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/322962 | 2 | Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I always thought that $S^0$ was the odd one out due to it not being connected however after reading page 4 of "H-spaces from ... | https://mathoverflow.net/users/54401 | Does the Hopf construction work for $S^0$? | This is completely elementary. From the H-space structure $X\times X\to X$ you construct the Hopf fibration $X\*X\to SX$ via $(x,t,y)\to (xy,t)$, where you think of $SX$ as the quotient of $X\times I$ by identifying both the $t=1$- and $t=-1$-level to one point, respectively. For $X=S^0$ you get a fibration $S^1\to S^1... | 7 | https://mathoverflow.net/users/39082 | 322972 | 139,272 |
https://mathoverflow.net/questions/322929 | 3 | It is well known that for a discrete random walk on the integers with a fair coin, the expected distance of the walker from the origin after $N$ time steps is $\sqrt{\frac{2N}{\pi}}$ if $N$ is large. For example, Wolfram Mathworld has a thorough explanation [here](http://mathworld.wolfram.com/RandomWalk1-Dimensional.ht... | https://mathoverflow.net/users/20838 | Expected distance from the origin for a recurrent 1D random walk in a random environment | The main result of the paper "The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium" by Y. B. Sinai is that the distance of the walker from the origin at time $n$ is of order $(\log n)^2$. He analysed a bigger class of transition probabilities but pointed out your case an example.
Moreover, the c... | 3 | https://mathoverflow.net/users/135648 | 322978 | 139,274 |
https://mathoverflow.net/questions/322982 | 1 | Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and $T : Y → Y$ be dynamical systems. Then the map $S × T$ is defined by:
$S × T : X × Y → X × Y$, $(S × T)(x, y) = (Sx,... | https://mathoverflow.net/users/74668 | Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive | If $\mathcal{X} = (\prod\_{i=1}^\infty X\_i ) \times X$ is endowed with the product topology, then to show that $F = (\prod\_{i=1}^N S\_i) \times f$ is topologically transitive (that is, given any pair of open subsets $U, V \subset \mathcal{X}$, there exists $n\in \mathbf{N}$ such that $F^n(U) \cap V \ne \emptyset$), i... | 2 | https://mathoverflow.net/users/14037 | 322985 | 139,275 |
https://mathoverflow.net/questions/322968 | 2 | Let $\alpha$ be a root of a polynomial $a\_nx^n + \ldots + a\_1x$ with integral coefficients.
I would like to determine $\varepsilon > 0$ depending on $a\_1, \ldots, a\_n$ so that $|\alpha| < \varepsilon$ implies $\alpha = 0$.
Is it possible to give a "formula" for such an $\varepsilon$ without refering to the com... | https://mathoverflow.net/users/3816 | Prove that given root of a polynomial is zero by approximation | Of course we must assume some $a\_j \ne 0$. Say $a\_j$ is
the one with least index. Then you want $\varepsilon$ such that
$p(x) = a\_n x^{n-j} + \ldots + a\_j \ne 0$ for $|x| < \varepsilon$.
You may use inequalities such as
$|p(x)| \ge |a\_j| - \sum\_{k=j+1}^{n} |a\_k| |x|^{k-j} \ge |a\_j| - m \sum\_{k=j+1}^n |a\_k|$
... | 3 | https://mathoverflow.net/users/13650 | 322992 | 139,277 |
https://mathoverflow.net/questions/322971 | 0 | Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}\_{\sigma \in [0,1]}$ such that
* For every $\sigma\_0 \in (0,1]$, $\Phi(t,x,\sigma\_0):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ ... | https://mathoverflow.net/users/36886 | Convergence of Stochastic Flow but not Flow | Take the solution to
$$
dx = \tanh^3(x)\,dt + \sigma x \, dW\;.
$$
For $\sigma > 0$, solutions diverge if you start anywhere except at $0$.
For any $\sigma \neq 0$ on the other hand, solutions converge to $0$ almost surely. (The cube makes sure that solutions behave like $dx = \sigma x \,dW$ once $x$
is sufficiently cl... | 3 | https://mathoverflow.net/users/38566 | 323001 | 139,284 |
https://mathoverflow.net/questions/322296 | 4 | Let $(X,x\_0)$ be a pointed simplicial set. Assume if you like that $X$ is the nerve of a category but *do not* assume that $X$ is a Kan complex.
Because $Ex^\infty X$ is a Kan complex, every homotopy class $\alpha \in \pi\_n(X,x\_0)$ may be represented by a map $sd^k \Delta[n] \to X$ such that the restriction $sd^k ... | https://mathoverflow.net/users/2362 | Representing simplicial homotopy classes cubically? | I've worked out a proof that a form of cubical $Ex^\infty$ functor is a fibrant replacement functor for cubical sets with connections. From this, it's basically immediate that a homotopy class can be represented as above. I hope to post this to the arxiv soon.
| 1 | https://mathoverflow.net/users/2362 | 323010 | 139,291 |
https://mathoverflow.net/questions/322881 | 3 | Every finite directed graph has a [majority coloring with $4$ colors](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p25/pdf). (The notion of majority coloring is defined below.)
**Question.** Can every infinite directed graph be majority-colored with $4$ colors?
---
If $X$ is a non-empty se... | https://mathoverflow.net/users/8628 | Does every directed graph have a directed coloring with $4$ colors? | Yes,$\DeclareMathOperator{In}{In}$ this seems to be true. Let $\deg^-(v)$ denote in-degree of $v$ and let $V$ denote the vertex set. We can partition $V$ as $V\_f\cup V\_<\cup V\_{\geq}$ where:
* $V\_f$ is the set of vertices $v$ with finite in-degree
* $V\_<$ is the set of vertices $v$ with infinite in-degree and $|... | 4 | https://mathoverflow.net/users/112284 | 323022 | 139,296 |
https://mathoverflow.net/questions/322981 | 9 | We have the result that $\mathsf{ZFCfin}$, the usual $\mathsf{ZFC}$ axioms with the axiom of infinity replaced by its negation, is bi-interpretable with $\mathsf{PA}$, first order Peano Arithmetic. We also know of a natural way to weaken $\mathsf{PA}$ into fragments, by restricting the induction axiom schema to forumla... | https://mathoverflow.net/users/102926 | Is restricting Replacement and Separation enough to make $Q+I\Sigma_n$ bi-interpretable with Set Theory? | First let me note that one should be careful with formulation of $\mathsf{ZFCfin}$, for it to be bi-interpretable with $\mathsf{PA}$ (see the paper "On interpretations of arithmetic and set theory" by Richard Kaye and Tin Lok Wong and the paper "$\omega$-models of finite set theory" by Ali Enayat, James H. Schmerl, and... | 10 | https://mathoverflow.net/users/36385 | 323034 | 139,301 |
https://mathoverflow.net/questions/322993 | 2 | Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:
$$
\Pi(\nu) = \{\pi: \pi(X,dy) = \nu(dy)\}.
$$
Let $f:X\times Y \mapsto \mathbb{R}$ be a measurable function (you can assum... | https://mathoverflow.net/users/101188 | optimal transport, measurable selection | The answer is affirmative if $f$ is Borel-measurable. It is suffices to prove the equality for $f$ having finitely many values. Let $V=f(X\times Y)$ be the finite set of values of $f$. By the Borel-measurability of $f$, for every $v\in V$ the set $D\_v=f^{-1}(v)$ is Borel and its projection $A\_v=pr\_X(D\_v)$ onto $X$ ... | 1 | https://mathoverflow.net/users/61536 | 323035 | 139,302 |
https://mathoverflow.net/questions/323030 | 1 | Let $P,Q,R$ be probability measures on the real line.
If these are discrete, we can show
\begin{equation}
D\_{\mathrm{KL}}(P\ast R\,\Vert\,Q\ast R)\le D\_{\mathrm{KL}}(P\,\Vert\,Q)
\end{equation}
by using the log sum inequality, where $D\_{\mathrm{KL}}(P\,\Vert\,Q)=\int \log\left(\frac{dP}{dQ}\right) dP$ is KL divergen... | https://mathoverflow.net/users/135516 | KL divergence and convolution of distributions | The KL divergence cannot increase after passing both distributions through the same Markov kernel (in your case, convolution with $R$). This is an immediate consequence of the data processing inequality:
<https://en.wikipedia.org/wiki/Data_processing_inequality>
| 1 | https://mathoverflow.net/users/12518 | 323040 | 139,305 |
https://mathoverflow.net/questions/323023 | 5 | Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic coordinates exist on a manifold that has doubling volume measure and supports Poincare inequality?
| https://mathoverflow.net/users/134172 | harmonic coordinates on non-compact manifolds | The Main Lemma 2.2 in "Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds" (<http://link.springer.com/article/10.1007/BF01233434>) says essentially that there is a uniform lower bound for the harmonic radius in term of Ricci curvature bound and lower bound of injectivity radius.
T... | 6 | https://mathoverflow.net/users/125531 | 323042 | 139,306 |
https://mathoverflow.net/questions/323037 | 3 | Let $|| = ||\_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert A \rVert \lVert B\rVert$?
| https://mathoverflow.net/users/134624 | When is the matrix norm multiplicative | At least, there is this important case: in $C^\*$-algebras, there is an involution and the norm has the property that
$$\|x\|=\|x^\*\|=\|xx^\*\|^{1/2}.$$
In the special case of ${\bf M}\_n({\mathbb C})$, this is
$$\|AA^\*\|=\|A\|\cdot\|A^\*\|$$
where the norm is the usual operator norm $\|\cdot\|\_{2,2}$.
Actually th... | 7 | https://mathoverflow.net/users/8799 | 323046 | 139,308 |
https://mathoverflow.net/questions/323049 | 0 | Let $F$ be a set consisting of some subsets of $[n]$, and any two sets in $F$ have at least one element in common. I think I read a result stating as following: there exists an element $x$, such that at least half of $F$ include $x$. But I do not remember the related reference. Do anyone know the reference or some simi... | https://mathoverflow.net/users/22925 | number of sets including the most popular elements in intersecting sets family | This is completely false. Consider the lines of a finite projective plane.
| 3 | https://mathoverflow.net/users/4312 | 323051 | 139,309 |
https://mathoverflow.net/questions/323056 | 4 | We say that a finite, simple, undirected graph $G=(V,E)$ is $\mathbb{R}^2$-*realizable* if there is an injective map $\varphi:V\to \mathbb{R}^2$ such that for $v\neq w \in V$ we have $\{v,w\} \in E$ if and only if $|\varphi(v)-\varphi(w)| < 1$ where $|\cdot|$ denotes the Euclidean distance.
