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/422670 | 10 | My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly.
Given a ringed space $(X,\mathcal{O}\_X)$ and ideal sheaves $\mathcal{I},\mathcal{J}\subset\mathcal{O}\_X$, we define the ideal product presheaf $\mathcal{I}\cdot\_p\mathcal{J}$ as th... | https://mathoverflow.net/users/101848 | Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't? | It need not be a sheaf. As an example, consider a space $X$ which is a disjoint union of open subspaces $X\_n$, and pick $\mathcal O\_X,\mathcal I,\mathcal J$ with the property that some element $c\_n$ of $\mathcal I(X\_n)\mathcal J(X\_n)$ cannot be written as a linear combination of fewer than $n$ products of elements... | 11 | https://mathoverflow.net/users/30186 | 422676 | 171,873 |
https://mathoverflow.net/questions/422666 | 0 | Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by
$$
\alpha(\mu,\Sigma,t) := \sup\_{\|w\| = 1}\frac{1}{\|w\|\_\Sigma}\varphi\left(\frac{w^\top \mu - t}{\|w\|\_\Sigma}\right),
$$
where $\|w\|\_\Sigma := \sqrt{w^\top \Si... | https://mathoverflow.net/users/78539 | Functional relationship between two quantities | In fact, even when $\Sigma=I$, we have
$$I\_1(t,\mu):=\inf\_{\|w\|=1}|w^\top\mu-t|=(t-\|\mu\|)\_+ \tag{1}\label{1}$$
(think of large $t$), whereas
$$I\_2(t,\mu):=\inf\_{\|z\|\le t}\|z-\mu\|=(\|\mu\|-t)\_+ \tag{2}\label{2}$$
(think of large $\|\mu\|$).
So, there is no functional relationship between your $\alpha$ and ... | 1 | https://mathoverflow.net/users/36721 | 422680 | 171,876 |
https://mathoverflow.net/questions/422402 | 7 | Recall that a [$q$-Pochhammer symbol](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol) is defined as
$$
(x)\_n = (x;q)\_n := \prod\_{l=0}^{n-1}(1-q^l x).
$$
I found the following curious $q$-series identity that seems to hold for any $n\geq 0$:
$$
(-1)^{n+1}q^{\frac{(n+1)(3n+2)}{2}}\sum\_{j\geq 0}q^{j}(q^{j+1})\_{n... | https://mathoverflow.net/users/45553 | A curious $q$-series identity on a truncated Euler function | Let ${n\choose k}\_q=\frac{(q)\_n}{(q)\_k(q)\_{n-k}}$ denote a $q$-binomial coefficient. We start with the following version of $q$-Vandermonde convolution identity:
$$
(x-y)(x-qy)\ldots(x-q^{n-1}y)\\=\sum\_{k=0}^n(-1)^{k}q^{k(k-1)/2}{n\choose k}\_q(y)\_{n-k}(xq^{1-k})\_k\quad\quad(\diamondsuit)
$$
A short proof of $(\... | 9 | https://mathoverflow.net/users/4312 | 422683 | 171,877 |
https://mathoverflow.net/questions/422689 | 1 |
>
> Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all orders is contained in any fractional Sobolev space $H^k(\mathbb R^n)$ ? Thanks in advance.
>
>
>
After all, at... | https://mathoverflow.net/users/78539 | Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? | One way to approach such questions is to start with an abstract situation (Hilbert scales). If $T$ is an unbounded s.a. operator with $T>1$ on a separable Hilbert space (in your case, the Schrödinger operator on the usual $L^2$-space), then it embeds in a natural way into an increasing family $H^\alpha$ (with $\alpha \... | 3 | https://mathoverflow.net/users/482310 | 422691 | 171,881 |
https://mathoverflow.net/questions/422646 | 9 | Let $F$ be the non-archimedean local field $\mathbb{Q}\_p$ for some prime $p$ and $D$ be a quaternion division algebra over $F$. Let $\mathcal{O}\_D$ and $\mathcal{P}\_D$ denote the ring of integers of $D$ and its unique maximal ideal (respectively). Then, what is the finite group
$$ \frac{D^\*}{F^\*(1+ \mathcal{P}\_D)... | https://mathoverflow.net/users/140574 | What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$? | Yes, we can.$\newcommand{\order}{\mathcal{O}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\prim}{\mathcal{P}}$ $\newcommand{\F}{\mathbb{F}}$
First, let me remind you of the following explicit description of $\order\_D$. I won't use it explicitly but it is convenient to check some of my claims below. Let $\pi$ be a un... | 4 | https://mathoverflow.net/users/40821 | 422694 | 171,882 |
https://mathoverflow.net/questions/422709 | 1 | Let $\mathbb{F}\_q$ be a finite field, $\psi$ be a non-trivial additive character over $\mathbb{F}\_q$, and $a, b \in \mathbb{F}\_q$ constants. Is there any known estimate for the gaussian sum
$$\sum\_{x \in \mathbb{F}\_q} \psi( a x^m + b x^n),$$
possibly for specific values of $m, n \in \mathbb{Z}\_{\ge 2}$, $m \n... | https://mathoverflow.net/users/269936 | Known estimate for gaussian sum $\sum_{x \in \mathbb{F}_q} \psi( a x^m + b x^n)$? | As Ofir Gorodetsky notes in the [comments](https://mathoverflow.net/questions/422709/known-estimate-for-gaussian-sum-sum-x-in-mathbbf-q-psi-a-xm-b-xn#comment1086256_422709), Weil's bound gives that the absolute value of the sum is most order $\max(m,n) \sqrt{q}$. This is non-trivial as long as $m$ and $n$ are $o(\sqrt{... | 3 | https://mathoverflow.net/users/630 | 422711 | 171,886 |
https://mathoverflow.net/questions/422190 | 1 | I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves.
The setting is the following: We have a DVR with uniformiser $t$, characteristic of the residue field not $2$ or $3$ and the arithmetic surface $\operatorname{Spec}(R[x,y]/(y^2-t(x^3+t^3))$. He claims the ... | https://mathoverflow.net/users/481865 | Singularities of arithmetic surface | There seems to be some confusion concerning "regularity" and "smoothness".
First of all, if $X$ is a noetherian integral scheme and $x\in X$, then $X$ is regular at $x$ if $$\mathrm{dim} \ m\_x/m\_x^2 = \mathrm{dim} X.$$ On the other hand, given a *morphism* of schemes $\mathcal{X}\to S$, one can also look at its smo... | 1 | https://mathoverflow.net/users/4333 | 422727 | 171,889 |
https://mathoverflow.net/questions/422718 | 2 | We say that a topological space $A$ is homotopy dominated by a topological space $X$ if there exist continuous maps $f:A\to X$ and $g:X\to A$ such that $g\circ f\simeq 1\_A$.
Let $X$ be $S^2 \times S^2 \times S^2$. I'm trying to show by the Whitehead Theorem that if $A$ is homotopy dominated by $X$, then $A$ is homot... | https://mathoverflow.net/users/114476 | Spaces homotopy dominated by $S^2 \times S^2\times S^2$ | Put
$$ R(n)=H^\*((S^2)^{\times n}) = \mathbb{Z}[x\_1,\dotsc,x\_n]/(x\_1^2,\dotsc,x\_n^2) $$
A key point is that if $u\in R(n)$ with $|u|=2$ then $u^2=0$ iff $u=0$ or $u=m\,x\_i$ for some $m\in\mathbb{Z}\setminus\{0\}$ and some $i$; this is easy to check.
Let $\phi\colon R(n)\to R(n)$ be a map of graded rings with $\p... | 6 | https://mathoverflow.net/users/10366 | 422739 | 171,891 |
https://mathoverflow.net/questions/422717 | 5 | As mentioned by Willie Wong, I modified to the following version:
Let $M$ be a closed smooth $4$ manifold.
**Q**
Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, such that the $\int\_MScal^2\_gdv\_g=c$, where $Scal\_g$ denotes the scalar curvature of $g$? If not, for any small ... | https://mathoverflow.net/users/95296 | Can we prescribe the $L^2$ norm of the scalar curvature on a four-manifold? | This is not always possible.
Let $M$ be a compact smooth manifold of dimension $n$. Consider the Einstein-Hilbert functional $\mathcal{E}$ given by
$$\mathcal{E}(g) = \dfrac{\displaystyle\int\_Ms\_g d\mu\_g}{\operatorname{Vol}(M, g)^{\frac{n-2}{n}}}.$$
If $\mathcal{C}$ is a conformal class, then by using the conf... | 11 | https://mathoverflow.net/users/21564 | 422742 | 171,893 |
https://mathoverflow.net/questions/422759 | 2 | For $n \geq 0$, let $a\_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum\_{\lambda \vdash n} s\_\lambda^2$, where $s\_\lambda$ are the symmetric Schur functions and the sum runs over all partitions $\lambda$ of $n$. These numbers can also be described via the generat... | https://mathoverflow.net/users/3621 | Conjectural congruences for numbers related to Littlewood-Richardson coefficients | This is true, if you replace $n-1$ in "the exponent in the highest power of 2 dividing $n−1$" to $n$.
First of all, we study the coefficients of the series $(1-4t)^{-1/2}=\sum C\_nt^n$ modulo 4. We have $C\_n=(-4)^n{-1/2\choose n}=2^n\frac{(2n-1)!!}{n!}$. We have for 2-adic valuation: $$\nu\_2(n!)=\sum\_{k=1}^\infty ... | 3 | https://mathoverflow.net/users/4312 | 422765 | 171,899 |
https://mathoverflow.net/questions/422761 | 0 | Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra.
1. Can an element of $L\_\infty(\Omega)\overline{\otimes}\mathcal M$ be regarded as a measurable map from $\Omega$ to $\mathcal M$? A reference where the properties of $L\_\infty(\Omega)\ov... | https://mathoverflow.net/users/136860 | Semi-commutative von Neumann algebras | Assuming $H$ is separable, $L^\infty(\Omega)\overline{\otimes} \mathcal{M}$ can be identified with the essentially bounded weakly measurable functions from $\Omega$ into $\mathcal{M}$. (Weakly measurable = its composition with any normal state on $\mathcal{M}$ is measurable.) This is a minor variation on Theorem 6.5.8 ... | 4 | https://mathoverflow.net/users/23141 | 422768 | 171,900 |
https://mathoverflow.net/questions/422737 | 2 | Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x\_0:x\_1:x\_2:x\_3]\in\mathbb{C}P^3\colon X\_0^4+x\_1^4+X\_2^4+X\_3^4=1\}$. It is well known that $H^2(Y,\mathbb{Z})\cong U^{\oplus 3}\oplus(-E\_8)^{\oplus 2}\cong \mathbb{Z}^{\oplus 22}$.
From the classification resul... | https://mathoverflow.net/users/136570 | Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface | You can say a fair amount about the topology of the total spaces of the different bundles, although I suspect none of them is a particularly well-known manifold that has a `name'. (Except of course for the trivial bundle; I guess you could say $S^1 \times$ a K3 surface is a well-known manifold.)
