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https://mathoverflow.net/questions/394813 | 4 | Let $X$ And $Y$ be smooth projective irreducible varieties over the complex numbers. Let $f:X \to Y$ be a non-constant morphism.
Assume that the dimension of $X$ is at least two.
>
> **Question.** Let $D\subset X$ be an ample divisor. Is the restriction $f|\_D : D\to X$ still non-constant?
>
>
>
This is fals... | https://mathoverflow.net/users/200661 | Restricting a non-constant map to an ample divisor | Let me expand my comment into an answer. I weaken my previous claim a bit to say that if $f \colon X \rightarrow Y$ is any nonconstant morphism and $D$ is an effective ample divisor on $X$, then $f(D)$ cannot be a point.
Suppose $X$ has dimension at least 2, $D$ is an effective ample divisor on $X$, and $f \colon X \... | 3 | https://mathoverflow.net/users/121595 | 394916 | 163,179 |
https://mathoverflow.net/questions/394903 | 1 | Let $M$ be a connected closed surface. Suppose $N$ is a connected closed surface embedded in the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and $M \times \{1\}$. Can we prove the region bounded by $M \times \{0\}$ and $N$ is homeomorphic to $M \times [0,1]$?
| https://mathoverflow.net/users/16323 | Surface in a product domain | This is not true in general.
One way to build examples is via "stabilisation". Take $N\_0 = M \times \{1/2\}$. Suppose that $\alpha$ is an arc in $M \times [0,1]$ with the following properties.
* $\alpha$ is simple (does not self-intersect).
* $\alpha \cap N\_0 = \partial \alpha$.
* $\alpha$ is isotopic, relative t... | 3 | https://mathoverflow.net/users/1650 | 394917 | 163,180 |
https://mathoverflow.net/questions/394924 | 5 | $\DeclareMathOperator\STop{STop}$I am interested in any information about the homotopy type of the groups $\STop\_{n,j}$ of homeomorphisms of $R^n$ preserving orientation and pointwise $R^j\subset R^n$. It is easy to see that $\STop\_{n,n-1}$ is contractible, being homeomorphic to the square of the group of relative-to... | https://mathoverflow.net/users/9800 | $\operatorname{STop}_{n,n-2}\simeq S^1$? | The general result in this direction is due to Kirby-Siebenmann, Theorem B of [Normal bundles for codimension 2 locally flat imbeddings](https://link.springer.com/chapter/10.1007/BFb0066125): the map
$$\mathrm{SO}(2) \longrightarrow \mathrm{STop}\_{n,n-2}$$
is $(n-2)$-connected. (They state this for $n \neq 4$ but ... | 7 | https://mathoverflow.net/users/798 | 394928 | 163,183 |
https://mathoverflow.net/questions/394934 | 16 | Question:
---------
On balance, with theoretical advances in algorithmic information theory and Quantum Computation it appears that the remarkable effectiveness of mathematics in the natural sciences is quite reasonable. By effectiveness, I am generally referring to Wigner's observation that mathematical laws have re... | https://mathoverflow.net/users/56328 | Revisiting the unreasonable effectiveness of mathematics | A 2013 [issue](https://www.tandfonline.com/toc/yisr20/36/3?nav=tocList) of *Interdisciplinary Science Reviews* was entirely devoted to this topic. One viewpoint, by Jesper Lützen, struck me:
>
> When Wigner claimed that the effectiveness of mathematics in the
> natural sciences was unreasonable it was due to a dogm... | 40 | https://mathoverflow.net/users/11260 | 394937 | 163,188 |
https://mathoverflow.net/questions/392198 | 3 | Let $A$ be an abelian variety over $\mathbb{C}$ and let $X\subset A$ be a closed subvariety. Let $X\to Y$ be the Ueno fibration. (That is, $Y$ is of general type and a closed subvariety of $A/B$ where $B$ is some abelian subvariety of $A$. Also, the morphism $X\to Y\to A/B$ equals the morphism $X\to A\to A/B$.)
>
>... | https://mathoverflow.net/users/200661 | Is the Ueno fibration smooth? | abx answered the question:
"Yes to the 3 questions. This is the content of Ueno's Theorem 10.9 in Classification theory of algebraic varieties and compact complex manifolds, LNM 439"
| 0 | https://mathoverflow.net/users/200661 | 394948 | 163,192 |
https://mathoverflow.net/questions/355650 | 11 | *This question was asked and bountied [at MSE](https://math.stackexchange.com/questions/3578756/where-did-the-language-in-this-proof-of-godels-incompleteness-appear), with no response.*
---
Many years ago I ran into the following proof of Gödel's first incompleteness theorem
*(here $T$ is an "appropriate" theory ... | https://mathoverflow.net/users/8133 | Where did this presentation of Gödel's theorem appear? | EDIT: I think I finally figured it out! At around the same time that I was reading Kreisel's invariant definability paper, I was *also* looking at [Kikuchi/Tanaka, *On formalization of model-theoretic proofs of Godel's theorems*](https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-35/issue-3/On... | 2 | https://mathoverflow.net/users/8133 | 394952 | 163,194 |
https://mathoverflow.net/questions/394927 | 4 | I just saw the following on wikipedia about Laplace transformations:
"In probability theory and applied probability, the Laplace transform is defined as an expected value. If $X$ is a random variable with probability density function $f$, then the Laplace transform of $f$ is given by the expectation: $L\lbrace f \rbr... | https://mathoverflow.net/users/277220 | Why is it possible to use the Inverse Laplace transform to get CDF? | Let $f$ be the density and $F$ the distribution function of $X \geq 0$. Then $F' = f$ (a.s.) and $\mathcal{L}\{F'\}(s) = \mathcal{L}\{f\}(s) = \int\_0^\infty f(x) e^{-sx} dx$ and $\mathcal{L}\{F\}(s) = \int\_0^\infty F(x) e^{-sx} dx$. Since $F(0) = 0$ and using partial integration we get
$$\int\_0^\infty F'(x) e^{-sx} ... | 4 | https://mathoverflow.net/users/100904 | 394954 | 163,195 |
https://mathoverflow.net/questions/394947 | 3 | [A survey by Nguyen Van Thé (2014)](https://arxiv.org/abs/1412.3254v2) has Conjecture 1,
which is that
"every closed oligomorphic
subgroup of $S\_∞$ should have a metrizable universal minimal flow with a generic
orbit." Later, it goes on to say that "it is even possible that this should be
true for a larger class of gr... | https://mathoverflow.net/users/277603 | Roelcke precompactness and Ramsey property | The isometry group of the ordered rational Urysohn space is extremely amenable but not Roelcke precompact, which gives a counterexample.
| 2 | https://mathoverflow.net/users/277832 | 394955 | 163,196 |
https://mathoverflow.net/questions/393947 | 10 | Numerical experiments suggest that the following integral identity holds for Bessel functions of the first kind,
$$J\_2(t) = 12 \int\_0^{1/2}\mathrm{d}x\,\cot \pi x \int\_0^x \mathrm{d}y\, \cot \pi y \, J\_0(ty)\big[J\_0(tx)\,J\_0(t(1-x-y))-J\_0(t(1-x))\,J\_0(t(x-y))\big],$$
but so far I have been unable to prove this.... | https://mathoverflow.net/users/47484 | An integral identity involving cotangents and Bessel functions | With hindsight the identity is not that magical: the Bessel functions play only a secondary role, in the sense that there is a more general identity for arbitrary differentiable functions $f : [0,1] \to \mathbb{R}$, namely
\begin{align}
&12 \int\_0^{1/2}\mathrm{d}x\,\cot \pi x \int\_0^x \mathrm{d}y\, \cot \pi y \, f(y)... | 10 | https://mathoverflow.net/users/47484 | 394961 | 163,199 |
https://mathoverflow.net/questions/394963 | 3 |
>
> **Question.** Is there any standard name for a (commutative or non-commutative) unital ring $R$ with the property that, for every $a \in R$, the (descending) chain $R, aR, a^2 R, \ldots,$ is eventually constant?
>
>
>
Let me refer to this condition as the DCCPRP, that is, the "DCC on chains of principal righ... | https://mathoverflow.net/users/16537 | Terminology for a ring satisfying the DCC on chains of principal right ideals generated by the powers of an element | Such rings are called *strongly $\pi$-regular* in the literature. The condition is left-right symmetric, as first proven by Dischinger.
| 4 | https://mathoverflow.net/users/3199 | 394964 | 163,200 |
https://mathoverflow.net/questions/394915 | 5 | Let $R$ be a (commutative or non-commutative) unital ring, fix $a \in R$, and denote by $r(\cdot)$ the right annihilator of an element.
>
> **Question.** If $r(a)$ is a (right) direct summand of $R$ and $r(a) = r(a^2)$, does there exist an idempotent $e \in R$ such that $r(a) = eR$ and $r(1-a) \subseteq (1-e)R$?
> ... | https://mathoverflow.net/users/16537 | Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $r(1-a) \subseteq (1-e)R$ for some idempotent $e$? | The answer is yes. Further, you don't need the condition $r(a)=r(a^2)$.
Given an idempotent $e\in R$ such that $r(a)=eR$, let $e'=e(1-a)$. It is easy to check that, since $ae=0$, we have $e'^2=e'$. Also $eR=e'R$ since $e'=ee'$ and $e=e'e$. Finally, if $(1-a)x=0$ then $(1-e')x=[1-e(1-a)]x=x$, so $r(1-a)$ is contained ... | 3 | https://mathoverflow.net/users/3199 | 394965 | 163,201 |
https://mathoverflow.net/questions/394935 | 3 | Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a **connected** finite $k$-group scheme acting on $X$.
Does there exist a maximal closed subscheme $T$ of $Y$ stable under the action of $G$?
If $G$ is étale (and not connec... | https://mathoverflow.net/users/105092 | Maximal closed subscheme stable under the action of a finite connected group scheme | $\newcommand{\cO}{\mathcal{O}}$It seems that your formula for the etale case indeed gives the answer in general, if it is paraphrased in terms of rings of functions.
Consider the coaction map $\Delta:\cO\_X\to \cO\_X\otimes\_k k[G]$ and denote by $I\subset\cO\_X$ the ideal sheaf of $Y$. Pick a basis $e\_1,\dots, e\_n... | 4 | https://mathoverflow.net/users/39304 | 394966 | 163,202 |
https://mathoverflow.net/questions/394979 | 0 | (apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a\_1, a\_2, ...,a\_n$.
