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/279548 | 7 | In the paper ''**A Nonstandard Model of Arithmetic Constructed by means of Forcing Method**'', Zhang Jinwen states the following in his abstract:
>
> The first nonstandard model of arithmetic was given by Skolem. A. Robinson has introduced the concepts of standard, internal and external objects (sets, relations, fu... | https://mathoverflow.net/users/11115 | On a theorem of Zhang Jinwen about models of arithmetic | (The final two sections are translated modulo tweaks...)
---
I do not know the answer to your question, but I can try to translate (part of) this paper. There are some terms that are unfamiliar to me (e.g., "infinite internal sets" as remarked in a comment) so I will indicate where I am quite confused by using br... | 5 | https://mathoverflow.net/users/22971 | 279722 | 123,927 |
https://mathoverflow.net/questions/29764 | 7 | Does the category of noetherian commutative rings have pushouts?
Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}\_{X/S}$ is only defined on the category of locally noetherian $S$-schemes (as far as I know). It is a group functor and in some situations it is representable. We ... | https://mathoverflow.net/users/2841 | Pushouts of noetherian rings | Let $k$ be a field, and let $\ell = k(x\_1,x\_2,\ldots)$ be the fraction field of $k[x\_1,x\_2,\ldots]$. Then $\ell \otimes\_k \ell$ is the localisation of $k[x\_1,x\_2,\ldots][y\_1,y\_2,\ldots]$ at the multiplicative set
$$S = \left\{fg\ \bigg|\ \begin{array}{ll}f \in k[x\_1,x\_2,\ldots]\setminus\{0\},\\g \in k[y\_1,y... | 7 | https://mathoverflow.net/users/82179 | 279727 | 123,928 |
https://mathoverflow.net/questions/279601 | 10 | Every game of Red-Blue Hackenbush represents a surreal number. Is the converse true? Assuming that it is false, what can be said about the class of surreal numbers that are representable by such games?
For instance, it is obvious that this class is a group under addition, but I can't visualize why it should even be c... | https://mathoverflow.net/users/113824 | What surreal numbers are representable by Red-Blue Hackenbush games? | All surreal numbers are representable by a Red–Blue Hackenbush game. This is discussed in *On Numbers and Games*, although it is left to the reader to fill in the details of the proof for the transfinite case. In Chapter 3 it is explained that every (surreal) number has a sign expansion. Then in Chapter 8 it is explain... | 4 | https://mathoverflow.net/users/3106 | 279729 | 123,930 |
https://mathoverflow.net/questions/279736 | 2 | If $ X $ is a toric variety and one has a closed sub-variety $ Y \subseteq X $, is the blow-up $ \operatorname{Bl}\_{Y}(X) $ a toric variety as well? I suspect not, but wanted to check with other users to see if someone had a counterexample or a proof. If my suspicions are correct, when is the blow-up a toric variety? ... | https://mathoverflow.net/users/113893 | When is the blow-up of a closed sub-variety of a toric variety a toric variety? | Usually the answer is no. Here are a couple of ways of proving $Bl\_Y(X)$ is not toric. I'll work over $\mathbb{C}$.
Example 1. Suppose $Bl\_Y(X)$ is a smooth and projective toric variety. Then the cycle class map
$$
CH^\*(Bl\_Y(X)) \rightarrow H^\*(Bl\_Y(X),\mathbb{Z})
$$
is an isomorphism. In particular, smooth tor... | 3 | https://mathoverflow.net/users/113898 | 279739 | 123,931 |
https://mathoverflow.net/questions/279737 | -2 | For any $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$ and let $S\_n$ be the set of all bijections (permutations) $\pi:[n]\to [n]$. For any set $X$ let $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big\}$. We let $\pi,\psi\in S\_n$ be connected by an edge if "they are one transposition away from each other", or more formally, set ... | https://mathoverflow.net/users/8628 | Chromatic number of transposition graph of permutations | For $n\geq 2$, we have
$$\chi(S\_n, E\_n) = 2.$$
It is at most $2$, since we can color permutations by their signs. It is obviously at least $2$.
| 6 | https://mathoverflow.net/users/39495 | 279744 | 123,932 |
https://mathoverflow.net/questions/255566 | 3 | The heat kernel in one dimension for the real line is given by the usual gaussian density function:
$$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$
In particular, by differentiating this function, one finds that for $|x-y|\leq \sqrt{T}$,
$$\sup\_{t\in [0,T]} g(t,x,y) =\frac{C}{|x-y|}\, ,$$
for some cons... | https://mathoverflow.net/users/46773 | Singularity of the heat kernel | Yes, away from the boundary: the heat kernel for the interval is given by $$\tag{1}g(t,x,y)=(2\pi t)^{-1/2}\sum\_{n\in\mathbb{Z}} (-1)^n \exp\left(-\frac{(x-y-n\pi)^2}{2t}\right),$$ and it is not difficult to show that the term corresponding to $n = 0$ is dominating for small time. It is in fact known that $$g(t,x,y) \... | 4 | https://mathoverflow.net/users/108637 | 279751 | 123,937 |
https://mathoverflow.net/questions/279734 | 1 | Let $m>n$ be positive integers. Consider the following sum:
\begin{equation}
S(m,n)=\sum\_{k=0}^n F\_{k+1} \frac{{m-1\choose{k}} {n-1\choose{k}}}{ {m+n-1\choose{2k+1}} {2k\choose{k}}},
\end{equation}
where $F\_k$ denotes the $k$th Fibonacci number. I would like to understand how $S(m,n)$ varies as a function of $m$ and... | https://mathoverflow.net/users/113490 | Series sum with coefficients that are Fibonacci numbers | First notice that
$$\frac{{m-1\choose{k}} {n-1\choose{k}}}{ {m+n-1\choose{2k+1}} {2k\choose{k}}} = \frac{(m-1)!(n-1)!(m+n-2-2k)!}{(2k+1)(n-1-k)!(m-1-k)!(m+n-1)!} = \frac{(2k+1)\binom{m+n-2-2k}{n-1-k}}{(m+n-1)\binom{m+n-2}{n-1}}.$$
Then $\binom{m+n-2-2k}{n-1-k}$ can be expressed as
$$\binom{m+n-2-2k}{n-1-k}=[x^{n-1-k}]\... | 4 | https://mathoverflow.net/users/7076 | 279766 | 123,941 |
https://mathoverflow.net/questions/279758 | 1 | Is Birkhoff-James orthogonality an orthogonality in the sense of Ratz?
**Orthogonality in the sense of Ratz:**
Suppose $X$ is a real vector space with $\dim X\geq2$ and $\perp$ is a binary relation on $X$ with the following properties:
1. *Totality* of $\perp$ for zero: $x\perp 0$ and $0\perp x$ for all $x$;
2. *... | https://mathoverflow.net/users/111987 | Birkhoff-James orthogonality and Ratz's orthogonality | Yes, indeed, Birkhoff-James orthogonality is an orthogonality in the sense of Rätz. A proof appears on p.36 of
>
> J. Rätz, On orthogonally additive mappings, Aequations Math. 28
> (1985), 35-49.
>
>
>
The argument there is a refinement of the proof given on p.188 of
>
> K. Sundaresan, Orthogonality and ... | 3 | https://mathoverflow.net/users/89429 | 279777 | 123,943 |
https://mathoverflow.net/questions/279705 | 7 | Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the surface is locally geodesically convex.
**Is it true that the number of connected components of the boundary is bounde... | https://mathoverflow.net/users/16183 | Estimate of number of boundary components of a compact Riemannian 2-surface | I think it follows from Gauss-Bonnet. Suppose $X$ has genus $g$ and $n$ boundary components. Gauss-Bonnet says that
$$\int\_X K\;dA+\int\_{\partial X}k\;ds=2\pi\chi(X)=2\pi(2-2g-n),$$
where $K$ is sectional curvature, $k$ is the curvature of the boundary. The local convexity implies that the boundary is positively curv... | 7 | https://mathoverflow.net/users/25051 | 279795 | 123,948 |
https://mathoverflow.net/questions/279784 | 3 | If $X$ is an Alexandrov space of curvature bounded below by a real number $k$, is it true that any geodesic in the tangent cone $T\_pX$ can be realized as a limit of geodesics when we view $T\_pX$ as the Gromov-Hausdorff limit of rescalings of neighborhoods of $p$?
More generally, if $X\_i$ is a sequence of Alexandro... | https://mathoverflow.net/users/52863 | Is any geodesic in the tangent cone of an Alexandrov space a limit geodesic? | The answer is "yes" assuming you are interested in *minimizing geodesics*.
Moreover the statement holds in the collapsing case as well.
Fix two points $p$ and $q$ in the limit space $X$.
If there is unique geodesic $[p,q]$ connecting these two points then we can choose arbitrary converging sequences $X\_n\ni p\_n\to ... | 4 | https://mathoverflow.net/users/1441 | 279799 | 123,951 |
https://mathoverflow.net/questions/279767 | 1 | The 3D fcc (face-centered-cubic) lattice, which has the same packing ratio as the 3D hexagonal close packed lattice, has the following 12 vectors connecting each vertex with its neighbors:
$(1,-1,0), (-1,1,0), (-1,-1,0), (1,1,0)$
$(1,0,-1), (-1,0,1), (-1,0,-1), (1,0,1)$
$(0,1,-1), (0,-1,1), (0,-1,-1), (0,1,1)$
... | https://mathoverflow.net/users/94774 | Closest vertex in a 3D fcc lattice | Recall that the face-centred cubic lattice comprises all vectors in $\mathbb{Z}^3$ whose coordinate sum is even.
Let $(x, y, z) \in \mathbb{R}^3$. For each coordinate, define the *discrepancy* to be the distance to the closest integer, i.e. $\delta(x) := |x - \lfloor x \rceil |$.
