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https://mathoverflow.net/questions/403617 | 1 | Given a complex vector space $V$ of dimension $n>2$, the universal conic $\mathcal C$ of $\mathbb P(V^\*)$ is a divisor in $\mathbb P(t^\*\mathcal E\_3)\overset{\pi}{\rightarrow}\mathbb P({\rm Sym}^2\mathcal E\_3^\*)\overset{t}{\rightarrow} Gr(3,V)$ where $\mathcal E\_3$ is the natural rank $3$ quotient bundle on $Gr(3... | https://mathoverflow.net/users/85595 | Class of the universal conic | The projective bundle $t \colon \mathbb{P}(\mathrm{Sym}^2\mathcal{E}\_3^\*) \to \mathrm{Gr}(3,V)$ comes with the tautological subbundle
$$
\mathcal{O}\_t(-1) \hookrightarrow t^\*\mathrm{Sym}^2\mathcal{E}\_3^\*.
$$
This embedding gives a global section in
$$
H^0(\mathbb{P}(\mathrm{Sym}^2\mathcal{E}\_3^\*),
\mathcal{O}\... | 1 | https://mathoverflow.net/users/4428 | 403622 | 165,575 |
https://mathoverflow.net/questions/403620 | 4 | It can be proven without any form of infinite choice that the product of two compact spaces (and thus any finite product) is compact, while on the other hand, it is well known that the general form of Tychonoffs theorem implies the axiom of choice.
So a general formulation of this question would be: Is there a proof ... | https://mathoverflow.net/users/174368 | Can Tychonoffs theorem for a countable number of spaces be proven with ZF plus the axiom of (countable) dependent choice? | From Herrlich's "The Axiom of Choice", Proposition 4.72 reads as follows:
>
> Each of the following conditions implies the subsequent ones:
>
>
> 1. $\sf DC$.
> 2. Countable products of compact spaces are compact.
> 3. $\sf CC$.
>
>
>
He goes on to remark that (2) is provable from countable choice + the Bool... | 6 | https://mathoverflow.net/users/7206 | 403623 | 165,576 |
https://mathoverflow.net/questions/403614 | 4 | Most often than not, the sheaves appearing in algebraic geometry (with the Zariski topology) are $\mathcal{O}\_X$-modules, instead of simple abelian sheaves.
Now, when dealing with topological spaces (for example, in Verdier duality) and in étale cohomology, it seems that abelian sheaves have the main role.
I wonde... | https://mathoverflow.net/users/131975 | Why abelian sheaves instead of $\mathcal{O}_X$-modules in topology and étale stuff? | We come to the question of what $\mathcal{O}\_X$ should mean, when $X$ is a manifold or topological space. If $\mathcal{O}\_X$ is smooth (or continuous) real valued functions, then we will always wind up studying $H^{\ast}(X, \mathbb{R})$. There is nothing wrong with this and, in fact, de Rham cohomology makes lots of ... | 4 | https://mathoverflow.net/users/297 | 403631 | 165,581 |
https://mathoverflow.net/questions/403643 | 3 | **Context.**
*Space of Lipschitz functions.* Denote by $Lip\_0(D)$ the space of all Lipschitz functions on a metric space $D$ vanishing at some base point $e \in D$. The norm in $Lip\_0$ is defined as follows
$$
\|f\|\_{Lip\_0} := Lip(f),
$$
where $Lip(f)$ denotes the Lipschitz constant of $f$.
*Radon-Nikodym pro... | https://mathoverflow.net/users/160379 | Does the space of Lipschitz functions have the Radon-Nikodym property? | Let $X$ be a metric space consisting of a countable set of points, the distance between any two of which is $2$, together with one additional point $e$ whose distance to any of the other points is $1$. Then ${\rm Lip}\_0(X)$ is isometrically isomorphic to $l^\infty$, which fails the RNP.
Another example: let $X = [0,... | 8 | https://mathoverflow.net/users/23141 | 403648 | 165,584 |
https://mathoverflow.net/questions/403629 | 20 | What sort of bounds (explicit of preference) can one give for
$$\int\_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$
Some obvious points:
* One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \leq C \log t$ (with $C\leq 42.9$ for $t\geq 2$) and deduce a bound of the form $\leq K T (\log T)^2$ on the integral ... | https://mathoverflow.net/users/398 | Bound on $L^2$ norm of $1/\zeta(1+i t)$? | Problems like this are classical (as noted in Terry's answer), and have been considered more recently with attention to uniformity in the moments. To give a quick indication, one can show that
$$
\zeta(1+it) \approx \prod\_{p\le y} \Big( 1-\frac{1}{p^{1+it}}\Big)^{-1}
$$
for all but a set of values $T\le t\le 2T$ of ... | 13 | https://mathoverflow.net/users/38624 | 403652 | 165,586 |
https://mathoverflow.net/questions/349409 | 6 | Let $\mathscr{S}$ be a [limit sketch](https://ncatlab.org/nlab/show/sketch) in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\mathcal{E} \to \mathbf{Set}$ which send the cones in $\mathscr{S}$ to limit cones) is ... | https://mathoverflow.net/users/2841 | Universal property of the cocomplete category of models of a limit sketch | Probably the earliest reference is Theorem 2.5 in
>
> A. Pultr, **The right adjoints into the categories of relational systems**, Reports of the Midwest Category
> Seminar IV. Springer, Berlin, Heidelberg, 1970
>
>
>
Pultr's "relational theories" are exactly small realized limit sketches.
Remark: I have rece... | 4 | https://mathoverflow.net/users/2841 | 403653 | 165,587 |
https://mathoverflow.net/questions/403635 | 5 | $\newcommand{\w}{\omega}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}{\mathfrak x}\newcommand{\cov}{\mathrm{cov}}\newcommand{\lac}{\mathrm{lac}}$[Taras Banakh](https://mathoverflow.net/users/61536/taras-banakh) and ... | https://mathoverflow.net/users/43954 | Bounds for a small cardinal | **EDIT:** In my original post, I showed that $\mathrm{cov}(\mathcal N) > \mathfrak{x}\_{lac}$ in the random model. Upon further reflection, I think we can prove a stronger result, with an arguably easier (but completely different) proof:
*Theorem:* $\mathfrak{x}\_{lac} \leq \mathrm{non}(\mathcal N)$.
Note that this... | 4 | https://mathoverflow.net/users/70618 | 403658 | 165,589 |
https://mathoverflow.net/questions/403465 | 1 | I am looking for ways to evaluate *exactly* (i.e. *analytically* or *semi-analytically*) integrals of the type:
$$
\int\_{-\infty}^{+\infty}B\_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,
$$
where $B\_i^k$ is a spline of order $k$, an element of the B-Spline basis for the linear space of splines of order $k$ on knots $\... | https://mathoverflow.net/users/156188 | Integrating a B-Spline basis function with respect to the standard normal PDF | Since a B-spline is a piecewise polynomial function, the question is whether there exists an exact equality formula for the integral $\int\_{-a}^{b}u^pe^{-u^2/2}du$. This integral equals an elementary function of $a$ and $b$ for $p$ an odd integer, while for $p$ an even integer it contains error functions. In general t... | 0 | https://mathoverflow.net/users/11260 | 403661 | 165,591 |
https://mathoverflow.net/questions/403153 | 10 | For a partition $\lambda$, let $P\_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p\_\lambda=p\_{\lambda\_1}p\_{\lambda\_1}\cdots p\_{\lambda\_k}$ be the power-sum symmetric functions.
It is known that the space $\Gamma$ spanned by the $P\_\lambda$ for $\lambda$ with... | https://mathoverflow.net/users/160416 | $2$-adic valuation of Schur $P$-functions in the power-sum basis | Here is a proof of the generalization suggested by Richard Stanley in the
comments and even of a more general result (with "odd" replaced by "not
divisible by a given prime $q$"). It is completely different from the argument
I sketched in the comments, and is completely elementary (using no Macdonald
polynomials). Unfo... | 3 | https://mathoverflow.net/users/2530 | 403672 | 165,593 |
https://mathoverflow.net/questions/403670 | 2 | Let $\mathcal{A}=\{A\_1, A\_2, \ldots, A\_m\}$ be a uniformly random set partition of $[n]$.
What can we say about $||\mathcal{A}||\_2 = \sqrt{\sum\_{i=1}^m |A\_i|^2}$? It is clearly upper bounded by $n$, but since the ``typical'' size of these $A\_i$ is more like $\log(n) - \log\log(n)$, it seems reasonable to expec... | https://mathoverflow.net/users/58551 | "Shape"/"norm" of a uniformly random set partition | Since $E[\|\mathcal{A}\|\_{2}^{2}]$ has no square roots, a good strategy for this problem will be to first compute $E[\|\mathcal{A}\|\_{2}^{2}]$ and then deal with $E[\|\mathcal{A}\|\_{2}]$ only after we are comfortable with $E[\|\mathcal{A}\|\_{2}^{2}]$.
We compute $$E[\|\mathcal{A}\|\_{2}^{2}]=B\_{n}^{-1}\sum\_{P\i... | 2 | https://mathoverflow.net/users/22277 | 403677 | 165,597 |
https://mathoverflow.net/questions/403502 | 1 | Write $A=(x\_{ij})$ for the generic matrix (comprised of indeterminates) defined over $\mathbb Z[x\_{11},\dots,x\_{nn}]$. In [their constructive commutative algebra book](https://arxiv.org/abs/1605.04832v3), Lombardi and Quitte write that the determinant of the family $(e\_1,Ae\_1,\dots,A^{n-1}e\_1)$ is nonzero. Hence ... | https://mathoverflow.net/users/69037 | Properties of the generic matrix - struggles with constructive proofs | Let me answer both of your explicitly asked questions in detail; if you have any further questions on the proof (which is indeed fast-going and slightly handwavy), please add them to your post.
