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https://mathoverflow.net/questions/394187 | 0 | Under what assumptions on $C$ and $X$ is the following true ? I was neither able to find a counterxample or prove this, though it appears that compactness, e.g. assuming $X$ is compactly generated, may be of help. Is $X$ being metrizable helpful?
Let $C$ be a closed subset of a normal Hausdorff space $X$. Any two ope... | https://mathoverflow.net/users/254885 | extending disjoint open subsets of a normal Hausdorff space | The first is true in any [hereditarily normal space](https://en.wikipedia.org/wiki/Normal_space#hereditarily_normal_space): separated sets have disjoint neighbourhoods.
It fails in the compact product $(\omega\_1+1)\times(\omega+1)$ ([Tychonoff's plank with corner point](https://en.wikipedia.org/wiki/Tychonoff_plank)).... | 2 | https://mathoverflow.net/users/5903 | 394193 | 162,948 |
https://mathoverflow.net/questions/394170 | 0 | There is a way to define the notion of prime number in the framework of group theory, thanks to the following observation:
*Observation*: A non-trivial group $G$ is cyclic of prime order iff for any subgroup $K \le G$ then $K=1$ or $G$.
Now the notion of group and subgroup can be [internalized](https://ncatlab.org... | https://mathoverflow.net/users/34538 | Are the prime number objects given by the prime numbers? | The question is very vague as it never says what it means by "given by the data of a prime number". Though I think the following example should convince anyone that classyfing such "prime number object" can be much more complicated than listing prime numbers.
Take $K$ to be a fixed group and $\mathcal{C}$ to be the c... | 6 | https://mathoverflow.net/users/22131 | 394199 | 162,950 |
https://mathoverflow.net/questions/394150 | 6 | I am currently reading the paper "[Holomorphic Differentials of Generalized Fermat Curves](https://arxiv.org/pdf/1710.01349.pdf)" by Rubén Hidalgo. The case I am interested in is that of a classical Fermat curve $F\_k$, which in his terminology is a generalized Fermat curve of type $(k, 2)$. This is the curve in $\math... | https://mathoverflow.net/users/253977 | A basis of holomorphic differentials on Fermat curves | The automorphisms $a\_1,a\_2,a\_3$ are part of a $\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$ action on $F\_k$, which in turn induces an action on the space $H^0(\Omega^1\_{F\_k})$ of holomorphic differentials on $F\_k$. Let me explain how the representation-theory of this group gives the desired independence.
... | 6 | https://mathoverflow.net/users/51424 | 394216 | 162,954 |
https://mathoverflow.net/questions/394160 | 7 | Recently while studying cubic residues modulo a prime $p$ with $p\equiv1\pmod 3$, I met the following character sum:
$$\sum\_{0\le x\le p-1}\left(\frac{x}{p}\right)\left(\frac{x+1}{p}\right)\left(\frac{x+\omega}{p}\right)\left(\frac{x+\omega^2}{p}\right)\chi\_p(x^3),$$
where $\omega\in\mathbb{Z}$ with $\omega$ mod $p$ ... | https://mathoverflow.net/users/254318 | on a strange character sum | Your sum is expressible as a linear combination of Jacobi sums. More precisely, let us notice that
$$
\left(\frac{x^3}{p}\right)=\left(\frac{x}{p}\right)
$$
for all $x$ and also that $(x+1)(x+\omega)(x+\omega^2)=x^3+1$. So, the sum $S$ in question is equal to
$$
S=\sum\_{0\leq x\leq p-1}\left(\frac{x^3}{p}\right)\left(... | 13 | https://mathoverflow.net/users/101078 | 394223 | 162,958 |
https://mathoverflow.net/questions/394194 | 2 | In the course of my research I'm confronted with performing a numerical approximation of the solution of an initial value problem
$$\begin{cases}
y'=f(y,t),\\
y(t\_{0})=y\_{0}
\end{cases}\quad\quad(1)$$
with $y:I\rightarrow \mathbb{R}^d,\quad I\subseteq\mathbb{R}$ and where $f$ satisfies standard regularity assumptions... | https://mathoverflow.net/users/14101 | Euler method (and others) for unbounded intervals | Regarding 1 and 2:
Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in **provably (pointwise) convergent schemes**. At least for traditional methods (Runge-Kutta, linear multistep, and many others) the step-by-step nature of the approximation **precludes the pos... | 5 | https://mathoverflow.net/users/20507 | 394233 | 162,961 |
https://mathoverflow.net/questions/394151 | 1 | A $b$-fold coloring of a graph G is an assignment of sets of size $b$ to vertices of a graph such that adjacent vertices receive disjoint sets. An $a:b$-coloring is a $b$-fold coloring out of $a$ available colors. The $b$-fold chromatic number ${\displaystyle \chi \_{b}(G)}$ is the least $a$ such that an $a:b$-coloring... | https://mathoverflow.net/users/148974 | $2$-fold edge $b$-coloring of graphs | The $(a:2)$-edge-coloring problem is equivalent to replacing each edge by two parallel edges. As there are many results on edge-coloring that take the multiplicity $\mu$ into account, they naturally give results on $(a:2)$-edge-coloring (or $(a:b)$-edge-coloring, more generally).
(You have to be careful about the fac... | 2 | https://mathoverflow.net/users/45855 | 394239 | 162,963 |
https://mathoverflow.net/questions/394198 | 1 | Start with a natural number $k$, and choose natural numbers $K=\{n\_1,\ldots,n\_k\}$ which are pairwise distinct. For each $1\leq j\leq k$, choose another integer $i\_j$ such that $0\leq i\_j\leq n\_j$.
>
> **Question :** What is the minimum size of the set $A=\{i\_1,n\_1-i\_1, i\_2,n\_2-i\_2,\ldots, i\_k,n\_k-i\_k... | https://mathoverflow.net/users/148866 | An intriguing inverse sumset problem | Your question is basically asking for sets where the sum set has maximal size, which is attainable by e.g. a geometric progression. More precisely, in your set-up:
Given $k$ the minimum possible size of $A$ is the smallest $n$ such that $\binom{n+1}{2}\geq k$ (so asymptotically $n\sim (2k)^{1/2}$).
This is an upper... | 2 | https://mathoverflow.net/users/385 | 394241 | 162,964 |
https://mathoverflow.net/questions/394237 | 7 | The rectification theorem says that near a regular point every vector field $v$ is standard, that is, there exists a local coordinate system such that $v=\frac{\partial }{\partial x\_1}$.
In the textbooks (e.g. of Arnold and Hartmann) the vector field is assumed to be $C^r$ with $r\ge 1$ and the local coordinate syst... | https://mathoverflow.net/users/14515 | Smoothness of coordinates in the rectification theorem for ODE | In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$.
Basically, the reason is this: If $V$ is a $C^r$ vector field on $\mathbb{R}^2$ and $V(0,0)\not=0$, then there exist local $C^r$ coordinates $(x^1,x^2)$... | 11 | https://mathoverflow.net/users/13972 | 394255 | 162,967 |
https://mathoverflow.net/questions/394254 | 4 | My question is "inspired" by the [uniformization theorem for Riemmannian surfaces](https://en.wikipedia.org/wiki/Uniformization_theorem) and [this post](https://mathoverflow.net/questions/45953/is-every-real-n-manifold-isomorphic-to-a-quotient-of-mathbbrn).
---
Suppose that $X$ is connected (finite-dimensional) t... | https://mathoverflow.net/users/36886 | Can every manifold be represented as a quotient | The answer to your first question is “no”. The first counter-examples appear in dimension four - for example the complex projective plane. There are many simply connected manifolds and that are not aspherical that will fit the bill.
Here is a somewhat wilder example. Suppose that $V$ is a genus two handlebody of dime... | 8 | https://mathoverflow.net/users/1650 | 394256 | 162,968 |
https://mathoverflow.net/questions/394269 | 7 | In Hatcher's Chapter 5 ([https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf](https://pi.math.cornell.edu/%7Ehatcher/AT/ATch5.pdf)) on page 574 (page 57 in the pdf), he states that $\iota\_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ is not in the image of $H^8(K(\mathbb Z/2,4);\mathbb Z)\to H^8(K(\mathbb Z/2,4);\mathbb Z/... | https://mathoverflow.net/users/256999 | Why does $\iota_4^2 \in H^8(K(\mathbb Z/2,4);\mathbb Z/2)$ not come from $H^8(K(\mathbb Z/2,4);\mathbb Z)$? | It is true, and follows from results of Browder on the mod 2 Bockstein spectral sequence for $K(\mathbb{Z}/2,4)$. (We can replace $4$ by any even integer $k$ and conclude that $\iota\_k^2$ ia not the reduction of an integral class.) An argument is spelled out in Section 3 of
*Grant, Mark; Szűcs, András*, [**On realiz... | 10 | https://mathoverflow.net/users/8103 | 394270 | 162,973 |
https://mathoverflow.net/questions/394238 | 3 | I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C\_1 \leq V(x) \leq C\_2$ for all $x \in [0,\infty)$ for some constants $C\_2 > C\_1 >0$.
I am interested to know that a solution (I have some more specific equation of this type, whose behaviour at $\infty... | https://mathoverflow.net/users/2095 | Second order differential equation with oscillating behavior | Under your conditions, not all solutions are bounded. Take $V=V\_0+\lambda$,
where $V\_0$ is a periodic function, and $\lambda$ is going to be large positive.
Then, for generic $V\_0$ we have the following picture: there is a sequence
of real numbers $\lambda\_k\to+\infty$ which divides the positive ray into
intervals.... | 6 | https://mathoverflow.net/users/25510 | 394279 | 162,977 |
https://mathoverflow.net/questions/394243 | 10 | It is well known that when $K$ is a local or global field the Galois cohomology group $H^{3}(K,K\_{\text{sep}}^{\times})=0$ where $K\_{\text{sep}}$ denotes the separable closure of $K$. Could someone give an example of a field $K$ where $H^{3}(K,K\_{\text{sep}}^{\times}) \neq 0$ and why it is non-zero in this case?
| https://mathoverflow.net/users/211978 | Third Galois cohomology group | The group $H^3(K,\bar{K}^\times)$ naturally arises when trying to calculate the Brauer group of a variety. Explicitly, the Hochschild-Serre sequence yields the exact sequence
$$0 \to \mathrm{Br}\_1(X)/\mathrm{Br}(K) \to H^1(K,\mathrm{Pic}(X\_{\bar{K}})) \to H^3(K,\bar{K}^\times)$$
for a projective variety $X$ over a pe... | 16 | https://mathoverflow.net/users/5101 | 394282 | 162,979 |
https://mathoverflow.net/questions/394302 | 4 | Let $(M, g)$ be a Riemanian manifold (or $\mathbb{R}^n$ if you prefer). A TT-tensor is a symmetric 2-tensor $\sigma\_{ab}$ satisfying
* $g^{ab} \sigma\_{ab} \equiv 0$ ($\sigma$ is trace free),
* $\nabla^a \sigma\_{ab} = 0$ ($\sigma$ is divergence free).