For the two coordinates with lowest... | 1 | https://mathoverflow.net/users/39521 | 279804 | 123,954 |
https://mathoverflow.net/questions/279542 | 2 | Let $p$ be a prime and consider the ($p$-deprived) Hecke algebra $\mathbb{T}$ which the projective limit of the Hecke $\mathbb{Z}\_p$-algebras $\mathbb{T}\_k$ which act on modular forms of level $1$ with coefficients in $\mathbb{Z}\_p$ and of weight at most $k$. (See $\S$2.1 of <https://www.math.uchicago.edu/~emerton/p... | https://mathoverflow.net/users/15899 | Krull dimension of Hecke algebra (level 1) for p = 2, 3 | When I posted this, I was under the impression that the Gouvea-Mazur infinite fern argument only holds when the residual representation is absolutely irreducible. Now I have learned that with this was because pseudo-deformations were not so well understood when GM wrote their article, but now the argument can be made t... | 1 | https://mathoverflow.net/users/15899 | 279816 | 123,959 |
https://mathoverflow.net/questions/279803 | 4 | I have shown that a solution to a nonlinear equation exists, and I am trying to show it is unique. Let *Y* > 0 be a continuous non-constant random variable, and $a\_1$, $a\_2$ real parameters. I have determined that if
$${E(Y^{a\_1+1})\over E(Y^{a\_1})}={E(Y^{a\_2+1})\over E(Y^{a\_2})}\Rightarrow a\_1=a\_2$$
then t... | https://mathoverflow.net/users/84415 | Implication from an equality in terms of expectations for uniqueness proof | Let $X$ and $Y$ be iid and non-constant with pdf $p$. Replace $a\_1$ and $a\_2$ by $a$ and $b$, and assume $a>b$. Then the equation is
$$\frac{E[X^{a+1}]}{E[X^{a}]} =
\frac{E[Y^{b+1}]}{E[Y^{b}]}$$
\begin{align}
0
&=E[X^{a+1}]E[Y^b]-E[X^a]E[Y^{b+1}]\\
&=E[X^{a+1}Y^b-X^a Y^{b+1}]\\
&= \iint (x-y)\,x^ay^b\, p(x)\,p(y)\... | 5 | https://mathoverflow.net/users/nan | 279819 | 123,960 |
https://mathoverflow.net/questions/279822 | 7 | I am interested in coverings of the (edge set of the) complete graph $K\_n$ by cycles of length $4$. It is clear that such coverings exist for each $n \ge 4$. I need to find the minimum number of $4$-cycles necessary to cover $K\_n$.
For example, $K\_5$ can be covered by following $4$-cycles $(1, 2, 3, 5), (2, 5, 4,... | https://mathoverflow.net/users/42586 | Minimum covers of complete graphs by $4$-cycles | If $n$ is odd, the answer is $\lceil \binom{n}{2}/4 \rceil$.
If $n$ is even, the answer is $\lceil \binom{n}{2}/4+n/8 \rceil$.
This follows from two special cases of a more general conjecture by Alspach.
For our purposes, we use a theorem of [Heinrich, Horák, and Rosa](https://www.researchgate.net/publication/26... | 6 | https://mathoverflow.net/users/2233 | 279830 | 123,963 |
https://mathoverflow.net/questions/279826 | 5 | Let $S$ be a submanifold of a real smooth manifold $M$. By a splitting of the normal bundle of $S$ I mean a sub-bundle $V$ of $TM|\_S$ such that $V\oplus TS = TM|\_S$.
*Question:* Given such a splitting, can I always find local coordinates $x\_1,\dots,x\_s,y\_1,\ldots, y\_r$ on $M$ around each point $p\in S$, such th... | https://mathoverflow.net/users/745 | Are all splittings of the normal bundle to a submanifold locally isomorphic? | Yes, one can always do this. Start with a $p$-centered local coordinate system $(x^\sigma,y^\rho)$ on an open $p$-neighborhood $U\subset M$ such that $S\cap U$ is given by $y^\rho=0$ $(1\le\rho\le r)$. Then there will exist functions $F^\sigma\_\rho(x)$ such that $V$ along $S\cap U$ is given by the equations
$$
\mathrm... | 8 | https://mathoverflow.net/users/13972 | 279837 | 123,965 |
https://mathoverflow.net/questions/279844 | 9 | Informally asking, can we step through all permutations of the set $\{1,\ldots,n\}$ by just using transpositions?
More formally: For any $n\in\mathbb{N}$ let $[n] = \{1,\ldots,n\}$ and let $S\_n$ be the set of all bijections (permutations) $\pi:[n]\to [n]$. For any set $X$ let $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big... | https://mathoverflow.net/users/8628 | "Gray code" of all permutations | From [V. L. Kompel'makher and V. A. Liskovets, "Sequential generation of arrangements by means of a basis of transpositions", *Kibernetika* **3**, 17, May-June, 1975](http://www.coga.tu-berlin.de/fileadmin/i26/coga/kompelmakherLiskovets.pdf):
>
> It is well known ([1], p. 28) that all $n!$ arrangements of $n$ symb... | 12 | https://mathoverflow.net/users/1847 | 279845 | 123,966 |
https://mathoverflow.net/questions/279842 | 9 | Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:
\begin{equation}\mathbf{P} = \begin{pmatrix}
1 & \beta & \alpha+\beta & \dots & (n-3)\alpha + \beta & (n-2)\alpha + \beta\\
0 & 1 & \beta & \dots & (n-4)\alpha + \beta & (n-3)\alpha + \beta\\
0 & 0 & 1 & \dots & (n-5)\alph... | https://mathoverflow.net/users/57020 | Inverse of special upper triangular matrix | Let $A$ be the nilpotent matrix $$\begin{pmatrix}0 & 1 & 1 & \cdots & 1 \\ & 0 & 1 & \cdots & 1 \\ & & \cdots & \cdots & \cdots \\ & & & 0 & 1 \\ & & & & 0\end{pmatrix},$$ then the matrix $P$ is equal to $1 + \beta A + \alpha A^2$.
This gives the inverse: \begin{eqnarray\*}P^{-1} & = & (1 + \beta A + \alpha A^2)^{-1}... | 20 | https://mathoverflow.net/users/76332 | 279850 | 123,969 |
https://mathoverflow.net/questions/279843 | 2 | I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information.
On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in Dynamical System - Lorenz Equations (<http://www.math.columbia.edu/~jb/bw-KPO-I.pdf>), Theorem 6.4 states that there are... | https://mathoverflow.net/users/113942 | Are all Torus Links in fact Lorenz links or not? | The point is that the two papers use slightly different definitions of "Lorenz Links".
The newer paper defines Lorenz links as links on the Lorenz template. With this definition all torus links are Lorenz links.
The older paper excluded links with a parallel cable around some component from the definition. So for e... | 6 | https://mathoverflow.net/users/39082 | 279851 | 123,970 |
https://mathoverflow.net/questions/279834 | 4 | Let $X$ be an affine spherical variety for some reductive algebraic group $G$. Let $X^0$ be the open orbit in $X$ under a fixed Borel subgroup $B \subseteq G$. Does there exists a function $f$ on $X$ such that $X^0$
is the same as $\{ \, x \in X \ | \ f(x) \neq 0 \, \}$?
I'm interested in any proof, counter-example o... | https://mathoverflow.net/users/108431 | Is the complement of the open $B$-orbit in a spherical variety cut out by one equation? | The answer is yes, even if $X$ is not normal. It even works for any connected solvable group acting on an affine variety with an open orbit.
To see this let $Y\_1,\ldots,Y\_r$ be the irreducible components of $X\setminus X^0$. Since the connected solvable group $B$ acts rationally on the ideal $\mathcal I(Y\_i)\subse... | 8 | https://mathoverflow.net/users/89948 | 279862 | 123,975 |
https://mathoverflow.net/questions/279870 | 28 | In the course of discussing [another MO question](https://mathoverflow.net/questions/279690/graph-to-bipartite-conversion-preserving-number-of-perfect-matchings?noredirect=1#comment689778_279690) we realized that we did not know the answer to a more basic question, namely:
>
> Is it true that for every positive int... | https://mathoverflow.net/users/3106 | Is every positive integer the permanent of some 0-1 matrix? | The answer to the question is yes. Given $k$, the 0-1 matrix given by
$1$ $1$ $\dotsc$ $1$ $0$ $0$ $\dotsc$ $0$ $0$ $0$ $0$
$0$ $1$ $1$ $0$ $\dotsc$ $0$ $0$ $\dotsc$ $0$ $0$ $0$
$0$ $0$ $1$ $1$ $0$ $\dotsc$ $0$ $\dotsc$ $0$ $0$ $0$
$\dotsc$
$1$ $0$ $0$ $0$ $0$ $\dotsc$ $\dotsc$ $0$ $0$ $0$ $1$
where the fi... | 27 | https://mathoverflow.net/users/108556 | 279872 | 123,978 |
https://mathoverflow.net/questions/279459 | 22 | Looking for an example of a symplectic manifold $(M,\omega)$ that is not symplectomorphic to $(M,-\omega)$.
In particular this means that $M$ must be chiral (i.e. doesn't admit an orientation-reversing diffeomorphism).
For a topological obstruction, I think it would be enough to find $(M,\omega)$ such that $\mathrm... | https://mathoverflow.net/users/91903 | $(M,\omega)$ not symplectomorphic to $(M,-\omega)$ | Let $n \geq 2$ be a natural number and $M$ a torus of dimension $2n$. Then a generic element of $H^2(M, \mathbb R)$ comes from a symplectic form, because we can take a $2$-form invariant under the torus action representing it, and it is a symplectic, and a generic such form is nondegenerate.
Thus it is sufficient to ... | 17 | https://mathoverflow.net/users/18060 | 279873 | 123,979 |
https://mathoverflow.net/questions/279875 | 2 | Let $R$ be a commutative semi-simple ring with unity (i.e. all modules over $R$ is semi-simple) , let $P \le N \le M$ be a chain of $R$ modules such that $M \cong M/N$ ; then is it true that $M \cong M/P$ ?
I can prove a kind of a dual version , that if $R$ is commutative semi-simple ring and $P \le N \le M$ is a ch... | https://mathoverflow.net/users/nan | A question on isomorphism between factor modules over commutative semi-simple ring | You can prove this using your dual version, for instance. By semisimplicity of the ring, we may fix a complementary submodule $N'$ for $N$ within $M$, as well as a complement $P'$ for $P$ within $N$, so that we have $$N = P \oplus P'$$ and $$M = N \oplus N' = P \oplus P' \oplus N'.$$ But then $N'$ is a submodule of $M$... | 1 | https://mathoverflow.net/users/778 | 279881 | 123,984 |
https://mathoverflow.net/questions/279835 | 2 | Let $X$ be a compact Riemann surface
and $D=\sum\_{j=1}^n\,(\theta\_j-1)\,P\_j$ be a ${\Bbb R}$-divisor on $X$ such that $\theta\_j\geq 0$ and $P\_1,\cdots,P\_n$ are $n$ distinct points on $X$.
We call $ds^2$ a conformal metric representing $D$ if $ds^2$ is a smooth conformal metric on $X\backslash {\rm Supp}\, D:=X\ba... | https://mathoverflow.net/users/104598 | Conformal hyperbolic metrics with mixed cone and cusp singularities | This is correct, and the same proof as in McOwen and Troyanov should work.
In fact they were not the first who proved this result. The story begins with E. Picard, who wrote several papers on this (also using PDE methods), and
the paper of M. Heins:
MR0143901 Heins, Maurice On a class of conformal metrics. Nagoya Mat... | 0 | https://mathoverflow.net/users/25510 | 279883 | 123,986 |
https://mathoverflow.net/questions/279764 | 6 | Let us work over $\mathbb{C}$ for the moment.
Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}\_K$.
