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https://mathoverflow.net/questions/448552 | 4 | The result of LMFDB claims (<https://www.lmfdb.org/EllipticCurve/Q/1640/c/1> )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. In particular, the order of $2$-Selmer group is larger than $16$.
But Magma calculates as following,
```
A:=Ellip... | https://mathoverflow.net/users/144623 | Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB | I don't see any contradiction: the Selmer group also has a contribution of rational points. Indeed, the group of 2-torsion rational points on this elliptic curve is isomorphic to $\mathbb Z/2\mathbb Z$ and generated by $(-54,0)$, as Magma will readily confirm (probably, I used GP/Pari).
UPDATE: As David suspected, th... | 5 | https://mathoverflow.net/users/2284 | 448554 | 180,589 |
https://mathoverflow.net/questions/448324 | 1 | I am searching a simple example of a finite group $G$, so that the Jacobson radical $J(FG)$, of group algebra $FG$ is commutative, where $F$ is a finite field. I know example for that if $G$ is any finite abelian group, or dihedral group when characteristic of field is $2$ and $p$-groups, or when group ring is semi sim... | https://mathoverflow.net/users/81355 | Example of a group algebra with commutative Jacobson radical | Here is a family of examples. Let $F$ have characteristic two, and let $G$ be a finite group of order $2n$ with $n$ odd. Then $J(FG)$ squares to zero, and is therefore commutative.
Another (similar) family of examples is as follows. Let $F$ have characteristic $p$, let $H$ be a $p'$-group, and let $t$ be an automorph... | 3 | https://mathoverflow.net/users/460592 | 448579 | 180,595 |
https://mathoverflow.net/questions/448524 | 1 | Given a topological space $X$ and a Banach space $V$, I wonder for which open sets $U$ it is possible to construct a continuous function $f: X \to V$ such that $f^{-1}[B(0, 1)] = U$ - or maybe there is some nonempty open $V \subseteq U$ such that $f^{-1}[B(0, 1)] = V$. Perhaps some separation properties can ensure this... | https://mathoverflow.net/users/161460 | Constructing a continuous function with a prescribed preimage | Open sets with the required property are exactly functionally open sets.
Let us recall that a subset $U$ of a topological space $X$ is *functionally open in* $X$ if $U=f^{-1}[V]$ for some continuous function $f:X\to \mathbb R$ and some open set $V\subseteq \mathbb R$. It is well-known that an open subset $U$ of a nor... | 2 | https://mathoverflow.net/users/61536 | 448581 | 180,596 |
https://mathoverflow.net/questions/448560 | 3 | The question is as in the title.
Let $H$ be a separable Hilbert space and $f : [0,1] \to H$ be a continuous mapping such that
\begin{equation}
f'(t):=\lim\limits\_{\alpha \to 0} \frac{f(t+\alpha)-f(t)}{\alpha} \in H
\end{equation}
exists for almost every $t \in [0,1]$ and $f' : [0,1] \to H$ is Borel-measurable. In th... | https://mathoverflow.net/users/56524 | If $f : [0,1] \to H$ has $t$-derivative with respect to the norm of $H$, and $H=L^2[0,1]$ itself, does the $t$-derivative exist in ordinary sense? | The convergence in $L^2$ of the variation ratios does not yield necessarily a pointwise convergence.
For example, consider the case where $f(t,x) = \mathbb{1}\_{x = 1/t - \lfloor 1/t \rfloor}$ for all $t>0$ and $x \in [0,1]$, and $f(0,x) = 0$. Each function $f(t,\cdot)$ is null almost everywhere, so $[f(t,\cdot)-f(0,... | 2 | https://mathoverflow.net/users/169474 | 448587 | 180,598 |
https://mathoverflow.net/questions/448594 | 4 | Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu\_1$ be an integer. We denote by $\lambda^\ast$, $\mu^\ast$, $\nu^\ast$ the conjugate partitions, and pad them with $0$'s so the conjugates have $m$ ... | https://mathoverflow.net/users/62135 | Is this simple symmetry of Littlewood-Richardson coefficients known? | I think you can understand this if you think of them as characters of $GL\_m$-representations, i.e., $s\_\lambda$ is the character of the irreducible representation which I'll denote by $V\_\lambda$. The convention is that the highest weight is $\lambda$.
First, conjugating (I assume this just means the usual transpo... | 5 | https://mathoverflow.net/users/321 | 448598 | 180,601 |
https://mathoverflow.net/questions/447645 | 3 | *Note: This is a strengthening of the following [result](https://mathoverflow.net/questions/438492/blow-up-limits-for-sde), motivated by the need for strong convergence in applications.*
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution to the SDE
$$dX\_t = \sigma(X\_t) \, dW\_t \, ... | https://mathoverflow.net/users/173490 | Strong blow up limits for SDE | Yes, this is true. Since $(X\_{t}^{c})\_{t\geq{0}}$ and $(W\_{t}^{c})\_{t\geq{0}}$ are both martingales for fixed $c$, applying Cauchy- Schwarz and then Doob's inequality tells us that:
$$
\mathbb{E}(\sup\_{0\leq{t}\leq{1}}|X\_{t}^{c}-\sigma(0)W\_{t}^{c}|)\leq{\big(\mathbb{E}(\sup\_{0\leq{t}\leq{1}}|X\_{t}^{c}-\sigma(0... | 2 | https://mathoverflow.net/users/80052 | 448602 | 180,603 |
https://mathoverflow.net/questions/448604 | -5 | imagine you have a list of ingredients. Each ingredient has five traits, let's call them A, B, C, D and E. You want a mix of ingredients for which the sum of each trait is known.
for example:
* ingredient 1 has 5 A, 2 B, 0 C, 0 D and 3 E.
* ingredient 2 has 0 A, 3 B, 2 C, 2 D and 0 E
* ingredient 3 has 0 A, 0 B, 6 ... | https://mathoverflow.net/users/506664 | optimization problem: find ingredients that add up to given totals | Let $b\_i$ be the demand for trait $i$. Let $a\_{ij}$ be the number of trait $i$ in ingredient $j$. Let nonnegative integer decision variable $x\_j$ be the number of times ingredient $j$ is used. The problem is to find a feasible solution to the linear equations
$$\sum\_j a\_{ij} x\_j = b\_i \quad \text{for all $i$}$$
... | 0 | https://mathoverflow.net/users/141766 | 448606 | 180,604 |
https://mathoverflow.net/questions/379369 | 3 | The following question appears, more or less, [here](https://math.stackexchange.com/questions/3955627/flatness-of-a-subring-generated-by-two-subrings):
Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra
(I do not mind to further assume that $S$ is an integral doma... | https://mathoverflow.net/users/72288 | Flatness of certain subrings | This is again false. The geometric interpretation is as follows: write $Y = \operatorname{Spec} S$ and $X\_i = \operatorname{Spec} R\_i$. Given étale morphisms $f\_1 \colon Y \to X\_1$ and $f\_2 \colon Y \to X\_2$ of affine schemes, the image factorisation
$$R\_1 \underset k\otimes R\_2 \twoheadrightarrow R \hookrighta... | 2 | https://mathoverflow.net/users/82179 | 448611 | 180,606 |
https://mathoverflow.net/questions/448612 | 0 | As the title suggests, I've been trying to create a lattice (Poset) generator from a positive integer parameter n which represents the number of nodes.
The reason why I'm trying to do this is because I want to create hasse diagrams of lattices for fun, but doing so by hand is a very tiring process and i want an algor... | https://mathoverflow.net/users/506667 | Algorithm to generate a lattice (Poset) from a positive integer parameter | Sage (<https://www.sagemath.org/>) has code to produce a random lattice on $n$ elements: see <https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/poset_examples.html#sage.combinat.posets.poset_examples.Posets.RandomLattice>
I have no idea what algorithm it uses or what distribution it aims to sam... | 1 | https://mathoverflow.net/users/25028 | 448613 | 180,607 |
https://mathoverflow.net/questions/448573 | 2 | Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}\_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected component of $S^n-S^{n-1}\_{1/2}$ is homeomorphic to the closed $n$-disk. In the last step of Morton Brown's proof (note tha... | https://mathoverflow.net/users/147463 | A detail in Brown's proof of the generalized Schoenflies theorem | **Update:** The argument I previously provided was incorrect. I will now outline a correct argument, followed by the incorrect argument and an explanation of what went wrong.
**Correct Explanation:**
By Alexander duality, we can write decompositions into exactly two connected components, for each of the following t... | 2 | https://mathoverflow.net/users/147463 | 448618 | 180,608 |
https://mathoverflow.net/questions/446652 | 12 | **Setting:**
Suppose $\{u\_i\}\_{i=1}^n \subset R^d$ is a collection of unit vectors such that $u\_i^Tu\_j < 0$ for all $i\neq j$, and $w$ is a unit vector such that $u\_i^T w> 0$ for all $i=1,\dotsc,n$. Let $a \in R^d$ be a unit vector.
**Goal:**
We want to show that the following hold or to find a counter-exa... | https://mathoverflow.net/users/128477 | Prove/disprove a linear algebra inequality | Notation:
$[n] = \{1,\dotsc,n\}$.
For symmetric $n \times n$ matrices $A,B$, $A \succ B$ and $A \succeq B$ means that $A-B$ is positive definite / positive semidefinite.
For a symmetric matrix $A$, $\lambda\_{\min}(A)$ and $\lambda\_{\max}(A)$ denote its minimal and maximal eigenvalues.
Claim:
If $n \ge 2$, and $u\_1... | 8 | https://mathoverflow.net/users/42355 | 448631 | 180,611 |
https://mathoverflow.net/questions/448632 | 1 | I need a good approximations for $H\_p$, for $p \in (0,1) \cap \mathbb{Q}$, the generalization of $H\_n=\sum\_{i=1}^n \frac{1}{i}$ to the real numbers.
I tried $H\_p = p \sum\_{k=1}^\infty \frac{1}{k (k + p)}$ and $H\_p = \sum\_{k=0}^\infty \sum\_{j=0}^\infty (-1)^j (1 + k)^{-2 - j} p^{1 + j}$ for $-\frac{1}{2}<p<\fr... | https://mathoverflow.net/users/506692 | Approximation for interpolation of harmonic numbers | We can compute $H\_p = p \sum\_{k=1}^\infty \frac{1}{k (k + p)}$ very fast and very accurately as follows. Let
\begin{equation\*}
f(x):=\frac p{(x+1)(x+1 + p)}=\frac{1}{x+1}-\frac{1}{x+1+p},
\end{equation\*}
so that
\begin{equation\*}
H\_p=\sum\_{k=0}^c f(k)+\sum\_{k=1}^\infty f\_c(k),
\end{equation\*}
where $c$ is a... | 5 | https://mathoverflow.net/users/36721 | 448649 | 180,614 |
https://mathoverflow.net/questions/448213 | 3 | I am interested in this non-convex mixed-integer program:
\begin{array}{cl}
\displaystyle\min\_{(x\_{i},y\_{i})\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}}&\displaystyle\sum\_{i=1}^{K}y\_{i}\left(\displaystyle\frac{x\_{i}}{y\_{i}}-\frac{X}{Y}\right)^{2} \\[0.2cm]
\mathrm{s.t.}&\sum\_{i}x\_{i}=X\;(\mathrm{fixed}) \\
&\sum\_{i... | https://mathoverflow.net/users/393675 | How to convexify or reformulate this non-convex MIP? | As @ManfredWeis noted, the objective is equivalent to minimizing $\sum\_{i=1}^K \frac{x\_i^2}{y\_i}$. You can reformulate this as a (convex) mixed integer second-order cone programming (MISOCP) problem by introducing a new variable $z\_i$ and minimizing $2\sum\_i z\_i$ subject to
\begin{align}
\sum\_i x\_i &= X \\
\sum... | 2 | https://mathoverflow.net/users/141766 | 448651 | 180,615 |
https://mathoverflow.net/questions/448658 | 14 | Suppose Suppose $A$ and $B$ are two entire, non-surjective, functions. This means
$$
A(z)=e^{f(z)}+c\_1
$$
and
$$
B(z)=e^{g(z)}+c\_2
$$
for some entire functions $f$ and $g$, and some complex constants $c\_1$ and $c\_2$.
