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/262056 | 4 | ncatlab, in its entry for $SEAR$ (<https://www.ncatlab.org/nlab/show/SEAR>), states that
>
> $SEARC$ ($SEAR$ augmented by an axiom of choice)...is strongly equivalent to $ZFC$.
>
>
>
Since for $SEAR$ (and $SEARC$, according to the ncatlab entry),
>
> The elements of a set have no "internal structure"; they... | https://mathoverflow.net/users/20597 | Forcing in $SEARC$ | Sheaves work in any structural set theory, because they work using any base topos. Note that SEAR-sets form a well-pointed boolean topos $\mathbf{Set}\_{SEAR}$ ([this](https://www.ncatlab.org/nlab/show/SEAR#heyting) Theorem and [this](https://www.ncatlab.org/nlab/show/SEAR#topos) Theorem on the nLab page on SEAR), and ... | 5 | https://mathoverflow.net/users/4177 | 262210 | 118,056 |
https://mathoverflow.net/questions/262209 | 2 | Given a set $A$ is there a known way to find a topological space $X$ such that $|A|=|X|<w(X)$?
Here $w(X)$ is the weight of the topological space.
This is clearly impossible for finite sets $A$. We know it is possible for $A=\mathbb Z$ because of the answers to [this question](https://mathoverflow.net/questions/105... | https://mathoverflow.net/users/24478 | Is there a known construction for heavy topologies of all sizes? | Let $U$ be a (free) ultrafilter on an infinite set $X$ which contains only sets of cardinality $|X|$ (that is, it is a normal ultrafilter). Put $\tau = U \cup \{\varnothing\}$; this is clearly a topology.
There is no base $B$ for $\tau$ with $|B|=|X|$ as every such base must have cardinality at least $|X|^+$ by a di... | 4 | https://mathoverflow.net/users/15129 | 262214 | 118,058 |
https://mathoverflow.net/questions/262184 | 15 | While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at:
$$k \cdot (f^k)^{(k-1)} = \sum\_{j=0}^{k-1} {{k} \choose {j}} (f^{j})^{(j)}(f^{k-j})^{(k-j-1)}. $$
Can somebody help?
| https://mathoverflow.net/users/104880 | Derivative formula | A detailed historical discussion of identities like this one can be found in Warren P. Johnson's paper *[The Pfaff/Cauchy derivative identities and Hurwitz type extensions](https://doi.org/10.1007/s11139-006-0246-0)*, The Ramanujan Journal 13 (2007) pp. 167–201.
In particular, his formula (1.3) is
$$\frac{d^n\ }{dx^n} ... | 29 | https://mathoverflow.net/users/10744 | 262215 | 118,059 |
https://mathoverflow.net/questions/262213 | 5 | As is well-known (see [here](https://mathoverflow.net/questions/220502/spin-structures-for-quaternionic-kaehler-and-hyper-kaehler-manifolds) for a M.O. question) all Kahler manifolds are $spin^c$. I would like to ask which are in fact $spin$.
Taking my motivation from the case of complex projective space, I make the... | https://mathoverflow.net/users/12653 | Which Kahler Manifolds Are Spin? | By a classical paper of Atiyah (<http://www.maths.ed.ac.uk/~aar/papers/atiyahspin.pdf>)
the spin structures on a **compact** complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism classes of holomorphic line bundles $\cal{L}$ such that $\cal{L}\otimes\cal{L}=K$ where ${\cal{K}}=\Lambda^{n}(T^{\*... | 5 | https://mathoverflow.net/users/20783 | 262219 | 118,061 |
https://mathoverflow.net/questions/262089 | 3 | What is the term for a matrix whose columns are mutually orthogonal, but not necessarily othonormal?
I can't name such a matrix "orthogonal" because that would imply that all columns are unit vectors. By the way, why don't we name these matrices "orthonormal" instead of "orthogonal"? But that fight is over, I guess.
... | https://mathoverflow.net/users/66043 | What is the term for a matrix whose columns are orthogonal? | Since it is a matrix of the form $OD$, with $D$ diagonal and $O$ orthogonal, in the need of a better term you may name it an "OD matrix", which at least is short and self-explanatory.
*Rmk:* Curiously, OD matrices already exist, in Transportation Planning, where $OD$ stands for "Origin\Destination."
| 3 | https://mathoverflow.net/users/6101 | 262222 | 118,062 |
https://mathoverflow.net/questions/262203 | 3 | Given a $n$x$n$ distance matrix of some undirected weighted tree graph, is it possible to infer the underlying tree and its edge weights?
For example, suppose we are given the following distance matrix
\begin{pmatrix}
0 & 1 & 4 & 5 & 6 \\
1 & 0 & 3 & 4 & 5 \\
4 & 3 & 0 & 1 & 2 \\
5 & 4 & 1 & 0 & 3 \\
6 & 5 & 2 & 3 & ... | https://mathoverflow.net/users/44464 | Inferring tree graph from distance matrix | The following greedy algorithm should reconstruct the tree corresponding to a given distance matrix $M$, assuming it exists.
Beforehand, one must show that $T$, if it exists, is unique, but unless you insist I will skip these details
(informally, you can use induction on $n$: from a tree $T$ for $M$, remove a leaf $l... | 3 | https://mathoverflow.net/users/56791 | 262229 | 118,065 |
https://mathoverflow.net/questions/262236 | 2 | Let $B(x,\delta)$ be an open ball centered at $x\in R^n$ with radius $\delta>0$. Let $F:R^n\rightarrow R^m$ be a vector-valued function. Then $F(B(x,\delta))$ would be a subset of $R^m$. Let $\overline{co}\{A\}$ be the convex hull of set $A$. My question is the following: under what conditions, the following relation i... | https://mathoverflow.net/users/97186 | interchange of infinite intersection and taking convex hull of a set | Just to show that $F:R^1\to R^1$ can be fairly nice without your relation holding, let $$F(x)=\begin{cases}1&\text{if }x=0,\\ x&\text{otherwise}. \end{cases}$$
Then $$\overline{\text{co}}\left\{\bigcap\_{\delta>0}F(B(0,\delta))\right\}=\overline{\text{co}}\{1\}=\{1\}\subsetneq [0,1]=\bigcap\_{\delta>0}\overline{\text{c... | 1 | https://mathoverflow.net/users/4600 | 262241 | 118,069 |
https://mathoverflow.net/questions/262239 | 1 | Let $\Omega \subset \mathbb{R}^n$ be some open set.
If, for all $\psi \in L^2(\Omega)$ and some fixed integral kernel $k \in L^2(\Omega\times \Omega)$ and $\ell>0$, it is true that both
$\int\_{\Omega} k(\cdot,y) \psi(y)\,dy$ and
$\int\_{\Omega} k(y,\cdot) \psi(y)\,dy$ are $H^\ell$-functions.
Does it follow hat $k \i... | https://mathoverflow.net/users/104662 | Regularity of integral kernel | The assumptions imply that $k\in H^\ell(\Omega)\hat\otimes L^2(\Omega) \cap L^2(\Omega) \hat\otimes H^\ell(\Omega)$, where $\hat\otimes$ is the (completed) Hilbert space tensor product. This space equals $H^\ell(\Omega\_x;L^2(\Omega\_y))\cap L^2(\Omega\_x;H^\ell(\Omega\_y))$ which, by interpolation, is contained in $H^... | 1 | https://mathoverflow.net/users/69194 | 262253 | 118,071 |
https://mathoverflow.net/questions/261789 | 1 | I need the connection between dense(see <https://ncatlab.org/nlab/show/dense+subcategory>) and reflective (<https://ncatlab.org/nlab/show/reflective+subcategory>) subcategories, i.e. If $C$ is a dense subcategory when $C$ also is reflective?
or if $C$ is reflective then $C$ is dense?
| https://mathoverflow.net/users/95695 | Dense and reflexive subcategories | There are dense subcategories (in the sense that every object is a colimit of objects in them) which are not reflective. Take R-mod, the finitely presented right R-modules. This is dense in R-Mod (the whole category of modules) as every module is a directed colimit of finitely presented modules. It is not reflective in... | 1 | https://mathoverflow.net/users/104910 | 262254 | 118,072 |
https://mathoverflow.net/questions/262237 | 9 | Let $A$ be an essentially small abelian category, and $D(A)$ it derived category.
Does $K\_{0}(D(A)) = 0$?
Thank you!
| https://mathoverflow.net/users/92487 | Grothendieck group of derived category | Yes, it's always zero, assuming $D(A)$ means the unbounded derived category.
My complexes will be cochain complexes, and $X[1]$ will be $X$ shifted down in degree.
First suppose $X$ is bounded below, and let $X'=\bigoplus\_{n\geq0}X[-2n]$. Even if $A$ doesn't have infinite direct sums, this is still an object of $D... | 14 | https://mathoverflow.net/users/22989 | 262255 | 118,073 |
https://mathoverflow.net/questions/262244 | 6 | In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem or Fourier transform to do so.
Also, the Schwartz-kernel theorem tells us that in a distributional sense there is alw... | https://mathoverflow.net/users/104662 | Resolvents of Schrodinger operators | There is a general answer for any dimension $n\geq1$. Let $\Omega\subseteq \mathbb R^n$ denote the underlying domain.
>
> For all $z$ in the resolvent set of $H$, $(H-z)^{-1}$ is a (classical) pseudodifferential operator of order $-2$, as $-\Delta+V$ is elliptic. Hence, its kernel $K\_z$ is a distribution on $\Omeg... | 6 | https://mathoverflow.net/users/69194 | 262256 | 118,074 |
https://mathoverflow.net/questions/262216 | 9 | In an article of Robert Friedman, I came up with a comment:
There are finitely many deformation types of Calabi-Yau threefolds for a given diffeomorhpic type if $b\_2 =1$.
And it is said that this is a special case of a result due to Kollar.
(p. 113, Friedman, Robert
{On threefolds with trivial canonical bundle}. C... | https://mathoverflow.net/users/38823 | Why are there finitely many deformation types of Calabi-Yau threefolds for a given diffeomorhpic type if $b_2 =1$? | I am just posting my comment as an answer. The original form of Matsusaka's Big Theorem is that for a pair $(X,H)$ of a smooth, projective $r$-fold $X$ and an ample divisor $H$, there exists an integer $n\_0$ that depends only on $r$ and the Hilbert polynomial $P(n) = \chi(X,\mathcal{O}\_X(nH))$ such that for all $n\ge... | 7 | https://mathoverflow.net/users/13265 | 262260 | 118,075 |
https://mathoverflow.net/questions/221665 | 16 | The game [Hanabi](https://boardgamegeek.com/boardgame/98778/hanabi) is a cooperative, hidden-information game. You can read the rules elsewhere, but broadly speaking the players are attempting to cooperatively build a fireworks display by playing cards from their hand. The goal is to get as high a score as possible, by... | https://mathoverflow.net/users/5010 | Is the game Hanabi NEXPTIME-complete? | There's an article at <https://arxiv.org/pdf/1603.01911.pdf> showing that a cheating solitaire variant is NP-complete. It's not very explicit about the membership of NP, but here's an argument that works: a witness is a sequence of moves; whether a sequence wins can be computed in linear time in the deck size (it's O(1... | 4 | https://mathoverflow.net/users/94914 | 262263 | 118,076 |
https://mathoverflow.net/questions/262206 | 0 | I am trying to find more generalized counterparts of some well-known results from modular group representations.
