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https://mathoverflow.net/questions/288384 | 4 | Let $G$ be a finite group of exponent $n$ and let $d\mid n$. Consider the class function
$$
f(g) =
\begin{cases}
1 & g^d =1\\0&\textrm{otherwise}.
\end{cases}
$$
As a class function $f$ can be written as $f= \sum\_{\chi} c\_{\chi} \chi$, with
$$
c\_{\chi} =\left<f,\chi \right> = \frac{1}{|G|} \sum\_{g^d=1} \chi(g)
$... | https://mathoverflow.net/users/2042 | How to explicitly expand a class function in terms of irreducible characters? | This is a partial answer. Frobenius investigated such Fourier coefficients to prove that the number of solutions to $x^d=1$ is divides the order of the group. He showed your class function is a virtual character, if I understood correctly, so all your Fourier coefficients are integers. The paper of Frobenius is in Germ... | 4 | https://mathoverflow.net/users/15934 | 288433 | 127,243 |
https://mathoverflow.net/questions/288271 | 3 | Let $E$ be a complex Hilbert space.
>
> Let $(A\_1,...,A\_n) \in \mathcal{L}(E)^n$, could you please help me to show that
> $$\displaystyle\sup\_{\|x\|=1}\bigg(\displaystyle\sum\_{i=1}^n\| A\_i^\*x\|^2\bigg)\leq (4n)\displaystyle\sup\_{\|x\|=1}\bigg(\displaystyle\sum\_{i=1}^n|\langle A\_ix\;,\;x\rangle|^2\bigg).$$... | https://mathoverflow.net/users/113054 | inequality involving tuple of operators on Hilbert spaces | Notice that
$$ \|[A\_1 \dots A\_n]\| = \|[A\_1 \dots A\_n]^\*\| = \left\| \left[\begin{smallmatrix} A\_1^\* \\ \\\vdots \\ A\_n^\* \end{smallmatrix}\right] \right\| = \sup\_{\|x\|=1} \left( \sum\_{i=1}^n \|A\_i^\*x\|^2 \right)^{1/2}.$$
By the answer to [this question](https://mathoverflow.net/questions/285471/a-numer... | 1 | https://mathoverflow.net/users/76593 | 288436 | 127,244 |
https://mathoverflow.net/questions/250752 | 1 | An interesting class of contact manifolds is the class of $K$-contact manifolds ($\mathcal{L}\_\xi g=0$) which have been studied by many authors. It is natural to study conformal Killing-contact manifolds. This means that the characteristic vector field satisfies the conformal killing equation; i. e. $\mathcal{L}\_\xi ... | https://mathoverflow.net/users/90655 | Conformal Killing vector field on contact manifolds | The reason is that the g-trace of the conformal Killing equation gives 2(divergence of characteristic vector field)=n(sigma). But divergence of characteristic vector field on a contact metric manifold is zero. Hence, sigma = 0. Thus it reduces to the K-contact manifold.
| 1 | https://mathoverflow.net/users/118539 | 288440 | 127,246 |
https://mathoverflow.net/questions/288429 | 7 | In Shioda's famous paper "[An Example of Unirational Surfaces in Characteristic $p$](https://link.springer.com/article/10.1007%2FBF01350715)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=374149)), the author proved that the Fermat surface over the characteristic-$p$ field $k$
$$
x\_1^n +x\_2^n + x\_3^n+x\_4^... | https://mathoverflow.net/users/101139 | Unirationality of Fermat varieties in characteristic $p$ | Let $F^r$ be the Fermat hypersurface of fixed degree $n$ in $\mathbb{P}^{r+1}$, so that $\dim(F^r)=r$ (in any characteristic). There is a rational dominant map $\varphi :F^r\times F^s \cdots\!\!\twoheadrightarrow F^{r+s}$: if you write the equation of $F^r$ in affine coordinates $x\_1^n+\ldots +x^n\_{r+1}=-1$, and that... | 6 | https://mathoverflow.net/users/40297 | 288441 | 127,247 |
https://mathoverflow.net/questions/288410 | 47 | Quoting his [Wikipedia page](https://en.wikipedia.org/wiki/Srinivasa_Ramanujan) ([current revision](https://en.wikipedia.org/w/index.php?title=Srinivasa_Ramanujan&oldid=815168011)):
>
> compiled nearly 3,900 results
>
>
> Nearly all his claims have now been proven correct
>
>
>
Which of his claims have been ... | https://mathoverflow.net/users/118526 | What did Ramanujan get wrong? | Hardy wrote some things about this, as I learned when writing [this blog post](https://qchu.wordpress.com/2016/05/08/the-man-who-knew-elliptic-integrals-prime-number-theorems-and-black-holes/). Here is a mistake which was even featured in the Ramanujan movie: in his letters to Hardy, Ramanujan claimed to have found an ... | 47 | https://mathoverflow.net/users/290 | 288448 | 127,250 |
https://mathoverflow.net/questions/288455 | 3 | Wikipedia gives [the following explicit formula](https://en.wikipedia.org/wiki/Matrix_function#Arbitrary_function_of_a_2x2_matrices) for the functional calculus of $2\times2$ matrices:
$$
f(A) = \frac{f(\lambda\_+) + f(\lambda\_-)}{2} I + \frac{\mathrm{tr}(A)/2 - \mathrm{adj}(A)}{\sqrt{\Delta(A)}} \frac{f(\lambda\_+) -... | https://mathoverflow.net/users/1849 | Explicit formula for the functional calculus of 2x2 matrices | A general procedure for $f(A)$ for any $n \times n$ matrix $A$, where $f$ is an analytic function in a neighbourhood of the spectrum of $A$, is this. Let
$p$ be a rational function such that $p(\lambda) = f(\lambda)$ for every
eigenvalue $\lambda$ of $A$, and $p^{(k)}(\lambda) = f^{(k)}(\lambda)$ for $k \le d(\lambda)... | 6 | https://mathoverflow.net/users/13650 | 288457 | 127,253 |
https://mathoverflow.net/questions/288462 | 3 | What is the number of terms of the unique multilinear polynomial $f\in\Bbb F\_2[x\_{1,1},\dots,x\_{n,n}]$ in $n^2$ variables such that $f$ vanishes only on matrices that are permutations?
Are there good bounds?
Note that such a polynomial should have $S\_n\times S\_n$ symmetry.
I think one can even guess how such... | https://mathoverflow.net/users/10035 | What does this permutation polynomial look like? | For any permutation $\sigma\in S\_n$ let's define $X(\sigma)=\prod\_{i=1}^n x\_{i\sigma(i)}$ and $Y(\sigma)=\prod\_{i=1}^n\prod\_{j\neq\sigma(i)}(1+x\_{ij})$.
---
**Lemma:** Our polynomial can be written explicitly as
$$f(x\_{ij})=1+\sum\_{\sigma\in S\_n}X(\sigma)Y(\sigma)$$
To prove this notice that $X(\sigma)Y... | 8 | https://mathoverflow.net/users/2384 | 288468 | 127,258 |
https://mathoverflow.net/questions/288452 | 3 | Suppose $X$ is a singular variety over a field $k$, which admits an action by a finite group $G$. Suppose the quotient $X/G$ is also a variety over $k$. If $Y\_G$ is a resolution of $X/G$, does there exist a variety $Y$ which is a resolution of $X$ such that the action on $X$ could be extended to an action on $Y$ and
... | https://mathoverflow.net/users/87910 | Is the quotient of resolution the same as resolution of the quotient? | As I said in the comments, the answer is no in general. Indeed, let $X = \mathbb{P}^2\times \mathbb{P}^2$ and let $G=\mathbb{Z}/2\mathbb{Z}$ act by permuting the factors. Then $X$ is smooth and $X/G$ is singular. If $(X/G)'\to X/G$ is a resolution of singularities, then $(X/G)' \not \cong X/G$ (clearly).
But, if $G$ ... | 3 | https://mathoverflow.net/users/4333 | 288480 | 127,259 |
https://mathoverflow.net/questions/288456 | 6 | So it's well-known that an alternative way to define a series-parallel (undirected graph) is by the forbidden minor $K\_4$. Is there a known analog of this definition for directed graphs — specifically, for DAGs?
| https://mathoverflow.net/users/9084 | Characterizing SP-DAGs by Forbidden Minors? | You might want to have a look at the paper
Jacobo Valdes, Robert E. Tarjan, Eugene L. Lawler; The recognition of Series Parallel digraphs; STOC 1979; doi:[10.1145/800135.804393](https://doi.org/10.1145/800135.804393)
Section 4 is called "Forbidden subgraph characterizations", and it contains the following two stat... | 4 | https://mathoverflow.net/users/12674 | 288482 | 127,260 |
https://mathoverflow.net/questions/288476 | 4 | I was reading the paper on [arixv](https://arxiv.org/abs/1312.5259v1).
I was confused the equation of nth moment of Poisson distribution.
The detail and partial paper as follow:
...
>
> For large N, this connection probability takes the form [28]
> $$\Pi\_T (a,a^\prime) = 1 − exp[−λ(a + a^\prime)]$$, (1)
> ... | https://mathoverflow.net/users/90343 | an application of nth moment of Poisson distribution with stirling number | It is the *conditional* distribution $g(k|a)$ that has a Poisson distribution, with mean $\lambda(T,a)=T(a+\langle a\rangle)$. So the *unconditional* average
$$\langle k^n\rangle = \sum\_a F(a)\langle k^n|a\rangle,$$
with
$$\langle k^n|a\rangle\equiv\sum\_k k^n g(k|a)=\sum\_m\genfrac{\{}{\}}{0pt}{}{n}{m}\lambda^m$$
i... | 2 | https://mathoverflow.net/users/11260 | 288484 | 127,261 |
https://mathoverflow.net/questions/288435 | 2 | Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and $\mu^u$ respectively. Given any simple closed curve $a$ let $I(\mu^s(a))=min\{\mu^s(\alpha): \alpha \text{ is in the hom... | https://mathoverflow.net/users/9485 | Length of a simple closed curve under Pseudo-Anosov maps | To answer the main question as well as the question in the comments, for every simply closed curve $a$ we have $I(\mu^s(a)) \in (0,\infty)$, and every $n$ we have $I(\mu^s(f^n(a))) = \lambda^{-n} I(\mu^s(a))$. It follows that $I(\mu^s(f^n(a)))$ limits to $0$ as $n \to+\infty$ and to $+\infty$ as $n \to -\infty$. So the... | 5 | https://mathoverflow.net/users/20787 | 288491 | 127,265 |
https://mathoverflow.net/questions/288469 | 3 | I'm not sure this question is suitable for MathOverflow. Currently, I'm reading a paper "[Inhomogeneous Dirichlet Problem in Lipschitz domain](https://doi.org/10.1006/jfan.1995.1067)" by Jerison and Kenig.
I have a question on some inequality on Bessel potential space and Besov space.
