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 |
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https://mathoverflow.net/questions/296520 | 16 | While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
1. When defining Dolbeault cohomology, one uses $\bar{\partial}$ but not $\partial$. I wonder whether there happens any problem if one define a cohomolo... | https://mathoverflow.net/users/122571 | Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology | 1. On differential forms, take complex conjugate to turn $\partial$ into $\bar\partial$, and holomorphic functions into conjugate holomorphic.
2. All of the proofs about differential forms then go through the complex conjugation effortlessly, including the Poincare lemma. We use $\bar\partial$ because we like holomorph... | 16 | https://mathoverflow.net/users/13268 | 296522 | 130,302 |
https://mathoverflow.net/questions/296478 | 3 | Notation: $k$ - field. "$k$-category" = $k$-linear abelian category. $Vect\_k$ - the $k$-category of $k$-vector spaces. For a field extension $K/k$ and a $k$-category $\mathcal{A}$, denote by $\mathcal{A}\_K$ the $K$-category consisting of objects $M \in \mathcal{A}$ together with a $k$-algebra homomorphism $K \to End(... | https://mathoverflow.net/users/2095 | Brauer group classifying some splitting categories | Everything you might dream about is true if you work with dg categories instead of abelian $1$-categories. This is proved in Toën's paper on derived Azumaya algebras. In particular, your question about local possession of a compact generator is dealt with and forms a crucial part of the proof. David Gepner and I proved... | 1 | https://mathoverflow.net/users/100 | 296528 | 130,306 |
https://mathoverflow.net/questions/296448 | 2 | I have studied the proof of the theorem of Torelli (Andreotti's proof) where the centerpiece is the definition of branch locus of a morphism to projective space.
My question is:
Let $f: X \longrightarrow Y$ be a morphism from a projective variety $X$ to projective space Y.
How to define the branch locus of $f$? ... | https://mathoverflow.net/users/29836 | branch locus of a morphism to projective space | The definition of branch locus of any morphism $f\colon X \to Y$ of irreducible varieties, with $Y$ normal, is given in I. Shafarevich, **Basic Algebraic Geometry**, Springer, volume 1, p. 144.
| 2 | https://mathoverflow.net/users/13268 | 296530 | 130,307 |
https://mathoverflow.net/questions/295046 | 28 | I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to use the word group. (**Edit:** Taking another look, at least some of Cassels' papers from this period do use the word gr... | https://mathoverflow.net/users/6518 | When did people start thinking of elliptic curves as groups? | The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was
Juel [Ueber die Parameterbestimmung von Punkten auf Curven zweiter
und dritter Ordnung. Eine geometrische Einleitung in die Theorie der
logarithmischen und elliptischen... | 18 | https://mathoverflow.net/users/3503 | 296532 | 130,309 |
https://mathoverflow.net/questions/296511 | 5 | I'm reading a book "Complex Geometry" by Daniel Huybrechts. In this book he says that a simply connected dga satisfying some conditions must be minimal. (p.147, Remark 3.A.13) I tried to prove this statement but eventually failed. It seems that there is a counter example.
The statement in the book is as follows:
>... | https://mathoverflow.net/users/122571 | A condition for a dga to be minimal | You are correct and the statement, as you cite it, is wrong. The counterexample you describe is well-known, and is usually cited to demonstrate precisely this failure.
| 7 | https://mathoverflow.net/users/3075 | 296535 | 130,311 |
https://mathoverflow.net/questions/296506 | 2 | Consider Propositional Lax Logic ($PLL$)
* <https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf>
The Hilbert system of $PLL$ takes as axiom schemata all theorems of (or a complete set of axioms for) the Intuitionistic propositional calculus plus the modal axi... | https://mathoverflow.net/users/122435 | Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle? | Notice that by the inference rule
$$\frac{M\supset N}{\bigcirc M\supset\bigcirc N}\tag{@}$$
we have
$$\frac{\bot\supset N}{\bigcirc \bot\supset\bigcirc N}$$
But $\bot\supset N$ always holds. So either we have $\neg\bigcirc\bot$, the case already covered in the linked answer; or we have $\bigcirc\bot$, in which case all... | 2 | https://mathoverflow.net/users/4600 | 296539 | 130,312 |
https://mathoverflow.net/questions/296345 | 2 | Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let
$\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere.
What can we say about the image of $\phi$? Of course in general is not an open subset (constant maps are harmonic). But i... | https://mathoverflow.net/users/86341 | Properties of harmonic maps into spheres | Two very good introductions to the subject are:
1) Harmonic maps, conservation laws, and moving frames by Frédéric Hélein
2) Analysis Of Harmonic Maps And Their Heat Flows by Changyou Wang, Fanghua Lin
I think the answer to your question is probably no, since a special case is given by minimal surfaces and there ... | 1 | https://mathoverflow.net/users/9253 | 296541 | 130,314 |
https://mathoverflow.net/questions/296461 | 3 | A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$
The mean of the probability distribution whose density this is, is
$$
\frac{a+b+c} 3
$$
and the variance is
$$
\frac{a^2+b^2+c^2-ab-a... | https://mathoverflow.net/users/6316 | Symmetry in the triangular distribution | The distribution function equals
$$
p(x)=\frac{|x-a|}{(a-b)(a-c)}+\frac{|x-b|}{(b-a)(b-c)}+\frac{|x-c|}{(c-b)(c-a)}.
$$
This is pretty symmetric. If you need a $k$-th moment, it equals
$\frac2{(k+1)(k+2)}h\_k(a,b,c)$, where $h\_k$ is complete homogeneous polynomial (sum of all monomials of degree $k$.)
| 3 | https://mathoverflow.net/users/4312 | 296544 | 130,315 |
https://mathoverflow.net/questions/296558 | 1 | Let $S$ and $T$ be sets of vectors from $\mathbb{R}^d$ such that $S$ and $T$ are at least different in one element.
Does there exist a random matrix $M \in \mathbb{R}^{d \times k}$, e.g., a gaussian matrix, such that the probability of $ \sum\_{s \in S} s M = \sum\_{t \in T} t M $ is small in terms of $k$?
| https://mathoverflow.net/users/56325 | Probability of collision of sums of vectors multiplied by random matrix | I hope this makes sense.
Let $v$ be the difference of the two vector sums. Since a randomly chosen Gaussian matrix will have maximuml rank $\min(d,k)$ with constant probability and almost maximum rank with overwhelming probability, the answer would be yes for most vectors $v \neq 0$.
So the answer would be dominate... | 1 | https://mathoverflow.net/users/17773 | 296561 | 130,320 |
https://mathoverflow.net/questions/296536 | 3 | Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim\_{p\rightarrow\infty}\frac {\ln \|f\|\_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)
Moreover generally I would like to know if... | https://mathoverflow.net/users/32470 | Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity | The limit doesn't always exist.
For convenience I will use the Lebesgue measure; similar construction can also be done for most $\mu$.
Let $c\_n$ be a sequence of increasing positive integers. Consider the function
$$ f(x) = \sum\_{n = 0}^\infty c\_n^2 \chi\_{[n, n + (c\_n!)^{-1}]}(x) $$
By the disjoint suppor... | 5 | https://mathoverflow.net/users/3948 | 296563 | 130,321 |
https://mathoverflow.net/questions/296567 | 6 | Here are some simple geometry problems I am unable to resolve to my satisfaction. I asked the question on Math Stack (<https://math.stackexchange.com/questions/2713754/a-problem-in-elementary-combinatorial-space-geometry>) but it has received no interest, so I ask here in a different format.
Let $\delta$ denote the ... | https://mathoverflow.net/users/49117 | Some Elementary Schubert Calculus Calculations | **Edit.** In the following parameter space, the "double line locus" $\Delta$, where $L$ equals $M$, appears multiply in the first intersection cycle $\delta \mu^2\nu^4\rho$. Indeed, if $L$ equals $M$, if $p$ equals the intersection of $L=M$ with the specified $\rho$-hyperplane, and if $L=M$ is one of the $2$ lines that... | 6 | https://mathoverflow.net/users/13265 | 296583 | 130,328 |
https://mathoverflow.net/questions/294995 | 2 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly harmonic maps between Riemannian manifolds. I would like to know if my approach makes sense. (can it be made valid unde... | https://mathoverflow.net/users/46290 | Is this approach for establishing regularity of harmonic maps between manifolds valid? | Regularity of harmonic map is true in dimension with $f\in W^{1,2}$ but it is subtle see Helein, Harmonic maps, conservation laws and moving frames.
It is false in higher dimension, even if f\in W^{1,n}$, see Rivière, Everywhere discontinuous harmonic maps into spheres.
Whet you need is to control the morrey norm, ... | 2 | https://mathoverflow.net/users/9253 | 296590 | 130,331 |
https://mathoverflow.net/questions/296596 | 1 | Let $q$ be a power of $2$.
Assume that elements of the finite field $\operatorname{GF}(q)$ are denoted by $\beta\_i$
for $0\leq i \leq q-1$. We divide elements of $\operatorname{GF}(q)$ in two parts as follows; $x\_i=\beta\_i$, $0\leq i \leq k-1$ and $y\_j=\beta\_{k+j}$, for $0\leq j \leq q-k-1$.