And we have information that partial sums of these elements are equal to something, say:
$$
a\_1 + a\_2 + ... + a\_{k\_1} = A\_{1} \\
a\_{k\_1+1} + a\_{k\_1+2} + ... + a\_n = B\_1 \\
a\_... | https://mathoverflow.net/users/4700 | How do you call a linear programming problem when the solution should be "constrained" to a norm? | If you are willing to replace $a\_i > 0$ by $a\_i \ge 0$,
then this becomes a **quadratic program**. Indeed,
it can be formulated as
\begin{align\*}
\text{Minimize}\quad & \frac12 a^\top Q a + q^\top a, \\
\text{such that} \quad & C a = d, \\
& a \ge 0.
\end{align\*}
Here, $Q$ and $C$ are matrices of appropriate size a... | 3 | https://mathoverflow.net/users/32507 | 394983 | 163,208 |
https://mathoverflow.net/questions/394973 | 0 | Consider a lower-semicontinuous convex function $f\colon \mathbb{R}^n \to \mathbb{R}$ with domain $C = \{x \in \mathbb{R}^d: f(x) < \infty\}$. I am interested in understanding under what conditions $f$ is continuous over $C$.
It is well known that this is true whenever $C$ is simplicial, but not otherwise (see the di... | https://mathoverflow.net/users/158537 | When is a convex function continuous on its domain? | I don't think that this is true.
Let us take
$$
C := \{ x \in \mathbb R^2 \mid x\_1^2 \le x\_2 \le 1\}$$
and
$$
f(x) = \frac{x\_1^2}{x\_2}
$$
for $x \in C \setminus \{0\}$, $f(0,0) = 0$.
This function is convex, lsc but discontinuous in $(0,0)$.
However, it is not strictly convex and not essentially smooth.
I think tha... | 1 | https://mathoverflow.net/users/32507 | 394986 | 163,209 |
https://mathoverflow.net/questions/394977 | 1 | Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and *uniformly transitive* in the sense that it satisfies the following two conditions:
i) The orbit of every point is dense in $X$.
ii) For every $\varepsil... | https://mathoverflow.net/users/173490 | Is a “uniformly minimal” dynamical system ergodic? | The answer is no, I think any of the usual examples works. Some argument below.
>
> Lemma. Suppose $X$ is a compact metric space, $\mu$ a nonatomic probability measure on $X$, and $T : X \to X$ is a minimal measure-preserving homeomorphism. Then $(X, \mu, T)$ is "uniformly transitive" (did you mean to write "unifor... | 3 | https://mathoverflow.net/users/123634 | 395000 | 163,213 |
https://mathoverflow.net/questions/394994 | 4 | It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f\_\* : C\_\*(X) \to C\_\*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to associated chain complexes.
When dealing with $\infty$ stuff, sometimes a bit more is needed. Let $\textrm{Sing}(\... | https://mathoverflow.net/users/140013 | Homotopy coherent space maps induces homotopy coherent chain complex morphisms | Be careful with this "simplicial structure on chain complexes". It's not really well-defined, as discussed in the comments below my answer [here](https://mathoverflow.net/questions/382151/homotopy-coherent-colimits-in-chain-complexes/382154#382154). Also see my remark at the end of this post.
In particular, the answe... | 4 | https://mathoverflow.net/users/102343 | 395003 | 163,214 |
https://mathoverflow.net/questions/394260 | 3 | Let $\mathcal{C}$ be the class of continuous functions that—
* map $[0, 1]$ to $[0, 1]$, and
* equal neither 0 nor 1 on the open interval $(0, 1)$.
A function $f(x)$ is *algebraic over the rational numbers* if—
* It can be a solution of a system of polynomial equations whose coefficients are rational numbers, or ... | https://mathoverflow.net/users/171320 | On the regularity of certain continuous algebraic functions | A continuous function $f : [0, 1] \to [0, 1]$ which is algebraic over the rational numbers in your sense is a semialgebraic function: its graph can be defined by a first-order formula in the language of an ordered field. Indeed, suppose $P(x, f(x)) = 0$ for a nonzero polynomial $P$. The zero locus $Z = \{\,(x,y) \in [0... | 1 | https://mathoverflow.net/users/126667 | 395018 | 163,220 |
https://mathoverflow.net/questions/394968 | 9 | As the title suggests, I am currently trying to understand Chebotarev's original proof of his density theorem, based on the proof in the appendix [here](http://pub.math.leidenuniv.nl/%7Elenstrahw/PUBLICATIONS/1994c/art.pdf). I am fully on-board with the cyclotomic extension case (which is essentially just a slightly mo... | https://mathoverflow.net/users/175051 | Original proof of Chebotarev's density theorem | Collecting comments into a community wiki answer:
Regarding why $L(ζ)=K(ζ)$: $L(ζ)⊆K(ζ)$ is the compositum of the subextensions $L⊆K(ζ)$ and $F(ζ)⊆K(ζ)$. Now (changing notation) when $F⊆K$ and $K'⊆L$ are subextensions of a Galois $L/F$, the compositum $K⋅K'⊆L$ has $\operatorname{Gal}(L/K⋅K')=\operatorname{Gal}(L/K)∩\... | 1 | https://mathoverflow.net/users/3106 | 395020 | 163,221 |
https://mathoverflow.net/questions/395012 | 7 | In $\mathbf{ZF}$, it is possible for a set $A$ to be infinite but not to admit a countable set. In other words, for any $\alpha\in\omega$, there is an injection from $\alpha$ into $A$, but there is no injection from $\omega$ into $A$. If we replace $\omega$ by a successor cardinal $\kappa^+$ in the above statement, any... | https://mathoverflow.net/users/138274 | Dedekind-"finiteness" for arbitrary limit cardinals | Start with your favourite model of $\sf ZFC$, your favourite regular cardinal $\mu$, and your favourite limit cardinal $\lambda>2^\mu$.
Now consider the ${<}\mu$-support product $\prod\_{\alpha<\lambda}\operatorname{Add}(\mu,\alpha)$. With automorphism groups that act on each individual component in the product, and ... | 5 | https://mathoverflow.net/users/7206 | 395025 | 163,223 |
https://mathoverflow.net/questions/395028 | 4 | Let $A$ be an $n\times n$ matrix with entries $a\_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b\_{i,j}=a\_{1,1}a\_{i+1,j+1}-a\_{1,j+1}a\_{i+1,1}$. Then $\det(B)=a\_{1,1}^{n-2}\det(A)$.
I can prove this by direct computation, but it seems like something that may be well known or follow from other pro... | https://mathoverflow.net/users/280363 | Determinant in terms of certain $2\times 2$ minors | A simple proof is given in the [Art of Problem Solving](https://artofproblemsolving.com/community/c7h1328834p7152628) (it is entered as an "olympiade problem").
I reproduce the two-line proof for the record, with the change that $a\_{11}\mapsto a\_{nn}$:
Add the $n$-th row of $A$ to the $i$-th row, multiplied by $-a... | 3 | https://mathoverflow.net/users/11260 | 395030 | 163,224 |
https://mathoverflow.net/questions/394993 | 1 | Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is *ample* if the relative twisting sheaf $\mathcal{O}\_{\mathbf{P}(E)}(1)$ is an ample line bundle on the projective bundle $\mathbf{P}(E)$ of hyperplanes in $E$. Now let $E$ be an arbi... | https://mathoverflow.net/users/nan | Making a vector bundle ample by twisting with ample line bundle | Because $L$ is ample, $E\otimes L^n$ is generated by global sections for $n\gg 0$, i.e., there is a surjective morphism $\mathscr O\_X^{\oplus r} \to E\otimes L^n$, which implies that there is a surjective morphism $L^{\oplus r} \to E\otimes L^{n+1}$. As $L$ is ample, so is $L^{\oplus r}$ and then so is its quotient $E... | 2 | https://mathoverflow.net/users/10076 | 395036 | 163,227 |
https://mathoverflow.net/questions/395042 | 3 | Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $M \times \{0\}$ and $M \times \{1\}$.
Can we show the region bounded by $M \times \{0\}$ and $i(M)$ is homeomorphic t... | https://mathoverflow.net/users/280895 | Embedded submanifold in a cylinder | In fact, there are "nice" counterexamples. There is a notion of an inertial h-cobordism on $M$. It is an h-cobordism with both boundaries homeomorphic to $M$. By the s-cobordism theorem, all h-cobordisms are invertible in the sense that we can stack one on top of the other and get the identity cobordism.
Given any in... | 7 | https://mathoverflow.net/users/134512 | 395048 | 163,232 |
https://mathoverflow.net/questions/395053 | 4 | Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$?
As far as I understand, this should be a consequence of the Bass-Quillen conjecture for $R$. Is it proven in this particular case?
| https://mathoverflow.net/users/66686 | Are finite projective modules over $R[t]$ free when $R$ is DVR? | The Bass-Quillen conjecture is known to be true for principal ideal domains (that is, if $A$ is a PID, all finitely generated projective modules over $A[T\_1,\dots,T\_n]$ are free). This was proven in theorem 4 of the paper
>
> *Quillen, Daniel*, [**Projective modules over polynomial rings**](http://dx.doi.org/10.1... | 7 | https://mathoverflow.net/users/43054 | 395057 | 163,234 |
https://mathoverflow.net/questions/395045 | 4 | Consider a simple (nearest neighbor) random walk on a lattice $\Bbb Z^2$ which starts at the origin, is constrained to $x\ge 0$ halfplane, and stops when it hits the line $x=n$. Denote by $p(n,k)$ the probability the walks stops at the point $(n,k)$, where $k \in \Bbb Z$.
**Question.** Does $p(n,tn)$ converge to a *G... | https://mathoverflow.net/users/4040 | Hitting probability of a line | As Timothy Budd has commented above, the limiting distribution is hyperbolic secant distribution. Here is a proof.
By the reflection principle, the random walk in question can be substituted with one that does not have $x\geq 0$ constraint, but is terminated upon hitting the line $x=n$ or $x=-n$. Let $X\_n$ be the ve... | 8 | https://mathoverflow.net/users/45902 | 395065 | 163,236 |
https://mathoverflow.net/questions/393019 | 6 | Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}\_L$ is free over its associated order $\mathfrak{A}\_{L/\mathbb{Q}}=\{x\in \mathbb{Q}[G]\mid x\mathcal{O}\_L\subseteq \mathcal{O}\_L\}$ if
1. $G$ is abelian (Leopoldt, 1959);
2. $G$ is dihedr... | https://mathoverflow.net/users/158845 | Galois module theory: from global to local | There are a number of things one can say about this.
