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https://mathoverflow.net/questions/313248 | 3 | Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$.
I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$-structure is equivalent to the Nijenhuis tensor of $J$.
Can some one give a suggestion to calculate this or a refe... | https://mathoverflow.net/users/105537 | Almost complex structure and intrinsic torsion | Not trying to beat M. G., I would rather provide an elementary explanation of what's essentially going on, trying to convince you that what you've asked is really something one might expect.
Let $\nabla$ be a connection preserving the field of endomorphisms $J$, i. e. $\nabla\_XJY = J\nabla\_XY$, and $T$ be its torsi... | 4 | https://mathoverflow.net/users/117251 | 313715 | 136,342 |
https://mathoverflow.net/questions/313698 | 1 | I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this [question](https://math.stackexchange.com/questions/1525110/relationship-between-bibo-marginal-and-asymptotic-stability-statemen... | https://mathoverflow.net/users/26106 | Marginal stability of discrete linear time-invariant system | The matrix $\mathbf{A}$ is similar to the matrix
$\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}$, which is not marginally stable. Hence the original matrix is not stable.
In general, one can apply Jordan decomposition on $\mathbf{A}$, and it is marginally stable if and only if there are no eigenvalues larger than $1$ ... | 3 | https://mathoverflow.net/users/125498 | 313718 | 136,343 |
https://mathoverflow.net/questions/313723 | 0 | Given a vector $\mathbf {x} =(x\_{1},\ldots ,x\_{n})$ the p-norm is defined as
$$
\left\|\mathbf {x} \right\|\_{p}:={\bigg (}\sum \_{i=1}^{n}\left|x\_{i}\right|^{p}{\bigg )}^{1/p}
$$
for $p \geq 1$.
For $p=\infty$ one obtains the maximum norm
$$\left\|\mathbf {x} \right\|\_{\infty }:=\max \left(\left|x\_{1}\right|,\... | https://mathoverflow.net/users/122887 | Existence of p=-infinity norm | Yes this limit is correct (subject to Yemon Choi's comment). See for example the section on means, in the book *Analytic Inequalities* by Mitrinovic.
| 3 | https://mathoverflow.net/users/17773 | 313724 | 136,344 |
https://mathoverflow.net/questions/313701 | 3 | I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for me if I knew that the center was non-trivial. Here is what I know about the group:
* $G$ is solvable.
* the derived su... | https://mathoverflow.net/users/128914 | Conditions for a solvable group to have a non-trivial center | There are such groups with trivial centre. One such (possibly the smallest) is a group $G$ of order $448$ with the shape $2^{3+3}:7$. It has derived group $G'$ of order $64$, and $G''$ has order $8$ and is the unique minimal normal subgroup of $G$.
This is $\tt{SmallGroup}(448,179)$ in the databases in GAP and Magma.... | 4 | https://mathoverflow.net/users/35840 | 313725 | 136,345 |
https://mathoverflow.net/questions/313730 | 2 | **Set-up and question.**
Let $\mathcal{X}$ be a complete separable metric space which is not locally-compact. Let $V: \mathcal{X} \to [0; +\infty]$ be a function and $(X\_t)\_{t\geq 0}$ a Markov process in $\mathcal{X}$ such that $P\{ V(X \_t) < \infty \text{ for all } t \geq 0 \} = 1$. Let $g: \mathcal{X} \to \mathbb{... | https://mathoverflow.net/users/41071 | Lyapunov-type function in a non locally-compact space and boundedness of the average | The conditions $g\ge0$, (2), and $0\le V(X\_t)<\infty$ almost surely (a.s.) imply $V(X\_t)\le c$ a.s. (Indeed, on the event $\{V(X\_t)<\infty\ \forall t\ge0\}$ -- which is of probability $1$, we have $0\le g(X\_t)\le c-V(X\_t)$ and hence $0\le c-V(X\_t)$ and $V(X\_t)\le c$.) So,
$$\limsup\_{t\to\infty}EV(X\_t)\le c<\i... | 1 | https://mathoverflow.net/users/36721 | 313732 | 136,347 |
https://mathoverflow.net/questions/300762 | 1 | The current best bound for the maximum possible density of an $n$-node graph with girth (shortest cycle length) $>2k$ is of the form
$$ex(n \ \mid \ C\_{\le 2k}) = O(n^{1 + 1/k}),$$
while the current best bound for the maximum possible density of a graph without a $2k$-cycle is of the form
$$ex(n \ \mid C\_{2k}) = O(k ... | https://mathoverflow.net/users/25121 | Extremal density of a graph without a non-backtracking $2k$-cycle | The best bounds on $ex(n, C^{\not \leftarrow}\_{2k})$ and on $ex(n, C\_{2k})$ that we can prove with the current techniques are going to be basically same. Below I explain why.
First, contrary to what the question states the best known bound on the number of edges in $C\_{2k}$-free graphs is $O(\sqrt{k}\log k\cdot n^... | 1 | https://mathoverflow.net/users/806 | 313740 | 136,351 |
https://mathoverflow.net/questions/313744 | 12 | Consider a non-empty set $S$ of primes, with the property that, for every finite subset $S'\subset S$, all the primes dividing $\left(\prod\_{k\in S'}k\right)+1$ are in $S$.
For instance, it can easily be proven that $2\in S$ (if not, then the smallest member $q$ of $S$ is odd, hence, $q+1$ is even, and thus $2\in S$... | https://mathoverflow.net/users/127150 | A set of prime numbers | If I'm not mistaken, it is true that $S$ must contain all primes.
First of all it is obvious that $S$ is infinite -- indeed, as Euclid teach us, if $S$ is finite then $\left(\prod\limits\_{p\in S} p\right) + 1$ is coprime to all primes in $S$ and therefore has at least one prime factor not in $S$.
Now let $p$ be a ... | 8 | https://mathoverflow.net/users/104330 | 313758 | 136,357 |
https://mathoverflow.net/questions/313703 | 7 | I know through Kirchoff's Theorem, one can calculate the number of spanning trees via the determinant of a Laplacian. This has complexity $O(N^{2.373}$). I was wondering if anyone was aware of a method which computes this faster, at least for planar graphs.
| https://mathoverflow.net/users/130484 | Counting spanning trees of a planar graph | The determinant of matrices whose support corresponds to incidence matrices of planar graphs (this includes the Laplacian matrix of a planar graph, or more precisely its cofactors) can be calculated in $O(n^{1.5})$ with the algorithm given in
>
> R.J. Lipton, D. Rose, R.E. Tarjan
> Generalized nested dissection
> ... | 9 | https://mathoverflow.net/users/2384 | 313760 | 136,359 |
https://mathoverflow.net/questions/313733 | 6 | I'm trying to write a constructive proof of the isomorphism ${Z}^{X \times Y} \equiv (Z^Y)^X$ in a category with exponential objects.
I can construct a map $(Z^Y)^X \rightarrow {Z}^{X \times Y}$, but I'm stuck trying to construct the reverse - ${Z}^{X \times Y} \rightarrow (Z^Y)^X$
Here's my inventory so far:
$$\... | https://mathoverflow.net/users/130507 | Constructive proof of exponential law in a category | There's a systematic way to to this -- it's an exercise in how to "unpack" the Yoneda lemma. After all, the quickest proof of this isomorphism goes like this:
$Hom(A,(Z^Y)^X) \cong Hom(A \times X, Z^Y) \\
\qquad\qquad\qquad\quad \cong Hom((A \times X) \times Y, Z) \\
\qquad\qquad\qquad\quad \cong Hom(A \times (X\time... | 10 | https://mathoverflow.net/users/2362 | 313765 | 136,360 |
https://mathoverflow.net/questions/313764 | 5 | Let $A\_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A\_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 & b & a & \cdots & \cdots & 0 & 0\\ \vdots & \vdots & \vdots &\ddots & \ddots & \vdots & \vdots \\ \vdots & \vdots & \vdots &\ddots & \ddots & \vdots & \vdots \\ 0 & 0 ... | https://mathoverflow.net/users/62673 | Eigenvalue density of a symmetric tridiagonal matrix | For large $n$ we may treat $x\equiv k/n+1$ as a continuous variable with a uniform density in the interval $0 <x< 1$. The corresponding eigenvalue $\lambda(x)=a+2b\cos\pi x$ ranges from $a-2|b|$ to $a+2|b|$. Since
$$|d\lambda/dx|=\pi\sqrt{4b^2-(\lambda-a)^2}.$$
The eigenvalue density follows from
$$\rho(\lambda)d\lambd... | 5 | https://mathoverflow.net/users/11260 | 313772 | 136,361 |
https://mathoverflow.net/questions/313780 | 4 | Let us define the basis of polynomials given by:
$$
\begin{array}\
P\_0=1, \\
P\_1=x, \\
P\_2=x(x-1), \\
P\_3=x(x-1)(x-2), \\
P\_4=x(x-1)(x-2)(x-3), \ldots\\
\end{array}
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!
| https://mathoverflow.net/users/41940 | Orthogonal basis of polynomials? | If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
\[
p\_{n+1}(t) = (t-a\_n)p\_n(t) - b\_n p\_{n-1}(t)
\]
where $b\_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
| 7 | https://mathoverflow.net/users/1266 | 313788 | 136,368 |
https://mathoverflow.net/questions/313761 | 2 | Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous homomorphism from $ G $ to the group of unitary operators on $ \mathcal{H} $, then we can define a strongly continuous act... | https://mathoverflow.net/users/50614 | Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space | The answer is no. Let $G$ be the Cartesian product of a sequence of copies of the unit circle, identified with the functions in $l^\infty$ whose modulus is constantly $1$. For $f \in G$ let $U\_f \in B(l^2)$ be multiplication by $f$. Let $G$ act on $K(l^2)$ by conjugation by these unitaries.
The action is strongly co... | 3 | https://mathoverflow.net/users/23141 | 313796 | 136,374 |
https://mathoverflow.net/questions/313808 | 13 | Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there.
**Edit.** Unsolved exercises are fine. Same with solved.
| https://mathoverflow.net/users/126532 | Reference request: Oldest calculus, real analysis books with exercises? | It depends if you mean exercises *with* or *without* solutions.