These tensors appear in general relativity to construct initia... | https://mathoverflow.net/users/24271 | Compactly supported transverse traceless tensors | The answer is Yes, at least under the reasonable conditions that (i) the number of conformal Killing vectors locally admitted by $(M,g)$ is constant and that (ii) the de Rham cohomology $H^{n-1}(M)=0$ (where the differential forms should be taken of whatever regularity in which you want to prove density). In fact, unde... | 3 | https://mathoverflow.net/users/2622 | 394311 | 162,989 |
https://mathoverflow.net/questions/394310 | 0 | Let $X\_t=2+t+W\_t$ for $t\ge 0$, where $(W\_t)\_{t\ge 0}$ is a standard Brownian motion. For every $n\ge 1$, set $X^n\_t:=X\_t-{\bf 1}\_{t\ge n}$. Denote respectively
$$\tau:=\inf\{t\ge 0:~ X\_t\le 0\}\quad \mbox{and} \quad \tau^n:=\inf\{t\ge 0:~ X^n\_t\le 0\}.$$
Could we prove or disprove $\lim\_{n\to\infty}\math... | https://mathoverflow.net/users/nan | Convergence of the probabilities that drifted Brownian motion with jump never hits zero | Just use the Markov property at time $n$.
Write $f(x)$ for the probability that $x+t+W\_t$ never hits $(-\infty, 0]$. Then
$$ \mathbb P[\tau = \infty] = \mathbb E[\mathbb 1\_{\{\tau > n\}} f(X\_n)] $$
and
$$ \mathbb P[\tau^n = \infty] = \mathbb E[\mathbb 1\_{\{\tau > n\}} f(X\_n - 1)] , $$
so the difference of the tw... | 0 | https://mathoverflow.net/users/108637 | 394312 | 162,990 |
https://mathoverflow.net/questions/394296 | 5 | **Motivation.** In their [paper](https://www.aumasson.jp/data/papers/AJN14a.pdf) about the cryptographic scheme NORX, the authors use a fast approximation of + by bitwise operations (taking fewer CPU cycles than proper addition) using the formula $$a+b "=" a \oplus b \oplus ((a \land b) \ll 1)$$ where $\oplus$ is bitwi... | https://mathoverflow.net/users/8628 | Hamming distance between $a+b$ and $a \oplus b \oplus ((a \land b) \ll 1)$ | The lim sup is exactly $1$. Almost certainly the exact value of $D\_n$ comes from $a = (1,1,1,\ldots,1)$ and $b = (1,0,0,\ldots,0)$, and even if not, it’s off by at most $O(1/n)$ which is inconsequential. In such case we have about the maximum number of carries which are undetected by the $a \land b$. So $a+b = 2^n$ bu... | 5 | https://mathoverflow.net/users/23373 | 394319 | 162,994 |
https://mathoverflow.net/questions/394069 | 6 | Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable but how bad can $P$ be? Is every separable path-connected metric space, homeomorphic to the path component of some compact... | https://mathoverflow.net/users/5801 | How complicated can the path component of a compact metric space be? | Not every path connected separable metric space is homeomorphic to a path component of a compact metric space. The following cardinality arguments can be used:
Fact 1. There is up to homeomorphism more than $|\mathbb R|$-many path-connected separable metric spaces:
Just consider all the subspaces of the form $X\_A=(\... | 4 | https://mathoverflow.net/users/128723 | 394350 | 162,999 |
https://mathoverflow.net/questions/394354 | 3 | Let $\pi: \mathcal{X} \to \mathbb{P}^1$ be a pencil of projective $\mathbb{C}$-varieties such that a general fiber is smooth. Let $\mathbf{P}$ be one of the properties: rational, unirational, stably rational. My questions is: is there a property $\mathbf{P}$ such that if a general fiber of $\pi$ satisfies $\mathbf{P}$ ... | https://mathoverflow.net/users/58203 | Rationality in pencil of projective varieties | For $\mathbf{P}=$ rational, the answer is no, even under your stronger assumption: take the Fermat cubic threefold $X\_0^3+\ldots+X\_4^3=0$, and consider the pencil of hyperplanes $X\_1=\lambda X\_0$. After blowing up the base locus you get a morphism $\mathscr{X}\rightarrow \mathbb{P}^1$ whose smooth fibers are isomor... | 7 | https://mathoverflow.net/users/40297 | 394361 | 163,002 |
https://mathoverflow.net/questions/394335 | 1 | Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (conformal field theory).
Similar to [Every mathematician has only a few tricks](https://mathoverflow.net/questions/363... | https://mathoverflow.net/users/111389 | What is a simplified intuitive explanation of conformal invariance? | An explanation to the level of high-school mathematics might be a bit too hard, but let's try undergrad mathematics.
On a space of $d$ dimension (take $d=1,2,3$ if that's easier to visualize), namely $\mathbb{R}^d$, the Euclidean group can defined as the group of bijective transformations $T$ which preserve the Eucli... | 3 | https://mathoverflow.net/users/7410 | 394363 | 163,003 |
https://mathoverflow.net/questions/394359 | 4 | In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps from the edge groups to the vertex groups are **n**ot necessarily **i**njective. I will call these n.i. graphs of group... | https://mathoverflow.net/users/99414 | Realizing groups as the fundamental group of graphs of groups allowing non-injective maps? | Let $\mathcal C$ be a class of groups closed under homomorphic images. Then any ni-graph of groups whose vertex and edge groups belong to $\mathcal C$ has the same fundamental group as an ordinary graph of groups with vertex and edge groups in $\mathcal C$ as per @HJRW's answer to [this question](https://mathoverflow.n... | 7 | https://mathoverflow.net/users/15934 | 394365 | 163,004 |
https://mathoverflow.net/questions/394373 | 1 | $$\text{Let } T\_n=\begin{Bmatrix}
a & b & \boldsymbol{0} \\
b & a & \ddots \\
\boldsymbol{0} & \ddots & \ddots
\end{Bmatrix}\text{ a symmetric tridiagonal Toeplitz matrix of size }n\text{.}$$
I need to find its determinant in terms of $n$.
We know $\det T\_1=a$ and $\det T\_2=a^2-b^2$.
We can prove that, $\for... | https://mathoverflow.net/users/248399 | Solving recurrence relation for symmetric Toeplitz matrices determinant | $$T\_n=\frac{1}{2^{n+1}} \left(c\_-^n+c\_+^n+\left(c\_+^n-c\_-^n\right)\frac{c\_++c\_- }{c\_+-c\_-}\right),\;\;\text{with}\;\; c\_\pm=a\pm\sqrt{a^2-4 b^2}.$$
| 2 | https://mathoverflow.net/users/11260 | 394381 | 163,012 |
https://mathoverflow.net/questions/394353 | 7 | I am looking for a proof of one of the integrals presented in Harry Bateman's *Tables of Integral Transforms*. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):
$$ \int\_0^{\infty} \left[x^{\nu+\frac{1}{2}} \mathrm{e}^{-\frac{1}{2} x^2} L^{\nu}\_n\!\left(x^2\right) \right]J\_{\... | https://mathoverflow.net/users/161011 | Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial | The question asks for a reference; the body asks for proof. Here's a proof that's really more of a verification. From the proposed formula make a generating formula:
$$\sum\_{n=0}^\infty z^n \int\_0^\infty x^{\nu+1}\exp{(-x^2/2)}L\_n^{\nu}(x^2) J\_{\nu}(xy)\,dx \overset{?}{=}$$
$$ y^{\nu}\exp{(-y^2/2)} \sum\_{n=0}^\inf... | 6 | https://mathoverflow.net/users/121836 | 394388 | 163,017 |
https://mathoverflow.net/questions/394391 | 17 | Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
| https://mathoverflow.net/users/105900 | Suspension of a topological space | It’s not true. The Poincare sphere $P$ is a manifold, and its suspension is not. But its double suspension is homeomorphic to $S^5$ by Cannon’s “Double Suspension Theorem”. I learned about this from Mark Grant in an answer to a different question of mine on MO.
| 31 | https://mathoverflow.net/users/3634 | 394392 | 163,019 |
https://mathoverflow.net/questions/394179 | 2 | Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added good node of residue $i$, the result will be the Mullineux involution $M(P)$ applied to $P$ to which we add a good node... | https://mathoverflow.net/users/198061 | Reference for the action of the Mullineux involution on a partition with an added good node | The equivalence of Kleshchev's algorithm and Mullineux's algorithm was proved by Ford and Kleshchev, but the result they prove is slightly weaker than you want. The result you're asking for is Corollary 4.12 in Bessenrodt & Olsson 'On Residue Symbols and the Mullineux Conjecture', J. Algebraic Combin 7.
| 1 | https://mathoverflow.net/users/6771 | 394410 | 163,023 |
https://mathoverflow.net/questions/394419 | 2 | Is there any closed form of
$$\prod\_{k=1}^{n}\left(\cos(kx)-1\right)?$$
I failed to find references on this problem in the internet.
| https://mathoverflow.net/users/159935 | Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$ | $$\prod\_{k=1}^{n}\left(\cos kx-1\right)=2^{-n} e^{-\frac{1}{2} i n (n+1) x} \left(e^{i x};e^{i x}\right)\_n^2,$$
with $(\cdot;\cdot)\_n$ the [q-Pochhammer symbol.](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol)
I guess this counts as a "closed form", but of course it's just a rewriting of the product in terms o... | 5 | https://mathoverflow.net/users/11260 | 394420 | 163,025 |
https://mathoverflow.net/questions/394418 | 8 | Let $\kappa$ be the supremum of ordinals first order definable in L without parameters. Assume $0^\sharp$ exists. Is $\kappa$ the least silver indiscernible ordinal?
| https://mathoverflow.net/users/170286 | Is the supremum of L-definable cardinals silver-indiscernible | **Updated.** The answer is no.
First, let me point out that in general, in ZFC we are not able to refer to the notion of *first-order-definable-in-$L$*, since definability is not expressible. But in your case, we have $0^\#$, from which we are able to define a truth predicate for first-order truth in $L$, and so your... | 8 | https://mathoverflow.net/users/1946 | 394421 | 163,026 |
https://mathoverflow.net/questions/394375 | 5 | $\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical gauge theory and principal bundles in general yet, therefore I need some help. To be precise, I have the following question:... | https://mathoverflow.net/users/259525 | Spin connection in the tetradic Palatini-formalism of general relativity | For a finite dimensional inner product space $(V,\eta)$, $\bigwedge^2 V \cong\_\eta \mathfrak{so}(\eta) \subset \operatorname{End}(V) \cong V\otimes V^\* \cong\_\eta V\otimes V$. The antisymmetry condition appears when expanding the identity $\eta(e^{tA}v,e^{t A}u) = \eta(v,u)$ to first order in $t$, to get $\eta(Av,u)... | 3 | https://mathoverflow.net/users/2622 | 394428 | 163,027 |
https://mathoverflow.net/questions/394426 | 3 | Let $l$ be a prime. Suppose that $M\_0^{!}(\Gamma\_0(l))$ donote the space of weakly holomorphic modular forms of weight $0$ for the congruence subgroup $\Gamma\_0(l)$. Does there exist a $f\in M\_0^{!}(\Gamma\_0(l))$ such that $f$ just has a simple pole at $\infty$? Here I require that $f$ is holomorphic on the upper ... | https://mathoverflow.net/users/228737 | weakly holomorphic modular forms with a simple pole at $\infty$ | No such $f$ exists, except for the five specific prime levels $2,3,5,7,13$ you describe.