For convenience, let us fix so... | https://mathoverflow.net/users/88462 | An inequality from Bessel potential space to Besov space | I think that your idea is completely correct, but the choice of $s$ and $q$ is indeed somewhat curious in the paper. However, the case $q=p$ should be sufficient for the proof to work:
We need $s$ and $q$ such that
1. $L^q\_s(\Omega)$ continuously embeds into $B^p\_\alpha(\Omega)$, and
2. Corollary 3.11 is useable... | 1 | https://mathoverflow.net/users/85906 | 288503 | 127,268 |
https://mathoverflow.net/questions/284242 | 3 | Let $1 \leq \nu \leq n+1$ and $M^n \subseteq \Bbb S^{n+2}\_\nu$ be a non-degenerate submanifold. Assume that $\renewcommand{\vec}[1]{{\bf #1}} \vec{L}\_0 \in \Bbb R^{n+3}\_\nu$ is lightlike and $M \subseteq \vec{L}\_0^\perp \cap \Bbb S^{n+2}\_\nu$.
>
> Is is true that $M$'s Second Fundamental Form is always lightl... | https://mathoverflow.net/users/54656 | Does every submanifold of $\Bbb S^{n+2}_\nu$ contained in a lightlike hyperplane have lightlike Second Fundamental Form? | Might as well answer it now. The point is that $\vec{L}\_0$ is not tangent to $\Bbb S^{n+2}\_\nu$ at every point, but it is tangent at the points of $\vec{L}\_0^\perp \cap \Bbb S^{n+2}\_\nu$. Because of this, $\vec{L}\_0$ being normal to $M$ ensures that the metric induced in the normal spaces to $M$ relative to $\Bbb ... | 0 | https://mathoverflow.net/users/54656 | 288519 | 127,273 |
https://mathoverflow.net/questions/288485 | 5 | In his biography of Gauss, G. Waldo Dunnington describes the Gauss-Bolyai episode and their correspondence. In particular, he describes the contents of one letter from Gauss to Janos-Bolyai:
>
> In the above mentioned letter Gauss gave as a sample of his own research a proof that in non-euclidean geometry the area ... | https://mathoverflow.net/users/118562 | How to derive from Gauss's results on the volume of hyperbolic orthoscheme tetrahedron the formula of Bolyai? | I am not sure of the notation, but I assume this can be derived from the Schlafli formula for the volume of a tetrahedron (so this seems to indicate that Gauss knew Schlafli's formula three quarters of a century prior to Schlafli):
$$
dV = -\frac12 \sum\_{ij}l\_{ij} d \alpha\_{ij},$$ where $l$ is the length of the ed... | 7 | https://mathoverflow.net/users/11142 | 288522 | 127,275 |
https://mathoverflow.net/questions/288529 | 3 | Let $n$ be a positive integer. For each $k = 1, \ldots, n$, let
$$
S\_k(n) := \sum\_{1 \le i\_1 < \cdots < i\_k \le n} i\_1 \cdots i\_k
$$
be the sum of the $k$-wise product of distinct integers from $1$ to $n$. Does this sum $S\_k(n)$ have a name and formula?
For example, $$S\_1(n) = 1 + \cdots + n = \binom{n + 1}{2... | https://mathoverflow.net/users/nan | Name of the sum of the $k$-wise product of distinct integers from $1$ to $n$ | These are the [Stirling numbers of the first kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind). They enumerate, among other things, the number of permutations with a given number of cycles.
| 12 | https://mathoverflow.net/users/2384 | 288531 | 127,278 |
https://mathoverflow.net/questions/261408 | 16 | I asked a [question](https://math.stackexchange.com/q/1373123/19661) at M.SE a couple of years ago about polylogarithms [$\!^{[1]}$](http://en.wikipedia.org/wiki/Polylogarithm)[$\!^{[2]}$](http://mathworld.wolfram.com/Polylogarithm.html)[$\!^{[3]}$](http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog)[$... | https://mathoverflow.net/users/9550 | Several conjectured identities for polylogarithms | ***(Updated answer)***:
Upon further research, it turns out your three equations involving $\phi$ are special cases of three polylogarithm ladders of index $12,\,20,\,24$ that can be found in "*[The Polylogarithm in Algebraic Number Fields](https://ac.els-cdn.com/0022314X85900526/1-s2.0-0022314X85900526-main.pdf?_ti... | 11 | https://mathoverflow.net/users/12905 | 288537 | 127,280 |
https://mathoverflow.net/questions/288526 | 3 | It is well known (see [here](https://math.stackexchange.com/questions/588/what-functions-can-be-represented-as-power-series) for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\mathbb{C}$ in the standard ... | https://mathoverflow.net/users/92164 | Functions on a field representable by Hahn series? | There are many papers, and even books, written on use of [transseries](https://en.wikipedia.org/wiki/Infinitesimal#Transseries) to represent functions. And yes, $e^{-1/x^2}$ is one example.
There is also the theory of *resurgence* proposed by Écalle ... transseries that represent a function, but not by convergence. ... | 3 | https://mathoverflow.net/users/454 | 288538 | 127,281 |
https://mathoverflow.net/questions/288546 | -2 | Let $p$ be a prime number. Is there a formula for the number of subgroups of
1. $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$
2. $$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$$
Thanks!
| https://mathoverflow.net/users/95750 | Number of subgroups of a group of orders $p^3$ | In the following bachelor thesis:
[On p-groups of low power order](https://people.kth.se/~boij/kandexjobbVT11/Material/pgroups.pdf)
there is a chapter on subgroups, which includes the results you're searching for.
It is an useful work for beginners on p-groups.
| 1 | https://mathoverflow.net/users/26380 | 288547 | 127,283 |
https://mathoverflow.net/questions/288524 | 7 | I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
>
> 1.- What is the idea behind his construction and what is one possible motivation for this?
>
>
> 2.- What are the b... | https://mathoverflow.net/users/108963 | A question on Voevodsky´s categories | One could say that the story begins with Beilinson's conjectures on the existence of a theory of *motivic cohomology*. In accordance with the insights of the Grothendieck school that cohomology theories in nature always come with a corresponding "category of coefficients", and that it is very often profitable to work a... | 12 | https://mathoverflow.net/users/2503 | 288576 | 127,290 |
https://mathoverflow.net/questions/288345 | 2 | Consider a non-singular, completely positive, unital map $\Psi: \mathbf M\_k(\mathbb C) \to \mathbf M\_h(\mathbb C)$. This map will have one or more retractions $\Phi: \mathbf M\_h(\mathbb C) \to \mathbf M\_k(\mathbb C)$.
For such $\Psi$, consider a choice of retraction $\Phi$ and integer $n > 0$. Consider a Hermitia... | https://mathoverflow.net/users/3723 | Retractions for completely positive unital maps, and their effect on spectral diameter | Define $\Phi\left(\left[\begin{matrix}a&b\\ c&d \end{matrix}\right]\right) = \left[\begin{matrix}3a - 2d& \frac{5}{2}c\\ \frac{5}{2}b&3d-2a \end{matrix}\right]$. Per [the previous question](https://mathoverflow.net/questions/285178/retractions-for-completely-positive-unital-maps-with-particularly-nice-norms), $\Phi$ is... | 2 | https://mathoverflow.net/users/76593 | 288579 | 127,291 |
https://mathoverflow.net/questions/288571 | 8 | [Cantor's Attic](http://cantorsattic.info/Cantor's_Attic "Cantor's attice website") is a really great website for the various descriptions of large finite numbers, large countable ordinals, and large cardinal axioms.
However, after looking through the archives of the website, I have found that originally, the follow... | https://mathoverflow.net/users/111429 | Abandoned LCAs on Cantor's Attic : Grand Reflection cardinals, universe cardinals, weak universe cardinals | Thanks for the question. I had briefly used the universe/weak-universe terminology in 2010 in my answer to a question on MathOverflow, [What interesting/nontrivial results in Algebraic geometry require the existence of universes?](https://mathoverflow.net/a/28913/1946)
My thinking at that time was that $\kappa$ shou... | 8 | https://mathoverflow.net/users/1946 | 288583 | 127,294 |
https://mathoverflow.net/questions/286860 | 6 | I am reading Artin's notes "Lipman's Proof of Resolution of Singularities for Surfaces" from the book "Arithmetic Geometry". I am very confused by the proof of Lemma $6.5.$ (I am formulating it below in a little bit different way than it appears in the text)
>
> Lemma 6.5: Let $(A,\mathfrak m, k)$ be a normal compl... | https://mathoverflow.net/users/115211 | Resolution of Gorenstein rational singularities on a surface | The explanation given in the text indeed seems to be too terse. What follows is taken from an insert I have in my copy of that book (since the argument I came up with when I read the article many years ago did not fit in the margin as other clarifications did). I hope it is helpful, and that I haven't made some blunder... | 4 | https://mathoverflow.net/users/81332 | 288585 | 127,295 |
https://mathoverflow.net/questions/288588 | 7 | For an uncountable regular cardinals $\kappa,$ let $\kappa$-Souslin hypothesis, denoted $SH(\kappa)$ be the assertion that there are no $\kappa$-Souslin trees.
By a result of Jensen, $GCH+SH(\aleph\_1)$ is consistent.
However, the problem of the consistency of $GCH+SH(\aleph\_2)$ is still open.
>
> **Question.** ... | https://mathoverflow.net/users/11115 | Historical question about the $\aleph_2$-Souslin hypothesis | First, in case your question suggests that you managed to prove the consistency of $GCH+SH(\omega\_2)$, then let me congratulate you wholeheartedly!
Second, to put things in context, let us recall that an $\omega\_2$-Souslin tree is a non-special $\omega\_2$-Aronszajn tree which is (obviously) an $\omega\_2$-Aronszajn ... | 13 | https://mathoverflow.net/users/20033 | 288609 | 127,302 |
https://mathoverflow.net/questions/288572 | 1 | This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.
Consider an $n \times n$ correlation matrix A such that all the off-diagonal entries are between [-1,0]. (**Note**: A correlation matrix is a positive semi-definite symmetric matrix, with di... | https://mathoverflow.net/users/118518 | Connection between weights in the last eigenvector (corresponding to least eigenvalue) and the corresponding column of a correlation matrix | This is false.
I randomly generated a few correlation matrices having non-positive off-diagonal elements and discovered a counterexample. I then rounded the elements of this counterexample correlation matrix to 2 decimal places, and still had a counterexample.
Here is the MATLAB output.