What is an example of a fi... | https://mathoverflow.net/users/8628 | Graphs that are not $\mathbb{R}^2$-realizable | How about the [star](https://en.wikipedia.org/wiki/Star_(graph_theory)) $S\_6$? To realize this graph in the way you describe, you would have to map the center point to some $c \in \mathbb R^2$; then you would need to map the $6$ other points of $S\_6$ inside the unit circle around $c$, but all at distance $\geq\!1$ fr... | 13 | https://mathoverflow.net/users/70618 | 323057 | 139,311 |
https://mathoverflow.net/questions/323063 | 1 | Let $Y\_1:=(X\_i)\_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.
We can then also define the shifted sequence $Y\_2:=(X\_{i+1})\_{i \in \mathbb Z}.$
If the $X\_i$ were also independent then $f(Y\_1)$ would have the same law as $f(Y\_2).$
In pa... | https://mathoverflow.net/users/135730 | Expectation of random variables coincides | Let $U$ and $V$ be two i.i.d. random variables having finite expectation. Let $X\_{3k}=X\_{3k+1}:=U$, $X\_{3k+2}:=V$ and $f\left(\left(x\_i\right)\_{i\in\mathbb Z}\right)=x\_0x\_1$. Then $\mathbb E\left[f\left(Y\_1\right)\right]=\mathbb E\left[U^2\right]$ and $\mathbb E\left[f\left(Y\_2\right)\right]=\mathbb E\left[UV\... | 1 | https://mathoverflow.net/users/17118 | 323067 | 139,315 |
https://mathoverflow.net/questions/323019 | 11 | In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^\*(G, \mathbb{Z}/2)$ was generated as an abelian group on the Stiefel-Whitney Classes of flat vector bundles over BG.
The out... | https://mathoverflow.net/users/21985 | Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$ | $\mathrm{H}^1(G,\mathbb{Z}/2) \cong \hom(G,\mathbb{Z}/2)$ consists precisely of first Stiefel–Whitney classes.
So your question is to decide whether there is a finite group with an element in $\mathrm{H}^2(G,\mathbb{Z}/2)$, i.e. with a double cover, which is not a second Stiefel–Whitney class, i.e. which is not pulle... | 9 | https://mathoverflow.net/users/78 | 323084 | 139,321 |
https://mathoverflow.net/questions/323073 | 2 | It is a normal practice to use a minimal set of operators in logical systems and construe the other operators as abbreviations.
---
Let's look at the propositional logic:
If $\mathcal{V}$ denotes the set of variables $v\_0, v\_1, \dots$ we can define the set of propositional formulas as the smallest set $L\_0 \... | https://mathoverflow.net/users/85040 | Termination of "unpacking" abbreviations | A simple idea which might neaten your proof is to define a notion of the "rank" of a formula, which measures the nesting non-primary operators used in the inductive definition of that formula.
This rank can be defined by induction on formulas by:
* $rank(\varphi) := 0$ for $\varphi \in L\_0$
* $rank(\langle i , \... | 2 | https://mathoverflow.net/users/45707 | 323089 | 139,322 |
https://mathoverflow.net/questions/323041 | 5 | Given a set of vectors $V = \{ \mathbf{v}\_1, \ldots, \mathbf{v}\_n \} \subset \mathbb{R}^d$, I want to project a point $\mathbf{x}\_0 \in \mathbb{R}^d$ onto the convex hull $\text{conv}(V)$ of the vectors in $V$.
I know this is a quadratic program, to find $\mathbf{z}^\*$ that minimizes $\frac{1}{2}\|\mathbf{x}\_0 -... | https://mathoverflow.net/users/24951 | Algorithms for projecting a point onto the convex hull spanned by a set of vectors | As you suggested, your problem can be formulated as a quadratic program:
$$\boxed{\begin{array}{rl}
\min & \|\mathbf x\_0-(\alpha\_1\mathbf v\_1+\cdots +\alpha\_n\mathbf v\_n)\|^2\\
\text{s.t.} & \alpha\_1+\cdots+\alpha\_n=1\\
& \alpha\ge 0
\end{array}}$$
The optimal solution to this program gives you the projecti... | 3 | https://mathoverflow.net/users/108884 | 323101 | 139,325 |
https://mathoverflow.net/questions/323093 | 3 | Below I am referring to complex representations.
We know that if $G$ is a finite group with $m=(G:Z(G))$, then every irreducible representation has size at most $\sqrt{m}$. One cannot hope for this to be exact up to $O(1)$ as $|G|\to \inf$ since there are finitely many groups with a given number of conjugacy classes.... | https://mathoverflow.net/users/135743 | How to construct groups and large dimension representations? How about faithful ones? | In the infinite family $G\_p=(\mathbb{Z}/p\mathbb{Z})\rtimes (\mathbb{Z}/p\mathbb{Z})^{\times}$, as $p$ runs over prime numbers, the bound is tight up to $O(1)$, and the representation in question is faithful. Indeed, the centre is trivial, and $\#G\_p=p^2-p$, so $p-1\leq \sqrt{m}\leq p$. By Clifford theory, the induct... | 9 | https://mathoverflow.net/users/35416 | 323102 | 139,326 |
https://mathoverflow.net/questions/323092 | 0 | Definition I am aware of for Lie groupoid is that (among other things) the source and target maps $s,t:\mathcal{G}\_1\rightarrow \mathcal{G}\_0$ are **submersions**.
On page 9 of Du Li's thesis *Higher Groupoid Actions, Bibundles,
and Differentiation* (arXiv:[1512.04209](https://arxiv.org/abs/1512.04209)) the author ... | https://mathoverflow.net/users/118688 | Requirement that source and target maps are surjective submersions | **Edit** I just realised your misconception: the source and target maps are automatically surjective since they both have a section, namely the unit map. So asking that they are surjective submersions or just submersions are equivalent.
---
>
> This is the only place where I saw this kind of requirement.
>
>
... | 2 | https://mathoverflow.net/users/4177 | 323105 | 139,329 |
https://mathoverflow.net/questions/323095 | 2 | The Fano threefold $X$ of index $2$, degree $5$ and Picard number $1$ is known to be a general codimension $3$ linear section of the $Pl\ddot{u}cker$ embedding of Gr(2,5).
My first question: what does 'general' mean specifically in the above definition?
If we use $p\_{ij}$ where $1\leq i<j\leq 5$ to denote the coor... | https://mathoverflow.net/users/48616 | Construction of Fano threefold of degree $5$ and its defining equations | To give a 3-dimensional linear section $X$ of $Gr(2,V)$ with $\dim V = 5$ is equivalent to giving a 3-dimensional subspace $A \subset \Lambda^2V^\vee$ (the space of linear equations of $X$). Then smoothness of $X$ is equivalent to the property
$$
\mathbb{P}(A) \cap Gr(2,V^\vee) = \varnothing.
$$
For the second questi... | 2 | https://mathoverflow.net/users/4428 | 323114 | 139,334 |
https://mathoverflow.net/questions/323129 | 8 | Let $\Pi:=\mathbb{P}(H^0(\mathbb{P}^5,\mathcal{O}\_{\mathbb{P}}(3)))$ be the space of cubic fourfolds in $\mathbb{P}^5$. It is well-known that those cubics which are *pfaffian*, i.e. defined by the pfaffian of a 6-by-6 skew-symmetric
matrix of linear forms, form a hypersurface in $\Pi$. Does anyone know the degree of ... | https://mathoverflow.net/users/40297 | The degree of the hypersurface of pfaffian cubic fourfolds | According to Example 3, p. 319 of
Li, Zhiyuan; Zhang, Letao: [*Modular forms and special cubic fourfolds*](http://dx.doi.org/10.1016/j.aim.2013.06.003), Adv. Math. **245**, 315-326 (2013). [ZBL1290.11077](https://zbmath.org/?q=an:1290.11077),
the degree of $\mathcal{C}\_{14}$ should be 915678.
| 7 | https://mathoverflow.net/users/7460 | 323133 | 139,339 |
https://mathoverflow.net/questions/323094 | 23 | I would like to know if I have discovered or merely rediscovered the following pretty fact.
A partition of $[0,1]$ into intervals of lengths $p\_{i, i=1\ldots n}$ induces a probability distribution with entropy $-\sum p\_i \log\_2 p\_i$; call this also the entropy of the partition.
For a given $n$, entropy gets max... | https://mathoverflow.net/users/10909 | The Euler-Mascheroni constant and entropy | The earliest reference I have found for this result is [Entropy and maximal spacings for random partitions](https://link.springer.com/article/10.1007/BF00533604) (E. Slud, 1978).
Theorem 2.2 states that the entropy $W\_n=-\sum\_{i=1}^n p\_i \ln p\_i$ of the random partition is asymptotically normally distributed for ... | 13 | https://mathoverflow.net/users/11260 | 323134 | 139,340 |
https://mathoverflow.net/questions/323132 | 3 | What is an example of a topological space $(X,\tau)$ on more than one point, with the following properties?
1. the only homeomorphism from $X$ to itself is the identity, and
2. given $x,y\in X$ there are open sets $U, V$ with $x\in U, y\in V$ and a homeomorphism $\varphi: U\to V$ such that $\varphi(x) = y$, where $U,... | https://mathoverflow.net/users/8628 | Rigid space, but with homeomorphic neighborhoods | For $n\in\mathbb N$ let $U\_n=\{m\in\mathbb N:m\geq n\}$. Then $\tau=\{\varnothing\}\cup\{U\_n:n\in\mathbb N\}$ is a topology on $\mathbb N$. This space is rigid because $n$ is characterized as the unique element contained in exactly $n+1$ open sets. However, for any $n,m$ the sets $U\_n,U\_m$ are homeomorphic through ... | 6 | https://mathoverflow.net/users/30186 | 323137 | 139,341 |
https://mathoverflow.net/questions/323140 | 2 | I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is).
