parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/149651 | 3 | Consider the space of $n \times n$ positive definite symmetric matrices and let $\Sigma$ be one such matrix. We make this space into a Riemannian manifold $M$ by means of the metric $$ds^2=tr(\Sigma^{-1}d\Sigma\Sigma^{-1}d\Sigma)$$
Now fix a $p<n$ and consider the collection $\Phi\_p$ of all positive definite symmetric... | https://mathoverflow.net/users/42140 | A submanifold of the space positive definite matrices | This Riemannian metric on the full space of all positive definite matrices turns up in the paper (and others)
* [Olga Gil-Medrano, Peter W. Michor: The Riemannian manifold of all Riemannian metrics. Quarterly J. Math. Oxford (2) 42 (1991), 183--202](http://www.mat.univie.ac.at/~michor/rie-met.pdf)
There are explici... | 1 | https://mathoverflow.net/users/26935 | 150271 | 80,311 |
https://mathoverflow.net/questions/150259 | 14 | Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of $\mathcal{U}$-small sets (which is also $\mathcal{V}$-small). Suppose that $\mathcal{E}$ is a category which can obtained as a left exact ... | https://mathoverflow.net/users/4528 | What properties do "large topoi" share with actual topoi? | By definition, a **$\mathbf{U}$-pretopos** is a category $\mathcal{C}$ that satisfies Giraud's axioms *except* for the existence of topological generators, i.e.
* $\mathcal{C}$ has finite limits,
* $\mathcal{C}$ is a $\mathbf{U}$-extensive category, i.e. $\mathcal{C}$ has coproducts for $\mathbf{U}$-small families of... | 9 | https://mathoverflow.net/users/11640 | 150273 | 80,312 |
https://mathoverflow.net/questions/150251 | 4 | Let $C$ be a category and assume either that $C$ has all binary pullbacks or that $C$ satisfies [right calculus of fractions](http://ncatlab.org/nlab/show/calculus+of+fractions). In both cases the localization of $C$ at every morphism (i.e. the groupoidification) can be represented by spans, i.e the objects are the sam... | https://mathoverflow.net/users/27923 | Relationship between pullbacks and the Ore condition | More generally: let $C$ have pullbacks and consider a class of morphisms $W$ containing the isos, pullback stable and satisfying 2 from 3. Benabou call this a pullback congruence. Lemma 1.2 of his paper "Some remarks on 2-categorical algebra" shows that such $W$ admits a calculus of right fractions.
Even better the c... | 5 | https://mathoverflow.net/users/8751 | 150275 | 80,314 |
https://mathoverflow.net/questions/150178 | 3 | For a $2$-edge-connected simple graph $G$ and a tree $T$ of $G$, let $C\_e$ be the unique cycle in $T + e$, $e \in E(G) - E(T)$. Define the set $\mathcal{C}(T) = \{C\_e | e \in E(G) - E(T)\}$.
Now given a set of cycles $\mathcal{C}$ of $G$, is it possible to decide if $\mathcal{C}$ = $\mathcal{C}(T)$ for some tree $T... | https://mathoverflow.net/users/23850 | Fundamental Cycles of a graphs | I had a complicated algorithm based on matroid intersection here earlier, but there's a much simpler answer: a cycle basis is fundamental if and only if each cycle includes at least one edge that is not part of any other cycle. And it's easy to see that any collection of cycles with this property is automatically linea... | 2 | https://mathoverflow.net/users/440 | 150279 | 80,317 |
https://mathoverflow.net/questions/150285 | 1 | This might be a easy question, but I couldn't get the point.
Let $F$ be a p-adic field, $\bar{F}$ a separable algebraic closure of $F$. Set $\Omega\_F=Gal(\bar{F}/F)$. Use $F\_{\infty}\subset \bar{F}$ to denote the maximal unramified extension of $F$. Use $E\_{\infty} \supset F\_{\infty} $ to denote the maximal tamel... | https://mathoverflow.net/users/1832 | a question about a result in Bushnell-Henniart book 'the local Langlands conjecture for GL(2)' | Your real question is : When we have a short exact sequence of (multiplicatively written) groups $1\to C\to E\to G\to1$ in which $C$ is commutative, how does $G$ act on $C$ "by conjugation" ?
The answer is that you make $g\in G$ act on $C$ by $c\mapsto\tilde g c\tilde g^{-1}$, where $\tilde g\in E$ is any lift of $g$... | 7 | https://mathoverflow.net/users/2821 | 150287 | 80,321 |
https://mathoverflow.net/questions/143531 | 2 | I'm using the following result in a computer science paper:
Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let
$$V^\perp = \{u \in (\mathbb{Z}/q\mathbb{Z})^n : \forall v \in V \quad v \cdot u = 0\}$$
where $v \cdot u = v\_1 u\_1 + \ldots + v\_n u\_n \te... | https://mathoverflow.net/users/40659 | Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$ | As explained in [this answer](https://mathoverflow.net/a/81938/42091), the property $V=(V^\perp)^\perp$ holds for submodules $V$ of $(\mathbb Z/q\mathbb Z)^n$ because you can use Smith normal form to find a suitable generating matrix for $V^\perp$. If you just want a reference for the result, I can only suggest the fol... | 2 | https://mathoverflow.net/users/42091 | 150306 | 80,327 |
https://mathoverflow.net/questions/150307 | 7 | Let $V =\{ f | \exists \_{\alpha<\omega\_1} ( f:\alpha \rightarrow \mathbb{N} \wedge f $ is $ 1-1) \}$. We define $E\subseteq [V]^2 $, such that $\forall\_{f,g\in V } (<f,g>\in E \longleftrightarrow ( f\subseteq g \vee g\subseteq f))$. We need to show that the chromatic number of $G=(V,E)$ is bigger than $\aleph\_ 0$. ... | https://mathoverflow.net/users/43455 | How to show that the chromatic number > aleph_0 | Consider any coloring $\varphi:V\to\omega$. Construct a strictly decreasing sequence $\langle X\_n:n\lt\omega\rangle$ in $\mathcal P(\mathbb N)$ and a strictly increasing sequence $\langle f\_n:n\lt\omega\rangle$ in $V$ so that for each $n\in\omega$ we have:
* $\text{range}(f\_n)\subset X\_n$,
* $|X\_n\setminus\text... | 10 | https://mathoverflow.net/users/43266 | 150317 | 80,330 |
https://mathoverflow.net/questions/150288 | 5 | **Definition 1:** A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has an "initial member" $M$ if each member of $\mathcal{K}$ is a forcing extension of $M$ for some partial order $\mathbb{P}\in M$ and some $\mathbb{P}$-generic $G$ over $M$.
**Definition 2:** An extension $T$ of $\text{ZF}$ has an... | https://mathoverflow.net/users/nan | Does ZF have an initial model? | Your question has a certain affinity with the concept of a *solid bedrock* model, which arises in the theory of [set-theoretic geology](http://jdh.hamkins.org/set-theoreticgeology/). Namely, $W$ is *bedrock* for $V$ if $V$ is a forcing extension of $W$ and $W$ satisfies the [ground axiom](http://en.wikipedia.org/wiki/G... | 7 | https://mathoverflow.net/users/1946 | 150319 | 80,332 |
https://mathoverflow.net/questions/139935 | 1 | Let $\alpha>0$ and $X$ be an $\alpha$-normal (meaning, for $x,y\in X$,
$0\leq x\leq y$ implies $\|x\|\leq\alpha\|y\|$) ordered Banach space
with closed generating cone $X\_{+}$. If $X$ is reflexive, then every
pair of elements from $X$ has a minimal upper bound, i.e., for $x,y\in X$,
there exists some $z\in X$ with $\{... | https://mathoverflow.net/users/12248 | Does every $\alpha$-normal ordered Banach space have minimal upper bounds? | The answer is that the reflexivity assumption cannot be dropped.
The following simple example (due to Tony Wickstead) is a 1-normal non-reflexive space with closed and generating cone, where there exists a pair of elements without minimal upper bound:
>
> Let $X$ be the space of all convergent sequences where $x\... | 1 | https://mathoverflow.net/users/12248 | 150323 | 80,333 |
https://mathoverflow.net/questions/150301 | 8 | I would like to have an explicit description of the left/right [Kan lift](http://ncatlab.org/nlab/show/Kan+lift) of a functor $F$ through $G$, $\text{Lift}\_GF$/$\text{Rift}\_GF$ in terms of coends/ends (this can be done for Kan *extensions*, so I hardly believe there is no way to dualize the argument).
Google is of ... | https://mathoverflow.net/users/7952 | (Fri)end(l)y way to express Kan liftings | Such a characterisation is not possible in $\mathbf{Cat}$ for trivial reason --- the dual of a functor generally cannot be thought of as a functor. In an analogical situation in $\mathbf{Set}$, for exactly the same reason, we do not expect that every colimit can be expressed in a canonical way as a limit.
For exampl... | 10 | https://mathoverflow.net/users/13480 | 150325 | 80,335 |
https://mathoverflow.net/questions/150305 | 6 | Makkai and Paré introduced the following binary relation on regular cardinals: given $\kappa$ and $\lambda$, $\kappa \vartriangleleft \lambda$ (read, $\kappa$ is **sharply less than** $\lambda$) when $\kappa < \lambda$ and, for every set $X$ of cardinality $< \lambda$, the set $P\_\kappa (X)$ of all subsets of $X$ of c... | https://mathoverflow.net/users/11640 | When are all greater cardinals sharply greater? | First note that if $\kappa \vartriangleleft \mu^+$ then $\mu^{\lt\kappa} = |P\_\kappa(\mu)| \leq 2^{\lt\kappa}\cdot\mu$. Therefore $\mu^{\lt\kappa} = \mu$ if $\kappa \vartriangleleft \mu^+$ and $2^{\lt\kappa} \leq \mu$. This is impossible if $\operatorname{cf}(\mu) \lt \kappa$ by [König's Theorem](http://en.wikipedia.o... | 4 | https://mathoverflow.net/users/2000 | 150332 | 80,338 |
https://mathoverflow.net/questions/150327 | 0 | Let $W(p,q,r,s)$ be the number of permutations of the letters which satisfy the following conditions :
**Condition 1** : The letters are consist of $P,Q,R,S$.
**Condition 2** : The number of letter $P,Q,R,S$ is $p,q,r,s$ respectively.
**Condition 3** : Any two adjacent letters are different from each other.
**... | https://mathoverflow.net/users/34490 | Permutations of letters under some conditions | Let $W\_P(p,q,r,s)$ be the number of such words starting by $P$ and *ending* by $P$, and let
$$
f\_P(x,y,z,t)=\sum\_{p,q,r,s}W\_P(p,q,r,s)x^py^qz^rt^s.
$$
Define $f\_Q$, $f\_R$, $f\_S$ similarly (all words start by $P$!). Then
$$
\pmatrix{f\_P\\f\_Q\\f\_R\\f\_S}
=
\pmatrix{0&x&x&x\\y&0&y&y\\z&z&0&z\\t&t&t&0}\pmatr... | 2 | https://mathoverflow.net/users/17581 | 150334 | 80,339 |
https://mathoverflow.net/questions/149511 | 21 | The $n$-th Mersenne number is $M\_n=2^n-1$. Write $M\_n=a\_n b\_n^2$ where $a\_n$ is positive and squarefree.