From elements $x\... | https://mathoverflow.net/users/64181 | An Extension of an $\operatorname{MDS}$ Code over $\operatorname{GF}(2^q)$ | Your notation is nonstandard. Code length is $n,$ dimension is $k,$ minimum distance is $d=n-k+1$ when a code is MDS. The paper by Alderson [available here](https://www.unb.ca/saintjohn/sase/dept/math/_resources/alderson/mds-codes.pdf)
proves the following:
**Theorem 2.** *A $q-$ary $(q+k−2,k)-$ MDS code can be exten... | 1 | https://mathoverflow.net/users/17773 | 296600 | 130,335 |
https://mathoverflow.net/questions/296555 | 5 | The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 theory of finite-dimensional representations of a semisimple Lie algebra. (This question was recently raised by a colleague... | https://mathoverflow.net/users/4231 | Difference of adjacent dominant weights is a root? | This is true. See "The partial order of dominant weights" by John Stembridge, 1998 (<https://www.sciencedirect.com/science/article/pii/S0001870898917364>). In particular look at Corollary 2.7. Alternatively, look at the second proof of Corollary 2.7 given there, due to Robert Steinberg, which the paper claims was commu... | 6 | https://mathoverflow.net/users/25028 | 296603 | 130,338 |
https://mathoverflow.net/questions/296581 | 5 | The set $${\cal B} = \big\{\emptyset\big\}\cup\big\{\{a + bn: n\in\omega\}: a\in\omega, b\in(\omega\setminus\{0\})\big\}$$ is a basis for a topology $\tau$ on $\omega$. Is there a surjective continuous map from $\mathbb{Q}$ with the Euclidean topology onto $(\omega,\tau)$, or the other way round, or in neither directio... | https://mathoverflow.net/users/8628 | Is $\mathbb{Q}$ the continuous image of a Golomb-like space, or vice versa? | Your topology is by definition the profinite topology of $\mathbf{Z}$, restricted to the nonnegative numbers $\mathbf{N}$. Hence it's a nonempty countable metrizable space. By a classical theorem of Sierpiński ([Dasgupta - Countable metric spaces without isolated points](http://at.yorku.ca/p/a/c/a/25.htm), [pdf](http:/... | 6 | https://mathoverflow.net/users/14094 | 296620 | 130,350 |
https://mathoverflow.net/questions/296608 | 2 | Let $N$ be a Poisson process with parameter $\lambda$, that is, for $a>b\geq0$, there is $$P[N(a,b)=k]=\frac{((a-b)\lambda)^k}{k!}e^{-(a-b)\lambda}.$$ Now denote $N\_t=N[0,t)$ and define
$$
M\_t=N\_t-\lambda t\qquad 0\leq t<\infty.
$$
we can easily show that $M$ is a Martingale. Now the question is, prove that for any... | https://mathoverflow.net/users/78326 | One question about compensated Poisson process | $\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$
If $(S\_t)\_{t\ge0}$ is a nonnegative submartingale, then the following [Dood inequalities](https:... | 2 | https://mathoverflow.net/users/36721 | 296621 | 130,351 |
https://mathoverflow.net/questions/296638 | 12 | This is a question about the proofs of Kazhdan-Lusztig's conjectures for category $\mathcal{O}$ using higher representation theory (avoiding Beilinson-Bernstein's geometric localization theory).
Using [Bernstein-Frenkel-Khovanov](https://arxiv.org/abs/math/0002087) and generalizations (Sussan, Stroppel-Mazorchuk, et... | https://mathoverflow.net/users/2623 | Questions about categorification (& combinatorial simplification of the Russian approach to Lusztig's conjectures, in zero & positive characteristic) | This is not a truly independent proof. Both of the papers of mine above use the decomposition theorem for various collections of algebraic varieties, which essentially include the proof of the original KL conjecture as special cases. You should think of the techniques in those papers as a generalization of Soergel's JA... | 20 | https://mathoverflow.net/users/66 | 296648 | 130,356 |
https://mathoverflow.net/questions/296647 | 5 | In the case of orientable closed $3$-manifolds, we have 4 ingredients that ensure parallelizability:
1a) Closed smooth $n$-manifolds have homotopy type of a $CW$-complex
1b) Closed smooth $n$-manifolds are Poincare duality spaces.
2) First Stiefel-Whitney class is $0$, from orientability
3) $BSU(2)$ is $3$-conn... | https://mathoverflow.net/users/62647 | Are open orientable 3-manifolds parallelizable via obstruction theory? | I think you are saying: for a closed $3$-manifold, vanishing $w\_1$ implies vanishing $w\_2$ by Wu's relations, but is this still true if the manifold is not closed? The answer is yes.
For a compact manifold with boundary, you can form the double (union of two copies of $M$ along $\partial M$). This will again be ori... | 11 | https://mathoverflow.net/users/6666 | 296649 | 130,357 |
https://mathoverflow.net/questions/296643 | 0 | **Motivation**: In the theory of harmonic maps between manifold, we often see the characterization
$$
\Delta\_g u = -g^{ij}A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right)
$$
when we are working in the extrinsic viewpoint.
>
> **Question**: Let $M,N$ be two Riemannian manifolds, $u... | https://mathoverflow.net/users/80191 | What does $A(u)\left(\frac{\partial u}{\partial x^i},\frac{\partial u}{\partial x^j}\right)$ mean, exactly? | You are correct that $\partial u/\partial x\_i$ is not a vector field on $N$. It is a section of $u^\*TN$, i.e. associating to each point $x \in M$ a vector $\partial u/\partial x\_i \in T\_{u(x)} N$. So we take the second fundamental form $A$ of $N$, which is a section of $S^2(T^\*N) \otimes T^{\perp} N$, and we pull ... | 2 | https://mathoverflow.net/users/13268 | 296657 | 130,358 |
https://mathoverflow.net/questions/296651 | 16 | Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one?
This question is motivated by the following observation. If $F$ satisfies the above and $X$ is uniformly distributed over $[0,1]$, then $F(X)\sim Unif[0,1]$, i.e., $F(X)$ seems t... | https://mathoverflow.net/users/71254 | Is measure preserving function almost surjective? | Yes, by Luzin's theorem. Fix $\varepsilon>0$ and take a compact subset $K$ of measure at least $1-\varepsilon$ such that $F$ is continuous on $K$. Then $F(K)$ is a compact set of at least the same measure as $K$, since $F^{-1}(F(K))\supset K$. So, for any $\varepsilon>0$, $F([0,1])$ contains a measurable subset of meas... | 21 | https://mathoverflow.net/users/4312 | 296658 | 130,359 |
https://mathoverflow.net/questions/295085 | 9 | Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T\_1}(\kappa)$ be the lattice of all $T\_1$-topologies on $\kappa$.
Is there an injective lattice homomorphism $\varphi: \text{Top}(\kappa)\to \text{Top}^{T\_1}(\ka... | https://mathoverflow.net/users/8628 | Does the lattice of all topologies embed into the lattice of $T_1$-topologies? | **Is there an injective lattice homomorphism $\varphi: \text{Top}(\kappa)\to \text{Top}^{T\_1}(\kappa)$?**
The answer is Yes, there is such an embedding.
I will argue that if $\kappa$ is an
infinite cardinal, then there is a complete lattice embedding
$\varphi: \text{Top}(\kappa)\to \text{Top}^{T\_1}(\kappa\times\k... | 10 | https://mathoverflow.net/users/75735 | 296664 | 130,363 |
https://mathoverflow.net/questions/296660 | 3 | Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L\_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ given by $\pi(g): \mathscr{H} \to \mathscr{H}, f \mapsto f \circ r\_g$, where $r\_g$ is right multiplication by $g$. He... | https://mathoverflow.net/users/122635 | Convergence of some object depending on functions with compact support | It's true: $\pi(\varphi)f\in L^2(G,\mu)$.
Let $M$ be the support of $\varphi$, so
$$(\pi(\varphi)f)(x) :=\int\_M \varphi(g)f(xg) \, d\mu(g).$$
By the Cauchy-Schwarz inequality, we have
$$|(\pi(\varphi)f)(x)|^2\le \|\varphi\|^2\_2\int\_M|f(xg)|^2d\mu(g)=\|\varphi\|^2\_2\int\_{xM}|f(g)|^2d\mu(g),$$
so
$$\int\_G|(\pi(... | 2 | https://mathoverflow.net/users/14094 | 296665 | 130,364 |
https://mathoverflow.net/questions/296655 | 4 | Is it possible for a natural number $k$ to exist such that for all primes $p$:
$$(p+1)k-1$$ is composite?
(i.e., can all $3k-1$, $4k-1$, $6k-1$, $8k-1$, $12k-1$, $14k-1$, $18k-1$, $20k-1$, $24k-1$, $30k-1$, $32k-1$, $38k-1$, $42k-1$, $44k-1$, $48k-1$, ... be composite?)
| https://mathoverflow.net/users/122633 | A question related to Dirichlet's theorem | Seva proved that Dickson's conjecture implies there is no such $k$.
One only needs a weak form of this conjecture. Such a weak form could potentially
be much easier than for example the problem on
infinitely Sophie Germain primes, as one does not need to know whether
for two fixed forms, such as
$f\_1(n)=n$ and $f... | 3 | https://mathoverflow.net/users/36707 | 296673 | 130,367 |
https://mathoverflow.net/questions/296682 | 5 | I am beginning to learn about automorphic forms, and stay perplex concerning the two languages of "forms" versus "representations" often used at the same time. As far as I understand,
* a modular/Maass form generates an automorphic representation (by considering the space of its right translations)
* conversely, an ... | https://mathoverflow.net/users/122645 | Modular forms, Maass forms and Automorphic representations | The archimedean component of the representation determines the type of the underlying automorphic form. Loosely speaking,
* if $\pi\_\infty$ is a discrete series (of index $k$), the underlying form is an holomorphic cusp form (of weight $k$, and level the arithmetic conductor of $\pi\_f$)
* if $\pi\_\infty$ is a pri... | 7 | https://mathoverflow.net/users/43737 | 296684 | 130,368 |
https://mathoverflow.net/questions/296685 | 4 | If $A$ is a cocomplete category and $C$ is small, we can say that for all functor $C \stackrel{l}{\to} A$ there exists the Kan Extension Lan$\_{y\_C}(l): \hat{C} \to A$. Moreover, the Kan extension is pointwise.
Is it known if the other implication is true? Namely:
>
> Conj: If for all functor $C \stackrel{l}{\to... | https://mathoverflow.net/users/104432 | Is this a characterization of cocompleteness? | Then answer is yes if you assume the extensions are pointwise.
Let $\ast$ be the terminal presheaf on $\hat C$. Because the extension is poinwise, we have $Lan\_{y\_C}(l)(\ast) = \varinjlim\_{c \in C} l(c)$. This can be seen, for example, by calculating the colimit using the category of elements of $\ast$, which is j... | 3 | https://mathoverflow.net/users/2362 | 296689 | 130,369 |
https://mathoverflow.net/questions/291821 | 8 | Good morning,
I've came across this question during my researches. It seems apparently very simple, however I Googled a bit and I couldn't find the answer I was looking for.