The main observation ... | 2 | https://mathoverflow.net/users/3460 | 422775 | 171,902 |
https://mathoverflow.net/questions/422721 | 0 | I am wondering if there is any literature on the following combinatorial optimization problem:
* **Input**: $n, k, T\in \mathbb{N}$ and positive integers $s\_1, \ldots, s\_n$.
For intuition, we may think that $n$ represents a number of players; $k$ represents the number of players that form a team; $s\_i$ represent... | https://mathoverflow.net/users/482421 | Maximum number of teams of fixed size over a score threshold | You can solve the problem via integer linear programming (ILP) as follows. Let $I=\{1,\dots,n\}$ be the set of players, and let $J=\{1,\dots,\lfloor n/k \rfloor\}$ be the set of potential teams. Let binary decision variable $x\_{i,j}$ indicate whether player $i$ is on team $j$, and let binary decision variable $y\_j$ i... | 0 | https://mathoverflow.net/users/141766 | 422780 | 171,904 |
https://mathoverflow.net/questions/422735 | 13 | I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$,
$$Z(a,b) = \sum\_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{2})^b},$$
which are converging very slowly. For these sums, there is also an alternative expression as an iterated ... | https://mathoverflow.net/users/10881 | Accelerating convergence for some double sums | This is easy to do with [PARI/GP](https://pari.math.u-bordeaux.fr/).
Here is my code
```
p(n) = binomial(n+2,2);
Y(k,b) = sumnum(l=0, (2*l+3)/(p(k)+p(l))^b,sumtable);
Z(a,b) = sumnum(k=0, (2*k+3)/p(k)^a*Y(k,b));
default(realprecision,57);
sumtable = sumnuminit();
print(2*Z(2,2)+4*Z(1,3))
/* 16.0000000000000000000000... | 10 | https://mathoverflow.net/users/113409 | 422790 | 171,908 |
https://mathoverflow.net/questions/422757 | 5 | There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
1. A functor $C\times D^o \to \text{Set}$
2. A co-continuous functor between presheaf categories $\hat C \to \hat D$
These are equivalent by the free co-completion property of the Yoneda emb... | https://mathoverflow.net/users/82445 | Strictness of two operations on proarrow equipments | I believe the answer to (2) is yes.
First, apply the strictification theorem for bicategories twice, to make composition of arrows and proarrows both strictly associative. Thus, when our equipment is regarded as a double category, we have a strict double category. (We could also probably apply some coherence theorem ... | 2 | https://mathoverflow.net/users/49 | 422798 | 171,909 |
https://mathoverflow.net/questions/422746 | 2 | Let $G$ be an adjoint algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra.
Let $\rho:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $g,g'\in G$ be two semisimple elements such that $\rho(g),\rho(g')$ are conjugated in $GL(\mathfrak{g})$, when do we have that $g$ and $g'$ are conjugated?... | https://mathoverflow.net/users/27398 | conjugacy in adjoint representation | Here is a partial answer. Let first $G=PGL(n)$. Let $g$ have eigenvalues $\lambda\_1,\ldots,\lambda\_n$ (up to a scalar). Then $\rho(g)$ has eigenvalues $\lambda\_i\lambda\_j^{-1}$ with $i,j=1,\ldots,n$. This set invariant under $z\mapsto z^{-1}$. So $\rho(g)$ is always conjugate to $\rho(g^{-1})$. But for $n\ge3$ ther... | 2 | https://mathoverflow.net/users/89948 | 422800 | 171,911 |
https://mathoverflow.net/questions/422534 | 7 | Here is a collection of facts that all seem true, but together seem to give a nonsensical solution:
1. After $T(n)$-localization, all natural transformations $F \sim G$ between homogenous functors $F,G:Sp \rightarrow Sp$ of different degrees appear trivial, in the sense that $\operatorname{cofiber}(\sim) \simeq G \ve... | https://mathoverflow.net/users/134512 | Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory | This is an elaboration on my comment above. Let us consider the natural transformation induced by the diagonal
$$\Sigma^\infty\Delta\colon \Sigma^\infty X \to \Sigma^\infty X\wedge X.$$
The natural transformation $\Sigma^\infty \Delta$ induces a map between the Goodwillie derivatives of the functors. Let us denote this... | 6 | https://mathoverflow.net/users/6668 | 422808 | 171,914 |
https://mathoverflow.net/questions/422781 | 1 | If we consider a finite set $A\subset\mathbb R^n$, uniqueness of the convex decomposition of points in $A$ is equivalent to the absence of $\mu\neq0$ signed measure supported on $A$ such that $\mu(\mathbb R^n) = 0$ and,
$$
\int\_{\mathbb R^n}x\mathrm d\mu(x)=0.
$$
My question is, what happens when $A$ is a measurable s... | https://mathoverflow.net/users/482476 | Measurable sets of $\mathbb R^n$ forming unique absolutely continuous convex combinations? | There is no such set. Given an $A\subseteq\mathbb R^n$, we can pick arbitrarily many disjoint positive measure subsets $A\_j$, $j=1,\ldots ,N$, and consider measures of the form
$$
d\mu = \left( \sum\_j c\_j \chi\_{A\_j}\right)\, dx .
$$
The conditions we're trying to satisfy lead to a homogeneous linear system on the ... | 0 | https://mathoverflow.net/users/48839 | 422826 | 171,918 |
https://mathoverflow.net/questions/422824 | 3 | Let $c>0$ and $T>0$ be fixed. Denote by $F$ the Gaussian CDF, i.e. $F:\mathbb R\to\mathbb R$ is defined by
$$F(x):=\int\_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2}dz.$$
For every $a\in [0,1)$, does there exist $C\equiv C\_a>0$ s.t.
$$\left|\int\_0^\infty \frac{1}{s-t}\left(F\left(\frac{y}{\sqrt{c(s-t)}}\right)-... | https://mathoverflow.net/users/261243 | On an integral of Gaussian CDFs | No. Indeed, let
$$u:=\frac x{\sqrt{c(s-t)}}.$$
Then, with the substitution $z:=\dfrac y{\sqrt{c(s-t)}}$, the inequality in question can be rewritten as
$$|I(u)|\le C|u|^a \tag{1}\label{1}$$
for all real $u$, where
$$I(u):=\int\_0^\infty dz\,(F(z)-F(z-u)).$$
For $u>0$,
$$I(u)=\int\_0^\infty dz\,\int\_{z-u}^z dF(t)
=\i... | 2 | https://mathoverflow.net/users/36721 | 422828 | 171,920 |
https://mathoverflow.net/questions/422838 | 1 | If $U$ is a $N\times N$ random unitary matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ from it has distribution given by
$$ \det(1-AA^\dagger)^{N-2M}.$$
If $O$ is a $N\times N$ random real orthogonal matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ ... | https://mathoverflow.net/users/78061 | Distribution of top left block from unitary symmetric matrices | The general formula in the circular ensembles follows from equation 2.10 of [arXiv:cond-mat/9612179](https://arxiv.org/abs/cond-mat/9612179):
$$P(H)\propto\text{det}\,(1-H)^{\tfrac{1}{2}\beta(N-2M+1-2/\beta)},\;\;N-2M\geq 0,\qquad\qquad(\ast)$$
where $H=AA^\dagger$ and $A$ is an $M\times M$ principal submatrix of an $N... | 1 | https://mathoverflow.net/users/11260 | 422844 | 171,923 |
https://mathoverflow.net/questions/422804 | 6 | Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology coming from the $I$-adic topology on $M$. I wonder how much this can be generalized to the case where $A$ is not necessar... | https://mathoverflow.net/users/476832 | On the Artin-Rees Lemma for non-commutative rings | There is some discussion of this in Rowen's "Ring Theory", volume I, Section 3.5, with additional references therein.
Exercise 19 on p. 462 in *op. cit.* states that a *polycentral* ideal $I$ of a noetherian ring $A$ has the Artin-Rees property, i.e., for every f.g. left module $M$ and a f.g. submodule $N\subseteq M$... | 4 | https://mathoverflow.net/users/86006 | 422845 | 171,924 |
https://mathoverflow.net/questions/422734 | 3 | Let $G=GL\_n$ be the general linear group (let's say over an algebraically closed field of char $=0$). Let's denote as $T$ the torus of diagonal matrices: is there an explicit description of the invariant functions $$\mathbb{C}[G \times G]^T $$ where $T$ acts by simultaneous conjugation?
| https://mathoverflow.net/users/146464 | Invariants of general linear groups under torus action | Call the two matrices ${}\_1A$ and ${}\_2A$.
Your ring can be expressed by taking the $T$-invariants of the free ring in $2n^2$ variables $\mathbb C[{}\_kA\_{ij}]\_{1\leq i,j\leq n, 1\leq k \leq 2}$ and then inverting $\det {}\_1 A$ and $\det {}\_2 A$.
The $T$-invariants of the free ring are the ring of functions o... | 3 | https://mathoverflow.net/users/18060 | 422847 | 171,925 |
https://mathoverflow.net/questions/422846 | 2 | Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let
$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$
be a local parametrization of $M$.
Assume that for all $p\in\mathcal{U}$ we have that $f(p)$ is a linear combination of $\frac{\partial f}{\partial x\... | https://mathoverflow.net/users/14514 | Manifolds whose tangent spaces have a special behavior | The answer to the first question is **No**.
The assumption you made is equivalent to stating that for every $q\in M$ that the vector $q\in T\_qM$.
This is satisfied whenever $M$ is a portion of a cone, which need not be affine.
| 6 | https://mathoverflow.net/users/3948 | 422848 | 171,926 |
https://mathoverflow.net/questions/422842 | 2 | In the article "[Concentration of the information in data with
Log-concave distributions](https://doi.org/10.1214/10-AOP592)" of Bobkov and Madiman, it is written that if $X$ is a positive random variable following a log concave distribution of order $p$, then one has $V(X) \leq \frac{E(X)^2}{p}$.
