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https://mathoverflow.net/questions/356749 | 14 | I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U\_i\}$ of $X$, an infinity local system on a connected space $X$ assigns:
-a chain complex $C\_i$ to every contractible open set $U... | https://mathoverflow.net/users/59235 | Infinity local systems | One way to discuss $\infty$-local systems over a space $X$ is in terms of the *fundamental $\infty$-groupoid* of $X$. To motivate this, recall that for classical local systems, one has the equivalence of categories
$$LocSys(X) \simeq Rep(\pi\_{1}(X)) $$
between the local systems on $X$ and representations of the fundam... | 7 | https://mathoverflow.net/users/165286 | 371611 | 155,356 |
https://mathoverflow.net/questions/370972 | 10 | Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the antidiagonal quotient: $(g.p, x)\sim (p, g.x)$.
**Q1:** When is $P\times^G X$ trivial? I.e., when do we have an isomorphism $P \t... | https://mathoverflow.net/users/86614 | When is a twisted form coming from a torsor trivial? | This lemma might hold in a more general topos-theoretic context, but for "safety" I'm going to formulate it in a more restricted setting.
**Lemma**:Let $G$ be a group scheme with an action $\rho: G \times X \to X$ corresponding to a morphism $\varphi: G \to \mathrm{Aut}(X)$. Assume $G, \mathrm{Aut}(X)$ are smooth ove... | 3 | https://mathoverflow.net/users/113296 | 371624 | 155,359 |
https://mathoverflow.net/questions/371560 | 5 | Let $S$ be a smooth surface and $\gamma\_1, \gamma\_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $\gamma\_1'$ on $S$ isotopic to $\gamma\_1$ and such that $\#(\gamma\_1\cap \gamma\_2)>\#(\gamma\_1'\cap \gamma\_2)$.
**Question.** Is it true that there is a ... | https://mathoverflow.net/users/13441 | Untangling two simple closed curves on a surface | This is also proved as Lemma 3.1 in
Joel Hass and Peter Scott,
Intersections of curves on surfaces, Israel Journal of Mathematics 51 (1985), 90–120. <https://doi.org/10.1007/BF02772960>
| 3 | https://mathoverflow.net/users/24076 | 371632 | 155,363 |
https://mathoverflow.net/questions/371536 | 5 | Free distributive lattices on a finite set exist and are finite, while free modular lattices on a finite set exist but are not finite when the set has at least 4 elements.
>
> **Question**: Is there a class (presumably, a variety in the sense of universal algebra) $C$ of lattices larger than the class of distributi... | https://mathoverflow.net/users/61949 | On free lattices | A variety is called *locally finite* if it has the property that its finitely generated algebras are finite. This is equivalent to the property that its finitely generated free algebras are finite. So, the questions may be rewritten as:
(1) Is there a locally finite variety of lattices larger than the variety of distri... | 7 | https://mathoverflow.net/users/75735 | 371641 | 155,364 |
https://mathoverflow.net/questions/371625 | 3 | For an $A\_\infty$ algebra (say I do everything over $\mathbb{Q}$) modeled by dg-algebras I can understand the definition of formality (that its homology groups is equivalent in a $A\_\infty$ way to the original algebra). I imagine that this definition also extends to the Lurie/spectra setting? As in take homotopy grou... | https://mathoverflow.net/users/136287 | What is an example of a non-formal $A_\infty$ algebra over a discrete ring? | Anyone with a non-trivial Massey product, e.g. the rational cochain algebra of the complement of the Borromean rings in $S^3$. Presented in this way, it is strictly associative. If you take a minimal model, the ternary operation $m\_3$ can't be trivial because it represents triple Massey products.
| 6 | https://mathoverflow.net/users/12166 | 371642 | 155,365 |
https://mathoverflow.net/questions/371335 | 3 | Let $\mathbb{G}$ be a connected reductive $\mathbb{F}\_q$ algebraic group over its algebraic closure $\bar{\mathbb{F}\_q}$, and $\mathbb{T}$ be an $\mathbb{F}\_q$-defined maximal torus. Let $\Phi$ be the root system of $\mathbb{G}$ wrt $\mathbb{T}$, and, given $g\in\mathbb{T}$, put $$\Phi(g)=\lbrace\alpha\in \Phi:\alph... | https://mathoverflow.net/users/14443 | Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type | As @LSpice already pointed out, you need $q$ to be sufficiently large even in the case of a Levi subgroup. Just take $G = \operatorname{GL}\_n(\overline{\mathbb{F}}\_q)$ and $G^F = \operatorname{GL}\_n(\mathbb{F}\_q)$ under the usual Frobenius endomorphism. If $T \leqslant G$ is the maximal torus of diagonal matrices t... | 5 | https://mathoverflow.net/users/22846 | 371645 | 155,367 |
https://mathoverflow.net/questions/371637 | 5 | Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E\_2$ algebra $HH^{\bullet}(A)$ which I'll just call $H\_A$. We know by HKR it carries $T\_A$ the tangent module in cohomological degree 1 and we... | https://mathoverflow.net/users/136287 | What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module? | (Incorporated David Ben-Zvi's comments below.)
An $E\_2$-$\mathrm{HH}^\bullet(A)$-module is the same as a left module over the algebra of Hochschild chains $\mathrm{HH}\_\bullet(\mathrm{HH}^\bullet(A))$.
Now assume $k$ has characteristic zero. It was shown by Tamarkin and Tsygan (see Theorem 2.7.1 in "The ring of d... | 5 | https://mathoverflow.net/users/18512 | 371647 | 155,369 |
https://mathoverflow.net/questions/371517 | 3 | Let $X$ be an algebraic space locally of finite presentation, and let $\tilde{X}$ denote the restriction of $X$ (as a functor on schemes) to the category of complete local rings. Is it true that the mapping $X \mapsto \tilde{X}$ (of algebraic spaces to functors on complete local rings) is a fully faithful functor?
I.... | https://mathoverflow.net/users/58651 | Algebraic spaces as functors on complete local rings | There is an exponential map from complex line to itself. This is not algebraic, but it is defined on complete local ring valued points. So the functor does not appear to be full.
| 5 | https://mathoverflow.net/users/148928 | 371651 | 155,371 |
https://mathoverflow.net/questions/371333 | 1 | Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C\_\alpha(\mu,\nu) = \inf\_\sigma \frac{W\_p(\mu+\sigma,\nu+\sigma)}{W\_p(\mu,\nu)},$$
where $W\_p$ is the $p$-Wasserstein distance and the infimum is taken over all nonnegative measures ... | https://mathoverflow.net/users/141188 | Scaling behavior of Wasserstein distances | Actually, we have $C\_\alpha =0$ for any $\alpha>0$. Indeed, let $\mu\_t = (1-t)\delta\_{x\_0} + t\delta\_0$ and $\nu\_t = (1-t)\delta\_{x\_0} + t\delta\_1$ for some $t\in (0,1)$ and $x\_0$ far away enough from $0$ and $1$. Then $W\_p^p(\mu\_t,\nu\_t)= t$, whereas if $\sigma = t\sum\_{k=1}^{n} \delta\_{k/(n+1)}$, then ... | 0 | https://mathoverflow.net/users/141188 | 371655 | 155,372 |
https://mathoverflow.net/questions/371654 | 4 | I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me:
We need to prove that if $A$ is a $C^\*$-algebra then $A^{\*\*}$ is isometrically isomorphic to $\pi\_u(A)''$, where $(H\_u,\pi\_u)$ is the... | https://mathoverflow.net/users/164203 | Normal linear functionals on bicommutants of C*-algebras |
>
> Doesn't this imply that all normal functionals on $\pi\_u(A)''$ are linear combinations of vector states, hence (SOT) continuous?
>
>
>
It sounds like you understand the proof, but are suspicious of this consequence. Not to worry, the proof is correct, and yes, in general ultraweakly continuous linear functi... | 10 | https://mathoverflow.net/users/23141 | 371664 | 155,375 |
https://mathoverflow.net/questions/371670 | 6 | This is a follow-up on [this (answered) question on math.SE](https://math.stackexchange.com/questions/3802588/are-lattice-operations-continuous-in-the-lipschitz-norm), but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited above.
Denote... | https://mathoverflow.net/users/160379 | Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology? | Yes. If $f\_n \to f$ weak\* then the sequence $(f\_n)$ must be bounded in ${\rm Lip}\_0(X)$ (Banach-Steinhaus), and for bounded nets weak\* convergence is the same as pointwise convergence. So $f\_n \to f$ boundedly pointwise, which easily implies the same of the positive parts.
Let me also correct a couple of inaccu... | 10 | https://mathoverflow.net/users/23141 | 371674 | 155,378 |
https://mathoverflow.net/questions/366510 | 4 | All manifolds considered here are compact and orientable. A 3-manifold (with possible boundary) is *irreducible* if any smooth sphere bounds a ball. Note that a closed irreducible 3-manifold is prime, and a closed prime 3-manifold is irreducible unless it's $S^1\times S^2$.
Suppose I remove a collection of thickened ... | https://mathoverflow.net/users/12310 | Irreducibility of 3-manifolds with (non)empty boundary | You're asking how reducibility/irreducibility behaves under drilling and [filling](https://en.wikipedia.org/wiki/Dehn_surgery). I think you've captured the essence of drilling: if a link is "sphere busting" in a reducible manifold (meets every essential sphere up to isotopy), and doesn't have components lying in a ball... | 6 | https://mathoverflow.net/users/1345 | 371688 | 155,381 |
https://mathoverflow.net/questions/359523 | 5 | The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" page 79.
These forms are used in Donaldson's "ANTI SELF-DUAL YANG-MILLS CONNECTIONS OVER COMPLEX ALGEBRAIC SURFACES AN... | https://mathoverflow.net/users/102114 | Derivative of the Bott-Chern forms | In order to construct the Bott-Chern class $\widetilde{ch}(h\_0,h\_1)$ we deform the Id complex over $\mathbb{P}^1$
\begin{equation\*}
0 \rightarrow (E,\tilde{h}) \rightarrow (E,h\_1) \rightarrow 0.
\end{equation\*}
so that $i\_0^\*\tilde{h}=h\_0$ and $i^\*\_{\infty}\tilde{h}\simeq h\_1$. We get then
\begin{equation\*}... | 0 | https://mathoverflow.net/users/102114 | 371726 | 155,395 |
https://mathoverflow.net/questions/371722 | 1 | I need a reference for field extension and Galois extension (like an introduction) please.
