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https://mathoverflow.net/questions/200804 | 3 | Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
| https://mathoverflow.net/users/5712 | Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$ | It was done by [Euler](http://eulerarchive.maa.org/pages/E251.html) . Here you can see [original text](http://eulerarchive.maa.org/docs/originals/E251.pdf) and [english tranclation.](http://home.sandiego.edu/~langton/eell.pdf) $\smile$
Now it is known as **Euler-Chasles correspondence.**
| 1 | https://mathoverflow.net/users/5712 | 201024 | 97,019 |
https://mathoverflow.net/questions/201002 | 2 | This question is an offshoot from the following [MSE post](https://math.stackexchange.com/questions/1205692). I hope that it is appropriate for this site.
Let $\sigma(x)$ be the sum of the divisors of $x$.
An integer $a$ is said to be solitary if there does not exist another integer $b \neq a$ such that
$$\frac{\si... | https://mathoverflow.net/users/10365 | Does there exist an integer that is both solitary and almost perfect? | An elementary sufficient condition for $n$ to be solitary is that $\gcd(n,\sigma(n))=1$. Indeed, suppose $\gcd(n,\sigma(n))=1$ and $\sigma(n)/n = \sigma(m)/m$. Since the left-hand fraction is in lowest terms, $n\mid m$. But then $\sigma(m)/m \ge \sigma(n)/n$ unless $n=m$.
Now if $n$ is almost perfect, then $\gcd(\si... | 10 | https://mathoverflow.net/users/16510 | 201037 | 97,024 |
https://mathoverflow.net/questions/201032 | 1 | In a paper of Cohn (see [here](http://www.fq.math.ca/Scanned/2-2/cohn2.pdf)), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and maybe even for more equations of this type)?
| https://mathoverflow.net/users/50081 | Source for equations involving congruences of Fibonacci and Lucas numbers | (13) follows easily by induction, namely it is true for $m=0$ and $m=1$ by inspection, and then the recursion yields it for all $m$. (11) and (12) hold more generally for all $k$ (including the odd ones) if the minus signs in them are replaced by $(-1)^{k-1}$. Indeed, it is easy to show by a double induction that
$$ F\... | 4 | https://mathoverflow.net/users/11919 | 201039 | 97,025 |
https://mathoverflow.net/questions/200955 | 4 | Recently I have been trying to find the definition of the subsystem $ATR\_0$ of second-order arithmetic. Only "definitions" I have found were quite vague, like informal definition on Wikipedia which says it's $ACA\_0$ plus statement that "any arithmetical functional can be iterated transfinitely along any countable wel... | https://mathoverflow.net/users/30186 | Formal definition of arithmetic transfinite recursion | To repeat Emil and Andreas's comments, it can be found in Stephen G. Simpson, "Systems of Second-Order Arithmetic", the first chapter of which is available here:
<http://www.personal.psu.edu/t20/sosoa/chapter1.pdf>
See Definition I.11.1 (p. 39, which is in the first chapter).
This first chapter also has informati... | 6 | https://mathoverflow.net/users/12978 | 201042 | 97,027 |
https://mathoverflow.net/questions/201047 | 1 | A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition of a Courant algebroid on the associated principal bundle? In other words, what is the structure that a principal bundl... | https://mathoverflow.net/users/66688 | Principal bundle associated to a Courant algebroid | Since the bracket (you call it product) is a bidifferential operator on the space of sections of the vector bundle, the translation to the principal bundles is not straightforward. You have to use the first jet bundle of a principal bundle and to associate to it the jet bundle of the associated vector bundle. Bilineari... | 1 | https://mathoverflow.net/users/26935 | 201050 | 97,030 |
https://mathoverflow.net/questions/200989 | 7 | We fix an integer $n$ and consider the stabilization map $O(n)\to O$.
Using rational methods one can easily check that the map
$\pi\_{4i-1}(O(n))\to \pi\_{4i-1}(O)\cong\mathbb{Z}$ vanishes for sufficiently large $i$. Is a similar fact known for the map
$$\pi\_{8i}(O(n))\to \pi\_{8i}(O)\cong\mathbb{Z}/2\mathbb{Z}\ ?$$
... | https://mathoverflow.net/users/58804 | Homotopy of orthogonal groups in the unstable range | This is false. In fact, for all sufficiently large n the maps $\pi\_{8i}(O(n)) \to \pi\_{8i}(O) \cong \Bbb Z/2$ are surjective for all $i$.
One possible tool for proving this has to do with $v\_1$-periodic homotopy theory. For sufficiently large $d$, the mod-2 Moore space $W = \Sigma^{d-1} \Bbb{RP}^2$, which is the c... | 7 | https://mathoverflow.net/users/360 | 201053 | 97,032 |
https://mathoverflow.net/questions/201059 | 9 | Freiling's Axiom of Symmetry says that for any function $f:[0,1]\to \mathcal{P}([0,1])$ such that for every $x\in [0,1]$ we have $|f(x)|=\aleph\_0$, then there exist $y,z\in [0,1]$ such that $z\notin f(y)$ and $y\notin f(z)$. This is equivalent to the negation of the continuum hypothesis (see [this wiki page](http://en... | https://mathoverflow.net/users/3199 | Freiling's Axiom of Symmetry Concretized | **Theorem.** The following are equivalent for a set $A$
1. $A$ has the property of Freiling's axiom. That is, if $a\mapsto
X\_a$ is any map from $A$ to the countable subsets of $A$, then
there are $a$ and $b$ with $a\notin X\_b$ and $b\notin X\_a$.
2. $A$ has size at least $\aleph\_2$.
**Proof.** ($1\to 2$) We prov... | 7 | https://mathoverflow.net/users/1946 | 201064 | 97,041 |
https://mathoverflow.net/questions/201057 | 7 | Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi\_1:\mathfrak{sl}\_2(\mathbb{C})\rightarrow\mathfrak{g}$ and $\phi\_{2}:\mathfrak{sl}\_2(\mathbb{C})\rightarrow\mathfrak{g}$ be complex Lie algebra morphisms. By composing $\phi\_1$ and $\phi\_2$ with the adjoint representation of $... | https://mathoverflow.net/users/25358 | Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation? | REVISED VERSION: On further reflection, I think the answer to your question is always "yes" (if $\mathfrak{g}$ is *simple*, to avoid complications of the type Dave indicates). At first I was confused by the example discussed in the question, but I think the main issue is how the automorphism group interacts with Dynkin... | 3 | https://mathoverflow.net/users/4231 | 201070 | 97,043 |
https://mathoverflow.net/questions/200409 | 5 | For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes\_\mathcal{C}$.
Let the fusion category $\mathcal{C}$ be a $G$-extension of the fusion category $\mathcal{D}$, that is:
$\mathcal{C} ... | https://mathoverflow.net/users/51008 | What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$? | If $\mathcal{C}$ is group-theoretical, then $\mathcal{D}$ is group-theoretical. So suppose $\mathcal{D}$ is group-theoretical and denote by $X$ the set of equivalence classes of pointed $\mathcal{D}$-module categories. The group $BrPic(\mathcal{D})$ acts on $X$ using the tensor product of module categories over $\mathc... | 1 | https://mathoverflow.net/users/6517 | 201080 | 97,048 |
https://mathoverflow.net/questions/201068 | 6 | Ramanujan's $\tau$ conjecture states that $$\tau(n)=O\_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in <https://math.stackexchange.com/questions/1205419/status-of-taun-before-deligne/1205516> tell that best exponent before Deligne reached $\frac{29}5$.
Wh... | https://mathoverflow.net/users/10035 | Etale cohomology approach on $\tau(n)$ | One of the goals of the development etale cohomology was to generate a cohomology theory that could successfully count points on varieties over finite fields, with one of the main goals of proving the conjecture that Andre Weil laid out in 1949 (see [his paper here](http://www.ams.org/journals/bull/1949-55-05/S0002-990... | 12 | https://mathoverflow.net/users/48142 | 201087 | 97,050 |
https://mathoverflow.net/questions/201075 | 2 | Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, like for examples $U(1)$ or $SU(2)$?
Thanks.
| https://mathoverflow.net/users/66688 | Moduli space of flat connections over a torus | Check out [Almost commuting elements in compact Lie groups](http://arxiv.org/abs/math/9907007) by Borel, Freedman, and Morgan.
"We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin dia... | 2 | https://mathoverflow.net/users/391 | 201091 | 97,052 |
https://mathoverflow.net/questions/201082 | 6 | Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus $C(\mathbb R)$ is a circle, which is not smply connected.
I'm then wondering if there is some more general connection between... | https://mathoverflow.net/users/37103 | UFD and fundamental group | It's the absolute Galois group that can be thought of as a fundamental group, since it is the [étale fundamental group](http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group) of $\text{Spec } K$. The ideal class group is instead a [Picard group](http://en.wikipedia.org/wiki/Picard_group) of line bundles, which for ... | 13 | https://mathoverflow.net/users/290 | 201095 | 97,055 |
https://mathoverflow.net/questions/200931 | 5 | Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For example a motivic decomposition of $F\_4/P$, with $P$ being a certain parabolic subgroup has been established by Semenov, N... | https://mathoverflow.net/users/51251 | Motives of a variety of type D4 | You can compute the motive of $SO(8)$ with rational coefficients in the category of derived mixed Tate motives with Biglari's theorem (after looking up the exponents of the Weyl group in Bourbaki's tables).