>
> **Question 1.** Why are the $s\_1, s\_2, \ldots, s\_n$ algebraically independent over $\mathbb{Z}$ ? (I am using the n... | 1 | https://mathoverflow.net/users/2530 | 403678 | 165,598 |
https://mathoverflow.net/questions/403507 | 1 | **I asked this question in <https://math.stackexchange.com/q/4236870/528430>, but did not get any help.**
I got stuck with the following while going through the proof of [Lemma 3.21](https://drive.google.com/file/d/1FqoY7rZhEV6miICQVGX-5a9M5b3SoTIB/view?usp=sharing) from the book 'Ergodic Theory: Independence and Dic... | https://mathoverflow.net/users/83956 | The mean ergodic theorem for weakly mixing extension | I suspect that your confusion stems from misinterpreting the norm on $L^2(X|Y)$ in your definition 2. You should note that $\mathbb{E}\_Y$ takes $L^\infty(X)$ to $L^\infty(Y)$ and the new norm taken on $L^\infty(X)$ is with respect to the $L^\infty$-norm on $L^\infty(Y)$, not the $L^2$-norm. That is $\|f\|:=\|\mathbb{E... | 1 | https://mathoverflow.net/users/89334 | 403695 | 165,605 |
https://mathoverflow.net/questions/403702 | 1 | Let $X$ be a smooth projective variety, let $E$ be a vector bundle of rank $4$ on $X$ and let $L$ be a line budnle on $X$. Consider the projectivization $\mathbb{P}\_X(E):=\mathrm{Proj}Sym(E^\*)$ of the vector bundle $E$. Denote the projection $\mathbb{P}\_X(E)\to X$ by $\pi$ and the Serre sheaf of the projectivization... | https://mathoverflow.net/users/98256 | Singular locus of a family of cubic surfaces | The discriminant of a degree $d$ polynomial in $n$ variables has degree $n (d-1)^{n-1}$, so the discriminant of a cubic in four variables is $4 \cdot 2^3 = 32$.
The discriminant is, by construction, invariant under $SL\_4$. If we look at scalars in $GL\_4$, they act on cubic polynomials by multiplication by the inver... | 3 | https://mathoverflow.net/users/18060 | 403705 | 165,606 |
https://mathoverflow.net/questions/403696 | 0 | **EDIT**: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\backslash \bar\Omega$ and vanishes on $\partial \Omega$. Let us also assume that $f$ vanishes at infinity.
**What ... | https://mathoverflow.net/users/16183 | Harmonic functions in infinite domain in Euclidean space | Assuming "vanish at infinity" means that
>
> for every $\epsilon > 0$ there exists a sufficiently large ball $B\_\epsilon$ such that $\big|f|\_{\mathbb{R}^n \setminus B\_\epsilon}\big| < \epsilon$
>
>
>
then you can just apply the maximum principle to $B\_\epsilon \setminus \Omega$ and conclude that $f$ is bou... | 2 | https://mathoverflow.net/users/3948 | 403709 | 165,608 |
https://mathoverflow.net/questions/403289 | 4 | One of the generalizations of algebraic geometry is provided by the theory of semiring schemes, cf. [Lorscheid 2012](https://arxiv.org/abs/1212.3261). The theory follows the same set up of scheme theory, but we use semirings instead of rings.
Given a semiring $R$, we have a semiringed space $\mathrm{Spec}(R)$, define... | https://mathoverflow.net/users/130058 | Are affine semiring schemes equivalent to semirings? | The thesis Algebraic geometry over semi-structures and hyper-structures of characteristic one by Jaiung Jun [accessible here](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/37850/JUN-DISSERTATION-2015.pdf?sequence=1&isAllowed=y) gives half of the proof in proposition 2.2.6, the remaining half being easy.
... | 4 | https://mathoverflow.net/users/130058 | 403710 | 165,609 |
https://mathoverflow.net/questions/403659 | 5 | I'm interested in Laplace Beltrami operators $$-\Delta\_g:\ \ D(-\Delta\_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $g$ on $M$.
For any other smooth metric $\widetilde g$ on $M$, we can identify the square integrable functions with r... | https://mathoverflow.net/users/366530 | Do Laplace-Beltrami eigenfunctions vary continuously with the metric? | EDITED: Added clarification, as pointed out by @TerryTao.
Let $g\_1$, $g\_2$ be Riemannian metrics and $\Delta\_1$, $\Delta\_2$ their respective Laplacians. Let $\lambda\_1$ be an eigenvalue of $\Delta\_1$ and $\lambda\_2$ an eigenvalue of $\Delta\_2$.
I think what can be proved is the following: Given an eigenfunc... | 3 | https://mathoverflow.net/users/613 | 403715 | 165,612 |
https://mathoverflow.net/questions/370814 | 2 | For $u \in L^\infty(\mathbb R)$ and $\eta\_\epsilon$ [mollifier](https://math.stackexchange.com/questions/1366268/standard-mollifier-comparing-the-definition-in-evans-and-wiki), it is well-known that for the (distributional) derivative it holds that $(u \ast \eta\_\epsilon)' = u'\ast \eta\_\epsilon$.
Is it also true fo... | https://mathoverflow.net/users/122620 | Fractional Laplacian and convolution $(-\Delta)^\alpha (u \ast \eta_\epsilon) = (-\Delta)^\alpha u \ast \eta_\epsilon$? | Answering the question asked in a comment recently: you can read more about distributional definition of the fractional Laplacian in the paper by Luis Silvestre [1] (or in his PhD thesis, if I remember correctly), in an excellent book by Stefan Samko [2], or in Section 5 of my survey [3]. I do not think any of these re... | 1 | https://mathoverflow.net/users/108637 | 403718 | 165,613 |
https://mathoverflow.net/questions/403674 | 2 | I seem to recall that the prime number theorem (PNT) is **equivalent** to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see <https://math.stackexchange.com/questions/1379583/why-is-zeta1it-neq-0-equivalent-to-the-prime-number-theorem> or <https://math.stackexchange.com/que... | https://mathoverflow.net/users/112504 | How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions? | **Theorem**: *Fix a positive integer $m$. The following two conditions are equivalent*:
(1) $L(1,\chi) \not= 0$ *for all nontrivial Dirichlet characters* $\chi \bmod m$.
(2) *For all $a \in (\mathbf Z/m\mathbf Z)^\times$, the set of primes $\{p \equiv a \bmod m\}$ has Dirichlet density $1/\varphi(m)$.*
*Proof*. W... | 9 | https://mathoverflow.net/users/3272 | 403725 | 165,615 |
https://mathoverflow.net/questions/403665 | 4 | Given two positive constants $c\_1,c\_2$ and two independent standard normal random variables $a,b$, how to calculate the following expected value
$$
\mathbb{E}\left[\frac{a^2}{c\_1a^2+c\_2b^2}\right]
$$
Thank you.
| https://mathoverflow.net/users/366173 | Expected value of a ratio of squared normal and linear combination of squared normal | $\newcommand\E{\mathscr E}$The expectation in question is
$I/c\_1$, where
$$I:=E\frac{a^2}{a^2+cb^2},\quad c:=c\_2/c\_1>0.$$
In turn, using polar coordinates, we get
$$I=\frac1{2\pi}\int\_0^{2\pi}\frac{\cos^2 t\, dt}{\cos^2 t+c\sin^2 t},\quad c:=c\_2/c\_1>0.$$
Further, writing
$$I=\frac4{2\pi}J$$
for
$$J:=\int\_0^{\pi/... | 4 | https://mathoverflow.net/users/36721 | 403735 | 165,620 |
https://mathoverflow.net/questions/403720 | 12 | The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the product of two Schur polynomials into a linear combination of Schur polynomials. In both cases, we label the components of ... | https://mathoverflow.net/users/12897 | Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule | Please excuse that I answer with a link, I only have a phone right now.
<http://www.findstat.org/MapsDatabase/Mp00192/>
| 4 | https://mathoverflow.net/users/3032 | 403740 | 165,622 |
https://mathoverflow.net/questions/403736 | 4 | Let $k$ be a field and $M$ the subgroup of $\operatorname{GL}(n,k)$ consisting of permutation matrices. We say a subgroup $G$ of $M$ linearly independent if $G$ is a linearly independent subset of $M\_n(k)$. It is well-known that there is an isomorphism between $M$ and the full symmetric group $S\_n$ with respect to th... | https://mathoverflow.net/users/134942 | Linearly independent subgroups of permutation matrices | This just deals with the complex case: Here is an initial remark: Since we know that the character afforded by the natural permutation representation of $S\_{n}$ is the sum of the trivial character an a degree $n-1$ irreducible, we know that the $n \times n$ permutation matrices span a space of dimension $1+(n-1)^{2}.$... | 1 | https://mathoverflow.net/users/14450 | 403751 | 165,624 |
https://mathoverflow.net/questions/403685 | 3 | Let $\mathcal P(\mathbb R)$ be the set of probability measures. Set for $\mu,\nu\in\mathcal P(\mathbb R)$
$$d(\mu,\nu) := \inf\left\{\varepsilon>0:~ F\_{\mu}(x-\varepsilon)-\varepsilon \le F\_{\nu}(x)\le F\_{\mu}(x+\varepsilon)+\varepsilon,~ \forall x\in\mathbb R\right\}$$
and
$$\rho(\mu,\nu) := \sup\left\{\int f... | https://mathoverflow.net/users/261243 | On the weak convergence of probability measures on $\mathbb R$ | For every $\varepsilon\in(0,1)$ there are $\mu$, $\nu$ such that $d(\mu,\nu)=\varepsilon$ and $\rho(\mu,\nu)=\varepsilon^2$.
For example, $\mu=\varepsilon\delta\_0 + (1-\varepsilon)\delta\_2$ and
$\nu=\varepsilon\delta\_\varepsilon + (1-\varepsilon)\delta\_2$.
Hence there is no $C>0$ such that $d(\mu,\nu)\leq C \rh... | 3 | https://mathoverflow.net/users/95282 | 403756 | 165,627 |
https://mathoverflow.net/questions/403758 | 0 | I think that Minimum weight vertex cover problem is NP-easy. However I don't know how to prove that. Does anyone know how to prove it?
| https://mathoverflow.net/users/178444 | Is minimum weight vertex cover problem NP-easy? | Consider the following decision problem:
>
> Given a vertex-weighted graph $G$, a subset $S$ of its vertices, and a number $k$, is there a vertex cover of $G$ extending $S$ with total weight at most $k$?
>
>
>
This problem is clearly in NP. Let us see how to solve the minimum weight vertex cover problem using ... | 1 | https://mathoverflow.net/users/30186 | 403761 | 165,630 |
https://mathoverflow.net/questions/403750 | 2 | I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \infty$
The calculation itself is an "exercise" and in particular it is surely well-known. I would like to know how number theoris... | https://mathoverflow.net/users/99732 | The growth of the number of Fano complete intersection families | Hardy and Ramanujan obtained the asymptotic $$P(n) \approx \frac{1}{4n \sqrt{3}} e^{ \pi \sqrt{\frac{2n}{3}}}$$ which can be summarized as saying that $P(n)$ grows roughly as the exponential of the square root of $n$.
Summing from $1$ to $n$ clearly increases the asymptotic by a factor of at most $n$, so we can still... | 2 | https://mathoverflow.net/users/18060 | 403764 | 165,631 |
https://mathoverflow.net/questions/403745 | 1 | Let $k$ be a field of characteristic zero, for example $k=\mathbb{R}$ or $k=\mathbb{C}$.