**Question 1:** What lower bound can be proved for $a\_n$?
Let $A$ be the set of all possible $a\_n$. The **natural density** of $A$ is defined as
$$
\delta\_A=\lim\_{X \rightarrow \infty} \frac{\# \{a \in... | https://mathoverflow.net/users/4140 | Squarefree parts of Mersenne numbers | Here's a simple proof using user43383's idea and a recent result of Andrew Granville: <http://arxiv.org/abs/1212.6306>
According to Theorem 1 of that paper, $2^n-1$ always has a primitive prime factor $p$ that occurs to an odd exponent, except when $n=1$ or $n=6$ (where there are no primitive prime factors at all). *... | 9 | https://mathoverflow.net/users/16510 | 150340 | 80,342 |
https://mathoverflow.net/questions/150333 | 1 | Let $X$ be a (smooth complex algebraic) surface. Suppose $\theta$ is an automorphism of order $2$ of $X$, such that its fixed locus is a disjoint union of smooth curves. I am trying to prove that the quotient
$$Y=X/\langle\theta\rangle$$
is in fact a smooth surface. (First of all: is this true/does it follow from some... | https://mathoverflow.net/users/40038 | Smoothness of the quotient surface by an involution with nice fixed locus | Francesco gave the answer to your first question. As for the second one, look at my answer to [this post](https://mathoverflow.net/questions/149875/k3-surface-with-a-non-symplectic-involution-a-basic-question).
| 1 | https://mathoverflow.net/users/40297 | 150341 | 80,343 |
https://mathoverflow.net/questions/150304 | 7 | I am wondering if the following generalization of van der Waerden's conjecture is true.
Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T\_i$. Then $per(A) \geq T\_1\ldots T\_n \frac{n!}{n^n}$. This obviously implies van der Waerden's conjecture. I can check i... | https://mathoverflow.net/users/43451 | A generalization of van der Waerden's conjecture | The conjecture is false. Here is a counterexample.
\begin{equation\*}
A = \begin{bmatrix}
\tfrac18 & \tfrac4{15} & \tfrac1{10}\\
\tfrac18 & \tfrac4{15} & \tfrac1{10}\\
\tfrac68 & \tfrac7{15} & \tfrac8{10}
\end{bmatrix}.
\end{equation\*}
For this matrix, $\text{per}(A)=\frac{21}{200}=0.105$. The row sums $T\_1,... | 13 | https://mathoverflow.net/users/8430 | 150349 | 80,346 |
https://mathoverflow.net/questions/134602 | 7 | I am sure that this is well known in the right places, but: Is the C\* completion of a star nuclear Fréchet algebra a nuclear C\* algebra? (Suppose that the C\* norm is continuous with respect to the Fréchet topology.)
[Basic, but commutative, example to explain topology - smooth functions on a compact manifold compl... | https://mathoverflow.net/users/29625 | Is a C* completion of a nuclear Fréchet algebra a nuclear C* algebra? | Counterexample.
I think the $C^\star$ algebra of the Free group on two generators $F\_2$ is not a nuclear $C^\star$ algebra, same for the reduced group $C^\star$ algebra of $F\_2$.
The finite support functions $c\_f (F\_2)$ is a dense subagebra of $C^\star(F\_2)$, but is not Frechet.
Let $\ell$ be the word length... | 3 | https://mathoverflow.net/users/43469 | 150353 | 80,348 |
https://mathoverflow.net/questions/150226 | 6 | Let X be a Hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does this imply that X is homotopy equivalent to a compact Hausdorf space? This question is a "compact version" of the following question;
[Paracompactness and inner product on vector bundles](https://mathoverflow.net/q... | https://mathoverflow.net/users/36688 | Spaces over which every vector bundle is a summand of the trivial bundle | Let $X$ be the wedge of infinitely many circles (equipped with the CW topology). Every vector bundle $\xi$ over $X$ is a summand of a trivial bundle, namely it is $\xi\oplus\xi$ is trivial because any vector bundle over a circle has this property (alternatively, one could appeal to the fact that $X$ is homotopy equival... | 13 | https://mathoverflow.net/users/1573 | 150363 | 80,354 |
https://mathoverflow.net/questions/150351 | 12 | Does [Tverberg's theorem](http://en.wikipedia.org/wiki/Tverberg%27s_theorem) hold for CAT(0) spaces of covering dimension $d<\infty$:
Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - 1) + 1$, there exists a point $x\in X$ and a partition of $E$ into $r$... | https://mathoverflow.net/users/21684 | Tverberg's theorem in CAT(0) spaces | No. Let $X$ be a tripod (three segments with one common endpoint), $d=1$, $r=2$ and $E$ the set of the 3 leaf points.
| 14 | https://mathoverflow.net/users/4354 | 150365 | 80,355 |
https://mathoverflow.net/questions/150342 | 3 | Let $G\_1$ and $G\_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G\_1\to G\_2$, with finite kernel contained in the center of $G\_1$.
By congruence subgroup of $G\_i$, for $i=1,2$, we means $K\cap G\_i(\mathbb{Q})$, with $K$ compact open subgroup of $G... | https://mathoverflow.net/users/4504 | isogeny and congruence subgroup | To deal with isogenies it is useful to start with some of the basic work done on the Congruence Subgroup Problem in the 1960s and later. This problem can be formulated for any connected linear algebraic group defined over a number field such as $\mathbb{Q}$. Much of the work focuses on the difficult case of an (almost)... | 3 | https://mathoverflow.net/users/4231 | 150370 | 80,357 |
https://mathoverflow.net/questions/150373 | 4 | It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$.
But if we consider $\text{GCH}$ as a theory with ordinal many statements like $\text{GCH}=\{\text{CH}\_{\alpha}~|~\alp... | https://mathoverflow.net/users/nan | Minimal Generalized Continuum Hypothesis & Axiom of Choice | All you need for AC in the standard argument from GCH is that the GCH holds for an unbounded class of cardinals. The reason is that this is sufficient to conclude that any set of sets of ordinals is well-orderable, and this is sufficient to imply AC.
So the answer to question 1 is yes; any unbounded class $C$ suffic... | 9 | https://mathoverflow.net/users/1946 | 150376 | 80,361 |
https://mathoverflow.net/questions/150367 | 5 | Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^\*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all infixes of length $n$ determine $\xi$ uniquely, meaning it is the only word with this prefix and infixes.
*Some context... | https://mathoverflow.net/users/37580 | What prefix and factors determine a ultimately periodic word uniquely | This is a more detailed version of Alessandro's comment. By a classical theorem of Morse-Hedlund, one has that an infinite word $\eta$ is ultimately periodic iff there exists $m\geq 0$ so that $\eta$ has the same number of factors of length $m$ and $m+1$. Moreover, if $M$ is the number of subwords of length $m$, then t... | 2 | https://mathoverflow.net/users/15934 | 150385 | 80,363 |
https://mathoverflow.net/questions/150247 | 14 | Is it true that the set $$S:=\{n\in \mathbb N\ |\ 2^n-1 \ \mbox{ is square free}\}$$ has positive density? What can we say when we replace $2^n-1$ with $\frac{a^n-1}{a-1}$?
| https://mathoverflow.net/users/18785 | Density of all n such that 2^n-1 is square free | User43383's answer linking the convergence of the sum of the reciprocals of the order of $2 \pmod{p^2}$ to the existence of a positive density is very nice. Let me add a little to this by saying what the density should be heuristically (approximately $0.754$), and by obtaining upper bounds for this density unconditiona... | 13 | https://mathoverflow.net/users/38624 | 150387 | 80,364 |
https://mathoverflow.net/questions/150400 | 1 | The topos ${\mathcal{Set}}$, at least as axiomatized in ETCS, is a well-pointed topos that satisfies the axiom of choice and has a natural numbers object.
Is there a characterization of the topos ${\mathcal{FinSet}}$ in the internal language?
| https://mathoverflow.net/users/nan | Is there a characterization of the topos of finite sets in the internal language? | Characterization is too strong; there are many models of $\operatorname{ETCS}$, so its axioms don't characterize $\operatorname{Set}$ either. But I'll interpret your question as asking how the axioms of $\operatorname{ETCS}$ might be changed if the intended model is $\operatorname{Fin}\operatorname{Set}$ instead of $\o... | 3 | https://mathoverflow.net/users/8508 | 150401 | 80,369 |
https://mathoverflow.net/questions/150235 | -1 | Suppose that $G$ is a simple $r$-regular graph with $n$ vertices.
We say $H$ is a dominating set for $T$, if
for every vertex $v\in T$, we have $v\in H$ or there is a vertex $u\in H$ such that $vu\in E(G)$.
It is easy to see that if $T$ is an independent set for $G$, then there exists an independent
dominating set $... | https://mathoverflow.net/users/42971 | A question about independent set in regular graphs | Clearly, if $G$ is $1$-regular or $2$-regular, and if $T$ is an independent set in $G$, then there is a maximal independent set $H$ in $G$ such that $T\cap H=\emptyset$.
Let $G$ be the $3$-regular graph of order $14$ with $V(G)=\{a,b,c,d,e,f,g,t,u,v,w,x,y,z\}$
and $E(G)=\{ab,bc,cd,de,ea,fb,fd,gc,ge,fg,tu,uv,vw,wx,xt... | 4 | https://mathoverflow.net/users/43266 | 150403 | 80,370 |
https://mathoverflow.net/questions/150402 | 12 | I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations of morphisms in $M$ do not extend to ones for $Pro-M$.