Inspired by [this](https://math.stackexchange.com/questions/1863314/when-is-the-universal-cover-of-a-riemannian-manifold-complete) and [this question](https://math.stackexchange.com/questions/1502865... | https://mathoverflow.net/users/135778 | $(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version) | Completeness of a pseudo-Riemannian manifold (or a manifold with affine connection, more generally) means precisely the completeness of its geodesic flow on its tangent bundle. But the tangent bundle of any covering space is a covering space of the tangent bundle. The geodesic flow is the flow of the geodesic vector fi... | 5 | https://mathoverflow.net/users/13268 | 323142 | 139,342 |
https://mathoverflow.net/questions/323139 | 1 | Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the log MGF $t \mapsto \log \mathbb E\_P[\exp(th(x)]$.
Now, using a simple Taylor expansion, one can carelessly write
$$
\lo... | https://mathoverflow.net/users/78539 | Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$ | To simplify notation, let $X$ be a random variable whose probability distribution is $P$, and then let $Y:=h(X)$. Your question is then is whether
$$\ln E e^{tY}=t\,EY+\frac{t^2}2\,Var\, Y+o(t^2),$$
apparently for $t\to0$.
Clearly, for this question to have meaning, we have to assume that the values $M(t):=E e^{tY}... | 3 | https://mathoverflow.net/users/36721 | 323145 | 139,344 |
https://mathoverflow.net/questions/323136 | 26 | **Edit of Feb. 14, 2019.** After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is
>
> Do binary coproducts always exist in the category of noetherian commutative rings?
>
>
>
**End of edit.**
I... | https://mathoverflow.net/users/461 | Is every commutative ring a limit of noetherian rings? | The answer is no to all questions except 4.
**Negative answers to 1,2 and 3**:
It is easy to construct a ring $A$ with an element $a$ satisfying:
(i) $a≠0$,
(ii) $a$ is nilpotent,
(iii) for each $n≥1$ there is $y\_n\in A$ such that $a=y\_n^n$.
For every morphism $\varphi:A\to C$, the image of $a$ inherit... | 36 | https://mathoverflow.net/users/7666 | 323146 | 139,345 |
https://mathoverflow.net/questions/322284 | 21 | I started a PhD in chromatic homotopy three years ago, but I had to quit it due to personal reasons after one year. Last week I was looking at all my notes from that period and I was wondering where are we now with this theory.
Are Lurie's lecture notes still the best way to approach the topic?
Could you please te... | https://mathoverflow.net/users/93775 | Latest results in chromatic homotopy theory | In 2017, the long-standing problem of whether the Brown–Peterson spectrum $\mathrm{BP}$ admits the structure of an $E\_\infty$-ring was shown in the *negative*.
The question dates back to 1975, formulated as Problem 1 of Peter May's “Problems in infinite loop space theory” [May75]:
>
> **Problem 1.** Does the Bro... | 11 | https://mathoverflow.net/users/130058 | 323154 | 139,348 |
https://mathoverflow.net/questions/323155 | 3 |
>
> If $(a\_n)\_{n \ge 1}$ is a non-negative sequence s.t., $$\sum\limits\_{n = c\_k}^\infty \frac{a\_n}{\log^{(k)} n} < \infty, \, \forall k \ge 1 \overset{?}{\implies} \sum\limits\_{n \ge 1} a\_n < \infty$$ where, $\log^{(k)} = \underbrace{\log \circ \log \circ \cdots \circ \log}\_{k \text{ times }}$ and $c\_k$ is ... | https://mathoverflow.net/users/62680 | A problem with sequences with composition of $\log$s | Let $l\_k:=\ln^{(k)}$. Let $(n\_k)$ be any strictly increasing sequence of natural numbers such that $l\_k(n\_k)>k^2$. Let $a\_n:=1$ if $n=n\_k$ for some $k$ and let $a\_n:=0$ otherwise. Then $\sum\_n a\_n=\infty$, but
$\sum\_n a\_n/l\_j(n)=\sum\_k 1/l\_j(n\_k)<\infty$ for all $j$, since $l\_j(n\_k)\ge l\_k(n\_k)>k^2$... | 5 | https://mathoverflow.net/users/36721 | 323159 | 139,349 |
https://mathoverflow.net/questions/323161 | 7 | I am looking for examples of sequences of polynomials $(p\_{k}(x))\_{k=1}^{\infty}$ with positive integer coefficients where $p\_{k}(0)=1$ for all $k\geq 1$ and where there is a positive integer $r$ where
$$\frac{1}{1-rx}=\prod\_{k=1}^{\infty}p\_{k}(x)$$
whenever $x\in(0,\frac{1}{r})$.
One can easily construct many ... | https://mathoverflow.net/users/22277 | Factoring $\frac{1}{1-rx}$ into an infinite products of polynomials | Slightly generalized from Ex. 1:
$$ \frac{1}{1-r x} = \prod\_{j=0}^\infty \sum\_{i=0}^{m-1} (r x)^{i m^j} $$
for integers $m \ge 2$.
| 2 | https://mathoverflow.net/users/13650 | 323165 | 139,351 |
https://mathoverflow.net/questions/323160 | 0 | Given a matrix $M\in\mathbb{N}^{n\times n}$, let $Z$ be the set of all the $M$'s entry subsets $S$ such that **(i)** no two entries of $S$ are on the same row or column of $M$ and **(ii)** $|S|=n$. Clearly we have $|Z|=n!$.
**Question**: How can we (efficiently) find the $M$'s entry subset $S^\* \in Z$ whose element... | https://mathoverflow.net/users/115803 | Finding the minimum sum of a subset of entries of a given matrix with combinatorial constraints | Here is a way to formulate it as a convex optimization problem, which can then be solved in polynomial time. Your variables are $a\_{i,j}$, one for each position in the matrix. The problem is then:
$$0\le a\_{i,j} \le 1$$
$$\forall\_j \sum\_i a\_{i,j} = 1$$
$$\forall\_i \sum\_j a\_{i,j} = 1$$
$$\textrm{Minimize } M\bul... | 2 | https://mathoverflow.net/users/97603 | 323174 | 139,354 |
https://mathoverflow.net/questions/323169 | 3 | The page 50 of (the arXiv version of) the above-mentioned paper of P. Scholze says "Now the Poincare duality pairing implies that $H^i(Y\_{\mathbb{C}\_p, et}, \bar{\mathbb{Q}}\_l)$ is a direct summand of $H^i(Z'\_{\mathbb{C}\_p^\#, et}, \bar{\mathbb{Q}}\_l)$." Surely I am missing something elementary but how exactly Po... | https://mathoverflow.net/users/135701 | A clarification of an argument in "Perfectoid spaces" | Let there be a map $f: X \rightarrow Y$ between two proper smooth varieties of the same dimension $n$, then we get a morphism $f^\*:H^i(Y) \rightarrow H^i(X)$, where we assume the coefficient field is of char $0$ and algebraically closed.
If $f^\*$ induces an isomorphism between top degree (i.e degree $2n$) cohomolo... | 3 | https://mathoverflow.net/users/102104 | 323176 | 139,355 |
https://mathoverflow.net/questions/323068 | 6 | I am working on some problems involving foliations and group actions and would be very nice to consider the second derivatives for the distance function of an orbit or a leaf.
So my question is: does anyone know a reference or an easy way to compute the hessian tensor associated to the function:
$d : M \to \mathbb{... | https://mathoverflow.net/users/94097 | Hessian formula for the sub manifold distance | I believe an answer to your question can be found in the book:
**A. Gray,** *Tubes.* Second edition. Progress in Mathematics, 221. Birkhäuser Verlag, Basel, 2004.
If you know how to search, you can find this book online.
The title *Tubes* refers to a tubular neighborhoods of radius $r$ of a submanifold. One of t... | 3 | https://mathoverflow.net/users/121665 | 323189 | 139,360 |
https://mathoverflow.net/questions/323048 | 16 | In [this](https://arxiv.org/pdf/1702.00679.pdf) paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a noetherian ring $A$, such that $(A,I)$ is $I$-adically complete: in particular, a hens... | https://mathoverflow.net/users/nan | GAGA for henselian schemes | OK, it turns out that $H^1((P^1\_A)^h, O^h)$ is nonzero in general. A counter example can be found in a blog post on the very blog you mention in your post. Here is a [link](https://www.math.columbia.edu/~dejong/wordpress/?p=4346).
| 6 | https://mathoverflow.net/users/135794 | 323190 | 139,361 |
https://mathoverflow.net/questions/323188 | 19 | (I asked this question on MSE, but someone suggested it would be better asked here.)
I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^\*)^n$ to a generic system of Laurent polynomials is the mixed volume of the polynomials' Newton polytopes), and I was intrigued ... | https://mathoverflow.net/users/135630 | Where to find some subset of Khovanskii's 15 proofs of the BKK theorem? | Khovanskii gives what he calls "the simplest proof" in section 4 of [Newton Polyhedron, Hilbert Polynomial, and Sums of Finite Sets](https://www.math.toronto.edu/askold/1992-Faa-4-english.pdf) (1992).
More proofs are in
* [Newton polyhedra and the genus of complete intersections](https://link.springer.com/article/... | 10 | https://mathoverflow.net/users/11260 | 323196 | 139,362 |
https://mathoverflow.net/questions/322940 | 7 | Since my question is related to sign convention, I want to define everything from the very beginning. $T\_{poly}^k(M)=\Gamma(\wedge^{k+1} TM)$ are the multi vector fields with shifted degree and with the Schouten bracket uniquely determined by
\begin{align\*}
[\![f,g]\!] =0\ \text{, } [\![X,f]\!]=X(f) \text{ and } [\!... | https://mathoverflow.net/users/135670 | Kontsevich Formality sign convention | Welcome to mathoverflow!
There is actually a whole paper (in French) about choices of signs for Kontsevich formality: <https://arxiv.org/pdf/math/0003003.pdf>
For instance, they define the Hochschild coboundary operator as $\delta(x)=-[m\_0,x]$ (see also their Remark on page 20).
I hope this will help.
| 2 | https://mathoverflow.net/users/7031 | 323209 | 139,367 |
https://mathoverflow.net/questions/323204 | 1 | We say that a simple, undirected graph $G = (\omega, E)$ on the vertex set $\omega$ is *tightly knit* if there is a positive integer $n>2$ such that for all $v,w\in \omega$ there is a cycle $C$ of length $\leq n$ in $G$ such that $v,w\in C$.