Thank you.
| https://mathoverflow.net/users/165372 | Reference book for Galois extension | There are many references for such topics. As @DieterKadelka suggested, the book by Lang is a really good reference. You may also be interested in the following ones:
*Bastida, J. R. (1984). Field Extensions and Galois Theory. Cambridge University Press*
*Edwards, Harold M. (1984). Galois Theory. Graduate Texts in ... | 0 | https://mathoverflow.net/users/160051 | 371727 | 155,396 |
https://mathoverflow.net/questions/371485 | 1 | In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$
$$R\_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1\_{x\geq y},$$
which is the Poisson distribution pdf. (This is also in semigroup form of the operator $I+\nabla^{-}$:
$$R\_{t}(x,y)=e^{-t(I+\nabla^{-})}(x,y),$$
where $If(x)=f(... | https://mathoverflow.net/users/99863 | Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme) | If you insist on $P\_t(x,y) = c(t,x-y) \mathbb{1}\_{1\geqslant x-y \geqslant 0}$, then it is not possible to factorise $R\_t$ as $P\_t Q\_t$ for whatever "reasonable" $Q\_t$. Indeed, this would mean that the corresponding Z-transforms satisfy
$$ \sum\_{x=0}^\infty R\_t(x,0) z^n = \sum\_{x=0}^\infty P\_t(x,0) z^n \times... | 1 | https://mathoverflow.net/users/108637 | 371729 | 155,397 |
https://mathoverflow.net/questions/371716 | 5 | I was recently trying to understand [generalized linear models](https://en.wikipedia.org/wiki/Generalized_linear_model) (GLMs) and after investing quite a few days, it still hasn't dawned on me what the fundamental benefit of the framework is. Normally, I am used to results like guarantees of convergence, limits for er... | https://mathoverflow.net/users/165368 | Generalized linear models: What's the benefit of the underlying theory? | What are the benefits of a unified framework? You are right that we are rapidly going into some much used special cases line logistic regression or Poisson regression, but there is still benefit in having a common framework.
* Technology transfer from the *general linear model* (**not** generalized!), that is with ga... | 8 | https://mathoverflow.net/users/6494 | 371742 | 155,400 |
https://mathoverflow.net/questions/371749 | -1 | As $M \to +\infty$, how could I make a good asymptotic analysis of this integral?
>
> $$\int\_0^1 \dfrac{\cos(M x)}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x$$
>
>
>
The exponential term shall dominate, yet I have no clue in who to deal with $\cos(Mx)$. I tried to apply geometric series for $\frac{1}{1+x^2}$ but ... | https://mathoverflow.net/users/88816 | Asymptotic expansion / analysis of this integral | For large $M$ the integrand contributes in the range $x\lesssim 1/\sqrt M$, so we can neglect the denominator $1+x^2$. The integral then has a closed form expression,
$$\int\_0^1 \cos (M x) e^{-M (x^2 - 1/9)}\ \text{d}x=\tfrac{1}{4}\sqrt{\pi }M^{-1/2}e^{M/9} e^{-M/4} \left[\text{erf}\left(\left(1+\tfrac{i}{2}\right) \s... | 2 | https://mathoverflow.net/users/11260 | 371758 | 155,404 |
https://mathoverflow.net/questions/371622 | 7 | Fermat famously showed that the only primes $p$ of the form $x^2 + y^2$ are the primes such that $p \equiv 1 \mod{4}$. Furthermore, we now know “effective” versions of Fermat's theorem, i.e. given a prime $p$ such that $p \equiv 1 \mod{4}$, we know how to find integers $x$, $y$ such that $x^2 + y^2 = p$ in time polynom... | https://mathoverflow.net/users/7400 | Expressing primes $p\equiv 1 \pmod 3$ in the form $p = x^2 + xy + y^2$ | This is an elaboration of the answer that Noam Elkies provided in the comments.
Suppose that $p=x^2 + xy + y^2$. Then note that $x$ and $y$ are small relative to $p$ (at most half as many digits). Note also that if $\zeta \not\equiv 1\pmod p$ satisfies $\zeta^3 \equiv 1\pmod p$ then $\zeta^2 + \zeta + 1 \equiv 0 \pmo... | 15 | https://mathoverflow.net/users/3106 | 371762 | 155,405 |
https://mathoverflow.net/questions/371732 | 3 | Let $R= k[x\_{11} , x\_{12} \dotsc , x\_{nm}]$ denote the coordinate ring of a generic $n \times m$ matrix, $M$. It is well known that under the standard diagonal term order, the ideal of maximal minors of $M$ forms a Grobner basis (I am aware that it is a universal Grobner basis, but I don't need that result).
I am ... | https://mathoverflow.net/users/73780 | "Classical" proof that maximal minors form a Grobner basis under diagonal term order | Here are a few articles that might be relevant:
[Narasimhan, *The irreducibility of ladder determinantal varieties*](https://mathscinet.ams.org/mathscinet-getitem?mr=853237), from 1986. It takes about 20 pages to prove the result (that the minors are a standard basis for the ideal they generate). I confess, I don't w... | 3 | https://mathoverflow.net/users/88133 | 371764 | 155,406 |
https://mathoverflow.net/questions/371792 | 2 | Let $X \overset{f^\prime}\longrightarrow S^\prime \overset\pi\longrightarrow S$ be morphisms of schemes such that $f^\prime$ is proper with geometrically connected fibers and $\pi$ is integral. Set $f = \pi \circ f^\prime$. Is it true that $S^\prime$ is the normalization of $S$ in $X$.
| https://mathoverflow.net/users/129738 | Uniqueness of Stein factorization | No. Take $S=S’$ a cuspidal curve, $\pi$ the identity, and $f’=f: X\to S’=S$ the normalization.
| 8 | https://mathoverflow.net/users/3847 | 371794 | 155,414 |
https://mathoverflow.net/questions/371798 | 0 | Let $(c\_{nr})$ be an $N\times R$ complex matrix, then $\forall z\_n \in \mathbb{C}$, we have
$$
\sum\_r \Big|\sum\_n c\_{nr}z\_n\Big|^2 \geq \frac{1}{\sigma\_{max}} \sum\_n |z\_n|^2
$$
where $\sigma\_{max}$ is the maximal sigular value of the complex matrix $(c\_{nr})$.
| https://mathoverflow.net/users/41499 | Could someone help me to prove or disprove the following inequality? | The $N\times R$ matrix $C$ has elements $c\_{nr}$, the $N\times N$ matrix $Z$ has elements $z\_n \bar{z}\_m$, and $C^\ast$ is the conjugate transpose of $C$. The Hermitian matrix product $CC^\ast$ has eigenvalues $\sigma\_n^2$, with $\sigma\_n\geq 0$, $n=1,2,\ldots N$ the set of singular values of $C$. Then we have
$$\... | 4 | https://mathoverflow.net/users/11260 | 371800 | 155,416 |
https://mathoverflow.net/questions/371812 | 2 | Does anybody know if there exists results on the probability distribution of the Hilbert Schmidt inner product of random unitary matrices?
To be more specific, given two random isotropically distributed unitary matrices $U\_1 \in \mathbb{C}^{n \times n}$ and $U\_2 \in \mathbb{C}^{n \times n}$, is something known abou... | https://mathoverflow.net/users/165441 | Hilbert Schmidt inner product of random isotropic unitary matrices | I assume the matrices $U\_1$ and $U\_2$ are independently uniformly distributed in the unitary group. The product $V=U\_1^HU\_2$ is then itself uniformly distributed in the unitary group. The distribution of the trace ${\rm Tr}\,V=\sum\_{p=1}^n e^{i\phi\_p}$ follows from the known joint distribution of the eigenvalues ... | 3 | https://mathoverflow.net/users/11260 | 371813 | 155,417 |
https://mathoverflow.net/questions/371817 | 0 | Assume $Q$ is a positive definite random matrix such that $0 < \lambda\_{\min}(Q)....\leq \lambda\_{\max}(Q) \leq 1$ holds. I want to show that
\begin{align}
E\left[\frac{\lambda\_{\min}(Q)}{\lambda\_{\min}(Q)+\lambda\_{\max}(Q)}\right] \geq \frac{\lambda\_{\min}(E[Q])}{\lambda\_{\min}(E[Q])+\lambda\_{\max}(E[Q])}
\end... | https://mathoverflow.net/users/165450 | Expectation of random matrix | $\newcommand\lmax{\lambda\_{\max}(P)}\newcommand\lmin{\lambda\_{\min}(P)}$Your first displayed inequality for all positive definite random matrices $Q$ means exactly that the function
$$P\mapsto\frac\lmin{\lmin+\lmax}$$
on the set of all positive definite matrices is convex. Looking at just the diagonal positive defini... | 2 | https://mathoverflow.net/users/36721 | 371823 | 155,421 |
https://mathoverflow.net/questions/371808 | 5 | How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?
Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}\_p$, there is an interesting non-toral elementary $p$-subgroup considered by Vistoli in [this paper](https://arxiv.org/abs/math/0505052).... | https://mathoverflow.net/users/100553 | Elementary $p$-subgroups of a compact Lie group | This is a topic with a long history going (at least) back to Borel, Serre, Steinberg and others. The existence of *non-toral* elementary abelian $p$-subgroups (i.e. subgroups not contained in a maximal torus of $G$) is equivalent to $H\_\*(G;\mathbb{Z})$ having $p$-torsion. Newer results include R. L. Griess' paper "[E... | 12 | https://mathoverflow.net/users/65801 | 371826 | 155,423 |
https://mathoverflow.net/questions/371788 | 3 | The standard definition of a power object seems to be: objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \hookrightarrow X \times B$, there is a unique morphism $\chi\_r: B \to \mathcal{P}X$ such that $r$ is the pullback of $\in$ along $\... | https://mathoverflow.net/users/136473 | Alternative definition of power object in a category | Let's first consider the analogous case of the exponential object $Y^X$. You might want to give an "alternative" definition that an exponential object should be an object $Y^X$ together with an evaluation map $\text{eval} : Y^X \times X \to Y$ such that for every morphism $f : X \to Y$ there is a point $r : 1 \to Y^X$ ... | 6 | https://mathoverflow.net/users/290 | 371832 | 155,425 |
https://mathoverflow.net/questions/371828 | 9 | Does there exists a knot $K\subset \mathbb{S}^3$ such that
1. $K$ is **not** slice
2. $\exists W^4$, $\partial W = \mathbb{S}^3$ *rational homology ball*
3. $\exists $ properly embedded smooth disk $(D,\partial D)\to (W,K)$. ?
In other words $K$ is not slice in $B^4$ but is slice in some rational homology ball.
| https://mathoverflow.net/users/99042 | Rational slice knot that is not slice | Yes. The figure-eight knot is an example: it bounds a smooth slice disk in a rational homology ball. This has been proven in a bunch of different ways, going back to the 1980s. Here are a couple of relevant references:
*Fintushel, Ronald; Stern, Ronald J.*, A (\mu)-invariant one homology 3-sphere that bounds an orien... | 12 | https://mathoverflow.net/users/65952 | 371833 | 155,426 |
https://mathoverflow.net/questions/371835 | 0 | Based on [this](https://mathoverflow.net/questions/309562/can-cardinality-be-defined-with-essentially-no-practical-restriction-on-non-well?answertab=active#tab-top) answer. Working in ZF-Reg., is it possible to have a non-well founded set that is strictly supernumerous to any well founded set?