The exponents are $1,3,5,3$, which gives the numbers $2,4,6,4$ as dimensions of certain cohomology groups of we... | 1 | https://mathoverflow.net/users/956 | 201115 | 97,059 |
https://mathoverflow.net/questions/201125 | 1 | Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq 2{G'(x)}^2.$$ Is there a constant positive bound from below on $G(1/2)$?
| https://mathoverflow.net/users/38889 | A differential inequality and a special value | Consider $G\_n(x) := (1-x)^n$. We have $G\_n(0) = 1$, $G\_n(1) = 0$ and $G\_n$ is monotonic (strictly) decreasing on $[0,1]$. Furthermore,
\begin{equation}
G\_n(x) G\_n''(x) = n(n-1) (1-x)^{2(n-1)} \leq 2n^2 (1-x)^{2(n-1)} = 2 G\_n'(x)^2.
\end{equation}
And witness that $G\_n(\frac 12) = 2^{-n}$.
| 4 | https://mathoverflow.net/users/62629 | 201131 | 97,064 |
https://mathoverflow.net/questions/200998 | 1 | I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
>
> Wirtinger's Inequality.
>
>
> Let $L$ be a complex linear space and let $M$ be a real
> even-dimensional subspace. Let $H$ be a positive definite Hermitian
> form on $L$ . Then $H = S + iA$ where $S$ is symmetric and $A$ is... | https://mathoverflow.net/users/32460 | Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$ | This is all "basic stuff" (this question should probably be asked on [Mathematics Stack Exchange](https://math.stackexchange.com/)).
Let $M$ be a manifold. A Riemannian metric $g$ and an almost complex structure $I$ are compatible (*i.e.* $I$ is $g$-orthogonal) if and only if $g$ is the real part of a (necessarily un... | 3 | https://mathoverflow.net/users/25590 | 201135 | 97,066 |
https://mathoverflow.net/questions/200940 | 1 | This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte Carlo or assume inequality encoded by each row of $A$ is independent for all the other inequalities; then there's an exp... | https://mathoverflow.net/users/69653 | Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s | You can determine the exact distribution of the Gaussian vector Ax but there is no closed formula for estimating its cumulative distribution function (see <http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Cumulative_distribution_function>).
You will have to estimate it numerically. Alan Genz's worked a ... | 1 | https://mathoverflow.net/users/43318 | 201139 | 97,068 |
https://mathoverflow.net/questions/201136 | 1 | For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q\_1\subset Q\_2\subset Q\_3$ is a chain of prime ideals of $S[X]$, then $Q\_1\cap S\subset Q\_3\cap S$.
Now let $R$ be a commutative ring of infinite dimension. Let $\mathcal{C}$ ... | https://mathoverflow.net/users/69591 | Arbitrary chains of prime ideals in $R[X]$ | From the fact $Q\_1 \cap R \subsetneq Q\_3 \cap R$, we see that the natural map from $\mathcal{C}$ onto $\mathcal{C}\_R$ is at most 2-to-1. If $\mathcal{C}$ is infinite, this implies that it has the same cardinality as its image $\mathcal{C}\_R$. Am I missing something?
| 2 | https://mathoverflow.net/users/68305 | 201148 | 97,072 |
https://mathoverflow.net/questions/201112 | 6 | Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times\_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? (I mostly care about the case of discrete stabilizers). More generally, if $G$ is any topological group acting on a CW... | https://mathoverflow.net/users/4528 | When does the Borel construction have the homotopy type of a CW-complex? | Yes, this is true. It suffices for $X$ to have the homotopy type of a CW-complex (this is true of smooth manifolds; see e.g. [here](https://mathoverflow.net/a/36841/360) or [here](https://math.stackexchange.com/a/4839/13952)).
I'm going to assume that you're using a definition of $EG$ that includes something like: $E... | 8 | https://mathoverflow.net/users/360 | 201150 | 97,073 |
https://mathoverflow.net/questions/201118 | 1 | After multiple plots I noticed that function $h(x)= (1-x)^b$ $\_2F\_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and $a=(K-1)d$, $b=K$, c=$Kd$; with $K \gg 1$, $K \gg d$ and $d>1$.
In other words, I need to prove that the first derivative of $h(x)$ ... | https://mathoverflow.net/users/51469 | How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case | The first few terms of the Maclaurin series can be obtained explicitly:
$$1-x-{\frac { \left( d-1 \right) \left( K-1 \right) }{2\,Kd+2}}{x}^{2
}-{\frac { \left( d-1 \right) \left( d-2 \right) \left( K-1 \right)
\left( K-2 \right) }{ \left( 6\,Kd+6 \right) \left( Kd+2 \right) }}{
x}^{3}+O \left( {x}^{4} \right)
$$
... | 1 | https://mathoverflow.net/users/13650 | 201152 | 97,074 |
https://mathoverflow.net/questions/200504 | 15 | 1) A $n$-dimensional homology manifold is a topological space $X$ such that for any $x\in X$, the homology groups
$$H\_p(X,X-x,\mathbb{Z})$$
are trivial unless $p=n$ where
$$H\_n(X,X-x,\mathbb{Z})\cong \mathbb{Z}$$
2) A $n$-dimensional pseudomanifold is a topological space together with a triangulation such that
* ... | https://mathoverflow.net/users/27816 | Pseudomanifolds and Poincaré duality | 1. Introduction
2. Easy example
3. Normality, Duality
4. Normal example
Introduction
------------
The pseudomanifold and homology manifold conditions are both local conditions, while Poincaré duality is a global condition. It is possible for a pseudomanifold to fail the homology manifold conditions in seve... | 3 | https://mathoverflow.net/users/4639 | 201188 | 97,085 |
https://mathoverflow.net/questions/201185 | 8 | Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any idempotent $\alpha: X\rightarrow X$ has a splitting in $\mathcal{C}$, i.e. there exists an object $Y$ together with morphisms $... | https://mathoverflow.net/users/24965 | Is the derived category of perfect complexes idempotent complete? | The derived category of perfect complexes is idempotent complete, because it is the sub category of compact objects in the derived category of quasi coherent sheaves (which is idempotent complete by the result you mention) and compact objects are stable under retracts.
| 14 | https://mathoverflow.net/users/43054 | 201189 | 97,086 |
https://mathoverflow.net/questions/201164 | -2 | 1. Are there any special properties known about the set of perfect matchings of $K\_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure on this set? Some group that is known to may be act nicely on this set? (this might be a broad question, so feel fre... | https://mathoverflow.net/users/36554 | About structure of the set of perfect matchings of $K_{n,n}$ | Maybe I'll summarise everything from the comments as an answer.
Firstly, a perfect matching $M$ of $K\_{n,n}$ can be identified with a permutation of the set $[n] = \{1,\ldots,n\}$ simply by numbering each side of the bipartition with $[n]$ and letting $\sigma$ be such that $(i,\sigma(i))$ is an edge of $M$.
Theref... | 4 | https://mathoverflow.net/users/1492 | 201192 | 97,089 |
https://mathoverflow.net/questions/201186 | 2 | Let $S$ be a complex surface with ample canonical class. Let $C\_1$ and $C\_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection of the homology class $[C\_1] + [C\_2]$ is positive. Is it true that the intersections of $C\_1$ and $C\_2$ can be sm... | https://mathoverflow.net/users/69766 | Smoothing transverse intersections | The answer is **no** in general, as the following example shows.
There exist complex surfaces $S$ with ample canonical class, $$p\_g(S):=h^0(S, \, K\_S)=2, \quad q(S):=h^1(S, \, K\_S)=0$$ and $|K\_S|$ *composed with a pencil*. This means that $$|K\_S|=M + |F|,$$
where $M$ is a fixed curve and $F$ is a divisor with $h... | 6 | https://mathoverflow.net/users/7460 | 201197 | 97,090 |
https://mathoverflow.net/questions/201195 | 0 | **Definition:** Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite index normal subgroup $G\_0$ in $G$ such that if $\pi$ is the projection of $G$ onto $G/G\_0$, then $$\pi(h)\not\in\pi (... | https://mathoverflow.net/users/9485 | Double coset separability and the existence of vanishing sequences for surface group | This is not correct (the stated conclusion implies $H \subset \langle \eta \rangle \cdot \langle \gamma \rangle$, which need not be true), so there must be a typographical error. Probably the conclusion should be $H \cap \Gamma\_n \subset \langle \eta \rangle \cdot \langle \gamma \rangle$. Since $\Gamma$ is countable, ... | 1 | https://mathoverflow.net/users/68305 | 201200 | 97,092 |
https://mathoverflow.net/questions/201174 | 1 | I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of Beilinson-Bernstein localization: we have a sheaf of (twisted) differential operators $$\mathscr{D}(\mathcal{L})=\mathcal{L}\o... | https://mathoverflow.net/users/34464 | Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf | If you want to "take derivative" with respect to an element $Y$ of a Lie algebra, you want to take a Newton quotient as usual. That means that you pick a path $\gamma(t)\colon [0,\epsilon) \to G$ in your Lie group such that $\dot{\gamma}(0)=Y$. If you want to differentiate a function with respect to this, for example, ... | 7 | https://mathoverflow.net/users/66 | 201201 | 97,093 |
https://mathoverflow.net/questions/201026 | 1 | Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong one are the same as $M$ is assumed to be compact.) Fix an embedding $j\in Emb(M,\mathbb{R}^n)$ and a tubular neighbourho... | https://mathoverflow.net/users/69525 | Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold` | Not to leave this question open:
The question as posed is wrong by the example I gave. However the example I gave is in some sense the maximum which can go wrong. More precisely small normal deformations as defined in the question are open if one factors out diffeomorphisms of $M$, e.g. they are open in $Emb(M,\mathb... | 1 | https://mathoverflow.net/users/69525 | 201207 | 97,096 |
https://mathoverflow.net/questions/201199 | 0 | *Definitions*: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the *fixed-point subspace* $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the *pointwise stabilizer subgroup* $G\_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.