First, some of the papers you mention **do** also cover the case of $p$-adic fields. For example, see Bergé's paper *Sur l’arithmétique d’une extension diédrale* Annales de l’institut Fourier, tome 22, no 2 (1972), p. 31-59. On page 32 you'll see that A can be a ... | 3 | https://mathoverflow.net/users/7443 | 395068 | 163,237 |
https://mathoverflow.net/questions/395072 | 2 | Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$. This is fairly easy to prove using the fact that $\mathbb{C}\mathbb{P}^n$ is monotone itself. But it is also mentioned that the... | https://mathoverflow.net/users/nan | $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology | The relative homology long exact sequence puts this group in between $H\_2(\mathbb{CP}^n)=\mathbb{Z}$ and $\mathbb{Z}/2$. It maps surjectively to $\mathbb{Z}/2$. Let D be a generator for relative homology. Then $2D$ generates $H\_2(\mathbb{CP}^n)$. The Maslov number of a relative class coming from $H\_2(\mathbb{CP}^n)$... | 4 | https://mathoverflow.net/users/10839 | 395074 | 163,238 |
https://mathoverflow.net/questions/392911 | -1 | If $(B\_1, \cdot\_1, +\_1, -\_1)$ is a complete atomic Boolean algebra (where the induced partial order is $\leq\_A$), and $(B\_2, \cdot\_2, +\_2, -\_2)$ is a complete atomic algebra (where the induced partial order is $\leq\_B$), do the class of functions from $B\_1 \to B\_2$ form a complete atomic Boolean algebra whe... | https://mathoverflow.net/users/122435 | A complete Boolean algebra on a function space | If you want to define an algebra of arbitrary subsets of $B\_1$, then indeed the function space from $B\_1$ to $2$ is what you want, and the boolean algebra structure of $B\_1$ need play no role (though it of course plays a role in defining $\sigma$).
So Andreas Blass's comment gives what you need: The algebra in que... | 0 | https://mathoverflow.net/users/18060 | 395080 | 163,241 |
https://mathoverflow.net/questions/395056 | 1 | Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M \times [-2,2]$ such that (a) $D$ is homeomorphic to $N \times [0,1]$; (b) $D$ contains $M \times [-1,1]$.
Can we prove the following statements?
1. $M$ and $N$ are homotopic.
2. $M$ and $N$ are homeomorphic.
If true,... | https://mathoverflow.net/users/280895 | Compatibility of two cylindrical regions | Yes.
For simplicity, we set $M\_1=M \times \{-2\}, M\_2=M \times \{0\},N\_1=N \times \{0\}$ and $N\_2=N \times \{1\}$ such that $N\_1$ is contained in $M \times (-2,0)$. Moreover, we denote the region bounded by $M\_1$ and $N\_2$ by $\Omega$.
There is a natural projection from $\Omega \to M\_1$ and the induced map ... | 1 | https://mathoverflow.net/users/16323 | 395082 | 163,242 |
https://mathoverflow.net/questions/395076 | 2 | In Engelking's *General topology*, in the exercises section, there is Ju. M. Smirnov's characterization of normal spaces:
A $T\_1$ space is normal iff the following properties hold (both):
1. Every closed $G\_\delta$ set is zero-set;
2. for every $F$ closed set and $G$ open set, such that $F$ is in $G$, there exist... | https://mathoverflow.net/users/175352 | An example of a $T_1$ space where all closed $G_\delta$ sets are zero-sets, but it isn't normal | Observe that every function $f:\omega\_{1}\rightarrow\mathbb{R}$ is eventually constant and $\omega\_{1}$ is normal. Observe also that if $A\subseteq\omega\_{1}$ is a closed $G\_{\delta}$ set, then the characteristic function $\chi\_{A}$ of $A$ is eventually constant.
Let $X=((\omega\_{1}+1)\times(\omega+1))\setminus... | 4 | https://mathoverflow.net/users/22277 | 395091 | 163,244 |
https://mathoverflow.net/questions/395086 | 9 | In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. In the quantized enveloping algebra $\mathrm{U}\_q(\mathfrak{g})$ where we define $\bracket{x}{y}\_q := xy-qyx$ is there ... | https://mathoverflow.net/users/64073 | Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra? | There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]\_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one:
$$
\big[A,[B,C]\_{q\_1}\big]\_{q\_2}+q\... | 15 | https://mathoverflow.net/users/85967 | 395097 | 163,246 |
https://mathoverflow.net/questions/395067 | 12 | Let $\text{Latt}$ denote the category of *lattices*, i.e., finitely generated free abelian groups. In the appendix to Lecture 4 of [Condensed.pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf), Scholze considers functors $F \colon \text{Latt} \to \mathcal D(\mathbb Z)$ that are additive: $F(A \oplus B) \co... | https://mathoverflow.net/users/21815 | Modules over the integral dual Steenrod algebra as linear functors | First I should say that Clausen-Scholze are *not* considering functors which preserve direct sums, but rather all functors. (This is likely why, in the end, one needs to know something about the homology of Eilenberg-MacLane *spaces* rather than the homology of the Eilenberg-MacLane spectrum. That guess/observation I l... | 11 | https://mathoverflow.net/users/6936 | 395102 | 163,249 |
https://mathoverflow.net/questions/395041 | 3 | Suppose that $X$ is a simplicial complex, and $f:X \rightarrow S^k$ a continuous map to a sphere. Is $f$ always homotopic to a simplicial map to the boundary of a $(n+1)$-simplex, $\partial \Delta^k$?
The simplicial approximation theorem implies that there is a barycentric subdivision of $X$, $X'$, and a simplicial m... | https://mathoverflow.net/users/165301 | Is every map from a simplicial complex to a sphere homotopic to a simplicial map to the boundary of a k-simplex? | Take $X = \partial \Delta^3$. Then homotopy classes of maps from $|X|$ to $S^2$ correspond to elements of $\pi\_2(S^2)\cong \mathbb{Z}$. But there are only finitely many maps of simplicial complexes $\partial \Delta^3 \to \partial \Delta^3$. So not all of them can be represented without further subdivision.
| 9 | https://mathoverflow.net/users/39747 | 395103 | 163,250 |
https://mathoverflow.net/questions/395062 | 3 | Let $M^2$ be a connected closed surface. Suppose there exists an smooth embedding from a connected closed surface $N$ into the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and $M \times \{1\}$.
If $N$ is homeomorphic to $M$, can we prove that the region bounded by $M \times \{0\}$ and $N$ is ... | https://mathoverflow.net/users/280895 | Surface separating the boundary of a cylinder | I am going to focus on the oriented case here. A similar argument should hold in the unoriented case but the degree argument ought to use twisted coefficients, which I don't want to go through here.
Write $W\_0$ and $W\_1$ for the closure of the two components of $M \times [0,1] \setminus N$, with $M \times \{i\} \su... | 4 | https://mathoverflow.net/users/40804 | 395126 | 163,259 |
https://mathoverflow.net/questions/395118 | 4 | Let $\mathbb F$ be a **finite-dimensional associative unital real algebra**. Consider $V:=\mathbb F^n$ and let's say $p \in V$ is ***good*** if $xp=0$ only has $x=0$ as solution.
>
>
> >
> > **Question: If $p\_1$ is good, are there $p\_2,\ldots, p\_n \in V$ such that $p\_1,\ldots,p\_n$ is a basis for $V$?**
> >
... | https://mathoverflow.net/users/43441 | Is it possible to complete a basis for a free module over a finite-dimensional associative unital real algebra? | Not in general, no.
Let $\mathbb{F}$ be the algebra of upper triangular $2\times 2$ matrices, let $n=2$, and let
$$p\_1=(x\_1,y\_1)=\left(\begin{pmatrix}0&0\\0&1\end{pmatrix},\begin{pmatrix}0&1\\0&0\end{pmatrix}\right),$$
so that
$$\mathbb{F}p\_1=\left\{\left(
\begin{pmatrix}0&b\\0&d\end{pmatrix},\begin{pmatrix}0&a\\... | 3 | https://mathoverflow.net/users/22989 | 395138 | 163,264 |
https://mathoverflow.net/questions/394908 | 4 | Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: *What can we say about the algebra structure of $Ext^{\ast}(E,E)$?*
Since this is a fairly general question, let me be more precise.
Let us for simplicity assum... | https://mathoverflow.net/users/124888 | $Ext$-algebra of stable vector bundles | There have been recently various results revolving around this question. Let me quote a few:
$\bullet$ For any line bundle $L$ on $X$, the graded algebra $\mathrm{Ext}^\*(L,L)$ is always graded-commutative. More generally, for any autoequivalence $\Phi$ of $\mathrm{D}^b(X)$, the graded algebra $\mathrm{Ext}^\*(\Phi(\... | 2 | https://mathoverflow.net/users/37214 | 395149 | 163,266 |
https://mathoverflow.net/questions/395179 | 3 | Are there only two solutions for $$\sum\_{k=0}^m3^k=2^n$$
Such as $3^0=2^0$ and $3^0+3^1=2^2$
*Note*
• If $m$ is even then $\sum\_{k=0}^m3^k$ will be odd.
• $$\sum\_{k=0}^m3^k=\sum\_{k=0}^m\binom{m+1}{k+1}2^k=\sum\_{k=0}^{m}\sum\_{l=0}^{k}\binom{m+1}{k+1}\binom{k}{l}$$
---
Edit: generalization may be more... | https://mathoverflow.net/users/149083 | Are there only two solutions for $1+3+9+...+3^m=2^n$ | Yes, those are the only solutions. To see this note that $$1+3+9+27 \cdots 3^m = \frac{3^{m+1}-1}{2}.$$
So we are looking for solutions of $\frac{3^{m+1}-1}{2}=2^n$, or equivalently looking for solutions of $3^{m+1} -1 = 2^{n+1}$. But this equation has only the obvious solutions, a result which one can prove with a l... | 9 | https://mathoverflow.net/users/127690 | 395181 | 163,274 |
https://mathoverflow.net/questions/395178 | 3 | Let $E\_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.
The unit point $e\in E\_q(K)$, the projection of $1\in \mathbf{G}\_m^{an}(K)$ to $E\_q(K)$ under the quotient map $$\mathb... | https://mathoverflow.net/users/nan | Geometric line bundles on the Tate curve | I think the action is $q\star z = -q^{-1} z$, so that you get the "correct" basic theta function. See p. 128 in Fresnel-van der Put or Roquette's book.
| 1 | https://mathoverflow.net/users/3847 | 395185 | 163,277 |
https://mathoverflow.net/questions/395143 | 9 | **Question.** Does the equality
$$\det\left[\sin 2\pi\frac{(j-k)^2}p\right]\_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}} $$
hold for every prime $p\equiv3\pmod4$?