```
>> disp(A), [eigenve... | 3 | https://mathoverflow.net/users/75420 | 288613 | 127,303 |
https://mathoverflow.net/questions/288554 | 2 | What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? It is a fundamental step in the proof of a famous theorem of Weierstrass that a non-rational meromorphic function which... | https://mathoverflow.net/users/62343 | What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$? | Let $\;g(n):=nf(n).\;$ If $x=0$ then the functional equation is $\;g(n)+g(m)=g(n+m)\;$ for all $n,m\in\mathbb{Z},\;$ which is [Cauchy's functional equation](https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation) for integers
and the general solution is $g(n)=cn$ which implies $\;c=f(n)\;$ for all $\;n\ne 0, n\in... | 4 | https://mathoverflow.net/users/113409 | 288626 | 127,307 |
https://mathoverflow.net/questions/288616 | 2 | Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
>
> **Problem**
> $M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse function on $M$, the number of critical points with $k$-th morse index of $f$ is $m\_k$ $0\leq k\leq n$,the st... | https://mathoverflow.net/users/91939 | Could an inverse of (weak) Morse inequality exists in some special case? | There are some weak things that one can say using elementary linear algebra (rank + nullity theorem). Eg $dim(H^k(M)) \geq m\_k -(m\_{k-1} + m\_{k+1})$.
With regard to your motivating problem: There are certainly (smooth) manifolds homotopy equivalent to $CP^3$ that are not homeomorphic to $CP^3$. You can find a comp... | 3 | https://mathoverflow.net/users/3460 | 288633 | 127,311 |
https://mathoverflow.net/questions/288634 | 1 | Suppose $f\_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f\_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d\_H$ be the [Hausdorff metric](https://en.wikipedia.org/wiki/Hausdorff_distance) on subsets of $X$.
Do these conditions guarantee Hausdorff c... | https://mathoverflow.net/users/99132 | Hausdorff convergence of preimages of discrete-valued functions | You can rephrase your question in the following way. Let $A\_n=f\_n^{-1}(y)$ and $A=f^{-1}(y)$, which are a generic sets basically. Your only relation between these sets is that $A=\liminf A\_n$ and $X\backslash A=\liminf (X\backslash A\_n)$, i.e. $A=\lim A\_n$. Then your question becomes:
>
> Is it correct that if... | 3 | https://mathoverflow.net/users/53155 | 288639 | 127,313 |
https://mathoverflow.net/questions/288630 | 5 | Let $P\_1,\ldots, P\_d, Q\_1, \ldots, Q\_k \in \mathbb{C}[x\_0,\ldots, x\_n]$ be homogenous polynomials of degree at most $r$.
Assume that $P\_1 \cdot P\_2 \cdots P\_{d-1} \cdot P\_d \in \langle Q\_1, \ldots, Q\_k \rangle$.
Here $\langle h\_1, \ldots, h\_s \rangle$ is the ideal with the generators $h\_1, \ldots, h\_... | https://mathoverflow.net/users/31356 | Generators of an ideal with small degree | Yes. The point is that many invariants of the ideal $I=(Q\_1,\dots,Q\_k)$ can be bounded depending only on $k$ and $r$. It is pure luck that the following paper has collected many of them in a very convenient Proposition 4.6.
<http://www-personal.umich.edu/~asnowden/papers/genstillman-071517.pdf>
In particular, you ... | 4 | https://mathoverflow.net/users/2083 | 288643 | 127,315 |
https://mathoverflow.net/questions/288637 | 4 | Suppose $X$ is a finite set of points in $\mathbb C^n$. Let $d\_r$ denote the minimum degree of a polynomial vanishing to order $r$ at each point of $X$. By linear algebra, we know find can find a polynomial of degree $d$ vanishing to order $r$ on $X$ provided $\binom{r+n-1}{n}|X|<\binom{n+d}{d}$. Now, suppose $d\_r$ i... | https://mathoverflow.net/users/118651 | Degrees of polynomials vanishing to various orders on a set of points | There is a conjecture of Chudnovsky that
$$\frac{d\_r}{r}\geq \frac{d\_1+n-1}{n}$$
and I believe the best current bound is the one by Esnault and Viehweg
>
> "Sur une minoration du degre d’hypersurfaces s’annulant en certains points" Math. Ann. 263 (1983), no. 1, 75–86
>
>
>
where it is proved that $$\frac{d\... | 4 | https://mathoverflow.net/users/2384 | 288647 | 127,317 |
https://mathoverflow.net/questions/288653 | 0 | Let consider a Galois transcendental field extension $T/K$, therefore for each subextension $L$ of $T/K$ we have $T^{\operatorname{Aut}(T/L)} = L$.
My question is how to prove that this conditions already imply that $T$ is a field of characteristic $0$.
| https://mathoverflow.net/users/108274 | Galois Transcendental Field Extension has characteristic Zero | Assume $char(K)=p>0$ and let $x\in T$ be transcendental over $K$ and consider $L=K(x^p)$. If $\alpha\in Aut(T)$ fixes $L$, i.e. $\alpha(x^p)=x^p$, then it also necessarily fixes $x$ because $x$ is the unique $p$-th root of $x^p$. Thus $T^{Aut(T|L)}$ is strictly bigger than $L$.
| 2 | https://mathoverflow.net/users/3041 | 288656 | 127,321 |
https://mathoverflow.net/questions/288655 | 7 | I ask this in mathematics for some days.it doesn't have an answer up to now. <https://math.stackexchange.com/questions/2565828/indecomposable-module-over-a-local-ring>
As we all know, for an arbitrary ring, any finite length module is indecomposable iff the endomorphism ring is local.
I have a question:
>
> If... | https://mathoverflow.net/users/106580 | indecomposable module over a local ring | If $M$ is allowed to be infinitely generated, then there are counterexamples even for finite dimensional local algebras.
Let $R=\mathbb{C}[x,y]/(x,y)^2$, a three-dimensional local $\mathbb{C}$-algebra.
Let $M=\mathbb{C}[t]\oplus\mathbb{C}[t]$ as a vector space, with $R$-module structure given by $\left(p(t),q(t)\ri... | 7 | https://mathoverflow.net/users/22989 | 288665 | 127,326 |
https://mathoverflow.net/questions/288673 | 0 | Given a positive integer $n\in\mathbb{N}$ we define the *zebra crossing* associated with $n$ to be the set $$Z\_n = \{[2kn, 2kn+(n-1)] \cap \mathbb{N}: k\in\mathbb{N}\}.$$
Is there an infinite set $A\subseteq\mathbb{N}$ such that $A\cap Z\_n$ is finite for all positive integers $n$?
| https://mathoverflow.net/users/8628 | Infinite subset of $\mathbb{N}$ almost avoiding all "zebra crossings" | Take the $k$th element of $A$ to be $a\_k:=2k!-1$, for all $k\ge 1$. If $k\ge n$, then $a\_k\equiv -1\!\!\pmod{2n}$, whence $a\_k\notin Z\_n$. Thus, for any fixed $n$, there are only finitely many elements of $A$ contained in $Z\_n$.
| 6 | https://mathoverflow.net/users/9924 | 288674 | 127,329 |
https://mathoverflow.net/questions/288667 | 5 | Let $S$ be a finite collection of finite non-abelian simple groups, for example $S=\{ A\_5 \}$. I am looking for infinite residually-finite finitely-generated groups $G$ such that the composition factors of each finite image of $G$ are in $S$. Are there such examples?
| https://mathoverflow.net/users/5034 | Infinite, residually-finite, finitely generated, groups with very limited composition factors of their finite images. | Theorem 1.27 of <https://arxiv.org/pdf/math/0510294.pdf> gives 63-generated, residually finite, just infinite groups with the property you want for any set of simple nonabelian groups. The construction is due to Dan Segal. See Dan Segal, The finite images of finitely generated groups, Proc. London Math. Soc. (3) 82 (20... | 6 | https://mathoverflow.net/users/15934 | 288679 | 127,331 |
https://mathoverflow.net/questions/288678 | 3 | Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G\_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for which $T(L)$ is not empty?
| https://mathoverflow.net/users/4690 | Triviality of torsors after a field extension of bounded degree | This is not even true for elliptic curves over $\mathbb Q$ or $\mathbb Q\_p$. For example, by Tate duality the group $H^1(\mathbb Q\_p,E)$ is dual to $E(\mathbb Q\_p)$, which is an infinite group. This shows that the period can be arbitrarily large, and then one can use the fact that the period divides the index, to sh... | 4 | https://mathoverflow.net/users/11926 | 288684 | 127,333 |
https://mathoverflow.net/questions/288465 | 5 | An **arc field** on a topological space $X$ is a continuous function $\Psi: X \rightarrow X^{[0,1]}$ such that for every $x \in X$, the path $\Psi(x): [0,1] \rightarrow X$
(1) starts at $x$,
(2) is either an embedding (in other words a simple path), or the constant path on $x$. In the latter case we call $x$ a *... | https://mathoverflow.net/users/21848 | Continuously varying the singularities of a vector field | My answer will only address the smooth case, and vector fields rather than arc fields. Partly because I think your question is already interesting in this setting, but mainly because I'm not familiar with the topological case!
So assume $X$ is a closed smooth $n$-manifold with nonzero Euler characteristic. You ask wh... | 1 | https://mathoverflow.net/users/8103 | 288687 | 127,334 |
https://mathoverflow.net/questions/288685 | 3 | Do there exist an amenable Beurling algebra that is neither Arens regular nor strongly Arens irregular? In his memoir **"The second duals of Beurling algebras"**, A. T. Lau proved that there exists a weight $\omega$ on $\mathbb Z$ such that $\ell^1(\mathbb Z,\omega)$ is neither Arens regular nor strongly Arens irregula... | https://mathoverflow.net/users/84700 | About Beurling algebras | It seems to be underappreciated by **too many** authors (this remark is not directed at the OP, but based on my frustrating experiences in several conferences and reading several papers) that what the older results of Gronbaek and White imply is the following: if $A=L^1(G,\omega)$ is amenable then $G$ is amenable and t... | 4 | https://mathoverflow.net/users/763 | 288693 | 127,336 |
https://mathoverflow.net/questions/288651 | 12 | *I'll be using homological grading throughout this question.*
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}(BG;\mathbb{R}) \cong Sym^{\bullet}(\mathfrak{g}^\*[2])^G$$
$$H^{\bullet}(G; \mathbb{R}) \cong Sym^{\bullet}(\mathf... | https://mathoverflow.net/users/22810 | Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG? | I don't know of a DAG mechanism that implies these statements, any more formally than the standard proofs -- i.e., we reduce to a maximal torus, where the statement follows from the shape of the cohomology of the circle.
There are higher mechanisms though of which this statement is a hint. On one side you have topolo... | 10 | https://mathoverflow.net/users/582 | 288699 | 127,337 |
https://mathoverflow.net/questions/288696 | 1 | Let $X$ and $Y$ be two subsets of $\ell^2$ space over $\mathbb{C}$ such that: $X \cup Y$ is linearly independent, $X \cap Y = \emptyset$ and $\inf\_{x \in X, y \in Y} \| x-y \|>0$ and such that each of them is: infinite, closed and bounded.