If that is possible, th... | https://mathoverflow.net/users/95347 | Can we have such an ill-founded model of ZF-Reg.? | No. Hartogs' theorem does not depend on the axiom of regularity. Once you have Replacement, and every well-ordered set is isomorphic to a von Neumann ordinal, which is a well-founded set, you cannot have that.
If you remove Replacement, sure. Start with a set of atoms, $A$, of size $V\_{\omega+\omega+1}$, then cut of... | 4 | https://mathoverflow.net/users/7206 | 371840 | 155,427 |
https://mathoverflow.net/questions/371672 | 8 | I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $A$ is a measurable function from a fixed probability space $(Ω,F,P)$ to a measurable space $(X,E)$. But that means that... | https://mathoverflow.net/users/50073 | Self-contained formalization of random variables? | **Proposition:** Let $\kappa$ be some infinite number cardinal number. There exists a probability space $(\Omega,\Sigma,\nu)$ that carries $\kappa$ independent random variables with uniform distribution on $[0,1]$ and such that
such for every family $\langle g\_i\rangle\_{i\in I}$ of real-valued random variables with $... | 2 | https://mathoverflow.net/users/35357 | 371843 | 155,429 |
https://mathoverflow.net/questions/371635 | 7 | Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along $Z$ is once again normal. Any reference will be most welcome.
| https://mathoverflow.net/users/32151 | General conditions for normality of blow-up | Let $X=Spec(R)$. Blowing-up $Z=V(I)$ is the same as to look at $Proj$ of the graded ring $R[It]=\oplus\_{j\geqslant 0} I^jt^j\subset R[t]$, the Rees ring associated to $I$.
Assume $R$ is a domain, integrally closed inside its fraction field $K$. In this case the integral closure of $R[It]$ inside its fraction field i... | 7 | https://mathoverflow.net/users/50468 | 371857 | 155,433 |
https://mathoverflow.net/questions/371602 | 6 | In my previous question [Set-theoretic geology: controlled erosion?](https://mathoverflow.net/questions/370411/set-theoretic-geology-controlled-erosion)
and the great answer by Jonas Reitz, I have learned a few things, starting from the awareness that I understand the fine-grain structure of Set Theoretic Geology even ... | https://mathoverflow.net/users/15293 | Set Theoretic Geology II: The structure of the directed partial order of grounds | Mirco, this is also a fantastic question - the structure of the grounds as a partial order seems to be a very basic aspect of forcing that is not entirely understood. Once again I don’t have a complete answer, but I can provide some background & a few observations.
**Intersection of grounds.** As pointed out in the c... | 5 | https://mathoverflow.net/users/10671 | 371862 | 155,436 |
https://mathoverflow.net/questions/371864 | 5 | In [Schwänzl and Vogt, *Strong cofibrations and fibrations in enriched categories*], the authors refer to an earlier preprint, [Schwänzl and Vogt, *Cofibration and fibration structures in enriched categories*] but give a URL that no longer works. Is this preprint still available somewhere online?
(It seems the prepri... | https://mathoverflow.net/users/11640 | Schwänzl and Vogt, Cofibration and fibration structures in enriched categories | Yes, it is also available here:
<https://www.math.uni-bielefeld.de/sfb343/preprints/pr97044.ps.gz>
| 7 | https://mathoverflow.net/users/402 | 371872 | 155,437 |
https://mathoverflow.net/questions/371863 | 1 | An agent $A$ is performing a random walk on the number line. Let $X\_t$ be his position at time $t$. $X\_{t+1}$ is calculated according to the following rules:-
$ X\_{t+1} =$
\begin{cases}
1 + X\_{t} & \text{with probability $a$} \\
\alpha(1 + X\_{t}) & \text{with probability $b$} \\
\beta(1+X\_t) & \text{with pro... | https://mathoverflow.net/users/120939 | A scaled random walk on the number line | It is known that:
* if $b\log \alpha+ c\log \beta>0$, then $X\_t\to\infty$ almost surely,
* if $b\log \alpha+ c\log \beta<0$, then $X\_t$ is recurrent and has a unique stationary measure, with (if $c>0$) tail $\mu([t,+\infty))\sim c t^\kappa$ for some $c>0$, $\kappa<0$. One can also derive a Central Limit Theorem, an... | 2 | https://mathoverflow.net/users/4961 | 371881 | 155,438 |
https://mathoverflow.net/questions/371883 | 2 | We say that two disjoint, non-empty subsets $S, T$ of an infinite cardinal $\kappa$ are *neighboring* if there is $\alpha\in \kappa$ such that $$S\cap\{\alpha,\alpha+1\} \neq \varnothing \neq T\cap\{\alpha, \alpha+1\}.$$
Given an infinite cardinal $\kappa$, is there a partition ${\cal B}$ of $\kappa$ with $|{\cal B}|=\... | https://mathoverflow.net/users/8628 | Partitioning an infinite cardinal $\kappa$ into pairwise neighboring subsets | Yes. List the pairs $(\alpha,\beta)$ with $\alpha<\beta<\kappa$ as $(\alpha\_\lambda,\beta\_\lambda), \lambda<\kappa$.
Then construct the sets $B\_\alpha\in\mathcal B, \alpha<\kappa$ as follows:
At stage 0, all $B\_\alpha=\emptyset$.
At limit stages just take unions.
At successor stages $\lambda+2n, n\in\omega, n... | 5 | https://mathoverflow.net/users/4600 | 371889 | 155,439 |
https://mathoverflow.net/questions/371715 | 13 | Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.
Is there a similar explicit, geometric definition of the addition law on (a family of?) abelian surfaces?
So the question is really: Give a n... | https://mathoverflow.net/users/58001 | A geometric definition of the addition law on abelian surfaces | Jacobians of genus-2 curves - and abelian surfaces in general, I suppose - can be realized as the variety of lines on the intersection of two quadrics in $\mathbb{P}^5$ (once you've chosen a line to act as the neutral element). This is analogous to seeing an elliptic curve as the variety of 0-dimensional spaces (i.e. p... | 8 | https://mathoverflow.net/users/156215 | 371890 | 155,440 |
https://mathoverflow.net/questions/371917 | 3 | Let $\mathcal{P}\_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum\_{i=1}^n k\_i \delta\_{x\_i}$. Then any measure in $\mathcal{P}\_n(\mathbb{R})$ is in the image of the map on $\Delta\_n \times \mathbb{R}^n$, where $\Delta\_n$ is the $n$-simplex, taking $(k\_1,\dots,k\_n)\times ... | https://mathoverflow.net/users/36886 | Continuous selection parameterizing discrete measures | $\newcommand\de\delta\newcommand\De\Delta$
The answer is no. Suppose for simplicity that $n=2$ (the case $n>2$ is handled similarly). Suppose that $g$ is a right inverse in question. Then $g(\de\_0)=((p,q),(0,0))$ for some $(p,q)\in\De\_2$. Take now any $(s,t)\in\De\_2\setminus\{(p,q),(q,p)\}$. Then for each natural $k... | 2 | https://mathoverflow.net/users/36721 | 371924 | 155,451 |
https://mathoverflow.net/questions/357753 | 3 | The following theorem can be found in Hardy-Wright (Theorem 459), except that they state it only for $d=2$. Do you know of a reference where the proof of this general statement is written?
**Theorem:** *Let $d\ge2$ be an integer. Let $F$ be a bounded subset of $\Bbb R^d$. For every positive real number $r$, denote by... | https://mathoverflow.net/users/56097 | Reference request: probability that d numbers are coprime | Let $F$ be a bounded subset of $\mathbb R^d$ with $d \geq 2$. We define $F\_r := rF \cap \mathbb Z^d$ for any real number $r>0$ and assume that the limit
$$
\mathcal V(F) := \lim\_{r \to + \infty} \frac{|F\_r|}{r^d}
$$
exists (for convex subsets, this is the Lebesgue volume of $F$). We rewrite the proof of Theorem 459... | 1 | https://mathoverflow.net/users/56097 | 371932 | 155,452 |
https://mathoverflow.net/questions/371926 | 2 | Let $\mathcal{P}\_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum\_{i=1}^n k\_i \delta\_{x\_i}$ where $k\_i>0$. Then any measure in $\mathcal{P}\_{n:+}(\mathbb{R})$ is in the image of the map on $\Delta\_n \times \mathbb{R}^n$, where $\Delta\_n$ is interior of the $n$-simple... | https://mathoverflow.net/users/36886 | Covering of discrete probability measures | This map is not a covering one, because the preimages of singleton sets under this map are not of the same cardinality. E.g., the cardinality of the preimage of the singleton set $\{\frac1n\,\sum\_{j=1}^n\delta\_j\}$ is $n!$, whereas the preimage of the singleton set $\{\delta\_0\}$ is of infinite cardinality.
---
... | 2 | https://mathoverflow.net/users/36721 | 371934 | 155,453 |
https://mathoverflow.net/questions/371753 | 5 | The category of commutative monoid objects in a symmetric monoidal category is cocartesian, with their tensor product serving as their coproduct. This sort of result seems to date back to here:
* Thomas Fox, Coalgebras and Cartesian categories, *Commun. Algebra* **4** (1976), 665–667.
I'm working on a paper with To... | https://mathoverflow.net/users/2893 | Is the tensor product of symmetric pseudomonoids their coproduct? | The result I wanted is Theorem 5.2 here:
Daniel Schäppi, [Ind-abelian categories and quasi-coherent sheaves](https://arxiv.org/abs/1211.3678), *Mathematical Proceedings of the Cambridge Philosophical Society*, **157** (2014), 391–423. [doi:10.1017/S0305004114000401](https://www.cambridge.org/core/journals/mathematica... | 4 | https://mathoverflow.net/users/2893 | 371941 | 155,455 |
https://mathoverflow.net/questions/278626 | 2 | Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:
$\operatorname{li}^{-1}(n)=n\sum \_{i=1} a\_i(-1)^{i+1} W\_{-1}\left(-e/n\right){}^{2-i} $
where $a\_i$ begins: $\small{-1, 0, 1, 3, 11, 1... | https://mathoverflow.net/users/45057 | Expansion of inverse logarithmic integral in terms of lambert w | Let me elaborate on reuns' idea and show how one can find coefficients $a\_i$ by solving a certain ODE.
Let's define:
$$f(z) := \sum\_{i\geq 3} a\_i z^{i-3}$$
so that we get a functional equation:
$$\mathrm{li}\big( -x(\frac{1}{t}+tf(t)) \big) = x,$$
where $t=t(x):=-\frac{1}{W\_1(-e/x)}$.
Differentiating this equatio... | 1 | https://mathoverflow.net/users/7076 | 371943 | 155,457 |
https://mathoverflow.net/questions/371946 | 3 | I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that oscilates, changes regularly signs, periodic if it is necessary...).
