parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/278405 | 0 | Let $X$ be an infinite set and suppose $\tau$ is an [ultraconnected](https://en.wikipedia.org/wiki/Ultraconnected_space) topology on $X$ without isolated points. Is there a topology $\sigma\supseteq \tau$ such that $(X,\sigma)$ is a connected $T\_2$-space?
| https://mathoverflow.net/users/8628 | Refining ultraconnected spaces to connected $T_2$ spaces | No, not always. Let $\tau$ denote the *initial segment topology* on $\mathbb N$: a subset of $\mathbb N$ is considered open if and only if it is an initial segment of $\mathbb N$. This topology is ultraconnected, but if $\sigma$ is any Hausdorff topology refining $\tau$, then it has isolated points: for example, $\{1\}... | 3 | https://mathoverflow.net/users/70618 | 278425 | 123,446 |
https://mathoverflow.net/questions/209595 | 2 | Is there an infinite $T\_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on $X^X$.)
What's the answer if we endow $\text{Aut}(X)$ with the compact-open topology instead of the product topology... | https://mathoverflow.net/users/8628 | $T_2$-space $X$ with $X\cong \text{Aut}(X)$ | In [this paper](http://www.ams.org/journals/tran/1983-280-02/S0002-9947-1983-0716833-2/S0002-9947-1983-0716833-2.pdf) Jan van Mill has constructed a separable metrizable Boolean topological group $X$ such that each homeomorphism $h:X\to X$
is equal to the translation $X\to X$, $x\mapsto x+h(0)$. This implies that the ... | 5 | https://mathoverflow.net/users/61536 | 278426 | 123,447 |
https://mathoverflow.net/questions/278409 | 5 | Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a strategy exploiting Stone's formula for the resolvent and then the Stone-Weierstrass theorem one can show this for $f \in C\_... | https://mathoverflow.net/users/112877 | Commuting with self-adjoint operator | Any bounded Borel function $f: \mathbb{R} \to \mathbb{R}$. If $TS = ST$ then (taking adjoint of both sides) $S^\*T = TS^\*$. Therefore both ${\rm Re}(S) = \frac{1}{2}(S + S^\*)$ and ${\rm Im}(S) = \frac{1}{2i}(S - S^\*)$ commute with $T$, and since they are self-adjoint it follows from standard spectral theory that the... | 4 | https://mathoverflow.net/users/23141 | 278439 | 123,451 |
https://mathoverflow.net/questions/278390 | 1 | Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain function has a fast decaying representation with respect to some basis. I encountered an example where I expected such ... | https://mathoverflow.net/users/112877 | Orthonormal basis and decay | Think in the opposite direction: start with an orthonormal sequence $g\_n$ and try to find appropriate $h\_n$. The conditions on $h\_n$ are: $h\_n$ is a linear combination of $g\_0, g\_1, \ldots, g\_n$, $\|h\_n\| = 1$, $\langle h\_n, h\_{n-1}\rangle = \alpha\_n$ for a given $\alpha\_n$ (in your question $\alpha\_n$ is ... | 1 | https://mathoverflow.net/users/108637 | 278459 | 123,456 |
https://mathoverflow.net/questions/278380 | 3 | Consider the variation of mixed hodge structures which generates at the origin:
$$
f:X = \text{Proj}\left( \frac{\mathbb{C}[t][x,y,z]}{(xy(x + y + tz))} \right) \to \mathbb{A}^1\_t
$$
How can I compute the monodromy of the cohomology groups $\mathbf{R}f\_\*(\underline{\mathbb{Q}}\_X^{\text{Hdg}})$?
| https://mathoverflow.net/users/78824 | How can I determine the monodromy of this variation of mixed hodge structures? | This monodromy is trivial because on the set $t\neq 0$ you can make a change of variables $(x,y,z)\to(x,y,z t^{-1})$, and your ideal becomes $x y (x+y+z)$, so it doesn't depend on $t$. So your family is just a constant family.
| 2 | https://mathoverflow.net/users/89514 | 278464 | 123,458 |
https://mathoverflow.net/questions/278465 | 3 | Let the operator vec($A$) unroll all the elements of $A$ into a single column vector in column-major order. Then, the elements of vec($A^T$) are a permutation of the elements of vec($A$). If I want to write this permutation as a matrix-vector product, I get
vec($A^T)$ = $P$ vec($A$).
I'm looking for a common name a... | https://mathoverflow.net/users/113174 | Better name for “vec transposition permutation matrix”? | This matrix is known as the *commutation matrix*. For more details, please see Chapter 3, Section 7 of: [Matrix differential calculus](http://www.janmagnus.nl/misc/mdc2007-3rdedition) by Magnus and Neudecker.
| 5 | https://mathoverflow.net/users/8430 | 278468 | 123,460 |
https://mathoverflow.net/questions/278421 | 10 | Let $G$ be a connected complex semisimple Lie group and $S$, $T$ two maximal tori in $G$. Is there a known upper bound on the number of connected components of $S\cap T$? For example, is it bounded by the cardinality of the centre $Z\_G$:
$$|\pi\_0(S\cap T)|\leq|Z\_G|?$$
| https://mathoverflow.net/users/113233 | Number of connected components of the intersection of two maximal tori | **Summary:** Let $X = \mathrm{Hom}(T,\mathbb{G}\_m)$ be the weight lattice, $\Phi \subset X$ the root system. Define a sublattice $L$ of $X$ to be a "root sublattice" if $L$ is generated as an abelian group by $L \cap \Phi$. Then the possible component groups of $S \cap T$ are the torsion subgroups of $X/L$, as $L$ ran... | 9 | https://mathoverflow.net/users/297 | 278472 | 123,461 |
https://mathoverflow.net/questions/278435 | 12 | Let $ [n] $ be the set $ \{1,2,\ldots n\}$.
A *summoid* is a subset $ A \subset [n] $ of the form $ \{a,b,a+b\} $ (you can choose a better name, if it doesn't exist already).
Now, I developed by accident this simple result:
>
> There is no partition of $ [9] $ into (disjoint) summoids.
>
>
>
I want to ask ... | https://mathoverflow.net/users/108318 | Partition of [3n] into summoids | I have the following results:
* N = 12:
(1, 11, 12)
(3, 7, 10)
(4, 5, 9)
(2, 6, 8)
* N = 15:
(1, 14, 15)
(3, 10, 13)
(4, 8, 12)
(5, 6, 11)
(2, 7, 9)
* N = 24:
(1, 23, 24)
(2, 20, 22)
(5, 16, 21)
(6, 13, 19)
(7, 11, 18)
(8, 9, 17)
(3, 12, 15)
* N = 27:
(1, 26, 27)
(2, 23, 25)
(4, 20, 24)
(7, 15, 22)
(8, 13, 21)
(9, 1... | 9 | https://mathoverflow.net/users/76332 | 278473 | 123,462 |
https://mathoverflow.net/questions/278117 | 12 | I'm looking for a lower bound for the probability that an arbitrary convex combination of iid Bernoulli (p) random variables is at least p. My guess is p/k (for some constant k; k must be at least e, as noted by Matt below), but I'm happy with any positive lower bound that depends only on p.
For example, if p is slig... | https://mathoverflow.net/users/85550 | Convex combination iid Bernoulli random variables | To complement my other answer, I will show
>
> **Proposition 1** Let $\xi\_k$ be a finite number of iid Bernoulli random variables of expectation $p > 1/2$, and let $a\_k > 0$ be real numbers. Then ${\bf P}( \sum\_k a\_k \xi\_k \geq p \sum\_k a\_k) \gg 1$.
>
>
>
By replacing $\xi\_k$ with $1-\xi\_k$ and $p$ wi... | 5 | https://mathoverflow.net/users/766 | 278474 | 123,463 |
https://mathoverflow.net/questions/278460 | 3 | I have the following problem:
A matrix $C\in \mathbb{R}^{2N}$, where
$C=\epsilon A+D$
$\epsilon A=(C-C')/2$ is skew symmetric with "block" anti-diagonal structure of size 4.
$ D=(C+C')/2$ (Diagonal matrix) with "block" diagonal structure of size 2.
$D=\begin{bmatrix}
0 & 0 & 0 & \dots \\
0 & 0 & 0 & \dots \\
... | https://mathoverflow.net/users/106076 | Inverting (via Taylor expansion) a sum of (rank-deficient) skew-symmetric matrix and (rank-deficient) Diagonal matrix | Just an idea.
I think the best way to start is to expand on the structure of the matrices $D, C$. For example, $D$ can be readily seen to be expressible in the form $$\alpha M\_1\otimes I\_2=\alpha\begin{bmatrix}
0 & \mathbf{0}^T\\
\mathbf{0} & J
\end{bmatrix}\otimes I\_2$$ where $J=diag(1,2,,\cdots)$. I think the st... | 4 | https://mathoverflow.net/users/64194 | 278478 | 123,466 |
https://mathoverflow.net/questions/278285 | 5 | Let $G$ be a complex reductive group acting linearly on a complex affine variety $X$, and let $K$ be the kernel of the action, i.e.
$$K:=\{g\in G:g\cdot x=x\text{ for all }x\in X\}.$$
Is
$$X\_K:=\{x\in X:\mathrm{Stab}\_G(x)=K\}$$
Zariski-open in $X$?
| https://mathoverflow.net/users/113147 | Is the set of points with smallest stabilizer open? | The answer is affirmative if $G$ is finite or abelian since in that case subgroups are rigid.
Otherwise, $X\_K$ may not even be dense, let alone open. The standard example is due to Luna from his slice paper: Let $G=SL(2,\mathbb C)$ act on the space $X=S^3\mathbb C^2$ of binary cubics. The action is effective so $K=1... | 4 | https://mathoverflow.net/users/89948 | 278485 | 123,470 |
https://mathoverflow.net/questions/277821 | 14 | Let $T^\*\_{\mathbb{C}}(Gr\_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^\*\_{\mathbb{C}}(Gr\_{n,r})$. Condsidering $Gr\_{n,r}$ as the homogeneous space $U\_n/(U\_r \times U\_{n-r})$, we have a unique representa... | https://mathoverflow.net/users/76104 | Schubert calculus expressed in terms of the cotangent space of the Grassmannians | The tangent space to the Grassmanian corresponds to the following representation of $U(r)\times U(n-r)$, call it $\rho$: it is the $r\times (n-r)$ matrices, with $U(r)$ acting on the left and $U(n-r)$ acting on the right, so if we denote by $A$ the standard representation of $U(r)$ and by $B$ the standard representatio... | 10 | https://mathoverflow.net/users/89514 | 278521 | 123,484 |
https://mathoverflow.net/questions/186373 | 13 | Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that
$$
\pi\_1(MO(2))=0,\quad \pi\_2(MO(2))=\mathbb{Z}/2,\quad\pi\_3(MO(2))=0,\quad \pi\_4(MO(2))=\mathbb{Z}.
$$
By the Pontrjagin-Thom construction $\p... | https://mathoverflow.net/users/8103 | Homotopy groups of $MO(2)$ | There are a couple of references which show that $\pi\_5(MO(2))=\mathbb{Z}/2$.