My question is if is it true that
$$
\overline{
\operatorname{span}
X
}
\cap
... | https://mathoverflow.net/users/108867 | The intersection of closure of span of infinite, linearly independent, closed, bounded, separated subsets of $\ell^2$ | No. Here's a simple example.
In $\ell\_2$ with orthonormal basis $(e\_n)\_{n\ge 1}$ choose $X=\{e\_n:n\ge 1\}$ (so $X$ is closed and bounded). Then choose a sequence $v\_n$ of vectors with $\|v\_n\|\le 1/2$ and define $f\_n=e\_{3n-2}+e\_{3n-1}+v\_n$, $Y=\{f\_n:n\ge 1\}$. Then $Y$ is also closed (discrete) and bounded... | 4 | https://mathoverflow.net/users/14094 | 288700 | 127,338 |
https://mathoverflow.net/questions/288659 | 9 | I am looking for explicit formulas for the four basic invariants $I\_4, I\_8, I\_{12}, I\_{18}$ of a generic binary quintic form, either given in the shape
$$\displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^3y^2 + 10dx^2y^3 + 5exy^4 + fy^5$$
or
$$\displaystyle F(x,y) = ax^5 + bx^4y + cx^3y^2 + dx^2y^3 + exy^4 + fy^5.$... | https://mathoverflow.net/users/10898 | Explicit formulas for invariants of binary quintic forms | Here's another way to do it that you might find useful:
Recall that $\mathrm{SL}(2,\mathbb{C})$
acts on the polynomial ring $\mathbb{C}[x,y]$
by linear substitution in $x$ and $y$,
making the subspace $V\_d\subset \mathbb{C}[x,y]$, consisting
of polynomials homogeneous of degree $d$ in $x$ and $y$, into
an irreducib... | 11 | https://mathoverflow.net/users/13972 | 288708 | 127,341 |
https://mathoverflow.net/questions/288695 | 3 | I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very obvious generalization: Let $P$ be a partition. Example: $31$. The symmetric polynomial corresponding to $P$ would be $f\_{3... | https://mathoverflow.net/users/11504 | Generalized Newton Identities | As far as I understand, what you call $f\_P$ is usually called monomial symmetric function and is denoted $m\_P$. So I interpret your question as asking for an algorithm converting monomial symmetric funtions to power sum symmetric functions. Such an algorithm seems to be described here:
<https://www.ncbi.nlm.nih.go... | 6 | https://mathoverflow.net/users/37190 | 288714 | 127,345 |
https://mathoverflow.net/questions/288723 | 19 | In his Midrasha Mathematicae lectures ("In Search of Ultimate $L$", BSL 23 [2017]: 1–109), Woodin notes that $V = \textit{Ultimate }L$ implies $\textrm{CH}$ (Theorem 7.26, p.103). Is it known whether $V = \textit{Ultimate }L$ implies $\textrm{GCH}$?
| https://mathoverflow.net/users/91635 | Does $V = \textit{Ultimate }L$ imply GCH? | In his slide [Absolutely ordinal definable sets](http://logic.berkeley.edu/logic@UCB/Steel_logic@UCB.pdf) John Steel writes:
>
> At the same time, one hopes that V = ultimate L will yield a detailed fine structure theory for V, removing the incompleteness that large cardinal hypotheses by themselves can never remov... | 24 | https://mathoverflow.net/users/11115 | 288725 | 127,350 |
https://mathoverflow.net/questions/288743 | 9 | I'm toying with the idea of using locales as a way to define topological manifolds without beginning with points, largely for philosophical reasons.
In this context I think I want to redefine a topological manifold as a locale that is paracompact and strongly Hausdorff in the sense of Johnstone *Stone Spaces* [p.82](... | https://mathoverflow.net/users/118701 | Which topological manifolds do not correspond to strongly Hausdorff locales? | Let me expand a bit my comment as this is a rather subtle property.
As I said any locally compact Hausdorff topological space is a strongly hausdroff locally compact locales. (and under the axiom of choice the two notion are completely equivalent) So this does not exaclty answer the precise question you ask, but as ... | 11 | https://mathoverflow.net/users/22131 | 288750 | 127,355 |
https://mathoverflow.net/questions/288742 | 2 | Denote when $k>2m$ $$f\_k(2m)=\sum\_{i=1,i\geq1}^m\binom{2m}{i}f\_k(i)f\_k(2m-i)$$
$$f\_k(2m+1)=\sum\_{i=1,i\geq1}^m\binom{2m+1}{i}f\_k(i)f\_k(2m+1-i)$$
$$f\_k(0)=1.$$
$$f\_k(1)=k.$$
1. What is a good bound for $f\_{2^{n+1}}(2^{n})$?
2. How does it compare with $\binom{2^{n+1}}{2^n}$? Is $f\_{2^{n+1}}(2^{n})>\binom... | https://mathoverflow.net/users/10035 | Is there a good bound for this double exponential recursion? | For brevity, rewrite the recursion as$$f\_k(n)=\sum\_{1\leqslant i\leqslant\frac n2}\binom nif\_k(i)f\_k(n-i).$$Now divide it by $n!k^n$ and rewrite like this:$$\frac{f\_k(n)}{n!k^n}=\frac1{n!}\sum\_{1\leqslant i\leqslant\frac n2}\binom nii!(n-i)!\frac{f\_k(i)}{i!k^i}\frac{f\_k(n-i)}{(n-i)!k^{n-i}}.$$
It follows that f... | 8 | https://mathoverflow.net/users/41291 | 288754 | 127,358 |
https://mathoverflow.net/questions/288761 | 4 | I failed to construct a process $N(t)$, $t\geq 0$, satisfying all the following three conditions for some positive $\lambda$:
1. $N(0)=0$,
2. for every $t$, $N(t)$ has Poisson distribution with parameter $\lambda t$
3. there are two number $s,t>0$ such that $N(t)$ and $N(t+s)-N(t)$ are **not** independent.
Does som... | https://mathoverflow.net/users/101832 | How to construct a Poisson-like process with dependent increments? | Take a two dimensional poisson process and and make N(t) the number of events in some shape where the increments in the shape overlap, and the shape at time time has area t.
You can find a discussion of 2 dimensional poisson processes in Wikipedia, <https://en.wikipedia.org/wiki/Poisson_point_process>. A particular imp... | 3 | https://mathoverflow.net/users/nan | 288763 | 127,362 |
https://mathoverflow.net/questions/288692 | 3 | Let $R$ be an infinite commutative Artinian ring such that for any two distinct ideals $I, J$ of $R$, $R/I$ and $R/J$ has different cardinalities; then is it true that $R$ is a PIR (principal ideal ring) ? If this is true, then can we reduce the Artinian hypothesis to Noetherian ?
| https://mathoverflow.net/users/nan | commutative, infinite, artinian ring (with unity) in which distinct ideals has distinct index | Let $R$ be an infinite, commutative, Artinian ring with the distinct-index property for distinct ideals. The claim is that $R$ is a field.
**Case 1.** $R$ has a nonzero ideal $I$ such that $I^2=0$.
Reasoning for this case: $I$ must be a finitely generated ideal, since Artinian rings are Noetherian. Hence $I$ is f.g... | 8 | https://mathoverflow.net/users/75735 | 288776 | 127,370 |
https://mathoverflow.net/questions/288755 | 5 | Let $\lambda \vdash nk$. Let $n^k$ denote the partiton with $k$ parts of size $n$. We can compute $\chi^\lambda(n^k)$ by using the [Murnaghan-Nakayama rule,](https://en.wikipedia.org/wiki/Murnaghan%E2%80%93Nakayama_rule) as a signed sum over border-strip tableaux, (shape $\lambda$ and border strips of size $n$).
Now... | https://mathoverflow.net/users/1056 | Murnaghan-Nakayama rule when all cycles have same size | This is corollary 10 in ["A bijection proving orthogonality of the characters of $S\_n$"](https://www.sciencedirect.com/science/article/pii/0001870883900385), by Dennis E. White.
| 3 | https://mathoverflow.net/users/2384 | 288777 | 127,371 |
https://mathoverflow.net/questions/287553 | 8 | Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?
| https://mathoverflow.net/users/nan | $p$-adic exponentials for $p$-adic Lie groups | Besides the books already mentioned, I highly recommend Michel Lazard's *Groupes analytiques p-adiques*, which is the original source for a lot of the material in both Dixon-DuSautoy-Mann-Segal's *Analytic pro-p groups* and Schneider's *p-adic Lie groups*. Lazard's text was most probably written in close collaboration ... | 5 | https://mathoverflow.net/users/27465 | 288783 | 127,373 |
https://mathoverflow.net/questions/288736 | 7 | **Note:** This question has a 1-categorical and an $\infty$-categorical versions. I am interested in the $\infty$-categorical one so this is the version that I write below, but an answer for the 1-categorical version would be interesting too.
Given an $\infty$-category $\mathcal{C}$ and a collection of simplciial set... | https://mathoverflow.net/users/50409 | Characterizing freely adjoining K-filtered colimits as K-continuous presheaves | For the 1-categorical case, it seems to be indeed a question of soundness. More precisely, the condition that $P\_{\cal I}({\cal C}) = P^{{\cal K}}({\cal C})$ is equivalent to the condition that every ${\cal K}$-limit preserving presheaf $F:{\cal C}^{op} \to {\rm Set}$ is an ${\cal I}$-colimit of representables, or, in... | 3 | https://mathoverflow.net/users/51164 | 288793 | 127,376 |
https://mathoverflow.net/questions/288801 | 1 | Is the following statement true: Any surjective homomorphism $f:G\to H$ of groups with equal rank maps every minimal generating system $x$ of $G$ to a minimal generating system $y$ of $H$? "Minimal generating set" means "generating set of minimal cardinality". Is it only valid for finite rank? How about infinite rank?
... | https://mathoverflow.net/users/114032 | Surjective group homomorphism sending minimal generating system to minimal generating system | I'm assuming that "minimal generating set" means "a generating set that does not properly contain a generating set" (that is, every proper subset does not generate). In that reading, the statement is false in both finite and infinite ranks.
For a counterexample in finite rank, let $G$ be infinite cyclic generated by ... | 3 | https://mathoverflow.net/users/3959 | 288803 | 127,381 |
https://mathoverflow.net/questions/288751 | 5 | I asked this question on stackexchange :
<https://math.stackexchange.com/questions/2570199/union-of-pairwise-almost-disjoint-sets>
but I got no answer after 24 hours, so I ask it here.
