parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/361183 | 10 | At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information Geometry.
Having said that, I would like to be advised as to which books should I study in order to prepare myself to get a... | https://mathoverflow.net/users/nan | Which books should I read in order to be prepared to study information geometry? | The [lecture notes by Frank Nielsen](https://arxiv.org/abs/1808.08271) are succinct and fairly self-contained and maybe good to get a first idea. The books [1,2] by Amari can serve for a more in-depth study and contain a fair bit of differential geometry background. In order to get a flavour for some of the application... | 8 | https://mathoverflow.net/users/69603 | 361325 | 152,057 |
https://mathoverflow.net/questions/361323 | 6 | I want to know whether the following inequality holds or not.
\begin{align}
(\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1}
\end{align}
where $A, B$ are Hermitian matrices of the dimension $D$.
Note that if $A$ and $B$ commute, we can see (1) holds using the simultaneously diagonalizing ... | https://mathoverflow.net/users/158608 | Matrix inequality : trace of exponential of Hermitian matrix | You can prove it using the Golden-Thompson inequality $Tr (e^{A+B}) \leqslant Tr(e^{A} e^{B})$ and then applying the Cauchy-Schwarz inequality.
| 10 | https://mathoverflow.net/users/24953 | 361326 | 152,058 |
https://mathoverflow.net/questions/361333 | -1 | Assume ${\bf x} \in \mathbb{R}^n$ denotes a real-valued bounded random variable with a distribution measured on the Borel space $(\mathbb{R}^n,\mathcal{B}^n)$. Let $f:\mathbb{R}^n\to\mathbb{R}$ denote a bounded Borel measureable function. Then, the following expectation value for any Borel set $B$
$$
E[f({\bf x}){\bf... | https://mathoverflow.net/users/158368 | Maximum of bounded expectations at a certain Borel set? | Yes, it's attained. Note that the desired expression can be written as $E[(f 1\_B)(\mathbf{x})]$. Then it's clear that we get the maximum by taking $B = \{f \ge 0\}$, so that $f 1\_B = f^+$, the positive part of $f$. Indeed, if $A$ is any other Borel set, then $E[(f 1\_A)(\mathbf{x})] = E[(f^+ 1\_A)(\mathbf{x})] - E[(f... | 1 | https://mathoverflow.net/users/4832 | 361336 | 152,059 |
https://mathoverflow.net/questions/361286 | 9 | I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of cobordisms to a suitable $\left(\infty,n\right)$-category of vector spaces.
The original Atiyah-Witten definition was a functo... | https://mathoverflow.net/users/104719 | Path integral derivation of extended TQFT | The physics motivation for extended QFTs (and not just TQFTs) comes from the locality principle (no spooky action at a distance).
The mathematical expression of locality is the descent property for extended QFTs.
For an early source, see, for instance, [Higher Algebraic Structures and Quantization](https://arxiv.org/... | 2 | https://mathoverflow.net/users/402 | 361339 | 152,060 |
https://mathoverflow.net/questions/361031 | 4 | There exists the following result in the literature: There exists a polarized $K3$ surface $(X, H)$ of genus $3$ and a smooth irreducible curve $C$ on $X$ satisfying $C^2 =4$, $C.H=6$ such that $\text{Pic}(X) \cong \mathbb Z[H] \oplus Z[C]$. The theorem follows from [<https://arxiv.org/pdf/math/9805140.pdf]> theorem $1... | https://mathoverflow.net/users/156533 | On the intersection numbers of the generators of $\text{Pic}(X)$ of a smooth quintic surface | A natural source of surfaces in $\mathbb{P}^{3}$ with Picard number $> 1$ is given by linear determinantal surfaces, i.e. zero sets of square matrices of linear forms. In what follows, let $X \subseteq \mathbb{P}^{3}$ be a smooth linear determinantal surface of degree $d \geq 2$ (smoothness can arranged by taking the $... | 4 | https://mathoverflow.net/users/5496 | 361341 | 152,061 |
https://mathoverflow.net/questions/361338 | 4 | It is known that in the category of sets the dualization of the notion of a subobject classifiers does not work because the only object admitting a morphism into an initial object is the empty set.
But if we look at the idea of a subobject classifier (to index subobjects), then we can see that in the category of sets... | https://mathoverflow.net/users/61536 | What does play the role of a subobject classifier for quotient objects? | One way to write the universal property of this object $E(a)$ is as follows:
a map $x \to E(a)$ is the same as an isomorphism class of epimorphism $x \times a \twoheadrightarrow k$ in $Set/x$, that is a diagram
$$ x \times a \twoheadrightarrow k \to x$$
whose composite is the first projection.
So it does "feel ... | 4 | https://mathoverflow.net/users/22131 | 361342 | 152,062 |
https://mathoverflow.net/questions/361328 | 12 | It's [known](https://oeis.org/A035120) that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
**Question**: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first few primes of form $4k+1$ (equivalently, $\mathbb{Q}(\sqrt{p})$ having odd conductor)?
To... | https://mathoverflow.net/users/125498 | Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor | We give a uniform approach to $p \leq 61$ by applying
analytic discriminant bounds to the Hilbert class field.
To be sure this is not entirely "conceptual", but then
some computation is needed even to deal with $p < 36$ using Minkowski.
If $p = 4k+1$ is prime then $K = {\bf Q}(\sqrt{p})$ has odd class number $h$,
so ... | 13 | https://mathoverflow.net/users/14830 | 361346 | 152,064 |
https://mathoverflow.net/questions/361343 | 13 | Let $a\_0>a\_1>\cdots>0$ have the property that, for each positive $a<\sum\_{n\in\Bbb N}a\_n$ (admitting $\infty$ for the sum), there is $A\subset\Bbb N$ such that $a=\sum\_{n\in A}a\_n$ . Are there known necessary and sufficient conditions on the $a\_n$ (not involving arbitrary partial sums) for this property? To illu... | https://mathoverflow.net/users/7458 | Is there a known condition for partial sums of a decreasing positive sequence to take all values up to the total sum? | It is necessary and sufficient that
>
> $\lim\_{n \rightarrow \infty}a\_n = 0$, and
>
>
> $a\_n \leq \sum\_{m > n}a\_m$ for all $n$.
>
>
>
In other words: the terms go to zero, and no term is bigger than the sum of all the following terms.
*Necessity*: First, it is necessary that $\lim\_{n \rightarrow \inf... | 17 | https://mathoverflow.net/users/70618 | 361348 | 152,066 |
https://mathoverflow.net/questions/360905 | 1 | Let $\overline{\mathbf{M}}\_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}\_{0,n}$ the universal family described by $\pi\_n:\overline{\mathbf{U}}\_{0,n}\longrightarrow\overline{\mathbf{M}}\_{0,n}$ with the n disjoints sections $\sigma\_i: \overline{\math... | https://mathoverflow.net/users/158337 | Pullback of boundary divisors under forgetful maps | Here are some examples.
1. Consider $\epsilon: \overline{M}\_{0,4} \to \overline{M}\_{0,3}$. Since $\overline{M}\_{0,3}$ is a point, $F\_3 = 0$. On the other hand, the boundary of $\overline{M}\_{0,4}$ consists of three points $0,1,\infty$. These are exactly the images of the three sections.
2. Consider $\epsilon : \... | 1 | https://mathoverflow.net/users/nan | 361354 | 152,068 |
https://mathoverflow.net/questions/361350 | 3 | Consider two skew-adjoint matrices $A$ and $A'$, i.e. $A^\*=-A$ and $A'^\*=-A'$. It is well-known that
$e^{-tA}$ and $e^{-tA'}$ are unitary operators.
I would like to know:
Is it true that $\sup\_{t \in \mathbb{R}} \Vert e^{-tA}-e^{-tA'} \Vert = 2(1-\delta\_{A,A'})?$
| https://mathoverflow.net/users/150549 | Distance of parametrized skew hermitian exponentials | This is not true in general. E.g., let
$$A:=\left(
\begin{array}{cc}
-i & 0 \\
0 & 0 \\
\end{array}
\right),\quad A':=\frac i2\,
\left(
\begin{array}{cc}
-1 & 1 \\
1 & -1 \\
\end{array}
\right).$$
Then, for real $t$,
$$e^{-tA}=\left(
\begin{array}{cc}
e^{i t} & 0 \\
0 & 1 \\
\end{array}
\right),\quad
e^{-tA'}=\f... | 6 | https://mathoverflow.net/users/36721 | 361356 | 152,069 |
https://mathoverflow.net/questions/361359 | 3 | Let $X$ be a (countably) infinite set and define an equivalence relation $\sim$ on the power set $P(X)$ of $X$ by defining two subsets $A$ and $B$ of $X$ to be equivalent if they differ by at most finitely many elements (i.e., $A \sim B$ if the symmetric difference $A \Delta B$ is finite).
Let $[-]\colon P(X) \to P(X... | https://mathoverflow.net/users/13356 | Constructing a section of an equivalence relation compatible with the intersection | No such section exists. The main point in the proof is that there exist uncountably many (in fact continuum many) infinite subsets of $\mathbb N$ such that the intersection of any two of them is finite. (Such sets are called *almost disjoint*.) In order to preserve $\cap$ and $\emptyset$, a section would have to send t... | 6 | https://mathoverflow.net/users/6794 | 361362 | 152,071 |
https://mathoverflow.net/questions/361352 | 1 | Let's say we have a several tetrahedrons $T\_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V\_{i}$.
Given a position $P$ inside the tetrahedron $T\_0$, and neighboring tetrahedron are labeled $T\_1, T\_2, T\_3, T\_4$.
How to compute the value $V(P)$ such that i... | https://mathoverflow.net/users/158630 | tetrahedral interpolation and integration along a segment | One common method is to assign values to the vertices of your central tetrahedron $T$,
and then use
[Barycentric coordinates](https://en.wikipedia.org/wiki/Barycentric_coordinate_system#Barycentric_coordinates_on_tetrahedra) to interpolate from the vertices of $T$ to any point $p \in T$.
The link shows how to convert b... | 1 | https://mathoverflow.net/users/6094 | 361365 | 152,072 |
https://mathoverflow.net/questions/361355 | 3 | Let $T$ be an unbounded self-adjoint operator.
Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi\_n)$ such that $$ \lim\_{n \rightarrow \infty} \Vert \varphi-\varphi\_n \Vert=0 $$
and $\Vert T \varphi\_n \Vert \le k(\varphi).$
Someh... | https://mathoverflow.net/users/108483 | Approximation of vectors using self-adjoint operators | No, this only holds when $\varphi \in D(T)$.