$\mathbf{Question:}$ Is there a smooth projective curve $C$ of genus $g=2$ such that $End(Jac(C))$ is a non-maximal order in $K$, that is $End(Jac(C))=\mathcal{O}\_{K,f}=\mathbb{Z}+f\... | https://mathoverflow.net/users/70593 | Are there curves of genus 2 with real multiplication by a non-maximal order? | Yes, for every quadratic ring of discriminant $f^2 D$ (where $D$ is the discriminant of $K$ in your notation), there's a Humbert surface's worth of such Jacobians. See David Gruenewald's thesis
(available at <http://echidna.maths.usyd.edu.au/~davidg/thesis.pdf>) and the references there for calculations of Humbert su... | 3 | https://mathoverflow.net/users/2698 | 279902 | 123,994 |
https://mathoverflow.net/questions/279854 | 1 | Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in S$, $E\_s \cong F\_s$, where $E\_s$ and $F\_s$ are the restriction of $E$ and $F$ respectively to the fiber $\mathcal{X... | https://mathoverflow.net/users/32151 | Isomorphism of sheaves in families of projective varieties | As *@nfdc23* points out, even in the simplest case of $\pi=\mathrm{id}\_X$ your suggestion would amount to saying that for any two locally free sheaves of the same rank (at every point) there would be a line bundle that twists one to the other. Obviously this fails.
On the other hand if one approaches the problem a l... | 2 | https://mathoverflow.net/users/10076 | 279904 | 123,995 |
https://mathoverflow.net/questions/279891 | 5 | *This is a problem which has been bothering me for a while now; it doesn't seem inherently too hard, but I haven't been able to make any real headway, so I'm putting it out in the open since at this point I just want to know the answer. I don't think it has any deep value, but it's a natural question (at least to me) w... | https://mathoverflow.net/users/8133 | A game with boldface strength | The proof of the Kechris-Solovay theorem produces a real $x\_0$ such that whenever $x\_0 \leq\_T x$, every game lightface definable in $M\_x = (\omega, \{y : y\leq\_T x\},\in)$ is determined in $M\_x$. This shows there are many principal Turing ideals in which *lightface* PD holds. We basically copy the argument given ... | 6 | https://mathoverflow.net/users/102684 | 279905 | 123,996 |
https://mathoverflow.net/questions/279920 | 15 | The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this conjecture first and how old is it?
The Cartan matrix of a finite dimensional algebra is defined as the matrix having entr... | https://mathoverflow.net/users/61949 | Who conjectured the Cartan determinant conjecture | This seems to have first been explicitly stated as an open problem by Dan Zacharia in 1983, in [On the Cartan matrix of an Artin algebra of global dimension two](http://www.sciencedirect.com/science/article/pii/0021869383901564)
| 12 | https://mathoverflow.net/users/11260 | 279924 | 124,001 |
https://mathoverflow.net/questions/279852 | 8 | Let $f:\mathbb{R}^d\to \mathbb{R}$ be real analytic. Define $S=\{x\in\mathbb{R}^d, \nabla f (x)=0\} $. Is it true that for any compact set $K\subset \mathbb{R}^d$, $f(S\cap K)$ is a finite set ?
| https://mathoverflow.net/users/107004 | Critical values of analytic functions of several variables | One can assume that $K$ is a cube, by enlarging it. Then $S \cap K$ is a bounded definable set in the $o$-minimal structure $\mathbb{R}^{\mathrm{an}}$ (obtained by adding restricted analytic function, cf [this paper](https://www.jstor.org/stable/2118545?seq=1#page_scan_tab_contents)).
By the Yomdin-Gromov parametriz... | 7 | https://mathoverflow.net/users/21724 | 279928 | 124,003 |
https://mathoverflow.net/questions/279932 | 5 | Let $G$ be a group scheme over a scheme $X$ with centre $Z(G)$, automorphism group $\mathrm{Aut}(G)$ and outer automorphism group $\mathrm{Out}(G)$ (viewed as group schemes on $X$).
>
> 1. If $G$ is *finite flat* over $X$, then are $Z(G), \mathrm{Aut}(G)$ and $\mathrm{Out}(G)$ also finite flat over $X$?
> 2. If $G$... | https://mathoverflow.net/users/5101 | centre and automorphism groups of finite group schemes | The answers to 1. are all no since a finite flat group scheme can have fibers of very different isomorphism type.
Here is an example of a finite flat group scheme where $Z(G)$ is not finite flat: Let $k$ be algebraically closed of odd characteristic $p$ and $X=\mathbf A^1$. Let $G$ be the closed subscheme of $GL\_{3... | 4 | https://mathoverflow.net/users/89948 | 279939 | 124,007 |
https://mathoverflow.net/questions/279914 | 27 | If definitions themselves are informally just maps from words to collections of other words. Then in order for one to define anything, they must inherently already have a notion of a function. I mean of course one could ask what the exact "things" our functions are mapping to and from, and then revert back to set theor... | https://mathoverflow.net/users/38626 | Why aren't functions used predominantly as a model for mathematics instead of set theory etc.? | Let me explain one sense in which using functions or sets provides
exactly equivalent foundations of mathematics, in a way that is
connected with some deep ideas in set theory. There is a
translation back and forth between these foundational choices.
For example, it is a standard exercise in set theory to consider
ho... | 42 | https://mathoverflow.net/users/1946 | 279942 | 124,008 |
https://mathoverflow.net/questions/279927 | 8 | I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference.
(The original Alexandrov's gluing theorem was proven in dimension 2 only but under much weaker assumptions on the regularity of the metrics.... | https://mathoverflow.net/users/16183 | Multidimensional gluing theorem for Riemannian manifolds | Just learned that the answer is positive, at least its main part saying that if the sum of second fundamental forms is non-negative then the curvature of $M$ is at least $\kappa$. The answer is published here:
N. N. Kosovski˘ı, “Gluing of Riemannian manifolds of curvature ≥ κ”, Algebra i
Analiz 14:3 (2002), 140–157.
| 4 | https://mathoverflow.net/users/16183 | 279944 | 124,009 |
https://mathoverflow.net/questions/279779 | 15 | **The Thompson conjecture:** in a finite simple non-abelian group, there exists a conjugacy class such that every element of the group can be expressed as a product of two elements from that conjugacy class.
The conjecture is still open despite much progress (see below). There are plenty analogies between conjugacy c... | https://mathoverflow.net/users/10446 | Analogy between product of conjugacy classes and irreps: is there analog of Thompson conjecture ? | In the following article
>
> Heide, Gerhard; Saxl, Jan; Tiep, Pham Huu; Zalesski, Alexandre E.
> *Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type*. Proc. Lond. Math. Soc. (3) **106**
> (2013), no. 4, 908–930.
>
>
>
Heide, Saxl, Tiep, and Zalesski show that if ... | 13 | https://mathoverflow.net/users/22846 | 279953 | 124,013 |
https://mathoverflow.net/questions/279106 | 6 | I've edited, just skip the first attempt and go to the second one.
**THE FRAMEWORK:** let us consider a real topological vector space $V$.
We denote with $\mathscr C\_k(V)$ the set of all continous functions $f:[0,T]^k\to V$ such that $f\_{t\_1\cdots t\_k}=0$ whenever $t\_i=t\_{i+1}$ for some $0\le i\le k-1$.
We ... | https://mathoverflow.net/users/70148 | Well definition of a function | Let us define then $\mathfrak {U}\_n:=\Pi\_n\cup\mathfrak S\_n$ and $u\_n+2$ as the cardinality of $\mathfrak U\_n$; then, relabeling the elements of $\mathfrak U\_n$, we can write $\mathfrak U\_n=\{s=t\_0^n<t\_1^n<\cdots<t\_{u\_n}^n<t\_{u\_n+1}^n=t\}$
We recall that
\begin{align\*}
M\_{ts}^{\Pi\_n}=B\_{ts}-\sum\_{i... | 0 | https://mathoverflow.net/users/70148 | 279961 | 124,017 |
https://mathoverflow.net/questions/279912 | 7 | The recent paper ["A Type Theory for Synthetic $\infty$-Categories"](https://arxiv.org/abs/1705.07442) proposes the syntax as the theory of the strict interval. In principle, any other suitable theory could be used instead. For instance, using Joyal’s theory of disks would yield a type theory in which to study $(\infty... | https://mathoverflow.net/users/43138 | Type Theory to Study $(\infty,n)$-Categories and $(r,n)$-Categories | I don't know of anyone who is specifically working on generalizing our paper to the $(\infty,n)$-case. But there have been other attempts to design a type theory for higher categories, such as Finster's [opetopic type theory](https://ncatlab.org/nlab/show/opetopic+type+theory).
| 6 | https://mathoverflow.net/users/49 | 279964 | 124,019 |
https://mathoverflow.net/questions/279966 | 2 | Questions:
1. Is there a module of complexity one that is not periodic over a selfinjective algebra over a finite field?
2. Is there a module of complexity one that is not periodic over a symmetric algebra over a finite field?
One may replace finite field by any field that consists only of roots of unity.
Here co... | https://mathoverflow.net/users/61949 | Complexity one modules that are not periodic | Over a finite field, there are only finitely many modules with dimension less than $d$ for fixed $d$, so for some $n$ $M\cong\Omega^n$ for some $n$ if the algebra is self-injective with complexity one.
If every element of the field is a root of unity, then the field is a union of finite fields, and so the algebra and... | 3 | https://mathoverflow.net/users/22989 | 279968 | 124,022 |
https://mathoverflow.net/questions/235838 | 2 | Given $f\in C^{\infty} (E)$, where $E\subseteq \mathbb{C}$, define $E\_{\rho} \subseteq \mathbb{C}$ as the maximal ellipse with foci at $\{-1,1\}$ where $f$ is analytic, and semi-minor + semi-major axis summing to $\rho$.
**Question:** Is there any connection between $\rho$ and the modulus of continuity $\omega (\de... | https://mathoverflow.net/users/42864 | Modulus of Continuity for an Analytic Function on an Ellipse | There is indeed a relation between $\rho$ and the modulus of continuity $\omega\_{f}$ of $f$ on $[-1,1]$ which is obtained via the rate of polynomial approximation to $f$ on $[-1,1]$. Denote by $E\_{n}(f)$ the distance from $f$ to polynomials of degree at most $n$ with respect to the uniform norm on $[-1,1]$. By the cl... | 1 | https://mathoverflow.net/users/89429 | 279978 | 124,025 |
https://mathoverflow.net/questions/279969 | 28 | I'm seeking for a *Certificate of Positivity* for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
>
> Can one write the LHS as a sum
> $\,\sum\_i h\_i\,s\_i\,$ with real polynomials
> $\,h\_i(a,b,c,d,e)\,$ and $\,s\_i(a,b,c,d,e)$, where
>
>
> * each $... | https://mathoverflow.net/users/89757 | Wanted: Positivity certificate for the AM-GM inequality in low dimension | The following paper:
>
> Fujiwara, Kazumasa, and Tohru Ozawa. [Identities for the Difference between the Arithmetic and Geometric Means,](http://m-hikari.com/ijma/ijma-2014/ijma-29-32-2014/ozawaIJMA29-32-2014.pdf) (2014).
