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|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/287172 | 11 | I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using Sagemath to perform some matrix rank computations. It turns up a few bizarre results occasionally.
For example, I had to f... | https://mathoverflow.net/users/7558 | How do computer algebra packages like Sagemath implement rank of a matrix | I don't know what algorithm Sage actually uses, but computing rank over the integers is fun and easy: [Complexity of computing matrix rank over integers](https://mathoverflow.net/questions/63278/complexity-of-computing-matrix-rank-over-integers) . It is NOT so easy if you want good running time, and for that there are ... | 8 | https://mathoverflow.net/users/11142 | 287204 | 126,770 |
https://mathoverflow.net/questions/287170 | 2 | This question is pretty self contained in the title. Let $G$ be a group regarded as a category with a single object and consider a functor $F:G \to Cat$. This exactly the data of a $G$-category that I will denote by $\mathcal{C}$. This has an obvious notion of $H$-fixed points category, $\mathcal{C}^H$ for $H \leq G$.
... | https://mathoverflow.net/users/117760 | $H$-objects of a category $\mathcal{A}$ as $H$-fixed points of a $G$-category | Yes, take $C = A$ with the trivial action of $G$. I will write $BG$ for the one-object category with automorphisms $G$; this is really a different object from $G$ and really should be indicated with different notation. Then we have
$$BG \cong \text{pt}/G$$
meaning that $G$ is the homotopy quotient of a point by the... | 3 | https://mathoverflow.net/users/290 | 287205 | 126,771 |
https://mathoverflow.net/questions/287189 | 5 | There are essentially two ways to impose extentionality on a type theory (I know, it is not very fashionable to impose extentionality these days, but please, bear with me) you can either have a "propositional extentionality axiom" like UIP (uniqueness of identity proof) which says that every two inhabitant of Id(x,y) a... | https://mathoverflow.net/users/22131 | Propositional vs Definitional extentionality in type theory | Martin Hofmann proves in [his thesis](http://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-327/) (theorem 3.2.5) that whenever we have a type $\Gamma \vdash A$ in intensional type theory (ITT) and a term $|\Gamma| \vdash a : |A|$ in ETT there is a term $\Gamma \vdash a' : A$ such that $|\Gamma| \vdash |a'| \equiv a : |A... | 3 | https://mathoverflow.net/users/62782 | 287208 | 126,773 |
https://mathoverflow.net/questions/282547 | 13 | The question may be trivial, but has eluded me, may be it is more appropriate for mathstack-exchange.
Let $B$, $C$ be boolean algebras and $i:B\to C$ be an homomorphism.
By Stone duality to each such $i$ corresponds a continuos map $\pi:St(C)\to St(B)$ defined by $G\mapsto i^{-1}[G]$ for any $G$ ultrafilter on $B$
(w... | https://mathoverflow.net/users/16645 | A curiosity on complete homomorphisms of boolean algebras | An example in which $i$ is a complete homomorphism but $\pi$ is not an open map:
Notation: For any Boolean algebra $A$ and any $a$ in $A$, $S(a)$ is the set of all ultrafilters $F$ such that $a$ is in $F$. If $f:A\to B$ is a homomorphism, then $f^d: \operatorname{ult}(B)\to\operatorname{ult}(A)$ is its dual, assigning ... | 8 | https://mathoverflow.net/users/90095 | 287210 | 126,775 |
https://mathoverflow.net/questions/287200 | 7 | This problem may be an embarrassing one, but I could not prove it even for the $1$ dimensional case. Here is the problem:
>
> **Question 1.** $M$ is a compact $n$-dimensional smooth manifold in $R^{n+1}$. Take a point $p\notin M$. Prove there is always a line $l\_p$ pass $p$ and $l\_p\cap M\neq \emptyset$, and $l\_... | https://mathoverflow.net/users/91939 | Could we always find a line to intersect transversally with a given compact manifold? | For the codimension 1 case.
===========================
### Using Thom transversality theorem.
Consider the maps $f\_s:\mathbb{R} \to \mathbb{R}^n$ parametrized by $s \in S^{n-1}$ and given by $f\_s(t) = p + t \cdot s$. The map $F(s,t) = f\_s(t)$, $F:S^{n-1} \times \mathbb{R} \to \mathbb{R}^n$ is clearly transverse... | 5 | https://mathoverflow.net/users/48261 | 287215 | 126,779 |
https://mathoverflow.net/questions/287206 | 10 | A space $X$ (by which I mean a CW complex) is **acyclic** if its reduced singular homology $\tilde H\_\ast(X;\Bbb Z)$ is trivial in all degrees.
A discrete group $\pi$ is said to be acyclic if its classifying space $B\pi$ is acyclic.
A space $X$ is **aspherical** if its universal cover is contractible. A space $X$... | https://mathoverflow.net/users/8032 | Acyclic aspherical spaces with acyclic fundamental groups | Yes, such things exist.
Take any finitely presented infinite acyclic group $G$, for example, Higman's group.
It is a theorem by Kervaire (''Smooth homology spheres and their fundamental groups'') that for each $n \geq 5$, there is an integral homology sphere $M^n$ with fundamental group $G$. Consider $X=M-\ast$. The i... | 16 | https://mathoverflow.net/users/9928 | 287218 | 126,782 |
https://mathoverflow.net/questions/287191 | 3 | Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p\_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X\_i,Y\_i)\}\_{i=1}^{N}$. Define $p\_{X0} = p\_{XY}(x,0)$ and $p\_{X1} = p\_{XY}(x,1)$. Define deterministic function $f: \mathbb{R}^d \mapsto \{0,1\}$. Define binary variable $Z\_i = \mathbb{1... | https://mathoverflow.net/users/74156 | Concentration inequality for sum of iid random variables that involve KL distance | Consider the sub-probability measures $\mu\_0$ and $\mu\_1$ defined by the conditions $\mu\_0(A):=P(Y=0,X\in A)$ and $\mu\_1(A):=P(Y=1,X\in A)$ for Borel sets $A\subseteq\mathbb R$, so that $\mu:=\mu\_0+\mu\_1$ is the probability distribution of $X$. Let
\begin{equation\*}
p:=p\_f:=P(Y\ne f(X))=EZ.
\end{equation\*}
... | 4 | https://mathoverflow.net/users/36721 | 287221 | 126,784 |
https://mathoverflow.net/questions/287224 | 9 | Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $.
The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials.
With the standard multiplication, $X$ is an Abelian semigroup with cancellation property.
Let $G$ be the Grothendieck group associated with $X$.
Is the... | https://mathoverflow.net/users/36688 | Semi group of polynomials which all roots lie on the unit circle | A complex polynomial is uniquely determined by its set of roots together with multiplicities. This means that the semigroup of your polynomials is freely generated by the set of point on the unit circle, aka by $\{z-k | k:\mathbb C, |k|=1\}$. Its Grothendieck group is the free abelian group on the continuum of generato... | 19 | https://mathoverflow.net/users/10605 | 287225 | 126,785 |
https://mathoverflow.net/questions/287194 | 20 | This is a question of mathematical writing. Let me know if it would be better suited to academia.SE.
I am writing a paper in invariant theory. It uses some slightly heavy commutative algebra. There are a few points where I use facts of which I am convinced, and I believe they are widely known, but I am not sure how t... | https://mathoverflow.net/users/12419 | Using a known result without a specific reference | I agree with RvDdB's answer, but want to add some thoughts. To quote from [another SE answer of mine](https://academia.stackexchange.com/a/66329/19607):
*First, my general philosophy is that one should try to make papers reasonably accessible to young people who have not spent months or years working on this specific... | 12 | https://mathoverflow.net/users/6518 | 287229 | 126,787 |
https://mathoverflow.net/questions/287233 | 8 | **Question 1**:Do there exist smooth projective Fano varieties over $\mathbb{C}$ with topological Euler characteristic $0$?
**Question 2**:If so what is the lowest dimension in which such examples occur?
| https://mathoverflow.net/users/99732 | Fanos with $\chi_{top} = 0$ | Every odd $n$-dimensional smooth (2,2)-complete intersection has Euler characteristic 0. I'll leave this website as evidence (<http://pbelmans.ncag.info/cohomology-tables/>), although this result should also follow from a careful Chern class calculation of $c\_n(T\_X)$.
When $n\ge 3$ these are Fano varieties. By Remy... | 10 | https://mathoverflow.net/users/117833 | 287236 | 126,789 |
https://mathoverflow.net/questions/287154 | 14 | Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This refers to the fact that they can be glued.
For instance, suppose that $X$ is a complex analytic space and $P\_1^{\bul... | https://mathoverflow.net/users/59235 | Counterexamples to gluing complexes of sheaves | One can ask whether the derived category forms a stack (of triangulated categories). The answer is no:
Let $S^2$ denote the two-sphere (or $\mathbb{P}^1$ if you prefer) and let $k$ denote a ring of coefficients (e.g. $\mathbb{Z}$). Let $k\_{S^2}$ denote the constant sheaf on $S^2$. Consider a map $\alpha: k\_{S^2}[-2... | 7 | https://mathoverflow.net/users/919 | 287238 | 126,790 |
https://mathoverflow.net/questions/287235 | 2 | I am reading [this](https://arxiv.org/abs/hep-th/0011256)
paper and am having a very hard time understanding the metric given on page six. The manifold we are working in "has the same topology as $\mathbb{R}^4 \times S^3$." The metric is given by
$$ds^2 = \alpha^2 dr^2 + \gamma^2 (\omega'^a)^2 + \beta^2(\omega^a - \fr... | https://mathoverflow.net/users/nan | Understanding a G2 metric | If you are unfamiliar with left invariant 1-forms, you might read Stillwell, **Naive Lie Theory**. The left invariant 1-forms are the 1-forms which are invariant when you think of the 3-sphere as $SU(2)$ (or as the unit quaternions), and you left translate on $SU(2)$ (or the unit quaternions) as a Lie group. To be spec... | 8 | https://mathoverflow.net/users/13268 | 287246 | 126,792 |
https://mathoverflow.net/questions/287249 | -5 | Let $G=(V,E)$ be a finite, simple, undirected graph. We say $G$ has the *singleton coloring class property* (SCCP) if for all $v\_0\in V$ there is a vertex coloring $c:V\to\{1,\ldots,\chi(G)\}$ such that no vertex other than $v\_0$ receives the same color as $v\_0$, that is, $c^{-1}\big(\{c(v\_0)\}\big) =\{v\_0\}$.
I... | https://mathoverflow.net/users/8628 | Graphs where every vertex can be its own color class | For a quick counter example look at odd cycles.