It's enough to assume that $T$ is closed and densely defined, so that $T^{\*\*} =T$. Let $\psi \in D(T^\*)$ be arbitrary. If the hypothesis holds then we have
$$ |\langle \varphi, T^\* \psi \rangle| = \lim\_{n \to \infty} |\langle \varphi\_n, T^\* \psi \rangle| = \lim\_{n ... | 3 | https://mathoverflow.net/users/4832 | 361366 | 152,073 |
https://mathoverflow.net/questions/361275 | 3 | Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space ($\mathbb P$ is countably additive). Let $\{p\_\omega: \omega \in \Omega\}$ be a family of (countably additive) probability measures on $(\Omega, \mathcal F)$, and assume that the mapping $\omega \mapsto p\_\omega$ is $\mathcal F$-measurable. Let $\mu$ be a *... | https://mathoverflow.net/users/96899 | A question about finitely additive integration | A measurable function $f$ taking values in $[-M,M]$ can be approximated above and below by simple functions: $f\_n \le f \le f\_n+1/n$ where $f\_n(x)=\lfloor nf(x) \rfloor/n$.
Note that $f\_n$ takes on less than $3Mn$ values. Now from the case of simple functions it follows that the LHS and RHS of (2) differ by at most... | 2 | https://mathoverflow.net/users/7691 | 361371 | 152,075 |
https://mathoverflow.net/questions/361279 | 4 | Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P\_m(z)$ be a partial sum of $P(z).$ How large $P\_m(z)$ can be on $|z|=1?$
| https://mathoverflow.net/users/128472 | A problem on polynomials | The trivial upper bound $\max\_{|w|=1}|P\_{m}(w)|\leq \|1+z+\cdots+z^{m}\|\_{L^{1}(\mathbb{T})} \asymp C \log(m)$ that I wrote in the comment is actually sharp in the regime $m=n/2$. Here is the proof.
Notice that $P\_{m}(z)$ is convolution of $P(z)$ with $D\_{m}(z) = 1+z+\cdots+z^{m}$ on the unit circle, therefore, ... | 5 | https://mathoverflow.net/users/50901 | 361378 | 152,076 |
https://mathoverflow.net/questions/361377 | 2 | The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we rather replace $\mathbb S^n$ by a compact orientable Homology sphere (without boundary) (<https://en.m.wikipedia.org/wik... | https://mathoverflow.net/users/135389 | Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex | You have to use Poincaré-Lefschetz duality : Let $M$ be a compact orientable $n$-manifold, $Y\subset M$ be a closed subset then we have an isomorphism
$$\check{\mathrm{H}}^p(M,Y)\cong H\_{n-p}(M-Y)$$
induced by the cap product with the fundamental class of $M$ (the left hand side is Cech cohomology).
You also have
$... | 3 | https://mathoverflow.net/users/27816 | 361384 | 152,080 |
https://mathoverflow.net/questions/361399 | -3 | This is a cross-post of [this](https://math.stackexchange.com/q/3691891/272127) MSE post that users commented that it is appropriate for MO.
>
> I want to know
>
>
>
> >
> > **Question:** Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions?
> > ... | https://mathoverflow.net/users/90655 | Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? | There are some applications of Gauß-Bonnet in foliations and laminations of 3-manifolds.
The first result that comes to mind is [Candel’s Uniformization Theorem](http://www.numdam.org/article/ASENS_1993_4_26_4_489_0.pdf), which gives necessary and sufficient conditions for a lamination to admit a leafwise hyperbolic ... | 3 | https://mathoverflow.net/users/39082 | 361402 | 152,084 |
https://mathoverflow.net/questions/361387 | 2 | I am referring to the proof of (4) implies (1) in Theorem 3.16 of Woodin's *The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal*. His proof leverages on the fact that if the sharp of every real exists, then $\delta^1\_2 = u\_2$, where $u\_2$ is the second uniform indiscernible (the least ordinal above... | https://mathoverflow.net/users/29231 | On a particular proof of "if the sharp of every real exists and every club contains a club constructible from a real, then $\delta^1_2 = \omega_2$" | Let $\kappa$ be a sufficiently large regular cardinal. Take a countable elementary substructure $H$ of $H(\kappa)$ containing $z$ and $<\_\alpha$ with $\omega\_1\cap H\in D$. Let $\pi : M\to H(\kappa)$ be the inverse of the transitive collapse of $H$. To see the fifth bullet point, it suffices to show that in $M$, $\pi... | 6 | https://mathoverflow.net/users/102684 | 361412 | 152,088 |
https://mathoverflow.net/questions/361411 | 0 | Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on it?
What about the multivariate case? That is, I have a multivariate complex normal, and would like to understand the ... | https://mathoverflow.net/users/17883 | Distribution of the direction of Gaussian random variable | In the 2D case, you can write $X=AZ$, where $A$ is a $2\times2$ nonsingular real matrix, $Z:=[Z\_1,Z\_2]^T$, and the $Z\_j$'s are iid standard normal. You want to find
$$p:=P(n\_1\cdot X>0,\;n\_2\cdot X>0),$$
where $n\_1$ and $n\_2$ are unit vectors in $\mathbb R^2$ and $\cdot$ is the dot product.
By the rotational... | 3 | https://mathoverflow.net/users/36721 | 361414 | 152,090 |
https://mathoverflow.net/questions/361334 | 1 | **General Problem:** Suppose $X\_1,\ldots,X\_n \sim \mathbb{P}\_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "admissible configurations".
*Are there efficient methods of sampling from the restriction of $\mathbb{P}\_X^{\otimes ... | https://mathoverflow.net/users/109191 | Sampling i.i.d. variables with restrictions | A natural approach would be [Markov chain Monte Carlo](https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo) (MCMC), which involves a Markov chain which has your desired measure as its equilibrium measure. At the very least you need this chain to be irreducible, and then you also want it to converge quickly to equili... | 1 | https://mathoverflow.net/users/5784 | 361418 | 152,091 |
https://mathoverflow.net/questions/361133 | 5 | Can you provide a proof or a counterexample for the claim given below?
Inspired by Theorem 5 in [this paper](https://arxiv.org/pdf/1011.4836.pdf) I have formulated the following claim:
>
> Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\operatorname{gcd}(k,6)=1$. Assume that $a \in \mathbb{Z}$ is a 6-th power non-residue .... | https://mathoverflow.net/users/88804 | Conjectured primality test for specific class of $N=k \cdot 6^n+1$ | In one direction (wnen $N$ is prime) the statement is trivial. In the reverse direction, it's false however.
Here is just one counterexample: $n=4$, $k=133$, and $a=11$ with $N=172369=97\cdot 1777$, where we already have
$$\Phi\_2(11^{\frac{172369-1}2})\equiv 0\pmod{172369}.$$
| 2 | https://mathoverflow.net/users/7076 | 361419 | 152,092 |
https://mathoverflow.net/questions/361400 | 12 | Let $D = {\rm Sp}\, \mathbb{C}\_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}\_p$.
Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\rm Sp}\, \mathbb{C}\_p\langle x, x^{-1}\rangle \subseteq D$ is disconnected?
| https://mathoverflow.net/users/3847 | Can a covering space of the $p$-adic disc split over the circle? | $\newcommand{\Sp}{\mathrm{Sp}\,}\newcommand{\cO}{\mathcal{O}}\newcommand{\fm}{\mathrm{m}}$There seem to exist such examples and this does not contradict de Jong's argument because his proof only shows that the homomorphism $G:=\pi\_1(\Sp C\langle x^{\pm 1}\rangle)\to H:=\pi\_1(\Sp C\langle x\rangle)$ satisfies the foll... | 5 | https://mathoverflow.net/users/39304 | 361428 | 152,095 |
https://mathoverflow.net/questions/360938 | 5 | I asked this question on [Math SE](https://math.stackexchange.com/q/3683600/385720) but didn't receive any response.
Let $(T\_t)$ be a $C\_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda\_0\in \mathbb{C}$ is such that $e^{\lambda\_0 t}$ is a pole of $R(\cdot,T(t)),$ then $\lambda\_0$ is a pole of $... | https://mathoverflow.net/users/119514 | Equality in spectral inclusion theorem | Here is a positive solution if the semigroup is eventually compact:
Consider, say, the open right halfplane
$$
H := \{\lambda \in \mathbb{C}: \, \operatorname{Re}\lambda > \operatorname{Re}\lambda\_0 - 1\}.
$$
Then $H$ contains only finitely many spectral values of $A$, and all these spectral values are poles of the... | 4 | https://mathoverflow.net/users/102946 | 361433 | 152,097 |
https://mathoverflow.net/questions/360651 | 30 | I stumbled across the following formula when working on a research problem in theoretical computer science. I am looking for a simple proof of it, or any idea which might prove useful.
I checked its correctness up to $N=5$ with a computer. Brendan McKay (see [comment](https://mathoverflow.net/questions/360651/sum-ove... | https://mathoverflow.net/users/157351 | Sum over 0-1 matrices | This is a result of joint efforts with Fedor Petrov.
First, we show that the L.H.S. of the general version does not depend on $P$ and $Q$, and then we compute that constant for some properly chosen $P$ and $Q$. The elements of $\mathcal M$ are called *admissible* matrices.
**Part 1.** We show that the L.H.S. does n... | 10 | https://mathoverflow.net/users/17581 | 361436 | 152,099 |
https://mathoverflow.net/questions/361430 | 3 | Given a set of roots in a root system, assume that the pairing of each two roots in this set is not positive. Then clearly the set gives a closed root subsystem. My question is, when this set extends to a system of simple roots in the original root system. Of course, it is not always true, for example taking closed sub... | https://mathoverflow.net/users/5082 | when a set of roots extend to a system of simple roots | Let $\alpha\_1,\ldots,\alpha\_6$ be the simple roots of $E\_6$, and suppose that $\alpha\_3$ is the root corresponding to the trivalent node of the Dynkin diagram. Let $\theta$ be the highest (positive) root. Then $\{\alpha\_1,\alpha\_2,\alpha\_4,\alpha\_5,\alpha\_6,-\theta\}$ is a system of simple roots for a root sys... | 4 | https://mathoverflow.net/users/25028 | 361443 | 152,100 |
https://mathoverflow.net/questions/351688 | 9 | Can you provide a proof or counterexample for the following claim?