Is there a collection of $2^{\aleph\_0}$ pairwise non-isomorphic tightly kni... | https://mathoverflow.net/users/8628 | Tightly knit graphs on $\omega$ | Consider the graph $G$ with vertices $ \left\{ (x,A) |x ∈\mathbb{N} \right\} \cup \left\{ (x,B) |x ∈\mathbb{N} \right\} \cup \left\{ (x,y) |x ∈\mathbb{N}, y ∈\mathbb{N},y \leq x \right\}$ with two vertices $(w,x)$ and $(y,z)$ connected iff $w=y$ or both $x$ and $z$ are letters.
For each $n\geq 2$, one can decide whet... | 2 | https://mathoverflow.net/users/125498 | 323215 | 139,368 |
https://mathoverflow.net/questions/323118 | 0 | Let $C = \{ e \_{1}, \cdots e \_{n} \}$, where each $e \_{i}$ are unit vectors in $\ell ^{p, \infty}$, and $1 < p < \infty$. I want prove that $\overline{conv}(C)$ is diametral. My doubt is: $\Vert x - y \Vert \_{p, \infty} \leqslant 1$ for all $x, y \in \overline{conv}(C)$, where
\begin{align\*}
\Vert x \Vert \_{p, \... | https://mathoverflow.net/users/134213 | $\overline{conv}(C)$, where $C = \{ e _{1}, \cdots e _{n} \}$, $e _{i} \in \ell ^{p, \infty}$ is diametral | Note that $\mathrm{diam}( \overline {\mathrm{conv}}(C) ) = \mathrm{diam}(C)$, and note that $\|e\_i-e\_j\|\_{p,\infty}=1$ for $i\neq j$. This proves your inequality $\|x-y\|\_{p,\infty}\le 1$.
However, you won't get diametrality from this. On page 39 of Goebel and Kirk's "Topics in metric fixed point theory" you wil... | 0 | https://mathoverflow.net/users/127871 | 323255 | 139,381 |
https://mathoverflow.net/questions/323240 | 5 | Let's say an $n$-manifold is *cubulated* if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} \times [0,1]^{n-k}$ of some cube must be glued to the face $[0,1]^{k-1} \times \{0\} \times [0,1]^{n-k}$ of some othe... | https://mathoverflow.net/users/78 | Are framed manifolds cubulatable? | If you glue together the Riemannian metrics on the various cubes you obtain a flat metric on your cubulated manifold. So e.g. $S^3\cong SU(2)$ is framed but not cubulated.
| 12 | https://mathoverflow.net/users/1310 | 323259 | 139,384 |
https://mathoverflow.net/questions/323212 | 2 | Sorry if this is too elementary, but when I was going to ask this question on math.stackexchange, I saw the same question with three up-votes and no answer. So I decided to post it here.
I am doing the following problem:
>
> Let $R$ be a ring, $x\in R$ a central non-unit non-zerodivisor. If $A\neq 0$ is an $R/xR$... | https://mathoverflow.net/users/124352 | Injective Change of Rings | The underlying idea of the below argument comes from the technique of spectral sequence presented in Chapter five of Weibel's book.
Let $A$ be an injective $R/xR$-module. Let $M$ be an arbitrary $R$-module. Let $P\_\*\to M$ be a projective resolution of $M$. One has
$$
\hom\_{R}(P\_\*,A) \cong \hom\_R(P\_\*, \hom\_{R... | 2 | https://mathoverflow.net/users/52982 | 323267 | 139,387 |
https://mathoverflow.net/questions/323260 | 2 | This was [asked](https://math.stackexchange.com/questions/3012437/eigenvectors-of-nth-root-of-identity-matrix) at math stackexchange a long time ago with no answers but some upvotes.
Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its characteristic polynomial is $x^n-1$... | https://mathoverflow.net/users/17773 | What do the eigenvectors of the $n$th roots of $I_n$ look like? | Since $A^n=I$, $A$ is diagonalizable and eigenvalues are $n$-th roots of unity. They don't have to be all $n$-th roots of unity and they don't have to be distinct.
Your condition that $A^k\neq I$ simply means that for all $k<n$ not all
$\lambda\_j^k$ are equal to $1$.
Speaking on eigenvectors, there are no restrictio... | 2 | https://mathoverflow.net/users/25510 | 323271 | 139,389 |
https://mathoverflow.net/questions/323262 | 1 | There are a number of plane curves listed on [mathworld](http://mathworld.wolfram.com/topics/AlgebraicCurves.html), described by implicit algebraic equations, including the [butterfly curve](http://mathworld.wolfram.com/ButterflyCurve.html), [ampersand curve](http://mathworld.wolfram.com/AmpersandCurve.html), and [bow ... | https://mathoverflow.net/users/133693 | Is there a process for finding an implicit representation for an approximation to an arbitrary plane curve? | I address your first question. Indeed, every desired shape can be approximated
by an algebraic curve: for example, using the Hilbert Lemniscate theorem.
It says that for every compact $K$ in the plane and every neighborhood $U$ of $K$,
there is a lemniscate which separates $K$ from the complement of $U$.
The lemnisca... | 2 | https://mathoverflow.net/users/25510 | 323272 | 139,390 |
https://mathoverflow.net/questions/323275 | 0 | I desperately need to read the paper [1] before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's *Algebra* and Churchill's *Complex Analysis*, not even one sentence of the paper is readable to me, so I even don't know what area of mathematics is it about!
What is the ... | https://mathoverflow.net/users/nan | On the 2002 paper "Dynamics of polynomial automorphisms of $\mathbb{C}^k$" by Guedj and Sibony | This paper is about holomorphic dynamics in high dimensions. I recommend to begin the
study of holomorphic dynamics from the one-dimensional case, and the best book on
the subject is J. Milnor, Dynamics in one complex variable, <https://arxiv.org/abs/math/9201272>. After that you may read the two survey articles mentio... | 6 | https://mathoverflow.net/users/25510 | 323277 | 139,392 |
https://mathoverflow.net/questions/323219 | 6 | A type $II\_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x\_{1}, x\_{2},..., x\_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $M$ with $\tau (u)=0$ and $||ux\_{j}-x\_{j}u||\_{2}<\epsilon$ for all $1 \leq j \leq n$. (Here $||T||\_2=(\tau(T^{\*}T))... | https://mathoverflow.net/users/6269 | Property $\Gamma$ in terms of Correspondences | A separable $\mathrm{II}\_1$ factor $M$ is non-$\Gamma$ iff $L^2(M)\prec H$ and $H\prec L^2(M)$ imply $L^2(M)\subset H$.
Proof:
Let $M$ be non-$\Gamma$ and take a finite critical set $F\subset M$.
By $L^2(M)\prec H$, there is a unit vector $\xi\in H$ which is $(F,\epsilon)$-central. Assume for contrapositive that ... | 6 | https://mathoverflow.net/users/7591 | 323282 | 139,394 |
https://mathoverflow.net/questions/323283 | -1 | Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e\_{n})=e\_{n-1}$, where $e\_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator so spectrum is on $S^{1}$. What is spectral measure of $U$? What is its spectral decomposition with respect to multipl... | https://mathoverflow.net/users/125816 | On spectral multiplicity of left shift operators | Identify $l^2(\mathbb{Z})$ with the space of square summable Fourier series $f(z)=\sum\_{n\in \mathbb{Z}} a\_n z^n$, $\sum |a\_n|^2<\infty$, on the unit circle $\mathbb{T}=\{z:|z|=1\}$. It is the space $L^2(\mathbb{T},\lambda)$ where $\lambda$ is Lebesgue measure on the circle and the operator $U$ maps the function $f(... | 3 | https://mathoverflow.net/users/4312 | 323293 | 139,398 |
https://mathoverflow.net/questions/323290 | 7 | There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties but let's forget about them for now.
There are several more or less precise incarnations of mirror symmetry, including... | https://mathoverflow.net/users/135701 | Multiple mirrors phenomenon from SYZ and HMS perspective | Some relation between these two ambiguities is indeed part of the expected but still largely conjectural big picture.
Let $X$ be a compact Calabi-Yau manifold, viewed in a symplectic way. Let $M\_X$ be the moduli
space of complex structures on $X$. The various maximally unipotent degenerations are the various ways to... | 7 | https://mathoverflow.net/users/25309 | 323296 | 139,399 |
https://mathoverflow.net/questions/323297 | 3 | Imagine a graph where the vertices and edges model an n dimensional hypercube (a line, a square, a cube and so on). A red vertex must have a minimum distance of 3 from every other red vertex. The problem is to maximise the number of red vertices for a given n.
I have absolutely no idea where to start. Any help is app... | https://mathoverflow.net/users/nan | Packing vertices on a hypercube graph? | This is the problem of finding not-necessary-linear codes in a Hamming cube: for odd dimensions, one could take [this table](https://www.eng.tau.ac.il/~litsyn/tableand/index.html#dist3) as a lower bound.
**EDIT**: The Hoffman bound gives a better bound that Fedor Petrov's for even $n$:
The Hoffman bound for a regul... | 5 | https://mathoverflow.net/users/125498 | 323300 | 139,401 |
https://mathoverflow.net/questions/323304 | 2 | Let $x=(z\_1,\ldots,z\_n)$ be real vector and $(p\_1,\ldots,p\_n)$ be a probability vector.
Question
========
$\log(\sum\_{i=1}^n p\_ie^{z\_i})-\sum\_{i=1}^np\_iz\_i \le ???$
Observation
===========
This paper allows us to upper-bound things like $\Delta\_f(z,p):=f(\sum\_{i}z\_i p\_i) - \sum\_i f(p\_iz\_i)$, th... | https://mathoverflow.net/users/78539 | Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector | One answer - subgaussian variables generalize this property.
Let $\mu = \sum\_i p\_i z\_i$, then the distribution is considered $\sigma^2$-subgaussian if for all $\lambda \in \mathbb{R}$,
$$ \log\left(\sum\_i p\_i e^{\lambda(z\_i - \mu)}\right) \leq \frac{\lambda^2 \sigma^2}{2} $$
i.e.
$$ \log\left(e^{-\lambda... | 1 | https://mathoverflow.net/users/29697 | 323305 | 139,404 |
https://mathoverflow.net/questions/323312 | 13 | For $0\leq d\leq 1$, let $\lambda\_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}\_d$ on $\mathbb{R}$, finer than the usual topology and compatible with the (additive) group structure (i.e., $+$ and $-$ are con... | https://mathoverflow.net/users/17064 | Are Hausdorff measures on the real line Haar measures for some locally compact topology? | The answer is NO because the Euclidean and the discrete topologies are the unique locally compact group topologies on $\mathbb R$, which are stronger that the Euclidean topology of the real line.