When is the sum $A(z)+B(z)$ an surjective entire function? More precisely, is there any condit... | https://mathoverflow.net/users/69275 | Sums of non-surjective entire functions | This sum is surjective, with some trivial exceptions. Here is
a general Theorem of Emile Borel: Let $g\_j$ be entire, and
$$e^{g\_1}+\ldots+e^{g\_n}=0.$$
Then the set of indices $\{1,\ldots,n\}$ can be partitioned
into disjoint subsets $I\_k$, such that for $i,j\in I\_k$ we have $g\_i-g\_j$ is constant, and
$$\sum\_{j\... | 15 | https://mathoverflow.net/users/25510 | 448678 | 180,621 |
https://mathoverflow.net/questions/448675 | 2 | Let $\beta$ be an element of $\overline{\mathbb F\_q(T)}\setminus\overline{\mathbb F\_q}$.
Is it true that the sequence $(\beta^{q^n}-T)\_n$ admits infinitely many zeros, that is there exist infinitely many distinct places $\mathcal P\_1,\cdots,\mathcal P\_n\cdots,$ of $\mathbb F\_q(T)(\beta)$ such that for every $n\in... | https://mathoverflow.net/users/33128 | Zeros of a sequence in $\overline{\mathbb F_q(T)}$ | The answer to the first question is yes:
An alternate way to phrase your question is: consider $X = \mathbb P^1\times \mathbb P^1$ with a given curve $C \subset X$ corresponding to $\beta$. We are interested in the geometric points of intersection of $I\_N = C\cap (\phi^n)^\*\Delta$ where $\Delta$ is the diagonal and... | 3 | https://mathoverflow.net/users/58001 | 448679 | 180,622 |
https://mathoverflow.net/questions/448697 | 2 | I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression ... | https://mathoverflow.net/users/41312 | Numerical methods for integral eigenvalue equation | To solve $\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$ I would discretize the coordinates $x\mapsto x\_n$, $y\mapsto y\_m$, $K(x,y,\lambda)\mapsto K(x\_n,y\_m,\lambda)\equiv K\_{nm}(\lambda)$ and solve the determinant equation
$$\text{det}\,[I-K\_{nm}(\lambda)]=0$$
for $\lambda$, when the indices $n,m$ vary over a finite r... | 3 | https://mathoverflow.net/users/11260 | 448698 | 180,630 |
https://mathoverflow.net/questions/448703 | 1 | I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda\_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $y$ a $(n-1)$ rank matrix, so $A$ is symmetric.
Is there any way to calculate the new eigenvalues and eigenvectors of... | https://mathoverflow.net/users/502666 | Eigenvectors of a non-symmetric rank-one update of a symmetric matrix | You can still use the [Bunch–Nielsen–Sorensen formula](https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen_formula), but you will need to distinguish between left and right eigenvectors. These are different, because the rank-one-update is not symmetric. So if you work in the basis where $A$ is diagonal... | 2 | https://mathoverflow.net/users/11260 | 448706 | 180,631 |
https://mathoverflow.net/questions/448704 | 9 | We say that a model $M$ of $\mathsf{ZF}$ satisfies *Small Violations of Choice* ($\mathsf{SVC}$) if all (any) of the following apply:
1. There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, and $M$ is a symmetric extension of $V$.
2. There is a forcing $\mathbb{P}\in M$ such that $\mathbb{P}\mathrel{\Vdas... | https://mathoverflow.net/users/478588 | Small Violations of Choice: Can we force AC without collapsing the cardinalities of ordinals? | The answer is no.
Firstly, let's get the obvious out of the way. The Feferman–Levy model is a symmetric extension in which $\omega\_1$ is singular. If we force the axiom of choice, we must collapse the ordinal to be countable.
Right. So perhaps we want to require that all successor cardinals are regular. This may b... | 9 | https://mathoverflow.net/users/7206 | 448711 | 180,635 |
https://mathoverflow.net/questions/397001 | 4 | The following surely is kind of a trivial question, but it keeps me confused. It concerns a detail in Lust and Stevens' paper "On depth zero L-packets for classical groups" London Math. Soc. (3) 121 (2020), more precisely about the discussion in the first paragraphs of the 3rd part on depth zero supercuspidal represent... | https://mathoverflow.net/users/125617 | Confusion with self-dual representations of $\mathrm{GL}_n$ over a $p$-adic field | You are right, there is an error in Lust-Stevens, and any unramified character provides a counter-example. Here is a way to fix their statement: $\rho=\mathrm{cInd}\_{KF^\times}^G\widetilde\tau$ is self-dual if and only if $\tau=\widetilde\tau|\_K$ is self-dual and the central character $\omega\_{\widetilde\tau}\colon ... | 2 | https://mathoverflow.net/users/123673 | 448714 | 180,636 |
https://mathoverflow.net/questions/448712 | 1 | We add a little bit to [On 'fair bisectors' of planar convex regions](https://mathoverflow.net/questions/356279/on-fair-bisectors-of-planar-convex-regions) and [Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia](https://mathoverflow.net/questions/437053/bisectors-and-part... | https://mathoverflow.net/users/142600 | A claim on the concurrency of area bisectors of planar convex regions | * A planar convex region is centrally symmetric if and only if its area bisectors are all concurrent.
Yes. Let all area bisectors of our region $K$ pass through a point $O$. Note that any line $\ell$ through $O$ is an area bisector of $K$: otherwise there exists a parallel area bisector (by continuity), and it does n... | 2 | https://mathoverflow.net/users/4312 | 448715 | 180,637 |
https://mathoverflow.net/questions/448717 | 1 | Lets extend $\mathcal L\_{\omega\_1, \omega\_1}$ with axioms of equality and of:
$\sf ZF + Definability+Ture$-$\sf Foundation+True$-$\sf Finiteness $
Where $\sf ZF$ is written, as usual, in $\mathcal L\_{\omega, \omega}$
$\sf Definability$ axiom is written in $\mathcal L\_{\omega\_1, \omega}$ as:
$\textbf{Defin... | https://mathoverflow.net/users/95347 | Can we have ZF + Definability + True Foundation + True Finiteness? Can it be Categorical? | The models of your theory are exactly the pointwise-definable (with respect to $\mathcal{L}\_{\omega,\omega}$ as usual) well-founded models of $\mathsf{ZF}$; incidentally, your true finiteness axiom is unnecessary. Assuming such models exist of course, this means that your theory is consistent and **incomplete** (hence... | 7 | https://mathoverflow.net/users/8133 | 448718 | 180,638 |
https://mathoverflow.net/questions/448259 | 7 | It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H\_{BC}^{\bullet,\bullet}:=\frac{\ker \partial\cap\ker\bar\partial}{\text{im }\partial\bar\partial}$ is isomorphic to the Dolbeault cohomology $H\_{\bar\partial}... | https://mathoverflow.net/users/99826 | When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$? | I'm not sure that the following answers the question, but at least it shows that it is not always possible to construct a map $H\_{\bar\partial}\to H\_{dR}$ by assigning a $d$-closed representative to each class in $H\_{\bar\partial}$.
Suppose $\alpha$ is a $\bar\partial$-closed form, and we want to find $\beta$ such... | 1 | https://mathoverflow.net/users/485324 | 448719 | 180,639 |
https://mathoverflow.net/questions/448727 | 4 |
>
> There are meaningful questions we can ask about Euclidean geometry which could not have been posed in the time of Riemann or even of Hilbert, and which would have made no sense at all to Euclid. For example, **does two-dimensional Euclidean geometry emerge as the large-scale limit of a quantum geometry**? The fac... | https://mathoverflow.net/users/159957 | What is Quantum Geometry supposed to be about? | I think it basically goes like this:
1. Take an algebra with a product and a skew product, for example a suitable algebra of functions with a Poisson bracket $\{f,g\}=-\{g,f\}$. You can start with a complex manifold, or mechanical system, or a Lie group, and get a naturally skew-symmetric bracket.
2. Replace multipli... | 8 | https://mathoverflow.net/users/38448 | 448728 | 180,640 |
https://mathoverflow.net/questions/448696 | 2 | While reading the well known book *Minimax Methods in Critical Point Theory with Applications to Differential Equations* by Paul Rabinowitz, in the proof of a generalisation of the Mountain Pass Theorem (Theorem 5.29 in the book), I encountered the following abstract result:
>
> Let $E$ be a real Hilbert space and ... | https://mathoverflow.net/users/124648 | If $b\in C^1(E, \mathbb{R})$ and $b'$ is compact, then $b$ is weakly continuous — a reference request | To obtain a contradiction, suppose that $b(u\_n)\not\to b(u)$. Passing to a subsequence, without loss of generality (wlog) we have $|b(u\_n)-b(u)|\ge c$ for some real $c>0$ and all $n$.
By the mean value theorem, $b(u\_n)-b(u)=b'(v\_n)(u\_n-u)$ for all $n$ and some $v\_n$ on the straight line segment from $u$ to $u\_... | 3 | https://mathoverflow.net/users/36721 | 448730 | 180,641 |
https://mathoverflow.net/questions/448639 | 2 | The *strong multiplicity one theorem* for newforms says the following. Suppose that $f\_1 \in S\_k(\Gamma\_0(N\_1))$ and $f\_2 \in S\_k(\Gamma\_0(N\_2))$ are newforms, where $N\_1, N\_2 \geq 1$ are arbitrary integers. If $a\_{\ell}(f\_1) = a\_{\ell}(f\_2)$ for all primes $\ell$ not dividing a fixed integer $L$, then $f... | https://mathoverflow.net/users/394740 | Multiplicity one for newforms modulo $p$ | If by $f\_1\equiv f\_2$ modulo $p$, you mean that $a\_n(f\_1)\equiv a\_{n}(f\_2)$ modulo $p$ for all $n\in\mathbb N$ or maybe for all except finitely many, then this theorem cannot be true.