My question is the following:
Suppose that $H$ is a finite $p$-group acting as automorphisms on a finite dimensional $k$-algebra $A$. Suppose moreover that the field $k$ has characteristic exactly this p... | https://mathoverflow.net/users/103474 | Representations of smash products with $p$-groups | Take $p=2$, $k$ algebraically closed, $A=kC\_3$, and $H=C\_2$ acting non-trivially on $C\_3$.
Then $A$ has three simple modules, but $A\# kH$ is the group algebra $kS\_3$, which has two simple modules.
| 2 | https://mathoverflow.net/users/22989 | 262270 | 118,078 |
https://mathoverflow.net/questions/262272 | 8 |
>
> **Question.** If $f(z)\in\mathbb{C}[z]$ is a [monic polynomial](https://en.wikipedia.org/wiki/Monic_polynomial) of degree $n$, is it true that
> $$\max\{\,\vert f(x)\vert: \, -1\leq x\leq 1\}\geq 2^{1-n} \,\, ?$$
>
>
>
*Context.* This came up while working on some [convexity problems](https://www.math.templ... | https://mathoverflow.net/users/66131 | How small (in modulus) can a polynomial get? | Yes. Proved by Chebyshev. The extreme case ($\max = 2^{1-n}$) is given by $2^{n-1}\cdot f(x)$ being a Chebyshev polynomial of first kind.
<https://en.wikipedia.org/wiki/Chebyshev_polynomials>
| 12 | https://mathoverflow.net/users/7076 | 262275 | 118,081 |
https://mathoverflow.net/questions/262168 | 14 | $\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
**Smooth Riemannian isometries are harmonic.** Can one conclude this just from "staring" at the Dirichlet energy functional? (Ideally without even computing the Euler-Lagrange eq... | https://mathoverflow.net/users/46290 | Tweetable way to see Riemannian isometries are harmonic? | Not exactly 'tweetable', but perhaps the identity (1) may help, if all you want to do is avoid the Euler-Lagrange equations. For simplicity, assume that $M^n$ is oriented. (One can write the identity (1) below as an identity on densities, so the orientability hypothesis is not essential, but I'll leave that detail for ... | 17 | https://mathoverflow.net/users/13972 | 262288 | 118,088 |
https://mathoverflow.net/questions/262284 | 3 | One of the most important achievements in analytic number theory is the establishment of the so-called large sieve inequality, which is formulated as follows. Let $\{a\_n\}$ denote a finite sequence of complex numbers, say supported on the segment $M \leq n < M + N$. For a real number $\alpha$ put
$$\displaystyle S(\... | https://mathoverflow.net/users/10898 | Higher dimensional large sieve inequality | There is a paper of Huxley from 1968 (<http://www.ams.org/mathscinet-getitem?mr=237455>) that answers this. If I am translating the notation correctly, Huxley shows that
$$F\_n(\delta, N) = (N^{1/2} + \delta^{-1/2})^{2n}$$
is allowable. More generally, if one restricts to $M\_i \leq v\_i < M\_i + N\_i$ and assumes that... | 5 | https://mathoverflow.net/users/2627 | 262291 | 118,090 |
https://mathoverflow.net/questions/262298 | 0 | I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to [this question](https://mathoverflow.net/q/262185/4149). Has any proof of this assertion been published?
| https://mathoverflow.net/users/4149 | Reference request: Any connected Lie group has a countable base for its topology | To summarize the easy argument given in MathSE by Cronus: if $V$ is a neighborhood of the unit homeomorphic to an open ball, $(B\_n)$ a countable basis of $V$ and $D$ a dense countable subset of $V$, then the $(g\_1\dots g\_kB\_n)$ when $k,n$ range over integers and $g\_i$ over $D$, form a countable basis for $G$.
| 2 | https://mathoverflow.net/users/14094 | 262303 | 118,093 |
https://mathoverflow.net/questions/253200 | 14 | It is not true that every map from $\mathbb CP^{\infty}$ to a finite complex is null-homotopic. For example it is a result due to Brayton-Gray that there are non-null maps $\mathbb CP^{\infty} \to S^3$, this story is told in P. May's "More Concise Algebraic Topology" (see cor. 2.4.3 on page 40). However, the examples c... | https://mathoverflow.net/users/14233 | Non-phantom maps from $\mathbb CP^{\infty}$ to a finite complex | All maps of $\mathbf CP^\infty$ to a finite complex are phantom maps; see
A. Zabrodsky, On phantom maps and a theorem of H. Miller,
Israel J. Math 58 (1987) 129-143. For maps out of classifying spaces of Lie groups, see Friedlander-Mislin, Locally finite approximations of Lie groups, I, Invent. Math. 83 (1986), 425-43... | 12 | https://mathoverflow.net/users/50509 | 262316 | 118,097 |
https://mathoverflow.net/questions/262265 | 2 | Let $X$ be a non-contractible, $(d-1)$-connected, $d$-dimensional simplicial complex. By the theorems of Hurewicz and Whitehead, $X$ is homotopy equivalent to a wedge of $d$-spheres. Does there exist a $d$-simplex that can be removed from $X$ without decreasing the connectedness?
| https://mathoverflow.net/users/90417 | Removing simplices from simplicial complexes without decreasing connectedness | For a counterexample, we glue two-dimensional Moore spaces for the groups $\mathbb Z/2$ and $\mathbb Z/3$ along a common $S^1$.
As a CW complex, $X$ can be realised by gluing two 2-disks into a closed
loop $\gamma$, such the boundaries of the disks wind around $\gamma$ two and three times, respectively.
To get a simp... | 3 | https://mathoverflow.net/users/70808 | 262320 | 118,098 |
https://mathoverflow.net/questions/262136 | 2 | Given a $p$-adic reductive group $G$ with Grothendieck group $R(G)$ and $f$ an element of the Hecke Algebra $H(G)$ we can consider the function $x: R(G) \to \mathbb{C}$ given by $\pi \mapsto trace \pi(f)$ (for $f \in H(G)$). It is stated
(condition (i) in 1.2 here, for instance: <https://publications.ias.edu/sites/d... | https://mathoverflow.net/users/97316 | Why are Trace characters regular functions on the Bernstein Variety? | As in [my comment](https://mathoverflow.net/questions/262136/why-are-trace-characters-regular-functions-on-the-bernstein-variety#comment646829_262136), Theorem 2 of [van Dijk - Computation of certain induced characters of $p$-adic groups](http://link.springer.com/article/10.1007%2FBF01429876) ([MR](http://www.ams.org/m... | 1 | https://mathoverflow.net/users/2383 | 262322 | 118,099 |
https://mathoverflow.net/questions/262301 | 2 | I know that eigenvalue estimates involving products of matrices are in general tricky, but probably this question has some hope:
Let $A$ and $B$ be two real symmetric positive semi-definite $n\times n$ matrices (with $A+B$ positive definite if needed). Let $\lambda(X)$ denote any eigenvalue of $X$.
I am fairly sure... | https://mathoverflow.net/users/9652 | Estimates of eigenvalues | Yes, and more may be said.
Assume that $(I+A+B+AB)^{-1}(A+B)z=\lambda z$, then $(I+c(A+B)+AB)z=0$, where $c=1-\lambda^{-1}$.
Denote $Bz=u$, where $u$ may be arbitrary vector such that $(u,z)>0$. Next, $A(cz+u)=-z-cu$ and this is possible whenever $0<(cz+u,-z-cu)=-c(z,z)-(1+|c|^2)(u,z)-{\bar c}(u,u)$. This is possi... | 4 | https://mathoverflow.net/users/4312 | 262325 | 118,100 |
https://mathoverflow.net/questions/261537 | 2 | Is there some way to express:
$$I(t) = \int\_{-\infty}^{t} e^{-2\mathrm{cosh}(x)}~\mathrm{d}x$$
From Bessel functions?
By substituting $y = \mathrm{cosh}(x)$ we get
$$I(t) = \int\_{1}^{\mathrm{cosh}(t)} \frac{e^{-2 y}}{\sqrt{y^2-1}}~\mathrm{d}y$$
In this form, Mathematica will give $I(\infty) = \mathrm{Besse... | https://mathoverflow.net/users/8737 | Integral of exp(-2cosh(x)) | By changing $s=e^{t-x}$ in the integral, one obtains $$I(t)=\int\_1^\infty e^{-e^{-t}s-e^t/s}\frac{ds}{s}$$ which corresponds to the definition of the incomplete Bessel function in [Harris](https://www.researchgate.net/publication/242980699_Incomplete_Bessel_generalized_incomplete_gamma_or_leaky_aquifer_functions):
$$... | 5 | https://mathoverflow.net/users/46744 | 262326 | 118,101 |
https://mathoverflow.net/questions/262321 | 5 | I am trying to better understand this nice [answer](https://mathoverflow.net/questions/262213/which-kahler-manifolds-are-spin) to a question of mine, which states
>
> Spin structures on a compact complex manifold $(M^{2n},J)$ are in bijective correspondence with isomorphism classes of holomorphic line bundles ${\ca... | https://mathoverflow.net/users/12653 | Lagrangian Grassmannian as a Spin Manifold | The complex Lagrangian Grassmannian $M=G/K=Sp(n)/U(n)$ is an isotropy irreducible Hermitian symmetric space, hence it admits a unique invariant complex structure. It occurs by painting black in the Dynkin diagram of $Sp(n)$ the last simple root $\alpha\_{n}$, hence the second Betti number of $M$ equals to 1. Now, $M$ a... | 6 | https://mathoverflow.net/users/20783 | 262327 | 118,102 |
https://mathoverflow.net/questions/262315 | 7 | Let $G$ be a a complex reductive algebraic group, together with an $\mathbb{R}$-form. Is it true that any continuous homomorphism $G(\mathbb{R}) \to \mathbb{R}^{\times}$ comes from an algebraic homomorphism $G \to \mathbb{C}^{\times}$?
I ask this because I want to see whether in the Langlands decomposition $P = MAN$,... | https://mathoverflow.net/users/2095 | Elementary question about Langlands decomposition | Now that the motivation for the question has emerged (algebraicity of $M$ inside $G$), here is how to handle it. Let $G$ be a connected reductive $\mathbf{R}$-group, $P$ a parabolic $\mathbf{R}$-subgroup of $G$, and $S$ a maximal split $\mathbf{R}$-torus in $P$ (so $S$ is also maximal as such in $G$). We may and do cho... | 9 | https://mathoverflow.net/users/81332 | 262328 | 118,103 |
https://mathoverflow.net/questions/262232 | 1 | Consider a real square matrix $A$, lets say $4$ by $4$. Each entry of this matrix $A(i,j)$ is a polynomial function of $x$. For example, $A\_{ij}=a\_{ij}x^3+b\_{ij}x^2+c\_{ij}x+d\_{ij}$. The parameters of $A\_{ij}$, namely $a\_{ij}$,$b\_{ij}$,$c\_{ij}$,$d\_{ij}$, are known explicitly.