These graphs are called [(vertex) critical](https://en.wikipedia.org/wiki/Critical_graph). You can look into the Hajós construction, which produces larger critical graphs from smaller ones without changing the chromatic number.
| 2 | https://mathoverflow.net/users/2384 | 287251 | 126,793 |
https://mathoverflow.net/questions/286629 | 1 | I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form
$$
\|u\|\_{H^{1+\epsilon}(\Omega)} \le C \|f\|\_{L^2(\Omega)}
$$
for some $\epsilon >0 $ where $u$ is the weak solution to either of the following PDEs.
\begin{align\*}
\nabla\cdot\nabla u &= f\quad x\in \Omega\... | https://mathoverflow.net/users/76602 | Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimensional polyhedron? | As discussed in the comments, such a result can be found in [Jochmann's "An $H^s$-Regularity Result for the Gradient of Solutions to Elliptic Equations with Mixed Boundary Conditions"](http://www.sciencedirect.com/science/article/pii/S0022247X99965186).
| 3 | https://mathoverflow.net/users/85906 | 287252 | 126,794 |
https://mathoverflow.net/questions/287237 | 7 | In his descent II Bourbaki paper, Grothendieck lays out some criteria for a functor $F:C\rightarrow\mathrm{Set}$ to be strictly pro-representable; i.e. isomorphic to an inductive limit of representable functors $Hom(X\_i,-)$ where X\_i form a projective system with transition morphisms which are epimorphims. Assume $C$... | https://mathoverflow.net/users/37110 | Pro-representability | If I try to prove 1:
Take objects $A,B$ given by a morphism $f:x\rightarrow y$ in $C$ and objects $u\in F(x)$, $v\in F(y)$ such that $v=F(f)(u)$. Consider another morphism $g:x\rightarrow y$ such that $v=F(g)(u)$. Then take $h:z\rightarrow x$ to be the equalizer of $f,g$. If I understand correctly $h$ is a typical ex... | 4 | https://mathoverflow.net/users/79968 | 287264 | 126,797 |
https://mathoverflow.net/questions/287166 | 6 | I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109
*Theorem (8.6.1)*. Let $X \subset \mathbb{R}^n$ be an open set, and let $P$ be an elliptic operator with constant coefficients. Then
$$\mathrm{singsupp}(u)=\mathrm{singsupp}(Pu)$$
As... | https://mathoverflow.net/users/86432 | The elliptic regularity theorem for differential operators with variable coefficients | I was able to find the right reference. This construction is present in the book "Introduction to pseudo-differential and Fourier Integral volume 1" by J.F. Treves. More specifically the sections are as follows
1. Parametrices of Elliptic Equations
2. Definition and Continuity of the "Standard" Pseudodifferential Ope... | 4 | https://mathoverflow.net/users/86432 | 287265 | 126,798 |
https://mathoverflow.net/questions/286880 | 4 | The [paper](https://www.cambridge.org/core/journals/compositio-mathematica/article/cohomologie-non-ramifiee-sur-une-courbe-p-adique-lisse-unramified-cohomology-on-a-smooth-p-adic-curve/282DCDCFCF1D6B8B07CE18B5D46AAFCA) of Ducros "Cohomologie non-ramifiée sur une courbe p-adique lisse" mentions a theorem (1.21) about th... | https://mathoverflow.net/users/2234 | finite number of vertices of the polyhedron of variation of an invertible function on a Berkovich curve | It is not true in general that $P$ is finite. To see this, take an open disk $D$ and a non-zero function $f$ on it with infinitely many zeroes. In this case, your polyhedron of variation is infinite: it is a tree where all those zeroes are leaves. If you insist on having an invertible function, just remove the zero loc... | 3 | https://mathoverflow.net/users/4069 | 287279 | 126,803 |
https://mathoverflow.net/questions/287273 | 2 | Let $X$ be a smooth and proper complex analytic space, or a Kahler complex manifold. Is the image $\Lambda$ of the finitely generated abelian group $H^{2i}(X,\mathbf{Z})$ into $J := H^{2i}(X,\mathbf{C})/F^iH^{2i}(X,\mathbf{C})$ a full lattice?, where $F^{\bullet}$ is the Hodge filtration. In other words, is $J/\Lambda$... | https://mathoverflow.net/users/nan | Full lattice images and Hodge decomposition | No, except in some exceptional cases. Take for instance $i=1$. Then the kernel of the composition $H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C})\rightarrow J$ is the Neron-Severi group $NS(X)\subset H^2(X,\mathbb{Z})$; its dimension is the Picard number $\rho $, and $\operatorname{rk}\Lambda=b\_2-\rho $.
If $\Lambda ... | 2 | https://mathoverflow.net/users/40297 | 287285 | 126,804 |
https://mathoverflow.net/questions/287286 | 6 | In 1926 Kneser showed that homeomorphisms of $\mathbf{S}^2$ admit a retraction into the orthogonal group $O(3)$. Smale extended this result to Diffeomorphisms of $\mathbf{S}^2$ in 1958; however, in that paper he assumes that the diffeomorphisms are of class $C^k$ for $k\geq 2$. So my question is: Has anyone established... | https://mathoverflow.net/users/68969 | Smale's theorem for $C^1$ diffeomorphisms of the sphere | Let $M$ be a closed $C^{\infty}$-manifold, then for any $k>0$ the canonical inclusion:
$$\mathrm{Diff}^{C^{\infty}}(M)\subset \mathrm{Diff}^{C^{k}}(M)$$
is a homotopy equivalence.
Embed $M$ in an euclidean space $\mathbb{R}^n$ as a $C^{\infty}$-submanifold and build a smoothing operator by using the convolution product... | 9 | https://mathoverflow.net/users/27816 | 287289 | 126,806 |
https://mathoverflow.net/questions/287222 | 1 | Let $X,Y$ be a bipartite graph and $X',Y'$ be two subsets of the vertices. Is there a Hall type theorem for the existence of matchings saturating both subsets simmultaneously?
Clearly necessary conditions are for a saturating matching to exist in $X'\cup Y$ and $X\cup Y '$ but I don't know if this is sufficient.
| https://mathoverflow.net/users/24478 | Hall type theorem for saturations of subsets of bipartite graphs | Your necessary condition is also sufficient. Let $M\_1$ and $M\_2$ be the matchings from $X'$ to $Y$ and from $Y'$ to $X$. The union of $M\_1$ and $M\_2$ is a bipartite graph of maximum degree at most $2$, so has as connected components paths and even length cycles.
On the cycle components, taking alternate edges aro... | 1 | https://mathoverflow.net/users/25485 | 287291 | 126,807 |
https://mathoverflow.net/questions/287298 | 16 | **[Migrated from [Math Stack Exchange](https://math.stackexchange.com/q/1987703/83875)]**
More than a year ago, I posted the following on the Math Stack Exchange.
>
> Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
> fact, the first ten natural numbers) we might "conclude" that "$n$ is
> pr... | https://mathoverflow.net/users/29316 | A conjecture in which both "if" and "only if" are near misses | **False claim:** A Hausdorff topological space is compact if and only if it is sequentially compact.
It's believable if your intuition of Hausdorff spaces comes entirely from metric spaces (where the claim is, in fact, true). However, both directions are false for different reasons:
**Counterexample I:** the Alexan... | 25 | https://mathoverflow.net/users/39521 | 287302 | 126,811 |
https://mathoverflow.net/questions/287300 | 3 | I am looking for a list of primes $\le N$ for which 2 is a primitive root, with $N$ as large as available. I know of a [table of smallest primitive roots for all primes below 1000](http://users.uoi.gr/abeligia/NumberTheory/NT2015/PrimitiveRoots.pdf), from which such a list can be extracted when $N=1000$, but I am inter... | https://mathoverflow.net/users/56920 | list of primes for which 2 is a primitive root | The sequence of primes for which 2 is a primitive root is [OEIS A001122](http://oeis.org/A001122). It contains a [list](http://oeis.org/A001122/b001122.txt) for the first 10.000 entries, which goes up to $N=310091$. I'm not sure if that is sufficient for you, but it's at least a first step.
| 8 | https://mathoverflow.net/users/70594 | 287303 | 126,812 |
https://mathoverflow.net/questions/287281 | 1 | Consider a finite set (of cardinality $\ge 2$) $S \subseteq \mathbb{C}$ and the holomorphic universal covering map $\pi: \ \mathbb{H} \rightarrow \mathbb{C} \setminus S$, where $\mathbb{H}$ denotes the upper half-plane. Take a half-line $l$ starting from an element of $S$ which goes to infinity without intersecting $S$... | https://mathoverflow.net/users/106337 | About some lines on the universal covering of the punctured plane | For the case where $|S|=2$ you can assume that $S=\{0,1\}$. It is then well-known that you can take $\pi$ to be the elliptic modular function $\lambda$. It follows easily that it works equally well to take $\pi(z)=1/\lambda(z)$. This induces a conformal isomorphism $\mathbb{H}/G\to\mathbb{C}\setminus S$, where $G$ is t... | 1 | https://mathoverflow.net/users/10366 | 287306 | 126,813 |
https://mathoverflow.net/questions/287268 | 2 | I have read in a paper that $\mu(0,1)\neq 0$, for the Möbius function $\mu$ on an atomistic (finite, modular) lattice with $0$ and $1$. Could someone provide me with a reference for this claim? Many thanks in advance.
| https://mathoverflow.net/users/117855 | Möbius function on atomistic (modular) lattices | More generally, the Möbius function of a geometric lattice is never 0. Although not stated explicitly in the text of *Enumerative Combinatorics*, vol. 1, 2nd ed., it is a simple consequence of equation (3.33) on page 277, as mentioned in the solution to Exercise 3.100(b). It also follows from Exercise 3.98 (where the d... | 4 | https://mathoverflow.net/users/2807 | 287310 | 126,815 |
https://mathoverflow.net/questions/287312 | 3 | I'm trying to understand the proof of Theorem 2.5.6 (chapter 2.5) in Amnon Pazy, *Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer 1983.
For the direction (a) $\implies$ (b) it would be helpful to have a proposition along the lines of:
An analytic semigroup of bounded lin... | https://mathoverflow.net/users/95408 | Does uniform boundedness carry over from the non-negative real axis to closed sectors of $\mathbb{C}$ for analytic semigroups? | There is a one-dimensional counterexample: Consider the analytic semigroup $z \mapsto e^{iz}$. This semigroup is bounded on the non-negative real line, but it is not bounded on any sector $\Delta\_\delta$.