>
>
>
proves the following representation for **odd $n$:**
\begin{equation\*}
\frac{1}... | 39 | https://mathoverflow.net/users/8430 | 279980 | 124,026 |
https://mathoverflow.net/questions/279931 | 18 | How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{\*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p\_{1}(M) = n a^{2}$? ($p\_{1}(M) \in H^{4}(M,\mathbb{Z})$ denotes the first Pontryagin class).
By Wall's classification of $6$-ma... | https://mathoverflow.net/users/99732 | A search for a sequence of $6$-manifolds | I looked at Wall's paper [Classification problems in differential topology. V
On certain 6-manifolds](https://link.springer.com/article/10.1007%2FBF01389738). In theorem 3 of that paper Wall describes some invariants of 6-mainfolds, and the relation between them. These invariants, in the case you are concerned with, ar... | 18 | https://mathoverflow.net/users/184 | 279982 | 124,027 |
https://mathoverflow.net/questions/144619 | 12 | $\newcommand{\C}{\mathbf{C}} \newcommand{\D}{\mathbf{D}}$ Let $\C$ be a category with pullbacks. Taking any choice of pullbacks gives us re-indexing functors $f^\* \colon \C /Y \to \C/X$, and these will be functorial in $f$ up to natural isomorphism, in that $g^\* \cdot f^\* \cong (f \cdot g)^\*$. However, these will u... | https://mathoverflow.net/users/2273 | Can we always make a strictly functorial choice of pullbacks/re-indexing? | **No, it is not always possible to make a strictly functorial choice of pullbacks.** Four years later, I found a simple (if contrived) counterexample for this:
Let $\newcommand{\C}{\textbf{C}}\C$ be any full subcategory of $\textbf{FinSet}$ containing infinitely many sets of size 2, and at least one set of every fini... | 9 | https://mathoverflow.net/users/2273 | 279985 | 124,029 |
https://mathoverflow.net/questions/245068 | 4 | Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and also a Frobenius monad? In [this paper](https://arxiv.org/abs/1408.5809) they give examples of List-like monads called Containers and they give one that is a Comonad, namely Trees. The comonad axiom takes each node in the tre... | https://mathoverflow.net/users/10007 | A List-Like Frobenius Monad | I claim the only Frobenius monad on $\mathrm{Set}$ is the trivial monad given by the identity functor (which I guess needless to say isn't "very much like" the List monad).
Frobenius monads on a category $C$ are essentially the same as monads on $C$ whose underlying endofunctor is left adjoint to itself. For details... | 5 | https://mathoverflow.net/users/2926 | 279989 | 124,033 |
https://mathoverflow.net/questions/279864 | 5 | Let $K\_g \le \mbox{Mod}\_g$ denote the Johnson kernel subgroup of the mapping class group of a closed surface of genus $g$. [Dimca-Hain-Papadima](https://arxiv.org/pdf/1101.1392.pdf) find an explicit presentation for $H\_1(K\_g; \mathbb{C})$ as a $\mbox{Mod}\_g/K\_g$-module. According to Theorem B of their paper, $H\_... | https://mathoverflow.net/users/960 | A question about the abelianization of the Johnson kernel | I have found an explicit construction in terms of the (higher) Johnson homomorphism. Briefly put, the third Johnson homomorphism is an $\mbox{Sp}\_{2g}(\mathbb{Z})$-equivariant map
$$
\tau\_3: K\_g \to M
$$
for some $\mbox{Sp}\_{2g}(\mathbb{Z})$-module $M$ which can be shown to be isomorphic to $\mbox{Sym}^2(V(2))$ (he... | 2 | https://mathoverflow.net/users/960 | 279994 | 124,034 |
https://mathoverflow.net/questions/280020 | 8 | GCT purports to provide a program to show that $NP \not \subset P/poly$.
>
> 1. At the high level what are the steps involved in the program and what stage is each step in?
> 2. What difficulties currently are known or envisioned to be roadblocks and what are the easy targets?
>
>
>
| https://mathoverflow.net/users/10035 | Steps in Geometric Complexity Theory | First off, GCT is usually stated as a way of showing that $NP \not \subset P/poly$, which would imply that $P \neq NP$. This strategy was proposed by Mulmuley and several collaborators in a series of papers Geometric complexity theory I-VIII. These papers and several survey articles are available on [Mulmuley's website... | 11 | https://mathoverflow.net/users/33089 | 280032 | 124,044 |
https://mathoverflow.net/questions/275868 | 5 | What is an example of a smooth vector field $V$ on an open set of the plane which is a geodesible vector field but there is no a conformal metric $g$ such that $V$ is geodesible vector field with respect to $g$.
A geodesible vector field is a non vanishing vector field for which there is a Riemannian metric $g$ such ... | https://mathoverflow.net/users/36688 | Non conformally geodesible vector field | Here is how one can construct an example: Consider the smooth, nonvanishing $1$-form
$$
\omega = y^3(1{-}y)^2\,\mathrm{d}x + \big(y^3-2(1{-}y)^2\bigr)\,\mathrm{d}y.
$$
Note: This $\omega$ came from Exercises 5 and 6 of Section 16 of Chapter XVIII of Volume IV of Dieudonné's *Treatise on Analysis*. These exercises sho... | 8 | https://mathoverflow.net/users/13972 | 280041 | 124,047 |
https://mathoverflow.net/questions/280051 | 3 | We know that heat equation is hypoelliptic but not analytic-hypoelliptic; and also operators such as Cauchy-Reimann,laplacian etc are elliptic. I would like to know what operator is analytic-hypoelliptic but not elliptic.
| https://mathoverflow.net/users/102092 | Example of partial differential Operator that is analytic-hypoelliptic but not elliptic | A canonical example is the sub-Laplacian $L$ on the real 3-dimensional Heisenberg group $\mathbb{H}^3$. If we realize the Heisenberg group as $\mathbb{R}^3$, we can write $L = X^2 + Y^2$ where
$$X = \frac{\partial}{\partial x} - \frac{1}{2} y \frac{\partial}{\partial z}, \quad Y = \frac{\partial}{\partial y} + \frac{1}... | 4 | https://mathoverflow.net/users/4832 | 280057 | 124,052 |
https://mathoverflow.net/questions/280036 | 1 | This is a continuation of the discussion in the mathoverflow, [Pushforward of semi-stable sheaves](https://mathoverflow.net/questions/279034/pushforward-of-semi-stable-sheaves-under-finite-field-extension). Let $X$ be a smooth projective variety over a field $k$ and $L$ be a finite field extension of $k$. Denote by $p:... | https://mathoverflow.net/users/43198 | Pushforward of coherent sheaves and field extensions | There are at least two things that can go wrong. To explain this, let's carefully prove the following lemma.
>
>
> >
> > **Lemma.** Let $X$ be a finite type $k$-scheme, let $k \to \ell$ be a finite extension, and let $\mathscr F$ be a coherent sheaf on $X\_\ell$. Let $p \colon X\_\ell \to X$ and $q \colon X\_{k^{... | 3 | https://mathoverflow.net/users/82179 | 280060 | 124,053 |
https://mathoverflow.net/questions/280062 | 4 | The real linear space of matrices $$A=\begin{bmatrix}a&-c\\c&a\end{bmatrix},$$ with $a$ and $c$ real numbers, which satisfy the conditions (i) $\det(A)\geq0$ and (ii) $\det(A)=0\to A=0$, is a representation of the complex numbers. Is that true for all (special) spaces of $n\times n$ matrices satisfying these two condit... | https://mathoverflow.net/users/114025 | Are all special linear spaces representations of the complex numbers? | No: consider the 3-dimensional space of matrices $A=\begin{pmatrix}
0 & a & b & c\\ -a & 0 & c & -b\\ -b & -c & 0 & a\\ -c & b & -a & 0
\end{pmatrix}$. Then $\det(A)=\operatorname{Pf}(A)^2=(a^2+b^2+c^2)^2 $.
| 8 | https://mathoverflow.net/users/40297 | 280064 | 124,055 |
https://mathoverflow.net/questions/280072 | 11 | Let $\mathscr{M}\_{1,1,\mathbb{Z}}$ denote the moduli stack of elliptic curves.
>
> Does there exist a scheme $X$ and a finite group $G$ acting on $X$ such that $\mathscr{M}\_{1,1,\mathbb{Z}}$ is isomorphic to the quotient stack $[X/G]$?
>
>
>
Remarks/thoughts: For any scheme $S$, set $\mathscr{M}\_{1,1,S} := ... | https://mathoverflow.net/users/15505 | Is $\mathscr{M}_{1,1,\mathbb{Z}}$ isomorphic to a quotient stack by a finite group? | I guess I'll post my comments as an answer.
The definition of the quotient stack makes $p : X\rightarrow [X/G]$ into a $G$-torsor (in whatever topology $\mathcal{T}$ one chooses). Since here we're working with a finite abstract group, $X\rightarrow[X/G] = \mathcal{M}\_{1,1,\mathbb{Z}}$ is $\mathcal{T}$-locally isomor... | 11 | https://mathoverflow.net/users/15242 | 280078 | 124,060 |
https://mathoverflow.net/questions/280003 | 5 | Let $V$ be a 2-dimensional vector space (over, say, $\mathbb{Q}$). Let $FL$ be the free lie algebra on $V$, then there is a natural action of the group $SL(V)$ on $FL$, such that the action of $-I$ is by multiplication by $(-1)^d$ on the $d$th graded component.
An article I'm reading seems to imply that the represent... | https://mathoverflow.net/users/15242 | $SL_2$-action on the free lie algebra on a 2-dimensional vector space | (This was meant to be a comment; at the OP's request I copy it as an answer. I leave it CW)
Every (finite-dim) irreducible rep of $\mathrm{SL}\_2$ is isomorphic to $[n]$ for some $n\ge 1$. Hence every finite-dim rep of $\mathrm{SL}\_2$ is a direct sum of such $[n]$. This is just what they say, and is not related to y... | 3 | https://mathoverflow.net/users/14094 | 280080 | 124,061 |
https://mathoverflow.net/questions/280048 | 7 | [I have posted this question on MSE some time ago, but received no answer.]
The title basically says all of it.
If a normed space $F$ is a dual of a normed space $E$, then $F$ is a Banach space. I wonder if the same holds for Frechet spaces.
The strong dual $F$ of a locally convex space $E$ is complete, once $E$ ... | https://mathoverflow.net/users/53155 | Is any dual metrizable locally convex space a Frechet space? | I am not 100% clear what you are asking, but I will answer according to two interpretations:
a) Suppose that $F$, a metrizable TVS, is the strong dual of $E$, a locally convex TVS. Need $F$ be complete?