A reference is given,... | https://mathoverflow.net/users/482426 | Random probability following a log concave distribution of order p | $\newcommand{\tla}{\tilde\lambda}\newcommand{\Ga}{\Gamma}$By Definition 4.1 in the paper by Bobkov and Madiman (BM), a positive random variable (r.v.) $\xi$ has a log-concave distribution of order $p\ge1$ if the pdf $f$ of $\xi$ is such that
\begin{equation\*}
f(x) = x^{p-1}g(x) \tag{1}\label{1}
\end{equation\*}
for $x... | 1 | https://mathoverflow.net/users/36721 | 422850 | 171,927 |
https://mathoverflow.net/questions/422840 | 16 | Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in [Kempf et al., New Dirac Delta function based methods with
applications to perturbative expansions in quantum
field theory](https://arxiv.org/pdf/1404.0747.pdf)):
\beg... | https://mathoverflow.net/users/6415 | Did Euler know (unconsciously) to integrate by differentiating? | I don't know about that particular integral, but Euler certainly knew about integrating by differentiating. He wrote about it in his *Exposition de quelques paradoxes dans le calcul integral* (1758). A recent summary of that work can be found in
A. Fabian and H.D. Nguyen,
Paradoxical Euler: integrating by differentia... | 21 | https://mathoverflow.net/users/159728 | 422861 | 171,929 |
https://mathoverflow.net/questions/422855 | 4 | Fix a prime $p >2$ and $q\_1$, $q\_2$ such that $q\_i - 1$ is exactly divisible by $p$. For any $n$, $a$, $b $, consider the sum
$$\sum\_{i=0}^{p^{n-1}-1}\zeta\_{p^n}^{aq\_1^i+bq\_2^i}.$$
Is this always divisible by $p^{n-1}$? In fact, perhaps it is always $0$ or all the summands are equal? I believe the following ... | https://mathoverflow.net/users/58001 | Has any one seen this sum of roots of unity before? | Using the sagemath code,
```
p = 5
q1 = p+1
q2 = 2*p+1
n = 3
a = 3
b = 1
k = CyclotomicField(p^n)
s = sum([k.gen()^(a*q1^i+b*q2^i) for i in range(0,p^(n-1))])
(s/p^(n-1)).norm()
```
it output 1/88817841970012523233890533447265625. Hence, the answer is no.
| 4 | https://mathoverflow.net/users/482554 | 422863 | 171,931 |
https://mathoverflow.net/questions/422867 | 3 | Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect?
Thanks @YCor for reformulating the question.
| https://mathoverflow.net/users/211682 | Perfect group that is split extension of a normal free subgroup of finite index | ~~The question should be clarified.~~ *(done)*
Let me assume the question is as follows (I couldn't think of another nontrivial interpretation):
>
> Does there exist a free group $F\neq 1$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect?
>
>
>
The answer is yes w... | 6 | https://mathoverflow.net/users/14094 | 422882 | 171,935 |
https://mathoverflow.net/questions/422479 | 2 | Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $g\in G$. Let $S$ be the set of reflection hypersurfaces. We denote by $c$ the vector space of
$G$-equivariant maps from... | https://mathoverflow.net/users/476832 | On the definition of the Cherednik algebra of a variety with a finite group action | Questions 2. and 3. are correct. By Cartan's Lemma $Y$ is smooth. For a point $p\in Y$ one has
$$T\_{p}Y=(T\_{p}X)^{g}\text{.}$$
If $1-\lambda\_{Y,g}(p)=0$ then the action of $g$ on $T\_{p}X$ is trivial. Hence $T\_{p}Y$ has the same dimension as $X$. But this can't be possible, for $Y$ is a smooth hypersurface. Therefo... | 1 | https://mathoverflow.net/users/123694 | 422899 | 171,939 |
https://mathoverflow.net/questions/422577 | 3 | $\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the duality argument I could not fully understand. The outline of the problem is as follows:
Consider we have a dispersive n... | https://mathoverflow.net/users/471464 | Duality argument | ### Context
Let me first clarify the context of the notation, since there are minor typos in the question posed.
Throughout $f$ is an $L^2(\mathbb{T}^2)$ function with restricted frequency support; this implies that $f$ is in fact $C^\infty$. For convenience let $X$ denote the (finite dimensional) subspace of $L^2(... | 4 | https://mathoverflow.net/users/3948 | 422907 | 171,941 |
https://mathoverflow.net/questions/422716 | 2 | Reading through [Modular specification of monads through higher-order presentations](https://arxiv.org/pdf/1903.00922.pdf), this paper includes the following lemma within set-truncated homotopy type theory:
Given a monad $R$ (they work on the type-theoretic universe $Set$) preserving epimorphisms and a collection of ... | https://mathoverflow.net/users/129807 | Well-behaved monad quotients | Steve Lack's paper [On the monadicity of finitary monads](https://doi.org/10.1016/S0022-4049(99)00019-5) shows that if $C$ is locally finitely presentable, then the category of finitary monads on $C$ is monadic over a power of $C$. Since monadic functors create certain coequalizers (this is part of the [monadicity theo... | 2 | https://mathoverflow.net/users/49 | 422926 | 171,944 |
https://mathoverflow.net/questions/422935 | 9 | This is a cross-post of [a question in MSE](https://math.stackexchange.com/questions/4448722/proof-of-proposition-a-2-6-13-in-higher-topos-theory).
---
I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15. (A.2.6.13 in the published version) This ... | https://mathoverflow.net/users/144250 | Proposition A.2.6.15 in HTT | Retracts of weak equivalences are weak equivalences.
Now if $f'$ is a retract of $f$ and you start with such a diagram with $f'$ on the left, you can create a new diagram with $f$, the same $X', X''$ but new $Y',Y''$, determined by cocartesianness of the two squares.
The claim is that the original diagram is a retr... | 10 | https://mathoverflow.net/users/102343 | 422936 | 171,947 |
https://mathoverflow.net/questions/422415 | 2 | I’m trying to understand Lurie’s proof that the homotopy category of a stable $\infty$-category is triangulated. In showing TR2, he constructs a diagram
$$\require{AMScd}
\begin{CD}
X @>f>> Y @>>> 0\\
@VVV @VVV @VVV \\
0’ @>>> Z @>>> W \\
@. @VVV @VuVV \\
@. 0’’ @>>> V
\end{CD}$$
in which every square is a pushout in s... | https://mathoverflow.net/users/37110 | TR2 for homotopy category of stable $\infty$-category | "This is equivalent to the assertion that the construction of the large diagram computes the suspension functor $\Sigma$"
My previous answer was based on me misreading this quote :)
You want to show that this large diagram *computes* $\Sigma$. Lurie says some words about why that is : he says that the large diagram... | 4 | https://mathoverflow.net/users/102343 | 422938 | 171,948 |
https://mathoverflow.net/questions/422915 | 3 | * For which (connected) two dimensional compact manifold M, oriented or not, the tangent bundle TM is trivial?
* For which of these manifolds the complexified tangent bundle $T^\mathbb{C}M = TM\otimes \mathbb{C}$ is trivial?
| https://mathoverflow.net/users/109905 | Tangent bundle of a compact two-dimensional manifold | I have an almost complete answer. I start with a summary of the different cases. In all the cases $M$ is assumed to be a closed 2-manifold.
1. $M$ is orientable. Then $TM$ is trivial if and only if $M$ is the torus (genus 1).
Proof: Euler characteristic $+$ Poincaré-Hopf theorem.
2. $M$ is orientable. Then $TM \otime... | 4 | https://mathoverflow.net/users/171215 | 422943 | 171,949 |
https://mathoverflow.net/questions/422919 | 1 | Let $\mathbb F\_2^n$ denote the set of binary vectors of length $n$. A $k$-sparse parity function is a linear function $h:\mathbb F\_2^n\to\mathbb F\_2$ of the form $h(x)=u\cdot x$ for some $u$ of Hamming weight (number of positive entries) $k$. Let $\text{SPF}\_k$ denote the set of $k$-sparse parity functions.
I am ... | https://mathoverflow.net/users/41669 | How far from a sparse parity function can a function be and still look like such a function on small sets? | For $k \geq m$, any linear function satisfies this condition. Any linear function that's not $k$-sparce has distance $2^{n-1}$ from the $k$-sparse functions.
For $m \geq 3$, any function satisfying this condition is linear (take $x, y ,x+y$, hence of the form $u \cdot x$ for some $u$). So if $n> k \geq m \geq 3$ then... | 2 | https://mathoverflow.net/users/18060 | 422944 | 171,950 |
https://mathoverflow.net/questions/422939 | 0 | Let $D \subseteq \mathbb{C} $ be bounded and simply connected, $\Gamma:= \partial D \in C^2 $, $\phi, \psi \in C^{0,\alpha}(\Gamma)$,
$$
f(z):= \frac{1}{2\pi i} \int\_{\Gamma} \frac{\phi(\zeta)}{\zeta - z} d\zeta,\quad g(z):= \frac{1}{2\pi i} \int\_{\Gamma} \frac{\psi(\zeta)}{\zeta - z} d\zeta，\quad z \in \mathbb{C} \s... | https://mathoverflow.net/users/114334 | Limiting behaviour of Cauchy integral near boundary | You did not say what $v$ is so I make a guess: $v(z)$ is the outer normal to $\Gamma$. Your $f,g$ are piecewise analytic:
inside $D$ they are equal to $f\_-, g\_-$ while outside $D$ they
are equal to $f\_+,g\_+$.
Then
$$\int\_{\Gamma}f\_-g\_-=0$$
by Cauchy theorem (apply it first to some small deformation of $\Gamma$
t... | 2 | https://mathoverflow.net/users/25510 | 422952 | 171,951 |
https://mathoverflow.net/questions/422767 | 7 | A set $S \subseteq [\kappa]^\omega$ is called projective stationary if for every stationary $A \subseteq \omega\_1$, and every algebra $F : \kappa^{<\omega}\to\kappa$, there is $z\in S$ such that $z$ is closed under $F$ and $z \cap \omega\_1\in A$.
One can show that projective stationarity is preserved by ccc forcing... | https://mathoverflow.net/users/11145 | Preservation of projective stationarity | It is possible that $\sigma$-closed forcing destroys projective stationarity:
Suppose $\mathcal A$ is a maximal antichain in $\mathrm{NS}\_{\omega\_1}^+$. Feng-Jech [have shown](https://arxiv.org/abs/math/9409202) that
$$\mathcal S=\{N\in [H\_{\omega\_2}]^\omega\mid N\prec H\_{\omega\_2}\wedge \exists S\in\mathcal A\... | 7 | https://mathoverflow.net/users/125703 | 422955 | 171,953 |
https://mathoverflow.net/questions/422929 | 3 | What are methods for proving nonnegativity of q-hypergeometric functions? Specifically, I have a function of the type 4-phi-3, it is a terminating series:
$$
{}\_{4}\phi\_3\left(\begin{matrix} q^{-i\_1},q^{-j\_1},zs\_1^{-1}s\_2
,q z^{-1}s\_1^{-1}s\_2\\
s\_2^{2},q^{1+j\_2-j\_1},s\_1^{-2}q^{1-i\_1-j\_2}\end{matrix}
\b... | https://mathoverflow.net/users/979 | Nonnegativity of q-hypergeometric series | In this paper (<https://arxiv.org/abs/1905.06815>) there is a similar 4-phi-3 nonnegativity statement which in fact can be utilized to get the nonnegativity of the function in question. In Proposition A.8 we prove something very similar for an expression defined in (A.14), and there Watson's transformation works indeed... | 1 | https://mathoverflow.net/users/979 | 422958 | 171,954 |
https://mathoverflow.net/questions/422963 | 1 | Is there a sound proof of or a counter example to the following conjecture:
>
> if $\boldsymbol{A}^T=\boldsymbol{A}$ is the cost-matrix of a bipartite assignment problem with unique optimal assignment,
>
> then the symmetry carries over to the solution, i.e., $$i\mapsto j\iff j\mapsto i$$ in the optimal assignme... | https://mathoverflow.net/users/31310 | Symmetry of optimal solutions to symmetric assignment problems | $\newcommand\si\sigma$Yes, this is true. Indeed, write $A=(a\_{i,j}\colon i\in[n],j\in[n])$, where $[n]:=\{1,\dots,n\}$, so that $a\_{i,j}=a\_{j,i}$ for all $i,j$ in $[n]$. The cost of an assignment $\si\in S\_n$ (where $S\_n$ is the symmetric group acting on $[n]$) is
$$c(\si):=\sum\_{i\in[n]}a\_{i,\si(i)}.$$
Since $A... | 2 | https://mathoverflow.net/users/36721 | 422970 | 171,957 |
https://mathoverflow.net/questions/422947 | 7 | Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}\_m$, and the symmetric group $S\_n$. A standard unitary representation of $S(m,n)$ is given by the semi-direct product of the $n\times n$ permutation matrices and $n \times n$ diagonal... | https://mathoverflow.net/users/482648 | Are generalized symmetric groups maximal finite groups (in a certain sense)? | I think the answer is "yes" when $m >6$. By arguments along the lines of Frobenius, Schur and Blichfeldt, if we set $G = \langle M(m,n), U^{\prime} \rangle $ and assume that $G$ is finite, then the non-scalar elements of $G$ whose eigenvalues all lies on an arc of length less than $\frac{\pi}{3}$ on the unit circle gen... | 5 | https://mathoverflow.net/users/14450 | 422975 | 171,958 |
https://mathoverflow.net/questions/422971 | 3 | Let $E$ a Banach space ($E$ is the space of continuous functions on $[0,T]$ for my case). Let $F, G: E\times E\to E$ be contraction maps of contraction constant $\epsilon>0$. Given $b\in\mathbb R$, consider the map
\begin{eqnarray}
(F,G\_b) : E\times E &\to& E\times E \\
(x,y) &\mapsto& (F(x,y),G(x,y)+bx).
