb{Z}$.
Is there an abelian torsion-free group which is not locally cyclic with such automorphism group?... | https://mathoverflow.net/users/127914 | Abelian torsion-free group with $\mathbb{Z}_2\times\mathbb{Z}$ as automorphism group | Here's one, a subgroup of $\mathbf{Q}^2$.
Below $\mathbf{Z}\_p$ means the $p$-adic ring.
Let $R$ be the matrix $\begin{pmatrix}0 & 1\\ 1 & 6\end{pmatrix}$. Let $a,b$ be the two roots $X^2-6X-1$ in $\mathbf{Q}\_3$, with $a\equiv 1$ (mod $3$). We have $\mathbf{Q}\_3^2=V\_a\oplus V\_b$, where $V\_a=\mathrm{Ker}(R-a)$ ... | 4 | https://mathoverflow.net/users/14094 | 330877 | 141,697 |
https://mathoverflow.net/questions/328080 | 2 | Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}\_K$ and let $f(X)\in \mathcal{O}\_K[X]$. Denote the Mahler measure by $M$.
Is there any known result about the comparison of the values $M(\alpha)$ and $M(f(\alpha))$?
| https://mathoverflow.net/users/127070 | Mahler measures of values of polynomials | This is all standard stuff about height functions. More generally, if we use absolute heights, then for any $f(x)\in\overline{\mathbb{Q}}[x]$ of degree $n$ there are constants $C\_1(f)>0$ and $C\_2(f)>0$ so that for any $\alpha\in\overline{\mathbb{Q}}$,
$$ C\_1(f)H(\alpha)^n \le H\bigl(f(\alpha)\bigr) \le C\_2(f)H(\al... | 3 | https://mathoverflow.net/users/11926 | 330883 | 141,700 |
https://mathoverflow.net/questions/330653 | 2 | Let $X$ and $Y$ be i.i.d. random variables with exponential distribution with mean $1$, and let $Z=(X-1)(Y-X)$. Let $Z\_1,...,Z\_n$ are i.i.d. copies of $Z$, and let $f(n)=P[\sum\_{i=1}^n Z\_i > 0]$. My question is to estimate $f(n)$. I am interested in asymptotic upper and lower bounds for large $n$, and also in effic... | https://mathoverflow.net/users/44407 | Estimating probability that a large sum of i.i.d variables is positive | The exact asymptotics of $f(n)$ for large $n$ follows by [Theorem 2.1, more specifically formula (2.4)](https://projecteuclid.org/euclid.aop/1176994938).
However, to use that formula (2.4), you will have to compute lots of asymptotics regarding the distribution of the random variable $Z$, and also a few moments of i... | 1 | https://mathoverflow.net/users/36721 | 330907 | 141,710 |
https://mathoverflow.net/questions/330723 | 3 | I'm considering a ratio of incomplete [Selberg integral](https://en.wikipedia.org/wiki/Selberg_integral):
$$f\_n(a,b)=\frac{\int\_{\Delta\_a}\prod\_{i=1}^nx\_i^{\alpha-\frac{n+1}{2}}\prod\_{i=1}^n(1-x\_i)^{-1/2}\prod\_{i<j}|x\_i-x\_j|}{\int\_{\Delta\_b}\prod\_{i=1}^nx\_i^{\alpha-\frac{n+1}{2}}\prod\_{i=1}^n(1-x\_i)^{-1... | https://mathoverflow.net/users/123075 | Ratio of Selberg integral | Denote the numerator by $I(a)$. The change of variables $x\_i=ay\_i$ gives
$$
I(a)=a^{n(\alpha-\frac{n+1}2)+\frac{n(n+1)}2}\times \\
\times\int\_{[0,1]^n}\prod\_{i=1}^ny\_i^{\alpha-\frac{n+1}{2}}\prod\_{i=1}^n(1-ay\_i)^{-1/2}\prod\_{i<j}|y\_i-y\_j|.