Any such form would give an isomorphism of Riemann surfaces between $X\_0(p)$ and $\mathbf{P}^1$. So it cannot exist if $X\_0(p)$ has genus $> 0$, which holds for all $p$ except these five.
EDIT. You asked about the minimal ord... | 8 | https://mathoverflow.net/users/2481 | 394438 | 163,030 |
https://mathoverflow.net/questions/394415 | 3 | I am reading the following article of [Berger, p8](https://mat.uab.cat/%7Emasdeu/files/padichodge/berger_intro_theory_padic_reps.pdf) and I don't understand the idea:
>
> $C\_p:=\widehat{\overline{\mathbb Q\_p}}$ does not contain the periods
>
>
>
The text seem to reason as follows
>
> (under some conditio... | https://mathoverflow.net/users/97321 | Why does $\mathbb C_p$ not contain the periods? | Consider the Tate motive $\mathbb Q(1)$. Its de Rham realization is simply $\mathbb Q$ (with the filtration $F^{-1}\mathbb Q=\mathbb Q$ and $F^{0}=0$) and its Betti realization is $2\pi i\mathbb Q$. The comparison theorem you recalled works because after extension of scalars to $\mathbb C$, you can multiply by $2\pi i$... | 9 | https://mathoverflow.net/users/2284 | 394449 | 163,034 |
https://mathoverflow.net/questions/374488 | 11 | It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F\_2$, the free group on two generators, cannot be embedded as a closed subgroup of a locally compact amenable group. However, by a result of Pestov, $F\_2$ embeds as a closed subgroup of $\mathrm{A... | https://mathoverflow.net/users/49381 | Does every topological group embed as a closed subgroup in an amenable group? | At least for Polish groups, which was the case I was most interested in, the answer is positive.
I mentioned this question to Ola Kwiatkowska yesterday and she immediately pointed out that one of the standard universal Polish groups, the group $\mathrm{Iso}(\Bbb U)$ of isometries of the Urysohn space, is in fact not ... | 2 | https://mathoverflow.net/users/49381 | 394459 | 163,036 |
https://mathoverflow.net/questions/393989 | 5 | For all ring with unit element $A$ let $W(A)$ be the ring of $p$-typical Witt vectors. Denote by $$\phi\;:\;W(A)\to A^{\mathbb{N}}$$
the ghost map, which is given by
$$\phi(a\_0,a\_1,a\_2,\ldots)\;=\;(\phi\_0,\phi\_1,\phi\_2,\ldots)$$
where $\phi\_n=\phi\_n(a\_0,\ldots,a\_n)$ is defined by
$$\phi\_n=a\_0^{p^n}+pa\_1^{p... | https://mathoverflow.net/users/21374 | Image of the ghost map of $p$-typical Witt vectors and $A$-ring structure of $W(A)$ | I don't have a full answer, however I also don't have enough reputation to just comment, so I will post this as an answer.
In what follows, we will overline all projections modulo $p$.
Let $A$ be any commutative ring with no $p$-torsion. Then the ghost map $\phi$ is injective. Let $a\in A$ be such that $\left(a,a,\... | 3 | https://mathoverflow.net/users/261563 | 394461 | 163,037 |
https://mathoverflow.net/questions/394471 | 4 | Let $(\mathcal{C}, \otimes , I)$ and $(\mathcal{C}, \otimes', I')$ be tensor categories. A tensor functor $F: (\mathcal{C}, \otimes , I)\to (\mathcal{C}', \otimes' , I')$ consists of a functor $F: \mathcal{C}\to \mathcal{C'}$ together with natural isomorphisms $J\_{X,Y}: F(X)\otimes' F(Y) \to F(X\otimes Y)$ and an isom... | https://mathoverflow.net/users/216007 | Equivalence of definitions tensor functor | Yes. Given (1) with quasi-inverse $F':\mathcal{C}'\to\mathcal{C}$, there is a unique way to make $F'$ a tensor functor such that $\eta$ and $\theta$ are tensor isomorphisms. Specifically, the tensor structure is
$$ F' X \otimes F' Y \cong F' F(F' X \otimes F' Y) \cong F'(F F' X \otimes' F F' Y) \cong F'(X\otimes' Y) ... | 7 | https://mathoverflow.net/users/49 | 394472 | 163,040 |
https://mathoverflow.net/questions/394477 | 0 | The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, [Are ITTM's necessary to compute Turing's "computable numbers" and what dos that mean for ordinary recursion theory](https://mathoverflow.net/q/393923/30186):
>... | https://mathoverflow.net/users/20597 | What is the smallest countable limit ordinal in which 'lost melodies' occur | I interpret this question as asking, what is the first ordinal $\alpha$ such that there is some lost melody (in the sense of [Hamkins-Lewis](https://arxiv.org/abs/math/9808093) Theorem 4.9) in $L\_\alpha$.
The answer is $\alpha=\Sigma+1$ (or $\Sigma+\omega$ if you really insist on it being limit). Indeed, the argumen... | 4 | https://mathoverflow.net/users/30186 | 394479 | 163,043 |
https://mathoverflow.net/questions/394367 | 4 | Let us fix $0 \neq \lambda \in \mathbb{R}$. Let us consider the following ODE, on $[0,\infty)$: $$ y^{\prime \prime} (x) + \frac{r e^{-x}}{(1+e^{-x})^2} y(x) = -\lambda^2 y(x).$$ Here $r \ge 1$ is a parameter. Let us consider the solution $e\_r (x)$ which satisfies $e\_r (x) \sim e^{i \lambda x}$ as $x \to +\infty$. Ho... | https://mathoverflow.net/users/2095 | Joint boundedness of solutions of a family of Sturm-Liouville ODE | It is bounded. Moreover, $|y(x)|\leq 1$ for $x\geq 0$ for all $r>0$ and $\lambda>0$ (so the estimate is uniform not only in $r$ but in $\lambda$ as well).
This is a special case of the following theorem due to User @Fedja.
Theorem. In the equation $y''+V(x)y=0$, let the potential $V$ be decreasing and
bounded from ... | 3 | https://mathoverflow.net/users/25510 | 394487 | 163,044 |
https://mathoverflow.net/questions/394484 | 1 | Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ and $z=F(x,y)$ is Lipschitz continuous. Assume that for any $x\in\mathbb R^d$, there is a unique $y$ such that $F(x,y)=0$. Let $y=y(x)$ be a function that is produced by the implicit function $F(x,y)=0$. Can we prove that $y=y(x)$ is also Li... | https://mathoverflow.net/users/120302 | Lipschitz continuity of an implicit function | The answer is **no**.
>
> **Theorem.** Let $f:\mathbb{R}^d\to\mathbb{R}$ be any continuous function. Then there is a Lipschitz function $F:\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R}^d$, $F(x,y)=0$ if and only if $y=f(x)$.
>
>
>
*Proof.* Let $G\subset\mathbb{R}^{d}\times\math... | 4 | https://mathoverflow.net/users/121665 | 394489 | 163,045 |
https://mathoverflow.net/questions/394405 | 0 | Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})$ and the open book decomposition induces a contact structure $\xi$ on $Y$. Is there any criterion for tightness/overtw... | https://mathoverflow.net/users/169890 | Tightness/Overtwistedness of genus one open book decomposition | [Honda–Kazez–Matić](https://arxiv.org/pdf/math/0609734.pdf) and [Baldwin](https://arxiv.org/pdf/math/0604580.pdf) proved that tight here is equivalent to right-veering, so you can sit down and work that out. The former paper shows that reducible monodromies always give you a tight contact structure, and periodic ones g... | 1 | https://mathoverflow.net/users/51178 | 394490 | 163,046 |
https://mathoverflow.net/questions/394482 | 1 | Let $(L^n)\_{n\ge 1}$ be a sequence of non-decreasing and right-continuous stochastic processes s.t. $0\le L^n\_t\le 1$ for all $t\ge 0$. Let $\ell:[0,\infty)\to [0,1]$ be a non-decreasing and right-continuous function. Assume that one has almost surely
$$\lim\_{n\to\infty}L^n\_t=\ell(t),\quad \mbox{for all the point... | https://mathoverflow.net/users/nan | Does convergence under expectation implies convergence almost surely? | $\newcommand{\ep}{\varepsilon}\newcommand{\om}{\omega}\newcommand{\Om}{\Omega}\newcommand{\Si}{\Sigma}$The answer is: not in general.
Indeed, let $(\Om,\Si,P)$ be the underlying probability space. Suppose that there exists a sequence $(\Om\_n)$ in $\Si$ such that
\begin{equation}
P(\Om\_n)\to0
\end{equation}
(as $n\... | 1 | https://mathoverflow.net/users/36721 | 394493 | 163,047 |
https://mathoverflow.net/questions/394494 | 1 | I noticed something with Mersenne numbers : you can write it with the form $a+b+c = 2^n-1$ and $a^2+b^2+c^2 = (4^n-1)/3$ when $n$ is a odd Mersenne exponent $(3, 5, 7, 13, \dotsc)$ or an exponent of a odd Mersenne exponent ($3^2, 5^4, 7^3,\dotsc $)
For example with Mersenne exponent :
* $4+2+1 = 7 = 2^3-1$ and $4^2... | https://mathoverflow.net/users/264367 | How to prove than $a+b+c = 2^n-1$ and $a^2+b^2+c^2 = (4^n-1)/3$ have integer solutions only with Mersenne exponent or exponents of Mersenne exponent? | Let $n\in\mathbb{N}$ be odd. The system of equations
$$a+b+c=2^n-1\qquad\text{and}\qquad a^2+b^2+c^2=\frac{4^n-1}{3}$$
is equivalent to
$$a+b+c=2^n-1\qquad\text{and}\qquad (a-b)^2+(b-c)^2+(c-a)^2=2^{n+1}-2.$$
It follows that any real solution $a,b,c\in\mathbb{R}$ satisfies
$$a,b,c\in\left[\frac{2^n-1}{3}-\sqrt{2^{n+1}-... | 4 | https://mathoverflow.net/users/11919 | 394497 | 163,048 |
https://mathoverflow.net/questions/394518 | 14 | In the issue **[Electronic Notes in Theoretical Computer Science](https://www.sciencedirect.com/science/article/pii/S1571066105803081?via%3Dihub)** *Volume 29, 1999, Page 79* there is a very intriguing abstract by *Peter Freyd*.
>
> **Path Integrals, Bayesian Vision, and Is Gaussian Quadrature Really Good?** Physic... | https://mathoverflow.net/users/104432 | Peter Freyd on path Integral? | In his late paper [**Algebraic real analysis**](http://www.tac.mta.ca/tac/volumes/20/10/20-10.pdf) (published on *Theory and Applications of Categories, Vol. 20, No. 10, 2008, pp. 215–306*) he writes in the very last page what follows.