Thanks in advance.
| https://mathoverflow.net/users/76102 | Arithmetical function comparable to sine function | The [Liouville function](https://en.wikipedia.org/wiki/Arithmetic_function#%CE%BB(n)_%E2%80%93_Liouville_function) ($-1$ to the power of the number of prime divisors of $n$) could be a candidate for a "sine-like" function. It's not periodic, but it does oscillate, changing sign infinitely often, and has the same range ... | 5 | https://mathoverflow.net/users/11260 | 371950 | 155,458 |
https://mathoverflow.net/questions/371940 | 16 | I've never seen an example of a category with a subobject classifier which didn't embed nicely into a topos. Is there a good reason for this?
**Question 1:** Let $\mathcal C$ be a category with a subobject classifier $\Omega$ (and whatever finite limits this entails -- namely, a terminal object and pullbacks along mo... | https://mathoverflow.net/users/2362 | Does every category with a subobject classifier embed into a topos? | Ivan's example in the comment actually proves that all the questions have negative answers.
As observed by Ivan, in the category of pointed set, there is a subobject classifier given by $\{\*\} \to \{\*,\bot \}$, where $\*$ is the special point.
Indeed, a subobject of $X$, is just a subset of $X$ containing $\*$ so... | 17 | https://mathoverflow.net/users/22131 | 371957 | 155,460 |
https://mathoverflow.net/questions/371929 | 6 | I've come up with the following piece of Python code (using the library Sympy):
```
def double_diagonalize(m1, m2):
V, _ = (m1.T * m2).diagonalize()
U, _ = (m1 * m2.T).diagonalize()
return U, V
```
What I've found is that given many (but not all) random pairs of $
n \times n$ matrices $M\_+$ and $M\_-$... | https://mathoverflow.net/users/75761 | Double-diagonalisation of nxn matrices? | It's the solution of the [product eigenvalue problem](https://doi.org/10.1137/S0036144504443110) $M\_1^T M\_2$, once you transpose the second relation.
| 3 | https://mathoverflow.net/users/1898 | 371982 | 155,467 |
https://mathoverflow.net/questions/371972 | 8 | Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:
* $|p|$ is strictly increasing on $[1,c]$
* and $|b \cdot p(c)| < |p(0)|$?
This might be satisfied by an interpolating polynomial, but how to actually construct it is beyond me.
| https://mathoverflow.net/users/157462 | Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? | No, it's not possible to construct such a $p$ whose degree $n$ is bounded independently of $b$ and $c$. In fact, it's not possible even if we fix the value of $c$. I'll prove this for $c=2$ below, but the same argument works in general.
Suppose to the contrary that it were possible. Then for every $b>1$ we could choo... | 11 | https://mathoverflow.net/users/126183 | 371986 | 155,469 |
https://mathoverflow.net/questions/371978 | 4 | I originally asked this on [Stack Exchange](https://math.stackexchange.com/questions/3821785/r-i-cong-r-textann-rr-i-but-i-neq-textann-rr-i) but with no luck. Upon doing research with some noncommutative rings, I thought of a curious question. Does there exist a noncommutative unital ring $R$ and left ideal $I$ such th... | https://mathoverflow.net/users/142858 | $R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$ | You might as well assume that $\mathrm{Ann}\_R(R/I)=0$ since if $S=R/\mathrm{Ann}\_R(R/I)$ then $R/I\cong S$ as $R$-modules if and only if $S/(I/\mathrm{Ann}\_R(R/I))\cong S$ as $S$-modules.
So now the question is whether you can have a ring $R$ and a non-zero left ideal $I$ such that $R/I\cong R$. Equivalently can t... | 4 | https://mathoverflow.net/users/345 | 371990 | 155,470 |
https://mathoverflow.net/questions/371992 | 3 | Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with metric $d\_X$. Let $x\_1,\dots,x\_n \in X$ for some natural $n>0$.
Let $\emptyset\neq Z\subseteq X$ be a compact subset of... | https://mathoverflow.net/users/36886 | Measurable selection for argmin to distance | Here is a simple direct argument. For $i=1,\ldots,n$, let $$C\_i=\{z\in Z\mid d(z,x\_i)\leq d(z,x\_j), j=1,\ldots,n\}.$$
Clearly, each $C\_i$ is closed and hence measurable. Let $M\_i=C\_i\setminus\bigcup\_{l=1}^{i-1}C\_l$. The nonempty sets of the form $M\_i$ form a finite measurable partition of $Z$. Now let $S$ map ... | 4 | https://mathoverflow.net/users/35357 | 371995 | 155,472 |
https://mathoverflow.net/questions/371963 | 5 | If one wants to understand representations of $\mathfrak{g}$ (a finite dimensional semisimple Lie algebra) of weight $\lambda$, the happiest you could be is if $\lambda+\rho$ is (integral) regular dominant, i.e. it's an element of the weight lattice whose product $\langle \lambda+\rho,\check{\alpha}\rangle$ with every ... | https://mathoverflow.net/users/119012 | Beilinson-Bernstein for nonintegral levels | Integrality doesn't really seem to play a role in the statement of Beilinson-Bernstein as far as I can tell. For a general weight $\lambda \in \mathfrak h^\ast$ it makes sense to talk about the notions of dominant and regular.
If $\lambda$ is regular, then there is a derived localization:
$D(D\_{G/B}^\lambda-mod) \... | 3 | https://mathoverflow.net/users/7762 | 372010 | 155,474 |
https://mathoverflow.net/questions/372006 | 1 | In a [paper](https://www.researchgate.net/publication/226613874_A_Simple_Proof_of_Two_Generalized_Borel-Cantelli_Lemmas) in *Lecture Notes in Mathematics vol. 1874*, Yan states the Kochen-Stone theorem in the following form, where $A\_n$ is a sequence of events such that $\sum\_{n=1}^\infty P(A\_n) = \infty$:
$$
P(A... | https://mathoverflow.net/users/8187 | Diagonal terms in the Kochen Stone inequality | Let
$$S\_n=\sum\_{k=1}^n P(A\_k),\quad T\_n:=\sum\_{1\le i<k\le n}P(A\_i)P(A\_k),$$
$$R\_n=\sum\_{i,k=1}^n P(A\_iA\_k),\quad U\_n:=\sum\_{1\le i<k\le n}P(A\_iA\_k).$$
Then $2T\_n\le S\_n^2\le2T\_n+S\_n$, and $S\_n<<S\_n^2$ (because $S\_n\to\infty$); we write $a<<b$ or, equivalently, $b>>a$ to mean $a=o(b)$; all the lim... | 2 | https://mathoverflow.net/users/36721 | 372011 | 155,475 |
https://mathoverflow.net/questions/371996 | 0 | Let $k$ be a field of characteristic zero.
Let $h=h(T) \in k[T]$ with $\deg(h)=d \geq 2$ and $h(0)=0$ (namely, $h$ has zero constant term).
Consider the following chain of $k$-algebras:
$$k \subseteq k[h(x),y] \subseteq k[h(x),y] + \langle h(x),y \rangle\_{k[x,y]} \subseteq k[x,y]$$
where $\langle h(x),y \rangle\_{k[... | https://mathoverflow.net/users/72288 | $k[h(x),y] \subseteq k[h(x),y] + \langle h(x),y \rangle \subseteq k[x,y]$ | You have the right idea. Every element of $B$ is of the form $r+ph+qy$ for some $r\in R$ and $p,q\in A$. Since $A$ is generated by $1,x,\ldots,x^{d-1}$ as an $R$-module, we see $$p=\sum\_{i=0}^{d-1}p\_ix^i$$ $$q=\sum\_{i=0}^{d-1}q\_ix^i$$ for some $p\_i.q\_i\in R$. But then $$r+ph+qy=r\cdot 1+\sum\_{i=0}^{d-1}p\_ihx^i+... | 1 | https://mathoverflow.net/users/142858 | 372016 | 155,477 |
https://mathoverflow.net/questions/371962 | 9 | Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper [A Segal conjecture for $p$-completed classifying spaces](https://www.sciencedirect.com/science/article/pii/S0001870807001223), it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$ is $p$-complete, but I have not succee... | https://mathoverflow.net/users/112348 | $p$-completeness of the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$ | Here is an argument (not as clean as Piotr's below). Use with caution; it's possible that I've made a mistake. We don't use any properties of $\Sigma^\infty BK$ -- this could be an arbitrary spectrum. But it's crucial that that we use $\Sigma^\infty BS$ with $S$ a finite $p$-group.
**Proposition:** If $S$ is a finite... | 3 | https://mathoverflow.net/users/2362 | 372027 | 155,479 |
https://mathoverflow.net/questions/372020 | 1 | I am reading a paper where the author derives the following Lagrangian dual problem :
$\min\_v \int\_R \frac{1}{4} \frac{\beta^2}{v-2\|x\|}dx+v\;\;\;\text{s.t.}\;\;\;v\geq 2\|x\|\;\;\;\forall x \in R$
from the primal problem :
$\max\_{f(.)} \int\_R (2\|x\| f(x) + \beta \sqrt{f(x)})dx\;\;\;
\text{s.t.}\;\;\;\int\... | https://mathoverflow.net/users/148279 | Dual problem with integrals | $\newcommand\R{\mathbb R}$
A convenient way to derive the dual problem from a primal one is by using the [minimax duality](https://people.eecs.berkeley.edu/%7Eelghaoui/Teaching/EE227A/lecture7.pdf) for the Lagrangian, which is given here by the formula $$L(f,v):=\int\_R\big[2|x|f(x)+b\sqrt{f(x)}\big]\,dx-v\Big(\int\_Rf... | 2 | https://mathoverflow.net/users/36721 | 372028 | 155,480 |
https://mathoverflow.net/questions/371733 | 5 | In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)|, \,\, \forall z \in \mathbb{D}$, $ Re F \geq 0$ and $ \Vert F \Vert\_p \leq c\_p \Vert f \Vert\_p $.
