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https://mathoverflow.net/questions/351410 | 2 | Let $k$ be an algebraically closed field with $\mathrm{char}(k)=p>0$. Let $U$ be a connected unipotent algebraic group over $k$.
**Question:** When $p$ is big enough, is it true that $Z\_U(u)$ is connected for any $u\in U$, or at least $u\in Z\_U(u)^o$ for any $u\in U$?
**Remark:** This is true when $U$ is the uni... | https://mathoverflow.net/users/56217 | On components of centralisers in unipotent groups | The following is a counterexample which can be defined for arbitrarily large $p$'s.
Consider $U=\left\{ \begin{pmatrix}1&a&b\\&1&a^p\\&&1\end{pmatrix}:a,b\in k\right\}\subseteq\mathrm{GL}\_3(k)$ and take $u=u\_\lambda=\begin{pmatrix}1&\lambda\\&1&\lambda\\&&1\end{pmatrix}$ with $0\ne \lambda\in\mathbb{F}\_p$ (i.e. $\... | 4 | https://mathoverflow.net/users/14443 | 351414 | 148,629 |
https://mathoverflow.net/questions/351346 | 4 | I have a question about unipotent group actions. I was referred to Rosenlicht's papers, but I had trouble getting much out of these because I don't understand the old algebraic geometry language very well. From what I can tell Rosenlicht's results do not imply what I'm looking for. I expect the answer or a reference sh... | https://mathoverflow.net/users/38145 | Rosenlicht's theorem and fundamental domain for unipotent group acting on $\mathbb A_k^n$ | The affirmative answer to the first two question is indeed well-known. The existence of the section $W$ boils down to the vanishing of $H^1(X,\mathbf G\_a)$ on an affine variety $X$.
A generic $n-d$-dimensional subspace will intersect a $G$-orbit in $D$ points where $D$ is the degree of that orbit. So, the answer to ... | 3 | https://mathoverflow.net/users/89948 | 351419 | 148,630 |
https://mathoverflow.net/questions/351424 | 6 | Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?
| https://mathoverflow.net/users/68983 | Conjectures and open problems in representation theory | The [Clemens conjecture](https://en.wikipedia.org/wiki/Enumerative_geometry#Clemens_conjecture) in enumerative geometry: a general quintic threefold has only finitely many rational curves in each positive degree.
| 6 | https://mathoverflow.net/users/13268 | 351428 | 148,632 |
https://mathoverflow.net/questions/351423 | 16 | For every positive real number $x$ we define $$E(x)= \int\_0^{\infty} x^t/t!\,\mathrm dt$$
where $t!=\Gamma(t+1)$. This is motivated by classical exponential function.
Is this function well defined (the problem of convergence)? Is there a real analytic extention of $E$ to all real numbers? What about a holomorphic ex... | https://mathoverflow.net/users/36688 | An analogue of the exponential function by replacing infinite series with improper integral | (Some obvious properties of $E$; too long for a comment, though).
The holomorphic extension of $E$ to $\mathbb{C} \setminus (-\infty, 0]$ (in fact, to the entire Riemann surface of the complex logarithm) is given by $$E(x) = \int\_0^\infty \frac{\exp(t \log x)}{\Gamma(t+1)}\, dt,$$ where $\log$ denotes the principal ... | 19 | https://mathoverflow.net/users/108637 | 351430 | 148,634 |
https://mathoverflow.net/questions/351422 | 1 | I am studying the matrix Completion problem, as well as its SDP relaxation. However, I am having trouble deriving the final SDP form of the matrix completion problem. I will give some background, which mainly comes from (ZHA) below.
Nuclear Norm Minimization
-------------------------
The matrix completion rank mini... | https://mathoverflow.net/users/128752 | Matrix Completion SDP Relaxation and Duality | The derivation of this dual is provided in example 8.8 "Sum of singular values revisited" of [section 8.6 "Semidefinite duality and LMIs" of Mosek Modeling Cookbook 3.2.1](https://docs.mosek.com/modeling-cookbook/duality.html#semidefinite-duality-and-lmis).
| 2 | https://mathoverflow.net/users/75420 | 351435 | 148,636 |
https://mathoverflow.net/questions/351413 | 0 | Let $X$ be a measure space, and suppose $\mu\_i$ are probability measures on $X$ that are absolutely continuous with respect to another probability measure $\mu$. Is strong convergence of $\mu\_i$ to $\mu$ equivalent to convergence in measure (wrt $\mu$) of the Radon nikodym derivatives $\frac{d\mu\_i}{d\mu}$ to $1$?
... | https://mathoverflow.net/users/132446 | Is strong convergence of measures equivalent to convergence in measure of the Radon Nikodym derivatives? | Let $A\_n:=\{x\colon f\_n(x)\le1\}$ and $B\_n:=\{x\colon f\_n(x)>1\}$, where $f\_n:=\frac{d\mu\_n}{d\mu}$. Then the total variation of $\mu\_n-\mu$ is
$$\|\mu\_n-\mu\|=\int\_{B\_n}(f\_n-1)d\mu+\int\_{A\_n}(1-f\_n)d\mu=2\int\_{A\_n}(1-f\_n)d\mu\to0$$
by dominated convergence if $f\_n\to1$ in measure wrt $\mu$; the latt... | 1 | https://mathoverflow.net/users/36721 | 351436 | 148,637 |
https://mathoverflow.net/questions/351357 | 0 | Let $A$ be a non-negative (entrywise) matrix such that $A(1,1)>0$. Set $u=(1,0,0,...,0)^T$. Is it always true that there exists a non-negative eigenvector $v$ of $A$ such that $\lim\_{n\rightarrow\infty}\frac{A^nu}{||A^nu||\_1}=\frac{v}{||v||\_1}$?
| https://mathoverflow.net/users/130361 | Matrix iteration for non-negative matrices. Does it converge to some eigenvector? | The statement is not true. Let $a>1$ and define
\begin{equation}
A:=\begin{bmatrix}1&0&0\\1&0&a\\0&a&0 \end{bmatrix}.
\end{equation}
Suppose there is $v\in\mathbb{R}^n\backslash\{0\}$ such that $\lim\_{n\rightarrow\infty}\frac{A^nu}{||A^nu||\_1}=\frac{v}{||v||\_1}$. By a trivial induction argument we can prove tha... | 1 | https://mathoverflow.net/users/130361 | 351448 | 148,638 |
https://mathoverflow.net/questions/351278 | 3 | I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E\_p$ be the fiber over $p$. Then $P(E)$, the projectivization of $E$ is a vector bundle with fiber $P(E\_p)\colon=\{\text... | https://mathoverflow.net/users/151571 | Chern classes of complex vector bundle | Bertram already mentioned this in the comments but I thought I'd write an answer for completness's sake.
The Leray-Hirsch theorem says that $H^{\*}P(E)\cong H^{\*}M\otimes H^{\*}(Fiber)\ \ $ $\textit{as $H^{\*}M$ modules}$.
So if $x$ is the first chern class of the tautological line bundle over $P(E)$, there's no r... | 1 | https://mathoverflow.net/users/148857 | 351467 | 148,643 |
https://mathoverflow.net/questions/351469 | 0 | Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu\_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ positive definite and $\nu\_{\mu,\Sigma}$ a normalizing constant to turn $d\lambda$ into a probability measure.
Let $m$... | https://mathoverflow.net/users/119875 | Sign of expectation value | The expectation is indeed never strictly positive: it is equal to zero.
The density of $\lambda$ is proportional to $\exp(-\tfrac{1}{2} \langle (x - m), A^{-1} (x - m)\rangle)$, where $A = \tfrac{1}{2} (\Sigma^{-1} + \operatorname{Id})^{-1}$ is a positive definite matrix. Thus, $Y = X - m$ is a centred Gaussian vecto... | 1 | https://mathoverflow.net/users/108637 | 351471 | 148,645 |
https://mathoverflow.net/questions/351480 | 1 | I am looking for an example of a factor $f\colon (X,T) \to (Y,T)$ between topological dynamical systems, where $(X,T)$ is a minimal subshift and $Y$ a connected topological space such that $(Y,T)$ is not a group rotation. Note that any eigenvalue $\lambda = \exp(2\pi i\alpha)$ of $(X,T)$ gives us a factor $(S^1,+\alpha... | https://mathoverflow.net/users/134135 | Example of connected factor of symbolic system that is not a rotation | Let $\alpha$ be irrational, let $Y$ be the two-dimensional torus equipped with the map $S(u,v)=(u+\alpha,v+u)$. Then the action of $S$ on $Y$ is minimal. Now partition the torus into two pieces, say $A\_0=S^1\times[0,\frac 12)$ and $A\_1=S^1\times[\frac 12,1)$ and let $j(y)=0$ if $y\in A\_0$ and $j(y)=1$ if $y\in A\_1$... | 1 | https://mathoverflow.net/users/11054 | 351485 | 148,650 |
https://mathoverflow.net/questions/351271 | 2 | Let $M$ be a $1$-dimensional complex manifold. Let $A$ be the space of all holomorphic functions $f:M\to \mathbb{C}$ such that either $f$ is a constant function or every level set $f^{-1}(c)$ is a finite (probably empty) set.
Is $A$ an algebra of functions? Is its closure, with respect to topology of uniform convergenc... | https://mathoverflow.net/users/36688 | A generalization of polynomial algebra on a Riemann surface | Counterexample to the first question ("is $A$ an algebra of functions?"):
Let $M$ be a vertical strip such as {$x + iy : 0 < x < 1$},
and define $f\_1,f\_2$ as the restriction to $M$ of the
entire functions
$$
f\_1(z) = \exp((1+i)z), \quad f\_2(z) = \exp((1-i)z).
$$
Then $f\_1,f\_2 \in A$ but $f\_1 f\_2 \notin A$.
In... | 4 | https://mathoverflow.net/users/14830 | 351488 | 148,651 |
https://mathoverflow.net/questions/351487 | 3 | While doing some exercises in Lie groups, I see that the Lorentz group $O(1,3)$ has four connected components and $\pi\_0(O(1,3))$ is the Klein four-group $\mathbb{Z}/2 \times \mathbb{Z}/2$. Not only that, but I can find explicit elements representing each connected component $\{1,a,b,ab\}$ which form a subgroup isomor... | https://mathoverflow.net/users/151664 | $\pi_0(G)$ as a subgroup of a Lie group $G$ | No. The first example that comes to mind is $\text{Pin}(2) \subset S^3$, given as $S^1 \cup jS^1$. This group has two components, but every element of $jS^1$ squares to $-1 \in S^1$. Thus every element of the non-identity component has order 4.