I have checked the equality numerically for $p=3,7,11$. I conjecture that the equality holds for each prime $p\equiv3\pmod4$, but I don't know how to pr... | https://mathoverflow.net/users/124654 | Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}}$ for each prime $p\equiv3\pmod4$? | We will use the notation $e\_p(t)=\exp\left(\frac{2\pi it}{p}\right)$. First, let us show that for any $1\leq m\leq \frac{p-1}{2}$ there is an eigenvector of your matrix $A\_p$ with eigenvalue
$$
\lambda\_m=\sqrt{p}\cos\frac{2\pi m^2}{p}.
$$
To do so, for any $m\in \mathbb Z$ denote by $v\_m$ the vector from $\mathbb C... | 6 | https://mathoverflow.net/users/101078 | 395196 | 163,279 |
https://mathoverflow.net/questions/395128 | 3 | $\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \Pic^0(X).$$ Suppose that $\Pic(X) = \Pic^0(X) \neq 0$. So all divisors are algebraically equivalent, and (by definition... | https://mathoverflow.net/users/174369 | Examples of complex manifolds with trivial Néron–Severi group? | There is a Kähler example constructed in [1, Section 1].
Let $\Gamma = \mathbb{Z}^{2n}$ be a lattice, $\phi: \Gamma \to \Gamma$ a $\mathbb{Z} $-linear map with characteristic polynomial $f(\lambda)=\prod\_{i=1}^n(\lambda - \lambda\_i)(\lambda - \overline{\lambda\_i}) $ where $\lambda\_1,\cdots,\lambda\_n,\overline{\l... | 5 | https://mathoverflow.net/users/192152 | 395200 | 163,280 |
https://mathoverflow.net/questions/395186 | 5 | Today I discovered this nice video of a lecture by Thurston:
<https://youtu.be/daplYX6Oshc>
in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot can be thought as a portal to Narnia, and when you pass again you switch back to the Earth. This forms in a sense a $\mathb... | https://mathoverflow.net/users/140013 | Thurston universe gates in knots: which invariant is it? | [Here](https://www.youtube.com/watch?v=IKSrBt2kFD4&t=327s) is a higher-quality video of the same material. My answer is a more algebraic version of Thurston's presentation, but I will tie this back to Thurston's "intention" at the end.
---
Suppose that $L$ is an oriented knot diagram of a knot $K$. Let $A\_i$ enu... | 6 | https://mathoverflow.net/users/1650 | 395206 | 163,283 |
https://mathoverflow.net/questions/395211 | 2 | Let $T\_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T\_2(R)$ is right Rickart, i.e., the right annihilator of every element is a (right) direct summand (and I think I can gener... | https://mathoverflow.net/users/16537 | The ring of upper triangular $n$-by-$n$ matrices over a skew field is (left and right) Rickart | By <https://encyclopediaofmath.org/wiki/Rickart_ring> , a left Rickart ring is characterised by all principal left ideals being projective.
But your ring (in the skew-field case as in the title of your question) is a hereditary ring so every submodule of a projective module is also projective.
Thus every principal left... | 2 | https://mathoverflow.net/users/61949 | 395212 | 163,284 |
https://mathoverflow.net/questions/395220 | 1 | Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\in \mathbb{R}$ and irrational?
What about such $z$ that $z^z\in\mathbb{R}$ and irrational?
I mean, we can rise $(-1)$ ... | https://mathoverflow.net/users/10059 | Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold? | I will address the question in the case $f(z) = (-1)^z = \exp(z \log(-1))$ and $A$ is a finite-dimensional commutative (associative unital) $\mathbb{R}$-algebra, where $\log(-1)$ is a suitable choice of the logarithm in $A$.
Wlog. $A$ is a local $\mathbb{R}$-algebra $(A,\mathfrak{m})$. There are two possibilities. Fi... | 2 | https://mathoverflow.net/users/1849 | 395238 | 163,292 |
https://mathoverflow.net/questions/395227 | 5 | I have asked this before on [MSE](https://math.stackexchange.com/questions/4152944/sufficient-condition-for-the-zero-set-of-an-analytic-function-to-be-still-immers), but received no answer yet.
Say I have a set in $\mathbb{R}^n$ defined to be the zero set of an analytic function $F:\mathbb{R}^n\to\mathbb{R}^k$, $k<n$... | https://mathoverflow.net/users/276879 | When is a real-analytic variety a union of non-singular subvarieties? | The general problem mentioned at the beginning of the question is extremely difficult, and, without more hypotheses, there is not that much that can be said.
The OP might be interested in [this answer of mine](https://mathoverflow.net/questions/98366/when-is-a-singular-point-of-a-variety-mathcalc-infty-smooth/98402#9... | 6 | https://mathoverflow.net/users/13972 | 395246 | 163,293 |
https://mathoverflow.net/questions/395199 | 7 | For [Sendov's conjecture](https://en.wikipedia.org/wiki/Sendov%27s_conjecture), the distance 1 appears in the conjecture is tight, if one consider the polynomials $f\_{n}(z) = z^{n} - 1$ for all $n\geq 2$. I wonder if this polynomial is the local optima for the conjecture. More precisely, I want to know if the followin... | https://mathoverflow.net/users/95471 | Local optimum for Sendov's conjecture | This follows from the work of
*Miller, Michael J.*, [**On Sendov’s conjecture for roots near the unit circle**](http://dx.doi.org/10.1006/jmaa.1993.1194), J. Math. Anal. Appl. 175, No. 2, 632-639 (1993). [ZBL0782.30007](https://zbmath.org/?q=an:0782.30007).
and independently
*Vâjâitu, Viorel; Zaharescu, A.*, [**I... | 6 | https://mathoverflow.net/users/766 | 395248 | 163,294 |
https://mathoverflow.net/questions/395247 | 8 | Let $G$ be a discrete group. For a $G$-CW complex $X$, let $H^G\_{\bullet}(X)$ denote the Borel equivariant homology of $X$. There are also relative versions of this.
Here's my question. Let $X$ be a $G$-CW complex. The suspension $\Sigma X$ is then a $G$-CW complex in a natural way, and has two $G$-invariant base po... | https://mathoverflow.net/users/286287 | Borel equivariant homology of a suspension | I assume that by Borel equivariant homology of $X$ you mean the ordinary homology of the "Borel construction" $X\times\_G EG$.
There is a homotopy cofibration sequence
$$
X\times\_G EG \to \*\times\_G EG \to \Sigma X\wedge\_G EG\_+.
$$
It induces a long exact sequence in homology, which you can interpret as a long ex... | 12 | https://mathoverflow.net/users/6668 | 395250 | 163,295 |
https://mathoverflow.net/questions/395233 | 2 | Let $A$ be an infinite-dimensional noncommutative algebra over a field, let $B$ be an infinite-dimensional subalgebra of $A$, and let $A$ be a direct sum of projective simple $B$-sub-bimodules. Then can one conclude that $A$, or indeed $B$, is a semisimple ring?
EDIT: I should highlight that I am interested only in t... | https://mathoverflow.net/users/176218 | An algebra which is a direct sum of simple sub-bimodules over a subalgebra | @BugsBunny answered the original version of the question. I'll answer the new version. The algebra $B$ must be finite dimensional and semisimple under these hypothesis, and even stronger, it must be separable meaning that it remains semisimple even under base extension.
Let $B^{e}=B\otimes\_k B^{op}$ be the envelopin... | 3 | https://mathoverflow.net/users/15934 | 395256 | 163,297 |
https://mathoverflow.net/questions/395223 | 11 | Let $S$ be the set of germs of riemannian metrics near $0$ on $\mathbb R^n$. It is acted on by the group $\textrm{Diff}$ of germs of diffeomorphisms of $\mathbb R^n$ preserving $0$.
Let's denote by $S^{(k)}$ the set of $k$-jets of riemannian metrics at $0$ (first $k$ terms of the Taylor expansion). The $k$-th [jet gr... | https://mathoverflow.net/users/13842 | Moduli space of germs of riemannian metrics | The answers to these questions are known, but, perhaps, not well-known. The typical approach is to first divide only by the local diffeomorphisms $\phi:\mathbb{R}^n\to\mathbb{R}^n$ that fix the origin and for which $\phi'(0):\mathbb{R}^n\to\mathbb{R}^n$ is the identity. This quotient is essentially sectioned by geodesi... | 21 | https://mathoverflow.net/users/13972 | 395258 | 163,298 |
https://mathoverflow.net/questions/395230 | 4 | *A topological space $X$ is $\mathbb Z$-formal, if the singular cochain complex $C^\*(X,\mathbb Z)$ is
quasi-isomorphic to $H^\*(X, \mathbb Z)$ as an augmented differential graded ring.*
---
It's quite simple to write down specific quasi-isomorphisms to show that the Spheres $S^n$ are $\mathbb Q$-formal spaces by... | https://mathoverflow.net/users/166050 | $\mathbb Z$-formality of spheres | Consider the simplicial set $\def\S{{\bf S}} \def\Sing{\mathop{\rm Sing}} \S^n=Δ^n/∂Δ^n$, which has exactly two nondegenerate simplices: a 0-simplex and an $n$-simplex.
Consider the map $\S^n→\Sing S^n$ that sends the only vertex of $\S^n$ to the given basepoint of $S^n$ and the only nondegenerate $n$-simplex of $\S^... | 4 | https://mathoverflow.net/users/402 | 395260 | 163,299 |
https://mathoverflow.net/questions/395240 | 0 | You are receiving a time series whose elements belong to a finite set. Assume the time series is distributed as a Discrete-Time Markov Chain. You receive one element at each time step.
For each time step, your goal is to produce the best possible approximation of the underlying Markov Chain, ideally through a minimal... | https://mathoverflow.net/users/286197 | How to detect, track and map a Markov chain | For a given state $i$, row $i$ of the transition matrix gives the transition probabilities $P\_{ij}$ from $i$ to $j$, $j=1..n$ (the number of states). This
is a probability distribution, and the minimum variance unbiased estimator of it is
the empirical distribution $\hat{P}\_{ij} = N\_{ij}/N\_i$, where $N\_i$ is the n... | 2 | https://mathoverflow.net/users/13650 | 395264 | 163,302 |
https://mathoverflow.net/questions/395205 | 9 | Let $V$ be a finite-dimensional real vector space with its Euclidean topology. Then all norms on $V$ are equivalent and consequently given two norms $\lVert-\rVert$, $\lVert-\rVert'$, the number
$$
d = d(\lVert-\rVert, \lVert-\rVert') := \sup\_{0 \neq v \in V}\big| \log\lVert v\rVert - \log\lVert v\rVert'\big|
$$
is fi... | https://mathoverflow.net/users/112369 | Continuously varying norms | I expand my comment where I claim that, on the space (call it $N(V)$) of all norms on $V$, the smallest topology making continuous the evaluations $\|\cdot\| \mapsto \|v\|$ (for $v \in V$) coincides with the topology defined by the distance $d$.