My question is: which arguments and statements of Hovey's book c... | https://mathoverflow.net/users/2191 | Which statements and arguments of Hovey's "Model categories" fail without functorial factorizations of morphisms? | Have a look at: Functorial Factorizations in Pro Categories by Ilan Barnea Tomer M. Schlank (ArXiv: <http://arxiv.org/abs/1305.4607>). This will in part answer your question. It is worth stating that Pro - M only fails to be fibrantly generated because there are too many generating fibrations, and Chorny has worked wit... | 5 | https://mathoverflow.net/users/3502 | 150411 | 80,372 |
https://mathoverflow.net/questions/150344 | 6 | It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where
$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum\_{i=1}^{n} z\_{i}^{2}$.
Two questions:
1) Is $M$ a symplectic submanifold of $\mathbb{C}^n\sim \mathbb{R}^{2n}$ (with the standard symplectic structure)?
If the answer is affirma... | https://mathoverflow.net/users/36688 | symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$ | Here is a formula for an explicit symplectomorphism $F$ from $T^\*S^{n-1}$ to the affine quadric $\{\sum z\_{j}^{2}=1\}$ in $\mathbb{C}^n$:
$$
F(p,q) = \left(\frac{1+\sqrt{1+4|p|^2}}{2}\right)^{1/2} q - i\left(\frac{1+\sqrt{1+4|p|^2}}{2}\right)^{-1/2}p
$$
Here I view $T^\*S^{n-1}$ as consisting of pairs $(p,q)\i... | 10 | https://mathoverflow.net/users/424 | 150412 | 80,373 |
https://mathoverflow.net/questions/150419 | 1 | Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let work on $P^1$ with field of rational functions in $t$ variable and $d:[d,t\*]=1, d(t)=1$. We can go to differential extens... | https://mathoverflow.net/users/35566 | Variety of factorizations of differential operator | A factorization would correspond to any flag such that the subspaces are invariant to the action of the differential Galois group and that each graded piece is an irreducible representation of the differential Galois group. It is easy to compute the space of such flags from the representation.
If the representation i... | 0 | https://mathoverflow.net/users/18060 | 150425 | 80,379 |
https://mathoverflow.net/questions/150428 | 21 | As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
$$3^6+19^6+22^6 = 10^6+15^6+23^6$$
among many other solutions are known, when the number of summands is increased
from $2$ ... | https://mathoverflow.net/users/6094 | State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$ | The short answer to your specific question is no, the resolution of FLT via modularity of elliptic curves does not seem to be helpful in dealing with rational points on higher dimensional varieties. The first two equations you list, $a^5+b^5=c^5+d^5$ and $a^6+b^6=c^6+d^6$, are surfaces of general type in $\mathbb{P}^3$... | 21 | https://mathoverflow.net/users/11926 | 150430 | 80,380 |
https://mathoverflow.net/questions/150418 | 5 | Let $C$ be a small site with fibre products. The (injective) [Čech model structure](http://ncatlab.org/nlab/show/%C4%8Cech+model+structure+on+simplicial+presheaves) on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this $(\infty,1)$-topos is *hypercomplete* or no... | https://mathoverflow.net/users/43339 | Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology | There is no relation between hypercompleteness and the property that Čech cohomology agrees with genuine cohomology, i.e., there is no implication either way. For example, étale cohomology of nice schemes can be computed using Čech cohomology even though the small étale (∞,1)-topos is typically not hypercomplete, and, ... | 6 | https://mathoverflow.net/users/20233 | 150431 | 80,381 |
https://mathoverflow.net/questions/150426 | 5 | Let $\Gamma\_g=< a\_1,...,a\_g,b\_1,...,b\_g | \prod\_{i=1}^g [a\_i,b\_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma\_g$ for $g>1.$ (I.e. free subgroups which are not properly embedded into any other free subgroup)?
For example,
1. Is $<a\_1,...,a\_g,b\_1,...,b\_{g-1},b\_g^2>$ free? ... | https://mathoverflow.net/users/23935 | What is known about maximal free subgroups of surface groups? | 1. The subgroup is free, and therefore not maximal (see 2.). To see this, the subgroup generated by all the generators except $b\_g$ is the fundamental group of a subsurface obtained by cutting along a curve (dual to $b\_g$). This surface lifts to a 2-fold cover dual to the curve, and the subgroup is obtained by adding... | 11 | https://mathoverflow.net/users/1345 | 150435 | 80,383 |
https://mathoverflow.net/questions/150432 | 12 | Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:
**It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c\_1(M)=0$ admits a Ricci-flat Kähler metric) that if a compact Kähler manifold $M$ has a complex symplectic form $\omega\_\mat... | https://mathoverflow.net/users/17294 | Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds? | The point is that a Bochner formula shows that, if the Kähler metric is Ricci-flat and the manifold is compact, then every global holomorphic $p$-form must be parallel.
In particular, if you have a complex, compact symplectic manifold that is Kähler, then you have a nonvanishing holomorphic volume form (the top powe... | 18 | https://mathoverflow.net/users/13972 | 150439 | 80,385 |
https://mathoverflow.net/questions/150442 | 0 | Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}\_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F\_{p}$ vector spaces. So we can talk of their dimensions. Now what can we say about the rank of $M$ from looking at the dimensions of $M[p]$ and $M/(p) ?... | https://mathoverflow.net/users/30999 | Rank of a $ \mathbb{Z}_{p}[[T]] $ module | Not much : take $M=\mathbb{Z}\_p[[T]]^r$. Then $M[p]$ is zero, and $M/pM=\mathbb{F}\_p[[T]]^r$ is infinite-dimensional...
| 5 | https://mathoverflow.net/users/40297 | 150443 | 80,386 |
https://mathoverflow.net/questions/150450 | 4 | Let $f : \mathbb{A}\_k^2 \to \mathbb{A}\_k^2$ be a regular self-map of the affine plane over a field $k$ of characteristic zero. Assume that the sequence $(\deg{f^n})\_{n \in \mathbb{N}}$ is bounded. Is this sequence eventually periodic?
[An embarrassing question... ]
Added: The Favre-Jonnson paper linked to in Gj... | https://mathoverflow.net/users/26522 | If the sequence of degrees of the iterates of a self-map of $\mathbb{A}^2$ is bounded, is it eventually periodic? | It's a theorem of Favre and Jonsson that the degree sequence satisfies a linear recurrence, so your claim follows. See their paper ["Dynamical compactifications of $\mathbb C^2$"](http://arxiv.org/abs/0711.2770).
| 4 | https://mathoverflow.net/users/2384 | 150456 | 80,389 |
https://mathoverflow.net/questions/150459 | 2 | **Question 1:** Is it consistent with $\text{ZF}$ that only countable subsets of $\mathbb{R}$ are well-orderable?
**Question 2:** Is it consistent that for some $\lambda$, $\aleph\_0 < \lambda < 2^{\aleph\_0}$, only those subsets of $\mathbb{R}$ are well-orderable which have size $\leq \lambda$?
**Question 3:** Is ... | https://mathoverflow.net/users/nan | Subsets of Real Numbers (Edited & Revised Version) | For the first question, yes. It is consistent to have that.
1. Solovay's model, or any model of $\sf AD$ for example.
2. Truss' models, which are similar to Solovay's model, only we start with a general limit cardinal (rather than an inaccessible). The result has that $\aleph\_1$ is singular, but still cannot be embe... | 11 | https://mathoverflow.net/users/7206 | 150460 | 80,392 |
https://mathoverflow.net/questions/150328 | 23 | Inspired by the discussion in the comments of [this question](https://mathoverflow.net/questions/150226/characterization-of-compact-space-up-to-homotopy-equivalent), I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak equivalent) to compact Ha... | https://mathoverflow.net/users/75 | Which spaces have the (weak) homotopy type of compact Hausdorff spaces? | Expanding on my comment, if there are measurable cardinals then it follows from the results of
A. Przeździecki, Measurable cardinals and fundamental groups of compact spaces.
Fund. Math. 192 (2006), no. 1, 87–92.
that there are spaces not weakly equivalent to any compact Hausdorff space, as Przeździecki proves th... | 15 | https://mathoverflow.net/users/22989 | 150464 | 80,394 |
https://mathoverflow.net/questions/150394 | 2 | How to extract the divergent part of the following integral simply as $u \rightarrow \infty$
$$g(u) = \frac{\sqrt{2u}}{\pi} \int^1\_{\frac{1}{u}} dz \frac{\sqrt{z-1}}{\sqrt{z^2-u^{-2}}} $$
| https://mathoverflow.net/users/25715 | How to extract the divergent part from the singular integral | it's an elliptic integral; a series expansion gives
$$g(u)=\frac{\sqrt{2u}}{\pi} \int^1\_{1/u} dz \frac{\sqrt{z-1}}{\sqrt{z^2-u^{-2}}}=i\frac{1}{\pi}(2u)^{1/2}\;[\ln (8u)-2]+{\cal O}(u^{-1/2})$$
so the integral diverges as $\sqrt{u}\ln u$
here is a plot of $-i(\pi/\sqrt{2u})g(u)$, evaluated numerically, and $\ln(... | 4 | https://mathoverflow.net/users/11260 | 150467 | 80,396 |
https://mathoverflow.net/questions/150474 | 3 | The notion of minimality in model theory is related to the existence of a *gap* in the size of definable subsets of a model. Now consider the following generalization:
**Definition 1:** Let $M$ be a $\mathcal{L}$ - structure. Define:
$Def(M):=\{X\subseteq Dom(M)~|~\exists n \in \omega~~~\exists \varphi (x ,y\_1 , .... | https://mathoverflow.net/users/nan | $(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$ | The answer is yes, one can always find models with as large a gap in their definable classes as desired.
**Theorem.** For every $\kappa\gt\lambda\geq\aleph\_0$, and for any consistent theory with an infinite model in a countable language, there is a $(\kappa,\lambda)$-minimal model.