Let $G$ be... | https://mathoverflow.net/users/34538 | A problem with pointwise stabilizer subgroups of fixed-point subspaces I | Let $G=C\_2\times C\_2$ and $H$ a subgroup of order $2$.
Let $U$ be the non-trivial irreducible module on which $H$ acts trivially, and $V$ one of the other non-trivial irreducible modules. Then $U\otimes V$ is irreducible, so $W$ can only be $U\otimes V$.
Then $G\_{(U^H)}=H$, and $G\_{(V^H)}=G\_{(W^H)}=G$, so $G\_... | 2 | https://mathoverflow.net/users/22989 | 201213 | 97,098 |
https://mathoverflow.net/questions/201101 | 1 | A matroid is said to be *strongly base-orderable* if for any two bases $B\_1,B\_2$ there is a bijection $f:B\_1 \to B\_2$ such that for any $S\subseteq B\_1$ set $(B\_1 \setminus S) \cup f(S)$ is also a base.
Are there strongly base-orderable matroids $M\_k = (E, \mathcal{I}\_k)$ for $k=1,2$ on the same ground set $E... | https://mathoverflow.net/users/8628 | Is the union of strongly base-orderable matroids strongly base-orderable? | Theorem 42.11 in A. Schrijver, *Combinatorial Optimization: Polyhedra and Efficiency*, 2003 (he references Brualdi, *Common Transversals and Strong Exchange Systems*, 1970):
>
> Any truncation of a strongly base orderable matroid is strongly base orderable again.
>
>
>
Given that, we may assume (w.l.o.g.) that... | 3 | https://mathoverflow.net/users/69775 | 201221 | 97,100 |
https://mathoverflow.net/questions/201170 | 12 | We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer.
Needed definitions may be found in, e.g., Blass's chapter in the Handbook of Set Theory.
For $f\in\omega^\omega$, let $K\_f:=\{g\in \omega^\omega... | https://mathoverflow.net/users/2415 | A classic cardinal characteristic of the continuum in disguise? | For every meager $M \subseteq \omega^{\omega}$ there exists $f\_M \in \omega^{\omega}$ such that for every increasing $f \in \omega^{\omega}$, $K\_f \subseteq M$ implies $f \leq^{\star} f\_M$. For a proof, see Theorem 2.2.2 in Bartozynski, Judah book. It follows that your invariant is the bounding number.
| 7 | https://mathoverflow.net/users/2689 | 201225 | 97,102 |
https://mathoverflow.net/questions/198886 | 17 | That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the vanishing of all Dolbeault cohomologies ($p \geq 0, q \geq 1$) implies Steinness? I could not find a reference, yet I noticed th... | https://mathoverflow.net/users/62130 | Vanishing of Dolbeault cohomologies and Steinness | The reference to the books of Hörmander and Gunning-Rossi only seems to cover a special case of this: a domain $X$ in $\mathbb{C}^n$ is a Stein manifold if and only if $H^i(X,\mathcal{O}\_X)=0$ for all $i$ with $1\leq i\leq n-1$. Another possible reference is Theorem 63.7 in
* L. and B. Kaup: Holomorphic functions o... | 9 | https://mathoverflow.net/users/50846 | 201229 | 97,105 |
https://mathoverflow.net/questions/201165 | 19 | I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of combinatorics and algebraic geometry, and gave some examples like the combinatorial Nullstellensatz, the affirmative answer to the conjecture of Read and Rota-Heron-Welsh and the graph-theoretic ... | https://mathoverflow.net/users/69757 | Are there any algebraic geometry theorems that were proved using combinatorics? | Jan Draisma's chapter "Noetherianity up to symmetry" in the book *Combinatorial Algebraic Geometry* (Springer LNM 2108) presents various finiteness theorems that are based on Kruskal's tree theorem (or actually the special case known as Higman's lemma).
The rest of the book also contains some potential examples, alth... | 5 | https://mathoverflow.net/users/3106 | 201231 | 97,106 |
https://mathoverflow.net/questions/201243 | 7 | If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms
$$\mathcal N\_p (\varphi)= \sum\_{|\alpha|, |\beta| \leq p} \sup\_{x\in \mathbb R^n} |x^\alpha \partial^\beta \varphi (x) |\, ,$$
Is it true that $\mathcal S(\mathbb R^n) \hat \... | https://mathoverflow.net/users/46773 | $\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$? | Yes, if you understand the tensor product topology in the right way.
Since the spaces are nuclear, inductive and projective tensor products coincide.
The result is theorem 51.6 of Treves: Topological vector spaces, distributions, and kernels.
Added later:
============
Attention: The description of seminorms on the... | 8 | https://mathoverflow.net/users/26935 | 201244 | 97,108 |
https://mathoverflow.net/questions/201250 | 42 | I know $\sum\_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum\_{k=1}^{n} \sin(k^2)$?
| https://mathoverflow.net/users/50874 | Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$? | No. If one selects a number $k$ at random from $1$ to a large number $n$, then for any fixed $h$, the random variables $\sin((k+1)^2), \dots, \sin((k+h)^2)$ asymptotically have mean zero, variance 1/2, and covariances 0, from standard Weyl sum estimates. Hence the variance of $\sum\_{i=1}^h \sin((k+i)^2)$ is asymptotic... | 63 | https://mathoverflow.net/users/766 | 201252 | 97,112 |
https://mathoverflow.net/questions/201193 | 4 | My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model.
Elementary excitations in the WW model can be point particles, loop defects and more general defects (see page 12 of arXiv:1104.2632). It is stated their description is closely related to the boundary condi... | https://mathoverflow.net/users/48552 | Loop defects in Walker-Wang model | In the original WW paper, we distinguish between "crude" and "topological" boundary conditions. (I think "algebraic" would be a better name than "topological" here, but perhaps it is too late to change the naming conventions. Or perhaps not.)
As you write, we associate a linear category $A(Y)$ to a 2-manifold $Y$. Th... | 4 | https://mathoverflow.net/users/284 | 201272 | 97,118 |
https://mathoverflow.net/questions/201204 | 11 | In a Euclidian space (Hermitian as well), say $\ell^2\_n$, the following inequality holds true
$$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2\_n.$$
In other words, the function
$$\delta:=\frac{|b-a|}{|a|\cdot|b|}$$
is a distance over $\ell^2\_n\setminus\{0\}$.
The proof consists... | https://mathoverflow.net/users/8799 | A "quadratic" triangular inequality | Metric space $(X,\rho)$ satisfying Ptolemy inequality $\rho(a,b)\rho(c,d)+\rho(b,c)\rho(a,d)\geq \rho(a,c)\rho(b,d)$ is called ptolemaic space. A normed ptolemaic space must be inner product space.
Reference: I.J. Schoenberg, A remark on M. M. Day’s characterization of innerproduct spaces and a conjecture of L. M. Blu... | 15 | https://mathoverflow.net/users/4312 | 201280 | 97,119 |
https://mathoverflow.net/questions/201259 | 1 | I have both a more general question (concerning stopping times), and then a more specific application (as described in the title).
Let $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t \geq 0},\mathbb{P})$ be a filtered probability space, which we can assume to satisfy the "usual conditions" if necessary.
Let $(X\_t)\_{t \... | https://mathoverflow.net/users/15570 | Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial "forward-invariant" set? | The answer to Q1 is Yes, and the answer to Q2 is No.
If $\mathbb{P}(X\_0 \in A) > 0$ then we are done by taking $\tau = 0$. So suppose $\mathbb{P}(X\_0 \in A^c) = 1$.
Call $y$ a *right endpoint* of $A$ if $y \in A$ and there exists $x < y$ with $(x,y) \subset A^c$. Let $A^+$ be the set of all right endpoints of $A... | 1 | https://mathoverflow.net/users/4832 | 201301 | 97,127 |
https://mathoverflow.net/questions/201296 | 9 | Let $Ell^\*(X)$ be the elliptic cohomology theory (associated to a given elliptic curve $E$) of a nice space $X$.
Recall the Landweber-Ravenel-Stong construction:
$MU^\*(X) \otimes\_{MU^\*} R \simeq Ell^\*(X)$, where $R \simeq Ell^\*$.
**How does the Landweber-Ravenel-Stong construction of $Ell^\*(-)$ tell us how ... | https://mathoverflow.net/users/56462 | Must we know $MU^*(X)$ in order to compute $Ell^*(X)$? | The AHSS is unlikely to be a good method for computing either $MU^\*(X)$ or $Ell^\*(X)$ except in cases where the AHSS collapses for easy reasons (the $E^2$ term is torsion free, or concentrated in even total degree). Even in those cases, it is usually desirable and possible to use other methods (usually Chern classes)... | 15 | https://mathoverflow.net/users/10366 | 201304 | 97,128 |
https://mathoverflow.net/questions/201228 | 1 | Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For a root $\alpha \in \Phi(\mathbf{G},\mathbf{S})$ let $\mathbf{U}\_\alpha \leq \mathbf{G}$ denote the associated root subg... | https://mathoverflow.net/users/3824 | Subgroups generated by opposite root groups | I believe that (2) and (3) will often fail. For example, let $\mathbf{G} = \mathrm{SL}(3,\mathbb{H})$, where $\mathbb{H}$ is the algebra of quaternions. This is an almost-simple algebraic group over $\mathbb{R}$. Consider
$$\mathbf{G}\_\alpha = \begin{bmatrix} \* & \* & 0 \\ \* & \* & 0 \\ 0 & 0 & 1 \end{bmatrix},
\ ... | 2 | https://mathoverflow.net/users/68305 | 201306 | 97,129 |
https://mathoverflow.net/questions/201308 | 5 | **Note from the answerer** : this question stems from [this article](https://www.sciencedirect.com/science/article/abs/pii/S0362546X96000223).