This clearly implies that, under your hypothesis, the maps $t \mapsto \|... | 4 | https://mathoverflow.net/users/10265 | 395278 | 163,305 |
https://mathoverflow.net/questions/395277 | 1 | **Definitions:**
We say a smooth Riemannian metric on $\mathbb R^n$ is smoothly equivalent to Lebesgue measure if the Radon Nikodym derivative of the associated Riemannian volume measure with respect to Lebesgue measure is smooth.
Given a smooth Riemannian metric $g$ on $\mathbb R^n$, and a point $x \in
\mathbb R^... | https://mathoverflow.net/users/173490 | Can two continuously differentiable functions be made $C^1$ close via a perturbation of the metric? | **Edit.** For an even simpler example, let $f \in C^1(\mathbf{R}^n)$ be non-constant and $h = 0$. Such a metric cannot exist when the constant is so small that $\epsilon < \lvert Df \rvert\_\infty$.
Let $f \in C^1(\mathbf{R}^n)$ be an arbitrary, non-constant function, and $h = -f$. Let moreover the metric $g$ be arbi... | 1 | https://mathoverflow.net/users/103792 | 395282 | 163,307 |
https://mathoverflow.net/questions/395183 | 3 | I have the following problem. Let $\Gamma\_{G\_1\times G\_2}$ be a full subcategory of the orbit category $\mathcal{O}\_{G\_1\times G\_2}$ consisting of graph subgroups of $G\_1\times G\_2$. Further, let $N$ be a dual (i.e., covariant) coefficient system over $G\_1$ and define the dual coefficient system $FN$ over $G\_... | https://mathoverflow.net/users/123432 | Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free | **TL; DR**: Suppose you have a functor between small categories $\mathcal C\_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories
$$
L:[\mathcal C\_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$
Where the right adjoint $R$ is th... | 3 | https://mathoverflow.net/users/6668 | 395285 | 163,308 |
https://mathoverflow.net/questions/395280 | 0 | Let $f:(0, +\infty)\to(0, +\infty) $ be a **monotone decreasing**, **right-continuous** function. Can we find a sequence $\{f\_{n}\}\_{n\in \mathbb{N}}$ of **strictly monotone decreasing**, **continuous functions**, such that $f\_{n}$ converges pointwise to $f$, that is, $f(x)=\lim\_{n\to \infty}f\_{n}(x)$ for all $x\i... | https://mathoverflow.net/users/163368 | Approximation of positive right-continuous function | Take $f\_n(x) =n\int\_{x}^{x+1/n} f(t) dt-x/n$
| 3 | https://mathoverflow.net/users/4312 | 395286 | 163,309 |
https://mathoverflow.net/questions/395287 | 2 | **Problem set up:**
Consider $C\_b$, the Banach space of continuous bounded functions on $[0, \infty)$ equipped with the sup norm. Denote by $M$ the set of probability measures on $[0, \infty)$, and for $r > 0$ denote by $M\_r$ the set of probability measures supported on $[r, \infty)$. We will consider $M$ as subset... | https://mathoverflow.net/users/173490 | Explicit example of a certain weak-* limit | (reading "sequence" as "net", as suggested in the comments)
Well, $C\_b(\mathbb{R}^+) \cong C(\beta\mathbb{R}^+)$, so any such $L$ will arise from a probability measure on the Stone-Cech remainder $\beta\mathbb{R}^+ \setminus \mathbb{R}^+$. You need some choice principle to know this set is nonempty, so no example ca... | 9 | https://mathoverflow.net/users/23141 | 395303 | 163,314 |
https://mathoverflow.net/questions/395302 | 0 | Let $(\mu\_n)\_{n=1}^{\infty}$ be a sequence in $\mathcal{P}\_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}\_1(X)\rightarrow \mathcal{P}\_1(X)$ satisfying:
$$
\mu\_{n+1} = F(\mu\_n) \qquad \forall n=2,\dots
.
$$
Then:
1. Must there exist a Markov proces... | https://mathoverflow.net/users/36886 | Do measure-valued dynamical systems correspond to marginals of Markov processes? | No. Any Markov operator is contracting in the total variation norm, whereas your function $F$ is subject to a much weaker condition of weak continuity. It is easy to construct a counterexample. For instance, take $X$ to be the two point set $\{0,1\}$, then the probability measures on $X$ are parameterized by a single p... | 2 | https://mathoverflow.net/users/8588 | 395306 | 163,316 |
https://mathoverflow.net/questions/394711 | 8 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say ${\cal A}\subseteq [\omega]^\omega$ is *almost disjoint* if $A \cap B$ is finite whenever $A\neq B \in {\cal A}$. Zorn's Lemma implies that every almost disjoint family is contained in a maximal one. Moreover, a diagonalization argument... | https://mathoverflow.net/users/8628 | Sunflowers in maximal almost disjoint families | The following is a ZFC example, ***due to Michael Hrušák***, of a MAD family without sunflowers of
cardinality $3$.
Start with the standard AD family $\mathcal{B}=\{B\_f:f\in{}^\omega2\}$ of
branches through the binary tree $2^{<\omega}$, so $B\_f=\{f|n:n\in\omega\}$.
Extend $\mathcal{B}$ to a MAD family by adding a ... | 7 | https://mathoverflow.net/users/5903 | 395323 | 163,321 |
https://mathoverflow.net/questions/395316 | 12 | $\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E\_\pm, H$ with commutation relations
$$
[H,E\_{\pm}]=\pm 2 E\_\pm,\quad [E\_+,E\_-]=H
$$
admits the well-known representation on $\mathbb{C}[x]$ with
$$
E\_+ = \partial\_x,\quad E\_- = -x^2 \partial\_x + s\,x,\quad H = -x \partial\_x - s
$$
where $\partial\_x = \frac{... | https://mathoverflow.net/users/288074 | Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators | Although i have some doubts as to what the OP is exactly looking for (see my comments above), i hope that the following will be of some interest for its purposes. In:
* [$U\_q(sl(n))$ Difference Operator Realization](https://arxiv.org/abs/hep-th/9408173v1), A. Shafiekhani,
the author introduces a unified scheme for... | 4 | https://mathoverflow.net/users/85967 | 395328 | 163,322 |
https://mathoverflow.net/questions/395320 | 17 | I am thinking about trying to formalise some parts of classical unstable homotopy theory in homotopy type theory, especially the EHP and Toda fibrations, and some related work of Gray, Anick and Cohen-Moore-Neisendorfer. I am encouraged by the successful formalisation of the Blakers-Massey and Freudenthal theorems; I w... | https://mathoverflow.net/users/10366 | Available frameworks for homotopy type theory | The HoTT libraries in Lean can be considered dead. Since Lean has moved away from HoTT, I don't think it's a more convenient system to do HoTT in than - for example - Coq.
There is still some formalization material in the Lean 2 library that hasn't been formalized in another proof assistant, but probably the best thi... | 12 | https://mathoverflow.net/users/112216 | 395330 | 163,323 |
https://mathoverflow.net/questions/395326 | 1 | I am looking for a reference (or a simple proof) of the fact that a free group is [sofic](https://en.wikipedia.org/wiki/Sofic_group). The preferred dynamical definition of a sofic group seems to be that
there is a sequence of finite sets $V\_n$ with $|V\_n|\to\infty$ and a sequence of maps $\sigma\_n\colon \Gamma\to \t... | https://mathoverflow.net/users/11054 | Direct proof that free groups are sofic | There is a very simple probabilistic proof. Begin with a large finite set - its elements are called "parents", and let each parent have $2d$ offspring labelled with the generators of our free group. Now let the kids go to a nightclub, where each of them randomly finds a partner from the opposite sex (or shall I say gen... | 1 | https://mathoverflow.net/users/8588 | 395332 | 163,324 |
https://mathoverflow.net/questions/395325 | 1 | There are two invariants for the type $III$ factor $M$, namely, $S(M)$ and $T(M)$.
When $S(M)=[0, \infty)$, $M$ is a factor of type $III\_{1}$.
My question : how to determine whether $M$ is a factor of type $III\_{1}$ by using the invariant $T(M)$?
| https://mathoverflow.net/users/153196 | Two invariants for type III factors | The invariants $S(M)$ and $T(M)$ of a type III factor $M$ are only partially related.
* If $M$ is of type III$\_1$, then $T(M) = \{0\}$.
* If $M$ is of type III$\_\lambda$ with $\lambda \in (0,1)$, then $T(M) = (2\pi/\log \lambda) \mathbb{Z}$.
* If $M$ is of type III$\_0$, then $T(M)$ ranges over a huge class of subg... | 4 | https://mathoverflow.net/users/159170 | 395345 | 163,331 |
https://mathoverflow.net/questions/395348 | 4 | In [this Wikipedia article](https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80#Infinite_series) the constant $\pi$ is represented by the following infinite series: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}+\frac{1}{12}-... | https://mathoverflow.net/users/88804 | The constant $e$ represented by an infinite series | Your sum actually equals $\frac{\pi\sqrt{3}}{2}$, so it's more like $\pi$ all over again, not $e$. To see this, note first that by definition $\mathrm{sgn}\_2$ is multiplicative, hence
$$
A=\sum\_{n=1}^{+\infty}\frac{\mathrm{sgn}\_2(n)}{n}=\prod\_p\left(1-\frac{\mathrm{sgn}\_2(p)}{p}\right)^{-1}=
$$
$$
=\left(1-\frac12... | 22 | https://mathoverflow.net/users/101078 | 395350 | 163,333 |
https://mathoverflow.net/questions/395284 | -3 | This question is a follow-up to [About Goldbach's conjecture](https://mathoverflow.net/questions/61842/about-goldbachs-conjecture?r=SearchResults) and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ and $n+r$ are prime.
So, considering... | https://mathoverflow.net/users/13625 | Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity? | This identity is false. To see that, notice first that
$$
\sum\_{p\leq \sqrt{2n-3}}a\_p(n)=2\sum\_{p\leq \sqrt{2n-3}}1-\sum\_{\substack{p\leq \sqrt{2n-3}\\ p\mid n}}1=2\mathrm{ord}\_C(n)-O(\ln n),
$$
so
$$
N\_2^{eq}(n)=n\prod\_{p\leq \sqrt{2n-3}}\left(\frac{\pi(n)}{n}\right)^{a\_p(n)/\mathrm{ord}\_C(n)}=n\left(\frac{1+... | 7 | https://mathoverflow.net/users/101078 | 395356 | 163,335 |
https://mathoverflow.net/questions/395364 | 4 | The title says it all. Let $A$ be a path connected $F\_\sigma$ subset of a plane (or more generally $\mathbb{R}^n$). Recall that a subset is called $F\_\sigma$ if it is a union of a sequence of closed sets.