Proof. Let $M$ be any $\kappa$-... | 9 | https://mathoverflow.net/users/1946 | 150477 | 80,399 |
https://mathoverflow.net/questions/150473 | 13 | Given positive integers $a$, $m$ and $n$, let $s\_{a(m)}(n)$ denote the
sum of the reciprocals of the prime numbers less than or equal to $n$
which are congruent to $a$ modulo $m$.
Is there an integer $n$ such that $s\_{1(3)}(n) > s\_{2(3)}(n)$?
For small $n$, the function $s\_{2(3)}$ is clearly ahead --
for exampl... | https://mathoverflow.net/users/28104 | Sums of reciprocals of prime numbers: $p \equiv a \!\! \mod m$ vs. $p \equiv b \!\! \mod m$ | Note that
$$
s\_{1(3)}(n)-s\_{2(3)}(n) = \sum\_{p\le n} \frac{\chi\_{-3}(p)}{p}
$$
where $\chi\_{-3}$ is the real Dirichlet character $\pmod 3$ (ie the Legendre symbol).
This sum converges (as in the proof of Dirichlet's theorem). So if it starts out being negative for a long while, it will continue to be negative.... | 17 | https://mathoverflow.net/users/38624 | 150481 | 80,401 |
https://mathoverflow.net/questions/150471 | 3 | Let $\gamma$ be Euler constant and $W$ Lambert W function.
One can show:
$$-2/3\,{\frac {\gamma+\ln \left( \pi \right) }{W \left(
-1/3\,{\frac { \left( \gamma+\ln \left( \pi \right) \right) {
{\rm e}^{-1/3\,\gamma}}}{\sqrt [3]{\pi }}} \right) }} = 2 \qquad (1) $$
This means at least one of $ \left(
-1/3\,{\frac... | https://mathoverflow.net/users/12481 | Identity between Euler gamma and pi | By the definition of the Lambert function
$W\left(-\frac13\alpha e^{-\alpha/3}\right)=-\frac\alpha3\,$ for any $\alpha$, so
$$
-2/3\frac\alpha{W\left(-\frac13\alpha e^{-\alpha/3}\right)}=2.
$$
Putting here $\alpha=\gamma+\ln\pi$ gives your formula.
| 5 | https://mathoverflow.net/users/14551 | 150485 | 80,403 |
https://mathoverflow.net/questions/150472 | 10 | Is well know that completely metrizable spaces are Baire's spaces. Reciprocally, if $X$ is a Baire's metric space, then $X$ is completely metrizable?
| https://mathoverflow.net/users/41166 | Completely Metrizable Space and Baire Theorem | No, in order for a subspace of a complete metric space to be completely metrizable it is necessary and sufficient for it to be $G\_\delta$. There are only $2^{\aleph\_0}$ many $G\_\delta$ subsets of $\mathbb{R}^2$ but there are $2^{2^{\aleph\_0}}$ sets in $$\{X \subseteq \mathbb{R}^2 : (0,1)\times(0,1) \subseteq X \sub... | 13 | https://mathoverflow.net/users/2000 | 150487 | 80,405 |
https://mathoverflow.net/questions/150488 | 2 | I already posted this question at MSE [here](https://math.stackexchange.com/questions/586058/random-algebraic-numbers-are-linearly-disjoint-almost-surely), but since it received no answer or
comment so far I cross-post it here.
It is well-known that if one considers a “random” monic polynomial of fixed
degree, say $X... | https://mathoverflow.net/users/10341 | Random algebraic numbers are linearly disjoint almost surely? | This should indeed be true and easy to prove using the large sieve approach of Gallagher (["The large sieve and probabilistic Galois theory", Proceedings of Symposia in Pure Mathematics 24, 1973, A.M.S., 91–101](http://www.ams.org/books/pspum/024/)) -- probably most other proofs of this result should also extend. Preci... | 3 | https://mathoverflow.net/users/39744 | 150493 | 80,408 |
https://mathoverflow.net/questions/150496 | 3 | **Definition 1:** Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define:
$Def\_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y\_1,...,y\_n)\in \mathcal{L}-Form~~\exists b\_1,...,b\_n \in A;X=\{a\in Dom(M)~|~M\vDash \varphi (a,b\_1,...,b\_n)\}\}$
**Definition 2:** Let $\kapp... | https://mathoverflow.net/users/nan | $(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem | The answer is no, because there are structures $M$ having subsets that are definable from parameters of arbitrary size up to $\kappa$, and this will include sizes in the forbidden region of the gap between $|\mathcal{L}+\aleph\_0|$ and $\kappa$.
For example, let $M=\langle\kappa,\lt\rangle$ be the usual order on the... | 4 | https://mathoverflow.net/users/1946 | 150500 | 80,411 |
https://mathoverflow.net/questions/150498 | 7 | All of the fast algorithms that I have seen which factor integers work by searching for smooth numbers. Are there any fast algorithms for factoring integers that don't work by searching for smooth numbers? Is there any reason to believe that it is possible to construct fast algorithms for factoring integers that don't ... | https://mathoverflow.net/users/7089 | Are there any fast algorithms for factoring integers that don't work by searching for smooth numbers? | This paper explains why smooth numbers play a key role in almost every modern integer factorization algorithm. Maybe it lists the exceptions, but I couldn't find them.
<http://www.math.dartmouth.edu/~carlp/PDF/paper104.pdf>
| 7 | https://mathoverflow.net/users/39754 | 150501 | 80,412 |
https://mathoverflow.net/questions/150497 | 4 | In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{\|\cdot\|^{d-2}}$ is non-negative). What can one say about positivity properties of the Coulomb energy in $d=2$? $$D(f,g):=-\displaystyle\int\_{\mathbb{R}^2}\int\_{\mathbb{R}^2}\overline{f(x)}g(y)\log{\|x-y\|}~dx... | https://mathoverflow.net/users/43528 | Positivity of the Coulomb energy in two dimensions | This is true when the support of $f$ is contained in the unit disc. If the support is contained in a disc $|z|<R$, then $(f,f)$ is bounded from below by a constant that depends
on $R$. This minor nuisance makes the logarithmic potential somewhat different from the Newtonian
potential, however most statements of potenti... | 6 | https://mathoverflow.net/users/25510 | 150510 | 80,418 |
https://mathoverflow.net/questions/150396 | 0 | Let $\Delta$ be an $n-1$-simplex in ${\mathbb R}^{n-1}$. For each vertex $v$ of $\Delta$ let $H\_v$ be the hyperplane through $v$ and parallel to the opposite facet. By 2-neighborhood of a simplex I mean a simplex which is the intersection of the half-spaces $H\_v^+$ containing $\Delta$. My question is:
Let $S$ be a ... | https://mathoverflow.net/users/14563 | 2-neighborhood of a simplex | $\def\Vol{\mathop{\rm Vol}}$
Take $a\_1,\dots,a\_n\in S$ such that $\Vol(a\_1,\dots,a\_n)$ attains the maximum. Now, if some $b\in S$ lies outside, say, $H\_{a\_n}^+$, then $\Vol(a\_1,a\_2,\dots,a\_{n-1},b)>\Vol(a\_1,\dots,a\_n)$ which is impossible. Thus $b$ lies in the intersection of all such halfspaces.
This work... | 0 | https://mathoverflow.net/users/17581 | 150513 | 80,419 |
https://mathoverflow.net/questions/150516 | 0 | Let $ M $ be a finite group of order $2^{a+1} $ and let $ M $ have a normal subgroup $ R $ such that $|M: R|=2 $. Also we know that $ R $ is an elementary abelian subgroup of $ M $.
For example $ D\_8 $ is an example for it. Is it true that for other cases $ M $ is abelian or what information is available for $ M $?... | https://mathoverflow.net/users/31045 | the structure of a 2-group | The elementary Abelian $2$-group $R$ is a ${\rm GF}(2)M/R$-module of dimension $a$. As such it is a direct sum of $t$ copies of the trivial module and $\frac{a-t}{2}$-copies of the free module of rank $1$. Then $Z(M)$ has order $2^{t},$ and there are examples of $M$ and $R$ for which every possibility of $t$ between $1... | 3 | https://mathoverflow.net/users/14450 | 150525 | 80,422 |
https://mathoverflow.net/questions/150508 | 1 | Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C\_e}$ the set of all cycles of $G$ containing the edge $e$.
For what set of edges does $\mathcal{C\_e}$ contain a basis of the cycle space of $G$?
| https://mathoverflow.net/users/23850 | Do the cycles containing a fixed edge generate the cycle space of a graph? | For all $e \in E(G)$, $\mathcal{C}\_e$ contains a basis of the cycle space.
**Proof.** Let $e=uv \in E(G)$, and let $C$ be a cycle of $G$. If $e \in E(C)$, then there is nothing to show. Thus, $e \notin E(C)$. By Menger's theorem, there are two disjoint paths $P\_u$ and $P\_v$ from $\{u,v\}$ to $V(C)$. By taking $P\... | 3 | https://mathoverflow.net/users/2233 | 150528 | 80,423 |
https://mathoverflow.net/questions/150519 | 3 | Let $f$ be a non-degenerate quadratic form with integral coefficients. The genus of $f$ is the set of quadratic forms up to integral equivalence which are equivalent to $f$ over the $p$-adic integers $\mathbb Z\_p$ for all primes $p$ and also over $\mathbb Q$. Let us denote it by gen(f).
A $S$-arithmetic analog is th... | https://mathoverflow.net/users/6836 | S genus of quadratic forms | Let $L=(\mathbf Z^n,b)$ be a bilinear module such that $L\otimes \mathbf Q$ is non-degenerate.
For a set of ultrametric places $S$, let us write $\mathbf Z[S^{-1}]$ for the set of rationals that are integral at each place out of $S$, and $\mathbf Z\_{(S)}$ for the set of rationals that are integral at each place in $... | 4 | https://mathoverflow.net/users/39552 | 150531 | 80,426 |
https://mathoverflow.net/questions/150520 | 4 | In order to construct the coarse moduli scheme of smooth projective curves of genus $g$, the classical results of Mumford (using the numerical criterion of stability) say that for large enough $m$, the $m$-th Hilbert point of a curve is stable. If I understand correctly, this implies that the Hilbert scheme $H\_m$ of $... | https://mathoverflow.net/users/17988 | A question on the existence of the quotient of the Hilbert scheme of tricanonical curves | I do not believe the result about stability of tricanonical curves is a *formal* consequence of asymptotic stability. Rather, I think that the explicit arguments that prove $m$-stability for $m\gg 0$, in fact, already apply ~~if m ≥ 3~~ if $m\geq 5$. The standard reference is the following.
