parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/323665 | 2 | Given the base case $a\_0 = 1$, does $a\_n = a\_{n-1} + \frac{1}{\left\lfloor{a\_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, 3, 3+\frac{1}{3}, 3+\frac{2}{3}, 4, 4+\frac{1}{4}, 4+\frac{2}{4}, 4+\frac{3}{4}, . . .$} and so forth. It seems lik... | https://mathoverflow.net/users/23508 | Does this deceptively simple nonlinear recurrence relation have a closed form solution? | The sequence $a\_n$ for $n\geq 1$ has the following formula:
$$a\_n=\left\lfloor \sqrt{2n}+\tfrac{1}{2}\right\rfloor +\frac{\left\lfloor \frac{1}{2} \left(\sqrt{8 n-7}+1\right)\right\rfloor-\left\lfloor \frac{1}{2} \left(\sqrt{8 n-7}+1\right)\right\rfloor ^2 +2 n}{2 \left\lfloor \sqrt{2n}+\frac{1}{2}\right\rfloor }.$$
... | 8 | https://mathoverflow.net/users/11260 | 323667 | 139,534 |
https://mathoverflow.net/questions/323672 | 1 | I read the Wikipedia article on [Percolation critical exponents](https://en.wikipedia.org/wiki/Percolation_critical_exponents). It says:
>
> In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the f... | https://mathoverflow.net/users/136074 | Percolation critical exponent $\nu$ does not depend on neighborhood connectivity. Does this follow from the universality principle? | For a reference on the independence of critical exponents on the type of two-dimensional lattice (square, triangular, honeycomb, bond/site percolation, nearest-neighbor+next nearest neighbor, ...) you could cite Staffer and Aharony's [Introduction to Percolation Theory](https://books.google.nl/books?id=v66plleij5QC) (p... | 2 | https://mathoverflow.net/users/11260 | 323676 | 139,536 |
https://mathoverflow.net/questions/260746 | 0 | Given
\begin{equation}\label{eq:definition\_of\_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}\_{11} & {z}\_{12} & \cdots & {z}\_{1P} \\
{z}\_{21} & {z}\_{22} & \cdots & {z}\_{2P} \\
{z}\_{31} & {z}\_{32} & \cdots & {z}\_{3P} \\
\vdots & \vdots & \ddots & \vdots \\
{z}\_{M1} & {z}\_{M2} & \cdots & {z}\_{MP... | https://mathoverflow.net/users/103291 | Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v | First, by the Cauchy-Schwarz inequality, we have
$$\mathbb{E} [ | \textbf{z}\_1^{H} \textbf{z}\_2 | ] \le\sqrt{\mathbb{E}[|\textbf{z}\_1|^2] \mathbb{E}[|\textbf{z}\_2|^2]} = \mathbb{E}[|\textbf{z}\_1|^2]<\infty,$$
hence, $\mathbb{E} [ \textbf{z}\_1^{H} \textbf{z}\_2 ]$ exists and is finite.
Therefore and because th... | 0 | https://mathoverflow.net/users/103291 | 323677 | 139,537 |
https://mathoverflow.net/questions/323685 | 0 | For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$,
$\lim\limits\_{n\rightarrow\infty}\sum\limits\_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$
Could it more than linear functions?
| https://mathoverflow.net/users/136078 | A functional equation in real analysis | A theorem of Sergei Bernstein says that if $u$ is continuous, then the sequence of functions on the left-hand side converges *uniformly* to $u$ on $[0,1]$. The polynomials on the left hand side are called the [Bernstein polynomials.](https://en.wikipedia.org/wiki/Bernstein_polynomial#Approximating_continuous_functions)... | 8 | https://mathoverflow.net/users/20302 | 323687 | 139,540 |
https://mathoverflow.net/questions/323690 | 1 | I have two points $x\_1,x\_2 \in \mathbb S^n $ which are distant $d$ from each other, where $d<<1$.
I also have a vector $v$ sampled uniformly at random from $\mathbb S^n$.
What is the probability that $x\_1$ and $x\_2$ lie on different sides of the hyperplane perpendicular to $v$?
Thank you!
| https://mathoverflow.net/users/93775 | Probability of two Points being divided by an high-Dimensional Hyperplane | Let us change notation somewhat: Let $x=(x\_1,\dots,x\_n)$ and $y=(y\_1,\dots,y\_n)$ be points in $\mathbb R^n$ such that $|x|=|y|=1$ and $|x-y|=d\in(0,1)$, where $|\cdot|$ is the Euclidean norm. Let $v$ be a random vector uniformly distributed on the unit sphere $S^{n-1}$ in $\mathbb R^n$. The probability in question ... | 0 | https://mathoverflow.net/users/36721 | 323697 | 139,542 |
https://mathoverflow.net/questions/322245 | 1 | For a set $X$ by $B(X)$ we denote the family of all finite subsets of $X$ endowed with the operation $\oplus$ of symmetric difference. This operation turns $B(X)$ into a Boolean group, which can be identified with the free Boolean group over $X$.
Let us recall that a group is *Boolean* if each its element has order a... | https://mathoverflow.net/users/61536 | The complexity on calculation of the Graev metric on the free Boolean group of a metric space | If I understand the problem correctly, then based on YCor's comment, your problem is equivalent to finding a minimum-weight perfect matching on $A \oplus B$, where the weights are given by the metric. If so, then a version of Edmond's "blossom algorithm" should be able to solve it in polynomial time.
Check out Kolmog... | 1 | https://mathoverflow.net/users/45707 | 323701 | 139,544 |
https://mathoverflow.net/questions/323707 | 3 | Suppose that $A$ is a $d^2$ dimensional algebra over $\mathbb{C}$ and we know the multiplication tensor $c\_{ij}^k$ and the unit $u^k$ in some basis. If $A$ is semi-simple and has a single simple representation, then computing an explicit isomorphism $A \cong M\_d(\mathbb{C})$ is often called **the explicit isomorphis... | https://mathoverflow.net/users/4002 | Algorithms for the explicit matrix isomorphism problem over $\mathbb{C}$ | When specifying an algorithmic problem "over $\mathbb{C}$", you should say precisely what the input and output should be.
The isomorphism problem is less studied over $\mathbb{C}$ because it is much easier than over a number field.
Here is the basic idea: if you can find an element $a\in A$ such that right multipli... | 3 | https://mathoverflow.net/users/40821 | 323714 | 139,547 |
https://mathoverflow.net/questions/323703 | 1 | The independence theorem says, roughly, that a family of $\sigma\in\mathrm{Aut}\_k(K)$ is $K$-linearly independent. In his Algebra, Serge Lang gave a proof of independence theorem following Artin (which is also the proof used in Jacobson's lecture notes), mentioning that Dedekind used a different approach, with a "suit... | https://mathoverflow.net/users/54774 | Dedekind's original proof of independence theorem | It seems that the original proof can be found in Volume 2 of "Gesammelte matematische Werke" [Complete mathematical works], published in 1930, which was compiled and edited by Fricke, Noether and Ore. More specifically, you can find the full proof -- two pages -- starting on Page 416, Section XLIV. This proof seems to ... | 3 | https://mathoverflow.net/users/120914 | 323716 | 139,548 |
https://mathoverflow.net/questions/320811 | 2 | Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$.
I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism of Lie groups (not sure if this has to be submersion) and that $S^1\subseteq Z(\hat{G})$. There is a local section $\sig... | https://mathoverflow.net/users/118688 | Central extension gives a gerbe over stack | As discussed in the comments, $[X/\hat{G}] \to [X/G]$ is the pullback of $[pt/\hat{G}] \to [pt/G]$ along the canonical map $[X/G] \to [pt/G]$, so it suffices to show that $[pt/\hat{G}] \to [pt/G]$ is a gerbe. Since every principal $G$-bundle is locally trivial, it can be locally lifted to a principal $\hat{G}$-bundle, ... | 1 | https://mathoverflow.net/users/4177 | 323725 | 139,550 |
https://mathoverflow.net/questions/298778 | 23 | The *diagonal Ramsey number* $R(n,n)$ is the least number $m$ for which the following holds: in any edge-colouring of the complete graph $K\_m$ in which each edge is coloured blue or red, there is a monochromatic $K\_n.$
The *list Ramsey number* (my name for it) $R\_\ell(n)$ is the least number $m$ for which the foll... | https://mathoverflow.net/users/43266 | List Ramsey numbers? | we stumbled upon the same concept in a different way and discovered this topic when looking if someone studied it already.
We have figured out some interesting things, such as:
* disproving your general conjecture (it is false for matchings),
* proving it is true for stars (except for very small stars),
* proving ... | 12 | https://mathoverflow.net/users/136102 | 323729 | 139,552 |
https://mathoverflow.net/questions/323286 | 2 | Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$.
A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\partial}\phi$ is a Kähler form.
**Question 1:** Given an arbitrary smooth function $f$ on $X$, is it possible to find two... | https://mathoverflow.net/users/135826 | Differences of $\omega$-plurisubharmonic functions | On your Question 1: the answer depends on the manifold.
For instance, if $X$ is the unit disk $D=\{z\in {\mathbb C}\vert |z|<1\}$, then, given $f\in {\mathcal C}^{\infty}(X,{\mathbb R})$, there exist two smooth plurisubharmonic functions $\phi$ and $\psi$ on $X$ such that $f=\phi-\psi$. Indeed, define $h$ a ${\mathc... | 2 | https://mathoverflow.net/users/48958 | 323731 | 139,553 |
https://mathoverflow.net/questions/323656 | 1 | Suppose that X is the subspace of the set of probability measures on the classical Wiener space $C[0,T]$, for some $T>0$, comprised of Gaussian measures.
In the finite-dimensional setting, the Wasserstein metric between two Gaussian random-variables has a very convenient form.... Are there any known analogues for th... | https://mathoverflow.net/users/36886 | Reference Request: 2-Wasserstein Metric on Wiener Space | This is discussed thoroughly in the following reference; see Section 3 and specifically Theorem 3.5.
*Gelbrich, Matthias*, [**On a formula for the $L^2$ Wasserstein metric between measures on Euclidean and Hilbert spaces**](http://dx.doi.org/10.1002/mana.19901470121), Math. Nachr. 147, 185-203 (1990). [ZBL0711.60003]... | 2 | https://mathoverflow.net/users/64449 | 323736 | 139,555 |
https://mathoverflow.net/questions/323738 | 0 | For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big\}$.