The former don’t differ much from example-driven textbooks (e.g. Euler’s *Institutiones* ([1755](https://www.e-rara.ch/zut/content/pageview/1174555), [1768](https://www.e-rara.ch/zut/content/pageview/9548569), [1769](https://www.e-rara.ch/zut/content/pag... | 20 | https://mathoverflow.net/users/19276 | 313810 | 136,380 |
https://mathoverflow.net/questions/285466 | 2 | Let $P(D)$ be hypoelliptic operator with constant coefficients in $\mathbb{R^n}$.Let $\Omega$ be an open subset of $\mathbb{R^n}$ and $\mathscr{N\_\Omega}$ denote space of distribution solutions of homogeneous equation $P(D)h=0 $. I need to prove that following topologies on $\mathscr{N\_\Omega}$ are identical:
(i) t... | https://mathoverflow.net/users/102092 | Equivalence of different topologies of Kernel of constant coefficient hypoelliptic operator | There are at least three published proofs: The first (for the result in this generality -- for particular operators it is of course older) I know is of Malgrange
[*Existence et approximation des solutions des equations
aux derivees partielles et des equations de convolution*, Ann. Inst. Fourier, Grenoble, 6 (1955)–(195... | 3 | https://mathoverflow.net/users/21051 | 313815 | 136,383 |
https://mathoverflow.net/questions/224330 | 5 | I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
H^2(\rho,\mu)=\int\_{\Omega}\left|\sqrt{\frac{d\rho}{d\lambda}}-\sqrt{\frac{d\mu}{d\lambda}}\right|^2 d\lambda
$$
shows up. Here $\Omega\subset R^d$ is a (possibly unbounded) smooth domain, $\rho,\mu$ are n... | https://mathoverflow.net/users/33741 | Lower semi-continuity of the Hellinger-Fisher-Rao distance | The lower semicontinuity of the Hellinger-Rao distance follows from the following more general lower semicontinuity result (see e.g. Buttazzo, *Semicontinuity, relaxation and integral representation in the calculus of variations*, Thm. 3.4.1):
If $f\colon \mathbb{R}^n\to [0,\infty]$ is a proper lower semicontinuous c... | 3 | https://mathoverflow.net/users/95776 | 313842 | 136,392 |
https://mathoverflow.net/questions/313816 | 6 | Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the number of days needed to reach my goal. What is the expected value of $S$?
| https://mathoverflow.net/users/8628 | Randomly picking $k$ members of $\{1,\ldots,n\}$ | If $X$ is a random variable with non-negative integer values, the expectation of $X$ equals $$\mathbb{E}(X)=\sum\_{m=1}^\infty {\rm Prob}\,(X\geqslant m).$$
In our situation the event $X\geqslant m$ means that after $m-1$ days there remains a not taken element. By exclusion-inclusion it equals $$\sum\_{i=1}^n (-1)^{i-1... | 6 | https://mathoverflow.net/users/4312 | 313846 | 136,394 |
https://mathoverflow.net/questions/313843 | 9 | Consider the field of real numbers $(\mathbb R,+,\cdot)$. An expansion of $(\mathbb R,+,\cdot)$ is a tuple $(\mathbb R,+,\cdot,S)$, where $S$ is a collection of subsets of $\mathbb R^n$ for $n \in \mathbb N$. A subset of $\mathbb R^n$ is said to be definable in $(\mathbb R,+,\cdot,S)$, when it is definable by a first o... | https://mathoverflow.net/users/8176 | Definable functions in $o$-minimal structures | The derivative of $e^{e^x}$ is $e^x e^{e^x}$. So
$$y=e^x\ \leftrightarrow\ y = \lim\_{h\rightarrow 0} \dfrac{e^{e^{x+h}} - e^{e^x}}{h e^{e^x}}$$
which is a first-order statement.
| 10 | https://mathoverflow.net/users/nan | 313848 | 136,395 |
https://mathoverflow.net/questions/313837 | 4 | Let $(X,d)$ be a metric space, $x\_1,\ldots,x\_N\in X$ and $x\_1',\ldots,x\_N'\in X$ be atoms, and $G=\sum\_{i=1}^Np\_i\delta\_{x\_i}$, $G'=\sum\_{i=1}^Np\_i'\delta\_{x\_i}$, and $G''=\sum\_{i=1}^Np\_i'\delta\_{x\_i'}$ be mixing measures. In words: $G$ and $G'$ have the same atoms, but different weights. $G'$ and $G''$... | https://mathoverflow.net/users/99132 | Effect of perturbing the atoms of a measure on the Wasserstein distance | There is a nontrivial counterexample for $N=2$, $p=1$, and $X=\mathbb{R}$. Pick $x\_1=-2$, $x\_2=2$, $x'\_1=-1$, $x'\_2=1$ and $p\_1=4/5$ and $p\_1'=1/5$. Then $2.2=W\_1(G,G'')<W\_1(G,G')=2.4$. (I hope I did not mess up the calculation).
The intuition seems clear:
In the counterexample, you have to move $1/5$ of th... | 6 | https://mathoverflow.net/users/69603 | 313850 | 136,397 |
https://mathoverflow.net/questions/313820 | 4 | Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base.
What are the relations between the following pairs?
$$f\text{ epi}\qquad f^\ast \text{ faithful}$$
$$f\text{ mono}\qquad f^\ast \text{ full}$$
$$f\text{ strong epi}\qquad f^\ast \text{ conservative}$$
I don't mind completeness assumpti... | https://mathoverflow.net/users/69037 | In a fibration, how do properties of arrows downstairs affect the base-change functors? | In short: There is absolutely no such relations, and in general essentially no properties of arrows of the codomain (except being a split epi/split mono or an iso) have any effect on the base change functor.
Indeed, because of Grothendieck's construction, absolutely any (pseudo)-functor from $X^{op}$ to $Cat$ is obta... | 8 | https://mathoverflow.net/users/22131 | 313853 | 136,400 |
https://mathoverflow.net/questions/313806 | 4 | I am interested in the following integral related to the Chebyshev polynomials
$$I\_{n,m}:= \int\_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$
where $n,m\in \mathbb{Z}^+.$
It is easy to see the following result.
For an even number $n \in \mathbb{Z}^+$ and an odd number $m \in \mathbb{N}$, we have
$$I\... | https://mathoverflow.net/users/42816 | A Conjecture about the integral related to Chebyshev polynomial | This is proven in [arXiv:1002.3844](https://arxiv.org/abs/1002.3844), see top of page 12. The quantity $A\_k^b(n)$ in that paper is a polynomial in $n$ of degree $\leq k$ and it is related to the integral $I\_{nm}=\pi P\_m(n)$ in the OP by $A\_{k}^b(n)=P\_{k+1}(2n+1)$.
The polynomial can be expressed as a terminating... | 6 | https://mathoverflow.net/users/11260 | 313854 | 136,401 |
https://mathoverflow.net/questions/313856 | 2 | Given independent Gaussian $d$ dimensional vectors $G\_i$,
Let $ \sigma^2\_n=\mathbb{E}(\sum\_{i \le n} G\_i) \cdot (\sum\_{i \le n} G\_i)^T$. $||\sigma\_n^2||$ is norm of $\sigma\_n^2$.
Is there a $d$-dimension Brownian motion $B\_t$ with covariance matrix $\sigma^2$ s.t.
$\sum\_{i \le n} G\_i=B\_{||\sigma\_n^2||... | https://mathoverflow.net/users/124254 | Gaussian sum VS Brownian motion | The answer is no. E.g., let $d=2$ and let $Z\_1,Z\_2,\dots$ be iid standard normal random variables. Let then $G\_i=(a\_iZ\_i,0)$ if $i$ is odd and $G\_i=(0,a\_iZ\_i)$ if $i$ is even, where the $a\_i$'s are positive real numbers increasing fast enough in $i$ so that $\sum\_1^{n-1}a\_i^2=o(a\_n^2)$; the convergence ever... | 2 | https://mathoverflow.net/users/36721 | 313864 | 136,402 |
https://mathoverflow.net/questions/313866 | 1 | This is a cross-post to [the question](https://math.stackexchange.com/questions/2952997/is-there-a-homeomorphism-between-the-sets-of-schur-stable-and-hurwitz-stable-mat) I asked at MSE.
---
The set of Schur stable matrices is
\begin{align\*}
\mathcal S = \{A \in M\_n(\mathbb R): \rho(A) < 1\},
\end{align\*}
where... | https://mathoverflow.net/users/103704 | Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms? | Essentially you're asking if there is a homeomorphism between the monic real polynomials of degree $n$ with roots in the open unit disk and those with roots in the left half plane. Just take $$\prod\_{j=1}^n (x - \alpha\_j) \mapsto \prod\_{j=1}^n \left(x - \frac{\alpha\_j+1}{\alpha\_j - 1}\right)$$
This should work bec... | 3 | https://mathoverflow.net/users/13650 | 313868 | 136,404 |
https://mathoverflow.net/questions/313860 | 5 | The Nielsen-Schreier theorem states that subgroups of a free subgroup are free.
Is this hold also for groups with operations?
Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action which is free (as a group with $G$-action). Let $F'\subset F$ be a subgroup closed by the $G$-action. Then, must $F$ be... | https://mathoverflow.net/users/49822 | Nielsen-Schreier with operations | Let $G$ be a two element group; the free group on two generators $x,y$ with the action of $G$ interchanging them is a free $G$-group (on one generator). Its subgroup generated by $xy^{-1}$ is closed under the $G$-action but is not a free $G$-group.
| 9 | https://mathoverflow.net/users/41291 | 313877 | 136,407 |
https://mathoverflow.net/questions/313805 | 0 | Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x\_0,x\_1,...,x\_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the unique interpolating polynomial $p\_n(x)$ of $f$ at the nodes $\{x\_0,x\_1,...,x\_n\}$ satisfy $p\_n(x)\ge 0,\forall x\in [a... | https://mathoverflow.net/users/127118 | For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve | Let $n \ge 2$. Given any points $x\_0 < x\_1 < \dots < x\_n$, there is a quadratic function positive at all those points, but negative somewhere in $[x\_0,x\_n]$.