The reason is that $\mathbb R$ endowed with such topology $\tau$ is a locally compact abelian topological group without s... | 15 | https://mathoverflow.net/users/61536 | 323317 | 139,406 |
https://mathoverflow.net/questions/323336 | 0 | The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$.
Informally speaking, the bit below is about how "close" a map between the vertex sets of graphs can become to being a graph homo... | https://mathoverflow.net/users/8628 | Minimizing the set of "faulty" edges in a map between the vertex sets of $2$ graphs | Let $G=K\_\omega$ be a clique on infinitely many vertices and $H$ a disjoint union of $K\_n$ for each $n\in\omega$. I claim $\text{Flt}(G,H)$ has no minimal elements.
Let $f:V(G)\to V(H)$ be arbitrary. Take some vertices $v,w\in V(G)$ with $f(v)\in K\_n$ and $f(w)\in K\_m$ with $n\neq m$. Consider a map $g:V(H)\to V(... | 1 | https://mathoverflow.net/users/30186 | 323338 | 139,412 |
https://mathoverflow.net/questions/323309 | 2 | I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 T^{\*}M)$ smooth, fixed and positive definite and then compute the time derivative for the divergence of $G(T)$:
$\bigg... | https://mathoverflow.net/users/119114 | Principal Symbol for the Ricci-DeTurck Flow | It looks like this is from the *Lecture notes on Ricci flow* from Peter Topping. He mentions that he uses the symbol $T$ as the symmetric, positive definite bilinear form and also for the map $\Gamma(TM^\ast) \to \Gamma(TM^\ast)$ induced by $T$ and the metric $g$ in the following way
$$
T(\alpha)(Z)=T(\alpha^\#,Z)
$$
w... | 3 | https://mathoverflow.net/users/20999 | 323341 | 139,414 |
https://mathoverflow.net/questions/323328 | 2 | Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that under the assumption that $H^{\bullet}(E) \rightarrow H^{\bullet}(F)$ is surjective, by the Leray-Serre spectral sequence... | https://mathoverflow.net/users/91572 | Leray-Serre spectral sequence for projective bundles | As you say, if the map $H(E)\to H(F)$ is surjective, the Leray-Serre spectral sequence degenerates at the $E^2$-page which is therefore isomorphic to the $E^\infty$-page. Unwrapping the definitions, this means that there is a filtration on $H^\*(E)$ such that the associated graded is $H^\*(X)\otimes H^\*(F)\cong H^\*(X... | 3 | https://mathoverflow.net/users/35687 | 323342 | 139,415 |
https://mathoverflow.net/questions/323351 | 2 | Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable.
| https://mathoverflow.net/users/12669 | parametrize triangles meeting certain conditions | See Konstantine Zelator, [Integral Triangles](https://arxiv.org/abs/1208.0497) with one angle twice another, and with the bisector(of the double angle) also of integral length. The abstract says, "In Result 2 in Section 5, we offer 3-parameter formulas that describe the entire family of integral triangles $ABC$ with $\... | 5 | https://mathoverflow.net/users/3684 | 323354 | 139,419 |
https://mathoverflow.net/questions/323369 | 3 | If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda\_\alpha(X)$ as the subset of $C\_b(X)$ satisfying that
$$
\sup\_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
Similarly, we can define the Little Hölder space $\lambda\_\alpha(X)$ as the subset of functions of $\Lambda\_\alpha(X)... | https://mathoverflow.net/users/12604 | Reference request: $\alpha$-Hölder spaces as double duals | First of all, I am not sure what you mean by $L^\infty(X)$ for a general complete metric space $X$. Don't you want $C\_b(X)$?
Secondly: when $X$ is compact and $0<\alpha<1$, the result you want is Theorem 3.5 in
>
> W. G. Bade, P. C. Curtis, Jr, and H. G. Dales,
> *Amenabilty and weak amenability for Beurling an... | 4 | https://mathoverflow.net/users/763 | 323370 | 139,425 |
https://mathoverflow.net/questions/323368 | 1 | Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary and that $M\subset \tilde{M}$ with $(\tilde{M},g)$ also a compact Riemannian manifold with smooth boundary.
Let us consider a one-form $\alpha \in L^2(M;T^\*M)$ with the additional property that $\nabla \cdot \alpha \in L^2(M)$, that is the divergen... | https://mathoverflow.net/users/50438 | Sobolev extension operators | **The answer is no**.
Consider the form
$$
\omega = \frac{-y}{x^2+y^2}\, dx + \frac{x}{x^2+y^2}\, dy
$$
on the annulus $\Omega=\{ (x,y):\, 1<x^2+y^2<2\}$. This form is closed, but not exact, because the integral along a circle around zero equals $2\pi$.
If you would manage to extend it to a closed form in the disc ... | 2 | https://mathoverflow.net/users/121665 | 323374 | 139,426 |
https://mathoverflow.net/questions/323356 | 2 | I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \mathbb{R}^+$, can be analytically continued to an entire function on $\mathbb{C}$.
Because of this, I wonder:
>
>... | https://mathoverflow.net/users/133882 | Must $q$ be analytic? | No. $F$ does not put enough constraints on $q$.
Suppose that $F$ is a given analytic function, and let $q\_0:[0, 1]\to \mathbb R\_+$ be continuous. Extend $q\_0$ to a function on $\mathbb R\_+$ using the recurrence
$$q(s+1) = (e^{q(s)}-e^s)F(s).$$
In general, $q$ will not be continuous or positive. One an write down ... | 3 | https://mathoverflow.net/users/82587 | 323375 | 139,427 |
https://mathoverflow.net/questions/323372 | 2 | Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{x}^{2}\textbf{I}\_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{y}^{2}\textbf{I}\_{M})$ are independent, what would be the p.d.f. of
$$Z = \left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2, $... | https://mathoverflow.net/users/103291 | p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians? | You can use the rotational invariance of the Gaussian distribution to take $\mathbf{y}=2^{-1/2}\sigma\_y(\sqrt \xi\_{2M},0,0,\ldots,0)$, with $\xi\_{2M}$ distributed independently of $\mathbf{x}$ according to a chi-squared distribution with $2M$ degrees of freedom. Then
$$Z=\tfrac{1}{2}\sigma\_y^2 \xi\_{2M}|z|^2\;\;\t... | 2 | https://mathoverflow.net/users/11260 | 323376 | 139,428 |
https://mathoverflow.net/questions/323343 | 6 | Motivated by [Parity of number of partitions of $n!/6$ and $n!/2$](https://mathoverflow.net/q/323308), I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions modulo 3. This is what it answered:
```
(1) -> s := [partition n for n in 1..]
... | https://mathoverflow.net/users/3032 | Number of integer partitions modulo 3 | The function $\,h(x) := \prod\_{n=1}^\infty (1 - x^n)\,$ is known as a Ramanujan theta function. It is essentially the Dedekind $\eta$ function. The connection is $\,f(q) := q h(q^{24}) = \eta(24\tau)\,$ where $\,q=\exp(2\pi i \tau).\,$ Differentiating we get
$\, dq = (2\pi i)\, q\, d\tau,\,$ and
$$ \frac{d}{d\tau} f(q... | 2 | https://mathoverflow.net/users/113409 | 323383 | 139,433 |
https://mathoverflow.net/questions/323386 | 2 | Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{x}^{2}\textbf{I}\_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}\_{M},\sigma\_{y}^{2}\textbf{I}\_{M})$ are independent, what would be the expectation
$$\mathbb{E} \left[ \left| \frac{\textbf{x}^{H} \textbf{y} }{\| \te... | https://mathoverflow.net/users/103291 | Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians? | Using the representation in this [answer](https://mathoverflow.net/a/323376/11260),
$$Z= \frac{|\textbf{x}^{H} \textbf{y} |^2}{ |\textbf{x} |^4} =\frac{\sigma\_y^2}{\sigma\_x^2} \frac{\xi\_{2M}\xi\_{2}}{(\xi\_{2}+\xi\_{2M-2})^2},$$
and integrating over
the independent chi-squared variables $\xi\_2$, $\xi\_{2M}$, and $... | 5 | https://mathoverflow.net/users/11260 | 323393 | 139,435 |
https://mathoverflow.net/questions/323392 | 5 | Where can I find a proof of the following fact?
>
> If
> $$w(u,x\_{0},r)=\sup \_{B\_{r}(x\_{0})}u-\inf \_{B\_{r}(x\_{0})}u$$
> for some function $u(x)$ satisfies
> $$ w\left(u,x\_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x\_{0},r\right)$$
> for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r... | https://mathoverflow.net/users/nan | Oscillation and Hölder continuity | Just prove it yourself:
Take $r=1$. Then
$$w(x\_0,2^{-n})\leq \lambda^n w(x\_0,1)=:C\lambda^n.$$
To estimate $|u(x\_0)-u(y)|$, where $y$ is close to $x\_0$, choose $n$ so that
$|x\_0-y|\in[2^{-n-1},2^{-n}].$
Then
$$|u(x\_0)-u(y)|\leq w(x\_0,2^{-n})\leq C\lambda^{n}=C. 2^{-hn}\leq 2^hC\_1|x\_0-y|^h,$$
where $h=-\log\l... | 5 | https://mathoverflow.net/users/25510 | 323398 | 139,440 |
https://mathoverflow.net/questions/323314 | 1 | Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\pi^{-1}(s)$ is a curve in $\mathbb{C}^2$ (under the restriction of the inclusion of $\mathcal{C}$ into $\mathbb{C}^2 \tim... | https://mathoverflow.net/users/58203 | Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities? | Without further hypotheses on $S$, there are counterexamples. For instance, begin with $\widetilde{S}$ equal to the affine line $\mathbb{A}^1\_k = \text{Spec} \ k[s]$. Inside of the affine space $\mathbb{A}^2\_S = \text{Spec}\ k[s,t,u]$, consider the closed subscheme $\widetilde{\mathcal{C}}$ equal to the zero scheme o... | 3 | https://mathoverflow.net/users/13265 | 323403 | 139,442 |
https://mathoverflow.net/questions/322362 | 5 | Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory.
I have the following questions.