I would like to solve the equat... | https://mathoverflow.net/users/104902 | Numerical ways of solving $\det(A(x)) = 0$ | This is called a polynomial eigenvalue problem, and the standard way to solve it is converting it into a standard eigenvalue problem with a technique called [linearization](http://www.maths.manchester.ac.uk/~higham/talks/talk_iwasep06.pdf): if $(Ax^3+Bx^2+Cx+D)v=0$ for some $x\in\mathbb{C}$ and a vector $v\in\mathbb{C}... | 3 | https://mathoverflow.net/users/1898 | 262333 | 118,106 |
https://mathoverflow.net/questions/261643 | 11 | I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
1. **Unsurprising results** confirm that the asymptotic behavior of a statistic behaves similarly to the finite sample behavior of a statistic. For example:
... | https://mathoverflow.net/users/40246 | What are some of the surprising results of finite sample statistical estimation? | First of all I have to express my opinion that the gap between large sample behavior and the finite sample behavior should be considered "unsurprising".
Basically speaking, the theory of asymptotics using the framework of decision theory is much tougher than most mathematicians think nowadays if they ever read into ... | 9 | https://mathoverflow.net/users/25437 | 262341 | 118,110 |
https://mathoverflow.net/questions/262349 | 28 | This might be easy, but let's see.
>
> **Question 1.** If $\mathfrak{S}\_n$ is the group of permutations on $[n]$, then is the following true?
> $$\sum\_{\pi\in\mathfrak{S}\_n}\prod\_{j=1}^n\frac{j}{\pi(1)+\pi(2)+\cdots+\pi(j)}=1.$$
>
>
>
**Update.** After Lucia's answer, I got motivated to ask:
>
> **Que... | https://mathoverflow.net/users/66131 | Sum over permutations is 1 | Yes. Write the sum as
$$
\sum\_{\pi \in S\_n} n! \int\_0^{\infty} e^{-\pi(1) x\_1} \int\_{0}^{\infty} e^{-(\pi(1)+\pi(2)) x\_2} \ldots \int\_0^{\infty} e^{-(\pi(1) + \ldots +\pi(n))x\_n} dx\_n \ldots dx\_1.
$$
Upon writing $y\_n = x\_n$, $y\_{n-1}=x\_n+x\_{n-1}$, etc this becomes
$$
\sum\_{\pi \in {S\_n} } n! \in... | 35 | https://mathoverflow.net/users/38624 | 262350 | 118,113 |
https://mathoverflow.net/questions/262217 | 7 | Let $A$ be a domain, not necessarily noetherian or normal, let $X = {\rm Spec}(A)$ and let $U\subseteq X$ be the complement of a prime divisor of $X$. Is it possible that $\mathcal O\_U(U) = \mathcal O\_X(X)$?
| https://mathoverflow.net/users/104201 | Complement of a divisor in an affine scheme | (Edited to add example) Yes, it may happen that $A=\mathcal{O}\_X(U)$: I will give an example at the end. On the other hand, this cannot happen if $A$ is noetherian. More generally:
>
> **Proposition.** If $U$ is quasicompact, then $A\subsetneq\mathcal{O}\_X(U)$.
>
>
>
Put $Y:= X\smallsetminus U$, and let $\ma... | 12 | https://mathoverflow.net/users/7666 | 262356 | 118,114 |
https://mathoverflow.net/questions/262357 | 1 | Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$.
The set of $\nabla$-compatible metrics on $E$ forms a *convex cone*.
This cone can be empty, however (see [here](https://mathoverflow.net/questions/54434/when-can-a-connection-induce-a-riemannian-metric-for-which-it-is-the-le... | https://mathoverflow.net/users/46290 | How large can the cone of $\nabla$-compatible metrics be? | The $\nabla$-compatible metrics on $E$ are the positive-definite $\nabla'$-parallel sections of $S^2(E^\*)$, where $\nabla'$ is the connection on $S^2(E^\*)$ induced by $\nabla$. When $M$ is connected and $x\in M$ is fixed, the space of $\nabla'$-parallel sections of $S^2(E^\*)$ is isomorphic to the set $Q\_x(\nabla)\s... | 6 | https://mathoverflow.net/users/13972 | 262362 | 118,118 |
https://mathoverflow.net/questions/262353 | 4 | Let $\mathbb{K}$ be a field and let $f \in \mathbb{K}[x]$ be a monic polynomial of degree $n$. Suppose that $\alpha\_1, \ldots, \alpha\_n$ are all the roots of $f$ (in some algebraic closure of $\mathbb{K}$) and define
$$\Delta\_f(y) = \prod\_{\substack{1 \leq i , j \leq n \\ i \neq j}} (y - (\alpha\_i - \alpha\_j)).$$... | https://mathoverflow.net/users/nan | A generalization of the discriminant of a polynomial | As to question (2): Suppose that $\Delta\_f(y)$ is separable. Then $\Delta\_f(y)$ is irreducible if and only if the Galois group of $f$ acts transitively on the set of differences $\alpha\_i-\alpha\_j$, $i\ne j$, which is equivalent to transitivity on the set of pairs $(\alpha\_i,\alpha\_j)$. This, however, is equivale... | 1 | https://mathoverflow.net/users/18739 | 262367 | 118,120 |
https://mathoverflow.net/questions/261677 | 5 | Maybe this is not the kind of question for this website, but nevertheless I believe it could be interesting for a large audience.
I am interested to know if there is some book where the subject of actuarial mathematics is treated in a rigorous "mathematical" way, somehow suitable for graduate math students or experie... | https://mathoverflow.net/users/88920 | Rigorous introductions to actuarial mathematics | Here are some references, where it is made precise that the rigorous mathematical framework leading to the theory is explained.
1)Term-Structure Models , a graduate course ,
by Damitir Filipovic (Springer 2009).
2)Risk Analysis in Finance and Insurance (2nd edition),
by Alexander Melkinov (Chapman & Hall/CRC, 2011).
... | 2 | https://mathoverflow.net/users/34304 | 262368 | 118,121 |
https://mathoverflow.net/questions/262370 | 6 | Suppose $X$ is a finite $d$-dimensional simplicial complex which is homotopy equivalent to a wedge of at least two $d$-spheres. Does $X$ contain a subcomplex which is homotopy equivalent to a single $d$-sphere?
| https://mathoverflow.net/users/90417 | Subcomplexes with homotopy type of a sphere in complexes with homotopy type of a wedge of spheres | EDIT: Wow, I worked a lot harder than necessary. I've added a simplified version of this proof. The original is at the bottom.
Let $A = S^1$, with fundamental group the free group $F$ on a generator $x$. Consider the elements
$$
\begin{align\*}
a &= x^6\\
b &= x^{10}\\
c &= x^{15}
\end{align\*}
$$
in $F$, and use the... | 12 | https://mathoverflow.net/users/360 | 262383 | 118,125 |
https://mathoverflow.net/questions/262342 | 6 | It is well known that if one uses the Schwarzschild coordinates (t, r, $\theta$, $\phi$) to solve Einstein's equations, the components of the metric tensor blow up at the "event horizon", r = 2M (in units where c = G = 1).
It is also well known that this is an "artifact of using bad coordinates", a so-called coordin... | https://mathoverflow.net/users/94232 | What exactly goes wrong with Schwarzschild coordinates at the event horizon? | Recall that a coordinate system is for $M$ is given by an open set $\Omega\in \mathbb{R}^n$ and a map $\psi: \Omega\to M$ that is a diffeomorphism between $\Omega$ and its image. Then the inverse $\psi^{-1}$ gives a local coordinate system $\psi^{-1}: \psi(\Omega) \to \mathbb{R}^4$.
In the Schwarzschild case it is u... | 3 | https://mathoverflow.net/users/3948 | 262386 | 118,127 |
https://mathoverflow.net/questions/262379 | 7 | If $E$ is an elliptic curve over $\mathbb{Q}$, and $\pi$ is the automorphic representation of $\mathrm{GL}\_2$ associated to $E$, then one can write $\pi = \otimes\_v \pi\_v$ with each $\pi\_v$ an irreducible representation of $\mathrm{GL}\_2(\mathbb{Q}\_v)$.
For a non-archimedean place $v$, how do we know the isomor... | https://mathoverflow.net/users/76332 | When is the local representation associated to an elliptic curve a Steinberg? | The local representation $\pi\_{v}$ attached to an elliptic curve is a Steinburg or a twisted Steinburg if and only if $E$ has potentially multiplicative reduction. This follows from the discussion in Section 15 of Rohrlich's paper "Elliptic curves and the Weil-Deligne group", where it is shown that the corresponding r... | 7 | https://mathoverflow.net/users/48142 | 262390 | 118,128 |
https://mathoverflow.net/questions/262314 | 2 | Let $R$ be a semi-algebraic, compact region in $\mathbb{R}^n$ with positive Lebesgue measure. Let $N(R) = \# (R \cap \mathbb{Z}^n)$. Davenport's lemma asserts that we have
$$\displaystyle N(R) = \operatorname{Vol}(R) + O(\max\{\operatorname{Vol} \overline{R}\}),$$
where the maximum is taken over all projections $\o... | https://mathoverflow.net/users/10898 | Counting prime points in a bounded region | The best known version of the prime number theorem in short intervals is $\pi(x+y)-\pi(x)\sim\frac{y}{\log x}$, provided that $x^{7/12}<y=o(x)$. So you can cut a connected set $C\subseteq[0,x]^k$ into cubes with side length $x^{7/12}$, and discard all cubes intersecting the boundary. In this way you obtain that the num... | 5 | https://mathoverflow.net/users/37555 | 262398 | 118,132 |
https://mathoverflow.net/questions/262105 | 0 | If M a monoid and $Set^M$ a topos associate to M, I found $Set^M$ have (epi, strong,mono) factorization system(<https://math.stackexchange.com/questions/541300/epi-mono-factorization-in-presentable-categories>), I think $Set^M$ has (epi,mono source) factorization [[http://katmat.math.uni-bremen.de/acc/acc.pdf](http://k... | https://mathoverflow.net/users/95695 | (Epi,mono source) factorization in M-Set | Based of the idea of [მამუკა ჯიბლაძე](https://mathoverflow.net/users/41291/%E1%83%9B%E1%83%90%E1%83%9B%E1%83%A3%E1%83%99%E1%83%90-%E1%83%AF%E1%83%98%E1%83%91%E1%83%9A%E1%83%90%E1%83%AB%E1%83%94) and $Set$ has (epi, mono source) factorization, if $(f\_i, (X, \lambda))$ is a source, the (epi, initial source factorization... | 0 | https://mathoverflow.net/users/95695 | 262409 | 118,138 |
https://mathoverflow.net/questions/262177 | 10 | Let $H\_n=\sum\_{k=1}^n\frac 1 k$ be the $n$-th harmonic number with $H\_0=0.$
**Question:** Is the following true?
$$\det\left(H\_{i+j}\right)\_{i,j=0}^n=(-1)^n \frac{2H\_{n}}{n! \prod\_{j=1}^n \binom{2j}{j} \binom{2j-1}{j}}.$$
**Edit:**
Comparing with the orthogonal polynomials whose moments are the numbers $\fra... | https://mathoverflow.net/users/5585 | Hankel determinants of harmonic numbers | I prove your identity $$\sum\_{j=0}^n (-1)^j\binom{n}{j}\binom{n+j}{j} H\_j= 2(-1)^n H\_{n}$$
which you claim to imply the result.