What is an example of a connected graph $G=(V,E)$ and a subset $S\subseteq V$ such that the subgraph $$G[S]:=(S, E\cap [S]^2)$$ is connected, and there is a surjective graph homomorphism $f:G[S]\to G$, but $G[S]\not\cong G$?
| https://mathoverflow.net/users/8628 | Connected subgraph of infinite graph with surjective homormophism, but no graph isomorphism | Let $G$ be the graph obtained by subdividing each edge of $K\_{1,\aleph\_0}$ with one subdivision point, and let $S$ be the set obtained by removing from $V(G)$ one vertex of degree $1$.
| 0 | https://mathoverflow.net/users/43266 | 323739 | 139,556 |
https://mathoverflow.net/questions/323666 | 3 | If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a\_1,\dotsc,a\_r)\mapsto m(a\_1,\dotsc,a\_r)$. Let $X$ be an algebra over $\mathcal{O}$. Then each choice of $m\in \mathcal{O}(2)$ gives us a binary product
$$X\times X\to X, (x,x')\mapst... | https://mathoverflow.net/users/124042 | $H$-space structure on coloured algebras | This is the homotopy version of the following data:
* a collection of spaces $\{X\_c\}\_c$ for all colors $c$;
* units $e\_c \in X\_c$;
* multiplications $- \cdot\_c - : X\_d \times X\_{d'} \to X\_c$;
satisfying the conditions:
* $x \cdot\_c e\_d = x$ for $x \in X\_c$;
* $(x \cdot\_b y) \cdot\_a z = x \cdot\_a (b... | 2 | https://mathoverflow.net/users/36146 | 323741 | 139,557 |
https://mathoverflow.net/questions/108406 | 12 | From [Isaacs et.al. 2005](http://www.uv.es/amoquin/35.pdf)
>
> Conjecture C. Let χ be a primitive
> irreducible character of an arbitrary
> finite group G. Then χ(1) divides |
> clG(g)| for some element g ∈ G.
>
>
> Here, of course, we have written
> clG(g) to denote the class of g in G.
> We have checked tha... | https://mathoverflow.net/users/10446 | Why would dim primitive irrep divide size of some conjugacy class ? | I have not noticed this question before, though it was posted several years ago. As a comment on the question as a whole, and especially Question 1 asked in the text, there are likely to be many such elements $g$ for many groups, and I would not expect there to be any "natural correspondence" between the $g$ associated... | 9 | https://mathoverflow.net/users/14450 | 323742 | 139,558 |
https://mathoverflow.net/questions/323478 | 4 | Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality:
$$
x'(t) \le \int\_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds,
$$
for any $t \ge 1$, where $k(t),t\in [1,\infty)$ is a nonincreasing positive function such that
$$
\int\_1^{+\infty}\frac{k(s)}{s}\,ds <\infty.
$$
Can we prov... | https://mathoverflow.net/users/105900 | A Riccati type integral inequality | The boundedness statement is true
---------------------------------
The general argument is similar to what I gave [in my previous answer](https://mathoverflow.net/a/323578/3948), which was essentially off by a log due to certain inefficiencies in the estimates. Here I rewrite the argument to get rid of the log loss.... | 3 | https://mathoverflow.net/users/3948 | 323749 | 139,560 |
https://mathoverflow.net/questions/323673 | 5 | Let $X$ be a smooth projective variety. Given fixed Chern classes with positive rank $r$, let $M$ be the moduli space of semistable sheaves with such Chern classes. In general we only know $M$ is a projective scheme. (if $\mathrm{dim}(X)$ is $1,2$ a lot more can be said)
1st Question: Is there an example where $M$ is... | https://mathoverflow.net/users/48616 | Connectedness of moduli space | The answer to the second question is also negative. Here is a quick review of the Hartshorne-Serre construction in the smooth case. Let $X$ be a smooth, connected, proper $k$-scheme of pure dimension $n\geq 2$. Let $C\subset X$ be a codimension $2$, closed subscheme that is a local complete intersection scheme such tha... | 9 | https://mathoverflow.net/users/13265 | 323751 | 139,561 |
https://mathoverflow.net/questions/323711 | 1 | There is a common argument used when investigating the concentration of the maximally loaded bin (say $X$ is the maximum load) when $m$ balls are thrown into $n$ bins under the uniform distribution. I give the argument for $m=n,$ showing that $X$ is approximately $\ln n/\ln \ln n$ with high probability. Using the union... | https://mathoverflow.net/users/17773 | Concentration of the load of the maximally loaded bin ($m$ balls $n$ bins) with nonuniform bin probabilities | If you only want upper bounds, there is a nice approach based on collisions. (Proving that the max-loaded bin is not too small w.h.p. seems to need completely different techniques.)
Define a $k$-way collision to be a case of $k$ different balls that land in the same bin. For example, if they all land in the same bin,... | 2 | https://mathoverflow.net/users/29697 | 323756 | 139,563 |
https://mathoverflow.net/questions/323419 | 3 | I've been studying the following singular PDE
$$
\mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$
in $\Omega \subset \mathbb{R}^{2}$.
Do you know any reference, where this kind of singularity has been studied?
| https://mathoverflow.net/users/127742 | Solution singular PDE | Assume for simplicity that $g$ is linear. Then the integrand $|\nabla f|^2 + const.|\nabla f|$ is smooth and uniformly convex (as a function of $\nabla f$) except at the origin. It is known that the minimizers of such functionals are at least $C^1$-regular.
Indeed, if $f$ minimizes $\int F(\nabla f)$ and $F$ is smoot... | 2 | https://mathoverflow.net/users/16659 | 323758 | 139,564 |
https://mathoverflow.net/questions/323764 | 10 | Let $S\_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S\_n$ by $\ell(\sigma)$.
>
> **QUESTION.** Assume $n>2$. Does this cancellation property hold true?
> $$\sum\_{\sigma\in S\_n}(-1)^{\ell(\sigma)}\sum\_{i=1}^ni(i-\sigma(i))=0.$$
>
>
>
| https://mathoverflow.net/users/66131 | A cancellation property for permutations? | Let $n$ be some integer greater than 2. Since the number of even and odd permutations in $S\_n$ is the same we have $\sum\_{\sigma\in S\_{n}}(-1)^{\ell(\sigma)}=0$ therefore the contribution of $\sum\_{\sigma\in S\_{n}}(-1)^{\ell(\sigma)}\left(\sum\_{i=1}^n i^2\right)$ is zero. It remains to show that
$$\sum\_{\sigma\i... | 23 | https://mathoverflow.net/users/2384 | 323765 | 139,566 |
https://mathoverflow.net/questions/323759 | 1 | I have an integral of the form
$$ I=\int\_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite (for example: $f(x)=x$); moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c\_0+c\_1a+... | https://mathoverflow.net/users/136119 | Expansion of an integral | Let $f(x)$ have the expansion $f(x)=x+c\_3 x^3+c\_4 x^4 +\cdots$, then define $y=x/a$ and you have
$$I=\int\_0^{\infty}{dx}\ln \bigg(1+\exp\left(-\frac{f(x)}{a}\right)\bigg)$$
$$=\int\_0^\infty \left[a\ln \left(1+e^{-y}\right)-\frac{a^3 c\_3 y^3}{e^y+1}-\frac{a^4 c\_4 y^4}{e^y+1}+{\cal O}(a^5)\right]\,dy$$
$$=\frac{1}{... | 1 | https://mathoverflow.net/users/11260 | 323767 | 139,568 |
https://mathoverflow.net/questions/323768 | 2 | Suppose that we have a (combinatorial if necessary) model category $M$, and let $F:\Delta^{op}\rightarrow M$ a simplicial object in $M$, such that for any natural number $n$, $F([n])$ is a fibrant object in $M$.
We define a new object $X= colim\_{n} F([n]) $. Is it true that $X$ is a fibrant object ?
| https://mathoverflow.net/users/136128 | simplicial objects in a model category | No, it is not.
If what you mean by $\rm colim\_n$ is the actual colimit of $F$ as a diagram of shape $\Delta$, then this colimit is isomorphic to the coequalizer of the two maps $F([1]) \rightrightarrows F([0])$. Coequalizers rarely preserve fibrancy, and with a little thought we can think of a counterexample that ex... | 7 | https://mathoverflow.net/users/49 | 323772 | 139,570 |
https://mathoverflow.net/questions/323727 | 7 | Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$.
For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they inherit the original metric of $G$?
I was inspired by this MSE question:
<https://math.stackexchange.com/questions/31210... | https://mathoverflow.net/users/36688 | Is every Lie subgroup of a Lie group isometric to all its conjugates? | Unless I miscomputed, the left-invariant metric
$Q(dg,dg)=\operatorname{Tr}\bigl(\overline{g^{-1}dg}\,g^{-1}dg\bigr)$ (bar $=$ transpose) on
\begin{equation}
G=\left\{g=\begin{pmatrix}a&b&c\\0&1&e\\0&0&1\end{pmatrix}:
\begin{matrix}a>0,\\b,c,e\in\mathbf R\end{matrix}\right\},
\qquad
N=\left\{n=\begin{pmatrix}a&b&0\\0&1... | 7 | https://mathoverflow.net/users/19276 | 323775 | 139,571 |
https://mathoverflow.net/questions/323774 | -1 | Suppose $X\_m\sim p\_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p\_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits\_{m\rightarrow\infty}p\_m(x)$, that is, $p\_m$ converge weakly to $p$ on $[0,1]$. (e.g., $p(x)$ is the uniform distrubtion on $[0,1]$... | https://mathoverflow.net/users/136078 | On probabilistic extension for Bernstein polynomials | The limit is not $u(\int\_0^1 x p(x)\,dx)$ but rather $\int\_0^1 u(x) p(x)\,dx$.