---
I ask this question in <https://math.stackexchange.com/questions/1206617>
I have a bounded sequence $(u\_n)$ from $W^{1,p}\_0(\Omega)$ so it weakly converge to $u\in... | https://mathoverflow.net/users/49045 | Weak convergence in $W^{1,p}_0$ | Let's call $I\_1$ and $I\_2$ your two integrals respectively.
You know that $u\_n \to u$ strongly in $L^p$. Because $\Omega$ is bounded, $u\_n$ also converges to $u$ strongly in $L^1$. As you assume that $f$ is bounded in its two arguments,
$$|I\_1| \leq \|f\|\_{L^{\infty}(\Omega \times \mathbb{R})} \|u\_n-u\|\_{L^... | 5 | https://mathoverflow.net/users/62629 | 201314 | 97,131 |
https://mathoverflow.net/questions/201324 | 6 | Let $V$ be an $n$-dimensional complex vector space. The stack $Coh^n(\mathbb C^2)$ of coherent sheaves on $\mathbb C^2$ supported on $n$ points (not necessarily distinct) is equivalent to the stack quotient $C\_n/GL\_n$, where $C\_n\subset End(V)^2$ is the variety of couples of commuting matrices.
In $Coh^n(\mathbb ... | https://mathoverflow.net/users/30827 | Coherent sheaves on $\mathbb C^2$ and commuting matrices | For any commuting pair of matrices there is a basis in which both are upper triangular. The eigenvalues give you $n$ points of $\mathbb{C}^2$ and this recovers the support of the corresponding sheaf. This leads to the following answers:
Question 1. Both $A$ and $B$ should be nilpotent.
Question 2. $(A',B')$ is a su... | 8 | https://mathoverflow.net/users/4428 | 201326 | 97,134 |
https://mathoverflow.net/questions/201320 | 6 | I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for non-oriented links?
Thank you for the help.
**Update**
I have found out that there exists another polynomial, not mentioned in t... | https://mathoverflow.net/users/47294 | Polynomial invariants for unoriented links | Colored Kauffman polynomials are independent of orientations of links. If you look at the skein relation of Kauffman polynomial, there is no arrow. This is true for colored cases. Kauffman polynomials are related to SO(N) quantum knot invariants by substitution $q^{N-1}=a$. Since the representations of SO(N) are pseudo... | 7 | https://mathoverflow.net/users/17644 | 201336 | 97,139 |
https://mathoverflow.net/questions/201254 | 3 | Let $\mathfrak{g}$ be a complex simple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^\*$ the set of roots. We fix $\Pi\subset R$ the set of simple roots, which induces the set of positive roots $R^+$. Let $W$ be the Weyl group which generator by the reflection $... | https://mathoverflow.net/users/16326 | Root in positive Weyl chamber | Rather than prolong the tangled comments, I'll try to provide a straightforward answer to the current formulation of the question. As noted already, there are two small cases of irreducible root systems (belonging to simple Lie algebras) in which the dominant (open) Weyl chamber denoted $K$ here contains a root: type $... | 3 | https://mathoverflow.net/users/4231 | 201344 | 97,141 |
https://mathoverflow.net/questions/201352 | 3 | My question considers the curve $E$ over the affine $j$-line $S$ given by
$$Y^2 - (j-1728)XY = X^3 - 36(j-1728)^3X - (j-1728)^5$$
This curve has the property that it's $j$-invariant is $j$ (see [Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728](https://mathoverflow.net/que... | https://mathoverflow.net/users/15242 | homological invariant of the "universal elliptic curve" over the punctured $j$-line | This is an illusion that the fiber is of type II, even though it looks like a cuspidal cubic. Your Weierstrass model is a **singular** surface with a type III fiber, where one of the two components has been blown down (so the other one, tangent to the first, does get a cusp).
| 4 | https://mathoverflow.net/users/44953 | 201354 | 97,146 |
https://mathoverflow.net/questions/201339 | 3 | My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom\_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains.
So the obvious map is $\phi(f+B^n(G,Hom\_{ZH}(ZG, A) ))(h\_1,\cdots,
h\_n)=f(h\_1,\cdots, h\_n)(1)+B^n(H, A)$.
QUESTION: What is the inverse o... | https://mathoverflow.net/users/57278 | When is finding an explicit inverse of an isomorphism not possible | Take a standard homogeneous $H$-resolution of $\mathbb{Z}$:
$$\ldots \to \mathbb{Z}[H^3] \to \mathbb{Z}[H^2] \to \mathbb{Z}H \to \mathbb{Z} \to 0 \qquad (1)$$
where $\phi\_n: \mathbb{Z}[H^{n+1}] \to \mathbb{Z}[H^n]$ for $n \geq 0$ takes $(h\_0, \ldots, h\_n)$ to $\sum\_{i = 0}^n (-1)^i (h\_0, \ldots, \hat{h\_i}, ... | 4 | https://mathoverflow.net/users/2926 | 201363 | 97,151 |
https://mathoverflow.net/questions/201383 | 0 | Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{\*}$ with $$T(\phi)(x)=\int\_{G} \phi(g.x)$$ The integration is based on the Haar measure. Since $H^{\*}$ is isomorphic to $H$ we actually have a linear operator on $H$. We denote this operastor with $T$, again... | https://mathoverflow.net/users/36688 | A $C^{*}$ algebra associated to a group | Because the group is compact one can assume the representation is isometric and the Haar measure is normalized. In this situation, $T$ is just the orthogonal projection on the space of $G$-invariant vectors.
(exercice: 1) $T(\phi)$ is $G$-invariant 2) Invariant linear forms are fixed by $T$ 3) $T$ is self-adjoint)
An... | 8 | https://mathoverflow.net/users/22131 | 201393 | 97,164 |
https://mathoverflow.net/questions/201381 | 13 | I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.
>
> Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(probably), I will have... | https://mathoverflow.net/users/33018 | number theory which is close to analysis | You could try the short book by Hugh Montgomery, which focuses closely on the interactions of harmonic analysis and number theory.
*Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis*
by Hugh L. Montgomery
Series: CBMS Regional Conference Series in Mathematics (Book 84)
Paperback: 220... | 18 | https://mathoverflow.net/users/11926 | 201394 | 97,165 |
https://mathoverflow.net/questions/201341 | 18 | I'm trying to understand better the relative Picard functor, as defined, for example, in [Kleiman's article](http://arxiv.org/pdf/math/0504020v1.pdf).
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are integral. Let
$$\mathrm{Pic}\_{X/S}(T) = \mathrm{Pic}(X\_T)/\mathrm{Pic}(T)$$
denot... | https://mathoverflow.net/users/5101 | Relative Picard functor for the Zariski topology | Let $Y$ be [Cayley's nodal cubic surface](http://en.wikipedia.org/wiki/Cayley%27s_nodal_cubic_surface) over the complex numbers, given in $\mathbb{P}^3$ by
$X\_0X\_1X\_2 + X\_0X\_1X\_3 + X\_0X\_2X\_3 + X\_1X\_2X\_3 = 0.$
This surface has four simple double points. It also contains six straight lines, each of which ... | 17 | https://mathoverflow.net/users/3753 | 201399 | 97,167 |
https://mathoverflow.net/questions/201406 | 4 | $(X, \mathcal{B}, \mu)$ is a measure space.
* Is there any well-known criteria for compactness of a closed set in $L^2(X, \mu)$?
* If the answer is negative what about $L^2(\mathbb{R}^n,\mu)$(in this part $\mu$ is Lebesgue measure)?
| https://mathoverflow.net/users/38805 | Criteria for Compactness of a Closed in $L^2$ Spaces | For Lebesgue measure, look at the [Fréchet-Kolmogorov theorem](http://en.wikipedia.org/wiki/Fr%C3%A9chet-Kolmogorov_theorem).
For a general measure space, see Theorem 4.7.28 of Bogachev's *Measure Theory*, which is attributed to Riesz. Let $\pi = \{E\_1, \dots, E\_k\}$ be a collection of disjoint measurable subsets o... | 5 | https://mathoverflow.net/users/4832 | 201408 | 97,171 |
https://mathoverflow.net/questions/200519 | 5 | Let $X$ be a projective variety over a perfect field $k$. Recall that a *twist* of $X$ is a variety $Y$ over $k$ such that $$X\_{\bar k} \cong Y\_{\bar k}.$$
The twists of $X$ are classified by the Galois cohomology set $\mathrm{H}^1(k,\mathrm{Aut} X\_{\bar k}).$
My question concerns what happens when one considers ... | https://mathoverflow.net/users/5101 | Twists of projective automorphisms | Thanks to the hint from Ulrich, I think I am now able to answer the question.