MR0450272 (56 #8568) Revie... | 4 | https://mathoverflow.net/users/13265 | 150541 | 80,427 |
https://mathoverflow.net/questions/150468 | 9 | Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of
rank $n$ have?
In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups.
I think I have got a algorithm to solve case $d=2$ for arbitrary $n$.
But the general case seems very difficult. For example in c... | https://mathoverflow.net/users/41853 | Number of subgroups of a given index of a free group | This calculation was performed by [Marshall Hall Jr](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=hall%2C%20m*&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=181&mx-pid=28836). Let... | 15 | https://mathoverflow.net/users/1463 | 150551 | 80,430 |
https://mathoverflow.net/questions/150547 | 20 | For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes\_R S\cong S\otimes\_R M$ satisfying the cocycle condition. This can be thought of as saying something about "agreeing on intersection... | https://mathoverflow.net/users/11546 | How is a descent datum the same as a comodule structure? | I find it easier to use geometric notation, so let $X = \mathrm{Spec}(S)$, $Y=\mathrm{Spec}(R)$, and $\phi: X \to Y$ be the morphism corresponding to $f$. We have adjoint functors
$$
\phi\_\ast : S\text{-}\mathrm{mod} \to R\text{-}\mathrm{mod}: \phi^\ast.
$$
Consider the (beginning of the) Cech simplicial set correpo... | 19 | https://mathoverflow.net/users/7762 | 150554 | 80,432 |
https://mathoverflow.net/questions/150437 | 0 | Suppose I have a smooth, noncompact manifold $M$ with metrics $g\_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that
$$ C^{-1} d\_1(x,y) \leq d\_2(x,y) \leq C d\_1(x,y)$$
where $d\_i$ are the induced distance metric of $g\_i$. Then, if the injectivity radius with respect to $g\_1$ satisfies $\mathrm{inj}(M,... | https://mathoverflow.net/users/11976 | Preservation of injectivity radius | This is an expansion of Anton Petrunin's comment.
Let me describe how to perturb the standard metric of the plane so that the resulting metric is bi-Lipschitz to the original with Lipschitz constant arbitrarily close to 1 and has arbitrarily small injectivity radius.
Locate the plane in $\mathbb R^3$ as the $xy$-pl... | 5 | https://mathoverflow.net/users/4354 | 150561 | 80,434 |
https://mathoverflow.net/questions/150335 | 39 | I am learning differential geometry and have a few questions on curvature. -- Background:
1. Gauss invented "Gauss curvature" to measure how surface curves.
2. Riemann gives an ingenious generalization of Gauss curvature from surface to higher
dimensional manifolds using the "Riemannian curvature tensor" (sectional ... | https://mathoverflow.net/users/39332 | Are there some other notions of "curvature" which measure how space curves? | in addition to these excellent examples of non-local curvature quantities and their extensions to the non-smooth setting (which I am not sure the OP was anticipating), I might add the 'original' non-local curvature measure: the holonomy.
Also, the OP was not precise about how to interpret the vague term 'space', so i... | 45 | https://mathoverflow.net/users/13972 | 150569 | 80,437 |
https://mathoverflow.net/questions/150586 | 5 | Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial. Is there an integer $a\neq 0$ such that $f(X)+a$ is also irreducible in $\mathbb{Z}[X]$?
Can this be also extended to $\mathbb{Q}[X]$?
| https://mathoverflow.net/users/43553 | Is there an integer a such that f(X)+a is irreducible in Z[X]? | Yes, and you don't need $f$ irreducible. The following irreducibility criterion suffices and shows that infinitely many $a$ work.
**Lemma:** Let $g(x) = a\_n x^n + ... + a\_0 \in \mathbb{Z}[x]$ be such that $a\_0$ is prime and
$$|a\_0| > |a\_1| + ... + |a\_n|.$$
Then $g(x)$ is irreducible.
*Proof.* The condit... | 23 | https://mathoverflow.net/users/290 | 150589 | 80,446 |
https://mathoverflow.net/questions/150597 | 1 | For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the equivariant setting? Do there exist equivariant sheafs that are not equivariant vector bundles?
| https://mathoverflow.net/users/41562 | Do there exist equivariant sheafs that are not equivariant vector bundles? | I assume that the algebraic group $G$ is smooth and connected, and that you are asking about equivariance for the natural action of $G$ on $H$. There is a quotient morphism $q:G\to H$ that is $G$-equivariant and faithfully flat. Thus, to prove that a $G$-equivariant coherent sheaf on $H$ is locally free, it suffices to... | 4 | https://mathoverflow.net/users/13265 | 150600 | 80,449 |
https://mathoverflow.net/questions/150606 | 12 | I'm interested in the cohomology ring $H^\*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to the coinvariants algebra of the associated root system. $H^\*(G/B)$ also has a distinguished basis given by Schubert cel... | https://mathoverflow.net/users/43555 | Cohomology ring of a flag variety and representation theory | Let $\mathfrak{g}$ be the Lie algebra of the group $G$. You might consider reading about the Springer resolution $$\mu:T^\*(G/B)\rightarrow\mathcal{N},$$ where $\mathcal{N}$ is the nilpotent cone of $\mathfrak{g}$. The fibres of this map are isomorphic over individual adjoint orbits of $G$. These are called Springer fi... | 14 | https://mathoverflow.net/users/25358 | 150610 | 80,455 |
https://mathoverflow.net/questions/149808 | 2 | Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ endowed with the topology of weak convergence. Let $\rho(p,q)$ denote the total variation distance between probability me... | https://mathoverflow.net/users/11768 | Particular neighborhoods of analytical sets | If I am correct, one can proceed as follows. Consider a set $A\subset X\times \mathcal P(X)^2$ given by
$$
A = \{(x,p,q):(x,p)\in \Gamma,\rho(p,q)\leq\varepsilon\}
$$
then we obtain $\Gamma^\varepsilon = \pi(A)$ where $\pi:(x,p,q)\mapsto (x,q)$. We obtain
$$
A = \left(\Gamma\times \mathcal P(X)\right)\cap \left(X\times... | 1 | https://mathoverflow.net/users/11768 | 150611 | 80,456 |
https://mathoverflow.net/questions/150537 | 8 | This is basically a repost of [this](https://math.stackexchange.com/questions/582835/an-inequality-related-to-the-number-of-binary-strings-with-no-fixed-substring) math.se question. At the time I was writing this I thought it has to have a straightforward solution so I posted it there. Now I am not so sure about it bei... | https://mathoverflow.net/users/1737 | An inequality related to the number of binary strings with no fixed substring | Your conjecture is true. Here is a proof.
Define $\operatorname{Av}\_n(w)$ to be the number of binary words of length $n$ which avoid the pattern $w$.
Let $u$ and $v$ be binary words with $|u| = k$ and $|v|=m$ with $k < m$. We will show that $\operatorname{Av}\_n(u) < \operatorname{Av}\_n(v)$ for all $n$.
Define ... | 9 | https://mathoverflow.net/users/36497 | 150618 | 80,459 |
https://mathoverflow.net/questions/64280 | 15 | **"Foreman`s maximality principle"** states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails everywhere,
2) there are no inaccessible cardinals,
3) there are no $\kappa-$Souslin trees,
4) Any non-trivial $c.c.c.$... | https://mathoverflow.net/users/11115 | Questions about $\aleph_1-$closed forcing notions | The answer to the second question is yes, without any large cardinals assumptions.
**Claim:** if $2^{\aleph\_0}$ is singular then every non-trivial $\sigma$-closed forcing of size $2^{\aleph\_0}$ collapses the continuum.
The reason is that such forcing must add a new set of ordinals of size $\lambda < 2^{\aleph\_0... | 8 | https://mathoverflow.net/users/41953 | 150619 | 80,460 |
https://mathoverflow.net/questions/150417 | 15 | **Question 1.** Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true that both are? (Perhaps, with different $w$.) - **answered in the negative**
**Question 1 (modified).** Assume that a... | https://mathoverflow.net/users/8131 | Two questions from combinatorics on words | I think Question 2 is true and the proof is as follows. Every Sturmian word is equivalent to the [cutting sequence](http://en.wikipedia.org/wiki/Sturmian_word#Cutting_Sequence_of_Irrational) of an irrational number (except, I think, those that have only one extra digit compared to a rational number, for which your stat... | 7 | https://mathoverflow.net/users/955 | 150624 | 80,462 |
https://mathoverflow.net/questions/150585 | 4 | It is known that continuous linear functionals on the space $C\_0({\mathbb{R}^n})$ are bounded Radon measures ${\cal M}({\mathbb{R}^n})$ where $C\_0({\mathbb{R}^n})$ is uniform closure of the space of continuous functions with compact support $C\_c({\mathbb{R}^n})$ (for reference e.g. Folland's book on real analysis ch... | https://mathoverflow.net/users/43552 | On Radon measures with values in Banach space | The formulation is not quite clear---do you mean linear functionals on a space of Banach space valued functions or linear operators on scalar-valued functions with valued in a Banach space? In the former case, the representing measure takes its value in the dual and is regular with respect to the topology of uniform co... | 2 | https://mathoverflow.net/users/43569 | 150629 | 80,464 |
https://mathoverflow.net/questions/150620 | 8 | Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $F$ be a local non-Archimedean field. Let $\rho$ be an irreducible smooth representation of $G(F)$. How does one define the conductor of $\rho$?
The conductor should be an integer canonically associated with the representation $\rho$ so that it matc... | https://mathoverflow.net/users/41301 | Conductor of a representation of a $p$-adic group | for GL\_n you could look at:
H.Jacquet,I.I.Piatetski-Shapiro,J.Shalika:Conducteur des representations du groupe lineaire, Math. Annalen 256, 199–214 (1981).
| 7 | https://mathoverflow.net/users/43571 | 150632 | 80,466 |
https://mathoverflow.net/questions/150250 | 2 | It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true?
Let A be a C\* algebra and X is a matrix which entries are positive elements of A.