>
> Is it true that there is a continuous surjection from $[0,1)$ onto $A$? Equivalently, can $A$ be represen... | https://mathoverflow.net/users/53155 | Is every path connected $F_\sigma$ subset of a plane an image of $[0,1)$? | No, this fails even for compact subsets of $\mathbb R^2$. Namely, let $X=C\times[0,1]\cup[0,1]\times\{0\}$, where $C$ is the Cantor set. It is clearly path connected. $X$ cannot be an image of $[0,1)$, because the image of any interval $[0,a],a<1$ by this map can contain only finitely many points of $C\times\{1\}$ (bec... | 10 | https://mathoverflow.net/users/30186 | 395365 | 163,337 |
https://mathoverflow.net/questions/395368 | -4 | * Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
* When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question was proved [Bertrand's postulate](https://en.wikipedia.org/wiki/Bertrand%27s_postulate)
| https://mathoverflow.net/users/122662 | A generalization Bertrand's postulate | First of all, as mentioned by Random above, this is a very strong conjecture, because it is stronger than Legendre's conjecture. As far as I know, it is not known even if we assume the truth of Riemann Hypothesis and also some reasonable conjectures on distribution of imaginary parts of zeros, such as the Montgomery's ... | 6 | https://mathoverflow.net/users/101078 | 395371 | 163,340 |
https://mathoverflow.net/questions/395389 | 2 | Let $F$ be a totally real number field having *at least* two different real embeddings $\sigma\_1 : F \hookrightarrow \mathbb{R}$ and $\sigma\_2 : F \hookrightarrow \mathbb{R}$.
Does a quaternion algebra $A = \left(\frac{a,b}{F}\right)$ over $F$ exist such that $A$ is not itself a matrix algebra, but which splits at ... | https://mathoverflow.net/users/98357 | Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others? | Let $a$ be any totally negative element and pick $b$ to be an element such that $\sigma\_1(b),\sigma\_2(b)$ are positive, while $\sigma(b)$ is negative for all other $\sigma:F\to\mathbb R$. Such elements exist by suitable approximation theorems in number fields. For $A$ defined using these elements, after tensoring by ... | 3 | https://mathoverflow.net/users/30186 | 395392 | 163,348 |
https://mathoverflow.net/questions/395397 | 2 | I am looking for the **Picard group of the moduli space of principal $G$-bundles** for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\mathrm{Sp}(2n,\mathbb{C})?$
Is there any reference for this?
| https://mathoverflow.net/users/139928 | Picard group of moduli of principal bundles | Theorem A in:
S. Kumar and M. S. Narasimhan. *Picard group of the moduli spaces of G-bundles.* Math. Ann., 308(1):155-173, 1997,
shows that when $G$ is a simple simply-connected connected complex affine algebraic group, $C$ is a complex smooth irreducible projective curve of genus at least 2, and $M$ is the moduli ... | 4 | https://mathoverflow.net/users/12218 | 395398 | 163,350 |
https://mathoverflow.net/questions/395399 | 11 | Suppose that $\mathcal{C}$ is a skeletally small additive category.
To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$, one may consider the ind-completion $\operatorname{Ind}\mathcal{C}$ of $\mathcal{C}$ in the sense of Grothendieck and Verdier.... | https://mathoverflow.net/users/36805 | Relation between Ind-completion and "additive"-ind-completion | The point is what Ivan hints at in his last paragraph, that additivity is a property rather than an extra structure.
In fact, we have:
>
> Suppose $C$ is an additive category. Then the forgetful functor $Fun^\times(C,\mathbf{Ab})\to Fun^\times(C,\mathbf{Set})$ is an equivalence of categories.
>
>
>
(if you o... | 7 | https://mathoverflow.net/users/102343 | 395405 | 163,354 |
https://mathoverflow.net/questions/395315 | 20 | $\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$*Originally [asked at MSE](https://math.stackexchange.com/questions/4164632/does-sine-equationally-interact-with-addition-or-with-multiplication) without success:*
For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\E... | https://mathoverflow.net/users/8133 | Does sine interact equationally with addition alone? | Here is the outline of a proof that $Eq(\mathbb{R},+)$ proves all the equalities in both $Eq(\mathbb{R},+,\sin)$ and $Eq(\mathbb{R},+,\exp)$.
**Step 1, defining $<$:**
For any term $t$ in $L(+,\sin)$, define a real function $t\_R$ by $$t\_R(r\_1,\ldots r\_n)=\sup(\{|t(z\_1,\ldots,z\_n)|: z\_1,\ldots,z\_n \in \mathb... | 11 | https://mathoverflow.net/users/nan | 395407 | 163,355 |
https://mathoverflow.net/questions/395355 | 6 | I am considering the following equation
$$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(2\pi x) \end{pmatrix}\varphi(x)=0$$
and I am wondering if there is an explicit characterization of $\lambda \neq 0$ for whic... | https://mathoverflow.net/users/150549 | Matrix-valued ordinary differential equation with symmetry | Your equation in fact admits a simple explicit solution. As noted in the previous answer, the general solution is encoded in a matrix ODE,
$$
M'(x) = \lambda A(x)M(x), \qquad M(0)=I\,.
$$
Now, the simplification becomes apparent if change to a new basis, using the $x$-dependent rotation
$$
R(x) = \begin{pmatrix} \cos(\... | 7 | https://mathoverflow.net/users/43462 | 395411 | 163,356 |
https://mathoverflow.net/questions/395410 | 1 | In "Hardy's Uncertainty Principle, Convexity and Schrödinger Evolutions" ([link](https://arxiv.org/abs/0802.1608v1)) on page 5, the authors state that they are using the Cauchy-Schwarz inequality to bound the derivative of the $L^2(\mathbb{R}^n)$ norm of a solution to a certain differential equations, but I am not sure... | https://mathoverflow.net/users/152473 | How is the Cauchy-Schwarz inequality used to bound this derivative? | You have a typo on the $\mathrm{Re}(Sv,v)$ term, the leading $A$ should be inside the integral. The formula from the paper reads
$$\mathrm{Re}\left(Sv,v\right) = \int -A |\nabla v|^2 + \left(A|\nabla \phi|^2+\partial\_t \phi \right) |v|^2 + \color{red}{ 2B \,\mathrm{Im}\, v^{\dagger} \nabla\phi\cdot\nabla v } + \left... | 3 | https://mathoverflow.net/users/3948 | 395413 | 163,358 |
https://mathoverflow.net/questions/395414 | 6 | Consider the sequence (of rational numbers) given by
$$a\_n=\sum\_{k=1}^n\binom{n}k\frac{k}{n+k}.$$
Let $s(n)$ be the sum of binary digits of $n$, i.e. the total number of $1$'s.
>
> **QUESTION.** Is it true that the $2$-adic valuation of the denominator of $a\_n$ equals $s(n)$?
> It seems so, experimentally.
>
> ... | https://mathoverflow.net/users/66131 | 2-adic valuation of a certain binomial sum | First we notice that
\begin{split}
a\_n & = n \int\_0^1 x^n (1+x)^{n-1}{\rm d}x \\
& = n \int\_0^1 (1-x)^n (2-x)^{n-1}{\rm d}x \\
& = n\sum\_{k=0}^{n-1} \binom{n-1}{k}2^k (-1)^{n-1-k} \int\_0^1 (1-x)^n x^{n-1-k}{\rm d}x \\
&= \sum\_{k=0}^{n-1} 2^k (-1)^{n-1-k} \binom{2n}{k} / \binom{2n}n. \\
\end{split}
Now, the numer... | 17 | https://mathoverflow.net/users/7076 | 395423 | 163,360 |
https://mathoverflow.net/questions/395415 | 10 | In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.
We have an adjunction $$\mathscr{l}: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)\_0,$$ induced by the *change of enrichment* as discussed for example between Ex. 3.2 and 3.3 [here](https://ncatlab.org/nl... | https://mathoverflow.net/users/104432 | Structural properties of $\mathcal{V}$-$\mathsf{Cat}$ | I'm going to assume that $\mathcal{V}$ is closed as well, or at least that its tensor product preserves colimits in each variable; I'm not sure that you get the left adjoint from $\mathsf{Cat}$ to $\mathcal{V}\text{-}\mathsf{Cat}$ otherwise. In this situation the underlying adjunction $\mathsf{Set} \rightleftarrows \ma... | 10 | https://mathoverflow.net/users/49 | 395434 | 163,361 |
https://mathoverflow.net/questions/395427 | 2 | Let $X$ be a locally Noetherian scheme and $K^{\bullet}$ a perfect complex of $\mathcal{O}\_X$-modules. We say $K^{\bullet}$ is "formal" if it is quasi-isomorphic to the complex $\bigoplus\_{n}H\_n(K^{\bullet})[n]$.
>
> Can formality of a perfect complex of $\mathcal{O}\_X$-modules be checked Zariski-locally on $X$... | https://mathoverflow.net/users/nan | Checking formality of a perfect complex Zariski-locally | Consider two vector bundles $V,W$ and $\alpha \in {\rm Ext}^2(V,W)$, $\alpha \neq 0$.
Then $$C:={\rm cone}(V[1] \to^\alpha W[-1])$$ is a nonsplit complex with $H^0(C) = V$ and $H^1(C) = W$. But $C$ splits locally, because locally $V$ and $W$ are projective.
E.g. we can take $X = \mathbb P^2, V =\mathcal O,$ and $W = ... | 3 | https://mathoverflow.net/users/131945 | 395436 | 163,362 |
https://mathoverflow.net/questions/395429 | 1 | Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?
This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.
| https://mathoverflow.net/users/159074 | Image of a projective variety is closed | To elaborate on Dori's comment, consider $[\mathbf{A}^1/\mathbf{G}\_m]$ which consists of two points: The closed point corresponding to the origin and the orbit of $1$ (which is open). Take $\operatorname{Spec} k \to [\mathbf{A}^1/\mathbf{G}\_m]$ corresponding to this open point. The image is not closed and certainly $... | 2 | https://mathoverflow.net/users/21278 | 395439 | 163,363 |
https://mathoverflow.net/questions/395438 | 4 | I am looking for the proof of the following claim:
First, define the function $\operatorname{sgn\_1}(n)$ as follows:
$$\operatorname{sgn\_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \equiv 3 \pmod{4}\\1 \quad \text{if } n \in \{2,3\} \text{ or } n \equiv 1 \pmod{4}\end{cases}$$
Let $n=p\_1^{\alp... | https://mathoverflow.net/users/88804 | The constant $\pi$ expressed by an infinite series | This can be proved similarly as [Alexander Kalmynin's method](https://mathoverflow.net/a/395350/156029) .