>
> Let $P\_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
> Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $ 4k<2^n$ , $p$ is a prime number and $n\ge3$ . Let $a$ be a natural number greater than two such that $\left(\fr... | https://mathoverflow.net/users/88804 | Conjectured primality test for specific class of $N=4kp^n+1$ | The "if and only if" statement fails for
$$[p,n,k,a] \in \{ [3, 4, 1, 100], [3, 4, 1, 225], [3, 6, 13, 2901] \}$$ and many others. In these cases, $N$ is not prime, but the congruence $S\_{n-2}\equiv 0\pmod{N}$ still holds.
| 7 | https://mathoverflow.net/users/7076 | 361444 | 152,101 |
https://mathoverflow.net/questions/361391 | 11 | Angelo Vistoli in the notes [Notes on Grothendieck topologies, fibered categories and descent theory](http://homepage.sns.it/vistoli/descent.pdf) starts the section of category theory with the following note:
>
> We will not distinguish between small and large categories. More
> generally, we will ignore any set-t... | https://mathoverflow.net/users/118688 | Size issues (small/large categories) when defining stacks in the Algebraic/differentiable/topological setting |
>
> Are there any places one has to be careful to not allow large categories?
>
>
>
No. For the purposes of forming the 2-category of algebraic/topological/differentiable stacks, or more generally, some kind of presentable stacks over a large category there are no size issues. Naively, the 2-category of stacks o... | 7 | https://mathoverflow.net/users/4177 | 361450 | 152,102 |
https://mathoverflow.net/questions/361454 | 0 | Does the following problem have a solution?
$$
\min\_X \mathbb{E} X
\quad\text{subject to}\quad
\mathbb{E} \log X = C.
$$
Here, the minimization is with respect to all integrable random variables $X$ and $C$ is some constant. Alternatively, instead of minimizing over random variables, one may equivalently view this a... | https://mathoverflow.net/users/158688 | Minimum mean over all random variables subject to logarithm constraint | Assume $X>0$ a.s. (so the constraint can be satisfied) and write $Y=\log X$.
By Jensen's inequality (<https://en.wikipedia.org/wiki/Jensen%27s_inequality>),
$\mathbb{E} X \ge e^{\mathbb{E} Y}=e^C$, so the minimum is attained for $X$ such that $\mathbb{P}(X=e^C)=1$.
| 2 | https://mathoverflow.net/users/7691 | 361456 | 152,104 |
https://mathoverflow.net/questions/361390 | 5 | What is known about the **duals of [cyclic polytopes](https://en.wikipedia.org/wiki/Cyclic_polytope)**, in particular, their *facets* (or equivalently, the vertex-figures of cyclic polytopes)?
* In *even* dimensions, all facets of the dual are combinatorially equivalent. Are these facets themselves duals of cyclic po... | https://mathoverflow.net/users/108884 | What is known about the duals of cyclic polytopes? | Many properties of the vertex figures of cyclic polytopes can be obtained from Gale evenness condition.
Let $P=C(n,d)$ be a cyclic $d$-polytope, and let $v\_1<\cdots<v\_n$ be its vertices ordered according to the moment curve. The following follows from Gale evenness condition.
1. In even dimensions, the vertex f... | 3 | https://mathoverflow.net/users/11134 | 361458 | 152,105 |
https://mathoverflow.net/questions/361464 | 2 |
>
> Let $G$ be a finitely generated group. Does there exist a constant $\kappa$ depending only on the rank of $G$ such that, if $G \simeq F\_1 \oplus \cdots \oplus F\_n$, then at most $\kappa$ factors are non-trivial free products?
>
>
>
**Motivation.** In order to optimize some of the results from [my preprint]... | https://mathoverflow.net/users/122026 | Rank of a sum with free products | No.
Indeed, write $C\_p\ast C\_p=\langle u\_p,v\_p\mid u\_p^p=v\_p^p=1\rangle$.
Let $J$ be any finite set of primes, and $G\_J=\prod\_{p\in J}C\_p\ast C\_p$. Then $G$ has generating rank two, regardless of $J$: indeed, it is generated by $u\_J=\prod\_{p\in J}u\_p$ and $v\_J=\prod\_{p\in J}v\_p$. While $G\_J$ is a... | 5 | https://mathoverflow.net/users/14094 | 361471 | 152,106 |
https://mathoverflow.net/questions/361467 | 15 | Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum\_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma\_z$ the set of all numbers $S(A)$. Remark that the cardinal of $\Sigma\_z$ is (likely) that of ${\cal P}({\mathbb N})$, the continuum.
>
> Is i... | https://mathoverflow.net/users/8799 | Partial sums of $\sum_0^\infty z^n$ | The number $z = i/\sqrt2$ seems to work!
Given $x \in [-2/3,4/3]$ we can find a "negabinary" expansion
$$x = \sum\_{k=0}^\infty (-1)^k\frac{b\_k}{2^k},$$ where each $b\_k \in \{0,1\}$.
Similarly, given $y \in [-2/3\sqrt2,4/3\sqrt2]$ we can find
$$y = \frac{1}{\sqrt2}\sum\_{j=0}^\infty (-1)^j\frac{c\_j}{2^j},$$
where ... | 12 | https://mathoverflow.net/users/2000 | 361478 | 152,108 |
https://mathoverflow.net/questions/361482 | 4 | What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions?
Is there a prototype of a counterexample?
| https://mathoverflow.net/users/135674 | BGG Category $\mathcal{O}$ is not closed under extension | You can usually extend two modules from $\mathcal{O}$ by a module which is not semisimple for the Cartan subalgebra (i.e. fails to be a weight module). See Exercise 3.1. in [J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$. Graduate Studies in Mathematics, 94].
| 4 | https://mathoverflow.net/users/15292 | 361485 | 152,111 |
https://mathoverflow.net/questions/361487 | -1 | **Motivating example.** Consider the graph $G=(V,E)$ with $V = \{0,1,2,3\}$ and $E = \big\{\{i,i+1\}: i\in \{0,1,2\}\big\}$. We have $\chi(G) = 2$, but if we collapse $0$ and $3$, we get the complete graph on $3$ points (having chromatic number $3$). Note that $0$ and $3$ have distance $3$.
**Problem statement.** If ... | https://mathoverflow.net/users/8628 | Effect of collapsing two vertices of distance $2$ | Add to your motivating example a fifth vertex called $4$ and make it adjacent to all of $0,1,2,3$. This graph is $3$-colorable (by the $2$-coloring of your example and a third color for vertex $4$) but if you collapse $0$ with $3$ you get the complete graph on $4$ vertices, which needs four colors. And the distance fro... | 3 | https://mathoverflow.net/users/6794 | 361490 | 152,112 |
https://mathoverflow.net/questions/361492 | 4 | Trying to characterize categories equivalent to the category of sets, I have discovered (for myself) that instead of requiring that the coprojection morphism $\mathsf{true}:1\to \Omega=1\sqcup 1$ is a *subobject* classifier, it suffices to require that this morphism is an *singleton* classifier, which means that for ev... | https://mathoverflow.net/users/61536 | Categories admitting singleton-classifiers and characterization of the category $\mathbf{Set}$ | I don't think this has been considered. Mainly I've never seen it, but also there are specific feature of this notions that makes it unlikely to be a relevant category theoretic notion independently of your other conditions:
* It is not really a universal property, in the sense that it does not characterize what are ... | 3 | https://mathoverflow.net/users/22131 | 361494 | 152,113 |
https://mathoverflow.net/questions/361495 | 2 | How should I estimate the following integral
$$I = \int\_0^1 \left( \sum\_{n=0}^{p-1} e(n^2t) \right)^2 dt $$
where $p$ is a prime?
Here is the method I followed:
\begin{align\*}
I & = \int\_0^1 \left( 1+ \sum\_{n=1}^{p-1} e(n^2t) \right)^2 dt \\
& = 1 + 2 \sum\_{n=1}^{p-1} \left( \int\_0^1 e(n^2t) dt \right) + \int\... | https://mathoverflow.net/users/73880 | Integral over an exponential sum with squares | The integral is equal to exactly one - just expand out the square and use orthogonality to get
$$ I = \sum\_{0\leq n,m<p} \int\_0^1 e((n^2+m^2)t) \mathrm{d} t = \sum\_{0\leq n,m<p}1\_{n^2+m^2=0}=1.$$
| 6 | https://mathoverflow.net/users/385 | 361498 | 152,115 |
https://mathoverflow.net/questions/361462 | 6 | From the famous book: Monopole and three manifold, Kronheimer and Mrowka(<https://www.maths.ed.ac.uk/~v1ranick/papers/kronmrowka.pdf>). It is known that:
Let $Y$ be a closed oriented $3$ manifold, choosing a spinc structure $\mathfrak s$ and metric $g$ and a generic perturbation $p$, one can construct the monopole Floe... | https://mathoverflow.net/users/95296 | Cobordism monopole Floer homology | The first bullet is definitely explained in the book! Surely around where it was introduced, it has to do with summing over all spin-c structures. We need to pass to the completion because the 4-manifold can have infinitely many spin-c structures that would need to be used.
The second bullet, yes. In general we shou... | 3 | https://mathoverflow.net/users/12310 | 361499 | 152,116 |
https://mathoverflow.net/questions/361491 | 5 | For $n\ge 1$ we write $[n]$ to denote the set $\{0,1,\ldots,n-1\}$. Let $2^{[n]}$ be the set of all functions from $[n]$ to $\{0,1\}$. Let $\mathcal{F}$ and $\mathcal{G}$ be two nonempty subsets of $2^{[n]}$. Fix $1\le k\le n$. Take $0\le j \le n-k$. We say that $f,g\in 2^{[n]}$ **disagree over $[j,j+k)$** if the restr... | https://mathoverflow.net/users/24676 | Size of a family of sets of $k$-separated functions over $\{0,1,\ldots,n-1\}$ | **1: Hopes.** Let me begin by taking away your hope - that is, disproving the conjectured asymptotic, $|{\mathscr{T}\_{n,k}}|/2^{2^n} \to 1$.
I will say that a family $\mathcal{F}$ is $k$-rich if for each $j \in [n]$ and $w \in \{0,1\}^k$, there is $f \in \mathcal{F}$ with $f|\_{[j,j+k)} = w$. Accordingly, I will say... | 3 | https://mathoverflow.net/users/14988 | 361506 | 152,118 |
https://mathoverflow.net/questions/361455 | 3 | This is a simplified version of a question I am looking at, but embarrassingly I can't do this one anyway.