Indeed, let $c \in [x\_0,x\_n]$ be any point other than those $n+1$ points. There is a quadratic $\phi(x) = -1+m(x-c)^2$ that is positive at those points.... | 6 | https://mathoverflow.net/users/454 | 313878 | 136,408 |
https://mathoverflow.net/questions/313882 | 4 | Let $\pi: \mathfrak{M}\_{(\theta,0)}(Q,\text{v},\text{w}) \rightarrow \mathfrak{M}\_{(0,0)}(Q,\text{v},\text{w})$ be the projective morphism between Nakajima quiver varieties when complex moment parameter is equal to zero $\zeta\_\mathbb{C}=0.$ We define the **Lagrangian core** to be $\Lambda\_{\theta}=\pi^{-1}(0).$
... | https://mathoverflow.net/users/114985 | Lagrangian cores of quiver variety in different GIT chambers | Any quiver variety where all the v\_i are 1's is a [hypertoric variety](https://arxiv.org/abs/0705.4236). They are determined combinatorially by an arrangement of affine hyperplanes and one can compute the core by looking at the compact chambers of the arrangement. Changing the git parameters corresponds to translating... | 3 | https://mathoverflow.net/users/333 | 313884 | 136,411 |
https://mathoverflow.net/questions/313669 | 6 | A game is played as follows. There is a set $X = \{1, \ldots, n\}$. Player 1 is trying to find a "locally minimal subset" $M \subseteq X$ - that is, player 2 has said that $M$ is good, and also that every subset $M - \{x\}$ for $x \in M$ is bad.
Formally, play proceeds as follows: A move from player $1$ is any subset... | https://mathoverflow.net/users/4959 | What is the minimum worst-case length of an element removal game? | I think that there's a simple adversary strategy:
Answer 0 unless it would block all paths from $X$ to the second level from below, i.e., to the $2$-element sets. (So for $\le 1$ element sets we always answer 0.)
Such a game can end only by receiving a 1 answer on the second level for some $(a,b)$. But since this s... | 3 | https://mathoverflow.net/users/955 | 313886 | 136,413 |
https://mathoverflow.net/questions/313894 | 2 | Suppose $G=(V,E)$ is a connected graph and $T=(V\_T, E\_T)$ is a subgraph of $G$ that is a tree.
If we further suppose that any pair of vertices $v,w \in V\_T$ that are not joined by a single edge in $E\_T$ are also not joined by an edge in $E$, is there a standard name for this condition? Informally, what I mean is ... | https://mathoverflow.net/users/23829 | Terminology for tree subgraphs where non-neighbouring vertices are not connected by single ambient edges | The subgraph $T$ is an [induced subgraph](https://en.wikipedia.org/wiki/Induced_subgraph) of $G$.
Some authors use the term [induced tree](https://people.maths.ox.ac.uk/scott/Papers/inducedtrees.pdf) meaning an induced subgraph which is a tree.
| 2 | https://mathoverflow.net/users/125498 | 313896 | 136,416 |
https://mathoverflow.net/questions/313904 | 5 | Let $\Omega\_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma\_1,\sigma\_2,\tau\_1,\tau\_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma\_j \tau\_k = \tau\_k \sigma\_j$ for each pair $(j,k) \in \{1,2\}^2$.
Any four permutations determine a $8$-regular edge-labeled directed graph on $\{1,... | https://mathoverflow.net/users/30721 | Random pairs of commuting permutations | This is not the case, in fact in both cases the limit is 1/2. This is because with probability going to 1 as $n \to +\infty$ a uniformly random morphism $\mathbb F\_2 \times \mathbb F\_2 \to \mathrm{Sym}(n)$ is trivial on one of the factors. The latter claim follows from Dixon's theorem that a random pair of independen... | 10 | https://mathoverflow.net/users/32210 | 313910 | 136,419 |
https://mathoverflow.net/questions/271043 | 1 | By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-parametric function in a local coordinate system defined by $p\_i$ as the origin and, $\frac{p\_{i+1}-p\_i}{\|p\_{i+1}-p\_i\|}$ as the ... | https://mathoverflow.net/users/31310 | Smoothness Conditions for Planar "Mock-parametric" Spline Interpolation | The "Wilson-Fowler" spline has the properties that you describe. Each segment is defined by a cubic polynomial in a rotated coordinate system. These splines were quite common in the days before parametric vector-valued ones became the standard. For example, they were used in most versions of [the APT system](https://en... | 1 | https://mathoverflow.net/users/50265 | 313912 | 136,420 |
https://mathoverflow.net/questions/313461 | 9 | For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E\_n)\_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To be precise, let $Topoi$ be the $\infty$-category of $\infty$-topoi and geometric morphisms (pointing in the direction ... | https://mathoverflow.net/users/2362 | What is a spectrum object in $\infty$-topoi? | Following up on the answer of Simon Henry, let us prove the following statement. For a pro-space $\hat{X} = \{X\_i\}\_{i \in I}$, we let $Spaces\_{/\hat{X}}$ denote the $\infty$-topos defined as the (cofiltered) limit in $Topoi$ of the $I$-family of étale topoi $Spaces\_{/X\_i}$. We will refer to such $\infty$-topoi as... | 7 | https://mathoverflow.net/users/51164 | 313944 | 136,429 |
https://mathoverflow.net/questions/313961 | 42 | First off I apologize if this question does not belong here, I would be happy to hear about any better locations to post this on.
I am a (first year) undergraduate mathematics student, and I recently discovered some interesting properties hidden in certain families of sequences. I ran these ideas past my math profess... | https://mathoverflow.net/users/129192 | Publishing a Simple Paper as an Undergraduate | After writing a manuscript (which it seems you may have already), go through it and revise it a few times until you feel that it is in a polished form. Then you could ask your professors to read it and provide some feedback and revise accordingly. This revision/feedback process will be a good experience for practicing ... | 34 | https://mathoverflow.net/users/68871 | 313963 | 136,435 |
https://mathoverflow.net/questions/313741 | 0 | Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we have the set $\{x \in X : f(x) \leq f(y)\}$ is closed.
I know this class is a Pure super-set of family of lower semi ... | https://mathoverflow.net/users/117299 | An extension for lower semi continuous lower bounded real valued functions class | I finally found a name for that mentioned class in some paper. **J. Morgan & F. Scalzo** have used this class in their paper( **Pseudocontinuity in Optimization and Nonzero-Sum Games**) in 2004, and called that class **lower pseudocontinuous** .
I should add this point that I am not sure if they have been first to us... | 0 | https://mathoverflow.net/users/117299 | 313973 | 136,441 |
https://mathoverflow.net/questions/313501 | 5 | I am reading the article "A convenient category for directed homotopy" by Fajstrup and Rosicky and I have a doubt about the proof of Proposition 3.5. The setting is the following:
let $\cal{C}$ be a concrete category, with forgetful functor $U\colon \cal{C}\to \mathsf{Set}$.
**Definition 1.** A full subcategory $\cal... | https://mathoverflow.net/users/49102 | Finally dense implies dense | You are right, in general $\mathcal I$ should contain the discrete object $D\_1$ on the singleton. A general result is in my paper Codensity and binding categories (Theorem 1.3). In Proposition 3.5 of the aforementioned paper, $D\_1$ is $\mathcal I$-generated and thus it can be added to $\mathcal I$ without changing $\... | 9 | https://mathoverflow.net/users/73388 | 313975 | 136,442 |
https://mathoverflow.net/questions/313955 | 0 | For $A=\begin{pmatrix} a\_1 & b\_1 \\ c\_1&d\_1 \end{pmatrix}, B=\begin{pmatrix} a\_2 & b\_2 \\ c\_2&d\_2 \end{pmatrix}\in M\_2(\mathbb Z)$, define
$A\*B:=a\_1L\_1BR\_1+b\_1L\_1BR\_2+c\_1L\_2BR\_1+d\_1L\_2BR\_2$, where
$L\_1=I\_2=\begin{pmatrix} 1 & 0 \\ 0&1 \end{pmatrix}, L\_2=\begin{pmatrix} 0 & 1 \\ 1&0 \end{p... | https://mathoverflow.net/users/127118 | A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity | There is an isomorphism $f:\mathbb{C\times C}\rightarrow (M\_2(\mathbb{R}),+,\*)$ given by
$$f((1,0))=\left(\begin{array}{cc}
-1 & -1\\-1 & -1
\end{array}\right)\!/2,\ \ \
f((i,0))=\left(\begin{array}{cc}
1 & -1\\1 & -1
\end{array}\right)\!/2\sqrt{3},
$$
$$f((0,1))=\left(\begin{array}{cc}
-1 & -1\\1 & 1
\end{array}... | 4 | https://mathoverflow.net/users/nan | 313984 | 136,446 |
https://mathoverflow.net/questions/313978 | 4 | Is there a heuristic argument behind the exponent in the [circle problem](https://en.wikipedia.org/wiki/Gauss_circle_problem)? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and let $N(R)$ denote the number of integer lattice points contained in ... | https://mathoverflow.net/users/8435 | Heuristics behind the Circle problem? | The heuristics for this problem is due to Gauss: the error term $\delta N(R)=N(R)-\pi R^2$ must scale with the circumference $2\pi R$ of the circle, because it is along the circumference that the ambiguity of lattice points just inside or just outside the circle appears; Gauss' estimate $\delta N(R)\propto R$ is an ove... | 8 | https://mathoverflow.net/users/11260 | 313985 | 136,447 |
https://mathoverflow.net/questions/313964 | 4 | Let $f:X\rightarrow Y$ be a smooth morphism of schemes such that all the fibres (for geometric points) are affine spaces. Let $F$ be a coherent sheaf on $X$. Is $R^i\_{et}~f\_\*F=0~~~\forall i>0$? Here "et" in the subscript means the etale higher direct image sheaf.
What if $F$ is the locally constant sheaf $\frac{\m... | https://mathoverflow.net/users/nan | etale higher direct image sheaf | It is convenient to formulate the hypotheses as a definition.
**Definition 1.** A morphism $f:X\to Y$ is an **almost affine fibration** if $f$ is quasi-compact, quasi-separated, smooth, surjective, and for every point $y$ of $Y$, the fiber $\text{Spec}\ \kappa(y)^{\text{sep}}\times\_Y X$ is $\kappa(y)^{\text{sep}}$-i... | 4 | https://mathoverflow.net/users/13265 | 313992 | 136,448 |
https://mathoverflow.net/questions/312676 | 11 | So here's my question:
Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is
1. minimal
2. uniquely ergodic with unique probability measure $\mu$
3. not ergodic with respect to the Lebesgue measure ?
