parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/336073 | 12 | Consider a non-compact manifold $M$.
Does there always exist a Riemannian metric on $M$ such that the isometry group is non-compact?
| https://mathoverflow.net/users/55948 | Possible isometry groups of open manifolds | Let $M$ be the triply punctured 2-sphere (i.e. $M=S^2-\{p\_1, p\_2, p\_3\})$); one can also take any noncompact connected surface of finite topological type as long as $\chi(M)<0$ but the proof is a bit more involved.
Suppose that $g$ is a Riemannian metric on $M$. To simplify matters, I consider only orientation-pr... | 13 | https://mathoverflow.net/users/21684 | 336092 | 143,517 |
https://mathoverflow.net/questions/336067 | 8 | Let $\text{Meas}$ be the category of measurable spaces with measurable functions as morphisms. Does $\text{Meas}$ have [exponential objects](https://en.wikipedia.org/wiki/Exponential_object)?
| https://mathoverflow.net/users/8628 | Exponential objects in the category of measurable spaces | As mentioned in the comments, Meas of course has *some* exponential objects $B^A$, but not for all $A$ and $B$, i.e., it is not cartesian closed.
This fact is discussed as Proposition 6 of [A Convenient Category for Higher-Order Probability Theory](http://arxiv.org/pdf/1701.02547.pdf) by Heunen, Kammar, Staton, and Yan... | 8 | https://mathoverflow.net/users/1015 | 336093 | 143,518 |
https://mathoverflow.net/questions/336071 | 5 | I once heard something like "inner forms of reductive groups have the same representation theory".
Is this assertion misguided?
If this assertion is not misguided, then is there a precise statement to this effect (perhaps in Tannakian terms)?
| https://mathoverflow.net/users/135687 | Representation theory of inner forms | I vote for "misguided". The representation theory of inner forms are certainly not the "same". What is true (over a local field) is they have the same L-group. A precise version is: the L-packets for G embed in the L-packets for the quasisplit form (assuming the local Langlands conjectures of course).
| 9 | https://mathoverflow.net/users/6030 | 336098 | 143,519 |
https://mathoverflow.net/questions/336104 | 0 | This question is pertaining to finite connected vertex-transitive graphs.
I recently read *"Transitive permutation groups without semiregular subgroup*" by Cameron, Giudici, Jones, Kantor, Klin, Marušič, Nowitz ([publisher link](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024610702003484); [MSN revi... | https://mathoverflow.net/users/52949 | Elusive groups and vertex-transitive graphs | Any elusive group of degree $n$ is a subgroup of the full automorphism group of the complete graph $K\_n$, so your second statement is not true.
The Polycirculant Conjecture asserts that the full automorphism group of a digraph contains a derangement of prime order, i.e, is not elusive.
But while there are various... | 4 | https://mathoverflow.net/users/1492 | 336111 | 143,523 |
https://mathoverflow.net/questions/336112 | 2 | Consider the ode
$$
f''(t)-e^{-2t} f(t)=0.
$$
What is the general behaviour of $|f|$ for large $t$s?
Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(t)|\sim A+Bt $ for all large enough $t$?
| https://mathoverflow.net/users/140146 | Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$ | The general solution of this differential equation is
$$ f \left( t \right) =a\,{{ I}\_{0}\left({{\rm e}^{-t}}\right)
}+b\,{{ K}\_{0}\left({{\rm e}^{-t}}\right)}
$$
where $I\_0$ and $K\_0$ are modified Bessel functions.
As $t \to +\infty$,
$$I\_0({\rm e}^{-t}) = 1 + O\left({\rm e}^{-2t}\right)$$ while
$$K\_0({\rm e}^{-... | 9 | https://mathoverflow.net/users/13650 | 336114 | 143,525 |
https://mathoverflow.net/questions/336117 | 4 | Let $\mathbb{N}$ denote the set of non-negative integers. If $f:\mathbb{N}\to\mathbb{N}$ is a map, we set for $k\in \mathbb{N}$:
* $f^{(0)}(k) = k$, and
* $f^{(n+1)}(k) = f(f^{(n)}(k))$ for all $n\in\mathbb{N}$.
We say $r\in \mathbb{N}$ is a *rocket element of $f$* if $f^{(n)}(r) < f^{(n+1)}(r)$ for all $n\in \math... | https://mathoverflow.net/users/8628 | "Rocket elements" in bijections $f:\mathbb{N}\to \mathbb{N}$ | Choose any infinite set $A\subseteq\mathbb N$ such that $\mu^+(A)=0$. Enumerate both $A$ and $B=\mathbb N\setminus A$ as
\begin{align\*}
A&=\{a\_1<a\_2< \dots < a\_n < \dots\}\\
B&=\{b\_1<b\_2< \dots < b\_n < \dots\}
\end{align\*}
Then you can definite the function $f\colon\mathbb N\to\mathbb N$
\begin{align\*}
f(b\_k... | 19 | https://mathoverflow.net/users/8250 | 336119 | 143,526 |
https://mathoverflow.net/questions/334793 | 8 | Let $A=:F(G)$ be the algebra of functions on a finite quantum groups aka a finite dimensional C\*-Hopf Algebra.
Suppose that $F(G)$ is neither commutative nor cocommutative.
In their 1966 paper Kac and Paljutkin, show that when we write $F(G)$ as a multimatrix algebra, one of which must be one-dimensional (to accou... | https://mathoverflow.net/users/35482 | Image of Comultiplication on Finite Quantum Groups/Hopf Algebras | I guess what you are looking for is the inclusion matrix for the unital inclusion of finite dimensional ${\rm C}^{\star}$-algebras $\Delta(A) \subset A \otimes A$. It is given by the fusion rules for $Rep(A)$, see Proposition 7.4 in my preprint [arXiv:1704.00745v5](https://arxiv.org/pdf/1704.00745v5.pdf).
*Example*:... | 3 | https://mathoverflow.net/users/34538 | 336123 | 143,529 |
https://mathoverflow.net/questions/336115 | 6 | $\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading [Strickland - Formal Schemes and Formal Groups](https://arxiv.org/abs/math/0011121), which is apparently the only article that deals with this topic in that way.
He defines (4.1) an formal ... | https://mathoverflow.net/users/142626 | Basic example of a formal affine scheme, functorial point of view | It might be illuminating to first work the example of (ordinary) affine space $\mathbb{A}^1\_\mathbb{Z}$ over the integers.
As a functor, $\mathbb{A}^1\_\mathbb{Z}$ is the forgetful functor $\mathit{Rings}^\mathrm{op}\rightarrow\mathit{Sets}$, sending a ring $R$ to its underlying set. It is representable by $\mathbb{... | 7 | https://mathoverflow.net/users/130058 | 336124 | 143,530 |
https://mathoverflow.net/questions/333095 | 7 | Suslin Rigidity conjecture states that motivic cohomology
$$
H\_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n))
$$
of the field $F$ coincides with motivic cohomology for the subfield of constants $F\_0$.
The fact that first motivic cohomology don’t change under pure transcendental extensions gives some evidenc... | https://mathoverflow.net/users/21620 | Motivation for Suslin’s Rigidity Conjecture | First off, I'm not sure that the assertion that ${\rm H}^1(-,\mathbb{Q}(n))$ doesn't change under purely transcendental extensions is known. The Gersten complex for the affine line provides the Milnor exact sequence
$$
0\to {\rm H}^1(F,\mathbb{Q}(n))\to {\rm H}^1(F(T),\mathbb{Q}(n))\to \bigoplus\_{x\in (\mathbb{A}^1)^{... | 4 | https://mathoverflow.net/users/50846 | 336127 | 143,531 |
https://mathoverflow.net/questions/336054 | 1 | How many involutions are there in $O\_7(11)$ and $PSp\_6(11)$ respectively? (Note that the sizes of the two groups mentioned here are the same.)
| https://mathoverflow.net/users/143059 | On the number of involutions in some groups | Let $q$ be an odd prime power and say that $q\equiv\epsilon\pmod4$, where $\epsilon=\pm1$.
Using Geoff's suggestion I calculated the number of involutions in $\Omega\_7(q)$ (the simple group; called $O\_7(q)$ in the Atlas) to be
$$
\frac12 q^5(q^4+q^2+1)(q+\epsilon) + \frac12 q^6(q^4+q^2+1)(q^2+1) + \frac12 q^3(q^3+... | 5 | https://mathoverflow.net/users/99221 | 336132 | 143,534 |
https://mathoverflow.net/questions/333480 | 2 | I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (<https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-covered-by-open-subfunctors>)
I want to prove Theorem 8.9 from Algebraic Geometry I ( U.Görtz, T.Wedhorn), which reads ... | https://mathoverflow.net/users/141600 | Proving the representability of a functor that is covered by open subfunctors | I found another way to prove this Theorem, which then also shows that this map is injective. Consider two Zariski sheaves $F,G:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ and for each sheaf a cover by open subfunctors $\alpha\_i:F\_i\rightarrow F$ and $\beta\_i:G\_i\rightarrow G$. Suppose we have isomorphisms $\varphi... | 2 | https://mathoverflow.net/users/141600 | 336133 | 143,535 |
https://mathoverflow.net/questions/317552 | 3 | If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in $V$ satisfying $\psi$ is non empty and is isomorphic on membership with $\small ORD$
Formally this is: $\psi(\small ... | https://mathoverflow.net/users/95347 | What is the consistency strength of this kind of reflection principle? | I claim that this is equiconsistent with "$ORD$ is Mahlo" (This is not the same as $Ord$ being actually Mahlo, as I have explained elsewhere). It turns out "$ORD$ is Mahlo" is a natural limit point for these kind of Ackermann/$KM$ based theories. Let $T$ be your theory.
First, the easy part. The consistency strength ... | 3 | https://mathoverflow.net/users/141402 | 336137 | 143,537 |
https://mathoverflow.net/questions/336129 | 2 | Let $T:\mathrm{dom}(T) \subseteq H \to H$ be a densely defined, self-adjoint operator on a Hilbert spaces $$. In general the range of $T$ is not guaranteed to be closed. What tools are available to check if the range is closed? More precisely, what are a list of equivalent formulations of closed range, or conditions th... | https://mathoverflow.net/users/128876 | Characterising closed range self-adjoint operators | This is a complete, but rather abstract characterisation.