$$
The latter integral may be estimated using the two-sided estimates... | 2 | https://mathoverflow.net/users/4312 | 330918 | 141,713 |
https://mathoverflow.net/questions/330711 | 11 | Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point set topology) that the fiber of that map is homotopy equivalent to $\Omega B \ast F$ hence we obtain a new fiber sequenc... | https://mathoverflow.net/users/22810 | The (fiber of the) cofiber of the fiber of a map of spaces | This sort of question was studied a lot by Ganea (see [here](https://link.springer.com/article/10.1007%2FBF02566956) and [here](https://link.springer.com/article/10.1007%2FBF02564393)). The first piece of bad news is that duality is not perfect: there is no formula for that cofiber that depends only on $\Sigma X$ and $... | 12 | https://mathoverflow.net/users/6936 | 330937 | 141,715 |
https://mathoverflow.net/questions/330914 | 2 | Given an integer partition $\lambda$ and its Young diagram $Y\_{\lambda}$, let $h\_{\lambda}(i,j)$ stand for the corresponding [hook length](https://en.wikipedia.org/wiki/Hook_length_formula) of the cell $(i,j)\in Y\_{\lambda}$. Write $\lambda\vdash n$ for $\lambda$ a partition of $n$.
Recall the [Gaussian binomials]... | https://mathoverflow.net/users/66131 | Reading off top hook-lengths in partitions | Let $j$ be the size of the $(1,1)$-hook, and $i+1$ the number of rows of this hook. To complete this hook to a partition of $n$, we must place to its southeast a partition of $n-j$ with at most $j-i-1$ columns and at most $i$ rows. The number of such partitions is $[q^{n-j}]\boldsymbol{{j-1\choose i}}$, and the proof f... | 6 | https://mathoverflow.net/users/2807 | 330938 | 141,716 |
https://mathoverflow.net/questions/330854 | 4 | Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^\*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for $\mathbb{C}\_\lambda$, the one-dimensional $\mathfrak{g}$-module with character $\lambda$.
I have calculated homolog... | https://mathoverflow.net/users/140292 | Homology of solvable (nilpotent) Lie algebras | Let me provide an elementary proof that for $\mathfrak{g}$ nilpotent finite-dimensional and $\lambda\neq 0$ we have $H\_\*(\mathfrak{g},V\_\lambda)=0$. (In general it seems to be particular case of results of Delorme in the 70s, and possibly known earlier.)
Recall that the homology of $V\_\lambda$ is the homology of ... | 4 | https://mathoverflow.net/users/14094 | 330949 | 141,720 |
https://mathoverflow.net/questions/330954 | -1 | How to find sum of the series Σ 1/ ((2n - 1)^2 (2n+1)^2) ?
| https://mathoverflow.net/users/140352 | How to find sum of series Σ 1/ ((2n - 1)^2 (2n+1)^2) | Mathematica gives the answer
$$\sum \_{n=1}^{\infty } \frac{1}{(2 n-1)^2 (2 n+1)^2}=\frac{\pi^2-8}{16}.$$
An elementary way to prove this is to write
$$\frac{4}{(2 n-1)^2 (2 n+1)^2}=
\frac{1}{2 n+1}-\frac{1}{2 n-1}+\frac{1}{(2 n-1)^2}+\frac{1}{(2 n+1)^2}
$$
and then
\begin{equation}
\sum \_{n=1}^{\infty }\Big(\fra... | 2 | https://mathoverflow.net/users/36721 | 330955 | 141,722 |
https://mathoverflow.net/questions/330929 | 6 | Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, which induces a positive root system $\Delta^+$, fundamental Weyl chamber, etc.
I wish a reference for the following r... | https://mathoverflow.net/users/20052 | Existence of a weight of a representation in the fundamental Weyl chamber | Here's maybe another (more conceptual?) way to think about it.
First of all, if $\mu\_1$ is a dominant weight which appears with nonzero multiplicty in $V^{\lambda\_1}$, and $\mu\_2$ is a dominant weight which appears with nonzero multiplicty in $V^{\lambda\_2}$, then $\mu\_1+\mu\_2$ is a dominant weight which appear... | 7 | https://mathoverflow.net/users/25028 | 330963 | 141,724 |
https://mathoverflow.net/questions/330916 | 12 | The computation of the unoriented bordism group of the point $N\_\*=\Omega\_\*^O$ is a classic result.
I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in H\_d(M,\mathbb{Z}^w)$ as part of the data. More precisely, I would like to consider the pairs
$$
(M, [M]\in H\_d(... | https://mathoverflow.net/users/5420 | Unoriented bordism with twisted orientation | I think this theory is the same as unoriented bordism, when one tries to make sense of it.