The method is the same as [here](https://mathoverflow.net/questions/238978/a-combinatorial-identity-involving-harmonic-numbers).
At first, use $(-1)^k\binom{n+k}k=\binom{-n-1}k$. Then ... | 7 | https://mathoverflow.net/users/4312 | 262422 | 118,141 |
https://mathoverflow.net/questions/262412 | 1 | Consider a prime power $q$ having the form $16t^2+1$, where $t$ is a positive integer. Numerical experiments show that when $t \leq 10^9$, each prime power $q$ with this form is indeed a prime.
In general, is it true that a prime power of this form must be a prime?
| https://mathoverflow.net/users/104986 | Must a prime power of the form 16t^2+1 be a prime? | [Catalan's Conjecture](https://en.wikipedia.org/wiki/Catalan's_conjecture) is a theorem (as of 2002):
With one exception, there are no solutions in positive integers of $$x^a+1=y^b$$ with $a,b \ge 2.$
The exception is, of course,
$$2^3+1=3^2$$
| 13 | https://mathoverflow.net/users/8008 | 262429 | 118,143 |
https://mathoverflow.net/questions/262400 | 5 | I was recent reading through Paul Erdos's classic [elementary proof](https://www.renyi.hu/~p_erdos/1934-01.pdf) of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p\_i$ represents the $i$th prime such that $p\_i \le n < p\_{i+1}$, then there are least $n - i$ ... | https://mathoverflow.net/users/15915 | Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$ | I like Larry Freeman's proof so much, I am going to rephrase it, first by weakening it.
Claim (Freeman): Given integer $n$ greater than 1, and given $x$ greater than $n!$, there are at most $\pi(n)$ many $n$-smooth integers in the interval $[x+1,x+n]$.
Indeed, his proof is an injection from the subset of $[x+1,x+n]... | 6 | https://mathoverflow.net/users/3402 | 262434 | 118,145 |
https://mathoverflow.net/questions/262427 | 2 | In the works of Spohn, Borodin and others, there are results on the height functions for random matrices showing they have fluctuations converging to the Gaussian free field. So I think we are nearing some statement of central limit theorem for random surfaces.
**Q1**: As with the original CLT, are there any first re... | https://mathoverflow.net/users/99863 | Central limit theorem for random surfaces | I think yes, it's implied by the following (from Wikipedia <https://en.m.wikipedia.org/wiki/Gaussian_free_field>):
>
> Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF ... | 1 | https://mathoverflow.net/users/4600 | 262443 | 118,148 |
https://mathoverflow.net/questions/262460 | 1 | Let $G$ be a compact connected Lie group and $T$ be a maximal torus in $G$. Then the homogeneous space $G/T$ is a simply connected orientable manifold. (See, e.g., Hofmann-Morris: The structure of compact groups, page 291). I would like to know, whether $G/T$ is expressible in terms of some familiar manifolds in case w... | https://mathoverflow.net/users/50457 | Flag manifolds for classical groups | As your title gives away, $G/T$ can be expressed as the manifold of full flags in $\mathbf C^n$ for $\mathrm{SU}(n)$, resp. full isotropic flags for the bilinear form defining $\mathrm{SO}(n)$ or $\mathrm{Sp}(n)$ — except that the one for $\mathrm{SO}(2l)$ splits into two open orbits. It can also be expressed as a coad... | 3 | https://mathoverflow.net/users/19276 | 262468 | 118,154 |
https://mathoverflow.net/questions/203755 | 3 | MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T\_p$ acting on this space can be written uniquely as a power series with zero constant term in $T\_3$ and $T\_5$.
I've been working o... | https://mathoverflow.net/users/6214 | Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism | It's much easier than I thought (granting a result about the Fricke involution in level Gamma\_0 (9)). Let M(odd) consist of those odd power series lying in Z/2[E]; it is the space of odd mod 2 modular forms of level Gamma\_0 (9). Suppose f in Z[1/3][[x]] is a modular form of weight w and the above level. Then the same... | 1 | https://mathoverflow.net/users/6214 | 262473 | 118,156 |
https://mathoverflow.net/questions/260528 | 4 | Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S\_X=R-\bigcup\_{\mathfrak{m}\in X}\mathfrak{m}.$$
The question is: $$Hom\_R(S\_X^{-1}R, E) =~\!\! ?$$
I know that this module is an injective cogen... | https://mathoverflow.net/users/104049 | Is $Hom_R(S_X^{-1}R, E)$ the minimal injective cogenerator of $S_X^{-1}R$? | I give a counter-example. Let $R$ be a semi-local domain with two maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$. So the minimal injective generator module is $E = E(R/\mathfrak{m}) \oplus E(R/\mathfrak{n})$. Choose $X = \{\mathfrak{m}\}$. If your question holds true then
$$\mathrm{Hom}\_R(R\_{\mathfrak{m}}, E) \con... | 2 | https://mathoverflow.net/users/17901 | 262487 | 118,162 |
https://mathoverflow.net/questions/225064 | 7 | The set $A\_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate
$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$
for infinitely many rationals $p/q$, has Hausdorff dimension $2/\tau$. This is the so called Jarník-Besicovitch theorem. What can be said about the $2/\tau$-H... | https://mathoverflow.net/users/72314 | Jarník-Besicovitch and outer measure | Given any interval $I$, the $2/\tau$-Hausdorff outer measure of the set $A\_\tau \cap I$ is always infinite for $\tau \neq 2$ and equal to $|I|$ for $\tau = 2$, independently of the constant $C$. This is a consequence of Jarník's theorem, which can be consulted here: <https://arxiv.org/abs/1601.01948>. Incidentally the... | 1 | https://mathoverflow.net/users/72314 | 262509 | 118,164 |
https://mathoverflow.net/questions/262496 | 1 | I have asked the following question on M.SE [here](https://math.stackexchange.com/questions/2147151/showing-using-exponentials-that-phi-is-a-jordan-morphism?__=1163220907), but I have not yet received a response.
I do apologize of this is not the correct site to post it on - if so, please do let me know and I will rem... | https://mathoverflow.net/users/60913 | Showing that $\phi$ is a Jordan morphism | A standard trick one sees in Banach algebra theory, relating additive and multiplicative structure, is to look at $\exp(\lambda a)$ for fixed $a\in A$ as a holomorphic function of $\lambda$, and then try to play games with power-series or Liouville's theorem or other complex-analytic techniques.
The given conditions ... | 4 | https://mathoverflow.net/users/763 | 262514 | 118,167 |
https://mathoverflow.net/questions/262490 | 9 | The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X\_i\stackrel{\phi\_i}{\hookrightarrow} X\}\in\mathrm{Cov}(X)$, $X\in\mathrm{ob}(\textbf{Set})$.
>
> Are there any other Grothendieck topologies on $\textbf{Set}$, not eq... | https://mathoverflow.net/users/4721 | Are all Grothendieck topologies on Set equivalent? | (Much of this has basically been said by someone in the comments already.)
Here is a way of making examples of topologies on Set. Let $\mathcal C$ be a class of sets. Define a topology on Set by saying that a sieve $S$ on an object $X$ is a cover if and only if for every $Y\in\mathcal C$ every morphism $Y\to X$ belon... | 12 | https://mathoverflow.net/users/6666 | 262515 | 118,168 |
https://mathoverflow.net/questions/142719 | 10 | I found an easy proof that the (levelwise) homotopy limit of a pointwise equivalence of finite diagrams of orthogonal spectra is an equivalence, without assuming that the spectra in the diagrams are fibrant. This makes me a bit nervous. Does anyone know if this is in fact true?
| https://mathoverflow.net/users/35183 | Are finite (levelwise) homotopy limits of spectra homotopy invariant? | It's perhaps a little strange to answer this question after three and a half years, but I've thought about this before too and couldn't resist posting. If the category $C$ indexing your diagram is finite in the sense that the classifying space $BC$ is a finite CW complex (or equivalently $C$ has finitely many composabl... | 7 | https://mathoverflow.net/users/1874 | 262516 | 118,169 |
https://mathoverflow.net/questions/260714 | 2 | The Kazdan-Warner Type equations I am talking about is the following:
Suppose X be a compact Riemannian manifold(of any dimension) and $A$, $B$, and $w$ to be smooth function, I hope to know more about the known result to the following equation:
$$\Delta u+ Ae^u-Be^{-u}-w=0 $$
Here is my questions:
1. Under some ... | https://mathoverflow.net/users/25054 | Any Good Reference for Kazdan-Warner Type Equations | Not sure what counts as a good reference, but a statement on the solvability of the equation you mention is contained in the paper by Bryant-Wentworth (Lemma 3.4) <http://www.math.ubc.ca/~jbryan/papers/kahlermonos.pdf>
| 3 | https://mathoverflow.net/users/105042 | 262525 | 118,171 |
https://mathoverflow.net/questions/261764 | 2 | Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^\*$ is regular at $0$, and $G$ action freely on $\mu^{-1}(0)$. Then $Ind^G(D\_M^L)=Ind(D^{L\_G}\_{M//G})$.
Is there an example?
Here is an i... | https://mathoverflow.net/users/95296 | An example of Guillemin Sternberg Conjecture | Since you are looking for an official source, I recommend the book [Symplectic Fibrations and Multiplicity Diagrams](http://dx.doi.org/10.1017/CBO9780511574788) by Guillemin, Lerman and Sternberg. It has a lot to say about symplectic reduction on homogeneous manifolds. However, I don't have it with me right now and can... | 1 | https://mathoverflow.net/users/70808 | 262526 | 118,172 |
https://mathoverflow.net/questions/262497 | 5 | Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ defined by $$V=\left\{f(x)=\sum\_{\alpha\in\mathbb{Z}^n}a\_\alpha x^\alpha\in\mathbb{C}[x\_1^{\pm},\cdots,x\_n^\pm] \mi... | https://mathoverflow.net/users/105027 | Zariski openness of Newton non-degenerate polynomials | The set described in the question is not Zariski open. Let $n=2$ and $\Delta = \mathrm{ConvexHull}((2,0), (0,2))$. So we are considereding polynomials $a x^2+bxy+cy^2$. Then your set is
$$\{ b^2 \neq 4 ac \} \cup \{ (t,0,0),\ (0,0,t) : t \neq 0 \}.$$
This contains the Zariski open set $b^2 \neq 4ac$, but is not Zariski... | 1 | https://mathoverflow.net/users/297 | 262535 | 118,174 |
https://mathoverflow.net/questions/262531 | 2 | For a positive integer $n$, let $p\_n$ denote the $n$-th prime number.
Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$
be the monomorphism which maps a permutation $\sigma$ to the permutation
$f(\sigma)$ which maps any prime number $p\_n$ to $p\_{n^\sigma}$ and which is
a homomorphism of the mo... | https://mathoverflow.net/users/28104 | Endomorphism of the symmetric group of the set of positive integers via action on the prime numbers | If $\sigma$ is the identity, then so is $f(\sigma)$.
Suppose that $\sigma$ is not the identity. Let $m\_k$ be the smallest non-fixed point of $f^{(k)}(\sigma)$.
It is clear that $m\_0\geq 1$ is some finite integer, and for any $k\geq 0$, $m\_{k+1} = p\_{m\_k} > m\_k$.
In particular, we have $m\_k\geq q\_k$, where $q... | 5 | https://mathoverflow.net/users/7076 | 262539 | 118,175 |
https://mathoverflow.net/questions/262486 | 0 | Let $H= L^2[0,1]$ be the space of measurable and square integrable functions from $[0,1]$ to $\mathbb{R},$ let $(\varepsilon\_k)\_{k \in \mathbb{Z}}$ denote the iid (or strict stationary) $H$-valued time series of innovations with $E(\varepsilon\_0(t)) = 0$ and $E(\varepsilon^2(t)) = 1, \forall t,$ let $\alpha\_0 = Id$... | https://mathoverflow.net/users/66236 | Estimating operators of functional linear processes | I came across the following paper from Aue and Klepsch from January:
<https://arxiv.org/pdf/1701.00770.pdf>
| 0 | https://mathoverflow.net/users/66236 | 262543 | 118,176 |
https://mathoverflow.net/questions/262392 | 18 | Let $x$ be an indeterminate and $n$ a non-negative integer.
>
> **Question.** The following seems to be true. Is it?
> $$x\prod\_{k=1}^n(k^2-x^2)=\frac1{4^n}\sum\_{m=0}^n\binom{n-x}m\binom{n+x}{n-m}(x+2m-n)^{2n+1}.\tag1$$
>
>
>
The problem came out of simplifying some work which reduced to the RHS of (1), bu... | https://mathoverflow.net/users/66131 | $\prod_k(x\pm k)$ in binomial basis? | It is sufficient to prove that the two sides agree when $x = n + \ell$ for $\ell \in \mathbb{N}$. As $k$ varies, the first term in $\prod\_{k=1}^n (k-n-\ell)(k+n+\ell)$ contributes
$$-\ell(-\ell-1) \ldots (-\ell-n+1) = (-1)^n (\ell+n-1)\ldots (\ell+1)\ell$$
and the second term contributes $(n+\ell+1)\ldots (2n+\e... | 7 | https://mathoverflow.net/users/7709 | 262547 | 118,178 |
https://mathoverflow.net/questions/262536 | 0 | Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a **little site** which is the full subcategory of the slice category $C\_{/X}$ whose objects are the morphisms that appear in some $J$-covering family of $X$ and whose morphisms are commuting triangles s.t. all edges ap... | https://mathoverflow.net/users/22810 | If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones) | The answer is yes. In fact the assumption that $X\mapsto J\_{/X}$ is a sheaf is not even relevant. First, let me point out that strictly speaking a Grothendieck topology $J$ does not determine a "small site" over an object $X$, because every morphism belongs to some $J$-covering sieve (the maximal sieve on any object i... | 2 | https://mathoverflow.net/users/20233 | 262565 | 118,182 |
https://mathoverflow.net/questions/262558 | 3 | Let $G$ be a semi-simple compact Lie group. Let $V$ be a real vector space and let:
$\rho : G \to Aut\_{\mathbb{R}}(V)$
be an irreducible real representation of $G$ on $V$. We say that $\rho$ is a real representation of complex type if and only if there exist a $J\in Aut\_{\mathbb{R}}(V)$ satisfying:
$J^{2} = - I... | https://mathoverflow.net/users/66688 | Real Adjoint representations of complex type | Irreducible *real* representations of *complex* type of a compact group correspond to irreducible complex representations that do not admit an invariant bilinear form. Irreducible real representations of *quaternionic* type correspond to irreducible complex representations that admit an alternating invariant bilinear f... | 1 | https://mathoverflow.net/users/4149 | 262566 | 118,183 |
https://mathoverflow.net/questions/262568 | 7 | Assume that $M$ is a smooth closed manifold and $E,F$ are fixed smooth vector bundles over $M.$
>
> Is there a number $C,$ such that for any elliptic operator $\mathcal{D}:\Gamma(E)\to\Gamma(F)$
> $$\dim\ker\mathcal{D}\leqslant C.$$
>
>
>
For simplicity, we may assume that $\Gamma(E)=\Gamma(F)=C^\infty(M).$
... | https://mathoverflow.net/users/62635 | Is there an upper bound on dimension of kernel of elliptic operator for a fixed closed manifold M | There are, at least, many examples where this is false. For instance, Hitchin showed (I think this was his thesis) that there is a sequence of metrics $\{g\_k\}\_{k=1}^{\infty}$ on the three sphere such that the kernel of the Dirac operator associated to $g\_{k}$ has dimension at least $k.$ It is stated as a conjecture... | 6 | https://mathoverflow.net/users/49247 | 262572 | 118,185 |
https://mathoverflow.net/questions/262570 | 3 | In case this question appeared here I apologize for the preposterous usage of computer space and attention of participants. I searched over there, for a similar or the same question for an answer but resigned. :)
I am reading:
S. B. Niefield, Cartesianness, topological spaces, uniform spaces, and affine schemes, *... | https://mathoverflow.net/users/73577 | Commutativity of triangle consisting of morphism and two adjuncts | If $f:X \to X'$ is a morphism, then we have an induced morphism $F(f) : F(X) \to F(X')$. By definition of $A/T$, this is a commutative diagram
$\require{AMScd}$
\begin{CD}
\Sigma\_T(F(X)) @>\Sigma\_T(F(f))>> \Sigma\_T(F(X')) \\
@V VV @VV V\\
T @>\mathrm{id}>> T
\end{CD}
This gives the desired diagram
\begin{C... | 5 | https://mathoverflow.net/users/98306 | 262573 | 118,186 |
https://mathoverflow.net/questions/262474 | 5 | This question is related to [Hankel determinants of harmonic numbers](https://mathoverflow.net/questions/262177/hankel-determinants-of-harmonic-numbers).
Let $f(n)=\sum\_{k=1}^n \frac{2^k}{k}$ and $r(n)=\sum\_{j=0}^n (-2)^{n-j}\binom{n}{j}\binom{n+j}{j}f(j).$
In order to compute the Hankel determinants $\det\left(... | https://mathoverflow.net/users/5585 | An identity related to Hankel determinants of $\sum_{k=1}^n \frac{2^k}{k}$ | Define the sequence $a\_n(x):=\sum\_{j=0}^n(-2)^{n-j}\binom{n}j\binom{n+j}jx^j$ so that
$r(n)=\int\_0^2\frac{a\_n(x)-a\_n(1)}{x-1}dx$.
We need the following fact which follows from the [Vandermonde-Chu identity](https://en.wikipedia.org/wiki/Vandermonde's_identity): for $n\geq1$,
\begin{align}\sum\_{j=0}^n(-1)^j\bin... | 7 | https://mathoverflow.net/users/66131 | 262574 | 118,187 |
https://mathoverflow.net/questions/262491 | 3 | Let $f : C\rightarrow S$ be a proper flat morphism whose geometric fibers are connected nodal curves.
Let $P\in C$ be a closed point with image $s\in S$, and suppose $P$ is a 'node' - that is, if $\widehat{\mathcal{O}}\_s$ is the completed etale local ring of $S$ at $s$, then we have:
$$\widehat{\mathcal{O}}\_P = \wi... | https://mathoverflow.net/users/88840 | Understanding the completed stalk of the dualizing sheaf of a family of nodal curves at a node | So as it turns out the key fact is the beautiful description of dualizing sheaves in terms of 'generalized determinants' given in Knudsen-Mumford's "The Projectivity of the Moduli Space of Stable Curves" (I and II).
There, in general given a scheme $X$ and a perfect complex of $\mathcal{O}\_X$-modules $F^\bullet$, wh... | 3 | https://mathoverflow.net/users/88840 | 262580 | 118,189 |
https://mathoverflow.net/questions/262567 | 4 | Let $F$ be, say, a non-archimedean local field. Let $G$ be a connected reductive (can be assumed simply connected) quasi-split group $G$ over $F$. Let
$X\in\operatorname{Lie}G$ be semisimple and $G\_X:=C\_G(X)$. How can we give an explicit example in which $G\_X$ is not quasi-split?
Thanks and pardon for the probab... | https://mathoverflow.net/users/31327 | Twisted Levi of a quasi-split group that is not quasi-split | The answer is no. Take $F$ to be reals, and consider the subgroup $K=U(g)\subset Sp\_{2g}(F)$. Then $K$ is the centraliser of "multiplication by $i$", and is not quasi-split since it is compact. The element multiplication by $i$ on $\mathbb{C}^g$ is to be viewed as the $g$-fold direct sum of the two by two matrix $\beg... | 3 | https://mathoverflow.net/users/23291 | 262582 | 118,191 |
https://mathoverflow.net/questions/262579 | 4 | Looking at [this](https://mathoverflow.net/questions/262321/lagrangian-grassmannian-as-a-spin-manifold) and [this](https://mathoverflow.net/questions/247585/canonical-bundle-of-the-lagrangian-grassmannian?noredirect=1&lq=1) question about the Lagrangian Grassmannian, and its linked [Wikipedia](https://en.wikipedia.org/... | https://mathoverflow.net/users/90430 | Explicit description of the Lagrangian Grassmannian as a homogeneous space | Like any normal person, I'm going to use the symplectic form where $\langle e\_i,e\_{j}\rangle =\pm \delta\_{j,2n-i+1}$ with $1$ if $i\leq n$ and $-1$ if $i>n$. The compact symplectic group you have in mind are the unitary matrices preserving this form. In these coordinates, you have to embed $U(n)$ as $A\mapsto \big[\... | 3 | https://mathoverflow.net/users/66 | 262585 | 118,192 |
https://mathoverflow.net/questions/262243 | 1 | **Question.** If $G$ is a Hamiltonian, does it contain a chromatic path visiting all the vertices? (I define the term "chromatic path" below.)