Let's start with a counterexample. Consider the two newforms
\begin{equation}
f(z)=\underset{n=1}{\overset{\infty}{\Sigma}}a\_{n... | 2 | https://mathoverflow.net/users/2284 | 448753 | 180,646 |
https://mathoverflow.net/questions/448751 | 4 | Consider based maps $f : X \to Z$ and $g : Y \to Z$, which induces the following map at the based loop space level : $$\theta := \mu\_Z \circ \big(\Omega f \times \Omega g) : \Omega(X\times Y) = \Omega X \times \Omega Y \to \Omega Z \times \Omega Z \to \Omega Z,$$
where $\mu\_Z$ is the standard choice of loop concatena... | https://mathoverflow.net/users/38824 | Delooping a weak $E_1$ map by bar construction | No, take $Z$ a connected space for which $\Omega Z$ is homotopy commutative but $Z$ has no $A\_\infty$ multiplication, e.g. $Z$ could be an $H$-space which doesn't have the homotopy type of a loop space. If $f=g=\mathrm{Id}\_Z$, then the multiplication $\Omega Z \times \Omega Z \rightarrow \Omega Z$ is an $A\_2$-map by... | 4 | https://mathoverflow.net/users/134512 | 448759 | 180,649 |
https://mathoverflow.net/questions/448512 | 0 | This posting is a continuation to an earlier one titled "[Can Foundation be captured in $\mathcal L\_{\omega\_1, \omega}$ ?](https://mathoverflow.net/questions/448486/can-foundation-be-captured-in-mathcal-l-omega-1-omega)"
It appears that capturing foundation is problematic at every $\mathcal L\_{\alpha, \omega}$ and... | https://mathoverflow.net/users/95347 | Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories? |
>
> **Lemma.** A model of ZF satisfies the “Foundation” axiom if and only if it is an $\omega$-model, i.e., iff it satisfies the simpler $\mathcal L\_{\omega\_1,\omega}$-sentence $\forall x\in\omega\,\bigvee\_{n\in\omega}x=n$.
>
>
>
**Proof:**
Right-to-left: Working inside the model, given $x$ and $K$, define ... | 1 | https://mathoverflow.net/users/12705 | 448767 | 180,651 |
https://mathoverflow.net/questions/448780 | 2 | **Background**: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K\_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K\_X^{-m} \cong i\_\*O(1)$. The standard (exponential) height $H$ on $\mathbb{P}^n$ is induced by standard metric on $O(1)$, and pull b... | https://mathoverflow.net/users/479911 | Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture | Your calculation is off in a couple places but neither removes the discrepancy.
First, the exponent of $(\log B)$ is $b(X,L)-1$.
Second, substituting $B^{1/k}$ for $B$ in the term $(\log B)^{ b(X,L)-1}$ produces $$(\log B^{1/k} )^{ b(X,L)-1}= (\log B)^{ b(X,L)-1} / k ^{ b(X,L)-1}$$ instead of $(\log B)^{ b(X,L)-1} ... | 4 | https://mathoverflow.net/users/18060 | 448789 | 180,655 |
https://mathoverflow.net/questions/448778 | 2 | Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v\_1, \cdots, v\_n)$ and $(w\_1, \cdots, w\_n)$ share an edge iff $\lvert v\_i - w\_i\rvert \leq 1$ for all $i$.
A *walk of length t* is a sequence $X\_0, \cdots, X\_t\in V$ where $(X\_i, X\_{i+1})\in E$. Among these walks of length $t... | https://mathoverflow.net/users/97160 | Randomly chosen walk of fixed length | The right setup here is that of **topological Markov chains**. This is essentially the same as a directed graph, i.e., a (finite, for simplicity) set of states (vertices) $A$ endowed with a $\{0,1\}$-valued $A\times A$ admissibility matrix $\Sigma$. Then one looks at the sequences (finite or infinite) of states $x\_0,x... | 1 | https://mathoverflow.net/users/8588 | 448791 | 180,656 |
https://mathoverflow.net/questions/283483 | 12 | As the title says, my question is on a specific argument in [Kirillov - Skew divided difference operators and Schubert polynomials](https://arxiv.org/abs/0705.4546) ([journal](http://www.emis.de/journals/SIGMA/2007/072), [MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2322799)) on positivity of divided differenc... | https://mathoverflow.net/users/66288 | Question on a reduction in Kirillov's paper on positivity of divided difference operators | I had the same concern about Kirillov's paper. For $b=n$, $x\_b$ can have power at most $1$, whereas for $b<n$ it could have a larger power. I have an alternative proof that uses different methods than Kirillov. It remains combinatorial, but it uses combinatorial results that appear not to be published (I'm trying to f... | 3 | https://mathoverflow.net/users/62135 | 448798 | 180,658 |
https://mathoverflow.net/questions/448529 | 1 | I'm reading "Topological entropy bounds measure-theorettic entropy", by L.W. Goodwyn.
[enter link description here](https://www.ams.org/journals/proc/1969-023-03/S0002-9939-1969-0247030-3/S0002-9939-1969-0247030-3.pdf)
After Proposition 2, he mentions that "finite closed cover can yield entropy strictly greater tha... | https://mathoverflow.net/users/506604 | Example of finite closed cover with entropy strictly greater than topological entropy | You can find an example of a cascade which has topological entropy equal to zero, and which has a finite closed cover having entropy equal to $\log 2$ in the [article](https://www.jstor.org/stable/1995916) "The Product Theorem for Topological Entropy" by L.W. Goodwyn.
Let $\hslash (T) = \sup h(\alpha, T)$, where the ... | 1 | https://mathoverflow.net/users/482407 | 448806 | 180,660 |
https://mathoverflow.net/questions/448805 | 4 | I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological groups with the same underlying groups.
**My question 1 is**: Are those classifying spaces associated to different topol... | https://mathoverflow.net/users/471160 | Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups | You may want to look at the classical paper of Jack Milnor, "On the homology of Lie groups made discrete." The Friedlander-Milnor conjecture states that the map $BG^\delta \to BG$ (where $G$ is a Lie group and $G^\delta$ is $G$ made discrete) is a mod $p$ cohomology equivalence. Also chase papers referring to this one,... | 8 | https://mathoverflow.net/users/460592 | 448807 | 180,661 |
https://mathoverflow.net/questions/448804 | 2 | I need help proving that the set $B \cap L^2\big( (0,T) \times (0,1)\big)$ is a closed subset of $L^2\big( (0,T) \times (0,1)\big)$, where $B$ is defined as:
$$B=\Big\{x \in L^{\infty}\big(0,T;L^1(0,1)\big) \ : \|x(t)\|\_{L^1(0,1)} \le 1 , \ \ \text{a.a in } \ (0,T) \Big\} $$
Any suggestions or proofs would be grea... | https://mathoverflow.net/users/473534 | Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$? | $\newcommand{\R}{\mathbb R}\newcommand{\C}{\mathbb C} $To attach a meaning to the intersection $C:=B\cap L^2\big((0,T)\times(0,1)\big)$ of the subset $B$ of the space $L^{\infty}\big(0,T;L^1(0,1)\big)$ with the space $L^2\big((0,T)\times(0,1)\big)$, we need to identify functions $(0,T)\times(0,1)\ni(t,s)\mapsto x(t,s)\... | 2 | https://mathoverflow.net/users/36721 | 448812 | 180,663 |
https://mathoverflow.net/questions/448735 | 6 | [On nlab](https://ncatlab.org/nlab/show/doctrine#as_2monads) I read
>
> For instance, there is a morphism of theories from the theory of commutative rings to the theory of abelian groups which sends a ring to its multiplicative group of units, but this is not induced by any morphism of monads because it does not pr... | https://mathoverflow.net/users/148161 | Multiplicative group of a ring as a morphism of theories | The functor sending a (not necessarily commutative) ring to its group of units is induced by a morphism of *cartesian* (= finite limit) theories.
More generally, suppose given (small!) cartesian theories $\mathcal{T}$ and $\mathcal{T}'$.
Let $\mathcal{M}$ and $\mathcal{M}'$ be the respective categories of models.
The... | 8 | https://mathoverflow.net/users/11640 | 448815 | 180,666 |
https://mathoverflow.net/questions/448817 | 5 | This question is about a technical imprecision which is easily avoidable but whose details I'd like to understand better. When we refer to "the consistency strength of $\mathsf{PA}$" (say) we aren't properly referring to $\mathsf{PA}$ as a set; rather, we care about the specific way in which $\mathsf{PA}$ is enumerated... | https://mathoverflow.net/users/8133 | Is the usual enumeration of $\mathsf{PA}$ "minimal for consistency strength"? | No (to the title), yes (to the question), see
Sy-David Friedman, Michael Rathjen, and Andreas Weiermann: *Slow consistency*, Annals of Pure and Applied Logic 164 (2013), no. 3, pp. 382–393, doi [10.1016/j.apal.2012.11.009](https://doi.org/10.1016/j.apal.2012.11.009).
The idea is that an axiom of usual PA of Gödel n... | 13 | https://mathoverflow.net/users/12705 | 448822 | 180,668 |
https://mathoverflow.net/questions/448835 | 0 | Let $X = \text{Spec}A$ be an affine scheme. It is well known that if $U$ is an open subset which contains $\text{SpecMax}A$, then $U\supseteq X$. The previous statement generalizes to arbitrary schemes, due to the nature of the Zariski topology of a scheme, being locally affine.
When talking about stacks, one typical... | https://mathoverflow.net/users/174655 | Covering a stack by an open substack that contains all points of finite type | Assuming you mean that $\mathscr X$ is an *algebraic* stack (e.g. in the sense of [Tag [026N](https://stacks.math.columbia.edu/tag/026N)]), then this is true, and follows relatively straightforwardly from the case of schemes.
Although you did not make a precise statement in the scheme case, it is indeed true that if ... | 4 | https://mathoverflow.net/users/82179 | 448847 | 180,673 |
https://mathoverflow.net/questions/448848 | 4 | I am aware that a Frechet space $V$ is a topological vector space whose topology is induced by countably many seminorms $\{ \lVert \cdot \rVert\_k \}$ such that
1. it must be Hausdorff
2. it must be complete
In the [relevant Wikipedia entry](https://en.wikipedia.org/wiki/Fr%C3%A9chet_space)
it is stated that when c... | https://mathoverflow.net/users/56524 | Confusion with the definition of a Frechet space regarding completeness and uniqueness of a limit | Consider the space $X$ of sequences $x=(x\_k)\_k \in \mathbf{C}^{\mathbf N}$ with finite support, and let $\|x\|\_k = |x\_k|$. $X$ is clearly not a Fréchet space (one not so elementary reason is that a Fréchet space cannot have countably infinite dimension by Baire's theorem). However, it is easy to see that your condi... | 6 | https://mathoverflow.net/users/10265 | 448850 | 180,674 |
https://mathoverflow.net/questions/448845 | 1 | Consider a multivariate normal distribution. What are necessary and sufficient conditions (esp. on the covariance matrix) under which this distribution is exchangeable?
| https://mathoverflow.net/users/506883 | Under what conditions is a multivariate normal distribution exchangeable? | $\newcommand\Si\Sigma\newcommand\si\sigma\newcommand\1{\mathbf1}$Let $D:=N(\mu,\Si)$ be the $n$-variate normal distribution with mean vector $\mu=[\mu\_1,\dots,\mu\_n]^\top$ and covariance matrix $\Si=[\si\_{ij}\colon i,j=1,\dots,n]$.