As it stands I don't think it makes sense, because I think that the expression "$\mathbb{Z}^w$" does not describe a local system on $M$, but rather an isomorphism class of local systems, and hence $H\_d(M ; \mathbb{Z}^w)$ is a... | 12 | https://mathoverflow.net/users/318 | 330970 | 141,726 |
https://mathoverflow.net/questions/330974 | 0 | Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s\_1\dots s\_n$ with $s\_i$ simple reflections, then $s\_i(\gamma)=\gamma$ for $i=1\dots n$?
I can't see a reason why this shouldn't... | https://mathoverflow.net/users/110074 | Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$ | This is definitely not true. For instance, already in $\Phi=B\_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group fixing it. But e.g. a simple root in $B\_2$ is not fixed by any simple reflection.
However, what you might want to know... | 4 | https://mathoverflow.net/users/25028 | 330975 | 141,727 |
https://mathoverflow.net/questions/330973 | 3 | I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f\_a(x)=\int\_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?
| https://mathoverflow.net/users/123075 | Non-asymptotic upper bound of right tail of Gamma function | In Gabcke thesis, that you can download at
<http://hdl.handle.net/11858/00-1735-0000-0022-6013-8>
page 84 you can find
Theorem: The incomplete gamma function
$$\Gamma(a,x)=\int\_x^\infty e^{-v}v^{a-1}\,dv,\qquad \text{$x>0$ and $a$ real numbers}$$
satisfies for $a\ge 1$ and $x>a$
$$\Gamma(a,x)\le a e^{-x} x^{a-... | 5 | https://mathoverflow.net/users/7402 | 330981 | 141,730 |
https://mathoverflow.net/questions/330977 | 36 | In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical:
$$ \sqrt{1+2\sqrt{1+3\sqrt{1+\ldots}}}.
\qquad(\*)
$$
Assuming the above is well-def... | https://mathoverflow.net/users/12518 | Examples where existence is harder than evaluation | Brownian motion is an example of this phenomenon in probability.
I am no expert on the history, but Einstein is often credited with having described, in 1905, the mathematical properties that Brownian motion ought to have: a continuous process with independent increments whose distribution at time $t$ is Gaussian wit... | 45 | https://mathoverflow.net/users/4832 | 330982 | 141,731 |
https://mathoverflow.net/questions/330948 | 7 | For $2\times 2$ matrices we have the following result.
>
> Any matrix in $\mathrm{SL}(2,\mathbb{Z})$ with nonnegative entries can be obtained from $\mathrm{Id}\_2$ by repeatedly adding one column to another.
>
>
>
*Proof:* It is enough to prove that if $$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\in\mathrm... | https://mathoverflow.net/users/117675 | Generators for the semigroup $\mathrm{SL}(n,\mathbb{N})$ | I believe that the answer is negative.
A positive answer would mean that for any $A\in\mathrm{SL}^\pm(n,\mathbb{N})$, $A\neq1$ there exists $i,j$ and some other element $B\in\mathrm{SL}^\pm(n,\mathbb{N})$ such that $A=L\_{ij}(1)\cdot B$ or $A=B\cdot L\_{ij}(1)$, possibly after some permutation of rows and/or columns ... | 3 | https://mathoverflow.net/users/5018 | 330983 | 141,732 |
https://mathoverflow.net/questions/330985 | 0 | I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness.
So suppose we are in Type $A\_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper triangular matrices in $G$ , $T=$ group of diagonal matrices in $G$. $W=N\_G(T)/T \cong \mathfrak S\_n$. Let $R=\{e\_i -e\_j ... | https://mathoverflow.net/users/127387 | In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$ | Unless I'm missing something obvious:
Consider the degree of the identity element $e$ in $\Gamma(w)$. On the one hand, by the assumption you've made it is $\ell(w)$. On the other hand, it is clearly equal to the number of reflections $s\_{\alpha}$ less than $w$, because the only things $e$ will be connected to are of... | 1 | https://mathoverflow.net/users/25028 | 330986 | 141,733 |
https://mathoverflow.net/questions/330987 | 3 | I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am currently approaching it is as follows:
**Step 1:** Let $i: N \hookrightarrow M$. Consider a almost complex structure $J^\p... | https://mathoverflow.net/users/92483 | Every symplectic submanifold is J-holomorphic | The normal bundle is a symplectic vector bundle (the fibres are symplectic vector spaces), and so, it has a compatible almost complex structure. Further, the normal bundle to $N$ can be realized as the symplectic orthogonal complement to $TN\subset TM|\_N$.