So there is no section of the map $\text{Pin}(2) \to \Bbb Z/2$ which sen... | 10 | https://mathoverflow.net/users/40804 | 351489 | 148,652 |
https://mathoverflow.net/questions/351398 | 4 | Let $\mathsf{Grp}$ be the category of groups. A [bifunctor](https://en.wikipedia.org/wiki/Functor#Bifunctors_and_multifunctors) $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an *addition bifunctor* if:
* $A(C\_n,C\_m) \simeq C\_{n+m}$,
* $A(C\_0,G) \simeq A(G,C\_0) \simeq G$,
for every group $G$ and ever... | https://mathoverflow.net/users/34538 | Existence of an addition bifunctor for the category of groups | The answer is no. Notice that $A(-,C\_1)$ is a functor $F : \mathsf{Grp} \to \mathsf{Grp}$ with $F(C\_n) \cong C\_{n+1}$. But there is no such $F$. There is a split monomorphism $C\_1 \to C\_2$, hence $F$ would induce a split monomorphism $C\_2 \to C\_3$, contradiction.
| 6 | https://mathoverflow.net/users/2841 | 351493 | 148,653 |
https://mathoverflow.net/questions/351507 | 1 | Does every [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) on $\omega$ contain an infinite directed path that doesn't visit any vertex twice?
| https://mathoverflow.net/users/8628 | Infinite directed paths in tournaments on $\omega$ | If $x\to y\to z\ldots$ and $x\leftarrow y\leftarrow z\leftarrow\dots$ are both called infinite paths, then yes. For two vertices $x<y$ color an edge $xy$ of the complete graph on $\omega$ red or blue in dependence of the direction of $xy$ in the tournament. By infinite Ramsey theorem, there exists an infinite monochrom... | 2 | https://mathoverflow.net/users/4312 | 351508 | 148,659 |
https://mathoverflow.net/questions/351514 | 3 | Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C\_F\sigma(F,m)m + O\_{F,\varepsilon}(m^{\frac{3}{4} + \varepsilon}),$$ where $C\_F$ is a positive constant depending only on $F$ and $... | https://mathoverflow.net/users/122635 | On quadratic forms in four variables | It follows from Deligne's bound for the Hecke eigenvalues of weight $2$ holomorphic cusp forms (which is really Eichler's theorem in this special case) that the error term is $O\_{F,\varepsilon}(m^{\frac{1}{2} + \varepsilon})$. Indeed, as proved by Siegel, the main term is the Eisenstein contribution of the underlying ... | 4 | https://mathoverflow.net/users/11919 | 351516 | 148,661 |
https://mathoverflow.net/questions/351502 | 5 | For $\mathsf{Grp}$ the category of groups, a [bifunctor](https://en.wikipedia.org/wiki/Functor#Bifunctors_and_multifunctors) $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a *multiplication bifunctor* if:
* $M(C\_n,C\_m) \simeq C\_{nm}$,
* $M(C\_1,G) \simeq M(G,C\_1) \simeq G$,
for every group $G$ and eve... | https://mathoverflow.net/users/34538 | Existence of a multiplication bifunctor for the category of groups | No. $C\_1$ is a retract of $C\_2$, so $M(C\_2,C\_1)\simeq C\_2$ would have to be a retract of $M(C\_2,C\_2)\simeq C\_4$, which it isn't.
| 8 | https://mathoverflow.net/users/22989 | 351518 | 148,662 |
https://mathoverflow.net/questions/351524 | 0 | If a compact group $G$ acts on vN algebra factor $M\subset B(L^{2}(M))$, what would be the index of subfactor $[M^{G}:M]$? KIndly explain the answer.
| https://mathoverflow.net/users/145907 | Index of a particular subfactor | Case $G=\mathbb Z/2$, generated by $g\in G$:
In that case, the action of $G$ on $M$ induces a grading $M=M\_0 \oplus M\_1$ where $M\_0=M^G=\{m\in M: gm=m\}$ and $M\_1=\{m\in M: gm=-m\}$.
Similarly, the action of $G$ on $L^2M$ induces a grading $L^2M=(L^2M)\_0\oplus (L^2M)\_1$. The subspace $(L^2M)\_0$ is isomorphic t... | 1 | https://mathoverflow.net/users/5690 | 351532 | 148,666 |
https://mathoverflow.net/questions/351535 | 0 | For a function $f:[0,1]\to\mathbb{R}$, define
$$ V(f)=\sup\_{0=x\_0<x\_1<\ldots<x\_n=1}\sum\_{i=1}^{n}|f(x\_n)-f(x\_{n-1})|.
$$
For $f$ with integrable derivative, the definition coincides with
$V(f)=\int\_0^1|f'(x)|dx$.
Now every continuous $f:[0,1]\to\mathbb{R}$ is uniformly approximable by a $C^\infty$ function (W... | https://mathoverflow.net/users/12518 | Characterization of bounded variation | This is quite simple: if $\|f - g\|\_\infty \leqslant \epsilon$, then clearly
$$ \sum\_{i = 1}^n |f(x\_i) - f(x\_{i-1})| \leqslant \sum\_{i = 1}^n |g(x\_i) - g(x\_{i-1})| + 2 n \epsilon \leqslant V(g) + 2 n \epsilon.$$
Therefore,
$$ \sum\_{i = 1}^n |f(x\_i) - f(x\_{i-1})| \leqslant \inf\_{g \in C\_\epsilon(f)} V(g) + 2... | 4 | https://mathoverflow.net/users/108637 | 351536 | 148,667 |
https://mathoverflow.net/questions/351546 | 14 |
>
> Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has **fixed distinct** singular values $0<\sigma\_1<\sigma\_2$ and an everywhere positive determinant (which is the product $\sigma\_1\sigma\_2$).
>
>
>
**Must $f$ be affine?**
My assumption is equivalent to $df\_x \in \text{SO}(2) \c... | https://mathoverflow.net/users/46290 | Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine? | **Answer modified on 1 February 2020:**
It's not true 'locally' in the sense that non-affine $f$'s satisfying this system of PDE can be constructed on some open sets in $\mathbb{R}^2$. This first order, determined PDE system is hyperbolic, so there are many local solutions. However, it turns out (see below) that all... | 20 | https://mathoverflow.net/users/13972 | 351550 | 148,669 |
https://mathoverflow.net/questions/351495 | 3 | I am looking for a function $\phi(x)$ such that
$\mathbb{E}\_{x\sim\mathcal{N}(0,1)}[\phi(x)^2] = \mathbb{E}\_{x\sim\mathcal{N}(0,1)}[\phi'(x)^2]$.
Obvious solutions are $\phi(x) = x$ and $\phi(x) = \exp(x)$. But do you know any other non-trivial solution?
| https://mathoverflow.net/users/151671 | When do $\phi^2$ and $\phi’^2$ have the same expectation under a Gaussian random variable? | In fact, every reasonable function can be made into an example by adding an appropriate constant.
I'll write $Z$ for a standard Gaussian random variable. Recall the Gaussian Poincaré inequality:
>
> **Theorem.** For every $f \in C^1(\mathbb{R})$ we have $\operatorname{Var}[f(Z)] \le E[f'(Z)^2]$.
>
>
>
Equiva... | 6 | https://mathoverflow.net/users/4832 | 351553 | 148,671 |
https://mathoverflow.net/questions/351418 | 3 | I have a sum of positive random variables, they are not identically distributed, but even that case would be interesting. They are not necessarily independent, but I already have a concentration bound for the individual random variables (that I got using one of the standard methods, such as Chernoff bound, Method of bo... | https://mathoverflow.net/users/128129 | Concentration of sum of concentrated random variables | There is a bad news and a good news. The bad one is that if you have no information other than that the probability of the $\varepsilon$-deviation is at most $p$ for each variable, then you can hardly do anything better than the union bound. Indeed, consider the random variables $X\_i$ that are exactly $E[X\_i]$ with p... | 5 | https://mathoverflow.net/users/1131 | 351554 | 148,672 |
https://mathoverflow.net/questions/351549 | 1 | I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems.
I would appreciate any example or also any other theory (not necessarily rough paths) that could give a formal framework to deal with non-Markovianity.
My main motivation is... | https://mathoverflow.net/users/151707 | Rough paths theory for Non-Markovian processes | It really depends on what sort of non-Markovian equations you have in mind, but it does certainly allow you to give solution theories for SDEs driven by fractional Brownian motion with Hurst parameter $H>1/4$.
| 3 | https://mathoverflow.net/users/38566 | 351556 | 148,673 |
https://mathoverflow.net/questions/351558 | 21 | Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P\_{m,n}(t) := \sum\_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $1\leq m \leq n \leq 80$ this polynomial has the property that all of its roots are real (negative, of course).
It s... | https://mathoverflow.net/users/147861 | Real rootedness of a polynomial | If you have two polynomials $f(x)=a\_0+a\_1x+\cdots a\_mx^m$ and $g(x)=b\_0+b\_1x+\cdots+b\_nx^n$, such that the roots of $f$ are all real, and the roots of $g$ are all real and of the same sign, then the Hadamard product
$$f\circ g(x)=a\_0b\_0+a\_1b\_1x+a\_2b\_2x^2+\cdots$$
has all roots real. This was proven original... | 24 | https://mathoverflow.net/users/2384 | 351562 | 148,676 |
https://mathoverflow.net/questions/351511 | 4 | For any natural number $k \in \omega$ and any set $X$ the set $$T \subseteq \bigcup\_{m \in \omega} (\omega^m)^k \times X^m $$ is a **tree on** $\omega^k \times X$ iff
$$(t\_o, \ldots, t\_k) \in T \: \Rightarrow \forall m \leq |t\_0| \: (t\_0 \upharpoonright m, \ldots, t\_k \upharpoonright m) \in T,$$
i.e. the eleme... | https://mathoverflow.net/users/97463 | Suslin representation of sets and limits to Shoenfield's Absoluteness | I don't know the answer in ZFC, but in ZF + DC the answer is consistently no. If $V = L$ and there is an inaccessible cardinal, then in the associated Solovay model the $\Pi^1\_2$ relation $\{ \langle x,y \rangle : y \notin L[x]\}$ has no uniformization, so it is not Suslin, meaning that it is not $X$-Suslin for any we... | 5 | https://mathoverflow.net/users/1682 | 351568 | 148,679 |
https://mathoverflow.net/questions/351412 | 4 | Let $X\neq \emptyset$ be a set and let $X^X$ denote the collection of all functions $f:X\to X$. We put a binary relation (reflexive and transitive), the *composition preorder* on $X^X$ by setting for $f,g\in X^X$:
$$ f\leq\_{\text{comp}} g \text{ if and only if } \exists h\in X^X (f = h \circ g).$$
It is easy to see th... | https://mathoverflow.net/users/8628 | Does the lattice of partitions map onto the lattice of subsets? | Comments:
I. This question reduces to the question of whether the lattice $\textrm{Eq}(X)$ of equivalence relations on $X$ has an order-preserving map onto the lattice ${\mathcal P}(X)$ of subsets of $X$.