The question is as follows: Is it true that **any** connected subgroup of $GL(d,\mathbb{R})$ admits at least one compact orbit on the sphere $... | https://mathoverflow.net/users/57571 | Existence of at least one compact orbit on the sphere | Yes, it is true that for every continuous real linear finite dimensional representation of a connected Lie group there exists a closed orbit for the associated projective representation.
To see this it is enough to assume the group is solvable (as every connected Lie group has a cocompact solvable subgroup) and the r... | 1 | https://mathoverflow.net/users/89334 | 296690 | 130,370 |
https://mathoverflow.net/questions/296674 | 1 | Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$)
$$\limsup\_{r \to 0} r^{-\alpha \beta}\frac{1}{|B\_{r}(x)|}\int\_{B\_{r}(x)} f(y)^\alpha dy < \infty$$
implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$... | https://mathoverflow.net/users/122620 | Morrey condition (integral condition) and (local) Holder condition | No. Take $n=2$, $x=0$ (and $\alpha=1$, wlog). Define $f$ in polar coordinates as $f(r,\theta)=r^\beta g\_r(\theta)$ where $\int\_0^{2\pi} g\_r(\theta)\ d\theta=1$ and $\lim\_{r\to 0}g\_r(0)=\infty$, which is clearly possible while keeping $f$ continuous. (For example, $g\_r$ could be piecewise linear on $[-r^\gamma,+r^... | 1 | https://mathoverflow.net/users/75422 | 296697 | 130,373 |
https://mathoverflow.net/questions/296693 | 16 | I am looking for some references related to gerbes and their differential geometry. Almost every article I have seen that is related to gerbes there is a common reference that is Giraud's book Cohomologie non-abelienne. For me, it is not readable as I can not read french.
Only references I am familiar with are
* <h... | https://mathoverflow.net/users/118688 | References on Gerbes | The book of Giraud is a fundamental reference on the subject, but you have to be used to the language of Grothendieck. A reference more accessible, for example for a differential geometer is the chapter 5 of the book of Brylinski which deals only with commutative gerbes.
J.L Brylinski Loop Spaces, Characteristic Clas... | 7 | https://mathoverflow.net/users/80891 | 296716 | 130,378 |
https://mathoverflow.net/questions/291718 | 8 | I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.
* I understand that the standard reference is Illusie's two books, unfortunately my French abilities are lacking
* I'm aware of the stacks project treatmen... | https://mathoverflow.net/users/120185 | Elementary (English) reference for the cotangent complex? | The homotopy-theoretic way to look at the cotangent complex is as the left derived functor of the Kahler differentials functor. Now:
* This statement makes sense just in the context of homological algebra. i.e. in some category of chain complexes of modules. This route to constructing the cotangent complex is the sub... | 4 | https://mathoverflow.net/users/74739 | 296728 | 130,383 |
https://mathoverflow.net/questions/296709 | 3 | Each topological space $A$ with fixed-point property is $T\_0$ space. Proof: suppose, two different points $a\_1$ and $a\_2$ belong to the same open subsets of $A$. Then the function
$$f(a)=
\begin{cases}
a\_1 \quad if\,\, a=a\_2\\
a\_2 \quad if\,\,a\neq a\_2
\end{cases}
$$
is a continuous function without fixed points... | https://mathoverflow.net/users/102487 | Fixed-point property and $T_0$ separation property | Consider in intuitionistic mathematics the example of $A$ being the unit interval $[0,1]$ with the trivial topology $\{\emptyset,[0,1]\}$.
Then $A$ is not $T\_0$ and yet $A$ still has the fixed-point property since any real function has to be continuous in the usual topology. The fixed-point property here should be r... | 1 | https://mathoverflow.net/users/101577 | 296731 | 130,384 |
https://mathoverflow.net/questions/296730 | 0 | Suppose $(X,d)$ is a metric space with the nearest point property and $a,b \in X$ with $a \ne b$. Suppose there is a path of finite length in $X$ from $a$ to $b$ and let $m$ be the infimum of the lengths of all paths from $a$ to $b$.Then, by Lipschitz reparametrization, there exists a path $g:[0,1] \rightarrow X$ from ... | https://mathoverflow.net/users/42134 | Is the function $g$ always injective where $g$ is obtained by lipschitz re-parametrization | It is injective because if $g(s)=g(t)$, $s<t$, then you can remove the interval $[s,t]$ from the domain of definition of $g$ and make the curve shorter. Then you rescale the domain to be $[0,1]$. Rescaling does not change the length of the curve. See also Lemma 3.10 in my notes linked to the answer to another related q... | 1 | https://mathoverflow.net/users/121665 | 296732 | 130,385 |
https://mathoverflow.net/questions/296688 | 4 | In their paper "Theories with recursive models" [1] Lerman and Schmerl used a version of Kruskal's tree theorem about finite n-augmented trees.
An n-augmented tree is a tree T together with $n$ unary relation symbols on its universe. The version of Kruskal's theorem is as follows
**Theorem 1.** Suppose $n<\omega$ a... | https://mathoverflow.net/users/94393 | Kruskal's tree theorem and $\Pi_1$ sentences of linear orderings with finitely many constants | Towards a contradiction, suppose not. Then we can find a sequence $\phi\_0, \phi\_1, \dots$ from $\Phi$ such that for all $i$, $LO, \phi\_0, \dots, \phi\_i \not \vdash \phi\_{i+1}$. So for each $i$, there is an $L^F$-model $M\_i$ with $M\_i \vDash LO \wedge \phi\_0, \dots, \phi\_i \wedge \neg \phi\_{i+1}$. Since these ... | 2 | https://mathoverflow.net/users/32178 | 296735 | 130,387 |
https://mathoverflow.net/questions/296705 | 2 | Writing the SVDs of symmetric positive-definite matrices $A=U\_AD\_AU\_A^T$ and $B=U\_BD\_BU\_B^T$, the following inversion can conveniently be simplified:
\begin{align\*}
(I + A \otimes B)^{-1}
&= ((U\_A \otimes U\_B)(U\_A \otimes U\_B)^T + (U\_A \otimes U\_B)(D\_A \otimes D\_B)(U\_A \otimes U\_B)^T)^{-1} \\
&=(U\_A \... | https://mathoverflow.net/users/nan | Inversion of the sum of an identity matrix and two Kronecker products | Solving a linear system with that matrix is equivalent to solving the linear matrix equation $X + B^TXA + D^TXC = E$. As far as I know, it is an open problem how to exploit the structure of that matrix product to solve it (with a direct algorithm) in fewer than $O(n^5)$ operations (which is the complexity of $n^2$ step... | 2 | https://mathoverflow.net/users/1898 | 296738 | 130,389 |
https://mathoverflow.net/questions/296741 | 2 | Given a poset $(P, \leq)$, is there a complete Boolean lattice $B$ and an order-preserving map $i\_P: P\to B$ such that for any complete Boolean lattice $B'$ and order-preserving map $f: P\to B'$ there is an order-preserving map $g: B\to B'$ such that $f = g\circ i\_P$?
| https://mathoverflow.net/users/8628 | Boolean completion of a partially ordered set | The answer is yes. Take $B := 2^P$ to be the power set of $P$, and $i\_P(x)$ to be the principal down-set generated by $x$. Now for $A\subseteq P$, put $g(A) := \sup f(A)$. This makes $f = g\circ i\_P$ hold by construction.
However, I don't think that this is a universal property, since the $g$ may not be unique. In ... | 6 | https://mathoverflow.net/users/27013 | 296747 | 130,391 |
https://mathoverflow.net/questions/296671 | 5 | Good day, everyone. I have been studying the paper "Formes modulaires de poid 1" written by Deligne and Serre. I am confused by lemma 6.11, or the so-called Deligne-Serre lifting lemma in the paper.
The original proof is somewhat short and rough, so I have found another note which gives a more detailed statement at ... | https://mathoverflow.net/users/122640 | A question on Deligne-Serre lifting lemma | I didn't read the article fully, but I guess it is a basic stuff in commutative algebra:
The quotient $\mathbb{T} / \mathfrak{p}$ being an integral extension of $R$, after base change, $\mathbb{T}\otimes\_R K / \mathcal{P}$ is a field extension of $K$. Thus $\mathcal{P}$ is a prime ideal. If $\mathcal{P}$ were not m... | 0 | https://mathoverflow.net/users/38052 | 296754 | 130,393 |
https://mathoverflow.net/questions/296717 | 27 | Let $(X\_n)$ be a sequence of i.i.d. random variables uniformly distributed in $[0,1]$; and, for $n\geq 1$, set
$$
S\_n = \sum\_{k=1}^n \frac{1}{\sqrt{X\_k}}\,.
$$
It follows from the generalized central limit theorem (as in [1] and [2, Theorem 3.1]) that
$$
\frac{S\_n-2n}{\sqrt{n\ln n}}
$$
converges in law to a Gaussi... | https://mathoverflow.net/users/37266 | Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$? | $\newcommand{\de}{\delta}
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\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$ ... | 12 | https://mathoverflow.net/users/36721 | 296762 | 130,397 |
https://mathoverflow.net/questions/296737 | 1 | Let $\mathbb R\_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R\_d[t]$ that are nonnegative on $[0, 1]$. Does the following equivalent hold for a polynomial $p \in \mathbb R\_d[t]$?
$$p \in S \iff \int\_0^1 p(t) q(t) \ge 0 \; \forall q \in S.$... | https://mathoverflow.net/users/46236 | For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials | Here is a counter-example for $d=1$. In this simple case $S$ is just the set of linear combinations of $x$ and $1-x$ with nonnegative coefficients.
$$p(x)=3x-1$$
$$\int\_0^1 p(x)x\ dx=\frac12$$
$$\int\_0^1p(x)(1-x)\ dx=0$$
Then $\int\_0^1pq\ge0\ \forall q\in S$, but $p\notin S$.
| 2 | https://mathoverflow.net/users/75422 | 296763 | 130,398 |
https://mathoverflow.net/questions/296538 | 4 | Given a separable metric space $X$, what are some ways of forming a new metric space $Y$ such that:
(i) $Y$ contains a ray $R\simeq [0,\infty)$ ($\simeq$ means homeomorphic);
(ii) $R$ is open and dense in $Y$;
(iii) $X\simeq Y\setminus R$.