| 1 | https://mathoverflow.net/users/110236 | 330988 | 141,734 |
https://mathoverflow.net/questions/330999 | 2 | In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), <https://www.sciencedirect.com/science/article/pii/S0001870898917443/pdf?md5=43f7ddfffa6e4e285eec1f7183f736c8&pid=1-s2.0-S0001870898917443-main.pdf>,
in Theorem 1.1 about ration... | https://mathoverflow.net/users/127387 | On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not | Expanded with lots of references at @JimHumphreys's [request](https://mathoverflow.net/questions/330999/on-a-criterion-for-rational-smoothness-of-schubert-varieties-and-an-ambiguity-of/331004#comment825502_331004). Borel is [Borel - Linear algebraic groups (2nd edition)](https://link.springer.com/book/10.1007/978-1-461... | 6 | https://mathoverflow.net/users/2383 | 331004 | 141,741 |
https://mathoverflow.net/questions/331003 | 0 | Let $\Gamma$ be a congruence subgroup of $\operatorname {SL}\_2 (\mathbb Z)$. Let $N$ be the smallest positive integer such that $\begin{pmatrix} 1 & N \\0 &1 \end{pmatrix}\in \Gamma$.
Is necessarily $\Gamma(N)\subset \Gamma$?
| https://mathoverflow.net/users/122104 | Congruence subgroups and translations | No. $\Gamma$ can be $\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname {SL}\_2 (\mathbb Z):\text{$M\mid c$ and $N\mid b$}\right\}$, which contains $\Gamma(N)$ if and only if $M\mid N$.
| 5 | https://mathoverflow.net/users/11919 | 331005 | 141,742 |
https://mathoverflow.net/questions/306821 | 2 | Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$ is a $\mathbb{R}$-divisor. Further assume that $(X, D+B')$ is klt and $(X, D+D')$ is log canonical but not klt.... | https://mathoverflow.net/users/80473 | Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic | We claim the following holds.
**Proposition** (cf. [Tan17, Prop. 3 and Idea of Thm. 1; Wan, Proof of Thm. 3.5, Case 2])**.** *Let $X$ be a complete normal variety $X$ of dimension $d$ over an infinite perfect field of characteristic $p > 0$. Let $\Delta$ be an effective $\mathbf{Q}$-Weil divisor on $X$ such that the ... | 3 | https://mathoverflow.net/users/33088 | 331008 | 141,745 |
https://mathoverflow.net/questions/331024 | 3 | $\newcommand{Tr}{\operatorname{Tr}}$
Is there a continuous map $(p,t) \mapsto \lambda(p,t)$ which, given a path $p: [0,1] \to M(2,\mathbb R)$ and a $t \in \mathbb [0,1]$, gives back an eigenvalue of $p(t)$? Additionally, I'd like it that if $p(a) = -p(b)$ for some $a, b \in [0,1]$, then $\lambda(p,a) = -\lambda(p,b)$.
... | https://mathoverflow.net/users/75761 | Eigenvalue-taking operator? | I originally gave a wrong candidate for an answer, but in fact no such path can exist (at least, assuming that your topologies are such that $t \mapsto \lambda(p, t)$ is continuous for all paths $p$). Indeed, define $p : [0, 1] \to \operatorname M(2, \mathbb R)$ by
$$
p(t) = \begin{pmatrix} \cos(\pi t) & \sin(\pi t) \\... | 6 | https://mathoverflow.net/users/2383 | 331026 | 141,756 |
https://mathoverflow.net/questions/331018 | 1 | A topological space $X$ has [Menger's property](https://en.wikipedia.org/wiki/Menger_space) $\textsf{S}\_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}\_1, \mathcal{U}\_2, \cdots $, we can select finite sets $\mathcal{F}\_1\subseteq\mathcal{U}\_1, \mathcal{F}\_2\subseteq\math... | https://mathoverflow.net/users/138770 | Countable union of Menger spaces | Copying @Taras Banakh's answer from the comments. The answer is correct and complete in my view.