II. The answer to this question is Yes if $X$ is infinite.
III. The argument in the infinite case can be appl... | 3 | https://mathoverflow.net/users/75735 | 351576 | 148,684 |
https://mathoverflow.net/questions/351221 | 1 | Suppose $U$ is an open set in $\mathbb{R}^{n}$ ($n\geq2$) whose complementary is not polar, and $f$ is a real-valued function defined at least on the boundary of $U$. We know that the generalized Dirichlet problem has a solution, i.e., if $f$ is continuous and all boundary points of $U$ are regular, there is a function... | https://mathoverflow.net/users/100746 | A question on the problem of Dirichlet | I found a book that talks about the problem of Dirichlet for unbounded regions. This is Lester L. Helms' book on " potential theory", Springer, 2009, Chapter 5. According to this book, the answer to the first question is yes; i.e. if $f$ is continuous on the boundary of $U$ then $H^{U}\_{f}(x)$ is harmonic and tends to... | 0 | https://mathoverflow.net/users/100746 | 351594 | 148,688 |
https://mathoverflow.net/questions/351555 | 2 | Is the following statement consistent?
$(\star)$ Let $K$ be a separable compact Hausdorff space containing the space $[0,\omega\_1]$ so that the complement $K\setminus[0,\omega\_1]$ is discrete. Then there exists an open neighborhood $U$ of $[0,\omega\_1)$ in $K$ such that $U$ contains no sequences convergent to $\om... | https://mathoverflow.net/users/61536 | Compactifications with remainder $[0,\omega_1]$ and convergent sequences | Here's an example, suggested by Alan Dow. Take a [Hausdorff Gap](https://en.wikipedia.org/wiki/Hausdorff_gap): a pair of sequences $\langle a\_\alpha:\alpha<\omega\_1\rangle$ and $\langle b\_\alpha:\alpha<\omega\_1\rangle$ of infinite subsets of $\mathbb{N}$ such that $a\_\alpha\subset^\*a\_\beta$ and $b\_\alpha\subset... | 3 | https://mathoverflow.net/users/5903 | 351600 | 148,691 |
https://mathoverflow.net/questions/351605 | 4 | I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following.
Let $\mu$ be a finite measure on $\mathbb{R}$. Is there an uncountable set of $p\_i \in [0,\infty)$ and finite Borel measures $\mu\_i$ on $... | https://mathoverflow.net/users/36886 | $L^{\infty}$ as colimit | The answer to this question is YES -- but it is useless!
In fact, a theorem of Valdivia (which you can find, e.g., as [Theorem 6.5.8](https://books.google.com/books?id=IKrsKlzLcKIC&pg=PA195) in the book *Barrelled Locally Convex Spaces* of Bonet and Perez-Carreras) states that given any infinite-dimensional separabl... | 9 | https://mathoverflow.net/users/21051 | 351609 | 148,694 |
https://mathoverflow.net/questions/351615 | 3 | Given two set $A,B$ we write $A\subset^\* B$ if the complement $A\setminus B$ is infinite.
---
A *Hausdorff gap* is a transfinite family $\langle A\_\alpha,B\_\alpha\rangle\_{\alpha\in\omega\_1}$ of infinite subsets of $\omega$ satisfying the following two properties:
$\bullet$ $A\_\alpha\subset^\* A\_\beta\sub... | https://mathoverflow.net/users/61536 | Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection? | Yes. This is a result due to Nyikos and Vaughan from 1983, appearing the paper
*Nyikos, Peter J.; Vaughan, Jerry E.*, [**On first countable, countably compact spaces. I: ((\omega\_ 1,\omega^\*\_ 1))-gaps**](http://dx.doi.org/10.2307/1999547), Trans. Am. Math. Soc. 279, 463-469 (1983). [ZBL0542.54004](https://zbmath.o... | 7 | https://mathoverflow.net/users/18128 | 351620 | 148,696 |
https://mathoverflow.net/questions/351478 | 3 | Let us consider a homotopy fibre sequence of connected spaces $A\rightarrow B\rightarrow C$
and let $K$ be a fixed field. Assume that the homology $H\_{\ast}(A, K)$ is trivial and that $C$ is a nilpotent space (but not simply connected).
Does $H\_{\ast}(B, K)\rightarrow H\_{\ast}(C, K)$ have to be an isomorphism ?
... | https://mathoverflow.net/users/128371 | Homologically trivial fibre | In your situation, the Serre spectral sequence looks as follows
\[H\_p(C,H\_q(A,K))\Rightarrow H\_{p+q}(B,K).\]
The left hand side is the $E^2$ term. There, $H\_q(A,K)$ carries an action of $\pi\_1(C)$ induced by the fibration, and $H\_p(C,H\_q(A,K))$ is the homology with local coefficients. Now, if $A$ has the hom... | 5 | https://mathoverflow.net/users/12166 | 351621 | 148,697 |
https://mathoverflow.net/questions/351617 | 10 | Is it true that any finite graph has a $K\_n$ minor, where $n$ is a minimal vertex degree?
| https://mathoverflow.net/users/148161 | Does minimal degree $n$ imply a $K_n$ minor | *No.*
The [edge-graph of the icosahedron](https://en.wikipedia.org/wiki/Regular_icosahedron#Icosahedral_graph) is regular of degree five, but does not have a $K\_5$ minor because it is planar ([Kuratowski's theorem](https://en.wikipedia.org/wiki/Kuratowski%27s_theorem)).
| 23 | https://mathoverflow.net/users/108884 | 351622 | 148,698 |
https://mathoverflow.net/questions/351597 | 3 | For two random variables $X$ and $Y$ taking values in $\mathbb{R}^m$, the convex distance $d\_c$ is defined as
$$d\_c(X,Y) = \sup\_{h} \lvert \operatorname{E}(h(X)) - \operatorname{E}(h(Y)) \rvert,$$
where the supremum is taken over all indicator functions of measurable convex subsets of $\mathbb{R}^m$.
For $m=1$... | https://mathoverflow.net/users/56931 | Does convergence in law imply convergence in convex distance? | The answer is no. For instance, let $Y$ be uniformly distributed on the unit sphere in $\mathbb R^m$ (not the ball, but the sphere) and, for each natural $n$, let $X\_n:=(1+1/n)Y$. Then the distribution of each of the random vectors $Y,X\_1,X\_2,\dots$ is [non-atomic](https://en.wikipedia.org/wiki/Atom_(measure_theory)... | 1 | https://mathoverflow.net/users/36721 | 351627 | 148,699 |
https://mathoverflow.net/questions/351523 | 3 | Recently I found a concentration inequality for infinite dimensional Gaussian r.v.s in [this paper](https://www.sciencedirect.com/science/article/pii/0047259X86900850). Specifically, Lemma 4 on page 307 states (without a proof) that
>
> There exists a universal constant $M$ such that for each Banach space valued Ga... | https://mathoverflow.net/users/101520 | Gaussian concentration inequality | This inequality is false. E.g., consider the random vector $X\_n:=(Z\_1,\dots,Z\_n)/\sqrt n$ in $\mathbb R^n$ with the Euclidean norm $\|\cdot\|$, where $Z\_1,Z\_2,\dots$ are independent standard normal random variables. Then $E\|X\_n\|^2=1$ and, by the law of large numbers,
$$\|X\_n\|^2=\frac1n\,\sum\_1^n Z\_i^2\to1$$... | 5 | https://mathoverflow.net/users/36721 | 351639 | 148,704 |
https://mathoverflow.net/questions/351642 | 2 | Consider the binary sequence $\{0,1\}^N$ where $N$ is an even integer (for simplicity). Let $M\_k := \{\beta\in \{0,1\}^N \rvert \sum\_{j=1}^N \beta\_j = k\}$ (i.e., $M\_k$ is the set that contains all binary sequences with length $N$ and with exactly $k$ 1's). The question is: **for general $k \le N/2$, is it always p... | https://mathoverflow.net/users/144546 | construct a bijective map between subsets of binary sequence | The answer is "yes, this is always possible." The bijection on ranks $k$ and $N-k$ induced from a symmetric chain decomposition of the Boolean lattice gives you what you're asking for. For basics on this topic, see [these slides](http://math.sjtu.edu.cn/conference/Bannai/2014/data/20141230B/slides.pdf).
| 2 | https://mathoverflow.net/users/25028 | 351643 | 148,705 |
https://mathoverflow.net/questions/351641 | 8 | Let $A$ be a dg- or $A\_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH\_\*(A)$ be the Hochschild homology of $A$.
Suppose that $HH\_n(A)=0$ for all $n \in \mathbb{Z}$. Does this imply that $A$ is the zero category?
If not, then what assumptions can I add to $A$ ... | https://mathoverflow.net/users/59235 | Vanishing of Hochschild homology of a category | This precise question was phrased as the vanishing conjecture in [Hochschild homology and semiorthogonal decompositions](http://arxiv.org/pdf/0904.4330v1). But we now know that there exist so called *(quasi)phantom categories*, which give counterexamples. These are categories with vanishing Hochschild homology, and van... | 14 | https://mathoverflow.net/users/6263 | 351647 | 148,706 |
https://mathoverflow.net/questions/351650 | 0 | Let $U$ be an open set in $\mathbb{R}^{n}$ with $n\geq2$ and $V$ an open set containing the boundary $\partial U$ of $U$. Suppose $u$ is subharmonic on $V$. We know that the generalized solution of Dirichlet problem $H^{U}\_{u}(x)$ exists, i.e. is a harmonic function on $U$ that $\to u(y)$ as $x\to y$, for all regular ... | https://mathoverflow.net/users/100746 | A question on the problem of Dirichlet 2 | The answer is no. E.g., let $U:=\{x\in\mathbb R^n\colon|x|<1\}$, $V:=\{x\in\mathbb R^n\colon1/2<|x|<2\}$ (or $V:=\mathbb R^n\setminus\{0\}$), $u(x):=|x|^{2-n}-1$ for $x\in V$ if $n\ge3$, and $u(x):=-\ln|x|$ for $x\in V$ if $n=2$. Then $u$ is harmonic and hence subharmonic on $V$. However, $u>0=H\_u^U$ on $V\cap U$.
| 1 | https://mathoverflow.net/users/36721 | 351654 | 148,708 |
https://mathoverflow.net/questions/351653 | 11 | As a follow-up to [this question](https://mathoverflow.net/questions/348418/do-spaces-admit-a-weak-cogenerating-set), let $\mathcal C$ be a category and $\mathcal S \subseteq \mathcal C$ a class of objects. Say that $\mathcal S$ is *weakly generating* if the functors $Hom\_{\mathcal C}(S,-)$ are jointly conservative, f... | https://mathoverflow.net/users/2362 | Does the homotopy category of spaces admit a weak generating set? | This paper by Kevin Carlson and Dan Christensen says that the answer to question one is no: *No set of spaces detects isomorphisms in the homotopy category*, arXiv:[1910.04141](https://arxiv.org/abs/1910.04141).