Moreover, can any non-compact separable metric space play the role of $R... | https://mathoverflow.net/users/91061 | Approaching a space with a ray | As @erz observes, if you do such construction for $X$ obtaining $Y$, for any $X' ⊆ X$ you may put $Y' := (Y \setminus X) ∪ X'$, and it still has all three properties. And since you can do this for $X$ being the Hilbert cube, you can to this for any separable metrizable $X$.
On the other hand, it is often additionally... | 1 | https://mathoverflow.net/users/112373 | 296765 | 130,399 |
https://mathoverflow.net/questions/296771 | 83 | I’m working on a paper that makes heavy use of colorful diagrams to supplement the text. For most of these it would probably not be possible to create grayscale versions that convey the same information as effectively. I’m a bit worried about this because (1) I imagine that some people like to print out papers to read ... | https://mathoverflow.net/users/25121 | What’s the etiquette on using diagrams that need color to be understood? | The concern with color figures for grey scale printing is less of an issue these days, when pretty much all displays are in color, so the reader can always check which color is which even if the document is printed in grayscale. (Many journals no longer insist that the figures should display well when printed in graysc... | 84 | https://mathoverflow.net/users/11260 | 296773 | 130,401 |
https://mathoverflow.net/questions/296759 | 4 | Let $E \to X$ be an oriented vector bundle over a CW complex $X$. Suppose that the second Stiefel-Whitney class $w\_2(E)=0$, then $E$ admits a spin structure. In terms of homotopy theory this means that the classifying map $X \to BSO(n)$ lifts to $X \to BSpin(n)$. Does this mean that the homotopy classes $[X,BSpin(n)]$... | https://mathoverflow.net/users/118622 | Understanding spinnable vector bundles with classifying spaces | Let me just expand what people written in the comments.
Given a fibration $F \to E \to B$ of pointed spaces and $X$ be a space with a map
$f: X \to E$. Then, the space of lifts $f' : X \to F$ is equivalent to the space of null-homotopies of the composition $X \to E \to B$.
Choosing
$F = BSpin\_n$, $E = BSO\_n$, $... | 5 | https://mathoverflow.net/users/115052 | 296774 | 130,402 |
https://mathoverflow.net/questions/296755 | 4 | Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation
$$
|\nabla^r \psi|^2 = \nabla\_{\alpha\_1}\cdots \nabla\_{\alpha\_r}\psi \nabla^{\alpha\_1}\cdots \nabla^{\alpha\_r}\psi
$$
where $\nabla\_\alpha$ is the covariant derivative and $\nabla^\alpha :=g^{\alpha\beta} \nabla\_\beta$.... | https://mathoverflow.net/users/80191 | Proving the inequality $|\nabla |\nabla^r \psi|| \le |\nabla^{r+1} \psi|$ | The line Aubin says to "develop" is the inner product of the $(0, 3)$ tensor $T\_{\nu\alpha\beta} = \nabla\_\nu \nabla\_\alpha \psi \nabla\_\beta \psi - \nabla\_\nu \nabla\_\beta \psi \nabla\_\alpha \psi$ with itself, i.e. $|T|^2\_{g}$. That's why it's positive.
By "develop" Aubin means "distribute out the multiplica... | 5 | https://mathoverflow.net/users/46591 | 296780 | 130,406 |
https://mathoverflow.net/questions/296670 | 5 | Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Is it true that the space contractible loops of this manifold can be contracted to the space of constant loops on $X$? What if $X$ is a finite dimensional $CW$-complex?
I understand that the answer is positive if $X$ is a negatively curved manifold, since in this c... | https://mathoverflow.net/users/13441 | The space of contractible loops of a finite dimensional $K(\pi,1)$ | The statement is true for a $K(\pi,1)$ but not true for other $X$. Finite dimensionality is not relevant.
Here's a sketch: let $X = K(\pi,1)$ and I will assume $X$ has the homotopy type of a CW complex. Let $\Omega\_0 X$ be the space of contractible based loops in $X$. It's easy to show that the homotopy groups of t... | 10 | https://mathoverflow.net/users/8032 | 296796 | 130,408 |
https://mathoverflow.net/questions/296807 | 3 | I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, generalized SVD, etc., none of them matching my problem.
Let $A$ and $B$ be two symmetric, positive semi-definite (but not po... | https://mathoverflow.net/users/88057 | « Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices | From the theoretical point of view: there is a Jordan-like canonical form for pairs of Hermitian matrices $(A,B)$ under the equivalence relation $(A,B) \sim (PAP^T, PBP^T)$ (with a nonsingular $P$), see <https://www.sciencedirect.com/science/article/pii/0024379576900215> . This is known today in my area as "(a variant ... | 4 | https://mathoverflow.net/users/1898 | 296809 | 130,409 |
https://mathoverflow.net/questions/296794 | 3 | I am trying to understand when a morphism defined in an open dense subset of a variety can be extended to the whole variety.
For curves, it is known that if $f:C \to C’$ is a rational morphism from the curve $C$ to the curve $C’$ then f can be uniquely extended to the whole curve $C$.
On the other hand, for higher ... | https://mathoverflow.net/users/43027 | Extension of morphism of quasiprojective varieties | Since $Y$ is projective, the question reduces to the case $Y = \mathbb{P}^n$.
In this case, a morphism to $Y$ is given by an epimorpism $\mathcal{O}^{\oplus n+1} \to L$ for a line bundle $L$. So, if you want to extend a morphism, you need to extend the line bundle and the epimorphism.
A line bundle $L$ always extends... | 4 | https://mathoverflow.net/users/4428 | 296811 | 130,410 |
https://mathoverflow.net/questions/296820 | 2 | The Bessel kernels $G\_{\alpha},\, \alpha>0$ are defined by their Fourier transform
$
\hat G\_{\alpha}(\xi):= \frac 1 { (1+4\pi ^{2}\vert \xi \vert ^{2})^{\alpha/2}}.
$
Bessel $(\alpha, p)$-capacity of a set $E\subset\mathbb{R}^n$ is defined by
$$
B\_{\alpha, p}(E):=\inf\{\|h\|\_{L^p}^p\colon\, g\_{\alpha}\star h\geq... | https://mathoverflow.net/users/122716 | Comparison of Bessel Capacities |
>
> **Theorem.** *If $\beta q<\alpha p\leq n$, then $$ B\_{\alpha,p}(E)=0 \quad \Rightarrow B\_{\beta,q}(E)=0. $$*
>
>
>
**Remark.** If $\alpha p>n$, then $B\_{\alpha,p}(\{ x\})>0$ (Remark 2.6.15 in [1]) so the only set with zero capacity is the exmpty set making the problem trivial. This is why in the statement... | 1 | https://mathoverflow.net/users/121665 | 296822 | 130,414 |
https://mathoverflow.net/questions/296830 | 3 | By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=x'^2+y'^2$.
**Question.** Suppose now we consider functions $f$ with Taylor series starting with $(x^2+y^2)^n$. For whi... | https://mathoverflow.net/users/13441 | Normal form of functions $(x^2+y^2)^n+$ higher terms | The expression $(x^2+y^2)^2 + x^5 + y^5$ cannot be written in the form $(z^2+w^2)^2$ for any smooth functions $z$ and $w$ of $x$ and $y$. (Just look at the Taylor series expansion.) Similarly, $n>1$, the function $(x^2+y^2)^n + x^{2n+1} + y^{2n+1}$ cannot be written in the form $(z^2+w^2)^n$ for any smooth functions $z... | 10 | https://mathoverflow.net/users/13972 | 296833 | 130,418 |
https://mathoverflow.net/questions/296723 | 2 | I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in [Hotta-Takeuchi-Tanisaki](http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf)'s book, which states that there exists a canonical isomorphism
$$
R\mathcal {H}om\_{D\_X}(M^\cdot, D\_X) \otimes^L\_{D\_X} N^\cdot ... | https://mathoverflow.net/users/69190 | A canonical isomorphism in derived categories of D-modules | A map of sheaves is an isomorphism if and only if it's an isomorphism on stalks. Thus, it's enough to check this for the stalk of these sheaves, and so on can just check that for a non-commutative ring $R$ and two complexes of modules, you have $RHom(M,R)\otimes^{L}\_RN\cong RHom(M,N)$. Since modules have free resoluti... | 3 | https://mathoverflow.net/users/66 | 296836 | 130,419 |
https://mathoverflow.net/questions/296823 | 5 | I am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\mathbb{R}^n$. Let $W $ be an algebraic set defined by homogeneous forms in $\mathbb{R}[x\_1, ..., x\_n]$ for which $W \subseteq \mathbb{A}... | https://mathoverflow.net/users/84272 | How can I prove that $(n-1)$-dimensional manifold is not contained in a $(n-2)$-dimensional affine variety? | You seem to be asking whether it is possible that the topological dimension of the set of real points of an algebraic variety is *greater* than its algebraic dimension (in your case, the former is $n-1$ and the latter is $n-2.$) It seems to be a standard fact that this is impossible (inequality in the opposite dimensio... | 4 | https://mathoverflow.net/users/11142 | 296837 | 130,420 |
https://mathoverflow.net/questions/296814 | 3 | What is the time complexity of the fastest known algorithm for the all-pair shortest paths in planar graphs?
| https://mathoverflow.net/users/91261 | A question regarding the all pair shortest paths in weighted planar graphs | This can be done in quadratic time using the linear-time single-source shortest path algorithm by [Henzinger et al.](http://theory.stanford.edu/~virgi/cs267/papers/planar-sssp.pdf)
| 7 | https://mathoverflow.net/users/4248 | 296838 | 130,421 |
https://mathoverflow.net/questions/293813 | 6 | Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U\_E)$ is relatively prime to class number $N$? ($U\_E$ denotes the group of units of $E$.)
(Indeed, I think if $E/F$ is an abelian (or at least is a cyc... | https://mathoverflow.net/users/98582 | Unramified non-abelian extension and Galois cohomology | If I understand your question correctly, every simple unramified extension of a quadratic number field with class number $1$ is an example. Artin constructed the first such extensions, now there are many examples known; see e.g.