\end{eqna... | https://mathoverflow.net/users/261243 | Question on the existence/uniqueness of the fixed point | $\newcommand\ep\epsilon$Yes, $H:=(F,G\_b)$ has a unique fixed point, for each real $b$. Indeed, let
$$\ep:=1/2$$
and then take any
$$a\in\Big(0,\frac\ep{\ep+|b|}\Big).$$
Let
$$\|(x,y)\|:=\|x\|+a\|y\|$$
for $(x,y)\in E\times E$. Then for all $(u,v)$ and $(x,y)$ in $E\times E$ we have
$$\begin{aligned}
\|H(u,v)-H(x,y)\|&... | 3 | https://mathoverflow.net/users/36721 | 422977 | 171,959 |
https://mathoverflow.net/questions/422984 | 6 | Let $f(x) = \sum\limits\_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$
If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange expression in terms of q-digamma functions.
But I would rather like to find it in terms of theta functions or ellipti... | https://mathoverflow.net/users/481251 | How to work out this elliptic function? | This is a divergent series. But if one applies summation in the sense of Eisenstein,
$$\lim\_{N\to\infty}\sum\_{n=-N}^N\left(\lim\_{M\to\infty}\sum\_{m=-M}^M\right)$$
then the sum is doubly periodic. Since the poles are at the lattice and residues are equal to $1$, it is equal $\wp(z)+C$. Looking at the Laurent expansi... | 9 | https://mathoverflow.net/users/25510 | 422986 | 171,962 |
https://mathoverflow.net/questions/422962 | 4 | Let $\mathcal C$ and $\mathcal H$ denote the Cauchy principal value and Hadamard finite part. According to the [Wiki](https://en.wikipedia.org/wiki/Hadamard_regularization):
$$
{\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int \_{{a}}^{{b}}{\frac {f(t)}{t-x}}\,\mathrm dt\right)={\mathcal {H}}\int \_{a}^{b}{\frac ... | https://mathoverflow.net/users/125801 | Derivative of Cauchy PV is equivalent to Hadamard regularization? | A derivation of the relation
$${\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int \_{{a}}^{{b}}{\frac {f(t)}{t-x}}\,\mathrm dt\right)={\mathcal {H}}\int \_{a}^{b}{\frac {f(t)}{(t-x)^{2}}}\,\mathrm dt$$
is given by W.T. Ang in [Notes on Cauchy principal and Hadamard finite-part integrals](https://www3.ntu.edu.sg/ho... | 5 | https://mathoverflow.net/users/11260 | 422988 | 171,964 |
https://mathoverflow.net/questions/422991 | 6 | What is the original reference where it was first proven that the generators and relations of the 2-dimensional cobordism category are those of commutative Frobenius algebras?
I've seen [this article by Abrams](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.4282&rep=rep1&type=pdf) being cited for it. But... | https://mathoverflow.net/users/115363 | Original reference for generators and relations of 2-dimensional TQFT | Have you looked at Joachim Kock's book "[Frobenius algebras and 2D topological quantum field theories](https://mat.uab.cat/%7Ekock/TQFT.html)"?
| 5 | https://mathoverflow.net/users/7031 | 422992 | 171,965 |
https://mathoverflow.net/questions/422985 | 15 | $\def\ZZ{\mathbb{Z}}$Call a function $f : \ZZ \to \ZZ$ "contracting" if
$$|f(j) - f(i)| \leq |j-i|$$
for all $i$, $j \in \ZZ$. The contracting functions form a monoid under composition; call it $C$. An element of a monoid is called a "unit" if it is invertible; the units of $C$ are the functions $x \mapsto \pm x + k$. ... | https://mathoverflow.net/users/297 | Indecomposable contracting maps on the integers | I'll solve question 2, on $C\_2$.
I will prove the irreducibles are those that only have one or two bends, verifying a prediction of [Nate](https://mathoverflow.net/a/422987/18060) (and disproving a prediction of [myself)](https://mathoverflow.net/questions/422985/indecomposable-contracting-maps-on-the-integers/42299... | 9 | https://mathoverflow.net/users/18060 | 422994 | 171,966 |
https://mathoverflow.net/questions/422989 | 1 | Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod\_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the sequence
$$a\_{n+1}=\frac{\operatorname{rad}(p\cdot a\_{n})}{p}+\frac{\operatorname{rad}(q\cdot a\_{n-1})}{q}$$
with $\,a\_1=a\_2=1\,$ and $\,p,\,q\,$ odd primes.
In some ... | https://mathoverflow.net/users/150698 | Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\frac{\operatorname{rad}(qa_{n-1})}{q}$ | For any odd $p$, $q$ (not necessarily prime) the values modulo $2$ follow a cycle of order 3.
| 4 | https://mathoverflow.net/users/46140 | 422997 | 171,967 |
https://mathoverflow.net/questions/423001 | 5 | Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler examples of such $X$?
If $X$ is Kähler and rationally connected, then $X$ is projective. So I might suspect that being r... | https://mathoverflow.net/users/471309 | Can a non-Kähler complex manifold be rationally connected? | I am writing my comment as a question. I have certainly explained these examples before on MathOverflow, since they show that the Kollár-Miyaoka-Mori conjecture cannot hold beyond Fujiki class $\mathcal{C}$ (roughly, in the setting of Kaehler manifolds).
Let $C$ be a copy of $\mathbb{CP}^1$. Let $E$ be a (geometric) ... | 3 | https://mathoverflow.net/users/13265 | 423019 | 171,971 |
https://mathoverflow.net/questions/423035 | 9 | As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting).
When it comes to computing $\log P(x)$, one may use the formula
$$
(\log P)' = \frac{P'}{P}
$$
to compute the expansion of the logarithm of $P(x)$ with $P(0)=1$ as
$$
\log P \equiv \int \frac{... | https://mathoverflow.net/users/116776 | Faster computation of p-adic log | There is a method with bit complexity $O(n \log^3 n)$, which is an adaptation of the "bit-burst algorithm" for real and complex functions. The idea here is to integrate the solution of a differential equation using binary splitting evaluation of power series, using integration steps that converge exponentially to the e... | 19 | https://mathoverflow.net/users/4854 | 423038 | 171,974 |
https://mathoverflow.net/questions/423034 | 7 | What is known about the classification of knots in a solid torus $S^1 \times D^2$? Is enumerating them a reasonable problem? Do we get a similar classification as for knots in $S^3$? Ideally there would be a simple description of the Seifert fibered knots in $S^1 \times D^2$ like for prime non-satellite knots in $S^3$ ... | https://mathoverflow.net/users/113402 | Classification of knots in solid torus | Up to Dehn twists, the class of knots in the solid torus is identical to the class of two-component links in the three-sphere, where the first component is an unknot.
For example, Seifert fibered knots in the solid torus give Seifert fibered links in the three-sphere. The base space is the orbifold $S^2(p,\infty,\inf... | 9 | https://mathoverflow.net/users/1650 | 423049 | 171,977 |
https://mathoverflow.net/questions/422981 | 3 | Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O\_X$-modules. Consider the following condition
* (#) For all containments $V \subseteq U$ of affine open subschemes of $X$, the natural map $O(V) \otimes\_{O(U)} F(U) \rightarrow F(V)$ of $O(V)$-modules i... | https://mathoverflow.net/users/19045 | Is this a true weakening of the quasi-coherence property? | Any submodule of a quasicoherent $O\_X$-module satisfies (#): this is clear via reduction to principal open sets, and the fact that localization is exact. More generally, as Neil observes, if $F$ satisfies (#) then so does every submodule of $F$.
For instance, if $F$ is quasicoherent on $X$ and $j:U\hookrightarrow X$... | 4 | https://mathoverflow.net/users/7666 | 423079 | 171,985 |
https://mathoverflow.net/questions/422382 | 2 | Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it *Grundsatz der einfachen Ungleichung* which may be translated as *fundamental theorem of simple inequalities* (where simple means linear, I know that *Grundsatz* can also be translated as *principle*, but read... | https://mathoverflow.net/users/21051 | Who called Farkas' fundamental theorem a lemma? | This is the earliest reference I have located:
[Minkowski-Farkas Lemma in Banach Spaces](https://cowles.yale.edu/sites/default/files/files/pub/cdp/m-0416.pdf), L. Hurwicz (1952).
The same result was also referred to as the Minkowski-Farkas-Weyl theorem in the 1950's, for example in
[The strong Minkowski-Farkas-We... | 2 | https://mathoverflow.net/users/11260 | 423094 | 171,990 |
https://mathoverflow.net/questions/423087 | 2 | Suppose $A \in \mathbb{R}^p$ is the adjacency matrix of a weighted directed acyclic graph $D$ with vertex set $\left\{v\_{1}, v\_{2}, \ldots, v\_{p}\right\}$, i.e.
$$
a\_{i j}=\left\{\begin{array}{lr}
w\left(v\_{i}, v\_{j}\right), & \text { if there is an arc from } v\_{i} \text { to } v\_{j} \\
0, & \text { otherwise... | https://mathoverflow.net/users/479490 | A property of directed acyclic graph | **Preliminary definition.** Let $\mathcal{S}$, $\mathcal{S}'$ be two complementary nonempty sets of indices, i.e., $\mathcal{S}\cup \mathcal{S}'=\left\{1,2,\ldots,p\right\}$ and $\mathcal{S}\cap \mathcal{S}'=\emptyset$.