>
> $T$ has closed range if and only if there is $H\_0\subseteq H$ closed and a bounded self-adjoint injective map $R:H\_0\rightarrow H\_0$ with $D(T) = \{ \xi+R(\eta) : \xi\in H\_0^\perp, \eta\in H\_0 \}$ and $T(\xi+R(\eta)) = \eta$.
>
>
>
If this hold... | 3 | https://mathoverflow.net/users/406 | 336141 | 143,539 |
https://mathoverflow.net/questions/336144 | 2 | My question is probably very basic, sorry about that.
Let $\{f\_i\},\{g\_i\}$ be two sequences converging to 0 **weakly** in $L^p[0,1]$ for **any** $p<\infty$. Can one conclude that $\int\_0^1f\_i(x)g\_i(x) dx\to 0$?
| https://mathoverflow.net/users/16183 | Weak convergence in $L^p$ | The answer is No.
Take $f\_{k}(x)=e^{ikx}$, and $g\_{k}(x)=e^{-ikx}$. Then by [Riemann--Lebesgue lemma](https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma) we have $\lim\_{k \to \infty} \int\_{0}^{1}f\_{k}(x)h(x)dx = \lim\_{k \to \infty}\int\_{0}^{1} g\_{k}(x)h(x)=0$ for any $h \in L^{q}([0,1])$, for any $1... | 9 | https://mathoverflow.net/users/50901 | 336145 | 143,541 |
https://mathoverflow.net/questions/336135 | 11 | I have recently been reading some of the literature on the Penrose inequality, especially the papers by Bray and by Huisken and Ilmanen.
One notices immediately that the existing proofs for the Penrose conjecture focus on a rather special case with special symmetry for the spatial hypersurface of the spacetime: this ... | https://mathoverflow.net/users/119114 | Are there currently any plausible approaches to proving the Penrose сonjecture? | The difficulty of a general proof was discussed in [A counter-example to a recent version of the Penrose conjecture](https://hal.archives-ouvertes.fr/hal-00578460/document) (2010): a general existence theorem cannot be expected with boundary conditions compatible with generalized apparent horizons.
| 6 | https://mathoverflow.net/users/11260 | 336168 | 143,547 |
https://mathoverflow.net/questions/336167 | 2 | Let $G$ be a Lie group (finite dimensional or Banach), and let $H$ be a Lie subgroup (in the Banach case we assume that $H$ is a submanifold which is also a Lie group). Let $\text{U}(1) \rightarrow \hat{G} \xrightarrow{\pi} G$ be a central extension of $\hat{G}$.
Suppose that $\hat{G}$ trivializes over $H$.
Write $f: G... | https://mathoverflow.net/users/99745 | Descending central extensions to homogeneous spaces | First up, $\sigma\colon G\times H \to G\times\_f G$ sending $(g,h)\mapsto (g,gh)$ is a diffeomorphism. For this, all you need is that $G\to G/H$ is a locally trivial $H$-bundle (so, with care, this works for beyond the Banach setting). Then since you have your descent data $\phi\colon p\_1^\*\hat{G}\to p\_2^\*\hat{G}$ ... | 3 | https://mathoverflow.net/users/4177 | 336171 | 143,549 |
https://mathoverflow.net/questions/336169 | 4 | Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S\_\omega$, the group of permutations of the set of non-negative integers $\omega$?
| https://mathoverflow.net/users/8628 | Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$? | See Theorem 4.3 of [this paper](https://pure.tue.nl/ws/portalfiles/portal/1866721/597514.pdf) by De Bruijn. Any abelian group of order $2^\kappa$ can be embedded in $Sym(\kappa)$ when $\kappa$ is infinite. (There is also an [addendum](https://core.ac.uk/download/pdf/82202896.pdf) to the paper which corrects some error ... | 15 | https://mathoverflow.net/users/38253 | 336172 | 143,550 |
https://mathoverflow.net/questions/335910 | 4 | Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?
Using general spectra stuff, one gets a Kuenneth spectral sequence for K-homology. The question is then, phrased this way, why this spectral sequence (in the case o... | https://mathoverflow.net/users/13356 | Kuenneth short exact sequence for K-homology | Let me answer my own question.
Markus Land referred me to a remark on top of page 62 in his PhD thesis ( <http://hss.ulb.uni-bonn.de/2016/4432/4432.htm> ), where he argues why we have a short exact UCT sequence relating K- and L-theory.
The same arguments also apply to the Kuenneth formula for K-homology: since the... | 3 | https://mathoverflow.net/users/13356 | 336179 | 143,551 |
https://mathoverflow.net/questions/336149 | 4 | I am learning Cartan's moving frame from chapter 6 of S.S. Chern's book "Lectures on differential geometry" ([Google books](https://books.google.ca/books?id=Mvk7DQAAQBAJ&printsec=frontcover&hl=zh-CN#v=onepage&q&f=false)). Suppose the moving frame in $E^N$ is denoted by $(p;e\_1,\cdots,e\_N)$, then we can apply an infin... | https://mathoverflow.net/users/143102 | About the Cartan's moving frame method | One way to make sense of this is to view $p, e\_1, \dots, e\_N$ as functions on the orthonormal frame bundle of $\mathbb{R}^N$, which is naturally isomorphic to the group of rigid motions, where there is a right action of the group $O(N)$ of rotations, which fixes the point $p$ and rotates the frame and $\mathbb R^N$ i... | 1 | https://mathoverflow.net/users/613 | 336185 | 143,553 |
https://mathoverflow.net/questions/336183 | 3 | Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. Does every normal tracial state on $S$ extend to a normal tracial state on $R$?
| https://mathoverflow.net/users/142953 | Extension of trace on von Neumann subalgebra | Let $C$ denote the complexes, and embed $C \times C$ in $M\_2 C$ ($2 \times 2$ matrices) as diagonal matrices. Then $M\_2 C$ has unique trace, but $C \times C$ has two extremal ones.
| 7 | https://mathoverflow.net/users/42278 | 336200 | 143,557 |
https://mathoverflow.net/questions/336163 | 2 | Let $k$ be a field and $(C,\mathcal{O}\_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, the functor
$$\mathcal{F}\longmapsto (\mathcal{F}(C\setminus\{x\}),\mathcal{F}\_x,\iota\_x)$$
from the category of cohe... | https://mathoverflow.net/users/66686 | Glueing modules over $\{x\}\times \operatorname{Spec} R$ | The Beauville-Laszlo theorem holds in much greater generality - see Tag [0BNI](https://stacks.math.columbia.edu/tag/0BNI "0BNI") on the Stacks Project.
Let $A$ be any ring and let $f\in A$ be a non-zero divisor. Then the category of $f$-torsion free $A$-modules $M$ is equivalent to the category of triples $(M\_1, M\_... | 2 | https://mathoverflow.net/users/56878 | 336213 | 143,563 |
https://mathoverflow.net/questions/336202 | 4 | The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-compact by <https://stacks.math.columbia.edu/tag/04XU>, so one may add this assumption if one wants.
In particular, is eve... | https://mathoverflow.net/users/63877 | Is a universally closed monomorphism a closed immersion? | There is a non-surjective epimorphism $B\to C$ where $B$ and $C$ are zero-dimensional local rings (D. Lazard, see <http://www.numdam.org/item/SAC_1967-1968__2__A8_0/>). Then $\mathrm{Spec}\,(C)\to \mathrm{Spec}\,(B)$ is a monomorphism but not a closed immersion, and it is universally closed because $B\_\mathrm{red}{\si... | 4 | https://mathoverflow.net/users/7666 | 336217 | 143,565 |
https://mathoverflow.net/questions/336191 | 59 | The succinct question
=====================
The conjecture of Birch and Swinnerton-Dyer (to take a random example) mentions L-functions and hence the complex numbers and hence the real numbers (because the complexes are built from the reals). Two naive questions which probably just indicate that I don't understand lo... | https://mathoverflow.net/users/1384 | Cauchy reals and Dedekind reals satisfy "the same mathematical theorems" | Here's a low-tech way to look at it, which to me seems perfectly convincing.
Let C be some implementation of the reals via Cauchy sequences and D be some implementation of the reals via Dedekind cuts. Here C is "really" something like a tuple consisting of the set of reals, a relation corresponding to addition, etc.;... | 21 | https://mathoverflow.net/users/80665 | 336226 | 143,567 |
https://mathoverflow.net/questions/336225 | 3 | Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence
$$ 1 \to T \to G \to A \to 1$$
of algebraic groups, where $T$ is an algebraic torus and $A$ is an abelian variety. I have heard somewhere that, given an algebraic group $G$, if $G$ is semi-abelian... | https://mathoverflow.net/users/142444 | Uniqueness of presentation for semi-abelian varieties | First of all, there are no non-trivial homomorphisms from a torus $T$ to an abelian variety $A$ (also true from additive group $\mathbb{G}\_a$ to $A$, or other unipotent like Witt groups). This is because there are no non-constant rational maps from $\mathbb{A}^1$ to $A$ (see for example [Milne's book on abelian variet... | 3 | https://mathoverflow.net/users/24442 | 336230 | 143,568 |
https://mathoverflow.net/questions/336203 | 6 | The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
**Question:** How do the nontrivial continuous idempotent functions (wrt convolution) look like? That is, functions $f\*f=f$, defined usi... | https://mathoverflow.net/users/48438 | Idempotent functions on Sp(1) | As Venkataramana says, this is a natural candidate for the Peter-Weyl theorem: Let $G$ be a compact group. Let $\{ V\_i \}\_{i \in I}$ be the set of isomorphism classes of irreducible complex representations of $G$ (for some index set $I$), and fix a $G$-invariant Hermitian inner product on each $V\_i$. Let $\{ e\_i^b ... | 7 | https://mathoverflow.net/users/297 | 336247 | 143,574 |
https://mathoverflow.net/questions/336182 | 3 | I am presently reading [this](https://arxiv.org/abs/1906.08616) paper on covariant phase space and I have difficulty understanding the following formalism developed:
In the paper (section $2.2$, pg. $12$), the authors have introduced the notion of pre-phase space and go on to reinterpret differential forms by their f... | https://mathoverflow.net/users/99716 | One-Forms in Functional Space? | There is no problem in defining the exterior differential $\delta$ on infinite-dimensional manifolds such as the function space. In particular, $\delta^2 = 0$ follows from a similar calculation as in finite dimensions.