The set $\mathrm{H}^1(k, \mathrm{Aut}(X\_{\bar k},L\_{\bar k}))$ classifes the following objects:
$k$-Isomorphism classes of pairs $(Y,E)$, where $Y$ is a projective variety over $k$ and $E \in \mathrm{Pic}\_{Y/k}(k)$, which become isomo... | 1 | https://mathoverflow.net/users/5101 | 201413 | 97,174 |
https://mathoverflow.net/questions/201382 | 5 | Let $P\_{n}(t)=\sum\_{k=0}^{n}\varepsilon\_{k}\cos(kt)$ where $\varepsilon\_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any $\delta\rightarrow 0$ we have
\begin{equation}
\int\_{-\pi}^{\pi}\mathbb{P}(|P\_{n}(t)|<\delta)\,dt \approx \frac{\d... | https://mathoverflow.net/users/24494 | Average probability that a random cosine polynomial with bernoulli coefficients is small | In general this is false, if $\delta$ is small enough. Indeed, let $Q\_n:=\sum\_{k=0}^{n}\varepsilon\_{k}$ and
$R\_{n}(t):=P\_{n}(t)-Q\_n$. Then $\mathsf{E}R\_{n}(t)=0$ and $\mathsf{Var}R\_{n}(t)\le\frac1{20}(n+1)^5t^4$.
Suppose now that $n$ is odd. Then
$$\mathsf{P}(|P\_{n}(t)|<\delta)\ge\mathsf{P}(Q\_n=0)-\maths... | 6 | https://mathoverflow.net/users/36721 | 201420 | 97,176 |
https://mathoverflow.net/questions/201116 | 3 | I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}\_p$). Let say that $\mathcal{O}\_D$ and $\mathcal{O}\_F$ are the maximal orders of $D$ and $F$, with uniformizers $P\_D$ and $P\_F = P\_D^d$. If $I\_D$ is the unique maxim... | https://mathoverflow.net/users/43737 | On conductors, levels and traces on quaternion algebras | I think I can give you at least a sketch of answer now, but I have not thought carefully about details, so you should check though things carefully.
Let $K$ be a cyclic extension of $F$ of degree $n$, and $\sigma$ a generator of the Galois group Gal($K/F$). Then we can write $D$ as a cyclic algebra $(K/F, \sigma, a)... | 1 | https://mathoverflow.net/users/6518 | 201437 | 97,182 |
https://mathoverflow.net/questions/201387 | 1 | Let $s \in \Sigma^\*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = \{\text{id}, \ (1,3), \ (2,4), \ (1,3)(2,4)\}$.
What about automorphisms of the associated grammars of strings. For in... | https://mathoverflow.net/users/40321 | The automorphism groups of smallest grammars of a language string are isomorphic | I think the grammars $A \to aaaaa$ and $A \to BaB$, $B \to aa$ are both minimal (5 symbols).
If I understand your definitions correctly, the automorphism group of the first one is $S\_5$ with $120$ elements, the automorphism group of the second is $S\_2\times S\_3$ with $12$ elements.
| 4 | https://mathoverflow.net/users/14915 | 201439 | 97,183 |
https://mathoverflow.net/questions/201424 | 26 | Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere differentiable?
| https://mathoverflow.net/users/6094 | Proof that no differentiable space-filling curve exists | The image of an interval under a Lipschitz map has finite $1$-dimensional Hausdorff measure.
EDIT: Here's a corrected version of Pablo Shmerkin's construction. Suppose $f: \mathbb R \to \mathbb R^d$ is differentiable.
For positive integers $m,n$ let $A\_{m,n} = \{x: |y -x| \le 1/n \implies \|f(y) - f(x)\| \le m |y ... | 24 | https://mathoverflow.net/users/13650 | 201442 | 97,184 |
https://mathoverflow.net/questions/201443 | 7 | Consider the following statements about a given set $X$ in in $\mathsf{ZF}$:
(1) There is $x\_0\in X$ such that there is a surjective map $\varphi: X\setminus\{x\_0\}\to X$.
(2) There is an injective map $\iota:\mathbb{N}\to X$.
It is easy to see that (2) implies (1) in $\mathsf{ZF}$, but are they equivalent?
| https://mathoverflow.net/users/8628 | Notions of infinity in $\mathsf{ZF}$ without choice | No, they are not equivalent.
It is a nice theorem that if there exists an infinite Dedekind-finite set (which is a set which satisfies the negation of (2)), then there is one which satisfies the first condition.
If $D$ is a Dedekind-finite set, then $S(D)$ which is the set of all injective finite sequences from $D$... | 14 | https://mathoverflow.net/users/7206 | 201447 | 97,186 |
https://mathoverflow.net/questions/201353 | 3 | I have asked this question in the Mathematics forum but I received no answer.
Let $E$ be an algebraic vector bundle of rank $r$ and degree $d$,
Then $\Lambda^2 E$ is of rank $r'=r(r-1)/2$, but is of determinant 0, because $$\Lambda^{r'}(\Lambda^2 E)\subset \Lambda^{r(r-1)}E=0$$
which I can't understand! because its de... | https://mathoverflow.net/users/66528 | Exterior product | Let me expand my comments into an answer.
Before thinking about vector bundles, it makes sense to see what happens with vector spaces. If $(a,b,c)$ is a basis for a $3$-dimensional vector space $V$, then $\bigwedge^{2}V$ has basis $(a \wedge b,b \wedge c,c \wedge a)$. Then the determinant $\bigwedge^{3}(\bigwedge^{2}... | 3 | https://mathoverflow.net/users/21815 | 201452 | 97,190 |
https://mathoverflow.net/questions/201431 | 3 | Faithfully flat Hopf-Galois extensions of rings: $A\to B$, with $H$ coacting on $B$ such that $B\otimes\_AB\simeq B\otimes H$, are often thought of as being accessible substitutes for $G$-torsors in the setting of noncommutative geometry, where $H$ is supposed to look like the Hopf-algebra of functions on a group schem... | https://mathoverflow.net/users/11546 | Classification of Hopf-Galois Extensions as Torsors | I think that the same proof that it is usually done for torsors will work for these "Hopf-torsors", at least when $H$ is of finite presentation over the base. Maybe you are able to remove that hypothesis too (if I understand correctly your motivation :))
Let me phrase that using the language of stacks. Let $H$ be an ... | 1 | https://mathoverflow.net/users/43054 | 201471 | 97,193 |
https://mathoverflow.net/questions/201459 | 6 | Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})\_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category (with the ordinary tensor product of functor $-\times -$, and an **internal hom** $sPh(\mathcal{C})[-,-]$) and a simplic... | https://mathoverflow.net/users/41970 | Monoidal structure on simplicial sheaves | Yes. This is well-known and can be deduced, for example, from the general statements in the last section of [this paper](http://arxiv.org/pdf/0708.2067v2.pdf) of Barwick. Take $V$ to be the symmetric monoidal model category of simplicial sets. Note that it is [tractable](http://ncatlab.org/nlab/show/tractable+model+cat... | 6 | https://mathoverflow.net/users/2503 | 201473 | 97,194 |
https://mathoverflow.net/questions/201469 | 4 | I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism.
I have a question about what he calls a "class III transformation", where, given a null tetrad $\lbrace l,n,m,\overline{m} \rbrace$ we can rescale the null directions and rota... | https://mathoverflow.net/users/51137 | Null tetrad transformation | From the equation you wrote down you have that the scalar product
$$(\nabla\_\ell \ell, n) = \pm (\epsilon + \bar{\epsilon})$$
(the $\pm$ is from the sign convention; I don't remember which one Chandrasekhar uses). So rescaling both $\ell$ and $n$ you get
$$ \nabla\_{A\ell} (A\ell) = A^2 \nabla\_\ell \ell + A (D A) \... | 3 | https://mathoverflow.net/users/3948 | 201474 | 97,195 |
https://mathoverflow.net/questions/201478 | 3 | We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.
Is there a maximal connected Hausdorff space with more than 1 point?
| https://mathoverflow.net/users/8628 | Is there a maximal connected Hausdorff space? | There is a maximal connected Hausdorff topology for the reals, according to [this paper](http://www.ams.org/journals/proc/1978-069-01/S0002-9939-1978-0467646-4/) of Guthrie, Stone and Wage.
I haven't read it, some googling led me to the paper "Problems on (ir)resolvability" of Oleg Pavlov in "Open problems in Topolo... | 5 | https://mathoverflow.net/users/29491 | 201484 | 97,198 |
https://mathoverflow.net/questions/201491 | 1 | I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of [Reference for Dieudonné modules](https://mathoverflow.net/questions/96507/reference-for-dieudonn%C3%A9-modules), but... the link proposed there are broken :'(
Tha... | https://mathoverflow.net/users/41314 | Dieudonné modules -reference request | Does <http://www.math.harvard.edu/~chaoli/doc/Dieudonne.html> help you (Chao Li's lecture notes)?
| 2 | https://mathoverflow.net/users/nan | 201495 | 97,201 |
https://mathoverflow.net/questions/201477 | 2 | We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.
If $(X,\tau)$ is connected, is there a topology $\tau' \supseteq \tau$ such that $(X,\tau')$ is maximal connected?
| https://mathoverflow.net/users/8628 | Maximal connected topologies | No. You can find a counterexample here: Baggs, Ivan. A connected Hausdorff space which is not contained in a maximal connected space. Pacific J. Math. 51 (1974), no. 1, 11--18
| 9 | https://mathoverflow.net/users/69907 | 201497 | 97,202 |
https://mathoverflow.net/questions/201489 | 10 | Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence [A130091](http://oeis.org/A130091) in OEIS. 24 such numbers are impossible because of $36n-6$ and $36n+6$.
| https://mathoverflow.net/users/60732 | Consecutive numbers with mutually distinct exponents in their canonical prime factorization | The answer to this question is almost certainly no.