**EDIT** in response to the comments: One can "modify" each analytic semigroup to obtain the following boundedne... | 6 | https://mathoverflow.net/users/102946 | 287313 | 126,817 |
https://mathoverflow.net/questions/287256 | 12 | Return to Frege's question, What justifies arithmetic? And consider the ur-proposition that counting a finite set always produces the same number, and ask whether this has a logical justification, that is a justification from assumptions which are necessarily true. Note that few of us actually first believed this asser... | https://mathoverflow.net/users/20716 | When we count the same set, must the number always be the same? | Such a model can be constructed using known independence results in bounded arithmetic as follows.
Let $I\Delta\_0(f)$ be a theory in the usual language of arithmetic ($0,S,+,\cdot,<$) augmented by a new function symbol $f$, axiomatized by Robinson’s arithmetic plus induction for all bounded formulas in the expanded ... | 17 | https://mathoverflow.net/users/12705 | 287314 | 126,818 |
https://mathoverflow.net/questions/287315 | 12 | Can someone provide some biographical details, especially the dates of birth and death, about Erich Stiemke? According to the Mathematics Genealogy Project, he obtained his Dr. phil at Universität Berlin in 1914 from Frobenius and Schottky.
| https://mathoverflow.net/users/2807 | Erich Stiemke biography | Erich Stiemke, born 12 April 1892, died in combat on 10 September 1915. His [Ph.D thesis](https://link.springer.com/article/10.1007/BF01283824) was published posthumously in 1925, with the following preface:
>
> Erich Stiemke, born on April 12, 1892, fell on September 10, 1915 in
> the eastern theater of war. On t... | 14 | https://mathoverflow.net/users/11260 | 287318 | 126,820 |
https://mathoverflow.net/questions/285548 | 1 | I asked the following question on math.SE (<https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d>) just over two months ago, and it only received one, rather unsatisfactory to me, answer there. I'm wondering if people here can have a look. Some rela... | https://mathoverflow.net/users/80370 | BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior? | By complementing [Deane Yang](https://mathoverflow.net/users/613/deane-yang)'s strategy with [Jochen Wengenroth](https://mathoverflow.net/users/21051/jochen-wengenroth)'s observations in the related question [Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold](https://ma... | 3 | https://mathoverflow.net/users/80370 | 287324 | 126,824 |
https://mathoverflow.net/questions/287272 | 5 | I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a discrete valuation ring $R$ of mixed characteristic $(0,p)$ and $K$ a closed subgroup of $G$ with corresponding lie algebras $\mathfrak{g}, \mathfrak{k}$ and $X$ be the flag variet... | https://mathoverflow.net/users/111845 | Beilinson-Bernstein localization, equivariant modules | I think the answer to your question might be partly contained in the paper "On irreducible representations of compact p-adic analytic groups" by Ardakov and Wadsley. You can find a link to a pdf on Simon Wadsley's [webpage](https://www.dpmms.cam.ac.uk/~sjw47/).
If you look at Proposition 5.15 there (putting $n=0$), y... | 2 | https://mathoverflow.net/users/106429 | 287330 | 126,826 |
https://mathoverflow.net/questions/287182 | 13 | Let $k \subset K$ be an extension of algebraically closed fields of characteristic $0$ (e.g. $\overline{\mathbb{Q}} \subset \mathbb{C}$).
>
> Let X be a smooth variety over $k$. Then is the natural map
> $$\mathrm{Pic}(X)\,/\,n \to \mathrm{Pic}(X\_K)\,/\,n$$
> and isomorphism for all $n \in \mathbb{Z}$?
>
>
>
... | https://mathoverflow.net/users/5101 | Picard group under base change for algebraically closed fields | $\require{AMScd}$
The following result holds in much more generality than asked by the OP. There is no characteristic zero/smoothness assumption.
>
>
> >
> > **Proposition:** Let $X/k$ be a qcqs scheme with $k$ separably closed. Let $K/k$ be a separable extension with $K$ separably closed. Let $n \in \mathbf{N}$ ... | 10 | https://mathoverflow.net/users/21278 | 287335 | 126,827 |
https://mathoverflow.net/questions/287240 | 2 | Let $E/\mathbb{Q}$ be an elliptic curve and $P \in E(\mathbb{Q})$. If $P$ is divisible by $p$ in $E(\mathbb{Q}\_p)$ for every prime $p$, does it follow that $P$ is a torsion point?
| https://mathoverflow.net/users/61621 | Local-Global divisibility of points on elliptic curves | Chris Wuthrich has already provided an answer when there are infinitely many anomalous primes, with the correction that the condition to be anomalous is $a\_p=1$, i.e. the number of points in the reduction modulo $p$ is $p$. In the case of non-anomalous prime, the condition that $P$ is $p$-adically divisible by $p$ is ... | 4 | https://mathoverflow.net/users/2290 | 287338 | 126,828 |
https://mathoverflow.net/questions/287261 | 1 | This is a question that is a bit outside my usual mathematical comfort zone, but I feel like an expert might know the answer.
---
Recall that a dimension group is an ordered abelian group $G$ with positive cone $G^+$ that arises as a limit of ordered groups of the form $\mathbb Z^d$ with the componentwise order. ... | https://mathoverflow.net/users/29404 | Realizing certain affine functions on Choquet simplices on dimension groups | Yes. Form Aff $(T)$; since $T$ is metrizable, Aff $(T)$ contains a countable dense set. Adjoin $f$ and the constant function $1$ to the countable dense set (creating a larger countable dense set), and let $G$ be the subgroup of Aff $(T)$ generated by this enlarged set. This is of course countable, and dense.
Therefo... | 1 | https://mathoverflow.net/users/42278 | 287340 | 126,830 |
https://mathoverflow.net/questions/287333 | 5 | Let $f,g \in \mathbb{C}[x,y]$.
There is a well-known result, that can be found for example
[here](http://www.math.wayne.edu/~lml/lmlnotes.pdf), pages 19-20, that says the following:
>
> $f,g$ are algebraically dependent over $\mathbb{C}$ if and only if their Jacobian $Jac(f,g):=f\_xg\_y-f\_yg\_x$ is zero.
>
>
>... | https://mathoverflow.net/users/72288 | A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero | The reverse implication is true in a considerably more general setting (Burchnall-Chaundy theory). Namely, for any pair $(U,V)$ of commuting meromorphic coefficient differential operators in one variable of order at least one, there is a two-variable polynomial $P(z,w)$ such that $P(U,V)=0$ (the polynomial evaluation i... | 7 | https://mathoverflow.net/users/5740 | 287345 | 126,833 |
https://mathoverflow.net/questions/275170 | 5 | Let $\mathcal{M}$ be a smooth 1-dimensional Deligne-Mumford stack with finite diagonal, and let $M$ be its coarse moduli scheme (all over some field $k$). Suppose $M$ is also smooth over $k$, and that all local deformation problems for $\mathcal{M}$ are pro-representable (I think this is already implied by the smooth +... | https://mathoverflow.net/users/15242 | How is a universal deformation ring in a 1-dimensional DM stack related to the complete etale local ring of coarse moduli scheme? | I believe that this is true - that is, "$p$ is finite flat totally ramified with ramification index equal to the order of the group of automorphisms of $X\_0$ modulo the subgroup of those which extend to an automorphism of $X^\text{univ}/\mathcal{R}$."
Specifically, let $G$ be the automorphism group of the object $x\... | 2 | https://mathoverflow.net/users/15242 | 287346 | 126,834 |
https://mathoverflow.net/questions/287275 | 8 | Let $E=\mathbb C/\Lambda$ be an elliptic curve,
and let $D\subset E$ be a very small disc.
($D$ is round for the usual flat metric on $E$)
By the main result of [1], there exists a holomorphic immersion $f:E\setminus D \to \mathbb C$.
The image $f(\partial D)$ is a closed curve in $\mathbb C$ that self-intersec... | https://mathoverflow.net/users/5690 | Image of boundary circle under map from punctured elliptic curve to ℂ | I don't know if this is the kind of answer which will satisfy. Write $f$ for the function on $\mathbb{C} - \Lambda$ and $z$ for the coordinate on $\mathbb{C}$. Write $D^{\ast}$ for the punctured disc.
[Forstneric and Ohsawa](https://arxiv.org/abs/1106.0936) show we can take $f$ to be of finite order as $z \to 0$, mea... | 3 | https://mathoverflow.net/users/297 | 287357 | 126,839 |
https://mathoverflow.net/questions/287368 | -3 | The *Hadwiger-Nelson graph* on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E\_n)$ where $$E\_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean distance in $\mathbb{R}^n$.
For $n>1$, is there a [matching](https://en.wikipedia.org/wiki/Matching_(graph_theory))... | https://mathoverflow.net/users/8628 | Does the Hadwiger-Nelson graph have a perfect matching? | You can find such a perfect matching on $\mathbb R$, by taking the pairs $\{x,x+1\}$ whenever $\lfloor x\rfloor$ is even. Next, you can put a copy of this matching on every parallel translate of some line in $\mathbb R^n$ to get a perfect matching in $E\_n$.
| 4 | https://mathoverflow.net/users/2384 | 287370 | 126,842 |
https://mathoverflow.net/questions/287356 | 6 | Let $X$ be the blowing-up of $\mathbb{P}^2\_{\mathbb{Q}}$ at four rational points on a line. Can one show that $X$ has bad reduction at 2? Or does $X$ secretly has good reduction at 2?
What one can be sure is that the closures in $\mathbb{P}^2\_{\mathbb{Z}}$ of any such four points cannot be disjoint over $(2)\in{\rm... | https://mathoverflow.net/users/3332 | bad reduction of a rational surface | You are completely right that for any 4 rational points on the projective line, two have equal reductions mod two. However as ulrich points out, this does not create a singularity of the underlying surface.