| 13 | https://mathoverflow.net/users/19230 | 351656 | 148,709 |
https://mathoverflow.net/questions/351124 | 0 | Let $d\_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to approximate the sum
$$\alpha=\sum\_{n=1}^{N}d\_n^{-3},$$
with the restricted sum
$$\hat{\alpha}=\sum\_{n\in\mathcal I}d\_n^... | https://mathoverflow.net/users/68835 | Limited sum for whole sum approximation | *(Edited after noticing an error which completely changes the answer.)*
*This is not a complete solution; however, it strongly suggests that the answer is **positive**.*
Let $U$ be a random variable with uniform distribution on $[0, 1]$. The random variable $U^{-3}$ has tail $\mathbb{P}[U^{-3} > x] = x^{-1/3}$, and... | 1 | https://mathoverflow.net/users/108637 | 351662 | 148,712 |
https://mathoverflow.net/questions/351545 | 2 | If we have a system of PDE of the form:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y\_{0}(x),\ x\in\Omega\end{cases}$$
with a unique solution $y=y^f$(mild solution)... | https://mathoverflow.net/users/61629 | Continuity of solution of a parabolic PDE w.r.t. system parameters | This is only a sketch of an argument that can be used. Assume that $F$ is Lipscthitz and let $y\_1,y\_2$ be the solutions corresponding to $f\_1, f\_2$. If $v=y\_2-y\_1$, then $$|v\_t-\Delta v|=|F(f\_2,y\_2)-F(f\_2,y\_1)+F(f\_2, y\_1)-F(f\_1,y\_1)| \le L(|v|+|f\_2-f\_1|)$$ with zero bc and initial value. If $T(t)$ is t... | 1 | https://mathoverflow.net/users/150653 | 351664 | 148,713 |
https://mathoverflow.net/questions/351666 | 2 | Let $X$ be an open domain in $R^n$. Let $E$ be a subspace of $X$ with Hausdorff dimension $m$. Fix $k$ and $p$. What are the optimal assumptions on $m$ and $n$ so that the trivial map $W^k\_p(X) \to W^k\_p(X \setminus E)$ becomes an isomorphism?
I am mostly interested in the case $k = 1$ and $p = 2$, and in that situ... | https://mathoverflow.net/users/151724 | Sobolev spaces complement of Hausdorff codimension 2, restriction theorem | If $E$ is a closed set such that Hausdorff measure $H^{n-p}(E)$ is $\sigma$-finite, then its capacity satisfies $Cap\_p(E)=0$ and it follows that $W^{1,p}(X)=W^{1,p}(X\setminus E)$. Sobolev $W^{1,p}$ functions simply `do not see' sets of capacity zero.
In particular if $E$ is a linear subspace of dimension $m\leq n-p... | 2 | https://mathoverflow.net/users/121665 | 351668 | 148,716 |
https://mathoverflow.net/questions/351657 | 3 | Let me **quote** en.wikipedia about the original Nikodym's example (the definitions above I've written myself just for MO):
*a Nikodym set is a subset of the unit square in $\ \mathbb R ^2\ $ with the complement of Lebesgue measure zero, such that, given any point in the set, there is a straight line that only inters... | https://mathoverflow.net/users/110389 | Generalized Nikodym sets | The answer to Question 1 is **yes**. This is a result of Falconer, Corollary 6.6 in
**K. J. Falconer,**
Sets with prescribed projections and Nikodým sets.
*Proc. London Math. Soc.* (3) 53 (1986), no. 1, 48–64.
The result states as follows:
>
> **Theorem.**
> Let $1\leq m<n$. Then there exists a set $K\subset\... | 2 | https://mathoverflow.net/users/121665 | 351670 | 148,718 |
https://mathoverflow.net/questions/351672 | 3 | Let $(M^n,g)$ be a complete flat Riemannian manifold. Suppose there exists a number $s \in (n-1,n]$ such that for some point $p \in M$
$$
\limsup\_{r \to +\infty} \frac{\text{Vol}\,B(p,r)}{r^s}>0.
$$
Can we prove that $(M^n,g)$ is isometric the Euclidean space?
| https://mathoverflow.net/users/105900 | Volume ratio of complete flat manifolds | Yes, it is true.
Note that $M$ is isometric to a quotient of the Euclidean space $\mathbb{E}^n$ by a totally discontinuous free isometric action of a group $\Gamma$.
Your condition implies that [the soul](https://en.wikipedia.org/wiki/Soul_theorem) of $M$ is a single point.
It follows that $\Gamma$ fixes a point in... | 4 | https://mathoverflow.net/users/1441 | 351682 | 148,722 |
https://mathoverflow.net/questions/351648 | 10 | The irreducible characters of the orthogonal group $O(2N)$ are given by
$$ o\_\lambda(x\_1,x\_1^{-1},...x\_N,x\_N^{-1})=\frac{\det(x\_j^{N+\lambda\_i-i}+x\_j^{-(N+\lambda\_i-i)})}{\det(x\_j^{N-i}+x\_j^{-(N-i)})}$$
I was playing with them as basis for the space of homogeneous symmetric polynomials. I wanted to write ... | https://mathoverflow.net/users/83671 | Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials | The coefficients do depend on $N$. A way to get around this and deal with "universal characters" was found by Koike and Terada ([Young-diagrammatic methods for the representation theory of the classical groups of type $B\_n$, $C\_n$, $D\_n$](https://www.sciencedirect.com/science/article/pii/0021869387900998)).
| 10 | https://mathoverflow.net/users/2807 | 351683 | 148,723 |
https://mathoverflow.net/questions/351587 | 2 | We are searching for the rank 8 elliptic curves with the torsion subgroup Z/6 using newly discovered families similar to Kihara's (Kihara's family is described in <https://arxiv.org/pdf/1503.03667.pdf>).
Today we came across a curve
$[0,8169768624655967629114128598,0,-45178755064731042061208646853636671586905440595... | https://mathoverflow.net/users/95511 | Resolved: Two more generators needed for a Z/6 elliptic curve | Just to give a more complete answer:
```
SetVerbose("cbrank",1);
E := EllipticCurve([0,8169768624655967629114128598,0,\
-451787550647310420612086468536366715869054405951830599,0]);
TwoPowerIsogenyDescentRankBound(E);
/---------------------------------------------------\
| SUMMARY TABLE Step No : 6 ... | 2 | https://mathoverflow.net/users/140496 | 351684 | 148,724 |
https://mathoverflow.net/questions/351652 | 7 | Let $P(n)$ be the statement "any $n$ coloring of $\mathbb{N}$ contains a monochromatic progression $a, a+d, a+2d$ such that $d>a$".
For which $n$ is $P(n)$ true?
It's easy to see that $P(2)$ is true by a simple modification of the color focusing argument that is used in the traditional proof of van der Waerden's th... | https://mathoverflow.net/users/nan | 3 term van der Waerden with large step size | $P(n)$ is false for all $n > 2$. To see this, it suffices to show that $P(3)$ is false, because if we start with a $3$-coloring that witnesses the failure of $P(3)$, then we can always add a few more colors, say by using each of the new colors on just one or two numbers each, to obtain a witness to the failure of $P(n)... | 4 | https://mathoverflow.net/users/70618 | 351685 | 148,725 |
https://mathoverflow.net/questions/351681 | 1 | Let $X\_1,...,X\_n$ be iid observations from $N(0,1)$. Let $\overline{X}=\dfrac{1}{n}\sum\_{i=1}^n X\_i$ and $S^2=\dfrac{1}{n}\sum\_{i=1}^n (X\_i-\overline{X})^2$. Then is it true that $\sqrt{n}\sup\_x |\Phi(\dfrac{x-\overline{X}}{S})-\Phi(x)|\stackrel{p}{\to}
0$? Here $\Phi(.)$ is standard normal cdf.
Seems somethin... | https://mathoverflow.net/users/66278 | Does $\sqrt{n}\sup_x |\Phi(\dfrac{x-\overline{X}}{S})-\Phi(x)|\to0$ in probability? | No, it is not true. First fix any $x$ and consider $\overline{X^2}=\frac1n\sum\_{i=1}^nX\_i^2$, $S^2=\overline{X^2}-\bigl(\overline X\bigr)^2$ and $h(s,t)=\Phi\left(\dfrac{x-s}{\sqrt{t-s^2}}\right)$.
Then use multivariate Delta method to prove that
$$
\sqrt{n} \left(\Phi\biggl(\dfrac{x-\overline{X}}{S}\biggr)-\Phi(x)... | 3 | https://mathoverflow.net/users/150967 | 351690 | 148,727 |
https://mathoverflow.net/questions/351270 | 7 | Fix a prime number $p$. Can there exist a continuous irreducible representation $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{GL}\_3(\mathbb{Q}\_p)$ that is unramified at almost all primes, is de Rham at $p$ and whose Hodge-Tate weights at $p$ are 0, 1 and 3?
| https://mathoverflow.net/users/nan | Irreducible global Galois representation with weights 0, 1, 3? | Here are two arguments for why such a representation $\rho$ cannot exist.
* Automorphic argument: Fontaine and Mazur have conjectured that any irreducible $n$-dimensional geometric representation $\rho$ of $Gal(\overline{\mathbf{Q}} / \mathbf{Q})$ comes from a cuspidal automorphic representation $\pi$ of $GL\_n(\math... | 6 | https://mathoverflow.net/users/2481 | 351699 | 148,733 |
https://mathoverflow.net/questions/351707 | -1 | Working in "[MK](https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory)-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom:
**Axiom of Super-Choice:**$$\forall \ relation \ R \ \exists F \subset R \ (F: dom(R) \to rng(R))$$ Where: $$R \text { is a re... | https://mathoverflow.net/users/95347 | Is Proper Class Choice equivalent to Global Choice? | Yes, this is still equivalent: given any relation $R$, consider the new relation $$R^{bigrows}=\{\langle x,y\rangle: \exists a,b(y=\langle a,b\rangle\wedge \langle x,a\rangle\in R)\}.$$ Basically, $R^{bigrows}$ just "pads out" the rows of $R$ with a dummy coordinate.