I guess the notation in the paper (as with almost every physics paper on this subject) should be un... | 2 | https://mathoverflow.net/users/17047 | 336250 | 143,576 |
https://mathoverflow.net/questions/336252 | 10 | The Cohen-Lenstra paper says the probability that the odd part of a class group being cyclic is close to 0.98. So I was thinking: can we find any conditions on a finite abelian group so that it cannot be a class group of any number field?
| https://mathoverflow.net/users/131448 | Is there any conditions on a finite abelian group so that it cannot be class group of any number field? | It follows from the Cohen-Lenstra heuristic that every finite abelian group is expected to be isomorphic to infinitely many class groups of real quadratic fields (even to a positive proportion of real quadratics, which is stronger), but nothing like this is known.
However, if you take the Galois action into account, ... | 16 | https://mathoverflow.net/users/35416 | 336260 | 143,580 |
https://mathoverflow.net/questions/336259 | 3 | Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 skeleton of $X$, such that $L$ is trivial on $X-Y$. Is it true in this case that $L$ is trivial on the entire $X$?
Th... | https://mathoverflow.net/users/109370 | Line bundles trivial outside of codimension 3 | This will hold for CW structures on manifolds coming from [handle decompositions](https://en.wikipedia.org/wiki/Handle_decomposition), e.g. induced by a [Morse function.](https://en.wikipedia.org/wiki/Handle_decomposition#Morse_theoretic_viewpoint) Complex line bundles on $X$ are classified by the homotopy class of map... | 5 | https://mathoverflow.net/users/1345 | 336264 | 143,582 |
https://mathoverflow.net/questions/336263 | 2 | Let $X$ be a variety over a base field $k$ of dimension $n$. Can there be non constant morphisms $P^m \to X$?
| https://mathoverflow.net/users/58001 | Morphisms from projective space to lower dimension spaces | Your question is answered negatively [here](https://mathoverflow.net/q/116398) (assuming $m>n$).
| 2 | https://mathoverflow.net/users/143179 | 336265 | 143,583 |
https://mathoverflow.net/questions/335470 | 5 | Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S\_1,S\_2\subset\{1,2,...,n\}$, define $U\_{S\_1S\_2}$ to be an $|S\_1|\times|S\_2|$ submatrix of $U$ which consists of the intersection entries of rows in $S\_1$ and columns in $S\_2$. Let $\odot$ be the Ha... | https://mathoverflow.net/users/123075 | A conjecture about the submatrix of orthogonal matrix | Neither conjecture is true in general, as seen from the following counterexample for $n = 4$, $k = 2$, and
\begin{align}
U = \begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{pmatrix}
\ \ \ \ \ \ \ \ \ \ \ \mathrm... | 3 | https://mathoverflow.net/users/113506 | 336299 | 143,595 |
https://mathoverflow.net/questions/336296 | 3 | (Cross-post from [Math Stackexchange](https://math.stackexchange.com/questions/3286716/different-definitions-of-a-relatively-compact-operator), where some work has been done in the comments)
Let $T,K$ be unbounded operators on a Hilbert space $H$.
I've seen the following definition of a relatively compact operator:
... | https://mathoverflow.net/users/117393 | Different definitions of a relatively compact operator | Both definitions aim to make the same statement -- the operator $KT^{-1}$ is compact. But we cannot really to do that, because $T^{-1}$ does not necessarily exist, so in (i) we replace the inverse by the resolvent and in (ii) we write the more usual definition of compactness; if $(Tx\_n)$ is bounded then $KT^{-1}(Tx\_n... | 6 | https://mathoverflow.net/users/24953 | 336302 | 143,597 |
https://mathoverflow.net/questions/336309 | 23 | In his 1841 article *[De determinantibus](https://eudml.org/doc/147138)*, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well of say $x,u$, then the coefficient $\frac{\partial z}{\partial x}$ appearing in the l... | https://mathoverflow.net/users/745 | Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? | An extensive review of the history is given by Florian Cajori, [The History of Notations of the Calculus](https://www.jstor.org/stable/1967725?seq=1#metadata_info_tab_contents).
**Q1:** Yes, it does appear that Jacobi was the first to explicitly state this ambiguity.
**Q2:** The German mathematician [Paul Stäckel](... | 34 | https://mathoverflow.net/users/11260 | 336310 | 143,599 |
https://mathoverflow.net/questions/336292 | 3 | What is known about the Bousfield localization of a left proper [accessible model category](https://ncatlab.org/nlab/show/accessible+model+category) by a set of maps ? (I mean not combinatorial which is already known)
| https://mathoverflow.net/users/24563 | Bousfield localization of a left proper accessible model category | When visiting Johns Hopkins this past April, I talked to Emily Riehl about this. It seemed like Smith's theorem should go through, with enough work (and this is the key input for the existence of localization in a combinatorial model category). I was planning to write up a short note to verify it, but haven't done so y... | 2 | https://mathoverflow.net/users/11540 | 336317 | 143,601 |
https://mathoverflow.net/questions/336315 | 4 | Gandy's basis theorem says that any nonempty $\Sigma^1\_1$ set $A$ contains a real $x$ with $\omega\_1^x=\omega\_1^{CK}$, the least nonrecursive ordinal.
Now the following question seems quite interesting to me:
>
> **Question**: Is it true that for any real $z$ and nonempty $\Sigma^1\_1(z)$ set $A$ containing a... | https://mathoverflow.net/users/14340 | A partial relativization of Gandy's basis theorem | Hmmm, it seems the answer to the question is no.
Fix any nonhyperarithmetic real $x$ so that the $\Sigma^1\_1(x)$ set $$A\_x=\{y\mid \forall n \in \mathscr{O} ( x\not\leq\_T y^{(|n|)})\}$$ is not empty, where $|n|$ is the $y$-recursive ordinal coded by $n$, and $y^{|n|}$ is the $|n|$-th Turing jump relative to $y$.
... | 0 | https://mathoverflow.net/users/14340 | 336323 | 143,603 |
https://mathoverflow.net/questions/336326 | 4 | Let $\mathbb{F}$ be a field, and denote with $\mathbb{F}[t,\sigma]$ the skew-polynomial ring, where $\sigma$ is an automorphim of $\mathbb{F}$. Recall that the multiplication of this ring is defined by the rule $t\cdot \alpha =\sigma(\alpha) t$ for every $\alpha\in\mathbb{F}$.
>
> If $\mathbb{F}$ is algebraically ... | https://mathoverflow.net/users/143213 | Irreducible skew polynomials over an algebraically closed field | There is no compelling reason for this proprty to be true in general, but it holds for quadratic polynomials in characteristic $p$ and the Frobenius automorphism.
Let us consider the special case of a monic reciprocal quadratic polynomial $p(t)=t^2-ct+1,\,$ to be factored as $(t-a)(t-b).\,$ Equating the coefficients... | 3 | https://mathoverflow.net/users/5740 | 336339 | 143,608 |
https://mathoverflow.net/questions/336338 | 1 | Let $\pi \in S\_n$ be an arbitrary permutation. By permutation graph, we refer to a simple graph with nodes $[n]$ and edges that connect pairs of nodes that appear sorted in $\pi$. Formally, $G=(V=[n],E=\{\{i,j\}\colon i<j\;\&\; \pi\_i<\pi\_j\})$. It is clear from the definition that an increasing subsequence in $\pi$ ... | https://mathoverflow.net/users/11363 | Spectral bound for maximum clique $k(G)$ in a permutation graph | [Permutation graphs](https://en.wikipedia.org/wiki/Permutation_graph) are perfect, therefore [Lovasz theta](https://en.wikipedia.org/wiki/Lov%C3%A1sz_number), which is essentially a spectral bound, computes the clique number in polynomial time in polynomial time for this class of graphs.
| 2 | https://mathoverflow.net/users/11100 | 336344 | 143,610 |
https://mathoverflow.net/questions/336308 | 11 | Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a measure, regularity of a measure and even the definition for a measure $\mu : \mathcal{R} \to R$ on its own (including domain... | https://mathoverflow.net/users/58682 | List of all known Riesz representation theorems | Such a list will always be based on subjective criteria but here is one suggestion, from a functional analytic rather than a probabilistic point of view.
In my view the ingredients for an extension of the standard Riesz theorem for $C(K)$-spaces are
1) a space with structure (topology, metric, uniformity, $\sigma$-al... | 3 | https://mathoverflow.net/users/131781 | 336350 | 143,612 |
https://mathoverflow.net/questions/317269 | 13 | Let $X$ be a finite set equipped with a partial order. (Not a preorder: $a < b$ implies $b \not< a$.) The corresponding *incomparability graph* has vertex set $X$ with an edge between two points iff they are incomparable.
I am interested in posets for which the incomparability graph is connected and are maximal for t... | https://mathoverflow.net/users/23141 | Connected incomparability graph | Proposition: Such posets have exactly two maximal elements, one of which lies above every non-maximal element, I call this one supermaximal (according with the original definition of Jeremy Rickard in the first link). Also, removing the supermaximal element leaves the incomparability graph connected (this is easy to se... | 6 | https://mathoverflow.net/users/128335 | 336353 | 143,613 |
https://mathoverflow.net/questions/336341 | 2 | Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \tilde{X}$ be the exceptional divisor.
Let $Z$ be a $k$-dimensional subvariety of $E$ such that $dim\,\pi(Z)=dim\,Z$. Now,... | https://mathoverflow.net/users/90911 | Question regarding Chow group of a blow-up | Say $Y$ is $\mathbb P^1$ and has codimension $2$, so $E$ is a $\mathbb P^1$-bundle on $\mathbb P^1$. Then this $\mathbb P^1$-bundle has many sections, which are not equivalent in the Chow group, but only equivalence class contains pullback of $Y$, which is the pullback of the pushforward of any section.
| 3 | https://mathoverflow.net/users/18060 | 336354 | 143,614 |
https://mathoverflow.net/questions/336347 | 5 | Suppose $\alpha$ is a countable ordinal and $U\_0,U\_1,\kappa$ are such that $L\_\alpha[U\_i] \models \mathrm{ZFC} + U\_i$ is a normal ultrafilter on $\kappa$. Does $U\_0 = U\_1$?