The reason is that if we blow up $[1,0,0], [0,1,0],[1,1,0]$ in $\mathbb P^2\_{\mathbb Z}$, say, we can then blo... | 5 | https://mathoverflow.net/users/18060 | 287374 | 126,843 |
https://mathoverflow.net/questions/287167 | 6 | I'm trying to close in on a definitive answer to my own question [BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?](https://mathoverflow.net/questions/285548/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d), and think I hav... | https://mathoverflow.net/users/80370 | Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold | Thanks to Jochen Wengenroth's [comments](https://mathoverflow.net/questions/287167/composition-of-a-smoothing-operator-with-an-l2-bounded-operator-non-compact/287375#comment710638_287167), I can now give the full answer: the idea is that $C\_\mathrm{c}^\infty(M) \hookrightarrow L^2(M, \mathrm{d} \mu\_g) \xrightarrow{G}... | 1 | https://mathoverflow.net/users/80370 | 287375 | 126,844 |
https://mathoverflow.net/questions/287277 | 2 | 1. I have a question about envelopes of surfaces. In a book I am reading the following:
>
> Suppose $S\_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on $x,y$ and the real parameter $a$. Consider also the equation $\partial\_a w(x,y;a)=0$. For a fixed values of $a... | https://mathoverflow.net/users/69441 | 1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones | 1. A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "adjacent" surfaces. If you don't mind infinitesimals this could be rephrased as: $(x\_0,y\_0,z\_0)$ is a point on the env... | 2 | https://mathoverflow.net/users/745 | 287377 | 126,845 |
https://mathoverflow.net/questions/287380 | 2 | Let $G=(V,E)$ be an infinite graph such that $|V| = \kappa$ for some infinite cardinal $\kappa$, and every $v\in V$ has degree $\kappa$. Does $G$ have a perfect matching?
| https://mathoverflow.net/users/8628 | Perfect matchings in infinite graphs | Well, it seems like the following should work:
Let us well order $V$ such that for every $v\in V$ set of $u$ such that $u < v$ has cardinality less then $|V|$. Now we are using transfinite induction to produce matching:
at each step if we are looking at vertex $v$ than if it is already in some pair with one of the ... | 6 | https://mathoverflow.net/users/104330 | 287384 | 126,847 |
https://mathoverflow.net/questions/287372 | 1 | A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.
Let $X$ be a Lindelof, perfectly normal, $\sigma$-space.
>
> Must $X$ be separable?
>
>
>
Thanks very much.
| https://mathoverflow.net/users/39873 | Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable? | If a $\sigma$-space $X$ is Lindelof, then it is paracompact and by Theorem 4.4 of Gruenhage's survey "Generalized Metric Spaces" in the "[Hanbook of Set-Theoretic Topology](http://www.sciencedirect.com/science/book/9780444865809)", $X$ has countable network and hence is hereditarily separable and hereditarily Lindelof.... | 4 | https://mathoverflow.net/users/61536 | 287417 | 126,857 |
https://mathoverflow.net/questions/287329 | 3 | Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}$ is at least $C$ ($f(\mathcal{R}x)$ is a rotation of $f$ on $S^2$). Does there exist $I\neq\mathcal{R}\_0 \in SO(3)$ such... | https://mathoverflow.net/users/42326 | Symmetry of functions on $S^2$ | Note that any eigen-function satisfies your condition for some $C>0$.
It remains to find a non-symmetric one. Take for example
$$f(x,y,z)=10\cdot x^3+y^3+\tfrac1{10}\cdot z^3.$$
| 1 | https://mathoverflow.net/users/1441 | 287432 | 126,858 |
https://mathoverflow.net/questions/287400 | 11 | A recent question [Why do we need model categories?](https://mathoverflow.net/q/287091/41291) reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I even dare not to use the [soft-question](/questions/tagged/soft-question "show q... | https://mathoverflow.net/users/41291 | The cofibration/fibration $\leftrightarrow$ epi/mono confusion | The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the homotopy theory of (-1)-truncated sets. One can also define a similar model structure on simplicial sets, where the weak eq... | 9 | https://mathoverflow.net/users/51164 | 287433 | 126,859 |
https://mathoverflow.net/questions/287420 | 2 | Consider probability measure $\mu\_{XY}$ defined on $\mathbb{R}^d \times \{1,2,3\}$, and sub-probability measures $\mu\_1$, $\mu\_2$, and $\mu\_3$ as $\mu\_1(A):=P(X\in A, Y=0)$ and $\mu\_2(A):=P(X\in A, Y=2)$ and $\mu\_3(A):=P(X\in A, Y=3)$ for Borel sets $A$, so that $\mu:=\mu\_1+\mu\_2+\mu\_3$ is the probability dis... | https://mathoverflow.net/users/74156 | Linking error probability based on total variation | The answer is no. E.g., let $(A\_1,A\_2,A\_3)$ be any Borel partition of $\mathbb{R}^d$ such that $A\_j\ne\emptyset$ for each $j$.
Let $\nu$ be any measure on $\mathbb{R}^d$ such that $\nu(A\_j)=1$ for each $j$.
For each $i$ and $j$ in $\{1,2,3\}$, let the density of the measure $\mu\_i(\cdot)=P(X\in \cdot,Y=i)$ with... | 2 | https://mathoverflow.net/users/36721 | 287436 | 126,860 |
https://mathoverflow.net/questions/287435 | 1 | For a given $k,n$ such that $0<k \leq n/2$, is there a number $N\_{n,k}$, such that if one has $N$ different $k$-dimensional subspaces $V\_1, V\_2,...,V\_N$ in $\mathbb{R}^n$ satisfying:
1) $\bigcap\_{i = 1}^N V\_i = \{0\}$
2) $dim (\bigoplus\_{i = 1}^N V\_i) \geq 2k$.
Then there are at least two subspaces $V\_i,... | https://mathoverflow.net/users/7894 | Maximal number of intersecting subspaces of a finite dimensional vector space | The number $N\_{6,3}$ does not exist:
let $V\_1=\mathrm{span}(e\_1,e\_2,e\_3)$, $V\_2=\mathrm{span}(e\_3,e\_4,e\_5)$, $V\_3=\mathrm{span}(e\_1,e\_6,e\_5)$, and $V\_j=\mathrm{span}(e\_1,e\_3, e\_4+j\cdot e\_5)$ for $j\ge 4$.
| 4 | https://mathoverflow.net/users/24076 | 287439 | 126,861 |
https://mathoverflow.net/questions/287440 | 7 | Presheaves on a category $C$ form the free co-completion, in the sense that every functor from $C$ to a cocomplete category extends in a unique way to the presheaf category.
If $C$ is equipped with a Grothendieck topology, then a similar statement holds if we ask that our cocontinuous extension also send some diagram... | https://mathoverflow.net/users/117964 | Why do sheaves embed in presheaves? | The universal properties you describe (by the way, you need to either restrict to the case that $C$ is essentially small or restrict to what are called small presheaves) tell you what cocontinuous functors out of sheaves or presheaves look like. But the embedding $\text{Sh}(C) \to \text{Psh}(C)$ of sheaves into preshea... | 8 | https://mathoverflow.net/users/290 | 287454 | 126,868 |
https://mathoverflow.net/questions/287458 | 1 | I was redirected from stackoverflow to ask here.
I want to count a number of all paths between two nodes in graph. The graph can contain cycles. I have read a lot of articles about this problem but for DAG.
[Stackoverflow: Number of paths between two nodes in a DAG](https://stackoverflow.com/q/5164719/5645978)
... | https://mathoverflow.net/users/117983 | Number of paths between two nodes | This is still a hard problem (#P-complete), so it almost certainly can’t be done in polynomial time. See Valiant, L. G. (1979). The Complexity of Enumeration and Reliability Problems. SIAM J. Comput., 8, 410-421.
As for the other part of your question, I wonder if you’ve fallen prey to some terminological confusion. ... | 2 | https://mathoverflow.net/users/8217 | 287462 | 126,871 |
https://mathoverflow.net/questions/287466 | -4 | Is there a connected infinite graph $G=(V,E)$ such that $\text{deg}(v) \geq 2$ for all $v\in V$, and $G$ possesses no perfect matching?
| https://mathoverflow.net/users/8628 | Connected infinite graph $G$ with $\delta(G)\geq 2$ and no perfect matching | What did you try? $K\_{32}$ has no perfect matching. So start with any infinite connected graph with minimum degree $2$, pick two vertices $u,v$ and add three new vertices $A,B,C$ each of degree $2$ with edges going to $u$ and $v.$ No matching can have edges on all three of the new vertices.
| 2 | https://mathoverflow.net/users/8008 | 287469 | 126,873 |
https://mathoverflow.net/questions/286501 | 6 | Let $0<a<1,\; \psi\_a(x)=\displaystyle \prod\_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set
$$f\_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi\_a(x).$$
**Question**. Is it true that, for mutually prime $j$ and $m$, where $1<j<m,$ the following inequality holds:
$$f\_{mk}(a^j;x)\leq f\_{jk}(a^m;x) \;\... | https://mathoverflow.net/users/78726 | Inequality for functions on [0,1], continued | OK, here goes.
We start with changing the notation ($z\to 20z^2$, $-z-3\to r$, $20rz\to y$ means that what was denoted by $z$ will be denoted by $20z^2$ from now on, $r$ is $-z-3$ with *new* $z$, so it is $-\sqrt{z/20}-3$ in terms of old $z$, and $y$ denotes $20rz$ with just (re)defined $r,z$).
So, execute the foll... | 6 | https://mathoverflow.net/users/1131 | 287479 | 126,878 |
https://mathoverflow.net/questions/287446 | 8 | The matchstick diagram is a really interesting and intuitive method of representing countable ordinals. However, because of how difficult it is to graphically represent ordinals with it, I started wondering about how large the ordinals can be.
**The "$\omega\_1$ of matchstick diagrams", or $\omega\_1^{md}$, is what I... | https://mathoverflow.net/users/115951 | Formalizations of The Matchstick Diagram Representation of Ordinals | Noah Schweber already gave a satisfactory answer, but let me make an extended comment that also gives an answer (not in the most efficient way, but in a way that is, I hope, both instructive and "constructive" to some extent), with a computer implementation of sorts.
I think that I'm somewhat responsible for the Inte... | 4 | https://mathoverflow.net/users/17064 | 287486 | 126,882 |
https://mathoverflow.net/questions/287461 | 4 | Let $M\in \mathcal{B}(F)^+$ and $S\_1,S\_2\in \mathcal{B}(F)$.
>
> I claim that if $S\_1$ and $S\_2$ are $M$-self adjoint (i.e. $MS\_1=S\_1^\*M$ and $MS\_2=S\_2^\*M$) such that $S\_1S\_2=S\_2S\_1$, then $\exists\,(X,\mu)$, $\varphi\_1,\varphi\_2\in L^\infty(\mu)$ and a unitary operator $U:F\longrightarrow L^2(\mu)$... | https://mathoverflow.net/users/116483 | Generalization to the spectral Theorem of self-adjoint operators | There is a 2x2 matrix counter-example.
$$ M = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, \quad
S\_1 = \begin{pmatrix} 1 & 2 \\ 1 & 4 \end{pmatrix},
S\_2 = \begin{pmatrix} a & 2c \\ c & d \end{pmatrix}. $$
Then
$$ MS\_1 = \begin{pmatrix} 1 & 2 \\ 2 & 8 \end{pmatrix}, \quad
MS\_2 = \begin{pmatrix} a & 2c \\ 2c & 2d... | 9 | https://mathoverflow.net/users/406 | 287490 | 126,883 |
https://mathoverflow.net/questions/287502 | 1 | Is there any "famous" family of trinomials over finite fields?