I prefer probabilistic notation, so let $X\_n \sim p\_n$ and $X \sim p$. We are then supposing that $X\_n \Rightarrow X$ in distribution. Let $Y\_n$ have conditional distribution $\mathrm{Bin}(n, X\_n)$ given $X\_n$. The question is then... | 1 | https://mathoverflow.net/users/4832 | 323787 | 139,575 |
https://mathoverflow.net/questions/323790 | 0 | Let $C\_n(x) = \frac{n!}{\Gamma(x)\cdot \Gamma(n-x)}$
Is the following true: $\int\_{0}^{n} [C\_n(x)\cdot y^x \cdot (1-y)^{n-x}dx] = 1$??
just wondering
In generality for continuous functions $f,g$ from the reals to the reals is it the case that:
$\int\_{0}^{n} [C\_n(x)\cdot f(y)^x \cdot g(z)^{n-x}dx] = [f(y)+... | https://mathoverflow.net/users/13953 | Integrals I am curious about | We have the classical [Euler integral](https://en.wikipedia.org/wiki/Beta_function)
$$B(x+1,n-x+1):=\int\_0^1y^x(1-y)^{n-x} dy=\frac{\Gamma(x+1)\Gamma(n-x+1)}{\Gamma(n+2)} $$
$$
=\frac{x(n-x)}{n+1}\frac{\Gamma(x)\Gamma(n-x)}{\Gamma(n+1)}=\frac{x(n-x)}{(n+1)C\_n(x)}
$$
Hence
$$
\int\_0^1\underbrace{C\_n(x)y^x(1-y)^{n-... | 4 | https://mathoverflow.net/users/20302 | 323791 | 139,576 |
https://mathoverflow.net/questions/323792 | 3 | given polish space $(X,d)$, consider weak\* topology of probability. optimal transport of probability $u,v$ is defined by $\pi(u,v)$ such that $\pi(u,v)$ minimizes:
$\{\int d(x,y) d \pi(dx,dy): \pi \text{ is the couple of } u,v\}$
if $u\_k \to u, v\_k \to v$, can we show $\pi(u\_k, v\_k) \to \pi(u,v)$?
and how is... | https://mathoverflow.net/users/124254 | continuity/ measurablity of optimal transport | Of course not! This is hopeless if $X$ has unbounded diameter.
As a counterexample, consider $X=\mathbb Z$ with the standard metric; $\mu\_k=(1-\frac 1k)\delta\_0+\frac 1k\delta\_k$; and $\nu\_k=\nu=\mu=\delta\_0$. Then $\nu\_k\to\nu$, $\mu\_k\to \mu$, but $\pi(\mu\_k,\nu\_k)=1$ for each $k$.
| 3 | https://mathoverflow.net/users/11054 | 323796 | 139,577 |
https://mathoverflow.net/questions/323800 | 15 | I have a friend who is having trouble in his math PhD career and is recently down with depression. He has decided to take a break from research and receive some medication. To help him recover faster and better, I am asking if anyone here with experience in helping depressed math PhDs could contribute his/her advice. T... | https://mathoverflow.net/users/37103 | Ways to help a math PhD out of depression | I do have something to share. I would ask the following:
1) Is he getting well with his advisor?
2) Does he have a daily routine (i.e wake up at 8AM everyday, preparing for teaching, going to gym at 11AM, reading papers from 1PM, attending weekly seminar at 3PM, etc). This can be *crucial* to get someone out of the... | 22 | https://mathoverflow.net/users/18850 | 323802 | 139,578 |
https://mathoverflow.net/questions/323753 | 9 | Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|\_U$ is given by a continuous function $f\in C(U)$, then for every $\phi\in C^\infty\_c(\mathbb R^n)$ with $\int\_{\mathbb R^n}\phi(x)d x=1$ we can define a Dirac seq... | https://mathoverflow.net/users/58125 | Defining the value of a distribution at a point | The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $\delta$-sequences, follows the path traced years ago by Stanisław Łojasiewicz in the paper [1], so I describe his approach to the problem below... | 5 | https://mathoverflow.net/users/113756 | 323805 | 139,579 |
https://mathoverflow.net/questions/323773 | 6 |
>
> Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits\_{n = 0}^{\infty} A\_n$ of finite type. ("Finite type" means that each $A\_n$ is a finite-dimensional $\mathbf{k}$-vector space. All algebras are understood to contain $1$.) Let $B = \bigoplus\limits\_{n = 0... | https://mathoverflow.net/users/2530 | If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis? | Q1: no (this makes Q2, Q3 obsolete)
Q4, Q5: yes (for $k$ a field)
---
**Example for Q1:** Let $B = k \oplus k$ (componentwise operations) be concentrated in degree zero and take
$$A = A\_0 \oplus A\_1 \oplus A\_2 := B \oplus k \oplus k$$
$B$ acts on $A\_i=k$ via the projection $p\_i:B \twoheadrightarrow k$. Thi... | 4 | https://mathoverflow.net/users/18571 | 323806 | 139,580 |
https://mathoverflow.net/questions/323804 | 3 | I have a die that produces uniformly distributed values in $\{1,\ldots, k\}$ for some integer $k\geq 2$. Now I play the following game.
I start rolling the die and produce one integer in $\{1,\ldots,k\}$ after another, $X\_1,X\_2,\ldots$, and I stop when my most recent integer $x$ lies between the previous integer $b... | https://mathoverflow.net/users/8628 | Expected value of length of interval game | Yes: Given that no stop has been achieved up to time $n$, the probability that a stop is achieved at time $n+3$ is at least $\frac 13$. To see this, notice that given $X\_1,\ldots,X\_n$, and the values of the (unordered) multiset $\{\{X\_{n+1},X\_{n+2},X\_{n+3}\}\}$, each ordering of the three terms is equally likely, ... | 4 | https://mathoverflow.net/users/11054 | 323808 | 139,581 |
https://mathoverflow.net/questions/323744 | 14 | Every [locally presentable category](https://ncatlab.org/nlab/show/locally+presentable+category) is [well-powered](https://ncatlab.org/nlab/show/well-powered%20category): since it is a full reflective subcategory of a presheaf topos, its subobject lattices are subsets of those of the latter.
Every [accessible categor... | https://mathoverflow.net/users/49 | Is every accessible category well-powered? | It seems to me that every category with a small set of *dense* generator is well powered. In particular accessible categories are well powered.
Dense generator means that you have a small full subcategory $C \subset A$ such that the induced nerve functor $A \rightarrow \widehat{C}$ is fully faithful. This functor sen... | 6 | https://mathoverflow.net/users/22131 | 323812 | 139,583 |
https://mathoverflow.net/questions/323824 | 5 | I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly negative, [see for example here](https://en.wikipedia.org/wiki/Hurwitz_matrix#Hurwitz_stable_matrices).
Usually the probl... | https://mathoverflow.net/users/nan | Stable matrices and their spectra | No, this is not true in general.
Note that in your condition you probably want to assume $\|u\|=1$ for otherwise you could always make the left hand side as small as you wish by making $u$ small. I will answer the question for the Euclidean norm and the induced spectral norm but this does not really make a huge diffe... | 4 | https://mathoverflow.net/users/85570 | 323830 | 139,590 |
https://mathoverflow.net/questions/323745 | 0 | I have a question about harmonic functions with respect to symmetric Markov processes.
Let $E$ be a locally compact separable metric space and $\mu$ a positive Radon measure on $E$.
Let $X=(\{X\_t\}\_{t \ge 0},\{P\_x\}\_{x \in E})$ be a $\mu$-symmetric Hunt process on $E$.
>
> **Def**. A Borel measurable funct... | https://mathoverflow.net/users/68463 | Continuity of harmonic functions | I do not know if a general answer to your question is known. This is an extended comment about a rather specific case.
---
In his article [*Doubly-Feller Process with Multiplicative Functional*](https://link.springer.com/chapter/10.1007/978-1-4684-6748-2_4), K.-L. Chung proved that if $X$ is both Feller and stron... | 1 | https://mathoverflow.net/users/108637 | 323846 | 139,599 |
https://mathoverflow.net/questions/323845 | 4 | If you have $n$ identically distributed Bernoulli trials whose sum is binomially distributed random variable, does it then follow that the $n$ Bernoulli trials are independent?
| https://mathoverflow.net/users/136167 | Can the sum of identically distributed dependent Bernoulli trials be binomially distributed? | That's false even for $n=3$.
Denote by $B(p)$ the Bernoulli distribution which has probability $p$ of being 1 and $(1-p)$ to be 0.
Define the three Bernoulli variables $(X\_1, X\_2, X\_3)$ by
$
X\_1 \sim B(0.5) \\
X\_2|(X\_1=1) \sim B(0.25);~ X\_2|(X\_1=0) \sim B(0.75) \\
X\_3|(X\_1+X\_2=2) \sim B(1);~ X\_3|(X\_1+X... | 9 | https://mathoverflow.net/users/106046 | 323848 | 139,600 |
https://mathoverflow.net/questions/323864 | 1 | Fix a positive integer $a>1$ and let $f\in\mathbb{Z}[x]$ be a polynomial with positive leading coefficient. We define a set $S$ of positive integers,
$$
S=\{n\in\mathbb{Z}^+:n\mid a^{f(n)}-1\}.
$$
Question is, can we compute the density of $S$, that is,
$$
\lim\_{n\to\infty}\frac{|S\cap \{1,2,\dots,n\}|}{n}?
$$
I belie... | https://mathoverflow.net/users/127150 | Density of a set of numbers dividing a fixed number with polynomial exponent | Fix $\varepsilon>0$ and choose large prime $q>5\varepsilon^{-1} \cdot \deg(f)$ such that $f$ is non-trivial modulo $q$ (then $f$ has at most $\deg(f)$ roots modulo $q$) and additionally such that $a$ is not a $q$-th perfect power. Say that a prime $p>q$ is $q$-appropriate if $a$ is not $q$-th power modulo $p$. By Chebo... | 2 | https://mathoverflow.net/users/4312 | 323874 | 139,608 |
https://mathoverflow.net/questions/323781 | 20 | In the [paper](https://arxiv.org/abs/1902.07321) by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in [PNAS](https://www.pnas.org/content/early/2019/05/20/1902572116)) the abstract includes
>
> In the case of the Riemann zeta function, this proves the GUE random
> matrix m... | https://mathoverflow.net/users/6756 | Jensen Polynomials for the Riemann Zeta Function | Just for clarification, what do you mean by the horizontal distribution? Are you asking about the distribution of the real parts of the zeros of $\zeta'(z):=\frac{\operatorname{d}}{\operatorname{d}s}\zeta(s)$? In that case, the answer is no. It is known that $\zeta'(s)$ does not satisfy the Riemann hypothesis, due to i... | 6 | https://mathoverflow.net/users/61910 | 323886 | 139,614 |
https://mathoverflow.net/questions/323828 | 0 | Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\Vert (N-i\lambda)u \Vert \le \varepsilon$
for some $\lambda \in \mathbb R.$
Since $N$ is normal this implies that... | https://mathoverflow.net/users/nan | Perturbing a normal matrix | $\epsilon = 0 \implies \exists v: v $ is eigenvector of N and $v$ is orthogonal to $e\_1 \implies Mv = Nv \implies v $ is an eigenvector of $M$ with same eigenvalue. So if $\epsilon = 0$ then $\delta$ can be anything. Now, since $\epsilon \ge 0$ does not rule out that $\epsilon = 0$ therefore $\delta$ can be anything i... | 0 | https://mathoverflow.net/users/29887 | 323895 | 139,616 |
https://mathoverflow.net/questions/323891 | 2 | Let $Z \subseteq \mathbb{C}$ without limit point. By the [Weierstrass factorization theorem](https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem) there is an entire function $h$ those zero set is $Z$. Let $a\_n > 0$ be a sequence where $\lim\_n \sqrt[n]{a\_n}=0$.