*Suzuki, H.*, [**On the realization of the Stiefel-Whitney characteristic classes by submanifolds**](http://dx.doi.org/10.2748/tmj/1178244748), Tohoku Math. J., II. Ser. 10, 91-115 (1958). [ZBL0107.17001](https://zbmath.org/?q=an:0107.1700... | 6 | https://mathoverflow.net/users/8103 | 278522 | 123,485 |
https://mathoverflow.net/questions/278525 | 3 | I would like to know if there is something I can read to compute the following:
Let $H$ be a hyperelliptic curve of genus $2$ given by $y^2=x^5 + 10$ and let $J$ be its Jacobian.
How can I prove for a fixed prime $p$ of good reduction that in $J(\mathbb{F}\_p)$ the element $D\_0:=[(-1,3)-\infty]$ is not divisible ... | https://mathoverflow.net/users/91023 | 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians | Since you know the group structure of $J({\mathbb F}\_p)$
(I guess $p \equiv -1 \bmod 5$ and $\alpha = p+1$), the simplest way
would be to compute $(\alpha/5) \cdot (D\_0 \bmod p)$ and check if this
is the zero element in $J({\mathbb F}\_p)$. Your point is divisible by 5
if and only if it is. Computation of such multip... | 4 | https://mathoverflow.net/users/21146 | 278528 | 123,486 |
https://mathoverflow.net/questions/278400 | 6 | A few years ago, I wanted to cite a result in a paper, for which I could not find a reference. I ended up not using the full strength of it, and the part that I needed could be easily proved. Still, I'd like to know where the full version appears. The result is as follows:
>
> The eigenvalues of the matrix
> $$\le... | https://mathoverflow.net/users/113161 | Result attribution for eigenvalues of a matrix of Pascal-type | I don't know a reference. One way to show the eigenvalues starts from the observation (which can be proved using
generating functions)
that $\sum\_{i=0}^n {i\choose k} A\_{i,j}={2n+1 \choose n-k} {j+k \choose k}={2n+1 \choose n-k}\,\sum\_{\ell=0}^k {k\choose \ell} { j \choose \ell}$ (where $A$ is the matrix above).
W... | 5 | https://mathoverflow.net/users/48831 | 278532 | 123,488 |
https://mathoverflow.net/questions/278517 | 5 | In some work I was doing with a colleague the following function of two natural number variables, defined by a recursion, came up and we have no clue how to solve it. Any suggestions or improvements on the upper bound given below would be appreciated. Asymptotics are also interesting.
The boundary conditions are
... | https://mathoverflow.net/users/15934 | A strange two-variable recursion | First, we note that $f(m,n) - f(m,n-1)$ does not depend on $n$, so for a fixed $m$ we can write $f(m,n) = \alpha\_m n + \beta\_m$ for some $\alpha\_m,\beta\_m$ which depend only on $m$. Imposing the boundary conditions lets us write
$$
f(m,n) = a\_m (n-1) + (2^m-1)
$$
Next, substituting that into the defining equation ... | 6 | https://mathoverflow.net/users/82067 | 278535 | 123,491 |
https://mathoverflow.net/questions/278502 | 3 | Assume that $V$ is a vector field on a
Riemannian manifold $(M,g)$ with natural volume form $\Omega$ arising from $g$.
Assume that the solution curves of $V$ are parametrized geodesics of the Riemannian metric $g$.
>
> Is it true to say that the space of harmonic functions is invariant under the derivational opera... | https://mathoverflow.net/users/36688 | Is the space of harmonic functions invariant under the derivational operator associated with a geodesible flow? | Take a [warped product](https://arxiv.org/pdf/1307.0236.pdf) $M=\mathbb S^1\times\_f\mathbb S^1$ for a nonconstant smooth function $f$.
The horizontal vector field $V$ satisfies your condition,
but for harmonic function is not invariant $f$, the function $Vf$ is not harmonic.
To see this, consider the $V$-flow $\Ph... | 1 | https://mathoverflow.net/users/1441 | 278536 | 123,492 |
https://mathoverflow.net/questions/278529 | 4 | Let $(\lambda\_1 , \cdots , \lambda\_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha\_1, \cdots, \alpha\_1}\_{\lambda\_1}, \cdots , \underbrace{\alpha\_d , \cdots, \alpha\_d}\_{\lambda\_d}) \in \mathbb{Z}^k: \, (\alpha\_1, \cdots , \alpha\_d) \in \... | https://mathoverflow.net/users/98093 | How to know if convex-hull of a set contains zero? | Like Emil noted in the comments, the question is equivalent to whether $0\in\mathrm{Conv}(\{\alpha\in\mathbb{Z}^d: \sum\_{i=1}^{d}\alpha\_i=0\}\setminus\{0\})$. To see that it is, note that it is the average of $(d-1,-1,-1,\ldots,-1)$ and its permutations.
| 4 | https://mathoverflow.net/users/20186 | 278537 | 123,493 |
https://mathoverflow.net/questions/263210 | 2 | A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An **edge** of a cube is a pair of points $\{\psi(\epsilon\_1),\psi(\epsilon\_2)\}$ in $X$ such that $\epsilon\_1$ and $\epsilon\_2 $ differ in exactly 1 coordinate. Denote the set of all edges by $E$.
An **diagonal** of a cube is a pair of points $\{\psi(\... | https://mathoverflow.net/users/80191 | An inequality about embedding of cube into metric spaces | By now surely you already know the answer, but here it is anyway. It follows from the assumption on $\theta$ that
$$
\text{diag}^2(\epsilon\_0)\ge k(1-\theta)^2 \sum\_E\text{edge}^2 - \sum\_{D\_0} \text{diag}^2
$$
where $D\_0$ is the set of diagonals excluding the one determined by $\epsilon\_0$.
We already know that f... | 3 | https://mathoverflow.net/users/3536 | 278544 | 123,494 |
https://mathoverflow.net/questions/278540 | 7 | Let $B, R\in M\_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite.
Let $s,t \in \mathbb{R}$ and $s,t \ge 0$ .
Does then hold $Tr[B^s (B R^2 B)^t] \ge Tr[B^s (R B^2 R)^t]$ ?
| https://mathoverflow.net/users/17261 | A conjectured trace inequality for some products of powers of matrices | I think the inequality is false. Consider for instance the choices
\begin{equation\*}
B = \begin{bmatrix} 5& 6& -2\\
6 & 13 & 2\\
-2 & 2 & 5\end{bmatrix},\quad
R = \begin{bmatrix}
-8 & 4 & 4\\
4 & -2 & -1\\
4 & -1 & 0
\end{bmatrix},\quad s=5,\ t=3.
\end{equation\*}
Then, we have (computed using Mathematica)
... | 5 | https://mathoverflow.net/users/8430 | 278549 | 123,497 |
https://mathoverflow.net/questions/278196 | 1 | Let $X$ be a random variable with the distribution $F$ (cdf).
What are the extreme points of the sets of the form:
\begin{align}
P\_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P\_2&=\left\{ F: |X| \le d \right\},\\
P\_3&=\left\{ F: \int |x|^k dF\le c, \, |X| \le d \right\}.\\
\end{align}
In [this question](https://... | https://mathoverflow.net/users/69661 | Extreme Points of a set of distributions with moment and/or support constraint | You already know (from the [previous question](https://mathoverflow.net/questions/277096/extreme-points-of-set-of-probability-measures-mathcalp-f-int-mathbbr) on MathOverflow, and in particular [this paper](https://www.jstor.org/stable/3689944)) that the extreme points of $P\_1$ form a subset of the set that consists o... | 0 | https://mathoverflow.net/users/108637 | 278552 | 123,498 |
https://mathoverflow.net/questions/278511 | 7 | What is the best unconditional upper bound for $p\_{n^2}-p\_{(n-1)^2}$ such that $p\_n$ is the $n$-th prime number?
Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this?
Edit: it's known that there is a prime between $[x,x+x^{13/23}]$, so one can bound $p\_{n^2}-p\_{(n-1)^2} \leq 2 n n^{13/23}... | https://mathoverflow.net/users/106239 | Upper bound for $p_{n^2} - p_{(n-1)^2}$? | By the results of
*Baker, R.C.; Harman, G.; Pintz, J.*, [**The difference between consecutive primes. II**](http://dx.doi.org/10.1112/plms/83.3.532), Proc. Lond. Math. Soc., III. Ser. 83, No.3, 532-562 (2001). [ZBL1016.11037](https://zbmath.org/?q=an:1016.11037).
the number of primes in the interval $[x, x+x^{0.52... | 12 | https://mathoverflow.net/users/766 | 278556 | 123,500 |
https://mathoverflow.net/questions/278559 | 3 | Let $p$ be a prime greater than 3, and let $M$ be a positive integer prime to $p$. Let $\Sigma$ be the set of isomorphism classes of pairs $(E, C\_M)$, where $E$ is a supersingular elliptic curve over $\overline{\mathbb{F}\_p}$ and $C\_M$ a cyclic subgroup of $E$ of order $M$. Then, by Deligne-Rapoport, $X\_0(pM)\_{\ov... | https://mathoverflow.net/users/46108 | Supersingular elliptic curves with extra automorphisms | If there is a supersingular elliptic curve $E / {\bf F}\_p$ with
an automorphism of order $4$ or $6$, then the automorphism is not
defined over ${\bf F}\_p$, but plenty of cyclic subgroups *are*
defined over ${\bf F}\_p$. This is because the Frobenius endomorphism $\varphi$
generates an imaginary quadratic ring and the... | 5 | https://mathoverflow.net/users/14830 | 278561 | 123,504 |
https://mathoverflow.net/questions/163472 | 8 | Consider the (reduced) homology functor $H\_\*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ from graded Abelian groups to spectra so that $H\_\*\circ F=Id$.
(I have very little experience in Algebraic Topolog... | https://mathoverflow.net/users/48932 | Section of the homology functor on spectra | I thought I will post the only reference I have found about non-existence of Moore space functor as an answer. It is due to Carlsson, "A counterexample to a conjecture of Steenrod" <https://link.springer.com/article/10.1007%2FBF01393939> and there is a nice discussion on nLab <https://ncatlab.org/nlab/show/Moore+space#... | 4 | https://mathoverflow.net/users/48932 | 278563 | 123,505 |
https://mathoverflow.net/questions/278208 | 1 | In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related to the derivatives w.r.t. the variance $\sigma^2$, this expansion is exactly the Taylor expansion.