The argument for uniqueness of $L[U]$ due to Kunen uses an iteration of length a $V$-regular cardinal above $\kappa$. This is not availabl... | https://mathoverflow.net/users/11145 | Uniqueness of countable version of $L[U]$? | Andrés's comment can be turned into an answer, but there is a slight subtlety. Suppose $M$ is a countable iterable model of ZFC + $V = L[U]$. Let $\kappa$ be its measurable cardinal and $\alpha$ be its ordinal height. Then there is a transitive model $N\neq M$ of height $\alpha$ satisfying ZFC + $V=L[U]$ + $\kappa$ is ... | 9 | https://mathoverflow.net/users/102684 | 336365 | 143,618 |
https://mathoverflow.net/questions/336295 | 3 | Let $\newcommand{\dbF}{\mathbb F}\dbF\_q$ be a finite field and let $G\subseteq\mathrm{GL}\_N(\bar{\dbF}\_q)$ be a connected reductive group defined over $\dbF\_q$. Let $F$ be the associated Frobenius map, such that $G(\dbF\_q)=G^F$.
Let $g\in G$ be a semisimple $F$-fixed element such that $C\_G(g)$ is disconnects. T... | https://mathoverflow.net/users/14443 | Image of the Lang-Steinberg map on disconnected centralizers of semisimple elements | One example is $G = \operatorname{PGL}\_4$, $g$ a semisimple lift of the Coxeter element of the Weyl group, say as $g = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}$, and $q \equiv 3 \pmod4$. The component containing $\operatorname{diag}(1, \zeta\_4, -1, -\zeta\_4)$ is ... | 2 | https://mathoverflow.net/users/2383 | 336370 | 143,619 |
https://mathoverflow.net/questions/336359 | 1 | Let $X$, $Y$, $Z$, $A$ be a set of random variables drawn from the Irwin-Hall distribution where $X$ is the sum of $c$ iid r.v.s, $Y$ is the sum of $c$ iid r.v., $Z$ is the sum of $n - c$ iid r.v.s, and $A$ is the sum of $n - c$ iid r.v.s.
I want to compare $\Pr[(X \leq x) \cap (Z \leq x - X) \cap (A \leq x - X)]$ w... | https://mathoverflow.net/users/143224 | Irwin-Hall Distribution relationship between two sets of events | $\newcommand{\R}{\mathbb{R}}$
Welcome to MathOverflow!
Your conditions on $X$, $Y$, $Z$, $A$, as I understood them, imply that $X$, $Y$, $Z$, $A$ are independent nonnegative random variables (r.v.'s), with $(X,Z)$ equal $(Y,A)$ in distribution -- which is all we need to verify your conjecture.
Indeed, since $Z,A\... | 2 | https://mathoverflow.net/users/36721 | 336371 | 143,620 |
https://mathoverflow.net/questions/336368 | 8 | Godel's Completeness Theorem is a straightforward consequence of Skolem 1922 and yet this conclusion was not drawn by Skolem himself. In a letter to Wang (Dec. 7, 1967 in Godel 2003) Godel gives an explanation for this oversight:
>
> At that time, nobody (including Skolem himself) drew this conclusion ... I think t... | https://mathoverflow.net/users/116705 | Infinitary reasoning in Godel's Completeness Proof | The best write-up I know of Godel's proof of the completeness theorem is by Avigad, in his paper [Godel and the metamathematical tradition](http://www.andrew.cmu.edu/user/avigad/Papers/goedel.pdf) (section $4$). Avigad divides the proof into $5$(ish) steps, and step $2$ crucially uses Konig's lemma:
>
> Step $2$: I... | 9 | https://mathoverflow.net/users/8133 | 336375 | 143,621 |
https://mathoverflow.net/questions/336387 | 2 | In a $d$ dimensional vector space defined over $\mathbb{C}$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V\_i\}$ which satisfies the following properties. Here $\epsilon$ is smaller but finite compared to 1.
$$\langle V\_i, V\_i\rangle = 1$$
$$|\langle V\_i, V\_j\rangle| \leq \epsilon, i \n... | https://mathoverflow.net/users/78150 | Neat/Approximate formula for maximum number of "almost orthogonal" vectors in a complex vector space? | The variant of the Johnson-Lindenstrauss lemma that you can use is derived by L. Welch in [Lower bounds on the maximum cross correlation of signals](https://ieeexplore.ieee.org/document/1055219) (1974). This paper is behind a paywall, I quote the result from [arXiv:0909.0206](https://arxiv.org/abs/0909.0206)
Conside... | 2 | https://mathoverflow.net/users/11260 | 336395 | 143,626 |
https://mathoverflow.net/questions/336393 | 2 | Let $H$ be the kernel of an action of a group $G$ on a space $X$. Is there a term for the actions with the property that the action of the quotient group $G/H$ on $X$ is free?
| https://mathoverflow.net/users/8588 | Actions that become free after quotienting out their kernel | Yes, it's called **effectively free** (see for instance [here, page 3](http://www-users.math.umn.edu/~olver/di_/rel.pdf), Fels/Olver, *On relative invariants*).
| 4 | https://mathoverflow.net/users/14094 | 336398 | 143,627 |
https://mathoverflow.net/questions/336400 | 5 | **Edit:** I revise the question according to the comment of Gabe Conant.
What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:
>
> For every $z\in \mathbb{C}$, the sequence $z,f(z),f^2(z),\ldots,f^n(z),\ldots$ is a bounded sequence but $f$ is not in the fo... | https://mathoverflow.net/users/36688 | An entire function all whose forward orbits are bounded | Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the *escaping set*, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the [Wikipedia article](https://en.wikipedia.org/wiki/Escaping_set#Properties), the escaping set of a non-linear entire function is nonempty.
The reference for this i... | 10 | https://mathoverflow.net/users/38253 | 336402 | 143,630 |
https://mathoverflow.net/questions/336396 | 1 | 1. Given an integer $T$ how many $n\times n$ matrices in $\mathbb Z\_{\geq0}^{n\times n}$ we have such that every row and column sum to same $T$?
2. Do the set of constant row and column sum matrices form any kind of algebraic structure (it is definitely definable by linear equations) by which I mean closedness, more t... | https://mathoverflow.net/users/10035 | Constant row-column sum matrices? | If you are looking for a simple formula, you are out of luck except for small $n$ or $T$. As Sam mentions in a comment, these are the integer points in a dilated Birkhoff polytope. Since the vertices of the polytope are lattice points, the number of matrices is a polynomial of degree $(n-1)^2$ in $T$ for any fixed $n$ ... | 2 | https://mathoverflow.net/users/9025 | 336403 | 143,631 |
https://mathoverflow.net/questions/336397 | 2 | Let $a\_i>0$ for $i=1,...,n$. It is well-known that $A\ge H$, where $A$ and $H$ are the arithmetic mean and harmonic mean of the vector $(a\_i)$, respectively. Is any lower bound on $H/A$ known?
| https://mathoverflow.net/users/26039 | Bound on the ratio of harmonic and arithmetic mean | If $(a\_k)$ and $(b\_k)$ are positive sequences of the same length, and
$$0<m\le \frac{a\_k}{b\_k} \le M<\infty$$
$$A=\frac{m+M}{2},\ \ G=\sqrt{mM}$$
then
$$(\Sigma{a\_k}^2)(\Sigma{b\_k}^2) \le (\frac{A}{G}\Sigma{a\_kb\_k})^2=\frac{A^2}{G^2}(\Sigma{a\_kb\_k})^2$$
This is a reverse of Cauchy-Schwarz which follows from ... | 3 | https://mathoverflow.net/users/133811 | 336404 | 143,632 |
https://mathoverflow.net/questions/336321 | 0 | This is a follow-up question to [this](https://mathoverflow.net/questions/335996/hadamard-ell-p-sum-of-two-symmetric-positive-semidefinite-matrices) and [this](https://mathoverflow.net/questions/336099/hadamard-ell-p-sum-of-two-symmetric-positive-semidefinite-matrices-follow-up).
Let $A=(a\_{ij})$ and $B=(b\_{ij})$ b... | https://mathoverflow.net/users/143037 | Hadamard $\ell_2$ sum of two symmetric positive semidefinite matrices | The answer is yes. If $I$ is an $n\times n$ matrix with all entries equal to $1$ and
$$
A=\left[\begin{array}{cccc}
I & I & 0 & 0 \\
I & I & 0 & 0 \\
0 & 0 & I & I \\
0 & 0 & I & I
\end{array}\right],\,\,
B=\left[\begin{array}{cccc}
I & 0 & I & 0 \\
0 & I & 0 & I \\
I & 0 & I & 0 \\
0 & I & 0 & I
\end{array}\right]
$$
... | 0 | https://mathoverflow.net/users/143037 | 336426 | 143,641 |
https://mathoverflow.net/questions/335975 | 4 | In a lot of fields in algebraic geometry (e.g. GIT or topics on étale cohomology) which make use of group scheme concepts (or in more tame way of algebraic groups), the class of *reductive* algebraic groups/group schemes seem to play a prominent role.
Presumably this is a quite broad question, but up to now I haven'... | https://mathoverflow.net/users/108274 | Reductive groups in algebraic geometry | Hilbert's 14th question roughly asks when the ring of invariants (under the rational action of an algebraic group $G$) of a finitely generated $k$-algebra $A$ ($k$ a field) is finitely generated. Nagata answered this affirmatively if $G$ is geometrically reductive and gave a counter-example if $G$ is not reductive (Pop... | 4 | https://mathoverflow.net/users/12218 | 336432 | 143,644 |
https://mathoverflow.net/questions/336408 | 16 | I have just read [Grayson's introduction](https://faculty.math.illinois.edu/~dan/Papers/ium.pdf) on homotopy type theory as a possible foundation for mathematics. It is very enlightening about what all the fuss is about, but I am left with some doubts. Forgive my naiveté, I am not expert in type theory at all.