**Question:** Can $h(z) = \sum\_{n \ge 0}... | https://mathoverflow.net/users/18571 | Coefficients of entire functions with specified zero set | The answer is in general no because the zero set's convergence exponent imposes a lower bound on the order of growth of the function, which in turn imposes a growth condition on a subsequence of the Taylor coefficients.
In detail:
Assume $Z$ infinite of course and order all its non-zero elements (obviously $Z$ has... | 6 | https://mathoverflow.net/users/133811 | 323896 | 139,617 |
https://mathoverflow.net/questions/323906 | 6 | Let $\mathbb{D}\_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}\_n^d)$ that fix the punctures. In particular I want to know what we know about all fundamental groups of this space
$$\pi\_k(Diff(\mathbb{D}\_n^d))$$
E.... | https://mathoverflow.net/users/22709 | Homotopy groups of Diffeomorphisms of punctured d-dim ball | Let me assume that you are interested in diffeomorphisms which also fix the boundary of the ball. If $F\_n(D^d)$ denotes the space of $n$ distinct ordered points in $D^d$, then there is a fibration sequence
$$Diff(D^d\_n) \to Diff(D^d) \to F\_n(D^d),$$
where the rightmost map is the orbit map given by acting on a fixed... | 11 | https://mathoverflow.net/users/318 | 323909 | 139,622 |
https://mathoverflow.net/questions/323898 | 6 | The following is a holdover from my math contest days that I never got around
to solve.
We will use the notation $\left[ k\right] $ for the set $\left\{
1,2,\ldots,k\right\} $ whenever $k$ is a nonnegative integer.
Let me first state my question in its general form, which is sadly not very
inviting. I recommend tak... | https://mathoverflow.net/users/2530 | An inequality for rearrangement-style sums | Let me prove it for $p=3$ (only because the notations look more friendly.) The maximum of left hand side is realized when the arrays $(a\_i),(b\_{\sigma(i)}),(c\_{\tau(i)})$ are equally sorted, so let us assume that it is the case.
I claim that not only the sum, but the $s$-th largest summand of RHS (clarification: ... | 2 | https://mathoverflow.net/users/4312 | 323913 | 139,623 |
https://mathoverflow.net/questions/323911 | 0 | Please help me with the following question.
What are some examples of Banach algebra $A$ satisfying the following two conditions?
$1$.$ A $ does not have an approximate identity.
$2$. $A^2=A$. That is, for any $a∈A$, there exist some $b,c∈A $ such that $ a=bc$.
A direct application of the Cohen factorization t... | https://mathoverflow.net/users/85784 | Banach algebra $A$ without an approximate identity but $A^2=A$ | For finite-dimensional algebra, to have an approximate unit is the same as having a unit.
The non-unital algebra $A$ of matrices $\begin{pmatrix}0 & x\\ 0 & y\end{pmatrix}$ has no unit (although it has a right unit), so over the reals has no approximate unit, and satisfies $A^2=A$.
The non-unital algebra $B$ of mat... | 7 | https://mathoverflow.net/users/14094 | 323915 | 139,624 |
https://mathoverflow.net/questions/323914 | 5 | Inspired by [A discontinuous construction](https://mathoverflow.net/questions/323713/a-discontinuous-construction):
Does there exist a function $a \colon [0,1] \to (0,\infty)$ and a family $\{D\_x \colon x \in [0,1]\}$ of countable, dense subsets of $[0,1]$ with $\bigcup\_{x \in [0,1]} D\_x = [0,1]$ and $\sum\_{r \in D... | https://mathoverflow.net/users/100904 | Existence of a strange function | The relation
$$
x\sim y \quad \iff \quad x-y\in\mathbb{Q}
$$
is an equivalence relation that partitions $[0,1]$ into countable sets of the form $[t]=(t+\mathbb{Q})\cap [0,1]$.
The set of all equivalence classes of the relation $\{[t]:\,t\in [0,1]\}$ has the same cardinality as $[0,1]$. Consider a bijection
$$
\psi:... | 7 | https://mathoverflow.net/users/121665 | 323918 | 139,626 |
https://mathoverflow.net/questions/323865 | 12 | Let $X\rightarrow Y$ be a morphism (may not be smooth) of varieties such that the fibres are vector spaces. Are the $l$-adic cohomologies of $X$ and $Y$ equal?
If not, under what condition (other than smoothness of the map) is sufficient to ensure that the cohomologies are equal?
What did I mean to ask?: Let $f:X\r... | https://mathoverflow.net/users/nan | Étale cohomology of morphism whose fibers are vector spaces | The answer below is slightly reorganized to incorporate the edits. I thank @PiotrAchinger and @S.D. for their comments correcting and clarifying this answer.
**Definition 1.** An **affine space morphism** is a separated, smooth morphism whose geometric fibers are affine spaces.
**Definition 2.** A morphism of sche... | 16 | https://mathoverflow.net/users/13265 | 323923 | 139,627 |
https://mathoverflow.net/questions/323922 | 8 | Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question is simply what is the minimum value of $k$ (as a function of $n$ and $p$), which I will call $k\_{n,p}$, such that this... | https://mathoverflow.net/users/136197 | What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices? | The quantity that you are trying to compute is called the *exponent* of the group $\text{GL}\_n(\mathbb F\_p)$. As you note by Lagrange ,it always divides the order of the group. More generally, one can ask about the exponent of the group $\text{GL}\_n(\mathbb F\_q)$, where $q$ is a power of $p$. You'll find a detailed... | 9 | https://mathoverflow.net/users/11926 | 323926 | 139,629 |
https://mathoverflow.net/questions/323925 | -2 | I desperately need to read [this paper](http://yoksis.bilkent.edu.tr/pdf/files/13182.pdf), before meeting a would-be supervisor but with limited undergraduate knowledge that I have like Aluffi's *Algebra* and Churchill's *Complex Analysis*, Rudin's *Analysis*, Rabenstein's *Ordinary Differential Equations*, etc. not ev... | https://mathoverflow.net/users/nan | On the 2018 paper "On the discretization of Laine equations" by K. Zheltukhin, et al | A couple hints:
* Equation (1.1) is an equation involving partial derivatives (that's what the subscripts denote), so you're in the area of PDE.
* Look up the [Mathematics Subject Classification](https://mathscinet.ams.org/msc/msc.html) codes on the first page. That is another way to identify the general area of math... | 0 | https://mathoverflow.net/users/4832 | 323931 | 139,631 |
https://mathoverflow.net/questions/318897 | 3 | Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the *average distance* of $S$ as
$$
\bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum\_{x,y\in S} d\_H(x,y)\tag{1}
$$
where $d\_H$ denotes the Hamming distance. For any $1\leq n \leq 2^d$, define the minimum average distance as
$$
\beta(d,n... | https://mathoverflow.net/users/37266 | A question of Ahlswede and Katona: known lower bounds on $\beta(d,n)$? | (0) Preliminaries:
(a) factors: above you should change the factors $\frac{1}{4}$ on the rhs of your equations (4) and (5)
to $\frac{1}{2}$ (Kündgens distance is half the distance of Ahlswede/Katona)).
(b) notation: in the sequel I use $n$ for the dimension , $s$ for the size of the set
$S$, $w\_H$ for the Hammin... | 3 | https://mathoverflow.net/users/48831 | 323942 | 139,634 |
https://mathoverflow.net/questions/323934 | 8 | Is there a collection of $2^{\aleph\_0}$ pairwise non-isomorphic countable Boolean algebras?
Equivalently, are there $2^{\aleph\_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
| https://mathoverflow.net/users/8628 | Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebra structures on $\omega$? | The answer is yes: there are $2^{\aleph\_0}$ countable Boolean algebras up to isomorphism, or equivalently $2^{\aleph\_0}$ homeomorphism class of metrizable totally disconnected compact Hausdorff spaces. This is the main result of:
*Reichbach, M.
The power of topological types of some classes of 0-dimensional sets.
... | 10 | https://mathoverflow.net/users/14094 | 323944 | 139,635 |
https://mathoverflow.net/questions/323946 | 4 | It is known that there exists a finitely additive translation invariant measure on $\mathbb{R}$ that extends the Lebesgue measure. I.e. a function $m:\mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$ that is
1. Finitely additive: $m(A \sqcup B) = m(A) + m(B)$
2. For a Lebesgue measurable set $A$, $m(A)$ is its Lebesgue... | https://mathoverflow.net/users/136187 | Finitely Additive Homogeneous Translation Invariant Measure on $\mathcal{P}(\mathbb{R})$ | [Corrected answer, entirely rewritten] Yes. And this also works on $V=\mathbf{R}^n$ (with homogeneity axiom rewritten as $\mu(tY)=|t|^n\mu(Y)$).
Let $\mu\_0:\mathcal{P}(V)\to [0,\infty]$ be a translation invariant, finitely additive mesure, extending the Lebesgue measure.
Let $I$ be the set of $Y\subset V$ such t... | 5 | https://mathoverflow.net/users/14094 | 323947 | 139,636 |
https://mathoverflow.net/questions/323941 | 3 |
>
> **Question:**
>
>
> * how did the classification of quadrilaterals come into being? Was there a single major contributor who coined terms like "rectangle", "square", "trapez/ium/oid", "kite", "deltoid", etc. or were these definitions contributed by several individuals over a longer stretch of time?