I don't really see why these requirements should contradic... | https://mathoverflow.net/users/25511 | Minimal, uniquely ergodic but not Lebesgue-ergodic? | I couldn't manage to find an online version, but [this paper](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=yoccoz&s5=denjoy-koksma&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSe... | 3 | https://mathoverflow.net/users/19393 | 314014 | 136,453 |
https://mathoverflow.net/questions/313871 | 1 | The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not even continuous functions are dense in $L^\infty$.
But, when $n>1$, could $X := \{ B \in L^\infty(\mathbb R^n,\mathbb R^... | https://mathoverflow.net/users/130570 | Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace? | Here is a sketch that I think could be turned into a counterexample.
Take $n=2$, let $\hat{r}$ be the outward-pointing radial unit vector field, and set $F = \cos(\pi r^2) \hat{r}$. Let $G$ be a vector field with $|\nabla \cdot G |\le M$. Now let $A\_n$ be the annulus defined by $\sqrt{2n-1} \le r \le \sqrt{2 n}$. Yo... | 1 | https://mathoverflow.net/users/4832 | 314019 | 136,455 |
https://mathoverflow.net/questions/313966 | 7 | [I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel free to close it if it's inappropriate. Also, I'm a physicist rather than a mathematician, so fancy answers might go over ... | https://mathoverflow.net/users/95043 | How do fractional tensor products work? | I'm not going to try to match Tao's notation.
I will use the word *line* to mean a one-dimensional real vector space.
Suppose that $L$ is a line. Consider the line $L^{\otimes 2}$. It has the following property: it has a well-defined notion of "positive" element. Indeed, for each $\ell \in L$, we declare that $\ell... | 6 | https://mathoverflow.net/users/78 | 314021 | 136,456 |
https://mathoverflow.net/questions/314020 | 4 | Let $n,p\geq 1$ be integers, and assume that $p$ is a prime.
**Question.** Does there always exist an integer $m\geq 1$ such that
$p^m\equiv m\pmod{n}$?
| https://mathoverflow.net/users/120165 | On the solvability of the congruence $p^m\equiv m\pmod{n}$ | The answer is affirmative when $(p,n)$=1, even without the assumption that $p$ is a prime.
Let us fix $p$ and proceed by induction on $n$. We can assume, without loss of generality, that $p\geq 2$. For $n=1$ the statement is clear. So let us assume that $n\geq 2$, and the statement holds for all proper divisors of $n... | 8 | https://mathoverflow.net/users/11919 | 314026 | 136,458 |
https://mathoverflow.net/questions/314025 | 7 | Let $G$ be a semisimple algebraic group of type $E\_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G\_{\bar{k}}$ is a semisimple algebraic group in the usual sense), and let $T$ be a $k$-subtorus of $G$ of rank $6$, so a (not necessarily split) maximal subtorus of $G$.
Does there exist ... | https://mathoverflow.net/users/130410 | Subtori of groups of type E6 | This question is precisely answered by [Borel–de Siebenthal theory](https://en.wikipedia.org/wiki/Borel%E2%80%93de_Siebenthal_theory). Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $E\_6$, $A\_1 + A\_5$, and $A\_2 + A\_2 + A\_2$ (as you ... | 10 | https://mathoverflow.net/users/2383 | 314027 | 136,459 |
https://mathoverflow.net/questions/314045 | 9 | Let $A$ be a finite-dimensional $k$-algebra and $U$ and $V$ two finite-dimensional projective $A$-modules (maybe neither the finiteness nor projectivity has to play a role, but these requirements are satisfied in my problem).
Now consider the $A$-bimodule
\begin{align}
M := \mathrm{Hom}\_A (U,A) \otimes \mathrm{Hom}... | https://mathoverflow.net/users/119240 | Hochschild homology with coefficients in a certain bimodule | (I'm assuming in the following that the base ring $k$ was a field.)
First, we note that for any $A$-bimodule of the form $M \otimes N$, where $M$ is a left $A$-module and $N$ is a right $A$-module, has an isomorphism
$$
HH\_\*(A;M \otimes N) \cong Tor^A\_\*(N,M).
$$
To see this, we note that there is an explicit *sim... | 8 | https://mathoverflow.net/users/360 | 314051 | 136,466 |
https://mathoverflow.net/questions/313917 | 35 | We fix $G=\mathrm{SL}\_3(\mathbf{R})$.
>
> Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b\_2(\Gamma)=0$?
>
>
>
Here the second Betti number $b\_2(\Gamma)$ is both the dimension of the cohomology group $H^2(\Gamma,\mathbf{Q})$ and the dimension of the de Rham cohomology in degree 2 of the local... | https://mathoverflow.net/users/14094 | Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$ | The arithmetic cocompact lattices constructed in (6.7.1) of [Witte-Morris' book](https://arxiv.org/abs/math/0106063) all have torsion-free finite index subgroups with arbitrarily large second Betti number.
I will briefly recall the construction because this is necessary for the answer. Let $F$ be a totally real numbe... | 18 | https://mathoverflow.net/users/54441 | 314073 | 136,474 |
https://mathoverflow.net/questions/314074 | 3 | Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where
* $X^{(0)}=0$
* If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)} = X^{(t)} + (1-\varepsilon)$, and with probability $.5$, $X^{(t+1)} = X^{(t)} - (1+\varepsilon)$
* If $X^{(t)} < 0$, ... | https://mathoverflow.net/users/17589 | Concentration of a modified random walk | Let me try an answer.
[Edit: simplified and (hopefully) corrected.]
Let $\alpha$ be the only positive solution of $\mathrm{ch}\alpha = \exp(\varepsilon\alpha)$, so that
$$ \exp(\alpha x) = \frac12\left(\exp(\alpha(x+1-\varepsilon))+\exp(\alpha(x - 1 - \varepsilon))\right)\text. $$
The purpose of the definition lies i... | 4 | https://mathoverflow.net/users/129074 | 314078 | 136,476 |
https://mathoverflow.net/questions/314083 | 1 | I have come across a lot of papers that are written about the palindromic polynomials, however, I am recently interested in polynomials satisfying
$$f(-x) = x^nf(1/x)$$
for $n\geq 1$ and for all $x\in \mathbb{R}$ except $0.$ Is there any reference where the roots of such polynomials are studied?
| https://mathoverflow.net/users/68232 | Reference request for anti-palindromic polynomials. | Assuming $n=\deg f$, it is easy to see that there is no such polynomial if $n$ is odd. So let us assume $n$ is even from now on. We may also assume $f$ is monic.
Let $S$ denote the multiset of roots of $f$ in $\mathbb{C}$ (note that $\# S$ is even). Then $f$ satisfies your condition if and only if
$$\prod\_{\alpha \i... | 7 | https://mathoverflow.net/users/6506 | 314088 | 136,479 |
https://mathoverflow.net/questions/313936 | 9 | Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows:
$$f(u,v) = \sum\_{1\leq i,j\leq n}|u\_iu\_j-v\_iv\_j|.$$
I want to find:
$$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=|v|=1, u\perp v\}.$$
An easy estimate using the triangle inequality gives $M(n) <2n$, but it seems there should... | https://mathoverflow.net/users/51663 | Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$ | Just use Cauchy-Schwarz and the identity
$$
\sum\_{i,j}(u\_iu\_j-v\_iv\_j)^2=\left[\sum\_i u\_i^2\right]^2+\left[\sum\_i v\_i^2\right]^2-2\left[\sum\_i u\_iv\_i\right]^2=2
$$
to get $f(u,v)\le \sqrt2 n$ for all $n$. As you have observed yourself, this is sharp for even $n$. For odd $n$ you can, probably, do a bit bette... | 8 | https://mathoverflow.net/users/1131 | 314095 | 136,480 |
https://mathoverflow.net/questions/314030 | 7 | Let $S\_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S\_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (equivalently think of it as the mapping class group of the once-punctured surface of genus $g$).
Is every element of $Mod(S\_{g... | https://mathoverflow.net/users/43097 | Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists? | Yes. As pointed out by Ian Agol, I actually answered my question.
I observed that the identity is a non-empty composition of right handed Dehn twists in $Mod(S\_{g,1})$. A priori this is not trivial. I was thinking about monodromies on Brieskorn-Pham singularities $(x^p+y^q)$ which are freely periodic and a compositi... | 4 | https://mathoverflow.net/users/43097 | 314097 | 136,481 |
https://mathoverflow.net/questions/301236 | 9 | Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation preserving periodic homeomorphisms, that is, $\phi^n = \psi^m = id$ with $n,m$ their periods. Suppose that $\phi$ is isotopic ... | https://mathoverflow.net/users/43097 | Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms | The answer is "yes". It follows as part of F. Bonahon's determination of the bordism group of surface diffeomorphisms. See
*Bonahon, Francis*, [**Cobordism of automorphisms of surfaces**](http://dx.doi.org/10.24033/asens.1448), Ann. Sci. Éc. Norm. Supér. (4) 16, 237-270 (1983). [ZBL0535.57016](https://zbmath.org/?q=an:... | 5 | https://mathoverflow.net/users/1822 | 314103 | 136,483 |
https://mathoverflow.net/questions/314107 | 2 | For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2\_H(F,G)=1-\int\_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ are two-component mixture densities with mixing probability $\pi$, such that
$$
f(x)=\pi f\_0(x)+(1-\pi)f\_1(x)\\
g(x)=... | https://mathoverflow.net/users/65953 | References for Hellinger distance/affinity involving mixture distributions | For $f=\pi f\_0+(1-\pi)f\_1$, $g=\pi g\_0+(1-\pi)g\_1$, and $\pi\in(0,1)$,
we have
\begin{equation}
\frac{\partial^2}{\partial\pi^2}\sqrt{fg}= -\frac{\left(f\_1 g\_0-f\_0 g\_1\right){}^2}{4 (fg)^{3/2}}\le0.
\end{equation}
So, $\sqrt{fg}$ is concave in $\pi$ and hence $d^2\_H(F,G)$ is convex in $\pi\,$:
$$
d^2\_H(F,G... | 1 | https://mathoverflow.net/users/36721 | 314109 | 136,485 |
https://mathoverflow.net/questions/314093 | 6 | In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen introduced obstructions to the Whitehead groups to satisfy a mayer-vietoris sequence.