>
> In September 1999 at an invited talk at the annual ctcs meeting (held that y... | 15 | https://mathoverflow.net/users/104432 | 394521 | 163,053 |
https://mathoverflow.net/questions/394509 | 4 | Let $A$ be an Artin algebra and $\text{mod}\,A$ the category of finite length modules. Further, let $X\_0 \longrightarrow X\_1 \longrightarrow X\_2 \longrightarrow ...$ and $Y\_0 \longleftarrow Y\_1 \longleftarrow Y\_2 \longleftarrow ...$ be chains of morphisms in $\text{mod}\,A$ such that $\underset{\longrightarrow}{\... | https://mathoverflow.net/users/145920 | Let A be an Artin algebra. What happens if the limit and inverse limit are the same in mod A? | Let me show that every such $Z$ has finite length.
First, note that being of the form $\varinjlim X\_i$ (with the $X\_i$ of finite length) is the same as being countably generated over $A$. So let us start with $Z=\varprojlim Y\_j$ (with the $Y\_j$ of finite length) and prove that $Z$ cannot be countably generated un... | 4 | https://mathoverflow.net/users/7666 | 394524 | 163,055 |
https://mathoverflow.net/questions/394483 | 6 | Let $ A := \mathbb{C}\langle x, y \rangle / ( xy-2 ) $ where $ \mathbb{C}\langle x, y \rangle $ is the free (non-commuative) $ \mathbb{C} $-algebra which is generated by $ x $ and $ y $ and $ ( xy-2 ) $ is the two-sided ideal which generated by the element $ xy-2 $.
Let us also write $ x $ resp. $ y $ for the classes... | https://mathoverflow.net/users/132492 | A power of a sum in a non-commutative algebra | Consider the sum of the coefficients of the degree-$(n-2k)$ "reduced" monomials $y^ix^j$ in $(x+y)^n$. The degree of a reduced monomial always differs from $n$ by an even number, so this will capture everything. This is clearly of the form $2^k\cdot T(n,k)$ for some unknown function $T$, since each time you reduce the ... | 4 | https://mathoverflow.net/users/141277 | 394538 | 163,056 |
https://mathoverflow.net/questions/394537 | 1 | Prove for the Bernoulli numbers $B\_n$, that for all $a \in \mathbb{N}$, that $ \sum\_{i=0}^{2a+1} {2a+1 \choose i} B\_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $. As much as this is a neat identity, it's a crucial step in another result concerning powersums. I have no idea how to prove it, aside from noting that it cannot be dir... | https://mathoverflow.net/users/265714 | Prove $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $ for Bernoulli numbers $B_{n}$ | Dividing your expression by $(2a+1)!$ and using the definition of binomial coefficients, we see that you would like to prove that
$$
\sum\_{i=0}^{2a+1} \frac{1}{(2a+1-i)!i!}B\_{2a+1-i}((n+1)^i+(-n)^i)=0
$$
or, in other words,
$$
\sum\_{b+c=2a+1}\frac{B\_b}{b!}\frac{(n+1)^c+(-n)^c}{c!}=0.
$$
Replacing $2a+1$ by arbitrar... | 8 | https://mathoverflow.net/users/101078 | 394539 | 163,057 |
https://mathoverflow.net/questions/392361 | 2 | I was wondering if geodesics are defined for all time on compact Finsler manifolds, or more generally, for any spray on a compact manifold (where by geodesics, I simply mean the integral curves of the spray).
I thought perhaps that the homogeneity condition of a spray would mean that the "size" of vectors couldn't gr... | https://mathoverflow.net/users/147463 | Completeness on the tangent bundle | These answers are primarily due to Juan Carlos Álvarez-Paiva's comments. I'm just recounting them:
1. If $(M,g)$ is a complete, then $TM$ is also complete under the Sasaki metric, as shown [here](https://math.stackexchange.com/questions/3116727/completeness-of-the-tangent-bundle-of-riemannian-manifold/4156094#4156094... | 1 | https://mathoverflow.net/users/147463 | 394548 | 163,059 |
https://mathoverflow.net/questions/394481 | 3 | If I consider a simple object $X$ in a fusion category and tensor it with its dual $X^\*$, and let $Y$ be a simple object in the decomposition $X \otimes X^\* = I + Y + \dotsb$. I want to say that $Y$ cannot be pseudo-real, i.e. $Y^{\*\*} = -Y$ (Frobenius-Schur indicator $= -1$), because $(X \otimes X^\*)^{\*\*} = (X^{... | https://mathoverflow.net/users/138208 | Pseudo-real (Frobenius-Schur indicator $= -1$) simple object in $X \otimes X^*$? | If $Y$ appears with multiplicity one, then it cannot be pseudo-real. The proof is that it has a symmetric pairing given by restricting the symmetric pairing on $X \otimes X^\*$.
However, if $Y$ appears with even multiplicity, then there are counterexamples, for example see [this answer](https://mathoverflow.net/a/554... | 4 | https://mathoverflow.net/users/22 | 394557 | 163,064 |
https://mathoverflow.net/questions/394558 | 7 | Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$.
Is it therefore also true that for any singular value $\sigma$ of $A$, in a domain where they are simple, turn $\log \sigma(A(z))$ i... | https://mathoverflow.net/users/150564 | Are $\log(\sigma(A(z))$ subharmonic functions? | First a positive result, which answers a different but related question. This paper of Aupetit
>
> On log sub-harmonicity of singular values of matrices, Studia Mathematica 122 (1997), 195-200; DOI 10.4064/sm-122-2-195-200
>
>
>
has the following abstract:
>
> Let $F$ be an analytic function from an open s... | 9 | https://mathoverflow.net/users/763 | 394561 | 163,066 |
https://mathoverflow.net/questions/394520 | 2 | I have never been interested in this before, and I have become interested in to find some answers and my teaching on the fundamental group has led me in this direction. Neither, I don't know if it suits for MO. So, please down-vote or vote-to-close after you provided some references.
I think if $X$ is semi-locally si... | https://mathoverflow.net/users/51223 | The topology of the Deck group of a covering map | I reiterate from my comments that a covering map is a discrete-sort-of-thing by definition, no matter what space you're looking at. If you are interested in non-discrete deck transformations or situations where such things are useful, there is a large literature on generalizations of covering space theory ([see my answ... | 4 | https://mathoverflow.net/users/5801 | 394562 | 163,067 |
https://mathoverflow.net/questions/394565 | 6 | **Definitions:**
Let $E$ be a subset of $X$. By an extension of a function $f: E \to \mathbb R$, I mean a function $\bar f: X \to \mathbb R$ such that $f = \bar f$ on $E$.
>
> **Question:** For every continuous function $f: \mathbb Q \to \mathbb R$, does there exist an extension $\bar f: \mathbb R \to \mathbb R$ ... | https://mathoverflow.net/users/173490 | Extending continuous functions from $\mathbb Q$ to $\mathbb R$ | You may extend by upper limit.
Details.
Denote $g(y)=\limsup\_{x\to y, x\in \mathbb{Q}} f(x)$ for all real $y$. So, possibly $g$ takes the value $+\infty$ or $-\infty$ at some points. But we have $g(x)=f(x)$ for rational $x$, and for each rational $x$ and each $\varepsilon>0$ there exists $\delta>0$ such that $|g(y... | 13 | https://mathoverflow.net/users/4312 | 394566 | 163,068 |
https://mathoverflow.net/questions/394491 | 4 | Question: Let $\varepsilon>0$ and $N\in\omega$ be sufficiently large (depending on $\varepsilon$).
Let $h:\subseteq N\rightarrow N$ be such that $h(B)\notin B$ for all $B\subsetneq N$.
Must there be $B\_0\subsetneq B\_1\subseteq N$ such that $|B\_1|\leq \varepsilon N$ and $h(B\_0)=h(B\_1)$?
More generally,
Let $a\i... | https://mathoverflow.net/users/74918 | Ramsey style theorem with unbounded colors | No, this is false if $\varepsilon<1/2$. Here is the construction for one $h$ that satisfies the conditions:
>
> Let $h(B)$ be the $|B|$-th smallest element of $N\setminus B$.
>
>
>
This way if $B\_1\subsetneq B\_2$, then $h(B\_1)<h(B\_2)$.
| 5 | https://mathoverflow.net/users/955 | 394567 | 163,069 |
https://mathoverflow.net/questions/394547 | 4 | I have a proof of the following fact related to ordinary generating functions, and I was curious if it was known, as it seems plausible it is classically known:
"Let $\lambda\_1,\ldots, \lambda\_k$ be algebraic numbers. Let $f(z)= \sum^\infty\_{n=0} c\_nz^n$ when each $c\_n\in \mathbb{Z}.$ Suppose $f$ analytically co... | https://mathoverflow.net/users/32470 | Ordinary generating functions with finitely many singularities at algebraic numbers are rational | It is not necessary to assume that $\lambda\_i$ are algebraic. This is a special case of the result in G. Pólya, Mathematische Annalen (1928), Volume: 99, page 687-706, [page 704](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0099?tify=%7B%22pages%22:%5B708%5D,%22view%22:%22help%22%7D) in particular:
>
> Der Sp... | 5 | https://mathoverflow.net/users/164965 | 394574 | 163,070 |
https://mathoverflow.net/questions/394579 | 10 | We reprint an old math SE question here (see <https://math.stackexchange.com/questions/1241224/composition-of-polynomials-and-galois-theory>):
>
> "
> Let $f(x)$ be a polynomial of degree $n$ over $\mathbb{Q}$, with Galois group
> isomorphic to the symmetric group $S\_n$. How do I show that $f$ cannot be expressed
... | https://mathoverflow.net/users/12826 | On the Galois group of the compositions of polynomials | Since $\mathcal{F} \le S\_k$, the wreath product of $\mathcal{F}$ with $S\_l$ is no larger than the wreath product of $S\_k$ with $S\_l$. This has cardinality
$$ (l!)^k k! < (k\cdot l)! = |S\_{k\cdot l}|.$$
(Perhaps the wreath product we need is the opposite one, with cardinality $(k!)^l l!$. The result is the same.)
... | 8 | https://mathoverflow.net/users/75344 | 394583 | 163,071 |
https://mathoverflow.net/questions/394532 | 2 | Let $D \subset \mathbf{R}^n$ be the unit ball, and $u \in C^2(D)$ be a solution of the linear elliptic PDE
\begin{equation}
a^{ij} D\_{ij} u + b^i D\_i u + cu = 0 \quad \text{in $D$},
\end{equation}
where the coefficients are regular, say of class $C^d$ for some integer $d \geq 1$.
Assume that $u(0) = 0$, $Du(0) = 0$... | https://mathoverflow.net/users/103792 | Does unique continuation also hold for derivatives of solutions? | In general there's no way what you want (as clarified [in this comment](https://mathoverflow.net/questions/394532/does-unique-continuation-also-hold-for-derivatives-of-solutions?noredirect=1#comment1009711_394532)) can be true.