The answer is well-known and is "yes": just divide the sequence of covers into infinitely many parwise disjoint sequences of covers and cover $n$-th set $X\_n$ by the union of finite subfamilies from the $n$-th subsequenc... | 3 | https://mathoverflow.net/users/2415 | 331029 | 141,758 |
https://mathoverflow.net/questions/331030 | 27 | I was surprised to see that On the construction of balanced incomplete block designs by [Raj Chandra Bose](https://fr.wikipedia.org/wiki/Raj_Chandra_Bose) was published (in 1939) in a journal named Annals of Eugenics (see [here](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-1809.1939.tb02219.x)) (published bet... | https://mathoverflow.net/users/140385 | How come mathematicians published in Annals of Eugenics? | Eugenics and agriculture were two areas of application that motivated much of early 20th century statistics. The paper was a continuation of research on the use of Latin squares in experimental design for those areas. It says so explicitly on the second page:
>
> It was, however, only about [1925] that the importan... | 42 | https://mathoverflow.net/users/nan | 331031 | 141,759 |
https://mathoverflow.net/questions/327496 | 1 | It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in [1](https://www.math.umd.edu/~yanir/OT/AmbrosioGigliDec2011.pdf)) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the internal energy functional is $\lambda$-convex where $\lamb... | https://mathoverflow.net/users/42644 | Strong convexity of internal energy with respect to Wasserstein metric | This is a very delicate topics and requires a curvature condition on the underlying Polish space $X$ (upon which the probability space $\mathcal P(X)$ is based). This is known nowadays as the "Sturm-Lott-Villani" synthetic theory of curvature, see part III in Villani's (big) book.
Long story short: requiring that $H(\m... | 2 | https://mathoverflow.net/users/33741 | 331041 | 141,764 |
https://mathoverflow.net/questions/330861 | 1 | I am trying to compute the expectation of $g(s,x)=s \ln \sigma(x)+(1-s)\ln(1-\sigma(x))$ with respect to the normal distribution $\mathcal{N}(x;m,v)$, where we have $\sigma(x)=\frac{1}{1+e^{-x}}$. If we define $$\langle g(s,x)\rangle\_q=\int\mathcal{N}(x;m,v)g(s,x)\mathrm{d}x$$
I would like to re-derive the formula w... | https://mathoverflow.net/users/140313 | The expectation of binary logistics regression with respect to Gaussian distribution | Notice that $g(s,x)=(s-1)x+\ln\sigma(x)$. So
$$\langle g(s,x)\rangle\_q=(s-1)m+\langle \ln\sigma(x)\rangle\_q$$
and the entire $s$-dependence is trivial. Now for the second identity we perform a partial integration (noting that, since ${\cal N}$ is a function of $x-m$, one has $\partial{\cal N}/\partial m=-\partial {\... | 0 | https://mathoverflow.net/users/11260 | 331056 | 141,770 |
https://mathoverflow.net/questions/330710 | 7 | A non-empty topological space without isolated points is called *maximal* if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple consequence of Zorn's Lemma.
Note that in a maximal space $(X, \tau)$, nowhere dense sets are closed (and discrete).... | https://mathoverflow.net/users/11647 | What's the minimal weight of a maximal space? | No, these cardinals are not provably equal.
This follows from a result of El'kin [1](https://carma.newcastle.edu.au/jon/Preprints/Books/Open%20Probs%20in%20Top/open2.pdf):
>
> Let $X$ be a set and let $x \in X$. If $\tau$ is a maximal topology on $X$, then $\{U \setminus \{x\} \,:\, x \in U \in \tau \}$ is a base... | 5 | https://mathoverflow.net/users/70618 | 331064 | 141,773 |
https://mathoverflow.net/questions/331069 | 4 | Let $P$ be a polytope given by some half-space description: $P=\{x\in\mathbb{R}^n: Ax\leq b\}$ for some $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$, $m\geq n$. Assume that $x\_0\in P$ for some given $x\_0$ (in particular, $P\neq\emptyset$). What is the complexity of finding **one** (i.e., any) vertex of $P$?
Obvi... | https://mathoverflow.net/users/67002 | Complexity of finding one vertex of a nonempty polytope | This is a classical problem in Linear Programming - to start a simplex method, one must find a vertex.
This is so-called "Phase I" of the simplex method, and without doubt the best ways to do this have been researched a lot.
See e.g. what
Brian Borchers [wrote in scicomp](https://scicomp.stackexchange.com/questions... | 8 | https://mathoverflow.net/users/11100 | 331072 | 141,775 |
https://mathoverflow.net/questions/330356 | 4 | An order-$m$ subplane of a finite projective plane of order $n$ is called a *Baer subplane* if $n=m^2$.