When o... | https://mathoverflow.net/users/828 | Practical example in using (homotopy) type theory | I know a bit about HoTT and have worked quite extensively with proof assistants, but I am certainly not a HoTT expert. Nonetheless, I think that the story is as follows. We can construct a type $L$ of bijections from $G$ to $\{1,\dotsc,n\}$ (where the inverse is explicitly given as part of the data of the bijection). G... | 4 | https://mathoverflow.net/users/10366 | 336435 | 143,646 |
https://mathoverflow.net/questions/336232 | 6 | Define a skew-symmetric form $(\cdot,\cdot)$ on $\mathbb{R}^{2k}$ by $$(e\_i,e\_j) = \begin{cases} 1 &\text{if $i<j$},\\ -1 &\text{if $i>j$},\\ 0 & \text{if $i=j$.}\end{cases}$$ Given a permutation $\pi:\{1,\dotsc, 2 k\}\to \{1,\dotsc, 2 k\}$, let $V\_\pi$ be the space spanned by $e\_{\pi(2i)}-e\_{\pi(2i-1)}$ for $1\le... | https://mathoverflow.net/users/398 | Permutations, skew-symmetric forms and degeneracy | I'll give it a try (clearly, comments aren't enough, and the references I found are too much).
Given the permutation $\pi$ of $\{1,2,\dots,2k\}$ one can build an orientable surface with boundary as follows.
Take a topological disk, say the rectangle $[0,2k+1]\times[0,1]$ in the complex plane, and consider its quoti... | 4 | https://mathoverflow.net/users/6451 | 336436 | 143,647 |
https://mathoverflow.net/questions/336415 | 9 | Let $p>q$ be two prime numbers. Let $\lambda$ be a partition whose $p$-core is $\lambda\_p$ and $q$-core is $\lambda\_q$. Assume that $|\lambda|>|\lambda\_p|+|\lambda\_q|$. Is it true that there always exists some partition $\mu\neq \lambda$ with $|\mu|=|\lambda|$, such that $\mu$ also has $p$-core $\lambda\_p$ and $q$... | https://mathoverflow.net/users/143264 | A conjecture on partitions | The answer is yes, and it follows from the results in my paper "A generalisation of core partitions", J. Combin. Theory 127. In that paper I define a class of partitions called $[p:q]$-cores. One characterisation of these partitions (Corollary 5.2 in the paper) is that $\lambda$ is a $[p:q]$-core if and only if $\lambd... | 9 | https://mathoverflow.net/users/6771 | 336454 | 143,654 |
https://mathoverflow.net/questions/336439 | 6 | Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^\*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional Hilbert $C^\*$-modules
Edit: By finite dimensional I mean that the $C^\*$-module is finite dimensional as a vector sp... | https://mathoverflow.net/users/125790 | Finite-dimensional Hilbert $C^*$-modules | As a vector space, okay. Let $A$ be a C\*-algebra and $E$ a Hilbert module over $A$ which is finite dimensional as a vector space. Then $A$ acts by left multiplication on $E$, i.e., we have a bounded homomorphism from $A$ into $B(E) \cong M\_n$. Letting $I$ be the kernel of this homomorphism, this shows that $A$ has a ... | 5 | https://mathoverflow.net/users/23141 | 336457 | 143,656 |
https://mathoverflow.net/questions/335415 | 5 | Given any function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ we define the *environment sum of $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ with respect to $f$* by
$$\text{es}\_f(x,y) = \sum\{f(x', y'): |(x',y') - (x,y)|=1\}.$$
**Question.** Is there a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ such t... | https://mathoverflow.net/users/8628 | "Uniformly continuous" environment sum of a bijection $\varphi:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}$ | There exist bijections $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ with $\text{es}\_f(x,y)$ vanishing identically, which directly answer the bonus question.
Note that the infinite set of equations that such a bijection must satisfy,
$$\begin{align}
{f(x,y+1)+f(x,y-1)+f(x+1,y)+f(x-1,y)=0,\\ \forall (x,y)\in \mathb... | 5 | https://mathoverflow.net/users/6101 | 336460 | 143,658 |
https://mathoverflow.net/questions/336453 | 6 | Let $\mathcal{O}$ be a number ring. Letting $r$ and $2s$ be the number of real and complex embeddings of $\mathcal{O}$, the number ring $\mathcal{O}$ is a lattice in $\mathcal{O} \otimes \mathbb{R} \cong \mathbb{R}^{r+2s}$, and thus is Zariski dense in this vector space.
Question: What is the Zariski closure of the g... | https://mathoverflow.net/users/143278 | Zariski closure of set of units in a number ring | Let $x\_1,\dots, x\_n$ be the embeddings of $\mathcal O$ into $\overline{\mathbb Q}$. Every monomial in $x\_1,\dots, x\_n$ restricts to a character of $\mathcal O^\times$. If some polynomial in $x\_1,\dots, x\_n$ vanishes on units, then we get a linear relation among characters of $\mathcal O^\times$, unless two of the... | 5 | https://mathoverflow.net/users/18060 | 336463 | 143,659 |
https://mathoverflow.net/questions/336373 | 12 | **The question**
My question is easy to state:
Is there a *Proj* construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’re free to interpret derived geometry in your favourite model for affines: (E-infinity/simplicial/dg)-algebras ... | https://mathoverflow.net/users/137364 | Proj construction in derived algebraic geometry | It is instructive to look at the simplest case of Proj: that of a free module, i.e. the projective space. Lurie works these out for us quite carefully in his Spectral Algebraic Geometry tome.
**Projective spaces in SAG**
*1. Projective space by gluing:*
In Section SAG.5.4, more specifically Construction SAG.5.4.1.3... | 16 | https://mathoverflow.net/users/39713 | 336465 | 143,660 |
https://mathoverflow.net/questions/336459 | 5 | Is the following statement true?
>
> ($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$, irreducible over $\mathbb{Z}[x]$, whose roots are all real and simple, with exactly $k$ roots of absolute value $<1$ and $n-k$ roots of absolu... | https://mathoverflow.net/users/1516 | Existence of algebraic integers with certain properties | You can even construct such a polynomial for any totally real field $K$ of degree $n$. By Dirichlet's theorem, the unit group $U$ of $K$ maps to
$$U \rightarrow (K^\* \otimes \mathbf{R}) \stackrel{\log| \cdot |}{\longrightarrow}
\mathbf{R}^{n},$$
and the image is a lattice $L$ (of rank $n-1$) in the co-dimension on... | 11 | https://mathoverflow.net/users/143286 | 336477 | 143,664 |
https://mathoverflow.net/questions/336440 | 9 | The category $Mod^{E\_n}\_A(\mathcal{C})$ of $E\_n$-modules for an $E\_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more general definition (just replace the little $n$-cubes operad with your favorite $\infty$-operad). The definition is r... | https://mathoverflow.net/users/101861 | Definition of $E_n$-modules for an $E_n$-algebra | $E\_n$ algebras have compatible multiplications for every way of placing a bunch of elements into a collection of balls in $\mathbb{R}^n$. A module for an $E\_n$ algebra has an action for every way of placing a bunch of elements of the algebra into balls not at the origin, and an element of the module into a ball at th... | 11 | https://mathoverflow.net/users/22 | 336481 | 143,666 |
https://mathoverflow.net/questions/336472 | 8 | Let $0<x < y < 1$ be given. Prove
$$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[
\sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$
I have been working on this problem for a while now and I have hit a wall. I have plotted this and it seems to be true. I also tried to prove that it is grea... | https://mathoverflow.net/users/143222 | Prove that this expression is greater than 1/2 | Let
$$f(x,y):=4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Bigl(
\sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}(\pi y-\pi x) \Bigr).$$
I will show that
$$\min\_{0\leq x\leq y\leq 1}f(x,y)=\min\_{0\leq x\leq 1/3}f(x,2x).\tag{$\ast$}$$
This suffices, because the minimum of the one-variable function $f(x,2x)$ is easy to analyze ... | 21 | https://mathoverflow.net/users/11919 | 336485 | 143,667 |
https://mathoverflow.net/questions/336483 | 6 | I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector.
It should be straight in dimensions $2$ and $3$.
| https://mathoverflow.net/users/121643 | Real orthogonal and sign | This is **false** already for $n=3$. A counterexample is given by the matrix
$$A=\frac{1}{3}\begin{bmatrix}
2 & 2 & -1\\
2 & -1 & 2\\
-1& 2 & 2
\end{bmatrix}.
$$
How did I construct this matrix? Start with $(a,b,c)$ such that $ab+bc+ca=0$, where $a,b,c\ne 0$. Observe that $a,b,c$ cannot be of the same sign. Renorm... | 13 | https://mathoverflow.net/users/5740 | 336490 | 143,670 |
https://mathoverflow.net/questions/336489 | 2 | If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (discrete) dynamical system?
| https://mathoverflow.net/users/36886 | Existence of topologically mixing (discrete) dynamical system on manifold | Yes it's true.
It follows from a result of Abdenur and Crovisier ([ArXiv link](https://arxiv.org/abs/1111.4206)) [AC]: given a volume form $\omega$ on $M$ (closed manifold — the authors seem to omit assuming $\dim(M)\neq 1$), inside the topological group $\mathrm{Diff}^1(M,\omega)\subset\mathrm{Diff}^1(M)$, there is ... | 4 | https://mathoverflow.net/users/14094 | 336496 | 143,671 |
https://mathoverflow.net/questions/336493 | 5 | This was a question I first asked on stack exchange, [here](https://math.stackexchange.com/questions/3277807/replacing-the-fibre-of-a-fibration). In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.
Let $p:E\rightarrow X$ be a Serre fibration over a p... | https://mathoverflow.net/users/54788 | Replacing the Fibre of a Fibration | You can make the construction as follows. There are three steps.