["Remark on infinite unramified extensions of number fields with class number one" by D. B... | 4 | https://mathoverflow.net/users/3503 | 296840 | 130,423 |
https://mathoverflow.net/questions/296835 | 1 | Can we interchange the integral order of this integral to start integration on $x$ ? (Taking $g$ and $f$ two functions of rapid decrease which are $o(x^2)$ near zero)
$$A=\int\_{0}^\infty \int\_0^{\infty} \int\_0^{\infty} \frac{1}{x} f(t) g(u)\sin(x(t-u)) \frac{1}{u-t} dt du dx$$
So can we write:
$$A=\int\_{0}^\i... | https://mathoverflow.net/users/38290 | Interchange of integration order (of a not absolutely convergent integral with sinus) | $\newcommand{\de}{\delta}
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\newcommand{\PP}{\operatorname{\mathsf P}}$
... | 3 | https://mathoverflow.net/users/36721 | 296856 | 130,428 |
https://mathoverflow.net/questions/296864 | 3 | I am reading through the text [*Analysis of Boolean Functions* by Ryan O'Donnell](http://www.cs.tau.ac.il/~amnon/Classes/2016-PRG/Analysis-Of-Boolean-Functions.pdf), and I am struggling with exercise 1.12, which is the following:
>
> A *Hadamard Matrix* is any $N \times N$ real matrix with $\pm 1$ entries and ortho... | https://mathoverflow.net/users/122741 | How do I use Walsh-Hadamard matrices to compute Fourier coefficients of a boolean function? | Let $n=3,$ for concreteness. As you said, the correspondence is:
$$
\begin{array}{c|c|l}
\mathbb{1}(S) & \leftrightarrow & S \\ \hline
(0,0,0) &\leftrightarrow & \{\} \\
(1,0,0) & \leftrightarrow & \{ 1 \}\\
(0,1,0) &\leftrightarrow & \{2\} \\
(1,1,0) & \leftrightarrow & \{ 1,2 \}\\
(0,0,1) &\leftrightarrow & \{3 \} \\... | 2 | https://mathoverflow.net/users/17773 | 296865 | 130,432 |
https://mathoverflow.net/questions/294746 | 9 | I am looking for a *constructive* proof of one of the following two statements. If they are not constructively provable, I would be very thankful for an explanation as to why that is so (i.e., at which point in a proof must non-constructive means be employed?).
1. There exists a normed space *X* such that for all *Y*... | https://mathoverflow.net/users/121659 | Constructive proof of existence of non-separable normed space | Andrew Swan and I proved that in the function realizability topos every metric space is separable (and that every object with decidable equality is countable). Therefore, it is not possible to prove constructively that a non-separable metric space exists. This result strengthens the answer by Matt Frank.
The note is ... | 3 | https://mathoverflow.net/users/1176 | 296882 | 130,439 |
https://mathoverflow.net/questions/296878 | 13 | Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M\_n(\mathbb{C})$ has no characters and also $B(H)$, where $H$ is a Hilbert space.
| https://mathoverflow.net/users/93709 | About the existence of characters on $B(X)$ | I guess that you mean that $B(H)$ has no character (=continuous unital algebra homomorphism into $\mathbf{C}$) if $H$ has dimension $\neq 1$ (idem for $M\_n(\mathbf{C})$ for $n\neq 1$), and thus that your question assumes $\dim(X)\ge 2$ (and hence $=\infty$).
Argyros and Haydon (Acta Math, 2011: [arXiv](https://arxiv... | 10 | https://mathoverflow.net/users/14094 | 296885 | 130,440 |
https://mathoverflow.net/questions/295249 | 5 | As we know that the transition matrix $P$ of a Markov chain with finite space is a stochastic matrix, and from Perron-Frobenius Theorem, we know that the spectral radius of the matrix $P$ is $1$, and the eigenvalue $1$ is simple. I was wondering if we consider ergodic Markov process with continuous spaces (from which w... | https://mathoverflow.net/users/121911 | Eigenvalue and eigenvector of ergodic Markov operator for continuous space Markov chain | With respect to slightly modified versions of your second and third questions (Is the *second* eigenvalue guaranteed to have module smaller than 1? Is the *first* eigenvalue simple as well?), the answers are both Yes, if the transition matrix is Harris recurrent and the associated operator self-adjoint and compact.
T... | 3 | https://mathoverflow.net/users/39115 | 296895 | 130,443 |
https://mathoverflow.net/questions/296900 | 7 | I have read that people think of a site as a presentation of the corresponding sheaf topos. For instance, on [page 7 of this text by Caramello](http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf): *as Grothendieck observed himself, a site of definition for a given topos can be see... | https://mathoverflow.net/users/39955 | Tietze transformations for sites of toposes | There are two observation to be made that limit a little this kind of analogy:
1) Site are indeed in some sense presentations, but an infinity theory (I mean with operation of infinite arity) something like the theory of ininitary pretopos, i.e. categories satisfying all of Giraud axioms except being presentable, but... | 6 | https://mathoverflow.net/users/22131 | 296901 | 130,444 |
https://mathoverflow.net/questions/296808 | 4 | I have asked this question [on Math StackExchange](https://math.stackexchange.com/questions/2717648/a-question-on-delignes-paper-valeur-de-fonctions-l-et-periodes-dintegrales), but have not got any reply.
In section 1.7 of Deligne's paper "Valeurs de fonctions L et périodes d'intégrales", which has an English transla... | https://mathoverflow.net/users/87910 | A question on Deligne's paper "Valeurs de fonctions L et périodes d'intégrales" | In the original French version of the article, Deligne actually defines $F^+$ and $F^-$ to be the subspaces of $H\_{\mathrm{dR}}(M)$ occurring in its Hodge filtration and having the same dimension as $H\_B^+(M)$ and $H\_B^-(M)$ respectively. (I agree that the translation is slightly imprecise as it may suggest these su... | 3 | https://mathoverflow.net/users/6506 | 296907 | 130,445 |
https://mathoverflow.net/questions/296915 | 11 | The [Wikipedia article](https://en.wikipedia.org/wiki/Axiom_of_determinacy) on the Axiom of Determinacy (AD) claims:
>
> Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is determined.
>
>
>
Is this claim true?
AD is usually s... | https://mathoverflow.net/users/2126 | The Axiom of Determinacy and the Banach-Mazur game | The claim is false. The Banach-Mazur game, also known as the $\*\*$-game shows (and is equivalent to) that every set of reals has the Baire property. What is true, as you've noted, is that if one has a pointclass $\Gamma$ which is adequate and closed under Borel ($\Delta^1\_1$) substitutions then we have $$Det(\Gamma) ... | 13 | https://mathoverflow.net/users/3859 | 296918 | 130,451 |
https://mathoverflow.net/questions/296911 | 10 | Q1:
For any given finite simple graph G with e edges, does there always exist an $n$ such that the edges of $K\_n$ can be partitioned into $\frac{\binom{n}{2}}{e}$ edge-disjoint copies of $G$? If so, can any upper bounds be placed on the minimal required $n$?
Q2:
Similarly for digraphs (with twice as many copies)?... | https://mathoverflow.net/users/24681 | Is every graph an isomorphic factor of some complete graph? | Q1: yes, this is a theorem by Wilson; see the first paragraph here: <https://arxiv.org/abs/1604.07282>
**Edit**: perhaps the book [Decomposition of graphs](https://books.google.com/books?hl=en&lr=&id=JQm7Nad3DuoC&oi=fnd&pg=PA3&ots=opWgR0u8Nd&sig=d1hkDAQ8kV7EVmyq6alqTLP43d4) by J. Bosak might be helpful (the preview o... | 11 | https://mathoverflow.net/users/24076 | 296928 | 130,455 |
https://mathoverflow.net/questions/296923 | 1 | Let us consider the following differential equation
$$
\dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}.
$$
>
> **My question.** Suppose $a>|b|$ and $x(0)=x\_0\in\mathbb{R}$. Can the solution to the above equation be written in the form
> $$
> x(t) = at + r(a,t),
> $$
> where the term $r(a,t)$ is such that $r(a... | https://mathoverflow.net/users/62673 | Behavior of a non-linear differential equation | Consider the case $a > |b|$, so the solutions are unbounded. The equation is separable, and we get implicit solutions of the form
$$ t = \int\_{x\_0}^x \frac{ds}{a - b \sin(s)} $$
which we can expand in a series in $1/a$ (uniformly convergent in $s$). Absolute convergence justifies interchanging sum and integral, s... | 4 | https://mathoverflow.net/users/13650 | 296929 | 130,456 |
https://mathoverflow.net/questions/296888 | 4 | Erdos conjectured that any set $ A $ of positive integers such that $ \sum\_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a special case of this conjecture, where $ A $ is the set of primes.
I would like to have references on this conjecture, and ... | https://mathoverflow.net/users/13625 | References on Erdos conjecture on arithmetic progressions | MR3203599 Gowers, W. Timothy, Erdős and arithmetic progressions. Erdős Centennial, 265–287, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, 2013. The review says the author gives a survey of progress on the conjecture (and on another conjecture of Erdős).
| 6 | https://mathoverflow.net/users/3684 | 296931 | 130,458 |
https://mathoverflow.net/questions/296898 | 4 | It seems that there is a nice inverse for matrices that can be written as a diagonal matrix plus a symmetric matrix consisting of scaled blocks of ones.
Consider a real matrix of the form:
$$\begin{pmatrix}a\_{11}{1}\_{r\_{1}\times r\_{1}}+b\_{1}I\_{r\_{1}} & a\_{12}{1}\_{r\_{1}\times r\_{2}} & a\_{13}{1}\_{r\_{1}\ti... | https://mathoverflow.net/users/7967 | Inverse of matrix with blocks of ones | $\newcommand{\de}{\delta}
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... | 2 | https://mathoverflow.net/users/36721 | 296933 | 130,459 |
https://mathoverflow.net/questions/296939 | 10 | I recently stumbled over the [large collection of mathematical constants](http://www.bitman.name/math/table/181) of Mauro Fiorentini; it is in Italian and appears to be something in the vein of the famous [OEIS](https://oeis.org/?language=english), however maintained by a single person in isolation.