Let $r$ and $n$ be two natural numbers, with $n \geq 2$. What is known about the least possible cardinality of the union of $r$ s... | https://mathoverflow.net/users/82840 | Union of pairwise almost disjoint sets | Your question is a special case of the set packing problem. My reference for this stuff (which I know nothing about) is A. E. Brouwer, Packing and covering of $\binom kt$ sets, in: A. Schrijver (ed.), *Packing and Covering in Combinatorics*, Mathematical Centre Tracts 106 (1979), 89–97. (This is obviously not the last ... | 5 | https://mathoverflow.net/users/43266 | 288806 | 127,382 |
https://mathoverflow.net/questions/288804 | 1 | I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$:
1. $v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and symmetric about $0$, and
2. $v(x, \theta) = g\_1 (x) f\_1 (\theta) + g\_2(x) f\_2 (\theta) + f\_3 (\theta)$
It is clear ... | https://mathoverflow.net/users/118723 | A functional equation with a quadratic solution | Let us slightly generalize your functional equation as $u(x,y)=v(x-y)$,
$$u(x,y)=\sum\_{j=1}^3f\_j(x)g\_j(y).$$
In your case, $f\_3=1$. According to a theorem of Rubel and Gauchman, for a sufficiently smooth function $u$, the necessary and sufficient condition
for such representation of sums of products of smooth funct... | 1 | https://mathoverflow.net/users/25510 | 288813 | 127,385 |
https://mathoverflow.net/questions/283926 | 8 | Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}
\max\_{S}~&\det(A+S) \\s.t.~&\sum\_{i}^{N}\sigma\_i(S)\,=\,c \\&S\geq0
\end{align}where $\sigma\_i(S)$ are the singula... | https://mathoverflow.net/users/27249 | Optimization problem with determinant as objective | One can verify that $U\_{A}=U\_{S}$ as follows. Note that $$\det(U\_{A}\Sigma\_{A}V\_{A}^{T}+U\_{S}\Sigma\_{S}V\_{S}^{T})=\det(\Sigma\_{A}+U\_{A}^{-1}U\_{S}\Sigma\_{S}V\_{S}^{T}V\_{A}^{-T})$$
Let $Y=U\_{A}^{-1}U\_{S}\Sigma\_{S}V\_{S}^{T}V\_{A}^{-T}$ so that the problem can be rephrased as:
$$\max\limits\_{Y} \det(\S... | 7 | https://mathoverflow.net/users/118731 | 288822 | 127,386 |
https://mathoverflow.net/questions/288810 | 3 | I'm interested in numerical algorithms for 1-dimensional Hamiltonians of the form
$$
H = -\frac{d^2}{dx^2} + V(x) \quad \quad (1)
$$
defined on the line ($x\in\mathbb{R}$) or on the circle. The potential $V$ is such that the lowest part of the spectrum is made of eigenvalues.
My definition of *best* (and of 'in... | https://mathoverflow.net/users/74539 | What is the best numerical algorithm for integrating the 1D Schrödinger equation? | To obtain the lowest eigenvalues by means of a [Krylov subspace](https://en.wikipedia.org/wiki/Krylov_subspace) method you could use the [Lanczos algorithm,](https://en.wikipedia.org/wiki/Lanczos_algorithm) as explained for example in:
[Solving the discretized time‐independent Schrödinger equation with the Lanczos pr... | 1 | https://mathoverflow.net/users/11260 | 288824 | 127,387 |
https://mathoverflow.net/questions/286605 | 6 | Who can give an example of an affine locally symmetric space that is not a Riemanian locally symmetric space?
| https://mathoverflow.net/users/110123 | Example of affine locally symmetric space | Ben's answer to the question is perfectly fine, but one might also want an example that is not even pseudo-Riemannian, i.e., for which the connection $\nabla$ does not admit any nondegenerate symmetric $2$-form that is $\nabla$-parallel.
The simplest such example is in dimension $2$: Let $M=\mathbb{R}^2$ with coordin... | 8 | https://mathoverflow.net/users/13972 | 288827 | 127,389 |
https://mathoverflow.net/questions/288840 | 7 | Let $A$ be a square matrix of order $n$, say with complex coefficients, and let $M$ be the plain matrix of minors of $A$ of order $n-1$ (no transpose, no sing changes). Let $I$ and $J$ be $r$-subsets of the index set $[n]$. Then, apparently
$$\det M\_{I\times J}=\det A\_{([n]\setminus I)\times ([n]\setminus J)}\det(A)^... | https://mathoverflow.net/users/6101 | Determinant of a sub-matrix of the classical adjoint | For me, a standard book to look for such things is Prasolov's linear algebra. This is Theorem 1.2.6.1. [Here](http://yandex.ru/clck/jsredir?bu=uniq1513695095152581690&from=yandex.ru%3Bsearch%2F%3Bweb%3B%3B&text=&etext=1640.ah88ixYOrS1RaE-8FQAFl2dCq8EWwxpqqPiGeK_YJ0K1Ebi3-a2TB5wpPqw82NkR42UkdRXxHsAPf4fCsmuWHw.799bfc877d... | 8 | https://mathoverflow.net/users/4312 | 288851 | 127,397 |
https://mathoverflow.net/questions/288841 | 13 | Let $C$ be the category of associative commutative rings with 1 and let $F:C\to C$ be a functor which commutes with the forgetful functor to abelian groups (i.e. $F$ is a functorial way to define another multiplication on every associative commutative ring with 1). Assume also that on $\mathbb Z$ the new multiplication... | https://mathoverflow.net/users/3891 | Functorial multiplication on commutative rings | Yes, $F$ is the identity functor.
The multiplication on $R$ would come in the form of a natural transformation $R \times R \to R$ which is bilinear in each variable. In commutative rings, the functor $R \mapsto R \times R$ is represented by the polynomial algebra $\Bbb Z[x,y]$, and so by the Yoneda lemma natural tran... | 15 | https://mathoverflow.net/users/360 | 288854 | 127,400 |
https://mathoverflow.net/questions/288837 | 3 | Let $\delta>0$. I am interested in obtaining a bound for the sum $\sum\_{1 \leq x\_1, ..., x\_n \leq N} \operatorname{lcm}(x\_1, ..., x\_n)^{- \delta}$ where lcm denotes the lowest common multiple of the numbers. I would appreciate any comments and suggestions! Thank you very much.
| https://mathoverflow.net/users/84272 | How to bound $\sum_{1 \leq x_1, ..., x_n \leq N} lcm(x_1, ..., x_n)^{- \delta}$? | For $\delta>1$ the sum is bounded, while for $\delta=1$ it grows by a power of $\log N$.
So let me focus on $0<\delta<1$ and $N\geq 2$. For a simple lower bound, we have
$$ \sum\_{1 \leq x\_1, ..., x\_n \leq N} \operatorname{lcm}(x\_1, ..., x\_n)^{- \delta}\geq \left(\sum\_{1 \leq x \leq N} x^{- \delta}\right)^n\gg\_... | 8 | https://mathoverflow.net/users/11919 | 288862 | 127,405 |
https://mathoverflow.net/questions/288732 | 3 | I am a physicist caught in the following situation:
I have two probability measures $\mathbb{P}\_1$ and $\mathbb{P}\_2$ and have to deal with the following integral where $X\_i$ are random iid:
$$\int\_{B} \mathbb{P}\_1\left(\frac{1}{N} \sum\_{i=1}^N X\_i \ge x\right) d\mathbb{P}\_2(x).$$
I was able to obtain a lar... | https://mathoverflow.net/users/118690 | Large deviations for integrands | It all depends on the shape of $P\_2$ and on the assumptions you put on $X\_i$. In what follows I'll assume that $\Lambda(\lambda)=\log E\_1 e^{\lambda X\_1}$ is finite
for all $\lambda$. I will also assume that $E\_1X\_i=0$. Further I will assume that
$P\_2$ is supported on $R$ with density $f$.
Case 1: $B\cap (-... | 2 | https://mathoverflow.net/users/35520 | 288866 | 127,408 |
https://mathoverflow.net/questions/288829 | 9 | I know that there are no solutions to $2^n\equiv 1\pmod{n}$ for $n>1$ and I can prove that there are infinitely many $n$ such that $2^{n+1}\equiv1\pmod{n}$.
My question is:
>
> Do we know other fixed values $k\in \mathbb{N}$ such that
> $2^{n+k}\equiv 1\pmod{n}$ holds *infinitely* often?
>
>
>
| https://mathoverflow.net/users/38851 | For which values of $k$ is it known that there are infinitely many $n$, such that $2^{n+k}\equiv 1\pmod{n}$? | For any $k\geq 1$, there are infinitely many solutions of the congruence $2^{n+k}\equiv 1\pmod{n}$. To see this, observe first that there is always a solution $n\geq 1$ satisfying $n+k\geq 7$. Indeed, for $k\geq 6$ this is verified by the trivial solution $n=1$, while for $1\leq k\leq 5$ it is verified by the pairs
$$ ... | 10 | https://mathoverflow.net/users/11919 | 288877 | 127,410 |
https://mathoverflow.net/questions/288880 | 3 | Originally I had asked this on mathstack exchange but seems possibly appropriate for overflow as well.
Let $k$ be a field. For the vector space $k^d$, a line in $k^d$ is a one dimensional subspace of $k^d$.
A *supplemented line bundle* of a free module $A^{d}$ over a commutative ring $A$ is a projective submodule... | https://mathoverflow.net/users/117411 | Why are supplemented line bundles the correct generalization of "lines"? | Supplemented line bundles are the things which correspond to maps $$Spec(A)\to \mathbb P^{d-1}$$
from the scheme $Spec(A)$ to the projective space of dimension $d-1$ (defined over $\mathbb Z$).
| 5 | https://mathoverflow.net/users/5690 | 288882 | 127,412 |
https://mathoverflow.net/questions/288847 | 25 | [Vassilev-Missana - A note on prime zeta function and Riemann zeta function](http://nntdm.net/volume-22-2016/number-4/12-15)¹ claims the following remarkable identity:
$$
P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-...}}}},
$$
for integer $s>1$, where $\... | https://mathoverflow.net/users/103722 | $P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-...}}}}$ | This has been answered in the [comments](https://mathoverflow.net/questions/288847/ps-1-sqrt-frac2-zetas-sqrt-frac2-zeta2s-sqrt-frac2-zeta#comment715366_288847) by [Lucia](https://mathoverflow.net/users/38624/lucia). The identity $$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqr... | 20 | https://mathoverflow.net/users/12176 | 288883 | 127,413 |
https://mathoverflow.net/questions/288105 | 6 | let $\pi:\omega\to\omega$ be permutation and $\mathcal{F}$ is Ramsey selective ultrafilter on $\omega$. There are uncountable many increasing subsequences of $\pi$. Can one proof that one of them has domain in $\mathcal{F}$ ?
| https://mathoverflow.net/users/118366 | Permutation on $\omega$ and Ramsey ultrafilter | Thanks to @AndreasBlass and @JingZhang for the answer. First of all saying "Ramsey ultrafilter" I meant "Every partition of $\omega$ into sets not in ultrafilter admits a selector in ultrafilter". So-defined ultrafilter better to be called selective. And Ramsey ultrafilter can be defined as follows "Every partition of ... | 1 | https://mathoverflow.net/users/118366 | 288902 | 127,421 |
https://mathoverflow.net/questions/288878 | 2 | Let $A$ be a semiperfect noetherian ring.