Let's assume $f(x)$ is a decreasing positive function, $f(0)$ is infinite and, moreover, $f(x)$ is comparable to $1/x$ near $x=0$. We look only at positive $x$'s.
**Question**. Assume we know that the limit
$... | https://mathoverflow.net/users/136925 | Integral convergence implies pointwise | If you are ok with $f$ being not continuous and not *strictly* decreasing, here's a simple enough counter-example. (With a bit more work you can make it continuous and strict, but it distracts from the point.)
Let $f(x)$ be the step-function defined by
$$ f(x) = \begin{cases}
2^{k} & x \in ( 2^{-k}, 2^{1-k}) \\
2^... | 4 | https://mathoverflow.net/users/3948 | 361509 | 152,119 |
https://mathoverflow.net/questions/361488 | 34 | For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x\_1, \dots, x\_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \quad \text{for} \quad t \in I.
$$
The mapping
$$
\varphi: I \ni t \mapsto \det(x\_1(t), \dots, x\_d(t)) \in \mathbb{R}... | https://mathoverflow.net/users/102946 | What do we learn from the Wronskian in the theory of linear ODEs? | Here is a typical use in an undergraduate textbook: to prove that for distinct $\lambda\_j$ the exponentials $e^{\lambda\_jt}$ are linearly independent. It has some applications on the more advanced level, but you were asking about undergraduate textbooks. Also notice: uniqueness theorem, even for linear ODE is rarely ... | 18 | https://mathoverflow.net/users/25510 | 361522 | 152,123 |
https://mathoverflow.net/questions/361518 | 14 | Let $F\_n$ be the free group on $n$ generators. Of course, every finitely-presentable group $G$ is a finite colimit of copies of $F\_n$, where $n$ is allowed to vary. But is $G$ a finite colimit of copies of $F\_2$?
Of course, because $F\_{2n}$ is a finite coproduct of copies of $F\_2$, we have that any finitely-pres... | https://mathoverflow.net/users/2362 | Is every finitely-presentable group a finite colimit of copies of $F_2$? | I believe the answer is yes. Assume, by way of contradiction, that some finitely presented group cannot be so expressed. Then we can choose such a group $G$ where for any generating set of the form $x\_1,y\_1, x\_2,y\_2,\ldots, x\_n,y\_n$ the number of non-permissible relations needed to define $G$ (together with some ... | 10 | https://mathoverflow.net/users/3199 | 361525 | 152,125 |
https://mathoverflow.net/questions/361368 | 2 | Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the completion of the localization at $\mathfrak{m}$ by $R^\wedge\_{\mathfrak{m}}$. Assume that for every $\mathfrak{m}$ there is an $R^\wedge\_{\mathfrak{m}}$-iso... | https://mathoverflow.net/users/nan | Varieties with everywhere good reduction that are isomorphic over every completion have isomorphic generic fibers | Here's an explicit example. Let $R=\mathbb{Z}[\sqrt{2}]$, let $X=\mathbb{P}^1\_R$, and let $Y$ be the smooth projective conic defined by the equation $$(2-\sqrt{2})x^2+y^2+(2-\sqrt{2})z^2+xy+yz+(3-2\sqrt{2})xz=0.$$
I claim this conic has good reduction everywhere; as smooth conics over the ring of integers of a $p$-adi... | 9 | https://mathoverflow.net/users/6950 | 361542 | 152,133 |
https://mathoverflow.net/questions/361538 | 2 | Let $J\_n$ be the Bessel function of the first kind. Let $J\_n^{(\max)} = \max\_{x>0} J\_n(x)$. What is known about the asymptotic behavior of $J\_n^{(\max)}$ at large $n$? Specifically, I am looking for a lower bound. It is ok if the result only holds for integer and half-integer $n$.
(It is potentially helpful to n... | https://mathoverflow.net/users/23252 | Asymptotic behavior of maximum of bessel function | The answer is given in Formula 9.5.25 of Abramowitz and Stegun, available here:
<https://pdfs.semanticscholar.org/1a2a/68cac86cb55abb9eb1858b3b58c4a1b16434.pdf>
| 4 | https://mathoverflow.net/users/12120 | 361543 | 152,134 |
https://mathoverflow.net/questions/280739 | 6 | As the title says, given a general $q$-binomial $\binom{n}{k}\_q$,
is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$?
| https://mathoverflow.net/users/1056 | Q-binomials at roots of unity | So, the canonical answer is the [q-Lucas theorem,](https://www.math.upenn.edu/%7Epeal/polynomials/q-analogues.htm#prelimQanaloguesQLucas) as pointed out in the comments.
This was proved in
*Olive, Gloria*, [**Generalized powers**](http://dx.doi.org/10.2307/2313851), Am. Math. Mon. 72, 619-627 (1965). [ZBL0215.07003](... | 1 | https://mathoverflow.net/users/1056 | 361550 | 152,135 |
https://mathoverflow.net/questions/361540 | 0 | In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) has an anti-symmetric Dirac operator.
1. Say, if the $\psi$ is a Dirac spinor, he wrote down an action
$$
\int d^2x \s... | https://mathoverflow.net/users/106497 | Anti-symmetric operators for the Dirac or Majorana spinors | In terms of the Hamiltonian $H$ the antisymmetry follows simply: Majorana fields $\Psi(x,t)$ are real, and they satisfy a real wave equation
$$\partial\Psi/\partial t =-iH\Psi,$$
where $iH$ is real, hence $H$ is imaginary. Since $H$ must also be Hermitian, it means that $H^T=-H$ (antisymmetric).
More explicitly, $H=\... | 4 | https://mathoverflow.net/users/11260 | 361553 | 152,137 |
https://mathoverflow.net/questions/361560 | 3 | Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum\_{A\subseteq N}\left(c\_A \prod\_{i\in A}x\_i \right) = 0$, where each $c\_A$ is a real number constant. (So each equation contains at most $2^n$ terms.) An example for $n=3$ is:
$$2x\_1x\_2x\_3 - 4x\_1x\_2+5x\_3+2=... | https://mathoverflow.net/users/79906 | Solving multilinear equations | Multilinear equations are hardly easier than general equations. For instance, the multilinear equations
$$
\begin{cases}
x\_0-x\_1=0,\\
x\_0x\_1-x\_2=0,\\
x\_0x\_2-x\_3=0,\\
\ldots\\
x\_0x\_{n-1}-x\_n=0
\end{cases}
$$
simply tell you that $x\_k=x\_0^k$ for all $k=1,\ldots,n$. Using this, it is very easy to replace a... | 8 | https://mathoverflow.net/users/1306 | 361561 | 152,140 |
https://mathoverflow.net/questions/361559 | 10 | In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a psd matrix $A$ is defined unambiguously via the spectral theorem.
Also, I use the conventional Frobenius scalar product a... | https://mathoverflow.net/users/33741 | A square root inequality for symmetric matrices? | The classical operator generalization of the scalar inequality $|\sqrt{a}-\sqrt{b}|^2 \leq |a-b|$ is the Powers-Størmer inequality, which involves two different norms : the trace norm $\|X\|\_1 = \operatorname{Tr}|X|$ and the Froebenius norm $\|X\|\_2 = (\operatorname{Tr}(X^\* X))^{\frac 1 2}$, where $|X| = (X^\* X)^{\... | 11 | https://mathoverflow.net/users/10265 | 361563 | 152,141 |
https://mathoverflow.net/questions/361567 | 4 | In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$:
$$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum\_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum\_{\substack{j+k=l\\ 0\leq j\leq n\\0\leq k \leq m}}(-1)^{j}{n \choose j}{m \choose k}.$$
Numerical evidence suggests that one has
$$
S... | https://mathoverflow.net/users/58125 | A binomial coefficient identity involving two parameters | Taking into account that $\binom pq=0$ for nonnegative integers $p$ and $q$ such that $q>p$, write
$$S(n,m)=\frac{(n+m+1)!}{n!m!}T(n,m),$$
where
\begin{align\*}
T(n,m)&:=\sum\_{l\ge0}\frac{1}{n+m-l+1}\sum\_{\substack{j+k=l\\ j\ge0,k\ge0}}(-1)^{j}\binom nj\binom mk \\
&=\sum\_{l\ge0}\int\_0^1 dx\,x^{n+m-l}\sum\_{\su... | 4 | https://mathoverflow.net/users/36721 | 361579 | 152,144 |
https://mathoverflow.net/questions/361556 | 6 | Let
$$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x\_i = 0\} $$
(one-way infinite eventually zero words). Let $\{0,1\}^\*$ denote the finite (not necessarily nonempty) words over $\{0,1\}$, and write $\{0,1\}^{\leq k} = \{w \in \{0,1\}^\* \;|\; |w| \leq k\}$ where $|w|$ denotes length.
Is there a... | https://mathoverflow.net/users/123634 | Is there a prefix-continuous bijection between finite words and eventually zero words? | I believe the following works, but I might be missing something.
If $x$ has at least two 1s, then $\phi(x)$ is the sequence cut just before the last 1:
$$\phi(0011101000\cdots) = 001110 $$
If $x$ has at most one 1, then $\phi(x)$ is the sequence cut before the one, with an additional zero:
$$ \phi(000\cdots) = \var... | 6 | https://mathoverflow.net/users/129074 | 361584 | 152,146 |
https://mathoverflow.net/questions/361520 | 9 | I am not working in the field of algorithmic algebraic geometry - yet, for my current work, I need some results from it.
More specifically, what is the state-of-the-art when it comes to solving (whatever "solving" means in this case) system of polynomials of fields that are not algebraically closed, whose ideal has ... | https://mathoverflow.net/users/43263 | What is the state-of-the-art for solving polynomials systems over fields that are not algebraically closed? | For the reals, I particularly like the book by Sturmfels mentioned by Alexandre Eremenko. For the rational numbers, you can hardly do better than Bjorn Poonen's book *Rational Points on Varieties*, which is available for browsing via [his homepage](http://www-math.mit.edu/~poonen/).