Suppose that $D$ is the distribution of an exchangeable random vector $X=[X\_1,\dot... | 2 | https://mathoverflow.net/users/36721 | 448858 | 180,675 |
https://mathoverflow.net/questions/448859 | 0 | Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K\_2$ does not contain any primitive $2^i$-th root of unity for $i \geq 3$ where $ K \_2 $ is the intermediate field.
If ... | https://mathoverflow.net/users/215016 | Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field | It's not quite clear from the post what $K\_2$ is, but I'll just pretend $K\_2=L$, which gives the strongest result.
Those infinitely many extensions constructed over $\mathbb{Q}$ are all linearly disjoint (e.g. since in each one, the respective prime $p$ is the only ramified prime), and thus there are in particular ... | 2 | https://mathoverflow.net/users/127660 | 448860 | 180,676 |
https://mathoverflow.net/questions/448856 | 0 | Let $W=(W\_t)\_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively measurable processes $\alpha=(\alpha\_t)\_{t\ge 0}$ such that
$$\alpha\_t\ge 0 \quad \mbox{and}\quad \int\_0^T\alpha\_s ds \le C.$... | https://mathoverflow.net/users/493556 | Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$ | The process you are describing is simply bigger than all the other: by the condition on $\alpha$
$$Z + W\_t + \int^t \alpha\_s ds \le Z + W\_t + C $$
| 3 | https://mathoverflow.net/users/143907 | 448868 | 180,679 |
https://mathoverflow.net/questions/448865 | 2 | Suppose $A \cong M\_n(\mathbb{D})$ where $A$ is a simple algebra over division ring $D$. We want to find an explicit isomorphism between $A$ and $M\_n(\mathbb{D})$. I read from Ivanyos et al. (2012) that if we have a rank 1 element $e \in A$ then the left action of $A$ on $Ae$ provides an isomorphism.
When I take a l... | https://mathoverflow.net/users/506897 | Solving the explicit isomorphism problem | The element $e\in A$ being of rank one is equivalent to asking $Ae$ to have dimension $n$ over $D$. Now, there is a homomorphism $A\cong End\_D(Ae)$ taking $a\in A$ to the endomorphism $be\mapsto abe$, which is readily seen to be $D$-linear. The map is injective, since otherwise the kernel $\{a\in A:abe=0,\forall b\in ... | 2 | https://mathoverflow.net/users/123673 | 448874 | 180,682 |
https://mathoverflow.net/questions/448863 | 1 | I know that it must be a simple consequence of the Kővári–Sós–Turán (and Erdős–Stone) theorem, but I am struggling to formulate a proof: Let $H$ be a fixed-size $r$-chromatic graph. Then there exists $\varepsilon = \varepsilon(H)$ s.t. if $G$ is an $n$-vertex graph that contains $\geq n^{r-\varepsilon}$ copies of $K\_r... | https://mathoverflow.net/users/506896 | Extremal graph theory - many copies of $K_r$ imply a copy of $r$-chromatic $H$ | This follows from Proposition 2.1 of the paper [Many $T$ copies in $H$-free graphs](https://arxiv.org/abs/1409.4192) by Alon and Shikhelman.
**Theorem** (Alon and Shikelman)
Let $T$ be a fixed graph with $t$ vertices. Then $ex(n,T,H)=\Omega(n^t)$ if and only if $H$ is not a subgraph of a blow-up of $T$. Otherwise, $e... | 3 | https://mathoverflow.net/users/2233 | 448876 | 180,683 |
https://mathoverflow.net/questions/448867 | 3 | $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant *finite* measure on $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
I wonder if there exists a bounded measurable subset $F$ of $\... | https://mathoverflow.net/users/506835 | Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$? | By the Iwasawa decomposition, $\text{SL}(d,\mathbb R)$ is homeomorphic to $\mathbb{R}^n\times\text{SO}(d,\mathbb R)$, where $n=(d^2+d-2)/2$. As $\mathbb{R}^n$ is homeomorphic to any open ball in $\mathbb{R}^n$, it follows that $\text{SL}(d,\mathbb R)$ is homeomorphic to a bounded Borel subset of $\text{SL}(d,\mathbb R)... | 3 | https://mathoverflow.net/users/11919 | 448877 | 180,684 |
https://mathoverflow.net/questions/448853 | 1 | Given $M$ a continuous local martingale, and $M^\text{\*} = \sup\_{0 \leq s \leq t} M\_s$ its running maximum, we consider the finite variation integral
$$
I\_T:= \int\_0^T (M^\text{\*}\_s - M\_s) \, \text{d}M^\text{\*}\_s
$$
I'm seeking to show **this integral is 0** a.s.. Intuitively, this is since $(M^\text{\*}\_s -... | https://mathoverflow.net/users/506885 | Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale | The integral with regard do $\mathrm{d}M^\*$ is a pathwise Stieltjès integral, so the question is an analysis problem.
Let $f : \mathbb{R}\_+ \to \mathbb{R}$ be any continuous function, $F$ its current maximum, and $\mu$ the Stieltjès measure associated to $F$
One checks that $F$ is also continuous, so $O := \{s \i... | 4 | https://mathoverflow.net/users/169474 | 448882 | 180,686 |
https://mathoverflow.net/questions/448879 | 1 | We consider the heat kernel
$$
g :\mathbb R\_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial\_t g(t, x) = \Delta g(t, x) = \left(\frac{|x|^2-2 d t}{4 t^2}\right) g(t, x).
$$
In a previous [thread](https://mathoverflow.n... | https://mathoverflow.net/users/477203 | Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(h(t), x)$? | This is impossible for **any** positive function $h$. Indeed, for $|x|=\sqrt t$, the inequality $|\partial\_t g| (t, x) \le f(t)g(h(t), x)$ implies
$$f(t)\ge\frac Ct\,u^{-d/2}e^{cu}\ge\frac Bt$$
for all real $t>0$, where $C,c,B$ are certain positive real numbers not depending on $t$, and $u:=t/h(t)$. So, $\int\_0^t f=\... | 3 | https://mathoverflow.net/users/36721 | 448885 | 180,688 |
https://mathoverflow.net/questions/448881 | 9 | Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V\_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum\_{x\in X}f(x)=0$ carries an action of $G$.
>
> Is $V\_X$ necessarily irreducible? Is it indecomposable?
>
>
>
When $G$ is finite, then there is an ea... | https://mathoverflow.net/users/123673 | Do doubly-transitive actions give rise to irreducible representations for infinite groups? | $V\_X$ can fail to be irreducible. If $X \subseteq {\bf C}$ is a field and $G$ is the group of affine linear transformations $x \mapsto ax+b$ (any $a,b \in X$ with $a \neq 0$) then $V\_X$ has the proper subrepresentation consisting of functions $f : X \to {\bf C}$ of finite support such that $\sum\_x f(x) = \sum\_x x f... | 17 | https://mathoverflow.net/users/14830 | 448887 | 180,689 |
https://mathoverflow.net/questions/448864 | 3 | I'm trying to understand module classes that are defined as the kernels of higher Ext functors (e.g., arising [here](https://www.sciencedirect.com/science/article/pii/S0168007207000632?via%3Dihub); as this paper suggests, I'm coming at this problem outside of module theory). One way to do this, of course, is to underst... | https://mathoverflow.net/users/178292 | Zeros of higher Ext functors | Elements of $\operatorname{Ext}^i(M,N)$ for $i \geq 1$ can be represented by *Yoneda extensions*: exact sequences $$E = \big(0 \to N \to Z\_{i-1} \to \ldots \to Z\_0 \to M \to 0\big)$$
modulo the equivalence relation $E \sim E'$ if there exists another Yoneda extension $E''$ with morphisms of chain complexes $E \leftar... | 5 | https://mathoverflow.net/users/82179 | 448889 | 180,691 |
https://mathoverflow.net/questions/448841 | 4 | I have a sequential, hereditarily Lindelöf topological space $\mathcal{X}$, and some subset $A \subseteq \mathcal{X}$. I am interested in the following properties:
1. There is some compact set $B$ with $A \subseteq B \subseteq \mathcal{X}$.
2. Every sequence $(a\_n)\_{n \in \mathbb{N}}$ in $A$ has a subsequence which... | https://mathoverflow.net/users/15002 | Being contained in a compact set | It seems that [Gutik's hedgehog](https://mathoverflow.net/questions/296395/what-is-the-genuine-name-for-the-gutik-hedgehog) is a required counterexample.
I recall that Gutik's hedgehog is the set $$X=\{(0,0)\}\cup\{(\tfrac1n,0):n\in\mathbb N\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\}$$
endowed with the topology ... | 2 | https://mathoverflow.net/users/61536 | 448893 | 180,693 |
https://mathoverflow.net/questions/448895 | 30 | For two variables, their maximum
$\max\{x\_1,x\_2\}$ can be expressed using one $|\cdot|$ operation:
$$
\max\{x\_1,x\_2\} = \frac12(x\_1+x\_2+|x\_1-x\_2|).
$$
For $3$ variables, it seems fairly clear that three $|\cdot|$ operations are necessary, though I can only demonstrate sufficiency:
$$
\max\{x\_1,x\_2,x\_3\}
=
\f... | https://mathoverflow.net/users/12518 | Minimum number of $|\cdot|$ operations necessary to express $\max$ | Let $f(k)$ be the minimum number of $|\cdot|$ operations to express the maximum of $2^k$ variables. Using the identities
$$\max\{x\_1,x\_2\} = (x\_1+x\_2+|x\_1-x\_2|)/2,$$
$$\max\{x\_1,\ldots,x\_{2^{k+1}}\} =
\max\{\max\{x\_1,\ldots,x\_{2^k}\},\max\{x\_{2^k+1},\ldots,x\_{2^{k+1}}\}\},$$
it is clear that $f(1)=1$ and $... | 27 | https://mathoverflow.net/users/11919 | 448896 | 180,694 |
https://mathoverflow.net/questions/448911 | 3 | Let $p\_i \in [0,1]$ and $\sum\_{i} p\_i = 1$, and furthermore let $a\_i$ and $b\_i$ be positive real numbers. Is the inequality
$$
\sum\_{i} p\_i \frac{a\_i}{b\_i} \leq \frac{\sum\_{i} p\_i a\_i}{\sum\_{i} p\_i b\_i}$$
true?
| https://mathoverflow.net/users/506925 | Is the inequality $\sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i b_i}$ true? | **No**. This reads
$${\mathbb E}[f(a,b)]\le f({\mathbb E}[a,b])$$
for an arbitrary discrete probability. This is true if and only if $f$ is a concave function (Jenssen Inequality). But $f(x,y)=\frac xy$ is not concave (nor convex).