First make a Serre fibration over the unit interval, pi:T->I=[0,1] such that the fibre over 0 (resp. 1) is F (resp. F'). You will find how to make this in the general case; for example, for F=1 point and F'=the interval, here is a proof that the triang... | 5 | https://mathoverflow.net/users/105095 | 336498 | 143,673 |
https://mathoverflow.net/questions/336504 | 4 | Assume we are given a scheme $X$ (feel free to add all the needed hypotheses, at this point I’m working with smooth schemes, but the fewer is needed, the better) and a vector bundle $E$ over $X$. I will denote $|E|$ the total space of this vector bundle. I know that if we blow up $|E|$ in the zero section, we get that ... | https://mathoverflow.net/users/91572 | Blowing up vector bundles in the zero section | You may have a look at remark 11.10 in Huybrechts' Fourier Mukai transform (or SGA6 or Fulton's Intersection Theory : appendix B5 and B6). I think your expectation is correct. In a nutshell a "complicated" proof would be the following:
i) $\mathrm{Pic}(|\mathcal{O}\_{\mathbb{P}(E)}(-1)|) = \mathrm{Pic}(\mathbb{P}(E))... | 4 | https://mathoverflow.net/users/37214 | 336505 | 143,676 |
https://mathoverflow.net/questions/336479 | 11 | In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set consists of $4$ points and the weights at these fixed points cannot be equal to any linear circle action (A linear circle ... | https://mathoverflow.net/users/99732 | Petries exotic circle action | This is only a partial answer. What follows was the result of a discussion with Silvia Sabatini, the author of this [paper](https://www.worldscientific.com/doi/abs/10.1142/S0219199717500432), which is related to the question of the OP.
The total Chern classes of possible almost complex structures on $\mathbb{CP}^3$ c... | 7 | https://mathoverflow.net/users/20999 | 336506 | 143,677 |
https://mathoverflow.net/questions/336495 | 7 | Given a probability measure $\mu$ on the interval $[0,1]$, the linear operator
$$
T\_\mu \! f(y) := \int\_0^1 f(yx) \, d\mu(x)
$$
takes the space of continuous functions $f: [0, \infty) \rightarrow \mathbb{R}$ such that $f(0) = 0$ to itself.
Assuming that $\mu$ is **not** the delta function at the origin, is this op... | https://mathoverflow.net/users/21123 | Injectivity of a class of integral operators | If the support of $\mu$ is contained in $[0, b]$ for some $b \in (0,1)$, then $T\_\mu f = 0$ for any $f$ that is $0$ on $[0,b]$, so it is not injective.
EDIT: Another example, where the support of $\mu$ is all of $[0,1]$: $d\mu(x) = g(x)\; dx$ where
$g(x) = 5/3$ for $0 \le x < 1/2$, $1/3$ for $1/2 \le x \le 1$. Let ... | 6 | https://mathoverflow.net/users/13650 | 336508 | 143,679 |
https://mathoverflow.net/questions/336511 | 5 | Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that normality is key here), and therefore every element of $u$ is stationary. My question is about the converse:
**Is it th... | https://mathoverflow.net/users/13059 | Stationary sets and $\kappa$-complete normal ultrafilters | Asaf's answer is totally right, but let me also point out that you don't even have to go to a special model to see that your conjecture fails. The point is that sets in any normal measure on $\kappa$ must reflect many properties of $\kappa$ itself (since these sets $X$ are exactly the ones for which $\kappa\in j(X)$, w... | 11 | https://mathoverflow.net/users/1058 | 336514 | 143,681 |
https://mathoverflow.net/questions/263370 | 17 | The [Dehn invariant](https://en.wikipedia.org/wiki/Dehn_invariant) of a polyhedron is a vector in $\mathbb{R}\otimes\_{\mathbb{Z}}\mathbb{R}/2\pi\mathbb{Z}$ defined as the sum over the edges of the polyhedron of the terms $\sum\ell\_i\otimes\theta\_i$ where $\ell\_i$ is the length of edge $i$ and $\theta\_i$ is its dih... | https://mathoverflow.net/users/440 | Are all Dehn invariants achievable? | In hyperbolic, spherical and Euclidean geometry the answer is no.
For Euclidean polytopes it is a result of B. Jessen (proved [here](https://www.mscand.dk/article/view/10888)). Namely, let $\Omega^1\_{\mathbb{R}/\mathbb{Q}}$ be a $\mathbb{Q}-$vector space of Kähler differentials. Then the image of Dehn invariant equ... | 4 | https://mathoverflow.net/users/21620 | 336515 | 143,682 |
https://mathoverflow.net/questions/336533 | 8 | I've seen that there was a [single-sorted definition](https://ncatlab.org/nlab/show/single-sorted+definition+of+a+category) of a category. In some ways, it seems more understandable than the original definition.
I don't know much about category theory. But I would like to know how each definition is useful.
| https://mathoverflow.net/users/143309 | Why is an object not defined as identity morphism? | They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally es... | 15 | https://mathoverflow.net/users/8133 | 336542 | 143,688 |
https://mathoverflow.net/questions/284065 | 3 | I wanted to see for which ranks Bernays' Reflection Principle holds; that is, for every class and every property (allowing quantification over all classes) which is true about that class, there is a transitive set $u$ for which $\mathcal{P}(u)$ satisfies that property about that class's intersection with $u$. Like Vope... | https://mathoverflow.net/users/115951 | Bernays' Reflection Principle Holding in Ranks? | I would like to add that, for $n>1$, the $\Pi\_n-$Bernays cardinals are precisely the $\Pi\_n^1-$indescribable cardinals. We can do this really simply by giving a $\Pi\_2$ definition of the Axiom of limitation of size:
$$ALZ\leftrightarrow\forall x(\exists W\ni x\leftrightarrow\exists F,y(F\text{ is a bijection } F:x... | 1 | https://mathoverflow.net/users/141402 | 336546 | 143,690 |
https://mathoverflow.net/questions/336545 | 14 | Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent?
References to articles in the literature if there are any will be much appreciated.
Thanks!
| https://mathoverflow.net/users/13904 | Which failed attempts have there been to find a contradiction in ZFC or ZF? | I think the most noticeable one was Nelson's attempt to prove the inconsistency of primitive recursive arithmetic. Terence Tao found a mistake in the proof, but Nelson's attempt was posthumously uploaded to arxiv (<https://arxiv.org/pdf/1509.09209.pdf>), together with an introduction by Sarah Jones Nelson and an afterw... | 22 | https://mathoverflow.net/users/12976 | 336547 | 143,691 |
https://mathoverflow.net/questions/333715 | 6 | Consider the finite field extension $\mathbb{F}\_{{q}^{d}}$ over $\mathbb{F}\_{q}$, where $q=p^a$ for some prime $p$. We assume $d\geq 2$. Let,
$$ S=\{ \alpha \in \mathbb{F}\_{q^d}\hspace{0.1 cm} | \hspace{0.1 cm} \mathbb{F}\_{q}(\alpha)=\mathbb{F}\_{q^d} \}$$
In other words, $S$ consists of all those elements in $\m... | https://mathoverflow.net/users/106299 | Enumeration of field generators of a finite field over $\mathbb{F}_{q}$, which are $m^{th}$ powers in the same field | $$| S \cap S^m | = \sum\_{n|d} \mu(n)
\frac{ \gcd ( m (q^{d/n}-1), q^{d}-1 )}{ \gcd (q^{d} - 1, m) } $$
First note that $|S \cap S^m | = |S \cap \mathbb F\_{q^d}^m |$ because every $m$th power that generates is an $m$th power of a generator.
We can count elements of $S$ by an inclusion-exclusion argument, subtracti... | 6 | https://mathoverflow.net/users/18060 | 336549 | 143,693 |
https://mathoverflow.net/questions/336551 | 4 | Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra.
Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie subalgebras?:
>
> There are $\mathfrak h\_1,\dots,\mathfrak h\_r$ proper Lie subalgebras of $\mathfrak g$ such that: (... | https://mathoverflow.net/users/20052 | On maximal closed connected subgroups of a compact connected semisimple Lie group? | It's true, and does not rely on classification of maximal subalgebras.
The first part of the statement is that there are finitely maximal subalgebras up to conjugation. It follows from the fact that in a semisimple real Lie algebra there are finitely many semisimple subalgebras up to conjugation, and that every maxim... | 7 | https://mathoverflow.net/users/14094 | 336560 | 143,697 |
https://mathoverflow.net/questions/336562 | 2 | I'm a beginner. When I searched for results in probabilistic number theory most of the results were asymptotic in nature. Are there any results like with probability 1-$\epsilon$ (w.h.p) some property $P(\epsilon)$ hold? Any references would be also helpful.
| https://mathoverflow.net/users/128058 | Non-asymptotic results in probabilistic number theory | [Chebyshev's bias](https://en.wikipedia.org/wiki/Chebyshev's_bias) says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).
| 4 | https://mathoverflow.net/users/4600 | 336564 | 143,698 |
https://mathoverflow.net/questions/336491 | 3 | When reading the characterization of Besov space with $L\_p$-modulus of continuity in the 7th chapter “Fractional Order Space” of Sobolev space written by Adams(Page 243), I encounter some small specific problem. We define
$$\Delta\_hw(x):=w(x)-w(x-h)$$
We want to prove
\begin{equation}||\Delta\_hw||\_p\leq |h||w|\_... | https://mathoverflow.net/users/141983 | Characterization of Besov space with Lp-modulus of continuity | 1) One way would be to apply Tonelli's theorem
\begin{align\*}
&\int\_{\mathbb{R}} \int\_{x-h}^{x}|w'(s)|^{p}ds dx = \int\_{\mathbb{R}} \int\_{\mathbb{R}} |w'(s)|^{p} \chi\_{[x-h,x]}(s) dsdx =\\
& \int\_{\mathbb{R}} \int\_{\mathbb{R}}|w'(s)|^{p}\chi\_{[x-h,x]}(s) dxds= |h| \int\_{\mathbb{R}}|w'(x)|^{p}dx
&
\end{alig... | 2 | https://mathoverflow.net/users/50901 | 336571 | 143,699 |
https://mathoverflow.net/questions/336577 | 1 | [Stephani](https://www.researchgate.net/publication/227258228_A_note_on_Killing_tensors) states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is
of Petrov type D or vanishes. Does this imply that the spacetime also cannot admit a Killing tensor for Petrov typ... | https://mathoverflow.net/users/142501 | Connection of the existence of Killing-Yano tensor and Killing tensor | Other Petrov types can admit Killing tensors, i.e. symmetric tensors $K\_{a b} = K\_{(a b)}$ such that $\nabla\_{(a} K\_{b c)} = 0$. A type N vacuum example is given in Example 3 of [A. J. Keane and B. O. J. Tupper, "Killing tensors in pp-wave spacetimes"](https://arxiv.org/abs/1011.6401), which has the metric
\begin{e... | 1 | https://mathoverflow.net/users/39284 | 336590 | 143,706 |
https://mathoverflow.net/questions/336585 | 4 | I need to compute the following integral
$$
I\_{n,m} := \int\_0^1 P\_n(x) P\_m(x) \; \mathrm{d}x
$$
where $P\_n$ is the [Legendre polynomial](https://en.wikipedia.org/wiki/Legendre_polynomials).