For dimension $1$ specifically, P... | 3 | https://mathoverflow.net/users/17907 | 361606 | 152,151 |
https://mathoverflow.net/questions/361612 | 7 | Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
1. $X,X'$ independent with uniform distribution on the sphere $S^{d-1}$
2. $X\in S^{d-1}$ deterministic, $X'$ uniformly distributed on $S^{d-1}$
?
| https://mathoverflow.net/users/158772 | Scalar product of random unit vectors | As noted in the comments, by the spherical symmetry, the distribution of the dot product in both parts of your question is the same that of $X\cdot(1,0,\dots,0)$. Moreover, the distribution of $X$ is the same as that of the random vector
$$\frac{Z}{\sqrt{Z\_1^2+\dots+Z\_d^2}},$$
where $Z=(Z\_1,\dots,Z\_d)$ is a standa... | 5 | https://mathoverflow.net/users/36721 | 361614 | 152,153 |
https://mathoverflow.net/questions/361620 | 0 | Suppose that $C^\*\_r(G)\cong C^\*\_r(H)$, can we conclude that $G\cong H$?
| https://mathoverflow.net/users/153196 | Two isomorphic reduced group $C^*$-algebras | The simplest counterexample is $G = \mathbb{Z}\_2\times \mathbb{Z}\_2$ and $H = \mathbb{Z}\_4$. These groups are not isomorphic but $C^\*(G) \cong \mathbb{C}^4 \cong C^\*(H)$.
| 6 | https://mathoverflow.net/users/23141 | 361623 | 152,154 |
https://mathoverflow.net/questions/361625 | 5 | Let $A$ be a commutative algebra over a field $k$ which is finite dimensional as a vector space over $k$. Let $M$ be a faithful $A$-module. Does it follow that $dim\_k(M)\geq dim\_k(A)$?
| https://mathoverflow.net/users/24483 | faithful modules over a finite dimensional commutative algebra | $A$ is a product of local Artinian rings, so the question is local. The best result I know is in this [paper](https://www.mscand.dk/article/view/11413/9430) of Gulliksen, who proved that if the socle dimension is at most $3$, then the length of any faithful module is at least the length of the ring. So the answer is ye... | 9 | https://mathoverflow.net/users/2083 | 361627 | 152,156 |
https://mathoverflow.net/questions/361636 | 3 | Let $G$ be a locally compact Hausdorff group. Assume that $\theta:G\to U\_d$ is a group homomorphism where $U\_d$ is a finite dimensional unitary group. Consider a action of $G$ on $U\_d$ by $g.u:=\theta(g)u,$ $u\in U\_d.$ Consider $U=\overline{\theta(G)}.$ Is it true that $G$ acts ergodically on $U$ where $U$ is equip... | https://mathoverflow.net/users/136860 | Ergodic action on unitary group | Since $U\_d$ is connected, every subgroup proper of it had infinite index, so unless $U=U\_d$ it must be a set of measure zero there, and the literal answer is "no".
On the other hand, the action of $G$ on $U$ is ergodic for the Haar measure of $U$. Indeed, let $f\in L^1(U)$ be $G$-invariant (e.g. the characteristic ... | 3 | https://mathoverflow.net/users/327 | 361639 | 152,158 |
https://mathoverflow.net/questions/361604 | 3 | Together with coauthors I'm working on a paper where we use the following Proposition:
>
> If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all its mass in a single point, its moment generating function $$ M(z) = E(e^{zX})$$ has a zero in the complex plane.
>
... | https://mathoverflow.net/users/14302 | Reference request: The transform of a bounded random variable has a zero in the complex plane | It is the content of Theorem 7.2.3 page 202 of Eugene Lukacs book "Characteristic Function".
| 4 | https://mathoverflow.net/users/69642 | 361662 | 152,162 |
https://mathoverflow.net/questions/361642 | 2 | This is a follow-up (but self-contained) question to my [previous one](https://mathoverflow.net/questions/361520/what-is-the-state-of-the-art-for-solving-polynomials-systems-over-fields-that-ar). There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields ... | https://mathoverflow.net/users/43263 | Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals | Just expanding my comments to this question and [the previous one](https://mathoverflow.net/questions/361520):
I assume that your polynomials have rational coefficients (which seem to be the case, since you mention they are floating point numbers with fixed precision, in particular they are decimals), and that you ar... | 2 | https://mathoverflow.net/users/6506 | 361663 | 152,163 |
https://mathoverflow.net/questions/358605 | 8 | Let $G$ be a locally compact Hausdorff topological group whose underlying abstract group is residually finite. Let $H\subset G$ denote the intersection of all finite-index, **closed** subgroups. Is there an example of such a $G$ where $H$ is not trivial? Is there an example where $H=G$ (i.e., every finite-index subgrou... | https://mathoverflow.net/users/44172 | Residually finite group with dense finite index subgroups | Yes it exists (at least the weaker version). Namely, here is a way to produce second-countable abelian, totally disconnected locally compact groups whose underlying abstract group is residually finite, but in which the intersection of finite index open subgroups is not trivial.
Fix $p$ prime and an infinite countable... | 2 | https://mathoverflow.net/users/14094 | 361667 | 152,164 |
https://mathoverflow.net/questions/361681 | 6 | In QFT and Statistical Mechanics the discrete Laplacian usually plays a key role when we want to discretize the theory. However, few books (at least to my knowledge) really work the properties of this operator in details, so I'm trying to figure out some of these properties myself.
Let $\Lambda := \epsilon Z^{d}/L\m... | https://mathoverflow.net/users/150264 | Invertibility of discrete Laplacian | Your reasoning is correct in that the discrete Laplacian for periodic boundary conditions has a zero mode. On the space of fields satisfying $\sum\_x\phi(x)=0$, its spectrum is, however, strictly positive, and it can be inverted on that space. This is all well known.
| 5 | https://mathoverflow.net/users/45250 | 361682 | 152,168 |
https://mathoverflow.net/questions/16410 | 15 | $\DeclareMathOperator\SL{SL}$[Casson's invariant](http://www.ams.org/mathscinet-getitem?mr=1154798) is an invariant of a homology 3-sphere, obtained by
“counting” representations of the fundamental group into $\operatorname{SU}(2)$.
I was wondering if there is an analogous invariant counting representations
into $\SL(... | https://mathoverflow.net/users/1345 | $\operatorname{SL}_2(\mathbb R)$ Casson invariant? | A 2020 arxiv posting of [Nosaka](https://arxiv.org/abs/2005.06132v1) (An $SL\_2(\mathbb{R})$-Casson invariant and Reidemeister torsions) defines an $SL(2,\mathbb{R})$ Casson invariant. As Charlie's answer suggests, the approach is inspired by Johnson's unpublished work.
| 5 | https://mathoverflow.net/users/3460 | 361683 | 152,169 |
https://mathoverflow.net/questions/361570 | 2 | Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that I'm not considering applications in quantum computing here.)
EDIT: To clarify what I mean, consider a simple system l... | https://mathoverflow.net/users/45250 | Physics applications of quantum logic | I'm fairly confident that there are no such derivations, and for good reason.
The paper I like on this topic is by Michael Dunn, [Quantum Mathematics](https://philpapers.org/rec/DUNQM). He concludes
>
> First-order Peano arithmetic formulated with quantum logic has the
> same theorems as classical first-order Pe... | 4 | https://mathoverflow.net/users/nan | 361687 | 152,171 |
https://mathoverflow.net/questions/361684 | 11 | In the book of Szabo "Algebra of Proofs", Definition 13.1.9 introduces an elementary topos as a cartesian closed category with a subobject classifier. On the other hand, many other sources including Johnstone add to this definition that the category should contain limits of finite diagrams. For the proof that the requi... | https://mathoverflow.net/users/61536 | Why do elementary topoi have pullbacks? | I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. **However**, this is based on the definition of sub-object classifier given on wikipedia (linked in the comment above) that I would consider as incorrect:
The wikipedia de... | 20 | https://mathoverflow.net/users/22131 | 361692 | 152,172 |
https://mathoverflow.net/questions/361691 | 2 | I am trying to find a reference (or, if it's false, a counterexample) for the following sort-of-intuitive fact: if $\tau$ is a stopping time with a subexponential probability distribution, and $(X\_n)\_{n\geq 1}$ are independent r.v.'s, also subexponential, then $\sum\_{n=1}^\tau X\_n$ aso has a subexponential distribu... | https://mathoverflow.net/users/37266 | Distribution of a stopped random sum, with subexponential stopping time | The hypothesis implies that $M\_k=[e^{\sum\_{n=1}^k X\_n}]$ is a supermartingale, with $M\_0=1$. Then the optional stopping theorem for positive supermartingales implies the requested inequality. (see e.g. Williams' book "Probability with martingales").
Note that no moment conditions on the stopping time $\tau$ are ne... | 2 | https://mathoverflow.net/users/7691 | 361697 | 152,174 |
https://mathoverflow.net/questions/361668 | 2 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\mathfrak{b}=$Lie$(B)$.
Let $M$ be a $U(\mathfrak{g})$-module with $N \subset M$ a finite dimensional $K$-module, which is ... | https://mathoverflow.net/users/135674 | Checking axiom of Category $\mathcal{O}$ | The module $U(\mathfrak{g})\otimes\_{U(\mathfrak{b})} N$ is locally $U(\mathfrak{b})$-finite and there is a surjective $U(\mathfrak{g})$-homomorphism $\varphi\colon U(\mathfrak{g})\otimes\_{U(\mathfrak{b})} N \to M$ given by $u \otimes n \mapsto u\cdot n.$ It is easy to see that every weight space of $M$ has only finit... | 1 | https://mathoverflow.net/users/6818 | 361702 | 152,175 |
https://mathoverflow.net/questions/361672 | 4 | Let $(X,d)$ be an $n$-dimensional $(n< \infty)$ complete geodesic metric space, where any two points in $X$ are joined by a unique shortest geodesic. Let $S$ be a sufficiently small metric $(n-1)$-sphere in $X$. Let $\epsilon>0$ be the radius of $S$. Pick a point $p \in S$. Find a metric $(n-1)$-sphere $S'$ of radius $... | https://mathoverflow.net/users/114032 | Is the intersection of two distinct sufficiently small metric spheres always empty, a point or a metric sphere of lower dimension? | Let $C\_\alpha$ denote the $n$-dimensional closed Euclidean solid cone with the cone angle $\alpha\in (0, \pi)$. Let $X\_\alpha$ be the metric space obtained by gluing two copies $C^\pm\_\alpha$ of $C\_\alpha$ at their tips, and equipped with the natural path-metric. Let $o\in X\_\alpha$ denote the common tip of the co... | 3 | https://mathoverflow.net/users/39654 | 361716 | 152,177 |
https://mathoverflow.net/questions/361715 | 0 | Some time ago, Kellogg communicated to Carmichael a result with an incomplete proof, which was soon after verified as correct. I do not recall the source but recall the result. Define
$$S\_n = \{ (x\_1,x\_2 \dots x\_n) \subset \mathbb{N}^n: \sum x\_i^{-1} = 1\}.$$
And regard elements of $S\_n$ identical up to a per... | https://mathoverflow.net/users/128941 | Maximum in solution set to a Diophantine equation related to unit fractions | $$1={1\over2}+{1\over2}={1\over2}+{1\over3}+{1\over6}={1\over2}+{1\over3}+{1\over7}+{1\over42}={1\over2}+{1\over3}+{1\over7}+{1\over43}+{1\over1806}=\cdots$$ so certainly the sequence $u\_n$, which is $1,2,6,42,1806,\dots$, is a lower bound as well for $l(n)$.