To see why this is t... | https://mathoverflow.net/users/153260 | A domination property for the Hardy space $H^1$ | In general it is impossible. We can take the square root, map the circle conformally to the half-plane, and arrive at the following problem: given any nonnegative $f\in L^2(\mu)$ with finite logarithmic integral, can we find $g\in H^2(\mu)$ with $\Re g\ge f$ and $|\Im g|\le \Re g$ where $d\mu(x)=\frac{dx}{1+x^2}$? Now,... | 3 | https://mathoverflow.net/users/1131 | 372032 | 155,482 |
https://mathoverflow.net/questions/372026 | 7 | Let $M\_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \mathbb{R}^{2n}$, we have $\langle Jx, Jx \rangle = \langle x, x \rangle$ and $\langle Jx, x \rangle = 0$. They can also be ... | https://mathoverflow.net/users/39521 | The space of skew-symmetric orthogonal matrices | Your $M\_n$ is (two copies of) the Riemannian symmetric space $\mathrm{SO}(2n)/\mathrm{U}(n)$ (which is $DI\!I\!I$ in Cartan's nomenclature). Its topology is well-studied from that point of view.
| 8 | https://mathoverflow.net/users/13972 | 372036 | 155,484 |
https://mathoverflow.net/questions/372051 | 1 | In this question, the notation $P^x(\alpha)$ denotes a situation where a particular [OTM-program](https://arxiv.org/abs/math/0502264) $P$ performs a computation on input $x$ with an ordinal parameter $\alpha$, assuming that $x$ is written on the initial segment of length $\omega$ (the smallest limit ordinal) of the tap... | https://mathoverflow.net/users/122796 | How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real? | Since there is disputation on how to interpret the problem, I think it would be better to clarify my interpretation:
>
> Let $P(x,\alpha)$ be a program, which takes a binary sequence $x\in 2^\mathbb{N}$ (also called a real, which is standard terminology in set theory) and an ordinal $\alpha$. Consider the set
> $$H... | 3 | https://mathoverflow.net/users/48041 | 372062 | 155,494 |
https://mathoverflow.net/questions/372066 | 2 | While examining the product of two upper triangular matrices, I've found that the $(m,n)$'th entry of the resulting matrix amounts to: $$\sum\_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $$ when $n \geq m$ (all other entries are zero).
Although I have found some summations of products of binomial coefficients [here](... | https://mathoverflow.net/users/93724 | Do identities exist for the binomial series $\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $? | $$\sum\_{k}{{k\choose m} {n+1\choose k-1}}=\frac{1}{m}\sum\_{k}k{k-1\choose m-1}{n+1\choose k-1} = \frac{n+1 \choose m-1}{m}\sum\_{k}k{n-m+2 \choose k-m} = \frac{n+1 \choose m-1}{m}(\sum\_{k}(k-m){n-m+2 \choose k-m}) + {n+1 \choose m-1}\sum\_{k}{n-m+2 \choose k-m} = \frac{{n+1 \choose m-1}(n-m+2)}{m}(\sum\_{k}{n-m+1 \c... | 3 | https://mathoverflow.net/users/88679 | 372075 | 155,497 |
https://mathoverflow.net/questions/371993 | 5 | Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense and $B$ positive. Then $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are closable. What are generic conditions so that $(A+iB)^\*$ is the closure of $(A-iB)|\_{\mathcal{D}(A)\cap\mathcal{D}(B)}... | https://mathoverflow.net/users/108697 | For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$? | In general this is not true. Let $\Omega$ be a smooth bounded domain, let $A$ be $-\Delta$ with Neumann conditions and let $B$ be $-\Delta$ with Dirichlet conditions. The $(A+iB)$ is $(-1-i)\Delta$ with domain $H^2\_0(\Omega)$. Its adjoint is $(-1+i)\Delta$ with domain $H^2(\Omega)$, which is not the closure of $A-iB$.... | 7 | https://mathoverflow.net/users/12120 | 372077 | 155,498 |
https://mathoverflow.net/questions/371254 | 1 | Let $H$ be a complex, infinite dimensional, separable Hilbert space. Fix any two nonzero operators $A,B \in B(H)$ such that $B$ is not a scalar multiple of $A$. It is well known that:
$$ \| R\_A (z) \| \rightarrow 0 \quad \text{when}\, |z| \rightarrow +\infty $$
This easily follows from:
$$\|R\_A(z) \| \leq \frac... | https://mathoverflow.net/users/160051 | On a limit for the resolvent norm | As Michael already observed: It is simple to construct counterexamples if $B$ has a nontrivial kernel $N(B)$ and $A$ maps $N(B)$ into itself, since on $N(B)$ the size of $c$ plays no role.
But even if $B$ is an isomorphism, even the identity $B=I$, the conjecture is false: $z-A-cB=(z-c)-A$. The maximum of the norm of... | 2 | https://mathoverflow.net/users/165275 | 372085 | 155,501 |
https://mathoverflow.net/questions/372076 | 10 | (edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
>
> There exists a join-irreducible element $a$ with $|[a,M]| \leq |P|/2$, when $M$ is the maximum of $P$.
>
>
>
Recall that an element a of a poset is join-i... | https://mathoverflow.net/users/61949 | Generalising the union-closed sets conjecture from lattice to a larger class of posets | Here is a counterexample of size 23.
Let $m=6$ and let $$P=\{0,a\_1,\dots,a\_m,1\}\cup\{b\_{ij}: 1\le i<j\le m\}$$
where $0<a\_i<b\_{jk}<1$ whenever $i$ is distinct from $j$ and $k$.
The cardinality of $P$ is $|P|=m+2+\binom{m}{2}=6+2+15=23$.
The join-irreducible elements are only the $a\_i$ and $0$, since
each $... | 11 | https://mathoverflow.net/users/4600 | 372095 | 155,504 |
https://mathoverflow.net/questions/372097 | 7 | Solutions to the differential equation $my'' + ky = F \sin \omega t$ show resonance when the driving frequency $\omega$ equals the natural frequency $\sqrt{k/m}$. That is, solutions are unbounded when $\omega = \sqrt{k/m}$ and periodic for all other frequencies. It seems that when the sine function is replaced by a saw... | https://mathoverflow.net/users/136 | Resonance arising when harmonic oscillator is excited using sawtooth | The sawtooth function $f$ has Fourier decomposition
$$
f(t) = \frac{1}{2}-\frac{1}{\pi}\sum\_{n=1}^\infty \frac{1}{n} \sin(n\omega t)
$$
Therefore, if $\omega=\frac{\omega\_0}{n}$, the $n$-th harmonic of $f$ will have angular frequency $n\omega=\omega\_0$, resulting in resonance.
| 18 | https://mathoverflow.net/users/45250 | 372098 | 155,505 |
https://mathoverflow.net/questions/372089 | -3 | The Euler equations are given as $$ \pmb{u}\_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u\_1,u\_2,\ldots u\_n]^T$$
Now I want to rewrite these same equations but with a new combination of variables $$\pmb{v} = [u\_1,u\_2,\ldots u\_n,p]^T $$
I want to do this just for the sake of getti... | https://mathoverflow.net/users/134538 | Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$ | The construction of adding $p$ as an additional element to the vector $\mathbf u$ only hides it in the vector $\mathbf v$, without actually eliminating it from the Euler equation. This can be easily done with a vector $\mathbf e=(0,0,\ldots 0,0,1)$, and a diagonal matrix $\Delta={\rm diag}\,(1,1,,\ldots 1,1,0)$ so that... | 5 | https://mathoverflow.net/users/11260 | 372107 | 155,507 |
https://mathoverflow.net/questions/372094 | 6 | Let $\mathcal{FG}$ be the category of finite groups. Let $S$ be a full subcategory of $\mathcal{FG}$.
Assume that $G\in \mathcal{FG}$ and $P\in S$ is a subgroup of $G$. We say that $P$ is $S$-maximal if there is no object $P'\in S$ with $P\subset P' \subset G$. Assume that $S$ satisfies the following conditions:
1. T... | https://mathoverflow.net/users/36688 | A possible characterization of the category of finite $p$-groups | We can get most of the way to an answer, without using all the conditions.
**Definition:** Say that $S \subseteq \mathcal{FG}$ is *nice* if it is nonempty and closed under subgroups, extensions, and isomorphy, and has the property that for any $G \in \mathcal{FG}$, the $S$-maximal subgroups of $G$ are all conjugate.
... | 6 | https://mathoverflow.net/users/2362 | 372110 | 155,509 |
https://mathoverflow.net/questions/372108 | 6 | If we have a category $\mathcal{C}$, then we can see it as an $\infty$-category. Furthermore, we can truncate and $\infty$-category $\mathcal{X}$ to get a category $\mathcal{X}\_{\leq 1}$. My question is if these functors are adjoint, i.e. if we have
$$\text{Hom}\_{\mathfrak{Cat}}(\mathcal{X}\_{\leq 1}, \mathcal{Y})\co... | https://mathoverflow.net/users/152554 | Truncation of infinity-categories | There is a bit of notation to be careful about here:
$\mathcal{X}\_{\leqslant 1}$ is often used to denote the full subcategory of $\mathcal{X}$ of set-truncated object. For example if $\mathcal{X}$ is an $\infty$-topos, then $\mathcal{X}\_{\leqslant 1}$ is its $1$-topos reflection.
with this definition, $\mathcal{X... | 13 | https://mathoverflow.net/users/22131 | 372111 | 155,510 |
https://mathoverflow.net/questions/371927 | 5 | I am computing the following integrals by numerical integration and this takes a lot of time, although I'm sure there is a general closed-form formula but I can't find it.
Let $t$ be a vector of $\mathbb R\_{+}^{d}$. For any integer $d \ge 1$, define $K\_d$ as a convex subset of $\mathbb R^{d}$ by :
$$x \in K\_d \i... | https://mathoverflow.net/users/143783 | Is there a closed-form expression for these integrals? | If $\Sigma\_t=\{x\_i \geq 0: x\_1+\cdots +x\_d=t\}$and $d\sigma\_t$ is its surface measure, then $I\_t=\int\_{\Sigma\_t}\prod\_{i=1}^d x\_i^{\alpha\_i}\, d\sigma\_t=t^{|\alpha|+d-1}I\_1$ (change variable $x\_i=ty\_i$). Then for $\alpha\_i>-1$
$$
\prod\_{i=1}^d \Gamma (\alpha\_i+1)=\int\_{[0,\infty[^d} \prod\_{i=1}^d x\... | 2 | https://mathoverflow.net/users/150653 | 372114 | 155,511 |
https://mathoverflow.net/questions/372040 | 7 | I'm looking for a reference on internal categories and externalization of internally defined notions.
The nlab has a stub on [externalization](https://ncatlab.org/nlab/show/externalization) (more details are available under [small fibration](https://ncatlab.org/nlab/show/small+fibration)) and the page on [internal ca... | https://mathoverflow.net/users/92164 | Reference on internal categories and externalization | For what it's worth, I should add my comment as an answer. Chapter 1 of Bart Jacobs' book *Categorical logic and type theory* (Studies in Logic and the Foundations of Mathematics **141** (1999), ([author's page](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html), [publisher page](https://www.elsevier.com/books/categorical... | 7 | https://mathoverflow.net/users/4177 | 372125 | 155,514 |
https://mathoverflow.net/questions/372129 | 25 | I am interested in any known results/empirical studies done on the average height of a poset with $N$ elements. Obviously this would depend on how that poset relation was randomly defined, however, at this point, I'll take any reasonable result on the topic regardless of how the poset was formed.