You might prefer the convex function $g(x,y)=\frac{x^2}y$, for which we thus have
$${\... | 7 | https://mathoverflow.net/users/8799 | 448913 | 180,701 |
https://mathoverflow.net/questions/448910 | 2 | Let $f:\mathbb{Z}\_{p}\rightarrow\mathbb{C}$ be a function such that $\delta\_{a}f:\mathbb{Z}\_{p}\rightarrow\mathbb{C}$ is a locally constant function for any $a\in\mathbb{Z}\_{p}$, where $\delta\_{a}f(x):=f(x+a)-f(x)$. Does it necessarily imply that $f$ is locally constant?
| https://mathoverflow.net/users/140298 | Smooth function on $\mathbb{Z}_{p}$ | No, $f$ does not have to be locally constant. Here is an example. Take a nonzero $\mathbb{Q}$-linear map $L \colon \mathbb{Q}\_p \to \mathbb{Q}$, and let $f = L|\_{\mathbb{Z}\_p}$. Then $\delta\_a f(x) = L(x+a)-L(x) = L(a)$, so $\delta\_a f$ is a constant function for every $a$. However $f$ is not locally constant: tak... | 5 | https://mathoverflow.net/users/42355 | 448916 | 180,702 |
https://mathoverflow.net/questions/448430 | 4 | For a proof for an article I would need the following result:
If $A\_\Gamma$ is an Artin group such that the $K(\pi,1)$-conjecture holds for it and $\Gamma'\subset\Gamma$ is an induced subgraph, then the $K(\pi,1)$-conjecture holds for $A\_{\Gamma'}$.
I've searched in all the references I know about this conjecture... | https://mathoverflow.net/users/482329 | $K(\pi,1)$-conjecture ofr Artin groups behave well with respect to special subgroups. Reference-Request | Yes, it has been proved by Godelle and Paris, it is cited as Theorem 5.5 in the article <https://arxiv.org/pdf/1211.7339> by Paris (and Theorem 3.1 for the equivalence with the K(pi,1) conjecture).
| 4 | https://mathoverflow.net/users/48519 | 448923 | 180,704 |
https://mathoverflow.net/questions/448849 | 1 | The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a number of applications it might have to the study of Hilbert polynomials in commutative algebra and certain combinatorial seq... | https://mathoverflow.net/users/147861 | Location of the negative real roots of certain integer-valued polynomials | Conjecture 1' looked a bit easier to test. Unless I've misinterpreted the question (not impossible), I believe the conjecture is false. If we look at the case $n=2$, the polynomial is
$$\begin{align}
P(x)&=c\_0\binom{x+2}{2}+c\_1\binom{x+1}{2}+c\_2\binom{x}{2} \\
&=\frac12\left(c\_0(x+2)(x+1)+c\_1 (x+1)x + c\_2 x(x-1)\... | 2 | https://mathoverflow.net/users/506194 | 448937 | 180,712 |
https://mathoverflow.net/questions/448920 | 1 | Let $(M,g)$ be a compact Riemannian manifold (e.g. $M=S^3$ the 3-sphere) and let $\Delta$ be the metric Laplacian on $M$. Then $\Delta$ has an $L^2(M)$ basis of eigenfunctions $\pi\_m$, $$ \Delta \pi\_m = - \lambda^2\_m \pi\_m. $$ Define the square root of the Laplacian, $(-\Delta)^{1/2}$, by $$ (-\Delta)^{1/2} \pi\_m ... | https://mathoverflow.net/users/104213 | Leibniz rule for square root of Laplacian | For the Leibniz rule, it is not true that
\begin{equation} (-\Delta)^{\frac12}(fg) = f (-\Delta)^{\frac12}g + g (-\Delta)^{\frac12}f \end{equation}
even in the simplest possible case, i.e. in $\mathbb{R}$. I think the easiest way to see it is to consider $f,g$ sufficiently "good" functions, i.e. Schwarz functions in th... | 2 | https://mathoverflow.net/users/153260 | 448938 | 180,713 |
https://mathoverflow.net/questions/448919 | 8 | Let $G$ be a topological (or simplicial) group, let $X$ and $Y$ be $G$-spaces, and let $f,f':X\to Y$ be $G$-maps which are homotopic as maps of spaces. In general, $f$ and $f'$ may (of course) fail to be equivariantly homotopic. For instance, we can just take $X$ to be a point and $Y$ to be any path-connected $G$-space... | https://mathoverflow.net/users/144250 | Homotopic but not equivariantly homotopic maps | For any $G$-space the $G$-equivariant maps $[EG,X]\_G$, also known as the homotopy fixed points $X^{hG}$, are a Borel homotopy invariant of $X$ meaning that it is an invariant of $G$-equivariant maps for which the underlying map is a weak equivalence. If $Z$ has the trivial action, this means we can compute $[EG,Z \tim... | 11 | https://mathoverflow.net/users/134512 | 448946 | 180,715 |
https://mathoverflow.net/questions/448795 | 2 | *This is a follow-up question to [Can I wrap a suitcase with hair ties](https://mathoverflow.net/questions/424215/can-i-wrap-a-suitcase-with-hair-ties).*
Now we know that it is possible to wrap a suitcase with hair ties without tying them together,
>
> but can you do it with large rotational symmetry?
>
>
>
... | https://mathoverflow.net/users/1441 | Wrapping a suitcase with large rotational symmetry | **Edit: my previous answer was incorrect**
No, this one you cannot do. If one had such an arrangement of circles in the solid torus inside $R^3$, and quotiented by a rotation, then by the [equivariant Dehn's lemma](https://link.springer.com/article/10.1007/BF02566211) there would be a collection of essential disks in... | 5 | https://mathoverflow.net/users/1345 | 448948 | 180,717 |
https://mathoverflow.net/questions/448949 | 4 | Let $A$ be an algebra over a field $K$. Loosely speaking, an algebra is said to be tame if for each $d \in \mathbb{Z}\_{>0}$ all but finitely-many of the indecomposable $A$-modules of $K$-dimension $d$ occur within a finite number of one-parameter families (indexed over $K$). (At least, this is often considered the def... | https://mathoverflow.net/users/95742 | Are polynomial algebras over fields (that are not algebraically closed) tame? | The definition of tame representation type usually assumes that you're dealing with a finite dimensional algebra $A$ over a field $k$. In this case, the definition of a one parameter family of modules is that there is an $A$-$k[T]$-bimodule $X$ which is finitely generated and free over $k[T]$, and then the modules in t... | 7 | https://mathoverflow.net/users/460592 | 448951 | 180,719 |
https://mathoverflow.net/questions/448871 | 5 | Are there modern good lecture notes/book about Brill-Noether theory of curves.
Most interesting theorems here are proved via limit linear series, which I found no lecture notes on (instead there is the original paper from the 80's which references several older papers and while well written is hard to read due to thi... | https://mathoverflow.net/users/135743 | Modern references to Brill-Noether theory on curves? | I recommend also Geometry of Algebraic Curves, Volume I, (1984), chs. V and VII, by Arbarello, Cornalba, Griffiths and Harris, as well as Volume II, (2011), ch. XXI, of the same title, by Arbarello, Cornalba, and Griffiths. (The authors present in Volume II, a "simplified" version, due to Pareschi, of Lazarsfeld's proo... | 6 | https://mathoverflow.net/users/9449 | 448958 | 180,721 |
https://mathoverflow.net/questions/448953 | 6 | *This is a follow-up to [this question](https://mathoverflow.net/questions/448881/do-doubly-transitive-actions-give-rise-to-irreducible-representations-for-infini).*
Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V\_X$ of functions $f\colon X\to\mathbb C$ with finite support such t... | https://mathoverflow.net/users/123673 | Do doubly-transitive actions give rise to indecomposable representations for infinite groups? | Yes.
Let $\varphi$ be a $G$-homomorphism $V\_X \to V\_X$. As mentioned in the question, it suffices to show $\varphi$ is a scalar.
For $x, y\in X$ with $x\neq y$, let $c\_{x,y}$ be obtained by evaluating the function $\varphi([x]-[y])$ at $x$. Then $c\_{g(x),g(y)}$ is the evaluation of $$\varphi([g(x)]-[g(y)]) = \v... | 8 | https://mathoverflow.net/users/18060 | 448960 | 180,723 |
https://mathoverflow.net/questions/448942 | 5 | For a Polish space $(X,d)$ its hyperspace $(K(X),d\_H)$ is also a Polish space. (Here $K(X)$ denotes the set of all nonempty compact subsets of $X$, and the Hausdorff metric $d\_H$ is defined by $d\_H(K,L)=\inf \{\varepsilon>0:\ K\subseteq L\_\varepsilon,\ L\subseteq K\_\varepsilon\}$, where $A\_\varepsilon$ is the ope... | https://mathoverflow.net/users/479121 | Polish space isometric to its hyperspace | It is much easier to deal with zero dimensional spaces and ultrametrics than connected metrics. $\omega^\omega$ with the standard ultrametric is isometrically isomorphic to its own hyperspace.
In particular, we give $\omega^\omega$ the ultrametric where if $f,g:\omega\rightarrow\omega$, then let $d(f,g)=2^{-n}$ where... | 2 | https://mathoverflow.net/users/22277 | 448961 | 180,724 |
https://mathoverflow.net/questions/448971 | 6 | Let $a\_n$ be a nonnegative sequence that Cesaro converges to $K > 0$. We recall this means
$$\frac{1}{N} \sum\_{n = 1}^N a\_n \to K$$
as $N \to \infty$.
Suppose $a\_{\phi\_n}$ with $\phi: \mathbb N \to \mathbb N$ is a rearrangement of the $a\_n$ such that for some $\varepsilon > 0$, we have
$$|\phi(n) - n| \le... | https://mathoverflow.net/users/173490 | Does control on the “magnitude” of the rearrangement give control of the rearranged Cesaro sums? | $\newcommand\ep\varepsilon$We have $\{\phi\_0,\dots,\phi\_{N-1}\}\subseteq\{0,\dots,\lfloor(N-1)(1+\ep)\rfloor\}$. Therefore and because $a\_n\ge0$ for all $n$,
$$\sum\_{n=0}^{N-1} a\_{\phi\_n}\le \sum\_{0\le n\le(N-1)(1+\ep)} a\_n.$$
So,
$$\limsup\_{N\to\infty}\frac{1}{N}\sum\_{n = 0}^{N-1} a\_{\phi\_n} \le (1+\ep)... | 5 | https://mathoverflow.net/users/36721 | 448973 | 180,726 |
https://mathoverflow.net/questions/448983 | 2 | In [this](https://en.wikipedia.org/wiki/Hele-Shaw_flow) Wikipedia article, Hele-Shaw flow is discussed in some detail. I'd like to find a textbook that discusses Hele-Shaw flow in greater detail. Thanks
| https://mathoverflow.net/users/506997 | Fluid dynamics textbook discussing Hele-Shaw flow | A mathematics-oriented text book is [Conformal and Potential Analysis in Hele-Shaw Cells](https://www.google.com/books/edition/Conformal_and_Potential_Analysis_in_Hele/y68nodht9RwC?hl=en&gbpv=0), by Gustafsson and Vasil'ev (2006).
>
> This monograph aims at giving a presentation of recent and new ideas
> that arise... | 3 | https://mathoverflow.net/users/11260 | 448987 | 180,730 |
https://mathoverflow.net/questions/448992 | 1 | Let $\mathcal{D}^b(X)$ be the bounded derived category of constructible $\mathbb{C}$-sheaves on $X$.