Let $\mathfrak{g}$ be a [basic classical simple Lie superalgebra](https://en.wikipedia.org/wiki/Lie_superalgebra)... | https://mathoverflow.net/users/33047 | Serre relations for Lie Superalgebras | The *Serre relations* (some authors also call them *Serre-Chevalley relations*) for the finite dimensional, complex, basic, classical, simple Lie superalgebras -**in analogy with the Lie algebra case**- read:
$$
(ad E\_i^\pm)^{1-\tilde{a}\_{ij}}E\_j^\pm=\sum\_{n=0}^{1-\tilde{a}\_{ij}}(-1^n)\binom{1-\tilde{a}\_{ij}}{n}... | 5 | https://mathoverflow.net/users/85967 | 323416 | 139,446 |
https://mathoverflow.net/questions/323385 | 12 | The Azuma-Hoeffding Inequality says that if $X\_1,X\_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X\_i - X\_{i-1}\| \le 1$ say, then we should not expect the difference $\|X\_N - X\_0\|$ to grow *too fast*. Formally we have
$$P\left(|X\_N -X\_0| > \epsilon N\right) \le \exp\Big ( \frac{... | https://mathoverflow.net/users/58082 | Can we do better than Azuma-Hoeffding when the variance is small? | Exponential inequalities for sums of independent random variables (r.v.'s) can be extended to martingales in a standard and completely general manner; see Theorem 8.5 or Theorem 8.1 for real-valued martingales, and Theorem 3.1 or Theorem 3.2 for martingales with values in 2-smooth Banach spaces in [this paper](https://... | 8 | https://mathoverflow.net/users/36721 | 323417 | 139,447 |
https://mathoverflow.net/questions/323447 | 0 | Given a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$.
A *linear hypergraph* is a hypergraph $H=(V,E)$ such that every edge has at least $2$ elements and for... | https://mathoverflow.net/users/8628 | Linear intersection number and chromatic number for infinite graphs | The answer is no. This is equivalent to stating that for any linear hypergraph $H=(\kappa,E)$ with $\kappa$ infinite (note that for $\kappa$ finite, $G=I(H)$ would be finite), we have $\chi(I(H))\leq\kappa$. This follows immediately once we show $|V(I(H))|=|E|\leq\kappa$.
For $\alpha<\kappa$ let $E\_\alpha=\{e\in E:\... | 3 | https://mathoverflow.net/users/30186 | 323448 | 139,458 |
https://mathoverflow.net/questions/323028 | 4 | Suppose we have a bounded Borel measurable vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. To make the question non-trivial, assume that $F\neq 0$ eveywhere.
>
> **Question.** Does there exist at least one Lipschitz integral curve? That is a Lipschitz function $\varphi:(a,b)\to\mathbb{R}^n$ such that
> $\varphi'(t)=... | https://mathoverflow.net/users/121665 | ODE with a measurable vector field | For completeness let me add some details (as suggested by @MateuszKwaśnicki).
Let $A\subseteq \mathbb R$ be [a Borel set of positive but not full measure in each interval](https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval), set $F(x) = -1 + ... | 2 | https://mathoverflow.net/users/44463 | 323449 | 139,459 |
https://mathoverflow.net/questions/322282 | 1 |
>
> Let $C$ be a proper smooth curve over a perfect field $K$ of positive characteristic $p$, $u: U \hookrightarrow C$ strictly open and $\mathfrak{F}$ a lisse (lcc) $\mathbb{F}\_l $-sheaf $(l \neq p)$ on $U$. Let $K'/K$ be **any** field extension and $x'$ a geometric point of $C'= C \times\_K K'$ that maps to $x \in... | https://mathoverflow.net/users/111259 | Swan-conductor and base change | Yes.
We have some finite etale Galois cover $D \to C$ with automorphism group $G$. We can base change this cover, obtaining a finite etale Galois cover $D' \to C'$ with the same automorphism group. Because the Swan conductor is determined by the lower numbering filtration on this Galois group, it suffices to show tha... | 3 | https://mathoverflow.net/users/18060 | 323461 | 139,462 |
https://mathoverflow.net/questions/323451 | 5 | Before giving a motivation let me ask the precise question firstly.
By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which transition functions are restrictions of functions belonging to $Aff(\mathbb{C}^{\dim\_\mathbb{C}M})$, the group of complex... | https://mathoverflow.net/users/85450 | Compact complex affine Kähler manifold is a torus | If a compact Kähler manifold $M$ admits a holomorphic affine connection, its Atiyah class and therefore all its Chern classes are zero. By Yau's solution of the Calabi conjecture, this implies that a finite covering of $M$ is a complex torus.
| 4 | https://mathoverflow.net/users/40297 | 323462 | 139,463 |
https://mathoverflow.net/questions/310211 | 8 | *Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is about the extent to which a second-countable space can have different subbases of (relatively) wildly different levels of... | https://mathoverflow.net/users/8133 | How much can complexities of bases of a "simple" space vary? | Welp, this wasn't my finest moment: unless I'm missing something, we can translate everything much more easily than I thought at first. *(I may indeed be missing something, however, and I'll wait a while before accepting this just in case.)*
For simplicity I'll look at *bases* only; it won't make a serious difference... | 1 | https://mathoverflow.net/users/8133 | 323468 | 139,465 |
https://mathoverflow.net/questions/323476 | 4 | Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
| https://mathoverflow.net/users/131790 | Are complete minimal submanifolds closed? | It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $\mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $\mathbb{R}^3$. In ... | 7 | https://mathoverflow.net/users/127803 | 323477 | 139,467 |
https://mathoverflow.net/questions/323475 | 7 | Recently I got interested in [predicative foundations](https://ncatlab.org/nlab/show/predicative+mathematics), mostly because of Laura Crosilla's work and because Agda employs a predicative type theory.
From the point of view of a predicative foundation to arithmetic, for instance as proposed in [Nelson's book](https... | https://mathoverflow.net/users/31233 | Explaining the consistency of PRA and ZF from predicative foundations | From the point of view of what you call predicative set theory --- I would say "predicativism given the natural numbers" --- I don't think there are any known arguments for the consistency of ZF, and such a thing seems very unlikely. The proof-theoretic strength of natural predicative theories are quite weak, generally... | 10 | https://mathoverflow.net/users/23141 | 323482 | 139,468 |
https://mathoverflow.net/questions/323485 | 6 | Let $Q$ be a quiver, and let $d=(d\_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \prod\_i GL(d\_i)$ acts naturally on Rep($Q,d$) in such a way that its orbits are the isomorphism classes of representati... | https://mathoverflow.net/users/468 | Closures of orbits in the space of representations of a quiver | It turns out the answer is "no". There is an example in section 3.4 of Riedtmann's paper "Degenerations for representations of quivers with relations", Ann. sci. Éc. Norm. Sup. v. 19 (1986), 275-301.
The quiver has vertices 1 and 2, with an arrow from 1 to 2 and a loop at 2.
The dimension vector is $d=(1,2)$. In $X$,... | 3 | https://mathoverflow.net/users/468 | 323486 | 139,469 |
https://mathoverflow.net/questions/323487 | 6 |
>
> Mathematics is not about numbers, equations, computations, or
> algorithms: it is about understanding.
>
>
>
Is this from Thurston? If yes, where and when it has been said. I've checked "ON PROOF AND PROGRESS IN MATHEMATICS" and it is not there.
| https://mathoverflow.net/users/29316 | William Thurston's quote? | This quote is from the book "Mathematicians: An Outer View of the Inner World" (Mariana Cook and Robert Clifford Gunning, Princeton University Press, 2009).
<https://www.jstor.org/stable/j.ctt2jc8h2>
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. I’ve loved mat... | 11 | https://mathoverflow.net/users/35306 | 323488 | 139,470 |
https://mathoverflow.net/questions/323481 | 8 | The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N\_\Box: sSet \to Set^{\Box^{op}}$.
* $N\_\Box$ is a right Quillen equivalence, so it preserves weak equivalences between Kan complexes.
* Cisinski also showed that $N\_\Box$ preserves weak equiva... | https://mathoverflow.net/users/2362 | Does the cubical nerve preserve weak equivalences of simplicial sets? | Yes. Proposition 8.4.28 of Cisinski's book *Les préfaisceaux comme modèles des types d'homotopie* states that the cubical nerve functor you describe preserves and reflects weak homotopy equivalences. By tracing through Cisinski's proof, one finds that the same is true for cubical sets with connections.
| 6 | https://mathoverflow.net/users/57405 | 323490 | 139,472 |
https://mathoverflow.net/questions/323471 | 7 | Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
>
> Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $R$?
>
>
>
**Remark 1:** It suffices to show that $S^+/p$ is flat over $R/p$, but I don't know how to see this.... | https://mathoverflow.net/users/115211 | Flatness of the integral closure | No. Take $R=\mathbb{Z}\_p[x,y]/(xy-p^2)$, $S^+=\mathbb{Z}\_p[u,v]/(uv-p)$, and map $R$ into $S^+$ by $x\mapsto u^2$, $y\mapsto v^2$. Note that $R$ is normal, $S^+$ is a finite extension of $R$, étale of degree 2 over $\mathbb{Q}\_p$, but not flat, e.g. because $S^+$ is regular and $R$ isn't (or because $\dim\_{\mathbb{... | 11 | https://mathoverflow.net/users/7666 | 323492 | 139,473 |
https://mathoverflow.net/questions/321344 | 1 | Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation\*}
f\_U(u)=\exp\Big\{{-\frac{1}{4 u^2}}\Big\} \,\frac{\left(8 n u^2-1\right) I\_n\left(\frac{1}{4
u^2}\right)+I\_{n+1}\left(\frac{1... | https://mathoverflow.net/users/103291 | MGF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs | As discussed in the [MO question](https://mathoverflow.net/questions/290092/distribution-of-ratio-between-complex-gaussian-and-chi-square-r-v-s) linked to in the OP, the variable $u$ is defined as $u=R^{1/2}\cos\phi$, where $\phi$ is uniformly distributed in $(0,2\pi)$ and $R=\xi\_{2}/(\xi\_2+\xi\_{2n-2})^2$ is compose... | 1 | https://mathoverflow.net/users/11260 | 323497 | 139,475 |
https://mathoverflow.net/questions/323503 | 5 | Let $\kappa>0$ be a cardinal, and let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality strictly less than $\kappa$. Is it consistent that $$|[\kappa]^{<\kappa}| > \kappa$$ for all cardinals $\kappa>\aleph\_0$?
| https://mathoverflow.net/users/8628 | Is it consistent that $|[\kappa]^{<\kappa}| > \kappa$? | Yes. First, let's just agree that $|[\kappa]^{<\kappa}|=\kappa^{<\kappa}$. One direction is immediate, in the other direction note that every function in $\kappa^{<\kappa}$ is an element of $[\kappa\times\kappa]^{<\kappa}$.