See <https://mathscinet.ams.org/mathscinet-getitem?mr... | https://mathoverflow.net/users/21985 | Example of nonvanishing Waldhausen Nil group | There are examples of non-vanishing nil groups due to Daniel Juan-Pineda. See Juan-Pineda, Daniel(MEX-NAMMO-IM) [On higher nil groups of group rings](https://projecteuclid.org/euclid.hha/1201127332). Homology Homotopy Appl. 9 (2007), no. 2, 95–100 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2366944)).
| 4 | https://mathoverflow.net/users/4042 | 314112 | 136,487 |
https://mathoverflow.net/questions/314117 | 7 | I have several questions regarding representability of matroids.
**Question 1.** Does there exist a finite matroid that is representable over an infinite field, but is not representable over any finite field?
**Question 2.** Does there exist a finite matroid that is representable over a field of characteristic $0$,... | https://mathoverflow.net/users/35484 | Representability of matroids over finite fields | The main results of Rado's [Note on Independence Functions](https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s3-7.1.300) settle all three questions. The first few lines of [Effective Versions of Two Theorems of Rado](http://homepages.ecs.vuw.ac.nz/%7Emayhew/Publications/BFKM.pdf) give a perfect recap of... | 8 | https://mathoverflow.net/users/94968 | 314136 | 136,493 |
https://mathoverflow.net/questions/314137 | 8 | This question arises in the context of a question asked on MSE: [Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$](https://math.stackexchange.com/questions/2961108/are-concrete-riemann-surfaces-riemann-domains-over-mathbb-c). Part of the answer to that question is the question above which is being asked ... | https://mathoverflow.net/users/124862 | Embedding open connected Riemann surfaces in $\mathbb{C}^2$ | This is an open problem, known as
>
> **Bell-Narasimhan conjecture**. Every open Riemann surface admits a proper holomorphic embedding into $\mathbb{C}^2$.
>
>
>
Look at F. Forstnerič's book *[Stein Manifolds and Holomorphic Mappings](https://www.springer.com/la/book/9783642222498)*, Problem 9.10.1 p. 446.
| 4 | https://mathoverflow.net/users/7460 | 314140 | 136,495 |
https://mathoverflow.net/questions/312608 | 6 | Let $f$ be a binary quadratic form with integer coefficients and non-zero discriminant $D$. Suppose for simplicity that $D$ is a fundamental discriminant (which in particular implies that $f$ is primitive). We say that $f$ *represents a square mod* $D$ if for any $a$ representable by $f$ we have
$$a \equiv \square \p... | https://mathoverflow.net/users/10898 | Distribution of 'square classes' of binary quadratic forms | The standard definition for a form to belong to the principal genus of forms with fundamental discriminant $d$ is that the primes $p$ coprime to $d$ that
the form $Q$ represents satisfy $(d\_1/p) = \ldots = (d\_t/p)$, where
$d =d\_1 \cdots d\_t$ is the factorization of $d$ into prime discriminants. In particular, $Q$ ... | 3 | https://mathoverflow.net/users/3503 | 314147 | 136,497 |
https://mathoverflow.net/questions/311637 | 6 | Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has *finitely many* orbits, and every point stabiliser $G\_x$ has *finite* orbits.
Can we always find a permutation $\tau\in\operatorname{Sym}(X)$ (not necessarily of finite order or finite support) such that $H=\langle G,\tau\rangle$ is trans... | https://mathoverflow.net/users/57533 | Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one? (follow-up) | It seems that a counterexample looks as follows (it is somewhat siimilar with the counterexample to your previous question). Let $k$ be a large integer. Take an infinite tree $T=(V,E)$, where all degrees equal $k+1$. Let $G$ be its group of automorphisms. $G$ acts on $V\cup E$ with two obvious orbits, and all orbits of... | 4 | https://mathoverflow.net/users/17581 | 314151 | 136,499 |
https://mathoverflow.net/questions/314148 | 10 | Let $A \subseteq \mathbb{N}$, define the *upper density* of $A$ as,
$$
\overline{\delta}(A) := \limsup\_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}.
$$
This naturally leads to a weaker form of convergence of a sequence $(x\_n)$ in $\mathbb{R}$:
A sequence $(x\_n)$ converges in *density* to $x\in \mathbb{R}$ if ... | https://mathoverflow.net/users/116999 | Density-$c_0$ in $\ell^\infty$ | This type of convergence is often called [statistical convergence](https://scholar.google.com/scholar?hl=en&q=%22statistical+convergence%22).
The paper Constantin P. Niculescu, Gabriel T. Prajitura: *Some open problems concerning the convergence of positive series*
([arXiv:1201.5156](https://arxiv.org/abs/1201.5156))... | 11 | https://mathoverflow.net/users/8250 | 314152 | 136,500 |
https://mathoverflow.net/questions/314146 | 8 | A semigroup $S$ is called
$\bullet$ *$n$-Shelah* for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ *Shelah* if $S$ is $n$-Shelah for some $n\in\mathbb N$.
>
> **Question.** Can an infinite Shelah semigroup be commutative?
>
>
>
This problem was motivated ... | https://mathoverflow.net/users/61536 | Can a Shelah semigroup be commutative? | An infinite Shelah semigroup must be a Jonsson semigroup (meaning that it is an infinite semigroup whose proper subsemigroups have lesser power). Therefore the following paper answers the question asked on this page:
McKenzie, Ralph
On semigroups whose proper subsemigroups have lesser power.
Algebra Universalis... | 13 | https://mathoverflow.net/users/75735 | 314162 | 136,503 |
https://mathoverflow.net/questions/314130 | 5 | Let $\left(U, d\right)$ be a finite [ultrametric space](https://en.wikipedia.org/wiki/Ultrametric_space) -- that is, $U$ is a finite set, and $d : U \times U \to \mathbb{R}\_{\geq 0}$ is a metric on $U$ such that every $x, y, z \in U$ satisfy $d\left(x, z\right) \leq \max\left\{d\left(x,y\right), d\left(y,z\right)\righ... | https://mathoverflow.net/users/2530 | Greedy simplices in an ultrametric space (generalized Bhargava $p$-orderings) | It looks that both claims are true.
For $V\subset U$ denote by $f(V)$ the sum of mutual distances between elements of $V$.
Exchange lemma. Let $V\subset U$ and $v\in U$, and let $w$ be a projection of $v$ on $V$ (that is, a point in $V$ that is closest to $v$). Then, $f(V\cup v\setminus w)\geqslant f(V)$ and $d(v,... | 2 | https://mathoverflow.net/users/4312 | 314166 | 136,505 |
https://mathoverflow.net/questions/314077 | 4 | The *odd girth* of a graph $G$ is defined as the minimum length of an odd cycle in $G$. Let $n\_g(k)$ denote the minimum number of vertices in a $k$-chromatic graph of odd girth $g$. What are the known upper and lower bounds on $n\_g(k)$?
It is known that the order of magnitude of $n\_5(k)$ is $k^2 \log k$; this corr... | https://mathoverflow.net/users/90417 | Minimum number of vertices in a $k$-chromatic graph of odd girth $g$ | In [this](https://link.springer.com/article/10.1007%2Fs10958-012-0882-4?LI=true) paper (also available [here](https://arxiv.org/abs/1201.3271)) we show that a graph having no odd cycles of length $\leq 2r-1$ and having at most
$$
\frac{(k+r)(k+r+1)\dots(k+2r-1)}{2^{r-1}r^r}
$$
vertices is properly $k$-colorable. This ... | 4 | https://mathoverflow.net/users/17581 | 314169 | 136,506 |
https://mathoverflow.net/questions/314168 | 2 | **Takahashi minimization theorem** says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded from below, $Z=\{x \in X: f(x)=\inf f\}$. Let for all $x \in X \setminus Z$, there exists $y\in X\setminus \{x\}$ su... | https://mathoverflow.net/users/117299 | Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces | The Takahashi Theorem holds also for lower pseudo-continuous functions.
To derive a contradiction, assume that $f:X\to [0,+\infty]$ a proper lower pseudo-continuous function such that for any point $x\in X$ with $f(x)<+\infty$ there exists a point $y\in X\setminus\{x\}$ such that $f(y)\le f(x)-d(x,y)<f(x)$.
**Claim... | 3 | https://mathoverflow.net/users/61536 | 314178 | 136,509 |
https://mathoverflow.net/questions/75238 | 15 | A colleague asked me to locate the following paper on the web:
Kovalenko, I.N.: On the reconstruction of an additive type of distribution based upon a sequence of independent trials.
Memoirs of the All-Union Conference on Probability Theory and Mathematical Statistics, Erevan 1958.
After a few failed attempts, I... | https://mathoverflow.net/users/12518 | How to locate an obscure paper? | University of Yerevan has a copy (<http://89.249.207.20/cgi-bin/koha/opac-detail.pl?biblionumber=156826&query_desc=kw%2Cwrdl%3A%20%D0%A1%D0%BE%D0%B2%D0%B5%D1%89%D0%B0%D0%BD%D0%B8%D1%8F%20%D0%BF%D0%BE%20%D1%82%D0%B5%D0%BE%D1%80.%20%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD.%20%D0%B8%20%D0%BC%D0%B0%D1%82%D0%B5%D0%BC.%20%... | 2 | https://mathoverflow.net/users/13268 | 314185 | 136,511 |
https://mathoverflow.net/questions/307944 | 3 | Let $V$ be a finite set, $G$ a simple graph with vertex set $V$, and $H$ a hypergraph (i.e., set of subsets) with vertex set $V$ satisfying the following three conditions:
* each pair of elements of $V$ is contained in a unique hyperedge of $H$;
* for $x, y, z \in V$, if $x$ is $G$-adjacent to $z$ and $y$ is $G$-adja... | https://mathoverflow.net/users/25028 | On the existence of a certain graph/hypergraph pair | I'm sorry, but I do not understand why you cannot model the same example over a finite field.