A module $M$ is called reflexive in case the canonical map $f\_M: M^{\*\*} \cong M$ is an isomorphism, when $(-)^{\*}:=Hom\_A(-,A)$. This is equivalent to say that $Ext\_A^i(Tr(M),A)=0$ for $i=1,2$ when $Tr$ denotes the Auslander-Bridger duality.
Question: Assume $M$ is fini... | https://mathoverflow.net/users/61949 | Characterisation of reflexive modules | This is true even with weaker assumptions (finitely generated modules for Noetherian rings, or for non-Noetherian semiperfect rings).
If $M\cong M^{\*\*}$ then $M$ is a dual, and for any dual the natural map $M\to M^{\*\*}$ is a split monomorphism, so if $M$ is not reflexive then $M\cong M\oplus N$ for some non-zero ... | 4 | https://mathoverflow.net/users/22989 | 288910 | 127,423 |
https://mathoverflow.net/questions/288680 | 1 | I am reading a paper in differential geometry, [Hitchin's Langlands duality and G2 spectral curves](https://arxiv.org/abs/math/0611524) (see the end of page 8 in the arxiv version), where $f: E \rightarrow F$ is a morphism of holomorphic vector bundles on a Riemann surface, and the kernel bundle $K$ is considered. A [s... | https://mathoverflow.net/users/91935 | Locus where a vector bundle is null | Mistery solved. The bundle $E$ is endowed with a bilinear, non-degenerate, symmetric form. In this context, "null" means that $K$ is isotropic at certain points.
| 1 | https://mathoverflow.net/users/91935 | 288917 | 127,426 |
https://mathoverflow.net/questions/273708 | 12 | I am looking for a reference of the following (true) fact:
**Theorem.** *For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a homeomorphism $h:2^\omega\to 2^\omega$ such that $h(\mathcal N\_\mu)=\mathcal N\_\lambda$.*
Here by $\mathcal N\_\mu... | https://mathoverflow.net/users/61536 | A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube | Finally I have found a good reference to this theorem, which can be easily derived from the following Theorem 2.12 in [this paper](http://topology.auburn.edu/tp/reprints/v24/tp24201.pdf) of Akin:
**Theorem** (Akin, 1999). For any strictly positive continuous measures $\mu,\nu$ on the Cantor cube $2^\omega$ and any $\... | 4 | https://mathoverflow.net/users/61536 | 288919 | 127,428 |
https://mathoverflow.net/questions/287633 | 2 | Fix some $r >0$ and let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d - \mathbb B(0,r)$. Let $W\_t = \cup\_{s \leq t} \mathbb B(B\_t,r)$ be the Brownian sausage around a Brownian motion $B\_t$ started from $\mathbf 0$. Run the process until the time $\tau = \inf \{ t \colon W\_t \cap \mathcal P... | https://mathoverflow.net/users/52896 | Brownian sausage surgery of Poisson point process | See the appendix of this paper on the Brownian frog model: <https://arxiv.org/abs/1710.05811>.
| -1 | https://mathoverflow.net/users/52896 | 288923 | 127,431 |
https://mathoverflow.net/questions/288907 | 4 | Let $s\_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character.
\begin{equation}
s\_\lambda(p\_1,p\_2,p\_3,\ldots) = \sum\_{\nu} \frac{\chi^\lambda\_\nu}{z\_\nu} p\_\nu = \sum\_{\rho=(1^{r\... | https://mathoverflow.net/users/45170 | Decompostion of hook schur function in terms of cauchy product of holonomic functions | It follows from the Cauchy identity and Exercise 7.43 of *Enumerative
Combinatorics*, vol. 2, that
$$ 1+ (u+t)\sum\_{1\leq l\leq d} s\_{l,1^{d-l}}(x)u^{l-1} t^{d-l} =
\prod\_i\frac{1+tx\_i}{1-ux\_i} $$
$$ \qquad = \exp \sum\_{n\geq 1}\frac 1n p\_n(x)(u^n-(-t)^n). $$
If the decomposition you want actually exists then... | 7 | https://mathoverflow.net/users/2807 | 288927 | 127,433 |
https://mathoverflow.net/questions/288920 | 5 | Let $X$ be a regular first countable space of cellularity at most $2^\omega$.
Is it true that the cardinality of $X$ is at most $2^\omega$?
>
> A cellular family is a family of pairwise disjoint non-empty open sets.
> The cellularity of a space $X$ (denoted by $c(X)$) is defined as the supremum of the cardinaliti... | https://mathoverflow.net/users/39873 | A regular first countable space of cellularity at most $2^\omega$ | First of all let me note that a first-countable space of *density* $2^\omega$ has cardinality $\leq 2^\omega$, so a counterexample $X$ to your question must satisfy $c(X) \leq 2^\omega < d(X)$. That leads us naturally to higher Suslin Lines.
A continuous linear order $X$ (endowed with the order topology) is called a... | 7 | https://mathoverflow.net/users/11647 | 288931 | 127,435 |
https://mathoverflow.net/questions/288925 | 5 | Let $L/K$ be a Galois extension of global fields with Galois group $G$. Assume that for a prime $p$ of $K$ we are given the datum $(L\_{p}(s,\rho))\_{\rho}$, where $\rho$ varies over the irreducible representations of $G$ and $L\_{p}(s,\rho)$ is the Euler factor in the sense of Artin; i.e. it is the reciprocal of the c... | https://mathoverflow.net/users/2042 | Can the relative degree and ramification index can be read off the characteristic polynomial? | Yes. The regular representation of $G$ decomposes as the sum of $\rho$'s with multiplicities $\dim\rho$, hence the product of $L(s,\rho)^{\dim\rho}$ over the various $\rho$'s equals the Dedekind zeta function $\zeta\_L(s)$. That is, if $e$ (resp. $f$) is the ramification (resp. inertia) degree in $L/K$ of a given prime... | 4 | https://mathoverflow.net/users/11919 | 288939 | 127,437 |
https://mathoverflow.net/questions/288901 | 3 | Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric matrices. Suppose that $AB$ is non-singular. Let us define
$$F := (AB)^{-1} CB (AB)^{-1}P(AB)^{-\top} + (AB)^{-1}P(AB)^{... | https://mathoverflow.net/users/62673 | Completing the square of a matrix expression | Nope. That would mean $c(X)$ is always nonnegative, but that can't be that way (unless it's always $0$) because it switches sign if you replace $C$ with $-C$.
| 2 | https://mathoverflow.net/users/1898 | 288949 | 127,440 |
https://mathoverflow.net/questions/288867 | 1 |
>
> Is it true that $$\| \Delta u \|\_{L^p(\Bbb R^d)} + \| u \|\_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|\_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?
>
>
>
This results is pretty easy and straightforward for $p=2$ using techniques via Fourier transform and Plancherel.
But what could we use in place of Fourier... | https://mathoverflow.net/users/112207 | Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms? | First, the classical Calderón–Zygmund estimate gives
$$\left\|D^2 u\right\|\_{L^p(\mathbb{R}^d)}\leq C(p,d) \left\|\Delta u\right\|\_{L^p(\mathbb{R}^d)}.$$
By interpolation, $\left\|u\right\|\_{W^{2,p}(\mathbb{R})}$ is equivalent to
$$\left\|\Delta^2 u\right\|\_{L^p(\mathbb{R}^d)}+\left\|u\right\|\_{L^p(\mathbb{R}^d... | 5 | https://mathoverflow.net/users/103093 | 288962 | 127,443 |
https://mathoverflow.net/questions/288960 | 9 | There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of the Handbook of Algebraic Topology (this chapter was written by C.A. McGibbon), of a phantom map $\Omega^2S^5\to \mathbb{H... | https://mathoverflow.net/users/11546 | Essential maps of spectra which are null when localized at any prime | Such maps exist, even when the target is the sphere $S^{0}$.
A map $X \rightarrow S^{0}$ is $p$-locally trivial for each $p$ if it vanishes when composed with each of the $p$-localizations $S^{0} \rightarrow S^{0}\_{(p)}$. This is the same as being in the kernel of the map $[X, S^{0}] \rightarrow [X, \prod S^{0}\_{(... | 10 | https://mathoverflow.net/users/16981 | 288963 | 127,444 |
https://mathoverflow.net/questions/288956 | 13 | Let $k$ be a field.
The proof that $Aut(\mathbb{P}\_k^n)=PGL(n+1:k)$ relies on the fact for any automorphism $\alpha$ of $\mathbb{P}\_k^n$, $\alpha^\*(\mathcal{O}\_{\mathbb{P}\_k^n}(1)) = \mathcal{O}\_{\mathbb{P}\_k^n}(1)$.
It is not necessarily true that $\pi^\*O\_{\mathbb{P}\_A}(1)\simeq O\_{\mathbb{P}\_A}(1)$ ... | https://mathoverflow.net/users/100155 | Automorphism of $\mathbb{P}_A^n$ | Let me summarize the comments of R. van Dobben de Bruyn. The $A$-automorphisms of $\mathbb{P}^n\_A$ correspond in a one-to-one way to couples $(u,L)$, where $L$ is an element of $\operatorname{Pic}(A) $ and $u:A^{n+1}\rightarrow L^{n+1}$ an isomorphism. In other words, there is an exact sequence
$$1\rightarrow \operato... | 12 | https://mathoverflow.net/users/40297 | 288965 | 127,446 |
https://mathoverflow.net/questions/288966 | 6 | This is not a homework problem, although I fear it may turn out to be at that level. For any nonnegative $x\in\mathbb{R}^n$, let $f\_k(x)$ be the sum of the $k$ largest values in $x$, and define $$f(x)=\max\_{k} \frac{f\_k(x)}{\sqrt{k}}$$ over all $k\in\{1,\dots,n\}$. It is easy to see that $f(x)\leq \|x\|\_2$ for all ... | https://mathoverflow.net/users/70190 | The Euclidean norm and $k$ largest elements | I assume that your question concerns only nonnegative elements of $\mathbb{R}^n$, since for instance $f$ vanishes on $(-1,0)\in \mathbb{R}^2$.
Now on $\mathbb{R}\_+^n$, $f$ coincides with $f\circ g$ where $g:\mathbb{R}^n \rightarrow \mathbb{R}\_+^n$ turns every coordinate to its absolute value.
$$f\circ g(x)=\operato... | 5 | https://mathoverflow.net/users/35609 | 288968 | 127,447 |
https://mathoverflow.net/questions/288969 | -2 | Let $n\in\mathbb{N}$ be a positive integer and let $G =(V,E)$ be a connected simple undirected graph with $|V| = 2n$. Is it true that if for the minimal degree $\delta(G)$ we have $\delta(G) \geq n$, then $G$ has a perfect matching?
| https://mathoverflow.net/users/8628 | Matching and minimal degree | Yes. This question was asked before, for example [here](https://math.stackexchange.com/questions/56063/minimum-degree-of-a-graph-and-existence-of-perfect-matching "Math StackExchange"), where you can find various proofs of your claim.