But from a choice function $F$ for $R^{bigrows}$ ... | 1 | https://mathoverflow.net/users/8133 | 351711 | 148,735 |
https://mathoverflow.net/questions/351702 | 14 | As far as I know, there is no "official" notation for the left adjoint of a functor $F : \mathcal{C} \to \mathcal{D}$ if it exists. I have seen the notation $F^\*$ sometimes, but this looks only nice when $F$ is already written as $F\_\*$, which is not practical. (This notation is then motivated by direct and inverse i... | https://mathoverflow.net/users/2841 | Notation for "the" left adjoint functor | In EGA 0.1.5.2-3 (from the 1971 Springer edition) the right adjoint and the left adjoint of a functor $F$ are denoted by $F^{\rm ad}$ and ${}^{\rm ad}\!F$, respectively.
| 18 | https://mathoverflow.net/users/11025 | 351714 | 148,736 |
https://mathoverflow.net/questions/351721 | 0 | Let $I\_{n,k} = \frac{(n+k)!}{k!(k-1)!(n-k)!}$. This is a sort of generalization of the [Apéry's numbers](http://mathworld.wolfram.com/AperyNumber.html), with $I\_{n,n} =$ the $n$-th Apéry number. I am studying integrals of the form:
$$f\_u(x)=\int \prod\_{j=1}^u \frac{x+j}{j-x}~dx.$$
Where $u$ is a natural number. For... | https://mathoverflow.net/users/128941 | Evaluation of $\int \prod_{j=1}^u \frac{x+j}{j-x}~dx$ | I may be mistaken, but I get
$$f\_u(x)=\int \prod\_{j=1}^u \frac{x+j}{j-x}~dx= C+ (-1)^ux+\sum\_{w=1}^{u}(-1)^w \log(x-w)I\_{u,w}$$
and this is correct for all $u=1,2,3,...$, so it seems issue 1 is resolved.
| 2 | https://mathoverflow.net/users/11260 | 351724 | 148,738 |
https://mathoverflow.net/questions/351640 | 99 | *This question is [cross-posted](https://academia.stackexchange.com/questions/143374/extent-of-unscientific-or-wrong-papers-in-research-mathematics) from academia.stackexchange.com where it got closed with the advice of posting it on MO.*
---
Kevin Buzzard's slides ([PDF version](http://www.andrew.cmu.edu/user/a... | https://mathoverflow.net/users/50912 | Extent of “unscientific”, and of wrong, papers in research mathematics | "Are most areas safe, or contaminated?"
Most areas are fine. Probably all important areas are fine. Mathematics is fine. The important stuff is 99.99999% likely to be fine because it has been carefully checked. The experts know what is wrong, and the experts are checking the important stuff. The system works. The sys... | 119 | https://mathoverflow.net/users/1384 | 351726 | 148,739 |
https://mathoverflow.net/questions/351709 | 0 | Let $G(V,E)$ be an undirekted $k$-vertex-connected, $k$-regular graph
and let $F$ be an $f$-factor of $G$ consisting of a set of $f$-vertex-connected components, $f<k$.
>
> **Question:**
>
> what is the vertex-connectivity of $G\setminus F$, is it $k-f$, resp., what is the highest lower bound on the resulti... | https://mathoverflow.net/users/31310 | Graph connectivity after deleting an f-factor | Make $G$ from two highly-connected pieces joined by a matching of $k$ edges. It has connectivity $k$. Now take $F$ to be a perfect matching that includes the edges of the cut. $G\setminus F$ is then disconnected. So there is no general lower bound except 0.
| 1 | https://mathoverflow.net/users/9025 | 351729 | 148,740 |
https://mathoverflow.net/questions/351742 | 3 | Let $E$ be a normed space and let $F\subset E^{\*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B\_E$ coincides with the weak topology.
Hence, if $F$ is dense and we have a bounded net $e\_\alpha$ and $e\in E$ such that $\left<e\_\alpha,f\right>\to \left<e,f\right>$, for every $... | https://mathoverflow.net/users/53155 | Criterion for weak convergence of sequences | In a nutshell, no, at least in the separable case. Let $F\subseteq E^\*$ be not norm dense, and with $F$ (norm-) separable. By Hahn-Banach there is $M\in E^{\*\*}$ which is non-zero and annihilates $F$. Let $f\_0\in E^\*$ with $\langle M,f\_0 \rangle=1$.
I shall use Helly's Lemma (which I have failed to find an onlin... | 5 | https://mathoverflow.net/users/406 | 351746 | 148,743 |
https://mathoverflow.net/questions/351747 | -1 | By diagonalization, it is possible to construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$, there exists an index $i \in \mathbb{N}$ such that $r\_i \neq q\_i$ (where $x\_i$ is the $i$'ith digit in $x$).
Can we make a stronger claim, and construct a real number $r \in [0,1]$ such that for... | https://mathoverflow.net/users/151810 | Real number which is different from all rationals | No. Consider the rational numbers 0, 0.111..., 0.222..., 0.333.., ..., 0.888...
Any real number either shares infinitely many digits with one of these, or it has only finitely many digits which are not 9. But then it is after some point only 9s, so is equal to a terminating rational and has infinitely many 0s.
If you... | 2 | https://mathoverflow.net/users/36212 | 351748 | 148,744 |
https://mathoverflow.net/questions/351731 | 6 | Among separable metrizable spaces:
Cantor set is the unique compact zero-dimensional space without isolated points.
$\mathbb Q$ is the unique countable space without isolated points
$\mathbb R \setminus \mathbb Q$ is the unique zero-dimensional, $G\_\delta$-space with no compact neighborhood.
$\mathbb Q ^\omeg... | https://mathoverflow.net/users/95718 | A classification of $G_{\delta\sigma}$ zero-dimensional spaces? | [This paper](https://www.ams.org/journals/tran/1981-264-01/S0002-9947-1981-0597877-9/S0002-9947-1981-0597877-9.pdf) by Van Mill from 1981 gives a characterisation of $\Bbb Q \times \Bbb P$ (where $\Bbb P$ is a common notation for the irrationals) in Thm 5.3:
>
> If $X$ is separable metrisable and zero-dimensional, ... | 5 | https://mathoverflow.net/users/2060 | 351752 | 148,745 |
https://mathoverflow.net/questions/351712 | 3 | **INTRODUCTION**
The *neoclassical [production function](https://en.wikipedia.org/wiki/Production_function)* is the main building block in [neoclassical growth theory](https://www.investopedia.com/terms/n/neoclassical-growth-theory.asp), and consequently the main building block of modern macroeconomic theory. Mathema... | https://mathoverflow.net/users/102651 | Can we characterize the set of neoclassical production functions? | Concerning your Question 1: "Can we characterize the set of neoclassical production functions?" -- This set is already characterized, tautologically, by its definition, as the set of all functions $F$ satisfying Assumptions 1 and 2. There seems to be no reason/way for there to exist a better characterization.
Concer... | 1 | https://mathoverflow.net/users/36721 | 351773 | 148,750 |
https://mathoverflow.net/questions/351710 | 7 | Let $A=(V, E)$ be a finite simple (no loops or multiple edges) graph. Let $G(A)$ be the following nilpotent group of class 2 and exponent $p$ (an odd prime). $G(A)$ as a set is $span(V)+span(E)$ where $span(X)$ is the elementary abelian group of exponent $p$ generated by the set $X$, and the commutator bracket is given... | https://mathoverflow.net/users/nan | Groups and graphs | I think it's time to write an answer.
Let $G = G(A)$, so $|G| = p^{|V|+|E|}$ with $|G'| = p^{|E|}$ and $G'$ and $G/G'$ are both elementary abelian ($p$-groups with that property are called *special* $p$-groups, and it is conjectured that almost all finite groups of order up to some bound are special $2$-groups, but t... | 1 | https://mathoverflow.net/users/35840 | 351779 | 148,752 |
https://mathoverflow.net/questions/351715 | 11 | This is inspired by [this question.](https://mathoverflow.net/questions/351704/which-finite-solvable-groups-have-solvable-automorphism-groups) Is there a description of finite groups without automorphisms of order $2$?
| https://mathoverflow.net/users/nan | Automorphism groups of odd order | New version (existence hinted in previous version): If $G$ is a non-trivial finite (solvable) group of odd order with $\Phi(G) = 1$, then $G$ has an automorphism of order $2$.
It is well-known and easy to check that $F = F(G)$ is a product of minimal normal subgroups of $G$, each an elementary Abelian $p\_{i}$-group ... | 4 | https://mathoverflow.net/users/14450 | 351781 | 148,753 |
https://mathoverflow.net/questions/351772 | 5 | What probability measure(s) maximize the quantity $\iiint\_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$?
The answer appears to be uniform measure, since informally it appears better to have more triangles in the support of $\mu$ which the function $|(x-z)\times(y-z)|$ computes the area of.
Is there an a... | https://mathoverflow.net/users/118731 | Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle | Yes, it is true for the circle (though the reason is not quite the one you suggested). We shall consider the discrete version of the problem, which is to put some odd number $n\ge 3$ of points (think of them as of being assigned the probability of $1/n$ each) on the circle so that the sum of triangle areas is maximized... | 6 | https://mathoverflow.net/users/1131 | 351803 | 148,760 |
https://mathoverflow.net/questions/351806 | 4 | This question is probably not research level that's why I asked it previously on [MSE](https://math.stackexchange.com/questions/3523959/classifying-space-bg-and-contractable-space-eg) a week ago. Unfortunately it doesn't get much attention there and I thought I would try it here.
Choose a arbitrary discrete group $G... | https://mathoverflow.net/users/108274 | Classifying space BG and contractable space EG | The easiest way to construct an explicit contracting homotopy
is to observe that EG is the geometric realization of the nerve of the groupoid G//G,
which has G as its set of objects and exactly one morphism between any pair of objects.
The nerve functor sends equivalences of groupoids
to homotopy equivalences of simp... | 17 | https://mathoverflow.net/users/402 | 351811 | 148,762 |
https://mathoverflow.net/questions/351809 | 1 | Let $X\_1,\dots,X\_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell\_2$ ball, but outside the ball of radius $r$ for some $r \in (0,1)$, i.e. $r \leq \|X\_i\| \leq 1$ . Let $P$ be a vector in the probability simplex $\Delta\_n$ with $P\_i>0$ for all $i$. Consider the second moment m... | https://mathoverflow.net/users/116451 | Is there a bound on the norm of the product of second moment matrix with random vector? | $\newcommand\Si{\Sigma}$
$\newcommand\X{\mathbf X}$
The answer is no. Indeed, let $p\_i:=P\_i$, $p:=P$, $\X:=(X\_1,\dots,X\_n)$, and $\Si\_\X:=\Si(p)$. At least one of the vectors $\Si^{-1}X\_j$ is nonzero, for some $j$, because otherwise the matrix
$$I=\Si\_\X^{-1}\Si\_\X=\sum\_1^n p\_i\Si\_\X^{-1}X\_i X\_i^T$$
would ... | 2 | https://mathoverflow.net/users/36721 | 351815 | 148,765 |
https://mathoverflow.net/questions/351807 | 10 | Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $X$ is nowhere dense.
| https://mathoverflow.net/users/95718 | Is the complement of a zero-dimensional subset of the plane path-connected? | If the zero-dimensional set $X$ is not closed, then the answer is "no".
To construct a suitable example, take any open bounded neighborhood $U\subset\mathbb R^2$ of zero, whose boundary $\partial U$ does not contain a topological copy of $[0,1]$. For example, for $U$ we can take a bounded connected component of the c... | 14 | https://mathoverflow.net/users/61536 | 351820 | 148,768 |
https://mathoverflow.net/questions/351614 | 7 | (Cross-posted from mathematics stackexchange.)