---
We denote by $\mathbb{N}$ the set of positive integers and set $[n] = \{1,\ldots,n\}$ for $n\in\mathbb{N}$.
Let $G= (V,E)$ be a simple undirected graph on $n\geq 1$ ... | https://mathoverflow.net/users/8628 | Chromatic paths in Hamiltonian graphs | No, here is a counterexample.
Take a cycle on 18 vertices and add three triangles between vertices 1,3,5 and 7,9,11 and 13,15,17. This graph has chromatic number 3, and all hamiltonian paths follow the cycle. Each of these hamiltonian paths will result in most degree 2 vertices receiving color 1, leading to the use o... | 2 | https://mathoverflow.net/users/12487 | 262586 | 118,193 |
https://mathoverflow.net/questions/262527 | 3 | Definition: A set $S \subset \mathbb {R^{n}}$ is **Jordan measurable** if it is bounded in $\mathbb {R^{n}}$ and its boundary is a set of Lebesgue measure zero.
The following conclusion has been proved to be correct.
>
> Let $\varphi : \Omega\subset \mathbb {R^{n}} \rightarrow \mathbb {R}$ be a $C^{1}$ function i... | https://mathoverflow.net/users/105043 | Whether $\varphi(E)$ is a Jordan measurable set? | This is the proof for the case when $E$ is **bounded** and Jordan measurable, but not necessarily closed. Also the proof seems to work for any $C^1$ map between manifolds.
We know that $\partial E$ is a null set. Let $F$ be the set of critical points of $\varphi$ in $int~E$ and let $U=int~ E\backslash F$. Then $\over... | 3 | https://mathoverflow.net/users/53155 | 262600 | 118,200 |
https://mathoverflow.net/questions/262583 | 4 | Let $f=\sum\_{n\ge1} a\_nq^n$ be an Hecke eigenform cusp form of integral weight $k$ on $\Gamma\_0(N)$ with character $\varepsilon\pmod N,$ and let $M\_f$ be the subfield of $\mathbb{C}$ generated by the image of $\varepsilon$ and all $a\_p$ ($p$ runs on primes)
$$M\_f:=\mathbb{Q}(\{a\_p\}\_p,\varepsilon)$$
On what co... | https://mathoverflow.net/users/101794 | The subfield of $\mathbb{C}$ generated by Fourier coefficients of prime index | When $k$ is even the answer is *always* since $p^{(k-2)/2} \in \mathbb Q$.
When $k$ is odd the answer is *never* since if this was true, then for all $p$ (prime not divdidig $N$), we would have $p^{1/2} \in M\_f$, so $M\_f$ would contain the field of infinite degree over $\mathbb Q$ generated by the square roots of tho... | 2 | https://mathoverflow.net/users/9317 | 262619 | 118,208 |
https://mathoverflow.net/questions/262593 | 1 | Suppose $\Omega\subset \mathbb{R}^n$ is bounded Lipschitz domain. When $1\leq p < n$, apparently $p< p^\* = np/(n-p)$, hence
$$
W^{1,p}(\Omega )\subset \subset L^{p}(\Omega ),
$$
and for any $u\in W^{1,p}(\Omega)$
$$
\|u\|\_{L^p}\leq C \|u \|\_{W^{1,p}}\;.
$$
To bound using the seminorm like the following Poincare ineq... | https://mathoverflow.net/users/13092 | Using the Rellich-Kondrachov theorem to prove Poincare inequality for a function vanishing at one point | A typical counterexample is like the following. For concreteness, let's take $p=1$ and $n=2$, and $\Omega = B(0,1) \subset \mathbb{R}^2$.
Let $$f\_n(t) = \begin{cases} nt, & 0 \le t \le 1/n \\ 1, & t > 1/n.\end{cases}$$
Set $v\_n(x) = f\_n(|x|)$, so $v\_n$ is continuous and $v\_n(0)=0$. (If you like you may modify t... | 4 | https://mathoverflow.net/users/4832 | 262623 | 118,210 |
https://mathoverflow.net/questions/262606 | 7 | Suppose we have a Schwartz distribution $\phi$ on $\mathbb{R}^d$ such that $$ \forall x, \ \lim\_{\lambda \to 0}| \langle\phi, \psi^{\lambda}\_x \rangle| =0$$
where $\psi^{\lambda}\_{x}=\lambda^{-d}{\psi\left(\frac{\cdot - x}{\lambda}\right)}$ approximates a delta in $x$. Here we assume that $\psi$ is in $C\_c^{\inf... | https://mathoverflow.net/users/33717 | Distribution that vanishes against approximated delta is zero | I think that the answer is yes. Assume by contradiction that the support $S$ of $\phi$ is nonempty. Since $S$ is closed, it is in particular a complete metric space.
Let $X:=\{\psi\in C^\infty(\mathbb{R}^d):\psi\equiv 0\text{ on }\mathbb{R}^d\setminus B\_1\}$, which is a Fréchet space.
Apply now, for any fixed $x\in ... | 2 | https://mathoverflow.net/users/36952 | 262624 | 118,211 |
https://mathoverflow.net/questions/262554 | 0 | Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q\_{D}(x,y)=x^2+Dy^2.$$
My questions are :
* What is the density of the set $\{\ell \hbox{ prime}| Q\_{D}(x,y)=\ell\quad\hbox{has a solution}\}$ if it is knowen ?
* How ... | https://mathoverflow.net/users/104396 | Diophantine equations and modular forms | In general, modular forms are very useful for understanding the number and distribution of representations by an integral quadratic form in three or more variables, but for binary quadratic forms these kinds of problems are usually treated by more classical algebraic number theory.
I recommend you to study Cox's book... | 8 | https://mathoverflow.net/users/11919 | 262632 | 118,212 |
https://mathoverflow.net/questions/262631 | 6 | Let $X$ be a smooth proper geometrically integral scheme over $\overline{\mathbb F\_p}$. Assume $X$ is the specialization of a smooth proper scheme over $\mathbb Z\_p^{nr}$. Let $L$ be an ample line bundle on $X$.
I would like to show that $L^{\otimes p}$ lifts to characteristic zero. However, the obstruction to lift... | https://mathoverflow.net/users/105062 | Lifting line bundles | This cannot always be done. If $X$ is a supersingular K3 surface then there are $22$ ample line bundles $L$
that give independent classes; if every $L^{\otimes p}$ lifted to char. zero, then you'd have a K3 surface in char. zero with $22$ independent line bundles, which is impossible. (Or $X$ could be any liftable unir... | 5 | https://mathoverflow.net/users/8726 | 262639 | 118,215 |
https://mathoverflow.net/questions/262626 | 4 | Let $P=\{1,\dots,n\}$ and $S\subseteq P$. The map $$\nu:\overline{\mathcal{M}}\_{i,S\cup\{q\}}\to \overline{\mathcal{M}}\_{g,P},$$ which attaches to a curve in the domain a pointed genus $g-i$ curve $[D,\{p\_i\}\_{i\in S^c},q]$ at the points labeled by $q$ is usually called a clutching map. The pullback map $$\nu^\*:H^... | https://mathoverflow.net/users/48522 | "Generalized" clutching maps between moduli spaces of curves | You should read Arbarello-Cornalba more carefully! Let's consider the gluing map
$$ h:\overline M\_{g,n+1} \times \overline M\_{g',n'+1} \to \overline M\_{g+g',n+n'}.$$
As you say we have $H^2(\overline M\_{g,n+1} \times \overline M\_{g',n'+1}) \cong H^2(\overline M\_{g,n+1}) \oplus H^2(\overline M\_{g',n'+1})$ by the... | 3 | https://mathoverflow.net/users/1310 | 262640 | 118,216 |
https://mathoverflow.net/questions/262646 | 8 | Let $f : X \rightarrow Y$ be a finite covering space. In [Quillen's paper](http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf) on the Adams conjecture in Topology, 1970, he gives the following argument for why
$$f\_\* \psi^p = \psi^p f\_\* \in KO[p^{-1}]$$
Quillen's argument is that the map $f\_\*$ can be interpr... | https://mathoverflow.net/users/84144 | Clarifying Quillen's comment on why Adams operations commute with transfer | Before I answer the question as stated, let me point out that you can also prove the claim as follows: the transfer is induced by a stable map $Y\_+\rightarrow X\_+$, and the pth Adams operation is a stable cohomology operation after inverting p, so the result is just naturality of cohomology operations.
Now for Quil... | 10 | https://mathoverflow.net/users/6936 | 262650 | 118,220 |
https://mathoverflow.net/questions/262578 | 8 | This question is motivated by [the MO problem here](https://mathoverflow.net/questions/262474/an-identity-related-to-hankel-determinants-of-sum-k-1n-frac2kk). Perhaps it is not that difficult.
>
> **Question.** Here is an cute formula.
> $$\frac1n\sum\_{k=0}^{n-1}\frac1{\binom{n-1}k}=\sum\_{k=1}^n\frac1{k2^{n-k}... | https://mathoverflow.net/users/66131 | The average of reciprocal binomials | As in the question
$$a\_n:=\sum\_{k=0}^{n-1}\frac{2^n}{n\binom{n-1}k} \qquad \text{and} \qquad
b\_n:=\sum\_{k=1}^n\frac{2^k}k.$$
It is clear that $a\_1=b\_1$ and $b\_{n+1}-b\_n=\frac{2^{n+1}}{n+1}$. But we have the same recursive relation for $a\_n$ because
\begin{align}
a\_n&=2^n\sum\_{k=0}^{n-1}\frac{k!(n-1-k)!(k+1+n... | 7 | https://mathoverflow.net/users/5712 | 262651 | 118,221 |
https://mathoverflow.net/questions/262467 | 8 | In the [paper](http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf)
>
> Cordier, Jean-Marc, and Timothy Porter. "Homotopy coherent category theory." Transactions of the American Mathematical Society 349.1 (1997): 1-54.
>
>
>
the authors define a notion of *coherent co/e... | https://mathoverflow.net/users/7952 | Co/fibrant replacements via coend calculus | Addressing this sort of question was one of the main goals of [math/0610194](https://arxiv.org/abs/math/0610194). The best answer I was able to give is that if $B$ has a suitable model structure with respect to which $F$ is *objectwise* fibrant (resp. cofibrant), then $\overline{F}$ (resp. $\underline{F}$) belongs to a... | 7 | https://mathoverflow.net/users/49 | 262661 | 118,224 |
https://mathoverflow.net/questions/262550 | 2 | A quick question about lax co/limits.
Strictly, when $F : J\to \bf A$ is a diagram and $J$ has an initial object $\varnothing$, then $\varprojlim F \cong F(\varnothing)$; dually, if $\cal J$ has a terminal object, then $\varinjlim F\cong F(\*)$.