Of course, $k(x^2,x^3)=k(x)$, since $x=\frac{x^3}{x^2}$.
Let $f\_1,\ldots,f\_n,g\_1,\ldots,g\_m \in k[x]$, $n,m \geq 1$.
Denote: $F(T)=f\_nT^n+\cdots+f\_1T+x^2$
and $G(T)=g\_mT^m+\cdots+g\_1T+x^3$.
Let $h \in k[x]$.
>
>... | https://mathoverflow.net/users/72288 | A variation on $k(x^2,x^3)=k(x)$ | Such $F, G$ do exist, unless I missed a criterion.
One possibility is to take $F(T)$ a polynomial such that $F(h)$ is never zero, such as $T^3 + x^2$, and then $G(T) = x F(T)$. Then you can use the same $x = G(T)/F(T)$ trick.
| 1 | https://mathoverflow.net/users/18060 | 403767 | 165,632 |
https://mathoverflow.net/questions/403699 | 0 | For any simple, undirected graph $G=(V,E)$, we denote by $\chi(G)$ the smallest cardinal $\kappa$ such that there is a coloring $c:V \to \kappa$.
We say that $v\neq w\in V$ are *incompatible* if $\{v,w\}\notin E$, and for any coloring $c: V\to \chi(G)$ we have $c(v) \neq c(w)$.
It is easy to see that if $v\neq w\in... | https://mathoverflow.net/users/8628 | "Incompatible" pairs with respect to graph coloring | If $R$ is a poset, then a function $C:R\rightarrow R$ is said to be a closure operator if $r\leq C(r)=C(C(r))$ and $(r\leq s)\rightarrow(C(r)\leq C(s))$ whenever $r,s\in R$. We say that a subset $C\subseteq R$ is a closure system if for all $r\in R$, there is a least $s\in C$ where $r\leq s$. If $R$ is a complete latti... | 2 | https://mathoverflow.net/users/22277 | 403771 | 165,633 |
https://mathoverflow.net/questions/403763 | 2 | A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. if there exists another $\epsilon'$ satisfying the counit axiom, then since
$$
\epsilon(a) = \epsilon(a\_{(1)}\epsilon'(a... | https://mathoverflow.net/users/352001 | What is a coalgebra? | Is it how we define a unital algebra? I’d define a unital algebra as coming endowed with a unit, rather than just asserting one exists. For one reason, then the natural notion of map is a unital map. But at any rate there’s just no distinction here between the definitions algebras and coalgebras, people who define alge... | 8 | https://mathoverflow.net/users/22 | 403776 | 165,636 |
https://mathoverflow.net/questions/403598 | 1 | By the central limit theorem (CLT), I mean the Lindeberg-Lévy CLT that says if $X\_1,X\_2,\ldots$ are i.i.d. random variables with $\mathbf{E}[X\_1] = 0$ and $\mathbf{E}[X\_1^2] = 1$, then
$$ \frac{X\_1+\cdots+X\_n}{\sqrt{n}}$$
converges in distribution to the standard normal random variable.
The de Moivre-Laplace th... | https://mathoverflow.net/users/365252 | Where should I submit a derivation of the CLT from the de Moivre-Laplace theorem? | I would be personally interested to see your proof. You should compare it to the Lindeberg proof that also does not use characteristic functions but replaces the variables one by one by Gaussians. You can find an exposition of this proof in the book [1] or in [Chin - A Short and Elementary Proof of the Central Limit Th... | 5 | https://mathoverflow.net/users/7691 | 403777 | 165,637 |
https://mathoverflow.net/questions/403697 | 3 | This question comes from P13 and P17 of the book [Andrei N.Borodin and Paavo Salminen](https://books.google.de/books?hl=en&lr=&id=8cZrllyXfOUC&oi=fnd&pg=PR9&dq=Andrei%20N.Borodin%20and%20Paavo%20Salminen&ots=VDvtxpIWm1&sig=_C4_7rZfM7zYWLroOkzG4NVQjyM&redir_esc=y#v=onepage&q=Andrei%20N.Borodin%20and%20Paavo%20Salminen&f... | https://mathoverflow.net/users/147009 | How to get speed measure $m(dx)$, scale function $s$, and killing measure $k(dx)$ of a diffusion from the infinitesimal generator? | Intuitively, you are equating coefficients in the two different representations of the generator:
$$
{1\over m(x)}\left[\left({f'(x)\over s(x)}\right)'-k(x)f(x)\right] = {1\over 2}a(x)^2f''(x)+b(x)f'(x)-c(x)f(x).
$$
The left side expands out to
$$
{1\over m(x)}\left[{f''(x)\over s(x)}-{s'(x)f'(x)\over s(x)^2}-k(x)f(x)\... | 2 | https://mathoverflow.net/users/42851 | 403781 | 165,639 |
https://mathoverflow.net/questions/403762 | 1 |
>
> Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that
>
> \begin{eqnarray}
> \left(\frac{1}{|B|}\int\_{B}|\triangledown u|^p dx\right)^{1/p}\leq C\left(\frac{1}{|2B|}\int\_{2B}|\triangledown... | https://mathoverflow.net/users/241460 | How to prove the reverse Hölder inequality for Laplace equations? | The inequality is scale invariant and holds for a ball of any radius. It follows by a standard argument that is the inductive step in what's known as Moser iteration.
The constant $C$ below can change from line to line but always depends only on the dimension. Let $B = B(0,r)$ and $2B = B(0,2r)$. Let $\chi$ be a smoo... | 3 | https://mathoverflow.net/users/613 | 403785 | 165,640 |
https://mathoverflow.net/questions/403726 | 1 | Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega\_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B\_{\epsilon/2}(0)$). Is it true that
$$\frac{1}{|\Omega\_\epsilon|}\int\_{\Omega\_\epsilon} \phi (-\Delta)^s u dx - \left( \frac{1}{|\Omega\_\epsi... | https://mathoverflow.net/users/122620 | Averaging and fractional Laplacian | (I believe this may be too basic for this site, but too long for a comment.)
All that we need to assume is that $\phi$ and $(-\Delta)^s u$ are uniformly continuous and bounded. (Continuity suffices if we additionally know that $\Omega\_\epsilon$ are all contained in a bounded region.)
If $|f(x)-f(y)|\leqslant\delta... | 0 | https://mathoverflow.net/users/108637 | 403790 | 165,642 |
https://mathoverflow.net/questions/403774 | 8 | Is it true that a finite group with squarefree order has periodic group cohomology (with trivial coefficients)?
I cannot see why this would be the case, but I'm looking at a paper which seems to implicitly say it's true [Vogel, ON STEENROD'S PROBLEM FOR NON ABELIAN FINITE GROUPS, p.1].
| https://mathoverflow.net/users/125639 | Finite group with squarefree order has periodic cohomology? | All subgroups are then squarefree, and by FTAG all such abelian subgroups are cyclic. Now, a group has periodic cohomology iff all its abelian subgroups are cyclic (Theorem VI.9.5 of Brown's book) -- this is proved by first reducing to Sylow p-subgroups.
| 10 | https://mathoverflow.net/users/12310 | 403793 | 165,643 |
https://mathoverflow.net/questions/403791 | 6 | Let $\chi\_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}\_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum\_{\lambda\vdash n}\chi\_{\mu}^{\lambda}=\delta\_\text{odd}(n).$$
I like to ask:
>
> **QUESTION 1.** Is there a formula for the enum... | https://mathoverflow.net/users/66131 | Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$ | It is an open problem to determine how many of the entries of the character table are zero — or, indeed whether the proportion of zeros tends to a positive constant, or to zero. One should be careful about the exact problem considered: if a character is picked at random, and a group element is picked at random (so $p(n... | 9 | https://mathoverflow.net/users/38624 | 403796 | 165,644 |
https://mathoverflow.net/questions/351751 | 8 | The standard simplices $\Delta^n \subset \{\mathbf{x}\in\mathbb{R}^{n+1}\mid x\_0 + \ldots + x\_n =1 \} =: \mathbb{A}^n$ carry two natural sorts of smooth differential forms:
1. Those differential forms on the interior of $\Delta^n$ that extend smoothly to a neighbourhood of $\Delta^n$ in $\mathbb{A}^n$ (this definit... | https://mathoverflow.net/users/4177 | Differential forms on standard simplices via Whitney extension vs diffeological structure | The two chain complexes are isomorphic for any $n≥0$.
Fix some $n≥0$, $k≥0$ and consider $k$-forms on the $n$-simplex.
I will use the notations $Ω\_e^k$ and $Ω\_d^k$ for forms of type 1 and 2 respectively.
I also use the notation $Δ\_d^n$ for the diffeological $n$-simplex
so that $\def\Hom{\mathop{\rm Hom}} \def\R{{\... | 3 | https://mathoverflow.net/users/402 | 403801 | 165,647 |
https://mathoverflow.net/questions/403802 | 2 | Suppose that $f : X \rightarrow Y$ is a morphism of schemes.
Let $Z \hookrightarrow Y$ the scheme-thereotic image of $f$.
Under what conditions is the morphism $X \rightarrow Z$ an epimorphism?
If both $X = \mathrm{Spec} \, B$ and $Y = \mathrm{Spec} \, A$ are affine, then $Z = \mathrm{Spec}\, A/I$, where $I$ is the k... | https://mathoverflow.net/users/122284 | Epi-mono factorisations in schemes via scheme-theoretic image | This already fails in the affine case. For example, if $Y = \mathbf A^1$ and $X = \mathbf A^1 \setminus \{0\}$, then $Y$ is the scheme-theoretic image of $X \hookrightarrow Y$. But if $Z$ is the line with two origins, then the two inclusions $Y \to Z$ agree on $X$ but are not identical.
There is a positive result whe... | 4 | https://mathoverflow.net/users/82179 | 403804 | 165,648 |
https://mathoverflow.net/questions/403803 | 4 | Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively irrational number is a number $x$ such that there is a known primitive recursive function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z... | https://mathoverflow.net/users/163672 | What is known about constructively irrational numbers? | All algebraic numbers are by this definition constructively irrational. You can adopt Liouville's proof that Liouville numbers are transcendental and turn it in the other direction to get a function of the sort you want given an algebraic number and its corresponding polynomial. Explicit versions of [Baker's theorem](h... | 5 | https://mathoverflow.net/users/127690 | 403807 | 165,650 |
https://mathoverflow.net/questions/403797 | 12 | Is there a fixed-point theorem that implies the following result?
>
> Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to left shifts and closed with respect to the pointwise convergence. Then $F$ contains a constant function.