Does anyone know if... | https://mathoverflow.net/users/113059 | Expand the pdf of Wishart distribution into power series via orthogonal polynomials | Note that the Laguerre orthogonal polynomials are in form of [1](bearing combinatoric interpretation) and [3]
\begin{align}
& L\_n^\nu(x)=(-1)^n\sum\_{m=0}^n \binom n m
\prod\_{i=1}^m (\nu+2(n-i))(-x)^{n-m} \\[8pt]
= {} & \sum\_{m=0}^n \frac{\Gamma(\nu+n+1)\frac{\Gamma(-n+m)}{\Gamma(-n)}} {n!m!(\nu+m+1)} x^k=\sum\_{m... | 2 | https://mathoverflow.net/users/25437 | 278583 | 123,512 |
https://mathoverflow.net/questions/278584 | 3 | Let $X\neq \emptyset$ be a set, and let ${\cal U}$ be a collection of subsets of $X$ such that
1. $\bigcup {\cal U} = X$, and
2. $U\_1\neq U\_2\in {\cal U}$ implies $|U\_1\cap U\_2| < \aleph\_0$.
Is there ${\cal U}\_0\subseteq {\cal U}$ such that
1. $\bigcup {\cal U}\_0 = X$, and
2. if $U\in{\cal U}\_0$ then $\bi... | https://mathoverflow.net/users/8628 | Minimal subcoverings of a cover with finite intersection | No, consider the covering of naturals by initial segments.
| 2 | https://mathoverflow.net/users/4312 | 278585 | 123,513 |
https://mathoverflow.net/questions/278484 | 5 | I am trying to prove that the integral
\begin{align}
\int\_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L\_n^{l-n}\left(\frac{r^2}{C}\right) I\_{l-n}\left(\rho r \right) r dr
\end{align}
has the form
\begin{align}
B^{l+1} e^{\frac{B}{2}\rho^2} \rho^{l-n} L\_n^{l-n}\left(\frac{\rho^2}{C} \right),
\end{align}
where ... | https://mathoverflow.net/users/112975 | Integral involving Laguerre, Gaussian and modified Bessel function | Mathematica can evaluate the integral$^\ast$
$$I\_{n,l}=\begin{align}
\int\_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L\_n^{l-n}\left(\frac{r^2}{C}\right) I\_{l-n}\left(\rho r \right) r dr
\end{align},\;\;B,C,\rho>0,$$
for any integer $n\geq 0$ as a function of $l>n-1$. The results are consistent with
$$I\_{n,l}=\b... | 4 | https://mathoverflow.net/users/11260 | 278593 | 123,516 |
https://mathoverflow.net/questions/278565 | 3 | The `Maybe` monad is based on the endofunctor $- + 1$ (coproduct with the singleton set). Its Lawvere theory $L$ is supposed to be generated by one nullary operation (see [Plotkin and Power](http://homepages.inf.ed.ac.uk/gdp/publications/Comp_Eff_Monads.pdf)), `raise`. How do I show that the `Maybe` monad is generated ... | https://mathoverflow.net/users/34546 | Lawvere theory and the Maybe monad | $L(n,1)$ consists of the $n$ projection operations $p\_i$ and the additional nullary operation $c$. So a typical element of $L(n,1)\times a^n$ has one of the two forms $(p\_i;x\_1,\dots,x\_n)$ or $(c;x\_1,\dots,x\_n)$. Th coend consists of the disjoint union of all these sets (over all $n$) modulo some identifications.... | 5 | https://mathoverflow.net/users/6794 | 278595 | 123,517 |
https://mathoverflow.net/questions/278568 | 9 | Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the spectrum of this operator is discrete and tends to infinity?
It almost looks like a result that would follow from t... | https://mathoverflow.net/users/112877 | Harmonic oscillator discrete spectrum | As already pointed out in the comments, $V\to\infty$ does imply discrete spectrum in general. In fact, this becomes an equivalence if a somewhat more general version of this condition is used:
>
> Theorem: Suppose that $V(x)$, $x\ge 0$, is bounded below (and locally integrable, as usual). Then the spectrum of $-d^2... | 8 | https://mathoverflow.net/users/48839 | 278598 | 123,518 |
https://mathoverflow.net/questions/278601 | 4 | In the [lecture notes](http://www-personal.umich.edu/~morilac/pdag/Positroid%20notes.pdf), it is said that (Theorem 3.1.3) the set of positroid cells in $Gr(k,n)$ are in one to one correspondence with the set of bounded affine permutations of type $(k,n)$. In Example 4.1.5, it is said that the permutation $\sigma$ corr... | https://mathoverflow.net/users/11877 | Big cells in a Grassmann and permutations | Positroid cells in $Gr(k,n)$ are indexed by many objects we often want to go between. The big cell will be given by the bounded affine permutation $i \mapsto i+k$. See Postnikov's original [preprint](https://arxiv.org/pdf/math/0609764.pdf) (section 16). Note the positroid for the big cell will the the positroid consist... | 4 | https://mathoverflow.net/users/51668 | 278602 | 123,519 |
https://mathoverflow.net/questions/278604 | 4 | Let $X$ be a degree $d$ hypersurface in $\mathbb{P}^n$. For $(d,n)=(3,4),(4,5)$, or $(5,6)$, Coskun and Starr proved in *Rational curves on smooth cubic hypersurfaces* that the Kontsevich space $\overline{\mathcal{M}}\_{0,0}(X,e)$ of rational curves has two irreducible components ($e>1$):
1) $e$ to 1 covers of a line... | https://mathoverflow.net/users/16356 | Conics on a cubic threefold | **Answer under revision.** The OP has now asked for the proof of irreducibility for all $e\geq 1$, not just for $e= 2.$ The argument for that uses the same basic ideas, but the edits are getting very long. As I have time, I will write this up as a PDF file and add a link to this post. What is written below is provision... | 4 | https://mathoverflow.net/users/13265 | 278610 | 123,521 |
https://mathoverflow.net/questions/154997 | 1 | In the following question, we defined the foliation values of an smooth manifold;
[Foliation values of a manifold](https://mathoverflow.net/questions/154975/foliation-values-of-a-manifold)
Let $S\_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological $S^{7}$.
According to the above definition,... | https://mathoverflow.net/users/36688 | Foliation values of Exotic spheres | The answer to your question is *yes*: For every $k\in\{0,\dots,7\}$, every smooth manifold homeomorphic to $S^7$ admits a $k$-dimensional $C^\infty$ foliation.
[Every exotic $7$-sphere $S\_i$ is parallelisable.](https://mathoverflow.net/questions/58131) Hence for every $m\in\{0,\dots,7\}$, every $S\_i$ admits a $C^\i... | 1 | https://mathoverflow.net/users/49655 | 278612 | 123,522 |
https://mathoverflow.net/questions/278578 | 2 | I'm concerned with the following problem:
We are given a set $\mathcal{A} \subseteq 2^S$ of subsets of the ground set $S=E(M)$ of a matroid $M$. I would like to know whether there is a basis $B$ of the matroid which intersects every set in $A$, meaning $B \cap A \neq \emptyset , \forall A \in \mathcal{A}.$
Is there ... | https://mathoverflow.net/users/113322 | Criterion for the existence of a basis in a matroid intersecting given sets | This is an elaboration of Fedor Petrov's comment. Suppose we could give a satisfactory answer to your question for uniform matroids. Then we would have a nice way to characterize solutions to the hitting set problem (which asks for the size of the smallest set that intersects every set in $\mathcal{A}$). But the hittin... | 1 | https://mathoverflow.net/users/3106 | 278616 | 123,526 |
https://mathoverflow.net/questions/278630 | 0 | Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime,
$N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian.
Does there exist an element $g\in G$ such that $\langle g\rangle$ is NOT normal in $G$
and $N\langle g\rangle$ is normal in $G$?(Note that $G$ and $\frac{G}{N}$ are ... | https://mathoverflow.net/users/27962 | Existence of a cyclic non-normal subgroup in a $p$-group | Let $G = \langle x,y,z \rangle$ be a $3$-generated group of order $p^6$ of exponent $p$ and class $2$. So $Z(P) = [P,P]=\Phi(P)$ is elementary abelian of order $p^3$, and so is $G/Z(P)$.
Let $N = \langle [x,y] \rangle$. So $N \lhd G$ with $|N|=p$. Then $Z(G/N) = Z(G)/N$, so for $g \in G$, we have $$\langle g \rangle ... | 2 | https://mathoverflow.net/users/35840 | 278631 | 123,528 |
https://mathoverflow.net/questions/278629 | 15 | I hope this is a suitable MO question. In a research project, my collaborator and I came across some combinatorial expressions. I used my computer to test a few numbers and the pattern was suggesting the following equation for fixed integers $K\geq n>0$.
$$\dfrac{K!}{n!K^{K-n}}\sum\limits\_{ \begin{subarray}{c} k\_1+... | https://mathoverflow.net/users/10333 | A combinatorial identity | This is the answer to the first question, I wrote a long answer to Question 2 as a separate answer.
Note that $A:=\sum\_{k\_i>0,k\_1+\dots+k\_n=K}\frac{K!}{n!k\_1!\dots k\_n!} \prod k\_i^{k\_i-1}$ is a number of forests on the ground set $\{1,2,\dots,K\}$ having exactly $n$ connected components and with a marked vert... | 28 | https://mathoverflow.net/users/4312 | 278640 | 123,532 |
https://mathoverflow.net/questions/278639 | 1 | In the stationary case, I know that if the chain is irreducible and aperiodic, it is Ergodic. But in the non-stationary case, i can not comprehend the content deeply. I want to know if Irreducibility holds for the Ergodic non-stationary Markov chain.
Generally, i want to know what are the main differences between Er... | https://mathoverflow.net/users/113357 | Does Irreducibility holds for the Ergodic non-stationary Markov chain? |
>
> Generally, i want to know what are the main differences between
> Ergodicity of a stationary Markov chain and non-stationary one?
>
>
>
This question could be a better question if formulated better.