For example, the condition of connectedness follows from $\delta(G) \geq n$. The cl... | 3 | https://mathoverflow.net/users/71028 | 288973 | 127,449 |
https://mathoverflow.net/questions/288952 | 2 | Call an indecomposable module $M$ over a ring $A$ (restrict to finite dimensional algebras if you like or if it helps) $n$-almost reflexive in case $M^{\*\*} \cong nM$, when $(-)^{\*}=Hom\_A(-,A)$ and $nM$ is the direct sum of $M$ $n$ times for a natural number $n \geq 1$.
Questions:
1. Is $n$ a square?
2. Can $n$ ... | https://mathoverflow.net/users/61949 | Reflexive modules up to multiplicity | Consider the quiver algebra with quiver
$\require{AMScd}$
\begin{CD}
\bullet@>>>\bullet@>>>\bullet@<<<\bullet@<<<\bullet\\
@.@.@VVV@.@.\\
@.@.\bullet@.@.
\end{CD}
and radical square zero.
The simple module $S$ at the central vertex has $S^{\*\*}\cong 2S$, and by taking $n$ paths of length two into the central vertex,... | 3 | https://mathoverflow.net/users/22989 | 288975 | 127,450 |
https://mathoverflow.net/questions/288784 | 6 | Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$.
Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is the one-point compactification, and the largest one is the Stone-Čech compactification.
Is $bG$ is isomorphic to one ... | https://mathoverflow.net/users/62739 | Bohr compactification as a topological compactification | As Francois Ziegler answered, for a locally compact group $G$, the Bohr compactification of $G$ is a compactification in the usual sense iff $G$ is compact. This is true with no further restrictions.
Recall that the forgetful functor from compact groups to topological groups has a left adjoint functor, denoted here $... | 4 | https://mathoverflow.net/users/89334 | 288977 | 127,451 |
https://mathoverflow.net/questions/288991 | 8 | The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces equivalences by general adjoint functors.
The [n-lab page on adjunctions](https://ncatlab.org/nlab/show/adjunction#general) says... | https://mathoverflow.net/users/85913 | "Equivalence" is to "group" as "adjoint" is to ....? | If everything has both a left and a right adjoint, then you're talking about a rigid monoidal category. (Here I've done the usual dimension shift where a 2-category with one object is the same as a monoidal category.)
If you only want adjoints on one side it's a bit more awkward because that's not left rigid (since t... | 5 | https://mathoverflow.net/users/22 | 288992 | 127,454 |
https://mathoverflow.net/questions/288980 | 4 | Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ between the unit interval $[0,1]$ and $B$
I would like to know if is it true that:
For every fixed $x \in (0,1)$
$$
\ove... | https://mathoverflow.net/users/108867 | A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space | It's not true.
---
Edit: thinking twice there's a simpler solution.
On $L^2([0,1])$, define $f(x)(t)=1+t^{1/2x}$ for $x\in\mathopen]0,1]$ and $f(0)=1$. It's free (because the $t\mapsto t^x$, $x\ge 0$, form a free family). It depends continuously on $x$ (check at $x=0$). For $x\le 1/2$ it contains all (restricti... | 9 | https://mathoverflow.net/users/14094 | 288994 | 127,455 |
https://mathoverflow.net/questions/287869 | 23 | (Edit #1 after Carlo's response)
It is often claimed that *the notion of natural transformations existed in mathematical vocabulary long before it had a definition*. In fact, I quoted the statement in *italic* from [1, p. 2]. As another example, in [2, p. 70] Ralf Kromer says: *The claim is that there was, at the tim... | https://mathoverflow.net/users/16046 | History of "natural transformations" | See Whitney's [paper from 1935 where he defined tensor products of abelian groups](http://projecteuclid.org/download/pdf_1/euclid.dmj/1077490789). There you will find the terms natural homomorphism and (especially) natural isomorphism. Whitney makes no attempt to give absolutely rigorous definitions of those concepts, ... | 11 | https://mathoverflow.net/users/3272 | 289005 | 127,462 |
https://mathoverflow.net/questions/289001 | 9 | In “A new infinite family in $\_{2}\pi^S\_\*$" (1976), Mark Mahowald constructs elements $\eta\_j \in \pi\_{2^j}(S^0)$ for $j \neq 2$ which come from permanent cycles in the Adams Spectral Sequence that are generated by $h\_1h\_j \in Ext\_A^{2, 2^j}(\mathbb{Z}\_2, \mathbb{Z}\_2)$. Let $H^\*$ denote reduced mod-2 cohomo... | https://mathoverflow.net/users/118831 | "Standard arguments" in Mahowald's eta_j paper | The short answer is that composition in Ext does correspond to composition of the maps, if nothing intervenes. In the case in hand, if $p$ is the projection of $X\_j$ onto its top cell, then $f\_j$ is represented by an element $a \in Ext^1$ such that $p\_\*(a) = h\_j$, and $g\_j$ is represented by $p^\*(h\_1)$, so that... | 12 | https://mathoverflow.net/users/102519 | 289006 | 127,463 |
https://mathoverflow.net/questions/289009 | 3 | If we consider the AdS-Schwarzschild manifold, defined by $M^n=[s\_0,\infty)\times\mathbb{S}^{n-1}$ equipped with the Riemannian metric
$$\overline{g}=\frac{1}{1-ms^{2-n}+s^2}ds\otimes ds+s^2g\_{\mathbb{S}^{n-1}},$$
where $m>0$ is a fixed positive number, $s\_0$ is the unique positive solution of the equation $1+s\_0^... | https://mathoverflow.net/users/39997 | Spin Structure on AdS- Schwarzschild manifold | The existence or nonexistence of a spin structure on a smooth manifold $M$ is a topological question, in that it does not depend on the choice of a Riemannian metric on $M$. Spin structures are obstructed by the second [Stiefel-Whitney class](https://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class) $w\_2\in H^2(M;\... | 6 | https://mathoverflow.net/users/97265 | 289019 | 127,468 |
https://mathoverflow.net/questions/289015 | 3 | Definition: A $C^\*$-algebra $A$ is called sub-homogeneous if there exists $n\in\mathbb{N}$ such that every irreducible representation of $A$ has dimension at most $n$.
I could not find a proof or a counter example for the following claim:
Let $A,B$ be sub-homogeneous $C^\*$-algebras and let $C$ be a $C^\*$-algebra... | https://mathoverflow.net/users/118836 | Extensions of sub-homogeneous $C^*$-algebras are subhomogeneous | Yes. Since $C^{\*\*}\cong A^{\*\*}\oplus B^{\*\*}$, uniform bounds on the dimension of the irreducible representations of $A$ and $B$ impose a uniform bound on the dimension of the irreducible representations of $C$
| 6 | https://mathoverflow.net/users/34640 | 289029 | 127,472 |
https://mathoverflow.net/questions/289031 | 17 | Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it's not known if they exist between finite complexes (Freyd's Generating Hypothesis). Unstably, it's easy to find maps which are the same on homotopy but not homotopic (even between finite complexes), however I don't know an example of a map whi... | https://mathoverflow.net/users/2362 | Example of an unstable map between finite complexes which is the identity on homotopy but not homotopic to the identity? | Pick a degree $1$ map $h: T^3 \to S^3$ from the $3$-torus to the sphere and define
$$f: T^3 \times S^3 \to T^3 \times S^3; \; f(x,y):=(x, yh(x)).$$
This map induces the identity on homotopy groups, but not on homology.
| 25 | https://mathoverflow.net/users/9928 | 289037 | 127,475 |
https://mathoverflow.net/questions/284114 | 21 | Suppose from distance $d$, while driving at speed $v\_0$, I notice that there's a red traffic light in front of me. Suppose that there are no other vehicles, my vehicle has perfect brakes, my maximum acceleration is $a$ and the red light will turn green according to some $\mu$ distribution. My goal is to get to my fina... | https://mathoverflow.net/users/955 | What is the optimal speed to approach a red light? | Similar to the linked variants on this problem, the optimal strategy takes the form of a function $v(t)$, which corresponds to the strategy in which you travel at velocity $v(t)$ until the light turns green, then you slam on the accelerator and accelerate at $a$ until you reach the speed limit $L$.
If $T$ is the time... | 7 | https://mathoverflow.net/users/34444 | 289049 | 127,479 |
https://mathoverflow.net/questions/289042 | 4 | Is there any global constant $C$ such that $$C<\frac{\sum\_{i=1}^{n}x\_{i}\log x\_{i}-x\_{i}+(1-x\_{i})\log(1-x\_{i})}{\sum\_{i=1}^{n}x\_{i}}+\log(\sum\_{i=1}^{n}x\_{i})-\log(\sum\_{i=1}^{n}x\_{i}^{2})$$ for all vectors $x\in\mathbb{R}^n$ such that $0<x\_i<1$ for all $i$? The main sticking point seems to be that the la... | https://mathoverflow.net/users/70190 | Is this function always bounded below? | Now I think that no, even if we remove the negative summands $-x\_i+(1-x\_i)\log(1-x\_i)$. Note that our inequality becomes homogeneous, so we may forget that $x\_i$ are less than 1. Choose $x\_i=1+t\_i$ so that $\sum t\_i=0$, then the inequality rewrites as
$$
\log\frac{\sum x\_i^2}{\sum x\_i}=\log\left(1+\frac{t\_1^... | 4 | https://mathoverflow.net/users/4312 | 289051 | 127,480 |
https://mathoverflow.net/questions/289032 | 1 | For two complex matrices $A,B \in \mathbb{C}^{n\times m}$ how to prove that:
\begin{equation}
\overline{\sigma}(B-A) \ge \underline{\sigma}(B) - \underline{\sigma}(A)
\end{equation}
where $\underline{\sigma}(A)$ and $\overline{\sigma}(A)$ denote the smallest and respectively the largest [singular value](https://en.wik... | https://mathoverflow.net/users/106923 | Inequality between the singular values for a sum of two matrices | We have $\underline{\sigma}(C)=\min\_{x:\|x\|=1} \|Cx\|$, $\overline{\sigma}(C)=\max\_{x:\|x\|=1} \|Cx\|$. Now the inequality is a partial case of a general inequality $\max (f-g)\geqslant \min(f)-\min(g)$ for any two real-valued functions $f,g$ on any set. This is seen from writing $g(x)=f(x)-(f-g)(x) \geqslant \min f... | 3 | https://mathoverflow.net/users/4312 | 289058 | 127,482 |
https://mathoverflow.net/questions/288085 | 26 | How to evaluate this integral:
$$\int\_0^1 \int\_0^1 \cdots \int\_0^1\frac{x\_{1}^2+x\_{2}^2+\cdots+x\_{n}^2}{x\_{1}+x\_{2}+\cdots+x\_{n}}dx\_{1}\, dx\_{2}\cdots \, dx\_{n}=?$$
I'm making use of the integral identity:
$$\int\_{0}^{+\infty }e^{-t(x\_{1}+x\_{2}\cdots +x\_{n})}dt=\frac{1}{x\_{1}+x\_{2}\cdots +x\_{n}}$$ an... | https://mathoverflow.net/users/117732 | Integral $\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$ | Here is another approach, which also gives the rational term.