Let, say, $V$ be the set of non-isotropic points in the projective space $P^3(\mathbb F\_p)$ for a large $p$, i.e., the points $(x:y:z:t)$ with $x^2+y^2+z^2+t^2\neq0$. Again, two points are connected in $G$ if they are ortho... | 1 | https://mathoverflow.net/users/17581 | 314203 | 136,520 |
https://mathoverflow.net/questions/314204 | 0 | Let $A$ be a set of integers with irrational natural density. That is, suppose that
$\lim\_{n\to\infty}\frac{\#(A\cap [-n,n))}{2n}$
exists and is irrational. Denote this value by $\alpha$. Now let $p$ be an odd polynomial with integer coefficients, and define the set $S\_{p}(k) = \{p(i) : i \in [-k, k)\cap\mathbb{Z... | https://mathoverflow.net/users/127904 | Irrational natural density set, intersected with odd polynomial | A slight modification of your example shows the answer is no. Let $A$ consist of the even numbers together with a set of odd numbers with irrational density, and take $p(x) = 2 x$.
| 2 | https://mathoverflow.net/users/13650 | 314207 | 136,522 |
https://mathoverflow.net/questions/314209 | 3 | I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive formula for the characters of $GL\_n(\mathbb{F}\_q)$ but there does not seem to be any literature on a Littlewood-Richardson ... | https://mathoverflow.net/users/130490 | Littlewood Richardson Rule for general linear group over finite field | The symmetric group $S\_n$ is the $q=1$ "limit" of $\mathrm{GL}\_n(q)$. We expect the $q=1$ case to be simpler than $q>1$. But decomposing tensor products of irreducible $S\_n$-characters is difficult (see [Ikenmeyer, Mulmuley, and Walter
- On vanishing of Kronecker coefficients](https://arxiv.org/abs/1507.02955) and ... | 4 | https://mathoverflow.net/users/2807 | 314212 | 136,523 |
https://mathoverflow.net/questions/314201 | 7 | Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space).
$T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T\_{\#\mu}$ and $\nu^\prime = T\_{\#\nu}$ be the resulting push-forward measures on the space $Y$.
Let $d(\mu, \nu)$ denote some di... | https://mathoverflow.net/users/23911 | How does a statistical divergence change under a Lipschitz push-forward map? | If $d(\mu,\nu)$ is taken to be the total variation metric, then Lipschitz and metric properties don't matter. This is due to a "data processing inequality" of sorts: applying a transformation can only make two distributions closer in TV. I'll illustrate this for discrete sets $X,Y$:
$$ d(\mu',\nu') =\sum\_{y\in Y}|\mu'... | 4 | https://mathoverflow.net/users/12518 | 314215 | 136,524 |
https://mathoverflow.net/questions/314098 | 5 |
>
> Is there a non-zero real number $t$ for which there exist infinitely
> many prime numbers $p$ with $p^{it}$ an algebraic integer?
>
>
>
I would even be surprised to find a real $t \neq 0$ with both $2^{it}$ and $3^{it}$ being algebraic integers.
| https://mathoverflow.net/users/38889 | Algebraic exponential values | The six exponentials theorem (look it up) states that if $x\_1$ and $x\_2$ are two complex numbers which are linearly independent over $\mathbb{Q}$ and if $y\_1,y\_2$ and $y\_3$ are three complex numbers linearly independent over $\mathbb{Q}$ then at least one of the numbers $e^{x\_i y\_j}$ will be transcendental.
T... | 8 | https://mathoverflow.net/users/nan | 314221 | 136,527 |
https://mathoverflow.net/questions/314079 | 0 | Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W\_J = \langle s: s \in J\rangle \subseteq W$.
Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$.
**Proposition.** For each $w \in W$ there is a unique $u \in W^J$ and $v \in W\_J$ such that $w = uv$.
Moreover, it holds f... | https://mathoverflow.net/users/110229 | About generator of minimal length coset representatives | In fact the minimal length coset representatives are exactly the elements $w$ such that for all reflections $t\in W\_J$ we have $\ell(wt) \gt \ell(w) $. So the answer to your question is yes. Specifically, suppose $w\in W^J$ has the reduced word $s\_1\cdots s\_k$. Define
$$t\_i=s\_ks\_{k-1}\cdots s\_i\cdots s\_{k-1}s\_... | 2 | https://mathoverflow.net/users/62135 | 314225 | 136,528 |
https://mathoverflow.net/questions/314222 | 13 | Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}\_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\mathbb{F}\_p$?
| https://mathoverflow.net/users/nan | Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$? | It is known that ${\rm PSL}\_{2}(\mathbb{F})$ can be realized for some $\mathbb{F}$ but definitely not all at present. According to David Zywina's note [here](http://pi.math.cornell.edu/~zywina/papers/smallGalois.pdf), ${\rm PSL}\_{2}(\mathbb{F}\_{27})$ is the smallest non-abelian finite simple group for which it's not... | 15 | https://mathoverflow.net/users/48142 | 314226 | 136,529 |
https://mathoverflow.net/questions/314181 | 5 | If I understand correctly, I can obtain the $O$-cobordism group of
$$
\Omega^{O}\_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4,
$$
The 3d cobordism invariants have 4 generators of mod 2 classes, are generated by
$$g^3,$$
$$g w\_2'(V\_{SO(3)}),$$
$$w\_3'(V\_{SO(3)}),$$
$$g w\_1(T)^2.$$
Denote:
The $T$ for the spacetime tang... | https://mathoverflow.net/users/106497 | Manifold generators of O-bordism invariants | $\newcommand{\RP}{\mathbb{RP}}\newcommand{\ang}[1]{\langle #1\rangle}\newcommand{\R}{\mathbb R}$A representative of
a class in $\Omega\_3^O(BO\_3)$ is a 3-manifold $M$ together with a principal $O\_3$-bundle $P\to M$. Principal
$O\_3$-bundles are equivalent to rank-3 real vector bundles, so I will use 3-manifolds $M$ t... | 7 | https://mathoverflow.net/users/97265 | 314228 | 136,530 |
https://mathoverflow.net/questions/314230 | 5 | Suppose $r$ integers $n\_1, \ldots, n\_r$ are given such that $0<|n\_i|<N$ for $1 \leq i \leq r$ and for a natural number $N.$ Is it possible to find a prime number $p$ such that all numbers $n\_1, \ldots, n\_r$ are quadratic nonresidue mode $p$? Can we say $p < N^6?$
I think the answer to my first question is yes, ... | https://mathoverflow.net/users/80245 | Quadratic Nonresidue | I don't think this is true in general, for instance, if there exists $1\leq i <j<k<r$ such that $n\_in\_j = n\_k$, then $(n\_i/p)=(n\_j/p)=-1\implies (n\_k/p)=1$ (where $(a/p)$ is the Legendre symbol). Moreover, if $n\_i$ is a perfect square, then trivially you have that for every prime $p$, $n\_i$ is a quadratic resid... | 6 | https://mathoverflow.net/users/127150 | 314231 | 136,531 |
https://mathoverflow.net/questions/314234 | 1 | Let $B(x\_{0},r)$ be the ball of center $ x\_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x\_{0},r)$. Let $\mu$ be the Riesz measure of $u$ on this ball. If $u$ is $C^{2}-$smooth, is it true that $\mu=\Delta u$, the laplacian of $... | https://mathoverflow.net/users/100746 | Riesz measure of a smooth subharmonic function on a ball | See [W.K. Hayman and P.B. Kennedy, Subharmonic functions, Vol. I, London-...-San Francisco, Academic Press. 1976](https://zbmath.org/?q=an%3A0419.31001), 3.5.4 ( Ch.3, Par. 5, Point 4) to this end.
| 2 | https://mathoverflow.net/users/35959 | 314236 | 136,532 |
https://mathoverflow.net/questions/314223 | 6 | I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$.
1. Since $\pi\_1(O(3)) = \mathbb{Z}\_2$, there are two possible framings up to equivalence. Is it possible to distinguish between them in a natural way (similar to how orientabili... | https://mathoverflow.net/users/130748 | Framings for 2-surgeries on 4-manifolds | You are right that there are two possible ways to perform a 1-surgery, but I don't think that there is a way to choose between them. Think for instance about the simple case where you do a 1-surgery to $S^3 \times S^1$ along the circle $\{p\} \times S^1$. There are two possible choices, but there is no way to choose be... | 5 | https://mathoverflow.net/users/6205 | 314241 | 136,534 |
https://mathoverflow.net/questions/314240 | 5 | In a math column in *Scientific American* many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any pointers or references.
Let $\mathbb{N}$ be the set of positive integers and let $T = \{2^n: n\in \mathbb{N}\cup \{0\}\}$ ... | https://mathoverflow.net/users/8628 | Number defined by a recursive binary sequence | Let $\{t(i)\}\_0^\infty$ be the [Thue-Morse sequence](https://oeis.org/A010060). (It starts $0,1,1,0,1,0,0,1,\ldots$.).
I claim that your sequence is described by $a(n)=1-t(n-1)$ where $n$ is a positive integer. (It is similar to sequence [A010059](https://oeis.org/A010059) in the OEIS, but its index starts at $1$ in... | 10 | https://mathoverflow.net/users/12357 | 314242 | 136,535 |
https://mathoverflow.net/questions/314232 | 12 | Let $A$ be a noetherian local ring with residue field $k$, one can consider $\operatorname{Ext}^i(k,k)$ for every natural number $i$. If it is zero for large $i$, then $A$ is regular and the converse is also true. Then, for which rings the dimension of $\operatorname{Ext}^i(k,k)$ is bounded with respect to $i$ ?
Exam... | https://mathoverflow.net/users/102104 | Noetherian local ring and the growth of $\dim_k \operatorname{Ext}^i(k,k)$ | The commutative noetherian rings such that the Betti numbers of $k$ eventually grow polynomially are precisely the complete intersections. This is a theorem of Gulliksen, see Theorem 2.3 [here](https://www.jstor.org/stable/24491379) .
The degree of the polynomial giving the growth (which is called the complexity if ... | 20 | https://mathoverflow.net/users/310 | 314245 | 136,536 |
https://mathoverflow.net/questions/314099 | 8 | Suppose $f: \mathbb{R}^d \to \mathbb{R}$ is a smooth convex function.
Consider the level sets of the function, namely $M\_s = \{x: f(x) = s\}$.
Is it true/known that the surface areas of $M\_s$ are log-concave as a function of $s$?
(This feels awfully like a Brunn-Minkowski style inequality, but I'm unsure if it... | https://mathoverflow.net/users/69071 | Log-concavity of areas of level sets | Yes, this is true, and you are right, this follows from a generalization of the Brunn-Minkowski inequality.
Let $K\_s = \{x \mid f(x) \le s\}$, so that $M\_s = \partial K\_s$. We have $K\_s \supseteq (1-s)K\_0 + sK\_1$, thus the surface area of the former is $\ge$ the surface area of the latter.