**(1)Stationary process**
You are right about the fact that when we have a stationary Markov process then... | 2 | https://mathoverflow.net/users/25437 | 278647 | 123,536 |
https://mathoverflow.net/questions/273937 | 8 | I have the following convex optimization problem:
$$\begin{array}{ll} \text{maximize}\_{{f,g}} & \displaystyle\int\_{\Omega} g^u{f}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int\_{\Omega} f \mathrm{d}\mu= 1,\quad \displaystyle\int\_{\Omega} g\mathrm{d}\mu =1 \\ & f\_L \leq {f} \leq f\_U\\ & g\_L \leq g \le... | https://mathoverflow.net/users/36356 | Uniqueness of a Solution for a Convex Optimization Problem | I tried to work on the problem and I think I am able to resolve some points. Here is my work:
Consider the Lagrangians:
$$L\_0=\int\_{\mathbb{R}} g^u{f}^{1-u}\mathrm{d}\mu+\int\_{\mathbb{R}}\left(\lambda\_0(f-f\_L)+\lambda\_{00}(f\_U-f)\right)\mathrm{d}\mu+\mu\_0\left(\int\_{\mathbb{R}} f\mathrm{d}\mu-1\right)$$
$$... | 1 | https://mathoverflow.net/users/36356 | 278652 | 123,540 |
https://mathoverflow.net/questions/278230 | 1 | Let $G$ be a finite group and $p$ be a prime. Suppose that every minimal normal subgroup of $G$ is isomorphic to $\mathbb{Z}\_{p}$. What possible structures does $G$ have?
| https://mathoverflow.net/users/97247 | Minimal normal subgroups of finite groups | Some easy observations (a bit too long for a comment):
Every $p$-group has this property, as does every quasisimple group whose centre is a non-trivial $p$-group. Conversely if $G$ has this property, the generalized Fitting subgroup has the form $O\_p(G)E(G)$ and the centre of every component is a non-trivial $p$-gro... | 1 | https://mathoverflow.net/users/4053 | 278662 | 123,541 |
https://mathoverflow.net/questions/278594 | 2 | I recently found the paper of Berarducci + Mantova [[1](https://arxiv.org/abs/1503.00315), [2](https://arxiv.org/abs/1703.01995)] saying that surreal numbers are equivalent to trans-series. These are very different objects:
* trans-series are used in physics to correct, Laplace transforms [[3](https://arxiv.org/abs/1... | https://mathoverflow.net/users/1358 | Are Surreal Numbers the same as Trans-series? | It seems there might be some confusion about the term "equivalent". The two structures in question are "equivalent" in a technical sense, i.e. elementarily equivalent (in the language of ordered differential rings).
Let $\mathcal{L\_d} = \{+,\times,0,1,<,\partial\}$ be the language of ordered differential fields. Th... | 5 | https://mathoverflow.net/users/51323 | 278683 | 123,547 |
https://mathoverflow.net/questions/278609 | 4 | For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which is also the field of fractions of the formal power series ring $k[[t]]$). The composite embedding $k[t] \hookrightarrow k... | https://mathoverflow.net/users/49 | An analogue of rational functions for Hahn series | The field $k(t^\Gamma)$ is sometimes called "the field of generalized rational functions". It is covered in section 2.9 of I. Efrat, "Valuations, Orderings, and Milnor $K$-Theory", AMS, 2006.
| 2 | https://mathoverflow.net/users/101929 | 278684 | 123,548 |
https://mathoverflow.net/questions/277769 | 11 | In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of *relative cycles* on $X\to S$, and, for integers $r\ge 0$, he defines the functor:
$$\text{Chow}\_r(X/S) : (\text{Sch}/S)^{\rm opp}\to\te... | https://mathoverflow.net/users/nan | Algebraic cycles, Chow spaces and Hilbert-Chow morphisms | Mathoverflow answer
===================
In my thesis [R4], I gave an ad hoc definition of a Chow functor ($\mathrm{Chow}\_r$ above) that was meaningful also in characteristic p and close to Barlet's and Angéniol's definitions. It is "ad hoc" because the definition involves specifying zero-cycles over every suitable p... | 11 | https://mathoverflow.net/users/40 | 278693 | 123,552 |
https://mathoverflow.net/questions/278624 | 0 | Assume that $n>1$.
The configuration space of $S^n$ is defined as follows $$M\_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$
We have two questions:
>
> 1.Is there a continuous function $f:M\_n \to S^{n-1}$ with $f(y,x)=-f(x,y)$, for all $x,y \in $S^{n}$?
>
>
> 2.Is there a continuos function $h: M\_n \to \mathb... | https://mathoverflow.net/users/36688 | A possible proof of the Borsuk Ulam theorem without "Homology-Cohomology" | Suppose such a function $M\_n \to S^{n-1}$ existed. Consider the composition $S^n \to M\_n \to S^{n-1}$, where the first map sends $x$ to the pair $(x,-x)$. That composition is an odd continuous function $S^n \to S^{n-1}$, hence can not exist by Borsuk-Ulam.
| 3 | https://mathoverflow.net/users/1310 | 278695 | 123,553 |
https://mathoverflow.net/questions/277628 | 1 | A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such
$$\rho=ag+b\alpha\otimes\alpha$$
where $a,b$ are smooth functions on $M$ and $\rho$ is ricci tensor of $g$. It is easy to see that if $b=0$ then $(M,g)$ reduce to Einstein manifold.
**Q1:** Is there a similar ve... | https://mathoverflow.net/users/90655 | Question on $\alpha-$Einstein manifolds | For **Q2** I found the following:
In [Ricci solitons and real hypersurfaces in a complex space form](https://www.jstage.jst.go.jp/article/tmj/61/2/61_205/_article) Cho and Kimura studied on Ricci solitons of real hypersurfaces in a non-flat complex space form and they defined $\alpha-$Ricci soliton $(g,V,\lambda,\mu,... | 1 | https://mathoverflow.net/users/90655 | 278705 | 123,555 |
https://mathoverflow.net/questions/278698 | 11 | In some cases, hexagons are used in percolation. Why do we use hexagons in percolation?
| https://mathoverflow.net/users/113382 | Why do we use hexagons in percolation? | Site percolation on the triangular lattice (or, equivalently, face percolation on the hexagonal lattice) is the only case for which conformal invariance of percolation at criticality has been [proven](http://www.sciencedirect.com/science/article/pii/S0764444201019917). This has to do with very special combinatorial pro... | 14 | https://mathoverflow.net/users/56624 | 278708 | 123,556 |
https://mathoverflow.net/questions/278689 | 10 | Suppose $X,Y$ are sets, and $f:X\to Y$ and $g: Y \to X$. Then there are disjoint subsets $X\_1,X\_2 \subseteq X$ with $X\_1\cup X\_2= X$ and disjoint subsets $Y\_1,Y\_2 \subseteq Y$ with $Y\_1\cup Y\_2= Y$ such that
* $f(X\_1) = Y\_1$, and
* $g(Y\_2) = X\_2$.
(This curious result is a consequence of the [Knaster-Ta... | https://mathoverflow.net/users/8628 | Curious decomposition between two sets | Yes, I think the exact same proof goes through for binary relations $R: X \nrightarrow Y$ and $S: Y \nrightarrow X$ (which of course includes the partial function case). Each induces a monotone operation between power sets, e.g. $\exists R: PX \to PY$ takes $A \subseteq X$ to $\{y: \exists\_{x \in A} R(x, y)\}$. Lettin... | 5 | https://mathoverflow.net/users/2926 | 278715 | 123,558 |
https://mathoverflow.net/questions/278720 | -4 | I am interested in asking the following question:
>
> What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation of his question from his slide presentation, ["Proof Theory for Set Theory"](http://www.helsinki.fi/lc2015/materia... | https://mathoverflow.net/users/20597 | Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? | $\sf ZF$ can prove that $\mathcal P(\omega)$ exists, but it cannot prove whether or not every subset of $\omega$ is constructible. In other words, there are sets which provably exist, but it is not provable that they are constructible.
At the same time, as Mohammad writes, if one can prove that there exists a non-co... | 11 | https://mathoverflow.net/users/7206 | 278723 | 123,560 |
https://mathoverflow.net/questions/267957 | 5 | Let $A$ and $B$ be unital $C^\*$-algebras. Let $u\in M\_n(A)$ be a unitary representing an element $[u]\in K\_1(A)$ and $p\in M\_m(B)$ be a projection representing an element $[p]\in K\_0(B)$. Then the unitary $$u\otimes p+1\_A\otimes (1\_B-p)$$ represets the product $[u]\times [p]\in K\_1(A\otimes B)$ (see in Nigel Hi... | https://mathoverflow.net/users/nan | explicit description of the product map in K-theory | I believe the correct formula is $$(u-1\_{A^+}) \otimes p + 1\_{(A \otimes B)^+}$$ which makes sense if you work this out in the following special case. Let $A$ be $C\_0(0,1)$ and let $B$ be $C(\{0,1\})$, and take $u(s)=e^{2\pi is}$ and $p(t)=t$. In this case $$((u-1\_{A^+}) \otimes p + 1\_{(A \otimes B)^+})(s,t)= (e^{... | 4 | https://mathoverflow.net/users/6133 | 278726 | 123,561 |
https://mathoverflow.net/questions/277586 | 5 | Let $\, f:M \to N$ be a smooth map, with rank $df \le r$ everywhere.
Does there exist a smooth map $\tilde f:M \to N$ of constant rank $r$, such that each level set of $\tilde f$ is contained in some level set of $f$?
Is it true at least locally?
(It is easy to see rank $df \le r$ is a necessary condition for the... | https://mathoverflow.net/users/46290 | Is every map of rank smaller than r dominated by a constant rank map? | No. The simplest case is $M = S^1$ and $N = \mathbb{R}$. Then any nonconstant map $f:M\to N$ has rank at most 1, but there is no smooth map from $M$ to $N$ that has constant rank $1$.
The question would be more interesting if you were considering $\tilde f: M\to \tilde N$ of rank at most $r$ instead of $\tilde f: M ... | 7 | https://mathoverflow.net/users/13972 | 278732 | 123,564 |
https://mathoverflow.net/questions/278675 | 3 | Consider $n$ iid observations $X\_1,X\_2,\dots ,X\_n$ from a $Uniform(a,b)$ distribution, where $a$ and $b$ are both unknown. How do we construct a joint confidence interval for $(a,b)$?
I would prefer a rectangular shape confidence interval, which can be obtained by using Bonferroni's method, so I guess the question... | https://mathoverflow.net/users/109950 | Confidence intervals for the endpoints of the uniform distribution | Let $Y\_i:=\frac{X\_i-a}{b-a}$, so that the $Y\_i$'s are iid from $U(0,1)$, and for the corresponding order statistics one has $X\_{(i)}=a+(b-a)Y\_{(i)}$. Let $R\_n:=X\_{(n)}-X\_{(1)}$, the ``sample range''. Then, for any real $c>0$
\begin{equation\*}
\alpha:=P(X\_{(1)}>a+cR\_n)
=P(Y\_{(1)}>(Y\_{(n)}-Y\_{(1)})c)=P(Y... | 4 | https://mathoverflow.net/users/36721 | 278743 | 123,568 |
https://mathoverflow.net/questions/278707 | 2 | Assume that $P\in \mathbb{C}[z]$ is a polynomial of degree $n$ with $n$ distinct roots $z\_1,z\_2,\ldots,z\_n$.
We identify $\mathbb{R}^3$ with $\mathbb{C}\times \mathbb{R}$. Put $a\_i=(z\_i,0)$.
Then $\mathbb{R}^3 \setminus \{a\_1,a\_2,\ldots,a\_n\}$ is the union of $\mathbb{R}^{3\geq0} \setminus \{a\_1,a\_2,\ldot... | https://mathoverflow.net/users/36688 | A line bundle on the wedge sum of spheres associated to a polynomial $P(z)\in \mathbb{C}[z]$ | OK, here's how I think it goes. No need for the residue theorem or anything like that.