In my case $N$ is a ... | https://mathoverflow.net/users/119995 | Expected height of a poset? | In [*Asymptotic Enumeration of Partial Orders on a Finite Set* (1975)](https://www.jstor.org/stable/1997200), Kleitman and Rothschild showed that almost all partial orders on an $n$-element set have a simple description: they have three "layers" $L\_1$, $L\_2$, and $L\_3$ of incomparable elements, of size $n/4$, $n/2$,... | 35 | https://mathoverflow.net/users/129595 | 372130 | 155,515 |
https://mathoverflow.net/questions/372135 | 2 | **Conjecture** For arbitrary integers $\ 0 \le k \le m\ $ there exists
integer $\ n\ge m\ $ such that for every natural number $\ s\ $ at
least one of the numbers $\ p(x)+s\ (\text{where}\ k\le x\le n)\ $ is not prime.
>
> Here, $\ p(0)=2, p(1)=3,\ldots\ $ is the strictly increasing sequence of all prime numbers.
>... | https://mathoverflow.net/users/110389 | A detail oriented prime conjecture | Let $q=p(k)$. Using Dirichlet's theorem on primes in arithmetic progressions, there is $n\geq m$ large enough so that for any $a$ not divisible by $q$, there is some $k\leq x\leq n$ such that $p(x)\equiv a\pmod q$. Also taking $a=0$ and $x=k$, we see this is true for *all* residue classes mod $q$.
Take any natural $s... | 4 | https://mathoverflow.net/users/30186 | 372136 | 155,517 |
https://mathoverflow.net/questions/372139 | 4 |
>
> I've grossly overstated things in my two posts before the last one. Thank you for providing references that have returned me to reality.
>
>
>
**Conjecture** For arbitrary integers $\ 0 \le k \le m\ $ there exists
integer $\ n\ge m\ $ such that for every natural number $\ s\ $ at
least one of the numbers
$... | https://mathoverflow.net/users/110389 | A typo-free detail oriented prime conjecture | Like in the previous post, let $q=p(k)$ and let $n\geq m$ be such that primes $p(x),k\leq x<n$ cover all residue classes mod $q$.
Suppose this $n$ doesn't work. This means that $p(x+s+1)-p(x+s)=p(x+1)-p(x)$ for all $k\leq x<n$. Adding up a bunch of such equalities we get $p(x+s)-p(k+s)=p(x)-p(k)$ for all $k\leq x\leq... | 5 | https://mathoverflow.net/users/30186 | 372142 | 155,519 |
https://mathoverflow.net/questions/372117 | 5 | Given $N$ points $X:=(x\_i)\_{i \in \{1,..,N\}}$, we now define a score function $S:X \rightarrow \mathbb{N}$ that is $S(X)= \sum\_{i=1}^N S(x\_i)$ where the score of $S(x\_i)$ is
$$S(x\_i) = 2\* \vert \{x\_j; \vert x\_i-x\_j \vert \in [1,2]\} \vert+ \vert \{x\_j; \vert x\_i-x\_j \vert \in [2,3]\} \vert$$
where $\ver... | https://mathoverflow.net/users/119875 | Packing in uniform domains | What follows below is the new answer to the modified question, where we assume in addition $|x\_i-x\_j|\ge 1$ (probably one can ask $|x\_i-x\_j|\ge 1-\varepsilon$ for sufficiently small $\varepsilon$). I want to propose a positive solution of this problem modulo the following guess, which, I hope, is correct.
**Guess... | 2 | https://mathoverflow.net/users/943 | 372154 | 155,524 |
https://mathoverflow.net/questions/372041 | 1 | I'm working on a project where I'm working with modulo functions. However, to continue, I need to integrate integral powers of a weighted sum of them (e.g of the form $\left(c+\operatorname{weighted sum} \right)^p$, with $c$ a real, positive constant, and $p \in \mathbb{Z}$). So, I first tried Fourier series. However, ... | https://mathoverflow.net/users/157462 | How do you make an accurate, integrable approximation of $a \operatorname{mod} \left(\frac xb,1 \right)$ with a scaling constant $N$? | Let's consider the concrete example given by the OP in comments,
$$
f(x) = \left( 4\ \mbox{mod} \left( \frac{x}{5} ,1\right) + 10\ \mbox{mod} \left( \frac{x}{33} ,1\right) \right)^{p} \ .
$$
$f(x)$ is periodic with period $5\cdot 33 = 165$. It is discontinuous at the points $\{ 5n : n\in \mathbb{Z} \} \cup \{ 33n : n\i... | 2 | https://mathoverflow.net/users/134299 | 372176 | 155,533 |
https://mathoverflow.net/questions/372173 | 1 | Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
* **1. Symmetry:** $\zeta \overset{d}{=} - \zeta$.
* **2. Small-ball probability:** there exists a constants $\alpha > 0$ and $u\_0 \in (0,\infty]$ such that $P(\|\zeta^\otimes\| \le u\sqrt{n}) \le (\alph... | https://mathoverflow.net/users/78539 | Characterization of random variables whose tensor powers have subexponential "small-ball" probabilities | Let $X:=(X\_1,\dots,X\_n)$, where the $X\_j$'s are iid copies of $\zeta$. Then the problem is about conditions for
$$P(\|X\|\le u\sqrt n)\le C^n u^n\tag{1}$$
for some real $C>0$, all natural $n$, and all small enough real $u>0$ (and then for all real $u>0$, possibly with a different $C>0$).
If the distribution of $X\... | 1 | https://mathoverflow.net/users/36721 | 372180 | 155,535 |
https://mathoverflow.net/questions/372179 | 8 | Recently, A. Carlotto and C. Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar curvature and mean-convex boundary. Namely, if $M^3$ is such a manifold, then there exist integers $A, B, C, D \geq 0$ s... | https://mathoverflow.net/users/85934 | Separating spheres in $3$-manifolds of positive scalar curvature and mean convex boundary | Every embedded 2-sphere will separate $M$ if and only if $C=0$.
Proof: Suppose $C=0$, and let $\Sigma$ be 2-sphere embedded in $M$. Let $\{S\_j\}$ be a collection of 2-spheres which decompose $M$ into prime summands. Look at the intersection of $\Sigma$ with $\{S\_j\}$. Let $\Delta$ be an innermost disk on some $S\_k... | 4 | https://mathoverflow.net/users/126206 | 372186 | 155,536 |
https://mathoverflow.net/questions/372175 | 3 | Fix a positive integer $n$. Let $X = \{X\_i\}\_{i \in \mathbb{N}}$ be a discrete time stochastic process such that each $X\_i$ is a $\{0,\dots,n-1\}$-valued random variable. Suppose that the joint probability distributions of any finite sequence of $X\_i$'s only depends on the order of their indices, or to be more prec... | https://mathoverflow.net/users/83901 | Characterizing 'very homogeneous' finitely valued stochastic processes | This only answers the first part of your question:
The strongly homogenous processes are , in particular, stationary. Extreme points of stationary processes are exactly the stationary ergodic processes. The i.i.d. measures are ergodic, hence they are extreme among stationary processes, which implies they are extreme am... | 4 | https://mathoverflow.net/users/7691 | 372189 | 155,537 |
https://mathoverflow.net/questions/372187 | 2 | We can get arithmetic lattices isomorphic to free groups in $\mathrm{SL}\_2\mathbb{R}$, so in general we can’t expect homomorphisms of lattices into semisimple Lie groups to say much about $\mathrm{SL}\_2\mathbb{R}$, even if we require the image to be discrete. In particular, unlike in higher-rank cases, such homomorph... | https://mathoverflow.net/users/58187 | Do discrete embeddings of surface groups not necessarily carry an embedding of SL_2? | Homomorphisms from surface groups are very flexible, so there are indeed such examples.
For instance, use: Breuillard–Gelander–Souto–Storm (Dense embeddings of surface groups.
Geom. Topol. 10 (2006), 1373–1389) ([DOI link](http://dx.doi.org/10.2140/gt.2006.10.1373))
They proved, for a surface group $\Gamma$ ($\pi\_... | 2 | https://mathoverflow.net/users/14094 | 372193 | 155,538 |
https://mathoverflow.net/questions/369644 | 3 | We can easily get the character table of $\mathrm{PSL}(2,q)$ for some fixed small prime power $q$, we can just do (for example):
```
gap> Display(CharacterTable(PSL(2,q)));
```
I don't know how the software is doing, I guess it uses some Atlas database for $q$ small enough or computes directly. The point is that t... | https://mathoverflow.net/users/34538 | A global code for the character table of PSL(2,q) | The quickest way to display the character table for a fixed prime-power $q$ is:
```
Display(CharacterTable( "PSL", 2, q)
```
The *generic* character table is in fact available on GAP4 (as pointed out in private by Frank Lübeck) with the following commands:
For $q$ even:
```
gap> Print(CharacterTableFromLibrar... | 2 | https://mathoverflow.net/users/34538 | 372194 | 155,539 |
https://mathoverflow.net/questions/371786 | 12 | Zeroth question - am I right that in the "ordinary" sense an abelian variety does not possess any representations at all?
More precisely, a representation of an algebraic group $G$ (over an algebraically closed field $K$, say, over complex numbers) is a homomorphism $G\to\operatorname{GL}(V)$ for some $K$-vector spac... | https://mathoverflow.net/users/41291 | What is the correct notion of representation for abelian varieties? | I am also very far from an expert here, but I think there's a case to be made that the "correct notion" involves actions on categories of sheaves, as Donu says in the comments.
Consider the following toy model: if $A$ is, say, a finite abelian group then its Pontryagin dual $\widehat{A}$ can be defined as the group $... | 6 | https://mathoverflow.net/users/290 | 372198 | 155,541 |
https://mathoverflow.net/questions/312809 | 21 | Let $p$ be a prime; $\mathbb{F}\_{p}$ is the field with $p$ elements
and $\mathbb{F}\_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}\_{p}$.
>
> Does $\mathrm{SL}\_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}\_{n}(\mathbb{F}\_{p}[t]/t^{2})$?
>
>
>
When $p$ does not d... | https://mathoverflow.net/users/2381 | Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$? | I would have preferred to not answer my own question, but here it goes. Yes, the two groups have the same number of conjugacy classes and in fact, the groups $\mathrm{SL}\_{n}(W\_{2}(\mathbb{F}\_{q}))$ and $\mathrm{SL}\_{n}(\mathbb{F}\_{q}[t]/t^{2})$, for $q$ a power of a prime $p$ dividing $n$, have the same number of... | 7 | https://mathoverflow.net/users/2381 | 372205 | 155,543 |
https://mathoverflow.net/questions/372200 | 6 | When $q$ is a prime power, then on the one hand the [$q$-binomial coefficient](https://en.wikipedia.org/wiki/Gaussian_binomial_coefficient) $\binom{n}{k}\_q$ equals the number of $k$-dimensional subspaces of $\mathbb{F}\_q^n$, and on the other hand it is the generating function of the sequence which sends $r$ to the nu... | https://mathoverflow.net/users/2841 | Enumerating subspaces of $\mathbb{F}_q^n$ in terms of words and inversions | You may act similarly as follows.