The surface area of... | 11 | https://mathoverflow.net/users/98590 | 314252 | 136,537 |
https://mathoverflow.net/questions/314250 | 9 | There are two somewhat widely known theorems which say
* if $A$ is a nonnegatively graded commutative algebra in char $0$, then forgetful map on operadic cohomology $H^\*\_{Harr}(A, A) \to H^\*\_{Hoch}(A, A)$ is injective
* if $L$ is a Lie algebra in char $0$, then $H^\*(L, L) \to H^\*(L, UL)$ is injective
Are they... | https://mathoverflow.net/users/81055 | "Exactness" of operadic cohomology | The second result you mention is significantly easier than the first. Indeed from the PBW theorem we know that $L$ is a direct summand of $UL$ as an $L$-module, and the second result follows. But in fact there is a strong connection between the two results, since the first theorem may also be seen as a consequence of a... | 13 | https://mathoverflow.net/users/1310 | 314254 | 136,538 |
https://mathoverflow.net/questions/314251 | 6 |
>
> Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
>
>
> **(1)** $\operatorname{Jac}(f,g)=f\_xg\_y-f\_yg\_x = 0$.
>
>
> **(2)** There exist no $h \in \mathbb{Z}[x,y]$ such that $f,g \in \mathbb{Z}[h]$.
>
>
>
Please see th... | https://mathoverflow.net/users/72288 | $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$? | $\def\ZZ{\mathbb{Z}}\def\QQ{\mathbb{Q}}$No. As explained in [this question](https://mathoverflow.net/questions/190920), in $\mathbb{Q}[x,y]$, the condition $\operatorname{Jac}(f,g)=0$ implies that there exists an $h \in \mathbb{Q}[x,y]$ such that $f$ and $g$ are in $\mathbb{Q}[h]$. We now need some lemmas that are basi... | 7 | https://mathoverflow.net/users/297 | 314264 | 136,543 |
https://mathoverflow.net/questions/311420 | 8 | Galaz-Garcia and Guijarro proved the geometrization of closed (compact, boundaryless) Alexandrov 3-spaces. Part of the strategy was to use the so-called ramified double cover $\tilde{X}$ of the space $X$. This ramified cover is a smooth $3$-manifold. Being this the case, the space $X$ would be isometric to a Riemannian... | https://mathoverflow.net/users/129376 | Does geometrization of Alexandrov 3-spaces follow from that of 3-orbifolds? | In principle, yes. Note, however, that topologically singular Alexandrov 3-spaces are homeomorphic to non-orientable orbifolds. We could not find an appropriate reference for the geometrization in the non-orientable case (where topological singularities are present). To the best of my knowledge, the geometrization stat... | 5 | https://mathoverflow.net/users/1944 | 314267 | 136,544 |
https://mathoverflow.net/questions/314279 | 9 | Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $\mathbb{C}$. Let $D^\*$ be the punctured open unit disc.
I am looking for an analogue of the valuative criterion of properness in complex analysis.
Is the following correct?
>
>
> >
> > The complex analytic space $X$ is co... | https://mathoverflow.net/users/130799 | What is the "analytic" analogue of the valuative criterion of properness | No, the open disk in $\mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).
| 11 | https://mathoverflow.net/users/35353 | 314283 | 136,550 |
https://mathoverflow.net/questions/314281 | 4 | Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell\_2$ such that the linear hull of $B$ is dense in $\ell\_2$.
>
> **Question.** Is there a non-compact linear bounded operator $T:\ell\_2\to \ell\_2$ such that $T(B)$ is compact?
>
>
>
| https://mathoverflow.net/users/61536 | Compact images of nowhere dense closed convex sets in a Hilbert space | Your revised assumption is that the norm (rather than semi-norm after the revision) on $\ell\_2$ given by sup-ing against vectors in $B$ is not equivalent to the usual norm on any finite codimensional subspace. So there is an ON sequence $x\_n$ in $\ell\_2$ s.t. $\sup \{\langle x\_n,b\rangle : b\in B \} \to 0$ as $n\to... | 4 | https://mathoverflow.net/users/2554 | 314284 | 136,551 |
https://mathoverflow.net/questions/314288 | 14 | Let $U$ be an affine open subscheme of an abelian variety $A$ over $\mathbb{C}$. Is $A-U$ an **ample** divisor?
If $\dim A =1$ this is true.
If $\dim A = 2$, the complement is a divisor $D\_1+\ldots + D\_n$. If all of these are elliptic curves, then the complement is not affine (as it will contain the translate of o... | https://mathoverflow.net/users/130799 | Is the complement of an affine open in an abelian variety ample? | Welcome new contributor. Yes, that is true. Let $k$ be any field, let $A$ be an Abelian variety over $k$, and let $U\subset A$ be a dense open affine. Denote by $D\subset A$ the complementary divisor with its induced reduced structure. Denote the invertible sheaf of this divisor by $\mathcal{L}:=\mathcal{O}\_A(D).$ Den... | 18 | https://mathoverflow.net/users/13265 | 314289 | 136,552 |
https://mathoverflow.net/questions/313233 | 6 | It is well-known that for a discrete group $G$ the following statements are equivalent:
* $C\_{red}^\*(G)$ nuclear
* $C\_{red}^\*(G) \cong C^\*(G)$ canonically i.e. there exists an \*-isomorphism between the full and the reduced group $C^\*$-algebras extending the identity on the corresponding conolution algebra.
A... | https://mathoverflow.net/users/64444 | Direct proof of "Nuclear implies $C_{red}^*(G) \cong C^*(G)$" | The following direct proof uses Fell's absorption principle, as well as its (easy) corollary: the "diagonal" map $C^\ast(G) \hookrightarrow C^\ast\_r(G) \otimes\_{\max{}} C^\ast\_r(G)$ is injective (see Theorem 8.2 in Pisier's book).
$(i) \Rightarrow (ii)$: If $C^\ast\_r(G)$ is nuclear, then the composition
\begin{eq... | 5 | https://mathoverflow.net/users/126109 | 314290 | 136,553 |
https://mathoverflow.net/questions/314049 | 1 | Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion of the open cone $Y = \left(L \times [0, 1)\right) / \left(L \times \{0\}\right)$.
Let $\boldsymbol{IC}^{\bullet}\_{\ov... | https://mathoverflow.net/users/88487 | What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure? | I just realized I made an error in my previous answer here, so I've updated it:
Do you mean $Ri\_\*$ or literally $i\_\*$? $Ri\_\*$ would be the more usual thing to ask about in this context. Then, in general, if $f:X\to Y$ we have $\mathcal H^i(Y;Rf\_\*S^\*)\cong \mathcal H^i(X;S^\*)$. So in this case you'd get agai... | 1 | https://mathoverflow.net/users/6646 | 314319 | 136,563 |
https://mathoverflow.net/questions/314308 | 5 | Let $G$ be a connected reductive group with maximal split torus $A\_0$, and $P = MN$ a parabolic subgroup with Levi $M$ containing $A\_0$. Let $A\_M$ be the split component of $\mathfrak a\_M^{\ast} = X(A\_M) \otimes \mathbb R$. Then the linear span of $\Phi(A\_M,G) \subseteq \mathfrak a\_M^{\ast}$ is usually not a gen... | https://mathoverflow.net/users/38145 | Reference for parabolic root systems | Try section "Relative roots" here: <http://math.stanford.edu/~conrad/249BW16Page/handouts/249B_2016.pdf>
They originate from the paper of Borel and Tits, I guess:
<http://www.numdam.org/item/PMIHES_1965__27__55_0/>
| 6 | https://mathoverflow.net/users/5107 | 314328 | 136,568 |
https://mathoverflow.net/questions/314321 | 6 | [This question](https://mathoverflow.net/questions/300253/t-2-spaces-where-all-non-empty-open-sets-are-homeomorphic) was about spaces in which all non-empty open sets "look alike".
Now I am interested in the opposite: Is there a $T\_2$-space $(X,\tau)$ with $|X|>1$ such that whenever $U\neq V$ are open subsets of $X$... | https://mathoverflow.net/users/8628 | $T_2$-spaces in which no two open sets are homeomorphic | There is an example in this paper, [A method for constructing ordered continua](https://fa.ewi.tudelft.nl/%7Ehart/37/publications/the_papers/published/ordered-continua.pdf), of an ordered continuum in which no two intervals are homeomorphic.
Because an open set is a union of a disjoint family of intervals a homeomorp... | 9 | https://mathoverflow.net/users/5903 | 314332 | 136,571 |
https://mathoverflow.net/questions/301650 | 2 | If $(X,\tau)$ is a $T\_2$-space such that [all non-empty open sets are homeomorphic](https://mathoverflow.net/questions/300253/t-2-spaces-where-all-non-empty-open-sets-are-homeomorphic) (with the subspace topology) to $X$, is there for all $x,y\in X$ with $x\neq y$ a clopen (closed and open) set containing $x$ but not ... | https://mathoverflow.net/users/8628 | Homeomorphic open sets and total disconnectedness | According to this paper, [Spaces of diversity two](https://doi.org/10.1016/0166-8641(95)00073-9), the paper [Spaces of diversity one](https://zbmath.org/?q=an%3A0724.54020) by Franklin and Rajagopalan contains examples of various spaces with just one open set, also one that is not totally disconnected. Unfortunately th... | 3 | https://mathoverflow.net/users/5903 | 314339 | 136,572 |
https://mathoverflow.net/questions/314259 | 13 | Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name for such a general construction? To me, it seems like the most straight-forward example of a semi-direct product.
2) A... | https://mathoverflow.net/users/94546 | Is there a name of semidirect product of a group with its automorphism group? | As a first remark, note that if $\tilde{H}\leq G\rtimes \operatorname{Aut}(G)$ is a finite subgroup and $G$ is torsion-free, then the projection $p: G\rtimes \operatorname{Aut}(G)\to \operatorname{Aut}(G)$ maps $\tilde{H}$ isomorphically to some finite subgroup $H=p(\tilde{H})\leq \operatorname{Aut}(G)$.
Now for eac... | 6 | https://mathoverflow.net/users/8103 | 314340 | 136,573 |
https://mathoverflow.net/questions/314338 | 3 | $Let f=o(g)$ stand for $\lim\_{x\to\infty} f(x)/g(x)=0$.
It is simple to prove the following fact.
**Proposition.** Let $f\_0,f\_1:\mathbb{R}\to(0,\infty)$ be such that $f\_0=o(f\_1)$. Then there is a family of functions $(f\_t)\_{t\in[0,1]}$ such that $f\_s=o(f\_t)$ if $0\leq s<t\leq1$.