The answer to this question is no, by the following counterexample, where we obtain an incomplete normed space as... | 9 | https://mathoverflow.net/users/61785 | 280082 | 124,062 |
https://mathoverflow.net/questions/275219 | 8 |
>
> Given a symmetric monoidal category $Q$, is there a construction of a (preferably full and faithful strong) monoidal embedding of $Q$ into some symmetric monoidal closed category $M$ which has all filtered colimits for which the functor $A\otimes -$ preserves cofiltered limits (i.e., limits with a directed poset ... | https://mathoverflow.net/users/100607 | Monoidal tensor product which preserves directed limits | **Public Service Announcement!**
It's very confusing -- I'd daresay *incorrect* -- to say "directed limit" to mean "limit indexed by a cofiltered diagram". Actually, historically the term "directed limit" has been used to mean "colimit indexed by a filtered diagram" or even just "colimit"-- this usage predates the in... | 5 | https://mathoverflow.net/users/2362 | 280085 | 124,063 |
https://mathoverflow.net/questions/280075 | 15 | By plotting the function and its derivatives, one can easily be convinced that the function
$$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and $p \in (0,1)$, is completely monotone (i.e., for all $x$, $f(x)>0$, $f'(x)<0$, $f''(x)>0$, etc). How can this statement be... | https://mathoverflow.net/users/7581 | Proof of complete monotonicity of a binomial function | The following is the completely monotonic claim that actually holds (also hinted by Iosif Pinelis).
>
> **Claim.** Let $f(x)=\log\binom{x}{px}$; then, $f''$ is CM.
>
>
>
We prove this claim as a corollary of the following impressive generalization.
>
> **Theorem.** [(Karp and Prilepkina, 2015)](https://ar... | 10 | https://mathoverflow.net/users/8430 | 280086 | 124,064 |
https://mathoverflow.net/questions/280081 | 3 | Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes\_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ nilpotent?
| https://mathoverflow.net/users/43198 | Tensor product of field extensions | In general, if $A$ is a field of characteristic $0$ and $B/A, C/A$ are two finite extensions of $A$, then the tensor product $B \otimes\_A C$ is isomorphic to the product (i.e. direct product of $K$-algebras) of all "composita" (plural of "compositum") of $B/A$ and $C/A$. Here a "compositum" of the two extensions is a ... | 7 | https://mathoverflow.net/users/76332 | 280090 | 124,066 |
https://mathoverflow.net/questions/280092 | 6 | Let $\{a(n)\}$ be a sequence satisfies $a(1)=1$, $a(2)=2$, and $a(n)=a(n-1)+a(\lfloor\ln(n)\rfloor)$ for $n\geq 3$.
According to the definition, it seems that $a(n)=\Omega(n\ln n\ln\ln n\ln \ln \ln n ...\ln^{(k)}n)$ for any constant $k$.
Does $\sum\_{i=1}^{\infty}\frac{1}{a(i)}$ still diverge? If yes, how fast it d... | https://mathoverflow.net/users/114036 | The sum of the reciprocals of a sequence that increases by its logarithmic terms | Consider the function $f(x)=1/x$ on $[1,e]$ and extend it on $[1,\infty)$ by equality $e^xf(e^x)=f(x)$ for $x\geqslant 1$. Then both functions $f(x),xf(x)$ decrease on $[1,\infty)$.
At first, I claim that $1/a\_n\geqslant f(n)$. This is true for $n=1,2$, and we induct in $n$. Assume that the claim is proved for $1,\d... | 13 | https://mathoverflow.net/users/4312 | 280098 | 124,068 |
https://mathoverflow.net/questions/280046 | 1 | [I have posted this question on MSE some time ago, but received no answer.]
It is known, that if two locally convex topologies on a vector space determine the same collection of continuous linear functionals, then the classes of closed convex sets are the same, as well as the classes of bounded sets. Consequently, th... | https://mathoverflow.net/users/53155 | Topologies of pairs and closed bounded convex sets | No. Let $E=\bigoplus\_{\mathbb N} \mathbb R$ be the space of scalar sequences with only finitely many non-zero entries, $F=\mathbb R^{\mathbb N}$ with the duality $\langle x,y\rangle=\sum x\_ny\_n$, and $H$ be an algebraic complement in $F$ of the one-dimensional space generated by $(1,1,1,\ldots)$. Every $\sigma(E,H)$... | 2 | https://mathoverflow.net/users/21051 | 280103 | 124,070 |
https://mathoverflow.net/questions/280108 | 4 | Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}\_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$?
A good reference will be very appreciated.
| https://mathoverflow.net/users/90655 | When a Killing vector field on Riemannian manifold $(M,g)$ is gradient? | It is possible for this to happen (for example, the unit translation vector fields in $\mathbb{R}^n$), but it does not necessarily hold (for example, there are unit Killing vector fields on the stanard Riemannian $3$-sphere, but, obviously, none of them could be gradients, since they never vanish and the $3$-sphere is ... | 9 | https://mathoverflow.net/users/13972 | 280111 | 124,073 |
https://mathoverflow.net/questions/279949 | 4 | Given two finite dimensional algebra $A$ and $B$ such that $A$ is Gorenstein and $B$ is not. Can the trivial extension algebras of $A$ and $B$ be isomorphic? See <http://www.sciencedirect.com/science/article/pii/0022404984900586> 1.3. for the definition.
Gorenstein means here that the injective dimension of the regular... | https://mathoverflow.net/users/61949 | Question on trivial extension algebras | If $S$ is a finite dimensional algebra, and $M$ a finite dimensional $S$-bimodule, and if we construct the algebras $A=S\ltimes M$ and $B=S\ltimes DM$, then the trivial extension algebras $T(A)=A\ltimes DA$ and $T(B)=B\ltimes DB$ are isomorphic. This is straightforward to check ($T(A)$ and $T(B)$ are naturally isomorph... | 4 | https://mathoverflow.net/users/22989 | 280112 | 124,074 |
https://mathoverflow.net/questions/270792 | 2 | I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\mapsto\gamma(-t)$ is a geodesic of $\tilde{F}$. Let $\tilde{\gamma}(t):=\gamma(-t)$
The first part is done. But I have s... | https://mathoverflow.net/users/110243 | Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$ | I solve this exercise using the properties of $\tilde{F}$. Indeed one can first prove that $\tilde{g}\_v$ is an inner product. Next step prove that $\tilde{F}$ is a Finsler metric and so on. Then in is not difficult to prove that $\gamma$ is a $f$-geodesic iff $\tilde{\gamma}$ is a $\tilde{F}$ geodesic.
| 0 | https://mathoverflow.net/users/110243 | 280119 | 124,078 |
https://mathoverflow.net/questions/280083 | -1 | Can anyone help me figure out how the identity below was obtained?
$ \frac{1}{\sqrt{(e\_1-e\_3)(e\_2-e\_3)}} = R \prod \limits\_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{2n}} \right)^{-4} = R\left(\sum\_{n=1}^{\infty} \frac{1}{R^{n(n-1)}}\right)^{-4}. \tag{1}$
(Komatu, "A coeffici... | https://mathoverflow.net/users/111697 | Infinite sum and product associated with the Weierstrass elliptic function | From standard definitions of $e\_1,e\_2,e\_3$ and $\theta\_2$ in reference works such as Abramowitz and Stegun, (e.g., $\theta\_2(0,x)=2x^{1/4}\sum\_{n>0} x^{n(n-1)}$), it turns out that the following identity holds
$$(e\_1-e\_3)(e\_2-e\_3)=(\pi/(4\omega\_1))^4\theta\_2(0,\sqrt{q})^8$$ which does agree with the stated ... | 4 | https://mathoverflow.net/users/113409 | 280125 | 124,080 |
https://mathoverflow.net/questions/280127 | 1 | Let $A \in \mathbb{R}^{n \times n}$ be a symmetric positive definite matrix, and let $B \in \mathbb{R}^{n \times n}$ be an arbitrary matrix.
Define the numerical range or field of values of $B$ as
\begin{align}
W(B) = \left\{\frac{(Bv,v)}{(v,v)}, 0 \ne v \in \mathbb{C}^n \right\}
\end{align}
where $(\cdot,\cdot)$ is... | https://mathoverflow.net/users/95387 | Norm numerical range | If $A=R^\*R$ is the Cholesky factorization of $A$, then $$\frac{(ABv,v)}{(Av,v)} = \frac{(R^\*RBv,v)}{(R^\*Rv,v)} = \frac{(RBv,Rv)}{(Rv,Rv)} = \frac{(RBR^{-1}w,w)}{(w,w)}$$ for $w=Rv$, hence $W\_A(B) = W(RBR^{-1})$ and has "all the properties" of a numerical range.
| 5 | https://mathoverflow.net/users/1898 | 280128 | 124,081 |
https://mathoverflow.net/questions/150310 | 3 | In closure spaces (thus, also in topological spaces), one may define the boundary of a set A as the closure of A minus the interior of A. This set is partitioned into "the closure of A minus A" and "A minus the interior of A", that are equal, respectively, to "the boundary of A intersected with the complement of A" and... | https://mathoverflow.net/users/7674 | Name of the concept "Topological boundary of A intersected with A" | **Q.** “Do these two sets partitioning a boundary have a common name?”
**A.** Yes. They are the *rims* of $A$ and its complement.
**Q.** “Who studied them?”
**A.** Kar-Ping Shum
**References**
1. [2017 Shum](http://www.mathtransit.com/cornucopia/2017_shum_a.php)
2. [1996 Shum](http://www.mathtransit.com/corn... | 0 | https://mathoverflow.net/users/5090 | 280137 | 124,085 |
https://mathoverflow.net/questions/280143 | 1 | Let $f:A\rightarrow B, g:B\rightarrow C$ be maps in the category CGWH (compactly generated weakly Hausdorff spaces). Do we have a homotopy fibration sequence
$$F(f)\rightarrow F(gf) \rightarrow F(g)$$
consisting of homotopy fibres? (And the dual statement for homotopy cofibres?)
| https://mathoverflow.net/users/42571 | Homotopy fibre of composition | This is a fundamental and basic property in homotopy theory, as is the dual statement for homotopy cofibers. These appear as Lemmas 1.2.7 and 1.2.5 in the recently published book *More concise algebraic topology* by May and Ponto. (And in stable model categories these merge and become the octahedral axiom.)
| 9 | https://mathoverflow.net/users/102519 | 280149 | 124,089 |
https://mathoverflow.net/questions/280154 | 1 | I'm looking at a paper by Arnold et al. (CPDE, 2001), in which they make use of convex functions in the context of relative entropies. There, they assume that on $(0,\infty)$, their entropy function satisfies $\psi(1)=0$, $\psi''>0$, and $(\psi''')^2 \leq \frac{1}{2}\psi'' \psi^{(4)}$ for all values in $(0,\infty)$. Fo... | https://mathoverflow.net/users/51335 | Properties of Relative Entropies | If the assumption is $(\psi''')^2\le\frac12\,\psi''\psi^{(4)}$, then it can be written as $(1/\psi'')''\le0$. Therefore $1/\phi''$ is concave. Since it is positive over $(0,+\infty)$, it must be non-decreasing, that is $(1/\phi'')'\ge0$. This is precisely $\phi'''\le0$.
| 2 | https://mathoverflow.net/users/8799 | 280157 | 124,093 |
https://mathoverflow.net/questions/280155 | 1 | Fermat's two squares theorem tell us that every prime number $p \equiv 1 \pmod 4$ can be written in a unique way as $p = a^2 + b^2$ for two positive integers $a < b$. In particular, we can associate to $p$ an angle $\theta\_p = \arctan \frac{a}{b}$.