Let $V$ be a $k$-dimensional subspace of $\mathbb F\_q^n$. Take any its base, put its elements into the rows of some matrix, and make it to the reduced row echelon form $B$ (which is unique). The rows of $B$ still form a base of $V$.
Let $r\_1$, $r\_2$, $\dots$, $r\_k$ be the indic... | 6 | https://mathoverflow.net/users/17581 | 372207 | 155,544 |
https://mathoverflow.net/questions/372191 | 6 | I asked [this](https://math.stackexchange.com/questions/3825588/a-rather-non-f-sigma-borel-set) question at MSE a week ago, but received no answer, so I cross-post it here.
I obtained a negative answer to [this](https://math.stackexchange.com/questions/3821820/cardinality-of-the-collection-of-measurable-subsets-of-me... | https://mathoverflow.net/users/43954 | A rather non-$F_\sigma$ Borel set | Here's an argument that the statement is false if the Continuum Hypothesis fails and the covering number for the null ideal is the same as the continuum. Wellorder the Borel sets of reals as $\langle B\_{\alpha} : \alpha < \mathfrak{c} \rangle$. Choose for each $\alpha < \mathfrak{c}$ an $F\_{\sigma}$ set $C\_{\alpha} ... | 2 | https://mathoverflow.net/users/31807 | 372209 | 155,545 |
https://mathoverflow.net/questions/372212 | 10 | Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?
| https://mathoverflow.net/users/99732 | 4-dimensional cohomology $\mathbb{CP}^2$'s | No. If $\Sigma$ is any homology 4-sphere with non-trivial fundamental group, $\mathbb{CP}^2 \# \Sigma$ is a homology $\mathbb{CP}^2$ with non-trivial fundamental group. (Here $\#$ denotes connected sum.) There are many examples of homology 4-spheres: for instance, Kervaire produced many examples in his paper *Smooth ho... | 17 | https://mathoverflow.net/users/13119 | 372213 | 155,548 |
https://mathoverflow.net/questions/372206 | 2 | The problem is
$b = (1, -1)^\top, c = (1, 1)^\top, A \in \mathbb{R}^{2 \times 2}$, suppose the sum of reverse diagonal elements of $A$ is zero (i.e., $A\_{12} + A\_{21} = 0$), prove that the sum of reverse diagonal elements of $\sum\limits\_{r=0}^{n-1} A^r c b^\top A^{n-1-r} $ is zero for any $n \in \mathbb{N}^{+}$.
... | https://mathoverflow.net/users/165697 | A problem of matrix polynomial expansion | Someone told me a simple method, I decide to post it here.
Note that for any $A \in \mathbb{R}^{2 \times 2}$, $A\_{12} + A\_{21} = 0$ if and only if
\begin{equation\*}
A^\top = \sigma^{-1} A \sigma
\end{equation\*}
with
\begin{equation\*}
\sigma = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\end{equation\*}
L... | 1 | https://mathoverflow.net/users/165697 | 372223 | 155,551 |
https://mathoverflow.net/questions/372208 | 1 | Let $X:x \mapsto X(x)$ be a centered stationary Gaussian process on the $\Omega:=\mathbb R^p$, such that $X(x) \overset{d}{=}X(x')$ for all $x,x' \in \Omega$. Set $\sigma^2 := \mbox{Var}(X(0)) = \mathbb E[X(0)^2]$. Let the random fields $X\_1,\ldots,X\_N$ be iid copies of $X$, and define a random process $Z\_N$ on $\Om... | https://mathoverflow.net/users/78539 | Central limit theorem for chi-squared random field on $\mathbb R^p$ | $\newcommand\si\sigma\newcommand\Om\Omega$To determine the limit distribution of the process $Z\_N$ (and even the distribution of the process $X$), it is not enough to know only $\si^2$; one also has to know the covariances $r\_{x,y}:=Cov(X(x),X(y))$ for $x,y$ in $\Om$.
Then, using e.g. the joint moment generating func... | 1 | https://mathoverflow.net/users/36721 | 372227 | 155,554 |
https://mathoverflow.net/questions/372181 | 10 | The theory of real closed fields is decidable. The [hyperreals](https://en.wikipedia.org/wiki/Hyperreal_number) satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.
If we add a unary predicate for "is a standard real number" to the language, is the theory ... | https://mathoverflow.net/users/65915 | Decidability of a first-order theory of hyperreals | Yes, the theory is decidable.
If $F$ is an ordered field and $R\subseteq F$ a non-cofinal subfield, then
$$O=\{x\in F:\exists u\in R\:(-u\le x\le u)\}$$
is a convex valuation ring of $F$, with maximal ideal
$$I=\{x\in F:\forall u\in R\_{>0}\:(-u\le x\le u)\}.$$
$R$ embeds as a cofinal subfield in the residue field $O... | 13 | https://mathoverflow.net/users/12705 | 372238 | 155,559 |
https://mathoverflow.net/questions/372222 | 6 | I am looking for natural groups with undecidable conjugacy problem. By natural, I mean that the word problem should be decidable, and the group should be given by some natural action. I know that $\mathbb{Z}^d \rtimes F\_m$ (with a suitable action of $F\_m$) has undecidable conjugacy problem. That's very nice, but I'd ... | https://mathoverflow.net/users/123634 | Examples of "natural" finitely generated groups with an undecidable conjugacy problem | Chuck Miller in [Miller, Charles F., III *On group-theoretic decision problems and their classification*. Annals of Mathematics Studies, No. 68. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971] proves the following two rather nice and natural examples.
**Theorem III.10.** The free ... | 4 | https://mathoverflow.net/users/120914 | 372239 | 155,560 |
https://mathoverflow.net/questions/372188 | 1 | Work over the complex numbers. Let $(B, \Delta)$ be a normal irreducible variety of log general type, i.e., $K\_B + \Delta$ is ample. Let $f : (\widetilde{B}, \widetilde{\Delta}) \to (B, \Delta)$ be a log resolution of $(B, \Delta)$, i.e., $\widetilde{B}$ is smooth and $\widetilde{\Delta}$ has simple normal crossing su... | https://mathoverflow.net/users/nan | Log resolution of a variety of log general type | If $K\_{\tilde B}+\tilde \Delta=f^\*(K\_B+\Delta)+E$ where $E$ is effective and exceptional, then $h^0(m(K\_{\tilde B}+\tilde \Delta))=h^0(m(K\_B+\Delta))$ for any $m\geq 0$ and hence also the Kodaira dimensions agree. In particular this works if $(B,\Delta )$ is log canonical and $\tilde \Delta$ is the strict transfor... | 3 | https://mathoverflow.net/users/19369 | 372242 | 155,561 |
https://mathoverflow.net/questions/372235 | 2 | Sorry if this question is maybe a bit basic, but it is on a rather specialized topic so I think it is more appropriate for MO than SE.
>
> Suppose that $X$ is a simplicial set that has finitely many
> non-degenerate simplices in every simplicial dimension. Then of course
> $X$ has finite $\mathbb{Q}$-type (i.e. $H\... | https://mathoverflow.net/users/44134 | Kan replacement of finite $\mathbb{Q}$-type simplicial set | Weak equivalences of simplicial sets induce isomorphisms on simplicial homology groups.
In particular, the property of having finite-dimensional rational
homology groups is preserved under weak equivalences.
The easiest way to prove this from scratch
is the observe that the free simplicial module functor
(in this cas... | 3 | https://mathoverflow.net/users/402 | 372245 | 155,562 |
https://mathoverflow.net/questions/372202 | 1 | Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f\_1,\ldots,f\_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f\_1(x),\ldots,f\_k(x)$ are independent $x \in \Omega$. Thus $f:=(f\_1,\ldots,f\_k)$ can be seen as a vector-valued Gaussian process with iid components at each poi... | https://mathoverflow.net/users/78539 | Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components | Consider the real-valued centered Gaussian process $(X\_{t,a}\colon(t,a)\in T\times B\_k)$, where
$$X\_{t,a}:=\sum\_{j\in[k]}a\_j f\_j(t),$$
$T:=\Omega$, $B\_k$ is the unit ball in $\mathbb R^k$, and $[k]:=\{1,\dots,k\}$. Then
$$\|\nu\|\_\infty=\|X\|\_\infty:=\sup\{|X\_{t,a}|\colon t\in T, a\in B\_k\}$$
and
$$EX\_{t,a}... | 5 | https://mathoverflow.net/users/36721 | 372246 | 155,563 |
https://mathoverflow.net/questions/372254 | 1 | I have a conjecture that a certain criterion is enough for two groups to be isomorphic. I tested it on all pairs of groups up to size 12, and it worked like a charm. I know, however, that groups are strange and that it is very likely that my conjecture will break for larger groups. I have made myself a python library t... | https://mathoverflow.net/users/159298 | What groups should I test my conjecture on? | Your conjecture is false. Probably an explicit counterexample is easy to write down but here's an existence proof that counterexamples are plentiful: asymptotically it's [known](https://en.wikipedia.org/wiki/Higman%E2%80%93Sims_asymptotic_formula) (Higman-Sims) that there are $p^{ \frac{2}{27} n^3 + O(n^{8/3})}$ groups... | 9 | https://mathoverflow.net/users/290 | 372256 | 155,566 |
https://mathoverflow.net/questions/371222 | 2 | Let $(M, \omega)$ be a connected closed symplectic manifold of dimension $2n$.
Assume there exist smooth covering maps $\phi\_k:M\to M$ such that $\phi\_k^\* \omega=\sqrt[n]{k}\omega$ for all $k\geq 1$. Is $n=1$ then?
| https://mathoverflow.net/users/nan | Smooth covers rescaling the symplectic form | It **is** the case that $n \in \{0,1\}$. (I include $n=0$ because $0$-manifolds with $\omega = 0$ are symplectic.)
**Two-sentence summary:** We know what integral elements of cohomology are, and pullback acts integrally. This allows us to think of the coefficients of $\omega$ as living in some fixed abelian subgroup ... | 5 | https://mathoverflow.net/users/66405 | 372272 | 155,569 |
https://mathoverflow.net/questions/372275 | 3 | We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-commutator to be zero.
The requirement of SUSY charge $Q$ includes that
1. $Q$ is a Hermitian operator.