Let $S\_k$ denote the sphere corresponding to $z\_k$. To compute the Chern class of the restriction of your line bundle to $S\_k$, it suffices to compute the degree of the map
$$
f:S^1\to S^1,\qquad f(z) = \frac{|P(z)|}{P(z)} = \f... | 1 | https://mathoverflow.net/users/8103 | 278746 | 123,570 |
https://mathoverflow.net/questions/278641 | 26 | My question essentially breaks down to
>
> How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you have?
>
>
>
For example, we can think of rotations by thinking, "Okay, let's just restrict to a plane fixed by th... | https://mathoverflow.net/users/58187 | Intuition for symplectic groups | In a different direction from Victor Protsak's answer, I will focus on your comment
>
> The next smallest example is for a 4-dimensional vector space, which
> isn't particularly easy for me to visualize.
>
>
>
It's true that the definition of the Lie group $\mathrm{Sp}(4,\mathbb{R})$ involves a $4$-dimensiona... | 14 | https://mathoverflow.net/users/1619 | 278748 | 123,571 |
https://mathoverflow.net/questions/278750 | 2 | Let $f:R\rightarrow S$ be a homomorphism of (commutative) rings with unity and suppose that $G$ is a group acting on $R$ and $S$ in such a way that $f\sigma=\sigma f$ for every $\sigma\in G$. Denote by $R^{G}$ and $S^{G}$ the respective rings of invariants of $G$, this is, $R^{G}=\left\{r\in R:\sigma r=r\textrm{ for ev... | https://mathoverflow.net/users/113244 | Is projectivity preserved by invariants? | That is not true. Let $G$ be a cyclic group with $2$ elements, $\{e,\sigma\}.$ Let $k$ be a field of characteristic different from $2$. Let $R$ be $k[x,y]$ with $$\sigma(x)=-x,\ \ \sigma(y)=-y.$$ Let $S$ be $R[z]/\langle z^2-1\rangle$ with $$\sigma(x)=-x,\ \ \sigma(y)=-y,\ \ \sigma(z)=-z.$$ Then $R^G$ equals $$k[u,v,w]... | 4 | https://mathoverflow.net/users/13265 | 278753 | 123,573 |
https://mathoverflow.net/questions/278699 | 20 | This question concerns the amount of information about a model $M$ that is contained in the collection of all reals Cohen over $M$.
Specifically, let $M$ and $N$ be countable transitive models of ZFC and suppose that they have the same collection of Cohen reals, i.e. any real $c\in V$ is Cohen over one of them iff it... | https://mathoverflow.net/users/1058 | Is a model of set theory determined by the Cohen reals over it? | Unless I'm missing something, don't you get a counterexample if $N=M[s]$ where $s$ is a Sacks real over $M$? The point is that every dense open subset of $2^{<\omega}$ in a Sacks extension includes such a dense open set in the ground model and so a Cohen real over $M$ remains Cohen over $M[s]$.
This is a standard fus... | 15 | https://mathoverflow.net/users/18128 | 278760 | 123,575 |
https://mathoverflow.net/questions/278762 | 6 | Consider the projectivization $\mathbb P\mathbb F\_p^n$ of $\mathbb F\_p^n$. How large a set $B \subseteq \mathbb P \mathbb F\_p^n$ can I pick so that no three points of $B$ lie on the same line?
| https://mathoverflow.net/users/113369 | Largest number of points one can pick in finite projective space without getting three on a line | The term for such sets is "caps". The problem you ask was posed by Bose ("Mathematical theory of the symmetrical factorial design", *Sankhyā* **8** (1947) 107–166), and is important in relation to coding theory: see Hill, *A first course in coding theory* (1986), around figure 14.9. At least at the time of Hill's book ... | 11 | https://mathoverflow.net/users/17064 | 278763 | 123,576 |
https://mathoverflow.net/questions/278741 | 1 | Let $0<t<1$ be a parameter. Let $n\in\mathbb{Z}$, $n>0$. For $i\in\{0,1,\dots,2^n-1\}$, we always consider $i$ as having $n$ binary digits (positions $1$ to $n$), putting $0$s if necessary. Let $f(i)$ be the number of times the $j+1$-th binary digit of $i$ is different from the $j$-th digit, $1\leq j\leq n-1$. Let $g(i... | https://mathoverflow.net/users/106551 | Limit of quotients of polynomials at fixed value | First, cancelling common factors we get
$$p(t,n) = \frac{ \sum\_{i=0\atop i\equiv 1\pmod{2}}^{2^n-1} \left(\frac{t}{1-t}\right)^{g(i)-f(i)} }{ \sum\_{i=0}^{2^n-1} \left(\frac{t}{1-t}\right)^{g(i)-f(i)} }$$
Clearly, $f(i)$ is the number of 2-mers "01" and "10", while $g(i)$ is the number of 2-mers $10$ and $11$ in the... | 5 | https://mathoverflow.net/users/7076 | 278764 | 123,577 |
https://mathoverflow.net/questions/278757 | 4 | Let $n$ be a positive integers and $T=T\_{n,n}$ be the $n\times n$ table in the first quadrant composed of $n^2$ unit squares, whose $(x,y)$-blank is locate in the $x^{th}$-column from the left and the $y^{th}$-row from the bottom hand side of $T\_{n,n}$ .
Put $D(n,n)$ be the number of all ... | https://mathoverflow.net/users/110804 | Hankel determinant evaluation of special lattice paths | The $d(n):=D(n,n)$ is OEIS sequence [A005773](http://oeis.org/A005773). The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd\_{n}=2nd\_{n-1}+3(n-2)d\_{n-2}$. A proof could come from a similar proof of the Hankel determinant property of the Catalan numbers. A related property is to us... | 6 | https://mathoverflow.net/users/113409 | 278765 | 123,578 |
https://mathoverflow.net/questions/278749 | 6 | It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. <https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples>). My question is:under which conditions does this spectral sequence degenerate at $E\_1$ (actually, I am more interested in neces... | https://mathoverflow.net/users/2191 | For which exact couples do associated spectral sequences degenerate at $E_1$? | Degenerating at $E\_1$ is, as you describe, equivalent to the image of the map $f\_1: D\_1 \to E\_1$ being contained in the image of $g\_1^i$ for all $i$. Roughly, this is because the definition of the $d\_r$-differential on a class $x$ is the equivalence class of any element $h\_1 g\_1^{1-r} f\_1(x)$, where $h\_1$ is ... | 3 | https://mathoverflow.net/users/360 | 278767 | 123,580 |
https://mathoverflow.net/questions/278706 | 4 | For $k>1$ ($k=2$ in particular for reason) and $\lim\limits\_{n \to \infty}\frac{a\_{n+1}}{a\_{n}}=1$ (it is $a\_{n} > 0$ as well, but that is not crucial) we compare these two series:
$\sum\limits\_{n=1}^{+\infty}(-1)^n ((\frac{a\_{n+1}}{a\_{n}})^k-1)$
$\sum\limits\_{n=1}^{+\infty}(-1)^n (\frac{a\_{n+1}}{a\_{n}}-1... | https://mathoverflow.net/users/nan | $\sum\limits_{n=1}^{\infty}(-1)^n ((\frac{a_{n+1}}{a_{n}})^2-1)$ converges.Does $\sum\limits_{n=1}^{\infty}(-1)^n (\frac{a_{n+1}}{a_{n}}-1)$ converge? | No, one can create sequences in which the first sequence converges but the second does not (or vice versa).
For sake of argument take $k=2$. To begin with let us ignore the requirement that the $a\_n$ be natural numbers. Let $\varepsilon\_m>0$ be a sequence of numbers tending very slowly to zero (e.g. $\varepsilon\_m... | 5 | https://mathoverflow.net/users/766 | 278773 | 123,584 |
https://mathoverflow.net/questions/278775 | 14 | For which $n$ is there a regular $n$-simplex with vertices in $\mathbb{Z}^n$ (or equivalently, in $\mathbb{Q}^n$)?
Some easy observations:
* Such an $n$-simplex exists with vertices in $\mathbb{Z}^{n+1}$: just take the $n+1$ points $(0, \ldots, 0, 1, 0, \ldots, 0)$.
* If $n$ is even and $n+1$ is not a perfect squar... | https://mathoverflow.net/users/644 | For which $n$ is there a regular $n$-simplex with vertices in $\mathbb{Z}^n$? | As shown in the accepted answer to [this question](https://mathoverflow.net/questions/38724/coordinates-of-vertices-of-regular-simplex), the Hadamard matrix condition is necessary and sufficent, so you have answered your own question...
**EDIT** As pointed out by Noam, I misread the linked-to question, and to atone: ... | 8 | https://mathoverflow.net/users/11142 | 278776 | 123,585 |
https://mathoverflow.net/questions/278786 | 24 | In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this moment, but this is what I understand:
**Anabelian geometry** tries to ask how much information about a variety is cont... | https://mathoverflow.net/users/85392 | How are motives related to anabelian geometry and Galois-Teichmuller theory? | Two clarifications:
For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group.
While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a spec... | 28 | https://mathoverflow.net/users/18060 | 278794 | 123,592 |
https://mathoverflow.net/questions/277445 | 9 | A morphism between schemes is a *universal homeomorphism* if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense.
It is well know that a universal homeomorphism between schemes induces:
(1) homeomorphism between their underlying topological spaces; ... | https://mathoverflow.net/users/112078 | Universal homeomorphism of stacks and etale sites | (1) is by definition. The standard proof of (2) is via descent of étale morphisms along universal submersions (see SGA1, Exp IX, Thm 4.10, or <http://stacks.math.columbia.edu/tag/04DY>, or my paper "Submersions and effective descent of étale morphisms", Thm 5.21). The point is that given a universal homeomorphism $f\co... | 4 | https://mathoverflow.net/users/40 | 278795 | 123,593 |
https://mathoverflow.net/questions/278782 | 3 | If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity?
$$
\operatorname{var} (X^T X)
=
\operatorname{var}(Z^T \Sigma Z)
$$
Moreover, if we are in the growing dimension regime, under what condition posed on $\Sigma$ (such... | https://mathoverflow.net/users/88033 | Multivariate normal concentration | Let $ Y := Z^T \Sigma Z $. We have $ \operatorname{var}(Y) = \mathbb{E}(Y^2) - \mathbb{E}(Y)^2 $ and $ \mathbb{E}(Y) = \sum\_i \sigma\_{i, i} $. We thus need to compute $ \mathbb{E}(Y^2) $. For this, write
$$ \mathbb{E}(Y^2) = \mathbb{E}\left( \sum\_{i, j, k, \ell} Z\_i Z\_j Z\_k Z\_\ell \sigma\_{i, j} \sigma\_{k, \ell... | 4 | https://mathoverflow.net/users/109373 | 278796 | 123,594 |
https://mathoverflow.net/questions/278755 | 1 |
>
> Let $A$ be a Banach algebra and let $\Gamma\_0, \Gamma\_1$ be circles of centres 0 and 1 respectively, each of radius less that $\frac{1}{2}$, which bound the two open disks $\Delta\_0$ and $\Delta\_1$.