For example over $F\_2$ we have
$$
f(x)=x^{2\times 3^k}+x^k+1
$$
| https://mathoverflow.net/users/118013 | Family of irreducible trinomials over finite fields | I don't know if it is famous, but there is a family due to Ore:
$x^q -x - a$ is irreducible over $F\_q$ whenever $a \in F\_q^\times.$ This is a result of O. Ore,
*Ore, O.*, [**Contributions to the theory of finite fields.**](http://dx.doi.org/10.2307/1989836), Transactions A. M. S. 36, 243-274 (1934). [ZBL60.0111.... | 1 | https://mathoverflow.net/users/11142 | 287505 | 126,890 |
https://mathoverflow.net/questions/287348 | 10 | Let $X$ be some space, $C^\*(X,R)$ its cochain complex. Then there is a multiplication
$$
\mu : C^\*(X,R) \otimes C^\*(X,R) \rightarrow C^\*(X,R)
$$
inducing the cup product, and a homotopy
$$H : C^\*(X,R) \otimes C^\*(X,R) \rightarrow C^{\*-1}(X,R)$$
"witnessing" the graded commutativity of the cup product, ... | https://mathoverflow.net/users/84144 | What is known about this cohomology operation? | I'm going to refer to $H$ as the cup-$1$ product.
So when $p = |x|$ is odd and $x$ is a cycle, the boundary formula says
$$d(x \cup\_1 x) = 2x^2$$
which is not necessarily zero. In this case, we don't get a cohomology operation unless (e.g.) when $2 = 0$ in $R$, and then the cohomology operation is "mostly like $Sq^{... | 11 | https://mathoverflow.net/users/360 | 287508 | 126,891 |
https://mathoverflow.net/questions/287415 | 6 | Fix $n\geq 2$ and let $S\_n$ be the symmetric group on $n$ letters with identity $e$. We consider elements of $S\_n$ to be bijections $[n]\to [n]$ as well as sequences (one line notation). For $1\leq i<n$, let $s\_i$ be the transposition exchanging $i$ and $i+1$. Consider the following algorithm that successively const... | https://mathoverflow.net/users/62135 | Upper bound on the number of permutations in a set during an algorithm | I have bad news. The (constant for large n for your) upper bound will never drop below 1/4. Thus my prediction in a comment above will hold.
A key observation is that every permutation sits in one of the A\_k, and so you will run through this n factorial times. One way to see this is that by induction every permutati... | 2 | https://mathoverflow.net/users/3402 | 287509 | 126,892 |
https://mathoverflow.net/questions/287503 | 6 | Is there a standard name for the type of singularities a codimension-$1$ subvariety of a smooth algebraic variety has when it looks locally (possibly analytically) like an arrangement of hyperplanes? «Normal crossings» refers to the special situation in which locally it looks like a subarrangement of the Boolean arrang... | https://mathoverflow.net/users/1409 | Singularities at worst like a hyperplane arrangement | [This paper](http://arxiv.org/abs/1706.00956) uses the term "arrangement of smooth, complex algebraic hypersurfaces", or simply, "arrangement of smooth hypersurfaces". To quote: "Our goal here is to further generalize these results to a much wider class of arrangements of hypersurfaces, by which we mean a collection of... | 5 | https://mathoverflow.net/users/17846 | 287510 | 126,893 |
https://mathoverflow.net/questions/287493 | 8 | Fontaine, J.-M.; Illusie, L.
p-adic periods: A survey. (English)
Ramanan, S. (ed.) et al., Proceedings of the Indo-French conference on geometry held in Bombay, India, 1989.
If anyone has the above paper can you please share it? Thank you.
| https://mathoverflow.net/users/109761 | Fontaine, J.-M.; Illusie, L. p-adic periods------Does any one have the following article? | It seems clear that there is no online version available, but a library can probably get a PDF copy through interlibrary loan if that's an option for you. The detailed listing is [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=1274494). Note too that Milne's short MathSciNet review of this two-author survey c... | 6 | https://mathoverflow.net/users/4231 | 287512 | 126,894 |
https://mathoverflow.net/questions/287211 | 3 | May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references?
In particular, let $S$ be a scheme that is torus invariant with finitely many zero and one dimensional orbits. What can you say about the singularity at a fixed point if the number of torus stable curves th... | https://mathoverflow.net/users/7780 | Singularity of torus fixed points from combinatorial data | I would recommend looking at Brion's paper "Rational Smoothness and Fixed Points of Torus Actions" (mostly section 1). A point in a variety is defined to be rationally smooth if local cohomology with constant coefficients is the same as for a point of a smooth variety. This is weaker than smoothness, yet it implies tha... | 4 | https://mathoverflow.net/users/2384 | 287519 | 126,897 |
https://mathoverflow.net/questions/287405 | 3 | I am looking for a reference of the proof of the above claim.
Basically it's the following claim:
>
> Consider a Wigner matrix $X\_N$ satisfying $r\_k\le k^{Ck}$ for some constant $C$ and all positive integers $k$. Then, $\lambda\_N^N$ converges to $2$ in $L^p$ norm.
>
>
>
where $r\_k$ is defined as: $r\_k :... | https://mathoverflow.net/users/13904 | Necessary and Sufficient Conditions for $L^p$ Convergence of the Largest Eigenvalue of a Wigner Matrix | I believe this follows from the standard estimates that show the convergence. For example, look at the proof in Anderson-Guionnet-Zeitouni's book, page 24.
Indeed, from the last display there, you get that
$P(\lambda\_N^N>(2+\delta))\leq e^{-\delta N^c}$ for an appropriate $c$, which is more than enough to conclude th... | 2 | https://mathoverflow.net/users/35520 | 287539 | 126,907 |
https://mathoverflow.net/questions/287391 | 4 | I don't know if this is an adequate question for MO. But I cannot understand many aspects of the said paper
<https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf>
by Krähmer.
He writes
$$S:=\{\psi \in \mathbb{C}\_q[G] \otimes \Sigma\_{2m}| X \rhd \psi = \sigma(S(X)) \psi \; \forall X \... | https://mathoverflow.net/users/88855 | Question on a paper by U. Krähmer ("Dirac operators on quantum flag manifolds") | The map $\sigma$ is a representation of $U\_q(\mathfrak{l})$ on the vector space $\Sigma\_{2m}$, let's say for the moment just any such representation.
To answer the second question, $\sigma(S(X))\psi$ should really be thought as $(1\otimes\sigma(S(X)))\psi$. Let $\psi$ be decomposable of the form $\psi\_1\otimes\psi... | 2 | https://mathoverflow.net/users/6032 | 287543 | 126,908 |
https://mathoverflow.net/questions/278080 | 1 | Edelstein and Kelly theorem states the following.
Let $A$, $B$ and $C$ be $3$ nonempty finite subsets of points in $\mathbb{R}^n$ such that affine-span $(A \cup B \cup C)$ has dimension at least $4$ and
$A \cap B \cap C$ is empty. Then there exists a line intersecting exactly $2$ of the sets $A$, $B$, $C$.
Where c... | https://mathoverflow.net/users/31356 | A proof of Edelstein and Kelly theorem | Thank to Victor:
cms.math.ca/openaccess/cjm/v18/cjm1966v18.0375-0380.pdf
| 0 | https://mathoverflow.net/users/31356 | 287545 | 126,909 |
https://mathoverflow.net/questions/275891 | 9 | [Kelly's theorem](https://link.springer.com/article/10.1007%2FBF02187687) states:
Every finite point set of complex space such that the line joining any two points from this set contains at least one more point from this set (every Sylvester-Gallai configaration) is confined to the plane.
The proof is rather short,... | https://mathoverflow.net/users/31356 | Kelly's theorem about a resolution of the Sylvester–Gallai problem | I have found: <https://arxiv.org/abs/1211.0330>
Also <https://arxiv.org/pdf/math/0403023.pdf> (thank to Mike Miller).
| 1 | https://mathoverflow.net/users/31356 | 287546 | 126,910 |
https://mathoverflow.net/questions/287537 | 2 | Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest eigenvalue of $A'A$?
| https://mathoverflow.net/users/118039 | Expected value of the largest singular value of a random matrix with entries in $N (0,1)$ | If $A$ is a Gaussian random matrix as you describe, then the ensemble of matrices given by $A^TA$ is known as the Wishart ensemble, or the Laguerre ensemble. It has been extensively studied, and you can find information in standard books about random matrix theory.
The average of the largest eigenvalue of $A^TA$ is ... | 5 | https://mathoverflow.net/users/78061 | 287555 | 126,911 |
https://mathoverflow.net/questions/287389 | 5 | I am trying to clarify how the archimedean admissible dual is classified in the $G=GL(n, \mathbf{R})$ case.
Fix a semistandard Levi subgroup $M$ in $G$, $\delta$ a square-integrable representation of $M^1$, and $\nu \in \mathfrak{a}\_\mathbf{C}^\star / W$, where $\mathfrak{a}\_\mathbf{C}^\star$ is the dual of the com... | https://mathoverflow.net/users/43737 | Archimedean Langlands classification | It is not necessarily the case that $\mathfrak a\_\mathbb C$ is a Cartan subalgebra of $\mathfrak g$. In general a Cartan subalgebra of $\mathfrak g$ has the form $\mathfrak h\_\mathbb C=\mathfrak t\_\mathbb C\oplus \mathfrak a\_\mathbb C$. The infinitesimal character is an element of $\mathfrak h\_\mathbb C^\*$, and $... | 4 | https://mathoverflow.net/users/6030 | 287557 | 126,912 |
https://mathoverflow.net/questions/287385 | 11 | suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero.
One has the de Rham complex of analytic differential forms on $X$, $\Omega^{\bullet}\_{X/k}$, say on $X\_{et}$.
We call $C$ the sheaf $\ker(\mathcal{O}\_X\xrightarrow{d}\Omega^1\_... | https://mathoverflow.net/users/nan | p-adic Poincaré Lemma | In the simplest case where $X$ is smooth and projective, and $k$ is discretely valued, then the answer to Q3 should be no.
**EDIT**: While waiting for the bus I realized there is a technical error here, which is that the etale topos of the Berkovich space is *not* the etale topos of the rigid/adic space. Instead it i... | 4 | https://mathoverflow.net/users/48362 | 287561 | 126,914 |
https://mathoverflow.net/questions/287465 | 11 | Are there known examples of finitely generated groups $G$ and $H$ that are quasi-isometric but do not admit finite-index subgroups $G'<G$ and $H'<H$ such that both $G'$ and $H'$ admit proper and cocompact actions on the same geodesic metric space $X$?