I am asking if it is known some result on the distribution of the va... | https://mathoverflow.net/users/114051 | A distribution related to Fermat's two squares theorem | Here is a good paper that can answer your question!
<https://arxiv.org/pdf/1705.07498.pdf>
| 3 | https://mathoverflow.net/users/76102 | 280160 | 124,095 |
https://mathoverflow.net/questions/280161 | 1 | Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f\_1,f\_2 \in C^2(\bar{\Omega})$. Suppose
$\int\_{\Omega}(f\_2-f\_1)\varphi \, dx=0$ and $\int\_{\Omega}(f\_2 \Delta^{-1} f\_2- f\_1 \Delta^{-1} f\_1)\varphi \, dx =0$
for all harmonic functions $\varphi$, where $\Delta^{-1}f=\int\_{\Omega}f(y)\Phi(x-y) \, dy... | https://mathoverflow.net/users/42326 | Functions orthogonal to harmonic functions | Assume without loss of generality that $0\in \Omega$.
Let $f\_1, f\_2$ be radial, with support contained in $\Omega$.
By the mean value property of harmonic functions, you have
$$ \int f\_2 \varphi ~\mathrm{d}x = 4\pi \varphi(0) \int\_0^{\infty} r^2 f\_2(r) ~\mathrm{d}r = \varphi(0) \int f\_2(r) ~\mathrm{d}x. $... | 4 | https://mathoverflow.net/users/3948 | 280164 | 124,097 |
https://mathoverflow.net/questions/280165 | 2 | Given $B>A>0$ and $C>0$. Let $\{X\_t\}\_{t=0}^{\infty}$ be a submartingale with $X\_0=A$ and
\begin{equation}
\mathbb{E}[X\_{t+1} | \mathcal{F}\_t] \geq X\_t + C.
\end{equation}
Let $ \tau := \min\{t:X\_t>B\}$. Under what condition can we upper bound $\mathbb{E}[\tau]$ roughly $\frac{B-A}{C}$?
For instance, define... | https://mathoverflow.net/users/94894 | Expected time for a submartingale increasing from A to B | I don't think your argument works as stated. You don't have $\eta\_{t \wedge \tau} \le B/C$ because $X\_\tau$ could be a lot larger than $B$. That is, maybe when $X\_t$ first exceeds $B$, it jumps a long way past it. An almost sure upper bound on the increment $|X\_t - X\_{t-1}|$ can fix it, but a bound on the expectat... | 3 | https://mathoverflow.net/users/4832 | 280172 | 124,099 |
https://mathoverflow.net/questions/280162 | 1 | In Miklos Bona's very nice "A Walk through Combinatorics", a following question is asked:
>
> Suppose you have two hundred balls placed in 100 urns, so that each
> urn contains at least one ball, and no urn contains more than 100
> balls. Then, there exists a subset of the urns which contains exactly
> 100 balls... | https://mathoverflow.net/users/11142 | counting ball in bin placements of a certain kind | As mentioned by Max Alekseyev in the comments, the number of subsets containing exactly $n$ balls is a coefficient of $x^n$ in the product $(1+x^{b\_1})\dots (1+x^{b\_n})$, where we denote $100=n$ and $b\_i$ denote the number of balls in $i$-th urn. Since $b\_i>0$ and $\sum b\_i=2n$, there are two more subsets with the... | 2 | https://mathoverflow.net/users/4312 | 280177 | 124,100 |
https://mathoverflow.net/questions/280176 | 7 | This [question](https://math.stackexchange.com/questions/547087/finitely-generated-group-which-is-not-finitely-presented/547144#comment4975221_547144) was answered by @Jim Belk And he defined $G\_n$ as follows:
$$
G\_n \;=\; \langle a,b \mid [a^{-1}ba,b] = \cdots = [a^{-n}ba^n,b]=1\rangle
$$
My question is:
Are $G\_... | https://mathoverflow.net/users/96862 | $G_n$ 's mutually non-isomorphic | Yes. For a group $G$, define $a(G)$ as the greatest $k$ such that $\mathbf{Z}^k$ embeds as a subgroup of $G$.
Then $a(G\_n)=n+1$ for all $n\ge 0$. Thus the $G\_n$, for $n\ge 0$, are pairwise non-isomorphic.
We have $a(G\_n)\ge n+1$ since $s\_k=a^{-k}ba^k$, $0\le k\le n$ generate a free abelian subgroup on $n+1$ gen... | 15 | https://mathoverflow.net/users/14094 | 280182 | 124,101 |
https://mathoverflow.net/questions/280186 | 8 | I have asked this question on [MathSE](https://math.stackexchange.com/questions/2341753/gaussian-distribution-maximum-entropy-and-the-heat-equation), but I got no replies, so I thought of trying here.
---
Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the exp... | https://mathoverflow.net/users/45285 | Gaussian distribution, maximum entropy and the heat equation | Both the Gaussian maximum entropy distribution and the Gaussian solution of the diffusion equation (heat equation) follow from the central limit theorem, that the limiting distribution of the sum of i.i.d. random variables with given average and variance is a Gaussian. The connection between the central limit theorem a... | 8 | https://mathoverflow.net/users/11260 | 280190 | 124,103 |
https://mathoverflow.net/questions/271576 | 0 | Let $(P,\leq)$ be a poset. We define the *order convergence topology*, denoted by $\tau\_o(P)$. By a *set filter* $\mathcal{F}$ on $P$ we mean a collection of subsets of $P$ such that:
* $\emptyset \notin \mathcal{F}$;
* $A, B\in \mathcal{F}$ implies $A\cap B\in \mathcal{F}$;
* $U\in \mathcal{F}$, $U'\subseteq P$ and... | https://mathoverflow.net/users/8628 | Product topology and order convergence topology | Please have a look at "Topologies on products of partially ordered sets I,II,III" by Marcel Erne. ([MR602017](http://www.ams.org/mathscinet-getitem?mr=602017), [MR602018](http://www.ams.org/mathscinet-getitem?mr=602018), [MR0631406](http://www.ams.org/mathscinet-getitem?mr=631406) (82m:54028c))
| 2 | https://mathoverflow.net/users/112206 | 280197 | 124,105 |
https://mathoverflow.net/questions/280168 | 1 | Let $d = (d\_1,...,d\_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with:
1. $V$ has orthonormal **columns**: $V^tV = I \in \mathbb{R}^{k \times k}$, and
2. $VDV^t \in \mathbb{R}^{m \times m}$ has un... | https://mathoverflow.net/users/89544 | Redistribute diagonal entries of a matrix | Yes, this works. Or, to be more honest, I'm fairly confident it does, but I'm only going to give a sketch.
The basic step is: a given symmetric $2\times 2$ matrix $A$ is unitarily equivalent to one with equal diagonal elements. This we can just check by direct computation.
Let's say
$$
A=\begin{pmatrix} a & b \\ b & ... | 1 | https://mathoverflow.net/users/48839 | 280203 | 124,107 |
https://mathoverflow.net/questions/280206 | 1 | This is a question that has emerged from a discussion about some topological properties of spacetimes.
Suppose that $M$ is a (Hausdorff, paracompact) smooth, orientable, connected, simply connected 4-dimensional manifold, which might be compact or noncompact, with or without the boundary. Is it true that the third de... | https://mathoverflow.net/users/114094 | Third de Rham cohomology group for the simply connected 4-manifolds | No. $\mathbb{R}^4 - \{ 0 \}$ obeys all your conditions, but it retracts onto the unit $3$-sphere, so $H^3(\mathbb{R}^4 - \{ 0 \}) \cong H^3(S^3) \cong \mathbb{R}$.
| 7 | https://mathoverflow.net/users/297 | 280207 | 124,108 |
https://mathoverflow.net/questions/280224 | 4 | I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly?
| https://mathoverflow.net/users/78863 | example of quintics with 5 ordinary triple point | Choose five points $P\_i$ in general linear position (all choices are
equivalent under ${\rm PGL}\_4$, so you might as well put four of them
at the coordinate vectors and the fifth at $(1:1:1:1)$); then
at each $P\_i$ the condition of a triple point imposes $1+3+6 = 10$
linear conditions on the space of quintics, which... | 6 | https://mathoverflow.net/users/14830 | 280226 | 124,114 |
https://mathoverflow.net/questions/280215 | 17 | **Question.** Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
| https://mathoverflow.net/users/61536 | Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$? | It looks like no.
Assume the contrary. We may start with two distinct rationals $q\_1,p\_1$ such that the sets $f(q\_1\times \mathbb{R}^{\omega-1})=f(\{(q\_1,\cdot,\cdot,\dots)\})$ and $f(p\_1\times \mathbb{R}^{\omega-1})$ intersect. Indeed, for any $p\_1$ the set $f(p\_1\times \mathbb{R}^{\omega-1})$ has non-empty ... | 16 | https://mathoverflow.net/users/4312 | 280228 | 124,115 |
https://mathoverflow.net/questions/280246 | 2 | I have a question about a passage from Springer's Linear Algebraic Groups (Birkhauser). The setup is: $F$ is an arbitrary field, $G$ is an $F$-split group, and $B$ is a Borel subgroup of $G$ containing a maximal $F$-torus $T$. This corresponds to a choice $R^+$ of positive roots. Let $D$ be the root basis. Let $\mathca... | https://mathoverflow.net/users/64244 | Automorphism of the Dynkin diagram | It is true that any automorphism stabilizing $B$ and $T$ induces an automorphism of $\mathcal D$. But Springer continues: *"Let $A\_0$ be the subgroup of $A$ of automorphisms of $\mathcal D$ obtained in this manner."*
So your quote is just the introduction for the definition of $A\_0$.
| 7 | https://mathoverflow.net/users/89948 | 280247 | 124,117 |
https://mathoverflow.net/questions/280251 | 8 | It is relatively easy (but sometimes quite cumbersome) to compute the minimal polynomial of an algebraic number $\alpha$ when $\alpha$ is expressible in radicals. For example, the simple query
>
> "minimal polynomial 2^(1/5)\*(1-exp(2\*pi\*i/5))"
>
>
>
to [Wolfram Alpha](https://www.wolframalpha.com/input/?i=m... | https://mathoverflow.net/users/22733 | Computation of a minimal polynomial | To compute the minimal polynomial of integer multiple of an algebraic integer is easy, so the only thing you need for linear combinations is the minimal polynomials of sums. Now, note that if $A$ is the companion matrix of $\alpha$ and $B$ is the companion matrix of $\beta,$ then
$A\otimes I + I\otimes B$ a companion ... | 9 | https://mathoverflow.net/users/11142 | 280253 | 124,121 |
https://mathoverflow.net/questions/280254 | 2 | Assume you have a set X (of points) and subsets $A\_i$ with the following conditions:
(1): For any two points in X exactly one of the sets contain them.