>
>
>
... | https://mathoverflow.net/users/1441 | A variation of the Ryll-Nardzewski fixed point theorem | The claim does not hold. Let $F$ be the free group on $\{a,b\}$ and $X⊂F$ be the words whose last letter is $a$ or $b$. Let $\xi=\delta\_a+\delta\_b−\delta\_{a^{-1}}−\delta\_{b^{-1}}\in\ell\_1(F)$. Then for any $s\in F$, one has $\langle 1\_X,s\xi\rangle\geq1$. (To see this, view $F$ as the $4$-regular tree and $s\xi$ ... | 12 | https://mathoverflow.net/users/7591 | 403815 | 165,654 |
https://mathoverflow.net/questions/403799 | 3 | My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector magnitudes and direction correlated up to some length scales $l$ (beyond which the correlation goes to zero). I prefer a const... | https://mathoverflow.net/users/115409 | Divergence-free Gaussian vector field with given mean magnitude and correlation function | One construction uses that divergence free fields are precisely the rotation fields:
Choose any isotropic matrix valued covariance function $C':\mathbb{R}^3\times\mathbb{R}^3\to \mathbb{R}^{3\times3}\_{\ge0}$. Then, the push forward $\nabla\times g=GP(0,\nabla\times C'\times\nabla)$ of the Gaussian process $g=GP(0,C'... | 1 | https://mathoverflow.net/users/25523 | 403818 | 165,655 |
https://mathoverflow.net/questions/403792 | 3 | In the paper "*Silting mutation in triangulated categories*" by *Aihara and Iyama*, I stumbled upon this nice definition( **Definition 2.1**) of a tilting/silting subcategory of a triangulated category $\mathcal{T}$. Let $\mathcal{M}$ be a subcategory of a triangulated category $\mathcal{T}$:
(1) We say that $\mathca... | https://mathoverflow.net/users/369159 | On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama | $\operatorname{Hom}\_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$ means that $\operatorname{Hom}\_{\mathcal{T}}\left(X, \Sigma^i(Y)\right) = 0$ for all objects $X,Y$ of $\mathcal{M}$ and all integers $i>0$.
A stalk complex is a complex with only one nonzero term. So "$A$ considered as a stalk complex" means the c... | 2 | https://mathoverflow.net/users/22989 | 403823 | 165,658 |
https://mathoverflow.net/questions/403752 | 10 | Something I've been thinking about for a while that I'm not sure I understand is why $\mathcal{Z}$ stability, as opposed to say $\mathcal{O}\_\infty$-stability or even $\mathcal{K}$-stability is so important to representation theory. I know that the Jiang-Su algebra has a lot of interesting properties such as being str... | https://mathoverflow.net/users/208800 | What is the significance of the Jiang Su algebra in classification of C$^*$ -algebras? | Diego's answer in the comments above is related to why we would expect any classifiable $C^\ast$-algebra to satisfy $A\cong A\otimes \mathcal Z$:
Since $\mathcal Z$ is separable, nuclear, unital, simple and UCT (the properties of $C^\ast$-algebras we wish to classify by $K$-theory and traces),
tensoring with any other ... | 15 | https://mathoverflow.net/users/126109 | 403830 | 165,661 |
https://mathoverflow.net/questions/403828 | 6 | In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
>
> Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is closed and the saturation of any open set is Borel. Then $E$ admits a Borel selector.
>
>
>
So under the hypotheses of t... | https://mathoverflow.net/users/120027 | A strong Borel selection theorem for equivalence relations | Let $X$ be the Cantor space $2^\omega$, and let $E$ be the relation of "equivalence mod $\mathrm{Fin}$" -- i.e., $xEy$ if and only if $\{n \in \omega :\, x(n) \neq y(n) \}$ is finite. The equivalence classes for this relation are countable (hence Borel, and even $F\_\sigma$). If $U \subseteq 2^\omega$ is open, then $U$... | 12 | https://mathoverflow.net/users/70618 | 403840 | 165,666 |
https://mathoverflow.net/questions/403825 | 4 | In the classical setting, we can define automorphic forms on $\text{SL}\_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups.
I have encountered the lifting of automorphic forms to the adeles (so to automorphic representations) for $\Gamma$ being a congr... | https://mathoverflow.net/users/168129 | Adelization for any classical arithmetic subgroup | For a subgroup to have a meaningful lift to the adeles, it is necessary and sufficient for the subgroup to be a congruence subgroup in the sense that for some $N$, the subgroup contains all elements congruent to the identity mod $N$.
Given an element of $SL\_n(\mathbb A\_{\mathbb Q})$ (or the same thing for the norm ... | 6 | https://mathoverflow.net/users/18060 | 403842 | 165,668 |
https://mathoverflow.net/questions/403410 | 8 | Let $T\_R$ be the first-order theory of [real closed fields](https://en.wikipedia.org/wiki/Real_closed_field). This is precisely the theory over the language $\{0,1,+,\times\}$ such that the theorems are the formulas that hold in $\Bbb R$. It can be effectively axiomatized by saying that it's a field in which every odd... | https://mathoverflow.net/users/4613 | Are these theories of real and complex number biinterpretable? | $\DeclareMathOperator\Th{Th}\def\R{\mathbb R}\def\C{\mathbb C}\DeclareMathOperator\Aut{Aut}$Basically, all the statements you mention are correct, we just have to be careful about the details to avoid the apparent contradiction. So let us review how these interpretations work:
* There is a 1-dimensional interpretatio... | 10 | https://mathoverflow.net/users/12705 | 403859 | 165,674 |
https://mathoverflow.net/questions/403858 | 3 | Cross-posted from Math Stackexchange.
In an older question to which I provided an [answer](https://math.stackexchange.com/questions/4119317/let-fx-fracx-ln-x1x-evaluate-lim-m-to-f-leftx-0-right-fracx-1/4135364#4135364) it was asked how to compute a particular limit involving the roots of a transcedental function arou... | https://mathoverflow.net/users/126536 | "Lagrange inversion" around an extremum | This is easily reduced to the ordinary Lagrange inversion.
Indeed, without loss of generality, $x\_0=0=f(x\_0)=f'(x\_0)<f''(x\_0)$, so that for all real $z$ close enough to $0$ we have $f(z)=z^2 h(z)$, where $h$ is a function analytic near $0$ such that $h(0)>0$. So, the equation $f(z)=w$ (for real $z$ near $0$ and s... | 3 | https://mathoverflow.net/users/36721 | 403864 | 165,676 |
https://mathoverflow.net/questions/403870 | 8 | Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that this would imply $2^{\aleph\_1}=2^{\aleph\_0}$.
| https://mathoverflow.net/users/4600 | VC dimension of standard topology on the reals | An uncountable $S\subseteq \mathbb{R}$ has an accumulation point $x\in S$. Then for $D=\{x\}$ there is no such open set $U$.
| 12 | https://mathoverflow.net/users/24076 | 403878 | 165,680 |
https://mathoverflow.net/questions/403868 | 2 | $\DeclareMathOperator\Ent{Ent}\newcommand{\prior}{\mathrm{prior}}\newcommand\Data{\mathrm{Data}}$I came across this paper on the optimality of Bayes' theorem
<https://sinews.siam.org/Portals/Sinews2/Issue%20Pdfs/sn_July-August2021.pdf>
and could not figure out where one inequality comes from.
Denote by $m$ the parame... | https://mathoverflow.net/users/157812 | An inequality in the optimality of Bayes' theorem | First, let us do some cleaning here.
1. Let $\pi:=\pi\_{\mathrm{prior}}$, $Y:=\rho/\pi$, and $F:=f(m)$.
2. You copied the inequality in question incorrectly. In your first-linked paper, the inequality is
$-\ln E\_\pi e^F\le E\_\rho(-F) + D\_{KL}(\rho|\pi)$, that is,
$$E\_\rho F-\ln E\_\pi e^F\le D\_{KL}(\rho|\pi). \t... | 2 | https://mathoverflow.net/users/36721 | 403880 | 165,681 |
https://mathoverflow.net/questions/403885 | 3 | Many examples of Calabi-Yau manifolds are constructed as algebraic varieties in weighted projective space, or more generally as complete intersection Calabi-Yau (CICY) manifolds. Are there such realizations of compact hyperkahler manifolds besides K3? If no, is there a fundamental obstruction?
| https://mathoverflow.net/users/138208 | Compact hyperkahler manifold as algebraic variety in weighted projective space? | If $X$ is a hypersurface and $\dim(X) > 2$ then by Lefschetz hyperplane theorem $H^{2,0}(X) = 0$, hence $X$ can't by hyperkahler.
| 8 | https://mathoverflow.net/users/4428 | 403892 | 165,686 |
https://mathoverflow.net/questions/403638 | 6 | Let $X$ be a preperfectoid space over $\mathrm{Spa}(\mathbb{Q}\_p,\mathbb{Z}\_p)$. It has several associated sites, with successively finer topologies: $$X\_{an} \subset X\_{et} \subset X\_{proet} \subset X\_v.$$
I was wondering: **what is the relationship between vector bundles on these different sites**? Here, by a... | https://mathoverflow.net/users/143589 | Vector bundles on the various sites of a preperfectoid | There is some relevant [work of Ben Heuer](https://arxiv.org/abs/2012.07918v2) on this.
In short, analytic and etale vector bundles agree on all sousperfectoid adic spaces (and much more generally, for all "etale sheafy" adic spaces), while proetale and v-vector bundles also agree (because proetale locally, the space... | 7 | https://mathoverflow.net/users/6074 | 403895 | 165,687 |
https://mathoverflow.net/questions/177232 | 10 | If a finite projective plane $\pi\_1$ of order $m$ contains, as a sub plane, a
finite projective plane $\pi\_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane. Otherwise $m \geq n^2 + n$. (This is a theorem of Bruck that can be found in Hall's Group Theory book, I believe.) M... | https://mathoverflow.net/users/39684 | Subplanes of Finite Projective Planes | No, the bound has not been improved. Even a subplane of order 3 of a projective plane of order 12 has not yet been ruled out. (You can see this
in the arguments in the work on possible collneations of a projective plane of order 12, such as Janko-Van Trung (1980) [On projective planes of order 12 which have a subplane ... | 1 | https://mathoverflow.net/users/369536 | 403902 | 165,689 |
https://mathoverflow.net/questions/403888 | 8 | Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is a Borel set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that this would imply $2^{\aleph\_1}=2^{\aleph\_0}$. This is a hopefully more interesting adaption of a recent [too easy quest... | https://mathoverflow.net/users/4600 | VC dimension of Borel sets | Yes! Martin's Axiom implies that if $S \subseteq \mathbb R$ and $|S| < \mathfrak{c}$, then every subset $D$ of $S$ is a relative $G\_\delta$ in $S$: i.e., there is a $G\_\delta$ set $X \subseteq \mathbb R$ with $X \cap S = D$. (And let me note that $2^{\aleph\_0} = 2^{\aleph\_1}$ is another consequence of Martin's Axio... | 9 | https://mathoverflow.net/users/70618 | 403903 | 165,690 |
https://mathoverflow.net/questions/403893 | 1 |
>
> Let $K$ be a number field and $v$ be it's one of $K$'s non-archimedian valuation.