For context: it is known and not hard to see that groups are quasi... | https://mathoverflow.net/users/5339 | Quasi-isometric groups without common virtual geometric model | The question was already answered by ThiKu and YCor.
I am adding this just to make a conceptual remark.
The OP asks about groups which are QI, but not in an "elementary fashion".
As YCor pointed out in his answer, if $G$ acts properly and cocompactly on a geodesic metric space $X$ then it could be seen as a cocompact... | 7 | https://mathoverflow.net/users/89334 | 287568 | 126,916 |
https://mathoverflow.net/questions/287523 | 4 | It is mentioned in [wikipedia](https://en.wikipedia.org/wiki/Pythagorean_quadruple#Relationship_with_quaternions_and_rational_orthogonal_matrices) that every single orthogonal $3 \times 3$ rational matrix is of the form
$$\dfrac{1}{m^2+n^2+p^2+q^2}\begin{pmatrix} m^2+n^2-p^2-q^2 & 2np-2mq & 2mp+2nq \\ 2mq+2np & m^2-n... | https://mathoverflow.net/users/116776 | Why is this mapping surjective? | This is an extended comment. (The answer is contained in the comment of Will Jagy:
indeed it is a very simple argument that every rational orthogonal matrix is obtained from a rational quaternion).
The paper of J. Cremona referred in the question gives an incorrect formula for the recovery of the quaternion. The corr... | 4 | https://mathoverflow.net/users/25510 | 287579 | 126,923 |
https://mathoverflow.net/questions/282621 | 10 | *This is a bit of an odd question, so I've included the motivation below the fold.*
Throughout we work in ZFC+"$\omega\_1^r$ is countable for all $r\in\mathbb{R}$:"
Say that a set $X\subset\omega\_1$ is *really climbable* if $X$ is countable *(that's the silly case)* or there is some real $r$ and some $f\in L[r]$ s... | https://mathoverflow.net/users/8133 | Climbing up subsets of $\omega_1$ using reals | "Every set is really climbable": this contradicts AC. By AC construct a set $X\subseteq \omega\_1$ such that $L[X]\models $"$\omega\_1 =\omega\_1^V$". Then for arbitrarily large $\gamma < \omega\_1$ we have that $L\_\gamma[X\cap \gamma] \models $"*Every set is countable*". But such a set cannot be climbable, for if $f\... | 6 | https://mathoverflow.net/users/6942 | 287580 | 126,924 |
https://mathoverflow.net/questions/286579 | 9 | Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is
$$\lambda\_{j}(X^{\*}X+Y^{\*}Y)\geq 2\sigma\_j(XY^\*)$$
for $j \in \{1, \ldots, n\}$, where $\lambda\_j(\cdot)$ and $\sigma\_j(\cdot)$ denote the $j$th largest eigenvalue and singular value, respectively, and $X$ and... | https://mathoverflow.net/users/54458 | A singular value-eigenvalue inequality | The conjecture is true.
**Lemma 1** : For every matrix $Z$ holds $\lambda\_j(Z^\* Z + Z^\* + Z ) \leq \lambda\_j(Z^\* Z + 2 (Z^\* Z)^{1/2})$ .
For a proof see the proof of $\lambda\_j(Z^\* + Z ) \leq \lambda\_j(2 (Z^\* Z)^{1/2})$ in Bhatia, Matrix Analysis, Proposition III.5.1 (Fan-Hoffman).
**Lemma 2** : For $a ... | 4 | https://mathoverflow.net/users/17261 | 287584 | 126,927 |
https://mathoverflow.net/questions/287538 | 1 | My problem is the following. Given $F$ and $G$ cumulative distribution functions, with densities $f,g$ (for example on $[0,1]$), what can I say on the monotonicity of $F(x)(1-G(x))$? More specifically: I would like to conclude that $F(1-G)$ should be increasing for low enough $x$ and decreasing for high enough $x$. I f... | https://mathoverflow.net/users/114507 | When a product of cdf and tail distribution is increasing? | One has $(F(1-G))'=(1-G)f-Fg$. So, for $F(1-G)$ to be increasing (that is, nondecreasing) in a right neighborhood (r.n.) of $0$, it is necessary and sufficient that $(1-G)f-Fg\ge0$ in a r.n. of $0$.
For the latter condition to hold, it is enough that
\begin{equation}
\liminf\_{x\downarrow0}\frac{f(x)}{F(x)g(x)}>1, ... | 0 | https://mathoverflow.net/users/36721 | 287593 | 126,930 |
https://mathoverflow.net/questions/287601 | 3 | I saw this recursive formula in a slide on algorithm design. It talks about matrix chain-multiplication, and its complexity is shown below. But according the recursive formula, I can't figure out the solution in the slides.
$$
P(n)=
\begin{equation}
\left\{
\begin{array}{lr}
1, & n=1.\\
\sum\_{k=1}^{n-1}P(k)P(... | https://mathoverflow.net/users/116387 | How to solve the recursive formula $P(n)=\sum_{k=1}^{n-1}P(k)P(n-k)$ | Well..., clearly $P(n) = n \frac{(2n-2)!}{n!^2}$ :).
Joke aside, here is how one can "solve" the recursion. Let's consider the generating formal power series
$ \Phi(z) = \sum\_{n \ge 1} z^n P(n) $.
Now, let's use the recursive relation to get an equation for $\Phi$.
$ \Phi(z) = \sum\_{n \ge 1} z^n P(n) =
z + \su... | 11 | https://mathoverflow.net/users/47322 | 287606 | 126,933 |
https://mathoverflow.net/questions/285386 | 1 | Are there algorithms for finding $\omega\_1\dots,\omega\_n$, so that
$\omega\_i+\omega\_j\le \|e\_{ij}\|\ \forall i,j\quad\wedge\quad\sum{\omega\_i}=max$
that are *not* based on linear programing, e.g. graph theoretic algorithms?
Remark:
the number of constraints in the $LP$ formulation can be reduc... | https://mathoverflow.net/users/31310 | Finding Optimal Vertex Weights without Linear Programing | See my recent paper "Maximizing the sum of radii of disjoint balls or disks", J. Computational Geometry 8 (1): 316–339, 2017, <http://doi.org/10.20382/jocg.v8i1a12>, on problems like this. It is the dual of the LP relaxation of a weighted matching problem, and (even though the relaxation allows fractional solutions ins... | 1 | https://mathoverflow.net/users/440 | 287612 | 126,935 |
https://mathoverflow.net/questions/287603 | 3 | Is there an atlas $\mathcal{A}$ for $\mathbb{C}P^2 \setminus \{pt\}$ such that all transition maps of this atlas are affine maps?
| https://mathoverflow.net/users/36688 | Can a puntured $\mathbb{C}P^2$ admit an affine structure? | The answer is 'no'. I don't have access to the right references right now, but I think an argument goes as follows. (Also, as Will points out below, you can use Stiefel-Whitney classes to get the same result.)
If there were such an atlas on $X^4 = \mathbb{CP}^2\setminus\{p\}$, then it would induce a torsion-free fla... | 9 | https://mathoverflow.net/users/13972 | 287614 | 126,936 |
https://mathoverflow.net/questions/287542 | 6 | There is [a version](http://lanl.arxiv.org/abs/math-ph/0008003v2) of the [Eilenberg-Watts theorem](https://ncatlab.org/nlab/show/Eilenberg-Watts+theorem) for $C^\ast$-algebras, where functors between appropriate module categories correspond to what are called '[correspondences](https://ncatlab.org/nlab/show/C-star-corr... | https://mathoverflow.net/users/4177 | Non-invertible version of unitary intertwiners between correspondences of $C^\ast$-algebras | As you note in your comment, a lot depends on which category of modules you choose to consider.
If you work with the category of Hilbert $C^\*$-modules, with adjointable operators as morphisms, then the appropriate kind of maps between correspondences would be the adjointable bimodule maps: for $C^\*$-algebras $A$ a... | 3 | https://mathoverflow.net/users/85913 | 287615 | 126,937 |
https://mathoverflow.net/questions/287599 | 3 | Let $Q$ be a wild quiver without oriented cycles and let $V$ be an indecomposable representation of $Q$. Assume that $V\_i\neq 0$ for each vertex $i$ of $Q$. The base field $k$ is algebraically closed. If $V$ is not a Schur representation, $\operatorname{End}V$ is a local $k$-algebra different from $k$, so there is a n... | https://mathoverflow.net/users/118079 | Endomorphism algebras of indecomposable quiver representations | Take the quiver $1 \rightarrow 2 \rightrightarrows 3$.
Let $V$ be the representation with $V\_1=k$, $V\_2=k^2$, $V\_3=k^2$, with arrows acting by $\pmatrix{0&1}$, $\pmatrix{0&1\\0&0}$ and $\pmatrix{1&0\\0&1}$.
Then, up to scalar multiplication, the only nilpotent endomorphism is zero at vertex $1$ and $\pmatrix{0&1... | 4 | https://mathoverflow.net/users/22989 | 287621 | 126,939 |
https://mathoverflow.net/questions/287608 | 46 | I've heard about two ways mathematicians describe Feynman diagrams:
* They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal closed category.
* They are combinatorial tools that allow one to give formulas for the asymptotic expansion of integrals... | https://mathoverflow.net/users/22131 | The two ways Feynman diagrams appear in mathematics | If I understand the question correctly, the search is for a calculation of the asymptotic expansion of Gaussian integrals using concepts and techniques from category theory. Here is one such calculation:
[Feynman diagrams via graphical calculus](https://arxiv.org/abs/math/0106001) (2001)
>
> There is a very close... | 37 | https://mathoverflow.net/users/11260 | 287622 | 126,940 |
https://mathoverflow.net/questions/287610 | 6 | [This question](https://mathoverflow.net/questions/190920/any-two-bivariate-algebraically-dependent-polynomials-are-always-in-the-same-rin?noredirect=1&lq=1) asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such t... | https://mathoverflow.net/users/72288 | If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$? | No. Choose a field $k$ and $D=k[u^2,u^3,v^2,v^3,uv]\subset k[u,v]$, so $D$ is a noetherian domain. In $D[x,y]$, choose $f=(ux+vy)^2$ and $g=(ux+vy)^3$; they are clearly algebraically dependent (but $ux+vy\notin D[x,y]$). Write $K=k(u,v)=\mathrm{Frac}(D)$.