> * what was... | https://mathoverflow.net/users/31310 | History of the Taxonomy of Quadrilaterals | **Q1.** If one wishes to single out one main early contributor to the systematic classification of convex quadrilaterals it would have to be the 9th century Indian mathematician [Mahavira](https://en.wikipedia.org/wiki/Mah%C4%81v%C4%ABra_(mathematician)), who divided quadrilaterals into five classes: those with unequal... | 3 | https://mathoverflow.net/users/11260 | 323948 | 139,637 |
https://mathoverflow.net/questions/323920 | 3 | Let $k$ be an algebraically closed field of positive characteristic. Let $G$ be an affine reductive group acting on a smooth projective variety $X$. Let $E$ be a vector bundle on $X$ such that the action of $G$ lifts to an action on $E$. I am looking for a criterion which tells me when $E$ descends to a bundle on the G... | https://mathoverflow.net/users/45758 | Criterion for vector bundle to descend to GIT quotient in positive characteristic | I have one question / comment. The notation $E\_x$ is used in some sources to mean the stalk of a sheaf $E$ at a point $x$, considered as a module over the local ring $(\mathcal{O}\_{X,x},\mathfrak{m}\_x).$ However, by the context here, I am guessing that $E\_x$ is intended to mean the "fiber" $E\_x/\mathfrak{m}\_x E\_... | 4 | https://mathoverflow.net/users/13265 | 323951 | 139,638 |
https://mathoverflow.net/questions/323957 | 3 | I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind all this was: If we take $W\_N$ being trace free skew-symmetric matrices of rank $N$ on an $N$-dimensional space. And we ... | https://mathoverflow.net/users/119875 | Eigenvalue estimates for operator perturbations | This problem of "non-Hermitian" quantum mechanics has been studied in the context of random-matrix theory (RMT), see for example the review [Random matrix approaches to open quantum systems](https://arxiv.org/abs/1610.05816). The particular case pointed to in the OP is when the Hermitian matrix ${\cal H}=iW\_{H}$ of hi... | 4 | https://mathoverflow.net/users/11260 | 323962 | 139,641 |
https://mathoverflow.net/questions/323966 | 7 | Let $A$ and $B$ be simplicial abelian groups, and let $N\_\ast(-)$ denote the normalized chain complex functor. Let
$$AW\_{A,B}\colon N\_\ast(A\otimes B)
\longrightarrow N\_\ast(A)\otimes N\_\ast(B)$$
and
$$ EZ\_{A,B}\colon N\_\ast(A)\otimes N\_\ast(B)
\longrightarrow N\_\ast(A\otimes B)$$
denote the Alexander-Whitney ... | https://mathoverflow.net/users/43574 | Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair | You have it in page 7 of [this paper](https://arxiv.org/pdf/math/0110308.pdf).
| 8 | https://mathoverflow.net/users/12166 | 323967 | 139,642 |
https://mathoverflow.net/questions/323778 | 3 | Let $K$ be a field of characteristic $0$, let $A = K[[t\_1, \ldots, t\_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes\_A \Omega^1\_{A/K} = \bigoplus\_{k=1}^n V \ dt\_k$ be an integrable connection, so $\nabla^2 = 0$. I want to show that there always ex... | https://mathoverflow.net/users/56878 | Reference request - existence of formal solutions for integrable connections | I hope you'll allow one or two slight adjustments to your question:
* I think you are interested in $(t\_1, \dots, t\_n)$-adically continuous connections rather than arbitrary connections (for which the desired solution principle would not generally hold).
* Judging from how you use "$r$," I think you intend $V$ to b... | 3 | https://mathoverflow.net/users/136220 | 323974 | 139,644 |
https://mathoverflow.net/questions/323982 | 1 | We call an topological space $(X,\tau)$ $n$-*product-periodic* for an integer $n\geq 3$ if $\prod\_{i=1}^n X \cong X$ but for all integers $k$ with $2\leq k\leq n-1$ we have $\prod\_{i=1}^k X \not\cong X$.
Is there an integer $n\geq 3$ such that there is an $n$-product-periodic space, and is there an integer $m\geq 3... | https://mathoverflow.net/users/8628 | $n$-product-periodic topological spaces | Garrett Ervin's answer to [When is $A$ isomorphic to $A^3$?](https://mathoverflow.net/q/10128#252553) mentions also some results on topological spaces. (Although the question was originally about abelian groups.)
The results mentioned there seem to answer your question - although I do hope that somebody can provide a... | 6 | https://mathoverflow.net/users/8250 | 323985 | 139,647 |
https://mathoverflow.net/questions/323964 | 9 | For completeness of MathOverflow and for clarity of the question, I will first recall a few things, including the the definition of Kleene realizability: experts can jump directly to the question below.
For natural numbers $n$ and first-order formulae $\varphi$ of Heyting arithmetic, the formula “$n$ realizes $\varph... | https://mathoverflow.net/users/17064 | Kleene realizability in Peano arithmetic | $\let\T\mathrm\def\kr{\mathrel{\mathbf r}}$Let me first answer a slightly modified question:
>
> **Proposition:** For any sentence $\phi$, there exists $n\in\mathbb N$ such that $\T{PA}\vdash\overline n\kr\phi$ if and only if $\T{HA+ECT\_0+MP}\vdash\phi$.
>
>
>
The right-to-left direction follows from $\T{HA+M... | 12 | https://mathoverflow.net/users/12705 | 323988 | 139,649 |
https://mathoverflow.net/questions/323814 | 2 | I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of [these notes](http://w3.countnumber.de/course_notes/A_Crash_Course_in_Lie_Algebras-v1.2.pdf) (pdf). I am having a difficult time understanding few parts of it and I would appreciate any exp... | https://mathoverflow.net/users/84272 | Question on the proof of any finite dimensional module of a semisimple Lie algebra is semisimple | Regarding your question (1), this follows from the way you defined $\mathcal{A}$ and $\mathcal{V}$, as the subspaces of $\hom(V,A)$ of maps $T:V\to A$ such that $T\in \mathcal{A}$ iff $T\mid\_A=0$ and $T\in\mathcal{V}$ iff $T\mid\_A$ is a homothety. Since $0$ is a homothety, it holds that $\mathcal{A}\subseteq\mathcal{... | 1 | https://mathoverflow.net/users/14443 | 323994 | 139,650 |
https://mathoverflow.net/questions/323989 | 2 | Let $H$ be an infinite dimensional Hilbert space over $\mathbb{C}$
Let $\{v\_n\}\_{n \in \mathbb{N}} \subset H$ be a sequence of linearly independent vectors in $H$ such that $v\_n \to u$
Let $\forall m \in \mathbb{N}: V\_m = \operatorname{span} \{v\_n\}\_{n \geq m}$ and $P\_m$ be the orthogonal projection on $V\_m... | https://mathoverflow.net/users/108867 | A sequence of orthogonal projection in Hilbert space | The answer is **no**, in general.
As a counterexample, let $H = L^2([0,1])$, let $(q\_n)\_{n \in \mathbb{N}}$ be your favourite enumeration of $[0,1] \cap \mathbb{Q}$ and define
\begin{align\*}
v\_n := 1 + \frac{1}{n} 1\_{[0,q\_n]}
\end{align\*}
for each $n \in \mathbb{N}$.
Then the span of $\{v\_n: \, n \ge m\}$ ... | 3 | https://mathoverflow.net/users/102946 | 323996 | 139,651 |
https://mathoverflow.net/questions/323810 | 6 | Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can expect any overlap between the fields. I have not really narrowed down my focus too much yet since I’m just starting out,... | https://mathoverflow.net/users/132446 | How are the fields of dynamical systems, stochastic processes and additive combinatorics, inter-related? | I'd strongly encourage you to read Terry Tao's blog posts on dynamics and relations to additive combinatorics. These are beautiful descriptions of the main ideas and intuitions behind them, and are much more accessible than, say, his book with Vu. To get started, read his course posts for ergodic theory, just search fo... | 8 | https://mathoverflow.net/users/8112 | 324007 | 139,655 |
https://mathoverflow.net/questions/324015 | 2 | Let $\mathbb S^n$ be the $n$-sphere: $$\mathbb S^n=\left\{x \in \mathbb R^{n+1}: \left\|x\right\|=1\right\}.$$The **hairy ball theorem** can be formulated as follows:
>
> If $n$ is even and $f\,\colon\, \mathbb S^n \to \mathbb S^n$ is a continuous function, then there exists at least one $x \in \mathbb S^n$ such th... | https://mathoverflow.net/users/124497 | Hairy ball theorem for odd-dimensional spheres | The Lefschetz fixed point theorem implies that any $f: S^n \to S^n$ without fixed points has degree $(-1)^{n+1}$. But an even map $S^n \to S^n$ has even degree, since it factors as
$$
S^n \xrightarrow{q} \mathbb{R}P^n \to S^n,
$$
and for odd $n$, $q$ has degree $2$, while for even $n$, $H\_n(\mathbb{R}P^n; \mathbb{Z})=... | 15 | https://mathoverflow.net/users/39747 | 324018 | 139,657 |
https://mathoverflow.net/questions/324014 | 4 | Each of the scalar random variables, $ Y $, $ X $, $ U $, and $ V $, is continuous and possibly has $ \mathbb{R} $ as its support. The random variable, $Z$, could be vector valued, but continuous.
I have the following
\begin{align}\nonumber
Y = \beta X + U\\\nonumber
X = \pi Z + V,
\end{align}
where $ Z \perp (U,V)... | https://mathoverflow.net/users/136237 | Proving Conditional Independence | The answer is yes. Indeed, $Z\perp(U,V)$ implies $\pi Z\perp(U,V)$. So, without loss of generality $\pi Z=Z$ and $X=Z+V$ (the condition $Y=\beta X+U$ is irrelevant and not needed here). So, the desired conditional independence $(U\perp X)|V$ can be rewritten as $(U\perp Z+V)|V$, which means that
\begin{equation}
E\Bi... | 1 | https://mathoverflow.net/users/36721 | 324027 | 139,659 |
https://mathoverflow.net/questions/324029 | 5 | I'm a graduate student just checking to make sure that what he's researching isn't already known.
Let $\mathbb{F}$ be a number field, and let $E$ be an elliptic curve defined over $\mathbb{F}$. Is it already known that, for any integer $r\geq1$, there exists a finite-degree extension $\mathbb{K}$ of $\mathbb{F}$ so ... | https://mathoverflow.net/users/120369 | Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions | This is (unfortunately for you) well-known. Theorem 10.1 of
*Frey, Gerhard; Jarden, Moshe*, [**Approximation theory and the rank of Abelian varieties over large algebraic fields**](https://doi.org/10.1112/plms/s3-28.1.112), Proc. Lond. Math. Soc., III. Ser. 28, 112-128 (1974). [ZBL0275.14021](https://zbmath.org/?q=an:0... | 8 | https://mathoverflow.net/users/30186 | 324030 | 139,660 |
https://mathoverflow.net/questions/324031 | 7 | Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero (see for instance <https://arxiv.org/pdf/math/0006107.pdf>).