Fix a finite dimensional vector space $V$ over a field of *characteristic zero*, and let $R=Sym(V[1])$ be the free graded commutative algebra generated by $V$ in cohomological degree $-1$, but thought of as a formal associative dg algebra (we forget that it is commutativ... | https://mathoverflow.net/users/151748 | Are exterior algebras intrinsically formal as associative dg algebras? | Here are examples of nontrivial $A\_\infty$-structures extending the product on $\operatorname{Sym}(\mathbb R^3[-1])$ and $\operatorname{Sym}(\mathbb R^5[-1])$, respectively. Below the fold, I have kept my original answer, which got the main ideas right but almost all degrees wrong.
Fix coordinates $\xi\_1,\xi\_2,\xi... | 8 | https://mathoverflow.net/users/35687 | 351832 | 148,772 |
https://mathoverflow.net/questions/351833 | 1 | This is a step of a proof in the book *Variational Problems in Geometry* by Seiki Nishikawa. I will ignore the background and change some of the statements and notations for simplicity.
>
> Let $(M,g)$ be a smooth Riemannian manifold. Suppose $w:M\to\mathbb R^q$ is an isometric embedding. Let $N$ be a tubular neigh... | https://mathoverflow.net/users/143284 | How does this orthogonality follow from the map being an isometry? | There are a number of ways to see this. One way is to take the covariant derivative of the isometric embedding equation $\partial\_iu\cdot\partial\_ju = g\_{ij}$ and "differentiate by parts". The calculation below is with respect to local coordinates, and $u$ is treated as a $q$-tuple of scalar real-valued functions. T... | 2 | https://mathoverflow.net/users/613 | 351843 | 148,774 |
https://mathoverflow.net/questions/351764 | 5 | The following question arises from trying to understand Lemma 1.3(ii) of [arXiv:math/0405063](https://arxiv.org/pdf/math/0405063.pdf). I believe a particular case of the proof (and in fact I think the proof is essentially equivalent to this claim) is:
>
> Let $G$ be a locally compact group. Let $C,D$ be cosets (*no... | https://mathoverflow.net/users/406 | Empty interior of union of cosets? | This is false. Take the (compact abelian) group $G=(\mathbf{Z}/2\mathbf{Z})^\mathbf{N}$ and let $H$ be a dense subgroup of index 2 (there are many, since $G$ has only countably many closed subgroups of index 2 but has $2^c$ subgroups of index $2$, and clearly a subgroup of index 2 is either closed or dense). Then $G=H\... | 5 | https://mathoverflow.net/users/14094 | 351846 | 148,775 |
https://mathoverflow.net/questions/351840 | 2 | Here: <https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf>, on pages: $ 1 $ and $ 2 $, we find the following paragraph:
For any scheme of finite type over a ground field and any integer $ k>0 $, we will define the so-called Chow groups $ A\_k (X) $ whose elements are formal linear combi... | https://mathoverflow.net/users/89900 | Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $ | If $C$ is a smooth projective curve of genus $g \geq 3$ and $J(C)$ is the Jacobian of $C,$ then an Abel curve $C \subset J(C)$ is not algebraically equivalent to its image $-C$ under the negation automorphism, even though $C$ is homologically equivalent to $C.$ This was proved by Ceresa in the paper
<https://www.jsto... | 6 | https://mathoverflow.net/users/5496 | 351849 | 148,776 |
https://mathoverflow.net/questions/347009 | 0 | I am looking for a generalisation of a modular form that transforms as something like:
$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$
I understand this cannot be literally true, as the multiplier c^k is not a root of unity, but does something like this arise in the context of modular forms? (or generalis... | https://mathoverflow.net/users/41940 | Generalisation of modular forms | The transformation should conform to the cocycle condition.
That is $f(\gamma\_2\gamma\_1\tau)=j(\gamma\_2\gamma\_1,\tau)f(\tau)$. But also, $f(\gamma\_2\gamma\_1\tau)=j(\gamma\_2,\gamma\_1\tau)f(\gamma\_1\tau)=j(\gamma\_2,\gamma\_1\tau)j(\gamma\_1,\tau)f(\tau)$. Thus, it must hold that $j(\gamma\_2\gamma\_1,\tau)= j... | 2 | https://mathoverflow.net/users/114143 | 351860 | 148,779 |
https://mathoverflow.net/questions/351856 | 1 | Let $x$ be an $\mathbb{R}$-valued random variable, then for any bounded and continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ one may write
$$
\mathbb{E}[f(X)] = \int\_{x \in \mathbb{R}} f(x)\pi(x)dx,
$$
where $\pi$ is the density of the law of $X$ *(granted that it exists)*.
In general, if $f:C([0,T];\mathbb... | https://mathoverflow.net/users/36886 | Writing path-dependent conditional expectation in terms of distribution | The answer is no. In general, to find $Ef(X\_\cdot)$, you need to know the distribution of the entire path $X\_\cdot\,$. So, as a minimum, you need to know the so-called finite-dimensional distributions of the process $X\_\cdot$, that is, the joint distributions of the random variables $X\_{t\_1},\dots,X\_{t\_k}$ for a... | 3 | https://mathoverflow.net/users/36721 | 351862 | 148,781 |
https://mathoverflow.net/questions/351863 | 2 | Let $C(\mathbb{R}^n;\mathbb{R}^d)$ be the space of continuous functions with the uniform-convergence on compacts topology. What are function spaces $X$ building on $C(\mathbb{R}^n;\mathbb{R}^d)$?
* $X$ properly contains $C(\mathbb{R}^n;\mathbb{R}^d)$,
* The subspace topology on $C(\mathbb{R}^n;\mathbb{R}^d)$ (with re... | https://mathoverflow.net/users/36886 | Topological spaces containing paths | Any such topology will be fairly unpleasant. For instance, the topology of $X$ cannot be induced by any translation-invariant metric $d$.
**Lemma.** Let $Y\_1, Y\_2$ be two topological vector spaces whose topologies are induced by translation-invariant metrics $d\_1, d\_2$, and let $T : Y\_1 \to Y\_2$ be a continuous... | 8 | https://mathoverflow.net/users/4832 | 351868 | 148,784 |
https://mathoverflow.net/questions/351870 | 5 | Let $(X\_n)$ be a sequence of $\mathbb{R}^d$-valued random variables converging in distribution to some limiting random variable $X$ whose CDF is absolutely continuous with respect to the Lebesgue measure.
Does it follow that $X\_n$ converges to $X$ in convex distance, i.e. that
$$\sup\_{h} \lvert \operatorname{E}(... | https://mathoverflow.net/users/56931 | Does convergence in law to absolutely continuous limit imply convergence in convex distance? | What is essential here is that the distribution of $X$ assigns little mass to sets which are essentially $(d-1)$-dimensional.
---
The standard approach to problems of this kind is to estimate
$$ \operatorname{P}(X\_n \in K) - \operatorname{P}(X \in K) $$
from above by
$$ \operatorname{E}(g(X\_n)) - \operatorname{... | 3 | https://mathoverflow.net/users/108637 | 351874 | 148,786 |
https://mathoverflow.net/questions/351855 | 8 | Up to now, what are the difference ways we know to define RSK correspondence? I already know:
1. By insertion and recording tableau.
2. Ball construction or Viennot's geometric construction.
3. Growth diagram proposed by Sergey Fomin.
Do you know other models?
| https://mathoverflow.net/users/68983 | RSK correspondence | Here, slightly edited, is the first paragraph of Steinberg's paper, [An occurrence of the Robinson–Schensted correspondence](https://doi.org/10.1016/0021-8693(88)90177-9).
>
> Let $V$ be an $n$-dimensional vector space over an infinite field, $\mathscr F$ the flag manifold of $V$, $u$ a unipotent transformation of ... | 4 | https://mathoverflow.net/users/3106 | 351879 | 148,789 |
https://mathoverflow.net/questions/351854 | 4 | Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.
Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and that each of these actions admit a quantum moment map. Then the associative algebra $A\otimes\_kB$ also has a $\mathfrak{g}... | https://mathoverflow.net/users/97652 | Quantum Hamiltonian reduction and tensor products | This is not true.
It is analogous to asking if $(X \times Y)/G \cong (X/G) \times (Y/G)$ for $G$ a group acting on spaces $X$ and $Y$, which is almost never the case. For example, take $X=Y=G$ with the action by left translation. Then the left hand side is isomorphic to $G$, but the right hand side is a point.
The ... | 3 | https://mathoverflow.net/users/7762 | 351888 | 148,792 |
https://mathoverflow.net/questions/351890 | 2 | Let $G$ be a connected, reductive quasi-split group over a field $k$, acting on an afffine $k$-variety $X$. Let $B = TU$ be a Borel subgroup of $G$ with maximal torus $T$ and unipotent radical $U$. Then $G$ acts on the $k$-algebra of global sections $\mathcal O\_X(X)$ of $X$. If necessary, we can assume $k$ has charact... | https://mathoverflow.net/users/38145 | Highest weight vector as a global section of an affine scheme | They are the same as the usual highest weight vectors but the $G$-representation is realized in the coordinate ring of some affine variety $X$. The usual argument goes if $\alpha:G \times X \to X$ is an algebraic action and $f \in \mathcal{O}\_{X}$ then $\alpha^\*(f) \in \mathcal{O}\_G \otimes \mathcal{O}\_X$ satisfies... | 4 | https://mathoverflow.net/users/136176 | 351892 | 148,794 |
https://mathoverflow.net/questions/351875 | 10 | I am looking for the asymptotic growth of the following sum
$$\sum\_{k=1}^{n}\frac{p\_{k+1}+p\_k}{p\_{k+1}-p\_k}$$
where $p\_k$ stands for the prime of index $k$.
Manual computations show, for small values of n, a behavior quite similar to that of the sum over naturals
$$\sum\_{k=0}^{n-1}(2k+1)=n^2$$
But more accurat... | https://mathoverflow.net/users/150698 | Asymptotic behavior of a certain sum of ratios of consecutives primes | It is elementary to prove that the sum grows at least as fast as $n^2$, and at most as fast as $n^2\log n$. The precise asymptotic behavior depends on the distribution of prime gaps $p\_{k+1}-p\_k$, on which we only have conjectures (see also my Added section below).
It is clear that
$$\#\{k\leq n: p\_{k+1}-p\_k>\lo... | 10 | https://mathoverflow.net/users/11919 | 351893 | 148,795 |
https://mathoverflow.net/questions/351891 | -3 | I refer to my previous question [Asymptotic behavior of a certain sum of ratios of consecutives primes](https://mathoverflow.net/questions/351875/asymptotic-behavior-of-a-certain-sum-of-ratios-of-consecutives-primes).