> Then, I would like to prove $K\_v(a^{1/m}) /K\_v$ is unramified if
> only if $v(a)≡0 \pmod m$.
>
>
>
This is from Silverman's 'the arithmetic of elliptic curves', p213.
I know unramified extension of local field is in bijecti... | https://mathoverflow.net/users/144623 | $K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$ | If $v(a)\not\equiv 0\pmod m$, ramification is easy: just consider the valuation of the element $a^{1/m}$.
The converse is a little subtler than you make it seem, and depending on how exactly you phrase it, it need not be true: for instance, if $m=p$ coincides with the residue characteristic of $v$, $a=1$ and we inter... | 3 | https://mathoverflow.net/users/30186 | 403905 | 165,692 |
https://mathoverflow.net/questions/403729 | 2 | Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R\_+$ be smooth bounded functions. Assume also $\phi' \ge 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of $u-v$. Let us assume $D\_\epsilon = \{x: u(x)-v(x) < \epsilon\} \subset B\_1(0)$. Under these assumptions, is it possi... | https://mathoverflow.net/users/122620 | Determine the sign (positive or negative) of an integral with the fractional Laplacian | The sign can be arbitrary already for $s = 1$. In this case we can take $u(x) - v(x) = 1 - \cos (\pi x)$ for $|x| \leqslant 1$ and $u(x) - v(x) = 2$ when $|x| > 1$, and $\epsilon = 2$. Then the integral becomes
$$ I := \int\_{-1}^1 \phi(x) (-2 \pi^2 \cos(\pi x)) dx = -2 \pi^2 \int\_{-1}^1 \phi(x) \cos(\pi x) dx .$$
Now... | 1 | https://mathoverflow.net/users/108637 | 403906 | 165,693 |
https://mathoverflow.net/questions/403898 | 2 | Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}\_X$-modules) are soft, hence acyclic for $\Gamma$.
**I wonder if the same holds for $\Gamma\_c$.**
| https://mathoverflow.net/users/131975 | Are vector bundles acyclic for $\Gamma_c$? | Yes, at least if $X$ is locally compact.
A useful reference is Bredon's book. In brief, soft sheaves are in particular $c$-soft (they satisfy the softness condition for *compact* subsets of $X$ as opposed to all closed subsets of $X$). On a locally compact space, $c$-soft sheaves are $\Gamma\_c$-acyclic. Bredon state... | 4 | https://mathoverflow.net/users/1310 | 403910 | 165,695 |
https://mathoverflow.net/questions/403926 | 3 | Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes\_R -$ from left $R$-modules to abelian groups that preserves and reflects exact sequences. Faithful flatness for a left $R$-module is defined analogously.
What is an example of an $... | https://mathoverflow.net/users/371382 | Faithful flatness for rings | Let $G$ be a non-trivial finite group and let $R=\mathbb ZG$. We can view $M=\mathbb ZG$ as an $R$-bimodule via the right regular module structure on the right and the trivial module structure on the left (so left multiplication by $g\in G$ fixes $M$). Then $M$ is faithfully flat as a right module because $M\otimes\_R ... | 3 | https://mathoverflow.net/users/15934 | 403948 | 165,707 |
https://mathoverflow.net/questions/403927 | 6 | There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them:
1. Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$, but in general there's no canonical way to identify them with $H^1$.
2. Spin structures on $TM$ are the same as t... | https://mathoverflow.net/users/113402 | Is a spin structure on a knot complement the same thing as an orientation of the knot? | Is the theorem true? There is an non-natural bijection. There is no natural bijection.
A link exterior is homotopy-equivalent to a $2$-complex, so a trivialization of the tangent bundle over the $2$-skeleton is simply a map:
$$S^3 \setminus L \to SO\_3.$$
This is because there is a canonical trivialization of $TS... | 7 | https://mathoverflow.net/users/1465 | 403950 | 165,708 |
https://mathoverflow.net/questions/403912 | 1 | This is a cross-post from Stackexchange Mathematics (<https://math.stackexchange.com/questions/3893961/h%c3%b6lder-continuous-dependence-on-parameters-for-solutions-of-ode>).
We have the following standard result for continuous dependence of the initial value for ODEs with a continuous right-hand side (Satz 8.18 in [... | https://mathoverflow.net/users/164084 | Hölder continuous dependence on parameters for solutions of ODE | No, what you want is not possible. (Basically, you can approximate the nonunique example with smooth examples which have slightly moved initial data. This gives you two nearby initial data points that move apart arbitrarily fast.)
Consider the case $n = 1$, let $F\_0(x) = \begin{cases} 0 & x \leq 0 \newline \sqrt{x} ... | 1 | https://mathoverflow.net/users/3948 | 403951 | 165,709 |
https://mathoverflow.net/questions/403919 | 2 | LLPO is the statement $\forall x \in \mathbb R. x \leq 0 \vee x \geq 0.$ The statement should be understood as a fragment of the Law of Excluded Middle, rather than a statement about the ordering of the real line. The fragment is usually considered large enough to not be constructive in any reasonable sense.
I have a... | https://mathoverflow.net/users/75761 | LLPO as constructivity/computability for dense subsets | A counterexample to "Stuff provable by $\mathrm{LLPO}$ is constructively true on some dense set" can be built as follows:
We start with two recursively inseparable disjoint c.e. set $A, B \subseteq \mathbb{N}$. Using $\mathrm{LLPO}$ countably many times, we can actually built a "decidable" set $C \subseteq \mathbb{N}... | 2 | https://mathoverflow.net/users/15002 | 403962 | 165,711 |
https://mathoverflow.net/questions/403939 | 30 | Some background first.
I recently graduated with a master's degree in applied mathematics. During graduate school I began working on a paper, which I continued to work on post-graduation. A complete working copy of the paper is done and I have posted it on the arXiv [here](https://arxiv.org/abs/2106.14958). The work ... | https://mathoverflow.net/users/125801 | How can I seek help in preparing a very long research article for publication? | First of all, I would consider it against the ethics of scientific publishing to accept an offer as a co-author when you were not involved in the research. So I don't think that is viable route.
What you have achieved is quite unusual, you have on your own identified and developed a research direction and produced a ... | 37 | https://mathoverflow.net/users/11260 | 403968 | 165,712 |
https://mathoverflow.net/questions/403958 | 4 | Following Takahashi (*"On the concrete construction of hyperbolic structure of 3-manifolds"*), I was able to construct the Euclidean cusp cross-section for the 5\_2 knot complement (please see <http://kias.dyndns.org/topogeoimages/5_2.cusp.png>), and determine the vertex invariants
```
x = 0.12256116687665364 + 0.74... | https://mathoverflow.net/users/371704 | Computation of cusp shape from vertex invariants | It looks like you have computed the "tetrahedra shapes" for the three ideal hyperbolic tetrahedra making up the knot complement. Each of these gives a (euclidian!) shape to the four "cusp triangles" that "cut off" the four ideal vertices of the tetrahedron. To compute the shape of the cusp, you need to understand how i... | 3 | https://mathoverflow.net/users/1650 | 403971 | 165,714 |
https://mathoverflow.net/questions/403957 | 4 | Let $\chi\_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}\_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum\_{\lambda\vdash n}\chi\_{\mu}^{\lambda}=\delta\_\text{odd}(n).$$
This time, I'm interested to know:
>
> **QUESTION 1.** Is there anyt... | https://mathoverflow.net/users/66131 | Total sum of characters of the symmetric group $\frak{S}_n$ | The sum $\sum\_\mu \chi^\lambda\_\mu$ over partitions $\mu$ of $n$ is the multiplicity of the irreducible $\chi^\lambda$ in the character afforded by $\mathfrak S\_n$ acting on itself by conjugation. If $\psi$ is the character for conjugation, then $\psi(g)$ is the size of the centralizer of $g$, so
$$\langle \chi^\lam... | 11 | https://mathoverflow.net/users/371843 | 403974 | 165,715 |
https://mathoverflow.net/questions/403937 | 15 | It is clear that the term [*Moore graph*](https://en.wikipedia.org/wiki/Moore_graph) was coined by Hoffman and Singleton in their paper [On Moore graphs with diameters $2$ and $3$](https://doi.org/10.1147/rd.45.0497), where they write
>
> E. F. Moore has posed the problem of describing graphs for which equality hol... | https://mathoverflow.net/users/2663 | Reference request: Moore graphs | Moore posed this problem to Hoffman at a conference, so it is not in print. Hoffman makes the following remark (from "Selected Papers of Alan Hoffman with Commentary", pp. 367):
>
> After I discussed the preceding paper at an IBM summer workshop, E.F. Moore
> raised the graph theory problem described in the paper, ... | 18 | https://mathoverflow.net/users/2384 | 403975 | 165,716 |
https://mathoverflow.net/questions/380983 | 1 | I am doing some research in combinatorics, and I found that I have to consider the following binomial coefficient :
$$ \binom{\binom{i}{j}}{k} $$
(In fact, I have to take the product for fixed $i,k$ and odd $j$’s, but to make this product, I have to manipulate those coefficients, and I don’t know how.)
Is there a... | https://mathoverflow.net/users/73667 | Binomial coefficient in a binomial coefficient | *(In order that this question appears as answered I'm converting Darij Grinberg's comment into an answer.)*
For $j$ and $k$ (that will remain fixed) in $\mathbb N$, and a set $X$, denote $\mathcal F(X):=\mathcal{P}\_k\mathcal{P}\_j(X)$, the set of all $k$-element sets $\mathcal U:=\{U\_1,\dots,U\_k\}$ of $j$-element ... | 2 | https://mathoverflow.net/users/6101 | 403981 | 165,718 |
https://mathoverflow.net/questions/310920 | 5 | Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given
$$ (a\_1 ,\ldots,a\_{d(\lambda)}|b\_1,\ldots,b\_{d(\lambda)}),$$
where $d(\lambda)$ denote the diagonal of $\lambda$.
Let $a\_i'= a\_i +\frac12$ and $b\_i'= b\_i +\frac12$ are modified Frobenius coordinates.