Let $A(n)\to S^n(A)$ be a resolution of singularities. Then... | https://mathoverflow.net/users/135215 | Is the symmetric product of an abelian variety a CY variety? | When $\dim A = 1$, $S^nA$ is a $\mathbb{P}^{n-1}$-bundle over $A$, so its Kodaira dimension is $-\infty$.
When $\dim A = 2$, the minimal resolution of $S^nA$ is given by the Hilbert scheme $A^{[n]}$, there is a natural map
$$
A^{[n]} \to A
$$
(summation of points), which is smooth with fiber $K\_{n-1}A$, so-called hi... | 11 | https://mathoverflow.net/users/4428 | 324032 | 139,661 |
https://mathoverflow.net/questions/324000 | 7 | Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such categories (and equivalent to some of them). For example, it implies that every cocomplete category with a dense small su... | https://mathoverflow.net/users/62782 | Locally presentable categories, universes, and Vopenka's principle | In the Grothendieck universe approach to category theory, as you say, we replace all small sets with $\mathcal{U}$-small sets. Let's look at the definition of a locally presentable $\mathcal{U}$-category, for example, since that's in the conclusion of the theorem you mentioned: a locally presentable $\mathcal{U}$-categ... | 8 | https://mathoverflow.net/users/126667 | 324039 | 139,663 |
https://mathoverflow.net/questions/323965 | 4 | Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not on the first element of the sequence. The cycle size is called the period of the polynomial.
**Definition** (cycle). A ... | https://mathoverflow.net/users/136217 | Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd? | If $gcd(e,\lambda(N))=1,$ the sequence is purely periodic. Otherwise it may have an initial segment followed by a cycle, as you observe. Its maximal period divides the Carmichael function $\lambda(N)$ which is $\textrm{lcm}(p-1,q-1)$ when $N=pq,$ with $p,q$ prime.
Note that $2$ divides $\lambda(N)$ in this case, and ... | 6 | https://mathoverflow.net/users/17773 | 324042 | 139,665 |
https://mathoverflow.net/questions/324025 | 6 | I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that the complex structure $j\_{\Sigma}$ and Riemann metric $\text{dvol}\_{\Sigma}$ is fixed once and for all.
My understand... | https://mathoverflow.net/users/48216 | Intuition about bubbling off a ghost bubble | Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of m... | 3 | https://mathoverflow.net/users/12310 | 324047 | 139,668 |
https://mathoverflow.net/questions/324044 | 8 | It is known that [sum-free](https://en.wikipedia.org/wiki/Sum-free_set)
subsets of $\mathbb{N}$ can have
[natural density](https://en.wikipedia.org/wiki/Natural_density) at most
$\frac{1}{2}$. This density is achieved by the odd numbers: the sum of two
odd numbers is even.
I ask now a similar question for XOR rather ... | https://mathoverflow.net/users/6094 | XOR-free sets: Maximum density? | Let $a\in S$ be some fixed element. Note that $a\oplus b \le a + b$. Let $N$ be some big number. Put $M = [1, \ldots , N]\cap S$. We have $a\oplus M \cap M = \varnothing$. We also have $a\oplus M \subset [1, \ldots , N + a]$. Therefore $2|M| \le N + a$ or $|M| \le \frac{N}{2} + \frac{a}{2}$. Since $a$ is fixed taking l... | 11 | https://mathoverflow.net/users/104330 | 324049 | 139,669 |
https://mathoverflow.net/questions/324017 | 0 | I'm looking for the cardinality of the set of symmetric matrices with entries in $\mathbb{Z}^\*$ (the nonnegative integers) and fixed margins $\mathbf{k}=(k\_1,k\_2,...,k\_q)$. I've also seen them called line sum symmetric matrices.
I have not found any known result, but I have found upper bounds.
One possibility i... | https://mathoverflow.net/users/41373 | Number of symmetric matrix with fixed margins | There is no simple exact formula known, nor prospect of one in my opinion.
Here are two relevant asymptotic results.
(A) In [this paper](https://arxiv.org/abs/1303.4218), Catherine Greenhill and I gave the asymptotic count in the case that $K=o(M^{1/3})$, where $K$ is the maximum row sum and $M$ is the total of all... | 2 | https://mathoverflow.net/users/9025 | 324061 | 139,672 |
https://mathoverflow.net/questions/322511 | 5 | A **Riemann surface** $X$ is a connected complex manifold of complex dimension one. A **homogeneous space** is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification of those Riemann surfaces which are also homogeneous spaces . . . however, I can't seem to find it.
| https://mathoverflow.net/users/126606 | Homogeneous Riemann Surfaces | If you only care about compact (oriented) surfaces, then it is easy to see that the only homogeneous examples are the sphere $S^2=\mathbb C P^1$ and the torus $T^2=S^1\times S^1$. Indeed, if $M$ is a compact homogeneous space, then its Euler characteristic is $\chi(M)\geq0$; so for an oriented surface this implies genu... | 4 | https://mathoverflow.net/users/15743 | 324064 | 139,674 |
https://mathoverflow.net/questions/324066 | 6 | Suppose we are given a cocomplete closed symmetric monoidal stable $(\infty,1)$-category $\mathcal{C}$ with suspension $\Sigma$, and let $X \in \mathcal{C}$ be dualizable. I'd like to create a new stable cocomplete symmetric monoidal category $\mathcal{C}[\Sigma X^{-1}]$ together with a symmetric monoidal exact and con... | https://mathoverflow.net/users/101861 | Inverting a suspension object in a stable monoidal category | In the case where $\mathcal{C}$ is presentable, this is constructed in proposition 2.9 of
>
> *Robalo, Marco*, [**$K$-theory and the bridge from motives to noncommutative motives**](http://dx.doi.org/10.1016/j.aim.2014.10.011), Adv. Math. 269, 399-550 (2015). [ZBL1315.14030](https://zbmath.org/?q=an:1315.14030).
>... | 4 | https://mathoverflow.net/users/43054 | 324069 | 139,676 |
https://mathoverflow.net/questions/324004 | 9 | I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has ex... | https://mathoverflow.net/users/131855 | Modern Algebraic Geometry and Analytic Number Theory | Do you consider $L$-functions of elliptic curves over $\mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much ... | 13 | https://mathoverflow.net/users/3272 | 324081 | 139,677 |
https://mathoverflow.net/questions/324077 | 0 | Let $X$ be an infinite set, and let ${\cal E}$ be a collection of non-empty subsets of $X$. We say that ${\cal E}$ has [property $\mathbf{B}$](https://users.renyi.hu/~p_erdos/1966-07.pdf) if there is $B\subseteq X$ such that $B\cap E\neq \emptyset$ and $E\not\subseteq B$ for all $E\in{\cal E}$. (This is equivalent to s... | https://mathoverflow.net/users/8628 | Maximizing set systems with property $\mathbf{B}$ | Any collection $\cal E$ satisfying property ${\mathbf B}$ (further: ${\mathbf B}$-collection) is contained in an inclusion-maximal ${\mathbf B}$-collection.
Proof: consider the corresponding 2-coloring of $\cal E$ and define the over-collection $\cal E\_0$ as the collection of all sets which contain elements of both ... | 1 | https://mathoverflow.net/users/4312 | 324084 | 139,678 |
https://mathoverflow.net/questions/324083 | 4 | I have trouble finding the proof of MacLane's statement that "(6) implies (5)" at page 101 of his Categories for the Working Mathematician. This is part of the proof of his Theorem 2 in the "Transformations of Adjoints" chapter (7).
The definition of *conjugate* natural transformations goes:
Let be two adjunctions
... | https://mathoverflow.net/users/29853 | Characterization of conjugate natural transformations (MacLane) | Use the naturality of $\sigma$ first, and then the naturality of $\eta$ :
$$\begin{align}G \epsilon'\_a \circ G F'U \circ G \sigma\_{x} \circ \eta\_{x} & = G (\epsilon'\_a) \circ G \sigma\_{G'a}\circ GFU \circ \eta\_{x}\\ & = G\epsilon'\_a \circ G \sigma\_{G'a} \circ \eta\_{G'a} \circ U.\end{align}$$
| 5 | https://mathoverflow.net/users/111486 | 324093 | 139,682 |
https://mathoverflow.net/questions/324088 | 2 | From Bóna's [A Walk through Combinatorics](https://books.google.com/books?id=x9tqMFkkTqQC):
>
> Prove or disprove that if we partition the positive integers into finitely many arithmetic progressions then there will be at least one arithmetic progression whose difference and initial term are equal.
>
>
>
A var... | https://mathoverflow.net/users/136283 | Partitioning the positive integers into finitely many arithmetic progressions | Any progression (if they are all infinite, of course, but otherwise the statement is clearly wrong) should have its initial term $a$ not greater than the difference $d$. Indeed, if $a>d$, and $a-d$ is covered by another progression $P$ with difference $d\_1$, then $a-d+dd\_1$ is covered twice --- a contradiction. There... | 7 | https://mathoverflow.net/users/4312 | 324101 | 139,685 |
https://mathoverflow.net/questions/324098 | 3 | We say that a $T\_2$-space has the *open extension property (OEP)* if for any open set $U$ and continuous map $f:U\to U$ there is a continous map $g:X\to X$ such that $g|\_U = f$.
The space $\mathbb{R}$ with the Euclidean topology does not have this property: consider $(0,1)\cup(1,2)$ and the map $f$ sending $(0,1)$ ... | https://mathoverflow.net/users/8628 | Extending a continuous self-map of an open subset to the whole space | Take any pair of distinct points $x, y$. Define a partial order on the set of pairs of disjoint open sets $U \ni x, V \ni y$ by double inclusion (i.e. $(U\_1, V\_1) \preceq (U\_2, V\_2)$ if $U\_1 \subseteq U\_2, V\_1 \subseteq V\_2$). The set of pairs is nonempty, as $X$ is $T\_2$. Then by Zorn's lemma, there is a maxi... | 7 | https://mathoverflow.net/users/44191 | 324102 | 139,686 |
https://mathoverflow.net/questions/324094 | 4 | In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited:
[1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti dell'accademia delle scienze fisico-matematiche di Napoli vol 3 ser 8, 132-166.