(I) To see how it works let $n\geq 2$ and consider first
the simpler case
\begin{align\*}
\mathbb{E}\bigg(\frac{1}{X\_1+\ldots+X\_n}\bigg)=\int\_0^\infty \bigg(\frac{1-e^{-t}}{t}\bigg)^n\,dt
\end{align\*}
Using $\frac{1}{t^n}=\int\_0^\infty \frac{z^{n-1}... | 12 | https://mathoverflow.net/users/48831 | 289068 | 127,486 |
https://mathoverflow.net/questions/289060 | 3 | I am afraid this question might be very naïve, but I find it hard to locate a reference that does not answer a slightly different question.
Consider the Cantor set $C$ and a continous map $f: C\to C$ with finite topological entropy. Is it true that there must exist a $k$ such that the one-sided full shift on an alpha... | https://mathoverflow.net/users/4961 | Topological universality for Cantor maps | No.
Your condition is called being a (topological) subshift. If $(C,f)$ is a topological subshift, then there exists a finite clopen partition $P$ of $C$ such that the family $(f^{-n}P)\_{n\ge 0}$ separates the points of $C$. (Indeed this is obvious of the shift on $k$ letters: take the partition into $k$ clopen subs... | 5 | https://mathoverflow.net/users/14094 | 289069 | 127,487 |
https://mathoverflow.net/questions/289064 | 10 | Let $M$ be a compact manifold with boundary. If we have two vector bundles $E, F \to M$ with inner products and a differential operator $D: C^{\infty}(E) \to C^{\infty}(F)$ then $D$ admits a formal adjoint $D^\*: C^{\infty}(F) \to C^{\infty}(E)$.
This satisfies, for smooth sections with compact support in the interio... | https://mathoverflow.net/users/43158 | Boundary terms of formal adjoints of differential operators | Yes, there is a generalization of such a "boundary term" for an arbitrary linear differential operator (smooth coefficients assumed, of course). My favorite way to define the formal adjoint $D^\*$ of a differential operator $D$ doesn't involve any integrals. Given $D$, $D^\*$ is defined by the requirement to satisfy an... | 12 | https://mathoverflow.net/users/2622 | 289070 | 127,488 |
https://mathoverflow.net/questions/289072 | 7 | Let $\mathbf{C}$ be a category with binary Cartesian products. Let's say that a map $f : X \to Y$ is *constant* if, for every pair of maps $g,h : A \to X$, maps $f \circ g$ and $f \circ h$ are equal. Now, we can define a "monoid object without points" in $\mathbf{C}$ as follows: it is an object $X$ together with a map ... | https://mathoverflow.net/users/62782 | "Monoid objects" without points | I don't believe there are non-trivial examples of this concept. Assume that $e: X \to X$ is a constant map, then $e$ is idempotent. Any category can be embedded fully faithfully into a [Cauchy complete](https://ncatlab.org/nlab/show/Cauchy+complete+category) category, so we can assume that $e$ splits as $X \xrightarrow... | 8 | https://mathoverflow.net/users/10605 | 289074 | 127,490 |
https://mathoverflow.net/questions/289089 | 6 | It [can be shown](https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space) that the infinite-dimensional rational projective space $\mathbb{QP}^\infty$ is a connected, Hausdorff topological space. What can be said about its homotopy type (is it simply connected, is there any hope to compute its coh... | https://mathoverflow.net/users/7952 | On the homotopy type of $\mathbb{QP}^\infty$ | Any countable Hausdorff space $Q$ is totally path-disconnected. Indeed, if $f:[0,1]\to Q$ is continuous, then its image $X$ is a countable connected compact Hausdorff space. By Urysohn's lemma, then, continuous maps from $X$ to $[0,1]$ separate points. But $X$ is connected, so the image of a continuous map from $X$ to ... | 19 | https://mathoverflow.net/users/75 | 289091 | 127,493 |
https://mathoverflow.net/questions/289084 | 9 | Is 47 the largest number which has a unique partition into five parts (15, 10, 10, 6, 6), no two of which are relatively prime?
| https://mathoverflow.net/users/60732 | A property of 47 with respect to partitions into five parts | Yes. Suppose $n>47$.
If $2\mid n$, we can take $(n-8,2,2,2,2),(n-10,4,2,2,2)$, which are distinct partitions for $n\geq 14$.
If $3\mid n$, we can take $(n-12,3,3,3,3),(n-15,6,3,3,3)$, which are distinct partitions for $n\geq 21$.
If $n\equiv 1\pmod 6$, we can take $(n-37,15,10,6,6),(n-43,15,12,10,6)$, which are d... | 22 | https://mathoverflow.net/users/30186 | 289093 | 127,495 |
https://mathoverflow.net/questions/289082 | 12 | Who first defined the class of locally convex topological vector spaces?
| https://mathoverflow.net/users/30395 | Who first defined locally convex topological vector spaces? | Attributed to von Neumann (1935) in Dieudonné ([1953, p. 496](http://www.ams.org/mathscinet-getitem?mr=62334); [1981, p. 218](http://www.ams.org/mathscinet-getitem?mr=605488)), Köthe ([1956](http://www.ams.org/mathscinet-getitem?mr=80262), [p. 20](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002134764)), Schaefer... | 11 | https://mathoverflow.net/users/19276 | 289103 | 127,500 |
https://mathoverflow.net/questions/289078 | 5 | Let $G$ be an algebraic group over an algebraically closed field of characteristic zero $K$ and let $L$ be another algebraically closed field, together with an embedding $K \hookrightarrow L$.
Why is it true that the extension of scalars is an equivalence of categories from finite dimensional $K$-representations of ... | https://mathoverflow.net/users/118861 | representations of an algebraic group and extension of scalars | First of all, even setting aside the issue with scalar automorphisms noted in comments, at the level of objects there is a "problem": the functor is *not* essentially surjective for unipotent groups (by a consideration of Ext$^1$-groups with their natural structure of *vector space* over the ground field). Probably nob... | 6 | https://mathoverflow.net/users/81332 | 289107 | 127,503 |
https://mathoverflow.net/questions/289066 | 0 | I have a sequence $a\_n$ such that $0 \leq a\_n \leq \log n$, and I am considering
$\sum\_{n \leq X} a\_n$. However, I prefer using smooth weights so I would like to approximate it with $\sum\_{n \geq 1} a\_n w(X)$.
I guess what I would like is a nice function $w$ such that
$$
E(X) = | \sum\_{n \leq X} a\_n - \sum... | https://mathoverflow.net/users/84272 | Approximating the sum $\sum_{n \leq X} a_n$ with a smooth sum $\sum_{n \geq 1} a_n w(X)$ | In Section 11.8 of Harman's book "Prime-Detecting Sieves", he introduced an infinitely differentiable function $\psi\_1(t)$ on $\mathbb{R}$ such that $\psi\_1(t)\in[0, 1]$ for all $t$ and
$$
\psi\_1(t)=\left\{
\begin{array}{lr}
1\text{ if }x-y+\Delta\_1\le t\le x-\Delta\_1,\\
0\text{ if }t\not\in(x-y, x)
\end{array}
\r... | 1 | https://mathoverflow.net/users/112214 | 289115 | 127,505 |
https://mathoverflow.net/questions/289041 | 7 | What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras?
In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping algebra is given. Such an algebra is called an $h$-deformation. In chapter XVII the Drinfeld-Jimbo algebras are introduc... | https://mathoverflow.net/users/103448 | The difference between $q$-deformations and $h$-deformations | While $\hbar$-quantum groups are really to be understood as deformations, $q$-quantum groups are somewhat different.
A quantum group in the $\hbar$-setting is a $\mathbb C[[\hbar]]$-Hopf algebra $\mathcal A\_\hbar$. As such, the deformation parameter can be given a specific value only at $\hbar=0$, by letting $\mathc... | 5 | https://mathoverflow.net/users/6032 | 289118 | 127,507 |
https://mathoverflow.net/questions/289099 | 3 | I study some definitions of accessible category (see [1](https://ncatlab.org/nlab/show/accessible+category)) and the applications of that notions; my question: exist a notion of accessible category in therm of enriched category theory? (in case of exist, what uses have?)
| https://mathoverflow.net/users/95695 | Accessible categories in enriched category theory | Yes, there is such a thing. Google for the papers "[a theory of enriched sketches](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)" and "[enriched accessible categories](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700021900)", both by J. Rosický and others. (btw, it's a duplicate o... | 3 | https://mathoverflow.net/users/7952 | 289121 | 127,509 |
https://mathoverflow.net/questions/289126 | 5 | Let $M,N$ be topological manifolds such that $M$ does not admit a $PL$ structure and $N$ does. Is $M\#N$ still a triangulable manifold?
| https://mathoverflow.net/users/117162 | Is the connected sum of a triangulable manifold with a non-triangulable manifold a non-triangulable manifold? | In high dimensions Galewski and Stern (and independently Matsumoto) proved that a manifold $M$ is triangulable iff $\beta \Delta(M)=0$. Here $\Delta \in H^4(M;\Bbb Z/2)$, and $\beta$ is the Bockstein corresponding to the short exact sequence $\ker \mu \to \Theta\_{\Bbb Z} \to \Bbb Z/2$. The middle term is the homology ... | 12 | https://mathoverflow.net/users/40804 | 289130 | 127,512 |
https://mathoverflow.net/questions/288971 | 2 | I have a question about the traces of functions in $W^{1,2}$.
Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by
\begin{align\*}
W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \partial f/\partial x\_i \in L^{2}(D,dx)\},
\end{align\*}
where $\partial f/\partial x\_i$ is the distributional derivative ... | https://mathoverflow.net/users/68463 | A Characterization of the traces of functions in $W^{1,2}$ | I think the most natural assumption is that $D$ is a **bilipschitzian image** of a smooth domain $D\_0$, since a change of variables $A:D\_0\to D$ with $a|x-y|\le|A(x)-A(y)|\le b|x-y|$ ($a>0$) preserves $H^1(D\_0)$ and the space of traces $H^{1/2}(\partial D\_0)$ as defined with $\int\int\frac{|f(x)-f(y)|^2}{|x-y|^d}\ ... | 1 | https://mathoverflow.net/users/75422 | 289141 | 127,515 |
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