Let $n = 3$, and fix $k = 3$. Consider the function $u = (x\_1)^2 - (x\_2)^2$ which is har... | 2 | https://mathoverflow.net/users/3948 | 394595 | 163,076 |
https://mathoverflow.net/questions/394589 | 2 | Let $1\leq{j}<p-1$ with $p$ a prime number. Is it true that for any positive integer $n$ with $n\not\equiv{j} \pmod{p-1}$ the congruence $$\sum\_{r>1}{n\choose r(p-1)+j}{r-1\choose j}\equiv 0 \pmod p$$ is valid?
| https://mathoverflow.net/users/169583 | Binomial congruence modulo prime number | Yes, this congruence is, in fact, true. To see this, notice first that
$$
{r(p-1)+j\choose j}=\frac{(r(p-1)+j)\ldots (r(p-1)+1)}{j!}\equiv
$$
$$
\equiv \frac{(j-r)(j-1-r)\ldots(1-r)}{j!}=(-1)^j{r-1\choose j}\pmod p
$$
so it's enough to show that
$$
\sum\_{r>1}{n\choose r(p-1)+j}{r(p-1)+j \choose j}\equiv 0 \pmod p.
$$
... | 5 | https://mathoverflow.net/users/101078 | 394598 | 163,078 |
https://mathoverflow.net/questions/394220 | 5 | Let
* $k$ be an algebraically closed field,
* $G$ a (smooth, connected) reductive algebraic group over $k$,
* $H$ a (smooth, connected) reductive group of semisimple rank 1, and
* $T$ a maximal torus in $H$.
I am specifically interested in the case where the characteristic of $k$ is a bad prime for $G$.
Suppose t... | https://mathoverflow.net/users/2383 | Lifting $\mathfrak{sl}_2$-triples | Typically not, even for good characteristic. This comes up indirectly in my paper with Adam Thomas [The Jacobson–Morozov theorem and complete reducibility of Lie subalgebras, JLMS](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms.12067) [arxiv version](https://arxiv.org/abs/1507.06234).
The nilpotent ... | 4 | https://mathoverflow.net/users/16185 | 394602 | 163,081 |
https://mathoverflow.net/questions/394608 | 1 | I am interested if there is a formula for these relationships. I was looking everywhere but I couldn't find anything. Is there some research regarding this topic or is there just no relationship? There is no specific reason for this question. Just curious.
Thank you for your help and time!
| https://mathoverflow.net/users/268222 | What is the relationship between the degree of a regular graph $k$, the number of verticies $v$ and the existence of cycles with length $n$? | The length of the smallest cycle in an $n$-vertex $k$-regular graph is no larger than $2 \log\_{k-1} n$; it is open as to whether this bound is tight, in the sense as to whether for $n$ arbitrarily large, is there a $k$-regular graph on $n' \ge n$ vertices where the length of the smallest cycle is at least $(2-o(1))\lo... | 2 | https://mathoverflow.net/users/122188 | 394613 | 163,085 |
https://mathoverflow.net/questions/394495 | 6 | For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ **spreads onto** $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$ the set $F[X]=\{F(r):r\in X\}$ is in $\mathbb{B}$.
Let $\mathbb{D},\mathbb{E}$ be the sets of dominating, escapi... | https://mathoverflow.net/users/8133 | Do escaping sets "uniformly" cover dominating sets under determinacy? | Here's something that seems to work. I can add more details if needed.
Suppose that every set of reals has the property of Baire. Then every function from $\omega^{\omega} \to \omega^{\omega}$ is continuous on a comeager set. Note that every comeager set is unbounded. Fix $F \colon \omega^{\omega} \to \omega^{\omega}... | 4 | https://mathoverflow.net/users/31807 | 394616 | 163,086 |
https://mathoverflow.net/questions/394621 | 2 | On [this](https://en.wikipedia.org/wiki/Shuffle_algebra) wikipedia page is stated that over the rational numbers, the shuffle algebra (over a set $X$) is isomorphic to the polynomial algebra in the Lyndon words (on $X$). I was wondering if you can show me a proof of this or give me a reference. Ideally I am looking for... | https://mathoverflow.net/users/226648 | The shuffle algebra over the rationals is isomorphic to the polynomial algebra in the Lyndon words | This is proven in Theorem 6.3.4 of [Hopf Algebras in Combinatorics](https://arxiv.org/abs/1409.8356) by Grinberg and Reiner.
They call it "Radford’s theorem on the shuffle algebra" citing Theorem 3.1.1(e) of [A natural ring basis for the shuffle algebra and an application to group schemes](https://doi.org/10.1016/002... | 2 | https://mathoverflow.net/users/51668 | 394622 | 163,089 |
https://mathoverflow.net/questions/394580 | 16 | Let $S$ be a smooth compact closed surface embedded in $\mathbb{R}^3$ of genus $g$.
Starting from a point $p$, define a random walk as taking discrete steps
in a uniformly random direction,
each step a geodesic segment of the same length $\delta$.
Assume $\delta$ is less than the injectivity radius and small with respe... | https://mathoverflow.net/users/6094 | Does a random walk on a surface visit uniformly? | This problem was first considered and solved by Sunada, see his 1983 paper [Mean-value theorems and ergodicity of certain geodesic random walks](http://www.numdam.org/item/?id=CM_1983__48_1_129_0). Alas, the authors of the quoted arxiv paper were not aware of this. Any assumptions on curvature and dimension are not nec... | 19 | https://mathoverflow.net/users/8588 | 394625 | 163,091 |
https://mathoverflow.net/questions/394642 | 8 | I've seen some works on the representation of fundamental groups, which are (at least for me) quite important topic in mathematics. For example, [Riemann-Hilbert correspondence](https://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_correspondence) relates representation of a fundamental group of complex algebraic varie... | https://mathoverflow.net/users/95471 | Representation theory of higher homotopy groups | There are many results that generalize the Riemann–Hilbert correspondence from the fundamental groupoid to the fundamental ∞-groupoid, for example:
* Jonathan Block, Aaron Smith. [A Riemann–Hilbert correspondence for infinity local systems](https://arxiv.org/abs/0908.2843).
* Joseph Chuang, Julian Holstein, Andrey La... | 12 | https://mathoverflow.net/users/402 | 394656 | 163,098 |
https://mathoverflow.net/questions/394647 | 4 | Is anything known about the set of concordance classes (also called pseudoisotopy classes) of the relative to the boundary diffeomorphisms of $D^4$?
| https://mathoverflow.net/users/9800 | Concordance classes of diffeomorphisms of $D^4$ | It is the trivial group.
Firstly, by isotopy extension the extension-by-the-identity homomorphism to the group of pseudoisotopy classes of orientation-preserving diffeomorphisms of $S^4$,
$$\widetilde{\pi}\_0(\mathrm{Diff}\_\partial(D^4)) \to \widetilde{\pi}\_0(\mathrm{Diff}^+(S^4)),$$
is surjective with kernel gener... | 5 | https://mathoverflow.net/users/798 | 394658 | 163,099 |
https://mathoverflow.net/questions/394660 | 2 | I've been using modular polynomials to compute isogeny vulcanoes with prime degree $l$ over finite fields $\mathbb{F}\_p$, excluding cases containing the $j$-invariants $0$ and $1728$ or $j$-invariants with double roots on the polynomial, building the graph by at each step using the polynomial to find the neighbors of ... | https://mathoverflow.net/users/269936 | Better way to compute elliptic curves over finite fields? | The paper [Isogeny volcanoes by Andrew V. Sutherland](https://arxiv.org/abs/1208.5370) contains various improvements for computing isogeny volcanoes. The published version is in *ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium*, 507–530,
Open Book Ser., 1, Math. Sci. Publ., Berkeley, CA, 2013.
| 2 | https://mathoverflow.net/users/11926 | 394661 | 163,101 |
https://mathoverflow.net/questions/394328 | 4 | Let $p$ be a prime number, $G=\textrm{Gal}(\overline{\mathbb{Q}}\_p/\mathbb{Q}\_p)$, and $\chi:G\rightarrow\mathbb{Z}\_p^\times$ the cyclotomic character. Let $\mathbb{C}\_p$ denote the completion of the algebraic closure of $\mathbb{Q}\_p$ and $\mathcal{O}\_{\mathbb{C}\_p}$ denote its ring of integers.
The Tate-Sen ... | https://mathoverflow.net/users/4181 | A Tate-Sen theorem mod $p$ | $\newcommand{\bQ}{\mathbb{Q}}\newcommand{\cO}{\mathcal{O}}\newcommand{\bC}{\mathbb{C}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\bF}{\mathbb{F}}$The open subgroup $Gal(\overline{\bQ}\_p/\bQ\_p(\mu\_p))$ acts trivially on the mod $p$ reduction of the cyclotomic character so the invariants in question can be computed as $$... | 4 | https://mathoverflow.net/users/39304 | 394666 | 163,103 |
https://mathoverflow.net/questions/394669 | 2 | If $X\neq\emptyset$ is a set, then ${\cal S}\subseteq {\cal P}(X)$ with ${\cal S}\neq \emptyset$ is said to be a *sunflower* if there is $K\subseteq X$ such that whenever $A\neq B\in{\cal S}$ ten $A\cap B = K$. ($K$ is sometimes said to be the *kernel* of ${\cal S}$, and it is allowed that $K = \emptyset$, in which cas... | https://mathoverflow.net/users/8628 | Sunflowers in $\omega$ consisting of infinite sets | There exist uncountable chains in the poset $([\omega]^\omega,\subseteq)$ (e.g. by bijecting $\omega$ with $\mathbb Q$ and taking all downwards-closed sets). Let $L$ be such a chain. The only sunflowers in $L$ have two elements, since if $A,B,C\in L$ and $A\subsetneq B\subseteq C$, then $A\cap B=A\neq B=B\cap C$.
| 6 | https://mathoverflow.net/users/30186 | 394671 | 163,104 |
https://mathoverflow.net/questions/394631 | 5 | Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?).
From what I have been able to gather from several lecture videos by Gaitsgory (such as [Gaitsgory - Singular support of cohe... | https://mathoverflow.net/users/143390 | Categorical-geometric Langlands for tori | In case $G=T$ is a torus and $G^\vee=T^\vee$ is the dual torus, the geometric Langlands conjecture — or “categorical geometric class field theory” (for a smooth projective curve $C$ over $\mathbb C$) becomes a theorem, and has the form you stated: a derived equivalence
$$\mathsf{DMod}(\mathsf{Bun}\_T)\simeq \mathsf{QCo... | 3 | https://mathoverflow.net/users/582 | 394673 | 163,106 |
https://mathoverflow.net/questions/394603 | 14 | Let $S\_n$ be the finite group given as $n \times n$ permutation matrices.