It is well known that the ABC Conjecture implies that there are only finitely many triples $(n,n+1,n+2)$ which are all powerful. A similar argument should work for tuples of the form $(2n+1,2n+3,2n+5)$. But, as pointed out by Mostafa and Adam (in the comments above)... | 12 | https://mathoverflow.net/users/3199 | 201499 | 97,204 |
https://mathoverflow.net/questions/201445 | 5 | Let $X\_i$ be a smooth projective variety with a smooth divisor $D\_i$ for $i=1,2$.
Suppose that $D\_1$ is isomorphic to $D\_2$. Then does it make sense to construct a normal crossing variety $X=X\_1 \cup\_D X\_2 $ by 'pasting' $D\_1$ and $D\_2$, where '$\cup\_D$' means pasting along $D\_1$ and $D\_2$?
If the answer ... | https://mathoverflow.net/users/69621 | Constructing normal crossing varieties | Q1: Yes. For affine schemes, the gluing construction you need is just the [fiber product of rings](https://mathoverflow.net/a/101178/18060). Because gluing two affines in this way produces an affine, you can glue two schemes and get a scheme by taking an affine cover of each.
Q2: Yes, and the proof is more or-less co... | 6 | https://mathoverflow.net/users/18060 | 201508 | 97,208 |
https://mathoverflow.net/questions/201496 | 6 | I would like to show that
$$
\lim\_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum\_{j=0}^{p-1}(-1)^j\binom{p-1}{j}
\left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1}
=\frac1{np+1}\binom{(n+1)p}{p},
$$
for $p,n=1,2,\ldots$.
**Background**
The reason we expect this equality to hold is the following: The left hand sid... | https://mathoverflow.net/users/27058 | Combinatorial identity and Fuss-Catalan numbers | From taking derivatives of $(x-1)^{p-1}$ and setting $x = 1$, we have
$$\sum\_{i=0}^{p-1} j^k (-1)^j \binom{p-1}{j} = 0$$
for $k < p - 1$. The same method gives
$$(-1)^{p-1}(p- 1)! = \sum\_{i=0}^{p-1} j^{p-1} (-1)^j \binom{p-1}{j}.$$
Now consider $$\left(\frac{\Gamma(N+ p - j)}{\Gamma(N - j)}\right)^{n+1} = (N + p - j ... | 4 | https://mathoverflow.net/users/69868 | 201512 | 97,212 |
https://mathoverflow.net/questions/201422 | 7 | I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every connection there is a curvature $F... | https://mathoverflow.net/users/66688 | Curvature of a principal bundle and the exterior covariant derivative | First, there is a canonical isomorphism $\Phi$ between $\Omega^k(M; Ad P)$ and $\Omega^k\_{Ad, h}(P; \mathfrak{g})$, where the subscripts signify that the form is horizontal and of type $Ad$ (i.e. equivariant). This isomorphism is canonical and does not need a local trivialization. To illustrate the idea, consider the ... | 5 | https://mathoverflow.net/users/17047 | 201523 | 97,219 |
https://mathoverflow.net/questions/201506 | 17 | By analogy:
The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell neatly into place.
Do you have examples from pure math? Again, you have a theory, it's OK in the sense that it gives ... | https://mathoverflow.net/users/11504 | "Epicycles" (Ptolemy style) in math theory? | Euler found values of the Riemann zeta-function by artful manipulations of divergent series, e.g., interpreting a function that's $(-1)^{n/2}$ at even $n > 0$ and $0$ at odd $n > 0$ as $\cos(\pi n/2)$. The calculations were later justified by analytic continuation of the zeta-function from the right half-plane ${\rm Re... | 10 | https://mathoverflow.net/users/3272 | 201524 | 97,220 |
https://mathoverflow.net/questions/201501 | 5 | Let $G$ be a connected, simply connected, semisimple, complex linear algebraic group with maximal torus $T$ and affine Grassmannian $\mathcal Gr$. It is well known that $\mathcal Gr$ admits a Bruhat decomposition
$$ \mathcal Gr = \bigsqcup\_{\lambda\in X\_\*(T)} \mathcal B \lambda $$
where $\mathcal B$ is the Iwahori ... | https://mathoverflow.net/users/34766 | Singular/Smooth locus of Schubert variety of the affine grassmannian | The smooth locus of a *spherical* orbit (i.e. $G(\mathcal O)$-orbit) on the affine Grassmannian is just the spherical orbit itself (but this orbit consists of several Iwahori-orbits). See
Malkin-Ostrik-Vybornov,
The minimal degeneration singularities in the affine Grassmannians,
Duke Math. J. 126 (2005), no. 2, 233–2... | 7 | https://mathoverflow.net/users/14154 | 201529 | 97,221 |
https://mathoverflow.net/questions/201536 | 42 | I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$:
**Question:** We have some totally-ordered set $T$ of mathematicians, each wearing a hat which is either red or blue. The mathematician $x$ can see the colour of $y$'s hat if and only if $x < y$. The mathem... | https://mathoverflow.net/users/39521 | Mathematicians wearing hats on arbitrary total orders | It's a great problem!
**Theorem.** The mathematicians have a winning strategy in the game for every ordinal $\alpha$.
**Proof.** Let's prove the theorem by transfinite induction. Suppose that the mathematicians have winning strategies in the games of any particular length $\beta$ less than $\alpha$, and let us fix ... | 48 | https://mathoverflow.net/users/1946 | 201539 | 97,224 |
https://mathoverflow.net/questions/201503 | 7 | Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta\_0$, say $P\_{\theta\_0}$, and we want to estimate the value of $\theta\_0$ using Bayesian method. Let $\pi(\theta)$ be the prior over the possible values of $\theta$, $\Theta$.
I understand that for almost all priors, if the ... | https://mathoverflow.net/users/64552 | Rate of convergence of Bayesian posterior | One can measure the rate of convergence of the posterior distribution with density $p\_n$ to the Dirac probability distribution at $\theta\_0$ by how large the ratios $p\_n(\theta\_0)/p\_n(\theta)$ are for $\theta\ne\theta\_0$, where $\theta\_0$ is the "true" value of the parameter. One has
$$
\frac{p\_n(\theta\_0)}{p... | 4 | https://mathoverflow.net/users/36721 | 201545 | 97,227 |
https://mathoverflow.net/questions/201444 | 5 | Let $R$ be a ring (not necessarily commutative or unital) that is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must hold. Here I use $\text{Ann}(R)$ to denote the set of all elements $r\in R$ such that $rR=Rr=0$. All I knew is that it holds when $R$ is commutative.
| https://mathoverflow.net/users/69184 | If $R$ is generated by idempotents, then $\text{Ann}(R)=0$? | No, $\mathrm{Ann}(R)$ does not necessarily hold when $R$ is generated as a ring by idempotents.
Let $K$ be a field, or more generally any commutative (associative) ring with 1. Let $R$ be the (associative, non-unital) $K$-algebra of matrices $m(e,a,b,c)=\begin{pmatrix}0 & a & c\\0 & e & b\\ 0 & 0 & 0\end{pmatrix}$ wi... | 9 | https://mathoverflow.net/users/14094 | 201562 | 97,231 |
https://mathoverflow.net/questions/201532 | 2 | I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph and let $b$ be its first Betti number. We choose $b$ directed cycles in $G$, which form a homology basis. Further, we wr... | https://mathoverflow.net/users/61532 | A certain matrix associated to graphs | For a connected $G$, in the case of the cycles being chosen in the usual way --- one for each edge of $G$ outside a spanning tree $\Theta$ of $G$, this matrix can be identified with a submatrix of $A^\top A$, for $A$ the oriented [incidence matrix](http://en.wikipedia.org/wiki/Incidence_matrix) of $G$. That is, $A$ is ... | 5 | https://mathoverflow.net/users/11100 | 201568 | 97,233 |
https://mathoverflow.net/questions/201576 | 0 | Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not.
The equation $x\_1^m+\cdots+x\_n^m=1$ has nonzero rational solutions iff $n\geq m$.
Here a nonzero rational solution means nonzero $y\_1,\cdots,y\_n\in\mathbb{Q}$ satisfying the above equation.
When $n=2$, the above conjec... | https://mathoverflow.net/users/7360 | A stronger version of Fermat's last theorem | This is a very well-known false conjecture [due to Euler](http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture).
| 20 | https://mathoverflow.net/users/14901 | 201577 | 97,235 |
https://mathoverflow.net/questions/201573 | 1 | Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}\_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}\_R(R[X])$ must satisfy $\sigma(X)=f(X)(X-a)$ (where the constant term of $f$ is invertible and the coefficients of other terms are... | https://mathoverflow.net/users/69184 | Structure of $\text{Aut}_R(R[X])$ | This is solved by R. Gilmer in his paper *$R$-automorphisms of $R[X]$* appared in the *Proceedings of London Mathematical Society*, (3) 18, pp. 328-33, (1968).
If you want to replace the polynomal ring $R[X]$ by the formal power series $R[[X]]$, look at M. O'Malley and C. Wood's *$R$-endomophisms of $R[[X]]$* in *Jou... | 4 | https://mathoverflow.net/users/18238 | 201592 | 97,242 |
https://mathoverflow.net/questions/201580 | 5 | Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H\_{2r}(S)$ be the fundamental class of $S$.
1. We have the integration current $T\_{S}$ associated to $S$ defined by
$$ \langle T\_{S}, \omega \rangle = \int\_{S} \omega $$
for any $... | https://mathoverflow.net/users/69938 | Integration currents vs Poincaré dual | Here is briefly the story. More details can be found in DeRham's monograph *Variétés differentiables*.