We can split the sum
$$\sum\_{k=1}^{n}\frac{p\_{k+1}+p\_k}{p\_{k+1}-p\_k}$$
where $p\_k$ stands for th... | https://mathoverflow.net/users/150698 | Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$ | My [response](https://mathoverflow.net/questions/351875/asymptotic-behavior-of-a-certain-sum-of-ratios-of-consecutives-primes) to your earlier question applies almost verbatim. The heuristic reasoning there gives that
\begin{align\*}
\sum\_{k=1}^{n}\frac{p\_k}{p\_{k+1}-p\_k}&\sim\frac{C}{2}\, n^2\log\log n,\\
\sum\_{k=... | 5 | https://mathoverflow.net/users/11919 | 351898 | 148,796 |
https://mathoverflow.net/questions/351897 | 0 | Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a [p-geometric rough path](https://en.wikipedia.org/wiki/Rough_path#Definition_of_a_rough_path) and let $\mathcal{G}\_p^d$ be the collection of all such paths. Does the [Lyons signature map](https://en.wikipedia.org/wiki/Rough_path#Signature) define a continuous bijection betwee... | https://mathoverflow.net/users/36886 | Signature Map From $p$-Geometric Rough Paths to $T(\mathbb{R})$ | The signature is continuous on the space of $p$-geometric rough paths, but it is not injective since it is parametrisation-independent and invariant under concatenation with "tree-like" pieces. Boedihardjo, Geng, Lyons and Yang showed in [this article](https://arxiv.org/abs/1406.7871/) that these are the only constrain... | 2 | https://mathoverflow.net/users/38566 | 351909 | 148,798 |
https://mathoverflow.net/questions/351884 | 2 | Let $X$ be a topological space, let $U \subset X$, and suppose that for every path $\gamma\colon [0,1] \to X$ the preimage $\gamma^{-1}(U)$ is open. Is it true that $U$ is open? Presumably not in general, but are there reasonable requirements we can put on $X$ to make it true?
To put it another way, the subsets $U$ a... | https://mathoverflow.net/users/16914 | Are open sets determined by paths? | A space $X$ is called $\Delta$-generated if $U$ is open in $X$ if and only if $\alpha^{-1}(U)$ is open in $[0,1]$ for every path $\alpha:[0,1]\to X$.
It's easy to see that a space $X$ is $\Delta$-generated if and only if $X$ is a quotient space of a disjoint union of copies of $[0,1]$. It follows that every $\Delta$-... | 4 | https://mathoverflow.net/users/5801 | 351912 | 148,800 |
https://mathoverflow.net/questions/351922 | 3 | Let $(X,\tau)$ be a topological space and $Y$ be a non-empty subset. Suppose that $Y$ is dense in $(X,\tau)$ and that there exists a topology $\tau^{\star}$ on $Y$ which is strictly finer than the subspace topology induced by restriction of $\tau$.
Does there exist a topology $\tau'$ on $X$ whose restriction to $Y$ ... | https://mathoverflow.net/users/36886 | Extension of refined subspace topology | Let $\mathcal{U} = \tau \cup \tau^\star$, and let $\tau'$ be the unique minimal topology on $X$ containing $\mathcal{U}$. Since $\tau$ and $\tau^\star$ are topologies, they are closed under finite intersection; and since $\tau^\star$ is finer than the subspace topology on $Y$, the intersection of a set in $\tau$ with a... | 4 | https://mathoverflow.net/users/3634 | 351924 | 148,805 |
https://mathoverflow.net/questions/351902 | -1 | In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R\_n$ (thus the sequence [*A104272*](https://oeis.org/search?q=A104272&language=english&go=Search) from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be potentially interesting due that Ramanujan primes ar... | https://mathoverflow.net/users/142929 | A conjecture about an inequality that involve Ramanujan primes | Not a complete answer, but a bit too long for a comment: Conjecture 1 is very likely to be very difficult if true. The corresponding conjecture for general primes is open. Let $p\_n$ be the $n$th prime and $g\_n$ be the $n$th prime gap. If one has $\sqrt{p\_{n+1}}-\sqrt{p\_n} <1 $ for sufficiently large primes then one... | 3 | https://mathoverflow.net/users/127690 | 351934 | 148,809 |
https://mathoverflow.net/questions/351908 | 10 | This is really a question about references. The [entry in Russian Wikipedia](https://ru.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0_%D0%93%D0%B8%D0%BB%D1%8C%D0%B1%D0%B5%D1%80%D1%82%D0%B0) about [Hilbert's axioms](https://en.wikipedia.org/wiki/Hilbert%27s_axioms) states, in part... | https://mathoverflow.net/users/148443 | Logical completeness of Hilbert system of axioms | The original is Alfred Tarski's book "The completeness of elementary algebra and geometry", which was due to appear in 1940 but never made it into print because of the outbreak of WW2. An edition appeared after all in 1967 (Institut Blaise Pascal, Paris), but is not easy to come by.
Essentially the same argument is p... | 13 | https://mathoverflow.net/users/31923 | 351936 | 148,811 |
https://mathoverflow.net/questions/351945 | 2 | I ask if the series
$$\sum\_{k=1}^{\infty}\frac{p\_{k+1}-p\_k}{(p\_{k+1}+p\_k)^\alpha}$$
where $p\_k$ stands for the prime of index $k$,
has the same properties of convergence of the series $$\sum\_{k=1}^{\infty}\frac{1}{k^\alpha}$$
that is convergent for all $\alpha \gt 1$ and divergent for all $\alpha \le 1$.
In th... | https://mathoverflow.net/users/150698 | Convergence of series $\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$ | Let $p$ denote a prime and $p'$ denote the next prime. Let $x>1$ be a large parameter.
By the positivity of $p'-p$ and the fact that $p'\sim p$,
$$\sum\_{x\leq p<2x}\frac{p'-p}{(p'+p)^\alpha}\asymp x^{-\alpha}\sum\_{x\leq p<2x}(p'-p)\asymp x^{1-\alpha}.$$
Hence, applying a dyadic decomposition, it follows that
$$\su... | 7 | https://mathoverflow.net/users/11919 | 351952 | 148,813 |
https://mathoverflow.net/questions/351953 | 0 | Is a totally ordered, separable and connected topological space metrizable (in the order topology)?
If we relax the assumption of connectedness, I know the counterexamples, but if we have a linear continuum that is also separable, can we say it is metrizable?
Thanks!
| https://mathoverflow.net/users/151925 | Is a totally ordered, separable and connected topological space metrizable (in the order topology)? | Yes, because it is regular and has a countable base, namely the family of open intervals with the ends at the dense countable set. The connectedness guarantees that each open interval $(a,b)$ is not empty and hence contains a point from the countable dense set.
| 3 | https://mathoverflow.net/users/61536 | 351959 | 148,816 |
https://mathoverflow.net/questions/351969 | 3 | What does the abbreviation "p.p." mean when referring to convergence? E.g. in [the following paper](https://www.jstor.org/stable/2037625) by Harry Pollard
>
> THEOREM. If $f \in L^p$ for some $p$ in the range $\tfrac{4}{3} <p < \infty$, then its Legendre
> series converges **p.p**. The result fails if $1 <p<\tfrac... | https://mathoverflow.net/users/112077 | What does the abbreviation "p.p." mean in the context of convergence | This appears to be an abbreviation for *presque partout*, meaning almost everywhere. In the article you cite, reference is made to a paper of Hunt; the MathSciNet review for Hunt's paper (MR0236019) is in French, and begins
>
> Il s'agit d'améliorations substantielles apportées au théorème de Carleson sur la conver... | 11 | https://mathoverflow.net/users/3753 | 351971 | 148,820 |
https://mathoverflow.net/questions/351758 | 0 | In the **book embedding** of a graph $G$ , each vertex of $G$ is placed on the spine and each edge is placed in the pages without crossing each other edge. If vertices have degree at most one in each page, the book embedding is **matching** . The minimum number of pages in which a graph can be matching book embedded is... | https://mathoverflow.net/users/42816 | Matching book embedding of Cartesian products of graphs | The general problem of matching book thickness for the Cartesian
product of a cycle and a complete graph is addressed in a preprint
which just popped up on RGate from Feb 2, 2020 by Z. Shao, Y. Liu and Z. Li [1] ([arXiv link](https://arxiv.org/abs/2002.00309))
It appears that they've answered your question. In fact, ... | 1 | https://mathoverflow.net/users/151955 | 351991 | 148,826 |
https://mathoverflow.net/questions/351995 | 8 | The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form
$$\sum\_{i=0}^k a\_i \cos b\_i \theta \geq 0,\;\;\;\;\;a\_\geq 0$$
such that $a\_{i\_0} = \sum\_{i\ne i\_0} a\_i$ for some $0... | https://mathoverflow.net/users/398 | Better trigonometrical inequalities for $\zeta(s)$? | Assuming the $b\_i$ are all distinct (or at least non-zero for $i \neq 0$), this is not possible. (Otherwise there are trivial examples, e.g. $1 + 2 \cos(0 \theta)+ \cos(0 \theta) \geq 0$ or $1 + 4 \cos \theta + \cos(2\theta) + 2 \cos(0 \theta) \geq 0$.)
Suppose that $\sum\_{i=0}^k a\_i \cos b\_i \theta \geq 0$. Sinc... | 17 | https://mathoverflow.net/users/766 | 352007 | 148,831 |
https://mathoverflow.net/questions/351958 | 3 | I would like to know an example of a projective variety over a totally real field where a complex conjugation is not odd on some of its étale cohomology.
*Edit:* I am looking for the most interesting statement. Namely, is there an example of a connected projective variety $X$ of dimension $>0$ over a totally real fie... | https://mathoverflow.net/users/140298 | Example of a non-odd motive appearing in cohomology of intermediate degree | How about the following construction?
Let $A$ be a principally-polarised abelian surface over $\mathbf{Q}$ which is "generic", i.e. $End\_{\overline{\mathbf{Q}}}(A) = \mathbf{Z}$. Then the Galois action on $H^1\_{\mathrm{et}}(A\_{\overline{\mathbf{Q}}}, \mathbf{Z}\_p)$ has to respect the polarisation, so we get a rep... | 1 | https://mathoverflow.net/users/2481 | 352012 | 148,833 |
https://mathoverflow.net/questions/352008 | 1 | Following this [question](https://mathoverflow.net/questions/345915/what-is-mathbbe-max-sigma-in-pm-1-n-sigmat-z-sigma-for-a-ra) I was thinking about ways to improve the upper bound and came up with the following argument. We want to find an upper bound for
\begin{equation}
\mathbb{E} [\max\_{\sigma \in \{ \pm 1\}^n}... | https://mathoverflow.net/users/130152 | Tighter upper bound for $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T W \sigma]$ | $\newcommand\si{\sigma}$
$\newcommand\Si{\Sigma}$
$\newcommand\R{\mathbb R}$
Let $\Si:=\{\pm 1\}^n$. The map
$$\R^{n\times n}\ni w\mapsto f(w):=(w\_\si)\_{\si\in\Si}\in\R^\Si, $$
where $w\_\si:=\si^T w\si$, is linear. Therefore and because $W$ is zero-mean Gaussian, we see that
$$(W\_\si)\_{\si\in\Si}:=f(W):=f\circ W... | 4 | https://mathoverflow.net/users/36721 | 352025 | 148,837 |
https://mathoverflow.net/questions/351787 | 2 | Let $G$ be a topological group, let $f:X\rightarrow Z$ be a $G$-equivariant map of (left) $G$-spaces such that
1. $X\rightarrow X/G$ and $Z\rightarrow Z/G$ are principal $G$-bundles.