Define for a given field $K$ the algebra $B\_n$ as the subalgebra of $M\_n(K)$ generated by all permutation matrices of $S\_n$. (more generally we can do this for any subgroup of $S\_n$ to associate to a finite group such a subalgebra. Often w... | https://mathoverflow.net/users/61949 | A finite dimensional algebra associated to the symmetric group | Let $K$ be a field. The literal answer has already been given by several people but let me try and get at the algebraic structure and provide a quiver with relations (***see the addition***). Let $J$ be the $n\times n$ all ones matrix. Then $J$ centralizes $S\_n$ and the centralizer of $J$ consists of all matrices whos... | 2 | https://mathoverflow.net/users/15934 | 394685 | 163,108 |
https://mathoverflow.net/questions/394684 | 3 | Let $\mathcal H\_r=\mathbb P (\mathcal O\_{\mathbb P^1}\oplus \mathcal O\_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$, $(0,-1)$, $(1,r)$, and $(0,1)$ in $N\_{\mathbb R}\cong \mathbb R^2$. When does the Hirzebruch surface $\mathcal H\_... | https://mathoverflow.net/users/69190 | When does the Hirzebruch surface have a nef anticanonical divisor? | If $r \ge 3$ the exceptional section has negative intersection with anticanonical divisors), so the answer is $r \le 2$.
| 4 | https://mathoverflow.net/users/4428 | 394687 | 163,109 |
https://mathoverflow.net/questions/394686 | 1 | I was going through [this](https://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00410-1/) paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a pole at $s=1$ where $\pi$ is a cuspidal representa... | https://mathoverflow.net/users/140336 | Behaviour of a certain $L$ function at $s=1$ | Let $n\geq 1$. The $L$-function of an automorphic representation of $\mathrm{GL}(n)$ is either (1) entire, or (2) holomorphic away from a pole of order $\leq n$ at $s=1+i\tau$ for some fixed $\tau\in\mathbb{R}$. Kim (Theorem B) proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}(2)$, then $L(s... | 8 | https://mathoverflow.net/users/111215 | 394689 | 163,110 |
https://mathoverflow.net/questions/394690 | 2 | One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as:
\begin{equation}
\rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle
\end{equation}
Where $\langle \dots \rangle$ denotes the average over the distribution of $H$.
In the large $N$ limit, it ... | https://mathoverflow.net/users/142153 | Two-level correlation function of eigenvalues for large random matrices | Let me address the issue raised by the OP of the universality of the two-point correlation function.
The universality of $\rho^{(2)}(\lambda,\mu)$ does exist if one considers the correlations locally, on the scale of the mean eigenvalue spacing. This is relevant for many applications, because the correlations decay q... | 1 | https://mathoverflow.net/users/11260 | 394696 | 163,112 |
https://mathoverflow.net/questions/394697 | 1 | Conjecture:
There is no $b,\{a\_n\}\_{n=1}^{\infty}$ such
that $b,a\_n \in \mathbb{N}^+, a\_{n+1}\ge a\_n$,
$$\lim\_{n\rightarrow \infty}\frac{a\_{n+1}}{a\_{n}}=\infty\qquad\text{and}\qquad\frac{1}{b}= \sum\_{n=1}^{\infty}\frac{1}{a\_{n}}.$$
This is just my guess, and it would be nice if someone could give a counter ... | https://mathoverflow.net/users/105551 | Under a condition, $\frac{1}{b } = \sum_{n=1}^{\infty}\frac{1}{a_{n}}$ will never happen | Actually, this happens for all natural $b$. Notice that
$$
\frac{1}{b}=\frac{1}{b+1}+\frac{1}{b^2+b}
$$
and iterate this identity. You will get
$$
\frac{1}{b}=\frac{1}{b+1}+\frac{1}{b^2+b+1}+\frac{1}{(b^2+b)(b^2+b+1)+1}+\ldots,
$$
i.e.
$$
\frac{1}{b}=\sum\_n \frac{1}{a\_n},
$$
where $a\_1=b+1$ and $a\_{n+1}=a\_n^2-a\_n... | 20 | https://mathoverflow.net/users/101078 | 394705 | 163,114 |
https://mathoverflow.net/questions/394721 | 1 | See [Wikipedia](https://en.wikipedia.org/wiki/Monsky%27s_theorem) for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.
**Questions:** Are there quadrilaterals that allow partition into any number of equal area triangles? On the other hand, are th... | https://mathoverflow.net/users/142600 | On a possible variant of Monsky's theorem | I showed in my (still unanswered) question [Is the equidissection spectrum closed under addition?](https://mathoverflow.net/questions/133777/is-the-equidissection-spectrum-closed-under-addition)
that a "kite" with corners in $(-1,0), (0,-1), (0,1), (2,0)$ can be equidissected into any number $n\geq 2$ of triangles. Tha... | 4 | https://mathoverflow.net/users/14302 | 394727 | 163,117 |
https://mathoverflow.net/questions/394691 | 1 | I have pure sheaves of dimension 1 on a ruled surface, in paticular the Hirzebruch surface **F**$\_e$=**P**($O \oplus O(-e)$) with linear Hilbert bipolynomial $P(x, y)=ax+by+c$.
A sheaf $E$ is pure of dimension d means that for all non trivial coherent subsheaves $F \subseteq E$, $\dim(F)=d$.
Also, $\dim(F)$ is the d... | https://mathoverflow.net/users/173120 | How can I get the scheme-theoretic support of coherent sheaf on a ruled surface with linear Hilbert bipolynomial ax+by+c? | **Question:** "Then, how can I get the scheme-theoretic support? Or where can I find the hint about this problem?"
**Answer:** If $A$ is a commutative ring and $M$ a finitely generated $A$-module, the support of $M$, denoted $Supp(M)$ is the closed subscheme $V(ann(M)) \subseteq S:=Spec(A)$. Here $ann(M) \subseteq A$... | 1 | https://mathoverflow.net/users/nan | 394728 | 163,118 |
https://mathoverflow.net/questions/394735 | 2 | Let $A\_1,..., A\_s \in M\_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that
$$
m = \min\_{(c\_1, ..., c\_s) \in \mathbb{R}^s \backslash \{0\}} rank( \sum\_{i=1}^s c\_i A\_i ) > 0
$$
I am interested in the question how large can $m$ be?
I am not sure where... | https://mathoverflow.net/users/84272 | rank of a linear combination of matrices | The best bound relating $m$, $n$, and $s$ (i.e., the best possible bound that does not take into account any structure of the $A\_j$ matrices) is
$$
s \leq \binom{n - m + 2}{2}.
$$
To see that this bound is tight (i.e., you can achieve $s = \binom{n - m + 2}{2}$), consider the matrices $A\_j$ that mostly consist of z... | 2 | https://mathoverflow.net/users/11236 | 394738 | 163,122 |
https://mathoverflow.net/questions/394729 | 2 | Suppose we have $A\_i$, $i=1\ldots n$, $n\times n$ complex matrices linearly independent. It may be conjectured that there exist $(a\_1,\ldots,a\_n) \in \mathbb{C}^n$ not all zero such that $\sum\_{i=1}^na\_iA\_i$ has a one eigenvalue of (algebraic) multiplicity $n$.
I didn't try many things as the linearly independent... | https://mathoverflow.net/users/121643 | An $n$ eigenvalue multiplicity | This is an elaboration on the comment of Alexandre Eremenko. Algebraic multiplicity $n$ means that we have the equality of polynomials
$$
\det(t I\_n -a\_1A\_1+\cdots+a\_nA\_n)=(t-\lambda)^n
$$
for some $\lambda$. Comparing coefficients of $t^{n-1},t^{n-2},\ldots,1$, we find a system of $n$ equations
$$
\begin{case... | 4 | https://mathoverflow.net/users/1306 | 394739 | 163,123 |
https://mathoverflow.net/questions/394745 | 21 | It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}\_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some conditions, meaning the moduli space $Z$ of all such lifts is *connected*. This is mentioned in Theorem 1.1 of Lurie's "... | https://mathoverflow.net/users/73372 | Why does elliptic cohomology fail to be unique up to contractible choice? | So the issue is with this:
>
> all of the groups occuring in the Goerss--Hopkins obstruction theory
> vanish
>
>
>
In "generic terms", for the obstruction theory that you're running in either the $K(1)$-local case or the rational case you care about some bigraded obstruction groups
$$
\mathfrak{E}xt^{s,t}(R;S)... | 18 | https://mathoverflow.net/users/360 | 394747 | 163,126 |
https://mathoverflow.net/questions/394529 | 4 | Suppose that $x\_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form
$$ z = \sum\_{i=0}^{n-1} x\_i \omega^{i}$$
>
> Are there any known **lowest** upper bounds for $|z|, |\operatorname{Re} z|, |\operatorname{Im} z|$?
>... | https://mathoverflow.net/users/13093 | Linear combinations of roots of unity | $\DeclareMathOperator\re{Re}\DeclareMathOperator\im{Im}\DeclareMathOperator\sgn{sgn}$Let me write
$$\begin{align\*}
A(n)&=\max\_{x\in[-1,1]^n}\biggl|\sum\_{j<n}x\_j\omega^j\biggr|,\\
R(n)&=\max\_{x\in[-1,1]^n}\re\sum\_{j<n}x\_j\omega^j,\\
I(n)&=\max\_{x\in[-1,1]^n}\im\sum\_{j<n}x\_j\omega^j.
\end{align\*}$$
First, it’s... | 6 | https://mathoverflow.net/users/12705 | 394757 | 163,128 |
https://mathoverflow.net/questions/394758 | 3 | Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix).
If you permute the columns and rows independently what is the probability resulting permutation matrix is a complete cycle?
| https://mathoverflow.net/users/10035 | Probability permutation in turned to cycle | If $M$ is $n\times n$, the probability is $1/n$.
Indeed, by symmetry, the probability in question is the probability that a random permutation of $[n]:=\{1,\dots,n\}$ is a complete cycle:
If the $(i,j)$-entry of $M$ is $M\_{i,j}=1(j=\pi(i))$ for some permutation $\pi$ of $[n]$ and all $i,j$ in $[n]$, then after apply... | 5 | https://mathoverflow.net/users/36721 | 394760 | 163,129 |
https://mathoverflow.net/questions/394742 | 7 | *Remark:* In this question I am first and foremost interested in a local problem and local solutions therefore I assume all functions are defined on open sets of real coordinate spaces and I will not bother with explictly considering domains. To simplify notation some maps will actually be partial maps defined on an op... | https://mathoverflow.net/users/85500 | Pfaffian systems that do not satisfy their integrability conditions | Since everything is local and $C^\infty$, it is not hard to derive *sufficient* conditions for there to exist solutions. Analyticity is not actually needed, but some assumption of regularity is necessary.
Here is one such result:
Let the Pfaffian system $\mathcal{I}$ be generated by $\theta^1,\ldots,\theta^s$ and c... | 6 | https://mathoverflow.net/users/13972 | 394765 | 163,130 |
https://mathoverflow.net/questions/394756 | 6 | Does anyone have a handy characterisation of *open* continuous surjections $X \to Y$ in terms of the corresponding injective $\*$-homomorphism $C(Y) \to C(X)$? (I'm only interested in the case where $X$ and $Y$ are compact Hausdorff spaces, but the question could be suitably modified for locally compact Hausdorff space... | https://mathoverflow.net/users/113152 | What is the Gelfand dual of an open surjection? | After more thought, I think the correct statement is the following:
**Theorem:** Let $\pi : X \to Y$ be a continuous map between compact Hausdorff spaces. Then the following condition are equivalents:
(a) $\pi$ is an open map.