By a classical th... | https://mathoverflow.net/users/45170 | Frobenius coordinate expansion of character | There is a generalisation of the character formula, although it is usually stated in terms of contents rather than Frobenius coordinates. Also, it applies to conjugacy classes labelled by partitions $(1^{m\_1} 2^{m\_2} \cdots)$, where $m\_2, m\_3, \ldots$ are fixed and $m\_1$ varies with $n$ (as in the example you gave... | 3 | https://mathoverflow.net/users/159272 | 403984 | 165,719 |
https://mathoverflow.net/questions/403953 | 2 | In the book **Neverending fractions** from Borwein, van der Poorten, Shallit and Zudilin, there is the so called **distance formula** (Theorem 2.45, p. 43) stated:
$$\alpha\_1\alpha\_2\cdot...\cdot\alpha\_n=\frac{(-1)^n}{p\_{n-1}-q\_{n-1}\alpha}$$
for $n\geq 0$ with $p\_{-1}=1$ and $q\_{-1}=0$, where $$\alpha=[a\_0;a... | https://mathoverflow.net/users/108459 | Distance formula for continued fractions | 1. This formula is valid for any continued fraction. But for quadratic irrationals it is especially useful because it allows to express fundamental unit of corresponding field in terms of continued fraction expansion of $\sqrt{n}$. For a reduced quadratic irrational $\omega=[0;\overline{a\_1,\ldots,a\_n}]$ with period ... | 4 | https://mathoverflow.net/users/5712 | 403990 | 165,721 |
https://mathoverflow.net/questions/403995 | 3 | If positive integer $n$ such $n\mid2^n-2$,where $n>1$, we called $n$ is Poulet number, see: <https://en.wikipedia.org/wiki/Super-Poulet_number>
I found if $n>2$ is Poulet number, then $\dfrac{2^n-2}{n}$ always is composite number
Is this a known result? if $n$ is odd number, so $\frac{2^n-2}{n}$ is even number, and... | https://mathoverflow.net/users/38620 | Prove that $\frac{2^n-2}{n}$ is composite number | Let $n=2k>2$ be a Poulet number, then $k$ divides $2^{2k-1}-1$ (in particular, $k$ is odd) and we must prove that $q:=\frac{2^{2k-1}-1}k$ is composite.
Let $s$ be minimal positive integer for which $k$ divides $2^s-1$ (i.e., $s$ is a multiplicative order of 2 modulo $k$). Then all numbers $t$ for which $k$ divides $2... | 17 | https://mathoverflow.net/users/4312 | 403996 | 165,724 |
https://mathoverflow.net/questions/403834 | 3 | Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V\_+$ denote the positive spin bundle. We know $Z$ admits more than one almost complex structure. So it can have canonical Spin$^{\mathbb{C}}... | https://mathoverflow.net/users/131004 | Pull back of Spin$^{\mathbb{C}}$ bundle | $\newcommand{\spinors}{\mathbb{S}}$The tangent bundle $TZ$ fits into an exact sequence
$$
0\to T\_{/M}Z\to TZ\xrightarrow{\pi\_\*}\pi^\*TM\to 0
$$
where the fiberwise tangent bundle $T\_{/M}Z$ is two-dimensional and equipped with a canonical complex structure. Explicitly, the fiber of $Z\to M$ over a point $m$ are the ... | 1 | https://mathoverflow.net/users/35687 | 403997 | 165,725 |
https://mathoverflow.net/questions/404011 | 1 | Consider the initial-value problem associated to the PDE $u^\epsilon\_t + u^\epsilon\_x - \epsilon u^\epsilon\_{xx} + \epsilon u^\epsilon\_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution $u^{\epsilon}$ converges in $L^2$ to the weak solution $u$ of the IVP for $u\_t + u\_x=0$ (with the same initial ... | https://mathoverflow.net/users/nan | Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$ | Since your equation is linear, this is sufficient. The convergence holds in the sense of distributions.
Notice the following: the limit, as a solution of the hyperbolic equation $u\_t+u\_x=0$ is unique, being actually $u(t,x)=u(0,x-t)$. In particular
$$\|u(t)\|\_{L^2({\mathbb R})}\equiv\|u(0)\|\_{L^2({\mathbb R})}.$$... | 4 | https://mathoverflow.net/users/8799 | 404017 | 165,731 |
https://mathoverflow.net/questions/403954 | 1 | Consider the notion of a split coequalizer (see the [nLab](https://ncatlab.org/nlab/show/split+coequalizer) for the definition). Note that the definition seems to be non-symmetric. Are there any conditions on the ambient category such that it becomes symmetric?
| https://mathoverflow.net/users/145805 | Conditions such that split coequalizers are a symmetric notion | The definition of split coequaliser is essentially a transcriptions of that of an algebra for a monad, ie an object in the Eilengberg--Moore construction of the maximal category with an adjunction that yields the given monad.
Yes, it's asymmetric. The construction is awkward for other reasons. For example, the compos... | 0 | https://mathoverflow.net/users/2733 | 404021 | 165,733 |
https://mathoverflow.net/questions/403914 | 1 | According to numerical simulation, the relationship
$$\sum^{\infty}\_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$
where $\Gamma$ is the Gamma function seems to be true.
Do you have any idea how to show this relationship using asymptotic or exact methods?
| https://mathoverflow.net/users/nan | Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true? | Put $\mu:=\alpha-1$; then $\sum^{\infty}\_{n=1}\frac{ x^{n-1}}{(n-1)!n^\alpha} = x^{-1}\sum^{\infty}\_{n=1}\frac{ x^n}{n!n^\mu}=x^{-3/2}I\_\mu(x)$ for the function $I\_\mu(x)$ given in Johannes Trost’s answer to [Asymptotic expansion of $\sum\limits\_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$](https://mathoverflow.... | 6 | https://mathoverflow.net/users/6101 | 404022 | 165,734 |
https://mathoverflow.net/questions/404009 | 5 | Forgive me if this question turns out to be too elementary-then feel free to move it to stack exchange. I believe that this should be very basic fact from topos theory nevertheless being not familiar with topos theory let me ask it.
One of the highlights of topos theory is the possibility to consider an analogue of the... | https://mathoverflow.net/users/24078 | Subobjects as an object in a topos | The general notion you're looking for is a **representable functor**. For example:
$${\mathcal E}(X\times{-},Y) \sim {\mathcal E}({-},Y^X)$$
$${\textsf{Sub}}({-}) \sim {\mathcal{E}}({-},\Omega)$$
The thing on the left is a general contravariant functor from the category to $\mathbf{Set}$.
The thing on the right... | 9 | https://mathoverflow.net/users/2733 | 404023 | 165,735 |
https://mathoverflow.net/questions/403978 | 2 | Let $G\_k$ be the graph obtained by applying the following procedure k-times:
1. Start with a graph with single vertex $v$ (Call this graph $H$)
2. Add a vertex $u$ such that $u$ is not adjacent to any vertex of $H$ (i.e., $K:= H \cup \{u\}$) union of two graphs
3. Add a vertex $w$ such that $w$ is adjacent to all th... | https://mathoverflow.net/users/33047 | Name of an inductively defined sequence of graphs | It's not quite the same question, but the graphs that can be obtained by repeating either of the two operations (add a disjoint vertex or a dominating vertex), not necessarily in strict alternation, are called [threshold graphs](https://en.wikipedia.org/wiki/Threshold_graph).
| 4 | https://mathoverflow.net/users/440 | 404031 | 165,738 |
https://mathoverflow.net/questions/404027 | 1 | I think the following inequality might be true and was hoping somebody might spot it or know a proof:
Suppose $f:\mathbb R\to \mathbb R$ is convex and suitably nice so that
$$\int\_{\mathbb R} e^{-f(x)} dx = 1$$
Then is it true that
$$\int\_{\mathbb R} f(x) e^{-f(x)} dx\ge 0$$
?
I also wonder what area of mathemati... | https://mathoverflow.net/users/9202 | Elementary inequality about integrals of exponentials of concave functions (possibly connected to log concave distributions) | The answer is no.
E.g., take any real $b>1$ and let $a:=2e^b$. Let $f(x):=a|x|-b$ for all real $x$.
Then $\int\_{\mathbb R}e^{-f(x)} dx=1$ but $$\int\_{\mathbb R} f(x) e^{-f(x)} dx
=1-b<0.$$
| 1 | https://mathoverflow.net/users/36721 | 404033 | 165,739 |
https://mathoverflow.net/questions/404029 | 1 | What is the Lp norm of the $N$-dimensional Hadamard matrix $H = ((-1)^{i \cdot j})\_{i,j}$ for $p > 2$? I know that $\|H\|\_1 = N$, $\|H\|\_2 = \sqrt{N}$, $\|H\|\_\infty = N$ but I can't figure out what it is for other values of $p$. Can we at least give a good upper-bound on it?
Here I consider the induced norm: $\|... | https://mathoverflow.net/users/92442 | Lp norm of Hadamard matrix | **Important Edit**: As J.J Green pointed out, the OP contains an incorrectly stated value for $\|H\|\_{\infty}$, which I copied without checking below. Interpolating between $(1,\infty)$ using the corrected version would give the trivial bound $\|H\|\_p \leq N$. You regain the sharp bound by interpolating instead betwe... | 6 | https://mathoverflow.net/users/3948 | 404034 | 165,740 |
https://mathoverflow.net/questions/404032 | 7 | In my [earlier MO post](https://mathoverflow.net/questions/403957/total-sum-of-characters-of-the-symmetric-group-fraks-n), I proposed the double sum $\sum\_{\mu\vdash n}\sum\_{\lambda\vdash n}\chi\_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}\_n$. Soon after, I started considering the sum ... | https://mathoverflow.net/users/66131 | Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$ | The sum $\sum\_\chi \chi(\alpha)^2$ is the size of the centralizer $z\_\mu=\frac{n!}{|K\_\mu|}=1^{m\_1}m\_1! 2^{m\_2}m\_2!\cdots n^{m\_n} m\_n!$ if $\alpha$ has cycle type $\mu=1^{m\_1}2^{m\_2}\ldots$, so
$$\frac{1}{n!} \sum\_{\alpha\in \mathfrak S\_n} \left(\sum\_\chi \chi(\alpha)^2\right)^2=
\frac{1}{n!} \sum\_{\mu\v... | 11 | https://mathoverflow.net/users/371843 | 404035 | 165,741 |
https://mathoverflow.net/questions/404039 | 3 | Suppose we have a sequence of entire functions $f\_n$ such that $$\text{$f\_n(z)\to0$ for each natural $z$}\tag{1}$$
(as $n\to\infty$).
Is it possible to give general additional conditions on the sequence $(f\_n)$ ensuring that (1) implies
$$\text{$f\_n(z)\to0$ for each complex $z$?}\tag{2}$$
As a minimum, I wou... | https://mathoverflow.net/users/36721 | On convergence of entire functions | Edited.
1. For your general question,
one sufficient condition is that your functions are of exponential type $<\pi$.