(1) For any closed embedding $i:Z\rightarrow X$ and $A\in \mathcal{D}^b(X)$, there is a natural morphism $i^!(A)\rightarrow i^\*(A)$ which induces a distinguished triangle in $\mathcal{D}^b(Z)$,$$i^!(A)\rightarrow i^\... | https://mathoverflow.net/users/507011 | Cones of natural morphisms $i^!\rightarrow i^*$ and $f_!\rightarrow f_*$ | For (2), every vector bundle can be compactified to a projective space bundle (the Proj of the graded ring of symmetric powers of the dual bundle plus a rank one trivial bundle). Let $j$ be the open immersion into the projective space bundle and $\pi$ the projection from the projective space bundle to $Y$. Then $f= \pi... | 4 | https://mathoverflow.net/users/18060 | 448995 | 180,733 |
https://mathoverflow.net/questions/448969 | 2 | This seems like something one might find in a book so I would be grateful for any references you think may be helpful.
I am interested in the rate at which of a function integrated against the $N$th Fejer Kernel approaches it's value at zero. In other words, I am looking to understand the big oh term below
$$\int\_{-... | https://mathoverflow.net/users/506972 | Rate of convergence of Fejer kernel to the Dirac delta function | For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F\_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$.
Two explicit examples, if $f(t)=\cos t$ one has
$$\int\_{-\pi}^\pi f(t)F\_N(t) dt =1-\frac{1}{N}+{\cal O}(N^{-2}).$$
and if $f(t)=\sin^2(t/2)... | 4 | https://mathoverflow.net/users/11260 | 448996 | 180,734 |
https://mathoverflow.net/questions/448998 | 12 |
>
> If we omit parameters in the definition of $L$ would the result still be $L$?
>
>
>
That is, we define a successor stage $L\_{\alpha+1}$ in the constructible universe $L$, without including parameters; as:
$L\_{\alpha+1} =\{\{ y \mid y \in L\_\alpha \land (L\_\alpha, \in) \models \phi(y) \}\mid \phi \text ... | https://mathoverflow.net/users/95347 | Can $L$ be defined without parameters? | Yes, the parameter-free version of $L$ gives rise to the same constructible universe $L$. You will still get all of $L$ this way, but it will come more slowly.
The reason is that at stage $\alpha+1$, you in effect have $\alpha$ as a parameter, since this is definable as the largest ordinal. So you can use $\alpha$ as... | 14 | https://mathoverflow.net/users/1946 | 449001 | 180,737 |
https://mathoverflow.net/questions/448972 | 0 | I am trying to understand why I am getting an almost singular matrix in a problem I have.
The problem is a simple as
$$
\min\_{X \in \mathbb{R}^{m,n}} \left\lVert AX - B \right\rVert\_F^2
$$
Obvioulsy in constructing $A$ and $B$ there're few steps. But I can overly simplify with a toy example the step I believe is ca... | https://mathoverflow.net/users/152487 | Least square error problem ill conditioning | At least when $c:=x\_i-y\_i$ does not depend on $i$, any $n\times n$ submatrix of $A$ is of the form $M=(f\_c(u\_i-u\_j)\colon i,j=1,\dots,n\}$, where $f\_c(x):=f(x+c)$. So, $\det M>0$, since the function $f\_c$ (being the characteristic function of a normal distribution) is positive definite.
However, $f\_c=f\_{c;2}... | 2 | https://mathoverflow.net/users/36721 | 449004 | 180,740 |
https://mathoverflow.net/questions/448977 | 3 | The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.
I’m interested in how it generalizes for fractional dimensional sets particular with dimensions between 2 and 3.
We can start by considering dimensions between 1 and 2. If... | https://mathoverflow.net/users/46536 | Do we have uniformization theorems for fractional dimensional spaces? | So I believe the situation is a bit complex. I think Vicsek-like fractals made from squares, cubes, hypercubes etc... all support a riemann mapping theorem and respectively have dimensions $\log\_3(5), \log\_3(7), \log\_3(9) ... \log\_3(2n+1) ... $ which grows without bound.
It seems like dimension isn't really the i... | 2 | https://mathoverflow.net/users/46536 | 449016 | 180,744 |
https://mathoverflow.net/questions/448982 | 2 | Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum\_{1\leq i\leq r}X\_i^2\equiv0\pmod{X\_k}(1\leq k\leq r)\end{equation}$$
**Example:** In the case that $r=3$, $(X\_1,X\_2,X\_3)=(1,5,13)$ is a solution, because $1^2+5^2+13^2=195=3\cdot 5\cdot 13$ is divisi... | https://mathoverflow.net/users/502468 | An arithmetic problem involving a system of equations | Even for $r=3$ the question isn't easy. For instance a special case is
[Markov's diophantine equation](https://en.wikipedia.org/wiki/Markov_number) $a^2+b^2+c^2=3abc$, for which one infinite family of solutions is $(1, F\_{2n-1}, F\_{2n+1})$, where $F\_i$ is the $i$th Fibonacci number.
| 2 | https://mathoverflow.net/users/18739 | 449020 | 180,745 |
https://mathoverflow.net/questions/449022 | 5 | Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$?
A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions, we came up with variations involving $$\int\_S|f|^{|p|}\arg f^p\,d\mu$$ and exponentiated versions. The argument wou... | https://mathoverflow.net/users/113397 | Is there a meaningful interpretation of an $L^i$-space? | Not only do $\def\L{{\sf L}}\L^p$-spaces make sense for all complex $p$, but their noncommutative generalizations play a crucial role in the Tomita–Takesaki modular theory.
One construction is described in the answer to the question [Is there an introduction to probability theory from a structuralist/categorical pers... | 8 | https://mathoverflow.net/users/402 | 449026 | 180,747 |
https://mathoverflow.net/questions/449024 | 28 | Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by ... | https://mathoverflow.net/users/68969 | Contractibility of the space of Jordan curves | It seems that the answer is "yes".
Let us identify $\mathbb{R}^2$ with $\mathbb{S}^2\setminus\{n\}$.
Each Jordan curve $\gamma$ bounds a disc containing $n$.
This disc admits a conformal parametrization by the unit disc $\mathbb{D}$ such that its center goes to $n$.
This parametrization is unique up to rotation of $\... | 25 | https://mathoverflow.net/users/1441 | 449029 | 180,749 |
https://mathoverflow.net/questions/449030 | 4 | *Note: All sets and functions defined below are assumed measurable. $\mu$ denotes the Lebesgue measure.*
Let $D$ be a dense subset of $[0, 1]$, and $f: D \to \mathbb R$ a function. Given $\varepsilon > 0$, say that $g: [0, 1] \to \mathbb R$ is an $\varepsilon$-almost extension of $f$ if
$$\mu(\overline{\{x \in D \,... | https://mathoverflow.net/users/173490 | Can a function that is continuous on a dense set be almost extended to a continuous function? | Let $A,B$ be disjoint countable dense subsets of $[0,1]$. Let $p:B\rightarrow(0,\infty)$ be a function where $\sum\_{b\in B}p(b)\leq 1$. Define a function $f:A\rightarrow[0,1]$ by letting $f(a)=\sum\_{b\in B,b<a}p(b)$. Then $f$ is a continuous function. Suppose now that $g:[0,1]\rightarrow\mathbb{R}$ is a continuous fu... | 10 | https://mathoverflow.net/users/22277 | 449032 | 180,751 |
https://mathoverflow.net/questions/449018 | 0 | Recall that a right module $M\_R$ is called **semiartinian** if every nonzero homomorphic image has nonzero socle. It's well known that the following two statements are equivalent:
1. $M$ is semiartinian.
2. Every nonzero homomorphic image of $M$ has essential socle.
It's clear that $(2)$ implies (1). But how can I... | https://mathoverflow.net/users/498775 | A characterization of semiartinian modules | Argue by contradiction.
Assume that $M$ is semiartinian and that $N$ is a nonzero homomorphic image of $M$ whose socle $A\_0$ is not essential. There will exist a nonzero submodule $B\leq N$ that is disjoint from $A\_0$. Enlarge the submodule $A\_0$ to a submodule $A\_1$ that is maximal for the condition $A\_1\cap B... | 1 | https://mathoverflow.net/users/75735 | 449036 | 180,752 |
https://mathoverflow.net/questions/448770 | 19 | $\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The **comass** of $\omega$ is defined to be $\max\_{|u|, |v| \leq 1} \omega(u,v)$. Here is the main question:
>
> If $\omega\_1$, $\omega\_2,\ldots,\omega\_n$ are $2$-forms of comass $\leq 1$ on $\RR^{2n}$,... | https://mathoverflow.net/users/28128 | How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be? | Yes! Here is a proof.
Define the norm $|\omega|$, for a $p$-form on $\mathbb R^m$, to be the least upper bound of $|\omega(v\_1,\dots ,v\_p)|$ for vectors $v\_i$ of length one. For any $p$ and $q$ we can ask for the smallest number $C$ such that for every $p$-form $\alpha$ and every $q$-form $\beta$ on $\mathbb R^m$ ... | 14 | https://mathoverflow.net/users/6666 | 449042 | 180,753 |
https://mathoverflow.net/questions/449049 | 3 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I am trying to find the commensurator of $\SL\_{2}(\mathbb{Z})$ on $\GL\_{2}^+(\mathbb{R})$.
So far I have been able to prove that $\GL\_{2}^+(\mathbb{Q})$ is included in the commensurator by looking at the congruence subgroups, and my intuition tells me that this... | https://mathoverflow.net/users/507084 | Commensurator of $\mathrm{SL}_2(\mathbb{Z})$ on $\mathrm{GL}_2^+(\mathbb{Q})$ | It's not harder to compute the commensurator in $\mathrm{GL}\_2(\mathbf{C})$. Since it contains the scalars, it is enough to compute the commensurator in $\mathrm{SL}\_2(\mathbf{C})$.
If a matrix $A=\begin{pmatrix}a & b\\c & d\end{pmatrix}$ of determinant 1 commensurates $\mathrm{SL}\_2(\mathbf{Z})$, then for some $n... | 5 | https://mathoverflow.net/users/14094 | 449055 | 180,755 |
https://mathoverflow.net/questions/449054 | 1 | What are classical/simple examples of smooth projective surfaces $S$ (over $\mathbb C$) for which all the $1$-forms have a common zero (and $h^{1,0}(S)>0$)?
| https://mathoverflow.net/users/85595 | reference quest: surface whose $1$-forms have a common zero | If $S$ admits a global holomorphic 1-form without zeros, $S$ would admit a $C^\infty$ real closed 1-form which has no zeros, equivalently, $S$ would admit a $C^\infty$ fibre bundle structure over the circle.
In this way, the Euler number $\chi(S)$ and the signature $\sigma (S)$ vanish: see [Holomorphic one-forms, fib... | 2 | https://mathoverflow.net/users/125498 | 449057 | 180,756 |
https://mathoverflow.net/questions/449053 | 5 | Let $X$ be a connected compact metric space.
**Question:** Is there a sequence of compact hyperbolic surfaces (the curvature may differ between surfaces) that converges to $X$ in the Gromov-Hausdorff metric?
| https://mathoverflow.net/users/125498 | Can hyperbolic surfaces approximate every connected compact metric space? | The answer is "yes".