>
> If $F$ is a diagram between 2-categories (same notation), and $J... | https://mathoverflow.net/users/7952 | Lax co/limit as evaluation on terminal/initial | No. If $\mathcal{J}$ is the interval category (two objects and one nonidentity morphism between them), which certainly has an initial and a terminal object, then the lax limits and colimits of such a diagram are comma objects of the morphism that determines its image, neither of which is usually equivalent to its domai... | 3 | https://mathoverflow.net/users/49 | 262663 | 118,226 |
https://mathoverflow.net/questions/262660 | 16 | A Sylvester-Gallai configuration in the the complex projective plane is a finite number of $n\ge 2$ points in the complex projective plane such that there is no line through exactly two of them. Trivial examples are obtained by taking $n\ge 3$ points on the same line. There is also the classical Hessian configuration, ... | https://mathoverflow.net/users/23758 | What are Sylvester-Gallai configurations in the complex projective plane? | Yes, there are other Sylvester-Gallai configurations in $\mathbb{P}^2(\mathbb{C})$. Apart from the Hesse configuration (that contains $9$ points) the minimum number of points for a non-collinear configuration is $12$.
A configuration with $12$ points actually exists over any field $\mathbb{K}$ of characteristic diff... | 14 | https://mathoverflow.net/users/7460 | 262678 | 118,230 |
https://mathoverflow.net/questions/261724 | 11 | Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ be a presheaf of sets on $C$ (i.e., an object of $\widehat{C}$) and let $a$ be the associated sheaf functor. Now consider... | https://mathoverflow.net/users/5641 | Sheaf associated to presheaf Aut | A small disclaimer: My knowledge of algebraic geometry is relatively basic so I will not discuss anything related to scheme directly. But it seems that most of what you are asking has very little to do with algebraic geometry so I will answer your question from a purely topos theoretic perspective.
Moreover, when I s... | 5 | https://mathoverflow.net/users/22131 | 262687 | 118,233 |
https://mathoverflow.net/questions/262630 | 3 | I'm reading through Lurie's Higher Topos Theory and I'm not conviced by the proof of corollary 2.3.2.4. it asserts that it $i:A\rightarrow A'$ is inner anodyne and $j:B\rightarrow B'$ is a cofibration, then $$(A\times B')\coprod\_{A\times B}(A'\times B)\rightarrow A'\times B'$$
is inner anodyne.
How does the fact that ... | https://mathoverflow.net/users/105094 | A smash product of an inner anodyne map with a cofibration is inner anodyne | Just to make it clear: in this context "smash product" of $i$ and $j$ means the map
$$
i\square j \colon (A\times B')\amalg\_{A\times B} (A'\times B)\to A'\times B'
$$
constructed from $i\colon A\to A'$ and $j\colon B\to B'$. (I can't find a place in the book where Jacob defines "smash product" properly, though this i... | 3 | https://mathoverflow.net/users/437 | 262697 | 118,236 |
https://mathoverflow.net/questions/262699 | 0 | Let $S$ be the set of all positive semidefinite Hermitian matrices of order $mn$ over $\mathbb{C}$. Any matrix $H$ can be partitioned into blocks $H\_{ij}$ of order $n$ that is $H\_{mn \times mn} = (H\_{ij})\_{m \times m}$.
Is there a function $f: S \rightarrow \mathbb{R}$ which may be defined as follws?
$$f(H) = \b... | https://mathoverflow.net/users/36977 | Define a matrix function with a specific property | Why not just take the sum of the norms of the commutators $[H\_{ij}, H\_{kl}]$ and $[H\_{ij}, H^\*\_{ij}]$?
| 3 | https://mathoverflow.net/users/13650 | 262704 | 118,238 |
https://mathoverflow.net/questions/262688 | 2 | I want to solve the folowing problem **B\*M=V**, where **B** is the unknown of size *3x3*, **M** of size *3xN* and **V** of size *3xN*. The difficulty is, that **B** has to be unitary.
*N* is in the range of 500. All matrices are real.
Solving the problem by multiplying from right with the pseudoinverse of **M** g... | https://mathoverflow.net/users/105118 | matrix regression under side conditions | For this to work, the singular value decompositions of $V$ and $M$ can be written in the form
$$ V = U\_1 \Sigma U\_2^\*, \ M = U\_3 \Sigma U\_2^\*$$
with the same $\Sigma$ and $U\_2$, and you have $B = U\_1 U\_3^\*$.
| 1 | https://mathoverflow.net/users/13650 | 262705 | 118,239 |
https://mathoverflow.net/questions/262706 | 1 | How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
| https://mathoverflow.net/users/94363 | Inequality with symmetric polynomials | This looks like a better fit for Math Stackexchange, because it's
the kind of thing one learns from Olympiad problem books . . .
One standard approach that has not been mentioned yet:
We may assume $a,b$ are both positive (if one is zero it's easy;
if they're of opposite sign then ${\rm LHS} > 0 > {\rm RHS}$;
and if ... | 9 | https://mathoverflow.net/users/14830 | 262717 | 118,245 |
https://mathoverflow.net/questions/262735 | 4 | Can any one provide some references which treat the relation between the class number of a biquadratic field and the class numbers of its sub-fields using the analytic class number formula ?
| https://mathoverflow.net/users/104396 | class number of biquadratic fields | See section VIII.7 (Brauer relations) in Fröhlich-Taylor: Algebraic number theory, especially Theorem 74 there.
| 9 | https://mathoverflow.net/users/11919 | 262736 | 118,254 |
https://mathoverflow.net/questions/262589 | 4 | I am looking for a relatively "elementary" proof that every variety in ${\mathbb R}^n$ contains at least one non-singular point.
So far I only have such a proof for the case of hypersurfaces. Hopefully this proof would also explain what I mean by an "elementary" proof. Consider a variety $V \subset {\mathbb R}^n$ of... | https://mathoverflow.net/users/17509 | Every real variety contains non-singular points | If you're willing to view a variety as a set of points rather than as a scheme, then it is fairly easy to show that every real algebraic variety $V$ in ${\bf R}^n$ is equal *as a set* to the finite union of smooth manifolds (of various dimensions, and typically not closed). Namely, one can view $V$ as the restriction o... | 12 | https://mathoverflow.net/users/766 | 262737 | 118,255 |
https://mathoverflow.net/questions/262553 | 2 | Let $A$ be a linear operator between two Hilbert spaces. Let $A^\*$ be its adjoint.
**Question.** Under what conditions the non-zero spectra of $A^\*A$ and $AA^\*$ coincide counting multiplicities?
In my situation $A$ is an elliptic differential operator between two complex line bundles over a compact smooth manifo... | https://mathoverflow.net/users/16183 | Equality of spectra of products of operators | If $A:C^\infty(M,F\_1)\to C^\infty(M,F\_2)$ is an elliptic operator on a closed manifold, then both $A^\*A$ and $AA^\*$ are self-adjoint elliptic operators. Hence they have discrete spectrum. For $\lambda\in \sigma(A^\*A)$ let $E\_\lambda(A^\*A)$ and $E\_\lambda(AA^\*)$ denote the eigenspaces of $A^\*A$ and $AA^\*$ wit... | 1 | https://mathoverflow.net/users/74307 | 262740 | 118,257 |
https://mathoverflow.net/questions/262721 | 6 | We know for $Y=\mathbb{P}^n$, the total space of the canonical sheaf $Tot(\omega\_Y)$ is the resolution of $\mathbb{C}^{n+1}/\mathbb{Z}\_{n+1}$ where the generator acts as scalar matrix of multiplying a primitive $n$-th root. I was told this result can be extended to certain types of del-pezzo surfaces but had a hard t... | https://mathoverflow.net/users/48616 | Total space of canonical bundle as resolution of singularity | I don't think this is special about the canonical bundle. All you need is a line bundle with a section which is contractible inside the total space of the line bundle.
Another way to generate such examples is the following: Take a smooth projective variety $Y$ (embedded in a projective space) and let $X$ be the cone... | 6 | https://mathoverflow.net/users/10076 | 262744 | 118,258 |
https://mathoverflow.net/questions/262739 | 2 | Let $\mathcal{F}:=\{f\_a\}\_{a\in\mathbb{Z}}$ be a set of symbols indexed by the integers and satisfying the rules:
$$f\_a=f\_{-a} \qquad \text{and} \qquad f\_af\_b=f\_{a+b}+f\_{a-b}.$$
Define a linear operator $\mathcal{L}$ on $\mathcal{F}$ according to
$$\mathcal{L}f\_a=\begin{cases} 1 \qquad \text{if $a=0$} \\ 0 \qq... | https://mathoverflow.net/users/66131 | valuation from $\mathbb{Z}^n$ into $\mathbb{Z}$. Easy way? | Suppose we have a knapsack of size $K = 10$, and items of size
$x\_1=4; x\_2 = 5; x\_3 = 6$. Clearly there is an exact solution to the decision version of the 0/1 knapsack problem: $K - x\_1 - x\_3 = 0$. If we set $A = 2K - x\_1 - x\_2 - x\_3$, this solution gives us $A - x\_1 + x\_2 - x\_3 = 0$.
If $K$ is the size o... | 3 | https://mathoverflow.net/users/30994 | 262747 | 118,259 |
https://mathoverflow.net/questions/262695 | 0 | I would like to have a basic models as a ground truth for the numerical solvers. I am looking for systems which have available analytic solution. As an example I know that the closed form solution of IVP:
\begin{align}
&\dot{y}(t) = -2y(t)\\
&y(0) = 1
\end{align}
is:
\begin{align}
y(t) = e^{-2t}
\end{align}
When usi... | https://mathoverflow.net/users/26106 | Benchmark Systems for ODE Solvers - Reference Request | There is a whole subfield of applied mathematics devoted to developing ODE solvers and understanding their properties. Consequently, there are thousands of relevant papers, and not much more can be said without a more specific question. The canonical reference, which includes lots of methods tested on lots of ODE syste... | 4 | https://mathoverflow.net/users/20507 | 262751 | 118,261 |
https://mathoverflow.net/questions/262755 | 2 | This question is inspired by the big MO question [here](https://mathoverflow.net/questions/45608/does-the-formal-power-series-solution-to-ffx-sin-x-converge); and also inspired by the big MO question [here](https://mathoverflow.net/questions/17605/how-to-solve-ffx-cosx?noredirect=1&lq=1). The premise of this question r... | https://mathoverflow.net/users/nan | What about the other $f$ such that $f(f(x)) = \sin(x)$? | Clearly, if $f\_+(z)$ is a functional square-root of $\sin(z)$, then so is $g(z) := -f\_+(-z)$; moreover, they have the same derivative at the origin. Hence, your post implies that $g(z) = f\_+(z)$, so $f\_+(z)$ is an odd function.
It follows that $-f\_+(z)$ is a functional square-root of $\sin(z)$, so $-f\_+(z) = f\... | 3 | https://mathoverflow.net/users/39521 | 262757 | 118,265 |
https://mathoverflow.net/questions/262657 | 6 | Suppose that $L\subseteq {\cal P}(\omega)$ has the following properties:
1. $\omega \notin L$, and for $e\in L$ we have $|e|\geq 2$;
2. if $e\_1\neq e\_2 \in L$ then $|e\_1\cap e\_2|\leq 1$;
3. if $m,n\in \omega$ there is $e\in L$ such that $\{m,n\}\subseteq e$.