My problem is that I can't find this... | https://mathoverflow.net/users/136276 | Cipolla's Prime numbers function: Computing the coefficients of the polynomial | According to the OP, the formula requested is the formula (4.7) in the paper :
*Arias de Reyna, Juan; Toulisse, Jérémy*, [**The $n$-th prime asymptotically**](http://dx.doi.org/10.5802/jtnb.847), J. Théor. Nombres Bordx. 25, No. 3, 521-555 (2013). [ZBL1298.11093](https://zbmath.org/?q=an:1298.11093).
| 4 | https://mathoverflow.net/users/469 | 324103 | 139,687 |
https://mathoverflow.net/questions/289046 | 3 | Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\mathcal E$.
Let $F$ be a lax monoidal functor from $\mathcal C$ to $\mathcal E$. Now let $\phi$ be a natural transformatio... | https://mathoverflow.net/users/70015 | What is the name for a natural transformation that has both lax and oplax monoidal properties? | I don't know a good name for such transformations, but I can give you a reference and some more information about them. First of all note that more generally you can define a monoidal transformation $f F \to G g$ when $F$ and $G$ are lax monoidal and $f$ and $g$ are colax monoidal. The axioms then become hexagons (with... | 6 | https://mathoverflow.net/users/49 | 324113 | 139,691 |
https://mathoverflow.net/questions/324026 | 2 | Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^\*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous theorem due to Johnson and Sinclair states that every derivation on a semisimple Banach algebra is continuous. Generalizing t... | https://mathoverflow.net/users/85784 | Continuity of the derivations from semisimple Banach algebras | Charles Read constructed a Banach space $E$ such that the algebra $\mathcal{B}(E)$ of all bounded linear operators on $E$ admits a discontinuous derivation. Certainly, $\mathcal{B}(E)$ is Jacobson-semisimple. Note that Read's space is quite exotic as this phenomenon does not occur on classical Banach spaces such as $L\... | 4 | https://mathoverflow.net/users/15129 | 324116 | 139,692 |
https://mathoverflow.net/questions/324108 | 23 | In addition to the Jordan-Holder theorem for groups, there are various Jordan-Holder Theorems for other categories:
1. Finite dimensional representations have filtrations whose associated graded consists of irreducible representations. Any other such associated graded is the same up to permutation of its elements.
2.... | https://mathoverflow.net/users/30211 | Categorical Unification of Jordan Holder Theorems | This does not really involve any category theory, but perhaps it is useful to note the following general setting for the Jordan-Hölder theorem.
For $G$ a group and $\Omega$ a set, a *group with operators* is $(G, \Omega)$ equipped with an action $\Omega \times G \rightarrow G$: $(\omega,g) \mapsto g^\omega$ such that... | 17 | https://mathoverflow.net/users/38068 | 324122 | 139,693 |
https://mathoverflow.net/questions/324119 | 3 | I've been reading about the Artin Spin operation. It's defined as taking the classical $n$-knot ($S^n\hookrightarrow S^{n+2}$) to an $(n+1)$-knot. For the $1$-knot case (in $\mathbb{R}^3$), I reproduce the procedure in [knot spinning](https://arxiv.org/pdf/math/0410606.pdf), p. 8,
1. We manipulate a knot $K$ so that... | https://mathoverflow.net/users/123309 | Is the Artin Spin construction related to the suspension functor? | This question is answered in section 4 of my first paper (with Alex Suciu)
Klein, John R.; Suciu, Alexander I.
Inequivalent fibred knots whose homotopy Seifert pairings are isometric.
Math. Ann. 289 (1991), no. 4, 683–701; [DOI: 10.1007/BF01446596](https://doi.org/10.1007/BF01446596).
The paper is available here:
... | 5 | https://mathoverflow.net/users/8032 | 324126 | 139,694 |
https://mathoverflow.net/questions/323866 | 9 | In "Grothendieck ring of pretriangulated categories", Bondal, Larsen and Lunts define a product of perfect (pretriangulated with Karoubian homotopy category) dg-categories as $A\bullet B:=Perf(A\otimes B)$ where $\otimes$ is the usual (non derived) tensor product of dg-categories and Perf(A) is the full subcategory of ... | https://mathoverflow.net/users/44499 | Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category? | Assume that every dg-category is over a field $k$. My guess is that there is a natural (I believe fully faithful) dg-functor
\begin{equation}
\Phi \colon \mathrm{Perf}(A) \otimes \mathrm{Perf}(B) \to \mathrm{Perf}(A \otimes B)
\end{equation}
which induces the equivalence $\mathrm{Perf}(\mathrm{Perf}(A) \otimes \mathrm{... | 5 | https://mathoverflow.net/users/20883 | 324157 | 139,705 |
https://mathoverflow.net/questions/324152 | 2 | Let us say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ is $T\_1$ if for $x\neq y$ there is $e\in E$ such that $e\cap\{x,y\} = \{x\}$.
Note that for any $T\_1$-[space](https://en.wikipedia.org/wiki/T1_space) $(X,\tau)$ the topology $\tau$ contains the cofinite topology (the collection of s... | https://mathoverflow.net/users/8628 | $T_1$-spaces vs $T_1$-hypergraphs | Counterexample:
For $a\in\mathbb Q$ let $L\_a=\{x\in\mathbb Q:x\lt a\}$, $R\_a=\{x\in\mathbb Q:x\gt a\}$.
Let $H=(\mathbb Q,E)$ where $E=\{L\_a:a\in\mathbb Q\}\cup\{R\_a:a\in\mathbb Q\}$.
Clearly $H$ is a $T\_1$-hypergraph. Consider any $E\_1\subseteq E$ such that $(\mathbb Q,E\_1)$ is $T\_1$. Choose $a\in\mathbb... | 4 | https://mathoverflow.net/users/43266 | 324158 | 139,706 |
https://mathoverflow.net/questions/324154 | 6 | Say I have compact Riemann surfaces $X$, $Y$. Is there necessarily a Riemann surface $Z$ which maps holomorphically onto both $X$, $Y$?
| https://mathoverflow.net/users/136313 | Existence of Riemann surface, holomorphic maps | Here is some construction. $X$ and $Y$ are smooth projective algebraic complex curves. So the complex surface $X \times Y$ is projective (e.g. by Segre embedding) and so admits a very ample line bundle $L$. By Bertini theorem, the vanishing locus of a general section of $L$ is a smooth projective complex curve $Z$ (so ... | 15 | https://mathoverflow.net/users/25309 | 324162 | 139,709 |
https://mathoverflow.net/questions/324110 | 1 | I thought of a certain point that seems to point to an apparent incongruity. Hopefully someone would promptly point out the logical mistake and/or some implicit assumption that may not hold under a closer look. Even though the question is a quite long in length, the actual point is very short. I have thought about it f... | https://mathoverflow.net/users/112385 | An Apparent Incongruity | This is an OK question, you just have not explained it very clearly. I think people are downvoting because they think you are confused about whether $f$ is computable or not (which you aren't).
What you're running into is something that ultrafinitists study in detail. It boils down to the fact that "having a short d... | 8 | https://mathoverflow.net/users/45707 | 324170 | 139,711 |
https://mathoverflow.net/questions/324133 | 4 | **Question.** Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\Delta + V$ on $L^2(\mathbb{R}^d)$ is an eigenvalue, but not an isolated point of the spectrum?
I would expect this to ... | https://mathoverflow.net/users/102946 | Non-isolated ground state of a Schrödinger operator | Yes, it is perfectly possible to have an embedded eigenvalue at the bottom of the spectrum. I do not have a reference (although I am quite sure there is one), but here is a simple example in dimension $d = 1$. Extension to higher dimensions is immediate.
---
Let
$$ \phi(x) = \frac{1}{1 + x^2} $$
and
$$ V(x) = \fr... | 7 | https://mathoverflow.net/users/108637 | 324177 | 139,714 |
https://mathoverflow.net/questions/324115 | 4 | Consider any three positive integers $a, b, n$. Is it true that $$\frac{(b-a)(b^2-a)\cdot\dotsb\cdot(b^n-a)}{(b-1)(b^2-1)\cdot\dotsb\cdot(b^n-1)}\in\mathbb{Z}\left[\frac{a-1}{b-1},\frac{a^2-1}{b^2-1},\dotsc ,\frac{a^n-1}{b^n-1}, 1/2 \right]?$$
Moreover, if the first question is true, then what is the minimal value of... | https://mathoverflow.net/users/82143 | The product of $\frac{b^i-a}{b^i-1}$ lies in a special ring (conjecture) | This may be understood $p$-adically as is done in my answer to your question on math.SE. It would be nice to see the algebraic formula proving the same statement (that is, the explicit polynomial not depending on $a$ and $b$ with integer coefficients for which $$
P\left(b,\frac{a-1}{b-1},\dots,\frac{a^n-1}{b^n-1}\right... | 1 | https://mathoverflow.net/users/4312 | 324181 | 139,715 |
https://mathoverflow.net/questions/323710 | 8 | Here all functions are $\mathbb R \to \mathbb R$.
Fix $M$ a positive integer. For $i = 0, 1, ..., M,$ let $f\_0 = Id$, and the other $f\_i$ be continuous functions such that for all $0 \leq k < M$, $f\_{k+1}$ is $o(f\_k)$ as their argument goes to $0$.
Suppose $x\_k$ is a sequence of reals such that $\sum\_{k=0}^{\... | https://mathoverflow.net/users/132446 | Simultaneous Riemann Rearrangement | Yes -- no matter what real numbers $n\_i$ you choose, there is a bijection $s: \mathbb N \rightarrow \mathbb N$ such that $\sum\_{k=0}^\infty f\_i(x\_{s(k)}) = n\_i$ for each $i$.
This follows from one form of the [Lévy-Steinitz rearrangement theorem](https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Steinitz_theorem).... | 7 | https://mathoverflow.net/users/70618 | 324185 | 139,717 |
https://mathoverflow.net/questions/324171 | -1 | Let $(\omega\_1, \omega\_2, \ldots)$ be iid in $\{-1, 1\}$ and $X\_k = \sum\_{i=1}^k \omega\_i$ be a simple one-dimensional random walk.
Let $\tau\_n = \min \{i\in\mathbb{N}: |X\_i|=n\}$ be the first time the random walk is $n$ steps from the origin. What I am interested in is the distribution of this hitting time --... | https://mathoverflow.net/users/58551 | Distribution of first time a 1D random walk hits n or -n | The order of magnitude of the sum of the first displayed series is indeed $\asymp n^2$ when the random walk is symmetric.