Moreover, given an integer $n\geqslant 2$, there a sequence of compact hyperbolic $n$-manifolds that converges to $X$ in the sense of Gromov--Hausdorff.
See V. Šahović. “Approximations of Riemannian manifolds with linear curvature constraints.” Thesis. Univ. Münster, 2009.
| 7 | https://mathoverflow.net/users/1441 | 449058 | 180,757 |
https://mathoverflow.net/questions/448956 | 5 | Crossposted from [MSE](https://math.stackexchange.com/questions/4690683/expositions-of-the-carlitz-scoville-vaughan-theorem-in-combinatorics):
I recently came across Ira Gessel's [slides](https://people.brandeis.edu/%7Egessel/homepage/slides/csv.pdf) on a theorem he says should "be considered one of the fundamental t... | https://mathoverflow.net/users/506963 | Intuitive explanations of the Carlitz-Scoville-Vaughan theorem | This is more of an extended comment than an answer, but maybe you will find it helpful. I agree with [Sam Hopkins](https://mathoverflow.net/questions/448956/intuitive-explanations-of-the-carlitz-scoville-vaughan-theorem#comment1160085_448956) that it is fruitful to think of the Carlitz–Scoville–Vaughan (CSV) theorem as... | 3 | https://mathoverflow.net/users/3106 | 449063 | 180,760 |
https://mathoverflow.net/questions/449060 | 6 | Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.
What is known about the kernel of $cores$? Are ... | https://mathoverflow.net/users/144623 | Ker of corestriction of Galois cohomology | (Not sure why this question comes up naturally. The more interesting question, and the one analogue to the kernel of restriction, is to ask what is the cokernel of corestriction. That turns up a lot. Like in class field theory where the cokernel of the norm map is a central question, while the kernel of the norm map is... | 7 | https://mathoverflow.net/users/5015 | 449064 | 180,761 |
https://mathoverflow.net/questions/449048 | 7 | Let $A$ be an associative algebra over $\mathbb{C}$ with irreducible finite-dimensional representations on $V$ and $W$. Then is the tensor product of representations on $V \otimes W$ semi-simple?
The below question is concerning representations of a group and in this case the answer is positive due to a result of Che... | https://mathoverflow.net/users/165204 | Tensor product of irreducible representations of an algebra | By a famous theorem of R. Steinberg, STEINBERG, R. Complete sets of representations of algebras. PROC. Am. Math. Soc. 13 (1962), 746-747, if $V$ is a faithful representation for a monoid $M$, then $T(V)=\bigoplus\_{n\geq 0}V^{\otimes n}$ is a faithful representation for the monoid algebra $KM$ for any field $K$. If $M$... | 10 | https://mathoverflow.net/users/15934 | 449067 | 180,762 |
https://mathoverflow.net/questions/449052 | 1 | Take the language $\mathcal L(=,\in)\_{\omega\_1,\omega\_1}$, if we restrict infinite quantification strings, i.e. of the forms $``(\forall v\_i)\_{i \in \omega}"; ``(\exists v\_i)\_{i \in \omega}"$, to bounded forms, that is $``\forall v\_0 \in x, \forall v\_1 \in x,..."$ and $``\exists v\_0 \in x , \exists v\_1 \in x... | https://mathoverflow.net/users/95347 | Is bounded-$\mathcal L_{\omega_1,\omega_1}$ significantly different from $\mathcal L_{\omega_1,\omega_1}$? | Your new formulation of foundation is equivalent to the corresponding unbounded form. The reason is that every failing instance of the unbounded formulation will lead to a failing instance of the bounded formulation, since if there is a countable collection of $v\_i$ with no $\in$-minimal element, then by the usual ZFC... | 5 | https://mathoverflow.net/users/1946 | 449073 | 180,764 |
https://mathoverflow.net/questions/449068 | 5 | In almost every introductory notes on [Tits buildings](https://en.wikipedia.org/wiki/Building_(mathematics)) these are motivated
as structures capturing/ sharing several features of [symmetric spaces](https://en.wikipedia.org/wiki/Symmetric_space). Could somebody elaborate what are precisely the main structural similar... | https://mathoverflow.net/users/108274 | Buildings as generalizations of symmetric spaces | (Wanted to post as a comment but didn't have enough reputation)
I have a partial answer only for your first question.
Let $F$ be a non-Archimedean field. For p-adic(or $\pi-$adic) symmetric space $\Omega^r \subset \mathbb{P}^{r-1}\_F$(now also called Drinfeld period domain), Drinfeld found a very clear relation bet... | 6 | https://mathoverflow.net/users/495875 | 449078 | 180,766 |
https://mathoverflow.net/questions/445569 | 2 | Is it easy to construct examples of smooth complex projective surfaces $S$ of general type such that $h^0(\Omega\_S)>1$, $alb\_S:S\rightarrow Alb(S)$ is generically finite (unto its image) and
$${\rm Bs}(\Omega\_S):=\{x\in S,\ \omega\_x(T\_xS)=0 \ \forall \omega\in H^0(\Omega\_S)\}$$
is non-empty?
| https://mathoverflow.net/users/85595 | Base locus of cotangent bundle of a surface | I am just posting my comments as one answer.
Let $A$ be an Abelian variety of dimension three. Denote by $\Theta$ an ample divisor class on $A$. Let $S’$ be a smooth effective Cartier divisor on $A$ in the algebraic equivalence class of $n\Theta$ for very positive $n$. Let $f:S\to S’$ be the blowing up of $S’$ at a p... | 0 | https://mathoverflow.net/users/13265 | 449087 | 180,769 |
https://mathoverflow.net/questions/449059 | 3 | Consider a discrete time Markov chain $(X\_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify with its probability mass function $(\pi(i))\_{i\in \mathcal{S}}$. In this case I know that, given any function $f:\mathca... | https://mathoverflow.net/users/143913 | "Ergodic theorem" for Markov kernels | I understand your setup in the following way (I will somewhat modify your notation). One is given a
measurable state space $S$ and two "kernels" $\{\pi\_s\}, \{\varphi\_s\}$ (i.e., families of probability measures indexed by $S$ and subject to the usual measurability conditions). The measures $\pi\_s$ are on $S$, and t... | 2 | https://mathoverflow.net/users/8588 | 449091 | 180,770 |
https://mathoverflow.net/questions/449094 | 0 | It is known since Bochner that for M compact with negative Ricci curvature, the group of isometries is discreet and hence finite. Are there any generalizations to compact non-positive sectional curvature manifolds? Of course, $T^n$ is a counterexample for strictly 0 curvature case. But what about sectional curvature <0... | https://mathoverflow.net/users/16877 | Isometries of manifolds with non-positive sectional curvature | Bochner's theorem extends to nonpositive Ricci to give:
>
> If $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ then any Killing vector $X$ is parallel and $\textrm{Ric}(X,X) = 0$.
>
>
>
See Petersen (3rd ed) Theorem 8.2.2.
Thus if $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ with $<0$ then $X = 0$.
| 3 | https://mathoverflow.net/users/1540 | 449097 | 180,772 |
https://mathoverflow.net/questions/449075 | 4 | I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite matrices.
For given $p, q \geq 1$, the function $N\_{(p,q)}$ is defined as follows:
$$ N\_{(p,q)} (A) := \inf \left\lbrac... | https://mathoverflow.net/users/105743 | A potential new norm for matrices and Horn's inequalities | By the standard classification of unitarily invariant norms (see e.g., [this blog post](https://nhigham.com/2021/02/02/what-is-a-unitarily-invariant-norm/)), the expression $N\_{(p,q)}$ is a norm if and only if the function
$$ \| (x,y) \| := \inf \left\{ \varepsilon > 0: \left(\frac{\max(|x|, |y|)}{\varepsilon}\right)^... | 8 | https://mathoverflow.net/users/766 | 449103 | 180,774 |
https://mathoverflow.net/questions/449089 | 3 | I have a Banach manifold $\mathcal{M}$ and I have a Lie group $G$, that is finite dimensional, such that $G$ acts freely on $\mathcal{M}$. I would like to know if $\mathcal{M} / G$ is a Banach manifold as well or not.
I know that this is true if $\mathcal{M}$ is a finite dimensional smooth manifold, and we have the q... | https://mathoverflow.net/users/151406 | Quotient by freely acting group on Banach manifold | In finite dimensions, properness of the action is all you need for a slice theorem (and thus for the manifold structure of the quotient). However, in the Banach realm, properness is not enough. For example, the translation action of $c\_0 \subseteq l\_\infty$ of sequences converging to zero on the Banach space $l\_\inf... | 5 | https://mathoverflow.net/users/17047 | 449112 | 180,775 |
https://mathoverflow.net/questions/448944 | 2 | The dihedral group $D\_8$ can be presented as $(\mathbb{Z}\_2\times \mathbb{Z}\_2)\rtimes \_{\rho}\mathbb{Z}\_2$, where the last factor acts on $\mathbb{Z}\_2\times \mathbb{Z}\_2$ as
$$
\rho\_1(a,b)=(b,a) \ .
$$
I am interested in explicit 3-cocycle of the group cohomology $H^3(D\_8,\mathbb{R}/\mathbb{Z})$. Let me be m... | https://mathoverflow.net/users/495347 | Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products | I think I found a solution to my question for any semidirect product $G\_1\rtimes \_{\rho}G\_2$, $G\_1,G\_2$ abelian. Essentially I was looking for the class of $H^3(B G\_1\rtimes \_{\rho}G\_2,\mathbb{R}/\mathbb{Z})$ corresponding to the subgroup $H^1\_{\rho}(BG\_2,H^2(BG\_1,\mathbb{R}/\mathbb{Z}))$. A class in this gr... | -1 | https://mathoverflow.net/users/495347 | 449116 | 180,776 |
https://mathoverflow.net/questions/449106 | 2 | The von Mises distribution has the highest entropy for a given first circular moment. It seems that the converse is true: for a fixed entropy, the von Mises distribution has the highest first circular moment (in magnitude). This seems like something that should already be proven somewhere, but I haven't been able to fi... | https://mathoverflow.net/users/507129 | von Mises Distribution property | Yes, this is true.