**Question:**
... | https://mathoverflow.net/users/31310 | Collection of Mathematical Constants | Steven Finch used to have a [site full of essays about his favourite constants](https://web.archive.org/web/20020603145930/http://algo.inria.fr:80/bsolve/constant/constant.html) including a [table](https://web.archive.org/web/20020618172325/http://algo.inria.fr:80/bsolve/constant/table.html), though the contents page i... | 5 | https://mathoverflow.net/users/12565 | 296968 | 130,469 |
https://mathoverflow.net/questions/286403 | 9 | In the very easy-to-read [[1]](https://arxiv.org/pdf/1112.0845.pdf), Kuperberg shows that, conditioned on the Generalized Riemann Hypothesis, knottedness is in $\mathsf{NP}$. As I understand the proof, given a knot-diagram of a knot $K$, the certificate is both a prime $p$ and a solution mod $p$ to a set of polynomial ... | https://mathoverflow.net/users/8927 | Is there a zero knowledge protocol for knottedness, similar to the GMW protocol for graph non-isomorphism? | I believe the answer is **yes**, such a zero-knowledge proof of knottedness to distinguish the unknot exists, by combining the results of [5], [6], and [7].
The standard Graph Non Isomorphism zero-knowledge protocol of [[5]](https://people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Zero%20Knowledge/Proofs_Th... | 1 | https://mathoverflow.net/users/8927 | 296974 | 130,471 |
https://mathoverflow.net/questions/296947 | 4 | Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F\_q$ and let $G = \mathbb G(\mathbb F\_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F\_q$-vector space with the adjoint $G$-action.
I want to understand the group cohomology $H^\*(G, \mathfrak g)$, and more generally $H^\*... | https://mathoverflow.net/users/101091 | cohomology of finite groups of lie type with coefficients in the adjoint module | This area of the subject is somewhat frustrating, since there is a lot of literature but not many satisfactory results involving the entire cohomology ring. For what it's worth, I'll point you to a survey I wrote around 2005 (not up to date at present) [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=2199819).... | 2 | https://mathoverflow.net/users/4231 | 296977 | 130,473 |
https://mathoverflow.net/questions/149224 | 5 | I'm reading a paper of Wise on cubulations and the following fact is used:
Let $H$ be a quasi-convex subgroup of a $\delta$-hyperbolic group and let $H\_i, i\in I$ be a finite family of translates of $H$ such that they are pairwise at bounded distance $D$. Then there is a constant $C$ and a point $x$ such that $d(x,H... | https://mathoverflow.net/users/40911 | Uniform incentre of collection of quasi convex subspaces in hyperbolic spaces | Yes, your statement is true. A proof can be adapted from Theorem 6.1 in this [paper](https://arxiv.org/abs/1804.00748). Since the argument is elementary, I write a proof below.
**Proposition:** *Let $X$ be a $\delta$-hyperbolic space and $\{C\_i : i \in I \}$ a finite collection of $Q$-quasiconvex subspaces such that... | 2 | https://mathoverflow.net/users/122026 | 296989 | 130,476 |
https://mathoverflow.net/questions/296988 | -4 | I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where
* $y$ is a $n \times 1$ vector.
* $A$ is a $m \times n$ matrix where $n \gg m$.
* $B$ is a $m \times m$ symmetric positive definite matrix; the Cholesky decomposition $B = LL^T$ is precomputed if it is needed.
Is it possible to calculate the above expressio... | https://mathoverflow.net/users/122810 | How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently? | $$\mathrm y^\top \mbox{diag}(\mathrm A^\top \mathrm B \,\mathrm A) \,\mathrm y = \sum\_{k=1}^n \mathrm e\_k^\top\mathrm A^\top \mathrm B \,\mathrm A \,\mathrm e\_k \, y\_k^2 = \sum\_{k=1}^n \mathrm a\_k^\top \mathrm B \, \mathrm a\_k \, y\_k^2$$
where $\mathrm a\_k \in \mathbb R^m$ is the $k$-th column of $\rm A$. Si... | 1 | https://mathoverflow.net/users/91764 | 296993 | 130,479 |
https://mathoverflow.net/questions/293840 | 6 | This question is a generalization of the question [Volume ratio of $\ell\_1$ balls and $\ell\_1$ surfaces](https://mathoverflow.net/questions/281611/volume-ratio-of-ell-1-balls-and-ell-1-surfaces)
For any $p\in[1,\infty]$ define $\|x\|\_p := (|x\_1|^p+\cdots+|x\_d|^p)^{1/p}$ for $p\in[1,\infty)$ and $\|x\|\_\infty :=... | https://mathoverflow.net/users/84299 | Volume ratio of general $\ell_p$ balls and surfaces | We have managed to obtain a complete solution to this problem, by applying the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem) and the classical results of [Naor and Romik](http://www.numdam.org/article/AIHPB_2003__39_2_241_0.pdf) on the affinity between cone and surface measures in $\ell\_p$ bal... | 2 | https://mathoverflow.net/users/84299 | 297003 | 130,480 |
https://mathoverflow.net/questions/296998 | 3 | Suppose that $(k,l,m) \in \mathbb{N\_0}^3$.
If $(k,l,m)=(0,0,0)$ then for $n=1,2$ there is an infinite number of solutions and, by the theorem of of Wiles there are no solutions when $n \geq 3$.
Is it known can there be an infinite number of solutions of $a^{n+k}+b^{n+l}=c^{n+m}$ if $k,l,m$ are not all equal and $n... | https://mathoverflow.net/users/122043 | What is known about equation $a^{n+k}+b^{n+l}=c^{n+m}$ and its set of solutions? | [Darmon and Granville](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms/27.6.513) proved, using [Faltings' Theorem](https://en.wikipedia.org/wiki/Faltings%27s_theorem), that your equation has finitely many primitive integer solutions for any fixed exponents which are at least $3$. In fact their result i... | 5 | https://mathoverflow.net/users/11919 | 297004 | 130,481 |
https://mathoverflow.net/questions/296996 | 0 | I am reading the following [book](https://rads.stackoverflow.com/amzn/click/0122165500) and there is a step in a derivation I don't quite follow. In equation 27 on page 409 it is claimed
$$
\int\_0^{\infty}\,\epsilon^{\alpha}(1+\epsilon/x)^{-1}e^{-\epsilon}d\epsilon=x^{\alpha+1}e^x\int\_x^{\infty}d\zeta\int\_0^{\inft... | https://mathoverflow.net/users/120521 | Doubt filling a step in a derivation | Working on the left-hand side, perform the change $\epsilon/x = \sigma$, in order to get
$$x^ {\alpha +1} \int \limits \_0 ^\infty \sigma^\alpha (1+\sigma)^{-1} \, \mathrm e^{-x \sigma} \, \mathrm d \sigma = x^ {\alpha +1} \mathrm e ^x \int \limits \_0 ^\infty \sigma^\alpha (1+\sigma)^{-1} \, \mathrm e^{-x (\sigma + ... | 2 | https://mathoverflow.net/users/54780 | 297007 | 130,482 |
https://mathoverflow.net/questions/296997 | 2 | Suppose $V$ is an affine algebraic set defined by real polynomials.
Let $\mathbb{A}\_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with Zariski topology where the closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials. Let $\mathbb{A}\_{\mathbb{C}}^n$ be the usual affine $n$ space.
Suppose $V(\mat... | https://mathoverflow.net/users/84272 | How to show $\dim_{\mathbb{A}_{\mathbb{R}}^n} V= \dim_{\mathbb{A}_{\mathbb{C}}^n} V$? | The upper bound on the real Dimension needs no assumption on the non-singular point. This frees us to perform induction on $n$. With finitely many exceptions, the fibers of the projection to the first coordinate have complex dimension one less than $\dim V$, and the remainder have dimension $\dim V$. Hence by induction... | 3 | https://mathoverflow.net/users/18060 | 297013 | 130,484 |
https://mathoverflow.net/questions/296978 | 1 | Let $(E,\mathscr{E})$ be a measurable space. Two transition of probabilities
$p, q\colon E\times\mathscr{E}\to [0,1]$ are said to be in duality relative to
a probability measure $m$ if for every pair of non-negative functions $f,g$ we have
$$
\int (Pf) \,g\, dm=\int f (Qg)\, dm
$$
where $P$ and $Q$ are the transfer... | https://mathoverflow.net/users/101233 | Transition of probability in duality and its properties | $\newcommand{\de}{\delta}
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... | 1 | https://mathoverflow.net/users/36721 | 297017 | 130,486 |
https://mathoverflow.net/questions/297030 | 1 | More than 6 months ago I submitted a paper to a very prestigious journal in Mathematics. Honestly I did not expect it to make it past the managing editor, I thought to give it a try, and then submit to a journal in my field. Surprisingly for me it was assigned a manuscript number, and no news since. I am assuming it we... | https://mathoverflow.net/users/122822 | Inquiring about the status of a submitted paper | It is perfectly okay to ask about the status of the paper. It often happens that referees and editors forget about their assignments and a reminder call is a right thing to do. You could ask the editor, but then be patient. It often takes about a year to get the reports. If this is a really prestigious journal, it is l... | 5 | https://mathoverflow.net/users/121665 | 297032 | 130,492 |
https://mathoverflow.net/questions/297014 | 2 | My set up is the following: I have an affine algebraic group $G$ over a $p$-adic field $F$, we assume that $G$ is semisimple and simply connected. I have an abstract subgroup $H\leq G(F)$ of the group of points of $G$. I know that $H$ is open and that $[G(F):H]<\infty$. I would like to conclude that $H=G(F)$, is this t... | https://mathoverflow.net/users/46157 | Open subgroups of finite index of p-adic semisimple groups | It's true if and only if every $F$-simple factor of $G$ is $F$-isotropic (i.e., has nonzero $F$-rank).