Let
\begin{equation}
M\_k:=\max\_{0\le j\le k}X\_j.
\end{equation}
Then, by the reflection principle (see e.g. [page 4](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahU... | 0 | https://mathoverflow.net/users/36721 | 324194 | 139,719 |
https://mathoverflow.net/questions/324121 | 2 | Let $X$ be the set of all points in $\ell^2$ with all rational coordinates. $X$ is known to be totally disconnected, but $X$ is not zero-dimensional. For instance, the empty set does not separate the point $\langle 0,0,0,...\rangle\in X$ from the closed set $\{x\in X:\|x\|\geq 1\}$ because $\{\|x\|:x\in A\}$ is unbound... | https://mathoverflow.net/users/95718 | Scattered separators in Erdős space | Observe that scattered subsets in metrizable separable (more generallity hereditarily Lindelof) spaces are at most countable.
On the other hand, any closed separator $S$ between zero and the set $F=\{x\in X:\|x\|\ge 1\}$ has cardinality continuum and hence cannot be scattered.
To see that $|S|=\mathfrak c$, write $... | 1 | https://mathoverflow.net/users/61536 | 324200 | 139,722 |
https://mathoverflow.net/questions/324198 | 4 | Let $X$ be the zero locus of $e\_1, \dots, e\_n$ sections of a vector bundle $\mathcal{E}$ of rank $r$ on $Y$. Assume that the codimension of $X$ is strictly less than $n$, then the Koszul complex associated to the sections is not exact. What can be said about the cohomologies of this complex in general? Are there nice... | https://mathoverflow.net/users/91572 | Koszul resolution with wrong dimension | Assume $X$ is a locally complete intersection of codimension $m < n$. Then the natural morphism
$$
E^\vee|\_X \to I\_X \otimes O\_X = I\_X/I\_X^2 = N^\vee\_{X/Y}
$$
is surjective, let $F$ be its kernel (it is a vector bundle on $X$ of rank $n - m$). Then
$$
H\_i(Kosz(E,e)) \cong \wedge^iF.
$$
| 5 | https://mathoverflow.net/users/4428 | 324205 | 139,724 |
https://mathoverflow.net/questions/324051 | 5 | Let $M$ be a [matroid](https://en.wikipedia.org/wiki/Matroid) with ground set $E$. If $t$ is a total order on $E$, and if $S$ is a nonempty subset of $E$, then $\max\_t S$ will mean the $t$-largest element of $S$ (that is, the maximum of $S$ with respect to the order $t$). If $t$ is a total order on $E$, then
* a *$t... | https://mathoverflow.net/users/2530 | Toggles for non-broken-circuit sets in matroids | In Concrete question 2, I think $\phi\_i$ and $\phi\_j$ commute, also due to matroid circuit change axiom. Let's see. Take an order $t$ and $K\in \operatorname{NBC}(t)$. We say that $K$ is *$i$-sensitive* for $t$, if there exists a circuit $C$ containing $[t]\_i$ (the $i$-th smallest element of $t$) and $[t]\_{i+1}$ bu... | 3 | https://mathoverflow.net/users/4312 | 324219 | 139,730 |
https://mathoverflow.net/questions/322439 | 8 | I would like to know the answers to the following two questions.
Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote
$$
\mathscr{H}=\{\mathcal{H}\subset 2^M: \mathcal{H}\mbox{ is a disjoint family of Borel sets of positive measure}\}
$$
Note that for mea... | https://mathoverflow.net/users/19593 | Measure support decomposition that "tends to infinity" | I think the following is a counterexample.
Consider the Stone–Čech compactification $\beta \mathbb{N}$. Fix some $x \in \beta \mathbb{N} \setminus \mathbb{N}$ and set $S = \beta \mathbb{N} \setminus \{x\}$, which is locally compact Hausdorff and not compact. Let $\mu$ be the Borel measure on $S$ which puts mass $2^{-... | 2 | https://mathoverflow.net/users/4832 | 324240 | 139,738 |
https://mathoverflow.net/questions/324243 | 3 | Euler’s totient function $\varphi$ is a function defined over $\mathbb{N}$ so that $\varphi(n)=|\{m\mid m<n\wedge (m,n)=1\}|$.
Now [Lehmer’s totient problem](https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem) asks whether $n$ is prime iff $\varphi(n)$ divides $n-1$.
I am curious whether the question can be ... | https://mathoverflow.net/users/14340 | Lehmer’s totient problem | Yes. Every computable relation on $\mathbb{Z}$ can be defined with a first-order formula in the language of rings.
The idea is to "arithmetize" computation: encode Turing machines and their states as natural numbers in such a way that the basic operations like changing the state of the head, writing a bit, etc. are ... | 5 | https://mathoverflow.net/users/45707 | 324247 | 139,741 |
https://mathoverflow.net/questions/323889 | 23 | What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
| https://mathoverflow.net/users/130058 | References and resources for (learning) chromatic homotopy theory and related areas | I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Emily's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this purpose chromatic homotopy theory pseudo-historically in different phases. The years denote the "main phase"; in each case... | 23 | https://mathoverflow.net/users/2039 | 324253 | 139,744 |
https://mathoverflow.net/questions/324254 | 12 | For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?
| https://mathoverflow.net/users/118366 | Selective ultrafilter and bijective mapping | No, this fails not only for selective ultrafilters but for all non-principal ultrafilters $\mathcal F$ on $\omega$.
The main ingredient in the proof is the theorem that, if an ultrafilter $\mathcal U$ on a set $X$ is sent to itself by some map $g:X\to X$ (meaning that $\mathcal U=g(\mathcal U):=\{A\subseteq X:g^{-1}(... | 15 | https://mathoverflow.net/users/6794 | 324261 | 139,747 |
https://mathoverflow.net/questions/323760 | 4 | Im looking for a reference that treats the Markov Chain defined by
$$W\_i=(W\_{i-1}-1)\vee X\_i$$
where $X\_i\geq 0$ are i.i.d discrete variables. In particular im interested in a reference that treats existence and moments of the invariant distribution, and gives moments of the return time to $0$ under some imposed ... | https://mathoverflow.net/users/92118 | Reference on a markov chain / Queue | This kind of process is sometimes called random exchange process. A starting point for a literature research might be the following article by Helland and Nilsen:
<https://www.jstor.org/stable/3212533?seq=1#metadata_info_tab_contents>
For a recent work with a characterization of nullrecurrence see
<https://arxiv.... | 2 | https://mathoverflow.net/users/115138 | 324264 | 139,748 |
https://mathoverflow.net/questions/324273 | 7 | I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: positive, negative, and zero.
Positive
--------
If $M$ is a connected $n$-manifold which admits a metric with constant... | https://mathoverflow.net/users/21564 | Can a hyperbolic manifold be a product? | Question 1:
in $\mathrm{Isom}(\mathbf{H}^n)$, the centralizer of any loxodromic element preserves its axis, and hence is contained in a closed subgroup isomorphic to $\mathrm{O}(n-1)\times\mathrm{Isom}(\mathbf{R})$. In particular, discrete subgroups of the latter are virtually cyclic.
In addition, if a subgroup has... | 9 | https://mathoverflow.net/users/14094 | 324279 | 139,752 |
https://mathoverflow.net/questions/324277 | 3 | Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric realisation of a simplicial set $S$.
Given $X$ and $Y$, (compactly generated) topological spaces, we use $TOP(X,Y)$ to den... | https://mathoverflow.net/users/99088 | Simplicial models for fibrations between mapping spaces | Yes, these agree.
The usual model structures on $C = sSet$ and on $C = Top$ are both cartesian monoidal. So the functor $[-,X] : C^{op} \to C$ is a right Quillen functor when $X$ is fibrant (where $[-,-]$ denotes the internal hom). Thus this functor is "already derived" when evaluated on cofibrant objects.
Moreover... | 4 | https://mathoverflow.net/users/2362 | 324284 | 139,754 |
https://mathoverflow.net/questions/120382 | 5 | Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The *oplax limit* of $F$ is the category whose objects consist of an object $x\_r \in F(r)$ for all $r$ together with a morphism $x\_s \to F(d)(x\_{r})$ for all morphisms $d:r\to s$ in $R$, satisfying obvious compatibility conditions.
I have ... | https://mathoverflow.net/users/49 | Reedy model structures on oplax limits | I think the paper I was thinking of when I asked this question was probably
* Markus Spitzweck, [Homotopy limits of model categories over inverse index categories](https://doi.org/10.1016/j.jpaa.2009.08.001), JPAA 214:6 (2010)
although he only considers the case of inverse categories rather than more general Reedy ... | 1 | https://mathoverflow.net/users/49 | 324287 | 139,755 |
https://mathoverflow.net/questions/323617 | 2 | For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds,
$$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||\_\*^2 - ||A||\_F^2$$
This is equivalent to the following inequality on the elementary symmetric polynomial,
$$\sum\_{i<j}\lambda\_i\lambda\_j \leq \sum\_{i<j} \sigma\_i \sigma\_j$$
where $\... | https://mathoverflow.net/users/135962 | An inequality on elementary symmetric polynomial of eigenvalues | As suggested by Darij in the comments to the OP, the compound matrix $\wedge^2 A$ has eigenvalues $\lambda\_i \lambda\_j$ and singular values $\sigma\_i \sigma\_j$. This result can be found, for example in Theorem 2.16 of
*Zhan, Xingzhi*, Matrix theory, Graduate Studies in Mathematics 147. Providence, RI: American M... | 2 | https://mathoverflow.net/users/135962 | 324302 | 139,763 |
https://mathoverflow.net/questions/324310 | 5 | Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to [this page](https://ncatlab.org/nlab/show/suspension+object) in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \mathbb{Z} \wedge X$, where $B \mathbb{Z}$ is the "classifying space of ... | https://mathoverflow.net/users/101861 | Smash product and the integers in a Grothendieck $(\infty, 1)$-topos | To summarize the comments so far:
* In an arbitrary $(\infty,1)$-topos $\mathcal{E}$, the integers can be defined as the loop space of the circle $S^1\_\mathcal{E}$, which itself is given as the (homotopy) pushout of two copies of the map $\mathrm{pt}\sqcup \mathrm{pt} \to \mathrm{pt}$ in $\mathcal{E}$. Otherwise, on... | 5 | https://mathoverflow.net/users/4177 | 324314 | 139,768 |
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