This is best possible since anything convergent to $\sin\pi z$ would be a counterexample, and $\sin\pi z$ has exponential type exactly $\pi$.
One can slightly improve this. For example when all $L^2... | 3 | https://mathoverflow.net/users/25510 | 404043 | 165,744 |
https://mathoverflow.net/questions/373336 | 5 | I'm sure it is well-known how many edges you must delete in a (highly linked) graph to destroy all cycles. Is it also known how many edges you must delete to destroy only all triangles? And even, how many you need if additionally, you may not use an edge to destroy more than one triangle? (Note the latter doesn't mean ... | https://mathoverflow.net/users/11504 | Deleting triangles in a graph | Given a graph $G$, the problem of determining the minimum size $\tau(G)$ of a set of edges $X$ such that $G-X$ is triangle-free is indeed NP-complete. This was proved by [Yannakakis](https://epubs.siam.org/doi/abs/10.1137/0210021?journalCode=smjcat). Regarding bounds for $\tau(G)$, we can consider the following dual pr... | 2 | https://mathoverflow.net/users/2233 | 404056 | 165,747 |
https://mathoverflow.net/questions/382050 | 1 | Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
Define the *abundancy index*
$$I(x)=\frac{\sigma(x)}{x}$$
where $\sigma(x)$ is the classical *sum of divisors* of $x$.
Since $q$ is prime, we have the bounds
$$\frac{q+1}{q} \leq I(q^k) < \f... | https://mathoverflow.net/users/10365 | Improving the lower bound $I(n^2) > \frac{2(q-1)}{q}$ when $q^k n^2$ is an odd perfect number | Here is a way to come up with an improved lower bound for $I(n^2)$, albeit in terms of $q$ and $n$:
We write
$$I(n^2) - \frac{2(q - 1)}{q} = \frac{I(n^2)}{q^{k+1}} = \frac{\sigma(n^2)}{q^k}\cdot\frac{1}{qn^2} > \frac{1}{qn^2},$$
from which it follows that
$$I(n^2) > \frac{2(q - 1)}{q} + \frac{1}{qn^2}.$$
This impro... | 0 | https://mathoverflow.net/users/10365 | 404066 | 165,748 |
https://mathoverflow.net/questions/404058 | 1 | According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if,
$$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$
and it is super-coercive if
$$\lim\_{\|x\| \to \infty} f(x)/\|x\| \to +\infty$$
My question is, does the Fenchel dual $f^\star$ share the same p... | https://mathoverflow.net/users/114734 | Does coercivity/supercoercivity conjugates? | No. The conjugate of the constant zero is the indicator of zero (and vice versa). More generally, the conjugate of the indicator of a closed convex set is positively one-homgeneous (and hence, not super coercive).
| 0 | https://mathoverflow.net/users/9652 | 404067 | 165,749 |
https://mathoverflow.net/questions/404076 | 4 | I am looking for a proofs of the following two claims:
>
> **Claim 1.**
> $$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum\_{n=1}^{\infty}\frac{(-1)^{\Omega\_1(n)}}{n}$$ where $\Omega\_1(n)$ is the number of prime factors of the form $p \equiv 1 \pmod{6}$ of $n$ .
>
>
>
The SageMath cell that demonstrates this claim ... | https://mathoverflow.net/users/88804 | Two conjectural infinite series for $\pi$ | You recently asked a similar [question](https://math.stackexchange.com/questions/4246647/a-conjectural-infinite-series-for-frac-pi2) for modulus $4$ on math.stackexchange. I just used the exact same technique.
---
For $i\in\{1,5\}$, define $f\_i:\mathbb{Z}\_{\ge 1}\to \mathbb{C}^\times$ by $f\_i(n)=(-1)^{\Omega\_... | 10 | https://mathoverflow.net/users/165478 | 404082 | 165,751 |
https://mathoverflow.net/questions/403980 | 5 | It is stated by Douglas Bridges in [Constructive mathematics: a foundation for computable analysis](https://doi.org/10.1016/S0304-3975(98)00285-0) that the following property, which I will call the zero product property:
If $x,y \in \mathbb{R}$ and $xy = 0$, then $x = 0$ or $y = 0$.
is equivalent to the Lesser Limi... | https://mathoverflow.net/users/312621 | How do working constructivists get by with out the zero product property? | There are a couple of very similar statements that all work out:
1. if $x$ and $y$ are both apart from $0$, then $xy$ is apart from $0$.
2. If $x$ is apart from $y$ and $xy = 0$, then $x = 0$ or $y = 0$ (as mentioned by Andreas Blass in the comments).
3. $xy = 0$ iff $\inf \{|x|,|y|\} = 0$.
A more round-about way w... | 3 | https://mathoverflow.net/users/15002 | 404116 | 165,758 |
https://mathoverflow.net/questions/404119 | 0 | For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every other source primarily defines matrices as being $m \times n$. Is the later more formal notation? Should I go through my... | https://mathoverflow.net/users/152336 | Is it improper to define matrices as being $n \times m$ rather than $m \times n$? | It is generally a good idea to keep things alphabetical, unless there is a good reason to do otherwise.
With matrices, it can be okay because an $n\times m$ matrix is really a linear transformation from $\mathbb{F}^m\to \mathbb{F}^n$ (where $\mathbb{F}$ is your field of definition), and so viewed this way, the $m$-di... | 5 | https://mathoverflow.net/users/3199 | 404120 | 165,760 |
https://mathoverflow.net/questions/401746 | 16 | Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's 1967 thesis *A Triple Miscellany: Some Aspects of the Theory of Algebras over a Triple* (Theorem 1.2.9). Manes cites Beck's 1967 th... | https://mathoverflow.net/users/152679 | Jon Beck's untitled manuscript containing the "tripleability theorem" (i.e. the monadicity theorem) | After reaching out to every researcher who cited the manuscript, John Kennison was kind enough to find and scan his copy of the untitled manuscript containing the crude and precise monadicity theorems. I have uploaded it to the nLab for posterity: [Jon Beck's untitled manuscript](https://ncatlab.org/nlab/files/Untitled... | 23 | https://mathoverflow.net/users/152679 | 404123 | 165,763 |
https://mathoverflow.net/questions/404094 | 12 | Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
For $n=1$ this means that a surjective entire function $\mathbb{C}\to\mathbb{C}$ without critical points assumes each value infinitely often. Is this obvious?
| https://mathoverflow.net/users/374739 | Are the fibers of a surjective holomorphic submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic? | The answer is no for $n=1$.
**Lemma**: Suppose $f$ is an entire function with $f^{-1}(z\_0)$ finite non-empty for some $z\_0 \in \mathbb{C}$. Then $f$ is surjective.
Proof: By Picard, $f$ misses at most one value. Up to translating $f$ by a scalar (which obviously preserves the hypothesis), we may assume $f$ misses... | 14 | https://mathoverflow.net/users/144469 | 404127 | 165,764 |
https://mathoverflow.net/questions/403088 | 8 | I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in:
*[1](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehme... | https://mathoverflow.net/users/357523 | Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?) | Yes, there are some omissions in the lists in the original BHV article. I think all of them were fixed by Mourad Abouzaid
Mourad Abouzaid, Les nombres de Lucas et Lehmer sans diviseur primitif, J. Théor. Nombres Bordeaux 18 (2006), no. 2, 299–313.
| 9 | https://mathoverflow.net/users/138069 | 404133 | 165,765 |
https://mathoverflow.net/questions/404131 | 7 | $\DeclareMathOperator\lcm{lcm}$Let $p\_k$ be the $k$th prime number. Set $$L(n) = \lcm(p\_1-1, p\_2-1, \dotsc, p\_n-1). $$
What can we say about the growth of $L(n)$? Trivially, one has that $L(n) < p\_1p\_2 \dotsb p\_n$.
One can do better than that since after $2$, every prime is odd. Thus, if $n \geq 3$, one has $$... | https://mathoverflow.net/users/127690 | Asymptotics of $\operatorname{lcm} ((2-1), (3-1), (5-1), (7-1), (11-1), \dotsc, p_n-1 )$ | It is true that $\ln L(n)=o(p\_n)$ for $n\to+\infty$. To prove it, let us get a bound for contribution of large primes into $\ln L(n)$ and then estimate the contribution of the rest trivially. Choose a parameter $R<\sqrt p\_n$. Let $M(n)=\mathrm{lcm}[1,2,\ldots,p\_n]$. Then we have $\ln M(n)=\psi(p\_n)\sim p\_n$, where... | 11 | https://mathoverflow.net/users/101078 | 404135 | 165,766 |
https://mathoverflow.net/questions/404130 | 7 | Giuga's conjecture (1950), which is still open and has strong numerical support, reads :
>
> Let $n$ be a positive integer. If $1+\sum\_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime.
>
>
>
What would an analog for function fields be?
| https://mathoverflow.net/users/469 | What is a function field analog of Giuga's conjecture? | The integers 1 to $n-1$ are the non-zero elements of $\mathbb Z/n\mathbb Z$. So one could take an ideal $I$ in a Dedekind domain $R$ and ask whether
$$
(\*)\qquad
1+\sum\_{\substack{a\in R/I\\ a\ne0\\}} a^{\#R/I - 1} \equiv 0 \pmod{I}
$$
is equivalent to $I$ being a prime ideal. I have no idea whether this is a reasona... | 10 | https://mathoverflow.net/users/11926 | 404141 | 165,768 |
https://mathoverflow.net/questions/404125 | 2 | Suppose we have a random walk $S\_n$ with i.i.d. steps $X\_i$. We assume that
$$\mathbb{E}[X\_i] = -\mu, \text{Var}[X\_i] = 1,$$
where $\mu$ is close (or going) to zero. We also assume that the moment generating function and its derivatives $M\_{X\_i}, M\_{X\_i}', M\_{X\_i}'', M\_{X\_i}'''$ are all bounded in $(-\epsil... | https://mathoverflow.net/users/49551 | Random walk always stays below a level $a$ | Let
$$p(a):=P\big(\max\_{n\ge0} S\_n\le a\big).$$
Assume that $c\_3:=E|X\_1-EX\_1|^3<\infty$.
By the improvement by [Sakhanenko](https://epubs.siam.org/doi/pdf/10.1137/1119047) of Lemma 8 by [S. Nagaev](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=1854&option_lang=eng) (the improvement consis... | 3 | https://mathoverflow.net/users/36721 | 404145 | 165,769 |
https://mathoverflow.net/questions/404097 | 1 | For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we say these probability measures constitute a compact subset in Wasserstein space?