For an even sum $n+m=2l$ it is easy to show that
$$
I\_{n,m} = \frac{1}{2} \int\_{-1}^1 P\_n(x) P\_m(x) \; \mathrm{d}x
= \d... | https://mathoverflow.net/users/143345 | Legendre Polynomial Integral over half space | Integration of Equation (34) in [MathWorld](http://mathworld.wolfram.com/LegendrePolynomial.html) gives the integral $I\_{nm}$ as a sum
$$I\_{nm}=\sum \_{q=0}^m \frac{2^{-q}}{q+1} \binom{-m-1}{q} \binom{m}{q} \, \_3F\_2\left(-n,n+1,q+1;1,q+2;\tfrac{1}{2}\right).$$
As noted by the OP, Mathworld also gives the explicit... | 6 | https://mathoverflow.net/users/11260 | 336594 | 143,708 |
https://mathoverflow.net/questions/335872 | 0 | Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t\_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial\_t+\Delta)u=(i\partial\_t+\Delta)v=0.$ Let us fix $j,k\in\mathbb{Z}$. For every $\delta\geq 0$, we have the following Bourgain-type bilinear estimate
$$\Vert... | https://mathoverflow.net/users/54552 | Bilinear Strichartz estimates for the Schrodinger equation | I don't understand why you have the $\delta$. One has the following estimate (see, for instance, Theorem 4.18 in the Clay lecture notes "Nonlinear Schrodinger Equations at Critical Regularity" by Rowan Killip and Monica Visan):
>
> **Bilinear Strichartz.** Let $j,k$ be integers such that $j\leq k$. Then
> $$\|(e^{... | 4 | https://mathoverflow.net/users/54316 | 336599 | 143,710 |
https://mathoverflow.net/questions/336607 | 12 | It's a [ZFC theorem of Freyd](https://ncatlab.org/nlab/show/complete+small+category), any small complete category is a preorder. Freyd's theorem continues to hold [in any Grothendieck topos](https://mathoverflow.net/questions/43433/small-complete-categories-in-a-grothendieck-topos "Small complete categories in a Grothe... | https://mathoverflow.net/users/2362 | Can there be a small complete category in ZF? | Yes, Freyd's theorem holds in ZF. The theorem relies on excluded middle, choice does not play a role. Just to be sure, let's work with excluded middle but without choice.
**Theorem:** *A small-complete small category is a preorder.*
*Proof.* Let $C$ be a small-complete small category, with $C\_0$ the set of objects... | 20 | https://mathoverflow.net/users/1176 | 336610 | 143,716 |
https://mathoverflow.net/questions/336606 | 7 | Let $n\in\mathbb{N}$ be a positive integer, and let $[n] = \{1,\ldots,n\}$. We set $S\_n$ for the set of all bijections $\varphi:[n]\to [n]$.
Let $G= ([n], E)$ be a simple, undirected graph, and let $\varphi\in S\_n$ We define the *label distance number of $G$* in the following way:
$$\lambda(G) = \min\big\{\max\{|\... | https://mathoverflow.net/users/8628 | Label distance number and chromatic number of a graph | The cycle on five vertices has chromatic number 3 and bandwidth 2. The complete bipartite graph with blocks 2 and 3 has chromatic number 2 and bandwidth 3.
| 7 | https://mathoverflow.net/users/3032 | 336617 | 143,718 |
https://mathoverflow.net/questions/336559 | 6 | I thought that this question is simple, and [asked it at Stackexchange](https://math.stackexchange.com/questions/3297283/dixmiers-lemma-as-a-generalisation-of-schurs-first-lemma). To my surprise, no one was able to answer it there. Now have to elevate it to Overflow.
---
What mathematicians call *Schur's lemma* i... | https://mathoverflow.net/users/137120 | Dixmier's lemma as a generalisation of Schur's first lemma | Yes, it indeed is correct that Dixmier's lemma and Schur's lemma, together, entail the stronger statement that $\,{\mathbb{M}}\,-\,c\,{\mathbb{I}}\,=\,0\,$ and, therefore, $\,{\mathbb{M}}\,$ is proportional to the identity operator.
See Lemma 99 in <https://arxiv.org/abs/1212.2578> , where that story follows N. R. Wa... | 3 | https://mathoverflow.net/users/137120 | 336623 | 143,721 |
https://mathoverflow.net/questions/336537 | 5 | This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question:
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$ with Cartan subalgebra $\mathfrak{h}$ and $V$ be an irredu... | https://mathoverflow.net/users/143312 | Weight spaces of representations of finite dimensional simple Lie algebras | The requisite property follows from the following key proposition:
>
> $U\_{\lambda}$ is a finitely generated right $U\_0$-module.
>
>
>
*Notation* The subscripts denote the grading of the universal enveloping algebra $U=U({\frak g})$ with respect to the adjoint action of the Cartan subalgebra ${\frak h},\, \... | 3 | https://mathoverflow.net/users/5740 | 336626 | 143,723 |
https://mathoverflow.net/questions/336553 | 2 | There is a [paper](https://doi.org/10.1023/A%3A1022669121502) by Rizzo in which tables are given that specify the local root numbers of an elliptic curve based on $(a, b, c)$ where $(a, b,c)$ are non negative and minimal so that $a\equiv c\_4\pmod{4}$, $b\equiv c\_6\pmod{6}, c\equiv\Delta\pmod{12}$ where $c\_4, c\_6, \... | https://mathoverflow.net/users/40983 | Statement of a result by Rizzo | It’s been a long while, but the definition of (a,b,c) should be the minimal non negative triplet of integers such that (a,b,c) + k(4,6,12) = (c4,c6,Delta).
| 7 | https://mathoverflow.net/users/143364 | 336627 | 143,724 |
https://mathoverflow.net/questions/336646 | 3 | Let $k$ be a field of characteristic 0 (not necessarily algebraically closed), let $G$ be a connected split reductive group over $k$ and let $\mathfrak{g}$ be the Lie algebra of $G$.
Let $X \in \mathfrak{g}$ be a nilpotent element. Does there exist a unipotent subgroup $U$ of $G$ such that $X$ is contained in the Li... | https://mathoverflow.net/users/138629 | Nilpotent elements of Lie algebra and unipotent groups | Indeed there's a much simpler way in characteristic zero, with $G$ an arbitrary $k$-defined linear algebraic group.
Let $X$ be nilpotent. Fix a faithful $k$-defined linear representation $\rho$ of $G$ and let $\rho'$ be the corresponding representation of $\mathfrak{g}$.
Now $\rho'(X)$ being nilpotent, it preserves... | 8 | https://mathoverflow.net/users/14094 | 336653 | 143,735 |
https://mathoverflow.net/questions/336618 | 9 | Let us work over $K = \mathbf{C}((t))$ for simplicity. We say that a smooth proper scheme $X/K$ has *good reduction* if it extends to a smooth and proper algebraic space $\mathcal{X}/\mathcal{O}\_K$ where $\mathcal{O}\_K = \mathbf{C}[[t]]$, and that it has *potentially good reduction* if for some finite extension $K' =... | https://mathoverflow.net/users/3847 | Good reduction of rational surfaces | As Ulrich commented, this is true. In fact, if $\mathcal{O}\_K$ is a Dedekind domain with fraction field $K$ and $X$ is a rational surface over $K$, then there is a finite field extension $L/K$ such that $X\_L$ has a smooth *projective* model over the normalization $\mathcal{O}\_L$ of $\mathcal{O}\_K$ in $L$.
I just... | 7 | https://mathoverflow.net/users/4333 | 336657 | 143,737 |
https://mathoverflow.net/questions/336634 | 4 | Let $F: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function that is coercive, that is, $\lim\_{||x|| \to \infty} F(x) = +\infty$.
Given any rectifable curve $c: [0, 1] \to \mathbb R^n$, define the $F$-arc length of the curve, $A(F, c)$ as $\sup \sum\_{i=1}^n |F(c(x\_i) - F(c(x\_{i-1}))|$, where the sup is t... | https://mathoverflow.net/users/132446 | Question on existence of “geodesic” curves | I assume you want a *continuous* minimiser $c:[0,1]\to\mathbb{R}^n$ (otherwise $c\_0:=a\chi\_{[0,1/2)}+b\chi\_{[1/2,1]}$ is a trivial solution with $A(F,c\_0)=|F(b)-F(a)|$).
In general the infimum is not attained by a continuous curve. Consider e.g. $n=2$ and let $F$ be the distance function from the [topologist sine... | 3 | https://mathoverflow.net/users/6101 | 336666 | 143,740 |
https://mathoverflow.net/questions/336669 | 2 | Let $\mu$ be the Mobius function, $\tau\_k(x)$ the number of ways to write $x$ as a product of $k$ natural numbers and $\phi$ the Euler totient function.