$u\_n$ is [tabulated](http://oeis.org/A007018) at the On... | 3 | https://mathoverflow.net/users/158000 | 361735 | 152,184 |
https://mathoverflow.net/questions/361605 | 3 | I have a question of some integrability of hitting times.
Let $X=(\{X\_t\}\_{t \ge0},\{P\_x\}\_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$.
We assume that there exist Borel measurable functions $f\colon E \to [1,\infty)$ and $g \colon E \to \mathbb{R}$ such that $\left\{f(X\_... | https://mathoverflow.net/users/68463 | Do Lyapunov functions imply exponential integrability of hitting times? | It may depend on what exactly you mean by a diffusion on a general metric space.
Here is a counterexample for a discontinuous process. Take $E = \mathbb{R}$, and let $(X\_t)$ be the process started at $x \in \mathbb{R}$ and jumping to the origin after a unit exponential time $\tau$. In other words, $(X\_t)$ jumps to ... | 1 | https://mathoverflow.net/users/41071 | 361738 | 152,185 |
https://mathoverflow.net/questions/359706 | 2 | In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\operatorname{rad}(n)=\prod\_{\substack{p\mid n\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(1)=1$. The abc co... | https://mathoverflow.net/users/142929 | On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means | abc implies your conjecture with $b-a$.
**Case 1** Let $a,b,c=a+b$ be bad abc triple,i.e. $c < rad(ab(a+b))$.
We have $rad(ab(a+b)) > c > b - a$.
**Case 2** Let $a,b,c=a+b$ be good abc triple,i.e. $c>rad(ab(a+b))$.
Then $T : (b-a)^2,4ab,(a+b)^2$ is good abc triple too.
The radical is divisor of $ab(a+b)(b-a)$... | 2 | https://mathoverflow.net/users/12481 | 361739 | 152,186 |
https://mathoverflow.net/questions/361709 | 8 | This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann algebras, but instead refer the reader to [Ching - Free products of von Neumann algebras](https://www.ams.org/journals/tran/197... | https://mathoverflow.net/users/6269 | Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor? | This should just be a comment- but for some reason I couldn't add a comment.
It seems to me that using Corollary 5.3 of [this paper](https://arxiv.org/pdf/funct-an/9211013.pdf) by Dykema, we indeed get a positive answer to your question.
Corollary 5.3 states that $L(G \ast H) \cong L(F(2-|G|^{-1}-|H|^{-1}))$, if $G... | 9 | https://mathoverflow.net/users/149852 | 361740 | 152,187 |
https://mathoverflow.net/questions/361743 | 5 | Take a genus $g$ surface $S$ standardly embedded in $\mathbb{R}^3$, by which I mean it is unknotted. Surface $S$ bounds a volume $V$ that deformation retracts on a standardly embedded planar graph $G$ with $\beta\_1 = g$, and that only has degree $3$ vertices.
Among the loops on $S$ that are null homotopic in $V$, t... | https://mathoverflow.net/users/112954 | "Basic" loops on standardly embedded surfaces | They are often called meridians of $G$. Note that there are many graphs $G$ to which $V$ deformation retracts (most nonplanar); if you are not particular about which graph $G$ then they are called meridians of $V$. $V$ is called a handlebody and $G$ is a spine of the handlebody. See, for example, Scharlemann's "Refilli... | 7 | https://mathoverflow.net/users/30679 | 361752 | 152,190 |
https://mathoverflow.net/questions/361755 | 0 | Let $\tau:\Omega\to \Omega$ be a measure-preserving transformation with $\mu(\Omega)<\infty$. Define $T:L\_p(\Omega)\to L\_p(\Omega)$ as $Tf:=f\circ \tau$. I want to prove that for all $1\leq p<\infty$, given $f\in L\_p$ there exists $\bar{f}\in L\_p$ such that $\|\bar{f}\|\_p\leq \|f\|\_p,$ $\bar{f}\circ \tau=\bar f$,... | https://mathoverflow.net/users/136860 | von Neumann ergodic theorem for $L_p$ | False for p infinite. True for finite. See e.g. the book by Krengel, Ergodic theorems. Other sources (that also go further) are [1, Sec. I.2.1] or [2, Theorem 8.8].
[1] T. Eisner, Stability of operators and operator semigroups, Operator Theory:
Advances and Applications, vol. 209, Birkh¨auser Verlag, Basel, 2010.
... | 5 | https://mathoverflow.net/users/7691 | 361756 | 152,192 |
https://mathoverflow.net/questions/360207 | 10 | The Hitchin fibration is a central topic of study in modern geometry. It seems to be folklore knowledge that the morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base (the direct sum of spaces of global sections of powers of the canonical bundle) is *flat*. Where is this proven?
| https://mathoverflow.net/users/940 | Flatness of the Hitchin system? | If you are okay working with the Hitchin map defined on the moduli stack of Higgs bundles rather than the coarse moduli space, then you can find a proof of this statement in the paper [The global nilpotent variety is Lagrangian](https://arxiv.org/pdf/alg-geom/9704005.pdf) by V. Ginzburg, at least in the case that the g... | 7 | https://mathoverflow.net/users/12402 | 361772 | 152,199 |
https://mathoverflow.net/questions/361783 | 4 | This is a similar question to <https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487>
Let f(n) be the maximum number of squares in an AP (arithmetic progression) of length n. For example, $f(3)=3$, as 1, 25, 49 is a 3-term arith... | https://mathoverflow.net/users/38744 | The number of perfect squares which can occur in an arithmetic progression of length n | It's [tabulated](http://oeis.org/A221671) out to $f(52)=12$ at the Online Encyclopedia of Integer Sequences.
A [reference](https://arxiv.org/abs/1301.5122v1) is given to Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions.
A conjecture is presented which give... | 3 | https://mathoverflow.net/users/158000 | 361786 | 152,205 |
https://mathoverflow.net/questions/361717 | 9 | Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am interested in analogues of this theorem for uncountable unions.
Specifically, suppose a compact Hausdorff space $X$
is... | https://mathoverflow.net/users/402 | Baire category theorem for uncountable unions | The hyperstonean case can be dealt with using a result from Fremlin's *Measure Theory*. For every hyperstonean space $X$, we can find a semi-finite measure $\mu$ defined on the sets with the Baire property whose nullsets are exactly the meagre sets and which is inner regular with respect to compact subsets. Therefore $... | 4 | https://mathoverflow.net/users/61785 | 361787 | 152,206 |
https://mathoverflow.net/questions/361693 | 4 | Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at the origin. Then we define the weighted length of curves in the family as
$$L(\gamma):= \int\_{\gamma} \frac{2}{1+|z|^2... | https://mathoverflow.net/users/51546 | minimizing weighted length of closed curves | I just found out two proofs. One is analytic, and the other is geometric.
For the analytic proof, one can use polar coordinates and apply some elemetary inequality. But such proof has limited applications. so I'm presenting a geometric proof.
Note that the metric of the unit sphere is given by $g=\frac{4}{(1+|z|^2)... | 0 | https://mathoverflow.net/users/51546 | 361795 | 152,210 |
https://mathoverflow.net/questions/361749 | 3 | All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers.
An algebra $A$ is *supercommutative-gradable* if it admits a grading $A=A\_0\oplus A\_1$ in $\mathbf{Z}/2\mathbf{Z}$ ($A\_iA\_j\subset A\_{i+j}$ for $i,j\in\mathbf{Z}/2\mathbf{Z}$) that makes it supercommutative: ... | https://mathoverflow.net/users/14094 | Polynomial identities of supercommutative-gradable algebras | I believe that the identity $(xy-yx)z-z(xy-yz)$ generates everything (in characteristic 0, at least). To show that no further identities are needed, it is enough to exhibit one algebra that has no further identities. It follows from [an old theorem of Krakowski and Regev](https://www.jstor.org/stable/1996643?seq=1#met... | 1 | https://mathoverflow.net/users/1306 | 361803 | 152,212 |
https://mathoverflow.net/questions/361788 | 2 | Suppose that $(\Omega,\mu)$ is a measure space. Let $\tau:\Omega\to\Omega$ is a measurable map such that $\mu\circ\tau^{-1}<<\mu$. Then $\tau$ s said to be null preserving. I want to prove the following. If $f:\Omega\to\mathbb R$ is measurable and $\mu(\{f\neq f\circ \tau\})=0,$ then there exists a measurable function ... | https://mathoverflow.net/users/136860 | Null preserving transformation | I'll rename your $\Omega$ into $X$ to simplify typing.
Let $D=\{x\in X,~\forall n\in\mathbb{N}:~ f(\tau^n(x))=f(x)\}=\bigcap\limits\_{n\in\mathbb{N}\_0} \{x\in X: f(\tau^{n+1}(x))=f(\tau^n(x))\}$ $=\bigcap\limits\_{n\in\mathbb{N}\_0}\tau^{-n}(A)=X\backslash \bigcup\limits\_{n\in\mathbb{N}\_0}\tau^{-n}(X\backslash A)$... | 2 | https://mathoverflow.net/users/53155 | 361813 | 152,215 |
https://mathoverflow.net/questions/361777 | 3 | I am considering a combinatorial argument which involves the following quantity. We use the prime counting function $\pi(n)$ and to save on exponents we set $h=\pi(n/2)$. The quantity as a function of integer $n \gt 7$ is
$$(\pi(n)!)^{1/(n-h)}$$
Computations for small $n$ suggest this is always less than $4$, as do r... | https://mathoverflow.net/users/3402 | Is this number theoretic quantity bounded above? | Let $k:=\pi(n)$, so that $p\_k\le n<p\_{k+1}$, where $p\_k$ is the $k$th prime. By the last displayed formula in [this section](https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number) of the Wikipedia article,
\begin{equation\*}
-1+\ln(k\ln k)<\frac{p\_k}k<\ln(k\ln k)
\end{equation... | 8 | https://mathoverflow.net/users/36721 | 361814 | 152,216 |
https://mathoverflow.net/questions/361805 | 0 | Let $H\_i, K\_i$ for $i=1,2,3$ be Hilbert spaces with horizontal exact sequences. Assume $T\_1, T\_2, T\_3$ have dense range, that $T\_1, T\_2, T\_3$ are trace class operators and that the squares commute. Assume that $S \colon H\_2 \to K\_2$ is a compact operator with dense range making the squares commute.