*Proof.* Consider $f\_t=f\_... | https://mathoverflow.net/users/59033 | Cardinality of growth rates | Let me rather define $f=o(g)$ as $\forall \varepsilon > 0, \exists x\_0, \forall x \geq x\_0, |f(x)| \leq \varepsilon |g(x)|$.
Let $f\_0,f\_1$ be such that $f\_0 = o(f\_1)$. Let us assume that we have a family $(f\_i)\_{i \in I}$ satisfying the properties $1,3,4$.
I am going to show that $\exists x\_0, \forall x \... | 5 | https://mathoverflow.net/users/21724 | 314341 | 136,574 |
https://mathoverflow.net/questions/314342 | 4 | (Apologies if this question is trivial, but I'm way outside my area here.)
Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived category. Suppose we have an object $X \in \operatorname{Obj} D^{\bullet}(R)$, and an endomorphism $t \in \operatorname{... | https://mathoverflow.net/users/2481 | Endomorphisms in the derived category | It slightly depends what you mean by "arrange that $X=Y$". As is the statement is not true. There are complexes with endomorphisms which are not realizable. However, what is true is the following:
Let $X\_\bullet$ be a complex. There exist a complex $X'\_\bullet$ and a morphism of complexes which is a quasi-isomorph... | 11 | https://mathoverflow.net/users/115052 | 314344 | 136,576 |
https://mathoverflow.net/questions/314255 | 3 | Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E\_1$ and $E\_2$ (Dirichlet BC $u=0$ on $E\_1$ and $u=1$ on $E\_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neum... | https://mathoverflow.net/users/73701 | Does current follow the path(s) of least (total) resistance? | Perhaps to resolve this issue it helps to work out a simple example.
Take a region $D$ consisting of the strip $|x|<1$, $0<y<1$, and a $y$-independent conductivity profile
$$\sigma(x)=\begin{cases}
1 &\text{for} -1<x<0,\\
2& \text{for}\;\;\;\; 0<x<1.
\end{cases}
$$
The solution of the Poisson equation $\nabla \cdot (... | 11 | https://mathoverflow.net/users/11260 | 314358 | 136,579 |
https://mathoverflow.net/questions/314353 | 5 | Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category.
There are two definitions of $\mathcal{U}$ category I've met.
>
> $(1)$ A category $\mathsf{C}$ is a $\mathcal{U}$-category if $\forall X,Y \in \mathsf{C}, \mathsf{Hom\_C}(X,Y) \in ... | https://mathoverflow.net/users/83143 | What is the definition of a $\mathcal{U}$-category? | I asked [a similar question](https://mathoverflow.net/questions/3409/how-should-we-define-locally-small) (note that there I called "$\mathcal{U}$-locally small categories" what you call "$\mathcal{U}$-categories"). I still don't have any strong opinion about this, but here are a few relevant points.
1. One generally ... | 8 | https://mathoverflow.net/users/126667 | 314367 | 136,584 |
https://mathoverflow.net/questions/312447 | 5 | Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the [Second Hardy-Littlewood Conjecture](https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture) which states that,
>
> For all $x,y\ge 2$ we have, $$\pi(x)+\pi(y)\ge \pi(x+y)$$where $\pi$ is the Pr... | https://mathoverflow.net/users/nan | Is the following weak version of second Hardy-Littlewood conjecture already known? | I have not seen your proposition before, but it follows routinely from the prime number theorem.
Indeed, there exists an absolute constant $c>0$ such that
$$\pi(ky)+\pi(y)-\pi((k+1)y)=\mathrm{Li}(ky)+\mathrm{Li}(y)-\mathrm{Li}((k+1)y)+O\_k\left(y e^{-c\sqrt{\log y}}\right),$$
where
$$\mathrm{Li}(ky)+\mathrm{Li}(y)-\... | 8 | https://mathoverflow.net/users/11919 | 314375 | 136,586 |
https://mathoverflow.net/questions/314195 | 1 | Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$.
Let $S \subset \mathbb{R}^n$ be a set with $\mathbb{P}[A \in S] = p$. I'm trying to show a lower bound on $\mathbb{P}[B \in S]$.
My intuition says that ... | https://mathoverflow.net/users/127773 | gaussian isoperimetric result for minimal measure under translation | Mateusz's comment is correct. Also, it turns out that this result is known in statistics as the Neyman-Pearson lemma.
| 0 | https://mathoverflow.net/users/127773 | 314384 | 136,590 |
https://mathoverflow.net/questions/314386 | 16 | **Question.** Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}\_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?
**Remark.** The [coherator](https://stacks.math.columbia.edu/tag/08D6) yields a way to find a quasicoherent she... | https://mathoverflow.net/users/31233 | Does every sheaf embed into a quasicoherent sheaf? | That already fails for $X$ equal to $\text{Spec}\ R$, where $R$ is a DVR with generic point $\eta = \text{Spec}\ K$. Since there are only two nonempty open subsets of $X$, namely all of $X$ and $\{\eta\}$, there is a straightforward equivalence between the category of $\mathcal{O}\_X$-modules and the category of triple... | 35 | https://mathoverflow.net/users/13265 | 314387 | 136,592 |
https://mathoverflow.net/questions/314372 | 4 | I wonder if the following inequality involving skew symmetric matrices is true:
Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ is positive-definite. Then,
$$\mbox{Tr}(B^2 \Sigma C^2) - \frac{1}{2}\mbox{Tr}((CB) \Sigma (CB) + (CB) \Sigma (BC)) \... | https://mathoverflow.net/users/130840 | Norm/trace of product inequality involving skew symmetric matrices | Something seems to be missing here, because the inequality is trivially seen to be false. Consider the following randomly picked matrices for instance:
\begin{equation\*}
B = \begin{bmatrix}0 & -4 & 4\\ 4 & 0 & -10\\ -4& 10 & 0\end{bmatrix},\
C = \begin{bmatrix}0 & -6 & 11\\ 6 & 0 & -12\\ -11& 12 & 0\end{bmatrix},\ ... | 1 | https://mathoverflow.net/users/8430 | 314407 | 136,596 |
https://mathoverflow.net/questions/314378 | 37 | Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\infty)$ (not identically zero) such that the convolution $f \* g$ is also non-negative. In other words, is there a convex ... | https://mathoverflow.net/users/766 | When can a function be made positive by averaging? | Here is the divisibility theorem for polynomials (and thus for Laurent polynomials in several variables), chapter 3 in *Positive polynomials, convex integral polytopes, and a random walk problem* SLN 1282 (1986 or so) [either this, or *Positive polynomials and product type actions of compact groups* in Memoirs AMS 320;... | 17 | https://mathoverflow.net/users/42278 | 314433 | 136,602 |
https://mathoverflow.net/questions/314429 | 2 | I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL\_n/B)$?
With respect to the [property of Kendall-Mann numbers](https://mathoverflow.net/questions/46368/the-property-of-kendall-mann-numbers) where the statement appeared I wonder if there are any simil... | https://mathoverflow.net/users/10903 | Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space? | More generally, for any sequence $0 < k\_1 < k\_2 < \cdots < k\_r < n$ of positive integers, let $F(k\_1, k\_2, \ldots, k\_r; n)$ be the set of flags $V\_1 \subset V\_2 \subset \cdots V\_r \subset \mathbb{C}^n$ with $\dim V\_j = k\_j$. We have a map $F(k\_1, k\_2, \ldots, k\_{r-1}, k\_r; n) \to F(k\_1, k\_2, \ldots, k\... | 3 | https://mathoverflow.net/users/297 | 314434 | 136,603 |
https://mathoverflow.net/questions/313948 | 2 | Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x\_{0,n},x\_{1,n}...,x\_{n,n}\}\_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that the sequence of interpolating polynomials $p\_n(x)$ of $f$ at those points satisfy $(-\infty,0]\cap p\_n([a,b])\ne \ph... | https://mathoverflow.net/users/127118 | For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often | Yes, it follows from the following result of S. Bernstein (Quelques remarques sur l'interpolation, Math. Ann. 79 (1918), 1-12):
>
> For an arbitrary scheme $X=\cup\_{n} A\_{n}$ of points in $[-1,1]$,
> there exist a continuous function $f$ and a point $x\_{0}$ in $[-1,1]$
> such that $$ { \limsup \_ { n \rightarr... | 1 | https://mathoverflow.net/users/89429 | 314457 | 136,611 |
https://mathoverflow.net/questions/314454 | 5 | The Klein Gordon equation of the form:
$\Delta u+ \lambda u^p=0$
is been studied for $p = 2$?
(i.e.$\Delta u+ \lambda u^2=0$)
If yes are there references?
| https://mathoverflow.net/users/111304 | Klein Gordon equation - references | This is a nonlinear Klein-Gordon equation of the type
$$ \Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., [sine-Gordon equation](https://en.wikipedia.org/wiki/Sine-Gordon_equation)... | 4 | https://mathoverflow.net/users/394 | 314461 | 136,612 |
https://mathoverflow.net/questions/314459 | 1 | A while ago I saved an internet reference to a work by P. Moree and G. Niklasch, published exclusively on a website, related to high-precision computations of constants related to prime numbers. I believe the paper was entitled
"*Ultraprecision number-theoretical constants*"
and was dated somewhere from 2002. The ... | https://mathoverflow.net/users/103722 | Looking for a reference by Moree and Niklash | The WaybackMachine has the page referenced saved here (from March 2018): <http://web.archive.org/web/20180325095137/http://www.gn-50uma.de:80/alula/essays/Moree/Moree.en.shtml>.
| 4 | https://mathoverflow.net/users/118731 | 314464 | 136,614 |
https://mathoverflow.net/questions/314467 | 5 | If $(X,\tau)$ is a connected space, then $\tau$ [need not be contained in a maximal connected topology](https://mathoverflow.net/questions/201477/maximal-connected-topologies).
Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?
| https://mathoverflow.net/users/8628 | Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology? | Yes, see *[Maximal connected expansions of the reals](https://doi.org/10.1090/S0002-9939-1978-0467646-4)* by J. A. Guthrie, H. E. Stone and M. L. Wage, *Proc. Amer. Math. Soc. 69 (1978), 159-165.*
| 8 | https://mathoverflow.net/users/5903 | 314468 | 136,615 |
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