2. $f$ is a fibration.
Let $\rho: G\rightarrow H$ be a morphism of topological groups. Is the induced map
$$ H\times\_{G}X \rightar... | https://mathoverflow.net/users/17895 | Fibration of principal bundles | The previous answer was getting a bit too complicated. Locally $f$ looks like $O\_i \times G \rightarrow U\_i \times G$ and this is a fibration since it is the restriction of the original fibration $f$. Now $\hat f$ locally looks like $O\_i \rightarrow U\_i$, and this is a fibration since it is a retract of a fibration... | 4 | https://mathoverflow.net/users/102519 | 352028 | 148,838 |
https://mathoverflow.net/questions/352015 | 13 | Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
| https://mathoverflow.net/users/136128 | Acyclic group and finite CW-complex | The Higman group with presentation
$$\langle{a,b,c,d}\mid{aba^{-1}b^{-2}},~bcb^{-1}c^{-2},~cdc^{-1}d^{-2},~
dad^{-1}a^{-2}\rangle$$
is perfect, and the 2-complex associated to this presentation
has Euler characteristic 0. Hence this complex is acyclic.
It is in fact aspherical, but it may be simpler to observe that
Hig... | 17 | https://mathoverflow.net/users/58488 | 352033 | 148,840 |
https://mathoverflow.net/questions/352030 | 0 | I have already addressed this problem on my [previous question](https://mathoverflow.net/questions/351921/expected-value-of-global-functions-in-renormalization-group) but I still have trouble understanding Brydges' RG maps on his [lecture notes](http://www.math.ubc.ca/~db5d/Seminars/PCMILectures/lectures.pdf), so I'll ... | https://mathoverflow.net/users/150264 | Renormalization group map on hierarchical models | I think Brydges is (tacitly) assuming that $\Omega=\mathbb{R}^{|\Lambda|}$. This, of course, is bound to create confusion between elements of $\mathbb{R}^{|\Lambda|}$ and random elements of $\mathbb{R}^{|\Lambda|}$.
| 2 | https://mathoverflow.net/users/36721 | 352039 | 148,842 |
https://mathoverflow.net/questions/352027 | 1 | Let $A$ be a domain and $K=\mathrm{Frac}(A)$.
The Zariski Riemann space $\mathrm{ZR}(K,A)$ is the set of all valuation rings of $K$ containing $A$. It comes with a natural center map
\begin{align}c\_A:\mathrm{ZR}(K,A)&\rightarrow \mathrm{Spec}\,A \\ (\mathcal{O},\mathfrak{m})&\mapsto\mathfrak{m}\cap A\end{align}
... | https://mathoverflow.net/users/114772 | Chain of closed irreducible sets on Zariski Riemann spaces | The answer to the problem is no. One reason is that the dimension of a valuation ring $V\in\mathrm{ZR}(K,A)$ may be greater than the dimension of $A$. (The supremum of the dimension of the elements of the Zariski space of $A$ is called the *valuative dimension* of $A$.)
For example, let $F$ be a field, $t,X$ indeterm... | 1 | https://mathoverflow.net/users/125073 | 352059 | 148,846 |
https://mathoverflow.net/questions/351977 | 4 | Let $A$ be an Artin algebra and $C$ a subcategory of mod-$A$ that contains all projective modules and is closed under finite direct sums (but not necessarily under direct summands).
Let $T:=add(C)$.
For an $A$-module $M$, let $f: X \rightarrow M$ be a right $C$-approximation with $X$ in $C$. Then it sounds plausible ... | https://mathoverflow.net/users/61949 | Right approximation in certain subcategories | I believe this is true. By Proposition 5.1.2 of *Relative Homological Algebra* by Enochs and Jenda, a minimal right $T$-approximation will be a direct summand of any right $T$-approximation.
Therefore it suffices to show that $f:X\to M$ is also a right $T$-approximation. Let $h:\tilde{Y}\to M$ be any morphism with $... | 3 | https://mathoverflow.net/users/151972 | 352070 | 148,849 |
https://mathoverflow.net/questions/352054 | 22 | The abc-conjecture is:
For every $\epsilon > 0$ there exists $K\_{\epsilon}$ such that for all natural numbers $a \neq b$ we have:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K\_{\epsilon}\cdot \text{rad}\left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)^{1+\epsilon} $$
I have two questions after doing some experiments with SAGEMATH:... | https://mathoverflow.net/users/nan | The abc-conjecture as an inequality for inner-products? | The matrix $L\_n$ is positive definite.
**Proof**. The matrix $G\_n$ with entries ${\rm gcd}(a,b)$ is positive definite because of $G=D^T\Phi D$ where $\Phi={\rm diag}(\phi(1),\ldots,\phi(n))$ ($\phi$ the Euler's totient function) and $d\_{ij}=1$ if $i|j$ and $0$ otherwise. Then the matrix $H\_n$ with entries $\frac1... | 15 | https://mathoverflow.net/users/8799 | 352086 | 148,857 |
https://mathoverflow.net/questions/352093 | 0 | For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has well-defined Dirichlet density (with respect to the set of all positive primes) and that density is equal to zero?
| https://mathoverflow.net/users/nan | A density zero set of primes dividing the values of a non-constant integer polynomial | No. The number of roots of $P(x)$ modulo a prime $p$, when averaged over $p$, asymptotically equals the number of irreducible factors of $P(x)$ by the prime ideal theorem. Together with the fact that this number of roots is at most the degree of $P(x)$, this shows that a positive density of primes $p$ have the property... | 2 | https://mathoverflow.net/users/5091 | 352098 | 148,860 |
https://mathoverflow.net/questions/351663 | 22 | *This question was previously [asked and bountied](https://math.stackexchange.com/q/3516216/28111) on MSE, with no response. [This MO question](https://mathoverflow.net/questions/350961/a-mathsfzf-example-of-two-baire-spaces-whose-product-is-not-baire) is related, but is also unanswered and the comments do not appear t... | https://mathoverflow.net/users/8133 | Undetermined Banach-Mazur games in ZF? | This is only a partial answer. ZF + DC + 'every Banach-Mazur game is determined' is inconsistent.
Let $X$ be the set of all functions of the form $f: \alpha \rightarrow \{0,1\}$, with $\alpha$ an ordinal such that for any ordinals $\beta < \gamma$ with $\omega \cdot \gamma + \omega \leq \alpha$, the sets $(n \in \mat... | 10 | https://mathoverflow.net/users/83901 | 352112 | 148,865 |
https://mathoverflow.net/questions/352074 | 12 | There are many recent papers on classification of Harish-Chandra bimodules for rational Cherednik algebras and, more generally, non-commutative algebras which are quantizations of symplectic singularities ([Losev](https://arxiv.org/abs/1810.07625)). What is the meaning of Harish-Chandra bimodules in terms of representa... | https://mathoverflow.net/users/12395 | What are Harish-Chandra bimodules used for? | Here is an answer from a mathematician who prefers me to post it here myself:
Harish-Chandra bimodules make sense in a very wide context. Take two
filtered algebras A, A' that quantize the same commutative algebra $C$,
and fix isomorphisms ${\rm gr} A \to C$, ${\rm gr} A^{'} \to C$. Then one can make sense
of the def... | 8 | https://mathoverflow.net/users/12395 | 352113 | 148,866 |
https://mathoverflow.net/questions/351986 | 7 | It is a well-known fact that the Henselization of the function field $\mathbb{F}\_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}\_{p}(t)^{alg} \cap \mathbb{F}\_{p}((t))$, so of course $\mathbb{F}\_{p}(t)^{h}$ embeds into $\mathbb{F}\_{p}((t))$, but is it known whether this embedding is elementary in the lan... | https://mathoverflow.net/users/118405 | Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$? | There is apparently not a very short answer to this. It is, I think, really not known whether this extension is elementary and it would be very surprising if it was known, since we do not even know if the theory of $\mathbb{F}\_{p}((t))$ is decidable. But there is the following result by Franz-Viktor Kuhlmann in his pa... | 3 | https://mathoverflow.net/users/118405 | 352125 | 148,869 |
https://mathoverflow.net/questions/352126 | 4 | Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U\_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+\infty\}$ is function which represents the potential energy of a system consisting in $N$ interacting particles. Now, ... | https://mathoverflow.net/users/150264 | Grand-canonical Gibbs measure for continuous systems | Looking at your formula (1), it appears that $\mu$ must be a measure defined on a $\sigma$-algebra $\mathscr F$ over the finite set $\Lambda$. The natural $\sigma$-algebra over the finite set $\Lambda$ is the largest $\sigma$-algebra over $\Lambda$, which is the (power) set $2^\Lambda$ of all subsets of $\Lambda$. By d... | 6 | https://mathoverflow.net/users/36721 | 352136 | 148,871 |
https://mathoverflow.net/questions/352140 | 2 | I'm reading [this book chapter](http://www.math.iit.edu/~fass/603_ch2.pdf), where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also using (2) the positive definiteness of the kernel matrix $K$ constructed fr... | https://mathoverflow.net/users/35936 | Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization) | Here is an answer to your Question 1: Let us actually prove a bit more: A function $\phi\colon[0,\infty)\to\mathbb R$ is non-constant and completely monotone if and only if
$$\phi(r)=\int\_{0}^{\infty} e^{-rt}d\mu(t)\quad \forall r\ge0\tag{1}$$
with $\mu\ne c\delta\_0$ for any real $c$.
The "only if" part: Suppose ... | 3 | https://mathoverflow.net/users/36721 | 352142 | 148,872 |
https://mathoverflow.net/questions/352067 | 1 | Given a prime $p$ should there always exist an elliptic curve over $\mathbb{Q}$ having super-singular reduction at $p$ ? I know examples with $p \equiv 2 \pmod 3,$ or $\equiv 3 \pmod 4$. But I am asking for all other $p$
| https://mathoverflow.net/users/100578 | Super-singular reduction at a given prime | This follows from Deuring's work on endomorphism rings (M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272). I couldn't find a free version online, but the content you need is summarized in sections 1-3 of [T. Yang, Minimal CM liftings of su... | 1 | https://mathoverflow.net/users/121 | 352143 | 148,873 |
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