Indeed, if this is the case, then we can reduce to the $F$-simple case. By Theorem 5.1 in Margulis' book (which uses the simple connectedness assumption), $G(F)$ is generated by its 1-dimensional unipotent subgroups... | 3 | https://mathoverflow.net/users/14094 | 297033 | 130,493 |
https://mathoverflow.net/questions/297006 | 3 | I have $n$
i.i.d samples from a unknown distribution. I want to prove or disprove that the mean is finite. Are there any statistical test for this hypothesis ?
| https://mathoverflow.net/users/122813 | Statistical test for boundedness of Expectation | $\newcommand{\al}{\alpha}
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\newcommand{\PP}{\o... | 3 | https://mathoverflow.net/users/36721 | 297046 | 130,497 |
https://mathoverflow.net/questions/297039 | 5 | Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}$. Let $$\mathfrak \kappa=\min\{|\mathcal K|:(\exists \text{ a connected compactum }X)(\mathcal K\in \mathscr K(X))\}.$$... | https://mathoverflow.net/users/95718 | How many disjoint compact sets are needed to form a connected compactum? | It is indeed axiom independent. In the Solovay random model (adding $\aleph\_2$ random reals over a model of ZFC+GCH) you have $2^\omega=\aleph\_2$ and there exists a $\aleph\_1$ partition of $[0,1]$ into nowhere dense closed sets (hence compact). The idea is to define such a collection in the ground model and use the ... | 5 | https://mathoverflow.net/users/23835 | 297047 | 130,498 |
https://mathoverflow.net/questions/296985 | 6 | Is the handlebody group of **genus two** surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple (not necessarily finite but conceptually simple)? For this part, really I'm asking the question for the handlebody... | https://mathoverflow.net/users/56571 | generators for the handlebody group of genus two | As stated in Ian Agol's answer, the mapping class group of a handlebody $B$ maps onto $Out(\pi\_1B)$. This is easy to show by lifting known generators for the automorphism group of a free group. Twists along disks lie in the kernel of the surjection to $Out(\pi\_1B)$, and it is a theorem of Luft (Math. Ann. 234 (1978) ... | 6 | https://mathoverflow.net/users/23571 | 297051 | 130,500 |
https://mathoverflow.net/questions/297049 | 15 | As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \rightarrow \tau+12$. Here comes the question, is $\eta(\tau)^2$ invariant on $\Gamma(12)$, where $\Gamma(12)$ denotes the ... | https://mathoverflow.net/users/66770 | Is $\eta(\tau)^2$ a modular form of weight 1 on $\Gamma(12)$? | Yes, it is. There are standard criteria (stated for example in Ken Ono's book "Web of Modularity" - Theorems 1.64 and 1.65) that indicate when an eta quotient is modular on $\Gamma\_{0}(N)$, and these show that $\eta(12\tau)^{2}$ is a modular form of weight $1$ on $\Gamma\_{0}(144)$ with Nebentypus $\chi\_{-4}$ (with o... | 15 | https://mathoverflow.net/users/48142 | 297053 | 130,501 |
https://mathoverflow.net/questions/297012 | 3 | Let $G = \left< a\_1,b\_1, ... , a\_g, b\_g | [a\_1,b\_1] \cdots [a\_g,b\_g] \right>$ be the fundamental group of a surface of genus $g$. Let $N\_1$ and $N\_2$ be two normal subgroups of $G$ that are described explicitly as the normal closures of two finite sets of words say $H\_1 = \left< x\_1,...,x\_n \right> ^N$ and... | https://mathoverflow.net/users/99414 | Normal generating set for the intersection of two normal subgroups of a surface group | I believe that the intersection is not always finitely generated as a normal subgroup. The factor-group $F\_g$ of $G$ over the normal subgroup $M=\langle a\_1,...,a\_g\rangle^N$ is freely generated by the images of $b\_1,...,b\_g$ which we shall denote by the same letters: $b\_1,...,b\_g$. Since $M$ is finitely normall... | 4 | https://mathoverflow.net/users/nan | 297055 | 130,502 |
https://mathoverflow.net/questions/296955 | 1 | Let $H = (V, E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardinal. We say that $c: V\to \kappa$ is a *weak coloring* if for all $e \in E$ the restriction $c|\_e$ is not constant. We call $c$ a *strong coloring*, if for all $e... | https://mathoverflow.net/users/8628 | Strong and weak chromatic number of infinite hypergraphs of finite rank | Since the OP did not require that the rank of the hypergraph be *bounded*, the **answer to the question is 'obviously yes'**. An example is any hypergraph $H$ which is(\*) the
>
> disjoint union of $\aleph\_0$-many hyperedges of *increasing cardinality*.
>
>
>
Then $\chi\_w(H)=2$ but $\chi\_s(H)=\aleph\_0$. ... | 2 | https://mathoverflow.net/users/108556 | 297059 | 130,504 |
https://mathoverflow.net/questions/296470 | 1 | *This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part.*
The first steps toward constructing a fundamental domain for a Hilbert-Blumenthal (aka Hilbert modular) surface were published ... | https://mathoverflow.net/users/14835 | How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's? | I'm not familiar with the Götsky--Cohn construction but Siegel's (as explained in van der Geer's book) seems clear:
* there is a "height function" $y$ on $X = \mathbb H^2 \times \mathbb H^2$ (the "distance to cusps") which is $\Gamma\_K$-invariant, and can be expressed as
$$
y = \max\_{\sigma \in \mathbb P^1(K)} y\... | 2 | https://mathoverflow.net/users/32210 | 297063 | 130,505 |
https://mathoverflow.net/questions/297064 | 2 | I have been reading some papers recently, in particular, Blanchard's paper *$\beta$-expansions and symbolic dynamics* which state that a $\beta$-shift $S\_{\beta}$ is a synchronised shift if and only if the orbit of the greedy $\beta$-expansion of $1$ is not dense in $S\_{\beta}$.
The reference given for this result ... | https://mathoverflow.net/users/10518 | Synchronised $\beta$-shifts | I do not know a reference to a published proof, most sources refer to Blanchard's paper. The proof is relatively easy once you remember the structure of a canonical graph representation of $\beta$-shifts. Assume that $\beta>1$ is such that the greedy $\beta$-expansion $(\omega\_i)\_{i\ge 1}$ of $\mathbf 1$ is not event... | 2 | https://mathoverflow.net/users/24676 | 297070 | 130,507 |
https://mathoverflow.net/questions/297065 | 1 | This is a follow-up of an [older question](https://mathoverflow.net/questions/296955/strong-and-weak-chromatic-number-of-infinite-hypergraphs-of-finite-rank).
Let $H = (V, E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardina... | https://mathoverflow.net/users/8628 | Strong and weak chromatic number of infinite bounded hypergraphs | Of course. Let $V$ be a set of non-zero integers, and $E$ consists of triples $(a,b,c)$ such that not all $a,b,c$ have the same sign. That is, the sign is a weak 2-colouring. On the other hand, for a strong colouring all elements must have different colours, since each pair of vertices is contained in a triple from $E$... | 4 | https://mathoverflow.net/users/4312 | 297077 | 130,510 |
https://mathoverflow.net/questions/297071 | 11 | Let $f: X \to S$ be a morphism of algebraic spaces, where $S$ is a scheme. If $f$ is separated and étale then Knutson's criterion says that $X$ is actually a scheme.
I have a some closely related questions.
>
> 1. Why does one require separatedness in Knutson's result? What is an example of an étale cover of a sc... | https://mathoverflow.net/users/5101 | Non-separated étale algebraic spaces | To answer question 2 (with $X$ a scheme), just take for $S$ the affine line over a field $k$, for $X$ the usual ``$S$ with the origin doubled'' (two copies of $S$ glued along $S\smallsetminus\{0\}$) and for $f$ the obvious projection (which is the identity on each copy).
For question 1, the best example I know is a `... | 20 | https://mathoverflow.net/users/7666 | 297089 | 130,514 |
https://mathoverflow.net/questions/296943 | 3 | Let $E\_1$ denote the infinite enumerated collection of two-symbol (`0` as blank symbol and `1` as non-blank symbol) one-tape (assuming that the tape is infinite in both directions) Turing machines: $${E\_1} = \{ 1,\;\;2,\;\;3,\;\; \ldots \},$$
where each element is simply the index of the corresponding Turing machi... | https://mathoverflow.net/users/122796 | Can all lengths of shortest non-halting inputs of all Turing machines be limited by the Busy Beaver applied to the corresponding numbers of states? | There exists a family of Turing machines $\{ \mathcal{T}\_n : n \in \mathbb{N} \}$ such that:
* $\mathcal{T}\_n$ has $k n$ states where $k$ is some fixed universal constant;
* $F(\mathcal{T}\_n) \geq BB(BB(n))$.
Specifically, $\mathcal{T}\_n$ simulates the $n$-state Busy Beaver in the left half of the tape, keeping... | 3 | https://mathoverflow.net/users/39521 | 297091 | 130,515 |
https://mathoverflow.net/questions/297079 | 4 | In one paper the author uses the statement without citation:
Let $(M,g)$ be a Riemannian manifold. The gradient $\nabla F$ of a proper function $F: M\rightarrow \mathbb{R}$ is integrable vector field, i.e. its integral curves are defined for all times.
In this particular case, $F$ is also bounded from below, and ... | https://mathoverflow.net/users/114985 | Gradient of a proper function is integrable | Unless there are some additional assumptions imposed, the claim is false. Consider $f(x)=x^4$ on $\mathbb{R}$. This function is bounded from below, $f'(x)=4x^3$, and solutions to $x'(t)=4x(t)^3$ have blowup in a finite time.
| 2 | https://mathoverflow.net/users/121665 | 297092 | 130,516 |
https://mathoverflow.net/questions/296965 | 10 | Let $G$ be a reductive algebraic group over a number field $k$. Weil's conjecture on Tamagawa numbers (now a theorem) tells us that the Tamagawa number $\tau(G)$ of $G$ is 1 if $G$ is semisimple and simply-connected. Are there known bounds for $\tau(G)$ for the general case?
Presumably one might be able to use the fo... | https://mathoverflow.net/users/48554 | Bounds on Tamagawa numbers of reductive groups | Yes, the formula is correct, see Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80, (10.1.2).
In the extreme cases (when G is semisimple or a torus) there are formulas for ${\rm Pic}(G)$ in Lemma 6.9 of Sansuc's paper.
Nam... | 4 | https://mathoverflow.net/users/4149 | 297093 | 130,517 |
https://mathoverflow.net/questions/297088 | 3 | I am working on an article based mainly on the notion of [Measure of non-compactness](https://en.wikipedia.org/wiki/Measure_of_non-compactness), to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
subsets of $E$.