$\requi... | https://mathoverflow.net/users/2258 | Extensions of trace class operators | The answer is no. Take $H\_2 = K\_2$ to be an infinite dimensional Hilbert space and take $H\_1 = K\_1$ to be an infinite dimensional subspace whose orthocomplement $H\_3 = K\_3$ is also infinite dimensional. Let $f\_1 = g\_1$ be the inclusion and $f\_2 = g\_2$ the orthogonal projection. Let $T\_1: H\_1 \to K\_1$ and $... | 1 | https://mathoverflow.net/users/23141 | 361816 | 152,217 |
https://mathoverflow.net/questions/361817 | 15 | Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
**Question 1:**
>
> What is known about $\chi(x)?$
>
>
>
1a) Are there criteria when $\chi(x)$ is positive / negative /... | https://mathoverflow.net/users/12826 | What is known about ordinary character values at involutions? | There are many results about the values of $\chi(t)$ when $t$ is an involution of a finite group $G$ and $\chi$ is an irreducible character: Isaacs' book on Character Theory has many such results collected from the literature, but there are many others scattered around:
For example, if $G = O^{2}(G)$ (equivalently, i... | 15 | https://mathoverflow.net/users/14450 | 361825 | 152,219 |
https://mathoverflow.net/questions/361747 | 5 | Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} \sum\_{x\in \mathbb{Z}^{d}}|\psi(x)|^{2}<\infty\}$ with inner product $\langle \psi, \varphi\rangle\_{l^{2}}:= \sum\_{x... | https://mathoverflow.net/users/150264 | Connections between two constructions of infinite dimensional Gaussian measures | The source of the confusion is *not saying* explicitly what are the sets and $\sigma$-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there exists a Gaussian measure $\nu\_C$ with covariance $C$ on $l^2(\mathbb{Z}^d)$ which is compatible with $\mu\_\Lambda$... | 3 | https://mathoverflow.net/users/7410 | 361829 | 152,221 |
https://mathoverflow.net/questions/361799 | 1 | Let $Y\_t,X\_t$ be $(\Omega,\mathcal{F},\mathcal{F}\_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \mathbb{R}^n$ such that
$$
\phi(X\_t)= Y\_t \mathbb{P}-a.s?
$$
More generally, since the continuous image (in the above s... | https://mathoverflow.net/users/36886 | Diffeomorphism for mapping one SDE into another | This is more of a long comment. As far as I know, the first question is a very difficult (and interesting) one. Here are a few examples of diffusions in $\mathbb R^3$ that are not diffeomorphic in your sense, and may give an idea of the difficulties ahead:
1. the standard Brownian motion in $\mathbb R^3$;
2. the Brow... | 2 | https://mathoverflow.net/users/129074 | 361846 | 152,228 |
https://mathoverflow.net/questions/361851 | 1 | Let $a(M)$ be the maximum absolute value of entries of matrix $M\in\mathsf{GL}\_k(\mathbb Z)$.
$M^{-1}\in\mathsf{GL}\_k(\mathbb Z)$ holds.
>
> What is a good upper bound for $|a(M)-a(M^{-1})|$?
>
>
>
I am thinking whether the dependence could be a little smaller than fully exponential in $k$ for $a(M)\cdot a... | https://mathoverflow.net/users/136553 | Upper bounds for difference of entries between matrices and their inverses in $\mathsf{GL}_k(\mathbb Z)$ | For $k=2$, the upper bound is zero.
For $k>2$, there is no upper bound. E.g., let $$M=\pmatrix{1&1&1\cr9&10&11\cr-n&n&3n-1\cr}$$ Then $$M^{-1}=\pmatrix{10-19n&2n-1&-1\cr38n-9&1-4n&2\cr-19n&2n&-1\cr}$$ and $a(M)-a(M^{-1})=(3n-1)-(38n-9)=-(35n-8)$ is unbounded.
| 2 | https://mathoverflow.net/users/158000 | 361854 | 152,230 |
https://mathoverflow.net/questions/361839 | 1 | Suppose that I'm given a sample from time-series $(x\_n)\_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?
So far, everything I've seen is hand-wavy...
| https://mathoverflow.net/users/36886 | Test for OU-Process | $\newcommand\th{\theta}$ $\newcommand\Si{\Sigma}$
The Ornstein--Uhlenbeck (OU) process is a Gaussian process, with the mean and covariance functions being [known functions](https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Mathematical_properties) depending on $s\le4$ unknown real-valued parameters of th... | 3 | https://mathoverflow.net/users/36721 | 361858 | 152,232 |
https://mathoverflow.net/questions/361820 | 1 | This is already posted here <https://math.stackexchange.com/questions/3695584/a-computation-in-a-commutative-frobenius-algebra> but I didn't get any answers.
Given a commutative Frobenius algebra (in the category of vector spaces over $\mathbb C$) $A$ with multiplication $m\colon A\otimes A\to A$, comultiplication $c... | https://mathoverflow.net/users/104719 | A computation in a commutative Frobenius algebra | I don't know what you mean when you ask if the expression simplifies. The generating relations for a commutative Frobenius algebra are equivalent to the fact that the only thing that matters is the diffeomorphism type of the corresponding surface (possibly with boundary) that you draw. (I like to think of this as the i... | 5 | https://mathoverflow.net/users/66405 | 361859 | 152,233 |
https://mathoverflow.net/questions/361661 | 5 | In [this paper](https://core.ac.uk/download/pdf/82458627.pdf), Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:
>
> An augmented differential graded algebra $R$ over $\mathbb{k}$ is *Gorenstein* if $\text{Ext}\_R(\mathbb{k},R)$ is concentrated in a single degree and has ... | https://mathoverflow.net/users/103150 | A generalization of integral Poincaré duality | Prior to Dwyer-Greenlees-Iyengar, Dwyer and myself (independently) defined Gorenstein conditions for group rings over the sphere $S[G]$, i.e., the suspension spectrum of a topological group.
The definition easily extends to the case of a morphism $R\to k$.
In the orientable case the definition goes like this:
... | 3 | https://mathoverflow.net/users/8032 | 361861 | 152,234 |
https://mathoverflow.net/questions/361773 | 1 | Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\iff Rg(x)=g(y)$.
Does $C$ imply $f=\pm g$?
[Update]: The answer is "yes" when $C$ is modified to $\forall x,y\in A, \f... | https://mathoverflow.net/users/158871 | Uniqueness of function with range $\mathbb{S}^2$ under a constraint | You are correct, if you change $C$ so $R$ is $\forall$-quantified, then $C$ implies $f = \pm g$.
Pairs $\{v, -v\}$ have distinct stabilizers under $SO(3)$ so setting $x = y$ we see $f = \pm g$ pointwise. But then if $f(x) = g(x)$ and $f(y) = -g(y)$, picking any $R$ such that $Rf(x) = f(y)$, if we were to have $Rg(x) ... | 0 | https://mathoverflow.net/users/123634 | 361877 | 152,238 |
https://mathoverflow.net/questions/361737 | 15 | I'm trying to learn Stokes data but can't find an example to get my teeth into it.
*Background*. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular holonomic D modules}\ \stackrel{\sim}{\longrightarrow} \ \text{perverse sheaves}$$
which for instance sends the regu... | https://mathoverflow.net/users/119012 | Examples of Stokes data | It helps to understand irregular singularities as the merging of regular singular points, say $$(x^2-a^2)y'+y=0$$ as $a\to0$.
For nonzero $a$ the data is encoded as monodromy (constant) matrices acting on your local solutions, given by the analytic continuation along loops generating the fundamental group.
Parts of ... | 8 | https://mathoverflow.net/users/24309 | 361880 | 152,239 |
https://mathoverflow.net/questions/361867 | 1 | As I was studying the Cauchy's integral formula, I tried to do the integral:
\begin{equation}
I = \int\limits\_{-\infty}^{\infty} \frac{1}{x - a} e^{(i A x^2 + i B x)} dx
\end{equation}
with $A>0, B>0$ and $a > 0$.
Consider an integral on a complex plan:
\begin{equation}
J = \int\limits\_{C + C\_R} \frac{1}{z - a} ... | https://mathoverflow.net/users/158914 | Cauchy's Integral with quadratic exponential term | Let me first remove the $Bx$ term by completing the square,
$$I=\int\limits\_{-\infty}^{\infty} \frac{e^{i A x^2+iBx}}{x - a}\,dx=e^{-iB^2/4A}\int\limits\_{-\infty}^{\infty} \frac{e^{i A x^2}}{x - a-B/2A}\,dx.$$
Mathematica evaluates the Cauchy principal value of the integral in terms of Meijer G-functions,
$$I=-\tfrac... | 3 | https://mathoverflow.net/users/11260 | 361889 | 152,241 |
https://mathoverflow.net/questions/361891 | 3 | Let $(M, \omega)$ be a symplectic manifold. A vector field $V: M \to TM$ is Liouville if $L\_{X}
\omega=\omega$. The existence of a Liouville vector field implies that $(M, \omega)$ is exact: the one-form $\lambda = i\_V \omega$ satisfies $d\lambda=d\circ i\_V\omega = L\_V\omega=\omega$. In particular, there is no Liou... | https://mathoverflow.net/users/11028 | Existence of Liouville vector fields on symplectic manifolds | If the symplectic form integrates to a nonzero quantity on a compact surface in your manifold, it is not exact. For example, on $M=S^2\times S^1\times [0,1]$ with symplectic form $dA\_{S^2} + d\vartheta \wedge dt$.
| 2 | https://mathoverflow.net/users/13268 | 361893 | 152,243 |
https://mathoverflow.net/questions/361871 | 2 | I asked this question in [MSE](https://math.stackexchange.com/questions/3694921/the-number-of-unitary-circulant-matrices-over-a-finite-field-mathbbf-q2) few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}\_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $... | https://mathoverflow.net/users/131819 | The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$ | Let $\tau$ denote the permutation matrix corresponding to $(1,2,\ldots,n)$. Consider it first as an element of $\mathrm{M}\_n(q^2)$.