2. $[Q,H]=0$, $Q$ com... | https://mathoverflow.net/users/27004 | Supersymmetry charge $Q$ as anti-linear and anti-unitary operator | Suppose you are given a super Hilbert space $\mathcal{H} = \mathcal{H}\_0 \oplus \mathcal{H}\_1$, with bosonic and fermionic subspaces $\mathcal{H}\_0$ and $\mathcal{H}\_1$ respectively. Define a new super Hilbert space $\mathcal{H}' = \mathcal{H}\_0 \oplus \overline{\mathcal{H}\_1}$, where you have complex-conjugated ... | 5 | https://mathoverflow.net/users/78 | 372280 | 155,573 |
https://mathoverflow.net/questions/372255 | 0 | Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
**(1)** $R$ and $S$ are integral domains.
**(2)** $Q(R)=Q(S)$, namely, their fields of fractions are equal.
**(3)** $S=R[w]$, for some $w \in S$.
**(4)** $S$ is [separable](https://en.wikipedia.org/wiki/Separable_algebra) over $R$, namely, ... | https://mathoverflow.net/users/72288 | Separable non-flat simple ring extension | Take $R$ to be the coordinate ring of the nodal curve $\mathbb{C}[t^2-1, t(t^2-1)]$ and $S$ to be its normalization $\mathbb{C}[t]$. It satisfies (1), ..., (4): The first three are immediate. For (4), note that since $R$ is noetherian, the projectivity of $S$ as an $S \otimes\_R S$-module is equivalent to $S$ being unr... | 2 | https://mathoverflow.net/users/14895 | 372290 | 155,576 |
https://mathoverflow.net/questions/372306 | 4 | Let $(h\_{ij})\_{i,j \in \mathbb N}$ be a sequence of real numbers (deterministic) and let $x\_1,\ldots,x\_n,\ldots$ be a sequence of iid $N(0,1)$ randm variables. For each positive integer $n$, consider the quadratic form $q\_n:=\dfrac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^nh\_{i,j}x\_ix\_j$.
>
> **Question.** Under what ... | https://mathoverflow.net/users/78539 | When does a gaussian quadratic form converge (in probability) to a constant? | First, notice that w.l.o.g. you can assume that the matrix $H\_n$ is diagonal (from rotational invariance of the isotropic Gaussian).
You thus are interested in
$$
\frac{1}{n} \sum\_{i=1}^n \lambda\_i(H\_n) X\_i^2.
$$
So the condition
$$\sum\_{i=1}^n \lambda\_i^2(H\_n)/n^2 \to 0$$
is the key.
Then, one has that... | 3 | https://mathoverflow.net/users/125260 | 372310 | 155,580 |
https://mathoverflow.net/questions/372313 | 1 | Let $Y$ be a symmetric random variable, $(X\_n)\_n$ be a sequence of nonnegative random variables, and set $p\_n = \mathbb P(Y \le X\_n)$. It is known from Slutsky's theorem that, if $c$ is a constant such that $X\_n \to c$ in probability, then $p\_n \to F\_Y(c)$, where $Y$ is the CDF of $Y$.
>
> **Question.** Can ... | https://mathoverflow.net/users/78539 | Compute limit of $\mathbb P(Y \le X_n)$ using limiting information on the sequence of random variables $X_n$ | Even if $X\_n\to c$ in probability for some real constant $c$, it is not necessary that $P(Y\le X\_n)\to P(Y\le c)$ -- you also need to require that $P(Y=c)=0$.
More generally, if the limit of $X\_n$ is not a constant, then you need to assume the convergence, not just of the distribution of $X\_n$, but of the joint d... | 5 | https://mathoverflow.net/users/36721 | 372315 | 155,582 |
https://mathoverflow.net/questions/372289 | 6 | By sum of two sets I mean $A+B := \{x+y:x \in A \quad y \in B\}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:
“If $A,B$ are Borel sets in $\mathbb{R}^{n}$, $A+B$ may not be a Borel set.”
I want to know some specific examples.(Maybe $\mathbb{R}^{1}$ ?)
Any comments will be helpful.
... | https://mathoverflow.net/users/165701 | An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space | This is a result of Erdos and Stone: <https://www.ams.org/journals/proc/1970-025-02/S0002-9939-1970-0260958-1/S0002-9939-1970-0260958-1.pdf>
| 9 | https://mathoverflow.net/users/6647 | 372326 | 155,587 |
https://mathoverflow.net/questions/372323 | 8 | Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p$ and $\ell$ be two distinct primes of good reduction. Let $T\_\ell = T\_\ell(E) = \varprojlim E[\ell^n](\overline{\mathbb{Q}})$ be the $\ell$-adic Tate module, and let $F\_p \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be a Frobenius element at $p$. Then... | https://mathoverflow.net/users/121225 | Is this lattice in the Tate module of an elliptic curve, coming from complex-analytic uniformization, stable under Frobenius? | There is a subtle problem with this idea, that causes serious problems. You observed that $\Lambda\_\ell \otimes \mathbb Z\_\ell = T\_\ell$ but didn't find any other information for it. There is a reason for that.
Let $K$ be the field generated by the coordinates of the $\ell$-power torsion points of $E$. Given an $\... | 13 | https://mathoverflow.net/users/18060 | 372329 | 155,589 |
https://mathoverflow.net/questions/372334 | 4 | Let $n$ be a nonnegative integer. It is well-known that the number of lattice paths from $(0, 0)$ to $(n, n)$ with steps $(0, 1)$ and $(1, 0)$ that are never rising above the line $y=x$ is given by the Catalan number $C\_n$. There are lots of generalisations of Catalan numbers in the literature; however, I have a diffe... | https://mathoverflow.net/users/165719 | A generalisation of the Catalan numbers | In the non-coprime case the answer is much more complicated. See ["Rational Dyck Paths in the Non Relatively Prime Case"](https://doi.org/10.37236/6901) by Gorsky, Mazin, and Vazirani. They explain that it is a result of Bizley that
$$ \sum\_{d\geq 0} C(da,db)z^d = \mathrm{exp}\left(\sum\_{d\geq 1} \frac{1}{d(a+b)}\bin... | 4 | https://mathoverflow.net/users/25028 | 372335 | 155,592 |
https://mathoverflow.net/questions/372244 | 6 | Let $A$ be a two-sided noetherian ring (which we should assume to be Gorenstein first so that everything is well defined, otherwise it is only well defined up to a conjecture, which states that every non-zero module has finite grade). For simplicity we can also assume first that $A$ is a finite dimensional algebra and ... | https://mathoverflow.net/users/61949 | An identity for Ext for rings | It seems that any module $M$ whose double $A$-dual $M^{\*\*}$ is not reflexive gives a counterexample. In this case $G(M) = M^{\*\*}$ is a summand of $G^2(M) = (M^{\*\*})^{\*\*}$ with non-trivial complement, and $G^2(M)$ can't be reflexive since this property is inherited by summands. Repeating this argument with $G(M)... | 1 | https://mathoverflow.net/users/165775 | 372340 | 155,594 |
https://mathoverflow.net/questions/370119 | 5 | Consider $n\times n$ symmetric Cauchy-like matrix $M$ with elements $(M\_{ij})\_{i,j=1}^{n}$ given by
$$M\_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int\_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{x^{n-j}}{(n-j)!}\:{\rm{d}}x.$$
Is there a way to compute the elements of the inverse $(M^{-1})\_{ij}$ analytical... | https://mathoverflow.net/users/18526 | Inverse of a Cauchy-like matrix | I was able to figure this out by viewing $M$ as a scaled Cauchy matrix.
>
> **Theorem.** $\left(M^{-1}\right)\_{ij} = \dfrac{(n-i)!(n-j)!}{2n-i-j+1}\dfrac{\displaystyle\prod\_{r=1}^{n}(2n-i-r+1)(2n-j-r+1)}{\left(\displaystyle\prod\_{\stackrel{r=1}{r\neq i}}^{n}(r-i)\right) \left(\displaystyle\prod\_{\stackrel{r=1}{... | 2 | https://mathoverflow.net/users/18526 | 372344 | 155,595 |
https://mathoverflow.net/questions/372339 | 2 | I am trying to get a better understanding of how the inner horn filling condition in higher cells corresponds to higher associativity laws. For instance, I am trying to understand how the difference of the inner horn filling condition and outer horn filling condition makes the difference of invertibility of cells.
Le... | https://mathoverflow.net/users/30211 | Filling condition for quasi-categories | There's a lot to stay here. There's a nice [blog post](https://golem.ph.utexas.edu/category/2009/10/associativity_data_in_an_1cate.html) by Emily Riehl which explores how to think about higher associativity in quasicategories.
---
To see why you don't want to fill outer horns, it suffices to think about what it w... | 4 | https://mathoverflow.net/users/2362 | 372356 | 155,598 |
https://mathoverflow.net/questions/372263 | 20 | The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric polynomial of the roots of a polynomial as a polynomial of its coefficients. We can apply this to the characteristic polynomial... | https://mathoverflow.net/users/6210 | Formula expressing symmetric polynomials of eigenvalues as sum of determinants | This is not a reference, but a short proof.
We use the following (probably known, but see later) lemma on representing a symmetric tensor as a linear combination of rank-1 symmetric tensors.
**Lemma.** Let $A$ be a finite set, $K$ an infinite field. Denote by $\mathcal S$ the set of symmetric functions $p:A^n\to K$... | 10 | https://mathoverflow.net/users/4312 | 372367 | 155,600 |
https://mathoverflow.net/questions/372316 | 0 | Let $\Omega\subset \mathbb{R}^m$ be a bounded Lipschitz domain. Let $D$ be a countable dense subset of $\Omega$, denoted as $D = \{p\_1,p\_2,p\_3\ldots \}$. Define the minimum seperation distance among first $n$ points of $D$ as
$$h(n) = \min\_{1\le i,j\le n,i\ne j}\|p\_i-p\_j\|\_2 .$$
I'd like to know the convergenc... | https://mathoverflow.net/users/14414 | What is the convergence rate of the minimum separation distance? | If $\Omega$ is any bounded domain in $\mathbb{R}^m$, Let $B\_0$ be a ball which contains it, and let $B$ be a ball with the same center but with its radius increased by $1$. Say $R$ and $V$ are the radius and volume of $B$.
Fix $r < {\rm min}(\frac{1}{2}h(n), 1)$. Then the $r$-balls about the points $p\_1$, $\ldots$,... | 1 | https://mathoverflow.net/users/23141 | 372369 | 155,602 |
https://mathoverflow.net/questions/372169 | 4 | Let $X$ be a finite partially ordered set, let $f\colon X\to X$ be an order-preserving map [edit: meaning $x\le y\implies f(x)\le f(y)$], and let $x\_0$ be an initial point. Define $x\_n = f(x\_{n-1})$ for all $n$; then the sequence $(x\_n)$ is ultimately periodic. What is its worst-case period? I.e. what are the minim... | https://mathoverflow.net/users/10481 | Maximal order of an order-preserving map | This is a comment about a special case that I think gives a negative answer to the last question. Let $\varOmega$ be a set of size $n$ and let $B$ be the lattice of its subsets with the inclusion order.
Now let $\phi:\varOmega\to\varOmega$ be any function. For $X\subseteq\varOmega$, define $X^\phi=\lbrace \phi(x) \mi... | 1 | https://mathoverflow.net/users/9025 | 372372 | 155,603 |
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