Let, the sum be $S$, then we can make the following identity because $\text{sgn}\_1$ of $2,3$ is defined to be $1$. So, $\text{sgn}\_1(ak)=\text{sgn}\_1(k), a=2,3,6$. Also, from the definition of $\text{sgn}\_2$ ... | 6 | https://mathoverflow.net/users/156029 | 395444 | 163,365 |
https://mathoverflow.net/questions/395408 | 2 | Let $f : X\to Y$ be a syntomic morphism of locally Noetherian $S$-schemes (i.e. flat and lci) and assume $X$ and $Y$ are smooth over a locally Noetherian scheme $S$.
>
> **Q1:** is $\Omega^1\_{X/Y}$ a flat $\mathcal{O}\_Y$-module?
>
>
>
The first answer here [Flatness of sheaf of relative Kahler differentials]... | https://mathoverflow.net/users/nan | Flatness of $\Omega^1_{X/S}$ | I think the calculation in the comment is incorrect. Let $S=\operatorname{Spec}(k)$ for a field $k$, let $m\geq 2$ be an integer invertible in $k$, and let $f\colon X\to Y$ be the $m$-th power map on $\mathbf{A}^1\_k$, i.e. $\operatorname{Spec} k[x] \to \operatorname{Spec} k[y]$ with $f^\*(y) = x^m$. Then $\Omega^1\_{k... | 1 | https://mathoverflow.net/users/3847 | 395454 | 163,369 |
https://mathoverflow.net/questions/395450 | 1 | Let $M$ be a closed subspace of a Banach space $X$. Then we can identify $(X/M)^\*$ with $M^\perp$ and $M^\*$ with $X^\*/M^\perp$.
Indeed, if $Q^\*:X\to X/M$ is the quotient map, then $Q^\*:M^\*\to X^\*$ is a linear isometry with range $M^\perp$. Moreover, if $J:M\to X$ is the embedding, then $J^\*:X^\*\to M^\*$ has ... | https://mathoverflow.net/users/39421 | Duality $(M/N)^*\equiv N^\perp/M^\perp$ for closed subspaces $N\subset M$ of a Banach space | As is so often the case, the isomorphism is the only reasonable map you can write down in the general case:
$N^\perp / M^\perp \to (M/N)^\ast, f+M^\perp \mapsto (m+N \mapsto f(m))$
All that's left to prove is that it actually works ;-)
| 5 | https://mathoverflow.net/users/3041 | 395458 | 163,370 |
https://mathoverflow.net/questions/393648 | 42 | Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf\_n d(T^n x, T^n y) = 0$ implies $x = y$.
Then it is true that $T$ is bijective.
**Question:** Is there an elementary proof of this fact? (Injectivity clearly follows, surjectivity is the issue.) The two proofs I know g... | https://mathoverflow.net/users/173490 | Is there an elementary proof that distal maps are invertible? | The answer is yesser than I thought. I mentioned this issue at <http://eventos.cmm.uchile.cl/edynamicsxiii/>, since the proximality lemma from my previous answer was discussed there. Someone pointed out that Hindman's original proof of his famous theorem is at least somewhat elementary in some technical sense, and impl... | 7 | https://mathoverflow.net/users/123634 | 395460 | 163,371 |
https://mathoverflow.net/questions/395472 | 2 | I was looking at the following paper by Tango:
<https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-14/issue-3/On-n-1-dimensional-projectlve-spaces-contained-in-the-Grassmann/10.1215/kjm/1250523169.full>
In Lemma 2.4 , at the end, he says that if $Y \subset \mathbb{G}(k,n)$ is a sub... | https://mathoverflow.net/users/146431 | Trivial subbundle of universal bundle on the Grassmannian $\mathbb{G}(k,n)$ | Let $Z\_p \subset \operatorname{Gr}(k,n)$ be the subscheme parameterizing all subspaces parameterizing all $k$-planes containing $p$. Then $Z\_p \cong \operatorname{Gr}(k-1,n-1)$ and the restriction of the tautological bundle to $Z\_p$ splits as the sum of $\mathcal{O}$ and the tautological bundle $S'$ of $\operatornam... | 6 | https://mathoverflow.net/users/4428 | 395473 | 163,373 |
https://mathoverflow.net/questions/395457 | 0 | By rank I imply rank over reals ($\mathbb R$).
I consider two $n\times n$ matrices $A,B$ having entries in $0/1$.
The product below follows usual matrix product rules except $xy$ is $AND(x,y)$ and $x+y$ is $OR(x,y)$.
Assume real rank of $AB$ is $n$ and assume $det(AB)=per(AB)=1$.
>
> 1. Would it follow $per(A... | https://mathoverflow.net/users/169713 | $\mathbb R$ and $\mathbb F_2$ rank in boolean matrix product | For $n = 3$, 1) is false: let $A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}$.
Some obvious modifications to the code below also show that the first question in 2) is false: let $A$ be as before and $B = \begin{pmatrix... | 0 | https://mathoverflow.net/users/1847 | 395479 | 163,374 |
https://mathoverflow.net/questions/395481 | 3 | Part of this question (asked by someone else) for semiperfect rings has circulated a few weeks [on math.se](https://math.stackexchange.com/q/4164710/29335) but without much attention. I think it might be above the threshold of difficulty to be on mathoverflow, but you can let me know if I should move it.
Let $R$ be a... | https://mathoverflow.net/users/19965 | Are corner rings of (semi)perfect rings (semi)perfect? | Rowen shows in Lemma 2.7.34 of his book Ring Theory that $R$ is right perfect iff for each idempotent $e$ one has both $eRe$ and $(1-e)R(1-e)$ are right perfect. Hence $R$ right perfect implies $eRe$ is right perfect.
Here is a proof for the semiperfect case. Note that $eR$ is a finitely generated projective module a... | 3 | https://mathoverflow.net/users/15934 | 395485 | 163,376 |
https://mathoverflow.net/questions/395492 | 4 | Let $A$ be a random matrix following multivariate normal distribution $N(\mu, \Sigma)$.
>
> What is the distribution of the eigenvalues of $A^TA$?
>
>
>
A reference to the literature would be most welcome.
| https://mathoverflow.net/users/292554 | What is the distribution of eigenvalues of $A^TA$, where $A \sim N(\mu, \Sigma)$? | Unlike in the case $\mu=0$, $\Sigma=I$, there is no simple closed-form expression for the eigenvalue distribution. In the limit of large matrices the eigenvalue density follows from the Pastur equation, see [Spectral density of the non-central correlated Wishart ensembles](https://arxiv.org/abs/1406.4184) (section VI a... | 3 | https://mathoverflow.net/users/11260 | 395496 | 163,380 |
https://mathoverflow.net/questions/395461 | 10 | I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that I've read focus on the compact case (e.g. Baez's approach via the octonions and the magic square constructions) while I... | https://mathoverflow.net/users/163024 | Viewing exceptional Lie algebras via the classical ones | Élie Cartan himself, recognized and used the following description of $\mathfrak{e}\_6$: Let $V$ be a vector space of dimension $6$ and let $W$ be a vector space of dimension $2$. Then there is a vector space splitting
$$
\mathfrak{e}\_6 = \mathfrak{sl}(V)\oplus\mathfrak{sl}(W)\oplus \bigl(\Lambda^3(V)\otimes W\bigr)
$... | 7 | https://mathoverflow.net/users/13972 | 395500 | 163,381 |
https://mathoverflow.net/questions/395501 | 8 | Bousfield, in his paper "The Boolean algebra of spectra" (Comm Math Helv 54, 368–377 (1979), <https://doi.org/10.1007/BF02566281>), defined $\mathbf{DL}$, a sublattice of the Bousfield lattice, to consist of all Bousfield classes $\langle X \rangle$ such that $\langle X \rangle \wedge \langle X \rangle = \langle X \ran... | https://mathoverflow.net/users/4194 | Bousfield's distributive lattice DL and non-ring spectra | My paper [*A combinatorial model for the known Bousfield classes*](https://arxiv.org/abs/1608.08533) defines an complete ordered semiring $\mathcal{A}$ and a homomorphism from $\mathcal{A}$ to the Bousfield lattice mod the telescope conjecture, whose image contains most of the Bousfield classes that have been named and... | 7 | https://mathoverflow.net/users/10366 | 395504 | 163,383 |
https://mathoverflow.net/questions/395509 | 9 | Let $G$ be a finite group and let $R$ be a commutative ring.
I'd like to ask, if there is a theorem of the following kind:
>
> The augmentation ideal $I\_G$ is projective as RG-module, if and only if ... ?
>
>
>
This should happen only in rare cases, but I was wondering, if there exists an if-and-only-if cri... | https://mathoverflow.net/users/12826 | When is the augmentation ideal projective as RG-module? | Okay, this happens precisely in the obvious case, namely if all primes dividing the order $|G|$ are invertible in $R$.
To see this, note that $\operatorname{Ext}^\*\_{R[G]}(R,R)$ is group cohomology of $G$ with coefficients in $R$. If $I\_G$ were projective, $I\_G\to R[G]$ would be a projective resolution of $R$ and ... | 16 | https://mathoverflow.net/users/39747 | 395512 | 163,388 |
https://mathoverflow.net/questions/395471 | -1 | Basic concepts question.
I am used to the Cartesian product of two sets: $A \times B = \{(a,b) \mid a \in A, b \in B\}$.
Is there an operator that produces sets instead of tuples? We might call it *set-product* and define $A \otimes B := \{a \cup b \mid a\in A, b\in B\}$.
What I am actually trying to do is "facto... | https://mathoverflow.net/users/143057 | Cartesian products and set-products | As far as I understand, you start with a "base set" $V$ and consider the monoid obtained by endowing the power set of the power set of $V$ with the binary operation that sends a pair $(A, B)$ of families of subsets of $V$ to $\{X \cup Y \colon X \in A,\, Y \in B\}$, which is still a family of subsets of $V$.
If my unde... | 4 | https://mathoverflow.net/users/16537 | 395513 | 163,389 |
https://mathoverflow.net/questions/395066 | 5 | I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example - consider the product of the big projective on $\mathbb{P}^1$ with itself (This is the projective cover of the skyscraper sheaf on the 0-dimensional stratum). Does this have a simple description? How ca... | https://mathoverflow.net/users/4477 | Tensor product of perverse sheaves on flag varieties | First a general comment: as Sasha alludes to, there are two tensor products of complexes of sheaves. Let $i : X \hookrightarrow X \times X$ denote the diagonal. We have, and given complexes of sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$ we can form their external tensor product $\mathcal{F} \boxtimes \mathcal{G}$ on... | 5 | https://mathoverflow.net/users/919 | 395520 | 163,392 |
https://mathoverflow.net/questions/395491 | 6 | I am looking for the sharpest known upper bound on $K(n, 1)$ as $n \rightarrow \infty$. This is the minimal cardinality of a (not-necessarily linear) *covering code* of $\{0, 1\}^n$ of radius 1.