This matrix has minimal polynomial equal to $X^n-1$, which is equal to its characteristic polynomial. It is therefore cyclic, and its centralizer is isomorphic to $\mathbb{F}\_{q^2}$-a... | 4 | https://mathoverflow.net/users/14443 | 361895 | 152,244 |
https://mathoverflow.net/questions/361824 | 7 | I've encountered the following sum:
$$
s\_n = \sum\_{j=1}^n {n \brace j}(\alpha n)\_j \beta^j.
$$
Here $\alpha$ and $\beta$ are positive constants, $(\alpha n)\_j$ is a falling power, and ${n \brace j}$ is the Stirling number of second kind. My computations suggests that $s\_n$ grows like $n^n C^n$ (where the constant... | https://mathoverflow.net/users/21690 | Asymptotics of sum involving Stirling numbers | The sum in question can be written (using Latin letters instead of Greek) as
$$ S\_n(a,b):= \sum\_{k=0}^n {n \atopwithdelims \{ \} k} k! \binom{na}{k} b^k $$
where the round brackets denotes an ordinary binomial. It will be shown as $n \to \infty$
$$ (1)\quad S\_n(a,b) \sim \frac{n!}{2}\,\exp{\Big(n\big( h(u\_0) + \fr... | 4 | https://mathoverflow.net/users/121836 | 361905 | 152,247 |
https://mathoverflow.net/questions/360323 | 1 | I am working on a problem in commutative ring theory, that deals with $p$-adic valuations. This leads to a number theoretical question that I want to explain in the following.
Let $n \in \mathbb{N}$ and $k$ an integer $\leq n/2$. Then, by the well-known result of Sylvester, there is an integer in $\{n, n-1,..., n-k+1... | https://mathoverflow.net/users/120027 | Strengthening Sylvester's theorem | With motivation from the original poster, a key idea from Sylvester, and technical inspiration from Iosif Pinelis, I contribute an observation that helps toward an answer.
I use m instead of n and n instead of k. I start with the inequality that p! is strictly less than 3^p for p less than 7, and less than (p/2)^p fo... | 1 | https://mathoverflow.net/users/3402 | 361906 | 152,248 |
https://mathoverflow.net/questions/361863 | 5 | Let $G=\mathrm U\_n$ or $\mathrm{GL}\_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n\_1+\dots+n\_k$. Is the normalizer $N=N\_G(H)$ computed anywhere in the literature?
I guess, but haven’t proved, that it is generated by $H$ and the permutations (“transpositions”) exchanging... | https://mathoverflow.net/users/19276 | The normalizer of block diagonal matrices | By [request](https://mathoverflow.net/questions/361863/the-normalizer-of-block-diagonal-matrices#comment912330_361863), from [my comment](https://mathoverflow.net/questions/361863/the-normalizer-of-block-diagonal-matrices#comment912275_361863): Your guess is correct. Because there are clearly elements in the normaliser... | 3 | https://mathoverflow.net/users/2383 | 361907 | 152,249 |
https://mathoverflow.net/questions/361273 | 3 | Let $M$ be a symplectic manifold with a symplectic action of a Lie algebra $\mathfrak{g}$. I am interested whether there exists a $\mathfrak{g}$-invariant symplectic connection on $M$. Where does the obstruction live and how to construct it?
| https://mathoverflow.net/users/109520 | Obstruction to the existence of an invariant symplectic connection | Let $\nabla$ be any symplectic connection. It is easy to check that the map $\mathfrak{g} \to \Omega^1 (M, \operatorname{End}\_{\omega} (TM))$ given by $X \to \mathcal{L}\_X (\nabla)$ is a Chevalley-Eilenberg 1-cocycle. Thus, we have an element in $H^1 (\mathfrak{g}, \Omega^1 (M, \operatorname{End}\_{\omega} (TM))$. Th... | 0 | https://mathoverflow.net/users/109520 | 361908 | 152,250 |
https://mathoverflow.net/questions/361784 | -1 | Is there an infinite cardinal $\kappa$ with a collection of subsets ${\cal E}$ of $\kappa$ with the following properties?
1. $\bigcup {\cal E} = \kappa$,
2. $e \neq f \in {\cal E}$ implies $|e \cap f|=1$
3. $|e|<\kappa$ for all $e\in {\cal E}$, and
4. not all members of ${\cal E}$ have the same cardinality.
| https://mathoverflow.net/users/8628 | Infinite complete linear hypergraphs with edges of different sizes | Partition $\omega \setminus \{0\}$ into infinitely many finite sets $I\_n$ such that $\vert I\_n \vert=n$. Define $\mathcal{E}:=\{I\_n \cup \{0\} \colon n \in \omega\}$.
| 2 | https://mathoverflow.net/users/134910 | 361917 | 152,252 |
https://mathoverflow.net/questions/361932 | 6 | this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system:
$$\Theta(\tau, z) = q^{1/12}(e^{\pi iz} - e^{-\pi iz}) \prod\_{n\ge 1} (1-q^n e^{2\pi iz})(1-q^ne^{-2\pi i z})$$
where $q=e^{2\pi i \ta... | https://mathoverflow.net/users/120548 | Jacobi forms and Kato's modular units | The triple product in the Jacobi theta function $\vartheta(\tau,z)$ can be rewritten
\begin{equation\*}
\prod\_{n \geq 1} (1-q^n) \prod\_{n \geq 0} (1-q^n e^{2\pi i(z+\frac{\tau}{2}+\frac12)}) \prod\_{n \geq 1} (1-q^n e^{-2\pi i(z+\frac{\tau}{2}+\frac12)})
\end{equation\*}
($2\pi i n$ should be replaced by $2\pi iz$ in... | 5 | https://mathoverflow.net/users/6506 | 361963 | 152,263 |
https://mathoverflow.net/questions/360876 | 8 | Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I\_X$ be the defining ideal. I am interested in knowing when:
>
> The syzygies of $I\_X$ contains no linear forms. Since we are in $\mathbb P^2$, this just says that the Hilbert-Burch matrix contains no (non-zero) linear entries. $(\*... | https://mathoverflow.net/users/2083 | Degrees of syzygies of points in $\mathbb P^2$ | I asked David Eisenbud and was told about Exercises 12 and 13 of Chapter 3 in his book "Geometry of Syzygies". Putting together, they show that for a generic set of $n$ points $X$ in $\mathbb P^2$, the syzygies of $I\_X$ has no linear forms if and only if $n= \binom{2s+1}{2}+s$, for some positive integer $s$. It is unc... | 3 | https://mathoverflow.net/users/2083 | 361978 | 152,267 |
https://mathoverflow.net/questions/361957 | 2 | Suppose I have $n$ independent 0-1 random variables $X\_1, \cdots, X\_n$ and I want to show a concentration of $X = \sum\_i X\_i$.
I can use either the Chernoff bound or the Hoeffding bound.
Suppose $E[X] = O(1)$. Then, I should use the Chernoff bound which will give me Poissonian tails. On the other hand, if $E[X]... | https://mathoverflow.net/users/128129 | Something between the Chernoff and Hoeffding bounds | $\newcommand\ep{\varepsilon}$ $\newcommand\si{\sigma}$ $\newcommand\Ga{\Gamma}$ $\newcommand\tPi{\tilde\Pi}$
It follows from Theorem 2.1 of [this paper](https://projecteuclid.org/download/pdfview_1/euclid.aihp/1388545263) or of its [better version](https://arxiv.org/pdf/0902.4058v1.pdf) that for a large class, say $\... | 2 | https://mathoverflow.net/users/36721 | 361992 | 152,270 |
https://mathoverflow.net/questions/361971 | 1 | Consider an irreducible non-Abelian subgroup $\mathrm{H}$ of group of unitary matrices $\mathrm{U}\_n(\mathbb{C})$, that contains the subgroup of diagonal matrices. Does there exist any result regarding the properties or structures of $\mathrm{H}$? The final motivations is to answer this [question](https://mathoverflow... | https://mathoverflow.net/users/152974 | Irreducible non-Abelian subgroup of $\mathrm{U}_n(\mathbb{C})$, containing diagonal matrices | 1) First, let $H$ be a closed connected subgroup with this property. Let $D$ the diagonal group in $U(n)$; denote Lie algebras with Gothic letters. Then $$\mathfrak{u}(n)=\mathfrak{d}\oplus \bigoplus\_{j<k}\mathbf{R}i(E\_{jk}+E\_{kj})\oplus\mathbf{R}(E\_{jk}-E\_{kj}).$$
Let $e\_j:D\to\mathbf{C}^\*$, $d\mapsto d\_j$ b... | 3 | https://mathoverflow.net/users/14094 | 361996 | 152,272 |
https://mathoverflow.net/questions/361986 | 4 | I found the interesting inequality when I study hypergraph 2-coloring
$$\sum\_{i+j=k} \binom{r-1}{i}\binom{r-1}{j}(1-p)^i(1+p)^j \leq \binom{2r-2}{k}$$
$0\leq i, j < r$, $0\leq p \leq 1$. I want to know how to proof it.
| https://mathoverflow.net/users/153948 | Combined identity perturbation | The left-hand side is the coefficient of $x^k$ in
$$
\left(1+(1-p)x\right)^{r-1}\left(1+(1+p)x\right)^{r-1}=\left(1+2x+(1-p^2)x^2\right)^{r-1}\ .
$$
This coefficient can be obtained via the Multinomial Theorem as
$$
\sum\_{a,b}\mathbf{1}\left\{
\begin{array}{c}
a,b\ge 0 \\
a+b\le r-1 \\
a+2b=k
\end{array}
\right\}\frac... | 2 | https://mathoverflow.net/users/7410 | 362001 | 152,274 |
https://mathoverflow.net/questions/361969 | 0 | $\def\Fin{\text{Fin}\_\*}
\def\Set{\text{Set}\_\*}
\def\dd{\mathop{\diamond\_\land}}$
>
> The present question is intimately related to [another question](https://mathoverflow.net/questions/361690/finitary-monads-on-operatornameset-are-substitution-monoids-finitary-monad).
>
>
>
Let $\Fin$ be the category of ... | https://mathoverflow.net/users/7952 | Substitution structure on pointed sets | In the same vein as my response to your other question, if pointed finite sets are an eleutheric system of arities, Lawvere theories over that system of arities will be equivalent to monads in a certain monoidal category. This is in section 11 of Lucyshyn-Wright [here](https://arxiv.org/pdf/1511.02920.pdf).
Edit: I b... | 2 | https://mathoverflow.net/users/75783 | 362002 | 152,275 |
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