If $2^\kappa=\kappa^{++}$ for all successor cardinals, and there are no inaccessible cardinals... | 16 | https://mathoverflow.net/users/7206 | 323505 | 139,478 |
https://mathoverflow.net/questions/323498 | 11 | In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x\_1\le\cdots\le x\_n$ and $\alpha\in[0,1]^n$ with $\sum\_{i=1}^n \alpha\_i=N\in\{1,2,\ldots,n\}$. Then $$\sum\_{i=1}^n\alpha\_i x\_i\ge\sum\_{i=1}^N x\_i.$$
While I already have found a (quite len... | https://mathoverflow.net/users/81999 | Does anyone recognize this inequality? | (In a way this is a rephrasing of Iosif's answer, but from a different perspective. My main intention is to point out the connection to matroids.)
The inequality follows from:
**Fact.** The polytope $K = \Bigl\{\alpha \in [0,1]^n: \sum\_{i = 1}^n\alpha\_i = N\Bigr\}$ is the convex hull of indicator vectors of subs... | 6 | https://mathoverflow.net/users/35733 | 323525 | 139,484 |
https://mathoverflow.net/questions/99613 | 14 | Let $n \geq 3$. The ring $H^\bullet(\overline{M}\_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D\_{A,B}$ corresponding to partitions $A \sqcup B = \{1,\ldots,n\}$ of the marked points with $|A|, |B| \geq 2$, and where $D\_{A,B} = D\_{B,A}$. All relations... | https://mathoverflow.net/users/1310 | Koszulness of the cohomology ring of moduli of stable genus zero curves | It is: <https://arxiv.org/abs/1902.06318> - this paper also explains how to use the Koszul dual algebra for something, where something is estimating Betti numbers of the free loop spaces of $\overline{M}\_{0,n}$; those, by Gromov & Ballmann-Ziller, allow one to estimate the number of closed geodesics of bounded length.... | 11 | https://mathoverflow.net/users/1306 | 323535 | 139,486 |
https://mathoverflow.net/questions/323543 | 6 | I am interested in obtaining an upper bound for $\prod\_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod\_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\varepsilon}
$$
for any $\varepsilon > 0$. I was wondering is it possibly to prove an upper bound $\ll \log N$ or some pow... | https://mathoverflow.net/users/84272 | How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free? | The $N$ for which this product is largest will be of the form $N = \prod\_{p<y} p$, so that $\log N \sim y$. For this $N$,
$$
\log \prod\_{p\mid N} \bigg( 1+\frac1{\sqrt p} \bigg) = \sum\_{p\mid N} \log \bigg( 1+\frac1{\sqrt p} \bigg) \sim \sum\_{p\mid N} \frac1{\sqrt p} = \sum\_{p<y} \frac1{\sqrt p} \sim \frac{\sqrt y... | 12 | https://mathoverflow.net/users/5091 | 323546 | 139,487 |
https://mathoverflow.net/questions/323548 | 0 | Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation\*}
f\_U(u)=\exp\Big\{{-\frac{1}{4 u^2}}\Big\} \,\frac{\left(8 n u^2-1\right) I\_n\left(\frac{1}{4
u^2}\right)+I\_{n+1}\left(\frac{1... | https://mathoverflow.net/users/103291 | CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs | The cumulative distribution function you are seeking is
$$F\_n(x)=2\int\_0^{x}\exp\left({-\frac{1}{4 u^2}}\right) \,\frac{\left(8 n u^2-1\right) I\_n\left(\frac{1}{4
u^2}\right)+I\_{n+1}\left(\frac{1}{4 u^2}\right)}{4 u^3}\, du.$$
It has the general form
$$F\_n(x)=e^{-1/z}\bigl(A\_n(z)I\_0(1/z)+B\_n(z)I\_1(1/z)\bigr),... | 2 | https://mathoverflow.net/users/11260 | 323553 | 139,489 |
https://mathoverflow.net/questions/323551 | 6 | Let $k$ be a totally real number field. Every quaternion algebra $B$ over $k$ can be written as
$$B = \left(\frac{a,b}{k}\right), $$
for some constants $a,b \in k^\times$. My question is, can I always choose these constants so that $\sigma(a) > 0$ for all real embeddings $\sigma \colon k \hookrightarrow \mathbb{R}$ at ... | https://mathoverflow.net/users/20140 | The Hilbert symbols of quaternion algebras over a totally real field | Yes.
First choose $a$. You can take any $a$ such that $K = k(\sqrt{a})$ is a splitting field of $B$, so by Grunwald--Wang (or elementary congruences and sign conditions) you can enforce $\sigma(a)>0$ at all places where $B$ splits.
Now pick any $b$ such that $(\frac{a,b}{k})\cong B$. Multiplying $b$ by any norm fro... | 8 | https://mathoverflow.net/users/40821 | 323557 | 139,490 |
https://mathoverflow.net/questions/323555 | 0 | Let $X$ be a Banach space. And let $X^\* $ be the dual space of $X$. Let $E\_X$ and $E\_{X^\*}$ denote the extreme points of the unit ball of $X$ and $X^\*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E\_{X^\*}.$ Does that imply $x\in E\_X?$
Can anyone suggest a text to study theory of extreme points of a convex set... | https://mathoverflow.net/users/127674 | Regarding extreme point in a Banach space | Suppose that $x$ satisfies the condition and is not extreme in the unit ball of $X$. It means that we can write $x = \frac{y+z}{2}$ for some $y,z \in B\_{X}$ different from $x$. Since $|\frac{1}{2}f(y)+\frac{1}{2}f(z)|=1$, we get that $f(y)=f(z)$, so $f(y-z)=0$ and $y-z\neq0$. It holds for any $f$ in $E\_{X^{\ast}}$, s... | 2 | https://mathoverflow.net/users/24953 | 323558 | 139,491 |
https://mathoverflow.net/questions/323506 | 0 | I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a\_1$, $a\_2$, $b\_1$ and $b\_2$ where $a\_1>b\_1$ and $a\_2>b\_2$. Prove that
$$
\frac{a\_1a\_2-b\_1b\_2}{a\_1a\_2-1}\geq\frac{(a\_1-b\_1)(a\_2-b\_2)}{(a\_1-1)(a\_2-1)}
$$
The ... | https://mathoverflow.net/users/135985 | Inequality involving product-of-minus vs minus-of-product for positive integers | We need to prove that
$$(a\_1a\_2-b\_1b\_2)(a\_1-1)(a\_2-1)\geq(a\_1-b\_1)(a\_2-b\_2)(a\_1a\_2-1),$$ which is a linear inequality of $b\_1$ and of $b\_2$ and since
$$1\leq b\_1\leq a\_1-1$$ and $$1\leq b\_2\leq a\_2-1,$$ it's enough to prove our inequality for $b\_1\in\{1,a\_1-1\}$ and $b\_2\in\{1,a\_2-1\},$
where $... | 1 | https://mathoverflow.net/users/135040 | 323560 | 139,492 |
https://mathoverflow.net/questions/323539 | 6 | I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read after Brin and Stuck’s very good general introduction? I like pretty much all the fields presented so far, especially to... | https://mathoverflow.net/users/132446 | Reference request: Dynamical systems | There is an encyclopedic book of A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995 which covers most of the subject, and requires minimal background.
Among the more specialized, smaller... | 12 | https://mathoverflow.net/users/25510 | 323563 | 139,493 |
https://mathoverflow.net/questions/321819 | 9 | Let $X$ be a normal variety over a finite field $F\_q$. Fix a prime number $l$ relatively prime to $q$. Let $\sigma$ be a irreducible lisse $l$-adic sheaf on $X$ whose determinant has finite order. It has been proved by L. Lafforgue that for every closed point $x\in |X|$, the roots of the polynomial $\mathrm{det}\_{\si... | https://mathoverflow.net/users/nan | Deligne conjecture without Langlands correspondence | I do not know any proof independent of the Langlands correspondence.
Anna Cadoret gave recently a Bourbaki talk on related works:
<http://www.bourbaki.ens.fr/TEXTES/Exp1156-Cadoret.pdf>
| 13 | https://mathoverflow.net/users/136017 | 323564 | 139,494 |
https://mathoverflow.net/questions/323522 | 4 | I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group.
Let $A$ be a noetherian graded $K$-algebra over a ring $K$. We write $C$ for the category of noetherian $A$-modules. Let $K\_0 (C)$ be the Grothendieck group... | https://mathoverflow.net/users/30211 | Graded Grothendieck Group and Hilbert Polynomial | Your construction is basically the natural homomorphism of $\mathbb{Z}[x^{\pm 1}]$-modules
$$K\_0(A\mathsf{-grMod}) \to \left\{\text{formal "Laurent series"} \sum\_{i=k}^\infty a\_i x^i : k\in\mathbb{Z}, a\_i\in K\_0(K\mathsf{-mod})\right\},$$
sending the class $[M]$ of a graded $A$-module $M=\bigoplus\_i M\_i$ to $\su... | 3 | https://mathoverflow.net/users/3041 | 323565 | 139,495 |
https://mathoverflow.net/questions/323348 | 1 | We consider the two distributions
$$
p\_t = p\_0 \* N(0, tI),\quad q\_t = q\_0 \* N(0, t I),
$$
where $\*$ denotes the convolution between two densities, while $p\_0$ and $q\_0$ have the same mean and variance. In particular, we assume that $q\_0$ is $N(0, I)$. In other words, we consider two random variables
$$
X\_t =... | https://mathoverflow.net/users/82358 | Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers | Write the KL divergence in terms of the [differential entropy](https://en.wikipedia.org/wiki/Differential_entropy) of the random variables $X\_t$ and $Y\_t$; the result quickly follows. Indeed, since $Y\_t \sim \mathcal{N}(0,1+t)$, we have \begin{align\*}
\operatorname{KL}(p\_t, q\_t) &= - h(p\_t) + \frac{1}{2} \int \f... | 1 | https://mathoverflow.net/users/64449 | 323569 | 139,497 |
https://mathoverflow.net/questions/323585 | 4 | Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The *von Neumann bicommutant theorem* says:
**Theorem.** Suppose that $\mathcal{A}$ is a $C^\*$-subalgebra of $\mathcal{B}(H)$ and that $\mathcal{A}$ contains the identity operator. If $\mathcal{A}$ is closed with respec... | https://mathoverflow.net/users/102946 | Bicommutant theorem for commutative operator algebras | The answer is negative. Consider the Hardy space $H^{\infty}(\mathbb{T})$ consisting of holomorphic functions admitting bounded extension to the unit circle and view it is a subalgebra of $B(L^{2}(\mathbb{T})$. Let us compute the commutant. Denote the operator of multiplication by $z$ by $M\_z$. If $T$ belongs to the c... | 12 | https://mathoverflow.net/users/24953 | 323593 | 139,507 |
https://mathoverflow.net/questions/323595 | 8 | If $G$ is a nonabelian finite simple group, $Aut(G)$ certainly contains a subgroup isomorphic to $G$, namely $Inn(G)$. Must this be the only subgroup of $Aut(G)$ isomorphic to $G$?