>
>
> Furthermore, let $a \in A$ with $\text{Sp}(a)=\{0,1\}$. Here $\text{Sp}(a)$ denotes the spectrum of $a$... | https://mathoverflow.net/users/87670 | Help trying to show that $p_0a_1 =0$ | Apply [holomorphic functional calculus](https://en.wikipedia.org/wiki/Holomorphic_functional_calculus) to the following functions which are defined on a disconnected open set in the plane containing $\Gamma\_0 , \Gamma\_1$
$f(z)=\begin{cases} 1& \text{Around 0}\\0& \text{around 1}\end{cases}$
$g(z)=\begin{cases} 0&... | 1 | https://mathoverflow.net/users/36688 | 278799 | 123,595 |
https://mathoverflow.net/questions/278724 | 5 | Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions about the equivariant cohomology ring $H^\*\_G(\mathbb{P}(V),\mathbb{Q})$.
1. Under what conditions on the representat... | https://mathoverflow.net/users/87718 | Equivariant cohomology ring is an integer domain | It is very rare for these rings to be integral domains. To see this, put
$$ f\_V(t)=\sum\_kc\_k(V)t^{\dim(V)-k} \in H^\*(BG)[t]. $$
(All cohomology here has rational coefficients.)
It is then standard that $H\_G^\*(PV)=H^\*(BG)[x]/f\_V(x)$, and also that $f\_{V\oplus W}(t)=f\_V(t)f\_W(t)$. From this it is clear that ... | 7 | https://mathoverflow.net/users/10366 | 278800 | 123,596 |
https://mathoverflow.net/questions/278780 | 8 | It is consistent with ZFC that $2^{\aleph\_1}=2^{\aleph\_0}$. This can be gotten easily via forcing; more interestingly, it is a direct consequence of forcing axioms (which also set this value at $\aleph\_2$). However, just because a bijection exists between the subsets of $\omega$ and the subsets of $\omega\_1$ doesn'... | https://mathoverflow.net/users/8133 | Conflating reals and sets of countable ordinals "nicely" | The technique of almost disjoint forcing was introduced in
>
> [MR0289291 (44 #6482)](http://www.ams.org/mathscinet-getitem?mr=289291). Jensen, R. B.; Solovay, R. M. *Some
> applications of almost disjoint sets*. In **Mathematical Logic and
> Foundations of Set Theory** (Proc. Internat. Colloq., Jerusalem, 1968)... | 6 | https://mathoverflow.net/users/6085 | 278825 | 123,602 |
https://mathoverflow.net/questions/278820 | 9 | Let $X$ be a modular curve and $\mathcal{L}$ a line bundle on $X$. I found some literature regarding modular forms as global sections on $\mathcal{L}$. Under this context, my question is:
For any line bundle $\mathcal{L}$ on $X$ (given modular curve), can any global section of $\mathcal{L}$ be considered as a modular... | https://mathoverflow.net/users/44005 | Line bundles and modular forms | Write your curve as $\Bbb{H}/\Gamma $, with $\Gamma \subset PGL(2,\mathbb{R})$ acting freely on $\Bbb{H}$. The pull back of $\mathcal{L}$ to $\Bbb{H}$ is the trivial line bundle $\Bbb{H}\times \mathbb{C}$; this implies that $\mathcal{L}$ is the quotient of $\Bbb{H}\times \mathbb{C})$ by $\Gamma $ acting by $\gamma \cdo... | 15 | https://mathoverflow.net/users/40297 | 278827 | 123,603 |
https://mathoverflow.net/questions/278840 | 3 | Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields?
What is the current status, more generally?
Related [What is the current status of the function fields Langlands conjectures?](https://mathoverflow.net/questions/10578/wha... | https://mathoverflow.net/users/nan | Global Langlands function fields | The abstract of V. Lafforgue's paper <https://arxiv.org/abs/1404.6416> says
>
> For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction "from automo... | 9 | https://mathoverflow.net/users/18060 | 278842 | 123,606 |
https://mathoverflow.net/questions/278826 | 1 | Let $T\_1,T\_2\in \cal{A}$ with $\cal{A}$ is an algebra.
>
> Let $n\_1,n\_2\in \mathbb{N}$. Is it true that
> $$[T\_1^{n\_1},T\_2^{n\_2}]=\displaystyle\sum\_{\substack{\alpha+\alpha'=n\_1-1 \\ \beta +\beta'=n\_2-1}}T\_1^{\alpha}T\_2^{\beta}[T\_1,T\_2]T\_2^{\alpha'}T\_1^{\beta'}?$$
>
>
>
If the formula is fals... | https://mathoverflow.net/users/113054 | Commutator of the power of two elements | First try the case $n\_2 = 1$:
$$\eqalign{[T\_1^{n\_1}, T\_2] &= \sum\_{j=0}^{n\_1-1} (T\_1^{n\_1-j} T\_2 T\_1^{j} - T\_1^{n\_1-j-1} T\_2 T\_1^{j+1})\cr
&= \sum\_{j=0}^{n\_1-1} T\_1^{n\_1-j-1} [T\_1,T\_2] T\_1^{j}}$$
Similarly,
$$ [T\_1, T\_2^{n\_2}] = \sum\_{k=0}^{n\_2-1} T\_2^{n\_2-k-1} [T\_1,T\_2] T\_2^{k}$$
Subst... | 6 | https://mathoverflow.net/users/13650 | 278843 | 123,607 |
https://mathoverflow.net/questions/278761 | 8 | In <https://arxiv.org/abs/1010.4252>, Szabo defines a link invariant $\hat{H}(L)$ which can be computed combinatorially from a link diagram and shows that there is a spectral sequence from Khovanov homology to $\hat{H}(L)$. Conjecturally, this spectral sequence is isomorphic to the spectral sequence (proved by Ozsvath-... | https://mathoverflow.net/users/44651 | Intuition for Szabo's geometric spectral sequence | 1. The first intuition is that the differentials should depend on a choice of orientation of the surgery arcs (the first differential eventually turns out to be independent, anyways).
Let's do a split, with a single circle $C$ splitting into two, $C\_1$ and $C\_2$, by an elementary saddle $S$. Then the monopoles for ... | 9 | https://mathoverflow.net/users/48932 | 278850 | 123,610 |
https://mathoverflow.net/questions/278832 | 2 | This question is a continuation of the discussion
[Normalization of Hochschild cocycles](https://mathoverflow.net/questions/244092/normalization-of-hochschild-cocycles)
but this time in the cyclic context. I would like to ask whether the following is true:
>
> The inclusion of normalized cyclic cochains into all c... | https://mathoverflow.net/users/24078 | Normalization of cyclic cocycles | If I'm not mistaken this is just the dual version of what Loday and Quillen proves in Proposition 4.4 (or Proposition 2.2.14 in Loday's book, with the same proof). And sorry for nitpicking, but it's not exactly a quasi-iso, more precisely you get an exact sequence
$$
H^\*(\bar{C}^\*\_\lambda(A)) \to HC^\*(A) \to HC^\*(... | 2 | https://mathoverflow.net/users/9942 | 278854 | 123,612 |
https://mathoverflow.net/questions/278805 | 2 | Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$
The computation by hand is very complicated and long.
I would like to apply this possible software to calculate the Gaussian curvature described in the following posts:
[... | https://mathoverflow.net/users/36688 | Computer algebra for calculating curvature when the tensor metric is very big | Try SageManifolds <http://sagemanifolds.obspm.fr/>
See this example (there are several others) for how to compute the curvature tensor from the metric
<http://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v1.0/SM_Schwarzschild.ipynb>
*Hint*: it's just
```
R = g.riemann()
R.dis... | 8 | https://mathoverflow.net/users/78645 | 278857 | 123,613 |
https://mathoverflow.net/questions/278638 | 10 | Given a smooth manifold $M$ and a *relatively compact exhaustion* $M=\bigcup\_{n\in\mathbb N} M\_n$ with open and relatively compact $M\_n\subseteq M\_{n+1}$ (hence $M=\lim\limits\_{\to}M\_n$) do we always have $$H^k(M)=\lim\limits\_{\leftarrow} H^k(M\_n)?$$
This looks so natural that the answer should be known in wh... | https://mathoverflow.net/users/21051 | Do de Rham cohomologies commute with direct limits? | $\def\RR{\mathbb{R}}\def\Hom{\mathrm{Hom}}$This seems too simple, so please tell me what I'm missing. We first prove the corresponding result in homology. Let $C\_i(X)$ be the group of singular $i$-chains in $X$. Then $C\_i(M) = \lim\_{j \to \infty} C\_i(M\_j)$ (because $i$-simplices are compact, so the image of any fi... | 7 | https://mathoverflow.net/users/297 | 278859 | 123,614 |
https://mathoverflow.net/questions/278807 | 8 | A version of this question was previously asked on [MSE](https://math.stackexchange.com/q/2390608/237). I'll mention progress below.
A geometric construction I'm exploring
leads to a set $R$ of $n$ positive real numbers, for example:
$$
R = \{ \pi, e, \sqrt{2} \} \approx \{3.14159, 2.71828, 1.41421\} \;.
$$
Given som... | https://mathoverflow.net/users/6094 | Scaling a set of reals to be nearly integers | Brute force is maybe not that brutal. Of course I don't know what $\epsilon$ and $|R|$ and $\max{R}$ you want for your application and all those things could affect the performance. However, since you have an application, it is worth checking the performance of a simple program to see if solvers (which are great) need ... | 5 | https://mathoverflow.net/users/8008 | 278860 | 123,615 |
https://mathoverflow.net/questions/278841 | 12 | In the published version of HTT (Def 4.4.5.2) and [on the nlab](https://ncatlab.org/nlab/show/idempotent+complete+%28infinity%2C1%29-category) one finds one definition of a split homotopy coherent idempotent corepresented by a quasicategory $Idem^+\_\mathrm{old}$. A few months ago, Gal Dor [found](https://mathoverflow.... | https://mathoverflow.net/users/2362 | Did Lurie's model of a homotopy coherent idempotent change? | A bijection of simplices is not too hard. Just note that a chain of composable non-identity morphisms in $A^{\overset{i}{\to}}\_{\underset{r}{\leftarrow}} X \overset{e}{\to} X$ can be recovered from the list of morphisms, or from the list of objects which can be specified as indices of $X$'s. So the 3-simplex example $... | 6 | https://mathoverflow.net/users/112284 | 278873 | 123,617 |
https://mathoverflow.net/questions/278872 | 1 | Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.