(b) The map $\pi^\* : C(Y)^+ \to C(X)^+$ has a left adjoint which is $C(Y)^+$-linear, i... | 6 | https://mathoverflow.net/users/22131 | 394769 | 163,132 |
https://mathoverflow.net/questions/394771 | 4 | Let $LG(n,2n)$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $n$) of $\mathbb{C}^{2n}$. Then $LG(n,2n)\subset G(n,2n)$, where $G(n,2n)$ is the Grassmannian of subspaces of dimension $n$ of $\mathbb{C}^{2n}$.
Fix a subspace $H\subset \mathbb{C}^{2n}$ of dimension $n+2$, and denote ... | https://mathoverflow.net/users/14514 | Subvarieties of Lagrangian Grassmannians | Note first that, if $L$ is a Lagrangian contained in $H$ then $L^\perp = L$ contains $H^\perp$. So $X\_n$ is non-empty only when $H$ is co-isotropic for your symplectic form.
When $H$ is co-isotropic, the symplectic form $\omega$ induces a symplectic form on $H/H^\perp$ (which has dimension $4$) and Lagrangians of $\... | 5 | https://mathoverflow.net/users/173096 | 394774 | 163,133 |
https://mathoverflow.net/questions/394761 | 2 | For a long time I've been under the impression that the Chevalley complex $\text{CE}(\mathfrak{g})$ of a semisimple (maybe can weaken this) Lie algebra $\mathfrak{g}$ can be extracted from the geometry $\text{BG}$, where $\mathfrak{g}$ is the Lie algebra of $\text{G}$. Does anyone know a statement like this with a proo... | https://mathoverflow.net/users/119012 | Chevalley complex and $\text{BG}$ | It’s the ring of functions on the formal completion at the base point:
CE is self ext of the trivial rep of the Lie algebra, equivalently of the formal group of G, ie global functions on B of the formal completion. This doesn’t seem to be a functor of BG as a stack, but only as a pointed stack..
| 8 | https://mathoverflow.net/users/582 | 394778 | 163,134 |
https://mathoverflow.net/questions/393689 | 4 | I am looking for compact Kähler manifolds of dimension $3$ with the following 2 properties:
**1.** $c\_1(K\_X)=c[\omega],c>0$ where $\omega$ is the Kähler form on $X$.
**2.** $1+h^{0,3}+h^{1,1}=h^{0,1}$
It's easy to find example satisfying the first condition. One can take $X=X\_1\times X\_2\times X\_3$ where $X\... | https://mathoverflow.net/users/131004 | Example of a Kähler manifold with certain properties | What about taking a product $X=Y\times \Sigma$, where $Y$ is a fake projective plane and $\Sigma$ is a surface of genus $3$? We have $h^{1,1}(X)=2$, $h^{0,1}(X)=3$, and $h^{0,3}(X)=0$ by Kuneth formula. Condition 1) holds since this is a $3$-fold of general type with ample canonical bundle.
| 2 | https://mathoverflow.net/users/943 | 394781 | 163,136 |
https://mathoverflow.net/questions/394725 | 10 | Given a compact closed surface $M$ (2-dim topological manifold) isometrically embedded in $\mathbb{R}^3$, are there 8 points $x\_i\in M(i=1,\dots,8)$ such that they are the vertices of a cube $C\subset\mathbb{R}^3$?
We may assume that (1)$M$ is smooth and homeomorphic to the 2-sphere $S^2$; (2)$M$ is piecewise-smooth... | https://mathoverflow.net/users/nan | Is there an inscribed cube for an arbitrary compact closed surface? | One can't inscribe cubes in generic surfaces by dimension reason. Indeed the space of cubes in $\mathbb R^3$ is $7=3+3+1$-dimensional, while a cube has $8$ vertices, and so a surface imposes $8$ conditions on the vertices of the cube.
To make this dimension reasoning rigorous one can do the following. Take the space ... | 7 | https://mathoverflow.net/users/943 | 394784 | 163,137 |
https://mathoverflow.net/questions/394488 | 0 | This is a rewording in combinatorial language of a question posed on another forum. The original was posed as a probabilistic problem.
**Problem set up:**
Consider for a fixed prime $p$, the multiplicative group $\mathbb Z\_{p-1}^{\times}$ of integers modulo $p$. Define, for each pair $j \neq k \in \mathbb Z\_{p-1}... | https://mathoverflow.net/users/173490 | On independence of multiples of $\mathbb Z_p$ | This is half an answer that I got by forgetting the outside absolute value signs; however, it shouldn't be hard to figure out how to correct it for that.
Fix $j, k$, and let $1 \leq n \leq p - 1$. Let $m = ord\_p(k j^{-1})$. Then I claim that if $m|n$, we can choose $E$ of size $n$ such that $jE = kE$; otherwise, I c... | 3 | https://mathoverflow.net/users/44191 | 394785 | 163,138 |
https://mathoverflow.net/questions/394526 | 7 | *This was originally part of [this older question of mine](https://mathoverflow.net/questions/252407/what-kind-of-compactness-does-expanding-mathbbr-by-constants-have), but in retrospect that question should have been broken into two parts - this is the still-unanswered part.*
Let $\Sigma$ be the language of ordered ... | https://mathoverflow.net/users/8133 | Is this compactness property for "satisfiability on $\mathbb{R}$" consistent? | It looks to me like under ZFC, $\mathbb{R}$-satisfiability is not (consistently) $(\omega\_2,\omega\_3)$-compact. To see this, we'll emulate your argument above for $\mathbb{R}\_{\mathbb{Z}}$. So basically, we want a theory with constants $c\_\eta$ for $\eta<\omega\_2$, which says:
* "$c\_\eta$ codes a wellorder of $... | 6 | https://mathoverflow.net/users/160347 | 394794 | 163,140 |
https://mathoverflow.net/questions/394772 | 3 | I tried to understand [this paper](https://arxiv.org/pdf/1606.01921.pdf) on page 31.
Let $K$ be an finite extension of $\mathbb Q\_p$ and $\overline{K}$ be its algebraic closure; $\mathcal{O}\_{\overline{K}}$ is the ring of integers of $\overline{K}$; $\mathcal{O}\_{\mathbb{C}\_K}$ is its $p$-adic completion. We have... | https://mathoverflow.net/users/119770 | The kernel from $A_\mathrm{inf}$ to $\mathcal{O}_{\mathbb{C}_K}$ | 1)Pick a sequence of elements $p^{1/p^n}\in \mathcal{O}\_{\overline{K}}$ such that $(p^{1/p^{n+1}})^p=p^{1/p^n}$. The ideal $\ker\phi$ is in fact principal and is generated by the element $p^{\flat}:=(\dots, p^{1/p^2},p^{1/p},0)$ (where we view $p^{1/p^n}$ as elements in the reduction $\mathcal{O}\_{\overline{K}}/p$). ... | 3 | https://mathoverflow.net/users/39304 | 394795 | 163,141 |
https://mathoverflow.net/questions/394798 | 2 | Let $M$ be a complex manifold, $N$ is a smooth immersed submanifold of $M$. If $T\_p M$ is invariant under the multiplication by $i$ for any $p\in M$, then can we conclude that $N$ is a complex immersed submanifold of $M$?
Since $C^1$ property somehow means analytic property in complex setting, can we drop the assump... | https://mathoverflow.net/users/167284 | Smooth submanifold of a complex manifold with invariant tangent space under multiplication by $i$ | Let $f : N \to M$ denote the immersion.
Since $f\_\*TN$ is invariant under $I$ where $I$ is the underlying almost complex structure of $M$, it induces an almost complex structure $I'$ on $N$.
Applying Newlander-Nirenberg theorem, we see that since
$I$ is integrable, $I'$ is also integrable.
Thus $I'$ is a complex struc... | 3 | https://mathoverflow.net/users/14037 | 394802 | 163,143 |
https://mathoverflow.net/questions/394706 | 6 | I just read the nice exposition [Fermionic Path Integral](https://ncatlab.org/nlab/show/fermionic+path+integral) on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the Osterwalder-Schrader approach to QFT:
* Is there a deeper analogy betwe... | https://mathoverflow.net/users/18936 | Fermions, their path integrals and effective actions | The fermionic path integral is not an integral in the analytic sense, and that's good so because therefore we can evaluate it rigorously. So, this is quite unrelated to measures or distributions, and I think that there is no analogy like the one you are looking for.
Your last point can be answered completely, more or... | 2 | https://mathoverflow.net/users/3473 | 394814 | 163,145 |
https://mathoverflow.net/questions/394825 | 4 | Let $\Sigma$ be a $d\times d$ semi-definite positive matrix (SDP). Let $I\subset\{1,\ldots, d\}\times \{1, \ldots, d\}$ be a symmetric subset of indices (i.e. if $(p,q)\in I$ then $(q,p)\in I$). We denote by $||.||\_{op}$ the operator norm over $\mathbb{R}^{d\times d}$.
Is it possible to find an absolute constant $c$ (... | https://mathoverflow.net/users/274436 | Operator norm of a masked SDP matrix | No, no such constant exists. For example, if $I = \{(i,j) \mid i<j\}$, then $\Sigma\mapsto \Sigma\_I$ is the usual triangular projection, and the norm is of order $\log n$, see for example [Norm of the upper triangular part of symmetric matrix](https://mathoverflow.net/q/177944) (the fact that you restrict to positive ... | 4 | https://mathoverflow.net/users/10265 | 394827 | 163,151 |
https://mathoverflow.net/questions/394773 | 4 | **Disclaimer**: I posted this question seven days ago [here on the Math.SE](https://math.stackexchange.com/questions/4157296/in-search-for-a-counterexample-related-to-the-abel-stolz-theorem), with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so... | https://mathoverflow.net/users/113756 | In search for a counterexample related to the Abel-Stolz theorem | In fact a trivial counterexample to the question, as now clarified, is just $a\_n:=(-1)^n$ with $s:=\frac12$. In your notation, $a\_n\to\frac12\;\bf (A)$, because the series $\sum\_{n=0}^\infty (-z)^n$ converges on the open disk to $f(z):=\frac1{1+z}$, which is holomorphic at $z=1$. But the series does not converges un... | 6 | https://mathoverflow.net/users/6101 | 394829 | 163,152 |
https://mathoverflow.net/questions/394803 | 0 | Let $V : [a,b] \to \mathbb{R}$ be smooth, strictly increasing and $V(a) = 0$. Suppose that $f : [a,b] \to \mathbb{R}$ is smooth and satisfies $f^{\prime \prime} (x) + V(x) f(x) = 0$ on $[a,b]$. Can we then bound $\sup\_{x \in [a,b]} |f(x)|$ in terms of $f(a) , f(b) , f^{\prime} (b), V(b)$? I intentionally don't put $f^... | https://mathoverflow.net/users/2095 | A bound on a solution of an ODE, given some bounds on endpoints | It will be more convenient (at least to me) to turn the problem right to left, restating it as follows:
>
> Let $V\colon[a,b]\to\mathbb{R}$ be smooth, strictly decreasing and
> $V(b) = 0$. Suppose that $f\colon[a,b]\to\mathbb{R}$ is smooth and
> satisfies $f''(x)+V(x) f(x)=0$ on $[a,b]$. Can we then bound $\sup\_{x... | 3 | https://mathoverflow.net/users/36721 | 394833 | 163,154 |
https://mathoverflow.net/questions/394799 | 6 | I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such approach is bound to fail.