>
> **Definition:**
>
>
> The function $... | https://mathoverflow.net/users/102228 | A particular measure of noncompactness? | $\newcommand{\de}{\delta}
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\newcommand{\PP}{\operatorname{\mathsf P}}$ ... | 3 | https://mathoverflow.net/users/36721 | 297094 | 130,518 |
https://mathoverflow.net/questions/297028 | 9 | The classical definition of an approximately differentiable function is as follows:
>
> **Definition.**
> *Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. We say that $f$ is
> approximately differentiable at $x\in E$ if there is a linear function
> $L:\mathbb{R}... | https://mathoverflow.net/users/121665 | Functions that are approximately differentiable a.e | It turns out this is something I did some literature research on when I was working on my dissertation. Except for **[6]** (which I discovered too late for inclusion) and **[7]** (appeared later, but I happened to think of looking at it just now), the following from p. 35 of my dissertation are the pre-1993 references ... | 6 | https://mathoverflow.net/users/15780 | 297099 | 130,520 |
https://mathoverflow.net/questions/73962 | 8 | Kendell-Mann numbers $M(n)$ ( see the sequence A000140 <http://oeis.org/A000140> ) have the simple property: $M(n+1) \approx (n-1/2)M(n)$.
The property can be proved by different methods.
For eg. [The property of Kendall-Mann numbers](https://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers)
W... | https://mathoverflow.net/users/10903 | A combinatorial proof for the property of KM numbers? | Here is a quick and dirty probabilistic analysis which gets the right answer. For a permutation $w \in S\_n$, define
$$I(w) = \sum\_{1 \leq i < j \leq n} \begin{cases} -1 & w(i) < w(j) \\ 1 & w(i) >w(j) \\ \end{cases}.$$
So $I(w) = 2 \# (\mbox{number of inversions of $w$}) - \binom{n}{2}$. If $w$ is chosen uniformly at... | 8 | https://mathoverflow.net/users/297 | 297103 | 130,522 |
https://mathoverflow.net/questions/297069 | 0 | In a [previous question](https://mathoverflow.net/questions/207452/eigenvectors-as-continuous-functions-of-matrix-diagonal-perturbations) I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) the results... | https://mathoverflow.net/users/13093 | Stability of eigenvectors for diagonal perturbations | Simple eigenvalues depend smoothly on the matrix by the implicit function theorem
applied to the characteristic polynomial (parameterized by the symmetric matrix).
For the general situation see [this paper](http://www.mat.univie.ac.at/~michor/DC-perturb.pdf) and references therein.
| 2 | https://mathoverflow.net/users/26935 | 297105 | 130,523 |
https://mathoverflow.net/questions/297043 | 7 | Just recently I've stumbled across Warren Dicks' book *Groups, trees and projective modules* (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" component is a special case of space studied in general group cohomology.
The results that caught my attention were:
>
> The aug... | https://mathoverflow.net/users/19965 | Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring | I'm adding a new answer since my other answer was the converse.
Question 1:
I can `simplify' Connell's argument using Dunwoody-Dicks. Assume that $RG$ is von Neumann regular. Note that if $g\in G$, then $(1-g)r(1-g)=1-g$ for some $r\in R$. So $(1-g)(1- r(1-g))=0$. As $r(1-g)$ is in the augmentation ideal, it follow... | 2 | https://mathoverflow.net/users/15934 | 297117 | 130,526 |
https://mathoverflow.net/questions/297120 | 7 | I am a PhD student at one university and an invited professor at a second, i.e. I do not have a permanent position in the second one. Now I need to indicate an affiliation in a journal paper but I do not know whether to indicate or not to indicate a university where I am an invited professor.
>
>
> >
> > What’s ... | https://mathoverflow.net/users/73577 | Which affiliation to use when publishing, when invited professor at second university | If you get support (financial and moral) from both places, you should list them both.
| 21 | https://mathoverflow.net/users/11142 | 297123 | 130,527 |
https://mathoverflow.net/questions/295763 | 10 | For the purposes of this question, a rank-$r$ (integral) *lattice* is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It is *even* if $\ell^2 = \langle \ell,\ell\rangle \in 2\mathbb Z$ for all $\ell \in L$ and *odd* if there is some $\ell... | https://mathoverflow.net/users/78 | Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24? | The discussion in the comments establishes the conjecture when $r$ is divisible by $24$. When $r$ is merely divisible by $12$, the comments establish that $\Delta^{-r/24} Z\_{RR}$ is divisible by $12$. (The latter case follows from the former because $Z\_{RR}$ is easily seen to be multiplicative.)
Specifically, when ... | 3 | https://mathoverflow.net/users/78 | 297130 | 130,531 |
https://mathoverflow.net/questions/297133 | 2 | Consider the following statement due to F.Behrend (1946)
**Theorem:** Let $N$ be a large integer. Then there exists a subset $A\subset [1,N]$ with $\frac{|A|}{N}\geq \exp(-4\sqrt{\ln N})$ which does not contain any arithmetic progression of length three.
By $[1,N]$ I mean the set of integers from this interval.
I... | https://mathoverflow.net/users/121924 | Behrend's Construction | The term "large" is subjective, of course, but arguably a subset of $\{1,2,\dots,N\}$ of size at least $N^{1-o(1)}$ can be regarded as "large". The set $A$ in the theorem is of that kind, since $\sqrt{\ln N}=o(\ln N)$. See also [small-o notation](https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation).
| 7 | https://mathoverflow.net/users/11919 | 297136 | 130,534 |
https://mathoverflow.net/questions/297118 | 6 | Let $r \leq s$ be fixed natural numbers. Then by the Kővári–Sós–Turán theorem, any graph on $n$ vertices with at least $cn^{2-\frac{1}{r}}$ edges contains a complete bipartite subgraph $K\_{r,s}$ for a constant $c.$
I was wondering if we can say that if we have at least $cn^{2-\frac{1}{r}}+1$ edges, then there are at... | https://mathoverflow.net/users/118765 | Kovari-Sos-Turan theorem | The appropriate search term is "supersaturation". I think Theorem 11.1 in
Furedi, Z., Simonovits, M. (2013). The history of degenerate (bipartite) extremal graph problems. In Erdos Centennial (pp. 169-264). [arxiv:1306.5167](https://arxiv.org/abs/1306.5167)
implies the existence of a constants $c\_1=c\_1(r,s)$ and... | 5 | https://mathoverflow.net/users/12674 | 297139 | 130,536 |
https://mathoverflow.net/questions/290904 | 3 | Let $m,n,t$ be some primes and $x$ and $y$ be some prime powers. Also suppose that $mn\mid(t-1)$. Which one of the following is correct?
1- If $t=\dfrac{x^{n}-1}{(x-1)(x-1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then $x=y$ and $m=n$.
2- If $t=\dfrac{x^{n}+1}{(x+1)(x+1,n)}=\dfrac{y^{m}-1}{(y-1)(y-1,m)}$, then there is n... | https://mathoverflow.net/users/119365 | Some equations similar to Goormaghtigh problem | There are counter-examples to all three claims in the question above, so none of the claims are correct.
1- A counter example is {x,n,y,m,t}={2,5,5,3,31} as was indicated in a comment above.
Another counter example is {x,n,y,m,t}={2,3,13,2,7}.
2- A counter example is {x,n,y,m,t}={3,3,13,2,7}.
Another counter ex... | 6 | https://mathoverflow.net/users/110710 | 297148 | 130,540 |
https://mathoverflow.net/questions/296990 | 7 | In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology
$$ H^{\bullet}\_{dR}(M \times N) \simeq H^{\bullet}\_{dR}(M) \otimes H^{\bullet}\_{dR}(N)$$
is proved under various finiteness hypothesis : one of the two manifolds is compact, or ... | https://mathoverflow.net/users/74372 | What is the scope of validity of Kunneth formula for de Rham? | Let me convert my comment into an answer and add something:
I think the assumptions (good cover / compact) you mention are simply there to provide a simpler proof. If it's true for singular cohomology, then it's true for de Rham cohomology, by de Rham's theorem. I'm pretty sure the most general statement you can get ... | 7 | https://mathoverflow.net/users/36146 | 297160 | 130,542 |
https://mathoverflow.net/questions/297173 | 2 | We are given a matrix $M \in \{0,1\}^{n\times n}$ satisfying the following property.
The rows and columns of $M$ can be partitioned into $k$ *rowgroups* and $k$ *colgroups* respectively, such that **in each** block $B \subseteq M$ induced by these partitions, we have $0$ or more rows and $0$ or more columns of $B$ co... | https://mathoverflow.net/users/115803 | Maximum rank in a class of $0\,$-$1$ partitioned matrices satisfying combinatorial constraints | The answer is $\min(n,k^2)$.
The upper estimate: rank of $n\times n$ matrix does not exceed $n$, and rank of the sum of at most $k^2$ matrices of rank 1 does not exceed $k^2$.
Example: enumerate rows and columns from 0 to $n-1$ and partition the columns from 0 to $N-1:=\min(n,k^2)$ by the value of remainder modulo... | 1 | https://mathoverflow.net/users/4312 | 297193 | 130,551 |
https://mathoverflow.net/questions/246812 | 12 | Asked once on SE-mathematics.
Let $U$ be an open subset in $\mathbb{R}^n$, $m\in\mathbb{N}$, $1\leq m<n$ and let
$$\mathcal{C}^k\_{\leq m}(U,\mathbb{R}^n):=\lbrace g\in\mathcal{C}^k(U,\mathbb{R}^n)\mid\dim \operatorname{im} Df(x)\leq m\:\forall x\in U\rbrace,$$
where $\mathcal{C}^k(U,\mathbb{R}^n)$ mean $k-$times con... | https://mathoverflow.net/users/95703 | Can $C^1$ mappings with derivative of low rank be approximated by smooth maps? | **There is a counterexample.**
**Example.** There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings
$g\in C^2(\mathbb{R}^5,\mathbb{R}^5)$ satisfying ${\rm rank}\, Dg\leq 3$.
**Example.** *There is $f\in C^1(\mathbb{R}^7,\mathbb{R}^7)... | 14 | https://mathoverflow.net/users/121665 | 297203 | 130,555 |
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