Does sp(X) has nonempty intersection with positi... | https://mathoverflow.net/users/36688 | the spectrum of matrix with positive entries | The answer is no. For example, the 4-by-4 matrix
$\begin{pmatrix} p & e \\ f & q \end{pmatrix}$,
with $p=\begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix}$, $q=1-p$, $e=\begin{pmatrix} 1/2 & 1/2\\ 1/2 & 1/2\end{pmatrix}$, and $f=1-e$ has the characteristic polynomial $x^4-2x^3+x^2+1/4=(x^2-x)^2+1/4\geq1/4$ and has no real ei... | 5 | https://mathoverflow.net/users/7591 | 150634 | 80,468 |
https://mathoverflow.net/questions/150635 | 2 | Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z\_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map $\pi: X\to V$ such that the restriction map $\pi: X\backslash \pi^{-1}(Z\_{sing})\to V\backslash Z\_{sing}$ is an isomo... | https://mathoverflow.net/users/5082 | Resolution of singularity of polynomials | For generic determinantal varieties, one log resolution is what is classically known as the "space of complete collineations". In the following article, Vainsencher computes the integers you want (although he uses slightly different notation than is common now).
MR0738261 (85f:14053) Reviewed
Vainsencher, Israel(... | 3 | https://mathoverflow.net/users/13265 | 150648 | 80,472 |
https://mathoverflow.net/questions/150660 | 0 | Denote $F\_2=\langle a, b\rangle$ to be the free group on two generators $a, b$.
Let $H\leq F\_2$ to be a subgroup with finite index $n$, so $H\cong F\_{n+1}$ by Nielsen–Schreier theorem, recall that $H$ is called self-normalizing if the normalizer of $H$ inside $F\_2$ to equal to $H$,
Question:
Can anyone give... | https://mathoverflow.net/users/9305 | finite index, self-normalizing subgroup of $F_2$ | Take any finite group generated by two elements with a subgroup of odd index that is self-normalizing, and lift it to $F\_2$.
For example, take $S\_3$, generated by $(1,2,3)$ and $(1,2)$, and consider the subgroup generated by $(1,2)$. It is self-normalizing. Now consider the induced homomorphism $F\_2\to S\_3$ mappi... | 4 | https://mathoverflow.net/users/3959 | 150663 | 80,476 |
https://mathoverflow.net/questions/150657 | 6 | Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that $\mathcal{G}r$ has a natural increasing filtration $$\mathcal{G}r\_0\subseteq\mathcal{G}r\_1\subseteq\ldots\subseteq\mathcal{G... | https://mathoverflow.net/users/25358 | Stratifications and Filtrations of the Affine Grassmannian | I do not really answer you question but maybe this helps:
Let $\mathcal{K} =\mathbb{C}((t))$ and $\mathcal{O}:=\mathbb{C}[[t]]$. For $n\geq 0$ denote the $\mathcal{K}\_n$ the $\mathcal{O}$ ideal in $\mathcal{K}$ generated by $t^{-n}$. Choose an embedding $G\hookrightarrow GL\_m$. Let
$$G(\mathcal{K}\_n):=\{A\in G(\ma... | 3 | https://mathoverflow.net/users/32972 | 150666 | 80,479 |
https://mathoverflow.net/questions/147211 | 6 | Let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$. When $G={\rm GL}\_n$, it is a classical result that $(G(F),G(\mathcal{O}))$ is a Gelfand pair. Is it actually a strong Gelfand pair? I am particularly interested in the case $n=2$.
| https://mathoverflow.net/users/42466 | Is $(G,K)$ a strong Gelfand pair? | $\DeclareMathOperator\Res{Res}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\GL{GL}$You mean restriction of irreducible smooth, admissible representations of $G(F)$ to $G(o)$ decomposes with multiplicity one? Then yes for $n=2$.
Here are some more exact references:
* One-dimensional representations are obvious.
... | 3 | https://mathoverflow.net/users/10400 | 150676 | 80,486 |
https://mathoverflow.net/questions/150614 | 4 | This is a question inspired by "[A question about independent set in regular graphs](https://mathoverflow.net/questions/150235/a-question-about-independent-set-in-regular-graphs)".
Suppose that $G$ is a simple $r$-regular graph with $n$ vertices.
We say $H$ is a dominating set for $T$, if
for every vertex $v\in T$, ... | https://mathoverflow.net/users/42971 | A new question about maximal independent sets in regular graphs | As Joro said, there is a counterexample for your question. In fact, C. Berge and C. Payan (independently) conjectured that any regular graph has two disjoint maximal independent sets of vertices, and this conjecture has shown to be false for some regular graphs. See the following papers:
1. C. Payan, A counter-exampl... | 2 | https://mathoverflow.net/users/43517 | 150683 | 80,488 |
https://mathoverflow.net/questions/150690 | 1 | I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO (see [How to detect a simple closed curve from the element in the fundamental group?](https://mathoverflow.net/questions... | https://mathoverflow.net/users/25511 | Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$ | The free subgroup generated by $a\_1,b\_1$ is carried by a one-holed torus $T$ embedded in the surface. Let $\gamma$ be a simple closed curve embedded in $T$, and consider $T\smallsetminus \gamma$. A simple argument with Euler characteristic shows that:
* either $T\smallsetminus\gamma$ is the union of a one-hold toru... | 6 | https://mathoverflow.net/users/1463 | 150694 | 80,492 |
https://mathoverflow.net/questions/150693 | 5 | As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, there is a chance, and even more: all limit ordinals less than $\omega^2$ come as good old natural numbers.
Let $\pi:\mat... | https://mathoverflow.net/users/2672 | Prime numbers and limit ordinals | The ordinals below $\omega^2$ are exactly those of the form $\omega\cdot n+k$ for natural numbers $n$ and $k$. Thus, these are the ordinals having two digits in base $\omega$, and counting to $\omega^2$ is much like counting to one hundred in this regard.
To order $\mathbb{N}$ in order type $\omega^2$, therefore, is... | 11 | https://mathoverflow.net/users/1946 | 150707 | 80,495 |
https://mathoverflow.net/questions/150478 | 33 | We know the followings :
$$\int\_{0}^{\infty}\frac{{\sin}x}{x}dx=\int\_{0}^{\infty}\frac{{\sin}^2x}{x^2}dx=\frac{\pi}{2},\int\_{0}^{\infty}\frac{{\sin}^3x}{x^3}dx=\frac{3\pi}{8}.$$
Also, we can get
$$\int\_{0}^{\infty}\frac{{\sin}^3x}{x^2}dx=\frac{3\log 3}{4},\int\_{0}^{\infty}\frac{{\sin}^4x}{x^3}dx=\log 2.$$
Then, I... | https://mathoverflow.net/users/34490 | Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$? | I'm posting an answer just to inform that the question has received an answer by Nick Strehlke on [MSE](https://math.stackexchange.com/questions/558014/can-we-simplify-int-0-infty-frac-sinpxxqdx/590849#590849).
\begin{align\*}
\int\_0^\infty {\sin^p x\over x^q}\,dx & = \left\{\begin{array}{ll}
\displaystyle{(-1)^{(... | 22 | https://mathoverflow.net/users/34490 | 150713 | 80,497 |
https://mathoverflow.net/questions/150698 | 2 | Suppose that $G$ is a reductive group defined over a field $k$ which is not quasisplit. Suppose that $S$ is a maximal $k$-split torus. Let $\mathcal{L}(S)$ be the centraliser of $S$ and $\mathcal{DL}(S)$ the commutator group of $\mathcal{L}(S)$ or the anisotropic kernel. Is it correct that $\mathcal{L}(S)=\mathcal{DL}(... | https://mathoverflow.net/users/15482 | centralisers of maximal split tori | The question needs a little more context, which I'll try to fill in. The comments by user76758 seem to answer the question correctly in the negative, but if I have time later I'll try to add a concrete counterexample based on the Tits classification (summarized by him in lecture ntoes for the 1965 AMS summer institute ... | 1 | https://mathoverflow.net/users/4231 | 150721 | 80,501 |
https://mathoverflow.net/questions/150722 | 6 | There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F\_n$. On the other hand, there seems not to be a publication in a journal about this, but several related publications. Is it possible to say which are the "best" resul... | https://mathoverflow.net/users/32332 | Is the conjugacy problem solvable in $Out(F_n)$? | One result in this direction is given by [Dahmani](https://arxiv.org/abs/1307.2108) . His algorithm will determine conjugacy for pairs of *atoroidal* outer automorphisms, ie automorphisms that do not fix a non-trivial conjugacy class. Note that he uses some very heavy machinery---his solution goes via the isomorphism p... | 4 | https://mathoverflow.net/users/1463 | 150726 | 80,502 |
https://mathoverflow.net/questions/150122 | 3 | I am perplexed by the following statement in the paper "A New Conformal Invariant and Its Applications to the Willmore Conjecture and the First Eigenvalue of Compact Surfaces" by Li and Yau found here <http://www.doctoryau.com/papers/NewConformalInvariant.pdf>
On page 275, in the derivation of (2.7) it seems that the... | https://mathoverflow.net/users/38468 | Minor technical question in diff geometry | The notation is potentially confusing, but the end result is correct.
Essentially this claim is the relationship between the area and the energy of of a conformal map. Here is a slightly expanded proof of their computation:
Let $X\_i:\mathbb{R}^{n+1}\to\mathbb{R}$ denote the $i$-th coordinate function.