Define $\mathcal{E}\_{\mathcal{S}}\overset{\Delta}=\left[\left(I-A\right)^{-1}x\right]\_{\mathcal{... | 1 | https://mathoverflow.net/users/138242 | 423103 | 171,993 |
https://mathoverflow.net/questions/423114 | 5 | Let $N(n,k)$ denote the moduli space of stable vector bundles of rank $n$ and degree $k$ over a compact Riemann surface $X$, and let $N\_0(n,k)$ denote the moduli space where we fix rank $n$ and some fixed determinant bundle of degree $k$. We know that the determinant map $det: N(n,k)\rightarrow Pic^k(X)$ is a proper s... | https://mathoverflow.net/users/90911 | A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface | Things are actually simpler. View $\Gamma \_n=H^1(X,\mathbb{Z}/n)$ as the group of line bundles $L\in \operatorname{Pic}^{0}(X) $ with $L^{{\tiny \otimes }n}=\mathscr{O}\_X$. The
map $N\_0(n,k) \times \operatorname{Pic}^{0}(X) \rightarrow N(n,k)\ $ given by $\ (E,L)\mapsto E\otimes L\ $ identifies $N(n,k)$ to the quoti... | 9 | https://mathoverflow.net/users/40297 | 423117 | 171,998 |
https://mathoverflow.net/questions/423125 | 2 | Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
I have seen many papers on Liouville's equation $\Delta u=K e^{ u}$ when $K>0$, such as [enter link description here](https://arxiv.org/pdf/0801.2866.pdf)
or the theorem
***If $\bar{M}$ is a connected, compact 2-manifold with nonempt... | https://mathoverflow.net/users/469129 | Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$? | The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is called the Nierenberg problem, and the complete answer is not known.
For the condi... | 5 | https://mathoverflow.net/users/25510 | 423132 | 172,002 |
https://mathoverflow.net/questions/423137 | 6 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are *almost disjoint* if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said to be an *almost disjoint family* if $a, b$ are almost disjoint for all $a \neq b \in A$. A standard application of [Zo... | https://mathoverflow.net/users/8628 | Gaps in cardinalities of MAD families | Yes, this is consistent.
Suppose we force to add $\kappa$ mutually generic Cohen reals to a model of $\mathsf{CH}$, where $\kappa$ is some cardinal with uncountable cofinality. In the extension, there are MAD families of cardinality $\aleph\_1$ and cardinality $\kappa = \mathfrak{c}$, but there are no MAD families of... | 8 | https://mathoverflow.net/users/70618 | 423139 | 172,003 |
https://mathoverflow.net/questions/422774 | 3 | In the book *The Bochner Integral*, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:
**Defn.** Let $X$ be a Banach space. A function $f:\mathbb{R}\to X$ is *Bochner integrable* if there exists a sequence of half-open intervals $[a\_i,... | https://mathoverflow.net/users/3948 | Mikusiński's approach to Bochner integrals; replace absolute by unconditional? | This was studied previously by James Brooks together with Jan Mikusiński; the relevant references are
* *Brooks, J. K.*, [**Representations of weak and strong integrals in Banach spaces**](http://dx.doi.org/10.1073/pnas.63.2.266), Proc. Natl. Acad. Sci. USA 63, 266-270 (1969). [ZBL0186.20302](https://zbmath.org/?q=an... | 0 | https://mathoverflow.net/users/3948 | 423148 | 172,004 |
https://mathoverflow.net/questions/412233 | 2 | Let $A$ be the path algebra of the quiver $\tilde{D}\_4$. I would like to find its exceptional regular representations with as little computation as possible.
Of course, we can compute the whole Auslander-Reiten quiver and compute all the $\text{Hom}\_A(X,X)$ and $\text{Ext}\_A^1(X,X)$ for all the regular representat... | https://mathoverflow.net/users/131868 | Finding exceptional regular representations of $\tilde{D}_4$ efficiently | The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B\_1, B\_2, \dots, B\_r$. Then, if I numbered them in the convenient way, there are another $r$ modules $C\_1, \dots C\_r$, and irredu... | 4 | https://mathoverflow.net/users/468 | 423156 | 172,007 |
https://mathoverflow.net/questions/423162 | 2 | Let $X\_1,X\_2,...$ be i.i.d variables with value in $\mathbb{N}$ (not necessarily finitely supported). Suppose $E(X\_1) < \infty$.
Denote :
$$R\_n = \textbf{Card}\{X\_1,...,X\_n\}$$
I must prove that $E(R\_n) = o(\sqrt{n})$, but do not really know how to proceed. Does anyone have a hint :)? Many thanks !
| https://mathoverflow.net/users/466576 | Estimation of the expected number of sites visited by i.i.d | Denote $p\_k=P(X=k)$. Then $E(R\_n)=\sum\_k P(k\in \{X\_1,\ldots,X\_n\})\leqslant \sum\_k \min(1,np\_k)$. We are given that $\sum kp\_k<\infty$. Fix $\varepsilon>0$. The sum of $\min(1,np\_k)$ over $k<\varepsilon \sqrt{n}$ is of course at most $1+\varepsilon \sqrt{n}$. The sum over $k\geqslant \varepsilon \sqrt{n}$ is ... | 5 | https://mathoverflow.net/users/4312 | 423165 | 172,008 |
https://mathoverflow.net/questions/423159 | 3 | What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, historical names, some naming attempts that haven't caught on?
* "[Conformal](https://en.wikipedia.org/wiki/Conformal... | https://mathoverflow.net/users/108884 | What do you call a scaled orthogonal map? | Wikipedia suggests "conformal orthogonal group" for the group of all such maps; see the articles
<https://en.wikipedia.org/wiki/Conformal_group>
<https://en.wikipedia.org/wiki/Orthogonal_group#Conformal_group>
The same term is used in Magma handbook:
<http://magma.maths.usyd.edu.au/magma/handbook/text/317>
and ... | 4 | https://mathoverflow.net/users/1306 | 423170 | 172,010 |
https://mathoverflow.net/questions/423175 | 0 | Be a non-empty set of primes $A $. Let us define $A^{\otimes}$ as the set of numbers smooth over $A$, that are the naturals having all their prime divisors in $A$ (where $1$ is arbitrarily considered as smooth over any set).
Using an elementary proof, I have established that the following sums
$$ \sum\_{p \in A} \fra... | https://mathoverflow.net/users/74910 | Series of reciprocals of smooth numbers | These are series of positive numbers, so they can be rearranged without affecting their values, whether or not they converge (look up Riemann rearrangement theorem).
Since $A^\otimes$ is the set of positive integers whose prime factorization is built from primes in $A$,
$$
\sum\_{n \in A^\otimes} \frac{1}{n} = \prod\... | 8 | https://mathoverflow.net/users/3272 | 423177 | 172,014 |
https://mathoverflow.net/questions/423157 | 2 | The Gaussian width of a set $S\subseteq \mathbb{R}^d$ is defined as $\mathbb{E} \sup\_{x\in S}|\langle x, g\rangle|$ where $g\sim \mathcal{N}(0,I\_d).$
I am interested in the subset $S$ of the sphere $\mathbb{S}^{d-1}$, of spherical measure $m$ for which the Gaussian width is minimized. It seems plausible that the mi... | https://mathoverflow.net/users/76094 | The minimum Gaussian width set of a fixed area | Not sure if this helps, but I *believe* it is possible to show that spherical caps minimize the Gaussian width up to a factor of 6, i.e. for any $\alpha\in [0,1]$ and any $A\subseteq S^{n-1}$ such that the $\sigma(A)=\alpha$, any spherical cap $B$ satisfying $\sigma(A)=\alpha$ satisfies
$$ \mathbb{E}\left[\sup\_{x\in B... | 2 | https://mathoverflow.net/users/170770 | 423184 | 172,018 |
https://mathoverflow.net/questions/423111 | 2 | Let $R$ be a ring, $d\_0, d\_1, d\_2, \dots \in R$ and $e\_0, e\_1, e\_2, \dots \in R$ be linear recurrence sequences, such that
* $d\_m = a\_1 d\_{m-1} + a\_2 d\_{m-2} + \dots + a\_k d\_{m-k}$ for $m \geq k$,
* $e\_m = b\_1 e\_{m-1} + b\_2 e\_{m-2} + \dots + b\_l e\_{m-l}$ for $m \geq l$.
It is possible to analyze... | https://mathoverflow.net/users/116776 | Hadamard product of linear recurrences with umbral calculus | Ok, I think I figured it out. For $k=l=1$ we have
$$
c(de) = de - \lambda \mu = de - d\mu + d\mu - \lambda \mu = d(e - \mu) + (d - \lambda) \mu.
$$
Rewriting it in the same way for arbitrary $k$ and $l$, we get
$$\begin{align}
c(de) = & \prod\limits\_{i=1}^k \prod\limits\_{j=1}^l (d(e-\mu\_j) + (d - \lambda\_i )\... | 2 | https://mathoverflow.net/users/116776 | 423186 | 172,019 |
https://mathoverflow.net/questions/423190 | 11 | Consider the quartic system in four variables $a,b,c,d\in\mathbb R$:
$$-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac).$$
>
> Does this system admit rational solution with $$abcd(c^2-d^2)(a^2-b^2)(a^2-c^2)(b^2-d^2)\neq0?$$
>
>
>
>
> Is there any easy way to compute for rational solution?
>
>
>
| https://mathoverflow.net/users/10035 | Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$? | There is no such solution. Let
$$
Q(a,b,c,d) = 2(ad-bc)(bd+ac) + (a^2-b^2)(c^2-d^2)
$$
be the difference between the two sides of the equation,
so we seek to solve $Q(a,b,c,d) = 0$. This is a quadratic equation
in each variable, so in a rational solution the discriminant of $Q$
with respect to each variable is a square... | 33 | https://mathoverflow.net/users/14830 | 423194 | 172,021 |
https://mathoverflow.net/questions/423181 | 5 | I need to prove or disprove that for a stochastic process $(X\_t)\_{t \in [0,1]}$ with marginals $(\mu\_t)\_{t \in [0,1]}$ on $\mathbb{R}$, if the sample paths of $(X\_t)\_{t \in [0,1]}$ are continuous, then $(\mu\_t)\_{t \in [0,1]}$ is weakly continuous in $t$. I see a similar question [Continuity of the densities of ... | https://mathoverflow.net/users/482859 | For stochastic process $X_t$ with marginals $\mu_t$, is it true that the sample-path continuity of $X_t$ implies $\mu_t$ is weakly continuous in $t$? | Yes, under mild assumptions.
If the state space $E$ is Polish (including $E = \mathbb{R}^n$ in particular), then the space $\mathcal{P}(E)$ of Borel probability measures on $E$, with the weak topology, is metrizable, and so it suffices to show weak sequential continuity. That is, for every sequence of times $t\_n \to... | 6 | https://mathoverflow.net/users/4832 | 423198 | 172,022 |
https://mathoverflow.net/questions/420897 | 25 | Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple polyominos overlap.)