>
> Claim: there is no $P\in D[x,y]$ such that $f,g\in D[P]$... | 9 | https://mathoverflow.net/users/14094 | 287625 | 126,941 |
https://mathoverflow.net/questions/287597 | 6 | Is it decidable whether a finite group presentation is diagrammatically aspherical (that is there is no reduced spherical diagram over this presentation)? Probably - not, but I cannot find a reference.
| https://mathoverflow.net/users/nan | Asphericity of 2-complexes | The answer is no.
This follows from a theorem of Collins and Miller, who constructed a recursive sequence of presentations $P\_n$ such that the set of $n$ for which $P\_n$ presents the trivial group is recursively enumerable but not recursive, and $P\_n$ is aspherical if and only if it presents a non-trivial group. I... | 9 | https://mathoverflow.net/users/1463 | 287629 | 126,943 |
https://mathoverflow.net/questions/287639 | 7 | Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an *infinite projective plane* on $\kappa$ if
1. $e\_1\neq e\_2\in E$ implies $|e\_1\cap e\_2| = 1$, and
2. whenever $n\neq m\in \kappa$, there is $e\in E$ with $\{n,m\}\subseteq e$.
Is it possible to find an infinite projective plane $E$... | https://mathoverflow.net/users/8628 | Infinite projective plane with small edges | **Update.** Here is a new simpler answer that works for all regular
$\kappa$, including $\kappa=\omega$. And I have omitted the use
of Fodor's lemma, using instead merely the pigeon-hole principle.
Suppose that $\kappa$ is infinite and we have a projective plane on
$\kappa$ many points, with all lines of size less th... | 9 | https://mathoverflow.net/users/1946 | 287642 | 126,945 |
https://mathoverflow.net/questions/287637 | 5 | Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated
($BP\_\*(X) = \Sigma^d BP\_\*/I$, where $I$ has the form $I=(v\_0^{i\_0}, \dots , v\_n^{i\_n})$ such that the homology is a $BP\_\*(BP)$ coalgebra).
Let $C\_nX$ the fiber of the map $ X \to L\_nX$ and let $\Sigma C\_n X$ be its cofiber.
Wha... | https://mathoverflow.net/users/93775 | Map between homology of spectra | For any $m$, there is a Kunneth spectral sequence
$$
Tor\_{BP\_\*} (K(m)\_\*, BP\_\*(X)) \Rightarrow K(m)\_\* X.
$$
For your $X$, $BP\_\*(X)$ is acted on nilpotently by $v\_m$ for $0 \leq m \leq n$, and so this spectral sequence starts with zero.
Thus $K(m)\_\* X = 0$ for $0 \leq m \leq n$, and so for any $d \leq n$ ... | 9 | https://mathoverflow.net/users/360 | 287644 | 126,946 |
https://mathoverflow.net/questions/287630 | 4 | I am considering to investigate on a variation of the cops and robber game where the robber is considered as an "invisible evader" for their location is unknown until one of the cops are at an adjacent node to the robber.
I was just wondering whether this has been previously studied by any individuals and if so, how ... | https://mathoverflow.net/users/118089 | Variation of Pursuit-evasion (Cops and Robbers) | A variation where the robber is invisible until at distance $\ell$ of some cop for a fixed parameter $\ell$ is considered in the preprint [Limited Visibility Cops and Robbers](https://arxiv.org/abs/1708.07179). The version you suggest is $\ell = 1$, and the preprint cites the following master's thesis as a source for t... | 5 | https://mathoverflow.net/users/51668 | 287650 | 126,947 |
https://mathoverflow.net/questions/287652 | 2 | Let $\Psi (x,y)$ denote the number of $y$-smooth integers less than or equal to $x$.
I know the precise results about $\Psi (x,y)$ in the literature (with good error terms), but what are some helpful "order of magnitudes" to keep in mind, that give a representation of what's going on when $y$ varies?
| https://mathoverflow.net/users/85239 | Order of magnitude for the number of y-smooth numbers less than or equal to x | Using the estimation
$$\Psi(x,y) = x \rho(u) \Big( 1 + O\Big(\frac{\log(u+1)}{\log y}\Big)\Big) \qquad (x \geq 3,\ e^{(\log\_2x)^{5/3+\varepsilon}} \leq y \leq x)$$ (see §III.5.5 of [T15])
or
$$\Psi(x,y) \ll x e^{-u\log u} + \sqrt{x} \qquad (x \geq2,\ y \geq 2)$$ (see §III.5.6 of [T15]), we can get the following... | 6 | https://mathoverflow.net/users/85239 | 287653 | 126,948 |
https://mathoverflow.net/questions/285806 | 3 | I have been working on my research as a student at Wilbur Wright College on Topological Complexity. We solved the problem of two robots moving on a circle and letter $T$ using Farber's theorem but having difficulty finding $TC$ for two robots moving on a number $8$. Our group founded $2 \leq TC \leq 3$ using Farber's t... | https://mathoverflow.net/users/117108 | Topological Complexity $TC$ of two robots moving on number $8$ | The topological complexity of $X$ is the minimum number of open sets needed to cover $X\times X$, on each of which the path fibration $X^I\to X\times X$ admits a local section. If I understood your question, then you are asking about the cases $X=S^1\vee S^1$, and $Y=F(S^1\vee S^1,2)$, the $2$-point ordered configurati... | 2 | https://mathoverflow.net/users/8103 | 287655 | 126,949 |
https://mathoverflow.net/questions/287649 | 10 | Assume given a pullback square of simplicial categories
$$\begin{array}[c]{ccc}
A&{\rightarrow}&B\\
\downarrow&&\downarrow\\
C&{\rightarrow}&D.
\end{array}$$
Suppose further that one of the induced arrows $Ho (B) \to Ho(D)$ or $Ho(C) \to Ho(D)$ is an isofibration, and for each couple of objects $x,y \in A$, the ind... | https://mathoverflow.net/users/42658 | Criterion for homotopy pullback square of simplicial categories | Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1.11 in [this paper](https://arxiv.org/abs/1612.02608).
| 5 | https://mathoverflow.net/users/51164 | 287661 | 126,950 |
https://mathoverflow.net/questions/287654 | 15 | Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I\_x/I\_x^2 \cong T^\*\_xM$, where $I\_x$ is an ideal of functions vanishing at $x$ in ring $C^\infty(M)$ (or $C\_x^\infty(M)$). One can prove this isomorphis... | https://mathoverflow.net/users/62635 | Where was $I_x/I_x^2$ first introduced? (DG or AG) | Zariski formulated the criterion for smoothness at a point in terms of the dimension of $I\_x/I\_x^2$ (to use your notation) as Theorem 3.2 in the paper <https://www.jstor.org/stable/pdf/2371499.pdf> from 1939 (see p. 260). See the paragraph and theorem preceding that also. He singles out Theorem 3.2 in the first parag... | 23 | https://mathoverflow.net/users/3272 | 287665 | 126,951 |
https://mathoverflow.net/questions/287663 | -1 | It's very standard to view rotations about the origin in $\mathbb{R}^2$ as a group $\mathbb{SO}(2)$, with the zero rotation as an identity and composition of rotations as addition. This can also be viewed as the interval $[0,2\pi)$ with $0$ as the additive identity and addition defined mod $2\pi$.
We can lift this to... | https://mathoverflow.net/users/92164 | A Rng of rotations? | The multiplication does not give a Rng structure because it does not distribute over addition:
$$
\pi\hat{\times}\big(\pi+\pi)=\pi \hat{\times} 0=0\neq \pi = \frac{\pi}{2}+\frac{\pi}{2}=\pi\hat{\times}\pi +\pi\hat{\times}\pi.
$$
| 7 | https://mathoverflow.net/users/5263 | 287666 | 126,952 |
https://mathoverflow.net/questions/287590 | 8 | I am trying to find a comprehensive reference on Cartan's construction, the comparison theorem of Moore and related subjects. I have the following references
* Cartan, H., Algebres d’Eilenberg-Mac Lane, Seminaire Cartan, ENS, 1954-55, expos´es 2 to 11.
(128, 180, 194, 197, 224, 232, 468)
* Moore, J.C., Cartan’s const... | https://mathoverflow.net/users/21326 | Is it possible to find a copy of the Cartan seminar (1954–1955) on homotopy and Eilenberg–Mac Lane spaces? | The Cartan Seminar is at: <http://www.numdam.org/actas/SHC>
Actually most French journals and seminars have been digitized and can be fount at: <http://www.numdam.org/>
Unfortunately, Astérisque has not been digitized completely and is not in the list; perhaps the only ones that are digitized are the volumes corre... | 9 | https://mathoverflow.net/users/3903 | 287672 | 126,956 |
https://mathoverflow.net/questions/287660 | 7 | Say $Q$ is a random variable which is sampling orthogonal matrices in $m$ dimensions using the Haar measure on $O(m)$. Let $A$ and $B$ be some (fixed) subset of rows and columns of $Q$ such that $\vert A \vert = \vert B \vert = k$.
Now is such an identity true? (If yes then could you kindly give the proof or a refer... | https://mathoverflow.net/users/89451 | About taking an expectation over orthogonal matrices | Let's denote $S=\{1,2,\dots m\}$ and $A\subset S$ with $|A|=k$. Then from orthogonality of $Q$ we have
$$Q\_{A,S}Q\_{A,S}^{T}=I\_k$$ therefore Cauchy Binet tells us that
$$\sum\_{B\in \binom{S}{k}}\det(Q\_{A,B})^2=1.$$
If you apply expectation on both sides and notice that $E(\det(Q\_{A,B})^2)$ doesn't depend on $B$ b... | 9 | https://mathoverflow.net/users/2384 | 287673 | 126,957 |
https://mathoverflow.net/questions/287647 | 1 | On Stein's ``Harmonic Analysis Real-variable methods, orthogonality, and oscillatory integrals'' (5.13, page 363) there is the following statement. Let $\phi$ be a real homogeneous polynomial on $\mathbb{R}^n$ of degree $k \geq 2$ that is non-degenerate, in the sense that $\det \left( \frac{\partial^2 \phi}{\partial x\... | https://mathoverflow.net/users/84272 | Bound of an oscillatory integral from Stein's Harmonic Analysis book | Something which may help to observe is that you can immediately restrict to the subset of $\mathbb{R}^n$ given by $\text{supp}(\psi)\cap \{\lambda\nabla \phi(x)+|\xi|^2=0\}$, by appealing to the non-stationary phase lemma. A version of this result is as follows: Given an oscillatory integral of the form $I\_{\psi,\varp... | 2 | https://mathoverflow.net/users/111338 | 287682 | 126,960 |
https://mathoverflow.net/questions/287419 | 3 | Let $V$ be a fixed vector space. Let $X$ be a smooth curve. Consider the Quot scheme $Q$ of Quotients of $V\otimes \mathcal{O}\_X$ of degree $d$ and rank $r$. Let $R$ be the open subscheme of $Q$ consisiting only vector bundles with following property: $H^1$ vanishes and $H^0\cong V$ via the natural map. Its clear that... | https://mathoverflow.net/users/nan | On a result of Sir Michael Atiyah "Vector bundles on Elliptic curves" Theorem 2 Page 426 | No, this is not true in general. Actually, the exact sequence on $X$ is not canonical, but is determined by an $(r-1)$-dimensional vector subspace in $V$. Moreover, to extend to such a sequence the first morphism should be a fiberwise monomorphism. For this, there is an obstruction given by $c\_2(E)$. Eventually, since... | 4 | https://mathoverflow.net/users/4428 | 287685 | 126,962 |
https://mathoverflow.net/questions/287364 | 8 | I am new to the study of unirational and rational varieties, but I want to know the motivation for why mathematicians started to study these conditions. The reasons that I could list to study unirational and rational varieties are the following.