Motivation
----------
Let's consider algebra $gl\_n$ with ''bosonic'' generators $B\_{ij}... | https://mathoverflow.net/users/273597 | Is there Z_n graded supersymmetry? | The realizations (of an algebra through another algebra) you are speaking about are actually homomorphisms. And as such they should map between algebraic structures of the same kind: that is from algebras to algebras, from Lie alg to Lie alg, from graded algebras to graded algebras (graded by the same group), etc
Sin... | 2 | https://mathoverflow.net/users/85967 | 394837 | 163,155 |
https://mathoverflow.net/questions/394844 | 11 | Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension
$h$ of a complex vector space of dimension $2h$.
For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is
a smooth quadratic $3$-fold. So $LG(2,4)$ contains no plane (linearly
embbeded $\mathbb{P}^2$).
Furthermore, by Lemma 2.5.... | https://mathoverflow.net/users/274889 | Planes in Lagrangian Grassmannians | This is, indeed, true.
To prove this, assume we have an embedding $\mathbb{P}^2 \to \operatorname{LGr}(V)$ (where $V$ is a symplectic vector space). Let $U \subset V \otimes \mathcal{O}$ be the tautological subbundle on $\operatorname{LGr}(V)$. Note that it extends to an exact sequence
$$
0 \to U \to V \otimes \mathc... | 13 | https://mathoverflow.net/users/4428 | 394848 | 163,157 |
https://mathoverflow.net/questions/394851 | 9 | $\DeclareMathOperator\SL{SL}$The volume of $\SL\_n(\mathbb{R})/\SL\_n(\mathbb{Z})$ can be computed under the natural measure that it inherits from $GL\_n(\mathbb{R})$. Two formulae seem to be known.
$$\operatorname{vol}(\SL\_n(\mathbb{R})/\SL\_n(\mathbb{Z}))= \zeta(2)\zeta(3) \dotsb \zeta(n).$$
This is available in not... | https://mathoverflow.net/users/94546 | Volumes of $\mathrm{SL}_n(K_\mathbb{R})/\mathrm{SL}_n(\mathcal{O}_K)$ | $\DeclareMathOperator\SL{SL}$The same argument (due to Siegel, in a classical form, of course), adelized, gives the analogous computation for any number field, and, yes, the corresponding Dedekind zeta appears. (To know that the adelic analogue computes the same thing, rather than the adelic quotient for $\SL\_n$ being... | 17 | https://mathoverflow.net/users/15629 | 394856 | 163,159 |
https://mathoverflow.net/questions/394865 | 6 | **Axiom of Countable Choice (CC)** states that for every countable family $\left\{A\_i\right\}\_{i=1}^\infty$ of nonempty sets there exists *choice function* $f \colon \mathbb{N} \to \bigcup\_{i=1}^\infty A\_i$ such that $f(i) \in A\_i$ for all $i \in \mathbb{N}$.
Consider the following axiom strongly related to the ... | https://mathoverflow.net/users/153916 | Countable choice for countable subfamily | They are equivalent. Given a countable family $B\_i$ of nonempty sets, let $A\_i=\prod\_{j=1}^iB\_j$. That these are nonempty is provable in ZF (we only use finite products). Now for any choice function $f$ with $f(k)\in A\_{i\_k}$, since $i\_k\geq k$, projecting $f(k)$ onto the $k$-th coordinate of $A\_{i\_k}=\prod\_{... | 8 | https://mathoverflow.net/users/30186 | 394866 | 163,162 |
https://mathoverflow.net/questions/394651 | 14 | The following little question has bugged me for a while.
Suppose $Z \subseteq \mathcal P(X)$. We say an ideal $I$ on $Z$ is *normal* when it is closed under diagonal unions, which means that if $\{ A\_x : x \in X \} \subseteq I$, then $\nabla\_x A\_x := \{ z \in Z : \exists x \in z ( z \in A\_x) \} \in I$. We say tha... | https://mathoverflow.net/users/11145 | For ideals, does normal imply countably complete? | I think the answer is yes. It helped to take generic ultrapowers, somehow.
We may assume without loss of generality that $I$ is a normal ideal on $P(\lambda)$ where $\lambda$ is an infinite cardinal. Assume towards a contradiction that $I$ is countably incomplete. Then there is an $I$-positive set $S\subseteq P(\lamb... | 6 | https://mathoverflow.net/users/102684 | 394879 | 163,165 |
https://mathoverflow.net/questions/394839 | 10 | ***[cross-posting] <https://math.stackexchange.com/q/4153692/522463>***
Let $G$ be an infinite group and $\varphi\colon G\longrightarrow\mathbb{C}$ such that
\begin{eqnarray\*}
\exists\,\delta>0, \forall\,a,b\in G,\, |\varphi(ab)-\varphi(a)\varphi(b)|\leq\delta\quad(\mathcal{P})
\end{eqnarray\*}
What can be said abou... | https://mathoverflow.net/users/172526 | On almost-(multiplicative)-morphisms between infinite groups and the complex numbers | If $\varphi$ is not bounded, there is a sequence $(x\_n)\_{n\in\mathbb{N}}\in G^{\mathbb{N}}$ such that $|\varphi(x\_n)|>2^n$ for all $n\in\mathbb{N}$. Let $x,y\in G$ and $n\in\mathbb{N}$.
\begin{eqnarray\*}
|\varphi(x)\varphi(y)-\varphi(xy)||\varphi(x\_n)|&=&|\varphi(x)\varphi(y)\varphi(x\_n)-\varphi(xy)\varphi(x\_n)|... | 3 | https://mathoverflow.net/users/23859 | 394881 | 163,167 |
https://mathoverflow.net/questions/392748 | 2 | I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the reductions except for the one using a rotation.
In this paper, $Q$ is a nonsingular quadratic form on $\mathbb R^n$, $n\... | https://mathoverflow.net/users/70388 | What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate? | It turns out that (1) Yes, I will need to use 'nonstandard' rotations, but (2) the hyperbolic ones suffice. This is because (as already in question body) the usual Euclidean rotations in $\mathbb R^j$ and $\mathbb R^{n-j}$ turns the vector $a\in \mathbb R^n$ essentially into a 2D vector which we can consider as living ... | 2 | https://mathoverflow.net/users/70388 | 394890 | 163,171 |
https://mathoverflow.net/questions/394665 | 2 | Cross-posted from [MSE.](https://math.stackexchange.com/q/4148811/272127)
---
I know that De Rham cohomology reveal some properties of the topology of smooth manifolds by finding closed differential $k$-forms $\mathsf{d}\omega=0$ that are not exact $\omega\neq\mathsf{d}\eta$. I wonder
>
> **Question:** Why no... | https://mathoverflow.net/users/90655 | Why non closed differential forms do not play important role for the topology of a manifold? | Actually, the non-closed forms on manifolds play an essential role in the definition of [*Massey products*](https://en.wikipedia.org/wiki/Massey_product), which are 'higher cohomology' operations.
Another place where they make an essential appearance is in the construction of the [*Hopf invariant*](https://en.wikiped... | 10 | https://mathoverflow.net/users/13972 | 394900 | 163,172 |
https://mathoverflow.net/questions/394894 | 7 | Let $W$ be a $3$-dimensional $h$-cobordism of closed surfaces $M\_0$ and $M\_1$. Can we prove that $W$ is trivial? That is, $W$ is homeomorphic to $M\_0 \times [0,1]$.
| https://mathoverflow.net/users/16323 | 3-dimensional h-cobordisms | For $M\_0=M\_1=S^2$, this follows from the 3-dimensional Poincaré conjecture: glueing in 3-balls, you get a simply connected $3$-manifold, that has to be $S^3$, so $W=S^2\times\left[0,1\right]$.
For surfaces of higher genus, the result follows from Waldhausens rigidity theorem, which says that homotopy-equivalent Hak... | 9 | https://mathoverflow.net/users/39082 | 394901 | 163,173 |
https://mathoverflow.net/questions/394877 | 3 | Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|\_{L^2}\le C \|\nabla u\|\_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$ independent of $u$, then $X$ needs to be such that it doesn't contain constant translates. That is, if we consider $... | https://mathoverflow.net/users/166785 | Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions | This is not true. Take $\Omega = (-1,1)$ and functions $u\_M$ like (I hope that I got the constants right)
$$
u\_M(x)
=\begin{cases}
-M^2 (|x|-1)(|x|-1+1/M) + M & \text{for } |x| > 1-1/(2M) \\
1/4 + M & \text{else.}\end{cases}
$$
Then, $\|u\_M\|\_{L^2}$ grows like $M$, whereas $\|\nabla u\_M\|\_{L^2}$ grows like $\sqrt... | 6 | https://mathoverflow.net/users/32507 | 394905 | 163,174 |
https://mathoverflow.net/questions/394899 | 15 | I am trying to understand whether there is a sense in which cohomology always relates to topology or whether this is the case only in particular examples. According to the Wikipedia [page](https://en.wikipedia.org/wiki/Chain_complex), a cochain complex is defined as:
“… a sequence of abelian groups or modules ..., $... | https://mathoverflow.net/users/36541 | Is cohomology always related to topology? | While it is always possible to introduce topology, it is not always the obvious or most useful thing to do. So at least in this sense, there are notions of cohomology that do not immediately connect to topology.
As an example group (co)homology, Lie algebra (co)homology, Hochschild (co)homology, ... all appear in alg... | 23 | https://mathoverflow.net/users/3041 | 394906 | 163,175 |
https://mathoverflow.net/questions/391630 | 7 | As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between sums of prime numbers and its distribution:
**Theorem 1**
*Let us define the prime counting function up to a given na... | https://mathoverflow.net/users/172800 | On the connection between sums of prime numbers and distribution of prime numbers | The [post that you quoted](https://mathoverflow.net/questions/383610/set-of-prime-numbers-q-such-that-sum-limits-p-leq-sqrtqp-piq-where) contains a proof that
$$\pi(x)-S(x)=\Omega\_\pm(x^c)\quad\text{for any}\quad c<3/4.$$
Equivalently, for any $c<3/4$ and any $C>0$, both $\pi(x)-S(x)\leq Cx^c$ and $\pi(x)-S(x)\geq -Cx... | 5 | https://mathoverflow.net/users/11919 | 394914 | 163,178 |
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