A braid monoid is a monoid generated by $T\_1, \ldots, T\_n$ subject to the relations: $T\_i T\_j = T\_j T\_i$, $|i-j|>... | https://mathoverflow.net/users/11877 | Reference for a proof of cancellation property of braid monoids | In the book Braids and Self-Distributivity by Patrick Dehornoy, Chapter 2, [Proposition 4.5](https://books.google.com/books?id=XfsHCAAAQBAJ&pg=PA80) states that the positive braid monoid $B\_{n}^{+}$ is left-cancellative. Dehornoy actually proves left-cancellativity for monoids constructed from a coherent complement (P... | 2 | https://mathoverflow.net/users/22277 | 278874 | 123,618 |
https://mathoverflow.net/questions/278878 | 0 |
>
> Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$
> which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $?
>
>
> If yes, are there two non homotopic maps with this property?
>
>
>
For a related post see this question
[Continuous maps $f:S^n \to \mathbb{C}P^... | https://mathoverflow.net/users/36688 | Continuous orthogonal preserving maps between projective space | Suppose such a map exists, let $x\in\mathbb{C}^n-\{0\}$, we denote by $[x]$ its class in $\mathbb{C}P^n$. Let $H\_x$ be the orthogonal of $x$, $f([H\_x])$ is orthogonal to $f(x)$, so $f$ induces a map $f\_1:\mathbb{C}P^{n-1}\rightarrow \mathbb{C}P^{m-1}$ which has the same property, recursively, you obtain a map $f\_m:... | 3 | https://mathoverflow.net/users/80891 | 278880 | 123,619 |
https://mathoverflow.net/questions/278868 | 11 | There is famous Kolmogorov-Arnold theorem for continuous functions composition - continuous function of several variables can be composed of continuous functions of two variables.
Specialization of such theorem into smooth functions is false: there is no similar composition obeying smoothness - that is there are smo... | https://mathoverflow.net/users/3811 | Kolmogorov-Arnold theorem for (just-)functions | The answer is **yes** for functions $f:[0,1]^n\to\mathbb{R}$: Any such function can be written as
$$
f(x\_1,\dots,x\_n) = g\Big(\sum\_{i=1}^n h\_i(x\_i)\Big).
$$
The proof is pretty simple: The $h\_i$ should "spread out" the digits of the $x\_i$ by $n$ places such that $\sum\_{i=1}^n h\_i(x\_i)$ contains all the digits... | 9 | https://mathoverflow.net/users/9652 | 278889 | 123,621 |
https://mathoverflow.net/questions/278824 | 3 | Let $V$ be a vector space over a field $k$. Consider the natural action of $SL(V)$ on Sym$^2 V$. Is there a easy formulation of Hilbert Mumford Criterion for semi-stablity of the points of Sym$^2 V$ under this action of $SL(V)$?
For example there is an easy formulation of the semi-stability of the points of $Gr(r,n)$... | https://mathoverflow.net/users/nan | Hilbert Mumford Criterion | Here is my attempt for a $d$-dimensional $V$ over an algebraically closed field, like $\mathbb{C}$. Let's identify $\text{Sym}^2(V)$ with the space of all symmetric, contravariant, rank-$2$ tensors or identically the space of all degree $2$ homogeneous polynomials in $d$ variables $(x\_1, \cdots , x\_d) \equiv \mathbf{... | 1 | https://mathoverflow.net/users/98093 | 278896 | 123,626 |
https://mathoverflow.net/questions/278901 | 30 | Motivation: Surfaces
--------------------
Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy equivalent. This means that somehow the homotopy type of the surface *contains essentially all information ... | https://mathoverflow.net/users/13767 | A manifold is a homotopy type and _what_ extra structure? | You are talking about the (much studied) Poincare duality spaces. For a survey, see the [very nice one by John Klein:](http://www.math.wayne.edu/~klein/survey-apr10.pdf) (seems to be unpublished, but dates to April 2010).
| 21 | https://mathoverflow.net/users/11142 | 278903 | 123,629 |
https://mathoverflow.net/questions/278862 | 5 | A wheeled properad is roughly, if I understand correctly, a properad (or PROP) with contraction maps $O\_i^j\to O\_{i-1}^{j-1}$ which contract an input with an output. There is a book, *[Infinity Properads and Infinity Wheeled Properads](https://arxiv.org/abs/1410.6716)* by Hackney, Robertson and Yau, which defines a c... | https://mathoverflow.net/users/7108 | Model structure on wheeled topological properads | I can only answer the first half of your question. I defined the notion of a topological wheeled properad in the preprint *[Dwyer-Kan Homotopy Theory of Algebras over Operadic Collections](https://arxiv.org/abs/1608.01867)*. As usual the ambient category can be any bicomplete symmetric monoidal closed category. You can... | 3 | https://mathoverflow.net/users/53034 | 278917 | 123,631 |
https://mathoverflow.net/questions/278679 | 11 | Suppose $x$ is a word over the alphabet $\{0,1\}$.
Let $a$, $b$ be elements of the group Dih$\_k$ for some $k$.
Let $\varphi=\varphi\_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the dihedral group Dih$\_k$ (having $2k$ elements) such that $\varphi(0)=a$, $\varphi(1)=b$, and $\varphi$ takes concatenat... | https://mathoverflow.net/users/4600 | Unique words in dihedral groups | The conjecture holds **true** and I don't know of anything similar.
Given a word $w$ over $\{0, 1\}$, we denote by $\overline{w}$ the word obtained from $w$ by interchanging $0$ and $1$.
Let us show first that the elements of $S \cup T$, where $T = \overline{S}$, are dihedrally simple.
>
>
> >
> > **Claim 1.** ... | 9 | https://mathoverflow.net/users/84349 | 278923 | 123,634 |
https://mathoverflow.net/questions/278922 | 20 | As we know, there are lots of consequences with the presupposition of the Riemann Hypothesis.
Similarly, are there any important consequences with the presupposition of $\mathbf{P} \neq \mathbf{NP}$ ?
An alternative statement of $\mathbf{P} \neq \mathbf{NP}$ is the extended Church-Turing Thesis. So if we have an sp... | https://mathoverflow.net/users/14024 | Any important consequences with presupposition of $\mathbf{P} \neq \mathbf{NP}$ | Because there are natural computational problems involving many mathematical objects, there are a bunch of implications of complexity class separations like $\mathrm{P} \neq \mathrm{NP}$. I think the first paper to investigate this idea is probably Mike Freedman's *[Complexity classes as mathematical axioms](http://ann... | 28 | https://mathoverflow.net/users/97414 | 278935 | 123,639 |
https://mathoverflow.net/questions/278906 | 2 | Let $G=(V,E)$ be a simple undirected graph. Define an *mmd$k$s in $G$* (for 'maximal minimum degree $k$ subset') to be any subset $S$ of $V$ such that
1. the subgraph induced by $S$ in $G$ has minimum degree $\geq k$,
2. $S$ is $\subseteq$-maximal w.r.t. property 1.
Moreover, for any $k$, let $\mathrm{smmds}(G,k):... | https://mathoverflow.net/users/113490 | Subsets of a graph, maximal w.r.t. the property of inducing a subgraph with minimum degree at least $k$ | According to [Wikipedia](https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)#k-Cores), a *$k$-core* of a graph $G$ is a maximal connected subgraph $K$ of $G$ such that every vertex of $K$ has degree at least $k$. Apart from the "connected" part of the definition, this is the same as your set $S\_k$. Your set $S\_k$... | 5 | https://mathoverflow.net/users/39146 | 278940 | 123,640 |
https://mathoverflow.net/questions/182679 | 7 | I originally posted [this on math.stackexchange](https://math.stackexchange.com/questions/867507/is-there-a-generalized-cauchy-binet-formula-where-some-rows-and-columns-are-forc), but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now... | https://mathoverflow.net/users/353 | Generalized Cauchy-Binet sum over a fixed subset of indices | This is probably a bit late, but the following formula is proved in [this paper](https://arxiv.org/pdf/1704.04405.pdf "this paper") (Eq. S4 in the Supplementary Material):
\begin{equation}
\sum\_{S \in \binom{[n - j]}{m - j}} \det(A\_{[m], S \cup T}) \det(B\_{S\cup T, [m]}) = (-1)^{j} \det \left(\begin{array}{c c} \m... | 6 | https://mathoverflow.net/users/113506 | 278941 | 123,641 |
https://mathoverflow.net/questions/278817 | 5 | Let $T$ be a r.e. and consistent extension of **PA** in the language of artihmetic. Are there examples of arithmetical sentences $\phi, \psi$ such that $T+\phi \equiv\_{\Pi^0\_1} T+ \psi$ but $T \not \vdash Con\_{\tau + \phi} \leftrightarrow Con\_{\tau + \psi}$, where $\tau$ numerates the $T$-axioms in $T$ and $\tau + ... | https://mathoverflow.net/users/113441 | Unprovable cases of $\Pi^0_1$-conservativity | I'd say that this situation is indeed possible. In general, the $\Pi\_1^0$-conservativity of one theory over another does not guarantee that a proof of their relative consistency can be formalised in $PA$. This observation is due to Guaspari ([Partially conservative extensions of arithmetic](https://doi.org/10.1090/S00... | 5 | https://mathoverflow.net/users/103227 | 278949 | 123,643 |
https://mathoverflow.net/questions/278845 | 1 | Given an ergodic Markov chain and 2 states $x$ and $y$, how may one algorithmically sample a path (of finite length) $x =: s\_0 \rightarrow s\_1 \rightarrow \ldots \rightarrow s\_T = y$ between $x$ and $y$ ?
**Poorman's solution:** Let $N$ be the transition operator for the chain.
Then the process $s\_{t + 1} \sim N(... | https://mathoverflow.net/users/78539 | How to sample a path between 2 states in a Markov chain | As observed by a MO user (Nate Eldredge), the naive solution that I provided alongside the question is essentially optimal as it's linear in the length of the chain.
**Solving the generalization:**
As observed by another user (Ori Gurel-Gurevich), the following simple procedure solves the second part of the question ... | 1 | https://mathoverflow.net/users/78539 | 278953 | 123,644 |
https://mathoverflow.net/questions/278899 | 2 | $\DeclareMathOperator{\Id}{Id}
\require{cancel}$
Jet Nestruev's "Smooth Manifolds and Observables" contains following exercise:
>
> **Exercise.** Show that $P$ is geometric if and only if the two modules $P$ and $\Gamma(P)$ are isomorphic.
>
>
>
It is clear that there is an injective map from $P$ to $\Gamma(P)... | https://mathoverflow.net/users/62635 | Why is $C^\infty(M)$-module homomorphism $P\mapsto\Gamma(P)$ surjective? | The exercise is fine. I misread the definition of $\Gamma(P).$ Jet Nestruev defines $\Gamma(P)$ as an image $\phi(P).$ Hence surjectivity follows obviously from the definition of $\Gamma(P)$.
| 2 | https://mathoverflow.net/users/62635 | 278956 | 123,645 |
https://mathoverflow.net/questions/278946 | 6 | This is a problem that I have been stuck for a few months.
Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that
$$\|(I+A+B)^{-1}A\|\le 1.$$
If it is, can you suggest any reference to me?
---
I was able to show $\|(I+A+B)^{-1}A\|\l... | https://mathoverflow.net/users/91196 | Norm estimation of identity plus two non-commuting self-adjoint operators | The claim is **false.** Consider the following matrix argument.
\begin{eqnarray\*}
\|(I+A+B)^{-1}A\| \le 1\quad\Leftrightarrow\quad \begin{bmatrix}I & (I+A+B)^{-1}A \\ A(I+A+B)^{-1} & I \end{bmatrix} \ge 0.
\end{eqnarray\*}
The latter inequality amounts to showing $I \ge (I+A+B)^{-1}A^2(I+A+B)^{-1}$, or in other wo... | 9 | https://mathoverflow.net/users/8430 | 278959 | 123,646 |
https://mathoverflow.net/questions/270177 | 2 | After answering another question ([The number of values of $f(x)/x$ when $f$ is a linearized polynomial](https://mathoverflow.net/questions/268339/the-number-of-values-of-fx-x-when-f-is-a-linearized-polynomial)), I stumbled upon an interesting polynomial in multiple variables. Let $\mathbb{F}\_q$ be the field of $q$ el... | https://mathoverflow.net/users/44191 | A polynomial in multiple variables with nice properties | This is a CW answer to remove this question from the unanswered list (once someone upvotes it). This is the determinant of the [Moore matrix](https://en.wikipedia.org/wiki/Moore_matrix) $\left( x\_i^{q^{j-1}} \right)\_{1 \leq i,j \leq n}$. This determinant can be expressed as a product of linear factors:
$$\det \left(x... | 6 | https://mathoverflow.net/users/297 | 278962 | 123,648 |
https://mathoverflow.net/questions/278955 | 6 | The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere.
I am wondering if this result can be generalized as follows. Let $X$ be an Alexandrov space with curvature bounded below of finite dimension $n$. It is well known that at almost e... | https://mathoverflow.net/users/16183 | Rademacher type theorem for Alexandrov spaces | I assume you are interested in the finite-dimensional case.
You can write this function in a distance chart (these charts cover almost all points). Apply the standard Rademacher theorem and notice that the distance chart is differentiable almost everywhere. Hence the result follows.
| 6 | https://mathoverflow.net/users/1441 | 278964 | 123,649 |
https://mathoverflow.net/questions/266648 | 20 | The collection of
*binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$*
forms a Borel set, neither closed nor open -- assuming Con(ZFC).
* Can you show it's not $F\_\sigma$ or $G\_\delta$?
* Is it actually complete for level $\omega$ of the Borel hierarchy?
| https://mathoverflow.net/users/4600 | Models of ZFC and the Borel hierarchy | @FrançoisG.Dorais commented:
>
> By reading the axioms (i.e. every axiom is satisfied, as a formalized statement) it's a $\Pi^0\_{\omega+1}$ set (since satisfying a first-order sentence is $\Pi^0\_n$ for some $n$). I think that it's not $\Pi^0\_n$ for $n<\omega$ amounts to the fact that ZFC is not finitely axiomati... | 1 | https://mathoverflow.net/users/4600 | 278997 | 123,656 |
https://mathoverflow.net/questions/278915 | 2 | Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f\_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f\_J(x)$. Suppose the initial distribution $\rho\_{X(0)}(x)$.
I would like to prove (or correct) the following intuitive expression for the survival probability above a ce... | https://mathoverflow.net/users/111000 | CTRW: solve a renewal equation | Using my previous remark (namely the correction of the operator $ \mathcal{O} $), an expression would just involve the "skeleton" of your continuous time random walk, namely the walk with initial law $ \rho\_{X\_0} \equiv \rho\_0 $ and jump law $ f\_J $, or, $ Y\_k = R\_0 + \sum\_{m = 1}^k J\_m $. Then, this survival p... | 2 | https://mathoverflow.net/users/109373 | 278998 | 123,657 |
https://mathoverflow.net/questions/278989 | 23 | Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be *of finite order* if $\phi^m = \mathrm{id}\_{\mathbf{R}^n}$ for some positive integer $m$.
**Question**: Does every finite order self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ h... | https://mathoverflow.net/users/38052 | Finite-order self-homeomorphisms of $\mathbf{R}^n$ | There is, naturally, a huge history regarding a basic question like this. This particular problem was figured out between 1930 and the early 1960's. The main names are P.A. Smith, Conner, Floyd. Here is a math review to get you going:
[MR0130929](http://www.ams.org/mathscinet-getitem?mr=0130929) (24 #A783) Reviewed
K... | 25 | https://mathoverflow.net/users/102519 | 279000 | 123,658 |
https://mathoverflow.net/questions/271878 | 3 |
>
> **Question 1.** Is a correct proof of Leray's theorem (the one that says that
> a connected graded Hopf algebra $H$ over a field of characteristic $0$ is
> isomorphic as an algebra to the symmetric algebra $\operatorname\*{Sym}\left(
> H^{+}/\left( H^{+}\right) ^{2}\right) $, where $H^{+}=\operatorname\*{Ker}
>... | https://mathoverflow.net/users/2530 | Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore? | I forgot to give this question some closure. Basically, it was answered by @JohnPalmieri in his comment; let me just expand this answer.
The proof of Proposition 4.17 in the Milnor/Moore paper [B] is correct (at least the part that I was having troubles with). The argument is rather surprising and seriously uses the ... | 1 | https://mathoverflow.net/users/2530 | 279002 | 123,660 |
https://mathoverflow.net/questions/278983 | 7 | The clasical Bers' theorem about pants decomposition says that any compact Riemann surface of genus $g \geq 2$ has a pants decomposition such that every cutting geodesic in this decomposition is of length $\leq \mathcal{B}\_g$, where $\mathcal{B}\_g$ is a constant (so-called Bers' constant) depending only on $g$. There... | https://mathoverflow.net/users/85466 | Bers' constant for compact hyperbolic surfaces with geodesic boundary | Balacheff, Parlier, and Sabourau proved that Bers's theorem also holds for arbitrary complete Riemannian metrics, as long as you rescale appropriately: there's a $C\_{g,n}$ such that a complete surface of genus $g$ with $n$ ends and area $A$ has a pants decomposition where each curve has length at most $C\_{g,n}\sqrt{A... | 6 | https://mathoverflow.net/users/25051 | 279003 | 123,661 |
https://mathoverflow.net/questions/278280 | 5 | Gromov's waist inequality for unit n-sphere $\mathbb{S}^{n}$ says:
For any continuous function $f: \mathbb{S}^{n} \rightarrow \mathbb{R}^{m} $,
there is some $y \in \mathbb{R}^{m}$ s.t. $Vol\_{n-m}(f^{-1}(y)) \geq Vol\_{n-m}(\mathbb{S}^{n-m}) $.
I'm wondering if there is an averaged version of the inequality,
comp... | https://mathoverflow.net/users/105627 | An Averaged Version of Gromov's Waist Inequality | My guess would be that there are ways to manipulate the average -- say you take the unit sphere in $\mathbb{R}^3$, and a function like $f(x,y,z)=x^{100}$. Then the image is the interval $[-1,1]$, but the preimage of a random point is small. The [coarea formula](https://en.wikipedia.org/wiki/Coarea_formula) is one way a... | 2 | https://mathoverflow.net/users/25051 | 279004 | 123,662 |
https://mathoverflow.net/questions/278968 | 4 | In a literature, I found the following object: let $k$ is a field of characteristic zero and $U=\mathbf{P}^1\_k\setminus\{0,1,\infty\}$, then there are *residue* maps
$\mathrm{Res}\_i:H^1(U,\mathbb{Q}\_p(1))\to\mathbb{Q}\_p$
for $i=0,1$. I could not imagine what these map should be. So, I would like to ask what the... | https://mathoverflow.net/users/44005 | General construction of residue map | Composing with pullback to the base change over the algebraic closure of $k$, the real content is with $k$ an algebraically closed base field (of char. 0), $U$ a non-empty proper open subset of a smooth proper connected curve $X$ over $k$, and $x \in X(k) - U(k)$ a choice of missing point: given such data one seeks to ... | 3 | https://mathoverflow.net/users/81332 | 279008 | 123,664 |
https://mathoverflow.net/questions/278954 | 10 | First let me define a few notions to phrase my question simply. Say a regular cardinal $\kappa$ is **threadable** if the threaded square $\Box(\kappa)$ fails, and **$\alpha$-threadable** if $\Box(\kappa,\alpha)$ fails, so 1-threadable is simply threadable. Now furthermore say that $\kappa$ is **$\alpha$-reflecting** if... | https://mathoverflow.net/users/38602 | Does/can threadability imply reflection principles? | To address your special case: threadable cardinals are not necessarily 1-reflecting. In fact, the assertion, "$\kappa$ is $\alpha$-threadable for every $\alpha$ such that $\alpha^+ < \kappa$" does not imply that $\kappa$ is 1-reflecting. To see this, suppose that $\kappa$ is a weakly compact cardinal whose weak compact... | 8 | https://mathoverflow.net/users/26002 | 279022 | 123,670 |
https://mathoverflow.net/questions/278888 | 8 | I'm interested in knowing more about the volume form canonically induced by a Finsler metric.
I've found some reasoning about it in this article <http://www.ams.org/journals/bull/1950-56-01/S0002-9904-1950-09332-X/home.html> but I was wondering if someone could point out a more recent source with results explained in... | https://mathoverflow.net/users/60675 | Volume form induced by a Finsler metric | In fact there are very many ways to provide a Finsler manifold with a "canonical" volume. Personally I've gone from thinking that this is a nuisance and trying to pin down which one is really the best to thinking that this is part of the landscape and should be accepted.
There is a very good notion of volume that go... | 10 | https://mathoverflow.net/users/21123 | 279026 | 123,673 |
https://mathoverflow.net/questions/279031 | 3 | Given Holder space $C^{\alpha}(\mathbb{R}^n)$, does there exist a Banach space $X$ such that the dual of $X$ is $C^{\alpha}(\mathbb{R}^n)$?
What I can imagine is that such $X$ must contain the fractional sobolev space $W^{-s,p}$ with $s>0$ and $sp' \ge n+\alpha$, where $p'$ is the conjugate number of $p$. This is bec... | https://mathoverflow.net/users/51546 | What's the predual of Holder continuous function spaces? | Yes, indeed every Lipschitz space is a dual space, a fact which has been rediscovered (in varying levels of generality) several times. The earliest proof is due to Arens and Eells.
Holder spaces are special cases of Lipschitz spaces because a function is $\alpha$-Holder continuous for a metric $\rho$ if and only if i... | 5 | https://mathoverflow.net/users/23141 | 279035 | 123,676 |
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