For example, considering exponential d... | https://mathoverflow.net/users/374718 | Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space? | Since the continuous image of a compact set is compact, it suffices to determine whether the mapping $p \to F(x,p)$ is continuous. This is the case for most natural parametrized families of distributions, but needs to be verified in each case. For instance, for the exponential family, the transformation $x \mapsto(1+\e... | 1 | https://mathoverflow.net/users/7691 | 404153 | 165,770 |
https://mathoverflow.net/questions/402482 | 4 | Let $k$ be a field of characteristic zero and $A$ be a graded commutative dg-algebra over $k$ with differential of degree $+1$ satisfying $H^0(A)=k, H^i(A)=0$ for $i<0$. Denote by $\mathcal J$ a dg ideal of generated by $A^k, k<0$ ($A^k$ denotes elements of degree $k$). Is it true that $\mathcal J$ is acyclic? In other... | https://mathoverflow.net/users/nan | $A$ is a commutative connective dg-algebra satisfying $H^0(A)=k$. Is it true that a dg ideal generated by elements of negative degree acyclic? | It is not necessarily true. I can provide a counterexample but don't know any general statement about when this is true.
Consider $$A=k[x,dx,y]/\langle x^2y, ydx\rangle$$ with $x$ in degree $-2$ and $y$ in degree $3$.
The obvious map from $k[x,dx]$ to $A$ is a dg-algebra map which induces an isomorphism below degre... | 2 | https://mathoverflow.net/users/3075 | 404160 | 165,772 |
https://mathoverflow.net/questions/404108 | 3 | Let $p$ and $q$ be integers.
Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a(0)=a(1)&=1\\
a(2n)& = pa(n)+qa(2n-2... | https://mathoverflow.net/users/231922 | Subsequence of the cubes | Experimenting with a CAS suggests an induction. In order to handle the induction, we need to consider the forms of the numbers involved. $\frac{4^m-1}{3} = 1 + 2^2 + 2^4 + \cdots + 2^{2m-2}$ alternates $1$ and $0$ bits. The map $2n + 1 \to n - 2^{f(n)}$ drops the rightmost bit (which is $1$) and clears the next least s... | 8 | https://mathoverflow.net/users/46140 | 404182 | 165,776 |
https://mathoverflow.net/questions/404169 | 4 | Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
| https://mathoverflow.net/users/374739 | Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic? | The following example is taken from Section 7 of Nguyen, "[A Remark on Polynomial Mappings from $\mathbb{C}^n$ to $\mathbb{C}^{n-1}$ and an Application of the Software Maple in Research](http://dx.doi.org/10.4236/am.2016.715154)": Map $\mathbb{C}^2$ to $\mathbb{C}$ by $f(z,w) = z+z^2w$.
We compute $\tfrac{\partial f}... | 11 | https://mathoverflow.net/users/297 | 404187 | 165,777 |
https://mathoverflow.net/questions/404183 | 2 | Consider the following process on $\mathbb{C}$:
* Start at the point 1.
* At each step, move by adding $e^{i\theta}$, where $\theta$ is uniformly drawn from $\mathbb{S}^1$.
* Stop at the first positive time $T$ where you move inside the *open* unit disc.
What is the distribution of the stopping times?
*Some notes... | https://mathoverflow.net/users/3948 | Distribution of stopping time for a 2D random walk | For the continuous counterpart, if a 2-D Brownian particle is started at $x$ with $|x| > 1$, the density function of the hitting time of the unit disk decays as
$$ \frac{1}{t (\log t)^2} $$
as $t \to \infty$. More precisely, it is comparable to
$$ \frac{|x|-1}{|x|} e^{-(|x|-1)^2/(2t)} \frac{(|x|+t)^{1/2}}{t^{3/2}} \fra... | 2 | https://mathoverflow.net/users/108637 | 404189 | 165,778 |
https://mathoverflow.net/questions/404191 | 1 |
>
> Dirichlet problem for Laplace equation as follows
> \begin{eqnarray}
> \Delta{u}&=&0\text{ in }B\_r(0)\\
> u&=&g\text{ on }\partial B\_{r}(0),
> \end{eqnarray}
> where $ g $ is continuous.
>
>
>
It is already known that $ u(x)=C\_n\int\_{\partial B\_{r}(0)}\frac{r^2-\lvert x\rvert^2}{\lvert x-y\rvert^n}g(y)d... | https://mathoverflow.net/users/241460 | Is the Poisson formula valid when the boundary condition is $ L^2 $? | This is clearly false as stated, since a necessary condition is that $g(x)\to g(\xi)$ as $x\to\xi$ a. e. in the sphere, but if $g$ is merely $L^2$, this may well fail for every point.
The convergence holds in a weaker sense. For example, it is true that if $g\_\rho(x)=g(\rho x)$ for $x\in\partial B\_r(0)$ and $\rho<1... | 4 | https://mathoverflow.net/users/56624 | 404195 | 165,779 |
https://mathoverflow.net/questions/404193 | 7 | Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
| https://mathoverflow.net/users/351872 | Different Bialgebra/Hopf algebra structures on coalgebras | Yes: if $k$ is a field of characteristic 2, let $C$ be the coalgebra over $k$ spanned by 1, $x$, $y$, and $z$ with $x$ and $y$ primitive, $\Delta z = z \otimes 1 + 1 \otimes z + x \otimes y + y \otimes x$ — in the algebra structure, $z$ is going to equal $xy = yx$, so $\Delta z$ has to equal $(\Delta x)(\Delta y)$. The... | 9 | https://mathoverflow.net/users/4194 | 404198 | 165,781 |
https://mathoverflow.net/questions/404194 | 1 | Let $t\_1 < t\_2 < \cdots <t\_m$ be real, and $X = \cup\_{i=1}^{m-1} (t\_i, t\_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form
$$
f(x) =
\begin{cases}
a\_1 & \text{ if } t\_1 < x < t\_2 \\
a\_2 & \text{ if }t\_2 < x < t\_3 \\
\vdots \\
a\_{m-2}... | https://mathoverflow.net/users/129192 | Determining polynomial approximations of piecewise constant functions | The polynomial $Q(y):=\frac43y-\frac13 {y^3}$ is increasing on the interval $[-1,1]$; it has fixed points $0$ and $\pm1$, and $\text{sgn }( Q(y)-y )=\text{sgn}y$. Thus the iterates of $Q$ starting from any $y\in [-1,1]\setminus\{0\}$ converge monotonically to $\text{sgn } y$ (in fact with exponential rate given by $Q'(... | 3 | https://mathoverflow.net/users/6101 | 404199 | 165,782 |
https://mathoverflow.net/questions/404213 | 11 | Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that no such $p$ exists. Does one exist? If so, is there an explicit example?
Failing a general yes or no answer, are there ... | https://mathoverflow.net/users/351164 | Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively? | No, this can't happen. One way to prove this is via Hilbert irreducibility: The polynomial $p(x) - t$ is irreducible over $\mathbb Q[x,t]$, so there are infinitely many specializations $t = c$ with $c \in \mathbb Q$ such that $p(x) - c$ is irreducible in $\mathbb Q[x]$. Since the degree of $p(x)$ is greater than 1, it ... | 32 | https://mathoverflow.net/users/66491 | 404217 | 165,787 |
https://mathoverflow.net/questions/404216 | 2 | I would like to construct a sequence of discrete random variable $X\_2, X\_3,...,X\_n,...$, where $X\_n \in\{0,1,2,...,n-1\}$. Given any $\epsilon \in (0,1)$, its Shannon entropy and min-entropy should satisfy the following relationships
\begin{cases}
H(X\_n)\geq(1-\epsilon)\log\_2(n)\\
H\_{min}(X\_n)=const
\end{c... | https://mathoverflow.net/users/376483 | Difference between Shannon entropy and min-entropy | Since the OP may be interested in what $\epsilon$ are achievable I am providing this alternative to the other answer, where it is correctly stated:
>
> If you fix the maximal atom (say, $p$) of a distribution $\mu$ supported by $n$ points, then its entropy is maximal when all the remaining atoms have the same weigh... | 3 | https://mathoverflow.net/users/17773 | 404223 | 165,789 |
https://mathoverflow.net/questions/404210 | 3 | **The classical setting.**
Given a monoid $A$, there's a category $\mathbf{B}A$, called the **[delooping](https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid)** of $A$, having a single object $\star$ and satisfying $\mathrm{Hom}\_{\mathbf{B}A}(\star,\star)\overset{\mathrm{def}}{=}A$, with compo... | https://mathoverflow.net/users/130058 | Delooping monoidal $\infty$-groupoids into $\infty$-categories | I assume that with ``monoidal ∞-groupoid'' you mean an $E\_1$-space. In this case the answer is yes. It is well known that $E\_1$-spaces can be modeled by functors
$$X:\Delta^{\mathrm{op}}\to \operatorname{Space}$$
satisfying the Segal conditions. Now if you are given an $\infty$-category $\mathcal{C}$ you can define... | 2 | https://mathoverflow.net/users/43054 | 404229 | 165,792 |
https://mathoverflow.net/questions/403965 | 4 | Let $A$ be a $C^\*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}\_A(E)$ be an adjointable operator satisfying $t=t^\*$. Is it true that
$$\|t\| = \sup\_{z \in E, \|z\| = 1} \|\langle tz,z\rangle\|\_A?$$
Obviously, if we denote the supremum on the right by $M$, we have $M \le \|t\|$ by the Cauch... | https://mathoverflow.net/users/216007 | $\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$ | I'll post an answer based on the comments above.
Since $t$ is self-adjoint, we can write $t= t\_+-t\_-$ where $t\_+$ and $t\_-$ are positive elements with $t\_+ t\_- = 0$. The latter condition ensures that $\|t\| = \max\{\|t\_+\|, \|t\_-\|\}$.
Assume, without loss of generality, that $\|t\| = \|t\_+\|$.
Let $\eps... | 2 | https://mathoverflow.net/users/216007 | 404233 | 165,793 |
https://mathoverflow.net/questions/404226 | 8 | It was known that the James space $J$ has separable second conjugate, is non-reflexive and isometric to its second conjugate. I want to know whether there are Banach spaces $X$ with separable second conjugates $X^{\*\*}$, but $X$ is not a dual space (the James space $J$ is a dual space). Furthermore, are there any refe... | https://mathoverflow.net/users/41619 | Banach spaces whose second conjugates are separable | Yes, there are such spaces. To see this, first note that Joram Lindenstrauss [showed](https://link.springer.com/article/10.1007/BF02771677) that for every separable Banach space $Y$ there exists a Banach space $X$ such that $X^{\ast\ast}$ is separable and $X^{\ast\ast}/X$ is isomorphic to $Y$. Apply this result with $Y... | 11 | https://mathoverflow.net/users/848 | 404243 | 165,798 |
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