For instance, for $n=3$, the fixed 3-polyominos are:
```
### #.. ##. ##. #.. .#.
... #.. #.. .#... | https://mathoverflow.net/users/16035 | What is the smallest size of a shape in which all fixed $n$-polyominos can fit? | It is actually $\ge cn^2$ with some $c>0$. The value of $c$ I'll obtain is pretty dismal but I tried to trade the precision for the argument simplicity everywhere I could, so it can be certainly improved quite a bit. I have no doubt that it is written somewhere (perhaps, in the continuous form: the $2$-dimensional meas... | 8 | https://mathoverflow.net/users/1131 | 423200 | 172,023 |
https://mathoverflow.net/questions/423191 | 0 | I am looking for a meromorphic [doubly periodic function](https://en.wikipedia.org/wiki/Doubly_periodic_function) such that the function is locally univalent.
A standard meromorphic doubly periodic funtion is the [Weirestrass $\wp$ function](https://en.wikipedia.org/wiki/Weierstrass_elliptic_function), defined as
$$\... | https://mathoverflow.net/users/51546 | Can a doubly periodic function be locally univalent? | If $f$ is a doubly-periodic meromorphic function on $\Bbb C$ then $f'$ necessarily has zeros – otherwise $1/f'$ would be an entire doubly-periodic function and therefore constant. (More precisely, the number of zeros equals the number of poles in the fundamental parallelogram, if counted with multiplicity. For a deriva... | 6 | https://mathoverflow.net/users/116247 | 423202 | 172,024 |
https://mathoverflow.net/questions/423204 | 3 | Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M\_G)\_{\rm Tors}$$
the torsion subgroup of the group of coinvariants $M\_G$ of $M$.
For a cyclic subgroup $i\colon C\hookrightarro... | https://mathoverflow.net/users/4149 | The torsion subgroup of the coinvariants for a $G$-module | Let $\mathbb Z[\mathbb Z/2\times \mathbb Z/2]$ be the free module of rank one over $G=\mathbb Z/2\times\mathbb Z/2$. Let $\mathbb Z$ be the trivial $G$-module. There is an obvious diagonal inclusion $\mathbb Z\hookrightarrow \mathbb Z[\mathbb Z\_2\times \mathbb Z\_2]$. Let $M$ be the quotient of this inclusion. As an a... | 5 | https://mathoverflow.net/users/6668 | 423206 | 172,025 |
https://mathoverflow.net/questions/423196 | 5 | $\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$
>
> **Question 1:** Let $S$ be a nonempty measurable subset of $\R^n$. Let $B$ be a closed ball in $\R^n$ such that $m(B)=m(S)$, where $m$ is the Lebesgue measure. Is there a bijective $1$-Lipschitz map from $S$ onto a dense subset of $B$?
>
>
>
>
> **Questi... | https://mathoverflow.net/users/36721 | Contracting a set to a ball | Trivial "No" to all: Take the union $S$ of two disjoint balls $B\_1$ and $B\_2$ of diameters $d$ and $\sqrt{1-d^2}$ respectively. If $f$ maps $S$ to the ball $B$ of diameter $1$, then $f(B\_1)$ has diameter $\le d$. If $d$ is small enough, then $B\setminus f(B\_1)$ still contains two opposite points on the circumferenc... | 7 | https://mathoverflow.net/users/1131 | 423225 | 172,030 |
https://mathoverflow.net/questions/423212 | 2 | Let $(f\_\epsilon)\_{\epsilon>0}$ be a family of positive measurable functions on $L\_p(\mathbb R)$ where $1<p<\infty.$ Assume that the pointwise supremum $f^\*(x)=\sup\_{\epsilon>0}|f\_\epsilon(x)|$ is in $L\_p(\mathbb R).$ Define $F:\mathbb R\to \ell\_{(0,\infty)}^\infty$ defined by $F(x)=(f\_{\epsilon}(x))\_{\epsilo... | https://mathoverflow.net/users/136860 | Measurability of a net | I assume "strongly measurable" is [in the sense of Bochner](https://en.wikipedia.org/wiki/Bochner_measurable_function). I define *nonnegative* measurable functions $f\_\epsilon$ for my example. See below$^\*$ for a modification with *positive* measurable functions.
---
Let $f\_\epsilon$ be defined by
$$
f\_\epsil... | 5 | https://mathoverflow.net/users/454 | 423241 | 172,033 |
https://mathoverflow.net/questions/423248 | 2 | Is it possible to count the number of conics in $\mathbb{P}^2$ that are fully tangent at one point to a given (generic) cubic curve using basic intersection theory calculations?
The corresponding Gromov–Witten invariant (virtually counting conics relative to a cubic divisor with maximal tangency at a point) is $135/4... | https://mathoverflow.net/users/5259 | Counting maximally tangent conics relative to a cubic | If $X$ is a cubic and $P \in X$ is a point such that there is totally tangent at $P$ conic then
$$
6P = 2H,
$$
where $H$ is the restriction to $X$ of the line class of $\mathbb{P}^2$. Thus, the set of such points is the fiber over $2H$ of the map
$$
X = \mathrm{Pic}^1(X) \stackrel{6}\to \mathrm{Pic}^6(X)
$$
over $2H$. ... | 3 | https://mathoverflow.net/users/4428 | 423250 | 172,037 |
https://mathoverflow.net/questions/423257 | 7 | The integral is:
$$f(a) = \int\limits\_{-\infty}^\infty \frac{x e^{-a^2 x^2}}{\tanh(x)}dx$$
which seems to converge for all $a>0$. But I don't know how to get a sense of the function $f(a)$ such as writing it as a convergent series. The usual Taylor series has infinity for each term. Any ideas?
**Edit:**
I beli... | https://mathoverflow.net/users/481251 | How would you work out this integral as a series? | A Taylor series exists in powers of $1/a$:
$$f(a) = \int\limits\_{-\infty}^\infty \frac{x e^{-a^2 x^2}}{\tanh x}\,dx=a^{-2}\int\limits\_{-\infty}^\infty x e^{-x^2}\,\text{cotanh}\, (x/a)\,dx$$
$$=\sum\_{n=0}^{\infty}\frac{2^{2 n} B\_{2 n} \,\Gamma \left(n+\frac{1}{2}\right)}{(2 n)!a^{2 n+1}},$$
with $B\_{2n}$ the [Bern... | 7 | https://mathoverflow.net/users/11260 | 423258 | 172,038 |
https://mathoverflow.net/questions/423249 | 5 | ''Baba is You'' is a recent puzzle game in which the player builds a set of rules by pushing squares with words written on them. If we leave aside the combinatorial difficulty of how to move the blocks around without getting stuck, the game seems to be unique insofar as it is determined entirely by rules which the play... | https://mathoverflow.net/users/119114 | Set theory / Formal logic of Baba is You | The developer has [this GDC talk](https://www.youtube.com/watch?v=Jf5O8S5GiOo) where he talks about the mechanics which you might find interesting. My impression is it's a lot of random hacks, which may fit with your description.
| 3 | https://mathoverflow.net/users/482917 | 423261 | 172,040 |
https://mathoverflow.net/questions/423263 | 1 | If $X \leq\_T Y + 0'$ does there always exist $Z \leq\_T Y$, $Z \leq\_T X$ with $X \leq\_T Z'$?
Obviously, we can find $Z \leq Y$ where the $y$-th column of $Z$ has a limit equal to $X(y)$. Just let $\langle y, s\rangle$ be given by the computation of $X$ from $0'\_s + Y$. However, I realized I wasn't sure if it was ... | https://mathoverflow.net/users/23648 | If $X \leq_T Y + 0'$ does there exist $Z \leq_T Y$, $Z \leq_T X$ with $X \leq_T Z'$? | Ohh, I think I'm being dumb. The answer is no.
Given $X \not\leq\_T 0'$ we build $Y$ using the finite extension method and modify the usual minimal pair construction by coding in the bits of $X$ into $Y$ between the minimal pair requirements.
Now, since $0'$ can figure out how the minimal pair requirements are met ... | 2 | https://mathoverflow.net/users/23648 | 423265 | 172,042 |
https://mathoverflow.net/questions/423262 | 3 | Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $F \supseteq E$ (continuous inclusion). I already know that the image $C= Q(E)$ is a cone in $F$. How do I prove that it ... | https://mathoverflow.net/users/85934 | Image of a quadratic form is a closed cone | In general, $Q(E)$ is not closed.
**Counterexample.** Let $E = F = \mathbb{R}^2$ and set
$$
B(x,y) :=
\begin{pmatrix}
x\_1y\_1 \\
x\_1y\_2 + x\_2y\_1
\end{pmatrix}.
$$
Then
$$
Q(x) :=
\begin{pmatrix}
x\_1^2 \\ 2x\_1x\_2
\end{pmatrix},
$$
so $Q(E)$ is the open right half plane together with the origin.
| 7 | https://mathoverflow.net/users/102946 | 423266 | 172,043 |
https://mathoverflow.net/questions/423160 | 2 | A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be *discrete harmonic* if it satisfies the discrete Laplacian equation
$$
\Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~.
$$
Let $\mathcal F$ be the set of all discrete harmonic functions on the square lattice.
>
> **Question:** Is there ... | https://mathoverflow.net/users/149337 | Difference equation satisfied by discrete harmonic functions on square lattice | $\DeclareMathOperator\u{\mathbf u}$$\DeclareMathOperator\e{\mathbf e}$I think I know how to answer my question. First, note that $p(x,y) = x^{-1}y^{-1} \tilde p(x,y)$, where $\tilde p(x,y) = (x-1)^2y+x(y-1)^2 \in \mathbb Z[x,y]$. More generally, any $q(x,y)\in\mathbb Z[x,x^{-1},y,y^{-1}]$ can be written as $q(x,y) = x^... | 0 | https://mathoverflow.net/users/149337 | 423269 | 172,044 |
https://mathoverflow.net/questions/117842 | 5 | I've been thinking for a while about different ways two Turing degrees might be "independent" of each other (from the point of view of computability theory). The simplest such notion would be to say that they have no information in common: $ d\_0\wedge d\_1= $ **0**. This is a very natural notion.
A different notion ... | https://mathoverflow.net/users/8133 | Notion of independence of Turing degrees | Rather belatedly, I should probably mention that Uri Andrews, Peter Gerdes, Steffen Lempp, Joe Miller, and I have written a paper on this: [*Computability and the symmetric difference operator*](https://people.math.wisc.edu/%7Ejmiller/Papers/symdiff_deg.pdf) ([DOI](https://academic.oup.com/jigpal/article-abstract/30/3/... | 1 | https://mathoverflow.net/users/8133 | 423272 | 172,046 |
https://mathoverflow.net/questions/423174 | 5 | Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is needed to prove that $\text{ker } T + C$ is closed.