I can prove this in many cases using the smallness of $Out(G)$, but I'm wondering if this is a general fact.
| https://mathoverflow.net/users/88840 | For nonabelian finite simple $G$, does $Aut(G)$ have a unique subgroup isomorphic to $G$? | [Schreier's conjecture](https://en.wikipedia.org/wiki/Schreier_conjecture) (a theorem provable with the classification of finite simple groups) tells you that $Out(G)$ is always solvable. Therefore $Inn(G)$ is in fact the *unique* subgroup isomorphic to $G$ inside $Aut(G)$.
| 20 | https://mathoverflow.net/users/3041 | 323596 | 139,509 |
https://mathoverflow.net/questions/323559 | 4 | Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).
I am a curious about potential or actual applications to other fields of mathematics and science of flat submanifolds of $\mathbb{R}^{d}$, where $d >3$.
By *flat* I mean locally iso... | https://mathoverflow.net/users/74033 | Applications of flat submanifolds to other fields of mathematics | 1. Crystallographic groups define flat compact manifolds and they are used to describe symmetries of crystals.
2. Flat tori are used in computational physics and chemistry: if you want to investigate dynamics, say of a gas and and for computational reasons you can only consider 1000 particles, you cannot place the part... | 3 | https://mathoverflow.net/users/121665 | 323607 | 139,512 |
https://mathoverflow.net/questions/323214 | 4 | Viterbo's theorem on cotangent bundles $M=T^\*N$ tells you in particular that singular cohomology $H^\*(M)$ gets embedded in $SH^\*(M)$ via the $c^\*$ map. Having a Weinstein manifold (or more generally Liouville manifold) $M$, are there any further examples when this occurs?
| https://mathoverflow.net/users/114985 | The singular cohomology embeds into the symplectic cohomology | There is a Morse-Bott spectral sequence computing the symplectic cohomology of affine varieties which are complements of normal crossing divisors in smooth projective varieties. For simplicity, let's assume that $M=X\setminus D$ is such an affine variety, and $D$ is smooth, then the first page of the spectral sequence ... | 4 | https://mathoverflow.net/users/43423 | 323609 | 139,513 |
https://mathoverflow.net/questions/323597 | 3 | Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
* matrix $B = \mbox{diag}\left(b\_1 ,0,0,\dots,0,b\_n \right)$ with $b\_1, b\_n > 0$.
* matrix $T$ is self-adjoint and positive semidefinite.
What can one say about the lowest... | https://mathoverflow.net/users/119875 | Spectrum of this block matrix | If $\lambda\_\max$ is the greatest eigenvalue of $T$, the least eigenvalue of $A$ is between $-\lambda\_\max$ and $\max(b\_1, b\_n) - \lambda\_\max$.
| 4 | https://mathoverflow.net/users/13650 | 323611 | 139,514 |
https://mathoverflow.net/questions/323604 | 5 | Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i\_s : X\_{\overline{s}}\to X$ the closed immersion and $h : X\_{\overline{\eta}}\to X$ the inclusion of the geometric gen... | https://mathoverflow.net/users/134203 | Nearby cycles and extension by zero | The statement is false without proper assumption.
Consider any "degeneration of a smooth elliptic curve to a nodal curve" and delete a singular point in a special fibre. This will give you a counterexample for the dimension reasons.
**Details:** Start with any proper morphism $f: \mathcal E' \to \operatorname{Spec... | 7 | https://mathoverflow.net/users/115211 | 323615 | 139,516 |
https://mathoverflow.net/questions/323575 | 16 | $\newcommand{\Spec}{\operatorname{Spec}}$
Cross-post from [Math.SE](https://math.stackexchange.com/q/3116195/127263), hopefully people more knowledgeable in the field will see the question here on MO.
It is a well-known fact that a smooth projective variety over $\mathbb Q$ has good reduction almost everywhere, i.e. ... | https://mathoverflow.net/users/30186 | Smooth proper variety over $\mathbb Q$ with everywhere bad reduction | As explained in the comments, there is no such variety.
This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the book:
Bjorn Poonen - Rational points on varieties.
I can't really give a clearer treatment than Poonen. But the proof... | 13 | https://mathoverflow.net/users/5101 | 323620 | 139,518 |
https://mathoverflow.net/questions/323614 | 7 | The [Fibonacci word](http://en.wikipedia.org/wiki/Fibonacci_word) is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S\_n= S\_{n-1}S\_{n-2}$ under initial conditions $S\_0 = 0, S\_1 = 01$. Let us denote this Fibonacci word a... | https://mathoverflow.net/users/20838 | Is the density of 1's in the Fibonacci word uniform? | Yes. By Proposition 2.1.10 in [Lothaire, Algebraic Combinatorics on Words](http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Auteurs/aaaDivers/Lothaire%20-%20Algebraic%20Combinatorics%20On%20Words.pdf), if $u$ is any substring of the Fibonacci word then
$$\left| \frac{\mbox{number of $1$'s in $u$}}{\mbox{length o... | 10 | https://mathoverflow.net/users/297 | 323621 | 139,519 |
https://mathoverflow.net/questions/323610 | 9 | Let $\mathbb F$ be an algebraic closure of the field of order $p$. Let $S=\textrm{Spec}(\mathbb F[[z]])$ with special point $s$ and generic point $\eta$. I'm looking for an example of a smooth projective morphism: $X\rightarrow S$
and integers $i,j$ so that $$h^i(X\_{\eta},\Omega^j\_{X\_\eta})<h^i(X\_{s},\Omega^j\_{X\_... | https://mathoverflow.net/users/791 | Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump | **Update:** The details of this construction are now available in my blog post with Sean Cotner on [Thuses](https://thuses.com/algebraic-geometry/the-torsion-component-of-the-picard-scheme/).
I was recently interested in exactly the same question. But I failed to find any reference where this issue is discussed. So I... | 17 | https://mathoverflow.net/users/115211 | 323622 | 139,520 |
https://mathoverflow.net/questions/323606 | 2 | For a (compact) Riemannian manifold $(M,g)$, can it happen that for a non-zero form $\text{d}^\*\omega$, and a smooth function $f$ such that $\text{d}f \neq 0$, we can have
$$
\text{d}f \wedge \text{d}^\*\omega = 0?
$$
Note that $\*$ denotes the codifferential with respect to $g$.
| https://mathoverflow.net/users/125790 | Vanishing product of a closed and coclosed form on a Riemannian manifold | Yes, this can happen: Take the flat torus $T=\mathbb R^2/\mathbb Z^2$, $$f\colon T\to \mathbb R; x\mapsto \sin(2\pi x),$$
$$g\colon T\to \mathbb R; y\mapsto \sin(2\pi y),$$
and $$\omega=g \text{vol}= g dx\wedge dy.$$ Then, $$df\wedge d^\* g=\pm \cos(2\pi x)\cos(2\pi y) dx\wedge dx=0,$$
where the actual sign does not ma... | 5 | https://mathoverflow.net/users/4572 | 323650 | 139,528 |
https://mathoverflow.net/questions/323639 | 0 | Let $W=(w\_{ij})\_{1 \leq i, j \leq N}$ and $\textbf{v}=(v\_j)\_{1 \leq j \leq N}$ be a random $N\times N$ matrix and N-vector, respectively, where all $w\_{ij}$ are jointly independent and have discrete distributions with non-zero variance. Likewise all $v\_j$ are jointly independent (and independent of $w\_{ij}$) and... | https://mathoverflow.net/users/136048 | Probability eigenvectors of discrete random matrix are orthogonal to discrete random vector | If there is no connection between the distributions for different $N$, it's not true.
Suppose $w\_{ij}$ and $v\_i$ all have Bernoulli distributions with parameter $1 - 2^{-N}$. Then with high probability, all entries of $\bf W$ and $\bf v$ are $1$, and $\bf v$ is orthogonal to all but one of the eigenvectors of $\bf W$... | 0 | https://mathoverflow.net/users/13650 | 323651 | 139,529 |
https://mathoverflow.net/questions/323638 | 3 | Just curious: This monday, I had an exam in Knowledge Processing. They asked what's the problem with FOL (compared to propositional), and I gave the textbook answer that iterating functions gives infinitely many ground terms and makes it undecidable. And since I'm a bigmouth, lavedida, I added that I strongly suspect t... | https://mathoverflow.net/users/11504 | Restricting First Order Logic to make it decidable | There are various decidable fragments of FOL. Beyond prefix classes which Emil Jeřábek mentioned, other common fragments are:
* Two variable logics. The fragment of FOL in which only two variables x and y are allowed is decidable. There are extensions of this logic which are decidable, e.g. with counting quantifiers ... | 4 | https://mathoverflow.net/users/136065 | 323655 | 139,530 |
https://mathoverflow.net/questions/323658 | 8 | Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows \*)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles.
Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)$ and stack $\underline{M}$ whose objects are smooth maps to $M$.
Given Lie group $G$, we have two Lie groupoids asso... | https://mathoverflow.net/users/118688 | Stack associated to Lie group and manifold | $\underline{G}$ is the homotopy loop space of $BG$.
More precisely, the two terminal maps $G\rightarrow pt$ and $G\rightarrow pt$
yield a weak equivalence $\underline{G} \rightarrow pt\times\_{BG} pt$,
where the right side denotes the homotopy pullback of the diagram $pt\rightarrow BG\leftarrow pt$
and $pt$ denotes t... | 5 | https://mathoverflow.net/users/402 | 323664 | 139,533 |
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