I would like to obtain an upper bound for
$$
\sum\_{x < X} \frac{\mu^2(x) \tau\_k(x)}{\phi(x)}.
$$
In the paper I am reading, this is bounded by
$$
\ll (\log X)^k
... | https://mathoverflow.net/users/84272 | How to obtain an upper bound for $\sum_{x < X} \frac{\mu^2(x) \tau_k(x)}{\phi(x)}$? | We can obtain an explicit upper bound using the identity (where $p$ is restricted to primes)
$$\frac{n}{\phi(n)}=\prod\_{p\mid n}\left(1+\frac{1}{p-1}\right)=\sum\_{d\mid n}\frac{\mu^2(d)}{\phi(d)}.$$
For $X\geq 1$, the above identity implies that
\begin{align\*}\sum\_{n\leq X}\frac{\tau\_k(n)}{\phi(n)}
&=\sum\_{n\leq ... | 5 | https://mathoverflow.net/users/11919 | 336696 | 143,750 |
https://mathoverflow.net/questions/336695 | 5 | Let $(C, \otimes, I)$ be a symmetric monoidal abelian category. For an object $M$ in $C$ with a map $\rho : M \rightarrow I$, we can form the chain complex $M\_\*$ where $M\_n = \otimes\_{i = 1}^n M$ and $d : M\_n \rightarrow M\_{n-1}$ is
$$ \sum\_{i = 1}^n (-1)^i 1 \otimes \cdots \otimes \rho \otimes \cdots \otimes 1... | https://mathoverflow.net/users/30211 | Simplicial Complex Induced by a Morphism | Have you looked at Illusie's *Complexe Cotangent et Deformations I*? Specifically Section 1.3, "The Theorem of Dold-Puppe" where there's a fairly formal formula for the simplicial object to which your complex corresponds. Also, Section 1.5, "The Standard Simplicial Resolution" which is Illusie's name for the Bar constr... | 4 | https://mathoverflow.net/users/39777 | 336698 | 143,751 |
https://mathoverflow.net/questions/336682 | 7 | It is known that a complex analytic function defined on an annulus, say, takes its maximum on the boundary. Does an analogue hold for $p$-adic analytic functions?
More precisely suppose we have a doubly infinite power series $f(z) = \sum\_{n\in \mathbb{Z}}a\_n z^n $ with coefficients $a\_n \in K$ where $K$ is a fini... | https://mathoverflow.net/users/142190 | P-adic functions on annuli | $\def\bQ{\mathbb{Q}}\def\bF{\mathbb{F}}\def\bZ{\mathbb{Z}}$This is false as stated because of the following important difference between $K$ and $\mathbb{C}$: the former is not algebraically closed. For example. consider $K=\bQ\_p(p^{1/k})$ with $k>1$ and the polynomial $$f(z)=z\prod\limits\_{a\in\bF\_p^{\times}}(z-[a]... | 10 | https://mathoverflow.net/users/39304 | 336699 | 143,752 |
https://mathoverflow.net/questions/336701 | 8 | The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times".
An example would be the existence of an unbounded class of inaccessible cardinals.
Now there are many examples of large cardinals $\kappa$ whose existence guarantees an unbounded ... | https://mathoverflow.net/users/94232 | "Bootstrapping" an unbounded class of inaccessible cardinals | $\kappa$ is superhuge if for any $\gamma$, there exists $j: V\to M$ such that $$crit(j)=\kappa,$$ $$\gamma<j(\kappa)$$ and $${}^{j(\kappa)}M\subset M.$$ But $j(\kappa)$ (inaccessible in M) must be inaccessible in $V$ as well.
| 10 | https://mathoverflow.net/users/119731 | 336705 | 143,755 |
https://mathoverflow.net/questions/336716 | 6 | Is it true that for every oriented vector bundle with odd-dimensional fibers, there is always a global section that vanishes nowhere?
| https://mathoverflow.net/users/140203 | Oriented vector bundle with odd-dimensional fibers | No.
Let us consider some vector bundles over $S^4$. Since $S^4$ is simply connected all vector bundles are oriented and rank $k$ vector bundles over $S^4$ are classified by $\pi\_{3}(SO(k))$ by the clutching construction (I am glossing over basepoint issues here, but I think it is correct in this setting). Now $\pi\_... | 17 | https://mathoverflow.net/users/12156 | 336722 | 143,758 |
https://mathoverflow.net/questions/336224 | 6 | The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of Neumann and Dirichlet heat kernels on inner uniform domains **by showing the Poincaré inequality and the volume doubling pr... | https://mathoverflow.net/users/68463 | Volume doubling, uniform Poincaré, counterexample | The answer is yes. The inequality (\*) is true. Eriksson-Bique, et al. [4] proved that on power cusp domains $M^{1,p}=W^{1,p}$, where $W^{1,p}$ is the classical Sobolev space and $M^{1,p}$ is the space of all $u\in L^p$ such that
$$
(1)\qquad |u(x)-u(y)|\leq d(x,y)(g(x)+g(y))
\quad
a.e.
$$
for some $0\leq g\in L^p$.
... | 3 | https://mathoverflow.net/users/121665 | 336730 | 143,760 |
https://mathoverflow.net/questions/335370 | 6 | *Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure.*
I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(X,d,\mu)$ which are defined as the collection of functions $f \in L^p\_{loc}(X)$ for which there exists some $g \in L^... | https://mathoverflow.net/users/36886 | Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space | The question is not stated in a very clear manner, but nevertheless, the answer is: **no**.
**Separability.** The space $M^{1,p}(X,d,\mu)$ is not separable even if $X$ is the standard ternary Cantor set, $d$ is the Euclidean metric $d(x,y)=|x-y|$ and $\mu$ is the natural Hausdorff measure. This was proved in [R]. The... | 5 | https://mathoverflow.net/users/121665 | 336731 | 143,761 |
https://mathoverflow.net/questions/336733 | -2 | I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B\_t \in A\}$. It seems to me that $T$ depends on random variable $\omega$ in the measure space $\Omega$, so, my question is the previous definition i... | https://mathoverflow.net/users/124426 | Brownian motion and Durret book | Yes: assuming $B\_t$ is a random variable, $T$ is also a random variable, and the inf is done pointwise with respect to the sample space.
| 0 | https://mathoverflow.net/users/13650 | 336734 | 143,762 |
https://mathoverflow.net/questions/336721 | 2 | I'm currently reading a paper of Georges Gras on the Reflection Principle.
The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting, but I want to get some practical motivation (from mathematicians working on the theory, or number theorists who use th... | https://mathoverflow.net/users/123226 | Motivation to study the order theory (ring theory) | Below are two [**Edit:** four] examples where orders in central simple algebras are useful. I might add more when I have the time.
Before giving them, let me first motivate in general why one would consider orders, and maximal orders in particular: They generalize rings of integers in field extensions.
In more det... | 3 | https://mathoverflow.net/users/86006 | 336743 | 143,764 |
https://mathoverflow.net/questions/336735 | 5 | Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch\_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes.
The standard ("Quillen") projective model structure on $\Ch\_{\ge 0}(R)$ has quasi-isomorphisms for weak equivalence, monomorp... | https://mathoverflow.net/users/36146 | Strøm model structure on nonnegatively graded chain complexes | Yes. This is contained in Section 6 of [this](https://arxiv.org/abs/math/0011216) paper by Christensen and Hovey. In the bounded case, *any* projective class gives rise to a model structure. In Section 1.4 they introduce a projective class whose weak equivalences are the chain homotopy equivalences. They don't discuss ... | 3 | https://mathoverflow.net/users/11540 | 336745 | 143,765 |
https://mathoverflow.net/questions/336741 | 8 | As far as I know, formulae involving rationals and basic arithmetic ($+$, $-$, $\cdot$ and $/$) have decidable equality. Is this still the case if we add floor rounding (or integer division)?
Define a "basic formula" by the following grammar (in Backus-Naur form):
$$
\begin{matrix}
e &= &q \\
&| &v \\
&| &e + e \... | https://mathoverflow.net/users/13767 | Is equality of formulas with floor rounding or integer division decidable? | Equality of formulas is undecidable. To prove this it is enough to show that an algorithm for equality of formulas would enable us to determine whether or not a multivariate polynomial with integer coefficients vanishes at at least one integer point, because the latter condition is undecidable according to the negative... | 11 | https://mathoverflow.net/users/5229 | 336748 | 143,766 |
https://mathoverflow.net/questions/336754 | 5 | This is related to [this](https://mathoverflow.net/questions/151292/number-of-curves-over-a-finite-field) question. I learnt about moduli problem mainly with the book Harris and Morrison. Therefore, I have only seen the construction of moduli spaces $M\_{g}$ over $\mathbb{C}$. But now I want to change the base field. I... | https://mathoverflow.net/users/90295 | Moduli of curves over finite field | The constructions of the Deligne–Mumford stack $\mathscr M\_g$ and its coarse moduli space $M\_g$ are very similar, and Deligne–Mumford's original article [DM69] is surprisingly readable. Note that Deligne and Mumford write $\mathscr M\_g$ (resp. $M\_g$) for what is now commonly known as the moduli of *stable* (rather ... | 10 | https://mathoverflow.net/users/82179 | 336763 | 143,770 |
https://mathoverflow.net/questions/336758 | 7 | Let $ S = \operatorname{Spec}A $ be an affine scheme, $ f : E \to S $ an elliptic curve and $\mathscr{I}$ the ideal sheaf of the $0$-section.
(This is invertible since the section defines the effective relative Cartier divisor.)
Assume that $f\_\* \Omega\_{E/S}, f\_\*\mathscr{I}^n$ are free over $\mathscr{O}\_S$.
($... | https://mathoverflow.net/users/128235 | Formal completion of an elliptic curve along the $0$ sectioin and the formal expansion of functions | Yes, what you are saying is true, at least over an algebraically closed base $k$ of characteristic $0$. In fact, all you need is that $S$ is affine and that the normal bundle $I/I^2$ is trivial (as a bundle over $S$), which in your case is equivalent to $f\_\*\Omega(E/S)$ being free. The rest follows essentially from d... | 4 | https://mathoverflow.net/users/7108 | 336764 | 143,771 |
https://mathoverflow.net/questions/336768 | 1 | Suppose we have a set $B\subseteq 2^\omega\times\omega^\omega$ and a sequence $(x\_n)$ in $2^\omega$ such that for each $n$, Player I (the one trying to get into the payoff set) has a winning strategy in the Gale-Stewart game where the payoff set is given by the corresponding slice: $B\_{x\_n}=\{f\in\omega^\omega:(x\_n... | https://mathoverflow.net/users/16107 | Convergence and winning strategies | After thinking about Jing and William's answers, as well as something else I already had in mind, here is a clopen counterexample:
Let $C\subseteq 2^\omega\times\mathbb{Z}^\omega$ be the set of all $(x,f)$ such if $f(0)>0$ and $s$ is the initial segment of $x$ having length that of the binary expansion of $f(0)$, the... | 0 | https://mathoverflow.net/users/16107 | 336773 | 143,775 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.