Let $M$ be a smooth, compact, oriented, $m$-dimensional manifold. Denote by $\Omega^k(M)$ the space of smooth degree $k$-forms on $M$ and by $\Omega\_k(M)$ its dual space, namely the space of $k$-dimensional current... | 8 | https://mathoverflow.net/users/20302 | 201595 | 97,244 |
https://mathoverflow.net/questions/84388 | 7 | The Clifford group $\mathcal{C}\_n$ is a matrix group on $\mathbb{C}^{2^n}$ generated by tensor products of the following matrices:
$$
P = \begin{pmatrix} 1 & 0 \\\\ 0 & i\end{pmatrix}
\quad
H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\\\ 1 & -1\end{pmatrix}
\quad
\wedge\\! X = \begin{pmatrix}1 & 0 & 0 & 0 \\\\ 0 & 1 ... | https://mathoverflow.net/users/2138 | Presentation of the Clifford group by generators and relations? | In the time since the question was asked this paper appeared which does exactly what I want:
[Generators and relations for n-qubit Clifford operators](http://arxiv.org/abs/1310.6813)
| 3 | https://mathoverflow.net/users/2138 | 201599 | 97,246 |
https://mathoverflow.net/questions/201561 | 1 | Let's call a space $(X,\tau)$ *minimal $T\_0$* if it is $T\_0$ and for every topology $\sigma\subseteq \tau$ with $\sigma \neq \tau$ we have that $(X,\sigma)$ is not $T\_0$ any more.
We say $x\leq y$ in a $T\_0$-space $(X,\tau)$ if and only if every open neighborhood of $x$ is an open neighborhood of $y$. It's easy t... | https://mathoverflow.net/users/8628 | Is the associated order of a minimal $T_0$ space always total? | This conjecture is correct. I claim that if the topological space is minimal $T\_{0}$, then the specialization ordering is total. We shall prove this result by contrapositive, so we shall show that if the specialization ordering is not linear, the the topology is not a minimal $T\_{0}$. The idea in this proof is simila... | 4 | https://mathoverflow.net/users/22277 | 201604 | 97,248 |
https://mathoverflow.net/questions/201538 | 0 | I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference:
Let $X$ be a $\mathbb{Q}$-factorial variety with log canonical singularities, and $Z \subseteq X$ be a subvariety. Then the set $$\{E \mid E {\rm{~is~ an~ exceptional~ divis... | https://mathoverflow.net/users/29730 | Finiteness of geometric valuations | Ok, let me show the the following which is well known to experts (although I'm not sure where the right reference is):
**Proposition:** *If $(X, \Delta)$ is KLT (or PLT), then there are at most finitely many divisorial valuations corresponding to divisors $E$ on birational models of $X$ such that the discrepancy of $... | 3 | https://mathoverflow.net/users/3521 | 201609 | 97,250 |
https://mathoverflow.net/questions/201608 | 3 | Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A\_n)\_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap\_{n=1}^\infty A\_n = \emptyset$. Given $\epsilon > 0$ does there exist an open set $U \subseteq [0,1]$ such that
(i) $\lambda(A\_n \setminus U) = 0$ for some $... | https://mathoverflow.net/users/30721 | Is every set of small measure contained in an open set of small measure with null boundary? | No, such a $U$ does not exist in general. You can have (i)+(ii), but (iii) is too much to ask. The problem is that the $A\_n$ might all be dense in $[0,1]$ so that $\partial U = \overline{U} \setminus U =[0,1]\setminus U$ has measure $\geq 1-\epsilon$ by (ii).
| 5 | https://mathoverflow.net/users/3041 | 201610 | 97,251 |
https://mathoverflow.net/questions/201606 | 6 | It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already provable in IMLL.
Who first proved this, and how? It doesn’t seem to be in Girard’s original Linear Logic paper, yet ... | https://mathoverflow.net/users/8217 | Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic | This is (a piece of) Proposition 3.8 in
* Harold Schellinx, Some Syntactical Observations on Linear Logic, J. Logic Compututa., Vol. 1 No. 4, pp. 537-559, 1991. ([pdf at oxford journals](http://logcom.oxfordjournals.org/content/1/4/537.full.pdf))
His proof-theoretic argument is that the only way to use a sequent w... | 7 | https://mathoverflow.net/users/1015 | 201612 | 97,253 |
https://mathoverflow.net/questions/201482 | 2 | Consider $(X,\sigma\_X)$ and $(Y, \sigma\_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma\_X)$ is a transitive subshift of finite type and $(Y, \sigma\_{Y})$ is a transitive sofic subshift such that $h\_{top}(\sigma\_X) < h\_{top}(\Sigma\_{Y})$.
My question is: What are the conditions... | https://mathoverflow.net/users/10518 | Embeddings of subshifts | Take a look at Klaus Thomsen's paper: On the structure of a sofic shift space. He discusses generalizations of Krieger's embedding theorem and gives conditions when it still holds respectively when it fails.
Not sure what your last condition (the intersection of periodic point sets is empty? for some $n$ or for all $... | 2 | https://mathoverflow.net/users/52920 | 201617 | 97,254 |
https://mathoverflow.net/questions/201368 | 6 | Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ are well-pointed compactly generated Hausdorff as topological spaces, where well-pointedness means that the inclusions of... | https://mathoverflow.net/users/68660 | Classifying spaces of topological groups whose underlying spaces are homotopy equivalent | As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.
Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.
We define $BG$ for a topological group $G$ to be the [fat realization](http://ncat... | 6 | https://mathoverflow.net/users/32022 | 201623 | 97,256 |
https://mathoverflow.net/questions/201613 | 1 | Given a non-acyclic quiver without loops with Kac's root system associated. When do we know there are infinitely many real roots?
| https://mathoverflow.net/users/69953 | Infinitely many real roots | Always. If the quiver has a cycle, then the root system is not of finite type (since the quivers corresponding to finite type root systems are trees), and any root system not of finite type has an infinite number of real roots (more or less by definition).
| 2 | https://mathoverflow.net/users/468 | 201633 | 97,261 |
https://mathoverflow.net/questions/201631 | 14 | I've been looking for any kind of universal coefficient theorem for group homology and cohomology, including dual universal coefficient theorems. However, the only things I can find are ones where the group action on the coefficients is trivial. As such, my question is:
Let $G$ be a group, $M$ be a $G$-module. Is the... | https://mathoverflow.net/users/44191 | Universal coefficient theorem for group homology and cohomology | You're only finding UCT in the literature for trivial group actions, *because there is no general UCT for nontrivial group actions*:
The general Kunneth formula does not hold for arbitrary groups and actions. But it does hold a good amount of times, and I elaborated on this here: [Kuenneth-formula for group cohomolog... | 10 | https://mathoverflow.net/users/12310 | 201645 | 97,262 |
https://mathoverflow.net/questions/201658 | -2 | Let $(X\_n)$ and $(Y\_n)$ be two sequences of random variables defined on the same probability space such that the variance of all components $X\_n$, $Y\_n$ is finite and the sequence of variances of $X\_n+Y\_n$ is non-zero and bounded. Does this imply that the variance sequences of $X\_n$ and $Y\_n$ are bounded as wel... | https://mathoverflow.net/users/57022 | If $(X_n+Y_n)$ has bounded variance, is the same true for $(X_n)$ and $(Y_n)$? | They do. Simply choose, say, $X\_n$ a Bernoulli random variable taking values $n$ or $-n$ with probability $\frac 12$ for each. Notice that $Var(X\_n) = n^2$. Define $Y\_n = - X\_n$. You have $Var(X\_n + Y\_n) = 0$ and, to answer your last interrogation, $Cov(X\_n,Y\_n) = - Var(X\_n) = -n^2$.
**Edit** : Let now $Z$ b... | 0 | https://mathoverflow.net/users/62629 | 201660 | 97,267 |
https://mathoverflow.net/questions/201557 | 3 | Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is *splittable* if there are $S,T\subseteq V$ with $S\cap T=\emptyset$ and $S\cup T = V$ such that
1. for all $s\in S$ we have $|N(s)\cap S| \leq |N(s) \cap T|$, and symmetrically
2. for all $t\in T$ we have $|N(t)\cap T| \l... | https://mathoverflow.net/users/8628 | Infinite non-splittable graphs | These partitions are normally called *unfriendly*. Every finite graph has an unfriendly partition (choose a partition that has the maximum possible number of cross edges). It was conjectured by Cowan and Emerson that every graph should have an unfriendly partition, but Milner and Shelah found an uncountable counterexam... | 5 | https://mathoverflow.net/users/25485 | 201666 | 97,270 |
https://mathoverflow.net/questions/201672 | 2 | A $k$-ary function $f$ on a bounded distributive lattice
$L$ is called compatible if for any congruence relation $\theta$ on $L$ and $(a\_i, b\_i)\in \theta$ for $i=1,\ldots,k$ we always have $(f(a\_1,\ldots,a\_k), f(b\_1,\ldots,b\_k))\in\theta$.
It's easy to see that all polynomials are compatible.