In elementary terms: Using how few (possibly non-disjoint) Hamming balls of radius 1 can we cover $\{0, 1\}^n$? I am intere... | https://mathoverflow.net/users/92003 | How many Hamming spheres of radius 1 does it take to cover the cube? | Clearly $K(n, 1) \le 2K(n-1, 1)$ by the construction of taking each code word of length $n-1$ and adding one copy with a suffix of $0$ and one with a suffix of $1$. (This is comment r to table 1 in the Cohen-Lobstein-Sloane paper referenced in the question).
Then by taking the largest $k$ such that $2^k - 1 \le n$ an... | 4 | https://mathoverflow.net/users/46140 | 395522 | 163,393 |
https://mathoverflow.net/questions/395521 | 2 | Let $L\_k = \mathbb{Q}(\zeta\_{2^k} + \zeta\_{2^k}^{-1})$ be the maximal real subfield of the cyclotomic field of conductor $2^k, k \ge 2$ and $f\_k(x)$ be the minimal polynomial of $\zeta\_{2^k} + \zeta\_{2^k}^{-1}$.
Define $L = L\_{k+1}, K = L\_{k}$ so $L/K$ has degree 2. Assume a prime ideal $\mathfrak{p} \subset ... | https://mathoverflow.net/users/106850 | How can I prove this claim about splitting of prime ideals in real cyclotomic fields? | Use that $\text{Gal}(L\_k/\mathbb{Q})$ is cyclic and look at the fixed field of the decomposition group.
| 2 | https://mathoverflow.net/users/96891 | 395523 | 163,394 |
https://mathoverflow.net/questions/395582 | 8 | Let $f: \mathbb{\mathbb{Z}^+} \to \mathbb{Z^+}$ be a function and suppose
$(\star)$ For all integers $x \geq 3$, if $f(x)$ is prime, then $x$ is prime.
A trivial example of such a function is the identity $f(x) = x$. However, a possible non-trivial example which I have come across is
\begin{align\*}
f(x) = \left\lf... | https://mathoverflow.net/users/171396 | Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime | As observed in comments, we have $f(n) = \lfloor g(n) \rfloor$ where $g(n) = \frac{\alpha^n + \alpha^{-n}}{4}$ and $\alpha = 2 + \sqrt{3}$. From the recurrence $g(n+1) = 4 g(n) - g(n-1)$ we see that $g(n)$ is a half-integer when $n$ is even and an integer when $n$ is odd. In fact we see from induction that for even $n$... | 15 | https://mathoverflow.net/users/766 | 395604 | 163,412 |
https://mathoverflow.net/questions/395602 | 2 | I am reading the paper Frames and Outer Frames for Hilbert $C^\*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following:
>
> "...Since in each $C^\*$-algebra, a convergent series of positive elements necessarily
> converges unconditionally..."
>
>
>
Unfortunately they did not gi... | https://mathoverflow.net/users/119788 | Unconditional Convergence of Positive Terms in a $C*$-algebra | Let $x = \sum x\_n$ be a convergent sum of positive elements of a C${}^\ast$-algebra $A$. Then for any state $\phi$ on $A$ we have $\sum \phi(x\_n) = \phi(x)$, converging unconditionally since it is a series of positive terms. So if some rearrangement of the series sums to $y$, we would have $\phi(x) = \sum \phi(x\_n) ... | 4 | https://mathoverflow.net/users/23141 | 395607 | 163,414 |
https://mathoverflow.net/questions/395606 | 8 | Does anyone has a simple example of a 1-category $\mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $\mathcal{C}\left[W^{-1}\right]$ is not a 1-category?
Of course there are obvious “big” examples like CW complexes / derived categories, I’m looking for a small e... | https://mathoverflow.net/users/125868 | Simple example of nontrivial simplicial localization | For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoid.
For example take $C$ to be the poset with 6 elements ordered as a,b < c,d < e,f and when you localize at all arrows you get the $2$-sphere $\mathbb{S... | 12 | https://mathoverflow.net/users/22131 | 395608 | 163,415 |
https://mathoverflow.net/questions/395464 | 1 | Let $R$ be a Noetherian local ring of dimension $d$, and $a\_1,\dots,a\_d$ is a system of parameters. I am wondering whether the following statement is true:
>
> $\mathrm{ht}(a\_1,\dots,a\_i)=i$ for all $i$, $1 \le i \le d$.
>
>
>
I am thinking about this because by definition, $\mathrm{ht}(a\_1,\dots,a\_d)=d$... | https://mathoverflow.net/users/119037 | Height of truncated system of parameter | This is true if $R$ is catenary and equidimensional. However, for example if $R$ is not equidimensional, then there exists a minimal prime $\mathfrak{p}$ of $R$ such that dim$(R/\mathfrak{p})<\text{dim}(R)$. Therefore any $x\in \mathfrak{p}$ which not in the union of the other minimal primes is a parameter element of h... | 2 | https://mathoverflow.net/users/127857 | 395630 | 163,423 |
https://mathoverflow.net/questions/395581 | 1 | I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}\_n)$ a filtration on it. Let $(\epsilon\_{n})$ be a sequence adapted to $(\mathcal{F}\_n)$ such that $\mathbb{E}[\epsilon\_{n+1} | \mathcal{F}\_n] = 0$ and
$$
... | https://mathoverflow.net/users/294260 | Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event? | I think that $e\_m - E(e\_m | \mathcal F\_{m-1}) $ is a square summable, martingale difference sequence, so $\Sigma e\_m - E(e\_m | \mathcal F\_{m-1}) $ converges a.s. If that is true, then $\Sigma e\_m $ converges on any set where $ E(e\_m | \mathcal F\_{m-1})$ is eventually 0.
| 1 | https://mathoverflow.net/users/143907 | 395644 | 163,430 |
https://mathoverflow.net/questions/395633 | 2 | Suppose $X$ is a smooth projective surface with a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ over a field $k$, where all the fibres of $\pi$ are conics (i.e. a conic bundle). If $\pi$ admits a section $s$ over $k$ (i.e. there exists $s:\mathbb{P}^{1}\_{k}\rightarrow X$ such that $\pi \circ s=\text{Id}\_{\math... | https://mathoverflow.net/users/211978 | Section of conic bundle | The differential of a section is right inverse to the differential of $\pi$, hence $d\pi$ is surjective and $\pi$ is smooth along the section.
| 6 | https://mathoverflow.net/users/4428 | 395651 | 163,431 |
https://mathoverflow.net/questions/394663 | 2 | Let $(W,S)$ be a finite and irreducible Coxeter Group. For $J \subseteq S$, let $W\_J = \langle s | s \in J \rangle$, a parabolic subgroup. For which $J$ is the action (group multiplication on the left) of $W$ permuting the (left) cosets of $W\_J$ faithful exactly?
Is there a good reference to find this result in the... | https://mathoverflow.net/users/145041 | When does a finite irreducible Coxeter Group act on the cosets of a parabolic subgroup faithfully? | Stumbled on this again, I should make my comment an official answer.
This is true for any proper $J\subseteq S$. Suppose $g\in W$ fixes every coset of $W\_J$, so $g$ lies in the intersection of all conjugates of $W\_J$. Any intersection of parabolic subgroups (meaning conjugates of standard parabolic subgroups) is a ... | 2 | https://mathoverflow.net/users/164670 | 395655 | 163,433 |
https://mathoverflow.net/questions/395657 | 30 | Let $P\_1,\dots,P\_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P\_1^N,\dots,P\_k^N$ are linearly independent?
| https://mathoverflow.net/users/908 | Are large powers of polynomials linearly independent? | The answer is yes. In fact, an even stronger claim is true: there exists some $N$ such that for all $n \geq N, \ P\_{1}^{n}, \dots, P\_{k}^n$ are linearly independent over $\mathbb{C}$.
For this we will use a generalization of the [Mason-Stother's theorem](https://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem)... | 25 | https://mathoverflow.net/users/88679 | 395664 | 163,437 |
https://mathoverflow.net/questions/395646 | 4 | Let $X$ be a Banach space. A bounded linear map $u:X\to\ell\_2$ is said to be $1$-summing if for all finite sequence $(x\_i)\subseteq X$ there is a constant $C>0$ such that $\sum\|ux\_i\|\leq C\sup\Big\{\sum|x^\*(x\_i)|\_2:\|x^\*\|\_{X^\*}\leq 1\Big\}.$ A Banach space is said to be satisfy Grothendieck's theorem (in sh... | https://mathoverflow.net/users/136860 | Banach space with dual not a GT space | The answer is yes (there exists a GT space whose dual is not a GT space), given by the very first test example that one might consider.
---
The form of Grothendieck's theorem that gives rise to the terminology "GT-space" is the fact that every bounded operator from $\ell\_1$ to $\ell\_2$ is 1-summing.
So the fi... | 11 | https://mathoverflow.net/users/763 | 395666 | 163,438 |
https://mathoverflow.net/questions/395645 | 4 | Theorem 1.1 in Tibor Bekes *Theories of presheaf type*
([pdf 1](https://sites.uml.edu/tibor-beke/files/2018/07/presheaf-24qtul3.pdf),
[pdf 2](http://faculty.uml.edu/tbeke/jsl.pdf))
looks like a convenient criterion for whether a given qotient $T^+$ of a geometric theory $T$ of presheaf type is again of presheaf type --... | https://mathoverflow.net/users/166281 | What about the enough points requirement in Bekes "Theories of presheaf type"? | It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exist, as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I mean is that almost all *natural*... | 6 | https://mathoverflow.net/users/22131 | 395667 | 163,439 |
https://mathoverflow.net/questions/395665 | 3 | Let $\sqrt{n}\mathbf{Z}$ be the one dimensional lattice, whose generator has length $2$. Associated to this is a lattice vertex algebra
$$V(\sqrt{2}\mathbf{Z}).$$
We also have the simple quotient of the affine vertex algebra associated to $\mathfrak{sl}\_2$ at level $1$:
$$L\_1(\mathfrak{sl}\_2).$$
A priori lattice ver... | https://mathoverflow.net/users/119012 | Geometric explanation for the $\widehat{\mathfrak{sl}}_2$ free field realisation (alias $L_1(\mathfrak{sl}_2)\to V(\sqrt{2}\mathbf{Z})$) | As Pavel Safranov commented, this is done in the paper [arxiv.org/abs/0710.5247](https://arxiv.org/abs/0710.5247) by Zhu. I've skimmed the paper and will sketch how I think it works.
Write $\mathcal{L}\_G$ for the determinant line bundle on the BD Grassmannian $(\text{Gr}\_{G,X^n})$, and $\pi\_n:\text{Gr}\_{G,X^n}\to... | 6 | https://mathoverflow.net/users/119012 | 395671 | 163,440 |
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