(2): Any two subsets intersect at most at one point.
The question is:
Can we find $|X|$ points in the plane such that the $A\_i$ are exactly the lines through ... | https://mathoverflow.net/users/45493 | Obstruction to embedding a point-line graph in $R^2$ | Note that the number of *ordinary lines* is [actually much bigger than one.](https://terrytao.wordpress.com/tag/dirac-motzkin-conjecture/) So if you have a design with few subsets of size $2,$ it cannot be realized.
| 2 | https://mathoverflow.net/users/11142 | 280255 | 124,122 |
https://mathoverflow.net/questions/280259 | 1 | Given two finite dimensional connected algebras A and B over a field $K$ with finite global dimension.
Their tensor product is not necessarily of finite global dimension when the field is not algebraically closed (seperable should be enough).
Is there an example where the tensor product of $A$ and $B$ is neither selfin... | https://mathoverflow.net/users/61949 | Tensor product of finite global dimension algebras | Let $K=k(t^p)$, where $k$ has characteristic $p>0$.
Then
$$k(t)\otimes \_K\pmatrix{k(t)&k(t)\\0&k(t)}\cong \pmatrix{K[s]/(s^p)&K[s]/(s^p)\\0&K[s]/(s^p)}.$$
| 4 | https://mathoverflow.net/users/22989 | 280264 | 124,125 |
https://mathoverflow.net/questions/280261 | 33 | Today ***homomorphism*** (resp. *isomorphism*) means what Jordan ([1870](https://archive.org/stream/traitdessubsti00jorduoft#page/56)) had called *isomorphism* (resp. *[holoedric](https://hsm.stackexchange.com/questions/3108/jordan-called-isomorphisms-iso-and-homomorphisms-iso-holoedriques-and-iso) isomorphism*). How d... | https://mathoverflow.net/users/19276 | Whence “homomorphism” and “homomorphic”? | I found this footnote on page 195 of Fricke and Klein's [Vorlesungen über die Theorie der automorphen Functionen](https://archive.org/details/vorlesungenber01fricuoft) (1897):
Translation:
>
> The term "homomorphic" seems more appropriate than the previous... | 49 | https://mathoverflow.net/users/11260 | 280271 | 124,129 |
https://mathoverflow.net/questions/280260 | 2 | Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence
$$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$
is exact. (If $H$ is a subgroup of $G$ this is equivalent to $K\cong G/H$)
Now, assume that $H,G,K$ are compact abelian topological groups, ... | https://mathoverflow.net/users/114118 | Profinite extension of a Lie group | Here is a counterexample.
Take $H=\prod\_{n\geq 1}\mathbb{Z}/n\mathbb{Z}$ (or any other profinite group into which $\mathbb{Z}$ injects). Let $\tilde{G}=H\times\mathbb{R}$, and let $G=\tilde{G}/\mathbb{Z}$, where $\mathbb{Z}$ sits inside $\tilde{G}$ diagonally. We get a short exact sequence
$$
1\to H\to G\to\mathbb{R... | 1 | https://mathoverflow.net/users/5263 | 280272 | 124,130 |
https://mathoverflow.net/questions/280071 | 4 | I would like to pick a random real normal (i.e. commuting with its transpose) matrix and I wonder if it can be done easily. I thought whether it would be possible to use a similar trick to drawing a symmetric matrix by first generating an arbitrary matrix A, and then setting $A=(A+A^T)/2$. Is there any similar normaliz... | https://mathoverflow.net/users/114031 | How to draw a random normal matrix? | Any real normal matrix $M$ can be written as $M=O\,\mathrm{diag}(B\_1,\ldots,B\_\ell)\,O^t$ where $O$ is orthogonal and where the blocks $B\_j$ are either $1\times 1$ real numbers or $2\times2$ matrices of the form:
$$
\left[\begin{matrix} a & b \\- b& a\end{matrix}\right],\qquad a\in\mathbb R,\qquad b>0.
$$
This provi... | 3 | https://mathoverflow.net/users/15517 | 280273 | 124,131 |
https://mathoverflow.net/questions/277021 | 7 | Let $A=A\_n$ be the algebra of upper triangular matrices over a field $K$ with $n$ simple modules.
It is a nice result that there are $C\_{n+1}=1,2,5,14,...$ (Catalan numbers for $n \geq 1$) tilting $A\_n$-modules, where a tilting module $T$ is a module with $n$ indecomposable summands (we assume all modules are basic)... | https://mathoverflow.net/users/61949 | Number of tilting modules | Let $e$ be the idempotent in $B$ such that $Be$ is the direct sum of the $n-l$ indecomposable projective-injectives *which do not have projective proper submodules*.
Then the two-sided ideal $BeB=Be$ is projective as a left $B$-module, so $B \to B/BeB$ is a homological epimorphism, see for instance
*Koenig, Steffe... | 6 | https://mathoverflow.net/users/18756 | 280277 | 124,132 |
https://mathoverflow.net/questions/280218 | 0 | Riemann's [prime counting function](http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html) is given as
$$J(n)=\sum\_{k=1}^{\infty}\frac{\mu(k)}{k}\operatorname{li}(n^{1/k})$$
the approximations
\begin{align}
\operatorname{li}(n)\sim J(n)\tag{1}\\
\operatorname{li}(n)-\sqrt{n}/\log n\sim J(n)\tag{2}\\
(1-... | https://mathoverflow.net/users/45057 | Asymptotic expansion of Riemann's prime counting function | To make it clear $\sum\_{k \ge 1} \frac{\mu(k)}{k} li(x^{1/k})$ is **not** a prime counting function.
The prime counting functions are $$\psi(x) = \sum\_{p^k \le x} \log p, \quad J(x) = \sum\_{p^k \le x} \frac{1}{k} = \int\_{2-\epsilon}^x \frac{\psi'(y)}{\log y} dy = \sum\_{k \ge 1} \frac{\pi(x^{1/k})}{k}, \quad \pi(... | 1 | https://mathoverflow.net/users/84768 | 280278 | 124,133 |
https://mathoverflow.net/questions/280193 | 9 | **Edited (after R. Bryant comment)**
Let $(M,\cal J,g)$ be a almost Hermitian manifold (*not necessary integrable*). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X\_i,{\cal J}X\_i\}$ be **any** local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$
$$g(Q{\cal... | https://mathoverflow.net/users/90655 | Almost Complex manifolds of constant curvature | Well, first of all, your conditions are vacuous if the dimension of $M$ is $2$, and the conclusion in that case is false. Thus, you must also assume, in order to get the conclusion, that the dimension of $M$ is $2n>2$.
It turns out that the answer is 'yes' if $M$ has dimension $4$, but my proof is not particularly si... | 8 | https://mathoverflow.net/users/13972 | 280290 | 124,136 |
https://mathoverflow.net/questions/280240 | 9 | Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of rational numbers, except in the rare cases where some iterate is a pole of $f$.
In the special case that $f(x) = \frac... | https://mathoverflow.net/users/6518 | Integrality of iterates of rational functions | As **Pasten** suggested in the comments, the key tool here is Siegel's theorem,
and this was already done by Silverman, see "Theorem A" in
>
> Joseph H. Silverman:
> Integer points, Diophantine approximation, and iteration of rational maps,
> *Duke Math. J.* **71** (1993) #3, 793--829.
>
>
>
**Proposition.**... | 10 | https://mathoverflow.net/users/14830 | 280292 | 124,137 |
https://mathoverflow.net/questions/280302 | 2 | The spaces $\mathbb{R}^n\setminus \mathbb{Q}^n$ and $(\mathbb{R}\setminus\mathbb{Q})^n$ with the Euclidean topology have the feeling of not being homeomorphic for $n>1$, because the "holes" in the former appear to be smaller in the former than in the latter, very informally speaking.
Is there any integer $n>1$ such t... | https://mathoverflow.net/users/8628 | $\mathbb{R}^n\setminus \mathbb{Q}^n$ vs $(\mathbb{R}\setminus\mathbb{Q})^n$ | For the usual Euclidean space $\mathbb{R}^n$, for the subset $X=\mathbb{R}^n\setminus \mathbb{Q}^n$ with the subspace topology, the space $X$ is path-connected. Indeed, for every $x=(x\_1,\dots,x\_n)\in X$, there exists some integer $1\leq i\leq n$ with $x\_i$ in $\mathbb{R}\setminus \mathbb{Q}$. Define the following f... | 10 | https://mathoverflow.net/users/13265 | 280304 | 124,141 |
https://mathoverflow.net/questions/280316 | 2 | I am looking at a particular integer sequence, the number of $n\times n$ Young Tableaus (see [OEIS](https://oeis.org/A039622)). In the comments at OEIS, Mitch Harris stated that the same sequence also defines the
>
> Number of linear extensions of the $n\times n$ lattice.
>
>
>
When looking for some more expl... | https://mathoverflow.net/users/41187 | Linear Extension of the $n\times n$ lattice | 1. The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
2. A linear extension of a poset $P$ with $m$ elements is just a bijection $f: P \to \{1,...,m\}$ such that $x \leq y$ in $P$ ... | 5 | https://mathoverflow.net/users/33089 | 280324 | 124,150 |
https://mathoverflow.net/questions/280328 | 4 | Let $f:\omega\to\mathbb N$ be a function such that $\sum\_{n=0}^\infty\frac{f(n)}{2^n}<\infty$.
We identify each natural number $n\in\mathbb N$ with the set $\{0,\dots,n-1\}$.
Then the map $$\sigma\_f:\prod\_{n=0}f(n)\to \mathbb R,\;\;\sigma\_f:(x\_n)\_{n\in\omega}\to\sum\_{n=0}^\infty\frac{x\_n}{2^n},$$ is well-d... | https://mathoverflow.net/users/61536 | What is the number of representations of a real number? | Consider the case where $f(n)=4$ for all $n$. If $0\leq a\leq 1$ then there is usually a unique sequence $x\in\{0,1\}^\omega$ with $\sigma\_f(x)=a$. Now for any subset $S\subseteq\mathbb{N}$ define $y^S$ by $(y^S\_{2i},y^S\_{2i+1})=(x\_{2i},x\_{2i+1})$ if $i\not\in S$, and $(0,x\_{2i+1}+2x\_{2i})$ if $i\in S$. This giv... | 4 | https://mathoverflow.net/users/10366 | 280330 | 124,154 |
https://mathoverflow.net/questions/280130 | 3 | Let $A \in \mathbb{R}^{n \times n}$ be nonsymmetric positive definite, if $A$ can be decomposed as $A = A\_1 \oplus A\_2$, where $A\_1 \in \mathbb{R}^{p \times p}$ and $A\_2 \in \mathbb{R}^{q \times q}$, $p+q=n$, it is known that
\begin{align}
W(A) = Co(W(A\_1) \cup W(A\_2))
\end{align}
where $W(A) = \left\{\frac{(Av,v... | https://mathoverflow.net/users/95387 | Numerical range in subspaces | The question is equivalent to search for the numerical range of overlapping block matrices.