Suppose tha... | 4 | https://mathoverflow.net/users/1540 | 150727 | 80,503 |
https://mathoverflow.net/questions/150674 | 3 | Let p be an integer, and let G be the graph $(V=Gr(k,\mathbb{F}\_q ^n),E)$
where: $Gr(k,\mathbb{F}\_q ^n)$ is the set of all subspace of $\mathbb{F}\_q$ of dimension k, and $E=\{ W\_1,W\_2 \in V | W\_1\cap W\_2 \neq \{0 \} \}$
Is there previous research about this and a standard name for this graph? I am looking m... | https://mathoverflow.net/users/43583 | Graph of Grassmannian | These are graphs in the Grassmann scheme, and are the $q$-analogs of the Kneser graphs. A clique will be a collection of subspaces of dimension $k$, any two of which have non-trivial intersection. An obvious class of cliques arise as the set of all $k$-subspaces that contain a given 1-dimensional subspace; such cliques... | 2 | https://mathoverflow.net/users/1266 | 150734 | 80,506 |
https://mathoverflow.net/questions/150571 | 5 | It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more interesting Poisson structures? For example, are there Poisson structures on $\operatorname{Spin}(5)$ (or $SO(5)$) such t... | https://mathoverflow.net/users/13767 | Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space? | Yes:
imho geometrically the most interesting one is obtained as a quotient $SO(5)/SO(4)$ where the Poisson stucture on $SO(5)$ is not the so-called standard one, but one determined by an element in the maximal torus (sometimes they are called twisted). This is the Poisson analogue of what is mentioned in Quantum symm... | 7 | https://mathoverflow.net/users/6032 | 150737 | 80,507 |
https://mathoverflow.net/questions/150687 | 1 | Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( \hat{\zeta })=0 $, where $\alpha$ here is connection form and $L^{\times}$ is $S^1$-bundle obtained from line bundle $L$,... | https://mathoverflow.net/users/nan | Unicity of a vector field on $S^1$-bundle | Let $(M,\omega)$ be a symplectic manifold. The first idea behind geometric quantization is that there exists a $S^1$-principal fiber bundle $\pi : Y \to M$, equipped with a connection $\lambda$ with curvature $\omega$, that is, $d\lambda = \pi^\*(\omega)$. Let $\xi$ be the vector field generating the $S^1$ action, then... | 1 | https://mathoverflow.net/users/11885 | 150743 | 80,508 |
https://mathoverflow.net/questions/150744 | 9 | Let $n>3$ be an odd integer and let $K\_n$ denote the complete graph on $n$ vertices.
For which integers $n$ the line graph $L(K\_n)$ is a Cayley graph?
For even $n$, it follows from a result of Watkins that $L(K\_n)$ is not a Cayley graph. For $n=5$, $L(K\_n)$ is the complement of the Petersen graph and so it is not a... | https://mathoverflow.net/users/19075 | The line graphs of complete graphs and Cayley graphs | If $L(K\_n)$ is a Cayley graph for the group $G$, then $G$ is 2-homogeneous on $V(K\_n)$, that is, it acts transitively on the set of unordered pairs of vertices
of $K\_n$. However it is not 2-transitive. Kantor "Automorphism groups of designs" determines the 2-homogeneous groups that are not 2-transitive. He finds th... | 13 | https://mathoverflow.net/users/1266 | 150749 | 80,509 |
https://mathoverflow.net/questions/150750 | 2 | Let $X$ be a manifold such that $dim(X)=n$. It is well-know that if $\mathcal{F}$ is a coherent sheaf $H^m(X,\mathcal{F})=0$ for all $m >n$ (where I denote with $H(-)$ Cech cohomology). But is there a sheaf such that $H^m(X,\mathcal{F}) \ne 0$ for $m>n$?
| https://mathoverflow.net/users/30618 | Vanishing Cech cohomology | I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by [this answer](https://mathoverflow.net/a/43405/75), the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that... | 3 | https://mathoverflow.net/users/75 | 150752 | 80,510 |
https://mathoverflow.net/questions/150733 | 5 | If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times\_C C^I \times\_C B$, where $I$ is the category with two objects and one isomorphism between them.
If we take $I$ to be the category with two objects and one **... | https://mathoverflow.net/users/42440 | 2 and 3 pullbacks | What you have described is a general construction of comma objects from pullbacks and cotensors with $2$. In your first construction you will get the comma object (comma category) $F \downarrow G$. Here is the full story.
The category $0 \rightarrow 1$ consisting of two objects and one morphism between distinct objec... | 7 | https://mathoverflow.net/users/13480 | 150754 | 80,512 |
https://mathoverflow.net/questions/150757 | 2 | Let $X$ be a scheme and let $U \subseteq X$ be an open subset of $X$. If $U$ is an affine chart, then $\mathrm{Spec}(\mathcal{O}\_X(U)) = U$. Suppose now that $U$ is not an affine chart. By this, I mean that $U \ne \mathrm{Spec}(A)$ for any commutative ring $A$. Then the ring $\mathcal{O}\_X(U)$ is a commutative ring; ... | https://mathoverflow.net/users/39777 | What is the significance of the Spec of non-affine structure sheaf sections? | Firstly, $\mathcal{O}\_X |\_U = \mathcal{O}\_U$ (restriction of sheaves), so you could just consider $U=X$. Then $\mathrm{Aff}(X) := \mathrm{Spec}\,\mathcal{O}\_X(X)$ is the universal affine scheme with a morphism from $X$, since we have an adjunction $\mathcal{O}^{op} \dashv \mathrm{Spec}$, i.e. $$\mathcal{S}ch(X, \ma... | 10 | https://mathoverflow.net/users/10605 | 150765 | 80,518 |
https://mathoverflow.net/questions/150740 | 5 | Consider a vector $V$ of length $2n-1$ whose entries are either $1$ or $0$, chosen independently and uniformly at random. Let $X\_i$ be the number of $1$s in subvector $V[i,i+n-1]$. We know $\mathbb{E}(X\_i)=n/2$ and we can consider the $X\_i$ as representing a 1d discrete random walk (with $i$ as the time variable). T... | https://mathoverflow.net/users/nan | Expected maximum distance of a random walk | Define $S\_i$, $i=1,\ldots,2n$ to be the number of $1$s in the interval $[0,i]$, minus $i/2$. Then,
with your notation, $Y\_i:=X\_{i+1}-n/2=S\_{n+i}-S\_i$. You are interested in
$M^\*\_n:=\max\_{i=1}^n |Y\_i|/\sqrt{n}$. Since $\{S\_{[nt]}\}/\sqrt{n}$, $t\geq 0$, converges to one half times a Brownian motion $\{W\_t\}$... | 2 | https://mathoverflow.net/users/35520 | 150774 | 80,525 |
https://mathoverflow.net/questions/150689 | 3 | Let $X$ be Banach space, and $\{Z(t)\}\_{t\geq 0}\subseteq B(X)$ be the $C\_0$-semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}\_{t\geq 0}$. A fractional power of any closed linear operator $F$ is defined, when $(-\infty,0)\subset \rho(F)$(the resolvent set.) and the ... | https://mathoverflow.net/users/43418 | Fractional power of operators in $C_0$-semigroup | Observe that for $r>0$ the semigroup law implies $Z(t)^r = Z(r t)$. If $Z(t)$ is a $C\_0$-group, then this is true for all $0\not=r\in\mathbb{R}$. The generator of $Z(t)^r$ is $(r A, D(A))$. You can find this construction sometimes called 'rescaled semigroup' e.g. in the book of Engel, Nagel, *One-parameter semigroups ... | 3 | https://mathoverflow.net/users/17879 | 150776 | 80,527 |
https://mathoverflow.net/questions/150680 | 4 | Let $ A $ be a set of non-zero integers. Then $A$ contains a sum-free subset $B$ of size $ |B|> \frac{|A|}{3} $ (a result of Erdős). It is conjectured that RHS can be improved to $\frac{|A|}{3} +10$. Is there any evidence/heuristic justification for this conjectured bound?
(A set $B$ is *sum-free* if it contains no s... | https://mathoverflow.net/users/30081 | a conjecture in sum-free sets | This is a notorious problem. It is my impression that a lot of talent people have looked at it, without much progress. Ben Green has an excellent talk on the topic (in part) which can be viewed here: <http://kva.screen9.tv/#gipkF4pTSbAL16tJjJzj8Q>.
There is nothing special about constant 10 in the statement in Tao an... | 16 | https://mathoverflow.net/users/630 | 150777 | 80,528 |
https://mathoverflow.net/questions/150778 | 3 | The name of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$ is one-parameter semigroup.
For example one writes $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$, where $f$ is a function of $\overrightarrow{v}$.
What is the meaning of this operator?
... | https://mathoverflow.net/users/27422 | What is the meaning of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$? | This is a symbol which reminds that the solution of linear system of ODEs is represented as an exponential function of a matrix. This symbol represents the solution family of the heat equation, and indeed called a one-parameter semigroup (or semi-dynamical system).
Great introduction in the subject are the books of E... | 4 | https://mathoverflow.net/users/12898 | 150781 | 80,529 |
https://mathoverflow.net/questions/150283 | 2 | I am interested in the following statement:
>
> Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
> domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\partial U$. Then $\int\_U div(F)=0$.
>
>
>
Naively, if $U\_n$ is a smooth domain approximating $U$ from inside and ... | https://mathoverflow.net/users/18207 | A special case of the Divergence theorem | I will show that if $U$ is open and bounded, $f \in C^1 (U) \cap C(\bar{U})$, $f=0$ on $\partial U$ and $v \in \mathbb{R}^n$, then
$$
\int\_U \partial\_v f = 0,
$$
where $\partial\_v f$ is the directional derivative of $f$ in the direction $v$.
The proof goes as follows. First, the statement can be proved in the one... | 3 | https://mathoverflow.net/users/42047 | 150785 | 80,531 |
https://mathoverflow.net/questions/150782 | 1 | Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is numerically stable?
An approach I could think of is to take a random vector, then perform a step in the Gram-Schmidt m... | https://mathoverflow.net/users/43066 | Augmenting orthonormal system into complete orthonormal system in a numerically stable way | Don't use Gram-Schmidt; simply take a QR factorization of your matrix $U$. If your starting matrix was exactly orthogonal, then the top $3\times 3$ block $R\_1$ of $R$ will be orthogonal as well (hence also diagonal, since it's upper triangolar -- if it's not the case and you know the initial data was correct, then you... | 2 | https://mathoverflow.net/users/1898 | 150788 | 80,533 |
https://mathoverflow.net/questions/150628 | 0 | Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to \mathbb{C}; f(cz)=c^{-1}f(z), c\in \mathbb{C}-\{0\} , z\in L^{\times}\}.$$
Here $L^{\times}$ is the structure bundle obtained fr... | https://mathoverflow.net/users/nan | An isomorphism on space of smooth sections | I want first to fix a confusion at the level of the two bundles involved in this construction. My way(\*) is to consider, first of all, a $S^1$-bundle over a symplectic manifold $(M,\omega)$, with curvature $\omega$ — assuming the symplectic form is integral. Let $(Y,\lambda)$ be this prequantization, and $\pi : y \map... | 1 | https://mathoverflow.net/users/11885 | 150792 | 80,536 |
https://mathoverflow.net/questions/150794 | 4 | Consider a countable transitive model $\mathfrak{M}$ of set theory.