1. The study of unirational varieties can be used to answer questions ab... | https://mathoverflow.net/users/113893 | Why study unirational and rational varieties? | For a given variety it is important to know whether it is rational or unirational, because if this is the case, then the variety admits a nice parameterization. I believe this is the historical reason for studying these notions.
| 3 | https://mathoverflow.net/users/4428 | 287686 | 126,963 |
https://mathoverflow.net/questions/163325 | 6 | Singular distributions are special mathematical objects. They have an interesting property of **not having** a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the typical example of such distribution.
I was wondering whether there are instances in nature that are explained using... | https://mathoverflow.net/users/39212 | Singular distributions: Applications and Instances | In the so-called *red and black*, a player starts with a given fortune and wants to reach a given target.
The reader may want to have a look at the exposition *[How to Gamble If You Must](https://www.maa.org/press/periodicals/loci/how-to-gamble-if-you-must)* by Kyle Siegrist for the further reference.
For concretenes... | 7 | https://mathoverflow.net/users/118083 | 287696 | 126,965 |
https://mathoverflow.net/questions/287556 | 3 | Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $(S\_1,S\_2)\in \mathcal{B}(F)^2$. We define
$$W(S\_1,S\_2)=\{(\langle S\_1 y\; ,\;y\rangle,\langle S\_2 y ,\;y\rangle):y \in F,\;\;\|y\|=1\}.$$
Consider the matrices
$$
S\_1 = \left[\begin{array}{ccc} 0&1&0 \\ 0& 0&0... | https://mathoverflow.net/users/116483 | example of convexity | In your specific case, $W(S\_1,S\_2)$ is the polydisk of radius $\frac12$. For
$$|b\bar a|^2+|b\bar c|^2=|b|^2(1-|b|^2)\le\frac14$$
and conversely if $|w|^2+|z|^2\le\frac14$, one takes $b=\sqrt t\in(0,1]$ a root of $t(1-t)=|w|^2+|z|^2$, then $\bar a=w/t$, $\bar c=z/t$.
| 3 | https://mathoverflow.net/users/8799 | 287702 | 126,967 |
https://mathoverflow.net/questions/287679 | 9 | I've run across a way of combining the integral cohomology of the real projective space $RP^\infty$ with its cohomology with twisted coefficients, that seems very simple and natural, but which I don't recall every seeing before, so my question is: Has this been noticed before and, if so, where is it published?
Let me... | https://mathoverflow.net/users/58888 | The integral cohomology of real projective space | This description is given in Lemma 1 of
* M. Cadek. The cohomology of $BO(n)$ with twisted integer coefficients. J. Math. Kyoto Univ. 39-2 (1999), 277-286.
(and Cadek goes on to give similar descriptions of the cohomology of infinite real Grassmannians)
| 9 | https://mathoverflow.net/users/50846 | 287706 | 126,969 |
https://mathoverflow.net/questions/287689 | 3 | I wonder if there is a general method for obtaining bounds on an analytic function using only its Taylor expansion (not using its special properties such as satisfying a good differential equation, etc.)
As a toy example, can we prove that $|sin(x)|\leq 1$ (or a weaker bound) only knowing that $sin(x) = x - \frac{x^3... | https://mathoverflow.net/users/51663 | Proving bounds on analytic functions using only the Taylor expansion | There is no such general method. You cannot see directly from the Taylor series
that $\sin x$ is bounded on the real line, or that $\exp z$ is bounded on the negative ray. Of course what I stated is not a theorem, but just think how this boundedness criterion could possibly look: ANY change in ONE coefficient of the
se... | 4 | https://mathoverflow.net/users/25510 | 287710 | 126,972 |
https://mathoverflow.net/questions/287700 | 4 | Let $d$ be a positive integer. My question is: can we then find a positive integer $r$ (dependent on $d$) with integers $\alpha\_1, \alpha\_2, ..., \alpha\_r$ and $\beta\_1, \beta\_2, ..., \beta\_r$ such that the Polynomials $P(X) = (X - \alpha\_1) \cdots (X - \alpha\_r)$ and $Q(X) = (X - \beta\_1) \cdots (X - \beta\_r... | https://mathoverflow.net/users/116289 | Polynomials $P$ with integer roots near to $X^{\mathrm{deg}(P)}$ | One can take $r = 1 + \frac{d(d-1)}{2}$.
Indeed, one can consider the map
$$
(\alpha\_i)\_{i=1}^r \in [|1,N|]^{r} \mapsto (\sum\_{i=1}^r \alpha\_i^k)\_{k=1}^{d-1} \in \prod\_{i=1}^{d-1} [|1,rN^k|].
$$
The source has cardinality $N^r$ while the target has cardinality $r^{d-1} N^{\frac{d(d-1)}{2}}$. If
$$
(\*) \quad N^... | 5 | https://mathoverflow.net/users/21724 | 287712 | 126,973 |
https://mathoverflow.net/questions/287468 | 3 | Let $q=p^f$ be a power of the prime $p$ and let $\alpha$ be an inner diagonal automorphism of ${^2}{\operatorname{E}\_6(q^2)}$, i.e., an element of $({^2}\operatorname{E}\_6)\_{\mathrm{ad}}(q^2)$ in the notation of [1, pp. 39ff.].
>
> **Question**:
> Is $|\{x\in{^2}{\operatorname{E}\_6(q^2)}\mid\alpha(x)=x\}|=|\op... | https://mathoverflow.net/users/57975 | A lower bound on the number of fixed points of an inner diagonal automorphism of ${^2}{\operatorname{E}_6(q^2)}$ | The answer is `yes'. The subgroup of $C\_{{^2}E\_6(q^2)}(s)$ generated by all its $p$-elements is a central product of groups of Lie type over extension fields of $\mathbb{F}\_q$, if $s$ is a semisimple inner diagonal automorphism. (This can be seen by writing $C\_{{^2}E\_6(q^2)}(s)=C\_{C\_G(s)}(\sigma\_q\tau)$ where $... | 4 | https://mathoverflow.net/users/99221 | 287718 | 126,974 |
https://mathoverflow.net/questions/287722 | 4 | Let $\Omega$ be a bounded domain of $R^d$ with Lipschitz boundary. If $m>\frac{d}{2}$, such that $H^m(\Omega)$ is continuously embedded in $L^\infty(\Omega)$. Is $L^1(\Omega)$ continuously embedded in the dual space of $H^m(\Omega)$? Thank you very much.
| https://mathoverflow.net/users/105782 | Is $L^1(\Omega)$ continuous embedded in the dual of $H^m(\Omega)$ $(m>\frac{d}{2})$? | Yes. This is a special case of the following result.
**Proposition.** Let $X$ and $Y$ be normed spaces and let $i: X \to Y'$ be a continuous embedding. Then the mapping $j: Y \to X'$, given by $\langle j(y), x\rangle = \langle y, i(x)\rangle$, is continuous.
If, moreover, $i(X)$ is weak${}^\*$-dense in $Y'$, then $j$... | 4 | https://mathoverflow.net/users/102946 | 287728 | 126,978 |
https://mathoverflow.net/questions/287725 | 11 | A well known consequence of the strong approximation theorem for semisimple simply connected algebraic groups over a number field is that certain reduction maps are surjective, for example, the canonical projection $Sp\_n({\mathbb Z}) \to Sp\_n({\mathbb Z}/m{\mathbb Z})$ is surjective for any modulus $m$. This latter f... | https://mathoverflow.net/users/8099 | Strong approximation for principal ideal domains | For $G(R) = SL\_n(R)$ (see, e.g., this [MO post](https://mathoverflow.net/questions/78404/when-is-sln-r-rightarrow-sln-r-q-surjective/241596#241596)) or $G(R) = Sp\_{2n}(R)$, there is a common line of reasoning in order to prove that the reduction modulo $\mathfrak{a}$, say $\varphi\_{\mathfrak{a}}: G(R) \rightarrow G(... | 8 | https://mathoverflow.net/users/84349 | 287748 | 126,986 |
https://mathoverflow.net/questions/285623 | 3 | Asked on math.stackexchange <https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde> but didn't get response (in fact got a negative vote without any comment), so trying here.
Is it true that in any polynomial Hopf algebra $K[X1,X2,...]$ over... | https://mathoverflow.net/users/117036 | What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates? | No, in general the claim is not true:
To see why, consider a field $k$ of characteristic $p$ and take the polynomial hopf algebra $k[x]$ (in a single variable). Then $x$ is primitive and so is $x^p$:
$$
\Delta(x^p)=1\otimes x^p+x^p\otimes 1
$$
(because in characteristic $p$: $\binom{p}{i}=0$, for $1\leq i\leq p-1 ... | 2 | https://mathoverflow.net/users/85967 | 287753 | 126,987 |
https://mathoverflow.net/questions/287741 | 12 | Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I remember having read a criticism thereof later, but remain ignorant of what happened since then. A quick googling only ... | https://mathoverflow.net/users/13625 | Has there been further work on Bender-Brody-Müller approach to RH? | It seems the only development so far is a response from Bender, Brody and Müller from 18 May 2017 to Bellissard's criticism of their work, also to be found on arxiv:
[Comment on 'Comment on "Hamiltonian for the zeros of the Riemann zeta function" '](https://arxiv.org/abs/1705.06767)
They seem to be addressing all ... | 11 | https://mathoverflow.net/users/1849 | 287754 | 126,988 |
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