The question is answered by studying overlapping matrices [here](https://math.stackexchange.com/questions/2416236/overlapping-positive-definite-block-matrices).
| 0 | https://mathoverflow.net/users/95387 | 280335 | 124,156 |
https://mathoverflow.net/questions/280314 | 12 | Are there simple, undirected graphs $G, H$ that are non-isomorphic, but there exist graph homomorphisms $f\_1: G\to H$ and $f\_2: H\to G$ which are *bijective* set-maps $V(G)\rightarrow V(H)$ and $V(H)\rightarrow V(G)$?
**Notes.**
* By the argument in Tobias Fritz's comment below, $G, H$ have to be infinite.
* As ... | https://mathoverflow.net/users/8628 | Non-isomorphic graphs with bijective graph homomorphisms in both directions between them | As vertex set, take $V=V'\cup V''$, the disjoint union of two infinite sets.
For $G$, take all edges except those joining pairs of vertices from $V''$.
For $H$, add one extra edge, between a pair of vertices $u,v\in V''$.
Then $G\not\cong H$, since if two vertices of $G$ are adjacent, then at least one of them is... | 24 | https://mathoverflow.net/users/22989 | 280338 | 124,158 |
https://mathoverflow.net/questions/280319 | 5 | An $\epsilon$-net of a closed hyperbolic surface $X$ is a finite set of points $p\_i$ such that the family of balls centered at $p\_i$ with radius $\epsilon$ is a cover of $X$, and the family of balls centered at $p\_i$ with radius $\epsilon/2$ are distinct pair by pair.
My question is that if there is a closed geode... | https://mathoverflow.net/users/105888 | Does there exist a closed geodesic go through a $\epsilon$-net of a hyperbolic surface? | Yes, such a closed geodesic always exists. See Theorem 1.1 of [this paper by Basmajian, Parlier, and Souto](https://arxiv.org/pdf/1610.08404.pdf) (which I found by searching under the term "density of closed geodesics on a hyperbolic surface").
Now, to be honest, what you want is simpler than what is proved in that p... | 5 | https://mathoverflow.net/users/20787 | 280339 | 124,159 |
https://mathoverflow.net/questions/280342 | 2 | Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
| https://mathoverflow.net/users/36688 | The number of limit cycles of a quadratic vector field with a unique singularity | [This survey](http://www.scholarpedia.org/article/Limit_cycles_of_planar_polynomial_vector_fields#Quadratic_Systems) seems to indicate that the answer is $1.$
| 2 | https://mathoverflow.net/users/11142 | 280346 | 124,162 |
https://mathoverflow.net/questions/280353 | 4 | Let $\mathscr{A} $ be a set of sets. Let's denote $\{A
\setminus B : A,B \in \mathscr{A}\}$ by $\mathscr{A} \setminus \mathscr{A} $.
The *Marica-Schönheim* theorem says that $|\mathscr{A} \setminus \mathscr{A}| \geq |\mathscr{A}|$ for every **finite** $\mathscr{A}$.
This immediately implies the result for counta... | https://mathoverflow.net/users/113612 | For every family $\mathscr A$ of sets, there are at least $|\mathscr{A}| $ sets of the form $A_1 \setminus A_2$ | The answer is yes, because from $A$ and $A-B$ and $B-A$, you can reconstruct $B$ via $$B=(B-A)\cup(A-(A-B)).$$ So if we fix $A$, we get a map from $(\mathscr{A}\setminus\mathscr{A})^2$ onto $\mathscr{A}$. So $\mathscr{A}\setminus\mathscr{A}$ must be at least as large as $\mathscr{A}$.
(Note that this argument used $... | 6 | https://mathoverflow.net/users/1946 | 280357 | 124,167 |
https://mathoverflow.net/questions/280320 | 3 | In Catalan's conjecture we have $$x^m-y^n=1$$ having solution $(3,2,1,1)$ and $(3,2,2,3)$.
Call $$ax^m-by^n=k$$ to be Pillai Diophantine equation.
>
> 1. Is it true no Pillai Diophantine equation exists with integer solutions $(x,y,m,n)$ and $(x,y,m+1,n+r)$ with $1<r$ true at $a=b=1\neq k$?
> 2. Is it also true a... | https://mathoverflow.net/users/10035 | Existence of Pillai equations with Catalan type solutions? | The answer to questions 1 and 2 are both no. In the example you consider, where $m = 1$, $n=1$ and $r = 2$, you are seeking solutions to $x-y = x^{2} - y^{3} = k$. The equation $x-y = x^{2} - y^{3}$ is an elliptic curve and the largest integral point on this curve gives you a solution for $k = -20$, namely
$$
-20 = (-... | 3 | https://mathoverflow.net/users/48142 | 280364 | 124,168 |
https://mathoverflow.net/questions/280220 | 7 | As is well known, the Saito-Kurokawa lifts maps (classical) cusp forms $f$ to Siegel (genus 2) cusp forms $SK(f)$.
>
> Is there an explicit formula for the Fourier expansion of a Saito-Kurokawa lift?
>
>
>
By explicit I mean something expressing the coefficients of $SK(f)$ in terms of the ones of $f$. Also, I ... | https://mathoverflow.net/users/98823 | Fourier expansion of the Saito-Kurokawa lift | Let's say $f$ is of weight $2k-2$ and $g$ is the associated form of weight $k - 1/2$. One can relate the Fourier coefficients of $g$ to those of the associated Jacobi form $J$, the Fourier coefficients of $J$ to those of the Saito-Kurokawa lift $F$ of $f$. See
>
> Agarwal, Mahesh; Brown, Jim. Saito-Kurokawa lifts o... | 3 | https://mathoverflow.net/users/6518 | 280370 | 124,171 |
https://mathoverflow.net/questions/280372 | 1 | I vaguely recall that resolution of singularity may be linked to continued fracton, possibly it is cusp that links to CF. Could any one give concrete reference and give example? Thanks.
| https://mathoverflow.net/users/14024 | How resolution of singularity is linked to continued fracton? | Discussed in [John Voight's paper.](https://math.dartmouth.edu/~jvoight/notes/cfrac.pdf) The paper does not appear to be published, I know not why.
| 2 | https://mathoverflow.net/users/11142 | 280374 | 124,172 |
https://mathoverflow.net/questions/280376 | -1 | [A Proposition from a book written by Benson Farb and Dan Margalit](https://i.stack.imgur.com/dbTFZ.png)
| https://mathoverflow.net/users/91288 | How to understand this isomorphism? | The question is terribly put , but the answer is: $S\_{0, 4}$ is the four times punctured sphere. You can think of this sphere as the ideal simplex in $\mathbb{H}^3$ (it is a theorem of mine that this is always possible). A simplex always has a Klein four-group worth of symmetries, hence the $\mathbb{Z}/2 \mathbb{Z} \t... | 3 | https://mathoverflow.net/users/11142 | 280377 | 124,174 |
https://mathoverflow.net/questions/280300 | 8 | If you are given an abelian group $\ (G, +)$, is there some algorithm to find **all** possible semigroups $\ (G, ×)$, such that $\ (G, +, ×)$ is a ring?
If not, can you at least decide, if the ring must have zero divisors?
| https://mathoverflow.net/users/114143 | Finding a compatible multiplication for a given group | **If not, can you at least decide, if the ring must have zero divisors?**
It would be interesting to know if there is a good answer to this question. Here I will just make three remarks. I will call an abelian group $G$ *good* if it can be equipped with a bi-additive multiplication which makes it a ring with no nontr... | 2 | https://mathoverflow.net/users/75735 | 280386 | 124,177 |
https://mathoverflow.net/questions/280388 | 8 | I have a question about Markov processes.
Let $\mathbb{M}=(X\_t,P\_x)$ be a Markov process on a locally compact separable metric measure space $(E,\mu)$.
$\mathbb{M}$ is called Feller process if its semigroup $\{p\_{t}\}\_{t>0}$ satisfies the following: for all $t>0$,
\begin{align\*}
(0)\quad p\_{t}(C\_{\infty}(E)... | https://mathoverflow.net/users/68463 | How to prove Feller property without using heat kernel estimates | In the case of a diffusion, (1) is implied for example by having bounded coefficients. This follows immediately from applying BDG to $X\_t-x$ and doesn't require (2) which is much harder to get.
Note by the way that (1) itself is quite a bit overkill since it rules out the OU process, which is the prototypical example... | 5 | https://mathoverflow.net/users/38566 | 280390 | 124,178 |
https://mathoverflow.net/questions/280359 | 6 | A topological space $\mathbf{X}$ is functionally Hausdorff, if for any two distinct $x, y \in \mathbf{X}$ there exists a continuous function $f\_{xy} : \mathbf{X} \to [0,1]$ with $f(x) = 0$ and $f(y) = 1$.
A space $\mathbf{X} = (X,\tau)$ is submetrizable, if there exists a topology $\tau' \subseteq \tau$ such that $(... | https://mathoverflow.net/users/15002 | Does second countable and functionally Hausdorff imply submetrizable? | **Fact.** Each second-countable functionally Hausdorff space is submetrizable.
This fact follows from a more general result:
**Theorem.** Each functionally Hausdorff space $X$ with hereditarily Lindelöf square $X\times X$ is submetrizable.
*Proof.* Denote by $\Delta$ the diagonal of the square $X^2:=X\times X$. F... | 5 | https://mathoverflow.net/users/61536 | 280401 | 124,182 |
https://mathoverflow.net/questions/280368 | 8 | Consider the "$m$-th power" map $f:K(\mathbb Z,n)\to K(\mathbb Z,n)$ given by $m\in \mathbb Z\cong H^n(K(\mathbb Z,n),\mathbb Z)\cong [K(\mathbb Z,n), K(\mathbb Z,n)]$. Is it true that in any degree the map $f^\*$ on integral cohomology sends any element to a multiple of $m$? It's obviously true in degree $n$ where $f^... | https://mathoverflow.net/users/18050 | Action on cohomology of the power map of $K(Z,n)$ | As you say, we can reduce to the case where $m=p$ is prime. By the universal coefficient theorem we know that $H^\*(K(\mathbb{Z},n);\mathbb{Z})/p$ injects in the ring $A^\*=H^\*(K(\mathbb{Z},n);\mathbb{Z}/p)$, so we just need to show that $f^\*$ acts as zero on $A^\*$ in positive degrees. The kernel of $f^\*$ is an ide... | 4 | https://mathoverflow.net/users/10366 | 280403 | 124,183 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.