It is not hard to see that $L$ is countable. Is there... | https://mathoverflow.net/users/8628 | Existence of a path in a set of subsets of $\omega$ | Clearly, $L$ is countably infinite. Thus it will suffice to prove the following lemma, which shows that any finite path $\langle e\_1,\dots,e\_k,a\rangle$ in $L$ can be extended to include any new element $b\in L.$
**Lemma.** Given a finite set $E=\{e\_1,\dots,e\_k\}\subseteq L$ and $a,b\in L\setminus E,$ we can find... | 2 | https://mathoverflow.net/users/43266 | 262759 | 118,267 |
https://mathoverflow.net/questions/262764 | 2 | The *index* and *period* of a finite monogenic semigroup $\langle x\rangle$ are the smallest numbers $i$ and $p$, respectively, satisfying $x^{i+p}=x^p$. The question is:
***Is there an algorithm to find the index/period of a finite monogenic semigroup of size $n$ in $o(n)$ (or $O(n^\epsilon)$ with $\epsilon<1$) step... | https://mathoverflow.net/users/40723 | Finding index/period of a semigroup element | Won't Pollard's rho method give you a collision $x^k=x^\ell$ with $k>\ell$ in time roughly $O(\sqrt{n})$ (and $O(1)$ storage). From there you should be able to work back to the index and period.
| 1 | https://mathoverflow.net/users/11926 | 262771 | 118,271 |
https://mathoverflow.net/questions/262763 | 6 | Let $n=\phi(l)$ to be the largest number definable by a first order arithmetic formula $f(x)$ having length at most $l$. By "$n$ is definable by formula $f(x)$" I mean $\mathcal{N}\vDash f(a)$ iff $a=n$, where $\mathcal{N}$ is a structure of natural numbers with the standard interpretation (if it's necessary to be spec... | https://mathoverflow.net/users/75935 | The set of largest numbers definable by formulas in different lengths | Let me interpret the question asking about what is true in the
standard model $\langle\newcommand\N{\mathbb{N}}\N,+,\cdot,0,1,<\rangle$, which
avoids the non-absoluteness issues mentioned in the comments. In
this case, the definition is well-defined and sensible. We define that $\Phi(n)$
is the largest number $m$ defin... | 12 | https://mathoverflow.net/users/1946 | 262772 | 118,272 |
https://mathoverflow.net/questions/262773 | 5 | Let $f$ be an automorphism of the algebra of octonions. Is it true that $f$ preserves some quaternionic subalgebra? Has the statement an elementary proof?
| https://mathoverflow.net/users/105159 | About some property of automorphism of octonions | Seen as a map of $8$-dimensional Euclidean vector spaces, $f$ is obviously (special) orthogonal, so we can find an orthonormal basis on which is has a block diagonal form of $2\times 2$ rotation matrices, and certainly at least $u,v$ unit octonions, orthogonal to each other and both orthogonal to $1$ (=pure imaginary) ... | 6 | https://mathoverflow.net/users/17064 | 262780 | 118,273 |
https://mathoverflow.net/questions/262756 | 11 | A well-known theorem of Stallings says that
*any finitely generated virtually free torsion-free group is free.*
>
> Is this true without **`finitely generated'** condition?
>
>
>
In other words,
>
> is every locally free virtually free group free?
>
>
>
| https://mathoverflow.net/users/24165 | Virtually free, torsion-free, and locally free groups | This is Theorem B of Swan's famous paper *Groups of cohomological dimension one*, in which he removes the 'finitely generated' hypothesis from Stallings' theorem. Theorem B states:
>
> Let $G$ be a torsion-free group. If $G$ has a free subgroup of finite index, then $G$ is free.
>
>
>
[Here](https://dx.doi.or... | 10 | https://mathoverflow.net/users/1463 | 262785 | 118,276 |
https://mathoverflow.net/questions/262782 | 2 | For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the [history of Fermat's Last Theorem](http://fermatslasttheorem.blogspot.com/).
I have made numerous mistakes and for the last seven of those 10+ years, I have, at a rate of once per ye... | https://mathoverflow.net/users/15915 | Are simplified elementary proofs if valid interesting to the professional mathematical community | I think such work is important and has its place. However, it is not clear that stackexchange is that place.
There are a lot of ideas to be considered. Where simple algebraic and combinatorial methods are concerned, many of those have been tried and published or discarded. However, such arguments have an educational ... | 3 | https://mathoverflow.net/users/3402 | 262797 | 118,278 |
https://mathoverflow.net/questions/262765 | 1 | Let A be a finite dimensional algebra. Call an indecomposable module M cool in case $\Omega^{i}(M)$ is nonzero and indecomposable for every $i \geq 1$ and $dim(\Omega^{i}(M))$ is bounded.
Questions:
1. In case every simple module is cool, is the algebra selfinjective?
Wrong by an answer of Jeremy Rickard.
2.In c... | https://mathoverflow.net/users/61949 | On some modules with bounded syzygies | For (1), take a quiver with two vertices, an arrow $\alpha$ from vertex $1$ to vertex $2$, a loop $\beta$ at vertex $2$, and relations $\alpha\beta=0$ and $\beta^2=0$.
For (2), take a quiver with four vertices, arrows $\alpha\_i$ from vertex $i$ to vertex $4$ for $i=1,2,3$, a loop $\beta$ at vertex $4$, and relations... | 2 | https://mathoverflow.net/users/22989 | 262803 | 118,279 |
https://mathoverflow.net/questions/262655 | 13 | This question arose from the recent one, [roots of a polynomial linked to mock theta function?](https://mathoverflow.net/q/262380/41291). Let
$$
g(x):=\sum\_{k=0}^\infty x^k\prod\_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+21 x^8+30 x^9+43 x^{10}+59 x^{11}+...;
$$
the sequence $1,1,1,3,4,6,10,15,2... | https://mathoverflow.net/users/41291 | A mystery sequence | The conjectured identity
$$
f(q)=(q;q)\_\infty\left(1+\sum\_{k=1}^\infty q^k(-q;q)^2\_{k-1}\right)=\sum\_{\substack{m,n\geqslant0\\n\ne1}}(-1)^mq^{\frac{(m+n)(3m+n+1)}2},\tag{1}
$$
using Euler's pentagonal number theorem $(q;q)\_\infty=\sum \_{m=-\infty}^\infty (-1)^m q^{\frac{1}{2} m (3 m+1)}$ can be brought to an equ... | 18 | https://mathoverflow.net/users/82588 | 262806 | 118,280 |
https://mathoverflow.net/questions/262810 | 0 | Let $S$ be a system of polynomial equations over $\mathbb{F}\_q$.
Assume that $S$ has a solution in $\overline{\mathbb{F}\_q}$.
Denote by $k$ the minimal number such that $S$ has $\mathbb{F}\_{q^k}$-rational solution.
How large can be $k$ (depends on $q$, the number of equations of $S$ and its degrees)?
| https://mathoverflow.net/users/31356 | Degree of a field extension with a rational solution | Let $f(x)$ be an irreducible polynomial in $\mathbb F\_q[x]$ of degree $d$. Then the smallest $k$ such that $f(x)$ has a root in $\mathbb F\_{q^k}$ is $k=d$. More generally, let $f\_i(x)$ be an irreducible polynomial in $\mathbb F\_q[x]$ of degree $d\_i$. Then for $S=\{f\_1(x\_1),\ldots,f\_r(x\_r)\}$, the smallest $k$ ... | 3 | https://mathoverflow.net/users/11926 | 262812 | 118,282 |
https://mathoverflow.net/questions/262811 | 2 | An **abstract polytope** is a poset $X$ (here finite), whose elements are called faces, satisfying these 4 conditions:
1. There is a least face and a greatest face.
2. All flags (i.e. maximal chains) have the same number of faces.
3. $X$ is strongly connected, i.e. for every interval $[F\_1,F\_2]$ in $X$ and $F, F' \... | https://mathoverflow.net/users/20391 | Is a finite abstract polytope of Euler characteristic 0 Eulerian? | The answer is no. Let $Y$ be the face poset of a triangulation of a
torus, with a top element $t$ adjoined. Let $\emptyset$ (the empty
face) be the bottom element of $Y$. Let $Z$ be the chain $0<1$. Then
the product $Y\times Z$ satisfies your conditions. However, the
interval from $(\emptyset,0)$ to $(t,0)$ does not sa... | 3 | https://mathoverflow.net/users/2807 | 262816 | 118,284 |
https://mathoverflow.net/questions/262795 | 10 | There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^\*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \mathfrak{g}^\*) \to H^1(\mathfrak{g}, S^2\mathfrak{g}),$$
where $\mathfrak{g}$ is a Lie algebra over a field $k$ and $H^i... | https://mathoverflow.net/users/43639 | Is this sequence of Lie algebra cohomology a part of spectral sequence? | It's part of the Pirashvili exact sequence, relating Lie algebra and Leibniz cohomologies of Lie algebras. This is discussed (in homology terms) in the end of p2 of this paper of mine on Koszul's homomorphism [(arxiv link)](https://arxiv.org/abs/1403.3895). Pirashvili's paper is freely accessible [here on Numdam](http:... | 6 | https://mathoverflow.net/users/14094 | 262822 | 118,286 |
https://mathoverflow.net/questions/262787 | 5 | It is well-known (at least well-known enough to be on [Wikipedia](https://en.wikipedia.org/wiki/Matching_polynomial)) that there are quite simple graphs whose matching polynomials
$$M(G;x) = \sum\_{m\geq 0} (-1)^m \#\{\text{matchings with $m$ edges}\}\, x^{\#V(G)-2m}$$
are essentially equivalent to the Chebyshev p... | https://mathoverflow.net/users/87683 | Are there graphs whose matching polynomials are Legendre? | Any family of orthogonal polynomials can be realized as the characteristic polynomials of a sequence of weighted paths, possibly with loops. If the implicit weight function is symmetric about the origin, loops are not needed. In this case the characteristic polynomial coincides with the matching polynomial. It follows ... | 6 | https://mathoverflow.net/users/1266 | 262826 | 118,287 |
https://mathoverflow.net/questions/262809 | 0 | The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.
>
> Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\boldsymbol{\nabla}\times\mathbf{A} = \boldsymbol{\nabla}\cdot\mathbf{P}$.
>
>
>
This isn't very hard if you don't requ... | https://mathoverflow.net/users/103864 | Relation between curl and tensor divergence | More generally, you can consider a given vector field v, not necessarily a curl. To solve $\nabla\cdot P=v$, set $P=\nabla u+(\nabla u)^T$. This results in the equation $\Delta u+\nabla(\nabla\cdot u)=v$, which is an elliptic system for u.
| 4 | https://mathoverflow.net/users/12120 | 262833 | 118,290 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.