A lattice $L$ i... | https://mathoverflow.net/users/nan | Is $[0,1]^\kappa$ an affine complete lattice? | The answer is yes, and a key ingredient is the following theorem due to George Grätzer:
>
>
> >
> > A bounded distributive lattice is affine complete if and only i
> > f it does not
> > contain a proper interval that is a Boolean lattice in the ind
> > uced order. (G. Grätzer, *Boolean functions on distributiv... | 2 | https://mathoverflow.net/users/8628 | 201673 | 97,271 |
https://mathoverflow.net/questions/201654 | 2 | Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
**Question 1.** Is it true that every point of $X$ has a neighborhood $U$ such that any two points from $U$ can be connected by at most one shortest path (which does not have... | https://mathoverflow.net/users/16183 | Shortest paths in Alexandrov spaces | Counterexample: the vertex p of the cone X over the circle of length less than 2π has no such neighborhood U.
| 6 | https://mathoverflow.net/users/1988 | 201679 | 97,273 |
https://mathoverflow.net/questions/201683 | 3 | Algebraic $K3$ surface means the $K3$ surface admits an ample line bundle. So the question is equivalent to asking whether every algebraic $K3$ surface can be embedded in $\mathbb{P}^3$.
| https://mathoverflow.net/users/43423 | Is every algebraic $K3$ surface a quartic surface? | No. Consider a K3 surface with a polarization of degree 2 and with Picard rank 1. Since the tautological line bundle on $\mathbb{P}^3$ pulls back to a degree 4 line bundle, it follows that such a K3 surface cannot be embedded in $\mathbb{P}^3$.
| 15 | https://mathoverflow.net/users/1703 | 201684 | 97,275 |
https://mathoverflow.net/questions/201421 | 11 | This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c\_n)\_n$ and we want to find an asymptotic formula for $S(x) = \sum\_{n \leq X} c\_n$. In favorable circumstances, standard Tauberian theorems, e.g. the one in Appendix A of Chambert-Loir and Tschinkel's paper "Fonctions zeta des haut... | https://mathoverflow.net/users/10458 | Tauberian theorem with better error term | If your zeta function $Z(s)$ has analytic continuation to the left of your pole $a$ (with some reasonable bounds etc.), then you can just use an inverse Mellin transform, no? The main term is the residue of $X^s Z(s)$ at $s=a$, so all you need is the Taylor series expansion of $X^s$ at $s=a$ (which gives the powers of ... | 6 | https://mathoverflow.net/users/11919 | 201685 | 97,276 |
https://mathoverflow.net/questions/201688 | 8 | There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
| https://mathoverflow.net/users/69999 | What's the name of this geometric mathematical modeling problem? | [The Moving sofa problem](http://en.wikipedia.org/wiki/Moving_sofa_problem), I believe.
| 8 | https://mathoverflow.net/users/60810 | 201690 | 97,279 |
https://mathoverflow.net/questions/201218 | 3 | *Definitions*: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the *fixed-point subspace* $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let the *pointwise stabilizer subgroup* $G\_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.
Let $G$ be... | https://mathoverflow.net/users/34538 | A problem with pointwise stabilizer subgroups of fixed-point subspaces II | **No**, GAP has found the following counter-example.
`gap> G:=TransitiveGroup(16,39);`
`gap> H:=Stabilizer(G,1);`
$G$ and $H$ are order $32$ and $2$.
`gap> R:=IrreducibleRepresentations(G);`
`gap> U:=R[9];`
`gap> V:=R[11];`
$U$ and $V$ are degree $2$.
Now $G\_{(U^H)}$ and $G\_{(V^H)}$ are order ... | 1 | https://mathoverflow.net/users/34538 | 201693 | 97,281 |
https://mathoverflow.net/questions/201682 | 0 | Let $(N \subset M)$ be a finite index irreducible subfactor, $P=P(N \subset M)$ its planar algebra.
*Notation*: For $a,b \in P\_{2,+}$ positive operators, then $\langle a,b \rangle$ is the *biprojection* they generate.
*Property (F)*: $\forall a , b \in P\_{2,+}$ min. proj., $\exists c \in P\_{2,+}$ min. proj. su... | https://mathoverflow.net/users/34538 | A short problem with minimal projections and biprojections | **No** there is a group-subgroup counter-example.
*Property (F')*: $\forall p , q $ min. central proj., $\exists r $ min. central proj. such that $\langle r,p \rangle , \langle q,r \rangle \ge \langle p,q \rangle$
*Lemma*: Let $p$ be a min. central proj., then $\exists a \le p$ min. proj. such that $\langle p \ra... | 0 | https://mathoverflow.net/users/34538 | 201695 | 97,282 |
https://mathoverflow.net/questions/200712 | 5 | I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from experts. It ended in a nice argument, again about the Jacobson Radical, proposed an proved in the following, but the main ... | https://mathoverflow.net/users/47638 | Why Jacobson, but not the left (right) maximals individually? | Dag has already answered the case where the quiver is finite and acyclic, and given a conjecture in the case that cycles are allowed. I will prove his conjecture.
Suppose we have an element $x$ of the Jacobson radical. We want to show that
it is generated by arrows not lying on any cycle. We can therefore throw
away... | 4 | https://mathoverflow.net/users/468 | 201698 | 97,285 |
https://mathoverflow.net/questions/201701 | 1 | Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as $x\to\infty$ uniformly for $u$ over $[0,a]$.
$f$ is actually called regularly (slowly) varying function. The usual assumpti... | https://mathoverflow.net/users/32660 | Local Uniform Convergence | Fix $\varepsilon>0$. For any positive integer $n$ consider the (closed) set $E\_n$ of $u\in [0,4a]$ such that $|f(x+u)-f(x)|\leq \varepsilon$ for all $x\geq n$. Then $[0,4a]=\cup\_n E\_n$, thus there exists $n$ such that $\mu(E\_n)>3a$. Then for any $b\in [0,a]$ we have $\mu(E\_n\cap [0,3a])>2a$, hence sets $B=E\_n\cap... | 1 | https://mathoverflow.net/users/4312 | 201710 | 97,289 |
https://mathoverflow.net/questions/201178 | 1 | I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum mechanics rigorously?
Is there any book on the market that you can recommend? Sometimes, I feel it is nice to have a rigor... | https://mathoverflow.net/users/69763 | Mathematical statistical qm book-recommendation | The two volumes by Bratteli and Robinson.
The book on lattice gases by Simon also has material on quantum statistical mechanics.
| 4 | https://mathoverflow.net/users/7410 | 201714 | 97,291 |
https://mathoverflow.net/questions/201700 | 5 | Is there any reference where I can find the character table of $\mathrm{SL}\_2(\mathbb{Z}/p^n\mathbb{Z})$? A simple search in google gave me this paper of Philip C. Kutzko on "[The characters of the binary modular congruence group](http://projecteuclid.org/euclid.bams/1183534742)". But this is just an announcement and ... | https://mathoverflow.net/users/8419 | Character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$ | Unlike the well-known case of these groups over a prime field, it's probably asking too much to exhibit full character tables over all such finite rings. (Note too that Kutzko limits his discussion to odd primes.)
There have been related discussions over the years of representations of the finite groups coming from $... | 3 | https://mathoverflow.net/users/4231 | 201721 | 97,293 |
https://mathoverflow.net/questions/201708 | 18 | Does there exist a probability distribution on $\mathbb{Z}$ such that
for every integer $n\geq 1$, the probability that a random integer $x$
is divisible by $n$ equals $1/n$?
Henry Cohn has an argument why this is not possible, but it is not
completely rigorous. First, it is easy to see that we can assume that
the di... | https://mathoverflow.net/users/2807 | Existence of a "quasi-uniform" probability distribution on $\mathbb{Z}$ | No. Let's restrict our attention to $\mathbb{N}$. The hypotheses imply that if $q$ is a prime, then the probability that a random positive integer is not divisible by $q$ is $1 - \frac{1}{q}$. They also imply that these events are independent. Now let $n$ be a positive integer. If $q\_1, q\_2, \dots$ is an enumeration ... | 25 | https://mathoverflow.net/users/290 | 201725 | 97,297 |
https://mathoverflow.net/questions/201718 | 33 | The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me while I explain the intended meaning.
I have taught multi-variable calculus several times. As usual of such repetition... | https://mathoverflow.net/users/29316 | Historical (personal) examples of teaching-based research | The first time I taught forcing, I wanted to mention, as motivation, the fact that the independence of the continuum hypothesis (CH) or even of the axiom of constructibility (V=L) cannot be proved by the method of inner models. That fact was proved in Cohen's book, "Set Theory and the Continuum Hypothesis" under the as... | 35 | https://mathoverflow.net/users/6794 | 201729 | 97,300 |
https://mathoverflow.net/questions/201728 | 37 | A brief description: I have written a paper which contains a new result which I believe is somewhat important but not vital to the field. It is a generalization of an existing proof to get significant new information, in a framework that did not exist at the time of the original paper (not by me). I do not believe the ... | https://mathoverflow.net/users/15735 | Should one post a paper on the arXiv if it is not intended to be published? | Based on what you say, your paper would be valuable and useful for the mathematical community. So I think you should put it on the arXiv, with the remark in the comment field that the paper is not intended for publication.
**Update.** I meant "the paper is not intended for publication in a journal". Thanks for the co... | 35 | https://mathoverflow.net/users/11919 | 201731 | 97,301 |
https://mathoverflow.net/questions/201674 | 2 | I'm trying to understand Böckle's proof of Theorem 2.1.1 in [his notes on deformation theory](http://www.iwr.uni-heidelberg.de/groups/arith-geom/boeckle/Deformations-Barca.pdf).
Let's start with some motivation. Let $\Gamma$ be a profinite group (I'm thinking of an absolute Galois group), $k$ a finite field, and $\b... | https://mathoverflow.net/users/6856 | Representability of deformation functors via SGA | It turns out that the only thing Böckle is using is the smoothness of $\widehat{\mathrm{PGL}}\_d$. I claim that the following more general result is true.
>
> Suppose $R\rightrightarrows X$ is an equivalence relation in $\mathsf{Vaf}$, for which one of the arrows is flat and one is smooth. Then $X/R$ (categorical ... | 1 | https://mathoverflow.net/users/6856 | 201733 | 97,303 |
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