It is known that the projective plane $PG(2,q)$ is a Baer subplane of the Desarguesian projective plane $PG(2,q^2)$.
**Question**: Is it the only Baer subplane? In other words, are all Baer subplanes of $PG(2,q... | https://mathoverflow.net/users/125498 | Is there a unique Baer subplane in a finite Desarguesian projective plane? | The answer is true.
Facts:
Any quadrangle can be mapped onto any other quadrangle by a collineation in a Desarguesian plane. See [this paper](https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-33.1.25).
Every Baer subplane contains a quadrangle.
Every quadrangle lies on a unique Baer subplane i... | 2 | https://mathoverflow.net/users/125498 | 331088 | 141,782 |
https://mathoverflow.net/questions/331098 | 0 | Let $\mathbb{R}^d$ be a the usual Euclidean space and let $Y$ be a fixed non-empty closed subset of $Ball(0,1)$ (the unit ball in $\mathbb{R}^d$ about $0$ of radius $1$).
Let $f$ be the map taking $x \in \mathbb{R}^d$ to the hyperplane passing through $0$ and $x$.
Is the map $g$ defined by
$$
g(x)\triangleq f(x)\... | https://mathoverflow.net/users/36886 | Intersection with a fixed set in Hausdorff metric space | Here is what is true: Let $h(p)$ be the hyperplane through $0$ with normal vector $p$ for each $p\neq0$. Then the function $p\mapsto H(p)=h(p)\cap B\_1(0)$ is continuous.
Indeed, $H\_p=\{x\in\mathbb{R}^d\mid px=0,\|x\|\leq 1\}$ and the Hausdorff distance of $H\_p$ and $H\_q$ is bounded by $\max\_{x\in B\_1(0)}|(p-q)... | 3 | https://mathoverflow.net/users/35357 | 331108 | 141,787 |
https://mathoverflow.net/questions/331104 | 20 | Cryptography sometimes uses elliptic curves over finite fields. Does cryptography also use elliptic curves over $\mathbb{Q}$ or rational points on them?
| https://mathoverflow.net/users/138729 | Cryptography and elliptic curves | Not directly, as far as I know, since explicitly computing large multiples of points in $E(\mathbb Q)$ is infeasible. However, people have considered lifting points from $E(\mathbb F\_p)$ to $E(\mathbb Q)$ or to the $p$-adics $E(\mathbb Q\_p)$ in order to devise algorithms to solve the discrete log problem in $E(\mathb... | 31 | https://mathoverflow.net/users/11926 | 331109 | 141,788 |
https://mathoverflow.net/questions/331106 | 1 | Consider a symmetric or arc-transitive graph except the odd cycle. Then, is it true that the graph could be partitioned into distinct parts such that each part has equal number of vertices except for a few singleton parts(which consists of single vertex) and such that each vertex in each part is adjacent to some other ... | https://mathoverflow.net/users/100231 | Uniform partitioning of regular graphs | Every connected vertex-transitive graph on an even number of vertices has a perfect matching, and each vertex in a connected vertex-transitive graph on an odd number of vertices is missed by a matching that covers all remaining vertices, see [here](http://mathworld.wolfram.com/PerfectMatching.html).
So, for the even ... | 4 | https://mathoverflow.net/users/125498 | 331111 | 141,790 |
https://mathoverflow.net/questions/257865 | 4 | I'll state the classic result in its density (rather than the more general differentiation) version. Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ and $A\subset \mathbb{R}^n$ a measurable set.
For $x\in\mathbb{R}^n$, define
$$ \chi\_{A,r}(x)=\frac{\mu(A\cap B\_r(x))}{\mu(B\_r(x))},$$
where $B... | https://mathoverflow.net/users/12518 | Lebesgue-Besicovitch theorem for partition elements rather than balls | The answer to both questions is positive (assuming the definition of aspect ratio for sets is modified to refer to the maximal inscribed and minimal circumscribed balls). The family of all sets $V \subseteq \mathbb{R}^n$ with aspect ratio at most $a$ has bounded eccentricity in the sense of the Wikipedia article on thi... | 1 | https://mathoverflow.net/users/30721 | 331114 | 141,791 |
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