Let $X$ be a definable collection of sets of reals.
My question is: is the type of nondefinable elements in $X$ is definable over $Th(\mathfrak{M})$ or not.
(I assume that $X$ is infinite.)
PS: note that the type of nondefinable elements of $X$... | https://mathoverflow.net/users/38200 | The type of nondefinable elements-2 | It depends on $\mathcal{M}$ and on $X$. I take your question to ask whether there is a formula $\phi(x)$ without parameters such that whenever $\mathcal{N}$ has the same theory as $\mathcal{M}$, then $\mathcal{N}\models\phi(A)$ just in case $A\in X^{\mathcal{N}}$ and $A$ is not definable in $\mathcal{N}$.
First, let... | 4 | https://mathoverflow.net/users/1946 | 150802 | 80,540 |
https://mathoverflow.net/questions/150803 | 7 | Let $\mathfrak{h}$ be a Cartan subalgebra of a $\mathbb{C}$-semi simple Lie algebra $\mathfrak{g}$. Given $\lambda \in \mathfrak{h}^\*$, $M(\lambda)$ the Verma module of highest weight $\lambda$ and $N(\lambda)$ its maximal submodule.
1. Do we have examples of $\lambda$ integral and regular for which $N(\lambda)$ is ... | https://mathoverflow.net/users/43653 | Maximal Submodule of a Verma Module | For your first question the answer is yes, but probably you need to look at specific examples worked out using Kazhdan-Lusztig theory. It should be enough to look at type $A\_3$ (as BGG did), where values $>1$ start to appear for some K-L polynomials evaluated at 1. I believe the hand computations by BGG were quicker f... | 9 | https://mathoverflow.net/users/4231 | 150806 | 80,542 |
https://mathoverflow.net/questions/150810 | 10 |
>
>
> >
> > Let $f \in \mathbb{Z}[x]$ be a non-zero polynomial which is *irreducible* over $\mathbb{Q}$. Suppose that $f$ has a root in $\mathbb{F}\_p$ for almost all primes $p$. Must $f$ be linear?
> >
> >
> >
>
>
>
Here by almost all, I mean all but finitely many.
Here is an equivalent way to state this q... | https://mathoverflow.net/users/5101 | Irreducible polynomials with a root modulo almost all primes | Yes, this is true and can be proved by applying Chebotarev's density theorem. For a hint, see exercise 7.2 in <http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf>
See also [Cassels-Fröhlich], p. 362, Exercise 6.2.
| 12 | https://mathoverflow.net/users/nan | 150812 | 80,543 |
https://mathoverflow.net/questions/106847 | 6 | It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see [previous question](https://mathoverflow.net/questions/106786/coordinate-free-derivation-of-the-einsteins-field-equation-from-the-hilbert-acti)). The standard deri... | https://mathoverflow.net/users/16464 | Geometric derivation of the Einstein’s field equation from the Hilbert action. | Comment: the following is a somewhat convoluted way of deriving the
Euler-Lagrange equation using Clairaut's theorem for the volume functional and
some standard, albeit not simpler, variation formulas (all is $C^{\infty}$ in
the following). Let $g\_{0}$ be a Riemannian metric and let $v$ be a symmetric
$2$-tensor. Let ... | 3 | https://mathoverflow.net/users/nan | 150820 | 80,549 |
https://mathoverflow.net/questions/150720 | 0 | Let $N,n$ be natural numbers.
Let us define $a\_n=m$ when $N!$ **can** be divided by $(n!)^m$ and it **cannot** be divided by $(n!)^{m+1}$.
For a given $N(\ge 2)$, let $\min(N)$ be the min of $na\_n\ (2\le n\le N)$.
Then, here is my question.
>
> **Question** : What is $\min(N)$?
>
>
>
**Example** :
$... | https://mathoverflow.net/users/34490 | Finding the min of a sequence related with factorials | Lemma: Let $n,a$, be positive integers with $r = n \bmod a$, the remainder after
dividing $n$ by $a$. Using Iverson notation [statement] is $1$ if true, $0$ otherwise,
$\lfloor 2n/a \rfloor = 2\lfloor n/a \rfloor + [2r \geq a]$.
From this, one shows that the exponent of a prime $p$ in a prime factorization of
$C\_n={... | 1 | https://mathoverflow.net/users/3206 | 150841 | 80,556 |
https://mathoverflow.net/questions/150725 | 4 | There are some discussion of shellable simplicial complexes here
[Testing simplicial complexes for shellability](https://mathoverflow.net/questions/121371/testing-simplicial-complexes-for-shellability). My question is the following:
Assume that $\Delta$ is a pure simplicial complex on a finite vertex set $V$. Assum... | https://mathoverflow.net/users/14563 | Cohen-Macaulay versus shellable simplicial complexes | The Dissertation of Frank Lutz
(<http://page.math.tu-berlin.de/~lutz/dissertation.ps>) has examples of
Cohen-Macaulay vertex-transitive simplicial complexes that are not
shellable. In particular, there is such a triangulation of Poincaré
dodecahedral space $X$ with 17 vertices. This triangulation is
Cohen-Macaulay sin... | 4 | https://mathoverflow.net/users/2807 | 150844 | 80,558 |
https://mathoverflow.net/questions/150840 | 0 | Let $E$ be a Frechet space and $\mathcal{F}$ be a non-degenerate bounded skew symmetric bilinear map $\mathcal{F}: E\times E\to \mathbb R$ on $E$. We can identify $TE$ with $E\times E$, with this identification, $\mathcal{F}$ defines 2 form $\omega$ on $E$, by the following:
$$\omega\_x: T\_x E\times T\_xE\to \mathbb R... | https://mathoverflow.net/users/23534 | Constant symplectic structure | Yes, it is. See Section 48 of
* [Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008](http://www.mat.univie.ac.at/~michor/dgbook.pdf)
| 2 | https://mathoverflow.net/users/26935 | 150849 | 80,559 |
https://mathoverflow.net/questions/150830 | 11 | It is well-known that the compactification $\overline M\_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with
weights $4$ and $6$. As far as I can tell, this is more or less directly linked to the fact that the ring of holomorphic modular forms for $SL\_2(\mathbb Z)$
is a ... | https://mathoverflow.net/users/38468 | Is the universal elliptic curve $\overline M_{1,2}$ a toric stack? | The fibers aren't rational curves when viewed as stacks, because they have four points with extra automorphisms. I think this will be problematic.
The universal family of elliptic curves is the quotient of the scheme with projective variablesz $x,y,z$, affine variables $g\_2,g\_3$, and equation $y^2z=x^3-g\_2xz^2-g\_... | 3 | https://mathoverflow.net/users/18060 | 150864 | 80,566 |
https://mathoverflow.net/questions/150863 | 4 | Let $f:\mathbb R\to\mathbb C$ be a sufficiently smooth and $1$-periodic function of average zero (i.e., $\int\_0^{1}f(x)\,dx=0$), and let $\alpha\in(0,1)\smallsetminus\mathbb Q$. We know that
$$
\lim\_{N\to\infty}\frac{1}{N} \sum\_{k=1}^N f(k\alpha)=0.
$$
Let $s\_n=\sum\_{k=1}^n f(k\alpha)$. Is it true that the se... | https://mathoverflow.net/users/43681 | Does equidistribution of zero average, due to irrationality, imply boundedness? | I believe the answer depends on the [irrationality measure](https://mathoverflow.net/questions/89600/numbers-with-known-irrationality-measures) of $\alpha$.
Per Fourier, Write $f= \sum\_{m=-\infty}^\infty a\_m e^{ 2 \pi i m x}$, then
$$s\_n = \sum\_{m=-\infty}^\infty a\_m \frac{ e^{2\pi i m (n+1) \alpha} - 1}{e^{2 ... | 10 | https://mathoverflow.net/users/18060 | 150866 | 80,567 |
https://mathoverflow.net/questions/150867 | 1 | I gather that the following two identities about $\xi(3)$ hold via some notion of zeta-function regularized integrals.
$\xi(3) = \frac{(2\pi)^3}{3}\int \_0 ^\infty d\lambda \frac{\sqrt{\lambda} }{1 + e^{2 \pi \sqrt{\lambda} } } = - \frac{4 \pi^3 }{3} \int \_0^\infty d\lambda \sqrt{\lambda}\text{ } tanh (\pi \sqrt{\l... | https://mathoverflow.net/users/36554 | zeta-function regularized integrals | We have
$$\tanh(x) = \dfrac{1 - e^{-2x}}{1 + e^{-2x}} = (1-e^{-2x}) \sum\_{k=0}^{\infty}(-1)^k e^{-2kx} = 1 + 2 \sum\_{k=1}^{\infty}(-1)^ke^{-2kx}$$
Now we have
$$\sqrt{x} \tanh(\sqrt{x}) = \sqrt{x} + 2 \sum\_{k=1}^{\infty}(-1)^k \sqrt{x}e^{-2k\sqrt{x}}$$
Now throwing away the divergent part, i.e., $\sqrt{x}$, as every... | 4 | https://mathoverflow.net/users/nan | 150869 | 80,568 |
https://mathoverflow.net/questions/150846 | 12 | Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in the case of Bridgeland and Maciocia's proof of Fourier-Mukai transforms on abelian fiberation etc.)
I want to know
(... | https://mathoverflow.net/users/29730 | Derived categories of singular varieties | Probably the main reason people avoid singular varieties is because of boundedness. Arguments using induction and, as you say, spectral sequences need boundedness of the complexes in play in one direction or the other (and sometimes both).
Related to this, is the failure of Serre duality, which lies at the heart of man... | 11 | https://mathoverflow.net/users/36922 | 150870 | 80,569 |
https://mathoverflow.net/questions/150801 | 4 | Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller than $\pi/2$?
| https://mathoverflow.net/users/28128 | hyperbolic orbifolds of small area | Yes. One can extend the Gauss-Bonnet theorem to 2-orbifolds since but area an orbifold Euler characteristic behave well under the covering map.
To find all hyperbolic 2-orbifolds with area smaller than $\pi/2$, we need to get a list of orbifolds $\{O\_i\}$ $0>\chi(O\_i)>-1/4$.
Thankfully, this is not a horrible tas... | 7 | https://mathoverflow.net/users/27453 | 150874 | 80,572 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.