My concern turns into to know whether there exist good examples of co... | https://mathoverflow.net/users/178198 | Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces? | The radial cone of $C$ is defined via
$$
\mathcal R\_C(x) := \bigcup\_{\lambda > 0} \lambda ( C - x)$$
for all $x \in C$
and we can show
$$
\mathcal R\_C(x) = C + \operatorname{span}(x),
$$
since $C$ is a cone.
By assumption $\mathcal R\_C(x)$ is closed for all $x \in C$, since $\operatorname{span}(x)$ is a subspace. B... | 6 | https://mathoverflow.net/users/32507 | 423284 | 172,052 |
https://mathoverflow.net/questions/422995 | 2 | A Boolean algebra $B$ is defined (e.g. in Jech) to be $\kappa$-saturated if there is no partition $W$ of $B$ where $|W|=\kappa$. He seems to assume that this implies $|W|<\kappa$ for any partition $W$. But why should this be the case?
For example, say that $B$ is $\aleph\_1$-saturated. Why does this imply that $B$ is... | https://mathoverflow.net/users/4133 | Why is a Boolean algebra being $\kappa$-saturated upward closed in $\kappa$? | Jech defines a partition of a Boolean algebra $B$ as a *maximal* antichain. Now the cardinalities of maximal antichains in $B$ and its completion can indeed differ: Take $B$ as the finite, cofinite subsets of $\omega\_1$ with the canonical Boolean algebra strucure. $B$ has maximal antichains of every nonzero finite car... | 4 | https://mathoverflow.net/users/125703 | 423297 | 172,054 |
https://mathoverflow.net/questions/423275 | 12 | Does the base-10 representation of $2^n$ contain all 10 digits for all sufficiently large integer $n$?
---
In general, let $x\_{k}$ denote the base-$k$ representation of the positive integer $x$. We say $x$ is **$k$-powerful** if $x^n\_{k}$ contains all of the $k$ digits for all sufficiently large integers $n$.
... | https://mathoverflow.net/users/75935 | Does the base-10 representation of $2^n$ contain all 10 digits for all sufficiently large integers $n$? | Heuristically, one would expect the answer to be yes. There's an existing partially explicit version of this mentioned in Richard Guy's "[Unsolved Problem in Number Theory](https://doi.org/10.1007/978-0-387-26677-0)" entry F24, which is that for $n> 86$, $2^n$ always contains a zero in its base 10 expansion. There are ... | 7 | https://mathoverflow.net/users/127690 | 423298 | 172,055 |
https://mathoverflow.net/questions/423268 | 4 | Let $k$ be a field of characteristic $0$ and $R = k[[x\_1, \dotsc, x\_n]]$. Suppose that $M$ is a faithful, finitely generated $R$-module and $\mathfrak{a} < R$ is an ideal such that $\mathfrak{a} M = \mathfrak{m} M$, where $\mathfrak{m} = (x\_1, \dotsc, x\_n)$. Is it true that $\mathfrak{a} = \mathfrak{m}$?
I know t... | https://mathoverflow.net/users/133916 | Faithful module cancellation with maximal ideal | Consider first the case where $\mathfrak{a}\subseteq \mathfrak{m}^2$. In this case we have $\mathfrak{a}\cdot M\subseteq \mathfrak{m}^2\cdot M\subseteq \mathfrak{m}\cdot M=\mathfrak{a}\cdot M$. In particulal, we have an equality $\mathfrak{m}\cdot (\mathfrak{m}\cdot M) = \mathfrak{m}\cdot M $. By Nakayama, we know that... | 2 | https://mathoverflow.net/users/41644 | 423299 | 172,056 |
https://mathoverflow.net/questions/423296 | 0 | Suppose you pick a random order of consecutive numbers from 1 to $n$.
The order for a series of $n$ numbers would then be:
$$
x\_1, x\_2, ... x\_{n-1} , x\_n
$$
The amount of unique combinations is of course $n!$.
e.g. for $n=3$ there are $3!$ (or 6) unique combinations, see table below.
We are interested in ... | https://mathoverflow.net/users/482958 | How many unique orders of length n are there where the index number is different from the number itself? | As @GordonRoyle pointed out: this is a description of derangements...
For a series $n$, the number of derangements are
$$
n!\sum\limits\_{k=0}^{n}\frac{(-1)^k}{k!}
$$
As @JukkaKohonen pointed out, there is a nice entry in the [OEIS database](http://oeis.org/A000166).
Thanks all!
| 1 | https://mathoverflow.net/users/482958 | 423302 | 172,057 |
https://mathoverflow.net/questions/423303 | 1 | Consider the following: fix a function $\bar{b} : \mathbf{R}\_+ \to [0, \infty]$, and define
\begin{align}
\mathcal{S} \left( \bar{b} \right) := \left\{ b : \mathbf{R}\_+ \to [0, \infty] \, \text{s.t.} \, b \leq \bar{b} \, \text{pointwise} \right\}.
\end{align}
Recall the set of totally-monotone functions $\mathcal... | https://mathoverflow.net/users/121692 | Can upper bounds on totally monotone functions be taken (WLOG) to be themselves totally monotone? | $\newcommand{\tb}{\tilde b}\newcommand{\bb}{\bar b}\newcommand{\S}{\mathcal S}\newcommand{\B}{\mathcal B}\newcommand{\T}{\mathcal T}$Note that any totally-monotone function is nonnegative and nonicreasing.
So, trivially, the constant function $\tb:=\bb(0)$ is a totally-monotone majorant of all functions $b\in\T(\bb):=\... | 2 | https://mathoverflow.net/users/36721 | 423309 | 172,060 |
https://mathoverflow.net/questions/421425 | 4 | The potential of the quiver associated to surfaces is the canonical one given by [Labardini-Fragoso's 2009 paper](https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/pdn051), who proved that the the QP associated to surfaces whose boundary is nonempty is rigid hence non-degenerate, in the sense of [this pa... | https://mathoverflow.net/users/480553 | How much is known about the non-degeneracy of Quiver-with-potential associated to closed punctured surfaces? | In [On cluster algebras from once punctured closed surfaces](https://arxiv.org/abs/1310.4454) another paper from Sefi Ladkani, it is shown in the theorem whose statement begins at the bottom of page 1 that if $Q$ is the adjacency quiver of a once punctured closed surface (so the once punctured torus as well as higher g... | 1 | https://mathoverflow.net/users/51668 | 423312 | 172,061 |
https://mathoverflow.net/questions/423317 | 1 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are *almost disjoint* if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said to be an *almost disjoint (AD) family* if $a, b$ are almost disjoint for all $a \neq b \in A$. [Zorn's Lemma](https://e... | https://mathoverflow.net/users/8628 | (Maximal) almost disjoint families of true cardinality ${\frak c}$ | No.
Suppose that there is a MAD family of size $\aleph\_1$ and $\sf CH$ fails. Let $\mathcal E=\{E\_\alpha\mid\alpha<\omega\_1\}$ be a MAD family on the even integers.
Suppose now that $\cal A$, your AD family, happened to be an almost disjoint family only on the odd integers. It can happen, who knows. Extend it to... | 4 | https://mathoverflow.net/users/7206 | 423319 | 172,064 |
https://mathoverflow.net/questions/419181 | 12 | Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the [question](https://mathoverflow.net/questions/414293/what-is-the-right-notion-of-a-functor-from-an-internal-topological-category-to-t) Greg Arone asks:
>
> What is the correct n... | https://mathoverflow.net/users/117088 | What is the right notion of a functor from an internal topological category to a topologically enriched category? | I don't believe it is possible to recover the "correct" notion of "functor $\mathcal{C}\to \rm Top$", as described at the other question you linked to, by viewing $\rm Top$ only as a topologically enriched category. But it is possible if you view $\rm Top$ as a more richly structured object called a [locally internal c... | 5 | https://mathoverflow.net/users/49 | 423329 | 172,069 |
https://mathoverflow.net/questions/423328 | 30 | There are many interpretations of arithmetic in set theory. The
Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor:
$$0=\{\ \}$$
$$1=\{0\}$$
$$2=\{1\}$$
$$3=\{2\}$$
and so on... The von Neumann interpretation, in contrast, is guided by the idea that every numb... | https://mathoverflow.net/users/1946 | Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences? | This is equivalent to the $\Sigma\_1$-soundness of $\mathsf{ZFC}$ (and this equivalence is highly robust to replacing $\mathsf{PA}$ with some other theory):
If $\mathsf{ZFC}$ is $\Sigma\_1$-sound then the answer is **yes** as follows: consider "the $\alpha$th (in the sense of $<\_L$) constructible model of $\mathsf{P... | 23 | https://mathoverflow.net/users/8133 | 423331 | 172,070 |
https://mathoverflow.net/questions/423314 | 4 | Consider the unit sphere $S^d$ in $\mathbb{R}^{d+1}$ with the antipodal action $\nu \colon x\mapsto -x$. This turns $S^d$ into a free $\mathbb{Z}/2\mathbb{Z}$-space.
Construct a CW-complex structure for $S^d$ with 2 cells in each dimension (which we think of as hemispheres) as follows: start with two vertices/$0$-cel... | https://mathoverflow.net/users/482978 | Special cell decomposition for spheres with free $\mathbb{Z}/p\mathbb{Z}$-action by orthogonal transformations? | This is a more explicit version of Ian's answer.
From the representation theory of $\mathbb{Z}/p$, we can assume that $V=\mathbb{C}^{m+1}$ with the generator $g$ of $\mathbb{Z}/p$ acting as $g.z=(\omega\_0z\_0,\dotsc,\omega\_mz\_m)$ for some primitive $p$-th roots of unity $\omega\_0,\dotsc,\omega\_m$. Put $I=[0,1]$ ... | 4 | https://mathoverflow.net/users/10366 | 423355 | 172,079 |
https://mathoverflow.net/questions/423342 | 1 | Suppose that $\{X^i\_t\}\_{1\leq i\leq n;\,t\in [0,T]}$ is an interacting particle system prescribed by certain SDEs, with each $X^i \in \mathcal{X}$ (the state space). Define the associated empirical measure as $$\mu^n\_t = \frac{1}{n}\sum\_{i=1}^n \delta\_{X^i\_t},\quad t\in [0,T].$$ It is mentioned in page 7 of [Dom... | https://mathoverflow.net/users/163454 | Distinction between $\mathcal{C}\left([0,T],\mathcal{P}(\mathcal{X})\right)$ and $\mathcal{P}\left(\mathcal{C}\left([0,T],\mathcal{X}\right)\right)$ | $\newcommand\om{\omega}\newcommand\Om{\Omega}\newcommand\C{\mathcal C}\newcommand\X{\mathcal X}\newcommand\Y{\mathcal Y}\newcommand\P{\mathcal P}$Apparently, here $\P(\Y)$ means the set of all probability measures on (a certain $\sigma$-algebra over) a set $\Y$.
First of all, it is not true that
$\mu^n:=(\mu^n\_t)\_{... | 1 | https://mathoverflow.net/users/36721 | 423356 | 172,080 |
https://mathoverflow.net/questions/423367 | 3 | This is a crosspost from MathStackExchange ([original question](https://math.stackexchange.com/q/4454178/789954)).
Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$).
Define a random bipartite graph on $X \sqcup Y$ in an Erdos-Renyi fashion by putt... | https://mathoverflow.net/users/160416 | Discrepancy of random bipartite graphs | The expected degree of a vertex is $k$, which we are keeping fixed as $n\to\infty$. As $n\to\infty$, the vertex degree distribution converges to Poisson($k$). In particular, a proportion roughly $e^{-k}$ of the vertices have degree $0$.
Let $A$ be the set of vertices in $X$ which have degree $0$. Let $B$ be the whole... | 7 | https://mathoverflow.net/users/5784 | 423378 | 172,084 |
https://mathoverflow.net/questions/423377 | 0 | Let $f\_w:\mathbb C \to \mathbb C$ be an entire function with $f\_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f\_w$ has infinitely many distinct roots and that $w\mapsto f\_w$ depends real-analytically on $w.$ Does $f\_w$ necessarily have infin... | https://mathoverflow.net/users/150549 | Zeros of entire functions with parameter | $$f\_w(z)=(1-z)e^z+(2w-1)(e^z-1)$$
is a counterexample.
More specifically, the only root of $f\_{1/2}$ is $1$, whereas for each $w\in(0,1)\setminus\{1/2\}$ the function $f\_w$ has infinitely many roots, of the form $2w+W\_k\left(e^{-2 w}(1-2 w)\right)$, where $k$ is any integer and $W\_k$ is the $k$th branch of the [... | 4 | https://mathoverflow.net/users/36721 | 423380 | 172,085 |
https://mathoverflow.net/questions/422372 | 2 | Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $M$ a finitely generated $R$-module. In this case, given $F\_{\bullet}$ a free resolution of $M$, what is the relation between $F\_{\bullet}$ and a minimal free resolution $G\_{\bullet}$ of $M$?
In this paper
<https://projecteuclid.org/journals/illinois-journal-o... | https://mathoverflow.net/users/482157 | Relation between free resolutions and minimal free resolutions | The proof is based on the following observation: If $A \in \mathrm{M}\_{n,m}(R)$ is an $n \times m$ matrix with coefficients $a\_{ij}$ such that there exists indices $i,j$ where $a\_{ij}$ is a unit, then there exist $B \in \mathrm{GL}\_n(R)$ and $C \in \mathrm{GL}\_m(R)$ such that
$$
B A C = \begin{bmatrix}
1 & 0 & \do... | 2 | https://mathoverflow.net/users/133916 | 423390 | 172,087 |
https://mathoverflow.net/questions/423346 | 4 | Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a *(vertex) cover* if $C \cap e \neq \emptyset$ for all $e\in E$. A *matching* is a set $M\subseteq E$ of pairwise disjoint edges (elements of $E$).
We say that $G$ has *König's Property* if there is a matching $M\subseteq E$ and a cov... | https://mathoverflow.net/users/8628 | Is König's Property for graphs inheritable from finite subgraphs? | (Just making my [comment](https://mathoverflow.net/questions/423346/is-k%c3%b6nigs-property-for-graphs-inheritable-from-finite-subgraphs#comment1088018_423346) an answer as [suggested](https://mathoverflow.net/questions/423346/is-k%c3%b6nigs-property-for-graphs-inheritable-from-finite-subgraphs#comment1088048_423346).)... | 4 | https://mathoverflow.net/users/17798 | 423391 | 172,088 |
https://mathoverflow.net/questions/423386 | 2 | I'm trying to follow the proof of proposition 7.22.7 from
*Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor*, [**Tensor categories**](http://dx.doi.org/10.1090/surv/205), Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343... | https://mathoverflow.net/users/44499 | Isomorphism between Davydov-Yetter complex and Hochschild complex of canonical algebra on a multitensor category | The category $$\mathsf{Vect}$$ behaves like a unit with respect to the Deligne tensor $$\boxtimes$$. I think the technical way to say it is that there is a canonical 2-natural equivalence $$\mathcal{C}\boxtimes\mathsf{Vect}\simeq\mathcal{C}$$. If we use this equivalence to identify $$\mathcal{C}$$ with $$\mathcal{C}\bo... | 3 | https://mathoverflow.net/users/125022 | 423393 | 172,089 |
https://mathoverflow.net/questions/423389 | 3 | Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int\_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}\_{[0,1]} f$ when $p \to \infty$.
I am interested in the case when $\operatorname{esssup}\_{[0,1]} f =+\infty$ but $f \in L^p$ for all $p ... | https://mathoverflow.net/users/17965 | Growth of $L^p$ norms as $p \to \infty$ | $N(p)$ can grow arbitrarily quickly. Given a sequence $a\_m \downarrow 0$ with $a\_0=1$ and $a\_m<a\_{m-1}/2$ for all $m$, define
$f(x)=x^{-1/m}$ for all $x \in (a\_m,a\_{m-1}]$ and $m \ge 1$. Then $N(p)<\infty$ for all $p<\infty$,
but for $p>2m$, we have
$$\int\_0^1 f^p \,dx \ge \int\_{a\_m}^{2a\_m} x^{-p/m} \ge \frac... | 9 | https://mathoverflow.net/users/7691 | 423395 | 172,090 |
https://mathoverflow.net/questions/423410 | 2 | Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$.
It is well known that the Grothendieck group $K\_0(\operatorname{coh} C)$ of the category of coherent sheaves on $C$ is the following (see, for example, Exercise 6.11 of Hartshorne):
$$
K\_0(\operatorname{coh} C)\sim... | https://mathoverflow.net/users/137654 | Does the Grothendieck group detect the Picard group? | I don't know about 2., but the answer to 1. is yes.
More generally, suppose $(S^1)^g\times\mathbb Z^k \cong (S^1)^h \times \mathbb Z^m$ as abstract groups, then $g=h, k=m$ (in fact, you only need $g=h$ for question 1., because the abstract isomorphism type of $\mathbb C^g/\Lambda$ only depends on $g$, and because you... | 4 | https://mathoverflow.net/users/102343 | 423414 | 172,095 |
https://mathoverflow.net/questions/423216 | 7 | After noticing that the determinant of an $n \times n$ matrix $A\_n$ with elements $a\_{i,j}=i^j$, $1 \le i \le n$, $1 \le j \le n$, is the superfactorial (product of the first $n$ factorials), I wanted to try other cases.
I noticed that with $a\_{i,j}=\binom{n+i-1}{j}$:
$$\lvert A\_n \rvert = \binom{2n-1}{n}$$
T... | https://mathoverflow.net/users/136218 | Determinant of matrix with Stirling numbers as elements | If we take a general sequence of (unsigned) Comtet numbers of the first kind [1, 2] $$c(n, k) = e\_{n-k}(\xi\_1, \ldots, \xi\_n)$$ then the $n \times n$ submatrix with a row offset of $k$ has determinant $$\det\_{0 \le i, j < n} \Big( e\_{k+i-j}(\xi\_1, \ldots, \xi\_{k+i}) \Big) = (\xi\_1 \cdots \xi\_k)^n$$
Similarly... | 3 | https://mathoverflow.net/users/46140 | 423415 | 172,096 |
https://mathoverflow.net/questions/422812 | 4 | In [the relevant Wikipedia entry](https://en.wikipedia.org/wiki/Direct_integral), I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or Banach algebras (in particular I am interested in the Schatten class... | https://mathoverflow.net/users/143779 | Why is it difficult to define a direct integral of Banach spaces or Banach algebras? | To see the problems one faces, start with the simplest example of a family of Banach spaces, the constant family $x\mapsto B$ with $x\in X$ and $X$ some measurable space. The direct integral will be some space of functions $X\to B$. If $B$ is a Hilbert space, then it has an orthonormal base $e\_i$, and we can consider ... | 5 | https://mathoverflow.net/users/2562 | 423418 | 172,097 |
https://mathoverflow.net/questions/423406 | 1 | Let $f\_w:\mathbb C \to \mathbb C$ be an entire function such that $(0,1) \ni w \mapsto f\_w$ is real-analytic.
Assuming that there is a dense subset $D \subset (0,1)$ such that for $w \in D$ the function $f\_w$ has infinitely many zeros. What does this imply for the number of zeros of $f\_w$ with $w\in(0,1) \setminu... | https://mathoverflow.net/users/150549 | Zeros of entire functions | The answer for 3.) is **True**.
Lemma. If $f\_w$ has at least $n$ roots, then there exists a neighbourhood $W$ of $w$ such that all functions $f\_z$ $(z \in W)$ has at least $n$ roots.
Let $L\_n$ denote the subset of $(0,1)$ for which $w \in L\_n$ iff $f\_w$ has at least $n$ zeros. By the lemma above, $L\_n$ is an ... | 1 | https://mathoverflow.net/users/125498 | 423422 | 172,098 |
https://mathoverflow.net/questions/423364 | 5 | For $K$ a number field, denote by $\mathcal{O}\_K$ its ring of integers and by $H\_K$ its Hilbert class field.
For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined over $H$, with complex multiplication by $\mathcal{O}\_K$ having everywhere good reduction (on $H$)?
>
> By Fontain... | https://mathoverflow.net/users/66686 | Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$? | Here is an example.
Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C\_2 \times C\_2$ and its Hilbert class field is $H\_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $H\_K$ is a CM-field and its maximal totally real subfield $H\_K^+$ is $\mathbf{Q}(\sqrt{3}, \sqrt{7})$.
The LMFDB... | 11 | https://mathoverflow.net/users/2481 | 423426 | 172,100 |
https://mathoverflow.net/questions/423403 | 1 | I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x\_1,x\_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial\_t f = {div} \left [\left( \begin{array}{rr}
1/4 & 0 \\
0 & 1 \\
\end{array}\right) \nabla f + \left( \begin{array}{rr}
1/4 & -4 \\
4 & 1 \\
\e... | https://mathoverflow.net/users/178225 | Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space | The solution to the linear PDE is just the PDF corresponding to $\mathcal{N}(\mu\_t, \Sigma\_t)$ where $$
\mu\_t = \exp(A t) x \;, \quad \text{and} \quad \Sigma\_t = 2 \int\_0^t \exp(A (t-s) \begin{bmatrix} 1/4 & 0 \\ 0 & 1 \end{bmatrix} \exp(A^T (t-s)) ds
$$ where $A= -\begin{bmatrix} 1/4 & -4 \\ 4 & 1 \end{bmatrix}$.... | 2 | https://mathoverflow.net/users/64449 | 423431 | 172,102 |
https://mathoverflow.net/questions/390541 | 4 | A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each geometric point has order prime to the characteristic of $k$.
I've heard that the theory of stacky curves and one of M-curves ... | https://mathoverflow.net/users/128235 | Relation between stacky curves and "M-curves" | This is a bit of a late answer, but if it is not useful to you anymore it may prove useful to somebody else.
For the first three questions the answer is yes. The crucial point, which you already mentioned, is that every stacky curve is a root stack.
First some notation, if $D$ is a Cartier divisor on $X$ then write... | 5 | https://mathoverflow.net/users/479261 | 423432 | 172,103 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.