Indeed, the first [circular moment](https://en.wikipedia.org/wiki/Von_Mises_distribution#Moments) for a probability density $f$ on the interval $[0,2\pi]$ is
$$m\_1(f):=\int\_0^{2\pi}e^{ix}f(x)\,dx.$$
So,
$$\begin{aligned}&|m\_1(f)| \\
&=\max\_{u\in[0,2\pi]}\Big(\cos u\,\int\_0^{2\pi}\cos x\,f(x)\... | 2 | https://mathoverflow.net/users/36721 | 449124 | 180,778 |
https://mathoverflow.net/questions/449114 | 1 | Consider two [multilinear](https://en.wikipedia.org/wiki/Multilinear_polynomial) Polynomials $A(x\_1,x\_2,x\_3,\dotsc,x\_n)$ and $B(x\_1,x\_2,x\_3,\dotsc,x\_n)$ of $n > 2$ variables $x\_i \in \mathbb{R}$ and their ratio
\begin{equation\*}
F(x\_1,x\_2,x\_3,\dotsc,x\_n) = \frac{A(x\_1,x\_2,x\_3,\dotsc,x\_n)}{B(x\_1,x\_2... | https://mathoverflow.net/users/507148 | Maximizing the ratio of multilinear polynomials | Not in general. E.g., let $n=3$, $A(x\_1,x\_2,x\_3):=1-x\_1 x\_2+2 x\_3 x\_2+x\_1 x\_3$, and $B(x\_1,x\_2,x\_3):=2-x\_1 x\_2+x\_3 x\_2+2 x\_1 x\_3$. Then for $x\_3\in(0,2\sqrt2)$
$$G(x\_3)=\max\_{t\in\mathbb R}\frac{A(t,-t,x\_3)}{B(t,-t,x\_3)}
=\frac{x\_3^2+2 \sqrt{6 x\_3^2+1}+6}{8-x\_3^2},$$
which is not the ratio of ... | 1 | https://mathoverflow.net/users/36721 | 449126 | 180,779 |
https://mathoverflow.net/questions/449128 | 3 | It is well-known that Chi-squared distribution $X\_n$ with degree-$n$ freedom has an approximate formula for its median as $n\left(1-\frac{2}{9n}\right)^3$. Or $(X\_n/n)^{\frac{1}{3}}$ is approximately normally distributed with mean $1-\frac{2}{9n}$ and variance $\frac{2}{9n}$. But unfortunately I cannot find any rigor... | https://mathoverflow.net/users/489992 | Quantitative results (with formal proof) on the median approximation of Chi-squared distribution | Such an approximation follows by the [Cornish--Fisher asymptotic expansion](https://en.wikipedia.org/wiki/Cornish%E2%80%93Fisher_expansion) for the quantiles of a probability distribution.
The error estimate can be obtained by noting that here the Cornish--Fisher asymptotic expansion can be obtained by inverting the ... | 4 | https://mathoverflow.net/users/36721 | 449129 | 180,781 |
https://mathoverflow.net/questions/449096 | 7 | Note 1. Early I posted a related question [Set-theoretic tautologies](https://mathoverflow.net/questions/176732/set-theoretic-tautologies). But the answer did not contain any concrete references to the literature. So I posted this, more precisely formulated question, hoping to receive a concrete reference to the litera... | https://mathoverflow.net/users/5761 | Formulas that are valid simultaneously in a power set Boolean algebra $B$ and the 2-element Boolean algebra $\mathbf2$ | The class of formulas you're looking for contains all quasi-identities:
$$a\_1 = b\_1 \land \cdots \land a\_k = b\_k \to a = b$$
where $k \geq 0$ and $a,a\_1,\ldots,a\_k,b,b\_1,\ldots,b\_k$ are terms formed using $\varnothing,V,{\complement},{\cap},{\cup}$. (We can omit ${\subseteq}$ because $a \subseteq b \iff a \cap ... | 7 | https://mathoverflow.net/users/2000 | 449138 | 180,783 |
https://mathoverflow.net/questions/449135 | 1 | While studying the "[coin-flipping degree](https://mathoverflow.net/questions/448538/bounds-on-the-coin-flipping-degree)" problem I have come across the following conjecture. It gives bounds on the power coefficients of a polynomial that maps the unit interval to itself. If true, this could contribute to solving the "c... | https://mathoverflow.net/users/171320 | Coefficient bounds for polynomials that map the unit interval to itself | This is correct. If $0=c\_0<c\_1<\ldots<c\_n=1$ are extrema of the polynomial $q$, that is, $q(c\_j)=(-1)^j$ for $j=0,1,\ldots,n$, you may interpolate $p$ at nodes $c\_0,\ldots,c\_n$ to get
$$
p(x)=\sum\_{j=0}^n p(c\_j)\frac{\prod\_{k\ne j}(x-c\_k)}{\prod\_{k\ne j}(c\_j-c\_k)},
$$
that gives for the coefficient $a\_m$ ... | 4 | https://mathoverflow.net/users/4312 | 449148 | 180,786 |
https://mathoverflow.net/questions/449139 | -4 | Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a coordinate free calculation of the first and second derivatives of functionals on the spaces of real valued functions o... | https://mathoverflow.net/users/492578 | Coordinate free computation of the second derivative of a functional | For
$$
F=\int d^4 z \ f^4 (z) \ ,
$$
the first functional derivative is
$$
\frac{\delta F}{\delta f(x)} = \int d^4 z \ 4 f^3 (z)\ \delta^4 (x-z)
=4 f^3 (x)
$$
and the second functional derivatives are
$$
\frac{\delta^{2} F}{\delta f(x) \delta f(y)} = 12 f^2 (x) \ \delta^4 (x-y)
$$
| 0 | https://mathoverflow.net/users/134299 | 449153 | 180,788 |
https://mathoverflow.net/questions/449123 | 5 | Let $k$ be an algebraically closed field of characteristic $p>0.$ How can I construct two projective curves $C\_1,C\_2$ of genus $ g \geq 2$ so that the abelianizations $\pi\_1(C\_i)^{ab},i=1,2$ are isomorphic, but $\pi\_1(C\_1) \not \equiv \pi\_1(C\_2)?$
One could try to construct curves $C\_1$ and $C\_2$ and try to... | https://mathoverflow.net/users/472750 | Two curves of genus $g \geq 2$ in characteristic $p >0 $ with isomorphic abelianizations | See
*Nakajima, Shoichi*, On generalized Hasse-Witt invariants of an algebraic curve, Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 69-88 (1983). [ZBL0529.14016](https://zbmath.org/?q=an:0529.14016).
The classical Hasse-Witt invariant, $\gamma$, is the rank of the $... | 5 | https://mathoverflow.net/users/297 | 449158 | 180,789 |
https://mathoverflow.net/questions/449147 | 5 | A function $F: \mathbb R\_+ \rightarrow \mathbb R\_+$ is said to be long tailed if $F(\infty)=0$ and for all $y \geq 0$ $$\frac{F(x+y)}{F(x)} \rightarrow 1, \quad x\rightarrow \infty.$$
Let $\mu$ be a measure on $[0, \infty)$ with full support and denote $f(x)=\mathcal L(\mu,x)$ its Laplace transform. I am trying to ... | https://mathoverflow.net/users/97651 | Long tail property of Laplace transforms | Yes, this is true, assuming that
\begin{equation\*}
f(s):=\int\_{[0,\infty)} e^{-sx}\mu(dx)<\infty
\end{equation\*}
for some real $s\_0$ and all real $s\ge s\_0$. As in the OP, we are also assuming that $\mu$ is a measure on $[0, \infty)$ with full support. We want to show that then for each real $y\ge0$ we have
\begi... | 4 | https://mathoverflow.net/users/36721 | 449160 | 180,790 |
https://mathoverflow.net/questions/449107 | 4 | Let $\Omega$ be a bounded domain, $f\in L^{\infty}(\Omega),$ and $0\leq u\in H\_0^1(\Omega)$ is a non-negative solution of $\Delta u=f$. My question is as follows:
* Can we conclude that $u\in C^0(\bar{\Omega})$ without any boundary smoothness assumption on $\Omega$? If not, could you please provide a counterexample?... | https://mathoverflow.net/users/166368 | Continuous up to the boundary without boundary smoothness | Here is an example in the plane which is genuinely discontinuous up to the boundary. I try to keep it as elementary as possible. There is an if and only if criterion for continuity up to the boundary of harmonic functions, due to Wiener: this example is essentially how one shows the only if part.
Take $B\_1$, and rem... | 4 | https://mathoverflow.net/users/378654 | 449161 | 180,791 |
https://mathoverflow.net/questions/449172 | 1 | We have the following term:
$$ (e^{-a h}+e^{-b h})^n / 2^n$$
Now we take the limit:
$$ h\to 0, n\to \infty $$
What relation of $h$ and $n$ must be satisfied for the following limit to hold?
$$\lim\_{h\to 0, n \to \infty}(\frac{e^{-a h}+e^{-b h}}{2})^n$$
$$=\lim\_{h\to 0, n \to \infty}(1-\frac{1}{2}ah-\frac{1}{2... | https://mathoverflow.net/users/503932 | calculating a double limit | For $h\to0$, we have
$$\frac{e^{-ah}+e^{-bh}}2=1-\frac{a+b}2\,h+O(h^2)
=\exp\Big(-\frac{a+b}2\,h+O(h^2)\Big)$$
and hence
$$\Big(\frac{e^{-ah}+e^{-bh}}2\Big)^n
=\exp\Big(-\frac{a+b}2\,nh+O(nh^2)\Big).$$
So, if $h\to0$ and $nh\to c\in\mathbb R$, then $nh^2\to0$ and hence
$$\lim\Big(\frac{e^{-ah}+e^{-bh}}2\Big)^n
=\lim\... | 1 | https://mathoverflow.net/users/36721 | 449181 | 180,799 |
https://mathoverflow.net/questions/449182 | 8 | I'm mainly following references such as Kelly, Loregian and the nLab, and it seems customary there to generalize (co)ends to the enriched context (over a symmetric monoidal category $\mathcal{V}$) by first defining the $\mathcal{V}$-valued (co)ends, for $\mathcal V$-functors
$$P\colon \mathcal C^\mathsf{op}\boxtimes\ma... | https://mathoverflow.net/users/170683 | Why are enriched (co)ends defined like that? | The $\newcommand{\V}{\mathcal{V}}\newcommand{\D}{\mathcal{D}}\newcommand{\C}{\mathcal{C}}$universal property of being an “initial/terminal cowedge” — even strengthened to “$\V$-initial/-terminal” — is weaker than the standard definition, and too weak for many purposes. In particular, it’s not strong enough to imply any... | 11 | https://mathoverflow.net/users/2273 | 449185 | 180,800 |
https://mathoverflow.net/questions/449156 | 2 | Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P\_d(x)
$$
where $P\_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
| https://mathoverflow.net/users/122182 | Fourier transforms of homogeneous functions | Your function is, with $P\_d$ homogeneous harmonic polynomial of degree $d$ in $n$ variables,
$$
u(x)=\frac{P\_d(x)}{\vert x\vert^{n+d}}.
\tag{1}
$$
This is an homogeneous distribution of degree $-n$ on $\mathbb R^n\backslash\{0\}$. The first thing to do is to extend that distribution to a distribution on $\mathbb R^n$... | 2 | https://mathoverflow.net/users/21907 | 449198 | 180,804 |
https://mathoverflow.net/questions/449208 | 4 | I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology:
Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex
$$\mathbb{R}\_X \rightarrow \Omega\_X^\bullet$$
is an acyclic resolution of $\mathbb{R}\_X$.
I am getting confused about in... | https://mathoverflow.net/users/131090 | Clarification on smooth de Rham theorem | The following statements are true:
* The complex $0 \to \mathbb{R}\_X \to \Omega^0\_X \to \Omega^1\_X \to \dots$ is an acyclic complex (of sheaves), i.e. it is exact at each step. This is the content of the Poincare lemma, and it is what it means for $\Omega^\bullet$ to be a resolution of $\mathbb{R}\_X$.
* Each indi... | 10 | https://mathoverflow.net/users/2481 | 449210 | 180,807 |
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